The ESWD, managed and maintained by the European Severe Storms Laboratory (ESSL), is the only multinational database and by far the largest archive of hail reports in Europe. Quality-checked reports of SCSs and related phenomena originate from storm chasers and trained spotters, sometimes supplemented by newspaper reports. In our study, we consider the reported maximum hail diameters of all quality levels (70.4 % of all reports were confirmed; 29.0 % were at least plausibility checked). This includes both large hail with a diameter of at least 2 cm usually given in increments of 1 cm (in rare cases of 0.5 cm) and hail layers with a depth of at least 10 cm, regardless of hail diameter. In those cases, and when a hail size is not specified (usually in the case of small hail), the diameter is set to 1 cm.

During the 10-year investigation period, a total of 4577 reports of severe hail in the study area are available. Most reports stem from Germany (76.5 %), followed by France (21.1 %), Belgium (1.7 %), and Luxembourg (0.7 %). This distribution does not reflect the occurrence probability of SCSs but is primarily due to the ESSL originally being a German initiative.

Because of the large spatial extent of the study area in a west–east direction, we converted the timestamps for the daily cycle analysis (only for that; cf. Fig. ) from UTC into local time (LT) by adding Δt=24 h/360∘ lat = 4 min per degree starting from 0∘ lat.

Atmospheric conditions prevailing over a larger area around the SCS tracks are studied using the ERA-Interim reanalysis from the European Center for Medium-Range Forecast (ECMWF). This dataset, which was also used for the detection of synoptic cold fronts (see Sect. ), is represented as spherical harmonics at a T255 spectral resolution (approx. 80 km) on 60 vertical levels from the surface up to 0.1 hPa with a temporal resolution of 6 h. In order to estimate the effects of the model resolution on the dynamic and thermodynamic environmental conditions, we additionally used high-resolution coastDat-3 reanalysis data for selected variables. This second reanalysis from the Helmholtz-Zentrum Geestacht (HZG) has a spatial and temporal resolution of 0.11∘ (approx. 10 km) and 1 h, respectively. It was produced by dynamically downscaling ERA-Interim using COSMO in climate mode CCLM;.

Mesoscale environments of the hailstorm tracks are characterized by severe-storm predictors representing both thermodynamical and dynamical conditions. We tested and applied several convection-related parameters but focus here only on those proxies with the highest prediction skill: surface lifted index (SLI) representing latent instability , lapse rate (LR) as the temperature difference between 700 and 500 hPa representing potential instability (only for coastDat-3), deep-layer shear (DLS) as the difference of the wind vectors between 500 hPa and the surface, and 0–3 km storm-relative helicity (SRH) quantified by 1SRH=∫vh-c⋅∇×vhdz,2=∫-u-cx∂v∂z+v-cy∂u∂zdz, where vh=(u, v) is the horizontal wind vector and c=(cx, cy) is the (constant) cell motion vector, which is usually estimated from a semi-empirical relation such as that from . As the convective cell-tracking algorithm directly computes c for each SCS or HS event (see next Sect. ), we used these values to quantify SRH in addition to the vertical wind shear provided by ERA-Interim. Helicity is a measure of the degree to which the direction of motion is aligned with the (horizontal) vorticity of the environment ωh=∇×vh . Only streamwise vorticity, which is a prerequisite for supercells bearing the largest hailstones, contributes to SRH .

Synoptic-scale cold fronts are detected in ERA-Interim based on the method outlined in , which is briefly summarized here. To identify and locate fronts in the reanalysis, we used the thermal front parameter TFP; defined as 3TFP=-∇|∇θe|⋅∇θe|∇θe|, where θe denotes the equivalent potential temperature at 850 hPa, a widely used choice in the forecasting community, which also neglects sea‐breeze fronts. The first term in Eq. () represents the gradient of the frontal zone (|∇θe|), which must be higher than 4 K (100 km)-1. The second term is the unit vector of the θe gradient. The TFP hence captures changes of the gradient of the frontal zone along the gradient itself. The frontal zone is strongest where TFP = 0, and its leading edge is where TFP = max. For the detection of propagating synoptic fronts, which are in the focus here because of their relevance for convection triggering, we require all fronts to have a length of at least 500 km and a minimum advection speed of 3 m s-1. These two thresholds may seem somewhat artificial or arbitrary. But as shown by , their implementation sufficiently removes the land–sea contrast and thermal boundaries from Alpine pumping from the dataset and limit the data to fronts typically associated with extratropical cyclones.

