TRANSITION FROM KINETIC TO MHD BEHAVIOR IN A COLLISIONLESS PLASMA

Tulasi N. Parashar, William H. Matthaeus, Michael A. Shay, and Minping Wan

Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE, USAReceived 2015 April 16; accepted 2015 August 20; published 2015 September 28

ABSTRACT

The study of kinetic effects in heliospheric plasmas requires representation of dynamics at sub-proton scales, but inmost cases the system is driven by magnetohydrodynamic (MHD) activity at larger scales. The latter requirementchallenges available computational resources, which raises the question of how large such a system must be toexhibit MHD traits at large scales while kinetic behavior is accurately represented at small scales. Here we studythis implied transition from kinetic to MHD-like behavior using particle-in-cell (PIC) simulations, initialized usingan Orszag–Tang Vortex. The PIC code treats protons, as well as electrons, kinetically, and we address the questionof interest by examining several different indicators of MHD-like behavior.

Key words: magnetohydrodynamics (MHD) – plasmas – solar wind – turbulence

1. INTRODUCTION

Many astrophysical plasmas are believed to be collisionlessand turbulent in nature. Prime examples are planetarymagnetospheres (e.g., Galeev 1975; Lysak & Carlson 1981;Saur et al. 2002; Borovsky & Funsten 2003; Retinòet al. 2007), solar corona (e.g., Crosby et al. 1993; Shi-mizu 1995; Dmitruk & Gomez 1997), solar wind (e.g.,Coleman 1968; Tu & Marsch 1995; Bruno & Carbone 2005;Marsch 2006), and galactic cooling flows (e.g., Sarazin 1986;Carilli & Taylor 2002; Quataert 2003; Fujita et al. 2004;Peterson & Fabian 2006). In the case of the terrestrialmagnetosphere, the principal energy source is the solar windflow, which buffets the system, creating a highly turbulentmagnetosheath (Retinò et al. 2007), and, when the configura-tion is correct, inducing magnetic reconnection and the transferof magnetic flux into the magnetosphere proper (Sonnerupet al. 1981). Energized in this way, the magnetosphere relaxesthrough sporadic reconnection in the magnetotail, producing ahighly dynamic and turbulent tail region (e.g., Borovsky &Funsten 2003). Likewise the dynamics of the corona areprobably driven mainly by photospheric motions that tanglemagnetic fields and launch Alfvén wave-like fluctuations(Beckers & Schneeberger 1977; Tomczyk et al. 2007) upwardinto the corona where turbulent motions with variations mainlyacross the large-scale magnetic field drive a cascade, includingsmall scale brightenings usually associated with reconnectionevents known as nanoflares, that brings energy to a muchsmaller scale where it can be dissipated. The solar wind mayalso be driven by an upward traveling wave emanating from thesub-Alfvénic corona (Tomczyk et al. 2007), as well as by shearfrom high speed streams (Belcher & Davis 1971). Again theenergy enters at relatively large scales, but cascades through aninertial range, which ultimately terminates and is capable ofsupplying sufficient energy to account for the observed heating(Coleman 1968; Tu et al. 1984; Hollweg 1986). In all of thesecases the dynamics are fundamentally cross-scale: the energyinjection is at scales comparable to the system size (outer scale)and the broadband cascade connects these larger “energy-containing” scales to the much smaller kinetic scales wheredissipation processes can cause conversion to internal energy.

The common theme in all of the above systems is thecascade, a fundamental feature of collisionless plasmaturbulence, and the eventual conversion of fluctuation energy

into thermal degrees of freedom at kinetic scales. Manytheoretical and observational studies have examined thedominant dissipative processes that heat protons (e.g., Howeset al. 2008; Parashar et al. 2009, 2010, 2011; Vasquez &Markovskii 2012; Wan et al. 2012a; Bourouaine & Chandran2013; Karimabadi et al. 2013; TenBarge et al. 2013; Xiaet al. 2013; Parashar et al. 2014b; Saito & Nariyuki 2014), thenature of the cascade and fluctuations at sub-proton scales (e.g.,TenBarge & Howes 1994; Shaikh & Zank 2009; Mithaiwalaet al. 2012; Boldyrev et al. 2013; Chang et al. 2013),intermittent structures at kinetic scales (e.g., Perri et al. 2012;Karimabadi et al. 2013; Wu et al. 2013a), and the electrontransport properties (e.g., Štverák et al. 2008; Rudakovet al. 2011; Wilson et al. 2013; Pulupa et al. 2014).Furthermore, various kinetic models and numerical schemes

