PACS numbers: Cv, Xx, Df, Xz Keywords: magnetohydrodynamics, stability, Hamiltonian, Poisson bracket

Transcription

1 Hamiltonian Magnetohydrodynamics: Lagrangian, Eulerian, and Dynamically Accessible Stability - Examles with Translation Symmetry T. Andreussi, 1 P. J. Morrison, and F. Pegoraro 3 1 SITAEL S..A., Pisa, 5611, Italy Institute for Fusion Studies and Deartment of Physics, The University of Texas at Austin, Austin, TX , USA 3 Diartimento di Fisica E. Fermi, Pisa, 5617, Italy Dated: 19 Setember 016 Because different constraints are imosed, stability conditions for dissiationless fluids and magnetofluids may take different forms when derived within the Lagrangian, Eulerian energy-casimir, or dynamical accessible frameworks. This is in articular the case when flows are resent. These differences are exlored exlicitly by working out in detail two magnetohydrodynamic examles: convection against gravity in a stratified fluid and translationally invariant erturbations of a rotating magnetized lasma inch. In this second examle we show in exlicit form how to erform the time-deendent relabeling introduced in Andreussi et al. Phys. Plasmas 0, that makes it ossible to reformulate Eulerian equilibria with flows as Lagrangian equilibria in the relabeled variables. The rocedures detailed in the resent article rovide a aradigm that can be alied to more general lasma configurations and in addition extended to more general lasma descritions where dissiation is absent. PACS numbers: 5.30.Cv, 0.30.Xx, Df, 5.5.Xz Keywords: magnetohydrodynamics, stability, Hamiltonian, Poisson bracket I. INTRODUCTION The early lasma literature on magnetohydrodynamics MHD is secked with traces of a general underlying structure: the self-adjointness of the MHD force oerator in terms of the dislacement ξ of the original energy rincile, the Woltjer invariants of helicity and cross helicity and their use in obtaining Beltrami states, and the reresentation of the magnetic and velocity fields in terms of Clebsch otentials being examles. All of these are symtoms of the fact that MHD is a Hamiltonian field theory, whether exressed in Lagrangian variables as shown by Newcomb 1 or in terms of Eulerian variables as shown by Morrison and Greene. General ramifications of the Hamiltonian nature of MHD were elucidated in our series of ublications, 3 6 while in the resent work we examine exlicitly the stability of stratified lasma and of rotating inch equilibria within each of the three Lagrangian, Eulerian, and dynamically accessible descritions. These articular two examles were chosen because they are at once tractable and significant. They dislay difficulties one faces in ascertaining stability within the three aroaches and rovide a means to comare and contrast stability results. The aer is designed to serve as a how-to guide for alication of the three aroaches, roviding a framework for what one might exect, and delineating the sometimes subtle differences between the aroaches. Here and in our revious aers the scoe was limited to MHD, but the same Hamiltonian structure exists for all imortant dissiation free lasma models, kinetic as well as fluid, and the story we tell for MHD alies to them as well. See e.g. Ref. 7 for review. Recently there has been great rogress in understanding the Hamiltonian structure of extended MHD, 8 14 the effect of gyroviscosity, 15 and relativistic magnetofluid models. 16,17 In addition, recent work on hybrid kinetic-fluid models 18,19 and gyrokinetics 0,1 now also lie within the urview. There are many concets of stability of imortance in lasma hysics see Sec.VI of Ref. for a general discussion here we will only be concerned with what could be referred to as formal Lyaunov stability, which has received wide attention in the fluid and lasma literature, both in the Hamiltonian and non-hamiltonian contexts see e.g. Refs. 3 and 4 and references therein for the latter. In the Hamiltonian context the Lyaunov stability we consider rovides at least a sufficient condition for stability, imlied by the ositive-definiteness of a quadratic form obtained from the second variation of an energy-like quantity. This kind of stability is stronger than sectral or eigenvalue stability: for finite-dimensional systems it imlies nonlinear stability, i.e., stability to infinitesimal erturbations under the nonlinear evolution of the system. Note, nonlinear stability should not be confused with finite-amlitude stability that exlores the extent of the basin of stability, a confusion that oft aears in the lasma literature. For infinite-dimensional systems like MHD there are technical issues that need to be addressed in order to rigorously claim that formal Lyaunov stability imlies nonlinear stability see e.g. Ref. 5 for an examle of a rigorous nonlinear stability analysis, but the formal Lyaunov stability of our interest is a most imortant ingredient and it does imly linear stability. A common ractice in the lasma literature, emloyed e.g. by Chandresekhar, 6 is to maniulate the linear equations of motion in order to obtain a conserved quadratic form that imlies stability. Although

