Pressure tensor in the presence of velocity shear: stationary solutions and self-consistent equilibria
S. S. Cerri, F. Pegoraro, F. Califano, D. Del Sarto, F. Jenko
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t Pressure tensor in the presence of velocity shear: stationary solutions andself-consistent equilibria
S. S. Cerri ∗ Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstr. 2, D-85748 Garching, Germany
F. Pegoraro
Physics Department “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
F. Califano
Physics Department “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy andMax-Planck/Princeton Center for Plasma Physics
D. Del Sarto
Institut Jean Lamour, UMR 7198 CNRS - Universit´e de Lorraine, BP 239 F-54506 Vandoeuvre les Nancy, France
F. Jenko
Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstr. 2, D-85748 Garching, GermanyDepartment of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA andMax-Planck/Princeton Center for Plasma Physics
Observations and numerical simulations of laboratory and space plasmas in almost collisionlessregimes reveal anisotropic and non-gyrotropic particle distribution functions. We investigate howsuch states can persist in the presence of a sheared flow. We focus our attention on the pres-sure tensor equation in a magnetized plasma and derive analytical self-consistent plasma equilibriawhich exhibit a novel asymmetry with respect to the magnetic field direction. These results arerelevant for investigating, within fluid models that retain the full pressure tensor dynamics, plasmaconfigurations where a background shear flow is present.
PACS numbers: 52.55.Dy, 52.30.Cv, 52.30.Gz, 95.30.Qd
I. INTRODUCTION
Sheared flows are frequently observed in space and laboratory plasmas. They are an important source of free energyand can drive various instabilities, such as the Kelvin-Helmholtz instability [1–10] (KHI) or the magnetorotationalinstability [11–17] (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 efficiently transferred to small scales. In this context, a relevantexample is given by the development of the KHI observed at the flanks of the Earth’s magnetosphere [18] driven bythe velocity shear between the solar wind (SW) and the magnetosphere (MS) plasma, and in general observed atother planetary magnetospheres. On the other hand, the MRI is considered to be a main driver of turbulence (andturbulent transport of angular momentum) in accretion disks around astrophysical objects, such as stars and blackholes. In addition, 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 whenthe scale length of the sheared flow is much larger than the typical ion microscales, i.e. when d i , ρ i ≪ L . However,in the case of the interaction of the SW with the MSP, satellite observations show that the typical scale length ofthe sheared flow is roughly comparable to the ion gyroradius and/or the skin depth, i.e. d i ∼ ρ i . L ( β ∼ gyrotropy , (ii) the system is symmetric with respect to the relative orientation of the magnetic field B and the fluid ∗ Electronic address: [email protected] vorticity Ω u ≡ ∇ × u , i.e. with respect to the sign of Ω u · B , and (iii) the equilibrium profiles are not dependent onthe velocity shear. These three points are substantially modified when the pressure tensor equation is retained in thefluid hierarchy or when kinetic models are adopted. Also retaining first-order finite Larmor radius (FLR) correctionsof the ions within a two-fluid (TF) model, the so called extended two-fluid ( e TF) model [23], was recently shown tosubstantially modify the previous picture.In this work, we investigate the role of retaining the full pressure tensor equation, still in the framework of afluid model. In order to simplify the picture for the sake of clarity, we will consider a configuration in which theinhomogeneity direction, the flow direction and the magnetic field are orthogonal to each other. The main resultof our approach is to prove that, in the presence of a shear flow, an additional anisotropy in the perpendicularplane ( agyrotropy ) and an asymmetry with respect to the