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Pressure tensor in the presence of velocity shear: Stationary solutions and self-consistent equilibriaS. S. Cerri, F. Pegoraro, F. Califano, D. Del Sarto, and F. Jenko Citation: Physics of Plasmas (1994-present) 21, 112109 (2014); doi: 10.1063/1.4901570 View online: http://dx.doi.org/10.1063/1.4901570 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Generation of coherent magnetic fields in sheared inhomogeneous turbulence: No need for rotation? Phys. Plasmas 18, 022307 (2011); 10.1063/1.3551700 Probability distribution function for self-organization of shear flows Phys. Plasmas 16, 052304 (2009); 10.1063/1.3132631 Nonlocal properties of gyrokinetic turbulence and the role of E × B flow shear Phys. Plasmas 14, 072306 (2007); 10.1063/1.2750647 Self-sustaining vortex perturbations in smooth shear flows Phys. Plasmas 13, 062304 (2006); 10.1063/1.2209229 Shear-flow driven current filamentation: Two-dimensional magnetohydrodynamic-simulations Phys. Plasmas 7, 5159 (2000); 10.1063/1.1322558

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Pressure tensor in the presence of velocity shear: Stationary solutionsand self-consistent equilibria

S. S. Cerri,1,a) F. Pegoraro,2 F. Califano,2,3 D. Del Sarto,4 and F. Jenko1,3,51Max-Planck-Institut f€ur Plasmaphysik, Boltzmannstr. 2, D-85748 Garching, Germany2Physics Department “E. Fermi,” University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy3Max-Planck/Princeton Center for Plasma Physics,4Institut Jean Lamour, UMR 7198 CNRS - Universit�e de Lorraine, BP 239 F-54506 Vandoeuvre les Nancy,France5Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA

(Received 3 October 2014; accepted 30 October 2014; published online 13 November 2014)

Observations and numerical simulations of laboratory and space plasmas in almost collisionless

regimes reveal anisotropic and non-gyrotropic particle distribution functions. We investigate how

such states can persist in the presence of a sheared flow. We focus our attention on the pressure

tensor equation in a magnetized plasma and derive analytical self-consistent plasma equilibria

which exhibit a novel asymmetry with respect to the magnetic field direction. These results are

relevant for investigating, within fluid models that retain the full pressure tensor dynamics, plasma

configurations where a background shear flow is present. VC 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4901570]

I. INTRODUCTION

Sheared flows are frequently observed in space and

laboratory plasmas. They are an important source of free

energy and can drive various instabilities, such as the

Kelvin-Helmholtz instability1–10 (KHI) or the magnetorota-

tional instability11–17 (MRI). The KHI, on the one hand,

leads to the formation of fully developed large scale vortices,

eventually ending in a turbulent state where energy is effi-

ciently transferred to small scales. In this context, a relevant

example is given by the development of the KHI observed at

the flanks of the Earth’s magnetosphere18 driven by the

velocity shear between the solar wind (SW) and the magne-

tosphere (MS) plasma, and in general, observed at other

planetary magnetospheres. On the other hand, the MRI is

considered to be a main driver of turbulence (and turbulent

transport of angular momentum) in accretion disks around

astrophysical objects, such as stars and black holes. In addi-

tion, small-scale sheared flows can emerge from turbulent

states and lead to kinetic anisotropy effects, as seen from

SW data and simulations.19–21

The standard magnetohydrodynamic (MHD) approach

to the study of shear flow configurations is justified when the

scale length of the sheared flow is much larger than the typi-

cal ion microscales, i.e., when di; qi � L. However, in thecase of the interaction of the SW with the MS, satellite

observations show that the typical scale length of the sheared

flow is roughly comparable to the ion gyroradius and/or the

skin depth, i.e., di � qi � L ðb � 1Þ. In general, a commonassumption in the framework of MHD modeling is to

consider an isotropic pressure tensor even in the presence of

a background magnetic field. However, a relatively simple

step to avoid such extreme simplification is to adopt the

Chew-Goldberger-Low (CGL) approximation,22 where the

pressure tensor is gyrotropic, i.e., the pressure can be differ-

ent along the magnetic field and perpendicular to it. In this

approach, three main features of the system appear to be rel-

evant: (i) the pressure is isotropic in the plane perpendicular

to the magnetic field, i.e., gyrotropy, (ii) the system is sym-metric with respect to the relative orientation of the magneticfield B and the fluid vorticity Xu � r� u, i.e., with respectto the sign of Xu � B, and (iii) the equilibrium profiles are notdependent on the velocity shear. These three points are

substantially modified when the pressure tensor equation is

retained in the fluid hierarchy or when kinetic models are

adopted. Also retaining first-order finite Larmor radius

(FLR) corrections of the ions within a two-fluid (TF) model,

the so called extended two-fluid (eTF) model,23 was recentlyshown to substantially modify the previous picture.

In this work, we investigate the role of retaining the full

pressure tensor equation, still in the framework of a fluid

model. In order to simplify the picture for the sake of clarity,

we will consider a configuration in which the inhomogeneity

direction, the flow direction, and the magnetic field are

orthogonal to each other. The main result of our approach is

to prove that, in the presence of a shear flow, an additional

anisotropy in the perpendicular plane (agyrotropy) and anasymmetry with respect to the sign of Xu � B arise even atthe level of the equilibrium configurations, which depend

also on the shear strength. Indeed, a sheared flow can induce

dynamical anisotropization of an initial isotropic pressure

configuration, together with an asymmetry with respect to

the sign of Xu � B, when, for instance, one retains first-orderFLR corrections23 or the full pressure tensor equation.24

Here, we focus our attention on the effect of the shear-

induced anisotropization at the level of a stationary state.

