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.

Prediction Error Growth in a more Realistic Atmospheric Toy Model with Three Spatiotemporal Scales

“…To summarize, we can test the validity of the following laws for scale dependent error growth rates and for the error growth over time: a constant Lyapunov exponent and hence an exponential error growth, as it is expected for the initial time of very small initial errors in a low dimensional chaotic system; the extension of this behavior with a saturation factor (1 / ) EE  − expected to be valid for all times in a low dimensional chaotic system; the (extended) quadratic law proposed in (Zhang et al 2019), and the (extended) power law growth proposed in Brisch and Kantz (2019).…”

“…The improvement of the numerical weather prediction systems raised the question of the intrinsic atmospheric prediction limit, i.e., for the maximal lead time into the future, after which every forecast will be useless. While the notion of seamless prediction (Shukla, 2009) and of seasonal prediction implies that it will be only a matter of technology to make forecasts far into the future, in recent years, there have been several publications whose authors assume a strict upper bound in time for making useful predictions (Brisch and Kantz, 2019;Zhang et al,2019, Palmer et al, 2014. Even if the numerical model were perfect, the uncertainty of the initial condition would give rise to prediction errors which grow over time.…”

“…A scale dependent error growth in the spirit of Lorenz (1996) was described by Brisch and Kantz (2019) using a power law, which successfully approximated the data of the National Center for Environmental Protect Global Forecast System (Harlim et al, 2005). Brisch and Kantz (2019) also introduced a theory connecting the power law exponent with the Lyapunov exponents and limit errors of the different scales, and its validity was demonstrated by a low-dimensional atmospheric system (Lorenz, 1996) extended to five spatiotemporal levels. However, the different scales in this model cannot be superimposed in order to gain a general signal in the real space, but the different scales were living in different subspaces of phase space.…”

Recalculation of error growth models’ parameters for the ECMWF forecast system

“…At the second time step (12 hours) values of predictability curves reach approximately the same values as it had initially. A possible explanation could be that initial errors set the initial state off the attractor and decrease occurs because the first tendency is to get on the attractor (Brisch and Kantz, 2019). With an increase of average errors, chaotic behavior becomes dominant.…”

“…7) can be found in the multiscale behavior of weather. If some events are predictable on a timescale longer than ten days (for example long-lived anomalies in sea surface temperature or soil moisture) than they wouldn´t be captured by medium-range weather forecast (Simmons et al, 1995;Brisch and Kantz, 2019). It is also possible that the overestimation is due to the different source of data used for calculation of EFS E  by Eqs.…”