Initially Forced Long Planetary Waves in the Presence of Nonzonal Mean Flow

Abstract

Abstract The purpose of this paper is to understand how long planetary waves evolve when propagating in a subtropical gyre. The steady flow of a wind-driven vertically sheared model subtropical gyre is perturbed by Ekman pumping that is localized within a region of finite lateral extent and oscillates periodically at about the annual frequency after sudden initiation. Both the background flow and the infinitesimal perturbations are solutions of a 2½-layer model. The region of forcing is located in the eastern part of the gyre where the steady flow is confined to the uppermost layer (shadow zone). The lateral scales of the forcing and of the response are supposed to be small enough with respect to the overall gyre scale that the background flow may be idealized as horizontally uniform, yet large enough (greater than the baroclinic Rossby radii) that the long-wave approximation may be made. The latter approximation limits the length of time over which the solutions remain valid. The solutions consist of (i) a forced response oscillating at the forcing frequency in which both stable (real) and zonally growing (complex) meridional wavenumbers are excited plus (ii) a localized transient structure that grows as it propagates away from the region of forcing. Application of the method of stationary phase provides analytical solutions that permit clear separation of the directly forced part of the solution and the transient as well as estimation of the temporal growth rate of the transient, which proves to be convectively unstable. The solutions presented here are relevant to understanding the instability of periodic (including annual period) perturbations of oceanic subtropical gyres on scales larger than the baroclinic Rossby radii of deformation.