Heat extremes have become more frequent and intense in Europe in recent decades, posing an increasingly severe burden on human health, economies, and ecosystems worldwide , particularly as even small increases in average temperature can cause disproportionately larger increases in the intensity and frequency of these extreme events . To mitigate their impacts, accurate heatwave predictions are essential for timely public warnings and informed policy decisions. These predictions rely on numerical weather prediction models (NWP) that must accurately represent the physical mechanisms driving weather and climate. However, in Europe – an emerging heatwave hotspot – uncertainties in both land-atmosphere interactions and atmospheric circulation remain . These uncertainties are further exacerbated by the observation that early and late summer heatwaves may have distinct physical drivers and impacts that are currently underexplored . Reducing such uncertainties is warranted since Europe recently experienced severe heatwaves unusually late in summer , in line with a recent study suggesting a potential shift of heatwaves into late summer and autumn in a warmer climate .

One of the most predominant land-atmosphere interactions is the amplification of heatwaves by dry soils, which in turn further deplete soil moisture, creating a positive feedback cycle. In particular, early summer heatwaves are often linked to droughts, which in turn amplify heatwaves in peak and late summer . The impact of this land-atmosphere feedback in amplifying surface temperatures depends on the amount of available soil moisture and is strongest under transitional conditions, where soils are neither very dry nor very wet . Heat extremes are associated with anomalous strong diabatic heating of near-surface air masses . While soil moisture determines the partitioning of energy at the surface into sensible and latent heating, the available energy at the surface is determined by the balance of radiative fluxes. In Central Europe, heatwaves typically occur under high pressure systems with decreased cloud cover, leading to enhanced surface net solar radiation (shortwave incoming minus outgoing radiation) . In urban environments, this enhanced solar radiation during heatwaves is absorbed by the urban canopy at daytime and its heat is released in the nighttime contributing to the urban heat island effect . Prolonged periods of high temperatures are especially hazardous when combined with elevated nighttime temperatures, as reduced cooling during the night – reflected in increased daily minimum temperatures – limits human recovery and amplifies health impacts . Therefore, soil moisture and surface net solar radiation are key variables to consider in understanding heatwave development, while daily maximum temperature is crucial for assessing impacts.

In addition to land-atmosphere processes, heatwave development is strongly influenced by large-scale atmospheric circulation patterns , which vary not only between individual heatwaves within the same season but also throughout the year. Summer heatwaves and wintertime warm spells are typically caused by significantly different atmospheric configurations and are characterized by a varying role of horizontal temperature advection. In summer, heatwaves are primarily associated with long-lasting blocking anticyclones. These systems block the westerlies and thus the movement of weather systems by typically splitting the jet into two branches around the system. This leads to extreme conditions in the affected regions . In summer heat extremes, geopotential height and the near-surface temperature anomaly patterns are usually close to being in phase , which hinders the advection of air masses . Thus, subsidence-induced adiabatic compression and sensible heating from surface radiation are the most relevant drivers for summer heatwaves . In contrast, winter warm spells are mostly driven by advection of warm air masses due to an anomalously amplified flow pattern in winter . The transitions between these patterns during spring and autumn are less well studied. The intricate nature of the relationship between large-scale flow anomalies and temperature and its seasonal variability highlights the need for a generalizable and objective quantification of the strength of this linkage. To this end, it is useful to first characterize the underlying seasonal atmospheric circulation patterns.

Typical seasonal atmospheric circulation patterns can be grouped into so-called weather regimes, which represent long-lasting, recurring quasi-stationary flow patterns based on 500 hPa geopotential height . As weather regimes are frequently used, processes determining their formation , transitions and connections to local weather patterns have been widely investigated and provide a multitude of characteristics associated with each regime. Different numbers of regimes have been proposed in order to account for the atmospheric variability that is observed in the Euro-Atlantic region. The most common approach involves the use of four reference states . To account for the disparate seasonal dynamics, the majority of weather regime definitions encompass a distinct set of weather regimes during the summer and winter months, with a notable absence of coverage in spring and autumn. proposed a year-round definition of seven weather regimes to describe the intraseasonal weather variability in the Euro-Atlantic region. It allows for a more detailed investigation of summer heatwaves, as well as the transition months. Though weather regimes provide a useful framework for describing atmospheric circulation patterns, quantifying their link to temperature requires additional diagnostic methods.

