Large-scale Kolmogorov flow on the beta-plane and resonant wave interactions

Abstract

The large-scale dynamics of the Kolmogorov flow near its threshold of instability is studied in the presence of the β-effect (Rossby waves). The governing equation, obtained by a multiscale technique, fails the Painlevé test of integrability when β ≠ 0. This “β-Cahn-Hilliard” equation with cubic nonlinearity is simulated numerically in various régimes. The dispersive action of the waves modifies the inverse cascade associated with the Kolmogorov flow (She, Phys. Lett. A 124 (1987) 161). For small values of β the inverse cascade is interrupted at a wavenumber which increases with β. For large values of β only resonant wave interactions (RWI) survive. An original approach to RWI is developed, based on a reduction to normal form, of the sort used in celestial mechanics. Otherwise, wavenumber discreteness effects, which are dramatic in the present case, are not captured. (The method is extendable to arbitrary RWI problems.) The only four-wave resonances present involve two pairs of opposite wavenumbers. This allows leading-order decoupling of moduli and phases of the various Fourier modes, so that an exact kinetic equation is obtained for the energies of the modes. It has a Lyapunov (gradient flow) functional formulation and multiple attracting steady-states, each with a single mode excited.