Sources of MJO teleconnection errors in the ECMWF extended‐range forecasts

The European Centre for Medium‐range Weather Forecasts (ECMWF) extended‐range forecasts display large errors in the representations of Madden–Julian Oscillation (MJO) teleconnections over the North Atlantic with a strong underestimation of the impact of the MJO on the North Atlantic Oscillation (NAO) following an MJO in Phase 2–3. The origin of this error was investigated using a large set of re‐forecasts covering 20 years where part of the atmosphere was relaxed towards the ECMWF Reanalysis v5 (ERA5) reanalysis. These relaxation experiments show that relaxing the Tropics significantly improves the MJO teleconnections associated with Phase 2–3, with a reduction of about 50% in the amplitude error of the teleconnections. However, model errors outside the Tropics also play an important role. Results also suggest that errors in local processes reduce by about 20% the amplitude of the MJO teleconnections over the North Atlantic. Another example of a source of MJO teleconnection errors is the near surface over the Tibetan and Mongolian Plateaux. Nudging this area towards ERA5 improves the representation of the Pacific subtropical jet and the amplitude of MJO teleconnections associated with Phase 2–3 in the extended‐range re‐forecasts. Nudging the stratosphere exerts a comparatively weaker impact on the MJO teleconnections 11–15 days after an MJO in Phase 2–3. Overall, these experiments indicate that there are multiple sources of MJO teleconnection errors, making the representation of realistic MJO teleconnection by dynamical models a particularly challenging task.


INTRODUCTION
The Madden-Julian Oscillation (MJO) (Madden & Julian, 1971;Madden & Julian, 1972) is a dominant source of global predictability at the subseasonal to seasonal (S2S) time range (e.g., Waliser, 2011).It has a significant impact on the local weather in the tropical regions where it is propagating, but it also impacts Indian and South Asian monsoons through northward propagation in boreal summer (e.g., Yasunari, 1979) and the Asian monsoon (Hendon & Liebmann, 1990).The MJO has a significant impact in the northern Extratropics (e.g., Ferranti et al., 1990) and southern Extratropics (e.g., Lee & Seo, 2019) through Rossby wave propagation.Therefore, the MJO has a global impact and modulates a large range of extreme events globally, such as tropical cyclones, tornadoes, floods, heat waves, fires … (Yoneyama & Zhang, 2020).Predicting accurately the MJO and its impacts is at the forefront of S2S prediction.This article will focus on the impact of the MJO on the Euro-Atlantic weather and most especially on the North Atlantic Oscillation (NAO).Using reanalysis data, Cassou (2008) and Lin et al. (2010) showed that the impact of the MJO on the European weather is the strongest about 10 days after the active phase of the MJO over the Indian Ocean (MJO Phase 2-3), or over the West Pacific (MJO Phase 6-7) through Rossby wave propagation.The probability of a positive (negative) phase of the NAO is significantly increased about 10 days after an MJO in Phase 2-3 (6-7) and significantly decreased after an MJO in Phase 6-7 (2-3).The impact of the MJO on the other main Euro-Atlantic regimes (e.g., Atlantic ridge and Scandinavian blocking) is much weaker than on the NAO.Vitart (2017) investigated the skill of the models from the joint World Weather Research Programme (WWRP)/World Climate Research Programme (WCRP) S2S database (Vitart et al., 2017) in predicting the MJO and in simulating its teleconnections over the northern Extratropics following an MJO in Phase 2-3 or 6-7.Although the S2S models displayed skill in predicting the evolution of the MJO 3-weeks in advance on average and more than four weeks in advance for the European Centre for Medium-range Weather Forecasts (ECMWF) model, all the S2S models failed to represent the MJO teleconnections adequately.All the S2S models produced an increased probability of a positive (negative) NAO 11-15 days after an MJO in Phase 2-3 (6-7), as observed, but the amplitude of the impact of the MJO on the NAO was strongly underestimated by all the S2S models.Garfinkel et al. (2022) also found that all the S2S models suffer from too weak MJO and ENSO teleconnections.This inability of the S2S models to reproduce the observed amplitude of the MJO teleconnections over the Euro-Atlantic region represents a major barrier for S2S prediction skill over Europe and might be partly responsible for the low prediction skill at weeks 3-4 in this region.Stan et al. (2022) applied a large range of diagnostics to improve the understanding of physical mechanisms controlling the MJO influence and confirmed that reproducing the magnitude of the extratropical response to MJO teleconnections remains a key challenge for the S2S forecast systems.
In Vitart (2017), the ECMWF model displayed the largest skill in predicting the MJO and was amongst the S2S models producing the most realistic MJO teleconnections, although significantly weaker than in the ERA-5 reanalysis (Hersbach et al., 2020).The ECMWF model used in that study was a version that was operational between April 2015 and March 2016 and referred to as Cycle 41R1.Since 2016, the ECMWF model has changed several times.Although its skill in predicting the MJO has slightly improved with a gain of about one day of predictive skill since 2016 (not shown), its ability to adequately represent MJO teleconnections has not improved since 2015 (Figure 1).The amplitude of the MJO teleconnection after an MJO in Phase 2-3 seems even weaker now than 2016 (Figure 1 top left and middle panels).Roberts et al. (2023) reported the weakness of these teleconnections using a longer (40-years) re-forecast period, confirming that the error is not affected by the temporal sampling.
Figure 1 shows clearly that the representation of the MJO teleconnections is still an issue in a state-of-the-art model like the ECMWF extended-range forecasting system.For improved S2S forecasts, it is urgent to understand the sources of this error in the representation of MJO teleconnections.Kim et al. (2023) and Stan et al. (2022) noticed that errors in MJO teleconnections evolved with lead time and were dependent on the regions and season, suggesting that the origin of errors in the representation of MJO teleconnections might be caused by model systematic errors.The main goal of this article is to shed light on the possible origin and location of the errors leading to the too weak MJO teleconnections.The method used is the nudging technique, where variables (e.g., temperature) over a specific region of the model (e.g., Tropics) are relaxed towards a predetermined state (in this article the ECMWF Reanalysis v5 [ERA5] reanalysis) to answer the question of how a 'perfect prediction' over a specific region would improve the representation of the MJO teleconnections.This will provide us information about the contribution of model errors in that region on MJO teleconnections.This information should be useful for guiding future model development and focusing on areas which have the strongest detrimental impact on MJO teleconnections.In this article, we will focus on the MJO teleconnection associated with Phase 2-3.The results obtained with the MJO teleconnections associated with Phase 6-7 were different due most likely to different mechanisms and pathways.These teleconnections will be discussed in a separate article.The impact of some of these nudging experiments on the probabilistic forecast skill scores will also be discussed.
The experimental setup and nudging techniques used in this study are described in Section 2. Section 3 describes the impact of the tropical nudging experiment on MJO extratropical teleconnections.Section 4 presents the results of experiments where other areas, such as the stratosphere, have been nudged.Section 5 discusses an experiment where the near-surface atmosphere has been nudged over the Tibetan and Mongolian Plateau.Section 6 concludes and discusses the main results of this article.

