Linear stability of magnetic vortex chains in a plasma in the presence of equilibrium electron temperature anisotropy
LLinear stability of magnetic vortex chains in a plasmain the presence of equilibrium electron temperatureanisotropy
C. Granier , E. Tassi Université Côte d’Azur, CNRS, Observatoire de la Côte d’Azur, Laboratoire J. L.Lagrange, Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France a r X i v : . [ phy s i c s . p l a s m - ph ] A p r inear stability of magnetic vortex chains in a plasma Abstract.
The linear stability of chains of magnetic vortices in a plasma isinvestigated analytically in two dimensions by means of a reduced fluid model assuminga strong guide field and accounting for equilibrium electron temperature anisotropy.The chain of magnetic vortices is modelled by means of the classical "cat’s eyes"solutions and the linear stability is studied by analysing the second variation of aconserved functional, according to the Energy-Casimir method. The stability analysisis based on a fluid model obtained from a gyrofluid model by means of a simpleHamiltonian reduction and is carried out on the domain bounded by the separatrices ofthe vortices. Two cases are considered, corresponding to a ratio between perpendicularequilibrium ion and electron temperature much greater or much less than unity,respectively. In the former case, equilibrium flows depend on an arbitrary function.Stability is attained if the equilibrium electron temperature anisotropy is bounded fromabove and from below, with the lower bound corresponding to the condition preventingthe firehose instability. A further condition sets an upper limit to the amplitude of thevortices, for a given choice of the equilibrium flow. For cold ions, two sub-cases haveto be considered. In the first one, equilibria correspond to those for which the velocityfield is proportional to the local Alfvén velocity. Stability conditions imply: an upperlimit on the amplitude of the flow, which automatically implies firehose stability, anupper bound on the electron temperature anisotropy and again an upper bound on thesize of the vortices. The second sub-case refers to equilibrium electrostatic potentialswhich are not constant on magnetic flux surfaces and the resulting stability conditionscorrespond to those of the first sub-case in the absence of flow.
1. Introduction
The identification of coherent structures and the investigation of their stability is aclassical subject in plasma physics. Among the various coherent structures that canform in plasmas, chains of magnetic vortices (also referred to as magnetic islands) areof considerable relevance for both laboratory and space plasmas, and the study of theirstability began already a few decades ago [1, 2, 3, 4, 5]. Such stability analysis was oftencarried out in the context of a magnetohydrodynamic (MHD) description of a plasmaand the modelling of the magnetic vortex chain took advantage from the existence ofa well known solution of the Liouville equation (explicitly given later in Eq. (22))which can be applied when investigating plasma equilibria with a symmetry [6]. Thissolution was adopted much earlier in fluid dynamics, where it is usually referred to asKelvin-Stuart "cat’s eyes" solution [7, 8]. In plasma physics, such equilibrium solutionproved to be a standard starting point for the investigation of problems related toisland coalescence (see, for instance Refs. [9, 10, 11, 12] and references therein). To the inear stability of magnetic vortex chains in a plasma inear stability of magnetic vortex chains in a plasma inear stability of magnetic vortex chains in a plasma
2. The general gyrofluid model and its reduction
The starting point of our analysis consists of a nonlinear two-field gyrofluid model forcollisionless plasmas, which assumes the presence of a strong component of the magneticfield (strong guide field assumption) along one direction. The model consists of the inear stability of magnetic vortex chains in a plasma ∂N e ∂t + [ φ, N e ] − [ B (cid:107) , N e ] − [ A (cid:107) , U e ] + ∂U e ∂z = 0 , (1) ∂∂t ( A (cid:107) − δ U e ) + [ φ − B (cid:107) , A (cid:107) − δ U e ] + 1Θ e [ A (cid:107) , N e ] + ∂∂z (cid:18) φ − B (cid:107) − N e Θ e (cid:19) = 0 , (2)complemented by the static relations N e + (1 − Γ i + Γ i ) B (cid:107) + (1 − Γ i − τ ⊥ i δ ∆ ⊥ ) φτ ⊥ i = 0 , (3) U e = b (cid:63) ∆ ⊥ A (cid:107) , (4) B (cid:107) = − β ⊥ e N e − (1 − Γ i + Γ i ) φ + (1 + 2 τ ⊥ i (Γ i − Γ i )) B (cid:107) ) , (5)which permit to express φ , A (cid:107) and B (cid:107) in terms of the dynamical variables N e and A (cid:107) − δ U e .Equation (1) is the continuity equation for electron gyrocenters, whereas Eq. (2)is the equation for the evolution of the momentum of electron gyrocenters in thedirection parallel to the guide field (which will be referred to as parallel direction,in the following, as opposed to "perpendicular", which, as customary, refers to theplane perpendicular to the guide field). Alternatively, Eq. (2) can be seen as ageneralized Ohm’s law in the parallel direction accounting for electron inertia andequilibrium electron temperature anisotropy. Eqs. (3), (4) and (5), on the other hand,correspond to the quasi-neutrality relation and to the components of Ampère’s lawparallel and perpendicular, respectively, to the direction of the guide field. The modelis formulated on a domain T n which, adopting a Cartesian reference frame xyz , is givenby T n = { ( x, y, z ) ∈ R | ≤ x ≤ πn, − L y ≤ y ≤ L y , − L z ≤ z ≤ L z } , with n anon-negative integer and L y and L z two positive constants. In Eqs. (1)-(5) the fields N e , U e , φ , A (cid:107) and B (cid:107) are all functions of the independent variables x, y, z, t , with t indicating time, and are all assumed to be periodic over the domain T n . We indicatedwith N e the fluctuations of the electron gyrocenter density, with U e , the fluctuations ofthe electron gyrocenter parallel velocity and with φ the fluctuations of the electrostatic inear stability of magnetic vortex chains in a plasma A (cid:107) and B (cid:107) are related to the magnetic field B by B ( x, y, z, t ) = ˆ z + B (cid:107) ( x, y, z, t )ˆ z + ∇ A (cid:107) ( x, y, z, t ) × ˆ z. (6)In Eq. (6), the first term on the right-hand side accounts for the (dimensionless) strongguide field, directed along the unit vector ˆ z , whereas B (cid:107) indicates the perturbation ofthe magnetic field along z and A (cid:107) is the fluctuation of the z component of the vectorpotential (also referred to as magnetic flux function). The strong guide field assumptionimplies B (cid:107) (cid:28) and |∇ A (cid:107) | (cid:28) in a sense that is made more precise in Appendix A.Also, evidently, the expression for B in Eq. (6) is not divergence-free. Indeed, it onlyrepresents the expression of the total magnetic field at the first order in the fluctuations.The higher-order contributions, which guarantee ∇ · B = 0 , are negligible at the orderretained in the model.The dimensionless variables adopted in Eqs. (1)-(5) are defined by x = ˜ xρ s ⊥ , y = ˜ yρ s ⊥ , z = ˜ zρ s ⊥ , t = ω ci ˜ t,N e = (cid:101) N e n , U e = (cid:101) U e c s ⊥ ,φ = e (cid:101) φT ⊥ e , B (cid:107) = (cid:101) B (cid:107) B , A (cid:107) = (cid:101) A (cid:107) B ρ s ⊥ , (7)where the tilde denotes the dimensional quantities. In Eq. (7) B is the amplitude of theguide field, n is the homogeneous equilibrium particle density (equal for both electronsand ions), T ⊥ e is the equilibrium electron temperature in the plane perpendicular to theguide field, e is the proton charge. Denoting with m i the mass of the ions present in theplasma and with c the speed of light, we also made use of the quantities ω ci = eB / ( m i c ) indicating the ion cyclotron frequency, c s ⊥ = (cid:112) T ⊥ e /m i indicating the sound speedbased on the perpendicular temperature and ρ s ⊥ , which is the sonic Larmor radius, alsobased on the perpendicular temperature. Four independent constant parameters arepresent in the system, and are given by δ = (cid:114) m e m i , β ⊥ e = 8 π n T ⊥ e B , τ ⊥ i = T ⊥ i T ⊥ e , Θ e = T ⊥ e T (cid:107) e , (8) inear stability of magnetic vortex chains in a plasma b (cid:63) defined