Inverse cascade and rossby waves in the kolmogorov flow on the beta-plane

Large-scale geophysical flows are subject to the competing effects of quasi-two-dimensional turbulence and Rossby waves. It is well known from phenomenological arguments and numerical simulations that the inverse cascade which characterizes the large-scale dynamics of two-dimensional turbulence can be halted by Rossby wave dispersion. A particular example for which this effect is amenable to a detailed numerical and theoretical understanding is the supercritical large-scale dynamics of the Kolmogorov flow on the β-plane. This flow is governed by the one-dimensional “β-Cahn—Hillird” equation [1], obtained by multiscale technique, with cubic non linearity ∂tυ(x,t)=∂((λ1υ2−λ2)∂xυ)−λ3∂4xυ−β∂−1xυ∂tυ(x,t)=∂((λ1υ2−λ2)∂xυ)−λ3∂x4υ−β∂x−1υ In the absence of β, the solutions to this equation live essentially within a slow manifold of soliton-like solutions characterized by alternating kinks and antikinks. With periodic boundary conditions of period L, fixed points are obtained with N pairs of regularly spaced kinks and antikinks. These fixed points are unstable saddle points of a Lyapunov functional excepted for the gravest mode N = 1 which is a stable absolute minimum. The temporal evolution is a succession of annihilations of kinks and antikinks, leading eventually to the gravest mode (see, e.g., Ref. [2]). The uniqueness of the final solution is ensured by the existence of a Lyapunov functional.