Daily mean surface temperature (TG) data were obtained from the fifth version (ERA5) of the atmospheric reanalysis of the European Centre for Medium-Range Weather Forecasts (ECMWF) . Data from 1950 to 2021 have been retrieved with a spatial resolution of 0.25°×0.25°. Daily temperature fields have been averaged over the smallest spatial domain including metropolitan France (42–52° N, 5° W–9° E). ERA5 was chosen for its large time coverage and its high horizontal resolution of 0.25°.

We compute running averages of temperature for four event durations (r=3, 10, 30 and 90 d) and determine the minimum value for each winter (from December to February). This corresponds to identifying the coldest r d period for each year, or TGrd. For an even value of r, we compute the minimum over all time steps t: 1TGrdt=1r∑i=-r/2(r/2)-1TGt+i, where TGt is the daily average temperature at time step t. For an odd value of r, the equivalent sum is centred around r. Figure a shows the variations in the TGrd time series for France. Additional information on the associated distributions is shown in Appendix . We observe an upward trend in TGrd at the four event durations r. The records for each event duration occurred before 1990. However, extreme cold events still happened in the 21st century: winter 2012 witnessed the fifth-coldest 10 d and the eighth-coldest 3 d cold spells of the 1950–2021 period. In the 30 d event duration, February 1956 and January 1963 were two exceptional events, with temperature anomalies relative to the 1950–2021 trend of -2.9σ and -2.8σ, respectively (see dashed black lines in Fig. a). One event stands out in the 90 d winter temperatures: winter 1963 is at 3.7σ from the trend of winter mean temperatures and 2.2σ under the second-coldest winter in 1950–2021, winter 1952–1953. For conciseness, this paper focuses on TG90d, i.e. DJF average temperatures. This assumption of the distribution of TGrd to be Gaussian is experimentally justified in Appendix .

For the computation of analogues of circulation, we use daily geopotential height at 500 hPa (z500) from the ERA5 reanalysis from 1950 to 2021. z500 was chosen over sea level pressure (SLP) because of its lower susceptibility to perturbations from the surface roughness and its common use in weather regime studies . also showed that it was better suited to simulate temperature anomalies, although that study investigated warm temperatures. z500 data were regridded on a 1°×1° grid to reduce computation time since a higher resolution has little impact on the analogue calculation because of the smooth spatial variability in the z500 fields. We considered the z500 field over the North Atlantic region (30–70° N, 20° W–30° E) to compute circulation analogues. This domain offers a compromise between spatial coverage large enough to study the role of the synoptic circulation but small enough to not drown out the signal in the too complex hemispheric circulation.

We first compute a database of analogues of circulation following the procedure of . For a given day t, we compute the Euclidean distance of the z500 fields between t and all days t′ that are not in the same winter and within a calendar distance to t of less than 30 d (i.e. the difference in days between 2 d, regardless of the year). The K analogue days of t are the K d for which the distance from t is the smallest. We chose K=20 analogues, as advocated in previous studies .

The circulation analogues were computed using the Blackswan Web Processing Service . We consider three different analogue datasets depending on the time period in which the analogues are selected:

1951–2021 – the whole length of available ERA5 data;

The analogue-based stochastic weather generator (hereafter referred to as SWG) developed by uses stochastic reshuffling of daily atmospheric fields to generate atmospherically consistent alternative trajectories of climate events. This algorithm was adapted by to simulate extreme heat waves using the principle of importance sampling. The goal is to simulate L d trajectories of a model while maximizing an observable variable (e.g. local temperature). Here we focus on a modified version of the dynamic type of simulations developed by , which computes alternative atmospheric trajectories starting with the same initial conditions as an observed event. Each time step is the combination of an atmospheric circulation (characterized by z500) and a variable of interest, such as temperature. The SWG starts at a given initial condition and goes from one time step to the next using analogues of circulation according to the process described hereafter. The resulting simulation is a constrained reshuffling of days of the input dataset. We start at an initial condition t0, which thus constitutes the first time step of the simulation. To simulate temperature for the day after t0 (t=t0+1), we randomly pick one analogue among the K best analogues of the observed z500 field on day t0+1. The geopotential height at 500 hPa (z500) and the 2 m temperature (TG) fields of this analogue day constitute the simulated day t′. For the next time step, t is replaced by t′+1, the day following t′. This random process is repeated sequentially for L time steps, the length of the simulation. This defines a Markov chain of TG, with latent states provided by the analogues of z500.

