On Kinetic Dissipation in Collisionless Turbulent Plasmas

ON KINETIC DISSIPATION IN COLLISIONLESS TURBULENT PLASMAS

by Tulasi Nandan Parashar

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 2011

by Tulasi Nandan Parashar

Approved: George Hadjipanayis, Ph.D. Chair of the Department of Physics & Astronomy

Approved: George Watson, Ph.D. Dean of the College of Arts & Sciences

Approved: Charles G. Riordan, Ph.D. Vice Provost for Graduate and Professional Education 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. 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: William H Matthaeus, Ph.D. Member of dissertation committee

Signed: Dermott Mullan, Ph.D. Member of dissertation committee

Signed: Stanley Owocki, 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: James MacDonald, Ph.D. Member of dissertation committee

Signed: I. Pablo Huq, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS

I have heard some people say that the acknowledgment section of a document should not be very personal but I believe in the opposite. My journey to this milestone of my life would not have been the same if not for the people around me. I consider myself to be one of the luckiest graduate students because I had the opportunity to work with my advisers Mike Shay and Bill Matthaeus. Whatever I know about plasma physics, turbulence theory, space physics and plasma compu- tation theory, I owe to them. They have an immense role in igniting and shaping my interest in a ﬁeld that I entered with a completely blank slate. I do not think a simple thank you or a simple statement of acknowledgment could do justice to their role in helping me to grow not just as a student and a potential scientist but also as a person. I hope I make them proud by putting whatever they have taught me to good use throughout my life. I was lucky to get a chance to collaborate with Paul Cassak and Sergio Servidio when they were postdoctoral fellows in our group. I bugged them a lot with millions of simple queries every single day they were here. Kittipat, John, Doug, Minping and Ben Breech were very helpful in countless discussions about basic physics and computations. I wish to especially thank my committee members for their invaluable input to this thesis. I am really indebted to them for pointing out a few aspects in the background material that I had glossed over. Their discussions have helped me make this thesis more streamlined and more relevant. I believe this has really helped in making this thesis a much better document than it originally was.

iv For the past few years, Daniel DeMarco very patiently listened to my com- plaints and problems about computers and very patiently and promptly fixed them, every single time, over and over again. I was never able to create a problem big enough that he could not fix quickly. I still remember that the first communication that I received, some time in the first week of February 2005, from UD about my application was from some Maura Perkins. She’s been a life saver ever since. Betty Baringer’s role in taking the pain out of the financial nuances is one which I think is hard to match by anyone. Debra and Mrs. Long, other than being my ”Good Morning Ladies”, were of great help with every thing else. Susan, Dennis, Dave Johnson, Tom Reed & Tom Reilly were very helpful whenever I went to them for any kind of help. This 6 year long journey would be almost impossible to finish if it was not for my friends. The list is so long that I would not attempt to include the names in here because this is no Oscar acceptance speech and I’m afraid I might miss one or two names. Friends, you know who you are and you know how much I value your support through the years!!! Thanks :-) Lastly but most importantly I should mention the real force behind my energies: my family. It is with their support, encouragement and love that I have made progress over the last 13 years of my life while living away from them. Trying to thank them would be like trying to describe the sun as a candle. So I’ll simply say: Ma, papa, sisters and Tintin: I love you all.

v Dedicated to:

The four people to whom I owe most of what I am and most of what I know

Ma, Papa, Mike & Bill

vi The voice on the solar wind breathed to them again. “I hope you are comfortable.” It said. ... “And I would like to congratulate you on the accuracy of your deductions.”

- The Hitchhiker’s Guide to the Galaxy.

vii TABLE OF CONTENTS

LIST OF FIGURES ...... xii LIST OF TABLES ...... xvii ABSTRACT ...... xviii

Chapter

1 INTRODUCTION ...... 6

1.1 A brief tour of the heliosphere ...... 6 1.2 Spacecraft Observations ...... 12 1.3 Coronal Observations ...... 13

1.3.1 Waves in solar corona ...... 13 1.3.2 Reconnection ...... 14 1.3.3 Heating ...... 16

1.4 Solar Wind Observations ...... 18

1.4.1 Waves in the solar wind ...... 18 1.4.2 Turbulence ...... 20 1.4.3 Reconnection ...... 22 1.4.4 Heating ...... 24

1.4.4.1 Protons ...... 26 1.4.4.2 Heavy ions ...... 28

1.5 Heating Mechanisms ...... 28

1.5.1 Wave Mechanisms ...... 30

1.5.1.1 Cyclotron Resonance ...... 31

viii 1.5.1.2 Waves in expanding box ...... 33 1.5.1.3 Dissipation or Dispersion? KAWs or Whistlers? ... 34

1.5.2 Non-wave Mechanisms ...... 38

1.5.2.1 Current sheets & reconnection sites ...... 38 1.5.2.2 Energization by trapping in magnetic islands .... 40 1.5.2.3 Stochastic heating ...... 40

1.6 Summary ...... 41

2 HYBRID CODE ...... 44

2.1 Various plasma descriptions ...... 45

2.1.1 Fluid description ...... 45 2.1.2 Fully kinetic description ...... 47

2.1.2.1 Vlasov ...... 47 2.1.2.2 Particle In Cell ...... 48

2.1.3 Compromise Models ...... 49

2.1.3.1 Gyrokinetics ...... 49 2.1.3.2 Hybrid Vlasov ...... 52 2.1.3.3 Hybrid PIC ...... 52

2.2 Hybrid Code P3D ...... 53

2.2.1 Electron Physics ...... 54 2.2.2 Code details ...... 55

2.2.2.1 Particle Stepping ...... 57 2.2.2.2 Particle Moments ...... 59 2.2.2.3 Field stepping ...... 60

2.2.3 Analysis ...... 61

3 PROBLEM SETUP ...... 65 4 THE ORSZAG TANG VORTEX: TURBULENT DECAY OF

ix ENERGY ...... 68

4.1 Hybrid Simulation Model ...... 69 4.2 Results and Discussion ...... 72

5 THE ORSZAG-TANG VORTEX: K-ω SPECTRA ...... 85

5.1 k − ω spectra of OTV ...... 88 5.2 Probability Density Functions (PDFs) ...... 89 5.3 Conclusions ...... 92

6 QUASI STEADY STATE TURBULENCE: TIME INDEPENDENT DRIVING ...... 94

6.1 Introduction ...... 94 6.2 Simulation details ...... 96 6.3 Results ...... 97 6.4 Conclusions ...... 101

7 QUASI STEADY STATE TURBULENCE: TIME DEPENDENT DRIVING ...... 105

7.1 Introduction ...... 105 7.2 Simulation details ...... 106 7.3 Results ...... 108 7.4 Conclusions ...... 113

8 CONCLUSIONS AND FUTURE DIRECTIONS ...... 117

Appendix

A TWO FLUID DISPERSION RELATION AND WAVES ...... 122 B HYBRID CODE RUN SEQUENCE ...... 125

B.1 Algorithm ...... 125 B.2 Normalization ...... 129

C PERMISSION LETTERS ...... 133

C.1 Permission Letter from Kazunari Shibata ...... 133 C.2 Permission Letter from Steven Cranmer ...... 134

x C.3 Permission Letter from Chi Wang ...... 135 C.4 Permission Letter from Jack Gosling ...... 137

BIBLIOGRAPHY ...... 141

xi LIST OF FIGURES

1.1 The heliosphere is a bubble of solar plasma in the local interstellar medium. Figure from NASA Pioneer spacecraft page...... 7

1.2 The solar wind solutions for various coronal temperatures. The speeds are in km/s and distances are in m...... 10

1.3 When oppositely directed field lines carrying plasma in the MHD limit are pressed against each other, strong current sheets develop. At one point, kinetic effects kick in and the plasma is no longer frozen-in. At this point, the field lines reconnect and change connectivity and the newly connected lines ”snap” outwards with Alfvénspeed. More plasma flows in to fill the void and the process continues until plasma inflows are available...... 15

1.4 A model of reconnection happening at a coronal loop. Used with permission from [Shi99]...... 16

1.5 Temperature proﬁles of various plasma species in quiet polar coronal holes derived from line emission measurements from UVCS and SUMER on board SOHO. Solid lines represent the electron temperatures, dotted lines represent neutral hydrogen, dashed and dashed-dotted lines represent the ionized oxygen temperatures. (Figure recieved with thanks from Steven Cranmer [CKA+10]) ... 17

1.6 Schematic showing the spectrum of magnetic energy. Based on Fig. 2 of [SGRK09]. The slope of the spectrum in the inertial range is −5/3, the spectrum steepens to −2.5 at proton inertial scales until electron inertial scales where it steepens again to −3.8. The steepening at the ion inertial scale (fρi) and the electron inertial scale (fρe) is associated with enhanced dissipation but the nature of dissipation and the nature of the ”cascade” below these scales is not well understood...... 21

xii 1.7 Reconnection events observed in the solar wind. The values are plotted in the RTN (radial, tangential and north) coordinate system. Vertical lines show the time interval in which the outﬂow is accelerated and is connected with a magnetic ﬁeld reversal. Used with permission from [GSMS05] ...... 23

1.8 Temperature proﬁle of solar wind protons up to 70 AU calculated from Voyager data. Adopted with permission from [WR01]. .... 27

2.1 Domain decomposition between diﬀerent processors in P3D. We denote the processors by their processors numbers in npex, npey notation as well as the processor number in the MPI numbering system with N = npex ∗ npey (bottom number in each cell) ..... 56

2.2 Flowchart describing the code P3D...... 63

2.3 Different grid points get part of the contribution from the particle based on weights as defined in this figure. For example, point A gets the weight equivalent of the ratio of area of green part to the area of the whole cell...... 64

4.1 Current density with magnetic ﬂux contours at t=1.96 in (a) ﬂuid and (b) hybrid simulations...... 71

4.2 Magnetic ﬁeld power spectra at t = 5.69 for the hybrid and MHD simulations. The Hall scale and MHD Kolmogorov dissipation scale (kd) are shown for reference. The Hall scale, where dispersive kinetic Alfv´enwaves arise, occurs when k di cs/cm ≥ 1, where cs is the sound speed and cm is the magnetosonic speed[RDDS01]...... 74

4.3 Hybrid and MHD comparison: (a) Magnetic energy EB, ﬂuid ﬂow energy Ev, their sum, the change in thermal energy ∆Eth, and total 2 energy Etot vs. time. (b) Flow enstrophy < ωv > and magnetic 2 enstrophy < ωB > vs. time...... 75

4.4 (a) ∆EB: change of EB in the hybrid run; ∆EvB: exchange between Ev and EB; ∆Ee: electron kinetic energy; DB: sum of these, total EB dissipated. (b) Dv and DB are cumulative dissipation through bulk ﬂow and magnetic channels, Dtot their sum, ∆Eth change in thermal energy. (c) Parallel and perpendicular proton temperatures vs. time...... 76

xiii 4.5 Trajectories of particles stepped through static electric and magnetic fields at two different times in the simulation. Left panels show the contour plots of out of plane current and the particle’s trajectory traced on top of it for the time window in which the magnetic moment is shown in the right panels. a),b) are the plots for the particle stepped in the electric and magnetic fields taken very early from the simulation. c),d) are the plots for the particle stepped in the turbulent fields taken from later in the simulation...... 81

4.6 Flow of energy through the turbulent hybrid simulations. Bold arrows denote signiﬁcant energy conversion through a channel. Light dashed arrows denote little or no energy conversion through a channel. Ωi represents cyclotron damping, DB represents magnetic dissipation, and Dv represents dissipation of proton bulk ﬂow. ... 82

4.7 Eﬀective (a) resistivity η and (b) viscosity ν vs. time...... 83

5.1 k − ω spectrum of magnetic fluctuations parallel to the mean magnetic field. We clearly see excess power in the Alfvénand slow modes...... 86

5.2 k − ω spectrum of magnetic ﬂuctuations perpendicular to the mean magnetic ﬁeld. We clearly see excess power in the fast magnetosonic mode. At the frequencies above the cyclotron frequency, we see the electromagnetic Bernstein modes...... 87

5.3 Shaded contours of energy in Bx plotted in the k − ω space on logarithmic scales. Contour lines show levels of energy. Dispersion relations are over-plotted (see [RDDS01]), calculated using a two fluid dispersion relation, using the k parallel to the in-plane magnetic field and ⊥ to Bg. Important length and time scales are marked. The spectrum shows no significant power in wave modes. 88

5.4 Shaded contours of energy in Bz in k − ω space. See Fig. 5.1 for details. This spectrum also does not show signiﬁcant power in wave modes. There is slight hint of magnetosonic activity but the power in that mode is about two orders of magnitude smaller than the maximum power available...... 90

xiv 5.5 The Eulerian frequency spectrum emphasizes the point again that the only recognizable wave mode is the magnetosonic mode. Also it clearly shows that there is no power at the cyclotron frequency to drive the perpendicular heating discussed in [PSCM09]. Note the −5/3 slope of the spectrum too which is a property of the hydrodynamic spectrum...... 91

5.6 The probability density functions (PDFs) of magnetic ﬁeld and velocity follow the Gaussians very nicely. The increments show tails, however, indicating strong intermittent structures...... 92

6.1 Plot of magnetic energy. The energy in the lowest wave mode (the largest scales) keeps growing monotonically throughout the simulation, indicating an inverse cascade of energy into the largest scales. The energy in higher wave modes achieves an approximate steady state after t ∼ 30. At times later than t = 70 slower relaxation processes start to dominate the time evolution and the steady state is lost...... 98

6.2 Time averaged omnidirectional k-spectrum of the system. The energy containing scale is at 2 ≤ k ≤ 3. The spectrum shows a bend at k = 1/ds where ds = cs/ωci is the scale where Hall physics becomes important. −5/3 line is shown for reference purposes. ... 99

6.3 Eulerian frequency spectrum of magnetic field. Solid line is the frequency spectrum for the total magnetic field, dashed line is the perpendicular component of the magnetic field and the dotted line is the frequency spectrum of the parallel component of the magnetic field (Bz). Cyclotron frequency and a −5/3 line ( dash-dotted) are shown for reference purposes. The only recognizable spectral peaks are roughly consistent with ion Bernstein and magnetosonic modes. The amount of energy in these modes is also many orders of magnitude smaller than the available free energy in the system. .. 103

xv 6.4 Energy spectrum of y-component of magnetic fluctuations as function of frequency and of wave vector component perpendicular to the mean magnetic field. The dispersion relation for monochromatic magnetosonic modes propagating across the mean magnetic field is shown, as well as the various important length scales (1/di, 1/de, 1/ds) and time scale (ωc). The spectrum is featureless without any recognizable wave modes present. In a similar energy spectrum of the z component of the magnetic field, the only recognizable feature is likely some combination of magnetosonic and ion Bernstein modes and the power in those modes is orders of magnitude smaller than the free energy in the system...... 104

7.1 a) Magnetic energy versus time for several forcing frequencies ωd. The two vertical lines denote the time period over which the total energy change is plotted in Figure 7.2. b) omnidirectional spectra for the same runs. −5/3 line and the ion inertial length scale kdi = 1 have been shown for reference purposes...... 114

7.2 Time averaged magnetic and thermal energies in the time window t ∼ 40 to t ∼ 50 as a function of driving frequency. Near the critical frequency of 0.4, points are labeled with the corresponding values of the driving frequency in NL time units based on individual runs. nl ∼ The critical frequency corresponds to ωd 1 (vertical arrow). ... 115

7.3 k − ω spectrum of the out-of-plane magnetic field Bz fluctuations for (a) ωd = 0.0 and (b) ωd = 13.4. Physical scales and cyclotron frequency denoted with solid white lines. Magnetosonic mode denoted with dashed white line. In (b), the first few Bernstein modes calculated from a linear Vlasov code [Gar05] shown as solid curves. (c) PDFs of magnetic field vector increments for various driving frequencies and vector increments of δs = 10∆x...... 116

xvi LIST OF TABLES

1.1 Properties of the sun ...... 8

2.1 Summary of diﬀerent numerical approaches for studying plasmas .. 51

xvii ABSTRACT

Plasma turbulence is a phenomenon that is present in astrophysical as well as terrestrial plasmas. The earth is embedded in a turbulent plasma, emitting from the sun, called the solar wind. It is important to understand the nature of this plasma in order to understand space weather. A critical unsolved problem is that of the source of dissipation in turbulent plasmas. It is believed to play a central role in the heating of the solar corona which in turn drives the solar wind. The solar wind itself is observed to be highly turbulent and hotter than predicted through adiabatic expansion models. Turbulence and its associated dissipation have been studied extensively through the use of MHD models. However, the solar wind and large regions of the solar corona have very low collisionality, which calls into question the use of simple viscosity and resistivity in most MHD models. A kinetic treatment is needed for a better understanding of turbulent dissipation. This thesis studies the dissipation of collisionless turbulence using direct numerical hybrid simulations of turbulent plasmas. Hybrid simulations use kinetic ions and ﬂuid electrons. Having full kinetic ion physics, the dissipation in these simulations at the ion scales is self consistent and requires no assumptions. We study decaying as well as quasi steady state systems (driven magnetically). Initial studies of the Orszag-Tang vortex [OT79] (which is an initial condition that quickly generates decaying strong turbulence) showed preferential perpendicular heating of protons (with T⊥/Tk > 1). An energy budget analysis showed that in the turbulent regime, almost all the dissipation occurs through magnetic interactions. We study the energy budget of waves using the k − ω spectra (energy in the wavenumber-frequency space). The k − ω

xviii spectra of this study and subsequent studies of driven turbulent plasmas do not show any significant power in the linear wave modes of the system. This suggests that in the strong 2D limit, contrary to the conventional belief, waves do not appear to play an important role in the heating of plasma. We also study the onset of turbulence and heating of plasma as a function of the driving frequency. We find that the onset of turbulence has a critical dependence on the relative size of the driving time scales and the nonlinear time scales of the system. The driving time scale has to be longer than the nonlinear time of the system or the intrinsic nonlinear time associated with the driving function. For smaller driving time scales (or higher driving frequencies) we do not generate turbulence and do not heat the plasma. This setup has a resemblance to the generation of turbulence and heating of the plasma in the solar corona. The driving frequency corresponds to the frequency of driving because of the foot point motions of the field lines. Our results are consistent with Parker’s picture for heating the corona (e.g. [Par01]). The time scale of the foot points has to be longer than the nonlinear time of the system in order to generate turbulence and heat the corona.

xix PREFACE

The Sun is the biggest source of energy for this planet that we live on and call our home. Some might argue that it is the only source of energy, tracing most of the energy sources ultimately back to the sun. It comes as no surprise that ever since the most ancient civilizations, the sun has been worshiped as a God. The Egyptians, Hindus, Nordics, Greeks, Incas, Aztecs all worshiped the Sun God. The following mantras from the oldest Hindu scripture Rig Veda clearly show the importance attached to the sun by the Hindus:

As a friend to another friend, the cosmic fire ripens the sap of three hundred fields and forests of people, and the Sun works for the destruction of widespread darkness over the three regions, celestial, inter-spatial and terrestrial. When the cosmic fire has ripened the sap of three hundred fields and forests, and the Sun has been able to destroy the darkness spread over the three regions, all Nature’s bounties express their gratitude to the Sun and offer the homage, for he has been of a great service in the struggle.1

1 Rig Veda, 5.29.7-8, From: Rigveda Samhita, Translated by Swami Satya Prakash Sarasvati & Styakam Vidyalankar, Published by Veda Prathishthana, New Delhi.

1 Kinic Ahau in Mayan religion, Ra in Egyptian, Tonatiuh in Aztec, the sun had an important place as the God who keeps darkness away and provides with warmth and security but still being feared for his wrath. Other than being the source of energy and the role in various religions, the sun affects us in not so obvious ways also. There is a constant flux of very hot plasma flowing outwards from the sun called the solar wind. The solar wind expands in the solar system and interacts with the planets, affecting their atmospheres. The first indications that the activity on the surface of sun could be related to events on earth were speculated by a British astronomer Richard Carrington when he related an unusual bright activity (a solar flare) that he observed in a group of sunspots [Car59], with the high auroral activity and the noise in telegraph networks. It was postulated by George Fitzgerald that the earth is being bombarded by matter accelerated by the sun. This was given the name “rays of electric corpuscles emitted by the sun” by a Norwegian physicist Kristian Birkland. Subsequent observations found the temperature of solar corona to be of the order of million degrees. In late 1950s Sydney Chapman calculated the properties of the corona and suggested it extends beyond earth’s orbit [CZ57]. In a series of papers, Eugene Parker developed the solar wind theory as we know it today and its interactions with Earth (e.g. [Par57, Par58b, Par58a]). A very good account of the development of ideas as well as historical references are given in chapter 1 of the book by N. Meyer-Vernet [MV07]. The solar wind interacting with earth’s outer atmosphere affects our satellites as well as activity on earth. Luckily our atmosphere protects us from these high energy particles and their radiation. But our satellites and astronauts out in space are not safe. Also if there is a large outburst of plasma from the sun (a solar storm) in the direction of earth, its effects can be felt at earth too. The magnetic field of the sun is carried with the plasma and close to earth, it continuously interacts

2 with earth’s magnetic field through a process called magnetic reconnection. When a solar storm occurs in the direction of earth, the interaction between the solar magnetic field and the earth’s magnetic field is very violent and creates what are called geomagnetic storms. The effects of such events on earth manifest in various ways. Apart from the breathtakingly beautiful aurora borealis (northern lights) there are serious issues involved too. The wireless communications are disrupted, electrical networks are compromised (e.g. Quebec2 1989 [MLKM89]), airplane safety becomes an issue because of communication disruption. The issues discussed above make it necessary for us to understand space weather, the dynamics of the solar plasma as well as the ionized outer atmosphere of the earth called the magnetosphere. Considerable observational as well as theoretical effort has gone into understanding space weather and developing models that could ultimately predict the space weather accurately enough for us to take preventive measures in time. A lot of satellite missions have been launched to study the sun, the solar corona, solar wind and the solar wind - earth interaction. A few missions to be named would be WIND, STEREO, CLUSTER, SOHO, SDO, Solar Probe Plus (to be launched by 2018). We shall discuss the details of relevant observations in the next chapter. A few important observations and the questions that they raise are: The solar corona is about a thousand times hotter than the solar surface. There are two kinds of solar wind. Slow wind which travels at about 400km/s and fast wind which travels at about 800 km/s. The solar wind is turbulent and nearly collisionless. Out of many questions that arise from observations, the ones relevant to this thesis are:

• The temperature of the solar corona is much higher than the photosphere. The temperature rises by three orders of magnitude in a few hundred kilometers.

2 Also see, http://www.nasa.gov/topics/earth/features/sun darkness.html

3 How does the corona get so hot? How is the energy transferred from the photosphere, which is at a lower temperature, to the corona which is at a higher temperature? What are the processes involved?

• The solar wind temperature profile falls off as the wind expands in the solar system, but it is higher than the temperature expected in an expanding system with no obvious source or sink of energy. There is some mysterious “source” which converts energy from the magnetic field fluctuations into the plasma thermal energy. What are these mechanisms? What are their relative strengths?

• In fast wind the thermal energy of the protons in the solar wind increases more in the direction perpendicular to the mean magnetic ﬁeld as compared to the thermal energy along the mean magnetic ﬁeld. How is this perpendicular anisotropy generated?

The solar wind is observed to be very weakly collisional at best. This calls into question the use of collisional viscosity and resistivity to heat the plasma. Hence, to answer these questions, we need to understand the dissipative mechanisms in collisionless turbulent plasmas. Many mechanisms have been proposed to address these problems. They can be put into two categories. Mechanisms that involves the natural wave modes present in the plasmas and ones which do not use these wave modes to explain the heating of the plasma. The usual ﬂuid picture conventionally used to describe the solar wind breaks down in the collisionless limit. We need to go to a kinetic description in which the details of physics at the molecular scales are important. In this thesis we use a kinetic computational model called the hybrid code to explore the question of dissipation in such systems.

4 We find that in a specific limit of turbulence in plasmas, wave modes do not seem to play an important role in heating the plasma. The current sheets and reconnection sites present in such turbulent systems are shown to be related to the heating processes. In the following chapters, we lay the background material with more technical details and then proceed to describe the specific projects done for this thesis.

