Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

TL;DR

This paper develops minimal finite-mode atmospheric models that preserve key geometric and Hamiltonian structures, demonstrating how to retain symmetries and conservation laws during discretization.

Contribution

It introduces finite-mode models that maintain symmetry and Hamiltonian structures, improving the structural fidelity of simplified atmospheric models.

Findings

Lorenz-1960 model preserves symmetries and Nambu form.

Lorenz-1963 model violates structural properties.

Six-component model retains symmetries and Hamiltonian structure.

Abstract

A typical problem with the conventional Galerkin approach for the construction of finite-mode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in some respect preserve the geometric attributes inherited from their continuous models: a three-component model of the barotropic vorticity equation known as Lorenz' maximum simplification equations [Tellus, \textbf{12}, 243--254 (1960)] and a six-component model of the two-dimensional Rayleigh--B\'{e}nard convection problem. It is reviewed that the Lorenz--1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical Hamiltonian form (Nambu form). In a similar fashion, it is proved that the famous Lorenz--1963 model violates the structural properties of the Saltzman equations and hence cannot be considered as the maximum simplification of the Rayleigh--B\'{e}nard convection problem. Using a six-component truncation, we show that it is again possible retaining both symmetries and the Nambu representation in the course of discretization. The conservative part of this six-component reduction is related to the Lagrange top equations. Dissipation is incorporated using a metric tensor.