Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane

Abstract

We study the nonlinear response of the equatorial shallow-water system at rest to a localized long-wave perturbation with small meridional to zonal aspect ratio. An asymptotic theory of such a response (adjustment) for small Rossby numbers is constructed. Possible scenarios of nonlinear adjustment are classified depending on the relation between the Rossby number and the aspect ratio. The calculations show that slow, geostrophically balanced Rossby and Kelvin waves and the fast inertia–gravity waves are dynamically split off. The fast component of motion exerts no drag on the slow one, which is proved by direct computation. Evolution equations are derived for both components confirming earlier results which were obtained by ad hoc filtering of one of the components of motion. A well-defined initialization procedure is developed for each component.

Due to the breaking of non-dispersive Kelvin waves, the asymptotic theory has obvious limits of validity. In order to go beyond these limits and to study strongly nonlinear effects during the adjustment process we undertook high-resolution shock-capturing numerical simulations based on recent progress in finite-volume numerical methods. The simulations confirm theoretical results but also reveal new effects such as fission of a strongly nonlinear Rossby-wave packet into a sequence of equatorial modons or jet formation in the wake of a breaking Kelvin wave.