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Particle-in-cell simulations of kinetic-scale turbulence in the solar wind
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Particle-in-cell simulations of kinetic-scale turbulence in the solar wind
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PARTICLE-IN-CELL SIMULATIONS OF KINETIC-SCALE TURBULENCE IN THE SOLAR WIND by R. Scott Hughes 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) August 2017 Copyright 2017 R. Scott Hughes ii Epigraph Epigraph If I have seen further, it is by standing on the shoulders of giants. – Isaac Newton, 15 February 1676 Abstract iii Abstract Plasma turbulence is defined as an ensemble of broadband, large amplitude, stochastic fluctuations in a hot, tenuous ionized medium. Turbulence is ubiquitous in the solar wind, and encompasses a hierarchical structure which ranges over many orders of magnitude in space and time. While it is believed that turbulence plays a key role in energy transport in the solar wind, the behavior of turbulent plasma on scales much smaller than the mean free path remains poorly understood. An important question, which has been the topic of debate in the heliophysics community for decades, addresses the nature of the fluctuations which compose small scale turbulence. If small scale turbulence can be roughly thought of as the superposition of a broad spectrum of normal mode fluctuations, which mode dominates the behavior of the system, and what are the relative contributions of the various possible modes to the dissipation of turbulent energy into heat? Two modes are of particular interest in the study of solar wind turbulence: the kinetic Alfvén wave, and the magnetosonic-whistler wave. The investigations presented here focus on characterizing the effectiveness of magnetosonic-whistler turbulence and kinetic Alfvén turbulence in the heating of protons and electrons in the solar wind. To address this topic large scale, fully kinetic, three-dimensional (3-D) electromagnetic (EM) particle-in-cell (PIC) simulations are carried out on a collisionless, homogeneous, magnetized, ion-electron plasma model. In this work, four major studies are conducted to investigate ion and electron heating by kinetic-scale plasma turbulence. The first three studies employ the 3-D EMPIC model in order to investigate ion and electron heating by whistler turbulence. The simulations use an initial ensemble of relatively long wavelength whistler modes with a broad range of initial propagation directions and parametrically vary initial physical properties of the system in order to obtain scaling laws for the maximum heating rates achieved by each plasma species. In the first study three simulations are performed corresponding to successively larger simulation boxes and successively longer wavelengths of the initial fluctuations. The computationsconfirmpreviousresultsshowingelectronheatingispreferentiallyparallel iv Abstract to the background magnetic field B 0 , and ion heating is preferentially perpendicular to B 0 . The significant results are that larger simulation boxes and longer initial whistler wavelengths yield stronger ion heating, weaker electron heating, and weaker overall dissipation. In the second study an ensemble of five simulations is performed with all parameters held fixed but the dimensionless fluctuating magnetic field energy density, 0 . The important results here are that, over 0:01 < 0 < 0:25, the maximum rate of electron heating (Q e ) scales approximately as 0 , and the maximum rate of ion heating (Q i ) scales approximately as 1:5 0 . The third study carries out two ensembles of simulations with all parameters held fixed but the electron beta, e . In Ensemble One, where each simulation began with 0 = 0:1, bothQ e andQ i scale approximately as 1 e , whereas over 0:1 e 1:0 in Ensemble Two, where each simulation was initialized such that 0 = 0:2 e , Q e is approximately constant while Q i scales approximately as 1=2 e . The results of these three studies suggest that sufficiently long wavelength and sufficiently large-amplitude magnetosonic-whistler turbulence at sufficiently large e may heat ions more rapidly than electrons. The fourth study of this work moves beyond investigations of whistler turbulence to investigate the role of kinetic Alfvén waves in the physics of kinetic scale dissipation of turbulence in the solar wind. Maximum electron and ion heating rates due to kinetic Alfvén turbulence are computed as functions of 0 . In contrast to the results for heating by whistler turbulence, the maximum ion heating rate due to kinetic Alfvén turbulence is substantially greater than the maximum electron heating rate. Furthermore, both ion and electron heating due to kinetic Alfvén turbulence scale approximately with 0 . Finally, electron heating leads to anisotropies of the type T ek > T e? , where the subscripts refer to directions parallel and perpendicular, respectively, to B 0 , whereas the heated ions remain approximately isotropic. A final study is conducted using the 3-D EMPIC model to investigate the whistler anisotropy instability (WAI) in three dimensions. Analysis of the magnetic field spectra show that the fluctuations generated by the WAI grow as relatively short- wavelength, quasi-parallel propagating whistler modes, as is predicted by linear dispersion theory. At late times, however, the field energy is transferred from small to large wavelength modes and forms a spectrum of fluctuations that exhibit quasi- perpendicular propagation. This inverse transfer process is reminiscent of an inverse cascade in a fully turbulent plasma. Entropy analysis is performed in order to verify that such a process obeys the second law of thermodynamics. Acknowledgments v Acknowledgments I am eternally grateful to the large number of friends, family, mentors and colleagues who helped me along my path toward earning my Ph.D.; only with their help, their advice, their inspiration and their support was I capable of pushing forward and stretching my limits farther than I ever thought was possible. I would first like to thank my advisors, Dr. Joseph Wang and Dr. S. Peter Gary. I was fortunate enough to have two advisors who both played essential roles in providing me opportunities and elevating me to the professional level in the science community. I am grateful to Dr. Misa Cowee, the director of the Los Alamos Space Weather Summer School, who put every effort into ensuring that the attending students were exposed to state of the art research in field of heliophysics and the near-Earth space environment. I am thankful to Dr. Viktor Decyk; I learned a great deal about the Particle-In-Cell (PIC) method, as well as efficient programming practices, through our discussions and the source code he was kind enough to share with me. I thank Dr. Tulasi Parashar, who helped me with my final investigation as a Ph.D. student and opened the door for the students who will continue my work. I owe a great deal to Dr. Ouliang Chang who helped introduce me to the topics of plasma turbulence and high performance simulation. His efforts to bring me up to speed significantly reduced the steep learning curves which are inherent in these complex topics. I would like to thank my dissertation committee for their useful help and suggestions; Dr. Mike Gruntman, Dr. Dan Erwin, Dr. Aiichiro Nakano, and Dr. Joseph Kunc. A highlight for me was learning from their expertise through the discussion and debate which took place during my qualifying exam and defense. I would like to thank my colleagues; Dr. Daoru Han, Dr. Vaibhav Gupta, Will Yu, Kevin Chou, Yuan Hu, and Guna Surrendra. Their support kept me motivated and our discussions over the years brought to light ideas that would have otherwise remained in the shadows. I am grateful for all of the hard work of the staff of the Astronautical Engineering department; Dell Causon, Linda Ly, Hayley Peltz, Marlyn Lat, Norma Orduna, Marrietta Penoliar, and Ana Olivares. Only through their efforts vi Acknowledgments can the department function and can the students thrive. I would also like to thank Dr. Mengu Cho and Dr. Arifur Khan of the Kyushu Institute of Technology. They were essential in my training as a Ph.D. researcher during my first year as a graduate student. They provided me the opportunity to work with state of the art equipment which helped allow me to gain a greater physical understanding of plasma physics phenomena. Finally, I would like to thank my family; Randy, Susan, Ryan, and Tobin. Through- out my academic career they have provided unconditional support to me in any way I may have needed it. Nothing is achieved alone, and this accomplishment belongs to all those who helped me along the way. The research presented here was supported largely by the National Science Foun- dation (NSF) grant AGS-1202603 under the NSF/DOE Partnership in Basic Plasma Science and Engineering. R.S.H. was also supported by the Los Alamos Space Weather Summer School, funded by the Institute of Geophysics, Planetary Physics, and Sig- natures (IGPPS) at Los Alamos National Laboratory. Computational resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center, as well as by the Yellowstone supercomputer sponsored by the NSF at the Computational and Information Systems Laboratory of the National Center for Atmospheric Research. TABLE OF CONTENTS vii Table of Contents Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii Chapter 1:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Turbulence in the Inertial Range . . . . . . . . . . . . . . . . . . . . 3 1.3 Turbulence in the Kinetic Range . . . . . . . . . . . . . . . . . . . . . 6 1.4 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Past Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 2:Numerical Modeling . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Particle-In-Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Applying the Electromagnetic Model to Particle-In-Cell . . . . . . . . 27 2.3 Applying the Darwin Model to Particle-In-Cell . . . . . . . . . . . . . 28 2.3.1 The Darwin Field Equations . . . . . . . . . . . . . . . . . . . 28 2.3.2 Scattering to the Mesh . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Solving on the Mesh . . . . . . . . . . . . . . . . . . . . . . . 32 viii TABLE OF CONTENTS 2.3.4 Resolving Force/Acceleration Coupling . . . . . . . . . . . . . 35 2.4 Advancing Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.1 Depositing Normalized Densities . . . . . . . . . . . . . . . . . 42 2.6 Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 3:Numerical Validation . . . . . . . . . . . . . . . . . . . . . . 44 3.1 The Electron Thermal Velocity . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.2 Linear Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.5 Effects of Particle Noise . . . . . . . . . . . . . . . . . . . . . 53 3.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Ion and Electron Heating . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2 Validation of Parameters: Numerical Heating . . . . . . . . . 67 3.2.3 Validation of Parameters: Ion to Electron Mass Ratio . . . . . 68 Chapter 4:Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength . . . . . . . . . . . . . . . . . . 72 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Chapter 5:Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density . . . . . . . . 91 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 6:Ion and Electron Heating by Whistler Turbulence: Variations with Electron . . . . . . . . . . . . . . . . . . . 108 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 TABLE OF CONTENTS ix 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Chapter 7: Kinetic Alfvén Turbulence . . . . . . . . . . . . . . . . . . 121 7.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Chapter 8:The Whistler Anisotropy Instability: Inverse Spectral Transfer . . . . . . . . . . . . . . . . . . . 140 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.4 Three-Wave Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Chapter 9:Summary and Conclusions . . . . . . . . . . . . . . . . . . . 157 9.1 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 9.2 Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.3 Study 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 9.4 Study 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.5 Study 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.6 Study 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 A Ion and Electron Maximum Heating Rates . . . . . . . . . . . . . . . 180 B Electromagnetics and Plasmas . . . . . . . . . . . . . . . . . . . . . . 183 B.1 Reference Tables . . . . . . . . . . . . . . . . . . . . . . . . . 183 B.2 Useful EM Relations in Gaussian Units . . . . . . . . . . . . . 189 B.3 Maxwell and Darwin Potential Fields . . . . . . . . . . . . . . 190 B.3.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . 191 B.3.2 Darwin Potential Equations . . . . . . . . . . . . . . 193 B.4 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . 196 x TABLE OF CONTENTS B.4.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.4.2 Additional Diagnostic Methods . . . . . . . . . . . . 199 C Parallel FFT on a 3-D Decomposed Array of Vector Data . . . . . . . 201 C.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 201 C.2 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 202 C.3 Divide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 C.4 Combine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 C.5 FFT on Real Data . . . . . . . . . . . . . . . . . . . . . . . . 204 C.6 FFT In Three Dimensions . . . . . . . . . . . . . . . . . . . . 206 C.7 Serial 3DFFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 C.8 Parallel FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 C.9 Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 C.10 Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 C.11 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . 210 C.12 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 C.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 LIST OF FIGURES xi List of Figures 1.1 Artists rendition of the solar wind flowing outward from the Sun and interacting with the Earth’s magnetosphere. . . . . . . . . . . . . . . 2 1.2 Magnetic structure of the solar wind showing variability over the 11 year solar cycle. Left: solar minimum, right: solar maximum. Image source: http://solarprobe.gsfc.nasa.gov/solarwind.jpg. . . . . . . . . . 3 1.3 Illustration depicting the hierarchical structure of turbulence and the forward cascade from large to small scales. . . . . . . . . . . . . . . . 4 1.4 Trace of the spectral matrix of magnetic fluctuation energy, illustrating the power law nature of magnetic fluctuations in the inertial range, 10 4 .f. 10 1 Hz, at parallel and perpendicular propagation. The spectrum at f. 10 4 Hz corresponds to the energy injection regime. Image source: [Alexandrova et al., 2013]. . . . . . . . . . . . . . . . . 7 1.5 A proposed process of energy transfer and dissipation from the inertial range to the electron scale. Image source: [Shaikh and Zank, 2009]. . 10 3.1 Solutions to the electromagnetic linear dispersion equation as functions of k k at k ? = 0. For all cases R T = 3. Solid lines represent the real frequencies, ! r , and dashed lines represent the growth rates, . The growth rates have been scaled for visual clarity and the scaling factors, F, are displayed. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:01; blue lines, ^ v te = 0:005. (a) e =! e = 0:0224, e is varied. (b) e = 0:1, e =! e is varied. . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Wavevector spectra of magnetic fluctuations at three simulation times, and varying ^ v te . The initial conditions in each case areR T = 3, e = 0:1. (a) ^ v te = 0:1, (b) ^ v te = 0:05, (c) ^ v te = 0:025. The vertical dashed lines bound the region predicted by Fig. 3.1 to give rise to fluctuation growth at parallel/antiparallel propagation. . . . . . . . . . . . . . . . . . . . 60 3.3 Magnetic fluctuation dispersion; ! r versus k k at k ? = 0 calculated over the simulation time range 50:3t e 167:7. The initial conditions in each case are R T = 3, e = 0:1. The black curves give the solution to the electromagnetic linear dispersion equation for R T = 3 under the realistic conditions e = 0:1, ^ v te = 0:005. (a) ^ v te = 0:1, (b) ^ v te = 0:05, (c) ^ v te = 0:025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 xii LIST OF FIGURES 3.4 Temporal characterization of the WAI. The initial conditions in each case are R T = 3, e = 0:1. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. (a) Fluctuating field energy densities (F 2 ): solid lines, magnetic energy; dotted lines, electrostatic energy. The dashed lines are fitted to the linear growth period of the magnetic fluctuations, with parameters as given in Table 3.1. (b) Temperature anisotropy ratio R T . . . . . . . . . . . . . . . . . . . . . 62 3.5 Fluctuating field quantities as functions of the number of particles per cell, N c . In each case ^ v te = 0:1; all other parameters are the same as in Sec. 3.1.3. Stars correspond to data points and each curve is fitted to each respective data set. The fitting equations are shown along with the curves. The quantities presented are: red, initial magnetic fluctuation energy; green, maximum magnetic fluctuation energy; blue, time-averaged electrostatic fluctuation energy; magenta, linear growth rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Temporal characterization of the WAI for simulations run with initial conditions R T = 3, e = 0:1, and N c = 36. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. (a) Fluctuating field energy densities (F 2 ): solid lines, magnetic energy; dotted lines, electrostatic energy. The dashed lines are fitted to the linear growth period of the magnetic fluctuations, with parameters as given in Table 3.2. (b) Temperature anisotropy ratio R T . . . . . . . . . . . . . . . . 64 3.7 Fluctuating field energy densities (F 2 ) at varying ^ v te in which the number of particles per square inertial length, n e 2 e , is held fixed. All other parameters are the same as in Sec. 3.1.3. Solid lines, magnetic energy; dotted lines, electrostatic energy. Red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. . . . . . . . . . . . . . . . . . 65 3.8 System component energies in a simulation with m i =m e = 400, e = 0:25, and 0 = 0. The components are: total ion thermal energy, W i , total electron thermal energy, W e , total ion plus electron energy, W i+e , magnetic field energy, K B , electric field energy, K E , total electromag- netic field energy, K B+E , and total field plus plasma energy, K total . (a) The temporal evolution of the equilibrium plasma energies with a Gaussian filter applied to the current density field at each time step, (b) the same simulation with no Gaussian filter applied. . . . . . . . 70 3.9 The red dots represent the ratio of the maximum value of the ion heating rate to the maximum value of the electron heating rate found during each simulation as a function of the ratio of the ion mass to the electron mass for the e = 0:25, 0 = 0:1, andm i =m e = 400 simulations. The dashed line represents the equation Q i;max =Q e;max = 35 (m e =m i ) drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 LIST OF FIGURES xiii 4.1 Reduced magnetic fluctuation energy spectra at three times as labeled forthethreesimulationswithdifferentsimulationboxsizes: (a)L! e =c = 25:6, (b) L! e =c = 51:2, and (c) L! e =c = 102:4. . . . . . . . . . . . . . 81 4.2 Simulation histories of the (a) parallel electron, (b) perpendicular electron, (c) parallel ion, and (d) perpendicular ion temperatures as functions of time from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2 Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Simulation histories of the (a) electron temperature, and (b) ion tem- perature averaged over the three spatial dimensions; (c) electron tem- perature anisotropy, and (d) ion temperature anisotropy defined as T j?=k T j? =T jk . The curves correspond to the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). . . . . . . . . . . . . . 84 4.3 Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Simulation histories of the total magnetic field fluctuation energy as functions of time from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Simulation histories of the (a) electron total heating rate, and (b) ion totalheatingrate,whereheatingisdefinedhereasQ j (dT j =dt)= [T j (t = 0)! e ]. The curves correspond to the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). . . . . . . . . . . . . . . . . . . . . . 87 4.6 The maximum values of the dimensionless time rates of change of the ion total temperature (red dots), and the electron total temperature (blue dots) as functions of L! e =c for the e = 0:05, 0 = 0:05 and m i =m e = 400 simulations. The dashed lines represent the equations Q i = (2:5L! e =c + 700) 10 7 and Q e = 45(L! e =c) 5=2 drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.7 Reduced electron velocity distributions from the three simulations at (a) t! e = 0 (b)t! e = 1000, and (c)t! e = 2000 as labeled for (left column) parallel and (right column) perpendicular component velocities. Results are from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells andL! e =c = 51:2 (green lines), and 1024 3 cells andL! e =c = 102:4 (blue lines). Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 xiv LIST OF FIGURES 4.8 Reduced ion velocity distributions from the three simulations at (a) t! e = 0 (b)t! e = 1000, and (c)t! e = 2000 as labeled for (left column) parallel and (right column) perpendicular component velocities. Results are from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells andL! e =c = 51:2 (green lines), and 1024 3 cells andL! e =c = 102:4 (blue lines). Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1 Temporal evolution of field quantities from the e = 0:25 andm i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustrated by the various panels are: (a) the total magnetic plus electric field fluctuation energy, K B+E , and (b) the wavevector anisotropy factor, Tan 2 ( B ), as defined by Eq. (1.10). The squares and circles represent the times of maximum heating rates for the electrons and ions, respectively. . . . . . . . . . . . . . . . . . . . 101 5.2 Temporalevolutionofplasmacomponenttemperaturesforthe e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustrated by the various panels are: (a) the electron parallel temperatures, (b) the electron perpendicular temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Temporalevolutionofplasmacomponenttemperaturesforthe e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustrated by the various panels are: (a) the ion parallel temperatures, and (b) the ion perpendicular temperatures.103 5.4 Temporal evolution of (a) electron and (b) ion total heating rates for the e = 0:25andm i =m e = 400simulationswith 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. Here (a) represents Q e and (b) shows Q i . . . . . 104 5.5 The maximum values of the dimensionless time rates of change of the ion total temperature (red dots), and the electron total temperature (blue dots) as functions of 0 for the e = 0:25 and m i =m e = 400 simulations. The dashed lines represent the equations Q i = 0:001 1:5 0 and Q e = 0:002 0 drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.3. . . . . . . . . . . . . . . . 105 5.6 Temporal evolution of (a) electron and (b) ion species entropy for the e = 0:25andm i =m e = 400simulationswith 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. Here (a) represents S e and (b) shows S i . . . . . . 106 5.7 Temporal evolution of total (ion plus electron) system entropy for the e = 0:25 and m i =m e = 400 simulations. . . . . . . . . . . . . . . . . 107 LIST OF FIGURES xv 6.1 Reduced velocity distributions for selected simulation times from the simulation with the initial conditions e = 0:5 and 0 = 0:10, which is a contributor to both Ensemble One and Ensemble Two. Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. (a) f e (v k ), (b) f e (v ? ), (c) f i (v k ), (d) f i (v ? ). . . . . . . . . . . . . . . . . . 117 6.1 Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Individual points represent results from Ensemble One. The maxi- mum values of the dimensionless time rates of change of the total ion temperatures (Q i ) and the total electron temperatures (Q e ) are plotted as functions of the initial value of e . The blue dashed line is Q e = 5:0( 1 e ) 10 5 and the red dashed line is Q i = 7:0( 1 e ) 10 6 ; both are drawn to guide the eye along the simulation data. The numer- ical values of the data presented are tabulated in Table A.4. . . . . . 119 6.3 Individual points represent results from Ensemble Two. The maxi- mum values of the dimensionless time rates of change of the total ion temperatures (Q i ) and the total electron temperatures (Q e ) are plotted as functions of the initial value of e . The blue dashed line is Q e = 1:0 10 4 and the red dashed line is Q i = 1:5( 1=2 e ) 10 5 ; both are drawn to guide the eye along the simulation data. The numerical values of the data presented are tabulated in Table A.5. . . . . . . . . 120 7.1 Reduced magnetic fluctuation energy spectra of KAW turbulence for the 0 = 0:50 simulation. The top panels show the initially loaded spectrum in the k x k y (perpendicular) plane and in the k y k z (perpendicular- parallel) plane; the parallel direction has been rescaled for improved visualization. The bottom panels show the same spectrum at a later time, t! e = 2400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Reduced 1-D magnetic fluctuation energy spectra of KAW turbulence as a function of perpendicular wavenumber for the 0 = 0:05; 0:1; 0:25; and 0:50 simulations. The black curves show the power law scaling of the fluctuation energy as a function of perpendicular wavenumber above and below the electron-scale spectral break. The time here is t i = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.3 The temporal evolution of field and plasma component energies for the kinetic Alfvén turbulence simulation with initial condition 0 = 0:50. The components are: the total ion thermal energy density, W i , the total electron thermal energy density, W e , the total ion plus electron energy density,W i+e , the magnetic field energy density,K B , the electric field energy density, K E , the total electromagnetic field energy density, K B+E , and the total field plus plasma energy density, K total . . . . . . 133 xvi LIST OF FIGURES 7.4 Reduced velocity distributions from the kinetic Alfvén turbulence simu- lation with initial condition 0 = 0:50. The blue lines represent velocity distributions at t = 0, and the red lines represent velocity distributions at t! e = 480; the black dotted lines are Maxwellian distributions with equivalent standard deviations as the respective late time velocity dis- tributions. The left-hand axis labels correspond to the solid lines, and the right-hand axis labels correspond to the dashed lines. (a) f e (v k ), (b) f e (v ? ), (c) f i (v k ), (d) f i (v ? ). . . . . . . . . . . . . . . . . . . . . . 134 7.4 Continued. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.5 The temporal evolution of plasma component temperatures for the kinetic Alfvén turbulence simulation with 0 = 0:50. The temperature curves are normalized by the respective initial temperature value. The red curve is parallel temperature, the green curve is perpendicular temperature, the yellow curve is total temperature, and the blue curve is temperature anisotropy factor T j? =T jk . . . . . . . . . . . . . . . . . 136 7.6 Temporal evolution of the (a) electron and (b) ion system entropy for the four kinetic Alfvén turbulence simulations as labeled. . . . . . . . 137 7.7 Temporal evolution of the (a) electron and (b) ion total heating rates for the four kinetic Alfvén turbulence simulations as labeled. . . . . . 138 7.8 The maximum values of the dimensionless time rates of change, Q j;max , of the ion total temperature (red circles) and the electron total tem- perature (blue squares) as functions of the initial magnetic fluctuation energy density of the four kinetic Alfvén turbulence simulations. The blue dashed line representsQ e;max (whistler) = 0:001 0 which is found by Eq. (6.3) to be the approximate scaling for maximum electron heating due to whistler turbulence at e = 0:50. The numerical values of the data presented are tabulated in Table A.6. . . . . . . . . . . . . . . . 139 8.1 Simulation results: time histories of (a) the normalized fluctuating mag- netic field energy density and (b) the electron temperature anisotropy (blue), the normalized parallel (red) and perpendicular (green) electron temperatures, and the “total” electron temperature (yellow), which is the temperature averaged over the three spatial dimensions. The initial values of the simulation are ek = 0:10 and T e? =T ek = 3:0. . . . . . . . 150 8.2 Reduced Electron velocity distributions for several simulation times as labeled. Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. (a) f e (v k ), (b) f e (v ? ). . . . . . . . . . . . . . . 151 8.3 Simulation results: Reducedk k k y magnetic fluctuation energy spectra jB(k k ;k y )j 2 =B 2 0 at 4 times as labeled. . . . . . . . . . . . . . . . . . 152 LIST OF FIGURES xvii 8.4 Time histories of (a) magnetic fluctuation energy density and (b) mag- netic fluctuation wavevector anisotropy factor Tan 2 ( B ). Red lines represent results summed over long wavelengths, green lines represent results summed over intermediate wavelengths, and blue lines represent results summed over short wavelengths with regions as defined in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.5 Magnetic fluctuation dispersion, real frequency ! r versus k k at a con- stant k ? = 0, calculated over the range 250<t! e < 1750. The lines overlaying the calculated dispersion of the simulation are numerical solutions to the full kinetic linear dispersion equation at frequencies corresponding to whistler fluctuations at e = 0:1. The purple curve is found under the condition T e? =T ek = 3:0, representing the initial state of the simulated system, while the blue curve is found at T e? =T ek = 2:0, representing the state of the simulated system at t! e 1750. . . . . . 154 8.6 Entropy of protons and electrons as a function of time during the evolu- tion of the whistler anisotropy instability. Two methods of calculating entropy are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.7 Linear dispersion theory results: Real frequency (solid lines) and damp- ing/growth rate (dotted lines) as functions of the wavenumber for the whistler mode and the whistler anisotropy instability at parallel propagation ( = 0 ) for three values of T e? =T ek as labeled. . . . . . . 156 8.8 Theory results: The frequency mismatch ! for whistler fluctuations over 0:20kc=! e 1:0 as a function of propagation angle for three values of T e? =T ek as labeled. . . . . . . . . . . . . . . . . . . . . . . . 156 B.1 Entropy of electrons as a function of time during the evolution of the whistler anisotropy instability in the 3D-EMPIC model. Two methods of calculating entropy are compared. . . . . . . . . . . . . . . . . . . 199 C.1 Butterfly technique applied to an FFT of length 8 . . . . . . . . . . . 203 C.2 Illustration of a communication group shuffling local data . . . . . . . 210 C.3 Compute time with increasing domain size. Number of processors fixed at N 3 proc = 512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 C.4 Speedup, comparing strong scaling with weak scaling. Strong scaling used a 1024 3 domain. Weak scaling used a fixed sized of 64 3 elements per processor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 C.5 Efficiency, comparing strong scaling with weak scaling. Strong scaling used a 1024 3 domain. Weak scaling used a fixed sized of 64 3 elements per processor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 xviii LIST OF TABLES List of Tables 3.1 Parameters for the curve fits shown in Fig. 3.4(a). Curves are fitted to the function B 2 =B 2 e 2 t +B 2 over the time period 0tt nl . is the linear growth rate, B 2 is the growing fluctuation energy, B 2 is non-resonant fluctuation energy, and t nl is the approximate time that the instability begins to saturate. . . . . . . . . . . . . . . . . . . . . 63 3.2 Parameters for the curve fits shown in Fig. 3.6(a). For a description of the parameters, see Table 3.1. . . . . . . . . . . . . . . . . . . . . . . 64 A.1 Whistler turbulence ion and electron maximum heating rates of Chapter 3.2.3, Figure3.9. L! e =c = 102:4,v te =c = 0:10, i = e = 0:25, 0 = 0:10, variable: m i =m e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 A.2 Whistler turbulence ion and electron maximum heating rates of Chapter 4, Figures 4.5 and 4.6. m i =m e = 400, v te =c = 0:10, i = e = 0:05, 0 = 0:10, variable: Domain Length. . . . . . . . . . . . . . . . . . . 180 A.3 Whistler turbulence ion and electron maximum heating rates of Chapter 5, Figures 5.4 and 5.5. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e = 0:25, variable: 0 . . . . . . . . . . . . . . . . . . . . . . . . 181 A.4 Whislter turbulence ion and electron maximum heating rates of Chapter 6, Figure 6.2. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e , 0 = 0:10, variable: e . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 A.5 Whislter turbulence ion and electron maximum heating rates of Chapter 6, Figure 6.3. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e , 0 = 0:20 e , variable: e . . . . . . . . . . . . . . . . . . . . . . . . . . 182 A.6 KAW turbulence ion and electron maximum heating rates of Chapter 7, Figures 7.7 and 7.8. L k ! e =c = 655:4, L ? ! e =c = 81:9, m i =m e = 100, v te =c = 0:32, i = e = 0:50, variable: 0 . . . . . . . . . . . . . . . . . 182 LIST OF TABLES xix B.1 Conversion factors, F, for converting formulae between Gaussian and SI forms. A parameter in Gaussian units, X Gs , relates to its SI form, X SI , by X Gs =FX SI . To convert from Gaussian (SI) to SI (Gaussian) replace each X Gs (X SI ) in the formula with FX SI (X Gs =F). Use the relation c = 1= p 0 " 0 where applicable. The value F for converting a numeric quantity from SI to Gaussian units can be calculated by absorbing the dimensions of the parameter X SI into the dimensions of its corresponding F (in SI), then replacing all (kg, m) dimensions with (1000g, 100cm). F will then have the correct numeric conversion value and Gaussian dimensionality. The inverse process gives an inverse conversion value of 1=F with the correct SI dimensionality. . . . . . . 183 B.2 EM parameters in SI units. . . . . . . . . . . . . . . . . . . . . . . . . 184 B.3 EM parameters in Gaussian units. . . . . . . . . . . . . . . . . . . . . 185 B.4 Physical Constants in SI . . . . . . . . . . . . . . . . . . . . . . . . . 185 B.5 Additional Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 186 B.6 Plasma parameters in Gaussian units. Each property is defined for the correspondingj-th plasma species;j =e for electronsj =p for protons and j = i for ions. When the Debye length does not specify j, the electron Debye length is assumed. . . . . . . . . . . . . . . . . . . . . 186 B.7 Useful dimensionless plasma parameter ratios. Here it is assumed that n e =n i =n 0 . ES denotes electrostatic, EM denotes electromagnetic. . 187 B.8 Characteristic solar wind parameters at 1 AU. Source for measured parameters [Bruno and Carbone, 2013]. Note that all quoted values are order-of-magnitude approximations, due to the substantial variability of solar wind quantities. . . . . . . . . . . . . . . . . . . . . . . . . . 188 xx LIST OF SYMBOLS List of Symbols k Wavenumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Power law index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ? Perpendicular with respect to the background magnetic field . . . . . 5 B 0 Background magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 5 k Parallel with respect to the background magnetic field . . . . . . . . 5 f Cyclic frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 f j j-th species distribution function . . . . . . . . . . . . . . . . . . . . 6 V A Alfvén speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 V sw Solar wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 c Speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 j j-th species inertial length . . . . . . . . . . . . . . . . . . . . . . . . 8 ! j j-th species plasma frequency . . . . . . . . . . . . . . . . . . . . . . 8 j j-th species thermal gyroradius . . . . . . . . . . . . . . . . . . . . . 8 v tj j-th species thermal speed . . . . . . . . . . . . . . . . . . . . . . . . 8 j j-th species gyrofrequency . . . . . . . . . . . . . . . . . . . . . . . . 8 j j-th species beta parameter . . . . . . . . . . . . . . . . . . . . . . . 8 K B Boltzmann constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 n 0 Particle number density . . . . . . . . . . . . . . . . . . . . . . . . . 14 LIST OF SYMBOLS xxi 0 Dimensionless magnetic fluctuation energy density . . . . . . . . . . 14 Angle of wave propagation w.r.t. B 0 . . . . . . . . . . . . . . . . . . 14 B Wavevector anisotropy factor . . . . . . . . . . . . . . . . . . . . . . 15 ! Angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ! r real frequency component . . . . . . . . . . . . . . . . . . . . . . . . 15 imaginary frequency component . . . . . . . . . . . . . . . . . . . . . 15 T j j-th species temperature . . . . . . . . . . . . . . . . . . . . . . . . . 15 Q j j-th species heating rate . . . . . . . . . . . . . . . . . . . . . . . . . 15 dt Computational time step . . . . . . . . . . . . . . . . . . . . . . . . . 16 Mesh spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 E Electric field fluctuation amplitude . . . . . . . . . . . . . . . . . . . 21 B Magnetic field fluctuation amplitude . . . . . . . . . . . . . . . . . . 21 Charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 J Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 D Debye length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ^ v te Electron thermal velocity normalized by the speed of light . . . . . . 25 R T Temperature anisotropy ratio . . . . . . . . . . . . . . . . . . . . . . 45 N c Number of particles per cell . . . . . . . . . . . . . . . . . . . . . . . 49 W j j-th species total kinetic energy density . . . . . . . . . . . . . . . . 125 K E Fluctuating electric field energy density . . . . . . . . . . . . . . . . 125 K B Fluctuating magnetic field energy density . . . . . . . . . . . . . . . 125 xxii Nomenclature Nomenclature Acronyms AU Astronomical Unit -D -Dimensional DFT Discrete Fourier Transform EM Electromagnetic ES Electrostatic FFT Fast Fourier Transform FDTD Finite Difference Time Domain IMF Interplanetary Magnetic Field KAW Kinetic Alfvén Wave MHD Magnetohydrodynamics MSW Magnetosonic Wave PIC Particle-in-Cell SW Solar Wind WAI Whistler Anisotropy Instability WHW Whistler Wave 1 Chapter 1: Introduction Plasma turbulence is defined as an ensemble of broadband, large amplitude, stochastic field fluctuations in a hot, tenuous, ionized medium. In the solar wind, turbulent fluctuations span orders of magnitude in both spatial and temporal scales. The primary drive of this dissertation is to improve the general understanding of turbulence in the solar wind on the kinetic scale. In this chapter, a brief background of this dissertation research is presented, past relevant studies are reviewed, and the specific objectives and contributions of this research are summarized. 1.1 The Solar Wind The solar wind is a variable stream of tenuous, magnetized plasma which is ejected from the Sun’s atmosphere and propelled outward in all directions. The solar wind engulfs the solar system and forms a distinct bubble in interstellar space known as the heliosphere. The characteristics of the solar wind can vary greatly as functions of heliocentric latitude and radial distance from the Sun as well as functions of characteristic time scales, such as the Sun’s rotation period of 24:5 days and the 11 year solar cycle. Figure 1.2 illustrates the contrast in solar wind features between the beginning of the solar cycle, solar minimum, and the peak of the cycle at solar maximum. The solar wind constitutes a highly dynamic system and many natural processes can occur which influence the transport of the mechanical and electromagnetic energy contained within that system. Characterizing the details of these processes, the conditions which may lead to their excitation, and the ensuing consequences on the surrounding environment is essential. Such knowledge can aid in anticipating how the solar wind will interact with the various bodies it encounters as it flows through the solar system; the body of principal importance being the Earth. 