Extended fluid models: Pressure tensor effects and equilibriaS. S. Cerri, P. Henri, F. Califano, D. Del Sarto, M. Faganello et al. Citation: Phys. Plasmas 20, 112112 (2013); doi: 10.1063/1.4828981 View online: http://dx.doi.org/10.1063/1.4828981 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v20/i11 Published by the AIP Publishing LLC. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

Extended fluid models: Pressure tensor effects and equilibria

S. S. Cerri,1,2 P. Henri,1,3 F. Califano,1 D. Del Sarto,4 M. Faganello,5 and F. Pegoraro1

1Physics Department “E. Fermi,” University of Pisa and CNISM, Largo B. Pontecorvo 3, 56127 Pisa, Italy2Max-Planck-Institut f€ur Plasmaphysik, EURATOM association, Boltzmannstr. 2, D-85748 Garching, Germany3Universit�e de Nice Sophia Antipolis, CNRS, Observatoire de la Cote d’Azur, BP 4229 06304, Nice Cedex 4,France4Institut Jean Lamour, UMR 7198 CNRS – Universit�e de Lorraine, BP 239 F-54506 Vandoeuvre les Nancy,France5International Institute for Fusion Science/PIIM, UMR 7345 CNRS Aix-Marseille University, Marseille,France

(Received 20 August 2013; accepted 15 October 2013; published online 25 November 2013)

We consider the use of “extended fluid models” as a viable alternative to computationally demanding

kinetic simulations in order to manage the global large scale evolution of a collisionless plasma while

accounting for the main effects that come into play when spatial micro-scales of the order of the ion

inertial scale di and of the thermal ion Larmor radius .i are formed. We present an extended two-fluidmodel that retains finite Larmor radius (FLR) corrections to the ion pressure tensor while electron

inertia terms and heat fluxes are neglected. Within this model we calculate analytic FLR plasma

equilibria in the presence of a shear flow and elucidate the role of the magnetic field asymmetry.

Using a Hybrid Vlasov code, we show that these analytic equilibria offer a significant improvement

with respect to conventional magnetohydrodynamic shear-flow equilibria when initializing kinetic

simulations. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828981]

I. INTRODUCTION

The theoretical modeling of the multi-scale dynamics of

a magnetized plasma is a top priority in laboratory and space

plasma research, as well as a frontier problem in computa-

tional plasma physics. Full kinetic Lagrangian and Eulerian

simulations based on the Vlasov-Maxwell system are still far

from being considered to be the best tools for multi-scale

plasma modeling because they are still too demanding in

terms of computational resources and, in addition, because

of the difficulty in constructing Vlasov equilibria capable of

representing the initial large scale systems to be investigated

in a realistic fashion.

An alternative approach with respect to a full kinetic

description is to develop fluid models that extend the large-

scale magnetohydrodynamic (MHD) (or multi-fluid)

approach towards regimes where the effects related to the

presence of fluctuations at scale lengths comparable to the

ion Larmor radius .i (or the ion skin depth di) are included.

These models can be used provided the characteristic fre-

quencies of the system remain smaller than the ion cyclotron

frequency and phenomena related to ion cyclotron resonan-

ces remain negligible. Extended models have a wide range

of applications, from laboratory and space plasma turbulence

to magnetic reconnection studies. More generally, they can

describe systems where the energy injected at large scales

cascades efficiently towards the ion Larmor radius (.i)

and/or the ion inertial scale (di) scale lengths, in some cases

up to electron micro-scales. This is the case in magnetic

reconnection events or after the development of strong fluid-

like instabilities such as the Kelvin-Helmholtz instability

(hereafter KHI).

The possibility of using “extended fluid models”

becomes crucially important when studying systems where

the large scale fields, defined as the “equilibrium fields,”

vary over typical scale lengths that do not exceed the kinetic

scales by too large a factor as is the case, for example, in the

region at the interface between the solar wind and the

Earth’s magnetosphere or the magnetotail. Such extended

fluid models could also play a key role when coupling fluid

and kinetic codes1–3 since they include some of the physics

bridging the low frequency MHD regime and higher fre-

quency regimes. Here, we define here as “kinetic effects”

first order Finite Larmor Radius (FLR) effects such as

“gyroviscosity”. We do not include the expansion of the

Bessel functions, e.g., J20ðk?.iÞ ’ 1� k2

?.2i =2, which are of

higher order in .i.

The need to extend the standard fluid approach, which is

closed in general with a polytropic equation of state, and to

include terms related to the pressure tensor, so as to avoid

the limitations of an isotropic pressure in the equations of

motion, is specially evident in the presence of a large scale

shear flow. This is due to the fact that in an almost collision-

less system the shear flow and the pressure tensor are

strongly coupled and interact on fast time scales. This is the

case when studying the dynamics at the low latitude bound-

ary layer between the Earth’s magnetosphere and the solar

wind where large-scale MHD-type vortices are generated by

the KHI driven by the velocity shear between the solar wind

flow and the magnetospheric plasma at rest (see Ref. 4 and

references therein). In this case the typical size of the MHD

vortices is of the order of several di (or .i), corresponding to

nearly five to ten times the typical scale length of the large

scale shear flow which is of the order of a few di (or .i). In a

theoretical description, this flow is considered as the

“equilibrium” configuration that drives the KHI. Until now,

in order to model such a system, MHD equilibrium configu-

rations have been considered where the shear flow and the

1070-664X/2013/20(11)/112112/13/$30.00 VC 2013 AIP Publishing LLC20, 112112-1

PHYSICS OF PLASMAS 20, 112112 (2013)

density variation between the two plasmas are represented by

a one dimensional profile, typically a hyperbolic tangent.5 The

total pressure balance is ensured either by a change in the am-

plitude of the magnetic field (that is nearly perpendicular to

the plane of the vortices)6 or in the temperature.7 An isother-

mal or an adiabatic polytropic equation of state is used.

However, such MHD equilibria with an isotropic pres-

sure are unsatisfactory for fluid simulations. In fact the inclu-

sion of, at least, first order corrections due to the pressure

tensor, the so-called FLR effects, alters the shape of the equi-

librium fields. As a consequence, not only is the system dy-

namics influenced by small-scale (non-MHD) effects but

also the very initial configuration is no longer an equilibrium

as it evolves on a short transient timescale. This obscures the

development of the “large-scale” KHI even in its linear stage

as it becomes mixed up with the non-equilibrium dynamics

of the initial profiles. In addition, the pressure tensor effects

depend on the relative orientation between the fluid vorticity

and the mean magnetic field. The importance of this problem

is underlined by the difficulty that is encountered when com-

paring the linear evolution of the KHI using different fluid

and kinetic models, as discussed in a recent paper8 where the

same initial MHD setup was adopted while running MHD,

two-fluid (TF), particle-in-cell (PIC)-Hybrid and full PIC

simulations. Hybrid and full kinetic simulations have shown

that compressible traveling perturbations are artificially

excited at the velocity shear layer when a MHD initial con-

figuration is imposed, and that no clear relaxation of such ar-

tificial disturbances is observed in the collisionless regime.

This emphasizes the importance of starting from a correct

equilibrium configuration when dissipation processes such as

collisional viscosity and resistivity are inefficient.

Our first aim is to present an “extended fluid model,”

including FLR microscopic effects, that captures the physics

of magnetized plasma phenomena approaching the ion cy-

clotron frequency. Since in this paper the focus is on ion dy-

namics the limit of massless electrons is considered.

Subsequently, this model is used for constructing novel

“extended fluid equilibria” starting from a MHD equilibrium.

Our second aim is to show, using a Vlasov-Hybrid numerical

code, that these “FLR equilibria” provide a better starting

point for kinetic simulations than MHD equilibria since they

include the main effects of the coupling between the pressure

tensor and the shear flow.