by b (cid:63) = 2 β ⊥ e + 1 − e , (9)to indicate the modification due to electron temperature anisotropy in the parallelAmpère’s law (4). Note that b (cid:63) = 2 /β ⊥ e in the isotropic case. We indicated with ∆ ⊥ the Laplacian operator in the perpendicular plane, so that ∆ ⊥ f = ∂ xx f + ∂ yy f fora function f . The operators Γ i and Γ i represent the standard operators associatedwith ion gyroaverage. We can define them in the following way. Let us considera function f = f ( x, y, z ) , periodic over T n and indicate with T n the lattice T n = { ( l/n, πm/L y , πp/L z ) : ( l, m, p ) ∈ Z } . We write the Fourier representation of f as f ( x, y, z ) = (cid:80) k ∈ T n f k exp( i k · x ) , where x and k are vectors of components ( x, y, z ) and ( k x , k y , k z ) , respectively, with k x = l/n , k y = mπ/L y , k z = pπ/L z , for ( l, m, p ) ∈ Z .It is also convenient to introduce the quantity b i = τ ⊥ i k ⊥ , where k ⊥ = (cid:112) k x + k y inadimensional variables is the perpendicular wave number (in dimensional variables onewould have b i = ˜ k ⊥ ρ th ⊥ i where ˜ k ⊥ is the dimensional perpendicular wave number and ρ th ⊥ i = (cid:112) T ⊥ i /m i /ω ci is the perpendicular thermal ion gyroradius). The action of theoperators Γ i and Γ i on the function f is defined by Γ i f ( x, y, z ) = (cid:88) k ∈ T n I ( b i )e − b i f k e i k · x , (10) Γ i f ( x, y, z ) = (cid:88) k ∈ T n I ( b i )e − b i f k e i k · x , (11)with I and I indicating the modified Bessel functions of the first kind, of order and , respectively. The canonical bracket [ , ] , on the other hand, is defined by [ f, g ] = ∂ x f ∂ y g − ∂ y f ∂ x g , for two functions f and g .In the light of the above definition, we can recognize in the first three terms of thecontinuity equation (1), the material derivative of the electron gyrocenter density, which inear stability of magnetic vortex chains in a plasma U ⊥ e = ˆ z ×∇ ( φ − B (cid:107) ) . The latteroriginates from the electron gyrocenter velocity in the perpendicular plane, induced byelectromagnetic perturbations. The last two terms, on the other hand, express thegradient of the parallel electron gyrocenter velocity along the magnetic field. Equation(2) expresses, with its first two terms, the material derivative of the fluid version of theparallel canonical electron momentum δ U e − A (cid:107) , advected by U ⊥ e . The third and lastterm of the equation express the force exerted by the parallel gradient of the electronparallel pressure and are affected by temperature anisotropy, whereas an additional forcecomes from the variation of φ − B (cid:107) along the guide field. Alternatively, one could thinkat the terms [ A (cid:107) , N e / Θ e + B (cid:107) ] − ∂ z ( N e / Θ e + B (cid:107) ) as coming from the projection of thedivergence of the anisotropic electron pressure tensor along the magnetic field. Theterms ∂ t A (cid:107) + [ φ, A (cid:107) ] + ∂ z φ , on the other hand, come from the projection of the electricfield along the magnetic field, whereas the remaining terms, i.e. those containing theparameter δ , are those associated with the projection of the electron inertia terms.In the quasi-neutrality relation (3), the first two terms and the last term indicate theelectron density fluctuations (recall that, in the limit of small electron finite Larmorradius (FLR) corrections, which is the case here, the relation n e = N e + B (cid:107) − δ ∆ ⊥ φ holds, where n e indicates the normalized electron density fluctuations, which differ fromthe electron gyrocenter density fluctuations N e [33]). The remaining terms accountfor the contributions due to electromagnetic perturbations, and depend on ion finiteLarmor radius effects, that arise when expressing the ion density fluctuations in termsof ion gyrocenter variables. Likewise, analogous contributions appear in Eq. (5), uponreplacing N e with n e − B (cid:107) (in Eq. (5), the above mentioned electron FLR contribution δ ∆ ⊥ φ , appearing in the relation between n e and N e , turn out to be negligible accordingto both scaling (A.10) and (A.11) ). Parallel Ampère’s law (4) expresses the fact thatthe parallel current density is proportional to the parallel electron gyrocenter velocity,the contribution of the gyrocenter ion velocity being negligible in the present model.This relation is also affected by the temperature anisotropy. inear stability of magnetic vortex chains in a plasma Θ e = 1 , for cold ions ( τ ⊥ i (cid:28) ), and with β ⊥ e small enough to neglectparallel perturbations B (cid:107) , the system reduces to the two-field model considered in anumber of works on collisionless magnetic reconnection such as those of Refs. [34] and[35]. In turn, such limit reduces to low- β reduced MHD [16, 15] when electron inertiaand the parallel pressure term in Eq. (2) are neglected. The system (1)-(5) can alsobe seen as an extension of the model for inertial kinetic Alfvén turbulence described inRef. [36], accounting also for ion finite Larmor radius effects, parallel electron pressureand equilibrium electron temperature anisotropy.The model (1)-(5) can be derived from the gyrokinetic model described in Ref. [37].The derivation procedure can be found in Ref. [13] (see, in particular Sec. 6 of the citedReference) and is summarized in Appendix A. As remarked in Ref. [13], the model is obtained by taking moments of gyrokineticequations and imposing a closure (A.9) that guarantees the existence of a Hamiltonianstructure. Such structure is of noncanonical type (see, e.g. Ref. [20]). The Hamiltonianfunctional is given by H ( N e , A e ) = 12 (cid:90) T n d x (cid:18) N e Θ e + δ U e + b (cid:63) |∇ ⊥ A (cid:107) | − N e ( φ − B (cid:107) ) (cid:19) , (12)with A e = A (cid:107) − δ U e . By means of the relations (3)-(5), it can be shown that theoperators permitting to express the functions A (cid:107) , B (cid:107) and φ in terms of the dynamicalvariables N e and A e are symmetric with respect to the inner product < f | g > = (cid:82) T n d x f g . This allows to show the conservation of H .We introduce the notation F f = δF/δf to indicate the functional derivative of afunctional F with respect to a function f . With this notation, the noncanonical Poisson inear stability of magnetic vortex chains in a plasma { F, G } = (cid:90) T n d x (cid:32) N e (cid:18) [ F N e , G N e ] + δ Θ e [ F A e , G A e ] (cid:19) + A e ([ F A e , G N e ] + [ F N e , G A e ]) + F N e ∂G A e ∂z + F A e ∂G N e ∂z (cid:33) . (13)The evolution equations (1) and (2) can then be written in the Hamiltonian form ∂A e ∂t = { A e , H } , ∂N e ∂t = { N e , H } , (14)with H and { , } given by Eqs. (12) and (13), respectively. The analysis of chains of magnetic vortices based on Kelvin-Stuart "cat’s eyes" solutionsleads us to consider a 2D reduction of the system (1)-(5). In the 2D limit, the applicationof the Energy-Casimir method becomes also particularly fruitful, due to the abundanceof Casimir invariants. Furthermore, chains of magnetic vortices observed, for instancein the solar wind, appear to have an essentially 2D structure [19]. Also, we introducea further simplification by neglecting corrections due to electron inertia (i.e. by takingthe limit δ → ). This amounts to neglecting effects occurring on scales comparable tothe electron skin depth. Magnetic chains on scales much larger than the electron skindepth are indeed those observed in the solar wind [19, 17].We perform this reduction of the two-field gyrofluid model acting on its Hamiltonianstructure, rather than directly on its equations of motion. This permits to preserve aHamiltonian character in the reduced model. We assume then that in the Hamiltonian(12) and in the Poisson bracket (13) the field variables N e and A e are invariant withrespect to the z coordinate. Also, we neglect in the Hamiltonian (12) the contributionsproportional to electron inertia, which amounts to neglecting the second term on theright-hand side of Eq. (12) (parallel electron kinetic energy) as well as the electronFLR correction corresponding to the last term of Eq. (3), which intervenes when oneexpresses φ and B (cid:107) in terms of N e . Analogously, we act on the Poisson bracket (13) by inear stability