Broken ergodicity in two-dimensional homogeneous magnetohydrodynamic turbulence

Two-dimensional (2-D) homogeneous magnetohydrodynamic (MHD) turbulence has many of the same qualitative features as three-dimensional (3-D) homogeneous MHD turbulence. These features include several ideal invariants, along with the phenomenon of broken ergodicity. Broken ergodicity appears when certain modes act like random variables with mean values that are large compared to their standard deviations, indicating a coherent structure or dynamo. Recently, the origin of broken ergodicity in 3-D MHD turbulence that is manifest in the lowest wavenumbers was explained. Here, a detailed description of the origins of broken ergodicity in 2-D MHD turbulence is presented. It will be seen that broken ergodicity in ideal 2-D MHD turbulence can be manifest in the lowest wavenumbers of a finite numerical model for certain initial conditions or in the highest wavenumbers for another set of initial conditions. The origins of broken ergodicity in ideal 2-D homogeneous MHD turbulence are found through an eigenanalysis of the covariance matrices of the modal probability density functions. It will also be shown that when the lowest wavenumber magnetic field becomes quasi-stationary, the higher wavenumber modes can propagate as Alfvén waves on these almost static large-scale magnetic structures. PACS numbers: 05.20.Jj, 47.27.Gs, 91.25.Cw, 95.30.Qd