Symmetry preserving parameterization schemes

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...

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Bibliographic Details
Published in:Journal of mathematical physics 2012-07, Vol.53 (7), p.1
Main Authors: Popovych, Roman O., Bihlo, Alexander
Format: Article
Language:English
Subjects:
Algebra
Classification
Differential equations
Exact sciences and technology
Invariants
Lie groups
Mathematical analysis
Mathematical methods in physics
Mathematical models
Mathematics
Nonlinear equations
Parameter estimation
Parametrization
Physics
Sciences and techniques of general use
Symmetry
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Summary: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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4734344