Normal Flow Boundary Conditions in Shallow Water Models — Influence on Mass Conservation and Accuracy

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Finite element solution of the shallow water wave equations has found increasing use by researchers and practitioners in the modeling of oceans and coastal areas. Wave equation models, most of which use equal-order, C 0 interpolants for both the velocity and the surface elevation, successfully eliminate spurious oscillation modes without resorting to artificial or numerical damping. An important question for both primitive equation and wave equation models is the interpretation of boundary conditions. Analysis of the characteristics of the governing equations shows that a single condition at each boundary is sufficient. Yet there is not a consensus in the literature as to what that boundary condition must be or how it should be implemented in a finite element code. Traditionally (partly because of limited data) surface elevation is specified at open ocean boundaries while the normal flux is specified as zero at land boundaries. In most finite element wave equation models, both of these boundary conditions are implemented as essential conditions. Our recent work focuses on alternate ways to numerically implement normal flow boundary conditions with an eye toward improving the mass-conserving properties of wave equation models. In particular, we have found that treating normal fluxes as natural conditions with the flux interpreted as external to the computational domain results in a mass conservative scheme for all parameter values. Use of generalized functions in the finite element formulation shows this is a natural interpretation. A series of twodimensional experiments demonstrates that this interpretation also improves the accuracy of primitive equation models by eliminating some of the spurious oscillation modes. BACKGROUND Shallow water equations are obtained by vertically averaging the microscopic mass and momentum balances over the depth of the water column. Early finite element solutions of the shallow water equations were often plagued by spurious oscillations. Various methods were introduced to eliminate the oscillations but all included some type of artificial damping. Lynch and Gray [1] and Gray [2] present the wave continuity equation as a means to successfully suppress spurious oscillations without resorting to numerical or artificial damping of the solution. Since the inception of the wave continuity formulation in 1979, the original algorithm has been modified in a number of substantial ways: a numerical parameter was introduced to provide a more general means of weighting the primitive continuity equations [3]; viscous dissipation terms were incorporated [4, 5]; and threedimensional simulations were realized by resolving the velocity profile in the vertical [6, 7]. The resulting algorithm has been extensively tested using analytical solutions and field data and is currently being used to model the hydrodynamic behavior of coastal and oceanic areas [8-10].