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.