Tracks of SCSs are identified from 2D radar reflectivity based on the precipitation scan at low elevation angles. Radar data with a spatial and temporal resolution of 1 km and 5 min, respectively, were provided by Météo France and by the German Weather Service (DWD) as entire radar composites. Whereas all 17 German radars operate in the C band, 19 radars in France are in the C band, and 5 each are in the S band and X band. The area in France covered by the S-band radars is rather small (<5 % of the total area) compared to that captured by the C band, and these are mainly restricted to the southwest (S-band radars at Opoul, Nîmes, Bollène, and Collobrières). Because of the dominance of C-band radars, we did not distinguish between the two radar types. X-band radars, exclusively operating in the Maritime Alps in southeastern France, are not considered due to their strong attenuation of the radar signal.

Storm tracks were reconstructed by applying a modified version of the cell-tracking algorithm TRACE3D originally designed for 3D reflectivity in spherical coordinates . Thus, TRACE3D has to be modified to rely on 2D radar reflectivity in Cartesian coordinates . The tracking algorithm first identifies all convective cells (reflectivity core; RC) embedded into larger “regions of intense precipitation” ROIP;. Afterward, the weighted center (barycenter) of all RCs is tracked spatially over subsequent time intervals dt by establishing a temporal connection between the detected RCs. For each RC, a 2D shift velocity vector vT is calculated in different ways, depending on whether and over what distance an RC has already been detected in previous scans. The new position of the RC is estimated from sT=vT⋅δt within a certain search radius r, which depends on the length of sT and the distance to the closest neighboring RC. This process is repeated for all subsequent scans until the complete track of a convective cell is reconstructed. The algorithm considers different processes such as cell splitting or merging. Correction algorithms are implemented for undesired radar effects such as the bright band or anomalous propagation (so-called anaprop). In addition, we eliminated all single grid points with high radar reflectivity but without lightning within a radius of 10 km. This filter is based on the assumption that SCSs are always accompanied by lightning. Note that the filter only eliminates single spurious signals but keeps the tracks that are composed of numerous radar grid points.

In our analyses, we considered only storm tracks above a threshold of Z≥55 dBZ, referred to as the criterion for hail detection. Several studies have provided evidence that this lower threshold is suitable to identify hail in radar data e.g.,. However, high radar reflectivity does not guarantee that there is hail on the ground, mainly because of potential melting hailstones and the relation Z∼D6, where D is the hail size diameter. For example, the evaluation of radar-derived cell tracks with damage data from two insurance companies by has shown that the criterion provides a satisfactory probability of detection (62 % and 55 %) but also a high false-alarm rate (35 % and 40 %). This means that our SCS sample based on this criterion consists mainly of hailstorms but also includes some heavy-rain events (see Sect.  for the definition of the HS sample).

Each SCS event, defined as an entire track reconstructed by the tracking algorithm, contains the following parameters: center (latitude and longitude) of the track including date and time, mean angle, width, total length, and duration; the latter two quantities allow us to compute the storm motion vectors c required for SRH (cf. Eq. ). For further details on the tracking and the results, see the study by .

The frequency of both SCS and HS events shows a rather high spatial variability but also some larger contiguous spatial patterns. In general, their frequency is lowest near the coast and highest inland. Most pronounced is the large hotspot of SCS events southeast of the center of France near the Massif Central. Other hotspots of SCS and HS events can be found in southwestern Germany between the Black Forest and Swabian Jura or in the southeast near the Ore Mountains. Given a southwesterly flow direction usually predominant on hail-prone days in both France and Germany , most of these hotspots are located over and downstream of the mountain ranges. Over France, SCS tracks are much more frequent compared to Germany (Fig. ). By contrast, HS tracks are far more frequently detected over Germany due to more available reports. Nevertheless, Fig.  suggests a relationship between the two records: regions with an increased SCS frequency also show an increased HS frequency and vice versa.

