June 2017

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA

C. Willmott was supported by MIT’s Charles E. Reed Faculty Initiative Fund. N.Loureiro was partially supported by the NSF-DOE partnership in basic plasma scienceand engineering, award no. DESC0016215, and by NSF CAREER award no. 1654168.Computations with the Viriato code were conducted at Massachusetts Green HighPerformance Computing Center on the Plasma Science and Fusion Center partition of theEngaging cluster, supported by the Department of Energy. Reproduction, translation,publication, use and disposal, in whole or in part, by or for the United States governmentis permitted.

PSFC/JA-17-25

Fully-Kinetic Versus Reduced-Kinetic Modelling of Collisionless Plasma Turbulence

Daniel Grošelj1,2, Silvio S. Cerri3, Alejandro Bañón Navarro2, Christopher Willmott4, Daniel Told1, Nuno F. Loureiro4, Francesco Califano3 , and Frank Jenko1,2

1Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstraße 2, 85748 Garching, Germany2Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA 3Physics Department “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy4Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Draft version June 8, 2017Preprint typeset using L

AT

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X style AASTeX6 v. 1.0

FULLY-KINETIC VERSUS REDUCED-KINETIC MODELLING OF COLLISIONLESS PLASMA TURBULENCE

Daniel Groselj1,2, Silvio S. Cerri3, Alejandro Banon Navarro2, Christopher Willmott4, Daniel Told1,Nuno F. Loureiro4, Francesco Califano3, and Frank Jenko1,2

1Max-Planck-Institut fur Plasmaphysik, Boltzmannstraße 2, 85748 Garching, Germany2Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA3Physics Department “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy4Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

ABSTRACT

We report the results of a direct comparison between di↵erent kinetic models of collisionless plasmaturbulence in two spatial dimensions. The models considered include a first principles fully-kinetic(FK) description, two widely used reduced models [gyrokinetic (GK) and hybrid-kinetic (HK) withfluid electrons], and a novel reduced gyrokinetic approach (KREHM). Two di↵erent ion beta (�

i

)regimes are considered: 0.1 and 0.5. For �

i

= 0.5, good agreement between the GK and FK modelsis found at scales ranging from the ion to the electron gyroradius, thus providing firm evidence fora kinetic Alfven cascade scenario. In the same range, the HK model produces shallower spectralslopes, presumably due to the lack of electron Landau damping. For �

i

= 0.1, a detailed analysis ofspectral ratios reveals a slight disagreement between the GK and FK descriptions at kinetic scales,even though kinetic Alfven fluctuations likely still play a significant role. The discrepancy can betraced back to scales above the ion gyroradius, where the FK and HK results seem to suggest thepresence of fast magnetosonic and ion Bernstein modes in both plasma beta regimes, but with amore notable deviation from GK in the low-beta case. The identified practical limits and strengths ofreduced-kinetic approximations, compared here against the fully-kinetic model on a case-by-case basis,may provide valuable insight into the main kinetic e↵ects at play in turbulent collisionless plasmas,such as the solar wind.Keywords: plasmas — solar wind — turbulence

1. INTRODUCTION

Turbulence is pervasive in astrophysical and spaceplasma environments, posing formidable challenges forthe explanation of complex phenomena such as theturbulent heating of the solar wind (Bruno & Car-bone 2013), magnetic field amplification by dynamo ac-tion (Kulsrud & Zweibel 2008), and turbulent magneticreconnection (Matthaeus & Lamkin 1986; Lazarian &Vishniac 1999; Loureiro et al. 2009; Matthaeus & Velli2011; Lazarian et al. 2015; Cerri & Califano 2017). Theaverage mean free path of the ionized particles in themajority of these naturally turbulent systems typicallygreatly exceeds the characteristic scales of turbulence,rendering the dynamics over a broad range of scales es-sentially collisionless. Driven by increasingly accuratein-situ observations, a considerable amount of currentresearch is focused on understanding collisionless plasmaturbulence at kinetic scales of the solar wind (Marsch2006; Bruno & Carbone 2013), where the phenomenol-ogy of the inertial range (magnetohydrodynamic) tur-bulent cascade is no longer applicable due to collision-

less damping of electromagnetic fluctuations, dispersiveproperties of the wave physics, and as a result of kinetic-scale coherent structure formation (Kiyani et al. 2015).The research e↵ort with emphasis on kinetic-scale tur-bulence includes abundant observations (Leamon et al.1998; Bale et al. 2005; Alexandrova et al. 2009; Sahraouiet al. 2009; Kiyani et al. 2009; Osman et al. 2011; He et al.2012; Salem et al. 2012; Chen et al. 2013; Lacombe et al.2014; Chasapis et al. 2015; Perrone et al. 2016; Naritaet al. 2016), supported by analytical predictions (Galtier& Bhattacharjee 2003; Howes et al. 2008a; Schekochi-hin et al. 2009; Boldyrev et al. 2013; Passot & Sulem2015), and computational studies (Howes et al. 2008b;Saito et al. 2008; Servidio et al. 2012; Verscharen et al.2012; Boldyrev & Perez 2012; Wu et al. 2013; Karimabadiet al. 2013; TenBarge & Howes 2013; Chang et al. 2014;Vasquez et al. 2014; Valentini et al. 2014; Told et al.2015; Wan et al. 2015; Franci et al. 2015; Cerri et al.2016; Parashar & Matthaeus 2016).Out of numerical convenience and based on physical

considerations, the simulations of collisionless plasma

2

turbulence are often performed using various types ofsimplifications of the first principles fully-kinetic plasmadescription. Among the most prominent reduced-kineticmodels for astrophysical and space plasma turbulenceare the so-called hybrid-kinetic (HK) model with fluidelectrons (Winske et al. 2003; Verscharen et al. 2012;Parashar et al. 2009; Vasquez et al. 2014; Servidio et al.2015; Franci et al. 2015; Kunz et al. 2016) and the gyroki-netic (GK) model (Howes et al. 2008a,b; Schekochihinet al. 2009; TenBarge & Howes 2013; Told et al. 2015; Liet al. 2016). Even though these models are frequentlyemployed, no general consensus concerning the valid-ity of reduced-kinetic treatments presently exists withinthe community, with distinct views on the subject be-ing favoured by di↵erent authors (Howes et al. 2008a;Matthaeus et al. 2008; Schekochihin et al. 2009; Ser-vidio et al. 2015). Fully-kinetic (FK) simulations, on theother hand, are currently constrained by computationalrequirements that limit the prevalence of such studies,the accessible range of plasma parameters, and the abil-ity to extract relevant information from the vast amountsof simulation data (Saito et al. 2008; Wu et al. 2013;Karimabadi et al. 2013; Chang et al. 2014; Haynes et al.2014; Wan et al. 2015; Parashar & Matthaeus 2016).In order to spend computational resources wisely andfocus fully-kinetic simulations on cases where they aremost needed, it is therefore crucial to establish a firmunderstanding of the practical limits of reduced-kineticmodels. Furthermore, the question concerning the va-lidity of reduced-kinetic approximations is not only im-portant from the computational perspective, but is alsoof major interest for the development of analytical pre-dictions based on a set of simplifying but well-validatedassumptions, and for the interpretation of increasinglyaccurate in-situ satellite measurements. Last but notleast, a thorough understanding of the range of param-eters and scales where reduced models correctly repro-duce the fully-kinetic results is instrumental in identify-ing the essential physical processes that govern turbu-lence in those regimes.In this work, we try to shed light on some of the above-

mentioned issues by performing a systematic, first-of-a-kind direct comparison of kinetic models in collision-less plasma turbulence with focus on ion and electronscales. Even outside the domain of kinetic-scale, colli-sionless plasma turbulence, studies of similar type havebeen relatively rare (Birn et al. 2001; Henri et al. 2013;TenBarge et al. 2014; Munoz et al. 2015). Consider-ing these circumstances, we choose here to investigatedi↵erent plasma regimes in order to give a reasonablycomprehensive view of the e↵ects of adopting reduced-kinetic approximations. On the other hand, the simu-lation parameters that we adopt per force result froma compromise between the desire for maximal relevance

of results in relation to solar wind turbulence and com-putational accessibility of the problem. To this end, weadopt a two-dimensional and widely-used setup of decay-ing plasma turbulence (Orszag & Tang 1979) and care-fully tailor the simulation parameters to reach—as muchas possible—the plasma conditions relevant for the so-lar wind. The particular type of initial condition waschosen here for its simplicity and popularity, allowingfor a relatively straightforward reproduction of our sim-ulations and/or comparison of the results with previousworks (Biskamp &Welter 1989; Dahlburg & Picone 1989;Politano et al. 1989, 1995; Parashar et al. 2009, 2015a;Loureiro et al. 2016; Li et al. 2016). On the other hand,it is still reasonable to expect that the turbulent sys-tem studied in this work shares at least some qualitativesimilarities with natural turbulence occurring at kineticscales of the solar wind (Servidio et al. 2015; Li et al.2016; Wan et al. 2016). We also note that the choice ofinitial conditions should not a↵ect the small-scale prop-erties of turbulence due to the self-consistent reprocessof the fluctuations by each model during the turbulentcascade (Cerri et al. 2017).The paper is organized as follows. In Section 2 we pro-

vide a short summary of the kinetic models included inthe comparison, followed by a description of the simula-tion setup in Sec. 3. The results presented in Sec. 4 aredivided into several parts. In the first part, we investigatethe spatial structure of the solutions by comparing thesnapshots of the electric current and the statistics of mag-netic field increments (Sec. 4.1). The analysis of turbu-lent structures in real space is followed by a comparisonof the global turbulence energy budgets for each model(Sec. 4.2). Afterwards, we perform a detailed comparisonof the spectral properties of the solutions (Sec. 4.3), fol-lowed by the analysis of the nonthermal ion and electronfree energy fluctuations in the spatial domain (Sec. 4.4).Finally, we conclude the paper with a summary and dis-cussion of our main results (Sec. 5).

