## 1. Forecast-error growth and predictability limits of a numerical weather prediction system

The first estimate of forecast-error growth was given by Charney *et al*(1966), who, using a simple model, concluded that a reasonable estimate of the forecast-error doubling time was 5 days. A few years later, Smagorinsky (1969), using a more refined, nine-level primitive equation model that contained also moist processes, gave a lower estimate of the forecast-error doubling time of 3 days. Lorenz (1982), analysing 500 hPa geopotential height forecasts from the operational model used at the European Centre for Medium-Range Weather Forecasts (ECMWF) at that time, a 15-level primitive equation model with moist processes and orography, further reduced the estimate to 1.85 days for the Northern Hemisphere (NH) winter. This estimate was obtained assuming that forecast error grows following a quadratic equation that was introduced earlier by Lorenz (1969), which assumes that the short-term errors grow linearly, while the long-term growth is dominated by a (negative) quadratic term that brings the long-range forecast error to an asymptotic value. Subsequently, Leith (1982) included in Lorenz's (1969) forecast-error growth model a constant term designed to simulate the effect of analysis errors on the short-range forecast error. Leith's (1982) forecast-error growth model was further modified by Dalcher and Kalnay (1987), who investigated both the chaotic and systematic component of model error, and modified the model to simulate the effects of nonlinear error saturation. Dalcher and Kalnay (1987) applied the newly modified model to the data analysed by Lorenz (1982), and introduced two measures of predictability, the *time limits* τ*(95%)* and τ*(50%)*, that are the times when the forecast error reaches 95% or 50% of the forecast-error asymptotic value. They suggested that the forecast-error doubling time that was used up to that time to assess predictability, was not a very good measure of error growth, but that better measures were the forecast-error growth rate, and time limits such as τ*(95%)* and τ*(50%)*. Using the same dataset of Lorenz (1982), they estimated that the forecast-error growth rate was about 0.43 d^{−1} (which corresponds to an error doubling time of about 1.7 days for the NH winter, a shorter value than the 1.85 days found by Lorenz (1982), longer for the long waves and shorter for the short waves. They concluded that, for the NH winter, the time limit τ*(95%)* was 12 days, and the time limit τ*(50%)* was 5.5 days.

Savijärvi (1995) applied the Dalcher and Kalnay (1987) model to assess the error growth of forecasts issued by the National Meteorological Centre (NMC) of Washington between 1988 and 1993. Savijärvi (1995) introduced a third time limit, τ*(71%)*, the time when the forecast error exceeds of the saturation value (see Savijärvi (1995) for more details on the definition of this threshold, which corresponds to the level of climatic variability), and named it the deterministic predictability time limit. He estimated that for the 500 hPa geopotential height during the 2-year period 1991–1993, the NMC forecasts reached this level at about day 9 for the large scales (i.e. waves with total wave number up to 3), and about day 3 for the small scales (i.e. waves with total wave number between 3 and 20). He indicated that if the model error and the initial condition error could be halved, this limit would be reached at day 10 for the large scales and at day 6.6 for the short scales. Savijärvi (1995) also estimated that the time limit τ*(95%)* for the NMC system in 1991–1993 was longer than what was given by Dalcher and Kalnay (1987), about 20 days for the large scales and 7 days for the short scales, and suggested that this limit will remain at these levels even if the model error and the initial condition error could be halved. Reynolds *et al*(1994) also used the Dalcher and Kalnay (1987) model for forecasts to investigate the three-dimensional structure of random error growth in the NMC system for the same period, and concluded that both the model and analysis uncertainties contributed to forecast-error growth.

More recently, Simmons and Hollingsworth (2002) used the Dalcher and Kalnay (1987) model to study forecast skill improvements of the ECMWF single, high-resolution forecasts. They showed that between winter 1996/1997 and winter 2000/2001, the forecast-error doubling times during the NH winter decreased by ∼30% in the short forecast range (say at forecast days 1–4). At days 1 and 2, for example, they quoted doubling times in winter 2000/2001 of 1.14 and 1.42 days, values that are ∼50% shorter than the ones quoted by Lorenz (1982). The shortening of the doubling times at forecast day 1 and 2 can be related to the more realistic model activity at all scales, due to the increased resolution of the forecast model and to the inclusion of more realistic schemes that simulate physical processes. From Fig. 11 of Simmons and Hollingsworth (2002), it is possible to estimate that for the ECMWF system in winter 2000/2001, the time limits τ*(71%)* and τ*(95%)* were about 9 and 13 days, respectively.

This work presents an updated assessment of the forecast-error growth of synoptic-scale features represented by instantaneous (as opposed to averaged over a period of time, e.g. 1 day) ECMWF forecasts of the 500 and 1000 hPa geopotential heights and the 850 hPa temperatures. As in Reynolds *et al*(1994) and Simmons and Hollingsworth (2002), the Dalcher and Kalnay (1987) model of forecast-error growth has been used. Through the years, increases in resolution, especially in horizontal resolution, have brought improvements in forecast accuracy. This work provides a clean (clean since forecasts have been produced using different horizontal resolutions but the same model version for the same period) assessment of the impact of horizontal resolution on the short- and the long-range forecast skill. In particular, the predictability times when the forecast root-mean-square error reaches different levels of the asymptotic limit (25%, 50%, 71% and 95%) are computed. To better understand the role of model uncertainty on forecast skill, error growth is studied not only in a realistic framework, with forecasts verified against analyses, but also in a perfect-model framework, using high-resolution forecasts as verification. The key questions addressed in this work are: how much do the time limits change when resolution is increased from T95 to T799? And how do the past estimates of the so-called predictability time limits compare with these more recent estimates based on winter 2007/2008 forecasts?

After this introduction, section 2 describes the methodology used in this study, section 3 discusses the impact of resolution on forecast-error growth, and finally some conclusions are drawn in section 4.