A theory for the evolution of wind-generated gravity-wave spectra due to dissipation

Abstract

In Naeser (1979), later denoted N, it was shown that a frequency shift of a wave spectrum had to be expected as a result of dissipation of energy and conservation of spin of the waves. While time-dependency was treated in N, mainly space-dependency is treated here. In order to do this, the velocity of the spin of the waves is calculated. It is shown that this can be made equal to the group velocity by a second order coordinate transformation. In the limits of deep and shallow water the transformation becomes the identity, and leaves the Stokes drift at its usual value if the moment point is located at the mean water level.

By supposing that the dissipation is proportional to the molecular dissipation, and that the entire wind-wave interaction takes place at extremely high frequencies and at a constant rate, it is shown that the energy density at the high frequency slope of a fetch-dependent spectrum is inversely proportional to the fifth power of the frequency, while a spectrum which is only a function of the time for which the wind has blown is inversely proportional to the fourth power. The theory is compared with observations which it fits within the accuracy of the method. It is also compared with existing theories and reasons for the discrepancies are pointed out.