2 Introduction Figure 1.1: Artists rendition of the solar wind flowing outward from the Sun and interacting with the Earth’s magnetosphere. Of practical interest is the development of reliable forecasting models which, in conjunction with real-time data provided by spacecraft located upstream of the Earth, are needed to accurately predict how the approaching solar wind will interact with the Earth’s magnetospheric system upon arrival; such interactions strongly influence conditions in the near-Earth plasma environment. In many cases the solar wind will induce magnetospheric storms which drive an influx of high energy particles toward the inner magnetosphere. These particles have the potential to disrupt and damage relatively sensitive electronic systems onboard Earth-orbiting spacecraft. Reliable solar wind forecasting allows time for these systems to be switched to a protected state prior to the onset of magnetospheric storms, minimizing the risk of irreparable damage. A thorough understanding of the nature of energy transport processes in the solar wind contributes to the knowledge-base on which space weather forecasting models are developed. One process believed to play a pivotal role in energy transport in the solar wind is turbulence. 1.2 Turbulence in the Inertial Range 3 Figure 1.2: Magnetic structure of the solar wind showing variability over the 11 year solar cycle. Left: solar minimum, right: solar maximum. Image source: http://solarprobe.gsfc.nasa.gov/solarwind.jpg. 1.2 Turbulence in the Inertial Range On relatively large scales turbulence in a plasma shares many common features with that in a neutral fluid. A characteristic process in turbulence is known as forward cascade. The theory of this process in a neutral fluid, referred to as the K41 theory, was formulated by Andrey Kolmogorov [Kolmogorov, 1941; Biferale et al., 2003]. According to the K41 phenomenology turbulent structures are driven at an outer scale length L, also called the energy injection scale. These large-scale structures break apart to form 4 Introduction smaller structures, which subsequently break apart to form yet smaller structures. This process continues, in effect transferring fluctuation energy from the outer scale, L, to the inner scale, , where the fluctuations are dissipated as heat. An illustration of the forward cascade is provided in Fig. 1.3. The range of spatial scales which span L ` is referred to as the inertial range. In this range the forward cascade is assumed to proceed in a localized manner; the interactions leading to energy transfer occur only between adjacent scale lengths. In addition, the time required to transfer a given amount of energy from a length ` i to an adjacent length ` j < ` i , known as the eddy turn-over time, is predicted to monotonically decrease with decreasing`. For two scale lengths` i and` k ` i , several eddy turn-over times elapse at ` i over the course of one turn-over time at ` k . The result is that the processes occurring at ` i lose all knowledge of the details of energy transfer at ` k . This implies a universality such that the statistical description of the forward cascade does not depend on the mechanism driving the turbulence at the energy injection scale L. Figure 1.3: Illustration depicting the hierarchical structure of turbulence and the forward cascade from large to small scales. 1.2 Turbulence in the Inertial Range 5 K41 theory predicts that the energy content of the turbulent quantities in the inertial range scales with wavenumber, k, according to a power law E(k)/k with a power law index = 5=3. The power law nature of the energy spectrum indicates scale invariance such that the same statistical description of the forward cascade is applicable at all scale lengths within the inertial range. Many studies have shown [Horbury et al., 2005; Bruno and Carbone, 2013, and citations therein] that plasma turbulence in the solar wind exhibits a cascade process similar to that of neutral fluid turbulence with a universally established inertial range. Moreover, long wavelength turbulence in magnetized plasmas, such as the solar wind, exhibits a self-similar, scale invariant behavior suggesting that, within the inertial range, the K41 model can largely be applied to plasma turbulence. Magnetized plasmas, however, are complicated by their inherent anisotropic nature; the motions of the plasma particles are constrained in the plane perpendicular (?) to the background magnetic field, B 0 , while they have a greater freedom of motion parallel (k) to B 0 . The anisotropy of magnetized plasmas results in distinct power law descriptions of the energy spectra. The magnetic energy fluctuations follow E(k ? )/k 5=3 ? andE(k k )/k 2 k . In the perpendicular plane the energy content of the fluctuating field properties follows the Kolmogorov scaling. Moreover, observational analysis has demonstrated [Narita et al., 2010, 2011b] that the multiscale spectrum of turbulent fluctuations primarily forms in the perpendicular plane, with energy spectra satisfying k ? > k k . Figure 1.4 [Alexandrova et al., 2013] illustrates the power law nature of the magnetic fluctuation energy in the parallel and perpendicular directions in terms of frequency. Frequencies of f . 10 4 Hz correspond to the energy injection scale, while Frequencies of 10 4 .f. 10 1 Hz correspond to the inertial range. Spatialscalescoveredbytheinertialrangetypicallycontainenoughplasmaparticles per unit volume to permit the use of magnetohydrodynamic (MHD) theory [Biskamp, 1993; Schekochihin et al., 2009] in the modeling of plasma fluctuations of appropriate scale length. MHD is a fluid model defined in terms of bulk quantities of the plasma; 6 Introduction the quantities are obtained by averaging moments of the particle distribution functions f j (x; v;t) over velocity space. On MHD scales a particular plasma wave pervades, the Alfvén wave. The characteristic speed in MHD solar wind is the Alfvén speed, given by V A =B 0 = p 4 p , where p is the proton mass density. The solar wind is super-Alfvénic, with a typical Alfvén mach number of M A 10. Single-point spacecraft are incapable of obtaining spatial structure measurements of plasma fluctuations, rather they have access to frequency information. However, because of the super-Alfvénic nature of the solar wind the propagation speed of the Alfvén-like fluctuations is negligible and, by the Taylor hypothesis [Taylor, 1938], the fluctuations can be assumed as “frozen in” to the flow of the solar wind which passes over thespacecraft. This leadstoamethodfordirectconversionfrom frequencyspectra to wavenumber spectra. If the fluctuations are assumed to be static in a reference frame moving with the solar wind, then the wavenumber of one such fluctuation is simply given byk = 2f=V sw , whereV sw is the solar wind speed. This method enables analysis of spatial structure in the solar wind by using measurements taken from, roughly, a single point in space. 1.3 Turbulence in the Kinetic Range The kinetic scale covers relatively small scale lengths, where the physical processes in a plasma are dependent on both the position space and velocity space details of the plasma species (i.e., the ions and electrons). This is in contrast to the MHD scale, where averages can be taken over velocity space in order to characterize the plasma in terms only of position space, leading to a model which describes plasma as a magnetized fluid (i.e., MHD). Because the velocity space details of the plasma are required to describe kinetic range dynamics, the fluid approximation breaks down and the MHD model can no longer be applied within this regime. 1.3 Turbulence in the Kinetic Range 7 Figure 1.4: Trace of the spectral matrix of magnetic fluctuation energy, illustrating the power law nature of magnetic fluctuations in the inertial range, 10 4 .f. 10 1 Hz, at parallel and perpendicular propagation. The spectrum at f. 10 4 Hz corresponds to the energy injection regime. Image source: [Alexandrova et al., 2013]. The point of transition between the fluid regime and the kinetic regime typically occurs at some characteristic scale length of the system. For neutral fluids the characteristic scale length is the mean free path; near this scale viscous forces begin to dominate nonlinear forces, leading to viscous dissipation of the fluctuation energy. Due to the fact that the solar wind is a hot and tenuous ionized medium, collisions do not constitute the dominant mechanism of particle interaction in this medium. As a result, characteristic scales at which the fluid approximation breaks down are found to be much smaller than the collisional mean free path; accordingly, particle-particle collisions are relatively uncommon on kinetic scales, and the solar wind plasma is typically treated as collisionless within this regime. In collisionless plasmas the kinetic range begins near characteristic scale lengths of the ion motion; there are two such scale lengths: the ion inertial length, i = c=! i 8 Introduction (where c is the speed of light, j is the j-th species inertial length, and ! j is the j-th species plasma frequency; j = i for ions, j = p for protons, and j = e for electrons), and the ion thermal gyroradius, i =v ti? = i (where j is the j-th species thermal gyroradius, v tj is the j-th species thermal speed, and j is the j-th species gyrofrequency). While i characterizes the scale length of ion interaction with the time-varying electromagnetic fields in a plasma, i characterizes the cyclotron motion of the ions about the background magnetic field. It remains unclear which scale length most appropriately determines the boundary of the kinetic range. The summary of Bourouaine et al. [2012] shows that observations and theories have reached a variety of conclusions including k i 1, k i 1, or as a combination of these two scales and the fluctuation amplitude at those scales [Markovskii et al., 2008]. Recent observations by Chen et al. [2014] indicate the break scales as k i 1 if i? 1 and as k i 1 if i? 1 (where j represents the ratio of the j-th plasma species thermal energy to the background magnetic field energy, Eq. 1.4). However, under typical conditions in the solar wind, i 1, these two lengths are similar in magnitude as i = i p 2= i . Within the region near these characteristic scale lengths distinct changes in the fluctuating field power spectra are observed; there is general agreement that these changes, referred to as the “spectral break”, mark the transition from the MHD regime to the kinetic regime. While in the inertial range, where the MHD model applies, turbulent fluctuation energy is transferred from large to small scale lengths via a forward cascade [Kol- mogorov, 1941; Alexandrova et al., 2013], the kinetic range is the region over which plasma turbulence fluctuations are dissipated [Sahraoui et al., 2009, 2010; Alexan- drova et al., 2009, 2012]; fluctuation energy is irreversibly transferred to the particles, resulting in a heating of the medium. The details of the dissipation process are not clear, however. At frequencies corresponding to MHD scales the frequency spectrum of magnetic fluctuations follows the power law E/ f , with = 5=3, over more than three 1.3 Turbulence in the Kinetic Range 9 orders of magnitude variation [Alexandrova et al., 2013, and citations therein]; this is indicative that an inertial-range forward cascade process is occurring within this regime. Near a spacecraft-frame frequency of 0:2Hz < f < 0:5Hz, corresponding approximately to kc=! p 1, a break in the frequency spectrum occurs; the power law spectral index increases to > 2 [Alexandrova et al., 2013], and strong spectral wavevector anisotropies of the type k ? k k are observed [Sahraoui et al., 2009, 2010, 2013; Narita et al., 2011a], possibly suggesting the onset of dissipation. This short-wavelength regime is frequently referred to as the dissipation range under the assumption that the spectral break corresponds to dissipation onset as fluctuation wavelengths become shorter. Dissipation in neutral fluids, however, is marked by exponential drop-off in the power spectrum. The power law character of the fluctuating field spectrum in the solar wind near ion scales may, instead, suggest a second forward cascade which is affected by the dispersion of the kinetic plasma waves. A minority view suggests that this regime should be called the dispersion range [Stawicki et al., 2001] as the inertial range break approximately corresponds to the onset of dispersion in magnetosonic-whistler waves. A third possibility is that some combination of cascade, dispersion, and dissipation are all operating at wavenumbers above the spectral break. As the relative roles of dissipation and dispersion are not yet well established in this regime, the convention used here will be to refer to this domain as the kinetic range. Models such as hybrid particle-in-cell (PIC) have been developed to study plasmas ontheionkineticscale, whileremovingthekineticdetailsoftheelectrons(i.e. electrons are modeled as an isothermal fluid) in order to reduce the complex multi-scale nature of a fully kinetic plasma. These models have been very useful in describing plasma turbulence on scales below that of the spectral break; they are, however, insufficient for describing the role of electrons in the dissipation process. At scales much smaller than the ion scales, i and i , the analogous electron characteristic lengths, e = c=! e and e = v te? = e , are reached and both ions as 10 Introduction well as electrons must be treated under a kinetic framework. At frequencies near f 50 100 Hz, corresponding to kc=! e 1 and/or k e 1 a second spectral break is observed, with still steeper wavenumber dependencies. Beyond this spectral break energy spectra have been fitted to both power law functions, with > 3 [Sahraoui et al., 2009, 2010, 2013], as well as exponential functions [Alexandrova et al., 2009, 2012, 2013] suggesting dissipation may occur primarily within electron scale lengths. Steep spectra at electron scales have also been observed recently in the terrestrial magnetosheath [Huang et al., 2014]. Shaikh and Zank [2009] have proposed a model for how the process of forward cascade and dissipation proceed, from the inertial range, through the kinetic range and down to the electron scale. This model is depicted in Fig. 1.5. Figure 1.5: A proposed process of energy transfer and dissipation from the inertial range to the electron scale. Image source: [Shaikh and Zank, 2009]. Explaining kinetic range plasma turbulence requires understanding not only the nonlinear processes which drive the field fluctuation cascade, but also linear dispersion 1.3 Turbulence in the Kinetic Range 11 (that is, the wavevector dependence of the fluctuation mode frequency) and both linear and nonlinear dissipation processes at short wavelengths. By the quasilinear premise [Klein et al., 2012; Howes et al., 2014] it has been argued that, at sufficiently small amplitudes, magnetized plasma turbulence can be roughly modeled as a large collection of randomly phased normal (linear) modes. Computer simulations provide substantial evidence that this premise is valid at short wavelengths [Howes et al., 2008a, 2011; Saito et al., 2008; Chang et al., 2013]. These normal modes have the potential to contribute to the spectrum of turbulence in the solar wind. Several such modes have been characterized under the approximations of linear dispersion theory: kinetic Alfvén waves, magnetosonic-whistler waves, the several proton Bernstein modes and the slow or ion acoustic wave. Comisel et al. [2016] has recently demonstrated that, although ion Bernstein modes can be driven in two-dimensional (2-D) hybrid PIC simulations, three-dimensional (3-D) hybrid simulations wash out the Bernstein modes. Svidzinski et al. [2009] earlier showed that 2-D fully kinetic PIC simulations did not sustain ion Bernstein modes. The slow mode is heavily damped unless T e T p ; as this condition is rarely observed in the solar wind, ion acoustic modes are usually not considered as an important component of turbulence in the interplanetary medium [see Howes et al., 2012, for a contrary opinion]. The two remaining modes, the kinetic Alfvén wave (KAW), and the magnetosonic- whistler wave (MSW) have remained of great interest in attempts to explain the physics of plasma turbulence at the kinetic scale. Both solar wind observations [Leamon et al., 1998; Bale et al., 2005; Sahraoui et al., 2009, 2010; He et al., 2012; Salem et al., 2012; Chen et al., 2013; Kiyani et al., 2013; Podesta, 2013; Roberts et al., 2013] and gyrokinetic simulations [Howes et al., 2008a,b, 2011; Matthaeus et al., 2008; TenBarge et al., 2013] of short-wavelength turbulence have suggested that incompressible kinetic Alfvén waves are the dominant constituent of solar wind turbulence in the wavenumber range 1<kc=! p < 10. Kinetic Alfvén waves propagate in directions quasi-perpendicular to the background magnetic field, B 0 , and at real 12 Introduction frequencies ! r < i . While it is likely that the turbulence measured at wavenumbers moderately larger than the inertial range spectral break does consist of kinetic Alfvén waves, there is debate as to whether or not such modes can cascade fluctuation energy down to the very short wavelengths of electron inertial (k e 1) or thermal electron gyroradius (k e 1) scales [Podesta et al., 2010; Smith et al., 2012; Sahraoui et al., 2013]. A second hypothesis is that magnetosonic-whistler fluctuations, with frequencies between the proton cyclotron and electron cyclotron frequencies, also contribute to such turbulence. Solar wind measurements provide evidence for magnetosonic-whistler and Bernstein mode contributions to short-wavelength turbulence [Gary and Smith, 2009; Narita et al., 2011a,b, 2016; Perschke et al., 2013, 2014]. In particular, four- point magnetometer observations from the NASA Magnetospheric Multiscale (MMS) mission have been used to demonstrate a kinetic range solar wind turbulence event that is primarily composed of fluctuations consistent with magnetosonic-whistler mode dispersion [Narita et al., 2016]. This new result gives fresh impetus to the computational study of whistler turbulence and its associated plasma dissipation processes in the solar wind. Kinetic Alfvén waves and magnetosonic-whistler waves possess distinctly different properties, such as dispersion (both frequency and damping rate as functions of wavenumber and angle of propagation) and magnetic and plasma compressibility [Gary and Smith, 2009]. It is possible that both modes make important contributions to energy transport in the solar wind; it is therefore an important endeavor that both modes be thoroughly studied in order to understand how and when each may contribute to the dynamics of turbulence in the solar wind. A third hypothesis is that turbulence is not composed of discernible normal modes, rather it enables interactions between fields and particles through the establishment of a hierarchy of intermittent current sheets which naturally arise [Karimabadi et al., 2013; Wan et al., 2015]. 1.4 Definitions 13 Before a satisfactory model explaining dissipation in solar wind turbulence can be developed, several open questions must be answered; what are the details of forward cascade through the kinetic range from ion scale lengths to the electron Debye length? At what scale of the kinetic range does energy dissipation dominate over forward cascade? To what degree do ions contribute to total energy dissipation and to what degree do electrons contribute? What plasma modes are most prevalent in kinetic range turbulence? What plasma modes are most efficient at dissipating fluctuation energy? What physical processes control wave-particle energy transfer in the solar wind? These are a few of the questions which remain under debate in the heliophysics community. 1.4 Definitions This section provides a collection of the notations used throughout this manuscript. The subscript j is often used to generalize species dependent plasma parameters, where j can represent ions (j =i), protons (j =p), or electrons (j =e). As the solar wind is composed of > 95% protons, the simulation studies presented are intended to represent a proton-electron plasma with no ion species of greater mass than the proton. In general, however, the term ion (subscript i) will be used in place of proton; the purpose here is to exercise correctness, as most of the following simulation studies employ a fictitious ion particle with mass less than that of the proton. The use of such a particle is necessary in order to make these studies computationally accessible. See Section 3.2.3 for a discussion arguing the acceptability of using an artificial ion mass in PIC simulations of plasma turbulence. We define the j-th species plasma frequency as ! j q 4n j e 2 j =m j ; (1.1) 14 Introduction the j-th species cyclotron frequency as j je j jB 0 =m j c; (1.2) the j-th species thermal speed as v tj q K B T jk =m j ; (1.3) the j-th species beta as j 8n 0 K B T j =B 2 0 ; (1.4) and the dimensionless initial magnetic fluctuation energy density as 0 X k jB k j 2 =B 2 0 ; (1.5) where B k is the wavevector-dependent magnetic fluctuation amplitude. We define, the angle of mode propagation, by k B 0 =kB 0 cos(): (1.6) The uniform background magnetic field is B 0 = ^ zB 0 ; (1.7) so that the subscripts z andk represent the same direction. Thus, k = ^ xk x + ^ yk y + ^ zk k ; (1.8) k ? = q k 2 x +k 2 y : (1.9) 1.4 Definitions 15 Following [Shebalin et al., 1983], the wavevector anisotropy factor, tan 2 ( B ), is defined here as tan 2 ( B ) P k k 2 ? jB k j 2 P k k 2 k jB k j 2 : (1.10) The plasma wave complex frequency is ! =! r +i (1.11) where < 0 corresponds to a damped fluctuation, while > 0 corresponds to an instability. Turbulent heating of the ions and electrons yields bi-Maxwellian-like velocity distributions, so we define the parallel, perpendicular, and total temperatures of the j-th species as n j T jk m j Z v 2 k f j (v)d 3 v (1.12) n j T j? 1 2 m j Z v 2 ? f j (v)d 3 v (1.13) n j T j 1 3 m j Z jvj 2 f j (v)d 3 v: (1.14) As j represents the ratio of thej-th plasma species thermal energy to the background magnetic field energy, and B 2 0 is used as a normalization factor throughout the studies presented here, the terms T j and j are taken to be synonymous. We further define the j-th species total heating rates by Q j (dT j =dt)= [T j (t = 0)! e ]: (1.15) As these heating rates vary with time, we choose the convention of reporting the maximum values, Q i;max and Q e;max , obtained over the duration of each simulation. 16 Introduction The computational time step is dtt! e ; (1.16) and the simulation domain dimensions are L k ; L ? (1.17) where the subscriptk indicates a direction parallel to B 0 , and the subscript? indicates directions perpendicular to B 0 . The mesh spacing, e.g. grid size, is denoted . 1.5 Past Numerical Studies In the past, the complexity of turbulent plasma processes on kinetic scales has hindered attempts to obtain an accurate depiction of solar wind turbulence. The current capability of high performance computers has allowed for new routes of exploring and understanding turbulence with far fewer simplifying assumptions than that required by analytical theory, and with more accessibility than space based missions allow. While results obtained from computer simulation must, of course, be supported by the other two methods, this approach has proven invaluable in advancing our understanding of the mechanisms which govern the behavior of the space plasma environment. Over the last few years a number of studies have taken place by means of elec- tromagnetic (EM) particle-in-cell (PIC) simulations which thoroughly characterize the behavior of kinetic turbulence under the assumption that the primary constituent of the fluctuations is whistler modes. Under this assumption the forward cascade of whistler turbulence near electron scale lengths and the means by which the cascade results in the dissipation of the fluctuation energy has been documented [Saito et al., 2008, 2010; Saito and Gary, 2012; Chang et al., 2011, 2013, 2014, 2015; Chang, 2013; Gary et al., 2008, 2012]. The large computational demands required to compare 1.5 Past Numerical Studies 17 relative contributions from all possible modes in the kinetic range by use of EMPIC confined these studies exclusively to high-frequency whistler modes. In these past studies, PIC simulations of collisionless, homogeneous, magnetized proton-electron plasmas were used to examine the forward cascade of initial ensembles of relatively long wavelength, approximately isotropic, relatively narrowband whistler fluctuations (kc=! e 1). Simulations have been carried out in both two spatial dimensions (2-D), with B 0 in the computational plane [Gary et al., 2008; Saito et al., 2008, 2010; Saito and Gary, 2012], and three dimensions (3-D) [Chang et al., 2011, 2013, 2014; Chang, 2013; Gary et al., 2012] which show forward cascades to shorter wavelengths (kc=! e 1) where the fluctuations form a broadband spectrum of whistler turbulence with characteristic wavevector anisotropies in the sense of k ? k k . These results are very general, taking place for a broad range of initial fluctuation amplitudes and for a substantial domain of e values. Chang et al. [2013] interpret this forward cascade in terms of nonlinear three-wave interactions, showing that such interactions favor the development of the k ? k k wavevector anisotropy. These studies addressed only electron heating, and demonstrated the following results. 1. Electron heating is substantially stronger in T ek than in T e? . 2. If the initial electron temperature is e = 0:10, the late-time electron heating increases with increasing initial fluctuating magnetic field energy 0 [Gary et al., 2012]. 3. Over 0:10< e < 1:0 the late-time heated electrons exhibit bi-Maxwellian-like velocity distributions, but at e = 0:01 turbulent heating produces extended tails in the suprathermal parts of the electron velocity distribution [Chang et al., 2013]. 18 Introduction 4. If the initial fluctuation amplitudes and the initial electron temperature are fixed so that e is varied due to changes in ! e = e , the late-time electron total heating rate normalized to e increases with increasing e [Chang et al., 2013]. 5. If the initial fluctuation amplitudes and the initial electron temperature are fixed so that e is varied due to changes in ! e = e , the late-time electron temperature anisotropy T ek =T e? decreases with increasing e [Chang et al., 2013]. 6. At small magnetic fluctuation amplitudes linear damping dominates the electron thermal dissipation process [Chang et al., 2014]. 7. As the magnetic fluctuation energy becomes large relative to the electron thermal energy, and in particular at small values of e , nonlinear damping becomes the dominant mode of thermal dissipation [Chang et al., 2014]. 8. The nonlinear mechanisms of thermal dissipation are associated with the inter- mittent formation of localized current sheets [Chang et al., 2014]. Ganguli et al. [2010] describe 2-D PIC simulations in which B 0 is oblique to the simulation plane. In these computations, a heavy electron velocity ring distribution leads to the growth of unstable whistler fluctuations at oblique propagation relative to B 0 withkc=! e ' 1 and!< e . At and after instability saturation, induced nonlinear scattering leads to the growth of lower-frequency waves with k k values much smaller than those of the early-time modes. Ganguli et al. [2010] interpret this inverse spectral transfer, i.e. the transfer of fluctuation energy from small to large wavelengths, as a wave decay process through the Landau resonance with thermal electrons. Chang et al. [2015] further investigated the role of inverse cascade in whistler turbulence. In this study an initial ensemble of whistler modes with kc=! e 1 were loaded in a ring distribution which allowed for observation of the energy transfer process to both longer and shorter wavelengths. The inverse cascade was then compared to 1.5 Past Numerical Studies 19 the forward cascade and it was determined that the forward cascade contained the dominant portion of the cascaded energy. Karimabadi et al. [2013] performed a study of fully developed turbulence in PIC simulations which cover spatial scales ranging from the fluid-like MHD regime down to kinetic electron scales. Their work investigated the evolution of plasma turbulence generated by a fluid shear flow. The turbulent cascade revealed a network of eddies and the intermittent formation of current sheets, both spanning a range of scale lengths. The primary results of their paper detailed the role these structures played in the dissipation of the initial flow energy. Analysis of wave generation due to the motions of the turbulent structures was conducted. Spectral analysis suggested that kinetic Alfvén waves, as well as magnetosonic waves were being launched into the surrounding ambient plasma. Numerical techniques which ease the computational demand relative to PIC, such as MHD; gyrokinetic; and hybrid PIC codes, have been used to better understand the nature of solar wind turbulence. While these techniques are more computationally tractable, they provide less detail than fully kinetic PIC codes. For instance, MHD codes [Cho and Lazarian, 2004, 2009] are based on a fluid model which can describe the bulk characteristics of a plasma but average out the detailed interactions between the individual particle velocities and the electromagnetic fields; gyrokinetic codes [Frieman and Chen, 1982; Howes et al., 2006] average over the gyroperiod of the particles, and therefore can only resolve frequencies less than the ion cyclotron frequency. Hybrid PIC codes are commonly used in numerical studies of astrophysical plasmas [Winske and Omidi, 1995; Vasquez et al., 2014]. These codes treat ions as kinetic particles and electrons as a typically massless, isothermal neutralizing fluid. As such, hybrid PIC omits much of the electron physics that occurs on small temporal and spatial scales. As a significant portion of thermal dissipation is expected to occur on electron scales, these codes cannot provide a complete description of the processes associated with kinetic range dissipation. PIC is best suited to studies which require a kinetic 20 Introduction description of both the electron and ion species; that is, cases where the details of velocity space have a significant effect on the behavior of the system. 1.6 Objectives and Contributions The goal of the work presented here is to contribute to our understanding of the complex processes associated with turbulent dissipation in homogeneous, collisionless, magnetized plasmas. This is accomplished by performing kinetic-scale simulations of plasma turbulence under conditions characteristic of the solar wind. In order to perform this task, the full-particle EMPIC numerical model is utilized to simulate the complete 6-dimensional phase-space (three dimensions in position-space and three dimensions in velocity-space) evolution of a homogeneous, collisionless, magnetized, kinetic plasma subjected to an initially quasi-linear spectrum of turbulent fluctuations. The studies presented here have yield several state of the art results which help advance our understanding of the heliospheric space environment. While the past studies described above have focused exclusively on the interaction of the electron species with the turbulent fluctuations, the details of the ion species interactions have been included here to provide a more comprehensive picture. The key contributions of this research include results of parameter dependent behavior under a model of whistler turbulence 1. When e = 0:05 and m i =m e = 400 ion heating is stronger in T i? than in T ik [Hughes et al., 2014]. 2. When e = 0:05 andm i =m e = 400, increasing the simulation domain size results in reduced electron heating but yields enhanced ion heating [Hughes et al., 2014]. 3. When e = 0:05 and m i =m e = 400, the late-time heated ions exhibit bi- Maxwellian-like velocity distributions [Hughes et al., 2014]. 1.6 Objectives and Contributions 21 4. When e = 0:25, m i =m e = 400 and 0 is varied, the ratio of ion heating to electron heating increases with increasing 0 [Gary et al., 2016]. 5. When e = 0:25, m i =m e = 400 and 0 is varied, the electron maximum heating rate scales as Q e / 0 while the ion maximum heating rate scales as Q i / 1:5 0 [Gary et al., 2016]. 6. When 0 = 0:1, m i =m e = 400 and e is varied, both ion and electron maximum heating rates scale as 1 e [Hughes et al., 2017a]. 7. Ifm i =m e = 400 and 0 and e are varied such that 0 = 0:2 e , then the electron maximum heating rate remains constant while the ion maximum heating rate scales as Q i / 1=2 e when e 1 [Hughes et al., 2017a]. 8. Under a simplified conservation of energy model such that E 2 B 2 and Q i Q e , the above results yield a general parameter dependent electron heating rate followingQ e = 0:0005 0 = e [Hughes et al., 2017a]. The symbolsE andB represent the electric and magnetic field fluctuation amplitudes, respectively. 9. Collisionless dissipation of all studies is consistent with the expectation that entropy increases at all times, suggesting that such dissipation can be interpreted as a heating process. Moreover, for the first time, a fully kinetic particle simulation study of kinetic Alfvén turbulence has been performed. This study applied the parameters m i =m e = 100, e = i = 0:5 and included several simulations at varying values of 0 . From this study, the following results obtained 1. The ion maximum heating rate exceeds the electron maximum heating rate over the range of values 0:05< 0 < 0:50 [Hughes et al., 2017b]. 2. Both the ion and electron maximum heating rates scale proportionally to 0 [Hughes et al., 2017b]. 22 Introduction 3. Electron heating leads to anisotropies of the form T ek > T e? [Hughes et al., 2017b]. 4. Ion heating remains relatively isotropic throughout the dissipation process [Hughes et al., 2017b]. 5. These results suggest that the Landau wave-particle resonance is the likely linear mechanism acting to heat the plasma species. In the following chapters, the details of these studies and the means by which they were carried out will be presented. Chapter 2 describes the particle-in-cell numerical model and details a less known PIC model which was considered for carrying out these investigations. Chapter 3 presents the results of several studies intended to validate the selected numerical model and justify the use of numerical parameters applied in the investigations presented thereafter. Chapter 4 presents the first study of whistler turbulence, in which the properties of ion and electron heating by whistler turbulence are investigated as a function of initial fluctuation wavelength. Chapter 5 presents the second study of ion and electron heating by whistler turbulence in which the character of plasma heating is studied as a function of the energy density of the turbulence. Chapter 6 presents the final study of ion and electron heating by whistler turbulence, where the plasma evolution is characterized as a function of the parameter e . Chapter 7 moves beyond the studies of whistler turbulence to investigate ion and electron heating by kinetic Alfvén turbulence. This represents one of the first fully kinetic studies of such phenomena in three spatial dimensions. Chapter 8 presents a study of the three-dimensional evolution of the whistler anisotropy instability, and demonstrates that, in three dimensions, this instability leads to an inverse transfer of whistler-like fluctuation energy toward large scale lengths. A review of the studies presented in this dissertation and the conclusions of those studies are summarized in Chapter 9. 23 Chapter 2: Numerical Modeling The simulation studies presented throughout this dissertation (with the exception of Chapter 3.1) are carried out using the three-dimensional (3-D) electromagnetic (EM) particle-in-cell (PIC) code 3D-USCPIC which is a further development of the code described by [Wang et al., 1995]. In this code, plasma particles are pushed using a standard relativistic particle algorithm, currents are deposited using a rigorous charge conservation scheme [Villasenor and Buneman, 1992], and the self-consistent electromagnetic field is solved using a local finite difference time domain solution to the full Maxwell’s equations. There are, however, several representations of the electromagnetic field that can be applied in PIC codes. Two representations are relevanttosimulationsofturbulenceinthemagnetizedsolarwind: electromagneticPIC (EMPIC), and Darwin PIC (DPIC). While the EMPIC algorithm is well established and is the standard method for magnetized PIC simulations, DPIC is less well known but has unique properties that can potentially benefit simulation studies of turbulence. For this reason the emphasis of this chapter is placed on presenting the details of the DPIC algorithm. Chapter 3.1 will then present a study that characterizes the properties of the DPIC model. 2.1 Particle-In-Cell Particle-in-cell (PIC) is a numerical technique which is used to simulate the temporal evolution of a plasma by modeling the plasma as a set of discrete, charged particle packets; each packet, referred to here as a macroparticle (or as simply a particle), represents a finite-sized spatial distribution of physical particles. PIC operates from first principles and describes the motion of the individual macroparticles subjected to forces which are generated by the charge and current density distributions formed 24 Numerical Modeling by the particles. The consistent set of equations which describe these interactions includes some sub-set from the full set of Maxwell’s equations r E = 4 (2.1) r B = 0 (2.2) r E = 1 c @B @t (2.3) r B = 4 c J + 1 c @E @t (2.4) along with the Lorentz force F =q(E + v c B) (2.5) and Newton’s second law of motion F = dP dt : (2.6) There are two typical representations of the electromagnetic field in PIC codes. One representation is based on the full set of Maxwell’s equations; models of this form are known as electromagnetic (EM) PIC. The second representation is applicable when the magnetic character of the plasma is unimportant and can be neglected. Codes which use this form are known as electrostatic (ES) PIC; under the assumption of no magnetic field Eqs. (2.2)-(2.4) vanish and only electrostatic fields, described by Eq. (2.1) along with the scalar potential function E =r, are involved in driving the dynamics of the system. The studies carried out here involve modeling solar wind plasma, which is inherently magnetized. The magnetic character of the plasma is, therefore, of great significance, and ESPIC is excluded from consideration for studies of solar wind turbulence. 2.1 Particle-In-Cell 25 EMPIC [Birdsall and Langdon, 1985; Wang et al., 1995] simultaneously solves Ampère’s (Eq. 2.4) and Faraday’s (Eq. 2.3) laws on a discretized spatial mesh at each time step in order to determine the electric and magnetic forces each simulated particle experiences, while Eqs. (2.1) and (2.2) are satisfied as initial conditions of the simulation. A drawback to this method is that in order to maintain numerical stability, light waves (i.e. EM waves in vacuum) must be resolved. These modes play an important role in calculating a consistent solution due to the codependency which arises in Maxwell’s equations between the electric and magnetic fields. The derivation of light waves from Maxwell’s equations is provided in Sec. B.3.1. In order to ensure light waves are resolved the Courant condition, cdt. (where dt is the time step and is the mesh spacing, also known as the cell size), must be satisfied. Because the cell size in PIC codes is constrained by the requirement that the Debye length, D =v te =! e , be resolved, assuming a cell size equal to the Debye length results in dt! e .v te =c: (2.7) In other words, the maximum value of the dimensionless time step, dt! e , is restricted by the choice of v te =c ^ v te . By applying a value of ^ v te 0:005 (a value representative of that found in the solar wind near 1 AU) in an EMPIC simulation the number of time steps required to run the simulation to completion (typically t! e > 1000) would result in an inordinate computation time. To counteract this difficulty EMPIC simulations often employ an artificially large electron thermal velocity with a value near ^ v te 0:1 [Gary et al., 2012, 2014; Chang et al., 2013, 2014; Hughes et al., 2014; Karimabadi et al., 2013; Wu et al., 2013]; at such a value the time step can become an order of magnitude larger. While this allows for an acceptable run time duration it raises the question: what effect does the value of ^ v te have on the relevant plasma physics? 