As a reference dataset, we use ERA5 , the latest generation of a global reanalysis product provided by the European Centre for Medium-Range Weather Forecasts (ECMWF). This dataset provides a gridded best possible estimate of the state of the atmosphere by combining short forecasts from ECMWF's Integrated Forecasting System (IFS) Cycle 41r2 and the so-called 4D-Var data assimilation scheme fed by numerous observations. For characterizing the observed 2018 heatwave in terms of near-surface temperature as well as the large-scale atmospheric flow, we used 2 m maximum temperature as well as 500 hPa geopotential at a daily temporal resolution on a spatial grid of 0.25° × 0.25° from the ERA5 post-processed daily statistics on single levels dataset .

In this study, we first use a storyline approach to estimate the impact of climate change on the July 2018 event. The dataset consists of 5 storylines, namely:

Counterfactual: a world without the influence of anthropogenic global warming, corresponding to pre-industrial times (1850–1920).

Each storyline represents a physically consistent, plausible scenario in which the event could have developed under an alternative thermodynamic background while holding the circulation fixed. The storylines were simulated as described in detail in using the atmospheric general circulation model ECHAM6.0 – atmospheric component of the MPI-ESM model – (T255, 95 levels), which integrates the land vegetation model JSBACH for land processes. The spectral nudging of vorticity and divergence (nudged for wave numbers n<38, >750 hPa) towards NCEP-NCAR reanalysis ensures that the large-scale weather pattern stays close to the observed one in all storylines. The storylines then mostly differ in their thermodynamics, diminishing the influence of internal variability. To impose different levels of global warming for each storyline, sea surface temperatures (SSTs) and greenhouse gases (GHGs) are prescribed according to the desired level of global warming (since these variables are influenced by anthropogenic forcing with a high level of certainty, ). As a result, all 5 storylines listed above are available for the same period, running from 2015 to the present day under different warming conditions. This means that each storyline contains the historical events occurring globally during this time period (at least those reproducible at this resolution). Each storyline has 5 members, such that the spin-up simulations for each member per storyline were started at different dates, in consecutive weeks, to account for model and initial condition uncertainty. We use the long-term simulation (ECHAM_SN) also spectrally nudged towards NCEP reanalysis data as a climatological period (1985–2014) consistent with our simulations.

Finally, the chosen model setup nudges the model to reproduce the observed evolution of the synoptic-scale flow (e.g, atmospheric ridges, troughs, and blockings), being able to react to imposed thermodynamic changes and allowing changes to realistic local weather events.

For the statistical analysis, we use transient simulations from the MPI-ESM-LR Grand Ensemble (50 members) climate model under the SSP5-8.5 emission pathway and historical run. The grand ensemble is simulated using the Max Planck Institute Earth System Model version 1.2 (MPI-ESM1.2), in the low resolution (LR) setup (T63, 1.8° atmosphere; GR15, 1.5° ocean) . Due to the availability of 50 ensemble members at a sub-daily resolution, the model provides a sample size large enough to work with flow-analogues and extreme events.

The dataset for each degree of global warming is defined as the 20 years centered around the year where the 20 year running average of global mean temperature surpasses the desired temperature anomaly above pre-industrial times (1850–1920). For example, if +2 K is reached in 2033, the period 2023–2043 is taken. This is done for each ensemble member, such that, in total, each global warming level has 1000 years (20 years × 50 members) of data.

As described Sect. , the spectrally nudged storylines reproduce a specific event of interest under different warming levels, in our case, the July 2018 heatwave. Since the only differences between them are given by the background thermodynamic conditions induced by anthropogenic global warming, all differences between them can be directly attributed to human-induced global warming. More specifically, all changes between the present (factual) or future (+2, +3, +4 K) storylines with respect to the counterfactual storyline address: What is the influence of anthropogenic global warming on the extreme event of study? This approach thus exploits communication of the consequences of global warming through historical events that are linked to personal experience .

