Symmetry preserving parameterization schemes

TL;DR

This paper develops methods for creating physical parameterization schemes that preserve symmetry properties, using group classification techniques to ensure invariance in averaged nonlinear differential equations, exemplified by vorticity flux parameterizations.

Contribution

It introduces a systematic approach for symmetry-preserving parameterizations using group classification, including inverse and direct methods, with applications to vorticity equations.

Findings

Computed differential invariants of subalgebras of the vorticity equation.

Constructed a hierarchy of normalized subclasses of vorticity equations.

Classified invariant parameterizations with minimal symmetry extensions.

Abstract

Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations. The demand that the averaged equation is invariant with respect to a subalgebra of the maximal Lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra. Given no prescribed symmetry group, the direct group classification problem is relevant. Within this framework, the algebraic method or direct integration of determining equations for Lie symmetries can be applied. For cumbersome parameterizations, a preliminary group classification can be carried out. The methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity equation. In particular, differential invariants of (infinite dimensional) subalgebras of the maximal Lie invariance algebra of the unaveraged vorticity equation are computed. A hierarchy of normalized subclasses of generalized vorticity equations is constructed. Invariant parameterizations possessing minimal symmetry extensions are described and a restricted class of invariant parameterization is exhaustively classified. The physical importance of the parameterizations designed is discussed.