This paper is organized as follows. In Sec. II, we present

our extended two-fluid model which we name the eTF

model. In Sec. III, we introduce the problem of the role of

the pressure tensor on a standard MHD fluid equilibrium in

the presence of a shear flow. In Sec. IV, we give explicit ana-

lytic equilibrium solutions using our eTF model including

FLR corrections. A Vlasov-Hybrid code is used in Sec. V to

show that our analytic equilibria can be considered as

“quasi-equilibria” useful for initializing full kinetic simula-

tions. Conclusions are given in Sec. VI.

In Appendix A, the eTF ordering is introduced in the

pressure tensor equations and the derivation of the eTF equa-

tions is detailed with attention to the role of the asymmetry

under inversion of the magnetic field vector. In Appendix B,

an improved derivation of the eTF equilibria is given. In

Appendix C, a procedure to construct exact self-consistent

kinetic plasma equilibria in the presence of shear flows is

illustrated.

II. EXTENDED TWO-FLUID MODEL EQUATIONS

We consider a plasma system composed of two species,

electrons and ions (protons), denoted by indexes e and i,respectively. Each fluid behaves as a collisionless fluid inter-

acting through the electromagnetic fields. We assume that

the quasi neutrality condition ne ’ ni � n holds. When a

standard isotropic pressure closure is adopted (for example,

the isothermal or adiabatic law) this model is simply referred

to as the two-fluid model of a plasma.5 In addition, for the

sake of simplicity, we take massless electrons.

More generally, depending on the highest order moment

that we choose to retain, the two-fluid theory variables are:

the number density n, the fluid velocity ua, the pressure ten-

sor Pa and the heat flux tensor Qa, with a¼ e, i. In the fol-

lowing, all quantities are expressed in dimensionless form by

using ion characteristic quantities: di, Xi, VA, minV2A where

VA is the Alfv�en velocity calculated at a given point in the

plasma (in our simulations at the right boundary x ’ Lx=2 of

our simulation domain where all fields, density, velocity,

etc., are assumed to become asymptotically constant). The

electric field and the magnetic field are normalized to �E ¼miVAXi=e and �B ¼ micXi=e, respectively.

In the limit of massless electrons the fluid velocities are

ui¼U and ue ¼ U� J=n, and the model equations become

@n

@tþ r � ðnUÞ ¼ 0; (1)

@ðnUÞ@t

þ r½nUUþPB þPð0Þe þPð0Þi þPð1Þi � ¼ 0; (2)

E ¼ �U� B þ J� B

n� rPð0Þe

n; (3)

@B

@t¼ �r� E ; r� B ¼ J; (4)

@pi?@tþ rðpi?UÞ ¼ �pi?ðI� bbÞ : rU � Pð1Þi : rU; (5)

@pe?@tþ r pe? U� J

n

� �� �¼ �pe? ðI� bbÞ :r U� J

n

� �� �;

(6)

@pik@tþ rðpikUÞ ¼ �2pik bb : rU; (7)

@pek@tþ r pek U� J

n

� �� �¼ �2pek bb : r U� J

n

� �� �;

(8)

where PB ¼ ðB2=2ÞðI�bbÞ �ðB2=2Þbb;Pð0Þa ¼ pa?ðI�bbÞþpakbb. Here pak and pa? are the parallel and perpendicular

components of the magnetic pressure tensor of the a species,

b¼B/B is the unit vector along the local magnetic field, and

I – bb is the projector in the perpendicular plane. Finally

112112-2 Cerri et al. Phys. Plasmas 20, 112112 (2013)

Pð1Þi is the first order ion gyroviscosity tensor. In the strong mag-

netic guide field limit, in our case B’ Bez, the ion gyroviscosity

tensor components Pð1Þi;lm, in dimensionless form, are given by9

Pð1Þi;zz ¼ 0; (9)

Pð1Þi;xx ¼ �Pi;yy ¼ �1

2

pi?Bð@yui;x þ @xui;yÞ; (10)

Pð1Þi;xy ¼ Pð1Þi;yx ¼ �1

2

pi?Bð@yui;y � @xui;xÞ; (11)

Pð1Þi;xz ¼ Pð1Þi;zx ¼ �1

B½ð2pik � pi?Þ@zui;y þ pi?@yui;z�

� 1

B@yqi?; (12)

Pð1Þi;yz ¼ Pð1Þi;zy ¼1

B½ð2pik � pi?Þ@zui;x þ pi?@xui;z�

þ 1

B@xqi?; (13)

where qi? and qik are the heat fluxes along the magnetic field

of the perpendicular and parallel thermal ion energy (here-

after neglected, as in pressure equations).

This system of equations conserves energy explicitly as

can be shown by taking the scalar product of the momentum

equation times U, of the generalized Ohm’s law times J, of

the Faraday law times B and summing together the perpen-

dicular and parallel pressure equations (the latter multiplied

by 1/2) of both species which finally leads to

@

@tðEU þ EB þ EP;i þ EP;eÞ þ r

�½ðEU þ EP;iÞIþPð0Þi

þPð1Þi � � Uþ ðEP;eIþPð0Þe Þ � U� J

n

� �þ E� B

�¼ 0;

(14)

where we have defined the following energy densities:

EU ¼nU2

2; EB ¼

B2

2; EP;a ¼

Tr½Pð0Þa �2

¼pak þ 2pa?

2:

(15)

In the following, we will define the set of Eqs. (1)–(13)

as the extended two-fluid model, or the eTF model, while we

define the (standard) two-fluid model, namely the TF model,

as that obtained from the eTF by assuming an isotropic

(scalar) pressure with pa? ¼ pak � Pa and Pð1Þi ¼ 0 with a

polytropic closure relation of the type dðPa � n�caa Þ=dt ¼ 0

with ca the polytropic index: ca¼ 1 (isothermal) or ca¼ 5/3

(adiabatic).

III. LARGE SCALE EQUILIBRIUM WITH A SHEARFLOW

We start by considering an initial MHD equilibrium

configuration in the presence of a shear flow within the

framework of the TF model. The MHD equilibrium is chosen

so as to represent at least schematically the interaction of the

solar wind with the Earth’s Magnetosphere at low latitude.5

This equilibrium and the geometry adopted in the following

can be considered as a standard model for equilibrium sys-

tems with a shear flow.

We take the plasma flow along the y-direction, varying

in the transverse x-direction, and a guiding magnetic field in

the perpendicular z-direction. We also allow for density var-

iations. The equilibrium reads

Uy ¼ U0 tanhðx=LUÞ; Beq ¼ B0ðxÞ ez;

n ¼ n0 �Dn

2½1� tanhðx=LUÞ�;

(16)

where LU is the equilibrium scale length. We define the

smallness parameter ei ¼ .i=LU . For a plasma with vA � vth;i

(i.e., b � 1), we obtain LU=di ¼ ð.i=diÞð1=eiÞ ’ 1=ei. In the

usual MHD approach, the parameter ei must be sufficiently

small so that terms of order ei can be neglected and the total

pressure equilibrium condition requires that thermal pressure

variations be compensated by a corresponding variation in

the magnetic field, B0(x), such that

Pe þ Pi þB2

0

2¼ cst; (17)

where Pe and Pi are the (isotropic) electron and ion pres-

sures. The same equilibrium conditions would hold by con-

sidering gyrotropic pressures and by substituting Pð0Þe;xx ¼ pe?for Pe and Pð0Þi;xx ¼ pi? for Pi, respectively.

Here instead we consider the case of an equilibrium

where the inhomogeneity scale length LU is not much larger

than the ion inertial skin depth, LU � 2 10 di (or .i), as

typically assumed in the problem of the interaction of the

Solar wind with the Earth Magnetosphere. In this case, ei is

not very small and ion kinetic terms such as gyroviscosity

can no longer be neglected even on the equilibrium scale

length. In the following, we investigate the importance of

including first order terms in ei in the rederivation of the

standard equilibrium configuration given above. For the sake

of simplicity, we shall assume .i � di.

