TRANSPORT OF CHARGED PARTICLES IN TURBULENT MAGNETIC FIELDS. Prachanda Subedi

Transcription

1 TRANSPORT OF CHARGED PARTICLES IN TURBULENT MAGNETIC FIELDS by Prachanda Subedi A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Spring 2017 c 2017 Prachanda Subedi All Rights Reserved

2 TRANSPORT OF CHARGED PARTICLES IN TURBULENT MAGNETIC FIELDS by Prachanda Subedi Approved: Edmund Nowak, Ph.D. Chair of the Department of Physics Approved: George Watson, Ph.D. Dean of the College of Sciences Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education

3 I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: William H. Matthaeus, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Michael A. Shay, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Bennett A. Maruca, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Ikramul Huq, Ph.D. Member of dissertation committee

4 I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: John Morgan, Ph.D Member of dissertation committee

5 ACKNOWLEDGEMENTS This thesis would not have been possible without the support and help from my advisor, Prof. William Matthaeus. His passion and motivation for physics have always inspired me. The experience to work with him for the past 6 years will be very beneficial to my future career. I would like to extend special thanks to Prof. David Ruffolo and Prof. Pasquale Blasi for useful physics discussions and countless exchanges that have helped shape several aspects of this thesis. The discussion that we had during Prof. David Ruffolo s visit once a year in spring has been invaluable. I am thankful to my collaborators and friends: Dr. Tulasi Parashar, Dr. Jeffrey Tessein and Rohit Chhiber for a number of useful physics discussions that we had over the course of 6 years. I wish to thank my friend Dr. Wirin Sonsrettee for useful collaboration during his visit from Thailand. Also, thanks to Prof. Minping Wan for helping me jumpstart my PhD career. I am indebted to my wife, Rebecca Chalise for her support and cheerfulness. I thank my family (father, mother and brothers) for their love, support and encouragement over the years. Last but not the least, I would like to thank all the wonderful friends of the Nepalese Student Association (NSA), the people of the noisiest office in the department (Sharp lab 223) and the amazing staff of DPA for making my PhD career very enjoyable. iv

6 TABLE OF CONTENTS LIST OF FIGURES viii ABSTRACT xvii Chapter 1 INTRODUCTION AND OVERVIEW Hydrodynamic Turbulence Kolmogorov Spectrum Cosmic Ray Transport Interplanetary Magnetic Field Diffusion of Magnetic Field Lines Transport of Charged Particles in Turbulent Magnetic Fields Diffusion Coefficient Charged Particle Diffusion in Isotropic Magnetic Field Turbulence Numerical Simulation and the Need for Intermittent Fields DIFFUSION Random Walk Fick s Law Time and Length Scales in the Diffusion of Particles Lagrangian Picture and Taylor-Green-Kubo (TGK) Formula Corrsin s Independence Hypothesis Fokker-Planck Equation POWER SPECTRUM AND CORRELATION LENGTH D and Slab Power Spectrum Power Spectrum in 3D Turbulence Fully 3-Dimensional Isotropic Turbulence Correlation Length v

7 4 MAGNETIC FIELD LINE DIFFUSION Magnetic Field Line Preliminary Work on Slab+2D Turbulence Diffusion of Magnetic Field Lines in Isotropic Turbulence Diffusive Decorrelation (DD) Random Ballistic Decorrelation (RBD) Self Closure Method Field Line Random Walk Model for Particles and its Limitations NUMERICAL SIMULATION Governing Equations of Motion Generating Slab and 2D Fields Generating 3D Random Gaussian Magnetic Fields Numerical code History Diffusion Coefficient Importance of Large Separation of Scales CHARGED PARTICLE DIFFUSION IN ISOTROPIC TURBULENCE Previous Studies and New Ideas Velocity Space Diffusion and Spatial Diffusion in Isotropic Turbulence Taylor-Green-Kubo Formula for Velocity Space Diffusion and Spatial Diffusion ISOTROPIC TURBULENCE: HIGH ENERGY THEORY Alternative Derivation Nonlinear Extended High Energy Theory ISOTROPIC TURBULENCE: LOW ENERGY THEORY Relation Between Parallel Spatial Diffusion and Pitch Angle Diffusion Generalized Pitch Angle Diffusion Coefficient Standard QLT Approach vi

8 8.4 Extended Low Energy Theory ISOTROPIC TURBULENCE: COMPARISON OF THEORETICAL RESULTS WITH NUMERICAL SIMULATION High Energy Relativistic Particles in extra-galactic Medium Non-Relativistic Example Discussion Implication of the Results SYNTHETIC GENERATION OF INTERMITTENT FIELDS Physical Motivation Extension of the Procedure to a Plasma Methods ANALYSIS AND APPLICATIONS OF SYNTHETIC INTERMITTENT FIELDS Intermittency Analysis of Synthetic Fields Scale dependent Kurtosis Tests in direct numerical simulation (DNS) of decaying turbulence Comparison with Solar Wind Observations Background Solar Wind Analysis Comparison Effect of intermittency on diffusion Discussion CONCLUSION AND FUTURE WORK REFERENCES Appendix A SOME VERIFICATION OF THE NUMERICAL CODE B ANGULAR DEFLECTION OF A FIELD LINE OVER AN INERTIAL RANGE OF TURBULENCE C PERMISSION LETTERS vii

9 LIST OF FIGURES 1.1 Laminar flow vs Turbulent flow. Laminar flow has parallel layers, with no disruption between the layers, whereas the turbulent flow has lateral mixing, eddies and swirls of fluids Turbulent Cascade. Larger eddies of outer scale break down into smaller eddies which break down into even smaller eddies until the smallest dissipation scales where the flow energy is converted to heat Kolmogorov spectrum at the inertial scale schematically presented as a power spectrum for magnetic fluctuations of the solar wind [54] Schematic representation of the magnetic field lines originated in the solar system and the transport of particles in the field. These field lines have irregularities at very small scales that influences the motion of particles. Solar energetic particles and galactic cosmic rays travel through the magnetic fields that have fluctuations at various scales before being detected on Earth Schematic representation of a slab field (left). It shows that the field is constant within an x y plane but varies along z. The figure to the right shows trajectories of field lines in the presence of a slab field. (Image credit: Piyanate Chuychai) Illustration of a 2D field. The solid arrows show the direction of a 2D magnetic field, b 2D, which lies in the contour of a(x, y) (solid lines). b 2D is along the equipotential of a(x, y) and a(x, y) is perpendicular to b 2D. The figure at the right shows an example of a trajectory of a 2D field line. (Image credit: Piyanate Chuychai) An example of random walk of a magnetic field line. The tracing of the field line was done numerically in a 3D isotropic field viii

10 1.8 The complexity of magnetic field turbulence causes charged particles to scatter from one field line to another. The particle initially following a field line B, scatters to field line C and then scatters to field line A Transport process of a charged particle in random magnetic field. a) Field line random walk limit (FLRW) where a particle keeps following a field line without parallel scattering. b) The particle is initially following a field line but parallel scattering causes the particle to backtrack along the same field line. c) Particle follows the field line and backtracks but due to the complexity of the magnetic field turbulence, the particle scatters from the field line Particle in a slab field. The particle keeps gyrating along the field line until resonant scattering causes particle to backscatter which causes subdiffusion in the perpendicular direction. Figure to the right shows continuous decrease in running perpendicular diffusion coefficient even during later times [118] Particle in a 2D+slab field. The addition of 2D field has enough complexity for the particle to scatter off the field line compared to the slab field (Figure 1.10). The particle has initial free streaming regime followed by subdiffusion due to backscatter then second diffusion due to scatter off the field line. Figure on the right shows stable value of running perpendicular diffusion coefficient at later times [117] The plot to the left shows high energy particle diffusion in a 3D isotropic field where the Larmor radius of the particle is greater than the correlation length of turbulence. The initial free streaming is followed by stable diffusive regime with no signs of subdiffusion. The figure on the right is an example of magnetic field line diffusion in a 2D+slab field where there is no subdiffusion at any point The yellow lines are the particles and the black lines represent the field lines. As shown the high energy particles deflect by a small angle as they travel about a coherence length of a field line whereas the low energy particles follow a local magnetic field line and backscatter along the field line due to small scale fluctuations of the field line Probability distribution of the z-component of current (solid line) in a synthetic intermittent field. Intermittency causes wider tails on the distribution compared to the Gaussian (dashed line) ix

11 1.15 Contour of the z-component of intermittent current in a selected plane from a MHD simulation. The intermittent field has thin and elongated structures of current sheets where most of the current is concentrated The plot at the top is the Microscopic Diffusion Coefficient vs time with the stable value of the diffusion coefficient at 7.2. The plot at the bottom is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. A total number of 2000 charged particles were used in isotropic turbulent magnetic field. The dotted line is the analytical curve with the diffusion coefficient Lagrangian coordinate system where the path of the marked fluid particle is defined by the dashed line The plot on the left shows longitudinal correlation in which the magnetic field components are in the same direction as the separation r. The plot on the right shows transverse correlation where the magnetic components are perpendicular to the separation r Longitudinal f(r) and transverse g(r) magnetic correlation functions. The correlation functions asymptotically approaches zero at later times but g(r) first goes slightly negative before nearing zero Slab correlation function R slab xx (z) vs distance z. The correlation is higher at smaller distance and decreases exponentially with the distance Parametrization s along a field line. ds is the infinitesimal arc length along the local magnetic field vector B Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. Here δb/b 0 = 0.3 and the fraction of slab energy in the fluctuation field is Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. Here δb/b 0 = 0.5 and the fraction of slab energy in the fluctuation field is x

12 4.4 Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. δb/b 0 = 0.9 and the fraction of slab energy in the fluctuation field is Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. δb/b 0 = 0.5 and the fraction of slab energy in the fluctuation field is Results of the diffusion coefficients of field lines from simulation using parametrizations s and τ. Since the turbulence is isotropic the three spatial coordinates are statistically identical, and therefore the diffusion coefficients with respect to different coordinates coincide Magnetic field line diffusion coefficients, averaged over the three spatial directions as a function of τ. The diffusion coefficient is calculated numerically over three different spectral index (Γ) of fluctuation power spectrum: a) Γ = 3/2, b) Γ = 5/3 and c) Γ = Slab spectrum D spectrum Omni-directional Spectrum of the magnetic field fluctuations Diffusion coefficient, macroscopic (top) and microscopic (bottom) when R L /l c = The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient which is close to the microscopic diffusion result on the bottom Diffusion coefficient, macroscopic and microscopic when R L /l c = 0.3. The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient which is close to the microscopic diffusion result on the bottom xi

13 5.6 Diffusion coefficient, macroscopic and microscopic when R L /l c = 1.1. The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient 49 which is close to the microscopic diffusion result on the bottom Microscopic vs macroscopic diffusion coefficients with good agreement Omni-directional spectrum with larger inertial range bandwidth Omni-directional spectrum with smaller inertial range bandwidth Pitch angle correlation of low energy particles with larger (solid line) and smaller (dotted) inertial range bandwidth. Larger inertial range bandwidth contains small scale fluctuation that resonate with particle motion which causes pitch angle scattering and therefore the pitch angle remains correlated only for a short period of time The effect of the resolution on low energy scaling. The inverted triangles are the results of numerical simulation, circles are those of simulation and the crosses are those of numerical simulation. The results of and simulation lay on top of each other but the low resolution simulation give almost constant diffusion coefficient for low energy particles, implying FLRW phenomenon due to the lack of resonant scales Mean free path of protons as a function of R L /l c (gyroradius divided by correlation scale). The solid line is the high energy scaling λ xx RL 2 and the dashed line is the low energy scaling λ xx R 1/3 L. The inverted triangles are the results of numerical simulation, circles are those of simulation and the crosses are the results of numerical simulation Theoretical vs. numerical results for cosmic rays in isotropic turbulence with zero mean field, showing good agreement. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for cosmic rays in our galaxy xii

14 9.3 Theoretical vs. numerical results for high energy relativisitc charged particles in isotropic turbulence for parameters as listed in Section 9.1. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and the dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for protons and magnetic field parameters for extra-galactic medium Parametrization of the diffusion coefficient with energy. κ E 1/3 for low energy particles and κ E 2 for high energy particles Theoretical vs. numerical results for non-relativistic or moderately relativistic charged particles in isotropic turbulence with zero mean field for parameters as listed in Section 9.2. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and the dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for protons and magnetic field parameters for interplanetary turbulence. The results also apply to any non-relativistic particles of a given R L /l c Plot of the energy spectrum being used and schematic representation of the hierarchy of nested levels Fluid particles for a particular filtering level, shown on the regular grid with the arrows representing the velocities of the fluid parcel at each node Fluid particles after being displaced from their original positions. Smaller dots represent the regular grid points. The velocities are interpolated back to each regular mesh point by using a weighted average over a sphere of radius R centered around the mesh point PDF of the longitudinal velocity field increment calculated from synthetic data with M = 5. With the increase in the increment r (direction of arrow) the PDF becomes more Gaussian, showing correct intermittent behavior. The smallest r is about 1/50 of the correlation length and the largest r is about a correlation length xiii

15 11.2 PDF of the transverse velocity field increment calculated from synthetic data with M = 5. With the increase in the increment r (direction of arrow) the PDF becomes more Gaussian showing correct intermittent behavior. The smallest r is about 1/50 of the correlation length and the largest r is about a correlation length PDF of longitudinal increment for synthetic magnetic field data produced after M = 3 iterations of the filtering and mapping method. The arrow indicates the direction of increase in separation r/l c for the various curves. Those closer to Gaussian are at larger r/l c. Intermittency is apparent in the fatter tails at smaller scale increment lag PDF of longitudinal increment for synthetic magnetic field data produced after M = 7 iterations of the filtering and mapping method. The arrow again indicates the ordering of the curves with increasing separation r/l c. Curves closer to Gaussian are at larger r/l c. The intermittent behavior is more pronounced as compared to the case with M = Scale dependent kurtosis of longitudinal magnetic increments for the synthetic field, after different total number of filtering/mapping iterations M. The kurtosis at small scales is seen to increase with M, indicating that increased number of iterations increases intermittency PDF of the z-component of current (J z ), calculated from synthetic data after M = 7 iterations, showing intermittent behavior The evolution of total energy (magnetic and kinetic) from MHD simulations using Gaussian initial conditions and using synthetic intermittent initial conditions. The total energy decays slightly faster using the synthetic intermittent fields as the initial condition in MHD simulations Contours of J z in a selected plane from a random phase Gaussian dataset. It has no significant structures Contours of J z in a selected plane from a synthetic intermittent field. The current is concentrated mostly in small volumes creating blob-like intermittent structures. The spatial structures can be considered to occupy an intermediate state between the Gaussian fields and the fields run through MHD simulations xiv

16 11.10 Kurtosis of longitudinal magnetic increment as a function of spatial lag r normalized by correlation scale l c, at different times, from MHD simulations. The run initialized with synthetic intermittent fields starts at higher kurtosis at time τ NL = 0. It is much higher at τ NL = 0.1, even though the run initialized with the Gaussian field shows intermittency at this time. At time τ NL = 0.5 the fields are highly intermittent using either of the initial conditions, however the case using synthetic intermittent fields as initial conditions maintains higher kurtosis Sample PDFs of kurtosis of longitudinal magnetic increments in the solar wind with varying lags: 10di, 100di and 1000di using 1 second ACE magnetic field data. The kurtosis increases with the decrease in lag and approaches three as the lag is increased because intermittency is reduced Comparison of kurtosis of longitudinal magnetic increment for the solar wind, synthetic intermittent field and the field run through a MHD simulation. The total number of levels used for the synthetic field is Diffusion coefficient vs the gyroperiod of the particles with R L /l c = 0.2, in intermittent and Gaussian fields. It is clear that intermittency enhances the diffusion coefficent Diffusion coefficient vs the gyroperiod of the particles with R L /l c = 0.5, in intermittent and Gaussian fields Diffusion coefficient vs the gyroperiod of the particles with R L /l c = 0.8, in intermittent and Gaussian fields. It is clear that intermittency enhances the diffusion coefficent but the effect is smaller when compared to R L /l c = 0.2 and R L /l c = Diffusion coefficient vs the gyroperiod of the particles with R L /l c = 1, in intermittent and Gaussian fields. The effect of intermittency is negligible for higher energy particles that have Larmor radius comparable to the correlation length of magnetic field A.1 Accuracy of particle orbit in a uniform magnetic field A.2 Accuracy of particle orbit in a constant magnetic field and constant electric field xv

17 A.3 Accuracy of particle orbit in isotropic magnetic field with root mean square field, δb = A.4 Diffusion Coefficient of magnetic field lines in slab turbulence, with solid line from simulation and dashed line from analytical theory. The ratio of root mean square field to the mean field, δb/b 0 = A.5 Diffusion Coefficient of magnetic field lines in slab turbulence, with solid line from simulation and dashed line from analytical theory. The ratio of root mean square field to the mean field, δb/b 0 = A.6 Diffusion Coefficient of magnetic field lines in slab turbulence, with solid line from simulation and dashed line from analytical theory. The ratio of root mean square field to the mean field, δb/b 0 = A.7 Diffusion Coefficient of magnetic field lines in slab turbulence, with solid line from simulation and dashed line from analytical theory. The ratio of root mean square field to the mean field, δb/b 0 = A.8 Diffusion Coefficient of magnetic field lines in slab turbulence, with solid line from simulation and dashed line from analytical theory. The ratio of root mean square field to the mean field, δb/b 0 = xvi

18 ABSTRACT Magnetic fields permeate the Universe. They are found in planets, stars, galaxies, and the intergalactic medium. The magnetic field found in these astrophysical systems are usually chaotic, disordered, and turbulent. The investigation of the transport of cosmic rays in magnetic turbulence is a subject of considerable interest. One of the important aspects of cosmic ray transport is to understand their diffusive behavior and to calculate the diffusion coefficient in the presence of these turbulent fields. Research has most frequently concentrated on determining the diffusion coefficient in the presence of a mean magnetic field. Here, we will particularly focus on calculating diffusion coefficients of charged particles and magnetic field lines in a fully three-dimensional isotropic turbulent magnetic field with no mean field, which may be pertinent to many astrophysical situations. For charged particles in isotropic turbulence we identify different ranges of particle energy depending upon the ratio of the Larmor radius of the charged particle to the characteristic outer length scale of the turbulence. Different theoretical models are proposed to calculate the diffusion coefficient, each applicable to a distinct range of particle energies. The theoretical ideas are tested against results of detailed numerical experiments using Monte-Carlo simulations of particle propagation in stochastic magnetic fields. We also discuss two different methods of generating random magnetic field to study charged particle propagation using numerical simulation. One method is the usual way of generating random fields with a specified power law in wavenumber space, using Gaussian random variables. Turbulence, however, is non-gaussian, with variability that comes in bursts called intermittency. We therefore devise a way to generate xvii

19 synthetic intermittent fields which have many properties of realistic turbulence. Possible applications of such synthetically generated intermittent fields are discussed. xviii

20 Chapter 1 INTRODUCTION AND OVERVIEW Turbulence is a feature of fluid flows. Very little fundamental research on turbulence was conducted until the earlier 20th century, mostly because of the difficulty that arises from the non-linearity of the problem. However, the study of turbulence has been greatly enhanced since the second half of the 20th century, thanks to some pioneering works by Von Kármán, Kolmogorov and Batchelor (see for e.g. [48] and [8]). The advancement in computer technology, data processing and signaling and experimental research has aided in the study of turbulence. Despite all these efforts, the fundamental problems of turbulence to this date remain unsolved, often referred to as the unsolved problem of classical physics. Our visible universe is mostly filled with electrically charged particles that move under the influence of electric and magnetic fields. They are abundantly found in the solar system and Earth s magnetosphere. The collective movement of these plasmas can be treated as fluids, but the physical laws they follow from the influence of the electromagnetic fields may be different compared to those followed by the neutral fluid. Nevertheless, turbulence exists in both cases and determines the dynamics of the flow. The study of the plasma flow as fluids is known as magnetohydrodynamics (MHD) and the turbulence associated with it as magnetohydrodynamic turbulence. The plasma found in the interplanetary system is accessible to the in situ measurements made by the spacecraft that helps in predicting space weather and testing scientific theories. Magnetic fields are present everywhere in space including the earth, sun, stars, interstellar and intergalactic space. These magnetic fields are distorted by the charged particles or plasma around it, and the turbulent motion of these particles is influenced by the magnetic field. Hence, the characteristics associated with these magnetic fields 1

21 are considered to be turbulent. The transport of charged particles in the presence of these turbulent magnetic fields has become an important problem in physics. The charged particles of astrophysical origin are often called cosmic rays, which have been studied for a long time. Their interaction with the magnetic field has been observed and analyzed intensively by the scientists. Cosmic rays could be of solar origin or galactic origin. From first principles, the motion of charged particles in an electromagnetic field is governed by the Lorentz force. If we could solve the Lorentz force equation of motion (Equation 1.1) for the charged particles [ ] d 1 dt [γmv(t)] = q v(t) B(x, t), (1.1) c (γ is the Lorentz factor) then practically our job would be done. But the motion of these charged particles has to be modeled in the presence of turbulent magnetic field, which is what makes Equation 1.1 very difficult to solve (analytically or numerically), so we have to use a statistical approach. The interaction of the turbulent magnetic fields with the motion of the charged particles is explored in this thesis. The heliospheric and galactic magnetic fields are all found to be turbulent. In the case of strong turbulence the root mean squared fluctuating (or turbulent) magnetic fields are of the order or greater than the background or the mean magnetic field. In some cases of galactic magnetic fields, there exist regions with negligible background magnetic fields where only turbulence prevails. We will mostly investigate the scenario where the turbulent field is dominant. The deflection of charged particles due to magnetic field turbulence can be described by diffusive phenomenon. Therefore, one of the important aspects of understanding the propagation of particles is to know the diffusion coefficients when turbulence parameters are given. The biggest contribution of the thesis work presented here, is the development of a complete theory that calculates the diffusion coefficients of charged particles in isotropic magnetic field turbulence. The study as presented is of highest importance to understanding the transport of cosmic rays in the galaxy. The 2

22 theories are also compared with a detailed numerical simulation of charged particles in isotropic turbulent field, yielding excellent agreement. This chapter provides an overview of several aspects of cosmic ray transport processes. Chapter 2 presents important fundamental and mathematical ideas of diffusion. Chapter 3 discusses turbulence parameters such as correlation length and energy spectrum. We discuss the magnetic field line diffusion process, particularly the field line diffusion theory in isotropic turbulence developed by us, in Chapter 4. The details of numerical simulation used in this thesis work is provided by Chapter 5. Chapters 6, 7 and 8 provide the details of the theory of charged particle diffusion in isotropic magnetic field turbulence, as a significant part of the thesis work. The theories are compared with the numerical simulations in Chapter 9. In Chapters 10 and 11 we provide a method of generating intermittent magnetic fields with several realistic properties of turbulence closely based on the method used by Rosales et al. [123] in hydrodynamic turbulence for velocity fields. Several implications of the proposed method on plasma physics and magnetohydrodynamics is discussed. Chapter 12 concludes the thesis work with some discussion and possible future expansion of the ideas presented in the thesis. 1.1 Hydrodynamic Turbulence Turbulence is a ubiquitous phenomenon. It is found in space, the atmosphere, oceans and basically everywhere around us. Turbulence is typically initiated by random perturbation and instabilities. For example, an instability is created on the interface between two fluids of different densities that occurs when one of the fluids is accelerated into the other. This type of instability is called Rayleigh-Taylor instability, which helps in generating turbulence. Similarly, when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids, an instability is created called Kelvin-Helmholtz instability, which initiates turbulence. In space we encounter the most naturally conducting fluid called plasma, where we observe instabilities and turbulence. 3

23 Figure 1.1: Laminar flow vs Turbulent flow. Laminar flow has parallel layers, with no disruption between the layers, whereas the turbulent flow has lateral mixing, eddies and swirls of fluids. 4

24 Turbulent flows exhibit three dimensional irregular rotational motion when compared to the smooth laminar flows. Figure 1.1, shows examples of a laminar flow and a complicated turbulent flow where there is rotation of fluids in different directions. The most important equation that describes hydrodynamic turbulence is the Navier-Stokes equation u t + u u = P + ν 2 u, (1.2) where u is the velocity vector, P is the pressure and and ν is the viscosity. incompressible fluids, u = 0. The Navier-Stokes equation is Newton s second law of motion applied to fluids. The term u u is the advection term and the term ν 2 u is the dissipation term. The ratio between the advection term and the dissipation term, which is called Reynolds number R given by For R = UL ν, (1.3) where L is the characteristic turbulence length scale. Typically, turbulence arises when R is large. Hence, Reynolds number is a dimensionless number that gives a measure of turbulence. The advection term u u is a nonlinear term that makes the Navier-Stokes equation extremely difficult to solve but interesting at the same time. It is accountable for transferring the energy from the mean flow to the turbulent flow and the breakdown of larger eddies to smaller eddies called turbulent cascade. The turbulent cascade is an important feature of turbulence shown schematically in Figure 1.2. The concept of energy cascade from the largest scales to the smallest scales in the turbulent flow was first introduced by Richardson (1922) [154]. The idea is that the energy fed into the turbulence at the largest scales of motion is then transferred to smaller and smaller scales until the smallest ones where the energy is dissipated to heat. 5

