Formal stability in Hamiltonian fluid models for plasmas

Abstract

We review the progress made, during the last decade, on the analysis of formal stability for Hamiltonian fluid models for plasmas, carried out by means of the Energy-Casimir (EC) method. The review begins with a tutorial Section describing the essential concepts on the Hamiltonian formalism for fluid models and on the EC method, which will be frequently used in the article. Subsequently, a nonlinear stability analysis applied to reduced magnetohydrodynamics (MHD) is described, as paradigmatic example for the application of the EC method. The review of the recent results begins with the equilibrium and formal stability analysis of MHD with general helical symmetry, followed by the treatment of extended MHD. Applications of the EC method to a hybrid MHD-Vlasov model with pressure coupling and to a reduced fluid model accounting for electron temperature anisotropy are described next. The formal stability analysis of compressible reduced MHD is then presented and used to show the connection between the EC method and the classical δW method for MHD stability. The concept of negative energy mode (NEM) is also briefly reviewed and applied to a model for electron temperature gradient (ETG) instability. In the context of the search for equilibria by a variational procedure, which is part of the EC method, we discuss a recent interpretation of the classical tearing modes in terms of singular equilibria of MHD linearized about Beltrami equilibria. Finally, we mention some possible directions for future developments.

Nonlinear feedback, double bracket dissipation and port control of Lie-Poisson systems

“…Moreover, the coordinate independence of the geometric formulation allows to apply Theorem 2.13 to the infinite dimensional problem of feedback stabilization of Hall MHD flow (Section 4). This system has been treated previously neither from the CL nor from the IDA-PBC point of view, although there are many works concerned with feedback control and (linear) stability of (Hall) MHD flow (e.g., [14,13,32,30,29]). The results of this paper are restricted to Lie-Poisson systems defined on direct product Lie algebras.…”