In the presence of a shear flow, even assuming an initial

uniform magnetic field (and pressure), the xx and yy-compo-

nents of the ion gyro-viscosity tensor do not vanish:

Pð1Þi;xx ¼ �Pð1Þi;yy ¼ �1

2

pi?;0B0

dUyðxÞdx

: (18)

Adopting the eTF model for the equilibrium configuration

given by Eqs. (16) and (17), Pð1Þi;xx and Pð1Þi;yy are the only non-

zero components of the gyroviscosity tensor Pð1Þi at t¼ 0. As

a result, the configuration spontaneously evolves because of

the force imbalance in the equation of motion, Eq. (2), due

to the diagonal components of the ion gyroviscosity tensor in

the (x, y) perpendicular plane.

We can estimate the order of magnitude of the gyrovis-

cosity time scale sG induced by this imbalance as follows:

@U

@t� �

dPð1Þi;xx

dx) U0

sG� 1

2

pi?;0Xci

d2Uy

dx2� pi?;0

Xci

U0

L2U

;

(19)

112112-3 Cerri et al. Phys. Plasmas 20, 112112 (2013)

which gives sG � L2UXci=pi?;0 (we recall that in dimension-

less units Xci ¼ jB0j). When LU approaches unity, sG also ap-

proaches unity, i.e., the cyclotron time (B0 ’ 1 and pi?;0 ’ 1

in dimensionless units), implying that the system reacts on a

fast dynamical time scale. We can compare this characteristic

time to the characteristic time scale of the growth of the KHI,

e.g., to the inverse of the growth rate of the Fast Growing

Mode (FGM) sKH � 1=cKHðFGMÞ � 10LU=U0.29 Assuming

U0 � 1, B0 � 1 and pi? � 1 as typical values for our system

(see, e.g., Refs. 5, 10, and 11), we obtain

sG

sKH� 0:1

B0U0LU

pi?

� 0:1LU � Oð0:1Þ; (20)

i.e., sKH is much larger than sG if the typical scale of the ve-

locity shear LU is not too large with respect to the ion scale, .i

or di. Thus, if in this case we start from a MHD equilibrium,

the system reacts to the absence of force balance on a very

short time scale, much shorter than the ideal KHI time scale.

This will influence the later development of the KHI even dur-

ing its linear phase. An example of this fast reaction to the ab-

sence of a proper equilibrium condition is shown in Fig. 1

where, using the eTF model, we draw the plasma density pro-

file at different times resulting from an initial configuration

given by a uniform MHD equilibrium with a superposed shear

velocity field, Eqs. (16) and (17) with B0(x)¼B0, Dn¼ 0,

U0¼ 1, LU¼ 3 and B0¼ 1. The left and right frames corre-

spond to inverse alignments of the fluid vorticity with respect

to the guide field, X � B > 0 and X � B < 0, respectively. We

see that density fluctuations (up to 4% in this case, but even

larger for different initial physical parameters) are generated

on a few ion cyclotron times. These perturbations are ejected

from the shear region in the form of soliton-like structures

traveling at constant speed and exiting the configuration

through the (open) left and right boundaries (see Ref. 5 and

references therein for details on the numerical scheme and

boundary conditions). As a consequence, the system evolves

towards a new, different density profile corresponding to a

new equilibrium configuration. Moreover, the change in the

length scale of the velocity gradient and the shape of the new

equilibrium profile depend on the direction of the fluid vortic-

ity field X with respect to the guide field, i.e., on the sign of

X � B. This is in agreement both with previous PIC simula-

tions8 and with our Vlasov-Hybrid simulations (see Sec. V).

Moreover, such an asymmetry seems to emerge even in the

nonlinear phase of the KHI11 by influencing the transport

properties and thus emphasizing the relevance of discerning

between the two configurations.

In Sec. IV, we construct the equilibrium configuration

analytically by solving the equations of the eTF model to

first order in ei.

IV. EQUILIBRIUM CONFIGURATION INCLUDING FLRTERMS

We consider an initial flow velocity field U directed

along the y-direction and varying in the perpendicular x-

direction, U ¼ UyðxÞey. A guide magnetic field is directed

along the z-direction with an amplitude that may depend on

x. As a result, the time derivatives of the density and of the

parallel and perpendicular pressures are identically zero.

Thus, the equation of motion (2) reduces to

d

dx½Pð0Þi;xx þPð1Þi;xx þPð0Þe;xx þPB;xx� ¼ 0: (21)

Let us assume that we have a valid equilibrium in the

MHD limit, described by three functions FðxÞ; GðxÞ, and

HðxÞ such that

pi?ðxÞ ¼ pi?;0FðxÞ; pe?ðxÞ ¼ pe?;0GðxÞ; B2ðxÞ ¼ B20HðxÞ:

Let us now take three new “correction” functions f(x), g(x),

and h(x) such that

FIG. 1. Frame. The density profile n(x)/n0 at different time instants resulting from an eTF simulation using an initial uniform MHD equilibrium (dotted line)

with a shear velocity field: We see a perturbation traveling towards the borders and eventually leaving the simulation domain. Note that the black continuous

line that corresponds to the final configuration partially covers the curves at previous times. The parameters of the equilibrium given in Eqs. (16) and (17) are:

B0(x)¼B0, Dn¼ 0, U0¼ 1, LU¼ 3, and B0¼ 1. The left and right frames correspond to the case X � B > 0 and X � B < 0, respectively.

112112-4 Cerri et al. Phys. Plasmas 20, 112112 (2013)

pi?ðxÞ ¼ pi?;0FðxÞ f ðxÞ; pe?ðxÞ ¼ pe?;0GðxÞ gðxÞ;B2ðxÞ ¼ B2

0HðxÞ hðxÞ

are now equilibrium profiles for the eTF model. Note that we

are considering an equilibrium velocity field Uy(x) in Eq.

(16) such that limx!61 dUyðxÞ=dx ¼ 0, i.e., the system is

homogeneous outside the shear layer. Therefore, since the

gyroviscosity depends on the derivative of the velocity field,

the three corrective functions must approach unity as we

move away from the central shear layer, thus recovering the

MHD profiles. We stress that this approach, with minor mod-

ifications, remains valid for any velocity profile. For the sake

of simplicity, in the following we discuss the case where

FLR corrections are computed in the limit of an “asymptotic

field” B0 defined as the value of the magnetic field at the

right boundary xþ of the simulation domain, i.e., B0¼B(xþ).

We define pi?;0 þ pe?;0 þ B20=2 as the asymptotic values

representing the MHD equilibrium away from the shear layer

where FLR corrections vanish. After some manipulations in

Eq. (21), we obtain

½1� ~u0ðxÞ�~bi?;0FðxÞf ðxÞ þ ~be?;0GðxÞgðxÞ

þ HðxÞ1þ bi?;0 þ be?;0

hðxÞ ¼ 1; (22)

where ~u0ðxÞ � ðdUyðxÞ=dxÞ=ð2B0Þ ¼ ðU0=2B0LUÞcosh�2

ðx=LUÞ; bs?;0 � 2ps?;0=B20 and ~bs?;0 � bs?;0=ð1þ bi?;0

þbe?;0Þ. We proceed by imposing the quasi-neutrality condi-

tion and by assuming the existence of a polytropic-like rela-

tion for each species. We obtain

GðxÞgðxÞ ¼ ½FðxÞf ðxÞ�1þ~c ;

where 1þ ~c ¼ ce?=ci?. Moreover, we assume that the per-

pendicular plasma beta bi?ðxÞ does not change with respect

to the MHD case giving

hðxÞ ¼ f ðxÞ:

Note that this implies that also b?ðxÞ ¼ bi?ðxÞ þ be?ðxÞdoes not change, if and only if ~c ¼ 0, which is true for this

model. This, in turn, implies we have the same electron and

ion perpendicular temperature profiles. In fact, splitting

FðxÞ ¼ F nðxÞF Ti?ðxÞ (and the same for electrons) using the

thermodynamic relation P¼ nT, we obtain F nðxÞ ¼ GnðxÞfrom quasi-neutrality and GTe?ðxÞ ¼ ½F Ti?ðxÞ�

1þ~cfrom the

polytropic hypothesis.