To characterize the large-scale circulation pattern associated with the 2018 heatwave over Central Europe (24 July–10 August), we use ERA5 geopotential height (GPH) anomalies at 500 hPa (with respect to the 1985–2014 mean climatology). A 5 d sequence (31 July–4 August) preceding the heatwave peak is extracted from ERA5 as the reference pattern (see Figs. S1–S2 in the Supplement). Analogous circulation patterns are identified at different global warming levels from the MPI-ESM-LR Grand Ensemble under SSP5-8.5. For each member and warming level, GPH anomalies are computed relative to the 1985–2014 mean climatology, linearly detrended within the warming level period, and restricted to summer months (June–August). Following recommendations to take an appropriate region for the flow-analogues, the reference flow pattern used is taken over an extended region (20° W, 20° E, 40, 65° N) surrounding the region of the event of interest (Central Europe: 3, 18° E, 44, 55° N). The region is large enough to capture either the synoptic-scale ridge and trough pattern or blocking anticyclones, which are mainly responsible for heatwave formation. On the other hand, it is not too large to ensure that the analogues are representative of Central European heatwave conditions. We further decided not to center the box around the Central European focus region, but to shift it by some degrees westward, as the important synoptic-scale features and high gradients in the geopotential field tend to be located further upstream (see Sect. S1.2 in the Supplement).

Analogue candidates are selected based on the smallest Euclidean distance calculated from every possible 5 d sequence within the MPI-ESM data that matches the reference 5 d flow pattern. To avoid using shifted time windows of the same flow-analogues, the N closest matches are taken with a minimum 15 d separation between events. From the N closest matches, only those with a spatial correlation greater than 0.8 compared to the time-averaged reference pattern are retained (See Fig. S4 for correlation matrices between analogues and ERA5). Conceptually, our approach asks: given this sequence and mean structure of the large-scale circulation, what is the spectrum of heatwaves that can occur?.

The analogue selection process is applied to subsets of the MPI-ESM-LR Grand Ensemble, grouped by global warming level (counterfactual, factual, +2, +3, +4 K), where each subset consists of 20 year time slices per ensemble member. With 50 ensemble members, a total of 1000 years of data is available per warming level, providing a large and robust sample for analogue selection. An analysis of the analogue detection performance and quality is provided in Figs. S3–S6.

We define a heatwave as an event in which the daily maximum temperature (TX) exceeds the climatology's 95th percentile (TX95) of the given calendar day for more than three consecutive days. The 95th percentile threshold is computed for each calendar day using a 15 d centered moving window over the 1985–2014 climatological period. This definition holds for both field-averaged and local analyses.

The heatwave intensity is defined as the temperature anomaly exceeding the 95th percentile of the climatology during the heatwave days (n) . HW intensity=TXi-TX95iHW mean intensity=∑i=1nTXi-TX95in

To attribute the likelihood of the event occurring under present and alternative warming levels, we calculate the return periods of the July 2018 heatwave using subsets of analogue events identified under different warming levels with the flow-analogue approach (Sect. ). The return period represents the inverse probability of an event surpassing a specific magnitude (return level). In our study, we are interested in the probability that a heatwave of magnitude M co-occurs with the circulation pattern of interest (flow-analogues) in the dataset representing a given global warming level.

To calculate this probability, we use the conditional probability definition P(E,D)=P(D)P(E|D), where E is the heatwave of interest of magnitude M, and D is the circulation pattern related to the event. Each term provides valuable information in our assessment:

P(D): Is the probability that we find an analogue of the circulation pattern (D) in the dataset. Example: D= “blocking over Scandinavia”. If the blocking occurs 25 % of the years, then P(D)=0.25.

Finally, to better represent return values within a 95 % confidence interval, the distribution of temperatures related to the flow-analogues are fitted using a generalized extreme value distribution and performing 1000 samples with bootstrap-resampling. Even though the analogues dataset is large enough for deriving empirical return periods, the generalized extreme value distribution provides a smooth interpolation and confidence intervals. The adequacy of the fit is supported by the agreement between empirical and fitted return levels (Fig. ), indicating a good representation of the upper tail with no systematic deviations.

