Nonlinear Instability of a Forced Baroclinic Rossby Wave

Abstract

Two-layer, quasi-geostrophic weakly nonlinear and low-order spectral models are developed and used to investigate the instability of forced baroclinic Rossby waves to finite-amplitude perturbations. The results are then applied to the interaction of planetary-scale stationary eddies with synoptic scale transient eddies. In the weakly nonlinear model, asymptotic series expansions are used in conjunction with the method of multiple time scales. The stability of a forced planetary-scale stationary baroclinic Rossby wave to synoptic-scale perturbations is first examined. The synoptic-scale perturbation modes initially grow exponentially after which they eventually settle into an amplitude vacillation cycle. This vacillation is driven by the linear interference between propagating and stationary synoptic-scale modes with the same zonal and meridional wavenumbers. During this vacillation, the time mean energy of the stationary planetary wave equals its initial value. This indicates that the transient synoptic-scale perturbation has neither an amplifying nor a dissipative influence on the stationary wave. A study of the energetics shows that eddy available potential energy is transferred from the planetary-scale stationary wave to the synoptic-scale perturbation, while eddy kinetic energy is simultaneously transferred in the reverse direction. The asymptotic series expansions are also used to determine the truncation for a fully nonlinear spectral model. The weakly nonlinear and spectral solutions are compared and are found to agree very well. In addition, by comparing spectral model solutions with and without the higher-order modes of the weakly nonlinear model present, it is found that the evolution of the basic wave and the perturbation are extremely sensitive to the presence of these modes. This suggests that the interaction between planetary-scale stationary eddies with synoptic-scale transient eddies is a nonlinear phenomenon that is very sensitive to the detailed structure of the eddies present.