LARGE-SCALE KOLMOGOROV FLOW ON THE BETA-PLANE AND RESONANT WAVE INTERACTIONS

The large-scale dynamics of the Kolmogorov flow near its threshold of instability is studied in the presence of the beta-effect (Rossby wave s). The governing equation, obtained by a multiscale technique, fails the Painleve test of integrability when beta not equal 0. This ''beta- Cahn-Hilliard'' equation with cubic nonlinearity is simulated numerica lly in various regimes, The dispersive action of the waves modifies th e inverse cascade associated with the Kolmogorov flow (She, Phys, Lett , A 124 (1987) 161). For small values of beta the inverse cascade is i nterrupted at a wavenumber which increases with beta, For large values of beta only resonant wave interactions (RWI) survive, An original ap proach to RWI is developed, based on a reduction to normal form, of th e sort used in celestial mechanics. Otherwise, wavenumber discreteness effects, which are dramatic in the present case, are not captured. (T he method is extendable to arbitrary RWI problems.) The only four-wave resonances present involve two pairs of opposite wavenumbers. This al lows leading-order decoupling of moduli and phases of the various Four ier modes, so that an exact kinetic equation is obtained for the energ ies of the modes. It has a Lyapunov (gradient Bow) functional formulat ion and multiple attracting steady-states, each with a single mode exc ited.