Symmetries in atmospheric sciences

TL;DR

This paper reviews the application of symmetry methods, especially Lie symmetries, in atmospheric sciences, demonstrating how they help derive exact solutions, relate different models, and simplify equations like the vorticity equation.

Contribution

It introduces the use of symmetry techniques to derive solutions, relate models, and simplify atmospheric equations, including the minimal finite-mode Lorenz model.

Findings

Derivation of exact solutions using Lie symmetries.

Construction of partially-invariant solutions for the vorticity equation.

Mathematically deriving the minimal finite-mode Lorenz model.

Abstract

Selected applications of symmetry methods in the atmospheric sciences are reviewed briefly. In particular, focus is put on the utilisation of the classical Lie symmetry approach to derive classes of exact solutions from atmospheric models. This is illustrated with the barotropic vorticity equation. Moreover, the possibility for construction of partially-invariant solutions is discussed for this model. A further point is a discussion of using symmetries for relating different classes of differential equations. This is illustrated with the spherical and the potential vorticity equation. Finally, discrete symmetries are used to derive the minimal finite-mode version of the vorticity equation first discussed by E. Lorenz (1960) in a sound mathematical fashion.