At each time step, the selection of the analogue day follows several constraints and weights controlled by the parameters that are described hereafter. To better follow the seasonal cycle of the simulated season, we use K weights ωcal(k) (k∈{1,…,K}) on the analogue selection that depend on a parameter αcal, which favours analogue days that are closest to the calendar date (i.e. the day of the year, regardless of the year) of time step t: 2ωcal(k)=Acale-αcaldk, where Acal is a normalizing constant, αcal≥0 is the calendar weight, and dk is the number of calendar days between the kth analogue day and t. The resulting seasonal cycle of the SWG is shown in Appendix .

To favour the simulation of the most extreme events, importance sampling weights ωT(k) are introduced, with a control parameter αT≥0. When the value of αT increases, the stochastic weather generator favours analogue days with extreme temperatures. The K analogues of t are sorted in ascending order of temperature with ranks Rk(k∈{1,…,K}) such that the coldest analogue day has a rank of 1. Hence the weight associated with the kth analogue day of t is 3ωT(k)=ATe-αTRk, where AT is a normalizing constant, αT is the importance sampling weight and Rk is the rank (in ascending order) of the kth analogue day among the K analogues. If αT=0, there is no importance sampling, and the SWG is the same as in . As explained by , using the ranks of temperature instead of their absolute values eliminates the need to re-scale variables and allows for an interpretation independent of the units of the variables. It also ensures that simulations are biased in the same way, as importance sampling weights are the same at each time step (the ranks are simply all integers between 1 and K=20).

The SWG with importance sampling is obtained by combining the weights on the calendar and importance sampling. The kth analogue day of day t has a probability of 4ωk=Ae-αcaldke-αTRk, where A is a normalizing constant such that ∑k=1Kωk=1.

We have refined the original methodology proposed by by introducing a new approach for simulating trajectories with the analogue-based SWG. Our primary objective is to generate extreme winter events that are characteristic of specific climate periods, essentially seeking to estimate the extremes attainable within a climate state characterized by a defined level of warming. Given a reference record-shattering event, we want to assess the likelihood of a similar event occurring within a given climate period, without specific information about the event itself.

To achieve this objective, we have made a crucial adjustment of the original SWG configuration (configuration 1, which considers the entire dataset in the simulations). This new configuration (configuration 2) no longer considers analogues that coincide with the reference event. At each time step, the selection process is based on the K best analogues, but the weights ωk of analogue days that fall within the observed event are set to zero. For example, if we initiate a simulation on 1 December 1962 spanning L=90 d, all analogue days falling between 1 December 1962 and 1 March 1963 are excluded from consideration in the simulation. Days of the same year were already excluded during the analogue computation process, but they can still be selected in simulations as analogues of days in other years. This procedural adjustment ensures that our simulations are solely driven by the initial conditions and the state of the climate and do not rely on any specific information pertaining to the observed event (apart from the first day). In other words, we are assessing whether a record-shattering event of 90 d can be inferred from information related to less extreme events.

We also adapted this stochastic weather generator from the one in , which was designed to simulate summer heat waves, to the version used in this analysis tailored for simulating winter cold spells. Consequently, the importance sampling weights now favour the coldest analogues instead of the warmest ones.

To control the consistency of the atmospheric trajectories produced by the SWG, we compute the derivatives of the resulting time series of z500 and temperature from first-order differences. The derivatives are compared on the one hand to the derivatives obtained in the observed winter and on the other hand to the time series computed by randomly picked winter days from the ERA5 dataset. The results are shown in Fig. a and b. Because of the approximation by analogues, the SWG loses continuity, which appears in the derivative calculation. But the ratio of the z500 derivative standard deviation between the simulations and the observations is ≈1.2, which is far less than the same ratio computed from random time series (≈3.5). This illustrates that the SWG atmospheric trajectories are mathematically consistent and realistic. The same analysis for the daily temperature derivative shows a larger difference between the ERA5 reanalysis and the SWG simulations. This can be explained by the fact that the analogue SWG trajectories are primarily based on an atmospheric trajectory constraint (and not a constraint on temperature). The resulting temperature time series are still more consistent compared to time series of random winter days (the ratios of standard deviation derivatives with the reanalysis are ≈1.8 and ≈3.2, respectively). Therefore, the short-term variability (i.e. a few days) is deemed to be relevant from a mathematical point of view.

