Transient error growth and local predictability: a study in the Lorenz system

Lorenz’s three-variable convective model is used as a prototypical chaotic system in order to develop concepts related to finite time local predictability. Local predictability measures can be represented by global measures only if the instability properties of the attractor are homogeneous in phase space. More precisely, there are two sources of variability of predictability in chaotic attractors. The first depends on the direction of the initial error vector, and its dependence is limited to an initial transient period. If the attractor has homogeneous predictability properties, this is the only source of variability of error growth rate and, after the transient has elapsed, all initial perturbations grow at the same rate, given by the first (global) Lyapunov exponent. The second is related to the local instability properties in phase space. If the predictability properties of the attractor are not homogeneous, this additional source of variability affects both the transient and post-transient phases of error growth. After the transient phase all initial perturbations of a particular initial condition grow at the same rate, given in this case by the first local Lyapunov exponent. We consider various currently used indexes to quantify finite time local predictability. The probability distributions of the different indexes are examined during and after the transient phase. By comparing their statistics it is possible to discriminate the relative importance of the two sources of variability of predictability and to determine the most appropriate measure of predictability for a given forecast time. It is found that a necessary premise for choosing a relevant local predictability index for a specific system is the study of the characteristics of its transient. The consequences for the problem of forecasting forecast skill in operational models are discussed.