Estuarine and Coastal Circulation: An Analytical Framework.

For the last several years, I have been developing a framework to help me understand the dynamics that drive steady and fluctuating circulation in estuaries, lagoons and over the continental shelf.  To me the great complexity in these flows results not so much from their non-linear behavior, as from the competing influences of rotation, stratification, friction, variable topography, etc...   My "method" is based on the well understood fact that if
the horizontal momentum equations can be solved analytically so that horizontal velocities are analytical functions of the depth times the pressure gradients or surface stress.  This provides a means of writing the flux divergence terms in a vertically integrated mass conservation equation in terms of sea level or interface gradients and whatever forcing exists at the surface, with the result that an elliptic system can be written that combines mass and momentum conservation into a single elliptic PDE for sea level (and interface for a two-layer system) depth.

In the simplest cases, for steady wind-forced flow or fluctuating tidal flow in a constant density basin, the governing equation can be solved analytically.  A full description of those results can be found here and here.  In more complicated cases, the governing equation can be solved numerically.  My favorite way to do this was suggested to me by Aurelien Ponte, and will be described here shortly, along with examples.