have been employed to study these fundamental questions,including hybrid codes (e.g., Parashar et al. 2014b), EulerianVlasov codes (e.g., Servidio et al. 2012), closures such asgyrofluids (Goswami et al. 2005) or gyrokinetic (Schekochihinet al. 2009) models, and particle-in-cell (PIC) models arefamiliar examples. The selection of an appropriate model can beboth important and subtle. For example, if proton cyclotronresonance is a central component of a question, a hybrid modelmight be the best choice, but if it is established that magneticmoments are conserved, then gyrokinetics might be preferred.Such choices are often made for efficiency. Including morephysics is always a priority, in particular when simplifyingassumptions cannot be persuasively defended. Similarly, higherdimensionality is another obvious preference—three-dimensions(3D) in space and in velocity space—although when justifiable,lower dimension, such as 2.5D (typically 2D in space and 3Dvelocity space; sometimes 3D in space and 2D in velocity spaceas in gyrokinetics) leads to much needed computationalefficiency. When affordable, a fully self consistent approachincluding kinetic treatment of both protons and electrons—eitherof PIC or the Eulerian Vlasov types—contain the most completetreatment of the kinetic physics of interest.With all of these choices for adding additional physics, and

ever more accurate treatments of microphysics, it is sometimespossible to overlook the possible effects of the system sizeitself. In fact, all of the natural systems alluded to above are insome sense (to be presently defined; see below) to beconsidered as “large.” On the other hand, the system sizes

The Astrophysical Journal, 811:112 (8pp), 2015 October 1 doi:10.1088/0004-637X/811/2/112© 2015. The American Astronomical Society. All rights reserved.

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simulated using kinetic simulation codes vary from very small(e.g., TenBarge et al. 2013) to at most tens or hundreds oftypical kinetic lengths (e.g., Karimabadi et al. 2013; Wanet al. 2015). High resolution 2.5D simulations (Karimabadiet al. 2013) are still an exception and not the norm. 3D fullykinetic simulations are even more expensive. There is,however, an additional consideration not explicitly accountedfor in the above discussion: heliospheric and astrophysicalplasma systems are typically large in the sense that much of theenergy resides in plasma flow and magnetic field at scales Lthat are much larger than typical scales where kinetic effectsbecome important, say, the inertial length di. For the naturalsystems of interest, the system sizes L vary from hundreds of difor magnetospheres to millions of di for the astrophysicalscenarios.

For large values of the ratio L d ,i4 3( ) which acts as an

effective Reynolds number (Matthaeus et al. 2005), the largeMHD scales and small kinetic scales are well separated, andtypically connected by energy transfer associated with aturbulent cascade. Therefore the conditions experienced bythe kinetic degrees of freedom are those determined largely byMHD dynamics. This cannot happen in a system too small toadmit MHD physics, at least approximately. While it is clearthat an extremely large system, such as the solar wind, with

~L di 104, will display this MHD-kinetic physics contrast. Itis not well established, as far as we are aware, how and whenMHD behavior emerges in a controlled plasma system as itssize, defined as L d ,i is increased in a series of experimentsbeginning from small system sizes for which MHD cannot beobtained. This is the question we address quantitatively in thispaper. Taking proper account of the system size issue will haveclear implications for strategies for kinetic simulations that areintended to be of relevance to high Reynolds number spaceplasmas. It is the main purpose of the present paper to examinethe transition of physical effects going from small to moderatesized systems. We discuss implications for selection of codesizes after presenting our main results.

To have a fluid description, many possible representations ofphysics are possible, ranging widely in the scales described andthe physics effects included. If we attempt to compare fullykinetic plasmas with “fluid models” we would have a plethoraof models to which a comparison might be made: electronreduced MHD (e.g., Schekochihin et al. 2009), electron MHD(e.g., Gordeev et al. 1994; Shaikh & Zank 2010), Landau-Fluid(e.g., Goswami et al. 2005), Hall-FLR MHD (e.g., Ghosh &Parashar 2015), Hall MHD (e.g., Ghosh & Goldstein 1997),compressible MHD (e.g., Ghosh et al. 1993), and incompres-sible MHD. All of these models might be applicable at differentstages of the study. A comparison at different system sizes tothe properties of various fluid models would become acumbersome diversion from the questions of interest.

To avoid these diversions, we instead concentrate on themodel applicable at largest scales, i.e., MHD, and define“MHD-like behavior” by the following two conditions:

1. The quantities of relevance in MHD, such as flow andmagnetic energies, decay rates, etc., can be directly andquantitatively compared with kinetic plasma results todefine an approach of the system to “MHD-like”behavior.

