We used the ECHAM6 atmospheric model (Stevens et al., 2013), with a horizontal truncation of T63 and 95 levels in the vertical with a model top at 0.02 hPa. It was run in bias-corrected and biased modes and with SST perturbation experiments. The bias correction process follows Kharin and Scinocca (2012) and has been used to study the effects of model biases on the Eurasian snow extent–polar vortex connection (Tyrrell et al., 2020), Quasi-Biennial Oscillation teleconnections (Karpechko et al., 2021), and ENSO–Northern Hemisphere winter teleconnections (Tyrrell and Karpechko, 2021) and involves two steps: first, the dynamic variables of the model (divergence, vorticity, temperature, and log of surface pressure) are nudged towards ERA-Interim reanalysis. During this step the nudging tendencies are recorded every 6 h. A total of 40 years of nudging tendencies are then composited and smoothed to create an annual climatology of the nudging tendencies. This climatology represents the inherent biases in the model. In the second step, the nudging tendency climatology is added to the model as an additional tendency at each time step, in order to correct the biases in the model's climatology. For the second step it was experimentally found that the biggest reduction in biases occurred when only the divergence and temperature were corrected. The dynamic variables of ECHAM6 are solved using a spectral decomposition of the globe, which allows for nudging and bias correcting on specific wavenumbers. Wavenumbers below n=21 were nudged and corrected, which means that features below about 1000 km were not corrected. The bias corrections can also be applied at different height levels, and three experiments were performed with bias corrections in the troposphere only, TropBC; stratosphere only, StratBC; and full atmosphere, FullBC (details in Table 1). The critical difference between the nudged and bias-corrected runs is that when the model is nudged it is very tightly constrained towards observations, whereas when the bias correction tendencies are applied the model can still respond realistically to perturbations. Additional details of the bias correction scheme are available in Tyrrell et al. (2020) and Tyrrell and Karpechko (2021).

Simplified ENSO SST sensitivity experiments were performed using the bias-corrected climatologies as described in Tyrrell and Karpechko (2021). For the ENSO SST pattern we used a regression of the Niño3.4 time series and HadISST SSTs from 1979–2009. Only the positive regression values between 30∘ S and 30∘ N and east of 150∘ E in the Pacific Ocean were used, and the regression values were multiplied by 1.5 to strengthen the response, corresponding to an El Niño or La Niña forcing magnitude of 1.5 K. There are also ENSO-related SST anomalies outside the tropical Pacific which were excluded from the perturbed SST forcing. Although they can be important for some ENSO teleconnections, they are primarily a response to the tropical Pacific forcing and occur at time lag (Tyrrell et al., 2015), so they were excluded to reduce the complexity of the forced ENSO signal. Climatological SSTs using HadISST data from 1979–2009 were used outside the tropical Pacific and for the control run (CTRL). The ENSO anomaly was kept constant in time (i.e. the anomaly did not vary seasonally), and each experiment was run for 100 years.

The ERA5 data from 1979–2019 (Hersbach et al., 2020) were used as a reference to compare to the model results. El Niño and La Niña years were defined by the DJF value of the Oceanic Niño of ERSST.v5 SST anomalies in the Niño 3.4 region (5∘ N–5∘ S, 120–170∘ W), from the NOAA CPC website (https://origin.cpc.ncep.noaa.gov/products/analysis_monitoring/ensostuff/ONI_v5.php, last access: 16 June 2021), and using a threshold of ±0.5 K; see Supplement Table S1. This results in 13 El Niño years with 9 SSWs, 13 La Niña years with 11 SSWs, and 15 neutral years with 6 SSWs. The relatively low number of El Niño and La Niña years and SSWs means that few of the reanalysis ENSO results have statistical significance, and they may be dependent on the temperature threshold for defining ENSO events. As such, the reanalysis is included as a reference, but a more in-depth analysis focusing on ERA5 – and other observational data sets – would be required to fully verify and explain those results.

The SSW central date is defined using the Charlton–Polvani criterion (Charlton and Polvani, 2007), defined as the first day when zonal mean zonal wind at 60∘ N and 10 hPa (uz60) is easterly (i.e. uz60<0 m/s). The reversal has to occur during 1 November–31 March. After an SSW has been detected, winds must return to westerlies for 20 consecutive days before another SSW is detected (as in Butler et al., 2017) to avoid multiple detection of the same event, and uz60 must return to westerlies for at least 10 consecutive days before 30 April to exclude final warming.

