Developments in Normal Mode Initialization Part I: A Simple Interpretation for Normal Mode Initialization

Abstract

An interpretation of the iterative schemes of nonlinear normal mode initialization (Machenhauer, Kitade, and Tribbia) is introduced, where the schemes are regarded as sequential applications of filters. The response functions of these filters provide a means of evaluating the convergence of the iterative methods. The filters annihilate that component of the signal corresponding to the theoretical frequency determined from the normal mode analysis. The actual frequencies are, of course, determined by the linear team in the differential equations (which are accounted for by the normal mode analysis) and the nonlinear terms which will change the actual frequencies. The interpretation is extremely simple, but has not appeared previously in the literature. It is primarily qualitative. It explains in a qualitative sense the convergence problems encountered when attempting to initialize modes with small equivalent depths, or models which include diabatic physical processes. It complements the analyses of Phillips and Errico. The analysis of Tribbia's higher order initialization suggests that higher order steps can be more sensitive to convergence problems than the earlier methods.