A Turbulent Constitutive Law for the Two-Dimensional Inverse Energy Cascade

The inverse energy cascade of two-dimensional (2D) 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 2D 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. 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 2D. 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 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 2D between the mean rates of differential strain-rotation and of vorticity-gradient stretching, analogous to a similar relation of Betchov for 3D. 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.