We analyse the features of the 2018 heatwave over Central Europe (blue box in Fig. b: [3, 18° E, 44, 55° N]) focusing on the core period of 24 July to 10 August (Fig. ). The flow-pattern for the entire event can be found in Fig. S7. In agreement with the literature, this event is characterized by persistent atmospheric blocking, strong geopotential height (GPH) anomalies, and thus an enduring disruption of the westerly zonal flow over Scandinavia and Northern Germany (Fig. a). These are the regions where the heatwave peaked in July, leading to high temperature anomalies compared to mean climatological values (Fig. b). Averaged over the heatwave duration, these anomalies reached up to 10 °C in Germany and up to 8 °C in some regions in Scandinavia. The time series in Fig. c corresponds to the field-averaged maximum near-surface temperatures over Central Europe (blue box) per storyline, which shows an increase in temperature and duration of the heatwave with global warming. The corresponding ERA5 temperatures for the region (black solid line) coincide with the factual storyline, showing an accurate representation of the observed event. The climatological period (1985–2014) was also simulated with spectral nudging as in the factual storyline for consistency. We limit most of our analysis to this region for two main reasons: the first one being to perform a regional study focused on Germany due to its large local impacts, and the other reason is methodological rather than impact-based; the use of a smaller region allows for a better chance to find close analogues of high quality over an extended region (green box in Fig. a) that captures the large-scale circulation pattern behind the heatwave event .

Even though the heatwave was particularly extreme over Central Europe and Scandinavia, it also affected other regions in Europe with less intensity. In Fig. b, it can be seen how in a factual world, the highest heatwave intensities occurred in Northern Germany. Here we define intensity as the exceedance (in °C) of the local 95th percentile of the climatological (factual 1985–2014) daily maximum temperature (see Sect. ). Local maximum intensities of around +9 °C (Fig. b), and a mean intensity of 2.2 °C (Fig. f) were reached over the region of interest during the days of the observed (factual) event. In the absence of anthropogenic global warming, the counterfactual storyline also shows the presence of a heatwave over a less extended region (Fig. a) and period of time (Fig. f). This counterfactual event would have also affected Northern Germany and Scandinavia, but the heatwave's mean intensity would be limited to 0.5 °C (Fig. f), and local maxima would have reached at most a 5 °C intensity (Fig. a). Hence, human activity amplified the observed heatwave's characteristics, but the heatwave would have still developed under pre-industrial conditions. As global mean temperature increases, the heatwave mean intensity increases from 0.5 °C in a counterfactual world to 6.6 °C under a +4 K level of global warming (Fig. a–e). The heatwave also expands from mainly affecting Northern Europe to affecting the entire European region, starting from a +2 K level.

The main contribution of anthropogenic global warming to temperature increase over the region of interest can be evidenced in Fig.  using warming rates (t2m increase per degree of global warming, computed as the linear trend across the storyline simulations). The 5 d running mean warming rate for maximum, mean, and minimum temperature increases towards the heatwave event, reaching values of ∼ +1.7 °C per degree of global warming at the center of the heatwave event for maximum near-surface temperature (Fig. a). The mean maximum temperature of the event (Fig. b) in a counterfactual world would be 26.7 °C, while the observed heatwave (factual world) had an average maximum temperature of 28.8 °C. This indicates a +2.1 °C increase in maximum temperature in the observed heatwave compared to the counterfactual, which can be directly attributed to anthropogenic global warming. Temperatures rise to 33.3 °C in a +4 K world, under a warming rate of 1.66 °C per degree of global warming. The local warming rate in maximum temperature (Fig. c) corresponds to the local trend per grid cell of the mean maximum temperature during the heatwave event for each storyline. Even if Northern Germany was the most affected region by the heatwave, Central and Southeastern Germany show the largest local warming rate for maximum temperatures, scaling by 2 °C per degree of global warming (Fig. b). In general, we find an amplified warming rate response to increased global mean temperature, meaning that the local trends are at least 1 °C per degree of global warming and more (blueish colors), with most of the region affected by a warming rate of 1.5 °C per degree of global warming or more (orange-redish colors). A similar behavior is seen for the 5 d running warming rate during summer months (Fig. a), an overall amplified response is seen for minimum, mean, and maximum near surface temperature for most time windows.

