Constant Vorticity Ekman Flows in the $$\beta $$-Plane Approximation

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Three dimensional Ekman flows with constant vorticity is considered in the $$\beta $$ -plane approximation. It is remarkable that we explore a totally new approach, which is much different from procedure in (Martin J Fluid Mech 865:762–774, 2019; Chu and Yang J Differ Equ 269:9336–9347, 2020), to show that any Ekman flow with a flat surface and constant vorticity vector is the stationary flow with vanishing velocity field.