Such new features are intrinsic properties of the system, and

their relevance is related, in general, to the plasma regime

under investigation. In particular, the deviation from an

MHD/CGL model becomes not negligible when the sheara)Electronic mail: [email protected]

1070-664X/2014/21(11)/112109/9/$30.00 VC 2014 AIP Publishing LLC21, 112109-1

PHYSICS OF PLASMAS 21, 112109 (2014)


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length scale is comparable with the ion microscales

(di; qi � L), as is the case for the SW-MS interaction. Suchdeviations, important at the level of equilibrium configura-

tions, can dramatically affect the study of one of the above

mentioned instabilities already in the linear phase and possi-

bly leading to very different nonlinear stages, when kinetic

models are adopted.10 This highlights the importance of a

correct modeling of the sheared flow equilibrium configura-

tion, in which the possibly relevant ingredients, such as the

pressure tensor, are retained. Moreover, as already antici-

pated, the approach presented here can give insights into

non-gyrotropic proton distribution functions that are

observed in SW data and Vlasov simulations.19–21

The remainder of this paper is organized as follows. In

Sec. II, we solve the stationary pressure tensor equation

without heat fluxes and in the presence of a sheared flow,

giving the solution in the form of traceless corrections to the

CGL gyrotropic pressure tensor, discussing the emergence of

the perpendicular anisotropy and of the Xu � B-asymmetry.In Sec. III, we consider the full non-gyrotropic ion pressure

tensor within the equilibrium problem. Implicit and exact

numerical solution for the equilibrium profiles are then

given, along with possibly useful explicit analytical approxi-

mations, underlining again the role of the asymmetry and the

perpendicular anisotropy. Finally, alternative approximated

and exact analytical equilibria are given in the Appendix.

II. STATIONARY SOLUTION OF THE PRESSURETENSOR EQUATION

Within a fluid description of a plasma, the pressure ten-

sor equation is given by23

@Pa;ij@tþ @@xk

Pa;ijua;k þ Qa;ijkð Þ þPa;ik@ua;j@xk

þPa;jk@ua;i@xk¼ qa

mac�ilmPa;jl þ �jlmPa;ilð ÞBm; (1)

where qa and ma are the charge and the mass of the speciesa, respectively, Pa;ij is the (ij-component of the) pressuretensor, ua;k is the (k-component of the) fluid velocity, Qa;ijk isthe (ijk-component of the) heat flux tensor, �ijk is the Levi-Civita symbol, and Bm is the (m-component of the) magneticfield. Now, we look for stationary solutions of Eq. (1) in the

limit of no heat fluxes, i.e., @Pa;ij=@t ¼ 0 and Qa;ijk ¼ 0 8 i,j, and k. Under these assumptions, Eq. (1) reduces to

@

@xkPa;ijua;kð Þ þPa;ik

@ua;j@xkþPa;jk

@ua;i@xk

¼ raXa �ilmPa;jl þ �jlmPa;ilð Þbm; (2)

where we have introduced the sign of the charge ra � signðqaÞand the cyclotron frequency Xa � jqajjBj=mac of the speciesa, respectively, and the magnetic field versor bm � Bm=jBj.

It is a well known result that, within a FLR expansion of

Eq. (2), the zero-order solution is given by the CGL pressure

tensor,22,23

Pð0Þa ¼ pa;?sþ pa;jjbb; (3)

where s � I� bb is the projector onto the plane perpendicu-lar to the magnetic field (b � B=jBj), pa;jj and pa;? are thepressure parallel and perpendicular to B, respectively. Thetensor Pð0Þa represents the kernel of the operator on the righthand side of the pressure tensor equation, i.e., it is an

approximated solution of the equation when the left hand

side is negligible.

We now consider a velocity shear configuration such

that the inhomogeneity direction, the flow direction, and

the magnetic field direction form a right-handed basis, e.g.,

u ¼ uyðxÞey and B ¼ BzðxÞez. Note that this configuration,despite its apparent simplicity, is actually commonly used in

various areas, e.g., for studying the KHI or the MRI. Without

loss of generality, we can write the full pressure tensor as

Pa ¼ Pð0Þa þ ~Pa, in which ~Pa represents a traceless correc-tion to the gyrotropic pressure tensor Pð0Þa . In fact, remainingwithin a fluid framework, one has unambiguous definitions

of pa;? � 12 Pa : s and pa;jj � Pa : bb, so the identity ~Pa :I ¼ 0 holds.25–30 The same conclusion can be directlyderived from the Vlasov equation.31 From the definition of

pa;jj, one obtains also ~Pa : bb ¼ 0, so in our configuration~Pa;zz ¼ 0 and the perpendicular components of the pressuretensor read

Pa;xx ¼ pa;? þ ~Pa;xx; Pa;yy ¼ pa;? þ ~Pa;yy with~Pa;yy ¼ � ~Pa;xx: (4)

Inserting the above expressions into Eq. (2) gives the follow-

ing non-gyrotropic diagonal pressure tensor:

Pa;zz ¼ pa;jj

Pa;xx ¼ 1�aa xð Þ

1þ aa xð Þ

� �pa;?

Pa;yy ¼ 1þaa xð Þ

1þ aa xð Þ

� �pa;?