Finally, we compare the mean temperatures of 1000 SWG trajectories with αT set to 0.5 to the mean temperatures of 107 control SWG trajectories with αT set to 0 (i.e. without importance sampling) and to analogues in 1950–1999 (Fig. c). In the control simulations, the SWG covers the range of observed winter TG90d, apart from extreme values in the tail of the distribution, as in winter 1963. The median of simulations with importance sampling is close to the coldest control simulation. The SWG with importance sampling simulates extreme events in the tail of the distribution and is able to reach values on the same order of magnitude as winter 1963. The difficulty of obtaining winters as cold as 1963 without importance sampling shows the added value of this method for record-shattering events. The resulting distributions are comparable to the ones obtained for heat waves with a rare-event algorithm by and a control simulation of a climate model.

We verify in Appendix  that the selection of analogues does not yield obvious biases towards the earlier parts of each analogue period and that the quality of analogues is stationary in time across the two periods. Appendix  compares the climatological variations in the SWG without importance sampling to the original ERA5 data.

First we tune the αT and αcal parameters for winter DJF simulations. We simulated ensembles of trajectories of 90 d starting on 1 December of each year, with different values of the parameter αcal. Figure a shows the percentage of simulations for which the last day of the simulation falls after 15 February. If the calendar weight value is too small (e.g. αcal=1), fewer than half of the simulations end with a calendar day after 15 February. This means that the trajectories of the simulated events with αcal=1 are less consistent with the seasonal cycle. With an αcal parameter greater than 5, more than 75 % of the simulations have their last day falling after 15 February. Therefore we use αcal=5 in the following.

We run the SWG with parameter values αT∈{0,0.1,0.2,0.5,0.75,1} starting on each 1 December, between 1950 and 2021. Figure b shows that for αT=0, the SWG simulates events covering the range of winter mean temperatures from 1950–1951 to 2020–2021 apart from winter 1962–1963. For αT=0.2, a few outlier simulations reach winter 1963 temperatures. A value of αT greater than 0.5 allows the simulation of a greater proportion of extreme events. The difference for αT greater than 0.5 being less significant, we chose for the following αT=0.5.

To compare the configuration of the SWG excluding information from the event of reference to the previous configuration of , we use both to simulate the worst-case winter scenario from 1950 to 2021. For each winter from 1950 to 2021, the simulation starts at t0 – 1 December – and runs for L=90 d over DJF (December, January and February). n=100 simulations are run per winter year; hence 100×71 events are simulated for each experiment.

Then we focus on the record-shattering event of winter 1963. Winter 1963 being our reference as a record-shattering event, we simulate alternative extreme winters in the counterfactual and factual climates without using the information from winter 1963 to evaluate to what extent such a winter can be extrapolated from available data in both climate periods. Simulations are also made using the two sets of analogues – the counterfactual and factual periods. Hence we simulate winters that could have occurred in the selected analogue period. This allows the simulation of worst-case events from the same initial conditions but considering different climate states. We run n=1000 simulations starting on 1 December 1962, using as previously the factual and counterfactual sets of analogues. This allows us to have a wider ensemble of possible winter temperatures starting from the initial conditions of winter 1963.

In this subsection, we simulate cold winters of 90 d starting on 1 December of each year from 1950 to 2021. Analogues can be selected in any year from 1950 to 2021. We evaluate the impact of the possibility of sampling analogues from the reference event on the winter average temperature: the SWG configuration in versus the one in the present paper.

Figure  shows the results of simulations from 1951 to 2021 using SWG configuration 1 (Fig. a) and SWG configuration 2 (Fig. b) in Sect. , with analogues sampled in 1950–2021. The SWG successfully simulates extremely cold winters, with simulations being 3.9 °C (Fig. a) and 3.1°C (Fig. b) colder overall compared to the long-term mean of ERA5 temperatures. Moreover, 40 % of all simulations reach a mean temperature as cold as the 1963 record with configuration 1, while only 13 % of them are as cold when using configuration 2.

The variability in the simulations performed with configuration 1 follows the variability in historical winter temperatures closely due to the possibility of selecting analogues falling in the observed event with this configuration. The medians of simulations made with configuration 1 are highly correlated (r=0.88) to the observed temperatures. With configuration 2, there is no correlation between the median of simulations and the observed temperature. This is a consequence of the fact that configuration 2 does not use information from the observed event apart from the initial conditions. The simulation length of 90 d being longer than the decorrelation time of atmospheric dynamics, the resulting events should not be highly influenced by their initial conditions. The standard deviation of the medians of the boxplots obtained with configuration 2 is also very low: 0.15 °C compared to the 0.44 °C obtained with configuration 1. This is coherent with the chaotic internal variability in the climate system, resulting in simulated events being representative of the climate analogue period used rather than the initial conditions.

