Topographically Forced Wave instability at Finite Amplitude

Richard C. Deininger Department of Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge 02139

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Abstract

The method of multiple time scales was used to study the weakly nonlinear effects on the instability of a basic state consisting of a topographically forced wave in an inviscid, barotropic beta-plane model. The results obtained differ substantially from those obtained when the basic state is a free Rossby wave. Here the basic-state wave is fixed in phase with respect to the mountain, while the amplitude of the topographic wave and perturbation evolve. The nonlinear feedback between the topographic wave and perturbation gives rise to an oscillation for a topographically subresonant zonal flow and an explosive nonlinear instability for a topographically superresonant zonal flow. In the subresonant case, the effect of the perturbation on the forced wave is a dissipative one, when averaged over the course of the nonlinear oscillation. The standing topographic wave interacts with the traveling instability on the topographic wave through the convergence of Reynolds' stresses which is suggestive of the way in which standing and traveling eddies interact in the atmosphere.

Abstract

The method of multiple time scales was used to study the weakly nonlinear effects on the instability of a basic state consisting of a topographically forced wave in an inviscid, barotropic beta-plane model. The results obtained differ substantially from those obtained when the basic state is a free Rossby wave. Here the basic-state wave is fixed in phase with respect to the mountain, while the amplitude of the topographic wave and perturbation evolve. The nonlinear feedback between the topographic wave and perturbation gives rise to an oscillation for a topographically subresonant zonal flow and an explosive nonlinear instability for a topographically superresonant zonal flow. In the subresonant case, the effect of the perturbation on the forced wave is a dissipative one, when averaged over the course of the nonlinear oscillation. The standing topographic wave interacts with the traveling instability on the topographic wave through the convergence of Reynolds' stresses which is suggestive of the way in which standing and traveling eddies interact in the atmosphere.

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