## Introduction

The hazards posed by heavy precipitation events increase with higher intensity and longer-duration storms. Flooding is most likely when prolonged intense precipitation falls on the same area1. In Europe, such multi-hour intense precipitation events often manifest in the form of mesoscale-convective systems (MCSs2,3; large areas of convective precipitation, defined here as contiguous precipitating areas exceeding 1000+ km2 with maximum precipitation exceeding 20 mm/h averaged over a 144 km2 grid box). MCSs are important contributors to extreme precipitation during the warm season (summer and autumn) in Europe4. Elsewhere, MCSs can contribute more than half of the annual total precipitation, like over the Great Plains and the La Plata Basin3,5.

MCSs have a large spatial extent, which can lead to high precipitation totals over a catchment. While most MCSs last for only a few hours, some last for more than 18 h6. MCSs may appear quasi-linear (e.g. squall lines) or quasi-circular. While we focus on the precipitation from MCSs, they cause other weather hazards such as intense wind gusts, tornadoes, hail, and lightning. Over Europe, MCSs have caused substantial casualties and damages. A selection of events is shown in Table 1 involving different MCS hazards, but the focus in this paper is on extreme precipitation.

The large-area and longevity of MCSs are tied to their interaction with the large-scale synoptic dynamic conditions at scales exceeding 1000 km. Large-scale winds control the vertical wind shear, which are crucial for MCS upscale growth and persistence. Weakly-sheared convective environments may lead to poorer-organised and typically slower-moving storms, but slow-moving storms cause longer precipitation durations at a given location and hence higher local precipitation accumulations1; this is in contrast with bigger and long-lived MCSs that have higher lifetime total precipitation volume, spreading over larger areas. Total precipitation volume is also closely linked to precipitation rate, which is controlled by the thermodynamics of the storm environment, fed by synoptic/mesoscale structures like low-level jets7. MCS area, lifetime, movement speed, storm precipitation rate and total volume can all be diagnosed by methods that track individual storms8.

Proper simulation of MCSs requires a model able to represent the underlying mesoscale dynamics. Some current-generation climate and forecast models have grid box sizes of ~1 km which enables the (deep) cumulus parameterisation to be switched off (convection-permitting models, hereby CPMs). CPMs are capable of partially-resolving convection and reduce some biases typically associated with the use of cumulus parameterisation9,10,11,12,13,14,15. CPMs have been shown to simulate realistic frequency and tracks of North American MCSs16 and more realistic MCS properties over Africa than lower-resolution parameterised-convection models17. CPMs have also been shown to reasonably represent key observed characteristics of intense precipitation events over the north-west Mediterranean, while underestimating the highest precipitation intensities14.

This study uses a series of the UK Met Office (UKMO) Unified Model and ETH-Zürich (ETHZ) COSMO Europe-wide 2.2 km CPM simulations11,18,19,20. These include ERA-Interim-driven hindcasts (1999–2008; UKMO:CPMUH, ETHZ:CPMCH) for both CPMs. However, the CPMs use different approaches to represent atmospheric changes due to global warming. The UKMO downscales a free-running global climate model for the present-day (driven by 1998–2007 sea surface temperatures; CPMUP) and for future-climate (same SST superposed with changes at the end of the century, 10 years long; CPMUF), while ETHZ uses a “pseudo-global warming” (PGW) simulation representing future-climate at the end of the century (10 years long; CPMCPGW). The PGW approach21 involves modifying the ERA-Interim boundary conditions used for the hindcast to capture the mean warming signal and associated changes in moisture and large-scale circulation, whilst the sequence of weather systems is unchanged. This allows the thermodynamic component of the future change22 to be robustly captured, with reduced influence from natural variability. Previous work over the US, using a PGW approach, showed increases in the frequency and precipitation volume of MCSs with global warming23. However, the PGW approach has limitations in representing variance changes driven by global warming21,24, which can be a critical control on MCS characteristics such as duration and area1; i.e., PGW cannot represent changes to the sequencing and timing of weather events and intra-seasonal/inter-annual modes such as the North Atlantic Oscillation (NAO). On the other hand, with the standard GCM downscaling approach, we assume that these changes are well represented by the driving GCM, noting that state-of-the-art GCMs still have significant biases in both the variance and the mean climate25,26.

