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 104 km and live on timescales of weeks and longer. Synoptic-scale structures (e.g., high and low pressure systems) have sizes of several thousands of kilometers and live on timescales of several days. Mesoscale structures (e.g., thunderstorms and weather fronts) have sizes from a few kilometers to several hundred kilometers and live on a timescale of a day or less. Microscale structures (e.g., turbulence) have sizes smaller than 1 km and live on a timescale of minutes. In addition to different spatiotemporal scales, structures also have different error growth rates and predictability. Error growth is faster, and predictability is lesser for smaller-scale structures.

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 hPa geopotential height).

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 N variables connected by governing equations: 1dXn/dt=-Xn-2Xn-1+Xn+1Xn-1-Xn+F, where n=1,…,N. Xn-2,Xn-1,Xn,Xn+1 are unspecified (i.e., unrelated to actual physical variables) scalar meteorological quantities (units), F is a constant representing external forcing, and t is time. The index is cyclic so that Xn-N=Xn+N=Xn and variables can be viewed as existing around a latitude circle. Nonlinear terms of Eq. () simulate advection. Linear terms represent mechanical and thermal dissipation. The model quantitatively, to a certain extent, describes weather systems, but, unlike the well-known Lorenz model of atmospheric convection (Lorenz, 1963), it cannot be derived from any atmospheric dynamic equations. The motivation was to formulate the simplest possible set of dissipative chaotically behaving differential equations that share some properties with the “real” atmosphere. One of the model's properties is to have 5 to 7 main highs and lows that correspond to planetary waves (Rossby waves) and several smaller waves corresponding to synoptic-scale waves. For Eq. (), this is only valid for N=30. Lorenz (2005), therefore, introduced spatial continuity modification (L05). Equation () is then rewritten to the following form: 2dXndt=[X,X]L,n-Xn+F, where [X,X]L,n=∑j=-JJ′∑i=-JJ′(-Xn-2L-iXn-L-j+Xn-L+j-iXn+L+j)/L2.

If L is even, ∑′ denotes a modified summation, in which the first and last terms are to be divided by 2. If L is odd, ∑′ denotes an ordinary summation. Generally, L is much smaller than N, J=L/2 if K is even, and J=(L-1)/2 if L is odd. To keep a desirable number of main highs and lows, Lorenz (2005) suggested a ratio N/L=30 and F=15. The choice of parameters F, and the setting of time unit as 5 d, is also made to obtain a similar value of the largest Lyapunov exponent to the ECMWF forecasting system (Lorenz, 2005).

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. (). There are N variables Xn plus J⋅N variables Yj,n defined for n=1,…,N and j=1,…,J. Governing equations are as follows: 3dXn/dt=-Xn-2Xn-1+Xn+1Xn-1-Xn+F-(c/b)∑j=1JYj,n,4dYj,n/dt=-cbYj-2,nYj-1,n+cbYj+1,nYj-1,n-cYj,n+(c/b)Xn, where c sets the rapidness of small scale compared to large scale, and b sets the small-scale amplitude size compared to large scale. Yj,n-N=Yj,n+N=Yj,n while Yj+J,n=Yj,n+1, and Yj-J,n=Yj,n-1. Xn represent the values of some quantity in N sectors of latitude circle, while the variables Yj,n (Y1,1,Y2,1,…,YJ,1,Y1,2,Y2,2,…,YJ,2,Y3,1,…) can represent some other quantity in JN sectors. In this system, one could construct a more general variable defined on all JN sectors, namely Zj,k=Xj+Yj,k.

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): 5dXn/dt=[X,X]L,n-Xn-cYn+F,6dYn/dt=b2[Y,Y]1,n-bYn+cXn.

Equation () is analogue to Eq. () (if we substitute F for Xn), and Eq. () is analogue to Eq. () (aside from the coupling where c is the coupling coefficient, and that Yn fluctuates b times as rapidly, and their amplitude is reduced by the factor b). L05-2 or L96-2 systems, however, have an unrealistic property compared to the numerical weather prediction systems. The large-scale and small-scale features are represented by separate sets of variables X and Y instead of appearing as superimposed features of a single set Z. Lorenz (2005) wanted to keep the system as simple as possible, so instead of, for example, Fourier analysis, a procedure for expressing variables Zn as sums of Xn and Yn was introduced: 7Xn=∑i=-II′(α-β|i||)Zn+i,8Yn=Zn-Xn.