5 Chapter 1

INTRODUCTION

In this chapter we lay the background to motivate the problems studied in this thesis. We start with a brief description of the heliosphere to motivate the extent of the sun’s influence on the solar system. In sections 1.3, 1.4 we discuss some observations and the questions that they pose. In section 1.5 we describe the basics of the theoretical efforts and conclude the chapter with the statement of the problem motivated by the observations and theoretical efforts.

1.1 A brief tour of the heliosphere The plasma that leaves the sun and expands in the solar system (aka solar wind) creates a bubble in the interstellar medium around the sun. This bubble is called the heliosphere. We start with a brief description of the heliosphere starting from the sun. For more details we refer the reader to excellent review books by David Alexander [Ale09] (a general overview in popular science manner) and by Kenneth Lang [Lan09] (an extensive overview from an observational point with a very up to date list of references). The sun is a typical star of spectral type G in the Orion arm of the Milky Way galaxy. Some properties of the sun are listed in table 1.1. Like any main sequence star, it is powered by fusion of hydrogen atoms into helium at the center,

3 the core. The core is about 0.2-0.25 R in size, has a density of about 150g/cm and a temperature of about 14 × 106 K. Outside the core is the radiative zone (∼

0.25 − 0.7 R ) where the energy emitted from the fusion in the core are transmitted

6 Figure 1.1: The heliosphere is a bubble of solar plasma in the local interstellar medium. Figure from NASA Pioneer spacecraft page. through the absorption and emission of photons by the atoms/ions in the region.

Outside the radiative zone (∼ 0.7 − 1.0 R ) is the convective zone where huge convection patterns take energy from the radiative zone and convect it to the surface (photosphere) where it escapes the sun as light. The temperature of the photosphere is about 5800K The solar atmosphere just above the photosphere is called the chromosphere and is about 5000 Km thick. The temperature in the chromosphere rises to about 50000 K. Above the chromosphere, the temperature rises very rapidly in a few hundred kilometers in what is called the transition region to about 2 MK in the corona. How the energy is transferred from photosphere which is at 6000 K to the corona which is at about 2 MK is still an open question. We describe the observations as well as the modeling approaches taken to address this question in

7 Property Value 30 Mass (M ) 2 × 10 Kg 8 Radius (R ) 7 × 10 m Surface Temperature (T ) 5800 K Magnetic ﬁeld ∼ 1 Gauss

Table 1.1: Properties of the sun the section 1.3. The solar corona is a very dynamic system and has complex structure with a plethora of processes and structures existing on at any given time. Chapman and Zirin gave a simplistic hydrostatic model of the corona [CZ57] but the model suﬀered from a problem that the thermal pressure at inﬁnity does not go to zero [Par58b] as would be expected. Parker corrected this problem by considering a hydrodynamic corona [Par58b]. A very simple model can be built from the basic continuity equation and the momentum equation [Par58b]:

∂ρ + ∇ · (ρv) = 0 ∂t ∂v ρ( + v · ∇v) = −∇p + J × B + F (1.1) ∂t g where ρ is the density, v is the ﬂow velocity, J is the current density, B is the

2 magnetic ﬁeld, Fg = ρGM /r rˆ is the gravitational force due to the sun, M is the solar mass and p is the thermal pressure. In order to understand the basic physics, we make a few simplifying assumptions:

1. We assume a time steady system ⇒ ∂t → 0

2. Spherical symmetry ⇒ ∂θ, ∂φ → 0. Only the radial component of the ﬂow v(r) is considered.

8 3. Isothermal equation of state p = 2nkbT , where n is the number density of

protons, kb is the Boltzmann constant and T is the temperature. The factor 2 comes because we have to consider both electrons and protons.

With these assumptions, the above equations reduce to:

1 ∂ρvr2 = 0 r2 ∂r ∂v ∂p ρGM ρv = − − (1.2) ∂r ∂r r2

The ﬁrst equation gives us ρvr2 = const. or 4πr2 × mnv = M˙ where m is the mass of one particle and M˙ is the total mass passing through a sphere per unit time. Using this and p = 2nkbT we can reduce the second equation to:

( ) 2k T 1 ∂v 4k T GM 2 − b b − v = 2 ( m ) v ∂r mr r 2 2 ( ) − vc ∂ v 4 − rc 1 2 2 = 1 (1.3) v ∂r vc r r √ where rc = GM m/4kbT is the critical radius and vc = 2kbT/m. It is reasonable to assume that the velocity is small at the surface of the sun. So the terms inside the brackets on both sides of the equation are negative and hence ∂v/∂r is positive. For the velocities starting at very small values close to the sun, we can have three kinds of solutions of eqn. 1.3.

⇒ ∂v 1. When r = rc occurs while v < vc ∂r = 0 at r = rc and hence velocity starts decreasing and would go to zero for r → ∞. For such solutions v → 1/r2 as

r → ∞. Using mass conservation, we see that ρ∞ is ﬁnite and hence p∞ is ﬁnite. Hence this solution is not physical.

∂v → ∞ 2. When v = vc occurs while r < rc, ∂r for the right hand side to be non-negative, which is physically not possible.

9 Figure 1.2: The solar wind solutions for various coronal temperatures. The speeds are in km/s and distances are in m.

∂v → ∞ → ∞ 3. When v = vc at r = rc, ∂r remains positive and hence v at r . The pressure at inﬁnity goes to zero in this case which is consistent with expectations. This solution is the solar wind solution.

The equation 1.3 can be readily solved to get:

2 2 v − v r rc − 2 log 2 = 4 log + 4 3 (1.4) vc vc rc r where the third condition v = vc at r = rc is used to get the constant of integration. The solutions are shown in figure 1.2. Even though there were earlier speculations about the existence of the solar wind through the observations of cometary tails [Bie51], Parker’s model was met with skepticism until measurements by a Soviet satellite Luna confirmed the existence of a supersonic wind. There are two types of solar wind that are observed. One is the slow wind originating from the close field regions. It flows at about

10 400km/s and has a density of about 30 particles/cm−3. The second kind of wind is observed emanating from the coronal holes which are open magnetic field regions with strong magnetic flux and lower density. It flows at about 800km/s and has a density of a few particles cm−3. Naively one would expect the expanding wind to cool adiabatically as there is no obvious ”source” or ”sink” of energy in the solar system. Observations show that the temperature of the solar wind falls as we go away from the sun but it does not drop as much as expected by adiabatic expansion. There is some unknown ”source” of energy which increases the thermal energy of the solar wind. This is called the “solar wind heating problem”. Where does the extra thermal energy come from? The velocity distribution functions of protons in the fast solar wind show that they are heated more perpendicular to the mean magnetic field. The details of how the solar wind gets heated and the cause of the temperature anisotropy are also unanswered at this stage. We describe the observations as well as the models used to address this problem in section 1.4. The expanding wind creates a bubble of this plasma of solar origins around the solar system. This is called the heliosphere. At about 80AU the solar wind slows down to subsonic speed because of the local interstellar medium. The region beyond that is called the heliosheath where the wind slows down more. At the heliopause the wind stops and local interstellar medium starts. The expanding wind interacts with the magnetospheres (the magnetic envi- ronments around planets) of various planets and affects their space weather. This makes it important to study the nature and origins of the solar wind. The origin of the solar wind requires an understanding of how the corona is heated to such high temperatures. In addition to that, there is the solar wind heating problem. Both of these problems require obtaining a good understanding of the kinetic dissipative processes in these systems which is the motivation for this thesis. Now let us turn our attention to the two sets of observations in this huge

11 system that motivate this thesis. Before going on with the discussion of observations, we brieﬂy describe the kind of spacecraft observations that are involved.

1.2 Spacecraft Observations Many spacecraft missions have been launched to observe the solar corona and solar wind (IMP, Mariner, Pioneer, Helios, Ulysses, Wind, Advanced Composition Explorer (ACE), Solar Heliospheric Observatory (SoHO), Yohkoh, Transition Region And Coronal Explorer (TRACE) to name a few). These spacecraft usually have instruments like spectrometers, plasma instruments, magnetic field instruments etc. Please see the book by Kenneth Lang for more details about the spacecraft and instruments [Lan09]. The plasma, electric and magnetic field instruments are used for in situ measurements of plasma e.g. in the solar wind. They measure the distribution functions for various particles, electric and magnetic fields. The particle distribution functions can then be used to calculate various moments to derive density, bulk flow velocities, temperature, heat flux etc. The electric and magnetic field data give us additional information about the nature of the plasma through the spectra. The spectrometers are used for remote measurements and are used to study the solar corona. For coronal observations, the coronagraphs create an artificial ”solar eclipse” by blocking out the solar disc to take the spectroscopic measurements of the corona. The spectroscopic observations are useful in many ways (see e.g. Cranmer’s review article in Living Reviews in Solar Physics [Cra09]):

• The Doppler shifts of the emission lines can provide information about the bulk ﬂows along the line of sight.

• Shapes of measured emission lines can provide information about line of sight distributions of the velocities of various particles.

12 • Integrated intensities of resonantly scattered lines can be used to infer the solar wind velocity and velocity distributions.

• Intensities of collision dominated lines, combined with emission measure distributions can be used to infer the electron temperature, densities of protons, ions etc. in the corona.

• High resolution measurements of UV spectral line proﬁles can be used to look at the departures of velocity distributions from Maxwellian and bi-Maxwellian distributions [Cra01].

1.3 Coronal Observations A number of modern spacecraft (Helios 2, Yokoh, SOHO, TRACE etc.) have been launched with speciﬁc instruments (e.g. soft X-ray telescope (SXT) on Yokoh, UVCS on SOHO, extreme UV (EUV) on TRACE and x-ray telescope (XRT) on HINODE) to look at the sun and understand the nature of the corona. Below we summarize some of the observations.

1.3.1 Waves in solar corona Unlike hydrodynamics which has only a few wave modes, MHD supports a number of different wave modes. Observations of various acoustic modes, kink mode and sausage modes in the solar corona have been summarized in [Asc06]. The energy budget of acoustic waves suggests that they do not carry enough energy to heat the corona [Asc06]. Arguably the most popular MHD wave mode for transporting energy out of sun and also dissipating it into the corona and solar wind is the Alfvén wave mode. We discuss the various proposed heating mechanisms involving Alfvén waves in section 1.5. Erdélyiet. al. [EDPW98] calculated coronal line widths using SOHO/SUMER data. Generally the line widths in the corona have two components, the thermal Doppler line width as well as the non-thermal component which could

13 be attributed to various mechanisms including the presence of waves or turbulence in the corona. Erdélyiet. al. estimated the excess line widths assuming magneto- acoustic waves as well as Alfvénwaves and found that the width estimates for Alfvén waves have the correct order of magnitude to match the observations. Tomczyk et. al. [TMK+07] analyzed the data from the Coronal Multi-Channel Polarimeter (CoMP) instrument in National Solar Observatory and found signatures of non- compressive waves propagating along the magnetic field and suggested these to be low frequency Alfvénwaves. They calculated the energy flux of these waves and found it to be insufficient to heat the corona.

1.3.2 Reconnection Magnetic reconnection is a process which results in the topological change in the magnetic field configuration. A simplified picture of reconnection is shown in fig 1.3. In the MHD limit plasma fluid elements are ”frozen-in” on the field lines [GB05]. A consequence of this is that the field lines can not intersect or break. If a situation arises that two oppositely directed field lines are pushed against each other by plasma flows, in the ideal fluid limit they will keep pressing against each other indefinitely, creating singular current sheets. In realistic systems at the kinetic scales (the scales at which the particle effects become important) the frozen-in theorem fails and the plasma is no longer bound to one field line. So at the kinetic scales, the field lines can ”reconnect” to change connectivity. The newly reconnected field lines have magnetic tension associated with them. This tension makes the field lines ”snap” outwards creating strong plasma flows having speeds of the order of Alfvén speed. This outflow creates a void behind it which is filled in by more in-flow of plasma and the process continues until there is supply of in-flowing plasma. A model based on simple scaling arguments was given by Sweet and Parker [Swe58, Par58b]. We refer the reader to an excellent review on the subject by Yamada et. al. [YKJ10] and the book by Priest [PF00] for more details on the subject.

14 Inflow Outflow Outflow

Inflow

Figure 1.3: When oppositely directed field lines carrying plasma in the MHD limit are pressed against each other, strong current sheets develop. At one point, kinetic effects kick in and the plasma is no longer frozen-in. At this point, the field lines reconnect and change connectivity and the newly connected lines ”snap” outwards with Alfvénspeed. More plasma flows in to fill the void and the process continues until plasma inflows are available.

Reconnection events can be observed by the specific signatures associated with them, the outflows and the correlations between velocity and magnetic field. Reconnection events have been observed and analyzed in the solar corona using the X-ray observations from the Yohkoh/SXT, SOHO/SUMER, SOHO/LASCO, TRACE and HINODE/XRT experiments. Reconnection is expected to happen throughout the braided jungle of field lines in the solar corona. But the most prominent reconnection events are the ones that occur in the coronal loops. Fig 1.4 (taken from [Shi99]) shows a model of reconnection in coronal loops. When in-flows pinch a coronal loop, an x-point is created, inducing reconnection and ejecting a plasma bubble outwards. The reconnection outflow plasma falls back towards the foot-points of the loop creating flares through bremsstrahlung. These reconnection events can be identified by signatures like identification of outward flowing plas- moids (e.g. [Tsu97]) and supra-arcade downflows (SADs, plasma voids created by downwards moving newly reconnected loops e.g. [MS09, SMR+10]).

15 Figure 1.4: A model of reconnection happening at a coronal loop. Used with permission from [Shi99].

Reconnection events add energy to the corona not only through the fast out- ﬂows but also by generating high frequency waves, which can propagae to high altitude, damp and give internal energy to the system, and also by generating turbulence (e.g. [MLO95]).

1.3.3 Heating The solar corona is observed to be much hotter than the photosphere. This much higher temperature of the corona is responsible for driving the solar wind into the solar system. The ﬁrst spectroscopic observations of the corona showed spectral lines that were not attributed to the existing elements. It was initially speculated that these lines were from yet undiscovered element called ”coronium”.

16 Figure 1.5: Temperature proﬁles of various plasma species in quiet polar coronal holes derived from line emission measurements from UVCS and SUMER on board SOHO. Solid lines represent the electron temperatures, dotted lines represent neutral hydrogen, dashed and dashed- dotted lines represent the ionized oxygen temperatures. (Figure recieved with thanks from Steven Cranmer [CKA+10])

In the 1940s, these spectral features were related to the spectral emissions from highly ionized elements (e.g. [Edl45, Paw46]). This implied that corona must be very hot (∼ 2 × 106 K). The temperature of the electrons and protons in the corona rises to about 106 K over a few thousand kilometers. The heavy minor ion (e.g. O+5) temperatures can even rise as high as 108 K. Figure 1.5 shows the temperature proﬁles of various charged species at the solar minimum (the period of low activity on the sun) in polar coronal holes. The temperatures were derived from UVCS and SUMER line emission measurements (from [CKA+10]). Kohl et. al. [KNA+98] empirically modeled the line widths of H0 Lyα and

+5 O emission lines. They varied the most probable thermal speeds wk, w⊥ and the macroscopic outflow velocity u as functions of radius over the polar coronal holes to find the best agreement with the observed line widths. The best fit model gave an indirect but a reasonable evidence that the temperatures of these ion species are

17 anisotropic with perpendicular temperatures being more than the parallel temperatures. How the temperature rises to such high values is still an unsolved problem. The photosphere of the sun is capable of providing such heating but how is the energy transported into the corona and how is it dissipated is still a matter of debate. Also the perpendicular anisotropies are an unsolved mystery. High perpendicular anisotropies have prompted explanations based on wave-particle resonances. We discuss these and some other heating mechanisms in the section 1.5.

1.4 Solar Wind Observations Ever since the Soviet spacecraft Luna discovered the solar wind, a lot of data has been gathered for the solar wind from a plethora of spacecrafts. In this section we summarize some of the ﬁndings that are relevant to the discussion of kinetic processes and dissipation in solar wind.

1.4.1 Waves in the solar wind The solar wind has also been observed to have fluctuations that can be interpreted as Alfvénwaves (e.g. [BDS69, BDJ71]). Belcher et.al. [BDS69] looked at the correlations between magnetic field and velocity in the Mariner 5 data to interpret the presence of Alfvénwaves. Matthaeus et. al. [MG82] calculated the cross helicity ∫ 3 Hc (= 1/2 d xv · b, where v is velocity and b is magnetic field) along with other rugged invariants for solar wind. The single sign of Hc over much of the spectrum indicates positive correlation between velocity and magnetic field fluctuations and hence implies presence of ’outward propagating Alfvénicfluctuations’. Matthaeus et. al. also reported a number of modes with negative cross helicity which would indicate the presence of ’inward propagating Alfvénicmodes’ too. Bruno et. al. [BBV85] analyzed Helios 1 and 2 data and found very low frequency (f < 10−4Hz)

18 Alfvénicfluctuations. Tu & Marsch [TM95] suggest these as well as the observations by Tu et. al. [TMT89] are relatively pure Alfvénwaves. The presence of such Alfvénicfluctuations has triggered a debate about whether the turbulence is present at all or whether it can be described by weak turbulence theory (e.g. [GNNP00, NBGG03, Cha05] ) which requires the nonlinear time scale (the time scale at which the nonlinear term in the fluid equation of motion becomes important) to be much longer than the time scales of the linear waves in the system. Simplisticlly, turbulence in this limit can be described as a mix of linear wave modes of the system. The other limit is that the turbulence is dominated by the highly nonlinear interactions. Dobrowolny et. al. [DMV80a, DMV80b] suggested that the turbulence evolves to a state of ”dynamic alignment” in which the only modes that are left are with a cross helicity +1 or -1. Recently Chandran [Cha08b] gave a model of strong incompressible MHD turbulence which also agrees with the dynamic alignment picture. A comment must be made regarding the debate on whether the solar wind turbulence can be described as a mix of linear wave modes. While there is evidence of existence of fluctuations that have characteristics of Alfvénwaves, there is no direct and definitive ”proof” that propagating Alfvénwaves have been observed in the solar wind. It appears that in the literature (both observational as well as theoretical) the terms like Alfvénicfluctuations and Alfvénwaves are used interchangeably and some times in a manner that can be interpreted as either waves in the linear theory sense or as nonlinear fluctuations that have Alfvéniccharacter. The same appears to be true for the use of term kinetic Alfvénwaves (KAWs) in the kinetic regime

(k & ρi where k is the wave vector and ρi is the proton inertial length). This kind of vague use of the terminology adds to the confusion while interpreting the results and trying to make comparisons between existing theories.

19 1.4.2 Turbulence The solar wind is observed to be turbulent (e.g. [Col68, GR99]). The magnetic energy spectra display a power law behavior E(k) ∝ k−α with the power law index 3/2 < α < 2 (e.g. [MG82]). The power law behavior suggests a cascade of energy from the largest energy containing scales to the smallest kinetic scales. De- pending on what approximation is used for triple correlation time τT (k) (the time scale of triple wave vector interactions), diﬀerent models of turbulence giving different power laws can be derived [ZMD04]. The transfer rate of energy ﬂux can be written as:

kE(k) = τT (k) 2 (1.5) τnl If the triple correlation time is taken to be the nonlinear time, the Kolmogorov −5/3 spectrum is obtained. If the triple correlation time is taken to be the Alfvén time (τT = 1/kVA), the Iroshnikov-Kraichnan scaling of −3/2 is obtained. For a detailed discussion of this topic, we refer the reader to the review article by Zhou et. al. [ZMD04]. The existence of a universal power law and hence a universal theory of solar wind turbulence has been a matter of debate. A few studies have concentrated on trying to find the power law for solar wind turbulence (e.g. [PRG06, Pod09, TSM+09, SZ09] etc. and the references therein) while some others have concentrated on the variability of the power law depending on various parameters of the system (e.g. [MG82, ZMD04, MWO+10] etc. and the references therein). A detailed discussion of the topic is out of the scope of this introduction and the reader is referred to the references cited here. One comment about the debate should be made though: It seems unreasonable to look for a ”universal” power law for a system as variable and dynamic as the solar wind. Different parts of the solar wind flow could correspond to different values of parameters that control turbulence.

20 104

102 f-1

100 f-5/3 10-2

-4 E 10 -2.5 f

10-6

-8 10 f-3.8

-10 10 fρi fρe

10-12 10-4 10-3 10-2 10-1 100 101 102 f

Figure 1.6: Schematic showing the spectrum of magnetic energy. Based on Fig. 2 of [SGRK09]. The slope of the spectrum in the inertial range is −5/3, the spectrum steepens to −2.5 at proton inertial scales until electron inertial scales where it steepens again to −3.8. The steepening at the ion inertial scale (fρi) and the electron inertial scale (fρe) is associated with enhanced dissipation but the nature of dissipation and the nature of the ”cascade” below these scales is not well understood.

21 Fig. 1.6 shows a schematic of solar wind spectrum following Sahraoui et. al. [SGRK09]. The spectrum shows a steepening at the proton inertial length scales. This steepening is generally interpreted as a signature of energy being deposited into the thermal degrees of motion of protons. Recently data from Cluster spacecraft observations has been analyzed to calculate the magnetic energy spectra well into the electron inertial range [SGRK09]. According to their analysis, the spectrum has a Kolmogorov like slope of −1.62 in the inertial range, the slope steepens to −2.52 at the proton inertial scale and then steepens again to −3.82 at the electron inertial length scales. One question that arises as to how much of the energy from the cascade goes into heating of protons. Another important unanswered question is if the power law behavior of the spectrum is because of a weak turbulence wave cascade or a highly nonlinear cascade.

1.4.3 Reconnection As we will discuss in section 1.5, reconnection is an important component of heating theories. It has been observed indirectly in the solar corona, as discussed in section 1.3. Here we discuss some observations of reconnection in the solar wind. In situ observations of reconnection are more straightforward as compared to coronal observations of reconnection as we can measure the magnetic field and flows in all the directions very precisely. We are not bound by remote observations as in the case of solar corona. A reconnection site can be identified by looking for a briefly accelerated flow within a bifurcated current sheet. When accompanied by a positive correlation between velocity and magnetic field on one edge and a negative correlation on the other, it can be taken as a positive sign for a reconnection event. Gosling et. al. [GSMS05] reported the first evidence of reconnection in the solar wind using ACE data. Figure 1.7 shows a plot from that paper showing the hallmark correlation between the outflow and the magnetic field reversal. Gosling et. al.

22 Figure 1.7: Reconnection events observed in the solar wind. The values are plotted in the RTN (radial, tangential and north) coordinate system. Vertical lines show the time interval in which the outﬂow is accelerated and is connected with a magnetic ﬁeld reversal. Used with permission from [GSMS05]

[GES+06] reported observation of 91 reconnection sites using data from the Ulysses spacecraft. Almost all the reconnection sites reported in this study were in either the slow wind or in the low proton β coronal mass ejection plasma. Phan et. al. [PGD+06] reported a reconnection X-line extending more than 390 earth radii in the solar wind using data from ACE, Cluster and WIND spacecraft. The observed reconnection site was Petschek like and the observed plasma acceleration from the reconnection exhaust from the three spacecraft matched to within 5o in direction and 10% in flow speed. Retinóet. al. [RSV+07] reported observations of many reconnection sites that were associated with turbulence in earth’s magnetosheath using data from the Cluster spacecraft. Stevens [Ste09] in his PhD work created a statistical approach to identify reconnection outflow jets in turbulent plasma and magnetic reconnection data. He used this approach to identify

23 138 reconnection outﬂow jets in the outer heliosphere using Voyager 2 data. Reconnection is observed almost everywhere in the solar system, from the solar atmosphere to outer heliosphere. This suggests that it is a regular feature of solar plasmas and hence can be expected to play a very important role in the dynamics of solar plasmas.

1.4.4 Heating The break in the solar wind spectrum at the proton inertial scales is usually attributed to heating of the protons. After leaving the sun, the solar wind does not encounter any obvious source or sink of energy. Naively we would expect the temperature of the solar wind to fall adiabatically. A simple heuristic argument can be used to see this. Consider a shell of solar wind plasma expanding into the solar system. At the leading order there is no input or loss of energy expected.