Finally, since HðxÞ and FðxÞ are related by the MHD

equilibrium condition and by the polytropic hypothesis, i.e.

HðxÞ1þ bi?;0 þ be?;0

¼ 1� ~bi?;0FðxÞ � ~be?;0½FðxÞ�1þ~c ;

the equilibrium condition including FLR corrections reads

f1� ~bi?;0FðxÞ~u0ðxÞ þ ~be?;0½FðxÞ�1þ~c ½ðf ðxÞÞ~c�1�gf ðxÞ ¼ 1:

(23)

In general, ~c < 1, and we solve Eq. (23) perturbatively in

this “small” parameter giving the convergence conditions

a posteriori. The zero-order solution for ~c ¼ 0 is given

by

f0ðxÞ ¼ ½1 � C0ðxÞaðxÞ��1; (24)

C0ðxÞ ¼1

2

U0

B0

~bi?;0LUFðxÞ;

where aðxÞ ¼ cosh�2ðx=LUÞ. Then, solving iteratively the

equation

f1 � ~bi?;0FðxÞ~u0ðxÞ þ ~be?;0½FðxÞ�1þ~c ½ðfn�1ðxÞÞ~c � 1�gfnðxÞ

¼ 1

(25)

and Taylor expanding fn�1ðxÞ~c ¼ ð1� OðeiÞaðxÞÞ�~c, we

obtain

fnðxÞ ¼ ½1 � CnðxÞaðxÞ��1; (26)

where

CnðxÞ ¼ C0ðxÞXn

m¼0

½ � ~c~be?;0ðFðxÞÞ1þ~c �m:

This is a geometric series, which converges if and only if

j� ~c~be?;0ðFðxÞÞ1þ~c j < 1. Since we can always choose

pi?;0 ¼ maxðpi?ðxÞÞ, we obtain 0 FðxÞ 1 and

~c; ~be?;0 < 1. Therefore, the series converge for m!1 to

C1ðxÞ ¼ ½1þ ~c~be?;0ðFðxÞÞ1þ~c ��1C0ðxÞ:

Note that the shape of the equilibria calculated here

depends on the relative orientation of the magnetic field

with respect to the fluid vorticity, i.e., on the sign of X � B(represented by the ratio U0/B0 in the C0 coefficient). We

underline that our model is based on the quasi-neutrality

assumption and on the polytropic closure (from which

we recover also the parallel pressure profiles,

pa;kðxÞ ¼ ½pa;?ðxÞ�ca;k=ca;? ). Moreover, the function f0 in

Eq. (24) is exact if ~c ¼ 0 irrespective of the polytropic rela-

tion if we require that ions and electrons follow the same

polytropic law. The solution in Eq. (26) for n¼1 is more

general and allows for a different treatment of the ion and

electron pressure equations. (In our eTF model, the ion and

the electron pressure equations are the same, but in general

these equilibria can be used by a very broad range of kinetic

models, as the hybrid ones, in which the two species are

treated in different ways.) Asymptotically, as required, the

solution fn(x) ! 1 for x! 61; 8n. Finally, if ~u0ðxÞ ¼ 0

we obtain fnðxÞ ¼ 1 8n, since the FLR correction vanishes.

The general equilibrium taking into account the local value

of the magnetic field is calculated in Appendix B.

Finally, since FLR equilibria can be relevant also for

mixed Kelvin-Helmholtz and Raileigh-Taylor unstable con-

figurations, we add the modified equilibria for the case in

which an external field g ¼ �exU0ðd/=dxÞ is present. We

will consider the ~c ¼ 0 case only, and thus the MHD equilib-

rium condition now reads

112112-5 Cerri et al. Phys. Plasmas 20, 112112 (2013)

HðxÞ1þ b?;0 � b/;0

¼ 1� ~b?;0FðxÞ þ ~b/;0/ðxÞ;

where we have defined b/;0 � 2U0=B20 and now ~b/;0 ¼

b/;0=ð1þ b?;0 � b/;0Þ and ~bs?;0 ¼ bs?;0=ð1þ b?;0 � b/;0Þ.In this case, the correction function f0(x) in Eq. (24) is modi-

fied as follows:

f0ðxÞ ¼1þ ~b/;0/ðxÞ

1þ ~b/;0/ðxÞ � C0ðxÞaðxÞ: (27)

V. VLASOV-HYBRID CODE RESULTS

In this section, we investigate the influence of the full

pressure tensor response by means of a Vlasov-Hybrid

model.

A. Initialization of a velocity shear flow in kineticplasma simulations

As already discussed in Sec. I, one of the main difficul-

ties in the kinetic modeling of plasma dynamics driven by

the presence of shear flows is the choice of the initial equilib-

rium conditions. Indeed few kinetic equilibria are known in

this case and may not always be useful for modeling specific

“realistic” configurations, in particular with open boundary

conditions. We recall in particular Ref. 12, where the mag-

netic field is taken to be uniform and not affected by the

plasma, and Ref. 13 where the magnetic field is instead self-

generated. This last case is extended in Appendix C. Note

that these analytic distribution functions are “agyrotropic,”

i.e., they lead to perpendicular pressure components that are

not equal, as consistent with the analysis in Sec. III in terms

of the contributions of the FLR corrections to the diagonal

terms of the pressure tensor.

Because the geometry of the available kinetic equili-

bria is generally too idealized, kinetic simulations with

non-trivial spatial configurations are almost always initial-

ized with MHD equilibria (single fluid force balance).

However, in the absence of dissipation processes, the sys-

tem immediately reacts to the absence of kinetic equilib-

rium, radiating spurious waves or generating large

amplitude oscillations that may strongly affect the evolu-

tion of the system.8

In this context, it is legitimate to ask (1) whether the

eTF equilibria discussed in Sec. IV, that take into account fi-

nite Larmor radius effects to first order in ei, represent a sat-

isfactory initialization for kinetic simulations and (2) how

they compare to a MHD equilibrium initialization when

modeling shear flows by means of a kinetic code. To address

these points, we have used a numerical code that solves the

Hybrid Vlasov-Maxwell system of equations for two differ-

ent shear flow initial conditions.

B. Model description and shear flow initializations

Our approach is based on the numerical solution of

the hybrid Vlasov-Maxwell equations in phase space. A

hybrid-Vlasov code14 is used in a 1D-3V configuration

(one dimension in physical space and three dimensions in

velocity space). This code has been extensively used in

1D-2V,15 1D-3V,16–18 and 2D-3V19,20 configurations.

Within this model, the kinetic dynamics of ions is investi-

gated through the Vlasov-Maxwell set of equations

@f

@tþ v � @f

@xþ Eþ v� Bð Þ � @f

@v¼ 0; (28)

@B

@t¼ �r� E; r� B ¼ J (29)

while the electron response is taken into account through the

generalized Ohm’s law as used in the eTF model (Eq. (3))

E ¼ �U� Bþ 1

nð J� BÞ � 1

nrPe: (30)

The same ion normalizations as in the TF and eTF models

are used. Here, f¼ f(x, v, t) is the ion distribution function

and the ion density n and the mean velocity u are computed

from the velocity space integration of the moments of f. As

in the TF and the eTF models, the displacement current is

neglected in Ampere equation since we consider the “low

frequency” plasma dynamics. Quasi-neutrality is also

assumed. An isothermal equation of state is chosen for the

electrons, with the electron to ion temperature ratio

Te/Ti¼ 1.