sign of Ω u · B arise even at the level of the equilibriumconfigurations, which depend also on the shear strength. Indeed, a sheared flow can induce dynamical anisotropizationof an initial isotropic pressure configuration, together with an asymmetry with respect to the sign of Ω u · B , when,for instance, one retains first-order FLR corrections [23] or the full pressure tensor equation [24]. Here, we focus ourattention on the effect of the shear-induced anisotropization at the level of a stationary state. Such new features areintrinsic properties of the system and their relevance is related, in general, to the plasma regime under investigation. Inparticular, the deviation from an MHD/CGL model becomes not negligible when the shear length scale is comparablewith the ion microscales ( d i , ρ i . L ), as is the case for the SW-MS interaction. Such deviations, important at the levelof equilibrium configurations, can dramatically affect the study of one of the above mentioned instabilities alreadyin the linear phase and possibly 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 configuration, in which thepossibly relevant ingredients, such as the pressure tensor, are retained. Moreover, as already anticipated, the approachpresented here can give insights into non-gyrotropic proton distribution functions that are observed in SW data andVlasov simulations [19–21].The remainder of this paper is organized as follows. In Sec. II, we solve the stationary pressure tensor equationwithout heat fluxes and in the presence of a sheared flow, giving the solution in the form of traceless corrections tothe CGL gyrotropic pressure tensor, discussing the emergence of the perpendicular anisotropy and of the Ω u · 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 explicitanalytical approximations, underlining again the role of the asymmetry and the perpendicular anisotropy. Finally,alternative approximated and exact analytical equilibria are given in Appendix A. II. STATIONARY SOLUTION OF THE PRESSURE TENSOR EQUATION
Within a fluid description of a plasma, the pressure tensor equation is given by [23] ∂ Π α,ij ∂t + ∂∂x k (Π α,ij u α,k + Q α,ijk ) + Π α,ik ∂u α,j ∂x k + Π α,jk ∂u α,i ∂x k = q α m α c ( ǫ ilm Π α,jl + ǫ jlm Π α,il ) B m , (1)where q α and m α are the charge and the mass of the species α , respectively, Π α,ij is the ( ij -component of the) pressuretensor, u α,k is the ( k -component of the) fluid velocity, Q α,ijk is the ( ijk -component of the) heat flux tensor, ǫ ijk isthe Levi-Civita symbol and B m is the ( m -component of the) magnetic field. Now, we look for stationary solutions ofEq. (1) in the limit of no heat fluxes, i.e. ∂ Π α,ij /∂t = 0 and Q α,ijk = 0 ∀ i, j, k . Under these assumptions, Eq. (1)reduces to ∂∂x k (Π α,ij u α,k ) + Π α,ik ∂u α,j ∂x k + Π α,jk ∂u α,i ∂x k = σ α Ω α ( ǫ ilm Π α,jl + ǫ jlm Π α,il ) b m , (2)where we have introduced the sign of the charge σ α ≡ sign( q α ) and the cyclotron frequency Ω α ≡ | q α || B | /m α c of thespecies α , respectively, and the magnetic field versor b m ≡ B m / | B | .It is a well known result that, within a finite Larmor radius (FLR) expansion of Eq. (2), the zero-order solution isgiven by the CGL pressure tensor [22, 23], Π (0) α = p α, ⊥ τ + p α, k bb (3)where τ ≡ I − bb is the projector onto the plane perpendicular to the magnetic field ( b ≡ B / | B | ), p α, k and p α, ⊥ arethe pressure parallel and perpendicular to B , respectively. The tensor Π (0) α represents the kernel of the operator onthe right hand side of the pressure tensor equation, i.e. it is an approximated solution of the equation when the lefthand side is negligible.We now consider a velocity shear configuration such that the inhomogeneity direction, the flow direction and themagnetic field direction form a right-handed basis, e.g. u = u y ( x ) e y and B = B z ( x ) e z . 