Abstract
Non-linear resonance of external Rossby wave in a continuously stratified atmosphere under cyclic condition in the meridional direction is studied by the perturbation expansion method. The feed-back effects of the mean zonal flow induced by the wave-topography (and/or heating) and wave-self interaction are included. As the forcings, a sinusoidal mountain and/or heating are assumed, while the Ekman damping, Ray Leigh friction and Newtonian cooling are included as the dissipations.
In the case of topographic forcing alone, it is shown that for long (short) waves the amplitude response curves incline toward negative (positive) side of the detuning parameter δ (deviation of the basic zonal flow from the linear resonant condition), to give the socalled multiple (three) equilibrium solutions for a certain range of δ for small damping coefficients (e.g., Charney and DeVore, 1979; hereafter referred to as CD). The positions of maximum response on δ-axis shift in proportion to 2nd power of the magnitude of forcing and -2nd power of the damping coefficients. At a 'critical' wave number, the amplitude response curve is similar to that of linear resonance as a result of the cancellation between non-linear terms; the multiplicity of equilibrium vanishes.
The stabilities of equilibrium solutions are examined by perturbation method, to show that the upper and the lower branches are stable, while the middle branch is unstable as has been pointed out by CD. All the equilibrium states outside the multiple equilibrium region are stable.
If thermal forcing is incorporated, the response becomes complicated, depending upon its magnitude and its phase relative to the topography; for example, the amplitude response curve forms a loop if plotted against δ and the largest amplitude state is shown to be unstable. This loop becomes larger and wider as the heating increases. It is noted that such a loop can be found also in CD's model if the mountain becomes high, though it seems to have been overlooked so far. Finally, we show the time-dependent behaviors of the solutions by numerical integrations of the equation obtained by the present method.