25 Figure 1.2: Turbulent Cascade. Larger eddies of outer scale break down into smaller eddies which break down into even smaller eddies until the smallest dissipation scales where the flow energy is converted to heat. 6

26 1.2 Kolmogorov Spectrum Kolmogorov s 5/3 law [82] is a simple scaling law for a high Reynolds number in fluid turbulence [8, 48] which is also applicable in plasma turbulence [80, 157]. Kolmogorov assumed that both the energy transfer and the interacting scales are local. This picture of energy transfer process is just like the one shown in Figure 1.2: force is applied to the fluid flow at a length scale l 0, injecting energy to the flow. l 0 is the largest scale of turbulence also called integral scale. The fluid motion at scale l 0 becomes unstable and loses its energy to neighboring smaller scales without directly dissipating it into heat (local energy transfer to inertial scales). This process repeats itself until one reaches the dissipation scale l d, where the energy transfer is directly dispersed into heat by viscous action. An important Kolmogorov s hypothesis is that in the limit of large Reynolds number, all the possible symmetries of the Navier-Stokes equation, usually broken by the mechanisms of producing turbulent flow; are restored in a statistical sense at small scales away from the boundaries [8]. If l 0 is the integral scales or the outer scale (the size of the largest eddies) that drives turbulence, then inertial scales l are of the order l << l 0 (Figure 1.3). Dissipation of turbulent energy into heat occurs at scales even smaller than inertial scales as shown in Figure 1.3, where Kolmogorov s law is not applicable. According to Kolmogorov s universality assumption, in the limit of high Reynolds number, all the inertial range properties are uniquely and universally determined by the scale l and the mean energy dissipation rate ɛ. Using this assumption a simple dimensional analysis can be performed to arrive at the highly celebrated result of the 5/3 power law. If u 2 is the amount of kinetic energy per unit mass of the eddies then the rate of transfer of energy is assumed to be proportional to u/l. The rate of energy supplied to the small scales is thus of the order of u 2 u/l = u 3 /l. This supply rate must be equal to the dissipation rate, so 7

27 Figure 1.3: Kolmogorov spectrum at the inertial scale schematically presented as a power spectrum for magnetic fluctuations of the solar wind [54]. 8

28 ɛ = u3 l. If the time scale is given by t, then the energy flow ɛ is of the order of l 2 /t 3. The wave number k is of the order of 1/l. Now the energy spectrum (E), which is defined as energy per unit mass per unit wavenumber of an eddy of length l, is given simply by dimensional analysis as of the order of l 3 /t 2. If we assume that E is mathematically only related to the energy flow ɛ and the wavenumber k and does not involve anything else, E must be of the order Cɛ x k y, where C is a dimensionless constant. Performing dimensional analysis to find the exponents x and y we get ( ) l 3 l 2 x ( ) y 1 t =. t 2 = t 3x x = 2 2 t 3 l 3, l 3 = l 2x y y = 5 3. Finally we arrive at Kolmogorov s 5/3 law E(k) = Cɛ 2/3 k 5/3. C is found to be close to 1 by experiments. The energy spectrum E(k) of interplanetary magnetic fluctuations (IMF) has been shown to follow Kolmogorov s 5/3 law closely at the inertial range [22, 88, 155]. The Power spectrum of the velocity fluctuations is also been shown to be closely related to the 5/3 law [113]. Figure 1.3 schematically shows the 5/3 rd slope of the inertial range. To this date, the spectral scaling laws in the solar wind and in MHD turbulence are still a subject of ongoing research. 1.3 Cosmic Ray Transport Cosmic rays are energetic charged particles, protons, electrons and ions that can have several origins, including solar cosmic rays, galactic cosmic rays, or extragalactic cosmic rays. Figure 1.4 shows cosmic rays being detected on earth, coming from several 9

29 sources ranging within the heliosphere and outside. The energy of cosmic rays can be as high as ev. There are some suggestions that the highest energy cosmic rays detected on earth are of extragalactic origin [36, 62, 164, 167]. The origin of cosmic rays in itself is an ongoing research with various observatories and satellites devoted to studying them. In this thesis we will mostly be interested in studying the interaction between the charged cosmic particles with turbulent magnetic fields in the interplanetary and interstellar medium, which causes the cosmic rays to deviate away from the original source. The prominent source of energetic charged particles originating within the solar system is attributed to various dynamical solar activities like the solar wind and Coronal Mass Ejection [64, 65, 76, 158]. The solar wind is an ionized gas, or plasma, continuously flowing out of the sun in all directions. Biermann et al. [15] suggested solar wind is based on the formation of enormous comet tails away from the Sun. Hence, the constant flux of hot plasma flowing outward from the sun interacts with planets and comets, affecting their atmospheres. There are two kinds of solar wind, a slow wind which travels at about 400 km/s, and a fast wind which travels at about 800km/s [40]. The solar wind is turbulent and nearly collisionless [53]. The region where the solar wind originates is the corona, the outer atmosphere of the sun. The Solar wind was discovered by the Soviet spacecraft Luna in Ever since then many spacecraft missions have been launched to observe the solar corona and solar wind (IMP, Marinar, Helios, Ulysses, Advanced Composition Explorer (ACE), Transition Region and Coronal Explorer (TRACE) to name a few). These satellites continuously provide us with various data that can enable us to understand better the solar wind, particle acceleration and diffusion, the solar corona, and solar activities. The turbulent solar wind magnetic energy spectrum displays a power law behavior, suggesting a cascade of energy from the largest energy containing scales to the smallest kinetic scales [2, 25]. A coronal mass ejection (CME) is a bright structure propagating outward through the solar corona more frequently during the solar maxima. The speed of coronal mass 10

30 ejections are highly variable, ranging from 10 km/s to greater than 2000 km/s. CMEs used to be viewed as a phenomenon accompanying large flares, but recently it has been suggested that CME is a primary energy release, while flare is a secondary process [77]. These activities including flares and CMEs provide an abundance of charged particles, and the knowledge of the propagation of these particles is not only essential to understanding the fundamental physical processes, but can also be used for practical purposes such as predicting space weather. Although the contribution of particles of solar origin is somewhat larger, there is a significant amount of cosmic rays which enter the solar system mostly believed to be originated in our own galaxy, with the exception of extremely high energy particles. The origin of the highest energy cosmic rays could also be related to the violent activities in the galaxy such as supernova explosions [21]. Figure 1.4 shows the schematics of the cosmic rays that originate within our own galaxy. The neutral particles are mostly traveling in a straight line, while the charged particles such as protons are affected by the electromagnetic fields and deviate from their original path before coming to the earth. This scattering of charged particles can be caused by both the fields in the galaxy and in the solar system. Because of the turbulent nature of the fields, the particle path is difficult to predict. As shown in Figure 1.4, although the field averaged over some large scale looks smooth, as we go down to smaller scales we can see more details of the small scale irregularities that are important in the scattering of charged particles [61, 109, 130]. In the presence of a constant electromagnetic field the motion of charged particles can be theoretically derived without assumptions using Equation 1.1, but turbulence in the interplanetary and interstellar medium causes charged particles or cosmic rays to undergo scattering parallel and perpendicular to the mean field and therefore their path is difficult to predict. Mean field or background field is referred to the average field over a long period of time, or over an ensemble of several realizations. The scattering of particles by turbulent magnetic field causes stochastically distributed path of particles. This leads to the suggestion that the scattering of the particles is 11

31 Figure 1.4: Schematic representation of the magnetic field lines originated in the solar system and the transport of particles in the field. These field lines have irregularities at very small scales that influences the motion of particles. Solar energetic particles and galactic cosmic rays travel through the magnetic fields that have fluctuations at various scales before being detected on Earth. 12

32 diffusive in nature and their transport in the interplanetary or interstellar medium can be described by a spatial diffusion coefficient, or alternatively by mean free paths. Understanding the diffusive effects due to a magnetic field is essential to study cosmic ray propagation [69]. Since diffusion is a stochastic process, it is unreasonable to discuss the motion of individual particles; instead one has to consider an ensemble of particles described by a distribution function. The turbulent nature of magnetic field also causes the field lines themselves to random walk in space. This random walk causes diffusion of magnetic field lines in space, which has been used to predict the diffusion of charged particles following these field lines [73]. In reality, however, the turbulent magnetic fields give rise to the perturbed motion of charged particles that deviate away from their original helical trajectory increasingly with time. This perturbation gives rise to diffusive propagation of charged particles different from the ones caused solely by the field line diffusion. The diffusive limit of these particles is achieved after some characteristic time, called the mean free time, analogous to other diffusive processes such as Brownian motion. The transport of cosmic rays in the interstellar medium is also an important topic in astrophysics. Ultra high energy particles (energies above ev, up to the ones that lie below the GZK cutoff) produced during extreme events in galaxies are extensively studied by astrophysicists [3, 11, 31]. The Greisen-Zatsepin-Kuzmin (GZK) cutoff [58, 169] is present if the UHE (protons, nuclei or photons) have energies greater than ev, when they start loosing energy to the photopion production off the cosmic microwave background (CMB). The nature, origin and the acceleration to extremely high energies of the UHE have triggered considerable interest in the astrophysics community. Also, the detection of cosmic rays above the GZK cutoff [16,17,87] has left scientists with many unanswered questions. This is beyond our scope. We will try to answer the question of how the propagation of high energy particles is affected by the turbulent magnetic fields present in the interstellar medium, which will be crucial to locate the source of cosmic rays by understanding the deflection of these particles by turbulent fields. 13

33 1.4 Interplanetary Magnetic Field The turbulent plasma of the solar wind drags the magnetic field of the sun as it propagates radially and fills the interplanetary space. This field is called the interplanetary magnetic field (IMF) which obtains its turbulent characteristic from the solar wind plasma. Since the sun is rotating and the solar wind is escaping radially outwards the IMF looks like an Archimedian spiral [108]. One of the models that has been used to describe the turbulence in the IMF is the 2D+slab model [98], also called the two component model. The magnetic field of the IMF can be written as the sum of the mean and a fluctuating field B = B 0 + b(x, y, z), (1.4) where the fluctuating field b is perpendicular to the mean field B 0. The magnetic field is considered to be static and homogeneous (statistical quantities are invariant under translation). In the two component model the fluctuating field can be written as the sum of the 2D-component b 2D which depends only on the perpendicular components x and y, and b slab which depends only on the z-coordinate along the mean field B 0, assuming the mean field is in the z-direction. Thus, the fluctuation can be written as The total field is given by b(x, y, z) = b 2D (x, y) + b slab (z). (1.5) B = B 0 + [b 2D x (x, y) + b slab x (z)]ˆx + [b 2D y (x, y) + b slab y (z)]ŷ. (1.6) Turbulent phenomena are often studied by working with the Fourier transform of the fields. The wave vector obtained after the Fourier transform of the slab field is parallel to the direction of the mean magnetic field. The 2D field has a wave vector perpendicular to the mean field and the magnetic fluctuation is perpendicular to both the wave vector and the mean field. 14

34 Figure 1.5: Schematic representation of a slab field (left). It shows that the field is constant within an x y plane but varies along z. The figure to the right shows trajectories of field lines in the presence of a slab field. (Image credit: Piyanate Chuychai) An example of a slab field is illustrated in Figure (1.5). The field is the same in the x, y plane but varies along z. Also, illustrated in Figure (1.5) are the trajectories of magnetic field lines in pure slab turbulence where the 2D component is zero. Figure (1.6) shows the 2D component of the field when the slab field is absent. It is easy to see that the 2D-component of the field can also be written in terms of the vector potential as b 2D (x, y) = [a(x, y)ẑ] = a(x, y) ẑ, (1.7) which implies that the 2D field must be in the direction perpendicular to the gradient of the potential and z- direction. Thus 2D field must be along the equipotential line of a(x, y), as shown in Figure 1.6. Figure 1.6 also shows the trajectory of a magnetic field line in pure 2D turbulence. The 2D-component without the slab component of a field line is non-diffusive (the trajectory in Figure (1.6) also suggests it), while the 2D+slab component of the magnetic field line is diffusive. 15

35 Figure 1.6: Illustration of a 2D field. The solid arrows show the direction of a 2D magnetic field, b 2D, which lies in the contour of a(x, y) (solid lines). b 2D is along the equipotential of a(x, y) and a(x, y) is perpendicular to b 2D. The figure at the right shows an example of a trajectory of a 2D field line. (Image credit: Piyanate Chuychai) 1.5 Diffusion of Magnetic Field Lines Understanding of the magnetic field line random walk is very important for the purpose of studying the transport of energetic particles in an astrophysical system. Turbulent magnetic fields are omnipresent in astrophysical plasmas, and the motion of the charged particles are determined by these magnetic fields in many ways. Since the particles in general gyrate around the magnetic field lines, they travel parallel to the magnetic field much more rapidly than perpendicular to the magnetic field. The solar magnetic field is dragged outward from the sun by the solar wind. The particles ejected outward from the violent events near the sun rapidly travel along those magnetic field lines (Figure 1.4). Hence, it is very useful to know the behavior of these field lines that guide the motion of the particles. The complex trajectories defined by an ensemble of turbulent magnetic field lines can be described in appropriate limit as a random walk. To study the random walk of magnetic field lines in space, we must first find out what is the spread of the 16

36 Figure 1.7: An example of random walk of a magnetic field line. The tracing of the field line was done numerically in a 3D isotropic field. field lines compared to the displacement along the mean field direction. In the presence of a mean field, say in the z-direction, the perpendicular magnetic field line diffusion is defined in terms of the ratio of the mean square displacement in the perpendicular direction to the displacement in parallel direction D xx = x2 2 z, D yy = y2 2 z. (1.8) The study of the random walk leading to the transverse displacement of a field line in the presence of a mean field was recognized by Jokipii [69]. Figure 1.7 illustrates a field line exhibiting a random walk in space. The statistics of the random walk of the magnetic field lines are central to understanding the diffusion of charged particles in turbulent magnetic fields [74, 97, 125, 159]. Field line diffusion, which is a consequence of a random walk, can be studied in slab, slab+2d or 3D isotropic fields, and may display different behaviors for different cases. Recently, I have co-authored two separate papers on the random walk of magnetic field lines in isotropic fluctuations with and without a mean field [147, 148], applicable to many astrophysical situations, some details of which are included in 17

37 Chapter 4. Field line random walk and diffusion are topics explored broadly by the astrophysics community. During the random walk of particles, for example Brownian motion, particles travel in a straight line before a collision. The straight trajectory before the first collision is called a ballistic trajectory, where the displacement of a particle is x = vt and the variance is x 2 = v 2 t 2. This behavior is also present during the random walk of field lines, where the field lines from the starting position of a study are initially straight before they reach the diffusive limit. The initially ballistic nature of field lines is used to derive a nonlinear theory of particle diffusion called Random Ballistic Theory (RBD [127]). The study of magnetic field lines, their topological structures, random walk phenomena, etc, can be vital in understanding the physical processes occurring in the sun, the Earth s magnetosphere and interplanetary space. For example, slab and 2D fluctuations of the solar wind have been used to describe the suppressed diffusive escape of field lines from 2D orbits due to strong or irregular 2D field, which is a possible candidate to explain the persistence of sharpness of dropouts of solar energetic particles observed near Earth s orbit [34, 131, 156]. 1.6 Transport of Charged Particles in Turbulent Magnetic Fields In this thesis we mostly look at the behavior of test particles where these particles move under the influence of the turbulent magnetic field but do not feed back to the field itself. This approach has earned a great deal of attention from the astrophysics community [69, 97, 125, 134, 139]. To this date, however, a lot of questions regarding test particle transport and diffusion have not been answered yet. In the last section we briefly discussed how the field lines random walk and spread diffusively. The standard idea of charged particle diffusion is that the gyrocenters of these particles follow field lines and their diffusive spread is governed by the diffusive spread of field lines. This is called the field line random walk limit (FLRW [74]). Although the FLRW limit provides a very physically appealing picture, 18

38 Figure 1.8: The complexity of magnetic field turbulence causes charged particles to scatter from one field line to another. The particle initially following a field line B, scatters to field line C and then scatters to field line A. it has proved to be inaccurate in most circumstances when compared to the numerical experiments [51, 93, 125]. Turbulent magnetic fields have complex spatial structure. Due to this complex structure a particle that is initially following a field line may scatter off to another field line and subsequently to other field lines as shown in Figure 1.8. This is the reason it has been hard to predict the statistics of particle transport. The effect of multiple parameters such as the strength of a mean field vs fluctuations, the energies of the particles, the correlation length of the magnetic field (in simple words this is the average distance up to which the magnetic field at one point is correlated with the magnetic field at another point) makes the problem even more difficult. Figure 1.9 demonstrates schematically different ways in which turbulent transport of particles take place. The top figure (a) illustrates a particle completely following a field line, and the transport of the particles can be described by the random walk of field lines. The figure in the middle (b) shows a particle following a field line for a while, but eventually backscatters by reversing its initial direction along the field line. The bottom figure (c) shows that the particle initially follows a field line, then 19

39 Figure 1.9: Transport process of a charged particle in random magnetic field. a) Field line random walk limit (FLRW) where a particle keeps following a field line without parallel scattering. b) The particle is initially following a field line but parallel scattering causes the particle to backtrack along the same field line. c) Particle follows the field line and backtracks but due to the complexity of the magnetic field turbulence, the particle scatters from the field line. 20

40 backscatters and finally abandons the field line. These three examples, which could be applicable in various circumstances, will be discussed in the next section. There is another instance that we have encountered in isotropic turbulence where the particles are of extremely high energy and have a gyro-radius larger than the correlation length of the magnetic field. The transport process in such a case does not typically follow any of the a), b) and c) of Figure 1.9. These particles do not complete a gyro-orbit before scattering from one field line to another. The process is almost like a Brownian random walk under the influence of a Lorentz force with random magnetic fields. The magnetic field causes small perturbations to the particle path. With enough time, the path of the particle deviates completely from its original path. Although we are discussing the behavior of an individual particle, it is important to note that this is a stochastic process where it is almost impossible to predict the path of a particular particle or a field line. The only possible way to study this problem, like other stochastic processes, is to average over an ensemble of different realizations; i.e. we average over the path, of many particles at different positions in the random field that varies over space, and try to come up with an explanation of the collective behavior of these particles. Therefore, Figures 1.8 and 1.9 are examples of how we think an average particle would behave under these stochastic conditions. 1.7 Diffusion Coefficient The diffusion coefficient κ appears in the diffusion equation [38] which for a particular dimension x can be written as f t = κ 2 f, (1.9) f t = κ 2 f xx x, (1.10) 2 where f is the distribution function and κ xx is the x-component diffusion coefficient. 21

41 Figure 1.10: Particle in a slab field. The particle keeps gyrating along the field line until resonant scattering causes particle to backscatter which causes subdiffusion in the perpendicular direction. Figure to the right shows continuous decrease in running perpendicular diffusion coefficient even during later times [118]. We define a running diffusion coefficient κ xx as half of the mean square displacement divided by time κ xx = x2 2 t, (1.11) where denotes ensemble average or the average over many realizations. The detailed derivation will follow later, but we can easily see the relevance of this definition to the diffusion equation (Equation 1.10) by simple dimensional analysis. The basic idea is that we are looking at the average spread (variance) of the particles with respect to time. The diffusion coefficient can be broken down into a parallel diffusion coefficient in the direction of the mean field and perpendicular diffusion coefficients perpendicular to the mean field. If the z direction is taken to be the mean field direction, then κ xx = x 2 /(2 t) and κ yy = y 2 /(2 t) are called the perpendicular diffusion coefficients and κ zz = z 2 /(2 t) is called the parallel diffusion coefficient. When the root mean square fluctuations and the correlation of the magnetic field are equal in the two perpendicular directions, then the turbulence is axisymmetric and we have κ xx = κ yy, which is the most commonly studied [97,117,134]. Ruffolo et al [125] is a good example of the study of particle diffusion in a non-axisymmetric turbulence. The 22

42 Figure 1.11: Particle in a 2D+slab field. The addition of 2D field has enough complexity for the particle to scatter off the field line compared to the slab field (Figure 1.10). The particle has initial free streaming regime followed by subdiffusion due to backscatter then second diffusion due to scatter off the field line. Figure on the right shows stable value of running perpendicular diffusion coefficient at later times [117]. diffusion coefficient can also be explored when the turbulence is completely isotropic and κ xx = κ yy = κ zz. As already mentioned, the simplest idea is that the particles are following field lines with constant pitch angle µ (the angle between the velocity of the particle and the mean field) and the diffusion is achieved by field line random walk. However, in practice the above process is quickly curtailed by parallel scattering dominated by resonant interaction between the field fluctuations and particle gyration leading to the random walk of µ with time, which eventually causes particle to backscatter along the field line. The resonant interaction occurs if the parallel distance the particle travels within an unperturbed gyro-period is equal to the parallel wavelength of the turbulence v Ω = 1 k, (1.12) where v is the parallel velocity, k is the parallel wave number and Ω is the gyrofrequency of the particle. The resonant scattering process first explained by Jokipii 1966 [69] has been successful in explaining the parallel diffusion of particles correctly in many circumstances. Perpendicular diffusion across the mean field, however, has been 23

43 Figure 1.12: The plot to the left shows high energy particle diffusion in a 3D isotropic field where the Larmor radius of the particle is greater than the correlation length of turbulence. The initial free streaming is followed by stable diffusive regime with no signs of subdiffusion. The figure on the right is an example of magnetic field line diffusion in a 2D+slab field where there is no subdiffusion at any point. a complicated subject of study for a long time. Recently a nonlinear theory developed by Matthaeus et al. in 2003 [97] called nonlinear guiding center theory (NLGC) has been successful for predicting the nonlinear perpendicular transport phenomenon of charged particles, particularly in 2D+slab turbulence. Since then several theories have been proposed for the improvement of NLGC [127, 132, 134]. In the following paragraph we will summarize important concepts relating to the perpendicular diffusion of charged particles. The random walk of a field line has an initially free streaming period, when the field line is approximately straight because it has not yet traversed a mean free path for directional changes. Since the trajectory of charged particles in a magnetic field is a helix around the field line, the guiding center of the particle which is averaged over a gyration also exhibits free streaming; i.e. x 2 t 2. The running diffusion coefficient defined as x 2 /(2 t) must be increasing at earlier times, which is evident both during field line and particle diffusion (Figure 1.10, Figure 1.11, Figure 1.12). Putting everything together, the initial free streaming regime of the particles 24

44 is followed by first diffusion, when the field lines spread diffusively. Then the resonant interaction causes the particles to reverse the guiding center direction along the field line, and finally if the magnetic field has enough transverse (perpendicular to mean field) complexity, particles scatter from one field line to another and achieve perpendicular diffusion again given enough time called the second diffusion. Most studies, however, do not find steady first diffusion [117, 118, 125]; i.e, the diffusion coefficient has a very short-lived maximum after the initial free streaming, then the particle immediately enters a subdiffusive regime and may or may not achieve diffusion depending upon the complexity of the field. For example, in the simplest case of slab diffusion in Figure 1.10, when the fluctuation is only dependent on the z direction, the running perpendicular diffusion coefficient keeps decreasing in time (subdiffusion) after the initial increase to a maximum. For the composite geometry 2D+slab, there is a clear second diffusion due to the transverse complexity provided by the addition of 2D field (Figure 1.11). In the isotropic turbulence with no mean field as described in the following section, when the Larmor radius of the particles is greater than the correlation length of the turbulent magnetic fields, the running diffusion coefficient of the particles is stable after the free streaming to a maximum value and does not show any sign of subdiffusion (Figure 1.12). As discussed in Section 1.6, this may be due to the fact that the diffusion process in this limit more resembles the collisional random walk process than the ones where there are clearly defined gyro-centers of the particles which follow the field lines. The diffusion coefficient of the field lines also tend to display a free streaming to a stable value without subdiffusion (Figure 1.12) in most circumstances, except that in the presence of strong 2D fields, trapping and eventual subdiffusion may occur [34]. 1.8 Charged Particle Diffusion in Isotropic Magnetic Field Turbulence Completely isotropic turbulence does not have any preferred direction and is spatially homogeneous. The presence of a mean field implies a preferred direction, so 25