2 this rocedure shows linear stability, it cannot be used to obtain nonlinear stability and may give a misleading answer. This is evidenced by the Hamiltonian system, which when linearized has both of the two Hamiltonians for two linear oscillators, H ± = ω q 1/ ± ω + q /. 1 Both signs of 1 are conserved by the linear system, yet only one arises from the exansion of the nonlinear Hamiltonian of the system. Nonlinear Hamiltonians that give rise to linear Hamiltonians of the form of H can in fact be unstable see Ref. 7 for an examle, and are rototyes for systems with negative energy modes. This examle shows why the formal Lyaunov stability, our subject, is stronger than sectral or eigenvalue stability. To reiterate, throughout by stability we will mean formal Lyaunov stability The remainder of the aer is organized as follows: in Sec. II we review basic ideas of the three aroaches, giving essential formulas so as to make the aer selfcontained. Of note is the new material of Sec. II D that summarizes various comarisons between the aroaches. This is followed by our convection examle of Sec. III and our inch examle of Sec. IV. These sections are organized in arallel with Lagrangian, Eulerian or so-called energy-casimr, and dynamically accessible stability treated in order, followed by a subsection on comarison of the results. Finally, we conclude in Sec. V. II. BASICS In what follows we will consider the stability of MHD equilibria that are solutions to the following equations: e v e v e = e + J e B e + e Φ e, v e B e = 0, 3 e v e = 0, 4 v e s e = 0, 5 for the equilibrium velocity field v e x, magnetic field B e x, current density 4πJ e = B e, density field e x, and entroy/mass field s e x. Here Φx, t reresents an external gravitational otential. The ressure field is assumed to be determined by an internal energy function U, s, where = U/ and the temerature is given by T = U/ s. For the ideal gas = c γ exλs, with c, λ constants and U = /γ 1. MHD has four thermodynamical variables, s,, and T. The assumtion of local thermodynamic equilibrium imlies that knowledge of two of these variables at all oints x is sufficient to determine the other two, once the U aroriate to the fluid under consideration is secified. For static equilibria with v e 0, the only equation to solve is e = J e B e + e Φ e. 6 Equation 6 is one equation for several unknown quantities; consequently, there is freedom to choose rofiles such as those for the current and ressure as we will see in our examles. If we neglect the gravity force by removing Φ e, Eq. 6 leads as usual to the Grad-Shafranov equation, e.g., by noting that B e = 0 imlies ressure is a flux function. However, unlike the barotroic case where only deends on, in general this does not imly that and s are flux functions, since their combination in, s could cancel out their variation on a flux surface. Thus, as far as static ideal MHD is concerned, because only occurs in the equilibrium equation, density and temerature on a flux function can vary while ressure is constant. The MHD static equilibrium equations give no information/constraints on this variation. When gravity is included, Eq. 6 still is only one constraining equation for several unknown quantities. In Sec. III we consider stratified equilibria both with and without a magnetic field and we will investigate there the role layed by entroy. For stationary equilibria the full set of Eqs. 5 must be solved. Because in general there are many ossibilities, we will restrict our analysis to the rotating inch examle of Sec. IV, where we describe the equilibrium in detail. A. Lagrangian formulae The Hamiltonian for MHD in Lagrangian variables is Hq, π = d 3 πi π i a + 0 U s 0, 0 /J 0 + q i q i B0 k B l 0 a k a l 8πJ + 0Φq, t, 7 where q, π are the conjugate fields with qa, t = q 1, q, q 3 denoting the osition of a fluid element at time t labeled by a = a 1, a, a 3 and πa, t being its momentum density. In 7 the quantities s 0, 0, and B 0 are fluid element attributes that only deend on the label a, and J := det q i / a j. Also, A i j qj / a k = J δk i, where Ai j denotes elements of the cofactor matrix of q/ a. In a general coordinate system π i = g ij q π i where g ij is the metric tensor. This Hamiltonian together with the canonical Poisson bracket δf {F, G} = d 3 δg a δq i δg δf δπ i δq i, 8 δπ i renders the equations of motion in the form π i = {π i, H} = δh δq i and q i = { q i, H } = δh, 9 δπ i where denotes time differentiation at constant label a and δh/δq i is the usual functional derivative. The results of these calculations can be found in Aendix A and further details can be found in Refs. 5 and.

3 3 In Ref. 5 we introduced the general time-deendent relabeling transformation a = Ab, t, with the inverse b = Ba, t, which gave rise to the new dynamical variables Πb, t = J πa, t, Qb, t = qa, t, 10 and the new Hamiltonian HQ, Π = H d 3 b Π V b Q, 11 = d 3 Πi Π i b Π i V j Qi 0 b j + 0 U s 0, 0 / J + Q i Q i Bk 0 Bl 0 b k b l 8πJ, = K + H f + W, 1 where K is the kinetic energy, H f is the fictitious term due to the relabeling, and W reresents the sum of the internal and magnetic field energies. In the first equality of 1, Vb, t := Ḃ B 1 = ḂAb, t, t, 13 which is the label velocity, b := / b, and H is to be written in terms of the new variables. In the second equality we used d 3 a = J d 3 b, with J := det a i / b j, 0 = J 0, J := det Q i / b j = J J, and 0 / J = 0 /J, which follows from mass conservation 0 d 3 a = 0 d 3 b. The relabeled entroy is s 0 b, t = s 0 Ab, t. From 9 it is clear that extremization of Hamiltonians give equilibrium equations. For the Hamiltonian Hq, π of 7 this gives static equilibria, while for HQ, Π of 1 one obtains stationary equilibria. This was the oint of introducing the relabeling: it allows us to exress stationary equilibria in terms of Lagrangian variables, which would ordinarily be time deendent, as time-indeendent orbits with the moving labels. The equilibrium equations are 0 = t Q e = Π e 0 V e b Q e, 0 = t Π e = b V e Π e + F e, 14 where F e comes from the W art of the Hamiltonian. From 14 the equilibrium equation follows, b 0 V e V e b Q e = F e. 15 Using b = Q e b = q e A e b, t, t = B e a, t and the definition of V of 13, Vb, t = ḂeA e b, t, t = v e b, where v e b denotes an Eulerian equilibrium state, we obtain uon setting b = x the usual stationary equilibrium equation, e v e v e = F e, 16 where e x is the usual equilibrium density. It can be shown that v e s e = 0, e v e = 0, and v e B e B e v e + B e v e = 0, follow from the Lagrange to Euler ma. Further details of this relabeling transformation are given in Ref. 5, while alication to our rotating inch examle of Sec. IV is worked out in Aendix B. For stability, we exand as follows: Q = Q e b, t+ηb, t, Π = Π e b, t+π η b, t, 17 and calculate the second variation of the Hamiltonian in terms of the relabeled canonically conjugate variables η, π η giving δ H la Z e ; η, π η = 1 1 d 3 x π η e v e η e +η V e η, 18 which deends on the time indeendent equilibrium quantities Z e = e, s e, v e, B e, i.e., the oerator V e has no exlicit time deendence. Again, see in Refs. 5 and for details. The functional δ W la Z e ; η := 1 d 3 x η V e η = 1 d 3 x e v e v e η η e v e η + δ W η, 19 is identical to that obtained by Frieman and Rotenberg 8, although obtained here in an alternative and more general manner. The energy δ W la can be transformed in the more familiar exression of Ref. 9, δ W la Z e ; η = 1 d 3 e x e η + η e η e + δb 4π + J e η δb e ηη Φ e, 0 where 4πJ e = B e is the equilibrium current and δb := η B e. For comleteness we record the first order Eulerian erturbations that are induced by the Lagrangian variation written in terms of the dislacement η: δ la = e η 1 δv la = π η / e η v e = η/ t + v e η η v e δs la = η s e 3 δb la = B e η 4 where δs la can be relaced by the ressure erturbation, δ la = γ e η η e, that is often used.