26 Numerical Modeling Many studies of kinetic scale plasma fluctuations concern the self-consistent inter- actions between plasma modes and plasma particles in resonance with those modes. For non-relativistic plasmas, there are very few particles energetic enough to interact with light waves. In this case light waves do not significantly contribute to the relevant physics. An alternate PIC model, known as Darwin particle-in-cell (DPIC) [Kaufman and Rostler, 1971; Nielson and Lewis, 2012; Busnardo-Neto et al., 1977; Hewett, 1985; Decyk, 2007a,b], operates on a set of equations which do not permit the propagation of light waves. DPIC is based on the Darwin approximation [Darwin, 1920] to a system of charged particles in which the effects of field potential retardation are neglected. This approximation leads to the modification of Maxwell’s equations such that the transverse displacement current (i.e., the contribution which is divergence- free), @E T =@t, vanishes in Ampère’s law. The Darwin potential fields are described in Sec. B.3.2. The Darwin approximation breaks the codependency of the electric and magnetic fields. While the transverse part of the electric field remains dependent on the magnetic field, the magnetic field depends solely on the current density. As a result light waves do not propagate under the Darwin approximation, while the interactions between the fields and the charge carriers remain intact. The elimination of light waves in the DPIC model removes the constraint set by the Courant condition. DPIC is therefore capable of maintaining a reasonably large time step as the value of ^ v te is reduced. The potential ability to model the magnetized solar wind under a set of parameters more closely representative of the real case makes DPIC worthy of consideration as an alternate to EMPIC. Sections 2.2 and 2.3 present the implementation details of the EMPIC and DPIC models respectively. In chapter 3 a case study is presented using an adapted open source two-dimensional (2-D) DPIC code, MDPIC2 (MDPIC2 is derived from a family of open source skeleton PIC codes which are maintained by the UCLA Plasma 2.2 Applying the Electromagnetic Model to Particle-In-Cell 27 Simulation Group and are accessible at http://picksc.idre.ucla.edu/software/skeleton- code/). The case study is inteded to characterize the physical effect of variations in the value of ^ v te on electromagnetic plasma phenomena. Moreover Chapter 3 highlights a number of numerical effects associated with varying ^ v te . 2.2 Applying the Electromagnetic Model to Particle- In-Cell The EMPIC model has been well developed over the past several decades. The model applied here, USCPIC, is a further development of the code described by Wang et al. [1995]. Here “three-dimensional” means that the simulations include variations in three spatial dimensions, as well as the full 3-D velocity-space response of each ion and electron particle. With this code, particles are advanced using a standard relativistic particle algorithm, currents are computed using a rigorous charge conservation scheme [Villasenor and Buneman, 1992], and the self-consistent electromagnetic fields are solved using a local finite difference time domain (FDTD) solution to the full Maxwell’s equations. USCPIC is built upon computational plasma physics and high performance com- puting research within research groups lead by Dr. Joseph Wang [Wang et al., 1995]. Dr. Joseph Wang developed the original version of the currently used 3-D parallel EMPIC code and carried out one of the largest 3-D PIC simulations of that time on the 512-processor Intel Touchstone Delta. Since then several optimizations have been introduced in the code, including a particle sorting scheme using a spatial prox- imity preserving algorithm which improves data and computation locality. As the overwhelming majority of computing time is applied to updating particle information, minimizing cache misses during particle updates is essential to running an efficient algorithm. The code is found to run with a parallel efficiency 96% on clusters with 28 Numerical Modeling more than 10 5 cores. The updated 3D-EMPIC code has been extensively applied to the study of whistler turbulence evolution in collisionless plasmas [Chang et al., 2011, 2013, 2014, 2015; Gary et al., 2012, 2014, 2016; Hughes et al., 2014, 2017a,b]. The reader is referred to [Wang et al., 1995] and [Chang, 2013] for further details of this state of the art EMPIC code. 2.3 Applying the Darwin Model to Particle-In-Cell While EMPIC is the standard particle-in-cell model for computational studies of magnetized plasma, DPIC is less well known and can potentially provide benefit over EMPIC in certain studies. It is therefore worthwhile to present here a detailed derivation of the DPIC algorithm. The following derivation is based on a general outline presented in [Decyk, 2007a]. 2.3.1 The Darwin Field Equations Darwin’s equations look very similar to Maxwell’s equations. They are the result of approximating the dynamics of a system of charged particles by expanding the terms that appear in the system Lagrangian in factors of 1=c and truncating all the terms greater than order (1=c) 2 . Once the dynamics are formulated under this approximation and the field equations are derived, Eqs. (2.1)-(2.3) are recovered; Ampère’s law, however, is modified, taking the new form r B = 4 c J + 1 c @E L @t : (2.8) Only the displacement current in Ampère’s law has been modified by the Darwin approximation. Here each vector field, , is viewed as being composed of two fun- damental components, = T + L . The first part is the solenoidal (or transverse) component which is divergence-free, i.e.r T = 0. The second part is the irrotational 2.3 Applying the Darwin Model to Particle-In-Cell 29 (or longitudinal) component which is curl-free, i.e.r L = 0. Darwin’s equations therefore neglect the transverse displacement current @E T =@t. It should be noted that the absence of @E T =@t in Ampère’s law only means that the transverse displacement current does not affect the magnetic field; the transverse electric field is still present and is still time dependent. By applying the continuity equation (Eq. B.7), Eq. (2.8) can be rewritten as r B = 4 c J T : (2.9) The derivation of this form is outlined in Sec. B.3.2. The strategy for determining the Darwin fields in practice begins as follows. By definitionr E L = 0,r E T = 0; applying this to Eq. (2.1) leads to the first field equation r E L = 4 (2.10) Taking the curl of Eq. (2.3), and applying identity (B.19) gives rr E T =r : 0 (r E T )r 2 E T = 1 c r @B @t r 2 E T = 1 c @ @t (r B): (2.11) Using Eq. (2.9) to replacer B results in the second field equation r 2 E T = 4 c 2 @J T @t (2.12) The solution to these two equations determines the complete electric field E = E L +E T . The magnetic field equation is given by taking the curl of Eq. (2.9) rr B =r : 0 (r B)r 2 B = 4 c r J T = 4 c r J 30 Numerical Modeling r 2 B = 4 c r J (2.13) The parameters, J and@J T =@t are functions of the particle positions, velocities, and accelerations respectively. A complication arises due to the need to use @J T =@t to obtain the electric field, because the electric field is needed to obtain the acceleration of a particle. The solution to this issue will be discussed in section 2.3.4. 2.3.2 Scattering to the Mesh The approach to finding the fields at the location of each particle, and the acceleration experienced by each particle, begins by using the known particle information (i.e. x j ; v j ) to collect the quantities , J and @J T =@t at mesh points distributed uniformly over the domain in each dimension with spacing . This is done with the help of a shape function S(x x j ). The shape function is the product of an interpolation function with a smoothing function. The former is used to interpolate the particle properties from the particle location, x j , to the mesh point location x; the later acts to reduce aliasing effects caused by variations in particle properties on scales smaller than the smallest resolvable length (i.e. ). The interpolation function has limited support; by this it is meant that the shape function, applied to a particular particle, only interacts with mesh points adjacent to that particle’s location. The bulk quantities, accumulated at the mesh points, are defined as follows. (x) = 1 V – c X j q j S(x x j ) (2.14) J(x;t) = 1 V – c X j q j v j (t)S(x x j (t)) (2.15) 2.3 Applying the Darwin Model to Particle-In-Cell 31 The definition of the final parameter, @J=@t, is found by taking the time deriviative of Eq. (2.15) @J @t = 1 V – c @ @t X j q j v j (t)S(x x j (t)) = 1 V – c X j q j dv j dt S(x x j ) + 1 V – c X j q j v j @S @(x x j ) d(x x j ) dt = 1 V – c X j q j dv j dt S(x x j ) + 1 V – c X j q j v j ( dx j dt ) @S @r @J @t = 1 V – c X j q j dv j dt S(x x j ) 1 V – c X j q j v j (v j rS) (2.16) where the mesh point location, x, is constant with time and r is the vector pointing from x j to x; V – c is the volume of the cell formed by the interior of adjacent mesh points, V – c = 3 . The parameter @J=@t can be decomposed into two quantities, each accumulated separately at each mesh point. The first quantity, represented by the first summation in Eq. (2.16), is the charge acceleration density a(x) = X j q j dv j dt S(x x j ) (2.17) The second quantity accumulated at the mesh points is the velocity flux tensor $ M(x) = X j q j v j v j S(x x j ) (2.18) where the second summation in Eq. (2.16) is equal to the divergence of $ M r $ M = X j q j r [v j v j S(x x j )] = X j q j [ : 0 Sr v j v j + v j v j rS] 32 Numerical Modeling r $ M = X j q j v j (v j rS): (2.19) With the quantities a and $ M stored at the mesh points, @J=@t can be constructed as @J @t = 1 V – c ar $ M (2.20) 2.3.3 Solving on the Mesh With the particle quantities collected at the mesh points, discretized solutions for B and E can be found on the mesh via finite difference. By taking the discrete Fourier transform (DFT) of Eqs. (2.10), (2.12) and (2.13), these partial differential equations become algebraic equations. The (spatial) discrete Fourier transform is defined as Ff(x)g = ~ (k) = 1 N X x (x)e ikx (2.21) where the vector summation represents a summation over each dimension d; for a given dimension, x d =n d , n d = 0!N d 1. N d is the number of mesh points in direction d and the total number of mesh points is N = Q d N d . The wavevector, k, is defined in a given dimension as k d = 2m d =N d , m d = 0!N d 1. The spatial operators in the field equations transform as follows 1 Ffr(x)g! ik ~ (k) (2.22) Ffr(x)g! ik ~ (k) (2.23) Ffr 2 (x)g!k 2 ~ (k): (2.24) 1 Because the functions are discretized on a mesh, it is implied that the gradient operator here represents discretized derivatives defined by the applied finite difference scheme. 2.3 Applying the Darwin Model to Particle-In-Cell 33 It is useful to note that in Fourier space the irrotational (longitudinal) component of a vector field, ~ L , is the component that is parallel to k at each value of k Ffr L = 0g! k ~ L = 0 (2.25) while the solenoidal (transverse) component, ~ T , is perpendicular to k at each value of k Ffr T = 0g! k ~ T = 0 (2.26) hence the alternative names longitudinal and transverse. Transforming Eq. (2.10) gives Ffr E L = 4g!ik ~ E L = 4~ : (2.27) Multiplying both sides by the unit vector in the direction of k and making use of (2.25), this can be rewritten as i(k ~ E L ) k k =ik ~ E L = 4~ k k : Solving for ~ E L gives ~ E L = ik k 2 4~ (2.28) Now transforming (2.12) Ffr 2 E T = 4 c 2 @J T @t g!k 2 ~ E T = 4 c 2 @ ~ J T @t ~ E T = 4 k 2 c 2 @ ~ J T @t (2.29) 34 Numerical Modeling Because @ ~ J=@t is what is known, the longitudinal component must be subtracted off to obtain @ ~ J T =@t @ ~ J T @t = @ ~ J @t k k 2 (k @ ~ J @t ) (2.30) And finally, Fourier Transforming (2.13) gives Ffr 2 B = 4 c r Jg!k 2 ~ B = 4 c ik ~ J ~ B = 4 c ik ~ J k 2 (2.31) Equations (2.28), (2.29), (2.30), and (2.31) provide the solutions for ~ E and ~ B in terms of the known Fourier transfromed quantities ~ , ~ J, and @ ~ J=@t at each mesh point k = (k x ; k y ; k z ) in Fourier space. The final step is to perform the inverse Fourier transform on ~ E and ~ B to convert the fields to physical space F 1 f ~ (k)g =(x) = X k ~ (k)e ikx : (2.32) With the fields now determined at the mesh points, the fields at each particle location can be approximated through an interpolation from the mesh to the particle E(x j ) =V – c X x E(x)S(x j x) (2.33) B(x j ) =V – c X x B(x)S(x j x) (2.34) The shape function is the same as before except the interpolation direction has been reversed. Once E and B are known at each particles’ location, the forces each particle is experiencing are known and each particle can be advanced by one time step dt. This procedure is discussed in section 2.4. 2.3 Applying the Darwin Model to Particle-In-Cell 35 2.3.4 Resolving Force/Acceleration Coupling In an EMPIC code the acceleration experienced by a particle at a particular time is dependent on the distribution of particle positions and velocities at that time. The Darwin PIC model, however, introduces a complication as, according to this model, to resolve the acceleration experienced by each particle, one needs the particle position and velocity distributions as well as the acceleration distribution. In other words, the acceleration experienced by each particle is required in order to solve for the acceleration experienced by each particle. This interdependency results in an unmanageably large system of coupled equations. The approach to handling this problem is by employing an iterative scheme which converges to the correct solution for the fields. This iterative scheme is combined with the standard leapfrog method of time discretization, discussed in section 2.4, in order to resolve the forces at the current time step n =t=t. A first attempt at an iterative solution would be to use the values of E T calculated at the previous time step, along with the values for E L and B calculated at the current time step to determine particle accelerations. These accelerations could then be used to calculate new E T and B values 2 , and this cycle would be repeated until convergence was achieved. At each iterative step, E T would be determined by E n T = X k h 4 k 2 c 2 ih @ ~ J o T @t i e ikx : (2.35) The superscripts n and o denote new and old values respectively. Unfortunately this approach has proved to become unstable when kc<! pe . Numerical stability can be achieved by subtracting a constant from both sides of (2.12) r 2 E n T ! 2 p0 c 2 E n T = 4 c 2 @J o T @t ! 2 p0 c 2 E o T : (2.36) 2 While B doesn’t directly depend on acceleration, determining B at time t requires knowledge of v j (t) which is obtained through knowledge of each particle’s acceleration. 36 Numerical Modeling The coefficient on E T , known as the shift constant, is the average plasma frequency ! 2 p0 = 4 V – X j q 2 j m j : (2.37) With this new term added (2.35) takes the form E n T = X k h 4 k 2 c 2 +! 2 p0 ih @ ~ J o T @t ! 2 p0 4 ~ E o T i e ikx : (2.38) Note that when the solution has converged, this equation reduces to the original one. Solvingforthefieldsrequiresknowledgeofthepositions, velocitiesandaccelerations of the particles at time t. The longitudinal electric field can be calculated correctly immediately as it depends only on particle positions, which are already known at time t. However, by the leapfrog scheme, velocities are known at increments of t + 1=2t, and accelerations at time t are yet to be resolved by the iterative method. Velocities and accelerations at the centered time are therefore approximated using v j (t) = v j (t + t=2) + v j (t t=2) 2 (2.39) dv j (t) dt = v j (t + t=2) v j (t t=2) t : (2.40) The procedure for resolving the particle accelerations starts by first calculating E L (t) from x(t). Next, rough initial estimates of B(t) and E T (t) are calculated by making the assumption that v j (t + t=2) = v j (t t=2). This is equivalent to assuming the forces are small and that changes in the currents are dominated by convection. With initial estimates of the fields the iteration loop proceeds. From these fields particle accelerations and velocities at time t are calculated and deposited to obtain improved J(t) and @J(t)=@t. This leads to improved solutions for E T (t) and B(t). This cycle is repeated until the fields and accelerations at time t become consistent. It has been found that as long as max(! 2 p (x))< 1:5! 2 p0 , the solution converges in one iteration. 2.4 Advancing Particles 37 2.4 Advancing Particles In order to advance particle positions and velocities according to the calculated accelerations in a finite state model, time must be discretized. The coupling between position, velocity and acceleration is typically dealt with by the use of a leapfrog scheme in which positions and accelerations are known at increments of nt, while velocities areknownatincrementsofnt+t=2. Withthisarrangementaccelerationscalculated at time t are used to advance particle velocities from v(t 1=2t)!v(t + 1=2t), then the new velocities advance particle positions from x(t)!x(t + t). This time staggering method is computationally inexpensive and is second order accurate. The position update is simply calculated as x(t + t) = x(t) + v(t + t=2)t: (2.41) The velocity update is more involved. This is because the acceleration must be calculated from two distinctly different forces, the electric force and the magnetic force, according to the Lorentz equation a(t) = q m E(t) + v(t) c B(t) : (2.42) While the electric force acts to change the particle’s kinetic energy, the magnetic force leaves kinetic energy unaffected but acts to change the direction of motion in a direction perpendicular to both the velocity and the magnetic field vectors. This describes, in the simplified case of no electric field and constant magnetic field, a circular motion of the particle about the magnetic field. The angular frequency of this motion is known as the cyclotron frequency, defined as = qB mc : (2.43) 38 Numerical Modeling The cyclotron frequency is a characteristic parameter of each species in a magnetized plasma. An accurate approach to discretely updating the particle’s velocity is to accelerate the particle half a time step by the electric force, then rotate the new velocity vector a full time step according to the magnetic force, and finally accelerate the particle another half time step by the electric force, v = v(t t=2) + q m E(t)t=2 (2.44) v + = v + q mc v(t) B(t)t (2.45) v(t + t=2) = v + + q m E(t)t=2: (2.46) It can be seen that the magnetic rotation is dependent on the velocity of the particle at timet, which is unknown; v(t) must be calculated by interpolation v(t) = 1 2 (v + v + ). Because of this (2.45) is rewritten v + = v + q mc v + v + 2 B(t)t: (2.47) The magnetic rotation is described by an implicit equation. Fortunately, this equation can be solved because the cyclotron frequency is independent of the particle’s velocity, but does depend on B(x j ;t), which is known. This makes it possible to know the angle through which the velocity vector should rotate over one time step based on the magnetic field strength at the location of each particle. Through geometric analysis of the process of updating the particle’s position and velocity it can be determined that the appropriate angle of rotation of the velocity vector over one time step t is = 2tan 1 t 2 : (2.48) 2.4 Advancing Particles 39 The direction of rotation is clockwise about . The task now is to transform the velocity vector from the standard basisf ^ i; ^ j; ^ kg into a convenient basis for rotation; rotate the vector through the angle ; then transform back to the standard basis. The vector gives the direction about which the velocity is rotated and only the velocity component perpendicular to , v ? , undergoes the rotation. This suggests these are good vectors from which to form the new basis. The final basis vector, forming the right handed coordinate system, is given by v ? . f ^ i; ^ j; ^ kg!f^ e 1 ; ^ e 2 ; ^ e 3 g =f v ? v ? ; v ? v ? ; g: (2.49) The velocity components in this coordinate system are v e1 = v ? = v 2 (v ); (2.50) v e2 = 0; (2.51) v e3 = v k = 2 (v ): (2.52) The rotation matrix for a clockwise rotation about ^ e 3 by an angle is given by R 3 = 2 6 6 6 4 cos() sin() 0 sin() cos() 0 0 0 1 3 7 7 7 5 : The rotated vector v + is then given as v + = R 3 v =v ? cos()^ e 1 v ? sin()^ e 2 +v k ^ e 3 : (2.53) Plugging in the definitions of ^ e 1 , ^ e 2 , and ^ e 3 in terms of the standard basis v + =v ? v ? v ? cos()v ? v ? v ? sin() +v k (2.54) 40 Numerical Modeling = v ? cos() + v ? sin() + v k (2.55) = v 2 (v ) cos() + v 2 (v ) sin() + 2 (v ): (2.56) The trigonometric functions can be removed due to the form of the angle , given by equation (2.48), along with the identities sin(tan 1 (x)) = x p 1 +x 2 ; cos(tan 1 (x)) = 1 p 1 +x 2 (2.57) sin(2) = 2sin()cos(); cos(2) = 2cos 2 () 1 (2.58) resulting in sin() = t 1 + t 2 2 ; cos() = 1 t 2 2 1 + t 2 2 : (2.59) Plugging (2.59) into (2.56) and rearranging terms gives v + = (1) v + + t v 1 + = t 2 2 = t 2 2 (v ) (2.60) representing the rotated velocity vector in terms of known quantities. 2.5 Normalization All parameters are typically normalized in PIC simulations. One advantage of working in dimensionless units is that all parameters become of order unity, 1. Many plasma parameters vary from one another by several orders of magnitude, and this can result in large roundoff errors during finite state calculations. Normalization reduces the significance of rounding errors by forcing all parameters to be closer to one another in magnitude. 2.5 Normalization 41 The normalization process begins by selecting parameters which are representative of the temporal and spatial scales of the system. Space is normalized by the grid spacing . In most cases the grid spacing is made to be equal to the plasma Debye length, D . This is typically the case as the simulation grid must resolve the Debye length, which represents the effective length of electric field interaction in a plasma, for numerical stability. Time is normalized by the electron plasma frequency ! e ; charge is normalized by the unit of charge e, equal in magnitude to the charge of one electron; mass is normalized by the mass of an electron m e . For added convenience, parameters involving volume are normalized by the uniform particle number density n 0 . All other parameters are normalized by some combination of these values. This leads to ^ x = x D ; ^ t =t! e ; ^ ! = ! ! e ; ^ q = q e ; ^ m = m m e (2.61) ^ v = v ! e D = v v te ; ^ c = c ! e D = c v te (2.62) ^ = n 0 e ; ^ J = J D ! e n 0 e (2.63) ^ E = eE m e ! 2 e D ; ^ B = eB m e c! e = e ! e : (2.64) Normalizing the Darwin equations under this framework results in ^ r ^ E = ^ (2.65) ^ r ^ B = 0 (2.66) ^ r ^ E = @ ^ B @ ^ t (2.67) ^ c 2 ^ r ^ B = ^ J + @ ^ E L @ ^ t : (2.68) 42 Numerical Modeling Including the equations of motion provides the full set of normalized equations to be solved d^ v d ^ t = ^ q ^ m ^ E + ^ v ^ B ; d^ x d ^ t = ^ v: (2.69) 2.5.1 Depositing Normalized Densities After the charge, current and acceleration of each particle is deposited at the mesh points, the cumulative values at each mesh point should roughly represent charge, current and current rate densities over each cell. A normalized volume is required to convert a cumulative value into a representation of the density of that value over a given cell. This normalized volume is hidden inside the definition of ^ , ^ J and @ ^ J=@t. For instance, the conversion from net charge to charge density can be revealed by expanding the definition of the normalized charge density ^ = en 0 = P Nc q j = 3 en 0 = P Nc ^ q j n 0 3 = P Nc ^ q j N 0c = N x N y N z N p Nc X ^ q j =A f Nc X ^ q j ; (2.70) where N c is the number of particles in a given cell, N 0c is the uniform number of particles per cell,N p is the total number of simulated particles andN x ,N y ,N z are the number of cells in the x, y, z directions respectively. What this reveals is a constant A f , which is equal to one over the average number of particles per cell. Once all charges are deposited at a mesh point, the cumulative value can be converted into a density by multiplying by A f . Similar formula result for current and current rate densities ^ J =A f Nc X ^ q j ^ v j (2.71) @ ^ J @ ^ t =A f ^ a ^ r ^ $ M : (2.72) 2.6 Computational Resources 43 2.6 Computational Resources The simulations performed here run on massively parallel compute clusters. Currently, a typical simulation requires 10 5 processor hours to complete, with a memory requirement of 8 TB. The facilities used to conduct these studies included the NASA Pleiades Supercomputer, located at the Ames Research Center, and the Yellowstone Supercomputer located at the NCAR-Wyoming Supercomputing Center. Pleiades is a 5.95-petaflops SGI ICE cluster consisting of 11,472 compute nodes (246,048 CPU cores), with a total memory of 938 terabytes. Yellowstone is a 1.5-petaflops IBM iDataPlex cluster consisting of 72,576 CPU cores with a memory capacity of 144.6 terabytes. 44 Chapter 3: Numerical Validation This chapter investigates several important numerical effects that can influence the results obtained from PIC simulations. In Section 3.1 the DPIC model is used to characterize the effect of the electron thermal velocity on the physics of the whistler anisotropy instability (WAI). A series of simulations are carried out using DPIC in which the electron thermal velocity is varied in order to make clear the physical and numerical variations that occur with changes inv te . The results show that the relevant physics are insensitive to the value of the thermal velocity, however nonphysical effects are enhanced with decreasing v te . It is found that for a fixed number of particles, the quantitative accuracy of the model improves with increasing values of the electron thermal velocity [Hughes et al., 2016]. The numerical validation studies of Section 3.2 investigate two additional numerical properties in order to ensure that the plasma turbulence studies of the following chapters produce reliable physical results. In Section 3.2.2 the effect of numerical heating in the EMPIC model is characterized. Section 3.2.3 then carries out an ensemble of simulations to demonstrate the effects of using an artificial ion-to-electron mass ratio in EMPIC simulations of whistler turbulence. 3.1 The Electron Thermal Velocity 3.1.1 Introduction The kinetic-scale behavior of a plasma is controlled by a number of physical parameters; the degree to which each parameter affects the evolution of the plasma depends on the physical processes the plasma is undergoing. One notable plasma parameter is the thermal velocity, defined for the j-th plasma species as v tj p k B T jk =m j . In general, achievable values for the thermal velocity in a plasma span orders of magnitude. To 3.1 The Electron Thermal Velocity 45 confine this range of values the emphasis here is placed on heliospheric plasma, i.e. the solar wind. The thermal velocity of the electron population, v te , is of interest in the solar wind as electrons provide the primary means of thermal energy transport [Montgomery et al., 1968] due to their superior mobility relative to the accompanying ion species. An effort of the heliophysics community has been focused on understanding how turbulent fluctuation energy is dissipated into thermal energy at small spatial scales. It has been suggested that electrons play a prominent role in turbulent energy dissipation [Alexandrova et al., 2009, 2012; Howes, 2010; Chang et al., 2014] and it would be instructive to better understand how the electron thermal velocity might influence the details of this process. Here, the electron thermal velocity will be referred to in terms of the ratio ^ v te = v te =c. Newbury et al. [1998] analyzed ISEE3 solar wind data to obtain a mean electron temperature near 1 AU of T e 141; 000 38; 000 K. This temperature range corresponds to ^ v te 0:0042 0:0055. Le Chat et al. [2011] applied the quasi- thermal noise spectroscopy technique to data obtained from the Ulysses spacecraft at high southern heliocentric latitudes and at distances between 1.5 and 2.3 AU. Their analysis yielded a power law for electron temperature with radial distance from the Sun. Extrapolating along this power law gives an estimate of T e 230; 000 K at 1 AU, corresponding to ^ v te 0:0063. These studies suggest that an appropriate value to apply in models of solar wind processes near 1 AU is ^ v te 0:005. The following sections will examine how the value of ^ v te affects the physical response of a plasma subjected to a kinetic plasma instability [Hughes et al., 2016]. The particular instability modeled is the whistler anisotropy instability (WAI). The WAI arises in the presence of a mean background magnetic field, B 0 , when the electron population develops a bi-Maxwellian velocity distribution such that R T > 1, where R T = T e? =T ek denotes the electron temperature anisotropy ratio. If R T becomes sufficiently larger than 1, exceeding some threshold value which is dependent on the state of the plasma [Gary, 1993], the WAI will be excited [Gary and Wang, 1996; Gary 46 Numerical Validation et al., 2014]. The electrons become cyclotron resonant, exciting the whistler mode across a narrowband spectrum of wavevectors with maximum growth at k B 0 = 0. Near instability onset electromagnetic fluctuation energy grows exponentially. Shortly after onset the growth in fluctuation energy is quenched by nonlinear processes and the instability saturates; the cyclotron resonance leads to electron pitch angle scattering which acts to reduce R T . Gary and Wang [1996] showed that wave-particle scattering results in a rapid decrease in the electron anisotropy at early times, while at later times the anisotropy remains approximately constant near a weak threshold condition. The numerical model used to perform this study is adapted from an open source two-dimensional (2-D) Darwin particle-in-cell (DPIC) skeleton code, MDPIC2. Section 3.1.2 conducts a linear dispersion analysis of the WAI to demonstrate that the details of how the plasma parameters vary with ^ v te are essential to understanding how the WAI’s dispersion will be affected. Section 3.1.3 presents results from a series of MDPIC2 simulations and characterizes the physical significance of ^ v te in a fully nonlinear evolution of the WAI. Section 3.1.5 investigates how electric and magnetic noise levels associated with the number of simulated particles alter the results obtained from the PIC simulation at varying ^ v te . Conclusions are summarized in section 3.1.6. 3.1.2 Linear Dispersion The dispersion properties of the WAI are described by linear dispersion theory [Gary, 1993]. Dispersion reveals how the frequency, ! =! r +i , of a normal mode varies as a function of wavevector (where ! r is the real frequency, is the growth/damping rate, and i = p 1). Dispersion is one of the primary defining properties of each type of mode that can exist in a plasma of specified physical conditions. The term ^ v te enters into the physical conditions of the system via the relationship between the electron thermal energy density and the magnetic field energy density. This 3.1 The Electron Thermal Velocity 47 relationship is characterized by the plasma , defined for the j-th plasma species as j 8n j K B T jk =B 2 0 . Focusing on the electron population, e can be rewritten as e = 2 (v te =c) 2 ( e =! e ) 2 (3.1) where e = eB 0 =m e c is the electron cyclotron frequency and e =! e represents the dimensionless magnetic field strength. From this definition it is clear that, as ^ v te is varied either e can be held fixed or e =! e can be held fixed, but one of these two parameters must change. The question now becomes, how is the response of the plasma affected by changing e , and by changing e =! e ? Schriver et al. [2010] performed linear dispersion analysis to investigate the effect of varying ^ v te . In their study the plasma was composed of two electron populations. The first population was low in density and consisted of mid-energy electrons with a large temperature anisotropy; the second population formed a low-energy, higher density core. The analysis showed that the dispersion was a strong function of ^ v te , and it was concluded that the particle-in-cell simulations subsequently carried out in that study required a realistic thermal velocity. Those calculations were based on parameters determined from in situ data obtained in the inner magnetosphere. The data provided values for both ^ v te and e =! e . For this reason e =! e was assumed fixed during their analysis. Presented below is a linear dispersion study applied to the current scenario, that is, a single electron population possessing a bi-Maxwellian velocity distribution. For this study the following values are taken to be representative of realistic conditions in the solar wind: ^ v te = 0:005, e = 0:1 (in the solar wind e 0:1 10 [Gary et al., 1998]), and (by Eq. 3.1) e =! e = 0:0224. Figure 3.1 provides the results of this study, obtained via numerical solution to the electromagnetic linear dispersion equation detjD(k;!)j = 0; where D is the dispersion tensor for electromagnetic fluctuations (for further details see Ch. 5 of Gary [1993]). The real frequency (solid lines) and the 48 Numerical Validation growth rate (dashed lines) of the WAI at k B 0 = 0 are shown for varying values of ^ v te . In panel (a) e =! e is held fixed at a realistic value while e varies with ^ v te . In panel (b) the analysis is repeated under the condition that e is held fixed at a realistic value and e =! e is allowed to vary; all cases use the condition R T = 3. Panel (a) shows that a similar conclusion as in Schriver et al. [2010] can be drawn. That is, when e =! e is held fixed and e is varied the dispersion properties of the WAI exhibit a large degree of variation. However, panel (b) reveals that when e is held fixed, the variation in dispersion over the range 0:1 ^ v te 0:005 is essentially negligible. Figure 3.1 indicates that e =! e is not a critical parameter; rather, the dispersion properties of the WAI are sensitive functions of e and relatively independent of the value of e =! e . Previously reported linear theory studies [Gary and Cairns, 1999; Gary et al., 2000] are consistent with this conclusion under the condition that e =! e is less than unity. The analysis presented suggests that simulations which use an artificial value of ^ v te can capture the relevant physics as long as they employ a realistic e . 3.1.3 Simulations Linear theory, of course, cannot provide a complete description of instability behavior. Shortly after instability onset the growth of the fluctuating fields leads to failure of the linear approximation and the subsequent processes require a nonlinear description. It would be instructive to investigate whether variations in the value of ^ v te affect nonlinear processes associated with the evolution of the WAI. This section presents the results of a series of fully nonlinear 2-D particle-in-cell simulations employing MDPIC2 at varying values of ^ v te , while the value of e is held fixed. The parameter ^ v te appears in the simulation initialization in two places. The first is in the relationship between e and e =! e (given by Eq. 3.1). The second is through the physical length of the cell. The numerical mesh must resolve the Debye 3.1 The Electron Thermal Velocity 49 length and the Debye length is a function of thermal velocity ( D = v te =! e ). As the thermal velocity is reduced the Debye length decreases; it follows that the cell size must decrease as well. Figure 3.1 displays the wavenumber region of fluctuation growth in terms of the electron inertial length e = c=! e . e represents the scale length of electromagnetic interaction in a plasma; spatial scales of electromagnetic waves are typically characterized in terms of this parameter. For this reason it is desirable to maintain a simulation domain size which is fixed relative to e . The effect of decreasing ^ v te under this constraint is made clear by the ratio D e =v te =c: (3.2) The consequence of decreasing ^ v te is that an increasing number of Debye lengths are required to simulate a fixed number of inertial lengths. Three simulations were carried out with respective initial conditions: ^ v te = 0.1, 0.05, and 0.025. Each run was initialized with R T = 3 and e = 0:1. The respective uniform background magnetic field strengths were e =! e = 0.447, 0.224, and 0.112. The domain was held at a fixed length ofL x =L y = 51:2 e with a cell spacing = D . By Eq. (3.2) the number of cells required to simulate a domain length L = 51:2 e must be doubled each time ^ v te is halved. Accordingly, the simulations at ^ v te = 0.1, 0.05, and 0.025 had 512, 1024, and 2048 cells in each direction respectively. The boundary conditions were periodic in both the x and y directions. The time step in all runs was dt! e = 0:05, and each simulation was run to 30,000 time steps, or to t! e = 1500. The number of electrons per cell in each simulation was N c = 144. All simulations presented assume a stationary neutralizing background of ions which do not directly participate in the system dynamics; according to linear theory [Gary, 1993] ion dynamics play a negligible role in the WAI. 50 Numerical Validation 3.1.4 Results Figure 3.2 shows the magnetic fluctuation energy density as a function of both parallel and perpendicular (relative to B 0 ) wavenumber for varying time and varying ^ v te . The vertical dashed lines overlaying each panel bound the region predicted by Fig. 3.1 to give rise to fluctuation growth at k B 0 = 0 when e = 0:1 and R T = 3. The instability results in a growth of magnetic fluctuations at a peak parallel wavenumber k k c=! e 1 which preferentially propagate along B 0 (k k >k ? ). This behavior remains consistent across all simulations and agrees with the predictions of linear theory that the region of instability growth is not a function of ^ v te when e is held fixed. At later times the instability saturates and the fluctuations are gradually reduced. The time t e = 167:2 roughly corresponds to the time of instability saturation for each case. The saturation level of the fluctuation energy near the region of maximum growth agrees well among the results. While the growth of the magnetic fluctuation spectra remains mostly consistent, at intermediate times the spectrum becomes slightly weaker and, near the time of saturation, it is somewhat less broad at smaller values of ^ v te . Moreover, there is a persistent elevated level of isotropic magnetic noise nearjkj 0 in all cases, and this noise level is reduced with ^ v te . The noise level of the magnetic fluctuations is not directly associated with the value of ^ v te ; rather it is controlled by the number of electrons per square inertial length which, when N c is held fixed, increases as ^ v te decreases. The cause of the variations observed in Fig. 3.2 is discussed in Sec. 3.1.5 on the basis of numerical modeling considerations. Figure 3.3 displays the real frequency dispersion properties of the magnetic fluctua- tions at parallel propagation for each value of ^ v te . The solution to the electromagnetic linear dispersion equation for realistic parameters (the solid blue line in each panel of Fig. 3.1) overlays each panel of Fig. 3.3. The results show that the majority of the fluctuation energy is contained along the WAI dispersion curve, accompanied 3.1 The Electron Thermal Velocity 51 by a low energy content of incoherent fluctuations which propagate at a continuum of frequencies inside the wavenumber region of instability. In all cases the results agree with the linear theory solution, confirming that the real frequency dispersion properties of the WAI are independent of variations in the electron thermal velocity. Figure 3.4 presents characteristic plasma properties with time at varying values of ^ v te . Panel (a) displays the energy densities, summed over all wavevectors, of the mag- netic and electrostatic fluctuating fields. The electrostatic fluctuations form a constant background noise, indicating that the character of this instability is fundamentally electromagnetic. The growing fluctuations are quasi-parallel propagating transverse waves resulting from the electron cyclotron resonance. The magnetic energy curves recap the evolution observed in Fig. 3.2. There is a period of wave growth, followed by instability saturation and eventual damping of the fluctuation energy. The evolution of the WAI on cyclotron time scales remains consistent among the three cases. Overlaying each magnetic energy curve in panel (a) are least squares fits which model the linear growth phase of the instability. The fitting function is of the form B 2 =B 2 e 2 t +B 2 , where is the instability growth rate, B 2 is the initial level of the resonant magnetic fluctuation energy which grows in time, and B 2 represents the long wavelength, non-resonant magnetic fluctuation energy which can be seen near jkj 0 in Fig. 3.2. Each curve is fitted over the time range 0 t t nl , where t nl represents the time that the linear growth phase ends and the instability begins to saturate. The parameters of each fit are provided in Table 3.1. The curve fits reveal that, when normalized to e , the linear growth rate is essentially independent of ^ v te ; this is in agreement with the linear theory study of Sec. 3.1.2. Comparing the growth rate obtained from the simulations with the linear theory maximum growth rate predicted by Fig. 3.1, lt max = e = 0:0437, it is seen that the simulation result is about 31% smaller than lt max . However, the simulation result represents a cumulative growth rate over all growing modes; it is therefore expected that this value will be somewhat less than lt max . Additional analysis reveals that in all 52 Numerical Validation simulations the growth rate of the fastest growing mode, k sim max c=! e = 1:1 ^ k k , remains near sim max = e = 0:040, in considerable agreement with the linear theory result. In addition to the linear growth rate, the nonlinear process of instability saturation is not sensitive to the value of ^ v te ; the instability saturates at roughly the same magnetic energy level in each case. Figure 3.4(b) shows the electron temperature anisotropy ratio, R T , with time. As the instability begins to saturate the cyclotron resonance leads to enhanced pitch angle scattering; this is observed as an elevated rate of decrease in R T near t e & 100. At ^ v te = 0.1 and 0.05 the process of pitch angle scattering proceeds in a very similar manner. In the case of ^ v te = 0.025,R T is reduced at a greater rate at early times but is approximately consistent with the former cases. This discrepancy in the results is attributed to nonphysical electrostatic effects which are described in Sec. 3.1.5. A noticeable difference among the results of Fig. 3.4(a) is the reduction in the initial magnetic energy level with decreasing ^ v te ; this was also observed in Fig. 3.2 as a decreasing level of magnetic noise. In connection with this variation is a mild dependence of the saturation time on ^ v te . While the saturation level remains unaffected by the variation in the magnetic noise level, the time required to reach the saturation level increases as the fluctuations grow at a fixed rate = e from a lower initial magnetic energy level. This explains the variation observed in Fig. 3.2 where, at t e = 74:4, the spectrum becomes weaker with decreasing ^ v te . It is apparent from Fig. 3.4 that, at a given time during the linear growth stage, the case with the smallest value of ^ v te is farthest from reaching the saturation level. Aside from causing a shift in the time of instability saturation, the details of the initial magnetic energy level do not appear to affect the physics of the WAI. 3.1 The Electron Thermal Velocity 53 3.1.5 Effects of Particle Noise Due to computational limitations, the number density in a PIC simulation is far less than that in the solar wind. For instance, the number of electrons per Debye cube in the solar wind at 1 AU can be estimated as n e 3 D 10 10 . A 2-D simulation running at a one-to-one particle count would require N c 10 6 . The relatively small number of particles in a PIC simulation leads to enhanced residual noise in the fluctuating electric and magnetic fields; this noise can have a significant effect on the results of the simulation. This section aims to characterize how the noise associated with the number of simulated particles affects the results of the WAI model as ^ v te is decreased. Figure 3.5 presents the results of a convergence study performed on the parameter N c forafixedvalueof ^ v te . Startingwithavalueof9,N c isincreasedbyafactoroffourin each successive simulation; in each case ^ v te = 0:1 and all other parameters are the same as in Sec. 3.1.3. Here, four characteristic field quantities are shown as functions of N c : the initial magnetic fluctuation energy density B 2 t=0 , the magnetic fluctuation energy density at instability saturation B 2 max , the time averaged electrostatic fluctuation energy density E 2 L , and the linear growth rate . The growth rate is strongly affected at small values of N c but rapidly converges to an asymptotic state. The rate of change in with N c is reduced substantially by N c 36. AtN c = 144, is within 5% of the asymptotic value. The saturation level is best fitted to a logarithmic function, which implies that the instability growth diverges withN c . However, a comparison betweenN c = 144 andN c = 10 6 yields an increase in the saturation level by only a factor of 2.8. The convergence study suggests that, for ^ v te = 0:1, a value of N c = 144 is more than sufficient to correctly model the physics of the WAI in the solar wind. An important result of this study is that both the initial magnetic fluctuation energy level and the electrostatic fluctuation energy level are inversely proportional to N c . 