45 in the isotropic case we drop the mean field and only fluctuations prevail. The isotropic field is mostly prevalent outside the solar system. The magnetic field, which is fully three-dimensional, is given by B = b(x, y, z). (1.13) 3D isotropic turbulence with zero background field may provide a reasonable description of magnetic fluctuations in the interstellar medium of our galaxy, which has a well-developed turbulent cascade [5] and a fluctuation field of the same order of magnitude as a galactic field that itself reverses on larger scales [102] and might be considered to have an average value near zero. Violent star formations in galaxies also can have intense magnetic fields without a clear mean field [171]. The study of charged particle diffusion in isotropic magnetic field turbulence is important to describe the transport processes in the galaxy and en-route to observation sites. Previous studies of diffusion processes in isotropic turbulence have mostly relied upon phenomenological and scaling arguments [31, 42, 107, 112]. This thesis makes an important contribution in the development of theories that agree very well with the numerical simulation at all energy ranges of particles. We believe this will significantly advance the understanding of the transport of cosmic rays in galaxies. In particular our theory at low energy predicts the energy scaling of the diffusion coefficient consistent with observations [1,104]. Since our theories give exact prediction of diffusion coefficient with particle energies given magnetic field turbulence in galaxies, they may be useful in predicting the turbulence parameters of the galaxies if the observation data of particle energies are provided. Our theories are very general and applicable to relativistic or non-relativistic particles. One important aspect in the developement of the theory of charged particle diffusion in isotropic magnetic field turbulence is the contrasting behavior of particles at low and high energies as shown in Figure The energies of the particles are defined using the ratio of Larmor radius (R L ) to the correlation length (l c ) of turbulence. When this ratio is much smaller than one it is the low energy limit and when the ratio 26

46 Figure 1.13: The yellow lines are the particles and the black lines represent the field lines. As shown the high energy particles deflect by a small angle as they travel about a coherence length of a field line whereas the low energy particles follow a local magnetic field line and backscatter along the field line due to small scale fluctuations of the field line. 27

47 is much greater than one it is the high energy limit. As shown by Figure 1.14 the highest energy particles deflect by a very small amount when they traverse aboout a correlation lengh of turbulence. The particles achieve diffusive limit after they have suffered multiple small angle deflections. As shown in the same Figure (1.14) the low energy particles that have Larmor radius much smaller than correlation length follow a nearby field line for a long time. The contrast between magnetic fields seen by low and high energy particles is clearly shown by the Figure (1.14) with a zoomed in view of one particular field line that the low energy particle is following. The diffusion process of the low energy particles is dominated by pitch angle scattering due to small scale fluctuations of the magnetic field. We also find that the mean free path of low energy particles are much smaller than the correlation length of magentic field (Chapter 8). Here, the particle actually scatter back and forth along a field line much before the field line effectively bends. The intermediate energy particles that lie between the low and high energies have a complicated nonlinear transport phenomenon that connects the two extremes. We extend the low and high energy theories developed using the ideas presented by Figure (1.14) to include the additional effects of cross field transport at low energies and the increased velocity decorrelation at high energies to develop the intermediate energy thoery. Chapters 6, 7, 8 and 9 provide the details of the important theory of transport of charged particles in isotropic magnetic field turbulence. The result has been published in Astrophysical Journal [150]. 1.9 Numerical Simulation and the Need for Intermittent Fields Numerical simulations are used to test the validity of analytical models, since they do not rely on numerous assumptions made by analytical calculations, and can provide an independent check of those assumptions. To calculate diffusion coefficients numerically, we should solve for the trajectories of the particles numerically using Equation 1.1. The right hand side of Equation 1.1 contains a magnetic field which we know is random and turbulent. So even before we solve for particle trajectories 28

48 using the Newton-Lorentz equation, our first step is to generate random magnetic field realization where we eventually perform time-stepping of particles. The magnetic field fluctuations we generate for simulation purposes are homogeneous (statistical quantities independent of position) and random in space with a specified energy spectrum. The random magnetic fields are generated first in Fourier or wave number space (k-space) before taking a Fourier transform to get the fields in configuration space. The fields in real space are at discrete positions inside a periodic box, and interpolation (linear or polynomial) is performed to get the fields at any position of the box. Obviously, the accuracy of the numerical simulation increases with resolution, but higher resolution makes the computations expensive and time consuming. Taking all the factors into account, we mostly perform simulations to push the particles in the field. The solenoidality of the magnetic field is always maintained, i.e. b(x) = 0, (1.14) which in Fourier space is given by k b(k) = 0. (1.15) The numerical code has freedom to use any energy spectrum in Fourier space, but we mainly follow the Kolmogorov scaling law, as shown in Section 1.2, because of the reasons mentioned therein, with an omnidirectional energy spectrum E(k) k 5/3. (1.16) The relationship between E(k) and the average energy density in real space is 1 2 b i(x)b i (x) = E(k)dk, (1.17) where the angular brackets imply an ensemble average over many realizations and repeated indices means summed over. 29

49 10 0 Gaussian 10-2 PDF Figure 1.14: Probability distribution of the z-component of current (solid line) in a synthetic intermittent field. Intermittency causes wider tails on the distribution compared to the Gaussian (dashed line). The magnetic field components for the purpose of numerical simulations are generated as Gaussian random variables. More precisely, the vector potential is first generated this way to ensure the solenoidal property (see Section 5.3). The distributions of magnetic field differences also show Gaussianity. Gaussian distributions are usually smooth but have continuous variations. Most of the numerical work on particle and field line diffusion has involved the generation of Gaussian random variables [93, 97, 117, 118, 124, 125, 138]. Turbulence, however, is non-gaussian with variability that comes in bursts called intermittency [4, 140]. Although turbulence is largely random, it contains isolated structures due to this intermittent nature (Figure 1.15) [47, 141]. Intermittency causes deviation from Gaussianity and longer or wider tails are observed in Probability Distribution Functions (Figure 1.13) [10, 149]. These wider tails of turbulent fields are very strong events that are absent in Gaussian fields. Hydrodynamic turbulence is typically characterized by the presence of very large velocity differences and gradients suggesting the presence of occasional very strong velocity jumps [84]. Naturally this leads to the departure from Gaussianity and creation of extended tails in the probability 30

50 Figure 1.15: Contour of the z-component of intermittent current in a selected plane from a MHD simulation. The intermittent field has thin and elongated structures of current sheets where most of the current is concentrated. distribution functions of velocity differences and gradients. Another important consequence of the non-gaussian feature of velocity gradients is that the vorticity, which is the curl of the velocity, v (describes the rotational motion of the fluid), is found to be concentrated in small volumes scattered over space, thus displaying intermittent characters [41, 66, 91]. The analog of hydrodynamic turbulence for plasma turbulence is that the magnetic field differences and gradients show departure from Gaussianity and strong current (curl of the magnetic field) sheets are formed that are concentrated in small volumes forming spatial structures (Figure 1.15). These bursty and intermittent forms of concentrated structures are often called coherent structures, where most of dissipation of large-scale energy to heat takes place. Intermittency is an important characteristic of fluid flows [48], plasma turbulence [18, 79, 89, 162, 163] and the solar wind [27, 95, 149, 163, 166], but this property of the magnetic field is relatively unexplored in cosmic ray scattering theories [3, 31, 42, 107]. 31

51 The Gaussian magnetic field generated for the purpose of studying cosmic ray scattering may have realistic features including Kolmogorov-like inertial range spectral behavior, but using the simple method will not produce higher order statistics compatible with the bursty properties of real intermittent turbulence. We have therefore devised a way to generate magnetic fields that are intermittent while controlling global parameters such as spectral distribution, total variances, means and correlation scales, etc. The procedure which involves generating the intermittent fields is similar to the one used by Rosales and Meneveau [123] for fluid turbulence called the Minimal Multiscale Lagrangian Map (MMLM) procedure, which is extended to include plasma turbulence. The synthetically generated intermittent fields can be eventually used to test the effects of intermittency present in astrophysical fields, in cosmic ray scattering. The Minimal Multiscale Lagrangian Map procedure (MMLM) developed in the context of neutral fluid turbulence (Rosales and Meneveau 2006 [123]) is a simple method to generate synthetic intermittent vector fields. Using a sequence of low pass filtered fields, fluid particles are displaced at their rms speed for some scale dependent time interval, and then interpolated back to a regular grid. Fields produced in this way are seen to possess certain properties of real turbulence. We extend this technique to plasmas by taking into account the coupling between the velocity and magnetic fields. We examine several possible applications to plasma systems. The studies of cosmic ray transport and modulation in the test particle approximation may benefit from improved realism in synthetic fields produced in this way. Synthetically generated intermittent fields can also be used as initial conditions for simulations, wherein these synthetic fields may efficiently produce a strongly intermittent cascade. We provide numerical method of generating synthetic intermittent fields in Chapter 10 and explore several applications of these intermittent fields in Chapter

52 Chapter 2 DIFFUSION The term diffusion is derived from the Latin verb diffundere which means to spread out. The diffusion process is the net movement or spread of quantities such as the density of particles, heat, etc. from a region of high concentration to the region of low concentration. Diffusion, therefore, mainly acts to reduce the differences of concentrations and spatial inhomogeneities. The two fundamental concepts in diffusion theory are the stochastic formulation of transport phenomena in terms of a random walk and the description through the deterministic diffusion equation [6,38]. In the latter case the flux of particles is assumed to be proportional to the negative gradient of concentration, which is a mathematical way of saying particles move from a region of higher density to a region of lower density. This is also known as Fick s law, which is derived in Section 2.2. From the particles point of view, when there are a number of random walkers starting from a concentrated region of space, then after a certain time the random walkers from the high concentration spread out or diffuse into regions of low concentration. In the limit, the step-size of the random walkers is close to zero, so this now becomes a continuous diffusion process described by Fick s law. Therefore, most of the studies of diffusion processes start with the study of random walk, which has a striking connection to Brownian motion. Brownian motion was named after the observation by the English botanist Robert Brown (1829 [23]), who noted the erratic motion of pollen grains suspended in fluids. Later, in 1905, when the existence of atoms and molecules was still open to objection, Einstein predicted the irregular motion of larger suspended particles which could be observed under microscope due to the random motion of impacting molecules [32, 45]. 33

53 Jean Perrin experimentally investigated Brownian motion in water and verified Einstein s explanation of these phenomena, thereby confirming the molecular nature of water. For this achievement he was honored with the Nobel Prize for Physics in 1926 [110]. First, we will discuss the microscopic description of the diffusion equation in terms of random walk of particles in space. That is followed by the macroscopic description in terms of densities and distribution of particles. We will employ notation similar to Bakunin 2008 [6]. We will start with a simple one-dimensional treatment and generalize to the three-dimensional problem. The equivalence of microscopic and macroscopic diffusion is shown using numerical simulation. We will then move to more advanced topics, such as the Lagrangian description and the Fokker-Planck equation. 2.1 Random Walk Suppose a group of particles begin their one-dimensional motion at time t = 0 at position x = 0, and they execute random walks; i.e. each particle steps to the right or to the left once every τ seconds, moving at velocity ±v x and covering a distance δ = ±v x τ. Let us consider τ and δ as constants. There is an equal probability of moving to the right or moving to the left. The particles are assumed to not interact with one another and therefore move independently of each other. This makes the successive steps statistically independent. Obviously, on average the particles are going nowhere ( x = 0). We are interested in the non-zero mean square displacement of particles x 2 (t). We will use an iterative procedure. An aggregate of N p particles is considered. The i th particle will remain at a position x i (N) after N steps, but the position of a particle after the N th step differs from its position after the (N 1) th step by ±δ: x i (N) = x i (N 1) ± δ. (2.1) 34

54 The + sign is for roughly half of the particles and the - sign for the other half. By summing over the particle index i and dividing by N p, the mean displacement of the particles after N th step is found to be x(n) = 1 N p N p i=1 Expressing x i (N) in terms of x i (N 1), we get x(n) = 1 N p N p i=1 x i (N). (2.2) [x i (N 1) ± δ]. Since the second term is positive for roughly half the particles and negative for the other half, it averages to zero to find This implies that x(n) = 0. x(n) = 1 N p N p i=1 x i (N 1). All the particles start at the origin and since the spreading of the particles is symmetrical about the origin, the mean must be zero. From Equation 2.1, rewriting x i (N) in terms of x i (N 1) and taking the square, The mean is simply x 2 (N) = 1 N p N p i=1 x 2 i (N) = x 2 i (N 1) ± 2δx i (N 1) + δ 2. (2.3) [x 2 i (N 1) ± 2δx i (N 1) + δ 2 ] = x 2 (N 1) + δ 2. (2.4) All the particles i have x i (0) = 0 and x 2 (0) = 0, so one obtains: x 2 (1) = δ 2, x 2 (2) = 2δ 2,..., x 2 (N) = Nδ 2. (2.5) This shows that the mean square displacement is directly proportional to the number of steps N. Since the particles step in every τ seconds and the total number of steps is N, we can say that the time taken is t = Nτ. Hence, N is proportional to t, x 2 (N) to t, and the root-mean-square displacement, x 2 1/2 to t 1/2. We note that N = t/τ and can denote x to be a function of t, so that ( ) t x 2 (t) = δ 2 = τ ( δ 2 τ ) t. (2.6) 35

55 Let us define a diffusion coefficient of the form κ = δ2 2τ, (2.7) which gives us a relationship between mean square displacement and diffusion coefficient and finally we arrive at x 2 = 2κ xx t, (2.8) κ xx = x2 2t. (2.9) This is called the microscopic diffusion coefficient which can be used in numerical simulations by tracking the path of each particle, and, then calculating the variance x 2 by averaging over many particles. Since the particles have free streaming regime before achieving the diffusive limit, this definition is used for larger times, much longer than the mean free time. If the transport of particles in the x, y and z directions are statistically independent, then x 2 = 2κ xx t and y 2 = 2κ yy t and z 2 = 2κ zz t. In the isotropic system when the statistics are equivalent in all three directions, κ xx = κ yy = κ zz = κ. Let r 2 = x 2 + y 2 + z 2, where r is the distance from the origin, then one finds r 2 = 6κt. In the derivation of the diffusion coefficient above, we can define the correlation length (equal to the mean free path in this particular example of random walk) l c = δ, and the correlation time or the mean free time τ c = τ. In general, if the value of the time and length are smaller than the correlation values, then the motion of the particle has a ballistic character, whereas if these values are larger than the correlation scale then we are dealing with the diffusion mechanisms. The one-dimensional transport model above looks elegant, but one has to be aware that the turbulent diffusion model may not be so simple. The key problem in 36

56 investigating the turbulent diffusion is the choice of the correlation scales responsible for the effective transport. Often, several types of transport phenomena are present simultaneously in turbulent diffusion. The diffusion coefficient could be different for different co-ordinate axes depending on several parameters. Moreover, turbulent transport could have a non-diffusive character, where the relation x 2 t may not be valid. This can be generalized by writing x 2 t σ, (2.10) where we characterize the particle s motion by using different parameter regimes of σ: 0 < σ < 1: subdiffusion σ = 1: normal (Markovian) diffusion 1 < σ < 2: superdiffusion σ = 2: ballistic motion (free streaming) 2.2 Fick s Law In this section we derive Fick s law, the spatial and temporal variations of particle distributions from the random walk model. We suppose, that each particle steps a distance δ once every τ seconds. It is assumed that the number of particles at each point along the x-axis at time t is known, so that the number of particles which move across a unit area in unit time from point x to x + δ can be known. Half the particles step to the right and half the particles step to the left. The net number of particles crossing to the right is 1 2 [N p(x + δ) N p (x)]. (2.11) If the area normal to the x-axis is S a, the net flux defined by the number of particles per unit time per unit area is q x = [N p(x + δ) N p (x)] 2S a τ [ = δ2 1 Np (x + δ) N ] p(x). (2.12) 2τ δ S a δ S a δ 37

57 The diffusion coefficient, κ xx = δ2 2τ (as defined in the previous section) gives 1 q x = κ xx [n(x + δ) n(x)], (2.13) δ where n(x + δ) = N p (x + δ)/s a δ is the density or the number of particles per unit volume at the point δ. In the continuous limit when the stepsize is very small, δ 0, and by the definition of partial derivative, one obtains Fick s Law: q x = κ xx n x. (2.14) Considering a box of volume S a δ, q x (x)s a τ particles will enter the box and q x (x+δ)s a τ particles will leave the box. Conservation of particles says that the particles are neither created nor destroyed, which means the particles per unit volume in the box must increase at the rate: 1 τ [n(t + τ) n(t)] = [q x(x + δ) q x (x)]s a τ τs a δ Again using the limit δ 0 and letting τ 0 = 1 δ [q x(x + δ) q x (x)]. (2.15) n t = q x x, (2.16) and finally combining equations 2.14 and 2.16 we get the classical diffusion equation n t = κ 2 n xx x. (2.17) 2 If particles of dye are injected into water at the rate Q s per second for an infinitesimal period of time dt at time t = 0, then the total number of particles that are injected is N = Q s dt, and the solution of the diffusion equation under these boundary conditions gives: n(x, t) = N p (4πκ xx t) 1/2 e x2 /4κ xxt. (2.18) The functional form of n(x, t) is the famous Gaussian. This is the simplest solution of the diffusion equation for a point source. In the numerical simulation we can inject N 0 particles and plot the normalized distribution of particles f(x) vs the distance 38

58 x and compare with the best fit Gaussian by varying the diffusion coefficient κ xx in Equation The diffusion coefficient we get using this method is called the macroscopic diffusion coefficient, for which intermediate particle steps are not needed. The equivalence of the macroscopic diffusion coefficient with the microscopic diffusion coefficient (Equation 2.9) is indeed shown by the numerical simulation where they are found to differ by less than 5% (Figure 2.1). The diffusion equation 2.17 can be easily extended to three dimensions to get: [ ] n(x, t) = κ 2 2 n n(x, y, z, t) = κ x + 2 n 2 x + 2 n. (2.19) 2 x 2 Let us summarize this section by deriving a three dimensional diffusion equation in a very simple and intuitive fashion. The difference of concentrations fluid particles causes the particles to flow from the region of higher concentration to the region of lower concentration. This implies that the flux of these particles must be proportional to the negative of the gradient of concentration: q = κ n, (2.20) where κ is the constant of proportionality. The Continuity equation or the conservation of particles says n t + q = 0. (2.21) Using Equations 2.20 and 2.21 we obtain the diffusion equation in three dimensions: n t = κ 2 n. (2.22) 2.3 Time and Length Scales in the Diffusion of Particles There are two different time scales in diffusive transport of particles: the scattering time scale and the transport time scale. The scattering time scale is related to the mean free time scale of the particle (the time the particle takes to reach the mean free path). The transport time scale is related to the bulk motion of a distribution of particles as they diffuse in space. The scattering and transport length scales are defined in a similar way. 39

59 Figure 2.1: The plot at the top is the Microscopic Diffusion Coefficient vs time with the stable value of the diffusion coefficient at 7.2. The plot at the bottom is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. A total number of 2000 charged particles were used in isotropic turbulent magnetic field. The dotted line is the analytical curve with the diffusion coefficient

60 Let s define τ as the scattering time scale of particles and Γ as the transport time scale. Similarly, define l as a scattering length scale and L as the transport length scale. The diffusion coefficient would then scale as: κ l2 τ. Since the transport length scale of a particle is much larger than the mean free path of the particle, one can write: l L ɛ << 1. The diffusion equation for a distribution of particles f is As simple scaling analysis implies that: Hence, τ = f t = f κ 2 x. 2 1 Γ l2 1 τ L. 2 ( ) 2 l Γ ɛ 2 Γ L This shows that the time and length scales have different ordering with respect to the small parameter ɛ. Often when a system involves several independent time scales and length scales, we resort to multiple scale analysis, which is a subset of perturbative analysis. We also assume that the distribution function can be expanded in powers of ɛ; i.e. f = f 0 + ɛf 1 + ɛ 2 f Multiple scale analysis will be used later to derive a closed-form solution to the diffusion equation in isotropic turbulence. 2.4 Lagrangian Picture and Taylor-Green-Kubo (TGK) Formula In fluid flows, the Eulerian or laboratory coordinate system focuses on specific locations in space through which fluid flows as time passes. We almost always use Eulerian coordinate frame in experiments. On the other hand, in the Lagrangian coordinate frame we follow the motion of a fluid particle as it moves in space and 41

61 time (Figure 2.2). Experimentally, it is not an easy thing to do and this is the main disadvantage of the method. Mathematically, however, Lagrangian coordinates can sometimes have a overwhelming advantage to the theoretical approach and we may get several profound insights by using them [100, 101]. Let us assume that we are able to follow the fluid particle somehow. We will mostly be looking at a homogeneous system where statistical quantities are independent of position. For example, consider an average of a correlation between velocity at two different positions v(x) and v(x + r) given by v(x)v(x + r). In a homogeneous system by definition this quantity should be independent of the position x and should only depend on the separation r to yield: v(x)v(x + r) = v(0)v(r). (2.23) Let X(t) be the Lagrangian position of the marked fluid particle (Figure 2.2). The Lagrangian velocity V(t), which is the instantaneous rate of change of position with respect to time, can be defined by the usual rules of differentiating a vector: V(t) = dx dt. The x-coordinate of the displacement can be written as x(t) = t 0 v x (τ)dτ. (2.24) We will derive the Taylor-Green-Kubo (TGK) formula similar to Shalchi s (2009) [139] derivation. The mean-square-displacement is given by ( t ) 2 x 2 (t) = dτv x (τ) = t 0 dτ τ 0 = t 0 dτ t dξ v x (τ)v x (ξ) dξ v x (τ)v x (ξ), t 0 dτ t τ dξ v x (τ)v x (ξ). 42

62 Figure 2.2: Lagrangian coordinate system where the path of the marked fluid particle is defined by the dashed line. Here homogeneity in time is also assumed where all the statistical quantities including velocity correlation depend only on the difference of times; for example, Hence x 2 (t) = t 0 dτ τ 0 v x (τ)v x (ξ) = v x (τ ξ)v x (0). (2.25) dξ v x (τ ξ)v x (0) + t 0 dτ t τ dξ v x (ξ τ)v x (0). (2.26) By using the integral transformation τ ξ ξ in the first and ξ τ ξ in the second integral, we find x 2 (t) = t 0 dτ τ 0 dξ v x (ξ)v x (0) + t 0 dτ t τ Inserting 1 = dτ/dτ in both integrals and by using partial integration, ( x) 2 (t) = + t 0 t dτ dτ dτ dτ dτ dτ 0 t τ 0 τ 0 0 dξ v x (ξ)v x (0) dξ v x (ξ)v x (0), dξ v x (ξ)v x (0). (2.27) 43

63 = t 0 dτ(t τ) v x (τ)v x (0) + = 2 t 0 t 0 dττ v x (t τ)v x (0), dτ(t τ) v x (τ)v x (0). (2.28) Note that the definition of the stable diffusion coefficient κ in x 2 = 2κt is only valid for t >> t d, where t d is the characteristic time-scale the particle needs in order to reach the diffusive behavior. We may define the running diffusion coefficient as Obviously, in the case of Markovian diffusion d xx (t) = 1 d 2 dt x2 d dt (tκ xx(t)). (2.29) d xx (t >> t d ) κ xx = constant. Using this definition of running diffusion coefficient in Equation 2.28 we get d xx (t) = 1 d 2 dt ( x)2 (t) = 1 d 2 dt 2 t 0 dτ(t τ) v x (τ)v x (0). For stable value of diffusion coefficient, we consider the limit of longer time and extend the upper limit of the integral to infinity to get the famous Taylor-Green-Kubo (TGK) formula: κ xx = lim t x 2 2t = 0 dτ v x (τ)v x (0). (2.30) In the above equation (2.30) we see that the spatial diffusion coefficient involves the integral over the velocity autocorrelation on the right hand side. Similarly, it can be shown that the diffusion in velocity space can be written as the integral of the autocorrelation over the time derivative of velocity (acceleration), and the TGK formula for velocity space diffusion is given by: D xx (v) = v x v x 2t = 0 dτ v x (τ) v x (0). (2.31) TGK formula has been very popular in calculating the diffusion coefficient in recent theories and will be used a lot in the subsequent chapters. 44

64 2.5 Corrsin s Independence Hypothesis In Eulerian coordinates the position x is measured from a reference point, and the velocity u(x, t) at the position x is measured as a function of time t. Therefore, the Eulerian velocity consists of a set of numbers associated with each space-time point (x, t). The average of an Eulerian quantity is the result of many fluid particles passing through the measuring point over a period of time, but we do not know the past or future of any of the individual particles [100, 101]. In the Lagrangian system we follow the path X(t) of one particular fluid particle about the system. Obviously, the Lagrangian velocity V(t) must be equal to the local Eulerian field value u(x, t) when the particle is at x. Hence, V(t) = u(x, t) if X(t) = x, or V(t) = u[x(t), t]. (2.32) Let us look at Lagrangian correlation function L αβ (t) where, L αβ (t) = u α (0, 0)u β (X(t), t). The Eulerian correlation E αβ (x, t) is given by, E αβ (t) = u α (0, 0)u β (x, t). The biggest obstacle in relating the two coordinate system is that the arguments, X(t) of Lagrangian correlation are randomly distributed and are dependent on the realizations considered but the Eulerian correlations have fixed non-statistical arguments x. It is obvious that the Lagrangian correlation can be written in terms of the Eulerian correlation as: L αβ (t) = u α (0, 0)u β (x, t)δ[x X(t)], where the Eulerian correlation picks out a certain path. The biggest problem with the term on the right hand side is that the average denotes joint average between velocity field and the particle path. 45

65 For large values of diffusion time t, Corrsin s independence hypothesis [37] is equivalent to the assumption that the joint average can be factored into two separate averages which disconnect the statistics of velocity field and the particle path to obtain L αβ (t) = u α (0, 0)u β (x, t) δ[x X(t)]. This can be further extended for single particle distribution functions [101] that given x in Eulerian system, there is a certain probability density P (x t) that the statistics of particles have displacements x in Lagrangian coordinates, then in one dimension one can express L(t) = where E(x, t) is the Eulerian correlation given by E(x, t) = u(0, 0)u(x, t). E(x, t)p (x t)dx, (2.33) Because of the diffusive nature of the displacement x, it is natural to use the Gaussian distribution function for P (x t) which is the solution of the diffusion equation P (x t) = ( ) 1 4πκxx t exp x2. (2.34) 4κ xx t If we assume the statistical independence of x, y and z coordinates then in three dimensions P (x t) = P (x t)p (y t)p (z t), (2.35) and Lagrangian correlation in terms of the Eulerian correlation is given by ( ) ( ) E(x, t) L(t) = 4πκxx t exp x2 1 4κ xx t 4πκyy t exp y2 4κ yy t ( ) 1 4πκzz t exp z2 dx dy dz. (2.36) 4κ zz t The mean square displacement in Equation 2.36 (e.g. x 2 ) has been approximated in the literature using three different closure schemes. One of them is that we assume the variance is diffusive, which is most common. But later we will find that sometimes it 46

66 is more accurate to assume that the variance is random-ballistic, which is true at the initial phase before the particle achieves the diffusive regime. For now we will defer this discussion until we actually get to the part where we calculate the field line and particle diffusion coefficient. The validity of Corrsin s independence hypothesis has been questioned numerous times. Nevertheless, this hypothesis remains popular and is vital in non-perturbative theories used in field line and particle transport. To a certain degree of accuracy Corrsin s independence hypothesis has been successful in a number of cases which we will consider. 2.6 Fokker-Planck Equation Fokker-Planck equation (Fokker 1914; Planck 1917) describes the time evolution of the probability density function of the velocity of particles, where we look at the distribution function f(v, t) as a function of velocity and time only. First we will give a very heuristic idea on what the Fokker-Planck equation is all about (as described by [33]), then provide more formal presentation on the subject. If in some initial situation, all the particles have nearly equal velocities, then the tips of all the velocity vectors for different particles lie within a small volume d 3 v of the velocity space. When stochastic forces which cause small changes of velocities are present, the tips of many velocity vectors would diffuse out of this small volume d 3 v after some time. Hence f(v, t) is expected to satisfy a diffusion equation in velocity space given by where D v is the diffusion coefficient. f t = D v 2 vf, (2.37) In Equation 2.37 the diffusion coefficient is taken to be a constant, but in some situations the diffusion coefficient is itself a function of velocity and cannot be taken out of the velocity space Laplacian. The coefficient D v is related to the statistical properties of the stochastic forces. 47

67 In the presence of the stochastic forces one also has to take account of the effect of drag forces which opposes the motion of the particles. There is certainly a diffusion in velocity space, but also the resulting deceleration due to the effect of drag forces which can be written in the form of negative acceleration as a(v) = γv, (2.38) where γ is the friction coefficient. To find out how this deceleration affects the distribution function, let s forget about the diffusion in the velocity space for time-being. When diffusion is absent, the effect of the acceleration a(v) can be incorporated by writing the continuity equation in velocity space: f t + v (fa) = 0. (2.39) We can combine Equation 2.37 and Equation 2.39 to treat both the acceleration a(v) and the diffusion in the velocity space to get the Fokker-Plank equation: f t = v (fγv) + D v 2 vf. (2.40) The main task is to find the drag coefficient γ and the diffusion coefficient D v from the interactions appropriate for a particular system. For completeness a more rigorous derivation of the Fokker-Plank equation following closely to that by Reif (1965) [121] is presented below. Let f(v, t) dv be the probability that a particle s velocity at time t lies between v and v + dv. The probability should not depend on the entire past history of the particle, but that it is determined if one knows that v = v 0 at an earlier time t 0. Hence f can be more explicitly written as a conditional probability which depends on v 0 and t 0 as parameters: fdv = f(vt v 0 t 0 )dv. (2.41) Equation 2.41 is the probability that the velocity is between v and v + dv at time t if the velocity v 0 at the earlier time t 0 is known. This problem is independent of the 48

68 origin from which time is measured, so f depends on the time difference s = t t 0 and one can simply write f(vt v 0 t 0 )dv = f(v, s v 0 )dv, (2.42) to denote the probability that if the velocity has the value v 0 at earlier time, then it takes a value between v and v + dv after a time s. v = v 0 if s 0 and f(v, s v 0 ) δ(v v 0 ). The right hand side is a Dirac δ function that vanishes unless v = v 0. The general condition that must be satisfied by the probability f(v, s v 0 ) can be readily written down. In any small time interval τ, the increase in probability that the particle has a velocity between v and v + dv must be equal to the increase in this probability owing to the fact that the particle, originally with a velocity in any range between v 1 and v 1 + dv 1, has the probability f(v, τ 1 v 1 )dv of changing its velocity to a value between v and v + dv plus the decrease in this probability owing to the fact that the particle, originally with velocity between v and v + dv, has a probability f(v 1, τ v)dv 1 of changing its velocity to any other value between v 1 and v 1 + dv 1. Mathematically this can be written as f s dv τ = f(v, s v 0 ) dv f(v 1, τ v) dv 1 v 1 + f(v 1, s v 0 ) dv 1 f(v, τ v 1 ) dv, v 1 (2.43) where the integrals include all possible values for velocities v 1. Since f(v, s v 0 ) does not depend on v 1 and one can write Equation 2.43 as f s τ = f(v, s v 0) + v 1 f(v 1, τ v)dv 1 = 1, f(v ξ, s v 0 )f(v, τ v ξ) dξ. (2.44) where v 1 v ξ is used. We are considering the case where the velocity v of the particle can only change by a small amount during the small time interval τ, therefore one can 49

69 assert that the probability f(v, τ v ξ) can only be appreciable when ξ = v v 1 is sufficiently small. For small values of ξ one can expand the integral in Equation 2.44 in a Taylor s series in powers of ξ to get, + n=0 where we have used the expansion f s τ = f(v, s v 0) [ ( 1) n n ] f(v, s v n! v n 0 ) dξ ξ n f(v + ξ, τ v), (2.45) f(v ξ, s v 0 )f(v, τ v ξ) = ( ξ) n n n! v [f(v, s v 0)f(v + ξ, τ v)]. (2.46) n n=0 The term n = 0 in Equation 2.45 is simply f(v, s v 0 ) by virtue of the normalization condition which gets canceled away by the first term in the equation. As for the other terms let us introduce the abbreviation M n 1 τ dξ ξ n f(v + ξ, τ v) = [ v(τ)]n, (2.47) τ where [ v(τ)] n = [v(τ) v(0)] n is the n th moment of the velocity increment in time t. Without the loss of generality set t 0 = 0 and so s = t t 0 = t. When τ is small one can ignore the higher order terms involving M n with n > 2 and finally arrive at the generalized form of the famous Fokker-Planck equation The first order moment is and the second order moment is given by f t = v (M 1f) v (M 2f). (2.48) 2 M 1 = 1 τ v(τ), M 2 = 1 τ [v(τ)]2. The first order moments describe the drag forces and the second order moments describe the diffusive process as explained intuitively using Equation This concludes our basic understanding of random walk and diffusion. Equation 2.48 with the given moments is extremely useful in calculating charged particle transport and diffusion. 50

70 Chapter 3 POWER SPECTRUM AND CORRELATION LENGTH The correlation function tells us how a vector field at two different points is correlated. For example, considering any two points in the system separated by r, the correlation of magnetic field at those two points has a high positive value when the magnetic field at the two points still point in the same direction and has a negative value when they are pointing in the opposite direction. The average of the magnetic field, b = 0 does not tell us much about the nature of the turbulent field and therefore the statistical quantities like the correlation function are essential to describe turbulence. Let us talk about the correlation function in isotropic turbulence. The normalized longitudinal correlation function is defined as f(r) = b p(x)b p (x + r), (3.1) b 2 p where b p denotes magnetic field component parallel to the separation r and x represents spatial position. (p is not a tensor suffix so the summation convention does not apply). The normalized transverse (or lateral) correlation function is defined as g(r) = b n(x)b n (x + r), (3.2) b 2 n where b n denotes the magnetic field component normal to the separation r. Figure 3.1 is an example of longitudinal and transverse magnetic correlation in space. The behavior of longitudinal and transverse correlation functions as a function of separation r is shown in Figure 3.2. It is obvious that the figure shows that correlation functions have the largest value when the separation distance is the least and approaches zero at larger distances. However, it may not be so obvious why the transverse correlation dips below zero. 51

71 Figure 3.1: The plot on the left shows longitudinal correlation in which the magnetic field components are in the same direction as the separation r. The plot on the right shows transverse correlation where the magnetic components are perpendicular to the separation r. Figure 3.2: Longitudinal f(r) and transverse g(r) magnetic correlation functions. The correlation functions asymptotically approaches zero at later times but g(r) first goes slightly negative before nearing zero. 52

72 The longitudinal correlation function cannot be less than zero, because then the divergence of b would be non-zero. The curl of the magnetic field, however, is not zero. A non-zero curl means that the magnetic field is rotational, which implies that the transverse correlation reverses its sign after a certain distance. This character of magnetic field causes the transverse correlation to fall below zero. Now we move on to generalized two-point correlation tensor which is defined as R ij (x, r) = b i (x)b j (x + r). (3.3) Almost always we deal with homogeneous turbulence where statistical quantities are independent of position to find R ij (r) = b i (0)b j (r). (3.4) This equation will be used to find the power spectrum and correlation length. If i = j and r is parallel to the ith component of magnetic field, then it is called the longitudinal correlation tensor, while i = j and r perpendicular to the magnetic field component implies transverse correlation tensor. When i = j, it is also called the auto-correlation. This chapter is divided into four sections. The first two sections introduce a power spectrum. The third section derives a generalized form of a fully three dimensional isotropic spectrum tensor. We will then introduce an important parameter or length scale in turbulence known as the correlation length D and Slab Power Spectrum The 2D and slab (1.4) correlation functions can be written as: R 2D ij (x, y) = b 2D i (0, 0)b 2D j (x, y) (3.5) 53

73 and R slab ij (z) = b slab i (0)b slab j (z). (3.6) The 2D power spectrum tensor Pij 2D (k x, k y ) is defined as the Fourier transform of the correlation tensor given by: Pij 2D (k x, k y ) = Rij 2D (x, y) exp[i(k x x + k y y]dxdy. (3.7) Similarly the slab power spectrum in terms of correlation function is: Pij slab (k z ) = Rij slab (z) exp[ik z z]dz. (3.8) To obtain slab or 2D magnetic field in real space it is very common to define the functional form of the power spectrum that follows a particular power law and get random magnetic fields in k-space before transforming the fields to real space. For 2D spectrum it is sometimes more convenient to work on the potential function a(x, y) where which yields and b 2D (k x, k y ) = ik a(k x, k y )ẑ (3.9) b x (k x, k y ) = ik y a(k x, k y ), (3.10) b y (k x, k y ) = ik x a(k x, k y ). (3.11) The relationship between P 2D ii and b i (k x, k y ) is P 2D ii (k x, k y ) = 1 V b i(k x, k y ) 2, (3.12) where V is the total volume. From Equations 3.10, 3.11 and 3.12 one finds P 2D xx (k x, k y ) = k 2 ya(k x, k y ), P 2D yy (k x, k y ) = k 2 xa(k x, k y ) (3.13) where A(k x, k y ) = a(k x, k y ) 2 /V. 54

74 3.2 Power Spectrum in 3D Turbulence In turbulence a magnetic field would correlate highly with itself than with the magnetic field at any other point. The highest value of the correlation function for the x-component is when R xx (0) = b 2 x. In three dimensions, the magnetic energy per unit mass is defined as R xx (0) + R yy (0) + R zz (0) = b 2 x + b 2 y + b 2 z. (3.14) The 3D spectrum tensor, P ij (k), is defined as the Fourier transform of R ij (r): P ij (k) = and therefore R ij (r) = From Equation 3.16 and Equation 3.4 it is obvious that b 2 x = When P is spherically symmetric then δb 2 = 0 R ij (r) exp (ik r)dr, (3.15) P ij (k) exp( ik r)dk. (3.16) P xx (k)dk. (3.17) 4πk 2 P ii (k)dk, (3.18) where δb 2 = b 2 x + b 2 y + b 2 z, and repeated index means summed over. A three dimensional energy spectrum E(k), also called the omnidirectional energy spectrum, is defined as E(k) = 4πk 2 P ii, (3.19) and one finds δb 2 = 0 E(k)dk. (3.20) 55

75 3.3 Fully 3-Dimensional Isotropic Turbulence Homogeneity implies the invariance of statistical quantities with respect to arbitrary translations. With isotropy, we also have invariance with respect to reflection and arbitrary rigid rotations of the configuration formed by two (or more) points and various directional velocity vectors. The symmetries associated with isotropy impose severe constraints on the general form of our tensors. Let us examine the consequences of symmetry. Group-theoretic methods lead to the following general form of isotropic tensors that are the functions of r alone (Batchelor 1953 [8]). Q i (r) = Ar i (3.21) Q ij (r) = Ar i r j + Bδ ij (3.22) Q ijk (r) = Ar i r j r k + Br i δ jk + Cr j δ ki + Dr k δ ij (3.23) where A, B..., are symmetric functions of r. The solenoidal condition for the magnetic field can be written as b = b i r i = 0. (3.24) This can be extended to correlations where one can write b i (x) b j(x + r) = 0. (3.25) r j From this and a similar relation we find the two continuity conditions R ij (r) r j = R ij(r) r i = 0. (3.26) By Fourier transforming Equation 3.26 it is easy to see that the solenoidal conditions for the spectrum tensor are k j P ij (k) = k i P ij (k) = 0. (3.27) 56

76 The corresponding second order isotropic spectrum tensor in wave-vector space for Equation 3.22 is given by P ij (k) = Ak i k j + Bδ ij, (3.28) where A and B are arbitrary even functions of k. Now applying solenoidality condition, Equation 3.27 to Equation 3.28 one finds Plugging it into Equation 3.28 we have P ij (k) = B A = B k 2. (3.29) ( δ ij k ) ik j. (3.30) k 2 Using the definition of omnidirectional spectrum (Equation 3.19): and 1 2 P ii = B = E(k) 8πk, (3.31) 2 P ij (k) = E(k) ( δ 8πk 2 ij k ) ik j. (3.32) k 2 This is an important form of isotropic power spectrum tensor which will be used later while calculating field line and particle diffusion both numerically and theoretically in the presence of isotropic random magnetic fields. 3.4 Correlation Length The correlation length is an important length scale in turbulence, since it is the approximate scale up to which the magnetic field is correlated. The one-dimensional slab correlation length is defined as l c = 0 R slab xx (z)dz R slab xx (z = 0), (3.33) which is the correlation length of the x -component magnetic field. Replace R slab xx R slab yy and we have the correlation length of the y -component. One interpretation of the correlation length is that the area of the rectangle l c R slab xx (0) is equal to the area by 57

77 under R slab xx (z) as shown in the Figure 3.3. In one dimension R slab xx (z = 0) is δb 2 x, the mean squared fluctuation of the x -component. When the field is axi-symmetric with respect to the z axis the correlation length in x and y direction are equal. In isotropic turbulence where the correlation lengths in all three directions are equal, it is better to work in spherical coordinates and find the correlation length in terms of the radial distance r. We also know from Figure 3.2 that the longitudinal and transverse correlation lengths are different. Hence the correlation length that we are going to calculate is approximately the average of them. In 3D isotropic turbulence the correlation length can be defined similar to the slab correlation length as l c3d = where R(r) = R ii (r) when R ii (r) is spherically symmetric. R(r)dr 0, (3.34) R(0) R(r) can be written in terms of the spherically symmetric power spectrum using Equation 3.16 as R(r) = Writing dk as k 2 sin(θ)dθdφ and integrating over φ one gets R(r) = 2π P ii (k) exp( ik r)dk. (3.35) k 2 P ii (k)e ikr cos(θ) sin(θ)dθdk. (3.36) Substituting sin(θ)dθ by d cos(θ) it is straightforward to find R(r) = 4πk 2 P ii (k) sin(kr) dk. (3.37) kr From the definition of the omnidirectional spectrum in Equation 3.19 one arrives at the relation Now simple integration yields R(r) = 0 R(r)dr = π 2 E(k) sin(kr) dk. (3.38) kr 0 E(k) dk, (3.39) k 58

78 Figure 3.3: Slab correlation function R slab xx (z) vs distance z. The correlation is higher at smaller distance and decreases exponentially with the distance. and since R(0) = δb 2 = 0 E(k)dk, the isotropic 3D correlation length (Equation 3.34) in terms of the energy spectrum is given by l c3d = π 2 0 E(k) dk k 0 E(k)dk. (3.40) 59

79 Chapter 4 MAGNETIC FIELD LINE DIFFUSION In an astrophysical plasma, the motions of charged particles are often guided by the magnetic fields that play a significant role to orchestrate the transport of particles and also the bulk plasma. The study of the random walk of magnetic field lines is an important issue in defining the topology and structure of magnetic fields in magnetohydrodynamic turbulence (MHD) [126]. The statistics of such random walk and diffusion of magnetic field lines is also central to understanding the transport of energetic particles [69, 73, 126]. The random walk behavior of field lines may depend upon the appropriate magnetic field model of the astrophysical systems. The two component model, in which magnetic field fluctuations are a mixture of one dimensional slab and two dimensional ingredients, is relevant to the heliospheric environments [98], while the isotropic turbulence model where the mean magnetic field is absent and the fluctuating fields are equally dependent in all three coordinate system is relevant to the interstellar magnetic fields [147]. Different magnetic field models may have a rich variety of topological effects on charged particles [34, 128, 156]. Scientists have studied various aspects of the effect of turbulence on magnetic field lines [73, 96, 143, 160]. Ruffolo et al [124] have investigated the diffusion and random walk of field lines in a 2D+slab model with non-axisymmetric x and y components, which can assist the investigation of the possible role of non-axisymmetric fluctuations in enhanced latitudinal transport of cosmic rays at high heliographic latitudes [26, 72]. In another study Ruffolo et al [126] have studied at the mutual separation of magnetic field lines as a substitute for the random walk of field lines (as 60

80 suggested by Jokipii 1973 [71]). Hence the behavior of magnetic field lines has garnered a lot of attention in recent years. In this chapter we first look at the preliminary work on the well-established result of field line diffusion in the two-component model, and then move on to our recent study of magnetic field line diffusion in isotropic turbulence. 4.1 Magnetic Field Line A magnetic field line is defined as a line that is parallel to the local magnetic field vector B, where B = b xˆx + b y ŷ + b z ẑ. (4.1) Let us define an infinitesimal arc length along the magnetic field line ds = dxˆx + dyŷ + dzẑ. (4.2) Since this arc length must be parallel to B (schematically represented in Figure 4.1), we must have ds B = 0. (4.3) Using Equations 4.1, 4.2, and 4.3 it is straightforward to show that dx b x = dy b y = dz b z. (4.4) Since ds = dx 2 + dy 2 + dz 2 and B = b 2 x + b 2 y + b 2 z, a simple calculation yields dx b x = dy b y = dz b z = ds B. (4.5) In the presence of a mean field, say in the z direction, the perpendicular magnetic field line diffusion is defined as D xx = x2 2 z D yy = y2 2 z, (4.6) which computes the spread of field lines in x and y coordinates with respect to the mean field direction. Equation 4.5 can be used to calculate diffusion coefficients using these definitions. 61

81 Figure 4.1: Parametrization s along a field line. ds is the infinitesimal arc length along the local magnetic field vector B. When the field is isotropic and 3D with no mean field, one should avoid special treatment of any coordinate. The turbulent fluctuations are both positive and negative in all directions and the field line trajectory moves back and forth in each direction. This suggests that for a given z there may be multiple points in the x-y plane on the same field line. Thus functions such as x(z) and y(z) may not be single-valued, and the z-coordinate (or any other Cartesian coordinate) would not be appropriate to parametrize the field line. One parameter that can be used to describe the motion of the field line in each direction is the parameter s along the field line (Figure 4.1). Equation 4.5 can also be written as dx ds = b x B dy ds = b y B dz ds = b z B. (4.7) Using the relation dτ = ds/ B, the field line can also be described in terms of a parameter τ: dx dτ = b x dy dτ = b y dz dτ = b z. (4.8) 62

82 Finally, we define the diffusion coefficients in terms of the parameter s as: D xx = x2 2 s and in terms of the parameter τ as: D xx = x2 2 τ D yy = y2 2 s D yy = y2 2 τ D zz = z2 2 s, (4.9) D zz = z2 2 τ. (4.10) In the following sections we proceed to calculate magnetic field line diffusion coefficients in various cases like the slab, slab+2d and isotropic 3D fields. 4.2 Preliminary Work on Slab+2D Turbulence To understand the flow of solar wind and the propagation of cosmic rays in it, we first need to know the geometry of the random component of the interplanetary magnetic field (IMF). The most commonly used approximation to describe the geometry of solar wind turbulence is the 2D+slab model [96,98,133,138] as described in Section 1.4. The mean magnetic field in the solar wind is of the same order as the root-mean-square fluctuations. The random walk of field lines in the 2D+slab geometry has been broadly studied by the scientific community [56, 96, 135, 136, 138]. The analytical method used in this section and the next will closely follow the previous works by Matthaeus et al [96], Ruffolo et al [126] and Ghilea et al [50]. When the mean field B 0 is in the z-direction, Equation 4.4 can be written as dx b x = dy b y = dz B 0. (4.11) The change in the x-coordinate over a distance z along a mean field is given by x x( z) x(0) = 1 z b x [x (z ), z )]dz, (4.12) B 0 where x = x, y is the random transverse component of the field line trajectory. x 2 can be expressed as 0 x 2 = 1 B 2 0 z z 0 0 b x [x (z ), z ]b x [x (z ), z ] dz dz, (4.13) 63

83 and since our system is statistically homogeneous, x 2 = 1 [ z ] z z b B0 2 x (0, 0)b x [ x (z), z] dz dz. (4.14) 0 z where x x (z ) x (z ) and z z z, are the Lagrangian coordinate points for locations along a field line trajectory. For long distances, i.e. large z, where the asymptotic limit has been reached for the diffusion coefficients, we can extend the limits of the z integration to ±, in which case the z integration is trivial and we obtain x 2 = 2 z B b x (0, 0)b x [ x (z), z] dz, (4.15) Using the definition D = x 2 /(2 z) and the fact that the slab and 2D fields are statistically independent, one finds D = 1 B b slab x (0)b slab x (z) + b 2D x (0)b 2D x [ x (z)] dz. (4.16) The first term on the right-hand-side is the slab contribution to the diffusion coefficient, and similarly the second term is the 2D contribution to the diffusion coefficient. The slab turbulence is obviously independent of the displacement x, but the 2D contribution to the diffusion coefficient involves x (z) which is influenced by both transverse and parallel aspect of the displacement. The slab contribution to the diffusion coefficient is simply given by D slab = 1 B b slab x (0)b slab x (z) = 1 B R slab xx (z)dz. (4.17) In terms of the power spectrum = P xx(k z = 0). (4.18) D slab Hence the slab diffusion coefficient is dependent on the power at zero wave number or the energy-containing scale. Using the definition of correlation length (Equation 3.33) one can also write: D slab 2B 2 0 = b2 x slab l B0 2 c, (4.19) 64

84 where l c is the slab correlation length. The comparison of the slab diffusion coefficient with numerical simulation is presented in Appendix A. For the 2D contribution to the diffusion coefficient the correlation function b 2D x (0)b 2D x [ x (z)], (4.20) is a Lagrangian correlation. We use Corrsin s independence hypothesis (Section 2.5) to relate the Lagrangian correlation function to the Eulerian correlation function, R xx b x (0, 0)b x (x, z), averaged using the conditional probability P (x z) of finding a displacement x after a distance z. We assume the independence between the Eulerian correlation and the probability distribution which is true for sufficiently long distances, where the random walk has no memory of its prior path: b 2D x (0)b 2D x [ x (z)] = R xx (x (z))p (x z)dx. (4.21) By assuming independece of x and y directions and that the conditional probability is Gaussian, one can write P (x z) = P (x z)p (y z), P (x z) = 1 (2πσ 2 x ) e x2 /(2σ 2 x ), P (y z) = 1 e y2 /(2σy 2). (4.22) (2πσy) 2 The axisymmetric case is considered where σ 2 x = σ 2 y. The third assumption is that there is diffusive spreading of field lines over the distance scales relevant to the decorrelation of random walk. We call this diffusive decorrelation (DD), which specifies σ 2 x and σ 2 y as a linear function of z, that is, σ 2 x = σ 2 y = 2D z. (4.23) From Equations 4.16, 4.21 and 4.22 the 2D contribution to the diffusion is D 2D = 1 B b 2D x (0)b 2D x [ x (z)] dz = 1 B R xx (x (z))p (x z)dx dz 65