8>>>>>><>>>>>>:

(5)

with

aa xð Þ �1

2

s3 raXa

dua;ydx

; (6)

where s3 � signðb3Þ is the relative orientation of the mag-netic field and the z-axis (see Ref. 23). The positivity condi-tion on the diagonal pressure terms in Eq. (5) gives

aa xð Þ � �1

2: (7)

It is interesting to note that here an asymmetry with respect

to the sign of aaðxÞ appears. In fact, not all the values of theshear strength are allowed when aa is negative, while in prin-ciple there is no limitation when it is positive. The sign of aadepends essentially on two physical factors: the species,

through ra, and the relative orientation of the magneticfield B and the fluid vorticity Xu, through the sign ofs3ðdua;y=dxÞ. For instance, if one considers ions (ri ¼ þ1)and a background magnetic field oriented in the positive

direction of the z-axis (s3 ¼ þ1), from Eq. (7) we find thatthere is no limitation in the velocity shear of ui if the

112109-2 Cerri et al. Phys. Plasmas 21, 112109 (2014)


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vorticity is aligned with the magnetic field (Xu � B > 0),while the velocity shear is limited to be maximum equal (in

absolute value) to the ion gyration frequency otherwise

(jXuj Xi). We note that this condition on the shearstrength limitation asymmetry is indeed in accordance with

Ref. 24, where it is found that shear configurations of the

type considered here become dynamically unstable when

X0=Xi � 1þ ð@xui;yÞ=Xi < 0, which corresponds exactly tothe condition aiðxÞ < �1=2 in our notation (in the case ofRef. 24, however, the non-stationary pressure tensor equa-

tion was considered).

The solution in Eq. (5) introduces an additional anisot-

ropy in the plane perpendicular to the magnetic field

Aa;? �jPa;xx �Pa;yyj

pa;?¼ 2 aa xð Þ

1þ aa xð Þ

��; (8)

which gives the maximum value of the anisotropy

(AðmaxÞ? ¼ 2) for a ¼ �1=2 and for a!1. A sketch ofPxx=p?; Pyy=p? and A? versus a is given in Fig. 1. Theasymmetry is thus found also in the perpendicular anisot-

ropy, which is here entirely due to the velocity shear.

For small jaj � 1, the corrections to the gyrotropicCGL pressure tensor are small and, if we expand the

solution in Eq. (5) to the leading order in a, the first-order FLR solution is found.23,25–30 However, in this

limit, the asymmetry with respect to the sign of a inEq. (7) is lost.

III. EQUILIBRIA WITH THE COMPLETE PRESSURETENSOR

We now want to use the stationary solution of Pxx inEq. (5) for solving the equilibrium condition23

d

dx

Xa

Pa;xx þB2

8p

" #¼ d

dxPi;xx þ pe;? þ

B2

8p

� �¼ 0; (9)

where here we are considering only the ions full pressure

tensor, while the electrons are gyrotropic, i.e.,

Pe ¼ pe;?sþ pe;jjbb. Note that in our configuration, sincethere are no parallel gradients, the parallel balance is auto-

matically satisfied.32 We define the gyrotropic pressure and

the magnetic field profiles as

pi;?ðxÞ ¼ pi;?0FðxÞf ðxÞ;pe;?ðxÞ ¼ pe;?0GðxÞgðxÞ;B2ðxÞ ¼ B20HðxÞhðxÞ;

8>><>>: (10)

where pi;?0 and pe;?0 are positive constants, FðxÞ; GðxÞ andHðxÞ correspond to the MHD equilibrium and f(x), g(x), andh(x) are the corrections to the relative MHD profile. Theequilibrium condition, Eq. (9), can then be conveniently

rewritten in dimensionless form as

~bi;?0 1�ai xð Þ

1þ ai xð Þ

� �F xð Þf xð Þ þ ~be;?0G xð Þg xð Þ

þ H xð Þh xð Þ1þ b?0

� 1 ¼ 0; (11)

where we have fixed the constant of integration to be

B20=8pþ pi;?0 þ pe;?0 and the quantities ~ba;?0 � ba;?0=ð1þb?0Þ; b?0 ¼

Pa ba;?0 and ba;?0 � 8ppa;?0=B20 are intro-

duced. The equation can be further simplified. First, we note

that the MHD equilibrium functions are related by the quasi-

neutrality requirement and the MHD equilibrium conditions,

i.e., quasi-neutrality gives23

GðxÞ ¼ ½FðxÞ1þ~c ; (12)

where ~c � ðce;?=ci;?Þ � 1 and a polytropic relation betweenthe pressure and the density is assumed,33,34 while the MHD

force balance condition gives23

HðxÞ ¼ 1þ b?0 � ½bi;?0 þ be;?0ðFðxÞÞ~c FðxÞ: (13)

Then, we require again quasi-neutrality for our modified

equilibrium, i.e.,

GðxÞgðxÞ ¼ ½FðxÞf ðxÞ1þ~c or; equivalently;gðxÞ ¼ ½f ðxÞ1þ~c ; (14)

where the equivalence is because of condition (12). We then

require that the perpendicular plasma beta bi;?ðxÞ remainsunchanged with respect to the MHD equilibrium (see the

Appendix for alternative requests), which then leads to the

condition

hðxÞ ¼ f ðxÞ: (15)

FIG. 1. Left: plot of the solutions

Pxx=p? (blue solid line) and Pyy=p?(red dashed line). Right: consequent

perpendicular anisotropy A? (solidline) versus the shear parameter a.