Here, to the best of our knowledge for the first time, we compare changes to MCSs in a full dynamical GCM-downscaled simulation (CPMUF) to those from a PGW simulation (CPMCPGW) at the convection-permitting scale for the late-C21 under the Representative Concentration Pathway (RCP) 8.527, using two different convection-permitting regional climate models. The influence of dynamical changes in circulation patterns on MCSs has not previously been considered at convection-permitting scales. Using a single-GCM downscaling approach, it is not possible to represent the uncertainty in projected changes, e.g. Coupled Model Intercomparison Project (CMIP) projections show good agreement in the reduction of summer synoptic variability over Europe but consensus is low for mean troposphere wind changes28,29. This uncertainty to mean dynamical changes is often used as a justification for the PGW approach with the drawback of not representing changes in intra-seasonal variability.

Most results are based on a Europe-wide analysis, but we do select three sub-domains for more detailed analysis. They are the British Isles, Germany, and “Alpine” region. They are described in the Methods and Supplementary Fig. 1.

We employ an automatic method for MCS detection and tracking8,17. Previous work applying a severe storm ingredients-based approach1 to the CPMUF simulation indicated a future increase in slow-moving storms with the potential for high precipitation accumulations, attributing that to weakening of upper tropospheric winds30. Here, to the best of our knowledge for the first time, we provide insight into the role of large-scale circulation changes in driving changes to MCSs. We explore changes to MCS sequencing across Europe, as well as characteristics of MCSs.

## Results and discussion

We first examine the percentage of total precipitation contributed by MCSs and the number of MCSs in the present-day and future simulations. Figure 1 indicates that MCSs currently cause ≈20–25% of the total precipitation (≈10–15 trillion m3), with agreement across CPMs. In a future warmer climate, both CPMs project a decrease in mean precipitation of ≈12%, largely due to warm season decreases (Supplementary Note 1) but a substantial increase in the contribution from MCSs. The CPMUF simulation projects almost a doubling of MCS contribution (from 20.6% → 35.3%); a smaller increase is projected from CPMCPGW (from 23.9% → 28.3%), with both increases significant compared to year-to-year variability.

Despite this agreement, the CPMs project different changes to the frequency of MCSs: CPMUF shows a near doubling, but CPMCPGW shows a ≈18% decline. This might be related to different spatial patterns in present-day MCS frequencies (Fig. 2a–c). Both models agree on a northward decrease in MCS frequencies consistent with observations5,31,32, but CPMUH and CPMUP favour MCSs over the Western Mediterranean while CPMCH favours storms over land. We explore this further in Supplementary Note 2 by comparing our results with recent satellite analysis estimates. The future changes to MCS frequency do not show any clear spatial pattern (Fig. 2d, e). The increase (decrease) projected by CPMUF (CPMCPGW) simulations is relatively consistent across the entire domain with better agreement over Central Europe (decreases for both CPMs) (panel f).

We now move on to examine the present-day simulation and changes to a number of key storm statistics—precipitation volume, maximum and average intensity, MCS movement speed, maximum area, and storm lifetime (Figs. 3 and 4). The maximum intensity, area, and movement speed of an MCS determines local precipitation accumulation and is hence important to localised flash flooding30,33. MCS lifetime precipitation volume is important for accumulations over larger catchments and hence wider flood impacts. We also examine temporal clustering across Europe, as well as national-scale regions. Temporal clustering is an important control on antecedent moisture conditions and hence the probability of flooding, but also to compounding impacts through increased vulnerability.

