On Ocean Currents with Constant Vorticity: Explicit Solutions and an Application to the ACC

Abstract

We study the three-dimensional, divergence-free, incompressible Euler equations in the $f$-plane approximation for off-equatorial oceanic flows of constant vorticity, where the fluid domain is bounded by a free surface and a flat bed. The major difference, compared to related earlier works, is that we refrain from any global restrictions on solutions with respect to the latitudinal coordinate $y$, which we justify by the $f$-plane approximation's locality. The resulting flows are necessarily steady (despite the time dependence of the governing equations), zonal, independent of the zonal coordinate $x$, and fully explicit; the corresponding free surface exhibits a nontrivial parabolic structure in $y$. We also provide an application to the Antarctic Circumpolar Current (ACC) for which we compare the sea surface height predicted by our constant vorticity model with satellite altimetry measurements available in the literature.