4 4 B. Eulerian formulae The Hamiltonian for MHD in Eulerian variables is HZ = d 3 x v + Us, + B 8π + Φ. 5 where Z =, s, v, B. When 5 is substituted into the noncanoncal Poisson bracket {F, G} nc of Ref. one obtains the Eulerian equations of motion in the form Z/ t = {Z, H} nc. Because the noncanonical Poisson bracket {F, G} nc is degenerate, i.e. three exist a functional C such that {F, C} nc = 0 for all functionals F, Casimir invariants C exist and equilibria are given by extremization of the energy-casimir functional F = H + C. For MHD with no symmetry the Casimirs are C s = d 3 x Ss, 6 and the magnetic and cross helicities, C B = d 3 x A B, and C v = d 3 x v B, 7 resectively. By maniulation of the MHD equations, the helicities were shown by Woltjer to be invariants C v requiring the barotroic equation of state and used by him to redict lasma states. Woltjer s ideas ertaining to magnetic helicity were adated by Taylor 34,35 to describe reversed field configurations. The invariant of 6 and Woltjer s helicities were shown to be Casimir invariants in Ref. 36. See Refs. 37 and 38 for further discussion. An imortant oint to note is that knowledge of the Casimirs determines this additional hysics, but this knowledge must come from hysics outside of the ideal model. Secial attention has been given to the equilibrium states obtained by extremizing the energy subject to the Woltjer invariants, erhas because these are the states for which Casimirs are at hand. See Refs. 39 and 40 for discussion of the Casimir deficit roblem. However, we will see in Sec. II C that all MHD equilibria are obtainable from the variational rincile with directly constrained variations, the dynamically accessible variations, rather than using Lagrange multiliers and helicities etc. In the case were translational symmetry is assumed, all variables are assumed to be indeendent of a coordinate z with B = B z ẑ + ψ ẑ, 8 M = M z ẑ + χ ẑ + Υ, 9 where χ, Υ and ψ are otentials, M = v, M z = v z and ẑ is the unit vector in the symmetry direction. The Hamiltonian then becomes M H T S Z s = d 3 x z + χ + Υ + + U + Φ, 30 Υ, χ + ψ 8π + B z 8π where Z s =, s, M z, χ, Υ, ψ, B z. With this symmetry assumtion, the set of Casimir is exanded and is sufficient to obtain a variational rincile for the equilibria considered here. However, because of this symmetry assumtion it is only ossible to obtain stability results restricted to erturbations consistent with this assumtion. In Refs. 3 and 4 the translationally symmetric noncanonical Poisson brackets were obtained for both neutral fluid and MHD dynamics. For the case of a neutral fluid, which we consider in Sec. III B for convection, the Poisson bracket for translationally symmetric flows was given in Ref. 3. This bracket with the Hamiltonian of 30, where the magnetic energy terms involving B z and ψ are removed, gives the comressible Euler s equations for fluid motion. The translationally symmetric fluid Poisson bracket has the following Casimir invariants: C 1 = d 3 x S s, v z, s, v z /,..., 31 C = d 3 x As χ + Υ, As / = d 3 x As ẑ v, 3 where f, g = ẑ f g. The second Casimir alies if v z deends only on s, which will suit our urose, i.e., the energy-casimir variational rincile δf = 0 will give our desired equilibria. For the case of MHD it was shown in Refs. 3 and 4 that the following are the Casimir invariants with translational symmetry: C s = d 3 x J s, ψ, s, ψ /, s, ψ /, ψ /, s, s, ψ / /,..., 33 C Bz = d 3 x B z H ψ, 34 C vz = d 3 x v z G ψ, 35 and, if the entroy is assumed to be a flux function, i.e., ψ, s = 0, then 33 collases to C s = d 3 x J ψ, 36 and there is the additional cross helicity Casimir, C v = d 3 x v z B z F ψ + 1 Υ, Fψ Fψ χ + = d 3 x v B F ψ. 37

5 5 where S, A, J, H, G, and F are arbitrary functions of their arguments J distinguished from the Jacobian with the same symbol by context with rime denoting differentiation with resect to argument. For both the neutral fluid and MHD equilibria that satisfy δf = 0 a sufficient condition for stability follows if the second variation δ F can be shown to be ositive definite. For MHD it was shown in Refs. 5 and 6 that δ F could be ut into the following diagonal form: δ FZ e ; δz s = d 3 x a 1 δs + a δq + a 3 δr z +a 4 δr + a 5 δψ, 38 where the variations δs, δr, δq, δψ are linear combinations of δv, δb, δ, δψ. The coefficients a i for i = 1 5 deend on sace through the equilibrium and were given first exlicitly in Ref. 5 and corrected in Ref. 6. Note, for these calculations the external otential Φ was omitted. See Refs. 41 and 4 for related work. Uon extremizing over all variables excet δψ and then back substituting the resulting algebraic relations, 38 becomes δ FZ e ; δψ = d 3 x b 1 δψ + b δψ +b 3 e ψ δψ, 39 where e ψ = ψ/ ψ and b 1 = 1 M c s M c s + c a, 40 4π c s M c s + c a + M4 4π ψ M b = ψ ψ 4π b 3 = 1 M 4π ψ + B z 8π + M 4π ψ, 41 b 1. 4 where the oloidal Alfvén-Mach number M := 4πF / < 1 has been assumed. Here c a = B / 4π and c s = / 43 are the Alfven and the sound seed, resectively. Thus, stability in this MHD context rests on whether or not 39 is definite, and for the neutral fluid equilibria we treat here, which include a gravity force, the same is true for the corresonding functional. C. Dynamically accessible formulae Extremizing the Hamiltonian of 5 without constraints gives trivial equilibria. With energy-casimir the constraints are incororated essentially by using Lagrange multiliers. Dynamically accessible variations, as introduced in Ref. 43, restrict the variations to be those generated by the noncanonical Poisson bracket and in this way assures that all kinematical constraints are satisfied. The first order dynamically accessible variations, obtained directly from the noncanonical Poisson bracket of Ref., are the following: δ da = g 1, 44 δv da = g 3 + s g + v g 1 + B g 4 / 45 δs da = g 1 s, 46 δb da = B g 1, 47 where the freedom of the variations is embodied in the arbitrariness of g 1, g, g 3, and g 4. Using these in the variation of the Eulerian Hamiltonian gives v δh da = d 3 x / + U + Φ δ da + v δv da +U s δs da + B δb da /4π, = d 3 x g 1 v v v / h + T s + J B g sv g 3 v + g 4 v B = 0, 48 whence it is seen that the vanishing of the terms multilying the indeendent quantities g 1, g, g 3, and g 4 gives recisely the Eulerian equilibrium equations 5. Next, stability is assessed by exanding the Hamiltonian to second order using the dynamically accessible constraints to this order see Refs. 5 and for details, yielding the following exression: δ H da Z e ; g = d 3 x δv da g 1 v + v g 1 +δ W la g If in 49 δv da were indeendent and arbitrary we could use it to nullify the first term and then uon setting g 1 = η, we would see that dynamically accessible stability is identical to Lagrangian stability. However, as we will see in Sec. II D, this is not always ossible. D. Comarison formulae In our calculations of stability we obtained the quadratic energy exressions of 18, 38, and 49, which can be written in terms of various Eulerian erturbation variables P := {δ, δv, δs, δb}. 50 In the case of the Lagrangian energy of 18, the set of erturbations P la as given by Eqs. 1 4 are constrained, while for the energy-casimir exression of 38