Many studies have reported increases in the intensity of extreme precipitation with warming22,34,35,36. Here we use an intensity threshold (20 mm h−1) to track MCSs (Fig. 3a). The 95th percentile peak intensity of those MCSs is ≈45 mm h−1 for both CPMs present-day simulations. We find that both models show an overall intensification, with CPMCPGW showing a lower intensification of the 95th percentile of ≈15%, compared to a 24% increase for CPMUF. Although CPMUF shows a significant increase in MCS frequency at all intensities (Fig. 3b), we find the largest increases in absolute counts for the highest peak intensity events (e.g. about a tripling for 40–45 mm h−1). Importantly, we find MCSs in the future simulation with unprecedented (70+ mm h−1) peak hourly intensities. For CPMCPGW, we find the same frequency increase for storms with the highest intensities (45+ mm h−1), but a projected decrease in the overall number of MCSs. Consistent with increasing peak intensities, both CPMs project an increase to MCS area-averaged intensity (Fig. 3c, d). Higher increases are found for CPMUF, with 95th percentile area-averaged precipitation intensity increased by 1.7 mm h−1 (+22%) respectively, relative to smaller 0.8 mm h−1 (+10%) increases from the PGW-CPM simulations.

We now analyse precipitation volume from individual MCSs (Fig. 3e, with the vertical line indicating the 95th percentile). CPMUF shows a large increase in low volume (Fig. 3f; PrVol ≤ 108 m3) storms, with the 95th percentile (109.4 ≈ 2.5 × 109 m3) of the future-climate simulation less than the present-day simulation. This means a future CPMUF MCS typically produces a lower volume of precipitation over its lifetime even though there is a much higher frequency of these MCSs (Fig. 1). In contrast for CPMCPGW, there is a decrease of MCSs with low volumes, but frequency changes for higher volume storms are small, resulting in an increase to the 95th percentile MCS volume. Both CPMs therefore show a future increase in the total precipitation resulting from MCSs, but changes in the underlying characteristics of the MCS population are quite different.

Importantly, we find that both modelling approaches indicate future increases to the absolute number of high-volume, and thus impactful, MCSs. Both UM and COSMO present-day 95th percentiles for precipitation volume per MCS (Fig. 3e) are just above 1.5 (precisely, 101.5 × 108 ≈ 3 billion m3). Future total precipitation volume above 1.5 are increased by about 52% and 17% for CPMUF and CPMCPGW respectively. The top 20% wettest MCSs in all simulations contribute close to 85–90% of the total annual precipitation from all MCSs, following the 80–20 Pareto principle, and hence explains agreement with satellite analysis estimates since they are biased in favour of larger-area MCSs. Weaker MCSs, despite being more numerous, have a smaller contribution. Hence, future changes to total MCS precipitation volume are dominated by changes at the tail of the distribution: for both CPMUF and CPMCPGW, there are increases of total precipitation volume attributable to the top 20% wettest MCSs (UKMO: 88%  90%; ETHZ: 86%  87%).

We now move on to examine the temporal variability of MCSs. We might expect MCSs to naturally cluster in time when conditions are favourable, and this is likely different in the future climate37. Temporal clustering on sub-seasonal timescales magnifies the impact of MCSs due to the larger spatio-temporal concentration of weather extremes. Here we use the index of dispersion38 to estimate temporal clustering for the whole simulation domain as well as regionally for Germany, the British Isles, and Alpine area. The domain-wide analysis is akin to estimating maximum simultaneous property losses, while the regional analysis is focused on indicating the potential for compound hazards.

