A Predictability Study of Lorenz's 28-Variable Model as a Dynamical System

Abstract

The dynamics of error growth in a two-layer nonlinear quasigeostrophic model has been studied to gain an understanding of the mathematical theory of atmospheric predictability. The growth of random errors of varying initial magnitudes has been studied, and the relation between this classical approach and the concepts of the nonlinear dynamical systems theory has been explored. The local and global growths of random errors have been expressed partly in terms of the properties of an error ellipsoid and the Lyapunov exponents determined by linear error dynamics. The local growth of small errors is initially governed by several modes of the evolving error ellipsoid but soon becomes dominated by the longest axis. The average global growth of small errors is exponential with a growth rate consistent with the largest Lyapunov exponent. The duration of the exponential growth phase depends on the initial magnitude of the errors. The subsequent large errors undergo a nonlinear growth with a steadily decreasing growth rate and attain saturation that defines the limit of predictability. The degree of chaos and the largest Lyapunov exponent show considerable variation with change in the forcing, which implies that the time variation in the external forcing can introduce variable character to the predictability. For sufficiently large initial errors, comparable to observational errors, the exponential growth phase is shown to be absent, indicating that the growth is governed completely by nonlinear error dynamics. During a part of the initial growth phase, the growth rate is higher than the largest Lyapunov exponent. We have shown that the estimations of the growth rates of small errors, obtained from the well-known empirical formula using the error data in their nonlinear growth phase, are inaccurate. The use of Lyapunov exponents to estimate growth rate and predictability is valid only for initially small errors. It is unlikely that the errors in weather prediction models, at the current level of exactness of the observed data, can be interpreted in terms of the Lyapunov exponents.