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 one 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 the model. In addition, we study the initial error growth for the ECMWF forecast system
(500

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 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 (Palmer et al., 2014; Brisch and Kantz, 2019; Zhang et al., 2019). Even if the numerical model were perfect, the uncertainty of the initial condition would give rise to prediction errors which grow over time. In the setting of classical low-dimensional chaos, one would observe an exponential error growth whose exponent is given by the largest Lyapunov exponent of the system, with some saturation when the error reaches the magnitude of the standard deviation of the quantity to be predicted. Although exponential error growth has been associated with the fact that a detailed forecast is meaningful only up to lead times of a few multiples of the Lyapunov time, which is the inverse of the Lyapunov exponent, in principle, with absolutely perfect knowledge of the initial condition, one could compute meaningful predictions up to arbitrary times.

In contrast to this, it has been observed by several authors in the past (Toth and Kalnay, 1993; Lorenz, 1996; Aurell et al., 1996, 1997; Boffetta et al., 1998) that the proper Lyapunov exponent of a dynamical system might not be relevant for the issue of predictability, in two ways. First, the Lyapunov exponent of, for example, atmospheric dynamics is so large that no useful weather prediction was possible. On the other hand, and this is the issue of the present paper, if the proper Lyapunov exponent is much larger than the error growth rate of large-scale errors, it might render every gain in resolution and in better precision of the knowledge of the initial condition useless and thereby impose a strict limit to the time into the future where prediction is meaningful. Brisch and Kantz (2019) and Zhang et al. (2019) predicted a strictly finite prediction horizon associated with a scale-dependent error growth, where tiny errors grow much faster than larger ones. In an idealized model, this could even mean that the proper Lyapunov exponent of the system was infinite, and that finite size approximations of the Lyapunov exponent (Aurell et al., 1996, 1997) were the larger, the smaller the scale of this finite size. In real physical systems, one would certainly expect some cut-off of such divergent small-scale instability, e.g., in turbulence at the Kolmogorov length, the lower end of the inertial range, but the Lyapunov exponent of the system then would still be so large that it could never be compensated by more precise measurements.

Palmer et al. (2014), referring to Lorenz (1969), call this growth the “real butterfly effect,” and it is not an exponential error growth defined by the largest positive Lyapunov exponent, but growth where the exponent is replaced by a scale-dependent quantity (scale-dependent error growth rate).

The atmosphere exhibits multi-scale dynamics both in space and time, and observations show a close linkage between spatial and temporal scales: the
smaller some structure, the shorter its lifetime and the faster its time evolution. Planetary-scale structures (e.g., semi-permanent pressure centers
or the westerlies and trade winds) have sizes on the order of 10

Lorenz (1996) gave a sketch of error growth in such a system: a typical quantity to be predicted is a superposition of the dynamics on different scales. 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. Therefore, Lyapunov exponents of structures of various spatiotemporal scales are taken as the previously mentioned scale-dependent quantity, and they determine the error growth on their respective scales.

Zhang et al. (2019) take a very different starting point and suggest the quadratic hypothesis to describe the scale-dependent error growth. It was originally designed to describe initial and model error growth (Savijarvi, 1995; Dalcher and Kalnay, 1987). Zhang et al. (2019) recently used a parameter previously specifying the model error to describe upscale error propagation from small-scale processes and showed the validity of this hypothesis on data of the numerical weather prediction systems of the European Centre for Medium Range Weather Forecasts (ECMWF) and the US Next Generation Global Prediction System.

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 Centers for Environmental Prediction 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. This lack of realism may limit the general acceptance of this theory and stimulated the present study.

