Variational Principle of Hall Magnetohydrodynamics

Abstract

The Hall current, which is written by a higher order derivative term, appears as a singular perturbation term to the magnetohydrodynamics (MHD) equations. The Hall MHD system has three invariants, the energy, the magnetic (electron) and ion helicities. The ion helicity is known to be “fragile” with respect to the energy norm of the magnetic and flow fields. In a sense of selective dissipation, the ion helicity may dissipate faster than the energy. Therefore a variational principle that gives minimumenergy state under two helicities constraints becomes an ill-posed problem. On the other hand, studying stability of a shear flow system, its non-Hermitian property invalidates the standard normal-mode analysis or energy principle. The Lyapunov stability analysis (Arnold method) is an effective way to that system. In this analysis, convexity (or coerciveness) of a functional, a linear combination of invariants, plays an important role. However the functional of Hall MHD is not a convex form. It is studied how thedifficulties appear in the variational principle of minimum energy state and Lyapunov stability analysis in the Hall MHD system. In both cases the difficulties stem from the fact that the highest order derivative term in the functional is not positive definite.