of magnetic vortex chains in a plasma δ U e due to electron momentum, in the expression for A e and suppress the term proportional to δ in the first line of Eq. (13). Note that we areallowed to carry out the latter operation because the Poisson bracket (13) maintains allthe properties of a Poisson bracket (and in particular the Jacobi identity) for all valuesof δ , and in particular also for δ = 0 .As a result, we obtain the following reduced Hamiltonian (where we reabsorbed inthe definition of H a factor L z due to the integration with respect to z between − L z and L z ) H ( N e , A (cid:107) ) = 12 (cid:90) D n d x (cid:18) N e Θ e + b (cid:63) |∇ ⊥ A (cid:107) | − N e ( φ − B (cid:107) ) (cid:19) , (15)and the reduced Poisson bracket { F, G } = (cid:90) D n d x ( N e [ F N e , G N e ] + A (cid:107) ([ F A (cid:107) , G N e ] + [ F N e , G A (cid:107) ])) . (16)We also introduced the 2D domain D n = { ( x, y ) ∈ R | ≤ x ≤ πn, − L y ≤ y ≤ L y } .Accordingly, the Fourier components for the fields in this model get defined over the 2Dlattice D n = { ( l/n, πm/L y ) : ( l, m ) ∈ Z } .The equations of motion resulting from the Hamiltonian (15) and the Poissonbracket (16) are ∂N e ∂t + [ φ − B (cid:107) , N e ] − b (cid:63) [ A (cid:107) , ∆ ⊥ A (cid:107) ] = 0 , (17) ∂A (cid:107) ∂t + [ φ − B (cid:107) , A (cid:107) ] + 1Θ e [ A (cid:107) , N e ] = 0 , (18)where we also made use of Eq. (4). The expressions for B (cid:107) and φ in terms of N e followfrom Eqs. (3) and (5) which, in the limit δ → , become N e + (1 − Γ i + Γ i ) B (cid:107) + (1 − Γ i ) φτ ⊥ i = 0 , (19) B (cid:107) = − β ⊥ e N e − (1 − Γ i + Γ i ) φ + (1 + 2 τ ⊥ i (Γ i − Γ i )) B (cid:107) ) . (20)We note that the Poisson bracket (16) is the same as that of 2D reduced MHD [38].Therefore, Eqs. (17)-(18), in addition to the Hamiltonian (15), possess infinite conserved inear stability of magnetic vortex chains in a plasma π π π π π π - - x y Figure 1: The figure shows a surface plot and some contour lines of the "cat’s eyes" function A eq . The domain D n , enclosed by the separatrices, indicated with black dotted curves, andthe domain R n , corresponding to the rectangle enclosed by black solid lines, are also shown.The figure refers to the case n = 3 and a = 1 . . functionals, given by C = (cid:90) D n d x N e F ( A (cid:107) ) , C = (cid:90) D n d x G ( A (cid:107) ) , (21)where F and G are arbitrary functions. The functionals C and C are Casimir invariantsof the Poisson bracket (16) and, as such, they satisfy { C , E } = { C , E } = 0 for everyfunctional E . As discussed in Ref. [38], the Casimir C includes, among others, theconservation of the integral of N e over an area bounded by contour lines of A (cid:107) . TheCasimir C , on the other hand, expresses, for G ( A (cid:107) ) = A (cid:107) , conservation of magnetichelicity at leading order. This arises as a consequence of removing electron inertia,which violates the frozen-in condition. We intend to analyse the linear stability of equilibria such that the equilibrium solutionfor the magnetic flux function A (cid:107) corresponds to the "cat’s eyes" solution A eq ( x, y ) = − log( a cosh y + √ a − x ) , (22)where a > .As shown in Fig. 1, the contour lines of the function A eq , in general describe chainsof magnetic vortices in the plane xy . As a → + , the equilibrium configuration tendstoward a uni-directional sheared magnetic field, with no magnetic vortices. inear stability of magnetic vortex chains in a plasma A eq is known to be a solution of the Liouville’s equation ∆ ⊥ A (cid:107) = − e A (cid:107) . (23)Inspired by the procedure followed in Ref. [32], we carry out the stability analysis ofthe magnetic vortex chains on the domain D n = (cid:110) ( x, y ) ∈ R | ≤ x ≤ πn, | y | ≤ cosh − (cid:16) √ a − a (cid:0) − cos x (cid:1)(cid:17)(cid:111) , (24) with a > .An example of such domain is depicted in Fig. 1. One can see that the domaincorresponds to the domain bounded by the separatrices of the vortex chain and thenumber n indicates the number of vortices in the domain. This choice for the domainallows, with appropriate boundary conditions, for the application of the Poincaréinequality, which is crucial for carrying out some stability estimates that will be requiredlater in the analysis. Of course, the choice of such domain rules out the effect ofperturbations coming from outside the vortex chain. This can indeed be seen as alimitation of the present analysis. However, numerical simulations [39, 40, 41] show that,for instance, secondary instabilities due to colliding jets, can originate inside a magneticisland, and the subsequent turbulent evolution of the instability remains confined withinthe island. Therefore, in addition to the above mentioned technical argument related tothe Poincaré inequality, restricting the analysis to the region enclosed by the separatricesdoes not appear to rule out all physically relevant processes. On the other hand, aspointed out in Ref. [32], this prevents from a direct comparison with classical resultson stability of magnetic island chains, such as those of Refs. [1, 2, 5].The domain D n differs from the domain D n introduced in Sec. 2.2 and differentboundary conditions will have to be adopted. In particular, the definitions (10) and(11) of ion gyroaverage operators Γ i and Γ i , are valid for a periodic domain. Variantsof gyroaverage operators, that permit to account for different boundary conditions (e.g.Dirichlet), have been discussed, for instance in Refs. [42, 43]. These variants are oftenbased on Taylor expansions or Padé approximants, and on the identification between the inear stability of magnetic vortex chains in a plasma b i and the operator − τ ⊥ i ∆ ⊥ . We follow the same practice and, in the two casesthat will be treated in the subsequent Sections, we will use (very rough) approximantsof the ion gyroaverage operators in two opposite limits.
3. Hot-ion case : τ ⊥ i (cid:29) We consider the model (17)-(20) in the limit τ ⊥ i (cid:29) , corresponding to anequilibrium perpendicular ion temperature much larger than the corresponding electrontemperature. This limit was adopted for instance in the model of Ref. [36] todescribe turbulence at kinetic scales in the magnetosheath. Because of the relation ρ th ⊥ i = √ τ ⊥ i ρ s ⊥ , considering the limit τ ⊥ i (cid:29) implies that the characteristic lengthof our model, corresponding to ρ s ⊥ , is much smaller than the perpendicular thermalion gyroradius ρ th ⊥ i . Therefore, in this sense, one can refer to this limit, also as toa sub-ion limit. Because I ( b i )e − b i = I ( τ ⊥ i k ⊥ )e − τ ⊥ i k ⊥ → , as τ ⊥ i → + ∞ , for all ( k x , k y ) ∈ D n \ (0 , and I ( b i )e − b i = I ( τ ⊥ i k ⊥ )e − τ ⊥ i k ⊥ → , as τ ⊥ i → + ∞ , for all ( k x , k y ) ∈ D n we simply take, for τ ⊥ i (cid:29) , the following approximated form for theoperators Γ i and Γ i : Γ i f ( x, y ) = 0 , Γ i f ( x, y ) = 0 , (25)for a function f ( x, y ) with ( x, y ) ∈ D n (for the mode ( k x , k y ) = (0 , , an agreementbetween the exact form for Γ i acting on functions over D n and the approximated form,written in Eq. (25), for τ ⊥ i (cid:29) and for functions over D n can be obtained if, in theformer case, one restricts to functions f ( x, y, t ) = (cid:80) ( k x ,k y ) ∈ D n f ( k x ,k y ) ( t ) exp( i ( k x x + k y y )) such that f (0 , = 0 , i.e. functions with zero spatial average).In the limit τ ⊥ i (cid:29) , the model (17)-(20) thus reduces to ∂N e ∂t − b (cid:63) [ A (cid:107) , ∆ ⊥ A (cid:107) ] = 0 , (26) ∂A (cid:107) ∂t − κ [ N e , A (cid:107) ] = 0 , (27)with B (cid:107) = − N e , φ = − β ⊥ e N e . (28) inear stability of magnetic vortex chains in a plasma κ = 2 β ⊥ e + 1Θ e − . (29)For Θ e = 1 , i.e. for isotropic temperature, the model is analogous to the 2D version ofthe reduced electron MHD model discussed in Refs. [44] and [45].Equations (26)-(27) are supplemented with the boundary conditions A (cid:107) | ∂D n = a A , (30) N e | ∂D n = a N , (31)with a A , a N ∈ R and where we indicated with ∂D n the boundary of D n . The boundarycondition (30) expresses the fact that the perpendicular magnetic field B ⊥ = ∇ A (cid:107) × ˆ z istangent to the boundary, i.e. B ⊥ · n = 0 , where n is the outward unit vector normal tothe boundary ∂D n . The condition (31), on the other hand, implies U ⊥ e · n = 0 , meaningthat the incompressible flow U ⊥ e = ˆ z × ∇ ( φ − B (cid:107) ) = (1 − /β ⊥ e )ˆ z × ∇ N e is tangent tothe boundary.The procedure we adopt to investigate the linear stability of magnetic vortex chainsis summarized in Appendix B. Detailed descriptions of the method can be found in Refs.