A novel framework from dynamical systems theory has recently been introduced by and that offers a means of quantifying the instantaneous linkage of multiple variables , as well as quantifying the persistence of an atmospheric circulation pattern. The latter quantity describes how long a pattern remains similar to itself and is directly related to the intrinsic predictability of the flow field . Further, as demonstrated by , it can also provide valuable insights into the practical predictability of, for instance, surface temperature. Both quantities are based on the concept of analogues , defined as similar states of a given variable that occurred at different times within the dataset. However, the definition of the analogues is fundamentally different from a regime-based definition of persistence in that no pre-defined number of classes of atmospheric configurations are established. Instead, each atmospheric configuration possesses its own distinctive set of analogues. It is then possible to conceptualize high persistence as a cluster of analogous occurrences over time. In case of the association of multiple variables, described by the so-called co-recurrence ratio, the analogues of all variables are determined independently. Subsequently, the frequency with which the analogues of different variables co-occur at the same time in the dataset is calculated to provide an objective measure of the link between several variables. The described quantities allow for an objective investigation of the relationship between atmospheric circulation and surface temperature, as well as the persistence of atmospheric circulation.

Recently, have shown that atmospheric states with high persistence are not a necessary prerequisite for the emergence of hot extremes in summer or winter. We are particularly interested in the changes of the persistence of the atmospheric flow and the linkage between the atmospheric flow and near-surface temperature extremes during the extended summer months. Previously, examined the co-recurrence ratio between temperature and atmospheric circulation over North America, concluding that summertime temperature extremes appear to be unrelated to joint dynamical properties in 2 m temperature and sea level pressure. However, to our knowledge, no study has investigated the co-recurrence ratio between temperature and atmospheric circulation for hot temperature extremes in Europe. Further, as both persistence and the co-recurrence ratio may possess the same value under markedly disparate configurations of the flow field, this approach is complemented by the inclusion of weather regimes, which serve to cluster the obtained information into a set of physically meaningful and distinct atmospheric circulation patterns . Combined, they offer a valuable and readily interpretable framework for analyzing the properties of heatwaves, including the linkage to the large-scale flow and the intrinsic predictability. Moreover, the co-recurrence ratio can be extended to additional variables relevant to heatwave development. For instance, investigated compound warm-dry and cold-wet extremes over the Mediterranean using the co-recurrence ratio between daily maximum temperature and daily total precipitation. found an increasing link in precipitation temperature association over the past 40 years, possibly due to lower soil moisture in a warming climate. Therefore, we further analyze the co-recurrence ratio between daily maximum temperature and soil moisture, as well as surface net solar radiation and daily minimum temperature associated with heatwaves in Central Europe.

This study aims at exploring intra-seasonal variations of heatwaves during the extended summer period from May to September, maintaining a monthly perspective throughout all analyses. In particular, it addresses the following research questions:

How does the large-scale atmospheric flow and its link to surface temperature vary throughout the year?

Section 2 defines the data used and introduces the stream function, the Euro-Atlantic weather regimes, the heatwave detection, as well as the dynamical system metrics persistence and co-recurrence ratio. Section 3 demonstrates the methodology for a case study on the July 2019 heatwave, before answering the first research question by presenting seasonal cycles, while Sect. 4 presents results on persistence and co-recurrence ratio during heatwave days, addressing the second research question. Section 5 investigates the final research question by presenting the co-recurrence ratio between daily maximum temperature and soil moisture, surface net solar radiation and daily minimum temperature on heatwave days. The results are discussed and summarized in Sect. 6.

Europe was affected by several heat waves between 20180–2022 with recurrent drought and strong legacy effects . To demonstrate the applicability of the inverse persistence and co-recurrence ratio in combination with weather regimes, we first examine the case study of the Central European heatwave of July 2019 . This exceptional heatwave resulted in numerous records being broken, in particular in France, Belgium and the Netherlands .

