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Vortex lattice theory: a linear algebra approach

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Vortex lattice theory: a linear algebra approach

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Content VORTEX LATTICE THEORY: A LINEAR ALGEBRA APPROACH by George C. Chamoun A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (AEROSPACE AND MECHANICAL ENGINEERING) August 2008 Copyright 2008 George C. Chamoun Dedication To my parents Chaouki Chamoun, Linda-Rose Myers and Leila Hawi Chamoun. ii Acknowledgments I am grateful and feel fortunate to have had Paul Newton as my advisor, as I have learned from him on both the academic and personal levels. He exudes a positive energy that invigorates sci- entific research and collaboration, and has been deeply committed to my work and development. I would also like to thank my co-advisor Eva Kanso for all her academic advice and financial support. Her presence was a motivating factor in taking my research a step further. The lessons I have taken from Paul and Eva will continue to benefit me for life. In addition, my academic experience has been enriched by several professors. Oussama Safadi has been a mentor to me since the first day I entered USC. Firdaus Udwadia stimulated my interest in dynamical systems. Through Geoffrey Spedding, I learned how to present scientific work in an accessible and concise manner. I would also like to thank my committee members: Igor Kukavica, Fred Browand, Susan Friedlander and Roger Ghanem. Several graduate students at USC provided me with valuable feedback on my research, and I sincerely thank Fangxu Jing, Babak Oskouei, Korkut Brown and Artan Sheshmani. I’d like to thank several people that have been a part of my personal life for the past few years and have endured with me the stresses and strains of graduate school: Russell Wald, Sonya Collier, David Crombecque, Valerie Pirktl, Alexandre Cerf, Larry Tsironis and Rachel Elias. Finally, I would like to thank my parents Chaouki, Linda-Rose and Leila, my grandparents Marvin and Janice, and my sisters Owana, Zara and Oriane for their faith in me and continuous support. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vii List of Figures viii Abstract xv Chapter 1. Introduction 1 1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 V ortex Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Symmetric Lattices: Superconductors and Superfluids . . . . . . . . . 4 1.3.2 Asymmetric Lattices: Bose Einstein Condensates . . . . . . . . . . . . 5 1.3.3 Lattices in Other Physical Systems . . . . . . . . . . . . . . . . . . . . 9 1.4 V on K´ arm´ an V ortex Streets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Where They Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 History of the V ortex Street Model . . . . . . . . . . . . . . . . . . . . 15 1.4.3 Bifurcation Problems and Streamline Topology . . . . . . . . . . . . . 16 Chapter 2. Fluid Mechanics Formulation 24 2.5 General Properties of Inviscid, Incompressible Flows . . . . . . . . . . . . . . 25 2.5.1 Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.2 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.3 Application of the Cauchy-Riemann Equations . . . . . . . . . . . . . 27 2.6 Point V ortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.1 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.2 The Point V ortex Model . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Condition for Relative V ortex Equilbrium . . . . . . . . . . . . . . . . . . . . 32 2.7.1 On the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.2 On the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 iv Chapter 3. Linear Algebra Methods 39 3.8 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.8.2 Computation of the SVD . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.8.3 SVD Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.8.3.1 Havelock’s Square (N = 4) . . . . . . . . . . . . . . . . . . 42 3.8.3.2 Havelock’s Square Plus Central V ortex (N = 5) . . . . . . . 45 3.9 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.9.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.9.2 Information Compression . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9.3 Shannon Entropy Examples . . . . . . . . . . . . . . . . . . . . . . . 52 3.9.3.1 Havelock’s Square (N = 4) . . . . . . . . . . . . . . . . . . 52 3.9.3.2 Havelock’s Square Plus Central V ortex (N = 5) . . . . . . . 53 3.10 Matrix Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.10.1 The Frobenius Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.10.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 4. V ortex Lattices on the Plane 58 4.11 Comparison to Physical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.12 Asymmetric Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.12.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.12.2 The Brownian Ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.12.3 Distance to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.13 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.14 The Reverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.14.1 Configuration Construction With and Without a Coordinate System . . 73 4.14.2 Number of Elements in the Configuration Matrix . . . . . . . . . . . . 74 Chapter 5. Single V on K´ arm´ an Streets on the Sphere 76 5.15 Construction and SVD Solution . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.15.1 Without Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.15.1.1 Number of V ortices Per Ring:n6= 3 . . . . . . . . . . . . . 77 5.15.1.2 Number of V ortices Per Ring:n = 3 . . . . . . . . . . . . . 79 5.15.2 With Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.15.2.1 Example:n = 2 . . . . . . . . . . . . . . . . . . . . . . . . 81 5.15.2.2 General Numeric Solution . . . . . . . . . . . . . . . . . . . 83 5.16 Angular Velocity and Explicit Solutions . . . . . . . . . . . . . . . . . . . . . 85 5.16.1 Without Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.16.2 With Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.17 Streamline Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.18 Shannon Entropy: Effect of Adding Pole V ortices . . . . . . . . . . . . . . . . 91 5.18.1 Number of V ortices Per Ring:n = 2 . . . . . . . . . . . . . . . . . . . 92 5.18.2 Number of V ortices Per Ring:n = 3 . . . . . . . . . . . . . . . . . . . 92 5.18.3 Number of V ortices Per Ring:n 4 . . . . . . . . . . . . . . . . . . . 93 v Chapter 6. Double V on K´ arm´ an Streets on the Sphere 95 6.19 Construction and SVD Solution . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.19.1 Without Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.19.2 With Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.19.3 Equilibrium Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.20 Angular Velocity and Explicit Solutions . . . . . . . . . . . . . . . . . . . . . 101 6.20.1 Without Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.20.2 With Pole V ortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.20.3 Shearing Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.21 Streamline Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.22 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Chapter 7. Conclusions 114 7.23 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.24 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Bibliography 119 Appendices 124 Appendix A. Examples of Symmetric Lattices 125 Appendix B. Structure of the Configuration Matrix: Single V ortex Street 129 B.1 Odd Number of V ortices per Ring . . . . . . . . . . . . . . . . . . . . . . . . 130 B.2 Even Number of V ortices per Ring . . . . . . . . . . . . . . . . . . . . . . . . 131 Appendix C. Viscous Point V ortex Model 134 vi List of Tables 4.1 Converged Brownian ratchet configurations of Figure 4.38 with (i) the Frobenius size of the con- verged equilibrium configuration, (ii) distance to exact equilibrium, (iii) Hamiltonian energy and (iv) the distance between the two converged configurations that had the same initial conditions. 69 4.2 Singular values of converged Brownian ratchet configuration of Figure 4.38. . . . . . . . . . 70 4.3 Comparison of the minimum number of data points required to construct a configuration of N points in the plane in: (a) the presence of a coordinate system, and (b) in the absence of a coordi- nate system. Note that forN < 5, less values are required to construct the system in the absence of a coordinate system, forN = 5 the number of values is the same, and forN > 5, less val- ues are required in the presence of a coordinate. The last column demonstrates the number of non-trivial entries in configuration matrixA associated with differentN. . . . . . . . . . . . 74 B.4 Oddn Combination Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 B.5 Evenn Combination Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 vii List of Figures 1.1 Left: the first Abrikosov triangular lattice image by [27], on the order of 1 micron. Black dots represent cylindrical magnetic flux tubes. Right: Image from [36] of anNbSe2 triangular lattice made using scanning-tunneling microscope imaging, showing high degree of symmetry, on the order of several thousand angstroms. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Image from [74] of superfluid Helium lattices. Images contain 1-11 equal-circulation vortices, with the vortex number depending on system parameters (rotational speed of capsule, etc.). . . . 3 1.3 Image from Abrikosov’s Nobel Prize Lecture ([3]) of velocity vs. distance from vortex center. The solid line represents the velocity flow around an isolated lattice site. The plot’s features include a potential vortex core, and constant decay proportional to the distance from the site. The curve- fitted dashed line 1=r demonstrates that the point vortex model makes a good fit for sufficiently spaced vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Images from [25] showing that lattices possess a nearly two-dimensional structure. (a) is a top view, (b) is cross-sectional intensity reading and (c) is a sideview of the lattice. . . . . . . . . 5 1.5 Bose Einstein condensate image from [1]. Lattices include 16, 32, 80, 130 vortices, with the vortices being the dark spots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Image from [26] of (a) an initially regular triangular lattice, (b) getting blasted by a laser, (c) resulting in a perturbed configuration that undergoes cooling, and the vortices exhibit random motion before (d) the vortices settle into a new irregular equilibrium. . . . . . . . . . . . . 6 1.7 Another image from [26] of (a) an asymmetric equilibrium lattice configuration. (b) Cross-section image of vortex intensity; lattice was initially symmetric before being blasted by laser, and settling into a final asymmetric configuration after cooling and random motion had settled. (c) Cross- sectional intensity reading showing the irregular distribution after the ‘random’ motion has settled. 7 1.8 Rapidly rotating BEC lattice perturbed by a laser-induced perturbation at the center of the lattice supports small sinusoidal waves referred to ‘Tykachencooscillations’ ([17]). . . . . . . . 7 1.9 Images from [25] showing that external forces on a lattice can change its shape and size. . . . . 8 1.10 Milli-sized magnetic structures floating on a rotating fluid. Magnets form lattice-like structures. Images above are for different numbers of magnets, from [34]. . . . . . . . . . . . . . . . 8 viii 1.11 Left, (a): magnets far from center of rotation possess asymmetric structure; and dye injected into fluid shows that magnets are themselves rotating, and form a prototype of point vortex equilibria. Right, (b): Subplots a, b: Lattices formed by variable-sized lattices. Interchangeable arrows indicate that system switches between the two configurations. Subplot c: Lattices formed on the curved surface of a drop. These configuration types never formed on the flat water-air interface. Images from [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.12 Trapped plasma in a cylindrical structure with a spiral electron source. V ortex lattice tubes form with different vortex numbers and configurations. Image from [30]. . . . . . . . . . . . . . 10 1.13 An image from the Los Alamos 2-dimensional vortex lattice catalogue of [14]. N = 18 vortices for all the configurations above, energies and vortex strengths are labeled below with the first (1 6 11) having the lowest energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.14 Side view schematic of vortex wake formation. The critical height, hc, above which air passes over island, and below which the flow separates around the island and forms the vortex street. Image from [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.15 Juan Fernandez islands off the coast of Chile. Island’s geophysical wake forms a vortex street. . 13 1.16 V ortex street on Jupiter. Points labeled C, E and G are cyclones while points D and F are anti- cyclones. These are part of a set of twelve cyclones coupled with twelve staggered anti-cyclones forming a loop about Jupiter. Image from [49]. . . . . . . . . . . . . . . . . . . . . . . 14 1.17 Symmetric and Staggered configurations for a vortex street with two vortices per periodic strip. Definition of vortex spacing for a staggered von K´ arm´ an street is present in (b). . . . . . . . . 15 1.18 An exotic wake with one singlet (S) and one pair (P ) per periodic strip, denoted ‘S +P ’, from [73]. 17 1.19 Three different streamline topology types for Jovian vortex streets (JVS) on the sphere identified in [37], labeled as (a) Type I, (b) Type II, and (c) Type III. In Type III, the sets of separatrices collapse, which entails that the region of flow that travels across the street disappears. Different types correspond to different parameter values. . . . . . . . . . . . . . . . . . . . . . . 18 1.20 Streamline topology for 2 vortices per periodic strip and velocityV = 0 (see Equation (1.1)). The images are for the strength ratios (a) 1:1 (b) 1:3 (c) (-1):3, from [69]. . . . . . . . . . . . . 19 1.21 Propagation angle vs. horizontal spacing (see Equation (1.6)) for (a) y = L=10; 000, (b) y =L=10, (c) y =L sinh 1 (1=) and (d) y =L. From [69]. . . . . . . . . . . . . 20 1.22 Heteroclinic orbits are presented as dashed lines, vortex streets with two opposite-strength vortices per strip. Upper subplot hasn = 1, lower hasn = 2, wheren denotes how many strips are passed with each heteroclinic connection. From [69]. . . . . . . . . . . . . . . . . . . . . . . 21 1.23 Streamline topology for two vortices per periodic strip, and with 1 = 2 = . Configurations (a) - (e) are linearly stable. y is determined for (a) x =L=2, (b) x = 6L=13, (c) x = 4L=9, (d) x = 3L=8, (e) x =L=4 and (f) x = 0 with y = sinh 1 (1=), from [69]. . . . . . 21 ix 1.24 Three vortex streets per period with V = 0 and G > 0. The vortex strengths of the above configurations are (a) 3:(-1):3, (b) (21/10):(-1):(21/10), (c) 2:(-1):3 and (d) 3:1:3. Image from [69]. 22 1.25 Left: V ortex positions and streamlines for three vortices withV = 0 andG < 0. Images above correspond to (from top to bottom) (a) j 2 >jGj = 2, (b) 1 2 = 1 <jGj = 7, (c) 1 2 = 1 <jGj = 6. Right: Three vortices per strip withS = 0, configurations have the strength ratio 1:2:(-3). From [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.26 Three generic vortices,, in the plane. The distance between the vortices and the area formed between them are labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.27 Image of three generic vortices on the sphere that show the definition of vortex strengths (’s), intervortical lengths (l12;l13;l23), and volumeV123. . . . . . . . . . . . . . . . . . . . 36 3.28 Four vortices on the plane in a square configuration, and without loss of generality, with side of unit lengthS = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.29 Solution to the 5-vortex square lattice configuration. The four outer edge vortices share one com- mon strength, while the central vortex has an independent strength ([56]). . . . . . . . . . . 46 3.30 Shannon Entropy vs. the probability of getting heads from a dice throw. Maximum entropy occurs when the probability = 1=2, which coincides with the proposition that the probability with the highest Shannon Entropy is also the most likely to occur. . . . . . . . . . . . . . . . . 49 4.31 Triangular lattice,N = 6. SVD solution produced for configuration was = (-1/4, -1/4, -1/4, 1, 1, 1) T . System resembles a plasma crystal configuration found in [30]. In (a), center of vorticity is labeled with an symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.32 V ortex lattices with N = 7 vortices. (a) Semi-circle model has null space of one, with = (1; 0; 0; 0; 0; 0) T . This implies that central vortex has arbitrary strength, while remaining 6 vortices on the circle of radius r are stagnation points orbiting the central vortex with veloc- ity u = =2r. (b) Hexagonal model has non-trivial null space of dimension 2 with = 1(1,0,0,0,0,0) T +2(0,1,1,1,1,1) T with 1 and 2 being unrelated arbitrary constants. This implies that central vortex has one independent strength, while the outer vortices have second independent strengths. (c) Comparison is made to milli-magnetized structures (from [34]) with different-sized magnets. Two configurations arise for different ranges of angular velocity (as denoted in figure), and we compare them to models (a) and (b) which support different-strength vortices with the selection16= 26= 0. (d) Uniform-sized milli-magnetized structures. Com- parison is made to (b), since the solution to (b) can support uniform-strength vortices with the selection1 =2. Center of vorticity coincides with location of central vortex. . . . . . . . 59 4.33 Triangular vortex lattice, N = 19 with solution. Lattice consists of two concentric hexagonal rings. Solution to system can be found in Equation (4.101). . . . . . . . . . . . . . . . . 60 4.34 Left: minimum eigenvalues of 10 6 random configurations, N = 10 (None are in equilibrium). Right: square root of sum of the squares of two smallest eigenvalues of 10 6 random configurations, N = 10 (None have a null-space of two). . . . . . . . . . . . . . . . . . . . . . . . . 63 x 4.35 Broken symmetries in triangular lattices of Bose Einstein Condensates. Left: Line defect. Right: Grain boundary defect (from [1]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.36 Schematic of the Brownian ratchet method. The vortices take a random walk, each within a circle scaled with the smallest singular value, (0) associated with its configuration matrix until the smallest singular value gets smaller , (1) . The vortices continue to take random walks in circles of ever decreasing radius until the smallest singular value drops below a prescribed threshold. . 65 4.37 (a): Smallest singular value squared vs. Brownian ratchet step number on lin-log scale forN = 6; 8; 10. The smallest singular value converges continuously to zero. (b): Close-up view of a vortex converging to an equilibrium point, scale is magnified10 6 the scale of configuration. . 65 4.38 Examples of converged asymmetric equilibrium configurations with one-dimensional null spaces using Brownian ratchet method. Total of six plots for 6, 8, 10 vortices, with one initial configu- ration (unfilled circles) and two corresponding final converged states (filled circles) with smallest singular value on the order of 10 6 . (a)N = 6, = (0.0040, -0.0033, -0.0254, -0.0079, -0.9996, -0.0001) T ; (b)N = 6, = (-0.2394, -0.1049, 0.1078, 0.0314, 0.7461, -0.6020) T ; (c)N = 8, = (0.0064, -0.0221, -0.0743, 0.0456, 0.1197, -0.1380, 0.0751, -0.9761) T ; (d) N = 8, = (0.0040, -0.0033, -0.0254, -0.0079, -0.9996, -0.0001) T ; (e) N = 10, = (-0.0364, -0.0666, - 0.0253, -0.0335, -0.1392, -0.5016, -0.0716, 0.0163, -0.8462, 0) T ; and (f)N = 10, = (0.1561, 0.0048,0.0970, -0.0078, 0.1108, 0.2844, -0.0047, -0.0410, 0.9334, 0.0004) T . . . . . . . . . . 66 4.39 Reverse Brownian ratchet scheme used to derive initial random configuration not in equilibrium from symmetric equilibrium configurations. (a)N = 6 triangular configuration with null-space 1 in open circles, with random non-equilibrium configuration that converges to it. (b)N = 7 hexag- onal equilibrium configuration with null space 2 in open circles, with random non-equilibrium configuration that converges to it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.40 Averaged 250 times + standard deviation. Comparison of Shannon Entropy associated with: ran- dom asymmetric configurations (not in equilibrium), random matrices (all entries are random numbers), converged asymmetric equilibria with null space 1, and the triangle lattice from Figure 4.31. Triangle lattice has significantly higher entropy than asymmetric equilbria, with entropy of asymmetric configurations not in equilibrium and in equilibrium being indistinguishable from one another within standard deviation. Random matrices have substantially lower entropy than any configuration matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 xi 4.41 Configuration Construction ofN points in the plane in the absence of a coordinate system. (a) N 1 relative distances from point 1 can all be chosen independently from R + , (b) N 2 relative distances from point 2 to = f3;:::;Ng can be chosen from the constricted range jl12l 1 j < l 2 <jl12 +l 1 j, (c)N 2 Relative orientations between points 1, 2 and = f3;:::;Ng can be chosen to be 12 = 1. This subfigure highlights the fact that the two possible choices for 12 result in two geometrically unique configurations (i.e., one configuration cannot be obtained from the other via rigid body rotations and translations), with vortex either having the relative position (1) or (2) , (d) The information corresponding to the remaining points, labeled =f + 1;:::;Ng, can be solved analytically by identifying their locations (in Regions I - VII) relative to three preceding fully-solved pointsf1;;g, such that<, < andN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.42 Schematic of a single von K´ arm´ an street on the sphere with an illustration of the vorticity vector J and the angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.43 Top and oblique view. Solution using SVD of single von K´ arm´ an street on the sphere forn = 3. Dimension of null space is three with same-shape vortices having equal and opposite strength. . 79 5.44 Computational result of time evolution of 6 vortices on the sphere with = 5=12. The ini- tial configuration is the same as that defined in the previous figure. Images of the first column represent degenerate case with single VKS VKS and vortex strengths [1; 2; 3] = [1; 1; 1]. Second column has strengths [1; 2; 3] = [1; 1:5; 1] with no resultant VKS, third case is [1; 2; 3] = [1; 1:5; 1:2], also with no resultant VKS. . . . . . . . . . . . . . . . . . . 81 5.45 (a) Angular velocity! as a function of vortex street latitude forn = 2 6. ! decreases from +1 to 0 as is increased from 0 to=2. (b) Additional angular velocity for anyn from pole vortices (p) with p = +1. As approaches=2, asymptotes at p=(2). . . . . . . . 85 5.46 Streamlines topologies for a single von K´ arm´ an street with 3 vortices per ring and pole vortices of equal and opposite strength. The pole strength increases from5 to +10. A total of three distinct topology types are identified within four different intervals, separated by three bifurcation topologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.47 Shannon Entropy sweeps for single vortex street with (solid line) and without (dashed line) pole vortices. (a) Two vortices per ring, (b) Three vortices per ring. . . . . . . . . . . . . . . . 91 5.48 Normalized Shannon Entropy sweeps for single von K´ arm´ an streets on the sphere for (a) n = 4; 5; 7; 13; 20 without poles, and (b)n = 4; 5; 7; 13; 20 with poles. The intention of these figures to compare the entropy of configurations with differentn. . . . . . . . . . . . . . . . . . 94 6.49 Double vortex street on Jupiter. Cyclones (anti-cyclones) are labeled ‘C’ (‘A’). Velocity profiles on the left demonstrate identification of cyclones & anti-cyclones, where the first have a positive slope and the latter have a negative slope in the y vs. v domain. Image from [37]. . . . . . . . 95 xii 6.50 Double V on K´ arm´ an Street schematic, illustrating SVD solution and ring definition. Dimension of the null space is 1. The angle associated with the two rings are1 and2 respectively. Note that which latitude is smaller is of no importance. Rather, we identify the different rings by the ordering in the state representation of Equation (6.139). . . . . . . . . . . . . . . . . . . 96 6.51 Double von K´ arm´ an Street with poles schematic, illustrating SVD solution. Dimension of the null space is 3, with the pole vortices having independent strengths of one another. Since the poles vortices affect the street vortices’ strength, their strengths are chosen to be 2; 2 = 2(1;2;n) in order to ensure 1 1 in Figure 6.50. . . . . . . . . . . . . . . . . . . 99 6.52 Equilibrium curves present for pole strength ( 2) vs. angle ratio (1=2) when n = [4; 5; 6]. Angle1 is held fixed at 3=8 in (a), and 5=12 in (b), while2 is the control parameter. Recall that 2 = 2(1;2;n). Recall that the pole strength can be any value for the degenerate case 1 =2 represented by the solid line in the graphs (single VKS). . . . . . . . . . . . . . 100 6.53 Angular velocity! vs. angle2 forn = 2 5. 1 held fixed at 3=8 in (a) and (b), and at 5=12 in (c) and (d). Singularity occurs since p!1 as2!1. . . . . . . . . . . . 105 6.54 Time evolution of point vortices of double street with poles given by solid black lines. Pole vortex strengths do not satisfy equilibrium condition given by Equation (6.143). Specifically, pole vortices have strength 0 instead of 2. Figures (a) and (b) show evolution of all vortices. Figures (c) and (d) show evolution of one isolated set of vortices marked by filled circles (one from each hemisphere), and demonstrate that the dynamics is periodic. Figures (e) and (f) show the evolution of a second set of vortices, also showing periodic dynamics. . . . . . . . . . . 