The storyline method enables specific, strong arguments about the absolute contribution of the thermodynamic influence of anthropogenic global warming to changing magnitudes of an extreme event's characteristics. Here, it indicates that the dynamical pathway leading to the 2018 heatwave with a mean intensity of 2.2 °C would have produced an event that was 6 d shorter and with a 0.5 °C mean intensity in the absence of human-induced global warming (Fig. a). Regarding future warming scenarios, an event developing under the same atmospheric conditions would result in a heatwave 1.9 °C more intense than in present times in a +2 K world, up to 4.4 °C more intense in a +4 K world, where average maximum 2 m-temperatures could reach 33.3 °C for the heatwave event. However, as compelling as these event-based narratives are, it is important to recognize that every heatwave is unique. While we may never witness the exact same event twice, we can search for analogue events with similar atmospheric conditions under future warming levels. In this way, the analogue approach aims to bridge the gap between storyline-driven projections and real-world probabilities.

Studies argue that it is challenging to find good analogues when the event is too intense , or due to changing dynamics in the future . This limitation of the analogue approach is particularly evident when used in the conventional way e.g.,, where the analogues aim to reproduce not only the flow pattern of interest but also the pattern of the variable of interest to reconstruct the observed event. In our case, we are interested in the resemblance of the large-scale flow only, featuring temperature fields that do not necessarily reproduce the observed one. We deliberately let the associated variable vary to be able to construct a distribution function out of all the possible temperatures fields related to the given circulation pattern, similar to .

In Fig. , we illustrate the similarity of the circulation pattern of interest across the storyline and flow-analogue approaches, relative to the observed pattern (based on ERA5). Both the simulated pattern in the storyline approach under present conditions and the mean of all analogues identified during the equivalent factual period across the 50 members of the MPI-ESM-LR GE using the analogue selection process described in Sect.  accurately represent the flow pattern of interest. Given the comparable dynamics in both approaches, we can attribute changes in the July 2018 heatwave more confidently to the thermodynamic component of anthropogenic global warming. In Figs. S3–S6, we further evaluate the quality of the identified analogues and their mean behavior at different levels of global warming, relative to the reference circulation. These results demonstrate the recurrence and robustness of the analogues, as well as their similarity to the reference flow, regardless of the warming level.

Based on the flow-analogue selection process described in Sect. , we built subsets of N best analogues for each level of global warming comparable to the ones in the storyline approach. These analogues represent events which feature a similar large-scale flow evolution during the 5 d preceding the heatwave peak (31 July–4 August) (See Figs. S1–S2). From the 1000 years (50 members × 20 years) dataset for each global warming level in the MPI-ESM-LR GE (see Sect. ), roughly N≈250 flow-analogues are consistently identified across all warming levels (Fig. : see label). This suggests that the circulation pattern linked to the 2018 heatwave remains roughly equally likely to occur in a warmer climate, regardless of increases in global mean surface temperature, in agreement with previously documented literature e.g.,. A common critique of the storyline method is its assumption that the same large-scale atmospheric circulation patterns will occur in future climate scenarios, even though studies project dynamical changes in the atmosphere , which could challenge the validity of such assumptions. However, our results provide evidence that, in this specific case study, the circulation pattern in question could be expected to occur with a ∼1-in-4 year probability (P(D)) regardless of the change in global mean surface temperature. In cases where the circulation pattern of a given case study can no longer be identified in future scenarios, this should not be regarded as a drawback of the method. Rather, it provides valuable information, allowing us to reject the possibility of a warmer storyline of the event occurring under future conditions.