Both HS and SCS events (the latter not shown) feature pronounced seasonal and diurnal cycles with a maximum in the afternoon in the warmest months of July and August. While the number of HSs is lowest in April and September and dominated by smaller-sized hail, the months of May to July are similar with the highest number of HS events of the diameter class D≥5 cm in June (Fig. a). A comparison of the 3 summer months shows that events with large hail are rarest in July. Reasons for this counterintuitive result might be a decrease in frontal events, which have low hail sizes on average (cf. Sect. ), or reduced reporting in this month due to summer vacations.

The diurnal cycle is much more pronounced than the seasonal cycle. The minimum number of HS events occurs in the early morning hours between 03:00 and 09:00 LT, and the maximum is in the afternoon between 15:00 and 18:00 LT (Fig. b). The largest increase occurs between 12:00 and 15:00 LT, and the largest decrease is after 21:00 LT. A total of 841 events, which correspond to 85.4 % of the HS sample, are registered in the period from 12:00 to 21:00 LT.

A separation of the diurnal cycle according to the hail diameter shows that during the first half of the day (00:00–12:00 LT), most events are associated with hail smaller than 5 cm. Especially from 03:00 to 09:00 LT, hailstones are the smallest of the entire day. This result, however, must be treated with care because of the low number of events in that period (26 events) in combination with the potential underreporting by spotters in the night. During noon and afternoon, the proportion of hail with a diameter of at least 5 cm increases, and the highest probability of occurrence is between 15:00 and 18:00 LT. In the evening and night (18:00–00:00 LT), the relative proportion of large hail remains almost constant.

The pronounced diurnal cycle of the HS probability (Fig. b) is closely linked to the warming of near-surface layers of air and the associated increase in lapse rate and CAPE together with a decrease in CIN . In addition, triggering mechanisms such as low-level flow convergence in the wake of thermally induced circulation over complex terrain or inhomogeneities in land cover are also connected to the diurnal temperature cycle. Studies using radar reflectivity or lightning detections found similar diurnal cycles for most of the area except for the Mediterranean e.g.,.

The investigation area is frequently affected by synoptic-scale cold fronts. The number of fronts per grid point of the size 1∘ × 1∘ during the 10-year investigation period ranges between 85 in eastern Germany and 175 near the Pyrenees (Fig. ). Overall, front density in France is larger than in Germany.

During their propagation, cold fronts tend to weaken over land mainly because of friction in the lowest layers and the horizontal mixing of air mass properties. Usually, they also dissolve when the air from the warm sector has entirely lifted (occlusion). As the largest fraction of fronts affecting central Europe propagates in eastern to southeastern directions, their detectable density gradually decreases in the same direction. In addition, an elevated front density can be found on the western and northern side (upstream) of large mountains such as the Pyrenees, Massif Central, and the Alps. These large mountain ranges tend to slow down the propagation of fronts, leading to an elevated frequency upstream when counting the time steps where a front prevails . Thus, slowly propagating fronts may be repeatedly detected and counted during the time steps of ERA-Interim (6 h). In contrast, fronts occur less frequently downstream of larger mountains as well as at a greater distance to the sea, where the increasing continentality acts to weaken or even dissolve the fronts.

To assess the role of synoptic cold fronts in the probability and properties of SCSs, we first discuss the spatial distribution of the ratio of frontal SCSs relative to all SCS events. This ratio is computed independently for each single grid point with a size of 0.5∘ × 0.5∘. Averaged over the entire area of Germany and over the 10-year study period, 18.9 % of all SCS tracks are related to a cold front; for France, the ratio is slightly higher with 22.4 % (Fig. ). The most conspicuous feature in the spatial distribution of the frontal streaks is the strong gradient in the south-to-north direction, particularly over Germany. For example, while in the German northeast (Mecklenburg Lake Plateau) the share of frontal SCSs reaches the highest value of 50 %, it decreases to less than 10 % in southern Germany over the Black Forest (southwestern Germany) and the region south of Nuremberg (southeastern Germany). Most striking in France is the extended maximum of the frontal share of around 45 % northeast of the domain's center and several minima with only a few percent near the coasts of both the North Atlantic and the Mediterranean.