107 6.55 Quasi-periodic time evolution of point vortices of double street with poles given by solid black lines. Pole vortex strengths do not satisfy equilibrium condition given by Equation (6.143). Specifically, pole vortices have strength2 2 instead of 2. . . . . . . . . . . . . . 108 6.56 Streamline topology sequence for the range of ratios 0:68 < R < 0:95 in ascending order. Stagnation points move either along constant longitude lines or the equator as the ratio varies. Four different topology types are illustrated (labeled I, II, III, IV), and are separated by five bifurcation scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.57 Equilibrium curve corresponding to the relation of the pole vortex strength factor 2 vs. angle ratioR forn = 5, = 13=40, and with2 as the control parameter. This plot also corresponds to the configurations found in Figures 6.56 and 6.58, and illustrates (with vertical lines) that some of the bifurcation points correspond to either an inflection in the curve or when 2 = 0. . . . . 110 6.58 Streamline topology sequence for the range of ratios 1:05<R< 2:25 in ascending order. Main- taining notation consistency with Figure 6.56, four different topology types are evident, labeled Types II, III, IV , V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.59 Shannon Entropy vs2 for1 = 3=8 andn = 3; 4; 8; 10. The figures llustrate the effect on entropy upon adding pole vortices to the configuration. . . . . . . . . . . . . . . . . . . 113 xiii A.60 Square lattices. Solutions are (a) null space of 1 with = (2, 1, 2, 2, 1, 2, 1, 1) T , (b) null space of two =(0, 1, 1, 0.2, 0 , 1, 0, 1, 0) T +(1, 0, 0, -0.1, 1, 0, 1, 0, 1) T . (c) has null space of one with = (1, 76/327, 1, -76/327, 1, - 76/327, -37/218, -69/295, -37/218, -76/327, 1, -69/295, 1, -69/295, 1, -76/327, 37/218, -69/295, -37/218, -76/327, 1, -76/327, 1, -76/327, 1) T . Fractions represent numerical approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.61 Triangular lattices: (a) null space of 1, solution is = (-11/9, - 3/2, -3/2, -3/2, -3/2, -3/2, -3/2, -34/9, -34/9, 1, 1, -34/9, -34/9, 1, 1, 1, 1, -34/9, -34/9, 1, 1, -14/9, -14/9, -14/9, -14/9, 1, 1, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, 1, 1) T , (b) null space of two, solution is =(-6/7, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0) T +(13/14, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1) T , (c) null space of 1, solution is = (-6/7, 0, 1, 1, 0, 1, 1, 0, 1, 1) T . Fractions represent numerical approximations. . . 127 A.62 Snowflake lattices. Characterized by concentric, rotational symmetry. The lattices above have a null space of one, with vortices that possess the same distance from the center having the same strength. (a) (0.1517, 0.1517, -0.1572, 0.1517, 0.1517, -0.1572, -0.0787) T , (b) (0.1134, 0.1134, -0.0741, 0.1134, 0.1134, -0.0741, 0.1134, 0.1134, -0.0741, -0.0973) T , (c) (0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, -0.0657) T , (d) (0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0640) T . . . . . . . . . . . . . 128 B.63 Combinations found in the configuration matrix of a single V on K´ arm´ an Street on the Sphere with n = 3 vortices per ring. Image shows top view of one ring (either northern or southern). There are six equations in the configuration matrix with 0 non-trivial terms per ring, and another nine with 1 vortex from each ring a with non-trivial coefficient . . . . . . . . . . . . . . . . . 130 B.64 Contrasting allowable and unallowable configurations forN = [7; 8]. Allowable configuration all have even symmetry with respect to a planar axis, while unallowable ones visibly do not have any symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.65 Combination types present in the configuration matrix forn = 9 vortices per ring. Number of filled vortices per ring are 7, 6 and 0. Each non-trivial combination type is present 18 times, in addition to 9 trivial ones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 B.66 Combination types present for N = 10. Number of filled vortices per ring are 10; 9; 8 and 6, repeated a total of 20; 100; 50 and 20 times respectively. . . . . . . . . . . . . . . . . . . 133 xiv Abstract V ortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics- based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we requirenullity(A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A ‘Brownian ratchet’ construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value ofA. Distances between configurations and equilibria were measured using the Frobenius normkk F and 2-normkk 2 , and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves. xv Chapter 1. Introduction 1.1 Purpose A vortex lattice is a set of point vortices that are in a state of relative equilibrium. The vortices remain equidistant with time, but the system as a whole can undergo rigid rotations and trans- lations. V ortex lattices have been observed in a variety of physical settings, each characterized by a different mathematical model. These include superconductors ([2], [3], [27]), superfluids ([74]), plasma gases ([23]), milli-sized magnets ([34]), in the wakes of islands ([28]), in geo- physical flows ([37], [33], [75], [48], [49], [50], [76]), and in the wakes of other bluff bodies. V ortex lattices observed in nature include highly symmetric ones (in particular: triangular and hexagonal structures ([74], [1])), structures with broken symmetries ([1]), structures that have elastic or periodic inter-vortical dynamics ([17], [25]), and structures that are completely irreg- ular and asymmetric ([26]). Lattices exist throughout a wide range of scales in the physical world including: nano scale (superfluids and superconductors), bacteria scale ([20]), laboratory macro-scale ([18]), planetary scale ([8], [16]), and on the interplanetary scale ([12], [68]). On the sphere, lattices can take the form of periodic vortex streets such as on the surface of Jupiter ([37]). A general coherent theory that describes lattices in all these shapes, sizes and forms does not yet exist, nor does a satisfactory model that describes the formation of irregular patterns. In this thesis, we describe an approach which reduces all of these systems to a classical problem in linear algebra for a non-normal matrixthat of characterizing the null space of the matrix as well as the distribution of singular values. This method opens the door to a series of possible investigations, specifically understanding and controlling properties of the lattices including: the 1 relative vortex strengths, the flow-field’s streamline topology, angular & translational velocities, and therobustness associated with vortex equilbria. 1.2 Organization In Chapter 1, we present the motivation for this thesis. Previous research on vortex lattices will be introducedwith an emphasis on the types of equilibrium geometries found in nature. Numeric and analytic models that have been developed to date will also be mentioned. The main mathematical techniques used in this thesis are split between Chapters 2 and 3. In Chapter 2, a general overview of fluid mechanics is presented that leads to the Hamiltonian point vortex model. The derivations for relative point vortex equilibria on the plane and sphere are presented in this chapter as well. In Chapter 3, the main tools from linear algebra that are imple- mented in the analysis are introduced. The first tool is Singular Value Decomposition (SVD), which provides rich geometric information for non-normal matrices, such as the null space and singular values. The next tool is the Shannon Entropy, which provides a scalar measure of a configuration’srobustness. The third tool is matrix approximation, which is used to measure the distance to equilibrium. Chapters 4, 5 and 6 contain the results of this thesis, with Chapter 4 addressing configura- tions on the plane, while Chapters 5 and 6 deal with those on the sphere. The contributions of this thesis can be classified into four main points. First, we present a model that describes the properties and formation of asymmetric vortex equilibria (i.e., lacking any discrete symmetries. See Chapter 4). Second, we implement ‘Shannon Entropy’, which we describe in Section 3.2, and interpret as a scalar measure of disorder, or likelihood that a lattice configuration will exist in a physical setting (see Chapters 4, 5 and 6). Third, we provide a means of deriving a fully analytic expression for the solution of vortex streets on the sphere (in Chapters 5 and 6), and highlight the importance of pole vortices in the formation and destruction of the streets. Fourth, we study the streamline topology of vortex streets on the sphere, and propose an analytic way of classifying bifurcations in the topology (in Chapters 5 and 6). 2 Figure 1.1: Left: the first Abrikosov triangular lattice image by [27], on the order of 1 micron. Black dots repre- sent cylindrical magnetic flux tubes. Right: Image from [36] of anNbSe2 triangular lattice made using scanning- tunneling microscope imaging, showing high degree of symmetry, on the order of several thousand angstroms. Figure 1.2: Image from [74] of superfluid Helium lattices. Images contain 1-11 equal-circulation vortices, with the vortex number depending on system parameters (rotational speed of capsule, etc.). In the next section, we will introduce several examples of vortex lattices and highlight their relevance to our work. 3 Figure 1.3: Image from Abrikosov’s Nobel Prize Lecture ([3]) of velocity vs. distance from vortex center. The solid line represents the velocity flow around an isolated lattice site. The plot’s features include a potential vortex core, and constant decay proportional to the distance from the site. The curve-fitted dashed line 1=r demonstrates that the point vortex model makes a good fit for sufficiently spaced vortices. 1.3 Vortex Lattices 1.3.1 Symmetric Lattices: Superconductors and Superfluids We will now outline several lattices that possess a high degree a symmetry. Triangular lattice structures were first predicted to exist by Abrikosov in 1957 (see [2]) during his research on a class of superconductors that are now referred to as type II superconductors. A study of super- conductivity, previously published by Ginzburg & Landau ([32]), used a wave-based theory to study phase transitions. An order parameter, now referred to as the Ginzburg-Landau parame- ter, was used to quantify the phase transitions in which the energy of the fluid became negative. When the parameter reached a critical value, Abrikosov then predicted the formation of quan- tized periodic arrays consisting of individual cylindrical magnetic flux tubes, and argued that the geometric structure which minimized the free energy should be triangular. The existence of these structures was not taken seriously until 10 years later, when the first image of a supercon- ductor triangular lattice was observed and published by Essmann & Trauble ([27]), see Figure 1.1. The second image in Figure 1.1 from Hessetal: ([36]) was made using scanning-tunneling microscope imaging on a much smaller length scale. Feymann then published a study in 1955 ([29]) on superfluid helium vortex tubes with quan- tized circulations =~=m 0:001cm 2 =s, where~ is Plank’s constant andm is the mass of a 4 Figure 1.4: Images from [25] showing that lattices possess a nearly two-dimensional structure. (a) is a top view, (b) is cross-sectional intensity reading and (c) is a sideview of the lattice. 4 He atom. The first good set of images of superfluid Helium lattices obtained by Yarmchucket al: ([74]) is shown in Figure 1.2. The model used in this thesis is one based on a collection of interacting point vortices. The vorticity at individual lattice sites is localized at a point inR 2 , and is purely radial (with no azimuthal component). The ‘core size’ of a vortex is measured by its visible diameter d (the spots), divided by the intervortical spacing,l. This ratiod=l must be sufficiently small ( 1). In the lattice structures mentioned, the ratio is on the order of 10 2 10 4 . Abrikosov’s lecture, [3], displayed two important features. First, the vortices have a core region in which the vortex has a flattened maximal strength. Second, the field exhibits a monotonic decay (inversely proportional to the distance from the vortex site), and the velocity tends to zero with increasing distance (see Figure 1.3). Since the vortices are sufficiently spaced, the behavior immediately inside the core is treated as unimportant. Therefore, the flow in an Abrikosov vortex lattice can be approximated quite well with the fluid mechanics model of point vortices presented in Chapter 2. 1.3.2 Asymmetric Lattices: Bose Einstein Condensates We now shift our focus to Bose-Einstein condensates, which in recent years have been the subject of intensive focus. The image in Figure 1.4(c) (from [1]) shows that BEC (Bose-Einstein con- densates) possess a nearly two-dimensional structure based on the fact that the vortex tubes are 5 Figure 1.5: Bose Einstein condensate image from [1]. Lattices include 16, 32, 80, 130 vortices, with the vortices being the dark spots. Figure 1.6: Image from [26] of (a) an initially regular triangular lattice, (b) getting blasted by a laser, (c) resulting in a perturbed configuration that undergoes cooling, and the vortices exhibit random motion before (d) the vortices settle into a new irregular equilibrium. nearly parallel. Figures 1.4 and 1.5 are excellent examples of the kind of imaging now available to study those lattices. The main feature of interest to us in these lattices is that some underlying structures exhibit broken symmetries, or entirely irregular patterns. These findings have exposed the current lack of knowledge on the formation of such configurations or of their robustness, dynamics and stability. Previous analytic and computational work that describes vortex lattice structures focused almost solely on highly symmetric geometries such as those of section 1.3.1. An interesting study in which a regular triangle lattice was blasted by a laser demonstrates that completely irregular equilibrium configurations can resultsee the work done by Engelset 6 Figure 1.7: Another image from [26] of (a) an asymmetric equilibrium lattice configuration. (b) Cross-section image of vortex intensity; lattice was initially symmetric before being blasted by laser, and settling into a final asymmetric configuration after cooling and random motion had settled. (c) Cross-sectional intensity reading showing the irregular distribution after the ‘random’ motion has settled. Figure 1.8: Rapidly rotating BEC lattice perturbed by a laser-induced perturbation at the center of the lattice supports small sinusoidal waves referred to ‘Tykachencooscillations’ ([17]). al: ([26]). From Figure 1.6, the vortex lattice begins as a highly symmetric triangular configura- tion (in subfigure 1.6(a)). The lattice is then blasted by a laser (in 1.6(b)), and appears to follow a random motion after the blast, (in 1.6(c)). As the cooling process takes place, the vortices seem to thermally fluctuate towards a final irregular pattern (see Figures 1.6(d) and 1.7). In Chapter 4, we introduce a Brownian ratchet method ([55]) which uses a random walk algorithm to mimic the process described in Figure 1.6 in order to find asymmetric equilibria. Two interesting studies have demonstrated shearing effects on lattices. In [25], Engelsetal. showed that adding external forces to a lattice can change its shape and size (see Figure 1.9). In a related work by Coddington et al. ([17]), it was shown that a controlled laser blast to 7 Figure 1.9: Images from [25] showing that external forces on a lattice can change its shape and size. the center of a rapidly rotating BEC lattice resulted in the removal of an atom from the center of the lattice, which can be interpreted as a change in the circulation of the central vortex. This perturbation triggers the propagation of small amplitude sinusoidal waves referred to as ‘Tykachenko oscillations’ (see Figure 1.8). Tykachenko oscillations appear to have a shearing effect on the angular velocity of the lattice. This phenomenon is explored in Chapter 6. In summary, Bose Einstein condensates produce the need for a mathematical model that describes asymmetric lattices in addition to symmetric ones, and one that can analyze shearing effects. 1.3.3 Lattices in Other Physical Systems Thus far, vortex lattices have solely been referenced in the context of superfluids and supercon- ductors. In this section, we highlight lattices that form in a variety of other physical settings. Grzybowskietal: ([34]) demonstrated lattices resulting from millimeter-sized magnetic struc- tures on an air-liquid interface. As shown in Figure 1.10, floating magnetic disks were rotated by a spinning magnet underneath the tank of water, and the disks interacted as particles with fluid (1=r velocity field) and magnetic interactions. Tests were done for a different number of magnets, each of which yielded a symmetric equilibrium configuration. Some of the systems have two resultant equilibrium configurations for a given number of magnets between which 8 Figure 1.10: Milli-sized magnetic structures floating on a rotating fluid. Magnets form lattice-like structures. Images above are for different numbers of magnets, from [34]. (a) Fluid Dye demonstrates rotation about magnets (b) Variable-sized magnets and surfaces Figure 1.11: Left, (a): magnets far from center of rotation possess asymmetric structure; and dye injected into fluid shows that magnets are themselves rotating, and form a prototype of point vortex equilibria. Right, (b): Subplots a, b: Lattices formed by variable-sized lattices. Interchangeable arrows indicate that system switches between the two configurations. Subplot c: Lattices formed on the curved surface of a drop. These configuration types never formed on the flat water-air interface. Images from [34]. 9 Figure 1.12: Trapped plasma in a cylindrical structure with a spiral electron source. V ortex lattice tubes form with different vortex numbers and configurations. Image from [30]. Figure 1.13: An image from the Los Alamos 2-dimensional vortex lattice catalogue of [14]. N = 18 vortices for all the configurations above, energies and vortex strengths are labeled below with the first (1 6 11) having the lowest energy. the system sporadically switched. Tests were also done with different-sized magnets, which can be interpreted as point vortices with different relative strengths. Figure 1.11(a) shows that the magnets themselves rotated since the dye injected into the fluid was advected around the mag- nets in a manner consistent with the streamline topology of point vortices ([53]). Figure 1.11(b) demonstrated two things. First, equilibria from different-sized magnets were investigated, and in the context of vortices, the size of the magnets can be interpreted as the strength of their circula- tion. Second, equilibria on the non-flat surface of a drop demonstrated that different equilibrium geometries can result on different-shaped surfaces. In a study from a different physical setting, vortex lattice formation in a confined cylindrical plasma tube (see Figure 1.12) was published by 10 Fine et al. in [30]. Again, depending on system parameters, different lattice patterns emerge. In the case of a central vortex, it occasionally had a larger diameter than the outer vortices. Campbell & Ziff ([14]) compiled the Los Alamos catalogue of equilibria for systems of equal strength vortices made up of circular, concentric vortex patterns along with their proper- ties, particularly the Hamiltonian energy. They showed that for the same number of vortices, there exist multiple equilibria, each of which generally has a different energy. The symmetric lattices in [14] can be found with classic gradient methods since the energy landscape associated with such structured configurations is generally not complex. But for large numbers of vortices (N > 100) or for asymmetric equilibria, classical gradients techniques become inconvenient since their associated energy landscapes are far more complex. Thus, the works cited in this section illustrate that vortex lattices exist in a variety of physical settings. Note that in [14], it was illustrated that more than one geometry is possible for the same number of vortices. Fur- thermore, certain geometry types appear to be more prevalent than others on a given surface. In the next section of this chapter, we move our discussion to vortex streets, which are an important kind of vortex lattice that are prevalent in a variety of physical settings. 11 Figure 1.14: Side view schematic of vortex wake formation. The critical height,hc, above which air passes over island, and below which the flow separates around the island and forms the vortex street. Image from [28]. 1.4 Von K´ arm´ an Vortex Streets The motivation for studying vortex streets is because of their presence in the physical world, their role in geophysical climates, and their possible usefulness in engineering applications. On the plane, idealized vortex streets are thought of as two infinite lines of evenly spaced vortices, where one line is shifted by a half-wavelength (skewed). The vortices in each line are equal in strength and opposite to those in the other line. Thus, the full system ofN = 2n vortices has vanishing circulation. Generally, they exist in nature as a dissipative set of vortices that remain in approximate relative equilibrium for a finite period of time. On the sphere, a street is formed by a closed set of two parallel skewed ringswith each ring having an equal number of vortices (see Figure 1.16 from [49]). The following section summarizes the most common physical settings in which vortex streets exist. 1.4.1 Where They Exist One place where vortex streets form is in the wake of bluff bodies due to the viscous boundary layera process that is well understood. In Figure 1.14 (from [28]), the island is modeled as a conic-shaped body. Above a critical altitude, the flow simply rises above the island and passes over it without separation or the formation of eddies. Below this altitude, the flow separates 12 Figure 1.15: Juan Fernandez islands off the coast of Chile. Island’s geophysical wake forms a vortex street. and goes around the island. Behind the island, the separated flow does not reconnect, and it is in this region that the vortex street is formed. See Figure 1.15 for a satellite image of a vortex street in the wake of an island. The wake as a function of the Reynolds number is a well- studied problemfor a thorough overview see [72]. V ortex shedding only commences when the Reynolds number Re > 45, and there are three different shedding regimes, characterized by different shedding frequencies, see [65] for a thorough empirical analysis of this problem. For a rationale of the relation between the Reynolds number and shedding frequency, see [62]. V ortex streets also affect aquatic life, and a recent study by Liao et al. ([47]) showed its potential usefulness by fish swimming in a stream. The study showed that the trout swimming in the wake of the cylinder (and interacting with a vortex street) modified their body motion, and that they used less locomotive energy than did the fish swimming in the free-stream. Therefore, fish can benefit from the presence of vortices in their ambient fluid. In the context of point vortex lattices, the methods described in Chapters 2 and 3 are applicable to point vortex equilibria in the absence of solid bodies. If one wishes to solve for lattices that contain solid bodies (such as a fish), it would be possible to expand the techniques in those chapters by using the method of images. In this thesis, we will be addressing vortex streets that appear in geophysical flows in the complete absence of bluff bodies, as is the case with Jupiter (see Figure 1.16). According to 13 Figure 1.16: V ortex street on Jupiter. Points labeled C, E and G are cyclones while points D and F are anti-cyclones. These are part of a set of twelve cyclones coupled with twelve staggered anti-cyclones forming a loop about Jupiter. Image from [49]. Humphreys & Marcus ([37]), a very clear vortex street that wraps around Jupiter was first formed around the year 1939, remained steady until around 1998, at which point it began to disintegrate. In the Jovian atmosphere, the full atmospheric layer consists of 12 major zonal jet streams and an additional 80 easily visible vortices, with the largest of these being the well-known Great Red Spot [48]. The physical mechanisms and processes which set the scale for the spacing and core size of vortex streets are less understood. The topic of ‘pattern formation’ is addressed in [37]. In Chapters 5 and 6, we propose that Shannon Entropy (defined in Section 3.2) can be used as a means to predict what street parameters are most likely (i.e, number of vortices per street, angle between the streets’ rings). 14 (a) Symmetric (b) Staggered Figure 1.17: Symmetric and Staggered configurations for a vortex street with two vortices per periodic strip. Definition of vortex spacing for a staggered von K´ arm´ an street is present in (b). 