Here we will present them for Cartesian coordinates and for constant friction coefficients only; the extension to spherical coordinates and non-constant friction will be left to the student. These equations are given in detail in Hurlburt and Thompson [31] and Preller and Hurlburt [55].
where,
Other quantities are as in the barotropic model.
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 
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 
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 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 replacing the free
surface elevation in the continuity equation (2)
and 
replacing 
in the pressure gradient terms in
(2) and (3).
However, also note that the variable upper layer thickness has
also replaced the topography 
in the momentum equations, so that
now the pressure terms have become nonlinear.