DATA AND EXPERIMENTAL SETUP
Large sets of extended-range re-forecasts have been produced using the ECMWF model versions CY47R1 and CY48R1.The horizontal resolution of the atmospheric model is Tco319 (about 32 km).The model has 137 levels in vertical.The experiments follow the practices employed at ECMWF, whereby extended-range forecasts are produced twice a week and each forecast is accompanied by a set of hindcasts initialised at the same calendar day as the operational forecast but during the preceding 20 winters.Here we use the hindcast ensembles corresponding to nine forecasts initialised twice a week in mid-December and early January (e.g., from 12 December 2018 to 9 January 2019).This sums up to 180 hindcast ensembles, referred to as control experiment (CTRL), covering the period between 12 December 1999 and 9 January 2019.The focus on the winter period is motivated by the MJO being more active during that season.Each hindcast ensemble consists of 11 members (an unperturbed forecast and 10 perturbed members).
Several relaxation experiments were initialised on the same dates as the control experiments.In the first one (TROP), temperature, divergence and vorticity in the Tropics were relaxed towards corresponding fields from ERA5 with relaxation time of 12 hr over the whole atmospheric column, including the stratosphere.Other experiments were produced to test the sensitivity of the simulations to the strength of the relaxation and the variables which are relaxed (e.g., relaxing humidity in addition to temperature and wind) with little impact on the results.Therefore, only the experiments with the same relaxation configuration as TROP will be discussed in this article.Additionally, to test the importance of the stratospheric state for tropospheric predictability, we conducted another experiment (STRAT), in which the stratosphere over the whole globe was relaxed towards ERA5.In this experiment the relaxation was done above 50 hPa, that is, only mid-to upper-stratospheric winds and temperatures were relaxed, to guarantee that the relaxation did not directly affect the tropospheric skill.
In this respect, STRAT differed from some other stratospheric relaxation experiments in which a relaxation was applied at lower stratospheric altitudes down to 70 hPa (Hitchcock & Simpson, 2014;Kautz et al., 2020) or even to 150 hPa (Huang et al., 2022).Additional relaxation experiments were performed.They are called: POLAR (north of 70 • N relaxed), TP (Tibetan Plateau relaxed), TMP (Tibetan and Mongolian Plateau relaxed) and JAPAN (area over Japan relaxed).The details of all the experiments are summarised in Table 1.
In order to investigate further the origin of these errors, additional experiments were produced, but with a more recent version of the ECMWF model, known as CY48R1, which became operational at ECMWF in June 2023.This change of model version was due to the old ECMWF model versions not being available in the new ECMWF computing system.The additional experiments included a new control experiment with CY48R1 (called CONTROL48R1) and an experiment where temperature, divergence and vorticity in the longitudinal band 40 • E-80 • W were nudged towards ERA5 with a 12-hr timescale.In this experiment almost the whole globe was nudged except the Atlantic sector.This experiment is referred to as NOATL.To provide a reference for NOATL, an experiment where the entire globe was relaxed was also produced (ALLGLOBE).This experiment TMP was repeated with CY48R1 and with the relaxation limited to only the lower troposphere (10 lowest vertical levels over 137, which corresponds to a pressure of about 850 hPa in low-orography areas).This experiment is referred to as TMP1.These additional experiments are summarised in Table 2.
In the relaxation experiments, a buffer zone was applied to avoid a too strong contrast between the regions being relaxed and those non-relaxed.Horizontally, the buffer zone extends over 10 • latitude across the border of the nudging domain where the relaxation term is multiplied by a factor evolving from 0 applied 5 • outside the nudging domain to 1 applied 5 • inside the nudging domain.For instance, in an experiment with tropical nudging over 10 • N-10 • S, the nudging is at 100% between 5 • S and 5 • N, 50% at 10 • N and 10 • S and 0% at 15 • N and 15 • S. In the stratosphere relaxation experiment, we applied a nudging profile that followed the logistic function, , where K b is the level specified from which to nudge and K is the vertical level considered.Therefore, at the level of nudging, the nudging strength was 0.5 of the full strength, so 24-hr relaxation time in these experiments, which all have a 12-hr relaxation time at the nudging level.

