The shallow water equations (2D, depth averaged) accurately capture estuarine hydrodynamics. Although the full equations can be solved, Ip et al. have shown that a kinematic approximation to these equations can be employed in shallow estuaries, which reduces computational overhead. We have adopted Ip et al.'s approach, which we briefly describe here (please see Ip et al. for details). What Ip et al. refer to as a kinematic approximation to the shallow water equations is actually a diffusive wave approximation (cf. Ferrick and Goodman) and is given by:

where is the water surface deviation measured from a datum, v is the water velocity vector, h is the bathymetry, and c_d the bottom drag coefficient. The simplified conservation of momentum eqn. (2) can be solved for v and substituted into the momentum eqn. (1) to give,

The kinematic approximation allows the hyperbolic advection eqn. (2) to be converted into a parabolic diffusion eqn. (3), which is numerically easier to solve.

To handle flooding and drying of the marsh without using a moving boundary, Ip et al. assume there exists a porous sub-layer (figure on right) where water flow, q, follows Darcy's Law:

where is the hydraulic conductivity and H is the total water depth. By combining (3) with (4), water elevation in the estuary is govern by:

where h_p is the depth of the porous layer that has a porosity of . Consequently, when water begins to "dry" the marsh, the water surface elevation is allowed to drop below the marsh surface, where the open water equation (3) is replaced by the groundwater model (4). This approach has several advantages: