Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary‐wave‐like development of small‐amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean‐flow velocity at a certain latitude in the β‐plane. We presented a general theory for the nonlinear critical‐layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D‐wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E‐wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical‐layer flow expansion diverged in the case of the E‐wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E‐wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg‐de‐Vries type‐equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E‐waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E‐wave during the critical‐layer formation is evaluated in the quasi‐steady régime assumption.