54 Numerical Validation Dawson [1983] provides the following relationships, derived as time averages from equilibrium statistical mechanics E 2 L (k) V – = 4K B T 1 +jkj 2 2 D (3.3) B 2 (k) V – = 4K B T 1 +jkj 2 2 e (3.4) whereV – is the volume of a finite size domain. These equations describe the equilibrium fluctuating field energies in a plasma as functions of wavevector, and can be used to obtain approximate scaling laws for the residual fields in the simulation model. Noting that, in the present study, e is held fixed so that n e K B T e / B 2 0 and integrating over all wavevectors in the 2-D plane fromjkj = 0! = D (the largest resolvable wavenumber) results in the relationships hE 2 L i B 2 0 / 1 n e 2 D = 1 N c (3.5) hB 2 i B 2 0 / log ( 2 2 e = 2 D + 1) n e 2 e (3.6) / (v te =c) 2 N c log(c=v te ): (3.7) The approximation in Eq. (3.7) results from noting that 2 2 e = 2 D 1. Equations (3.5) - (3.7) demonstrate that, while the electrostatic noise level is inversely proportional to the number of particles per square Debye length (i.e. particles per cell), the magnetic noise level is roughly inversely proportional to the number of particles per square inertial length. In the convergence study, ^ v te is held fixed and so, by Eq. (3.7), the magnetic noise level should scale inversely with N c in this particular case. This simple model precisely explains the behavior observed in Fig. 3.5. An enhancement in the electrostatic noise level leads to an elevated electrostatic “collisional” effect, referred to here as electrostatic scattering. This effect, reviewed in 3.1 The Electron Thermal Velocity 55 depth by Okuda and Birdsall [1970], causes particle trajectories to have an additional Brownian-like motion. The induced motion acts to scatter particles in velocity space towards an equilibrium state. Okuda and Birdsall [1970] show that the scattering rate, , is directly proportional to both the plasma frequency and the electrostatic potential energy level. As, by Eq. (3.5), the electrostatic energy level is inversely proportional to N c , the scattering rate should scale as / ! e =N c , or equivalently = e / 1=(N c e =! e ). The scattering rate can be written in terms of ^ v te by use of Eq. (3.1) e / p e N c v te =c : (3.8) As the WAI evolves at a rate proportional to the cyclotron frequency, this ratio indicates the degree to which electrostatic effects alter the natural evolution of the WAI. In order to obtain meaningful results from the simulations = e 1 must be satisfied; however, for a fixed number of particles per cell and a fixed e , = e grows with decreasing ^ v te . This means that the minimum acceptable value of N c grows as ^ v te is decreased at a fixed e . To illustrate the effects of electrostatic scattering on the evolution of the WAI, a set of simulations was run with the same parameters as in Sec. 3.1.3, but with fewer particles per cell, N c = 36. Figure 3.6(a) shows the magnetic and electrostatic fluctuation energy densities with time for these simulations. As in Fig. 3.4(a) the electrostatic energy forms a constant background noise. The level of the electrostatic energy in this case, however, is greater by roughly a factor of four. The effect of the increased noise level is apparent; the linear growth rate and the saturation level of the WAI are no longer independent of ^ v te . In each case the linear growth rate is reduced relative to the runs of Sec. 3.1.3; however, as ^ v te decreases the growth rate is reduced to a successively larger degree. A larger reduction in growth rate is accompanied by a lower instability saturation level. In the case of ^ v te = 0:025 the electrostatic noise level exceeds the magnetic saturation level. 56 Numerical Validation Panel (b) of Fig. 3.6 shows that R T is reduced with time in all cases; however, at lower ^ v te it is reduced at a higher rate as electrostatic scattering becomes increasingly influential. This trend is consistent with panel (a) in that, a reduction in the anisotropy ratio during the linear growth phase, i.e. t e . 100, weakens the driving mechanism of the WAI. When = e becomes sufficiently large, R T is reduced too quickly by electrostatic scattering for the WAI to fully develop, leading to a diminished linear growth rate and a weaker saturation level. While every effort was made to remove nonphysical effects from the simulations of Sec. 3.1.3, both the early time reduction in R T due to electrostatic scattering and the subsequent decrease in the magnetic energy saturation level can be seen in Fig. 3.4 for the case of ^ v te = 0:025. This suggests that a value larger than N c = 144 is necessary to completely remove scattering effects when ^ v te 0:025; such a value exceeds the computational limitations of the present study. However, by observing the convergence in the results when N c is increased from 36 to 144, it is reasonable to conclude that at a sufficiently large number of electrons per Debye volume, the physical response of the WAI is independent of ^ v te when e is held fixed. Both Figs. 3.4(a) and 3.6(a) show a decreasing level of initial magnetic fluctuation energy and an increasing time of instability saturation as ^ v te is decreased. In each series of simulations, N c is held fixed. Equation (3.6) predicts that the magnetic fluctuation noise will be approximately proportional to 1=n e 2 e = ^ v 2 te =N c . This prediction is consistent with those figures, where each time ^ v te is halved the initial magnetic energy level is reduced by roughly a factor of four. To further illustrate this dependency, Fig. 3.7 shows results from a set of simulations at varying ^ v te where the number of particles per square inertial length is held fixed; all other parameters are the same as in Sec. 3.1.3. Under these conditions, the initial magnetic fluctuation energy level is shown to converge among all runs. When the initial values of B 2 =B 2 0 are corrected by the log term appearing in Eq. (3.6) the agreement further improves, with < 6% variation 3.1 The Electron Thermal Velocity 57 among the values. The time of WAI saturation is also shown to converge in Fig. 3.7 for the cases of ^ v te = 0.1 and 0.05. It is worth noting that, because the domain lengths are held fixed relative to the inertial length, the total number of simulated particles is a fixed quantity in this series of simulations. The number of cells, however, grows with decreasing ^ v te . As the number of cells increases each cell holds fewer particles, resulting in elevated electrostatic noise levels. While the case of ^ v te = 0:1 has many more particles per cell than is necessary to remove electrostatic scattering effects, the case of ^ v te = 0:025 has far too few particles per cell. It is clear that the latter case is strongly suffering from electrostatic scattering in Fig. 3.7. In PIC, a fixed number of simulated particles roughly corresponds to a fixed computational cost. Figure 3.7 illustrates that, for a fixed computational cost, an increase in ^ v te results in a significant reduction in the adverse effects of electrostatic scattering and a more quantitatively accurate model of the WAI. 3.1.6 Conclusions In this study the response of a plasma to the whistler anisotropy instability over a range of thermal velocities 0:1 ^ v te 0:025 was investigated using a two-dimensional Darwin particle-in-cell model. It was found that, with e held fixed, both the linear dispersion and the nonlinear saturation characteristics of the WAI are insensitive to variations in ^ v te . In particular, two-dimensional magnetic energy spectra show that the fluctuations grow inside a fixed region of wavevector space at a peak parallel wavenumberk k c=! e 1 in all cases; dispersion analysis reveals that the real frequency dispersion remains largely invariant of ^ v te ; time histories of the magnetic fluctuation energy show that the linear growth rate remains constant with decreasing ^ v te . These results are in agreement with the predictions of linear dispersion theory. Moreover, the nonlinear magnetic energy saturation level is maintained as ^ v te is decreased, and 58 Numerical Validation a time history of the temperature anisotropy ratio demonstrates that pitch angle scattering rates remain similar at all values of ^ v te . These results suggest that the nonlinear properties of the WAI do not depend on the value of ^ v te in the solar wind when e remains constant. An investigation concerning how the WAI model is affected by the noise levels of the fluctuating electric and magnetic fields at varying ^ v te was conducted. It was demonstrated that, with N c held fixed, the simulations are more strongly influenced by electrostatic scattering as ^ v te is decreased. This results from the reduction in the parameter e =! e as ^ v te is decreased at constant e . The WAI grows at a rate proportional to e , while the electrostatic scattering rate is proportional to ! e . As ^ v te is decreased the plasma particles are subjected to an extended duration of electrostatic scattering during the period of WAI growth, leading to a more pronounced change in the evolution of the WAI. The electrostatic scattering rate can be maintained with decreasing ^ v te by increasing the number of simulated particles per cell, which reduces the electrostatic noise level responsible for the effect. However this can quickly lead to an unreasonably large computational cost. The PIC study presented here demonstrates that, as long as a realistic value of e is used, particle-in-cell simulations running at ^ v te = 0:1 accurately reproduce the response of a cooler plasma to the WAI. Moreover, the results demonstrate that the computational cost required to obtain an accurate model is substantially reduced by the use of a relatively large value of ^ v te . Due to these considerations, the DPIC model has been determined to provide little benefit to large scale simulations of plasma turbulence. All following studies, therefore, are conducted by use of the USCPIC particle-in-cell code. 3.1 The Electron Thermal Velocity 59 0 0.5 1 1.5 2 −0.5 0 0.5 1 0 0.5 1 1.5 2 −0.5 0 0.5 1 Figure 3.1: Solutions to the electromagnetic linear dispersion equation as functions of k k at k ? = 0. For all cases R T = 3. Solid lines represent the real frequencies, ! r , and dashed lines represent the growth rates, . The growth rates have been scaled for visual clarity and the scaling factors, F, are displayed. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:01; blue lines, ^ v te = 0:005. (a) e =! e = 0:0224, e is varied. (b) e = 0:1, e =! e is varied. 60 Numerical Validation −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 Figure 3.2: Wavevector spectra of magnetic fluctuations at three simulation times, and varying ^ v te . The initial conditions in each case areR T = 3, e = 0:1. (a) ^ v te = 0:1, (b) ^ v te = 0:05, (c) ^ v te = 0:025. The vertical dashed lines bound the region predicted by Fig. 3.1 to give rise to fluctuation growth at parallel/antiparallel propagation. 3.1 The Electron Thermal Velocity 61 1.5 −13 −12 −11 −10 −9 −8 −7 −6 Figure 3.3: Magnetic fluctuation dispersion; ! r versus k k at k ? = 0 calculated over the simulation time range 50:3t e 167:7. The initial conditions in each case are R T = 3, e = 0:1. The black curves give the solution to the electromagnetic linear dispersion equation for R T = 3 under the realistic conditions e = 0:1, ^ v te = 0:005. (a) ^ v te = 0:1, (b) ^ v te = 0:05, (c) ^ v te = 0:025. 62 Numerical Validation 0 100 200 300 400 2 2.5 3 3.5 0 100 200 300 400 10 −6 10 −4 10 −2 Figure 3.4: Temporal characterization of the WAI. The initial conditions in each case are R T = 3, e = 0:1. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. (a) Fluctuating field energy densities (F 2 ): solid lines, magnetic energy; dotted lines, electrostatic energy. The dashed lines are fitted to the linear growth period of the magnetic fluctuations, with parameters as given in Table 3.1. (b) Temperature anisotropy ratio R T . 3.1 The Electron Thermal Velocity 63 Table 3.1: Parameters for the curve fits shown in Fig. 3.4(a). Curves are fitted to the function B 2 =B 2 e 2 t +B 2 over the time period 0tt nl . is the linear growth rate, B 2 is the growing fluctuation energy, B 2 is non-resonant fluctuation energy, and t nl is the approximate time that the instability begins to saturate. ^ v te = 0:1 ^ v te = 0:05 ^ v te = 0:025 t nl e 80 100 120 = e 0.0300 0.0308 0.0301 B 2 =B 2 0 1.08E-6 2.77E-7 5.63E-8 B 2 =B 2 0 1.30E-5 3.16E-6 9.42E-7 0 200 400 600 10 −6 10 −4 10 −2 Figure 3.5: Fluctuating field quantities as functions of the number of particles per cell, N c . In each case ^ v te = 0:1; all other parameters are the same as in Sec. 3.1.3. Stars correspond to data points and each curve is fitted to each respective data set. The fitting equations are shown along with the curves. The quantities presented are: red, initial magnetic fluctuation energy; green, maximum magnetic fluctuation energy; blue, time-averaged electrostatic fluctuation energy; magenta, linear growth rate. 64 Numerical Validation Table 3.2: Parameters for the curve fits shown in Fig. 3.6(a). For a description of the parameters, see Table 3.1. ^ v te = 0:1 ^ v te = 0:05 ^ v te = 0:025 t nl e 80 90 100 = e 0.0236 0.0220 0.0160 B 2 =B 2 0 6.78E-6 2.29E-6 8.08E-7 B 2 =B 2 0 3.98E-5 1.32E-5 2.98E-6 0 100 200 300 400 2 2.5 3 3.5 0 100 200 300 400 10 −6 10 −4 10 −2 Figure 3.6: Temporal characterization of the WAI for simulations run with initial conditions R T = 3, e = 0:1, and N c = 36. For each panel: red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. (a) Fluctuating field energy densities (F 2 ): solid lines, magnetic energy; dotted lines, electrostatic energy. The dashed lines are fitted to the linear growth period of the magnetic fluctuations, with parameters as given in Table 3.2. (b) Temperature anisotropy ratio R T . 3.1 The Electron Thermal Velocity 65 0 100 200 300 400 10 −6 10 −4 10 −2 Figure 3.7: Fluctuating field energy densities (F 2 ) at varying ^ v te in which the number of particles per square inertial length, n e 2 e , is held fixed. All other parameters are the same as in Sec. 3.1.3. Solid lines, magnetic energy; dotted lines, electrostatic energy. Red lines, ^ v te = 0:1; green lines, ^ v te = 0:05; blue lines, ^ v te = 0:025. 66 Numerical Validation 3.2 Ion and Electron Heating 3.2.1 Introduction Assuming conservation of the adiabatic invariant i K B T i? =2B 0 , it would be expected that solar wind ions become strongly anisotropic in the sense T i? T ik as they flow away from the Sun in a slowly decreasing magnetic field. Analysis of ion velocity distributions near 1 astronomical unit (AU), however, are typically observed to be relatively isotropic indicating that one or more plasma processes are acting to scatter such particles from parallel toward perpendicular velocities. Several candidate mechanisms of ion heating have been proposed to explain the unexpected character of the observed ion velocity distributions. Very few studies, however, have considered ion heating by whistler turbulence as whistler fluctuations typically propagate at frequencies much greater than that of ion characteristic frequencies. On the other hand, the interactions of kinetic Alfvén turbulence with ions has been extensively studied while little research has been conducted characterizing the electron response to KAW turbulence. In the following chapters, ion and electron heating by whistler turbulence and kinetic Alfvén turbulence is considered. In each study the behavior of the heating process for each species is characterized as a function of a fundamental system parameter: the characteristic wavelength of the initial turbulence (Chapter 4), the magnetic energy density of the initial turbulence (Chapters 5 and 7), and the initial plasma (Chapter 6). In order apply a computationally feasible model to the study of ion heating in plasma turbulence some parameters must be selected which are not realistic of the solar wind plasma. It is, of course, essential that differing such parameters from the real case must be shown to either have little to no effect on the physics under investigation, or affect the system in a predictable manner such that the realistic physics can be recovered. As was shown in Section 3.1, the value of the electron thermal velocity is one such parameter that is expected to leave the physics relatively 3.2 Ion and Electron Heating 67 unaltered when given an artificial value. Before proceeding with the investigations of ion and electron heating, the validity of the choices of numerical parameters used in these studies must first be discussed. 3.2.2 Validation of Parameters: Numerical Heating The 3D-USCPIC particle-in-cell code uses a discredited mesh as well as a relatively small number of particles per cell. In the simulation of an equilibrium plasma these properties lead to thermal fluctuation levels that are substantially enhanced over the amplitudes which arise in a physical plasma at thermal equilibrium. As a result of the numerical noise, electrons and ions are heated artificially; to quantify this unphysical heating, a test run was performed with 0 = 0, that is, with no initial fluctuations imposed on the system. The system parameters were as follows; a cubic simulation box with L x =L y = L z =L was used with a length ofL! e =c = 102:4. The grid spacing was = 0:10c=! e , the time step was t! e = 0:05, and the number of particles per cell was 64 with 32 electrons and 32 ions. The initial conditions for this test run included v te =c = 0:10, i = e = 0:25, and Maxwellian velocity distributions for both species. The results of this test run are shown in Fig. 3.8a. Note that the y-scale of this figure is set to be equal to that of the smallest value of 0 used in any of the studies of the following chapters; the smallest value being 0 = 0:01. It can be seen that, due to the use of finite particles, numerically induced electromagnetic noise leads to a slight growth in the electric and magnetic fields with time, with a corresponding decrease in the particle energies. The calculated heating rates of this test run yielded an electron heating rate of Q e =2:0 10 6 and an ion heating rate ofjQ i j< 10 7 , where Q j is defined by Eq. (1.15). Here Q i = (m e =m i ) 1=2 Q e . The negative sign on the electron heating rate indicates an artificial cooling. 68 Numerical Validation This cooling effect results from the use of a Gaussian filter applied to the calculated current density field at each time step. This technique acts to smooth the calculated current density and helps to reduce the consequences of numerical heating. A side effect of the Gaussian filter, however, is a slight loss of particle energy with time, accompanied by a loss in total system energy, made apparent by Fig. 3.8a. In order to illustrate the effect of the Gaussian filter on the system evolution, a second test run was performed with the same parameters as the first, but with the Gaussian filter turned off. The energy results of this test run are illustrated in Fig. 3.8b. This figure makes clear the benefits of applying the filtering technique; the noise level is amplified, resulting in larger growth of non-physical electric and magnetic fields and greater particle cooling by an order of magnitude. The energy conservation is very good at early times, but the onset of numerical heating can be seen near t! e 1000. At later times this heating effect becomes unstable and the electron and total energies rapidly diverge. In contrast, Fig. 3.8a shows no signs of numerical heating. While there is a small loss of total energy in this case, the effect remains stable out to late simulation times. For the runs with nonzero 0 values described in the following chapters, the artificial cooling in the presence of the filtering technique is at least an order of magnitude smaller than the maximum simulated heating rates; additionally, the loss in system energy is nearly an order of magnitude smaller than the smallest value of 0 = 0:01; it is therefore reasonable to neglect such effects in the following studies, as the filtering technique has been applied in all cases. 3.2.3 Validation of Parameters: Ion to Electron Mass Ratio Simulations of the studies described in Chapters 4, 5, and 6 use an artificial ion/ electron mass ratio of m i =m e = 400 in order to make the simulations computationally accessible. In Chapter 7 a further reduction of the mass ratio, m i =m e = 100, is 3.2 Ion and Electron Heating 69 required in order to obtain characteristic scales of the kinetic Alfvén wave in the PIC model. The use of an artificial mass ratio in these studies raises the question: How does the choice of mass ratio affect the physics of the problem under investigation? To help justify the validity of the results obtained under this computational approximation, a supplementary study has been carried out on an ensemble of PIC simulations at varying values of m i =m e . This series of simulations represents whistler turbulence evolution in a homogeneous, magnetized, collisionless plasma; the initialization process is, therefore, the same as that described in the studies of the following chapters. In this study the following parameters were held fixed; the simulation box was cubic in shape with a length L! e =c = 102:4, and the physical parameters included i = e = 0:25, v te =c = 0:1, and 0 = 0:10. The variable parameter was the mass ratio, m i =m e , which was given the values 25, 100, 400, 900, and 1836. There are two important results from this ensemble. First, the maximum electron heating rate, Q e;max , during each simulation remains essentially independent of the mass ratio. This is consistent with the picture that the heating of the relatively light electrons reaches its maximum value quickly, before the heavier ions develop a significant response to the turbulence. The second result is illustrated in Fig. 3.9 which shows Q i;max =Q e;max as approximately inversely proportional to the mass ratio over 25 < m i =m e < 1836. This is consistent with the explanation that the greater inertia of the heavier species makes it more difficult for a given fluctuation spectrum to heat that species. The results of this study demonstrate that, as long asm i =m e 1 is satisfied, ion heating rates, relative to those of electrons, scale linearly with the mass ratio. This suggests that a simple extrapolation differentiates results obtained at m i =m e < 1836 from the realistic case of m i =m e = 1836. 70 Numerical Validation (a) 0 500 1000 1500 2000 2500 tω e -0.01 0 0.01 [K - K(t = 0)]/B 0 2 W i W e W i+e K B K E K B+E K total (b) 0 500 1000 1500 2000 2500 tω e -0.01 0 0.01 [K - K(t = 0)]/B 0 2 W i W e W i+e K B K E K B+E K total Figure 3.8: System component energies in a simulation with m i =m e = 400, e = 0:25, and 0 = 0. The components are: total ion thermal energy, W i , total electron thermal energy, W e , total ion plus electron energy, W i+e , magnetic field energy, K B , electric field energy, K E , total electromagnetic field energy, K B+E , and total field plus plasma energy, K total . (a) The temporal evolution of the equilibrium plasma energies with a Gaussian filter applied to the current density field at each time step, (b) the same simulation with no Gaussian filter applied. 3.2 Ion and Electron Heating 71 25 100 400 900 1836 m i /m e 10 -2 10 -1 10 0 Q r Q r = Q i,max /Q e,max Figure 3.9: The red dots represent the ratio of the maximum value of the ion heating rate to the maximum value of the electron heating rate found during each simulation as a function of the ratio of the ion mass to the electron mass for the e = 0:25, 0 = 0:1, and m i =m e = 400 simulations. The dashed line represents the equation Q i;max =Q e;max = 35 (m e =m i ) drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.1. 72 Chapter 4: Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength In this chapter the first simulation study of ion and electron heating by whistler turbulence is presented. The 3-D EMPIC code USCEMPIC is used to simulate whistler turbulence in a collisionless, homogeneous, magnetized, electron-ion plasma model. The simulations use an initial ensemble of relatively long wavelength whistler modes with a broad range of initial propagation directions and follow the temporal evolution of the fluctuations as they cascade into broadband turbulent spectra at shorter wavelengths where they ultimately suffer dissipation and heat the plasma species. In this study three simulations are carried out corresponding to successively larger simulation boxes and successively longer wavelengths of the initial fluctuations. The temperature profiles and velocity distribution functions of both the electron and ion populations are analyzed in order to characterize the relative contributions of each species to whistler turbulence dissipation. The new results here are that larger simulation boxes and longer initial whistler wavelengths yield weaker overall dissipation, consistent with linear dispersion theory predictions of decreased damping, stronger ion heating, consistent with a stronger ion Landau resonance, and weaker electron heating [Hughes et al., 2014]. 4.1 Introduction Solar wind ions, as they flow away from the Sun in a slowly decreasing magnetic field, should become strongly anisotropic in the sense ofT ? T k due to conservation of their magnetic moments. But ion velocity distributions near 1 AU are typically observed to 4.1 Introduction 73 berelativelyisotropic, indicatingthatoneormoreplasmaprocessesareactingtoscatter such particles from parallel toward perpendicular velocities [Hellinger et al., 2011, 2013]; an exhaustive list of possible such processes is presented in section 2 of Cranmer [2014]. Two of the more popular of these processes are nonresonant ion scattering by low-frequency, long-wavelength magnetohydrodynamic (MHD) fluctuations including a nonlinear mechanism called “stochastic heating” [Chandran et al., 2010; Xia et al., 2013, and citations therein], and quasilinear ion cyclotron resonant scattering by Alfvén-cyclotron fluctuations at frequencies less than the proton cyclotron frequency and k k c=! i 1 [Hollweg and Isenberg, 2002; Gary and Saito, 2003; Cranmer, 2014, and citations therein]. Recently Saito and Nariyuki [2014] used two-dimensional PIC simulations to show that decaying whistler turbulence can transfer energy to ions preferentially in directions perpendicular to B 0 , thereby arguing that whistler turbulence can also be a contributing mechanism to ion heating in the solar wind. Particle-in-cell simulations have been used to address the forward cascades of magnetosonic turbulence [Svidzinski et al., 2009] and whistler turbulence [Saito et al., 2008,2010;Saito and Gary,2012;Chang et al.,2011,2013,2014;Gary et al.,2008,2012]. Whistler PIC simulations show that forward cascades give rise to short-wavelength turbulence with a preference for quasi-perpendicular propagation relative to B 0 . This k ? k k wavevector anisotropy implies that whistler modes at kc=! e < 1 should have substantial parallel electric field components, so that the Landau resonance is the primary means of wave dissipation on the magnetized electrons and the consequent heating should yieldT ek >T e? as the body of PIC whistler turbulence simulations have demonstrated. The same wavenumber anisotropy further implies that the electrostatic component of the fluctuating electric fields should provide the dominant contribution to E [e.g., Fig. 6.8 of Gary, 1993], so that the primary heating on the relatively unmagnetized ions should yield T ik <T i? , as has been argued by Saito and Nariyuki [2014] and as their simulations have demonstrated. 74 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength Whistler turbulence PIC simulations have, except for Saito and Nariyuki [2014], addressed only electron heating, but Howes [2010] used a cascade model to conclude that kinetic Alfvén turbulence should preferentially heat ions at i > 2:5, and heat electrons more strongly at smaller values of the ion beta [TenBarge and Howes, 2013]. Comisel et al. [2014] have recently studied wave properties of plasma turbulence in hybrid simulations of different system sizes at ion kinetic scale wavelengths (0 < kc=! i < 6). The following study describes three-dimensional PIC simulations of decaying whistler turbulence in three different box sizes near electron scale wavelengths with analyses of how the field fluctuation energy is dissipated on both the electrons and the ions. Consistent with the results of Saito and Nariyuki [2014], the ions are heated with T i? > T ik , although more weakly than the electrons with T ek T e? . The simulations further demonstrate that the relative ion heating increases as larger simulation systems admit longer wavelengths which allow the ion Landau resonance to become stronger. Section 4.2 describes the computational setup, Section 4.3 provides the results of the study, and Section 4.4 summarizes the findings and provides conclusions. 4.2 Simulations The simulations were carried out using the 3D-USCPIC code to model a collisionless, homogeneous, magnetized ion-electron plasma. In this study an ensemble of relatively long-wavelength, approximatelyisotropicwhistlerfluctuationsareimposedatt = 0[see Chang,2013, Sections2.1and2.3.1], andthesimulationsfollowthesubsequenttemporal evolution of the fluctuating fields and the particles. The whistler modes undergo a forward cascade to a broadband turbulent spectrum at shorter wavelengths, while at the same time the plasma is heated. An important difference between the simulations ofChang et al. [2011, 2013, 2014] and Gary et al. [2012] and the computations described here concerns the initial conditions on the ions. Prior simulations were concerned 4.2 Simulations 75 only with the interactions between the fluctuations and the electrons, so that the initial ion velocity distributions were taken to be Maxwellian. Here the intention is to compare electron and ion heating, so that care is taken to initialize both the electron and ion velocity distributions to reflect both species responses to the initial spectrum of whistler waves. Earlier PIC simulations of whistler turbulence evaluated the transfer of field fluctuation energy only to the electrons. Here, following Saito and Nariyuki [2014], the ion heating is examined as well. To facilitate the comparison of dissipation on both species, an artificial mass ratio, m i =m e = 400, is used; other physical initial conditions include isotropic Maxwellian velocity distributions for both species with e = i = 0:05, v te =c = 0:1, and 0 = 0:10 at t = 0. The small values of used here are appropriate for solar wind conditions close to the Sun, but not for 1 AU where typical values are 1. The whistler turbulence simulations of Chang et al. [2013] show that low e values yield relatively large wavevector and electron temperature anisotropies, so that the consequences of wave-particle heating are very clear. The studies of plasma heating by whistler turbulence presented in Chapters 5 and 6 move toward larger e values appropriate to the interplanetary medium near 1 AU. It is noted that the commonly assumed initial condition v te =c = 0:10 was used here [Chang et al., 2011, 2013, 2014, 2015; Gary et al., 2012; Karimabadi et al., 2013]. This choice of electron thermal speed is large compared to that typically observed in the solar wind near Earth. However, the use of such an artificially large value was shown in Chapter 3 to reduce the unphysical electrostatic noise levels in PIC simulations. Moreover, in Chapter 3 it was determined that such relatively large values of v te =c accurately reproduce the response of a cooler plasma as long as v te =c is varied through variations in e =! e rather than by variations in e . The computational parameters are as follows: the grid spacing is = 0:10c=! e , the time step ist! e = 0:05, and the number of particles per cell is 64 with 32 electrons and 32 ions. The simulations are carried out on a cube of size L x = L y = L z = L. 76 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength Three simulations are executed at varying box length L! e =c = 25:6; 51:2 and 102:4, corresponding, respectively, to 256 3 ; 512 3 , and 1024 3 simulation cells and fundamental wave numbers of, respectively, k 0 c=! e = 0:2454; 0:1227, and 0:0614. The initial spectra then correspond to arrays of 150 whistler modes that are relatively isotropic (the k ? = 0 whistler modes have well-defined dispersion properties, but there are no whistler fluctuations at k k = 0), distributed as described in Gary et al. [2012]. 4.3 Results Figure 4.1 shows reduced magnetic fluctuation spectra at three times for each simula- tion. These reduced spectra are obtained by summing power over the perpendicular wavevector component corresponding to k x : jB(k k ;k y )j 2 X kx jB(k)j 2 : (4.1) In each case the relatively isotropic initial spectrum undergoes a forward cascade to larger wavelengths and primarily to quasi-perpendicular propagation. These short- wavelength oblique modes are strongly damped [e.g. Saito et al., 2008] and have the potential to transfer energy to the perpendicular velocities of the ions. Figure 4.1 suggests that the cascade grows stronger with decreasing initial wavelength. This is not necessarily the case; as the wavelength increases a large number of electrons fall out of resonance with the initial fluctuations. Because the forward cascade of whistler turbulence is largely dependent on the electron motions [Gary, 1993], a larger number of resonating electrons results in a faster forward cascade, as is suggested by Fig. 4.1. In the case of the smallest initial wavelengths, the cascade proceeds slower as fewer electrons are in resonance with the initial spectrum, however, given enough time the fluctuation energy should mostly cascade to the same level as that of the case of the shortest initial wavelengths. 4.3 Results 77 Figure 4.2 illustrates the relative temperature profiles of electrons (4.2a and 4.2b) versus ions (4.2c and 4.2d). The temperatures illustrated here are calculated differently than those shown in Figure 2 of Hughes et al. [2014]. In that paper, species temperatures were determined by computing the variance in velocity over all particles of a particular species; this gave rise to temporal oscillations in T i? andT ek , indicating that particle responses to large-amplitude, reversible field fluctuations were making spatially inhomogeneous contributions to the computed temperatures. As a better measure of irreversible temperature, the calculation here takes the variance in velocity over each cell and then averages these numbers over all cells in the 3-D computational grid. Figure 4.2 confirms earlier simulation results that whistler turbulence heats elec- trons preferentially in directions parallel to B 0 [Saito et al., 2008; Gary et al., 2012; Chang et al., 2013], and heats ions preferentially in directions perpendicular to the background magnetic field [Saito and Nariyuki, 2014]. Figure 4.3 illustrates the same results as Fig. 4.2 but from a different perspective; here, the total electron and ion temperatures are shown along with the temperature anisotropy factor T j? =T jk . As the simulation box size is made bigger and the wavelengths become longer, electron Landau damping weakens leaving more fluctuation energy available for ion heating; although electron heating still dominates for the parameters chosen here. Furthermore, Fig. 4.4 shows that, for these three simulations, the total dissipation of the magnetic field fluctuations decreases as the simulation box size (and the overall wavelengths of the turbulence) increases. This is consistent with Fig. 7b of Saito et al. [2008] which shows linear theory damping of whistlers at quasi-perpendicular propagation decreasing as wavelengths increase. The linear theory damping is due to the Landau wave-particle resonance, and it is inferred here that the same mechanism is heating the electrons and the ions in these simulations although nonlinear processes certainly contribute to the heating as the fluctuation amplitudes increase [e.g. Chang et al., 2014]. 