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

## TL;DR

This paper introduces a hierarchical model explaining power law error growth in chaotic systems, providing insights into the finite prediction horizon in weather forecasting due to scale-dependent error dynamics.

## Contribution

The paper presents a novel hierarchical modeling approach that captures power law error growth, explaining finite forecast horizons in weather models.

## Key findings

- Power law error growth observed in weather models.
- Hierarchical models replicate scale-dependent error dynamics.
- Finite prediction horizon explained by diverging error growth rate.

## Abstract

We propose a dynamical mechanism for a scale dependent error growth rate, by the introduction of a class of hierarchical models. The coupling of time scales and length scales is motivated by atmospheric dynamics. This model class can be tuned to exhibit a scale dependent error growth rate in the form of a power law, which 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 NCEP Global Forecast System published by Harlim et al.[13] we show that such a power law error growth rate can indeed be found in numerical weather forecast models.