2. Accompanying the above sequence, other quantities—those not readily represented in MHD—such as plasma

heating, are compared at varying system sizes and areseen to approach stable values.

Together these two tests qualify the system behavior to beMHD-like or non-MHD-like.To understand the transition from a fully kinetic to MHD-

like regime we employ the PIC code P3D (Zeiler et al. 2002) toperform simulations with varying system size with the initialconditions, scaled to the global system size, as close to eachother as possible. In order to characterize the transition tomagnetofluid behavior, we focus on most common turbulencediagnostics: global energetics, spectra and intermittencystatistics for the various cases.

2. PROBLEM SETUP AND DIAGNOSTIC TOOLS

We study the evolution of the Orszag–Tang Vortex (OTV;Orszag & Tang 1979; Dahlburg & Picone 1989; Parasharet al. 2009; Vasquez & Markovskii 2012) in a kinetic setup.OTV is a 2D MHD initial condition that involves a system sizevortex stretching two large-scale magnetic islands in periodicgeometry. The initial velocity v and magnetic field B are

=- + +=- + +

v

B

y e x e e

y e x e B e

sin sin 0

sin sin 2 1x y z

x y g z

( ) ˆ ( ) ˆ ˆ( ) ˆ ( ) ˆ ˆ ( )

where Bg is the strength of the out-of-plane mean magneticfield (guide field) that is added to the original OTV initial stateto reduce compressibility effects. These equations are written,in effect, in Alfvén speed units, with the magnetic fieldnormalized by pr4 , with mass density ρ. Note that the initialstate fluctuations reside entirely at long wavelengths in thesimulation box of dimensionless size p2 in each coordinatedirection. The wave numbers of the initial state will remainscaled to the box size as that parameter is changed in differentruns; in this way the ratio of energy containing scale to ioninertial length L di will vary across the different runs.OTV is a strongly nonlinear initial condition that evolves

into turbulence very quickly. It is often employed to test therobustness of numerical schemes. Physical properties of OTVhave been studied in great detail by Dahlburg & Picone (1989),Parashar et al. (2009), and Vasquez & Markovskii (2012) in thecompressible as well as kinetic regime. The latter studies wereperformed using hybrid codes that treat protons kinetically andelectrons as a neutralizing fluid. Here we take that initialcondition one step further by studying it in a particle-in-cellcode P3D. This code has been used to study many plasmaphysics problems like reconnection as well as turbulence (e.g.,Zeiler et al. 2002; Malakit et al. 2013; Wu et al. 2013a, 2013b).The details of the equations solved and the computationalmethodology can be found in Zeiler et al. (2002).Given the potential computational expense of such a study,

we choose a strategy with some care. Recent hybrid and PICsimulations that are very large compared to kinetic scales (e.g.,Karimabadi et al. 2013, 2014) appear to display large-scalebehavior close to the expectations of MHD, but a comparisonto MHD was not the prime focus of those studies. For exampleKarimabadi et al. (2013) found amplification of magnetic fieldsby large-scale shear, and the driving of secondary Kelvin–Helmholtz instabilities, both of which are expected in the MHDregime. Similarly Karimabadi et al. (2014) found the establish-ment of large-scale MHD-like magnetospheric structure in ahybrid simulation. A full quantitative comparison of MHD

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versus kinetic plasmas at very large system sizes, a grandchallenge project itself, remains to be done, but is not ourfocus here.

We are interested here in examining the transitionalbehavior from purely kinetic behavior to MHD-like behavior,and therefore require more numerous runs, but of relativelymodest system sizes as measured in units of di. We also workmainly with 2.5D as the first step, with two control runs in 3Dincluded for comparison.

Adopting this approach, multiple simulations with variedsystem sizes were performed. The run sizes and otherparameters are shown in Table 1. Magnetic field fluctuationshad an amplitude of B0.2 .g The plasma β was 0.08 for bothprotons as well as electrons. There were 200 particles per gridcell for each species. The grid spacing was d =x d0.02 i with∼2.5 cells across a Debye length. Speed of light was 20V ,A0

=m m 0.04,e i and a time step of w= -dt 0.002 ci1 was used.

The only parameters that changed between simulations werethe total number of grid points and accordingly the system size.For the 3D case, the 2.5D OTV initial condition is embedded in3D, and although no MHD scale perturbation in the third (z)direction is included, the system is now free to exhibitadditional kinetic instabilities and waves not present in the2.5D case.

