Finite-Time Lyapunov Stability Analysis and Its Application to Atmospheric Predictability

Abstract

Finite-time Lyapunov stability analysis is reviewed and applied to a low-order spectral model of barotropic flow in a midlatitude ? channel. The tangent linear equations of the model are used to investigate the growth of small perturbations superposed on a reference solution for a prescribed time interval. Three types of reference solutions of the model, stationary, periodic, and chaotic, are investigated to demonstrate usefulness of the analysis in the study of the atmospheric predictability problem. The finite-time Lyapunov exponents, which give the growth rate of small perturbations, depend upon the reference solution as well as the prescribed time interval. The finite-time Lyapunov vector corresponding to the largest Lyapunov exponent gives the streamfunction field of the fastest growing perturbation for the time interval. In the case of the chaotic reference solution, the streamfunction field has large amplitudes in limited areas for a small time interval. The areas of the large perturbation growth have some relation to the reference streamfunction field. A possible application of the finite-time Lyapunov exponents and vectors to the atmospheric predictability problem is discussed. These quantities might be used as several forecast measures of the time-dependent predictability in numerical weather predictions.