1.4.2 History of the Vortex Street Model The first known result that described a vortex street appeared in 1928 ([31]). The authors’ result was that the constant velocity ofN repeating vortices is given by _ z =UiV = 1 2Li N X =1 cot( L (z z )); (1.1) whereU,V are the Cartesian velocity components of the system, is the strength of the th vortex,L is the width of the periodic strip,N is the number of vortices in the periodic strip, and z =x +iy is the complex coordinate of vortex (see [45] for a comprehensive overview of von K´ arm´ an’s work). V on K´ arm´ an used this idea for a strip withN = 2 and with the strengths of the two vortices being 1 = 2 = . The street then propagates with velocity UiV = 2Li cot( L (z + z )); (1.2) wherez + andz are the complex locations of any two vortices in the same period. For a street that moves along the real axis,V = 0, which implies that the cotangent portion of Equation (1.2) 15 must be purely imaginary. For this condition, the requirement is thatR(z + z ) = 0 orL=2, which corresponds to symmetric and staggered configurations respectively (see Figure 1.17). V on K´ arm´ an then performed a linear stability analysis on the streets. He did so by per- turbing the position of one vortex. The result was that the symmetric configuration was always linearly unstable, and that the staggered configuration was linearly stable for a specific ratio of the spacing for the two vortices in each strip, namely that sinh a h = 1; (1.3) wherea andh are defined in Figure 1.17(b). Domm went on to show that even streets satisfy- ing this special aspect ratio failed the stability test in second order perturbation theory ([21]). Although this thesis will not be addressing the issue of stability of either vortex streets or of vortex lattices, we will direct the reader in the next paragraph to several references that do. Both linear and non-linear studies have been made for evenly distributed vortex cores in rectilinear planar settings (see [66]). This problem is reasonably well understood and used as a basis to predict the persistence and lifetimes of vortex streets found in physical systems. The stability of more general point vortex configurations on the sphere is not a very well studied or understood problem, but there is presently a small body of work available on the subject. One study can be found in Laurent-Polzetal: ([46]). Pekarsky & Marsden ([61]) studied the nonlinear stability of three point vortices on the sphere by considering the momentum map (i.e., center of vorticity vector in Equation (2.53)), and referred to their technique as the energy- momentum method. Studies on the nonlinear stability ofN evenly distributed point vortices on the same latitude can be found in the works of [13], [10] and [44]. 1.4.3 Bifurcation Problems and Streamline Topology The observed vortex street patterns on Jupiter have had long, but finite lifetimes, and their pres- ence affects the overall mixing regimes on the planet and its climate. Therefore, it is funda- mentally important to understand how fluid transport changes with the vortex configuration, and 16 Figure 1.18: An exotic wake with one singlet (S) and one pair (P ) per periodic strip, denoted ‘S +P ’, from [73]. the associated bifurcations. The techniques presented in Chapters 2 and 3 make it possible to thoroughly and analytically investigate bifurcations in the vortex strengths and streamline topol- ogy when smoothly varying a parameter linked with the geometry of the lattice. This analysis is fleshed out in Chapters 5 and 6 in the context of vortex streets on the sphere. In this section, we will reference some of the work that has analyzed bifurcations in the street geometry and streamline topology. The references will include both streets formed in the wake of a bluff body, and streets in the absence of a bluff body. Although our work deals exclusively with vortex equilibria formed in the absence of solid bodies, we reference bluff body wakes in order to fully understand the mechanisms creating vortex streets. Furthermore, we focus on the geometry of the street itself, and ignore the presence of the bluff body by confining our analysis to the vortex street pattern that lies sufficiently far from the bluff body. Computational models of different shedding regimes in the wake behind a stationary cylinder can be found in the works of [38], [64], [77], [24], [58], [59], and most recently by [11]. V ortex shedding also takes place behind non-stationary bluff bodies. When the cylinder in the flow is allowed to oscillate, an “exotic” wake results, which consists of both single shed vortices ‘S’also referred to as singlets, and vortex pairs ‘P ’. For instance, in Figure 1.18, the wake is labeled ‘S +P ’, for one singlet and one doublet. The wake is affected by two parameters: the amplitude (normal to the free stream) and wavelength of oscillation, with the wavelength compensating for both the frequency of oscillation and the free stream velocity. For an empirical 17 Figure 1.19: Three different streamline topology types for Jovian vortex streets (JVS) on the sphere identified in [37], labeled as (a) Type I, (b) Type II, and (c) Type III. In Type III, the sets of separatrices collapse, which entails that the region of flow that travels across the street disappears. Different types correspond to different parameter values. study of all the different possible exotic wakes, see [73]. In [63], a rationale for how the exotic wake changes with the parameters of oscillation was formulated. In Chapters 5 and 6, two sets of analytic equilibrium curves for vortex streets on the sphere that undergo bifurcations are derived. The first set (Figures 6.52, 6.57) illustrates the relative vortex strengths of the streets’ vortices as a function of the vortex street’s geometry (i.e., angle between the rings, or number of vortices per ring). The second set (Figures 5.45, 6.53) represents the rigid rotational velocity as a function of its geometry. The analogy is made to the bifurcations encountered in the aforementioned experiments of this section, in which shedding regimes (or street geometries) encounter bifurcations and change with the system parametersjust as the exotic wake changes with the parameters of oscillation (Figure 1.18). Next, we consider bifurcations in the streamline topology of vortex streets. In Figure 1.19 from [37], three different topology types for a vortex street are identified and labeled, each corresponding to different parameters. In our work (see Figures 5.46, 6.56 and 6.58), we identify 18 Figure 1.20: Streamline topology for 2 vortices per periodic strip and velocityV = 0 (see Equation (1.1)). The images are for the strength ratios (a) 1:1 (b) 1:3 (c) (-1):3, from [69]. a broader class of topology types corresponding to single and double vortex streets on the sphere, and note that any two topology types are separated by a bifurcation topology point. In addition, we show that the topology type in Figure 1.19(c) corresponds only to a bifurcation point within the configuration space, and not to that of a range of parameters. In a second relevant study for inviscid point vortex streets by Stremler ([69]), the streamlines of vortex streets on the plane containing 2 and 3 vortices per periodic strip are studied. Using Equation (1.1) forN repeated vortices per strip, and multiplying both sides by , the author derives 1 2Li N X =1 cot L (z z ) =SV = 0; (1.4) where S = N X =1 : (1.5) Therefore, the vortex street is in equilibrium (i.e., all vortices remain equidistant) if eitherV = 0 (the street has no velocity) orS = 0 (the strengths of the vortices in each street sum to zero). Revisiting von K´ arm´ an’s configuration for the case of two vortices per periodic strip (N = 2): 19 Figure 1.21: Propagation angle vs. horizontal spacing (see Equation (1.6)) for (a) y = L=10; 000, (b) y =L=10, (c) y =L sinh 1 (1=) and (d) y =L. From [69]. ifV = 0 and for anyS, the cotangent must be zero which requires that all vortices are collinear on the real line (see Figure 1.20). For the caseS = 0, this implies that the two vortices have equal and opposite strength (i.e., 1 = 2 = ). For this case, any choice of vortex positions results in an equilibrium configuration that is translating with a constant velocity. The angle of propagation of the street (from the real axis) is determined from the ratio R(V ) = sinh( 24y L ); I(V ) = sin( 24x L ); (1.6) where4x and4y are the Cartesian distances between the two vortices in each periodic strip. Off-axis propagation occurs unless the spacing4x = 0 or L=2. Figure 1.21 illustrates the propagation angle as a function of vortex spacing. Furthermore, from [51], the vortex street is linearly stable if it satisfies the criterion sinh( 24y L ) = sin( 24x L ): (1.7) An interesting observation for the case of two vortices per strip is the existence of heteroclinic orbits that connect different stagnation points. In Figure 1.22, two such configurations are pre- sented. In the first, a heteroclinic orbit connects stagnation points from two adjacent periodic 20 Figure 1.22: Heteroclinic orbits are presented as dashed lines, vortex streets with two opposite-strength vortices per strip. Upper subplot has n = 1, lower has n = 2, where n denotes how many strips are passed with each heteroclinic connection. From [69]. Figure 1.23: Streamline topology for two vortices per periodic strip, and with 1 = 2 = . Configurations (a) - (e) are linearly stable. y is determined for (a) x =L=2, (b) x = 6L=13, (c) x = 4L=9, (d) x = 3L=8, (e) x =L=4 and (f) x = 0 with y = sinh 1 (1=), from [69]. 21 Figure 1.24: Three vortex streets per period withV = 0 andG> 0. The vortex strengths of the above configura- tions are (a) 3:(-1):3, (b) (21/10):(-1):(21/10), (c) 2:(-1):3 and (d) 3:1:3. Image from [69]. Figure 1.25: Left: V ortex positions and streamlines for three vortices withV = 0 andG < 0. Images above correspond to (from top to bottom) (a) j 2 >jGj = 2, (b) 1 2 = 1 <jGj = 7, (c) 1 2 = 1 <jGj = 6. Right: Three vortices per strip withS = 0, configurations have the strength ratio 1:2:(-3). From [69]. 22 strips, and this configuration is denotedn = 1. In the lower subplot, a heteroclinic orbit con- nects stagnation points from two periodic strips that are two strips apart (i.e., separated by a single strip), and this configuration is denoted as n = 2. Heteroclinic orbits have interest- ing implications for mixing. Fluid in these configurations passes from above the street, swirls through it and exits below. Asn increases, less fluid passes through the street until finally, when n1, no fluid passes through the street at all, and the configuration forn1 can be found in Figure 1.23(a). For the case of three repeated vortices per periodic strip, a similar analysis was conducted, and highlights of the results will be mentioned here. An important quantity for these configura- tions is the geometric strength sum G = 1 2 + 2 3 + 1 3 : (1.8) For the static case V = 0, if G > 0, all three vortices lie on the real line, and the relative positions depend on the vortex strengths (see Figure 1.24). If V = 0 and G < 0, either all three vortices lie on a vertical line if 2 >jGj, or two lie on a vertical line while the third satisfies the condition 2 >jGj and is offset by L=2 (see Figure 1.25). In the second case when S = 0 (i.e., 1 + 2 + 3 = 0), two of the vortex positions can be chosen arbitrarily while the third is determined analytically (Figure 1.25). Furthermore, these configurations will, in general, translate with a constant speedV . Configurations can be chosen to have a particular speed. To summarize, the works that have been referenced in this chapter presented several impor- tant features that describe vortex lattices: prevalent configuration geometry types, relative vortex strengths, and bifurcationsspecifically within the streamline topology. The methods (Chap- ters 2 & 3) and results (Chapters 4, 5 & 6) presented in the remainder of this thesis provide a thorough, and unified approach to the aforementioned problems referenced in this chapter. 23 Chapter 2. Fluid Mechanics Formulation In this chapter, we detail all the required mathematical background that leads to the point vortex model. We begin by introducing the Navier-Stokes equation which is the most general formula- tion in fluid mechanics, given by Du Dt u t + uru =rp + f + r 2 u; (2.9) where u is the velocity field.D()=Dt =@()=@t+(;r) is the absolute derivative with respect to time, f is a set of conservative external forces, is the viscosity of the fluid, and its density. In our work, we will be dealing exclusively with inviscid (i.e., = 0), incompressible fluid. The condition for incompressibility is characterized by r u = 0; (2.10) and this is referred to as the continuity equation for an incompressible fluid. The properties for this class of fluids will be expanded in the following section. 24 2.5 General Properties of Inviscid, Incompressible Flows The balance of linear momentum of an inviscid, incompressible fluid acted upon by conservative forces is given by the Euler equation Du Dt u t + uru =rp + f; (2.11) where the fluid density is normalized to one. The vorticity field !2 R 3 associated with a velocity field u2R 3 is defined as ! =r u: (2.12) Substituting Equation (2.12) into (2.11) gives the vorticity equation D! Dt ! t +!r! =!ru: (2.13) The fluid is said to be irrotational if! is zero. Taking the dot product and curl of Equation (2.12), yields respectively r! = r (r u) = 0; (2.14) r! = r (r u) =r 2 (u): (2.15) Using Stokes’ theorem and integrating the divergence of the vorticity in a finite volume gives Z V r!dV = Z S ! ndS = 0; (2.16) whereS is a closed surface with normal vector n bounding a volumeV of fluid with vorticity field!. This relation tells us that flux of the vorticity acrossS is zero. The same conclusion can be made regarding the flux of the velocity. 25 2.5.1 Hodge Decomposition The velocity field can be decomposed into a superposition of translational and rotational motion components using Helmholtz, or Hodge Decomposition, and this is given by u = u + u ! =r +r : (2.17) If a fluid is irrotational (i.e.,r u = 0), the fluid velocity is given solely by u u and can be derived from the gradient of a scalar field . Since our velocity field is divergence free, it follows that satisfies Laplace’s equation:r 2 = 0. If we have a rotational component to the velocity field, we define this as u ! , which can be derived from the curl of a vector field . It follows from Equation (2.12) for of Equation (2.17) that the vorticity scalar field is given by the Poisson equation r 2 =!: (2.18) A standard solution ([19]) to this Poisson equation in terms of the Green’s function for the Laplacian is (x) = Z G(x z)!(z)dz; (2.19) where G(x) = 8 > > > > < > > > > : 1 2 logkxk; x2R 2 1 4 1 kxk ; x2R 3 9 > > > > = > > > > ; ; (2.20) with r 2 G(x) +(x) = 0: (2.21) Finally, since u ! =r , we can substitute into Equation (2.19) to get u ! = r Z G(x z)!(z)dz (2.22) = Z K(x z)!(z)dz; (2.23) 26 whereK is the singular Biot-Savart kernel defined for 2D and 3D as K(x) = 8 > > > > > > > > > > > < > > > > > > > > > > > : + 1 2 1 kxk 2 ( y x ); x2R 2 1 4 1 kxk 3 0 B B B B @ 0 z y z 0 x y x 0 1 C C C C A ; x2R 3 9 > > > > > > > > > > > = > > > > > > > > > > > ; : (2.24) 2.5.2 Circulation A fundamental quantity associated with vorticity (and with the point vortex model in the next section) is circulation, given by the scalar via Stokes’ theorem = I C uds = Z A r u ndS = Z A ! ndS: (2.25) C is a curve that encloses a surface A. In an ideal, incompressible fluid, the circulation of a body of fluid bounded by its moving material boundaryC(t) remains constant (i.e., d=dt = 0 =) (t) = 0 ). Equation (2.25) can be used to prove that for a point vortex, the circulation is located at a single point (i.e., the vortex site), and remains unchanged for any selection ofC orA for the integral in Equation (2.25). 2.5.3 Application of the Cauchy-Riemann Equations The final definition we will reference is the general Cauchy-Riemann identity for analytic func- tions ([53]), and we will demonstrate its usefulness in the context of the point vortex model in the following section. Definition 1. The complex analytic function w(z) = +i 27 is called the complex potential for the flow, wherez = x +iy: The real part,, is called the velocity potential, while the imaginary part, , is the streamfunction. Since the functionw(z) is analytic, and are related to each other via the Cauchy-Riemann equations @ @x = + @ @y ; @ @y = @ @x : Thecomplexvelocity is defined by dw dz =uiv = _ xi _ y _ z : 2.6 Point Vortex Dynamics 2.6.1 Two Dimensions In deriving an expression for point vortices on the plane, we first consider Equation (2.13) in two dimensions. The right hand side is equal to zero, therefore D! Dt 0; (2.26) with! = (0; 0;! z ) T (0; 0;!) T and u = (u;v; 0) T = ( _ x; _ y; 0) T . We now deal with the scalar vorticity ! = @v @x @u @y : (2.27) Another key feature is that the vector potential from Equation (2.17) becomes the scalar streamfunction found in Definition 1. Therefore, u =r 0 @ 0 1 1 0 1 A 0 @ @ x @ y 1 A = ( y ; x ) T : (2.28) 28 We now define the Hamiltonian as the streamfunction,H , since we can write _ x = @H @y (x;y;t); _ y = @H @x (x;y;t): (2.29) The Cartesian positions (x;y) are the canonical conjugate variables ofH. When the flow is time-independent, it is referred to as steady state. The streamlines arising fromH are tangent to the velocity lines for all time. The normal component to the velocity field, n, becomes the gradient of the streamfunction, namely that n =r =rH: (2.30) Since we can state that u n = ur = 0, we can conclude that the streamline curves form solid boundaries which the fluid cannot pass. This is the Hamiltonian that is used for the point vortex formulation. In the next section, we will introduce the model forN interacting discrete point vortices. 2.6.2 The Point Vortex Model For the flowfield around a system ofN discrete vortex sources, we assume the vorticity distribu- tion !(x) = N X i=1 i 2 (x x i ); (x) = 1 2 x : In this definition, is a small number such that 1. is a normalized, radially symmetric function (i.e., R dx = 1). In taking the limit! 0, we generate what is called the point vortex, and our function becomes the Dirac delta function (i.e.,!(x) = (x) (x)). Using the 29 solution to Poisson’s equation via Green’s function for the Laplacian (Equation (2.19)), we derive that the streamfunction for vortex with position x = (x (t);y (t)), where 1N to be (x;t) = 1 2 Z log(x z)(x z)dz (2.31) = 2 log(x x ): (2.32) The velocity of a passive particle in the fluid that is being advected byN point vortices can be obtained via linear superposition. That is, with the velocity being the curl of the streamfunction as in Equation (2.28), the velocity of a passive particle in the flow becomes _ x = N X =1 r (x;t): (2.33) The net velocity at the site of vortex therefore is _ x = N X =1 0 r (x ;t): (2.34) The prime in the summation implies that the terms = are excluded since a vortex does not advect itself. Plugging Equation (2.32) into (2.34) gives us an expression for the Cartesian velocities in the plane ( _ x (t); _ y (t)) as _ x = 1 2 N X =1 0 (y y ) l 2 ; (2.35) and _ y = 1 2 N X =1 0 (x x ) l 2 ; (2.36) 30 wherel 2 =kx x k 2 is the square of the inter-vortical distance of vortices and. See Figure 2.26 for an illustration. In more compact form usingz = x +iy , the equations of motion can be expressed as _ z = i 2 N X =1 0 z z jz z j 2 ; 6=; ( = 1;:::;N); (2.37) or alternately as _ z = 1 2i N X =1 0 z z ; 6=; ( = 1;:::;N): (2.38) The result in Equation (2.37) was first presented by Helmholtz in 1857. Kirchhoff in 1887 used Helmholtz’s equations of motion to derive the Hamiltonian in canonical form: _ x = @H @y ; _ y = @H @x ; (2.39) whereH, the Hamiltonian energy from Equations (2.35) and (2.36) ([53]) is given by H = 1 4 N X =1 N X =1 log(z z ): (2.40) The Hamiltonian is also referred to as the interaction energy in the literature. The phase space, therefore, is described by the Cartesian coordinates with contours corresponding to passive fluid streamlines. The next set of constants for vortex systems is given by Q +iP = N X =1 z ; (2.41) where the complex quantityQ+iP is also referred to as the center of vorticity or linear impulse, and for the planar setting defines the complex point about which the system is instantaneously 31 rotating. This quantity is also the discrete equivalent of the linear momentum for point vortex systems. In vector form, the center of vorticity is commonly written as J = (Q;P ) T : (2.42) The last conserved quantity is I = N X =1 jz j 2 ; (2.43) whereI is referred to in the literature as the angular impulse, and is the discrete equivalent of the angular momentum for point vortex systems. The next major result for point vortex dynamics was the solution to the three-vortex problem. The results were first published by Gr¨ obli in 1877, but were not given much notice. Fourteen years later, however, Poincar´ e published his results for the three-vortex problem. For more background on theN-vortex problem, see [53], [5] and [52]. In the following section, we derive the condition for equilibrium on the plane and sphere. 2.7 Condition for Relative Vortex Equilbrium 2.7.1 On the Plane Several references for vortex equilibria on the plane can be found in [7], [69], [54], [55], [56], [6], and [4]. In this section, the condition of relative vortex equilibria will be derived. We begin with the vector form of Equation (2.37) which is given by _ x = 1 2 N X =1 0 ^ e z (x x ) l 2 ; ( = 1:::N); (2.44) in which _ x is the Cartesian coordinate of vortex, andl 2 is the square of the distance between vortices and , see Figure 2.26. The equilibrium condition, therefore is that these squared distances are invariant with timenamely: d dt (l 2 ) = 0 = 2(x x ) ( _ x _ x ): (2.45) 32 Figure 2.26: Three generic vortices,, in the plane. The distance between the vortices and the area formed between them are labeled. Note that net translation and rotation of the system as a whole is allowed. The condition is that the distance between the vortices themselves remains unchanged with time. Rewriting the difference of velocities in terms of summations as in Equation (2.44) and simplifying, _ x _ x = 1 2 N X =1 0 ^ e z (x x ) l 2 1 2 N X =1 0 ^ e z (x x ) l 2 = N X =1 00 ^ e z (x x ) l 2 ^ e z (x x ) l 2 ! + + 2 ^ e z (x x ) l 2 : The first of the two summations on the first RHS of the equation above excludes the = term, and the second excludes the = term. When these two summations are combined in the second RHS, an additional term needs to be removed from each summation in order to make the merger possible. This entails removing the term from the first summation, and the term from the second summation. The last quantity in the equation above corresponds to the removed 33 and terms, and the double prime implies that6= and also that6= . The next step would be to plug the above result into Equation (2.45): d dt (l 2 ) = 2(x x ) ( _ x _ x ) = x x N X =1 00 ^ e z (x x ) l 2 ^ e z (x x ) l 2 ! + + (x x ) ^ e z (x x ) l 2 = 1 N X =1 (x x + x x + x x ) 1 l 2 1 l 2 ! ; with + (x x ) ^ e z (x x ) l 2 = 0; (2.46) since (x x ) is clearly perpendicular to ^ e z (x x ). The sum of the three cross products above can be rewritten in the following way: (x x ) + (x x ) + (x x ) = (x x ) (x x ) (x x ) + (x x ) = x (x x ) x (x x ) = (x x ) (x x ) = 2A ^ e z = 2 A ^ e z ; whereA is the area of the parallelogram generated by the cross product of the two vectors (which also corresponds to the sides of the triangle in Figure 2.26. is the sign of the 34 expression ( 1), which is dictated by the clockwise (+1) or counter-clockwise (1) ordering of vortices; and. The governing equilibrium condition then becomes d dt (l 2 ) N X =1 00 A 1 l 2 1 l 2 ! ; 6=6=; (2.47) or N X =1 00 A 1 l 2 1 l 2 ! 0: (2.48) The idea is to cycle through all the possible intervortical distances and come up withN(N1)=2 equations, which can be rewritten in matrix form: A = 0; (2.49) whereA2R N(N1)=2N is the configuration matrix and 2R N is a vector containing the vortex strengths of the system. The system has a non-trivial solution if the covariance matrix A T A is singular, namely that det (A T A) 0: (2.50) 2.7.2 On the Sphere The equilibrium condition for vortices on the sphere is solved in a similar fashion as that for vortices on the plane. The coupled equations of motion ofN vortices on the surface of a unit sphere are given by _ x = 1 2 N X =1 0 x x 1 x x ; ( = 1:::N); (2.51) where x is the three dimensional Cartesian unit vector that points from the center of the sphere to a fluid point on the surface of the sphere and 1N is the vortex number. The term is 35 (a) 3 vortices on sphere (b) V olume view Figure 2.27: Image of three generic vortices on the sphere that show the definition of vortex strengths (’s), intervortical lengths (l12;l13;l23), and volumeV123. excluded from the summation since a vortex does not advect itself. The denominator of Equation (2.51) is the square of the chord distance between vortices and: l 2 =jjx x jj 2 = 2(1 x x ): (2.52) Additional relevant formulations for vortex systems on the sphere in our analysis include the center of vorticity vector J which describes the instantaneous axis about which the system is rigidly rotating, and is given by J = N X =1 x : (2.53) Furthermore (from [53]), the equations of motion in terms of the longitudinal and latitudinal angles ( and respectively) are given by _ = 1 4 N X 6= sin sin cos ( ) 1 cos ; (2.54) 36 and _ = 1 4 sin N X 6= sin cos cos sin cos ( ) 1 cos ; (2.55) with cos = cos cos + sin sin cos ( ): (2.56) As for the condition for an equilibrium configuration, as before, is that the distance between the vortices remains equidistant, or that d dt l 2 = 2(x x ) ( _ x _ x ) = 0: (2.57) The difference of velocities can be written as _ x _ x = 1 2 N X 6= 0 x x l 2 1 2 N X 6= 0 x x l 2 = 1 2 N X 6=;6= 00 x x l 2 x x l 2 ! 1 2 ( + ) x x l 2 : Using this result in Equation 2.57, d dt l 2 = 1 (x x ) N X 6=;6= 00 x x l 2 x x l 2 ! 1 ( + )(x x ) x x l 2 : The last portion on the second line of the equation above is equivalent to zero since the cross product of x and x is perpendicular to the plane containing them (and the inner product is zero). Simplifying, one gets d dt l 2 = N X 6=;6= 00 x x x l 2 x x x l 2 ! : (2.58) 37 Note that x x x is the scalar triple product that gives the volume of the parallelepiped formed between the three vectors; this volume is denoted asV . The condition for equilibrium becomes: d dt l 2 = 0 N X 6=;6= 00 V 1 l 2 1 l 2 ! : (2.59) Using this equation, as before, it is necessary to cycle between all the possible lengths in the configuration. Therefore, we will have a totalN(N 1)=2 equations. The system can be placed in matrix form: A = 0; (2.60) whereA and are synonymous to those of Equation (2.49). The condition for equilibrium is identical to Equation (2.50). More detailed results for vortex equilibria on the sphere can be found in [39] and [15]. 