The Number of Traveling Wave Families in a Running Water with Coriolis Force

In this paper, we study the number of traveling wave families near a shear flow under influence Coriolis force, where speeds lie outside range u. Under $$\beta $$ -plane approximation, if u has critical point at which attains its minimal (resp. maximal) value, then unique transitional value exists in positive negative) half-line such that changes suddenly from finite to infinite when passes through it. On other hand, no points, is always for values. This true general flows mildly technical assumptions, and large class including cosine jet $$u(y) = {1+\cos (\pi y)\over 2}$$ (i.e. sinus profile) analytic monotone unconditionally. The sudden change indicates long time dynamics around much richer than non-rotating case, exist.