85 = 1 B 2 0 P 2D xx (k x, k y ) 0 ( ) ( ) e ikxx P (x z)dx e ikyy P (y z)dy dzdk x dk y where we converted the Eulerean correlation to the power spectrum in k-space. Equations 4.23 and 4.22 can be used to get e ikxx P (x z)dx = e ikxx 4πD z e x2 /(4D z ) dx = e D k 2 x z (4.24) Using similar formula to the y integral the 2D contribution to diffusion is: D 2D = 1 P 2D B0 2 xx (k x, k y )dk x dk y 0 e D k 2 z dz where k 2 = k2 x + k 2 y. = 1 B0D 2 P 2D xx (k x, k y ) dk x dk y, (4.25) Considering the limit of vanishing slab turbulence, we have D = D 2D. Let us denote the 2D contribution to the diffusion coefficient in this limit as D 2D. Then D 2D 1 Pxx 2D (k x, k y ) = dk B0D 2 2D k 2 x dk y. (4.26) The final expression for D 2D D 2D = k 2 without the slab component is given by: 1 Pxx 2D (k x, k y ) B 2 0 When the slab component is non-negligible, Using Equation 4.25 Equation 4.27 gives: D = D slab k 2 + D 2D D = D slab + 1 B0D 2 D = D slab + P 2D dk x dk y. (4.27) xx (k x, k y ) dk x dk y k 2 ( D 2D ) 2 D (4.28) 66

86 When D slab coefficient is: = 0 then D = D 2D. The final solution to the perpendicular diffusion D = Dslab 2 + (D ) slab 2 + (D 2D 2 )2. (4.29) The expression for D in terms of slab and 2D contribution (Equation 4.29) has been derived in several research articles in different ways [50, 96, 126]. In place of the assumption of diffusive spreading corresponding to Equation 4.23, Ghilea et al [50] considered the case where there is ballistic spreading in random directions over the distance scales relevant to decorrelation of the random walk. For the ensemble of ballistic (straight-line) trajectories, we use σx 2 = σy 2 = b2 x z 2, (4.30) B0 2 where b 2 x refers to the sum of 2D and slab components since both contribute to the random ballistic decorrelation (RBD). Using the procedure similar to the diffusive decorrelation (DD), for random ballistic decorrelation (Equation 4.30) we present the final solution of the 2D contribution to the diffusion coefficient as: π D 2D 1 P 2D (k x, k y ) = dk x dk y, (4.31) 2 B 0 b k where b = b 2 x + b 2 y and the trace of the spectral matrix P 2D Pxx 2D + Pyy 2D. of D 2D In contrast with the calculation for DD, for RBD there is no implicit dependence on D, and D 2D = D2D. Hence for RBD: D = D slab + D 2D = D slab + D 2D. (4.32) 67

87 Simulation result Theoretical result D x /l c z/l c Figure 4.2: Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. Here δb/b 0 = 0.3 and the fraction of slab energy in the fluctuation field is 0.2. Now we will present a comparison of theory in diffusive decorrelation formalism (Equations 4.19, 4.27 and 4.29) with the numerical simulation. The details of the numerical simulation is presented in Section 5.2. We use different ratios of rms field δb to the mean field B 0. The slab correlation length is 10 times the 2D correlation length. The ratio of slab energy to the total fluctuation enery δb 2 slab /δb2 is 0.2 for Figures 4.2, 4.3 and 4.4 and 0.5 for Figure 4.5. One of the reasons the numerical diffusion coefficients give higher values when compared to the analytical results is due to the discretization error of the numerical simulations [35]. 68

88 Simulation result Theoretical result 0.04 D x /l c z/l c Figure 4.3: Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. Here δb/b 0 = 0.5 and the fraction of slab energy in the fluctuation field is Simulation result Theoretical result 0.12 D x /l c z/l c Figure 4.4: Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. δb/b 0 = 0.9 and the fraction of slab energy in the fluctuation field is

89 Simulation result Theoretical result D x /l c z/l c Figure 4.5: Theoretical vs numerical results for magnetic field line diffusion coefficient in 2D+slab fields using diffusive decorrelation approximation. δb/b 0 = 0.5 and the fraction of slab energy in the fluctuation field is Diffusion of Magnetic Field Lines in Isotropic Turbulence The turbulent magnetic field leads to random walk and diffusion of magnetic field lines in space. Most of the studies of diffusion of magnetic field lines have been in the presence of a mean field [73,96,124]. These studies have focused in the fluctuation field b much smaller than the mean field B 0, or for transverse turbulence (i.e. perpendicular to the mean field) with b z = 0 [96, 139] where z is along B 0. In this section we will expand the study of random walk and diffusion of magnetic field lines in fully three dimensional isotropic turbulence with no mean field which may serve as a reasonable model in interstellar, galactic and intergalactic turbulence [5]. This work [147] is antecedent to the study of charged particle diffusion in turbulent isotropic field with no mean field presented in chapters 6, 7, 8 and 9. The turbulent magnetic field is 3D and isotropic and fluctuates in all directions given by: B(x, y, z) = b x (x, y, z)ˆx + b y (x, y, z)ŷ + b z (x, y, z)ẑ. (4.33) 70

90 We assume there is no presence of mean field. Since the x, y and z directions are statistically identical the diffusion coefficients are equal in all directions, i.e. D = D x = D y = D z. (4.34) From Equation 4.10 we can express the change in x-coordinate of the field line over τ by x x( t) x(0) = τ The ensemble average of ( x) 2 is then given by x 2 = τ τ b x [x(τ ), y(τ ), z(τ )]dτ. (4.35) b x (x, y, z )b x (x, y, z ) dτ dτ, (4.36) where we introduce the notation x for x(τ ), x for x(τ ), etc. Setting τ τ τ, and with the assumption of homogeneity x 2 = τ τ τ 0 τ b x (0, 0, 0)b x ( x, y, z) d τ dτ. (4.37) Here x = x x, etc. The asymptotic field line diffusion is determined for large τ, which occurs when the correlation vanishes after a very long distance, so we can extend the limits of τ to ±. The integrand has a Lagrangian correlation function. To convert the Lagrangian correlation to Eulerian correlation b x (0, 0, 0)b x (x, y, x) with fixed non-statistical arguments we use conditional probability of finding a displacement ( x, y, z) after a given τ b x (0, 0, 0)b x ( x, y, z) = R xx ( x, y, z) P ( x τ )P ( y τ )P ( z τ )d xd yd z, (4.38) where we invoke the statistical independence of x, y and z. Next we write R xx in terms of the Power spectrum in k-space: R xx ( x, y, z) = P xx (k)e ikx x e iky y e ikz z dk x dk y dk z, (4.39) 71

91 and the mean square displacement, Equation 4.37 is given by: τ x 2 = 0 P xx (k)dτ d τ dk e iky y P ( y τ )d y e ikx x P ( x τ )d x e ikz z P ( z τ )d z. (4.40) We assume that the conditional probability distributions are Gaussian (see also 2.5). The probability P would then give e ikx x P ( x τ )d x = e ikx x 2πσ 2 x e x2 /(2σ 2 x ) d x = e 1 2 σ2 xk 2 x. (4.41) Analogous formulae for y and z can be used and since for isotropic variances σ 2 x = x 2 = y 2 = z 2 = σ 2, using Equations 4.40 and 4.41 we get τ x 2 = 0 P xx (k)e 1 2 σ2 k 2 dτ d τ dk. (4.42) To further solve the problem we have to specify the variance σ 2. Since we are dealing with diffusive mechanisms it is self-evident to apply the diffusion approximation for σ 2. But this also means we are considering that diffusion governs the displacement distribution at earlier times during the decorrelation process, which is obviously not the case. At earlier times the field lines are ballistic (straight) before reaching the mean free path, where decorrelation of the path of the field line begins. Therefore another method is to assume ballistic trajectory of field lines while specifying σ 2. The third method is to use the evolution of random walk process that has initial ballistic character and diffusive behavior at later times. In the following sub-sections we will describe three different closure schemes for the variance σ Diffusive Decorrelation (DD) The diffusive decorrelation model assumes that the magnetic field lines spread diffusively over the decorrelation scale of the random walk. Thus, the variances, σ 2 = 72

92 σ 2 x = x 2 = σ 2 y = σ 2 z, are diffusive and statistically isotropic with Equation 4.42 would then give τ x 2 = 0 σ 2 = 2D τ. (4.43) P xx (k)e Dk2 τ dτ d τ dk. (4.44) Using the definition x 2 = 2D τ, Equation 4.44 can be simplified to obtain 1 P (k) D = dk. (4.45) 3 k 2 P (k) = P xx (k) + P yy (k) + P zz (k), is the modal energy spectral density Random Ballistic Decorrelation (RBD) The second method is to use Random Ballistic Decorrelation [50, 127] where instead of diffusive spreading, we assume that the magnetic field lines ballistically spread in random directions over the decorrelation distance. The mean square displacements read σ 2 = σ 2 x = σ 2 y = σ 2 z = b 2 x τ 2, (4.46) With isotropic distribution of magnetic field components b 2 x = b 2 y = b 2 z = b 2 /3, for large τ, x 2 (Equation 4.42) can be written as τ x 2 = P xx (k)e 1 6 b2 k 2 τ 2 dτ d τ dk. (4.47) 0 Finally, using the fact that x 2 = 2D τ, the simplified random ballistic diffusion coefficient is given by π kp (k)dk 0 D = b 6 k 0 2 P (k)dk. (4.48) The omni-direction energy spectrum is given by E(k) = 4πk 2 P (k) (Equation 3.19). Using Equation 3.40, we can simplify the diffusion coefficient in the RBD model in terms of the correlation length l c as D = 2 3π l cb. (4.49) 73

93 4.3.3 Self Closure Method The diffusive decorrelation and random ballistic decorrelation method determine the magnetic field line diffusion coefficients in the asymptotic limit when t and the mean square displacements are linear in time. Using Ordinary Differential Equation (ODE) now we will consider the evolution of mean square displacements and diffusion coefficients. We start by considering the running diffusion coefficient (see Equation 2.29) in terms of the τ parameter D(τ) = 1 d x 2 2 dτ where we identified x 2 = V. From Equation 4.36, D(τ) = 1 d x 2 2 dτ = Now, differentiating (4.51) will give Equations 4.38, 4.39 and 4.41 gives τ 0 = 1 dv 2 dt, (4.50) b x (0, 0, 0) b x ( x, y, z) dτ. (4.51) 2 dd dτ = d2 V dτ 2 = 2 b x (0, 0, 0) b x ( x, y, z). (4.52) b x (0, 0, 0) b x ( x, y, z) = Then, for ordinary differential equation (4.52) one finds 2 dd dτ = d2 V dτ 2 = 2 P xx (k)e k2 σ 2 2 dk. (4.53) P xx (k)e k2 σ 2 2 dk. (4.54) To form the self-closure equation and study the evolution of the random walk process, we can identify σ 2 (τ) in (4.54) by V (τ) to yield 2 dd dτ = d2 V dτ 2 = 2 P xx (k)e k2 V 2 dk. (4.55) In terms of the modal energy spectral density P (k) = P xx (k) + P yy (k) + P zz (k), we have dd dτ = = P (k) 3 e k 2 V 2 dk P (k)e k2 V 2 dk P (k)dk b2 3. (4.56) 74

94 Since the omnidirectional spectrum E(k) = 4πk 2 P (k), dd dτ = E(k)e k2 V 2 dk E(k)dk b2 3. (4.57) Combining (4.50) and (4.57), we yield a system of two first-order differential equations, dv dτ dd dτ = 2D (4.58) = E(k)e k2 V 2 dk E(k)dk b2 3. (4.59) Now we can use Equations (4.58) and (4.59) to determine the running diffusion coefficients at any values of τ and compare the results with DD, RBD and ODE models. In general, there are no analytic solutions of Equations (4.58) and (4.59) [145], so we solve them numerically. The Fourier integral over k is evaluated with the QUADPACK library [111] with initial values V (0) = 0 and D(0) = 0 to be consistent with ballistic field line trajectories (V τ 2 ) at low τ. The theoretical models presented above is compared with numerical simulation in Figures 4.6 and 4.7. The effect of the power spectrum on the random walk of field lines is explored using three different values of the spectral index Γ. These spectral indices are chosen corresponding to different models of magnetohydrodynamic turbulence phenomenology [170]. Γ = 3/2 is called Iroshnikov-Kraichnan scaling [67, 83], Γ = 5/3 is the familiar Kolmogorov scaling [8,82], and Γ = 2 is the weak turbulence scaling [49, 103]. 500,000 field lines are traced for b = 1 and λ = 1. The details of the numerical method are shown in Section 5.3. For the Kolmogorov spectrum, Figure 4.6 gives an example of D i (τ) and D i (s). The statistical equivalence of the three directions in isotropic turbulence is evident. Figure 4.7 shows the comparison of the three theoretical formalisms with the numerical method. All three methods give diffusion coefficients within 20%, with none having a significant advantage. The ODE model overestimates diffusion coefficient coefficient, whereas the DD and RBD models underestimate the diffusion coefficient. 75

95 Figure 4.6: Results of the diffusion coefficients of field lines from simulation using parametrizations s and τ. Since the turbulence is isotropic the three spatial coordinates are statistically identical, and therefore the diffusion coefficients with respect to different coordinates coincide. 76

96 Figure 4.7: Magnetic field line diffusion coefficients, averaged over the three spatial directions as a function of τ. The diffusion coefficient is calculated numerically over three different spectral index (Γ) of fluctuation power spectrum: a) Γ = 3/2, b) Γ = 5/3 and c) Γ = 2. 77

97 In conclusion, we have devised three different theoretical models of the magnetic field line diffusion coefficient in isotropic turbulence. The diffusion coefficient is predicted well by the theories when compared to the numerical simulation. 4.4 Field Line Random Walk Model for Particles and its Limitations The random walk of magnetic field lines begin with free-streaming, where the field line is approximately straight until it reaches its mean free path. The corresponding feature of particle motion is that over short distances and times, the particle trajectory is a helix around that field line at a roughly constant pitch angle, the angle between particle velocity and magnetic field direction. Hence, the guiding center of particles which is averaged over a gyration follows straight field lines over short distances and the initial ballistic character of gyrocenter exhibits x 2 t 2. After the field line reaches its mean free path, it begins to random walk, which in most cases is diffusive in nature if the ensemble is properly defined. The field line random walk (FLRW) model (Jokipii and Parker 1968 [73]) of particle diffusion assumes that a particle follows magnetic field lines with constant pitch angle and constant gyrocenter velocity at all times. This also implies that given the velocity, the diffusion of particles is completely dictated by the random walk of fieldlines. given by In the case of a mean field in z -direction the diffusive nature of a field line is x 2 = 2D xx z. (4.60) If the particle takes time t to reach the distance of z then in the FLRW limit Equation 4.60 can be rewritten as x 2 = 2D xx z t t = 2 v z D xx t = 2κ xx t. (4.61) If we assume an isotropic distribution of particles then the particle diffusion coefficient is given by where v is the average speed of the particle. κ xx = v z D xx = 1 2 vd xx, (4.62) 78

98 The above relation can be demonstrated using the fact that after a long time when particles relax to a Maxwellian distribution of velocity, then v z = v z f(v z )dv z = 1 v, (4.63) 2 where f(v z )dv z is the probability distribution of a component of velocity, m f(v z ) = 2πkT e mv2 z /(2kT ). The average speed v is given by v = 0 vf (v)dv, (4.64) where F (v) is the speed probability distribution, which in a spherically symmetric distribution is ( m ) 3/2 F (v)dv = 4πv 2 e mv 2 /(2kT ) dv. 2πkT In light of above equations it is pretty straightforward to show that v z = 1 2 v, and Equation 4.62 describes the diffusion coefficient of particles in terms of the velocity v and magnetic field line diffusion coefficient D xx, assuming an isotropic particle distribution. to write For isotropic fluctuating field with no mean field similar arguments can be used x 2 = 2D xx t s t = vd xx t = 2κ xx t, (4.65) where D xx (s) is the field line diffusion coefficient defined as D xx = x2 2 s parametrization along the field line. and s is a The final expression for the particle diffusion coefficient in terms of the field line diffusion coefficient in the isotropic FLRW limit is then κ xx = 1 2 vd xx. It should be emphasized that the FLRW limit is insufficient in describing particle diffusion because resonant scattering of particles due to small scale magnetic field violates the assumption of constant pitch angle. The FLRW process may become a 79

99 transient first diffusive process where particles could follow the field lines initially before resonant scattering dominates. Most of the studies, however, show that the resonant scattering occurs before the particle achieves stable diffusive regime which is confirmed by numerical simulations [117]. Quasilinear theory is a strong candidate to describe the resonant scattering (e.g. Chapter 8) where particles suffer changes in pitch angle due to magnetic disturbances of the scale of the Larmor radius of the particles. Due to resonant scattering the particles can backtrack along the field line which is absent in the FLRW phenomenon. Quasilinear theory, although applicable to some resonant scattering phenomena, has its limitations. Quasilinear theory gave incorrect results for perpendicular diffusion of charged particles [117, 118, 120, 152]. In general, quasilinear theory is also incorrect for parallel transport if we consider composite geometry [120,152]. Parallel diffusion in slab geometry is an example where quasilinear theory works reasonably [152], but for perpendicular transport in slab turbulence quasilinear theory provides a diffusive behavior, while subdiffusion is found in simulations [118, 152]. In a pure two dimensional geometry quasilinear theory gives the value of all Fokker-Planck coefficients equal to zero or infinity [134]. These theoretical results indicate that quasilinear theory is not valid for pure two-dimensional geometry and many non-slab geometries. In quasilinear theory the particle orbits are replaced by unperturbed trajectories. But, after a long time the particle trajectory increasingly deviates from the unperturbed trajectory and nonlinear effects set in. Therefore, instead of assuming unperturbed motion along the background magnetic field, we have to formulate nonlinear theory to have the particles, gyrocenters move perpendicular to the background field. Theories involving nonlinear effects are, however, difficult to develop and may involve assumptions, hypotheses (like the Corrsin s independence hypothesis) and ansatzes. One of the older models to include nonlinear terms is the method of Owens [105,106], which uses the nonlinear closure approximation (NCA) to the quasilinear type equation. The idea was similar to the earlier methods by Dupree [43, 44]. Later, the Nonlinear Guiding Center theory [97] emerged as a strong candidate for explaining perpendicular 80

100 diffusion when compared to the simulations. We will describe the numerical methods used in this thesis in the next chapter and then move on to charged particle diffusion in isotropic turbulent magnetic fields which uses a combination of nonlinear theory and quasilinear theory to explain particle diffusion in all possible energy ranges. 81

101 Chapter 5 NUMERICAL SIMULATION Numerical simulations are used to provide an independent check of the validity of the theoretical models. The nonlinear analytical method used in the calculation of diffusion coefficients make a lot of assumptions and use Ansätze that are not present in the numerical simulations which are very reliable when properly performed. In this chapter we provide details of the numerical methods used in the thesis work. There are two types of numerical approaches considered in the study of plasma turbulence: fluid approach and kinetic approach. Fluid description is a reasonable description of the flow of plasma (eg. solar wind) where the spatial and temporal scales are much larger than the kinetic scales (like the gyromotion of particles). At smaller scales kinetic effects such as wave-particle interactions start becoming important. In this thesis we mostly use the kinetic approach; in particular, we test particle simulations, which are useful tools to study interaction of charged particles with the electromagnetic fields. In the test particle approach the electromagnetic fields are treated as prescribed and the particles are treated fully independent of one another in the simulation. These particles are affected by the electromagnetic fields around them but they do not feed back to the fields. Test particle calculations have been used in a broad class of problems in space physics and astrophysics, such as particle transport, energizations and dynamics in complex systems. The test particle model is not self-consistent in the sense that the particles do not interact with one another and do not feed back to the electromagnetic fields. However, because of the complexity of the fully self consistent dynamic models, the test particle method is a useful bridge between the fluid model applicable only to the macroscopic aspect of plasma and the complex self-consistent kinetic approach. 82

102 We consider the transport of charged particles in a turbulent magnetic field with no temporal variation. The electric field in the plasma system is also ignored. In general, since the plasma is highly conducting, they arrange themselves in such a way that the electric fields are negligible inside a plasma. We only consider the velocities (v) of the particles, that are much greater than the Alfvén speed, the speed at which the disturbances in magnetic field travel in space, so that the magnetostatic assumption is valid. The electric field of the order v A B/c, where c is the speed of light, created by the change in magnetic field in time is negligible. Before we start our test particle simulation we should first have the magnetic field with turbulent properties and specified energy spectrum. We prescribe two different ways in which magnetic field can be generated. The most prevalent method of generating the magnetic field is to create a Gaussian random magnetic field vector in space with solenoidal property and a prescribed spectrum which is usually the standard Kolmogorov (k 5/3 ) spectrum. The numerical simulation of the propagation of charged particles in three-dimensional isotropic turbulence is also performed using these Gaussian random magnetic fields. However, using this simplest method will not produce higher order statistics compatible with the bursty properties of intermittent turbulence found in the real astrophysical systems like the solar wind. Therefore, we describe an approach to generating synthetic magnetic field with intermittency, generalizing a method due to Rosales and Meneveau (2006) [123] for synthesizing intermittent hydrodynamic velocity fields, in Chapter 10. This method involves using Gaussian random magnetic fields in k space which goes through a sequence of low pass filtered fields at which fluid particles are displaced at their rms speed for some scale-dependent time interval, and then interpolated back to a regular grid. The magnetic field is also carried along with the fluid parcel and eventually interpolated back. The velocity and magnetic fields produced in this way are seen to possess intermittent properties of real turbulence. Effect on the transport of charged particles due to the introduced intermittency in the fields will be a subject of future study. 83

103 5.1 Governing Equations of Motion For the numerical simulation we solve the most fundamental equation: The Newton-Lorentz equation of motion, which is also used in our theoretical calculations, given by: dr dt = v, (5.1) dv dt = q (v b). (5.2) mγc where q and m are the particle s charge and mass, respectively, γ is the Lorentz factor, c is the speed of light and b is the magnetic fluctuation field. Use distance in the units of correlation length l c, time in the units of l c /c and velocity in the units of c. The magnetic field is normalized in terms of the rms magnetic field δb. The physical parameters can now be written as: r = r l c, (5.3) v = v c, (5.4) b = b δb, (5.5) t = t l c c. (5.6) The Newton-Lorentz equation in terms of the normalized units is given by where dv dt = β v B, (5.7) β = q δb l c γmc = Ω l c 2 0 c. (5.8) Ω 0 = qδb/(mγc) is the gyrofrequency of the particle and β is dimensionless. The Larmor radius of the particle is then: R L = v Ω 0 = cmγv qδb. (5.9) 84

104 The ratio of the Larmor radius to the correlation length of turbulence is given by R L l c = cmγv qδbl c = v βc. S.I. units are used and the particle is taken to be a proton. We study the dependence of the mean free path of the particles with the ratio R L /l c. Particularly for isotropic magnetic fields, the mean free path is dependent only on this ratio (Chapter 6), and the results of transport of particles in isotropic fields is invariant of the system under study. For example, the physical parameters can be chosen to be applied for three different astrophysical systems, intergalactic, galactic and interplanetary. For the galactic system we choose the root mean square magnetic field δb = 0.1 nt and the correlation length l c = 10 pc. The Larmor radius of the protons is equal to the correlation length of the magnetic field fluctuations at Energy E ev. For the interplanetary system the root mean squared magnetic field is taken to be 5 nt and the correlation length is chosen to be l c = 0.01 AU, for which R L = l c at 1.5 GeV. We solve the Newton-Lorentz equation of motion for protons in several astrophysical systems mentioned above. The energy of the particles is varied and we average over several particles to calculate statistical quantities such as the mean free path, velocity autocorrelation, etc. In the next section we will show in detail how the turbulent magnetic fluctuation fields are generated in space where the particle trajectories are solved using the Newton-Lorentz equation. 5.2 Generating Slab and 2D Fields We choose the Kolmogorov (5/3 rd ) spectrum in the inertial range of turbulence (Section 1.2). To fulfill this power law behavior the power spectrum for slab turbulence is given by Pxx slab (k z ) = Pyy slab (k z ) = C slab, (5.10) [1 + (k z l z ) 2 ] 5/3 where C slab is the normalization constant to set the desired level of slab energy and l z is the parallel coherence length. Figure 5.1 shows the typical form of the slab spectrum. 85

105 Figure 5.1: Slab spectrum. The slab components of the magnetic fluctuations can be written in terms of the power spectrum as (see also Section 3.1): b slab x (k z ) = Pxx slab (k z ) exp[iφ(k z )] (5.11) b slab y (k z ) = Pyy slab (k z ) exp[iφ(k z )], (5.12) where φ is the random phase number generated using a Gaussian random number generator. For 2D turbulence A(k ) as defined by Equation 3.13 is set to A(k ) = C 2D, (5.13) [1 + (k l ) 2 ] 7/3 so that we again have a Kolmogorov-like power spectrum as shown in Figure 5.2. The magnetic field components of 2D fluctuations in k space are b 2D x (k x, k y ) = ik y A(k ) exp [iφ(k x, k y )] (5.14) 86

