On the stability of MHD equilibria with flow

Three kinds of energy principles arising from the Hamiltonian structure of the (MHD) equations are used to determine sufficient stability conditions. The Lagrangian energy principle of Ref.[1] is presented and the stability conditions for symmetric and non-symmetric perturbations are introduced. Exploiting the noncanonical Hamiltonian formulation of MHD [2] plasma flows are analyzed in terms of Eulerian variables. An energy principle in Eulerian form is deduced for equilibria with a geometric symmetry and sufficient conditions for stability are obtained by expanding a functional F composed of the sum of the Eulerian energy plus Casimir invariants to second order. Next, an energy principle based on dynamically accessible variations [3] that preserve the invariants of the system explicitly is considered. Dynamically accessible variations do not rely on any symmetry and thus give general criteria for stability. Finally, the conditions obtained from the three different approaches are compared and implications about nonlinear stability are discussed.\\[4pt] [1] E.A. Frieman and M. Rotenberg, Rev. Mod. Phys., 32 898 (1960).\\[0pt] [2] P.J. Morrison and J.M.~Greene, Phys. Rev. Lett., 45 790 (1980).\\[0pt] [3] P.J. Morrison, Rev. Mod. Phys., 70 467 (1998).