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

If ~c ¼ 0 holds, this condition is equivalent to the requirementthat the total perpendicular plasma beta b?ðxÞ remainsunchanged. Thus, using the previous relations, the equilib-

rium condition (11) reads

1� ~bi;?0ai xð Þ

1þ ai xð Þ

� �F xð Þ þ ~be;?0 F xð Þð Þ1þ~c f ~c xð Þ � 1

� �� f xð Þ � 1 ¼ 0; ð16Þ

which is intrinsically nonlinear in f, not only because of theparameter ~c, but also because aiðxÞ itself contains the mag-netic field profile which must be derived self-consistently

from the equilibrium, i.e.,

ai xð Þ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H xð Þh xð Þp s3 mi c

2 e B0

dui;ydx� a0i xð Þffiffiffiffiffiffiffiffiffi

h xð Þp ; (17)

where for convenience we have separatedffiffiffiffiffiffiffiffiffihðxÞ

pand a0iðxÞ,

since the latter term does not depend on the self-consistent

solution h(x). This intrinsic nonlinearity is the reason thatjustifies our approach of extending an MHD/CGL equilib-

rium to the “corresponding” full pressure tensor equilibrium:

in the MHD/CGL case, we do not have aiðxÞ, so the equilib-rium condition is easily solvable and we find FðxÞ; GðxÞ andHðxÞ. Then, we can compute a0iðxÞ and give the solution interms of it. Moreover, the MHD equilibria are the most com-

monly adopted for various simulation initialization, even in a

kinetic framework,10 and thus using them as a starting point

may be convenient.

In the following, when we give explicit examples, a

velocity profile described by a hyperbolic tangent will be

adopted

ui;y xð Þ ¼ u0 tanhx� x0

Lu

� �; (18)

which is often used for the study of KHI,4–6 and all the quan-

tities will be given in units of ions quantities (mi, e, Xi; di)and Alfvèn velocity (vA).

A. Discussion of the fully self-consistent equilibria

Let us consider the complete problem in which the aiðxÞfunction is computed with the actual self-consistent magnetic

field profile, Eq. (17). For a0i the positivity condition of thepressure, ai � �1=2, reads

a0i xð Þ � �ffiffiffiffiffiffiffiffiffih xð Þ

p2

8 x: (19)

Then, under the assumption of quasi-neutrality and the

request that the perpendicular plasma beta bi;?ðxÞ remainsunchanged with respect to the MHD equilibrium, Eqs. (14)

and (15), the equilibrium condition (16) can be recast in the

following form:

~be;?0FðxÞ1þ~c f ðxÞ3=2þ~c þ ½1� ~be;?0FðxÞ1þ

~c f ðxÞ3=2

þ ~be;?0FðxÞ1þ~ca0iðxÞf ðxÞ1þ~c þ ½1� ~bi;?0FðxÞ

� ~be;?0FðxÞ1þ~c a0iðxÞf ðxÞ � f ðxÞ1=2 � a0iðxÞ ¼ 0; (20)

which is absolutely non trivial, since the parameter ~c canchange the order of the equation in a non obvious way. We

thus restrict the problem to the case ~c ¼ 0, i.e., to the case ofan equal polytropic law for the electrons and the ions in the

plane perpendicular to B. This assumption is physicallyreasonable and leaves total freedom for what concerns the

parallel polytropic laws for electrons and ions. The equilib-

rium condition for ~c ¼ 0 readsffiffiffiffiffiffiffiffiffiffif ðxÞ3

qþ ½1� ~bi;?0FðxÞa0iðxÞf ðxÞ �

ffiffiffiffiffiffiffiffif ðxÞ

p� a0iðxÞ ¼ 0;

(21)

which can be interpreted as a cubic equation for wðxÞ �ffiffiffiffiffiffiffiffif ðxÞ

p(f(x) cannot be negative since it is related to the pres-

sure—see Eq. (10)). In order to gain some insights from

Eq. (21), we can solve it for a0iðf Þ, i.e.,

a0i fð Þ ¼ffiffiffiffif 3

p�

ffiffiffifp

1� 1� ~bi;?0F xð Þh i

f; (22)

together with the condition in Eq. (19), which in our case,

hðxÞ ¼ f ðxÞ, becomes

a0i fð Þ � �ffiffiffifp

2: (23)

The implicit solution a0iðf Þ is shown in Fig. 2 for the case ofF ¼ H ¼ 1 and jB0j ¼ 1, with different values of ~bi;?0 andvelocity shear strength, assuming a hyperbolic tangent

profile of the type in Eq. (18) with Lu¼ 3. Three cases areshown: bi;?0 ¼ be;?0 ¼ 0:1 and u0 ¼ 2=3 (left panel),bi;?0 ¼ be;?0 ¼ 1 and u0 ¼ 2=3 (center), and bi;?0 ¼ be;?0 ¼2 and u0 ¼ 1:5 (right panel). The red continuous line repre-sents the positivity condition border set by Eq. (23), i.e.,

a0i ¼ �ffiffiffifp

=2, and only solutions above that curve are physi-cal. The red dashed lines represent instead the maximum

value of a0iðxÞ for the chosen velocity profile.From Fig. 2 several interesting features emerge: (i) the

parameter ~bi;?0F (here F ¼ 1) essentially determines how“fast” the solution f deviates from unity when a0i deviatesfrom zero, (ii) the strength of the velocity shear, i.e., the

parameter u0=B0Lu, determines how far from gyrotropy thesystem is allowed to go, (iii) the presence of an asymmetry

with respect to the sign of a0i (due to the plot scale, this ismore evident in the right panel, but it is true in general), and