106 Figure 5.2: 2D spectrum. b 2D y (k x, k y ) = ik x A(k ) exp [iφ(k x, k y )]. (5.15) 5.3 Generating 3D Random Gaussian Magnetic Fields The isotropic 3D fluctuating magnetic field is generated on a regular Fourier grid (k space) in a periodic box. It is more convenient to work in wave number space because the magnetic field we generate must have the superposition of fluctuations at many different scales. Eventually we use the Fourier transform to obtain the magnetic field realization in configuration space. It is important to have several correlation lengths in the box to satisfy the homogeneity and isotropy condition. Similarly, the small scale fluctuations become very important when the Larmor radius of the particle is much smaller than the correlation length of turbulence. Therefore, we need a large separation of scales in our magnetic field realization. 87

107 The magnetic field is solenoidal; i.e., B = 0 in real space and ik B = 0 in wave number space. This is the reason we can specify the magnetic field fluctuation in k-space in terms of two polarizations: b(k) = [a 1 (k)ˆb 1 (k)e iφ1(k) + ia 2 (k)ˆb 2 (k)e iφ2(k) ], (5.16) where ˆb 1 = k ẑ/k and ˆb 2 = k (k ẑ/(k k) are unit vectors perpendicular to each other and also to k, which ensures that the constructed field is divergence-free. For 3D isotropic fluctuations at a given k, the omnidirectional energy spectrum E(k) is chosen to have a Kolmogorov spectrum at large k E(k) = 4πk 2 P (k) k 5/3, (5.17) where P (k) = P ii = P xx (k) + P yy (k) + P zz (k) is the spectral density (see Chapter 3 for details). To satisfy this condition, for isotropic turbulence we have chosen E(k) = Ck 4, (5.18) (1 + (kλ) 2 17/6 ) where C is the normalization constant, which determines root mean square value of the fluctuations and λ is the bendover scale where the spectrum bends to k 5/3 (Figure 5.3). By assuming equal polarization along ˆb 1 (k) and ˆb 2 (k) and using the definition of P (k) one finds a 2 1 = a 2 2 = P (k)/2 Re [ a 1 (k)e ] iφ 1(k) P (k) = cos[φ 1 (k)] 2 Im [ a 1 (k)e ] iφ 1(k) P (k) = sin[φ 1 (k)] 2 Re [ a 2 (k)e ] iφ 2(k) P (k) = cos[φ 2 (k)] 2 Im [ a 2 (k)e ] iφ 2(k) P (k) = sin[φ 2 (k)]. 2 88

108 We employ the inverse on Fourier transform b(k) to obtain b(x) in real space. The magnetic field in real space is generated on a periodic box with a specified number of grid points. The fluctuations are usually constructed over the simulation box with dimensions N x N y N z = grid points. Generating magnetic field in a periodic box with grid points is computationally more expensive but not so advantageous than the field in terms of capturing the physics required to compare with the analytical results. The root mean square field δb and bendover scale λ are set to be 1. The correlation length, calculated using Equation 3.40 gives l c 0.5λ = 0.5. With these parameters, we use smaller box size of 2π for low energy particles to have larger inertial range bandwidth and resonant scales, and larger box size of 50 for high energy particles to have about 100 correlation lengths in the box where particles deflect from their path at the scale l c and homogeneity and isotropy are important. We use Runge-Kutta method with adaptive time stepping regulated by fifth-order error estimate to solve the Newton-Lorentz Equation for tracing the trajectories of the particles in the random field particles are traced, to help ensure sufficient sampling. 5.4 Numerical code History The version of the code used is a part of a series of codes developed at the University of Delaware under the supervision of Dr. Matthaeus s group. The original code, also called the streamline code was a versatile Fortran 77 and C++ program for calculating charged particle trajectories or field lines using test particle method [39]. The streamline code solved the first order ODE, to calculate particle trajectories or static field lines. The user maintained control over the initial conditions of the trajectories or field lines and the boundary conditions. However, because of the need to advance from the aging Fortran 77 to the newer Fortran 90, and also to get rid of the convolution between Fortran and C++, Douglas Rodgers developed a new streamline code version 5.0 in 2011 consisting entirely of Fortran 90, which is the one used in 89

109 Figure 5.3: Omni-directional Spectrum of the magnetic field fluctuations. this thesis. The code has a serial version and an MPI parallelized version for the computation of streamlines, magnetic field lines or particle trajectories. In all of the version of the codes, the basic idea is to solve the first order ODE and the main integration relies on the code from Numerical Recipes [114]. The code can handle analytical fields or tabulated fields. The interpolation of the tabulated fields to evaluate at a test particle position can be performed with a linear interpolation scheme or a polynomial interpolation scheme. For the calculation of particle trajectories in this thesis we mostly use linear interpolation. The integration of ODE is performed using fourth order Runge-Kutta method with adaptive time-stepping and fifth order error estimate. The accuracy of the code is presented in Appendix A Diffusion Coefficient After the magnetic field line is generated we solve the Newton-Lorentz equation of particle motion using the Runge-Kutta method. We trace 2000 particles in a periodic box in isotropic magnetic field turbulence, so that we have enough statistics to study the average collective motion of the particles. The particles are initialized at random 90

110 positions with random directions in velocity in the simulation box to analyze their trajectories and to calculate the diffusion coefficient. The omni-directional spectrum goes like k 4 at small k and k 5/3 at large k. This functional form is chosen so as to be consistent with the strict homogeneity [8], and for high k to represent Kolmogorov scaling in the inertial range of turbulence. The form of the omni-directional spectrum used in our numerical simulations is given by Equation For 3D isotropic turbulence, we will also look at how the resolution effects the diffusion coefficients of particles by using different grid points of 512 3, and simulation and conclude that captures most of the physics required. After we run the simulation for the particles, we can directly compute x 2 for each t from the particle trajectories data. denotes the averaging over the 2000 particles. Obviously for small t we have more samples for averaging than for large t. We calculate the particle trajectories well beyond the time where we can see the clear asymptotic diffusion coefficient. This way of calculating diffusion coefficient is also called the microscopic diffusion coefficient (as described in Section 2.1). Figures 5.4, 5.5 and 5.6 are the examples of clear asymptotic limits of the running diffusion coefficients for low energy, intermediate energy and high energy particles respectively as defined by the ratio of Larmor radius to the correlation length of turbulence R L /l c. The microscopic diffusion coefficient and macroscopic diffusion coefficient (as described by Gaussian distribution in Section 2.2) give results that are very close to each other. This is further demonstrated by Figure 5.7 for a range of particle energies. The macroscopic diffusion coefficient displacement is calculated for the number density f(x, t) of the particles after a sufficiently long period of time so that the diffusive limit is assumed to have occurred. All particles (protons) are high energy with a speed close to the speed of light c. A simulation is used. The correlation length l c = 1Mpc, and root mean squared magnetic field δ b = 10 ng, corresponding to the extra-galactic system (see Section 9.1 and [107]). 91

111 Figure 5.4: Diffusion coefficient, macroscopic (top) and microscopic (bottom) when R L /l c = The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient which is close to the microscopic diffusion result on the bottom. 92

112 Figure 5.5: Diffusion coefficient, macroscopic and microscopic when R L /l c = 0.3. The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient which is close to the microscopic diffusion result on the bottom. 93

113 Figure 5.6: Diffusion coefficient, macroscopic and microscopic when R L /l c = 1.1. The plot on the top is the number density of the particle f(x, t) normalized by the injection number N 0 vs the displacement. The dotted line is the analytical curve (Equation 2.18) with the diffusion coefficient 49 which is close to the microscopic diffusion result on the bottom. 94

114 Figure 5.7: Microscopic vs macroscopic diffusion coefficients with good agreement. 95

115 5.4.3 Importance of Large Separation of Scales The elementary notion of particles following field lines emerges in a single particle orbit, where the gyrocenter of charged particle motion remains on a certain magnetic field line, when that field is uniform and constant and the electric field is negligible. In the isotropic turbulence problem, at low energies, when the Larmor radius of the particles are much smaller than the correlation length of turbulence, a similar phenomenon dominates where particles are tied to their initial field lines for a much longer period of time. If small fluctuations on the scales comparable to the Larmor radius of the particles are absent then locally particle sees smooth magnetic field line. This means that the particles pitch angle will be constant for a long period of time and the diffusion of the particle can be explained by magnetic field line diffusion which is called the Field Line Random Walk (FLRW) model. However, when the small scale fluctuations comparable to the Larmor radius of the particles are present, then the particle experiences pitch angle scattering. This pitch angle scattering will eventually turn the particle around and the diffusive process becomes completely different from the simple FLRW model. The astrophysical systems have large separation of scales, and therefore the fluctuation of magnetic field contains the superposition of very small scales to large scales, so the FLRW limit becomes inaccurate in explaining the scattering of charged particles. It can be demonstrated using numerical simulations that if the power at larger wave-numbers that pitch angle scatter the particles, is absent then the pitch angles of the particles are correlated for a much longer period of time. Figure 5.10 shows a pitch angle correlation for an ensemble of particles in magnetic field fluctuations that have larger (Figure 5.8) and smaller (Figure 5.11) inertial range bandwidth. For a larger inertial range bandwidth which has small scale fluctuations that resonate with particle motion, the pitch angle is correlated for a much shorter period of time. When the field has a smaller inertial range bandwidth, the particles pitch angle remains highly correlated even after a long period of time. The effect of numerical resolution is clearly seen in Figure 5.11, which shows 96

116 Figure 5.8: Omni-directional spectrum with larger inertial range bandwidth. Figure 5.9: Omni-directional spectrum with smaller inertial range bandwidth. 97

117 Figure 5.10: Pitch angle correlation of low energy particles with larger (solid line) and smaller (dotted) inertial range bandwidth. Larger inertial range bandwidth contains small scale fluctuation that resonate with particle motion which causes pitch angle scattering and therefore the pitch angle remains correlated only for a short period of time. 98

118 Numerical Numerical Numerical R L λ xx /l c Low Energy Intermediate Energy R 1/3 L High Energy 10-1 R 2 L R L /l c Figure 5.11: The effect of the resolution on low energy scaling. The inverted triangles are the results of numerical simulation, circles are those of simulation and the crosses are those of numerical simulation. The results of and simulation lay on top of each other but the low resolution simulation give almost constant diffusion coefficient for low energy particles, implying FLRW phenomenon due to the lack of resonant scales. 99

119 a low resolution simulation (512 3 ) with the same box size and correlation scale l c as high resolution simulations ( and ). With a smaller number of grid points, the grid scale, which is the smallest fluctuation scale, is large when compared to a simulation with larger number of grid points, since the box size and the correlation length are unchanged. The smallest scale at which a particle can resonate is also larger, and the lowest energy particles do not have a resonance scale. Without resonance the low energy particles in simulation keep following the field lines and their deviation from the initial path is solely dependent upon field line random walk and independent of energy (Figure 5.11). This is why we get incorrect mean free paths (λ xx ) at low energy for particles in a simulation. 100

120 Chapter 6 CHARGED PARTICLE DIFFUSION IN ISOTROPIC TURBULENCE The transport of charged particles in many astrophysical systems is governed by the highly turbulent magnetic field which leads to the diffusion of charged particles in space. Some of the most notable ideas in the study of diffusion of charged particles in magnetic turbulence are quasilinear theory [69], field line random walk theory [73], and nonlinear guiding center theory [97]. These theories focus on calculating the diffusion of charged particles in directions parallel and perpendicular to a constant mean magnetic field B 0, under influence of a fluctuating magnetic field b(x) that depends on position x but not on time. In the presence of a mean field, the diffusion tensor is calculated separately for parallel and perpendicular directions with respect to the ordered field. For the 2D and slab model applicable to heliospheric magnetic field turbulence, there have been numerous studies on the calculation of parallel diffusion [116,165] dominated by the slab field and perpendicular diffusion [12,97,137,159,168] dominated by the 2D field. A considerable modification to these theories is required when the fluctuations are isotropic with zero (or very small) mean field. Such a modification turns out to be of the highest importance when describing the transport of cosmic rays (CRs) in the galaxy, as well as for the modeling of diffusive CR acceleration in astrophysical sources. In this chapter we consider the problem of magnetostatic scattering with B 0 = 0, and for fluctuations b that are statistically isotropic, in terms of both the polarization and spectral distributions. The propagation of CRs in the galaxy is usually modeled as diffusive in a turbulent magnetic field, where the rms fluctuations of strength δb are of the same order of magnitude as the large scale field B 0 [68], δb/b 0 1, although it is not clear whether this condition is fulfilled both in the disc and the halo of the galaxy. On the other hand, supernova shocks, usually supposed to be the main sites 101

121 of acceleration of galactic CRs [19], are observed to possess intense turbulent magnetic fields, where the large scale field, if any, only affects the development of the fast growing instabilities that lead to the existence of intense turbulent fields [29]. At the shock itself, the field is probably well modeled as isotropic with a negligible mean field. The turbulent magnetic field is not the only complication that appears in a realistic scenario of cosmic ray transport in the Galactic system. The complex composition of background plasma and the presence of neutral hydrogen atoms that suppress the propagation of waves through ion-neutral damping are examples of factors that could influence the charged particle propagation. Including these complexities is beyond the scope of this thesis. Here we will develop a systematic derivation of charged particle diffusion in the presence of an isotropic turbulent magnetic field and test its validity using Monte-Carlo simulation of particles in a synthetically generated turbulent magnetic field. In fact, we would like to point out that incorporating the several factors mentioned above will probably require MHD, hybrid or particle in cell (PIC) simulations that provide a self-consistent picture of the development of the cosmic ray transport phenomenon [30,122]. When the field is purely turbulent, the combination of particle diffusion and random walk of magnetic field lines is expected to play a crucial role. Diffusion of magnetic field lines in isotropic turbulence with zero mean field was examined by Sonsrettee et al. [147], where I am also a co-author, and that paper is in some ways an antecedent of the present work. The transport of charged particles in interplanetary space and the interstellar medium, including in regions of particle acceleration, is highly influenced by the presence of turbulent magnetic fields and their spectral distribution. The nature of particle transport in these fields also depends on particle energy. In general, higher energy particles, with a gyro-radius larger than the correlation length of the magnetic field, will sample many uncorrelated field lines within one gyration. Lower energy particles with a gyro-radius much smaller than the magnetic field correlation length, on the other hand, will see a relatively coherent large scale field, and their transport will be heavily affected by resonances with local magnetic fluctuations. In the next section we provide 102

122 a short account on previous related works and summarize the ideas to be presented in the subsequent sections. 6.1 Previous Studies and New Ideas There have been some studies of diffusion of charged particles in magnetic turbulence without a mean field [31,42,107,112,146], but a clear theory covering all ranges of particle energies was still lacking. In particular, the simulations carried out in [42] and [107] clearly showed that in the limit of large δb/b 0 1 the diffusion coefficient of particles with Larmor radius much smaller than the energy containing scale of the turbulent field closely resembles the one naively estimated from quasilinear theory, a rather curious result since such theory applies to cases where mean field is significant. Although, the quasilinear scaling of low energy particles was known [3,31,42,107], there was a lack of a theory that closely agreed with the numerical results. The simulation result of [42,107] has been verified using our code and we have developed a quasilinear theory applicable to isotropic turbulence at low energies that has an excellent agreement with the numerical results. The quasilinear theory as developed involves nonlinear corrections at 90 0 pitch angle and the use of Maxwellian magnetic field distribution in space. As far as we know, the theory of diffusion of charged particles in isotropic turbulence that describes the intermediate energy range of particles did not exist. Other than a simple curve fitting model provided by [107], the nonlinear theory of intermediate energy particles was largely unknown. We will particularly extend the low and high energy theories to develop the intermediate energy theory. It is however important to note that the high energy theory we have developed is not new to the astrophysics community, although the theory we provide is systematic and detailed than the known intuitive derivation. Basically, we classify the diffusive behavior of charged particles into three different regimes based on the ratio of the Larmor radius R L of the charged particle to the characteristic outer length scale of turbulence l c : a high energy region with 103

123 R L /l c 1, an intermediate energy region with R L /l c 1, and the low energy region with R L /l c 1. Corresponding to the two extreme inequalities, we will develop two corresponding theoretical approaches for particle diffusion in the extreme energy ranges. First, in the high energy limit, the path of the particle experiences only small deviations from its initial trajectory due to small angle scattering on magnetic field irregularities. In this case velocity diffusion is achieved when the magnetic fluctuations probed by the particles become uncorrelated. At the opposite extreme, particles with a very small gyro-radius will gyrate about the local field produced by the large scale fluctuations while experiencing perturbations, the most effective of which will be at scales comparable to the gyroradius. This leads to resonant interactions and a random walk of the particle pitch angle. The particle s guiding center will eventually reverse direction and parallel spatial diffusion is achieved. The range of validity of both the high energy and the low energy asymptotic theories will be extended by building in additional decorrelation effects in the relevant Lagrangian correlation functions, thereby providing an accurate description of the particle mean free paths also in the intermediate range of energies. To calculate the spatial diffusion coefficient it is important to understand the behavior of velocity space diffusion coefficient which we will explore in the following section. 6.2 Velocity Space Diffusion and Spatial Diffusion in Isotropic Turbulence Since the turbulence is isotropic, and the particle speed v is constant in the absence of electric fields, the velocity space diffusion tensor is expected to have an isotropic form [8] (see also Chapter 3), D ij (v) = (δ ij ˆv iˆv j )D v, (6.1) which implies that D v = 1 2 Tr[D ij(v)]. (6.2) 104

124 The distribution function of test particles obeys the Fokker-Planck equation in velocity space as described in Section 2.6, which when spatial gradients are present can be written as f t + v f = f D ij. (6.3) v i v j Plugging in the isotropic form of the velocity space diffusion (Equation 6.1) into Equation 6.3 we get, f t + v f = D v 2 f, (6.4) where 2 stands for that part of the velocity space Laplacian in spherical co-ordinates tha involves no radial derivatives. To make a connection with spatial diffusion, we construct a multiple time scale solution of Equation 6.4, by breaking down time and spatial scales into fast (τ, ξ) and slow variables (T, X). We seek a solution with f = f (0) + ɛf (1) + ɛ 2 f (2) +..., where ɛ 1. The q th order distribution function can be expanded in spherical harmonics as: f (q) = l,m C (q) lm (x, v, t)y lm(θ, φ). (6.5) We proceed using spatial and temporal variables divided into slow variables X and T, and fast variables ξ and τ, respectively. The slow and the fast variables can be written in terms of a small parameter ɛ as: T = ɛ 2 t, X = ɛx τ = t, ξ = x. The motivation for different time ordering when compared to the spatial ordering comes from the fact that, if we let ɛ be the ratio of scattering length scale to the transport length scale, then from the spatial diffusion equation a simple dimensional analysis shows that the ratio of scattering time scale to the transport time scale should be of the order ɛ 2 [48] (also shown in Section 2.3). 105

125 The derivatives can then be written as t = τ + ɛ2 T, x = ξ + ɛ X. Considering Equation 6.3 (Fokker-Plank equation), we seek a solution of the distribution function f with f = f (0) + ɛf (1) + ɛ 2 f (2) +..., (6.6) where each order of f can be expanded in terms of the spherical harmonics Y lm (θ, φ) (Equation 6.5), which satisfy v 2 2 Y lm (θ, φ) = [ l(l + 1)] Y lm (θ, φ). (6.7) The expansion in Equation 6.6 is inserted into Equation 6.4. With the help of Equations 6.5 and 6.7 we get separate equations for each of the different orders of f. The O(ɛ 0 ) terms give: f (0) τ + v ξ f (0) = D v 2 f (0). (6.8) Now we introduce an operator which is the space-time average over the fast variables. Since all variables will be assumed to have finite variations in time and space, the averaging operator gives a zero result when operating on any quantity that may be written as a derivative with respect to fast variables. This is also known as the solvability condition so that we have a closed form of equations. Also, let f (q) = F (q). Equation 6.8 averaged over the fast variables gives: D v 2 F (0) = 0, (6.9) which implies that F (0) is isotropic in v. Averaged over the fast variables, the O(ɛ 1 ) term is v X F (0) = D v 2 F (1). (6.10) 106

126 Without loss of generality we may pick the direction of the gradient to be the z-direction. The left-hand-side of Equation 6.10 has a projection only onto the Y 10 term of the spherical harmonics. The right hand side must also have a projection onto the Y 10 term. With the help of Equations 6.5 and 6.7 in Equation 6.10, one finds v X F (0) = 2 D v v 2 F (1) F (1) = v2 2D v v X F (0). (6.11) When averaged over the fast variables the O(ɛ 2 ) term gives: F (0) T + v XF (1) = D v 2 F (2). (6.12) Substituting the relation between F (1) and F (0) from Equation 6.11 into Equation 6.12 one gets F (0) T v2 2D v ( vi v j Xi Xj ) F (0) = D v 2 F (2), (6.13) where a repeated index indicates summation. To study the diffusion of the bulk distribution, we consider the average over all directions. From Equation 6.7, we see that the right hand side of Equation 6.13 averages to zero. Using we then obtain [v i v j ] direction averaged = v2 3 δ ij, (6.14) F (0) T = v4 6D v 2 XF (0). (6.15) This is a diffusion equation in configuration space with diffusion coefficient κ xx = v4 6D v, (6.16) and in isotropic turbulence: κ xx = κ yy = κ zz = κ The mean free path (λ) in terms of the diffusion coefficient is simply λ = 3κ v. (6.17) 107

127 6.3 Taylor-Green-Kubo Formula for Velocity Space Diffusion and Spatial Diffusion Particle statistics may be related to the spatial diffusion coefficient by the Taylor-Green-Kubo (TGK) formula [57, 86, 154] (see Section 2.4): κ xx lim t x 2 2t = 0 dt v x (0)v x (t). (6.18) This formula, with v x interpreted as the guiding center velocity in the x-direction, was used to calculate the diffusion coefficient in theories that have a mean magnetic field, e.g., BAM theory [12], the original NLGC theory [97], etc. as: The velocity space diffusion coefficient can also be written in a TGK formulation dvi D ij (v) = dt dv j 0 dt 0 dt. (6.19) t To calculate the rate of change of velocity in Equation 6.19 we use the Newton- Lorentz equation of motion dv dt = q (v b) = α(v b), (6.20) mγc where q and m are the particle charge and mass, respectively, γ is the Lorentz factor, c is the speed of light, b is the magnetic fluctuation field and we define α = q/(mγc). We apply the Newton-Lorentz equation in Equation 6.19 to find D ij (v) = α 2 dt ɛ iαβ ɛ jγη v α (0)v γ (t)b β (0)b η [x(t)]. (6.21) 0 Assuming that the particle velocity and magnetic field are uncorrelated (as would be the case, e.g., for an isotropic particle distribution) and that the turbulence is statistically homogeneous, one finds D ij (v) = α 2 ɛ iαβ ɛ jγη dt v α (0)v γ (t) b β (0)b η [x(t)]. (6.22) 0 Equation 6.1 is used to simplify Equation The high energy theory and nonlinear theory explained in the following chapters differ in the treatment of the correlation tensors in Equation

128 Chapter 7 ISOTROPIC TURBULENCE: HIGH ENERGY THEORY Very high energy particles with Larmor radii much larger than the correlation scale will experience only minor deflections from their original path as they complete a distance equivalent to the correlation length of the magnetic field. The high energy theory developed in this section is applicable when R L /l c 1. The appropriate simplifying assumptions in this case are that the displacement follows a straight line x(t) = vt, and that the velocity autocorrelation is simply v α (0)v γ (t) = v α v γ. Using these, Equation 6.22 can be written as: D ij (v) = α 2 ɛ iαβ ɛ jγη v α v γ dt b β (0)b η (x(t) = vt). (7.1) 0 Assuming that the magnetic field cross-correlation is zero and writing the contraction of the Levi-Civita symbols in terms of the Kroneker deltas, in view of Equation 6.2 and computing the trace of Equation 7.1 gives a simple expression for the high energy velocity space diffusion coefficient, D v = α2 vδb 2 l c, (7.2) 3 where δb is the rms magnetic field. The correlation length l c is given by (Section 3.4) l c = v 0 dt b i (0)b i (vt) δb 2. (7.3) Notice that D v is independent of the spectrum of turbulence, because the only thing that matters is the fact that most of the energy is in magnetic fluctuations at a fixed scale l c, and at that scale we may assume that the effective magnetic field 109

129 seen by the particle is the rms field δb. Plugging D v in Equation 6.16, in terms of gyrofrequency Ω 0 = αδb, the spatial diffusion coefficient is then 7.1 Alternative Derivation κ xx = κ yy = κ zz = v3 2Ω 2 0l c. (7.4) The result given by Equation 7.4 is not entirely new to the astrophysics community [3]. An intuitive derivation of Equation 7.4 proceeds as follows. Suppose that most of the power in the magnetic field spectrum is on a spatial scale l c which is much smaller than the Larmor radius of particles. Then one can think of a very large but otherwise arbitrary distance l d as being tiled with cells of size l c. In each cell, when R L l c, one has a deflection of the order l c /R L 1. The average angle of deflection when there are n scatterings satisfies ( ) 2 θ 2 lc = n, (7.5) R L where n = l d /l c. Now, the average deflection angle is of order unity when l d = R2 L l c. (7.6) At that point the distance is also l d vt (because the displacement from the unperturbed trajectory is small), and the diffusion coefficient is: which is identical to Equation 7.4. κ = l2 d 2t = l dv 2 = R2 L v 2l c = v3 2Ω 2 0l c, (7.7) 7.2 Nonlinear Extended High Energy Theory For rigidity decreasing towards unity from large values, the assumption above that the unperturbed particle trajectory is a straight line with constant velocity becomes less accurate. 110

