Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

We are concerned here with an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates, which incorporates discontinuous fluid stratification. This solution represents a steady, purely azimuthal equatorial flow with an associated free surface and an interface separating two fluid regions, each of which has its own continuous distribution of density. However, the two density functions do not match along the interface. Following the derivation of the solution, we demonstrate that there is a well-defined relationship between the imposed pressure at the free surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. Interestingly, it turns out that the interface defining function has infinite regularity.