Geophysical water flows with constant vorticity and centripetal terms.

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity. To be more specific, we show that the components of the velocity field (with respect to the rotating coordinate system) vanish. We also determine a formula for the pressure function and we show that the surface, denoted z = η ( x , y , t ) , is time independent, but is not flat and can be explicitly determined. We conclude our analysis by converting to the fixed inertial frame, the solutions we obtained before in the rotating frame. It is established that, in the fixed frame, the velocity field is non-vanishing and the free surface is non-flat, being explicitly determined. Moreover, the system consisting of the velocity field, the pressure and the surface defining function represents explicit and exact solutions to the three-dimensional water waves equations and their boundary conditions.

A steady stratified purely azimuthal flow representing the Antarctic Circumpolar Current.

We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.

On periodic geophysical water flows with discontinuous vorticity in the equatorial f-plane approximation.

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f-plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ1 adjacent to the surface situated above another layer of constant non-zero vorticity γ2≠γ1 adjacent to the bed. For certain vorticities γ1,γ2, we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows.This article is part of the theme issue 'Nonlinear water waves'.