Some Basic Formalisms in Numerical Variational Analysis

This study aims at the theoretical development of a method of "four-dimensional analysis," namcly the numerical variational analysis. The three basic types of variational formalism in the nurncrical variational analysis method are discussed. The basic formalisms are categorized into three areas: (1) "timewise localized" formalism, ( 2 ) formalism with strong constraint, and (3) formalism with weak constraint. Exact satisfaction of selected prognostic equations were formulated as constraints in the functionals for the first two formalisms. However, only the second formalism contains explicitly the time variation terms in the Euler equations. Thc third formalism is characterized by thc subsidiary condition which requires that the prognostic or diagnostic equations must be approximately satisfied. The variational formalisms and the associated Euler-Lagrangc equations are obtained in the form of finite-difference analogs. I n this article, the filtering of each formalism and the uniqueness of solutions of the Euler equations are discussed for a limit that time and space increments (At and Ax) approach zero. The results from the limited case study can be applied, with some modification, for the cases where these increments are finite. I n addition, a numerical method of solving the Euler equations is discussed. The discussion is facilitated, merely for the sake of simplicity, by choosing a linear advection equation as a dgnamical constraint. However, the discussion can bc applied t o morc complicated and realistic cases.