1. Introduction

2. Data

Figure 1. Annual mean distribution of lightning (a, d), hail ≥ 2 cm (b, e) and hail ≥ 5 cm (c, f), Across Europe, the annual mean refers to the period 2008–2020 while across the U.S. to 2010–2020. The black squares highlight the training regions for the different hazard models: lightning Europe (34.5° – 63.5°N, -9.0° – 46.0°W), lightning U.S. (29.0° – 41.5°N, -109.0° – -79.0°W), hail Europe (45.0° – 54.0°N, 5.0° – 22.0°W), hail U.S. (30.5° – 41.5°N, -105.0° – -82.0°W).

Figure 1. Annual mean distribution of lightning (a, d), hail ≥ 2 cm (b, e) and hail ≥ 5 cm (c, f), Across Europe, the annual mean refers to the period 2008–2020 while across the U.S. to 2010–2020. The black squares highlight the training regions for the different hazard models: lightning Europe (34.5° – 63.5°N, -9.0° – 46.0°W), lightning U.S. (29.0° – 41.5°N, -109.0° – -79.0°W), hail Europe (45.0° – 54.0°N, 5.0° – 22.0°W), hail U.S. (30.5° – 41.5°N, -105.0° – -82.0°W).

2. Model development

a. Model setup

We developed Additive Logistic Regression Models that, once trained with lightning observations and hail reports, yield a probability of hail (or lightning) as a function of the values that each of the predictor parameters from the atmospheric reanalysis take at any grid point at a given time. We followed Rädler et al. 2019 who named the resulting models Additive Regression Convective Hazard Models (AR-CHaMo). The probability of hail ≥ 2 cm P_hail$\ge$_2cm at a given time and place is computed as the product of the probability of a thunderstorm occurring P_lightning and the conditional probability of hail given a storm P_hail$\ge 2cm$_|lightning:

P_hail≥2cm = P_lightning ∙ P_{hail≥2cm|lightning}

To sample the pre-convective environment, the models were developed using ERA5 conditions sampled one hour before the lightning and hail reports occurred, as done by Taszarek et al. (2020b). Compared to Rädler et al. (2019), the model training regions were significantly extended. This, summed with the high spatio-temporal resolution of the ERA5 reanalysis, resulted in a high number of data points being sampled to train the logistic models, more than a billion for the Europe lightning model (Table 1).

Table 1. Number of data points, events, and ratio of events to all data points for lightning, hail ≥ 2 cm and hail ≥ 5 cm models. The number of hail ≥ 2 cm and hail ≥ 5 cm events occurring in presence with lightning is shown in brackets.

Table 1. Number of data points, events, and ratio of events to all data points for lightning, hail ≥ 2 cm and hail ≥ 5 cm models. The number of hail ≥ 2 cm and hail ≥ 5 cm events occurring in presence with lightning is shown in brackets.

	Number of data points		Number of events		Ratio
	Europe	U.S.	Europe	U.S.	Europe	U.S.
Lightning	1 607727264	882965952	10872890	13088628	0.68 %	1.48 %
Hail ≥ 2 cm*	1710071	5678261	5879 (5493)	39619 (38516)	0.32 %	0.10 %
Hail ≥ 5 cm*	1710071	5678261	848 (805)	3789 (3723)	0.05 %	0.07%

* Only reports in the presence of lightning within 1 hour and 50 km (in brackets) were considered within the training dataset.

c. Regional differences in instability-shear parameter performance for conditional hail models

The Europe and the U.S. models were developed under the hypothesis that convective storms require the same ingredients regardless of the geographical region where they form, as they are controlled by the same physical processes. This, however, does not mean that predictor parameters are equally skillful in every region or subregion. For example, if a lack of wind shear is the limiting factor for hailstorm development in one region, wind shear will be a good discriminator between situations with and without hail there, while in another region where wind shear is ubiquitous, but a lack of instability limits hailstorm development, it may be a poor predictor. While developing the conditional hail models, we found the skill of predictors that represent instability to differ substantially geographically (Table 2). The difficulty to find a skillful predictor for both Europe and the different regions of the U.S. is supported by findings from Zhou et al. (2021) who showed that the conditions supportive for hailstorms are highly geographically dependent. While in Europe CAPE variants are most skillful, mid-level temperature lapse rates discriminate better between hail and non-hail situations in the U.S. CAPE is a particularly poor predictor of hail in the U.S. Southeast (Figure 2b). Here, sizable CAPE is common, but often occurs with modest vertical lapse rates and with a moist lower troposphere that promotes high condensate loading. This combination negatively affects the probability of hail, since relatively weak convective updrafts with modest parcel buoyancy occur under such circumstances according to cloud model simulations (Storer and van den Heever 2014; Kirkpatrick et al. 2009).

