Action Principles and Conservation Laws for Chew-Goldberger-Low Anisotropic Plasmas

Abstract

The ideal CGL plasma equations, including the double adiabatic conservation laws for the parallel ($p_\parallel$) and perpendicular pressure ($p_\perp$), are investigated using a Lagrangian variational principle. An Euler-Poincar\'e variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux ${\bf M}$, the density $\rho$, three entropy variables, $\sigma=\rho S$, $\sigma_\parallel=\rho S_\parallel$, $\sigma_\perp=\rho S_\perp$ ($S_\parallel$ and $S_\perp$ are the two scalar entropy invariants), and the magnetic induction ${\bf B}$. Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, center of mass, and angular momentum. Cross helicity conservation arises from a fluid relabeling symmetry, and is local or nonlocal depending on whether the entropy gradients of $S_\parallel$, $S_\perp$ and $S$ are perpendicular to ${\bf B}$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.