Finite-Time Evolution of Small Perturbations Superposed on a Chaotic Solution: Experiment with an Idealized Barotropic Model

Abstract

Fundamental principles of finite-time evolution of small perturbations in chaotic systems are examined by using an idealized barotropic model on a rotating sphere, which is a forced-dissipative system of 1848 real variables. A time-dependent solution that is investigated is a chaotic solution with four nonnegative Lyapunov exponents. Attention is focused on the subspace spanned by the first four backward Lyapunov vectors. It is found that the time variations of the subspace Lorenz index, which is the mean amplification rate of perturbations defined in the subspace, are highly correlative with those of the Lorenz index, which is the mean amplification rate defined in the whole phase space, when the time interval of the Lorenz index is several days longer than that of the subspace Lorenz index. The first forward singular vector in the subspace has a property that its amplification rate is insensitive to the measuring norm, like the first backward Lyapunov vector, and has a tendency that its evolved pattern becomes similar to that of the first forward singular vector in the whole phase space. Application of the method introduced in this study to construct initial members in ensemble forecasts is discussed.