Action principles and conservation laws for Chew–Goldberger–Low anisotropic plasmas

Abstract

The ideal Chew–Goldberger–Low (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–Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux${\boldsymbol {M}}$, the density$\rho$, the entropy variable$\sigma =\rho S$and the magnetic induction${\boldsymbol {B}}$. Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, centre of mass and angular momentum. Cross-helicity conservation arises from a fluid relabelling symmetry, and is local or non-local depending on whether the gradient of$S$is perpendicular to${\boldsymbol {B}}$or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.