University of Southern California Dissertations and Theses

Grid-based Vlasov method for kinetic plasma simulations

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Grid-based Vlasov method for kinetic plasma simulations

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Content GRID-BASED VLASOV METHOD FOR KINETIC PLASMA SIMULATIONS by Chen Cui A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY (ASTRONAUTICAL ENGINEERING) December 2023 Copyright 2023 Chen Cui Epigraph ii Epigraph Ad Astra per Aspera Dedication iii Dedication To my family and friends Acknowledgments iv Acknowledgments This dissertation has been an extensive and exhilarating journey, and I cannot imagine achieving this milestone without the invaluable help, advice, inspiration, and support I received from my family, friends, mentors, and colleagues. I would like to express my gratitude to each and every one of them. First and foremost, I would like to express my deepest appreciation to my Ph.D. advisor, Prof. Joseph J. Wang, for his continuous support throughout my entire doctoral study. His guidance, expertise, and invaluable advice have played a pivotal role in shaping my academic and research skills. I am truly grateful for his trust in allowing me to pursue my research interests while maintaining a high standard of excellence and providing insightful guidance. His mentorship has been instrumental in my growth as a researcher, and I am really grateful for his support and encouragement in realizing my dreams and career goals. I will always cherish the kindness and unwavering support from him that has shaped me into both a better researcher and a better individual. Thanks very much, Prof. Wang, for supporting me all the journey. I would like to express my gratitude to all the members of my dissertation committee for their time and valuable feedback, which have greatly enhanced my research skills and contributed to the quality of my dissertation. I am particularly grateful to Prof. Aiichiro Nakano for introducing me to the world of high-performance computing. His prestigious CSCI-596/653 high-performance computing classes have been immensely enlightening, and I deeply appreciate his guidance not only in the field of high-performance computing but also in various aspects of academic careers and daily life. Additionally, I am truly grateful Acknowledgments v for his kindness in inviting me for post-work drinks, fostering a friendly and supportive environment. I am sincerely thankful to Prof. Mike Gruntman for his unwavering support during my studies in the Astronautical Engineering department at the University of Southern California. Serving as his teaching assistant for his spacecraft propulsion class has been an invaluable experience. I also want to express my gratitude to him for his encouragement and valuable suggestions regarding my pursuit of an academic career. Moreover, I greatly enjoy his humor and jokes, especially his description of me as a “machine gun” during presentations. I am really grateful to Dr. Lubos Brieda for his invaluable contributions to my academic and personal growth. His insightful discussions and extensive knowledge in plasma simulations have been instrumental in expanding my understanding of computational plasma physics. I have greatly benefited from our engaging conversations, which have deepened my insights into various topics within this field. I would also like to extend my sincere appreciation to Dr. S. Peter Gary. I am truly grateful to him and Prof. Wang for introducing me to a new research area that I had not previously explored. I have gained extensive knowledge of instabilities and turbulence in the solar wind through my interactions with Dr. Gary, greatly enriching my understanding of plasma physics. Our collaboration has undoubtedly made me a better researcher. Furthermore, I am grateful to Dr. Keith Goodfellow, whom I have had the privilege of assisting in his electric propulsion classes multiple times. I deeply appreciate his encouragement and recognition throughout the teaching process. I am deeply grateful for the support and guidance of my esteemed colleagues throughout this dissertation journey. I would like to express my sincere appreciation to my labmates, Prof. Daoru Han, Prof. Yuan Hu, Prof. Yinjian Zhao, Dr. Robert Antypas, Dr. Jeffery Asher, Mr. Daniel Depew, Mr. Ziyu Huang, Ms. Elana Helou, Mr. Kevin Sampson, Mr. Jose Ferreira, and Mr. Jacob Meyer. Their unwavering support and guidance have been invaluable in my personal and academic growth. The shared experiences and fruitful discussions between us have not only made this journey enjoyable, but they have also contributed significantly to my research growth. The daily interactions and collaborative atmosphere Acknowledgments vi have been a source of inspiration and motivation, and I am grateful for the friendship we have developed. Furthermore, I would like to acknowledge the hardworking staff in the Department of Astronautical Engineering, including Ms. Dell Cuason, Ms. Marlyn Lat, Ms. Linda Ly, Mr. Luis Saballos, Ms. Prisila Vasquez. Their tireless efforts and dedication have created a welcoming and supportive environment that truly feels like a second home. I am sincerely thankful for their assistance, kindness, and the sense of belonging they have fostered. I would like to express my appreciation to all of my friends at the University of Southern California who have played a significant role in making my journey memorable and enjoyable. Specifically, I want to extend my gratitude to my friends in the giant house on 23rd street. I have great memories of our time together living in the giant house on 23rd street, where we shared countless moments of laughter and fun. Their friendship has been a source of joy and support throughout my academic journey. I would also like to express my appreciation to my roommate. The late-night conversations about robotics and control have been both intellectually stimulating and enjoyable. While I am unable to mention every friend, I want to express my sincere gratitude to all of my friends for being an integral part of my life. Their friendship and support have made this journey all the more meaningful and memorable. I would like to acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this dissertation. I would also like to acknowledge Cheyenne cluster provided by NCAR’s Computational and Information Systems Labora- tory, sponsored by the National Science Foundation. This dissertation also used Anvil cluster at Purdue Rosen Center for Advanced Computing through allocation NSF-ACCESS- PHY230064 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants 2138259, 2138286, 2138307, 2137603, and 2138296. Finally, I would like to express my deepest and most heartfelt appreciation to my beloved mother and father. Their unwavering love, support, and care for me have been the foundation of my success. To all other family members, I extend my gratitude for their constant presence Acknowledgments vii and care. I am immensely grateful for my mother and father’s endless and unconditional support. I really appreciate them for continuously motivating me to strive for excellence and be the best version of myself. I am indebted to them for bringing me into this beautiful world and for always being there for me, providing guidance and encouragement. They have consistently given their utmost efforts to ensure my happiness and success. Their unwavering belief in me has been a source of strength and inspiration. I cannot express enough gratitude for the selflessness sacrifices they have made to bring me up. They have wholeheartedly supported my dreams without hesitation, giving me everything they have and always having my back. I am truly blessed to be their son, and I am forever grateful for the love and guidance they gave me. This dissertation is dedicated to my dear father and mother. From the curious young boy who looked at the sky with wonder and dreams of exploring the space, I have now become the first Dr. Cui in our family. It is with immense pride and gratitude that I share this accomplishment with my father and mother both. TABLE OF CONTENTS viii Table of Contents Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi Chapter 1:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Plasma Expansion: Analytical and Semi-Analytical Solutions . . . . . 5 1.2.2 Electric Propulsion Beam . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Kinetic-Range Turbulence and Microinstability in the Solar Wind . . 16 1.2.4 Grid-based and Particle-based Method for Kinetic Plasma Simulations 20 1.3 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Dissertation Organization and Outline . . . . . . . . . . . . . . . . . . . . . 29 Chapter 2:Physical Models and Numerical Methods . . . . . . . . . . . . . . 32 2.1 Kinetic Theory, First-Principle Governing Equations . . . . . . . . . . . . . 32 2.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 Normalizations and Normalized Models . . . . . . . . . . . . . . . . . . . . . 39 2.4 Numerical Methods for Vlasov Equation . . . . . . . . . . . . . . . . . . . . 42 2.5 Numerical Methods for Poisson Equations . . . . . . . . . . . . . . . . . . . 48 TABLE OF CONTENTS ix 2.6 Numerical Methods for Darwin Equations . . . . . . . . . . . . . . . . . . . 51 2.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Chapter 3:Grid-based Vlasov Solver for Electrostatic Plasma Simulations . 56 3.1 Numerical Implementation and Parallelization Strategy . . . . . . . . . . . . 57 3.2 Vlasov Equation Boundary Condition Implementations . . . . . . . . . . . . 60 3.3 Parallel Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Verification of Vlasolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.1 Linear Advection and Gyro-Motion . . . . . . . . . . . . . . . . . . . 69 3.4.2 Landau Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.3 Two-Stream Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 Comparison with PIC: Plasma Wake Expansion . . . . . . . . . . . . . . . . 75 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping . 82 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 4:Grid-based Vlasov Simulation of Collisionless Plasma Expansion 94 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 Comparison with the Self-Similar Solution . . . . . . . . . . . . . . . . . . . 102 4.4 The Electron Langmuir Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 The Perturbations Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 111 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Chapter 5:Grid-based Vlasov Simulation of Electric Propulsion Beam . . . 121 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Comparisons with Full Kinetic PIC Simulations . . . . . . . . . . . . . . . . 129 5.4 Electron Velocity Distribution Function . . . . . . . . . . . . . . . . . . . . . 134 5.5 Electron Collisionless Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Chapter 6:Grid-basedVlasovSolverforElectromagneticPlasmaSimulations154 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.2 Method for Coupling the Vlasov and Darwin Equations . . . . . . . . . . . . 155 6.3 Parallel Efficiency of the Electromagnetic Vlasolver . . . . . . . . . . . . . . 159 TABLE OF CONTENTS x 6.4 Verification of the Electromagnetic Vlasolver: Simulation of Whistler Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.5 Verification of the Electromagnetic Vlasolver: Simulation of Electron Weibel Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Chapter 7:Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 170 7.1 Conclusions and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.1.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.1.2.1 Contributions in Computational Physics . . . . . . . . . . . 175 7.1.2.2 Contributions in Plasma Physics and Engineering . . . . . . 175 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.2.1 Further Development of Grid-based Vlasov Method and the Vlasolver Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.2.2 Impact of Initial Non-Maxwellian Electron Velocity Distribution Func- tion on Electric Propulsion Beam Expansion . . . . . . . . . . . . . . 178 7.2.3 Evaluating the Significance of Heat Flux Closure in the Expansion of Electric Propulsion Beams . . . . . . . . . . . . . . . . . . . . . . . . 179 7.2.4 Investigation of the Effects of Finite Ion Temperature on Plasma Expan- sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2.5 Comparative Analysis of Grid-based Vlasov Method and PIC Method for Short Wavelength Turbulence and Microinstability in the Solar Wind180 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability . . . . 204 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.2 Simulation Model and Setup . . . . . . . . . . . . . . . . . . . . . . . 207 A.3 Fluctuating Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . 211 A.4 Electron Temperature Anisotropy . . . . . . . . . . . . . . . . . . . . 216 A.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 LIST OF TABLES xi List of Tables 3.1 Signal-noise ratio, units in dB, unfiltered/filtered . . . . . . . . . . . . . . . 90 4.1 Representative Simulation Cases (Additional fully kinetic simulations are also carried out using m i =m e =900 and 400) . . . . . . . . . . . . . . . . . . . . . 102 5.1 Physical parameters for the beam . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Simulation Domain/Time Parameters . . . . . . . . . . . . . . . . . . . . . . 129 A.1 Summary of simulation cases. In all cases, e = 0:1 and v te =c = 0:1. . . . . . 210 A.2 R e (i;j;k) value range of figure A.6. . . . . . . . . . . . . . . . . . . . . . . . 220 LIST OF FIGURES xii List of Figures 1.1 A schematic plot on different kinetic modeling methods of plasmas. (a) Particle- in-cell (PIC) method. (b) Grid-based Vlasov method. . . . . . . . . . . . . . 4 1.2 Comparisons of the results from PIC and Vlasov method on a linear Landau damping problem. (a) Error v.s. CPU time. (b) Efficacy v.s. CPU time. . . 24 3.1 Schematic illustration on the 2D2V phase space discretization. . . . . . . . . 57 3.2 Domain decomposition schematic plot. . . . . . . . . . . . . . . . . . . . . . 58 3.3 Schematic plot for the two-dimensional physical space communications. . . 59 3.4 Schematic plot for the solving field. . . . . . . . . . . . . . . . . . . . . . . 60 3.5 Flowchart for Vlasolver numerical algorithm. In the flowchart the red ecliptic box stands for the start and end control command, green diamond box stands for the decision process, blue rectangular box stands for the computing process and yellow rectangular process stands for the communication process between processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 Weak scaling of Vlasolver’s efficiency with dependency to core numbers. . . . 65 3.7 Weak scaling of the Vlasov equation solver module in Vlasolver. (a) Weak scal- ing of Vlasov module only. (b) Weak Scaling of Vlasov and MPI communication module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.8 Weak scaling of the Poisson equation solver module and gather-scatter com- munication module in Vlasolver. . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.9 Profiling results of Vlasolver. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.10 Linear advection in physical space using Vlasolver. (a) Linear advection along x direction. (b) Linear advection along x and y direction. . . . . . . . . . . . 70 3.11 Linear advection in velocity space using Vlasolver. (a) Linear advection along v x direction. (b) Linear advection along v x and v y and v z direction. The iso-surface is plotted for f = 0:3. . . . . . . . . . . . . . . . . . . . . . . . . 71 3.12 Comparison of the average bulk velocity of distribution function at different time moment between numerical and analytical results. . . . . . . . . . . . 72 3.13 Electric field energy history for Landau Damping cases. (a). Linear Landau Damping. (b). Non-linear Landau Damping. . . . . . . . . . . . . . . . . . 74 LIST OF FIGURES xiii 3.14 Contour of integrated velocity distribution function R fdydv y at different time moment. (a) t = 0. (b) t = 8. (c) t = 16. (d) t = 24. (e) t = 32. (f) t = 40. 76 3.15 Comparisons between Vlasov and PIC simulations on the particle number densities in plasma wake. (a) Ion number density. (b) Electron number density. 79 3.16 Comparisons between Vlasov and PIC simulations on the electron temperature in plasma wake. (a) T ex . (b) T ey . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.17 Comparisons between Vlasov and PIC simulations on the electron heat flux in plasma wake. (a) Q ex . (b) Q ey . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.18 Comparisons between Vlasov method and PIC method on the Landau damping electric field energy history. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.19 Comparative analysis of electron number density contours in Landau damping simulations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simulation, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. . . . 86 3.20 Comparative analysis of electron temperature contours in Landau damping simulations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simulation, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. . . . 88 3.21 Comparative analysis of electron heat flux contours in Landau damping simu- lations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simula- tion, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. . . . . . . 89 3.22 Comparisons of electron number density, temperature, and heat flux between the benchmark Vlasov simulation case and the PIC cases with ppc= 10 5 and ppc= 10 6 at different time instances. . . . . . . . . . . . . . . . . . . . . . . 91 4.1 Problem Setup. (a) Initial plasma density profile. (b) Initial ion and electron velocity distribution function. (c) Simulation domain. . . . . . . . . . . . . . 98 LIST OF FIGURES xiv 4.2 Potential profile at different time. (a) Case 1 (hybrid). (b) Case 2A (fully kinetic, m i =m e = 1600). (c) Case 2B (fully kinetic, m i =m e = 100). The analytical solutions of the ion rarefaction wave front, the potential slot, quasi- neutral region front and plasma expansion front are also shown for comparison.103 4.3 Zoomed in comparisons of potential profile at different time. a) ! pe0 t = 100 (Case 1: ! pi0 t = 2:5; Case2A:! pi0 t = 2:5; Case 2B:! pi0 t = 10.) b)! pe0 t = 400 (Case 1: ! pi0 t = 10; Case2A: ! pi0 t = 10; Case 2B: ! pi0 t = 40). . . . . . . . . . 105 4.4 Spectrum contour of ~ E(~ x; ~ t) after Fourier transform in the ~ ! ~ k plane. (a) Case 2A (m i =m e = 1600). (b) Case 2B (m i =m e = 100). . . . . . . . . . . . . 107 4.5 Spatial-temporal contours of ~ E(~ x; ~ t). (a) Case 2A (m i =m e = 1600). (b) Case 2B (m i =m e = 100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 Electron temperature profile at different time. (a) Case 2B (m i =m e = 100). (b) Zoomed in plot for Case 2B (m i =m e = 100 at ~ t = 400). . . . . . . . . . . 110 4.7 Electron temperature, electron heat flux and corresponding charge density. (a) Case 2B (m i =m e = 100) at ! pe0 t = 200. (b) Case 2B (m i =m e = 100) at ! pe0 t = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.8 Local electron velocity distribution functions (VDFs) at selected locations for Case 2B ! pe0 t = 400. A fit of the Maxwellian VDF using the local ~ T e (~ x) is also plotted for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.9 Electron phase space (~ v e vs. ~ x) contour for Case 2B and electric field profile ~ E(~ x). (Left axis: Electron velocity ~ v e . Right axis: Electric field strength ~ E.) (a) ! pe0 t = 10. (b) ! pe0 t = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.10 Electron phase space (~ v e vs. ~ x) contour for Case 2B. (a) ! pe0 t = 200. (b) ! pe0 t = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.11 ~ T e;max vs. p m e =m i in Region B. . . . . . . . . . . . . . . . . . . . . . . . 117 4.12 vs. p m e =m i in Region C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1 Schematic plots of simulation model. (a) Simulation setup. (b) Phase space configuration and discretization. . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2 Initial velocity distribution functions for electrons and ions. . . . . . . . . . . 128 5.3 Comparisons of simulation results between PIC and Vlasov at t = 1000. (a) Ion density contour. (b) Electron density contour. (c) Potential contour. . . 131 5.4 Comparisons of simulation results att = 1000 on electron temperature on both x and y direction between grid-based Vlasov method and particle-based PIC methods. (a) electron x direction temperature T ex . (b) electron y direction temperature T ey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.5 Comparisons of simulation results at t = 1000 on electron heat flux on both x and y direction between grid-based Vlasov method and particle-based PIC methods. (a) electron x direction heat flux Q ex . (b) electron y direction heat flux Q ey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 LIST OF FIGURES xv 5.6 Density normalized electron velocity distribution function contours att = 1000. (a) (x;y) = (40; 0), a position selected from the region upstream of the self- similar region. (b) (x;y) = (100; 0), a position inside the self-similar region. (c) (x;y) = (160; 0), a position close to the quasi-neutral front calculated by eq. (5.19). (d) (x;y) = (220; 0), a position in the “pure electron gas” region. 135 5.7 Density normalized v x direction electron velocity distribution function plots at t = 1000. (a)-(d) On axis (y = 0) positions with x = 40, x = 100, x = 160, x = 220 respectively. (e)-(h) positions alongy = 20 with withx = 40,x = 100, x = 160, x = 220 respectively. The black dashed lines in all figures are the local reconstructed Maxwellian distribution by eq. (5.14). . . . . . . . . . . . 137 5.8 Density normalized v y direction electron velocity distribution function plots at t = 1000. (a)-(f) On axis (y = 0) positions with x = 20, x = 40, x = 60, x = 100, x = 160, x = 220 respectively. The black dashed lines in all figures are the local reconstructed Maxwellian distribution by eq. (5.15). The blue dotted lines in (a)-(d) are the semi-analytical fitted distribution functions by eq. (5.16). Note that only the positive part of the fitted distribution function is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.9 Contour of Q ex at different time moments. Green dash dotted lines: iso- contour lines of the ion density value at the on-axis ion rarefaction wave fronts. Magenta dashed lines: iso-contour lines of the ion density value at the on-axis quasi-neutral fronts. (a) t = 1000. (b) t = 1500. (c) t = 2000. . . . . . . . . . 142 5.10 Contour of Q ey at different time moments. Magenta dashed lines: iso-contour lines of the ion density value at the on-axis quasi-neutral fronts. (a) t = 1000. (b) t = 1500. (c) t = 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.11 Q ex , Q e;xxx , Q e;yyx profiles along axis at different time. (a) t = 1000. (b) t = 1500. (c) t = 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.12 (a) Flux limited model coefficient along axis at various time. (b) Region division at t = 1500. (c) Region division at t = 2000. In (b) and (c), the red dashed line is x =t(v d C s ). . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.13 Representative electron trajectory in phase space (x;v x ). The trajectory is plotted for an electron with an initial condition of (x;y) = (200; 0); (v x ;v y ) = (0; 0:9) from time t = 979 to time t = 2000. . . . . . . . . . . . . . . . . . . 150 6.1 Schematic illustration on the 2D3V phase space discretization. . . . . . . . . 160 6.2 Weak scaling of the electromagnetic Vlasolver’s efficiency with dependency to core numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.3 Profiling results of the electromagnetic Vlasolver. . . . . . . . . . . . . . . . 162 6.4 Spatial-temporal spectrum of the B y component. . . . . . . . . . . . . . . . 165 6.5 Time history of the longest wavelength mode magnetic field energy of B y and B z . The theoretical linear growth rate is obtained by iteratively solving eq. (6.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 LIST OF FIGURES xvi A.1 Time history of B(t) 2 =B 2 0 for all simulation cases. (a) Simulation Groups A and B. (b) Simulation Group C. (c) Simulation Group D. . . . . . . . . . . . 211 A.2 Time history of tan 2 B for all simulation cases. (a) Simulation Groups A and B. (b) Simulation Group C. (c) Simulation Group D. . . . . . . . . . . . . . 213 A.3 Comparison of the probability distribution function of B k along Y axis at e t 111:80. (a)-(d): Run A2 to A5. (e)-(h): Run B1 to B4. (i)-(l): Run C2 to C5. (m)-(p): Run D1 to D4. . . . . . . . . . . . . . . . . . . . . . . . . . 215 A.4 Average electron temperature anisotropy R e v.s. e at e t = 0 and at e t 447:2 for Group A and Group C. The initial anisotropies are shown as transparent circle for Group A and transparent square for Group C, respective. The final anisotropies for Run A1 through A5 are color circles with increas- ingly dark shades, and that for Run C1 through C5 are color squares with increasingly dark shades. The upper bound predicted by eq. (A.6) is shown as the dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A.5 Contours of R e (i;j;k) on an x-y plane in the middle of the simulation box at time e t 223:60 for Run D1 (a), D2 (b), D3 (c), and D4 (d). . . . . . . . . 218 A.6 Contours of R e (i;j;k) on an x-y plane in the middle of the simulation box at time e t 223:60 for Run B3 (a), C2 (b), C3 (c), C4 (d), and C5 (e). . . . . 219 LIST OF SYMBOLS xvii List of Symbols Kn Knusden number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 mfp Mean free path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 L c Characteristic length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Z Charge number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 n Number density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Self-similar variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 C s Ion acoustic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Electric potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 k b Boltzmann constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 T e Electron temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 T i Ion temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 d i Ion inertial length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Plasma thermal pressure to magnetic pressure ratio . . . . . . . . . . . . 17 Initial spectrum strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 e Electron cyclotron length . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 N vlasov Qualitative numerical costs of grid-based Vlasov simulations . . . . . . . 26 N PIC Qualitative numerical costs of PIC simulations . . . . . . . . . . . . . . . 26 g PIC Inverse of the number of super-particles per cell . . . . . . . . . . . . . . 26 LIST OF SYMBOLS xviii N d v Number of total grid cells in the discretized velocity space; d: number of dimensions in velocity space . . . . . . . . . . . . . . . . . . . . . . . . . 26 Particle species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 E Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 B Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 f The velocity distribution function for particle . . . . . . . . . . . . . . 34 Charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 0 Vacuum permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 J Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 0 Vacuum permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 v Velocity space coordinate vector . . . . . . . . . . . . . . . . . . . . . . . 36 x Physical space coordinate vector . . . . . . . . . . . . . . . . . . . . . . . 36 q Charge number of particle . . . . . . . . . . . . . . . . . . . . . . . . . 37 n Number density of particle . . . . . . . . . . . . . . . . . . . . . . . . . 37 t Numerical time step length . . . . . . . . . . . . . . . . . . . . . . . . . . 37 x Numerical grid size length . . . . . . . . . . . . . . . . . . . . . . . . . . 37 c Light speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 E T Transverse electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 E L Longitudinal electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ! pe;0 Electron initial plasma frequency . . . . . . . . . . . . . . . . . . . . . . 40 a Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Numerical flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 + Slope correctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Slope correctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A Sparse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 LIST OF SYMBOLS xix ^ n Unit normal vector to the boundary surface . . . . . . . . . . . . . . . . 54 E p Parallel Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v 0 Drifting velocity/bulk velocity . . . . . . . . . . . . . . . . . . . . . . . . 77 M 0 Initial Mach number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 L x Numerical domain length in x direction . . . . . . . . . . . . . . . . . . . 78 L y Numerical domain length in y direction . . . . . . . . . . . . . . . . . . . 78 ppc Particles per cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 SNR Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 P signal Power of signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 P noise Power of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 ! pi0 Ion plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 D0 Debye length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 ! Wave frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 k Wave vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 ~ Q e Normalized electron heat flux . . . . . . . . . . . . . . . . . . . . . . . . 111 e Electron polytropic coefficient . . . . . . . . . . . . . . . . . . . . . . . . 119 N p,cell Particles per cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Q e;kmn Electorn heat flux tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Q e;n Electorn heat flux vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 u e Electron bulk velocity vector . . . . . . . . . . . . . . . . . . . . . . . . . 140 p e Electron pressure tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Q e Electron heat flux vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Q e Electron heat flux tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 J Adiabatic invariant of the electron transverse direction momentum . . . . 150 LIST OF SYMBOLS xx d e0 Initial electron inertial length . . . . . . . . . . . . . . . . . . . . . . . . 164 Z Plasma dispersion function . . . . . . . . . . . . . . . . . . . . . . . . . . 167 th Linear growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 T ?e Electron temperature perpendicular to the background magnetic field . . 205 T ke Electron temperature parallel to the background magnetic field . . . . . . 205 R e Local temperature anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 216 Abstract xxi Abstract Many kinetic plasma physics problems in astronautical engineering and space physics require accurate solutions of the higher order moments of particle velocity distribution or small amplitude micro-scale plasma perturbations. Such problems often can not be adequately resolved by the commonly used Particle-in-Cell (PIC) method due to the inherent statistical noise. This dissertation develops a fully kinetic grid-based Vlasov simulation model for kinetic plasma simulations. This simulation model directly solves the velocity distribution function at every physical space location through phase space discretization without the interference of particle noise. The Vlasov algorithm employs the semi-Lagrangian Positive and Flux Conservation scheme, and is integrated with an electrostatic Poisson solver utilizing the velocity distribution function for charge density computation and a Darwin electromagnetic field solver utilizing the first three orders of Vlasov moments equations. The simulation code is implemented on parallel supercomputers using multi-dimensional domain decomposition in the physical space and Message Passing Interface (MPI). Weak scaling tests show the code runs at over 75% parallel efficiency in electrostatic simulations with 360 cores and 80% parallel efficiency in electromagnetic simulations with 1280 cores, demonstrating an excellent scalability of the code. Electrostatic Vlasov simulation studies are carried out to investigate first electron-scale physics in one-dimensional collisionless plasma expansion and then electron thermodynamics in two-dimensional electric propulsion beam emission. Simulation results on plasma expansion show that the expansion generates both an ion-acoustic rarefaction wave mode and an electron Langmuir wave mode that propagate into the unperturbed plasma upstream, and that the Abstract xxii assumption used in the classical expansion solution that the electrons are an isothermal fluid is accurate within a quasi-neutral, self-similar expansion region but fails in both the upstream and downstream of that region due to electron timescale perturbations. Simulation results on plasma beam emission demonstrate the significant role from the anisotropic 3rd order electron velocity moment, i.e. the collisionless electron heat flux, on beam expansion. Two distinct regions in the beam, an electron trapped region and a self-similar expansion region, are identified using the on-axis flux-limited coefficient. A connection is also established between the coefficient in the flux-limited model of electron collisionless heat flux and the features of the electron velocity distribution function, which can be utilized to develop a macroscopic electron heat flux model incorporating the correct microscopic electron kinetic effects in the future. Darwin Vlasov simulation studies are carried out on whistler wave propagation and Weibel instability to demonstrate the capability of the Vlasov method on modeling kinetic plasma processes. Simulation results show that the Darwin-Vlasov simulation model is advantageous over electromagnetic PIC in resolving the effects from small amplitude waves and slow growing microinstabilities because it eliminates the interference of particle noise and numerical heating in PIC, and can be applied as an efficient tool in future simulation studies of kinetic range plasma turbulence. 1 Chapter 1: Introduction This chapter presents a summary of the motivation and objectives of the dissertation, reviews relevant literature, and provides an overview of the structure of the dissertation as well as the research topics it covers. 1.1 Overview The field of space science and engineering encompasses various types of plasma flows, each requiring distinct modeling methods. Plasma flows can be classified into different types based on several criteria. One of the most crucial criteria is whether the plasma is collisional or collisionless. In collisional plasmas, particles tend to relax to an equilibrium state and follow the Maxwellian distribution when there are a sufficient number of elastic collisions [1–3]. In contrast, collisionless plasmas experience infrequent collisions and are primarily influenced by long-range interactions involving electrostatic and magnetic fields. Due to the lack of collisions in such collisionless plasmas, particles do not easily attain an equilibrium state and typically deviate from the Maxwellian distribution. The collisionality of plasmas is determined by two key physical factors: the mean free path and the characteristic length of the plasmas. The Knudsen number is utilized to characterize different regimes of plasma collisionality Kn = mfp L c (1.1) where mfp is the mean free path and L c is the characteristic length. In the regime where Kn 1, the mean free path is much shorter than the characteristic length. Therefore, the distance particles travel between collisions is small compared to the characteristic distance and the flow is collisional. In the regime where 0:01< Kn< 10, the mean free path is comparable to the characteristic length, and the flow is characterized as transitional. In the regime where Kn 1, the flow is referred to as collisionless. In the transitional and collisionless flow 1.1 Overview 2 regimes, the lack of collisions may prevent particles from reaching a equilibrium state, and the velocity distribution function can significantly deviate from the Maxwellian distribution. Consequently, kinetic modeling methods, which do not assume plasma equilibrium, are necessary to study collisionless plasma flows. This study considers the plasma flow problems with characteristic lengths much greater than the mean free path of the plasmas, i.e. the collisionless plasma flows. In addition to the collisionality of plasmas, another crucial characteristic is the thermal velocity (v t ). The thermal velocity characterizes the random motion of particles. If v t is much smaller than the average velocity of the flow v d , i.e., v t v d , the flow is commonly referred to as “cold”. In “cold” plasmas, kinetic effects are usually not important. In contrast, in “thermal” plasmas, interactions between particles with different energies can give rise a variety of kinetic effects. Moreover, it is noteworthy that plasmas can exhibit different thermal velocities along different directions, leading to a rich array of physical phenomena that require kinetic investigation [4]. Even when the plasma is relatively “cold”, anisotropic thermal energy can still induce kinetic effects. To accurately capture these potential kinetic effects, it is necessary to employ kinetic modeling methods for plasma flow problems in the field of space science and engineering. The standard method for the kinetic modeling of plasma is the Particle-in-cell (PIC) method [5]. PIC utilizes macro-particles to represent an ensemble of real particles, and it solves self-consistently the particle dynamics, space charge and current carried by the particles, and the electromagnetic fields. The PIC method is relatively straightforward to implement and has been widely applied. However, the discrete nature of the macro-particles introduces statistical noise. This inherent noise not only persists throughout the simulation but also grows over the course of the simulation. As a result, any physical information with amplitudes comparable to the noise level may be overshadowed. Numerical heating due to particle noise can also lead to unphysical results [5–7]. Therefore, while the PIC method has achieved success in many applications, there are areas where its effectiveness is limited. 1.1 Overview 3 In the following sections, we discuss two representative examples relevant to the research presented in this dissertation. One example pertains to collisionless plasma dynamics in electric propulsion. In recent years, electric propulsion (EP) has attracted considerable interest from researchers and engineers due to its potential capabilities and advantages over traditional propulsion methods for tasks such as orbit-keeping and deep space exploration [8–14]. Although various electric propulsion devices have been designed and manufactured, numerous physical mechanisms related to plasma flows both inside and outside these devices remain unclear and require further investigation [15, 16]. One of the critical issues in the electric propulsion community is the dynamics of electric propulsion plumes [17–19]. The plasma beam emitted from electric propulsion thrusters can interact with the spacecraft, leading to contaminations [20, 21] and operational interference. To minimize the negative influences on spacecraft and enhance thruster performance, extensive modeling and experimental studies have been conducted to understand the physical mechanisms that govern the plasma beam injected from thrusters. Prior modeling studies of plasma beam emission have primarily centered on ion dynamics, with electrons primarily assumed to provide neutralization. Consequently, a common assumption involves treating electrons as a massless equilibrium fluid, such as a massless, isothermal ideal gas. This assumption has formed the basis for the Boltzmann relation, n e =n 0 exp e k b T e0 (1.2) which has been widely employed in nearly all previous electric propulsion plume models. However, recent fully kinetic PIC simulations conducted by Wang et al. [22], Hu and Wang [23], and Wang and Hu [24] have unveiled that electrons within the beam emitted by an EP thruster exhibit nonequilibrium and anisotropic behavior. Notably, a standard hybrid particle-ionBoltzmann-electronmodelmayoverestimatetheplumepotentialbyapproximately 30%-40% [24]. Simply replacing the Boltzmann relation with the more general polytropic thermodynamic relation for electrons is unlikely to significantly enhance accuracy. A thorough 1.1 Overview 4 understanding of electron thermodynamics and its role in beam emission necessitates the accurate resolution of the heat flux carried by the electrons. However, obtaining the heat flux is currently beyond the capabilities of PIC due to particle noise [25]. Anotherexamplepertainstothesmall-scalemicroscopicprocesseswithincollisionlessspace plasmas. It is widely acknowledged that these microscopic plasma processes have considerable influence over energy and momentum transport on a macroscopic scale. Consequently, understanding the complex interplay between microscopic phenomena and the macroscopic characteristics of collisionless plasmas has remained a crucial objective across numerous sub-fields within plasma and space science. The exploration of microscopic kinetic physics within plasmas has primarily occurred within the realms of plasma waves, instabilities, and turbulence. The standard numerical modeling approach involves the use of Particle-in-Cell (PIC) simulations. Although PIC has yielded notable success in addressing numerous such problems, its efficacy becomes significantly constrained when dealing with situations in which the amplitude of the microscopic fluctuations of interest is not substantially greater than thermal fluctuations or when instability growth rates are exceptionally slow. This constraint arises due to the inherent particle noise in PIC and the introduction of artificial numerical heating, factors that can obscure the physical processes being simulated. For instance, discrete particle noise in numerical simulations, as discussed in [26], can impact localized effects within plasma turbulence. Consequently, the ability to resolve small-scale localized effects and complex fine-scale phase space structures is currently beyond the reach of PIC due to particle noise. (a) (b) Figure 1.1: A schematic plot on different kinetic modeling methods of plasmas. (a) Particle- in-cell (PIC) method. (b) Grid-based Vlasov method. 1.2 Literature Review 5 The grid-based Vlasov method is an alternative method in kinetic plasma simulation [27]. As depicted in Fig. 1.1, the schematic illustrates both the PIC method and the grid- based Vlasov method. To address the aforementioned unresolved physical problems, a novel grid-based Vlasov method coupled with both Poisson and Darwin equations is developed in this dissertation. A parallel, multi-dimensional solver for Vlasov-Poisson/Darwin equations is designed and validated. The simulation code is implemented and deployed on parallel distributed-memory supercomputers utilizing domain decomposition and the MPI library. In contrast to other implementations, such as Vlasiator [28], which is designed for global space weather near Earth, and Gkeyll [29], which is dedicated to gyrokinetic modeling of fusion devices, the simulation code developed in this dissertation is specifically designed for modeling collisionless plasma flows in space engineering. Throughout this dissertation, the code are applied to three applications related to space plasma flows: collisionless plasma expansion, electronthermodynamicsinelectricpropulsionbeamemission, andelectromagnetic wave/instabilities. 1.2 Literature Review This section presents a review of previous research that pertains to the focal areas addressed in this dissertation. 1.2.1 Plasma Expansion: Analytical and Semi-Analytical Solutions Collisionless plasma expansion is a classical subject in plasma dynamics. Exploring the basic plasma expansion problem in a simplified, reduced-dimension one-dimensional geometry can contribute to the understanding of the physical process underlying more complex problems. This subsection will focus on the review of previous investigations on collisionless plasma expansion under the reduced-dimension one-dimensional model. 1.2 Literature Review 6 Extensive theoretical studies have been conducted on the collisionless plasma expansion process, particularly within the framework of the one-dimensional model. One commonly used approach in studying one-dimensional plasma expansion is the semi-infinite assumption, which assumes an infinite energy supply during the expansion process. The initial unperturbed plasma extends infinitely in the region with x < 0, while a sharp boundary is present at x = 0. The perturbed expansion region consumes energy, which can be replenished by the infinite region of unperturbed plasma. Plasma expansions that exhibit similar characteristics are referred to as semi-infinite plasma expansions. The pioneering work on the semi-infinite plasma expansion was undertaken by Gurevich et al. [30], followed by subsequent studies by the same authors [31–34]. Notably, a well-known self-similar solution has been derived, which can be expressed as follows: N = exp ( 1)!n e =Zn i =Zn 0 exp ( 1) u = + 1!v i =C s ( + 1) =( + 1)! = k b T e e ( + 1) (1.3) The self-similar solution (1.3) characterizes the behavior of plasma expansion over the ion time scale within the region bounded by the upstream propagating ion acoustic wave and the expansion front. This solution is derived under the following assumptions: 1. The solutions presented in this study are applicable within the region defined by an upstream propagating ion acoustic wave and a downstream propagating expansion front. 2. Cold ions (i.e. T i 0 or T i T e ). 3. Electrons in equilibrium and with Maxwellian distribution (i.e. electrons can be described with Boltzmann relations). 4. Quasi-neutrality is assumed (i.e. no charge separation). 1.2 Literature Review 7 Similar results, based on the aforementioned assumptions, were also obtained simultaneously by other researchers [35–37]. Although the self-similar solution has proven successful in explaining various experimental results, its validity is restricted by the four assumptions/constraints mentioned earlier. While the ion temperature can typically be neglected in many plasma expansion applications, there are specific scenarios where the ion and electron temperatures are comparable. Consequently, the effects of assumption 2 on the self-similar solution have become a compelling research topic, with several studies focusing on this area. Gurevich et al. [30] investigated the potential self-similar solution under non-zero ion temperature, although the characteristics of the solution were not extensively explored. Medvedev [38, 39] addressed the problem from a hydrodynamic standpoint and derived a solution to the hydrodynamic equations in the form of a self-similar rarefaction wave. The solution was found to be in good agreement with both numerical simulations and experimental results [39]. Mora [40] also studied the expansion problem with non-zero ion temperature. The solution obtained was more accurate compared to Gurevich’s work. It was discovered that ion cooling results in a substantial ion heat flux due to the distortion of the ion distribution function. Additionally, the electron heat flux exhibited a significant increase compared to the self-similar model. It is important to note that all the studies mentioned in this paragraph employed the quasi-neutral approximation and the electron Boltzmann model. Considering assumption 4, it is observed that eq. (1.3) implies the ions are accelerated to infinite velocity, which contradicts experimental and numerical observations. Several attempts have been made to resolve this contradiction. Crow et al. [41] quantified the effects of assumption 4 and found that charge separation effects should be taken into account. They identified criteria for when assumption 4 becomes invalid, but did not provide a quantitative mechanism to describe the charge separation effects at the expansion front. Mora [42] conducted a quantitative study on plasma expansion using a theoretical and numerical approach, incorporating assumptions 1, 2, and 3, but not assumption 4. Quasi-neutrality was not assumed, and instead, the Poisson equation was utilized. A double layer was observed 1.2 Literature Review 8 to develop at the forward propagating plasma expansion front. The location at the center of the double layer defined the plasma expansion front, and an semi-analytical expression for the electric field E front was derived based on modifications of expressions from previous works [41, 43]. No ion bump near the expansion front was identified, a finding further supported by extensive numerical investigations conducted by Allen et al. [44]. Notably, Mora’s work employed an isothermal model for the electrons, neglecting electron effects. The study suggested that the region bounded by the rarefaction wave and the expansion front could be largely treated as quasi-neutral, but the assumption failed at the left edge of the double layer. Consequently, the determination of the downstream boundary where the self-similar solution remains valid became a question of interest. Medvedev [45] addressed this problem using a similar initial setup to Mora’s model [42], with a focus on the boundary of the quasi-neutral region. Assumptions 2, 3, and 4 were employed to establish a model [38], and a characteristic line analysis of the model was conducted. The analysis revealed three regions where quasi-neutrality breaks down. The first region occurs when t is small in the characteristic plane. The second region arises due to the discontinuity between the unperturbed region and the rarefaction wave front. The third region corresponds to the double layer near the expansion front. y adopting the criterion that quasi-neutrality fails when the Debye length equals the scale length of the rarefaction wave, Medvedev derived an analytical expression for the downstream boundary of the quasi-neutral region. The study verified that as time progresses, the expansion front calculated from Mora’s work [42] deviates from the quasi-neutral front [45], and the self-similar solution is bounded by the rarefaction wave front and the quasi-neutral downstream front. The mechanisms of the electron cloud in front of the expansion front were investigated by Rhodes [46, 47] under a slightly different setup. Nearly all of the aforementioned works have neglected electron kinetics (assumption 3). However, to accurately study the plasma expansion process, it is necessary to include electron kinetics. Mora [48] addressed electron kinetics by relaxing assumption 3. Their results demonstrated that the self-similar solution should be modified when considering 1.2 Literature Review 9 electron kinetics, as electrons transfer energy to ions leading to electron cooling. This suggests that the isothermal model may not be sufficient. Numerical studies are required to further quantify these effects. Denavit [49] investigated the effects of electron kinetics in the one- dimensional expansion of semi-infinite collisionless plasmas. Fully kinetic simulations were conducted, considering both ions and electrons, while assumptions 3 and 4 were dropped, and assumption 2 was retained in the simulations. It was found that electron cooling followed a polytropic relation instead of the isothermal model, with the polytropic factor dependent on the electron-to-ion mass ratio. However, due to computational limitations, Denavit’s work only simulated the region close to the region perturbed by the ion rarefaction wave. Additionally, Denavit’s work exhibited noisy data due to the discrete noise of the particle simulations. Mora and Grismayer [50, 51] investigated electron kinetics in plasma expansion and found that electrons did not maintain a Maxwellian distribution during the expansion process; instead, their velocity distribution function transformed into a “top-hat” shape. The distortion of the electron velocity further accelerated the ion-scale dynamics of the expansion. Kiefer et al. [52] quantified the effects of the electron distribution, comparing them to the previous assumption of electron equilibrium (Maxwellian distribution). Their results demonstrated that the electron distribution significantly influenced the expansion process. Kiefer’s subsequent work [53] examined cases with a step-like distribution and found significantly different results from those assuming an electron Maxwellian distribution, highlighting the importance of resolving electron kinetics. Medvedev [54] investigated plasma expansion using two species with a small mass ratio and observed hump structures in the heat flux of the lighter species outside the region perturbed by the rarefaction wave, along with the formation of a “cooling” wave. However, the formation mechanism of the “cooling” wave was not explored, and the cases of electron-ion plasmas were not investigated. In summary, while extensive research has been conducted on the one-dimensional semi- infinite collisionless plasma expansion, most studies have primarily focused on the dynamics at the ion scale and the effects within the region perturbed by the ion rarefaction wave. Detailed investigations are required to understand the cooling effects of electrons and the 1.2 Literature Review 10 region upstream of the rarefaction wave front. Given the temporal and spatial scale differences betweenelectronsandions, simulationsutilizingmethodsthatexhibitlownoiseandencompass large domains should be performed. 