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In this paper, we devise a new exact and partially explicit solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible azimuth flow with a discontinuous density distribution and subjected to forcing terms in terms of cylindrical coordinates. The obtained solution represents a steady, purely azimuthal, stratified flow with an associated free surface and an interface that is suitable for describing the Antarctic Circumpolar Current. Resorting to a functional analysis, we demonstrate that the relationship between the imposed pressure at the free surface and the resulting surface deformation is well-defined and show that the continuity of the pressure along the interface generates an equation that describes implicitly the shape of the interface. Moreover, a particular example is considered to show that the interface can be determined explicitly. Finally, we derive an infinite regularity about the interface and obtain the expected monotonicity properties between the surface pressure and its distortion.