Table 2 presents the JJA and SON pentad index of dispersion ($$\hat{\phi }$$); we find clear evidence of temporal clustering of MCSs as all simulations have dispersion indices above 1 (indicating clustering). CPMC simulations show a lower level of clustering than the UKMO simulations. Europe-wide, both CPMs project MCS warm-season clustering to increase, but only SON increases are significant. For SON, dispersion indices increase from 25.5 (15.2) to 45.4 (24.4) for the UKMO UM (ETHZ COSMO) simulations, which are outside the confidence intervals of the timeslice estimates. However, JJA increases can be robust regionally (e.g. over Germany and the British Isles). There are no clear JJA changes for the Alpine region, although this region has large summer decreases to the precipitation mean and frequency19,39, but a robust SON increase is found. For CPMUF, SON shows large increases in precipitation extremes19,39. The increase of MCS temporal clustering diagnosed here for both models will likely amplify the flood impacts of increases in precipitation volume projected from the strongest MCSs. An alternate approach to assess clustering is by diagnosing the number of simultaneous objects-of-interest (i.e. MCSs) passing near a fixed point40; this is examined in Supplementary Note 3, and results are consistent with the dispersion analysis here.

We now focus on identifying why the CPMs agree on projected increases in the total precipitation from MCSs, their contribution to the mean, and MCS peak intensity, whilst disagreeing on changes to the underlying number of MCSs and precipitation volume per individual storm. We start by considering storm speed and its relationship to storm area and lifetime. Movement speed and steering winds are closely linked to the vertical wind shear of the large-scale environment, which is crucial for the development and persistence of these deep convective storms (i.e. storm organisation)41, suggesting a movement speed (Fig. 4a, b) relationship with area and lifetime (Fig. 4c–f). Comparing the two hindcast simulations, storms are faster-moving and larger but more short-lived in the ETHZ simulations. Given the hindcasts have highly similar lateral boundary conditions (LBCs), differences in model numerics and physics appear to be a factor in determining MCS speed, but this is beyond the scope of this study (see “Methods” for LBC caveats and model differences).

In CPMUF, we find that MCS speed decreases from 7.0 to 6.3 m s−1 30 with maximum area and lifetime responding similarly (median $$\sqrt{{{{{{{{\rm{AreaMax}}}}}}}}}$$: 77.3 → 61.9 km; median lifetime: 8 → 6 h). In contrast, CPMCPGW projects somewhat faster (median speed: 9.0 → 9.6 m s−1) and larger (median $$\sqrt{{{{{{{{\rm{AreaMax}}}}}}}}}$$: 103.9 → 106.0 km) MCSs, resulting in a projected increase to average precipitation volume per MCS (panel e).

Robust aspects of MCS changes in both CPMs, including increases to total storm precipitation volume, peak storm intensities, and MCS temporal clustering, are important to climate impacts. Increases in peak precipitation intensities are well understood in terms of increases to saturation water vapour pressure with temperature22. However, CPMUF and CPMCPGW disagree on the magnitude of these changes. Changes to total storm volume, representing the combined effects of changes to the number of MCSs and precipitation volume per MCS, are driven by different underlying processes. In particular, precipitation volume per MCS is linked to storm movement speed (e.g., a proxy to vertical wind shear) which controls both convective organisation3 and the amount of time a storm precipitates over a catchment. CPMCPGW indicates a different change in storm movement speed—faster MCSs—to the fully-dynamically downscaled CPMUF simulation—storm slowdown. We note here that our results also suggest that different model physics may play a role in storm movement, as there are some differences between CPMUH and CPMCH hindcasts (Fig. 4). Model physics may also explain the differences in land-sea contrast in the spatial density of MCSs between the two models (Supplementary Note 2).

Our results highlight the importance of including circulation change uncertainties through sampling different driving GCMs and different downscaling approaches in projecting changes to MCS characteristics, particularly precipitation volume per MCS and number of MCSs, both critical for flood impacts. Additional experiments, for example a PGW-experiment with CPMU and standard GCM downscaling with CPMC, are needed to quantify the relative importance of the large-scale dynamics compared to other factors (e.g. differences in model dynamics and physics) that may contribute to differences in different CPM projections. Nevertheless, large-scale circulation changes have been identified as a key driver of some aspects of MCS changes, e.g. storm movement speed30. These large-scale changes may be different between CPMUF and CPMCPGW due to the different driving GCM and/or downscaling approach; we cannot disentangle these contributing factors with the experiments here. However, this study does identify the importance of large-scale circulation changes for impact-relevant changes in organised convective storms.