(iv) the existence of a second real solution for a0i 0 andnot for a0i > 0, which is however unphysical (i.e., below thepositivity border - see below). The above plots allow us to

represent solutions in implicit form. Considering a hyper-

bolic tangent velocity shear profile as in Eq. (18) and H ¼ 1,the function a0iðxÞ reads

a0i xð Þ ¼s32

u0B0Lu

cosh�2 x=Luð Þ; (24)

where we took x0 ¼ 0 for simplicity. We consider the caseshown in the central panel of Fig. 2 for a0iðf Þ. This plot ofa0i versus f, zoomed in the region around a0i ¼ 0 and f¼ 1,is reproduced in the right panel of Fig. 3. In the left panel of



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

Fig. 3, we show the plot of a0iðxÞ in Eq. (24), for both signsof s3. The aim is to visualize how f(x) should look by draw-ing a0iðxÞ and a0iðf Þ next to each other, with the values of a0ion the y-axis of both plots (and with the same scale), so onecan go back and forth from one plot to the other in three

steps: go along the x-axis of left panel of Fig. 3 (x coordi-nate), (ii) look at the value that a0iðxÞ takes in the left paneland then move to the same level of right panel (in both pan-

els, the y-axis has the same scale and the values of a0i arerepresented on it) and (iii) after reaching the curve a0iðf Þ onthe right panel in the point that corresponds to the value of

a0i found at point (ii), go down to the x-axis of the rightpanel in order to find the value of f that corresponds to thevalue of x at point (i).

Imagine going along the whole x-axis, from �1 toþ1. For sufficiently large jxj, we have a0i ¼ 0, so the solu-tion is asymptotically f¼ 1 (e.g., for jxj� 4 in left panel ofFig. 3). Then, still referring to Fig. 3, as we approach x¼ 0from negative values a0i starts to deviate from zero (leftpanel) and thus also f(x) starts to deviate from unity (rightpanel), becoming bigger or smaller if a0i becomes positive ornegative, respectively. In passing through x¼ 0 (left panel),we pass through the global maximum (minimum) of the pos-

itively (negatively) valued a0i, corresponding to the maxi-mum deviation of f(x) from unity in the right panel (i.e., inthe point where the curve a0iðf Þ intercepts the horizontal reddashed line, above f¼ 1 or below it accordingly with thesign of a0i). Then, leaving x¼ 0 behind and proceeding toincreasing positive x-values in the left panel of Fig. 3, a0istarts to decrease (increase) and so does f(x) in the right

panel, until it comes back to unity for sufficiently high x-val-ues (note that for x> 0 we are going on the curve a0iðf Þ allthe way back compared to how it had been covered for x< 0,the point x¼ 0 being the turning point).

Explicit numerical solutions f(x) of Eq. (21) for a0iðxÞgiven in Eq. (24) are plotted in Fig. 4 for the three cases in

Fig. 2. The corresponding profiles PxxðxÞ (s3 ¼ þ1: bottomblue solid line and s3 ¼ �1: top blue dashed line) andPyyðxÞ (s3 ¼ þ1: top red solid line and s3 ¼ �1: bottom reddashed line) are also given (from Eqs. (33) and (34)—see

below). The asymmetry with respect to the sign of Xu � B ismore evident on the right panel due to the choice of the pa-

rameters, but it is present in all cases.

In order to trace back the origin of the double solution

for a0i 0 and to show that one of the two solutions isunphysical because of the positivity condition in Eq. (23),

we consider the equilibrium condition, Eq. (21), and look for

solutions which deviate from gyrotropy weakly. For this pur-

pose, we treat ~bi;?0 as a small parameter (here, we considerthe case F ¼ 1, but the bound 0 F 1 always holds). For~bi;?0 ¼ 0, Eq. (21) is exactly solvable

ðf ðxÞ � 1Þðffiffiffiffiffiffiffiffif ðxÞ

pþ a0iðxÞÞ ¼ 0; (25)

and admits two real roots

f0ðxÞ ¼ 1 8 x~f 0ðxÞ ¼ a20iðxÞ 8 x 2 fxja0iðxÞ 0g;

((26)

where f0 ¼ 1 means gyrotropy, while ~f 0 represents the~bi;?0 ! 0 limit of the second solution in Fig. 2 for a0i 0.

FIG. 2. Plot of the implicit solution a0iðf Þ in Eq. (22) for different plasma parameters: bi;?0 ¼ be;?0 ¼ 0:1 and u0 ¼ 2=3 (left), bi;?0 ¼ be;?0 ¼ 1 and u0 ¼ 2=3(center), bi;?0 ¼ be;?0 ¼ 2 and u0 ¼ 1:5 (right). The other parameters are F ¼ H ¼ 1; jB0j ¼ 1, Lu¼ 3 for all cases. Red continuous line represents the borderabove which the pressure is positive, Eq. (23). Red dashed lines represent the bounds of a0i values (for both s3 ¼ þ1 and s3 ¼ �1), while the dotted anddashed-dotted lines are the reference for f¼ 1 and a0i ¼ 0, respectively.

FIG. 3. Left: plot of a0iðxÞ in Eq. (24)for both sign of s3 (see inset). Right:zoom around f¼ 1 and a0i ¼ 0 of thecenter plot in Fig. 2.