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 Π α = Π (0) α + e Π α , in which e Π α represents a tracelesscorrection to the gyrotropic pressure tensor Π (0) α . In fact, remaining within a fluid framework, one has unambiguousdefinitions of p α, ⊥ ≡ Π α : τ and p α, k ≡ Π α : bb , so the identity e Π α : I = 0 holds [25–30]. The same conclusion canbe directly derived from the Vlasov equation [31]. From the definition of p α, k one obtains also e Π α : bb = 0, so in ourconfiguration e Π α,zz = 0 and the perpendicular components of the pressure tensor readΠ α,xx = p α, ⊥ + e Π α,xx , Π α,yy = p α, ⊥ + e Π α,yy with e Π α,yy = − e Π α,xx . (4)Inserting the above expressions into Eq. (2) gives the following non-gyrotropic diagonal pressure tensor: Π α,zz = p α, k Π α,xx = (cid:16) − a α ( x )1+ a α ( x ) (cid:17) p α, ⊥ Π α,yy = (cid:16) a α ( x )1+ a α ( x ) (cid:17) p α, ⊥ (5)with a α ( x ) ≡ s σ α Ω α du α,y dx , (6)where s ≡ sign( b ) is the relative orientation of the magnetic field and the z -axis (see Ref. [23]). The positivitycondition on the diagonal pressure terms in Eq. (5) gives a α ( x ) ≥ − . (7)It is interesting to note that here an asymmetry with respect to the sign of a α ( x ) appears. In fact, not all the valueof the shear strength are allowed when a α is negative, while in principle there is no limitation when it is positive.The sign of a α depends essentially on two physical factors: the species, through σ α , and the relative orientation ofthe magnetic field B and the fluid vorticity Ω u , through the sign of s ( du α,y /dx ). For instance, if one considers ions( σ i = +1) and a background magnetic field oriented in the positive direction of the z -axis ( s = +1), from Eq. (7) wefind that there is no limitation in the velocity shear of u i if the vorticity is aligned with the magnetic field ( Ω u · B > | Ω u | ≤ Ω i ). We note that this condition on the shear strength limitation asymmetry is indeed in accordance withRef. [24], where it is found that shear configurations of the type considered here become dynamically unstable whenΩ ′ / Ω i ≡ ∂ x u i ,y ) / Ω i <
0, which corresponds exactly to the condition a i ( x ) < − / A α, ⊥ ≡ | Π α,xx − Π α,yy | p α, ⊥ = 2 (cid:12)(cid:12)(cid:12)(cid:12) a α ( x )1 + a α ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , (8)which gives the maximum value of the anisotropy ( A (max) ⊥ = 2) for a = − / a → ∞ . A sketch of Π xx /p ⊥ ,Π yy /p ⊥ and A ⊥ versus a is given in Fig. 1. The asymmetry is thus found also in the perpendicular anisotropy, whichis here entirely due to the velocity shear.For small | a | ≪
1, the corrections to the gyrotropic CGL pressure tensor are small and, if we expand the solutionin Eq. (5) to the leading order in a , the first-order FLR solution is found [23, 25–30]. However, in this limit, theasymmetry with respect to the sign of a in Eq. (7) is lost. III. EQUILIBRIA WITH THE COMPLETE PRESSURE TENSOR
We now want to use the stationary solution of Π xx in Eq. (5) for solving the equilibrium condition [23] ddx "X α Π α,xx + B π = ddx (cid:20) Π i ,xx + p e , ⊥ + B π (cid:21) = 0 , (9)Figure 1: Left: plot of the solutions Π xx /p ⊥ (blue solid line) and Π yy /p ⊥ (red dashed line). Right: consequentperpendicular anisotropy A ⊥ (solid line) versus the shear parameter a .where here we are considering only the ions full pressure tensor, while the electrons are gyrotropic, i.e. Π e = p e , ⊥ τ + p e , k bb . Note that in our configuration, since there are no parallel gradients, the parallel balance is automaticallysatisfied [32]. We define the gyrotropic pressure and the magnetic field profiles as p i , ⊥ ( x ) = p i , ⊥ F ( x ) f ( x ) p e , ⊥ ( x ) = p e , ⊥ G ( x ) g ( x ) B ( x ) = B H ( x ) h ( x ) (10)where p i , ⊥ and p e , ⊥ are positive constants, F ( x ), G ( x ) and H ( x ) correspond to the MHD equilibrium and f ( x ), g ( x )and h ( x ) are the corrections to the relative MHD profile. The equilibrium condition, Eq. (9), can then be convenientlyrewritten in dimensionless form as e β i , ⊥ (cid:20) − a i ( x )1 + a i ( x ) (cid:21) F ( x ) f ( x ) + e β e , ⊥ G ( x ) g ( x ) + H ( x ) h ( x )1 + β ⊥ − , (11)where we have fixed the constant of integration to be B / π + p i , ⊥ + p e , ⊥ and the quantities e β α, ⊥ ≡ β α, ⊥ / (1+ β ⊥ ), β ⊥ = P α β α, ⊥ and β α, ⊥ ≡ πp α, ⊥ /B are introduced. The equation can be further simplified. First, we note thatthe MHD equilibrium functions are related by the quasi-neutrality requirement and the MHD equilibrium conditions,i.e. quasi-neutrality gives [23] G ( x ) = [ F ( x )] e γ , (12)where e γ ≡ ( γ e , ⊥ /γ i , ⊥ ) − H ( x ) = 1 + β ⊥ − h β i , ⊥ + β e , ⊥ ( F ( x )) e γ i F ( x ) . (13)Then, we require again quasi-neutrality for our modified equilibrium, i.e. G ( x ) g ( x ) = [ F ( x ) f ( x )] e γ or , equivalently , g ( x ) = [ f ( x )] e γ , (14)where the equivalence is because of condition (12). We then require that the perpendicular plasma beta β i , ⊥ ( x )remains unchanged with respect to the MHD equilibrium (see Appendix A for alternative requests), which then leadsto the condition h ( x ) = f ( x ) . (15)If e γ = 0 holds, this condition is equivalent to the requirement that the total perpendicular plasma beta β ⊥ ( x ) remainsunchanged. Thus, using the previous relations, the equilibrium condition (11) reads (cid:20) − e β i , ⊥ (cid:18) a i ( x )1 + a i ( x ) (cid:19) F ( x ) + e β e , ⊥ ( F ( x )) e γ (cid:16) f e γ ( x ) − (cid:17)(cid:21) f ( x ) − , (16)which is intrinsically nonlinear in f , not only because of the parameter e γ , but also because a i ( x ) itself contains themagnetic field profile which must be derived self-consistently from the equilibrium, i.e. a i ( x ) = 1 p H ( x ) h ( x ) s m i c e B du i ,y dx ≡ a ( x ) p h ( x ) , (17)where for convenience we have separated p h ( x ) and a ( x ), since the latter term does not depend on the self-consistentsolution h ( x ). This intrinsic nonlinearity is the reason that justifies our approach of extending an MHD/CGL equi-librium to the “corresponding” full pressure tensor equilibrium: in the MHD/CGL case, we do not have a i ( x ), so theequilibrium condition is easily solvable and we find F ( x ), G ( x ) and H ( x ). Then, we can compute a ( x ) and givethe solution in terms of it. Moreover, the MHD equilibria are the most commonly adopted for various simulationinitialization, 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: u i ,y ( x ) = u tanh (cid:18) x − x L u (cid:19) , (18)which is often used for the study of KHI [4–6], and all the quantities will be given in units of ions quantities ( m i , e ,Ω i , d i ) and Alfv`en velocity ( v A ). A. Discussion of the fully self-consistent equilibria
Let us consider the complete problem in which the a i ( x ) function is computed with the actual self-consistentmagnetic field profile, Eq. (17). For a the positivity condition of the pressure, a i ≥ − /
2, reads a ( x ) ≥ − p h ( x )2 ∀ x . (19)Then, under the assumption of quasi-neutrality and the request that the perpendicular plasma beta β i , ⊥ ( x ) remainsunchanged with respect to the MHD equilibrium, Eqs. (14)-(15), the equilibrium condition (16) can be recast in thefollowing form: e β e , ⊥ F ( x ) e γ f ( x ) / e γ + h − e β e , ⊥ F ( x ) e γ i f ( x ) / + e β e , ⊥ F ( x ) e γ a ( x ) f ( x ) e γ + h − e β i , ⊥ F ( x ) − e β e , ⊥ F ( x ) e γ i a ( x ) f ( x ) − f ( x ) / − a ( x ) = 0 , (20)which is absolutely non trivial, since the parameter e γ can change the order of the equation in a non obvious way. Wethus restrict the problem to the case e γ = 0, i.e. to the case of an equal polytropic law for the electrons and the ionsin the plane perpendicular to B . This assumption is physically reasonable and leaves total freedom for what concernsthe parallel polytropic laws for electrons and ions. The equilibrium condition for e γ = 0 reads p f ( x ) + h − e β i , ⊥ F ( x ) i a ( x ) f ( x ) − p f ( x ) − a ( x ) = 0 , (21)which can be interpreted as a cubic equation for w ( x ) ≡ p f ( x ) ( f ( x ) cannot be negative since it is related to thepressure - see Eq. (10)). In order to gain some insights from Eq. (21), we can solve it for a ( f ), i.e. a ( f ) = p f − √ f − h − e β i , ⊥ F ( x ) i f , (22)together with the condition in Eq. (19), which in our case, h ( x ) = f ( x ), becomes a ( f ) ≥ − √ f . (23)The implicit solution a ( f ) is shown in Fig. 2 for the case of F = H = 1 and | B | = 1, with different values of e β i , ⊥ and velocity shear strength, assuming a hyperbolic tangent profile of the type in Eq. (18) with L u = 3. Three cases areshown: β i , ⊥ = β e , ⊥ = 0 . u = 2 / β i , ⊥ = β e , ⊥ = 1 and u = 2 / β i , ⊥ = β e , ⊥ = 2and u = 1 . a = −√ f /
