  In this paper, we aim to derive an exact solution to the three-dimensional
governing equations for wind-induced equatorial flows in the $\beta$-plane
approximation with a depth-dependent density distribution and subjected to
centripetal terms. The obtained explicit solution represents a steady purely
azimuthal stratified flow with a flat surface and an impermeable flat bed that
is suitable for describing the Equatorial Current. Resorting to a functional
analysis, we show that the thermocline exhibits some monotonicity properties.