Parameters α, β, and I are chosen so that X is a low-pass filtered version of Z, and Y represents the difference between the full signal Z and the filtered signal. By this procedure, Y has a much smaller amplitude than X, and also its time evolution should be faster since the temporal derivative is related to the spatial derivative via the difference (Xn+1-Xn-2), which for the low-pass-filtered signal X is typically smaller than for the signal Y.

More precisely, Lorenz's (2005) idea is that the parameters α, β are chosen so that X equals Z whenever Z changes quadratically over the longitudes (variables) n-I through n+I. It is when ∑i=-II′(α-β|i|)=1 and ∑i=-II′i2(α-β|i|t)=0. By solving these equations, we get the following: 9α=(3I2+3)/(2I3+4I),10β=(2I2+1)/(I4+2I2).

The procedures (Eqs.  and ) are functions of the length of the interval [-I,I]. When creating a system dZ/dt as the sum of dX/dt and dY/dt (sum of Eqs.  and ), the coupling term cXn in Eq. (), which enables short waves to develop, is combined with the dissipation term -Xn in Eq. (). Therefore the coupling term can be completely canceled, or it can appear in X rather than Y when Z is analyzed, and there might be nothing to enable the short waves in Y to grow. Lorenz (2005) reformulated the coupling process by adding a small fraction of X to Y, so small waves in Y can amplify, and proved that this is done by replacing b2[Y,Y]1,n+cXn by [Y,Y+c′X]1,n in Eq. (), and a new form of L05-2 system would be as follows: 11dZn/dt=[X,X]L,n+b2[Y,Y]1,n+c[Y,X]1,n-Xn-bYn+F, where c=c′⋅b2.

Based on the L05-2 system (Eqs. –), we designed a three-level (three-scale) system (L05-3): 12dXtot,n/dt=[X1,X1]L,n+b12[X2,X2]1,n+b22[X3,X3]1,n+c1[X2,X1]1,n+c2[X3,X2]1,n-X1,n-b1X2,n-b2X3,n+F, where c1, c2, b1, and b2 are parameters and the procedure for expressing the variables: 13X1,n=∑i=-I1I1′(3I12+3/2I13+4I1-(2I12+1/I14+2I12)|i|)Xtot,n+i,14X2,n=∑j=-I2I2′(3I22+3/2I23+4I2-(2I22+1/I24+2I22)|j|)⋅(Xtot,n+j-X1,n+j),15X3,n=Xtot,n-X2,n-X1,n, where I1 and I2 set the length of the intervals [-I,I].

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. ), the chaotic behavior is determined by the value of F, and the number of variables N. Lorenz (2005) states that for N≥12 chaos is found when F>5 (for N=4 it is when F>12 and for N>6 when F>8). In cases such as the L96-2 system (Eqs.  and ), where the forcing F acts only on the largest scale, the chaotic behavior of smaller scales is created by coupling. The size of the coupling is cascaded from the largest scale to the smaller ones. Because the values of the largest-scale variables are determined by the forcing F, the F value indirectly affects the smaller scales' chaotic behavior and must be chosen large enough to ensure chaotic behavior through coupling for all scales (levels). For the L05-2 system (Eq. ), variables are superposed features of a single set calculated by Eqs. () and (). In addition to those mentioned above, this procedure affects the chaotic behavior, amplitude, and fluctuation rate of the levels, and the choice of I between 10 and 20 may be optimal (Lorenz, 2005). In order to maintain the required properties of the two-scale L05-2 system, Lorenz (2005) chose N = 960, L = 32, I = 12, F = 15, b = 10, and c = 2.5 (note that for L05-2 and L05-3 systems it is not possible to directly determine the amplitude and fluctuation rate of smaller scales using spatiotemporal scaling factors b, because these values are mainly determined by the procedure for expressing variables and the length of the intervals [-I,I]).

