Power law error growth in a more realistic atmospheric Lorenz system with three spatiotemporal scales

Inspired by the Lorenz (2005) system, we mimic an atmospheric variable in one dimension, which can be decomposed into three spatiotemporal scales. This is motivated by and consistent with scale phenomena in the atmosphere. When studying the initial error growth in this system, it turns out that small scale phenomena, which contribute little to the forecast product, significantly affect the ability to predict this product. In other words, a more precise knowledge of the initial condition does not translate into a longer closeness of the forecast to the truth. Lorenz gave a sketch of such error growth. After a fast growth of the small scale errors with saturation at these very same small scales, the large scale errors continue to grow at a slower rate until even these saturate. We will present that scale dependent error growth can be translated into power law error growth. We will explain how parameter values of the power law are related to the error growth properties of the individual scales. We apply the results to the initial error growth of numerical weather prediction systems and show that the validity of the power law would imply a finite prediction horizon.