γ−1 γ−1 T1V1 = T2V2 T (4πr2dr)γ−1 = T (4πr2dr)γ−1 1 1 2 ( 2) 2(γ−1) r1 T2 = T1 (1.6) r2 where T and V are temperature and volume of the plasma shell at positions 1 and 2. With adiabatic gas constant γ = −5/3, we would expect the temperature to fall

−4/3 6 as T ∝ r . This means a temperature of about 10 K at 3 R would fall off to a temperature of 106/(215/3)4/3 ∼ 3355 K at 1 AU which is off by a couple of orders of magnitude. So we can say that the solar wind does not expand adiabatically. Parker in 1964 [Par64] studied the role of heat conduction by electrons in various limits of coronal temperatures and densities. The stationary heat flow equation for a corona can be written as [Par64]:

[ ] [ ] 1 d dT dT 2kT dN r2κ(T ) = Nv 3k − (1.7) r2 dr dr dr N dr

24 where T is the temperature, N is the numver density of protons, κ(T ) is the conductivity and v is the expansion velocity. In the very low density approximation,

5/2 along with the Spitzer thermal conductivity κ(T ) = κ0T [Cha54, Spi62] we can write the above equation as:

[ ] 1 d dT r2κ(T ) = 0 (1.8) r2 dr dr −2/7 6 This gives us T ∝ r . So a temperature of about 10 K at 3 R would fall oﬀ to 106/(215/3)2/7 ∼ 3 × 105 K which is the right order of magnitude. The argument of using heat conduction appears to be a promising argument but there are a few considerations that bring the use of thermal conductivity into question:

• Richardson et. al. [RPLB95] fitted the proton temperature profile from Voy- ager 2 data and found that the temperature falls off as r−1/2. This fall off is slightly steeper than the r−2/7 fall off expected using heat conduction only.

• From Fig. 1.5 we see that the electron and proton temperatures are not in thermal equilibrium close to the sun. Even at 1 AU, proton temperature is about twice the electron temperature. This violates one of the underlying assumptions (thermal equilibrium) in the derivation of thermal conductivity.

• The solar wind is almost completely collisionless. Using the simple Spitzer deﬂection time as a measure of time between two successive collisions, we can ﬁnd the collision frequency for the solar wind to be ∼ 10−6 sec−1 for the slow wind and ∼ 10−7 sec−1 for the fast wind (see e.g. [BC05], appendix A). These frequencies are much smaller than the other frequencies of physical importance (plasma frequency, cyclotron frequency etc.) in the system. Comparing the collision time (e.g. for the slow wind ∼ 106 sec) to the expansion time of the solar wind (∼ 3 × 105 sec) we see that the slow wind protons undergo at most one collision before reaching the earth. we can make a statement that for

25 all practical purposes, the solar wind is collisionless. This calls into question the use of any collisional transport processes like the viscosity, resistivity and thermal conduction etc.

A caveat should be mentioned here. The above discussion relies on Coulomb collisions being the only collisional process. Even in plasmas without Coulomb collisions, we could have processes like mirroring of particles in local magnetic bottles or micro-instabilities isotropizing the distribution function in a way similar to collisions along with other complex processes. We might be able to write some kind of “effective” transport coefficients describing the effects of such “collisions”. But this has not yet been done in the literature and the role of instabilities etc. is being actively studied in the literature at present. Now we go on to discuss some observations related to solar wind heating in detail in the following subsections. We concentrate mostly on protons and heavy ions as we address the problem of dissipation at those scales in this thesis. Before talking about the heating of protons and heavy ions, we mention some features of electron distribution functions that can be used as constraints on fully kinetic models. The electron velocity distribution functions have been measured by various instruments on-board spacecrafts. The distribution functions generally have three components, one nearly isotropic core, a halo and a hotter narrow forward beam called the strahl (e.g. [Mar06, CMBK09] and the references therein). A fully kinetic model would have to produce these features in electron velocity distribution functions, among other observables, to be applicable to the solar wind.

1.4.4.1 Protons Fig. 1.8 shows measurements of solar wind proton temperatures calculated from Voyager 2 data. The temperature drops farther away from the sun but it is always higher than the drop expected from adiabatic expansion. This suggests some

26 Figure 1.8: Temperature proﬁle of solar wind protons up to 70 AU calculated from Voyager data. Adopted with permission from [WR01].

indirect source of energy. Magnetic fluctuations of the solar wind are one obvious candidate and the spectral break at the proton inertial scales is also consistent with this loss of magnetic energy. The details of how the energy is transferred to the protons is still a matter of debate which we discuss in detail in the section 1.5. Velocity distribution function of protons calculated from the in-situ measurements of fast solar wind show an anisotropic core (e.g. [Mar06] and the references therein). The average temperature of the core is about twice the temperature of the core of electron distribution functions. The proton core can be represented by a bi-Maxwellian with different temperatures for the directions parallel and perpendicular to the mean magnetic field. The temperature, as in the corona, still has perpendicular anisotropy (i.e. T⊥ > Tk) but the value of anisotropy is smaller than it is closer to the sun. This perpendicular heating is also suggested by the increase of mean magnetic moment of the protons away from the sun (see e.g. [Mar06]).The magnetic moment is proportional to the perpendicular temperature (µ = T⊥/B).

27 So the increasing magnetic moment in turn implies increasing perpendicular temperature. The perpendicular anisotropy of the proton distribution core suggests the use of ion cyclotron resonances with Alfv´enwaves as the heating mechanism. We talk about this and other heating mechanisms used to explain the observations in detail in the section 1.5.

1.4.4.2 Heavy ions The heavy ions (e.g. He+2, C, O, Ne etc.) are much hotter than the protons. In the fast wind the relative heavy ion temperatures are more than simple mass proportional (i.e. Ti/Tp > mi/mp where i denotes the ion and p denotes protons). The distribution functions of the alphas also show a forward beam in the direction of magnetic field. Often a differential streaming between two ionic species is also observed. The differential alpha-proton speed approaches but rarely exceeds the local Alfvénspeed. To quote Marsch [Mar91]: ”It appears as if protons carry the waves whereas alpha particles tend to ”surf” or ride the waves.” Analysis of data from ACE spacecraft has been used to find correlations between the differential speed and the temperature anisotropies of the alphas and protons [GSS05, GYW+06, KLG08]. This has been related to the ion cyclotron resonances. We discuss this in detail in section 1.5. In the following section we describe some of the theoretical as well as observational studies aimed at explaining the observations of solar coronal as well as solar wind heating described above.

1.5 Heating Mechanisms The observations of solar coronal and solar wind heating have prompted many theoretical studies. It is generally agreed that the source of energy is the magnetic ﬁeld. The details of how the energy is transferred from the magnetic ﬁeld to the plasma thermal energy are still a matter of debate. The break at the proton inertial

28 scales is also of relevance to this question. It is still an open question as to how much of the energy goes into heating the protons and also what is the nature of the power law below protons inertial scales. Whether the cascade below the proton inertial scales is another wave cascade of kinetic Alfv´enwaves or whistlers or if it is a fully nonlinear cascade is also a matter of debate. Observations put many constraints on the theories. Here we mention some of these constraints:

• The theory should be able to explain the preferential heating of the protons as well as minor ions.

• The theory should be able to explain the power laws observed in the inertial as well as sub proton scales.

• The theory should be able to explain the details of the observed velocity distribution functions like the core anisotropy and forward beam of ion distributions.

• The solar corona and slow solar wind are weakly collisional and the fast wind is almost completely collisionless. This means that the basic assumption of the mean free path being much smaller than the typical length scale of the system (e.g. [Bra65]) is no longer valid and hence the use of conventional viscosity and/or resistivity in the fluid models is not justified to address the question of dissipation. More complete studies with kinetic effects are required. For a discussion of modeling the dissipative effects in MHD and more references, we refer the reader to the nice review article by Goldstein et. al. [GGSJ99].

Moreover the theory should include the nonlinear processes and hence the turbulent behavior self consistently and not as an empirical model. Considerable amount of literature has been devoted to the problem of dissipation in context of solar plasmas. Broadly these mechanisms can be divided into two categories - (i) wave mechanisms and (ii) low frequency mechanisms. It is not possible to cover all

29 the work done on the topic in this discussion. We attempt to summarize the works that give a better perspective to the place of our research in the big picture.

1.5.1 Wave Mechanisms Wave mechanisms rely on the existence of waves to either transport energy down to smaller scales and/or dissipate the energy into the thermal energy of the plasma. These models rely on the observations of Alfvénwaves in the solar corona and solar wind as a preliminary suggestion of validity of such theories. Theories relying on wave-particle resonances require fluctuation energy at high parallel wavenumbers kk. But theories and simulations of magnetohydrodynamic turbulence in the presence of a strong guide field predict a cascade to high k⊥

(e.g. [SMM83, OPM94, MGOR96]). The power in kk in such theories may be very small and would require replenishment by some mechanism other than the nonlinear cascade. The anisotropy can be used to idealize the decomposition of fluctuations into purely 2D (perpendicular to mean field) and purely slab (parallel to the mean field) [MGR90]. Bieber et. al. [BWM96] used Helios data to suggest that 80% or more of the fluctuation energy is in the 2D mode while 20% or lower is in the slab mode. This lack of energy in kk calls into question the applicability of theories using wave-particle interactions to dissipate energy. Some source of significant fluctuation energy at kk is required.

One such source for fluctuating energy at high kk in the weak turbulence limit can be achieved by including compressive effects [Cha05, Cha08a]. Interaction between three different wave modes of Alfvénwaves (A) and fast magnetosonic waves (F ) can be used to explain transfer of fluctuating energy to high kk. AAF (AlfvénAlfvénFast) and AF F (AlfvénFast Fast) interactions transfer energy from the fast waves into the Alfvénwaves. For small propagation angles, the transfer is effective and can result in significant power in Alfvénwaves at high kk.

30 Cranmer et. al. [CvB03] used a phenomenological model of the turbulent cascade in terms of the advection and diﬀusion strengths (β and γ in their terminology) in k space and suggested that when γ & 4β a parallel cascade of energy could occur supporting a possibility of cyclotron resonances. For γ < 4β electron phase space holes could be produced which we discuss later in this section. Markovskii et. al. [MVSH06] suggested that a velocity shear instability at high k can take energy from the quasi 2D ﬂuctuations and transfer it into proton resonant cyclotron waves that can be dissipated. Now we go on to discuss a few of the wave mechanisms believed to be active at the kinetic scales.

1.5.1.1 Cyclotron Resonance After the discovery of Alfvénwaves in solar wind [BDJ71], it was suggested that non-resonant Alfvénwave pressure gradients [Hol74] as well as resonant wave particle interactions [HT78, MGR82] play an important role in heating the minor ions. It was suggested that Alfvénwaves generated at the bottom of the corona travel outwards. Further away from the sun, the magnetic field weakens and the waves encounter cyclotron resonant frequencies corresponding to successively heavier charge-to-mass ratio ions, heating them preferentially. The basic idea is that when the frequency of gyration of a particle matches with the Doppler-shifted frequency of the wave, the particle can resonate with the wave.

ω = kkVk − nΩ where ω is the frequency of the wave, kk is the wave number along the mean ﬁeld,

Vk is the particle’s velocity along the mean field, n is an integer and Ω is the gyro- frequency of the particle. When this condition is matched, the electric field that the particle sees in its rest frame does not change sign and the particle gets accelerated perpendicular to the mean magnetic field (see e.g. [Hol06] or chapter two in [Gar05]).

31 There is evidence of cyclotron resonance being active in the solar wind. Anal- ysis of data from ACE satellite [GSS05, GYW+06] has been used to show the relation between the differential flow speed of alphas and protons ∆Vαp = Vα − Vp and the large temperature anisotropy for alphas. When the differential flow is small

∆Vαp ∼ 0, the alphas heat more and when ∆Vαp/VA is increased, the alphas come out of resonance. Using linear Vlasov theory and hybrid simulations Gary et. al. [GSS05, GYW+06] showed that plasma, consisting of protons and alphas, interacting with Alfv´enwaves also shows the same correlation between Vαp/VA. Kasper et. al. [KLG08] took this study further and studied the relative temperature anisotropies of alphas and protons as a function of ∆Vαp/VA and the Coulomb collisional age

Ac = R/(Vswτc, where R is the distance from the sun, Vsw is the solar wind speed and τc is the time scale for α−p energy exchange due to Coulomb scattering. Larger collisional age tends to isotropize the anisotropy but even at larger Ac the effect of stronger resonance with alphas is apparent. They also found signatures of a new heating mechanism leading to Tkα/Tkp for ∆Vαp/Vs > 1. They speculated it to be some kind of electrostatic ion acoustic fluctuations. Gary & Saito [GS03] did Particle In Cell (PIC - see the next chapter for details) simulations of very low proton β plasma. They presented two cases. In the first case, they started with a high value of T⊥p/Tkp. This anisotropy drives a proton cyclotron instability which generates a broad spectrum of waves. The proton distributions maintain their bi-Maxwellian shape in the long run. In the second case, they imposed an initial spectrum of Alfvénwaves. In this case pitch angle scattering of ions developed non-Maxwellian features. Pitch angle scattering becomes weaker as parallel velocity goes to zero. Another formalism suggested by Isenberg et. al. [IV07] is to heat the mi- nor ions through second order Fermi mechanism in low β plasmas. The ultimate

32 source of dissipation in this mechanism is also ion cyclotron resonances. The basic idea is that for the minor ions there exists a range of velocities at which the ions can simultaneously resonate with both sunward as well as anti-sunward waves. This simultaneous interaction of minor ions with multiple wave modes eﬀectively diﬀuses the ions in the perpendicular direction, heating them preferentially. This is equivalent of second order Fermi acceleration.

1.5.1.2 Waves in expanding box In principle the theory dealing with heating of the solar wind should take into account the fact that it is expanding. It is computationally very expensive to model the solar wind at a global scale even with fluid codes. One way around this problem is to simulate an expanding parcel of plasma in the solar wind (e.g. [VGM89, GVM93, RVED05]). A box flowing with a parcel of solar wind plasma is considered. The small volume element of the spherical shell, surrounding the plasma is approximated as being a parallelepiped with a Cartesian coordinate system. The equations are then transformed into the moving plasma frame of reference. The effects of expansion are included in the transformed equations to the first order. A non inertial term arises which acts as a force on the plasma because of the expansion. The dissipation mechanism in this model is still cyclotron resonance. Liewer et. al. [LVG01] implemented this model in a 1D hybrid code. They studied the evolution of the plasma consisting of protons and alpha particles with monochromatic Alfvénwaves as well as spectra of Alfvénwaves as initial conditions. As the plasma expands, the alphas come into resonance with waves first and are heated. Hellinger et. al. also implemented the expanding box model in a hybrid code [HTMG03, HVT+05]. They studied the evolution of plasma consisting of protons and minor ions with a spectrum of waves in an expanding box using 1D and 2D simulations. They found that in a relatively high beta plasma, parallel fire-hose instability becomes active in the long time evolution of the system. In the 1D case,

33 this instability generates waves which keep the system close to marginal stability. In the 2D case, oblique fire-hose instability becomes important and the system departs from marginal stability. They also found that as the system expands, the ions come into resonance with waves, heavier ions coming into resonance before the lighter ones. The oxygen ions are essentially transparent to ion cyclotron waves and do not block most of the power in these waves as the box expands. They also report generation of small differential oxygen/proton velocity via Alfvéncyclotron interaction. Camporeale et. al. [CB10] implemented the expanding box in cylindrical geometry in a PIC code and studied the effects of a variety of speeds on the evolution of the plasma. They found that the expansion increases parallel anisotropy Tk > T⊥ and instabilities try to reduce the anisotropy. They also found signatures for growing electron fire-hose instability.

1.5.1.3 Dissipation or Dispersion? KAWs or Whistlers? Another related puzzle is the nature of the cascade at small scales scales. It is expected that part of the energy goes into the thermal energy of the protons and the rest cascades down to electron inertial scales. In weak turbulence theories relying on nonlinear interactions of the waves to cascade energy down to smaller scales, this would be another cascade resulting from mode conversion to modes like kinetic Alfvénwaves (KAWs) or whistler waves. In theories of strong turbulence relying on highly nonlinear couplings to cascade energy down, it could be another such highly nonlinear cascade. A better understanding of the nature of this cascade could have a great impact on our understanding of the heating mechanisms not only at the proton scales but also at the electron scales. The phase velocity of waves can be shown to be related to the ratio of electric field spectrum to the magnetic field spectrum. Bale et. al. [BKM+05] calculated the power spectra for electric as well as magnetic field power spectra using data from the Cluster spacecraft. They plotted the ratio of electric to magnetic spectra

34 in the plasma frame and fitted it with the approximate dispersion relation of kinetic Alfvénwaves (KAWs). They used this fit to suggest that the cascade turns into a KAW cascade at the ion inertial lengths. Self consistent gyrokinetic simulations [HDC+08] driven with Alfvénicsolutions of the Langevin equation were used to simulate the cascade of energy into the proton inertial range. The calculated electric and magnetic field spectra showed the behavior similar to seen in the solar wind by e.g. [BKM+05]. They also suggested that this is a sign of KAW cascade and discounts the presence of cyclotron resonances at ion inertial scales. These claims of KAW cascade being active below the ion inertial scales were challenged by Matthaeus et. al. [MSD08, MSD10]. They presented various cases of MHD runs in 1D, 2D and 3D with and without the Hall term. The runs with the Hall term produced the electric field spectrum enhancement at scales smaller than the ion inertial lengths. They suggested that the results of Bale et. al. and Howes et. al. are not sufficient to claim that KAWs are active at those scales and that we can not rule out the possibility of Whistlers or even low frequency eddy dynamics which can not be described by linear theory. In a related study (not in the solar wind) Eastwood et. al. [EPBT09] looked at turbulence generated by reconnection in earth’s magnetotail using data from the Cluster spacecraft. The reconnection site was identified by the Hall electric and magnetic field signatures. They calculated the electric and magnetic field spectra and saw the enhancement of electric field spectrum in the kinetic regime. To understand if there is a wave mode active at those scales, they derived a dispersion relation from data. The dispersion relation derived from the data showed consistency with whistler waves than KAWs. Saito, Gary and others have concentrated on the dynamics at the sub proton inertial scales. In a series of papers they have used PIC simulations to study the self consistent evolution of whistler waves and its effect on electron velocity distributions as well turbulence at the kinetic scales. In [SG07a, SG07b] they set up simulations

35 which were effectively 1D and studied the effects of whistler anisotropy instability, monochromatic whistler waves as well as broadband spectrum of whistlers on the broadening of the strahl in the electron distribution functions (for details of strahl see [Mar06] and the references therein). The spectrum of waves was found to be more efficient at pitch-angle scattering the electrons and hence broadening the strahl as the plasma evolves. Smaller wavelength whistlers were found to be more efficient than larger wavelength whistlers at pitch angle scattering. In [GSL08, SGLN08, GSN10] they used 2D PIC simulations (with ∼ 67M particles) of a low β plasma to look at the turbulence generated by whistler waves at sub proton inertial scales. The fluctuations parallel to the mean magnetic field were found to have more energy at the parallel wave vectors and perpendicular fluctuations were found to have more energy at the perpendicular wave vectors. The electrons heated preferentially in the direction parallel to the mean magnetic field which was interpreted as the Landau damping of oblique whistler waves. The power laws in the range between proton inertial scales and electron inertial scales was

−2 −3 −4 found to be k⊥ to k⊥ and at sub electron inertial scales was found to be k⊥ . These power laws along with high compressibility observed in these simulations indicates that in the wave description of turbulence, the observed solar wind spectrum can be explained using not only Kinetic Alfv´enwaves but whistlers also. In another study Svidzinsky and others [SLR+09] looked at the properties of turbulence generated by fast magnetosonic waves in 2D geometry with a few billion particles. The cascade of energy at the sub proton inertial scales was found to be highly anisotropic with more energy in the perpendicular wave vectors than parallel wave vectors. They used k − ω spectra to show that most of the energy remains in the fast modes at high k and hence the coupling of these modes with slow Alfv´en and Bernstein modes is weak. Most of the energy damped went into heating the electrons.

36 Recently Valentini et. al. have developed a hybrid Vlasov code (kinetic protons and fluid electrons) [VTC+07] and in a series of papers have looked at the evolution of kinetic plasma. In [VVCM08, VCV11] they studied 1D-3V (1 space and 3 velocity dimensions hybrid Vlasov simulations with only three modes of velocity and magnetic spectrum excited at t = 0 and in [VCV10] they studied the turbulence generated by the initial spectrum of Alfvénicfluctuations in a 2D-3V box. Enhanced electrostatic activity was observed at high wave numbers late in the evolution of the simulations. The protons heated preferentially in the direction perpendicular to the mean magnetic field and the proton distribution function develops a beam similar to observed in the solar wind. The k − ω spectra of electric fluctuations showed a new kind of acoustic like mode which is a solution of hybrid Vlasov equations excited by resonant particle trapping [VV09]. One problem with these studies is the ion- electron temperature ratio of Te/Ti = 10. Such temperatures are not observed in the solar wind. For lower values of Te/Ti = 1. the energy level of the fluctuations dropped by 10 orders of magnitude because of Landau damping of waves by electrons. Matthaeus et. al. [MMD+03] and Cranmer et. al. [CvB03] have explored the idea of electron phase-space holes. [MMD+03] motivated by observations in earth’s magnetosphere suggested that strong electric fields in turbulent reconnection sites could trap electrons and accelerate them. These electron beams in the corona give rise to electron phase space holes. These holes have very high frequency transverse electric field. Protons when bombarded with these very high frequency uncorrelated transverse impulses would be heated perpendicularly. Cranmer et. al. [CvB03] suggested that KAWs at high k⊥ can be Landau damped by electrons. This Landau damping produces parallel electron beams and electron phase space holes required for heating the protons perpendicularly.

37 1.5.2 Non-wave Mechanisms Theoretical as well as observational constraints (from solar wind observations) in anisotropic MHD turbulence raise questions about the applicability of wave dissipation mechanisms in the solar wind. Mechanisms to heat the plasma without linear wave modes could equally well be at work. Li et. al. [LGS01] used the diffusion approximation model of Zhou and Matthaeus [ZM90] to compare the cutoff of magnetic spectra at the damping scales with the cutoffs expected from linear damping of Alfvénwaves and magnetosonic waves. They did not find any regime of parameter space in propagation angle and proton β which would produce a broken power law behavior for the wave modes considered. Based on these results they suggested that the role of linear modes in heating the solar wind is question- able. In this section we discuss some of the non-wave mechanisms present in the literature.

1.5.2.1 Current sheets & reconnection sites As discussed in the observations section, reconnection is observed almost everywhere in the solar system. In a series of papers (see e.g. [Par01] and the references therein) Parker developed the theory of magnetic field line topology and equilibrium. The basic idea is that the foot point motions of the field lines embedded in the solar photosphere produce an entangled mesh of these lines. Parker showed [Par72] that the lowest energy state of this mesh of magnetic field lines exhibits large gradients in magnetic field. These large gradients are manifested as topologically singular discontinuities in magnetic field in the ideal case (infinite current sheets). In a realistic system like the solar atmosphere, there is small but finite resistivity. This resistivity dissipates the current heating the plasma and also creating reconnection sites. This way the magnetic field topology moves perturbatively away from the equilibrium.

38 To summarize, the constant shuffling of field lines because of foot point motions entangles the field lines. The system relaxes to equilibrium states exhibiting strong current sheets. When the current sheets become very strong, resistivity dissipates energy into heat and also creates reconnection sites. The system moves perturbatively away from static equilibrium and goes into dynamical quasi-equilibrium, constantly creating strong current sheets and reconnection sites. Lather, rinse, repeat! One natural requirement of this picture is that the time scales of the foot point motions have to be very slow in order for the above dynamics to be effective. Fast foot point motions would not give the system enough time to attain a quasi-equilibrium before dissipating the energy through current sheets. In [Par88] Parker suggested that a large number of small reconnection events, which he called nanoflares, generated in this way and with very small energy happening all the time in the corona could explain the heating budget for the corona. The energies released in the nanoflares are still too small to be detected by the present day instruments and hence there is no direct observational evidence of this mechanism. The idea of current sheets and reconnection sites heating the plasma has been studied using empirical modeling (e.g. [HP84]) and also supported by simulations of MHD turbulence (e.g. [Hv96, DG99, RDEV06]). Heyvaerts and Priest [HP83] suggested a heating mechanism which involves Alfvénwaves but uses ohmic dissipation in current sheets at the end to dissipate the energy. They suggested that shear Alfvénwaves grow out of phase with each other while propagating for long times (e.g. multiple reflections in a coronal loop). These phase differences generate large gradients and hence current sheets which can dissipate energy into heating the plasma. Dmitruk et. al. [DMS04] did test particle simulations to test this scenario. They updated test particles in a snapshot of fields from a 2563 MHD simulation. They found that the particles get energized at the current sheets and reconnection

39 sites. Moreover they found that the protons get heated perpendicular to the mean field and electrons get heated parallel to the mean field. The electrons being lighter have very small gyroradii. When they get trapped in the electric field of a current sheet, they essentially experience only one kind of electric field, and are accelerated along it. Protons having larger gyroradii may sample variations of electric field that are in phase with their own velocity. Under such circumstances, some protons gain substantial perpendicular energy. Osman et. al. [OMGS11] analyzed data from Wind and ACE spacecraft to identify heating processes in the solar wind. They used normalized partial variance of increments of magnetic field. Using data analysis and comparisons with MHD simulations, they correlated the heating to the intermittent non Gaussian tails of the probability density functions (PDFs) which have been shown to be associated with current sheets and reconnection sites [GMS+09]. This suggests that the dissipation in the solar wind occurs in concentrations and is associated with coherent structures.

