保持对称性的参数化格式
Symmetry preserving parameterization schemes
摘要：
讨论了具有一定不变性的物理参数化方案的设计方法。这些方法是基于不同的技术组分类，并提供手段，以确定表达式的平均过程中产生的非线性微分方程的未封闭的条款。平均方程的要求是不变的关于一个子代数的最大李不变性代数的非平均方程导致逆组分类的问题，这是解决所选择的子代数的微分不变量的描述。如果没有规定的对称群，直接群分类问题是相关的。在这个框架内，代数方法或直接积分确定方程的李对称性可以应用。对于繁琐的参数化，可以进行初步的组分类。以平均涡度方程中的涡度通量参数化为例，说明了本文方法的有效性。特别地，计算了非平均涡度方程的极大李不变性代数的（无穷维）子代数的微分不变量。构造了广义涡度方程的规范化子类族。描述了具有最小对称扩张的不变参数化，并对一类受限制的不变参数化进行了详尽的分类。的物理意义的参数化设计进行了讨论。
英文摘要：
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.
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