[20] and [21]. The first step consists of finding a functional F given by a combination ofconstants of motion of the system (26)-(27). To this purpose we can use the Hamiltonian(15) and the Casimir invariants (21), with φ and B (cid:107) given by Eq. (28). Indeed, suchfunctionals are also conserved by the system (26)-(27) on the domain D n . This can beshown by direct computation making use of the identities (cid:90) D n d x f ∆ ⊥ g = − (cid:90) D n d x ∇ f · ∇ g + (cid:90) ∂D n f ∂g∂n ds, (32) (cid:90) D n d x f [ g, h ] = (cid:90) D n d x h [ f, g ] − (cid:90) ∂D n hf ∇ g · d l , (33)for functions f, g and h , and of the boundary conditions (30) and (31). In Eq. (32)we indicated with ds the scalar infinitesimal arc element and with ∂g/∂n = ∇ g · n the gradient normal to the boundary. In Eq. (33) we indicated with d l the vectorialinfinitesimal arc element. inear stability of magnetic vortex chains in a plasma We consider then the conserved functional F = H + C + C , explicitly given by F ( N e , A (cid:107) ) = (cid:90) D n d x (cid:18) b (cid:63) |∇ A (cid:107) | κ N e N e F ( A (cid:107) ) + G ( A (cid:107) ) (cid:19) . (34)Adopting, for the variations δA (cid:107) and δN e , the boundary conditions δA (cid:107) | ∂D n = 0 , δN e | ∂D n = 0 , (35)the first variation of F is given by δF ( N e , A (cid:107) ; δN e , δA (cid:107) ) = (36) (cid:90) D n d x (cid:0) ( − b (cid:63) ∆ ⊥ A (cid:107) + F (cid:48) ( A (cid:107) ) N e + G (cid:48) ( A (cid:107) )) δA (cid:107) + ( κN e + F ( A (cid:107) )) δN e (cid:1) , where the prime denotes derivative with respect to the argument of the function.Setting the first variation δF equal to zero for arbitrary perturbations, leads to thesystem ∆ ⊥ A (cid:107) = F (cid:48) ( A (cid:107) ) N e b (cid:63) + G (cid:48) ( A (cid:107) ) b (cid:63) , (37) F ( A (cid:107) ) = − κN e , (38)Solutions of Eqs. (37)-(38) are equilibrium solutions of the system (26)-(27). Eq. (37)can be seen as a Grad-Shafranov equation for the current density − ∆ ⊥ A (cid:107) , whereas Eq.(38) expresses the fact that the electron gyrocenter density fluctuations N e (and, byvirtue of Eq. (28), the electrostatic potential and the parallel magnetic perturbations)are constant on perpendicular magnetic field lines identified by A (cid:107) = constant . For suchequilibria F ( A (cid:107) ) = κ ( β ⊥ e / φ , and in particular F ( A (cid:107) ) = φ for isotropic temperature.Therefore, for F = 0 we obtain an equilibrium with no perpendicular equilibrium flow.For F ( A (cid:107) ) = ± (cid:112) /β ⊥ e A (cid:107) and assuming isotropic temperature, on the other hand,one obtains Alfvénic solutions, in which the equilibrium E × B velocity field, given by ˆ z × ∇ φ , equals, in dimensional units, the local Alfvén velocity field (or its opposite). Inthe more general case with Θ e (cid:54) = 1 , the Alfvén velocity will be modified by an effect due inear stability of magnetic vortex chains in a plasma F is taken as a linear function of A (cid:107) , clearly also theperpendicular equilibrium flow exhibits the "cat’s eyes" pattern.The system is characterized by the two arbitrary functions F and G . Because we areinterested in solutions for A (cid:107) given by the "cat’s eyes" function (22), we constrain Eq.(37) to equal the Liouville equation (23) (we consider here non-propagating solutionsbut a generalization to account for a constant propagation velocity could be carriedout). This occurs if the following condition on the function G is fulfilled: G ( A (cid:107) ) = − b (cid:63) A (cid:107) + F ( A (cid:107) )2 κ + c , (39)with c arbitrary constant.Our analysis will then focus on the class of equilibria given by A (cid:107) = A eq , (40) N e = − F ( A eq ) κ , (41)for κ (cid:54) = 0 , with A eq given by Eq. (22) and arbitrary F . The corresponding expressionsfor φ and B (cid:107) at equilibrium are given by φ = 2 F ( A eq ) / ( β ⊥ e κ ) and B (cid:107) = F ( A eq ) /κ ,respectively. Therefore, we note that, for τ ⊥ i (cid:29) , equilibria obtained from the abovevariational principle and possessing a magnetic vortex chain, admit a whole class offlows (or, equivalently, of electron gyrocenter density or parallel magnetic perturbations)depending on an arbitrary function. The second variation of F , making use of the boundary conditions (35) and rearrangingterms, can be written as δ F ( A (cid:107) , N e ; δA (cid:107) , δN e ) = (cid:90) D n d x (cid:16) b (cid:63) |∇ δA (cid:107) | + ( F (cid:48)(cid:48) ( A (cid:107) ) N e + G (cid:48)(cid:48) ( A (cid:107) ) − F (cid:48) ( A (cid:107) )) | δA (cid:107) | +( κ − | δN e | + ( F (cid:48) ( A (cid:107) ) δA (cid:107) + δN e ) (cid:1) (42)We intend to find conditions for which δ F , evaluated at the class of equilibrium ofinterest, is positive for arbitrary perturbations. If we impose b (cid:63) > and κ > , it is only inear stability of magnetic vortex chains in a plasma | δA (cid:107) | that can provide a negative contribution, and thus indefiniteness,to δ F . Using the relation (39) one finds that, for the class of equilibria of interest, suchcoefficient is given by − b (cid:63) e A eq + (1 /κ − F (cid:48) ( A eq ) . For κ > this coefficient is alwaysnegative, so the second variation has no definite sign. This indefiniteness seems to reflecta feature of "cat’s eyes" equilibria that was already pointed out in Ref. [46] in the caseof the 2D Euler equation for an incompressible flow. This difficulty can be overcome, asindicated in Ref. [46], by making use of a Poincaré inequality. In our specific case, therequired Poincaré inequality reads (cid:90) D n d x |∇ δA (cid:107) | ≥ k min (cid:90) D n d x | δA (cid:107) | , (43)with δA (cid:107) | ∂D n = 0 . In the inequality (43), k min is the minimal eigenvalue of the operator − ∆ ⊥ acting on the functions defined over D n and vanishing on the boundary of D n .The inequality (43) can be derived with a straightforward modification of the procedurefollowed in Ref. [32]. Following this same Reference, we make use of the fact that k min > k R , where k R is the minimal eigenvalue of the operator − ∆ ⊥ on the functionsdefined over R n and vanishing on the boundary of R n . The domain R n ⊃ D n is definedby R n = (cid:110) ( x, y ) ∈ R | ≤ x ≤ nπ, | y | ≤ l = cosh − (cid:16) √ a − a (cid:17)(cid:111) (44) and corresponds to the rectangle of width nπ and height l equal to the magneticisland width. The rectangle R n is depicted in Fig. 1. For perturbations vanishing onthe boundary of R n , one has k R = 14 n + π l . (45)With regard to this point, we remark that the expression for the minimal eigenvalue(45) differs by a factor from the one used in Ref. [32] for the fluid case. The reasonfor this difference is due to the fact that in Ref. [32], in order to obtain the equilibriumequation, the perturbations of the stream function were assumed to vanish on theboundary and to have zero circulation along the boundary. In our case, in order toobtain the desired equilibrium equations for the magnetic field, it is sufficient to impose inear stability of magnetic vortex chains in a plasma A (cid:107) and N e vanish on the boundary.With the help of the above reasoning, we can state that, for b (cid:63) > δ F ( A eq , F ( A eq ); δA (cid:107) , δN e ) ≥ (cid:90) D n d x (cid:16)(cid:16) b (cid:63) k R − b (cid:63) e A eq + (1 /κ − F (cid:48) ( A eq ) (cid:17) | δA (cid:107) | +( κ − | δN e | + ( F (cid:48) ( A eq ) δA (cid:107) + δN e ) (cid:1) . (46)The coefficient of | δA (cid:107) | on the right-hand side of Eq. (46) can be made positive bychoosing appropriate bounds for F (cid:48) ( A eq ) . In particular, noticing that min ( x,y ) ∈ D n ( − b (cid:63) e A eq ( x,y ) ) = − b (cid:63) e A eq ( π, = − b (cid:63) ( a − √ a − , (47)one can write that, for ( x, y ) ∈ D n : b (cid:63) k R − b (cid:63) e A eq ( x,y ) + (cid:18) κ − (cid:19) F (cid:48) ( A eq ( x, y )) ≥ b (cid:63) (cid:18) k R − a − √ a − (cid:19) + (cid:18) κ − (cid:19) F (cid:48) ( A eq ( x, y )) . (48)Making use of the relations (46), (48), (45) as well as of the previously mentionedconditions b (cid:63) > and κ > , we can conclude that the linear stability of the family ofequilibria (40)-(41) is attained if the following three conditions are satisfied: b (cid:63) > , (49) κ > , (50) b (cid:63) (cid:18) n + π l − a − √ a − (cid:19) ≥ max ( x,y ) ∈ D n (cid:18) − κ (cid:19) F (cid:48) ( A eq ( x, y )) . (51)Note that the right-hand side of Eq. (51) is not negative when the condition (50) isfulfilled.In order to get some physical insight from these conditions we resort first to thedefinitions (9) and (29). In terms of the perpendicular electron beta parameter β ⊥ e and on the electron temperature anisotropy parameter Θ e , the conditions (49) and (50)imply Θ e > β ⊥ e β ⊥ e , if < β ⊥ e ≤ , (52) β ⊥ e β ⊥ e < Θ e < β ⊥ e β ⊥ e − if < β ⊥ e < . (53) inear stability of magnetic vortex chains in a plasma β ⊥ e = 4 for the perpendicular electron plasma beta parameter. This boundappears not to be too restrictive for typical solar wind or magnetospheric parameters.We also observe that the condition Θ e > β ⊥ e / (2 + β ⊥ e ) , that emerges in our analysisin both Eq. (52) and (53) (and which corresponds to b (cid:63) > ), is the condition thatsuppresses the firehose instability in the stability analysis of spatially homogeneousequilibria based on linear waves (see, e.g. Ref. [47]). Although our conditions aresufficient but not necessary, we could argue that also magnetic vortex chains could besubject to the same instability. For < β ⊥ e < an upper bound for temperatureanisotropy also appears. This is due to the condition (50). However, unlike the lowerbound, this bound does not appear to be related to instability thresholds familiar fromwave linear theory and in particular to those, such as mirror instability (see, e.g. Ref.[47]), occurring when the temperature anisotropy parameter Θ e is too large.The condition (51), on the other hand, involves directly the structure ofthe magnetic vortex chain and of the equilibrium electron gyrocenter density (or,equivalently, of the equilibrium electrostatic potential or of the parallel magneticperturbations). Inserting the expression for the length l in terms of a , which can beextracted from Eq. (44), the condition (51) can be reformulated as b (cid:63) n + π (cid:16) cosh − (cid:16) √ a − a (cid:17)(cid:17) − a − √ a − ≥ max ( x,y ) ∈ D n (cid:18) − κ (cid:19) F (cid:48) ( A eq ( x, y )) . (54)Obviously, this condition depends on the choice of the arbitrary function F . For thechoice F = 0 , which corresponds to φ = 0 at equilibrium, and thus no perpendicularflow, the right-hand side of Eq. (54) vanishes. If we consider a single vortex ( n = 1 ) inthe absence of flow (i.e. the most favorable situation for stability), then, for b (cid:63) > , onecan verify numerically that the condition (54) is satisfied for < a < . .. (55) inear stability of magnetic vortex chains in a plasma n , makes it more difficult to satisfy the stability condition. For instance, for n = 4 ,always in the absence of perpendicular flow, one has that the stability condition issatisfied for < a < . .. .Considering the expression (22), this implies a ratio √ a − /a , between the amplitude of the vortices and that of the background shearedmagnetic field, equal at most to approximately . , in order to fulfill the stabilitycondition.When F ( A (cid:107) ) is chosen as a linear function F ( A (cid:107) ) = V A (cid:107) , with constant V , thecondition Eq. (54) becomes b (cid:63) n + π (cid:16) cosh − (cid:16) √ a − a (cid:17)(cid:17) − a − √ a − ≥ (cid:18) − κ (cid:19) V . (56)Because, at equilibrium F ( A eq ) = κ ( β ⊥ e / φ , from interpreting φ as a stream functionfor the equilibrium flow, it follows that V is proportional to the ratio between thesquare of the amplitude of the equilibrium flow and that of the local Alfvén velocity.The condition (56) can then be seen as an upper bound on the speed of the equilibriumflow. This condition is similar to the sub-Alfvénic condition emerging from the Energy-Casimir method applied to other plasma models [48, 21, 49].
4. Cold-ion case : τ ⊥ i (cid:28) In this Section we consider the opposite limit, i.e. τ ⊥ i (cid:28) . This limit is adopted mainlyfor laboratory plasmas [45]. In terms of scales, it implies that the characteristic scale ρ s ⊥ is much larger than the perpendicular ion thermal gyroradius ρ th ⊥ i .Based on the relations I ( τ ⊥ i k ⊥ )e − τ ⊥ i k ⊥ = 1 − τ ⊥ i k ⊥ + O ( τ ⊥ i ) and I ( τ ⊥ i k ⊥ )e − τ ⊥ i k ⊥ → , as τ ⊥ i → , for all ( k x , k y ) ∈ D n , we consider the followingapproximations for the ion gyroaverage operators for the cold-ion limit : Γ i f ( x, y ) = (1 + τ ⊥ i ∆ ⊥ ) f ( x, y ) + O ( τ ⊥ i ) , Γ i f ( x, y ) = 0 , (57) inear stability of magnetic vortex chains in a plasma f defined over the domain D n . With this prescription, the model (17)-(20) in thecold-ion limit becomes ∂N e ∂t + [ φ, N e ] − b (cid:63) [ A (cid:107) , ∆ ⊥ A (cid:107) ] = 0 , (58) ∂A (cid:107) ∂t + [ φ, A (cid:107) ] + λ [ N e , A (cid:107) ] = 0 , (59)with B (cid:107) = − β ⊥ e β ⊥ e N e , ∆ ⊥ φ = N e (60)and the parameter λ defined by λ = β ⊥ e β ⊥ e − e . (61)The parameter λ is associated with the terms coming from the divergence of theanisotropic electron pressure tensor. In the limit of isotropic temperature ( Θ e = 1 )and when B (cid:107) is negligible, this model can be seen as the two-field model studied inRef. [35] in the limit of vanishing electron inertia. If, furthermore, the third term onthe left-hand of Eq. (59) is also neglected, the model becomes analogous to 2D low- β reduced MHD.We adopt the following boundary conditions: A (cid:107) | ∂D n = a A , (62) φ | ∂D n = a φ , (63)with a A , a φ ∈ R . The boundary condition (62) is identical to Eq. (30) and implies B ⊥ · n = 0 . Equation (63), analogously to Eq. (31), refers to a condition of a velocity fieldtangent to the boundary. However, in the hot-ion case, because of the proportionalitybetween φ and B (cid:107) , the condition applied to the entire field U ⊥ e = ˆ z × ∇ ( φ − B (cid:107) ) . In thecold-ion case, φ and B (cid:107) are no longer proportional, so that the condition (63) expressesthe fact that the normalized E × B velocity field, given by U E × B = ˆ z × ∇ φ , is tangentto the boundary, i.e. U E × B · n = 0 .With the help of the identities (32)-(33) and applying the boundary conditions(62)-(63), it is possible to show that the functionals given in (15) and (21), with B (cid:107) and inear stability of magnetic vortex chains in a plasma φ related to N e by Eq. (60), are conserved by the system (58)-(59) on the domain D n .Therefore, we can consider the constant of motion F = H + C + C given by F ( N e , A (cid:107) ) = (cid:90) D n d x (cid:18) b (cid:63) |∇ A (cid:107) | |∇ φ | − λ N e N e F ( A (cid:107) ) + G ( A (cid:107) ) (cid:19) . (64)We remark that, although F is a functional of N e and A (cid:107) , we also used, for convenience,the variable φ for its expression on the right-hand side of Eq. (64). We point out that φ has to be intended as the unique solution of the problem ∆ ⊥ φ = N e , with φ | ∂D n = a φ .In this way, the field φ can be interpreted as φ = ∆ − ⊥ N e and is unambiguously definedfor a given N e . We impose the following boundary conditions for the perturbations of A (cid:107) and φ : δA (cid:107) | ∂D n = 0 , δφ | ∂D n = 0 , (cid:90) ∂D n ∂δφ∂n ds = 0 . (65)Analogously to the case of the field φ , also the perturbation δφ has to be interpretedas the solution of the problem ∆ ⊥ δφ = δN e , with δφ | ∂D n = 0 , with δN e indicating theperturbation of the dynamical variable N e . The two boundary conditions concerning δφ correspond to those also adopted in Ref. [32]. Indeed, in the cold-ion case, the secondterm on the right-hand side of Eq. (64) is analogous to the kinetic energy term in theconserved functional of the 2D Euler equation for an incompressible fluid.Subject to the boundary conditions (65), the first variation of F reads δF ( N e , A (cid:107) ; δN e , δA (cid:107) ) = (66) (cid:90) D n d x (cid:0) ( − b (cid:63) ∆ ⊥ A (cid:107) + F (cid:48) ( A (cid:107) ) N e + G (cid:48) ( A (cid:107) )) δA (cid:107) + ( F ( A (cid:107) ) − λN e − φ ) δN e (cid:1) . Setting the first variation equal to zero leads to the following equilibrium equations: ∆ ⊥ A (cid:107) = F (cid:48) ( A (cid:107) ) N e b (cid:63) + G (cid:48) ( A (cid:107) ) b (cid:63) , (67) F ( A (cid:107) ) = φ + λN e , (68) inear stability of magnetic vortex chains in a plasma A (cid:107) satisfies Liouville’s equation implies that Eq. (67) becomes G (cid:48) ( A (cid:107) ) = − N e F (cid:48) ( A (cid:107) ) − b (cid:63) e A (cid:107) . (69)We consider first the case where F (cid:48) ( A (cid:107) ) (cid:54) = 0 . In this case, from Eq. (69), one has N e = − b (cid:63) e A (cid:107) + G (cid:48) ( A (cid:107) ) F (cid:48) ( A (cid:107) ) , (70)from which it follows that, at equilibrium, N e = N e ( A (cid:107) ) . Equation (68) thus impliesthat also φ = φ ( A (cid:107) ) , for the equilibria of interest. Using this fact in Eq. (69), togetherwith the relation ∆ ⊥ φ = N e , leads to the equation φ (cid:48)(cid:48) ( A (cid:107) ) |∇ A (cid:107) | = − G (cid:48) ( A (cid:107) ) F (cid:48) ( A (cid:107) ) + e A (cid:107) F (cid:48) ( A (cid:107) ) ( φ (cid:48) ( A (cid:107) ) F (cid:48) ( A (cid:107) ) − b (cid:63) ) . (71)We specialize now to the solution of interest A (cid:107) = A eq . Because the right-hand sideof Eq. (71) is a function of A (cid:107) only, so has to be the left-hand side. In particular, for A (cid:107) = A eq one has to verify if |∇ A eq | is a function of A eq only. In order to test this, weconsider the function Υ( x, y ) = |∇ A eq ( x, y ) | = ( a −
1) sin x + a sinh y ( a cosh y + √ a − x ) . (72)If |∇ A eq | were a function of A eq only, then, upon the local change of coordinates ( x, y ) ↔ ( x (cid:48) , A eq ) given by x = x (cid:48) , (73) y = cosh − (cid:18) e − A eq a − √ a − a cos x (cid:48) (cid:19) , (74)(invertible, for instance, for < x < π and < y < cosh − (1 + ( √ a − /a )(1 − cos x )) )one would have Υ( x, y ) = ¯Υ( x (cid:48) , A eq ) = ¯Υ( A eq ) , for every x (cid:48) in the domain of invertibility.However, Υ( x, y ) = ¯Υ( x (cid:48) , A eq ) = 1 − e A eq − √ a − A eq cos x (cid:48) . (75)Because ∂ ¯Υ /∂x (cid:48) = 2 √ a − A eq ) sin x (cid:48) (cid:54) = 0 (for instance for < x (cid:48) < π ), weconclude that ¯Υ is not constant with respect to x (cid:48) and thus |∇ A eq | is not a function inear stability of magnetic vortex chains in a plasma A eq only on D n . As a consequence, in order for Eq. (71) to hold for "cat’s eyes"equilibria, one has to set φ (cid:48)(cid:48) ( A eq ) = 0 , which implies φ = K A eq + K , (76)with K and K arbitrary constants. As a consequence, using N e = ∆ ⊥ φ , we obtainthat the equilibria supporting magnetic vortex chains are given by A (cid:107) = A eq , (77) N e = K ∆ ⊥ A eq = − K ( a cosh y + √ a − x ) , (78)with K (cid:54) = 0 . From Eqs. (68) and (69) one obtains that the corresponding choice forthe arbitrary functions F and G are given by F ( A (cid:107) ) = − λK e A (cid:107) + K A (cid:107) + K , (79) G ( A (cid:107) ) = − λK A (cid:107) + K − b (cid:63) A (cid:107) + G , (80)with arbitrary constant G .We recall that, in the case λ = 0 , the problem of determining equilibrium solutionswith flow can be circumvented [50, 51], in the case of sub-Alfvénic flows, by rewritingEq. (67) in terms of the new variable u ( A (cid:107) ) = (cid:90) A (cid:107) dg (cid:113) − F (cid:48) ( g ) /b (cid:63) . (81)This transformation leads to a Grad-Shafranov equation (i.e. without flow) for theindependent variable u . Once solutions for this equation are found, the correspondingequilibrium magnetic and velocity fields can be constructed. This procedure was appliedalso in Ref. [29]. However, it was applied to a magnetic field different from the one weobtain from Eq. (22), although it shares the same magnetic surfaces.The expressions for φ and B (cid:107) at equilibrium, on the other hand, are given by Eq.(76) and by B (cid:107) = − K β ⊥ e / (2 + β ⊥ e )∆ ⊥ A eq , respectively. For these equilibria, the U E × B velocity is locally proportional to the perpendicular Alfvén velocity. The corresponding inear stability of magnetic vortex chains in a plasma − ∆ ⊥ A eq .In the case F (cid:48) ( A (cid:107) ) = 0 the equilibrium equations (67)-(68) decouple. The "cat’seyes" solutions for the magnetic flux function are obtained with the choice G (cid:48) ( A (cid:107) ) = − b (cid:63) e A (cid:107) . (82)On the other hand, given that F (cid:48) ( A (cid:107) ) = 0 implies F ( A (cid:107) ) = F , with F arbitraryconstant, Eq. (68) yields λ ∆ ⊥ φ + φ = F . (83)Therefore, in this case, φ and N e are not constrained to be constant, at equilibrium, onthe contour lines of A eq , as in the previous case. We remark that in this case, unlikelow- β reduced MHD (formally retrieved by setting λ = 0 and b (cid:63) = 2 /β ⊥ e ), the choice F (cid:48) ( A (cid:107) ) = 0 does not necessarily lead to zero E × B flow. Indeed, the presence of theadditional contribution due to the first term on the left-hand side of Eq. (83), originatedfrom the electron pressure tensor, makes it possible to obtain non trivial flows in thepresence of magnetic vortex chains.The equilibria considered in this case are thus given by A (cid:107) = A eq , (84) N e = ∆ ⊥ φ eq , (85)where φ eq is a solution of Eq. (83) with boundary condition (63). Clearly, one cantransform this problem into an equivalent problem for a homogeneous equation withDirichlet boundary conditions, by introducing the new variable ¯ φ = φ − F and imposingthe boundary condition ¯ φ | ∂D n = a φ − F . Analytical solutions of this problem can besought for, for instance with the method described in Ref. [52]. inear stability of magnetic vortex chains in a plasma The second variation of the functional (64) reads δ F ( A (cid:107) , N e ; δA (cid:107) , δN e ) = (cid:90) D n d x (cid:0) b (cid:63) |∇ δA (cid:107) | + |∇ δφ | − λ | δN e | + 2 F (cid:48) ( A (cid:107) ) δN e δA (cid:107) + ( G (cid:48)(cid:48) ( A (cid:107) ) + N e F (cid:48)(cid:48) ( A (cid:107) )) | δA (cid:107) | (cid:1) . (86)Considering δ ( F ( A (cid:107) )) = F (cid:48) ( A (cid:107) ) δA (cid:107) and using the boundary conditions (65), theexpression (86) can be reformulated in the following way (see also Refs. [32, 49]): δ F ( A (cid:107) , N e ; δA (cid:107) , δN e ) = (cid:90) D n d x (cid:16) ( b (cid:63) − F (cid:48) ( A (cid:107) )) |∇ δA (cid:107) | + |∇ δφ − ∇ δ ( F ( A (cid:107) )) | +( F (cid:48) ( A (cid:107) )∆ ⊥ F (cid:48) ( A (cid:107) ) + G (cid:48)(cid:48) ( A (cid:107) ) + N e F (cid:48)(cid:48) ( A (cid:107) )) | δA (cid:107) | − λ | δN e | (cid:1) . (87)We specialize now to the equilibria of interest and consider first the case F (cid:48) ( A (cid:107) ) (cid:54) = 0 .Making use of the expressions (77)-(78), as well as of the relations (79)-(80), in Eq.(87), we obtain that the second variation, evaluated at the equilibrium of interest, canbe rearranged to give δ F ( A eq , K ∆ ⊥ A eq ; δA (cid:107) , δN e ) = (cid:90) D n d x (cid:0) ( b (cid:63) − K (1 − λ e A eq ) ) |∇ δA (cid:107) | + |∇ δφ − K (1 − λ e A eq ) ∇ δA (cid:107) | (88) +( K − b (cid:63) + 4 λK |∇ A eq | (2 λ e A eq − − λ K e A eq )2e A eq | δA (cid:107) | − λ | δN e | (cid:1) , where we also used the equilibrium relation ∆ ⊥ A eq = − exp(2 A eq ) .The coefficients of |∇ δA (cid:107) | , | δA (cid:107) | and | δN e | in the integrand have indefinite sign.Identifying conditions for which they are positive will make the integrand, and in turn δ F , positive, thus providing stability conditions for the equilibria under consideration.We begin by noticing that λ < makes the coefficient of | δN e | positive. With regardto the coefficient of |∇ δA (cid:107) | , we observe that it is positive, on the domain, if b (cid:63) > max ( x,y ) ∈ D n K (1 − λ e A eq ( x,y ) ) . (89)For λ < , the maximum of the function on the right-hand side of Eq. (89) is attainedat x = π and y = 0 . Evaluating the function on the right-hand side of Eq. (89) at this inear stability of magnetic vortex chains in a plasma b (cid:63) > K (cid:18) − λ ( a − √ a − (cid:19) . (90)For λ < the coefficient of | δA (cid:107) | is positive if b (cid:63) < K (1 − λ exp(4 A eq )) . Thiscondition, however, is in conflict with the condition (89). Again, we can resort to thePoincaré inequality (43) which, if the condition (89) holds, when applied to the firstterm on the right-hand side of Eq. (88), provides the following bound: δ F ( A eq , K ∆ ⊥ A eq ; δA (cid:107) , δN e ) ≥ (cid:90) D n d x (cid:0) |∇ δφ − K (1 − λ e A eq ) ∇ δA (cid:107) | +( k R ( b (cid:63) − K (1 − λ e A eq ) ) + ( K − b (cid:63) + 4 λK |∇ A eq | (2 λ e A eq − (91) − λ K e A eq )2e A eq ) | δA (cid:107) | − λ | δN e | (cid:1) . The coefficient of | δA (cid:107) | on the right-hand side of Eq. (91) can be made positiveconsidering that, for λ < , the terms proportional to |∇ A eq | are non-negative andnoticing that k R ( b (cid:63) − K (1 − λ e A eq ) ) + ( K − b (cid:63) − λ K e A eq )2e A eq ≥ k R ( b (cid:63) − max ( x,y ) ∈ D n { K (1 − λ e A eq ( x,y ) ) } ) + min ( x,y ) ∈ D n { ( K − b (cid:63) − λ K e A eq )2e A eq } (92) = k R (cid:32) b (cid:63) − K (cid:18) − λ ( a − √ a − (cid:19) (cid:33) − b (cid:63) − K ( a − √ a − − λ K ( a − √ a − . (93)We can therefore conclude that the second variation is positive, and consequently theequilibria (77)-(78) are linearly stable, if the following three conditions are satisfied: b (cid:63) > K (cid:18) − λ ( a − √ a − (cid:19) , (94) λ < , (95) (cid:18) n + π l (cid:19) (cid:32) b (cid:63) − K (cid:18) − λ ( a − √ a − (cid:19) (cid:33) > b (cid:63) − K ( a − √ a − + 8 λ K ( a − √ a − . (96) inear stability of magnetic vortex chains in a plasma β ⊥ e , Θ e and a , on theamplitude K of the E × B flow. This condition also suppresses the firehose instability.The condition (95), on the other hand, can be reformulated as Θ e < β ⊥ e (97)and, analogously to Eq. (50) of the hot-ion case, provides an upper bound on electrontemperature anisotropy. One can note that, for Θ e = 1 , this condition is always satisfied.In this limit, the term − λ | δN e | in Eq. (88) always provides a positive contribution tothe second variation. This suggests that, for isotropic electron temperature, the electronpressure term associated with λ has a stabilizing role, with respect to the reduced MHDcase where λ = 0 . The condition (96) can be fulfilled by sufficiently reducing the widthof the islands letting the parameter a approach . Indeed, the left-hand side of Eq.(96) can be made arbitrarily large letting a → + , in which limit l → + and the term π / (4 l ) goes to infinity. In the same limit, on the other hand, the denominators on theright-hand side tend to , so that the right-hand side remains bounded.In the case F (cid:48) ( A (cid:107) ) = 0 , the second variation, evaluated at the equilibria (84)-(85),and using Eq. (82), reads δ F ( A eq , ∆ ⊥ φ eq ; δA (cid:107) , δN e ) = (cid:90) D n d x (cid:0) b (cid:63) |∇ δA (cid:107) | + |∇ δφ | − b (cid:63) e A eq | δA (cid:107) | − λ | δN e | (cid:1) . (98)The second variation (98) actually corresponds to the second variation (88) in the limit K = 0 , i.e. with no E × B flow. However, as we pointed out in Sec. 4.1, for F (cid:48) ( A (cid:107) ) = 0 ,the potential φ eq can correspond to non-trivial flows. Nevertheless, stability conditionsfor this case can be directly obtained from Eqs. (94)-(96) by setting K = 0 and can beformulated as β ⊥ e β ⊥ e < Θ e < β ⊥ e , (99) (cid:18) n + π l (cid:19) > a − √ a − . (100) inear stability of magnetic vortex chains in a plasma b (cid:63) > and λ < and preventsinstabilities due to temperature anisotropy. The condition (100), on the other hand,implies restrictions on a and is amenable to the same considerations discussed for thecondition (54) in the case with no perpendicular flow.
5. Concluding remarks
In this work we studied the existence and the stability of stationary solutions, of areduced fluid model, describing chains of magnetic vortices. The formation of chainsof magnetic vortices, due to the reconnection of magnetic field lines, is a frequentphenomenon in laboratory and space plasmas. Observational evidence shows, inparticular, the existence of chains of magnetic vortices, for instance in the plasma of thesolar wind. The presence of such structures can have a strong impact on the turbulentspectrum of magnetic and kinetic plasma energy.We first reduced the general gyrofluid model, by acting on its Hamiltonian structure, toa 2D version without electron inertia effects. Subsequently, we considered the resultingmodel in the asymptotic limit in which the equilibrium ion temperature, referred to theplane perpendicular to the direction of a strong magnetic guide field, is much greaterthan the electron one, i.e. τ ⊥ i (cid:29) . In this limit we found equilibrium equationsadmitting solutions describing magnetic vortex chains supporting a class of non-trivialperpendicular flows, constant on the magnetic flux function contour lines, and dependingon an arbitrary function. We obtained that such magnetic vortex chains equilibriaare linearly stable if three conditions are fulfilled. Two of these conditions imposebounds on the electron temperature anisotropy, which, as expected, can be a sourcefor instabilities. Depending on the range of values for β ⊥ e , the temperature anisotropyhas only a lower bound or is bounded from above and from below. Interestingly, thelower bound corresponds to the bound for firehose instability known for homogeneousequilibria according to linear wave stability analysis. Upper and lower bounds dependon the electron beta parameter. The third condition depends explicitly on the choice inear stability of magnetic vortex chains in a plasma β ⊥ e and Θ e , it can be seen as acondition on the maximum island width and on the length of the chain. Shorter chainswith thin islands favor stability.In the opposite, cold-ion case, with τ ⊥ i (cid:28) , a slightly more intricate situationoccurs, presenting two sub-cases. In one sub-case, the magnetic vortex chain supportsan electrostatic potential φ linear with respect to the magnetic flux function. Thisrestricts the equilibrium E × B velocity to be proportional to the local Alfvén velocity.The electron gyrocenter density N e and the parallel magnetic perturbations B (cid:107) , on theother hand, are proportional to the current density associated with the vortex chain. Inthis sub-case, one stability condition suppresses the firehose instability but is strongerthan the aforementioned condition, due to the presence of the equilibrium flow. A secondcondition sets an upper bound to temperature anisotropy and a third condition, againconcerns also the size and the length of the chain. In the second sub-case, the fields φ and N e are no longer constrained to be constant on contour lines of the magnetic fluxfunction and satisfy the relation N e = ( − φ + F ) /λ . In principle this can provide non-trivial flows. Stability conditions bound temperature anisotropy from above and frombelow, with the lower bound again corresponding to the firehose stability condition. Thethird condition, on the other hand, turns out to correspond to the one found for thehot-ion case in the absence of flows. If a non-trivial solution for the flow can be foundin this case, the characteristics of such solution appear not to be crucial for stability.Our analysis suggests that, in both hot and cold-ion regimes, several parametersof the system have to be controlled to attain the stability conditions. Such conditionsappear to be rather compelling, and favor short chains with thin vortices and moderateanisotropy. The condition on the maximum vortex width is analogous to the conditionfor nonlinear stability of "cat’s eyes" vortex chains derived in Ref. [32]. We point outagain, that our analysis is carried out over the domain enclosed by the separatrices andthus rules out external perturbations. It is well known that magnetic island chains areactually unstable on larger domains including regions outside the separatrices [1, 2, 5]. inear stability of magnetic vortex chains in a plasma
6. Acknowledgments
CG and ET acknowledge fruitful discussions with the members of the Fluid and PlasmaTurbulence Team of the Laboratoire Lagrange.
