Steady azimuthal stratified flows over a sloping beach

In this paper, we construct an exact solution to the governing equation for geophysical fluid dynamics in cylindrical coordinates. The obtained solution represents a steady, purely azimuthal equatorial flow with a general density distribution over a sloping beach, and this solution can be used to model the equatorial undercurrent. Moreover, we establish a Bernoulli-type relationship between the imposed pressure at the free surface and the resulting distortion of the surface's shape. A further employment of the implicit function theorem demonstrates the existence of surface function solutions, and functional analytic considerations provide the expected monotonicity properties. Finally, we use the short-wavelength perturbation approach to prove that the equatorial flow is linearly stable under a specific density selection.