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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



tex2html_wrap_inline2491 are the horizontal Laplacian terms representing the horizontal mixing of momentum; tex2html_wrap_inline2493 is the horizontal Laplacian operator; tex2html_wrap_inline2351 is longitude; tex2html_wrap_inline2353 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, tex2html_wrap_inline2513 is a representative density for seawater, t is time, and tex2html_wrap_inline2517, tex2html_wrap_inline2519 are the lateral and vertical eddy viscosities. The model is assumed hydrostatic, so that
where tex2html_wrap_inline2523 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, tex2html_wrap_inline2537, and tex2html_wrap_inline2539 are the lateral and vertical diffusivities, and tex2html_wrap_inline2541, tex2html_wrap_inline2543 are Kronecker delta functions defined as
where k is the vertical depth level, and
the case tex2html_wrap_inline2551 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 tex2html_wrap_inline2559 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 tex2html_wrap_inline2563, tex2html_wrap_inline2451, tex2html_wrap_inline2453, and tex2html_wrap_inline2441 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 tex2html_wrap_inline2575 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 tex2html_wrap_inline2589 is the drag coefficient, and tex2html_wrap_inline2365 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 up previous contents
Next: Numerical Considerations Up: Modeling the Climate System Previous: The Ice Model

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