Derivation and Solution of the Omega Equation Associated With a Balance Theory on the Sphere - John F. Dostalek; Wayne H. Schubert; Mark DeMaria (Journal of Advances in Modeling Earth Systems)
Abstract The quasi‐geostrophic omega equation has been used extensively to examine the large‐scale vertical velocity patterns of atmospheric systems. It is derived from the quasi‐geostrophic equations, a balanced set of equations based on the partitioning of the horizontal wind into a geostrophic and an ageostrophic component. Its use is limited to higher latitudes, however, as the geostrophic balance is undefined at the equator. In order to derive an omega equation which can be used at all latitudes, a new balanced set of equations is developed. Three key steps are used in the formulation. First, the horizontal wind is decomposed into a nondivergent and an irrotational component. Second, the Coriolis parameter is assumed to be slowly varying, such that it may be moved in and out of horizontal derivative operators as necessary to simplify the derivation. Finally, the mass field is formulated from the nondivergent wind field. The resulting balanced set of equations and the omega equation derived from them take a similar form to the quasi‐geostrophic equations, yet are valid over the whole sphere. A method of solution to the global omega equation using vertical normal modes and spherical harmonics is presented, along with a middle‐latitude and low‐latitude example.