Appendix A. Summary of the derivation of the model
In this Section we summarize some assumptions underlying the model, which determinealso its limits of validity, and we describe qualitatively the main steps of its derivation.First, we recall the gyrokinetic system from which the gyrofluid model (1)-(5) can bederived. This gyrokinetic system corresponds, in turn, to the gyrokinetic model derivedin Ref. [37] when equilibrium drifts are neglected and a bi-Maxwellian distribution ischosen as equilibrium distribution function. Such gyrokinetic system, in dimensionalvariables, reads ∂ (cid:101) g s ∂t + cB (cid:34) J s (cid:101) φ − v (cid:107) c J s (cid:101) A (cid:107) + 2 µ s B q s J s (cid:101) B (cid:107) B , (cid:101) g s (cid:35) + v (cid:107) ∂∂z (cid:32)(cid:101) g s + q s T (cid:107) s F s (cid:32) J s (cid:101) φ − v (cid:107) c J s (cid:101) A (cid:107) + 2 µ s B q s J s (cid:101) B (cid:107) B (cid:33)(cid:33) = 0 , (A.1) (cid:88) s q s (cid:90) d W s J s (cid:101) g s = (cid:88) s q s T ⊥ s (cid:90) d W s F s (cid:0) − J s (cid:1) (cid:101) φ − (cid:88) s q s (cid:90) d W s µ s B T ⊥ s F s J s J s (cid:101) B (cid:107) B , (A.2) (cid:88) s q s (cid:90) d W s v (cid:107) J s (cid:32)(cid:101) g s − q s T (cid:107) s v (cid:107) c F s J s (cid:101) A (cid:107) (cid:33) = − c π ∆ ⊥ (cid:101) A (cid:107) + (cid:88) s q s m s (cid:90) d W s F s (cid:32) − s v (cid:107) v th (cid:107) s (cid:33) (1 − J s ) (cid:101) A (cid:107) c , (A.3) (cid:88) s β ⊥ s n (cid:90) d W s µ s B T ⊥ s J s (cid:101) g s = − (cid:88) s β ⊥ s n q s T ⊥ s (cid:90) d W s µ s B T ⊥ s F s J s J s (cid:101) φ − (cid:32) (cid:88) s β ⊥ s n (cid:90) d W s F s (cid:18) µ s B T ⊥ s J s (cid:19) (cid:33) (cid:101) B (cid:107) B . (A.4) inear stability of magnetic vortex chains in a plasma s indicates the particle species ( s = e for electrons and s = i for ions, whenassuming a single ion species). Equations (A.2)-(A.4) express quasi-neutrality, paralleland perpendicular components of Ampère’s law, respectively. The gyrokinetic equation(A.1) describes the evolution of the function (cid:101) g s ( x, y, z, v (cid:107) , µ s , t ) = (cid:101) f s ( x, y, z, v (cid:107) , µ s , t ) + q s T (cid:107) s v (cid:107) c F s ( v (cid:107) , µ s ) J s (cid:101) A (cid:107) ( x, y, z, t ) . (A.5)In Eq. (A.5) (cid:101) f s is the perturbation of the distribution function, whereas F s is the bi-Maxwellian equilibrium distribution function defined by F s ( v (cid:107) , µ s ) = (cid:16) m s π (cid:17) / n T / (cid:107) s T ⊥ s e − msv (cid:107) T (cid:107) s − µ sB T ⊥ s , (A.6)The independent variables are the time coordinate t ∈ [0 , + ∞ ) , the spatial coordinates ( x, y, z ) ∈ T n , the velocity coordinate parallel to the guide field v (cid:107) ∈ ( −∞ , + ∞ ) andthe magnetic moment µ s ∈ [0 , + ∞ ) of the particle of species s , in the presence ofthe unperturbed magnetic guide field. We indicated with q s the charge of the particleof species s and with d W s = (2 πB /m s ) dµ s dv (cid:107) the volume element in velocity space,integrated over the particle gyration angle. The gyroaverage operators J s and J s aredefined as J s f ( x, y, z ) = (cid:88) k ∈ T n J ( a s ) f k exp( i k · x ) , (A.7) J s f ( x, y, z ) = (cid:88) k ∈ T n J ( a s ) a s f k exp( i k · x ) , (A.8)where J and J are zero and first order Bessel functions of the first kind and a s = k ⊥ (cid:112) µ s B /m s /ω cs is the perpendicular wave number multiplied times the gyroradiusof the particle of species s .For the gyrofluid model we consider a plasma composed by electrons and a singleionized species of ions. Because the model was conceived mainly as an extended modelfor kinetic Alfvén waves, the coupling with ion gyrocenter fluctuations is removed byneglecting contributions of the perturbations of all ion gyrocenter moments in the quasi-neutrality relation and parallel as well as perpendicular projections of Ampère’s law. inear stability of magnetic vortex chains in a plasma δ (cid:28) and (cid:15) (cid:28) , the latter correspondingto the normalized characteristic frequency of the fluctuations under consideration.Assuming these two parameters to be small, amounts to considering electron inertiaas a small perturbation and to considering phenomena with frequencies much lowerthan the ion cyclotron frequency, the latter being one of the fundamental hypotheses inthe gyrokinetic ordering.A Laguerre-Hermite expansions is adopted for the perturbation of the electrongyrocenter distribution functions. This expansion is truncated by imposing the relations T (cid:107) e = 0 , T ⊥ e = − B (cid:107) , (A.9)where T (cid:107) e and T ⊥ e correspond to the electron gyrocenter parallel and perpendiculartemperature fluctuations, respectively. The closure relations (A.9), correspond, in termsof particle temperature fluctuations, to isothermal closures for both the parallel andperpendicular electron temperatures (recall that, when electron FLR are neglected, asin this case, the perpendicular temperature fluctuations of particles t ⊥ e are related tothose of gyrocenters T ⊥ e by t ⊥ e = T ⊥ e + B (cid:107) , as explained in Ref. [33]). Such truncatedexpansion for (cid:101) f e is inserted into Eqs. (A.2)-(A.4), whereas all the fluctuations of the iongyrocenter moments are neglected.Two different scalings are then adopted, one of them valid for τ ⊥ i = O (1) and asecond one valid for τ ⊥ i = O (1 /δ ) , as δ → . The two scalings read τ ⊥ i ∼ Θ e ∼ ∇ ⊥ = O (1) , β ⊥ e = O ( δ ) ,A (cid:107) ∼ ∂ z = O ( δ / (cid:15) ) , U e = O (cid:16) (cid:15)δ / (cid:17) , (A.10) ∂ t ∼ φ ∼ N e = O ( (cid:15) ) , B (cid:107) = O ( δ(cid:15) ) inear stability of magnetic vortex chains in a plasma τ ⊥ i = O (1 /δ ) , ∇ ⊥ ∼ Θ e = O (1) , β ⊥ e = O ( δ ) ,A (cid:107) ∼ ∂ z = O ( δ / (cid:15) ) , U e = O (cid:16) (cid:15)δ / (cid:17) , (A.11) ∂ t ∼ φ = O ( (cid:15) ) , B (cid:107) ∼ N e = O ( δ(cid:15) ) , respectively. Such two scalings are applied to the quasi-neutrality relation and to theparallel and perpendicular projections of Ampère’s law (A.2)-(A.4), after inserting for (cid:101) f e its truncated Laguerre-Hermite expansion, integrating with the help of orthogonalityrelations for Hermite and Laguerre polynomials and neglecting the perturbation of theion gyrocenter distribution function. For each scaling, all terms at leading order, as wellas corrections of order δ , are retained. This yields Eqs. (3)-(5).On the other hand, from the zero and first order moments, with respect to v (cid:107) , ofthe evolution equation for the electron gyrocenter distribution function (Eq. (A.1)), oneobtains Eqs. (1)-(2), upon imposing the closure relation (A.9) and considering only theleading order terms in the expansion of the electron gyroaverage operators as δ → (the next order terms being of order at least δ smaller, and thus negligible). Appendix B. The stability algorithm
In this Section we briefly summarize the steps required for determining linear stabilityconditions according to the Energy-Casimir method.We consider a dynamical system ∂χ i ∂t = X i ( χ , · · · , χ N ) , i = 1 , · · · , N, (B.1)evolving N fields χ , · · · , χ N all of which depend on time and on space variables x , · · · , x m belonging to some domain U ⊂ R m , with m and N positive integers.We suppose the system admits a family of s constants of motion C , · · · , C s , i.e.functionals C ( χ , · · · , χ N ) , · · · , C s ( χ , · · · , χ N ) such that d C i /dt = 0 , for i = 1 , · · · , s .The functional F = (cid:80) si =1 C i is then a constant of motion as well. For noncanonical inear stability of magnetic vortex chains in a plasma F is given by F = H + (cid:80) s − i =1 C i , where H is the Hamiltonian of the system and C , · · · , C s − are Casimir invariants. This is whythe method is referred to as Energy-Casimir method.Solutions of the equation δF ( χ , · · · , χ N ; δχ , · · · , δχ N ) = 0 , (B.2)where δF is the first variation of F , correspond to equilibria of the system (B.1). Suchequilibrium points, denoted as ( χ e , · · · , χ eN ) , can then be related to constants of motionby requiring that ( χ e , · · · , χ eN ) be a point where δF vanishes. In this way, classes ofequilibria (although in general not all the equilibria of the system) can be associatedwith different choices of constants of motion.An equilibrium ( χ e , · · · , χ eN ) solution of δF ( χ , · · · , χ N ; δχ , · · · , δχ N ) = 0 isformally stable (which implies linearly stable) if the second variation of F , evaluated atsuch equilibrium, i.e. δ F ( χ e , · · · , χ eN ; δχ , · · · , δχ N ) (B.3)has a definite sign. If this is the case, in fact, the expression (B.3) (or its opposite)can be taken as a conserved norm for the system (B.1) linearized about the equilibrium ( χ e , · · · , χ eN ) . References [1] Finn J M and Kaw P K 1977
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