TROPICAL NUDGING EXPERIMENT
A prime suspect for the too weak MJO teleconnection in the S2S models is the representation of the MJO itself in the S2S model integrations.MJOs in the ECMWF model, as well as in most S2S models, are too weak and propagate too slowly (e.g., Figure 3 in Vitart, 2017).In ERA5, teleconnections associated with very strong MJOs (amplitude larger than 1.5 standard deviations) are significantly stronger than MJOs associated with weaker MJOs (amplitude between 1 and 1.5) (not shown).There seems to be statistically a strong relationship between the amplitude of the MJO and the amplitude of its teleconnections, which the model seems to capture (see for example Figure S1), although the impact might vary for individual cases.The MJO propagation speed can also affect the extratropical response: Yadav and Strass (2017) and Yadav et al. (2024) showed that slow MJO events have stronger teleconnections.Therefore, errors in the propagation speed of the MJO in S2S models are also likely to impact the representation of its extratropical teleconnections.Could the systematic errors in the representation of the MJO in S2S models explain the difference in MJO teleconnections displayed in Figure 1?To answer this question, a tropical relaxation experiment was performed.In this experiment (TROP), temperature, vorticity and divergence over the tropical band between 10 • S and 10 • N were relaxed towards ERA5 with a relaxation scale of 12 hr.Figure 2 shows the forecast skill of the MJO, as a function of lead time.The MJO forecast skill score was measured by computing the Wheeler and Hendon (2004) MJO index from the forecasts and ERA5 and by computing the bivariate correlation between their time series as a function of the forecast lead time.According to the left panel of Figure 2, the prediction of the MJO in TROP was almost perfect with a bivariate correlation close to 1 during the whole duration of the forecast, which was expected since the MJO propagates over the area where the relaxation had been applied, although humidity was not relaxed.The amplitude of the MJO is also significantly improved in TROP, with an error relative to ERA5 close to 0, while CONTROL displays an MJO about 20% too weak after 10 days (right panel of Figure 2).The model biases in the tropical region are significantly reduced in the relaxation experiment (not shown), indicating that the nudging performed as expected and produced a simulation of the MJO in the Tropics very close to ERA5.
According to Figure 3, the tropical nudging leads to stronger MJO teleconnections over the northern hemisphere.In particular, the positive anomaly of 500-hPa geopotential height (Z500) over northeast Canada and western Europe is significantly stronger in TROP than in CONTROL, as well as the negative anomaly over the North Atlantic.These stronger teleconnections are consistent with the stronger MJO in TROP than in CONTROL.However, since the whole tropical band, and not only the MJO, was relaxed, it is not possible from this experiment alone to attribute the improvement in MJO teleconnections to the better representation of the MJO.Nevertheless, a striking feature of Figure 3 is that the MJO teleconnections in the tropical relaxation experiments, although improved, still have an intensity significantly weaker than in ERA5.If we consider individual ensemble members rather than computing the composite over the 11 members, none of the ensemble members produced a teleconnection close to ERA5.This result suggests that the Tropics are not the only source of error for the MJO teleconnections and that errors at higher latitudes must play a stronger role.These errors could be in the representation of the subtropical jet stream, or in the representation of Rossby waves and   their breaking in the ECMWF extended-range re-forecasts.
Further relaxation experiments are needed to identify the source of these errors.
In the experiment TROP, the nudging domain was limited to 10 • N and 10 • S.This was enough to properly capture the MJO, but this might not be enough to capture the patterns of the divergence outflow that is directed away from the deep Tropics and ultimately triggers Rossby waves (Sardeshmukh & Hoskins, 1988).Garfinkel et al. (2022) found that S2S models represented well the deep tropical omega and divergence response, but getting the subtropical response to the MJO might be more challenging.To assess the potential impact of the subtropics, an additional experiment called EXTENDED_TROPICS (described in Table 2), run with model CY48R1 and with a nudging domain extended to 25 • N-25 • S has been produced.Since this experiment was produced with model CY48R1, the TROP experiment was repeated with this new cycle (experiment called DEEP_TROPICS) for comparison.Figure S2 shows the MJO teleconnections produced by DEEP_TROPICS (top panel) and EXTENDED_TROPICS (bottom panel).The MJO teleconnections produced by EXTENDED_TROPICS are slightly stronger over the North Pacific and North America, with a significant increase of the projection of the MJO teleconnection into the Pacific North-American teleconnection pattern (PNA) pattern (PNA index of 0.13 in EXTENDED_TROPICS instead of 0.10 in DEEP_TROPICS), but the MJO teleconnections over the North Atlantic are not improved in EXTENDED_TROPICS, with the same value of the projection on an NAO pattern (NAO index of 0.15) which is much weaker than in ERA5 (0.26).Therefore, extending the tropical nudging domain did not have a significant impact on the MJO teleconnections over the Euro-Atlantic sector.We note that although the tropical relaxation experiments are efficient in constraining the MJO index, and the overall tropical circulation, some differences still remain when comparing with ERA5 (Figure S3).These differences with ERA5 are similar in DEEP_TROPICS and EXTENDED-TROPICS.