2. KINETIC MODELS INCLUDED IN THECOMPARISON

For the sake of clarity, we provide below brief descrip-tions of the kinetic models involved in the comparison.However, we make no attempt to give a comprehensiveoverview of each model and we refer the reader to thereferences given below for further details.We start with a quick summary of the most com-

plete model—the fully-kinetic, electromagnetic model(Lifshitz & Pitaevskii 1981; Klimontovich 1997). At themost fundamental level, the kinetic properties of a colli-sionless plasma are governed by the Vlasov equation for

3

each particle species s:

@f

s

@t

+ v ·rf

s

+ q

s

✓E+

v ⇥B

c

◆· @fs@p

= 0, (1)

where fs

(r,p) is the single-particle distribution function,q

s

is the species charge, v is the velocity, and p is the(relativistic) momentum. The electromagnetic fields, Eand B, are obtained self-consistently from the full set ofMaxwell equations,

@E

@t

= cr⇥B� 4⇡J,@B

@t

= � cr⇥E, (2)

r ·E = 4⇡⇢, r ·B = 0, (3)

where J =P

s

q

s

Rvf

s

d3p is the electric current and⇢ =

Ps

q

s

Rf

s

d3p is the charge density. The aboveequations form a closed set for the kinetic descriptionof collisionless plasmas.The hybrid-kinetic Vlasov-Maxwell model (Byers et al.

1978; Harned 1982; Winske et al. 2003) is a non-relativistic, quasi-neutral plasma model where the ionsare fully kinetic and the electrons are treated as a back-ground neutralizing fluid with some underlying assump-tion for their equation of state (typically an isothermalclosure). The fundamental kinetic equation solved by theHK model is the Vlasov equation for the ions:

@f

i

@t

+ v ·rf

i

+q

i

m

i

✓E+

v ⇥B

c

◆· @fi@v

= 0, (4)

where fi

(r,v) is the ion single-particle distribution func-tion and m

i

is the ion mass. The B field is advancedusing Faraday’s law

@B

@t

= � cr⇥E. (5)

Since the HK model assumes the non-relativistic limit,the displacement current is neglected in the Ampere’slaw, which thus reads

r⇥B =4⇡

c

J. (6)

Finally, the electric field is obtained from the generalizedOhm’s law (Valentini et al. 2007):

�1� d

2

e

r2

�E = �u

e

⇥B

c

� rp

e

ne

+ ⌘J

� m

e

m

i

⇢ui

⇥B

c

� 1

ne

r ·hm

i

n(ui

ui

� ue

ue

) +⇧i

i�,

(7)

where d

e

, m

e

/m

i

, ⌘, n, ui

, ue

= ui

� J/en, p

e

, and⇧

i

are the electron skin depth, electron-ion mass ratio,(numerical) resistivity, plasma density, ion fluid velocity,electron fluid velocity, electron pressure, and ion pressuretensor, respectively. The HK model is a quasi-neutraltheory (Tronci & Camporeale 2015), n

i

' n

e

⌘ n, andtherefore it deals only with frequencies much smaller

than the electron plasma frequency, ! ⌧ !

pe

. We alsonote that the finite electron mass in expression (7) is inpractice often set to zero, which consequently neglectselectron inertia and results in a much simplified versionof Ohm’s law:

E = �ue

⇥B

c

� rp

e

ne

+ ⌘J. (8)

The gyrokinetic model (Frieman & Chen 1982; Brizard& Hahm 2007) orders fluctuating quantities according toa small expansion parameter ✏. In particular, the follow-ing ordering is assumed for the fluctuating quantities:

�f

s

/F

0,s

⇠ �B/B

0

⇠ q

s

��/T

0,s

⇠ ✏, (9)

where F

0,s

is the background particle distribution func-tion, �f

s

is the perturbed part of the distribution, ��

is the perturbed electrostatic potential, T0,s

is the back-ground temperature, and B

0

is the background magneticfield. A similar ordering is assumed for the characteris-tic macroscopic length scale L

0

and the frequencies ofthe fluctuating quantities !:

!/⌦cs

⇠ ⇢

s

/L

0

⇠ ✏, (10)

where ⌦cs

is the species cyclotron frequency and ⇢

s

isthe species Larmor radius. For the problems of inter-est, such as kinetic-scale turbulence in the solar wind(Howes et al. 2006, 2008a; Schekochihin et al. 2009), themacroscopic length scale can be typically associated witha characteristic wavelength of the fluctuations measuredalong the magnetic field. Time scales comparable to thecyclotron period are cancelled out by performing an av-eraging operation over the particle Larmor motion. Thephase space is then conveniently described in terms of thecoordinates (R

s

, µ

s

, vk), where Rs

= r+ (v⇥b0

)/⌦cs

isthe particle gyrocenter position, µ

s

= m

s

v

2

?/2B0

is themagnetic moment, and vk = v ·b

0

the velocity along themean field, where b

0

= B0

/B

0

. Instead of evolving theperturbed distribution �f

s

, the gyrokinetic equations aretypically written in terms of the ring distribution:

h

s

(Rs

, µ

s

, vk, t) = �f

s

(r,v, t) +q

s

��(r, t)

T

0,s

F

0,s

(v). (11)

The distribution h

s

is independent of the gyrophase angleat fixed R

s

. The electromagnetic fields are obtained self-consistently from the ring distribution under the assump-tion (9) and (10). For the electrostatic potential, the as-sumptions result in a quasineutrality condition, which isused to determine ��. The magnetic field, on the otherhand, is obtained from a GK version of Ampere’s law.Finally, we summarize the main features of the

so-called Kinetic Reduced Electron Heating Model(KREHM) (Zocco & Schekochihin 2011). KREHM isa fluid-kinetic model, obtained as a rigorous limit of gy-rokinetics for low-beta magnetized plasmas. In particu-

4

lar, the low beta assumption takes the following form:p�

e

⇠pm

e

/m

i

⌧ 1, (12)

where �

e

is the electron beta ratio. Due to its low betalimit, the addition of KREHM to this work clarifies theplasma beta dependence of our results. Instead of solvingfor the total perturbed electron distribution function �f

e

,KREHM is written in terms of

g

e

= �f

e

� (�n/n0

+ 2vk�uk,e/v2

th,e

)/F0,e

, (13)

where �n

e

is the perturbed density, �uk,e the perturbedelectron fluid velocity along the mean magnetic field, andv

th,e

=p2T

0,e

/m

e

is the electron thermal velocity of thebackground distribution F

0,e

. The modified distributiong

e

has vanishing lowest two (fluid) moments, thus high-lighting the kinetic physics contained in KREHM, lead-ing to small-scale velocity structures in the perturbeddistribution function. As the final set of equations forg

e

does not explicitly depend on velocities perpendic-ular to the magnetic field, the perpendicular dynamicsmay be integrated out, leaving only the parallel veloc-ity coordinate. As such, the model is by far the leastcomputationally demanding of the four that we employ.As shown in Zocco & Schekochihin (2011), a convenientrepresentation of g

e

in the vk space can be given in termsof Hermite polynomials.

