The homogeneous turbulent dynamo

Abstract

Ideal, homogeneous, magnetohydrodynamic turbulence is represented by finite Fourier series whose coefficients form a canonical ensemble. Here, the relevant statistical theory is substantially extended. This includes finding eigenvalues and eigenvectors of the covariance matrix for each modal probability density. The eigenvectors allow for a special unitary transformation of phase space coordinates into a set of eigenvariables. The smallest eigenvalues occur at the lowest wavenumber and are associated with three dominant eigenvectors. The lowest wavenumber eigenvariables, in statistical equilibrium, are seen to have large mean values containing significant energy and thus define a homogeneous turbulent dynamo. These large mean values arise because the symmetry of phase space is dynamically broken. Nonzero viscosity and magnetic diffusivity are expected to have minimal effect, since this coherent structure exists at the lowest wavenumber, where dissipation is least.