Geophysical water flows with constant vorticity and centripetal terms

Calin Iulian Martin

doi:10.1007/s10231-020-00985-4

Abstract

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=eta (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.

Highlights

We focus here on a problem of paramount importance in the study of geophysical fluid dynamics (GFD), namely, the determination of explicit analytical solutions to the full governing equations and their boundary conditions
In regard to the two-dimensionality, a somewhat similar result was achieved by Martin [45], where it was proved that a water flow satisfying the water wave equations with full Coriolis terms has a two-dimensional character, but of different structure
Writing the family of solutions we obtained in terms of the fixed inertial frame, we find that it has non-vanishing velocity field, the pressure and the free surface are explicitly determined, the latter being non-flat

Summary

Introduction

We focus here on a problem of paramount importance in the study of geophysical fluid dynamics (GFD), namely, the determination of explicit analytical solutions to the full governing equations and their boundary conditions. In regard to the two-dimensionality, a somewhat similar result was achieved by Martin [45], where it was proved that a water flow satisfying the water wave equations with full Coriolis terms (but without centripetal ones) has a two-dimensional character, but of different structure.

Results

Conclusion