6 6 the erturbations P ec are entirely unconstrained rovided they satisfy the translation symmetry we have assumed. Similarly the erturbations for the energy exression 49, P da of 44 47, are constrained. In our revious work of Ref. 5 we showed that the three energy exressions are equivalent if restricted to the same erturbations, and we established the inclusions P da P la P ec, which led to the conclusions stab ec stab la stab da, viz., dynamically accessible stability is the most limited because its erturbations are the most constrained, while energy-casimir stability is the most general, when it exists, for its erturbations are not constrained at all. The choice between the three aroaches should be based on the hysics of the situation, which determines the relevant constraints that need to be satisfied by the erturbations. Our goal is to exlore further the differences between these kinds of stability by exloring, in articular, the differences between Lagrangian and dynamically accessible erturbations. From 49 it is clear that if δv da is arbitrary, indeendently of g 1, then the first term of this exression can be made to vanish. This would reduce δ H da to the energy exression obtained for Lagrangian stability, making the two kinds of stability equivalent. Given that there are five comonents of g, g 3 and g 4, in addition to g 1, one might think that this is always ossible. However, as ointed out in Ref. 5 this is not always ossible and whether or not it is deends on the state or equilibrium under consideration. We continue this discussion here. Consider first a static equilibrium state that has entroy as a flux function and no equilibrium flow. Thus, for this case, the cross helicity C v of 7 vanishes. For a dynamically accessible erturbation δc v = d 3 x δv da B e = d 3 x g 3 + s e g B e = d 3 x g B e s e = 0, 51 where the last equality assumes g 3 is single-valued and the vanishing of surface terms, as well as s e being a flux function. The fact that δc v = 0 for this case is not a surrise since it is a Casimir, but we do see clearly that if s were not a flux function, then a erturbation δv da could indeed create cross helicity. Because of the term η/ t of, which can be chosen arbitrarily, it is clear that δv la can create cross helicity for any equilibrium state, sulying clear evidence that δv da is not comletely general. Although δv da is not comletely general, it was noted in Ref. that for static equilibria the first term of 49 becomes d 3 x δv da 5 and this can be made to vanish indeendent of g 1 by choosing g = g 3 = 0 and g 4 = 0. Thus, for static equilibria the Lagrangian and dynamically accessible aroaches must give the same necessary and sufficient conditions for stability, i.e. stab la stab da, As another examle consider the variation of the circulation integral Γ = v dx on a fixed closed contour c c for an equilibrium with v e 0 and B e 0. Clearly δv la can generate any amount of circulation. However, for a dynamically accessible variation δγ = δv da dx 53 c = s e g dx + g 4 dx B e / e c c and we can draw two conclusions: in the case where c is a closed magnetic field line dx B and δγ becomes δγ B = s e g dx = s e g g s e dx c c = g s e dx, 54 c whence we see clearly that if s e is everywhere arallel to B e, then δγ B = 0 and otherwise this is not generally true. Alternatively, suose the contour c lies within a level set of s e, for which it need not be true that B e s e = 0 along c. For this case δγ s = g 4 dx B e / e 55 c which in general does not vanish. If a magnetic field line were to lie within a surface of constant s e, then in the general case, B e s e = 0 otherwise surfaces of constant s e would be highly irregular, i.e., if B e s e 0, then B e cannot lie within a level set of s e. We oint out that similar arguments can be sulied for cases where v e 0, e.g., variation of the fluid helicity δc ω = d 3 x ω δv da for an equilibrium with B e 0 becomes δc ω = d 3 x ω e s e g + ω e g 1 56 = d 3 x ω e g s e = d 3 x g ω e s e which vanishes if ω e is erendicular to s e or if the entroy is everywhere constant. In summary, the general conclusion is that δv da, unlike δv la, is not comletely arbitrary and the degree of arbitrariness deends on the equilibrium. We also oint out that although we are here interested in erturbations away from equilibrium states, for the urose of assessing stability, the conditions we have described aly to erturbations away from any state, equilibrium or not. Now we turn to our examles. For the remainder of this aer we dro the subscrit e on equilibrium quantities, so as to avoid clutter.

7 7 III. CONVECTION For this first examle we consider thermal convection in static equilibria, both with and without a magnetic field. This examle has been well studied by various aroaches, e.g., heuristic arguments that mix Lagrangian and Eulerian ideas were given in Ref. 44 for the neutral fluid. Here our analysis will be done searately in urely Lagrangian and urely Eulerian terms, and it will illustrate the role layed by entroy in determining stability. We suose the equilibrium has stratification in the ŷ-direction due to gravity, i.e. Φ = gy, with and s deendent only on y. Thus the only equation to be solved for the neutral fluid is d dy = dφ = g. 57 dy If a magnetic field of the form B = Byˆx is suosed, then the equilbirum equation is the following: d dy = db B dy 4π dφ = JB g. 58 dy For barotroic fluids, s is constant everywhere and is eliminated from the theory, i.e., U alone. Thus, 57 together with U determines comletely the thermodynamics at all oints y by integrating d = g 59 dy giving y and consequently y. For this secial case, no further information is required. However, in the general case where, s, 57 is not sufficient and one needs to know more about the fluid, since now we have d dy + s ds = g, 60 dy which is insufficient because we have only one equation for the two unknown quantities and s. Thus, knowledge of additional hysics is required, which could come from boundary or initial conditions, solution of some heat or transort equation with constitutive relations, etc. Next consider the case of MHD where d dy = d dy + ds s = JB g. 61 dy If gravity is absent MHD differs from that of the stratified fluid because only the ressure enters and the thermodynamics of and s do not exlicitly enter the equilibrium equation. We will consider the case where gravity is resent. Thus, in general, equilibria deend on two kinds of conditions: force balance, as given in our cases of interest by 57 or 58 and thermodynamics. For latter convenience we record here several thermodynamic relations: = U and T = U s 6 c s = = U = U 63 s s = s c s = U s 64 where, without confusion, we use subscrits on U to denote artial differentiation with the other thermodynamic variable held constant and the subscrit of c s denotes sound. The coefficient of thermal exansion, α, is given by α = 1 T 65 and for tyical fluids s = α > 0 and s < If the ressure is given by = c γ exλs, then c s = γ/, as it is often written. A. Lagrangian convection 1. Lagrangian convection equilibria From 9 Lagrangian equilibria must satisfy π i = δh δq i = 0 and qi = δh = 0, 67 δπ i whence if follows from 7 that π i 0 and 0 = π i = A j 0 i a j J U + 1 q k J a l +B j 1 q i 0 a j J a l Bl 0 q k a m Bl 0B m 0 0 Φ q i, 68 which is the Lagrangian variable form of the static Eulerian equilibrium equation 6. See e.g. Refs. 1,, and 37 for further details. Because we are investigating equilibria that only deend on the variable y and have magnetic fields of the form B = Byˆx, we only consider the ŷ-comonent of 68, which is the Lagrangian variable form of the static Eulerian equilibrium equation 58.. Lagrangian convection stability The second variation of the energy about this equilibrium is the usual exression given in Ref. 9. For static equilibria this is obtained by setting η ξ in 0, and we know that the stability of such configurations is determined by this second variation of the otential energy.

