The last 2 decades have been characterized by an increased number of summer heat waves (HWs), some of them of unprecedented magnitude and impact e.g.,. HWs are the most visible sign of ongoing global warming in central Europe , which lead to an increased awareness in our society and stakeholders . As a result, both government agencies and the private sector have developed plans not only for long-term investments towards climate protection but also for the development of sustainable adaptation strategies, which are now regularly finding their way into policy agendas . In Germany, local governments are key actors implementing adaptation strategies . Nearly one-fourth of the German cities had climate adaptation plans in place by 2018 , documenting an increasing interest in the subject. Moreover, the German federal government has launched large research activities like the RegIKlim consortium (regional information for action on climate change) to further strengthen this development.

From the perspective of administrations in municipalities in southern Germany, the greatest need for action lies indeed in adapting to heat extremes . HWs – increased temperature over several consecutive days – are a threat to ecosystems, the economy, and human health e.g.,. HWs are in fact the weather hazard causing the highest number of deaths in Europe ; e.g., for the European HW in 2003 alone, up to 80 000 additional deaths were recorded in over 12 European countries concerned by excess mortality . However, there is no unified definition of a HW. Different thresholds for, e.g., length and temperature can be found in the literature, and a variety of indices have been developed for classification, e.g., warm spell duration index (WSDI) , heat wave magnitude index (HWMId) , or excess heat factor (EHF) . Recent efforts have gone towards quantitative approaches and a higher comparability between methods e.g.,, leading to a better understanding of the strengths, weaknesses, and range of applicability of the individual indices. Irrespective of the index used, there is a clear consensus in the scientific community that HWs will become more severe in terms of duration, frequency, and magnitude with increasing global warming, also in central Europe.

Climate information on the regional to local scale is needed for the development of tailored climate adaptation measures. This can be achieved with regional climate models (RCMs) which perform a downscaling of the climate projections from global climate models (GCMs) to the required spatial scales and timescales, as is done in the Coordinated Regional Downscaling Experiment (CORDEX) e.g.,. Novel developments include RCM simulations performed with a grid spacing under 4 km, which resolves convection-permitting scales, thus making parametrizations of deep convection not required (convection-permitting models, CPMs) . Due to the very high resolution on the scale of urban districts, either relevant data fields can be derived directly or a direct coupling with impact models can be allowed. Several recent studies have documented the advantages of these convection-permitting simulations, in terms of both dominant convective precipitation and regions with strong spatial heterogeneity as present in mountainous or urban areas . Regarding the representation of temperature, there is not yet a consensus on added value in convection-permitting simulations. Whereas and attribute improvements in the temperature output to the better resolution of orography, even found an increasing bias on the convection-permitting scale but improvements in the diurnal cycle of temperature in a domain covering the alpine region. In contrast, an improvement in mean temperature was found in for most of the study area and in investigations by over Germany. In addition, found an added value for temperature extremes. Mixed results with a regional dependency were found in , concluding a gain for temperature due to an improved spatial representation of local atmospheric circulations and land–atmosphere interactions.

To quantify the associated uncertainties of the regional climate projections, ensemble simulations are required. As the computational costs of CPMs are (very) high, many climate studies are based on single-model projections, and only a few studies using CPM ensembles exist . The very first ensembles of convection-permitting climate projections exist, e.g., from the CORDEX Flagship Pilot Study on Convection (FPSConv; ). There, several GCMs from the Coupled Model Intercomparison Project (CMIP5) using the RCP8.5 scenario were downscaled by multiple RCMs to a common grid with 3 km resolution covering the larger Alpine area (ALP-3). They used 10-year time slices for the historical period (1996–2005) and two future periods (2041–2050 and 2090–2099). The current study applies a different ensemble approach, which is a four-member ensemble of convection-permitting climate projections performed by a single RCM, downscaling four GCMs under the scenario RCP8.5. All simulations cover the period from 1971 to 2100 in a quasi-transient manner, where the projection is composed of several time slices. To our best knowledge, an ensemble of this temporal extent is currently unique. Such a long simulation period allows for a better statistical representation of extremes and the application of approaches used for typical coarser-scale transient GCM or RCM ensembles, e.g., the analysis for different global warming levels (GWLs) as is used in the IPCC AR6 to compare climate change signals for GCMs with a different climate sensitivity or between different emission scenarios.

