TY - JOUR

T1 - Wind set-down relaxation on a sloping beach

AU - Gelb, Anne

AU - Gottlieb, David

AU - Paldor, Nathan

N1 - Funding Information:
The authors express their appreciation to Eitan Tadmor for his insight and ideas, which were essential for the completion of this project. The authors were supported by AFOSR Grant F49620-95-1-0074, NSF Grant DMS 9500814, and DARPA/ONR AASERT Grant N00014-93-1-0985. Nathan Paldor also acknowledges the Israel Academy of Sciences for making this research possible by providing a grant to the Hebrew University.

PY - 1997/12

Y1 - 1997/12

N2 - 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.

AB - 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.

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U2 - 10.1006/jcph.1997.5837

DO - 10.1006/jcph.1997.5837

M3 - Article

AN - SCOPUS:0031537297

VL - 138

SP - 644

EP - 664

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -