A similarity relation for the nonlinear energy transfer in a finite-depth gravity-wave spectrum

Abstract

The energy transfer in a finite-depth gravity-wave spectrum is investigated in the approximation of a narrow spectrum. It is shown that for ocean depths larger than approximately one tenth of the wavelength (kh [ges ] 0·7) the finite-depth case can be reduced to Longuet-Higgins’ (1976) result for an infinitely deep ocean by a similarity transformation involving changes in scale of the angular spreading function and the transfer rate. For shallower water (kh < 0·7) Longuet-Higgins’ expansion technique is no longer applicable without modification, as the nonlinear coupling coefficient develops a discontinuity at the origin of the expansion. In the range kh [ges ] 0·7 both the magnitude and the two-dimensional frequency-directional distribution of the energy transfer are found not to differ significantly (to within variations by a factor of 2) from the case of an infinitely deep ocean. The transformation rules relating the infinite-depth and finite-depth cases may provide a useful guide for constructing parametrizations of the nonlinear transfer for finite-depth wave prediction models.