Zonal flows in two-dimensional decaying magnetohydrodynamic turbulence on a $\beta$-plane

A scale for two-dimensional $\beta$-plane magnetohydrodynamic turbulence is proposed that characterizes an upper bound of the energy containing interval similar to a classical Rhines scale for neutral fluid turbulence on a $\beta$-plane. It is found that only unsteady zonal flows with complex temporal dynamics are formed in two-dimensional magnetohydrodynamic turbulence on the $\beta$-plane. It is shown that flow nonstationarity is due to the appearance of isotropic magnetic islands caused by the Lorentz force in the system. The characteristic dimensions of a flow are in agreement with the proposed scale.