Asymptotic Law of Decay of Homogeneous Magnetoturbulence

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Abstract
Solutions of the magnetohydrodynamic equations for the spectral energy tensors are obtained by assuming that the turbulent field is homogeneous and weak enough for triple correlations to be negligible. The total turbulent magnetic and kinetic energies are derived by integrating over all wave number space. The asymptotic law of decay (for times larger than the ``inhibition time'') is obtained in an analytical form by the method of steepest descent, and it is shown that there are two important contributions to the total energy, one which decays as t−5/2, as in ordinary turbulence and corresponds to equipartition between magnetic and kinetic modes, and one which decays as t−3 and leads to the partition of energy between the two modes in the inverse ratio of their respective diffusivities. In the course of the decay, the latter becomes negligible, but the time at which the t−5/2 component dominates depends upon the ratio n = ν/λ of the two diffusivities and is found to fall beyond any reasonable time of observation if n is much less or much larger than one; in which cases, equipartition is not approached before the turbulence is damped out or has attained such a large scale that, in experimental situations, the limit of validity of the homogeneity hypothesis is reached.