The bias corrections are applied globally at different pressure levels. The reductions in biases have a three-dimensional structure which has relevance to the ENSO teleconnection to the stratospheric vortex and the Northern Hemisphere, and this was explored in Tyrrell and Karpechko (2021). As this paper focuses on SSWs, the reduced model biases in the wintertime polar vortex are of particular interest. In Fig. 1a we show the seasonal progression of uz60 using the mean daily values for the 100-year model runs and 41 years of ERA5 data. The standard deviation for ERA5 and CTRL is also shown as shading. The CTRL run (blue) has a too weak vortex compared to ERA5 from October to January, and this bias is reduced by approximately half in the FullBC and StratBC runs. The bias corrections in TropBC actually increase the bias in the polar vortex in November–December. All model runs effectively capture the polar vortex strength during February and March. As shown in Fig. 1b the interannual variability of the vortex strength is relatively well simulated in CTRL, and the bias corrections do not significantly change the variance. The largest difference between the reanalysis and the model is in January when ERA5 exhibits increased variance, which is not simulated by any of the model runs. The mean difference in daily uz60 between El Niño and neutral years and La Niña and neutral years is shown in Fig. 1c and d for the model (i.e. daily mean of 100 El Niño or La Niña years minus 100 neutral years) and ERA5 (15 El Niño years, 13 La Niña years, minus 13 neutral years). The CTRL, FullBC, and StratBC runs have the strongest mean El Niño response throughout winter, although only the StratBC has a statistically significant response in January (as shown at the 5 % level by bold lines). CTRL and FullBC show a weaker response in January. TropBC has only a weak El Niño response throughout winter. The CTRL has the weakest La Niña response and the StratBC the strongest, with a persistent response from December to March. The mean daily ERA5 response to both El Niño and La Niña shows large variability with little significance in the response. For certain months the ERA5 response is opposite to that seen in the models, e.g. the February–March La Niña response, and at times it shows a similar magnitude and sign, e.g. the La Niña response in January or the El Niño response in March. The mean ENSO response was studied in more detail in Tyrrell and Karpechko (2021), where seasonal mean values indicated that in early winter the models and ERA5 disagreed on sign of the El Niño response and agreed on the La Niña; then in late winter they agreed on the El Niño response and disagreed on La Niña.

Before analysing the SSW responses, we assess the ability of the model to capture the timescales of variability. This is explored in Fig. 2, following Fig. 1 from Baldwin et al. (2003). Using the geopotential height averaged over the polar cap (60–90∘ N) (Zcap), and at pressure levels from 1000 to 1 hPa, the figure shows the time in days when the autocorrelation function drops to 1/e. The day-to-day variability is smoothed with a Gaussian filter (σ=26 d). The CTRL run captures the timescales of the variability in the winter stratosphere reasonably well, but it is slightly too weak in early winter. The timescales are shorter in the FullBC, and again in TropBC runs, and are slightly longer in StratBC. In all experiments, the timescales are shorter in El Niño experiments and longer in La Niña experiments than in the corresponding neutral experiments. The relationship between the strength of the polar vortex and the timescales of variability was tested in Fig. 3, which plots the DJF UZ 60∘ N 10 hPa against the DJF timescales of variability averaged from 150 to 50 hPa. Figure 3a shows each ENSO phase for each model separately, so a weaker or stronger vortex strength may be due to the ENSO phase or the bias corrections. A stronger vortex corresponds to longer timescales of variability, and a weaker vortex corresponds to shorter timescales, with a correlation coefficient of r=0.62. We examine this more closely in Fig. 3b by averaging each ENSO phase (i.e. the mean of CTRL_EN, FullBC_EN, TropBC_EN, and StratBC_EN; El Niño is upward-pointing triangles, La Niña is downward-pointing triangles, and neutral is the circles) and each bias-corrected run (i.e. the mean of FullBC, FullBC_EN, FullBC_LN; coloured crosses). We see that as the vortex strengthens and weakens by ENSO phase, the timescales of variability change accordingly. However, changes to the vortex strength due to the bias corrections do not correspond neatly to changes to timescales of variability. A stronger vortex has weaker dynamical variability and is driven by slow radiative processes, which may explain the vortex–variability timescale relationship between ENSO phases. On the other hand, at least in the case of CTRL–FullBC, the relationship does not hold, because FullBC has both stronger vortex and shorter variability; therefore, application of the bias correction technique may have affected the timescales of the variability.