Equations (28)–(30) are solved numerically in 1D-3V

phase space configuration, using an Eulerian algorithm based

on the coupling between the so-called splitting scheme21 and

the current advance method22 for the evolution of the elec-

tromagnetic fields, generalized to the hybrid case.14 The

length of the physical domain is Lx=di ¼ 20p, using

Nx¼ 512 uniformly distributed grid points, corresponding to

a grid spatial mesh Dx=di ’ 0:12. Periodic boundary condi-

tions are imposed in the spatial domain. The velocity domain

range is limited by vmax;x ¼ vmax;z ¼ 65vth;i (resp.

vmax;y ¼ 66vth;i) in the directions vx and vz (resp. vy), using

Nvx ¼ Nvz ¼ 51 (resp. Nvy¼ 61) uniformly distributed grid

points, corresponding to a grid mesh Dv¼ 0.2 in each veloc-

ity direction. The time step Dt is constrained by the Courant-

Friedrics-Levy condition for the numerical stability of the

hybrid Vlasov algorithm.

We have considered two different initializations, the

first one is a MHD equilibrium, the second an eTF equilib-

rium. In both cases, the initial ion-electron plasma flow is

characterized by a double velocity shear, imposed by setting

the mean ion velocity to

U ¼ U0 tanhx� x1

L

� �� tanh

x� x2

L

� �� 1

� �ey (31)

with x1¼ 0.25Lx and x2¼ 0.75Lx the central positions of the

two velocity shears and L¼ 3 their scale length. In the MHD

initialization the ion distribution function is set to a drifting

Maxwellian velocity distribution, using a uniform density.

The plasma is embedded in a uniform background magnetic

field B¼B0ez. In the eTF initialization, the ion distribution

function is set to

112112-6 Cerri et al. Phys. Plasmas 20, 112112 (2013)

f ðx; vÞ ¼ zðxÞ exp � 1

2

vx

vthx ðxÞ

� �2""

þ vy � UðxÞvth

y ðxÞ

!2

þ vz

vthz ðxÞ

� �2##; (32)

where the profile of the thermal velocities in the three direc-

tions vthi ðxÞ; i ¼ x; y; z, is set according to the profiles of the

diagonal elements of the pressure tensor in the eTF equilib-

rium, calculated in Sec. IV. The density profile n(x) is

ensured by the choice of zðxÞ in Eq. (32). The initial mag-

netic field is set to B ¼ ðB0 þ dBðxÞÞ ez, with dB(x) the cor-

rection due to FLR effects as calculated in Sec. IV. The

velocity shear layers at x1 and x2, Eq. (31), are characterized

by a different orientation of the fluid vorticity X with respect

to the mean magnetic field direction B. In this case, the

response of the ion distribution function is expected to be

different on the two velocity shear layers. Finally, the ion

plasma beta, defined as bi ¼ 2v2th;i=v

2A, is set to bi¼ 2, corre-

sponding to a ion Larmor radius .i ¼ di. The same simula-

tions have been repeated with bi¼ 0.5 and finally with the

addition of a small component of the magnetic field in the

direction of the flow By¼ 0.05 Bz; in all these cases, our

results were not qualitatively modified.

C. Kinetic evolution of the shear flow initializations

We discuss the self-consistent evolution of the ion distri-

bution function initialized either with the MHD or with the

eTF setup. In Fig. 2, we show the diagonal components, Pxx

Pyy and Pzz ¼ Pk (top to bottom) of the ion pressure tensor

vs. x at different times as obtained by integration of the

Hybrid Vlasov model, Eqs. (28)–(30). The left panels repre-

sent the results obtained using the MHD equilibrium as ini-

tial conditions while the right panels those obtained using

the eTF initial equilibrium. In the figure, the dashed

red-green line corresponds to the eTF profile. Note that the

MHD initial profile corresponds to the black straight con-

stant lines, Pii¼ 1 while the eTF profile is agyrotropic (near

the shear layer regions).

First, we observe that the plasma observables, initialized

with the MHD or with the eTF set ups, oscillate around the

same mean value corresponding to the eTF profile (red-green

line). In particular, the plasma initialized with the MHD

setup jumps very rapidly, on a time scale of the order of the

ion cyclotron time to a “dynamical” state around the eTF

equilibrium, becoming agyrotropic in the regions with veloc-

ity shear where now Pxxðx ¼ x1; x2Þ 6¼ Pyyðx ¼ x1; x2Þ6¼ Pzzðx ¼ x1; x2Þ. This reaction of the system when initial-

ized with the MHD equilibrium is not a relaxation towards a

kinetic Vlasov equilibrium but a transition to a regime of

undamped oscillations around a mean value. An additional

important point is that the reaction of the pressure tensor

changes the gradient of the velocity shear layers that readjust

their slope in a different way depending on the alignment of

the fluid vorticity X with respect to the mean magnetic field.

Since the change of the shear gradient occurs in a few ion cy-

clotron times, this readjustment has important consequences

on the growth rate of the KH instability that depends primar-

ily on the gradient of the velocity field.8

On the other hand, when initialized with the FLR setup,

all diagonal components Pii of the pressure tensor remain

almost constant with very small oscillations in the vicinity of

the eTF equilibrium profile. Again, the system does not relax

toward a kinetic equilibrium but instead it oscillates without

damping around a profile that is very close to the analytic so-

lution of the eTF equilibrium. This is true for all the varia-

bles of interest in this system: magnetic field component Bz,

parallel pressure Pzz, density, etc. The absence of a relaxa-

tion of the system toward a kinetic equilibrium is due to the

fact that the hybrid Vlasov algorithm is dissipation-free at

the wave-lengths of interest.

In order to make it more evident that the system oscillates

around the eTF equilibrium solution, in Fig. 3 we plot the

time evolution of the two components of the perpendicular

FIG. 2. Initial evolution of the ion

pressure tensor elements Pxx, Pyy, Pzz

(top to bottom panels) as obtained by

integration of the Vlasov equation for

the ion distribution function fi in the

presence of an initial velocity shear

flow for the MHD setup (left panels)

and for the FLR-corrected setup (right

panels). The profiles are shown by

black lines at different times during

0 < tXci < 30, while the mean profile

and the analytic solution of the FLR

equilibrium are shown by green and

red lines, respectively. The same verti-

cal scale is used in the left and right

panels to ease comparison.

112112-7 Cerri et al. Phys. Plasmas 20, 112112 (2013)

pressure, Pxx and Pyy, at the center of the two shear layers

x¼ x1, x2, left and right frame, respectively. A very similar

behavior is observed for the parallel pressure component Pzz

(not shown here). The continuous and dashed black lines rep-

resent Pxx and Pyy, respectively, in the case of the MHD initi-

alization. We see that they separate immediately and oscillate

around a mean value, the red line, corresponding to the eTF

equilibrium. The typical amplitude of these oscillations is

larger than 10% of the mean value. The same figure also

shows that the same quantities obtained from the simulation

with the eTF initialization oscillate around the eTF equilib-

rium with the same frequency. However, in this case the am-

plitude is reduced to much lower values, of the order of 1%.

Finally, we note that the oscillations observed in the two

layers are different both in amplitude and in frequency.

Indeed, the frequency of the observed oscillations in Fig. 3

corresponds to the local ion cyclotron frequency after taking

into account the value of the magnetic field at the shear layer

position which is different in the eTF equilibrium from the

value of the background magnetic field, B0¼ 1.

These results allow us to emphasize that, when using a

MHD initialization for kinetic simulations of the KHI, a

“see” of high amplitude oscillations would fill the simulation

box thus strongly modifying both the evolution of the KHI,

even at the linear stage, and the nature of the turbulence in

the late non-linear stage of the KHI. This is particularly true

in the case of periodic boundary conditions, as is the case in

our numerical experiments, but the same conclusions are

expected to be valid in the case of any boundary conditions

that would not efficiently enable these artifact oscillations to

escape the simulation box. Note that our results remain valid

in the presence of a “small” in-plane component of the mag-

netic field, as for example By¼ 1/20Bz.

As mentioned before, gyroviscosity effects at the shear

layer positions correspond to the ion velocity distribution

function being agyrotropic, as consistent with the eTF equa-

tions. The distribution function evolves according to two

competing processes: strain and gyration. The ion gyration

around the magnetic field direction tends to rotate the per-

pendicular distribution function in an ion cyclotron time,

while the strain due to the velocity shear tends to maintain

the agyrotropy.