3. PROBLEM DESCRIPTION AND SIMULATIONSETUP

In the following section we provide details regardingthe choice of initial condition and plasma parameters.Further technical details describing the numerical set-tings used for each type of simulation are given in Ap-pendix A.The initial condition used for all simulations is the

so-called Orszag-Tang vortex (Orszag & Tang 1979)—awidely-used setup for studying decaying plasma turbu-lence (Biskamp & Welter 1989; Dahlburg & Picone 1989;Politano et al. 1989, 1995; Parashar et al. 2009, 2015a;Loureiro et al. 2016; Li et al. 2016). The initial fluidvelocity and magnetic field perturbation are given by

u? = �u

�� sin(2⇡y/L) e

x

+ sin(2⇡x/L) ey

�, (14)

B? = �B

�� sin(2⇡y/L) e

x

+ sin(4⇡x/L) ey

�, (15)

where L is the size of the periodic domain (x, y 2 [0, L)),and �u and �B are the initial fluid and magnetic fluctu-ation amplitudes, respectively. A uniform magnetic fieldB

0

= B

0

ez

is imposed in the out-of-plane (z) direction.In addition to magnetic field fluctuations, we initializea self-consistent electric current in accordance with Am-pere’s law:

J

z

=c

4⇡

2⇡�B

L

�2 cos(4⇡x/L) + cos(2⇡y/L)

�. (16)

For the FK and GK simulations, we explicitly initializethe parallel current via a locally shifted Maxwellian forthe electron species. Similarly, we prescribe the perpen-dicular fluid velocities for the FK and HK models by lo-cally shifting the Maxwellian velocity distributions in v

x

and v

y

. For GK and KREHM, the same approach is notpossible due to the gyrotropy assumption imposed on theparticle perpendicular velocities. Instead, the in-planefluid velocities for GK and KREHM are set up with aplasma density perturbation, resulting in self-consistentelectrostatic field that gives rise to perpendicular fluidmotions via the E⇥B drift (Numata et al. 2010; Loureiroet al. 2016). In the large-scale and strong guide fieldlimit, the leading order term for the electric field is

E? ⇡ �1

c

u? ⇥B0

(17)

and the perturbation is electrostatic. For better consis-tency with reduced-kinetic models used here, the E fieldfor the FK model is initialized according to Eq. (17) witha corresponding (small) electron density perturbation tosatisfy the Poisson equation for the electrostatic poten-tial. Apart from minor density fluctuations that accountfor the electrostatic field, the initial ion and electron den-sities, as well as temperatures, are chosen to be uniform.Finally, it is important to mention that we perform theHK simulations with the generalized version of Ohm’slaw given by Eq. (7) which includes electron inertia, andas such features a collisionless mechanism for breakingthe magnetic field frozen-in flux constraint.A list of simulation runs with the corresponding plasma

parameters and box sizes is given in Table 1. The keydimensionless parameters varied between the runs are theion beta

�

i

= 8⇡n0

T

i

/B

2

0

(18)

and the initial turbulence fluctuation strength

✏ = �B/B

0

= �u/v

A

, (19)

where vA

= B

0

/

p4⇡n

0

m

i

is the Alfven speed. Through-out this article, the species thermal velocity is defined asv

th,s

=p2T

s

/m

s

and the species Larmor radius is givenby ⇢

s

= v

th,s

/⌦cs

, where ⌦cs

= eB/(ms

c) is the speciescyclotron frequency. Unless otherwise stated, lengths arenormalized to the ion inertial length d

i

= ⇢

i

/

p�

i

, veloc-ities to the Alfven speed v

A

, masses to the ion mass mi

,magnetic field to B

0

, the electric field to v

A

B

0

/c, anddensity to the background value n

0

. We take the (inte-gral scale) eddy turnover time ⌧

0

as the basic time unit,which we define here as

⌧

0

=L/2

�u

. (20)

The eddy turnover time, being independent of kineticquantities, is a robust measure allowing for a direct com-

5

parison of simulations obtained from di↵erent plasmamodels and for variable fluctuation levels and box sizes(Parashar et al. 2015a). Further simulation detailspertaining to each kinetic model are provided in Ap-pendix A.

Main plasma parameters

Run �

i

✏ m

i

/m

e

T

i

/T

e

L/d

i

A1 0.1 0.2 100 1 8⇡

A2 0.1 0.1 100 1 8⇡

B1 0.5 0.3 100 1 8⇡

B2 0.5 0.15 100 1 8⇡

Table 1. List of simulation runs with their correspond-ing plasma parameters: ion beta (�

i

), initial turbulencefluctuation level (✏), ion-electron mass ratio (m

i

/m

e

),ion-electron temperature ratio (T

i

/T

e

), and box size inunits of the ion inertial length (L/d

i

).

4. RESULTS

Below we present a detailed analysis of the simulationruns listed in Sec. 3. All diagnostics are implementedfollowing equivalent technical details for all models andthe numerical values from all the simulations are normal-ized in the same way. For the intermittency and spectralanalysis of the fully-kinetic, particle-in-cell (PIC) simu-lations, we use short-time averaged data as is often donein the PIC method (Daughton et al. 2014; Roytershteynet al. 2015; Munoz et al. 2017). Technical details re-garding the time averaging are provided in Appendix A.The e↵ect of short-time averaging is to highlight the tur-bulent structures and reduce the amount of fluctuationsarising from PIC noise that are mainly concentrated athigh frequencies. In our case, the averaging roughlyretains frequencies up to ! . ⇡⌦

ci

for the �

i

= 0.1runs and ! . 2⇡⌦

ci

for �

i

= 0.5. The possibility tosearch for phenomena that could potentially reach fre-quencies much higher than the ion cyclotron frequency[e.g. whistler waves (Saito et al. 2008; Gary & Smith2009; Chang et al. 2014; Gary et al. 2016)] is thereforelimited in the time averaged data. We note however thatwithout the use of short-time averaging the amount ofPIC noise in our simulations is too high to allow for adetailed analysis of turbulent structures at electron ki-netic length scales. The limitations set by the PIC noiseare most strict for the electric field, the detailed knowl-edge of which is an important piece of information foridentifying the small-scale nature of the turbulent cas-cade. Furthermore, by comparing the turbulent spectracalculated from raw data with those obtained from time-averaged data, and with the spectra of the backgroundparticle noise, we found no evidence that any significantphysics is lost due to short-time averaging.

4.1. Spatial field structure

We begin the discussion of our results by comparingthe spatial structure of the turbulent fields. In Figure 1we compare the out-of-plane electric current J

z

at aroundone eddy turnover time ⌧

0

in the simulation. The con-tour plots are shown for variable fluctuation strengths✏ = �B/B

0

= �u/v

A

and for both values of the ion beta(�

i

= 0.1, 0.5). The electric current obtained from FKsimulations is plotted using the raw data without anyshort-time averaging or low-pass filtering to provide thereader with a qualitative measure of the strength of back-ground thermal fluctuations stemming from PIC noise.To highlight the ✏ dependence, the FK and HK data havebeen rescaled by 1/✏, whereas in the GK and KREHMmodels the current is already naturally rescaled by 1/✏.With the exception of the KREHM solution correspond-ing to the �

i

= 0.5 case, a relatively good overall agree-ment is found between all models. Since KREHM relieson the low plasma beta assumption, the disagreementfor �

i

= 0.5 is to be expected and highlights the in-fluence of �

i

on the turbulent field structure. On theother hand, KREHM gives surprisingly accurate resultsalready for �

i

= 0.1 even though the formal requirementfor its validity is given by

p�

e

⇠pm

e

/m

i

⌧ 1 (Zocco &Schekochihin 2011), which gives �

i

. 0.01 in our case (us-ing m

e

/m

i

= 0.01 and T

0,i

= T

0,e

). By looking at Fig. 1we also see that the field structure changes relatively lit-tle with ✏. Moreover, the field morphology depends muchmore on the plasma beta than it does on the turbulencefluctuation level. It is worth noticing that for even higher✏ the field structure is expected to change significantly;at least when the sonic Mach number M

s

⇠ ✏/

p�

i

ap-proaches unity (Picone & Dahlburg 1991).In Table 2 we list the root-mean-square values of the

out-of-plane current density at 1.5 eddy turnover times.Short-time averaged data were used here to computethe root-mean-square values for the FK model. Whilethe spatial field structures are overall in good qualita-tive agreement, the root-mean-square current numericalvalues di↵er. More specifically, the numerical values de-viate from the FK model up to 9% for GK, up to 12% for KREHM, and up to 27% for the HK model. Wenote that minor deviations can occur not only as a resultof physical di↵erences but also due to di↵erences in thenumerical grid size and low-pass filters, as well as dueto PIC noise in the FK runs that may slightly a↵ect thesmall-scale dynamics. On the other hand, regarding therelatively large disagreement between the HK and FKmodels, it is reasonable to assume that the main causefor the di↵erence are electron kinetic e↵ects absent in theHK approximation, as discussed in the following sections.In agreement with previous works (see review

by Matthaeus et al. (2015) and references therein), the

6

Figure 1. Contour plots of Jz

for �i

= 0.1 (left) and �

i

= 0.5 (right). A doubly-logarithmic color scale (one for positiveand one for negative values, with a narrow linear scale around 0 to connect the two) is used here to reveal also thefluctuations of weaker intensity. The color range has been adjusted according to the maximum value for any data set.