The dynamical evolution of the heatwave is illustrated in Fig.  by means of 500 hPa streamlines averaged over 5 d periods one week before prior to HW onset (Fig. a) and around its peak (Fig. b). Over the period 17–21 July, the large-scale flow setup over the Euro-Atlantic sector displayed no notable anomalies. Europe was still influenced by a weakening and eastward-moving long-wave trough, whereas the Atlantic region was characterized by zonal westerly flow. Overall, the large-scale circulation was rather close to summer climatology, which is further demonstrated in Fig. e, showing no anomalously elevated Iwrs for any regime during that period. In line with the large-scale flow, near-surface temperatures were close to summer climatology across the whole of Europe. The large-scale flow situation changed around 22 July when warm air advection ahead of two cyclones over the North Atlantic led to a rapid increase of geopotential heights over Europe. Over the next two days, the further amplification of the ridge over Europe resulted in the formation of an atmospheric blocking in the shape of an Omega (See Appendix ).

Large areas of western and central Europe were now experiencing temperature anomalies of more than 10K as visualized in Fig. d. At the same time, the soils begin to desiccate considerably. These dry conditions were preceded by soil moisture deficits in Central Europe (Fig. c), initiated by a heatwave at the end of June 2019. The pre-existing deficits significantly contributed to the intensity of the July 2019 heatwave.

Finally, Fig. f illustrates the temporal evolution of the July 2019 heatwave from the dynamical system metrics perspective. From the 18 July onward, the inverse persistence of the 500 hPa stream function starts to decline, reaching a minimum around the 26 July. This coincides very well with the peak development of the Omega blocking, which is a large-scale flow anomaly known to be rather persistent. Moreover, the onset phase of the heatwave is marked by a sharp increase in the co-reccurrence ratio, which underlines in this case the substantial linkage between the large-scale flow and near-surface temperature. We hypothesize that the subsequent decrease in the co-recurrence ratio around the 25 July likely reflects less common atmospheric dynamics with the Iwr of EuBL, and later GL, also exceeding 1 and/or the exceptionally high Iwr values of ScBL. This could explain the reduced frequency with which this atmospheric circulation and its corresponding temperature pattern occur simultaneously throughout the time series. In summary, this case study suggests that inverse persistence and co-recurrence may be linked to weather regimes and, therefore, yield physically meaningful insights, while co-recurrence may also provide valuable information on the relationship between atmospheric circulation and other variables.

We now assess the characteristics of heatwaves across the entire historical period from 1950 to 2023. Given our focus on heatwaves from early to late summer, understanding the seasonal cycle of inverse persistence and co-recurrence is crucial for two reasons: first, to establish a baseline understanding of general seasonal variability, and second, to enable the identification of anomalies relative to this seasonal cycle. Figure  displays the daily mean across the entire time series, along with its standard deviation (dashed line) and a 21 d running window mean (lighter curve), for both the co-recurrence ratio between stream function and tasmax (Fig. 3a), and the inverse persistence of the stream function (Fig. 3b). The seasonal cycle of the co-recurrence ratio reveals a peak in July and August and a second smaller peak in January, with minima in April and November. The inverse persistence follows a similar seasonal cycle, with atmospheric circulation highly persistent during the summer months, while November and April exhibit the least persistent atmospheric circulation. Considering the extended summer months May to September, it is remarkable that the dynamical system metric values in the end of September are similar to the ones in the beginning of May, while the commonly considered summer months, June to August, display significantly lower values of both persistence and co-recurrence ratio in the beginning of June compared to end of August. This supports the use of the extended summer months as a more representative period of summer climatology. Anomalies of co-recurrence and persistence relative to their seasonal cycles will be examined in Fig.  for the extended summer months. To interpret those seasonal cycles, Fig. 3c displays the 21 d running window mean Iwr of all seven weather regimes. It becomes apparent that the high persistence of the atmospheric circulation in July and August coincides with a maximum in Iwr of the blockings ScBL, GL, and relative minima in the cyclonic regime ZO. Typically, the regions under atmospheric blocking experience only little advection, as the atmospheric circulation and the temperature patterns are mostly in phase and similar patterns in geopotential anomalies should therefore often result in similar patterns in temperature anomalies. Thus, we hypothesize that the more frequent occurrence of persistent blockings in summer might be associated with the summer peak in co-recurrence between temperature and stream function.

