Liouville-type results for time-dependent stratified water flows over variable bottom in the $\beta$-plane approximation

TL;DR

This paper proves Liouville-type results for time-dependent stratified water flows over variable bottoms, showing that bounded solutions require specific velocity conditions and a flat interface, with implications for geophysical fluid dynamics.

Contribution

It extends previous work by considering non-flat bottoms and different constant vorticities in each layer, providing new conditions for bounded solutions in stratified water flows.

Findings

Bounded solutions exist only under specific velocity conditions.

The interface must be flat, and the free surface exhibits traveling behavior.

Pressure is hydrostatic in both layers.

Abstract

We consider here time-dependent three-dimensional stratified geophysical water flows of finite depth over a variable bottom with a free surface and an interface (separating two layers of constant and different densities). Under the assumption that the vorticity vectors in the two layers are constant, we prove that bounded solutions to the three-dimensional water waves equations in the $\beta$-plane approximation exist if and only if one of the horizontal components of the velocity, as well as its vertical component, are zero; the other horizontal component being constant. Moreover, the interface is flat, the free surface has a traveling character in the horizontal direction of the nonvanishing velocity component, being of general type in the other horizontal direction, and the pressure is hydrostatic in both layers. Unlike previous studies of three-dimensional flows with constant vorticity in each layer, we consider a non-flat bottom boundary and different constant vorticity vectors for the upper and lower layer.