Geophysical water flows with constant vorticity and centripetal terms

• Published in: Annali di matematica pura ed applicata 2021, Vol.200 (1), p.101-116
• Main Author: Martin, Calin Iulian
• Format: Article
• Language: English
• Subjects: Boundary conditions; Coordinates; Coriolis force; Exact solutions; Free surfaces; Geophysics; Mathematical analysis; Mathematics; Mathematics and Statistics; Rotation; Three dimensional flow; Velocity distribution; Vorticity; Water waves; Wave equations
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-020-00985-4

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.