Impact of the stratosphere
Another important potential source of errors in MJO teleconnections are model errors in the stratosphere, which can be a pathway for MJO teleconnections (e.g., Garfinkel et al., 2012).In the STRAT experiment, the stratosphere above 50 hPa has been relaxed towards ERA5.The hindcasts from the STRAT experiment have a representation of weak and strong stratospheric events which is in excellent agreement with ERA5 (not shown).However, the impact of the stratospheric nudging on the MJO teleconnections is very limited, with no statistically significant differences between the Z500 composites 10-15 days after an MJO in Phase 2-3 (Figure 4) in CONTROL and STRAT.This small impact from the STRAT experiment is not surprising since this article focuses on a lead time (day 10-15) and phase of the MJO (Phase 2-3) where the stratospheric response to the MJO is weak (e.g., Garfinkel et al., 2012).The stratospheric pathway is stronger following an MJO in Phase 6-7 than in Phase 2-3.Several articles (e.g., Barnes et al., 2019;Green & Furtado, 2019;  1.For the model integrations, the composites have been computed for each individual ensemble member and then averaged from the 11-member ensemble.Schwartz & Garfinkel, 2017) emphasized that the stratospheric pathway was important for longer lead than day 10-15, since the delayed impact of the stratosphere on the NAO increases the lead time of this teleconnection.In addition, the ECMWF model displays significant skill in predicting polar stratospheric variability (e.g., Figure S4 which shows that the skill in predicting the probability of zonal wind in the upper tercile is significantly higher at 10 hPa than at 500 hPa in agreement with Domeisen et al., 2020) and the nudging experiments may not improve the stratospheric circulation enough to have a significant impact on the NAO.

Impact of high and mid-latitudes
The possible role of model errors at high latitudes was also explored.In order to assess the impact of errors at high latitudes, a relaxation experiment (POLAR) was run where the latitudes north of 70 • N are relaxed towards ERA5.This experiment produced a better representation of the MJO teleconnections, which was not surprising since the relaxation domain covers a large portion of the teleconnection pattern.However, as with TROP, POLAR underestimated the amplitude of the negative Z500 anomaly near South Greenland in the teleconnections following an MJO in Phase 2-3 (Figure 4).
Since model errors from the Tropics and stratosphere do not seem to be the only causes for errors in the representation of MJO teleconnections, we focussed on the representation of the Pacific jet stream in the model.The Pacific subtropical jet stream is a waveguide for Rossby wave propagation, including those originating from the MJO.Therefore, it plays a key role in the representation of MJO teleconnections.Lee et al. (2019) hypothesised that the impact of ENSO on the Pacific subtropical jet, which retreats to the west during La Niña and extends more to the east during El-Niño years, could explain why the MJO teleconnections in ERA5 are weaker during La-Niña years than El-Niño years.Errors in the representation of the jet stream are likely to generate errors in the representation of MJO teleconnections.Indeed, Wang et al. (2020) linked biases in the West Pacific jet to biases in MJO teleconnections in models from the World Climate Research Programme (WCRP) Coupled Model Intercomparison project (CMIP, Eyring et al., 2016) and confirmed this hypothesis using a linear baroclinic model.
In the ECMWF extended-range forecasts, the jet eastward extension retracts to the west as the lead time increases (Figure 5).By week 4, the jet stream retreats by about 15 • longitude to the west.This model bias, common to most S2S models and analysed in Vitart et al. (2022), is similar to the impact of La Niña which moves the jet F I G U R E 5 Zonal wind climatology at 300 hPa computed from European Centre for Medium-range Weather Forecasts (ECMWF) operational 20-year and 11-member re-forecasts and verifying in January for four different lead times; day 0-7 (week 1), 8-14 (week 2), 15-21 (week 3) and 22-28 (week 4).The climatology has been computed using the 11-member ensemble mean.The black vertical line represents the position of the eastward extension of the Pacific subtropical jet stream, identified as the most eastern location with a wind speed larger than 40 m/s.The green vertical line represents this location in the ERA5 reanalysis.
stream westward and could explain the too weak MJO teleconnections.In Vitart et al. (2022), several relaxation experiments aimed at improving the representation of the jet stream were run: TP, TMP, and JAPAN described in Table 1.These experiments improved the representation of the Pacific jet stream, but they did not completely remove the bias shown in Figure 5 (Vitart et al., 2022).According to Figure 4, the amplitude of the MJO teleconnections is improved in these three experiments.In particular, TP and TMP display a stronger negative Z500 anomaly than CONTROL over the Greenland region.Despite these improvements none of these three experiments produced MJO teleconnection over the North Atlantic as intense as in ERA5.Still their impact on the teleconnection amplitude over the North Atlantic is commensurable to that of the tropical relaxation.
In Figure 4, the composites were computed from MJOs produced by the re-forecasts up to forecast day 20.
In some relaxation experiments, the link between predicted MJOs and predicted Z500 might be broken by the relaxation.This could be the case for the stratosphere relaxation experiment, where the impact of the relaxed stratosphere on Z500 is likely to be more in phase with ERA5 MJOs than with the MJOs produced by the model.Therefore, Figure S5 displays the teleconnections using ERA5 MJOs instead of model MJOs.The teleconnections are significantly stronger for POLAR, particularly over the Arctic region which has been nudged.Despite this significant improvement, POLAR still displays weaker teleconnections than ERA5 over the Euro-Atlantic sector.For the other experiments, the composites in Figure S5 are close to the composites from the MJOs produced by the re-forecasts (Figure 4), but the difference is not large.This could be due to the fact that this model being skilful in predicting the propagation of the MJO, there is generally a relatively good match between the predicted phases of the MJO in the re-forecasts and in ERA5 during the first 20 days of the model integrations.