If we compare the proportion of frontal SCSs both with the distribution of all SCS tracks (Fig. ) and with the frontal density (Fig. ), the opposite behavior is often observed. In several regions with an increased number of fronts and/or SCS events, the number of frontal SCSs is low and vice versa. This is especially true for Germany but also for parts of France. Over complex terrain such as in southwestern Germany (Black Forest) or southern France (Massif Central), where frontal SCSs are comparatively rare, it can be assumed that orographically induced vertical lifting is often sufficient to trigger convection so that a front is not necessary.

Considering HS instead of SCS events, we found that an even higher number, namely 25 % of all HS tracks across the entire study domain, are connected to a synoptic cold front. Because of the small number of HS track detections, especially in France (cf. Fig. ), we do not show this relation here. Note, however, that if only areas with a sufficient number of events are considered, the spatial distributions of frontal HS and SCS tracks are quite similar.

For the HS events, a relation is found between the length of the tracks as detected by the radar algorithm and the maximum observed hail diameter (Fig. a). While the mean diameter for a length of L<50 km is around 2 cm, it increases to around 3 cm for 50≥L<150 km and to 4 cm for L≥150 km. Furthermore, the distributions of both quantities, maximum diameters and track lengths, differ between frontal and non-frontal streaks. Mean diameters are 3.3 cm in for frontal events and 2.73 cm for the others (Fig. b, left). For hail size diameter classes of <2, 2–3.5, 4–5.5, and ≥6 cm, the ratio between frontal and non-frontal events is 16.7 %, 23.1 %, 35.8 %, and 34.7 %, respectively (not shown; note that the finer classes are used only in this example). This means that the higher the probability of a nearby front is, the larger the hailstone diameter is on average.

Differences between frontal and non-frontal HS events are also found for the length and mean propagation direction of the tracks. While frontal HS tracks have a mean length of 96.2 km (interquartile range of 40–125 km), non-frontal tracks are almost half shorter with 56.8 km (25–65 km; Fig. b, right part). Non-frontal HS events have a mean propagation angle of 215∘ (interquartile range 185–255∘), whereas those connected to a front propagate slightly more to the east with a direction of 232∘ (interquartile range 217–258∘; not shown). In that latter range of angles, also the largest hailstones can be observed.

Averaged over all classes of HS events, SLI around the center of the tracks has a mean value of -3.8 K (Fig. a), indicating a high potential for convective storms e.g.,. SLI has its absolute minimum about 140 km southeast of the events, but the difference to the center, on average of 0.2 K, is almost negligible. Overall, a significant increase in convection-favoring conditions can be observed from the northwest of the HS center to the southeast. While these conditions prevail over 400 km to the south and east of the center, the area to the north and west sees higher and positive values of SLI, thus stable conditions, at approximately 100–200 km distance already. The SLI field occurs rather smooth mainly because of the low resolution of ERA-Interim (cf. Sect. ).

The vertical wind shear (DLS) has its maximum about 250 km to the west of the HS centers in an upstream direction (Fig. b). This spatial difference is plausible because a trough frequently prevails to the west of the events. Since DLS is dominated by the wind speed aloft (500 hPa), a trough with an associated jet manifests itself by a maximum in DLS. Considering the magnitude of DLS, it is found that the values are quite low with a mean of 12.5 m s-1 around the HS events. Several authors have shown that organized convection capable of producing larger hail develops only in sheared environments above around 10 m s-1 e.g.,. This is one of the reasons to further subdivide the whole sample as mentioned above and shown in the next paragraph.

Separating the hail events according to their intensity allows for a detailed view of the prevailing environmental conditions. The SLI composites show a slight decrease (higher instability) around the center of the HS events from small hail with shorter tracks (SLI ≈ -3.7 K) to large hail with longer tracks (SLI ≈ -4.5 K; Fig. ). The strongest decrease in stability occurs for increasing hail diameter, while the composites are less sensitive to variations in track lengths. In all cases, the lowest instability prevails to the southeast of the hail events as was already found for the mean composite (cf. Fig. ). Despite favorable environments for SCSs, which predominate all classes, the highest instability in the case of larger hail is an indicator of higher updraft speed within the thunderstorm cloud, which is a prerequisite for the growth to large hailstones.