J

rms

z

/✏

�

i

= 0.1 �

i

= 0.5

✏ = 0.2 ✏ = 0.1 ✏ = 0.3 ✏ = 0.15

FK 1.08 1.24 0.85 0.93

HK 1.31 1.44 1.08 1.24

GK 1.13 0.91

KREHM 1.12 0.82

Table 2. Comparison of the root-mean-square values ofJ

z

/✏ around 1.5 eddy turnover times in the simulation.

snapshots of Jz

in Fig. 1 also reveal a hierarchy of cur-rent sheets which di↵erentiate with respect to the back-ground Gaussian fluctuations, leading to scale-dependentstatistics also known as intermittency. To investigate in-termittency in our simulations, we study the statistics ofthe perpendicular magnetic field increments �B

i

(`) =B

i

(r + `) � B

i

(r), where i 2 {x, y}. In Figure 2, wecompare the PDFs of magnetic field increments for the�

i

= 0.1 runs. Short-time averaged fields are used herefor the FK model. The PDFs are averaged over two or-thogonal directions for the lag ` and over the x and y

component of the magnetic field. Additionally, we alsoaverage the PDFs over a time window from roughly 1.4to 1.6 eddy turnover times, when the turbulence is al-ready well developed. The departure from a GaussianPDF (black dashed lines in Fig. 2) increases from large

to small scales. Good agreement is found between allmodels, except for very large values at the tails of thePDFs. Even though the level of intermittency in oursimulations, compared to the solar wind, is likely under-estimated due to a limited system size, the analysis showsthat the turbulent fields still maintain a certain degreeof intermittency, which does not appear to be particu-larly sensitive to di↵erent types of reduced-kinetic ap-proximations. Our results are also in qualitative agree-ment with some observational studies and kinetic sim-ulations (Leonardis et al. 2013; Wu et al. 2013; Franciet al. 2015; Leonardis et al. 2016), even though it shouldbe mentioned that some contradictory results have beenreported based on observational data (Kiyani et al. 2009;Chen et al. 2014), for which the authors found non-Gaussian but scale-independent PDFs at kinetic scales.To compare the level of intermittency in detail, we

also compute the scale-dependent flatness, K(`) =h�B

x,y

(`)4i/h�B

x,y

(`)2i2, of magnetic field fluctuations(Fig. 3). The flatness is close to the Gaussian value of3 (black dashed lines in Fig. 3) at large scales and in-creases well above the K = 3 threshold at kinetic scales.Furthermore, Fig. 3 also shows that the intermittencyis slightly underestimated in the HK simulations and inthe KREHM run corresponding to the �

i

= 0.5 case.Here we point out that intermittent properties can be

7

very sensitive to the choice of numerical resolution, whichmight be responsible for the slightly lower level of inter-mittency in the HK simulations. In particular, the HKflatness is closer to the FK and GK ones for the �

i

= 0.5run, for which the smallest relevant scale of the FK andGK models (⇢

e

) is closer to the limits of the HK reso-lution than in the �

i

= 0.1 case. Further investigationswill be necessary to establish a thorough understandingof kinetic-scale intermittency with respect to di↵erenttypes of reduced-kinetic approximations. The need toperform a detailed study of intermittency with empha-sis on kinetic model comparison has already been recog-nized by several authors and a detailed plan to fulfil this(among others) ambitious goal has been recently out-lined in the so-called “Turbulent dissipation challenge”(Parashar et al. 2015b).

Figure 2. PDFs of magnetic field increments for di↵erentseparation lengths `. The data is shown for the �

i

= 0.1case with ✏ = 0.1 for the FK and HK models. The blackdashed line corresponds to the (normalized) GaussianPDF.

4.2. Global time evolution

Next, we consider the global turbulence energy budgetof the decaying Orszag-Tang vortex. For this purpose,we split the bulk energy into magnetic M = hB2

/8⇡i,kinetic ion K

i

= hnmi

u

2

i

/2i, kinetic electron K

e

=hnm

e

u

2

e

/2i, internal ion I

i

= h3nTi

/2i, and internal elec-tron I

e

= h3nTe

/2i energy density, where h. . . i denotesa space average. To measure the energy fluctuations oc-curring as a result of turbulent interactions, we consider

Figure 3. Scale-dependent flatness of perpendicular mag-netic field increments for �

i

= 0.1 (top) and �

i

= 0.5(bottom).

only the relative changes for each energy channel, nor-malized to the sum of the initial magnetic and kinetic ionenergy density, E

0

= M(t = 0) + K

i

(t = 0). The timetraces for the FK model are obtained by space-averagingthe raw simulation data without the use of short-time av-eraged fields. For the GK model and KREHM, we definethe perturbed internal energy similar to Li et al. (2016)as

�I

s

= �T

0,s

�S

s

� �K

s

+

Zt

0

Ds

dt0, (21)

where �S

s

= � 1

V

Rd2r d3v�f2

s

/2F0,s

is the perturbedspecies entropy and D

s

is the mean species dissipationrate, occurring as a result of (hyper)collisional and (hy-per)di↵usive terms in the equations. The first two termson the right-hand side correspond to free energy fluctua-tions without contributions from bulk fluid motions. Thelast term represents the amount of dissipated free energyof species s. In full f models, the “dissipated” nonther-mal energy stemming from (numerical) collisional e↵ectsin velocity space remains part of the distribution functionin the form of thermal fluctuations. On the other hand,in �f models such as GK and KREHM the thermalizedenergy is not accounted for in the background distribu-tion F

0,s

and has to be added explicitly to the internalenergy expression. The dissipative term is significant forboth ions and electrons. In the GK simulations, hyper-collisionality contributes to around 56% and 67% of �I

e

and to about 5% and 14% of �Ii

at t ⇡ 2⌧0

for �i

= 0.1and �

i

= 0.5, respectively.The time traces of the mean energy densities are shown

in Fig. 4. The observed deviations from the FK curvesappear to be consistent with the kinetic approximationsmade in each of the other three models. In particular,

8

the HK model exhibits an excess of magnetic and elec-tron kinetic energy density, which can be understood inthe context of the isothermal electron closure that pre-vents turbulent energy dissipation via the electron chan-nel. For the GK model, the result shows that GK tendsto overestimate the electron internal energy and under-estimate the ion one, with the deviation from the FKmodel shrinking as ✏ is lowered. The convergence of theFK internal energies towards the GK result as ✏ ! 0 isconsistent with the low ✏ assumption made in the deriva-tion of the GK equations. Similarly, the improved agree-ment of KREHM with the FK model for the lower beta(�

i

= 0.1) runs can be understood in the context of thelow beta assumption made in KREHM. We also notethat ion heating is ordered out of the KREHM equations,which appears to be reasonable given the fact that theion internal energy increases more rapidly in the higherbeta (�

i

= 0.5) run.In summary, from the point of view of the global energy

budget, all models deliver accurate results within the for-mal limits of their validity and in some cases even wellbeyond. The HK model resolves well ion kinetic e↵ectsbut falls short in describing electron features, whereas theGK model becomes increasingly inaccurate for large tur-bulence fluctuation strengths. In addition to the ✏ ⌧ 1assumption of GK, the accuracy of KREHM also dependson the smallness of the plasma beta. For the partic-ular setup considered, KREHM gives surprisingly goodresults already for �

i

= 0.1, which is well beyond its for-mal limit of validity (�

i

. 0.01 for m

e

/m

i

= 0.01 andT

0,i

= T

0,e

).

