Plasma relaxation and the turbulent dynamo

Abstract

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series whose independent coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. When the eigenvalues at lowest wave number are very small, a large-scale coherent structure appears: a turbulent dynamo, which is seen in computations. A theoretical explanation is given and contains Taylor’s theory of force-free states. Numerical effects are examined and it is shown that larger grid sizes and smaller time steps provide for better resolution of coherent structure. Ideal hydrodynamic (HD) turbulence is examined and the results are compared and contrasted with those of ideal MHD turbulence. In particular, coherent structure appears in ideal MHD turbulence at the lowest wave number, but can occur in ideal HD turbulence only at the highest wave numbers in a simulation. In the case of real, i.e., dissipative flows, coherent structure and broken ergodicity are expected to occur in MHD turbulence at the largest scale. However, real HD turbulence at all scales and real MHD turbulence at all scales but the largest are expected to be ergodic.