Symmetry justification of Lorenz' maximum simplification

TL;DR

This paper rigorously justifies Lorenz's three-mode model for the barotropic vorticity equation using symmetry transformations, establishing it as the maximal simplified form derived through symmetry-based reduction.

Contribution

It provides a symmetry-based justification for Lorenz's maximum simplification, connecting it to a hierarchy of models obtained via symmetry reductions.

Findings

Lorenz's three-mode model is derived as a symmetry reduction.

The model is the final step in a hierarchy of symmetry-based reductions.

The justification confirms the model's status as the maximum simplification.

Abstract

In 1960 Edward Lorenz (1917-2008) published a pioneering work on the `maximum simplification' of the barotropic vorticity equation. He derived a coupled three-mode system and interpreted it as the minimum core of large-scale fluid mechanics on a `finite but unbounded' domain. The model was obtained in a heuristic way, without giving a rigorous justification for the chosen selection of modes. In this paper, it is shown that one can legitimate Lorenz' choice by using symmetry transformations of the spectral form of the vorticity equation. The Lorenz three-mode model arises as the final step in a hierarchy of models constructed via the component reduction by means of symmetries. In this sense, the Lorenz model is indeed the `maximum simplification' of the vorticity equation.