Wind set-down relaxation on a sloping beach

Consider a long and narrow basin where both the Coriolis force and the dynamics in the cross-basin direction can be neglected. The bottom of the basin is assumed to slope linearly from the head of the basin toward its mouth, where the water is infinitely deep. The shape of the sea surface in the steady-state solution of the "wind set-down" problem is determined by the balance between the wind which blows over the basin from the shore seaward and the pressure gradient which results from the slope of the sea surface. This study addresses the time-dependent problem encountered when the wind in the wind set-down solution suddenly relaxes and the water gushes landward under the influence of the pressure gradient force. We call this problem the "relaxation of the wind set-down." The difficulty in solving this problem is due to the moving singularity associated with the ever-changing location of the point where the sea surface intersects the sloping bottom. At this point the problem is onlyweakly hyperbolic, thus onlyweakly well posed. We solve this problem numerically using two completely different types of numerical solvers, finite difference schemes and spectral methods. Both types of solvers are successfully tested on a similar problem where the analytical solution is known. Both the MacCormack finite difference scheme and the Chebyshev spectral method concurred in their results, strongly suggesting the validity of the numerical solution. Our results indicate that no wave breaking occurs and that the water will slosh up and down the sloping bottom, similar to the behavior of a nonlinear gravity wave. The spectrum of this wave motion consists of peaks associated with the motion of regular gravity waves in a triangular basin, as well as frequency beatings associated with the movement of the singular point of the present problem.