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One strategy for solving the shallow water equations in the finite element community is to reformulate the primitive continuity equation into the generalized wave continuity equation (GWCE). The GWCE contains a parameter, "G," as well as the second derivatives in space and time that are generally associated with the classic wave equation (hence its name). G is a numerical parameter: as G goes to zero, the equation reduces to the form of a pure wave equation, and as G goes to infinity, the equation takes the form of the primitive continuity equation. Atkinson et al. [1] found that the GWCE formulation is equivalent to the quasi-bubble discretization (as applied to the linearized shallow water equations) for a specific value of G, which depends on the wave frequency and the bottom friction parameter. Additionally, they theorize that this relationship for G which makes them equivalent also yields optimal dispersion properties. Numerical simulations using such a space and time-variable G parameter based on the quasi-bubble/GWCE comparison have yielded favorable initial results: the variable G implementation substantially reduces wave constituent and mass balance errors for highly non-linear cases. In particular, for a 1-D linear sloping beach with wetting and drying and an M2 forcing, the error with the variable G is two orders of magnitude less for the steady and M2 constituents than with a constant G. The M4 and M6 constituent errors are an order of magnitude less and the mass balance errors decrease by 20%. Due to such promising results from the 1-D study, the variable G formulation is being implemented in a 2-D depth-averaged production code; simulations will be run on a variety of idealized and real basins and selected results will be presented. [1] J.H. Atkinson, J.J. Westerink, and J.M. Hervouet. Similarities between the Quasi-Bubbl and the Generalized Wave Continuity Equation solutions to the Shallow Water Equations. International Journal for Numerical Methods in Fluids, 45, 2004; 689-714.

Proper specification of boundary conditions in shallow water models have been the subject of much research over the years and still remains an active area of research. When coupled to transport models, as in baroclinic or water quality simulations, the number of conditions on the boundary increases. Yet, few studies have looked at boundary conditions for coupled models. In many instances, the following "traditional" conditions have been used: a strong boundary condition for ocean boundaries in the continuity equation; for the land boundaries, a strong boundary condition is employed in the momentum equation, while a weak formulation of the boundary condition for the velocities is used in the continuity equation; finally for the transport equation, a strong boundary condition is utilized on the ocean boundary with a weak boundary conditions applied for the land boundary. However, our experience is that such implementations of the boundary conditions can lead to instabilities in both the flow and transport models, or they can degrade the accuracy of the solution for the continuity, momentum and transport equations. In this study, we examine an alternative treatment to the strong formulation at the ocean boundary for both the continuity and transport equation, utilizing a weak form of these boundary conditions for both the surface elevation field and the baroclinic fields. This implementation allows for both inflow and outflow boundaries to exist on the ocean boundary for the transport equation. We will assess the impact of the weak implementation by evaluating the stability and accuracy changes within the shallow water model, ADCIRC (ADvanced CIRCulation). ADCIRC, which is based on over 20 years of research and applications, is a hydrodynamic model capable of simulating water surface elevation and velocity fields in lakes, bays, estuaries, and oceans. Model applications range from determining the effects of dredging on circulation to determining the storm surge that accompanies the landfall of a hurricane. The model is based on the full non-linear St. Venant (shallow water) equations, using the traditional hydrostatic pressure and Boussinesq approximations; the equations are discretized in space using linear finite elements, while time is discretized using an efficient split-step Crank-Nicolson algorithm. The transport equation utilized in the study also uses linear finite elements and utilizes a split-step Crank- Nicolson algorithm in the temporal discretization.

Most prominent three-dimensional (3D) shallow water models invoke the Boussinesq and hydrostatic approximation to reduce the dimensionality of the problem and simplify the governing equations. Furthermore, a scaling analysis allows the problem to be decomposed into external and internal mode solutions, which allows a computationally-efficient sequential solution procedure. More specifically, the depth-averaged continuity equation is first solved for the free surface elevation, and the 3D momentum equation (subject to the assumptions above) is then solved for the depth-varying horizontal velocity field. Finally, the 3D, incompressible continuity equation is used to resolve the vertical velocity component. It has long been recognized that the latter system is overconstrained, in that it is a first order equation is subject to two boundary conditions (bottom and surface). In ADCIRC, the shallow water model that is the subject of this research, the overconstrained system is solved in a least-squares sense through the use of adjoint procedures. The weights introduced in the objective function allow one to place emphasis on either the interior equation or the boundary conditions. When coupling a hydrodynamic model to a transport algorithm (for diagnostic or prognostic simulations), a velocity field that does not fully satisfy the 3D continuity equation can introduce sources or sinks of mass into the model. Yet, depending on the weights chosen in the vertical adjoint problem, the continuum equation may not be satisfied exactly. To our knowledge, the impact of this overconstrained problem on a coupled flow and transport model has not been studied. Herein, through analyses and numerical simulations, we have found that the weights in the objective function can affect results significantly.

Most prominent three-dimensional (3D) shallow water models invoke the Boussinesq and hydrostatic approximation to reduce the dimensionality of the problem and simplify the governing equations. Furthermore, a scaling analysis allows the problem to be decomposed into external and internal mode solutions, which allows a computationally-efficient sequential solution procedure. More specifically, the depth-averaged continuity equation is first solved for the free surface elevation, and the 3D momentum equation (subject to the assumptions above) is then solved for the depth-varying horizontal velocity field. Finally, the 3D, incompressible continuity equation is used to resolve the vertical velocity component. It has long been recognized that the latter system is overconstrained, in that it is a first order equation is subject to two boundary conditions (bottom and surface). In ADCIRC, the shallow water model that is the subject of this research, the overconstrained system is solved in a least-squares sense through the use of adjoint procedures. The weights introduced in the objective function allow one to place emphasis on either the interior equation or the boundary conditions. When coupling a hydrodynamic model to a transport algorithm (for diagnostic or prognostic simulations), a velocity field that does not fully satisfy the 3D continuity equation can introduce sources or sinks of mass into the model. Yet, depending on the weights chosen in the vertical adjoint problem, the continuum equation may not be satisfied exactly. To our knowledge, the impact of this overconstrained problem on a coupled flow and transport model has not been studied. Herein, through analyses and numerical simulations, we have found that the weights in the objective function can affect results significantly.

Includes bibliographical references., 23rd annual AGU hydrology days was held on March 31 - April 2, 2003 at Colorado State University., This article presents a criterion for limiting source dimensions of the instantaneous point source model. The source dimensions are required for applying superposition of the point sources in space to represent the non-point sources. The criterion generates less than one percent error when compared to the existing finite source model. A comparison between the existing point and the finite source models is conducted. It is found that the finite source model does not give acceptable output at all times, if series distribution is used to compute the error function. The front and tail of the concentration breakthrough curve from the finite source model shows uncharacteristic output depending on the source dimension and the distance of the observation point. On the contrary, the point source model generates smooth breakthrough curves at all time.