8 8 We will maniulate the energy exressions to facilitate comarison with results obtained in Sec. III B. Cases with and without B = 0 are considered. Case B = 0: By exloiting the equilibrium equation we obtain δ W la = 1 1 d 3 x ξ + ξ ξ s + ξ + 1 s ξ + ξ s ξ. In conventional δw stability analyses one would consider conditions for ositivity of the above as a quadratic exression in terms of ξ. However, for our resent uroses we rewrite it in terms of δ la = ξ δs la = s ξ, 69 which with 1 s ξ s ξ = 1 ξ s s ξ δs la, yields δ W la = 1 1 d 3 x δ la + s s δ la δs la 1 ξ s s ξ δs la. 70 Now, using 64 we can rearrange this equation as δ W la = 1 d 3 x c s δ la s δs la + ξ s s ξ δs la. 71 s We will see that 71 is of the same form as that of 98 of Sec. III B, obtained via the energy-casimir functional, yet here the erturbations δ and δs are both constrained to deend on ξ according to 69. Examination of 71 reveals that ositivity of the second term is sufficient for ositivity of δ W la, viz. ξ s ξ < s 7 Given that the equilibrium only deends on the variable y, in which the systems is stratified, 7 gives the following sufficient condition for stability d/dy ds/dy < s < If the equilibrium is stably stratified, i.e., d/dy < 0, then ds/dy must be ositive and we would have a threshold involving the density and entroy scale lengths. However let us roceed further. Define = s d/dy ds/dy = s d ds 74 where in the second term of the second equality we have relaced the coordinate y by s, which is ossible if ds/dy does not vanish. Observe in the definition of of 74 this second term deends on the equilibrium rofiles, while the first term is of a thermodynamic nature. So far, the sufficient condition for stability > 0 does not account for the fact that d/dy and ds/dy are not indeendent but are related through the equilibrium equation 57. To address this we first rewrite the exression for using d dy = ds s dy + d s dy resulting in = c s = 1 c s ds s dy + c s d dy 75 d/dy ds/dy = 1 d c s ds. 76 where use has been made of 63 and 64. Now inserting 57 into 76 yields for the B = 0 case the following condition: = 1 c s g ds/dy > 0, 77 and because g > 0 we obtain the comact sufficient condition for stability ds dy > We will see that an identical condition is obtained in the Eulerian energy-casimir context see Eq Now, given that ds/dy > 0 we can use 75 to obtain a condition on d/dy, which imlies c s d dy + g = c s ds s dy < 0 d dy < g c < s Uon defining the scale height L 1 = 1 d/dy, 79 is seen to be equivalent to c s > Lg. Thus the system is stable to convection if the free fall kinetic energy is smaller than twice the kinetic energy at the sound seed. Or, equivalently, if the free fall seed through a distance L is smaller than c s.

9 9 The above rocedure leading to 78 and 79 was designed for comarison with Sec. III B. However, the conventional δw stability analysis roceeds with an extremization over ξ that takes account of any ossible stabilization effect due to the first ositive definite term of 71. To this end we let ξx = ξ x y, ξ y y, ξ z y e ikz+lx / + c.c. 80 and rewrite 71 as δ W la = 1 g dy 0 c + g d ξ y s dy + c dξ y s dy + ilξ x + ikξ z g ξ y 81 Given any ξ y y, one can choose ξ x and ξ z that make the second term vanish. Thus the smallest value of δ W la is given by δ W la = 1 0 c s g dy c + g d ξ y 8 s dy which yields 79 as a necessary and sufficient condition for stability. Thus 79 is in fact a counterart equivalent to ds/dy > 0. Another equivalent condition exists in terms of the temerature: dt dy > gt c T, 83 which follows in a manner similar to 79. Lastly, for an ideal gas, 79 and 83 become, resectively, where again all equilibrium quantities deend only on y, which we use together with 80 to rewrite this as δ W la = 1 B dy k ξ y + ξ x + dξ y 0 4π dy + ilξ x + c dξ y s dy + ilξ x + ikξ z g d dy ξ y dξy g ξ y dy + ilξ x + ikξ z, 85 where, following Ref. 49, the dislacements ξ y, ilξ x, and ikξ z can be taken to be real-valued. By minimizing this functional the following necessary and sufficient condition for interchange stability of Tserkovnikov 48, can be obtained: d dy < g c s + c < 0, 86 a where recall c a = B /4π. In Ref. 49 Newcomb rearranges 85 and minimizes it in the limit k 0 by choosing iξ z gξ y /kc s for arbitrary ξ y. With this aroach he obtains the more stringent stability condition of 79, the condition for the case without B. Newcomb s singular aroach allows dislacements that interchange lasma elements containing long segments along magnetic field lines, relieving local fluid ressures. In Ref. 50 it is shown that this amounts to the lasma being least stable against these long quasi-interchange dislacements because the restoring force due to the magnetic field tension vanishes. B. Eulerian convection d dy < g γ and dt dy > g c. 1. Eulerian convection equilibria Observe, 73 could be satisfied with ds/dy < 0 and d/dy > 0. But, the stability condition ds/dy > 0, which came from 77, imlies d/dy < 0. Thus it is not ossible to have stability unless the fluid density is stably stratified. Case B 0: The case with B 0 has been studied extensively, e.g. in the early works on interchange instability of Refs For this alication, Eq. 0 can be written as follows: δ W la = 1 d 3 x c s ξ + ξ ξ + δb + J ξ δb gη ŷ ξ, 84 4π Case B = 0: Using the Casimir invariants of 31 and 3, hydrodynamic equilibria with translational symmetry are obtained as extrema of the following energy-casimir functional: 1 F = d 3 x v + U, s + Φ + S s A s ẑ v, 87 where v z = v z s. Variation of 87 will automatically yield equations that are cases of 5 with B e 0. Because v z s, we have v s = 0 and v v z = 0. Variation with resect to v yields v = A ẑ, 88