Note that the energy density (per units mass) of the initialvelocity and magnetic fluctuations given by Equation (1) isá ñ = á ñ =v b 1,1

22 1

22∣ ∣ ∣ ∣ where the angle brackets denote a

volume average; the magnetic fluctuation is = -b B B z.g ˆ It isstraightforward to verify that the initial energy density per unitmass is independent of the system size. Therefore theelementary estimate of the nonlinear timescale tnl will increaseas system size increases, i.e., t l= Znl c increases with l ,c theenergy containing scale, where Z is the turbulence amplitude inspeed units. The first column of Table 1 lists the grid, secondcolumn the system size in di and the third column lists the sizeof a characteristic large eddy (lc) in these simulations.

An important note should be made here about initializingPIC codes. Electrons are given an extra initial flow speed tomake the displacement current zero and hence to enforce ´ =B J as an initial condition, appropriate to MHDbehavior. If this condition is not satisfied, the system willlaunch fast time scale electromagnetic waves (light). Thisprocedure can influence the energetics in small systems at earlytimes in the evolution. Apart from the additional electron flowkinetic energy needed for this procedure, all the runs weemploy in this study have the same initial condition, scaled todifferent system sizes.

3. RESULTS

3.1. Decay Rates

First we study the time evolution of the Elsässer energies= á + ñ = á ñ+ +v b zZ 2 1

22 1

22∣ ∣ ∣ ∣ and = á - ñ =- v bZ 2 1

22∣ ∣

á ñ-z ,1

22∣ ∣ where v and b are the fluid velocity and magnetic

fluctuations, respectively. The latter is measured in Alfvénspeed units (computed from mean density). These energies aredirect analogs of the fluctuation energy per unit mass

= á ñvu1

22 1

22∣ ∣ in incompressible hydrodynamics. The signifi-

cance of such energy-like quantities in the fluid turbulence limitis that typically in hydrodynamics or MHD the decay of energyin time is governed by just a few parameters, such as turbulenceamplitude and a length scale. In hydro these would beamplitude u and an energy containing a length scale λ such that

l~du dt u .2 3 In the simplest extension to MHD, with zerocross helicity, the amplitude is ~ ~+ -Z Z Z and

l~dZ dt Z .2 3 This type of decay is expected based on vonKarmán similarity analysis of the fluid equations (De Karman& Howarth 1938; Wan et al. 2015). Numerical simulationsevidence this type of behavior, at least approximately, in fluid(Pearson et al. 2002) and MHD (Hossain et al. 1995) systemsthat have at least moderately large Reynolds numbers (orsystem sizes). Recent evidence (Wu et al. 2013a) shows thatthis behavior also may be found in magnetized collisionlessplasmas of sufficiently large size. Given that a system size isrelated to an effective Reynolds in a kinetic plasma (Matthaeuset al. 2005), one might then surmise that emergence of vonKarmán similarity decay is evidence of approach to a highReynolds number MHD-like description. Therefore it isrelevant to the present goals to examine the time histories ofenergies in the set of kinetic simulations shown in Table 1.The first result of this examination is fundamental, although

not surprising from the perspective of turbulence theory, inwhich global dynamics are controlled by macroscopic, ratherthan microscopic, timescales. Usually in studies of isolatedkinetic plasma physics processes, one would plot quantities ofinterest such as the fluctuation energies Z t2 ( ) versus t indimensionless time units tied to kinetic properties. A familiarexample is the custom of examining time histories in units of adimensionless cyclotron time W t.ci The top panel of Figure 1shows the time history of total incompressive fluctuationenergy = ++ -E Z t Z tt

2 2( ) ( ) versus W t,ci for the collection ofruns of varying size shown Table 1. It is immediately apparentthat the curves do not even approximately superpose. Largerruns decay much more slowly in these units, and it is evidentthat the effect shown is due to the progressive separation ofcyclotron and nonlinear timescales, alluded to above. Thisdisparity increases as system size (and energy containing lengthscale) increases. However, interpreting the evolution in termsof (even the simplest) estimated nonlinear timescale betterorganizes the time evolution data.Plotting Et versus tt ,nl where t l p= ~Z L Z2nl c 0 box 0( )

with = ++ -Z Z Z ,0 02

02 1 2( ) for all the runs (Figure 1) shows

this effect rather clearly. This simple exercise demonstrates thatenergy decay is controlled by the timescales of the large energycontaining fluctuations, and not by the fast small scale kineticprocesses. This becomes particularly important when examin-ing larger systems in which one expects MHD to become aprogressively better description. To illustrate this point, thefluctuation energy versus time Et(t) for an incompressiblespectral method MHD simulation with the same OTV initial

Table 1Parameters for All Runs Described in This Paper

N N N, ,x y z Lbox lc k dmin i k dmin e

64, 64, 64 1.28 di 0.2 di 4.91 0.9864, 64, 1 1.28 di 0.2 di 4.91 0.98

128, 128, 128 2.56 di 0.41 di 2.45 0.49128, 128, 1 2.56 di 0.41 di 2.45 0.49256, 256, 1 5.12 di 0.85 di 1.22 0.24512, 512, 1 10.24 di 1.63 di 0.61 0.12

1024, 1024, 1 20.48 di 3.26 di 0.31 0.061024, 1024, 1 MHD h n= = 0.001 L L

Note. Here lc is the characteristic length scale and p=k l2min box is theminimum wave-number in the box.