Our focus in this study is heat extremes and related impacts under global warming compared to recent climate conditions. Specifically, we were motivated by three guiding questions:

What are the benefits of convection-permitting models for temperature extremes in Germany (Sect. 3)?

An evaluation of the uncorrected raw output of the ERA40-driven CCLM simulations compared to the observations shows a cold bias in the simulations over Germany. Figure a shows that in the reanalysis-driven simulation, the median monthly temperature over the evaluation domain in the 7 km simulation (thick solid blue line) is always lower than in the observations (thick solid grey line). This deviation is larger in the summer months. A similar pattern is found for further percentiles of the distribution, as shown for example for the 10th and 90th percentiles (thin lines in Fig. a), as they are generally underestimated, especially in summer. However, the 7 km output occasionally exceeds the observation in single autumn and winter months (October for the 90th and January for the 10th percentile). In the convection-permitting simulation (2.8 km), the monthly median temperature in the warm season is comparably higher than in the coarser simulation, leading to a reduced cold bias. In autumn it even exceeds the observation by 0.6 ∘C. However, there is no strong improvement in the mean temperature during the winter months, and the cold bias persists. A consistent reduction in the cold bias is found for the 10th and 90th percentiles, but a possible overestimation of higher percentiles seems to become more frequent, especially in late summer and autumn. In the convection-permitting ensemble, monthly mean temperature is similarly improved, as shown by dashed lines in Fig. a. Again, the largest improvement is in the summer. However, the mean bias in the ensemble median is larger than in the reanalysis, especially in the winter.

Averaged over all grid points, the mean error in the reanalysis-driven simulations is reduced from -1.1 to -0.13 ∘C in the summer half year (Fig. b). Moreover, the spread is increased. In the winter half year the median is reduced from -0.69 ∘C for 7 km to -0.56 ∘C for 2.8 km. This is a smaller but still significant reduction confirmed by a Wilcoxon signed-rank test. The test was applied to the two fields of mean error in the coarser-resolution (7 km) and convection-permitting (2.8 km) simulations. The null hypothesis of zero difference between the errors was rejected by the test based on a significance level of 0.05. Those patterns in the temperature output from coarse to high resolution are similar in the ensemble as in the reanalysis-driven run. For further information on the performance of the single ensemble members, please refer to the Supplement (Fig. S1).

To reveal spatial patterns, the mean summer half-year temperatures of the second 7 km nest (Fig. a) and the third 2.8 km nest (Fig. b) of the reanalysis-driven run are compared to the observations (Fig. a). Whereas there is a negative bias at nearly all grid points for the coarser nest (Fig. a), local differences are visible for the convection-permitting simulation (Fig. b). Here, a negative bias is still present in the north of the domain, especially in the hilly regions. In the south of Germany, predominantly positive anomalies are visible. Even though the regions with positive bias are not correlated with altitude, they do not seem to be independent of orography. The largest positive bias is found in the South German Scarplands (long ≈ 9.0∘, lat ≈ 48.5∘) – located directly between two major mountain ridges: the Black Forest and the Swabian Alps.

For nearly all grid points, there is an improvement with the convection-permitting simulation, which is indicated by a positive MSESS in Fig. c comparing the second and third nest with respect to the reference dataset HYRAS. There are a few grid points with negative MSESS. Those are associated with a positive bias and an overshoot of the convection-permitting simulation.

The density distribution of daily summer temperature shows nearly perfect agreement of observation and the convection-permitting reanalysis run (Fig. d). In comparison, the distribution for the reanalysis-driven 7 km simulations is shifted towards colder temperatures and has a lower spread. Especially the highest summer temperatures are better resolved by convection-permitting simulations. An improvement is also visible for the 2.8 km median of the ensemble simulations compared to the 7 km output. However, especially the high summer temperatures are still underestimated by the CPM. Low temperatures, from approximately -3 to 10 ∘C, are overestimated.

Overall, we identify a significant reduction in the mean bias for the convection-permitting resolution, which is especially pronounced during summer. Over Germany, the convection-permitting simulation reproduces a realistic frequency distribution of daily 2 m temperature. The remaining mean errors show a trend from negative bias in the north toward positive bias in the south. Other local patterns are partly associated with the predominant landscape regions. Based on the added value found in the 2.8 km resolution, its output is used for the following analysis.