1.5.2.2 Energization by trapping in magnetic islands Another way to energize particles is for the particles to be trapped in magnetic islands created by reconnection. Matthaeus et. al. [MAG84] did test particle runs in the ﬁelds produced by spectral MHD simulations of reconnection and found that the most energetic particles are the ones that are trapped in O points (or magnetic islands). Drake et. al. [DSCS06] suggested that energetic electrons can be produced when they get trapped in magnetic islands created by reconnection. The electrons get accelerated by ﬁrst order Fermi acceleration that they encounter when the island contracts, gaining large amounts of energy.

1.5.2.3 Stochastic heating Chandran et. al. [CLR+10] suggested a stochastic heating mechanism relying on random walk of the particles in the potential variations. When the size of rms

40 fluctuations in velocity and magnetic field becomes large, the ion orbits become chaotic and ions undergo stochastic perpendicular heating. Chandran [Cha10] used this method to calculate the heating rates of ions in the lower corona and found the temperature profiles from their theory in good agreement with observations.

1.6 Summary The solar corona and solar wind are observed to be heated. A huge amount of literature has been devoted to the observations of corona and solar wind and also to theoretical studies to understand these observations. We have discussed a few key observations and theoretical studies which are relevant to lay a background for this thesis. The corona is about 100-1000 times hotter than the photosphere. How energy is transported from cooler photosphere to the hotter corona and dissipated away is still a matter of debate. Protons are observed to be hotter than electrons. Heavy ions are seen to be heated much more than the protons, more than expected from the mass ratio. The protons as well as heavy ions are seen to be heated preferentially in the direction perpendicular to the mean magnetic field. The anisotropy is much larger for the minor ions. This suggests cyclotron resonances in action but the details of how this anisotropy occurs are still not agreed upon. The solar wind is also ”heated”, which means that the temperature of the solar wind does not fall as much as would be expected from adiabatic expansion. Also the power spectrum of magnetic fluctuations in the solar wind shows a power law behavior consistent with hydrodynamic turbulence. The spectrum steepens at ion inertial scales suggesting conversion of some fluctuation energy into heating. The power law behavior below the proton inertial scales suggests another cascade to electron scales but the details of the nature of this power law are also not well understood.

41 What is agreed upon is that the magnetic field is the ultimate carrier of energy from the sun to the corona and outwards. The details of the specific processes are not well understood/ agreed upon. Understanding these issues requires going to the ”kinetic” limit as the solar corona as well as the solar wind are observed to be almost completely collisionless. This leaves the use of simple viscosity and resistivity in fluid models unjustified. Most of the kinetic studies done to address this issue are recent (just before or contemporary to this thesis) and have concentrated on presence of waves in the system to dissipate the energy through wave particle interactions. Even though this is a good first step in enhancing our understanding of the system, it is not clear as to how much linear waves are present. Also almost all the kinetic studies have the wave modes in the plane of the simulation which would correspond to slab like geometry. Observational constraints suggest that the slab mode has less than 20% of the energy budget of the solar wind. More than 80% of the fluctuation energy is in the 2D fluctuations perpendicular to the mean magnetic field. In this thesis we attempt to address the question of dissipative mechanisms in this 2D limit of turbulence. We concentrate on the dissipative processes at the proton inertial scales. For this matter the use of a hybrid code (particle protons and fluid electrons) is optimal as in such a code we have the complete kinetic physics at the proton scales. Hence all the dissipative processes at the ion scales are self consistent and there is no need to assume any kind of ad hoc dissipative mechanisms. We study decaying as well as driven turbulence and use diagnostics such as energy budget analysis, wavenumber- frequency (k − ω) spectra and PDFs etc. to identify the possible role of waves and current sheets - reconnection sites in heating the plasma in this limit. We find that in the 2D limit, waves do not seem to play any significant role in heating the plasma. The intermittent non Gaussian features are shown to have a correlation with the

42 heating of plasma too. In the chapter 2 we mention various models used to describe the plasmas and describe the hybrid code used in this thesis. In chapter 3 we describe the set up of the speciﬁc problems and subsequent chapters 4, 5, 6, 7 contain the details of the research work carried out. The last chapter contains the conclusions and possible future extensions of the present work.

43 Chapter 2

HYBRID CODE

In this chapter we brieﬂy introduce various approaches used to study plasmas and describe in detail the hybrid code used in this thesis. Based on what system we’re interested in studying, a plasma can be treated in various limits. Ideally the most detailed and statistically accurate description of the plasma would require solving the Louiville’s equation. However it is very hard to deal with N particle distribution functions. With the low density plasma approximation, we can reduce the equation to the Boltzmann equation. The Boltzmann equation is an equation for the time evolution of the single particle distribution function that has the eﬀects of two particle collisions modeled in it in terms of the single particle distribution function itself. The Boltzmann equation along with the Maxwell’s equations is a self consistent set of equations to describe a plasma in the low density limit.

∂f α + v · ∇f + a · ∇ f = C ∂t α α v α ∑ ∫ ∇ · E = 4π nαqα fαdv + 4πρext α ∫ 1 ∂E 4π ∑ 4π ∇ × B = + n q vf dv + J c ∂t c α α α c ext α 1 ∂B ∇ × E = − c ∂t ∇ · B = 0 (2.1)

44 where fα is the distribution function for the α species (e.g. electron, proton, heavy ions etc.) of charged particles, nα is the average density of α species ,aα is the acceleration due to various forces acting on the species α (e.g. electromagnetic, gravitational etc.) and C is the term representing the collisional eﬀects. nα is the ∫ average density of the entire system and hence the normalization is nα fαdxdv =

N α = total number of particles of species α in the system. There are two major problems with solving the Boltzmann equation:

• The collision term is very hard to model. Arguments based on intuition and empirical observations can be used to approximate this term but still the conditions within one system might change drastically to make the use of one simple expression unphysical.

• It is impossible to solve the Boltzmann equation analytically with the present day mathematics. The 6 dimensional phase space requires N 6 sized grid to solve it numerically. This means that even a small system of 256 grid points per dimension would require ∼ 1015 calculations every time step. This is only within the reach of the few largest (peta-scale) supercomputers available today. A realistic sized system is still out of reach of even the largest supercomputers at present to solve it in reasonable time.

Because of these problems with solving the Boltzmann equation, various approximate models are used by plasma physicists to study the system of their interest. Below we brieﬂy describe various approaches and their pros and cons to study e.g. the solar plasmas.

2.1 Various plasma descriptions 2.1.1 Fluid description The dynamics of the plasmas at the largest length and time scales can be described reasonably accurately using the ﬂuid equations obtained by taking the

45 moments of the Boltzmann equation. The zeroth moment of the Boltzmann equation gives the continuity equation but the equation requires calculation of the first moment. The equation of the first moment gives the momentum equation. The momentum equation in turn requires the knowledge of the second moment. It turns out that the equation for any moment requires the knowledge of the next higher moment. The chain of equations is generally closed by assuming an equation of state. This set of equations along with the Maxwell’s equations constitutes the fluid equations describing the behavior of a plasma. The detailed derivation of magnetohydrodynamic (MHD) equations and their different limits can be found in any standard textbook on plasma physics. The equations of single fluid, isothermal, incompressible, resistive MHD can be written as:

∂u 1 1 + u · ∇u = ∇p + (∇ × B) × B + ν∇2u ∂t ρ 4πρ ∂B = ∇ × (u × B) + µ∇2B ∂t ∇ · u = 0

∇ · B = 0 (2.2) where u is the ﬂuid velocity, p is the scalar pressure, B is the magnetic ﬁeld, ρ is the density, ν is the viscosity and µ is the resistivity.1

1 A note to future graduate students reading this thesis: The form of fluid equations describing plasmas varies vastly depending on what limits the equations are being written for. Depending on whether the equations are being written for one fluid or multiple fluids, what kind of kinetic effects are being modeled and for what parameter regime are the equations being written could end up giving you a set of equations (Ideal/Resistive, Single/Multi-Fluid MHD, Com- pressible/Incompressible MHD, Hall-MHD, Reduced MHD, Electron MHD to name a few). Care should be taken to understand what limit the equations have been written for.

46 Advantages: MHD being applicable to large space-time scales is very useful in describing the dynamics of large systems where the details of kinetic effects are not important. The set of equations is in 3 spatial dimensions. This means that with the present day computational power it is feasible to simulate very large systems (∼ 10243) using MHD modeling. A number of global models of the solar wind and magnetosphere have been built up (e.g. BATS-R-US developed by the group at the University of Michigan) with appropriate boundary conditions, realistic initial conditions and empirically modeled dissipation functions (e.g. [EOJPG09]). Limitations: The act of closing the set of equations by assuming an equation of state is equivalent to discarding the kinetic effects. Attempts have been made to estimate higher order effects (e.g. [GGSJ99] but most efforts are either heuristic or empirical. Many MHD based studies use a simple constant viscosity and resistivity. Solar wind and upper corona have been observed to have very low collisionality (e.g. [Mar06]). The mean free path of a particle in the fast wind is about 1AU. Under such circumstances, the very assumption of the mean free path of particles being much smaller than the system size breaks down. To simulate such systems, we need a more complete and consistent kinetic description.

2.1.2 Fully kinetic description 2.1.2.1 Vlasov To study the time evolution of a collisionless plasma expanding in the solar system, the Vlasov equation is the appropriate limit of the Boltzmann equation. The collision term is set to zero, gravitation is ignored and only electromagnetic forces are considered. With these approximations, the evolution equation for the distribution function becomes:

( ) ∂fα qα v × B + v · ∇fα + E + · ∇vfα = 0 (2.3) ∂t mα c

47 The rest of the equations remain the same as in eqn. 2.1. Advantages: The equation self consistently describes the time evolution of collisionless plasmas in the absence of gravity which is a very good approximation for the upper corona and solar wind. The equation includes all the kinetic effects and hence there is no need of approximating the dissipative functions. This method does not suffer from the ”noise” that will be discussed in the context of PIC codes. Limitations: As discussed in the context of the Boltzmann equation, the Vlasov equation is still a 6 dimensional equation. So it is computationally very expensive to solve this equation. The equation is generally studied in the linearized limit (e.g. [KT73, Gar05]) but the full numerical solution for a reasonably big system is still out of the reach of present day computers. Also the solution of the Vlasov equation suffers from very sharp gradients in the velocity space, called velocity space filamentation.

2.1.2.2 Particle In Cell A clever way of solving the Vlasov equation resembles the molecular dynamics technique. The electromagnetic fields and moments are defined on a grid. A large number of “pseudo particles” (phase space points) are loaded within each grid cell (hence the name Particle In Cell - PIC) with the properties of plasma particles and a specific distribution function. The equations of motion for these “pseudo-particles” are updated individu- ally. This effectively means that we’re evolving a randomly discretized (using Monte Carlo methods) distribution function. The values of the fluid moments like effective density, velocity etc. at the grid points are calculated from these pseudo particles. We describe the process of calculating the moments in the next section. These moments in turn used to update the Maxwell’s equations. We refer the reader to excellent book by Birdsall & Langdon [BL85] for further details of the method.

48 Advantages: This method is much less computationally expensive compared to Vlasov. It might not be an exaggeration to call it “a poor man’s Vlasov solver”. It has self consistent dynamics of the plasma with complete kinetic physics. This means there is no need for artificial viscosity etc. With increasing computational power, this technique is becoming more common in space sciences to study reconnection, turbulence etc. (e.g. [SG07a, DSCS06, MSCB10]). Limitations: Even though much less expensive than Boltzmann/Vlasov, PIC codes are still very expensive. Only tiny 3D boxes or reasonable sized 2D boxes with ∼ 109 particles are doable in reasonable time. There is still no scope for any global model like the MHD ones. Also the finite number of particles introduces a noise in the calculation of the moments like density, currents etc. This noise can make analysis of the simulation much more difficult and makes studies of very small amplitude oscillations nearly impossible. In certain situations, it can also lead to artificial numerical heating.

2.1.3 Compromise Models Given the limitations of MHD at the kinetic scales and fully kinetic models at the large scales, we can devise “compromise” models which have some aspects of kinetic physics and are less computationally expensive.

2.1.3.1 Gyrokinetics In many situations with a strong background magnetic ﬁeld, the time scales of interest are much longer than the time scales of the gyro motion of the particles but the length scales shorter than the gyro scales are important. It is possible to write for such systems a set of equations that captures the details of the low frequency phenomena of interest. We present the equations as described by Brizard & Hahm [BH07]. By eliminating the gyroangle from the equations, a new gyrocenter magnetic momentµ ¯ can be constructed as an adiabatic invariant and the gyrocenter

49 gyroangle ζ¯ becomes an ignorable coordinate. The equation for the gyrocenter

Vlasov distribution F¯(X¯ , v¯k, t;µ ¯) in the gyrocenter phase space (X¯ , v¯k;µ, ¯ ζ¯) can be written as:

∂F¯ dX¯ dv¯k ∂F¯ + · ∇¯ F¯ + = 0 (2.4) ∂t dt dt ∂v¯k where X¯ denotes the position of the gyrocenter,v ¯k = (B/|B|) · dX¯ /dt is the velocity along the magnetic field. dµ/dt¯ = 0 and dF¯ /dζ¯ = 0 are satisfied by definition and gyrocenter equations of motion are independent of ζ¯. Gyrokinetic equations have been extensively used to study instabilities and turbulence in fusion plasmas (see e.g. Table I in Brizard [BH07] and references therein). Recently gyrokinetics has been used to study turbulence in astrophysical plasmas including the solar wind (e.g. [HCD+06, HDC+08, SCD+09]). Advantages: The gyrokinetic equations are fully kinetic for time scales slower than the gyromotion of the particles. The equations are essentially 5D and there is no requirement of resolving time scales shorter than the cyclotron time. This means that the codes can run much faster than Vlasov codes. When implemented without particles, gyrokinetic codes do not suffer from noise like PIC codes. This makes them well suited to study instabilities. Also the implementation of collisions is much easier than PIC which might be important in collisional systems like the lower corona or the fusion plasmas. Limitations: In the act of averaging over gyro motions of the particles we lose physics for time scales faster than the gyro time scales. One example would be whistler wave physics which would be absent from the dynamics of gyrokinetic equations. Also the gyrokinetic codes are still very expensive.

50 Model Assumptions Feasible Kinetic physics Issues system size Boltzmann only binary colli- Very tiny Complete Modeling collisions sions Vlasov collisionless, EM Very tiny Complete Velocity filamen- forces, Fg = 0 tation PIC Randomly dis- Small Complete Particle noise cretized Vlasov (>Vlasov) Gyrokinetics Averaged out Small ∼ p+s & e−s up to no physics for gyro motions PIC gyro time scales ω > ωc Hybrid Vlasov Vlasov p+s, fluid > Vlasov p+s only Velocity filamen- e−s tations Hybrid PIC PIC p+s, fluid > above p+s only particle noise e−s Fluid fluid p+s & e−s Big, no kinetic no kinetic Global physics physics models

Table 2.1: Summary of diﬀerent numerical approaches for studying plasmas

51 2.1.3.2 Hybrid Vlasov Another compromise solution is to treat ions kinetically and electrons as a neutralizing fluid. This approximation is valid to study the processes at length and time scales larger than the ion length and time scales. Given the high mass ratio of ions to electrons, the length and time scales of electron motions are very small as compared to the time scales of interest. Hence electrons behave essentially like a fluid at those length and time scales. Adopting a fluid model for the electrons brings the computational time down to at least half as compared to fully kinetic descriptions. If we treat the electrons as a “massless” neutralizing fluid we can neglect electron inertia, and we do not need to resolve the electron scales at all and the computational time drops by many factors. Recently a hybrid Vlasov code has been developed by Valentini et. al. (e.g. [VTC+07, VVCM08]) in which they solve the Vlasov equation for the ions. They use the fluid equation of motion as the electrons as the Ohm’s Law and keep electron inertia effects in it. Advantages: The model contains complete ion kinetic physics. As discussed in context of Boltzmann/Vlasov, this model does not have the problem of noise because of finite number of particles like the PIC models. The codes are much faster to run if electrons are assumed to be massless. Limitations: As discussed earlier, even though less expensive than full Boltzmann/Vlasov, this model is still very expensive and hence runs are generally limited to 4D (1 space + 3 Velocity dimensions) or smaller 5D (2 space + 3 velocity dimension) runs. Also as discussed in case of full Vlasov, there is still the problem of velocity space filamentation.

2.1.3.3 Hybrid PIC As with PIC simulations, we can also solve the hybrid Vlasov equations by particle in cell methods (hybrid PIC). The ion equations of motions are updated

52 for a large number of pseudo-particles and electron equation of motion is used as Ohm’s law. In this thesis we use a hybrid code which we describe in detail in the next section. Advantages: Full ion kinetic physics means self consistent dynamics at the ion scales. Hence no assumptions needed to model the kinetic processes at those scales. The hybrid codes are much less expensive than Vlasov & PIC. This means it is possible to simulate relatively bigger system sizes. Limitations: The hybrid model does not have electron kinetic physics. Most of the hybrid models actually do not even have electron inertia eﬀects (e.g. [LVG01, WWG+09]). These codes like the PIC codes have noise because of ﬁnite number of particles. Even though the system sizes are bigger than PIC, it’s still tiny (in computational as well as physical units) as compared to MHD simulations. The various models discussed above have been summarized in the table 2.1. We now go on to describe in detail the code used in this thesis.

2.2 Hybrid Code P3D For the studies in this thesis, a fully parallel hybrid PIC code P3D was used, which models protons as (pseudo-)particles and electrons as ﬂuid. It is a parallel version of the code described in [SDRD01]. P3D is a 3 dimensional electromagnetic code (which means that it solves for the full electromagnetic hybrid plasma equations). For the studies done in this thesis, the code was run in 2.5D limit (ﬂuc- tuations are 3D but the dynamics occurs only in the x, y plane and d/dz = 0). The code has been used extensively to simulate reconnection (e.g. [MCSD09]). The code advances the following equations:

dx i = v dt i dvi 1 1 = (−ue × B + vi × B) − ∇Pe dt H n

53 ( ) ( ) ( ) ∂ P v P γ − 1 { e −∇ · e e ∇2 γ−1 = γ−1 + γ−1 χ (Te) ∂t n n } n ν − e |∇2B|2 n0 ( ) 0 ∂B J 0 1 Ji 1 = −H ∇ × × B − × B − ∇Pe ∂t n H n n

1 νe 2 2 − ∇ ∇ B + H αFb cos(ωdt) H n0 0 J 0 Ji 1 E = × B − × B − ∇(Pe) (n n ) n m 1 0 − e ∇2 ∇ × B = 1 2 B; J = B (2.5) mi H where γ = 5/3 is the ratio of speciﬁc heats, J = ∇×B is the current density, Ji is ion

flux, Pe electron pressure, Te = Pe/n is the electron temperature, H ≡ c/(L0ωpi) is the normalized proton inertial length, me and mi are the electron and proton masses, xi and vi are the positions and velocities of the individual protons. Magnetic field is normalized to a characteristic magnetic field b0 and density to a characteristic 1/2 density n0. Length is normalized to L0, velocity to V0 = b0/(4πmn0) , time to 2 t0 = L0/V0, and temperature to b0/(4πn0). νe is the electron viscosity and χ is the thermal diffusivity. Fb is the forcing term for the magnetic field in the plane of the simulation to which we assign a specific spatial structure (described in the research chapters). α is a dimensionless constant to control the strength of driving. The νe term transfers energy from the magnetic field into the electron thermal energy. Different problems discussed in this thesis used different limits of the above equations. We present the specific limits of the above equations used for the specific problems in the corresponding chapters.

2.2.1 Electron Physics Unlike most of the hybrid codes, our code includes the electron inertia eﬀects. The electron equation of motion is used as the Ohm’s law. Under the assumptions

54 that at electron scales Ji << J and n does not vary signiﬁcantly,the Faraday’s law can be written as (with the plasma normalizations described in appendix): ( ) 0 ∂B J 0 Ji 1 νe 2 2 = −∇ × × B − × B − ∇Pe − ∇ ∇ B + forcing ∂t n n n n0

0 me 2 where B = (1 − ∇ )B includes the electron inertia effects indirectly. νe term mi represents the effects of an electron viscosity. The electron pressure equation as originally implemented in the code was numerically unstable. To stabilize the equation at high k, we added a simple electron diffusivity term to the pressure equation.

2.2.2 Code details The code is written in a combination of FORTRAN, M4 & CPP. It uses CPP and M4 pre-processors to create FORTRAN compilable files for each run, based on the specific parameters defined in a parameter file. The parameter file is used to define the grid size for each processor, the number of processors, time step, size of the domain in c/ωpi, number of particles per grid and physical parameters like the temperatures, background magnetic field, mass ratio me/mi etc.

When running on computing clusters, hybrid P3D utilizes npex ∗ npey processors where npex and npey are the number of processors in x and y directions. The parallelization involves a standard domain decomposition among processors and echanging boundary cells using MPI. Fig. 2.1 shows the distribution of the domain on the processors. Each processor creates its part of the grid, nx+2, ny +2 points. The physical part of the grid is nx ∗ ny such that the total grid size of the physical domain is nx ∗ npex ∗ ny ∗ npey. To calculate the derivatives correctly at the boundary, each processor needs information about the row/column next to the boundary from the neighboring processors. This information is stored in the temporary ”guard cells” (hence the +2 in each direction). Each processor updates the equations on its

55 Figure 2.1: Domain decomposition between diﬀerent processors in P3D. We denote the processors by their processors numbers in npex, npey notation as well as the processor number in the MPI numbering system with N = npex ∗ npey (bottom number in each cell)

56 grid and at the end of each time step, the boundary cells are exchanged with the neighboring processors. The code uses periodic boundary conditions. Hence the processors at the simulation domain boundaries exchange the guard cells with the processors at the opposite boundary. For example processor 0, 0 would exchange the guard cells in the x direction with processors 1, 0 & npex − 1, 0 and guard cells in the y direction with processors 0, 1 and 0, npey − 1 (see fig. 2.1 for processor numbering). The initialization depends on the problem of interest and is done using the routine init hybrid. This routine creates the initial magnetic field configuration as well as loads the particles on processors. Each processor then writes the magnetic field, electron pressure and particle data to unformatted ”dump files”. The main program reads the initial conditions from the dump files and evolves the hybrid equations described above for those initial conditions. The main loop essentially involves the following steps:

• Advance the particles.

• Calculate the moments (density, velocity etc.) on the grid points.

• Advance the ﬁelds based on these moments

• Repeat

The algorithm of the code is described in the ﬂowchart 2.2 and the detailed scheme of execution in given in appendix. Below we discuss a few important numerical aspects of the code.

2.2.2.1 Particle Stepping The particles are advanced in time using the Boris stepping algorithm (e.g. [BL85]) which follows the following steps:

57 • Advance the particles by half time step dt/2 using the old value of velocity.

• Advance the velocity by a full time step dt

1. Advance velocity by half time step using only the electric ﬁeld vnew = q vold + m Edt/2

2. Advance velocity by full time step using the magnetic force vnew−dt/2(vnew×

B) = vold + dt/2(vold × B) 3. Advance the new velocity by half time step using the electric force.

