On the interaction of surface and internal waves

Abstract

The steady-state interaction between surface waves and long internal waves is investigated theoretically using the radiation stress concepts derived by Longuet-Higgins & Stewart (1964) (or Phillips 1966). It is shown that, over internal wave crests, those surface waves for which c_g0cosϕ₀ > c_i experience a change in direction of propagation towards the line of propagation of the internal waves and their amplitudes are increased. Here c_g0 is the surface-wave group speed at U = 0, ϕ₀ is the angle between the propagation direction of the surface waves at U = 0 and the propagation direction of the internal waves, and c_i is the phase speed of the internal waves. If c_g0cos ϕ₀ < c_i the direction of the surface waves is turned away and their amplitudes are decreased. Over troughs the opposite effects occur.

At positions where the local velocity of surface-wave energy transmission measured relative to the internal wave phase velocity is zero, i.e. c_g + U − c_i = 0, there is a singularity in the energy of the surface waves with resulting infinite amplitudes. It is shown that at these critical positions two wavenumbers which were real and distinct on one side coalesce and become complex on the other. The critical positions are thus shown to be barriers to the propagation of those wave-numbers. It is also shown that there is a critical position representing the coalescence of three wavenumbers. Surface-wave crest configurations are shown for three numerical examples. The frequency and direction of propagation of surface waves that exhibit critical positions somewhere in an internal wave field are shown as a function of the maximum horizontal surface current. This is compared with measurements of wind waves that have been reported elsewhere.