Dispersive and friction-induced stabilization of an inverse cascade. The theory for the Kolmogorov flow in the slightly supercritical regime

Abstract

We discuss the stabilisation of the inverse cascade in the large scale instability of the Kolmogorov flow described by the complete Cahn-Hilliard equation with inclusion of $\beta$ effect, large-scale friction and deformation radius. The friction and the $\beta$values halting the inverse cascade at the various possible intermediate states are calculated by means of singular perturbation techniques and compared to the values resulting from numerical simulation of the complete Cahn-Hilliard equation. The excellent agreement validates the theory. Our main result is that the critical values of friction or $\beta$ halting the inverse cascade scale exponentially as a function of the jet separation in the final flow, contrary to previous theories and phenomenological approach.