1.2.2 Electric Propulsion Beam Though the one-dimensional reduced model can help understand the basic plasma expansion mechanisms in electric propulsion plume dynamics, studies incorporating higher dimensions are required to further comprehend the coupling of expansions in different directions. Only a few analytical/semi-analytical models have been established based on fluid assumptions over the past decades. Typically, these models employ a simplified equilibrium electron model and a set of steady-state cold ion fluid equations. Park and Katz [55] developed a self-similar solution for the steady-state neutralized beam. In their work, the beam’s axial velocity is assumed to vary only along the axis direction and remain constant in the transverse direction. Their solution is valid within a certain range of transverse distances, and beyond this range, the solution becomes unphysical. Ashkenazy and Fruchtman [56] presented a different approach to obtaining a semi-analytical solution. In their approach, the beam’s axial velocity is allowed to vary with transverse distance. Their solution exhibits a similar feature compared to Park and Katz’s solution. However, it is also invalid for large transverse distances. It should be noted that these solutions only work within a certain propagating region. Korsun and Tverdokhlebova [57, 58] provided a self-similar solution that resolves this issue. In their solution, the beam’s axial velocity is allowed to change differently in the transverse direction based on both the distance to the axis and the adiabatic constant. Their solution provides physically meaningful values for large transverse distances and does not require a constrained propagating region. Additionally, their solution better satisfies the conservation of momentum in the axial direction. However, their solution exhibits a larger error compared to the previously mentioned solutions. 1.2 Literature Review 11 Camporeale et al. [59] generalized the self-similar model and presented a more compre- hensive solution that encompasses the previously mentioned solutions as specific cases. It does not require a specific propagating region while reducing the error. This new solution can also be applied to cases with non-zero azimuthal direction velocity. Merino et al. [60] established an asymptotic solution and compared it to the self-similar solution and exact numerical integration of the control equations. They found that the asymptotic solution can provide a reusable solution for different Mach numbers and tolerate more flexible initial plume profiles but with a larger error downstream. On the other hand, the self-similar solution is more accurate but limited to specific initial profiles and requires re-initialization for different Mach numbers. It is also noted that the selection of the electron cooling mechanism (such as the choice of the polytropic index) plays a significant role in determining the expansion, and the details of electron kinetics need further investigation. While these analytical/semi-analytical solutions can provide reasonable estimations of plume dynamics, they are merely approximations of the exact solution and only offer steady- state estimates. Additionally, several assumptions, such as cold ions and adiabatic/isothermal electrons, are employed without rigorous assessment. Under more realistic conditions, these solutions may lead to inaccurate representations of the underlying physics. To gain a better understanding of the physical mechanisms involved in plume expansion, numerical modeling becomes an essential approach. There are several numerical modeling methods available for studying plasma plume expansion, including fluid, kinetic, and hybrid modeling approaches. Among these options, fluid modeling has the lowest computational resource requirements. In fluid modeling, plasmas are treated as being in an equilibrium state, allowing the fluid equations to be derived from the Vlasov/Boltzmann equations. This reduced-order treatment of plasmas significantly reduces computational costs by ignoring the velocity space. Ortega and Mikellides have utilized a multi-fluid code in their works [61–63] to investigate the plume dynamics in the near-region of Hall thrusters. However, the application of the fluid approach is primarily limited to the near-region plume, where the plasma is mostly collisional and can be considered in equilibrium. 1.2 Literature Review 12 Conversely, in the far-region plume expansion, where collisions can be neglected, the plasmas are typically not in equilibrium, raising questions about the validity of using fluid equations. Another important issue that restricts the use of fluid equations is the closure problem. As the fluid equations are derived from kinetic equations, there is always one more unknown variable than the number of equations. Therefore, an appropriate closure relation is required to close the fluid equations. Typically, the closure can be implemented at either the pressure tensor level or the heat flux tensor level. However, the widely-used closures are typically valid only under the assumption that the plasmas are fully ionized and collisional. Since far-region plasma plumes from electric propulsion devices are typically collisionless, the applicability of these widely-used closures becomes questionable. It is necessary to conduct kinetic analyses to justify the use of these closures. As a result, the fluid approach is rarely used for studying plasma plume expansion problems. The hybrid model resolves the kinetics of heavier species such as ions and neutrals while treating electrons as a fluid. Due to significant differences in time scale and spatial scale between electrons and heavier species, electrons can be considered empirically to be in a steady and equilibrium state, allowing them to be treated as a fluid. The ions and neutrals are typically solved kinetically using Particle-in-Cell methods. Over the past few decades, hybrid modeling has been successfully employed to study various plasma plume-related problems and realistic engineering scenarios. Wang et al. have developed a hybrid Particle-in-Cell (PIC) code for investigating plasma plume dynamics [64–66]. In this code, ions are treated as particles, while electrons are considered as an isothermal massless fluid. The code has been applied to the study of charge exchange ions in the plume of NSTAR ion thruster [67] as well as the charge exchange ions in the plume environment of the Deep Space 1 mission [18, 20]. Subsequently, the code was further enhanced by incorporating an immersed finite element solver to accommodate more complex spacecraft geometries [68–70]. In the same decade, Oh et al. [71] developed a hybrid Particle-in-Cell code that employed the electron Boltzmann relation to investigate the plume of Hall thrusters. Roy et al. utilized a hybrid Particle-in-Cell code with Boltzmann 1.2 Literature Review 13 relation for electrons [72] and a modified Boltzmann relation for electrons [73] to study the backflow and contamination in ion thruster plumes. Recently, Korkut and Levin [74–76] have developed a hybrid code that treats ions as particles and electrons as an isothermal massless fluid and couples with the DSMC method to model neutral dynamics. Although the hybrid models mentioned above have been successful in various applications, there is a fundamental issue with these models: they do not adequately capture the electron cooling process due to the use of electron isothermal assumption. The assumption of electrons as a massless isothermal fluid contradicts the electron temperature observed in experimental measurements. Several studies have been conducted to develop improved electron models. Boyd et al. developed a detailed electron model [77], but it is based on the collisional assumption, making it questionable for simulating far-field plume dynamics. An alternative approach is the use of polytropic models for electrons, which have been employed in various studies due to their ability to better represent the electron cooling process compared to assuming a constant electron temperature. Taccogna et al. [78] utilized a polytropic relation in their studies to examine the plume emitted from a multi-channel Hall thruster. Brieda and Wang developed a hybrid PIC code that incorporates both the Boltzmann relation and polytropic relation to investigate plume expansions and their interaction with spacecraft [79–81]. Experimental investigations [82] were also conducted to explore the polytropic coefficients in the Hall thruster plume and compare them with results obtained from the DRACO code. The experimental findings confirmed the validity of the DRACO code and established it as the primary tool used by the Air Force Research Lab (AFRL) for simulating plume-spacecraft interactions during that period. Martin et al. and the AFRL team [83– 85] developed a code utilizing polytropic relations to investigate the dynamics of electric propulsion plumes, building upon the previously employed DRACO code [86]. Cichocki et al. [87, 88] developed a polytropic electron model in their studies of plume interactions with nearby objects. While the improved simplified model provides better resolution of the physics compared to the isothermal model, it still represents an empirically derived ad hoc model, requiring verification through fully kinetic analysis and further investigations. In summary, 1.2 Literature Review 14 although hybrid models have demonstrated success in engineering applications, the adoption of different electron closure models raises concerns regarding the accurate representation of plume physics. Appropriate electron closures must be investigated and incorporated into the hybrid models. In order to account for the electron cooling process in the expansion of plasma plumes, it is essential to utilize fully kinetic analysis and modeling techniques. However, the intrinsic complexity of kinetic processes restricts the availability of analytical and semi-analytical results in this regard. Merino et al. [89] developed a semi-analytical model for electrons in plasma plumes using a paraxial approximation. It is observed that electrons can be categorized into different regions in phase space, and each region contributes differently to the cooling mechanism. A lumped cooling model is developed by studying the cooling mechanisms of different electron groups [89, 90]. This model offers a computationally efficient alternative to numerical simulations, but its validity is limited by the use of the paraxial approximation. Furthermore, since the model assumes steady-state electrons, it cannot self-consistently determine the filling process of the double trapped region, highlighting the need for time-dependent modeling approaches [91, 92]. Fully kinetic numerical modeling is a valuable approach for comprehending the intricacies of plasma plume dynamics. Brieda and Wang [80] made significant contributions to the field by developing the fully kinetic code DRACO for studying plasma plumes generated by ion thrusters. This work marked one of the earliest instances of utilizing three-dimensional fully kinetic simulations for plume analysis. Subsequently, Wang et al. employed fully kinetic simulations to examine the neutralization and subsequent expansion processes in a vacuum [22]. Their findings revealed that the neutralization process primarily occurs through electron-ion coupling rather than instabilities. Additional studies conducted by Wang et al. [93] investigated the effects of ground facilities, which revealed that the potential significantly differs from that observed in vacuum conditions. Recently, Jambunathan and Levin have developed a fully kinetic Particle-in-Cell code that is coupled with the DSMC method and parallelized using MPI and CUDA [94]. This code has 1.2 Literature Review 15 been employed to investigate the dynamics and backflow region of plasma plumes [94] and the interactions between plumes and spacecraft [95]. The aforementioned studies have explored various plume properties, including electron kinetics, and have achieved successful results. However, the validity of the electron Boltzmann relation and its impact on the underlying physics mechanisms have not been quantified. In order to address this issue, Hu and Wang conducted a comprehensive investigation using a fully kinetic particle-based code. Their research revealed that electrons deviate from a Maxwellian distribution during the expansion process [96] and demonstrated the existence of anisotropic properties in electron temperature and cooling during the expansion [23]. In the study conducted by Hu and Wang [97], direct comparisons between hybrid Particle-in-Cell and fully kinetic Particle-in-Cell codes were performed. The results revealed that the expansion characteristics varied in different directions for fully kinetic simulation results, primarily due to electron temperature anisotropy. It was also determined that the polytropic relation did not improve the ability to address electron temperature anisotropy. These findings highlight the inadequacy of both the electron Boltzmann relation and the polytropic relation for plume simulations, as they fail to capture the underlying physics mechanisms. Wang and Hu [24] quantified the effects of the electron Boltzmann relation by comparing the results obtained from hybrid and fully kinetic models. The comparison revealed significant differences in the potential between the two models, amounting to approximately 40%. Hu and Wang’s extensive research series explored the discrepancies between hybrid and fully kinetic models, emphasizing that current hybrid models are unable to accurately resolve electron cooling anisotropy. To employ an electron fluid assumption, a model capable of addressing electron temperature anisotropy must be developed. The first step towards developing such a model involves a deeper understanding of the cooling mechanism and energy transfer mechanisms in plume expansion. As the physical properties involved are high-order moments of the velocity distribution function, the study of energy transfer processes in plume expansion is hindered by the statistical noise inherent in particle-based codes. Thus far, no similar investigations focusing on the energy transfer process related to electric propulsion 1.2 Literature Review 16 plume study have been identified. To overcome this limitation, a low-noise method must be employed to comprehensively examine the energy transfer process in the electric propulsion plume expansion and its relationship with electron kinetics. In summary, analytical and semi-analytical solutions for plasma plume expansion problems are limited due to their inherent complexity. Fluid modeling approaches for plasma plumes are questionable as they do not adequately account for the collisionless nature of the plasma plume, and the search for an appropriate closure remains ongoing. While hybrid models have shown success in engineering applications, recent studies have revealed their inability to capture electron anisotropies in the cooling process, and no simple electron closure, such as the polytropic model, has been found. Fully kinetic Particle-in-Cell studies offer insights into electron temperature anisotropies, but they are impeded by statistical noise. The energy transfer processes and their mechanisms in plume expansion from a kinetic perspective remain an open question, requiring the development of a low-noise method to address this challenge. 1.2.3 Kinetic-Range Turbulence and Microinstability in the Solar Wind A critical aspect of plasma physics in space that requires understanding is solar wind plasmas. Solar wind (SW) refers to the continuous flow of magnetized plasma emitted by the Sun. Instabilities and turbulence are important factors governing energy and momentum transfer processes in solar wind plasma. Consequently, understanding the complex dynamics of these phenomena within solar wind plasma holds significant implications for ensuring the success of future space missions. Plasma turbulence can be defined as an ensemble of turbulent and incoherent electric and magnetic field fluctuations with broadband features in the field spectrum. Here we use the term space plasma turbulence for the plasma turbulence in the solar wind. Despite the complex nature of energy transport in plasma turbulence, its behavior on a larger spatial scale can be analogous to fluid turbulence [98]. Energy injection at a larger scale results in 1.2 Literature Review 17 the development of significant large-scale structures that subsequently fragment into smaller components, continuing until energy dissipation generates heat (forward cascade). At long wavelengths, specifically when kd i 1, the magnetic fluctuation spectra of space plasma turbulence adheres to the “5/3” scaling law [99], denoted asP (k ? )k ? 5=3 . This corresponds to the “inertial range” of space plasma turbulence. In-situ observations reveal that magnetic fluctuation spectra become steeper as the wavelength decreases [100], and this transition to steeper spectra typically occurs at the length scale kd i 1 for solar wind plasmas in the low plasma pressure to magnetic pressure ratio () regime [101]. As the wavelength decreases, the space plasma enters the “kinetic range”, characterized by a different energy transport process. In the kinetic range, most of the energy cascades and dissipates as heat due to wave-wave and wave-particle interactions, and can be influenced by the formation of current sheets and magnetic reconnection [102–104]. This mechanism is distinct from fluid turbulence, where dissipation relies on particle collisions. The transition from non-dispersive MHD modes to dispersive waves occurs as the wavelength crosses the boundary between these two ranges [105], with potential candidates for these waves being kinetic Alf´ ven waves and higher frequency magnetosonic-whistler fluctuations [106], as observed in recent solar wind observations [107–111] and simulations [106, 112–115]. Given the nonlinear characteristics of cascade and dissipation in space plasma turbulence, numerical kinetic studies are crucial. Particle-in-Cell (PIC) methods are commonly used due to their implementation simplicity and manageable computational requirements. Considering the observed anisotropic magnetic field fluctuations in space plasma turbulence [114, 116, 117],kinetic simulations exceeding one spatial dimension are necessary. Initial studies by Gary et al. [118, 119] and Saito et al. [120–122] involved 2.5D PIC simulations of Whistler turbulence, confirming anisotropy in magnetic fluctuations and steep spectral slopes during the forward cascade process. This exploration was extended by Chang et al. [123–126] and Gary et al. [127, 128] to encompass 3D kinetic PIC simulations, validating the earlier 2.5D findings. These investigations highlighted that energy dissipation relies on the initial spectrum strength, with linear Landau damping dominating for small and nonlinear processes taking 1.2 Literature Review 18 precedence for larger . While the mentioned studies focused solely on electrons with fixed ion backgrounds, Hughes et al. [105, 129] and Gary et al. [130] included ions to explore ion and electron heating in kinetic Whistler turbulence. Notably, electron heating exceeded ion heating, and electron and ion temperature anisotropies were observed. Hughes et al. [105] also investigated ion and electron heating in the context of kinetic Alf´ ven waves. Recent computational advances enabled 3D simulations of kinetic Alf´ ven wave turbulence by Hughes et al. [131], exploring ion and electron heating scaling in relation to the initial spectrum strength. Grošelj et al. [132] also explored this problem with 3D simulations, validating sub-ion scale cascade patterns aligned with theoretical predictions for kinetic Alf´ ven wave turbulence and uncovering associated heating mechanisms. The reviewed studies predominantly explore particle energy transport and heating mech- anisms, primarily focusing on global field-averaged parameters. Yet, plasma turbulence’s capability to generate pronounced gradients, thereby enhancing local temperature anisotropies [133–135] and other localized phenomena, emphasizes the significance of investigating these effects. These local influences have a direct impact on turbulence dissipation processes, underscoring the necessity to delve into their characteristics and mechanisms. PIC’s discrete particle noise rises challenge to model plasma phase space. While even a small number of particles can capture coarse plasma dynamics, inherent statistical noise of PIC make it difficult to study small-scale phenomena or recover local distribution functions. Recent studies by Camporeale et al. [136] quantify the effects of the PIC-induced noise in space plasma turbulence. The presence of particle noise has been identified to lead to a spurious inverse cascade of energy, generated numerically, which consequently introduces inaccuracies in regions characterized by large wavenumber modes. High signal-noise ratios are essential in these areas to ensure accurate resolution of underlying physical mechanisms. Haynes et al. [26] further demonstrated the implications of particle noise on investigating localized effects within plasma turbulence. Their findings reveal that the reliability of the spectrum is compromised above a threshold of k e 4 due to the noise floor. Over 6400 particles per cell are required to mitigate such noise, significantly challenging computational demands. As noise 1.2 Literature Review 19 levels scale inversely with the particle count (Error 1= p N p ), noise suppression through particle addition is inefficient. An alternative approach involves employing a grid-based method, intrinsically free from noise. Valentini et al. employ an ion-Vlasov, electron fluid model to investigate the influence of small-scale fluctuations on larger-scale space plasma turbulence [137, 138], uncovering substantial impacts of these small-scale structures on the broader turbulent plasma dynamics. Sevidio et al. [139–141] analyzed the influence of local kinetic effects on space plasma turbulence using the aforementioned hybrid models. Their outcomes emphasize the significant role of local kinetic effects in elucidating turbulence intermittency and its association with local temperature anisotropy. It’s noteworthy that their model exclusively resolves ions on grids in the phase space and thus ignoring electron kinetic effects. These inquiries highlight the importance of employing low-noise, high-fidelity methodologies for studying space plasma turbulence. Local kinetic effects emerge significance in explaining the origin of local temperature anisotropy, where turbulence generates sharp gradients leading to enhanced anisotropy. However, an alternative mechanism could also be in play. Predicted by linear theory [4], temperature anisotropy can trigger kinetic instabilities, which, in contrast to turbulence, typically impose an upper limit on anisotropy. This prompts a fundamental question: How can a heterogeneous nonlinear phenomenon like turbulence coexist with anisotropy constraints derived from the linear theory of homogeneous plasmas and what is the role of local kinetic effects in this competing process? Qudsi et al. [142] and Bandyopadhyay et al. [143] conducted 2D PIC simulations of Alf´ venic turbulence, investigating turbulence-induced ion temperature anisotropy. Their results demonstrated potential micro-instability emergence due to this anisotropy, have global impacts on the plasma. A clear linkage between linear instability theory and highly intermittent turbulence was established, and a similar process on the electron scale was predicted. The specific role of local kinetic effects remains unexplored. To address this fundamental problem, a method characterized by minimal noise and high accuracy should be adopted to examine the involvement of local kinetics in the complex interplay between turbulence and instabilities. This is particularly important since the 1.2 Literature Review 20 inherent noise within the widely accepted PIC method could potentially introduce unphysical phenomena. To investigate the aforementioned effects, it is imperative to develop a low-noise, high-fidelity method. 1.2.4 Grid-based and Particle-based Method for Kinetic Plasma Simulations As mentioned previously, the plasma flow problems studied in this dissertation necessitate a numericalkineticapproachthatensureslownoiseandhighfidelity. Suchanapproachiscrucial for investigating the underlying physics mechanisms and gaining a deeper understanding of these applications. Therefore, the selection of appropriate methods holds significant importance. In this section, the significance of grid-based methods is discussed in relation to the two primary kinetic approaches, namely the particle-based approach and the grid- based approach. The discussion primarily focuses on comparing the noise effects and fidelity associated with these approaches. Particle-based methods, particularly Particle-in-Cell (PIC) methods, have gained widespread acceptance as a numerical approach for investigating various problems [96, 131, 132, 144–148]. These methods are favored due to their ease of implementation and relatively low computational costs. By employing discrete macro-particles, the computational burden associated with solving complex problems is significantly reduced. However, the introduction of statistical noise is an inevitable consequence of this approach. The presence of statistical noise poses a challenge, as any physical properties with magnitudes smaller than the noise level remain obscured. While employing a large number of particles can mitigate the effects of statistical noise, it also diminishes the computational advantages offered by PIC methods. Moreover, it is worth noting that even when the inherent statistical noise is reduced, it can still lead to unphysical phenomena in diverse physical problems. Denavit conducted pioneering work comparing grid-based methods and particle-based methods [149], specifically examining the two-stream instability and electron oscillations. 1.2 Literature Review 21 The study revealed that discrete particle effects in particle-based methods introduce beaming instabilities that can interfere with the accurate representation of physical mechanisms. Subsequently, researchers across various communities have identified limitations and the generation of unphysical phenomena associated with Particle-in-Cell (PIC) methods. Nevins et al. [6] investigated the discrete particle issues related to electron temperature gradient turbulence. They observed significant discrepancies between the results obtained using particle-based methods [7] and those obtained using grid-based methods [150]. Nevins et al. demonstrated that particle noise alters the growth rate of initial-stage instabilities, thereby invalidating estimations of heat transport at later times and resulting in an unphysical mechanism associated with particle-based methods. Tavassoli and Smolyakov [151] compared the influence of Particle-in-Cell (PIC) methods and grid-based Vlasov methods on the growth rate of the Buneman instability. Their findings revealed significant inconsistencies between PIC results, even with a quiet start, and the theoretically predicted linear growth rate. The inherent particle noise in PIC methods hindered the resolution of small-scale phase space structures, while the small-scale distortion of the electron velocity distribution function greatly affected the linear growth rate of the Buneman instability. Consequently, the particle-based method yielded significantly different predictions for the linear growth rate of the Buneman instability, suggesting that the insufficient resolution of microscopic phase space structures due to particle noise leads to unphysical macroscopic properties. Additionally, investigations of the electron cyclotron drift instability (ECDI) in the Hall thruster community revealed that PIC simulations may lack a saturation stage [152], while recent grid-based Vlasov simulations demonstrated energy saturation results for ECDI [153]. Juno et al. [154] recently discovered that inherent noise in PIC methods causes fluctuations in current density, which in turn lead to unphysical magnetic field saturation. Furthermore, studies comparing PIC and grid-based Vlasov simulations of the Hall thruster discharge chamber found that PIC results exhibit plasma with non-correlated features, while grid-based Vlasov methods reveal more coherent structures [155]. Collectively, these findings suggest that the failure to resolve small-scale structures may impede an accurate understanding of the underlying physical mechanisms. 1.2 Literature Review 22 Moreover, an insufficient resolution in phase space can lead to inaccuracies in the high- order moments of the kinetic equation in inhomogeneous plasma. These high-order moments, often represented as kurtosis, skewness, and other attributes of the velocity distribution function (VDF), are highly sensitive to the numerical reconstruction of this function. In low-density regions of inhomogeneous plasma, particle numbers are typically low, resulting in substantial statistical noise. This noise leads to poorly resolved VDFs and consequently, significantly impacts the high-order moments. One approach to improving the fidelity of the VDFs in these regions is to drastically increase the particle numbers. However, due to the system’s inhomogeneous nature, this could result in an excessively high particle count in the high-density region, complicating simulation efforts. The plasma sheath serves as an example of such an inhomogeneous system. Within the sheath region (the region of interest), the plasmas exhibit low-density profiles, making it challenging for the particle-based method to accurately resolve the physical properties. Sanchez-Arriaga [156] examined the sheath region near a Langmuir probe using a grid-based Vlasov method. The study found that accurately evaluating a transient current in the sheath region using the PIC method is challenging, whereas the grid-based Vlasov method can resolve it without any associated noise, emphasizing the capability and advantages of the grid-based Vlasov method in studying inhomogeneous plasma systems. Given the significant concern with particle noise issues in correctly resolving physical mechanisms, several attempts have been made to address the inherent statistical noise issues in the PIC method. Techniques such as f-PIC [157], the remapping method [158], and the high order PIC method [159] have been developed to decrease the noise. These methods offer a means to mitigate the noise inherent in PIC methods while retaining the computational efficiency advantages of PIC methods. The grid-based Vlasov method, also known as the Vlasov method, is an alternative kinetic approach for studying plasma dynamics. Unlike particle-based methods, the grid-based Vlasov method deterministically solves the discretized kinetic equations in phase space, making it inherently noise-free. This feature makes the grid-based Vlasov method an excellent choice for studies requiring low noise levels and high resolutions of fine phase space structures. 1.2 Literature Review 23 The grid-based Vlasov method has been known in the plasma physics community since the pioneering works of Knorr and his colleagues in the 1970s [27, 160, 161]. In 1976, Cheng and Knorr published their groundbreaking work on the split-form grid-based Vlasov method [27], which sparked significant interest and subsequent research in both algorithm development and applications. The Vlasov method has achieved success in various one-dimensional problems, providing insights into physical mechanisms without being influenced by numerical noise in fusion [162] and astrophysics [163] communities. However, due to computational limitations in the last century, most studies using Vlasov methods were restricted to one-dimensional systems (1D1V), one dimension in physical space and one dimension in velocity space. As a result, many researchers in the plasma dynamics community turned to particle-based methods due to their ease of implementation and relatively low computational cost, making the PIC method suitable for studying high-dimensional problems. However, recent advancements in high-performance computing techniques have enabled pioneering works in the community of astrophysics [28, 164–166] and fusion [167] that explore high-dimensional problems using the grid-based Vlasov method. Despite the renewed interest in Vlasov simulations in recent years, there are still only a few studies utilizing this method due to the challenges associated with developing and utilizing a Vlasov code. Additionally, Vlasov simulations are rarely employed in the space science and engineering community to study plasma flow dynamics. Given that plasma flows in space are typically inhomogeneous systems involving various kinetic processes, there is value in developing the grid-based Vlasov method and a high-performance parallel multi-dimensional Vlasov solver coupled with different field solvers. This approach would enable noise-free investigations of these problems and advance our understanding of plasma dynamics in space. Developing a high-performance parallel multi-dimensional Vlasov solver, coupled with different field solvers, is a challenging and time-consuming task. To determine the necessity and appropriate applications of the grid-based Vlasov solver, it is essential to comprehend the trade-off between computational cost and accuracy inherent in both PIC and Vlasov methods. Furthermore, an understanding of the capabilities of these methods in reducing numerical 1.2 Literature Review 24 (a) (b) Figure 1.2: Comparisons of the results from PIC and Vlasov method on a linear Landau damping problem. (a) Error v.s. CPU time. (b) Efficacy v.s. CPU time. errors is crucial. To examine these characteristics, we conducted a series of grid-based Vlasov simulations with varying velocity space grid cells, as well as a series of PIC simulations with varying particles per cell. These simulations were applied to a linear Landau damping problem [168]. We quantitatively compared the outcomes obtained from the PIC and grid-based Vlasov methods, and the results are presented in Fig. 1.2. In order to mitigate the statistical noise inherent in the particle-based method, we employed the “quiet start” technique [5] 1.2 Literature Review 25 in our PIC simulation. As for the grid-based Vlasov method, we utilized a 3rd order PFC scheme [169] for the simulations in the comparisons. In the physical space, both species utilize 128 cells, while the number of velocity space cells and particles per cell are varied in both ensembles. The reference case for this study is determined to be the Vlasov simulations, which use 8192 cells in the velocity space. In this study, the error is quantified as the L 1 norm of the difference between the electric field strength at t! pe = 35 in the reference case and the given case. Furthermore, an indicator referred to as “efficacy” [168, 170] is employed to assess the combined impact of both the cost and accuracy across different methods. In this context, “efficacy” is defined as the inverse of the product of the error level and CPU time. A higher efficacy implies an algorithm that utilizes fewer computational resources to achieve a lower error level. Given that CPU time is linearly proportional to the computational load, it is used here as an indicator of this load, denoted as t CPU . Fig. 1.2a presents a comparison between two methods regarding error versus CPU time. The results indicate that in order to reach an error level of O(10 4 ), PIC methods require CPU time on the order of O(10 2 ), whereas the grid-based method only needs O(10 0 ) level CPU time. Given the same CPU time, at a level of O(10 1 ), the grid-based Vlasov method can achieve an error level of O(10 7 ), while the PIC method can only attain O(10 3 ). This quantitatively demonstrates the superior performance of the grid-based method in resolving fine scale structures in phase space, small scale perturbations, and achieving low error level numerical solutions compared to the particle-based method. Fig. 1.2b provides a quantitative assessment of the performance of both the PIC and Vlasov methods using efficacy as a measure. The results indicate that the grid-based Vlasov method utilizes computational resources more efficiently and with higher efficacy when the computational cost is higher, whereas the PIC method exhibits decreasing efficiency as more particles are employed. This finding suggests that increasing the number of particles does not lead to a favorable accuracy-cost trade-off [168, 170]. Based on these quantitative analyses, it is evident that increasing the number of particles to reduce statistical noise is not an optimal choice for studying inhomogeneous plasma systems with a 1.2 Literature Review 26 low-noise requirement. Instead, the grid-based method emerges as the preferred approach for such studies. Although the grid-based Vlasov method has demonstrated advantages over the PIC method in terms of numerical efficacy and the ability to resolve small-scale perturbations and fine structures, it is essential to consider the computational cost associated with both methods. In this regard, assuming that the computational resources and time required for a single step of operation on a single grid cell in the grid-based Vlasov method are equivalent to those for the super-particles in the PIC method, we introduce the ratio N vlasov N PIC =g PIC N d v (1.4) as defined in Ref. [171], where N represents the required numerical effort, g PIC is the inverse of the number of super-particles per cell, and N d v is the total number of grid cells in the discretized velocity space. In previous PIC studies focused on short wavelength turbulence and micro-instabilities in the solar wind [125, 131, 172, 173], a typical choice was to use a total of 64 super-particles per cell. Specifically, electrons were represented by 32/48 super-particles per cell, while protons were represented by 32/16 super-particles per cell. These studies were conducted in a full 3D space, encompassing three dimensions in physical space and three dimensions in velocity space. It is important to note that for problems related to short wavelength turbulence and micro-instabilities in the solar wind, velocity space typically consists of three dimensions. Using eq. (1.4), it can be observed that in order to achieve the same computational cost as the aforementioned PIC studies [125, 131, 172, 173], the velocity space would need to be discretized with a total of only 4 grid cells in each dimension (equivalent to 2 grid cells for electrons and 2 grid cells for protons). Clearly, such a small number of grid cells is insufficient to adequately resolve the phase space dynamics of the plasmas in any meaningful way. If we consider a reasonable number of 10 2 grid cells per direction in a three-dimensional velocity space for grid-based Vlasov simulations, which is necessary to resolve small-scale perturbations and capture the dynamics of the phase space, 1.3 Motivations and Objectives 27 the computational cost would be equivalent to using 10 6 super-particles per cell in a PIC simulation. Compared to the previously mentioned PIC studies [125, 131, 172, 173], there is a significant increase in computational costs. It is important to note that, as discussed in the previous paragraph on numerical efficacy, the PIC method offers computational cost advantages over the Vlasov method, although they do not achieve the same level of numerical noise and accuracy. To achieve comparable accuracy and low noise levels, a significant increase in the number of super-particles per cell is required, which in turn diminishes the computational cost advantages of the PIC method. Therefore, currently, when considering computational resource savings, the PIC method remains a favorable choice for 3D problems unless there is a specific need to resolve fine-scale physics [171, 174]. In summary, the particle-based method has demonstrated success in various applications; however, the presence of statistical noise can lead to unphysical phenomena. Conversely, the Vlasov method offers a promising approach for studying problems that require low noise and accurate representation of fine structures in phase space, albeit with a high computational resource requirement. Increasing the number of particles in the PIC method does not effectively reduce the noise and instead leads to significantly higher computational costs compared to the Vlasov method. Therefore, despite the challenges associated with understanding and implementing the Vlasov method, it is worthwhile to develop such an approach to gain a deeper understanding of the physics underlying the plasma flow problems of interest in this study. 1.3 Motivations and Objectives The objectives of this dissertation research encompass the development of a Vlasov plasma simulation model to address application problems that require accurate solutions for high- order moments of particle velocity distributions and/or small-amplitude micro-scale plasma perturbations. Additionally, this research aims to conduct three application studies in the 1.3 Motivations and Objectives 28 fields of space engineering and space physics, including: 1) 1-dimensional plasma expansion, 2) 2-dimensional EP beam emission, and 3) plasma waves and instabilities. The first application study is one-dimensional plasma expansion. The one-dimensional semi-infinite plasma expansion model is a valuable tool for comprehending the dynamics of axial plasma beam expansion. Previous studies focusing on this model have primarily concentrated on ion-scale dynamics and have solely considered the region perturbed by the ion rarefaction wave. Although it has been observed that cooling effects can occur outside the rarefaction region in an ion-ion plasma expansion, no studies have specifically examined the electron cooling effects in the region upstream of the rarefaction front. Furthermore, the presence of particle noise in simulations has overshadowed certain physical properties in previous plasma expansion studies, underscoring the need for low-noise simulations with a larger simulation scale. This will enable a better understanding of electron-scale dynamics and physical properties in the region upstream of the region perturbed by the ion rarefaction wave. Additionally, a low-noise model is required to investigate the cooling effects of electrons and the underlying mechanisms for such phenomena. The second application study is two-dimensional electric propulsion beam emission. A two-dimensional model of plasma beam expansion can facilitate the coupling of axial and transverse expansion effects, leading to a more comprehensive understanding of electron energy transfer processes. Recent studies have raised concerns about the validity of the Boltzmann electron fluid model and the polytropic electron fluid model. Electrons have been found to exhibit anisotropic temperature distributions. However, research on the energy transfer process in plasma expansion remains limited, as energy fluxes and heat fluxes are high-order moments of velocity distribution functions and are heavily influenced by particle noise. Furthermore, there is a lack of investigations into the physical relationship between electron temperature anisotropy and microscopic velocity distribution function properties, mainly due to the presence of particle noise. Consequently, the development of a low-noise methodology is crucial for addressing these issues within an inhomogeneous plasma system. 1.4 Dissertation Organization and Outline 29 The third application study focuses on microscopic processes within collisionless plasma. This preliminary investigation aligns with the goal of understanding kinetic-range plasma turbulences and their impact on energy transfer and regulation in the solar wind. Specifically, our focus is on the complex velocity-space physics, which includes a variety of wave-particle interactions between electrons and ions and the various fluctuation modes of the system. Our prior research [175] has showed the tremendous challenge of resolving such physics using the Particle-in-Cell (PIC) method due to the interference of particle noise. At present, the research in this dissertation has not progressed to the point of achieving our eventual goal. Instead, the objective of this preliminary study is to showcase the feasibility of simulating electromagnetic instabilities and waves using the Vlasov simulation method. Within this dissertation, the Vlasov Solver is coupled with the Darwin field solver. Darwin Vlasov simulations are conducted on whistler and Weibel instabilities, with results compared to theoretical predictions. The outcomes demonstrate a high degree of agreement, suggesting the feasibility of employing the Vlasov solver for studying instability and turbulence-related problems in the future. 1.4 Dissertation Organization and Outline This dissertation focuses on the development and implementation of a grid-based Vlasov method in a multi-dimensional parallel solver, with the objective of investigating plasma flow problems in the space science and engineering communities while achieving low noise levels. The first part of the dissertation entails the selection of numerical schemes for the grid-based Vlasov method and the development of a novel coupling method between the Vlasov equation and the field equations. The development and implementation process of a multi-dimensional parallel numerical Vlasov-Poisson solver, referred to as Vlasolver, is presented, and its parallel efficiency is evaluated. Verification and validation procedures are conducted, and comparisons between the solver’s results and those obtained from a benchmarked PIC code are performed. 1.4 Dissertation Organization and Outline 30 In the second part of the dissertation, the focus shifts towards utilizing the Vlasolver for investigating plasma expansion problems. The 1D expansion model is extended to regions upstream of the ion rarefaction region by incorporating electron dynamics. The analysis includes the examination of electron energy transfer processes and the explanation of new phenomena from both microscopic and macroscopic perspectives. The investigation is then expanded to a 2D expansion model, where the electron energy transfer process is further examined. The correlations between microscopic velocity distribution functions (VDFs) and macroscopic electron heat flux are elucidated based on the results obtained from the Vlasolver. The third part of this dissertation is dedicated to the extension of the Vlasolver for electromagnetic simulations. This part focuses on solving the Vlasov-Darwin equations in the electromagnetic version of Vlasolver. A unique coupling technique is devised to effectively incorporate the Vlasov and Darwin equations. Subsequently, an assessment of the parallel efficiency and profiling of the electromagnetic Vlasolver is carried out. To test its accuracy, verification processes are undertaken, with notable agreements observed between the numerical outcomes and theoretical predictions. At the end of this dissertation, an appendix features a Particle-in-Cell study, intended to serve as a benchmark for the future development of fully three-dimensional (3D3V) grid-based Vlasov simulations focused on exploring space plasma turbulence and instabilities. This dissertation comprises a total of seven chapters and one appendix. The remaining chapters of this dissertation are outlined as follows: 1. Chapter 2 provides an introduction to the physical models and numerical methods employed in this study. It provides a novel coupling method between the Vlasov equation and the Darwin equation. Additionally, a summary of the boundary conditions used in this work is provided. 2. Chapter 3 provides a detailed account of the development of the large-scale parallel multi-dimensional grid-based Vlasov code, referred to as Vlasolver. The chapter includes comprehensive discussions on the implementation details of the code. Furthermore, the 1.4 Dissertation Organization and Outline 31 chapter presents the results from a series of verification and validation tests conducted on both electrostatic and electromagnetic problems. 3. Chapter 4 utilizes the Vlasolver developed in this dissertation to explore the electron time scale physics in the one-dimensional semi-infinite collisionless plasma expansion. 4. Chapter 5 leverages the Vlasolver developed in this dissertation to investigate the two- dimensional expansion process of an electric propulsion beam. The emphasis is placed on exploring the relationships between electron microscopic kinetics and macroscopic collisionless heat flux. 5. Chapter 6 presents the development of the electromagnetic version of Vlasolver. It encompasses the introduction of a novel coupling algorithm for the Vlasov and Darwin equations, as well as assessments of the scalability of the electromagnetic Vlasolver. To validate its functionality, two classical physics problems are employed as verification cases. 6. Chapter 7 provides a comprehensive summary of this research and presents the conclu- sions drawn from the studies conducted in this dissertation. The main findings and key insights are highlighted. Additionally, future research directions stemming from this dissertation are outlined. 7. AppendixAutilizesEMPIC(ElectromagneticParticle-in-Cell)simulationstoinvestigate the interplay between whistler turbulence and whistler anisotropy instability in the solar wind. The primary focus is to establish a benchmark for the future development of the Vlasolver. 32 Chapter 2: Physical Models and Numerical Methods Plasma flows in space often exhibit deviations from equilibrium due to the absence of collisions. In such cases, the fluid approach may not be sufficient to accurately describe the plasma behavior, necessitating the application of kinetic theory. However, the kinetic theory equations are inherently complex and nonlinear, making it challenging to analytically study the time evolution of phase space dynamics. Consequently, numerical methods play a crucial role in investigating the behavior of plasma systems. In this chapter, an overview is first given on the governing equations that govern plasma dynamics. These equations capture the evolution of particle distribution functions in phase space. Additionally, the physical model employed in this study is introduced, encompassing the relevant physical phenomena and boundary conditions specific to the problems in this dissertation. Finally, the numerical method used and developed in this work will be introduced. 