10 10 while variation with resect to and s, resectively, yield 1 v + Φ + U + U + S = 0, 89 v z v z + U s + S A ẑ v = 0,. 90 For our case of interest with v = 0, we merely set A 0, whereuon the first variation, δf = d 3 x U + U + Φ + S δ + U s + S δs, gives rise to 91 Φ + U + U + S = 0 9 U s + S = 0, 93 where recall for our analyses we choose Φ = gy. Case B 0: For case with equilibrium magnetic field we choose the following secial case for the Casimir of 33: C s = d 3 x S s, ψ, 94 which with the Hamiltonian 1 H = d 3 x v + U + ψ 8π + gy, 95 gives uon varying F = H + C s, which imly δf δv = v = 0 δf δψ = ψ + S ψ = 0 δf δs = U s + S s = 0 δf δ = U + U + gy + S = 0 U + U + 1 ψ ψ U s s = g. 96 Equation 96 gives for our case with stratification in y the equilibrium equation 58.. Eulerian convection stability Now we examine δ F for our two cases and look for conditions that make this quantity ositive definite, conditions that will be sufficient conditions for stability. Case B = 0: The second variation is δ F = d 3 x U + U δ + U s + U s + S δ δs + U ss + S δs. 97 By exloiting the equilibrium equations, 97 can be rewritten as δ F = d 3 x c s δ s δ δs + d dy s dy ds δs, 98 where we used 6 and 64, and the derivative of the equilibrium equation U s + S = 0 with resect to y, Next, we use S d dy + U ss + U s dy ds = δ s δδs = δ s δs 100 δs s obtaining δ F = d 3 x c s δ s δs d/dy s ds/dy δs, s an exression of the form of 71. Thus, as in Sec. III A, stability is again determined by ositivity of the quantity of 74 and all of the conditions of that section are reroduced as sufficient stability conditions. In Eq. 101, unlike the case of 71, δ and δs are indeendent so a sharer sufficient condition cannot be ursued by relying on the ositivity of the first term, even though in the δ W la formulation this did not materialize. Also, the aroach here gives ds/dy > 0 as a sufficient condition for stability or equivalently 79, while the δ W la formulation shows that this condition is both necessary and sufficient Case B 0: Now consider the second variation of F = H + C s with H given by 95 and C s given by 94 with S ψ, s = K ψ + Ls,

11 11 which is general enough to describe the equilibria of our interest as given by 58. This leads to δ F = d 3 x U δ + U s + L s δ δs + U ss + L ss δs + δψ + K ψ δψ δ +K ψψ δψ. 10 Rewriting 10 in terms of equilibrium quantities and maniulating then gives c δ F = d 3 x s δp + s δs + J s c δs δψ s + K ψψ J c δψ, 103 s where use has been made of the definition of of 74, the current density J, defined by J = ψ = K ψ, 104 the thermodynamic exressions of 6 and the following, which is a consequence of the equilibrium equation, which imlies U ss + K ss = U s d ds, 105 s U ss + K ss 1 c s = U d s ds 1 c s = s d ds + s c s s = s. 106 In addition we have introduced the new variable δp defined by δp = δ + s c δs J s c δψ. 107 s Next, we collect the terms with δs to obtain δ F = d 3 c s x δp + δψ s δs J c s δψ + K ψψ J c J s s c 4 δψ. s If we introduce the variation δq = δs and we use the gradient of 104 J c δψ 109 s J = J K ψψ ψ, 110 which for equilibria that deend only on the y coordinate can be written as or K ψψ = J K ψψ = J d dψ dj dψ = dj/ dψ 111 d/ds dψ/ds dj/ds dψ/ds, 11 then the last term of Eq. 108 can be rewritten as K ψψ J c J s s c 4 s = dj/ dψ + J d c s ds. 113 Then, finally δ F = d 3 c s x δp + s δq + δψ + dj/ dψ + J d c δψ. 114 s ds From the energy exression of 114 we can immediately read off the following sufficient conditions for stability: d 0 < = ds + s c 115 s 0 < dj//dy + J d/dy dψ/dy c s ds/dy, 116 where recall the form of of 115 is equivalent to that of 74. In the case with B = 0 we had the two free functions, and s and one stability inequality. Thus we were able to obtain searate conditions on the equilibrium rofiles of and s for stability. In the resent case we again have one equilibrium equation, but now with three rofiles, s, and B and two inequalities. Again we should exect to obtain indeendent conditions on the rofiles, s, and B. However, even the condition of 79, which has clear hysical meaning, is not immediately imlementable because c s deends on y through both and s. Similarly, the inequalities 115 and 116 require the rofiles for their determination. In ractice one may construct a family of equilibria with rofiles that deend on one or more arameters and then seek thresholds in arameter sace. Inequalities 115 and 116 can be written in various ways. For examle, using the equilibrium equation 61, d d dy = c s dy ds s = g B /8π, 117 dy the inequality > 0 can be rewritten as = 1 c s d ds = g + B / c > s ds/dy

12 1 Consequently, if d/dy is negative for stability we must have ds/dy > 0 and, conversely, we must have ds/dy < 0 if, due to B decreasing sufficiently fast with height, we have d/dy > 0. This is effectively the threshold against the magnetized Rayleigh-Taylor instability. Thus, as for the case with B = 0, d/ds < 0 ensures stability. Also note, as in the B = 0 case, a critical oint arises if for some y we have d/dy = 0 unless at the same oint we also have ds/dy = 0, in which case one then has to look deeer into the limit. If d/dy < 0 and ds/dy > 0 we obtain from 115 an inequality for d/dy analogous to the inequality 79, in articular, d/dy must be negative because s /c s > 0; however, this inequality is different from the Tserkovnikov inequality of 86. If d/dy > 0 and ds/dy < 0 we obtain a reversed inequality, i.e., d/dy must be ositive. This imlies that in the inequality 116, if is ositive, the second term is always negative and thus for B > 0 we obtain the condition dj//dy < 0, or dj/dy < J/d/dy. 119 Consider the two cases of decreasing and increasing magnetic fields: for a magnetic field decreasing with height, J = db/dy > 0, so d ln J/dy < d ln /dy 10 and if d/dy < 0 we can use the inequality obtained before for d/dy and obtain an inequality that involves the second derivative of the magnetic field and the density rofile. Similarly, if J = db/dy < 0, d ln J /dy > d ln /dy 11 and if d/dy > 0 we can use the reverse inequality obtained before for d/dy and again obtain an inequality that involves the second derivative of the magnetic field and the density rofile. These cases above do not exhaust all ossibilities. It is erhas best to consider families of equilibria and investigate arameter deendencies as mentioned above. C. Dynamically accessible convection 1. Dynamically accessible convection equilibria In Sec. II C we showed how the general dynamically accessible variations of 44 47, when inserted into the first variation of the Hamiltonian 48, give rise to the general MHD equilibrium equations of 5. Thus, equilibria that are solutions of 58, with or without the magnetic field, are extremal oints of this kind of variation, and we can roceed to assess stability by examination of the energy exression of 49.. Dynamically accessible convection stability For static equilibria the first term of 49 reduces to the form of 5. As noted in Sec. II D this term vanishes if g 3 = g = g 4 0. Thus, choosing g 1 roortional to ξ, the condition for dynamically accessible stability in the case of static equilibria is determined by δ W la, viz. the Lagrangian energy exression. In both the cases with and without a magnetic field this is the usual δw energy, for each case resectively, and thus dynamically accessible stability in both cases is identical to that for Lagrangian stability. D. Convection comarisons Results for the case with equilibria B = 0 can be summarized succinctly: the Lagrangian and dynamically accessible aroaches both give the simle necessary and sufficient condition for stability, ds/dy > 0, or equivalently the inequality of 79 on d/dy, while the Eulerian energy-casimir aroach gives this same result, but only as a sufficient condition for stability and only alicable to the case with the imosed translational symmetry. For case of equilibria B 0 the situation is more comlex, although it again must be true, in light of the general discussion of Sec. II D, that the Lagrangian and dynamically accessible aroaches must give the same necessary and sufficient condition for stability, viz. that of 86. However, this necessary and sufficient condition is much simler than the inequalities of 115 and 116 obtained by the energy-casimir method and, again, these inequalities are only alicable to the case with the imosed translational symmetry and only give sufficient conditions for stability. Moreover, the energy-casimir inequalities deend on an extra derivative with resect to y of at least one of the equilibrium rofiles; e.g. 115 contains a derivative of the current J, which can be eliminated in terms of two derivatives of the ressure, but cannot easily be eliminated entirely. If one inserts the Lagrangian variations of 1, adated to the convection examle, into δ F of 10, then dj/dy is removed. In the context of our convection examle the relevant connection is rovided by δψ la = ξ ψ = ξ y ψ, with rime denoting y-differentiation. Whence, the line-bending term of 10 becomes δψ = ξ yψ + ξ y ψ = ξ yψ + ξ yψ + ξ y ξ yψ ψ 1 and one finds uon integrating the last term of 1 by arts, a term roortional to J. This term cancels the J term of δψ the same cancellation was shown to occur in the context of the magnetorotational instability in Ref. 5. As noted in Sec. II cf. Refs. 5 and such a corresondence by constraining the Eulerian variations in general connects energy-casimir and Lagrangian stability.

13 13 IV. ROTATING PINCH Now we investigate the stability of the azimuthally symmetric rotating inch, again within the Lagrangian, Eulerian energy-casimir, and dynamically accessible frameworks. This examle is chosen to illustrate two features introduced in Sec. II associated with the inclusion of an equilibrium velocity field: the relabeling transformation that removes time deendence from a Lagrangian state associated with a stationary Eulerian equilibrium and the origin of the difference between Lagrangian and dynamically accessible stability. As in Sec. III, we begin by discussing the lasma equilibrium configurations of interest by solving directly the Eulerian MHD equations 5 without referring secifically to any of the three frameworks. We use cylindrical coordinates r, ϕ, z and consider lasma equilibrium configurations where all equilibrium quantities including entroy deend only on the radial coordinate r: B = B z rẑ + B ϕ r ˆϕ, 13 v = v z rẑ + v ϕ r ˆϕ, 14 = r, s = sr, 15 B ϕ = ˆϕ ψ ẑ = dψr dr. 16 Equation 15 imlies that = e r. From Eqs. 5 we obtain the generalized Grad-Shafranov equation for the flux function ψr where 1 r d 1 M dr 4π Mr = d + B z dr 8π + d M dr 4π B ϕ = 0, 17 rb ϕ 1 ψ r 4πrv ϕ r B ϕ r 1/ is the oloidal Alfvèn Mach number. Note that v z r does not aear in 17 and in the following it will be set equal to zero. In 17 we need to assign three free functions. We will assign B z r, B ϕ r, and v ϕ r and treat 17 as an equation for r that can be written as 1 B ϕ d + B z + d dr 8π dr Bϕ + 1 M 4π 4π B ϕ r = For the sake of simlicity we will examine the case of an isothermal lasma configuration as it makes the relationshi between and linear, and also makes M linear in. A further simlification is obtained by taking the current density J z to be uniform. By defining a dimensionless radial variable r in terms of a characteristic length r 0, the latter assumtion leads to B ϕ = B 0 r and 18 becomes d dr ˆ + ˆB = 1 ˆr wr /, 19 where we have set r = c sr = ˆr B0/4π, ˆBr = B z /B 0, and M r = 4π/B0 vϕ/rc s = ˆr w r, with ˆB being the dimensionless magnetic field, ˆ the dimensionless ressure, wr = v ϕ /rc s the dimensionless rotation rate, and c s the sound velocity in the isothermal case. For a configuration where B z is uniform and the lasma rotation is rigid with rotation frequency Ω, Eq. 19 takes the elementary form d ˆr d r = 1 ˆr w /, 130 where w = Ωr 0 /c s. While a uniform B z field does not alter these equilibrium configurations, it will be shown to affect their stability. Assuming w / < 1 we obtain ˆr = w 1 1 w w r ex, 131 where ˆ0 = 1, ˆ r = 0 for r = /w ln 1 w /. Equation 131 describes a one-arameter family of equilibria. In the absence of rotation this configuration reduces to the standard arabolic inch with r = 1 and ˆr = 1 r, while for w we have r = and ˆr 1. A. Lagrangian inch 1. Lagrangian inch equilibria For the rotating inch the aroriate Hamiltonian is that of 7 with Φ 0 and, as before, the inch equilibrium equations should follow from Eqs. 9 adated to the inch geometry. In articular, with the cylindrical coordinate system with indices i, j {r, ϕ, z}, a = a r, a ϕ, a z, q = q r, q ϕ, q z, with π = g ij q π i π j = π i π i, and From 13 we obtain g ij q = q r π r = g rr π r = π r π ϕ = g ϕϕ π ϕ = π ϕ /q r, π z = g zz π z = π z,