In addition to the pronounced peak summer months characteristics, Fig.  reveals the intra-seasonal variations during the extended summer months May to September. In September, the atmospheric circulation differs significantly from the peak summer months. Cyclonic regimes become more prevalent again during September and the blocking regimes GL and ScBL no longer exhibit an elevated frequency. From a dynamical system metric perspective, the overall atmospheric circulation during September becomes less persistent and the temperature and atmospheric circulation patterns are less likely to occur together. The dynamical system metrics at the beginning of May display similar values than in the end of September. However, the Iwr shows that different weather patterns are dominant in May and September. In May, until mid/end May, ScBL and GL are together with AT the dominant regimes, like in August, while those three regimes are minimal in September. Mid May until mid June is quite variable in the prevalence of the regimes. EuBL becomes dominant while GL loses some of its importance and ScTr and AT has a minimum. Interestingly, from mid June until July, a peak in zonal regime and ScTr appears while the other regimes have mostly minima. However, the atmospheric circulation overall becomes more persistent and has a higher co-recurrence.

In summary, the analysis highlights the necessity of considering May to September to fully capture the summer peak of co-recurrence and persistence. The Iwr reveals notable differences in dominant weather regimes both across and within months, demonstrating the benefit of combining weather regimes with dynamical system metrics. Consequently, those methods will be applied to heatwaves in the extended summer months (May–September) in the next analysis.

To investigate the characteristics of heatwaves and identify systematic differences in co-recurrence and inverse persistence compared to the summer climatology, we analyze heatwave days and non-heatwave days separately during the extended summer months. Figure a, b displays daily means of co-recurrence between stream function and tasmax, and of inverse persistence of stream function on heatwave days and non-heatwave days, with their corresponding standard error to account for the varying number of days considered. The co-recurrence of tasmax and stream function shows anomalously high values on heatwave days compared to non-heatwave days from the end of May to mid-September. For most days from the end of May to mid-September, the differences are statistically significant. Further, the magnitude of the difference is approximately constant over the significant part, except for a slightly lower anomaly in June. The findings are robust against the choice of domain (Appendix ). Interestingly, the anomalies of the inverse persistence of stream function show a different pattern than the co-recurrence ratio. A significantly higher persistence of the atmospheric circulation relative to non-heatwave days is only observed for a few scattered days from the end of June to the beginning of September. The differences between heatwave and non heatwave days are further less pronounced than for the co-recurrence ratio. This supports the findings of , which suggest that the persistence of the large-scale atmospheric circulation is not a necessary criterion for the appearance of heatwaves in Europe. Moreover, our results reveal that only in July and August does the average atmospheric circulation associated with Central European heatwaves exhibit slightly increased persistence relative to non-heatwave days. We further report that those anomalies enhance for a smaller domain and vanish for a larger domain with a similar size than , highlighting the importance of choosing a suitable region for the specific use case (Appendix ). In conclusion, heatwaves are characterized by an anomalous high co-recurrence ratio between tasmax and stream function end of May to mid-September, while a higher persistence of atmospheric circulation is only present in July and August. Therefore, a higher persistence of the atmospheric circulation does not necessarily lead to more similar temperature and stream function patterns.

This finding is examined in more detail in Fig. , which presents a monthly breakdown (rows) of co-recurrence and persistence anomalies relative to their seasonal cycles (Fig. ), combined with the corresponding weather regimes and daily maximum temperature anomalies.

