Non-Linear Behaviors of Forced Baroclinic Wave in a Continuous Zonal Flow

Abstract

The nonlinear behaviors of Eady-type baroclinic waves forced quasi-resonantly by topography are investigated. The complex amplitude equation which is a Landau equation with forcing terms is derived by using the perturbation method; the effect of viscosity is taken into account in the leading order as the Ekman layers at the top and the bottom of the fluid layer.
For a given wave number and a basic zonal flow with a constant vertical shear and with zero vertical mean, the resonance due to the topographical forcing occurs just at the critical thermal Rossby number (β=β0) of baroclinic instability. However, when the wave self-interaction is taken into account, the maximum response occurs at a value smaller than β0. Three equilibrium solutions, say A, B, and C, are found near the maximum response; the largest amplitude state, A, is stable, while B, the smallest amplitude, and C with amplitude almost equal to A are both unstable. It is shown that B is unstable for amplitude and phase perturbation, while C is unstable for phase perturbation only. It is noted that C is identical with A except phase difference 180° and stable if the bottom is flat. In addition, by plotting flow vectors composed of the time change of amplitude and phase on amplitude-phase plane, it is suggested that the potential surface characterizing the stabilities would have a minimum around state A, a maximum around B and a saddle point around C.
At a slightly off-resonant condition of the basic zonal flow, too, we can have three equilibrium solutions for a certain narrow range of β. The stabilities of these solutions are similar to those obtained under the resonant condition. In addition, it is shown by the numerical integration of the complex amplitude equation that baroclinic waves propagate, interacting with the forced waves when the height of topography is small, and hence the amplitude and phase velocity oscillate with time. When the height is large, the phasee velocity becomes small, and if it exceeds some critical value, the baroclinic waves are trapped by the topography.