• Advance the particles by half time step using the new value of velocity.

The fields required to advance the particle are extrapolated from the nearest grid points to the position of the particle. The basic weighting scheme is shown in fig. 2.3. The value of the field at a grid point gets weight according to area of the rectangle diagonally opposite to that grid point. In the figure, we code the related areas and grid points by alphabetical letters and colors. With this coding, the magnetic field at the particle’s position would be (we write the equation only for one component of magnetic field): ∗ ∗ ∗ ∗ p (Bx(i, j) Ad + Bx(i + 1, j) Ac + Bx(i, j + 1) Aa + Bx(i + 1, j + 1) Ab) Bx = (Aa + Ab + Ac + Ad) (2.6)

p where Bx represents the value of x component of magnetic field at the particle’s position, Aa etc. donate the area of the rectangle denoted by the corresponding alphabet. After stepping the particles, some particles might jump processor boundaries and go onto another processor. Such particles have to be moved to the particle buffer of the appropriate processor. The redistribution routine finds out the particles that jump the processor boundaries, creates a temporary memory buffer and sends that buffer to the appropriate processor. After a few time steps, the particle buffers on each processor are sorted to for better usage of cache.

58 2.2.2.2 Particle Moments The moments (density, velocity etc.) at the grid points are also calculated using the weighting scheme described in fig 2.3, but the other way around now. Each particle contributes part of its mass and charge to the surrounding grid points based on the weight the grid point gets. For example, grid point A in fig. 2.3 gets q ∗ Aa/Acell contribution to the charge density from the particle shown. A sum over all the particles in the cell would give the total charge density at the point A. The densities and bulk flow velocity calculated at the grid points using this weighting scheme have a ”noise” associated with them. This noise appears because of the finite number of particles, the moments calculated at the grid points have finite √ discontinuities. This noise is proportional to 1/ N and can reduce the accuracy of simulation results. This noise can be overcome in a number of ways.

• Complicated weighting schemes can be used to calculate the moments at the grid points. A good discussion of diﬀerent weighting schemes can be found in books by Birdsall et. al. [BL85] and Lipatov [Lip02]. More complicated weighting schemes are computationally more expensive.

• More particles per simulation cell can be used. But as the noise is inversely √ proportional to N, doubling the number of particles (and hence doubling √ the run time) brings the noise down only by a factor of 1/ 2. For really big system size and/or really long run time, this is not a feasible choice.

• The moments can be smoothed at the grid points. P3D uses wighted averaging with the nearest neighboring points to smooth the noise in moments.

1 1 f(x, y) = f(x, y) + [f(x + 1, y) + f(x − 1, y) + f(x, y + 1) + f(x, y − 1)] 4 8 1 + [f(x + 1, y + 1) + f(x − 1, f(y − 1) + f(x + 1, y − 1) + f(x − 1, y + 1)] 16

59 2.2.2.3 Field stepping The field equations, i.e. the Faraday’s Law and the electron pressure equation, are advanced using the leapfrog method. The fields are essentially stepped by ∆t/2 before the particles are advanced and stepped by ∆t/2 after the particles are advanced. Each of these ∆t/2 jumps is done over a loop with substeps number of iterations. Hence the time step for the fields is smaller than the particle time step by ∆tf = ∆t/(2 ∗ substeps), where ∆t is the particle time step and substeps is defined in the parameter file. In each iteration, the fields are stepped by the leapfrog method described below:

• The advancing of ﬁelds is done in two steps. To implement the leapfrog

method, we need the value of the ﬁeld at t − ∆tf /2 and t to advance the

ﬁelds to t + ∆tf . In the code, the ﬁelds at the regular time steps are denoted

as B1, pe1 and the ﬁelds at 1/2 time steps are denoted as B2, pe2.

• First advance the half step ﬁelds (B2 and pe2) to t + ∆tf /2. We ﬁrst calculate 0 0 γ−1 0 0 B2 from B2 and pe2 = pe2/ρ , advance the equation for B2 and pe2 by half

step using the values of B1 and pe1

0 − 0 − B2(t + ∆tf /2) B2(t ∆tf /2) − ∇ × 0 − = H E1(t) visc term + forcing ∆tf 0 − 0 − pe2(t + ∆tf /2) pe2(t ∆tf /2) −∇ · 0 = (vepe1)(t) + visc & diﬀusive terms ∆tf (2.7)

• 0 0 0 Invert the B2 and pe2 to get B2 and pe2. Inverting B2 requires inverting the Poisson’s equation. It is done by using a multigrid solver (for an introduction to multigrid methods, we refer the reader to the book by Pieter Wesseling [Wes92] and a very resourceful website http://www.mgnet.org).

60 • Use the new values of B2 and pe2 to update B1 and pe1 from t to t + ∆tf . 0 0 Calculate B1 and pe1 and step forward:

0 − 0 B1(t + ∆tf ) B1(t) − ∇ × 0 − = H E2(t + ∆tf /2) visc term + forcing ∆tf 0 − 0 pe1(t + ∆tf ) pe1(t) −∇ · 0 = (vepe2)(t + ∆tf /2) + visc & diﬀusive terms ∆tf (2.8)

• 0 0 Invert B1 and pe1 to get B1 and pe1.

• Update the value of B2 and pe2 by averaging B1 and pe1 at t and t + ∆tf

B2(t + ∆tf /2) = 0.5 ∗ [B1(t) + B1(t + ∆tf )]

pe2(t + ∆tf /2) = 0.5 ∗ [pe1(t) + pe1(t + ∆tf )] (2.9)

2.2.3 Analysis The code outputs the data in unformatted binary ﬁles in byte scaled as well as double precision. The data was output in double precision at very high time cadence. Double precision was required to have accurate Fourier transforms. Byte scaling the data loses precision which can translate into an error in the high k coeﬃcients when using FFTs. High time cadence was required to resolve frequencies well above the cyclotron frequency in the FFTs. Huge amounts of data are generated in each run. A small 2562 run for about 75000 time steps generates about 50Gb of data. One big run (20482) done to check the scaling of the code generated about 6T b of data. The data was analyzed using codes that were developed in IDL as well as FORTRAN. Visualizations were done using IDL as well as Gnuplot. The following diagnostics were done as part of the analysis:

1. Visualization of various ﬁelds including movies of time evolution.

61 2. Calculation of energies, parallel and perpendicular temperatures.

3. Calculation of effective dissipation coefficients through magnetic interactions and flow interactions.

4. Calculation of omnidirectional spectra of electric, magnetic as well as ﬂow energies including time evolution movies of the same. The ratio of electric to magnetic spectra was also analyzed.

5. Particle stepping routines to step test particles in a snapshot of electric and magnetic ﬁelds output from the hybrid code. This was used to look at the magnetic moments of individual particles as they traverse current sheets and reconnection sites.

6. Calculation of k − ω spectra. Because of huge amounts of data this required tricks in writing the spatial FFT’d data to ﬁles and then doing the temporal FFT by reading parts of the data from the ﬁles.

7. Calculation of probability density functions (PDFs) and kurtosis.

In the next chapter we lay out the problems discussed in this thesis and present the results in the following chapters.

62 Figure 2.2: Flowchart describing the code P3D.

63 A B (i,j+1) (i+1,j+1) grid point

c d

particle

b a

(i,j) (i+1,j) D C

Figure 2.3: Different grid points get part of the contribution from the particle based on weights as defined in this figure. For example, point A gets the weight equivalent of the ratio of area of green part to the area of the whole cell.

64 Chapter 3

PROBLEM SETUP

In this thesis, we address the issue of kinetic dissipative processes at the ion scales in collisionless turbulent plasmas. In the introduction, we discussed the observations and theories which have been used to study the kinetic processes in the solar wind. The heating mechanisms proposed can be divided into two broad categories: i) wave mechanisms ii) non-wave mechanisms. The solar wind has been shown to have an approximate representation 80% and 20% decomposition of energy budget in the 2D (perpendicular) and slab (parallel) (to the mean field) fluctuations respectively. Most of the kinetic simulations done so far (most of them contemporary to this thesis) have concentrated on the wave mechanisms. They have a strong background magnetic field in the plane of the simulation, start with a spectrum of waves in that plane and study the evolution of the particle distributions, energy spectra etc. to find out mechanisms which can be used to explain the solar wind observations. Many of these studies correspond to the slab part of the solar wind as they start with waves that have k parallel to the mean magnetic field. We concentrate on the 2D part of fluctuations. All the studies done in this thesis are approximately quasi-2D. It means that the system has fluctuations in all directions but the wave vectors are predominantly perpendicular to the mean magnetic field. There is very little power in kk. We approximate this siuation by running the simulations in 2.5D mode. All the vectors have 3 components but the

65 variations in the out of plane direction (z in our notation throughout rest of this thesis) are suppressed by having only one grid point in the z direction. This ensures d/dz = 0 and hence the dynamics of the system is essentially 2D even though the fluctuations are 3D. Particle in cell codes have a problem that particles tend to diffuse along the field lines quickly. Also the particle codes are inherently highly compressible unless the plasma beta is very high. We use a strong ”guide field” out of the plane of the simulation. The strong field out of the plane of simulation suppresses the parallel diffusion of particles along the in plane component of the magnetic field. The diffusion along the magnetic field does not affect the dynamics as there is no variation in the z direction. Another important aspect of strong magnetic field out of the plane of simulation is that the presence of a background magnetic field makes plasma turbulence anisotropic (e.g. [SMM83]). The fluctuations evolve to become mostly perpendicular to the background magnetic field and hence we get a quasi 2D system. In chapter 4 we study freely decaying turbulence. We start with an initial condition called the Orszag-Tang vortex (OTV) [OT79]. This initial condition quickly develops into strong turbulence and is usually used to study the numerical stability of plasma codes. We analyze the energy budget of this system and calculate effective dissipation coefficients and viscosity. We present a simple energy budget picture. This work was published in Physics of Plasmas 2009 [PSCM09]. In chapter 5 we present the magnetic energy in the Fourier wavenumber- frequency (k − ω) space. We also present the Eulerian frequency spectrum and probability density functions (PDFs) which show nonGaussian features. This work was published in the proceedings of Solar Wind 12 conference [PSS+10]. A decaying system like OTV loses energy to the heating of the plasma. It is

66 not possible to run a decaying system to study the long term steady state behavior of turbulence. We need to go to driven systems which are supported by continuous supply of energy. In chapter 6 we present results from a driven 2.5D system. We present the k − ω analysis to support the results of our previous study being applicable to quasi-steady state systems. This work has been published in Physics of Plasmas 2010 [PSB+10]. In chapter 7 we generalize the results of chapter 6 and introduce a simple time dependence in forcing. This simulation setup is qualitatively similar to 2D turbulence in the solar corona. We study the onset of turbulence as a function of the driving frequency and also use k − ω analysis to understand the implications for onset of turbulence and heating of plasma in the solar corona. We present PDFs of magnetic ﬁeld increments to relate the heating to current sheets and reconnection sites which manifest as intermittent non-Gaussian features on the PDFs. This work has been submitted to Physics of Plasmas for publication. We conclude by summarizing the central results of our ﬁndings and future directions that can be taken to address the problem of kinetic dissipation in chapter 8.

67 Chapter 4

THE ORSZAG TANG VORTEX: TURBULENT DECAY OF ENERGY

In this chapter, we report a demonstration of turbulent anisotropic proton heating in the Orszag-Tang vortex[OT79] using a hybrid simulation, which includes all proton kinetic effects. The hybrid simulation results are very similar at large length scales to MHD simulations of the same system (having very similar magnetic power spectra), but show significant differences at small scales where kinetic effects are important. The magnetic power spectra of the hybrid simulations show a break

−1 at k = (di cs/cm) (where cs is the sound speed and cm is the magnetosonic speed), where linear two-fluid theory predicts that the Hall term becomes significant, leading to dispersive kinetic Alfvénwaves([RDDS01] and references therein). Analysis of the hybrid results shows that energy is dissipated into proton heating almost exclusively through the magnetic field and not through the proton bulk velocity. The proton heating occurs preferentially in the plane perpendicular to the mean magnetic field. This simulation, to our knowledge, is the first self consistent demonstration of turbulent anisotropic proton heating associated with a quasi-incompressible nonlinear MHD cascade. Finally, effective transport coefficients from the hybrid simulations are calculated, showing that the approximation of constant resistivity η is potentially reasonable (although it cannot reproduce the proton temperature anisotropy), but a constant viscosity ν is untenable.

68 4.1 Hybrid Simulation Model The Orszag-Tang vortex[OT79] is a well studied MHD initial conﬁguration given by

B = − sin y xˆ+sin 2x yˆ + Bg zˆ (4.1) v = − sin y xˆ+sin x yˆ (4.2)

with B the magnetic field (Bg a uniform guide field) and v the proton bulk velocity in normalized units described later. This configuration leads immediately to strong nonlinear couplings, producing cascade-like activity that might reasonably approximate the highest wavenumber decade of the inertial range. These couplings, which are dominantly local in wavenumber, in turn drive the dissipation range. The physics of the Orszag-Tang vortex has been previously studied using incompressible [OT79] and compressible [DP89] MHD simulations. Its robust production of nonlinear activity is a motivation for its frequent use in validating numerical schemes (see e.g. [RPM07]). Simulating kinetic dissipation is difficult and computationally expensive due to the requirement of treating a wide range of length scales. By choosing a computational domain with approximately one decade of scale in the MHD range and another in the kinetic range, we can study the conversion of strongly driven MHD fluctuations into kinetic motions. A related earlier study of the Orszag Tang vortex compared global behavior of hybrid and Hall MHD simulations [MGML95], but included a mean in-plane magnetic field while not adequately resolving the proton inertial length. We use the hybrid code P3D in 2.5D (a parallel version of the code described in [BDS+01]), which models protons as individual particles and electrons as a fluid. This code has been used extensively to simulate magnetic reconnection([SDDB98,

69 SDRD99, SDRD01, MCSD09] and references therein). The code advances the following equations:

dx i = v (4.3) dt i dvi 1 0 = (E + vi × B) (4.4) dt H ( ) 0 ∂B J 0 = ∇ × (v × B) − H ∇ × × B (4.5) ∂t ( ) n m 0 − e 2 ∇2 B = 1 H B (4.6) (mi ) J E0 = B × v− (4.7) H n where J = ∇ × B is the current density, H ≡ c/(L0ωpi) is the normalized proton inertial length, me and mi are the electron and proton masses, xi and vi are the positions and velocities of the individual protons, and v is the proton bulk ﬂow speed.

1/2 Length is normalized to L0, velocity to V0 = B0/(4πmn0) , time to t0 = L0/V0, 2 and temperature to B0 /(4πn0). The average density is n0, and B0 is the root mean square in-plane magnetic field. The magnetic field B is determined from B0 using the multigrid method. The fields are extrapolated to the particle positions using a first-order weighting scheme, which is essentially linear interpolation[BL85]. This allows the smooth variation due to particle motion of the fields felt by the particle. A similar first-order weighting scheme is used to determine the fluid moments at the grid locations from the particles. The code assumes quasi-neutrality. The electron temperature is zero and is not updated. Hybrid simulations are ideally suited for exploring dissipation and proton heating in collisionless plasmas because they include a complete kinetic description of protons. Due to the finite temperature of the protons, kinetic Alfvénwaves are present in this set of hybrid simulations ([RDDS01, HCD+06] and references therein), as well as parallel proton bulk flows.

70 2 (a) 6

1 5

0 4

y 3 −1

2 −2

1 −3

−4 1 2 3 4 5 6 x

2 (b) 6

1 5

0 4

y −1 3

2 −2

1 −3

−4 1 2 3 4 5 6 x

Figure 4.1: Current density with magnetic ﬂux contours at t=1.96 in (a) ﬂuid and (b) hybrid simulations.

71 The simulation domain is a square box of side length 2 π × 2 π with 512 × 512 grid points. About 105 million protons are loaded with an initial Maxwellian distribution having a uniform temperature = 8, and H = 2π/25.6 and me = 0.04mi. No artiﬁcial dissipation is present other than grid scale dissipation. Choosing a

2 guide field Bg = 5 (total β = 2nT/B ≈ 0.62) reduces the system compressibility. Incompressibility is further promoted by adding perturbations to the background density n0 that enforce ∂(∇ · v)/∂t = 0 at t = 0. Simulations without the added perturbation show only small differences. While the present model is idealized and not intended to be interpreted as a representation of the solar wind, the parameter regime and geometry are roughly compatible with solar wind conditions. For example, the solar wind is usually viewed as a β ∼ 1 plasma, and solar wind fluctuations are frequently characterized as quasi- two dimensional[BWM96]. Furthermore solar wind turbulence is typically viewed as driven by large scale fluctuations or velocity shears that predominantly occur at scales much larger than the ion inertial scale[TM95]. Each of these characteristics is represented in a very approximate way in the present simulation model.

4.2 Results and Discussion The hybrid simulation results are compared to those of a compressible 2.5 D MHD version of the code F3D [SDSR04] with constant and uniform resistivity η = 0.0048, zero viscosity ν, and ratio of specific heats γ = 5/3. (We motivate values for η and ν later.) In both cases, the magnetic islands initially centered on the midplane (y = π) begin a clockwise rotation. The initial velocity profile shears the magnetic islands until t ∼ 2 as the islands approach and undergo a brief period of magnetic reconnection from t ∼ 2 − 4. After t ∼ 4, the system is dominated by strong turbulence. A comparison of out-of-plane current density Jz and magnetic field lines at t = 1.96 is shown in Fig. 4.1. The hybrid and MHD results show strong

72 similarities at large scales, but significant differences at small scales where kinetic effects become important. In the magnetic field power spectra at t = 5.69, shown in Fig. 4.2, the MHD and hybrid spectra are nearly identical at small k, showing a power-law roughly consistent with a Kolmogorov (−5/3), shown for reference. The two spectra diverge when the Hall scale is reached (denoted by a vertical dashed line), i.e., when the Alfvénwave becomes dispersive due to the Hall term in Ohm’s law[RDDS01] at 2 2 2 2 ≈ 2 k di (cs/cm) 1, where di is the ion inertial length, cs = Ti/mi is the sound speed, 2 2 2 and cm = cs + cA is the magnetosonic speed. This effective gyroradius scale corresponds to k ≈ 8.3. That the linear theory of Alfvénwaves should so accurately predict the spectral break is surprising, because the interactions are decidedly nonlinear. It should be noted that this Hall scale does not correspond to k di ≈ 1, which is typically used for the scale at which parallel propagating whistlers occur. A higher k is required for the dispersive kinetic Alfvénwave to become active because the electron velocities for this wave are slower than that for a whistler at the same k. The oblique whistler is part of the high frequency magnetosonic branch, and arises when kkdicA/cm ≥ 1 [RDDS01]. However, for the simulations in this study, dicA/cm < de, so electron inertia effects become important before whistler become active, meaning that there are no oblique whistlers in this study. It is instructive to examine the energy and dissipation budgets for the MHD

2 and hybrid simulations, where flow energy Ev = hρ|v |/2i, magnetic energy EB = 2 h|B| /2i, thermal energy Eth (total proton kinetic energy minus flow energy), and total energy Etot. For the duration of this chapter, h...i represents a spatial average over the entire simulation domain. Grid scale fluctuations in the hybrid data are smoothed using a standard local, weighted iterative averaging.

Figure 4.3(a) shows Ev,EB, their sum, ∆Eth (where ∆ means the change since t = 0) and Etot, as a function of time from the hybrid and MHD simulations,

73 100

10−1

−2 10 −5/3

10−3 E(k) 10−4

10−5

Hybrid 10−6 MHD

Hall Scale kd 10−7 1 10 100 K

Figure 4.2: Magnetic ﬁeld power spectra at t = 5.69 for the hybrid and MHD simulations. The Hall scale and MHD Kolmogorov dissipation scale (kd) are shown for reference. The Hall scale, where dispersive kinetic Alfv´enwaves arise, occurs when k di cs/cm ≥ 1, where cs is the sound speed and cm is the magnetosonic speed[RDDS01].

with EB and Etot shifted down by a constant for convenience. Note that Etot changes very little over the course of the hybrid run, demonstrating good numerical energy conservation. During the initial phase (t < 2), bulk flow energy is converted strongly into magnetic energy as field lines are stretched, but with little proton heating. The magnetic energy converts back to flow energy (with some heating) during the reconnection event (t ∼ 2 − 4). Until t ∼ 4, the energetics of the hybrid and MHD results are very similar. However, in the turbulent phase (t > 4), the hybrid and MHD codes show significant differences, and more dissipation occurs in the MHD case. Notably, the proton thermal energy increases monotonically during the turbulent phase in the hybrid simulation. In MHD, mean square gradients of v and B are proportional to the energy dissipation rate. Although the hybrid code lacks explicit dissipation coefficients in the ion momentum or magnetic induction equations, it is instructive to compare in h 2i h| · ∇× |2i Fig. 4.3(b) the out-of-plane enstrophy (mean square vorticity) ωv = zˆ ( v) h 2 i h| · and out-of-plane magnetic enstrophy (mean square current density) ωB = zˆ

74 (a) 1.2 Etot−24.35

1.0 EB+Ev−12.5

Hybrid 0.8 MHD

E −12.5 0.6 B E(t)

0.4 Ev

0.2

∆Eth 0.0 (b) 2 <ωB > 15

Hybrid MHD 10 Enstrophies 2 <ωB > 5 2 <ωv >

2 <ωv >

0 0 2 4 6 8 10 12 t

Figure 4.3: Hybrid and MHD comparison: (a) Magnetic energy EB, ﬂuid ﬂow energy Ev, their sum, the change in thermal energy ∆Eth, and to- 2 tal energy Etot vs. time. (b) Flow enstrophy < ωv > and magnetic 2 enstrophy < ωB > vs. time.

75 0.3 (a) 0.2 −∆EvB 0.1 ∆E B −∆Ee 0.0

−0.1 Energy

−0.2 DB −0.3 −0.4

0.4 (b) |Dtot| 0.3

(t) ∆E 0.2 th |D |

D B

|D | 0.1 v

0.0

8.3 (c)

8.2

T⊥

8.1 Temperature T|| 8.0

0 2 4 6 8 10 12 t

Figure 4.4: (a) ∆EB: change of EB in the hybrid run; ∆EvB: exchange between Ev and EB; ∆Ee: electron kinetic energy; DB: sum of these, total EB dissipated. (b) Dv and DB are cumulative dissipation through bulk ﬂow and magnetic channels, Dtot their sum, ∆Eth change in thermal energy. (c) Parallel and perpendicular proton temperatures vs. time.

76 ∇ × |2i h 2i ( B) = Jz in the hybrid and MHD simulations. At early time (t < 4), the hybrid and MHD enstrophies peak at about same time, but their magnitudes are very different, indicating that the length scales in the hybrid case are larger, probably due to finite Larmor radius effects. During the turbulent phase (t > 4), the enstrophies continue to be different but the magnetic enstrophies are surprisingly similar. This suggests that the kinetic dissipation may have some resemblance to a classical resistivity, which we revisit later. Note that the value of enstrophy in the hybrid case is necessarily sensitive to the averaging that defines the fluid scales. In order to understand the nature of the collisionless dissipation occurring during the hybrid simulations, consider the flow of magnetic energy in the system. Dotting the induction equation [Eq. (4.5)] with B, averaging over space, and integrating over time gives ∫ t 2 0 de 2 ∆EB(t) = − hv · (J × B)idt − h∆J (t)i − DB(t), (4.8) 0 2 where any “∆” refers to the change since t = 0. The v · (J × B) term, which will be denoted as ∆EvB, is the exchange in energy between bulk flow and the magnetic 2 field, and the de term is essentially the electron kinetic energy. Because the total energy conservation is very good in the hybrid simulations, the first three terms in Eq. 4.8 can be combined to yield the cumulative energy change in the magnetic channel, denoted as DB(t). Included in DB(t) are kinetic dissipative processes and grid scale diffusion. Because the first three terms of Eq. 4.8 can be calculated directly from the simulations, it is possible to determine DB(t). For t < 4, the energy terms in Eq. 4.8 are characterized by a reversible transfer of energy back and forth between the magnetic field and ion flow, as seen in

Fig. 4.4 (a). The magnetic energy EB first rises and then falls, and this change is closely matched by the ion flow/magnetic energy conversion term −∆EvB. This reversible oscillation is an Alfvénwave-like response of the system as magnetic islands are stretched and then release their tension. Starting around t ≈ 4 when the

77 system becomes turbulent, substantial magnetic dissipation occurs (DB decreases).

