Relaxation processes in a low-order three-dimensional magnetohydrodynamics model

Abstract

The selective decay and dynamic alignment relaxation theories are used to interpret the time asymptotic behavior of a Galerkin model of three-dimensional (3-D) magnetohydrodynamics (MHD). A large number of simulations are performed that scan a parameter space defined by the rugged ideal invariants: energy, cross helicity, and magnetic helicity. Ranges of the initial parameters are found where one or both of the relaxation theories are needed to describe the time asymptotic properties of the system, as previously found in analogous studies of two-dimensional (2-D) MHD [Ting et al., Phys. Fluids 29, 3261 (1986)]. In many cases, the time asymptotic state can be interpreted as a relaxation to minimum energy. For certain parameter ranges spectral back transfer of cross helicity can lead to growth in velocity-magnetic field correlation [Stribling and Matthaeus, Phys. Fluids B 2, 1979 (1990)]. A simple decay model, based on absolute equilibrium theory, predicts a mapping of initial onto time asymptotic states, and accurately describes the long time behavior of the runs when magnetic helicity is present. We also discuss two processes, operating on time scales shorter than selective decay and dynamic alignment, in which the ratio of kinetic to magnetic energy relaxes to values O(1). The faster of the two takes states initially dominant in magnetic energy to a state of near-equipartition between kinetic and magnetic energy through power law growth of kinetic energy. The other process takes states initially dominant in kinetic energy to the near-equipartitioned state through exponential growth of magnetic energy.