4.3. Spectral properties

The nature of the kinetic-scale, turbulent cascade is in-vestigated by considering the one-dimensional (1D) tur-bulent spectra, which we compute as follows. We dividethe two-dimensional perpendicular wavenumber plane(k

x

, k

y

) into equally-spaced shells of width�k = 2kmin

=4⇡/L and calculate the 1D spectra, P (k?), by summingthe squared amplitudes of the Fourier modes containedin each shell. The wavenumber coordinates for the 1Dspectra are assigned to the middle of each shell. To com-pensate for the global energy variations between di↵er-ent models (see Fig. 4) and to account for the fact thatthe turbulence is decaying, we normalize the spectra foreach simulation and for each time separately, such thatP

k?P (k?)/�k = 1. Finally, we average the (normal-

ized) spectral curves over a time interval from 1.4 to 1.6eddy turnover times in order to smooth out the statisticalfluctuations in P (k?).The turbulent spectra of the magnetic, perpendicu-

lar electric, and electron density fields are compared inFig. 5. The spectra are shown for all simulation runs,using short-time averaged data for the FK model. For

reference, we also plot the FK spectra obtained from theraw PIC data in the higher ✏ runs (A1 and B1 in Ta-ble 1). Since di↵erent numerical resolutions were used fordi↵erent models, the scales at which (artificial) numeri-cal e↵ects become dominant di↵er between the models.For the HK model, it can be inferred from the spectrumthat numerical e↵ects due to low-pass filters (Lele 1992)become dominant for k? & 1/d

e

. On the other hand, forthe FK, GK model, and KREHM, the physically well-resolved scales are roughly limited to k? . 1/⇢

e

. In theGK model and KREHM, the collisionless cascade is ter-minated by hyperdi↵usive terms. In the FK runs, themain limiting factor are the background thermal fluctu-ations, which dominate over the collisionless turbulenceat high k?, as indicated by the grey shading in Fig. 5.Two notable features stand out when comparing the

spectra over the range of well-resolved scales. First,good agreement is seen between the FK and GK spec-tra, and secondly, the HK spectra have shallower spec-tral slopes at ion scales. Under the assumptions madein the derivation of GK, the kinetic-scale, electromag-netic cascade can be understood in the context of kineticAlfven wave (KAW) turbulence—the nonlinear inter-action among quasi-perpendicularly propagating KAWpackets (Howes et al. 2008a; Schekochihin et al. 2009;Boldyrev et al. 2013). Thus, the comparison between theGK and FK turbulent spectra suggests a predominantlyKAW type of turbulent cascade between ion and elec-tron scales without major modifications due to physicsnot included in GK, such as ion cyclotron resonance (Li &Habbal 2001; Markovskii et al. 2006; He et al. 2015), highfrequency waves (Gary & Borovsky 2004; Gary & Smith2009; Verscharen et al. 2012; Podesta 2012), and largefluctuations in the ion and electron distribution func-tions. We emphasize here that our results are, strictlyspeaking, valid only for the particular setup consideredand could be modified in three-dimensional geometryand/or for a realistic value of the ion-electron mass ratio.Concerning the HK results, the shallower slopes can beregarded as manifest for the influence of electron kineticphysics on the ion-scale turbulent spectra. In particular,several previous works demonstrated that electron Lan-dau damping could be significant even at ion scales of thesolar wind (Howes et al. 2008a; TenBarge & Howes 2013;Told et al. 2015, 2016b; Banon Navarro et al. 2016), andtherefore, we explore this aspect further in what follows.The role of electron Landau damping is investigated

with KREHM, for which electron Landau damping is theonly process leading to irreversible heating.1 All other

1 The use of the term irreversible implies here the productionof entropy, which is ultimately achieved by collisions. For weaklycollisional plasmas, such irreversible heating can only become sig-nificant if there exists a kinetic mechanism, such as Landau damp-ing, able of generating progressively smaller velocity scales in the

9

Figure 4. Global energy time traces for �

i

= 0.1 (left) and �

i

= 0.5 (right). Shown from top to bottom are therelative changes in the magnetic, kinetic ion, kinetic electron, internal ion, and internal electron energy densities. Allquantities are normalized to the initial turbulent energy E

0

, which we define as the sum of magnetic and ion kineticenergy density.

dissipation channels are ordered out as a result of the low� and low ✏ limit (Zocco & Schekochihin 2011). To testthe influence of electron heating on the KREHM spectra,we performed a new set of simulations in the isothermalelectron limit with g

e

in Eq. (13) set equal to g

e

= 0.In Figure 6 we compare the FK and HK magnetic fieldspectra with the KREHM spectra obtained from simula-tions with and without electron heating. Only the curvescorresponding to the low ✏ runs (A2 and B2) are shownfor the FK and HK models. Over the range of scaleswhere numerical dissipation in the HK runs is negligible(k? . 1/d

e

), excellent agreement between the isother-mal KREHM spectra and the HK spectra is found forthe �

i

= 0.1 case. On the other hand, when the isother-mal electron assumption is relaxed, the KREHM resultis much closer to the FK than to the HK spectrum. Asimilar trend can be inferred for the �

i

= 0.5 regime, al-beit with some degradation in the accuracy of KREHM,which is to be expected given its low � assumption. Fur-

perturbed distribution function (Howes et al. 2008a; Schekochihinet al. 2009; Loureiro et al. 2013; Numata & Loureiro 2015; Pezziet al. 2016; Banon Navarro et al. 2016).

thermore, to demonstrate that electron heating indeedleads to small-scale parallel velocity space structures,reminiscent of (linear) Landau damping, we compute theHermite energy spectrum of g

e

(Schekochihin et al. 2009;Zocco & Schekochihin 2011; Loureiro et al. 2013; Hatchet al. 2014; Schekochihin et al. 2016), which is shownFig. 7. The Hermite energy spectrum can be regardedas a decomposition of the free energy among parallel ve-locity scales (the higher the Hermite mode number m,the smaller the parallel velocity scale). Due to a lim-ited number of Hermite polynomials used in the KREHMsimulations (M = 30), no clean spectral slope can be ob-served. However, over a limited range of scales (m . 7),the Hermite spectrum is shallow and suggests a ten-dency towards the asymptotic limit ⇠ m

�1/2 predictedfor linear phase mixing, i.e. Landau damping (Zocco &Schekochihin 2011). We also mention that a recent three-dimensional study of the turbulent decay of the Orszag-Tang vortex demonstrated a clear convergence towardsthem�1/2 limit for much larger sizes of the Hermite basis(up to M = 100), thus indicating a certain robustness ofthe result (Fazendeiro & Loureiro 2015).

10

Figure 5. One-dimensional k? energy spectra for �i

= 0.1 (left) and �

i

= 0.5 (right). Shown from top to bottom arethe �B spectra, the E? spectra, and the electron density spectra. The grey-shaded regions indicate the range wherePIC noise dominates over the turbulent fluctuations for the short-time averaged fully-kinetic PIC data. The grey linesshow the FK spectra for the raw data.

Finally, the importance of electron Landau dampingat ion scales might be somewhat overestimated in ourstudy due to the use of a reduced ion-electron mass ratioof 100, resulting in a ⇠ 4.3 times smaller scale sepa-ration between ion and electron scales (compared to areal hydrogen plasma). In this work, no definitive an-swer concerning the role of the reduced mass ratio can begiven and further investigations with higher mass ratioswill be needed to clarify this aspect. We note, however,that a recent GK study of KAW turbulence, as well asa comparative study of linear wave physics contained inthe FK, HK, and GK model found that ion-scale electronLandau damping can be relevant even for a realistic massratio (Told et al. 2016b; Banon Navarro et al. 2016). Wealso mention that, in principle, one should not overlookthe fact that electron Landau damping is absent in theHK model not only for KAWs but for all other waves aswell, such as the fast/whistler, slow, and ion Bernsteinmodes.To compare the spectral properties of the solutions in

even greater detail, we examine the ratios of the 1D k?spectra (Gary & Smith 2009; Boldyrev et al. 2013; Salemet al. 2012; Chen et al. 2013; Franci et al. 2015; Cerriet al. 2016). The ratios are often considered in literatureas a diagnostic for distinguishing di↵erent types of wave-

Figure 6. Comparison of the FK, HK, and KREHMmag-netic field spectra with the isothermal electron limit ofKREHM for �

i

= 0.1 (top) and �

i

= 0.5 (bottom).

like properties (e.g. KAWs versus whistler waves). Weemphasize that the reference to wave physics should notbe considered as an attempt of demonstrating the domi-

11

Figure 7. Hermite spectrum of the electron free energy,obtained from the KREHM simulation corresponding tothe �

i

= 0.1 set of runs. The m

�1/2 slope is shown forreference.

nance of linear dynamics over the nonlinear interactions.Instead, the motivation for considering linear propertiesshould be associated with the conjecture of critical bal-ance (Goldreich & Sridhar 1995; Howes et al. 2008a; Ten-Barge & Howes 2012; Boldyrev et al. 2013), which allowsfor strong nonlinear interactions while still preservingcertain properties of the underlying wave physics. It isalso worth mentioning that a nonzero parallel wavenum-ber kk along the magnetic field is required for the ex-istence of many types of waves (such as KAWs), rele-vant for the solar wind. Wavenumbers with |k

z

| > 0along the direction of the mean field are prohibited inour simulations due to the two-dimensional geometryadopted in this work. While k

z

= 0, the magnetic fluc-tuations �B

<

K?above some reference scale `

0

⇠ 1/K?may act as a local guide field on the smaller scales(` ⇠ 1/k? < `

0

), giving rise to an e↵ective parallelwavenumber kk ⇠ k? · �B<

K?/B

0

(Howes et al. 2008a;Cerri et al. 2016; Li et al. 2016). In this way, certain waveproperties may survive even in two-dimensional geome-try, even though a fully three-dimensional study wouldsurely provide a greater level of physical realism.Four di↵erent ratios of the 1D spectra are considered:

