Numerical Study of the Equations of Nonlinear Swell
in the Directional Approximation

V. G. Polnikov

State Oceanographic Institute, Kropotkinskii per. 6, Moscow, 119838 Russia

Received May 14, 1998; in final form, September 17, 1998

Abstract—Numerical methods are employed to study equations of directional approximation, previously
derived by Zakharov and Smilga [10], for Hasselmann’s kinetic equation and transformed by Zaslavskii [12]
into a form that admits self-similar solutions. Numerical solutions to Zaslavskii’s equations are obtained in a
limited band of wave numbers. It is shown that, in the region of a spectral peak, the numerical solutions are in
close agreement with the analytic solutions derived by Zaslavskii [12]. However, these solutions differ in the
region away from the spectral peak. A comparison of the above solutions with the numerical solution of the
initial kinetic equation previously obtained by the author [9] testifies to their good agreement and to a wide
range of validity of the directional approximation. The results of this comparison of solutions make it possible
to give an efficient analytic approximation for a self-similar spectrum of swell, suitable for use in numerical
models of wind waves.

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