Properties of Time-Dependent Solutionsof the Hasselmann Kinetic Equation amg

The Hasselmann kinetic equation for deep-water nonlinear gravity waves is studied analytically and numerically to elucidate the existence of a self-similar form of the spectrum resulting from a time-dependent solution of this equation on large time scales. It is shown analytically that, due to the presence of three integrals of motion, the complete self-similarity of solutions to the kinetic equation is impossible. However, this fact does not forbid the existence of an "incomplete self-similarity," which is defined as the establishment of fixed values of the integral parameters of the spectrum in the course of its long-term evolution. This inference is confirmed by the results of a numerical study carried out by invoking both an improved method of calculating the kinetic integral and modern schemes for numerically solving the kinetic equation. The results of numerical experiments are used to determine the limiting characteristics of the form of the energy spectrum of waves on evolutionary scales when the form of the spectrum is completely controlled by nonlinear processes in waves.

Vladislav Polnikov. Properties of time-dependent solutions of the Hasselmann kinetic equatio. September 2001, Izvestiya Atmospheric and Oceanic Physics 37(5):661-670