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

Abstract. This article studies the growth of the prediction error over lead time in a schematic model of atmospheric transport. Inspired by the Lorenz (2005) system, we mimic an atmospheric variable in 1 dimension, which can be decomposed into three spatiotemporal scales. We identify parameter values that provide spatiotemporal scaling and chaotic behavior. Instead of exponential growth of the forecast error over time, we observe a more complex behavior. We test a power law and the quadratic hypothesis for the scale dependent error growth. The power law is valid for the first days of the growth, and with an included saturation effect, we extend its validity to the entire period of growth. The theory explaining the parameters of the power law is confirmed. Although the quadratic hypothesis cannot be completely rejected and could serve as a first guess, the hypothesis’s parameters are not theoretically justifiable. In addition, we study the initial error growth for the ECMWF forecast system (500 hPa geopotential height) over the 1986 to 2011 period. For these data, it is impossible to assess which of the error growth descriptions is more appropriate, but the extended power law, which is theoretically substantiated and valid for the Lorenz system, provides an excellent fit to the average initial error growth of the ECMWF forecast system. Fitting the parameters, we conclude that there is an intrinsic limit of predictability after 22 days.