Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas

Abstract

Dedicated to Professor Harold Weitzner on the occasion of his retirement “Say to wisdom ‘you are my sister,’ and to insight ‘you are my relative.’”—Proverbs 7:4 Based on an extension to plasmas of Ertel's classical vorticity theorem in fluid dynamics, it is shown that for each species in a multi-fluid plasma there can be constructed a set of nested surfaces that have this species' fluid particles confined within them. Variational formulations for the plasma evolution and its equilibrium states are developed, based on the new surfaces and all of the dynamical conservation laws associated with them. It is shown that in the general equilibrium case, the energy principle lacks a minimum and cannot be used as a stability criterion. A limit of the variational integral yields the two-fluid Hall-magnetohydrodynamic (MHD) model. A further special limit yields MHD equilibria and can be used to approximate the equilibrium state of a Hall-MHD plasma in a perturbative way.