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## The Ocean General Circulation Model

In the climate model we employ the Geophysical Fluid Dynamics Laboratory (GFDL) Modular Ocean Model (Pacanowski, 1995), which is based upon the primitive equations for a Boussinesq, hydrostatic fluid under the rigid lid approximation. Turbulent closure is parameterized as a Laplacian mixing process. The horizontal momentum equations (including nonlinearities) in spherical geometry are then  where    are the horizontal Laplacian terms representing the horizontal mixing of momentum; is the horizontal Laplacian operator; is longitude; is latitude; and z is the vertical coordinate. Here, the velocity components are u, v, and w in the zonal, meridional and vertical directions respectively, f is the Coriolis parameter, a is the radius of the earth, p is pressure, is a representative density for seawater, t is time, and , are the lateral and vertical eddy viscosities. The model is assumed hydrostatic, so that where is density and g is the acceleration due to gravity. The continuity equation is given by The conservation laws for heat and salt may be written as  where T is temperature, S is salinity, , and are the lateral and vertical diffusivities, and , are Kronecker delta functions defined as where k is the vertical depth level, and the case represents instantaneous convective adjustment that restores a neutral stratification whenever unstable stratification occurs (here parameterized explicitly by Rahmstorf's convection scheme, see Pacanowski, 1995, sec. 11.11). The equation of state for seawater can be written as with the density a nonlinear equation of temperature, salinity and pressure (UNESCO, 1981).

At the surface, the model is driven by both wind stress (applied as a body force over the depth of the first grid box) and surface buoyancy forcing (see the array sbcocn, Appendix A). The surface boundary conditions are therefore  where are the zonal and meridional surface wind stresses. The surface buoyancy forcing is applied directly to the tracer conservation equations (2.3.5, 2.3.6). The net heat flux into/out of the ocean is defined as The definitions of , , , and are the same as those in (2.1.6), (2.1.7), (2.2.3), and (2.2.4); while the latent heat flux out of the ocean takes the form the freshwater flux over the ocean takes the form where are representative salinities for the ocean and ice respectively, R is the runoff from the landmass, and B is the freshwater flux from ice formation or melt given by At the lower boundary of the ocean model we specify a no flux condition on tracers (heat/salt), and no normal flow We further apply a quadratic bottom friction (in cases with topography) as  where is the drag coefficient, and is the turning angle. At lateral walls, no flux of tracers is permitted, and a no slip condition is applied to the horizontal flow: where n is a unit normal to the boundary.    Next: Numerical Considerations Up: Modeling the Climate System Previous: The Ice Model

Daniel Robitaille
Mon May 5 14:22:13 PDT 1997