Nonlinear Equilibration of Two-Dimensional Eady Waves

Abstract

The initial-value problem for Eady's model is reexamined using a two-dimensional (x–z) primitive equation model. It is generally accepted that a finite amplitude instability of Eady's basic state will produce a frontal discontinuity in a finite time. When diffusion prevents the frontal discontinuity from forming, the wave amplitude eventually stops growing and begins to oscillate. We analyze this equilibration and suggest that it is a result of enhanced potential vorticity in the frontal region that is mixed into the interior from the boundaries. The dynamics of equilibration is crudely captured in a modified quasi-geostrophic model in which the zonal-mean static stability is allowed to vary. The magnitude of the meridional wind speed of the equilibrated wave is O(N0H), where N0 is the initial buoyancy frequency and H is the depth of the fluid. This is of the same order as the amplitude of the wave predicted by semigeostrophic theory at the point of frontal collapse. Scaling arguments are presented to determine the three-dimensional flows for which this equilibration mechanism should be important. It is argued that this mechanism is likely to be of some importance for shallow cyclones forming in regions of weak low-level static stability.