Stratified Ocean Currents with Constant Vorticity

We analyze vertically stratified three-dimensional oceanic flows under the assumption of constant vorticity. More precisely, these flows are governed by the f-plane approximation for the divergence-free incompressible Euler equations at arbitrary off-equatorial latitudes. A discontinuous stratification gives rise to a freely moving impermeable interface, which separates the two fluid layers of different constant densities; the fluid domain is bounded by a flat ocean bed and a free surface. It turns out that the constant vorticity assumption enforces almost trivial bounded solutions: the vertical fluid velocity vanishes everywhere; the horizontal velocity components are simple harmonic oscillators with Coriolis frequency f and independent of the spatial variables; the pressure is hydrostatic apart from sinusoidal oscillations in time; both the surface and interface are flat. To enable larger classes of solutions, we discuss a forcing method, which yields a characterization of steady stratified purely zonal currents with nonzero constant vorticity. Finally, we discuss the related viscous problem, which has no nontrivial bounded solutions.