Wave Over-reflection and Baroclinic Instability of the Eady Problem

Abstract

The concept of wave over-reflection is applied to the unstable normal mode of the Eady problem. In order to consider propagation properties of the Rossby waves trapped at the boundaries, two thin boundary layers are introduced at the top and bottom of the fluid layer of the original model. It is shown that the Rossby waves in the boundary layers are always over-reflected as long as their critical levels exist in the constant-shear evanescent region and the waves are transmitted to the opposite boundary layer. The dispersion relation obtained by using laser formula and quantization qualitatively coincides with that of the normal mode. Although the growth rate is systematically overestimated, the short-wave cutoff is well described by the over-reflection solution. The mechanism of over-reflection obtained in this study is understood by conservation of momentum. Since the pseudomomentum of the transmitted and the incident waves have opposite signs to each other, the amplitude of the reflected wave should become larger than that of the incident wave to satisfy constant momentum flux constraints. This mechanism corresponds to that of over-reflection in a linear shear flow of the shallow water model shown in a paper by Takehiro and Hayashi.