Finally, we stress that we have chosen a Vlasov Eulerian

algorithm for this study because this kind of codes is charac-

terized by a very low noise even during the non linear re-

gime. This feature allows us to identify the kinetic response

to the shear flow initializations with great accuracy. On the

other hand, particle-in-cell algorithms, which are much less

consuming in terms of computational resources, are much

more noisy. When applied to a PIC code, the MHD initializa-

tion has been shown to be significantly modified and to be

unsatisfactory, the kinetic relaxation being well above the

noise level of the PIC simulations.8 On the contrary, we may

expect the eTF initialization to be very useful for PIC simu-

lations since the tiny oscillations of the Vlasov-Maxwell sys-

tem around the eTF shear equilibrium would be of the order

of the intrinsic thermal noise level of PIC simulations.

In conclusion, this kinetic investigation shows that tak-

ing into account the first order corrections in the ratio

between the ion Larmor radius and the equilibrium scale

lengths in the ion pressure tensor provides a sufficiently

accurate starting point for kinetic simulations, at least in the

case of shear flows with scale lengths not smaller than the

ion Larmor radius. This will make it possible to study the ki-

netic evolution of shear flow configurations without intro-

ducing spurious effects that persist in time as they are not

dissipated in a collisionless plasma.

VI. CONCLUSIONS

The eTF model developed in this article allows us to

include microscopic ion dynamics effects related to the ion

thermal Larmor radius (taken to be of the order of the ion in-

ertial skin depth) in a TF plasma description. This model

provides a viable alternative to computationally demanding

kinetic simulations in multiscale collisionless plasma

regimes where the evolution of global large spatial scales

and the dynamics occurring at microscopic scales cannot be

treated independently. In the present article, we have re-

stricted ourselves to ion microscales and considered the limit

of massless electrons. This limitation will be relaxed in a

future paper in view of the interest in performing fully non-

linear fluid-type simulations involving microscopic electron

scales as is the case, e.g., in collisionless magnetic reconnec-

tion events. In the derivation of the eTF equations, we have

taken special care in ensuring energy conservation within the

model equations and in stressing the role of the asymmetry

of the gyroviscosity contribution to the ion pressure tensor

under inversion of the magnetic field orientation.

The relative simplicity of the (eTF) model allowed us to

obtain analytic expressions for a one-dimensional plasma

equilibrium configuration in the presence of a shear flow that

includes first order FLR effects and where the ion pressure

tensor is explicitly agyrotropic.

Hybrid Vlasov simulations have then been performed

using, as initial configuration with a shear flow, either a

FIG. 3. Time evolution of Pxx (full

lines) and Pyy (dashed lines) in kinetic

simulations of a velocity shear flow in

the case of a MHD setup (black lines)

and an eTF setup (blue lines) at posi-

tions x¼ x1 (left panel) and x¼ x2

(right panel).

112112-8 Cerri et al. Phys. Plasmas 20, 112112 (2013)

standard MHD equilibrium or its “corrected” version

within the eTF framework. Neither initialization corre-

sponds to fully kinetic selfconsistent Vlasov equilibria

which, however, are difficult to obtain for a plasma with a

shear flow and are often too restricted in form to be of

practical use.

However, we have shown that the eTF initialization pro-

vides a significant improvement as it strongly reduces the

initial absence of force balance at the particle level that

make MHD initializations unsuitable for “controlled” kinetic

investigations. In fact, in a low-noise Vlasov simulation of a

collisionless plasma, an initial absence of force balance at

the particle level does not relax towards a new (even if

uncontrolled) Vlasov equilibrium but leads to relatively large

amplitude, long lasting oscillations. These oscillations affect

the development of the phenomena of interest (such as the

development of the Kelvin-Helmholtz instability driven by

the velocity shear in the plasma) even in their linear phase.

In the case of the eTF initialization, oscillations around the

eTF equilibrium values are still present but their amplitude is

strongly reduced.

ACKNOWLEDGMENTS

The research leading to these results has received fund-

ing from the European Commission’s Seventh Framework

Programme (FP7/20072013) under the grant agreement

SWIFF (Project No. 263340, www.swi.eu). We are pleased

to acknowledge T. Passot, P. L. Sulem, P. J. Morrison, and

B. Scott for useful discussions, in particular on energy con-

servation, and C. Cavazzoni for technical discussion about

code implementation and parallelization. This work was

supported by the Italian Super-computing Center CINECA

under the ISCRA initiative. We acknowledge the access to

the HPC resources of CINECA made available within the

Distributed European Computing Initiative by the PRACE-

2IP, receiving funding from the European Community’s

Seventh Framework Programme (FP7/2007-2013) under

Grant Agreement No. RI-283493. This work was supported

by the PRACE Super-computing initiative, Project No.

2012071282.

APPENDIX A: THE FLR EXPANSION OF THEPRESSURE TENSOR

The eTF model can be summarized as an extension of

the so-called two-fluid model5 to which we add a gyrotropicpressure for both species and FLR agyrotropic corrections.

We also allow in principle for heat fluxes along the magneticfield lines, supposed to be almost straight (see the so-called 3

þ 1 model23).

Let us define the smallness parameterea � .a=L � x=Xc;a, where Xc;a = jeajB0=mac and .a ¼vth;a=Xc;a are the cyclotron frequency and the Larmor radius

of the a-species, respectively, L and x ¼ vth;a=L the hydrody-

namic characteristic length and frequency, and vth;a the ther-

mal velocity of the a-species. Actually, the smallness

parameter is defined as the ratio of the Larmor radius to the

hydrodynamic length scale, ea � .a=L. Then, we assume the

ordering ba � 1 and ua=vth;a � 1, from which we recover the

frequency ordering x= X c;a � ea. Note that this ordering is

different from previous orderings, such as the one adopted by

Chew, Goldberger and Low24 in which u=vth � 1 or the

usual FLR ordering u=vth � e (and thus x= X � e2, see e.g.

Rosenbluth and Simon25). We call our ordering the eTF order-

ing. The full pressure tensor equation for the species a, still in

dimensionless form, reads

ð�ilmPa;lj þ �jlmPa;liÞBm ¼ raeaL

vth;as

� �@Pa;ij

@t

"

þ @

@xkðPa;ijua;kÞ þ Pa;ik

@ua;j

@xkþPa;jk

@ua;i

@xk

� �þ @Qa;ijk

@xk

#;

(A1)

where Pa,ij is the (ij-component of the) pressure tensor, Bm

the (m-component of the) magnetic field, ua,k the (k-compo-

nent of the) fluid velocity, Qa,ijk the (ijk-component of the)

heat flux tensor, ra � signðeaÞ the sign of the charge species

(and thus r�1a ¼ ra).

In the limit of the maximal ordering, L=s � vth;a, we

assume x=Xc;a � ea � 1 and we expand the pressure tensor

and heat flux tensor in powers of ea, i.e.

Pij ¼X1r¼0

er PðrÞij ; Qijk ¼X1r¼0

er QðrÞijk ; (A2)

where (and hereafter where its suppression does not generate

confusion) the species index a is omitted. The nth-order pres-

sure tensor then reads:

LB½PðnÞij � ¼ Ru½Pðn�1Þij � þ D½Qðn�1Þ

ijðkÞ �; (A3)

where LB½PðnÞij � and Ru½Pðn�1Þij � and are the nth-order contri-

butions to the left-hand side and to right-hand side of the pres-

sure tensor equation involving only the pressure tensor and

D½Qðn�1ÞijðkÞ � is the contribution to its right-hand side involving

the heat flux tensor. These three operators depend only on B,

u, and Q, respectively. At zero order, n¼ 0, (Ru½Pð�1Þij � � 0

and D½Qð�1ÞijðkÞ � � 0), and at first order, n¼ 1, we obtain

ð�ilmPð0Þlj þ �jlmPð0Þli ÞBm ¼ 0; (A4)

ð�ilmPð1Þlj þ �jlmPð1Þli ÞBm ¼ ra

"dPð0Þij

dtþPð0Þij

@ua;k

@xk

� �

þ Pð0Þik

@ua;j

@xkþPð0Þjk

@ua;i

@xk

� �

þ@Qð0Þijk

@xk

#;

(A5)

where @=@tþ ua;k@=@xk has been replaced by the total time

derivative d/dt. Note that the pressure tensor equation, Eq.

