# 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 vertical momentum balance is taken to be hydrostatic (long
waves)
- vertical friction terms dominate over lateral terms
- the system can be considered as weakly non-linear.

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.