Imbalance of energy and momentum source terms of the sea wave transfer equation for fully developed seas

Abstract

In the concept of full development, the sea wave spectrum is regarded as a nearly stationary solution of the wave transfer equation, where source and sink terms should be in balance with respect to both energy and momentum. Using a two-dimensional empirical sea wave spectral model at full development, this paper performs an assessment of the compatibility of the energy and momentum budgets of sea waves over the whole spectral range. Among the various combinations of model functions for wave breaking and wind source terms tested, not one is found to fulfill simultaneously the energy and momentum balance of the transfer equation. Based on experimental and theoretical grounds, wave breaking is known to contribute to frequency downshift of a narrow-banded wave spectrum when the modulational instability is combined with wave breaking. On those grounds, it is assumed that, in addition to dissipation, wave breaking produces a spectral energy flux directed toward low wavenumbers. I show that it is then possible to remove the energy and momentum budget inconsistency, and correspondingly the required strength of this spectral flux is estimated. Introducing such a downward spectral flux permits fulfilling both energy and momentum balance conditions. Meanwhile, the consistency between the transfer equation and empirical spectra, estimated by means of a cost function K, is either improved or slightly reduced, depending upon the wave breaking and wind source terms chosen. Other tests are performed in which it is further assumed that wave breaking would also be associated with azimuthal diffusion of the spectral energy. This would correspondingly reduce the required downward spectral flux by a factor of up to 5, although it would not be able to remove it entirely.