This section characterizes HWs in the future based on their different features – length, temperature, magnitude, and frequency. Be aware that throughout this section we are focusing on a relative definition of these events – an anomaly versus the 90th percentile from the reference period (Sect. ). The relationship between HW magnitude, duration, and excess temperature is examined for the most severe HWs in each year in terms of the cumulative HW magnitude in the evaluation area (Fig. ). The corresponding figure providing absolute HW temperatures is provided in the Supplement (Fig. S3).

Firstly, the observed HW characteristics in the reference period (1971–2000) are analyzed, as shown in black in Fig. a. The average duration of the strongest HWs per year ranges from 3 to 12 d. The temperature excess ΔTmax above the 90th percentile ranges from 1.5 to 6.3 ∘C. The longest observed HW reaches up to 12 d; however, the correlation with excess temperature is weak (r=0.22). The observed HWMId has a range of 5 to 22, with an average of 8.8. HWMId increases with HW length (r=0.99).

To evaluate the representation of the three HW characteristics in the ensemble projection, the results of observation and simulations in the reference period are compared (Fig. a). The HW duration is reproduced well by all ensemble members, and no significant deviation from the observed distribution of duration (marginal distribution on the abscissa) is visible, which is confirmed by a two-sample Kolmogorov–Smirnov test at the level of significance of 0.05. Also for the HWMId, the test confirms no significant deviation between simulation and observation. For the excess temperature, no significant deviation from the observed distribution is found for three out of four ensemble members. However, significant differences for the results of the CNRM-CM5-driven simulation and an underestimation of the simulated excess temperature are visible in the marginal distribution on the ordinate (Fig. a). Moreover, there is a peak around ΔTmax=4 ∘C in the observation that is not reproduced by any ensemble member. The ensemble of climate simulations shows no significant deviation from the observed distributions in the reference period for the characteristic duration and HWMId, confirmed by a two-sample Kolmogorov–Smirnov test at a level of significance of 0.05. For the excess temperature, the test results support no significant deviation from the observed distribution for three out of four ensemble members but significant differences for the results of the CNRM-CM5-driven simulation. The underestimation of the modeled excess temperature is shown in Fig. a. Moreover, a peak around ΔTmax=4 ∘C is visible in the observation that is not reproduced by any ensemble member.

The climate change signal of excess temperature (1), duration (2), and magnitude (3) develops differently in the four ensemble members (Fig. b and c), but when superimposing the three GWLs, a clear picture of HW intensification emerges (Fig. d). (1) All ensemble members agree on an increase in average excess temperature and an increased width of its distribution compared to the reference period. The highest excess temperatures are found for MPI-ESM-LR-driven simulation up to ΔTmax=15 ∘C in GWL2 and GWL3. The lowest excess temperatures are projected by the simulation driven by CNRM-CM5 that already showed an underestimation in the reference period. In the ensemble median, there is a shift towards higher HW excess temperatures up to 5.3 (GWL2) and 6.9 ∘C (GWL3) (Fig. d). This implies that with HW excess temperatures from 2.6 to 4.5 ∘C (25 % and 75 % confidence interval) in the reference period, hardly any HWs are occurring today that will be a common scenario in the future. (2) Also for HW duration, a future increase in mean and spread of the distribution is detected by all ensemble members. Again, the smallest changes are projected by the simulation driven by CNRM-CM5. The simulation driven by HadGEM2-ES projects in general the longest HWs. These discrepancies in HW duration indicate different dynamics in the driving models. In fact, HadGEM2-ES is described as one of the best-performing CMIP5 GCMs for past climates and weather types , as well as blocking, which is underestimated in CMIP5 models in general . The extremely long HWs presented should therefore not be discounted as outliers but treated with caution. In the ensemble average, there is a clear shift towards longer HWs. The average duration increases from 4.3 (reference) over 5.1 (GWL2) to 7.5 d (GWL3). Moreover, the spread increases drastically, which leads to maximum HW duration up to 21 d. (3) The development of HWMId is strongly correlated with HW duration in the simulations. In the ensemble, a 26 % (100 %) increase in the median of HWMId is expected from the reference to GWL2 (GWL3) (HWMId in the reference: 8.2; GWL2: 10.3; and GWL3: 16.5). The significance of the increase in duration, excess temperature, and HWMId is confirmed by a two-sample Kolmogorov–Smirnov test at the level of significance of 0.05.

