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 tow ard its mouth, where the water is infinitely deep, The shape of the se a surface in the steady-state solution of the ''wind set-down'' proble m is determined by the balance between the wind which blows over the b asin from the shore seaward and the pressure gradient which results fr om the slope of the sea surface. This study addresses the time-depende nt problem encountered when the wind in the wind set-down solution sud denly 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 poi nt the problem is only weakly hyperbolic, thus only weakly well posed. We solve this problem numerically using two completely different type s of numerical solvers, finite difference schemes and spectral methods . Both types of solvers are successfully tested on a similar problem w here the analytical solution is known. Both the MacCormack finite diff erence scheme and the Chebyshev spectral method concurred in their res ults, 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 non linear gravity wave. The spectrum of this wave motion consists of peak s associated with the motion of regular gravity waves in a triangular basin. as well as frequency beatings associated with the movement of t he singular point of the present problem.