Starting with the first column, it becomes clear that most heatwave days (colored dots) are found in the upper half for all months and in the upper left quadrant in July and August (Fig. g, j) indicating anomalously high co-recurrence (vertical axis) and persistence (horizontal axis). This finding is in line with Fig. . Additionally, the highest tasmax anomalies (red colors) do not necessarily appear in clear clusters at specific persistent or co-recurrence anomalies. However, in August a strong tendency to the upper left quadrant can be observed (high co-recurrence, high persistence). July shows a similar pattern to August, but with a less pronounced concentration of strong heat anomalies in the upper-left quadrant. In September (Fig. m), a slight tendency of high tasmax anomalies on anomalous high co-recurrence values can be observed, but with a smaller anomaly in persistence. In contrast, the strongest tasmax anomalies in May (Fig. a) occur under less persistent atmospheric circulations, with no clear association to co-recurrence anomalies. In June (Fig. d), the strongest tasmax anomalies are either low persistent or highly persistent and with a positive co-recurrence anomaly (like in July and August). It thus reflects a transition from spring to peak summer. When considering all days instead of heatwave days, June, July and August (Fig. d, g, j) are similar in their overall distributions of anomalies in co-recurrence and inverse persistence. Those months have a oval shape with a slight tilted axis from upper left to lower right, suggesting some degree of correlation between co-recurrence and persistence. In contrast, September (Fig. m) has only few days in the upper left with a distance to the overall distribution, which distribution is tilted from bottom left to upper right. In May (Fig. a), in particular the co-recurrence has less low minimal values leading to the distribution being more densely packed. Thus, the degree of correlation between inverse persistence and co-recurrence differs in the extended summer months.

To relate the persistence and co-recurrence anomalies to the weather regimes, the second column of Fig.  shows all heatwave days colored in their active weather regime. It further indicates the monthly upper tercile of co-recurrence and lower tercile of inverse persistence on heatwave days (dashed black lines). The monthly values of the terciles reflect the findings of Fig.  that July and August are overall more persistent on heatwave days than in early and late summer and the co-recurrence anomaly on heatwave days is the highest in August, but varies only slightly from June to August.

Starting with May (Fig. b), the weather regimes appear clustered: the blocking regimes EuBL, ScBL, and GL lie on the left side, indicating higher persistence. GL is associated with low co-recurrence, while EuBL and ScBL show strong positive co-recurrence anomalies. In contrast, the more cyclonic regimes, ZO and AR, are less persistent and exhibit a wider range of co-recurrence anomalies. In June (Fig. e), these clusters become less distinct. EuBL and ScBL remain persistent, as observed consistently across all months, but the other regimes are more scattered, particularly across the full range of co-recurrence values. This pattern continues in July (Fig. h), although it's notable that most heatwave days occur under EuBL and ScBL, especially those with high co-recurrence and persistence (upper left quadrant). August (Fig. k) resembles July, but the AT regime becomes more frequent and also appears among the high co-recurrence, high persistence anomalies. In September (Fig. n), other regimes such as the ZO and AR become more frequent. However, the outliers with both high persistence and high co-recurrence values are primarily associated with EuBL and ScBL, while those with slightly lower persistence also include ZO.

The results show that heatwaves in the peak summer months, July and August, occur frequently under persistent blockings (EuBL, ScBL; Appendix ), while early and late summer heatwaves form under a greater variety of weather regimes. We detect a notably increased presence of the zonal regime and the “no regime” case for both May and September heatwaves. Despite those regimes excluding atmospheric blocking, they may still feature atmospheric ridges over Central Europe, and have been previously linked to heatwaves in the region .

Further, the weather regimes show tendencies for persistence anomalies, but vary greatly in their co-recurrence anomalies within and between the months. Considering the co-recurrence ratio of all months, it becomes apparent, that no particular regime(s) can be attributed to the anomalous high co-recurrence ratio on heatwave days.

To translate those qualitative findings into a quantitative result, we further considered the monthly weather regime frequency in panels c, f, i, j, o, comparing all days, heatwave days and heatwave days with the highest co-recurrence and lowest inverse persistence anomalies (terciles in the second column). In particular, we are aiming at answering the question which regimes dominate the persistence and co-recurrence anomalies on heatwave day and how the weather regime frequency at the most anomalous heatwave days differ from the weather regime frequency on heatwave days and all days.

First, the no regime category is the most frequent category on all days and particularly frequent during heatwave days with 28.2 %–34.2 % in every month. Interestingly, the no regime category occurs less often during the most persistent heatwaves days in every month, while it is more frequent in May and July (Fig. c, i) for the highest co-recurrence ratio heatwave days.