In order to put the results into perspective, they are compared with an actual reference event for Germany – the HW in 2003. The HW had a strong economic and environmental impact, caused a thousand deaths, and is referred to as a record HW e.g.,. Performing an analog analysis on 2003 HYRAS data, this HW had an average length of 12.7 d, maximum excess temperature of 7.4 ∘C, and HWMId of 26.7. It is visualized in black in Fig. d. As expected the event is extremely unlikely in the reference period. Only one simulated summer in the reference period by HadGEM2-ES exceeds the measured event in 2003. In a warmer world, events with such a strength occur with higher probability. In GWL3 such an event is in the 25 % confidence interval of 5.3 to 8.4 ∘C. For duration, the HW 2003 exceeds the 25 % confidence interval of 5.1 to 10.4 d in GWL3, and its duration is ranked 16th in the projections of 4×30 years, corresponding to an 8-year return period in GWL3. For HWMId, its rank of 21 in GWL3 leads to a 6-year period. It should be noted that in this case no distinction is made between ensemble members. The variations between ensemble members discussed earlier indicate the range of uncertainty in this projection. Moreover, the analysis considers only the local observations of 2003 limited to the simulation area. Summing up, an event like HW 2003 will become more likely but is projected to stay an extreme event with a return period of 5 to 10 years.

To assess regional patterns, the cumulative number of HW days as a measure of HW frequency is analyzed (Fig. ). Again the summer half year is considered only. In the reference period, averaged over 30 years, few HW days are observed per summer half year. In the evaluation area, the number of HW days ranges from 8.7 to 9.9 (5th to 95th percentile) and is distributed relatively uniformly across space. An overall increase from 18.8 to 23.0 (5th to 95th percentile) HW days is simulated in GWL2. With even more warming in GWL3, spatial features become visible. The increase predominantly affects the southwest. Moreover, slightly enhanced HW occurrence is projected in regions with higher elevation like the Black Forest. Over the domain, 28.7 to 36.9 (5th to 95th percentile) HW days are expected in GWL3.

The analysis of the seasonal changes reveals that HW severity is distributed inhomogeneously over the summer (Fig. ). In the reference period the occurrence of large HWs, defined as HWs with a coverage of at least 50 % of the study area, is relatively flatly distributed. There is a declining trend of the probability throughout the summer. From GWL2 to GWL3, it is apparent that there is an increased HW occurrence in late summer, around August and September. All members of the ensemble agree on this trend. However, the magnitude and timing of the adjustment vary. The highest HW probabilities are projected by EC-EARTH and HadGEM2-ES. Some ensemble members even depict a decrease in the occurrence of large HWs in early summer in GWL2 (May to June).

The analysis of both regional and seasonal patterns supports HW frequency as being closely linked to the future temperature increase, for which a similar spatial pattern and annual cycle of the change signal were found. However, it should be noted that, while the average temperature increases by only 2.6 ∘C from the reference period to GWL3 (Sect. ), this translates into an enormous increase in HW frequency of more than a factor of 3. This amplified increase in HW frequency is attributed mainly to a higher change of higher percentiles compared to the increase in mean temperature (Sect. ). Furthermore, more persistent weather patterns potentially enhance the severity of HWs in a future climate .

In summary, future HWs are characterized by significantly higher temperatures and longer HW duration. Thus, the magnitude of HWs increases dramatically in a warmer future, namely by 26 % (100 %) in GWL2 (GWL3). Furthermore, enhanced variability is projected for the HW characteristics. While the increase in HW days is spatially largely homogeneous, there is clear seasonality, with a strong increase in HW occurrence in late summer.

The meteorological perspective leaves open the question of the impacts of heat extremes, which will be addressed in the following section. The focus is first on human heat stress, and then the analysis is extended to further heat-related climate parameters.

In the presented analysis of heat extremes and related impacts in a convection-permitting climate ensemble for Germany, we can draw three main conclusions:

We found an added value for simulated temperature in the convection-permitting ensemble, especially for hot temperatures, that goes beyond better representation of the topography only. The improvement is particularly prominent in the summer half year.

