Turbulence of capillary waves — theory and numerical simulation

Abstract

An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the 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 K → -space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost istropic Kolmogorov spectrum in the large k → region. Besides, at small amplitudes of the pumping, an esentially 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 k → . This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.