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data is also shown in Figure 1. While the sequence of PIC runsdoes not strictly approach the MHD result, the smallersimulations are much further from the MHD case, while thelarger PIC runs are beginning to cluster near the MHD curve.

The separate Elsässer energies +Z t2 ( ) and -Z t2 ( ) alsodisplay interesting behavior in this sequence of simulations.When these quantities are equal, the cross helicity

º á ñ ~ -+ -u bH Z t Z tc2 2· ( ) ( ) is zero. There is a theoretical

expectation that the relative cross helicity H Etc may beamplified. The original theory of dynamic alignment(Dobrowolny et al. 1980) postulated that the cascade (decay)rates of the two Elsässer energies are typically equal, evenwhen there is non-zero cross helicity. This leads to increasein dominance of the larger Elässser energy (the majorityspecies) over the small, minority species. An alternativemodel (Matthaeus et al. 1983) notes that the nonlinear timefor the minority species may be much shorter than that of themajority species, so the weaker field may be transferred to

high wavenumber more rapidly than the stronger field. Thisaccelerates dynamic alignment, by transferring the minorityspecies preferentially to higher wavenumber, where it maybe more rapidly dissipated. The minority species in the OTVis -Z and indeed it is found to decay faster than +Z in bothour PIC and MHD systems (in terms of percentages; notshown). Furthermore both the +Z and -Z energies evolve inprogressively more MHD-like fashion for the larger systems,with the family of +Z curves very near the MHD case forsystems as small as d2.56 .i The -Z energies saturate towardMHD-like behavior somewhat more slowly with increase ofsystem size. While this approach to MHD energy decay isexpected, it is a bit surprising that it is clearly the case for +Zfor systems that are well in kinetic regime with

~k d 2.45.min i

The decay rate of fluctuation energy ++ -Z Z2 2 may be moreformally analyzed by adopting a normalization based on theextension to MHD of von Karmán hydrodynamic similaritydecay (De Karman & Howarth 1938; Hossain et al. 1995; Wanet al. 2012b). In this context, it is reasonable to expect that

a= - ++ - - +dE dt Z Z Z Z Lt2 2( ) for the special case in which

the two Elsässer fields have equal values for their respectiveenergy containing scales L and their von Karmán constants α.We recall again that some systematic departures are expectedbecause, among other things, here we are analyzing a systemthat is compressible, as well as admitting distinctive plasmadynamics. So, a close correspondence might not be expectedhere. To account for some of the higher frequency fluctuations,while maintaining the overall decay behavior, we fit the totalenergy curves with a polynomial that captures the global trendof the energy decay. We then normalize the fit decay ratedE dt as - ++ - + -dE dt Z Z Z Z L ,( ) ( ( ) ) where L is anestimate of the average correlation length (here very close tothe largest available length scale). These normalized rates areillustrated in the final panel of Figure 1. We omit depiction ofthe the initial and final transients that are artifacts of both thefitting procedure and the startup of the turbulence cascade. To areasonable level of approximation, the the computed normal-ized decay rates for all the kinetic runs are the same, with avariability similar to what is seen in moderate Reynoldsnumber MHD simulations (Hossain et al. 1995). The relativelytight clustering of the normalized decay rates for system sizesas small as d2.56 i is a bit of a surprise. Decay rates offluctuations were shown previously to cluster around MHD-von Karmán rates for somewhat larger system sizes, i.e., d25.6 iin Wu et al. (2013b). However, here we show that the decayrates are approximately the same for a system that is smaller byan order of magnitude as compared to Wu et al. (2013b) and iswell outside the traditional “inertial range” and inside the“kinetic dispersive/dissipative” regime with ~k d 2.45.min i

3.2. Spectra

Next we consider the behavior of the energy spectra. Themagnetic spectral densities (energy per wavenumber) shown inFigure 2 are time-averaged for each run, with the averagecalculated for ten roughly equally spaced time slices betweennonlinear times 4.9 and 10. To plot the MHD spectra on thesame figure, we assumed the Kolmogorov scale( h n= 3 1 4( ) ) aligns with the geometric mean of di and de.With this assumption, the MHD spectrum matches reasonablywell with other runs in the inertial range. Spectra show similarqualitative behavior, and agree very well at the highest

