The role of angular momentum conservation in homogeneous turbulence

Abstract

Loitsyanky's integral I = − ∫ r2〈u ⋅u′〉dr is known to be approximately conserved in certain types of fully developed, isotropic turbulence, and its near conservation controls the rate of decay of kinetic energy. Landau suggested that this integral is related to the angular momentum H = ∫ (x × u)dV of some large volume V of the turbulence, according to the expression I = 〈H2〉/V. He also suggested that the approximate conservation of I is related to the principle of conservation of angular momentum. However, Landau's analysis can be criticized because, formally, it applies only to inhomogeneous turbulence evolving in a closed domain. So how are we to interpret the near conservation of I? And what is its relationship, if any, to angular momentum conservation? We show that the key to extending Landau's analysis to strictly homogeneous turbulence is to rewrite Loitsyansky's integral in terms of the vector potential of the velocity field, i.e. I = 6 ∫〈A ⋅ A′〉dr, where ∇ × A = u. This yields I = 6〈[∫VAdV]2〉/V for any large spherical volume V of radius R. Crucially, J = 3∫VAdV can be rewritten as the weighted integral of the angular momentum density throughout all space. This fundamentally changes the way in which we interpret the dynamical behaviour of I. For example, we show that the conservation of 〈J2〉/V, and hence of I, which occurs when the long-range correlations are weak, is a direct consequence of the decorrelation of the flux of angular momentum out through a spherical control surface S and the local angular momentum in the vicinity of S. Thus, within the framework of strictly homogeneous turbulence, we provide the first self-consistent interpretation of Loitsyanky's integral in terms of angular momentum conservation. We also show that essentially the same ideas carry over to certain types of anisotropic turbulence, such as magnetohydrodynamic (MHD), rotating and stratified turbulence. This is important because conservation of angular momentum, which manifests itself in the form of a Loitsyansky-like invariant, places a fundamental restriction on the way in which the integral scales can evolve in such turbulence. This, in turn, controls the rate of decay of energy. We illustrate this by deriving new decay laws for MHD and stratified turbulence. The MHD decay laws are consistent with the available numerical evidence, but further study is required to verify, or otherwise, the predictions for stratified turbulence.