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 ). 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
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.