4.1 Assumptions

MOM is a finite difference version of the ocean primitive equations, which govern much of the large scale ocean circulation. As described by Bryan (1969), the equations consist of the Navier-Stokes equations subject to the Boussinesq and hydrostatic approximations. The equation of state relating density to temperature, salinity, and pressure can generally be nonlinear, thus representing important aspects of the ocean's thermodynamics. Prognostic variables are the two active tracers potential temperature and salinity, the two horizontal velocity components, any number of passive tracer fields, and optionally the height of the free ocean surface. The discretization consists of spatial coordinates fixed in time (fully Eulerian), with surfaces of constant depth determining the vertical discretization and a spherical (latitude/longitude) grid for the horizontal.

As discussed by many authors, including the original paper by Boussinesq (1903), as well as Spiegel and Veronis (1960), Chandrasekhar (1961), Gill (1982), and Müller (1995), the Boussinesq approximation is justified for large-scale ocean modeling on the basis of the relatively small variations in density within the ocean. The mean ocean density profile $\rho_\circ(z)$ typically varies no more than 2% from its depth averaged value $\rho_\circ=1.035\,\mbox{g}\,\mbox{cm}^{-3}$ (page 47 of Gill 1982). The Boussinesq approximation consists of replacing $\rho_\circ(z)$ by its vertically averaged value $\rho_\circ.$^4.1 In order to account for density variations affecting buoyancy, the Boussinesq approximation retains the full prognostic density $\rho=\rho(\lambda,\phi,z,t)$ when multiplying the constant gravitational acceleration. Equivalently, the vertical scale for variations in the vertical velocity is much less than the vertical scale for variations in $\rho_\circ(z)$, and fluctuating density changes due to local pressure variations are negligible. The latter implies that the fluid can be treated as incompressible, which excludes sound and shock waves.

In addition to the Boussinesq approximation, Bryan imposed the hydrostatic approximation, which implies that vertical pressure gradients are due only to density. When horizontal scales are much greater than vertical scales, the hydrostatic approximation is justified and, in fact, is identical to the long-wave approximation for continuously stratified fluids. According to Gill (1982), the ocean can be thought of as being composed of thin sheets of fluid in the sense that the horizontal extent is very much larger than the vertical extent^4.2. Therefore, kinetic energy is largely dominated by horizontal motions.

Consistent with the above approximations, Bryan also made the thin shell approximation because the depth of the ocean is much less than the earth's radius, which itself is assumed to be a constant (i.e., a sphere rather than an oblate spheroid). The thin shell approximation amounts to replacing the radial coordinate of a fluid parcel by the mean radius of the earth, unless this cordinate is differentiated. Correspondingly, the Coriolis component and viscous terms involving vertical velocity in the horizontal momentum equations are ignored on the basis of scale analysis. These assumptions form the basis of the Traditional Approximation. For a review and critique of the Traditional Approximation, as well as for a review of the typical approximations made in ocean modeling, see Marshall et al. (1997). Additionally, the thesis by Adcroft (1995) provides added details regarding the different dynamical processes omitted upon making the various approximations.

For the handling of subgrid scale (SGS) processes, Bryan made an eddy viscosity/diffusivity hypothesis. This hypothesis says that the affect of sub-grid scale motion on larger scale motions can be accounted for in terms of eddy mixing coefficients, whose size is many orders of magnitude larger than the molecular values. This hypothesis is controversial, and likely will remain so as long as turbulence remains a fundamentally unsolved problem. Pragmatically, however, some form of this approximation appears necessary in order to maintain numerical stability. Much of the research since Bryan revolves around SGS parameterizations. The hope is that such work will yield more physically based and consistent SGS assumptions.

Bryan made the rigid lid approximation to filter out external gravity waves. The speed of these waves places a severe limitation on economically solving the equations numerically. As noted above, displacements of the ocean surface are relatively small. Their affect on the solution is represented as a pressure against the rigid lid at the ocean surface. Two options in MOM relax the rigid lid approximation by allowing a free surface.