Until now we have not included effects of lateral or vertical transfer of momentum (or vorticity) by net advectional effects associated with stochastic vortex modes of scales less than the radius of deformation (baroclinic) or vertical scales less than the surface or bottom Ekman layer thickness. We adopt the zero order eddy viscosity closure hypothesis for the lateral transfer; for vertical transfer of momentum by such processes we merely allow for horizontal shear stresses within the top and bottom Ekman layers and close the system by stipulating the stress at the sea surface (the wind stress) and relate the bottom stress to the bottom velocity.
The governing equations, which are a generalization of Eqs. (1.1) to (1.7) of Chap. III are:
where (4) is a combination of (1.3) and (1.5) of Chap. II,
are x,y-components of the shear stress
vector 
(divided by 
), and K is the
lateral eddy viscosity with dimensions 
which
we will consider constant for simplicity.
The boundary conditions are
where the actual sea bed is at 
and B is considered small relative to the mean
depth D. In (6b) 
is a resistance coefficient
with dimensions of L/T.
It can be shown from an analysis of the energetics of the
above system that
where supply and dissipation of the non-turbulent energy E
are identified ( 
being considered positive).
The quasi-geostrophic vorticity equation obtained from (1), (2) and (3) is
where
and 
is the biharmonic operator
. Following Flierl (1978,
Dyn. of Atmos. and Oceans, pp. 341-384) we consider
that p can be represented in the form
where the vertical structure functions 
satisfy the
eigenvalue problem
These functions possess the orthonormality condition
Using (12), relation (10) can be inverted to to give
which is analogous to the relation for Fourier coefficients with (10) being analogous to a Fourier series.
We can convert (8) to a sequence of equations for the
following the technique given by Flierl (1977). This
involves multiplication of (8) by followed by
integration over z; integration by parts is carried
out for terms on the right followed by use of (4), boundary
conditions (5) and (6) and the quasi-geostrophic condition
relating 
and p at the sea bed. The result is
for 
.
The various terms in this equation can be interpreted as follows for a given mode:
is proportional to the potential vorticity anomaly 
;
is the planetary vorticity tendency associated with meridional
flow across f contours;
is the stretching term associated with flow at the sea bed
over bottom topography;
is the lateral resisting ``torque'';
is the bottom Ekman layer resisting ``torque'';
is the driving ``torque'' due to wind curl, i.e. Ekman pumping.
and
.
All surface values 
will be taken positive, which
implies that 
will be positive for even n (0, 2, 4, ...)
and negative for odd modes.
Both the stretching term and the bottom friction term introduce
a coupling of the modes. The stretching term will vanish
either if the sea bed is flat ( 
) or if
, which requires that
The bottom friction term will vanish if (15) holds or if
. If these conditions are not met than all
modes are coupled, the strongest coupling for interior regions
of the ocean (regions away from lateral boundary layers) being
due to bottom topography.
For the coupled problem, what this means is that one must
seek solutions for the entire set of normal modes
simultaneously; the other option is to solve the
three-dimensional forced vorticity equation given by
(8) with (4) and appropriate boundary conditions.
However, the model approach is attractive both for
pedagogical and practical reasons; pedagogical, since
we can relate to the history of the development of our
understanding of ocean circulation more easily via the
modal approach; practical, because even if we truncate
the modal series representation at only two or three
terms it is enough to fully appreciate the physics.