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Richard C. Deininger

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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Richard C. Deininger

Abstract

The finite-amplitude evolution of the instability of a nonparallel basic-state flow and the basic state are studied. The basic state consists of a free Rossby wave in an inviscid, barotropic beta-plane model. The method of multiple time scales is used to obtain the weakly nonlinear evolution of the system on the long time scale corresponding to the slow growth of the slightly unstable perturbation.

The results of the analysis show that, as the perturbation grows, both the amplitude and phase of the Rossby wave are modified, producing a nonlinear feedback which acts to stabilize the perturbation. Feedback due to nonlinearly produced harmonics can be either stabilizing or destabilizing to the perturbation. The total feedback is usually stabilizing and leads to an oscillatory exchange between the Rossby wave and perturbation. The mechanism responsible for the nonlinear feedback is the tilted trough mechanism.

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Richard C. Deininger
and
Arthur Z. Loesch

Abstract

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Arthur Z. Loesch
and
Richard C. Deininger

Abstract

Nonlinear dynamics of equatorial waves, interacting resonantly in coupled triad configurations which form closed systems, are investigated in the context of the divergents β-plane model. Closure is attained by demanding that spatial structures of the modes obey the atmospheric constraints. At larger fluid depths the wave systems are relatively small and concentrated at the smaller wavenumbers; at small depths the systems are larger and spread more widely in the wavenumber domain. Strong energy transfers in a system are consistently associated with modes characterized by the maximum frequency in individual triads. The lower frequency modes are energetically less active, especially when their frequencies are much less than and amplitudes greater than those of the maximum frequency modes in the same triads.

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