A Parameterization Technique for Nonlinear Spectral Models

Abstract

The feasibility of developing an objective parameterization technique is examined for general nonlinear hydrodynamical systems. The typical structure of these hydrodynamical systems, regardless of their complexity, is one in which the rates of change of the dependent variables depend on homogeneous quadratic and linear forms, as well as on inhomogeneous forcing terms. As a prototype of the generic problem containing this typical structure, we apply the parameterization technique to various three-component subsets of a five-component nonlinear spectral model of forced, dissipative quasi-geostrophic flow in a channel. The results obtained here lead to specification of the necessary data coverage requirements for applying the technique in general. The emphasis of this work is on preserving some behavior of the steady states by incorporating in the parameterized models information concerning the topological structure of the original solutions. By formulating the parameterization in terms of the steady states, we intend primarily to illustrate the general technique, but not to suggest that the preservation of temporal behavior can be achieved by addressing the steady solutions alone. The parameterized spectral components are expressed as power series involving the retained components, and it is found that the optimum parameterization is obtained when these series are terminated at quadratic terms. The values of the coefficients in these series are determined from the moments of the original set of spectral components over some range of forcing. For testing convenience, the moments are computed using the steady solutions to the original five-component model as data. This is accomplished by assuming that the values of the zonal forcing rate obey some standard statistical distributions. In regions of phase space in which multiple steady solutions occur, the likelihood of the occurrence of any one solution may be weighted according to its stability. Thus, the datasets can be viewed as simulating either idealized data, in which both stable and unstable solutions are permitted, or observational data, in which only stable solutions are permitted. Special attention is paid to the sensitivity of the parameterization to data coverage requirements, and to the relation of these requirements to the general structure of the solution surfaces. Significantly, it is shown that with sufficient data coverage, a successful parameterization may be obtained even in the more restrictive case when only stable (observable) solutions are used as data.