38 Chapter 3. Linear Algebra Methods In this chapter, three main methods from linear algebra will be discussed. In 3.1, we will define Singular Value Decomposition. In 3.2, Shannon Entropy will be introduced, and in 3.3 a matrix approximation technique will be presented. In each section, examples showing either their appli- cation to the vortex equilibrium condition in equations (2.49) (on the plane) and (2.60) (on the sphere) or to matrices will be presented. 3.8 Singular Value Decomposition We begin with Singular Value Decomposition (a definitive reference on this topic can be found in [41]). Now that an equilibrium condition has been formulated (see sections 2.7.1 and 2.7.2), the solution to the dynamical system has to be found (i.e., the strength of the vortices, or vector from (2.49) and (2.60)). 3.8.1 Definition Singular Value Decomposition (SVD) is the equivalent of matrix decomposition for non-normal matrices (i.e.,A T A6=AA T ). Instead of having eigenvalues and eigenvectors, SVD has singular values and singular vectors. SVD is utilized in a variety of applicationsfrom robotics & kine- matics, to graphics ([71]), information compression ([22]) and retrieval ([9]), and the following 39 chapter will illustrate its usefulness in the context of point vortex equilbria. For anMN non- normal real rectangular matrixA withM > N, the singular values must satisfy the following conditions: Av (i) = (i) u (i) ; A T u (i) = (i) v (i) ; i = 1N; (3.61) where (i) is thei th singular value, u (i) 2R M is thei th left singular vector and v (i) 2R N is thei th right singular vector. The decomposition is expressed as A =UV T ; (3.62) whereU andV are square unitary matrices (i.e.,U T U =I2R M ;V T V =I2R N ) composed of the left and right singular vectors respectively. is anMN diagonal matrix containing the singular values in the form = 0 B B B B B B B B B B B B B B B B B B B B @ (1) 0 0 0 0 (2) 0 0 . . . . . . . . . 0 0 (N1) 0 0 0 0 (N) 0 0 0 0 . . . . . . . . . . . . 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C A : (3.63) The singular values are ordered diagonally from largest to smallest in the first N rows of , and the remainingMN rows are zeros. The rank ofA is ‘r’ and is equal to the number of non-zero (i) ’s (i.e., (max) (1) (2) (r) > 0 = (r+1) = = (N) ). As can be seen from multiplying the first equation in Eq. (3.61) byA T and the second byA, the following relations emerge: (A T A (i)2 )v (i) = 0; (AA T (i)2 )u (i) = 0; (3.64) 40 which indicates that the squares of the singular values ofA are also the eigenvalues of the square covariance matricesA T A andAA T and their eigenvectors are v (i) and u (i) respectively. Note also that the ranks ofA,A T A andAA T are all the same. With the knowledge thatnullity(A) + rank(A) =N, a nontrivial solution to Equations (2.49) and (2.60) exists when the rank ofA is less thanN, which in (3.61) implies that (i) =f0; i>rg, and gives Av (i) = (i) u (i) = 0: (3.65) Therefore, the columns ofV associated with (r+1) ; (r+2) ; ; (N) correspond to to the non- trivial solution of the homogeneous system of equationsA = 0, or namely by the linear super- position =c 1 v (r+1) +c 2 v (r+2) + +c Nr v (N) ; (3.66) wherec 1 ;c 2 ;c Nr are constants inR that can be chosen arbitrarily. In the context of point vortex equilibria and with null spaces larger than one, it is desirable to convert the solution in Equation (3.66) into a rational basis of reduced echelon form by using a series of matrix operations. In other words, v (r+1) v (r+2) v (N) 0 B B B B B B B @ c 1 c 2 . . . c Nr 1 C C C C C C C A X 0 B B B B B B B @ 1 2 . . . Nr 1 C C C C C C C A = 0: (3.67) MatrixX = x (r+1) x (r+2) x (N) is in reduced echelon form, 1 Nr are the new arbitrary constants, and the new solution is given by = 1 x (r+1) + 2 x (r+2) + + Nr x (N) : (3.68) Note that in the event that vectors v (r+1) ; v (r+2) ; ; v (N) are already in reduced echelon form, no linear transformation needs to be made. 41 3.8.2 Computation of the SVD A basic way of computing the singular values is by first forming the square covariance matrices AA T orA T A, and then solving for their eigenvalues. From Equation (3.64), the singular values ofA are then the square root of the eigenvalues. The problem with this approach is that it can be numerically unstable ([70]). From [41], an alternative way of computing the singular values is by solving the left and right singular vectors directly from 0 @ 0 A A T 0 1 A 0 @ u (i) v (i) 1 A = (i) 0 @ u (i) v (i) 1 A (3.69) The benefit of this approach is that if the matrix A is perturbed by a small amount, it can be shown that the perturbed singular value ^ (i) satisfies j^ (i) (i) j =O(kAk): For further efficient algorithms for deriving the decomposition, see [60] and [42]. 3.8.3 SVD Example Examples of how this method works in the context of point vortex equilibria will be given for two square configurations withN = 4 andN = 5. 3.8.3.1 Havelock’s Square (N = 4) The square vortex configuration was studied by Havelock ([35]), and was shown to be lin- early stable. The configuration contains four vortices placed at the corners of a square (see Figure 3.28). See Figures 1.2 and 1.10 for examples of this lattice configuration in physical set- tings. Recall that forN vortices, there existN(N 1)=2 intervortical distances. The number of vorticesN = 4 for this case gives a total of 4(3)=2 = 6 intervortical distances, and hence there are 6 equations and 4 unknown vortex strengths to be solved. Using equation (2.45), and 42 Figure 3.28: Four vortices on the plane in a square configuration, and without loss of generality, with side of unit lengthS = 1. considering that the sides of the square have unit length, the first of the six equations for this system has the form: d dt (l 2 12 ) = 0 + 0 + 3 A 132 132 1 l 2 13 1 l 2 23 + 4 A 142 142 1 l 14 2 1 l 2 24 = 3 1 2 (1) 1 2 1 1 + 4 1 2 (1) 1 1 1 2 = 1 4 3 1 4 4 = 1 4 0 0 1 1 0 B B B B B B B @ 1 2 3 4 1 C C C C C C C A = 0: The next five equations in A, which correspond to rate of change of the squares of the inter- vortical distancesl 2 13 ,l 2 14 ,l 2 23 ,l 2 24 , andl 2 34 can be solved in a similar fashion, and placing them 43 in matrix form (see Equation (2.49)) as well as reordering them diagonally, the configuration matrix for this particular system becomes: 0 B B B B B B B B B B B B B B @ A 341 1 l 2 31 1 l 2 41 A 342 1 l 2 32 1 l 2 42 0 0 A 241 1 l 2 21 1 l 2 41 0 A 243 1 l 2 23 1 l 2 43 0 A 231 1 l 2 21 1 l 2 31 0 0 A 234 1 l 2 24 1 l 2 34 0 A 142 1 l 2 12 1 l 2 42 A 143 1 l 2 13 1 l 2 43 0 0 A 132 1 l 2 12 1 l 2 32 0 A 134 1 l 2 14 1 l 2 34 0 0 A 123 1 l 2 13 1 l 2 23 A 124 1 l 2 14 1 l 2 24 1 C C C C C C C C C C C C C C A : (3.70) Replacing all the symbolic forms above with their numerically equivalent results, and placingA in Equation (2.49), we get A = 1 4 0 B B B B B B B B B B B B B B @ 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 C C C C C C C C C C C C C C A 0 B B B B B B B @ 1 2 3 4 1 C C C C C C C A = 0 B B B B B B B @ 0 0 0 0 1 C C C C C C C A (3.71) As in Equation (3.62), Singular Value Decomposition ofA gives the following three matrices: U = 0 B B B B B B B B B B B B B B @ 0:5 0:5 0:5 0:1015 0 0 0 0 0 0:9792 0 0 0:5 0:5 0:5 0:1015 0 0 0:5 0:5 0:5 0:1015 0 0 0 0 0 0 0 0 0:5 0:5 0:5 0:1015 0 0 1 C C C C C C C C C C C C C C A ; (3.72) 44 = 0 B B B B B B B B B B B B B B @ 0:5 0 0 0 0 0:3536 0 0 0 0 0:3536 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C C C C C C C C C C C C C C A ; (3.73) V = 0 B B B B B B B @ 0:5 0:7071 0 0:5 0:5 0 0:7071 0:5 0:5 0:7071 0 0:5 0:5 0 0:7071 0:5 1 C C C C C C C A : (3.74) The singular values are ( (1) , (2) , (3) , (4) ) = (1/2, 1=2 p 2, 1=2 p 2, 0). The 4 th singular value in matrix is equal to zero, which indicates that the rank of the configuration matrixA is 3, giving a null space of 1. Using Equation (3.68), the solution of the system lies in the 4 th right-most eigenvector v (4) : = v (4) = 0 B B B B B B B @ 0:5 0:5 0:5 0:5 1 C C C C C C C A 1 0 B B B B B B B @ 1 1 1 1 1 C C C C C C C A ; 1 2R: (3.75) This solution means that the four vortices must all have the same strength for the configuration to remain in equilibrium. Note also that this strength, given by 1 , can be any real number. 3.8.3.2 Havelock’s Square Plus Central Vortex (N = 5) We now consider the same square configuration with an additional vortex located at the center of the square, thus having a total of five vortices (see the schematic of the solution in Figure 3.29). This type of configuration appears in the milli-sized magnets lattices from [34], and they appear 45 Figure 3.29: Solution to the 5-vortex square lattice configuration. The four outer edge vortices share one common strength, while the central vortex has an independent strength ([56]). on the curved surface of a water drop in Figure 1.11. The configuration matrix from Equation (2.49) for this case will be of dimension 5(4)=2 5 = 10 5 and when solved gives A = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C A : (3.76) 46 Matrix decomposition as in (3.62) for this configuration gives U = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 0:5774 0 0 0:2224 0 0 0 0 0 0 0 0:5774 0 0:7195 0 0 0 0 0 0 0:2887 0:2887 0:5 0:0908 0 0 0 0 0 0 0:5774 0 0 0:2993 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0:2887 0:2887 0:5 0:1677 0 0 0 0 0 0 0 0:5774 0 0:5481 0 0 0 0 0 0 0:2887 0:2887 0:5 0:0806 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0:2887 0:2887 0:5 0:0037 0 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C A ; (3.77) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B @ p 3 2 p 2 0 0 0 0 0 p 3 2 p 2 0 0 0 0 0 1=2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C A ; (3.78) V = 0 B B B B B B B B B B @ 0:7071 0 0:5 0:5 0 0 0:7071 0:5 0:5 0 0:7071 0 0:5 0:5 0 0 0:7071 0:5 0:5 0 0 0 0 0 1 1 C C C C C C C C C C A : (3.79) 47 The singular values for this configuration are 0 B B B B B B B B B B @ (1) (2) (3) (4) (5) 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ p 3 2 p 2 p 3 2 p 2 1 2 0 0 1 C C C C C C C C C C A : (3.80) The two smallest singular values are zero, which indicates that the dimension of the solution space of this configuration is 2 and is found in the 4 th and 5 th right singular vectors. The solution, therefore, becomes = 0 B B B B B B B B B B @ 1 2 3 4 5 1 C C C C C C C C C C A = 1 v (4) + 2 v (5) = 1 0 B B B B B B B B B B @ 1 1 1 1 0 1 C C C C C C C C C C A + 2 0 B B B B B B B B B B @ 0 0 0 0 1 1 C C C C C C C C C C A ; 1 ; 2 2R; (3.81) where 1 and 2 are arbitrary constants. This implies that the four vortices forming the square have equal strengths ( 1 = 2 = 3 = 4 = 1 ), and that the central vortex has a different, independent strength ( 5 = 2 ). This 5-vortex configuration can be reduced to the square lattice by choosing 2 = 0. In this section of this chapter, our focus has solely been on the role of the zero singular values associated with the configuration matrix, as they determined whether a given configuration has a non-trivial solution. In the following two sections, we will look at the information encoded in the non-zero singular values. 48 Figure 3.30: Shannon Entropy vs. the probability of getting heads from a dice throw. Maximum entropy occurs when the probability = 1=2, which coincides with the proposition that the probability with the highest Shannon Entropy is also the most likely to occur. 3.9 Shannon Entropy 3.9.1 Definition Shannon in [67] introduced a scalar measure for entropy, which is referred to as the Shannon Entropy ‘H’. We describe the entropy, or modal energy distribution of a system ~ P with proba- bilities (P 1 ;P 2 ;:::;P N ) by H = N X =1 P lnP ; P 6= 0; 0H lnN: (3.82) Alternately, to normalize the system, log N can be chosen instead of ln log e in Equation (3.82). In other words, the normalized Shannon Entropy becomes H = N X =1 P log N (P ); P 6= 0; 0H 1: (3.83) The Shannon Entropy of a system can be used to determine how likely it is for a certain system ~ P to occur. An intuitive example is the throwing of a dice. The system has two likely outcomes, 49 and is defined as ~ P = (P (heads);P (tails)) = (P 1 ;P 2 ), whereP 1 is the probability of landing heads, andP 2 is the probability of landing tails withP 1 +P 2 = 1. Note that we can rewrite the system as ~ P = (P 1 ; 1P 1 ). Intuitively, we know that the most likely outcome of the coin toss would be that we land heads half the time (P 1 = 1=2), and tails during the other half. Therefore, we expect the system ~ P = (1=2; 1=2) to be the most likely or most robust, while the systems ~ P = (1; 0) and ~ P = (0; 1) to be least likely. Since there are only two possible outcomes to the coin toss, we can use log 2 in the formulation, and with Equation (3.83) the Shannon Entropy of our system becomes H = N X =1 P log 2 P = P 1 log 2 P 1 (1P 1 ) log 2 (1P 1 ): Figure 3.30 illustratesH vs.P 1 , and shows us that the maximum entropy occurs whenP 1 = 1=2. This supports the claim that there is a positive correlation between the Shannon Entropy and likelihood of an event to occur. In the context of a non-normal matrix A, we interpret the normalized eigenvalues of the corresponding covariance matrices AA T and A T A from Equation (3.64) as the probabilities, which gives us P = ^ = P N =1 = () 2 P N =1 () 2 ; = 1N: (3.84) An alternate way of interpreting the Shannon Entropy is by the information compression present in a system, which we will briefly define in the following subsection. 50 3.9.2 Information Compression Consider the two general square diagonal matrices of dimensionN below: V min = 0 B B B B B B B B B B @ 1 1 . . . 1 1 1 C C C C C C C C C C A ; V max = 0 B B B B B B B B B B @ 1 0 . . . 0 0 1 C C C C C C C C C C A : (3.85) Since both matrices are diagonal, the eigenvalues are simply the diagonal elements. V min has eigenvalues (1; 1;:::; 1; 1). All its eigenvalues are equal in magnitude, and the matrix is said to have minimum information compression since all its numerical information is equally distributed among itsN modes, or eigenvalues. We can also state that the probability for numerical infor- mation to lie on any given mode is (100/N)%. MatrixV max ’s eigenvalues are (1; 0;:::; 0; 0). Since all the numerical information is compressed into its first mode,V max is said to have max- imum information compression. For this matrix, there is a 100% probability that the numerical information will lie on the first mode, and a 0% probability for it to lie on any of the remaining (N 1) modes. If a system has minimum compression, it is said to have maximum entropy since its infor- mation is as distributed as possible throughout the modes of the system. On the other hand, if a system has maximum compression, it is said to have minimum entropy since its information is condensed into a single mode. Referring back to the two matricesV min andV max (which rep- resent the two extreme possible scenarios) defined in Equation (3.85), the probabilities (derived from Equation (3.84)) associated with V min are (1=N; 1=N;:::; 1=N; 1=N). Using Equation (3.82), H max = 1 N ln 1 N 1 N ln 1 N ::: 1 N ln 1 N = lnN: (3.86) 51 On the other hand, the minimum possible Shannon Entropy, H min can be derived from ~ P (V max ) = ~ P (1; 0;:::; 0). From Equation (3.82), the minimum entropy is simply H min = ln 1 = 0: (3.87) This tells us that the entropyH of any system with dimensionN is bounded by 0<H < lnN. Therefore, from [67], any system has the following properties: 1. H = 0 if and only if all theP but one are zero, this one having the value unity. Thus only when we are certain of the outcome doesH vanish. OtherwiseH is positive 2. For a givenN,H is maximum and equal to lnN when all theP are equal (i.e., 1 N ). This is also intuitively the most uncertain situation. With this in consideration, we can derive an expression for the percentage of information com- pression in a system as lnNH lnN 100%; (3.88) which for the systemfV max ;H min g is 100%, and 0 % forfV min ;H max g. 3.9.3 Shannon Entropy Examples Let’s now apply the principle of Shannon Entropy to the same vortex lattice examples introduced previously in Section 3.8.3. 3.9.3.1 Havelock’s Square (N = 4) Consider again Havelock’s square. The (four) singular values were given as (1) ; (2) ; (3) ; (4) = 1=2; p 2=2; p 2=2; 0 . Squaring and normalizing using Equation (3.84) gives us the probabilities of the system (P 1 ;P 2 ;P 3 ;P 4 ) = 1 2 ; 1 4 ; 1 4 ; 0 : (3.89) 52 This gives us a Shannon Entropy of H = N X =1 P lnP = 1 2 ln 1 2 1 4 ln 1 4 1 4 ln 1 4 = 1:0397: (3.90) Note that the maximum possible entropy forN = 4 is ln 4 = 1:3863. Using Equation (3.88), the percentage of information compression for this system 25 %. Remark 1. Note that the Shannon Entropy of point vortex systems is simply a function of their geometry, and is completely unaffected by the allowable choice of the vortex strengths. 3.9.3.2 Havelock’s Square Plus Central Vortex (N = 5) The normalized eigenvalues for this configuration are (P 1 ;P 2 ;P 3 ;P 4 ;P 5 ) = 3 8 ; 3 8 ; 1 4 ; 0; 0 ; (3.91) which gives us an entropy of H = N X =1 P lnP = 3 8 ln 3 8 3 8 ln 3 8 1 4 ln 1 4 = 0:3890: (3.92) Compared to the maximum entropy forN = 5, which is ln 5 = 1:6094, Equation (3.88) tells us that the percentage of information compression for this lattice configuration is 76 %. From this, we conclude that the square lattice without the square (N = 4) has a higher entropy than the lattice with the central vortex (N = 5), which means that the energy is more evenly distributed among the modes when there is no central vortex. This also tells us that for vortices on the Cartesian plane, the configuration without the central lattice has a higher likelihood of existing than does the lattice with the central lattice. 53 An alternative way for comparing two different systems with the same number of particles (or vortices in our case), is by computing the Kullback-Leibler divergence given by [43]. Con- sider two systems with modal probabilities P = (P 1 ;:::;P N ) and Q = (Q 1 ;:::;Q N ). The value is defined as D KL (P;Q) = N X =1 P i ln P Q : (3.93) An equivalent way of writing the equation is D KL (P;Q) = N X i=1 P i ln(P i ) N X i=1 P i ln(Q i ) = N X i=1 P i ln(Q i ) + N X i=1 P i ln(P i ) = H(P;Q)H(P ); whereH(P;Q) is defined as the cross-entropy between systemsP andQ, andH(P ) is simply the Shannon Entropy ofP . Such a formulation can be used to contrast two configurations with the same number of vortices. Note that our application of Shannon Entropy was for systems with one set of events, which for our case corresponded to the set of normalized eigenvalues ofAA T (i.e., H = H(P ()), with = 1N). The Shannon Entropy formulation, can be extended to systems in which more than one event can occur simultaneously, for instance if two events can occur simultaneously, H(x;y) = X ; P (;) lnP (;); H(x) = X ; P (;) X lnP (;); H(y) = X ; P (;) X lnP (;): 54 3.10 Matrix Approximation Matrix Approximation is the third linear algebra tool that we use in our analysis. It is primarily used in Chapter 4 when dealing with asymmetric equilibria. We begin this section by introducing the Frobenius norm. 3.10.1 The Frobenius Norm The Frobenius normkAk F gives us a measure of the ‘size’ of a matrix. It is a function of a matrix’s singular values, and is given by kAk 2 F = (1) 2 + (2) 2 +::: + (r) 2 trace(A T A); (3.94) wherer is the rank of covariance matrixA T A. Note also that the 2-norm,kAk 2 is given by kAk 2 = (1) : (3.95) For instance, the size of Havelock’s square configuration without a central vortex is given by kAk 2 F = 1 2 2 + 1 2 p 2 2 + 1 2 p 2 2 = 1 2 ; kAk 2 = 1 2 : Upon adding the central vortex, the size the conguration becomes kAk 2 F = p (3) 2 p 2 ! 2 + p 3 2 p 2 ! 2 + 1 2 2 = 1; kAk 2 = p 3 2 p 2 : Clearly, the configuration with the central vortex is larger than the configuration without the central vortex. We can also express the distance between two matricesA andB from (3.94) as kABk F . 55 3.10.2 Reduction For a given matrix, it might be desirable to reduce its size, given bykAk F . This is a process commonly done in graphics and information compression ([22]). Since any non-normal matrix A2R MN with rankr can be decomposed by Equation (3.62), the matrix can equivalently be expressed as the linear sum of sub-matrices given by A = (1) u (1) v (1)T + (2) u (2) v (2)T +::: + (r) u (r) v (r)T ; (3.96) or alternatively as A = r X =1 () () ; () = u () v ()T : (3.97) The sub-matrix () is scaled with the corresponding th singular value, which gives a measure of its size relative toA. A rank-1 approximation entails the removal of ther th term from the summation in Equation (3.97). From an information storage perspective, we have effectively reduced the size of our matrix. The net size reduction betweenA and a rank-1 approximation A 1 , is given via the Frobenius norm kAA 1 k 2 F = (r) 2 ; (3.98) Note that if (r) is sufficiently small, we have effectively approximated the matrix with a rela- tively small loss of information. A generalized rank-k approximation is given by kAA k k 2 F = (r) 2 +::: + (rk+1) 2 : (3.99) We can quantify the amount of information reduction with the ratio kAA k k F kAk F 100%: (3.100) 56 Example 1. (From [41]) Find a rank-one approximation to the matrix A = 0 B B B B @ 1 1 0 1 1 0 1 C C C C A : Begin by decomposing the matrix as in (3.62): U = u (1) u (2) u (3) = 0 B B B B @ 2= p 6 0 1= p 3 1= p 6 1= p 2 1= p 3 1= p 6 1= p 2 1= p 3 1 C C C C A ; V = v (1) v (2) = 0 @ 1= p 2 1= p 2 1= p 2 1= p 2 1 A ; = 0 B B B B @ (1) 0 0 (2) 0 0 1 C C C C A = 0 B B B B @ p 3 0 0 1 0 0 1 C C C C A : Next, decompose the matrix into the form of Equation (3.97), and make the estimation A 1 (1) (1) = (1) u (1) v (1)T = p 3 0 B B B B @ 1= p 3 1= p 3 1=2 p 3 1=2 p 3 1=2 p 3 1=2 p 3 1 C C C C A = 0 B B B B @ 1 1 1=2 1=2 1=2 1=2 1 C C C C A : 57 With Equation (3.99), the change betweenA andA 1 iskAA 1 k F = (2) = 1. The amount of information reduction from (3.100) becomes kAA 1 k F kAk F 100% = 1 2 100% = 50%: 58 Chapter 4. Vortex Lattices on the Plane In this chapter, we apply the tools from Chapters 2 and 3 to systems ofN vortices on the Carte- sian plane. The first two of the four main contributions of this thesis are presented in this chapter. The first contribution is the study of asymmetric equilibria, and the second is the application of Shannon Entropy on vortex equilibria. Before we proceed to these main results, a few compar- isons to physical lattices will be made in the following section, and at the close of this chapter we will address the reverse problem of Equation (2.49). (a) N = 6 Triangle Model (b) Plasma Crystal Figure 4.31: Triangular lattice,N = 6. SVD solution produced for configuration was = (-1/4, -1/4, -1/4, 1, 1, 1) T . System resembles a plasma crystal configuration found in [30]. In (a), center of vorticity is labeled with an symbol. 59 (a) N = 7 Semi-circle Model (b) N = 7 Hexagon Model (c) N = 7 Different-Sized Milli-Size Magnets (d) N = 7 Uniform- Sized Milli-Size Mag- nets Figure 4.32: V ortex lattices with N = 7 vortices. (a) Semi-circle model has null space of one, with = (1; 0; 0; 0; 0; 0) T . This implies that central vortex has arbitrary strength, while remaining 6 vortices on the circle of radiusr are stagnation points orbiting the central vortex with velocityu = =2r. (b) Hexagonal model has non-trivial null space of dimension 2 with =1(1,0,0,0,0,0) T +2(0,1,1,1,1,1) T with1 and2 being unrelated arbitrary constants. This implies that central vortex has one independent strength, while the outer vortices have second independent strengths. (c) Comparison is made to milli-magnetized structures (from [34]) with different-sized magnets. Two configurations arise for different ranges of angular velocity (as denoted in figure), and we compare them to models (a) and (b) which support different-strength vortices with the selection16=26= 0. (d) Uniform- sized milli-magnetized structures. Comparison is made to (b), since the solution to (b) can support uniform-strength vortices with the selection1 =2. Center of vorticity coincides with location of central vortex. 4.11 Comparison to Physical Lattices Figure 4.31 is a triangular lattice withN = 6, and the comparison is made to the plasma lattice in Figure 1.12 ([23]). Note that this configuration also appears in the milli-magnets on the curved water drop surface in Figure 1.11 from [34]. The configuration was formulated as in (2.49), and a solution satisfying (2.50) was found. The dimension of the null space for this configuration is 60 (a) N = 19 Double Hexagon Model (b) Superfluid lattice Figure 4.33: Triangular vortex lattice,N = 19 with solution. Lattice consists of two concentric hexagonal rings. Solution to system can be found in Equation (4.101). 