# Coupled Shallow Water Flow and Transport Models: A ...

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 XVI International Conference on Computational Methods in Water Resources (CMWR-XVI) Ingeniørhuset Coupled Shallow Water Flow and Transport Models: A Look at the Vertical Equations and B oundary Conditions Author: Randall Kolar (University of Oklahoma) Clint Dawson (University of Texas-Austin) Presenter: Randall Kolar (University of Oklahoma) Date: 2006-06-18     Track: General Sessions     Session: General DOI: 10.4122/1.1000000381 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.