Figure 1.
Growth of the global forecast errors simulated by the ECMWF ensemble forecasting system in (a) 7-day forecasts in December 2014 and (b) 15-day forecasts in May 2015. Presented quantity is defined by Equation (6) as a function of the zonal scale L which is defined as (km), where k is the zonal wavenumber. The zonal (x) axis is logarithmic and the meridional axis (time) is linear.
Figure 2.
As in Fig. 1 but the errors are normalized with their values at initial time in the same scale. Presented quantity is . Notice that y axis is also logarithmic in order to show more clear the relative growth early in forecasts.
Figure 3.
Function fitting of the normalized error growth in different scales in (a) December 2014 and (b) May 2015. Empirical data are presented by markers and their fit based on Equation (9) by full lines. The data and fitting solutions for planetary scales () is shifted along the x-axis for 5 days, synoptic scales () are shifted for 10 and subsynoptic scales () for 15 days along the x axis. Dashed lines correspond to the asymptotic values of the normalized forecast errors as defined by Equation (16).
Figure 4.
As in Fig. 3(b) but with the exponential solution (21) for the error dynamics near saturation added (red curves).
Figure 5.
Forecast time when the difference between F(t) and becomes smaller than 10% of for each zonal wavenumber k. For F(t) is less than 10% different from from the beginning of forecasts.
Figure 6.
As in Fig. 2(a) but modelled for each scale independently over (a) 2-day period, (b) 30 days, and (c) small scales during 2 days. Presented quantity in (a) and (b) is and in (c) F(k, t). Notice that the zonal (x) axis is logarithmic whereas the meridional axis (time) is linear.
Figure 7.
The growth of global forecast errors towards the saturation based on ECMWF data in May 2015. The errors F(t) are computed independently for each k and normalized by their values at day 60 in each . The isolines are every 0.1 and thick black isolines correspond to 0.6 (bottom), 0.9 (middle) and 0.99 (top) values. The small figure in the upper right corner is the growth over the first 2 days of the forecast.
Figure 8.
Parameters of the model (3) by Dalcher and Kalnay (1987) computed from Equations (23) to (24) based on ECMWF data in May 2015.
Figure 9.
Comparison between the two terms of Dalcher and Kalnay (1987) model with ECMWF data for May 2015 in several zonal wavenumbers. (a) wavenumbers 2, 7 and 14, (b) wavenumbers 20, 30 and 40. Shown are solutions for the growth if only term is used (red curves), growth obtained with the term (blue curves) and the results for the sum of the two terms (black curves). Empirical data are added as circles. Solutions for various zonal wavenumbers are shown by curves of different thickness. The thickest line is for and , the medium thickness applies to and whereas the thin curves are solutions for and .
Figure 10.
Sketch of the evolution of forecast errors of geopotential height at 500 hPa. Top line is based on the curves from Lorenz (1982) and errors estimated from forecast differences which show the error growth in the perfect model. The red curve shows the forecast errors based on Simmons and Hollingsworth (2002) with analysis errors estimated around 10 m in late 1990s. More recent curves found in various studies indicate S-shape in relation to a slow initial growth and a faster growth lateron.