The distance between the location of the events and the location of the highest instability is greater for longer tracks than for shorter ones but only in the case of small- to medium-sized hail. At this point one may speculate that the reason for this shift might be related to the role of cold fronts, considering that longer tracks and larger hailstones are more often connected to a cold front as discussed in the previous section (cf. Fig. ). The role of cold fronts vs. environmental conditions will be investigated in the next section.

In contrast to the thermodynamical proxy SLI, the dynamical quantity DLS shows significantly pronounced differences between the nine HS categories (Fig. ). Even though DLS also distinguishes between the diameter classes, the largest differences are found for the three length classes. For example, DLS has a mean value of 17 m s-1 for long tracks in the smallest diameter class (D<3 cm), which is almost twice as high compared to short tracks with the same diameter class (8.5 m s-1; Fig. , upper row). The same applies to the other diameter classes. For long tracks with large hail, DLS reaches values of about 20 m s-1 and is thus in the range of the values given in the literature e.g.,. The area of the highest DLS values is located several hundred kilometers to the west of the HS events on average. For large hail, the DLS maxima are even higher and further away from the HS events. These events are usually triggered by upper-level troughs to the west, associated with higher wind speed at mid-troposphere levels. One may argue that a relationship between DLS and track length prevails per se, since both are dominated by the wind speed aloft. Note, however, that the separation of DLS applies not only to track length but also to storm duration not shown here, but see.

In addition to DLS, SRH has been suggested by several authors e.g., to be an important proxy not only for the prediction of tornadoes but also for large hail. In our composite analyses, SRH (Fig. ) shows even more pronounced differences between the nine HS categories compared to DLS. Hail events with shorter tracks on average are in a range between 0 and 50 m2 s-2. By contrast, longer tracks have much higher mean values between 84 and 116 m2 s-2. According to the investigations of proximity soundings by , such environments favor the development of weakly tornadic and nontornadic supercells – provided that sufficient CAPE is present. In addition, there is also an increase in SRH from small to large hail, which is weaker compared to the trend in the length classes. Interestingly, the highest SRH values occur directly at or near the location of the hail event and not on the upstream side as was the case for DLS.

To further investigate which of the dynamical parameters, SRH or DLS, best distinguishes the HS intensity, we consider only the two categories that correspond to the highest and lowest damage potentials: smaller hail with D<3 cm combined with short track length of L<50 km and large hail with D≥5 cm combined with longer tracks of more than 100 km (high-intensity events). Environmental parameters are computed by the mean of the 3×3 ERA-Interim grid points centered around the HS locations.

Overall, the scatterplots presented in Fig.  show a much clearer separation between the events when SRH is considered (Fig. a) instead of DLS (Fig. b). About 50 % of the high-intensity events have values of 100 m2 s-2 or greater for SRH, while only 3 % of the low-intensity events display these values. Furthermore, most of the latter events have values between -50 and 50 m2 s-2. It can also be seen that SLI for all events in these two categories varies between 0 and -10 K, with only a few exceptions having positive values. Approximately 70 % of the high-intensity events have values of -2.5 K or less. Unlike DLS (Fig. b), splitting the events into two different categories is not possible. Even if most of the high-intensity events form in an environment with DLS of at least 15 m s-1 (approx. 60 % of these events), there are still many low-intensity events for larger DLS values.

Subgrid-scale spatial variations of the environmental conditions, for example, as a result of diabatic heating or temperature and moisture advection , cannot be expected to be reproduced by the coarse ERA-Interim reanalysis. For this reason, we additionally considered the high-resolution coastDat-3 reanalysis. Due to the hourly resolved model fields, the maximum time difference between the HS events and the environments is 30 min. The purpose is not to reproduce the above analyses but to investigate exemplarily the influence of the model resolution on the results. Since SLI and SRH are not available or quantifiable from coastDat-3, we used LR as a thermodynamical proxy and DLS as a dynamical proxy (cf. Sect. ). Because the two proxies do not show significant differences between the nine intensity categories (cf. Figs.  to ), we discuss only the most severe HS category with L≥100 km and D≥5 cm.