Figure  shows the distribution of 5 d mean temperature anomalies of the analogues, defined as the field averaged daily maximum temperatures relative to the climatological (factual 1985–2014) 95th percentile (mean(TXi-TX95i)) for Central Europe. We refer to these values as t2m anomalies rather than heatwave intensities (although they share the same definition; see Sect. ), since not all identified flow-analogues develop into a heatwave. In Fig. , the black line defines the limit between analogues that evolve into a heatwave (anomalies >0), and those that do not (anomalies <=0). As expected, the temperature distributions shift towards higher temperature anomalies as the global mean temperature increases. Compared to an alternative counterfactual climate where a heatwave has only a 1.8 % probability to emerge from such a circulation pattern, in a factual world, the probability rises to 16.5 %, in a +2 K world 48.7 %, in a +3 K world 87.3 %, and in a +4 K world 97.1 %. This suggests that the atmospheric blocking system enabling the 2018 heatwave shifts from being a necessary but insufficient factor to an becoming an increasingly sufficient condition for heatwaves to occur under global warming. By +3 K, the thermodynamic background alone would be likely enough to trigger a heatwave when such a circulation pattern appears, regardless of additional contributing factors . Therefore, in +3 and +4 K worlds, under such atmospheric circulation pattern, what is currently considered a heatwave would merely represent a regular day, while the absence of a heatwave would become increasingly rare under these large-scale atmospheric circulation conditions.

The stars in Fig.  denote the magnitudes of the heatwave mean intensities projected in the storyline approach (as in Fig. ). The area of the distributions surpassing one of these thresholds corresponds to its probability of occurrence in the given global warming level. Hence, it is evident how the probability of an analogue to exceed the intensity of the factual 2018 heatwave (2.2 °C) increases with global mean temperature. Similarly, there is an increase in the probability of a flow-analogue to surpass the magnitude of projected heatwaves in future levels of global warming. These values correspond to the probability of the event of interest occurring within the atmospheric circulation pattern of interest for a given global warming level, denoted as P(E|D)GWL. The overall probability of the analogue events to occur in each level of global warming has to be scaled by the probability of the atmospheric circulation pattern to occur P(D), resulting in PM,GWL (see Sect. ), presented in Fig.  in terms of return periods.

The key question in statistical attribution methods is to assess the change in the likelihood of the observed event, in this case, the 2018 heatwave, occurring in the present climate compared to counterfactual or future warming scenarios. We denote the probability of the factual heatwave of magnitude 2.2 °C to occur in a given global warming level as P2.2°C,GWL. In Fig. , the blue star markers provide the answer to such a question; the counterfactual world has been omitted in Fig. a since a heatwave of such magnitude was not identified (see Fig. ). This is also evidenced in Fig. b, where the counterfactual dataset does not reach the observed magnitude, meaning that it was not possible to occur in the absence of human-induced warming. In a factual climate, roughly ∼25 % of years exhibit a similar blocking pattern (P(D)), and only ∼1.6 % of these analogues exceed the observed mean heatwave intensity of 2.2 °C (P(E|D)) (Fig. 6). Hence, in the factual period, the 2018 heatwave conditioned on this atmospheric circulation pattern was a rare event, occurring with a probability of about 1-in-277 years (1/P°C,Factual) according to the fitted value (Fig. ). Under continued global warming, the return period of a heatwave with the same intensity would decrease exponentially (Fig. a), eventually becoming a common 1-in-5-year event in a +4 K world.

