On the Necessity of the Energy Principle of Kruskal and Oberman for Stability

Abstract

Kruskal and Oberman have developed an energy principle for stability of hydromagnetic systems in which it is assumed that the particles move adiabatically in the magnetic field. They established the sufficiency of the energy principle but left the necessity an open question. Newcomb has established the necessity of a slightly different principle which takes into account charge neutrality. He shows that the stability equations may be written in a self‐adjoint form. A simpler proof is now given which is based on the conservation of energy so that it is not necessary to establish the self‐adjointness by an explicit calculation. The proof is the direct analog of that given by Bernstein, Frieman, Kruskal, and Kulsrud on an energy principle for stability from the fluid equations. In both the proof first given by Newcomb, and the present one, the assumption is made that the geometry is not toroidal, but that the particles are trapped in a magnetic field as far as their longitudinal motion is concerned.