Error growth and predictability in operational ECMWF forecasts

We study the forecast error growth in the 100-day ECMWF data set of 10-day forecasts previously utilized by Lorenz, separating the square of the error into systematic and random components. The random error variance is approximately equally distributed among all zonal wavenumbers m corresponding to the same total wavenumber n. Therefore, we combine the random error variance for all zonal wavenumbers and study its dependence on total (2-dimensional) wavenumber rather than on zonal wavenumber.We extend the Lorenz model for global r.m.s. error growth by including the effect of errors associated with deficiencies in the forecast model, and apply this new parameterization to the error variance of 10-day ECMWF operational forecasts obtaining an excellent fit. We point out that the commonly used parameter “doubling time of small errors” is not a good measure of error growth because it has to be determined by extrapolation to small errors and it is very sensitive to the method of extrapolation. On the other hand, the error growth at finite times is a better measure because it is well defined by the data. The results of the error fit to each total wavenumber n are the basis for the main conclusions of the paper. In the northern hemisphere winter data set, the error growth rate α increases almost monotonically with wavenumber, from about 0.3 day-1 at long waves to about 0.45 day-1 at medium and short waves. The saturation error variance V∞ is about 30% larger than the error variance at day 10 for long waves, which have not yet reached saturation. The scaled source of external error S/V∞ (due to model deficiencies) varies from about 3% day-1 at long waves to about 20% day-1 at short waves. This increase of relative model error may be due to the second-order finite differences used in the ECMWF model during 1980/1981.Finally, we estimate the theoretical limit of dynamical predictability as a function of total wavenumber n. We define it as the time at which the error variance reaches 95% of the saturation value. This time decreases monotonically with n. In the winter, waves n ≤ 10 have not reached saturation at 10 days. In a perfect model with error growth similar to the present model, the predictability limit would be extended by about a week. In the summer, error growth rates and model deficiencies are larger than in winter, and the limit of predictability is close to 10 days even for long waves.