The mean of the boxplot median is also higher by 0.77 °C with configuration 2 compared to configuration 1, which can be explained by the fact that configuration 1 allows more days to be selected during the importance sampling process, so the coldest days can be selected intentionally during the simulations, while some analogue days are excluded when using configuration 2.

In this subsection, we simulate winters starting on 1 December 1962 and consider circulation analogues in the counterfactual and factual periods. We simulate 1000 winters that could have been in 1962–1963, with z500 analogues in two periods. Figure a focuses on those simulations for both the factual and counterfactual analogue periods. The first quartile of simulations does not reach 1963 mean values in both the factual and counterfactual periods (boxplots in Fig. a). The year 1963 was already a very rare event in the 1950–1999 climate but remains reachable using only 1971–2021 analogues in a climate with more global warming. The observed increase in mean winter temperatures between the counterfactual and factual periods is not reproduced if we consider extreme winters as simulated by the SWG, even if we exclude winter 1963 from the counterfactual period (configuration 2). We observe an increase of 0.44 °C in TG90d between the two periods, while the difference is only 0.13 °C between the two ensembles of extreme winters. In other words, extremely cold winters do not warm at the same rate as winter mean temperatures.

Figure b summarizes the time series of daily average temperatures associated with winter 1963 simulations. The temperatures of winter 1963 are below the seasonal cycle for most of the season. The temperatures of simulated events for both the factual and counterfactual periods are also overall below the seasonal mean for the whole length of the event. The median fluctuates around 2 °C, while the 5th percentile of all simulations reaches -5 °C during most of the winter. The 95th percentile stays under 2 °C above the seasonal average, while the 5th percentile reaches -4 °C during most of the event, which corresponds to the coldest daily mean temperatures observed in 1963. Overall the range of daily winter temperatures as simulated for extreme winter temperatures in the factual and counterfactual periods matches the range of daily mean temperatures observed in the reference event in 1963. Hence a winter like the one of 1962–1963, even with a low probability, can still occur using data from the warmer climate of the 21st century. We find that the likelihood of such cold winters barely decreases with time, as the distributions of simulated temperatures are very similar for the two analogue periods.

During winter 1963, a strong and persistent anticyclonic anomaly prevailed over Iceland. It was associated with a negative z500 anomaly over continental Europe, the Azores and the glacial Arctic Ocean, leading to a weakening of the westerlies and advection of cold air from the Arctic (Fig. a).

Here, we compute the composites of z500 and z500 anomalies over a region that is larger than the region for which the analogues are obtained. The z500 composite over DJF does not correspond directly to a North Atlantic weather pattern (the negative phase of the North Atlantic Oscillation (NAO)), as, for instance, obtained by . The low over Europe is located more to the east than for an NAO- weather pattern, while the positive z500 anomaly over Iceland is located more to the north than it would be in an Atlantic Ridge weather regime. The respective positive and negative z500 anomalies over Iceland and the Azores are, however, characteristic of NAO-, which is often, even if not systematically, an indicator of colder-than-usual winter temperatures over Europe .

The z500 anomalies of SWG simulations for winter 1963 are spatially smoother than the ERA5 field for the same winter. This can be explained by the fact that the map is averaged over 100 (10 % of 1000 simulations) different simulations and that simulations are less auto-correlated than observed events would be, thus having more spatial variability. Hence, we compute the z500 and z500 anomaly composites of SWG simulations for the coldest 10 % of members of the ensemble starting on 1 December 1962. For comparison purposes with winter 1963, this selection is reasonable because 75 % of simulations are warmer than this record event (Fig. a), and we want to focus on the coldest members of the ensemble. Appendix  also shows that the average maps are representative of individual events. We find that the pattern of a strong negative anomaly over western Europe and a positive anomaly over Iceland is still reflected in the coldest 10 % of events simulated with the SWG in both the counterfactual (Fig. b) and factual (Fig. c) simulations. The intensity and position of the z500 low over the Barents Sea and the z500 high over western Russia, as seen in ERA5, seem to have a lower contribution to the intensity of the event, as they are weaker and less marked in the SWG-simulated events.