1 (i.e., (6) = 0). From Equation (3.68), the three outer corner vortices have equal strengths to one another; the three inner vortices have equal strengths to one another and are one-fourth as strong as the outer vortices with opposite circulation. The milli-sized magnets in Figure 1.11(b) possess two special features ([34]). First, the magnets have different sizes (we interpret different-sized magnets as having different vortex strengths). Second, more than one configuration geometry occurs for the same number of mag- nets. In Figure 4.32, we specifically modeled the two geometry types for the caseN = 7, and implemented Equation (2.49). The first model presented in Figure 4.32(a) is of a configuration consisting of a central vortex coupled with six vortices evenly spaced along a semi-circle cen- tered at the central vortex. The SVD solution yields a null space of one for this configuration given by = (1; 0; 0; 0; 0; 0) T . This tells us that the central vortex has arbitrary strength, while the six vortices on the semi-circle have zero strength, and are passive non-rotating parti- cles being advected around the central vortex with constant angular speedu = =2r, where r is the radius of the semi-circle. Note that for any6= 0, this model’s vortex strength distri- bution agrees with magnet size distribution in Figure 4.32(c) when! < 800 r.p.m.. The second 61 geometry type is presented in 4.32(b). It consists of a central vortex and six vortices distributed evenly about a full circle around the central vortex, resulting in a hexagonal structure. The SVD solution to this model returns a solution with a null space of two with Equation (3.68), being = 1 (1,0,0,0,0,0) T + 2 (0,1,1,1,1,1) T , where 1 and 2 are independent and unrelated arbi- trary constants. This model agrees with the different-sized magnetic system of Figure 4.32(c) (! > 800 r.p.m) for the selection 1 6= 2 , as well as the uniform-sized magnet system illus- trated in 4.32(d) for the selection 1 = 2 . In Figure 4.33, we have a full numerical solution to a configuration with two radially con- centric hexagonal ringscontaining 19 vortices in total (1 central vortex, 6 vortices in the inner hexagonal ring, and 12 in the outer hexagonal ring). The SVD method’s solution found that the null space of this configuration is 2. In Equation (4.101) below, horizontal lines in the vector dis- tinguish the central vortex (top-most entry), the inner-hexagon vortices (next set of six entries), and the outer-hexagon vortices (remaining twelve entries). The solution is given by 62 = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 2 . . . 7 8 9 10 11 12 13 14 15 16 17 18 19 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A = 1 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ +7=6 1=6 . . . 1=6 1 0 1 0 1 0 1 0 1 0 1 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A + 2 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 5=9 3=7 . . . 3=7 0 1 0 1 0 1 0 1 0 1 0 1 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A : (4.101) The corner vortices of the outer ring share a common strength of 1 ; the mid-edge vortices of the outer ring share a different common strength of 2 ; the vortices of the inner ring all share a single strength which is a linear combination of the strength of the outer ring’s corner and mid- edge vortices ((1=6) 1 + (3=7) 2 ). Finally, the strength of the central vortex is also a linear combination of the strength of the outer ring’s corner and mid-edge vortices ((7=6) 1 +(5=9) 2 ). The micro-scale image in the figure is of a superfluid vortex crystal that possesses the same type of radial hexagonal structure. 63 Figure 4.34: Left: minimum eigenvalues of 10 6 random configurations,N = 10 (None are in equilibrium). Right: square root of sum of the squares of two smallest eigenvalues of 10 6 random configurations,N = 10 (None have a null-space of two). 4.12 Asymmetric Equilibria The study of asymmetric equilibria corresponds to the first main contribution of this thesis, which will be fleshed out in the following subsections. 4.12.1 Existence In the former section, we only made comparisons to symmetric lattices. As was discussed in Section 1.3.2 of the introduction (see Figures 1.6 and 1.7), vortex lattices with broken symmetries also exist in nature, see Figure 4.35 from [1]. In the investigation asymmetric equilibria, we began by depositing a finite number of point vortices at random from a uniform distribution in the compact planar region x2 (1; 1);y2 (1; 1). Next, the configuration matrix from Equation (2.49) was derived, and the minimum eigenvalue of the covariance square matrixA T A satisfying (3.64) was found. If the smallest eigenvalue is a non-zero number, then the matrix A has full rank and (2.50) is not satisfied. This implies that the configuration has only a trivial equilibrium solution (i.e., = 0). 64 Figure 4.35: Broken symmetries in triangular lattices of Bose Einstein Condensates. Left: Line defect. Right: Grain boundary defect (from [1]). A million random configurations containing N = 10 vortices were generated, and their covariance matrices’ minimum eigenvalues were computed. Figure 4.34 shows that the min- imum eigenvalue associated with all 10 6 random configuration is always non-zero, indicating that an asymmetric configuration with a non-trivial equilibrium cannot be chosen at random, and these configurations form a set of measure zero in the space of all possible configurations. Also from Figure 4.34, we conclude that a configuration with a null space of two is even less likely to be chosen at random. In the next section, an algorithm that utilizes the minimum singular value as a scaling factor to drive the random asymmetric configuration to a converged non-trivial equilibrium is introduced. 4.12.2 The Brownian Ratchet Recall from Figures 1.6 and 1.7 from Section 1.3.2, that when a regular triangular lattice was blasted by a laser, the vortices exhibited an apparently random motion that resembled a thermal cooling process which eventually homed in on a new irregular / asymmetric equilibrium con- figuration ([26]). (Another good example with lattices with broken symmetries is illustrated in Figure 4.35). The Brownian ratchet method starts off with a random configuration ofN vortices. Using the SVD method, the ratchet process has the following algorithm: 65 Figure 4.36: Schematic of the Brownian ratchet method. The vortices take a random walk, each within a circle scaled with the smallest singular value, (0) associated with its configuration matrix until the smallest singular value gets smaller , (1) . The vortices continue to take random walks in circles of ever decreasing radius until the smallest singular value drops below a prescribed threshold. Figure 4.37: (a): Smallest singular value squared vs. Brownian ratchet step number on lin-log scale for N = 6; 8; 10. The smallest singular value converges continuously to zero. (b): Close-up view of a vortex converging to an equilibrium point, scale is magnified10 6 the scale of configuration. 66 Figure 4.38: Examples of converged asymmetric equilibrium configurations with one-dimensional null spaces using Brownian ratchet method. Total of six plots for 6, 8, 10 vortices, with one initial configuration (unfilled circles) and two corresponding final converged states (filled circles) with smallest singular value on the order of 10 6 . (a)N = 6, = (0.0040, -0.0033, -0.0254, -0.0079, -0.9996, -0.0001) T ; (b)N = 6, = (-0.2394, -0.1049, 0.1078, 0.0314, 0.7461, -0.6020) T ; (c)N = 8, = (0.0064, -0.0221, -0.0743, 0.0456, 0.1197, -0.1380, 0.0751, -0.9761) T ; (d)N = 8, = (0.0040, -0.0033, -0.0254, -0.0079, -0.9996, -0.0001) T ; (e)N = 10, = (-0.0364, -0.0666, -0.0253, -0.0335, -0.1392, -0.5016, -0.0716, 0.0163, -0.8462, 0) T ; and (f) N = 10, = (0.1561, 0.0048,0.0970, -0.0078, 0.1108, 0.2844, -0.0047, -0.0410, 0.9334, 0.0004) T . 67 Figure 4.39: Reverse Brownian ratchet scheme used to derive initial random configuration not in equilibrium from symmetric equilibrium configurations. (a)N = 6 triangular configuration with null-space 1 in open circles, with random non-equilibrium configuration that converges to it. (b)N = 7 hexagonal equilibrium configuration with null space 2 in open circles, with random non-equilibrium configuration that converges to it. 1. We identify the smallest singular value (0) min associated withA. 2. The vortices are then randomly fluctuatedeach within a circle that is scaled with (0) min . During this fluctuation, the singular values of the ‘fluctuated’ configuration also fluctuate, and we continuously monitor the smallest singular value associated with the fluctuated configuration, 0 min . 3. As soon as 0 min becomes smaller than (0) min , we consider that we have converged one step closer to an equilibrium, and the fluctuation is paused. The current configuration is saved as the new one, and the smallest singular value now becomes (1) min = 0 min . 4. The random fluctuation recommences, and is repeated for each of the vortices, but this time within a smaller circle scaled with the new smallest singular value (1) min . This process is repeatedK times until (K) min drops below a prescribed threshold thresh 10 6 . Figure 4.36 provides a schematic of the Brownian ratchet’s convergence process. In Figure 4.37, we track the square of the smallest singular value of A as the Brownian ratchet step number increases, and we show that it converges continuously to zero. Also in that figure, a close-up 68 image that is magnified to a millionth the area of the overall configuration shows the convergence process of a single vortex with individual ratchet steps. Figure 4.38 illustrates configurations of vortices that converged to an asymmetric equilibrium with a null space of 1. The important results here are that 1. Asymmetric equilibria do exist. 2. An initial configuration not in equilibrium will converge to completely different asym- metric equilibrium configurations each time the ratchet is repeated (i.e., there are many neighboring equilibrium configurations, possibly lying arbitrarily close to one another). 3. It is quite curious that configurations always numerically converge to purely asymmetric patterns as did the Bose-Einstein condensates upon being blasted by a laser. Hence, the Brownian ratchet scheme appears to have a convergence bias towards asymmetric, rather than symmetric configurations. Figure 4.39 uses a reverse Brownian ratchet process to show that a configuration can in theory converge to a symmetric configuration, and can also converge to a configuration with a null space greater than one. 4.12.3 Distance to Equilibrium The next natural question to ask is: how close is the nearest equilibrium to a random configura- tion? In other words, the question is whether the set of asymmetric equilibrium configurations is dense in the broader set of asymmetric configurations. The comparison is made to rational and irrational numbers on the real line. Both rational and irrational numbers are dense in the sense that a rational number exists between any two irrational ones, and an irrational number exists between any two rational ones. That is, a rational number is arbitrarily close to an irra- tional one. The difference between the two is that, if a number is picked at random from the real number line, it will always be an irrational number. Therefore, irrational numbers form a full- measure set, while the rational numbers have measure zero on the real line. The question now 69 Table 4.1: Converged Brownian ratchet configurations of Figure 4.38 with (i) the Frobenius size of the converged equilibrium configuration, (ii) distance to exact equilibrium, (iii) Hamiltonian energy and (iv) the distance between the two converged configurations that had the same initial conditions. N jjA final jj F jjA init A final jj F H jjA 1 A 2 jj F N = 6(a) 53.6412 3:15 10 05 0:0047 59.0357 N = 6(b) 23.4493 6:14 10 06 0:0816 59.0357 N = 8(a) 22.0578 1:00 10 05 0:0104 55.5116 N = 8(b) 51.4244 9:99 10 06 0:05100 55.5116 N = 10(a) 22.0578 9:96 10 06 0:0166 51.0580 N = 10(b) 44.6184 9:99 10 06 0:0157 51.0580 is whether randomly deposited vortices (that are always not in equilibrium) can be compared to the irrational numbers, while the converged asymmetric configurations (with null space one) can be compared to rational numbers. The answer to this question is no. The ‘minimum distance’ between a rectangular matrixA 0 with full rank to a rank-1 deficient matrixA 1 (i.e., the smallest singular value is zero) is given via the Frobenius norm in (3.94) as equal to the smallest singular value(N): SinceA 0 is not rank deficient, this distance is not arbitrarily small. The Frobenius norm tells us that an asymmetric equilibrium configuration does not lie arbitrarily close to a ran- dom configuration, but rather has a minimum and finite distance to it, namely min (N). In other words, the rank-1 deficient matrixA 1 is also the closest equilibrium to full rank matrixA 0 . Tables 4.1 and 4.2 contain a set of numerical information associated with the converged equilibria of Figure 4.38. The matrixA init corresponds to the converged configuration,A final corresponds to the equilibrium approximation, andjjA init A final jj F is the converged con- figuration’s closest distance to a configuration whose minimum singular value is exactly zero. Remark 2. Consider two configuration matrices A and B such that their dimensions are the same. IfkAk F =kBk F , we conclude thatA andB correspond to configurations of the same size, but in general the configurations can foreseeably have different locations, or even different geometries, whereas ifkABk F = 0, we conclude that the two matrices have the same size 70 Table 4.2: Singular values of converged Brownian ratchet configuration of Figure 4.38. N ’s N = 6(a) (36.8899, 34.0091, 16.8357, 8.0756, 3.3529, 3:15 10 05 ) N = 6(b) (16.4495, 15.1967, 6.6437, 2.0431, 0.1741, 6:42 10 06 ) N = 8(a) (11.8714, 11.344, 9.347, 8.4775, 5.43, 4.5957, 2.6626, 1:00 10 05 ) N = 8(b) (34.6657, 34.5454, 11.1811, 9.6584, 4.1464, 2.9317, 2.2993, 9:99 10 06 ) N = 10(a) (13.2665, 12.1482, 10.7102, 8.7633, 7.4807, 5.1647, 4.7517, 2.4412, 0.8549, 9:96 10 06 ) N = 10(b) (28.3155, 27.8314, 12.5556, 10.5146, 7.3477, 7.1772, 5.1661, 3.6358, 0.9141, 9:99 10 06 ) and, if each configuration matrix corresponds to a single unique configuration, that the vortices ofA andB have the same relative locations as well, with respect to rotations and translations. 4.13 Shannon Entropy We now move our focus to the second main contribution of this thesis: Shannon Entropy. We claim that this measure can predict the likelihood of formation for a given lattice geometry in the physical systems referenced in the introduction (plasma lattices, magnetic lattices, helium lattices). We begin the analysis by comparing two configurations that have the same number of vortices, but different inter-vortical distances. For four-vortex equilibra, we compare the square configuration in Section 3.8.3.1 to an equi- lateral triangle lattice with a central vortex. The square configuration is more prevalent than the equilateral triangle configuration (Figures 1.2 and 1.10). As it turns out, the Shannon Entropy from Equation (3.82) of the square configuration is H 1:0397, while that of the triangle is H 0:6931. Therefore, the square configuration has a higher entropy, and arguably, a higher likelihood to exist. Note also that from Equations (2.40) and (3.68) that the square configuration has a lower Hamiltonian energy. For five-vortex equilbria, consider the pentagon configuration (5 vortices evenly distributed on a circle) and the Havelock’s square with a central vortex (see 71 (a) Normalized eigenvalues ofA T A (b) Shannon Entropy Figure 4.40: Averaged 250 times + standard deviation. Comparison of Shannon Entropy associated with: ran- dom asymmetric configurations (not in equilibrium), random matrices (all entries are random numbers), converged asymmetric equilibria with null space 1, and the triangle lattice from Figure 4.31. Triangle lattice has significantly higher entropy than asymmetric equilbria, with entropy of asymmetric configurations not in equilibrium and in equi- librium being indistinguishable from one another within standard deviation. Random matrices have substantially lower entropy than any configuration matrix. the example in Section 3.8.3.2). The pentagon configuration is more prevalent in the referenced physical systems than the square lattice with a central vortex (Figures 1.2, 1.10 and 1.12). The Shannon Entropy of the pentagon configuration isH 1:3662, while that of the square configu- ration isH 0:3890. Again, the configuration that is more prevalent in physical systems has the higher entropy. Comparing the Hamiltonian of these two configurations is not as clear-cut, since the dimension of the null space for the square lattice with a central vortex is two. This makes the Shannon Entropy analysis more definitive in this case since it relies solely on the system’s geometry, and not its null space (i.e., relative vortex strengths). An interesting observation is that the square lattice with a central vortex appears to be the most likely geometry type on the curved surface in Figure 1.11. This tells us that the entropy of a given configuration may change with its surface’s curvature. In Figure 4.40, four sets of matrix modal probabilities (i.e., normalized squared eigenval- ues ofA T A) corresponding to different geometry types are computed and compared. Namely 72 these four include: averaged probabilities of 250 random configurations (not in equilibrium), averaged probabilities from random matrices (all elements chosen from uniform distribution), probabilities of 250 converged asymmetric equilibria (via the Brownian ratchet), and the prob- abilities of the triangular lattice from Figure 4.31. The triangular lattice comparatively had the highest entropy of all. This supports the claim that higher entropy equates to higher likelihood for existence, since triangular geometries are the most common configuration types found in many different lattice types (specifically in Figure 1.12). It is important to note that asymmet- ric configurations both in equilibrium and not in equilibrium had indistinguishable entropies. This could be due to the fact that the Brownian ratchet searches for equilibria in the vicinity of the initial configuration, and can in theory, home in on a configuration within the same entropy range. In the context of Figure 1.6, in which a regular triangle lattice converged to an asym- metric equilibrium after the blast, it is foreseeable from the distribution in Figure 4.40, that the Shannon Entropy is quantized. In other words, after the laser blast the entropy of the system changed. The Brownian ratchet’s search can be broadened or condensed (i.e., the vortices can be controlled to fluctuate within circles of variable radius), and this potentially could have an effect on the standard deviation of the entropy when averaged over many runs. We will revisit Shannon Entropy in Chapters 5 and 6 in the context of vortex streets on the sphere. 4.14 The Reverse Problem With the closing of this chapter, we pose the question: can a single configuration matrix cor- respond to more than one configuration? Moreover, looking at the problem in reverse: can a configuration be derived from a configuration matrix? And if so, can a single configuration matrix correspond to more than one configuration? A scenario in which the answer to the latter is yes was found. For the caseN = 3, the associated configuration matrix has three columns and 3(2)=2 = 3 rows. Consider a zero matrixA 0 with the same dimensions as a system with N = 3 vortices (i.e.,A 0 2R 33 ). This implies that either (a) all the intervortical distances are equal, or that (b) the intervortical area is zero. The first corresponds to an equilateral triangle, 73 and the second corresponds to three collinear vortices. Therefore, it was possible to identify two configurations that correspond to a single configuration matrix. This can be thought of as a degenerate scenario, so in the next subsections, the combinatorial structures of the configura- tion and configuration matrices will be studied and compared in order to provide a more general answer for non-degenerate scenarios. 4.14.1 Configuration Construction With and Without a Coordinate System In the presence of a coordinate system, a minimum of 2N values are required to fully character- ize a system ofN points in the plane (i.e., (x ;y ) or (r ; ), where =f1;:::;Ng). In the absence of a coordinate system, one way to construct a configuration is with the use of relative distances l ( =f1;:::;N 1g and =f + 1;:::;Ng) and relative orientations ( =f1;:::;N 2g, =f + 1;:::;N 1g and =f + 1;:::;Ng) . Let’s begin by choosing the relative distances between point 1 and points =f2;:::;Ng. These distances can be chosen arbritrarily such thatl 1 > 0, and this gives us a total ofN 1 independent values (Figure 4.41(a)). Next, we consider the distancesl 2 between point 2 and the remainingN 2 points defined as =f3;:::;Ng. Sincel 12 andl 1 have already been defined, the distancel 2 is constrained byjl 12 l 1 j < l 2 <jl 12 +l 1 j, see Figure 4.41(b) for an illustration of this constraint. This gives us an additional N 2 values. Finally, the relative orientation between vortices 1, 2 and ( =f3;:::;Ng) can be chosen to be 12 = 1; this gives us an additionalN 2 values. Note that a triangle with lengthsl 12 ;l 13 ;l 23 can either have relative orientation 123 = +1 or 123 =1. The two possible orientations give us two geometrically different triangles with respect to rigid rotationssee Figure 4.41(c) for a schematic of this. The remaining relatives distances l with =f3;:::N 1g and =f + 1;:::Ng, and orientations ( ; =f1;:::;N 2g; =f3;:::;N 1g (with > ), and = f + 1;:::;Ng) between the points are fully determined. The area between any three points in the configurations (A ) can then be solved using the relative distances and orientations. 74 Table 4.3: Comparison of the minimum number of data points required to construct a configuration ofN points in the plane in: (a) the presence of a coordinate system, and (b) in the absence of a coordinate system. Note that for N < 5, less values are required to construct the system in the absence of a coordinate system, forN = 5 the number of values is the same, and for N > 5, less values are required in the presence of a coordinate. The last column demonstrates the number of non-trivial entries in configuration matrixA associated with differentN. Number of With Without Non-trival points coordinate system coordinate system entries inA N 2N 3N 5 N(N 1)(N 2)=2 N = 2 2 1 0 N = 3 6 4 2 N = 4 8 7 8 N = 5 10 10 20 N = 6 12 13 40 N = 7 14 16 70 N = 8 16 19 168 Therefore, in the absence of a coordinate system, a total of 3N 5 values are required to fully characterize the system, see Table 4.3. 4.14.2 Number of Elements in the Configuration Matrix For a general configuration, A has N 2 non-trivial entries per row (recall that each row is associated with the rate of change of the relative distance between two unique vortices = f1;:::;N 1g and =f + 1;:::;Ng, and the non-trivial entries correspond to the entries where 6= ;6= ). With N 2 non-trivial entries per row and a total of N(N 1)=2 rows, in general we will have a total of N(N 1)(N 2)=2 non-trivial matrix entries. The non-trivial entries of A are a function of the configuration’s intervortical lengths, orientations and areas. As described in the previous section, there are 3N 5 unique independent values that describe the coordinate-free configuration. The reverse problem remains an open question since the analysis in this section is not fully conclusive. However, using the information presented, we can conclude whether the configuration matrix is over-determined or under-determined for different values ofN (Table 4.3). 75 (a) Relative Distancesl12;:::;l 1 (b) Relative Distancel 2 (min)<l 2 <l 2 (max) (c) Relative Orientation 12 =1 (d) Identification of Remaining Points Figure 4.41: Configuration Construction ofN points in the plane in the absence of a coordinate system. (a)N 1 relative distances from point 1 can all be chosen independently fromR + , (b)N 2 relative distances from point 2 to =f3;:::;Ng can be chosen from the constricted rangejl12l 1 j < l 2 <jl12 +l 1 j, (c)N 2 Relative orientations between points 1, 2 and =f3;:::;Ng can be chosen to be 12 =1. This subfigure highlights the fact that the two possible choices for 12 result in two geometrically unique configurations (i.e., one configuration cannot be obtained from the other via rigid body rotations and translations), with vortex either having the relative position (1) or (2) , (d) The information corresponding to the remaining points, labeled =f + 1;:::;Ng, can be solved analytically by identifying their locations (in Regions I - VII) relative to three preceding fully-solved points f1;;g, such that<, < andN. 76 Chapter 5. Single Von K´ arm´ an Streets on the Sphere In this chapter, we move our focus to single vortex streets on the sphere. From the four main contributions of this thesis (outlined in Section 1.2 of the introduction), three are present in this chapter. First, this chapter includes a derivation of the analytic solution to the single street using the center of vorticity and angular velocity equations. Second, we will flesh out the streamline topology and bifurcations for this class of configurations, and third, we will apply the Shannon Entropy formulation. The analysis is repeated for configurations with and without vortices at the poles. 5.15 Construction and SVD Solution 5.15.1 Without Pole Vortices In this subsection, we investigate the existence of single von K´ arm´ an streets (VKS) consisting of two symmetrically skewedn-vortex rings without pole vortices. That is, we write (2.60) for a total ofN = 2n vortices on the sphere as illustrated in Figure 5.42, and seek configurations satisfying (2.50). Admissible solutions are those that also satisfy the VKS requirement, that is, two rings of vortices with equal and opposite strength that rotate rigidly about a fixed center of vorticity axis. 77 J Γ Γ Γ Γ −Γ −Γ −Γ −Γ Γ φ Figure 5.42: Schematic of a single von K´ arm´ an street on the sphere with an illustration of the vorticity vector J and the angle. 