This article expands Lorenz's (2005) system from two spatiotemporal levels to three and discusses the setting of parameters and its advantage over
other systems. In this system, the scale-dependent growth of the initial error is calculated and is approximated by the power law and the quadratic
hypothesis. The results are discussed, the power law is modified, and the theoretical justifications of the approximations' parameter values are
sought and verified. The findings are then applied to the initial error growth of the ECMWF
numerical weather prediction system over the 1986 to 2011 period (500

This article is divided into five sections. The second describes the system with three spatiotemporal scales based on Lorenz (2005). In Sect. 3, we present the numerical error growth behavior and fit it with previously suggested laws such as a power law growth and the so-called quadratic law, where we also introduce extensions into the saturation regime where there are large errors. In Sect. 4, we trace back the empirically found parameters of the power law error growth to properties of the system and show that we can explain these findings self-consistently with the different error growth rates at different scales. In the fifth section, we perform a similar analysis for the ECMWF forecast system data. Conclusions and discussions are then presented in the final section.

The designed system with three spatiotemporal levels (L05-3) is based on systems created by Lorenz (2005). The first and simplest of this type is the
low-dimensional atmospheric system (L96) presented by Lorenz (1996). It is a nonlinear model, with

If

A two-level (scales) system (L96-2) was introduced by Lorenz (1996) by coupling two such systems, each of which, aside from the coupling, obeys a
suitably scaled variant of Eq. (

Similarly to the L96-2 systems, Brisch and Kantz (2019) created L-level systems (L96-H). Their model, however, lacks an essential property of atmospheric variables: the different levels of their hierarchy are different variables, occupying different subspaces of the phase space, as if it were different Fourier modes of some system but were defined in real space. Therefore, we introduce here a model which is closer to reality.

We start from a system L05-2 like L96-2 (Lorenz, 2005):

Equation (

Parameters

More precisely, Lorenz's (2005) idea is that the parameters

The procedures (Eqs.

Based on the L05-2 system (Eqs.

The parameters of any multi-level Lorenz's system (L96-2, L96-H, L05-2, L05-3) should be set so that all levels behave chaotically (the largest
Lyapunov exponent of each level is positive) and that all levels have a significant difference in amplitudes and fluctuation rates. For the L-96
system (Eq.

For the L05-3 system (Eqs.

In the pioneering papers of Aurell et al. (1996, 1997), a mathematical concept and a numerical scheme for the calculation of scale-dependent error growth was introduced, called the “finite size Lyapunov exponent”. Actually, Aruell et al. (1996) have already referred to the atmospheric predictability problem and named this as one motivation for their work. These papers triggered a set of follow-up publications where it was studied how scale-dependent error growth manifests itself in different models with intrinsic hierarchies such as models for fully developed turbulence like the shell model (Aurell et al., 1996) or how they can be used to study the issue of predictability (Boffetta et al., 1998; for a review see Cencili and Vulpiani, 2013). In brief, a finite size Lyapunov exponent can be defined as the ergodic average over phase space of the growth rate of perturbations of a given magnitude, where the growth rate is defined as the inverse of the time needed for the error magnitude to increase by a pre-defined factor.

In order to be able to investigate the data from the ECMWF forecasts as well, we use here a less rigorous approach: we measure the error amplitude after fixed time intervals. We then calculate the mean error amplitudes after fixed times, calculate the average growth rates during the last time interval, and can report the mean growth rates versus mean error magnitudes. Due to the fact that the conditioning is different, this is not the same as the calculation of the finite size Lyapunov exponent, but it reports similar properties of the system. However, our quantity is more appropriate for the forecast problem: the initial condition of a forecast has some small error compared to the unknown truth, and assuming a perfect model, we therefore calculate the average forecast error after a finite time.

