Constant vorticity two-layer water flows in the $\beta$-plane approximation with centripetal forces

TL;DR

This paper analyzes constant vorticity two-layer water flows on the $eta$-plane with centripetal forces, providing explicit solutions and convergence results, extending previous single-layer models to multi-layer scenarios.

Contribution

It introduces a two-layer water wave model with explicit velocity and pressure expressions, and proves convergence properties as the number of layers increases.

Findings

Pressure depends only on depth and surface when pressure variation is bounded.

Inner waves do not influence pressure if layer densities are equal.

Pressure sequence converges uniformly as layers increase.

Abstract

The constant vorticity {\bf two-layer water wave} in the $\beta$-plane approximation with centripetal forces is investigated in this paper. Different from the works (Chu and Yang\cite[JDE, 2020]{chu} and Chu and Yang \cite[JDE, 2021]{chu2}) on the singe-layer wave flows, we consider the two-layer water wave model containing a free surface and an interface. The interface separates two layers with different features such as velocity field, pressure and vorticity. We prove that if the change in pressure in the $y$-axis direction is bounded, then the pressure is a function only related to depth and the surfaces of the water flows. And the inner wave will not affect the pressure function, if the water flow densities in each layer are equal. Furthermore, the explicit expressions of the velocity, pressure are given for the two-layer water flows. It is interesting that our method and results are also valid for the multi-layer water waves. Let the number of layers of water waves $n$ tend to infinity, we prove that the squence of pressure in the lowest layer $\{P_1^n(x,y,z,t)\}_{n\geq1}$ is uniformly convergent, if the density of each layer is bounded and the each surface of wave flows is uniformly convergent.