Then, the weather regimes most associated with heatwaves are EuBL, ScBL and ZO, as they are more frequent during heatwave days. EuBL (14.6 %–24.9 %) and ScBL (11.7 %–28.4 %) are particularly notable, as they are among the most frequent weather regimes in every month during heatwave days. In addition, these two regimes have increased occurrence during the most persistent heatwave days. Interestingly, EuBL shows a stronger frequency increase than ScBL on the most persistent heatwave days in May (Fig. c), June (Fig. c, f) and September (Fig. o). However, ScBL slightly exceeds EuBL in frequency above the heatwave average during peak summer (Fig. i, l). In terms of high co-recurrence ratio, EuBL is more frequent in May and June but declines in peak and late summer. ScBL shows fewer anomalies overall but is less common in May and especially in September with -7.3 percentage points. In contrast, the zonal regime, which is particularly common during early and late summer heatwaves, appears more often on high co-recurrence heatwave days in September (+6 %), but otherwise at similar frequencies as on average heatwave days. It is less persistent than the blockings and thus less frequent during high persistent heatwave days in all months. These differences point to the fact that the high persistent and high-co recurrent days are not necessarily coupled.

Considering the other weather regimes, all of them occur less frequently relative to heatwave days on the most persistent heatwave days except for GL in May and September. However, during the highest co-recurrence heatwave days AR is less frequent across all months. In early summer, GL is generally not associated with the most co-recurrent days. AT has an above-average heatwave day frequency on high co-recurrence heatwave days during June, August, and September (4.6 %–7.7 % percentage points), while ScTr occurs more frequently with 2.7 more percentage points in June and 6 more percentage points in September. This shows, that depending on the months, the weather regimes are more or less anomalous in co-recurrence, while they have tendencies for specific persistence values throughout all summer months.

As the anomalous high co-recurrence ratio on heatwave days cannot be attributed to specific weather regimes, we investigate the role of land-atmosphere interactions associated with heatwaves. Therefore, Fig.  depicts the co-recurrence ratio of tasmax and stream function at 500 hPa to daily minimum temperature (tasmin), soil moisture in the upper soil layer (0–7 cm (swvl1)) and surface net solar radiation (ssr). Conceptually Fig.  is similar to Fig. , showing daily means of heatwave and non heatwave days, as well as their standard error.

As the daily minimum temperature (tasmin) is related to radiation trapping during the night and has a large impact on perception of heat stress, its co-recurrence with tasmax and the stream function is analyzed. Figure a shows a high anomaly (Note the different y-scale) on heatwave days in July and August with a minor anomaly end of June. In September, tasmin and tasmax have a smaller co-recurrence ratio on heatwave days than on non heatwave days. During the core summer months, particularly warm days are expected to coincide with above-average nighttime temperatures due to limited time for nocturnal cooling, and the release of heat trapped during daytime. Consequently, patterns of maximum and minimum temperatures are closely linked, resulting in high co-recurrence values. However, later in the season, in September, day length decreases substantially, and areas experiencing high daytime temperatures may now tend to show average or even below-average nighttime temperatures, thus weakening the link between tasmax and tasmin. Additional factors might further influence nighttime radiative cooling, such as cloud cover and wind. The co-recurrence between stream function and tasmin (Fig. b) has a similar seasonal cycle, with statistically significant anomalies in July and August. However, the magnitude of the anomalies is highest end of July and in the beginning of August and drops faster than the co-recurrence anomalies between tasmin and tasmax. Moreover, no negative anomaly in September is observed.

Then, the link between soil moisture in the upper soil layer and tasmax is examined in Fig. c, as a lack of soil moisture availability can lead to more severe heatwaves, while high temperatures in turn also lead to a drying of soils. It shows an anomalous high link on heatwave days, which is particularly pronounced in July and the beginning of August. In general, from June to mid-September, there is a significantly higher linkage of soil moisture to temperature during heatwaves, while in May and in mid-to-late September, no significant anomaly is observed. A large intra-monthly variability of the co-recurrence can be observed, which could be associated with different spatial and temporal scales, as well as spatial heterogeneity. The link of soil moisture and stream function at 500 hPa (Fig. d) shows almost continuous statistically significant anomalies from mid-June to mid-September. This behavior is similar to the link of tasmax-swvl, but with an overall smaller magnitude and later start of the anomalies (mid June vs beginning of June). This indicates a high importance of soil moisture leading to heatwaves in particular in July and August. However, examining soil moisture availability prior to the heatwaves as a potential explanation for the high co-recurrence ratio anomalies between stream function and tasmax showed no clear link (Appendix ).