The statistics of SSWs are detailed in Table 1 and Fig. 4. The years used for compositing ERA5 are shown in Supplement Table S1. The number of SSWs per year is overestimated in CTRL (1.12 SSWs per year) in comparison to ERA5 (0.63 SSWs per year). This statistic is made more realistic in FullBC (0.71 SSWs per year), but there is only a small improvement in the StratBC (1.04 SSWs per year) and TropBC (1.10 SSWs per year) runs. The significance of the change in the number of SSWs due to the bias corrections was tested using a Monte Carlo simulation (Fig. 4). Specifically, in each experiment, 100 years were randomly chosen with replacement, and the number of SSWs in the sample was calculated. The procedure was repeated 10 000 times to obtain a distribution. The 5–95th percentiles of the distributions are shown in Fig. 4 with solid error bars. We see that FullBC is significantly different from CTRL for neutral (green markers) and El Niño years (red markers), while TropBC differs only for El Niño years. The StratBC runs do not differ significantly from the CTRL for any ENSO phase, although in all cases its distributions lie between those of CTRL and FullBC.

The difference between the models and ERA5 was also tested with Monte Carlo simulations. This was done by choosing the same number of years from the model experiments as there were in the corresponding ERA5 samples. For example, 13 years were taken from the El Niño experiments to match the number of El Niño years in ERA5. The 5–95th percentiles from those simulations are also shown in Fig. 4 as the dashed vertical lines that extend beyond the solid error bars (since there is greater uncertainty with fewer chosen years). The ERA5 statistics are shown as dashed horizontal lines. Two lines are shown to compare the experiments without an ENSO forcing with ERA5. The green dashed line is the number of SSWs per year for the full ERA5 period (41 years), and the black dotted–dashed line is for the 15 neutral years. Although the full period includes El Niño and La Niña years, it is not dependant on one phase and includes a large number of years, whereas the neutral-only years have a small sample size. The model distribution is again estimated by choosing either 41 years (green dashed vertical line which corresponds to the green dashed horizontal line) or 15 years (black dotted–dashed vertical line which corresponds to the black dotted–dashed horizontal line). The results show that FullBC is the only run that consistently captures the SSW statistics of ERA5 in all experiments. Note all La Niña experiments are consistent with ERA5 as well as with each other.

Consistent with previous modelling studies (e.g. Polvani et al., 2017), SSW frequency is increased during El Niño years and decreased during La Niña years in all model experiments. For all model experiments, except TropBC, the number of SSWs during El Niño years is nearly twice as large as that during La Niña years. In TropBC, the exceedance is 40 %. For ERA5 we find that SSW frequency is increased during both El Niño (0.69 SSWs per year) and La Niña (0.85 SSWs per year) years, consistent with previous studies. The years used for compositing ERA5 are shown in Supplement Table S1.

The seasonal evolution of SSW frequency is shown in Fig. 5. To explore the differences in seasonal evolution more clearly, the number of SSWs in each month is divided by the total number of SSWs for each experiment, similarly for ERA5. This gives the percentage of the annual total SSWs in each month. Compared to ERA5 there is not enough seasonal variation in CTRL, with too many SSWs in November and March and too few in January and February. The seasonal variation is improved slightly in FullBC, although the seasonal cycle is still underestimated. In StratBC and TropBC the SSW seasonal statistics are not improved as much as in FullBC. In particular, TropBC almost has an inverse of the seasonal relationship of SSWs compared to ERA5, with the most SSWs in November. There are no consistent changes to the seasonality of SSWs with El Niño or La Niña years in the model or ERA5. Note that the model does not have a seasonally evolving ENSO pattern, which may affect the seasonality of SSWs in the ENSO experiments but not in experiments with neutral ENSO conditions which have seasonally evolving SSTs. Yet the neutral ENSO experiments similarly lack seasonal SSW variations.

Figure 6 shows the SSW composite of normalized Zcap and then the anomalous values in El Niño an La Niña years relative to neutral years. For ERA5 the ENSO phases are normalized using the standard deviation for all ERA5 years. The CTRL run simulates the downward propagation of stratospheric anomalies after an SSW reasonably well, although it underestimates the tropospheric response in comparison to ERA5. For neutral years (Fig. 6d, g, j, m) the CTRL and StratBC runs have the weakest tropospheric response, and the FullBC run has the strongest response, which is most similar to that of ERA5. All runs show a weaker stratospheric response during El Niño years in comparison to neutral years, both before and after SSWs (i.e. Fig. 6e, h, k, n); however this does not always correspond to a weaker tropospheric response. This is more clearly shown in Fig. 7, which shows a 30 and 90 d time mean of Fig. 6. We see that in the CTRL and FullBC runs the weaker stratospheric response in El Niño years corresponds to a weaker tropospheric response (Fig. 7b and c). In the TropBC and StratBC runs (Fig. 7d and e) the change in the stratospheric response due to El Niño is not as pronounced (i.e. the SSW response is only slightly weaker than in neutral years). Correspondingly, there is only a small difference in the tropospheric SSW response in TropBC, while in StratBC the response is actually stronger than in neutral years (for the 30 d mean) or very similar (for the 90 d mean).

