Scaling laws in magnetohydrodynamic turbulence

Abstract

We analyze the decay laws of the kinetic and magnetic energies and the evolution of correlation lengths in freely decaying incompressible magnetohydrodynamic (MHD) turbulence. Scale invariance of MHD equations assures that, in the case of constant dissipation parameters (i.e., kinematic viscosity and resistivity) and null magnetic helicity, the kinetic and magnetic energies decay in time as $E\ensuremath{\sim}{t}^{\ensuremath{-}1}$, and the correlation lengths evolve as $\ensuremath{\xi}\ensuremath{\sim}{t}^{1/2}$. In the helical case, assuming that the magnetic field evolves towards a force-free state, we show that (in the limit of large magnetic Reynolds number) the magnetic helicity remains constant, and the kinetic and magnetic energies decay as ${E}_{v}\ensuremath{\sim}{t}^{\ensuremath{-}1}$ and ${E}_{B}\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ respectively, while both the kinetic and magnetic correlation lengths grow as $\ensuremath{\xi}\ensuremath{\sim}{t}^{1/2}$.