We expect the PGW approach to remain popular due to its lower computation cost and its advantage in constraining natural variability in climate change simulations. A standardisation of the PGW approach would be helpful; for instance, the PGW perturbations have been derived from multi-model means16 or a single GCM42. Initiatives like the recent Europe CPM collaboration39 would help such a standardisation, with the broader goal of creating a favourable environment for climate model intercomparisons. We also recommend a coordinated set of GCM/PGW driven pairs are carried out, allowing a clear assessment of the importance of capturing changes in sub-seasonal circulation patterns for future changes in local climate extremes and limitations of the PGW approach. This will be valuable for informing future CPM experimental design.

Future large-scale circulation changes over Europe are uncertain28,29, however, there is some consensus across models of a significant weakening of the summer circulation, with a shift of the polar jet and thus midlatitude cyclone tracks towards the north due to Arctic Amplification43,44. This is manifested over Europe by CMIP simulations as reductions to synoptic variability29,43. PGW approaches do not fully capture such changes, although new developments to PGW such as the one applied here21 do account for mean changes at synoptic time scales by superposing day-by-day annual cycle changes onto present-day variability. Since MCSs and other high-impact precipitation extremes are often triggered by synoptic variability (for example, changes to the frequency of cut-off lows that triggered the 2021 European floods45), a multi-model downscaling of different GCMs is needed to assess uncertainty in the large-scale circulation change and to link these changes to organised convective storms. We therefore recommend that assessments of changes in convective storms must include a range of circulation change scenarios and downscaling approaches if possible. Given model physics and numerics are also important, a multi-model and/or parameter-perturbed ensemble of fully dynamically downscaled CPM future scenarios (like UKCP Local46) would be recommended; this is not different to what the global climate modelling community has been doing47,48 but is a technical challenge at convection-permitting scales. The even longer-term goal is to achieve global climate CPM49. Future work should also consider other (lower) emission scenarios, as RCP8.5, although plausible, is not the most likely scenario50.

The significant increased contribution to European precipitation by intense, high-volume and clustered convective storms shown here has important social-economic implications, with (flash and fluvial) flooding, landslide and drought risks all expected to increase at the same time if hazard management remains as status-quo. Increased contributions to total precipitation from such storms under a drier European climate19,39 imply that water supply from precipitation will be distributed more unequally in space and time, increasing stresses to future water supply. Efforts to incorporate climate change into risk management approaches need to be fully embraced to ensure that robust climate adaptation measures are realised. It is important that results such as these are used to help society prepare for the impacts of more extreme future precipitation events.

## Methods

### Model simulations

The UKMO climate model simulations span Europe (Supplementary Fig. 1) at 2.2 km resolution11,19. Three simulations with different LBCs are used: ERA-Interim51 driven and 25 km HadGEM3 present- and future-climate GCM driven simulations52. No intermediate nests are used. The model is based on the UKMO operational UKV model9. Cumulus parametrisation is disabled; a new semi-implicit semi-Lagrangian dynamical core and standard UKMO land-surface and boundary layer physics are used53,54,55. Precipitation data are available hourly, and are regridded to 12 km grid boxes before carrying out any analysis.

The 10-year 25 km GCM present-climate simulation (CPMUP) and ERA-Interim hindcast (CPMUH) are for the period 1998–2007 and 1999–2008 respectively, and use daily observed SSTs56. The future-climate end-of-century (2099–2108; CPMUF) simulation uses the RCP8.5 greenhouse gas scenario27. Future SSTs are observed SSTs plus 20-year mean changes that are derived from coupled GCM simulations52; prescribed global mean SST change is ≈4K52 but varies with time and regions.

