Bounds for the number of degrees of freedom of incompressible magnetohydrodynamic turbulence in two and three dimensions

We study incompressible magnetohydrodynamic turbulence in both two and three dimensions, with an emphasis on the number of degrees of freedom N. This number is estimated in terms of the magnetic Prandtl number Pr, kinetic Reynolds number Re, and magnetic Reynolds number Rm. Here Re and Rm are dynamic in nature, defined in terms of the kinetic and magnetic energy dissipation rates (or averages of the velocity and magnetic field gradients), viscosity and magnetic diffusivity, and the system size. It is found that for the two-dimensional case, N satisfies N <= Pr Re-3/2 + Rm(3/2) for Pr > 1 and N <= Re-3/2 + Pr-1 Rm(3/2) for Pr <= 1. In three dimensions, on the other hand, N satisfiesN <= (Pr Re-3/2 + Rm(3/2))(3/2) for Pr > 1 and N <= (Re-3/2 + Pr-1 Rm(3/2))(3/2) for Pr <= 1. In the limit Pr -> 0, Re-3/2 dominates Pr-1 Rm(3/2), and the present estimate for N appropriately reduces to Re-9/4 as in the case of usual Navier-Stokes turbulence. For Pr approximate to 1, our results imply the classical spectral scaling of the energy inertial range and dissipation wave number (in the form of upper bounds). These bounds are consistent with the existing predictions in the literature for turbulence with or without Alfven wave effects. We discuss the possibility of solution regularity, with an emphasis on the two-dimensional case in the absence of either one or both of the dissipation terms.