Symmetry preserving parameterization schemes

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.

38 pages, 2 figure, minor corrections

References in corpus (8)

• New results on group classification of nonlinear diffusion-convection equations
• Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
• More common errors in finding exact solutions of nonlinear differential equations. I
• Enhanced preliminary group classification of a class of generalized diffusion equations
• Invariant discretization schemes for the shallow-water equations
• Lie symmetries and exact solutions of the barotropic vorticity equation
• Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
• Consequences of Symmetries on the Analysis and Construction of Turbulence Models

Cited by in corpus (24)

• Complete group classification of a class of nonlinear wave equations
• Equivalence transformations in the study of integrability
• Enhanced group classification of Gardner equations with time-dependent coefficients
• Group classification of linear evolution equations
• Group analysis of general Burgers-Korteweg-de Vries equations
• Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
• Algebraic method for finding equivalence groups
• Lie symmetries of two-dimensional shallow water equations with variable bottom topography
• Complete point symmetry group of the barotropic vorticity equation on a rotating sphere
• Group classification of variable coefficient generalized Kawahara equations
• Extended symmetry analysis of two-dimensional degenerate Burgers equation
• Inverse problem on conservation laws
• Equivalence groupoid of a class of variable coefficient Korteweg--de Vries equations
• A new gauge-invariant method for diagnosing eddy diffusivities
• Equivalence groupoids of classes of linear ordinary differential equations and their group classification
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
• Conservative parameterization schemes
• On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis
• Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
• Group analysis of variable coefficient generalized fifth-order KdV equations
• Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation
• Equivalence groupoid of generalized potential Burgers equations
• Group analysis of Benjamin-Bona-Mahony equations with time dependent coefficients