The ETHZ COSMO CPM (aka COSMO-crCLIM) simulations20,57 have the same horizontal resolution and simulation domain. Different versions of COSMO are used in operational weather forecast and climate research across Europe20. Unlike the UKMO simulations, the ETHZ simulations have a 12 km nested simulation58 in between the driving data and the CPM. Substantial model physics differences exist between the ETHZ and UKMO simulations; for instance, the inclusion of a shallow-convection scheme in the ETHZ simulations59, the use of a different land-surface and boundary layer scheme60,61, and a split-explicit dynamical core for model numerics20. There are two simulations: ERA-Interim51-driven hindcast (CPMCH) and a PGW (CPMCPGW) simulation21. Strictly speaking, due to the use of an intermediate nest, the ETHZ and UKMO hindcasts do not have the same LBCs; however, we expect the differences to be small away from the model lateral boundaries as the intermediate nest domain is not much larger than the CPM one57. There are different flavours of the PGW approach, but they are all performed by adding some form of the global warming signal on top of reanalysis62. For the PGW simulation presented here, the seasonal cycle of changes to temperature, circulation, and stratification are first diagnosed from a specific CMIP5 RCP8.5 2070–2099 GCM simulation42, and are then superimposed onto 1999–2008 ERA-Interim data; this retains the multi-year mean seasonal changes but does not account for the changes to intra-seasonal and inter-annual circulation variability (i.e. the sequencing of weather events and the timing of NAOs are unchanged, etc.). Underlying global mean temperature change is ≈4K42 but varies with time and regions. Like the UKMO data, data are available hourly, and are regridded to 12km grid boxes.

RCP8.5 is a high emission scenario50. Given CPMs are computationally expensive, RCP8.5 is often chosen to increase the robustness of the climate change response over natural variability.

All results are annual unless stated otherwise. The season of an individual MCS is determined by its initiation month. Only MCSs with their mean centroid location within the latitude-longitude box of $$\left[1{1}^{\circ }{{{{{\rm{W}}}}}}-2{5}^{\circ }{{{{{\rm{E}}}}}},35.{5}^{\circ }{{{{{\rm{N}}}}}}-59.{5}^{\circ }{{{{{\rm{N}}}}}}\right]$$ are analysed.

### DYMECS precipitation tracking algorithm

The precipitation system detection and tracking algorithm was originally developed for use with sub-hourly radar and forecast model precipitation data8. Since then, it has been applied to hourly climate model data over Africa with resolution as coarse as 25 km17.

The algorithm has two main parts: the detection of objects-of-interest for each image and the tracking of these objects-of-interest between consecutive images. The detection algorithm is based on the local table method63: this is done by labelling pixels-of-interest (defined to be pixels with precipitation exceeding 1 mm/h) by line-by-line scanning. The tracking component is based on the windowed cross-correlation between consecutive images (hereby images at time t − 1 and t)64. Each image is divided into 18-by-18 grid box windows with the centres of each window a half-window width apart. The correlations between each window at time t − 1 and t are computed. Velocities for a particular window are estimated using the distance to the window that has the maximum correlation with at time t. The labels that are identified at time t − 1 are then moved by those estimated velocities. Their areal overlap with labels at t are computed. If the overlap fraction exceeds 0.6, the overlapping labels are considered part of the same track, with splitting and merging allowed. A label with no valid predecessor is identified as a new track.

### Definition of MCS

The definition here is inspired by previous work17,32. We require MCSs to have maximum contiguous area of interest exceeding 1000 km2, but they can have much greater maximum areas (100,000+ km2)32. Due to the relatively low precipitation threshold (1 mm h−1) used to identify precipitation area, we only examine tracks with lifetime maximum intensity of at least 20 mm h−1, thereby removing weaker precipitation tracks from the analysis.

