The role of angular momentum in the magnetic damping of turbulence

Abstract

Landau & Lifshitz showed that Kolmogorov's E∼t^−10/7 law for the decay of isotropic turbulence rests on just two physical ideas: (a) the conservation of angular momentum, as expressed by Loitsyansky's integral; and (b) the removal of energy from the large scales via the energy cascade. Both Kolmogorov's original analysis and Landau & Lifshitz's reinterpretation in terms of angular momentum are now known to be flawed. The existence of long-range velocity correlations means that Loitsyansky's integral is not an exact representation of angular momentum, nor is it strictly conserved. However, in practice the long-range velocity correlations are weak and Loitsyansky's integral is almost constant, so that the Kolmogorov/Landau model provides a surprisingly simple and robust description of the decay. In this paper we redevelop these ideas in the context of MHD turbulence. We take advantage of the fact that the angular momentum of a fluid moving in a uniform magnetic field has particularly simple properties. Specifically, the component parallel to the magnetic field is conserved while the normal components decay exponentially on a time scale of τ=ρ/σB². We show that the counterpart of Loitsyansky's integral for MHD turbulence is ∫x²_⊥Q_⊥dx, where Q_ij is the velocity correlation. When the long-range correlations are weak this integral is conserved. This provides an estimate of the rate of decay of energy. At low values of magnetic field we recover Kolmogorov's law. At high values we find E∼t^−1/2, which is a result derived earlier by Moffatt. We also show that ∫x²_⊥Q_∥dx decays exponentially on a time scale of τ. We interpret these results in terms of the behaviour of isolated vortices orientated normal and parallel to the magnetic field.