Constant vorticity water flows in the modified equatorial $$\beta $$-plane approximation

In this paper, we study three-dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the modified $$\beta $$ -plane approximation. It is proved that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity, due to the incorporation of a gravitational-correction term in the tangent plane approximation.