P dJpSp VdSJp

Equations (2.152) and (2.153) combined with Eqns. (2.144) and (2.147) constitute a complete set of basic equations of a dynamical system under the so-called Boussinesq approximations, i.e., the following simplifications are assumed:

• Using the volume conservation to replace the mass conservation

• Retaining the buoyancy forcing associated with small density deviation

• Using volume conservation in the tracer (temperature and salinity) prognostic equations.

2.7.2 Potential problems associated with Boussinesq approximations

Boussinesq approximations have been widely used in the study of atmospheric and oceanic circulations. In particular, using the volume conservation to replace the continuity equation filters out the sound waves and greatly simplifies the dynamics. The momentum equation is linearized and is therefore much easier to handle. However, these approximations can also cause some problems, which require caution:

• In reality neither a nor P is constant, depending nonlinearly upon temperature and pressure.

• Mass conservation is violated; thus, gravitational potential energy is not conserved, owing to the existence of an artificial source/sink of gravitational potential energy in the model.

• Sea surface elevation and bottom pressure due to surface heating/cooling and freshwater fluxes are incorrectly simulated in such models.

• The accurate calculation for both temperature and salinity should be based on mass conservation, as stated in Eqn. (2.8); thus, using the volume conservation approximation in the advection term, such as Eqns. (2.147a) and (2.147b), can introduce errors on the order of a few percent.

2.7.3 Buoyancy fluxes

Since Boussinesq approximations have been widely used in oceanography, the equivalent dynamical effects of surface heat and freshwater fluxes are simulated in terms of two related fluxes: density and buoyancy fluxes. Density flux is defined as

Fp = -po (aFj - PFs), a =--|p,S , P = — |p,r (2.154)

where FT = Q/p0cp, Q is the net heat flux into the ocean, cp is the heat capacity of water; FS = (E - P)S/(1 - S), where E — P is evaporation minus precipitation.

Buoyancy is defined as b = -g Ap/ p°; thus, light water is more buoyant. In terms of the air-sea fluxes, therefore, buoyancy flux is defined as

and its units are m2/s3.

Although the concepts of both density flux and buoyancy flux are very useful in describing dynamical effects due to surface thermohaline forcing, these fluxes are artificial or conceptual and are applicable to Boussinesq models only. We need to be clear-headed about the possible problems.

First, density flux associated with surface thermal forcing is an artifact associated with the Boussinesq approximations. When water is cooled down, its density increases, so the water parcel should shrink, but the total mass should be the same as before. In a Boussinesq model, density is increased; however, owing to the volume conservation approximation, the increase in density gives rise to more mass in the ocean, and thus an artificial density flux from the atmosphere to the ocean.

Second, density flux associated with evaporation and precipitation through the sea surface is also incorrect. In the ocean, precipitation and evaporation add a mass flux of

Owing to precipitation, surface salinity tends to decline and the corresponding density decreases. This decline of density due to precipitation in a Boussinesq model gives rise to a loss of mass because the total volume is assumed to be unchanged; thus precipitation is interpreted in terms of a negative density flux through the sea surface. According to Eqn. (2.154), the dynamical effect of precipitation is interpreted as the equivalent density flux

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