78 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength Figure 4.5 illustrates the electron and ion rates of energy gain Q j , defined by Eq. (1.15), over the course of each simulation. Electrons gain energy rapidly at early times, while ions gain energy at a maximum rate at later times. As the initial wavelengths in each simulation are increased the time at which maximum ion energy gain occurs is pushed to later times, and the value Q i;max increases marginally but remains near maximum for a longer period. In contrast, while electron heating is at maximum at early times for all simulations, the maximum rate, Q e;max , decreases significantly with increasing initial wavelength. This can be seen in Figure 4.6, where the electron maximum heating rate drops off with increasing wavelength following a power law of the form Q e /L 5=2 . The average electron heating rate becomes increasingly steady with increasing initial wavelength; because fewer electrons are resonating with the longer wavelength initial fluctuations, the electrons within the resonance condition can continue to gain energy more steadily. Figures 4.7 and 4.8 illustrate the reduced electron and ion velocity distributions, respectively, in the parallel and perpendicular directions at selected times for each of the three simulations. The figures show that the transfer of fluctuation energy to both the electrons and ions is indeed a heating process, because the late-time velocity distributions of both species for the most part retain their thermal, Maxwellian-like character even as they gain energy. The primary late-time departures from Maxwellian forms are on the electron parallel velocity distributions in the presence of enhanced high-speed “tails” for the runs at 256 3 and 512 3 . This feature is a typical electron response to obliquely propagating whistlers at e 1, and is discussed in detail in Chang et al. [2013]. 4.4 Conclusions Fully three-dimensional, fully kinetic particle-in-cell simulations have been used to examine how decaying whistler turbulence in a low- collisionless plasma dissipates 4.4 Conclusions 79 energy on both electrons and ions. These computations confirm previous results showing that electron heating is preferentially parallel to the background magnetic field B 0 , and ion heating is preferentially perpendicular to B 0 . The new results here are that larger simulation boxes and longer initial whistler wavelengths yield weaker electron heating, stronger ion heating, and weaker overall dissipation. Anisotropic electron and ion heating by whistler turbulence in the weak fluctuation limit is interpreted here based on kinetic linear dispersion theory. The k ? k k wavevector anisotropy which results from the whistler turbulence forward cascade implies the Landau resonance dominates wave-particle interactions at kc=! e < 1 and yields preferential parallel heating on the electrons and perpendicular heating on the ions. Furthermore, the overall decrease in fluctuation damping with increasing box size is consistent with linear theory predictions of decreasing whistler damping with increasing wavelengths. In the limits of kc=! i 1 and kc=! e 1, linear dispersion theory predicts that the whistler frequency is ! r = e =kk k c 2 =! 2 e : (4.2) Heating of the ions, which are essentially unmagnetized in response to the relatively high-frequency whistler fluctuations, is by means of the Landau resonance at v ? =! r =k ? = (k k c=! e )c e =! e (4.3) so a larger simulation box and longer wavelengths correspond to a smaller resonant ion velocity. This implies that the resonant modes move from the tail of the perpendicular velocity distribution toward the thermal part of the distribution, resonating with a larger number of ions and therefore leading to stronger ion heating, as this study has shown. Howes [2010] has proposed a scenario for plasma heating by kinetic Alfvén wave turbulence in which the maximum dissipation on the ions is at wavelengths of the order of the thermal ion gyroradius; the turbulent energy which remains is 80 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength then carried by the forward cascade down to electron dissipation at thermal electron gyroradii. Although the simulations presented here are limited to relatively short- wavelength turbulence, the result that electron heating increases at shorter wavelengths is consistent with the Howes [2010] scenario. Further simulations are necessary to gain a more complete understanding of how short-wavelength turbulence dissipates its energy in collisionless plasmas. Chapters 5 and 6 examine the relative heating of electrons and ions by whistler turbulence as functions of the dimensionless parameters e and 0 , with parameter values more appropriate for the solar wind near 1 AU, that is, e = 1 and 0 < e . Figures 81 (a) -3 -2 -1 0 1 2 3 k y c/ω e tω e = 0 tω e = 1000 tω e = 2000 -8 -6 -4 -2 log 10 (δB 2 /B 0 2 ) (b) -3 -2 -1 0 1 2 3 k y c/ω e -8 -6 -4 -2 log 10 (δB 2 /B 0 2 ) (c) -3 -2 -1 0 1 2 3 k || c/ω e -3 -2 -1 0 1 2 3 k y c/ω e -3 -2 -1 0 1 2 3 k || c/ω e -3 -2 -1 0 1 2 3 k || c/ω e -8 -6 -4 -2 log 10 (δB 2 /B 0 2 ) Figure 4.1: Reduced magnetic fluctuation energy spectra at three times as labeled for the three simulations with different simulation box sizes: (a) L! e =c = 25:6, (b) L! e =c = 51:2, and (c) L! e =c = 102:4. 82 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength (a) 0 500 1000 1500 t ω e 1 1.5 2 2.5 T e|| Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (b) 0 500 1000 1500 t ω e 1 1.5 2 2.5 T e⊥ Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.2: Simulation histories of the (a) parallel electron, (b) perpendicular electron, (c) parallel ion, and (d) perpendicular ion temperatures as functions of time from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). Figures 83 (c) 0 500 1000 1500 t ω e 1 1.02 1.04 1.06 1.08 1.1 T i|| Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (d) 0 500 1000 1500 t ω e 1 1.02 1.04 1.06 1.08 1.1 T i⊥ Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.2: Continued. 84 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength (a) 0 500 1000 1500 t ω e 1 1.2 1.4 1.6 1.8 2 T e Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (b) 0 500 1000 1500 t ω e 1 1.02 1.04 1.06 1.08 1.1 T i Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.3: Simulation histories of the (a) electron temperature, and (b) ion tempera- ture averaged over the three spatial dimensions; (c) electron temperature anisotropy, and (d) ion temperature anisotropy defined asT j?=k T j? =T jk . The curves correspond to the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). Figures 85 (c) 0 500 1000 1500 t ω e 0.4 0.6 0.8 1 1.2 T e⊥/|| Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (d) 0 500 1000 1500 t ω e 0.99 1 1.01 1.02 1.03 1.04 T i⊥/|| Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.3: Continued. 86 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength 0 500 1000 1500 t ω e 0 0.02 0.04 0.06 K B+E /B 0 2 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.4: Simulation histories of the total magnetic field fluctuation energy as functions of time from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). Figures 87 (a) 0 500 1000 1500 t ω e 10 -5 10 -4 10 -3 10 -2 Q e Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (b) 0 500 1000 1500 t ω e 10 -6 10 -5 10 -4 Q i Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.5: Simulation histories of the (a) electron total heating rate, and (b) ion total heating rate, where heating is defined here as Q j (dT j =dt)= [T j (t = 0)! e ]. The curves correspond to the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells and L! e =c = 102:4 (blue lines). 88 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength 25.6 51.2 102.4 Lω e /c 10 -4 10 -3 10 -2 Q max Q e,max Q i,max Figure 4.6: The maximum values of the dimensionless time rates of change of the ion totaltemperature(reddots), andtheelectrontotaltemperature(bluedots)asfunctions of L! e =c for the e = 0:05, 0 = 0:05 and m i =m e = 400 simulations. The dashed lines represent the equations Q i = (2:5L! e =c + 700) 10 7 andQ e = 45(L! e =c) 5=2 drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.2. Figures 89 (a) t! e = 0 0 1 2 3 4 5 6 7 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (b) t! e = 1000 0 1 2 3 4 5 6 7 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v e⊥ /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (c) t! e = 2000 0 1 2 3 4 5 6 7 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v e⊥ /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.7: Reduced electron velocity distributions from the three simulations at (a) t! e = 0 (b) t! e = 1000, and (c) t! e = 2000 as labeled for (left column) parallel and (right column) perpendicular component velocities. Results are from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells andL! e =c = 102:4 (blue lines). Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. 90 Ion and Electron Heating by Whistler Turbulence: Variations with Wavelength (a) t! e = 0 0 1 2 3 4 5 6 7 v i|| /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v i|| /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (b) t! e = 1000 0 1 2 3 4 5 6 7 v i|| /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v i⊥ /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 (c) t! e = 2000 0 1 2 3 4 5 6 7 v i|| /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 0 1 2 3 4 5 6 7 v i⊥ /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 Lω e /c = 256 Lω e /c = 512 Lω e /c = 1024 Figure 4.8: Reduced ion velocity distributions from the three simulations at (a) t! e = 0 (b) t! e = 1000, and (c) t! e = 2000 as labeled for (left column) parallel and (right column) perpendicular component velocities. Results are from the runs with 256 3 cells and L! e =c = 25:6 (red lines), 512 3 cells and L! e =c = 51:2 (green lines), and 1024 3 cells andL! e =c = 102:4 (blue lines). Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. 91 Chapter 5: Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density The simulation study of this chapter continues the investigation of ion and electron heating by whistler turbulence. An ensemble of five simulations is carried out with conditions chosen to be more representative of those in the solar wind near 1 AU. In each successive simulation the value of the dimensionless fluctuating magnetic field energy density, 0 , is increased. The simulations then follow the evolution of the fluctuations as they cascade to broadband, anisotropic turbulence which dissipates at shorter wavelengths, heating both electrons and ions. The electron heating is shown to be stronger and preferentially parallel/antiparallel to B 0 ; the ion energy gain is weaker and is preferentially in directions perpendicular to B 0 . The important new results of this study are that, over 0:01< 0 < 0:25, the maximum rate of electron heating scales approximately as 0 , and the maximum rate of ion heating scales approximately as 1:5 0 [Gary et al., 2016]. 5.1 Introduction In Chapter 4, 3-D PIC simulations were carried out at e = 0:05 with computation domains large enough to accommodate magnetosonic waves as well as whistler mode turbulence. This study showed 1. Ion heating is stronger in T i? than in T ik . 92 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density 2. Increasing the simulation domain size yields reduced electron heating. 3. Increasing the simulation domain size yields enhanced ion heating. 4. The late-time heated ions exhibit bi-Maxwellian-like velocity distributions. This study extended the work of Saito and Nariyuki [2014], who were the first to address how whistler turbulence may heat ions as well as electrons. Using 2-D PIC simulations with B 0 in the simulation plane Saito and Nariyuki [2014] showed that forward cascading whistler fluctuations could transfer energy to ions in directions perpendicular to the background magnetic field. Several different calculations have been used to examine plasma species heating by fluctuations in turbulent collisionless plasmas. Markovskii and Vasquez [2010] and Markovskii et al. [2010] carried out 2-D hybrid simulations (PIC ions, fluid electrons) of magnetosonic turbulence in collisionless plasmas with B 0 in the simulation plane and i = 0:02. More recently Vasquez et al. [2014] and Vasquez [2015] executed fully 3-D hybrid simulations of the forward cascade of an initial spectrum of Alfvénic turbulence primarily with p = 0:10. The forward cascades developed the same k ? k k wavevector anisotropy as in the whistler turbulence PIC simulations mentioned above, and in the recent computations the ion heating rate scales as 0 where = 1:6 and 0 represents the initial energy density of the fluctuations. Howes [2010] used a cascade model for Alfvénic turbulence to calculate the ratio of ion and electron heating rates, Q i =Q e , using damping rates derived from linear gyrokinetic dispersion theory. For typical solar wind parameters the heating rates are weak functions of T e =T i . If i = e < 1, only the electrons are Landau-resonant with Alfvén waves, and electron heating dominates; at i = e = 3 the two heating rates are approximately equal, and as further increases, ions become Landau resonant and that species gains the stronger heating. 5.1 Introduction 93 Karimabadi et al. [2013] carried out several 2-D PIC simulations on very large simulation domains (L x ! i =c = 50, L y ! i =c = 100) where the long-wavelength fluc- tuations are excited by a Kelvin-Helmholtz velocity shear instability. The inertial range turbulence generated by the instability cascades down to the kinetic range where both Alfvén-cyclotron and magnetosonic waves are excited. For initial values e = i = 2, the electrons are heated about twice as much as the ions, and the electrons are preferentially heated in the parallel/antiparallel direction, indicating that the energization of that species is primarily due to parallel electric fields associated with the nonlinear generation of current sheet structures, rather than to quasilinear heating by magnetosonic or other small-amplitude waves. The 2-D PIC simulations of Wu et al. [2013] used forward cascading Alfvénic turbulence with B 0 oblique to the simulation plane in a e = i = 0:1 plasma to show that at relatively weak turbulence amplitudes the electrons are preferentially heated, whereas at larger amplitudes ion heating becomes more important. Here, in contrast to the low- computations of Chapter 4, 3-D PIC simulations are carried out using the USCPIC code on a simulation domain of dimension L = 102:4c=! e = 5:12c=! i to study electron and ion heating rates due to the forward cascade of whistler turbulence at e = 0:25, a value more appropriate to typical solar wind conditions, and at different values of the initial magnetic fluctuation energy density (Eq. 1.5). To quantify the plasma species heating rates, the definition of Eq. (1.15) is applied; that is, the dimensionless time rate of electron total energy gain is Q e (dT e =dt)= [T e (t = 0)! e ], and the dimensionless time rate of ion total heating is Q i (dT i =dt)= [T i (t = 0)! e ]. 94 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density 5.2 Simulations As in the whistler turbulence study described in Chapter 4, an ensemble of relatively long-wavelength, narrowband approximately isotropic whistler fluctuations correspond- ing to 150 normal modes with random phases were initially imposed upon the system, with the amplitudes of each field component of each mode derived from solutions of the kinetic electromagnetic linear dispersion equation [Gary, 1993]. Many parameters remain similar to those in the runs of Chapter 4, with m i =m e = 400 within a cube of shape L x =L y =L z =L and a fixed simulation domain size L! e =c = 102:4 (the largest of the three domains used in Chapter 4) with a minimum wavenumber of kc=! e = 0:0614, that is, kc=! i = 1:23. The grid spacing was = 0:10c=! e , the time step was t! e = 0:05, and the number of particles per cell was 64 with 32 electrons and 32 ions. The initial conditions for all simulations described here were v te =c = 0:10, i = e and Maxwellian velocity distributions for both species. However, two initial parameters were chosen to be significantly different from those in Chapter 4. While those runs began with e = 0:05, the simulations of the present study begin with e = 0:25 (that is, ! 2 e = 2 e = 12:5); furthermore the previous study used only 0 = 0:10, whereas here 0 is chosen to have several different initial values; specifically 0 = 0:01; 0:025; 0:05; 0:1, and 0:25. 5.3 Results The ensemble of runs consisted of five whistler turbulence PIC simulations with all initial parameters constant except for the initial fluctuating magnetic field energy density, 0 . The field properties of all five runs are consistent with the runs at e = 0:10 from Gary et al. [2012] and at e = 0:05 from Chapter 4: the magnetic field total fluctuation energy exhibits an exponential-like temporal decay (Figure 5.1a), and the fluctuations undergo a forward cascade to shorter wavelengths developing spectra with 5.3 Results 95 anisotropies in the sense of k ? k k which, at 0 1, become more anisotropic as the initial fluctuating filed energy density increases (Figure 5.1b). Figures 5.2 and 5.3 illustrate the temporal evolution of the parallel and perpen- dicular components of the electron and ion temperatures for these five simulations. After some early-time adjustments, the electron and ion temperatures display almost completely monotonic increases with time, indicating that the improved temperature diagnostic described in Chapter 4.3 is working successfully to remove any energy contributions of the plasma oscillations and is instead illustrating the energy gains associated with irreversible heating. The plasma properties illustrated in Figs. 5.2 and 5.3 are also consistent with the anisotropy results of Chapter 4: the electrons are heated more strongly than the ions and obtain the condition T ek >T e? , whereas the ions are heated such that T ik <T i? . The quasilinear premise in conjunction with the wavevector anisotropy illustrated in Fig 5.1b implies that the dissipation at weak fluctuation amplitudes is due to the Landau resonance ! r k k v k = 0 (5.1) of whistler fluctuations at propagation oblique to B 0 . Linear theory of whistler modes at quasi-perpendicular propagation predicts that 0<jE k j 2 jE ? j 2 which, in turn, implies that such modes preferentially heat the magnetized electrons in directions parallel and anti-parallel to the background magnetic field while heating the unmagnetized ions preferentially in directions perpendicular to B 0 . Although not shown here, the late-time electron velocity distributions for all five runs are very similar to the e = 1:0 distributions shown in Fig. 5 of Chang et al. [2013]; that is, they are essentially bi-Maxwellian. The extended “tails” at high speeds of f e (v k ) which are driven by whistler turbulence at e < 0:10 (Fig. 4.7; as well as Fig. 5 of Chang et al. [2013]) appear to be a consequence of electron acceleration by the Landau resonance of high phase speed fluctuations at low e as discussed in 96 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density Chang et al. [2011]. The late-time character of the ion velocity distributions is also bi-Maxwellian-like. Figure 5.4 plots Q e (t) and Q i (t) for each of the five runs. The electron total heating rate reaches its maximum value (which is denoted here as Q e;max ) at relatively early times (t! e 200), whereas the ion total heating rate attains its maximum value (denoted as Q i;max ) at relatively later times (400 < t! e < 1000). This is of course consistent with the picture that the lighter electrons respond more quickly to the turbulent fluctuations. From this point onward, the primary focus will be on the maximum heating rates of the two species. The purpose of this is to facilitate the comparison between the initial value simulations of decaying turbulence in an isolated system, presented here, with a more realistic steady-state scenario in which the fluctuation energy is continuously replenished at longer wavelengths, and then, after forward cascade, is dissipated by processes such as Landau damping at short wavelengths. Figure 5.5 summarizes the values of Q i;max and Q e;max from each of the five runs. This figure shows that, over 0:01< 0 < 0:25, the following results are obtained: 1. The electron total heating rate scales approximately as 0 , and 2. The ion total heating rate scales approximately as 1:5 0 . There are several important conclusions here. The scaling of the maximum electron heating rate with initial fluctuating field energy density at small values of 0 is consistent with predictions of quasilinear-like second order perturbation theories [Gary and Paul, 1971; Gary and Feldman, 1978]. This suggests that quasilinear-like electron heating rate calculations such as those of Howes [2010] are likely to be valid for sufficiently weak magnetic field fluctuations. Second, the maximum ion heating rate scaling with 1:5 0 compares favorably to the 1:6 0 scaling for the ion heating rate due to the Alfvénic turbulence forward cascades in the hybrid simulations of Vasquez et al. [2014] and Vasquez [2015]. Third, this ion heating rate is also consistent with the 5.4 Entropy 97 approximately 1:5 0 scaling for the rate of perpendicular ion heating by Alfvén waves and kinetic Alfvén waves in the phenomenological theory of Chandran et al. [2010]. The fourth conclusion is that, for sufficiently large 0 , the maximum ion heating rate increases faster than the quasi-linear rate proportional to 0 . This transition to an apparently nonlinear ion heating rate is consistent with the results of Wu et al. [2013] who used 2-D PIC simulations in much larger simulation domains to show that at sufficiently large amplitudes of Alfvénic turbulence ion heating dominates electron heating. 5.4 Entropy Howes et al. [2006] and others have claimed that particle-particle collisions are the only processes which can act to thermalize a plasma. This argument brings into question the claims of the studies presented here as, in the present case, collisions are absent from the system dynamics. The argument is whether the heating observed in these collisionless plasma studies is truly irreversible heating, or whether it is a reversible exchange of energy between fields and particles. There is validity to the claim of Howes et al. [2006] because, although kinetic range turbulence can energize electrons and ions, this process does not fully thermalize the velocity distributions. Low- whistler turbulence produces suprathermal “tails” in f e (v k ); whistler turbulence in general preferentially heats ions in the perpendicular direction, and both whistler turbulence and kinetic Alfvén wave turbulence produce T k > T ? on f e . On the other hand, in weakly collisional plasmas such as the solar wind, particle-particle collision times (and presumably collisional heating rates) are much longer than the wave-particle interaction rates which are presented here for whistler turbulence. Using a simple 1-D model, Pezzi et al. [2016] showed that collsionality increases in the presence of greater structure in the velocity distribution. They infer this by 98 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density showing that entropy increases at a greater rate when some structured element is superimposed on a bi-Maxwellian velocity distribution. This enhanced rate of entropy gain reverts to the usual rate after the structure is smoothed via collisions. Note that an enhancement in collisionality is not observed due to the bi-Maxwellian structure of the velocity distribution. In order to gain insight into the nature of the heating processes being obeserved in the simulations of Section 5.2, an entropy study is conducted across the series of simulations. The method used for calculating entropy generation, defined here as S =k B N Z f(v)ln[f(v)]dv; (5.2) is described in Section B.4.1. If the processes observed are in fact reversible, they should not result in an increase in system entropy. Figure 5.6 shows the temporal change in entropy for the electron (5.6a) and ion (5.6b) species. The entropy increases with time monotonically at most times, with larger entropy gains as 0 is increased, in agreement with the greater energy gains with increasing 0 seen in Figs. 5.2 and 5.3. glso shown in these figures is the entropy change for the case of 0 = 0, described in Chapter 3:2:2. In this case entropy slowly decreases as a result of the Gaussian filter applied in the simulations, but remains close to zero, indicating that the plasma remains at thermal equilibrium, as would be expected in the absence of numerical heating (see Chapter 3:2:2 for a further explanation). It is noted that, neart! e 1700, there is a small oscillation in the entropy for each case. From Fig. 5.7, however, which shows the total electron plus ion system entropy, it can be seen that this oscillation is removed and the curves, in general, become much smoother (there remains plateaus, in which entropy holds steady, in some cases). This smoothing of the oscillations suggests a particle-particle interaction between the ions and the electrons, where one species experiences a decrease in entropy at the expense of an increase in entropy of the other species. 5.5 Conclusions 99 While the study of Pezzi et al. [2016] can help explain heating rates greater than what would be expected due to collisions on scales of the Sun-Earth distance, the study here provides evidence that collisionless wave-particle dissipation processes can also contribute in bringing the plasma toward thermal equilibrium. This is not to say that these collisionless processes can bring the plasma to complete thermal equilibrium, only that they can move the system closer to thermal equilibrium; this is in the same sense as the claim made by Pezzi et al. [2016], where the enhanced collisionality subsides before complete thermal equilibrium is reached. The entropy study presented here therefore supports the claim that the processes observed in the collisionless whistler turbulence simulations can in fact be considered irreversible heating of the plasma species. 5.5 Conclusions 3-D particle-in-cell simulations were used to examine the temporal evolution of initially narrowband spectra of whistler fluctuations in homogeneous, collisionless plasmas with initial e = 0:25; m i =m e = 400; and L! i =c = 5:12. The fluctuations undergo a forward cascade to shorter wavelengths, developing a broadband, anisotropic spectrum of turbulence which dissipates energy on both electrons and ions. The simulations confirm earlier results for whistler turbulence that the electrons gain more energy than the ions, that the electrons gain energy primarily in the directions parallel/antiparallel to the background magnetic field, and that ions are heated primarily in directions perpendicular to B 0 . This study carried out an ensemble of simulations on a range of initial fluctuation energy densities; the important new results here are that, over 0:01< 0 < 0:25, the maximum rate of electron heating scales approximately as 0 and the maximum rate of ion heating scales approximately as 1:5 0 . These results apply only to the initial parameters stated above; a particular limitation of these calculations are that they are carried out on a cubic domain 100 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density approximately five ion inertial lengths on a side. This limits the physics to that of heating by whistler fluctuations; in order to obtain a more complete picture of dissipation of the magnetosonic-whistler cascade it will be necessary to carry out computations on considerably larger simulation domains (see, for example, the 3-D hybrid simulations of Vasquez et al. [2014] and Vasquez [2015]). Although relatively weak compared to electron heating, the perpendicular heating of the ions demonstrated in these simulations is consistent with the hypothesis that the obliquely propagating component of magnetosonic-whistler turbulence contributes to the persistent perpendicular heating of protons observed in the solar wind. Figures 101 (a) 0 500 1000 1500 2000 2500 t ω e 0 0.1 0.2 0.3 K B+E /B 0 2 ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 (b) 0 500 1000 1500 2000 2500 t ω e 0 2 4 6 8 Tan 2 (θ B ) ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.1: Temporal evolution of field quantities from the e = 0:25 andm i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustratedbythevariouspanelsare: (a)thetotalmagneticpluselectricfieldfluctuation energy, K B+E , and (b) the wavevector anisotropy factor, Tan 2 ( B ), as defined by Eq. (1.10). The squares and circles represent the times of maximum heating rates for the electrons and ions, respectively. 102 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density (a) 0 500 1000 1500 2000 2500 t ω e 1 1.1 1.2 1.3 1.4 1.5 T e|| ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 (b) 0 500 1000 1500 2000 2500 t ω e 1 1.1 1.2 1.3 1.4 1.5 T e⊥ ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.2: Temporal evolution of plasma component temperatures for the e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustrated by the various panels are: (a) the electron parallel temperatures, (b) the electron perpendicular temperatures. Figures 103 (a) 0 500 1000 1500 2000 2500 t ω e 1 1.05 1.1 1.15 1.2 1.25 T i|| ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 (b) 0 500 1000 1500 2000 2500 t ω e 1 1.05 1.1 1.15 1.2 1.25 T i⊥ ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.3: Temporal evolution of plasma component temperatures for the e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. The quantities illustrated by the various panels are: (a) the ion parallel temperatures, and (b) the ion perpendicular temperatures. 104 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density (a) 0 500 1000 1500 2000 2500 t ω e 10 -7 10 -6 10 -5 10 -4 10 -3 Q e ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 (b) 0 500 1000 1500 2000 2500 t ω e 10 -7 10 -6 10 -5 10 -4 10 -3 Q i ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.4: Temporal evolution of (a) electron and (b) ion total heating rates for the e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. Here (a) represents Q e and (b) shows Q i . Figures 105 0.01 0.025 0.05 0.1 0.25 ǫ 0 10 -6 10 -5 10 -4 Q max Q e,max Q i,max Figure 5.5: The maximum values of the dimensionless time rates of change of the ion total temperature (red dots), and the electron total temperature (blue dots) as functions of 0 for the e = 0:25 and m i =m e = 400 simulations. The dashed lines represent the equations Q i = 0:001 1:5 0 and Q e = 0:002 0 drawn to guide the eye. The numerical values of the data presented are tabulated in Table A.3. 106 Ion and Electron Heating by Whistler Turbulence: Variations with Fluctuation Energy Density (a) 0 500 1000 1500 2000 2500 t ω e 0 0.1 0.2 0.3 S e ǫ 0 = 0.000 ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 (b) 0 500 1000 1500 2000 2500 t ω e 0 0.1 0.2 0.3 S i ǫ 0 = 0.000 ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.6: Temporal evolution of (a) electron and (b) ion species entropy for the e = 0:25 and m i =m e = 400 simulations with 0 = 0:01; 0:025; 0:05; 0:10; and 0:25 as labeled. Here (a) represents S e and (b) shows S i . Figures 107 0 500 1000 1500 2000 2500 t ω e 0 0.1 0.2 0.3 S tot ǫ 0 = 0.000 ǫ 0 = 0.010 ǫ 0 = 0.025 ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 Figure 5.7: Temporal evolution of total (ion plus electron) system entropy for the e = 0:25 and m i =m e = 400 simulations. 108 Chapter 6: Ion and Electron Heating by Whistler Turbulence: Variations with Electron This chapter concludes the studies of ion and electron heating by whistler turbulence. In this study two ensembles of whistler turbulence simulations are carried out and analysed. From these ensembles, each consisting of six simulations, electron and ion dissipation rates are computed as functions of the initial electron beta, e , over a range of values. As e increases the plasma species gain energy relative to that of the background magnetic field, and are afforded a greater freedom of motion; this suggests that the heating process should be a sensitive function of e . Ensemble One holds the value of 0 (Eq. 1.5) constant while Ensemble Two follows solar wind observations, imposing the initial condition 0 = 0:2 e . In Ensemble One, both dissipation rates are found to scale approximately as 1 e . The results of Ensemble Two, however, show Q e to remain approximately constant whileQ i scales approximately as 1=2 e . These results, when combined with the conclusions of Chapters 4 and 5, suggest that sufficiently long wavelength and sufficiently large-amplitude magnetosonic-whistler turbulence at sufficiently large e may heat ions more rapidly than electrons [Hughes et al., 2017a]. 6.1 Introduction The study described here examines the e dependence of electron and ion dissipation rates due to the forward cascade and dissipation of whistler turbulence. In these simu- lations, initially narrowband, relatively isotropic field fluctuation spectra at relatively long wavelengths are loaded into the domain as an initial condition. These fluctuations 6.1 Introduction 109 cascade into braodband, anisotropic spectra extending to shorter wavelengths where they transfer energy to both electrons and ions. Saito and Gary [2012] used 2-D PIC simulations in an initial value problem to examine the e dependence of forward-cascading whistler turbulence. In that study the initial values of T i and T e were equal and fixed, so that variations in e would be due only to variations in the dimensional value of B 0 . Then choosing 0 = 0:10, the authors ran three simulations with respective initial conditions e = 0:01; 0:10; and 1:0. In these three runs, increasing values of e led to magnetic spectra with successively smaller wavevector anisotropies in the sense of k ? > k k , successively weaker electron anisotropies in the sense of T ? <T k , and successively smaller values of Q e . Subsequently, Chang et al. [2013] carried out 3-D PIC simulations of whistler turbulence in which the initial value of e was varied by changes only in the value of B 0 . In contrast to Saito and Gary [2012], the initial magnetic fluctuation energy was chosen to satisfy 0 = 2 e . Under these assumptions, the simulations of Chang et al. [2013] indicated that the electron heating rate decreased (on dimensional timescales) as e was increased from 0:10 to 1:0, in agreement with the trend for Q e found in Saito and Gary [2012]. Saito and Nariyuki [2014] used 2-D PIC simulations to show that ions, as well as electrons, could be heated by the forward cascade of whistler turbulence. Building from this work, Chapter 4 showed that larger simulation domains, corresponding to longer initial wavelength whistler turbulence, yields weaker electron heating but stronger ion dissipation. In Chapter 5 approximate scaling relations for the changes in maximum electron and ion dissipation rates due to whistler turbulence as functions of 0 were obtained. In particular it was shown that, at e = 0:25 and over 0:01 0 0:25, the maximum Q e is approximately proportional to 0 , indicating a quasilinear-like heating, whereas the maximum Q i scales approximately as 1:5 0 ; the latter is a scaling similar to the nonlinear dissipation rates derived from both analytic [Chandran, 2010; Xia et al., 2013] and hybrid simulation [Vasquez et al., 2014; Vasquez, 2015] studies 110 Ion and Electron Heating by Whistler Turbulence: Variations with Electron of turbulent ion dissipation. These similarities suggest that ion heating by short- wavelength turbulence may be in some sense independent of the details of the energy transfer process. The 2-D PIC simulations of Wu et al. [2013] showed that relatively weak Alfvénic turbulence preferentially heats electrons, whereas larger amplitudes of such turbulence lead to stronger heating of the ions. The recent 2-D PIC simulations of Alfvénic turbulence by Matthaeus et al. [2016] further demonstrated that the ratio of proton heating rate to electron heating rate increases in proportion to the total heating. Here 3-D PIC simulations are used to address how plasma species’ dissipation rates associated with forward-cascading whistler turbulence change as the value of e is varied. An analysis of magnetic field measurements from the ACE spacecraft has shown [Smith et al., 2006] that, within the relatively long-wavelength inertial range of solar wind turbulence near Earth, 0 0:10 p . In the study presented here the ion and electron is varied by changes in the value of B 0 . However, in order to give a broader picture of the dissipation rates for both plasma species, two simulation ensembles are carried out with two distinct initial conditions. The runs of Ensemble One are subjected to the initial condition 0 = 0:10 (6.1) whereas the simulations of Ensemble Two are initialized with 0 = 0:20 e : (6.2) Equation (6.1) was the initial condition used in the 2-D simulations of Saito and Gary [2012], whereas Equation (6.2) is of the same form as the initial condition used by Chang et al. [2013] and as suggested by the measurements of Smith et al. [2006] in the inertial range near Earth. 6.2 Simulations 111 6.2 Simulations Two ensembles of 3-D PIC simulations, each consisting of six runs, were carried out using the USCPIC code. As before, each simulation is initialized with a collection of relatively long-wavelength, narrowband, approximately isotropic whistler fluctuations corresponding to 150 normal modes with random phases. The amplitudes of each field component of each mode were derived from solutions of the kinetic electromagnetic linear dispersion equation [Gary, 1993] subject to the appropriate plasma parameters (namely the species and v A =c where v A is the Alfvén speed). The initial parameters of these simulations were similar to those used in Chapters 4 and 5. Thus, fully kinetic electron and ion species with m i =m e = 400 were represented within a cube of dimension L x =L y =L z =L and simulation domain size L! e =c = 102:4; the selected domain size corresponds to a minimum resolvable wavenumber of kc=! e = 0:0614, that is, kc=! i = 1:23. The grid spacing was ! e =c = 0:10, the time step was t! e = 0:05, and the number of particles per cell was 64 with 32 electrons and 32 ions. The initial conditions for all simulations included i = e and Maxwellian velocity distributions for both species. Superimposed on the velocity distributions of each species was a distribution of velocity perturbations that served to improve consistency between the initial particle motions and the initially imposed field fluctuations. The perturbation amplitudes were derived from solutions to the kinetic linear dispersion equation alongside the field component amplitudes, and the perturbation phases were determined by field-velocity phase relations at the location of each particle inside the domain. Each ensemble consisted of six simulations with initial values e = 0:1; 0:25; 0:50; 1:0; 2:5; and 5:0, so that, under the condition v te =c = 0:1, the corre- sponding dimensionless frequency ratios were ! e = e = 2:24; 3:54; 5:0; 7:07; 11:2; and 15:8; where the ratio ! e = e indicates the separation between characteristic electron time scales. 