As shown in Fig. , the higher model resolution (right column) has little influence on the spatial distribution of the environmental parameters even though coastDat-3 composites show a much larger spatial variability compared to ERA-Interim. In the case of LR, the maximum is located to the southwest; in the case of DLS, it is located northeast of the HS events as was already found in the above analyses. Also the distance between the maxima and the events remains almost the same. The coastDat-3 values around the maxima show a slight increase of approximately 10 % for both parameters. In the vicinity of the HS centers, the increase is only marginal but larger for LR compared to DLS. In particular the LR increase is a consequence of the higher temporal resolution of coastDat-3 leading to an improved representation of the diurnal temperature and moisture cycles. Note that this finding does not only apply to LR but also to other thermodynamic quantities such as the precipitable water (not shown).

As already discussed in Sect. , the characteristics of HS tracks having a front nearby substantially differ from non-frontal events, especially with regard to the maximum hail size and the track lengths (cf. Fig. ). This suggests that prevailing environmental conditions may likewise differ for the two kinds of events. Therefore, we further subdivided the HS sample into frontal and non-frontal types. To ensure that enough events enter the subsamples, we made a further separation by considering only two length classes (L<75 and ≥75 km) and two diameter classes (D<3 and ≥3 cm; the former not shown).

Whereas the mean SLI composites are almost similar for frontal and non-frontal events (not shown), DLS shows significant differences between the four classes (Fig. ). Overall, DLS reaches higher values with larger gradients for frontal compared to non-frontal events (Fig. , panels a and c vs. b and d). However, when considering additionally the track lengths, much larger differences in DLS can be found, but only for non-frontal events (Fig. b and d). While short non-frontal tracks form at a DLS value of 10.9 m s-1 on average, long tracks require medium-sheared environments, here with values of 15.9 m s-1. A similar result is obtained for small hail sizes (D<3 cm) with DLS even rising from 9.0 to 16.7 m s-1 (not shown). Furthermore, while the DLS maximum for non-frontal events is located to the west of the center, it is more northwest for frontal events at a distance of about 200 km. Since almost all synoptic fronts in Europe propagate in a west–east direction, this location is a clear indication that frontal HS events preferably develop in prefrontal environments (and not postfrontal).

It is well-known that supercells due to specific conditions, such as a strong and spatially extended updraft, high amounts of supercooled liquid water, or their longevity, are capable to produce the largest hailstones . The propagation of these highly organized convective systems can substantially deviate from the horizontal wind at mid-tropospheric levels mainly because of the dynamics of the cold pools and induced vertical pressure deviations .

In the last step, therefore, we want to investigate whether our samples show a relation between the storm motion relative to the mean wind and the hail size. The storm motion vector c follows from the radar tracking of the individual HS events; the wind direction is estimated from the 500 hPa mean wind from ERA-Interim (3×3 grid point around the HS centers). The cell-tracking algorithm (cf. Sect. ) yields very reliable shift vectors of individual hailstorms. The wind field in 500 hPa, on the other hand, is mainly determined by the setting of the synoptic systems and only marginally affected by local-scale flow deviations. Positive differences in the analyses indicate right-moving storms; negative values indicate left-moving storms.

Most of the events with smaller hail (D<3 cm) propagate approximately parallel to the wind vectors in 500 hPa; the mean difference between the tracks and the wind vectors is only 8∘ (Fig. a). About 13 % of all HS events have a deviation between 30 and 60∘ to the right, while only 6 % of the events show deviations to the left for this interval (-30 to -60∘). Hail events with maximum diameters between 3 and 4.5 cm show a deviation of the propagation direction preferably to the right of the wind vectors (Fig. b); 23 % of all HS events propagate with the wind in 500 hPa (decreasing by 8 % compared to small hail), while 38 % of the tracks show a deviation between 10 and 30∘.

HS events of the largest hail class not only show an increased spread of the propagation deviation but also the entire histogram is shifted to more right-moving storms (median of 17∘; Fig. c). An angle difference between 10 and 30∘ is observed for 35 % of all events. The largest difference to the other hail size classes is the comparatively high number of HS events between 30 and 60∘ (21 %). In contrast, 27 % of the events propagate with the wind in 500 hPa, and only 10 % have a negative deviation to the left of the wind in 500 hPa. In summary, the larger the hailstone diameters are, the stronger the deviation of the cell's propagation direction from the flow at 500 hPa is.