The magnetic energy which is dissipated comes both directly from EB and from

ﬂow energy which is converting to magnetic energy (EB decreases and (−∆EvB) increases). Of the dissipated magnetic energy DB, about half comes directly from

EB, and the other half from the ion flow/magnetic energy conversion. A similar analysis is performed for energy flow in the bulk flow channel. Dotting the MHD momentum equation with v and integrating over time and space gives ∫ t 0 ∆Ev(t) = hv · (J × B)idt − Dv(t), (4.9) 0 where Dv is the cumulative energy converted into heat from the flow channel through non-fluid effects and compression. Fig. 4.4(b) shows Dv (dashed line), DB (dot- dashed), Dtot = DB + Dv (solid), and ∆Eth (dotted). In the turbulent phase (t >

4), there is very little energy dissipated through the flow channel (Dv is relatively constant). Nearly all of the change in Dtot for t > 4 occurs due to DB, so the main source of dissipation in this system is through magnetic interactions. The small departure between Dtot and ∆Eth remains relatively constant during the turbulent phase and can be accounted for by the change in total energy Etot [see Fig. 4.3(a)]; this is only about 10 % of the total dissipated energy. We wish to emphasize that total energy is not an explicit constant in our equations of motion. Grid scale diffusion of magnetic energy would lead to a decrease in total energy. Therefore, the total energy is only conserved when grid scale diffusion and other numerical effects are kept at a minimum. Upon reflection of some basic physical arguments, it seems reasonable on physical grounds that little dissipation occurs through the flow channel. In the MHD regime (low wavenumber k), the dynamics of the hybrid simulation are at most weakly compressible, so the majority of energy is in oblique Alfvénwaves that are weakly damped[Gar99]. As energy cascades to smaller scales, from Fig. 4.2 it is clear

78 that the hybrid simulation reaches the Hall scale (eﬀective proton gyroradius shown in Fig. 4.2) and diverges from the MHD spectra before the MHD system reaches the

Kolmogorov dissipation scale (kd in Fig. 4.2). Thus, there is little dissipation in the hybrid simulation in the MHD scales where the proton flows are significant. Below the effective proton gyroradius, the ions decouple from the magnetic field and only weakly participate with the non-MHD waves in this region. Consequently, we would expect the Alfvénratio Ev/EB to go to zero in the kinetic regime as evidenced by the structure of kinetic Alfvén. A central result of this study is that the dissipated magnetic energy preferentially heats the protons perpendicular to the mean magnetic field, as shown in

Fig. 4.4(c). The perpendicular and parallel temperatures T⊥ and T|| are calculated relative to the guide field. The total kinetic energy of all the “particles” is added and the bulk flow energy is subtracted from it. This gives us the component of energy in the random fluctuations about the mean flow. We treat this as the thermal energy and calculate the temperature from it using the ideal gas law. In the turbulent phase

(t > 4), T⊥ increases monotonically, while T|| remains relatively steady. The relative anisotropy is small because the available magnetic free energy in the system (from

Bx and By) is small compared to the proton temperature, i.e. β⊥ 1. Preliminary simulations with the proton temperature reduced to 1.5 (β⊥ ∼ 3) have been performed, and initial results indicate a sizeable temperature anisotropy ( T⊥/T|| − 1) of around 0.303, as opposed to 0.0494 in the current study. The perpendicular heating occurs without any obvious connection to classical cyclotron resonances, as the latter generally are construed [ILH01] to involve gyroresonance with waves propagating parallel to a background ﬁeld (along the invariant direction in this study). Resonance can also involve wave frequencies near the cyclotron frequency; however, dominance in this simulation of incompressible modes and relatively low frequency kinetic Alfv´enwaves make the connection to standard cyclotron resonance [ILH01]

79 uncertain. We note that the perpendicular heating seen in this study is evidence that

2 the first adiabatic invariant, µ = (miv⊥)/ωc is not conserved in our simulation for at least some of the protons. As a preliminary test , we have stepped test particles through the static fields of the hybrid simulation and verified that often µ is not conserved when particles cross the simulation current sheets. To justify the use of static fields, we have verified that there is not significant power in the magnetic field at the cyclotron frequency. This can be seen explicitly in the Eulerian frequency spectrum shown in the next chapter. In fig. 4.5 we show two the paths of two particles and also their instantaneous magnetic moments stepped through the initial (at timet = 0.02) electric and magnetic fields and through turbulent electric and magnetic fields (at t = 5.03). The trace is shown only for a small time window around the time of crossing the reconnection site in the turbulent case. For the particle stepped in initial fields, the magnetic moment fluctuates around the mean value of about 7. For the particle stepped in turbulent fields, we show the case of a particle that crossed a reconnection site. The crossing is marked by an asterisk in the particle trace and a vertical line in the magnetic moment plot. We clearly see a jump, from a mean value of about 0.5 to a mean value of about 1.5, in the particle’s magnetic moment as the particle crosses the reconnection site. The in-plane electric fields and magnetic fields change sign across the current sheets. For simplicity, we consider only the changing magnetic fields. In order for µ to be conserved, the following parameter must be small [KT73]:

1 ∂B/∂t τc δB = ∼ , (4.10) ωc B τ B where τc is the cyclotron time and τ is the time over which the particle feels a change of δB. For reconnecting current sheets with a guide ﬁeld with small electron temperature, the current sheet width is comparable to the proton Larmor radius based on the thermal speed[KDW95, RDDS01]. Therefore τc/τ ∼ 1 and δB/B ≈

80 b)12 a) 6 10

5 8 4 6

y 3 4

2 Magnetic Moment

1 2 0 0 0 1 2 3 4 5 6 0.5 1.0 1.5 2.0 2.5 3.0 x t

4 d) c) 6 3 5

4 2

y 3

2 Magnetic Moment 1 1 0 0 0 1 2 3 4 5 6 0.5 1.0 1.5 2.0 2.5 3.0 x t

Figure 4.5: Trajectories of particles stepped through static electric and magnetic fields at two different times in the simulation. Left panels show the contour plots of out of plane current and the particle’s trajectory traced on top of it for the time window in which the magnetic moment is shown in the right panels. a),b) are the plots for the particle stepped in the electric and magnetic fields taken very early from the simulation. c),d) are the plots for the particle stepped in the turbulent fields taken from later in the simulation.

81 V V . J x B B

Ω DV i DB

Proton Thermal Energy

Figure 4.6: Flow of energy through the turbulent hybrid simulations. Bold arrows denote signiﬁcant energy conversion through a channel. Light dashed arrows denote little or no energy conversion through a channel. Ωi represents cyclotron damping, DB represents magnetic dissipation, and Dv represents dissipation of proton bulk ﬂow.

2/5, giving = 0.4. There are a considerable number of protons with speeds greater than the thermal speed, which will have larger τc/τ and thus larger . As the guide field becomes larger, , will decrease. However, to maintain µ conservation for the large majority of particles it would be necessary to have an extremely large guide field in our hybrid simulations. The energy analysis described in this paper is summarized in Fig. 4.6, where the primary energy exchange is denoted with bold arrows and the dashed arrows denote very small or no energy exchange. In short, energy that was initially in bulk flows and magnetic fields is converted into proton heating during the turbulent phase. The magnetic energy dissipates directly, while the bulk flow energy is first converted into magnetic energy and then dissipated. If we assume that the classical functional forms for the dissipation rates h 2i h 2i η Jz and ν ωv are valid, we can compute effective transport coefficients ηeff and

82 0.008 (a) 0.006

eff 0.004 η 0.002 ηeff = 0.0043 0.000 0.02 (b) 0.01 νeff = 0.002

eff 0.00 ν −0.01 −0.02 6 8 10 12 t

Figure 4.7: Eﬀective (a) resistivity η and (b) viscosity ν vs. time.

D h 2i νeff from the hybrid simulations. Figure 4.7 shows ηeff = (∂ B/∂t)/ Jz and νeff = D h 2i (∂ v/∂t)/ ωv versus time. Surprisingly, the spatially averaged ηeff is fairly constant in time, as is often assumed in MHD models. The mean value is ηeff = 0.0043, corresponding to a magnetic Reynolds number of Seff = L/ηeff ≈ 1461, which motivated the value used in the MHD simulation. In classical turbulence theory, the length scale at which dissipation occurs λd is related to the Reynolds number and energy 4/3 containing scale L through Seff ∼ (L/λd) [Bat53]. For the hybrid simulation,

λd ∼ 0.0266, which is of the order of the electron skin depth, c/ωpe ∼ 0.049. The spatially averaged νeﬀ , on the other hand, shows oscillations much larger than the mean, calling into question the assumption of a non-zero viscosity assumed in many MHD models. The eﬀective resistivity being fairly constant does not imply that the dissipa-

2 tion is of the form ηJ . The MHD simulations performed with this ηeﬀ show more dissipation of B than the hybrid simulations, and cannot reproduce the preferential heating of T⊥. This study presents one of the ﬁrst self-consistent simulations to show anisotropic

83 heating of protons, and is an important first step to understanding collisionless dissipation in turbulence on the sun and in the solar wind. There are limitations to the scope of this study, however. Primarily, the simulations in this study are 2D, with no variation allowed along the direction of the mean magnetic field. As such, Landau damping and transit-time damping due to particle motion along this mean magnetic field are not present. It is possible, therefore, that parallel heating from these two effects could boost the parallel proton temperature and minimize or even reverse the anisotropy seen in these simulations.

84 Chapter 5

THE ORSZAG-TANG VORTEX: K-ω SPECTRA

In the previous chapter, we suggested that the current sheets and reconnection sites might play an important role in heating the plasma. We take that study further and examine the distribution of energy at various length and time scales. To do this, we look at the energy in the Fourier (wavenumber-frequency k − ω) space. If any wave modes dominate the energy budget, they should show up as dispersion relations on the k − ω diagram. The Eulerian frequency spectrum shows no enhanced power at the proton cyclotron frequency. Also the PDFs of magnetic ﬁeld increments show non-Gaussian peaks which have been related to intermittent current sheets and reconnection sites in the solar wind (e.g. [GMS+09, OMGS11]). To make the point about dominant wave modes appearing as dispersion relations on k − ω diagrams clear, we ﬁrst show k − ω spectra from a test run which was designed to have some wave modes. We set up a huge hybrid run, 20482 grid points,

2 128c/ωpi in size, about 840M particles, Te = 0., Ti = 2., strong in plane guide ﬁeld

Bg = 10.yˆ with an effective plasma beta of β = 0.04. This system extends about two decades into the inertial range and about two decades into the sub inertial range. The very low plasma beta makes it good for the cold plasma approximation to be valid. We force the magnetic field with a forcing function that has power at wavenumbers k = 2, 3 in the x, y plane (the effective set of equations evolved in this case was the one given in the next chapter 6). Because the code was run on a small cluster of about 300 nodes, the system was evolved only up to 2.25 Alfvéncrossing times in about two days. The k − ω

85 Figure 5.1: k −ω spectrum of magnetic fluctuations parallel to the mean magnetic field. We clearly see excess power in the Alfvénand slow modes.

86 Figure 5.2: k − ω spectrum of magnetic fluctuations perpendicular to the mean magnetic field. We clearly see excess power in the fast magnetosonic mode. At the frequencies above the cyclotron frequency, we see the electromagnetic Bernstein modes. spectra are shown in figs. 5.1 and 5.2. We see clear power at the driving wave numbers k = 2, 3. The only dynamic features recognizable in these figures are the familiar wave solutions for a cold MHD plasma. Fig 5.1 clearly shows excess of energy along the Alfvénmode and the slow mode. Fig 5.2 shows an excess of power following the fast magnetosonic mode. The fast magnetosonic mode turns into discrete Bernstein modes at frequencies higher than the ion cyclotron frequency ωc. It should be noted that the Bernstein modes are a solution of Vlasov equation and are not present in the fluid dispersion relation. It is a purely kinetic effect captured by the hybrid code. This also shows nicely that if there are any kinetic wave modes (like the KAWs), it should also show up on the k − ω diagram. This system was specifically designed to have waves and the waves showed

87 up in the k − ω spectra. Now we go ahead and analyze the data from our OTV run. If there are any wave modes which dominate the energy budget, they should show up on the k − ω diagram as dispersion relations.

5.1 k − ω spectra of OTV

Figure 5.3: Shaded contours of energy in Bx plotted in the k−ω space on logarithmic scales. Contour lines show levels of energy. Dispersion relations are over-plotted (see [RDDS01]), calculated using a two fluid dispersion relation, using the k parallel to the in-plane magnetic field and ⊥ to Bg. Important length and time scales are marked. The spectrum shows no significant power in wave modes.

This section discusses the spectra of the OTV in hybrid code in detail. Many mechanisms proposed earlier (e.g. [MGR82, TM97, HDC+08]) involve wave particle

88 interactions like cyclotron resonances or kinetic Alfvénwaves (KAWs). The classical cyclotron resonance condition requires a k|| to the mean magnetic field for the resonance to occur. Our system does not have a k|| and hence there should not be any cyclotron resonances. However there could be a kind of resonance relative to the ”local mean field” when the finite amplitude fluctuations are taken into account. To further support this argument, we look at the energy in the k − ω space.

Figure 5.1 shows the k − ω spectrum of energy in Bx. It shows no signs of any wave like activity. The spectrum is completely featureless. The k −ω spectrum of Bz (fig. 5.1) has slight hint of magnetosonic activity but the energy in that mode is about two orders of magnitude smaller than the available energy in the system. There is no sign of cyclotron wave activity. The Eulerian spectrum (fig. 6.3) has a slope of −5/3 which is often discussed in hydrodynamics as a consequence of random sweeping in strong turbulence (e.g. [Ten75]). This result being valid in plasma turbulence is not necessarily straightforward. It also clearly emphasizes the small amount of power in the magnetosonic mode and the absence of cyclotron resonance. The k − ω spectra show no obvious signatures of significant wave activity in this system. This study suggests that it is not necessary to have strong discrete frequency resonant wave particle interactions to dissipate energy in turbulent plasmas. However some form of nonlinear or non-resonant wave particle interactions are implied.

5.2 Probability Density Functions (PDFs) The absence of signiﬁcant wave modes in our system calls for another expla- nation of heating. One possibility is heating caused by energization of particles as they sample current sheets and reconnection sites [DMS04]. It would require frequent intermittent current sheets and reconnection sites for the cumulative heating to be signiﬁcant.

89 Figure 5.4: Shaded contours of energy in Bz in k−ω space. See Fig. 5.1 for details. This spectrum also does not show signiﬁcant power in wave modes. There is slight hint of magnetosonic activity but the power in that mode is about two orders of magnitude smaller than the maximum power available.

The normalized PDFs of velocity and magnetic fields (fig. 5.2) coincide with the unit Gaussian very closely, signaling the statistically homogeneous turbulent behavior at large scales (e.g. [MI71]). The PDFs of velocity field vector increments as well as magnetic field vector increments, (e.g. |∆B| = |B(s+∆s)−B(s)|), at points separated by ∆s show departures from Gaussianity for small ∆s. This is a clear sign of intermittent structures in the system [GCM+08, GMS+09]. These intermittent structures in our system happen to be the current sheets and the reconnection sites. The particles can be energized by the electric field variations in these sites [DMS04].

90 ωc

10−2

−4 −5/3

) 10 E (ω)

ω B E ⊥(ω)

E( B ω EB||( )

10−6

10−8 1 10 100 ω

Figure 5.5: The Eulerian frequency spectrum emphasizes the point again that the only recognizable wave mode is the magnetosonic mode. Also it clearly shows that there is no power at the cyclotron frequency to drive the perpendicular heating discussed in [PSCM09]. Note the −5/3 slope of the spectrum too which is a property of the hydrodynamic spectrum.

We are studying correlations between various quantities to further investigate this conjecture. Another preliminary study of stepping particles through a time snapshot of electric and magnetic ﬁelds from the hybrid code has been done. The particles that pass through a current sheet or a reconnection site get a jump in the magnetic moment. This increases the energy of the particle in the perpendicular direction. This study is being further pursued to step particles through the time evolving electric and magnetic ﬁelds outputted from the hybrid code. Another possible mechanism involved in the heating could be the second order Fermi acceleration

91 of the protons.

Figure 5.6: The probability density functions (PDFs) of magnetic ﬁeld and velocity follow the Gaussians very nicely. The increments show tails, however, indicating strong intermittent structures.

5.3 Conclusions This highly nonlinear system shows no obvious signs of signiﬁcant wave activity but still dissipates energy and heats the ions preferentially. This suggests that we do not need to have strong wave modes and resonant wave particle interactions to heat ions in a collisionless plasma. One possible mechanism for heating is through energization of particles by the electric ﬁeld variations in the intermittent reconnection sites and current sheets. Presently we are examining the correlation of particle energization with proximity to current sheets and reconnection sites to

92 further investigate this conjecture. Another possibility is the stochastic second order Fermi heating of particles which will also be investigated in near future.

93 Chapter 6

QUASI STEADY STATE TURBULENCE: TIME INDEPENDENT DRIVING

6.1 Introduction Strong turbulence, a difficult problem in hydrodynamic and magnetohydrodynamics (MHD) regimes, is an even more challenging subject in a kinetic (or low collisionality) plasma. In all turbulent systems the problem involves complex nonlinear interactions and a potentially very large number of fluctuation degrees of freedom. Plasma turbulence additionally involves wave-particle interactions that are responsible for crucial effects such as plasma dissipation. In addition, plasmas support new types of normal modes, including waves not found in fluid models. Substantial effort has been made to arrive at useful descriptions of plasma dynamics in terms of wave modes, postulating that turbulence might be described, in a leading order fashion, as an ensemble of waves. Under this assumption, coherent waves interact non linearly transmitting energy to small scales. This kind of “weak turbulence” approach has been employed in both high frequency and low frequency regimes [Tsy72, MRZ95], as well as in the case of low frequency magnetofluid (MHD) cases [GNNP00, Cha05, Cha08a]. Properly speaking normal mode analysis emerges from small amplitude perturbations about a known state, and for the collisionless case linear analysis about a Vlasov equilibrium has gained favor as a method to compute damping rates of specific wave modes [Bar79]. Linear damping rates are then integrated into an analysis of turbulence as a mechanism for terminating a

94 cascade (e.g., [LSNW99, GB04, GS95]). While there is considerable variation in the details of implementation of these approaches in the cited works and others, a typical common feature is the assumption that the normal mode analysis applies as first approximation. While the accuracy of this procedure will vary according to circumstances, the present chapter focuses on examination of the applicability of the linear wave description for a simple but relevant model of a kinetic plasma. In particular we examine the space-time (wavenumber-frequency) structure of a two dimensional hybrid plasma, in a geometry that occupies an important role in the anisotropic cascade favored by MHD turbulence in the presence of a moderate to strong applied magnetic field. The central role of two dimensional(2D) turbulence in various systems has been studied experimentally [RR71, ZMT79] and in simulations [SMM83, OPM94]. In brief, low frequency magnetofluid turbulence shows a preference for spectral transfer to wave vectors perpendicular or nearly-perpendicular to a mean magnetic field. Theoretical models that build in this spectral anisotropy include Reduced MHD and related models [Str76, Mon82, GS95]. The core driver of the anisotropic low frequency cascade is population of 2D Fourier modes, which are non propagating and do not experience the global Alfvénicdecorrelation that weakens transfer. In- deed studies have shown that the non propagating 2D turbulence component must be present to sustain strong turbulence, as seen, e.g., in coronal heating models driven by waves [DMMO01]. Furthermore there is evidence that plasmas with even moderately strong guide fields exhibit substantial anisotropy [OPM94]. An example is the solar wind in which simple parametrization are in accord with observations when as much as 80% of the energy is placed in 2D degrees of freedom [BWM96]. The present study employs a kinetic hybrid simulation in a 2.5D geometry to study the space-time structure of driven turbulence. The space time structure as revealed by a “k-ω” (wavenumber frequency) analysis will reveal any evidence,

95 if present, of excited normal modes and point spectral features. This approach was first used by the authors to study the space-time structure of the Orszag-Tang vortex [PSS+10] and is complementary to a recent study [DM09] of driven three dimensional (3D) MHD turbulence with varying mean field strength. In the latter case the frequency spectra were found to be dominated by broad band signals with little evidence of point spectral features at all values of applied magnetic field. In contrast the present model begins with 2D and does not follow the development of anisotropy (for a complementary case see [VCV10]). By including kinetic effects and a spectral range that spans the ion inertial scale, we can examine the relative roles of purely nonlinear continuous spectra and point spectral features that arise from wave dispersion relations.

6.2 Simulation details We study kinetic turbulence starting with a Maxwellian plasma, driven in time with large scale magnetic vortices. We use the hybrid code P3D-Hybrid in 2.5D (a parallel version of the code described in [BDS+01]), which models protons as individual particles and electrons as a ﬂuid. This code has been used extensively to simulate magnetic reconnection ([SDDB98, SDRD99, SDRD01, MCSD09] and references therein). The code advances the following equations:

dx i = v dt i dvi 1 1 = (−ue × B + vi × B) − ∇Pe ( dt) H ( ) ( ) n ∂ P v P γ − 1 { e −∇ · e e ∇2 γ−1 = γ−1 + γ−1 χ (Te) ∂t n n } n ν − e |∇2B|2 n0 ( ) 0 ∂B J 0 1 Ji 1 = −H ∇ × × B − × B − ∇Pe ∂t n H n n

96 1 νe 2 2 − ∇ ∇ B + H αFb H n0 0 J 0 Ji 1 E = × B − × B − ∇(Pe) (n n ) n m 1 0 − e ∇2 ∇ × B = 1 2 B; J = B (6.1) mi H where γ = 5/3 is the ratio of speciﬁc heats, J = ∇ × B is the current density, Ji is ion ﬂux, Pe electron pressure,Te = Pe/n is the electron temperature,

H ≡ c/(L0ωpi) is the normalized proton inertial length, me and mi are the electron and proton masses, xi and vi are the positions and velocities of the individual protons. Magnetic ﬁeld is normalized to some b0 and density to some n0. Length 1/2 is normalized to L0, velocity to V0 = b0/(4πmn0) , time to t0 = L0/V0, and 2 temperature to b0/(4πn0). νe is the electron viscosity and χ is the thermal diﬀusivity.

Fb is the forcing term for the magnetic field in the plane of the simulation. α is a dimensionless constant to control the strength of driving. The magnetic field B is determined from B0 using the multigrid method. The fields are extrapolated to the particle positions using a first-order weighting scheme, which is essentially linear interpolation [BL85]. This allows the smooth variation due to particle motion of the fields felt by the particle. A similar first-order weighting scheme is used to determine the fluid moments at the grid locations from the particles. The code assumes quasi- neutrality. The νe term transfers energy from the magnetic field into the electron thermal energy.

6.3 Results The numerical results are for system size 2π × 2π and 256 × 256 grid points. There are about 52.4 × 106 protons having an initial Maxwellian distribution and a uniform temperature of Ti = 20. The initial electron temperature is Te = 2.0. The 4 Hall parameter H = 2π/12.8 and the electron mass me = 0.04mi.A ∇ equivalent −6 electron viscosity with νe = 3 × 10 is present, which damps magnetic ﬂuctuations

97 Figure 6.1: Plot of magnetic energy. The energy in the lowest wave mode (the largest scales) keeps growing monotonically throughout the simulation, indicating an inverse cascade of energy into the largest scales. The energy in higher wave modes achieves an approximate steady state after t ∼ 30. At times later than t = 70 slower relaxation processes start to dominate the time evolution and the steady state is lost. at electron ﬂuid scales. The thermal diﬀusivity is χ = 5 × 10−2, which regulates a

2 Fick’s law electron heat flux. A guide field of Bz = 8.0 (β = 2nT/B = 0.625) lies out of the plane and reduces the compressibility of the system. The driving function was chosen to have non zero wave vectors (kx, ky) with 2 ≤ |k| ≤ 3 and no time dependence. These associated amplitudes were assigned with a flat modal spectrum

2 having amplitudes such that < Fb >= 0.5 (<> denotes the spatial averaging) and random phases. The strength of the driving was controlled by setting the parameter α = 0.05. The system was evolved for ∼ 147 nonlinear times. The magnetic energy shown in Fig. 6.1 indicates that the system attains

98 an approximately steady state at about t = 30 which suﬃces to analyze the high frequency ﬂuctuations that are the emphasis of the present study. At later times, the slower relaxation processes start dominating the time evolution of the system and the steady state is lost. A time averaged omnidirectional spectrum in the wavenumber space (between time t = 40.84 and t = 147.25) is shown in Fig. 6.2. The spectrum shows steepening at the Hall scale ds = cs/ωci (cs is the sound speed and ωci is the ion cyclotron frequency) where Hall physics becomes important [RDDS01].

100 −5/3

10−2

−4 (k)

b 10 E

10−6

1/ds 10−8 1 10 100 k

Figure 6.2: Time averaged omnidirectional k-spectrum of the system. The energy containing scale is at 2 ≤ k ≤ 3. The spectrum shows a bend at k = 1/ds where ds = cs/ωci is the scale where Hall physics becomes important. −5/3 line is shown for reference purposes.

The Eulerian frequency spectrum (Fourier transform of the single point two time magnetic autocorrelation function), which is shown in Fig. 6.3, follows a broadband power law distribution with index not very diﬀerent from −5/3 [Ten75]. One notable feature is that the total Eulerian spectrum does not show prominent peaks at any frequency. Another missing feature to be emphasized is a discernible peak in power at or near the ion cyclotron frequency. This suggests the apparent absence of cyclotron resonance or any harmonics of it in the plasma dynamics. The smaller peaks seen at the higher frequencies are roughly consistent with ion Bernstein and

99 magnetosonic modes and have very small energy. We will elaborate on the space- time structure by resolving the Eulerian correlation into spectral contributions of individual wavenumbers. The k − ω (wavenumber-frequency) spectra were calculated during the time t = 40.84 to t = 53.91. The guide field being out of the plane of simulation, does not allow for any global parallel wave vectors. Hence we look for dispersion relations corresponding to propagation of waves perpendicular to the mean magnetic field. These could correspond to Alfvénicwaves propagating along the local magnetic field, local kinetic Alfvénicwaves having highly oblique wave vectors, local whistlers, magnetosonic modes propagating oblique or perpendicular to the mean magnetic field, cyclotron waves, or many possible types of global oscillations of magnetic flux tubes and magnetic islands. (One simple type of wave that is absent is the global Alfvénwave relative to the uniform vertical field, as there is no vertical wavevector component in this model.) In spite of the apparently many possible wave modes that this system can support, each of which could contribute to point spectral features, the k − ω spectra are practically featureless. Top figure in Fig. 6.4 shows the k − ω spectrum of the magnetic field in the y direction with wave vectors in the x direction.

The spectrum of Bx (not shown) is similar also. There are no recognizable wave modes present in the perpendicular components of the magnetic field. Most of the energy is in the driving wave numbers and some energy can be seen in k = 1 mode also. This happens through inverse cascade driven by back transfer of the magnetic potential [FMJ77] which is a familiar feature of 2D magnetofluid turbulence. Inverse transfer of energy to the longest wave length modes is also indicated by the growth of magnetic energy in the wave vector k = 1. Bottom figure in Fig. 6.4 shows the k−ω spectrum of the parallel component of the magnetic field Bz. We can see a very small amount of power in the some wave modes (indicated by the peaks following the dispersion relation curve) but these have energy which is about two orders of

100 magnitude smaller than the available free energy of the system.

6.4 Conclusions These numerical results provide a picture of the space-time structure of 2D kinetic turbulence in the regime spanning both magnetofluid scales and proton kinetic scales. Apart from the specialization to the two dimensional geometry, and the lack of representation of the electron kinetic scales, we have made very little assumption about the nature of the dynamical degrees of freedom that operate at these scales that are crucial in providing an interface between MHD and kinetic plasma dynamical regimes. In particular the hybrid simulations allow the presence of local Alfvénicfluctuations that propagate along local magnetic field direction as well as the cyclotron motions of the ions. The system should be able to support global Alfvénicoscillations of islands in addition to local couplings akin to Alfvén, Kinetic Alfvén,whistler, cyclotron and magnetosonic modes. Of these only the magnetosonic and ion Bernstein modes would have a classical dispersion relation, but a number of the other cases would contribute to discrete frequency oscillations that give rise to point spectral features on the Eulerian spectrum. With the exception of Bernstein and magnetosonic activity with at most a percent of available energy, none of these discrete frequency modes are observed in any prominent feature of the spectrum. This suggests that the kinetic plasma in 2D geometry in a range of scales spanning the ion inertial scale behaves essentially as a complex nonlinear system, when driven statically or at low frequencies. (We present the time dependent forcing cases in a later chapter.) The dynamics appears to be essentially broad band with little evidence of wave activity in the cascade. In spite of many kinetic degrees of freedom available, this feature is reminiscent of zero frequency hydrodynamic behavior, and is closely related to the similar result found recently in incompressible MHD with a varying strength guide field [DM09].

101 We conclude tentatively that in the limit of strong turbulence, 2.5D hybrid kinetic behavior is dominated by featureless “zero frequency” behavior. It is not clear to us how relevant to this case will be the descriptions of turbulence that rely on iteration about a leading order wave solution. It is diﬃcult to see how weak turbulence approaches can be made relevant to this plasma regime, or to a more complex higher dimensional plasma regimes in which the 2D dynamics forms the core turbulent solution. The quest to identify relevant dissipation mechanisms become more problematic when wave modes are not the dominant dynamical feature. First the vocabulary for describing channels for dissipation becomes less precise – many studies discuss dissipation in terms of which wave modes are involved [LSNW99, GB04]. Second, the familiar method of computing the rates of conversion of ﬂuid to internal energy based in linear Vlasov damping rates of wave modes seems likely to require revision when the core solutions are of the type described here. A more fundamen- tally nonlinear picture of dissipation may be required, either by extension of the uniform Vlasov approach to a nonlinear regime (e.g., [VCV10] or by examination of inhomogeneous nonlinear mechanisms perhaps based on particle energization at vortices [MVSH06] or at current sheets and reconnection sites[DMS04, SMS+10]. Further study of the space-time structure of the kinetic plasma will need to examine three dimensional cases as well as a wider range of plasma parameters.

102 Figure 6.3: Eulerian frequency spectrum of magnetic field. Solid line is the frequency spectrum for the total magnetic field, dashed line is the perpendicular component of the magnetic field and the dotted line is the frequency spectrum of the parallel component of the magnetic field (Bz). Cyclotron frequency and a −5/3 line ( dash-dotted) are shown for reference purposes. The only recognizable spectral peaks are roughly consistent with ion Bernstein and magnetosonic modes. The amount of energy in these modes is also many orders of magnitude smaller than the available free energy in the system.

103 Figure 6.4: Energy spectrum of y-component of magnetic fluctuations as function of frequency and of wave vector component perpendicular to the mean magnetic field. The dispersion relation for monochromatic magnetosonic modes propagating across the mean magnetic field is shown, as well as the various important length scales (1/di, 1/de, 1/ds) and time scale (ωc). The spectrum is featureless without any recognizable wave modes present. In a similar energy spectrum of the z component of the magnetic field, the only recognizable feature is likely some combination of magnetosonic and ion Bernstein modes and the power in those modes is orders of magnitude smaller than the free energy in the system.

104 Chapter 7

QUASI STEADY STATE TURBULENCE: TIME DEPENDENT DRIVING

7.1 Introduction In the previous chapter we studied the distribution of energy in space and time for a driven turbulent system. In this chapter we generalize the system studied in the previous chapter to have a time dependent forcing. We examine the dependence of the onset of turbulence as a function of the frequency of driving. This picture has qualitative similarities to the turbulence in the solar corona. In the case of solar corona, this driving frequency would correspond to the frequency of field line driving due to foot-point motions. Using 2.5D hybrid simulations, we drive the system magnetically and examine the evolution of various components of the energy (thermal, magnetic, etc.) as a function of driving frequency ωd. Because hybrid simulations have full kinetic ion physics including Kinetic AlfvénWaves, our simulations include self-consistent kinetic dissipation at ion scales. Building on previous hybrid simulation studies [PSS+10, PSB+10], we examine the distribution of energy in various spatial and temporal scales using wavenumber-frequency k − ω spectra. The above method, together with a standard intermittency analysis based on the Probability Distribution Functions (PDFs) of the increment of the magnetic field vector, provides information on the dissipative processes that generally occur in turbulent plasmas.

105 We ﬁnd that in order to excite turbulence and heat the plasma, the forcing time scale must be longer than the nonlinear time of the system. k−ω spectra for the turbulent case show only a very small amount of energy following the magnetosonic mode indicating the relatively negligible role of waves in heating the plasma. The PDFs of the magnetic ﬁeld increments show vanishing super-Gaussian tails with increasing driving frequency, implying that non-Gaussian intermittent events are playing an important role in the heating of the plasma.

7.2 Simulation details We study kinetic turbulence starting with a Maxwellian plasma, driven in time with large scale magnetic vortices. We use the hybrid code P3D-Hybrid in 2.5D (a parallel version of the code described in [SDRD01]), which models protons as individual particles and electrons as a ﬂuid. This code has been used extensively to simulate magnetic reconnection ([SDDB98, SDRD99, SDRD01, MCSD09] and references therein). The code advances the following equations:

1 νe 2 2 − ∇ ∇ B + H αFb cos(ωdt) H n0 0 J 0 Ji 1 E = × B − × B − ∇(Pe) (n n ) n m 1 0 − e ∇2 ∇ × B = 1 2 B; J = B (7.1) mi H

106 where γ = 5/3 is the ratio of speciﬁc heats, J = ∇×B is the current density, Ji is ion

flux, Pe electron pressure, Te = Pe/n is the electron temperature, H ≡ c/(L0ωpi) is the normalized proton inertial length, me and mi are the electron and proton masses, xi and vi are the positions and velocities of the individual protons. Magnetic field is normalized to a characteristic magnetic field b0 and density to a characteristic 1/2 density n0. Length is normalized to L0, velocity to V0 = b0/(4πmn0) , time to 2 t0 = L0/V0, and temperature to b0/(4πn0). νe is the electron viscosity and χ is the thermal diffusivity. Fb is the forcing term for the magnetic field in the plane of the simulation to which we assign a specific spatial structure (described in the next paragraph). α is a dimensionless constant to control the strength of driving. The

νe term transfers energy from the magnetic field into the electron thermal energy. The numerical results are for system size 2π × 2π and 256 × 256 grid points. There are about 52.4 × 106 protons having an initial Maxwellian distribution and a uniform temperature of Ti = 4.. The initial electron temperature is Te = 2.0. The 4 Hall parameter H = 2π/12.8 and the electron mass me = 0.04mi.A ∇ equivalent −6 electron viscosity with νe = 3 × 10 is present, which damps magnetic fluctuations at electron fluid scales. The thermal diffusivity is χ = 5 × 10−2, which regulates a

2 Fick’s law electron heat flux. A guide field of Bz = 4.0 (β = 2nT/B = 0.75) lies out of the plane and reduces the compressibility of the system. Fb was chosen to have non zero wave vectors (kx, ky) with 2 ≤ |k| ≤ 3. The associated amplitudes were h 2i h i assigned with a flat modal spectrum having amplitudes such that Fb = 0.5 ( ... denotes the spatial averaging) and random phases. The strength of the driving was controlled by setting the parameter α = 0.05. Various simulations were performed with values of ωd varying from ωd = 0.0 to ωd = 13.04. The system was evolved for t ∼ 147 Alfvéntimes.

107 7.3 Results The amount of turbulence generated has a strong dependence on the forcing frequency ωd. From the total magnetic energy Eb versus time shown in Figure 7.1 a), it is evident that Eb decreases with increasing ωd. For ωd > 0.4, in fact, the magnetic energy is practically zero. Fig. 7.1 b shows the time averaged (t ∼ 41 to t ∼ 73) omnidirectional magnetic ﬁeld spectra for the same runs. We see the same trend of decreasing input of energy into the magnetic ﬁeld in these spectra. For

ωd < 0.4, the spectra have the tendency to follow the Kolmogorov −5/3 behavior in the inertial range (k < 1/di). For ωd > 0.4 the spectral amplitude drops off, first at the lowest wavenumbers and then across the entire spectrum, leaving an enhancement near the driving wavenumbers. The frequency ωd = 0.4 seems to be the “critical frequency” below which turbulence can be generated. Note that in the computational units, the proton cyclotron frequency based on the guide field is

Ωci = 8; therefore this transition is not closely connected with that characteristic parameter. The effect of forcing frequency on turbulence generation can be illuminated further by examining a time period of relatively steady turbulence, denoted by the vertical lines in fig 7.1 a). The average magnetic energy < Eb > and thermal energy < Eth > during the time period 40 . t . 50 is plotted as a function of the driving frequency ωd. Each point in Figure 7.2 represents a separate simulation. The behavior observed in fig. 7.1 a) of decreasing magnetic energy input with increasing frequency is clearly evident. The average turbulent energy changes abruptly near the critical frequency of ωd = 0.4. For ωd < 0.4 the energy input saturates very quickly to its value at ωd = 0.0 and for ωd > 0.4 the energy input drops very quickly. The thermal energy shows similar behavior. For higher frequencies, the heating is reduced the level associated with the numerics (due to finite particle number). Although not shown, the fluid kinetic energy shows similar behavior. This analysis

108 suggests that the critical frequency ωd = 0.4 is related to an important dynamical timescale. The critical frequency can be understood in terms of the nonlinear (NL) time of the system. Nonlinear time for a given system is the time scale at which the nonlinear term in the equation of motion becomes important, which can be estimated √ ∼ h|∇ × |2i by τ 1/ v z . We calculate the average nonlinear time for each run within the time window of consideration. When the driving frequencies are normalized to

nl the NL time of the corresponding run (denoted ωd ), the critical frequency is near nl ∼ nl ωd 1. The numerical values of ωd for several simulations straddling the critical point are shown in fig. 7.2 as bold text adjacent to their respective data point. It is evident, therefore, that if the time scale of forcing is shorter than the nonlinear time scale, there is insufficient time for the dynamical nonlinear terms to cascade the driving energy to smaller scales. No turbulence is generated and the plasma does not heat because there is little energy at the smallest length scales where dissipation occurs. Another potentially important time scale to consider is the intrinsic nonlinear f time of the forcing τNL, the nonlinear time scale associated with the structure of the forcing. It should be noted that the intrinsic nonlinear time of the forcing is different from the driving time (1/ωd), the time scale arising because of the driving frequency. f Starting from a zero in-plane magnetic field, τNL is the nonlinear time of an eddy f created by a half-cycle of forcing. This means that τNL has to be smaller than

1/ωd so that the forcing eddies can cascade energy and create turbulence before f the direction of stirring changes, τNLωd < 1. The typical velocity of the eddies in forcing can be estimated to be the Alfvénvelocity based on the magnetic field generated by forcing in one half cycle, V ∼ αFb/(2ωd). This velocity yields the f nonlinear time τ ∼ 2λdωd/αFb where λd is the size of a typical eddy in forcing. NL √ The condition for efficient transfer of energy therefore becomes ωd < αFb/(2λd).

109 With the parameters of our system, the condition is ωd < 0.25. Hence for the speciﬁc forcing we have, driving frequencies of smaller than 0.25 give the forcing enough time to introduce energy to the system and transfer it down to smaller scales before the direction of forcing reverses. Consistent with this picture, the amount of turbulent magnetic ﬁeld energy in Figure 7.2 sharply decreases for frequencies greater than 0.25.

An important question regards the primary dissipation mechanisms which convert the turbulent energy from driving into heat. We ﬁrst consider wave damping as a possible source and show that this possibility is unlikely because the energy present in wave modes is quite small in the simulations. Examining ω − k diagrams [PSS+10, SLR+09, PSB+10] are quite revealing because energy present in wave modes should lie on dispersion curves ω = ω(k), as has been seen in fully kinetic PIC simulations [SLR+09]. Fully developed turbulence occurs as long as the driving frequency is less than the nonlinear time, so we focus on the ω = 0 case. The ω−k spectra for the in-plane

+ magnetic field (Bx,By) has been published previously for both decaying [PSS 10] and the driven case here [PSB+10]. These spectra do not show any recognizable wave modes. Fig 7.3 a) shows the spectrum for the out of plane component of the magnetic field. We see slight activity along the magnetosonic branch but the amount of energy in these peaks observed is orders of magnitude smaller than the free energy in the system (which is the same order of magnitude as the heating, see fig 7.2). This suggests that wave-particle resonances might not be the dominant mechanism in heating the plasma in this case. To contrast the relative energy budget of waves in different driving frequency regimes, we show the k − ω diagram of out of plane magnetic field for the highest frequency, ωd = 13.04, case in fig 7.3 b). We can clearly see that most of the energy is

110 in the electromagnetic Bernstein modes as indicated by the over plotted dispersion relations calculated from linear Vlasov theory [Gar05]. It should be noted that this case the plasma gained negligible amounts (∼ 4 orders of magnitude smaller than the zero frequency case) of energy and the plasma did not heat (other than the numerical heating). Also note that the spectrum in fig 7.3 b) lacks the strong continuous enhancements at low ω and k associated with the turbulence cascade, which is seen clearly in Fig 3a. The absence of strong linear wave modes in the turbulent regime suggests that it is unlikely that conventional wave particle interactions are involved in heating the plasma. A possible alternative mechanism is the energization of plasma by intermittent structures like current sheets and reconnection sites as suggested by various observational as well as theoretical studies [BN68, Par72, Par88, VH01, OMGS11]. Examination of the probability distribution functions (PDFs) of the magnetic field increments has revealed that the statistics of the solar wind magnetic discontinuities are strongly non-Gaussian, resembling the intermittent character of simulations of MHD turbulence [GCM+08, GMS+09]. In fig. 7.3 c) we plot the PDFs of the normalized vector increments of magnetic field, defined by:

∆B I = (7.2) σ where σ = h∆B2i1/2 is the standard deviation and ∆B = B(x + δs) − B(x) is the magnetic ﬁeld vector increments. Several driving frequencies are shown and an increment values δs = 10∆x ∼ 0.5λc is used where λc is the correlation length. For low frequency forcing, super-Gaussian tails are present. As the driving frequency ωd is increased these tails gradually disappear and the PDFs approach the unit Gaus- sian. The strong non-Gaussian tails at low frequency are associated with current sheets formed during the turbulent cascade. These current sheets disappear with higher forcing frequency which has been veriﬁed by examination of the out-of-plane

111 current evolution in these simulations. Consistent with plasma heating in current sheets and reconnection sites, the increase in thermal energy is much reduced for higher ωd in Figure 7.2. The correlation between the super-Gaussian tails and the heating of plasma is consistent with earlier studies suggesting a relation between heating and non-Gaussian features in the solar wind [OMGS11]. The presence of intermittent magnetic field structures, i.e., current sheets, in the case of slow forcing of the plasma is strongly reminiscent of Parker’s picture of coronal heating. In Parker’s scenario, very slow foot-point motions (with timescales much longer than the system Alfvéntime) generate an entangled magnetic field structure which quickly relaxes to a quasi-static equilibrium state characterized by a web of current sheets. Over longer timescales, dissipation and reconnection in these current sheets heat the solar corona. The key point is that the foot-point motions must be slow enough to allow the system to relax through nonlinear couplings, create the current sheets, and then dissipate them. Fast foot-point motions, reminiscent of the large ωd simulations, continuously disrupt these current sheets, preventing the heating process from occurring. The problem of finding the nonlinear time associated with the corona is a very difficult one because of the lack of precise measurements of velocity fluctuations perpendicular to the mean magnetic field in the corona. For a simple estimate of numbers, we can use the Alfvéncrossing time of a typical flux tube as an order of magnitude estimate of the nonlinear time. Using the typical numbers for magnetic field (∼ 100G) and density (∼ 1010cm−3) we can estimate the Alfvénspeed to be ∼ 2 × 108cm/s. Taking the typical size of a flux tube to be ∼ 109cm the crossing time is 5 sec. The typical time scale of the foot-point motions can be estimated from the power spectrum of the foot point velocities (e.g. from Fig. 2 of Matsumoto & Shibata [MS10]). Most of the power lies in foot point motions slower than 3 mins, giving us an estimate of typical foot point motion time scale of 180 sec. This is

112 consistent with our ﬁndings given that most of the power is in time scales longer than 180 sec. It should be kept in mind though that a more robust comparison of time scales would require a proper calculation of nonlinear time in the corona which requires more advanced observational results than are available at present.

7.4 Conclusions This study demonstrates that the excitation of turbulence and heating of plasma depends critically on the frequency of driving. The time scale of the driving has to be longer than the nonlinear time of the system. For driving time scales shorter than the nonlinear time, there is insufficient time for the forcing to excite turbulence; the energy input plummets with increasing driving frequency. For longer driving time scales, the energy input quickly saturates to the zero frequency value. Another potentially important time scale is the intrinsic nonlinear time of the forcing f τNL which has to be smaller than the driving time scale 1/ωd in order for the forcing to be efficient. For the turbulent case, the k−ω spectra of magnetic field fluctuations do not show any significant energy in wave modes. The presence of super-Gaussian tails of PDFs of magnetic field vector increments on in the turbulent case indicate a role of current sheets and reconnection sites in heating the plasma.

113 a) 0.4 <∆ E> 0.3

b 0.2 E

0.1

0.0 20 40 60 t b) 0 10 −5/3 10−2

10−4

−6 (k)

b 10

E ωd=0.00 −8 kd =1 ω 10 i d=0.102

ωd=0.407 −10 ω 10 d=0.815 ω =13.04 10−12 d 1 10 100 k

Figure 7.1: a) Magnetic energy versus time for several forcing frequencies ωd. The two vertical lines denote the time period over which the total energy change is plotted in Figure 7.2. b) omnidirectional spectra for the same runs. −5/3 line and the ion inertial length scale kdi = 1 have been shown for reference purposes.

114 Figure 7.2: Time averaged magnetic and thermal energies in the time window t ∼ 40 to t ∼ 50 as a function of driving frequency. Near the critical frequency of 0.4, points are labeled with the corresponding values of the driving frequency in NL time units based on individual runs. The nl ∼ critical frequency corresponds to ωd 1 (vertical arrow).

115 Figure 7.3: k − ω spectrum of the out-of-plane magnetic field Bz fluctuations for (a) ωd = 0.0 and (b) ωd = 13.4. Physical scales and cyclotron frequency denoted with solid white lines. Magnetosonic mode denoted with dashed white line. In (b), the first few Bernstein modes calculated from a linear Vlasov code [Gar05] shown as solid curves. (c) PDFs of magnetic field vector increments for various driving frequencies and vector increments of δs = 10∆x.

116 Chapter 8

CONCLUSIONS AND FUTURE DIRECTIONS

The questions: ”What are the dissipative processes in a collisionless turbulent plasma like the solar wind? What processes are dominant (under what conditions)??” are of central importance to our understanding of the inner heliosphere. The solar corona and the solar wind have a huge separation of important physical length scales. On one hand, we should be able to simulate the whole system in its entirety (at least 1AU in its radius) and on the other hand, we should be able to resolve at least the ion inertial length scale which is at least 6 orders of magnitude smaller than the system size (∼ 100Km at 1AU). This great disparity in scales makes it hard to simulate the entire system even with fluid codes. Many mechanisms to heat this turbulent plasma have been proposed. Wave particle interactions (like cyclotron resonance, Landau damping etc.) as well as current sheets and reconnection sites have been suggested to be of central importance in heating the solar plasmas. Models have been built to study the dissipation coefficients using empirical as well as statistical methods. Much observational as well as theoretical effort is still required to understand the relative importance of various dissipative processes. Also the fluid models are inapplicable at the kinetic scales as they use some kind of viscosity and/or resistivity. The solar plasmas in the inner heliosphere (like the upper solar corona and the solar wind) are very weakly collisional and hence can not be described at the kinetic scales using simple viscosity or resistivity. Self consistent kinetic studies are required to understand the dissipative processes.

117 The need to go to kinetic scales requires use of more expensive tools like PIC codes. This means that the range that we can simulate is at maximum only a few thousand kilometers in each direction. Even though this is a very limited size as compared to the whole system of interest, these studies can still prove to be invaluable by studying the scales on both sides of the spectral breaks, e.g. hybrid codes can be used to simulate a decade above and below the spectral break at the ion inertial scales. Most of the kinetic studies done so far (other than this thesis) have concentrated on studying the importance of wave mechanisms in collisionless plasmas. We study the quasi-2D limit of turbulence in the solar plasmas using a hybrid code. Energy budget analysis of decaying turbulence in the Orszag-Tang vortex showed that the fluctuations involved in the kinetic dissipative processes are predominantly magnetic fluctuations. The central finding of this thesis is that in the quasi-2D limit of strong turbulence, there is very little energy in the linear wave modes. This suggests that in this limit, classical wave particle interactions can not be the dominant dissipation mechanism. The PDFs of the magnetic field increments suggest that the non-Gaussian intermittent current sheets and reconnection sites are connected with heating of the plasma. We also find that the onset of turbulence in a simple time dependent driven system has a critical dependence on the relative size of driving time scale and the nonlinear times involved (the nonlinear time of the system as well as the intrinsic nonlinear time of the forcing function). The driving time scale has to be longer than the nonlinear time scales mentioned above for the forcing to be efficient, generate turbulence and heat the plasma. The studies presented in this thesis are limited in scope of course. The model used, the hybrid code, has kinetic physics only up to the ion inertial length scales. The physics below ion inertial scales is represented by a fluid model. A complete

118 kinetic study like PIC/Vlasov is required to address the problem self consistently. Also the studies were done in the quasi-2D limit. Ideally 3D simulations are required to incorporate all the possible physical effects to be self consistently present. With the present day computational power, reasonably size 3D hybrid models are still very expensive. Below we discuss a few directions that can/will be taken in order to advance our understanding of kinetic dissipative processes. One aspect that has not been addressed in the literature yet is a comparison of relative strengths of the wave and non-wave processes in heating the plasma. Ideally a 3D kinetic study is required to do so. A simple compromise solution to this problem could be as follows: Two runs with exactly the same physical parameters but with different guide field and initial magnetic configuration would be performed. One run with an in plane guide field and a spectrum of linear waves (like Alfvénwaves or magnetosonic waves) and another run with the same physical parameters but the guide field out of the plane of simulation and a highly nonlinear initial condition. As suggested by the existing studies including the work in this thesis, in the former case, the dominant dissipative mechanism would be wave-particle interactions and in the latter, it would be the low frequency current sheets and reconnection sites. By comparing the heating rates of these two runs, we can get an idea of the relative efficiency of wave vs. non-wave mechanisms. Once we have a good understanding of the relative importance of wave vs. non-wave processes, we can design the 3D runs judiciously based on the knowledge we get from the above studies. The 3D studies even though more expensive, when properly designed, would be invaluable to extend our understanding of more realistic systems. Another important aspect for future study is to look at the effects of expansion in the solar wind. The expanding box model [GVM93, RVED05] is a very

119 promising model to simulate the expansion eﬀects in the solar wind without having to simulate the whole system. Implementing this in the hybrid code would give us a better understanding of how the dynamics of the plasma change because of the expansion of the solar wind. A few kinetic studies have been done with expanding box [HTMG03, HVT+05, CB10] but a lot more needs to be done including a study of quasi-2D systems in expanding box. A few diagnostics, which were not performed in this thesis, would also be of great help in understanding the dissipative processes. Here are few ideas to be looked at:

1. If the runs develop strong local temperature anisotropies (T⊥/Tk), then we can compute the distribution of temperature anisotropy with respect to the parallel

+ plasma beta βk like Bale et. al. [BKH 09]. This can help in understanding the role of instabilities in the simulations.

2. Heating rates can be computed and then compared to existing models like Chandran’s stochastic heating model [CLR+10, Cha10]. A good agreement would validate the usability of appropriate models. This could be very helpful in designing more realistic dissipation functions for the ﬂuid models.

3. Test/tagged particles will be implemented in the hybrid/PIC code to follow the self consistent dynamics of the particle through the simulation. This would help us track the dynamics of most energetic particles and hence help in pin- pointing the most common places where acceleration happens. This has uses in studying solar energetic particles also.

Until the computers get powerful enough to handle huge global kinetic simulations, or our mathematics advances enough to deal with nonlinear partial diﬀerential equations analytically (Oh how I wish it were possible!), we will have to keep ﬁnd- ing clever compromises to increase our understanding of turbulent plasmas and the

120 universe around us. The road to reality is a tough one to travel but we have to keep pushing forward and keep exploring! That is the only way to unravel the mysteries of mother nature.

121 Appendix A

TWO FLUID DISPERSION RELATION AND WAVES

The two ﬂuid dispersion relation can be derived starting from two ﬂuid equations:

∂n = −∇ · Ji ∂t ( ) ∂J J J i = −∇ · i i + J × B − T ∇n ∂t n ∂B0 = −∇ × E0 ∂t J J E0 = × B0 − i × B n n 0 − 2∇2 ∇ × B = (1 de )B; J = B (A.1)

where Ji = nvi is the ion current density, n is the number density, de = c/ωpi is the electrom inertial length. The length is normalized to the ion inertial length √ 2 −1 −1 di = c mi/4πne , time to t0 = Ωi = (eB/mic) and velocity to the Alfv´en √ velocity Va = B0/ 4πmin0, where B0 is some normalizing magnetic ﬁeld and n0 is some normalizing density. The equations can be linearize by assuming small ˜ perturbations about the mean e.g. B = B0 + B etc.

∂n˜ = −n ∇ · v˜ ∂t 0 ∂v˜ n = (∇ × B˜ ) × B − T ∇n˜ 0 ∂t 0 0 ∂ 1 − 2∇2 ˜ − ∇ × ∇ × ˜ × ∇ × × (1 de )B = [( B) B0] + (v˜ B0) (A.2) ∂t n0

122 ˜ ˜ i(k·x−ωt) Assuming plane wave solutions B = B0e etc. we can write:

ωn˜0 = n0k · v˜0 ˜ −ωn0v˜0 = (k × B0) × B0 − T0kn˜0 1 2 2 ˜ × × ˜ × − × × ω(1 + dek )B0 = k [(ik B0) B0] k (v˜0 B0) (A.3) n0

With B0 = B0zˆ and k = k sin θ, 0, k cos θ, these equations can be solved to get the following dispersion relation (e.g. [RDDS01]): [ ( )] [ ( )] c2 1 k2d2 1 c2 c2 k2d2 c2 6 − mk i 4 mk s i 2 − s v 2 + 1 + v + 2 + 2 1 + v 2 2 = 0 cAk D D D cAk cAk D D cAk (A.4)

2 2 · 2 2 where v = ω/(kcAk) = ω/(kkcA), cAk = Bk/(4πn0), Bk = k B/k, D = 1 + k de, 2 2 2 2 2 2 2 2 2 cA = B /(4πn0), cs = (Te + Ti)/mi, cmk = cA/D + cs and cm = cA + cs. From this, the usual MHD waves can be derived using the approximations kde << 1 and kdi << 1. With these approximations the dispersion relation reduces to: 02 − 2 2 04 2 2 2 02 − 2 4 v (cm + cak)v + (cm + cs)cakv cscak = 0 (A.5) where v02 = ω2/k2. This can be easily solved to get the three MHD modes:   2 2 2  k cA cos θ shear Alfv´en√ c2 + c4 −4c2c2 cos2 θ ω = k2c2 c = m m s A Fast  f f √ 2  2 − 4 − 2 2 2  2 2 cm cm 4cscA cos θ k cs cs = 2 Slow

Two important modes that arise in the intermediate frequency (Ωci < ω <

Ωce) regime are the whistler and the kinetic Alfv´enwave (KAW). The whistler root can be shown to have the dispersion relation:

2 ω 2 ω cos θ 2 = cA 2 2 k Ωci(1 + k de)

123 and the kinetic Alfv´enwave arises in the regime where the phase velocity is

2 2 2 much smaller than the sound speed (1 << v << cs/cAk) and corresponds to the dispersion relation: ω2 c2 ≈ s 2 2 2 2 2 cAk di cos θ k cm The whistler propagates mainly almost parallel to the mean field and the KAW propagates mainly almost perpendicular to the mean magnetic field. A very good treatment of the two fluid dispersion relation and various wave modes can be found in papers by Stringer (low frequency waves in cold plasmas [Str63]), Formisano & Kennel ([FK69] waves in high β plasmas), Rogers et. al. ([RDDS01] a nice discussion of the parameter regime where we can have whistlers and KAWs) and the book by Cramer ([CJWS01] detailed discussion of kinetic Alfvénwaves). Both whistlers and kinetic Alfvénwaves have quadratic dispersion relation (ω ∝ k2) and can produce enhancement of electric field signal as compared to the magnetic field. This has triggered a debate in the space physics community. The electric field spectrum in the solar wind is seen to be enhanced as compared to the magnetic field spectrum in the kinetic regime. It has been suggested that this steepening could be because of kinetic Alfvénwaves (e.g. [BKM+05, HDC+08] etc.) and this argument has been countered that this could also be because of the whistler waves also and it the elctric field spectrum steepening is not a conclusive “proof” of the presence of KAWs in the solar wind (e.g. [EPBT09], C.W. Smith at SHINE workshop 2011, Chadi Salem at the SHINE workshop 2011).

124 Appendix B

HYBRID CODE RUN SEQUENCE

B.1 Algorithm The code executes the following steps during its run:

1. Initialization:

• Define the initial magnetic field configuration

• Load the particles with initial Maxwellian distribution

• Give them the initial velocity of the ﬂuid

• Redistribute the particles for best cache usage

• Write the dump data

2. Include modules that define the processor information, particle and field variables and timing routines and define some temporary variables.

3. Deﬁne Movie variables for byte as well as double precision data

4. Initialize MPI, set processor ranks and set processor numbers in x, y, z style

(mypex, mypey, mypez)

5. Based on the allowed run time and start time, set the stop time so that the code does not terminate paprt-way on machines with alloted time slots.

6. Read the arguments from the command line, if provided

125 7. Write the run speciﬁc parameters to stdout.

8. If driven system, read driving function from a ﬁle. Based on mypex, mypey each processor loads part of the driving function read from a ﬁle.

9. Read the input data from the dump ﬁles created by the initialization routine.

The dump ﬁles contain bx, by, bz, Pe and particle information (in that order)

10. Sort particles for better cache usage.

11. Print the statistics of loaded particles on all processors: minimum, maximum and average number of particles on processors

12. Enforce the divergence condition for B, ∇ · B = 0

13. Initialize b2 and pe2

14. Calculate the charge density due to particles using the simple linear interpolation described in ﬁg. 2.3.

15. Smooth out the ﬂuctuations in density because of the ﬁnite number of particles. The smoothing routine does weighted averaging over the nearest neighbors. Maybe diagram?

16. Add some uniform base density everywhere. This is to avoid zero density problems if a ”cavity” of very few or no particles is created.

17. Calculate the current density becuase of particles Ji and smooth it.

18. Calculate the current density because of electrons using total current and ion current density.

19. Calculate the pressure because of the particles

20. Open movie ﬁles for both byte as well as double precision data

126 21. Write the starting time slices of data to the ﬁles.

22. Initialize the dump data output timer

23. Initialize the time tracking variables for various subroutines. This way we have a good idea of which subroutine takes the most time.

24. Main loop:DO n=1,nsteps

(a) Calculate various energies like bulk ﬂow, magnetic, thermal, total etc.

(b) Step ﬁeld equations:

• DO i=1,substeps *call stephybrid(.false.)

• First half step

• calculate J using J = ∇ × b1

• calculate electric ﬁeld

• calculate b2’; advance b2’ half a step

• call helmholtz(b2,....): invert the helmholtz equation to calculate b2

• add the eﬀect of electron viscosity

• calculate the electron velocity

• calculate temporary variable pe2’=ve*pe1*rho**(1-gamma)

• advance the above temporary variable

• invert the above temporary variable to get pe

• Second half step

• Calculate J

• calculate electric ﬁeld

• calculate b1’; advance b1’ half a step

• save b1 to connect leapfrog grids

• call helmholtz(b1,....)

127 • add electron viscosity eﬀects

• calculate electron velocity

• save pe1 to connect leapfrog steps; calculate pe1’=ve*pe2*rho**(1- gamma)

• advance pe1’

• invert pe1’ to get pe1

• connect leapfrog grids b2=(b2+b1)/2 pe2=(pe2+pe1)/2

• ENDDO

(d) Calculate density, smooth it, add base density

(e) Calculate Ji, smooth it

(f) Calculate pressure because of the particles.

(g) Calculate the electric ﬁeld to advance particle velocities (J × B − ∇Pe)

(h) Step particle velocities

(i) Step particle positions by half step

(j) Calculate density, smooth it and add base density

(k) Calculate Ji and smooth it

(l) Calculate the pressure because of particles

(m) DO i=1,susbsteps; call stephybrid(.false.); ENDDO

(n) Calculate electron current

(o) t = t+dt

(p) write movie ﬁles

128 (q) check and print load balance statistics

(r) check if time limit reached, if yes, broadcast the message and exit run

(s) check if time to write dump data, do it

25. ENDDO MAIN LOOP ENDS

26. Write dump ﬁles

27. close movie ﬁles

28. call MPI Finalize(mpi err)

B.2 Normalization The code solves the equations in the standard plasma normalizations. We present the results in this thesis in ”turbulence normalizations” described in chapter 4. In the table below we give the plasma normalizations as well as turbulence normalizations and then show the transformations to go from plasma to turbulence normalizations. We present the equations without the forcing term but generalizing these transformations to the forcing term is simple and the equations are described in chapters 6 and 7.

129 Variable Code Normalization Turbulence Normalization

n (Density) n0c n0t = n0c

B (Mag. Field) B0c B0t = B0c √ √ V (Velocity) V0c = B0c/ 4πn0c V0t = V0c = B0c/ 4πn0c

J (Current) J0c = B0c/L0c J0t = B0t/L0t 2 2 P (Pressure) P0c = B0c/4π P0t = P0c = B0c/4π 2 2 T (Temperature) T0c = B0c/4πn0c T0t = T0c = B0c/4πn0c

L (Length) L0c = di L0t = L0(s.t. Lbox = 2π)

t (Time) t0c = L0c/V0c t0t = L0t/V0t 3 3 ν (Viscosity) ν0c = V0L0/4π ν0t = V0tL0t/4π

χ(Heat Flux) χ0c = L0n0V0 χ0t = L0tn0tV0t

The set of equations in the code normalizations is (without the forcing term):

dxic = vic dtc dvic 1 = −uec × Bc − ∇cPec + vic × Bc ( dt)c ( n)c ( ){ } ∂ P v P γ − 1 ν ec −∇ · ec ec ∇2 − ec |∇2 |2 γ−1 = c γ−1 + γ−1 χc c (Tec) c Bc ∂tc nc ( nc n ) n0c ∂B0 J J 1 ν c ∇ × c × 0 − ic × − ∇ − ec ∇2∇2 = c Bc Bc cPec c c Bc ∂tc ( nc ) nc nc n0c m 0 − e ∇2 ∇ × Bc = 1 c Bc; Jc = c Bc (B.1) mi here we have used the subscript c to highlight the fact that these equations are in the code normalizations. Writing the code normalizations explicitly, we can write the equations as: t dx v 0c i = i V0c dt V0c t0c dvi ue × B L0cn0c 1 vi × B = − − ∇Pe − (V0c dt) V0cB0c (P0c n ) V0(cB0c ) γ−1 γ−1 − t0cn0c ∂ Pe −L0cn0c ∇ · veP e γ−1 γ 1 × γ−1 = γ−1 + n0c γ−1 P0c ∂t n V0cP0c n n

130 { } L2 L4 ν 0c ∇2 − 0c e |∇2 |2 χ (T e) 2 B χ0cT0c( ν0cB0c n0 ) 0 t0c ∂B n0c J 0 1 Ji L0cn0c 1 = L0c∇ × × B − × B − ∇P e B0c ∂t J0cB0c n V0cB0c n P0c n L4 n ν − 0c 0c e ∇2∇2B ( ν0cB0c n0 ) B0 m B J L − e 2 ∇2 0c ∇ × = 1 L0c ; = B (B.2) B0c mi b0c J0c B0c

And we can convert the above equations to ”turbulence normalization by:

t V t dx V v 0c 0t 0t i = 0t i V0c t0t V0t dt V0c V0t t0c V0t t0t dvi V0tB0t ue × B L0cn0c P0t L0tn0t 1 = − − ∇Pe V0c t0t V0t dt V0cB0c V0tB0t P0c L0tn0t P0t n × − V0tB0t vi B V0cB0c V0tB0t − − ( ) − − ( ) t nγ 1 P t nγ 1 ∂ P L nγ 1 V P L nγ 1 v P e 0c 0c 0t 0t 0t e = − 0c 0c 0t 0t 0t 0t ∇ · e P γ−1 P ∂t nγ−1 V P γ−1 V P nγ−1 0c t0tn0t 0t 0c 0c (L0tn0t ){0t 0t n γ − 1 L2 χ T L2 0c γ−1 0c 0t 0t 0t ∇2 + n0t γ−1 2 χ (T e) n0t n χ0cT0c L0}t χ0tT0t L4 ν B2 L4 ν − 0c 0t 0t 0t e |∇2 |2 2 4 2 B ν0cB0c L(0t ν0tB0t n0 0 t0c B0t t0t ∂B L0c n0c J0tB0t n0t J 0 = L0t∇ × × B B0c t0t B0t ∂t L0t J0cB0c n0t J0tB0t n ) V B 1 J L n P L n 1 − 0t 0t i × B − 0c 0c 0t 0t 0t ∇P e V0cB0c V0tB0t n P0c L0tn0t P0t n L4 n ν B L4 n ν − 0c 0c 0t 0t 0t 0t e ∇2∇2 4 B ( ν0cB0c L0tn0t ν0tB)0t n0 B B0 m L2 B 0t − e 0c 2 ∇2 = 1 2 L0t B0c B0t mi L0t B0c J J L B L 0t = 0c 0t 0t ∇ × B (B.3) J0c J0t B0c L0t B0t

131 This allows us to write (we use an explicit subscript t to emphasize that these equations are in turbulence normalization):

dxit = vit dtt dvit 1 1 = (−uet × Bt + vit × Bt) − ∇tPet ( dt)t H ( ) ( ){nt } ∂ P v P γ − 1 ν et = −∇ · et et + χ ∇2(T ) − et |∇2B |2 ∂t γ−1 t γ−1 nγ−1 t t et n0 t t t nt (nt ) t ∂B0 J 1 J 1 1 ν t ∇ × t × 0 − it × − ∇ − et ∇2∇2 = H t Bt Bt tPet t t Bt ∂tt ( nt ) H nt nt H n0t m 1 0 − e ∇2 ∇ × Bt = 1 2 t Bt; Jt = t Bt (B.4) mi H where H = L0c/L0t.

132 Appendix C

PERMISSION LETTERS

C.1 Permission Letter from Kazunari Shibata Kazunari Shibata Tue, Aug 9, 2011 at 1:37 PM To: Tulasi Parashar Dear Dr Tulasi Parashar

Yes I am happy to give you a permission to use the following figure in your thesis.

Sincerely, Kazunari Shibata

Tulasi Parashar-san wrote (2011/08/09 4:34): Dear Dr. Shibata,

I am a graduate student in the department of Physics and Astronomy at the University of Delaware, USA. I worked on hybrid simulations of the kinetic dissipative processes in collisionless plasmas. In the introduction to my thesis I would like to use a figure, from one of your papers, that describes the reconnection process in the solar coronal loops. It is the figure two

133 from the following paper:

K. Shibata. Evidence of magnetic reconnection in solar flares and a unified model of flares. Astrophysics and Space Science, 264(1):129144, 1999. and I use it as an example to illustrate one of the reconnection processes in the solar corona as Fig 1.4 on page 16 of my thesis. A copy of my thesis is available at: http://www.physics.udel.edu/~tulasi/tmp/Parashar_PhD_Thesis.pdf

To use figures from already published papers, we need permission from the authors to do so. Please let me know if it is fine with you.

Regards, Tulasi

C.2 Permission Letter from Steven Cranmer Steven R. Cranmer Tue, Aug 9, 2011 at 11:35 AM Reply-To: [email protected] To: Tulasi Parashar Cc: [email protected], "Steven R. Cranmer" Hi Tulasi,

I’m always glad to help out a fellow Bartolean!

A slightly more jazzy version of the figure in question can be found in a white paper that we submitted to the recent solar/heliophysics

134 decadal survey. See Figure 2 of: http://arxiv.org/abs/1011.2469

I am attaching a postscript version of this figure below. I think you can use it freely, since this white paper is not officially "published" anywhere.

Alternately, I would think it’s probably okay to use the figure from the Kohl et al. (2006) paper in your thesis. If the rule is just that the author’s permission is needed, you now have two of the four! :-)

I look forward to reading your thesis.

Steve

C.3 Permission Letter from Chi Wang Chi Wang Wed, Aug 10, 2011 at 2:41 AM Reply-To: Chi Wang To: Tulasi Parashar HI Tulasi,

It is fine with me.

--Chi Wang

From: Tulasi Parashar Sent: Tuesday, August 09, 2011 11:06 AM To: [email protected] Subject: Fwd: Permission to use figure

135 Dear Dr. Wang,

I am a graduate student in the department of Physics and Astronomy at the University of Delaware, USA. I worked on hybrid simulations of the kinetic dissipative processes in collisionless plasmas. In the introduction to my thesis I would like to use a figure, from one of your papers, that describes the temperature profile of the solar wind protons. It is adopted from the figure 1 from the following paper:

C. Wang and JD Richardson. Energy partition between solar wind protons and pickup ions in the distant heliosphere: A three-fluid approach. Journal of Geophysical Research, 106:2940129408, 2001. and I use it as an example to illustrate the solar wind heating problem as Fig 1.8 on page 27 of my thesis. A copy of my thesis is available at: http://www.physics.udel.edu/~tulasi/tmp/Parashar_PhD_Thesis.pdf

To use figures from already published papers, we need permission from the authors to do so. Please let me know if it is fine with you.

Regards, Tulasi -- Tulasi Nandan Parashar, 217 Sharp Laboratory,

136 Department of Physics & Astronomy, University of Delaware, Newark, DE 19716. Cell: +1-302-465-3670 http://www.flickr.com/photos/tulasinandan

C.4 Permission Letter from Jack Gosling Dear Dr. Gosling,

Thanks a lot for pointing out the mistake! I did take it from the GRL and thought I had referred to the GRL. I guess I mixed it up and cited the JGR instead, while citing it. I have fixed that mistake.

Regards, Tulasi - Hide quoted text -

On Mon, Aug 8, 2011 at 6:13 PM, Jack Gosling wrote:

Dear Tulasi,

I have just downloaded your thesis and started to browse through it. I discovered that the figure of mine that you are including in your thesis is not from the paper you mentioned in your original message. Rather, your Figure 1.7 is a reproduction of Figure 2 from our paper: J. T. Gosling, R. M. Skoug, D. J. McComas, and C. W. Smith, Magnetic disconnection from the Sun: Observations of a reconnection exhaust in the solar wind at the

137 heliospheric current sheet, Geophys. Res. Lett., 32, L05105, 2005. You have my permission to use that figure in your thesis.

Regards,

Jack Gosling

On 8/8/11 4:02 PM, Tulasi Parashar wrote: Thanks a lot for the quick reply Dr. Gosling!

Regards, Tulasi

On Mon, Aug 8, 2011 at 4:14 PM, Jack Gosling wrote:

Dear Tulasi,

You have my permission to use Figure 2 from our paper JT Gosling, RM Skoug, DJ McComas, and CW Smith. Direct evidence for magnetic reconnection in the solar wind near 1 AU. Journal of Geophysical Research, 110(A1):A01107, 2005.

Best regards,

Jack Gosling

138 On 8/8/11 1:50 PM, Tulasi Parashar wrote: Dear Dr. Gosling,

JT Gosling, RM Skoug, DJ McComas, and CW Smith. Direct evidence for magnetic reconnection in the solar wind near 1 AU. Journal of geophysical research, 110(A1):A01107, 2005. and I use it as an example to illustrate the observations of reconnection in the solar wind as Fig 1.7 on page 23 of my thesis. A copy of my thesis is available at: http://www.physics.udel.edu/~tulasi/tmp/Parashar_PhD_Thesis.pdf

To use figures from already published papers, we need permission from the authors to do so. Please let me know if it is fine with you.

Regards, Tulasi

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