For the L05-3 system (Eqs. –), it is necessary to specify eight parameters. We tested that the values of coupling coefficients c1 and c2 do not affect the L05-3 system compared to the values of other parameters, and therefore for simplification c1 = 1 and c2 = 1. The parameter F=15 is set the same as for other L05 systems. For the medium-scale amplitude to be approximately 10 times smaller than the large-scale amplitude and the small-scale amplitude to be approximately 10 times smaller than the medium-scale amplitude and for the scales to have different oscillation rates (Fig. 1), the spatiotemporal scale factors are chosen as b1 = 1 and b2 = 10 and interval lengths I1 = 20 and I2 = 10. N = 390 turned out to be most suitable for the chaotic behavior of all three levels. In the following, we will present numerical results of the L05-3 system (Eqs. –) with the parameters N = 390, L = 13, J = 6, F = 15, b1 = 1, b2 = 10, c1 = 1, c2 = 1, I1 = 20, and I2 = 10, calculated by a fourth-order Runge–Kutta method with a time step Δt = 1/240 or 0.5 h. Initial conditions (Xtot,0,n, X1,0,n, X2,0,n, X3,0,n), which should be free of transient effect, are chosen as final values of arbitrary values integrated for 175 200 steps or 10 years. Initial conditions and values of variables at days one, two, and three can be seen in Fig. 1.

The error growth EEFS(t) of the ECMWF forecasting system's 500 hPa geopotential height values (Magnusson, 2013) is calculated (Magnusson and Kallen, 2013) as annual averages over the Northern Hemisphere (20–90∘) obtained daily from 1 January 1986 to 31 December 2011. As “errors” one uses the differences between two operational forecasts issued with 1 d lag for the same day, in order to eliminate the effects of model errors. Specifically, we evaluate these for 27 different lead times and used the following time intervals in hours: 0–24, 6–30, 12–36, 18–42, 24–48, 30–54, 36–60, 42–66, 48–72, 54–78, 60–84, 66–90, 72–96, 78–102, 84–108, 90–114, 96–120, 108–132, 120–144, 132–156, 144–168, 156–180, 168–192, 180–204, 192–216, 204–228, 216–240. Detailed information about calculating the error growth of the ECMWF forecasting system can be found in Lorenz (1982). The error growth rate λEFS(E):=dln⁡E/dt≈ln⁡(E(t+Δt)/E(t))/Δt with Δt=6 h for the first 17 time steps and Δt = 12 h for the remaining 10 time steps and the error growth EEFS(t) are calculated between the first and tenth day. The differences 0–24, 6–30, 12–36 are discarded because λEFS(E) for these differences is either increasing or constant due to transient behavior; see Fig. 5.

