In this chapter, we further investigate the influence of steps on finite difference models and, in particular, we consider the accuracy of model vorticity budgets for wind-driven circulations under the free-slip dynamic boundary condition. Free-slip circulations are typically more energetic than no-slip circulations, e.g. Pedlosky pedlo96. He considered the vorticity budget for a quasi-geostrophic (QG) model. Simple scaling arguments reveal that vorticity is more easily fluxed out of the basin when no-slip conditions are employed. When there is a net vorticity forcing under free-slip conditions, therefore, stronger gyres are needed to achieve the necessary viscous flux of vorticity across the basin boundary. The vorticity budget is also an interesting diagnostic tool because all the terms are in the form of domain integrals that can be transformed into boundary integrals. This suggests that values of these integrals may be very sensitive to coastline representation and that careful consideration of the vorticity budget may give further insight into the effect of steps on the overall strength of the gyres. The difficulty is in deriving a vorticity budget consistent with the model's numerical formulation.
We propose to test different formulations for the advective and diffusive terms for the shallow water C-grid model detailed in Section 4.2. Additionally, we use vorticity budgets to investigate problems we found with the B-grid model in Section 4.3. Finally, we draw some similarities with results from a quasi-geostrophic (QG) FD model in Section 4.4. Indeed, it may seem reasonable that vorticity budgets are more accurate in QG models since the vorticity equation is solved instead of the primitive equations. Section 4.1, 4.2 and 4.4 are excerpts from a paper we intend to submit to Tellus (the authors are Frédéric Dupont, David N. Straub and Charles A. Lin).
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