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.

On Bell polynomials approach to the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

“…1,2 Nowadays, there has been growing interest in studying the variable-coefficient NLEEs, which are often considered to be more realistic than their constantcoefficient counterparts in modeling a variety of complex nonlinear phenomena under different physical backgrounds. [3][4][5][6][7][8][9][10][11] Since those variable-coefficient NLEEs are of practical importance, it is meaningful to systematically investigate completely integrable properties such as bilinear forms, Lax pairs, bilinear Bäcklund transformations, infinite conservation laws and various exact analytic solutions. The Bell T.-T. Zhang et al polynomials are found to play an important role in the characterization of the integrability of NLEEs.…”

Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials

“…The representation (11) implies that X 1 1 = X 2 2 = |T t | 1/2 cos θ and X 1 2 = −X 2 1 = −|T t | 1/2 sin θ, which means that transformation components X 1 and X 1 satisfy the Cauchy-Riemann system…”

“…A number of classes of differential equations that are important for applications are normalized or specifically semi-normalized or can be partitioned into normalized classes and hence were classified within the framework of the algebraic method. See, e.g., [1,7,11,14,15,16].…”

“…Form-preserving transformations under the name allowed transformations also arose in the course of symmetry analysis of variable-coefficient Korteweg-de Vries equations [4] and variable-coefficient (1+1)-dimensional cubic Schrödinger equations [5]. Later on, formalizing the framework of admissible transformations was initiated [6] and the range of applicability of admissible transformations was extended to various classes of differential equations being important for applications, including nonlinear Schrödinger equations [7], variable-coefficient reaction-diffusion equations [8,9,10], eddy vorticity flux parameterizations of the inviscid barotropic vorticity equation [11] and nonlinear wave equations from the theory of elasticity [1].…”

Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report

“…The investigations of Hamiltonian dynamics and Lie algebra are major theoretical developments under the present COST Action [106][107][108][109][110]. In general, symmetries of differential equations are fundamental constraints on how physically self-consistent parameterizations must be constructed for a given system.…”