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

However, ~f 0 is below the positivity condition ai � �1=2 (forf ! ~f 0 we obtain ai ! �1). Keeping the first order in~bi;?0F � 1 for the physical solution near unity leads to

f xð Þ ’ 1þ ~bi;?0F xð Þa0i xð Þ

1þ a0i xð Þ; (27)

which corresponds to the first-order Taylor expansion for

small C0a0i=ð1þ a0iÞ of the solution in Eq. (30) (see below).This means that, at least in the limit of small-~bi;?0, the twosolutions are close to each other. Note that the small-~bi;?0limit does not necessarily mean small-bi;?0, but it can bereached also in the large perpendicular temperature ratio,

s? ¼ Te;?0=Ti;?0 � 1. Finally, physical solutions of Eq. (21)only exist within restricted domains of (~bi;?0; a0i). In partic-ular, this turns out to be the case only when a0i is negative,i.e., for Xu � B < 0. An example of this asymmetric behavioris given in Fig. 5, where we show that solutions for a0i < 0may disappear because of the positivity constraint depending

on the value of ~bi;?0. For a0i > 0, i.e., for Xu � B > 0, a

solution is instead always present, regardless of the value of~bi;?0.

B. Approximate analytical solution of the equilibriumcondition

In the limit of small corrections to the MHD equilib-

rium, we can give analytical solutions for f(x) with ~c 6¼ 0. Inthis limit, the ion cyclotron frequency Xi in the ai functioncan be computed with the MHD profile of the magnetic field,

jBðxÞj ’ B0ffiffiffiffiffiffiffiffiffiffiHðxÞ

p, i.e.,

ai xð Þ ’ a0i xð Þ ¼s3

2B0ffiffiffiffiffiffiffiffiffiffiffiH xð Þ

p dui;ydx

; (28)

which is an approximation that one should check a poste-

riori (see the Appendix for a simple case in which this

case is exact), and leads to the following equilibrium

condition:

FIG. 4. Top row: plot of the explicit numerical solution f(x) of Eq. (21). Bottom row: corresponding Pxx and Pyy profiles (see in the text). The three casesabove correspond to the three cases in Fig. 2: bi;?0 ¼ be;?0 ¼ 0:1 and u0 ¼ 2=3 (left), bi;?0 ¼ be;?0 ¼ 1 and u0 ¼ 2=3 (center), bi;?0 ¼ be;?0 ¼ 2 and u0 ¼ 1:5(right).

FIG. 5. Plot of the implicit solution

a0iðf Þ for the case bi;?0 ¼ be;?0 ¼ 5(left) and for the case bi;?0 ¼ 10;be;?0 ¼ 1 (right), corresponding to~b i;?0 ¼ 5=11 and 5/6, respectively.The other parameters are F ¼ H ¼ 1;u0 ¼ 2:45; jB0j ¼ 1, Lu¼ 3 for bothcases.



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

1� ~bi;?0F xð Þa0i xð Þ

1þ a0i xð Þ

�

þ~be;?0 F xð Þð Þ1þ~c f ~c xð Þ � 1� �i

f xð Þ ¼ 1: (29)

Treating ~c as a small parameter, we solve the above equilib-rium problem iteratively. The solution of Eq. (29) for ~c ¼ 0is straightforward, i.e.,

f0 xð Þ ¼ 1� C0 xð Þa0i xð Þ

1þ a0i xð Þ

� ��1; (30)

where C0ðxÞ ¼ ~bi;?0FðxÞ. Noting that Eq. (29) is equivalentto the equilibrium condition Eq. (23) in Ref. 23, with the

substitution

~u0 xð Þ ! a0i xð Þ1þ a0i xð Þ

;

we can derive the iterative solution (cf. Eqs.(24)–(26) in

Ref. 23), i.e.,

f xð Þ ¼ 1� C xð Þ a0i xð Þ1þ a0i xð Þ

� ��1; (31)

with

CðxÞ ¼ ½1þ ~c~be;?0ðFðxÞÞ1þ~c �1C0ðxÞ: (32)

In the above, we have assumed that j~c~be;?0ðFðxÞÞ1þ~c j < 18x

for the convergence of the resulting series, which can be

shown to be always the case (see Ref. 23). Moreover, the

solution passes through a Taylor expansion in which we

consistently assume that the correction to the MHD profile is

small, i.e., jCa0i=ð1þ a0iÞj � 1. In this regard, the relationja0i=ð1þ a0iÞj 1 8 a0i � �1=2 holds, ~bi;?0 < 1 by defini-tion and we can always choose pi;?0 such that FðxÞ < 18x.Thus, the condition jCa0i=ð1þ a0iÞj � 1 is valid for most ofthe parameter range commonly adopted. Finally, the solution

in Eq. (31) reduces to the first-order FLR solution given in

Ref. 23 if we retain only the first order in a0i inside the squarebrackets.