Figure 1. (a) Total incompressive fluctuation energy E t E 0t t( ) ( ) for the PICruns, plotted vs. normalized kinetic timescale W t;ci (b) again E t E 0t t( ) ( ) nowplotted vs. the nominal nonlinear time tt ;nl (c) Change in majority speciesElsässer energy + +Z t Z 02 2( ) ( ) vs. nonlinear time; (d) Change in minorityspecies Elsässer energy - -Z t Z 02 2( ) ( ) vs. nonlinear time; horizontal dottedlines in blue show the initial value of 1 in these panels; (e) decay rates of totalenergy normalized to the form expected in MHD for von Karmán–Howarthlike energy decay (see Hossain et al. 1995), ignoring start-up and final decayperiods. The normalized decay rates are surprisingly similar for all the runs,with the exception of d1.28 i simulations.

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The Astrophysical Journal, 811:112 (8pp), 2015 October 1 Parashar et al.

wavenumbers. This agreement is attributed to the noise floor,which is expected to be equal for all the runs, given that thegrid spacing and the number of particles per cell are the same inall cases. There also are apparent breaks in slope near thephysical scales kdi = 1 and kde = 1. It is evident that thedifferences in system size produce noticeable dissimilarities atlarger scales. Note the initial condition places the same energyper unit volume Z2 in all simulations, and this is peaked at thelargest scale present l. Therefore, the spectral density atwavenumber l1 is ~Z l,2 which accounts for the spectralpeaks moving to the right and downward for the smallersystems.

To obtain a different view of spectra that emphasizes scaledynamical similarity governed by the largest (energycontaining) scales, we return briefly to the similarity theoryof von Karmán and Howarth (De Karman & Howarth 1938),and its self-similar extension introduced by Kolmogorov(1941). The von Karmán–Howarth hypothesis allows that thespectra may be written as =E k Z lE k2( ) ˆ ( ) for energycontaining scale l and a dimensionless function E k .ˆ ( )Adapting the Kolmogorov self-similarity hypothesis to aplasma, and assuming that the role of dissipation scale istaken by the ion inertial scale di, we may further express thespectrum as = -E k cZ l kl g kd ,2 5 3

i( ) ( ) ( ) where c is a constantand g is a dimensionless function such that g 1 for kd 1iand g 0 for kd 1.i This family of spectra will “open up”a −5/3 spectral inertial range when l d ,i and corresponds

at least qualitatively to the long wavelength and shortwavelength ranges of the simulation spectra described above.In this perspective, it is of interest to examine the behavior ofthe numerically computed spectra for different system sizes,when they are rescaled by a factor l .2 3 The scaled spectra areshown in the bottom panel of Figure 2. Almost all the spectralie on top of each other including the MHD spectrum in theinertial range. The MHD spectrum is not expected to matchwith kinetic spectra in the kinetic regime as the dissipationfunction in MHD is artificial. The kinetic simulations line upvery well down to the scales ~kd 3.0.e The 3D cases showconsiderably less noise than its 2.5D counterparts. This isunderstandable as the 3D case has more particles per Debyeunit (sphere versus circle). Interestingly, the slopes in threeregions <k kd ,i < <kd k kdi e, and <kd ke are comparableto-5 3,-8 3, and-11 3 as indicated by the reference lineson the plot. This is another confirmation of Von-Kármandecay rates being same for these runs. Moreover, as far ascapturing the beginnings of an MHD inertial range, a systemsize of 20.48 di appears sufficient.

3.3. Energetics

Next, we examine issues related to exchange of energies,and, in particular, the onset of Alfvénic activity, by which wemean exchanges of energy between the magnetic field andplasma fluid flow velocity. First we show, in Figure 3, thateasily identifiable Alfvénic activity of the OTV magneticenergy Eb and proton flow energy Eif appears for system size assmall as d5.12 .i This interchange of energy between themagnetic field and the flow is absent in the smaller system sizesof d2.56 i and d1.28 ,i although in those smaller systems, smallamplitude activity at much higher frequencies is evident,presumably associated with electrostatic exchanges of energy.That the Alfvénic exchange occurs at scales =k d 1.22min i issomewhat surprising, since, in principle, this is outside the“inertial range” and one would not expect any MHD-like likebehavior. However the magnetic energy does not convergetoward a particular curve quickly even though, at larger sizes,its evolution shows the expected typical Alfvénic activity.An important feature that is frequently studied using kinetic

codes is the heating of protons and electrons. As discussedabove, the heating of electrons converges slowly with systemsize, but as shown in Figure 3, proton heating converges almostcompletely for system size d20.48 .i