C

A

= |E?|2/|�B?|2, C

p

= �

2

i

|�ne

|2/|�Bz

|2,C

e

= |�ne

|2/|�B|2, Ck = |�Bz

|2/|�B|2,

where C

e

and Ck are known as the electron and mag-netic compressibility, respectively. Large scale Alfvenicfluctuations have C

A

⇠ 1, whereas C

p

⇠ 1 implies abalance between the perpendicular kinetic and magneticpressures. In the asymptotic limit 1/⇢

i

⌧ k? ⌧ 1/⇢e

,the ratios for KAW fluctuations are expected to be-have as C

A

⇠ (k?⇢i)2/(4 + 4�i

), C

e

⇠ 1/(�i

+ 2�2

i

),Ck ⇠ �

i

/(1 + 2�i

), and C

p

⇠ 1, assuming k? � kk,singly charged ions, and T

0,i

= T

0,e

(Schekochihin et al.2009; Boldyrev et al. 2013). Thus, the C

A

ratio is ex-pected to grow with k?, while the other three ratios are

in the first approximation roughly independent of k? onion scales. In the regime �

i

. 1 relevant to this work, wealso expect C

e

& Ck.The spectral ratios are compared in Fig. 8. Focusing

first on the range of scales 1/⇢i

. k? . 1/⇢e

, we findgood agreement between the GK and FK models, albeitwith moderate disagreements between the C

p

ratios inthe �

i

= 0.1 regime. Thus, the ratios provide firm evi-dence for a KAW cascade scenario for �

i

= 0.5, whereasfor �

i

= 0.1, even though apparently dominated by KAWfluctuations, the cascade is also influenced by phenomenaexcluded in GK. Considering the large-scale dynamics inthe k? . 1/⇢

i

range, very good agreement is found be-tween the FK and HK results, whereas the GK C

p

andCk ratios significantly deviate. It is worth noticing thatthe C

p

ratio depends only on the magnetic fluctuationsparallel to the mean field �B

z

, which represent a fractionof the total magnetic energy content as evident from theresults for Ck.The general trends exhibited by the FK and HK ra-

tios seem to suggest the presence of high frequency waves(e.g. fast magnetosonic modes) excluded in GK. To checkif the FK and HK solutions in the k? . 1/⇢

i

range (andeven beyond for �

i

= 0.1) are indeed consistent with amixture of modes present (KAWs, slow modes, entropymodes) and excluded (e.g. fast modes) in GK, we numer-ically solve the HK dispersion relation in the �

i

= 0.1regime using a recently developed HK dispersion rela-tion solver (Told et al. 2016a). The qualitative findingsdiscussed below for �

i

= 0.1 can be also applied to the�

i

= 0.5 case (not shown here).A collection of least damped modes identified with the

HK linear dispersion relation solver is shown in Fig. 9.Two fixed propagation angles with respect to the mag-netic field are considered; 89 and 85 degrees. The iden-tified branches can be interpreted as KAWs, fast modes,and generalized ion Bernstein modes. For the sake of sim-plicity, we do not consider slow modes and entropy modesbecause our goal here is to study the properties of wavesexcluded in GK [for a detailed discussion on GK wavephysics see Howes et al. (2006)]. The identified ion Bern-stein modes are generalized in the sense that they arenot purely perpendicularly propagating nor electrostaticand they also split into multiple branches around the cy-clotron frequency. As pointed out by Podesta (2012), thefine splitting of the ion Bernstein modes may increase theopportunity for wave coupling with the fast and KAWbranch.Having identified the main branches of interest, we

now proceed to show the corresponding spectral ratios(Fig. 10). In addition to the spectral ratios obtainedfrom the HK solver, we also show for reference theGK prediction for the spectral ratios of KAWs (usingm

i

/m

e

= 100), as well as the linear damping rates of the

12

Figure 8. Ratios of the one-dimensional turbulent spectra. The left panel shows the ratios for the �

i

= 0.1 runs andthe right panel shows the ratios for �

i

= 0.5.

Figure 9. Numerical solutions of the hybrid-kinetic dis-persion relation for �

i

= 0.1. The labels KA, F, and IBare used to denote kinetic Alfven waves, fast waves, and(di↵erent branches of) ion Bernstein modes, respectively.

HK branches. Comparing the linear predictions with theturbulent ratios shown in the left panel of Fig. 8, we findthat the directions in which the FK and HK turbulent ra-tios are “pulled away” from the GK curves are consistentwith linear properties of fast and ion Bernstein modes. Inparticular, the magnetic compressibility Ck is order unityfor the fast and ion Bernstein modes, whereas their C

p

ratio is below the pressure balance regime C

p

⇠ 1. Sim-ilarly, the electron compressibility C

e

exceeds the one of

KAWs at low wavenumbers. All these properties are inqualitative agreement with the general trends exhibitedby the FK and HK turbulent solutions for k? . 1/⇢

i

. Re-garding the C

A

ratio, we note that �B?/�B ⌧ 1 for thefast and ion Bernstein modes, meaning that these modescannot significantly influence the total |E?|2/|�B?|2 ra-tio when mixed together with the Alfvenic fluctuations.Looking at the damping rates, linear theory predicts cy-clotron damping of the fast mode already at k? ⇠ 1/d

i

.However, the FK and HK turbulent ratios deviate fromthe GK model beyond the k? ⇠ 1/d

i

scale. Therefore,the coupling of the fast modes to the ion Bernstein modesis likely significant for the HK and FK ratios. The mostnatural candidate for the conversion of the fast wavewould be the mode denoted as IB1 in Fig. 9, which isvery weakly damped in the HK model and allows for acontinuation of the fast branch above the ion cyclotronfrequency. In the FK model, ion Bernstein waves aresubject to additional damping due to electron Landauresonance which could potentially explain why all theFK ratios show a tendency for converging onto the GKcurves with increasing wavenumbers, whereas in the HKmodel, the turbulent ratios (in particular C

p

and Ck) donot share the same trend.In summary, we conclude that the deviation of the GK

model from the FK and HK turbulent solutions in therange k? . 1/⇢

i

, and possibly even beyond for �i

= 0.1,

13

Figure 10. Spectral ratios corresponding to di↵erentsolutions of the HK dispersion relation (top four plots)and their linear damping rates (bottom). For reference,we also show the GK prediction for the spectral ratios ofKAWs.

can be reasonably well explained with the presence offast magnetosonic and ion Bernstein modes, coexistingtogether with Alfvenic fluctuations. On the other hand,we admit that alternative explanations involving, for ex-ample, non-wave-like phenomena could be in principlepossible. As a side note, it is also worth mentioningthat according to linear theory, KAWs should undergocyclotron resonance around k?di ⇠ 10 (see bottom plotin Fig. 10), resulting in an abrupt change in the spectralratios. No such variation is found in the FK turbulentratios. This suggests that cyclotron resonance has in ourcase only a minor e↵ect on the turbulent cascade, in con-sistency with the arguments against cyclotron resonancegiven by Howes et al. (2008a).