In the following, we describe in more detail our numerical approach. By “error growth”, we denote the growth of errors in the initial conditions, which limit predictability if a system is chaotic. A numerical error growth experiment in a model system, therefore, consists of repeatedly generating a reference trajectory, which is considered to be the “truth” or verification, and a perturbed trajectory which is the numerical solution of the system for a perturbed initial condition (perfect model assumption). The time evolution of the difference vector between these two trajectories averaged over many repetitions then gives insight into the growth of prediction errors. In order for this scheme to be meaningful, we have to ensure that the reference trajectory is on the attractor of the system, that the repetition of this scheme samples the whole attractor with correct weights (the invariant measure), and that initial perturbations point already into the locally most unstable direction, since otherwise errors might even shrink on short timescales (this is also a relevant issue in ensemble forecasts and there its solutions are found in using bred vectors; Toth and Kalnay, 1997). We solve these issues in the following way: we first integrate the system over 10 years, starting from arbitrary initial conditions, and assume that after discarding this transient, the trajectory is on the attractor. We continue to integrate this single trajectory and consider segments of it as reference trajectories for error growth, i.e., the many reference trajectories are simply segments of one very long trajectory, which ensures not only that all these segments are located on the attractor, but that in addition, they sample the attractor according to the invariant measure. For the perturbed trajectories, we start with a random perturbation of the reference trajectory of very small amplitude and let this trajectory evolve over time before we start to determine its distance towards the reference trajectory. In other words, we discard some initial time interval of error growth from our study since this is affected by some complicated transient behavior before it starts to grow with the maximum Lyapunov exponent. However, due to the hierarchical nature of our model, the error growth with the maximal Lyapunov exponent will saturate already for rather small error amplitudes and will be replaced by slower error growth on larger scales, an effect which we will study in detail. The above-described scheme was originally introduced by Lorenz (1996).

In our system, the three spatial scales

Let us first consider a classical low-dimensional chaotic system. If the initial errors were infinitesimal, one could follow their growth for infinite
times and define the maximal Lyapunov exponent of the system as

We calculate the error growth rate in our L05-3 system using the method of Sprott (2006) (

The L05-3 system is designed with three spatial and temporal scales, so the error growth rate

Similar to the exponential law, this power law does not take the saturation of errors at their largest scale into account. We, therefore, multiply the
right hand side by

Unfortunately, the time integration in order to arrive at an expression of

A different description of scale-dependent error growth rate was proposed by Zhang et al. (2019), namely a quadratic model which we write down
directly in its extended form containing the saturation effect:

When removing the saturation factor

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 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

In Figs. 2–4, we present the numerical results of the error growth over time of the errors

The power law

We also fit the quadratic model

Brisch and Kantz (2019) derived the value of the exponent

The model of Brisch and Kantz (2019) was constructed of weakly coupled identical sub-systems, with additional scaling parameters for the amplitude and
time of the different subsystems. Therefore, there were rather well defined error growth rates

In the L05-3 model, we have determined the error growth of a given level by the study of the distance between reference trajectories and perturbed
trajectories in the corresponding coordinates

By Sprott's method (Sprott, 2006) we calculate the error growth rates of the three levels separately and the total error growth rate. As one can see
in Fig. 4,

These derived values for

The coupling of the levels has the consequence that one cannot define the Lyapunov exponents for the individual levels in a mathematically rigorous
way. We calculated them as

Values

The error growth rate

The error growth

Values of coefficients

Values of parameters

Values of limit (saturated) values

Despite these missing parts of the error growth

We designed a three-level (three-scale) system (L05-3; Eqs.

The quadratic hypothesis does not provide an as good a fit as the extended power law does. Its parameter

We also checked the appropriateness of the extended quadratic hypothesis and the extended power law to describe the error growth rate

The ECMWF forecasting system dataset was obtained from the personal repository of Linus Magnusson (Magnusson, 2013). The L05-3 system dataset, products from the ECMWF forecasting system dataset, codes, and figures were conducted in Wolfram Mathematica, and they are permanently stored at

HB proposed the idea, carried out the experiments, and wrote the paper. HK supervised the study and co-authored the paper.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to Linus Magnusson for offering a dataset (ECMWF forecasting system) from his personal repository.

This research has been supported by the Czech Science Foundation (grant no. 19-16066S).The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Ignacio Pisso and reviewed by two anonymous referees.