Impact on indices
To quantify the impact of the MJO on the mid-latitude teleconnections, five indices associated with the MJO teleconnections displayed in Figure 4 were considered: • NAO index computed by projecting the composites on the empirical orthogonal function (EOF) pattern of Z500 associated with the NAO, which was pre-computed from National Centers for Environmental Prediction re-analysis (second column in Table 3).
• The amplitude of the teleconnections (second column in Table S1) which is the sum of the absolute values of the Z500 anomalies from the composites 11-15 days after an MJO in Phase 2-3 (as shown in Figure 4 for example) over each grid point north of 40 • N. The value obtained is then normalised by dividing it by the value obtained from ERA5.
• The amplitude of the teleconnections over the Euro-Atlantic sector (40 3).
• The spatial correlation of MJO teleconnections with ERA5, north of 40 • N (third column in Table S1).
• The spatial correlation of MJO teleconnections with ERA5 over the Euro-Atlantic sector (40 3).
According to Table 3, the control and all the relaxation experiments produced a lower NAO index than ERA5 following an MJO in Phase 2-3, confirming the too low impact of the MJO on the NAO in the ECMWF model.Although the experiment TROP does not produce MJO teleconnections as strong as in ERA5, it reduces the error in the representation of the MJO teleconnections over the Euro-Atlantic associated with Phase 2-3 by 40%-60% depending on the index (Tables 3 and S1).For instance, the amplitude of the NAO index is almost twice as large in TROP as in CONTROL.The impact of tropical relaxation on spatial correlation over the Euro-Atlantic sector is the largest of all the experiments.JAPAN, TP, and TMP have also a strong impact on the NAO index, sometimes even stronger than in TROP, but these experiments include the impact of a perfect representation of the MJO on the subtropical jet and are therefore more difficult to interpret.The values of Tables 3 and S1 also confirm that the stratosphere exerts a comparatively weak impact on the MJO teleconnections 11-15 days after an MJO in Phase 2-3, suggesting the importance of the tropospheric pathway at this timescale for these teleconnections.Overall, Tables 3  and S1 indicate that addressing tropical errors is a very important step towards improved MJO teleconnections.
So far, the MJO teleconnections in the ECMWF extended-range forecasts have been computed by including all the 11 ensemble members.Therefore, the sampling is much larger than in ERA5, and might create weaker teleconnections.This issue was considered in Vitart (2017) by computing the teleconnections for each individual ensemble member, in order to estimate the uncertainty due to sampling.We applied the same methodology in Figure 6, by computing three MJO teleconnection indices (NAO index, amplitude over the Euro-Atlantic sector and spatial correlation over the Euro-Atlantic sector) for each individual ensemble member.Figure 6 shows the indices averaged over the 11 ensemble members (circles in Figure 6) and two standard deviations computed from the ensemble distribution.Figure 6  statistically significant within the 5% level of confidence using a Wilcoxon-Mann-Whitney test (Wilcoxon, 1945).

Impact on forecast skill
In agreement with previous analogous studies (e.g., Dias et al., 2021;Jung et al., 2010) the tropical relaxation experiment improves the extended-range forecast skill over the North Pacific, the North Atlantic and part of the United States (Figure 7).POLAR improves the skill over continental masses more than over the North Atlantic and Pacific.The stratosphere relaxation experiment has a more modest impact than the two other experiments but produces a stronger impact over Europe than the tropical relaxation experiment.However, an improved representation of tropical-extratropical teleconnection in the ECMWF model would increase the impact of tropical relaxation over Europe.

