On the Form of the Momentum Equation and Lateral Stress Closure Law in Shallow Water Modeling

Previous studies have shown that utilizing the conservative form of the momentum (CM) equation improves the mass conserving ability of the ADCIRC (ADvanced CIRCulation) finite element model, especially in areas of steep bathymetry gradients. Local spatial accuracy also improves, due to a decrease in the truncation error, yet does so without significantly impacting temporal or global spatial accuracy. That study raised other research questions, two of which are examined herein. The first deals with the lateral stress closure scheme utilized in the model. As the ADCIRC community begins to incorporate more near shore physics and baroclinic capabilities, a more sophisticated closure scheme is needed that allows both spatial and temporal variation, such as the Smagorinsky formula. The second question explores the treatment of the advective terms in the governing equations. In this paper, we examine various algorithms using a variety of metrics in order to determine the "best" formulation for the lateral stress and advective terms in the governing equations. Initial results indicate that mass conservation errors are impacted more by the treatment of the advective terms than the lateral stress formulations, with algorithms based on the CM equation producing the lowest error. Results also indicate that the lateral stress formulation in the continuity equation only slightly impacts the elevation results; other experiments were unaffected by the different formulations. When analyzing the CM equation, we observe that the native treatment (i.e., derivatives not expanded) of the advective terms produces smaller differences in spatial accuracy for most of the elevation and velocity components. Taken together, results do not single out a "best" algorithm, so additional work is needed, particularly with the comparison of velocity-based versus flux-based lateral stress formulations.