(A5), is not invariant under magnetic field inversion,

B! �B.

At zero order, we recover the gyrotropic pressure tensor

112112-9 Cerri et al. Phys. Plasmas 20, 112112 (2013)

Pð0Þ ¼ p?ðI� bbÞ þ pkbb or

Pð0Þxx ¼ Pð0Þyy � p?; Pð0Þzz � pk:(A6)

Here b¼B/B is the unit vector along the magnetic field and,

without loss of generality, we have assumed the z-axis aligned

with the magnetic field. Note that the zero-order solution for

the pressure tensor is invariant under magnetic field inversion.

Let us now consider the first order equation

LB½Pð1Þij � ¼ Ru½Pð0Þij � þ D½Qð0ÞijðkÞ�: (A7)

If we assume, following the standard approach (see, e.g.,

Refs. 26 and 27), invariance with respect to magnetic field

inversion B! �B of Pð1Þ, we obtain

LB½Pð1Þij � ! �LB½Pð1Þij �; Ru½Pð0Þij � ! Ru½Pð0Þij �;

D½Qð0ÞijðkÞ� ! �D½Qð0ÞijðkÞ�;

so that Pð1Þ is no more solution and thus the invariance

assumption is wrong. As a result, Pð1Þ is not invariant with

respect to the inversion B!�B. For these reasons, we intro-

duce a coefficient that takes into account the relative orientation

of the magnetic field with respect to the coordinate axes sm �signðb � emÞ ¼ signðbmÞ (such that s�1

m ¼ sm), where em is the

unit vector along the m-axis of the reference system. Then, by

defining LB½Pð1Þij � � smLB½Pð1Þij � and D½Qð0ÞijðkÞ� � smD½Qð0ÞijðkÞ�,Eq. (A7) can be now cast in a invariant form, at first-order, that

correctly takes into account the gyromotion of the particles

with respect to the coordinate axes, LB½Pð1Þij � ¼ Ru½Pð0Þij �þD½Qð0ÞijðkÞ�, which, using s�1

m ¼ sm, explicitly reads

ð�ilmPð1Þlj þ �jlmPð1Þli ÞBm ¼ smra

"dPð0Þij

dtþPð0Þij

@ua;k

@xk

� �

þ Pð0Þik

@ua;j

@xkþPð0Þjk

@ua;i

@xk

� �#

þ ra

@Qð0Þijk

@xk: (A8)

In order to evaluate the right-hand side of Eq. (A8), we calculate

the total time derivative of the zero-order pressure tensor Pð0Þij

dPð0Þij

dt¼ d

dt½p?dij þ ðpk � p?Þbibj�

¼ dp?dt

� �sij þ

dpkdt

� �bibj þ

ðpk � p?ÞjBj2

� dBi

dt

� �Bj þ Bi

dBj

dt

� �� 2BiBj

jBjdjBjdt

" #; (A9)

where we have introduced sij � dij � bibj for shortness. Then,

we take the curl of the generalized Ohm’s law, Eq. (3), and

retaining only terms that are consistent with the adopted ordering.

In fact, remember that in this limit ui ¼ U and ue ¼ U� J=n,

thus the generalized Ohm’s law reads either Eþ ue � B ¼� 1

nr �Pð0Þe or Eþ ui � B ¼ 1

n ðJ� B�r �Pð0Þe Þ. In both

cases, after taking the curl, the righthand side is of order ee �de and thus zero, for consistency with the massless electron

limit that we adopt in the present paper. Thus, we evaluate the

total time derivative of the i-component of the magnetic field:

dBi

dt¼ Bk

@ua;i

@xk� @ua;k

@xk

� �Bi; (A10)

where we used Eq. (4). Then, we finally rewrite Ru½Pð0Þij � as

Ru½Pð0Þij � ¼ radp?dt

� �gij þ

dpkdt

� �bibj þPð0Þij

@uk

@xk

� �(

þp?@uj

@xiþ @ui

@xj

� �þ 2ðpk � p?Þ

� bibk@uj

@xkþ bjbk

@ui

@xk� bibj blbk

@ul

@xk

� �� ��;

(A11)

where we used the fact that djBj=dt ¼ biðdBi=dtÞ. To evalu-

ate the divergence of the zero-order heat flux tensor, we use

the gyrotropic expression:28

Qð0Þ ¼ q?½sb�S þ qkbbb; ½sb�Sijk � sijbk þ sjkbi þ skibj:

We now project the first order equation on the two orthogo-

nal spaces using the projectors bb and s, which are the kernel

of the LB operator. In this way we obtain two equations for

the Pð0Þ only, which determine the “secular” evolution of the

zero order quantities, pk and p?

dpakdtþ pakð$ � uaÞ þ 2pakðbb : $uaÞ

þ $ðqakbÞ � 2qa?ð$ � bÞ ¼ 0; (A12)

dpa?dtþ pa?ð$ � uaÞ þ pa?ðs : $uaÞ

þ $ðqa?bÞ þ qa?ð$ � bÞ ¼ 0; (A13)

where we the orthogonality relationships. s : bb ¼ s � b ¼ 0

together with bb : bb ¼ 1; s : s ¼ 2. In deriving the heat flux

terms, we used the property bb : $b ¼ blbk@kbl ¼ 0. Note

that, when written in terms of the U and J/ n variables and the

heat fluxes are neglected, Eqs. (A12) and (A13) coincide with

Eqs. (5)–(8), except for the gyroviscosity term in the ion per-

pendicular pressure equation. Indeed, in order to conserve

energy, we must add a second order term to the perpendicular

pressure equation, Eq. (A13), when a¼ i. Formally, such a

term comes from projecting the equation for Pð0Þ þPð1Þ on

the two eigenspaces bb and s. Neglecting the second order

changes in the magnetic field direction, we obtain the following

equation for the pak and pa? evolution with FLR corrections:

dpakdtþ pakð$ � uaÞ þ 2pakðbb : $uaÞ

þ $ðqakbÞ � 2qa?ð$ � bÞ ¼ 0 (A14)

dpa?dtþ pa?ð$ � uaÞ þ pa?ðs : $uaÞ

þ $ðqa?bÞ þ qa?ð$ � bÞ þPð1Þa : $ua ¼ 0; (A15)

112112-10 Cerri et al. Phys. Plasmas 20, 112112 (2013)

where Pð1Þa : rua term will be retained for the ions only

since we are considering the massless electron limit.

Retaining this term is necessary for energy conservation in

the model, which is a fundamental consistency property of a

model.30 We can now solve Eq. (A8) for Pð1Þ. Without loss

of generality, we assume again that the z-axis is aligned to

the magnetic field, so b¼ s3ez. Since the parallel component

Pð1Þzz remains undetermined, we can choose it to be zero,

Pð1Þzz ¼ 0. Then, using the traceless property of the first-order

tensor Tr½Pð1Þ� ¼ Pð1Þxx þPð1Þyy ¼ 0, we solve Eq. (A8) for the

Pð1Þ components

Pð1Þa;xx ¼ �Pð1Þa;yy ¼ �s3ra

2

pa?B

@ua;x

@yþ @ua;y

@x

� �; (A16)

Pð1Þa;xy ¼ Pð1Þa;yx ¼ �s3ra

2

pa?B

@ua;y

@y� @ua;x

@x

� �; (A17)

Pð1Þa;xz ¼ Pð1Þa;zx ¼ �s3ra

Bð2pak � pa?Þ

@ua;y

@zþ pa?