14 14 and similarly B r = B r, B ϕ = B ϕ /q r, and B z = B z. As shown in Aendix A, the equations of motion in terms of the Lagrangian variables q r, q ϕ, q z follow from Eqs. 9, and are q i = g ij π j and π i = δi r π ϕ π ϕ q r J 0 0 q i U 134 J δi r J B ϕ B ϕ q r 4π + J Bi Bj q j J B 4π q i. 8π Transforming 133 and 134 to Eulerian variables, we first obtain the intermediate form 0 q r = g rr π r = π r 0 q ϕ = g ϕϕ π ϕ = π ϕ /r and π r = π ϕ r 3 J + B 0 8π π ϕ = rj + B 8π π z = J + B 8π from which, using 0 q z = g zz π z = π z 135 ˆr + J B B 4π ˆϕ + rj B B 4π ẑ + J B B 4π D π r = 0 Dt v r q, t, D π ϕ = 0 q r v ϕ q, t, Dt D π z = 0 Dt v z q, t, ˆr, 136 ˆϕ, 137 ẑ, 138 with D/Dt = t + q i / q i and qa, t = vx, t, we recover the cylindrical comonents of the Eulerian equation of motion, v t + v v = + B + B B π 4π The rotating inch equilibrium configuration of this section corresonds to π r = π z = π ϕ = 0, q r = q z = 0, q ϕ = Ω, 140 with 0 b r = r/c s where r is given by 131. Because q ϕ = Ωt+a ϕ, we see exlicitly that stationary Eulerian equilibria corresond to time-deendent Lagrangian trajectories. Next, we consider the relabeling transformation introduced in Ref. 5 and described in Sec. II, a = U b, t b = B a, t where a = A b, t is given by a r = b r, a ϕ = b ϕ Ω b r t, a z = b z V z b r t 141 and b = B a, t is given by b r = a r, b ϕ = a ϕ + Ω a r t, b z = a z + V z a r t, 14 with J := a i / b j = 1, with given by V b, t := Ḃ B 1 = Ḃ A b, t, t V r = 0, V ϕ = Ω b r, V z = V z b r. 143 By inserting 143 into the transformed Hamiltonian of 1 see Aendix B we obtain the time-relabeled equations of motion corresonding to 133 and 134 see B3 and B4. Then in the relabeled variables by exlicitly setting / t = 0, Q i = b i and by assigning the functions B0 i and 0 U as functions of b i consistently with the choices made in Sec. IV, these equations yield the equilibrium equations in the relabeled form of B6 B9. Thus, we have shown that the equilibrium equation of 19 describes the reference state Q e, Π e that follows from δ H δπ = 0 and δ H δq = Given that our equilibrium corresonds to the vanishing of the first variation of the Hamiltonian H of 1, we can exand as in 17 to address stability via the energy rincile described in Sec. IV A.. Lagrangian inch stability Now, to address stability we exand H by inserting 17 see also Eq. 7 of Ref. 5, where the reference state is our inch equilibrium of Sec. IV A 1. This leads to the second variation of the Hamiltonian H written in terms of the canonically conjugate variables η, π η as given by 18 with δ W la η defined by 19 with 0. Due to the arbitrariness of π η we can make the first term of 18 vanish, so that a sufficient stability condition for the configuration 14 is given by δ W la η > 0. We will roceed further by minimizing δ W la for our inch examle. In order to be able to comare the Lagrangian stability conditions with those obtained in the Energy-Casimir framework we restrict our analysis to erturbations η that do not deend on z. Working out terms of 19 with 0 for our examle,

15 15 we obtain in cylindrical curvilinear coordinates v ϕ v ϕ η η = v ϕ/r η r r η r η ϕ /r ϕ η r η ϕ/r, v ϕ η = v ϕ /r ϕ η r η ϕ ϕ η ϕ + η r + ϕ η z, / η = c s /r r rη r ϕ η ϕ, where in 150 the isothermal equation of state / = = c s has been used, η r r η = η r /r r r rη r + ϕ η ϕ, 148 η B /4π = B 0/4π r rη r + ϕ η r +B 0/4π ϕ η z ˆB/r r rη r + ϕ η ϕ, 149 J η δb = B 0/π η r r rη r η ϕ ϕ η r, 150 where the restriction that ˆB = B z /B 0 and J z be indeendent of r has been used in accordance with the derivation in Sec. IV. In the above we used the notation r := / r etc., which we use throughout the resent section. In the following we will refer exlicitly to the rigid rotation equilibrium given by 131 and adot the dimensionless variables used there. Also, we suose η eximϕ and consider azimuthally symmetric m = 0 and azimuthally asymmetric m 0 erturbations searately. Case m = 0: If ϕ = 0 the functional δ W la deends only on the radial comonent η r and its radial derivative: δ W la η =π rdr w ˆ η r r rη r + ˆ/r r rη r +η r /r r ˆ r rη r + r rη r 1 + ˆB/r η r r rη r ; 151 then using the equilibrium 17 this reduces to δ W la η = π rdr 4η r r rη r 15 + ˆ + r + ˆB /r r rη r. The first term of 15 is a divergence and vanishes by integration with the roer boundary conditions, while the second term is ositive definite. Thus we conclude our inch equilibrium is stable to azimuthally symmetric erturbations. Case m 0: In this case, besides η r and η ϕ, the functional δ W la deends also on η z if ˆB 0. We use the orthogonality of the different m-comonents and consider the m th comonent. The resulting exressions, as obtained from , are given in Aendix C. Case B z = 0: If ˆB = 0 the dislacement η z along the symmetry axis of the erturbation decoules, and minimization with resect to η z gives η z = 0, rovided m 1 w ˆ > 0 w < Combining C1 C5 and using 130 we can write the integrand of the functional δ W la in the following matrix form: η ϕ ηr η ϕ, ηr, r rηr W, 154 r rη r where W is the 3x3 matrix given by W = m ˆ ς/r imˆw imˆ/r imˆw m ϖ 0. imˆ/r ˆ/r where for convenience we have defined ϖ := 1 w ˆ and ς := 1 w r. 155 Then, to ascertain stability we use Sylvester s criterion on the matrix W. This criterion states that a necessary and sufficient criterion for the ositive definiteness of a Hermitian matrix is that the leading rincial minors be ositive. The first rincial minor of W is seen to be ositive if 1 w r > 0, i.e., w < 1 ex 1/, 156 while the second rincial minor of W is ositive if for m = 1 which is the worst case which imlies ˆ 1 w r 1 ˆw r ˆ w 4 > 0, 157 w 1 r <, ˆ 1 r and coincides with the condition given by 156. Finally, the determinant of W is ositive for the worst case m = 1 if r + ˆ 1 w r 1 ˆw 159 which imlies ˆ 1 ˆw ˆr w 4 r + ˆ > 0, w < 1 r + ˆ, 160 and yields the stronger condition w < 1/. Alternatively we can first minimize δ W la with resect to η ϕ in order to to obtain a quadratic form involving