The models show a stronger stratospheric response during La Niña years (Fig. 6f, i, l, o). In FullBC and StratBC in particular, this corresponds with a strong tropospheric response. The ENSO response in ERA5 differs from the models. During El Niño years there is a stronger response (relative to neutral years) before SSW events, with a slightly stronger stratospheric response and weaker tropospheric response after SSW events. Whereas during La Niña years the normalized Zcap response is weaker before and stronger after SSW events.

The sea level pressure response to SSWs is well represented in all model experiments and is similar across different climatologies, i.e. bias correction does not greatly affect the surface response. Figure 6 shows the composites of absolute sea level pressure anomalies averaged over 30 d after the central dates of SSWs, and the differences between this quantity in El Niño minus neutral years (middle column) and La Niña minus neutral years (right column). A negative Arctic Oscillation (AO) pattern following SSWs is seen in all runs. The negative AO pattern is stronger in La Niña experiments for the FullBC and StratBC runs. These runs both have a stronger La Niña stratospheric Zcap response (Figs. 6i, o and 7h and j); however, there is not a linear relationship between the stratospheric ENSO response and the surface response. TropBC has a smaller La Niña stratospheric response and surface pressure response (Fig. 6l), but CTRL has a large stratospheric La Niña response (Figs. 6f and 7g) without a surface pressure response (Fig. 6f). Similarly, the weaker Zcap response in Fig. 6 during El Niño years can be seen in the weaker negative AO response in CTRL and TropBC, but not FullBC or StratBC (Fig. 8e, h, k, n). The 2 m temperature response was expected to be weak in the model runs, since the same climatological SSTs were used for all runs (except SST anomalies prescribed in the tropical Pacific in El Niño and La Niña experiments), which dampens the near-surface temperature anomalies. However, there was a La Niña – El Niño difference of 0.4 K across Eurasia in the monthly averaged 2 m temperature (not shown).

We now look at the wave forcing that causes SSWs. Figure 9 shows the SSW composite anomalies for the heat flux at 100 hPa, 45–75∘ N. The black lines show all wave numbers, the red lines are wave 1, and the blue lines are wave 2. Solid lines indicate dates when the composite anomalies are significantly different from zero at the 90 % confidence level. The ratios of wave 1 to wave 2 SSW events are also listed in Table 1, where each event is defined based on the average heat flux for the 10 d preceding an SSW. The CTRL run has a too small wave 1 / wave 2 flux ratio of 62/38 compared to the ERA5 ratio of 77/23; i.e. there are too many wave 2 events in CTRL. ERA5 has 0.15 wave 2 SSWs per year, and CTRL has 0.41 wave 2 SSWs per year. This ratio is improved in the bias correction experiments, with the FullBC (76/24) being most similar to ERA5, and a smaller improvement in StratBC (67/33) and TropBC (68/32). As expected, in ERA5 the wave 2 flux is weaker in El Niño years and stronger in La Niña years; hence, La Niña events have a smaller wave 1 / wave 2 ratio than El Niño events (e.g. Garfinkel and Hartmann, 2008). This is simulated reasonably well in the experiments, but the relationship is weaker. For all climatologies the El Niño years have a larger wave 1 / wave 2 ratio than La Niña years; however, in CTRL the La Niña experiment has a larger wave 1 / wave 2 ratio than the neutral experiment. The total heat flux anomaly before an SSW is smallest in El Niño and largest in La Niña in all climatologies. Since the anomalies are calculated with respect to each experiment's own background flux, which is largest in El Niño and smallest in La Niña experiments, the result explains the larger frequency of SSWs in El Niño and small frequency in La Niña. It happens because during El Niño years, an SSW can be induced by a weaker wave activity pulse, which happen more frequently. However, a larger wave activity pulse that occurs more rarely is required to induce an SSW during La Niña years. Note that in all experiments as well as in ERA5 the larger flux during La Niña years is due to increased wave 2 contribution; however, only in StratBC is the wave 2 increase larger than that of wave 1, which is also seen in ERA 5.