In general, ocean models describe
the response of a variable density ocean to atmospheric momentum
and heat forcing. This response can very simply be represented in
terms of eigenmodes of a linearized system of equations. We will
attempt now to give a simplified physical description of these
modes (at this point the student should also read Chapter 6 of
Gill
[22]).
The zeroth mode is equivalent to the vertically-averaged
component of the motion, also known as the *barotropic*
mode. The higher modes are called *baroclinic* modes and are
associated with higher order components of the vertical density
profile. All ocean models described in this chapter will make the
*hydrostatic* shallow water approximation, in which the pressure
depends only on the depth , i.e. it's given by the classic
hydrostatic relation

Figure 4 Barotropic Model where a region of the ocean's water column is vertically integrated to obtain one value for the vertically different horizontal currents. View Figure

This relation holds if the horizontal
dimensions of the ocean volume under consideration are much larger
than the vertical dimension, hence the *shallow water* designation.

As we will see in section 2.1, the pressure gradients associated with the free surface elevation are constant with depth. Thus they form part of the zeroth mode or the vertically-averaged mode, and appear only in the barotropic mode equations. Consequently, the baroclinic system representing the higher-order modes has no surface elevation associated with it, and the corresponding surface boundary condition is that of a rigid lid.

A particular form of the baroclinic models are the so-called
*reduced gravity* models. These are essentially isopycnal models of
several deformable layers where the lowest layer has infinite
depth and zero velocity. Clearly no barotropic mode is associated
with such models, and although the motions are driven by the
density differences between layers, there are restrictions on the
motions and interface deformations. For example, in a two-layer
reduced gravity (RG) model the deviations of the interface between
the two layers (representing excursions of the pycnocline) are
multiples of the free surface elevation, by a factor which is
proportional to the ratio ,
being the density and the
density difference between the two layers (the lower one being
heavier, of course).

We will discuss the barotropic models in the next section and the reduced gravity models in section 3.