Mathematical study of the $\beta$-plane model for rotating fluids in a thin layer

TL;DR

This paper mathematically analyzes a rotating fluid model in a thin layer, revealing how boundary layers and spatial variations influence wave propagation and energy speed, with implications for oceanographic dynamics.

Contribution

It provides a rigorous mathematical analysis of the $eta$-plane model, highlighting the effects of boundary layers and spatial variations on wave behavior and energy propagation.

Findings

Boundary layer solutions exhibit strong singularities due to rotation vector variations.

The thin layer modifies Poincaré wave propagation, creating small scales.

Energy propagates at slower speeds than in classical models.

Abstract

This article is concerned with an oceanographic model describing the asymptotic behaviour of a rapidly rotating and incompressible fluid with an inhomogeneous rotation vector; the motion takes place in a thin layer. We first exhibit a stationary solution of the system which consists of an interior part and a boundary layer part. The spatial variations of the rotation vector generate strong singularities within the boundary layer, which have repercussions on the interior part of the solution. The second part of the article is devoted to the analysis of two-dimensional and three-dimensional waves. It is shown that the thin layer effect modifies the propagation of three-dimensional Poincar\'e waves by creating small scales. Using tools of semi-classical analysis, we prove that the energy propagates at speeds of order one, i.e. much slower than in traditional rotating fluid models.