An algebraic approach to the spontaneous formation of spherical jets

TL;DR

This paper introduces an algebraic method to analyze the spontaneous formation of jets on spherical domains, using the Zeitlin model and Lie algebra decomposition to classify jet structures.

Contribution

It presents a novel algebraic framework for understanding jet formation in geophysical flows, extending from a discrete model to the continuous Euler equations.

Findings

Jet structures classified via Lie algebra decomposition.

Discrete Zeitlin model effectively analyzes jet formation.

Results extend to original Euler equations.

Abstract

The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.