In the 10-year hindcast and present-climate UKMO simulation data, there are ≈1183 and ≈860 tracks per April–September (AMJJAS) annually that satisfy the above criteria. Similarly, the 10-year ETHZ hindcast has ≈1184 tracks per AMJJAS. This is comparable to the number of tracks identified with outgoing longwave radiation (OLR) over a similar European domain for the same months32—6311 tracks in five AMJJAS seasons (1993–1997; i.e. ≈1262 tracks per AMJJAS). However, we note an OLR-only analysis has severe limitations5. This is further explored in Supplementary Note 4, where we show model OLR can falsely indicate areas of precipitation. Comparisons of the spatial pattern for AMJJAS are in Supplementary Note 2 as part of the discussion showing the land-sea differences between the two models.

### Precipitation volume

The time-integrated precipitation volume from track i is its areal sum over all times:

$${{{{{{{{\rm{PrVol}}}}}}}}}_{i}=\mathop{\sum }\limits_{{{{{{\rm{life}}}}}}=1}^{{{{{{{{{\rm{life}}}}}}}}}_{i}}{{{{{{{{\rm{MeanPr}}}}}}}}}_{i,t}\times {{{{{{{{\rm{A}}}}}}}}}_{i,t}$$
(1)

The PrVoli has units of volume L3. The total volume follows naturally as:

$${{{{{{{\bf{V}}}}}}}}=\mathop{\sum }\limits_{i=1}^{{{{{{{{\rm{nTracks}}}}}}}}}{{{{{{{{\rm{PrVol}}}}}}}}}_{i}$$
(2)

Total precipitation volume, from MCSs and non-MCSs, is calculated by spatially summing precipitation, either annually or seasonally, across the entire model simulation domain. Annual (seasonal) totals only compare with annual (seasonal) MCS totals.

### Temporal clustering

Temporal clustering of MCS is examined using count data statistics. Count clustering is measured by the index of dispersion $$\hat{\phi }$$38, which is the quotient between the variance $${\sigma }_{{{{{{{{\rm{nMCS}}}}}}}}}^{2}$$ and the mean (E(nMCS)):

$$\hat{\phi }=\frac{{\sigma }_{{{{{{{{\rm{nMCS}}}}}}}}}^{2}}{E({{{{{{{\rm{nMCS}}}}}}}})}$$
(3)

$$\hat{\phi }\approx 1$$ if counts are randomly distributed (i.e. a Poisson Process); values much greater than 1.0 suggest temporal clustering. The above is calculated by aggregating daily counts into pentad (5-day) sums; weather systems generally persist for a few days, and the 5-day aggregation reduces serial correlation between the intervals. We examine the pentad index of dispersion in JJA and SON, accounting for the seasonal cycle and the higher frequency of MCSs during the warm season. Confidence intervals are estimated by recomputing the index 10,000 times by pentad bootstrapping (see “Bootstrapping method” for details).

### Bootstrapping method

Confidence intervals are estimated by block bootstrapping data, assuming independence between blocks (pentad or year). Blocks are resampled with replacement, with the same sample size as in the original data, and the metric of interest is recomputed. This process is repeated, generating new estimates to the same metric. The 95 confidence interval is estimated by taking the 2.5 and 97.5 percentile of the bootstrap.

### Regional analysis

The clustering index and certain precipitation indices are calculated regionally—British Isles (an ex-Mediterranean maritime region), Germany (an ex-Mediterranean continental region), and the CORDEX Alpine domain which is subject to multi-CPM study39,65. The “Alpine” region covers central and southern Europe, including Switzerland, Austria, southern France and Germany, northern Italy, Adriatic Sea, Ligurian Sea, and the Gulf of Genoa; for this region, heavy precipitation is common for both summer over land and autumn over sea. The regions are illustrated in Supplementary Fig. 1. Finally, we note that temporal clustering and impacts occur at even smaller scales (river catchment or a city), but changes at that these scales are hard to diagnose with just 10-year climate simulations.