112 Ion and Electron Heating by Whistler Turbulence: Variations with Electron 6.3 Results All 12 runs exhibited fluctuating field responses qualitatively similar to earlier whistler turbulence simulations: there is a forward cascade to shorter wavelengths, leading to the development of a wavevector anisotropy in the sense of k ? k k , and the increase in Landau damping with increasing wavenumber [Saito et al., 2008] leads to a gradual dissipation of the fluctuating field energy. The late-time plasma heating properties are also similar to the results of Chapters 4 and 5: the electrons are heated more strongly than the ions, the electrons are heated such that T k >T ? , and the ions are heated in the opposite anisotropic sense. Under the assumption that the fluctuations are sufficiently weak that linear damping constitutes the primary dissipation mechanism [Chang et al., 2014], the wavevector anisotropy implies that Landau damping of obliquely propagating whistlers [Saito et al., 2008] is the source of both the parallel heating of the magnetized electrons and the perpendicular heating of the ions that are responding to the relatively high-frequency whistler fluctuations as though they were unmagnetized. Figure 6.1 shows reduced electron and ion velocity distributions for the simulation with e = 0:50 and 0 = 0:10; this run is a member of both Ensemble One and Ensemble Two. Electron heating is primarily in the parallel/antiparallel directions, with f e (v k ) retaining a Maxwellian-like character. In contrast, ion energy gain is concentrated in directions perpendicular to B 0 , with the primary change occurring in the suprathermal tail off i (v ? ). This is consistent with the idea that the parallel phase speed of the whistler fluctuations is much larger than the ion thermal speed. Thus, use of thermal terms in describing ion dissipation should be understood in terms of such suprathermal tails of the ion velocity distribution. Figure 6.2 displays the maximum values of the electron and ion dissipation rates as functions of e computed from the runs of Ensemble One. Note that these quantities are obtained by plotting the temporal derivatives of the total T e andT i as functions of 6.3 Results 113 time and then reporting the maximum values for each run as Q e and Q i , as was done in Chapters 4 and 5. In this case the electron energy gain rates reach maxima within the time range 100<! e t< 500, whereas theQ i reach maxima at later times, typically 500<! e t< 1000; but both Q e and Q i decrease approximately as 1 e . Comparing the scaling of Q e = 0:002 0 for the ensemble of runs at e = 0:25 shown in Figure 5.5, and the scaling of Q e = 0:00005= e from Figure 6.2, a general scaling for electrons is obtained as Q e = 0:0005 0 = e : (6.3) It is pleasing to note that Equation (6.3) follows directly from the conservation of energy if one uses the simulation result Q i Q e and assumes bothjEj 2 jBj 2 and the quasilinear condition that @ 0 =@t is proportional to 0 . The derivation of this claim is presented in Section 6.4. Figure 6.3 shows the maximum electron and ion dissipation rates from the simula- tions of Ensemble Two as functions of e . For these runs, over 0:1 e 1:0, the maximum value of Q e scales approximately as a constant and the maximum value of Q i scales approximately as 1=2 e . The constant scaling of Q e is consistent with the substitution of 0 = 0:20 e into Equation 6.3, leading to the result Q e = 1:0 10 4 . Both Figures 6.2 and 6.3 show that electron maximum dissipation rates decrease more rapidly at e > 1 than at smaller values of e potentially leading to ion-dominated dissipation in high- plasmas. It is well established that the linear damping rate of whistler fluctuations increases with increasing e [e.g., Figure 6.7, Gary, 1993]. Furthermore, as shown by Chang et al. [2014], linear damping can provide most of the collisionless dissipation of whistler turbulenceatsmallvaluesoftheinitialfluctuationenergy, withatransitiontononlinear dissipation at larger values of 0 . It is possible that the increase in dissipation with increasing e shown in Figures 6.2 and 6.3 may, analogously, be due to the transition from weak to strong turbulence. 114 Ion and Electron Heating by Whistler Turbulence: Variations with Electron Note that simulated Q e values in both Figures 6.2 and 6.3 fall below the blue dashed lines at e 1. This condition on e also corresponds to the onset of the electron firehose instability [e.g., Gary, 1993, Section 7.3.12] driven by theT e? =T ek < 1 anisotropy characteristic of electron heating due to whistler turbulence as shown in Figure 6.1. It may be possible that the excitation of this instability plays some role in whistler turbulence heating at high e . 6.4 Conservation of energy In this section it is shown that Eq. (6.3) follows directly from the conservation of energy. The total change in system energy, with the negligible terms canceled out, is written 1 8 * 0 d P k jEj 2 dt + 1 8 d P k jBj 2 dt + d(n 0 T e ) dt + > 0 d(n 0 T i ) dt = 0: (6.4) Dividing by B 2 0 =8 ! d 0 (t) dt + d e (t) dt = 0: (6.5) Assuming that 0 (t) = 0 e t leads to d 0 (t)=dt = 0 e t , ! d e (t) dt = 0 e t : (6.6) Under this model, maximum heating occurs at t = 0 ! d e (t) dt j max = 0 : (6.7) Now dividing both sides by e0 ! e , where e0 = e (t = 0), gives dT e =dt T e0 ! e j max =Q e;max = ! e 0 e0 : (6.8) 6.5 Conclusions 115 So the maximum electron heating rate should be proportional to 0 and inversely proportional to e0 , as is shown to be the case by Figures 5.5 and 6.2. Solving for the damping rate using Figure 6.2, where 0 = 0:1 0:1 ! e = 0:00005 ! ! e = 0:0005 (6.9) and comparing with the damping rate of Figure 5.5, where e0 = 0:25 4 ! e = 0:002 ! ! e = 0:0005: (6.10) These values are in remarkably good agreement with one another. This model also predicts that electron heating should remain constant when 0 / e0 . Figure 6.3 confirms this result; when e0 1 the electron heating rate remains approximately constant. 6.5 Conclusions Three-dimensional PIC simulations were used to study the forward cascade of whistler turbulence to shorter wavelengths, where the fluctuations dissipate energy on both electrons and ions. These simulations show that the rate of plasma dissipation is not only a function of the electron , but also depends upon the dimensionless parameter 0 that characterizes the initial value of the magnetic fluctuation energy density. Two ensembles of simulations were presented within a range of initial values 0:10 e 5:0, where each ensemble varied in how 0 was selected. The simulations of Ensemble One used Equation (6.1) as an initial condition and yielded maximum dissipation rates of both the electrons and ions that scaled approximately as 1 e . The runs of Ensemble Two used Equation (6.2) as an initial condition, and demonstrated that over 0:1 e 1:0 the maximum electron heating rate Q e scales approximately as a constant, whereas the maximum ion dissipation rate Q i scales approximately as 1=2 e . 116 Ion and Electron Heating by Whistler Turbulence: Variations with Electron Gyrokinetic simulations of kinetic Alfvén wave turbulence at i = 1 and T e =T i = 1 [Told et al., 2015] in a weakly collisional plasma imply that electron dissipation dominates ion dissipation at most wavenumbers of the kinetic regime. Further insight into the plasma physics of kinetic Alfvén turbulent dissipation will be obtained in Chapter 7 by following the initialization suggested by Parashar et al. [2015]. The PIC simulations of Chapters 4 and 5 have demonstrated that larger simulation domains and larger values of the initial magnetic field fluctuation amplitudes each lead to increasing values of Q i =Q e . Combining those results with the consequences presented above suggests that sufficiently long wavelength, sufficiently large amplitude, and/ or sufficiently large e magnetosonic-whistler turbulence may heat solar wind ions more rapidly than electrons. Figures 117 (a) 0 1 2 3 4 5 6 v e|| /v te0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 1250 tω e = 2500 (b) 0 1 2 3 4 5 6 v e⊥ /v ti0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 1250 tω e = 2500 Figure 6.1: Reduced velocity distributions for selected simulation times from the simulation with the initial conditions e = 0:5 and 0 = 0:10, which is a contributor to both Ensemble One and Ensemble Two. Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. (a) f e (v k ), (b) f e (v ? ), (c) f i (v k ), (d) f i (v ? ). 118 Ion and Electron Heating by Whistler Turbulence: Variations with Electron (c) 0 1 2 3 4 5 6 v i|| /v ti0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 1250 tω e = 2500 (d) 0 1 2 3 4 5 6 v i⊥ /v ti0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 1250 tω e = 2500 Figure 6.1: Continued. Figures 119 0.1 0.25 0.5 1 2.5 5 β e 10 -6 10 -5 10 -4 Q max Q e,max Q i,max Figure 6.2: Individual points represent results from Ensemble One. The maximum values of the dimensionless time rates of change of the total ion temperatures (Q i ) and the total electron temperatures (Q e ) are plotted as functions of the initial value of e . The blue dashed line is Q e = 5:0( 1 e ) 10 5 and the red dashed line is Q i = 7:0( 1 e ) 10 6 ; both are drawn to guide the eye along the simulation data. The numerical values of the data presented are tabulated in Table A.4. 120 Ion and Electron Heating by Whistler Turbulence: Variations with Electron 0.1 0.25 0.5 1 2.5 5 β e 10 -6 10 -5 10 -4 Q max Q e,max Q i,max Figure 6.3: Individual points represent results from Ensemble Two. The maximum values of the dimensionless time rates of change of the total ion temperatures (Q i ) and the total electron temperatures (Q e ) are plotted as functions of the initial value of e . The blue dashed line isQ e = 1:010 4 and the red dashed line isQ i = 1:5( 1=2 e )10 5 ; both are drawn to guide the eye along the simulation data. The numerical values of the data presented are tabulated in Table A.5. 121 Chapter 7: Kinetic Alfvén Turbulence This chapter presents the final study of ion and electron heating by kinetic-scale turbulence. Thisstudyisintendedtomovebeyondinvestigationsofwhistlerturbulence and begin to explore the role of kinetic Alfvén waves (KAW) in the physics of turbulece dissipation in the solar wind. The 3-D EMPIC code USCPIC is used to model the forward cascade of decaying kinetic Alfvén wave turbulence. Due to the restrictive computational demands which accompany studies of this scale, these simulations are among the first fully kinetic 3-D PIC simulations of kinetic Alfvén turbulence to be carried out. The maximum electron and ion heating rates are computed as functions of 0 . In contrast to the results for heating by whistler turbulence, the maximum ion heating rate due to kinetic Alfvén turbulence is found to be substantially greater than the maximum electron heating rate. Moreover, both ion and electron heating rates are found to scale approximately with 0 [Hughes et al., 2017b]. 7.1 introduction Homogeneous turbulence is a challenging research topic in the theory and computations of collisionless plasmas. Understanding plasma turbulence requires fully nonlinear calculations which are not yet well understood at the fundamental level, and are therefore an appropriate subject for numerical study in homogeneous plasmas. Intheinertialrangeofplasmaturbulencethefluctuations, aswellastheinteractions among the fluctuations, must both be calculated using nonlinear methods. However, turbulentfluctuationamplitudesgenerallydecreasewithdecreasingwavelengthssothat in the kinetic regime nonlinear processes become weaker and the quasilinear premise Klein et al. [2012] may provide an appropriate representation. The typical application 122 Kinetic Alfvén Turbulence of the quasilinear premise is that field fluctuations are assumed to satisfy dispersion theory; that is, one may use linear theory [Gary, 1993] to derive an approximate expression relating the real frequency and the wavevector of a fluctuation, and then use such dispersion properties to nonlinearly compute the interactions among the ensemble of fluctuations. Following Gary and Smith [2009], the viewpoint here is that the primary linear modes contributing to kinetic range turbulence are whistler and kinetic Alfvén waves. As Chapters 4, 5, and 6, along with the work of [Chang et al., 2011, 2013, 2014; Gary et al., 2012], have produced a substantial body of PIC simulation research on whistler turbulence, attention is now turned toward the possible contributions of kinetic Alfvén waves to kinetic range turbulence in collisionless plasmas such as the solar wind. Kinetic Alfvén waves (KAWs) are constrained to real frequencies ! r < p and are strongly damped with increasing e or i unless k ? k k [Stawicki et al., 2001; Gary and Smith, 2009]; Landau damping increases substantially with increasing proton or electron [Gary, 1993]. Measurements of solar wind turbulence in the kinetic range show that kinetic Alfvén waves have the largest amplitudes over 1kc=! p 5 [Leamon et al., 1998; Bale et al., 2005; Sahraoui et al., 2009, 2010; Salem et al., 2012; Chen et al., 2013; Kiyani et al., 2013; Podesta, 2013; Roberts et al., 2015]. However, more recent solar wind and magnetotail observations indicate that magnetosonic-whistler waves and ion Bernstein modes [Perschke et al., 2013, 2014; Narita et al., 2011a, 2016] also contribute to turbulence over 1kc=! p 10, although usually with weaker amplitudes than the kinetic Alfvén waves. Modeling the forward cascade of KAW turbulence by full PIC presents difficulties. In particular, to model KAW turbulence by the PIC method the simulation domain must extend beyond the outer boundary of the kinetic range, while the very small Debye length, D , must be resolved to avoid artificial numerical heating. Additionally, as KAW have relatively low frequencies, simulations may require an extended number 7.2 Simulations 123 of time steps in order to span time scales of the forward cascade process. Previous computational studies of kinetic Alfvén wave (KAW) turbulence have been limited in their representations of electron dynamics. Gyrokinetic codes [Howes et al., 2008a, 2011; Told et al., 2015] describe only electron responses to fluctuations much below the proton cyclotron frequency; hybrid PIC codes [Vasquez et al., 2014; Vasquez, 2015] assume electron dynamics may be represented by a fluid model, and the simulations of Wu et al. [2013] and Matthaeus et al. [2016] were computed in a 2.5D configuration which represents only two of the three spatial dimensions. If a comprehensive model of KAW turbulence in the solar wind is to be constructed, the full kinetic and three- dimensional behavior of the electrons must be represented; such a calculation is described here for the first time. 7.2 Simulations In order to carry out simulations of this scale, parameters similar to those suggested by Parashar et al. [2015] are used. Parashar et al. [2015] proposed a standardized set of simulation parameters and diagnostics for performing studies of solar wind turbulence dissipation, with the intention of promoting direct comparison of results among the variety of plasma models in use. The suggested physical parameters include i = e = 0:6, ! e = e = 1:5, v te =c = 0:37, and m i =m e = 100. The value e = 0:6 is a value typically found in the fast solar wind; moreover, this value was found by Howes et al. [2008a] to correspond to particularly low damping rates of kinetic Alfvén waves, so that using a value near e = 0:6 provides the best chance of observing the forward cascade before damping can remove the fluctuation energy. The initial physical parameters used in the simulations described here were selected to be similar to those of Parashar et al. [2015]: i = e = 0:50, m i =m e = 100, v te =c = 0:32, v A =c = 0:064 and ! e = 156:25 i . 124 Kinetic Alfvén Turbulence With the above physical parameters selected the required simulation domain size, time step, and run-time for this study can be determined. Kinetic Alfvén waves propagate at highly oblique angles to the mean magnetic field [Podesta et al., 2010], requiring domain dimensions which should scale as L k =L ? p m i =m e , and L x =L y =L ? ,L z =L k . Taking the spectral boundary of the kinetic range to be equal to the inverse ion inertial length [Chen et al., 2014], k min ! i =c, the required box length isL ? 2c=! i . The domain dimensions are, therefore, selected to beL ? = 256 computational cells, L k = 2048 cells. Using these values leads to L ? = 8:19c=! i , L k = 65:5c=! i . The time step was selected to be t = 0:08; this choice was intended to facilitate the increased simulation run-time requirements while maintaining a numerically stable model. Four simulations with different values of the initial magnetic fluctuation energy density were carried out with 0 = 0:05; 0:10; 0:25; and 0:50. For all four runs, the computational parameters wereL k = 655c=! e = 65:5c=! i ; L ? = 81:9c=! e = 8:19c=! i and ! e t = 0:08. The corresponding values of the minimum wavevector components are k k c=! i = 0:096 and k ? c=! i = 0:767. The numerical approach followed the same setup as for the whistler turbulence computations of Chapters 4, 5, and 6, while applying the parameters described above. Thus in each simulation a collection of relatively long-wavelength, narrowband, fluctuations corresponding to 150 normal modes with random phases were initially imposed upon the system. The initial amplitudes of each field component of each mode were derived from solutions of the kinetic electromagnetic linear dispersion equation [Gary, 1993] subject to the appropriate plasma parameters. A difference from the simulations of whistler turbulence is that, for these simulations the fluctuations are loaded initially as a highly anisotropic spectrum in the sense of k ? k k . As linear theory predicts [Gary, 1993] that KAW are strongly damped at all but the most oblique directions of propagation, B > 80 , the simulations here begin with a 7.3 Results 125 spectrum that provides the least damped modes, and the greatest potential to lead to forward cascade. 7.3 Results The evolution of the kinetic Alfvén turbulent fields in these simulations is similar to the development of whistler turbulence in the earlier PIC simulations as well as to that of kinetic Alfvén turbulence in gyrokinetic simulations [Howes et al., 2008a, 2011]: As shown in Figs. 7.1 and 7.2, the fluctuations cascade to shorter wavelengths developing magnetic spectra that exhibit power-law dependence on wavenumber and anisotropic wavevector spectra in the sense of k ? k k . Figure 7.3 shows the various terms in the run with 0 = 0:50 which contribute to the total plasma energy as functions of time. The total plasma energy density is K total =W i +W e +K B +K E (7.1) where W j is the total kinetic energy density of the j-th species, that is, [Gary, 1993] W j (m j =2) Z jvj 2 f j (v)d 3 v (7.2) and K j is the fluctuating field energy density for electric and magnetic fields K E jEj 2 =8 (7.3) K B jBj 2 =8: (7.4) In each simulation the magnetic field energy density monotonically decreases with increasing time due to wave-particle damping in this collisionless plasma. Damping of the fluctuating fields transfers wave energy into plasma kinetic energy with the ions being heated more rapidly than the electrons in this case. 126 Kinetic Alfvén Turbulence Figure 7.3 from the run with 0 = 0:5 indicates, just as the other three runs not illustrated here, three distinct time domains. Over 0<t! e < 20, there are no clear trends in the magnitudes of the fluctuating field energies, the electron temperatures or the ion temperatures. This phase is interpreted as a period of organization during which the various modes develop into a fully self-consistent ensemble of fluctuations. During the second interval, corresponding approximately to 20 < t! e < 500, this ensemble develops into a forward cascading, rapidly dissipating spectrum of turbulence; the transfer of energy is evident from the monotonically decreasing fluctuating fields and appears to correspond to collisionless wave-particle interactions. The third interval corresponds to 500<t! e ; during this period the rates of plasma component heating diminish. The analyses here is concentrated on the second interval which corresponds not only to the best-defined period of forward cascade but also to the time of the strongest plasma heating rates. Figure 7.4 compares the electron and ion reduced velocity distributions at t = 0 and at t! e = 480 for the run with the initial condition 0 = 0:50. Also shown in the four panels of Fig. 7.4 are Maxwellian distributions (black dotted lines); each Maxwellian has been given a standard deviation equal to the thermal velocity of each respective velocity component distribution at the timet! e = 480, providing a basis for evaluation of the Maxwellian-like character of each distribution. The electron f e (v ? ) distribution shows little change, whereas both ion distributions show modest heating and the strongest thermal energy change appears on f e (v k ). There are fundamental differences in how energy is transferred in each respective direction; while in the parallel direction ions experience bulk distribution broadening and remain close to Maxwellian, corresponding to an increasing v ti and an increase in parallel temperature in the traditional sense, the perpendicular distribution shows a relatively large energy gain in the suprathermal tail. Comparing the ion perpendicular velocity distribution to the equivalent Maxwellian it is clear that the temperature has not increased in the traditional sense as the Maxwellian continues to closely follow the 7.3 Results 127 initially loaded Maxwellian distribution; rather ion velocities have been scattered from the core into the suprathermal tail of the distribution. The behavior in the parallel direction indicates a Landau resonance with low speed ions and ions falling out of resonance near the thermal velocity. In the perpendicular direction the indication is either that ions remain in a cyclotron resonant condition over a larger range of velocities, ultimately shifting the distribution into the suprathermal tail, or the ions are gaining energy through a nonresonant “stochastic” heating process [Dmitruk et al., 2004; Chandran et al., 2010]. Electron heating leads to a characteristic flattening of the parallel velocity distribution at v ek <v te [e.g. Rudakov et al., 2011], whereas the ion distribution retains a more nearly Maxwellian shape with enhanced heating primarily near v ik v ti . As kinetic Alfvén fields oscillate at frequencies below i , the quasilinear premise impliesthatbothelectronsandionsareprimarilyheatedthroughtheLandauresonance which corresponds to a transfer of energy via E k . So, as for electron heating by whistler turbulence, the simulation results here are consistent with preferential parallel heating on the electron species. The ion heating produces a similar result, however there is a nonlinear effect in which ions are slowly gaining energy in the perpendicular direction across a large range of v ? . Figure 7.5 shows the parallel, perpendicular and total temperatures as well as the temperature anisotropy for the electrons and the ions for the run with 0 = 0:50. The total temperature of the ion population grows at a substantially larger rate than that of the electrons, indicating a stronger heating Q i relative to Q e . This can be attributed to the fact that the electrons are being heated only in the parallel direction, while the ion population is gaining energy approximately isotropically. As one might expect, monotonically decreasing values of 0 correspond to monotonically decreasing values of T e and T i . Figure 7.6 demonstrates that the entropy gains of both species are monotonic and abrupt, with ions reaching a maximum entropy by t! e 1000. The electrons 128 Kinetic Alfvén Turbulence have the largest entropy gains by the same time, but show a gradual and continuous increase in entropy to late times. The results here suggest that, even under the most favorable conditions for KAW turbulence forward cascade, thermal dissipation on both electrons and ions occurs very rapidly, which largely prevents the establishment of a steady-state forward cascade down to electron scales. In particular, the fluctuations are strongly damped on the ion population, and have significantly dissipated before the transient establishment of a broadband spectrum has completed. Figure 7.7 illustrates the electron and ion total heating rates as functions of time for all four simulations. Just as Fig. 5.4 showed that both species heating rates due to whistler turbulence increase with increasing 0 and that the maximum values of Q e and Q i arise at relatively early simulation times, so does Fig. 7.7 demonstrate the same properties for electron and ion heating due to kinetic Alfvén turbulence. At most times, the Q i values are substantially larger than the corresponding Q e . The individual points of Fig. 7.8 illustrate the maximumQ e and maximumQ i from the four kinetic Alfvén turbulence simulations shown in Fig. 7.7 as functions of 0 . The dashed line represents Eq. (6.3) which is the best fit to the corresponding maximum electron heating rate for whistler turbulence. Best fits to the four kinetic Alfvén turbulence points yield Q e = 0:0010 1:10 0 and Q i = 0:0025 1:08 0 . For the parameters chosen here, the maximum heating rates of both electrons and ions due to wave-particle damping of kinetic Alfvén turbulence scale approximately as the initial value of 0 . As in Chapter 5, such dependence can be associated with quasilinear-like analytic perturbation theories [Gary and Paul, 1971; Gary and Feldman, 1978]. Indeed, as kinetic Alfvén turbulence corresponds to frequencies well below both the ion and electron cyclotron frequencies, it is likely that heating of both electrons [TenBarge and Howes, 2013] and ions is due at least in part to Landau and/or transit-time damping in association with the Landau resonance at ! r k k v k = 0. Finally, it can be noted from Fig. 7.8 that the slopes of both the Q e (KAW) andQ i (KAW) plots increase as 0 7.4 Conclusions 129 increases, indicating that stronger initial fluctuation amplitudes correspond to more strongly nonlinear dissipation processes on both species. 7.4 Conclusions An ensemble of four fully three-dimensional particle-in-cell simulations of the forward cascade and damping of kinetic Alfvén turbulence has been carried out in a collisionless, homogeneous, magnetized plasma model. The spectral properties of the fluctuating magnetic fields are similar to the magnetic spectral properties which result from gyrokinetic simulations of kinetic Alfvén turbulence. The new results here are the maximum electron and ion heating rates, Q e and Q i , due to the turbulent decay of kinetic Alfvén turbulence which are computed as functions of 0 . The results show maximum values such thatQ e <Q i ; this is the opposite case from that of the previous whistler turbulence PIC simulations. Furthermore, these simulations show that both the electron and ion heating rates vary approximately as 0 . Again these scalings stand in contrast to the heating rates due to whistler turbulence where electron heating varies with 0 but ion heating increases as 1:5 0 . It is not yet clear whether these differences are due to fundamentally different plasma processes associated with the two different modes, or simply to the longer ion-kinetic-scale wavelengths of the kinetic Alfvén waves favoring ion heating against the shorter electron-kinetic-scale wavelengths of the whistlers favoring electron heating. By analogy with the computations of whistler turbulence, there are a number of topics in kinetic Alfvén turbulence worthy of further study. These would include learning the consequences of initializing the simulations on larger computational domains, and of varying the initial values of e , i , T e =T i , and m i =m e . A primary conclusion of this KAW turbulence study is that, for the parameters chosen here, Q i >Q e . At the values e = i = 0:50 used here, the model of Howes [2010] predicts the opposite result Q e Q i . The studies of chapters 4, 5, and 6 130 Kinetic Alfvén Turbulence have also yielded Q e Q i , although these studies involved whistler-type fluctuations rather than the KAW-type fluctuations of Howes [2010]. A more complete study of turbulent heating will require a larger PIC simulation domain that will permit both magnetosonic-whistler turbulence and KAW turbulence to develop and, if possible, interact through forward cascade wave-wave processes. In contrast to expectations, the cascade/dissipation process occurs very rapidly when considering the very low frequencies of the initial fluctuations. While it was expected that 10 or more ion cyclotron periods would be required for forward cascade to fully establish, the results show that the majority of the fluctuation energy is dissipated within a few ion cyclotron periods, with only a mild cascade to kc=! e . 1 accompanying the dissipation. This may be explained by linear theory [Gary, 1993], which predicts the high damping rates of such modes at small wavelengths. This may imply that the kinetic Alfvén waves observed below the inertial range spectral break are not suited to carry fluctuation energy to electron-scale wavelengths [Podesta et al., 2010; Smith et al., 2012; Sahraoui et al., 2013], suggesting that the observed kinetic Alfvén waves make way to a whistler turbulence cascade at scales smaller than the ion scale [Shaikh and Zank, 2009], which then terminate as whistler fluctuation energy is dissipated on solar wind electrons and, to some extent, ions. Figures 131 -3 -2 -1 0 1 2 3 k x c/ e -3 -2 -1 0 1 2 3 k y c/ e -0.3 0 0.3 k || c/ e t e = 0 -7 -6 -5 -4 -3 -2 log 10 ( B 2 /B 0 2 ) -3 -2 -1 0 1 2 3 k x c/ e -3 -2 -1 0 1 2 3 k y c/ e -0.3 0 0.3 k || c/ e t e = 2400 -7 -6 -5 -4 -3 -2 log 10 ( B 2 /B 0 2 ) Figure 7.1: Reduced magnetic fluctuation energy spectra of KAW turbulence for the 0 = 0:50 simulation. The top panels show the initially loaded spectrum in the k x k y (perpendicular) plane and in the k y k z (perpendicular-parallel) plane; the parallel direction has been rescaled for improved visualization. The bottom panels show the same spectrum at a later time, t! e = 2400. 132 Kinetic Alfvén Turbulence 0.2 0.4 0.6 0.8 1 2 3 k ⊥ c/ω e 10 -5 10 -4 10 -3 10 -2 δ B 2 /B 0 ϵ 0 = 0.05 ϵ 0 = 0.10 ϵ 0 = 0.25 ϵ 0 = 0.50 Figure 7.2: Reduced 1-D magnetic fluctuation energy spectra of KAW turbulence as a function of perpendicular wavenumber for the 0 = 0:05; 0:1; 0:25; and 0:50 simulations. The black curves show the power law scaling of the fluctuation energy as a function of perpendicular wavenumber above and below the electron-scale spectral break. The time here is t i = 2. Figures 133 0 500 1000 1500 2000 2500 ω pe t -0.5 -0.25 0 0.25 0.5 [K - K(t = 0)]/B 0 2 W i W e W i+e K B K E K B+E K total Figure 7.3: The temporal evolution of field and plasma component energies for the kinetic Alfvén turbulence simulation with initial condition 0 = 0:50. The components are: the total ion thermal energy density,W i , the total electron thermal energy density, W e , the total ion plus electron energy density, W i+e , the magnetic field energy density, K B , the electric field energy density,K E , the total electromagnetic field energy density, K B+E , and the total field plus plasma energy density, K total . 134 Kinetic Alfvén Turbulence (a) 0 1 2 3 4 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 t e = 0 t e = 480 (b) 0 1 2 3 4 v e /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 t e = 0 t e = 480 Figure 7.4: Reduced velocity distributions from the kinetic Alfvén turbulence simula- tion with initial condition 0 = 0:50. The blue lines represent velocity distributions at t = 0, and the red lines represent velocity distributions at t! e = 480; the black dotted lines are Maxwellian distributions with equivalent standard deviations as the respective late time velocity distributions. The left-hand axis labels correspond to the solid lines, and the right-hand axis labels correspond to the dashed lines. (a) f e (v k ), (b) f e (v ? ), (c) f i (v k ), (d) f i (v ? ). Figures 135 (c) 0 1 2 3 4 5 6 v i|| /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 t e = 0 t e = 480 (d) 0 1 2 3 4 5 6 v i /v it0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 t e = 0 t e = 480 Figure 7.4: Continued. 136 Kinetic Alfvén Turbulence (a) 0 500 1000 1500 2000 2500 ω pe t 0.6 0.8 1 1.2 1.4 1.6 T e T e|| T e⊥ T e,tot T e,anis (b) 0 500 1000 1500 2000 2500 ω pe t 0.8 1 1.2 1.4 1.6 T i T i|| T i⊥ T i,tot T i,anis Figure 7.5: The temporal evolution of plasma component temperatures for the kinetic Alfvén turbulence simulation with 0 = 0:50. The temperature curves are normalized by the respective initial temperature value. The red curve is parallel temperature, the green curve is perpendicular temperature, the yellow curve is total temperature, and the blue curve is temperature anisotropy factor T j? =T jk . Figures 137 (a) 0 500 1000 1500 2000 2500 ω pe t 0 0.05 0.1 0.15 0.2 S e ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 ǫ 0 = 0.500 (b) 0 500 1000 1500 2000 2500 ω pe t 0 0.1 0.2 0.3 0.4 S i ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 ǫ 0 = 0.500 Figure 7.6: Temporal evolution of the (a) electron and (b) ion system entropy for the four kinetic Alfvén turbulence simulations as labeled. 138 Kinetic Alfvén Turbulence (a) 0 500 1000 1500 2000 2500 ω pe t 10 -5 10 -4 10 -3 Q e ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 ǫ 0 = 0.500 (b) 0 500 1000 1500 2000 2500 ω pe t 10 -5 10 -4 10 -3 Q i ǫ 0 = 0.050 ǫ 0 = 0.100 ǫ 0 = 0.250 ǫ 0 = 0.500 Figure 7.7: Temporal evolution of the (a) electron and (b) ion total heating rates for the four kinetic Alfvén turbulence simulations as labeled. Figures 139 0.05 0.1 0.25 0.5 ǫ 0 10 -4 10 -3 Q max Q e,max (whistler) Q e,max (KAW) Q i,max (KAW) Figure 7.8: The maximum values of the dimensionless time rates of change, Q j;max , of the ion total temperature (red circles) and the electron total temperature (blue squares) as functions of the initial magnetic fluctuation energy density of the four kinetic Alfvén turbulence simulations. The blue dashed line represents Q e;max (whistler) = 0:001 0 which is found by Eq. (6.3) to be the approximate scaling for maximum electron heating due to whistler turbulence at e = 0:50. The numerical values of the data presented are tabulated in Table A.6. 140 Chapter 8: The Whistler Anisotropy Instability: Inverse Spectral Transfer In the final study of this dissertation the USCPIC code is used to simulate the whistler anisotropy instability (WAI) in a collisionless, homogeneous, magnetized plasma model. The WAI is driven by an anisotropic electron velocity distribution with T e? =T ek > 1. This study represents the first simulation of the WAI carried out in three spatial dimensions. Reduced two-dimensional (2-D) magnetic wavevector spectra are analysed and show that the magnetic fluctuations generated by the WAI grow as relatively short-wavelength, quasi-parallel propagating whistler modes; this result is predicted by linear dispersion theory. An interesting result appears at late simulation times; magnetic fluctuation energy is transferred from small wavelength to large wavelength modes, and begins to form a secondary spectrum of fluctuations that exhibit quasi- perpendicular propagation. This process of inverse energy transfer is reminiscent of the concept of inverse cascade in plasma turbulence, in which fluctuations in a broadband turbulent spectrum interact such that fluctuation energy flows from small to larger wavelength modes. Entropy analysis is performed in order to verify that such a process obeys the second law of thermodynamics, i.e., total system entropy increases at all times. Finally, nonlinear three-wave coupling theory is used to argue the physical consistency of this inverse transfer process [Gary et al., 2014]. 8.1 Introduction If the electron velocity distribution is approximately bi-Maxwellian and if the tem- perature anisotropy T e? =T ek is sufficiently greater than unity, the whistler anisotropy 8.1 Introduction 141 instability arises in a collisionless, homogeneous, magnetized plasma. Linear dispersion theory [e.g. Gary, 1993] predicts that this instability grows at real frequencies! r which lie between the proton cyclotron and electron cyclotron frequencies, with maximum growth rate m at propagation strictly parallel or antiparallel to B 0 when e & 0:025; the corresponding wave number of maximum growth is k m c=! e 1. When e . 0:025 maximum wave growth shifts to oblique directions of propagation [Gary et al., 2011]. Previous particle-in-cell (PIC) simulations of the whistler anisotropy instability using a single anisotropic hot electron component have utilized codes with either one-dimensional (1-D) spatial variations [Ossakow et al., 1972; Cuperman and Salu, 1973; Pritchett et al., 1991; Lu et al., 2010] or two-dimensional (2-D) spatial variations [Devine et al., 1995; Gary and Wang, 1996; Gary et al., 2000]. These simulations typically show that the fluctuating magnetic field energy density has exponential growth at early times, followed by saturation and relatively weak changes at later times. Wave-particle scattering yields a reduction of the electron anisotropy at early times, while at later times T e? =T ek remains close to a weak threshold condition [e.g. Gary and Wang, 1996, Figure 2]. The emphasis here is on the inverse transfer, that is, the change from shorter to longer wavelengths, of whistler fluctuation energy. Most computational studies of this process have been done in the electron magnetohydrodynamic (EMHD) model. Some of these studies show the presence of both forward and inverse cascades [Cho and Lazarian, 2004, 2009; Kim and Cho, 2015; Shaikh and Zank, 2005]. A pair of EMHD computations demonstrated inverse cascades under both 2-D [Wareing and Hollerbach, 2009] and 3-D [Wareing and Hollerbach, 2010] conditions. Furthermore, Wareing and Hollerbach [2010] conclude that their 3-D inverse cascade is considerably weaker than the 2-D inverse cascade illustrated in their earlier paper. Cho [2011] used an incompressible EMHD model to show that, for wave packets initially traveling in one direction, both inverse and forward energy cascades can develop. 142 The Whistler Anisotropy Instability: Inverse Spectral Transfer The work of Ganguli et al. [2010] employed a 2-D PIC model with B 0 at an angle relative to the simulation plane. The plasma was driven unstable by an electron velocity ring distribution and spectral signatures of inverse energy transfer were observed. Chang et al. [2015] carried out very-large-scale PIC simulations with an initial spectral configuration which allowed for the observation of both forward and inverse spectral transfer. The conclusion of this investigation was that, while present, the inverse transfer process remained much weaker than that of the forward cascade. Kinetic instabilities grow at relatively short wavelengths and, as in Ganguli et al. [2010] and Winske and Daughton [2012], are appropriate sources for the study of inverse spectral transfer (i.e., from short to long wavelengths). In the present study a 3-D PIC simulation is used to address the temporal development of enhanced whistler fluctuations in a collisionless, homogeneous, magnetized plasma. An initial temperature anisotropy is imposed on the electron velocity distribution, leading to the self-consistent generation of enhanced fluctuations through the excitation of the whistler anisotropy instability. Bale et al. [2009] used solar wind observations to show correlations between the thresholds of instabilities driven by proton temperature anisotropies and magnetic fluctuation amplitudes at frequencies consistent with such unstable modes. Their observations also indicate that these instabilities enhance magnetic spectra across a broad range of short-wavelength fluctuations, providing further motivation for the study of the relationship between anisotropy-driven instabilities and the associated wavevector spectra. Section 8.2 discusses the PIC simulation, Section 8.3 presents the results and investigates entropy generation during the inverse cascade process, Section 8.4 treats possible contributions of three-wave coupling to the spectral transfer process, and Section 8.5 provides conclusions of the investigation. 8.2 Simulation 143 8.2 Simulation In this simulation, the plasma is collisionless and homogeneous, with a uniform background magnetic field B 0 . The initial plasma conditions are; m p =m e = 1836, p = e = 0:10, and v tek =c = 0:1. The initial proton velocity distribution is an isotropic Maxwellian. The computational parameters are as follows; the grid spacing is = 0:10c=! e , the time step is t! e = 0:05, and the number of particles per cell is 64, with 32 electrons and 32 protons. The system has a spatial length of L = 102:4! e =c in each direction. In contrast with all previous studies of this dissertation, which were initialized with a large number of relatively long-wavelength (kc=! e 1) plasma modes, this simulation begins with a strongly anisotropic bi-Maxwellian (T e? =T ek = 3:0) electron velocity distribution. This nonthermal condition excites the whistler anisotropy instability [e.g. Gary and Wang, 1996, and citations therein] with maximum growth rate at kc=! e < 1 [Gary, 1993, Fig 7.7] and maximum growth at k B 0 = 0 when e & 0:025. 