2, and only solutions above that curve are physical. The red dashed lines represent instead the maximumvalue of a ( x ) for the chosen velocity profile.Figure 2: Plot of the implicit solution a ( f ) in Eq. (22) for different plasma parameters: β i , ⊥ = β e , ⊥ = 0 . u = 2 / β i , ⊥ = β e , ⊥ = 1 and u = 2 / β i , ⊥ = β e , ⊥ = 2 and u = 1 . F = H = 1, | B | = 1, L u = 3 for all cases. Red continuous line represents the border above whichthe pressure is positive, Eq. (23). Red dashed lines represent the bounds of a values (for both s = +1 and s = − f = 1 and a = 0, respectivelyFrom Fig.2 several interesting features emerge: (i) the parameter e β i , ⊥ F (here F = 1) essentially determines how“fast” the solution f deviates from unity when a deviates from zero, (ii) the strength of the velocity shear, i.e. theparameter u /B L u , determines how far from gyrotropy the system is allowed to go, (iii) the presence of an asymmetrywith respect to the sign of a (due to the plot scale, this is more evident in the right panel, but it is true in general)and (iv) the existence of a second real solution for a ≤ a >
0, which is however unphysical (i.e.,below the positivity border - see below). The above plots allow us to represent solutions in implicit form. Consideringa hyperbolic tangent velocity shear profile as in Eq. (18) and H = 1, the function a ( x ) reads a ( x ) = s u B L u cosh − ( x/L u ) , (24)where we took x = 0 for simplicity. We consider the case shown in the central panel of Fig. 2 for a ( f ). This plotof a versus f , zoomed in the region around a = 0 and f = 1, is reproduced in the right panel of Fig. 3. In the leftpanel of Fig. 3, we show the plot of a ( x ) in Eq. (24), for both signs of s . The aim is to visualize how f ( x ) shouldlook by drawing a ( x ) and a ( f ) next to each other, with the values of a on the y -axis of both plots (and with thesame scale), so one can go back and forth from one plot to the other in three steps: go along the x -axis of left panelof Fig. 3 ( x coordinate), (ii) look at the value that a ( x ) takes in the left panel and then move to the same level ofright panel (in both panels, the y -axis has the same scale and the values of a are represented on it) and (iii) afterreaching the curve a ( f ) on the right panel in the point that corresponds to the value of a found at point (ii), godown to the x -axis of the right panel in order to find the value of f that corresponds to the value of x at point (i).Imagine going along the whole x -axis, from −∞ to + ∞ . For sufficiently large | x | , we have a = 0, so the solution isasymptotically f = 1 (e.g. for | x | & x = 0from negative values a starts to deviate from zero (left panel) and thus also f ( x ) starts to deviate from unity (rightpanel), becoming bigger or smaller if a becomes positive or negative, respectively. In passing through x = 0 (leftpanel), we pass through the global maximum (minimum) of the positively (negatively) valued a , corresponding tothe maximum deviation of f ( x ) from unity in the right panel (i.e., in the point where the curve a ( f ) intercepts thehorizontal red dashed line, above f = 1 or below it accordingly with the sign of a ). Then, leaving x = 0 behindand proceeding to increasing positive x -values in the left panel of Fig. 3, a starts to decrease (increase) and so doesFigure 3: Left: plot of a ( x ) in Eq. (24) for both sign of s (see insert ). Right: zoom around f = 1 and a = 0 ofthe center plot in Fig. 2. f ( x ) in the right panel, until it comes back to unity for sufficiently high x -values (note that for x > a ( f ) all the way back compared to how it had been covered for x <
0, the point x = 0 being the turningpoint).Explicit numerical solutions f ( x ) of Eq. (21) for a ( x ) given in Eq. (24), are plotted in Fig. 4 for the three casesin Fig. 2. The corresponding profiles Π xx ( x ) ( s = +1: bottom blue solid line, s = −