4.4. Ion and electron nonthermal free energy

fluctuations

At last, motivated by previous works on non-Maxwellian velocity structures in HK simulations (Grecoet al. 2012; Valentini et al. 2014; Servidio et al. 2015) andfree energy cascades in GK and KREHM (Schekochi-hin et al. 2009; Banon Navarro et al. 2011; Zocco &Schekochihin 2011; Told et al. 2015; Schekochihin et al.2016), we investigate the spatial distribution of the non-

thermal species free energy fluctuations:

�Es

=

ZT

0,s

�f

2

s

2F0,s

d3v, (22)

where �f

s

is the perturbed part of the distribution func-tion with vanishing lowest three moments (density, fluidvelocity, and temperature), T

0,s

is the equilibrium tem-perature, and F

0,s

is the background Maxwellian. Ex-pression (22) can be regarded as a measure for quanti-fying the deviations from local thermodynamic equilib-rium, characterized by non-Maxwellian fluctuations invelocity space that set the stage for irreversible plasmaheating to occur. Due to limited data availability forthis diagnostic, we present here only the results froma subset of all simulations and models. In particu-lar, we focus on the �

i

= 0.1 regime and considerthe ion nonthermal fluctuations calculated from the HKmodel and the electron nonthermal fluctuations calcu-lated from the v? integrated distribution functions inthe FK model and KREHM. In the HK and FK mod-els, we define �f

s

as �f

s

= f

s

� F

0,s

, where F

0,s

isthe local Maxwellian with matching density, fluid veloc-ity, and temperatures of the total distribution f

s

. InKREHM, the equivalent of expression (22) can be definedas �Ek,e = n

0

T

0,e

PM

m=3

g

2

m

/2, where gm

are the Hermiteexpansion coe�cients of g

e

(Zocco & Schekochihin 2011;Schekochihin et al. 2016). By skipping the lowest threecoe�cients (m 2) in the above sum, the contributionsfrom density, fluid velocity, and temperature fluctuationsare explicitly excluded from the KREHM free energy.Furthermore, we only consider the vk electron fluctua-tions since g

e

does not depend on v? in KREHM.The nonthermal ion and electron free energy fluctua-

tions are shown in Fig. 11. The snapshots correspondto the �

i

= 0.1 runs around 1.5 eddy turnover times,with ✏ = 0.2 in the HK and FK simulations. For ref-erence, we also show in Fig. 12 the corresponding ionand electron temperature fluctuations �T

s

= T

s

� hTs

i,plotted at same time in the HK and FK simulation. Ahighly nonuniform spatial distribution of the nonthermalfree energy is found for both ions and electrons. Fur-thermore, by overplotting the contours of the vector po-tential A

z

we find that the nonthermal fluctuations aremostly concentrated around small-scale magnetic recon-nection sites, corresponding (in 2D) to the X-points ofA

z

. The observed correlation between reconnection sitesand nonthermal fluctuations is in good agreement withprevious works (Greco et al. 2012; Valentini et al. 2014;Haynes et al. 2014; Servidio et al. 2015) and as such rein-forces the idea that reconnection might play a significantrole in the turbulent heating of kinetic-scale, collisionlessplasma turbulence. Compared to Fig. 12, it is seen thatthe nonthermal free energy peaks are well correlated withtemperature fluctuations, the main di↵erence to the lat-

14

ter being that the nonthermal fluctuations are spatiallymore di↵used. This seems to suggest that the energy sup-plied to the particles at reconnection sites is progressivelycascaded to smaller velocity scales in the reconnectionoutflows, thus allowing the nonthermal fluctuations tospread over a larger volume compared to the temperaturefluctuations. The generation of non-Maxwellian velocityspace fluctuations can be, on the other hand, linked withthe presence of wave-particle interactions such as Lan-dau damping, as well as with kinetic instabilities (Hayneset al. 2014).While the role of Landau damping in kinetic-scale, col-

lisionless plasma turbulence is somewhat less known forthe ions, the significance of electron Landau dampinghas been demonstrated in many previous studies (Howeset al. 2008a; TenBarge & Howes 2013; Told et al. 2015;Banon Navarro et al. 2016) and in Sec. 4.3 of this work.Conversely, we find—at the same time—that reconnec-tion could as well be a major process leading to heatingin collisionless plasma turbulence. Similar as for Lan-dau damping, the latter idea is also well-supported by anumber of previous studies (Sundkvist et al. 2007; Osmanet al. 2011; Karimabadi et al. 2013; Chasapis et al. 2015;Matthaeus et al. 2015). Some of the previous works triedto establish a sharp distinction between heating due towave-particle interactions versus heating occurring as aconsequence of reconnection, often favouring one of thetwo possibilities as the main cause for turbulent heat-ing. As one additional option, we would like to suggesta rather di↵erent view, which is that these two seem-ingly distinct processes might work hand in hand andare spatially entangled with each other, making a cleardistinction between heating due to Landau damping andreconnection a somewhat ill-posed task. Instead, it mightbe more instructive to consider these two processes in aunified framework and investigate, for example, how the(nonuniform) damping via Landau resonance might bea↵ected by the presence of reconnection outflows, whichcould (locally) modify the Landau resonance condition.The view proposed here is supported by previous studiesof weakly-collisional reconnection (Loureiro et al. 2013;Numata & Loureiro 2015), and by a recent study of GKturbulence (Klein et al. 2017) which showed that, con-trary to naive expectations, Landau damping in a turbu-lent setting can be spatially highly nonuniform and mightbe responsible for the intermittent heating observed inthe vicinity of current sheets (Sundkvist et al. 2007; Os-man et al. 2011; Wu et al. 2013; Karimabadi et al. 2013;Chasapis et al. 2015).

5. CONCLUSIONS

In this work, we performed a detailed comparison ofkinetic models in two-dimensional, collisionless plasmaturbulence with emphasis on kinetic-scale dynamics.

Four distinct models were included in the comparison:the fully-kinetic (FK), hybrid-kinetic (HK) with fluidelectrons, gyrokinetic (GK), and a reduced gyrokineticmodel, formally derived as a low beta limit of gyrokinet-ics (KREHM). Two di↵erent ion beta (�

i

) regimes andvariable turbulence fluctuation amplitudes (✏) were con-sidered (�

i

= 0.1 using ✏ = 0.1, 0.2 and �

i

= 0.5 using✏ = 0.15, 0.3). The main findings can be summarized asfollows:

• For �

i

= 0.5, the kinetic-scale spectral propertiesof the FK and GK solutions were found to be ingood agreement, thus suggesting a kinetic Alfvencascade scenario from ion to electron scales with-out significant modifications due to physics not in-cluded in GK (Figs. 5 and 8).

• A detailed comparison between the GK and FKspectral ratios at �

i

= 0.1 reveals a slight deviationbetween the two models at kinetic scales (Fig. 8).However, given that the disagreement is clearly ob-served only for two out of four di↵erent spectralratios considered, nor is it clearly seen in the tur-bulent spectra themselves (Fig. 5), it is reasonableto assume that kinetic Alfven fluctuations still playa significant role even in the low-beta regime.

• At the largest scales above the ion gyroradius, theratios of the GK turbulent spectra notably devi-ate from the FK and HK approaches, the likelycause for it being the lack of the fast magnetosonicand ion Bernstein modes in the GK approximation(Figs. 8, 9, and 10). The disagreement is largerfor higher fluctuation amplitudes and for the lowervalue of the ion beta.

• The ion-scale HK spectra were found to be shal-lower than those obtained from the FK and GKsimulations, presumably due to the lack of electronLandau damping (Figs. 5, 6, and 7).

• The real space turbulent field structures are in goodqualitative agreement (Fig. 1). Furthermore, allmodels considered give rise to intermittent statis-tics at kinetic scales, albeit with some minor quan-titative di↵erences between the models (Figs. 2 and3). The main reason for the observed deviationsmight be related to the fact that di↵erent numericalresolutions were used for di↵erent models. Furtherstudies will be necessary to identify if the observedquantitative di↵erences are indeed physical.

• The spatial profiles of the nonthermal ion and elec-tron free energy fluctuations suggest that kinetic-scale reconnection might play an important role in

15

Figure 11. Nonthermal electron and ion free energy fluctuations. In the left and middle plots, we compare the FKand KREHM electron free energy, stemming from small scale structures in vk. The plot on the right shows the (total)nonthermal ion free energy obtained from the HK simulation. Overplotted are the contours of the vector potential A

z

.The color map is normalized to the maximum value of the free energy, separately for each plot.

Figure 12. Ion and electron temperature fluctuations cor-responding to the HK and FK nonthermal free energiesshown in Fig. 11.

the heating of the plasma (Figs. 11 and 12). Fur-thermore, our results also suggest that reconnec-tion might be closely entangled with (nonuniform)Landau damping, making the distinction betweenheating due to reconnection versus heating due toLandau damping an ill-posed task.

• KREHM delivers surprisingly accurate results al-ready for �

i

= 0.1, even though its range of valid-ity is formally limited to �

i

. 0.01 for our choice ofthe reduced mass ratio and equal ion and electronbackground temperatures (Figs. 1, 2, 3, 4, and 5).

We emphasize that, strictly speaking, our findings arevalid only for the particular setup considered and couldbe modified in a more realistic setting. In particular, theresults could be modified in a full three-dimensional ge-ometry, which allows for arbitrary wave propagation an-gles with respect to the mean magnetic field (Howes et al.

