Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence

A comprehensive picture of three-dimensional (3D) isotropic magnetohydrodynamic (MHD) turbulence is presented based on the first ${512}^3$ -mode numerical simulations performed. Both temporal and spatial scaling properties are studied. For finite magnetic helicity H the energy decay is governed by the constancy of H and the decrease of the ratio of kinetic and magnetic energy ${\mathrm{\Gamma =E}}^K{\mathrm{/E}}^M.$ A simple model consistent with a series of simulation runs predicts the asymptotic decay laws ${\mathrm{E\sim t}}^{\text{−1/2}},$ $E^K{\mathrm{\sim t}}^{\text{−1}}.$ For nonhelical MHD turbulence, $\text{H≃0,}$ the energy decays faster, ${\mathrm{E\sim t}}^{\text{−1}}.$ The energy spectrum follows a $k^{\text{−5/3}}$ law, clearly steeper than $k^{\text{−3/2}}$ previously found in 2D MHD turbulence. The scaling exponents of the structure functions are consistent with a modified She–Leveque model ${\zeta }_p^{\mathrm{MHD}}{\text{=p/9+1−(1/3)}}^{\text{p/3}},$ which corresponds to a basic Kolmogorov scaling and sheet-like dissipative structures. The difference between the 3D and the 2D behavior can be related to the eddy dynamics in 3D and 2D hydrodynamic turbulence.