Power law error growth in multi-hierarchical chaotic systems—a dynamical mechanism for finite prediction horizon

Abstract

We propose a dynamical mechanism for a strictly finite prediction horizon, i.e. a scenatio of chaotic motion where asymptotically a more precise knowledge of the initial condition does note translate into a longer closeness of the forecast to the truth. For this, we propose a class of hierarchical dynamical systems which possess a scale dependent error growth rate in the form of a power law. Actually, this is motivated by and consistent with well known hierarchies of patterns in atmospheric dynamics. This scale dependent error growth rate in form of a power law translates in power law error growth over time instead of exponential error growth as in conventional chaotic systems. The consequence is a strictly finite prediction horizon, since in the limit of infinitesimal errors of initial conditions, the error growth rate diverges and hence additional accuracy is not translated into longer prediction times. By re-analyzing data of the National Center for Environmental Protect Global Forecast System, a weather prediction model, published by Harlim et al (2005 Phys. Rev. Lett. 94 228501) we show that such a power law error growth rate can indeed be found in numerical weather forecast models and estimate it average maximal prediction horizon to about 15 d.