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 Painleve test of integrability when β ≠ 0. This “β-Cahn-Hilliard” equation with cubic nonlinearity is simulated numerically in various regimes. 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.

[1]  Z. She,et al.  Metastability and vortex pairing in the Kolmogorov flow , 1987 .

[2]  Takao Ohta,et al.  Kink dynamics in one-dimensional nonlinear systems , 1982 .

[3]  U. Frisch.  Turbulence: The Legacy of A. N. Kolmogorov , 1996 .

[4]  E. Hille,et al.  Ordinary di?erential equations in the complex domain , 1976 .

[5]  Sophie Kowalevski,et al.  Sur le probleme de la rotation d'un corps solide autour d'un point fixe , 1889 .

[6]  Gregory I. Sivashinsky,et al.  Weak turbulence in periodic flows , 1985 .

[7]  Frisch,et al.  Eddy viscosity of parity-invariant flow. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[8]  S. Kuksin.  KAM-Theory for Partial Differential Equations , 1994 .

[9]  J. Pedlosky.  Geophysical Fluid Dynamics , 1979 .

[10]  L. Segel,et al.  Nonlinear aspects of the Cahn-Hilliard equation , 1984 .

[11]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[12]  Sophie Kowalevski.  Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe , 1890 .

[13]  Gregory Falkovich,et al.  Kolmogorov Spectra of Turbulence I , 1992 .

[14]  L. D. Meshalkin,et al.  Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid , 1961 .

[15]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .

[16]  J. Henrard,et al.  Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location , 1991 .

[17]  James Gerard Williams,et al.  Secular Perturbations in the Solar System. , 1969 .

[18]  P. Painlevé,et al.  Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (septembre, octobre, novembre 1895) sur l'invitation de S. M. le roi de Suède et de Norwège , 1897 .

[19]  K. Hasselmann.  On the non-linear energy transfer in a gravity wave spectrum Part 2. Conservation theorems; wave-particle analogy; irrevesibility , 1963, Journal of Fluid Mechanics.

[20]  V. Zakharov,et al.  Stationary Nonlinear Theory of Parametric Excitation of Waves , 1971 .

[21]  D. J. Benney,et al.  RANDOM WAVE CLOSURES , 1969 .

[22]  U. Frisch,et al.  Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics , 1994, Journal of Fluid Mechanics.

[23]  K. Hasselmann.  On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory , 1962, Journal of Fluid Mechanics.

[24]  J. Holton.  An introduction to dynamic meteorology , 2004 .