Supercritical Dynamics of Baroclinic Disturbances in a Free-Surface Model

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Abstract
Nonlinear dynamics of unstable baroclinic disturbances are examined in the context of the Eady model modified by Ekman dissipation at the lower boundary while the upper boundary remains stress-free. Three approaches are used: the asymptotic approach which pivots about the constraints of strong bottom dissipation and weak supercriticality, the ad hoc approach which neglects wave-wave interactions by truncating the wave field to a single wave, and the spectral numerical approach. The time evolution of the disturbance is generally characterized by a “single hump” pattern consisting of a growth stage to a maximum amplitude followed by a decay stage. During the decay stage, the spectral solution develops an amplitude vacillation which, for most parameter settings, becomes chaotic in nature and persists at a mean level substantially below the “hump” maximum and of the order of the initial amplitude. The exceptions are moderate or long waves in a strongly viscous fluid, for which the vacillation decays yielding a wave free final state, and short waves in a strongly viscous and weakly stratified fluid, for which both the initial “hump” and the subsequent vacillation are of the order of the initial amplitude. In the weak viscosity limit, a different kind of vacillation also appears during the early evolution stage of the disturbance. The asymptotic and the ad hoc solutions qualitatively capture the growth and the decay of the disturbance but fail to predict its vacillatory final state.

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