Turbulence of capillary waves-theory and numerical simulation

An ensemble of weakly-interacting capillary waves on a free surface of deep ideal fluid is described statistically by the methods of weak turbulence. The stationary kinetic equation for capillary waves has an exact Kolmogorov solution which gives, for a spatial spectrum of elevations, asymptotics I_k = C (P ^1/2 /σ^3/4 k^-19/4. The Kolmogorov constant C is found analytically together with the interval of locality in k̄-space. Direct numerical simulation of dynamical equations, in the small surface angles approximation, confirms the presence of an almost isotropic Kolmogorov spectrum in the large k̄ region. In the pumping region, the spectrum is defined by non-resonant processes of nonlinear damping. This fact can be explained by the narrowness of the inertial interval. Moreover, at small amplitudes of the pumping, an essentially new phenomenon is found: `frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k̄. This phenomenon is connected with the finiteness of the region (or, in other words, the discreetness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.