A high-latitude quasi-geostrophic delta plane model derived from spherical geometry

Abstract

For quasi-geostrophic models, the beta plane approximation is well established and can be derived from spherical geographic coordinates. It has been argued that such a connection does not exist for a higher-order approximation, the so-called delta plane. Here it will be demonstrated that a quasi-geostrophic potential vorticity equation on the delta plane can formally be derived using rotated geographic instead of geographic coordinates. The rigorous derivation of such a model from the shallow-water equations leads to a correction of previous more intuitive-based formulations of the delta plane model. Some applications of the corrected delta plane model are given. It is shown that the delta plane model describes well the low-frequency basin modes of a polar plane shallow-water model. Moreover, it is found that the westward phase speed of the delta plane model shows a dependency on latitude comparable to a model on the sphere. The ratio of delta to beta plane zonal phase speed decreases monotonically with increasing latitude, in qualitative agreement with the phase speed ratio obtained by comparing a spherical to a beta plane model. Finally, it is demonstrated analytically that Rossby wave energy rays are curved on the delta plane, in contrast to the beta plane. Ray curvature is important for a realistic description of energy dispersion at high latitudes. The results suggest that the quasi-geostrophic delta plane model is a suitable tool for conceptual studies on Rossby wave dynamics at high latitudes.