The co-recurrence between surface net solar radiation (difference between shortwave incoming and shortwave reflected radiation) and tasmax is assessed in Fig. e. On heatwave days, this link is anomalously high from May to August with particular high anomalies in June and July. As the solar radiation is strongest end of June, this might explain the observed asymmetry. The large anomalies are most likely attributable to the predominantly clear sky conditions associated with high pressure systems that mostly lead to heatwaves. This is confirmed by the co-recurrence between ssr and stream function (Fig. f), which shows a constant small anomaly on heatwave days during the extended summer months. Comparing the link between ssr and tasmax to the link between swvl and tasmax reveals that ssr is associated with a particular anomaly in early summer until mid August, while soil moisture shows the highest anomalies only in July and August. In addition, the surface net solar radiation is the only quantity which shows an anomaly on heatwave days in May, as the link between stream function and tasmax (Fig. ) does not become anomalous until June. This indicates that heatwaves are driven by different processes and surface net solar radiation is particularly relevant for early summer heatwaves.

Thus, the surface net solar radiation on heatwave days is strongly related to daily maximum temperature in early summer, while soil moisture and temperature are strongly linked in July and August. However, the ssr has a constant link to atmospheric circulation at 500 hPa due to the predominantly clear sky conditions, while an anomaly in co-recurrence between soil moisture and stream function appears starting from mid June only. In addition, daily maximum and daily minimum temperatures are linked strongly in July and August, similar to daily minimum temperature and mid-tropospheric atmospheric circulation.

In this study we investigated the relationship between the large-scale atmospheric circulation and hot temperature extremes in Central Europe, with an emphasis on whether this relationship may also change throughout the summer season. With this aim, we used metrics derived from dynamical system theory, that allow both to objectively identify the persistence of any given atmospheric state and to further obtain an instantaneous measure of the association between two quantities. In addition, we use the concept of Euro-Atlantic weather regimes, which are physically meaningful modes of large-scale flow variability on subseasonal time scales. To exemplify the utilized methods, we first analyzed the July 2019 heatwave in Central Europe as a case study, followed by a climatological perspective. The main conclusions can be formulated as answers to our research questions as follows:

How does the large-scale atmospheric flow and its link to surface temperature vary throughout the year?

We find similar variations of both the co-recurrence ratio and atmospheric persistence throughout the year, with relative minima in spring and autumn and a maximum in July and August. Their high values in July and August are associated with an increased occurrence of Scandinavian blocking (ScBL) and reduced frequency of the zonal regime (ZO). While the inverse persistence and co-recurrence ratio display similar magnitudes in early and late summer, the dominant weather regimes differ. In May, the Scandinavian and Greenland blockings (ScBL, GL), as well as the Atlantic Trough (AT) dominate, while these are the least prevalent regimes in September. This highlights the added value of combining both approaches to better understand the atmospheric circulation – hot temperature link and capture the atmospheric dynamics even if the heatwaves are not associated with any regime.

Unlike previous studies , which characterized extreme events using dynamical system metrics, we combined persistence, co-recurrence ratio between various variables and weather regimes to gain a broader understanding of Central European heatwaves and their relationship to the large-scale atmospheric circulation. We further placed a strong emphasis on seasonal variations during the extended summer months, May to September. This focus is particularly interesting, given that Europe recently experienced severe heatwaves unusually late in summer/early autumn and that projections from high-resolution climate models suggest that Central Europe may experience more frequent heatwaves particularly in late summer and early autumn in a warmer climate . With this unique combination of methods, we provide a novel perspective on Central European heatwaves during the extended summer months.

have shown that mid-tropospheric atmospheric states with high persistence are not a necessary prerequisite for the emergence of hot extremes, neither in summer nor winter. Our results support and expand this finding for Central European heatwaves by considering the persistence and weather regimes on individual heatwave days during the extended summer months. For July and August, our results indicate a tendency for higher persistence on heatwave days due to the enhanced frequency of European and Scandinavian blockings. This finding supports the important role of blockings leading to heatwaves, which has been highlighted by several studies e.g. . However, the size of this anomaly varies with the chosen domain size (Appendix ). In May, June and September, no persistence anomaly is observed, and cyclonic regimes such as the zonal regime are more frequent on heatwave days than in July and August.