2 This suggests that tocapture self consistent proton and electron heating along withother dynamics, in a kinetic code, we must have a system sizethat is at least ~ d40 i in length. The fifth panel of Figure 3shows the interesting result that the sum of magnetic energyand the electron thermal energy does converge as quickly as theproton flow energies, but this is as yet unexplained. The finaltwo panels show the anisotropy for protons ( = ^ A T Tp p ) andfor electrons ( = ^ A T Te e ). Just like plasma heating, theanisotropies also converge well for protons, but not well forelectrons. For the parameter regime in this figure, theanisotropy is weakly parallel for protons and is approachingweakly parallel from strongly parallel for the electrons. The 3Dcases show almost the exact same energetics as 2.5D cases.The details of the observed variation in heating behavior

seen in Figure 3 will require further detailed analysis. Howeverit is clear that as the system size increases the effectiveturbulent frequencies will decrease as the nonlinear timeincreases relative to the plasma time scales such as proton

Figure 2. Top panel shows the spectral densities directly from each simulationand the bottom panel shows the same spectra multiplied by l .2 3 The unscaledspectra show similar behavior but the similarity is particularly evident in therescaled spectra.

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cyclotron time (e.g., Matthaeus et al. 2014). This is expected tovary the impact of waves and instabilities that might affect theprotons, thus influencing the distribution of energy among theplasma species.

3.4. Intermittency

Finally we consider the intermittency statistics for thesesimulations. In Figure 4 we show the out of plane current forthe 2.5D PIC simulations. The top panel shows the boxes inunits of di and the bottom panel shows the boxes rescaled to p2

in order to highlight the dynamical current structures. Thecolors are the same in all panels. Minimum and maximumvalues have been forced to be equal to those for the d20.48 i

2

system. Smaller systems have stronger “core currents”—thecurrents in the magnetic bubbles, because of the initial ´ =B J enforcement at t = 0. However, the “dynamicalcurrent structures” that arise because of the system’s nonlinearevolution, for example, the current sheets at center of the box,are stronger in larger systems. We further quantify this bycomputing the probability density functions (PDFs) of Jz andkurtosis for the magnetic field increments.Figure 5 shows the PDFs of out of plane current Jz for

various runs. The PDFs were calculated for total data of 20snapshots of the current between t = 5, 10 to get betterstatistics. Smaller runs show PDFs that are close to Gaussian.Larger systems show stronger super-Gaussian tails that areassociated with strong dynamical currents (e.g., Grecoet al. 2009). The strength of the current sheets (rms current)is larger for larger systems. MHD shows the strongestintermittent structures as expected.The final Figure 6 shows kurtosis (k d d= á ñ á ñb b4 2 2) of the

magnetic field increments (d = - +b r b x b x r( ) ( ) ( )) of asingle cartesian component of the magnetic field. The kurtosisis 3 for a Gaussian field and is larger for more stronglyintermittent fields. The three panels of Figure 6 show kurtosisfor increments of each of the magnetic field components,evaluated for two values of increment =r di and =r d .e Thekurtosis increases monotonically for increasing system sizes,the dependence being a power law. The relative difference inthe power-law behavior of Bx and By kurtosis is likely becauseof the large scale inhomogeneity of the OTV. The power-lawfits were done on runs with effective Reynolds numbers largerthan 10. These power-law fits (k ~ R 0.1,0.5{ }) have an indexsimilar to the power law observed for the kurtosis of velocityderivatives obtained from high Reynolds number hydrody-namics experiments (k ~ R3 16; Van Atta & Antonia 1980;Pope 2000). The similarity in the power-law behavior forcompressible kinetic plasmas and incompressible hydrody-namics is striking despite all the differences in the two systems.Moreover, this is a quantitative manifestation of the increase instrength of intermittency, and in the importance of coherentstructures, for larger systems.Figures 5 and 6 clearly show that smaller simulations cannot

realistically capture intermittency statistics unless the systemsize is large. Consequently, to capture self consistentintermittent heating of a plasma, a suitably large system sizeis required. This conclusion supports previous similar findings(Wan et al. 2010; Wu et al. 2013a). Intermittent structures havebeen suggested to be associated with plasma heating (e.g.,Parashar et al. 2009, 2011; Osman et al. 2011; Wanet al. 2012a). It is interesting that the PDFs of current densityand the kurtosis of increments do not change significantly ingoing from 2.5D to 3D when systems of similar size arecompared.

