The Eigensolutions of the Linearized Balance Equations over a Sphere

Abstract

Solutions of the linearized balance equations over a sphere are presented and compared with the Laplace's tidal equations results obtained by Longuet-Higgins. On these lines, this study searches for a partial answer concerning the accuracy of the balance system for describing slow, large-scale motions in the atmosphere. The solutions corresponding to Hough's second class waves [small values of ε = (2 Ωa)2/c2] are well described by the balance system. At large values of ε there are apparent discrepancies for the Rossby symmetric modes as compared to Longuet-Higgins type 2 solutions. Nevertheless, for the antisymmetric modes the agreement is good. The linearized version of the motions studied by Burger is also a solution of the balance equations, corresponding to small frequencies and negative values of ε. There are also unrealistic solutions (in the light of the balance approximation) with high frequencies and ε < 0. An integral theorem shows that for ε > 0 only westward propagating waves are solutions of the balance system. In particular, it shows that the equatorial Kelvin wave is not a solution. The westward propagating part of the mixed Rossby-gravity mode is a solution, but with slightly higher frequency when compared to Longuet-Higgins’ results. A study of the “modified balance” equations derived by Charney shows that they describe well all the equatorial Rossby modes. They also describe the equatorial Kelvin wave at large values of ε. Unfortunately, they have additional unrealistic high-frequency eastward-propagating free wave solutions. An interative numerical method is suggested in the hope of avoiding these spurious solutions. A two-layer spherical model is used to study the instability properties of a basic state of solid rotation. It shows that the balance and the quasi-geostrophic equations have unstable solutions which are remarkably alike for realistic values of the parameters involved.