Other complementary experiments
As described in Section 2, a series of additional relaxation experiments were performed with a more recent version of the ECMWF model: model CY48R1 which became operational at ECMWF in June 2023.These experiments are described in Table 2.They include a control experiment with CY48R1 (CONTROL48R1), NOATL where almost the whole globe was nudged except the Atlantic sector, and ALLGLOBE where the entire globe was relaxed.In CONTROL48R1 (Figure 8), the MJO teleconnections are significantly weaker than in CONTROL (Figure 4) over the North Pacific, and similar over the North Atlantic.ALLGLOBE MJO teleconnections are very similar to ERA5, confirming that by nudging vorticity, divergence and temperature with a 12-hr timescale, we obtain MJO teleconnections similar to ERA5.In NOATL, the teleconnections are very close to ERA5 over most of the North Pacific, North America and Asia, which is expected since the atmospheric circulation was nudged in those regions.However, over the North Atlantic, which is a region without nudging, the MJO teleconnections are still significantly weaker than in ERA5.In particular, the very strong low near South Greenland in ERA5 teleconnections 11-15 days after an MJO in Phase 2-3 is considerably weaker in NOATL than in CONTROL48R1.This could be an artefact due to the much larger sample in NOATL which has 11 ensemble members and therefore a sample 11 times larger than in ERA5. Figure S6 shows the teleconnections in NOATL computed for each individual ensemble member.All of them produce a low in the Greenland region significantly weaker than in ERA5.In addition, ERA5 teleconnections computed over older periods (e.g., 1980-1999) show a similar strong low over Greenland (not shown), suggesting that this pattern is quite robust in the ERA5 reanalysis.As expected, experiment NOATL captures the extratropical Rossby wave train, diagnosed from meridional wind averaged between 35 • N and 60 • N, following the MJO in Phase 2-3 (Figure 9).However, the stationary patterns over the North Atlantic are visibly damped and slightly delayed.Therefore, these results indicate that part of the teleconnection errors in the ECMWF models are likely to originate from model errors over the North Atlantic rather than from remote regions.According to Table 3, by comparing the NAO index between NOATL (0.80) and ERA5 (1), we can estimate that errors in the Atlantic region reduce the NAO index by about 20%.The NAO index is 58% too weak in CONTROL48R1 compared to ERA5 (NAO index of 0.42 compared to 1 in ERA5).Therefore the 20% too weak NAO index in the experiment NOATL explains about 34% of the NAO index error in CONTROL48R1.A similar calculation indicates that the error in the amplitude of the teleconnections over the North Atlantic in NOATL is about 30% of the error in CONTROL48R1.This suggests that model errors over the Atlantic contribute to a significant portion of the MJO teleconnection errors.This could possibly come from the difficulties of the S2S models to adequately represent Rossby wave breaking in this region (e.g., Quinting et al., 2019).POLAR 0.12 0.08 0.04 0.02 -0.02 -0.04 -0.08 -0.12

NUDGING TIBETAN AND MONGOLIAN PLATEAU NEAR SURFACE
A general conclusion of all these relaxation experiments is that it is unlikely that there is a unique source of errors for the MJO teleconnections.This section will discuss one possible source of errors over the Tibetan and Mongolian Plateau.According to Figures 4 and S5, one of the most successful relaxation experiments in representing the MJO teleconnections was the experiment where a domain covering the Tibetan and the Mongolian plateau was relaxed towards ERA5, with an amplitude of the teleconnections over the North Atlantic (0.67) about 30% stronger than in the control experiment (Table 3), although the experiments failed to capture the amplitude of the Greenland low.In this experiment the whole troposphere and stratosphere between longitudes 80 • E and 130 • E and latitudes 30 • N and 60 • N was nudged towards ERA5 with a 12-hr timescale.However, relaxing the whole atmosphere column might interact directly with extratropical planetary waves and wavenumber and as a consequence might impact Euro-Atlantic weather regimes.This experiment was repeated with CY48R1 and with the relaxation limited to only the lower troposphere (10 lowest levels over 137, which corresponds to a pressure of about 850 hPa in low-orography areas).This experiment is referred to as TMP1 (see Table 2 for more details).TMP1 produces slightly stronger MJO teleconnections over the Euro-Atlantic sector following an MJO in Phase 2-3 than CONTROL48R1 and the difference with CONTROL48R1 is generally statistically significant within the 5% level of confidence (Figure 10).The amplitude of the MJO teleconnections over the North Atlantic is about 35% higher in TMP1 than in CONTROL48R1 (Table 3), which is similar to the impact of the nudging in TMP compared to CON-TROL.MJO teleconnections over the Euro-Atlantic sector are also improved following an MJO in Phase 6-7 (not shown).These improvements could be because errors near the surface in the Tibetan and Mongolian Plateau region can impact the position and strength of the subtropical jet, which plays a key role for MJO teleconnections as a waveguide for Rossby waves, due to the very high altitude of this region.Indeed, TMP1 produces a slightly stronger jet over the western Pacific, extending more eastward (Figure 11) than in CONTROL48R1 but still too westward compared to ERA5 (right vertical black line in Figure 11).This is likely to favour the propagation of the Rossby waves generated by an MJO in Phase 2-3 and the generation of stronger teleconnections.Figure 9   although still significantly weaker than in ERA5.The extended-range forecast skill scores of 500 hPa geopotential height are significantly improved in TMP1 over large areas of the northern Extratropics, including western Europe, by week 3 (Figure 12).Table 3 shows an increase in NAO index and amplitudes, but not in spatial correlations when comparing TMP1 to CONTROL48R1.These results are consistent with Xue et al. (2023) who showed that changes to Tibetan Plateau land temperature initialization had remote impacts on subseasonal to seasonal prediction, although they focussed on the spring season instead of winter as in the present article.