Our finding that the co-recurrence ratio between tasmax and stream function at 500 hPa is anomalously high during heatwave days differs from the result of a study over North America examining the co-recurrence ratio between temperature and atmospheric circulation . The authors concluded that summertime temperature extremes appear to be unrelated to joint dynamical properties in 2 m temperature and sea level pressure. However, our study differs from not only in the considered region, but also in the considered variables. Instead of sea level pressure, we used mid-tropospheric stream function, as reported a weak relationship between summertime heatwaves and surface persistence, but rather a significant link to mid-tropospheric persistence. Moreover, we use daily maximum instead of daily mean 2 m temperature. In addition, both our domain for atmospheric circulation and for temperature are smaller than the ones in , as we aimed at targeting only the relevant features for heatwaves in Central Europe instead of covering the entire continent. The anomalous high link is however robust to changes in domain size (Appendix ).

In the last step of our study, we computed the co-recurrence ratio to assess the link between daily maximum temperature and soil moisture, daily minimum temperature and net surface solar radiation. Previously, found an increasing co-recurrence ratio of daily precipitation and daily maximum temperature over the past 40 years in the Mediterranean, possibly associated with lower soil moisture in a warming climate. While we did not examine trends in our study, we found a strong link between soil moisture and daily maximum temperature in the summer months. Moreover, did not find particular synoptic structures leading to the warm-dry extremes, hypothesizing that this is due to different sets of weather circulation regimes. This is in line with our results: For the co-recurrence ratio between stream function and daily maximum temperature, the highest anomalies on heatwave days are driven by different atmospheric flow fields, whose relative frequency further varies over the extended summer months.

While providing a novel perspective on heatwave dynamics, some limitations arise from the choices and properties inherent to the methods themselves. One choice is the size of the considered region for both co-recurrence and persistence. It determines how much the analogues resemble each other and should thus be chosen to match the specific use case. Nonetheless, our result of anomalous high co-recurrence ratio on heatwave days is robust against varying domain sizes (Appendix ), even though some sensitivity on the choice of model domain is found for the persistence. As the analogues are computed for daily data, some caution has to be applied when interpreting the co-recurrence ratio of slowly varying variables, such as soil moisture. However, this methodological characteristic is not expected to compromise the analysis of the other variables considered in this study or their relationship with temperature. Further, from the co-recurrence ratio between atmospheric circulation and temperature patterns, the role of advection cannot be directly inferred, as a high association between the two variables may occur both under advection-driven and non-advection-driven conditions. Additionally, to ensure a comparable number of analogues across decades, it is necessary first to remove the global warming trend. Otherwise, similar states in the past with slight differences in spatial distribution or magnitude may not be identified as analogues. While weather regimes allow a more intuitive understanding of the state of the large-scale atmospheric circulation, difficulties in interpretation can arise due to the frequent occurrence of the no regime category. The absence of a weather regime may result from a variety of atmospheric circulation patterns. For instance, the no regime phases may still feature short-lived atmospheric ridges, and thereby allow heatwave formation over Central Europe, as evidenced by our results showing comparable high frequencies of no regime for both heatwave and non-heatwave days.

Several open questions and future directions arise from this work. First, it would be valuable to extend the analysis to other heatwave domains in Europe to test the generality of the observed links. Second, investigating the full life cycle of heatwaves might reveal further characteristics. Third, to better capture the temporal dynamics and predictability, the recently developed local predictability index could be applied. While we have not addressed the impacts of heatwaves, future work could also examine links to vulnerability, such as effects on agriculture (e.g., blooming or frost risk) and mortality. Lastly, evaluating how these relationships may shift under future climate conditions would provide essential insights into potentially changing atmospheric dynamics leading to heatwaves. This last step is planned as future work using CMIP6.