# Rigidity of Stratospheric Travelling Waves

## TL;DR

This paper studies how stratospheric traveling waves behave on outer planets, showing how their speeds are constrained by the surrounding atmospheric flow.

## Contribution

The paper introduces new rigidity results for stratospheric traveling waves under different planetary conditions and coordinate systems.

## Key findings

- Wave speeds in the f-plane setting are constrained within the range of the zonal flow velocity.
- In the β-plane setting, wave speeds cannot exceed the maximum zonal velocity but may be less than the minimum.
- Symmetry breaking in zonally traveling waves can occur due to planetary rotation.

## Abstract

We present some rigidity results for travelling waves that propagate zonally in bands of latitude in the stratosphere of the outer planets. Such waves arise as perturbations of a background zonally shear flow. The f-plane and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\beta $$\end{document}β-plane approximations are adequate for the small and moderately wide bands on Jupiter and Saturn, while for the broad zonal jets on Uranus and Neptune one must use spherical coordinates. We show that within the f-plane setting the wave speeds must lie in the range of the flow’s zonal velocity, while in the \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\beta $$\end{document}β-plane setting they cannot exceed the maximum of the zonal velocity but may be slightly less than its minimum. In spherical coordinates we obtain a rigidity result for travelling waves near a zonal flow. By means of some examples, we also show that such zonally travelling waves do not have to be zonally symmetric, and symmetry breaking may occur due to the rotation.