Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid
Starting from general equations describing potential surface waves in inviscid fluids of arbitrary depth, the author develops an asymptotic analysis for a small-amplitude case. The resultant equations of motion are obtained in the canonical Hamilton's forms for properly defined complex amplitudes expressed in terms of two-dimensional spatial Fourier transforms of surface perturbations. To simplify the equations, the author uses a canonical transformation aimed at the elimination of quadratic nonlinearities. The final equation has the form of an integral first-order evolutional equation with linear and cubic terms (the integrations are performed in the space of the Fourier-transform wave-vectors, which are continuous parameters of complex canonical variables). Various special cases, which are of interest for applications, are considered in detail, including deep- and shallow-water limits. In the latter case, the kernel of the nonlinear integral term is formally singular, but it is possible to prove that all the singularities mutually cancel. It is stressed that the canonical transformation which eliminates the quadratic terms changes the definition of physically relevant action-type variables, which may have important consequences in the application to the shallow-water limit. The author also develops a statistical description of the weak wave turbulence, based on a combination of the integral evolution equation and Gaussian statistical correlations imposed on physically correct action-type variables. This analysis produces an effective kinetic equation, which, in particular, allows one to find a nonlinear correction to the dispersion equation for weakly nonlinear plane waves. Stationary analytical solutions to this kinetic equation are found in the form of Kolmogorov's spectra. Finally, the author demonstrates that the applicability of the above weakly nonlinear theory to the shallow-water case is strongly limited, especially in the limit of long surface waves.
two-dimensional spatial Fourier transforms
integral first-order evolutional equation
scientific article; zbMATH DE number 1357998