130 Instead, we assume the velocity autocorrelation is exponential with a characteristic decorrelation time τ, v α v γ (t) = (v 2 /3)δ αγ e t/τ. (7.8) Making use of the TGK formula, Equation 6.18, one sees that the spatial diffusion coefficient κ xx is related to the time scale τ by κ xx = 1 3 v2 τ. (7.9) That is, from Equation 6.16 τ = Then, using Equations 6.2, 6.22 and 7.8 one finds that D v = 1 2 Tr[D ij(v)] = α 2 where there is an implied summation over i. v2 2D v. (7.10) 0 dt v2 3 e t/τ b i (0)b i [x(t)], (7.11) The closure for the Lagrangian correlation in this equation should now be reconsidered. In particular, we question the approximation of a straight line unperturbed trajectory x(t) = vt that was previously used in the high energy theory. To allow for the effect of perturbations of this trajectory, x(t) may be treated as a random variable. Following the familiar procedure used in other nonlinear diffusion theories [97], Corrsin s independence hypothesis ( [37] explained in Section 2.5) is used to write the magnetic correlation in terms of the power spectrum in Fourier space to obtain b i (0)b i [x(t)] = dkp ii (k) e ik x(t). (7.12) In order to evaluate the characteristic functional e ik x(t), we now use the approximation of random ballistic decorrelation (RBD) [127]. The latter was first introduced in describing the magnetic field line random walk by [50]. Here, the result is e ik x(t) = e ikµvt µ = dµ e ikµvt = sin(kvt). (7.13) kvt 111

131 The random ballistic model is justified since the particles undergo ballistic motion at earlier times before they reach the asymptotic diffusive regime. With τ given by Equation 7.10, substituting Equation 7.13 into Equation 7.12 and then into Equation 7.11 gives D v = α 2 v2 3 dke(k) 0 dt sin(kvt) e 2tDv/v2, (7.14) kvt where E(k) = 4πk 2 P ii (k) is the omnidirectional energy spectrum. If we take the limit τ = v 2 /(2D v ) in Equation 7.14, Equation 7.2 is recovered, which is the high energy limit. Hence, the nonlinear theory allows departures from the high energy limit, but also recovers the exact form of the high energy limit as τ becomes large. More generally, Equation 7.14 is an implicit equation for D v and Equation 6.16 gives the spatial diffusion coefficient. It is worth pointing out that another standard approximation would be to treat the trajectories of the particles as having a diffusive distribution (DD) for the purpose of calculating the Lagrangian magnetic correlation function [13,97,132,133]. In that case, the ensemble average on the left hand side of Equation 7.13 would then be e ik x(t) = e k2 κ xxt. This alternative approach, however, fails to predict the high energy behavior of the particles where the particles are ballistic for a long time before undergoing multiple deflections to reach the diffusive limit. For the intermediate range DD converges with the RBD model. 112

132 Chapter 8 ISOTROPIC TURBULENCE: LOW ENERGY THEORY Low energy particles (R L /l c 1) behave entirely differently because they experience a local mean magnetic field due to large scale fluctuations. These small gyroradius particles typically scatter before moving far enough for the local mean field to average to zero. In such circumstances, there are two dominant effects that are expected to contribute to transport and diffusion: Field Line Random Walk (FLRW) and a kind of resonant wave particle scattering in which the local field acts as a mean field that organizes the particle gyro-motion. In the presence of FLRW alone (Section 4.4), particles would follow magnetic field lines and achieve spatial diffusion as the magnetic field lines diffuse in space. Decorrelation of the particle trajectories due to this mechanism [60, 73] gives rise to an energy-independent mean free path. This would not explain the results of our numerical simulations, as discussed in Chapter 9. One may, however, estimate from a simple Kolmogorov turbulence theory the degree to which the field lines bend for distances shorter than a correlation scale, as is done in Appendix B. The conclusion is that the angular deflection of the mean field seen by a particle in moving over a scale l is small provided that l/l c is small, i.e., the scale l lies deep in the inertial range. This conclusion is also empirically supported, in that the numerical results (Figure 9.2) show that at low energy the particle s mean free path is smaller than the correlation length of the magnetic field. For lower energy the local mean field, due to the large scale parts of the spectrum, becomes increasingly coherent. Assuming that the local mean field remains well defined for long enough distance, we may examine whether the resonant scattering of particles on fluctuations with wavenumber around the inverse of the Larmor radius [46, 69] provides effective 113

133 scattering in the regime in which turbulence is weak compared with a locally evaluated large scale mean magnetic field. Of course if the small scale power is suppressed, then one recovers the FLRW result [78, 92]. In the presence of a guide field, the original quasilinear theory [69] represents the standard tool for calculating resonant pitch angle diffusion coefficients of particles. It is successful in predicting the resonant scattering of the particles in the case of a slab fluctuation field (wave vectors parallel to the mean field) but may require some modification in other fluctuation geometries [118]. Non-linear theories are also useful in attaining better agreement with the results of numerical simulations of particle propagation [43, 44, 97, 106, 132, 134]. In the present case of isotropic turbulence, with no guide field, the assumptions of quasilinear theory (QLT) can no longer be applied, although we are encouraged to proceed based on the above reasoning concerning the local mean field. To be specific, it is useful first to put forward a physical explanation of the approach we propose to describe the low energy regime: if the power spectrum of magnetic fluctuations is such that most power is on scales of order l c, then particles with Larmor radius much smaller than l c move following a roughly ordered magnetic field line at least until a distance of l c has been covered. On such scales the propagation is diffusive in the direction of the local magnetic field, although particles suffer little motion in the direction perpendicular to that of the local field. Let us refer to this parallel diffusion coefficient as κ, although the physical meaning of this quantity should be kept in mind. The effective velocity of particles in the direction of the local field is v p κ /l c. When particles move over many times the coherence scale l c, their transport in the directions perpendicular to the original local field becomes evident (due to isotropic turbulence) and one can estimate the global diffusion coefficient as: κ(p) = 1 3 L cv p 1 3 L κ c 1 L c 3 κ. (8.1) In other words, provided the propagation is diffusive on small scales, the global diffusion coefficient on large scales proceeds with a similar diffusion coefficient to that calculated using the local magnetic field as an effective local guide field. The problem 114

134 is now reduced to calculating the diffusion coefficient experienced by particles along the local magnetic field. This can be done by using a close analogy to the case of QLT. In the following section we will derive a relationship between parallel spatial diffusion and pitch angle diffusion along the local mean field. 8.1 Relation Between Parallel Spatial Diffusion and Pitch Angle Diffusion Before we proceed to derive the details of the pitch angle diffusion coefficient in the subsequent sections, we will first derive a relationship between the parallel spatial diffusion coefficient and the pitch-angle diffusion coefficient. Let us consider the z motion which is governed by the Fokker-Planck equation: f t f + µv z = 1 [ µ 2 2 µ t f µ ] (8.2) Scattering causes an evolution of the distribution function to a diffusive regime that relaxes towards isotropy. But there is also a relatively small anisotropy that leads to a slow temporal change in isotropy. This implies that we can expand the distribution function as a sum of the isotropic part and the smaller anisotropic part: f(µ, z, t) = F 0 (z, t) + F 1 (µ, z), (8.3) where F 1 is an odd function of µ. Since 1 f dµ = 0, there are no even functions 1 in the expansion. This is analogous to the expansion of the distribution function in Legendre polynomials by Jokipii (1968 [70]). By substituting Equation 8.3 into the Fokker-Planck Equation 8.2, independent equations are obtained for the odd and even components: Equation 8.4 can now be written as: F 0 t + µv F 1 z = 0 (8.4) µv F 0 z = 1 [ µ 2 2 µ t ] F 1. (8.5) µ F 0 t + µv F 1 z = 0 (8.6) 115

135 Integrating over µ from -1 to +1 gives F 0 t + S z This is the continuity equation where the flux S is given by S = (v/2) +1 1 µf 1 dµ = v = 0. (8.7) 1 0 µf 1 dµ. (8.8) Applying an indefinite integral over µ ( dµ with no limits) to both sides of Equation 8.5 simply gives: φ F 1 µ = (1 µ2 )v F 0 z, (8.9) where φ = µ 2 / t and the constant of integration was chosen to satisfy the condition φ(+1) = φ( 1) = 0. A second integration from 1 dµ to both sides of Equation gives F 1 = v F 0 z µ 0 1 ν 2 dν. (8.10) φ(ν) Integrating by µdµ from -1 to 1 on both sides of 8.10 and using the results of Equation 8.8 one finds S = v 2 F 0 z 1 0 µ µdµ 0 1 ν 2 dν. (8.11) φ(ν) Let us look at Equation 8.7. Fick s law says that the flux must be proportional to the gradient of concentration or density, and the constant of proportionality is the diffusion coefficient κ, in which case which when used in Equation 8.11 gives 1 κ = v 2 µdµ S = κ F 0 z, (8.12) 0 µ 0 1 ν 2 dν. (8.13) φ(ν) Interchanging the order of integration over µ and ν, in terms of the pitch angle diffusion coefficient, Equation 8.13 now becomes κ = v (1 µ 2 ) 2 dµ. (8.14) D µµ (µ) 116

136 Equation 8.14 completes the derivation of the parallel spatial diffusion coefficient in terms of the pitch-angle diffusion coefficient D µµ. The diffusion coefficient along a fixed direction (e.g., the x-direction) is given by κ xx = κ cos 2 θ + κ sin 2 θ, (8.15) where θ is the angle between the magnetic field direction and the x-axis. Since the particles are expected to travel parallel to the local mean field and undergo resonant pitch angle scattering, which eventually causes a random walk in the parallel direction with little average perpendicular motion, we neglect κ in Equation 8.15 and average over all directions to obtain κ xx = κ yy = κ zz = κ /3, (8.16) consistent with Equation Generalized Pitch Angle Diffusion Coefficient given by Using the Taylor-Green-Kubo formula, the pitch angle diffusion coefficient is D µµ (µ) = 0 dt µ µ, (8.17) where µ is the pitch angle at time t = 0 and µ is the pitch angle at a later time t. Particles are assumed to follow a local field line. The initial direction of the field line can be assumed without loss of generality to be the z-direction and using v z = µv, we get Using Equation 6.20 one can write D µµ (µ) = α2 v 2 0 D µµ (µ) = 1 dt v v 2 z v z. 0 dt [ v x v xb y b y + v y v yb x b x v x v yb y b x v y v xb x b y ]. (8.18) The Cartesian x and y components of the velocities are given by v x = v sin φ 0 v y = v cos φ 0 v x = v sin(ωt + φ 0 ) v y = v cos(ωt + φ 0 ), (8.19) 117

137 where v = v 1 µ 2 is the perpendicular component of the velocity, Ω is the local gyrofrequency and φ 0 is the initial gyrophase of the particle motion. Assuming the product b i b j is independent of φ 0, assuming axisymmetry of magnetic fluctuations along the local mean field ( b x b x = b y b y and b x b y = b y b x ), using elementary identities and after averaging over the initial gyrophase φ 0, Equation 8.18 reduces to D µµ (µ) = α2 v 2 v 2 [ 0 ] dt cos(ωt) b y b y. (8.20) The Lagrangian two-time correlation function b y b y = b y (0, 0)b y (x(t), t) is greatly simplified using the QLT-like assumption that the particles locally follow straight magnetic field lines in the z-direction with constant pitch angle, so that z = vµt. Using Corrsin s hypothesis [37], which is exact for slab fluctuations that vary only along z, the inverse Fourier transform of the correlation function b y b y in terms of the power spectrum P yy (k) becomes b y b y = dk x dk y dk z P yy (k)e ikzvµt e ikxx e ikyy. (8.21) In the following two sections, we proceed to evaluate Equation 8.21 by adopting different approximations for the characteristic functional e ikxx e ikyy which describes the statistics of the local random perpendicular displacements x(t) and y(t). 8.3 Standard QLT Approach When the perpendicular displacements x(t), y(t) are very small, we may adopt the approximation e ikxx e ikyy 1. This can be viewed as an approximation that the particle is at its guiding center, or that the fluctuations b x and b y are independent of x and y. This approximation is implemented in standard QLT, and is exact in one-dimensional slab geometry [69]. Using this in Equation 8.21, one finds [ ] b y b y = dk z dk x dk y P yy (k) e ikzvµt. (8.22) In the usual way we define E y (k z ) = P yy (k)dk x dk y as a one dimensional reduced transverse spectrum function. Substituting Equation 8.22 in Equation 8.20, and 118

138 carrying out the time integral yields a Dirac delta function that defines the resonance. Then, the integral over the parallel wave number (k z ) gives ( ) D µµ = πα2 (1 µ 2 ) E y k z = Ω v µ, (8.23) v µ which is the standard QLT result. One modification applicable to isotropic turbulence which we would like to discuss in detail is related to the µ dependence of D µµ in Equation It is well known that quasilinear theory does not provide correct pitch angle diffusion coefficients for pitch angles close to 90 (µ 0) where nonlinear effects are important [14, 139, 153]. This problem was discovered in the years after quasilinear theory had been proposed [52, 75, 106]. The strict quasilinear calculation using Equation 8.23 (for typical spectra of turbulence) has D µµ 0 as µ 0, which is applicable in slab turbulence [119] where nonlinear effects are weak. In most other geometries, however, for any realistic finite amplitude fluctuations, nonlinear orbit effects [78] allow particles to scatter more easily through the neighborhood of µ = 0 than would be expected from QLT calculations [119]. Our numerical results also indicate that the quasilinear theory for isotropic turbulence as derived using Equations 8.23, 8.14 and 8.16 gives mean free paths much larger than those obtained from numerical simulation, since the parallel diffusion coefficient is unrealistically amplified by D µµ near zero in the denominator. To account for such nonlinear effects, we consider an Ansätze that the pitch angle dependence of D µµ is of the form D µµ 1 µ 2 and set µ = 1 inside the square bracket of Equation 8.23 so that D µµ has a finite nonzero value at µ = 0. Before implementing this, we checked our numerical simulation results for the pitch angle distribution. For D µµ 1 µ 2 (isotropic scattering) the eigenfunctions of the diffusion operator are Legendre polynomials, and at later times when the pitch angle distribution is nearly isotropic, the distribution should be dominated by the most slowly evolving eigenfunctions, P 0 = 1 and P 1 = µ, yielding a nearly linear pitch angle distribution. We indeed found this in our simulation data. Thus, we implemented the Ansätze, which is more appropriate for our case of isotropic turbulence with B 0 = 0 than for a 119

139 perturbative situation with b B 0 where nonlinear effects are weak. The Ansätze is also justified a posteriori in Chapter 9, where the theoretical predictions now provide a much better fit to the numerical results. With this physically motivated modification, the simplified pitch angle diffusion coefficient can be written as D µµ = πα2 (1 µ 2 ) v ( E z k z = Ω ). (8.24) v If we assume that the local mean field is constant throughout the system with magnitude B local then in Kolmogorov turbulence E z k 5/3 z and (from Equation 6.20) α = Ω/B local with Ω = v/r L, it is straightforward to show that the substitution of Equation 8.24 into 8.14 gives κ vl c (R L /l c ) 1/3 and the mean free path (Equation 6.17) scales as λ l c (R L /l c ) 1/3. A further correction that we apply here is related to the variability of the magnetic field strength in an isotropic random field. In the realization of turbulence used here, the components of the magnetic field have a Gaussian distribution (this is also usually a reasonable approximation for fully developed turbulence). Therefore the gyrofrequency Ω = qb/(γmc) = αb and Larmor radius R L = γmvc/(qb) = v/(αb) vary in space as the magnetic field strength b changes. In fact b is likely to undergo major changes during the parallel scattering process described by κ. It is fairly simple to show that b has a Maxwellian distribution given by F(b)db = 3 3/2 2 π b 2 3b 2 δb 3 e 2δb 2 db, (8.25) where δb is the root mean square field strength. With the above Maxwellian distribution, we use Equation 8.24 to compute the average D µµ (µ) over the local field b before substitution in Equation 8.14 to obtain κ = v dµ (1 µ2 ) 2 D µµ. (8.26) Equation 8.16 gives the spatial diffusion coefficient along a particular axis. The mean free path scaling of λ l c (R L /l c ) 1/3 is also maintained. 120

140 8.4 Extended Low Energy Theory As an extension to the above quasilinear approach, we now take into account the perpendicular displacements that enter into consideration in Equation Assuming a Gaussian distribution of the perpendicular displacements x and y implies that e ikxx e ikyy = e 1 2 [ x2 kx 2+ y2 ky 2]. (8.27) Since we are assuming locally an unperturbed orbit about a well defined local mean magnetic field, it is clear that x 2 = y 2 RL 2. In particular, for pitch angle θ and gyrophase φ, be written as x = R L sin θ cos φ, y = R L sin θ sin φ. (8.28) After omnidirectional averaging of x 2 and y 2 over θ and φ, Equation 8.27 can e ikxx e ikyy = e k2 R2 L /6, (8.29) where k 2 = k2 x + k 2 y. This statistical description of the perpendicular displacement is consistent with our earlier assumption that b i b j is independent of the initial gyrophase. Using Equation 8.20 along with Equations 8.21 and 8.29 one finds 0 D µµ (µ) = 2π2 α 2 (1 µ 2 ) v ( ) k dk P yy k, k z = Ω v µ e k2 R2 L /6. µ (8.30) According to the arguments used in Section 8.3, we take into account the presence of nonlinear orbit effects near 90 pitch angle and modify the pitch angle dependence so that D µµ (1 µ 2 ) and set µ = 1 inside the square bracket of Equation For the extended case, the pitch angle diffusion coefficient is now given by 0 D µµ (µ) = 2π2 α 2 (1 µ 2 ) ( v k dk P yy k, k z = Ω ) e k2 R2 L /6. v (8.31) 121

141 Note that when the exponential term is set to unity (e.g., for R L 0), this reduces to Equation As described in Section 8.3 we average Equation 8.31 over a Maxwellian distribution of the magnetic field magnitude b before substitution in Equation 8.14 to obtain our final result for the spatial diffusion coefficient. The exponential term can be seen as another modification to the original quasilinear theory appropriate for the pitch angle diffusion of low energy particles in isotropic turbulence. Hence we use this extended low energy theory in Chapter 9 to compare with numerical simulation results. 122

142 Chapter 9 ISOTROPIC TURBULENCE: COMPARISON OF THEORETICAL RESULTS WITH NUMERICAL SIMULATION In this chapter we present results of numerical simulations of charged particle propagation in synthetic magnetostatic turbulence with a specified spectrum. numerical results are compared with the theoretical formulations described in Chapters 6, 7 and 8. The numerical simulations make use of a homogeneous and isotropic magnetic fluctuation field, generated on a spatial grid with a specified energy spectrum. The trajectories of 2000 particles are obtained by numerical solution of the Newton-Lorentz equation, using a fifth-order Runge-Kutta method with adaptive time-stepping. To satisfy the magnetostatic assumption, the velocity v of the particles is chosen so that v v A, where v A is the Alfvén speed. The electric field is ignored as it is of order v A B/c, where B is the magnetic field and c is the speed of light. The random magnetic field realization is generated in a periodic box as described by [147] and Section 5.3. The functional form of the omni-directional spectrum used in the numerical simulation is given by E(k) = Cλ c (kλ c ) 4 [1 + (kλ c ) 2 ] 17/6, with a normalization constant C used to control the magnetic field strength b. The parameter λ c is the bendover scale, which is of the same order as the correlation length. This form of E(k) is chosen so that E(k) k 4 for low k to be consistent with strict homogeneity [7], and E(k) k 5/3 for high k to represent Kolmogorov scaling in an inertial range of turbulence. We have chosen the Kolmogorov spectrum, as is often The 123

143 Numerical Numerical Numerical λ xx /l c R 1/3 L 10-1 R 2 L R L /l c Figure 9.1: Mean free path of protons as a function of R L /l c (gyroradius divided by correlation scale). The solid line is the high energy scaling λ xx RL 2 and the dashed line is the low energy scaling λ xx R 1/3 L. The inverted triangles are the results of numerical simulation, circles are those of simulation and the crosses are the results of numerical simulation Numerical High Energy Theory Nonlinear Theory Low Energy Theory λ xx /l c Low Energy ev Intermediate Energy High Energy ev R L /l c Figure 9.2: Theoretical vs. numerical results for cosmic rays in isotropic turbulence with zero mean field, showing good agreement. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for cosmic rays in our galaxy. 124

144 assumed in scattering theory [59, 107], but emphasize that the theoretical approach can be applied to any reasonable spectrum. The diffusion coefficient is calculated from the asymptotic rate of increase of the mean square displacement of the particles κ xx = lim t x 2 2 t. (9.1) The results of the numerical simulations are shown in Figure 9.1. The general asymptotic trends for the mean free path at low energies (R L l c ) and high energies (R L l c ) are easy to identify. In the low energy regime the mean free path scales as λ xx R 1/3 L, while in the high energy regime λ xx RL 2, and these scalings are valid whether the particles are relativistic or non-relativistic. The high energy scaling is obtained irrespective of the power spectrum, provided most power is concentrated on scales around l c. These results confirm previous functional forms proposed by [3] and [107] and the numerical results of [31], [42] and [146]. Observations of isotopic composition [1, 104] also support a scaling of λ xx R 1/3 L at R L l c. The resonant nature of particle scattering in the low energy regime makes numerical simulation rather challenging, in that the resulting mean free path is sensitive to the resolution in real space, which is equivalent to the number of independent degrees of freedom in k-space used to represent the magnetic power spectrum [146]. In Figure 9.1 we show the results of our simulations for 512 3, , and real space grid points, from which it is possible to see that the lower resolution (512 3 ) case does not match correctly the scaling of the mean free path with Larmor radius. This is because the simulation does not numerically resolve the resonant scales corresponding to the Larmor radius of the particles. On the other hand, no appreciable difference can be seen in the two higher resolution cases with and simulations. Hence, we will use the realization for subsequent discussion of the results. The mean free path calculated using the theoretical framework we have developed is compared with the results of simulations in Figure 9.2. These results may be considered as representative of propagation of very high energy CR protons in the 125

145 Numerical High Energy Theory Nonlinear Theory Extended Low Energy Theory λ xx /l c Low Energy ev Intermediate Energy High Energy ev R L /l c Figure 9.3: Theoretical vs. numerical results for high energy relativisitc charged particles in isotropic turbulence for parameters as listed in Section 9.1. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and the dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for protons and magnetic field parameters for extra-galactic medium. galactic magnetic field [129] with correlation length l c = 10 pc and root mean square magnetic field δb = 0.1 nt. The Larmor radius of the particles equals the correlation length of the magnetic field fluctuations at energy E ev, somewhat above the knee in the all-particle spectrum of galactic CRs. However, we note that when plotted this way, as mean free path vs. Larmor radius, with both quantities normalized to the correlation scale of the turbulence, the actual curves are identical for any energy range. As shown by Figure 9.2, the theoretical results are in a very good agreement with numerical simulation. 9.1 High Energy Relativistic Particles in extra-galactic Medium The mean free path from theory is also compared with numerical simulation in Figure 9.3 for high energy protons in the extra-galactic medium. The physical parameters chosen for our numerical simulation are: l c = 1Mpc (Megaparsec) and the root mean square magnetic field δb = 10 ng (nano gauss). Similar parameters have 126

146 Figure 9.4: Parametrization of the diffusion coefficient with energy. κ E 1/3 for low energy particles and κ E 2 for high energy particles. 127

147 been used in previous numerical studies (Aloisio and Berezinsky (2004) [3]; Parizot (2004) [107]), mostly applicable to the extra-galactic space where the regular magnetic field is expected to be negligible and the fluctuations are also expected not to be more than a few tens of nano Gauss [20, 85]. The low energy particles have energies approximately less than ev, and the high energy particles have energies approximately greater that ev. The numerical results agree well with the theoretical predictions. The diffusion coefficient (κ) for charged particles in the extra-galactic medium scales as κ E 1/3 (corresponding to R 1/3 ) for the low energy resonant regime, and L κ E 2 (corresponding to RL 2 ) for the high energy limit. A very good fit of the diffusion coefficient with energy is obtained using the scaling relation [ ( ) 1/3 ( ) ( ) ] 2 E E E κ = D 0 + +, E 0 for some arbitrary constant D 0 and critical energy E 0. Hence, the diffusion coefficient can be obtained straightforwardly from one single parameter, E 0, which basically gathers the relevant information about the magnetic field intensity and coherence length. Although high energy particles in the extra-galactic medium were used in Figure 9.4, the result is in fact very general and can be applied to any system when the turbulence is isotropic with no mean field. E 0 E Non-Relativistic Example To provide a broader context, in Figure 9.5 we evaluate the theory for the case of non-relativistic particles or moderately relativistic particles. For convenience we adopt parameters that closely resemble interplanetary space in our solar system, except that here no mean magnetic field is included. For this example, the particles are non-relativistic at low energies with R L /l c < 1, but become moderately relativistic at higher energies. The correlation length is chosen to be l c = 0.01 AU and the root mean square magnetic field to be 5 nt [28]. Then R L = l c at 1.5 GeV. The theoretical mean 128