5.15.1.1 Number of Vortices Per Ring:n6= 3 We first consider configurations containingn = 2; 4; 5; 6; 7;::: (excluding 3) vortices per ring. As an example, the configuration matrix for the casen = 2, using Equation (2.49) becomes A 2 = 0 B B B B B B B B B B B B B B @ a 2 0 a 2 0 a 2 0 0 a 2 0 a 2 a 2 0 0 a 2 0 a 2 0 0 0 0 0 0 0 0 1 C C C C C C C C C C C C C C A ; (5.102) wherea 2 =a 2 (), and is the co-latitude of the ring in the Northern hemisphere (Figure 5.42). In our notation, the subscript associated with a configuration matrix and its entries’ names cor- responds ton. The entrya 2 is given by a 2 = cos 3 cos 2 1 2(cos 2 + 1) : (5.103) 78 Note that a 2 is the only unique non-trivial entry in A 2 associated with the case n = 2 (In general, there are multiple distinct non-trivial entries whenn 4 in the configuration matrix. For example, in the configuration matrix associated with n = 4, one finds 9 distinct entries which would be labeleda 4 ;b 4 ;c 4 ;:::;i 4 .). From (3.62), the singular values of (5.102) are 2 = 2 p 2a 2 ; 2 p 2a 2 ; 2a 2 ; 0 T : (5.104) Since one of the singular values is zero, the null space of this configuration is one (i.e., nullity(A 2 ) = 1). From (3.68) the solution to this configuration has the general form = 1; 1; 1; 1 T : (5.105) This solution can be generalized, and the general form of for anyn6= 3 becomes = 1; :::; 1; 1; :::; 1 T ; (5.106) where has length 2n, with the first and second set ofn vortices corresponding to upper (north- ern) and lower (southern) vortex rings respectively, see Figure 5.42. Therefore, the general solution to the SVD problem is that the two rings have equal and opposite strengths. For these configurations, the center of vorticity given by Equation (2.53), is aligned with the z-axis, J = 2n 0 B B B B @ 0 0 cos 1 C C C C A : (5.107) 79 Γ 1 Γ 2 Γ 3 −Γ 1 −Γ 2 −Γ 3 −Γ 2 Γ 1 −Γ 1 Γ 3 −Γ 3 Γ 2 Figure 5.43: Top and oblique view. Solution using SVD of single von K´ arm´ an street on the sphere forn = 3. Dimension of null space is three with same-shape vortices having equal and opposite strength. 5.15.1.2 Number of Vortices Per Ring:n = 3 The case of 3 vortices per ring (n = 3) is a special degenerate case worthy of separate treatment. For this case, the configuration matrix from Equation (2.49) takes on the form A 3 = 0 B B B B B B B B B B B B B B B B B B B B B B B B @ a 3 0 0 0 a 3 0 a 3 0 0 0 a 3 0 0 a 3 0 0 0 a 3 0 a 3 0 0 0 a 3 0 0 a 3 a 3 0 0 0 0 a 3 a 3 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C C C C C A ; (5.108) 80 where the lower 9 rows are all zeros rows, and,a 3 is given by a 3 = 3 p 3 2 sin 2 cos( 1 sin 2 + 4 cos 2 1 3 sin 2 ): (5.109) From (3.62), the singular values ofA 3 are 3 = 2a 3 ; 2a 3 ; 2a 3 ; 0; 0; 0 T : (5.110) Clearly, the nullspace of the configuration matrix has dimension 3 and the corresponding vector from Equation (3.68) has the form ~ = 1 0 B B B B B B B B B B B B B B @ 1 0 0 1 0 0 1 C C C C C C C C C C C C C C A + 2 0 B B B B B B B B B B B B B B @ 0 1 0 0 0 1 1 C C C C C C C C C C C C C C A + 3 0 B B B B B B B B B B B B B B @ 0 0 1 1 0 0 1 C C C C C C C C C C C C C C A ; (5.111) where 1 , 2 , and 3 are three independent constants. One vortex in the northern ring is equal and opposite to the vortex in the southern ring that subtends it longitudinally by an angle equal to (see Figure 5.43 for an illustration). With (2.53), the center of vorticity of the system satisfies J = 0 B B B B @ sin 0 0 0 sin 0 0 0 cos 1 C C C C A 0 B B B B @ 1 0 B B B B @ 2 0 2 1 C C C C A + 2 0 B B B B @ 1 p 3 2 1 C C C C A + 3 0 B B B B @ 1 p 3 2 1 C C C C A 1 C C C C A ; (5.112) which, in general, does not align with the z-axis, hence does not correspond to a von K´ arm´ an street (i.e., the street was constructed about the polar axis, and we require the center of vorticity to be aligned with the polar axis). One obtains a von K´ arm´ an street only when 1 = 2 = 3 (= , say) in which case = (1; 1; 1;1;1;1) T and the center of vorticity aligns with the 81 Figure 5.44: Computational result of time evolution of 6 vortices on the sphere with = 5=12. The initial configuration is the same as that defined in the previous figure. Images of the first column represent degenerate case with single VKS VKS and vortex strengths [1; 2; 3] = [1; 1; 1]. Second column has strengths [1; 2; 3] = [1; 1:5; 1] with no resultant VKS, third case is [1; 2; 3] = [1; 1:5; 1:2], also with no resultant VKS. z-axis. In this case, when integrating (2.51), the vortices evolve as a single VKS (maintaining constant latitudes), see Figure 5.44. Whereas for 1 = 3 6= 2 , the vortices no longer evolve as a von K´ arm´ an Street (do not maintain constant latitude) and move on three orbits about a common axis that is not aligned with the z-axis. When 1 6= 2 6= 3 , each vortex rotates about its own orbit (total of six). 5.15.2 With Pole Vortices Pole vortices exist in geophysical flows, such as on the surface of Jupiter, and have been shown to play a critical role in the stability of co-latitudinal vortex rings on the sphere ([13], [10] and [44]). 5.15.2.1 Example:n = 2 As an example, for a configuation with two vortices per ring (n = 2), there will be a total of N = 2(2) + 2 = 6 vortices in the system. The configuration matrix in (2.60) has dimension 82 R (65=2)6 =R 156 . We label this configuration matrix ^ A 2 (i.e., In our notation, the overhead ^ symbol indicates the addition of pole vortices) and it has the form ^ A 2 = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ ^ d 2 ^ d 2 0 0 0 0 ^ c 2 ^ c 2 0 0 0 0 ^ d 2 ^ d 2 0 0 0 0 ^ c 2 ^ c 2 0 0 0 0 ^ a 2 0 ^ a 2 0 ^ b 2 ^ b 2 ^ a 2 0 0 ^ a 2 ^ b 2 ^ b 2 0 ^ a 2 ^ a 2 0 ^ b 2 ^ b 2 0 ^ a 2 0 ^ a 2 ^ b 2 ^ b 2 0 0 ^ c 2 ^ c 2 0 0 0 0 ^ d 2 ^ d 2 0 0 0 0 ^ c 2 ^ c 2 0 0 0 0 ^ d 2 ^ a 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A ; (5.113) with ^ a 2 = cos(3 cos 2 1) 2(cos 2 + 1) ; (5.114) ^ b 2 = cos; (5.115) ^ c 2 = cos(cos 2 + 2 cos 1) 2(cos 2 + 1) ; (5.116) 83 and ^ d 2 = cos(cos 2 + 2 cos + 1) 2(cos 2 + 1) : (5.117) In the state representation of ^ A 2 , the first (second) two elements correspond to the northern (southern) street’s vortices, and the last two elements correspond to the north and south pole vortices respectively. From (3.62), the singular values of this configuration are ^ 2 = 0 B B B B B B B B B B B B B B @ 2 q ^ a 2 2 + 2 ^ b 2 2 q 2^ a 2 2 + 4^ c 2 2 + 4 ^ d 2 2 q 2^ a 2 2 + 4^ c 2 2 + 4 ^ d 2 2 0 0 0 1 C C C C C C C C C C C C C C A : (5.118) Since three of the singular values are zero, the dimension of the null space is three (i.e., nullity( ^ A 2 ) = 3). The solution to is at best numeric for this class of configurations when using SVD. In the next subsection, we will highlight the general numeric solution for single vortex streets with pole vortices for anyn. 5.15.2.2 General Numeric Solution For a general single vortex withn vortices per ring, the vector has lengthN = 2(n + 1). The first and second set ofn values correspond to the upper and lower vortex rings respectively; the (2n + 1) and (2n + 2) components correspond to the values of the north and south pole vortices. 84 Using the SVD method, the general solution forn 2 has a null space of 3 and is of the general form = 0 B B B B B B B B B B B B B B B B B B B B @ 1 . . . 1 1 . . . 1 0 0 1 C C C C C C C C C C C C C C C C C C C C A + north 0 B B B B B B B B B B B B B B B B B B B B @ f(;n) . . . f(;n) 0 . . . 0 1 0 1 C C C C C C C C C C C C C C C C C C C C A + south 0 B B B B B B B B B B B B B B B B B B B B @ f(;n) . . . f(;n) 0 . . . 0 0 1 1 C C C C C C C C C C C C C C C C C C C C A : (5.119) The solution that interests us is when the poles have equal and opposite strength (i.e., north = south ) since it is the only case that will produce a single von K´ arm´ an street with equal and opposite rings. For this selection of pole vortices, thef(;n) terms cancel out, the vortex street will have equal and opposite rings with strength , and the two poles will have an independent equal and opposite strength p , given by the following: = 0 B B B B B B B B B B B B B B B B B B B B @ 1 . . . 1 1 . . . 1 0 0 1 C C C C C C C C C C C C C C C C C C C C A + p 0 B B B B B B B B B B B B B B B B B B B B @ 0 . . . . . . . . . . . . 0 1 1 1 C C C C C C C C C C C C C C C C C C C C A : (5.120) 85 0 0.5 1 1.5 1 3 5 7 ω n=2 n=3 n=4 n=5 n=6 φ (a) V ortex Street Component 0 0.5 1 1.5 0 2 4 6 8 10 φ Ω (b) Pole V ortices Component Figure 5.45: (a) Angular velocity! as a function of vortex street latitude forn = 2 6. ! decreases from +1 to 0 as is increased from 0 to=2. (b) Additional angular velocity for anyn from pole vortices (p) with p = +1. As approaches=2, asymptotes at p=(2). Furthermore, from (2.53), the center of vorticity for the single street with poles becomes J = 2 0 B B B B @ 0 0 n cos + p 1 C C C C A ; (5.121) and is parallel to the polar axis. With SVD, the solution tof(;n) is numeric. In the following section, we will build on SVD, and provide analytic solutions to the angular velocity of the street and tof(;n). 5.16 Angular Velocity and Explicit Solutions From Equations (5.107) and (5.121), the longitude of vortex , given by in Section 2.7.2, rotates rigidly about the polar axis. The angular velocity, therefore, is simply _ for any vortex such that its latitude 6= 0. We will solve for the angular velocity of single single streets, and show that this formulation can provide an analytic solution to Equation (5.119), specifically with an explicit expression forf(;n) in that equation. 86 5.16.1 Without Pole Vortices For the single vortex street without poles, we require that _ 1 = _ 2 =::: = _ 2n for relative equi- librium. The two rings are symmetrically skewed (by an angle of=n). Therefore, the longitudes of then vortices in the northern ring are: 0; 2(1)=n; 2(2)=n; ; 2(n 1)=n. The longi- tudes of then vortices in the southern ring are=n; 2(1)=n+=n; 2(2)=n+=n; ; 2(n 1)=n +=n. The longitudinal spacing between any two vortices and in the same ring (i.e., 0<f;gn orn<f;g 2n) is given by = 2 n (): (5.122) If the two vortices are in different rings, with 0 < n andn < 2n, the longitudinal spacing is given by = 2 n ( (n)) n 2 n () n (5.123) As for the latitudes, the firstn vortices have a latitude of, while the secondn vortices have a latitude of (). Note that sin() = sin() and cos() = cos(). As for the vortex strengths, the rings must have equal and opposite strengths from (5.106) for anyn6= 3 1 . Evaluating 1 in Eq. (2.56) gives cos 1 = cos 2 + sin 2 cos 2 n () ; 1<n; (5.124) cos 1 = cos 2 + sin 2 cos 2 n () n ; n< 2n: (5.125) 1 We do not solve the angular velocity for configurations withn = 3 when the vortex strengths of the streets’ rings do not have equal and opposite strengths, since this system does not evolve as a vortex street (i.e., the polar axis and center of vorticity axis are not aligned). In order to solve for the angular velocity, one must derive the angular coordinates of the system with respect to the center of vorticity axis. For the treatment of a similar problem, see [57]. 87 Substituting these properties into Eq. (2.55) we get ! _ 1 = 4 sin n X =2 sin cos cos sin cos ( 2 n (1)) 1 cos 1 (5.126) + 4 sin 2n X =n+1 sin cos cos sin cos ( 2 n (1) n ) 1 cos 1 (5.127) = cos 4 n X =2 1 cos ( 2 n (1)) 1 + cos 2 sin 2 cos ( 2 n (1)) (5.128) + cos 4 2n X =n+1 1 cos ( 2 n (1) n ) 1 cos 2 sin 2 cos ( 2 n (1) n ) : (5.129) For example, if we haven = 4 vortices per ring, the angular velocity of the system becomes ! = cos 4 cos 4 10 cos 2 15 cos 6 + 5 cos 4 5 cos 2 1 : (5.130) 5.16.2 With Pole Vortices From Eq. (2.55), the overall change to the rotational velocity of the single vortex street system from the addition of vortices at the poles with equal and opposite strengths p 2R is denoted as (see Figure 5.45(b)), and given by = p 4(1 cos) + p 4(1 + cos) (5.131) = p 2 sin 2 : (5.132) Furthermore, we can derive an analytic solution tof(;n) in Equation (5.119) by supplementing the SVD solution with a constraint on Equation (2.55). We contrast the angular velocity of the north and south rings with the addition of north and south pole vortices. Using the solution in (5.119), we will approach the problem by considering a configuration that has a north pole vortex 88 and no south pole vortex. That is, the selected vortex strengths are (; north ; south ) = (1; 1; 0), which gives us vortex strengths = 0 B B B B B B B B B B B B B B B B B B B B @ 1 +f . . . 1 +f 1 . . . 1 f 0 1 C C C C C C C C C C C C C C C C C C C C A : (5.133) Plugging (5.133) into (2.55), we derive the angular velocity,! 1 , for the vortices in the northern ring to be ! 1 _ 1 = _ 2 =::: = _ n = 1 +f 4 sin n X =2 sin cos cos sin cos ( 1 ) 1 cos 1 + 1 4 sin 2n X =n+1 sin cos cos sin cos ( 1 ) 1 cos 1 + f 4(1 cos) ; or ! 1 = 1 (1 +f) + 2 + 3 f: (5.134) The angular velocity of the vortices in the southern ring,! 2 , is given by ! 2 _ n+1 = _ n+2 =::: = _ 2n = 1 +f 4 sin n X =1 sin cos + cos sin cos ( n+1 ) 1 cos (n+1) + 1 4 sin 2n X =n+2 sin cos + cos sin cos ( n+1 ) 1 cos (n+1) + f 4((n + 1) + cos) ; 89 or ! 2 = 4 (1 +f) + 5 + 6 f: (5.135) For the system to obtain relative equilibrium, we require that ! 1 ! 2 , which gives us the analytic expression forf: ff(;n) = ( 4 + 5 ) ( 1 + 2 ) ( 1 + 3 ) ( 4 + 6 ) : (5.136) Note that we have obtained this result by using only a north pole vortex in the formulation. The procedure can be repeated for the selection of vortex strengths in Equation (5.119) to be (; north ; south ) = (1; 0; 1). That is, the same result for f is derived when adding only a south pole vortex. The conclusion here is that the SVD formulation provides a simple means to solving the structure of the null space (which would be difficult to produce without SVD). The setback is that for large N, the SVD solution is at best numerical. When supplementing the SVD’s numerical result with a constraint on the angular velocity of the system, we can fully solve the system analytically. 5.17 Streamline Topology In this section, we present the fourth main contribution of this thesis. The analysis of the stream- line topology of vortex equilibria and the associated bifurcations is greatly facilitated by the methods presented in Chapters 2 and 3. We will revisit this topic again in the context of dou- ble vortex streets, and commence this section by presenting a basic tool in understanding the streamline patterns on the sphere in the index theorem of Poincar´ e: Theorem 1. (PIT): The index I f (S) of a two-dimensional surface S relative to any C 1 vec- tor field f on S with at most a finite number of critical points, is equal to the Euler-Poincar´ e characteristic of S, denoted(S), i.e.I f (S) =(S). We know for a sphere, (S) = 2. The index for a center is +1, while that for a saddle is 1. Hence ifc denotes the number of centers present (point vortices plus other centers), ands 90 (a) Type I (b) p = 0 (c) Type II (d) p = 1:2 (e) Type I (f) p = 7:5 (g) Type III Figure 5.46: Streamlines topologies for a single von K´ arm´ an street with 3 vortices per ring and pole vortices of equal and opposite strength. The pole strength increases from5 to +10. A total of three distinct topology types are identified within four different intervals, separated by three bifurcation topologies. denotes the number of saddles, then one hascs = 2. See [40] for more on applications of the PIT to the understanding of streamline patterns on the sphere. Consider the casen = 3, = 3=8 and pole vortices of equal and opposite strength p . In this case, the VKS vortices are equal and opposite, with the vortices in the northern ring having strength =1. We vary p and study the change in the streamline topology. The sequence of distinct streamline topologies when p is increased from5 to 10 is shown in Figure 5.46. For =5, one gets three stagnation points of hyperbolic or saddle type in each hemisphere. The streamlines associated with these hyperbolic points are plotted in Figure 5.46(a). These stream- lines are referred to as separatices because they separate regions with different fluid behavior. For example, in a region containing a point vortex and bounded by a separatix, the fluid orbits around the vortex, whereas in the middle region void of point vortices, the fluid is transported globally around the entire sphere. The streamline topology remains the same until p reaches the critical value p = 0. At this bifurcation value, the stagnation points collapse to two points of hyperbolic or saddle type at the poles as shown in Figure 5.46(b). As p becomes positive, three hyperbolic points reappear around each pole as shown in Figure 5.46(c). As p increases, the three hyperbolic points of the north and southern hemispheres approach each other until the 91 0 0.5 1 1.5 0.5 0.6 0.7 0.8 φ H W/out Poles With Poles (a) n=2 0 0.5 1 1.5 0.5 0.6 0.7 φ H (b) n=3 Figure 5.47: Shannon Entropy sweeps for single vortex street with (solid line) and without (dashed line) pole vortices. (a) Two vortices per ring, (b) Three vortices per ring. associated separatrices coincide at a critical pole strength p 1:2, see Figure 5.46(d). At this bifurcation value, the overall streamline topology changes and region characterized by global fluid transport disappears. As p increases beyond p = 1:2, the separatices detach and the streamline topology changes again with a region of global fluid transport reemerging, see Fig- ure 5.46(e). Here, the separatices of the stagnation points of the northern hemisphere encircle vortices from the southern hemisphere and vice-versa. The streamline topology remains the same until p reaches another critical value p 7:5 where the separatrices associated with the hyperbolic points of northern and southern hemispheres coincide and the region of global trans- port disappears again, see Figure 5.46(f). As p increases beyond p = 7:5, the separatrices split again forming three distinct regions characterized by global fluid transport and six regions characterized by fluid orbiting around a local vortex. 5.18 Shannon Entropy: Effect of Adding Pole Vortices Here we analyze the Shannon Entropy of single vortex streets upon adding pole vortices. Since the total number of vortices in the system changes by two when adding pole vortices, we will use Equation (3.83), which makes it possible to compare systems consisting of different number of vortices. 92 5.18.1 Number of Vortices Per Ring: n = 2 We begin our analysis for streets with two vortices per ring. Recall from (5.104) that the sin- gular values for this configuration are = (2 p 2a 2 ; 2 p 2a 2 ; 2a 2 ; 0) T , where a 2 is given by Equation (5.103). From (3.84), the probabilities (normalized eigenvalues of covariance matrix A T A) are ~ P = (2=5; 2=5; 1=5; 0) T , and are independent ofa 2 (). The Shannon Entropy of this configuration becomes H = 1 2 (log 4 2 + 2) =const: (5.137) Upon adding pole vortices to this system, the singular values are given by (5.118). From Equa- tion (3.84), the probabilities for this system become a non-linear function of. In Figure 5.47(a), we illustrate the change of Shannon Entropy upon introducing pole vortices to the vortices, and we see that for any latitude, the configuration without pole vortices has a higher entropy. 5.18.2 Number of Vortices Per Ring: n = 3 The dimension of the null space for a single vortex street with three vortices per ring, and no pole vortices is three (see section 5.15.1.2),That is, three of the six singular values are equal to zero. From (5.110), the singular values are (2a 3 ; 2a 3 ; 2a 3 ; 0; 0; 0) T , witha 3 () defined in (5.109). The probabilities for this configuration with Equation (3.84) become ~ P = (1=3; 1=3; 1=3; 0; 0; 0) T , and are independent of. The normalized Shannon Entropy from (3.83) forn = 3 becomes H =H = log 6 3 =const: (5.138) Upon adding pole vortices, the probabilities become a non-linear function of the angle. In Figure 5.47(b), we compare the normalized Shannon Entropy with and without poles. Interest- ingly, the two lines cross, which tells us that certain intervals of have a higher entropy pole vortices are present. 93 5.18.3 Number of Vortices Per Ring: n 4 In Figure 5.48, we contrast the normalized Shannon Entropy of configuration with differerentn, namely for the selectionsn = 4; 5; 7; 13; 20. We repeat the analysis for configurations without (5.48(a)) and with (5.48(b)) pole vortices. We observe that all curves are nonlinear functions of , and that there exist local maxima and minima on the curves. Values of which correspond to local maximizers of the Shannon Entropy are deemed to be more robust with respect to their singular value distributions, than those corresponding to local minimizers. Interestingly, the most likely number of vorticesn per ring changes with. That is in Figure (5.48), for any given , onen curve has the highest entropy. 94 0 0.5 1 1.5 .6 0.7 0.8 0.9 φ H n=4 n=5 n=7 n=13 n=20 (a) No Poles 0 0.5 1 1.5 0.65 0.7 0.75 0.8 0.85 0.9 φ H n=4 n=5 n=7 n=13 n=20 (b) With Poles Figure 5.48: Normalized Shannon Entropy sweeps for single von K´ arm´ an streets on the sphere for (a) n = 4; 5; 7; 13; 20 without poles, and (b)n = 4; 5; 7; 13; 20 with poles. The intention of these figures to compare the entropy of configurations with differentn. 95 Chapter 6. Double Von K´ arm´ an Streets on the Sphere We now expand on the previous chapter by adding a second vortex street to our model. As shown in Figure 6.49 from [37], two adjacent vortex streets on the surface Jupiter lie in close proximity to one another. In this chapter, we investigate the effects that the additional street has on the equilibrium solution, role of the pole vortices, streamline topology, and Shannon Entropy. Figure 6.49: Double vortex street on Jupiter. Cyclones (anti-cyclones) are labeled ‘C’ (‘A’). Velocity profiles on the left demonstrate identification of cyclones & anti-cyclones, where the first have a positive slope and the latter have a negative slope in the y vs. v domain. Image from [37]. 96 Γ Γ Γ Γ −γ 1 Γ −γ 1 Γ −γ 1 Γ γ 1 Γ γ 1 Γ γ 1 Γ Γ Γ Γ Γ φ 2 φ 1 Figure 6.50: Double V on K´ arm´ an Street schematic, illustrating SVD solution and ring definition. Dimension of the null space is 1. The angle associated with the two rings are1 and2 respectively. Note that which latitude is smaller is of no importance. Rather, we identify the different rings by the ordering in the state representation of Equation (6.139). 6.19 Construction and SVD Solution 6.19.1 Without Pole Vortices The double von K´ arm´ an street model used in this chapter consists of one vortex street in the northern hemisphere, and a second in the southern hemisphere, where each street consists of two symmetrically skewed n-vortex rings. One ring in each hemisphere has a latitude of 1 from its respective hemisphere’s polethese are referred to as the 1 -rings. The second ring in each hemisphere has an angle of 2 , and these are referred to as the 2 -rings. See Figure 6.50 for an illustration. We write (2.60) for a total ofN = 4n vortices on the sphere and seek configurations satisfying (2.50). Using the SVD method, it was found that the dimension of the null space for this system is one. The 2 rings have equal and opposite strengths, while the 1 -rings have equal and opposite strengths 1 , where 1 = 1 ( 1 ; 2 ;n)2 R. Note that the SVD method can at best provide a numerical result for 1 . As for the state representation, the firstn state variables corresponds to the northern 2 -ring’s vortices. The second and third set ofn state variables correspond to the northern and southern 1 -ring’s vortices respectively. Finally, the 97 lastn state variables correspond to the southern 2 -ring’s vortices. In other words, the solution is = (1;:::; 1; 1 ;:::; 1 ; 1 ;:::; 1 ;1;::: 1) T : (6.139) Furthermore, the general center of vorticity for this class of configurations from Equation (2.53) is given by J = 2 0 B B B B @ 0 0 n(cos 2 1 cos 1 ); 1 C C C C A ; (6.140) which implies that the center of vorticity is aligned with the polar axis for any choice of param- eters. 98 6.19.2 With Pole Vortices A double VKS with equal and opposite vortex rings (i.e., 1 = 1) can be achieved by adding vortices at the north and south poles. The dimension of the null space for the system solved using (2.49) is three. The additional dimensionality of the solution comes from the pole vortices that have independent strengths north and south , and the general form is = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 . . . 1 1 . . . 1 1 . . . 1 1 . . . 1 0 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A + north 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ f 1 ( 1 ; 2 ;n) . . . f 1 ( 1 ; 2 ;n) f 2 ( 1 ; 2 ;n) . . . f 2 ( 1 ; 2 ;n) f 3 ( 1 ; 2 ;n) . . . f 3 ( 1 ; 2 ;n) 0 . . . 0 1 0 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A + south 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ f 1 ( 1 ; 2 ;n) . . . f 1 ( 1 ; 2 ;n) g 2 ( 1 ; 2 ;n) . . . g 2 ( 1 ; 2 ;n) g 3 ( 1 ; 2 ;n) . . . g 3 ( 1 ; 2 ;n) 0 . . . 0 0 1 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A ; (6.141) where numerically (f 2 g 2 ) =(f 3 g 3 ) = p . Clearly, the pole vortices affect the strength of the street vortices, and the poles can be chosen so to make the constant 1 from Equation (5.106) equal to one, thus achieving the double VKS shown in Figure 6.51. One solution would be to choose the pole strengths to be equal and opposite with north = south = 2 , where 2 = 2 ( 1 ; 2 ;n) = 1 1 p : (6.142) 99 Γ γ 2 Γ −γ 2 Γ −Γ −Γ Γ Γ Γ Γ −Γ −Γ −Γ Γ Γ −Γ −Γ Figure 6.51: Double von K´ arm´ an Street with poles schematic, illustrating SVD solution. Dimension of the null space is 3, with the pole vortices having independent strengths of one another. Since the poles vortices affect the street vortices’ strength, their strengths are chosen to be 2; 2 = 2(1;2;n) in order to ensure 1 1 in Figure 6.50. The solution collapses into one dimension with the pole vortices stacked at the end of the state representation vector, such that = (1;:::; 1;1;:::;1; 1;:::; 1;1;::: 1; 2 ; 2 ) T 2R 4n+2 : (6.143) Note that this solution to 2 is numeric. In Section 6.20.1 we will couple this solution with the angular velocity formulation to produce an analytic result. It is also relevant to note that the center of vorticity of the double VKS is J = 2 0 B B B B @ 0 0 n(cos 2 cos 1 ) + 2 1 C C C C A : (6.144) 100 (a) 1 = 3=8 (b) 1 = 5=12 Figure 6.52: Equilibrium curves present for pole strength ( 2) vs. angle ratio (1=2) whenn = [4; 5; 6]. Angle 1 is held fixed at 3=8 in (a), and 5=12 in (b), while2 is the control parameter. Recall that 2 = 2(1;2;n). Recall that the pole strength can be any value for the degenerate case1 = 2 represented by the solid line in the graphs (single VKS). 