2008b; Chang et al. 2014; Vasquez et al. 2014; Howes2015; Wan et al. 2015). Another aspect which might haveimpacted our results is the reduced ion-electron mass ra-tio of 100. For a realistic mass ratio, the increased sep-aration between ion and electron scales might improvethe agreement between the HK and FK models at ionscales as well as potentially reveal certain deviations fromthe phenomenology of KAW turbulence at electron scales(Podesta et al. 2010). No definitive answer regarding therole of these limitations can be presently given. How-ever, based on a number of previous works, it is stillreasonable to expect that the simplified setup adoptedin this work can qualitatively capture at least some ofthe key features of natural turbulence at kinetic-scales ofthe solar wind (Servidio et al. 2015; Li et al. 2016; Wanet al. 2016). Furthermore, many of the turbulent prop-erties found in this work, such as the dominance KAWfluctuations (Salem et al. 2012; Chen et al. 2013), non-Gaussian statistics at kinetic scales (Kiyani et al. 2009;Wu et al. 2013; Chen et al. 2014; Perrone et al. 2016), andkinetic-scale reconnection (Sundkvist et al. 2007; Osmanet al. 2011; Chasapis et al. 2015) are supported by in-situspacecraft observations.Lastly, apart from exposing the pros and cons of

reduced-kinetic treatments, the outcome of the studydemonstrates that detailed comparisons between fully-kinetic and reduced-kinetic models can provide signif-icant advantages for elucidating the nature of kinetic-scale dynamics and may also serve as a much needed aidfor interpreting the results of fully-kinetic simulations,as well as guide the interpretation of experimental data,acquired from present and (potential) future spacecraftmissions (Burch et al. 2016; Fox et al. 2016; Vaivads et al.

16

2016). An ambitious plan to perform a study similar tothe one presented in this work, but involving an evenlarger participation from the community, has been re-cently outlined by Parashar et al. (2015b). For the pro-posed comparative study, and perhaps others to come,our results could provide valuable guidelines.

We gratefully acknowledge helpful discussions withO. Alexandrova, P. Astfalk, V. Decyk, T. Gorler,Y. Kawazura, W. Mori, P. Munoz, F. Tsung, andM. Weidl. The research leading to these results hasreceived funding from the European Research Coun-cil under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013)/ERC Grant AgreementNo. 277870. C. Willmott was supported by MIT’sCharles E. Reed Faculty Initiative Fund. N. Loureirowas partially supported by the NSF-DOE partnershipin basic plasma science and engineering, award no. DE-SC0016215, and by NSF CAREER award no. 1654168.Furthermore, this work was facilitated by the Max-Planck/Princeton Center for Plasma Physics. Computer

resources for the hybrid-kinetic and fully-kinetic simula-tions have been provided by the Max Planck Computingand Data Facility and by the Gauss Centre for Super-computing/Leibniz Supercomputing Centre under grantpr74vi, for which we also acknowledge Principal Inves-tigator J. Buchner for providing access to the comput-ing resource. We also acknowledge the Italian super-computing center CINECA (Bologna) under the ISCRAsupercomputing initiative. The gyrokinetic simulationspresented in this work used resources of the NationalEnergy Research Scientific Computing Center, a DOEO�ce of Science User Facility supported by the O�ce ofScience of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231. Computations with the Viri-ato code were conducted at Massachusetts Green HighPerformance Computing Center on the Plasma Scienceand Fusion Center partition of the Engaging cluster, sup-ported by the Department of Energy. The authors wouldlike to acknowledge the OSIRIS Consortium, consist-ing of UCLA and IST (Lisbon, Portugal) for the use ofOSIRIS, for providing access to the OSIRIS framework.

APPENDIX

A. NUMERICAL DETAILS

For the fully-kinetic, electromagnetic simulations, we use the particle-in-cell (PIC) code OSIRIS (Fonseca et al.2002, 2008). The reader is referred to Dawson (1983) and Birdsall & Langdon (2005) for a detailed overview of thePIC method. Parameters specific to the PIC simulations are given in Table 3. A 2nd order compensated binomialfilter is applied on the electric current at each time step and we perform all simulations using cubic particle shapefactors (Birdsall & Langdon 2005). For the problem type considered here, preliminary tests have shown that cubicshape factors deliver superior numerical results, compared to lower-order quadratic and linear shapes, due to betterenergy conservation properties and a significant reduction in the amount of so-called PIC noise. However, even withthe use of large numbers of particles and smooth shape factors, the background thermal fluctuations may still maskthe fluctuations arising from the turbulent cascade which are of main interest here. Therefore, to further reduce PICnoise we also employ short-time averages of the turbulent fields. The grid data is averaged over a time window oflength �t ⇡ ⌦�1

ci

for A type runs and �t ⇡ 0.5⌦�1

ci

for B type runs, resulting in a sample of over 2,000 snapshots foreach time average.

FK (PIC)

Run N

x

N

ppc

�x/�

D

v

th,e

/c !

pe

/⌦ce

A1 20482 625 1.0 0.174 1.822

A2 20482 625 1.0 0.174 1.822

B1 19202 1024 1.0 0.185 3.820

B2 19202 1024 1.0 0.185 3.820

Table A3:. Numerical parameters for the FK PIC simulations: the real space resolution (Nx

), number of particles percell per species (N

ppc

), the grid spacing in units of the Debye length (�x/�

D

), the electron thermal velocity comparedto speed of light (v

the/c), and the plasma frequency to cyclotron frequency ratio (!pe

/⌦ce

).

The HK simulations are performed with the Eulerian hybrid Vlasov-Maxwell solver HVM (Valentini et al. 2007).For a detailed description of numerical algorithms employed by the HVM code, we refer the reader to Matthews(1994) and Mangeney et al. (2002). Parameters specific to the HK simulations are given in Table 4. An isothermalequation of state is assumed for the electrons. No explicit resistivity is used, but one e↵ectively exists due to thelow-pass filters that are used by the code’s algorithm (Lele 1992). For better consistency with the FK, GK model, and

17

KREHM, the HK simulations are performed using the generalized Ohm’s law given by Eq. (7), which retains electroninertia (Valentini et al. 2007).

HK (Eulerian)

Run N

x

N

v

�x/d

e

v

max

/v

th

A1 5122 513 0.49 3.54

A2 5122 513 0.49 3.54

B1 5122 513 0.49 3.54

B2 5122 513 0.49 3.54

Table A4:. Parameters for the Eulerian HK simulations: the real space resolution (Nx

), the velocity space resolution(N

v

), grid spacing (�x/d

e

), and the maximally resolved velocity (vmax

/v

th

).

The GK simulations are carried out with the Eulerian GK code Gene (Jenko et al. 2000). Parameters describingthe numerical settings for the Eulerian GK simulations are given in Table 5. Only one simulation is performed for typeA and type B runs because ✏ does not appear as a parameter in the normalized GK equations. For the Orszag-Tangvortex initialization, we follow the implementation details described in Numata et al. (2010). Numerical instabilities atthe grid scale are eliminated by adding high-order hyperviscous (⇠ k

8

?) and hypercollisional terms to the perpendiculardynamics in spectral space and to the parallel dynamics in velocity space, respectively. With Gene being a grid-basedspectral code, the (artificial) symmetry of the Orszag-Tang vortex is reflected in the numerical solution at all scaleswith high precision. In the FK model, on the other hand, minor asymmetries are naturally present due to PIC noise.In order to incorporate slight deviations from the ideal symmetry into the initial condition, we use an approach similarto the one employed by TenBarge et al. (2014) and add a small amount of random perturbations to the electron andion distribution functions. The perturbations are limited to the wavenumber range k? . 1/d

e

.

GK (Eulerian)

Run N

x

N

vk N

µ

�x/⇢

e

v

max,s

/v

th,s

A 15362 32 16 0.73 3.5

B 7682 32 16 0.65 3.5

Table A5:. Parameters for the Eulerian GK simulations: total number of grid points in spectral space (Nx

), parallelvelocity resolution (N

vk), µ space resolution (Nµ

), grid spacing (�x/⇢

e

), and the maximally resolved species velocity(v

max,s

/v

th,s

). Note that the number of fully-dealiased modes in spectral space is given by (2/3)2Nx

.

For the KREHM simulations we use the Viriato code (Loureiro et al. 2016). The chosen numerical parametersare listed in Table 6. The implementation details for the Orszag-Tang vortex are described in the abovementionedreference. Similar to Loureiro et al. (2013), hyperviscous and hyperresistive terms are added to the perpendiculardynamics in order to terminate the turbulent cascade before it reaches the limits of the numerical grid. For theparallel velocity space, a model hypercollision operator is adopted. Even though KREHM is a low-beta limit of GK,we still perform separate simulations for the �

i

= 0.1 and �

i

= 0.5 case by varying the ⇢

i

/d

e

ratio for the two cases.

KREHM

Run N

x

M �x/d

e

A 15362 30 0.16

B 7682 30 0.33

Table A6:. Parameters for the KREHM Fourier-Hermite spectral simulations: total grid size in spectral space (Nx

),size of the Hermite basis for vk (M), and the grid spacing (�x/d

e

).

18

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