On the validity of the surface quasi-geostrophic model

Abstract: We discuss the limitations of the popular surface quasi-geostrophic model in applications to realistic atmospheric and oceanic flows. This model, derived at first order in Rossby number (assumed small) of the three-dimensional primitive equations, reduces the dynamics to the two-dimensional inviscid conservative transport (advection) of surface buoyancy or potential temperature by assuming that the interior potential vorticity remains uniform for all time. The surface field enables one to construct the entire three-dimensional instantaneous flow structure via geostrophic and hydrostatic balance. However, the conservative transport of the surface field typically results in the development of increasingly sharp gradients in that field, implying an unbounded growth in Rossby number, or more precisely the ratio of the Rossby number to its initial value. The model equations are independent of Rossby number, but the second-order balance relations needed for instance to diagnose the vertical velocity or the static stability are not. Any comparison with observational data, or with simulations of the parent primitive equation model, requires one to specify an initial Rossby number (and Reynolds number, which is another issue). Thus, in any such comparison, the surface quasi-geostrophic model will fail to correspond to observational data or primitive equation simulations once the Rossby number exceeds order unity. Here, we show that this violation can happen early in the flow evolution, after a time of the order of a single eddy turn-around time, for Rossby numbers typical of large-scale atmospheric and oceanic dynamics. A potential fix is to add viscosity of some form to the surface quasi-geostrophic model, but this is inconsistent with the derivation of the model (the potential vorticity would then not remain uniform in the interior). Even disregarding this inconsistency, we show that the model still diverges from its parent primitive equation model so long as the (numerical) diffusion is sufficiently weak.