1: top blue dashed line) andΠ yy ( x ) ( s = +1: top red solid line, s = −
1: bottom red dashed line) are also given (from Eqs. (33) and (34) - seebelow). The asymmetry with respect to the sign of Ω u · B is more evident on the right panel due to the choice of theparameters, but it is present in all cases.Figure 4: Top row: plot of the explicit numerical solution f ( x ) of Eq. (21). Bottom row: corresponding Π xx and Π yy profiles (see in the text). The three cases above correspond to the three cases in Fig. 2: β i , ⊥ = β e , ⊥ = 0 . u = 2 / β i , ⊥ = β e , ⊥ = 1 and u = 2 / β i , ⊥ = β e , ⊥ = 2 and u = 1 . a ≤ e β i , ⊥ as a small parameter (here, weconsider the case F = 1, but the bound 0 ≤ F ≤ e β i , ⊥ = 0, Eq. (21) is exactly solvable:( f ( x ) − (cid:16)p f ( x ) + a ( x ) (cid:17) = 0 , (25)and admits two real roots (cid:26) f ( x ) = 1 ∀ x ˜ f ( x ) = a ( x ) ∀ x ∈ { x | a ( x ) ≤ } (26)where f = 1 means gyrotropy, while ˜ f represents the e β i , ⊥ → a ≤ f is below the positivity condition a i ≥ − / f → ˜ f we obtain a i → − e β i , ⊥ F ≪ f ( x ) ≃ e β i , ⊥ F ( x ) a ( x )1 + a ( x ) , (27)which corresponds to the first-order Taylor expansion for small C a / (1 + a ) of the solution in Eq. (30) (seebelow). This means that, at least in the limit of small- e β i , ⊥ , the two solutions are close to each other. Note thatthe small- e β i , ⊥ limit does not necessarily mean small- β i , ⊥ , but it can be reached also in the large perpendiculartemperature ratio, τ ⊥ = T e , ⊥ /T i , ⊥ ≫
1. Finally, physical solutions of Eq. (21) only exist within restricted domainsof ( e β i , ⊥ , a ). In particular, this turns out to be the case only when a is negative, i.e. for Ω u · B <
0. An exampleof this asymmetric behavior is given in Fig. 5, where we show that solutions for a < e β i , ⊥ . For a >
0, i.e. for Ω u · B >
0, a solution is instead alwayspresent, regardless of the value of e β i , ⊥ .Figure 5: Plot of the implicit solution a ( f ) for the case β i , ⊥ = β e , ⊥ = 5 (left) and for the case β i , ⊥ = 10, β e , ⊥ = 1 (right), corresponding to e β i , ⊥ = 5 /
11 and 5 /
6, respectively. The other parameters are F = H = 1, u = 2 . | B | = 1, L u = 3 for both cases. B. Approximate analytical solution of the equilibrium condition
In the limit of small corrections to the MHD equilibrium, we can give analytical solutions for f ( x ) with e γ = 0. Inthis limit, the ion cyclotron frequency Ω i in the a i function can be computed with the MHD profile of the magneticfield, | B ( x ) | ≃ B p H ( x ), i.e. a i ( x ) ≃ a ( x ) = s B p H ( x ) du i ,y dx , (28)which is an approximation that one should check a posteriori (see Appendix A for a simple case in which this case isexact), and leads to the following equilibrium condition: (cid:20) − e β i , ⊥ F ( x ) a ( x )1 + a ( x ) + e β e , ⊥ ( F ( x )) e γ (cid:16) f e γ ( x ) − (cid:17)(cid:21) f ( x ) = 1 . (29)Treating e γ as a small parameter, we solve the above equilibrium problem iteratively. The solution of Eq. (29) for e γ = 0 is straightforward, i.e. f ( x ) = (cid:20) − C ( x ) a ( x )1 + a ( x ) (cid:21) − , (30)where C ( x ) = e β i , ⊥ F ( x ). Noting that Eq. (29) is equivalent to the equilibrium condition Eq. (23) in Ref. [23], withthe substitution e u ′ ( x ) → a ( x )1 + a ( x ) , we can derive the iterative solution (cf. Eqs.(24)-(26) in Ref. [23]), i.e. f ( x ) = (cid:20) − C ( x ) a ( x )1 + a ( x ) (cid:21) − , (31)with C ( x ) = h e γ e β e , ⊥ ( F ( x )) e γ i − C ( x ) , (32)In the above, we have assumed that | e γ e β e , ⊥ ( F ( x )) e γ | < ∀ x for the convergence of the resulting series, which canbe shown to be always the case (see Ref. [23]). Moreover, the solution passes through a Taylor expansion in whichwe consistently assume that the correction to the MHD profile is small, i.e. |C a / (1 + a ) | ≪
1. In this regard, therelation | a / (1 + a ) | ≤ ∀ a ≥ − / e β i , ⊥ < p i , ⊥ such that F ( x ) < ∀ x . Thus, the condition |C a / (1 + a ) | ≪ a inside the square brackets.