The added value of this combined approach lies in its ability to quantify the likelihood that an alternative, physically consistent storyline of the heatwave would emerge under its corresponding level of global warming (Fig. b). Based on the combined storyline-statistical analysis, the 2018 heatwave would have a mean intensity of 4.1 °C in a +2 K world, with a 1-in-112 years probability to occur (P4.1°C,+2K). In a +3 K world, the mean intensity would increase to 5.4 °C, with a return period of 1-in-58 years (P5.4°C,+3K). Finally, in a +4 K world, the heatwave's mean intensity would reach 6.6 °C, with a return period of just 1-in-26 years (P6.6°C,+4K). Since we are using a fixed climatology (1985–2014) as a baseline to define present and future heatwaves, it is no surprise that the 2018 heatwave, as experienced in the present, undergoes an exponential increase in likelihood until it becomes a common event in a +4 K world. However, one might initially assume that the warmer storylines simulated for future climates would remain as unlikely in their respective levels of global warming as the original heatwave was in the present. Instead, the results indicate that these projected events also become more common at their corresponding global warming levels, following an exponential trend. These return periods already take into account the probability of the atmospheric circulation pattern occurring under future climate conditions, whose frequency stays roughly constant with increasing global mean surface temperature. This indicates that due to dynamical conditions, future heatwaves have the same chance to occur. Despite that, anthropogenic global warming seems to intensify rare heatwave events in warming scenarios at a higher rate than expected and projected by the storyline approach.

We hypothesize that the increase in extreme heatwaves could come from a combination of several factors: (1) the conservative definition of global warming in the storyline approach (which only imposes changes in SST and GHGs, as they are certain to have a human-induced contribution) may restrict the heatwave intensities. (2) the integration of more complex interactions in the MPI-ESM GE (coupled earth system model) used for the flow-analogue analysis, compared to the nudged storylines simulated with ECHAM (atmospheric model with an integrated land component JSBACH). (3) The role of soil moisture as a source of variance in the temperature distributions for future warming levels .

The 2018 heatwave was very extreme not only because of its atmospheric circulation but also because of the exceptionally dry conditions that preceded it . In Fig. , we explore the role of soil moisture in both approaches. In Fig. a, we see how soil moisture is also affected by global warming in the storylines, decreasing at a faster rate in future warming levels. The counterfactual storyline had a mean soil moisture of 62.9 kg m⁻² during the main heatwave event, while in a +4 K storyline, the soil moisture dropped to 54.9 kg m⁻². In Fig. b, we show the bivariate distribution of soil moisture (SM) and temperature anomaly (T) associated with the flow-analogues. The lines in this plot show the corresponding magnitudes in the storyline approach per global warming level. The bivariate comparison demonstrates that the analogues at each warming level occupy temperature (T)-soil moisture (SM) states consistent with the corresponding storylines and that also their number of extreme heatwaves increases in their own warming levels when soil moisture is included as an additional condition in the analogue attribution (area of bivariate (SM-T) threshold exceedance shown as darker dots in Fig. b). Here, we can provide evidence that despite having similar atmospheric circulation patterns, the analogues do portray a larger spectrum of events that could emerge due to other underlying conditions, like enhanced soil moisture deficit. It is possible that due to the increasing thermodynamic drying of the future simulations, the role of the atmospheric circulation becomes less important, as more and more of the analogues also have drought conditions.

Furthermore, the increase of extreme heatwaves in the upper tail of the temperature (Fig. ) and bivariate SM-T (Fig. ) distributions does not come only from the shift in the mean of the distributions towards warmer temperatures due to global warming but also from an increase in variance in future warming levels (see label Fig. ), which could be associated to a strengthened feedback with soil moisture in the analogue catalogue. If one could condition the temperature distributions on the soil moisture of a storyline (i.e. p(SM =60, T)), the variance would be largely reduced. This would lead unfortunately also to a smaller sample size, too small to perform significant attribution.

In sum, both attribution methods are conditioned on the observed dynamics and were made as comparable as possible. The storyline approach provides a representation of how this single event would change in response to anthropogenic thermodynamic forcing (SSTs and GHGs). The flow-analogue approach includes a wider range of heatwaves developing under similar dynamics but different interacting processes, including more extreme heatwaves, probably resulting from a reduced soil moisture availability. Even though these results may be model-dependent, it is worth emphasizing that both approaches rely on the same atmospheric model. The storylines are simulated with ECHAM6, the atmospheric component of the MPI-ESM model, while the flow-analogues are extracted from scenario simulations of MPI-ESM. Including additional models for comparison in future work could provide more detailed insights on this matter.