Baroclinic Theories of the Wind Driven Circulation

Abstract

This article reviews, some idealized baroclinic theories of the wind driven ocean circulation. The two layer quasigeostrophic model, where the layers represent the upper thermocline waters rather than the full depth of the ocean, is used throughout. The emphasis is on interior solutions and the role of mesoscale eddies. Western boundary layers, which close the flow patterns, are ignored. This heavily idealized model is a convenient expository vehicle for important concepts which re-emerge in more complicated multilayer (and continuously stratified), nonquasigeostrophic theories.

Sections 1 and 2 give a brief review of the scaling arguments and physical assumptions which are used to simplify the equations of motion. Section 1 shows how rapid rotation ensures vertical velocities are much smaller than naive scale analysis of the mass conservation equation suggests. This is responsible for a major simplification: vortex tilting and twisting cannot effectively produce vertical vorticity. Hence the privileged position of vertical vorticity in the theory of rapidly rotating fluids. Section 2 develops the two layer model and potential vorticity dynamics. Tractable models are obtained by simplifying the potential vorticity equation using one of two complementary approximations: quasigeostrophy or planetary geostrophy. A particular example, the propagation of a long nonlinear baroclinic Rossby wave, is used to illustrate the connection between these approximations.

Section 3 introduces the concept of a geostrophic contour. In the absence of forcing and dissipation fluid cannot cross geostrophic contours. Thus the geometry of geostrophic contours (closed, blocked by eastern boundaries, impinging on the base of the mixed layer) constrains the fluid motion. This is illustrated with two examples: flow around topographically closed contours in a one layer model and closure of the lower layer geostrophic contours in a two layer model. In both these examples the ideal fluid equations have an infinite number of solutions and it is necessary to consider the effects of small dissipation to select a unique one. Different types of dissipation select different solutions from the infinity admitted by the ideal fluid equations.

Section 4 and 5 take up this last point. In section 4 it is argued that the mesoscale eddy field is the dominant dissipation mechanism which retards the large scale wind driven flow. Its mean field effect is plausibly a down-gradient flux of potential vorticity. Section 5 uses an extension of the Prandtl-Batchelor theorem to conclude that this downgradient flux leads to the expulsion of potential vorticity gradients from closed geostrophic contours. Thus lateral diffusion of potential vorticity has selected a solution in which the potential vorticity is homogenized inside closed geostrophic contours. This selection principle allows us to construct a complete picture of the baroclinic circulation in the Sverdrup Interior.

It is emphasized, using passive scalar advection-diffusion models, that homogenization occurs only if diffusion is weak. Thus the term “potential vorticity mixing” is misleading when applied to homogenization since it has the erroneous connotation that the stronger the diffusivity the more homogeneous the potential vorticity. These passive scalar problems also allow one to examine the departures from the homogenized state. These corrections are exponentially small with distance from the nonhomogeneous region.