2.1 Kinetic Theory, First-Principle Governing Equations Plasmas can be investigated using two fundamental approaches: the macroscopic (fluid, thermodynamics) approach and the microscopic (kinetic, statistical) approach [176]. The macroscopic approach assumes that all particles in the plasma are in a single thermal equilibrium state, enabling the use of fluid models to describe the overall behavior of the plasma. While this approach is effective in capturing the collective properties of plasmas, it does not account for the detailed distribution of particles in velocity space or the correlation between velocity and physical space. It has been observed that phenomena arising from the properties of velocity space and its correlation with configuration space can significantly impact plasma behavior, necessitating the adoption of a microscopic approach. In the microscopic approach, velocity space and physical space are combined to form phase space, 2.1 Kinetic Theory, First-Principle Governing Equations 33 wherein the particles composing the plasma are analyzed. The microscopic approach relies on kinetic theory to study plasmas, treating them as a large ensemble of particles evolving in phase space over time [177]. The kinetic theory of gases and plasmas focuses on the time-dependent evolution of a vast number of particles in phase space, considering their equations of motion and statistical characteristics. To fully characterize a plasma composed ofN particles, information about the position and velocity of each particle at a given moment in time is required [176] N (x;v;t) = X 1iN [xx i (t)][vv i (t)] (2.1) where stands for the species . Eq. (2.1), in conjunction with the motion equations for each individual particle and the electromagnetic theory forms the Klimontovich equation @N (x;v;t) @t +v @N (x;v;t) @x + q m (E +vB) @N (x;v;t) @v = 0 (2.2) which is a mathematical equation describing the status of all particles rather than a statistical equation. The kinetic theory encompasses the examination of systems composed of numerous parti- cles. The statistical mechanics approach is commonly employed to investigate such systems. The microstate of these multi-body systems is fully determined by the coordinates of the particles within the phase space. Consequently, the distribution of particles entirely deter- mines the characteristics of such systems. Thus the many bodies particle distribution func- tion F N (x 1 ;x 2 ;:::;x N ;v 1 ;v 2 ;:::;v N ;x 1 ;x 2 ;:::;x N ;v 1 ;v 2 ;:::;v N ;:::;t) can be defined and interpreted as the probability density in the 6D phase space. Within this particular space, the distribution function F N adheres to the principles outlined in Liouville’s theorem. Liouville’s theorem conceptualizes F N as an incompressible fluid residing within the 2.1 Kinetic Theory, First-Principle Governing Equations 34 phase space. Consequently, eq. (2.2) can be reinterpreted, within the framework of Liouville’s theorem, to assume a new form pertaining to F N . @F N @t + X i v i @F N @x i + X i a i @F N @v i = 0 (2.3) From the aforementioned many-body distribution function, it is possible to derive the corre- sponding few-body distribution function. Eq. (2.3) can be expressed as a series of equations for different few-body distributions. This sequence of equations establishes a hierarchical struc- ture known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [176, 177]. In order to study the dynamics of the system, it becomes necessary to close the infinite chain of equations. In plasma dynamics, where the plasma parameter g = (1=n 3 D ) 1, the Mayer expansion can be employed as a series expansion in terms of the plasma parameter g. By terminating the infinite chain of equations at order g s and deriving (s-1) equations, closure can be achieved [176]. For instance, considering the case of order s = 2, the equation f = f f can terminate the infinite chain of equations. This implies that only binary collisions are considered, and the distribution function of two particles is assumed to be uncorrelated [178], ultimately leading to the Boltzmann equation. @f @t +vr x f +ar v f =S = f @t coll (2.4) where f is the velocity distribution function for particle andx andv are physical and velocity space coordinate in phase space respectively. The term a represents acceleration, which in the case of plasma dynamics, is typically influenced by the electromagnetic force and can be written asa =q (E +vB)=m . The termS corresponds to the collision term, which quantifies the effects of particle collisions. In the scenario of a collisionless plasma, the collision term can be neglected (S = 0), resulting in the collisionless Boltzmann equation, commonly referred to as the Vlasov equation. @f @t +vr x f +ar v f = 0 (2.5) 2.1 Kinetic Theory, First-Principle Governing Equations 35 The Vlasov equation serves as the fundamental governing equation for describing the dynamics of collisionless plasmas. Several remarks can be made regarding the Vlasov equation. Firstly, if the acceleration terma is known, the Vlasov equation itself represents a linear hyperbolic partial differential equation (PDE). The velocity distribution function f remains constant along the characteristic lines of the Vlasov equation. Secondly, the Vlasov equation conserves particles [176]. This implies that the rate of change of the total number of particles with respect to time is zero. Thirdly, in the Vlasov equation, the velocity distribution functionf is always non-negative 1 . Lastly, the acceleration term cannot be determined solely by the Vlasov equation. The electromagnetic field equations must be solved to obtain the acceleration term. However, the input for the electromagnetic field equations must be obtained through integrating the velocity distribution function. Conversely, the velocity distribution function itself needs to be obtained from the solution of the Vlasov equation. This demonstrates that although the Vlasov equation is a linear PDE, in order to study plasma dynamics self-consistently, it must be coupled with the electromagnetic field equations, thereby giving rise to a nonlinear PDE. Lastly, the acceleration term cannot be determined solely by the Vlasov equation. The electromagnetic field equations must be solved to obtain the acceleration term. However, the input for the electromagnetic field equations must be obtained through integrating the velocity distribution function. Conversely, the velocity distribution function itself needs to be obtained from the solution of the Vlasov equation. This demonstrates that although the Vlasov equation is a linear PDE, in order to study plasma dynamics self-consistently, it must be coupled with the electromagnetic field equations, thereby giving rise to a nonlinear PDE. 1 Further details and explanations can be found in reference [176]. 2.2 Physical Model 36 2.2 Physical Model The Vlasov equaton needs to be coupled with electromagnetic equations to self-consistently describe the collisionless plasma dynamics. The electric fieldE and magnetic fieldB can be solved self-consistently from the Maxwell’s equations, rE = 0 (2.6a) rB = 0 (2.6b) rE = @B @t (2.6c) rB = 0 J + 0 0 @E @t (2.6d) where the term 0 and 0 denote the permeability and the permittivity of vacuum, respectively. is the charge density andJ is the current density. andJ can be determined from the VDF directly and will illustrated in the following contents. Eq. (2.5) and eq. (2.6) together form a coupled system known as the Vlasov-Maxwell equations. This system serves as the fundamental physical model for describing collisionless plasmas. Depending on the specific approximations made, the Maxwell equations can be simplified further into either Poisson equations or Darwin equations. In these simplified cases, the coupled system is referred to as the Vlasov-Poisson system or the Vlasov-Darwin system, respectively. The physical models used in this work are Vlasov-Poisson and Vlasov-Darwin system. These models work as the kinetic descriptions of the electrostatic and low-frequency electro- magnetic collisionless plasma system. The simulation model can be written as: @f @t +vr x f + q (E +vB) m r v f = 0 (2.7) where q is charge and m is mass for the corresponding particle specie . E and B is electric field and magnetic field respectively. 2.2 Physical Model 37 In the context of an electrostatic model, the magnetic field is considered as a static external field. The electric field is determined by solving the gradient of the electric potential, denoted as E =r. It is necessary to solve the electric potential, denoted as , in a self-consistent manner through the Poisson equation r ( 0 r) = X q n = (2.8) where q , n and denote the charge number, number density for specie and net charge density respectively. In the case of an electromagnetic model, the determination of both the electric field E and the magnetic field B is achieved through a self-consistent solution of the Maxwell equations. When employing explicit numerical integration schemes to solve the Maxwell equations, it becomes necessary to satisfy the CFL (Courant-Friedrichs-Lewy) condition in order to maintain stability of the numerical methods t< x=c (2.9) where c is the light speed in vacuum. This condition arises due to the propagation of light waves being governed by the Maxwell equations, imposing stringent restrictions on the numerical time step. The grid-based method poses significant computational demands due to the direct integration of the Vlasov equation in the discretized phase space. The presence of the CFL condition, imposed by the Maxwell equations, introduces a large number of time steps during the simulation process. Consequently, achieving the simulation objectives within a reasonable computational time frame becomes challenging when employing the grid- based method. One approach to address this issue is to employ the Darwin approximation. This approximation effectively eliminates the light wave mode by neglecting the transverse displacement current in the Maxwell equations. Given that this work primarily focuses on scenarios where velocities are considerably smaller than the speed of light and radiation effects 2.2 Physical Model 38 are not considered, the Darwin approximation is employed. This allows for the self-consistent solution of the electromagnetic fields without the constraints imposed by the CFL condition. In the Darwin approximation, the electric fieldE is decomposed into two components: the transverse part E T and the longitudinal part E L . Both components must satisfy the following requirement: rE T = 0 (2.10a) rE L = 0 (2.10b) Under these requirements, the original Maxwell equations can be rewritten into the following form rE L = 0 (2.11a) rB = 0 (2.11b) rE T = @B @t (2.11c) rB = 0 J + 0 0 @E L @t (2.11d) It can be seen the important physical variables that need to couple Vlasov equations and Darwin equation are the charge density and current densityJ. The macroscopic properties of the plasmas are described by the moments of the VDF <M n >= Z 1 1 M n ^ fdv (2.12) where ^ f = f=n is the density normalized VDF, and n = R 1 1 fdv is the density. Using eq. (2.12), the charge density can be determined by = X q <n >= X q Z n ^ fdv = X q Z fdv (2.13) 2.3 Normalizations and Normalized Models 39 the current density can be determined by J = X q <n v >= X q Z n v ^ fdv = X q Z v fdv (2.14) It should be noted that various other physical variables, such as pressure, average velocity, heat flux, and energy flux, can be derived by integrating different order moments of the velocity distribution function. This implies a connection between microscopic properties, such as velocity distribution functions, and macroscopic variables. 2.3 Normalizations and Normalized Models Within the realm of scientific computing, a common practice is to employ a normalization scheme. This scheme involves scaling the physical variables and governing equations with respect to predefined reference values. By adopting a normalization scheme, the inherent limitations of finite precision in representing floating-point numbers on computers, which give rise to rounding errors, can be mitigated. Consequently, the utilization of a normalization scheme serves to reduce the impact of rounding errors. In this study, all the physical variables have been normalized based on a selected set of normalization factors. It is observed that the equations eq. (2.7) and eq. (2.8)/(2.11) exhibit a characteristic where all physical quantities can be expressed as combinations of five fundamental units: length, time, unit charge, mass, and amount of substance (or alternatively, density). To convert a physical model into a dimensionless normalized model, it is necessary to employ five distinct normalization relationships ~ x = x x 0 ; ~ t = t t 0 ; ~ q = q q 0 ; ~ m = m m 0 ; ~ n = n n 0 (2.15) in which the variable with the subscript \0" means the chosen normalization factor. The remaining physical parameters can be normalized by utilizing the unique normalization relations presented in eq. (2.15). Considering our objective of studying the plasma dynamics 2.3 Normalizations and Normalized Models 40 in a fully kinetic manner, the chosen normalization factors are based on the electron time scale. Hence, the normalization factors are set as follows: x 0 =d; t 0 =! 1 pe0 ; q 0 =e; m 0 =m e ; n 0 =n e0 (2.16) where d;! pe;0 ;e;m e ;n e0 are system characteristic length, electron plasma frequency at initial time moment ! pe;0 = p n e0 e 2 = 0 m e , unit charge, mass of electron, electron density at initial stage of simulation. The determination of the system’s characteristic length is based on the scale of the specific physical problems being simulated. Once these factors are determined, the following normalizations can be applied to the physical variables appearing in the governing equations ~ x = x d ; ~ t =t! pe0 ; ~ v = v d! pe0 ; ~ m = m m e ; ~ n = n n e0 ; ~ T = k b T m e d 2 ! 2 pe0 ; ~ = e m e d 2 ! 2 pe0 ; ~ E = eE m e d! 2 pe0 ; ~ B = eB m e ! pe0 ; ~ f = (d! pe0 ) Nv f =n 0 (2.17) wherethek b istheBoltzmannconstant. T isthetemperatureforspecie. Thesuperscription N v denote the velocity space dimensions. Utilizing the dimensionless normalized variables presented in eq. (2.17), the Vlasov equation can be expressed as follows: @ ~ f @ ~ t + ~ v ~ r ~ x ~ f + ~ q ~ m ~ E + ~ v ~ B ~ r ~ v ~ f = 0 (2.18) The Poisson equation can be rewritten into: ~ r 2 ~ = X ~ q ~ n = ~ (2.19) where ~ ==(en 0 ). The Darwin approximations of the Maxwell equations can be rewritten into: ~ r ~ E L = ~ (2.20a) 2.3 Normalizations and Normalized Models 41 ~ r ~ B = 0 (2.20b) ~ r ~ E T = @ ~ B @ ~ t (2.20c) ~ r ~ B = 1 ~ c 2 ~ J + @ ~ E L @ ~ t ! (2.20d) where the normalized net current ~ J = J=(en 0 d! pe0 ). The two requirement of Darwin approximation can also be written into the dimensionless form ~ r ~ E L = 0 and ~ r ~ E T = 0. The requirements of the Darwin approximation can also be expressed in a dimensionless form as ~ r ~ E L = 0 and ~ r ~ E T = 0. Additionally, in eq. (2.20d), the vacuum speed of light c = 1= p 0 0 is utilized to eliminate the vacuum permittivity and permeability. It can be observed that in the dimensionless form, the vacuum speed of light can be represented by a dimensionless constant ~ c =c=(d! pe0 ). By appropriately selecting a normalization factor 2 , the moment of the velocity distribution functioncanbeexpressedinthefollowingdimensionlessnormalizedformusing ~ M n =M n =M n0 : < ~ M n >= R 1 1 ~ M n ~ fd~ v R 1 1 ~ fd~ v = Z 1 1 ~ M n ^ ~ fd~ v (2.21) Upon comparing eqs. (2.18), (2.19), (2.20), and (2.21) to eqs. (2.7), (2.8), (2.11), and (2.12), it becomes evident that the dimensionless normalized form equations are exactly equivalent to the original equations, with only certain constants being altered. Hence, in this work, the equations solved are specifically eqs. (2.18), (2.19)/(2.20), and (2.21). For the sake of convenience in writing and reading, throughout the subsequent sections of this chapter, all variables are expressed in their dimensionless normalized form, while omitting the use of the symbol “” above the dimensionless normalized variables. 2 This can be accomplished by employing the five unique normalization factors 2.4 Numerical Methods for Vlasov Equation 42 2.4 Numerical Methods for Vlasov Equation As discussed in Section 2.2, in order to investigate the self-consistent dynamics of plasma, it is necessary to couple the Vlasov equation with the field equations to determine the acceleration terma. This coupling renders the Vlasov equation a nonlinear hyperbolic partial differential equation (PDE), making analytical solutions challenging to obtain for most time-dependent problems. Consequently, numerical methods are typically employed to study time-dependent problems involving the Vlasov equation. There are primarily two types of numerical methods used to solve the Vlasov equation: particle-based methods and grid-based methods. Particle-based methods employ a Lagrangian scheme to solve eq. (2.7) by discretizing the particles. The particle trajectories are then computed by considering the equations of motion within the self-consistent electromagnetic field. dx dt =v ; m dv dt =q E +q v B (2.22) In particle-based methods, macroscopic physical quantities are statistically calculated by sampling discrete particles. There are three main categories of particle-based methods: particle-particle, particle-mesh, and particle-particle-particle-mesh. The particle-particle method directly calculates the interactions between particles without employing a mesh system [1, 179, 180]. While this approach avoids the finite mesh resolution issues, it is computationallyexpensiveduetothecomputationalcostscalingwiththesquareofthenumber of particles. The particle-particle-particle-mesh method [181] addresses the computational cost issue by introducing a mesh system. It computes interactions between nearby particles directly and interactions between more distant particles through the mesh. The particle-mesh method further reduces computational cost by fully employing a mesh system. In the plasma community, this method is commonly known as the Particle-in-Cell (PIC) method [5]. The PIC method utilizes macro-particles to represent an ensemble of “real” particles, enabling sampling of the velocity distribution functions. Interactions between these macro-particles are calculated through a mesh in physical space. The position and velocity information 2.4 Numerical Methods for Vlasov Equation 43 of these macro-particles are updated following eq. (2.22), allowing reconstruction of phase space information. The PIC method has been extensively developed in plasma physics since the 1960s, thanks to its ease of implementation and computational efficiency. Despite its popularity and widespread adoption, the PIC method has its drawbacks. One major issue is the presence of statistical noise, where unphysical phenomena can be introduced due to the numerical particles [182]. To mitigate this noise, the number of macro-particles needs to be increased. However, it is observed that the statistical noise in the PIC method is inversely proportional to the square root of the number of macro-particles, denoted as N p . In non-homogeneous plasmas, a PIC-based model typically requires a significant number of macro-particles to suppress noise in low-density regions. Consequently, this diminishes the computational advantage of the PIC method. The grid-based method directly solves the Vlasov equation for the velocity distribution function (VDF) in the discretized phase space. Both physical and velocity spaces are discretized using a finite mesh system. The Vlasov equation, being a partial differential equation, can be directly solved on the mesh points using finite difference [183], finite volume [169, 184, 185], or finite element [186] methods. Numerous numerical schemes have been developed for solving hyperbolic PDEs. These schemes can be divided into two groups based on whether they employ dimension splitting. The first group utilizes operator splitting methods to convert the high-dimensional Vlasov equation into a set of advection-type PDEs [27]. The second group directly solves the Vlasov equation in the phase space without any operator splitting operations [187]. Split methods are generally easier to implement and achieve second-order time accuracy, with the Strang splitting method being commonly used. Higher-order time accuracy can also be attained relatively easily by employing alternative split methods [188]. Consequently, split methods are more frequently adopted. In comparison to particle-based methods, the grid-based method does not exhibit inherent statistical noise issues and can readily achieve high-order schemes to resolve complex small perturbations. As a result, the grid-based method is chosen as the numerical approach to solve the Vlasov equation in this dissertation. 2.4 Numerical Methods for Vlasov Equation 44 The Vlasov equation is solved in the phase space, where the position x and velocity v serve as independent variables. In the case of a two-dimensional physical space and a three-dimensional velocity space (2D3V), the eq. (2.18) can be expressed as follows: @f @t +v x @f @x +v y @f @y +a x @f @v x +a y @f @v y +a z @f @v z = 0 (2.23) in which the term a x and a y and a z is determined through the eq. (2.19) or eq. (2.20) depending on whether the system is ES or EM. Eq. (2.23) is further split into a group of advection equations using the Strang splitting schemes and solved. @f @t +vr x f = 0 (2.24a) @f @t +ar v f = 0 (2.24b) Eqs. (2.24a) and (2.24b) can be further split into a group of one-dimensional linear advection equations using Lie splitting method. Different time stepping schemes can be applied to solve eq. (2.24a) and eq. (2.24b). In this work, the Semi-Lagrangian method is used. The numerical scheme used to solve Vlasov equation in this work is based on the Positive Flux Conservation (PFC) method [169, 184]. Under the PFC scheme, the split one-dimensional linear advection equation on x direction @f=@t +v@f=@x = 0, can be written into following conservation form: Z x i+1=2 x i1=2 f(t n+1 ;x)dx = i1=2 (t n ) + Z x i+1=2 x i1=2 f(t n ;x)dx i+1=2 (t n ) (2.25) where i1=2 and i+1=2 are the numerical fluxes in the cells adjunct to mesh point x i : i+1=2 (t n ) = Z x i+1=2 X(t n ;t n+1 ;x i+1=2 ) f(t n ;x)dx (2.26) 2.4 Numerical Methods for Vlasov Equation 45 In eq. (2.26),X(t n ;t n+1 ;x i+1=2 ) is the start of a characteristic line at time t n level which ends at x i+1=2 (t n+1 ). See Ref. [169, 184] for details of constructing the numerical flux . Let f n i denote the function at mesh point x i and time step t n . Eq. (2.25) is further converted to f n+1 i =f n i + 1 x i 1 2 (t n ) i+ 1 2 (t n ) (2.27) The PFC scheme used here is chosen to be the third-order PFC to balance the accuracy requirement and computational cost. This scheme has a third order of accuracy in space. For eq. (2.24a), the third-order PFC scheme under the condition v x > 0 is f n+1 i =f n i + i1 X k=j f n k i X k=j+1 f n k + i1 x " f n j1 + + j1 6 1 i1 x 2 i1 x (f n j f n j1 ) # + i1 x " f n j1 + j1 6 1 i1 x 1 + i1 x (f n j1 f n j2 ) # i x " f n j + + j 6 1 i x 2 i x (f n j+1 f n j ) # i x " f n j + j 6 1 i x 1 + i x (f n j f n j1 ) # (2.28) wheresubscriptj denotesthenumberofthecellwherethecharacteristiclineX(t n ;t n+1 ;x i+1=2 ) for eq. (2.24a) starts, i =x j+1=2 X(t n ;t n+1 ;x i+1=2 ), and + and are the slope correctors [169, 184]. The function at an arbitrary position x inside a cell is obtained through the following interpolation [169, 184] f h (x) =f i + + i 6x 2 2(xx i )(xx i3=2 ) + (xx i1=2 )(xx i+1=2 ) (f i+1 f i ) + i 6x 2 2(xx i )(xx i+3=2 ) + (xx i1=2 )(xx i+1=2 ) (f i f i1 ) (2.29) The PFC method guarantees the conservation of the flux Z x i+1=2 x i1=2 f h (x)dx = xf i (2.30) 2.4 Numerical Methods for Vlasov Equation 46 and ensures that f h (x) is always positive, up to the forth order accuracy [185]. The PFC scheme used here is chosen to be the third-order PFC to balance the accuracy requirement and computational cost. The method used to solve the Vlasov equation follows the algorithm proposed by Cheng and Knorr [27], which is a variant of the leapfrog algorithm. The update process in physical space is straightforward, as one simply follows the Lie splitting method to update each direction sequentially. However, in the presence of a magnetic field, the update of the velocity space velocity distribution function (VDF) is more complex compared to the update in physical space. In the work by Mangeney and Califano [189, 190], a second-order update method is proposed to address this issue. v (t) = vx (t=2) vyvz (t) vx (t=2) = vx (t=2) vy (t=2) vz (t) vy (t=2) vx (t=2) (2.31) where x and v denote for solving eq. (2.24a) and eq. (2.24b) using the 3rd order PFC scheme respectively. The Back-Substitution method, developed by Schmitz and Grauer [191, 192], is another approach for updating the velocity space velocity distribution function (VDF). v (t) = vx (t) vy (t) vz (t) (2.32) The advantage of this method lies in the reduction of velocity space updates required. It necessitates only 3 updates, compared to the 5 updates in the method proposed by Mangeney and Califano [189, 190]. However, the essential nature of this method necessitates a rigorous implementation of the characteristic line calculations, rendering it complex. Following the 2.4 Numerical Methods for Vlasov Equation 47 notations by Rieke et al. [192], the characteristic lines calculation process can be written as following. v z =v + x (t z s x s y ) +v + y (t z s y +s x ) +v + z (1t x s x t y s y ) v y =v + x s x t y +s z (t z s x s y )(t y s z s x ) 1t x s x t y s y +v + y 1t x s x t z s z (t z s y +s x )(t y s z s x ) 1t x s x t y s y +v z t y s z s x 1t x s x t y s y v x = v + x v y (t x s y +s z )v z (t x s z s y ) 1t y s y t z s z (2.33) Here the termt ands can be calculated as the following t = q m t 2 B s = 2t 1 +t 2 (2.34) Thev + andv is calculated as the following. v + =v n q m t 2 E n1=2 v =v n1 + q m t 2 E n1=2 (2.35) The calculation of the characteristic lines should be sequenced as v z , followed by v y , and then v x . Conversely, the update of the Vlasov equation should adhere to the sequence of the v x direction, followed by the v y direction, and finally the v z direction. Another noteworthy aspect regarding the implementation of the back-substitution method is that when calculating v x and v y in the code, all quantities with the sign “” should be computed following the rules of the corresponding quantities with the sign “+”. Further details of the method can be found in Ref. [191]. Both methods are implemented in this dissertation. Consequently, the numerical algorithm for solving the Vlasov equation can be summarized in Algorithm 1. An additional comment 2.5 Numerical Methods for Poisson Equations 48 Algorithm 1 Vlasov Equations Solving Process 1. Update the VDF f =f(x;v;t) in the physical space by half time step. f ? = x (t=2)(f) Note the velocity space information is now half time retarded to the information of VDF in physical space for f ? . 2. Get the required moments of VDF at t = t + t=2, feed this information into field equations to get the field information to calculate the acceleration. Use acceleration to update the VDF by t in velocity space. f ?? = v (t)(f ? ) Note the velocity space information is now half time ahead to the information of VDF in physical space for f ?? . 3. Update the VDF f ?? in the physical space by half time step. f(x;v;t + t) = x (t=2)(f ?? ) should be made regarding the numerical time step length for the Vlasov equation solver. The useofthePFCmethodensuresthattheCFLconditionisnotatheoreticalrestriction. However, in practice, due to the implementation of non-periodic boundary conditions, adherence to the variation of the CFL conditions is still necessary. For cases involving magnetic fields, to further minimize numerical errors, a condition of q Bdt=m < 0:1, as suggested by Schmitz and Grauer [191], should also be followed. 2.5 Numerical Methods for Poisson Equations Depending on the type of problem, whether it is an electrostatic (ES) or electromagnetic (EM) problem, either the Poisson or Darwin equations need to be solved to determine the field properties. These field properties are then used self-consistently to obtain the acceleration in the advection of the Vlasov equation in velocity space. For the electrostatic problems, in the current work, the direct LU decomposition method is employed to solve the Poisson equation. The Poisson equation is discretized directly using the 2.5 Numerical Methods for Poisson Equations 49 second-order central finite difference method. For instance, in the case of a two-dimensional physical space, the discretization can be expressed as follows: ~ i+1;j 2 ~ i;j + ~ i1;j ~ x 2 + ~ i;j+1 2 ~ i;j + ~ i;j1 ~ y 2 =~ i;j (2.36) wherei;j means theith index on thex direction andjth index on the y direction respectively. Eq. (2.36) can be written as a linear algebra equation Ax =b (2.37) whereA is a large sparse matrix with size m 2 n 2 assuming a discretized computational domain with m points on x direction and n points on y direction. It should be also noted that the boundary condition can also be written into the matrix A. Various methods such as direct matrix inverse, iterative methods, or spectral methods can be employed to solve eq. (2.37). However, due to the presence of multiple boundary conditions in Vlasolver, a non-spectral method is adopted. Notably, the sparse matrix A remains unchanged once determined at the beginning of the simulation. Hence, the direct LU decomposition method is employed to solve eq. (2.37). Although the matrix inverse process can be time-consuming, it only needs to be performed once. Subsequently, in each iteration, only a simple matrix multiplication is required. In comparison to iterative methods that may necessitate several tens of loops to solve eq. (2.37) within one iteration, the direct LU decomposition method offers computational speed advantages. The process of solving the Poisson equation can be described in the following manner as shown in algorithm 2. For solving the Poisson equation, the charge density needs to be provided from the Vlasov equation solver. Examining algorithm 1, it is evident that in Step 2, the macroscopic variables are computed using the moment integration eq. (2.21) with f(x;v;t =t + t=2). However, the available information of the velocity distribution function (VDF) at the beginning of Step 2 is f ? , given by f ? =f(x(t + t=2);v(t);t =t + t=2). Comparing this with the required VDF information, f ? has a velocity space that is retarded by half a time step. 2.5 Numerical Methods for Poisson Equations 50 Algorithm 2 Poisson Equations Solving Process 1. Get the A 1 by LU decomposition before entering the main loop of Vlasov-Poisson solver and store it. 2. Update from Vlasov equation’s integration and copy values to the vectorb, also apply the corresponding boundary conditions. 3. Solve eq.(2.37) by doing a matrix multiplication x =A 1 b. 4. Copy the value ofx to . 5. In the main loop, perform steps 2-4. In algorithm 2, one needs the net charge density(x;t =t+t=2) which can be calculated from (x;t =t + t=2) = X (x;t =t + t=2) = X q Z f(x;v;t =t + t=2)dv (2.38) here we define ? which can be obtained exactly from eq. (2.38) with onlyf(x;v;t =t+t=2) substitute byf ? . In order to advancef ? tof(x;v;t =t + t=2), one needs to use eq. (2.24b). Eq. (2.24b) for specie can be integrated for 0th moment on both side as Z @f @t dv = Z q m (E +v B)r v f dv (2.39) It is straight forward to sum up eq. (2.39) to get the expression for net charge density @ @t = X q Z @f @t dv = X q 2 m Z (E +v B)r v f dv = 0 (2.40) Thus we have @=@t = 0. This can immediately be written into discrete form as ((x;t =t + t=2) ? ) t 2 = 0 (2.41) 2.6 Numerical Methods for Darwin Equations 51 which means that (x;t =t + t=2) = ? . Thus, for the coupling of Vlasov equation and Poisson equation, in order to save computational resource, one does not need to perform any advancement off ? since only the net charge density is needed to be computed from the VDF. In Algorithm 1’s step 2, is directly computed from ? and then the net charge density is given as input for Algorithm 2. Once the poisson solver return back the electric field and magnetic field information, f ? is advanced to f ?? following step 2. 2.6 Numerical Methods for Darwin Equations An advantage of the Darwin approximation of the Maxwell equations is its ability to eliminate the time integration of the fields. Following the methods employed in previous works [172, 193– 196], eq. (2.20) can be transformed into the following form (where the “” signs are omitted for convenience): r 2 =; E L =r (2.42a) r 2 B = 1 c 2 rJ (2.42b) r 2 E T = 1 c 2 @J T @t (2.42c) where @J T =@t is the time derivative of the transverse current which can be written as [197] @J T @t = @J @t r( @ 2 @t 2 ) (2.43) It is evident that regardless of the numerical scheme employed, the information of J and @J=@t from the Vlasov equation is always required. Both of these variables can be obtained by performing moment integrations of the Vlasov equation with terms involvingv and@v=@t. As mentioned earlier, the update process of the Vlasov equation follows the method of Cheng and Knorr, which is a variant of the leapfrog scheme. In this scheme, the position-related information and velocity-related information are retarded by half a time step with respect to each other. However, solving the transverse component of the electric field necessitates 2.6 Numerical Methods for Darwin Equations 52 the information of @v=@t, which is retarded by half a time step. Updating such information requires the knowledge ofE T , resulting in a loop that necessitates iterations for solution. The process for solving such a system can be expressed in the following manner: In the process of Algorithm 3 Darwin Equations Solving Process 1. Solve eq. (2.42a) for longitudinal component of electric field, E L . 2. Solve eq. (2.42b) for “old” magnetic field,B 0 . Note the current information used here is half time step fall back. 3. Use the transverse electric fieldE 0 T from last step, get electric field informationE 0 by E 0 =E 0 T +E L . 4. Use theE 0 andB 0 to update the derivative of transverse current @J 0 T =@t to @J 1 T =@t. 5. Solve eq. (2.42c) with @J 1 T =@t to getE 1 T . Then getE 1 =E 1 T +E L . 6. Inner loop: (a) Use theE 1 andB 0 to update @J 0 T =@t to @J 2 T =@t andJ 0 toJ 1 . (b) Solve eq. (2.42b) for “new” magnetic field,B 1 . Note the current used here should beJ 1 . (c) Solve eq. (2.42c) with @J 2 T =@t to getE 2 T . Then getE 2 =E 2 T +E L . 7. OutputE 2 asE,B 1 asB for Vlasov equation, storeE 2 T for next solving process use. solving the eq. (2.42c), the equation can become unstable if not approximately dealt with. Here in practice, in the process of solving equation eq. (2.42c), it is rewritten into [194] r 2 E n T ! 2 c 2 E n T = 1 c 2 @J T @t ! 2 c 2 E o T (2.44) where ! = P q =m . In practice, this constant is computed only once during the initial stage of the simulation, and the actual constant used is defined as ! = 0:5 (!(x;t = 0)max +!(x;t = 0)min). This approach helps stabilize the iteration process, particularly when there is a significant variation in plasma density. Utilizing this approach contributes to the algorithm’s stability and typically enables convergence within a single inner loop. It should also be noted that the termr(@ 2 =@t 2 ) in eq. (2.43) is omitted, as it only introduces a curl-free component to the transverse component of the electric field [193]. However, a 2.7 Boundary Conditions 53 coupling method should be developed to solve the problem of time step retard, which will be presented in Chapter 6. 2.7 Boundary Conditions The solution of the Vlasov-Poisson/Darwin system is influenced by both the initial conditions and the boundary conditions. Implementing general boundary conditions for the Vlasov equation is not a trivial task and remains an active area of research in applied mathematics [198–200]. In this section, we summarize specific boundary conditions that have been used in other chapters of this work and provide detailed explanations of their implementation. Additionally, we introduce the boundary conditions utilized in this work and describe how they are implemented for both the Poisson and Darwin equations. The most commonly used and straightforward boundary condition for the Vlasov equation in physical space is the periodic boundary condition. Although this boundary condition needs to be implemented in phase space, it is essentially the same as the periodic boundary condition in physical space. When particles cross the boundary, their velocities remain unchanged. As a result, the distribution function in velocity space experiences no changes. For a one-dimensional periodic domain with length A, the periodic boundary condition can be expressed as f(0;v) =f(A;v). In the physical problems investigated in this work, inflow and outflow boundary conditions are employed for the Vlasov equation in physical space. As the Vlasov equation is a hyperbolic partial differential equation, its solution can be traced back along characteristic lines. The initial value in phase space only affects the region swept by the characteristic lines originating from that point. For the inflow boundary condition, only the positive velocity branch of the velocity distribution function (VDF) needs to be specified, while the negative branch of the VDF is determined self-consistently by the particle flow out of the computational domain. Specifically, we set f(x;v> 0) =f 0 , wheref 0 represents the specified value of the VDF. The detailed implementation of injecting plasmas at the specific injection plane is provided in 2.8 Summary 54 subsequent sections. As for the outflow boundary condition, this work exclusively employs the vacuum boundary condition. In this boundary condition, the VDF is set to be zero at the boundary, indicating that no particles are allowed to leave the computational domain. Another boundary condition employed in this work for the physical space is the symmetric boundary condition. At the symmetric boundary, the “particles” undergo specular reflection, where there is no energy loss and only the velocity component perpendicular to the surface is reversed without any change in magnitude. For the velocity distribution function at the symmetric boundary, we utilize the equation provided in Vogman’s work [201], which is expressed asf(x;v) =f(x;v2(v ^ n)^ n), where ^ n is the unit normal vector to the boundary surface. In the numerical solution of the Vlasov equation, it is necessary to impose a boundary in the velocity space, as the velocity space is infinite. In this work, a “cut-off” boundary condition is employed in the velocity space. An arbitrary velocity value, typically chosen to be several times the particle thermal velocity, is selected such that the velocity distribution function (VDF) becomes sufficiently small at this point. Generally, the normalized VDF value is set to be smaller than 10 12 . Beyond this point, the VDF is truncated, and its value is set to zero. For the Poisson equation, various boundary conditions are used, including Dirichlet, Neumann, Mixed, and Periodic boundary conditions. The implementation details for these boundary conditions can be found in several works [178, 201, 202], and are not elaborated upon here. As for the Darwin equations, only the periodic boundary condition is employed. 2.8 Summary The objective of this work is to develop a grid-based Vlasov method for studying plasma flows at the kinetic level, and the efforts made in developing such a method are presented in this chapter. The chapter begins by introducing the Vlasov equations, derived from first principles, anddiscussestheVlasov-PoissonandVlasov-Darwinsystemsusedinthisstudy. Thenumerical 2.8 Summary 55 schemes employed for solving the Vlasov equations, as well as the Poisson/Darwin equations, aredescribedindetail. Furthermore, specificconsiderationsregardingtheboundaryconditions for the Vlasov and Poisson/Darwin equations are discussed. 56 Chapter 3: Grid-based Vlasov Solver for Electrostatic Plasma Simulations With the fast advance of the high-performance computing in recent years, grid-based method attracts the sights of researchers in different communities. The grid-based Vlasov simulations was carried out in the community of fusion plasma [182, 203], astrophysical plasma [29, 164, 165, 204] and electric propulsion [155, 205–208] in recent years. However, due to the huge computational resources needed in discretizing the high-dimensional phase space, most of the implementations of the grid-based method are restricted to low dimensional problems. Currently only a few grid-based code can handle the high dimensional problems by using parallelization techniques [29, 164, 209] and most of them focus on the space weather and fusion aspects. The goal of this chapter is to develop and utilize a new multi-dimensional parallel electrostatic grid-based Vlasov solver for kinetic studies of plasma flows in the field of space science and engineering, namely, Vlasolver(Vlasov Solver). The Vlasov-Poisson system is solved in the Vlasolver by using conservative schemes. Parallel techniques are used to accelerate the code for efficiently solving multi-dimensional problems. In this chapter, the numerical implementation of Vlasolver is introduced and the parallelization techniques used are illustrated. The parallel efficiency of the Vlasolver is also introduced in this chapter. Several classical problems are used for the verification of Vlasolver and the results are presented in this chapter. One case simulating more complex “real-world” applications using Vlasolver are also presented here. The results from Vlasolver are compared with the results from some well benchmarked PIC code to validate the Vlasolver. The effects of discretized particle noise are thoroughly discussed, and the simulation results from the PIC method are compared with the results obtained from the grid-based Vlasov method using a linear Landau damping problem. Finally a summary is presented at the end of this chapter. 3.1 Numerical Implementation and Parallelization Strategy 57 3.1 Numerical Implementation and Parallelization Strat- egy Figure 3.1: Schematic illustration on the 2D2V phase space discretization. Fig. 3.1 illustrates the discretization of the phase space in Vlasolver, taking the 2D2V case as an example. The discretization of both physical and velocity space poses significant computational challenges, as even simple physical problems can require a substantial amount of computing resources. For instance, the well-known two-stream instability in a 2D2V simulation utilizes approximately 128 4 mesh points, corresponding to approximately 0.26 billion mesh points in total. In practice, this case typically requires around 2000 computational steps for the instability to saturate. On a workstation with a single Intel i7-8700 processor, such a case would typically take around two days to complete. To tackle more complex physical problems, parallel computation techniques must be employed to accelerate the computation process of Vlasolver. Vlasolver has been parallelized using the domain decomposition technique. The physical space simulation domain is divided into computational sub-domains, as depicted in Fig. 3.2. 3.1 Numerical Implementation and Parallelization Strategy 58 Figure 3.2: Domain decomposition schematic plot. Eachsub-domainisassignedtoadifferentprocessforparallelexecution. Informationregarding the velocity distribution functions (VDFs) and field quantities is stored and exchanged in the ghost cells of each sub-domain, utilizing the MPI library. However, the velocity space itself is not decomposed in Vlasolver. This is because there are frequent operations that require global information of the entire velocity space, such as moment integration. Therefore, each process retains a complete set of the velocity space, allowing for efficient computation and data exchange among the processes. The PFC method is employed to solve the hyperbolic PDE Vlasov equation in Vlasolver. This method is known for its conservative nature, requiring the update of local velocity distribution functions (VDFs) using incoming and outgoing flux information. The flux calculation relies on neighboring cell information, making communication between cells essential. To facilitate communication and minimize modifications to the original serial Vlasolver, the MPI (Message Passing Interface) library is utilized. The MPI functionality is encapsulated within a C++ class, which incorporates new MPI data types for handling the communication of VDFs data containers. Within the class, buffer arrays are created to store 3.1 Numerical Implementation and Parallelization Strategy 59 boundary data from the data container. The “MPI_Sendrecv” function is then employed to exchange data between the buffers using a point-to-point communication mode, as illustrated in Fig. 3.3. Figure 3.3: Schematic plot for the two-dimensional physical space communications. Under the 1D/2D physical domain, the computational time for solving the field equations is not the dominant factor compared to the time required for solving the Vlasov equation. Therefore, for the field equation solver, the “Gather-Solve-Scatter” communication mode, as depicted in Fig. 3.4, is adopted. In each step, every process calculates the local information (such as moment integration of VDFs) required by the field equations. These local information are then gathered by the “solver” process, which solves the field equations to obtain the global field information. The results of the global field information are subsequently scattered to each process to form the local field information. Based on the parallel strategy mentioned here and Algorithm 1, the numerical algorithm for Vlasolver can be summarized as the following form in Fig. 3.5. 3.2 Vlasov Equation Boundary Condition Implementations 60 Figure 3.4: Schematic plot for the solving field. 3.2 Vlasov Equation Boundary Condition Implementa- tions Boundary conditions are crucial in ensuring that simulation results accurately reflect the desired physics. In this work, the theoretical aspects of the boundary conditions used are discussed in Section 2.7. However, it is equally important to address the numerical implemen- tation details of these boundary conditions in Vlasolver. The numerical implementation of boundary conditions is a topic of great interest in various scientific computing communities. In this section, we delve into the specific details of how the boundary conditions for the Vlasov equation are implemented in Vlasolver. Boundary conditions can be implemented using different approaches, each with its own advantages and considerations. One approach is to modify the numerical algorithm or stencil at the boundary points. For instance, in a system employing a central second-order finite difference scheme, the stencil can be changed to a forward or backward second-order finite 3.2 Vlasov Equation Boundary Condition Implementations 61 Start Constrcut the C++ classes MPI start New run or restart? Initialize VDF and field Reload the stored restart files Update the VDF in the phys- ical space by half time step. f = Λ x (Δt/2)(f(x,v,0)) Update the VDF in the phys- ical space by half time step. f ⋆ = Λ x (Δt/2)(f(x,v,t)) Get the required moments of VDF at t = t + Δt/2 Gather local moments to global container Is this process the “field solver” process? Solve the field equation and get global field information Scatter the global field infor- mation to every local process Update the VDF by Δt in velocity space. f ⋆⋆ = Λ v (Δt)(f ⋆ ) Send/Recv VDF information Update the VDF in the phys- ical space by half time step. f(x,v,t+Δt) =Λ x (Δt/2)(f ⋆⋆ ) t+Δt≤ T? MPI End Destruct the C++ classes End no yes yes no yes no Figure 3.5: Flowchart for Vlasolver numerical algorithm. In the flowchart the red ecliptic box stands for the start and end control command, green diamond box stands for the decision process, blue rectangular box stands for the computing process and yellow rectangular process stands for the communication process between processes. 3.2 Vlasov Equation Boundary Condition Implementations 62 difference scheme at the boundary points. This eliminates the need for information from exterior mesh points. Another approach is to introduce additional mesh points adjacent to the boundary of the original system, known as “ghost cells” or “guard cells”. These ghost cells contain the necessary information from the exterior points of the original mesh system, allowing the numerical scheme and stencils to remain unchanged. While this approach incurs additional computational cost due to the presence of extra mesh points, it is generally easier to implement. In the case of Vlasolver, which is a parallel solver employing domain decomposition, buffer regions are already defined outside each sub-computational domain (as shown in Fig. 3.2). These buffer regions can be directly used as ghost cell regions for the sub-domains located on the boundary of the global domain. Therefore, in this work, all boundary conditions are implemented using these “ghost cell” regions. The periodic boundary condition in physical space is straightforward to implement in Vlasolver. The communication between sub-domains through the “MPI_Sendrecv” function facilitates the exchange of VDF values between sub-domains located on opposite sides of the global domain. This direct exchange ensures that no additional steps are required for implementing periodic boundary conditions. On the other hand, implementing the inflow and outflow boundary conditions in physical space is more complicated in Vlasolver. In the case of the inflow boundary condition, the ghost cells adjacent to the sub-domains located on the inflow boundary of the global domain serve as a “virtual” source. The VDF values in these ghost cells are prescribed and remain constant throughout each iteration of the main loop. To ensure accurate calculation of influx using the scheme employed in Vlasolver, it is necessary to have at least three layers of ghost cells. The inflow boundary condition can be expressed as follows (under a 2D3V phase space setup, for example): f [(x=3; y); (vx>0; vy; vz )] =f 0 ; f [(x=2; y); (vx>0; vy; vz )] =f 0 ; f [(x=1; y); (vx>0; vy; vz )] =f 0 (3.1) The implementation of the outflow boundary condition in physical space is relatively straight- forward in Vlasolver, as it involves a vacuum boundary condition. The ghost cells adjacent 3.3 Parallel Efficiency 63 to the sub-domains located on the outflow boundary of the global domain act as a “virtual” sink. Any phase space flow entering this sink region is effectively removed, mimicking the behavior of particles entering the free region outside the simulation domain and escaping. Consequently, all VDF values within this sink region are set to zero. f [(x=Nx+1; y); (vx; vy; vz )] = 0; f [(x=Nx+2; y); (vx; vy; vz )] = 0; f [(x=Nx+3; y); (vx; vy; vz )] = 0 (3.2) The implementation of symmetric boundary condition in physical space is the most complex one. Here the specular reflection method from Vogman’s work [201] is adopted as the symmetric boundary condition here. The boundary condition can be implemented as 1 f [(x=3; y); (vx=ivx; vy; vz )] =f [(x=2; y); (vx=Nvxivx; vy; vz )] f [(x=2; y); (vx=ivx; vy; vz )] =f [(x=1; y); (vx=Nvxivx; vy; vz )] f [(x=1; y); (vx=ivx; vy; vz )] =f [(x=0; y); (vx=Nvxivx; vy; vz )] (3.3) The velocity space “cut-off” boundary condition is implemented by setting the velocity space ghost cells’ VDF values equal to 0 2 . f [(x; y); (vx=1; vy; vz )] = 0 (3.4) 3.3 Parallel Efficiency ThescalingtestsoftheVlasolver codewereconductedtoevaluateitsweakscalingperformance. In this context, “weak scaling” refers to measuring how the computation time changes with 1 Here we use the expression under 2D3V phase space set-up. Assuming that the velocity normal to surface is y direction. 2 Here we use the v x direction’s “cut-off” boundary condition as an example. 3.3 Parallel Efficiency 64 the number of processes while keeping the problem size fixed for each process. The parallel efficiency in this context can be defined as E p = S p P = T (w; 1) T (Pw;P ) (3.5) where S p , P, T, w is Speedup, number of processes, computational time and constant workload per processor respectively [210]. The weak scaling test of the Vlasolver code was conducted using the linear Landau damping problem in a 2D2V phase space geometry. The computational domain in the physical space was set to range from 0 to 4 in both the x and y directions, using normalized units. The computational domain in the velocity space was set to range from6 to 6 in both the v x and v y directions, also using normalized units. For the purpose of the weak scaling test, the number of computational cells per process in the physical space was kept fixed. A grid size of 20 20 cells was chosen for the 2D physical space per process. Since only the physical space is decomposed for parallel computing, each exchanged mesh cell in the communication process contains the complete set of velocity space information. However, the number of cells in the velocity space may affect the scalability of the code. Therefore, three groups of velocity space cell numbers were chosen: 32 32, 64 64, and 128 128 cells for the two-dimensional velocity space, respectively. By varying the number of processes and keeping the problem size per process fixed, the weak scaling test allows us to evaluate the code’s performance and efficiency as the computational resources are scaled up. The scaling tests were conducted on the NCAR Cheyenne supercomputer, which consists of nodes equipped with dual 18-core 2.3 GHz Intel Xeon E5-2697v4 processors and 64GB DDR4 memory. Each Cheyenne computational node can accommodate a total of 36 processes. The code was compiled using the GNU 8.3.0 compiler, and the communication environment was set to OpenMPI 4.0.5. The linear algebra computations in the field equation solving process were performed using the Eigen 3.3.9 library. The code for all scaling tests was compiled with the “-O3” optimization flag. When choosing the referencing benchmark for the scaling test, there are two options to consider: single process performance and single node performance. In this work, the single-node performance was chosen as the referencing 3.3 Parallel Efficiency 65 benchmark for the scaling test, taking into account the modern computer architecture. With multiple cores within a CPU unit sharing the same L2 cache, there can be a bottleneck in the speed of communication between CPUs and memory. Additionally, within a single computational node, different processes working in a distributed memory parallelization mode read and save data from the same physical memory, which can result in memory bottlenecks due to limited memory bandwidth. It is important to consider these factors as they can impact the performance and communication overhead. Considering single-process performance alone may lead to disregarding the communication overhead induced by inter-node communication. 