6.19.3 Equilibrium Curves Each double street configuration with its unique set of parameters 1 ; 2 and n has a single corresponding pole strength factor 2 ( 1 ; 2 ;n) associated with its pole vortices. Figures 6.52(a) and 6.52(b) illustrate two sets of equilibrium curves present in the coordinate system 2 vs. the angle ratio 1 = 2 . The angle 1 is kept fixed at 3=8 in 6.52(a), and 5=12 in 6.52(b), with a sweep of 0< 2 <=2. Three curves are displayed in each of the two subplots corresponding ton = 4, 5 and 6 vortices per ring. A degenerate case occurs when 1 = 2 , which entails the merger of the two rings in each hemisphereand this results in the collapse of the system into a single vortex street. As was described in the previous section, the configuration will be an equilibrium single VKS for any north = south (hence the vertical line in the figure). When 2 is perturbed by an infinitesimal amount from 1 = 2 , the magnitude of the pole strength factor 2 required to create a double VKS must be infinitely large. It is also relevant to point out that for certain ratio values, the pole strength 2 0, which implies that some configurations are double vortex streets only in the absence of pole vortices. 101 6.20 Angular Velocity and Explicit Solutions 6.20.1 Without Pole Vortices Using the angular velocity formulation, we will derive an analytic solution to 1 ( 1 ; 2 ;n) in Equation (6.139). Assuming that the center of vorticity is parallel to the polar axis (i.e., Equa- tion (6.140) applies), we conclude that the rate of change of the vortices’ longitudes in Equa- tion (2.55) is equivalent to the angular velocity of the configuration 2 . We begin our analysis by choosing the same vortex strengths found in Equation (6.139), and assume that 1 is an unknown parameter to be solved. Next, we derive one expression for the angular velocity of the vortices in the 1 rings, and a second for the vortices in the 2 rings. For simplicity in writing, we define sin cos cos sin cos ( ) 4 sin (1 cos ) : (6.145) Let! 1 be the angular velocity of the 1 rings’ vortices. Therefore, it follows that ! 1 _ n+1 = _ n+2 =::: = _ 2n (6.146) = _ 2n+1 = _ 2n+2 =::: = _ 3n : Next, we come up with a general expression for ! 1 from the angular velocity _ n+1 of vortex n + 1. From Equations (2.55) and (6.145), we have _ n+1 = n X =1 (n+1) + 2n X =n+2 (n+1) + 3n X =2n+1 (n+1) + 4n X =3n+1 (n+1) : 2 The longitudes of the vortices in the2 rings are 0; 2(1)=n; 2(2)=n; ; 2(n 1)=n , while those of the 1 rings are=n; 2(1)=n +=n; 2(2)=n +=n; ; 2(n 1)=n +=n. 102 The four summations correspond to the four vortex rings, with an exclusion of then + 1 term from the 2 nd summation. Next, we replace the terms with their values from (6.139) to get _ n+1 = n X =1 (n+1) 1 2n X =n+2 (n+1) + 1 3n X =2n+1 (n+1) 4n X =3n+1 (n+1) = 1 3n X =2n+1 (n+1) 2n X =n+2 (n+1) ! + n X =1 (n+1) 4n X =3n+1 (n+1) ! ; or simply that ! 1 = 1 # 1 ( 1 ; 2 ;n) + # 2 ( 1 ): (6.147) We now repeat the process for the vortices on the 2 of rings by defining ! 2 as their angular velocity. Therefore, ! 2 _ 1 = _ 2 =::: = _ n (6.148) = _ 3n+1 = _ 3n+2 =::: = _ 4n : We use the angular velocity _ 1 of vortex 1 to form an expression for ! 2 . From Equa- tions (2.55) and (6.145), _ 1 = n X =2 1 + 2n X =n+1 1 + 3n X =2n+1 1 + 4n X =3n+1 1 : Note the exclusion of the 1 st term in the 1 st summation. Again, we replace the terms with their values from (6.139) to get _ 1 = n X =2 1 1 2n X =n+1 1 + 1 3n X =2n+1 1 4n X =3n+1 1 = 1 3n X =2n+1 1 2n X =n+1 1 ! + n X =2 1 4n X =3n+1 1 ! : 103 Therefore, ! 2 = 1 # 3 ( 1 ; 2 ;n) + # 4 ( 2 ): (6.149) Our condition for relative equilibrium is that the 1 and 2 rings have the same angular velocity (i.e., ! 1 = ! 2 ). By equating (6.147) and (6.149), we derive an explicit expression for 1 of Equation (6.139) to be 1 = # 4 # 2 # 1 # 3 : (6.150) This result can be used to generate analytic curves for the angular velocity, or of 1 , as a function of the parameters. 6.20.2 With Pole Vortices Maintaining the state representation in Equation (6.143), the desired vortex strengths of the system are given by = (;:::; ;;:::;; ;:::; ;:::; ; p ; p ) T ; (6.151) where p is the pole vortex strength that we will solve (to match the solution in (6.143)). We can also assume that Equation (6.144) applies (i.e., the system is rotating rigidly about the polar axis). Therefore, as in Equation (6.146), we define! 1 to be the angular velocity of the vortices in the 1 ring. We derive an expression for! 1 by substituting (5.131), (6.145) and (6.151) into (2.55) to get _ 1 = n X =1 (n+1) 2n X =n+2 (n+1) + 3n X =2n+1 (n+1) 4n X =3n+1 (n+1) ! + p 2 sin 2 1 ; 104 or ! 1 = 1 ( 1 ; 2 ;n) + p 2 ( 1 ): (6.152) Similarly, if define! 2 to be the angular velocity of the vortices in the 2 rings, and using the same substitutions in conjunction with the relations of Equation (6.148), we derive that ! 2 = 3 ( 1 ; 2 ;n) + 4 ( 2 ) p ; (6.153) where 3 = n X =2 1 2n X =n+1 1 + 3n X =2n+1 1 4n X =3n+1 1 ! ; and 4 = 1 2 sin 2 2 : Since the angular velocity of the two rings should be equal for relative equilibrium (i.e., ! ! 1 ! 2 ), we derive the analytic expression for the required pole vortex strength to be p = 2 ; 2 = 2 ( 1 ; 2 ;n) = 1 3 4 2 : (6.154) Example 2. Consider a double vortex street withn = 2 per ring. Derive the angular velocity. The angular velocity! 1 in terms of 1 ; 2 . From (6.152), we have ! 1 = 1 + 2 p ; (6.155) with 1 = 1 4 cos 1 1 cos 2 1 cos 2 2 + 4 cos 3 1 cos 2 4 cos 1 cos 2 cos 2 1 + cos 4 1 cos 2 2 + 1 cos 2 1 cos 2 2 ; (6.156) 105 0.2 0.4 0.6 0.8 1 1.2 −5 −4 −3 −2 −1 0 1 2 φ 2 ω 3π/8 n=2 n=3 n=4 n=5 n=6 (a) 1 = 3=8; 0<2 <1 1.2 1.3 1.4 1.5 −2.5 2.5 7.5 12.5 17.5 21.5 φ 2 ω 3π/8 n=2 n=3 n=4 n=5 n=6 (b) 1 = 3=8; 1 <2 <=2 0.2 0.4 0.6 0.8 1 1.2 −15 −10 −5 0 φ 2 ω 5π/12 n=2 n=3 n=4 n=5 n=6 (c) 1 = 5=12; 0<2 <1 1.3 1.35 1.4 1.45 1.5 10 20 30 40 50 60 φ 2 ω 5π/12 n=2 n=3 n=4 n=5 n=6 (d) 1 = 5=12; 1 <2 <=2 Figure 6.53: Angular velocity! vs. angle2 forn = 2 5. 1 held fixed at 3=8 in (a) and (b), and at 5=12 in (c) and (d). Singularity occurs since p!1 as2!1. and just as in Equation (5.131), 2 = 1 2 sin 2 1 : (6.157) The angular velocity! 2 from (6.153) becomes ! 2 = 3 + 4 p ; (6.158) with 3 = 1 4 cos 2 1 cos 2 2 cos 2 1 + 4 cos 3 2 cos 1 4 cos 2 cos 1 cos 2 2 + cos 4 2 cos 2 1 + 1 cos 2 2 cos 2 1 ; (6.159) 106 and with Equation (5.131) we have 4 = 1 2 sin 2 2 : (6.160) The required pole vortex strength factor 2 to ensure that! 1 and! 2 are equivalent can then be solved analytically using Eq. (6.154). In Figure 6.53, analytic curves for the angular velocity vs. 2 with constant 1 are presented forn = 2 6, and for the two scenarios 1 = 3=8 and 1 = 5=12. The angular velocity approaches either +1 or1 when 2 ! 1 or 2 ! =2 since the required pole vortex strength factor 2 to ensure relative equilibrium also becomes infinite. Note also that the equi- librium curves in Figure 6.52 can be derived analytically. In the next subsection, we address the result of choosing different angular velocities for each set of vortex rings in each hemisphere (i.e., Eq. (6.154) is not satisfied). 6.20.3 Shearing Structures In Section 1.3.2 of the introduction, we referenced an investigation by Coddington et al: in Figure 1.8 from [17]. To recapitulate, small amplitude sinusoidal perturbations resulted from laser-zapping the central vortex site, and effectively modifying its circulation strength. In the context of vortex streets on the sphere, the pole vortices can be interpreted as the ‘central’ vor- tex, while the vortex rings can be interpreted as concentric bands. As was discussed in the previous section, the selection of the pole vortices’ strengths plays a crucial role in attaining a relative equilibrium. If 2 from Eq. (6.154) is not chosen correctly, the different vortex rings will rotate with different angular velocities, resulting in a non-equilibriumshearing-like flow. Numerical experiments of vortex street configurations with an incorrect selection of parameter 2 demonstrated the existence of periodic and quasi-periodic dynamics with respect to the inter- vortical distances (see Figures 6.54 and 6.55). These types of configurations exhibit periodic and quasi-periodic oscillations, and can be more generally classified as shearing vortex lattices. 107 (a) All vortices (side view) (b) All vortices (top view) (c) V ortex set I (side view) (d) V ortex set I (top view) (e) V ortex set II (side view) (f) V ortex set II (top view) Figure 6.54: Time evolution of point vortices of double street with poles given by solid black lines. Pole vortex strengths do not satisfy equilibrium condition given by Equation (6.143). Specifically, pole vortices have strength 0 instead of 2. Figures (a) and (b) show evolution of all vortices. Figures (c) and (d) show evolution of one isolated set of vortices marked by filled circles (one from each hemisphere), and demonstrate that the dynamics is periodic. Figures (e) and (f) show the evolution of a second set of vortices, also showing periodic dynamics. 108 Figure 6.55: Quasi-periodic time evolution of point vortices of double street with poles given by solid black lines. Pole vortex strengths do not satisfy equilibrium condition given by Equation (6.143). Specifically, pole vortices have strength2 2 instead of 2. 6.21 Streamline Topology The stagnation points for a double street naturally have a more elaborate structure than those of a single street. Consider the case withn = 5, 1 = 13=40, pole vortices with equal and opposite strength 2 such as in (6.143), and with 2 is the control parameter. We distinguish streamline topologies based on the angle ratioR = 1 = 2 . Figures 6.56 and 6.58 contain a sequence of streamline topologies for ratiosR < 1 andR > 1 respectively. Figure 6.57 is a curve of pole vortex strength 2 vs. angle ratioR for the configuration at hand, and provides a supplementary insight into the 2 R and changes in the streamline topology. First consider Figure 6.56 in which R < 1 (which means that the 2 -rings are closer to the equator). When R < 0:691, one gets ten stagnation points of hyperbolic or saddle type in each hemisphereeach of which is aligned longitudinally with one of the street vortices. In addition, there are two sets of separatrices in each hemisphere that separate five different regions of global fluid transport; this is the first topology type and is illustrated in Figure 6.56(a). The streamline topology remains the same untilR 0:691. At this point, the two separatices in each hemisphere collapse into one, and two of the regions associated with global fluid transport disappearsee Figure 6.56(b). Next, as soon asR > 0:691, the separatrix in each hemisphere 109 splits into a set of two, and two regions with global fluid transport reemerge; this marks topology type II illustrated in Figure 6.56(c). As we increaseR, the five stagnation points that are aligned with the 1 -rings in each hemisphere approach the equator until they merge at the bifurcation valueR 0:755. At this point, there are five stagnation points at the equator that are aligned with the 1 -rings. In addition, there are still five remaining stagnation points of hyperbolic or saddle type in each hemisphere that are longitudinally aligned with the 2 -rings, see Figure 6.56(d). As soon as R > 0:755, each of the stagnation points at the equator splits into two, and they move away from each other asR increases. The topology remains unchanged (labeled type III in Figure 6.56(e)) until the next bifurcation value of R 0:801 is reached. At this bifurcation, two sets of separatrices in each hemisphere coincide and two regions associated with global fluid transport disappearsee Figure 6.56(f). Note also from Figure 6.57 that 2 = 0 at this bifurcation value. As soon as R > 0:801, the separatrices that had just merged split and the two regions with global fluid transport reemerge; this range ofR is identified as topology type IV in Figure 6.56(g). As R increases, the five stagnation points in each hemisphere that are longitudinally aligned with the 2 -rings approach their respective hemisphere’s poles. The next bifurcation point is reached when R 0:843 at which point the ten stagnation points collapse into two of hyperbolic or saddle type at the poles (Figure 6.56(h)). Observe from Figure 6.57 that this bifurcation point corresponds to an inflection point in the 2 R relation. As soon as R > 0:843, five hyperbolic points reappear around the poles, and begin moving towards the 1 -rings’ vortices; this returns us to topology type III shown in Figure 6.56(i). The next bifurcation that occurs results from the ten stagnation points at the equator. AsR increases, the stagnation points continue to move on the equator untilR 0:893, at which point they collapse into a total of five stagnation points of hyperbolic or saddle typewith each one longitudinally aligned with one of the vortices from the 2 rings. See Figure 6.56(j). As soon asR > 0:893, the five stagnation points at the equator split, reappear as a total of ten, and asR increases, they begin to move away from the equator and towards the 2 -rings. The range 0:893<R< 1 marks the final topology interval (which is back to Type II) for the range 0<R< 1, and is illustrated in Figure 6.56(k). 110 (a) Type I (b) R = 0:691 (c) Type II (d) R = 0:755 (e) Type III (f) R = 0:801 (g) Type IV (h) R = 0:843 (i) Type III (j) R = 0:893 (k) Type II Figure 6.56: Streamline topology sequence for the range of ratios 0:68<R< 0:95 in ascending order. Stagnation points move either along constant longitude lines or the equator as the ratio varies. Four different topology types are illustrated (labeled I, II, III, IV), and are separated by five bifurcation scenarios. 0.8 1 1.2 1.4 1.6 1.8 −4 −2 0 2 4 6 R γ 2 γ 2 R=0.801 R=0.843 R=1.308 Figure 6.57: Equilibrium curve corresponding to the relation of the pole vortex strength factor 2 vs. angle ratio R forn = 5, = 13=40, and with2 as the control parameter. This plot also corresponds to the configurations found in Figures 6.56 and 6.58, and illustrates (with vertical lines) that some of the bifurcation points correspond to either an inflection in the curve or when 2 = 0. 111 (a) Type II (b) R = 1:225 (c) Type III (d) R = 1:308 (e) Type IV (f) R = 1:417 (g) Type V Figure 6.58: Streamline topology sequence for the range of ratios 1:05 < R < 2:25 in ascending order. Main- taining notation consistency with Figure 6.56, four different topology types are evident, labeled Types II, III, IV , V . We now describe the results of Figure 6.58 in whichR > 1 (this means that the 2 rings are closer to the poles). We start with the first topology interval illustrated in Figure 6.58(a). To be consistent with the notation from Figure 6.56, the first topology interval is Type II. There are ten stagnation points of hyperbolic or saddle type in each hemisphereeach of which is aligned longitudinally with one of the street vortices. In addition, there are two sets of separatrices in each hemisphere that separate five different regions of global fluid transport. As R increases, the stagnation points move away from the street vortices. The first five stagnation points in each hemisphere are longitudinally aligned with the 1 -rings and are approaching the equator; the second five are longitudinally aligned with the 2 -rings and are approaching the poles. The topology remains unchanged until R 1:225, at which point the first five stagnation points from each hemisphere (total of ten) approach each other, and merge to form five stagnation points of hyperbolic or saddle type at the equator. At this bifurcation value, one of the regions associated with global fluid transport disappears. As soon asR> 1:225, each of the five stagnation points at the equator splits into two, giving a total of ten stagnation points of hyperbolic or saddle type at the equator. AsR increases, these stagnation points continue to move apart on the equator 112 towards the longitudes associated with the 2 -rings. This range ofR is described by topology type III, and is illustrated in Figure 6.58(c). Meanwhile, the remaining five stagnation points in each hemisphere are longitudinally aligned with the 2 -rings and are approaching the poles as R is increasing. The next bifur- cation occurs whenR 1:308 (see Figure 6.58(d)) when the stagnation points collapse to two points of hyperbolic or saddle type at the poles. Note that from Figure 6.57 that this bifurcation point corresponds to the case when 2 = 0. AsR increases, five stagnation points reappear around the poles and begin moving towards the 2 -rings as shown in Figure 6.58(e), and described as topology type IV . The next bifurcation takes place whenR 1:417 as shown in Figure 6.58(f). At this point, the ten stagnation points at the equator merge (at the longitude associated with the 2 -rings) into five of hyperbolic or saddle type. As soon asR > 1:417, we get our last topology type (see Figure 6.58(g)), labeled type V . Each of the five stagnation points at the equator split into two to give a total of ten, and begin to move away from the equator and towards the 2 -rings asR increases. It can be observed from looking at Figure 6.57 that one can extract information about the streamline topology by looking at the 2 vs. R curve. In Figure 6.52, we can see that each set of parametersn and 1 has a unique 2 vs. R curve, and will therefore also have a unique set of streamline topologies over the range R. Therefore, the streamline topologies discussed for Figures 6.56 and 6.58 are specific only to the aforementioned parameters, and cannot be generalized as being the result for any double vortex street. 6.22 Shannon Entropy In this section, we will illustrate a few results for the normalized Shannon Entropy (from (3.83)) of double vortex streets. In Figure 6.59, we have the Shannon Entropy vs. angle 2 , for 1 = 3=8 andn = 3; 4; 8; 10. In each subfigure, we illustrate the effect of adding pole vortices to the configurations. The entropy curves for the double vortex street are much richer in structure than were those for the single vortex street. There is a set of local minima and maxima associated with 113 0 0.5 1 1.5 0.7 0.8 0.9 φ 2 H W/out Poles With Poles (a) n = 3 0 0.5 1 1.5 0.7 0.8 φ 2 H W/out Poles With Poles (b) n = 4 0 0.5 1 1.5 0.65 0.7 0.75 0.8 0.85 0.9 φ 2 H W/out Poles With Poles (c) n = 8 0 0.5 1 1.5 0.65 0.7 0.75 0.8 0.85 0.9 φ 2 H W/out Poles With Poles (d) n = 10 Figure 6.59: Shannon Entropy vs2 for1 = 3=8 andn = 3; 4; 8; 10. The figures llustrate the effect on entropy upon adding pole vortices to the configuration. the curves. For the casen = 3, the curves with and without pole vortices cross several times, while when n > 3 the curves for configurations without pole vortices have a higher entropy than do the curves for configurations that do not for all 2 . We interpret values of 2 which correspond to local maximizers of the Shannon Entropy as more robust with respect to their singular value distributions, than the remaining values of 2 . In addition, those at local maxima would tend to get ‘selected’ more often than others in a random search setting as distributions with larger entropy have more combinatorial ways of being chosen at random [56]. 114 Chapter 7. Conclusions 7.23 Summary This report was a study of several key properties of relative vortex equilbriaor vortex lattices. To recapitulate, we used a condition for relative vortex equilbrium derived from the Hamiltonian point vortex model (Chapter 2). In Chapter 3, we introduced three tools from linear algebra to facilitate our analysis. At the starting point of our work, the techniques presented in Chapters 2 and 3 made it possible to find all vortex strengths, given by , for a given configuration that will ensure that the vortices remain in relative equilbrium. The main contributions of this thesis can be classified into four categories: 1. Asymmetric equilibria: It was shown that asymmetric lattices with a non-trivial equilib- rium cannot be chosen at random (i.e., when the vortex locations are selected at random from a uniform distribution). Instead, it is possible to begin with a general random asym- metric lattice, and converge to an asymmetric geometry that has a non-trivial solution via a ‘Brownian ratchet’ scheme that mimics the thermal cooling observed in Bose Einstein Condensates ([25], [26]). Via Singular Value Decomposition (SVD), the ratchet scheme utilized the singular values associated with the configuration matrixA (see (2.49)) to con- verge to a non-trivial equilibrium (i.e., nullity(A) > 0). That is, the smallest singular value must be zero. If the convergence scheme is repeated for the same initial configura- tion, the ratchet converges to a different asymmetric equilibrium with each run. Although non-trivial equilibria lie arbitrarily close to one another, there is a non-zero, finite distance 115 between a trivial and non-trivial equilibrium configuration, and is given by the Frobenius normkk F . In other words, the set of non-trivial equilbria is not dense within the set of all possible configurations. For results, see Chapter 4. 2. Shannon Entropy: The Shannon Entropy associated withA was studied, and it was found that geometries that are more prevalent in physical settings have higher entropies than those that are less common. Therefore, we interpret the Shannon Entropy as a scalar mea- sure of the robustness, or likelihood of existence of a given configuration geometry. With the Brownian ratchet scheme, the system tended to converge to an equilibrium configu- ration whose entropy was within the same neighborhood as the initial configuration. By normalizing the Shannon Entropy, we proposed that we can be compare configurations containing a different number of vortices. In the context of geophysical flows, the Shan- non Entropy could be used as a means to predict the most likely parameters of vortex streets (i.e., latitude of the streets, number of vortices in the street, etc.). Shannon Entropy results are present in Chapters 4, 5 and 6. 3. Model of vortex streets of the sphere: We presented solutions to both single (chapter 5) and double (Chapter 6) vortex streets on the sphere. The SVD technique provided a sys- tematic way of solving vortex equilbria for largeN. The solution to with this approach was numeric since the dimension ofA was large. When coupled with the center of vortic- ity and angular velocity formulations, it was possible to build on SVD and extract analytic results for the vortex strengths. It was found that non-trivial solutions to symmetric vor- tex equilibria lie along smooth equilibrium curves within the configuration space, which means that it is possible to modify the inter-vortical geometry smoothly from one con- figuration with a nontrivial solution that possesses a certain symmetry to a second while maintaining relative equilibrium. This also implies that certain symmetric equilibria solu- tions lie arbitrarily close. One example of such a smooth transition was for the double 116 vortex street on the sphere with pole vortices, where the relative strength of the pole vor- tices 2 ( in Equation (6.143)) was given by an analytic function of the system geometry, and varied smoothly with the angles of the vortex streets from the poles. It was found that pole vortices play an important role in fixing, or in other words, assuring that the vorticity vector remained constant. That is, pole vortices have a central role in the creation or destruction of vortex streets. Studies in which the strength of the pole vortices was perturbed from equilibrium resulted in shearing, which we define as periodic and quasi-periodic orbits of the inter-vortical distances about the center of vorticity vector. We made the comparison to [17], in which the central vortex of a BEC lattice was blasted by laser. The blast effectively changed the strength of the central vortex, and resulted in small amplitude sinusoidal perturbations. 4. Streamline topology of vortex streets on the sphere: With the techniques presented, we demonstrated that the streamline topology bifurcation analysis is greatly facilitated. Streamline topology analysis is a crucial feature of vortex equilibria, as different topolo- gies entail different fluid transport and mixing regimes. For single and double vortex streets, we presented the different topology types, and linked bifurcations to critical points on the equilibrium curves. 7.24 Future Work Several important results and conclusions were attained in this study. However, with the closing of this thesis, many open questions were discovered along the way, and there is now a rich variety of topics that are ripe for investigation. Several topics include: 1. Stability of vortex lattices: In particular, the stability of vortex streets on sphere. A start- ing point for this investigation would be to follow the work done by [13], in which they examined the nonlinear stability of a single vortex ring with a fixed latitudeboth with and without the presence of pole vortices. The general Hessian matrix for the configuration 117 was solved, and the authors were able to derive a bifurcation diagram that distinguished stable and unstable regimes with respect to the parameters affecting the configuration’s geometry and vortices’ relative strengths. Since a vortex street consists of two symmetri- cally skewed rings, stability results could follow from the techniques used in this work. 2. Vortex lattices interacting with solid boundaries: This could be incorporated into the aforementioned techniques by modeling boundaries via the method of images. A start- ing point could be the introduction of symmetric arrays of boundaries and vortices, and developed into cases in which solid boundary breaks the symmetry of lattices. 3. Viscous vortex lattices: The Navier Stokes equation can produce a model for a single viscous point vortex (see Appendix B.2). Modeling systems with more than one viscous point vortex becomes much more difficult, since the natural radial symmetry associated with the single viscous vortex breaks. It might possible to formulate a model for approxi- mate viscous vortices, and derive whether the types of geometries solved for inviscid point vortices translate as solutions to viscous vortex configurations. 