Our results show an improved representation of the 2 m temperature in the raw CPM output compared to the coarser 7 km grid with parametrized convection. The improvement found is largest in the summer, when the cold bias in the coarser simulation was substantially reduced. This applies to both the median temperature and the more extreme percentiles (10th and 90th) of the temperature distribution over the model domain in the historical period. The improvement found in the temperature output on the convection-permitting scale confirms the findings by , , and . However, recent studies have shown that this temperature bias, especially in daily minimum and maximum, can still be addressed in CCLM with an improved formulation of the 2 m temperature in the land surface scheme . Moreover, it needs to be clarified whether the improved temperature output in the convection-permitting simulation justifies the higher computational cost for high-resolution simulations. While systematic biases between raw model temperature output and observations remain in our CPM ensemble, we show a clear benefit from a relatively large simulation area across different landscapes. We find a dependency of the remaining error in the landscape type and an association with orography – especially in transition areas between different major landscape types. Therefore we support a region-specific magnitude of the added value as in . In order to provide information for climate change impact studies or user-oriented studies, in this case focusing on heat stress, there is still a need for bias correction. Especially for threshold-based parameters, bias correction is a necessity to obtain meaningful values. Nevertheless, we expect that such studies will benefit from the better representation of high temperatures on the convection-permitting scale due to the smaller impact of bias correction and thus a smaller source of error.

The analysis allows for the first time a very high-resolution projection of temperature and temperature extremes over Germany in a 2 and 3∘C warmer world. The regional, high-resolution analysis confirms general warming over the whole region and a slightly higher change signal in mountainous regions. As in , the smallest temperature increase was found in spring. Indeed, the peak of summer temperatures in a warmer climate shifts to later in the summer. Moreover, the analysis confirms a wider distribution of temperature with global warming, implying a greater change in extreme temperature compared to the average warming in the future (e.g., ). Our study shows that HW probability is expected to increase significantly over Germany, and especially in late summer large HWs are anticipated. HW severity is projected to rise dramatically, indicated by a 26 % (100 %) percent increase from 1971–2000 to a 2∘C (3∘C) warmer world. Increasing variability in HW characteristics is projected for the future. This is consistent with past trends of HW temperature and duration derived from observational data . Our study thus suggests that the trend is likely to continue in the future.

Apart from meteorological insights, a closer look at human heat stress and other tailored climate parameters shows the potential of using convection-permitting simulations in different fields of application and highlights the importance of individual consideration. Strong human heat stress – parametrized via UTCI > 32∘C and associated with very hot days or tropical nights – is prevalent in the flat regions such as the Rhine Valley. Moreover, the largest absolute increase is expected for these regions, comparable to . The change signal of tailored climate parameters does not always scale linearly with global warming – as is the case for the relative quantity of dry hot summers or growing days, a quantity that applies to moderate conditions. Especially for extreme heat stress (UTCI > 32∘C, very hot days, or tropical nights) we see a nonlinear but rather exponential increase with global warming. In particular for specialized applications – expressed, e.g., over more complex climate parameters – behavior depends crucially on the prevailing landscape and might even lead to opposing trends. Therefore, the analysis supports previous results of spatial patterns and shows the benefit of CPMs, which allows the representation of distinct characteristics in clearly defined areas.

The limitation of the study is that the assessment of uncertainty is restricted with four GCMs and only one RCM. However, the magnitude of the uncertainty associated with the RCM choice is typically smaller than from the large-scale GCM forcing . In the future, larger ensembles on the convection-permitting scales are expected to be available, enabling assessment of GCM and RCM uncertainty. Currently, ongoing downscaling of the CMIP6 GCMs is a promising source of future driving data for high-resolution climate simulations. In particular, the improved representation of Northern Hemisphere blocking in the new generation of climate models will necessitate additional analysis of HWs and is anticipated to provide complementary insights to the results shown. Moreover, long convection-permitting projections would profit from the implementation of variable land surface characteristics over time, as, e.g., recently provided by FPS-LUCAS . Moving from constant to variable input fields could yield valuable information for heat stress in impact studies. Especially for climate adaptation studies, development is still anticipated for urban areas and the evaluation of corresponding urban parametrization schemes. Since no parametrization is used in this study, further improvements for urban areas are to be expected e.g.,.

Heat extremes and related impacts derived from a convection-permitting ensemble document that the climate change signal depends on major landscape regions. Therefore, such convection-permitting projections have the potential to facilitate tailored impact studies and can help to narrow down the gap between climate research and the requirements of stakeholders, e.g., for sustainable risk management and climate adaptation. This presented finding stresses the need for climate adaptation strategies on a local level and supports the regional approach in climate adaptation research, e.g., in the German Federal Ministry of Education and Research (BMBF) RegIKlim project: basic research is done in a pilot region, concentrating on region-specific key issues to develop, evaluate, communicate, and test the implementation of adaptation strategies with the aim of an upscaling in the concerned region in the future.