1. Explicit equilibrium profiles
We give also the explicit equilibrium profiles of the physical quantity of interest, for the sake of clarity:Π i ,xx = p i , ⊥ (cid:18) − a i ( x )1 + a i ( x ) (cid:19) F ( x ) f ( x ) , (33)Π i ,yy = p i , ⊥ (cid:18) a i ( x )1 + a i ( x ) (cid:19) F ( x ) f ( x ) , (34)Π i ,zz ≡ p i , k = p i , k ( F ( x ) f ( x )) γ i , k /γ i , ⊥ , (35) n ( x ) = n ( F ( x ) f ( x )) /γ i , ⊥ , (36)Π e ,xx = Π e ,yy ≡ p e , ⊥ = p e , ⊥ ( F ( x ) f ( x )) e γ , (37)0Π e ,zz ≡ p e , k = p e , k ( F ( x ) f ( x )) γ e , k /γ i , ⊥ , (38) B z ( x ) = B p H ( x ) f ( x ) . (39)If the parallel and perpendicular temperatures are of interest [32], instead of the corresponding pressures, fromEqs. (33)-(36) we have T i , ⊥ ≡ n Π i ,yy + Π i ,yy T i , ⊥ ( F ( x ) f ( x )) γ i , ⊥− γ i , ⊥ , (40) T i , k ≡ p i , k n = T i , k ( F ( x ) f ( x )) γ i , k− γ i , ⊥ , (41) T e , ⊥ ≡ p e , ⊥ n = T i , ⊥ ( F ( x ) f ( x )) γ e , ⊥− γ i , ⊥ , (42) T e , k ≡ p e , k n = T i , k ( F ( x ) f ( x )) γ e , k− γ i , ⊥ . (43)We define the MHD profiles with the superscript (0), e.g. T (0)i , ⊥ ≡ T i , ⊥ F ( x ) ( γ i , ⊥ − /γ i , ⊥ , and the overlined quantitiesas the ratio between the actual and the MHD profiles, e.g. ¯ T i , ⊥ ≡ T i , ⊥ /T (0)i , ⊥ . Note that taking the perpendicular andparallel polytropic indices to be γ ⊥ = 2 and γ k = 1 for both species, one obtains¯ T k = 1 , (44)¯ T ⊥ = ¯ n = ¯ B , (45)which mean ¯ p k ¯ B / ¯ n = const . and ¯ p ⊥ / ¯ n ¯ B = const . . Thus, if the initial MHD profiles are such that p (0) k ( B (0) ) / ( n (0) ) = const . and p (0) ⊥ /n (0) B (0) = const . (as, e.g., for the case F = H = 1), then also p k B /n = const . and p ⊥ /nB = const . , as expected from the double adiabatic laws [22].Finally, in order to visualize the asymmetry due to the sign of Ω u · B , we plot the equilibrium profiles for a given case.In Fig. 6 we show the velocity shear u i ,y ( x ) and the function C ( x ) a ( x ) / [1 + a ( x )] (left panel) and the correspondingequilibrium profiles for Π i ,xx and Π i ,yy (right panel). The parameters used for the profiles in Fig. 6 are u = 2 / L u = 3, B = ± s = ± β i , ⊥ = β e , ⊥ = 1 and e γ = 0. For the sake of clarity we have chosen the simplestMHD case of F = G = H = 1. By an inspection of the plot, some considerations emerge. First, while the condition |C ( x ) a ( x ) / [1 + a ( x )] | ≪ ∼ − s = ± a i , which gives us an idea of how far from gyrotropy ( a = 0) the system is, only reaches a ≃ .
11 ( a ≃ − .
