Three-dimensional surface waves propagating over long internal waves

A non-uniform current, such as may be generated by long internal waves, int eracts with short surface waves and causes patterns on the sea surface that are of interest. In particular, regions of steep breaking waves may be rel evant to specular radar scattering. A simple approach to modelling this problem is to take a set of short, surf ace waves of uniform wavenumber on the sea surface, as may be caused by a g ust of wind. The direction of propagation of the surface waves is firstly t aken to be the same as that of the current, and surface tension and viscous effects are neglected. We have a number of methods of solution at our disp osal: linear (one-dimensional) ray theory is simple to apply to the problem , a nonlinear Schrodinger equation for the modulated wave amplitude, modifi ed to include to effect of the current, can be used and solutions can be fo und using a fully nonlinear irrotational flow solver. Comparisons between t he 'exact' nonlinear calculations for two dimensions (which are too complic ated/ computationally intensive to be extended to three dimensions) compare well with the two approximate methods of solution, both of which can be ex tended, within their limitations, to model the full three-dimensional probl em; here we present three-dimensional results from the linear ray theory. By choosing such a simple (although we consider physically realistic) initi al state of uniform wavenumber short waves and assuming a sinusoidal surfac e current, we can reduce the two-dimensional problem to dependence on three non-dimensional parameters. In three-dimensions, we consider an initial condition with a uniform wavetr ain at an angle alpha say, to the propagating current, thus introducing a f ourth parameter into the problem. Extension of the linear ray theory from o ne space to two space dimensions is numerically quite simple since we maint ain uniformity, in the direction perpendicular to the current, and the only difficulty lies with the presentation of results, due to the large number of variables now present in the problem such as initial wavenumber, angle o f propagation, position in (x, y, t) space etc. In this paper we present ju st one solution in detail where waves are strongly refracted and form two d istinct foci in space-time. There is a collimation of the short waves with the direction of the propagating current. (C) Elsevier, Paris.