0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100 110 Figure 3.6: Weak scaling of Vlasolver’s efficiency with dependency to core numbers. Fig. 3.6 shows the weak scaling test results of Vlasolver under three different set-ups as mentioned above. It can be found from the figure that the weak-scaling efficiencies all show a slow decreasing pattern. The parallel efficiency for 32 32 velocity space cells, 64 64 velocity space cells, 128 128 velocity space cells are 76:38%, 80:51%, 85:35% respectively when the core number is 360. The efficiency clearly show a slower decreasing trend when 3.3 Parallel Efficiency 66 the velocity space mesh becomes more dense. In an ideal scenario, if the problem size is fixed for a single process, the total computational time should remain constant regardless of the number of processes employed. However, in practice, there are overheads that arise due to various factors in parallel computing. The actual computational time can be defined as T actual =T work +T overhead =T work +T overhead; s +T overhead; c . The overhead of a parallel program typically arises from two main sources. Firstly, there is overhead resulting from communication between different processes. Secondly, there is overhead stemming from the parts of the program that cannot be parallelized, often referred to as “sequential parts”. Vlasolver is composed of two main modules: the Vlasov equation solver and the field equation solver. The Vlasov equation solver is parallelized through domain decomposition. Fig. 3.7a displays the weak scaling results for the Vlasov equation solver module alone. It can be observed that the Vlasov equation module demonstrates a nearly constant efficiency with respect to the core numbers. After accounting for communication time, a slight decreasing trend appears in all three cases, as depicted in Fig. 3.7b. Nonetheless, even with a large number of cores, the efficiency can still reach around 90%. Moreover, it can be observed that as the number of cores increases, the decreasing trend becomes less pronounced for all three cases. In particular, for the 32 32 case, the efficiency reaches an almost constant value once the core number becomes sufficiently large. By comparing this efficiency with the results shown in Fig. 3.6, it becomes evident that other factors are impacting the performance in a dominant manner. On the other hand, the field equation solver utilizes a “gather-solve-scatter” mode and only solves the field equations on the “solver” process. This solving process is therefore the main contributor to the sequential overhead. Fig. 3.8 illustrates the weak scaling results for the Poisson equation solver module and the gather-scatter communication module in Vlasolver. It can be observed that the efficiency decreases nearly linearly with the core number for all three cases. When the core number reaches 360, the efficiency drops to around 60%. This sequential overhead has a substantial impact on the overall performance and scalability of the code. 3.3 Parallel Efficiency 67 0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100 110 (a) 0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100 110 (b) Figure 3.7: Weak scaling of the Vlasov equation solver module in Vlasolver. (a) Weak scaling of Vlasov module only. (b) Weak Scaling of Vlasov and MPI communication module. Fig. 3.9 presents the profiling results for selected core numbers from all three groups of scaling tests. It is evident that the field equation solver module and its corresponding communication module play a less significant role as the number of velocity space mesh cells increases. As discussed earlier, the field equation solver and its associated communication module contribute to the sequential overhead of the code. The code performs better when 3.3 Parallel Efficiency 68 0 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 100 110 Figure 3.8: Weak scaling of the Poisson equation solver module and gather-scatter communi- cation module in Vlasolver. Figure 3.9: Profiling results of Vlasolver. 3.4 Verification of Vlasolver 69 the percentage of sequential overhead is low. Based on the profiling and scaling test results, it can be concluded that when Vlasolver has a larger number of velocity space cells, it is less affected by the overhead from the field solver, leading to improved scalability. This explains the better scaling efficiency observed when the velocity space mesh becomes more dense in Fig. 3.6. In summary, Vlasolver exhibits better performance and scalability when it has a dense velocity space mesh. Currently, although there is a sequential overhead introduced by the field solver module, Vlasolver can still achieve good scalability with efficiency above 75%. Future plans include replacing the sequential field solver with a parallel field solver to further enhance the scalability of Vlasolver. 3.4 Verification of Vlasolver Verification and validation are crucial steps in ensuring the accuracy of numerical software in representing the desired physics. In this section, we present multiple simulation cases using the Vlasolver to solve various physical problems. These problems have been extensively studied in the literature and possess well-established theoretical results, making them suitable benchmarks for verifying and validating the numerical outcomes. For the three problems considered here, we normalize the spatial length using the Debye length at the beginning of the simulation, denoted as d = D0 = p 0 k b T e0 =n 0 e 2 . The reference velocity is given by d! pe0 =v te0 , where v te0 represents the electron thermal velocity at the beginning of the simulation. With these normalization factors, the simulations are conducted on the electron time scale, allowing for the resolution of electron-scale dynamics. 3.4.1 Linear Advection and Gyro-Motion In this study, we conduct simulations to verify the implementation of the PFC scheme and Algorithm 1. The focus of this verification is to validate the update process of the Vlasov equation in the physical space. For this purpose, we consider a periodic boundary two-dimensional physical domain coupled with three dimensions in the velocity domain. 3.4 Verification of Vlasolver 70 we use a distribution function that exhibits a Gaussian shape in the physical space while maintaining a uniform shape in the velocity space. The distribution function is illustrated below f(x;y;v) = exp (x 2 y 2 ) (3.6) Eq. (2.24a) is solved numerically to advance eq. (3.6) in physical space. It can be found the eq. (2.24a) is a linear advection with analytical solution f (t+t) (x;y;v) =f (t) (xv x t;yv y t;v) (3.7) By comparing the numerical results with the analytical solution given by eq. (3.7), one can readily verify the accuracy and correctness of the PFC scheme and the implementation of the algorithm. A computational grid with 100 cells in each direction for x and y, and 21 -5 -4 -3 -2 -1 0 1 2 3 4 4.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) -5 -4 -3 -2 -1 0 1 2 3 4 4.9 -5 -4 -3 -2 -1 0 1 2 3 4 4.9 (b) Figure 3.10: Linear advection in physical space using Vlasolver. (a) Linear advection along x direction. (b) Linear advection along x and y direction. cells in each direction for the velocity domain is used, resulting in approximately 0.1 billion mesh cells in phase space. The physical space is defined as [5; 4:9] and the velocity space as [1; 1]. A time step length of t = 0:05 is chosen, and a total of 80 steps are performed. Periodic boundary conditions are applied to all boundaries of the physical space, while cut-off boundary conditions are used for all boundaries in the velocity space. The simulation is executed on four processes, and the entire test case can be completed within 3 minutes on 3.4 Verification of Vlasolver 71 an Intel i7-7700 workstation with a total memory usage of 8 GB. Fig. 3.10 presents the results of the verification test. It is evident that the numerical solution closely matches the analytical solution for the x direction advection test. Additionally, the results demonstrate good agreement with the analytical solution when performing advection in both the x and y directions. A similar test is also conducted in the velocity space. In this case, a distribution function with a uniform shape in physical space but a 3D Gaussian shape in velocity space is utilized. f(x;v x ;v y ;v z ) = exp (v 2 x v 2 y v 2 z ) (3.8) eq. (2.24b) is solved numerically with the acceleration term set to be a constanta = (1; 0; 0) (advection only on x direction) anda = (1;1;1) (advection on all directions of physical space). The analytical solution for eq. (2.24b) can be written as f (t+t) (x;v x ;v y ;v z ) =f (t) (x;v x a x t;v y a y t;v z a z t) (3.9) In this test case, the length of the domain is adjusted to be [1; 1] for all directions in physical -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) Figure 3.11: Linear advection in velocity space using Vlasolver. (a) Linear advection alongv x direction. (b) Linear advection along v x and v y and v z direction. The iso-surface is plotted for f = 0:3. space and [5; 5] for all directions in velocity space. Eight cells are used for all directions in physical space, and 101 cells are used for all directions in velocity space. All other parameters 3.4 Verification of Vlasolver 72 and boundary conditions remain the same as in the previous test case. Fig. 3.11 illustrates the results, demonstrating good agreement between the analytical solution and the numerical solution of eq. (2.24b). In order to assess the capability and accuracy of handling problems involving magnetic fields, a pure gyro-motion test similar to the test problem described in refs. [193, 211] is conducted. It is important to note that a pure rotational test represents a rigorous assessment, as in most real-world scenarios, the rotational effects of the magnetic field are typically counterbalanced by the electric field. In this test, a distribution function is proposed that is uniform in physical space but Gaussian-shaped in velocity space. f (t+t) (x;v) = exp ((vv d ) 2 ) (3.10) wherev d is the drifting velocity and it is set to be equal tov d = (1; 0; 0). In this test case, a Figure 3.12: Comparison of the average bulk velocity of distribution function at different time moment between numerical and analytical results. two-dimensional physical domain with three dimensions in the velocity domain is utilized. The magnetic field is aligned along the z direction with a magnitude of 1, and electrons are 3.4 Verification of Vlasolver 73 considered with a charge-to-mass ratio of1. The electric field is set to zero. The length of the domain is adjusted to be [1; 1] in all directions of the physical space and [5; 5] in all directions of the velocity space. Eight cells are used for each direction in the physical space, while 101 cells are used for each direction in the velocity space. The time step length is set to t = =240, resulting in a total of 480 steps and a simulation time length of 2. The boundary conditions are set to be the same as in the previous two test cases. In order to visualize the rotational test, the numerical average bulk velocity of the distribution function is calculated using eq. (2.21) with M =v. The analytical solution for this problem can be expressed as follows: v x (t =Nt) =jv(t = 0)j cos ( N 240 ) v y (t =Nt) =jv(t = 0)j sin ( N 240 ) v z (t =N t ) =v z (t = 0) (3.11) Fig. 3.12 shows a comparison between the numerical results and the analytical results for the rotational test. It can be observed that the numerical results are in good agreement with the analytical results, indicating that the implementation of rotational motion in Vlasolver has been successfully verified. 3.4.2 Landau Damping Landau damping is employed as a validation case to verify the solving process of the coupled Vlasov-Poisson system in Vlasolver. The simulation setup follows that of the reference [169]. The initial velocity distribution function (VDF) is set to be f = 1 2 exp((v 2 x +v 2 y )=2)(1 + 0:05 cos (0:5x) cos (0:5y)) (3.12) In the Landau damping verification case, the simulation setup consists of a physical computa- tional domain spanning [0; 4] in both the x and y directions, and a velocity computational 3.4 Verification of Vlasolver 74 domain spanning [6; 6] in both the v x and v y directions. The phase space is discretized using a grid with 64 computational cells in each direction. The simulation is parallelized using 4 processes, and for a run of 10,000 steps, the computational time is approximately 1 hour and 3 minutes on an Intel i7-7700 workstation. Fig. 3.13a displays the electric field energy in a normalized scale. The red dashed line represents the slope predicted by linear theory. The simulation results closely match the theoretical prediction, confirming the accuracy of the implementation. A nonlinear Landau damping test case is also performed under 2D2V phase 0 5 10 15 20 25 30 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 (a) 0 10 20 30 40 50 60 70 80 90 100 10 -8 10 -6 10 -4 10 -2 10 0 10 2 (b) Figure 3.13: Electric field energy history for Landau Damping cases. (a). Linear Landau Damping. (b). Non-linear Landau Damping. space. The initial VDF is set to be [212] f = 1 2 exp((v 2 x +v 2 y )=2)(1 + 0:5 cos (0:5x) + cos (0:5y)) (3.13) n this nonlinear Landau damping test case, the perturbation in the initial velocity distribution function leads to the growth of Landau damping along the diagonal direction of the 2D physical domain. The simulation setup includes a physical computational domain with a length range of [0; 4] in both the x and y directions (normalized units), and a velocity computational domain with a length range of [6; 6] in both the v x and v y directions (normalized units). The discretization of the phase space involves 32 cells in each physical direction and 128 cells in each velocity direction. The simulation is parallelized using 4 processes, and for a run of 2500 steps, the computational time is approximately 0.33 hours on an Intel i7-7700 workstation. Fig. 3.13b shows the electric field energy in the normalized 3.5 Comparison with PIC: Plasma Wake Expansion 75 scale, and the obtained data is in good agreement with the theoretical predictions, validating the accuracy of the simulation results. 3.4.3 Two-Stream Instability In this validation case, a two-dimensional simulation of the two-stream instability is performed to validate the 2D2V version of Vlasolver. The initial set-up is based on the configuration described in Ref. [212]. For this simulation, only electrons are included, and they are divided into two counter-streaming populations in the v x direction. The initial velocity distribution function is defined as follows: f = 1 12 exp((v 2 x +v 2 y )=2)(1 + 0:05 cos (0:5x))(1 + 5v 2 x ) (3.14) The simulation is carried out in a 2D physical domain with periodic boundary conditions in both the x and y directions. The velocity domain extends from9 to 9 in both the v x and v y directions, and “cut-off” boundary conditions are applied. The computational domain is discretized into 128 cells in each direction, resulting in a total of approximately 0:26 billion cells in phase space. To visualize the four-dimensional velocity distribution function, a reduced-order operation is performed by integrating along they andv y directions. This allows for a clearer visualization of the growth of the instability along the x direction. Fig. 3.14 depicts the evolution of the integrated velocity distribution function R f;dy;dv y over time. The figure clearly shows the growth of a vortex-like structure, which corresponds to the nonlinear saturation of the two-stream instability. This visualization provides evidence of the accurate implementation of the Vlasolver in capturing the dynamics of the instability. 3.5 Comparison with PIC: Plasma Wake Expansion Validation is an essential process to ensure that numerical solvers are capable of accurately simulating complex problems without analytical solutions. Unlike verification, which compares 3.5 Comparison with PIC: Plasma Wake Expansion 76 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (a) 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (b) 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (c) 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (d) 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (e) 0 2 3 4 -9 -6 -3 0 3 6 9 0 0.5 1 1.5 2 2.5 3 3.5 (f) Figure 3.14: Contour of integrated velocity distribution function R fdydv y at different time moment. (a) t = 0. (b) t = 8. (c) t = 16. (d) t = 24. (e) t = 32. (f) t = 40. numerical results with analytical solutions in idealized cases, validation focuses on comparing numerical results with results obtained from other well-established numerical solvers or experimental data. In this section, one real-world practical problems are considered to validate the performance of Vlasolver. The results obtained from Vlasolver are compared with those from other well-benchmarked Particle-in-Cell (PIC) codes. By conducting these comparisons, we can confidently assess the validity and reliability of Vlasolver for real-world applications. 3.5 Comparison with PIC: Plasma Wake Expansion 77 The formation of wakes behind a large objects immersed in a collisionless mesothermal plasma flow (characterized by v ti v 0 v te , where v ti , v 0 , and v te are the ion thermal velocity, plasma drifting velocity, and electron thermal velocity, respectively) is relevant to many applications in space physics and spacecraft-plasma interaction. Hybrid Particle-in-Cell (PIC) method is frequently used for the modeling of collisionless plasma wakes. In the hybrid Particle-in-Cell (PIC) method, ions are treated as macro-particles and electrons are treated as a equilibrium massless fluid with assumed thermodynamic process. A common assumption for the electrons is the isothermal equilibrium assumption using Boltzmann relation and the hybrid Particle-in-Cell method using this assumption has been successfully applied to various of plasma wake modeling works [213–215]. Recently, Wang and Hu [216, 217] carried out both hybrid and fully kinetic Particle-in-Cell simulations of plasma wakes and found that the hybrid PIC model using Boltzmann electron relation fails in the deep wake region due to the non-equilibrium and non-Maxwellian distributed electrons. In order to further develop a more appropriate electron fluid model in the plasma wake, one has to have a clear understanding of electron kinetics and thermodynamics in the deep wake region. However, the inherent numerical noise in PIC simulations is a limitation that makes it difficult to resolve the electron velocity distribution function and its higher order moments. This prevents further understanding of the macroscopic electron thermodynamic processes. The numerical noise effects become more severe in the deep wake region due to the lack of macro-particles. In this study, we utilize the grid-based Vlasov method (Vlasolver) as an alternative approach to investigate the plasma wake and compare its results to those obtained from the PIC method. Our goal is to demonstrate the effectiveness of this method in studying non-periodic boundary electrostatic problems and mitigating the numerical noise that is inherent in PIC method. In this study, we investigate the plasma wake behind a thin conductive plate that is immersed in a collisionless, unmagnetized mesothermal plasma flow. We utilize setups that are comparable to those employed in previous research by Wang and Hu [216, 217]. Specifically, the plasma inflow comprises electrons and protons with n e =n i =n 0 , v ti :v 0 :v te 0:0023 : 3.5 Comparison with PIC: Plasma Wake Expansion 78 0:1867 : 1, and an initial Mach number of M 0 = 8. The ambient plasma’s electrons and ions are assumed to follow Maxwellian distribution. The conductive plate has a length of L = 140 and are set to have no potential difference with the ambient plasma (i.e. = 0). To optimize computational efficiency, we analyze only one-half of the physical domain, leveraging its inherent symmetry. Boundary conditions are set to be the same with Refs. [216, 217]. In this study, we utilize a 2D2V computational domain for the grid-based Vlasov simula- tions. The physical domain spans a length of L x L y = 2000 300, and we set the mesh resolution to x = y = 1. This yields a total ofN x = 2000 andN y = 300 cells in the spatial domain. For the electron velocity domain, we set the range from v ex =7:5 to v ex = 8:5 and v ey =8:0 to v ey = 8:0 with a mesh resolution of v ex = v ey = 0:125. This leads to a cell number of N e;vx = 128 and N e;vy = 128. Similarly, for the ion velocity domain, we set the range from v ix = 0:15 to v ex = 0:35 and v ey =0:02 to v ey = 0:12 with a mesh resolution of v ix = v iy = 0:0015625. This results in cell numbers of N i;vx = 128 and N i;vy = 128. To satisfy the CFL condition, we set the time step to t = 0:05. All simulations are run until t = 4000, which corresponds to approximately 93 ion time periods. The set-ups for the fully kinetic PIC simulation are identical to those used in the grid-based Vlasov simulations. To control the numerical noise, we use 400 particles per cell and sample the flow field 1200 times. All simulations in this section are conducted on the USC CARC’s Discovery cluster. The grid-based Vlasov simulations employ Vlasolver and utilize 1500 AMD epyc-64 CPU cores, typically finishing within 50 hours. The fully kinetic PIC simulations are executed with 64 threads and usually complete within 30 hours. Fig. 3.15 compares the ion and electron densities in the plasma wake region obtained from Vlasov and PIC simulations. The results show good agreement between the two methods, with minor discrepancies observed only in the low-density region. This discrepancy may arise due to particle noise in the PIC simulation, which may accumulate over time, especially when the number of particles is small. Furthermore, it is worth noting that despite using 1200 samples, the electron densities in the deep wake region still exhibit noisy features. 3.5 Comparison with PIC: Plasma Wake Expansion 79 0 50 100 150 200 250 300 0 50 100 150 (a) 0 50 100 150 200 250 300 0 50 100 150 (b) Figure 3.15: Comparisons between Vlasov and PIC simulations on the particle number densities in plasma wake. (a) Ion number density. (b) Electron number density. Fig. 3.16 presents a comparison between the electron horizontal and transverse direction temperatures in the plasma wake region, as obtained from Vlasov and PIC simulations. The results show good agreement between the two methods. However, noisy features are visible inside the wake region, particularly in front of the expansion front of the ambient plasma flow. 3.5 Comparison with PIC: Plasma Wake Expansion 80 (a) (b) Figure 3.16: Comparisons between Vlasov and PIC simulations on the electron temperature in plasma wake. (a) T ex . (b) T ey . Fig. 3.17 compares the horizontal and transverse direction electron collisionless heat flux between the Vlasov and PIC simulations. Both results show good agreement with each other. However, it is worth noting that the collisionless electron heat flux data obtained from the PIC simulation exhibit noisy features in both high density and low density regions, even with 1200 samples used to reduce statistical noise. Previous works [218, 219] have shown that even slight noise can hinder the accurate identification of heat flux features and 3.5 Comparison with PIC: Plasma Wake Expansion 81 (a) (b) Figure 3.17: Comparisons between Vlasov and PIC simulations on the electron heat flux in plasma wake. (a) Q ex . (b) Q ey . 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 82 closures. In contrast, the collisionless heat flux data obtained from Vlasov simulations show smooth features throughout the entire domain, making them well-suited for investigating heat flux features in wake expansion. It is also worth noting that in our previous work [220], we compared heat flux data obtained from Vlasov simulations and PIC simulations with a large number of particles but without sampling (i.e., 10000 particles per cell at the injection plane but no samples taken). The results showed much stronger noise than the noise level observed in Fig. 3.17. Therefore, in order to control the noise in PIC simulations, both the sampling method and a large number of particles should be used, but this will result in a computational cost comparable to that of Vlasov simulations. Thus, Vlasov simulations still have advantages for studying high-order thermodynamic properties. In summary, this section investigates the plasma wake expansion problem using Vlasov simulations and compares the results with those obtained from PIC simulations. The findings suggest that, for this non-periodic boundary electrostatic problem, Vlasov simulations can effectively remove numerical noise, making them a valuable tool for investigating thermody- namic processes in wake expansion. 3.6 Discussions of Discrete Particle Noise Effects: Com- parisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping In Section 3.5, we investigate a non-periodic boundary problem, specifically the plasma wake expansion, using both PIC simulations and the grid-based Vlasov simulations. The problem is considered as a steady-state scenario, allowing for the application of sample average techniques to mitigate particle noise. The results demonstrate that, even after 1200 sample averages, the high-order moments of the velocity distribution function obtained from PIC simulations still exhibit noticeable noise compared to the results derived from the grid-based Vlasov simulations for this particular problem. The steady-state nature of this problem 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 83 allows for a large number of sample averages. However, for unsteady problems such as plasma wave/instabilities and turbulence, the use of sample average techniques becomes impractical. The PIC method has been extensively utilized to investigate plasma waves, instabilities, and turbulence for several decades. It is capable of capturing physical phenomena even with a small number of particles per cell. However, this leads to noisy features in the simulated data, and the noise introduced by PIC may occasionally result in purely numerical artifacts that can interfere with the accurate representation of physical phenomena. Recently, data-driven approaches have gained prominence in the study of inverse problems in plasma physics. These methods leverage data obtained from kinetic simulations to train neural networks or other data-driven frameworks. Alves and Fiuza [221] employed the Sparse Identification of Nonlinear Dynamics (SINDy) method [222] to investigate the problem of identifying differential equation relations from the available kinetic data. They employed PIC simulations to generate the training sets and test sets for the SINDy method. In their verification study, aimed at identifying the Vlasov equation from the two-streaming instability phase space data, they discovered that the presence of noise in the PIC simulation data resulted in significant deviations in the inferred differential equation coefficients [221]. To address this issue, they opted to identify the partial differential equations in their integral form instead. This study illustrates the significance of data noise levels in data-driven investigations. Cheng et al. [223] employed multi-moment fluid modeling to investigate the classical Landau damping problem. Their primary objective was to demonstrate the potential of moment-based fluid modeling for reducing computational time in modeling kinetic plasma physics problems. The key aspect of successful moment-based fluid modeling lies in selecting an appropriate collisionless heat flux closure. To achieve this, Cheng et al. utilized the mPDE-net method to identify the coefficient for the localized Hammet-Perkins closure [224, 225]. It is important to note that the kinetic data set they prepared for this analysis was obtained from noise-free grid-based Vlasov simulations. By successfully identifying a localized closure, they observed excellent agreement between the results obtained from fluid modeling and kinetic modeling. Qin et al. [226] employed the Physics-Informed Neural Network (PINN) method [227] to 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 84 model the Landau damping problem. They trained the neural network within the PINN framework using noise-free data obtained from grid-based Vlasov simulations. The PINN model trained with these noise-free data demonstrated promising capabilities in accurately predicting the Landau damping process. Both the physical-driven and data-driven studies highlight the significance of discrete particle noise effects, as it can not only lead to occasional unphysical phenomena but also impede the construction of reliable datasets. In this regard, we delve deeper into the matter and conduct a assessment and comparison of the discrete particle noise effects on simulation results. To achieve this, we focus on a classical periodic-boundary unsteady plasma problem: electron linear Landau damping. By comparing the results obtained from PIC simulations and Vlasov simulations for this scenario, we aim to gain valuable insights into the impact of discrete particle noise. We investigate a classical Landau damping problem in a 1D1V simulation domain. In this scenario, ions are treated as a fixed neutralizing background, while the electrons are initialized with a Maxwellian distribution and subject to initial density perturbations given by the following equation. f e = (1 +A 1 sin(kx)) 1 p 2v th exp v 2 2v 2 th (3.15) The perturbation strength is set to A 1 = 0:36, and the wavelength is set to k = 0:35. The length of the simulation domain in physical space is L = 2=k. Both the PIC simulation and Vlasov simulation employ 512 mesh cells in the physical space. As a benchmark case, the Vlasov simulation uses 1024 cells in the velocity space. Additionally, 128 cells are used in the velocity space (no obvious difference is observed), but the results are not presented here. The PIC simulation involves several simulations with varying particles per cell (ppc). Specifically, we consider ppc values of 10 2 , 10 3 , 10 4 , 10 5 , and 10 6 . The PIC method employed in this study is the classical first-order interpolation PIC method. The initial particle load follows the evenly loaded method in physical space, while the sampling of particles in the 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 85 velocity space does not utilize the Sobol sequence. It is worth noting that there are several denoising techniques available, such as high-order interpolation, particle-splitting, and the f method. However, for the purposes of this study, we have chosen to use the most widely adopted classical approach to illustrate the effects of discrete particles. 0 5 10 15 20 25 30 -9 -8 -7 -6 -5 -4 -3 -2 Figure 3.18: Comparisons between Vlasov method and PIC method on the Landau damping electric field energy history. Fig. 3.18 presents the electric field energy history for various cases of PIC and Vlasov simulations. The results demonstrate that the PIC simulation exhibits good numerical convergence for cases with particles per cell (ppc) greater than 1000. Notably, for the ppc= 100 case, the initial electric field energy agrees well with the other cases, although the negative dips do not match. However, for t > 15, the ppc= 100 case deviates from the other cases due to strong numerical noise. On the other hand, the PIC simulations with a larger number of particles per single cell display a better match with the benchmark Vlasov simulation. The cases with ppc= 10 5 and ppc = 10 6 exhibit the best match with the benchmark Vlasov simulation, as expected. 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 86 Next, we investigate the flow field of the electron Landau damping and compare the results between the PIC simulations and the Vlasov simulation. In data-driven studies, the macroscopic variables play a crucial role in constructing the training data sets. Therefore, four variables are of particular importance: density, velocity, temperature, and heat flux. In this analysis, we compare the density, temperature, and heat flux between the Vlasov and PIC simulations. To further reduce noise in the PIC results, a low-pass moving smoothing filter is applied, and both the unfiltered and filtered results are compared with the Vlasov simulation results. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Figure 3.19: Comparative analysis of electron number density contours in Landau damping simulations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simulation, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 87 Fig. 3.19 illustrates a comparative analysis of electron number density contours in Lan- dau damping simulations utilizing Vlasov and PIC methods. Particularly noteworthy are Figs. 3.19a and 3.19b, which display different colorbar contour ranges compared to the other figures. This disparity arises due to strong noise perturbations present in these specific cases. The examination of Fig. 3.19a highlights that prominent numerical noise substantially obscures the representation of physical phenomena, resulting in wave structures that are only vaguely discernible. This observation confirms the conclusion drawn from the electric field energy history in Fig. 3.18, indicating that the case with ppc= 10 2 cannot accurately capture the damping characteristics. This observation confirms that, in the case with ppc=10 2 , the application of smoothing filters is not effective in enhancing the quality and accuracy of the physical properties. Figs. 3.19c and 3.19d show the electron number density contour for the PIC simulations with ppc= 10 3 . In these instances, while the wave structure can be identified, it is accompanied by substantial noise. Unfortunately, the application of a filter does not effectively mitigate this noise, thus failing to yield smoother results. Figs. 3.19e and 3.19f exhibit the electron number density contours for PIC simulations with ppc= 10 4 . Notably, distinct wave structures are clearly observed and closely resemble the benchmark Vlasov simulation case, although some noise-induced features are still apparent. In Figs. 3.19g to 3.19j, a compelling alignment with the benchmark Vlasov case is evident, which is also validated by the similar electric field energy histories shown in Fig. 3.18. Fig. 3.20 illustrates a comparative analysis of electron temperature contours in Landau damping simulations employing both Vlasov and PIC methods. Similar to the number density, the cases with ppc= 100 particles per cell are unable to accurately capture the temperature behavior. However, increasing the particle count to ppc= 10 3 leads to improved results, and the application of a filter smoothing technique reduces the impact of noise. In both cases with ppc=10 2 and ppc=10 3 , the presence of significant numerical noise results in discrepancies in the magnitude of the temperature when compared to the benchmark Vlasov case. For the cases with ppc= 10 4 , noise effects are reduced. The magnitude of the temperature is qualitatively similar to the benchmark case. Subsequently increasing the ppc to 10 5 and 10 6 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 88 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Figure 3.20: Comparative analysis of electron temperature contours in Landau damping simulations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simulation, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. effectively diminishes the noise level, although slight numerical noise is still observable. The temperature magnitudes in the cases with ppc=10 5 and ppc=10 6 are in good agreement with the benchmark Vlasov case. The heat flux data is considered crucial in characterizing physical properties, as it closes the fluid moment equations at an appropriate level and introduces kinetic wave-particle effects into these equations [224]. However, obtaining accurate heat flux data poses a significant challenge compared to number density and temperature, given that heat flux involves 3rd order moments of the velocity distribution function. Any noise associated with the distribution 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 89 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Figure 3.21: Comparative analysis of electron heat flux contours in Landau damping simulations between Vlasov and PIC methods. (a) PIC simulation, particles per cell (ppc) = 10 2 , unfiltered. (b) PIC simulation, particles per cell (ppc) = 10 2 , filtered. (c) PIC simulation, ppc = 10 3 , unfiltered. (d) PIC simulation, ppc = 10 3 , filtered. (e) PIC simulation, ppc = 10 4 , unfiltered. (f) PIC simulation, ppc = 10 4 , filtered. (g) PIC simulation, ppc = 10 5 , unfiltered. (h) PIC simulation, ppc = 10 5 , filtered. (i) PIC simulation, ppc = 10 6 , unfiltered. (j) PIC simulation, ppc = 10 6 , filtered. (k) Grid-based Vlasov simulation. functions is amplified in cubic order, making it highly sensitive to noise when obtained via the PIC method. Fig. 3.21 presents a comparison of electron heat flux contours in Landau damping simulations employing both Vlasov and PIC methods. In cases with ppc= 10 2 , the numerical noise is prominent, obscuring the underlying physical signals. Only a qualitative wave-like structure is observable due to this noise, and the magnitude of the heat flux significantly deviates from that of the benchmark case. For cases with ppc= 10 3 , the wave-like structure of the heat flux more closely resembles that of the benchmark Vlasov simulation, yet the magnitude still noticeably deviates from the benchmark. In the case with ppc= 10 4 , 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 90 Table 3.1: Signal-noise ratio, units in dB, unfiltered/filtered ppc= 10 2 ppc= 10 3 ppc= 10 4 ppc= 10 5 ppc= 10 6 n e 21.867/27.631 31.874/37.659 41.871/47.622 51.833/57.472 61.387/66.972 T e 18.615/24.054 28.652/34.119 38.640/44.069 48.476/53.492 56.726/59.756 Q e -15.581/-10.224 -5.451/0.092 4.518/9.884 14.454/19.662 23.234/26.975 the heat flux contours exhibit a wave-like structure and magnitude similar to the benchmark case, but they still display notable noise. Starting from pcc= 10 5 , the heat flux data begins to show qualitative agreement with the data obtained from the grid-based Vlasov simulations. The case with ppc= 10 6 shows the best agreement, although with slight residual noise. Fig. 3.22 illustrates quantitative comparisons of electron number density, electron temper- ature, and electron heat flux at different specific time moments. Here, we only present the two low-noise PIC simulation cases with a high number of particles per single cell. These figures provide a clearer illustration of the noise level present in the PIC simulations with varying numbers of particles per single cell. A satisfactory agreement is observed between the PIC simulation and the grid-based Vlasov simulation. The results of the PIC simulation for all three quantities exhibit oscillations around the benchmark provided by the grid-based Vlasov simulation. This assessment allows us to quantitatively evaluate the noise level present in the three quantities obtained through the PIC simulations. In this context, we use the concept of signal-to-noise ratio for quantification [151, 228]: SNR = P signal P noise (3.16) In terms of SNR values, a higher value indicates lower noise, while a lower value indicates higher noise. A positive SNR value signifies that the signal power surpasses the noise power, whereas a negative SNR value indicates that the noise power exceeds the signal power. As shown in Fig. 3.22, the data derived from the benchmark Vlasov simulation can serve as the signal, while the noise is defined as the discrepancy between the Vlasov simulation data and the PIC simulation data. Table 3.1 presents the computed Signal-to-Noise Ratio (SNR) values for the spatio-temporal flow fields in various PIC simulation cases. The SNR values 3.6 Discussions of Discrete Particle Noise Effects: Comparisons between PIC Simulations and Grid-based Vlasov Simulations of Linear Landau Damping 91 0 /4k /2k 3 /4k /k 5 /4k 3 /2k 7 /4k 2 /k 0.97 0.98 0.99 1 1.01 1.02 1.03 (a) 0 /4k /2k 3 /4k /k 5 /4k 3 /2k 7 /4k 2 /k 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 (b) 0 /4k /2k 3 /4k /k 5 /4k 3 /2k 7 /4k 2 /k -0.06 -0.04 -0.02 0 0.02 0.04 0.06 (c) Figure 3.22: Comparisons of electron number density, temperature, and heat flux between the benchmark Vlasov simulation case and the PIC cases with ppc= 10 5 and ppc= 10 6 at different time instances. 3.7 Summary 92 clearly indicate that, within the same case and with the same number of particles per cell, the density can be resolved most effectively, followed by the temperature, while the ability to resolve the heat flux is considerably lower. The SNR value for the temperature is only slightly smaller than that for the density. In contrast, the SNR value for the heat flux experiences a significant drop, underscoring the challenge of accurately resolving heat flux using particle-based simulations. This observation aligns with expectations. Even in the case with ppc= 10 6 , the ability to resolve the heat flux data only reaches a level comparable to the ability of resolving the density profile in the ppc= 10 2 case. The cases with ppc= 10 2 and ppc= 10 3 exhibit a significantly stronger noise profile compared to the signal profile for the heat flux, evident by the negative SNR values. This observation suggests that cases with ppc< 10 4 struggle to effectively resolve the heat flux, as the substantial numerical noise tends to dominate the physical quantities. This observation is consistent with the findings presented in Fig. 3.21. The quantified SNR values highlight the significance of employing a noise-free method when aiming to accurately resolve moments of the velocity distribution function beyond the second order. 3.7 Summary In this chapter, we have presented the development of Vlasolver, a parallel multi-dimensional Vlasov equation solver. We have discussed the numerical implementation details of Vlasolver, including the discretization of the phase space, the treatment of boundary conditions, and the parallelization strategy using domain decomposition. The effectiveness of the parallelization strategy has been demonstrated through tests of parallel efficiency, showing that Vlasolver exhibits good scalability with increasing core numbers. To ensure the accuracy and reliability of Vlasolver, both verification and validation processes have been conducted. In the verification process, classical physical problems with known analytical solutions were used to verify the correctness of the numerical solver. The 3.7 Summary 93 results obtained from Vlasolver were compared with analytical solutions, and good agreement was observed, confirming the accuracy of the solver. In the validation process, one practical problem, plasma wake expansion, was considered. The results generated by Vlasolver were compared with those obtained from well-benchmarked PIC codes. Additionally, qualitative comparisons with theoretical predictions were made to further validate the correctness of Vlasolver. The comparisons and analyses performed in the validation process demonstrated that Vlasolver is capable of generating reliable physical results for real-world applications. Moreover, we investigate a classical electron Landau damping problem employing both Vlasov and PIC simulations with varying numbers of particles per single cell, and conduct a comparative analysis on the impact of discrete particle noise. The results clearly demonstrate that, in the context of a periodic-boundary unsteady problem, particle-based simulations encounter challenges in accurately resolving high-order moments of the velocity distribution function. Notably, even in the case of an impractical particle count of ppc= 10 6 , efficient resolution of the heat flux is not achieved. These observations, in conjunction with our assessment, suggest that particle-based methods could impede the creation of highly precise training datasets for novel AI/learning-based techniques. Investigating the consequences of particle noise on constructing training datasets and its influence on the predictive capabilities of trained neural networks using noise-affected datasets remains a compelling project for future research. Overall, the development, implementation, verification, and validation of Vlasolver have been presented in this chapter, establishing its effectiveness as a parallel multi-dimensional Vlasov equation solver. The reliability and scalability of Vlasolver make it a valuable tool for studying complex plasma phenomena and simulations in various scientific and engineering fields. 94 Chapter 4: Grid-based Vlasov Simulation of Collisionless Plasma Expansion In this chapter, grid-based Vlasov simulations using the 1D1V version Vlasolver are carried out to re-evaluate the one-dimensional collisionless plasma expansion into vacuum. The grid-based method eliminates the inherent statistical noise in particle-based methods and allows us to extend the solution beyond the self-similar expansion region and resolve small electron time scale wave perturbations. It is shown that the expansion generates both an ion acoustic rarefaction wave mode and and an electron Langmuir wave mode that propagate into the unperturbed plasma upstream. The assumption used in the classical expansion solution that the electrons are an isothermal fluid is accurate within a quasi-neutral, self-similar expansion region but fails in both the upstream and downstream of that region due to electron time scale perturbations. 4.1 Introduction The expansion of a collisionless, unmagnetized plasma has been studied extensively both as a basic plasma dynamics problem and within the context of various applications, such as plasma wake formation by spacecraft and plasma plume emission from electric propulsion thrusters (see [20, 30, 35, 41, 42, 48–50, 89, 97, 213, 214, 229–232], and references therein). Most previous studies focused on the ion time scale physics and considered the role of the electrons as to simply provide charge neutralization. Hence, a common approach was to treat the ions kinetically while simplifying the electrons as a massless, perfect gas with an ad hoc equilibrium thermodynamic closure relation, such as the isothermal assumption. Within this framework, the one-dimension (1D) expansion process is described by the self-similar solution 4.1 Introduction 95 first presented by Gurevich et al. [30]. It was shown that, when a stationary plasma freely expands into a vacuum, an ambipolar electric field at the expansion front accelerates the ions downstream into the vacuum while an ion acoustic rarefaction wave propagates upstream into the unperturbed plasma [30, 35, 41]. Because of the massless electron fluid model, the classical expansion solution [30, 35, 41] only includes wave perturbations on ion time scale. In the rest of the paper, we shall refer the location of the vacuum as “downstream” and that of the unperturbed plasma as “upstream”, the direction of plasma expanding into vacuum as “forward” and that of ion acoustic rarefaction wave propagation towards upstream as “backward”, and the region between the forward plasma expansion front and backward ion rarefaction wave front as the “self-similar expansion” region. Several studies have examined the validity of the fluid assumption for electrons in colli- sionless plasma expansion [23, 24, 48, 97, 216, 217]. In a collisionless plasma, particle-particle collisions are largely absent, while the long-range Coulomb interaction is not an effective mechanism to establish a thermal equilibrium for electrons. Mora and Pellat [48] showed that the classical self-similar solution was not able to correctly predict the electron density and the energy transfer between electrons and ions in 1D expansion. Wang and Hu [24, 216] compared fully kinetic Particle-in-Cell (PIC) with hybrid PIC with the isothermal electron fluid model for two-dimensional 2D plasma expansion and showed that there exists a fluid-like expansion region and a kinetic expansion region for electrons. In the kinetic expansion region, where the plasma density is reduced to below about 10% of the initial density, the local electron velocity distribution functions (VDFs) becomes non-Maxwellian; the electron temperature is highly anisotropic; the errors resulting from using the electron fluid assumption strongly correlates with the deviation of the local electron VDF from the Maxwellian VDF; and neither the isothermal assumption nor the more general polytropic law represents an accurate approximation of the macroscopic electron thermodynamic closure relation [24, 216]. In previous studies, the focus has been on the self-similar expansion region. The interaction characteristics outside the self-similar region and possible wave perturbations on electron time scale were not considered. 4.1 Introduction 96 This chapter presents a more complete study of 1D free expansion of a semi-infinite collisionless, unmagnetized plasma into a vacuum. In addition to electron kinetics from finite electron mass, we extend the solution to include the perturbations beyond the self-similar expansion region and the effects from electron waves. There are two general numerical approaches to solve a collisionless plasma expansion problem. The first one is particle-based, such as the Particle-in-Cell (PIC) method. A particle-based method uses macro-particles to represent the plasma, follows the trajectories of the macro particles in the self-consistent electromagnetic field using the Lagrangian scheme, and obtains macroscopic physical variables from the statistics of the particles. The second one is grid-based. A grid-based method directly solves the Vlasov equation for the velocity distribution function (VDF) in the discretized phase space. The particle-based method is computationally efficient but may suffer from inherent numerical noise unless an extremely large number of macro-particles are used. A grid-based method has no inherent statistical noise but the computational cost is typically higher than PIC. Previously, Hu and Wang carried out fully kinetic PIC simulations using the real ion-to-electron mass ratio to include electron kinetic effects during plasma plume expansion and plasma wake formation [23, 24, 97, 216, 217]. They find that, while fully kinetic PIC can resolve most of the physics associated with expansion, the interactions in the very low density region beyond the expansion front and the effects from small electron plasma wave perturbations are generally beyond the reach of the PIC method due to statistical noise. For instance, Cui et al. [25, 233] compared PIC and the Vlasov method and found that one may need to use up to 10 4 macro-particles per cell in PIC to suppress the particle noise in local electron heat flux calculations. This diminishes the computational advantage of the PIC method. In this chapter, we apply the grid-based Vlasov simulation method to investigate colli- sionless plasma expansion. While the grid-based Vlasov simulation method has been applied in studies of fusion plasmas [182, 203], astrophysical plasmas [28, 165], and electric propul- sion [155, 205, 208, 234], to our knowledge, this method was rarely applied to address the collisionless expansion problem considered in this chapter. Previously, Manfredi et al. [235] 4.2 Simulation Model 97 carried out a grid-based simulation to investigate the application of the polytropic relation for electrons in 1D self-similar expansion. In this chapter, we carry out grid-based simula- tions using a higher mesh resolution than that in Ref. [235], and investigate the expansion characteristics both on ion time scale and on electron time scale and both within and beyond the self-similar expansion region. Section 4.2 discusses the formulation and simulation model. Section 4.3-4.5 presents the simulation results, extends the classical expansion solution, and discusses the effects from electron kinetic interactions. Section 4.6 contains a summary and conclusions. 4.2 Simulation Model The problem setup is similar to that in Gurevich et al. [30] (Fig. 4.1a). A stationary, collisionless, unmagnetized, homogeneous plasma with density n 0 is initially confined in the space of x 0 and the space at x> 0 is a vacuum. The plasma undergoes free expansion into x> 0 at t = 0. The expansion process can be solved from the Vlasov-Poisson system of equations: @f @t +vr x f +ar v f = 0; a = q r m (4.1) r ( 0 r) =e(n i n e ) (4.2) where subscript = i or e denotes the ions or electrons, respectively; f (x;v;t) is the velocity distribution function; is the electric potential; and n i , and n e denote the ion and electron number density, respectively. The simulations presented here only consider singly charged ions although the model can be easily extended to include multi-charged ions. The macroscopic properties of the plasma are described by the moments of the VDF <F >= Z 1 1 F ^ fdv (4.3) where ^ f =f=n is the normalized VDF, andn = R 1 1 fdv is the density. Since the plasma is collisionless, the particles’ velocity in the y and z direction will remain unchanged during the 4.2 Simulation Model 98 -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 Plasma Vacuum (a) -8 -6 -4 -2 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 30 35 40 (b) (c) Figure 4.1: Problem Setup. (a) Initial plasma density profile. (b) Initial ion and electron velocity distribution function. (c) Simulation domain. 4.2 Simulation Model 99 1D expansion process. Hence, we only need to consider 1D in physical space and 1D in velocity space when solving eq. (4.1). The simulation domain is (LxL,v max v x v max ) (Fig. 4.1c). The discretized mesh points are x i = -L +i x; i = 0; 1;:::;N x ; v j = -v max +j v; j = 0; 1;:::;N v where the mesh resolution is x = 2L=N x and v = 2v max =N v . The numerical scheme for solving eq. (4.1)-(4.2) and description of the numerical solver Vlasolver used here in this chapter can be found in Chapter 2 and 3 respectively and will not be introduced in detail here. In the simulation, the plasma at t = 0 follows the Maxwellian VDF (Fig. 4.1b) f (x;v;t = 0) =f m (v) =n 0 1 v t0 p 2 exp( v 2 2v 2 t0 ); Lx 0 (4.4) where subscript “0" denotes the initial condition, v t0 = p k b T 0 =m is the initial thermal velocity,T 0 the initial temperature, andk b the Boltzmann constant. The electric potential is (x;t = 0) = 0 in the entire domain. The physical domain sizeL is chosen to be far away from the expansion front through out the simulation. The space at x>L is vacuum. The space at x<L is a reservoir of the unperturbed initial plasma during the simulation. The velocity domain size v max is also chosen to be sufficiently large so that the VDF approaches zero at v max . In the simulation presented here, the normalized VDF atv max is less than 1 10 12 . From the numerical scheme used in this work, the update of f n+1 0 requires f n 2 , f n 1 , f n 0 , and f n 1 ; and the update of f n+1 Nx requires f n Nx1 , f n Nx , f n N x+1 , and f n N x+2 . Hence, the boundary conditions for the VDF are implemented using 3 guard cells outside of the simulation domain as shown in Fig. 4.1c. In the guard cells at x = L and v =v max , the VDFs are set to be 0. In the guard cells at x =L, the VDFs are fixed as the initial Maxwellian VDF for inflow particles, f m (v> 0) =n 0 1 v t0 p 2 exp( v 2 2v 2 th ). The boundary conditions for the electric potential are set to be = 0 at x = 0 and @=@x = 0 at x =L. 4.2 Simulation Model 100 To compare with the classical expansion solution, both fully kinetic and hybrid Vlasov simulations are carried out. In the fully kinetic simulation, both the electron and ions VDFs are solved from eq. (4.1). In the hybrid simulation, the ions are treated kinetically but the electrons are assumed to be a massless, isothermal fluid. Hence, instead of solving the electron VDF from eq. (4.1), the electron density is taken to follow the Boltzmann relation as in the classical solution [30, 35, 41]: n e =n 0 exp e k b T e0 (4.5) and the electric potential is solved from r ( 0 r) =e(n i n 0 exp e k b T e0 ) (4.6) In the fully kinetic simulation, all variables are normalized with respect to the initial electron parameters: ~ x = x D0 ; ~ t =t! pe0 ; ~ m = m m e ; ~ v = v v te0 ; ~ n = n n 0 ; ~ = e k b T e0 ; ~ f = f n 0 v te0 (4.7) where D0 = p 0 k b T e0 =n 0 e 2 , ! pe0 = p n 0 e 2 = 0 m e , v te0 = p k b T e0 =m e . In the hybrid simu- lation, as the electrons are treated as a massless fluid, the variables are normalized with respect to the initial ion parameters. Thus, the normalizations for time step, mass, velocity, and VDF in eq. (4.7) are changed as ~ t =t! pi0 ; ~ m i = 1; ~ v i = v i C s0 ; ~ f i = f i n 0 C s0 (4.8) where! pi0 = p n 0 e 2 = 0 m i , andC s0 = p k b T e0 =m i is the ion acoustic speed in the unperturbed plasma. The details of implementation of normalization scheme can be found in Section 2.3. It should be noted that different than Chapter 2 and Chapter 3, the sign in this chapter is not dropped for the normalized variables to distinguish it from the variables with physical units. 