# Shallow water equations on a fast rotating surface

## TL;DR

This paper analyzes the behavior of rotating shallow water equations on a surface of revolution, establishing uniform estimates and convergence results as the Rossby and Froude numbers approach zero, and characterizing the limiting zonal flows.

## Contribution

It provides a rigorous mathematical analysis of the asymptotic behavior of solutions to the rotating shallow water equations with variable Coriolis parameter, including convergence to zonal flows.

## Key findings

- Uniform estimates on solutions independent of small parameters
- Strong convergence to a limit governed by a simplified equation
- Time-averages approximate zonal flows and height fields

## Abstract

We prove that for rotating shallow water equations on a surface of revolution with variable Coriolis parameter and vanishing Rossby and Froude numbers, the classical solution satisfies uniform estimates on a fixed time interval with no dependence on the small parameters. Upon a transformation using the solution operator associated with the large operator, the solution converges strongly to a limit for which the governing equation is given. We also characterize the kernel of the large operator and define a projection onto that kernel. With these tools, we are able to show that the time-averages of the solution are close to longitude-independent zonal flows and height field.