Galerkin approximations for dissipative magnetohydrodynamics

A Galerkin approximation scheme is proposed for voltage-driven, dissipative magnetohydrodynamics. The trial functions are exact eigenfunctions of the linearized continuum equations and represent helical deformations of the axisymmetric, zero-flow, driven steady state. In this paper, the lowest nontrivial truncation is explored: one axisymmetric trial function and one helical trial function each for the magnetic and velocity fields. The system resembles the Lorenz approximation to Bénard convection, but in the region of believed applicability its dynamical behavior is rather different, including relaxation to a helically deformed state similar to those that have emerged in the much higher resolution computations of Dahlburg et al. [Phys. Rev. Lett. 57, 428 (1986); J. Plasma Phys. 37, 299 (1987); 40, 39 (1988)]. In the region of applicability, the dynamical behavior is to seek out the steady state of lowest energy dissipation rate.