8.3 Results The evolution of the fluctuating fields and the electron temperatures is similar to that from previous simulations of this instability [e.g. Gary and Wang, 1996; Hughes et al., 2016]. As shown in Fig. 8.1, the fluctuating magnetic field energy increases with exponential growth to saturation, followed by a slow decay; the electron temperature anisotropy shows a rapid decrease during the exponential growth phase, followed by a much more gradual decrease at late times. Figure 8.2 shows the parallel (8.2a) and perpendicular (8.2b) electron velocity dis- tributions at four times during the simulation. It is apparent that the reduction in the temperature anisotropy is due to the theramlization of the parallel electron velocities 144 The Whistler Anisotropy Instability: Inverse Spectral Transfer with the most significant heating occurring in the region v ek & 3v te0 . A slight cooling relative to the initial state can be seen in the perpendicular distribution; however, because the particles are initially distributed over a larger region of velocity space, the change appears less pronounced. Near the end of the simulation the anisotropy has dropped to a value of T e? =T ek 2, and the plasma has nearly approached a stable condition. Figure 8.3 shows the reduced k k k y magnetic fluctuation energy spectra at four times as labeled. During the early-time exponential temporal growth phase the spectrum is what one would expect from the predictions of linear theory for the whistler anisotropy instability [e.g. Gary, 1993, Fig. 7.7]; the most intense fluctuations exhibit narrowband spectra with maximum intensity at kc=! e ' 1 at both parallel and antiparallel propagation relative to B 0 . At and after saturation, the spectra show substantial broadening due to both forward and inverse spectral transfer in k k but little extension of fluctuation energy in k y , that is, k ? . This result is consistent with both forward and inverse spectral transfer being due to three-wave interactions which satisfy k 1 +k 2 = k 3 ; if both k 1 and k 2 have relatively small perpendicular components, then k 3 must also have a small perpendicular component, as indicated by Fig. 8.3. At late times, the most strongly damped regions of the magnetic spectra correspond to fluctuations at k k c=! e > 1, where electron cyclotron damping becomes strong, as well as regions such that k ? =k k & 1, where Landau damping becomes significant; this results in the cone-shaped spectra along k k observed at t! e = 2000. This does not apply to the wavelengths at k k 1; here the fluctuations have become highly anisotropic and propagate relatively undamped due to the fact that they remain outside an electron resonance condition at such long parallel wavelengths. Figure 8.4 shows two magnetic fluctuation quantities summed over three ranges of jkj. Eachofthesewavenumberregimescorrespondstoadifferentphysicalprocess. The green curves represent intermediate wavelengths (0:5jkjc=! e 1:4), approximately spanning the wavenumbers of linear instability growth. The blue curves represent 8.3 Results 145 short wavelengths (1:4 <jkjc=! e 2:0) corresponding to fluctuations driven by forward spectral transfer, and the red curves illustrate results at long wavelengths (0<jkjc=! e < 0:5) driven by an inverse spectral transfer. Figure 8.4a shows the temporal evolution of the total fluctuation energy in each regime. The instability growth regime, as would be expected, exhibits an early and rapid increase in fluctuation energy and contains the overwhelming majority of that energy at all simulation times. The other two wavenumber regimes show significant growth only after instability-driven fluctuations have reached appreciable amplitudes, suggesting that these two regimes are the likely result of fluctuation energy transfer from the instability regime. After instability saturation the forward transfer regime exhibits rapid damping and drops below the background noise level by t! e 800. In contrast, the fluctuations of the inverse transfer regime exhibit weak damping rates and, like the fluctuations of the instability growth regime, persist to late simulation times. Figure 8.4b illustrates as functions of time the wavevector anisotropy factors, defined by Eq. (1.10), for the three different wavenumber ranges. At intermediate wavelengths, the primary spectral contribution at early times is due to growth of the whistler instability with maximum growth at k B 0 = 0, so k ? k k . The short wavelength fluctuations are damped at t! e > 700, but the long wavelength modes persist indicating, as implied by the late-time panels of Fig. 8.3, that the inverse spectral transfer accumulates fluctuating field energy at the smallest k k which fits in the simulation box. Thewhistlerfluctuationsatk k c=! e > 1arestronglydissipatedbyelectroncyclotron damping [e.g. Saito et al., 2008, Fig. 7]. But the long wavelength fluctuations at late times in Fig. 8.3 are weakly damped and persist with an anisotropy in the sense of k ? k k . This is the same type of wavevector anisotropy developed by the forward cascades of whistler turbulence simulated by Chang et al. [2011, 2013] and Gary et al. [2012], as well as in the studies of Chapters 4, 5, and 6. The implication here, however, 146 The Whistler Anisotropy Instability: Inverse Spectral Transfer is not that k ? k k is a universal property of the spectral transfer, but rather only that the late-time wavevector anisotropy is a sensitive function of the early-time wavevector anisotropy. Figure 8.5 shows the dispersion (that is, frequency as a function of wavevector) of the magnetic fluctuations at parallel propagation calculated over the time range 250 ! e t 1750. The lines overlaying the dispersion plot represent numerical solutions to the full kinetic linear dispersion equation at frequencies corresponding to whistler fluctuations and e = 0:1. Two curves have been calculated, as over the course of the simulation the temperature anisotropy decreases; the purple curve was calculated under the condition T e? =T ek = 3:0, corresponding to the initial state of the system, while the blue curve was calculated for the condition T e? =T ek = 2:0, corresponding to the state of the system at the end of the given time range. The agreement between linear theory and fully nonlinear simulation is remarkable, with the curves very closely overlaying the region of maximum energy density. Figure 8.6 shows the proton and electron entropies as functions of time for the whistler anisotropy instability. Entropy is defined here as S =k B N Z f(v)ln[f(v)]dv: (8.1) Two methods of calculating entropy have been carried out and compared; the details of these calculations are presented in Section B.4.1. The proton entropy remains constant, as the protons began from a state of thermal equilibrium. Figure 8.6 demonstrates that the electron entropy monotonically increases throughout the instability process with a maximum rate of increase during the same time period that isotropization of velocity space is at a maximum (Fig. 8.1b), near t! e 300. While the process of inverse spectral transfer may or may not be accompanied by a local decrease in entropy, the system entropy on a whole continues to obey the second law of thermodynamics. This means that, if entropy is reduced by the generation of long wavelength fluctuations, it 8.4 Three-Wave Interactions 147 is at the expense of higher entropy generation on small scales, which provides the work necessary to drive this inverse transfer process. This explanation is also consistent with the observations of Chang et al. [2015] who found that the inverse cascade of fluctuation energy remains considerably weaker than the forward cascade, meaning that entropy will, overall, increase monotonically with time. 8.4 Three-Wave Interactions Galtier and Bhattacharjee [2003] used analytic EMHD theory to argue that the evolution of whistler turbulence is dominated by nonlinear three-wave interactions. Here, following the discussion in section 3 of Chang et al. [2013], the frequency and wavevector matching conditions of three-wave interaction theory [Shebalin et al., 1983; Gary, 2013] are considered; these conditions are necessary (but not sufficient) for the exchange of energy among an ensemble of whistler modes in a homogeneous, magnetized plasma. The amplitudes of the individual modes here are very small compared to the magnitude of the background magnetic field, so that the use of the three-wave matching conditions should be a valid approximation. In contrast to Chang et al. [2013], however, it is assumed here that the initial electron velocity distribution is a bi-Maxwellian with T e? =T ek sufficiently greater than unity so that the whistler anisotropy instability is excited. Assuming ek = 0:10 so that Landau damping is weak, three anisotropy conditions are selected: T e? =T ek = 1:0; 3:0, and 5:0. Numerical solution of the kinetic linear dispersion equation [Gary, 1993] yields !(k) for all three cases as illustrated in Fig. 8.7. As for the case when the electron velocity distribution is Maxwellian [Chang et al., 2013], there is a wavenumber regime (0:20kc=! e 1:0) on which the whistler frequency approximately satisfies ! j e j =a +b kc ! e : (8.2) 148 The Whistler Anisotropy Instability: Inverse Spectral Transfer Forsimplicity, itisassumedthatthree-wavecouplingfollowsaone-dimensionalspectral transfer in wavenumber at fixed propagation angle , and that the frequency matching equation has a mismatch !=j e j defined by the equation ! 1 +! 2 =! 3 + !: (8.3) The frequency mismatch can then be computed by fitting Eq. (8.2) to the computed !(k); the results of this procedure are illustrated in Fig. 8.8 as a function of . At quasi-parallel propagation ( 45 ), the strongest electron anisotropy corre- sponds to the strongest frequency mismatch. This means that, at the early stages of the instability, when T e? =T ek 1, quasi-parallel spectral transfer is not favored, and indeed the simulation results show very little forward transfer. As the instability grows to saturation and beyond, the anisotropy is reduced, the quasi-parallel mismatch is diminished, and the inverse spectral transfer becomes more favored, consistent with the simulation results. 8.5 Conclusions This study carried out a three-dimensional particle-in-cell simulation of the whistler anisotropy instability in a homogeneous, magnetized, collisionless plasma with e = 0:10. This is the first 3-D PIC simulation of such growing modes driven by an initial anisotropy on a bi-Maxwellian electron velocity distribution. The early-time magnetic fluctuation spectrum grows with properties reflecting the predictions of linear theory with narrowband maxima at kc=! e ' 1 and k B 0 = 0, and a wavevector anisotropy in the sense of k ? k k . At later times the fluctuations undergo both a forward spectral transfer to shorter wavelengths, also with k ? k k , and an inverse spectral transfer to longer wavelengths with wavevector anisotropy k ? k k . The former fluctuations are strongly damped, but the latter are weakly damped and persist to late 8.5 Conclusions 149 simulation times. It is found that these inverse spectral transfer results are consistent with the second law of thermodynamics as well as with the predictions of nonlinear three-wave coupling theory. A possible application of this phenomenon is to the scattering of fast electrons in the magnetosphere. Energetic electrons can, at sufficiently high flux levels, damage or destroy spacecraft electronics. Relatively fast electrons are cyclotron resonant with waves at relatively small k k values, so that the inverse spectral transfer shown here can provide enhanced long-wavelength fluctuations to scatter such electrons into their loss cone, thereby reducing the threat they pose to spacecraft operation. 150 The Whistler Anisotropy Instability: Inverse Spectral Transfer (a) 0 500 1000 1500 2000 t ω e 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 δ B 2 /B 0 2 (b) 0 500 1000 1500 2000 t ω e 0.5 1 1.5 2 2.5 3 T e T e|| T e⊥ T e,tot T e⊥/|| Figure 8.1: Simulation results: time histories of (a) the normalized fluctuating magnetic field energy density and (b) the electron temperature anisotropy (blue), the normalized parallel (red) and perpendicular (green) electron temperatures, and the “total” electron temperature (yellow), which is the temperature averaged over the three spatial dimensions. The initial values of the simulation are ek = 0:10 and T e? =T ek = 3:0. Figures 151 (a) 0 1 2 3 4 5 6 v e|| /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 480 tω e = 1300 tω e = 2000 (b) 0 1 2 3 4 5 6 v e⊥ /v et0 0 0.1 0.2 0.3 0.4 10 -8 10 -6 10 -4 10 -2 10 0 tω e = 0 tω e = 480 tω e = 1300 tω e = 2000 Figure 8.2: Reduced Electron velocity distributions for several simulation times as labeled. Solid lines are represented on a linear scale corresponding to the left-hand ordinates of each panel, while dashed lines are the same respective curves but on a logarithmic scale corresponding to the right-hand ordinates. (a) f e (v k ), (b) f e (v ? ). 152 The Whistler Anisotropy Instability: Inverse Spectral Transfer -3 -2 -1 0 1 2 3 k y c/ω e tω e = 250 tω e = 480 -9 -8 -7 -6 -5 -4 log 10 (δB 2 /B 0 2 ) -3 -2 -1 0 1 2 3 k || c/ω e -3 -2 -1 0 1 2 3 k y c/ω e tω e = 1300 -3 -2 -1 0 1 2 3 k || c/ω e tω e = 2000 -9 -8 -7 -6 -5 -4 log 10 (δB 2 /B 0 2 ) Figure 8.3: Simulation results: Reduced k k k y magnetic fluctuation energy spectra jB(k k ;k y )j 2 =B 2 0 at 4 times as labeled. Figures 153 (a) 0 500 1000 1500 2000 t ω e 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 δ B 2 /B 0 2 (0≤kc/ω e ≤0.5) (0.5≤kc/ω e ≤1.4) (1.4≤kc/ω e ≤2.0) (b) 0 500 1000 1500 2000 t ω e 10 -2 10 -1 10 0 10 1 10 2 Tan 2 (θ B ) (0≤kc/ω e ≤0.5) (0.5≤kc/ω e ≤1.4) (1.4≤kc/ω e ≤2.0) Figure 8.4: Time histories of (a) magnetic fluctuation energy density and (b) mag- netic fluctuation wavevector anisotropy factor Tan 2 ( B ). Red lines represent results summed over long wavelengths, green lines represent results summed over intermediate wavelengths, and blue lines represent results summed over short wavelengths with regions as defined in the legend. 154 The Whistler Anisotropy Instability: Inverse Spectral Transfer 0 0.5 1 1.5 k || c/ω e 0 0.5 1 ω r /Ω e -13 -12 -11 -10 -9 -8 -7 -6 log 10 (δB 2 /B 0 2 ) Figure 8.5: Magnetic fluctuation dispersion, real frequency ! r versusk k at a constant k ? = 0, calculated over the range 250 < t! e < 1750. The lines overlaying the calculated dispersion of the simulation are numerical solutions to the full kinetic linear dispersion equation at frequencies corresponding to whistler fluctuations at e = 0:1. The purple curve is found under the condition T e? =T ek = 3:0, representing the initial state of the simulated system, while the blue curve is found atT e? =T ek = 2:0, representing the state of the simulated system at t! e 1750. Figures 155 0 500 1000 1500 2000 t ω e 0 0.02 0.04 0.06 0.08 0.1 S S i S e Figure 8.6: Entropy of protons and electrons as a function of time during the evolution of the whistler anisotropy instability. Two methods of calculating entropy are compared. 156 The Whistler Anisotropy Instability: Inverse Spectral Transfer Figure 8.7: Linear dispersion theory results: Real frequency (solid lines) and damp- ing/growth rate (dotted lines) as functions of the wavenumber for the whistler mode and the whistler anisotropy instability at parallel propagation ( = 0 ) for three values of T e? =T ek as labeled. Figure 8.8: Theory results: The frequency mismatch ! for whistler fluctuations over 0:20kc=! e 1:0 as a function of propagation angle for three values of T e? =T ek as labeled. 157 Chapter 9: Summary and Conclusions In this research dissertation five major studies were presented which focused on enhancing our understanding of turbulence in the solar wind on kinetic scales. In particular, understanding the process by which field fluctuation energy is dissipated on both the ion and electron plasma species was pursued. This process was examined under both models of whistler turbulence as well as kinetic Alfvén turbulence, and under a variety of parametric variations. Each of the studies presented has helped to advance our understanding of turbulent dissipation in the solar wind. The following sections summarize the major findings of these studies. 9.1 Numerics In Chapters 2 and 3 the numerical model used to carry out these studies was inves- tigated. Two particle-in-cell models, Darwin (D) PIC and electromagnetic (EM) PIC were considered in order to determine an optimal approach to the numerical investigation of solar wind turbulence. The important difference between these models is associated with the value of the electron thermal velocity which can be applied in each. DPIC has a superior capability to use physically realistic values of the thermal velocity, while EMPIC demands an artificially large value in order to achieve an acceptably large time step. The first study of Chapter 3 then investigated how the value of the thermal velocity affects the plasma physics when the plasma is subjected to the whistler anisotropy instability. The results of the study revealed that the relevant physics is not significantly affected by the choice of thermal velocity value. Moreover, it was found that reducing the thermal velocity toward realistic values enhances numerical 158 Summary and Conclusions noise in the PIC model which leads to less reliable results. The conclusion of this study was that the use of an artificial velocity is not only physically acceptable, but is in fact preferable from a numerical standpoint. As DPIC uses a more complex algorithm than EMPIC that requires significantly more computation per time step, it was determined that the DPIC model provides no benefit to the studies considered here; EMPIC was therefore selected as the particle-in-cell model to carry out the plasma turbulence studies. Chapter 3 then carried out two additional studies in order to characterize the nonphysical effects of the numerical model on the physical system. The first study was intended to characterize the contribution of numerical heating to the total observed heating. It was found that the use of a Gaussian filter at each time step on the current density field removed the numerical heating effect for the time range of interest. It was concluded that numerical heating could be ignored in the results of the turbulence studies. The second study investigated the effects of using an artificially small proton- to-electron mass ratio, as the use of an artificial mass ratio would be necessary in order to make the turbulence studies numerically feasible. By performing a series of simulations it was found that the ratio of the ion to electron maximum heating rates scaled inversely with the mass ratio. This result suggests that a simple linear extrapolation can recover the heating rates at a realistic mass ratio. 9.2 Study 1 The study of Chapter 4 carried out the first investigation of ion and electron heating by whistler turbulence in a low- plasma. In this study an initial total energy density 0 = 0:1, an initial beta i = e = 0:05 and a mass ratio m i =m e = 400 characterized the plasma. The results demonstrated the forward cascade of the turbulent fluctuations into broadband turbulent spectra at shorter wavelengths with a preference for quasi-perpendicular propagation. Three simulations were carried out 9.2 Study 1 159 with cubic computational domains of three different sizes, namely L! e =c = 25:6; 51:2; and 102:4. These domain sizes correspond to successively longer wavelengths of the initial fluctuations. The simulation results confirmed the results of previous studies showing that electron heating is preferentially parallel to the background magnetic field B 0 , and ion heating is preferentially perpendicular to B 0 . In each case the maximum heating rates satisfiedQ e Q i . The new contribution of this study was the result that larger simulation boxes and longer initial whistler wavelengths yield weaker overall dissipation, consistent with linear dispersion theory predictions of decreased damping, stronger ion heating, consistent with a stronger ion Landau resonance, and weaker electron heating. Anisotropic electron and ion heating by whistler turbulence in the weak fluctuation limit is interpreted here based on kinetic linear dispersion theory. The k ? k k wavevector anisotropy which results from the whistler turbulence forward cascade implies the Landau resonance dominates wave-particle interactions at kc=! e < 1 and yields preferential parallel heating on the electrons and perpendicular heating on the ions. Furthermore, the overall decrease in fluctuation damping with increasing box size is consistent with linear theory predictions of decreasing whistler damping with increasing wavelengths. In the limits of kc=! i 1 and kc=! e 1, linear dispersion theory predicts that the whistler frequency is ! r = e =kk k c 2 =! 2 e : (9.1) Heating of the ions, which are essentially unmagnetized in response to the relatively high-frequency whistler fluctuations, is by means of the Landau resonance at v ? =! r =k ? = (k k c=! e )c e =! e (9.2) so a larger simulation box and longer wavelengths correspond to a smaller resonant ion velocity. This implies that the resonant modes move from the tail of the perpendicular 160 Summary and Conclusions velocity distribution toward the thermal part of the distribution, resonating with a larger number of ions and therefore leading to stronger ion heating, as this study has shown. 9.3 Study 2 In the study of Chapter 5 an ensemble of five simulations was conducted under conditionsmorerepresentativeofthoseinthesolarwindnear1AU.Thefivesimulations were initialized with the fixed physical parameters e = 0:25 and m i =m e = 400 in a cubic domain of length L = 102:4c=! e = 5:12c=! i . The fluctuation energy density, 0 , wasvariedwitheachsuccessivesimulation, takingthevalues 0 = 0:01; 0:025; 0:05; 0:1, and 0:25. From these simulations ion and electron heating rates were calculated. The electron heating was found to be stronger and preferentially parallel/antiparallel to B 0 , while the ion energy gain is weaker and is preferentially in directions perpendicular to B 0 . Using the maximum heating rates, Q i and Q e , calculated for the ion and electron species, respectively, scaling relations for each species as functions of 0 were derived. The important new results of this study were that, over 0:01< 0 < 0:25, the maximum rate of electron heating scales approximately as 0 , and the maximum rate of ion heating scales approximately as 1:5 0 . Several important conclusions were drawn from these results. The scaling of the maximum electron heating rate with initial fluctuating field energy density at small values of 0 is consistent with predictions of quasilinear-like second order perturbation theories[Gary and Paul,1971;Gary and Feldman,1978]. Thissuggeststhatquasilinear- like electron heating rate calculations such as those of Howes [2010] are likely to be valid for sufficiently weak magnetic field fluctuations. Second, the maximum ion heating rate scaling with 1:5 0 compares favorably to the 1:6 0 scaling for the ion heating rate due to the Alfvénic turbulence forward cascades in the hybrid simulations of 9.4 Study 3 161 Vasquez et al. [2014] and Vasquez [2015]. Third, this ion heating rate is also consistent with the approximately 1:5 0 scaling for the rate of perpendicular ion heating by Alfvén waves and kinetic Alfvén waves in the phenomenological theory of Chandran et al. [2010]. The fourth conclusion is that, for sufficiently large 0 , the maximum ion heating rate increases faster than the quasi-linear rate proportional to 0 experienced by the electron population. This transition to an apparently nonlinear ion heating rate is consistent with the results of Wu et al. [2013] who used 2-D PIC simulations in much larger simulation domains to show that at sufficiently large amplitudes of Alfvénic turbulence ion heating dominates electron heating. Although relatively weak compared to electron heating, the perpendicular heating of the ions demonstrated in these simulations is consistent with the hypothesis that the obliquely propagating component of magnetosonic-whistler turbulence contributes to the persistent perpendicular heating of protons observed in the solar wind. 9.4 Study 3 Chapter 6 presented two ensembles of whistler turbulence simulations; the first ensemble held fixed 0 = 0:10, while the second ensemble, following solar wind observations, used the relation 0 = 0:2 e . In each simulation the parametersm i =m e = 400 and L! e = 102:4 were held fixed and e was varied across the simulations of each ensemble, taking the values e = i = 0:1; 0:25; 0:5; 1:0; 2:5; and 5:0. The electron and ion dissipation rates were then computed as functions e . These simulations showed that the rate of plasma dissipation is not only a function of the electron , but also depends upon the parameter 0 . Once again the evolved species anisotropies were found to be T ek > T e? and T i? > T ik . Moreover, in both ensembles the maximum dissipation rate of the electrons and the maximum dissipation rate of the ions continued to satisfy Q e Q i . 162 Summary and Conclusions The new results revealed by this study were the following; the simulations of Ensemble One yielded maximum dissipation rates of both the electrons and ions that scaled approximately as 1 e , while the runs of Ensemble Two demonstrated that over 0:1 e 1:0 the maximum electron heating rate scales approximately as a constant, whereas the maximum ion dissipation rate scales approximately as 1=2 e . From these results, along with those of Chapter 5, a general scaling relation for the electrons was obtained as Q e = 0:0005 0 = e . Gyrokinetic simulations of kinetic Alfvén wave turbulence at i = 1 and T e =T i = 1 [Told et al., 2015] in a weakly collisional plasma imply that electron dissipation dominates ion dissipation at most wavenumbers of the kinetic regime. The PIC simulations of Chapters 4 and 5 demonstrated that larger simulation domains and larger values of the initial magnetic field fluctuation amplitudes each lead to increasing values of Q i =Q e . Combining those results with the scaling relations calculated in Chapter 6 suggests that sufficiently long wavelength, sufficiently large amplitude, and/ or sufficiently large e magnetosonic-whistler turbulence may heat solar wind ions more rapidly than electrons. 9.5 Study 4 The study of Chapter 7 moved beyond investigations of whistler turbulence in order to explore the role of kinetic Alfvén waves (KAW) in the physics of kinetic scale dissipation of turbulence in the solar wind. In this study an ensemble of four 3-D EMPIC simulations were carried out which modeled the forward cascade of decaying kinetic Alfvén wave turbulence. The physical parameters of this study included e = i = 0:50 and m i =m e = 100. From these simulations, maximum electron and ion heating rates were computed as functions of the initial fluctuating magnetic field energy density 0 , with values 0 = 0:05; 0:10; 0:25; and 0:50 assigned in each successive simulation. 9.5 Study 4 163 The spectral properties of the fluctuating magnetic fields agreed well with the magnetic spectral properties which result from gyrokinetic simulations of kinetic Alfvén turbulence. Several new results were found from this study. In contrast to the results for heating by whistler turbulence, the maximum ion heating rate due to kinetic Alfvén turbulence was consistently greater than the maximum electron heating rate. Furthermore, again in contrast to the case of whistler turbulence, both ion and electron heating due to kinetic Alfvén turbulence were found to scale approximately with 0 . Finally, electron heating was found to lead to anisotropies of the typeT k >T ? , while the ions showed approximately isotropic energy gain. The implication is that the Landau resonance is the primary heating mechanism for the electrons, and may contribute to ion heating as well. Again, the scalings found for the ion and electron maximum heating rates due to kinetic Alfvén turbulence as functions of 0 stand in contrast to the heating rates due to whistler turbulence where electron heating was shown to vary with 0 but ion heating increases as 1:5 0 . It is not yet clear whether these differences are due to fundamentally different plasma processes associated with the two different modes, or simply to the longer ion-kinetic-scale wavelengths of the kinetic Alfvén waves favoring ion heating against the shorter electron-kinetic-scale wavelengths of the whistlers favoring electron heating. A primary conclusion of this KAW turbulence study is that, for the parameters chosen, Q i >Q e . At the values e = i = 0:50 used here, Howes [2010] predicts the contrasting resultQ e Q i . The studies of chapters 4, 5, and 6 have yielded Q e Q i , although these studies involved whistler-type fluctuations rather than the KAW-type fluctuations of Howes [2010]. 164 Summary and Conclusions 9.6 Study 5 The final study, presented in Chapter 8, took a divergence from the topic of ion and electron heating by kinetic scale turbulence to present results of a study intended to investigate the spectral properties of the whistler anisotropy instability (WAI). The plasma properties included e = 0:10, m i =m e = 1836, and T e? =T ek = 3. This was the first 3-D PIC simulation of such growing modes driven by an initial anisotropy on a bi-Maxwellian electron velocity distribution. The results showed that the early-time magnetic fluctuation spectrum grows with properties reflecting the predictions of linear theory with narrowband maxima at kc=! e ' 1 and k B 0 = 0, and a wavevector anisotropy in the sense of k ? k k . At later times the fluctuations undergo both a forward spectral transfer to shorter wavelengths, also withk ? k k , and an inverse spectral transfer to longer wavelengths with wavevector anisotropy k ? k k . The former fluctuations are strongly damped, but the latter are weakly damped and persist to late simulation times. Entropy analysis was performed and showed these inverse spectral transfer results to be consistent with the second law of thermodynamics as well as with the predictions of nonlinear three-wave coupling theory. The process of inverse energy transfer is reminiscent of the concept of inverse cascade in plasma turbulence. As inverse cascade is a poorly understood phenomenon, the observation of inverse energy transfer resulting from a plasma microinstability can help shed light on the underlying physics of inverse cascade. 9.7 Future Work PIC simulations to study the physics of long-wavelength plasma turbulence require calculations which approach the limit of currently available computing resources. The study of Chapter 7 demonstrated that the use of artificial parameters can significantly 9.7 Future Work 165 relieve the computational constraints which limit the PIC method in scale. That study represents the first study of KAW turbulence to be carried out using the USCPIC code, and one of the first fully kinetic 3-D simulations of KAW turbulence ever to be performed. This opens the door for a number of additional studies. For instance, a series of simulations should be performed to characterize the plasma species heating rates as functions of e . The spectral properties of KAW turbulence and the possible couplings between various normal modes during the cascade process should be characterized. Intermittency analysis would be helpful in determining the degree of nonlinear damping in KAW turbulence due to the intermittent formation of coherent structures, such as current sheets, relative to linear processes such as the Landau resonance. As Chapters 4, 5, and 6 have shown that the efficacy of ion perpendicular heating by whistler fluctuations substantially increases with increasing simulation domain size, it is important that further PIC simulations of magnetosonic-whistler and kinetic Alfvén turbulence be carried out at larger computational systems. Ion heating rates calculated from large scale simulations could be used for comparison against competing candidates for ion energization in the solar wind. While simulations in two spatial dimensions do not fully represent the 3-D character of magnetized plasma turbulence, 2-D PIC simulations of turbulence conducted with B 0 in the simulation plane can qualitatively reproduce much of the plasma physics of fully 3-D simulations [Wan et al., 2015]. For example, 2-D PIC simulations of whistler turbulence [Saito et al., 2008; Saito and Gary, 2012] anticipated many properties which were later confirmed by 3-D computations [Gary et al., 2012; Chang et al., 2013] including the forward cascade to shorter wavelengths, development of the k ? k k wavevector anisotropy, preferential heating of electrons in the directions parallel/antiparallel to B 0 , and decreasing anisotropies of both magnetic wavevector spectra and electron velocity distributions with increasing e . The reduced computational requirements for 2-D in-plane PIC simulations suggest that they can be useful precursors to 3-D PIC simulations as these 166 Summary and Conclusions studies move to successively longer wavelengths for both magnetosonic-whistler and Alfvénic turbulence. Finally, a topic that should be addressed is the one of thermal dissipation in nearly collisionless plasmas, such as the solar wind. Is thermal dissipation of the turbulent fluctuation energy necessarily the result of collisions over long enough periods of time, or do the long-range interactions of the fluctuating fields with the plasma particles contribute to the thermalization of the solar wind plasma? If the collisionless wave- particle interactions are moving the plasma toward thermal equilibrium, what are the details of this process and to what extent can the process continue as thermal equilibrium is approached? 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Appendices 179 Appendices 180 Appendices A Ion and Electron Maximum Heating Rates The tables which follow provide the numeric values of the ion and electron maximum heating rates for the studies described in Chapters 3.2, 4, 5, 6, and 7. The heating rate is defined by Eq. (1.15): Q j (dT j =dt)= [T j (t = 0)! e ] where j =i for ions and j =e for electrons. Table A.1: Whistler turbulence ion and electron maximum heating rates of Chapter 3.2.3, Figure 3.9. L! e =c = 102:4, v te =c = 0:10, i = e = 0:25, 0 = 0:10, variable: m i =m e . m i =m e Q i Q e 1836 4:10 10 6 1:91 10 4 900 8:81 10 6 1:91 10 4 400 2:27 10 5 1:91 10 4 100 7:64 10 5 1:92 10 4 25 3:26 10 4 3:01 10 4 Table A.2: Whistler turbulence ion and electron maximum heating rates of Chapter 4, Figures 4.5 and 4.6. m i =m e = 400, v te =c = 0:10, i = e = 0:05, 0 = 0:10, variable: Domain Length. L! e =c Q i Q e 25:60 7:87 10 5 1:35 10 2 51:20 7:97 10 5 2:22 10 3 102:4 9:74 10 5 4:11 10 4 A Ion and Electron Maximum Heating Rates 181 Table A.3: Whistler turbulence ion and electron maximum heating rates of Chapter 5, Figures 5.4 and 5.5. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e = 0:25, variable: 0 . 0 Q i Q e 0:010 1:14 10 6 1:64 10 5 0:025 3:19 10 6 4:80 10 5 0:050 7:36 10 6 9:85 10 5 0:100 2:27 10 5 1:91 10 4 0:250 1:28 10 4 3:91 10 4 Table A.4: Whislter turbulence ion and electron maximum heating rates of Chapter 6, Figure 6.2. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e , 0 = 0:10, variable: e . e Q i Q e 0:10 9:40 10 5 4:27 10 4 0:25 2:27 10 5 1:91 10 4 0:50 1:13 10 5 9:63 10 5 1:00 5:71 10 6 4:60 10 5 2:50 2:50 10 6 1:50 10 5 5:00 1:59 10 6 4:65 10 6 182 Appendices Table A.5: Whislter turbulence ion and electron maximum heating rates of Chapter 6, Figure 6.3. L! e =c = 102:4, m i =m e = 400, v te =c = 0:10, i = e , 0 = 0:20 e , variable: e . e Q i Q e 0:10 4:59 10 6 9:17 10 5 0:25 7:36 10 6 9:85 10 5 0:50 1:13 10 5 9:63 10 5 1:00 1:43 10 5 8:21 10 5 2:50 1:58 10 5 5:44 10 5 5:00 1:71 10 5 4:51 10 5 Table A.6: KAW turbulence ion and electron maximum heating rates of Chapter 7, Figures 7.7 and 7.8. L k ! e =c = 655:4, L ? ! e =c = 81:9, m i =m e = 100, v te =c = 0:32, i = e = 0:50, variable: 0 . 0 Q i Q e 0:05 9:10 10 5 3:93 10 5 0:10 1:90 10 4 7:35 10 5 0:25 5:50 10 4 2:08 10 4 0:50 1:21 10 3 5:03 10 4 B Electromagnetics and Plasmas 183 B Electromagnetics and Plasmas Also see NRL Plasma Formulary for a more exhaustive tabulation of parameters/rela- tions relevant to plasma physics. B.1 Reference Tables Table B.1: Conversion factors, F, for converting formulae between Gaussian and SI forms. A parameter in Gaussian units, X Gs , relates to its SI form, X SI , by X Gs =FX SI . To convert from Gaussian (SI) to SI (Gaussian) replace each X Gs (X SI ) in the formula with FX SI (X Gs =F). Use the relation c = 1= p 0 " 0 where applicable. The value F for converting a numeric quantity from SI to Gaussian units can be calculated by absorbing the dimensions of the parameter X SI into the dimensions of its correspondingF (in SI), then replacing all (kg, m) dimensions with (1000g, 100cm). F will then have the correct numeric conversion value and Gaussian dimensionality. The inverse process gives an inverse conversion value of 1=F with the correct SI dimensionality. Name Symbol F Electric Field, Electric Potential (E; ) p 4" 0 Electric Flux Density D p 4=" 0 Charge, Charge Density, Current, Current Density, Electric Dipole Moment, Polarization Density (q; ; I; J; p; P) 1= p 4" 0 Magnetic Flux Density, Magnetic Flux, Magnetic Vector Potential (B; m ; A) p 4= 0 Magnetic Field Intensity H p 4 0 Magnetic Moment, Magnetization (m; M) p 0 =4 Permittivity, Permeability ("; ) (1=" 0 ; 1= 0 ) Electric Susceptibility, Magnetic Susceptibility ( e ; m ) 1=4 Conductivity, Conductance, Capacitance (; S; C) 1=4" 0 Resistivity, Resistance, Inductance (; R; L) 4" 0 184 Appendices Table B.2: EM parameters in SI units. Name Symbol Unit Dimensions Alt. Force F Newton (N) kg m=s 2 Energy E Joule (J) kg m 2 =s 2 N m Electric Current I Ampère (A) A Electric Charge q Coulomb (C) A s Electric Potential Volt (V) kg m 2 =A s 3 J=C Magnetic Flux m Weber (Wb) kg m 2 =A s 2 J=A Capacitance C Farad (F) A 2 s 4 =kg m 2 C=V Inductance L Henry (H) kg m 2 =A 2 s 2 V s=A Resistance R Ohm ( ) kg m 2 =A 2 s 3 V=A Conductance S Siemens (S) A 2 s 3 =kg m 2 1= Resistivity kg m 3 =A 2 s 3 m Conductivity A 2 s 3 =kg m 3 S=m Electric Field Intensity E V=m N=C Electric Flux Density D C=m 2 F V=m Electric Dipole Moment p C=m Magnetic Flux Density B Tesla (T) kg=A s 2 Magnetic Field Intensity H A=m T m=H Magnetic Vector Potential A kg m=A s 2 T m Magnetic Moment m A m 2 N m=T B Electromagnetics and Plasmas 185 Table B.3: EM parameters in Gaussian units. Name Symbol Unit Dimensions Alt. Force F dyne (dyn) g cm=s 2 Energy E erg g cm 2 =s 2 dyn cm Charge q statC g 1=2 cm 3=2 =s dyn 1=2 cm Current I g 1=2 cm 3=2 =s 2 statC=s Electric Potential statV g 1=2 cm 1=2 =s erg=statC Magnetic Flux m Maxwell (Mx) g 1=2 cm 3=2 =s Capacitance C cm statC=statV Inductance L s 2 =cm statV s 2 =statC Resistance R s=cm statV s=statC Electric Field Intensity E g 1=2 =cm 1=2 s statV=cm Electric Flux Density D g 1=2 =cm 1=2 s statC=cm 2 Magnetic Flux Density B Gauss (Gs) g 1=2 =cm 1=2 s Mx=cm 2 Magnetic Field Intensity H Oersted (Oe) g 1=2 =cm 1=2 s dyn=Mx Table B.4: Physical Constants in SI Name Symbol Value Dim Notes Speed of Light c 2:99792458 10 8 m=s c = 1= p 0 " 0 Permeability 0 4 10 7 H=m H = 1 0 B Permittivity " 0 8:8541878 10 12 F=m D =" 0 E Electron Mass m e 9:1093836 10 31 kg Proton Mass m p 1:6726219 10 27 kg m p = 1836:2m e Neutron Mass m n 1:6749275 10 27 kg Elementary Charge e 1:6021766 10 19 C Electronvolt eV 1:6021766 10 19 J 1eV 11605K Gas Constant R 8:3144598 J=mol K PV – =N mol RT Avogadro Constant N A 6:0221409 10 23 1=mol Boltzmann Constant K B 1:3806485 10 23 J=K K B =R=N A Planck Constant ~ 1:0545718 10 34 J s E = ~!; p = ~k Gravitational Constant G 6:6741 10 11 N m 2 =kg F =m 1 m 2 G=r 2 Astronomical Unit AU 1:4959787 10 11 m 186 Appendices Table B.5: Additional Parameters Name Symbol Dimensions Notes Number Density n 1=m 3 Temperature T Kelvin (K) 3K B T =m j <jv j j 2 > Pressure P N=m 2 (Pa) P =nK B T Wavelength m=cyc Wavenumber k 1=m k = 2= (Angular) Frequency (!) f (1=s) Hz ! = 2f Phase Velocity v m=s v =!=k Group Velocity v g m=s v g =@!