4.3 Comparison with the Self-Similar Solution 101 The simulation cases are summarized in Table 4.1. Case 1 is carried out using hybrid Vlasov simulation (ion to electron mass ratio m i =m e =1) . Cases 2A and 2B are carried out using fully kinetic Vlasov simulations, with an ion to electron mass ratio at m i =m e = 1600 and m i =m e = 100, respectively. The results from these 3 cases are also combined with that from additional simulations usingm i =m e = 900 and 400 to exam scaling relations. In all cases, the initial temperature ratio between the ions and electrons is taken to be T i0 =T e0 = 0:01; the physical domain size is from2000 D0 to 2000 D0 with mesh resolution x = D0 and number of the cells N x = 4000; and the velocity domain size for the ion Vlasov equation is from0:4v te0 to 0:4v te0 with mesh resolution v i = 1 640 v te0 and number of cells N v;e = 512. In Cases 2A and 2B, the velocity domain size for the electron Vlasov equation is from8v te0 to 8v te0 , with mesh resolution v e = 1 32 v te0 and number of cells N v;e = 512. All cases are run to 100 ion plasma time period, t! pi0 = 100. The time step used in the hybrid simulation is t! pi0 = 0:025. The time step used in the fully kinetic simulation is t! pe0 = 0:025. In all cases, the time step used satisfy v max t x 1; a max t v 1 (4.9) where v max and a max are the maximum velocity and acceleration, respectively. The simulations in this chapter were carried out using the serial version of Vlasolver on a Xeon- 2640v3 processor node on USC CARC’s Discovery cluster. The code is run serially. The clock time for the representative cases are about 1.8 hours for Case 1 (4000 time steps, ! pi0 t = 100), 86 hours for Case 2A (160000 time steps, ! pi0 t = 100, ! pe0 t = 4000), and 21 hours for Case 2B (40000 steps, ! pi0 t = 100,! pe0 t = 1000), respectively. The total memory requirements are about 1.2GB for Case 1, 2GB for Case 2A, and 2GB for Case 2B, respectively. 4.3 Comparison with the Self-Similar Solution 102 Table 4.1: Representative Simulation Cases (Additional fully kinetic simulations are also carried out using m i =m e =900 and 400) m i =m e T i0 =T e0 v ti0 =v te0 v ti0 =C s0 Hybrid 1 1 0.01 - 0.1 Fully kinetic 2A 1600 0.01 2:5 10 3 0:1 2B 100 0.01 0:01 0:1 4.3 Comparison with the Self-Similar Solution We first consider the expansion process at the ion time scale. Fig. 4.2 shows the potential profile, ~ (~ x), at ! pi0 t = 10, 40. We use two different mass ratios(Case 2A, m i =m e = 1600 and Case 2B, m i =m e = 100) for fully kinetic simulations. After the expansion starts, both the hybrid and fully kinetic simulation results show that a linearly decreasing potential region starts to develop at the plasma-vacuum interface x = 0. The potential perturbation expands in both the forward and backward direction. The classical solution was derived under the assumption that the electrons are isothermal and massless. Gurevich et al. [30] showed that the expansion process is self-similar, and the potential perturbation generated by the expansion is e k b T e (x;t) =1x=C s t; or ~ (~ x;t) =1 ~ x=(t! pi0 ) (4.10) where C s = p k b T e =m i = C s0 if the electrons are isothermal. The potential perturbation propagates at the ion acoustic speed in thex direction. The location of the backward propagating ion rarefaction wave front is given by ~ x ion ( ~ t) = ~ C s ~ t; or ~ x ion (! pi0 t) = ~ C s r m i m e t! pi0 =t! pi0 (4.11) where ~ C s = C s =v te0 is the normalized ion acoustic speed. A double layer develops at the forward propagating plasma expansion front. Mora [42] defined the plasma expansion front by the location at the middle of the double layer, and derived that location analytically. 4.3 Comparison with the Self-Similar Solution 103 -200 -150 -100 -50 0 50 100 150 200 250 300 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 (a) -200 -150 -100 -50 0 50 100 150 200 250 300 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 (b) -200 -150 -100 -50 0 50 100 150 200 250 300 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 (c) Figure 4.2: Potential profile at different time. (a) Case 1 (hybrid). (b) Case 2A (fully kinetic, m i =m e = 1600). (c) Case 2B (fully kinetic, m i =m e = 100). The analytical solutions of the ion rarefaction wave front, the potential slot, quasi-neutral region front and plasma expansion front are also shown for comparison. 4.4 The Electron Langmuir Wave 104 Under the isothermal assumption for electrons, ~ C s = p m e =m i . The asymptotic solution of the location of the expansion front at ! pi0 t 1 is given by ~ x Mora + (! pi0 t)' r m i m e ~ C s ! pi0 t [2 ln (! pi0 t) + ln 2 3] =! pi0 t [2 ln (! pi0 t) + ln 2 3] (4.12) As the self-similar characteristics is only valid under the quasi-neutral condition, Medvedev [45] examined the boundary of the quasi-neutral plasma region behind the double layer. Based on parametric fitting of hybrid PIC simulation results, Medvedev [45] derived the location of the quasi-neutral plasma expansion front as ~ x Medvedev + (! pi0 t)' 2 r m i m e ~ C s ! pi0 t ln (! pi0 t)= 2! pi0 t ln (! pi0 t): (4.13) where ' 0:19. We find that the location of the quasi-neutral expansion front in our fully kinetic Vlasov simulations can also be fit using eq. (4.13) but the fitting parameter is' 0:15. Fig. 4.2 also shows the slope of ~ (~ x), and the propagation of the boundaries of the linearly decreasing ~ (~ x) region. In all cases, the ~ (~ x) slope agrees well with that of eq. (4.10), the left boundary of the linearly decreasing ~ (~ x) region agrees well with ~ x ion (! pi0 t) from eq. (4.11), and the right boundary of the linearly decreasing ~ (x) region agrees well with ~ x Medvedev + (! pi0 t) from eq. (4.13) with ' 0:15. When the t! pi0 is same, the characteristics of the self-similar expansion are the same regardless of the ion to electron mass ratio. Hence, within the self-similar expansion region, the fully kinetic simulation results confirm the ion time scale solution derived using the massless electron model. For comparison, the location of ~ x Mora + (! pi0 t) from eq. (4.12) is also shown in Fig. 4.2. As time increases, the difference between ~ x Mora + and ~ x Medvedev + starts to increase, indicating a spatial spread of the double layer expansion front. 4.4 The Electron Langmuir Wave 105 -2000 -1600 -1200 -800 -400 0 400 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 (a) -2000 -1600 -1200 -800 -400 0 400 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 (b) Figure 4.3: Zoomed in comparisons of potential profile at different time. a) ! pe0 t = 100 (Case 1: ! pi0 t = 2:5; Case2A: ! pi0 t = 2:5; Case 2B: ! pi0 t = 10.) b) ! pe0 t = 400 (Case 1: ! pi0 t = 10; Case2A: ! pi0 t = 10; Case 2B: ! pi0 t = 40). 4.4 The Electron Langmuir Wave Both the analytical solution and the hybrid Vlasov simulation (Case 1) show that the plasma remains unperturbed in the region upstream of the ion acoustic rarefaction wave front, 4.4 The Electron Langmuir Wave 106 ~ x< ~ x ion ( ~ t). However, the fully kinetic Vlasov simulation results (Case 2A and 2B) exhibit additional small perturbations in that region. This new perturbation patten can be seen clearly in Fig. 4.3, where we compare ~ (~ x) at ! pe0 t = 100 and 400. A large amplitude oscillation in ~ (~ x) is generated at the plasma vacuum interface. The oscillation magnitude gradually decreases as the perturbation propagates ahead of the ion acoustic rarefaction wave front into the unperturbed plasma. Fig. 4.4 shows the Fourier transform of the electric field, ~ E(~ !; ~ k), for the time period ~ t = 0 1000 and over the entire computational domain. Here, ~ ! =!=! pe0 and ~ k =k D0 . The ~ E(~ !; ~ k) contours show that the waves are concentrated around two distinctive wave modes. For the first wave mode, the ~ E(~ !; ~ k) contour concentration obtained from simulation corresponds well with the dispersion relation of the ion acoustic wave, as shown by the red dashed line ~ != ~ k' ~ C s0 (4.14) This wave model is the ion acoustic rarefaction wave described by the classical self-similar solution. The second wave mode is associated with perturbations on electron time scale. The dispersion relation for 1-D electron Langmuir wave derived from the kinetic moment equations can be written as an asymptotic series [236] ! 2 =! 2 pe m X n=0 (2n + 1)!! k 2n v 2n t ! 2n ! +O k 2m+2 v 2m+2 t ! 2m+2 ! (4.15) Note that, in the electron fluid limit, eq. (4.15) is reduced to the familiar form of ! 2 =! 2 pe + 3k 2 v 2 t ; or ~ ! 2 = 1 + 3 ~ k 2 4.4 The Electron Langmuir Wave 107 -3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 -7 -6.5 -6 -5.5 -5 -4.5 -4 (a) -3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 -7 -6.5 -6 -5.5 -5 -4.5 -4 (b) Figure 4.4: Spectrum contour of ~ E(~ x; ~ t) after Fourier transform in the ~ ! ~ k plane. (a) Case 2A (m i =m e = 1600). (b) Case 2B (m i =m e = 100). We also compare eq. (4.15) with the wave model obtained numerically. In Fig. 4.4, the horizontal red dashed line corresponds to ~ ! pe = 1, i.e. the cut-off frequency of the electron Langmuir wave, and the purple line shows eq. (4.15) truncated after m = 3: ! 2 =! 2 pe 1 + 3k 2 v 2 t ! 2 + 15k 4 v 4 t ! 4 + 105k 6 v 6 t ! 6 +O k 8 v 8 t ! 8 (4.16) 4.4 The Electron Langmuir Wave 108 (a) (b) Figure 4.5: Spatial-temporal contours of ~ E(~ x; ~ t). (a) Case 2A (m i =m e = 1600). (b) Case 2B (m i =m e = 100). ! 2 pe 1 + 3k 2 v 2 t ! 2 + 15k 4 v 4 t ! 4 + 105k 6 v 6 t ! 6 pe where we have approximated the higher order terms ! 2 pe ( 105k 6 v 6 t ! 6 +O k 8 v 8 t ! 8 ) by 105k 6 v 6 t ! 6 pe for !>! pe . In the normalized form, eq. (4.16) is ~ ! 2 1 + 3 ~ k 2 ~ ! 2 + 15 ~ k 4 ~ ! 4 + 105 ~ k 6 ! (4.17) 4.5 The Perturbations Characteristics 109 Fig. 4.4 shows that the contour concentration region on the spectrum of ~ E(~ !; ~ k) obtained from simulation corresponds well with eq. (4.17). Thus, the electron time perturbations observed in the simulation is the electron Langmuir wave. Fig. 4.4 shows that the backward propagating electron Langmuir waves ( ~ k 0) are concentrated within (1 ~ ! 1:55,0:45 ~ k 0) along the dispersion curve. From eq. (4.17), we find the maximum group velocity is around d~ !=d ~ k'3:5 ( evaluated at ~ ! ' 1:55; ~ k '0:45). Fig. 4.5 further shows the spatial-temporal plot of the electric field ~ E(~ x; ~ t). The blue dashed line overlaid over the contours separates the unperturbed region from the perturbed region and shows the time history of the backward propagating electron wave front location. The slope of the blue dashed line shows the speed of the wave front. Fig. 4.5 shows that the wave front propagation speed observed in the simulation is approximately3:5, which matches well with the group velocity the dispersion relation. Thus, ~ x ele ( ~ t) =~ v ele g ~ t; ~ v ele g ' d~ ! d ~ k ' 3:5 (4.18) Hence, when the finite electron mass is included in the solution, the result show that the free expansion of plasma into vacuum generates both the ion acoustic rarefaction wave and the electron Langmuir wave. Both wave modes propagate backward into the unperturbed plasma. For m i =m e =1600 and 100, the ion acoustic speed is ~ C s0 ' 0:025 and ~ C s0 ' 0:1 respectively, and is much smaller than the group velocity of the electron Langmuir wave, ~ v ele g ' 3:5. Thus, the electron Langmuir wave propagates ahead of the ion rarefaction wave into the unperturbed plasma, as shown in Fig. 4.3. 4.5 The Perturbations Characteristics 110 -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (a) -1000 -800 -600 -400 -200 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (b) Figure 4.6: Electron temperature profile at different time. (a) Case 2B (m i =m e = 100). (b) Zoomed in plot for Case 2B (m i =m e = 100 at ~ t = 400). 4.5 The Perturbations Characteristics 111 4.5 The Perturbations Characteristics We next examine the characteristics of the perturbations generated by expansion. Fig. 4.6 shows the electron temperature normalized by T e0 , ~ T e (~ x), at ! pe0 t =100, 200, 300 and 400. Here, ~ T e (~ x) is calculated from the local electron velocity distribution f e : ~ T e = 1 ~ n e Z ~ w 2 ~ f e d~ v (4.19) where w = v x < v x > is the random electron velocity and ~ w = w=v te0 . Fig. 4.7 further correlates ~ T e (~ x), with the local electron heat flux ~ Q e (~ x), and the space charge density ~ n i (~ x) ~ n e (~ x) for Case 2B. Here, the electron heat flux ~ Q e (~ x) is calculated from ~ Q e = 1 2 ~ m~ n e < ~ w 3 >= 1 2 ~ m e Z ~ w 3 ~ f e d~ v (4.20) and is normalized by Q e0 =n 0 m e v 3 te0 =n 0 T e0 v te0 . Figs. 4.6 and 4.7 show that ~ T e (~ x) is not isothermal but exhibits four different regions in the simulation domain. For convenience of discussions, the region around ~ x = 0 where ~ T e (x) decreases linearly is denoted as Region C; the region upstream of Region C where ~ T e (x) also decreases but at a much slower rate is denoted as Region B; the region further upstream of Region B where ~ T e (x) = 1 is denoted as Region A; and the region downstream of Region C is denoted as Region D, where ~ T e (x) exhibits a hump immediately downstream of the expansion front before decreasing to ~ T e (x) = 0. Region C is the quasi-neutral (~ n i (~ x)' ~ n e (~ x)), self-similar expansion region. The pertur- bations in Region C expands in both the forward and backward direction. The backward propagation of Region C boundary coincides with that of the ion rarefaction wave front given by eq. (4.11), ~ x ion ( ~ t) = ~ C s ~ t. Since there is a slight drop in the electron temperature in the region swapped by the rarefaction wave, the propagation speed is slightly smaller than ~ C s0 . In the classical solution, the exact location of the self-similar expansion front is somewhat ambiguous. Here, a comparison of the ~ T e (~ x) with ~ n i (~ x) and ~ n e (~ x) shows that the turning 4.5 The Perturbations Characteristics 112 -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.03 0 0.03 0.06 0.09 0.12 0.15 0.18 -2000 -1500 -1000 -500 0 500 1000 1500 2000 -4 -3 -2 -1 0 1 2 3 4 10 -3 (a) -2000 -1500 -1000 -500 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.03 0 0.03 0.06 0.09 0.12 0.15 0.18 -2000 -1500 -1000 -500 0 500 1000 1500 2000 -2 -1 0 1 2 10 -3 (b) Figure 4.7: Electron temperature, electron heat flux and corresponding charge density. (a) Case 2B (m i =m e = 100) at ! pe0 t = 200. (b) Case 2B (m i =m e = 100) at ! pe0 t = 400. point on the ~ T e (~ x) profile closely matches the boundary of the quasi-neutral plasma region as well as the quasi-neutral expansion front defined by eq. (4.13), ~ x Medvedev + (! pi0 t). Hence, the location of Region C is defined by ~ C s ~ t< ~ x ~ x Medvedev + (! pi0 t). In Region C, the ~ Q e (~ x) quickly decreases to 0, and the self-similar expansion is at the expanse of the electron thermal energy. 4.5 The Perturbations Characteristics 113 In all the previous studies, the region upstream of the ion rarefaction wave front has been considered as the unperturbed plasma. The fully kinetic simulation results show that perturbations in both the electric potential and electron temperature profile propagates ahead of the ion rarefaction wave front in the backward direction. Figs. 4.6 and 4.7 show that the left-side boundary of Region B propagates towards the upstream at the same speed as the group velocity of the electron Langmuir wave, ~ v ele g . Thus, the left boundary of Region B coincides with that of the electron Langmuir wave front given by eq. (4.18), ~ x ele ( ~ t) =~ v ele g ~ t, and the location of Region B is defined by~ v ele g ~ t< ~ x< ~ C s ~ t. The ~ n i ~ n e vs. ~ x profile in Fig. 4.7 also shows that, while the plasma is overall quasi-neutral during the expansion process, there are small density oscillations on electron time scale upstream of the expansion front. These small density oscillations correspond to the perturbations caused by the electron Langmuir wave. In Region B, the electron temperature decreases slightly from upstream toward Region C, the heat flux is non-monotonic, and the electron Langmuir wave dominates. The region upstream of the electron Langmuir wave front (~ x<~ v ele g ~ t), Region A, is the true unperturbed plasma region. Region D is downstream of the quasi-neutral plasma expansion front, ~ x> ~ x Medvedev + (! pi0 t). This region contains the double layer downstream of the quasi-neutral expansion and the vacuum region. Previous studies have not considered the electron temperature in this region because the plasma density quickly decreases to 0. Fig. 4.8 shows the local electron VDFs for Case 2B at selected locations in Regions B, C, and D. The Maxwellian electron VDFs based on local ~ T e (~ x) are also plotted for comparison. Fig. 4.8 shows that the electron VDF in Region C is close to Maxwellian but deviates from Maxwellian in Regions B and D. In the backward direction, the electron VDF shows a “dip” in Region B and a “peak” in Region D. To understand the electron VDF, Figs. 4.9 and 4.10 further show the evolution of the electrons in thev x vs. x phase space for Case 2B. In Fig. 4.9, the electric field profile is also overlayed with the phase space contour for comparison. As the expansion starts, the much lighter electrons leave the ions behind, creating a charge separation and the initial electric field at the plasma-vacuum interface. The electric field at 4.5 The Perturbations Characteristics 114 -6 -4 -2 0 2 4 6 -3 -2.5 -2 -1.5 -1 -0.5 0 -6 -4 -2 0 2 4 6 -3 -2.5 -2 -1.5 -1 -0.5 0 -6 -4 -2 0 2 4 6 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 -6 -4 -2 0 2 4 6 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 Figure 4.8: Local electron velocity distribution functions (VDFs) at selected locations for Case 2B ! pe0 t = 400. A fit of the Maxwellian VDF using the local ~ T e (~ x) is also plotted for comparison. the expansion front does not affect those electrons that leave the bulk plasma at the start of the expansion but attracts some of the electrons to backflow during subsequent expansion. The stream of the backflow electrons creates the peak in the electron VDF in Region D. The electron VDF at the expansion front does not spread out but has a non-Maxwellian component. For such a VDF, the electron temperature will be estimated high from eq. (4.19) This peak corresponds to the temperature enhancement shown in the ~ T e (~ x) profile in Fig. 4.7. The “dip” shown in electron VDF in Region B corresponds to the void region shown in the phase space contours (Fig. 4.9 and Fig. 4.10). We find that the slope of the void region (marked by the green dashed inclined line) is ~ v=~ x' 3:5 ~ t, corresponding to the value of ~ v ele g . The location of the void region in the phase plot corresponds to that of the backward propagating electron Langmuir wave front, ~ x ele . Thus, the “void" regions in Fig. 4.9 and Fig. 4.10 are caused by those electrons in resonance with the electron Langmuir wave at the wave front location. Those electrons with a velocity of ~ v e '~ v ele g lose the energy to 4.5 The Perturbations Characteristics 115 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 -8 -6 -4 -2 0 2 4 6 8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -3 -2.5 -2 -1.5 -1 -0.5 0 (a) -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 -8 -6 -4 -2 0 2 4 6 8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -3 -2.5 -2 -1.5 -1 -0.5 0 (b) Figure 4.9: Electron phase space (~ v e vs. ~ x) contour for Case 2B and electric field profile ~ E(~ x). (Left axis: Electron velocity ~ v e . Right axis: Electric field strength ~ E.) (a) ! pe0 t = 10. (b) ! pe0 t = 40. the wave as the wave front propagates upstream. The energy lost by the electrons sustains the propagation of the electron Langmuir wave. This resonant wave-particle interaction depletes the number of electrons with a velocity at vv ele g . Behind the wave front, the void in the electron VDF is gradually refilled. This leads to the slight decrease of ~ T e in 4.5 The Perturbations Characteristics 116 -2000 -1750 -1500 -1250 -1000 -750 -500 -250 0 250 500 -8 -6 -4 -2 0 2 4 6 8 -3 -2.5 -2 -1.5 -1 -0.5 0 (a) -2000 -1750 -1500 -1250 -1000 -750 -500 -250 0 250 500 -8 -6 -4 -2 0 2 4 6 8 -3 -2.5 -2 -1.5 -1 -0.5 0 (b) Figure 4.10: Electron phase space (~ v e vs. ~ x) contour for Case 2B. (a) ! pe0 t = 200. (b) ! pe0 t = 400. 4.5 The Perturbations Characteristics 117 Region B as observed in Fig. 4.7. Previously, Medvedev [54] showed that an ion “cooling” wave with a spatial size much longer than the rarefaction wave perturbed region exists in the ion-ion plasma expansion. It was also speculated that the cooling observed should also exist in an electron-ion plasma. The results here showed that a similar “cooling” perturbation on electrons in the electron Langmuir wave region and the heat flux has a similar structures as in Ref. [54]. However, we conclude that this “cooling” is not due to bulk cooling but a reflection of the change in electron VDF caused by those electrons in resonance with the electron Langmuir wave. Figure 4.11: ~ T e;max vs. p m e =m i in Region B. Fig. 4.11 summarizes the maximum normalized electron temperature change at steady state in Region B, ~ T e;max = max(T e T e0 )=T e0 , for all the electron to ion mass ratios m e =m i considered. The results show that ~ T e;max scales quasi-linearly with p m e =m i ~ T e;max ' 0:0035 + 0:8906 p m e =m i (4.21) 4.5 The Perturbations Characteristics 118 However, we note that the electron temperature as calculated from eq. (4.19) is not a good representation of the average electron thermal properties in Region B because the local electron VDF is strongly non-Maxwellian. A key assumption in the classical solution [30] is that the electrons are isothermal during the expansion. Many previous studies have also assumed that the electrons in collisionless plasma expansion follow the more general polytropic law as in classical gas expansion T e =T e0 = (n e =n e0 ) e1 (4.22) where e is the electron polytropic coefficient, n e0 and T e0 represent the electron density and temperature at a reference point, respectively, and = 1 corresponds to the isothermal process. As the electrons in Region C is approximately Maxwellian, it is also interesting to examine whether ~ T e and ~ n e obtained from Region C can be fit into the polytropic law, eq. (4.22). Fig. 4.12 summarizes this fit at steady state for all the electron mass ratios Figure 4.12: vs. p m e =m i in Region C. 4.6 Summary 119 considered. The result shows that, in the self-similar expansion region, the scaling of the polytropic coefficient e over the mass ratio is e ' 1:00 + 0:45 p m e =m i (4.23) The electrons are isothermal in the limiting case of massless electrons. For a proton plasma, m i =m e = 1836:15, the deviation from the the isothermal condition is only e ' 0:01. Hence, the assumption used in the classical expansion solution that the electrons can be assumed as an isothermal fluid is quite accurate within the quasi-neutral, self-similar expansion region for 1D plasma expansion. 4.6 Summary Grid-based Vlasov simulations using the 1D1V version of Vlasolver are carried out to re- evaluate 1-D collisionless plasma expansion into vacuum. The Vlasov method eliminates the inherent statistical noise in particle-based methods and allows us to extend the expansion solution beyond the self-similar expansion region described by the classical solution and resolve the electron scale time wave perturbations during the expansion. It is shown that the expansion generates both an ion acoustic rarefaction wave mode and an electron Langmuir wave mode. Both wave modes propagate into the unperturbed plasma upstream. As the electron Langmuir wave propagates at a speed of ~ v ele g 3:50 ~ C s , the region upstream of the ion acoustic rarefaction wave front is also perturbed. We find the perturbations characteristics can be characterized as following when the effects of finite electron mass is included: around the plasma vacuum interface within the boundaries defined by the backward propagating ion acoustic wave front and the forward propagating quasi-neutral expansion front, ~ C s ~ t< ~ x ~ x Medvedev + (! pi0 t), the plasma is quasi-neutral, the electron VDF is close to Maxwellian, and the expansion process is self-similar and can be described quite accurately by the classical solution using the isothermal electron fluid assumption. The classical solution 4.6 Summary 120 is sufficient to describe the self-similar expansion process but cannot be extended beyond the boundary defined by the backward propagating ion acoustic rarefaction wave front and the forward propagative quasi-neutral expansion front. Upstream of the ion acoustic wave front, the plasma within the region of~ v ele g ~ t< ~ x< ~ C s ~ t (Region B) is perturbed by a small amplitude electron Langmuir wave. In this region, those electrons with a velocity of~ v ele g lost their energy to the wave front and sustains wave propagation. Such interactions creates a small “cooling” in electron temperature. Downstream of the quasi-neutral expansion front, ~ x > ~ x Medvedev + (! pi0 t) (Region D), a double layer develops immediately downstream of the quasi-neutral front. The electric field at the double layer attracts some electrons to backflow, creating an enhancement in electron temperature around the double layer front. However, the changes in electron temperature (as calculated from the second order velocity moment) observed in both Regions B and D are not a good representation of the average electron thermal property because the local electron VDFs are strongly non-Maxwellian. 121 Chapter 5: Grid-based Vlasov Simulation of Electric Propulsion Beam This chapter presents an investigation of the two-dimensional expansion of electric propulsion beam into vacuum, employing grid-based Vlasov simulations. The main aim of this research is to investigate the relationship between the macroscopic thermodynamic properties of electrons and the microscopic electron kinetic properties during the expansion process of unmagnetized plasma beam. The utilization of the grid-based method effectively mitigates the inherent statistical noise encountered in particle-based approaches, enabling a focused analysis of the higher order moments of the velocity distribution function. Specifically, this methodology facilitates the examination of crucial parameters, including electron temperature and electron heat flux. The analysis reveals that the collisionless electron heat flux during beam expansion is primarily dominated by the axial direction collisionless electron heat flux. Furthermore, to explore the relationship between heat flux characteristics and electron microscopic kinetics, the skewness of the electron velocity distribution function (VDF) is considered. The density- normalized electron velocity distribution function (VDF) exhibits a skewness that shares the same form with the coefficient in the flux-limited model of electron collisionless heat flux. The analysis identifies two distinct regions in the on-axis flux-limited coefficient, which exhibit hump and tail structures, respectively, at different time intervals. Notably, these structures exhibit self-similarity in terms of their evolution over time. Through these investigations, valuable insights are gained into the underlying physical mechanisms governing collisionless electron heat flux during electric propulsion beam expansion. Furthermore, these findings establish a solid foundation for future developments of predictive reduced-order models. 5.1 Introduction 122 5.1 Introduction The plasma plume emitted by an electric propulsion (EP) thruster is a fundamental problem in EP, and has been studied extensively. Most modeling studies focused on the physics on the ion plasma time scale. Earlier modeling studies mostly concern with plume induced spacecraft interaction and contamination which resolve only the dynamics of the charge-exchange (CEX) ions generated in the beam. The majority of such studies are based on hybrid Particle-in-Cell (PIC) simulations where the CEX ions are treated as macro-particles, the electrons are modeled as a massless, ideal gas with an assumed thermodynamic closure relation, and the beam ions are modeled by an empirical density profile (see, for example, [19, 20, 68, 73, 78, 87] and references therein). Many recent modeling studies start to focus more on the propellant ion beam which resolve the dynamics of the beam ions and the electrons. Such studies require fully kinetic PIC simulations where both the beam ions and the neutralizing electrons are treated as macro-particles (see, for example [22–24, 96, 97, 237] and references therein). An important unresolved issue in plume modeling studies so far is the thermodynamics of the electrons in the plume, and their effects on the overall plume characteristics [24]. A common assumption used in most plume models is to simplify the electrons as a massless ideal gas undergoing either the isothermal process (which leads to the commonly used Boltzmann relation n e = n 0 exp [e( 0 )=k b T e0 ]) or the more general polytropic process, T e n 1 e e =T e0 n 1 e e0 process, to save computational time. Such an assumption leads to at least two issues. First, the electron temperature (for the isothermal assumption) or the polytropic coefficient e (for the polytropic assumption) and the plume potential with respect to ambient must be provided as an input. The choice of these values can have a great influence on the plume modeling results. For example, the excellent agreement between PIC simulation [20] and in-flight measurement [18] of the Deep Space I ion thruster plume environment is due in no small part to the availability of in-flight data of the electron temperature and plume potential. Second, and more importantly, the assumption of an equilibrium gas model for electrons in a collisionless plasma is in general not valid because the long-range Coulomb 5.1 Introduction 123 interaction between charged particles is not an effective equilibrating mechanism. Indeed, recent fully kinetic PIC simulations of unmagnetized plasma beam emission [23, 24, 96, 97] showed that the electrons in an unmagnetized beam are non-equilibrium and anisotropic, the electron velocity velocity distribution (VDF) significantly deviates from the Maxwellian distribution, the adoption of an equilibrium gas model for electrons can lead to significant errors in ion thruster plume results, and simple variations of the Boltzmann or the polytropic relation for electrons will not lead to a significant improvement in accuracy [24]. To develop an accurate model for electrons requires one to have a clear understanding of both the microscopic kinetics and the macroscopic thermodynamic properties of the electrons in the beam. The classical transport mechanisms [238] are not applicable in a collisionless plasma [239]. For the plasma under collisionless conditions, if the electrons are strongly magnetized, the small electron gyro-radius plays the role of an equivalent collisional mean free path, and thus can enable the construction of electron fluid models [240–243] and heat flux dynamical evolution [244]. However, if the electrons are completely unmagnetized, the construction of fluid and thermodynamic properties for electrons is not trivial due to the non-local electron kinetics [224, 232, 245, 246]. Previous full PIC simulations of ion thruster beam and collisionless plasma wake have demonstrated that the macroscopic electron properties are sensitively influenced by the electron VDF [23, 24, 216]. However, the inherent numerical noise in PIC makes it difficult to resolve higher order electron velocity moments in a simulation, preventing further understanding of the macroscopic electron thermodynamic processes (For example, Cui and Wang [25] tested the effect of particle noise on electron temperature in a 1D full PIC simulations of axial beam expansion, and found that even utilizing 25600 macro-particles/cell in the simulation, a computation parameter that is not realistic for PIC application studies, the noise level is still not acceptable). This chapter concerns with the physics of primary beam emitted by an electrostatic thruster (e.g. ion thruster, Hall thruster), with a focus on the electron thermodynamic properties in an unmagnetized plasma beam. To eliminate the interference from the inherent 5.2 Simulation Model 124 numerical noise of the PIC method, we present an alternative approach for EP simulation using grid-based Vlasov simulation. The grid-based Vlasov simulation method has been applied in studies of fusion plasmas [167] and astrophysical plasmas [28, 165]. However, other than a few studies on Hall thruster discharge [205, 247], hollow cathode instabilities [208, 248, 249] and one-dimensional plasma expansion [250–252], it has not been applied extensively by the EP community. This chapter applies a recently developed Vlasov model, Vlasolver [25, 233, 250, 253, 254], to simulate the emission and expansion of a neutralized beam of cold beam ions and thermal electrons in a two-dimensional (2-D) domain. To simplify the problem, the simulation setup does not include the initial ion and electron mixing region near the thruster exit, the charge-exchange collisions between the beam ions and the un-ionized neutrals, and any effects from the magnetic field of the thruster or the ambient magnetic field. To the best of our knowledge, this paper represents the first Vlasov simulation of EP beam. The rest of this chapter is organized as follows. Section 5.2 discusses the simulation model setup. Section 5.3 compares the results from Vlasov simulation and full particle PIC. Section 5.4 investigates the electron microscopic kinetics. Section 5.5 and Section 5.6 discusses the electron heat flux in the beam and its connection to electron microscopic characteristics. Section 5.7 contains a summary of this chapter. 5.2 Simulation Model A collisionless plasma with only the electrostatic field can be solved from the following Vlasov-Poisson system of partial differential equations (PDEs): @f @t +vr x f +ar v f = 0; a = q r m (5.1) r ( 0 r) =e(n i n e ) (5.2) 5.2 Simulation Model 125 where subscript = e or i denotes the electron or ion, respectively; q and m denote the species charge and mass, respectively;x andv denote the position and velocity vector in phase space, respectively; f (x;v;t) is the velocity distribution function (VDF) for the species ; is the electric potential; n i and n e denote the ion and electron number density, respectively. The plasma macroscopic properties are obtained through the moments of the velocity distribution function <F >= Z 1 1 F ^ f dv (5.3) where ^ f =f =n is the density normalized velocity distribution function, andn = R 1 1 f dv. We will apply the following dimensionless variables in eqs. (5.1)-(5.3): ~ x = x d ; ~ t =t! pe0 ; ~ v = v d! pe0 ; ~ m = m m e ; ~ n = n n e0 ; ~ T = k b T m e d 2 ! 2 pe0 ; ~ = e m e d 2 ! 2 pe0 ; ~ E = eE m e d! 2 pe0 ; ~ f = (d! pe0 ) Nv f =n 0 (5.4) where e and m e denote the electron charge and mass, respectively; n e0 , d = De0 = p 0 k b T e0 =n e0 e 2 , and! pe0 = p n e0 e 2 = 0 m e are the electron density, the Debye length, and the electron plasma frequency at t = 0, respectively; k b is the Boltzmann constant; and T is the temperature for specie . The superscription N v denotes the velocity space dimension. In the rest of this chapter, we will drop the \ " sign for the dimensionless variables. We consider the emission of an unmagnetized, collisionless, mesothermal plasma beam consisting of cold beam ions and thermal electrons in a 2-D planar domain. The flow characteristics are v ti0 v d v te0 , where v ti0 , v d , v te0 denote the ion thermal velocity, beam drifting velocity and electron thermal velocity respectively. The simulations are carried out using an ion to electron mass ratio of m i =m e = 1836. Fig. 5.1 shows the problem setup. Only half of the physical domain is considered due to symmetry [253, 254]. The plasma beam is injected from an emission surface on the x min domain boundary, (x;y) = (0; 0 20), with a drifting velocity along the x direction. The electric potential of the emission surface is set to be = 0 and the potential boundary condition at all other boundaries is the Neumann boundary condition with @=(@xn) = 0). 5.2 Simulation Model 126 (a) (b) Figure 5.1: Schematic plots of simulation model. (a) Simulation setup. (b) Phase space configuration and discretization. Fig. 5.1b shows the discretized 2D2V phase space. Both the physical and the velocity space are discretized into uniform meshes. The VDF is obtained by directly solving Vlasov-Poisson equations numerically on the discretized mesh points. In the simulation, the plasma beam parameters at the emission plan are similar to that of a typical ion thruster beam: the temperature of the emitted electrons and ions are set to 5.2 Simulation Model 127 Table 5.1: Physical parameters for the beam Species m T 0 v t0 v d C s0 Ma Electron 1:0 1:0 1:0 0:1 0:0233 4.29 Ion 1836 0:01 0:00233 0:1 0:0233 4.29 T i = 0:01T e , and the beam velocity is set to have a Mach number of 4.29 (v d = (0:1; 0; 0)). Other relevant parameters of the plasma beam are shown in Table 5.1. ThephysicaldomainhasalengthofL x L y = 800400withmeshresolution x = y = 1 and thus the cell numbers are N x = 800 and N y = 400. The velocity domain length for the ions is from v i;x =0:1 to v i;x = 0:3 and from v i;y =0:2 to v i;y = 0:2 with mesh resolution v i;x = v i;y = 1=640. This results in cell numbers N i;vx = 256 and N i;vy = 256. The velocity domain length for the electrons is from v e;x =7:9 to v e;x = 8:1 and from v e;y =8:0 to v e;y = 8:0 with mesh resolution v e;x = v e;y = 1=16. This results in cell numbers N e;vx = 256 and N e;vy = 256. All of the simulations are performed until t = 2000 with a time step length t = 0:05. Here the time steps are chosen in order to fulfill the CFL condition v max t x 1; a max t v 1 (5.5) wherev max anda max are the maximum velocity and acceleration, respectively. The parameters mentioned above are summarized in the Table. 5.2. The implementation of the boundary conditions in Vlasov equation is not trivial. An electricpropulsiondeviceemitsequalionandelectronfluxeswhichquicklyformaquasi-neutral beam outside the thruster exit. In this study, we consider the beam to be a quasi-neutral (n i =n e ) and current free (J i =J e ) at the emission plane. To achieve this emission condition, a virtual reservoir is placed outside the simulation domain adjacent to the emission plane through the use of ghost cells. Inside the reservoir, the plasma is pre-loaded with the semi- Maxwellian VDF (Fig. 5.2). Only the “particles” with v 0 are injected inside the domain. While the cold ions rarely backflow from the emission surface, a large populations of the electrons may backflow due to their much large mobility. Hence, the number density of the 5.2 Simulation Model 128 (a) (b) Figure 5.2: Initial velocity distribution functions for electrons and ions. injected thermal electrons is dynamically adjusted based on the electron to ion density ratios at just downstream of the emission place f i (v x;i 0;x =2;1; 0;y2 ) =n i 1 p 2v th;i 2 exp (vv d ) 2 2v 2 th;i ! (5.6a) f e (v x;e 0;x =2;1; 0;y2 ) =n ? e 1 p 2v th;e 2 exp (vv d ) 2 2v 2 th;e ! (5.6b) with n ? e = (n i;inlet =n e;inlet )n 0 ; n i =n 0 (5.6c) In the equations above, denotes the injection plane. At all other physical boundaries except the boundary at y = 0, we apply the vacuum boundary condition. The boundary conditions is implemented by setting zero values in all of the ghost cells adjacent to the boundaries. f [(x=Nx+1; y); (vx; vy )] = 0; f [(x=Nx+2; y); (vx; vy )] = 0; f [(x=Nx+3; y); (vx; vy )] = 0 (5.7) 5.3 Comparisons with Full Kinetic PIC Simulations 129 Table 5.2: Simulation Domain/Time Parameters Length Cells Left Bound Right Bound Total Time Time Step Size X 800 800 0 800 t = 2000 t = 0:05 Y 400 400 0 400 t = 2000 t = 0:05 V x;e 16 256 7:9 8:1 t = 2000 t = 0:05 V y;e 16 256 8:0 8:0 t = 2000 t = 0:05 V x;i 0:4 256 0:1 0:3 t = 2000 t = 0:05 V y;i 0:4 256 0:2 0:2 t = 2000 t = 0:05 For the boundary at y = 0, the specular reflection method from [201] is adopted as the symmetric boundary condition. The boundary condition can be implemented as [201, 255] f [(x; y=3); (vx; vy =ivy)] =f [(x; y=2); (vx; vy =Nvyivy)] f [(x; y=2); (vx; vy =ivy)] =f [(x; y=1); (vx; vy =Nvyivy)] f [(x; y=1); (vx; vy =ivy)] =f [(x; y=0); (vx; vy =Nvyivy)] (5.8) The “cut-off” boundary is implemented in the velocity space. Sufficient largev x;max and v y;max are selected so that the VDF approaches zero at these velocity values. Then all of the VDFs with velocity larger than these values are set to zero. f [(x; y); (vx>vx;max; vy; vz )] = 0 (5.9) The Vlasolver code is fully parallelized. The simulations are carried out on the USC CARC’s Discovery cluster. A typical run uses around 40 billion mesh points and requires about 800 CPU cores. The runs are performed for around 40000 numerical steps and can be typically finished within 98 hours. 5.3 Comparisons with Full Kinetic PIC Simulations Previously, Wang et al. [22] and Hu and Wang [23, 97] carried out full particle PIC simulations of mesothermal plasma beam emission. In this section, we first carry out a comparison with 5.3 Comparisons with Full Kinetic PIC Simulations 130 full particle PIC simulations using the same simulation setup as that in [23, 97]. In [23, 97], PIC simulations were carried out using N p,cell = O(10 2 ) particles per cell at the beam exit. Here, to suppress numerical noise, we rerun the simulations presented in [23, 97] with N p/cell = 10 4 per cell at the injection plane. As the statistical noise in PIC scales inversely with p N p , the use of 10000 particles per cell at injection plane will reduce the noise level in the results of [23, 97] by one order of magnitude. Fig. 5.3 shows the comparisons of ion density, electron density, and potential between the results from Vlasov and PIC simulations at t = 1000. The ion density contours from Vlasov and PIC are almost identical. This is to be expected because the cold, heavy ions have a low thermal velocity and, thus, low statistical noise as compared to the electrons. The electron density contours from Vlasov and PIC agree well in the core beam region, where there are a large number of macro-particles per cell in PIC which suppresses the statistical noise efficiently. However, in the outer low-density region, the electron density contour in PIC becomes noisy. On the other hand, the result from Vlasov shows very smooth contours in both the high and the low density region. The electric potential contours from Vlasov and PIC agree well. The only discrepancies are in the region ahead of the beam expansion front, where the electron density from PIC is very noisy due to a lack of macro-particles which results in small differences in charge density. The blue dash dotted lines shown in Fig. 5.3a and Fig. 5.3b are the first Mach line drawn from the top edge of the injection plane: y = top tan arcsin 1 Ma x (5.10) (where top = 20.) Previous hybrid PIC simulation using the Boltzmann relation for electrons and analytical solution for mesothermal plasma expansion [213, 214] showed the first Mach line is the boundary that separates the unperturbed region and from the plasma expansion region. The simulation result agrees with this theoretical solution. Here, the blue dashed line approximately divides the beam into an unperturbed core region and an expansion region. 5.3 Comparisons with Full Kinetic PIC Simulations 131 Vlasov 20 40 60 80 100 120 140 160 180 200 PIC 0 40 80 120 160 200 240 280 320 360 400 200 180 160 140 120 100 80 60 40 20 0 -2 -1.5 -1 -0.5 0 (a) (b) (c) Figure 5.3: Comparisons of simulation results between PIC and Vlasov at t = 1000. (a) Ion density contour. (b) Electron density contour. (c) Potential contour. 5.3 Comparisons with Full Kinetic PIC Simulations 132 (a) (b) Figure 5.4: Comparisons of simulation results at t = 1000 on electron temperature on both x and y direction between grid-based Vlasov method and particle-based PIC methods. (a) electron x direction temperature T ex . (b) electron y direction temperature T ey . Fig. 5.4 compares electron temperature in both the x andy direction between Vlasov and PIC simulation results. Here, the electron temperature in Vlasov simulation is obtained by directly integrating the electron velocity distribution function T e;k = 1 n e ZZ m e (v k <v k >) 2 f e dv x dv y (5.11) 5.3 Comparisons with Full Kinetic PIC Simulations 133 where the subscript k is the indicator of direction, x or y. The electron temperature in PIC simulation is obtained by averaging the macro-particles within each cell as in Ref. [23]. The results show general good agreement between the two methods. The Vlasov simulation eliminates the statistical noise of PIC and shows smooth temperature contours in the low- density region where the PIC simulation fails to do so even with N p/cell = 10 4 per cell for beam injection. (a) (b) Figure 5.5: Comparisons of simulation results att = 1000 on electron heat flux on bothx and y direction between grid-based Vlasov method and particle-based PIC methods. (a) electron x direction heat flux Q ex . (b) electron y direction heat flux Q ey . 5.4 Electron Velocity Distribution Function 134 Fig. 5.5 compares electron heat flux vectors on both x andy direction between the Vlasov and PIC simulation results. Here, the electron heat flux vector Q e;n and heat flux tensor Q e;kmn is defined as Q e;kmn = ZZ m e (v e;k <v e;k >)(v e;m <v e;m >)(v e;n <v e;n >)f e dv x dv y Q e;n = ZZ 1 2 m e (v<v e >) 2 (v e;n <v e;n >)f e dv x dv y (5.12) where the subscript k, m and n are the indicators of direction, x or y, respectively. We find that the heat flux contours are qualitatively similar between the two methods. However, the heat flux obtained from PIC is quite noisy in both the high density and the low density region. On the other hand, the heat flux obtained from Vlasov is smooth in the entire simulation domain. As heat flux is the third order velocity moment, the discrete particles noise are amplified in the PIC result. In general, the statistical noise in higher order velocity moments from particle-based methods prevents an accurate description and identification of the thermodynamic processes in a system [218, 219]. The comparisons show that the Vlasov method is more suitable to investigate high order moments such as temperature and heat flux. 5.4 Electron Velocity Distribution Function One of the advantages of the grid-based Vlasov method is its capability to directly obtain the complete velocity distribution function (VDF) information at each grid point in the computational domain. We next carry out a detailed study of the electron VDF. Fig. 5.6 displays the density contours of the normalized electron distribution function, denoted as ^ f (eVDF), at various spatial locations. ^ f = f RR fdv x dv y (5.13) 5.4 Electron Velocity Distribution Function 135 (a) (b) (c) (d) Figure 5.6: Density normalized electron velocity distribution function contours at t = 1000. (a) (x;y) = (40; 0), a position selected from the region upstream of the self-similar region. (b) (x;y) = (100; 0), a position inside the self-similar region. (c) (x;y) = (160; 0), a position close to the quasi-neutral front calculated by eq. (5.19). (d) (x;y) = (220; 0), a position in the “pure electron gas” region. Fig. 5.6a specifically illustrates ^ f within the “unperturbed” beam core region, located at (x;y) = (40; 0). At the injection plane reservoir, only half of the velocity space is filled with the VDF defined by eq. (5.6). As electrons are reflected from the downstream expansion process, the initially empty half of the velocity space in the reservoir gradually becomes 5.4 Electron Velocity Distribution Function 136 populated in the velocity distribution function. It can be observed that the high-energy tail of the electron velocity distribution function (eVDF) in the v y direction starts to deplete in comparison to the semi-Maxwellian distribution in the injection plane reservoir. Fig. 5.6b illustrates the electron velocity distribution function (eVDF) within the self-similar region. In thev x direction, the electron velocity distribution function (eVDF) exhibits a near-Maxwellian shape. However, in the v y direction, the eVDF displays a reduced population of high-energy tail and an increased population in the low-energy regions. Fig. 5.6c presents the electron velocity distribution function (eVDF) near the on-axis quasi-neutral front. The shape of the eVDF is similar to that shown in Fig. 5.6b, but with smaller high-energy tails in the v y direction within the (v x > 0;v y ) velocity space. This observation can be explained as follows: A double layer forms near the on-axis quasi-neutral front [42, 45], leading to an enhanced axial electric field. Consequently, as electrons pass through the self-similar region and quasi-neutral front, only those with higher axial velocities can reach the quasi-neutral front. Due to the conservation of electron energy in the electrostatic field, electrons with larger axial velocities exhibit smaller transverse velocities in the paraxial region [89]. Fig. 5.6d illustrates the electron velocity distribution function (eVDF) in the “pure electron gas” region beyond the ion beam front. The eVDF exhibits a shape similar to the loss cone. Wang et al. [22] demonstrated that a potential well forms between the ion beam front and the beam emission plane, effectively trapping the majority of emitted thermal electrons. Only electrons with sufficiently high energy are able to overcome the potential well and escape, reaching the “pure electron gas” region. Fig. 5.7 shows the normalized electron VDF in the v x direction, R 1 1 ^ fdv y , at different positions. For comparison, a Maxwellian VDF based on local temperature T e;x (x;y) and local bulk velocity u e;x (x;y) is also plotted, Z 1 1 ^ f e;M dv y = m e 2T e;x (x;y) 1=2 exp m(v x u e;x (x;y)) 2 2T e;x (x;y) (5.14) 5.4 Electron Velocity Distribution Function 137 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (a) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (b) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (c) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (d) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (e) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (f) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (g) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (h) Figure 5.7: Density normalized v x direction electron velocity distribution function plots at t = 1000. (a)-(d) On axis (y = 0) positions with x = 40, x = 100, x = 160, x = 220 respectively. (e)-(h) positions along y = 20 with with x = 40, x = 100, x = 160, x = 220 respectively. The black dashed lines in all figures are the local reconstructed Maxwellian distribution by eq. (5.14). Figs. 5.7a to 5.7d show the electron velocity distribution function (eVDF) along the beam axis in the v x direction. Within the unperturbed beam region and the self-similar region, the electron velocity distribution function (eVDF) exhibits an almost Maxwellian shape, characterized by a depleted high-energy tail in thev x direction. In the vicinity of the ion beam front, a stronger depletion is observed in thev x tail of the electron velocity distribution function (eVDF) (see Fig. 5.7c). This phenomenon is expected since a fraction of electrons is reflected by the ambipolar field at the beam front. Additionally, a slight increase is observed in the +v x tail of the electron velocity distribution function (eVDF). This can be seen in Fig. 5.7d. In the “pure electron gas” region, the eVDF significantly deviates from the local Maxwellian distribution. This discrepancy arises because only high-energy electrons are able to enter this region. Figs. 5.7e to 5.7h show the electron velocity distribution function (eVDF) in the v x direction at the same x position along the line y = 20. The shapes of these distributions are similar to those observed along the axis. However, there is a slight increase in the degree of depletion in thev x tail, which can be attributed to the stronger off-axis electric field [97]. 