Despite these missing parts of the error growth EEFS(t), Bednář et al. (2021) showed that the extended quadratic model Eq(t) Eq. () is more accurate than the extended exponential growth (solution of E˙/E=Λ(1-E/E∞)). Because EEFS(t) calculated in the abovementioned way should be insensitive to any model error which therefore can not affect the error growth, the extended quadratic model parameter β describes the effect of small-scale processes, which justifies considering the error growth rate λEFS(E) for all annual averages as scale-dependent and also approximating it by the extended power law λw(E) Eq. (). The quadratic model λr(E) (Eq. ) and the power law λp(E) (Eq. ) cannot be used due to the lack of an early part of the error growth EEFS(t). The extended power law λw(E) (Eq. ) matches well with the annual averages of the error growth rate λEFS(E) of the ECMWF forecasting system (Fig. 5) with the values of the exponent σEFS displayed in Fig. 6b (σ‾EFS = 0.21 ± 0.07), the value of the coefficient aEFS shown in Fig. 6a (a‾EFS = 0.93 ± 0.18 m0.21d-1), and with the limit (saturated) values EEFS,∞w displayed in Fig. 8 (E‾EFS,∞w = 114 ± 7 m). Exponents σEFS, coefficients aEFS, and limit values EEFS,∞w do not have significant trends over the years (Figs. 6 and 8). The extended quadratic model λq(E) Eq. () approximates annual averages of the error growth rate λEFS(E) of the ECMWF forecasting system (Fig. 5) with the values of the parameter αEFS shown in Fig. 7a (α‾EFS = 0.35 ± 0.04 d-1), the value of the parameter βEFS displayed in Fig. 7b (β‾EFS = 2.8 ± 0.9 md-1), and with the limit (saturated) values EEFS,∞q displayed in Fig. 8 (E‾EFS,∞q = 111 ± 7 m). Again, parameters αEFS and βEFS and limit values EEFS,∞q do not have significant trends over the years (Figs. 7 and 8). The approximations by the extended quadratic model λq(E) and the extended power law λw(E) differ in parts where data are not available (Fig. 5). The limit values EEFS,∞w and initial errors EEFS,0w of the extended power law are greater than the limit values EEFS,∞q and initial errors EEFS,0q of the extended quadratic model (Figs. 5 and 8). If we set the initial errors of both approximations to the same value, the error growth of the extended quadratic model EEFS,q(t) approximation would grow faster and would reach the limit value EEFS,∞ earlier than the error growth of the power law EEFS,w(t). This difference is more significant when E0→0. Of particular interest is the limit of predictability. We, therefore, study the time when the error E(t) reaches 95 % of the limit error E∞ for the size of the initial error E0 ≈ 3 m, which is an accepted value for current global operational numerical weather prediction (NWP) models (Zhang et al., 2019). The limit time is 14 d for the extended quadratic model E‾EFS,q(t) when using the abovementioned parameter values for α‾EFS, β‾EFS, and E‾EFS,∞q (Fig. 9a). For the extended power law E‾EFS,w(t), using the above listed values for the exponent σ‾EFS, coefficient a‾EFS, and limit values E‾EFS,∞w, it is 15 d (Fig. 9b). When the size of the initial error E0 is reduced to E0 ≈ 0.1 m, which is realistic for the current global experimental NWP models (Zhang et al., 2019), this limit time is 15 d for the extended quadratic model E‾EFS,q(t) (Fig. 9a), and 18 d for the extended power law E‾EFS,w(t) (Fig. 9b). Both models have an intrinsic predictability limit even if the size of the initial error E0→0, which is 15 d for the extended quadratic model (Fig. 9a) and 22 d for the extended power law E‾EFS,w(t) (Fig. 9b).

We designed a three-level (three-scale) system (L05-3; Eqs. –) with the parameters N=390, L=13, J=6, F=15, b1=1, b2=10, c1=1, c2=1, I1=20, I2=10. These parameters were chosen in such a way that all levels behave chaotically, i.e., the largest Lyapunov exponents of each level is positive, and that all levels have a significant difference in amplitudes and fluctuation rates (Fig. 1). In this system, the error growth rates λτ(E) and the error growths as a function of time Eτ(t) were calculated for the system as a whole (τ=tot) and for the different levels (τ=1,2,3). We fitted the numerically obtained results by the power law λp(E) (Eq. ), the extended power law λw(E) (Eq. ), the quadratic hypothesis λr(E) (Eq. ), the extended quadratic hypothesis λq(E) (Eq. ), and their corresponding time integrations E(t). Without the saturation terms (1-E/E∞) both the power law and the quadratic hypothesis fail to provide good fits for larger errors or longer times (Figs. 2 and 3) but are reasonable for the early parts of error growth when the initial error is small. Indeed, with our type of numerical analysis, we are unable to reach the very small scales where the error growth rate saturates at the proper Lyapunov exponent of the system which we estimate to be about 2.5 d-1 but instead are faced by an overshooting of the initial error growth.

The quadratic hypothesis does not provide an as good a fit as the extended power law does. Its parameter β that according to Zhang et al. (2019) and Zhang and Sun (2020) should describe the upscale error growth rate from small-scale processes could not be identified or described in the L05-3 system. In contrast, the extended power law λw(E) and Ew(t) with the exponent σ=0.47, the coefficient a=0.46 (units0.47d-1), and E∞=7.36 best describes the error growth rate λtot(E) and the error growth Etot(t) of the L05-3 system on the whole range of times and error magnitudes. The Brisch and Kantz (2019) definition of the exponent σ was confirmed and extended to cases when c1≠c2 (c is the ratio of the rapidness of a smaller scale compared to the rapidness of a larger scale) and b1≠b2 (b is the ratio of a smaller-scale amplitude compared to a larger-scale amplitude) for σ=ln⁡c1/ln⁡b1=ln⁡c2/ln⁡b2. It was shown that the coefficient a determines the degree of the system's coupling because the same value of the exponent σ is valid for the same system with and without coupling.