1. Explicit equilibrium profiles

We give also the explicit equilibrium profiles of the

physical quantity of interest, for the sake of clarity

Pi;xx ¼ pi;?0 1�ai xð Þ

1þ ai xð Þ

� �F xð Þf xð Þ; (33)

Pi;yy ¼ pi;?0 1þai xð Þ

1þ ai xð Þ

� �F xð Þf xð Þ; (34)

Pi;zz � pi;jj ¼ pi;jj0ðFðxÞf ðxÞÞci;jj=ci;? ; (35)

nðxÞ ¼ n0ðFðxÞf ðxÞÞ1=ci;? ; (36)

Pe;xx ¼ Pe;yy � pe;? ¼ pe;?0ðFðxÞf ðxÞÞ1þ~c ; (37)

Pe;zz � pe;jj ¼ pe;jj0ðFðxÞf ðxÞÞce;jj=ci;? ; (38)

BzðxÞ ¼ B0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHðxÞf ðxÞ

p: (39)

If the parallel and perpendicular temperatures are of interest,32

instead of the corresponding pressures, from Eqs. (33)–(36)

we have

Ti;? �1

n

Pi;yy þPi;yy2

¼ Ti;?0 F xð Þf xð Þð Þci;?�1ci;? ; (40)

Ti;jj �pi;jjn¼ Ti;jj0 F xð Þf xð Þð Þ

ci;jj�1ci;? ; (41)

Te;? �pe;?

n¼ Ti;?0 F xð Þf xð Þð Þ

ce;?�1ci;? ; (42)

Te;jj �pe;jjn¼ Ti;jj0 F xð Þf xð Þð Þ

ce;jj�1ci;? : (43)

We define the MHD profiles with the superscript (0), e.g.,

Tð0Þi;? � Ti;?0FðxÞ

ðci;?�1Þ=ci;? , and the overlined quantities asthe ratio between the actual and the MHD profiles, e.g.,�T i;? � Ti;?=Tð0Þi;?. Note that taking the perpendicular andparallel polytropic indices to be c? ¼ 2 and cjj ¼ 1 for bothspecies, one obtains

�T jj ¼ 1; (44)

�T? ¼ �n ¼ �B; (45)

which mean �pjj �B2=�n3 ¼ const: and �p?=�n �B ¼ const:. Thus,

if the initial MHD profiles are such that pð0Þjj ðBð0ÞÞ

2=ðnð0ÞÞ3 ¼const: and p

ð0Þ? =n

ð0ÞBð0Þ ¼ const: (as, e.g., for the caseF ¼ H ¼ 1), then also pjjB2=n3 ¼ const: and p?=nB¼ const:, as expected from the double adiabatic laws.22

Finally, in order to visualize the asymmetry due to the

sign of Xu � B, we plot the equilibrium profiles for a givencase. In Fig. 6, we show the velocity shear ui;yðxÞ andthe function CðxÞa0iðxÞ=½1þ a0iðxÞ (left panel) and the cor-responding equilibrium profiles for Pi;xx and Pi;yy (rightpanel).

The parameters used for the profiles in Fig. 6 are

u0 ¼ 2=3, Lu¼ 3, B0 ¼ 61 (s3 ¼ 61), bi;?0 ¼ be;?0 ¼ 1;and ~c ¼ 0. For the sake of clarity, we have chosen the sim-plest MHD case of F ¼ G ¼ H ¼ 1. By an inspection of theplot, some considerations emerge. First, while the condition

jCðxÞa0iðxÞ=½1þ a0iðxÞj � 1 holds, as assumed in thederivation on the profiles, the effects on the anisotropy

are actually big (�20%� 30%). The same considerationremains true even for more extreme cases. Second, the asym-

metry between the two cases s3 ¼ 61 is impressive, evenfor this moderate case: the two set of profiles are remarkably

different, while the value of ai, which gives us an idea ofhow far from gyrotropy (a0i ¼ 0) the system is, only reachesa0i ’ 0:11 (a0i ’ �0:11) for the configuration with s3 ¼ þ1(s3 ¼ �1). The fundamental difference between the two con-figurations, s3 ¼ 61, may then lead to very different dynam-ical evolution of the system and thus of the instability under

study already in the linear phase.1,2,10 In Fig. 7, we show the

same quantities as in Fig. 6 for a different set of parameters:

F ¼ G ¼ H ¼ 1; u0 ¼ 3=2, Lu¼ 3, B0 ¼ 61 (s3 ¼ 61),



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

bi;?0 ¼ be;?0 ¼ 2; and ~c ¼ 0, which correspond toa0i ’ 60:25.

IV. CONCLUSIONS AND DISCUSSION

We have presented a study of the role of the pressure

tensor in the presence of a sheared velocity field within a

fluid plasma framework. The heat fluxes are neglected.

Solutions of the stationary pressure tensor equation are given

for a simple, but commonly adopted configuration, and the

properties of such equilibrium solutions are discussed. In

particular, we have shown that, in addition to the well known

parallel-perpendicular anisotropy (pjj 6¼ p?), the system isalso anisotropic in the plane perpendicular to the magnetic

field, i.e., Pxx 6¼ Pyy 6¼ Pzz. The magnitude of the perpen-dicular anisotropy turns out to depend on the strength of the

velocity shear and on its scale length of variation. Moreover,

the system is strongly asymmetric with respect to the relative

orientation of the background magnetic field and of the fluid

vorticity, i.e., with respect to the sign of Xu � B. These prop-erties of the system are present even at the level of the equi-

librium state representing the starting point for the study of

shear-flow instabilities.

A method for deriving equilibrium profiles is presented

and both numerical and approximated analytical solutions

are provided for some representative cases. The profiles

derived in the present paper are shown to be different with

respect to the usual MHD or even CGL equilibria. In particu-

lar, they depend on the velocity shear and are asymmetric

with respect to the sign of Xu � B. These features, arising al-ready at the level of the equilibrium configuration, turn out

to be relevant when fluid models that retain the pressure

tensor equation and/or kinetic models are adopted, as for the

study of the KHI and the MRI.

