File last modified 3 December 1998
We will begin our discussion of general circulation models per se with a description of three very coarse parts of the model: the primitive equations, the grid, and the bathymetery underneath the model.
Primitive equations are a mathematical description of the local balance of heat and salt. They allow the simulation of wind driven and thermohaline (density) driven circulation as a response to forcing of heat, wind and freshwater at the surface of the ocean. The primitive equations are the following:
The hydrostatic equation is written:
which can be used to give the pressure at any depth through the following integration:
where ps is the pressure at the surface (it is held constant). The Boussinesq approximation is
accomplished by replacing 
in the Navier-Stokes equations with 
kg/m3 (depth
averaged). Essentially the buoyancy term has the form of 
but the continuity equation is
approximated by:
i.e. the fluid is incompressible.
Additional simplifications are made. The rigid lid approximation is made since the vertical range of motion of the free surface is much smaller than that of internal waves, hence the pressure effect of surface displacement can be approximated by the pressure exerted by a rigid lid. Put all of this together and you get:
where:
And that's, basically, the primitive equation(s) used to calculate the general circulation of the ocean.
In this discussion of GCMs we are going to use the coarse resolution GCM used in Toggweiler et al. (1989a&b) as a basic example. In this case, we will talk about the grid used in these GCM simulations and give you an idea of what we mean when we say "coarse". In his model integrations Toggweiler used a grid that was 4.5o in latitude and 3.75o in longitude, with 12 vertical levels. This comes to only 46,080 grid points before land masking. Although Toggweiler learned a lot from this coarse grid, today finer and finer grids are being used with the current politically correct grid size goal for global long-term integrations appears to be about 1o x 1o. Investigators like Bert Semtner are pushing the grid resolution to scales as small as 1/16o (or even smaller) for limited ocean basin scale integrations or near-shore simulations.
Once the grid has been established temperature, salinity and chemical tracers are placed into the grid, but in different ways. Figure 23.2.1 shows, in plan view, how these components are placed on the underlying grid.
Like the plan view grid, the "levels" of the GCM also have to be adjusted to allow for some semblance to the real bathymetry. When the grid is coarse this can present some problems Fig. 23.2.2 shows Drake Passage in cross-section.
In Toggweiler's coarse resolution model Iceland was eliminated and the Greenland-Scotland sill was deepened. The Straight of Gibraltar was left as it is, but no flow between the Mediterranean and the Atlantic was allowed. Additionally the horizontal viscosity of the water was increased to give rise to realistic western boundary currents (Gulf Stream, Kuroshio, etc.).
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