The long-time evolution of mean field magnetohydrodynamics

The equations of mean field magnetohydrodynamics with constant mean velocity are proved to posses solutions bounded in the $H^{1}$-norm for all time, and a compact attractor whose dimension is estimated. It is shown that depending on the functional form of the so-called alpha term the attractor may reduce to zero or be a larger set. If, as usual in physical situations, there exists a set of solutions with a minimum size $N$, the dimension of this set decreases rapidly with increasing $N$. Finally, the dependence of the dimension on the magnetic diffusivity is analyzed, suggesting that the evolution of a magnetic field under the mean field equation is much more restricted than the one deduced from the full magnetohydrodynamic system.