@ua;z

@y

� �

� ra

B

@qi?@y

; (A18)

Pð1Þa;yz ¼ Pð1Þa;zy ¼s3ra

Bð2pak � pa?Þ

@ua;x

@zþ pa?

@ua;z

@x

� �

þ ra

B

@qa?@x

; (A19)

Pð1Þa;zz ¼ 0: (A20)

The above expressions for the FLR corrections take into

account both the orientation of the magnetic field with

respect to the z-axis and heat flux contributions. We under-

line that, contrary to previous works, the derivation of

Eqs. (A16)–(A20) presented here accounts for the orienta-

tion of the magnetic field with respect to the z-axis via the

s3 coefficient, making the physical asymmetry under the

inversion B! �B explicit. In fact, if for the sake of sim-

plicity one neglects heat fluxes this inversion causes a

change of sign in the Pð1Þ tensor, i.e., Pð1Þ ! �Pð1Þ. This

asymmetry has a direct consequence on the dynamical

evolution of the system and on its equilibrium configura-

tion, as shown in this article and pointed out by recent ki-

netic simulations.8

Finally, we remark that the expressions for the Pð1Þ

components obtained here are derived assuming B along

the z-axis and thus they can be considered valid as long

as the magnetic field component in the xy-plane remains

negligible with respect to the z-component, i.e.,

ðB2x þ B2

yÞ1=2 � jBzj.

APPENDIX B: EQUILIBRIA INCLUDING FLRCORRECTIONS COMPUTED WITH THE LOCAL FIELD

In general, a formal treatment of the FLR correction

should take into account the local value of the magnetic field

in their expressions, if it is not spatially homogeneous. This

corresponds to allow for a spatial dependence of the

cyclotron frequency (and thus of the Larmor radius propor-

tional to the ratioffiffiffiTp

=B). In this case, it is easy to show that

Eq. (21) should be rewritten as

1� ~u0ðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðxÞhðxÞ

p !

~bi?;0FðxÞf ðxÞ þ ~be?;0GðxÞgðxÞ

þ HðxÞ1þ bi?;0 þ be?;0

hðxÞ ¼ 1: (B1)

Then, by using the same relations between f(x), g(x) and

h(x), as in Sec. IV, we obtain

1� ~u0ðxÞffiffiffiffiffiffiffiffiffiffiHðxÞ

p !

~bi?;0FðxÞffiffiffiffiffiffiffiffif ðxÞ

pþ ~be?;0½FðxÞf ðxÞ�1þ

~c

þ ½1þ ~bi?;0FðxÞ þ ~be?;0ðFðxÞÞ1þ~c �f ðxÞ ¼ 1:

(B2)

We now consider only the case ~c ¼ 0, since it is easier and

supported by the simulations results. In this case, we obtain a

quadratic equation for wðxÞ ¼ffiffiffiffiffiffiffiffif ðxÞ

pwith non-constant

coefficients:

AðxÞw2ðxÞ þ BðxÞwðxÞ � 1 ¼ 0; (B3)

where the coefficients are given by

AðxÞ ¼ 1� ~bi?;0FðxÞ

BðxÞ ¼ 1� ~u0ðxÞffiffiffiffiffiffiffiffiffiffiHðxÞ

p !

~bi?;0FðxÞ:

Since we are considering MHD functions such that 0 FðxÞ 1 and ~bi?;0 < 1; AðxÞ > 0 for all x and we can solve

for w6(x), retaining only the meaningful wþ(x) solution

wþðxÞ ¼1

2AðxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2ðxÞ þ 4AðxÞ

q� BðxÞ

� �: (B4)

One can easily verify that w�ðxÞ 0 8x, while

wþðxÞ > 0 8x. This is based on the fact that, for all x,

AðxÞ > 0 and jBðxÞj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2ðxÞ þ 4AðxÞ

q. We finally

remark that the solution wþðxÞ ! 1 for x! 61 and if

~u0ðxÞ ¼ 0 we obtain wþ(x)¼ 1, as to be expected.

APPENDIX C: EXACT SELF-CONSISTENT KINETICEQUILIBRIA

The distribution function discussed in Ref. 13 can be

rewritten as a function of the integrals of the motion

� ¼ mðv2x þ v2

y þ v2z Þ=2þ q/, and Py ¼ mvy þ qAy=c in gen-

eral terms in the form

f ð�;PyÞ ¼ C exp � �h�

c1 P2y

mh� v Py

h

� �; c1 > �1=2;

where � is the particle energy, Py is the y-component of the

canonical momentum of a particle of mass m and charge q;

v* and h are parameters; and /ðxÞ and Ay(x) are the electro-

static and the y component of the vector potential, respec-

tively. It can be rewritten as

112112-11 Cerri et al. Phys. Plasmas 20, 112112 (2013)

f ð�;PyÞ ¼ C0 exp � �h� c1 ðPy þ mv =ð2c1ÞÞ2

mh

� �:

The well known Harris distribution31 (with in general a non-

vanishing electrostatic potential) is obtained by setting

c1¼ 0. In terms of the particle velocities and density, the dis-

tribution function can be rewritten as

f ðvx; vy; vz; xÞ ¼noð1þ 2c1Þ1=2

ð2h=mÞ3=2exp � q/þ ½1=ð1þ 2c1Þ� ½c1ðqAyÞ2=ðmc2Þ þ qAyv =c�

h

� �

� exp �mðv2x þ v2

z Þ2h

� �exp �mð1þ 2c1Þðvy � Vy0Þ2

2h

� �; with Vy0 ¼ �

2c1qAy

mcð1þ 2c1Þ� v

1þ 2c1

:

Note that this distribution function is agyrotropic, i.e., the two perpendicular pressure components are not equal for c1 6¼ 0, as

consistent with the analysis in Sec. III in terms of the contributions of the diagonal terms of the gyroviscosity. Reintroducing

the species index a we obtain the charge and current densities . and jy and the xx component of the total pressure tensor Pxx

(which plays the role of the Sagdeev potential) as

. ¼ Raqanoa exp � qa/þ ½1=ð1þ 2c1aÞ� ½c1aðqaAyÞ2=ðmac2Þ þ qaAyv a=c�ha

( );

jy ¼ Raqanoa�2c1aqaAy

macð1þ 2c1aÞ� v

1þ 2c1a

� �exp � qa/þ ½1=ð1þ 2c1aÞ� ½c1aðqaAyÞ2=ðmac2Þ þ qaAyv a=c�

ha

( );

Pxx ¼ Ra2noaha

maexp � qa/þ ½1=ð1þ 2c1aÞ� ½c1aðqaAyÞ2=ðmac2Þ þ qaAyv a=c�

ha

( ):

More general distributions can be constructed following the

procedure devised in Ref. 32 by using a proper (positive)

superposition of distribution functions with different values

of the parameters c1 e v*. This can lead to a Sagdeev poten-

tial Pxxð/;AyÞ that has the required dependence on / and Ay

in order to obtain through Poisson and Ampere equations a

physically relevant equilibrium configuration.

An interesting case is obtained by considering a neutral

plasma with no electric field, / ¼ 0, and v a ¼ 0. This

implies that noa exp½�c 1aðqaAyÞ2=ðmac2haÞ� must be inde-

pendent of a for all x, i.e., no e ¼ no i; and c 1 eq2e=

ðmec2heÞ ¼ c 1 ipq2i =ðmic

2hiÞ where c 1a � ½c1a=ð1þ 2c1aÞ�. If

we take equal temperatures, he¼ hi, we obtain equal and op-

posite x-dependent velocities for the two species that lead to

a fluid velocity Uy(x) and to a current density jy(x). If we

require the magnetic field to be even in x, B2 must be suffi-

ciently large compared to Pxx. In this case, the vector poten-

tial Ay(x) is odd in x and so is the fluid velocity Uy(x). Note

that in this equilibrium the plasma flow is not due to an

E�B drift and is fully related to the agyrotropicity of the

distribution function.

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