4. CONCLUSIONS

How big of a plasma system is big enough to capture selfconsistent MHD-like behavior? The answer is not simple andwill depend on the quantity of interest. Some features of MHD-like behavior appear to persist down to scales much smallerthan expected. However, some features start to converge slowly

Figure 3. Evolution of various plasma energies with time, for kineticsimulations of varying size, and for the top two panels, for a roughly equivalentsized MHD simulation. From top to bottom the panels are (a) the change fromt = 0 in magnetic energy DEb vs. time; (b) the change from t = 0 in protonflow energyDEif vs. time; (c) the change from t = 0 in proton thermal energyas a fraction of initial free energy DU ;pˆ (d) the change from t = 0 of electronpressure energy as a fraction of initial free energyDU ;eˆ and (e) the change fromt = 0 of sum of magnetic and electron thermal energies. (f) Thermal anisotropy( = ^ A T Tp p p ) for protons (g) Thermal anisotropy ( = ^ A T Te e e ) forelectrons.

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for large system sizes. The main conclusions of this study areas follows.

1. The evolution of proton flow energies and +Z Elsässerenergy show MHD-like behavior up to scales >kd 1.i

This is unexpected.2. The normalized decay rates were found to be comparable

to MHD decay rates even for systems with ~L d2.56 ,iextending the conclusion of (Wu et al. 2013b) to systemsnearly an order of magnitude smaller.

3. Proton thermal energies start showing saturated behaviorat ~kd 0.3i , which is still significantly smaller than usualexpectations.

4. The -Z energies, magnetic energies, and electron heatingstart converging very slowly for increasing system sizes.

5. The intermittency becomes stronger for larger systems,indicating that to capture realistic intermittency statistics,the system size to be simulated has to be suitably large.The control parameter in this regard is the ratio of system

energy containing scale to ion inertial length, L d ,i wherewe recall that =R L deff i

4 3( ) is an effective large scaleReynolds number.

Based on the above conclusions, the requirements foraccurate simulation of a large, strongly intermittent turbulentplasma become quite reminiscent of the well known require-ments for computing high Reynolds number turbulent flows.The diagnostics discussed here are not the only parameters to

gauge MHD-like behavior. The problem of interest will dictatethe diagnostics to be used. The present results document howthe most common turbulence diagnostics of a kinetic plasmaapproach fluid-like behavior, beginning from a standard initialcondition that leads rapidly to nonlinear behavior. In problemssuch as the “Turbulent Dissipation Challenge” (Parashar &Salem 2013; Parashar et al. 2015, 2014b), the goal is toaccomplish comparisons of various models in computing

Figure 4. Out of plane current Jz for the 2.5D PIC simulations with varying system size. The top panels show the box in di units. The system size increases from left toright. The bottom panels show the same information with each box rescaled to p2 , highlighting the dynamical current structures. The smaller simulations have stronger“core currents” as part of the initial condition because of the ´ =B J enforcement at t = 0 as discussed above. However, the larger systems have stronger“dynamical currents” that are the consequence of dynamical evolution of the system.

Figure 5. Probability density functions (PDFs) for out of plane current Jz forthe simulations. Larger systems have stronger non-Gaussian tails, indicatingstronger dynamical current sheets.

Figure 6. Kurtosis for the magnetic field component increments at spatial lagsof di and de. The smallest system is too small to have an increment of di, hencewe do not plot the blue point for that system. Triangles represent the 3D cases.We note that the kurtosis at a fixed small scale lag increases as a power lawwith increasing system size. We plot the power-law fits to k =r di( ) as a reddashed line, and to k =r de( ) as a cyan dashed line. The fit values are given inthe same color text in each panel.

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turbulence properties that are relevant to observed solar windfluctuation dynamics (e.g., Tu & Marsch 1995; Bruno &Carbone 2005; Marsch 2006). Since solar wind fluctuations arewell known to display many MHD-like features, the presentresults indicate that model systems that are too small will not berelevant in this comparison. Our present conclusion is that thesystem size must be - d40 50 i in length to adequately andself-consistently represent the evolution of all the energies,inertial range spectra, and proton and electron heating.

Research supported by NSF AGS-1063439, AGS-1156094(SHINE), Solar Probe Plus science team (ISIS/SWRI sub-contract D99031L), NASA grants NNX14AI63G (Heliophy-sics Grand Challenge Theory), NNX08A083G-MMS, andNNX13AD72G. We would like to thank UCAR for providingus with computer time on the Yellowstone supercomputer tofacilitate this study.

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