11) for the configuration with s = +1 ( s = − s = ±
1, may then lead to very different dynamical evolution of the system and thus of the instabilityunder study already in the linear phase [1, 2, 10]. In Fig. 7 we show the same quantities as in Fig. 6 for a differentset of parameters: F = G = H = 1, u = 3 / L u = 3, B = ± s = ± β i , ⊥ = β e , ⊥ = 2 and e γ = 0, whichcorrespond to a ≃ ± . 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 afluid plasma framework. The heat fluxes are neglected. Solutions of the stationary pressure tensor equation are givenfor a simple, but commonly adopted configuration, and the properties of such equilibrium solutions are discussed. Inparticular, we have shown that, in addition to the well known parallel-perpendicular anisotropy ( p k = p ⊥ ), the systemis also anisotropic in the plane perpendicular to the magnetic field, i.e. Π xx = Π yy = Π zz . The magnitude of theperpendicular 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 field1Figure 6: Left: velocity profile u i ,y ( x ) (blue solid line) and the function C ( x ) a i ( x ) / [1 + a i ( x )] (red dashed line).Right: plot of the approximated equilibrium profiles Π i ,xx ( x ) ( s = +1: bottom blue solid line, s = −
1: top bluedashed line) and Π i ,yy ( x ) ( s = +1: top red solid line, s = −
1: bottom red dashed line). Here, the parameters are: F = G = H = 1, u = 2 / L u = 3, B = ± s = ± β i , ⊥ = β e , ⊥ = 1 and e γ = 0.Figure 7: Left: velocity profile u i ,y ( x ) (blue solid line) and the function C ( x ) a i ( x ) / [1 + a i ( x )] (red dashed line).Right: plot of the approximated equilibrium profiles Π i ,xx ( x ) ( s = +1: bottom blue solid line, s = −
1: top bluedashed line) and Π i ,yy ( x ) ( s = +1: top red solid line, s = −
1: bottom red dashed line). Here, the parameters are: F = G = H = 1, u = 3 / L u = 3, B = ± s = ± β i , ⊥ = β e , ⊥ = 2 and e γ = 0.and of the fluid vorticity, i.e. with respect to the sign of Ω u · B . These properties of the system are present even atthe level of the equilibrium 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 solutionsare provided for some representative cases. The profiles derived in the present paper are shown to be different withrespect to the usual MHD or even CGL equilibria. In particular, they depend on the velocity shear and are asymmetricwith respect to the sign of Ω u · B . These features, arising already at the level of the equilibrium configuration, turnout to be relevant when fluid models that retain the pressure tensor equation and/or kinetic models are adopted, asfor the study of the KHI and the MRI.Finally, despite the relative simplicity of the system configuration adopted, it seems plausible that our results canbe used for the interpretation of satellite data where non-gyrotropic distribution functions are observed. This couldbe the case, for instance, of solar wind data, since, as pointed out by recent studies, one expects that the turbulencespontaneously generates local velocity shear flows.2 Acknowledgments
Appendix A: 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 and h , one can compute very different equilibrium profiles.In the following, we give some examples.
1. Preserving the total perpendicular plasma beta
Requiring that the total perpendicular plasma beta β ⊥ ( x ) remains unchanged in passing from MHD to full pressuretensor equilibria, the relation in Eq. (15) is substituted by h ( x ) = ξ e γ ( F , f ; x ) f ( x ) , (A1)where the function ξ e γ ( F , f ; x ) ≡ β i , ⊥ + β e , ⊥ ( F ( x ) f ( x )) e γ β i , ⊥ + β e , ⊥ ( F ( x )) e γ , (A2)has been defined, such that it reduces to ξ ( F , f ; x ) = 1 for e γ = 0. For e γ = 0 we indeed recover Eq. (15) and thus theequilibrium condition in Eq. (16) and its solution. For e γ = 0, the equilibrium condition reads ( ξ e γ ( F , f ; x ) + e β i , ⊥ (cid:20) − ξ e γ ( F , f ; x ) (1 + a ( x ))1 + a ( x ) (cid:21) F ( x ) + e β e , ⊥ ( F ( x )) e γ (cid:16) f e γ ( x ) − ξ e γ ( F , f ; x ) (cid:17) ) f ( x ) = 1 , (A3)which, Taylor expanding f e γ and ξ e γ as in Sec.III B and after some algebra, admits the solution f ( x ) = (cid:20) − C ( x ) a i ( x )1 + a i ( x ) (cid:21) − , (A4a) C ( x ) = h e γ e β e , ⊥ e F e γ ( x ) i − C ( x ) . (A4b) e F e γ ( x ) ≡ ( F ( x )) e γ e β i , ⊥ + e β e , ⊥ ( F ( x )) e γ . (A4c)
2. Preserving the magnetic field configuration
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