Finally, despite the relative simplicity of the system con-

figuration adopted, it seems plausible that our results can be

used for the interpretation of satellite data where non-

gyrotropic distribution functions are observed. This could be

the case, for instance, of solar wind data, since, as pointed

out by recent studies, one expects that the turbulence sponta-

neously generates local velocity shear flows.

ACKNOWLEDGMENTS

The research leading to these results received funding

from the European Research Council under the European

Unions Sevenths Framework Programme (FP7/2007-2013)/

ERC Grant Agreement No. 277870. The research leading to

these results has received funding from the European

Commission’s Seventh Framework Programme (FP7/2007-

2013) under the Grant Agreement SWIFF (Project No.

263340, www.swiff.eu).

APPENDIX: ALTERNATIVE ANALYTICAL EQUILIBRIA

A relevant feature of Eq. (11) is that it is general and

versatile. Depending on the physical requirements, whichthen translate into mathematical relations between f, g, andh, one can compute very different equilibrium profiles. In thefollowing, we give some examples.

1. Preserving the total perpendicular plasma beta

Requiring that the total perpendicular plasma beta b?ðxÞremains unchanged in passing from MHD to full pressure

tensor equilibria, the relation in Eq. (15) is substituted by

FIG. 6. Left: velocity profile ui;yðxÞ(blue solid line) and the function

CðxÞaiðxÞ=½1þ aiðxÞ (red dashed line).Right: plot of the approximated equi-

librium profiles Pi;xxðxÞ (s3 ¼ þ1: bot-tom blue solid line, s3 ¼ �1: top bluedashed line) and Pi;yyðxÞ (s3 ¼ þ1: topred solid line, s3 ¼ �1: bottom reddashed line). Here, the parameters are:

F ¼ G ¼ H ¼ 1; u0 ¼ 2=3, Lu¼ 3,B0 ¼ 61 (s3 ¼ 61), bi;?0 ¼ be;?0 ¼ 1and ~c ¼ 0.

FIG. 7. Left: velocity profile ui;yðxÞ(blue solid line) and the function

CðxÞaiðxÞ=½1þ aiðxÞ (red dashed line).Right: plot of the approximated equi-

librium profiles Pi;xxðxÞ (s3 ¼ þ1: bot-tom blue solid line, s3 ¼ �1: top bluedashed line) and Pi;yyðxÞ (s3 ¼ þ1: topred solid line, s3 ¼ �1: bottom reddashed line). Here, the parameters are:

F ¼ G ¼ H ¼ 1; u0 ¼ 3=2, Lu¼ 3,B0 ¼ 61 (s3 ¼ 61), bi;?0 ¼ be;?0 ¼ 2and ~c ¼ 0.



130.183.100.95 On: Tue, 07 Apr 2015 12:47:42

www.swiff.eu

hðxÞ ¼ n~cðF ; f ; xÞf ðxÞ; (A1)

where the function

n~c F ; f ; xð Þ �bi;?0 þ be;?0 F xð Þf xð Þð Þ~c

bi;?0 þ be;?0 F xð Þð Þ~c; (A2)

has been defined, such that it reduces to n0ðF ; f ; xÞ ¼ 1 for~c ¼ 0. For ~c ¼ 0 we indeed recover Eq. (15) and thus theequilibrium condition in Eq. (16) and its solution. For ~c 6¼ 0,the equilibrium condition reads

n~c F ; f ; xð Þ þ ~bi;?01� n~c F ; f ; xð Þ 1þ a0i xð Þð Þ

1þ a0i xð Þ

" #F xð Þ

(

þ ~be;?0 F xð Þð Þ1þ~c f ~c xð Þ � n~c F ; f ; xð Þ

�)

f xð Þ ¼ 1; (A3)

which, Taylor expanding f ~c and n~c as in Sec. III B and aftersome algebra, admits the solution

f xð Þ ¼ 1� C xð Þ ai xð Þ1þ ai xð Þ

� ��1; (A4a)

CðxÞ ¼ ½1þ ~c~be;?0 ~F~cðxÞ�1C0ðxÞ; (A4b)

~F~c xð Þ � F xð Þð Þ~c

~bi;?0 þ ~be;?0 F xð Þð Þ~c: (A4c)

2. Preserving the magnetic field configuration

We may want to fix the magnetic field configuration:

this is equivalent to requiring that the magnetic field profile

remains unchanged with respect to the MHD profile, i.e.,

h xð Þ¼ 1; H xð Þ1þb?0

¼ 1� ~bi;?0þ ~be;?0 F xð Þð Þ~ch i

F xð Þ; (A5)

and thus we need to solve the following equilibrium

condition:

~bi;?0 1�ai xð Þ

1þ ai xð Þ

� �þ ~be;?0F~c xð Þf ~c xð Þ

� �f xð Þ

¼ ~bi;?0 þ ~be;?0F~c xð Þ; (A6)

which, considering the case ~c ¼ 0 for simplicity, has thesolution

f0 xð Þ ¼1þ s?

1þ s? � a0i xð ÞffiffiffiffiffiffiffiffiH xð Þp þa0i xð Þ; (A7)

where we have defined the perpendicular temperature ratio

s? � Te;?0=Ti;?0 for brevity and now the function aiðxÞ iscomputed with the actual local magnetic field, without

approximations (i.e., aiðxÞ � a0iðxÞ).

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