=@k Table B.6: Plasma parameters in Gaussian units. Each property is defined for the corresponding j-th plasma species; j =e for electrons j =p for protons and j =i for ions. When the Debye length does not specifyj, the electron Debye length is assumed. Name Symbol Definition Alt. Beta j 8n j K B T jk =B 2 0 2(v tj =c) 2 =( j =! j ) 2 Cyclotron Frequency j jq j jB 0 =m j c Plasma Frequency ! j q 4n j q 2 j =m j Thermal Speed v tj p K B T jk =m j c ( j =! j ) p j =2 Alfvén Speed V A p B 2 0 =4n i m i c ( i =! i ) Debye Length Dj q K B T jk =4n j q 2 j v tj =! j Inertial Length j q m j c 2 =4n j q 2 j c=! j Cyclotron Radius j (c=jq j jB 0 ) p m j K B T j? (v tj = j ) p T j? =T jk B Electromagnetics and Plasmas 187 Table B.7: Useful dimensionless plasma parameter ratios. Here it is assumed that n e =n i =n 0 . ES denotes electrostatic, EM denotes electromagnetic. Ratio Definition Significance e =! e (v te =c) p 2= e Electron EM/ES time ratio e = D c=v te Electron EM/ES length ratio e = e p T e? =T ek p e =2 Influence of electron magnetization ! i =! e p m e =m i ES Ion/electron time scaling Di = D p T ik =T ek ES Ion/electron length scaling i = e ( e = e ) p T i? =T e? p m i =m e influence of ion magnetization i =! e ( e =! e )(m e =m i ) Kinetic range time ratio i = D ( e = D ) p m i =m e Kinetic range length ratio V A =v te p 2= e p m e =m i Kinetic range speed ratio 188 Appendices Table B.8: Characteristic solar wind parameters at 1 AU. Source for measured parameters [Bruno and Carbone, 2013]. Note that all quoted values are order-of- magnitude approximations, due to the substantial variability of solar wind quantities. Name Symbol Slow Wind Fast Wind Dimensions Plasma Parameters Number density n 0 15 4 1=cm 3 Magnetic field B 0 6 6 nT Magnetic Fluctuations ( P B 2 ) 1=2 4:0 0:6 nT Bulk speed V sw 350 600 km=s Alfvén speed V A 34 65 km=s Debye length D 8 11 m Proton Parameters Temperature T p 5 10 4 2 10 5 K Thermal speed v tp 20 41 km=s Plasma frequency ! p 5100 2600 1=s Cyclotron frequency p 0:6 0:6 1=s Inertial length p 59 110 km Gyro-radius p 35 71 km Beta p 0:72 0:77 P-P collision frequency pp 2 10 6 2 10 7 1=s Distance per P-P collision L pp 1:2 20 AU Electron Parameters Temperature T e 2 10 5 1 10 5 K Thermal speed v te 1700 1200 km=s Plasma frequency ! e 2:2 10 5 1:1 10 5 1=s Cyclotron frequency e 1100 1100 1=s Inertial length e 1:4 2:7 km Gyro-radius e 1:6 1:2 km Beta e 2:9 0:39 B Electromagnetics and Plasmas 189 B.2 Useful EM Relations in Gaussian Units Maxwell’s Equations Gauss’s Law: r E = 4 (B.1) Gauss’s Law for Magnetism: r B = 0 (B.2) Faraday’s law: r E = 1 c @B @t (B.3) Ampère’s law: r B = 4 c J + 1 c @E @t (B.4) Darwin’s Equations Darwin’s set of equations is composed of Eqs. (B.1), (B.2), and (B.3) along with the modified form of Eq. (B.4) in which the transverse component of the displacement current, @E T =@t, is neglected Darwin’s equation: r B = 4 c J + 1 c @E L @t (B.5) More Useful Equations Lorentz Force: F =q[E + v c B] (B.6) Continuity: r J + @ @t = 0 (B.7) Electric Field Potential: E =r 1 c @A @t (B.8) Magnetic Field Potential: B =r A (B.9) Coulomb’s law: E =qr=r 3 (B.10) Biot-Savart law: B =q v c r r 3 (B.11) Field Energy Density: u = 1 8 (E 2 +B 2 ) (B.12) Energy flux (Poynting): S = c 4 E B (B.13) Entropy: S = Z v f(v)log[f(v)]dv (B.14) 190 Appendices Useful Relations ; represent generic vector, scalar fields respectively. `; S; V – represent length, area, volume respectively. Stokes Theorem: Z S (r)ds = I c d` (B.15) Divergence Theorem: Z V – rdV – = I S ds (B.16) Vector Identities: r (r ) = 0 (B.17) r (r) = 0 (B.18) rr =r(r)r 2 (B.19) r ( ) =r + r (B.20) . B.3 Maxwell and Darwin Potential Fields The effect of electric/magnetic field coupling via Faraday’s and Ampère’s laws can be made clear by specifying the fields in terms of potentials, and decoupling by proper choice of potential functions. The potentials associated with each field are determined by use of identities (B.17) and (B.18). Equation (B.17) says that the curl of the gradient of any scalar field is zero; therefore if the curl of a vector field equals zero, that vector field can be described by the gradient of some scalar field. Equation (B.18) says that the divergence of the curl of a vector field is zero; if some given vector field is divergence-free, it can be described by the curl of some other vector field. The definition of the magnetic field, B, in terms of a vector potential (A) results from comparing Eq. (B.2) with Eq. (B.18) B =r A (B.21) B Electromagnetics and Plasmas 191 A vector field is fully determined by the specification of its curl and its divergence. Note that the vector A is not fully determined because its divergence hasn’t been specified. This property is free to be chosen, and is selected for a given situation based on convenience (of course, once chosen this property must remain consistent). For now, it will be seen shortly that the most convenient specification for A will come in the form of the Lorentz gauge r A + 1 c @ @t = 0 (B.22) The electric potential can be derived by plugging Eq. (B.21) into Eq. (B.3) r E + 1 c @ @t (r A) =r (E + 1 c @A @t ) = 0 Comparing with (B.17) the quantity inside the parentheses can be represented by a scalar field E + 1 c @A @t =r this provides the definition of the electric field, E, in terms of potentials E =r 1 c @A @t (B.23) where is the scalar electric potential. The negative sign placed onr is required for conformance with conservation of mechanical energy. B.3.1 Electromagnetic Waves With the fields specified in terms of potentials, the wave-nature of time varying electromagnetic fields can be revealed. Plugging Eqs. (B.21) and (B.23) into Eq. (B.4) yields rr A = 4 c J + 1 c @ @t (r 1 c @A @t ) 192 Appendices Applying the identity (B.19) and rearranging leads to r 2 A = 4 c J 1 c 2 @ 2 A @t 2 r(r A + 1 c @ @t ) By applying the Lorentz gauge, Eq. (B.22), the final term on the right vanishes, resulting in r 2 A 1 c 2 @ 2 A @t 2 = 4 c J (B.24) The scalar potential can be arranged into an analogous form by plugging Eq. (B.23) into Eq. (B.1) r (r 1 c @A @t ) =r 2 1 c @ @t (r A) = 4 Making further use of the Lorentz gauge to replacer A r 2 1 c @ @t ( 1 c @ @t ) = 4 r 2 1 c 2 @ 2 @t 2 =4 (B.25) These equations take the form of non-homogeneous wave equations with a propagation speed equal to c, the speed of light. This demonstrates that the two fields couple in a way such that electromagnetic radiation propagates in the form of waves originating from the motions of charged particles. Electromagnetic wave propagation does not, however, require charged particles to be sustained. This can be seen by setting and J to zero, and solving the above equations by the method of separation of variables on periodic boundary conditions. Doing so produces a non-vanishing, periodic solution of the form (; A)/e i(kxckt) (B.26) B Electromagnetics and Plasmas 193 The source of this self-sustaining effect is the inductive coupling between the electric field and the magnetic field. An important implication of Eq. (B.26) is that the fields, at some point far away from the source charges, are a function of the charge/current distribution at an earlier time, e.g. (x;t) =(t x=c). In other words, the change in potential field at some point in space isn’t realized until the change has had enough time to travel to that point. This is known as field retardation. This effect suggests the importance of resolving light wave propagation in numerical simulations that make use of the complete set of Maxwell’s equations. B.3.2 Darwin Potential Equations The Darwin B and E fields remain defined by Eqs. (B.21) and (B.23), as these equations were derived from Eqs. (B.2) and (B.3), which remain consistent between Darwin’s and Maxwell’s equations. For the case of Darwin’s equations, however, it is more convenient to define the divergence of A by the Coulomb gauge r A = 0 (B.27) Beginning by looking at how the scalar potential behaves under the Darwin model, and proceeding in the same way as before r E =r (r 1 c @A @t ) =r 2 1 c @ @t (r A) = 4 The second term on the left hand side vanishes under the Coulomb gauge resulting in r 2 =4 (B.28) This is the standard Poisson equation of electrostatics. 194 Appendices The first step in examining the vector potential involves satisfying conservation of charge, Eq. (B.7). Using Eq. (B.1) to replace in Eq. (B.7), and noting that the divergence of the transverse part of any vector field is zero gives r J L + @ @t ( 1 4 r E L ) = 0 r (J L + 1 4 @E L @t ) = 0 (B.29) Because the term in parenthesis is irrotational, it can be expressed as the gradient of a scalar field J L + 1 4 @E L @t =r (B.30) which converts Eq. (B.29) into a Laplace equation r 2 = 0 (B.31) For a strictly periodic system (which is the system of interest here) the solution to Eq. (B.31) is = Const:!r = 0 (B.32) and so the divergence operator in Eq. (B.29) can be removed J L + 1 4 @E L @t = 0 (B.33) In conformance with continuity, Eq. (B.5) can be modified as follows r B = 4 c (J T + J L ) + 1 c @E L @t = 4 c J T + 4 c : 0 (J L + 1 4 @E L @t ) r B = 4 c J T (B.34) B Electromagnetics and Plasmas 195 It can be seen that, under the Darwin approximation, the curl of the magnetic field is independent of the time rate of change of the electric field, and only depends on the transverse part of the current density. Making use of Eq. (B.21) to replace B in Eq. (B.34) rr A =r(r A)r 2 A = 4 c J T By applying the Coulomb gauge this becomes r 2 A = 4 c J T (B.35) The vector potential is described by a Poisson equation analogous to that of the scalar potential. Because of this similarity with the electrostatic field equation, methods employing the Darwin equations are sometimes referred to as magnetostatic. While the form of the Darwin equations is very similar to that of Maxwell’s equations, the implications of the small change are quite large. A first note is that the equations no longer take the form of wave equations. In particular, electromagnetic waves cannot propagate independent of the charge and current distributions; the Darwin equations eliminate electromagnetic radiation. Further, the solution to these equations does not depend on retarded times, meaning that the effects of a change in the charge or current distributions is felt immediately. The radiation effect of Maxwell’s equations is responsible for the retardation of the potential fields, and hence also responsible for the need to satisfy the Courant condition. At large distances from the source charges the Darwin approximation could potentially result in calculation error, however near the sources the static effects of Maxwell’s equations dominate over the radiation effects. For this reason, another term for the Darwin model is a near field model. The accuracy of the Darwin model can also be seen from the fact that 196 Appendices it is second order accurate in (v=c). So long as v=c 1, it is a very good model for describing the behavior of a system of charged particles. B.4 Methods of Analysis B.4.1 Entropy The classical concept of energy transfer in a fluid medium is that of a transfer of energy from large scales to small scales; this is analogously thought of as a transition from an initial state to a less ordered final state. In other words it is expected that the movement of energy from large to small scales results in an increase in the system entropy. It is, therefore, curious to consider how the state of the system entropy evolves during the transfer of energy from small- to large-scale fluctuations. Entropy of the j-th plasma species is defined as S j =k B N j Z f j (v)f1ln[h 3 n j f j (v)=m 3 j ]gdv (B.36) [Schrock et al., 2005b,a] wheref j denotes thej-th species velocity distribution function, k B is Boltzmann’s constant, and h is Planck’s constant. Equation (B.36) can be rewritten as S j =k B N j (1ln[h 3 n j =m 3 j ])k B N j Z f j (v)ln[f j (v)]dv (B.37) This form can be interpreted as consisting of a term describing entropy transport (the leading term), proportional to the log of the number of particles in some region of space, and a term describing entropy generation (the following term). For the closed system calculations carried out here the number density calculated on the entire domain is constant, resulting in a constant leading term; as absolute entropy has no B Electromagnetics and Plasmas 197 meaning, only entropy relative to some reference value is important, the leading term can be neglected. This simplifies the calculation to S j =k B N j Z f j (v)ln[f j (v)]dv (B.38) It is a fairly simple procedure to carry out this calculation in order to quantify how the entropy of the plasma system considered here evolves. One issue arises, however, in the calculation of Eq. (B.38); how to accurately model the form of the velocity distribution. While the shape of f j (v) comes directly from the particle information provided by the PIC simulation, it becomes impractical to calculate a fully three-dimensional velocity distribution in practice. One assumption that can be made to quantify entropy generation in the presented numerical simulations is to consider each of the three velocity-space dimensions to be independent of one another [Schrock et al., 2005b,a]. In this case, the velocity distribution function can be written as f j (v) = f xj (v x )f yj (v y )f zj (v z ). Under this assumption it can be shown that the calculation of entropy becomes much more manageable numerically; that is, only one-dimensional velocity distributions are required as inputs into Eq. (B.38) and entropy can be calculated as S j =k B N j X i=x;y;z Z f ij (v i )ln[f ij (v i )]dv i (B.39) One dimensional velocity distributions are a standard diagnostic of the simulations and so it is possible, under this assumption, to use currently available data to calculate entropy generation for all previous simulation studies. A concern, however, is whether it is a valid assumption that the three velocity dimensions remain decoupled from one another. After all this assumption requires Maxwellian velocity distributions and, while the initial simulated distributions are very near Maxwellian, they are slightly perturbed from thermal equilibrium. In order 198 Appendices to prove/disprove the validity of this assumption an improved attempt to calculate entropy generation is carried out using the full three-dimensional velocity distribution. As it is computationally impractical to calculate a 3-D velocity distribution, advantage can be taken of the symmetries present in a magnetized plasma to reduce the distribution to a 2-D distribution from which the 3-D distribution can be recovered. All studies of this manuscript indicate that there is no preferential direction of structure in the plane perpendicular to the magnetic field; it is a valid assumption that the velocity distribution remains gyrotropic at all simulation times. In place of the 3-D distribution, 2-D velocity distributions of the form f j (v k =v z ;v ? = q (v 2 x +v 2 y )) can be used to calculate a coupled solution to Eq. (B.38). By switching from Cartesian to cylindrical coordinates, and imposing the assumption that the distribution is gyrotropic, Eq. (B.38) can be written as S j =k B N j Z v k Z v ? f j (v k ;v ? )ln[f j (v k ;v ? )=(2v ? )]dv ? dv k (B.40) The results of calculating the entropy by Eq. (B.40) are compared with the method of Eq. (B.39) in Fig. B.1 for the whistler anisotropy instability simulation. It is seen that, for this case, the 1D method very closely agrees with the 2-D method. The results suggest that, for studies of plasmas slightly perturbed from thermal equilibrium, the 1D approximation is valid; all entropy results of the following chapters will therefore employ the simpler 1D method (Eq. B.39). B Electromagnetics and Plasmas 199 0 500 1000 1500 2000 t ω e 0 0.02 0.04 0.06 0.08 0.1 S e S e (1D) S e (3D) Figure B.1: Entropy of electrons as a function of time during the evolution of the whistler anisotropy instability in the 3D-EMPIC model. Two methods of calculating entropy are compared. B.4.2 Additional Diagnostic Methods In addition to the analysis previously described, the following analytical techniques can be applied in select studies in order to better characterize physical process associated with kinetic range plasma turbulence. Identification of plasma modes. Toanalyzetherelativestrengthsof themodesthat may contribute to short wavelength plasma turbulence, we have developed a dispersion diagnostic which enables visualization of frequency versus wavenumber during the evolution of the simulations (e.g., see Fig. 3.3). Additional normal mode identification diagnostics include plots of magnetic compressibility C k (k)B k (k) 2 =jB(k)j 2 [Gary and Smith, 2009], and polarization analysis. Polarization can be determined by a number of methods outlined in [Lacombe et al., 2014]. Analysis of wave-wave interactions. We have developed an analysis technique based upon the frequency and wavevector matching conditions of wave-wave interaction 200 Appendices theory, namely k 1 + k 2 = k 3 and ! 1 (k 1 ) +! 2 (k 2 ) =! 3 (k 3 ) [Gary et al., 2014]. These equations, in conjunction with solutions of the kinetic linear dispersion equation, can be used to test the likelihood of three-wave interactions contributing to the forward cascade of various plasma modes. Analysis of the linear damping rate. The overall linear wave-particle damping rate can be calculated, which is the linear theory damping rate times the magnetic energy density integrated over the wavevector spectrum. Comparison against the full magnetic field dissipation rate will yield an estimate of the fraction of wave damping and particle heating due to Landau and cyclotron damping (e.g. [Chang et al., 2014]). Analysis of intermittency. As was determined in the work of [Karimabadi et al., 2013], a major part of turbulent dissipation in the short-wavelength regime can be due to the intermittent formation of coherent current sheets. Probability density functions of the increments of magnetic fluctuations can be constructed to obtain information about the nonlinear and presumably intermittent qualities of the turbulent spectra. C Parallel FFT on a 3-D Decomposed Array of Vector Data 201 C Parallel FFT on a 3-D Decomposed Array of Vec- tor Data C.1 The Fourier Transform The Fourier transform is ubiquitous across virtually all scientific disciples. As such it is the subject of much research. This function, defined as F (f(t)) =F (!) = Z 1 1 f(t)e 2i!t dt (C.1) takes a continuous signal and maps it to a continuous spectrum of frequencies which compose the signal. Because continuous signals are measured by finite state systems, the equation must be discretized if it is to be useful. The discrete Fourier transform (DFT) is defined as F k = N1 X j=0 e i2jk=N f j (C.2) and maps N sample points of a signal into N frequency points. If the signal is sampled with a frequency of t, then the frequency resolution will be 1 Nt , and the range of valid frequencies is f = 1 2t ! 1 2t . Frequencies outside of this range cannot be uniquely identified (this is known as aliasing) and so cannot be resolved. Because of the wide range of application and immense usefulness of this function, an efficient algorithm for computing the DFT of a set of data is highly desired. A Naive approach to designing an algorithm is to follow the definition and solve for each F k one at a time. The time complexity of such an approach is O(N 2 ). It so happens that there is a far faster way to compute the DFT, known as the fast Fourier transform (FFT). The procedure for performing the FFT is described in section C.2. As efficient as the FFT algorithm is, when it is carried out on a 3-dimensional set of data, the time required can become very large, simply because such a large number of FFT’s must be performed. To further improve on execution time, advantage can be 202 Appendices taken by distributing the data to be processed across an array of processors [Huang et al., 1993]. Efficient parallelization of the 3D FFT can be difficult. However, despite the lack of efficiency, a significant speedup can be achieved with enough processors and parallelization can, in many cases, be justified. C.2 Fast Fourier Transform The FFT is an extremely clever approach to solving for the DFT of a function. The strategy is known as divide and conquer. First the problem, of length N, is systematically subdivided into smaller problems until each sub-problem is of a constant size, regardless of what N may be. The subproblems are then solved and recombined with one another until the solution, of length N, is acquired. Each part of this approach, applied to the FFT, is described below. C.3 Divide The divide step is achieved by rearranging the input into bit reversed order. What this means is that the input at index j should be moved to the position of index k, where k is the value obtained by reversing the order of the binary representation of the value j. This converts the input f j into its transform of length 1, f k . The reason for this is that the recursion of the divide step requires that the full summation be successively split into groups of summations of half length, where the first group contains the even indexed f 0 j s and the second group contains the odd indexed f 0 j s. Once the groups of f j ’s have been split down to groups of length 1 each summation reads F k;l=1 = 0 X j=0 e i2jk=N f k =f k (C.3) which is just the input value at the kth index after all the even/odd sorting has been performed. The final index of a value which started at index j turns out to be the C Parallel FFT on a 3-D Decomposed Array of Vector Data 203 bit reversal of j. This is because, whether a value is shuffled to the front half of its current set, or the back half depends on whether it is even (next lowest order bit is 0) or odd (next lowest order bit is 1). If it is even, it will go to the front half, in which case the next highest order bit in the new index will be 0, otherwise it will be in the latter half, in which its next highest order bit will be 1. Ultimately this dependence results in a copying of the original bit sequence from lowest to highest into the new bit sequence from highest to lowest. C.4 Combine The combining phase employs the butterfly technique, illustrated in figure C.1. At a given level there are a total of O(N) operations (l operations per group, N=l groups), where two members of a group are paired up, an odd member and its even compliment. The operation performed for each of these members is F k =f e +W k f o where W = e i2=l . W is periodic in the length of the subset l (each subset spans k l = 0!l 1) and so W k =W k+ln . Therefore W k for a given F k can also be used Figure C.1: Butterfly technique applied to an FFT of length 8 204 Appendices for F k+ln . Reusing W k in this way increases the efficiency of the algorithm. Efficiency is further increased by observing that k e and k o share a relationship)k o =k e +l=2 W ko =e i2(ke+l=2)=l =e i2(ke=l+1=2) =e i2ke=l e i =W ke (C.4) and so, with one calculation F ko and F ke can be updated. The final part to this algorithm is how the phase angle, =2k=l, updates as k spans from 0 to l=2 1. A clever way of updating the phase angle without recalculating it on every iteration is to form the constant =2=l. The first pair in the set always has W real = 1;W imaginary = 0 (k = 0! 2k=l = 0). The phase for the following pairs can then be found by the recurrence e i k+1 =e i k e i (C.5) As there is O(N) operations at each level, and there is log(N) levels, the time complexity of the FFT algorithm is O(Nlog(N)), resulting in an incredible speedup for large N. C.5 FFT on Real Data In my case, the input data is always real. It is, then, inefficient to represent the input data as complex which, at first glance, appears to be required to do the FFT in place, as the output data will almost exclusively be complex. However, there is a trick that can be performed to use the space required to store the real data, and have enough space there to store all the unique complex output data, unique being the key word. C Parallel FFT on a 3-D Decomposed Array of Vector Data 205 This is achieved by leveraging a symmetry property which is valid when the input data is real F Nk =F k (C.6) The only independent values are F 0 ! F N=2 . The length of the input array is N, and, to store the independent output data, the length of the output array must be (accounting forf = 0) (N=2+1)2 =N +2. This length is slightly larger because there is N=2 + 1 unique output values. Now the question is how to exploit this symmetry. The idea is to assume the array of real data f, length N, is instead an array of complex data h of length N=2, with the even indices representing the real part and the odd indices representing the imaginary part. So the DFT under this assumption is H n = N=21 X j=0 e i2jn=(N=2) (f 2j +if 2j+1 ) (C.7) looking at the complex conjugate of H N=2n H N=2n = [ N=21 X j=0 e i2j(N=2n)=(N=2) (f 2j +if 2j+1 )] = N=21 X j=0 [e i2j e i2jn=(N=2) ] (f 2j if 2j+1 ) = N=21 X j=0 e i2jn=(N=2) (f 2j if 2j+1 ) (C.8) adding and subtracting these equations results in H n +H N=2n = 2F e n H n H N=2n = 2iF o n (C.9) with n = 0! N=2 1. These are the even and odd sub-transforms which, when combined, give the first half of the final data set. The second half can be re-obtained 206 Appendices whenever it may be needed through the symmetry relation for real input data. The final operation required to obtain the correct values is F n =F e n +F o n e i2n=N (C.10) Combining equations C.9 and C.10 provides the formula for converting from the H values to the F values F n = 1 2 (H n +H N=2n ) i 2 (H n H N=2n )e i2n=N (C.11) . C.6 FFT In Three Dimensions The logic of extending the 1D transform to 3D is very straight forward. The DFT in 3 dimensions is F (k 1 ;k 2 ;k 3 ) = N 3 1 X j 3 =0 N 2 1 X j 2 =0 N 1 1 X j 1 =0 e i2j 3 k 3 =N 3 e i2j 2 k 2 =N 2 e i2j 1 k 1 =N 1 f(j 1 ;j 2 ;j 3 ) = N 3 1 X j 3 =0 e i2j 3 k 3 =N 3 [ N 2 1 X j 2 =0 e i2j 2 k 2 =N 2 [ N 1 1 X j 1 =0 e i2j 1 k 1 =N 1 f(j 1 ;j 2 ;j 3 )]] =F 3 [F 2 [F 1 (f)]] (C.12) It is apparent that the input for the FFT in one dimension is the output of the FFT from the preceding dimension. It is also clear that the order of operations is not unique. Any dimension can be considered first, second and third. The logical approach is to perform 1-D FFT’s in the first dimension at every index (j 2 ;j 3 ), followed by FFT’s in the second dimension at every index (k 1 ;j 3 ), and finally do FFT’s in the third dimension at every index (k 1 ;k 2 ) . For the case ofN 1 =N 2 =N 3 this results in a time complexity of O(N 3 log(N)) for the 3D FFT. C Parallel FFT on a 3-D Decomposed Array of Vector Data 207 C.7 Serial 3DFFT The first half of my project was to write my own serial 3D fast Fourier transform. The data I process is typically vector data; each point in 3D space contains three vector components. For each vector component a 3D FFT must be performed. To extend the FFT algorithm to accommodate vector data, an additional loop is introduced that iterates for each vector component set. The number of vector components is specified as an input parameter. Selecting a value of 3 implies 3D vectors, while specifying this value as 1 implies scalar data (any size vector can be accommodated). Each spatial dimension to be FFT’d has its own subroutine. Depending on the domain lengths specified in the input file, the program can perform a 3D, 2D, or 1D FFT. If all three domain lengths are greater than 1 then all three subroutines are called, and a 3D FFT is performed. If the z dimension is specified as 1, then the z-FFT subroutine is not called and the FFT is 2D. If y and z are set to 1, then only the x-FFT is called and the FFT is 1D. The program can also perform inverse FFT’s. This is determined by the input parameter isign. If set to 1 a forward FFT will be carried out, if set to -1 an inverse FFT will be carried out. Finally, the FFT routine can accommodate real or complex input. If the input is real, then the code performs the optimization described in section C.5. Each of the three subroutines are optimized to reduce cache misses. This is done by maintaining the loop order on the reduce and combine steps in the following way, from outermost loop to innermost: loop over each vector component set loop over the z-dimension loop over the y-dimension loop over the x-dimension 208 Appendices Thishelpsinsurethatelementswhichareadjacentinmemoryareaccessedinsuccession and greatly reduces the number of large strides in the array between elements required for a given calculation. C.8 Parallel FFT The second part to my project was to extend my serial FFT into a parallel version. The FFT does not greatly lend itself to parallelism. A typical method for parallelizing the 3D FFT is to distribute the data among the processors in ’slabs’ that contain the full array in two dimensions but only a fraction in the last dimension. With this distribution 1D FFT’s are carried out independently on each processor. The array is then transposed, so that each processor has a new 2D slab, with the array divided along the direction that was already FFT’d. Each array can then independently carry out a 2D FFT, and complete the process. The parallel transposition is very time consuming and greatly reduces the efficiency of the process. In my case, I am restricted in algorithm design to a specific data distribution. The purpose of creating a parallel FFT is so that I can implement it in the code I use to conduct research, and that code is designed around a 3D decomposition of the data such that each processor owns a sub-cube of the full 3D domain. As this is the case, my intention is to design a parallel FFT which operate specifically on a three dimensionally decomposed data set. Two methods for parallelizing the 3D FFT are presented and analyzed in order to determine which approach proves more efficient. C.9 Method 1 Consider a 1D FFT on an array which is split amongst N proc processors and has a total of N e elements (N proc ;N e are powers of 2) where n =N e =N proc is the number of elements per processor. C Parallel FFT on a 3-D Decomposed Array of Vector Data 209 The FFT method, in some sense, lends itself to this type of decomposition, where the FFT is now performed on the element level in each processor, then on the processor level. The bit reversal can be executed as follows each processor does bit reversal on its local elements each processor does a bit reversal on an array holding the processor indices in the order specified by this array gather the elements in the range myid n=N proc ! (myid + 1)n=N proc 1 from every processor (including itself) and place them in the local array in strides of N proc beginning at the current loop index. The next step is to perform the regular combine algorithm until each processor has combined its local data into one set. This proceeds exactly how it would in the serial version. Once this is complete, the butterfly continues, except now across processors. At each level two processors will pair, communicate their elements, and combine the data locally. The extension of this procedure to 3D is similar to the extension of the serial FFT to 3D. C.10 Method 2 The second approach is far less complicated. In this method, each one dimensional row of processors forms a communication group. The communication groups shuffle their local data so that they each have an array that extends the full length of their dimension, but a reduced extent in one of the other two dimensions. Each processor can then independently FFT their new set of local data in that dimension. Following the FFT, the processors send the data back to its original source. This procedure is carried out once for each dimension. An illustration of the shuffle procedure is shown in figure C.2. 210 Appendices Figure C.2: Illustration of a communication group shuffling local data C.11 Complexity Analysis Accumulatingtheoperationcountoneachpartofmethod1’sprocedure, thecomplexity is shown to beO(n 3 log(N e )+n 3 log(N proc )), where and represent typical floating point operation time and communication time, respectively. Performing the analysis for method 2 results in a time complexity ofO(n 3 log(N e )+n 3 ). Method 2 proves to require a significantly fewer number of communications. This is fortunate as method 2 is conceptually simpler. On this analysis method 2 was selected for implementation. Furtheranalysisisrequiredtocharacterizethespeedupandefficiencyofthismethod. With a serial time complexity ofO(N 3 e log(N e )) the speedup can be manipulated into the form speedup =N 3 proc (1 + 1 log(N e ) ) 1 (C.13) As a 100% speedup would be by a factor of the total number of processors (N 3 proc ) the form of equation C.13 reveals the efficiency as efficiency = (1 + 1 log(N e ) ) 1 (C.14) The value for efficiency is likely to be quite low, as communication times are much longer than floating point operation times. C Parallel FFT on a 3-D Decomposed Array of Vector Data 211 C.12 Results The implementation of method 2 into a parallel algorithm for computing a 3D FFT on vector data was successful. The code was run on the NASA Pleiades super-cluster, using up to 256 compute nodes each housing 16 2.6-GHz sandy-bridge processors with 32 GB of memory per node. Test cases were run on data sets of size 256 3 , 512 3 , and 1024 3 , and the output was verified for correctness against the output of the data from Matlab’s FFTN routine. The timing results are presented below. Figure C.3: Compute time with increasing domain size. Number of processors fixed at N 3 proc = 512 Figure C.4: Speedup, comparing strong scaling with weak scaling. Strong scaling used a 1024 3 domain. Weak scaling used a fixed sized of 64 3 elements per processor. Figure C.3 compares the serial execution time to the parallel execution time at a fixed number of processors and increasing domain size. The difference increases 212 Appendices Figure C.5: Efficiency, comparing strong scaling with weak scaling. Strong scaling used a 1024 3 domain. Weak scaling used a fixed sized of 64 3 elements per processor. a significant amount and shows that, for a fixed number of processors, the parallel implementation becomes more worthwhile at larger problem sizes. Figures C.4 and C.5 compare the strong scaling of the algorithm to the weak scaling. It is very interesting that the two curves almost lay on top of one another. This could possibly be coincidence as the number of data points is very small. The efficiency in both cases is on the order of 0:1. From this value, the ratio of communication time to floating point operation time can be calculated from equation C.14 to be in the range 100 500. More runs are necessary to quantify this more accurately. C.13 Conclusions In this project, I wrote a serial implementation of the fast Fourier transform (following the outline for the 1D FFT given in Numerical Recipes). I then extended the algorithm to 3D (my own design), taylored it to handle vector data, and optimized the loops to reduce cache misses. From there I selected an approach for parallelization from two candidate methods, and successfully applied the approach to the FFT. The results show that the efficiency is poor. This, however, can be attributed to the difficulty of creating an efficient parallel FFT in general, and to the restriction of the data distribution required by my research code. The FFT will serve its purpose well C Parallel FFT on a 3-D Decomposed Array of Vector Data 213 when implemented in my code. The efficiency is low however the execution time is insignificant compared to the execution time of my simulation.
Abstract (if available)
Abstract
Plasma turbulence is defined as an ensemble of broadband, large amplitude, stochastic fluctuations in a hot, tenuous ionized medium. Turbulence is ubiquitous in the solar wind, and encompasses a hierarchical structure which ranges over many orders of magnitude in space and time. While it is believed that turbulence plays a key role in energy transport in the solar wind, the behavior of turbulent plasma on scales much smaller than the mean free path remains poorly understood. An important question, which has been the topic of debate in the heliophysics community for decades, addresses the nature of the fluctuations which compose small scale turbulence. If small scale turbulence can be roughly thought of as the superposition of a broad spectrum of normal mode fluctuations, which mode dominates the behavior of the system, and what are the relative contributions of the various possible modes to the dissipation of turbulent energy into heat? Two modes are of particular interest in the study of solar wind turbulence: the kinetic Alfvén wave, and the magnetosonic-whistler wave. The investigations presented here focus on characterizing the effectiveness of magnetosonic-whistler turbulence and kinetic Alfvén turbulence in the heating of protons and electrons in the solar wind. To address this topic large scale, fully kinetic, three-dimensional (3-D) electromagnetic (EM) particle-in-cell (PIC) simulations are carried out on a collisionless, homogeneous, magnetized, ion-electron plasma model. In this work, four major studies are conducted to investigate ion and electron heating by kinetic-scale plasma turbulence. ❧ The first three studies employ the 3-D EMPIC model in order to investigate ion and electron heating by whistler turbulence. The simulations use an initial ensemble of relatively long wavelength whistler modes with a broad range of initial propagation directions and parametrically vary initial physical properties of the system in order to obtain scaling laws for the maximum heating rates achieved by each plasma species. In the first study three simulations are performed corresponding to successively larger simulation boxes and successively longer wavelengths of the initial fluctuations. The computations confirm previous results showing electron heating is preferentially parallel to the background magnetic field B₀, and ion heating is preferentially perpendicular to B₀. The significant results are that larger simulation boxes and longer initial whistler wavelengths yield stronger ion heating, weaker electron heating, and weaker overall dissipation. In the second study an ensemble of five simulations is performed with all parameters held fixed but the dimensionless fluctuating magnetic field energy density, ε₀. The important results here are that, over 0.01 < ε₀ < 0.25, the maximum rate of electron heating (Qₑ) scales approximately as ε₀, and the maximum rate of ion heating (Qᵢ) scales approximately as ε₀^1.5. The third study carries out two ensembles of simulations with all parameters held fixed but the electron beta, βₑ. In Ensemble One, where each simulation began with ε₀ = 0.1, both Qₑ and Qᵢ scale approximately as βₑ⁻¹, whereas over 0.1 ≤ βₑ ≤ 1.0 in Ensemble Two, where each simulation was initialized such that ε₀ = 0.2 βₑ, Qₑ is approximately constant while Qᵢ scales approximately as βₑ^1/2. The results of these three studies suggest that sufficiently long wavelength and sufficiently large-amplitude magnetosonic-whistler turbulence at sufficiently large βₑ may heat ions more rapidly than electrons. ❧ The fourth study of this work moves beyond investigations of whistler turbulence to investigate the role of kinetic Alfvén waves in the physics of kinetic scale dissipation of turbulence in the solar wind. Maximum electron and ion heating rates due to kinetic Alfvén turbulence are computed as functions of ε₀. In contrast to the results for heating by whistler turbulence, the maximum ion heating rate due to kinetic Alfvén turbulence is substantially greater than the maximum electron heating rate. Furthermore, both ion and electron heating due to kinetic Alfvén turbulence scale approximately with ε₀. Finally, electron heating leads to anisotropies of the type Tₑ∥ > Tₑ⊥, where the subscripts refer to directions parallel and perpendicular, respectively, to B₀, whereas the heated ions remain approximately isotropic. ❧ A final study is conducted using the 3-D EMPIC model to investigate the whistler anisotropy instability (WAI) in three dimensions. Analysis of the magnetic field spectra show that the fluctuations generated by the WAI grow as relatively short-wavelength, quasi-parallel propagating whistler modes, as is predicted by linear dispersion theory. At late times, however, the field energy is transferred from small to large wavelength modes and forms a spectrum of fluctuations that exhibit quasi- perpendicular propagation. This inverse transfer process is reminiscent of an inverse cascade in a fully turbulent plasma. Entropy analysis is performed in order to verify that such a process obeys the second law of thermodynamics.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Hughes, Randall Scott
(author)
Core Title
Particle-in-cell simulations of kinetic-scale turbulence in the solar wind
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Astronautical Engineering
Publication Date
07/21/2017
Defense Date
07/18/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Alfvén,dissipation,instability,kinetic range,OAI-PMH Harvest,plasma,solar wind,turbulence,Whistler
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Wang, Joseph (
committee chair
), Gary, S. Peter (
committee member
), Gruntman, Mike (
committee member
), Kunc, Joseph (
committee member
), Nakano, Aiichiro (
committee member
)
Creator Email
rshughes.sc@gmail.com,rshughes@usc.edu
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https://doi.org/10.25549/usctheses-c40-407346
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UC11264486
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etd-HughesRand-5572.pdf (filename),usctheses-c40-407346 (legacy record id)
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407346
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Hughes, Randall Scott
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texts
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University of Southern California Dissertations and Theses
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
Alfvén
dissipation
instability
kinetic range
plasma
solar wind
turbulence