Fundamental principles of finite-time evolution of small perturbations in chatic systems are examined by using a simple barotropic model on a rotating sphere, which is a forced-dissipative system of 1848 real variables. A time-dependent solution which we investigate is a chaotic one with four non-negative Lyapunov exponents. The properties of the finite-time evolution of small perturbations superposed on the solution are investigated by means of two indices: Lorenz index, which is the root-mean-square amplification rate of the perturbations distributed equally in the whole phase space, and subspace Lorenz index, which is the root-mean-square amplification rate of the perturbations distributed equally in the subspace spanned by the first four Lyapunov vectors.

The Lorenz index decreases initially for a finite time due to the dissipative structure of the system before an exponential increase due to the nature of chaos, while the subspace Lorenz index tends to increase from the beginning. It is found in this model that the time variations of the Lorenz index for a finite time interval are highly correlative with those of the subspace Lorenz index when the time interval of the Lorenz index is several days longer than that of the subspace Lorenz index.