An Analysis of Interactions between Geostrophic and Ageostrophic Modes in a Simple Model

Abstract We examine some properties of the extratropical atmosphere which act to maintain quasi-geostrophic balance. A nonlinear, f-plane, primitive equation, two-layer model is used. The momentum and temperature fields are described in terms of normal modes of the system given by the model's linear terms. These modes are classified as either geostrophic or ageostrophic depending on their associated eigenvalues. The original nonlinear equations are transformed into a system in which the modulations of the modal amplitudes by nonlinear effects are explicitly expressed in terms of the modal amplitudes themselves. This transformation facilitates a multiple-time-scale analysis. The stability of simple finite-amplitude geostrophic solutions with respect to infinitesimal perturbations in other modes is investigated. Results are discussed for nondimensional unperturbed-state amplitudes of magnitude ϵ < 1. These geostrophic solutions may be unstable with respect to further geostrophic perturbations, with growth r...