4. The reverse problem and uniqueness: The problem in reverse is whether it is possible to derive or estimate a configuration from a configuration matrix. As a numeric solution, it might be possible to do this via a Brownian ratchet, in which the geometry compo- nents (inter-vortical distances and orientations) are fluctuated until they converge towards a desired A. In addition, does a given configuration matrix A correspond to a single configurationX, or can multiple configurations with unique geometries possess the same configuration matrix? This problem was introduced in a Section 4.4, in which the number of independent variables corresponding the positions of a configuration withN points (or vortices) were solved, both in the presence and absence of a coordinate system, and were compared to the number of non-trivial elements in the configuration matrixA for different N. The point to be made was whether the system can be described as undetermined or over-determined for differentN. 118 5. Biased ratchets: The Brownian ratchet presented converges with no consideration to the form of the solution. Biased ratchets can be designed that possess constraints on the geometry (for example: an allowable range of distances between the vortices, or bounded domains), or on the solution (example: one might aspire to solve for configurations in which the vortices all have equal strength.) 6. Shannon Entropy: In a work to appear by Newton & Sajako, vortex lattices are perturbed from equilibrium, and the Shannon Entropy of A will be tracked as the vortex system evolves dynamically with time. The motivation is to determine the correlation between the dynamics, entropy and evolution of point vortex systems. 119 Bibliography [1] J. R. Abo-Shaer, C. Raman, J. M. V ogels, and W. Ketterle. Observation of vortex lattices in bose einstein condensates. Science, 292:476–479, 2001. [2] A. A. Abrikosov. On the magnetic properties of superconductors of the second group. Sov. Phys. JETP, 6:1174–1178, 1957. [3] A. A. Abrikosov. Type ii superconductors and the vortex lattice. Rev. Mod. Phys., 76, 2004. [4] H. Aref. A note on the energy of relative equilibria of point vortices. J. Fluid Mech., 39:1–79, 2007. [5] H. Aref. Point vortex dynamics: A classical mathematcs playground. Jour. Math. Phys., 48:065401, 2007. [6] H. Aref. V ortices and polynomials. Fluid Dynamics Research, 39:5–23, 2007. [7] H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein. V ortex crystals. Advances in Applied Mechanics, 39:1–79, 2003. [8] R. A. Barkley. The kuroshio-oyashio front as a compound vortex street. J. Mar. Res., 26:83–104, 1968. [9] M. W. Berry, S. T. Dumais, and G. W. O’Brien. Using linear algebra for intelligent infor- mation retrieval. University of Tennessee Preprint, pages 1–24, 1994. [10] S. Boatto and H. E. Cabral. Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere. SIAM J. Appl. Math., 64 (1):216–230, 2003. [11] M. Brons, B. Jakobsen, K. Niss, A. V . Bisgaard, and L. K. V oigt. Streamline topology and vorticity patterns in the near-wake of a circular cylinder at moderate reynolds numbers. J. Fluid Mech., 584:23–43, 2007. [12] L. F. Burlaga. A heliospheric vortex street. J. Geophys. Res., 95:4333–4336, 1990. [13] H. E. Cabral, K. R. Meyer, and D. S. Schmidt. Stability and bifurcations for the n+1 vortex problem on the sphere. Regular and Chaotic Dynamics, 8:259–282, 2003. [14] L. J. Campbell and R. Ziff. A catalog of two-dimensional vortex patterns. Informal Report, Los Alamos Scientific Laboratory, 7384-MS:1–40, 1978. 120 [15] G. C. Chamoun, E. Kanso, and P. K. Newton. V on k´ arm´ an vortex streets on the sphere. under consideration for publication in J. Fluid. Mech., 2008. [16] J. Y-K. Cho and L. M. Polvani. The emergence of jets and vortices in freely evolving shallow-water turbulence on a sphere. Physics of Fluids, 8 (6):1531–1552, June 1996. [17] I. Coddington, P. Engels, V . Schweikhard, and E. A. Cornell. Observations of tkachenko oscillations in rapidly rotating bose-einstein condensates. Physical Review Letters, 91 (10):100402, September 2003. [18] Y . Couder and C. Baddevent. Experimental and numerical study of vortex couples in two- dimensional flows. J. Fluid Mech., 173:225–251, 1986. [19] R. Courant and D. Hilbert. Methods of Mathematical Physics, volume I,II. Wiley, New York, 1953. [20] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler. Self- concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett., 93:098103, 2004. [21] U. Domm. ¨ Uber dir wirbelstrassen von geringster instabilit¨ at. Z. Angewandte Math. Mech. (ZAMM), 36:367–371, 1956. [22] P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V . Vinay. Clustering large graphs via the singular decomposition. Machine Learning, 56:9–33, 2004. [23] C. F. Driscoll and K. S. Fine. Experiments on vortex dynamics in pure electron plasmas. Phys. Fluids, 2:1359–1366, 1990. [24] J. Dusek, P. Le Gal, and P. Fraunie. A numerical and theoretical study of the first hopf bifurcation in a cylinder wake. J. Fluid Mech., 264:59–80, 1994. [25] P. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell. Nonequilibrium effects of anisotropic compression applied to vortex lattices in bose-einstein condensates. Phys. Rev. Lett., 89:100403–1–100403–4, 2002. [26] P. Engels, I. Coddington, P. C. Haljan, V . Schweikhard, and E. A. Cornell. Observation of long-lived vortex aggregates in rapidly rotating bose-einstein condensates. Phys. Rev. Lett., 90:170405–1–170405–4, 2003. [27] U. Essmann and H. Trauble. The direct observation of individual flux lines in type ii superconductors. Phys. Lett., 24A:256–530, 1967. [28] D. Etling. On atmospheric vortex streets in the wake of large islands. Met. Atmos. Phys., 41:157–164, 1989. [29] R. P. Feynman. Progress in Low Temperature Physics. D.F. Brewer, North-Holland, 1 edition, 1955. 121 [30] K. S. Fine, A. C. Cass, and W. G. Flynn C. F. Driscoll. Relaxation of 2d turbulence to vortex crystals. Phys. Rev. Lett., 75:3277, 1995. [31] A. Friedmann and P. Poloubarinova. ¨ Uber forschreitende singularit¨ aten der ebenen bewe- gung einer inkompressiblen fl¨ ussigkeit. Recueil de G´ eophysique, V:9–23, 1928. [32] V . L. Ginzburg and L. D. Landau. On the theory of superconductivity. Zh. Eksp. Theor. Fiz., 20:1064, 1950. [33] D. Godfrey, G. E. Hunt, and V . E. Suomi. Some dynamical properties of vortex streets in saturn’s atmosphere from analysis of voyager missions. Geo. Res. Lett., 10:865–868, Sept 1983. [34] B. A. Grzybowski, H. A. Stone, and G. M. Whitesides. Dynamic self-assembly of mag- netized millimeter-sized objects rotating at a liquid-air interface. Nature, 405:1033–1036, 2000. [35] T. H. Havelock. The stability of motion of rectilinear vortices in ring formation. Philos. Mag., 11:617–633, 1931. [36] H. F. Hess, R. B. Robinson, R. G. Dynes, J. M. Valles, and J. V . Waszczak Jr. Scanning- tunneling microscope observation of the abrikosov flux lattice and the density of states near and inside a fluxoid. Phys. Rev. Lett., 62:214–216, 1989. [37] T. Humphreys and P. S. Marcus. V ortex street dynamics: The selection mechanism for the areas and locations of jupiters vortices. Jour. Atmos. Sci., 64:1318–1333, 2007. [38] C. P. Jackson. A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid. Mech., 182:23–45, 1987. [39] M. I. Jamaloodeen and P. K. Newton. The N-vortex problem on a rotating sphere: II. heterogeneous platonic solid equilibria. Proc. Roy. Soc. London Ser. A, 462:3277–3299, 2006. [40] R. Kidambi and P. K. Newton. Streamline topologies for integrable vortex motion on a sphere. Physica D, 140:95–125, 2000. [41] M. Kirby. Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction. Wiley-Interscience, 1 edition, 2001. [42] V . C. Klema and A. J. Laub. The singular value decomposition: Computation and some applications. IEEE Transactions on Automatic Control, 25 (2):164–176, 1980. [43] S. Kullback and R. A. Leibler. On information and sufficiency. Ann. Math. Stats., 22:79– 86, 1951. [44] L. G. Kurakin. On nonlinear stability of the regular vortex systems on a sphere. Chaos, 14 (3):592–602, 2004. [45] H. Lamb. Hydrodynamics. Cambridge University Press, 6 edition, 1932. 122 [46] F. Laurent-Polz, J. Montaldi, and M. Roberts. Stability of relative equilibria of point vor- tices on the sphere. Preprint, 0:1–30, 2006. [47] J. C. Liao, D. N. Beal, G. V . Lauder, and M. S. Triantafyllou. The krmn gait: Novel body kinematics of rainbow trout swimming in a vortex street. Oceanology, 1:12–16, 2003. [48] P. S. Marcus. V ortex dynamics in a zonal shearing flow. J. Fluid Mech., 215:393–430, 1990. [49] P. S. Marcus. Jupiter’s great red spot and other vortices. Ann. Rev. Astron. Astrophys., 31:523–573, 1993. [50] P. S. Marcus. Prediction of global climate change on jupiter. Letters to Nature, 428:828– 831, 2004. [51] A. W. Maue. Zur stabilitot der karmanschen wirbelstrasse. Z. Angew. Math. Mech., 20:129, 1940. [52] V . V . Meleshko and H. Aref. A bibliography of Vortex Dynamics 1858-1956. available online, 2007. [53] P. K. Newton. The N-Vortex Problem: Analytical Techniques. Applied Math. Sci., Springer- Verlag, 2001. [54] P. K. Newton and G. Chamoun. On regular triangular vortex lattices. USC preprint, 2006. [55] P. K. Newton and G. Chamoun. Construction of point vortex equilibria via brownian ratch- ets. Proc. Roy. Soc. London Ser. A, 463:1525–1540, 2007. [56] P. K. Newton and G. Chamoun. V ortex lattice theory: Harnessing thermal fluctuations to produce patterns. SIAM Review, 2008 (in press). [57] P. K. Newton and H. Shokraneh. The N-vortex problem on a rotating sphere: I. multi- frequency configurations. Proc. Roy. Soc. London Ser. A, 462:149–169, 2006. [58] B. R. Noack and H. Eckelmann. A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech., 270:297–330, 1994. [59] B. R. Noack and H. Eckelmann. A low-dimensional galerkin method for the three- dimensional flow around a circular cylinder. Phys. Fluids, 6:124–143, 1994. [60] C. C. Paige and M. A. Saunder. Towards a generalized singular value decomposition. SIAM Journal on Numerical Analysis, 18 (3):398–405, 1981. [61] S. Pekarsky and J. E. Marsden. Point vortices on a sphere: Stability of relative equilibria. J. Math. Phys., 39 (11):5894–5907, 1998. [62] F. L. Ponta and H. Aref. The strouhal-reynolds number relationship for vortex streets. Phys. Rev. Lett., 93:84501, 2004. 123 [63] F. L. Ponta and H. Aref. V ortex synchronization regions in shedding from an oscillating cylinder. Phys. Fluids, 17:011703, 2005. [64] M. Provansal, C. Mathis, and L. Boyler. Benard-von k´ arm´ an instability: Transient and forced regimes. J. Fluid Mech., 182:1–22, 1987. [65] A. Roshko. Experiments on the flow past a circular cylinder at very high reynolds number. Journal of Fluid Mechanics, 10:345–356, 1961. [66] P. G. Saffman and J. C. Schatzman. Stability of a vortex street of finite vortices. J. Fluid Mech., 117:171–185, 1982. [67] C. E. Shannon. A mathematical theory of communication. Bell. Sys. Tech. J., 27:379–423, 1948. [68] P. R. Shukla, G. T. Birk, and R. Bingham. V ortex streets driven by sheared flow and applications to black aurora. J. Geophys. Res. Lett., 22:671–674, 1985. [69] M. A. Stremler. Relative equilibria of singly periodic point vortex arrays. Phys. Fluids, 15:3765–3775, 2003. [70] L. N. Trefethen and D. Bau III. Numerical Linear Algebra. SIAM, Philidelphia, 1 edition, 1997. [71] S. Umeyama. Least-squares estimation of transformation between two point patterns. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4 (3):376–380, 1991. [72] C. H. K. Williamson. V ortex dynamics in the cylinder wake. Ann. Rev. Fluid Mech., 28:477–539, 1996. [73] C. H. K. Williamson and A. Roshko. V ortex formation in the wake of an oscillating cylin- der. J Fluids Struct., 2:355–381, 1988. [74] E. J. Yarmchuck, M. J. V . Gordon, and R. Packard. Observation of stationary vortex arrays in rotating superfluid helium. Phys. Rev. Lett., 43:214–217, 1979. [75] S. Yoden and M. Yamada. A numerical experiment on two-dimensional decaying turbu- lence on a rotating sphere. J. Atmos. Sci., 50 (4):631–643, 15 Feb. 1993. [76] A. Youssef and P. S. Marcus. The dynamics of jovian white ovals from formation to merger. Icarus, 162 (1):74–93, March 2003. [77] A. Zebib. Stability of viscous flow past a circular cylinder. J Engng Maths, 21:155–165, 1987. 124 Appendices 125 Appendix A. Examples of Symmetric Lattices Figures A.60, A.61 and A.62 are a compact catalogue of various configurations that possess symmetries in addition to and their solutions (which is listed in the captions). 126 (a) N = 8, hollow contour (b) N = 9 (c) N = 25 Figure A.60: Square lattices. Solutions are (a) null space of 1 with = (2, 1, 2, 2, 1, 2, 1, 1) T , (b) null space of two =(0, 1, 1, 0.2, 0 , 1, 0, 1, 0) T +(1, 0, 0, -0.1, 1, 0, 1, 0, 1) T . (c) has null space of one with = (1, 76/327, 1, -76/327, 1, - 76/327, -37/218, -69/295, -37/218, -76/327, 1, -69/295, 1, -69/295, 1, -76/327, 37/218, -69/295, -37/218, -76/327, 1, -76/327, 1, -76/327, 1) T . Fractions represent numerical approximations. 127 (a) N = 37, Triple Hexagon (b) N = 13, Star (c) N = 10, Triangle Figure A.61: Triangular lattices: (a) null space of 1, solution is = (-11/9, - 3/2, -3/2, -3/2, -3/2, -3/2, -3/2, -34/9, -34/9, 1, 1, -34/9, -34/9, 1, 1, 1, 1, -34/9, -34/9, 1, 1, -14/9, -14/9, -14/9, -14/9, 1, 1, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, -14/9, 1, 1) T , (b) null space of two, solution is =(-6/7, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0) T +(13/14, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1) T , (c) null space of 1, solution is = (-6/7, 0, 1, 1, 0, 1, 1, 0, 1, 1) T . Fractions represent numerical approximations. 128 (a) N = 7, Two branches (b) N = 10, Three branches (c) N = 13, Four branches (d) N = 16, Five branches Figure A.62: Snowflake lattices. Characterized by concentric, rotational symmetry. The lattices above have a null space of one, with vortices that possess the same distance from the center having the same strength. (a) (0.1517, 0.1517, -0.1572, 0.1517, 0.1517, -0.1572, -0.0787) T , (b) (0.1134, 0.1134, -0.0741, 0.1134, 0.1134, -0.0741, 0.1134, 0.1134, -0.0741, -0.0973) T , (c) (0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, 0.0918, 0.0918, 0.0500, -0.0657) T , (d) (0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0265, 0.0265, 0.1341, 0.0640) T 129 Appendix B. Structure of the Configuration Matrix: Single Vortex Street We now comment on the structure of the configuration matrix corresponding to single vortex streets on the sphere for the case of a single vortex street with 3 vortices per ring. The configu- ration matrix takes the following form From the structure of the configuration matrix found in (5.108) forn = 3, one can readily conclude thatA = 0 gives rise to a total of fifteen equations: six equations that link exactly one vortex in the northern ring to one in the southern ring and nine equations that are identically zero. This information can be represented pictorially as done in Figure B.63. The number of filled vortices per ring is either 1 or 0 which means that one has 2 or 0 non- trivial entries in a given row of the configuration matrixA. In the case of 2 non-trivial entries, one entry is associated with a vortex from the northern hemisphere and the second is associated with a vortex from the southern hemisphere. For vortex streets withn> 3 vortices per ring, two distinct patterns emerge in the configu- ration matrix: one pattern forn odd and another pattern forn even. These patterns share some properties. The first similarity is that if, in a given row ofA, there aremn non-trivial entries associated with vortices from the northern hemisphere, there is an equal number of non-trivial entries associated with vortices from the southern hemisphere in that row. The second similarity 130 (a) 0/3 Filled, 9 eqs. (b) 1/3 Filled, 6 eqs. Figure B.63: Combinations found in the configuration matrix of a single V on K´ arm´ an Street on the Sphere with n = 3 vortices per ring. Image shows top view of one ring (either northern or southern). There are six equations in the configuration matrix with 0 non-trivial terms per ring, and another nine with 1 vortex from each ring a with non-trivial coefficient (a) N=7 Allowed (b) N=7 Not Allowed (c) N=8 Allowed (d) N=8 Not Allowed Figure B.64: Contrasting allowable and unallowable configurations forN = [7; 8]. Allowable configuration all have even symmetry with respect to a planar axis, while unallowable ones visibly do not have any symmetries. is that if there arem<n non-trivial entries in a given row ofA associated with vortices from the northern hemisphere, the ordering of these entries must possess a certain symmetry. For exam- ple, consider the configuration matrixA associated withn = 7 and say that one row ofA has 4 non-trivial entries associated with vortices from the northern hemisphere, these 4 entries can be ordered as shown in Figure B.64(a) but cannot appear in the order shown in Figure B.64(b). B.1 Odd Number of Vortices per Ring Forn 5, the number of vortices per ring corresponding to non-trivial entries in the configu- ration matrix is (n 2), (n 3) and (nn). For example, forn = 9, the number of filled vortices per ring is either 7, 6 or 0 as shown in Figure B.65, which means that one has 14, 12 or 0 non-trivial entries in a given row of the configuration matrixA. In the case of 14 non-trivial 131 Table B.4: Oddn Combination Distribution # Filled # Equations # of Total # of per ring per combination combinations equations n 2 2n (n 1)=2 n(n 1) n 3 2n (n 1)=2 n(n 1) nn n 1 n entries, 7 entries are associated with vortices from the northern hemisphere and 7 are associ- ated with vortices from the southern hemisphere – similarly in the case of 12 non-trivial entries. Rows for which the entriesA 1 ;A 2 ;A 4 ;A 5 ;A 7 ;A 8 are non-trivial correspond to the vortices [1; 2; 4; 5; 7; 8] being filled which is shown in Figure B.65(b). Note that the pattern of the con- figuration matrix is independent of the labeling of the point vortices. For example, one could choose to relabel vortices [1; 2; 4; 5; 7; 8] as [2; 3; 5; 6; 8; 9] or [3; 4; 6; 7; 9; 1], that is, to rotating the label about the polar axis – this does not affect the pattern of the configuration matrix. There are (n 1)=2 distinct combinations in the case of (n 2) or (n 3) filled vortices per ring. For example, whenn = 9, there are (9 1)=2 = 4 different combinations that contain six filled vortices per ring as depicted in Figures B.65(b-e), which means that one has 4 distinct rows ofA containing 12 non-trivial entries. Each combination pattern containing either n 2 or n 3 filled vortices is present 2n times (note in the labeling of Figure B.65 that any pattern with 7 or 6 filled vortices is repeated 2(9) = 18 times). Next, each non-trivial equation type in the configuration matrix is repeated exactly twice. This means that there are exactly two different equations containing non-trivial coefficients for, say, the case when [1,2,4,5,7,8] are filled vortices. Finally, there are n trivial equations present (i.e., nn filled vortices, or zero vectors in the configuration matrix). The generalized description is tabulated in B.65. B.2 Even Number of Vortices per Ring The pattern present for evenn (withn 4) is more elaborate. The possible number of vortices per ring with non-trivial entries in the configuration matrix aren, (n 1), (n 2) and (n 4). 132 (a) 1/9, 9x (b) 6/9, 18x (c) 6/9, 18x (d) 6/9, 18x (e) 6/9, 18x (f) 7/9, 18x (g) 7/9, 18x (h) 7/9, 18x (i) 7/9, 18x Figure B.65: Combination types present in the configuration matrix forn = 9 vortices per ring. Number of filled vortices per ring are 7, 6 and 0. Each non-trivial combination type is present 18 times, in addition to 9 trivial ones. For example, ifn = 10, the number of filled vortices per ring is either 10, 9, 8 or 6. For the first case of n non-trivial entries per ring, there are ( P (n=2)1 =1 2) corresponding rows in the configuration that have this structure, and this applies to the cases n 4. For example, in Figure B.66(g) whenn = 10, there are 20 rows in the configuration matrix that contain 10 non- trivial entries per ring. Second, there are ( P (n=2)1 =0 4 + 8) rows withn 1 filled vortices per ring, and this applies to the casesn 2. For instance, in Figure B.66(f) whenn = 10, there are 100 equations that contain 9 non-trivial entries from each ring. Third, there are ( P (n=2) =0 (42)) rows in the configuration matrix withn 2 non-trivial entries per ring, and this applies when n 2. However, the difference between then 2 case and the previous ones is that there is only one unique kind of combination possible forn orn 1 filled vortices, while there are more than one different combinations found forn 2 (andn 4) filled vortices, and the total number of rows containingn 2 non-trivial entries per ring is distributed amongst the different possible combination types according to a second sub-pattern. Figures B.66(c-e) show three different 133 Table B.5: Evenn Combination Distribution #Filled #Equations n P (n=2)1 =1 2;n 4 n 1 P (n=2)1 =0 4 + 8;n 2 n 2 P (n=2) =1 4 2;n 2 n 4 P (n=2) =1 2 2;n 4 (a) 6/10, 10x (b) 6/10, 10x (c) 8/10, 30x (d) 8/10, 10x (e) 8/10, 10x (f) 9/10, 100x (g) 10/10, 20x Figure B.66: Combination types present forN = 10. Number of filled vortices per ring are 10; 9; 8 and 6, repeated a total of 20; 100; 50 and 20 times respectively. types of combinations that have 8 filled vortices per ring, and these patterns are repeated in the configuration matrix 30, 10 and 10 times respectively, giving us a total of 50 rows in the configuration matrix that have 8 filled vortices per ring. Lastly, there are ( P (n=2) =1 (2 2)) rows that containn 4 non-trivial entries per ring, and this applies to the casesn 4. Table B.2 contains the generalized description for evenn. 134 Appendix C. Viscous Point Vortex Model This thesis has dealt exclusively with invcisid point vortices. As a supplemental reference, we now describe the incompressible viscous point vortex model as a starting point for the investigation of viscous lattices. An expression for a viscous point vortex can be derived by looking at the Navier Stokes equations in cylindrical coordinates, where the velocity field u(r;t) = (u r ;u ;u z ) (0;u ; 0). We consider the Navier Stokes equation corresponding to the component, with theLHS andRHS of the equation being LHS = @u @t +u r @u @r + u r @u @ +u z @u @z + u r u r ; (C.161) RHS = 1 r @p @ + 1 r @ @r r @u @r + 1 r 2 @ 2 u @ 2 + @ 2 u @z 2 + 2 r 2 @u r @ u r 2 +f : (C.162) Assumptions made about the system include: 1. The flow is axisymmetric:@=@ =@ 2 =@ 2 = 0. 2. Two-dimensional:@=@z =@ 2 =@z 2 = 0 &u z = 0: 3. No pressure gradient:@p=@ = 0. 4. No body forces:f = 0. Plugging these assumptions into our the NV equation we get 135 LHS = @u @t ; RHS = 1 r @ @r r @u @r : Therefore, our governing equation becomes @u @t = 1 r @ @r r @u @r : (C.163) The assigned boundary conditions for our problem are: 1. At t = t 0 , the point vortex has the same velocity field as an inviscid point vortex: u (r;t 0 ) = =2r. 2. As r!1, the inviscid point vortex resembles the inviscid point vortex8t : u (r! 1;t) = =2r. 3. The angular velocity at the center of the inviscid point vortex is zero:u (0;t) = 0. Plugging the assumptions above into the continuity equation in cylindrical coordinates (see Equation (2.10)), we have r u 0 = 1 r @ @r (ru r ) + 1 r @u @ + @u z @z = 0 + 0 + 0 0: Note that this is also the heat equation in cylindrical coordinates. One of way solving this equa- tion is by considering a similarity solution. We proceed by non-dimensionalizing the velocity in Equation (C.163) by defining 136 ^ u =u 2r : (C.164) With the chain rule, Equation (C.163) becomes @^ u @t = 1 r 1 r @^ u @r + @ 2 ^ u @r 2 : (C.165) The boundary conditions in terms of the non-dimensional velocity are 1. ^ u(r;t 0 ) = 1, 2. ^ u(r!1;t) = 1, 3. ^ u(0;t) = 0. Defining the variable =(r;t;) as = r 2 4t ; (C.166) we can find a similarity solution ^ u() when plugging Equation (C.166) into Equation (C.165) ^ u 00 () + ^ u 0 () = 0: (C.167) Settingt 0 = 0, the new boundary conditions 1. ^ u( = 0) = 1, 2. ^ u(!1) = 0, Our solution becomes ^ u() = 1e ; or 137 ^ u(r;t) = 1 exp r 2 4t ; or u (r;t) = 2r 1 exp r 2 4t : (C.168) The solution, therefore, coincides with a point vortex whose overall vorticity is preserved, but is advected outward away from the center of vorticity. The above construct for a visous point vortex contains radial symmetry reminiscent of the inviscid vortex model due to the face that there are no other sources of rotation in the flow. Introducing a second viscous vortex into the flow would break this symmetry, and make a clean formulation such as that in Equation (C.168) impossible. Therefore, in order to approximate viscous point vortex systems, restrictions on the inter-vortical viscous effects would have to be introduced. Although viscous vortex lattices will not be addressed in this thesis, the problem has been introduced and discussed briefly since it is an open topic worthy of notice and investigation. 138

Abstract (if available)

Abstract Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity(A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A "Brownian ratchet" construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm || . ||F and 2-norm || . ||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.

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Creator Chamoun, George C. (author)

Core Title Vortex lattice theory: a linear algebra approach

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace

Publication Date 07/28/2008

Defense Date 05/28/2008

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

Tag fluid mechanics,Hamiltonian,lattice,OAI-PMH Harvest,SVD,vortex

Language English

Advisor Newton, Paul K. (committee chair), Browand, Frederick (committee member), Friedlander, Susan (committee member), Ghanem, Roger (committee member), Kukavica, Igor (committee member)

Creator Email chamoun@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1418

Unique identifier UC1286877

Identifier etd-Chamoun-20080728 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-199701 (legacy record id),usctheses-m1418 (legacy record id)

Legacy Identifier etd-Chamoun-20080728.pdf

Dmrecord 199701

Document Type Dissertation

Rights Chamoun, George C.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu