A turbulent constitutive law for the two-dimensional inverse energy cascade

Abstract

The inverse energy cascade of two-dimensional turbulence is often represented phenomenologically by a Newtonian stress–strain relation with a ‘negative eddy viscosity’. Here we develop a fundamental approach to a turbulent constitutive law for the two-dimensional inverse cascade, based upon a convergent multi-scale gradient (MSG) expansion. To first order in gradients, we find that the turbulent stress generated by small-scale eddies is proportional not to strain but instead to ‘skew-strain,’ i.e. the strain tensor rotated by $45^\circ$. The skew-strain from a given scale of motion makes no contribution to energy flux across eddies at that scale, so that the inverse cascade cannot be strongly scale-local. We show that this conclusion extends a result of Kraichnan for spectral transfer and is due to absence of vortex stretching in two dimensions. This ‘weakly local’ mechanism of inverse cascade requires a relative rotation between the principal directions of strain at different scales and we argue for this using both the dynamical equations of motion and also a heuristic model of ‘thinning’ of small-scale vortices by an imposed large-scale strain. Carrying out our expansion to second order in gradients, we find two additional terms in the stress that can contribute to the energy cascade. The first is a Newtonian stress with an ‘eddy-viscosity’ due to differential strain rotation, and the second is a tensile stress exerted along vorticity contour lines. The latter was anticipated by Kraichnan for a very special model situation of small-scale vortex wave-packets in a uniform strain field. We prove a proportionality in two dimensions between the mean rates of differential strain rotation and of vorticity-gradient stretching, analogous to a similar relation of Betchov for three dimensions. According to this result, the second-order stresses will also contribute to inverse cascade when, as is plausible, vorticity contour lines lengthen, on average, by turbulent advection.