CONCLUSIONS AND DISCUSSION
The inability of S2S models to adequately represent MJO teleconnection is a major blockage for future improvements in subseasonal prediction over Europe.Although the ECMWF model has improved over the past years with, in particular, significantly improved MJO forecast skill scores, the representation of the MJO teleconnections has not improved.It seems to have even slightly regressed since 2015, when considering the amplitude of the teleconnections.In order to better understand the origin of the errors in the representation of MJO teleconnections in the ECMWF model, relaxation experiments where several variables over specific geographical areas were nudged towards ERA5 reanalysis have been conducted.Although these experiments do not indicate the exact source of model errors, they are helpful to understand the geographical origin of these errors.The relaxation experiments include a large set of re-forecasts with a focus on December and January forecast start dates.Several relaxation experiments were produced where different regions have been nudged (e.g., Tropics or stratosphere).
All the relaxation experiments produced improved but still weaker MJO teleconnections over the Euro-Atlantic sector than in ERA5.Relaxing the Tropics produced stronger, and more realistic MJO teleconnections over the North Atlantic than the control experiment, reducing the NAO index and teleconnection amplitude errors by about 50% according to Figure 6.This indicates that errors in the Tropics play an important role and should be a major focus for improved representation of MJO teleconnections over the North Atlantic.However, this experiment still produced too weak MJO teleconnections indicating that the Tropics is not the only source of MJO teleconnection errors, and that errors in the mid-and high latitudes  are also likely to impact the representation of the MJO teleconnections in the ECMWF model.A relaxation experiment focusing on different regions in the Extratropics also displayed improvements in MJO teleconnection, supporting the idea that model errors in the Extratropics are also an important source of errors for MJO teleconnections.Stratospheric relaxation had little impact on the quality of the MJO teleconnection 11-15 days following an MJO in Phase 2-3, except for a slight improvement in the spatial correlations.Another intriguing result was the misrepresentation of the strong low over South Greenland in the teleconnections following an MJO in Phase 2-3 when relaxing the whole globe, except the Atlantic region.This showed that, although the largest part of the teleconnection errors come from remote regions, about 30% of these teleconnection errors originate from local errors in the Atlantic and might be caused by misrepresentation of Rossby waves breaking over the Atlantic in S2S models.
Overall, these results suggest that almost all the amplitude error of these MJO teleconnections originates from errors in the representation of the MJO, in the propagation along the jet, and in the North Atlantic local amplification.
A main conclusion from this study is that there is not a single origin for errors in the representation of MJO teleconnections in the ECMWF model.This is not surprising, since a good representation of the impact of the MJO on European weather implies a good representation of the MJO, a good representation of the subtropical jet stream and Rossby wave propagation, a good representation of weather regimes and a good representation of the impact of the weather regime on local weather.Each of these steps can be an important source of error.As an example, an experiment where the near-surface troposphere over the Tibetan and Mongolian Plateau was relaxed towards ERA5 showed a slight improvement of MJO teleconnections, consistent with an improvement in the representation of the Pacific subtropical jet and improved forecast skill scores at weeks 3 and 4 and beyond.This result suggests that fixing the MJO teleconnections will require addressing multiple model errors, such as over the Tibetan and Mongolian Plateau.This is likely to be a long and slow process to address all these errors in dynamical models.This could be an area where machine-learning methods could help improve the teleconnection through online or a posteriori flow-dependent bias correction.
This article presented results from the ECMWF model only.The question remains open of how general these results are.Different models might have different sources of MJO teleconnection errors.For instance, stratospheric model errors might have a larger contribution in S2S models with poor representation of the stratosphere or low vertical stratospheric resolution or low top vertical level.This question could be addressed through coordinated relaxation experiments.This article presented results only for MJO teleconnections associated with Phase 2-3.The teleconnections associated with Phase 6-7 are also too weak in the ECMWF re-forecasts.However, the results were quite different than for Phase 2-3, with for instance a much weaker impact of tropical relaxation than for Phase 2-3 teleconnections, which might be explained by different teleconnection pathways.This will be discussed separately in another publication.
Composite of 500-hPa geopotential height, 11-15 days after a Madden-Julian Oscillation (MJO) in Phase 2-3 (top panels) and Phase 6-7 (bottom panels) for the European Centre for Medium-range Weather Forecasts (ECMWF) model versions operational in 2015/16 (Cycle 41r1, middle panels) and 2022/23 (Cycle 47r3, left panels).Verification from ERA5 is displayed in the right panels.For the model integrations, the composites have been computed for each individual ensemble member and then averaged from the 11-member ensemble.
U R E 2 Madden-Julian Oscillation (MJO) bivariate correlation of ensemble mean (left panel) as a function of forecast lead time and MJO amplitude error averaged over ensemble members (right panel), relative to ERA5 for TROP (blue line) and CONTROL (red line).In the right panel, positive (negative) values indicate a too strong (weak) MJO amplitude.The orange shading and red vertical bars indicate the 10% level of significance.In the right panel, the continuous line indicates the evolution of the amplitude of the ensemble mean, while the dotted lines represent the amplitude error averaged over all the ensemble members.

