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3.1 Model Equations for a Reduced Gravity Ocean     continued...

The interfacial stress can be related in a linear or a nonlinear way to the velocity jump across the layers. For the formulation of these terms and a discussion of the typical values used in modeling efforts see section 2.2.1. The boundary conditions at the lateral walls (coastlines) are essentially the same as were used for the barotropic velocities discussed at the end of section 2.1. We have defined h to be the total upper layer thickness, and it is equal to

where is the mean (initial undisturbed) layer thickness, is the free surface elevation, and P is the pycnocline deviation. As noted before,

where is the density jump between the upper and lower layers, (with the lower being heavier).

The student should note that the two deformations and P constitute only one dependent variable, because of the relation (20). Also, equations (18) are very similar to equations (2)--(4) for the barotropic ocean, with the total upper layer thickness h replacing the free surface elevation in the continuity equation (2) and replacing g in the pressure gradient terms in (2) and (3). However, also note that the variable upper layer thickness h has also replaced the topography H in the momentum equations, so that now the pressure terms have become nonlinear.