5.4 Electron Velocity Distribution Function 138 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (a) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (b) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (c) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (d) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (e) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 (f) Figure 5.8: Density normalized v y direction electron velocity distribution function plots at t = 1000. (a)-(f) On axis (y = 0) positions with x = 20, x = 40, x = 60, x = 100, x = 160,x = 220 respectively. The black dashed lines in all figures are the local reconstructed Maxwellian distribution by eq. (5.15). The blue dotted lines in (a)-(d) are the semi-analytical fitted distribution functions by eq. (5.16). Note that only the positive part of the fitted distribution function is shown. Fig. 5.8 shows the density-normalized electron velocity distribution function (eVDF) in the v y direction, denoted as R 1 1 ^ fdv y , at various positions. For comparison, a Maxwellian velocity distribution function (VDF) based on the local temperature T e;y (x;y) and local bulk velocity u e;y (x;y) is also plotted. The Maxwellian VDF in the v y direction is given by: Z 1 1 ^ f e;M dv x = m e 2T e;y (x;y) 1=2 exp m(v y u e;y (x;y)) 2 2T e;y (x;y) (5.15) Figs. 5.8a to 5.8d show the electron velocity distribution function (eVDF) in the v y direction along the beam axis. It can be observed that the eVDFs gradually deviate from a Maxwellian shape along the beam direction, evolving into a non-Maxwellian form. To investigate the formation process of the non-Maxwellian distribution in thev y direction during two-dimensional plasma beam expansion, we can explore the mapping between the spatial evolution of the on-axis v y direction electron velocity distribution function (eVDF) in 5.4 Electron Velocity Distribution Function 139 two-dimensional beam expansion and the temporal evolution of the slab-center point eVDF in one-dimensional finite-size plasma expansion. This mapping can be established by introducing the parameter t ? =x=v d , where t ? represents the evolution time in the one-dimensional thin foil expansion process, x denotes the axial location, and v d represents the drifting velocity in the two-dimensional beam expansion. By utilizing this mapping, insights can be provided into the underlying mechanisms responsible for the emergence of non-Maxwellian distributions in the v y direction. In Figs. 5.8a to 5.8d, the blue dotted lines represent the fitted distribution function R ^ f e dv x obtained from Mora’s semi-analytical solution for the electron velocity distribution function at the center of the one-dimensional finite-size plasma [50, 51]. These fitted lines provide a comparison between the observed eVDFs in the v y direction and Mora’s theoretical solution. Z ^ f e dv x = 1 + v 2 y 2 v 2 y t ? Z ^ f ? e0 dv x ; = L C s0 Z ^ f ? e0 dv x = m e 2T e;y (0; 0) 1=2 exp m(v y u e;y (0; 0)) 2 2T e;y (0; 0) (5.16) whereL is the initial beam length in the transverse direction, and are control parameters that influence the shape and characteristics of the distribution function. In the original work by Mora, the ratio of = was set to 1.5. However, in this particular study, we have used = = 1 instead. In Mora’s study of one-dimensional finite-size plasma expansion, eq. (5.16) is derived by adiabatically calculating the variation of electron energy in the linear self-similar potential field formed at the edge of the thin foil before the inward-marching rarefaction wave reaches the center. It is assumed that all electrons are trapped by the potential field, and therefore, high-energy escaping electrons cannot be described accurately by Equation (5.16). In the 2D plasma beam expansion process, it can be observed from Figs. 5.8a to 5.8c that the low and medium energy range electron velocity distribution functions (eVDFs) align well with the fitted distribution function, as predicted by eq. (5.16). However, the high energy range eVDFs deviate from the fitting, indicating that these electrons do not follow the same behavior as predicted by eq. (5.16). Specifically, in the high energy range, some 5.5 Electron Collisionless Heat Flux 140 electrons lose energy and escape from the potential field, resulting in a depleted tail of the eVDF. Simultaneously, in the low energy range, electrons gain energy, leading to an increased population in this region. These combined effects contribute to a flattened top feature and a depleted tail in the electron velocity distribution function. Similar to the 1D expansion case, the distorted eVDF in the v y direction exhibits a “top-hat” (i.e. super Maxwellian distribution), indicating a departure from a purely Maxwellian shape. Indeed, according to Mora’s study, the two inward-marching rarefaction waves in the transverse direction meet at the position x =v d t ?0 =v d 0:88 75 [51]. Beyond this point, the eq. (5.16) no longer accurately describes the evolution of the electron velocity distribution function (eVDF). This observation is evident in Fig. 5.8d, where the fitted distribution function deviates significantly from the actual distribution function. At this stage, all electrons are losing energy, and the eVDF further evolves towards a super Maxwellian distribution. The evolution of the electron distribution function in this region provides an explanation for the near-adiabatic cooling process of T e;y observed in the plasma beam expansion. 5.5 Electron Collisionless Heat Flux In fluid theory, the electron fluid moments equations are obtained by integrating eq. (5.1) and the first three integrated electron moment equations are @n e @t +r (n e u e ) = 0 m e @n e u e @t +r (m e n e u e u e + p e ) +en e E = 0 @ @t 1 2 n e m e u 2 e + 1 2 Tr( p e ) +r 1 2 n e m e u 2 e u e + 1 2 Tr( p e )u e + p e u e +Q e +en e Eu e = 0 or @ @t (n e m e u e u e + p e ) +r n e m e u e u e u e + ( p e u e ) s + Q e +en e (Eu e +u e E) = 0 (5.17) 5.5 Electron Collisionless Heat Flux 141 where n e = R f e dv e is the electron number density,u e = R v e f e dv e =n e is the electron bulk velocity vector, p e = R m e (v e u e )(v e u e )f e dv e is the electron pressure dyad tensor, Q e and Q e are the electron heat flux vector and tensor defined in eq. (5.12). The sign “Tr" denotes taking the trace of a dyad tensor. ( p e u e ) s is a third order symmetric tensor and it can be defined as ( p e u e ) s ijk =u k p ij +u j p ik +u i p jk . Eq. (5.17) has more unknown variables than the equations and an appropriate closure relation need to be provided to close the eq. (5.17). For most of the previous works, the electrons are assumed to be isothermal (i.e. Q e =1), massless and in steady state. Under this condition eq. (5.17) can be reduced to the well-known Boltzmann relation. Recent studies on magnetic nozzle plasma flows [92, 256, 257] have indicated that if one can assume the heat flux to be proportional to the enthalpy flux and assume steady state and massless electron together with isotropic pressure tensor, eq. (5.17) can be reduced to the well- known polytropic relation (or adiabatic relation, when Q e = 0). However, for unmagnetized collisionless plasma flow, no models on the electron heat flux have been proposed which can capture the anisotropic electron kinetic features shown in fully kinetic PIC simulations [23, 24, 97]. In this section, we apply the results from Vlasov simulation to investigate electron heat flux. Fig. 5.9 shows the contour of the x direction electron heat flux Q ex . Two ion density iso-contour lines shown as green dashed dotted line and magenta dashed line are plotted in addition to the x direction electron heat flux contour. The green dashed-dotted line represents the ion density value at the on-axis ion rarefaction wave front, which can be determined using eq. (5.18). x = t(Ma 1) p (m i =m e ) (5.18) The magenta dashed line represents the ion density value at the on-axis quasi-neutral front determined by eq. (5.19). x + ' 2t p m i =m e ln (0:15 t p m i =m e ) + Mat p m i =m e (5.19) 5.5 Electron Collisionless Heat Flux 142 (a) (b) (c) Figure 5.9: Contour of Q ex at different time moments. Green dash dotted lines: iso-contour lines of the ion density value at the on-axis ion rarefaction wave fronts. Magenta dashed lines: iso-contour lines of the ion density value at the on-axis quasi-neutral fronts. (a) t = 1000. (b) t = 1500. (c) t = 2000. 5.5 Electron Collisionless Heat Flux 143 The electron heat flux contour in the x direction reveals the presence of three distinct regions. The first region, located in close proximity to the injection plane, exhibits a triangular shape and can be identified as the core region. Notably, this region maintains a quasi-steady shape over time. In addition to the core region, an adjacent fan region can be observed. The fan region extends along the axial direction over time while maintaining a consistent shape in the transverse direction. Notably, the heat flux magnitude (Q ex ) within the fan region increases with time and becomes comparable to the heat flux magnitude in the core region. The approximate outer boundary of the fan region can be represented by the green dashed-dotted iso-contour line, as illustrated in Fig. 5.9. Fig. 5.9 clearly demonstrates that the magenta dashed line serves as an approximate boundary between the non-zero heat flux region and the zero heat flux region. This observation aligns with expectations, as the quasi-neutral front in the plasma expansion process can be considered as an approximate outer boundary of the beam expansion fan. The region enclosed by the green dashed-dotted line and the magenta dashed line is referred to as the corona region. Notably, the magnitude of Q ex within this region is significantly smaller compared to other regions. Fig. 5.10 shows the contour of the y direction electron heat flux Q ey . Similar to the Fig. 5.9, the magenta dashed line approximately separate the non-zero y direction electron heat flux with the zero heat flux region. Two distinct regions can be found in the Q ey contour: a core region and a corona region. The Q ey component demonstrates a relatively high amplitude in the vicinity of the injection surface, comparable to the magnitudes of Q ex observed in the core and fan regions in Fig. 5.9. Consequently, this region can be identified as the core region for the Q ey heat flux. The shape of the Q ey core region remains constant over time, and there are no significant changes in the magnitude of the Q ey heat flux. The region outside the core region, but still within the beam, is referred to as the corona region. In the corona region, the magnitudes of the Q ey heat flux are approximately one order of magnitude smaller compared to the magnitudes observed in the core region. From Figs. 5.9 and 5.10, it can be observed that the y direction electron heat flux (Q ey ) only plays a significant role in the core region near the injection plane. However, in other regions within the beam, the 5.5 Electron Collisionless Heat Flux 144 (a) (b) (c) Figure 5.10: Contour of Q ey at different time moments. Magenta dashed lines: iso-contour lines of the ion density value at the on-axis quasi-neutral fronts. (a) t = 1000. (b) t = 1500. (c) t = 2000. 5.5 Electron Collisionless Heat Flux 145 x direction electron heat flux vector (Q ex ) dominates as the major component of the total electron heat flux. 0 50 100 150 200 250 300 350 400 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (a) 0 50 100 150 200 250 300 350 400 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (b) 0 50 100 150 200 250 300 350 400 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (c) Figure 5.11: Q ex , Q e;xxx , Q e;yyx profiles along axis at different time. (a) t = 1000. (b) t = 1500. (c) t = 2000. 5.6 Discussions 146 In the two-dimensional setup investigated in this study, the x direction electron heat flux vector can be decomposed into the sum of two electron heat flux tensors. These individual heat flux tensors exhibit a close relationship with the microscopic electron kinetic features. Q ex = 0:5 Q e;xxx + 0:5 Q e;yyx (5.20) In Fig. 5.11, the on-axis profile of the heat flux vector Q ex and its two heat flux tensor components, Q e;xxx and Q e;yyx , is presented. It is observed that Q e;xxx exhibits a similar trending feature to Q ex after a short distance from the injection plane. The magnitude of Q e;xxx is about 2 6 times larger than the magnitude of Q e;yyx in this region. The Q e;yyx only has a large contribution to the Q ex in the region adjacent to the injection plane but remains a ignorable magnitude in the region downstream. From the results above, it can be concluded that the electron heat flux tensor Q e;xxx is the major contribution to the total electron heat flux and dominates the electron energy transfer in the beam expansion process. The on-axis profile of the heat flux vector and tensor also verified the fact that the heat transfer process considered here does not fulfill the Fourier type heat conduction law since the electron x direction temperature T ex remains quasi-constant inside the beam and the gradient of the electron x direction temperature T ex can not explain the large magnitude of heat flux shown in Fig. 5.11. 5.6 Discussions In light of the dominance of the Q e;xxx component in the electron energy transfer process, the subsequent discussions pertaining to the connection between electron microscopic kinetic features and macroscopic collisionless electron heat flux will focus on the Q e;xxx . As shown in Fig. 5.7, the electron velocity distribution functions (eVDFs) normalized by density in the v x direction exhibit a similar degree of depletion in the high-energy tail of the negative branch at various positions within both the upstream “unperturbed” region and the 5.6 Discussions 147 self-similar region. This indicates that the eVDFs in these regions share similar skewed shapes. In statistics, the skewnessS is a measure used to quantify the asymmetry of a distribution. It provides information about the relative sizes of the two tails in the distribution function [258] S [V ] =E " V 3 # = Z v 3 gdv (5.21) where V is the random variable,E is the expectation operator. The expectation of V, is defined as =E [V ] = R vgdv. The standard variation is defined as = p E [(V) 2 ] = q R (v) 2 gdv. The probability function g(v) satisfies the property R gdv = 1. If one considers that g = R ^ f e dv y and V =v x , the statistical moments and can be interpreted as x electron direction bulk velocity u e;x and x direction electron thermal velocity v te;x respectively. The skewness of thev x direction marginal electron velocity distribution function can then be written as S [v x ] = Z v x u e;x v te;x 3 Z ^ f e dv y dv x = 1 n e m e v 3 te;x ZZ m e (v x u e;x ) 3 f e dv x dv y = Q e;xxx n e m e v 3 te;x (5.22) eq. (5.22) expresses the skewness of the marginal eVDF in the v x direction as the ratio between the heat flux tensor component Q e;xxx and the local variable n e m e v 3 te;x . This equation establishes a relationship between the collisionless heat flux tensor Q e;xxx and the relative asymmetry of the distribution function’s tails. By utilizing this equation, the behavior and characteristics of the heat flux tensor can be described and explained in terms of the relative tailedness of the distribution function. The right-hand side of eq. (5.22) corresponds to the coefficient in the flux-limited closure model for collisionless electron heat flux, which has been utilized in studies related to the solar wind [239] and fusion scrape-off layer research [259]. This indicates that studying the relative tailedness of the electron velocity distribution 5.6 Discussions 148 function can aid in selecting the appropriate coefficient for establishing a potential collisionless heat flux closure model. 0 50 100 150 200 250 300 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 (a) 0 50 100 150 200 250 300 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 (b) 0 50 100 150 200 250 300 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 (c) Figure 5.12: (a) Flux limited model coefficient along axis at various time. (b) Region division at t = 1500. (c) Region division at t = 2000. In (b) and (c), the red dashed line is x =t(v d C s ). 5.6 Discussions 149 In Fig. 5.12a, the right-hand side term Q e;xxx =(nm e v 3 te;x ) of eq. (5.22) (i.e., the flux-limited model coefficient) is plotted as a function of axial distance along the beam axis at different time instances. It is observed that, after a short distance from the injection plane, the value of Q e;xxx =(nm e v 3 te;x ) exhibits time-dependent features. A hump structure and a nearly monotonically increasing tail structure can be observed in Fig. 5.12a. Interestingly, these structures exhibit self-similar features at different time moments. This suggests that the evolution of Q e;xxx =(nm e v 3 te;x ) along the axial direction in the beam demonstrates certain patterns that remain consistent over time. Based on these distinct self-similar features, the value of Q e;xxx =(nm e v 3 te;x ) can be split into two distinct regions, region A and B, as shown in Figs. 5.12b and 5.12c. The boundary between region A and region B is represented by the red dashed line, which is defined as x =t(v d C s ). The time-dependent self-similar hump structure is confined within region A. It is noteworthy that the boundary x = t(v d C s ) corresponds to the on-axis ion rarefaction wave front. Region B is situated downstream of region A and partially overlaps with the self-similar expansion region of the beam expansion process. The quantity Q e;xxx =(nm e v 3 te;x ) exhibits a nearly monotonically increasing behavior in this region. This increasing trend can be attributed to the following explanation: In the self-similar region, a quasi-constant strong electric field arises due to the linear decrease of the potential with distance [30]. As the electrons are pulled upstream by the electric field, a significant portion of them undergo reflection. As the position progresses downstream, only electrons with sufficiently large positive velocities are able to pass through. Consequently, the skewness of the marginal velocity distribution function in the x direction increases due to the depletion of the negative velocity tail. This phenomenon is reflected in the increasing flux limited coefficient observed in region B. The validity of this process is demonstrated in Figs. 5.6b5.6d and Figs. 5.7b5.7d. The relationship between electron microscopic kinetics and the value of Q e;xxx =(nm e v 3 te;x ) in region A is not straightforward. The potential field exhibits quasi-constant and slightly monotonically decreasing profiles along the axis upstream of the on-axis ion rarefaction wave front. Additionally, an axial weak electric field that pulls electrons upstream is present in 5.6 Discussions 150 region A. However, in contrast to region B where the flux limited coefficient (i.e., the skewness of the v x direction electron velocity distribution function or the value of Q e;xxx =(nm e v 3 te;x )) exhibits a nearly monotonically increasing behavior, region A displays hump structures in the value of flux limited coefficient. The observed phenomena in the on-axis potential profile and the on-axis flux limited coefficient profile may initially appear contradictory. However, these discrepancies can be explained by taking into account the influence of transverse direction electron kinetics. Since the skewness is primarily influenced by the electron’s axial motion, our focus in this study is on the electron’s axial kinetics. The transverse direction electron kinetics and potential can be considered as an “effective potential” by utilizing the 1D electron par-axial kinetic model developed by Merino et al. [89]. In the presence of a potential field, the electrons’ transverse action integral J = H m e v y dy is considered as an adiabatic invariant. This adiabatic invariant allows us to analogize the effects of electron transverse motion in the unmagnetized beam expansion to the magnetic mirror effects observed in magnetized beam expansions [260]. 0 20 40 60 80 100 120 140 160 180 200 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 5.13: Representative electron trajectory in phase space (x;v x ). The trajectory is plotted for an electron with an initial condition of (x;y) = (200; 0); (v x ;v y ) = (0; 0:9) from time t = 979 to time t = 2000. Previously, Wang et al. [22] studied the dynamics of the electrons in an unmagnetized, collisionless mesothermal plasma beam using full particle PIC, and showed that electron-ion coupling in a collisionless beam is achieved through interactions between electrons and a potential well established between the emission surface and the propagation of the ion beam front along the beam direction. The dynamics of individual electrons can also be obtained 5.7 Summary 151 from Vlasov simulation. Fig. 5.13 shows a representative electron trajectory in the phase space (x;v x ). This trajectory is obtained by post-processing the potential data from the grid-based Vlasov simulation for a test particle initially at (x;y) = (200; 0) and (v x ;v y ) = (0; 0:9). The trajectory is integrated backward following the method in Ref. [91]. Consistent with the findings presented in [22], our results show that a competition between electron acceleration resulting from the conservation of the electron’s transverse motion integral and electron deceleration due to the electric potential leads to the trapping of certain electrons within the beam. The works of Perkins [261] and Hollweg [239] have identified the asymmetric portion of the electron velocity distribution function (VDF) (i.e., the free electrons) as the cause of collisionless heat flux in the solar wind corona. This asymmetry is determined by the presence of trapped electrons resulting from the competing processes of magnetic mirror effects and the monotonically decreasing electric potential. In the context of the unmagnetized beam expansion, the conservation of electron transverse direction motion plays a similar role to the magnetic mirror effects. Therefore, the trapping of electrons in region A can be considered a significant factor in determining the asymmetric part of the on-axis electron velocity distribution function. It is also noteworthy that the trapping loop of the electron trajectory shown in Fig. 5.13 exhibits a growing size over time while maintaining a similar loop shape. Hence, the qualitative explanation for the self-similar hump structures observed in region A can be attributed to the trapping of electrons in this region. 5.7 Summary In this chapter, we conduct a two-dimensional grid-based Vlasov simulation to explore the electric propulsion plasma beam expansion process using our newly developed multi- dimensional parallel grid-based Vlasov code, Vlasolver. To the best of our knowledge, this study represents the first full 2D grid-based Vlasov simulation of plasma beam expansion. We devise a novel specific injection scheme to address the challenge of injecting quasi-neutral, current-free plasma flows in grid-based methods. 5.7 Summary 152 To validate our approach, we conduct a benchmark by comparing the results obtained from the grid-based Vlasov method with those from the Particle-in-Cell (PIC) method. This benchmark serves to demonstrate the suitability of the grid-based Vlasov method for modeling electric propulsion beam expansion. Good agreement can be found between the results obtained from the two methods. Notably, the grid-based Vlasov method yields smooth results in both low and high-density regions, while the PIC method exhibits significant numerical noise, particularly in the low-density region due to a lack of particles. Furthermore, the results obtained from the PIC method display strong noisy features even in the high- density region when examining higher-order moments. Considering these observations from the benchmark, we leverage the grid-based Vlasov method’s inherent noise-free nature and its convenient access to velocity distribution functions (VDFs) to investigate the evolution of electron VDFs, as well as higher-order moments such as temperature and heat flux. The electron velocity distribution function are investigated through grid-based Vlasov simulations. The results reveal that the v x direction eVDF exhibits near-Maxwellian shapes but has a depleted high-energy tail in the negative velocity branch at different positions within the beam. On the other hand, the v y direction eVDF evolve towards a top-hat shape. The evolution of thev y direction eVDF followed Mora’s semi-analytical 1D finite-size thin-foil plasma expansion adiabatic solution and this provides insights into the near-adiabatic cooling process of the electron temperature in the v y direction. The electron heat flux in the x direction exhibits three distinct regions: a time-steady core region, a time-dependent fan region, and a time-dependent corona region. In contrast, the electron heat flux in the y direction can be divided into two regions: a time-steady core region and a time-dependent fan corona region. The magnitude of the electron heat flux in the x direction is significantly larger than that in the y direction in all regions, except for the near-injection plane region. The main contributor to the electron heat flux is the pure x direction heat flux tensor, denoted as Q e;xxx . Thefeaturesof Q e;xxx andelectronmicroscopickineticsareconnectedthroughtheskewness of the electron velocity distribution function (VDF). The skewness of the v x direction electron 5.7 Summary 153 VDF has the same form to the coefficient in the flux limited model of the electron collisionless heat flux. The flux limited coefficients demonstrate a time-dependent self-similar hump structure and a self-similar tail structure. These structures can be further divided into two regions: region A (x<t(v d C s )) and region B (x>t(v d C s )), wheret(v d C s ) represents the on-axis ion rarefaction wave front. Region A is formed by the trapping of electrons, while region B is formed due to the presence of a strong electric field within the self-similar region. These findings provide valuable insights towards the future development of physical heuristic closures for collisionless heat flux. In conclusion, this study has provided valuable insights into the characteristics and behaviors of electron collisionless heat flux in the context of unmagnetized multi-dimensional beam expansion. Notably, we have established a link between the collisionless heat flux and electron kinetics by examining the skewness of the electron velocity distribution function (VDF). This connection offers a quantitative understanding of the fundamental physical mechanisms governing the collisionless electron heat flux. The implications of our findings are significant for the development of a collisionless heat flux closure that can effectively account for the anisotropic thermodynamic features exhibited by electrons. Such a closure would facilitate accurate and computationally efficient reduced-order multi-fluid modeling of electric propulsion beam. Future studies will focus on further developing physical heuristic heat flux closures for electric propulsion beam modeling, building upon the features and mechanisms uncovered in this paper. 154 Chapter 6: Grid-based Vlasov Solver for Electromagnetic Plasma Simulations Chapter 3 presents the development and specific features of the electrostatic version of Vlasolver. This chapter presents the development and detailed aspects of the electromagnetic version of Vlasolver. 6.1 Introduction In this chapter, driven by the necessity for a method with high fidelity and low noise to investigate electromagnetic plasma phenomena such as space plasma instabilities and turbulence as reviewed in Section 1.2.3, we introduce the development of an electromagnetic version of the Vlasolver as an extension to the electrostatic Vlasolver established in Chapter 3. To eliminate the computational constrains on the simulation time step imposed by the light wave from the Maxwell’s equation, the electromagnetic Vlasolver uses the Darwin equations. To couple the Vlasov solver with the Darwin solver, a novel reduced-order moment-based coupling method is formulated to accurately couple the Vlasov and Darwin equations. The electromagnetic Vlasolver is expanded into a 2D3V formulation to capture fundamental physics. Detailed methodologies for solving the Vlasov equation and the Darwin equation are available in Sections 2.4 and 2.6, respectively. The subsequent sections present a novel moment-based reduced-order algorithm for coupling the Vlasov and Darwin equations in Section 6.2, followed byan assessment of the parallel efficiency of the electromagnetic Vlasolver in Section 6.3. Two verification cases are employed in Sections 6.4 and Sections 6.5 to validate the accuracy of the electromagnetic Vlasolver. Finally, a summary of this chapter is provided in Section 6.6. 6.2 Method for Coupling the Vlasov and Darwin Equations 155 6.2 Method for Coupling the Vlasov and Darwin Equa- tions As mentioned in Section 2.6, the Darwin equations rely on input information regarding plasma velocity and acceleration obtained from the Vlasov equation. However, due to the utilization of Cheng and Knorr’s splitting algorithm [27] in this dissertation, the acceleration and velocity are half time step retarded. In Step 2, the macroscopic variables are computed using the moment integration eq. (2.21) with f(x;v;t =t + t=2). However, the available information of the velocity distribution function (VDF) at the beginning of Step 2 is f ? , given by f ? =f(x(t + t=2);v(t);t =t + t=2). Comparing this with the required VDF information, f ? has a velocity space that is retarded by half a time step. The leapfrog scheme in the Particle-in-Cell (PIC) method also encounters a similar issue. To address this, the technique used for solving the problem involves “pseudo” updating of particle information without writing the data into memory [194]. However, for the grid-based Vlasov method, this approach is not suitable. While one could advance f ? using eq. (2.31) by half a time step, this approach is computationally inefficient in grid-based methods. Integrating the Vlasov equations in high-dimensional phase space consumes a significant portion of computational time, and employing such a method would result in a decrease in explicit efficiency. Furthermore, this method would require additional memory space, which presents a critical challenge as the storage of information in discretized high-dimensional phase space requires substantial memory capacity. Consequently, a novel coupling method must be developed to establish the connection between the Vlasov and Darwin equations for the grid-based Vlasov method. Motivated by the advanced method proposed by Valentini et al. [262], we adopt a reduced-order integration approach for eq. (2.18) to advance the macroscopic variables. This allows us to avoid directly numerically integrating the Vlasov equations. 6.2 Method for Coupling the Vlasov and Darwin Equations 156 Both net current and the net time derivation of current needs to be obtained from the Vlasov equations for solving the Darwin equations. Net currentJ can be calculated from the first order moment of VDF J = X J = X q Z v f dv (6.1) andnettimederivationofcurrent@J=@tcanbeobtainedbythefirstordermomentintegration on both side of eq. (2.18) [193, 263] @J @t = X @J @t = X @ @t q Z v f dv = X q Z v @f @t dv = X Z v v r x f dv X q m Z v (E +v B)r v f dv =r X <v v > + X q 2 n m E + X q 2 n m u B (6.2) whereu =<v> is the average bulk velocity for specie . In order to couple Vlasov equation to Darwin equations, at the beginning of step 2 of algorithm 1 one needs to get the information of P (q 2 =m ) R f ? dv , P q R v f ? dv , P (q 2 =m ) R v f ? dv , P q R v v f ? dv from f ? and then input these values to eq. (6.1) and eq. (6.2) to get J 0 and @J 0 =@t and then input these two variables along together with P q R f ? dv to the algorithm 3’s steps 1-3. Step 4 of Algorithm 3 requires the update of current derivative from @J 0 =@t to @J 1 =@t. Here, @J 1 =@t stands for the current derivative obtained at time moment t = t + t=2 by advancing f ? in velocity space with the “old” field information to f(x;v;t =t + t=2). Now one needs to solve eq. (6.2) with the information of P (q 2 =m ) R f(x;v;t =t + t=2) ? dv , P (q 2 =m ) R v f(x;v;t =t + t=2) ? dv , P q R v v f(x;v;t =t + t=2)dv . In order to avoid solving Vlasov equation directly, we adopt the similar methods with eq. (2.39) to (2.41). The coupling can be done by integrating the eq. (2.24b) on both side with 0th, 1st, 6.2 Method for Coupling the Vlasov and Darwin Equations 157 2nd order moments and multiply the corresponding charge and mass information respectively for each species and then sum up. By doing 0th order integration one can find @ @t X q 2 m Z f dv ! = 0 (6.3) and eq. (6.3) can be further written into discrete form as X q 2 m Z f(x;v;t =t + t=2) dv = X q 2 m Z f ? dv (6.4) The first order integration of the eq. (2.24b) can be written as @ @t X q 2 m Z v f dv ! = X E q 3 m 2 Z f dv + X q 3 m 2 Z v f dv B (6.5) and eq. (6.5) can be written in discrete form as X q 2 m Z v f(x;v;t =t + t=2) dv = X q 2 m Z v f ? dv + t 2 X E q 3 m 2 Z f ? dv + X q 3 m 2 Z v f ? dv B ! (6.6) The second order integration of the eq. (2.24b) requires to have the termv v rather than termv 2 inside the integration. Since the term P q R v v f dv is a tensor, here we only show the advance method for the ijth term of it @ @t X q Z v ;i v ;j f dv ! = X E i q 2 m Z v ;j f dv + X E j q 2 m Z v ;i f dv + X B k q 2 m Z v ;j v ;j f dv X B j q 2 m Z v ;j v ;k f dv + X B i q 2 m Z v ;i v ;k f dv X B k q 2 m Z v ;i v ;i f dv (6.7) 6.2 Method for Coupling the Vlasov and Darwin Equations 158 and eq. (6.7) can be further written into the discrete form as X q Z v ;i v ;j f(x;v;t =t + t=2) dv = X q Z v ;i v ;j f ? dv + t 2 X E i q 2 m Z v ;j f ? dv + t 2 X E j q 2 m Z v ;i f ? dv + t 2 X B k q 2 m Z v ;j v ;j f ? dv t 2 X B j q 2 m Z v ;j v ;k f ? dv + t 2 X B i q 2 m Z v ;i v ;k f ? dv t 2 X B k q 2 m Z v ;i v ;i f ? dv (6.8) By substituting the corresponding electric and magnetic fieldE 0 andB 0 inside eqs. (6.3)-(6.8) and solve them, the advancement in step 4 of algorithm 3 can be done with out advancing the Vlasov equation. Step 6(a) of algorithm 3 requires the update of both the time derivative of current and current. The update of the time derivative current follows eqs. (6.3)-(6.8) withE =E 1 and B =B 0 . Since net currentJ = P q R v f dv , the update process of current is similar to eq. (6.5) and can be written as @ @t X q Z v f dv ! = X E q 2 m Z f dv + X q 2 m Z v f dv B (6.9) and eq. (6.9) can be written in the discrete form as X q Z v f(x;v;t =t + t=2) dv = X q Z v f ? dv + t 2 X E q 2 m Z f ? dv + X q 2 m Z v f ? dv B ! (6.10) 6.3 Parallel Efficiency of the Electromagnetic Vlasolver 159 In summary, it can be found that in order to coupling Vlasov equation and Darwin equations, one needs to have in total 8 moment quantities computed from the f ? at the beginning of step 2 of Algorithm 1: X q Z f ? dv ; X q 2 m Z f ? dv ; X q 3 m 2 Z f ? dv ; X q Z v f ? dv ; X q 2 m Z v f ? dv ; X q 3 m 2 Z v f ? dv ; X q Z v v f ? dv ; X q 2 m Z v v f ? dv (6.11) WiththesequantitiesintegratedandcalculatedfromVlasovequations, Algorithm3canoutput the needed E andB information for the velocity space update of VDFs in Algorithm 1’s step 2. Thus two algorithms can be coupled together without any additional half time step advancing of Vlasov equation and avoid the additional computational resources consumption. 6.3 Parallel Efficiency of the Electromagnetic Vlasolver Indeed, modeling studies of space plasma instabilities and turbulence typically necessitate the inclusion of all three dimensions of velocity space [264]. However, the current limitations of computational resources pose significant challenges in conducting fully kinetic 3D3V (three dimensions in physical space and three dimensions in velocity space) grid-based Vlasov simulations. As previously discussed in Ref. [264], the 2D3V setup is considered a suitable approximation. Hence, taking into account the feasible computational costs, this chapter presents the extension of Vlasolver into a 2D3V version with a focus on the parallel efficiency. Fig. 6.1 depicts the phase space discretization in Vlasolver under the 2D3V setup. The parallelization strategies employed are consistent with those presented in the electrostatic version of Vlasolver, as illustrated in Section 3.1. For the Vlasov equation, the discretization of physical space takes place, and subsequently, it is allocated to distinct processors. Each processor contains the entire set of velocity space, and relevant information is exchanged at 6.3 Parallel Efficiency of the Electromagnetic Vlasolver 160 Figure 6.1: Schematic illustration on the 2D3V phase space discretization. the boundary of each sub-domain. The field equation (specifically, the Darwin equations) is solved sequentially or in a serial manner. Scaling tests of the electromagnetic Vlasolver code were conducted to assess its weak scaling performance. The chosen example problem for the weak scaling evaluation is the Weibel instability, which is described in detail in the subsequent Section 6.5. However, to ensure comprehensive performance testing, we deviate from the numerical setup mentioned in Section 6.5 by not utilizing the quasi-1D setup. Instead, a 2D3V set-up is adopted and the y direction is configured with 32 cells. As discussed in Section 3.3, the single-node performance is used as the referencing benchmark. All scaling tests in this section are performed on the NSF-ACCESS Anvil supercomputer using Anvil-A series nodes equipped with dual 64-core 2.45 GHz AMD epyc-64 “Milan” 7763 processors and 256GB of memory. Each Anvil-A computational node can accommodate a total of 128 processes. 6.3 Parallel Efficiency of the Electromagnetic Vlasolver 161 The code was compiled using the GNU 11.2.0 compiler, and the communication environ- ment was set to OpenMPI 4.0.6. Linear algebra computations in the field equation solving process were performed using the Eigen 3.3.9 library, while fast Fourier transform computa- tions were executed using the FFTW 3.3.8 double precision version library. For all scaling tests, the code was compiled with the “-O3” optimization flag. Additionally, considering the specific architecture of the AMD “Milan” CPU, an extra “-march=znver3” optimization flag is employed to further enhance the automatic optimization by the GNU compiler. For the referencing benchmark case, a single node with 128 cores is utilized. The physical space domain is decomposed in 2D, with 16 cores in the x direction and 8 cores in the y direction. Three groups of velocity space cell numbers are selected: 30 30 30, 50 50 50, and 70 70 70 cells for three-dimensional velocity space, respectively. The weak scaling test is carried out by changing the number of processes while keeping the problem size per process constant. 0 200 400 600 800 1000 1200 1400 0 10 20 30 40 50 60 70 80 90 100 110 Figure 6.2: Weak scaling of the electromagnetic Vlasolver’s efficiency with dependency to core numbers. 6.3 Parallel Efficiency of the Electromagnetic Vlasolver 162 Fig. 6.2 illustrates the weak scaling results of the electromagnetic Vlasolver for up to 10 computational nodes (1280 CPU cores). A gradual decrease is observed in the figure. At 1280 CPU cores, the parallel efficiency for velocity space cells of 30 30 30, 50 50 50, and 70 70 70 is found to be 80:77%, 85:33%, and 91:47%, respectively. Similar to the trend demonstrated in Section 3.3, as the velocity space becomes denser, the parallel efficiency improves. The underlying reason is akin to that observed in the electrostatic version of Vlasolver. As the velocity space becomes denser, the computational time for parallelized Vlasov equations increases, and the significance of serial overhead decreases accordingly. Figure 6.3: Profiling results of the electromagnetic Vlasolver. Fig. 6.3 presents the profiling results of the electromagnetic Vlasolver for selected core numbers, considering three different velocity space mesh points groups. In contrast to the results depicted in Fig. 3.9 of Section 3.3, where the field equation constituted a substantial portion of the calculations, in this case, the field equation contributes only minimally to the total calculation time. This can be attributed to two main factors. Firstly, the adoption of the FFT method to solve the field equations reduces their computational burden. Secondly, 6.4 Verification of the Electromagnetic Vlasolver: Simulation of Whistler Wave Propagation 163 in comparison to the two-dimensional velocity space in the electrostatic Vlasolver, the three- dimensional velocity space in the electromagnetic Vlasolver necessitates more time, leading to a reduced time portion occupied by the field solver. It is noteworthy to observe that in the profiling of the electromagnetic Vlasolver, the moment integration part occupies a substantial portion of the total computation time compared to its counterpart in the electrostatic Vlasolver. This can be attributed to the fact that during the process of coupling the Vlasov equation with the Field equations, integration of the charge density, current, and moment flux tensor from the velocity distribution function necessitates a total of 10 integrations of the velocity distribution function within a single loop. As mentioned earlier, the parallelization of the Vlasolver is accomplished through domain decomposition in physical space. In order to further optimize the code’s performance, shared-memory parallelization schemes need to be employed for the velocity space. 6.4 Verification of the Electromagnetic Vlasolver: Sim- ulation of Whistler Wave Propagation In the following two sections, we present two simulation cases that employ the Vlasolver to address two distinct physical problems: the propagation of Whistler wave and Weibel instability. These problems have been extensively studied in the literature and this them appropriate benchmarks for verifying and validating the numerical outcomes in the following two sections. In both electromagnetic problems presented, we normalize the spatial length using the electron inertial length, denoted as d =c=! pe0 , wherec represents the speed of light, and ! pe0 denotes the electron plasma frequency at the beginning of the simulation. Employing these normalization factors ensures that the simulations are conducted on the electron time scale, enabling accurate capturing of electromagnetic waves and instabilities. 6.4 Verification of the Electromagnetic Vlasolver: Simulation of Whistler Wave Propagation 164 In the first test case, we simulate the propagation of a Whistler mode wave, which is an electromagnetic wave propagating under the electron-cyclotron frequency. The simulation is conducted in a two-dimensional physical space and a three-dimensional velocity space. The physical domain consists of 512 cells in the x direction, while only 4 cells are used in the y direction, making it a quasi-1D computational domain in physical space. To ensure accurate simulation, we set the speed of light in the simulation, denoted as c, to an arbitrary value of 10v te0 , where v te0 represents the initial electron thermal velocity. Under this condition, we define the initial electron inertial length d e0 as Ref. [172] d e0 = c v te0 De0 = 10 De0 (6.12) In this test case, we use the initial electron inertial length d e0 and the light speed c as the normalization factors. The computational parameters are set as follows. Only electrons are considered in this test case, as the ions are treated as a fixed background due to their larger mass. An initial fixed background magnetic field with a strength of B 0 = 1 is set along the x direction. The length of the physical computational domain is set to be in the range [0; 51:1d e0 ] in the x direction and [0; 0:3d e0 ] in the y direction. The grid sizes are set to be x = 0:1d e0 = De0 and y = 0:1d e0 . The length of the velocity computational domain is set to be in the range [0:5c; 0:5c] in the x, y, and z directions. 80 cells are used on each velocity direction. This setup results in approximately 1 billion mesh cells. The initial electron velocity distribution function is set to be f e (x;v;t = 0) = m e 2kT e0 3 exp ( m e (vv d ) 2kT e ) (6.13) where v d is the initial perturbation on drifting velocities of electrons v d = (0; A sin (k 1 x); A cos (k 1 x)). The parameter A and k 1 are set to be the same with the Ref. [195] so A = 0:001 and k 1 ==256. The time step length is defined as t = 0:025, and the total simulation duration is set to t = 1000. Consequently, the simulations consist of a total of 40,000 numerical steps. The simulations were conducted using the USC CARC’s 6.4 Verification of the Electromagnetic Vlasolver: Simulation of Whistler Wave Propagation 165 Discovery clusters, employing 128 AMD epyc-64 CPU cores distributed across two nodes. The simulations were completed within a period of 52 hours. The computational costs for these simulations significantly increased compared to the previous electrostatic verification cases for two main reasons. Firstly, the introduction of three dimensions in velocity space limited the extent to which computational resources could be fully leveraged, as velocity space parallelization is not applied in the current version of Vlasolver. Secondly, the solving process of the Vlasov-Darwin solver necessitates multiple numerical integrations of the distri- bution function to obtain its moments, in contrast to the electrostatic Vlasov-Poisson solver where only one integration is required. In future work, considering these characteristics of the Vlasov-Darwin solver, shared-memory parallelization techniques should be employed to parallelize the velocity space. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Figure 6.4: Spatial-temporal spectrum of the B y component. 6.5 Verification of the Electromagnetic Vlasolver: Simulation of Electron Weibel Instability 166 Fig. 6.4 displays the spatial-temporal spectrum of the B y component. The black dashed line in Fig. 6.4 is the theoretical dispersion relation for the whistler wave under the cold ion condition [176, 195] ! = B 0 k 2 1 +k 2 (6.14) The excellent agreement between the highlighted regions in the numerical results and the theoretical dispersion relation, as shown in Fig. 6.4, validates the accuracy of the electromag- netic Vlasov-Darwin simulations performed using Vlasolver. The highlight regions in the numerical results exhibit some noise in the high wavenumber range. This noise is attributed to the mesh size, which is only one tenth of the electron inertial length when considering our reduced light speed. 6.5 Verification of the Electromagnetic Vlasolver: Sim- ulation of Electron Weibel Instability In this test case, we simulate the electron Weibel instability, which has been extensively studied in laboratory and astrophysical plasmas. The generation of a magnetic field in initially unmagnetized plasmas is a significant phenomenon in plasma physics [265, 266]. Electron Weibel instability can be triggered by either counter-streaming plasma jets or anisotropic temperature plasmas. Recent studies have shown that counter-streaming triggering can result in a combination of both two-stream and Weibel instabilities [267]. To isolate the Weibel mode and avoid interference from the two-stream mode, we employ an anisotropic temperature plasma setup in this study. The set-up here follows the set-up in Ref. [195, 268]. The simulation is conducted in a two-dimensional physical space and a three-dimensional velocity space. The wavevector k of the Weibel instability is set to along the x direction. In order to avoid the more complex behaviors of Weibel instability in the 2D set-up, a quasi-1D set-up is used here which makes the physical space simulation box a thin and long box along the x direction. The 2D3V 6.5 Verification of the Electromagnetic Vlasolver: Simulation of Electron Weibel Instability 167 version of the Vlasolver is used here for the simulation. The normalization here adopts the electromagnetic normalization which normalizes the length by the electron inertial length d e and normalizes the time by the electorn frequency ! pe and normalizes the velocity by the speed of light c. Both electrons and ions are included in the simulation, with a reduced mass ratio of m i =m e = 100 to capture the disparate masses. The thermal velocities in the x direction for electrons and ions are equal, given by v tex =v ti = 1. The initial electron distribution exhibits temperature anisotropy and follows a bi-Maxwellian distribution with T ey;z =T ex = 2:56. To reduce computational costs, the electron thermal velocity is scaled with respect to the speed of light by setting v th;x =c = 1=40. It is important to note that the ions in this simulation act as a fixed neutralizing background. The simulation domain is set to have a length of L = 32d e0 , with the largest wavenumber allowed being k 1 d e0 = 0:1963495. Following the configuration in Ref. [195], the domain is discretized into 64 cells in the x direction and 4 cells in the y direction. The small number of cells in the y direction results in a quasi-1D simulation domain in physical space. The velocity space is divided into 50 cells in each direction, with the lower and upper bounds set as7v te;x and 7v te;x , respectively. These specifications lead to a total of 16 million simulation cells. Initially, no external magnetic field is present in the simulation domain. To trigger the Weibel instability in the noise-free grid-based Vlasov simulation, initial perturbations are added to the electron bulk velocity following the method outlined in Ref. [195]. = 1 k 2 x c 2 ! 2 ! 2 pe ! 2 1 + T y;z 2T x Z 0 ! k x p 2T x =m !! (6.15) The theoretical dielectric function of the Weibel instability, as shown in eq. (6.15), follows the formulation presented in Ref. [176, 265]. Based on the setups described earlier, the theoretical linear growth rate is determined to be th = 0:004! 1 pe0 . All simulations were conducted on the Discovery cluster at the USC CARC center, utilizing a total of 16 AMD epyc-64 CPU cores. The simulations were run until t = 2000, with a time step length of 6.6 Summary 168 t = 0:025, resulting in a total of 80000 numerical steps. The simulations could be completed within approximately 12 hours. 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Figure 6.5: Time history of the longest wavelength mode magnetic field energy of B y and B z . The theoretical linear growth rate is obtained by iteratively solving eq. (6.15). Fig. 6.5 illustrates the comparison between the theoretical and numerical growth rates of the magnetic field energy for the longest wavelength mode. A remarkable agreement is observed during the linear growth stage of the Weibel instability, confirming the accuracy of Vlasolver. Furthermore, the oscillation characteristics observed after saturation align with the findings in Ref. [195, 268]. 6.6 Summary This chapter presents the development and expansion of the Vlasolver introduced in Chapter 3 into its electromagnetic counterpart. To address the time step discrepancy between velocity and acceleration information, a novel coupling approach has been developed for integrating 6.6 Summary 169 the Vlasov equation solver and the Darwin equation solver. This efficient coupling technique circumvents the need for a “pseudo update”, thus significantly reducing computational costs. The parallel efficiency and profiling of the electromagnetic version of Vlasolver have been examined. The electromagnetic Vlasolver demonstrates good parallel efficiency, exceeding 80% with over 10 3 CPU cores, indicative of its favorable scalability. Notably, a comparative analysis with the electrostatic Vlasolver reveals that the predominant computational time in the electromagnetic variant is attributed to the moment integration process. This arises from the requirement of computing 8 quantities, encompassing both vectors and second-order tensors, essential for the coupling method. It is envisaged that the current computational bottleneck could be resolved by implementing shared-memory parallelization techniques in the velocity space, which offers potential pathway for future enhancement. Two classical electromagnetic plasma problems, the propagation of whistler waves and the Weibel instability, are examined as benchmark cases to validate the accuracy of the electromagnetic Vlasolver. Remarkable agreements are observed between the numerical simulation results and the theoretical dispersion relations and growth rates. Overall, this chapter introduces the electromagnetic version of the Vlasolver, highlighting its scalability. Two fundamental electromagnetic plasma problems are investigated and their results are rigorously compared with theoretical solutions, revealing substantial agreement. Theseoutcomesestablishtheelectromagnetic Vlasolver asapromisingtoolfortheforthcoming exploration of solar wind turbulence and instabilities. 170 Chapter 7: Summary and Conclusions This chapter provides a summary of the key conclusions drawn from the research conducted in this study, the major contributions of this dissertation, and discusses the recommended areas for future work. 7.1 Conclusions and Contributions 7.1.1 Conclusions This dissertation presents the development and application of a fully kinetic, large-scale, parallelized, multi-dimensional grid-based Vlasov solver, known as Vlasolver, along with its applications in three plasma engineering and physics studies. This dissertation first discusses the development of the electrostatic Vlasolver. Several key features are implemented in Vlasolver to ensure accurate and efficient simulations. For time stepping, the semi-Lagrangian scheme is employed, while the third-order positive flux conservation (PFC) scheme is utilized for updating the velocity distribution function in the phase space. The Vlasolver supports both periodic and non-periodic boundary conditions, providing flexibility in simulating different physical scenarios. In terms of parallelization, the Vlasolver adopts domain decomposition in the physical space using the Message Passing Interface (MPI). Each sub-domain contains a complete set of velocity space. The parallel efficiency and weak scalability of the Vlasolver are carefully assessed. Despite the sequential overhead introduced by the field solver, the Vlasolver demonstrates good scalability, with an efficiency exceeding 75%. To ensure the accuracy of the Vlasolver, the Vlasolver model is first benchmarked against problems that have well-established theoretical solutions. Specifically, linear advection and 7.1 Conclusions and Contributions 171 gyro-motion problems are employed to validate the correct implementation of the Vlasov equationsolver. Subsequently, twoclassicalelectrostaticproblems, namelylinearandnonlinear Landau damping, as well as the two-stream instability, are utilized to verify the accuracy of the Vlasov-Poisson solver within the Vlasolver. Remarkably, the numerical results obtained from the Vlasolver exhibit a high level of agreement with the corresponding theoretical predictions. The accuracy of the Vlasolver is also validated by comparing it with a well-established fully kinetic Particle-in-Cell (PIC) simulation for the plasma wake expansion process [216]. The simulation results demonstrate a remarkable agreement between the two methods. Moreover, even with 1200 sample averages, the PIC method still exhibits statistical noise in low-density regions and higher-order moments. In contrast, the grid-based Vlasov method smoothly resolves all small-scale structures and higher-order moments of the distribution function throughout the entire computational domain. Moreover, the influence of numerical noise in particle-based simulation methods is examined through an unsteady periodic scenario. An electron Linear Landau damping problem is chosen as the model case. Vlasov and PIC simulations with varying numbers of particles per single cell are carried out. Surprisingly, the outcomes demonstrate that even with a considerable number of particles per cell (10 6 ), the heat flux cannot be accurately resolved, suggesting a significant challenge in achieving smooth results. These comparisons not only establish the reliability of the Vlasolver for real-world applications but also emphasize the significance of utilizing the grid-based Vlasov method to investigate small-scale physics phenomena and higher-order moments of the velocity distribution function. Consequently, these findings motivate the use of the Vlasolver to study the following physical mechanisms in the plasma expansion process. This dissertation then applies the Vlasolver to investigate the electron-scale physics in the one-dimensional semi-infinite collisionless plasma expansion process, analogous to axial direction electric propulsion beam expansion. Traditionally, this process was considered ion-scale, neglecting electron kinetics. Recent observations challenged this view, showing anomalous electron cooling beyond predicted regions. To address this problem, noise-free 7.1 Conclusions and Contributions 172 grid-based Vlasov simulations are used to re-evaluate the expansion process, extending the simulation domain beyond classical self-similar regions and enabling the study of small- scale perturbations. Hybrid simulation results aligned with classical solutions, but fully kinetic simulations unveiled small-scale perturbations beyond the classical self-similar regions, highlighting their significance in electron-scale dynamics. The electric field spectrum in !k space reveals two distinct modes: the ion acoustic mode and the electron Langmuir mode. The first mode corresponds to the rarefaction wave predicted by classical self-similar theory, while the second mode is not anticipated by classical theory and is found for the first time in the plasma expansion process. Both modes propagate upstream, with the speed of the electron Langmuir mode wavefront significantly exceeding the sound speed, resulting in electron-scale perturbations ahead of the ion acoustic wavefront. The observed anomalous electron “cooling” and the hump in the heat flux outside the self-similar region can be attributed to the presence of the electron Langmuir wave and the wave-particle interactions involved in its propagation. Downstream of the quasi-neutral expansion front, a double layer forms, attracting backflow electrons and leading to an observed enhancement in electron temperature. Finally, we extend the classical solution beyond the self-similar region and provide an explanation for the observed anomalous electron cooling. This dissertation then investigates the relationship between electron microscopic kinetics and macroscopic heat flux in the electric propulsion beam using two-dimensional setup. Traditional models often treat electrons as massless equilibrium fluids, assuming isothermal processes. However, recent studies reveal non-equilibrium electron behavior and anisotropic temperature, significantly impacting plasma dynamics and leading to over 40% errors in potential estimation [23, 24, 97]. This investigation seeks answers to three key questions: the electron heat conduction mechanism in a collisionless plasma, the influence of electron kinetics on macroscopic plasma properties, and the possible appropriate fluid closure for electrons. To address these, high-order moments of the electron velocity distribution need to be examined, necessitating noise-free grid-based Vlasov simulations due to statistical noise 7.1 Conclusions and Contributions 173 in Particle-in-Cell (PIC) methods. This study is, to the best of our knowledge, the first 2D grid-based Vlasov simulation of plasma beam expansion. We compare grid-based Vlasov, theoretical predictions, and PIC simulations. Grid-based Vlasov closely aligns with PIC and theory, exhibiting smooth profiles and accurately resolving physical properties across the entire domain. In contrast, PIC displays significant noise, especially in low-density regions, and shows strong noise in higher-order moments even in high-density areas. The electron velocity distribution function are found to have near- Maxwellian shapes in the v x direction with depleted high-energy tails in the negative velocity branch. In the v y direction, velocity distribution functions evolve toward a top-hat shape, offering insights into the near-adiabatic cooling process of electron temperature. The study examines electron heat fluxes, identifying three distinct regions in the x direction: core, fan, and corona. In the y direction, two regions, core and corona, are observed, each with different heat flux characteristics. The x-direction electron heat flux is generally greater than the y-direction heat flux in the beam, except near the injection plane. The primary contributor to the electron heat flux is found to be Q e;xxx . It reveals that the skewness of the density-normalizedv x direction electron VDF corresponds to the coefficient in the flux-limited model of electron collisionless heat flux Q e;xxx and identifies two distinct regions for this coefficient at different times: region A (0<x<t(v d C s )) and region B (x>t(v d C s )). Region A shows a time-dependent structure characterized by a hump, while Region B displays a nearly monotonically increasing tail structure in the flux-limited coefficients. These regions exhibit self-similar behavior with respect to time evolution, with Region A formed by electron trapping and Region B resulting from a strong electric field within the self-similar region. By establishing a connection between the skewness of the velocity distribution function and the flux-limited coefficient of the collisionless heat flux, a groundwork is set up for the development of collisionless heat flux closure models that take into account the anisotropic thermodynamic characteristics of electrons. The last part of the dissertation extends the Vlasolver into electromagnetic version by incorporating a Vlasov-Darwin solver. A novel coupling strategy is developed to address 7.1 Conclusions and Contributions 174 challenges arising from the half-time step retardation so the Vlasov and Darwin equations can be connected effectively. The parallel efficiency and weak scalability of this electromagnetic iteration of the Vlasolver are evaluated. In comparison to the electrostatic Vlasolver, the electromagnetic variant displays a modestly improved scalability, achieving an efficiency exceeding 80% across more than 10 3 CPU cores. Parallel profiling outcomes reveal that the primary computational bottleneck in the Vlasolver resides in the moment integration component. Future improvements could involve implementing shared-memory parallelization techniques within the velocity space. To validate the Darwin Vlasov solver, two electromag- netic problems, the propagation of whistler waves and the electron Weibel instability, are explored. The numerical results from the Vlasolver exhibit a satisfied level of agreement with corresponding theoretical predictions. An electromagnetic PIC simulation study regarding the interrelation between whistler anisotropy instability and whistler turbulence is provided in the appendix. While the central emphasis of this dissertation remains on the grid-based Vlasov method, resource limitations currently prevent the execution of a fully 3D3V grid-based Vlasov simulation. Consequently, weundertakethissimulationthroughtheParticle-in-Cellmethod, preservingitasabenchmark for the prospective advancements of the Vlasolver. The investigation reveals that the presence of whistler anisotropy instability in a plasma concurrently experiencing whistler turbulence and global electron temperature anisotropy yields minor effects compared to those induced by whistler turbulence alone. This discrepancy arises as the energy fluctuations associated with the narrowband whistler instability are overshadowed by those stemming from the broadband whistler turbulence. Nonetheless, considering that the field fluctuations arising from the growth of a single instability mode might facilitate a more efficient wave-particle scattering process than those starting from a spectrum of whistler modes, the whistler instability effectively dampens the intermittency of electron temperature anisotropy generated by turbulence. This conclusion implies that micro-instabilities could potentially serve as regulatory mechanisms for turbulence. 7.1 Conclusions and Contributions 175 7.1.2 Contributions The contributions of this dissertation can be outlined in two main aspects: computational advancements and physical insights. 7.1.2.1 Contributions in Computational Physics The contributions in computational physics of this dissertation research include the following aspects Development of a fully kinetic large-scale parallelized electrostatic/electromagnetic grid-based Vlasov computational framework, Vlasolver. The code is parallelized using domain decomposition in physical space via the MPI library and achieves excellent parallel efficiency. Development of a coupling method between the Vlasov equation and the Darwin equations using the first three-order Vlasov moment equations to save computational time. Code verification and validation of Vlasolver using various physics and engineering problems, including comparisons with the PIC method to illustrate the advantages of the grid-based Vlasov method in obtaining high-order moments of the velocity distribution function and resolving small-amplitude physical properties. Showcase the capability of Vlasolver in studying electromagnetic problems without numerical noise. 7.1.2.2 Contributions in Plasma Physics and Engineering The contributions in plasma physics and engineering of this dissertation research include the following aspects 7.2 Future Work 176 Investigation of the electron scale dynamics in the collisionless plasma expansion process. Find both ion acoustic mode and electron Langmuir mode wave exists in the expansion process and explain the previous observed anomalous electron “cooling” effects by the wave-particle interaction between electron and electron Langmuir mode wave. Extend the classical solution to regions beyond tje self-similar region. Investigation of the electric propulsion beam expansion. Establish a connection between the electron velocity distribution function skewness and flux limited model coefficient of the electron collisionless heat flux. Provide pathways for the future collisionless electron heat flux closure. 7.2 Future Work The following recommendations are proposed for future research. 7.2.1 Further Development of Grid-based Vlasov Method and the Vlasolver Framework Inthisdissertation, afullykineticmulti-dimensionalparallelgrid-basedVlasovcode, Vlasolver, is developed for the modeling of plasma dynamics in collisionless space plasma flow. The time stepping methodology employed in Vlasolver is the Semi-Lagrangian method, while the discretization of phase space is achieved through the adoption of a uniform Cartesian mesh. This choice of methodology and discretization scheme aligns with the specific scope of the research problems of interest in this study. However, the combination of Semi-Lagrangian time stepping and uniform Cartesian mesh may not suitable for other physical problems interested and thus the further development of grid-based Vlasov method is required. In certain physical scenarios, such as plasma-wall interaction and the dynamics of plasma- beam interactions, the focus is often limited to a specific region within the physical/velocity 7.2 Future Work 177 space. In such cases, utilizing a uniform Cartesian mesh can result in unnecessary compu- tational expenses. Consequently, the adoption of a non-uniform or adaptive mesh proves to be more advantageous in effectively addressing these particular problems. Hence, future research efforts should focus on further developing and implementing the semi-Lagrangian grid-based Vlasov method to accommodate non-uniform and adaptive mesh configurations. The use of cylindrical mesh is often found to be more suitable than Cartesian mesh for applications in plasma technology. Therefore, as part of the future research plans, the development of the semi-Lagrangian grid-based Vlasov method under cylindrical coordinates will be undertaken. The grid-based Vlasov method is known for its high computational cost, highlighting the significance of parallelization and optimization in the code implementation of this method. As computationalresourcescontinuetoadvance, effortsshouldbemadetoparallelizeandoptimize the Vlasolver framework to fully leverage the capabilities of modern computing architectures. Load balancing strategies, and efficient memory management techniques can significantly enhance thecomputationalperformance ofthe framework. The currentparallelization strategy implemented in Vlasolver focuses on domain decomposition in the physical space, utilizing MPI for communication. However, the velocity space is not parallelized in the current implementation. Given the frequent integration requirements in the velocity space, shared- memory parallelization techniques are better suited for parallelizing this aspect compared to distributed memory parallelization. In future work, shared-memory parallelization techniques, such as OpenMP or OpenACC, will be explored in future work to parallelize the velocity space more effectively. Moreover, the recent advancements in large-memory GPUs and the availability of CUDA-aware MPI libraries have made GPU parallelization highly promising for grid-based Vlasov codes. In upcoming research, the computational performance of Vlasolver willbeassessedbyincorporatingheterogeneousGPUparallelization. Ifsignificantperformance enhancements are observed, Vlasolver will be transformed into a heterogeneous parallelization framework, leveraging the benefits of GPU acceleration to enhance computational performance and enable the investigation of high-dimensional problems. 7.2 Future Work 178 By focusing on these areas of further development, the grid-based Vlasov method and the Vlasolver framework can evolve into more powerful tools for studying plasma dynamics, facilitating more accurate and efficient simulations, advancing our understanding of complex physical processes, and driving innovations in plasma technology. 7.2.2 Impact of Initial Non-Maxwellian Electron Velocity Distribu- tion Function on Electric Propulsion Beam Expansion In this dissertation work, the noise-free grid-based Vlasov method was employed to examine the connections between electron kinetics and thermodynamics during the expansion of an electric propulsion beam (see Chapter 5). The research revealed that the skewness of the electron velocity distribution plays a crucial role in linking the microscopic electron kinetics with the macroscopic heat flux. Furthermore, it was found that electron trapping serves as a significant physical mechanism contributing to the heat flux within the electric propulsion beam expansion process. Although the study conducted in Chapter 5 provides insights into the heat flux within the electric propulsion beam expansion process, it focuses solely on the scenario where both the initial ions and electrons follow a Maxwellian distribution, which is a typical assumpation in previous theoretical and computational works [22, 72, 89, 94]. However, in practical engineering applications, such as electric propulsion systems that employ electrons for charge neutralization of positively charged ions in the beam, the electron distribution emitted from an external cathode often deviates from a Maxwellian distribution, which assumes thermal equilibrium [248, 249, 269–271]. This non-Maxwellian distribution can exhibit characteristics such as a high-energy tail or other deviations from a state of thermal equilibrium. Chapter 5 has highlighted the significant role of high-energy tail electrons in the formation of collisionless heat flux. Consequently, investigating the impact of initial non-Maxwellian electron distributions on electric propulsion beam expansion becomes a crucial research problem. To tackle this issue, future research endeavors will employ the 7.2 Future Work 179 Vlasolver framework. This computational tool will be utilized to explore and analyze the aforementioned problem, shedding light on the effects of non-Maxwellian electron distributions on beam expansion in electric propulsion systems. 7.2.3 Evaluating the Significance of Heat Flux Closure in the Expansion of Electric Propulsion Beams Existingstudiesonplasmaplumeexpansionpredominantlyfocusontheiontimescale, treating electrons as an massless equilibrium charge-neutralizing fluid with assumed thermodynamic relations such like isothermal and polytropic relations. While the simplified electron fluid models are frequently employed, recent studies have revealed that electrons in such processes exhibit non-equilibrium and anisotropic behavior. Consequently, utilizing an equilibrium gas model for electrons can result in substantial inaccuracies in the characterization of plume properties. To enhance the accuracy of hybrid modeling in electric propulsion plasma flow, one potential improvement is to incorporate additional terms, such as electron inertia, electron velocity, electron pressure, and electron energy, into the macroscopic fluid equations. By integrating this additional information, the fluid equations can yield more precise results compared to simplistic assumptions like massless isothermal or polytropic relations. However, the successful implementation of this approach relies on the development of an appropriate heat flux closure. In this dissertation, a thorough investigation is conducted to analyze the heat flux occurring in the expansion process of electric propulsion beams and establish the relationships between electron macroscopic heat flux and microscopic kinetics (see Chapter 5). This research provides valuable insights towards the design of a suitable heat flux closure. Nevertheless, the importance of the quality of the heat flux closure on the accuracy of the fluid equations remains uncertain. Therefore, the forthcoming research in this dissertation aims to evaluate the significance of a high-quality heat flux closure in order to highlight the importance of incorporating such closures. 7.2 Future Work 180 7.2.4 Investigation of the Effects of Finite Ion Temperature on Plasma Expansion This dissertation exclusively focuses on studying the plasma expansion under the cold ion limitation, where T i T e . However, in some practical applications, the ion temperature can be comparable to the electron temperature. Previous one-dimensional theoretical studies conducted by Gurevich et al. [30] and Mora [40] have examined the effects of finite ion temperature on semi-infinite plasma expansion. Their research demonstrates that considering the kinetics of ions with finite temperature can result in a significant increase in ion heat flux, subsequently leading to an increased electron heat flux compared to the case with cold ions. In future research, following the completion of this dissertation, the impact of finite ion temperature on the electric propulsion beam expansion process will be investigated. 7.2.5 Comparative Analysis of Grid-based Vlasov Method and PIC Method for Short Wavelength Turbulence and Microinstabil- ity in the Solar Wind This dissertation employs the Vlasov-Darwin model within the Vlasolver, intending to investigate short wavelength turbulence and microinstability in the solar wind. However, due to the significant computational resources demanded by the specific problem of interest, typically requiring 2D3V or 3D3V simulations, the Particle-in-Cell (PIC) method is utilized instead. 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However, due to current limitations in computational resources and the granularity of parallelization in the implementation of the grid-based Vlasov code developed in this dissertation, the PIC method can be employed instead to study this problem. This PIC study can serve as a benchmark for future development and application of the grid-based Vlasov code to address this research question. Particle-in-Cell simulations and statistical analysis are carried out to study the dynamic evolution of a collisionless, magnetized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has almost no effects on the fluctuating magnetic field energy and intermittency generated by turbulence. However, it leads to a small reduction of the magnetic field wavevector anisotropy and a major reduction of the intermittency of electron temperature anisotropy. Hence, while the overall effect from whistler instability is minor as compared to that of whistler turbulence due to its much smaller field energy, the whistler instability may act as a regulation mechanism for kinetic-range turbulence through wave-particle interactions. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 205 A.1 Introduction High-frequency short-wavelength whistler turbulences are often observed in space plasma [114, 272–274]. We define whistler turbulence as a broadband ensemble of incoherent field fluctuations in a magnetized plasma at frequencies between the lower hybrid and electron cyclotron frequencies and at wavelengths much shorter than the ion inertial length. There have been significant debates about the possible sources of whistler turbulence in recent years. One possible scenario is the cascade of fluctuations from the longer wavelength inertial range. Kinetic Alfvén waves and higher frequency magnetosonic-whistler fluctuations have been considered as the two candidates [106]. Recent solar wind observations [107–111] and numerical simulations [112, 113] have identified the existence of the kinetic Alfvén fluctuations with a wavelength around the ion inertial length or ion thermal gyro-radius. However, the mechanism on how such modes cascade fluctuation energy down to electron scale remains unclear. For instance, as the inertial range cascade preferentially transfers fluctuation energy topropagationdirectionsrelativelyperpendiculartobackgroundmagneticfield, wherewhistler fluctuations can be damped [275, 276], the cascade processes may not be able to to provide a sufficiently large amplitude to feed to whistler turbulence. Another possible scenario is kinetic whistler instabilities. A specific growing mode which can be a source for whistler turbulence at relatively long electron-scale wavelengths is the whistler anisotropy instability. We use subscripts \? " and \k " to denote the directions perpendicular and parallel to the background magnetic field B 0 , respectively, subscripts e and i to denote electrons and ions, respectively, and ~ k to denote the wave-vector. This instability is driven by electron temperature anisotropy T ?e =T ke > 1 and propagates at ~ kB 0 = 0 in a homogeneous plasma. Observations have indicated that this instability is operating in the terrestrial magnetosheath [277]. In the solar wind, while adiabatic expansion of the solar wind would typically lead to T ke >T ?e which can excite the firehose instability, there is evidence that local compressions and turbulence in the solar wind may also create T ?e > T ke [4, 278, 279], which can excite the whistler instability. Particle-in-Cell (PIC) A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 206 simulations [120, 128, 280, 281] have demonstrated that this mode can generate enhanced whistler fluctuations and spectral transfer [120, 125, 126, 128, 280, 281]. To further investigate the aforementioned scenarios, one must first understand the com- peting effects from whistler turbulence and whistler anisotropy instability. For instance, one of the primary consequences of plasma turbulence is to produce sharp spatial gradients in the plasma which can produce enhanced anisotropies locally and forward cascade to dissipate the energy from long wavelength to short wavelength [133–135]. On the other hand, simulations of whistler anisotropy instabilities driven by a bi-Maxwellian velocity distribution for electrons in a homogeneous plasma showed that the instability imposes an upper bound or constraint on that anisotropy uniformly across the plasma [128, 172, 280, 281]. The electron anisotropy upper bound derived by Gary and Wang [280] was verified by observations in the solar wind and magnetosphere [277, 282–284]. However, past studies have mostly addressed the effects from whistler turbulence and whistler anisotropy instability separately. Gary et al. [118, 119], Saito et al. [120, 121], and Saito and Gary [122] presented the first 2-dimensional (2D) PIC simulations of whistler turbulence, and Gary et al. [127, 128] and Chang et al. [123–126] presented the first 3-dimensional (3D) PIC simulations of whistler turbulence. These simulations considered a homogeneous, magnetized, collisionless plasma upon which an initial spectrum of relatively long wavelength whistler fluctuations is imposed. The results showed that the forward cascade leads to fluctuations which are consistent with the linear dispersion solution for whistler fluctuations. Electron temperature anisotropy were also found to form during forward cascade. Hughes et al. [105, 129] and Gary et al. [130] further investigated electron/ion heating due to whistler turbulence as a function of the initial fluctuating magnetic field energy density, and found the maximum electron heat rate scales approximately linearly with the fluctuating field energy density. This suggests a quasi-linear type heating due to electron Landau damping [130]. In this chapter, we consider a collisionless, magnetized plasma with co-existing whistler turbulence and the electron temperature anisotropy as the initial condition, and investigate the consequences of both microinstability growth and turbulent cascade using 3D fully kinetic A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 207 PIC simulations. Qudsi et al. [142] and Bandyopadhyay et al. [143] carried out 2D PIC simulations of the Alfvénic turbulence, where ion temperature anisotropy is generated by the development of the turbulence. The results showed that microinstabilities can develop locally in response to ion temperature anisotropies generated by turbulence and may affect the plasma globally, and that there is an apparent correlation between linear instability theory and strongly intermittent turbulence. It was also speculated that a similar process might also occur on electron scale. In this chapter, in order to evaluate the effect from whistler instability at a given temperature anisotropy, we prescribe electron temperature anisotropy as the initial condition to drive the instability [285]. We carry out four different ensembles of PIC simulations: 1) an initially quiet, anisotropic plasma with prescribed initial electron temperature anisotropy; 2) an isotropic plasma with prescribed initial whistler fluctuations; 3) an anisotropic plasma with initial whistler fluctuations (varying initial electron temperature anisotropy and fixed initial fluctuation field energy); and 4) an anisotropic plasma with initial whistler fluctuations (fixed initial electron temperature anisotropy and varying initial fluctuation field energy). Results from PIC simulation are linked with a statistical analysis to understand whether there is any interplay between whistler turbulence and whistler instability, and what are the competing effects from these two processes. A.2 Simulation Model and Setup We consider a collisionless electron-ion plasma with a uniform background magnetic field B 0 =B 0 ^ z. For the jth (j=e, i) species, we denote the plasma frequency as! pj = p 4n j e 2 =m j , the cyclotron frequency as j = eB 0 =m j c, the thermal speed as v tj = p T kj =m j , and j = 8n j T kj =B 2 0 . We denote the angle of mode propagation bykB 0 =kB 0 cos(). The physical and numerical parameters are chosen to assure that the consequences of both micro-instability growth and turbulent cascade can be accurately resolved in the simulation. In this chapter, the initial electron plasma beta is taken to be of a typical value for the A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 208 solar wind plasma, e = 0:1. To study the effects of whistler anisotropy instability, we consider a range of initial T ?e =T ke values that are below and above the instability threshold in the simulation. Through a sequence of test runs with varying initial T ?e =T ke values, we find the threshold to excite whistler anisotropy instability for the parameters considered is T ?e =T ke ' 2:3, close to the calculation using the linear thoery from [4]. In this chapter, we present simulations with initial temperature anisotropy of T ?e =T ke = 1, 2, 3, 5, 7, 9. To study the effects of whistler turbulence, an ensemble of whistler fluctuations are imposed at t = 0. The initially loaded whistler fluctuations are set to be relatively long-wavelength with approximately isotropic wavevectors. The spectrum is the same as that used in our previous simulation studies on whistler turbulence [123, 124, 126, 127, 129]. The initial whistler modes includen = 0,1,2, and3 of the fundamental wavenumber in the perpendicular direction, andn =1,2, and3 of the fundamental wavenumber in the parallel direction, where the fundamental wavenumber corresponds to the maximum wavelength that can be contained in the domain. This leads to a total of N = 150 normal modes with random phases [124]. The simulations will consider initial total fluctuating magnetic field energy density = N X n=1 jB n (t = 0)j 2 =B 2 0 (A.1) at = 0, 0:05, 0:25, and 0:5. We apply a three-dimensional (3D) full particle electromagnetic Particle-in-Cell code, 3D-EMPIC by [286], to simulate the evolution of plasma under four different sets of initial conditions. In Simulation Group A, the ions are set to follow an isotropic velocity distribution while the electrons follow an anisotropic bi-Maxwellian velocity distribution function with different initial values of T ?e =T ke , at T ?e =T ke =2, 3, 5, 7, and 9. The plasma has no initial field fluctuations, = 0. Simulation Group A is a typical setup for simulations of whistler anisotropy instability [280]. In Simulation Group B, both the ions and electrons are set to have an isotropic velocity distribution. An ensemble of whistler fluctuations are imposed at t = 0, with the initial total fluctuating magnetic field energy density at = 0, 0:05, 0:25, and A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 209 0:5. The initial condition in Simulation Groups C and D is a combination of that of Groups A and B, where the electrons follow an anisotropic bi-Maxwellian velocity distribution function and the plasma is initially loaded with an ensemble of whistler fluctuations. In Group C, we take the initial field fluctuation at = 0:25 and change the initial temperature anisotropy at T ?e =T ke =2, 3, 5, 7, 9. In Group D, we take the initial temperature anisotropy at T ?e =T ke = 3 and change the initial field fluctuation field density at = 0, 0:05, 0:25, 0:50. The simulation groups are summarized in Table A.1. All the simulations are run using an artificial ion to electron mass ratio of m i =m e = 400. The ion initial temperature is set to be T i =T ke . The ratio of the electron gyro-frequency to plasma frequency is e =! pe ' 0:447. The simulation box is a cube with a size in each direction at 51:2d e , where d e =c=! e is the electron inertial length. The grid spacing is set to be = 0:10d e , and hence the mesh size is 512 512 512. The time step is set to be t! pe = 0:05. All the simulations are run for t! pe > 1000 (t pe > 447:20), i.e. more than 20,000 steps. The macro-particles used is 48 ions and 48 electrons per cell or about 3:1 10 11 total macro-particles. We use the Probability Density Function (PDF) in statistical analysis of magnetic fluctuations. The PDF of a random field B(x) may be defined as [287] PDF(B)dB = probability that the random value lies between B and B +dB (A.2) Then the increments of the field components are B(x) =e r [B(x +r)B(x)] (A.3) where r is the spatial separation length vector along the direction of any unit vector e r . By summing over all the cells of a PIC simulation, one may construct a PDF for each component of the fluctuating fields as a function of the spatial separation r. If the random variable r is subject to a central limit theorem, the distribution is expected to be a Gaussian, whereas any departure from a Gaussian corresponds to a more strongly intermittent ensemble A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 210 Table A.1: Summary of simulation cases. In all cases, e = 0:1 and v te =c = 0:1. Group Initial Condition Run No. T ?e =T ke A quiet anisotropic electrons 1 2 3 4 5 2 3 5 7 9 0.00 0.00 0.00 0.00 0.00 B isotropic electrons with whistler fluctuations 1 2 3 4 1 1 1 1 0.00 0.05 0.25 0.50 C anisotropic electrons with whistler fluctuations 1 2 3 4 5 2 3 5 7 9 0.25 0.25 0.25 0.25 0.25 D anisotropic electrons with whistler fluctuations 1 2 3 4 3 3 3 3 0.00 0.05 0.25 0.50 of fluctuations. An important advantage of the PDF analyses is that, by statistically averaging over a large body of observational and/or computational data, one may draw general conclusions which are less readily available via other means of data analysis. For example, the statistical analysis of solar wind magnetic fluctuations measured from the Cluster and ACE spacecraft by [110] shows that the PDFs of both B k and B ? exhibit the same functional form in the kinetic range but not in the inertial range. The PDF analysis of the solar wind data from the Helios spacecraft [288] shows that, as the heliospheric distance of the spacecraft increases, the distribution of the local mean magnetic field vectors gradually broadens in the radial direction and becomes more scattered. The PDF analysis of 3D PIC simulations of whistler turbulence [125] shows distinct non-Gaussian “tails" in both the B k and B ? distributions as well as distinctly different functional forms between the two magnetic polarizations. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 211 A.3 Fluctuating Magnetic Fields 0 200 400 600 800 1000 10 -6 10 -4 10 -2 10 0 (a) 0 200 400 600 800 1000 10 -6 10 -4 10 -2 10 0 (b) 0 200 400 600 800 1000 10 -6 10 -4 10 -2 10 0 (c) Figure A.1: Time history of B(t) 2 =B 2 0 for all simulation cases. (a) Simulation Groups A and B. (b) Simulation Group C. (c) Simulation Group D. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 212 Fig. A.1 shows the normalized fluctuating magnetic field energy density averaged over all mesh points,jB(t)j 2 =B 2 0 , as a function of time for all the simulation runs. The results from Group A show that the fluctuating magnetic fields in Run A2 through A5 (T ?e =T ke 3) grow rapidly to saturation, with the growth rate in the linear phase matching the value calculated from the linear theory [4]. This is similar to that in previous simulations of whistler anisotropy instability [280], where it showed pitch-angle scattering of the electrons by fluctuating magnetic fields reducesT ?e and increasesT ke , and thus the temperature anisotropy. In Run A1 (T ?e =T ke = 2), the fluctuating magnetic field stays relatively unchanged because the initial anisotropy is below the instability excitation threshold and thus the instability is not excited. The results from Group B show that the fluctuating magnetic field energy decreases with time. This is similar to that in previous simulations of whistler turbulence [127], where it showed that the decrease of fluctuating magnetic field corresponds to an increase in the electron thermal energy with the parallel electron temperatures gaining more energy than the perpendicular electron temperatures. Gary et al. [127] showed that such energy dissipation is primarily through wave-particle interactions via linear Landau damping at relatively small initial fluctuating field energy (0:02 0:2) and fully nonlinear processes at large initial fluctuating field energy (> 0:2). Recent studies have suggested that both nonlinear Landau damping [124, 289, 290] and current structure dissipation [102, 103, 279] contribute to the nonlinear dissipation processes. Chang et al. [125] suggested current structure dissipation as the dominant nonlinear dissipation process in whistler turbulence. The results from Group C show that, for a plasma with co-existing whistler fluctuations and electron temperature anisotropy as the initial condition, the time history ofjB(t)j 2 =B 2 0 is almost identical to that from Run B3 ( = 0:25, T ?e =T ke = 1). The initial temperature anisotropy, whether below the instability threshold (Run C1) or above the instability threshold (Run C2 through C5), has little effect onjB(t) 2 j=B 2 0 . Comparing the results from Group D with that from Group B further shows that the time history ofjB(t)j 2 =B 2 0 is influenced only by the initial fluctuating field energy density . The whistler anisotropy instability has almost no effect on the evolution of the overall magnitude of the fluctuating magnetic fields. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 213 0 100 200 300 400 500 600 700 0 0.5 1 1.5 2 2.5 3 3.5 (a) 0 100 200 300 400 500 600 700 0 0.5 1 1.5 2 2.5 3 3.5 (b) 0 100 200 300 400 500 600 700 0 0.5 1 1.5 2 2.5 3 3.5 (c) Figure A.2: Time history of tan 2 B for all simulation cases. (a) Simulation Groups A and B. (b) Simulation Group C. (c) Simulation Group D. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 214 Fig. A.2 shows the magnetic fluctuation wavevector anisotropy averaged over all mesh points tan 2 B = P k k 2 ? jB(k) 2 j P k k 2 k jB(k) 2 j (A.4) as a function of time for all the simulation runs. The tan 2 B history in Run A1 is nearly constant as no whistler anisotropy instability is excited. The results from Run A2 through A5 follow the predictions of the linear theory [4]: the maximum growth rate happens in the direction ofkB = 0; the energy in the perpendicular direction is quickly damped due to resonance scattering of pitch angle, and thus the energy perturbation is mostly along B 0 . The results from Group B shows that the wavevector anisotropy increases rapidly. Larger initial fluctuating magnetic field energy density leads to a more rapid increase in wavevector anisotropy. This reflects the effect of forward cascade of whistler turbulence, which transfers the energy preferentially for k ? k k , thus leading to the expansion of wavevector in the perpendicular direction. The forward cascade of whistler turbulence was discussed in detail in Gary et al. [127] and Chang et al. [123–125]. The tan 2 B history from Group C qualitatively follows that of Run B3 ( = 0:25, T ?e =T ke = 1). However, the initial temperature anisotropy also has a limited effect, showing that an increase of T ?e =T ke reduces the growth rate of tan 2 B . This may be explained as a result of the action by the whistler anisotropy instability. A larger initial temperature anisotropy leads to a larger growth rate of whistler anisotropy instability, which in turn leads to stronger scattering of pitch angle, and thus faster damping of the energy in the perpendicular direction. Comparing the results from Group D with that from Group B, we find that, at a given initial temperature anisotropy, the effect of the whistler instability diminishes as the initial fluctuating field energy increases. This suggests that forward cascade from whistler turbulence has a far more dominating effect over pitch angle scattering from whistler instability on wavevector anisotropy. To further investigate the effects of the whistler anisotropy instability on the intermittency generated by whistler turbulence, we calculate the probability density function (PDF) of A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 215 -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (a) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (b) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (c) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (d) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (e) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (f) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (g) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (h) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (i) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (j) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (k) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (l) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (m) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (n) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (o) -6 -4 -2 0 2 4 6 10 -8 10 -6 10 -4 10 -2 10 0 (p) Figure A.3: Comparison of the probability distribution function of B k along Y axis at e t 111:80. (a)-(d): Run A2 to A5. (e)-(h): Run B1 to B4. (i)-(l): Run C2 to C5. (m)-(p): Run D1 to D4. the local fluctuating magnetic B(i;j;k) for each cell. Fig. A.3 shows the PDF along the y direction. For Groups A and C, the PDFs from Run A1 and C1 are not shown because the whistler anisotropy instability is not excited in these two cases. Figs. A.3e (Run B1) shows the result for a quiet isotropic plasma. Figs. A.3a to A.3d (Run A2-A5 from Group A) show the result for a quiet anisotropic plasma. The initial temperature anisotropy excites whistler anisotropy instability. As there is little change in the tail region, the whistler anisotropy instability did not generate enhanced whistler fluctuations for the simulation parameters considered. Figs. A.3e to A.3h (Group B) show the result for an isotropic plasma with whistler fluctuations. Similar to previous simulations of whistler turbulence [125], an increase in A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 216 leads to an enhanced tail region and the increased deviation from the Gaussian distribution. Figs. A.3i to A.3l (Group C) show the PDFs for Group C are qualitatively similar to that in Group B, indicating that increasing the initial electron temperature anisotropy has very little effect on the fluctuating magnetic fields. Figs. A.3m to A.3p (Group D) further show that the PDFs are only influenced by the initial fluctuating field energy. The results from Figs. A.1 to A.3 are not surprising. As the magnetic field energy from the broad band whistler turbulence dominates over that from the narrow band, the operation of whistler anisotropy instability will have a very minor effect on the evolution of the fluctuating magnetic field. A.4 Electron Temperature Anisotropy We next compare the effects of whistler turbulence and whistler instability on electron temperature anisotropy. We calculate both the local temperature anisotropy R e (i;j;k) = T ?e (i;j;k)=T ke (i;j;k)frommacro-particlesineachcellandtheaveragetemperatureanisotropy over all the cells of the simulation domain R e = 1 N x N y N z R e (i;j;k) = 1 N x N y N z X i;j;k T ?e (i;j;k) T ke (i;j;k) (A.5) where N x , N y , N z are the total mesh points each direction, and subscripts i, j, k denote the cell number. Fig. A.4 compares R e vs. e for Groups A and C at ! pe t = 1000 ( e t' 447:2), when both the whistler turbulence and the whistler instability are developed (see Fig. A.1). Gary and Wang [280] showed that the wave-particle scattering from whistler instability imposes an upper bound on T ?e =T ke commensurate with that predicted by linear theory: R e 1 = S e e ek (A.6) where S e is the dimensionless scalar conductivity of electrons [4, 280]. The anisotropy upper bound in the form of eq. (A.6) was numerically fitted in [280] for parameters similar to that used here, and is shown as the dotted line in Fig. A.4. The results show that the average A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 217 Figure A.4: Average electron temperature anisotropyR e v.s. e at e t = 0 and at e t 447:2 for Group A and Group C. The initial anisotropies are shown as transparent circle for Group A and transparent square for Group C, respective. The final anisotropies for Run A1 through A5 are color circles with increasingly dark shades, and that for Run C1 through C5 are color squares with increasingly dark shades. The upper bound predicted by eq. (A.6) is shown as the dotted line. temperature anisotropy R e at the end of the simulations from Run A2 through A5 lay under the upper bound of eq. (A.6). TheR e from Run A1 is almost unchanged, as expected, as the whistler instability is not excited in this case. It is interesting to observe that the R e points from Group C are further below the upper bound from the linear theory prediction. As turbulence produces strong inhomogeneity in plasma, we next examine the contours of local electron temperature anisotropy, R e (i;j;k). Figs. A.5 and A.6 show the contours of R e (i;j;k), on an x-y plane in the middle of the simulation domain, where we compare R e (i;j;k) for increasing initial fluctuating field energy (from left to right) for a fixed initial temperature anisotropy (T ?e =T ke = 3). In Fig. A.6, we compare R e (i;j:k) for increasing initial T ?e =T ke (from left to right) for a fixed initial fluctuating field energy ( = 0:25). Both Figs. A.5 and A.6 are plotted for ! pe t = 500 ( e t' 223:6), when all the cases are starting to approach an asymptote. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 218 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (a) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (b) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (c) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (d) Figure A.5: Contours of R e (i;j;k) on an x-y plane in the middle of the simulation box at time e t 223:60 for Run D1 (a), D2 (b), D3 (c), and D4 (d). Fig. A.5a (Run D1) has no initial whistler fluctuation and thus the temperature anisotropy distribution is homogenous. As the initial spectrum strength increases, Fig. A.5 shows increasing intermittent fluctuations in temperature anisotropy due to turbulence. Fig. A.6a (Run B3) shows that turbulence produces strong anisotropies in an initially isotropic plasma. All of these results are to be expected. However, in Fig. A.6, as the initial anisotropy increases, A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 219 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (a) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (b) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (c) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (d) 0 10 20 30 40 50 0 10 20 30 40 50 0 0.5 1 1.5 2 2.5 3 3.5 4 (e) Figure A.6: Contours of R e (i;j;k) on an x-y plane in the middle of the simulation box at time e t 223:60 for Run B3 (a), C2 (b), C3 (c), C4 (d), and C5 (e). A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 220 Table A.2: R e (i;j;k) value range of figure A.6. Sub figure No. T ?e =T ke min(R e (i;j;k)) max(R e (i;j;k)) a 1 0.1738 4.8790 b 3 0.2415 4.8039 c 5 0.2975 3.8607 d 7 0.3531 3.0551 e 9 0.3942 2.8061 we find that the intermittent fluctuations in the R e (i;j;k) contours start to diminish. The range of R e (i;j;k) in Fig. A.6 (from left to right) is summarized in Table A.2, showing the temperature becomes more homogenous as the initial T ?e =T ke increases. Figs. A.5 and A.6 show that, in contrast to the minor effects on the fluctuating magnetic field, the whistler anisotropy instability has a major effect on the intermittency of temperature anisotropy generated by turbulence. The whistler anisotropy instability acts to reduce electron temperature anisotropy through wave-particle scattering. Wave-particle scattering is a microscopic process. Wave-particle scattering of electrons are affected more efficiently by local field fluctuations and are less dependent on the overall field energy. The results suggest that the fluctuation from the growth of a single mode of whistler anisotropy instability is more efficient in wave-particle scattering of the electrons than that of a spectrum of whistler modes. A stronger initial temperature anisotropy leads to a larger whistler instability growth rate, and stronger wave-particle scattering effect, and thus reducing the intermittency in temperature anisotropy generated by turbulence. A.5 Summary 3D PIC simulations are carried out to study the dynamic evolution of a collisionless, magne- tized plasma with co-existing whistler turbulence and electron temperature anisotropy as the initial condition, and the competing consequences of whistler turbulence cascade and whistler anisotropy instability growth. The results show that the operation of the whistler instability within whistler turbulence has no obvious effects on the fluctuating magnetic field. A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 221 We find the overall fluctuating magnetic field energy and intermittency generated by turbu- lence are not influenced by the inclusion of an initial electron temperature anisotropy, while wavevector anisotropy is reduced somewhat by increased electron temperature anisotropy. In contrast, the results show that whistler instability has a major effect on electron tempera- ture anisotropy. While whistler turbulence produces sharp gradients and enhanced electron temperature anisotropies locally, we find that increasing the initial electron temperature anisotropy actually leads to a reduction of the intermittency in the electron temperature anisotropy generated by turbulence, and the reduction of the average electron temperature anisotropy further below the upper bound as predicted by the linear theory for a homogeneous anisotropic plasma. The results suggest the small reduction of wavevector anisotropy and major reduction of electron temperature anisotropy are apparently due to whistler instability growth. Comparing to an isotropic plasma with whistler turbulence, an increase in the initial electron temperature anisotropy leads to a larger growth rate of the whistler anisotropy instability, resulting in faster damping of the energy in this perpendicular direction, and thus a small reduction of the growth rate of the wavevector anisotropy tan 2 B . While turbulence produce sharp gradients and local enhanced anisotropies, the whistler anisotropy instability acts to reduce electron temperature anisotropy through wave-particle scattering. The results suggest that the fluctuation from the growth of a single mode of whistler anisotropy instability is more efficient in wave-particle scattering of the electrons than that of a spectrum of whistler modes. Thus, a larger initial global electron temperature anisotropy, when combined with enhanced local electron temperature anistropy, would lead to even stronger wave-particle scattering effects locally, thus leading to local temperature anisotropy reduction and a more homogeneous electron temperature landscape. In conclusion, we find that the overall effect of whistler anisotropy instability on a plasma with co-existing whistler turbulence and global electron temperature anisotropy is minor comparing to that of whistler turbulence. This is because the fluctuating energy associated with the narrowband whistler instability is dominated by that from the broadband whistler turbulence. However, as field fluctuations from the growth of a single instability mode A Three-dimensional Particle-in-Cell Simulation of Space Plasma Instability and Turbulence: Whistler Turbulence v.s. Whistler Anisotropy Instability 222 may be more efficient in the wave-particle scattering process than that from a spectrum of whistler modes, the whistler instability can significantly reduces the intermittency of electron temperature anisotropy generated by turbulence. This suggests that microinstability may act as a regulation mechanism on turbulence development. In this study, the whistler instability is generated by imposing a bi-Maxwellian electron velocity distribution as the initial condition. The competing consequences of whistler turbulence cascade and whistler anisotropy instability growth will need to be further evaluated in a more realistic setup where the instability develops naturally from turbulence in future study.

Abstract (if available)

Abstract Many kinetic plasma physics problems in astronautical engineering and space physics require accurate solutions of the higher order moments of particle velocity distribution or small amplitude micro-scale plasma perturbations. Such problems often can not be adequately resolved by the commonly used Particle-in-Cell (PIC) method due to the inherent statistical noise. This dissertation develops a fully kinetic grid-based Vlasov simulation model for kinetic plasma simulations. This simulation model directly solves the velocity distribution function at every physical space location through phase space discretization without the interference of particle noise. The Vlasov algorithm employs the semi-Lagrangian Positive and Flux Conservation scheme, and is integrated with an electrostatic Poisson solver utilizing the velocity distribution function for charge density computation and a Darwin electromagnetic field solver utilizing the first three orders of Vlasov moments equations. The simulation code is implemented on parallel supercomputers using multi-dimensional domain decomposition in the physical space and Message Passing Interface (MPI). Weak scaling tests show the code runs at over 75% parallel efficiency in electrostatic simulations with 360 cores and 80% parallel efficiency in electromagnetic simulations with 1280 cores, demonstrating an excellent scalability of the code.

Electrostatic Vlasov simulation studies are carried out to investigate first electron-scale physics in one-dimensional collisionless plasma expansion and then electron thermodynamics in two-dimensional electric propulsion beam emission. Simulation results on plasma expansion show that the expansion generates both an ion-acoustic rarefaction wave mode and an electron Langmuir wave mode that propagate into the unperturbed plasma upstream, and that the assumption used in the classical expansion solution that the electrons are an isothermal fluid is accurate within a quasi-neutral, self-similar expansion region but fails in both the upstream and downstream of that region due to electron timescale perturbations. Simulation results on plasma beam emission demonstrate the significant role from the anisotropic 3rd order electron velocity moment, i.e. the collisionless electron heat flux, on beam expansion. Two distinct regions in the beam, an electron trapped region and a self-similar expansion region, are identified using the on-axis flux-limited coefficient. A connection is also established between the coefficient in the flux-limited model of electron collisionless heat flux and the features of the electron velocity distribution function, which can be utilized to develop a macroscopic electron heat flux model incorporating the correct microscopic electron kinetic effects in the future.

Darwin Vlasov simulation studies are carried out on whistler wave propagation and Weibel instability to demonstrate the capability of the Vlasov method on modeling kinetic plasma processes. Simulation results show that the Darwin-Vlasov simulation model is advantageous over electromagnetic PIC in resolving the effects from small amplitude waves and slow growing microinstabilities because it eliminates the interference of particle noise and numerical heating in PIC, and can be applied as an efficient tool in future simulation studies of kinetic range plasma turbulence.

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Creator Cui, Chen (author)

Core Title Grid-based Vlasov method for kinetic plasma simulations

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Astronautical Engineering

Degree Conferral Date 2023-12

Publication Date 09/11/2023

Defense Date 07/18/2023

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag collisionless plasma expansion,electric propulsion beam emission,electromagnetic plasma waves/instabilities,grid-based Vlasov method,high-performance computing,kinetic plasma simulations,OAI-PMH Harvest

Format theses (aat)

Language English

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Advisor Wang, Joseph (committee chair), Gruntman, Mike (committee member), Nakano, Aiichiro (committee member)

Creator Email cuichen@usc.edu,cuichenbuaa@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113305032

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