148 Numerical High Energy Theory Nonlinear Theory Low Energy Theory λ xx /l c Low Energy 30 MeV Intermediate Energy High Energy 4GeV R L /l c Figure 9.5: Theoretical vs. numerical results for non-relativistic or moderately relativistic charged particles in isotropic turbulence with zero mean field for parameters as listed in Section 9.2. The circles represent the numerical results, solid line represents the high energy theory, dashed line is the nonlinear theory and the dotted dashed line is the theoretical estimate for low energies. The energy ranges are shown for protons and magnetic field parameters for interplanetary turbulence. The results also apply to any non-relativistic particles of a given R L /l c. free paths depend only on rigidity and test particle simulation is in close agreement whether the particles are relativistic or non-relativistic. It is well known that the interplanetary turbulent magnetic field is represented by an anisotropic spectral model in which the fluctuations are of the same order of magnitude as the mean field. Numerous realistic studies have also been performed and compared with observational results in the past [151], so our study with isotropic turbulence and no background field should be interpreted as simply a test case. 9.3 Discussion Referring to Figures 9.2, 9.3 and 9.5, we can see that both the asymptotic low and high energy limits are fitted well by our theoretical approach. The high energy theory fails in the intermediate and low energy regimes because the particle velocity changes over a shorter period of time, contrary to the high energy assumption. 129

149 The extended low energy theory compares well with the numerical data even in the intermediate energy regime when R L /l c 0.5. However, it fails when R L /l c 0.5 as a consequence of the fact that the effective guide field becomes ill-defined. The nonlinear theory best describes the scattering of the intermediate energy particles in the range R L /l c 0.5. The two extended theories give equal results at R L /l c 0.3 where they each differ from the numerical results by about 30%. In conclusion we have developed theories that can describe diffusive propagation of charged particles at all energies in an isotropic turbulent medium without a mean field. 9.4 Implication of the Results There are several important problems and unanswered questions in Astrophysics. One of them relates to the identification of astrophysical sources of very high energy cosmic rays and the acceleration mechanisms to such high energies. The other important independent problem relates to the propagation of these cosmic rays in the universe as a whole. An important aspect of the propagation is the deflection of the cosmic rays in the presence of the ambient magnetic fields. This deflection of cosmic rays play an important role during the transport of cosmic rays and significantly modify the original path that must be taken into account to locate the source of cosmic rays. In the galactic system the background magnetic field is expected to be very low and the turbulent fields heavily influence the cosmic ray spectrum. These turbulent fields scatter the particles from their original path and the particles are often completely isotropized. In such a circumstance, it is essential to abandon the idea of individual particles and treat the propagation as a diffusive process. Therefore it is very essential to have a complete theoretical idea of the diffusive process of particles in the presence of highly turbulent magnetic fields. The flux of the cosmic rays observed is sharply reduced due to the fact that the diffusion process may prevent significant propagation of cosmic rays and the particles may remain confined in a galaxy for a long time. Figure 9.2, represents a complete picture of the theoretical model we have developed for 130

150 charged particle propagation in intense turbulent fields with no background field. All the energies of the particles are accounted for and the mean free path is only dependent upon the rigidity of the particle. The theory is also applicable to any spectral index of turbulence. As such, Figure 9.2 can be extensively used to predict the turbulence parameters of galaxies given we can identity the energies of the particles we observe. The diffusion coefficient as formulated is also essential to understand the energy dependence of the observed ratio of primary to secondary galactic cosmic ray abundances [1, 104]. The composition of cosmic rays can change while they propagate from the source to the observer, along with the production of secondary particles, and the abundance of these secondary particles when compared to the primary particles reflects the span of confinement of these particles in the galaxy. The observed data of primary and secondary particles can be used to test the diffusion model. The availability of a working diffusion model can also be useful in predicting the ratio of primary and secondary cosmic rays and understand the cosmic ray decay process as a whole. In conclusion, the diffusive propagation of cosmic rays has been an important topic of discussion for astrophysicists. We believe that our work has significantly advanced the understanding of the cosmic ray diffusion in highly turbulent magnetic fields that is present in galactic and extra-galactic medium. 131

151 Chapter 10 SYNTHETIC GENERATION OF INTERMITTENT FIELDS Plasma turbulence is a non-linear dynamical phenomenon found in the solar wind, Earth s magnetosphere and several other astrophysical systems. Studies of turbulence and its effects ordinarily involve analysis of observational or experimental data, or data derived from direct numerical simulations of dynamical models such as magnetohydrodynamics (MHD), or data derived from simplified models. In the latter class a particularly useful and simple approach is to construct synthetic realizations of turbulence data. A familiar procedure would involve construction of a random Gaussian vector field (Section 5.3) by controlling global parameters such as spectral distribution, total variances, means and correlation scales. It is straightforward in this way to construct synthetic surrogates for records of turbulence data accumulated for example in a wind tunnel or in the solar wind in circumstances in which the Taylor frozenin hypothesis is valid. Such synthetic samples have some realistic features including Kolmgogorov-like inertial range spectral behavior (as outlined in 5.3), but using the simplest methods will not produce higher order statistics compatible with the bursty properties of real intermittent turbulence. Here we describe a numerical method to generate synthetic magnetic fields with intermittency, generalizing a method due to Rosales and Meneveau 2006 [123] for synthesizing intermittent hydrodynamic velocity fields Physical Motivation The basis for this technique is a simple dynamical system, derived from the Navier-Stokes equation, for the Lagrangian evolution of velocity increments [90]. This system exhibits important features of three dimensional hydrodynamic turbulence. 132

152 / E(k) / Level 1 Level 2 Level 3 Level 4 Level 5 Level k 2 Figure 10.1: Plot of the energy spectrum being used and schematic representation of the hierarchy of nested levels. Even when all interactions between fluid particles are removed (by neglecting the effects of pressure and viscosity), the probability density of functions (PDFs) of the increments retain skewness towards negative values and stretched exponential tails. The equation of motion that describes incompressible fluid flow is the Navier- Stokes equation: u t + u u = P + ν 2 u, (10.1) where u is the fluid velocity, P is the pressure, and ν is the viscosity. On eliminating all interactions, a model equation emerges t u + (u. )u = 0, (10.2) which describes fluid parcels moving purely as a consequence of self-advection. If this advection maps the element at position a at time t = 0 to the position x in time t, then the solution to Equation 10.2 may be written as u(x, t) = u(a, 0). (10.3) 133

153 Figure 10.2: Fluid particles for a particular filtering level, shown on the regular grid with the arrows representing the velocities of the fluid parcel at each node. Figure 10.3: Fluid particles after being displaced from their original positions. Smaller dots represent the regular grid points. The velocities are interpolated back to each regular mesh point by using a weighted average over a sphere of radius R centered around the mesh point. 134

154 Making the simplifying assumption that the velocities are constant over the range of space and time in question, this mapping may be written as x(t) = a + tu(a, 0). (10.4) To proceed, the working hypothesis is that (at least some) dynamical effects of Equation 10.2 that generate intermittency are captured in the linear map L : a x, called the minimal Lagrangian map. This map has been called the naive Lagrangian map [9, 48]. It is used iteratively in the MMLM formulation to distort an initially Gaussian vector field to introduce realistic features of turbulence Extension of the Procedure to a Plasma While considering the extension of the procedure to a plasma system, we are particularly interested in generation of synthetic magnetic fields in addition to turbulent velocity field. One possibility would be to simply generate two independent fields and take one to be the velocity field, and the other to be the magnetic field. However, in this approach, we would be neglecting the coupling between the two fields in a plasma system. The induction equation of magnetohydrodynamics is where b is the magnetic field, and ν is the resistivity. b t = u b + µ 2 b, (10.5) In ideal MHD, when the conductivity is infinite the above equation becomes b t = u b. (10.6) According to Alfvén s frozen-in theorem [33], the above equation implies that the magnetic field is advected along with the fluid parcels. In view of these considerations, while applying the MMLM procedure to a plasma, rather than performing an independent mapping for the magnetic field, we employ the following Lagrangian mapping equation for the magnetic field: b(x, t) = b(a, 0), (10.7) 135

155 where the relation between a and x is given by Equation That is, the Lagrangian transport of the magnetic field is due to the local value of the fluid velocity, under the approximation that the velocity is uniform over space and time over the region spanned by the local mapping operation. We may outline the magnetic implementation of the MMLM procedure as follows: we have two independent fields to begin with, which are filtered sequentially to different scales. At each level the Lagrangian mapping is applied at every node, which displaces the fluid parcels to the irregular grid, employing the velocity at that node, as shown in Figure The magnetic field at each node is assumed to be carried along with the fluid parcel with the velocity at that node, and after its transport onto the irregular grid, both magnetic and velocity fields are interpolated back to the regular grid using the averaging procedure schematically illustrated in Figure Here we would like to allude to a complication that arises from the coupling of the velocity and magnetic fields in MHD. The momentum equation for a plasma is u t + u u = (P + b2 2 ) + b b + ν 2 u, (10.8) which means that the velocity field is also dependent on the magnetic field, even in the absence of fluid interactions. However, in the current procedure, we regard the velocity as being independent of the magnetic field, with the magnetic field being advected passively along with the fluid Methods The technique begins with a vector field that is initialized in the usual way, by generating a synthetic isotropic solenoidal field with a prescribed energy spectrum. This involves generating random Fourier modes ˆη(k), with real and imaginary parts having standard unit-variance Gaussian distributions. Solenoidality is enforced by projecting the modes to a plane perpendicular to the wavevector k. The amplitudes are scaled to match the desired energy spectrum. The number of grid points in each direction is denoted as N, and the maximum resolved wavenumber is k max. The spatial 136

156 resolution, which corresponds to the mesh spacing on the computational grid, is h = π/k max. In order to capture the multi-scale character of turbulence, the original field ˆη, is filtered sequentially to several different scales, and a mapping procedure is applied at each scale. If M is the total number of filtering levels, then for the reduced mesh at the level n of the sequence, the number of modes is reduced by a factor m = 2 M n. At each filtering level, the filtered field is represented on a reduced mesh in physical space with N/m nodes in each direction, where m is an integer associated with the filtering level, and the spatial resolution at that level is mh. The process begins at the coarsest mesh and moves on to finer meshes that include smaller scales. The Lagrangian map is applied at each step with displacements of the order of the current mesh size mh. For example, if we begin with grid points and have the total number of levels, M = 7, then the procedure is applied recursively for The filtered field at each level is computed from the complete field from the previous level, which contains structures created by the Lagrangian mapping at scales greater than its filter scale, and Gaussian modes at the undistorted subfilter scales. The procedure thus acts on a hierarchy of nested levels, as illustrated in Figure 10.1, and the effects of the mapping at each level are superimposed onto the features at larger scales. The procedure is illustrated for a particular filtering level in Figures 10.2 and As shown in Figure 10.2, the fluid particles are displaced from their original positions on the regular rectangular grid using the mapping in Equation 10.4, with the arrows representing the magnitude and direction of the fluid particles at each grid point. In Figure 10.3, the fluid particles are shown displaced from their original positions, and the velocities are interpolated back to the regular grid (small dots) using a weighted average over the area (shown as a circle with radius R in the two dimensional figure) surrounding each grid point. Hence, the method consists of a sequence of filtering to coarsen, Lagrangian mapping, interpolation back to the coarse grid, and reassembly of a full resolution representation, prior to the next stage of filtering to a more refined representation. In 137

157 the following, we employ a notation similar to [123]. We begin with Gaussian velocity and magnetic fields in Fourier-space: û 0 (k) and ˆb 0 (k). If M is the total number of filtering levels, then in the reduction to a coarsened mesh at the level n of the sequence, the number of Fourier modes is reduced by a factor of m 3 where m = 2 M n. The original grid spacing h, where h = π/k max, is expanded to h n = mh. Filtering at level (n 1) yields the filtered velocity field ˆū n (k) = Ĝn(k)û n 1 (k) (10.9) and magnetic field ˆ b n (k) = Ĝn(k)ˆb n 1 (k) (10.10) at the nth level. Here Ĝn(k) is the transfer function of the filter corresponding to the scale h n = mh. In the current work we use a sharp Fourier cutoff filter at wave-number k c,n, and Ĝ n (k) = 1, Ĝ n (k) = 0, k k c,n k > k c,n with k c,n = π/h n. The velocity ū n (x n ) and magnetic field b n (x n ) in physical space are obtained by performing an inverse Fourier transform using the filtered (reduced) Fourier representation. The next step is to apply the Lagrangian map x n r n. That is, for each point at position x n = a n on the reduced computational grid, a new position r n is calculated as r n (a n ) = a n + t n ū n (a n ). (10.11) The new position r n (a n ) is the same for the velocity and the magnetic field. The time parameter t n is calculated based on the r.m.s. velocity for the current level, u rms,n, as t n = h n u rms,n, (10.12) 138

158 as suggested by [123]. In this way the r.m.s. displacement for the ensemble of particles is about one mesh increment. The velocities and the magnetic field are transported by the map from the original positions to the new locations: ū n (r n ) = ū n (a n ) and b n (r n ) = b n (a n ). They are represented on an irregular mesh defined by the points r n. For brevity we now employ the symbol v to mean either the velocity field or the magnetic field, as the following operational steps are identical for both of these. We next interpolate the field from the irregular mesh to the nodes of the regular mesh to obtain a new field v(x n ) on the regularly spaced coarsened mesh. The interpolation to each regular mesh point is performed by calculating weighted averages over nearby irregular points: v n (x n ) = r n(a n) x n R r n(a n) x n R W (x n, r n (a n )) v n (a n ) W (x n, r n (a n )), (10.13) where W (x n, r n ) are the weighting functions and R is the radius of the sphere around the node containing the irregular points used for interpolation. We take R to be the mesh increment h n. The weighting function is the inverse of the distance of the irregular points to the node, W (x n, r n ) = x n r n 1. (10.14) After this, the new field is transformed back to Fourier space, and made solenoidal by use of an algebraic projection operator. This yields ˆ v (s) such that ˆ v (s) n (k) = 0. The Fourier amplitudes associated with the part of ˆv n 1 that was filtered out in the prior step are ˆ v n(k) = (1 Ĝn(k))ˆv n 1 (k), (10.15) which are defined at all wavenumbers above the last used Fourier filter. The lower wavenumber Fourier amplitudes obtained after the interpolation and enforcement of solenoidality are the set ˆ v n (s) (k), which we now combine with ˆ v n(k) to produce a new full resolution field ŵ n (k) = ˆ v (s) n (k) + ˆ v n(k). (10.16) 139

159 These procedures, especially the mapping, have modified the energy spectrum, so the Fourier modes ŵ n are rescaled to recover the original spectrum: ˆv (a) n (k) = ŵ n (k) p =k E(k) ŵ n (p) ŵn(p) 1/2. (10.17) Note that the spectra are different for the magnetic field and the velocity field. The spectral renormalization step Equation completes the n level of the iterative procedure. Recalling that v in Equation refers to either the velocity or magnetic field, we next make the velocity field û (a) n and magnetic field ˆb (a) n the initial fields for the n + 1 level. After the procedure is carried out for M levels, we have generated non-gaussian turbulent fields, the features of which will be explained in Chapter

160 Chapter 11 ANALYSIS AND APPLICATIONS OF SYNTHETIC INTERMITTENT FIELDS This chapter presents detailed analysis and applications of the synthetic intermittent field generated using the procedure described in Chapter 10. Intermittency is a fundamental feature of turbulence that has been studied extensively in fluid flows [48] and MHD [18, 79]. Satellite observations of solar wind fluctuations have been used to study intermittency in the solar wind [27, 95, 162, 166]. Intermittency is related to the development of non-gaussian features at the small scales which lead to increased dissipation, emergence of characteristic scaling laws, and departure from self similarity. The usual method to generate synthetic vector fields is to use random phase Fourier modes (Section 5.3). These fields have Gaussian PDFs without the extended tails that characterize real turbulence and do not have turbulent coherent spatial structure. It is, however, desirable to have these features in synthetic fields for various reasons. For example, numerical simulations that use Gaussian synthetic fields typically require an adaptation period for a realistic turbulent field to develop. An initial field that already possesses turbulent structure would eliminate the adaptation period and save on computational time. Further, in order to have a more realistic representation of boundary and initial data for realistic smaller scale kinetic plasma simulation (e.g., using the Particle in Cell, or PIC, methodology), it is desirable to specify and emulate the intermittent, structured electromagnetic fields that are imposed at near-kinetic scales by decades of larger scale MHD activity. For example, in the lower solar corona MHD or related fluid models may well describe activity from the supergranulation scale (30,000 km) down to scales approaching the ion inertial scale ( 1 m). The turbulent 141

161 Figure 11.1: PDF of the longitudinal velocity field increment calculated from synthetic data with M = 5. With the increase in the increment r (direction of arrow) the PDF becomes more Gaussian, showing correct intermittent behavior. The smallest r is about 1/50 of the correlation length and the largest r is about a correlation length. cascade that spans more than seven decades in scale is expected to be highly structured and intermittent, and not simply random-phased and unstructured. Another area of application is in numerical studies of suprathermal particles and cosmic ray transport, which often employ Monte Carlo test particle orbit studies based on realizations of the turbulence present in the system under study. This usually requires specification of a background spectrum, and then generation of fields by imposing either random phases, or a Gaussian distribution of components. The effect of intermittency on these systems has not been well investigated, and synthetic intermittent fields would be useful for that purpose Intermittency Analysis of Synthetic Fields We perform statistical analysis of a sample velocity and magnetic field generated using the MMLM procedure. We initialize the fields with random phases (Gaussian components) on a discrete mesh which is processed though the MMLM procedure to 142

162 Figure 11.2: PDF of the transverse velocity field increment calculated from synthetic data with M = 5. With the increase in the increment r (direction of arrow) the PDF becomes more Gaussian showing correct intermittent behavior. The smallest r is about 1/50 of the correlation length and the largest r is about a correlation length. generate intermittency. We use nodes, and up to M = 7 levels of filtering and mapping are employed. The spectrum has a 5/3 slope in the inertial range and a 8/3 slope in the smallest scales. See Figure 10.1 and Chapter 10 for more details. To study the intermittent structure of the synthetic vector field p α (x), we calculate the longitudinal and transverse increments at different lags, r, as follows: δ p(r) = [p(x + rˆr) p(x)] ˆr, (11.1) δ p(r) = [p(x + rˆr) p(x)] ê, (11.2) where the vector lag is rˆr and ê is a unit vector perpendicular to ˆr. The shorthand δ p x indicates that ˆr is in the Cartesian x direction, while δ p x indicates that the lag is perpendicular to the x direction. The probability density functions (PDFs) of the velocity increments, δ u x (r) and δ u x (r), for total number of filtering levels M = 5, are shown in Figures 11.1 and 143

163 Figure 11.3: PDF of longitudinal increment for synthetic magnetic field data produced after M = 3 iterations of the filtering and mapping method. The arrow indicates the direction of increase in separation r/l c for the various curves. Those closer to Gaussian are at larger r/l c. Intermittency is apparent in the fatter tails at smaller scale increment lag. 11.2, respectively. It is clear that the PDFs exhibit qualitatively correct intermittent behavior; the statistics deviate increasingly from Gaussian as the lag r is decreased. This deviation from non-gaussianity at small scales implies the presence of extreme events that characterizes intermittency. Wider (fatter) tails are observed in the PDFs for both longitudinal and transverse increments. The longitudinal increments of the magnetic field obtained by the mapping method exhibit statistical features very similar to those of the velocity field. The associated PDFs are displayed for two values of the iteration number in Figure 11.3 (M = 3 iterations) and in Figure 11.4 (M = 7 iterations). The production of increased intermittency with increasing iteration number is evident. It is also once again clear that the mapping method generates the property that tails of the PDFs are stronger for shorter spatial lags, which is necessary for realism. 144

164 Figure 11.4: PDF of longitudinal increment for synthetic magnetic field data produced after M = 7 iterations of the filtering and mapping method. The arrow again indicates the ordering of the curves with increasing separation r/l c. Curves closer to Gaussian are at larger r/l c. The intermittent behavior is more pronounced as compared to the case with M = Gaussian Kurtosis M=7 M=6 M=5 M=4 M= r/l c Figure 11.5: Scale dependent kurtosis of longitudinal magnetic increments for the synthetic field, after different total number of filtering/mapping iterations M. The kurtosis at small scales is seen to increase with M, indicating that increased number of iterations increases intermittency. 145

165 Figure 11.6: PDF of the z-component of current (J z ), calculated from synthetic data after M = 7 iterations, showing intermittent behavior Scale dependent Kurtosis The non-gaussian property of the increments can be analyzed using the fourth moment of the distribution. We compute the kurtosis of the longitudinal increment, defined as K = δ b 4 δ b 2 2. (11.3) where K is kurtosis, and the suffix denotes a longitudinal increment. The kurtosis is a measure of non-gaussian features, such as extended tails in the PDF. A Gaussian field has a kurtosis equal to three. In Figure 11.5 we examine the kurtosis of the magnetic increments at different lags, also known as the scale-dependent kurtosis (SDK). We plot the SDK as a function of the lag normalized by the correlation scale, and we see that the kurtosis increases to non-gaussian values at smaller scales. In the same figure, we also look at how the total number of iteration levels in the MMLM procedure affects the kurtosis. The results quantify the effect alluded to above, that the procedure generates stronger intermittency as a higher number of iterations are employed. Hence we get some flexibility to obtain the desired kurtosis for the synthetic field experiencing the iterative Lagrangian map. We recall that the 146

166 longitudinal magnetic increment (Figure 11.3 and Figure 11.4) displayed increased intermittency for larger iteration number, and also for smaller spatial lag. At smaller spatial lags the increments may behave similarly to derivatives. This is confirmed in Figure 11.6, where we plot the PDF of the z-component of current density and observe the expected wider tails. The intermittency in the current density will be seen again in Sec. 11.3, where the two-dimensional contour plot of the current density has patchy and blob-like structures Tests in direct numerical simulation (DNS) of decaying turbulence We now look at the evolution of the synthetic fields under the influence of the incompressible, viscous and resistive MHD equations. We perform DNS of freely decaying MHD turbulence in a periodic cubic domain with mesh points. We initialize the simulation with fields generated using the MMLM technique (Synthetic Intermittent I.C.) and for comparison, we perform another run initialized with Gaussian fields (Gaussian I.C.). It is to be noted that we use fields only in this section, and that the nomenclature refers to the runs initial properties only. A standard Fourier pseudospectral method with second order Runge-Kutta time integration is used. Figure 11.7 illustrates the evolution of the total energy (magnetic plus kinetic) in the two runs during the first eddy turnover time, τ NL, of the dynamics. The initially intermittent run is seen to decay faster than the initially Gaussian field from the very beginning. The Gaussian field requires some time to generate the structures that lead to locally enhanced dissipation, while the intermittent field already possesses these features, which means that it starts with a higher energy decay rate. From a practical point of view this may be advantageous, as it may be desirable to obtain the appropriate decay rate for the initial spectral conditions near the beginning. Not only may this strategy save computing time in very large and demanding runs, but it also may suppress startup transients that might confuse interpretation of results. Once the Gaussian field develops structure and the prevailing spectral conditions have changed, the subsequent evolution may be more realistic (and intermittent). To deal with this 147

167 Energy (Kinetic+Magnetic) Gaussian Synthetic Intermittent τ NL Figure 11.7: The evolution of total energy (magnetic and kinetic) from MHD simulations using Gaussian initial conditions and using synthetic intermittent initial conditions. The total energy decays slightly faster using the synthetic intermittent fields as the initial condition in MHD simulations. adjustment, a familiar procedure is to rescale the spectrum after an initial period, and the simulation is considered to commence at that moment. The use of intermittent synthetic fields may reduce this computationally expensive presimulation period that the Gaussian fields require. We also compare the evolution of the scale-dependent kurtosis for the Gaussian and intermittent initial conditions for spatial lags r normalized to correlation scale l c. Figure shows that at τ NL = 0 the Gaussian fields have kurtosis 3 and intermittent fields start with higher kurtosis. Immediately after, at time τ NL = 0.1, the Gaussian I.C.s have formed some non-gaussian features, but the kurtosis in the intermittent case is much higher. At time τ NL = 0.5 the kurtosis of both the fields has increased but the intermittent initial fields maintain a higher value. To visualize the spatial structure of the field, we examine contours of the z- component of the current density (J z ) in physical space in a selected x y plane from 148

168 Figure 11.8: Contours of J z in a selected plane from a random phase Gaussian dataset. It has no significant structures. Figure 11.9: Contours of J z in a selected plane from a synthetic intermittent field. The current is concentrated mostly in small volumes creating bloblike intermittent structures. The spatial structures can be considered to occupy an intermediate state between the Gaussian fields and the fields run through MHD simulations. 149