A new algorithm for three-dimensional tidal and storm surge computations

Abstract

A computational algorithm is proposed for the solution of the hydrodynamical equations appropriate for motions in shallow seas and estuaries. The surface elevation and depth-averaged velocity components are first computed using one of the standard two-dimensional algorithms for the solution of the shallow-water equations. These values are then used to compute the vertical profile of the velocity field by solving the two horizontal momentum equations. These equations form a coupled parabolic system, and a generalized Crank-Nicholson scheme is developed to solve them. In the case of constant eddy viscosity this scheme can be shown to be unconditionally stable, so the only stability restriction is the usual CFL condition on the shallow-water equations. Thus the time step is limited only in terms of the horizontal grid spacing, not by the vertical spacing. The algorithm is tested on a model problem previously used by Davies and Owen 1 involving the development of circulation in a rectangular sea under the influence of a constant surface wind-shear. The velocity profiles at various times are presented, and the final steady-state flow agrees very well with that computed by Davies and Owen. For constant eddy-viscosity the complete three-dimensional computation is only twice as slow as the two-dimensional computation that forms part of it.