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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
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where
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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
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where tex2html_wrap_inline2523 is density and g is the acceleration due to gravity. The continuity equation is given by
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The conservation laws for heat and salt may be written as
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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
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where k is the vertical depth level, and
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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
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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
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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
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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
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the freshwater flux over the ocean takes the form
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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
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At the lower boundary of the ocean model we specify a no flux condition on tracers (heat/salt), and no normal flow
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We further apply a quadratic bottom friction (in cases with topography) as
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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:
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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