Growth of finite errors in ensemble prediction
We study the predictability of chaotic conservative and dissipative maps in the context of ensemble prediction. Finite-size perturbations around a reference trajectory are evolved under the full nonlinear system dynamics; this evolution is characterized by error growth factors and investigated as a function of prediction time and initial perturbation size. The distribution of perturbation growth is studied. We then focus on the worst-case predictability, i.e., the maximum error growth over all initial conditions. The estimate of the worst-case predictability obtained from the ensemble approach is compared to the estimate given by the largest singular value of the linearized system dynamics. For small prediction times, the worst-case error growth obtained from the nonlinear ensemble approach is exponential with prediction time; for large prediction times, a power-law dependence is observed the scaling exponent of which depends systematically on the initial error size. The question is addressed of how large an ensemble is necessary to reliably estimate the maximum error growth factor. A power-law dependence of the error in the estimate of the growth factor on the ensemble size is established empirically. Our results are valid for several markedly different chaotic conservative and dissipative systems, perhaps pointing to quite general features.