A cylindrical coordinates approach to constant vorticity geophysical waves with centripetal forces at arbitrary latitude

Abstract

We consider three-dimensional geophysical flows with centripetal forces and at arbitrary latitude, with constant vorticity beneath a wave train and above a flat bed. We consider the f-plane approximation as well as the β-plane approximation. For the f-plane approximation, we determine all possible flow solutions that exhibit constant vorticity vector. For the β-plane approximation, we show that the flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity. Our results reveal some essential differences from those results in the literature, due to the presence of centripetal forces. Moreover, for the case exhibiting the surface tension, we prove that there are no flows exhibiting constant vorticity.