In the setting of the thin-shell approximation of the Euler equations in spherical coordinates for oceanic flows with variable density on the spinning Earth, we study a vorticity equation for a pseudo stream function \(\psi \) , whereby the assumption of incompressibility allows us to express the density as a function of \(\psi \) . Via an elliptic comparison argument, we show that, under certain assumptions, the (explicit) solution in the case of zero rate of rotation (i.e., on a fixed sphere) in a bounded region with smooth boundary contained either in the Northern or in the Southern Hemisphere is an approximation, in a suitable sense, of the corresponding solution of the equation with positive rate of rotation in the same region. This provides new insight into the dynamics of ocean gyres.
