Influences of the earth's sphericity in the Quasi-Geostrophic theory

Abstract

It is shown that a model characterised by the quasi-geostrophic momentum and thermodynamic equations incorporating the full effects of the earth's sphericity is governed by an extended form of the potential vorticity equation for a single unknown, the geopotential. That equation contains several additional terms that represent the vortex stretching due to the horizontal divergence of the quasigeostrophic flow and the advection of planetary vorticity by the nongeostrophic component of the flow. They are shown to have significant quantitative impacts in the context of two simple nontrivial dynamical problems. First, the analytical and numerical investigations of the fundamental modes of oscillation verify that both the frequency and structure of the normal modes in this system closely approximate those of the corresponding Hough modes. The counterpart frequency obtained with the additional approximations as in a traditional quasi-geostrophic model overestimates by about 25 %. The relative error is larger for the longer waves. Second, the difference between Matsuno's eq. for planetary wave propagation and the linearised form of the governing eq. derived in this study is identified. The wave propagation governed by the two equations is structurally quite similar and Matsuno's eq. yields a somewhat stronger response under the same parameter condition. The sample calculations reveal that the forced wave field governed by the corresponding conventional Q-G eq. is structurally more distorted than that obtained with Matsuno's equation, although the maximum amplitude is closer to that obtained with the equation derived in this study. The corresponding conventional Q-G equation has an intrinsic deficiency in that the wave propagation has a spurious dependence upon the constant of integration in the expression for the basic geopotential field associated with a given geostrophic basic flow.