60°W
Composites of 500-hPa geopotential height, 11-15 days after a Madden-Julian Oscillation (MJO) in Phase 2-3 for ERA5 (left panel), CONTROL (middle panel) and TROP (right panel).For the model integrations, the composites have been computed for each individual ensemble member and then averaged from the 11-member ensemble.
Composites of 500-hPa geopotential height anomalies from ERA5 (top left panel) or re-forecast experiments (other panels than top left), 11-15 days after a Madden-Julian Oscillation (MJO) in Phase 2-3 in ERA5 (top left panel), CONTROL (second from the left, top), and several relaxation experiments described in Table confirms the results discussed previously: none of the relaxation experiments produces MJO teleconnections as large as in ERA5, but the Tropics represent a major source of errors for MJO teleconnections associated with Phase 2-3.The difference in MJO teleconnection indices between CONTROL and TROP is North Atlantic Oscillation (NAO) index (blue), amplitude over the Atlantic sector (red) and spatial correlation over the Atlantic sector (green) relative to ERA5 (value of 1) for each experiment.The indices have been computed for each ensemble member individually.The circles represent the average value averaged over the 11-member ensemble and the vertical bars represent two standard deviations.
Impact of POLAR (top panel), TROP (bottom left panel) and STRAT (bottom right panel) on the Continuous Ranked Probability Skill Score (CRPSS) of 2-m temperature at week 4. Red (blue) colours indicate an improvement (degradation) of the probabilistic skill score.
Same as Figure 4 but for the new control experiment with cycle 48R1 (top right panel), ALLGLOBE (bottom left panel) and NOATL (bottomright panel).For the model integrations, the composites have been computed for each individual ensemble member and then averaged from the 11-member ensemble.
Meridional wind anomaly at 850 hPa, averaged between 35 • N and 60 • N, 0 to 20 days (y-axis) following a Madden-Julian Oscillation (MJO) in Phase 2-3 in (a) ERA5, (b) Control, (c) TMP1 and (d) NOATL.The x-axis represents the longitude.Contours are 0.08 m⋅s −1 starting from 0.08 for positive anomalies (yellow/red) and −0.08 for negative anomalies (green/blue).U R E 10 Composite of Z500 anomalies 3 pentads after a Madden-Julian Oscillation (MJO) in Phase 2 or 3, in ERA5 (top left pane),CONTROL48r1 (bottom left panel), TMP1 (middle bottom) and the difference between TMP1 and CONTROL48R1 (right panel).Grid points where the difference is statistically significant within the 5% level of confidence are indicated by crosses in the right panel.For the model integrations, the composites have been computed for each individual ensemble member and then averaged from the 11-member ensemble.

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years, the nine start dates and 11-member ensemble).The bottom panel (b) shows the difference with CONTROL48R1.The right black vertical line in the top panel indicates the longitude of the eastward extension of the Pacific subtropical jet (defined as the most eastern location with a zonal wind speed at 300 hPa larger than 40 m⋅s −1 ) in ERA5.The left black vertical line indicates the extension at week 4 in the TMP1 experiment.

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I G U R E 12 Difference between TMP1 and CONTROL48R1 of receiver operating characteristic (ROC) scores of the probability of Z500 to be in the upper tercile for different lead times: day 5-11 (top left panel), day 12-18 (top right panel), day 19-25 (bottom left panel) and day 26-32 (bottom right panel).Red (blue) colour indicates an improvement (degradation).

TA B L E 2 Additional experiments with model cycle 48R1. Experiment name Relaxation Region where the nudging has been applied
NAO index, amplitude (north of 40 N over North   Atlantic)and spatial correlation (north of 40 • N over north Atlantic) relative to ERA5 associated with the composites of Z500 anomalies 10-15 days after an MJO in Phase 2-3 for ERA5 (second row), and the 7 experiments displayed in Figure4(black colour) and 4 new experiments produced with IFS cycle 48R1 (blue colour).
TA B L E 3Note: These numbers are relative to the value computed from ERA5, which now has a value of 1 by definition.Abbreviations: MJO, Oscillation; NAO, North Atlantic Oscillation.
confirms that the Extratropical Rossby wave train is stronger over North America and Europe in TMP1 than in CONTROL48R1,