Table 2. Deviance explained (higher values are better) for a selection of instability and shear candidate predictor parameters for large (≥ 2 cm) hail across Central Europe, the U.S., and two U.S. subregions, the Southeast and th e Southern Plains. The acronyms are explained in the text and in Table A1, Table A2 which include a list of all tested predictor parameters for the lightning, hail ≥ 2 cm and hail ≥ 5 cm models.

Table 2. Deviance explained (higher values are better) for a selection of instability and shear candidate predictor parameters for large (≥ 2 cm) hail across Central Europe, the U.S., and two U.S. subregions, the Southeast and th e Southern Plains. The acronyms are explained in the text and in Table A1, Table A2 which include a list of all tested predictor parameters for the lightning, hail ≥ 2 cm and hail ≥ 5 cm models.

	Central Europe	U.S.	U.S. Southeast	U.S Southern Plains
	Deviance explained (%)
MU_CAPE	6.38	4.54	2.02	7.35
ML_CAPE	6.26	3.25	0.80	5.67
SB_CAPE	6.39	2.96	0.60	5.97
MU_LI	6.11	3.91	1.98	5.87
MU_CAPE_HGL	5.99	2.77	0.90	4.76
MU500_CAPE	6.26	5.94	3.45	7.69
MU_CAPE-10°	7.34	7.01	6.04	9.15
MU500_CAPE-10°	7.43	7.28	6.35	9.95
LR_3-6km	1.50	6.86	6.17	5.57
BS_0-6km	1.39	4.10	4.12	3.47
EFF_MU_BS	4.81	6.84	5.49	6.53

Figure 2. Training region for hail ≥ 2 cm and hail ≥ 5 cm for (a) Europe and (b) the U.S. Across the U.S. the hail models were tested also across two sub-regions: the Southern Plains (30.00° – 38.00°N, 94.25° – 102.50° W) and the Southeast (29.25° – 36.00°N, 83.00° – 93.25° W).

Figure 2. Training region for hail ≥ 2 cm and hail ≥ 5 cm for (a) Europe and (b) the U.S. Across the U.S. the hail models were tested also across two sub-regions: the Southern Plains (30.00° – 38.00°N, 94.25° – 102.50° W) and the Southeast (29.25° – 36.00°N, 83.00° – 93.25° W).

We investigated variants of CAPE to find a better predictor for large hail. First, CAPE for the most-unstable parcel (defined as the one with the highest theta-e value in the 0–3 km AGL layer) originating above 500 m AGL (MU500_CAPE) was tested. This was done to eliminate the possibility of a very shallow near-surface layer, unrepresentative of the storm inflow, to be selected as source of the lifted parcel. Although this variant outperformed standard MU_CAPE, across the U.S. Southeast it still had considerably lower skill than other instability-related parameters such as the Lapse Rates between 3 and 6 km (LR_3–6km) AGL. Possibly, large hail in the Southeast is strongly tied to the presence of an Elevated Mixed Layer (EML) as this results in a lower fraction of the CAPE being released in the bottom portion of the updraft - which Lin and Kumjian (2022) show to favor large hail growth. Next, we tested parameters that consider the amount of CAPE in the cold, upper, portion of the storm. CAPE released in the hail growth layer (CAPE_HGL; between the -10°C and -30°C isotherms; Knight and Knight 2005) was tested but yielded a lower predictive skill than MU_CAPE (Table 1). MU_CAPE in the layer above the -10 °C isotherm (MU_CAPE-10°), more specifically the variant computed for the most unstable parcel above 500 m (MU500_CAPE-10°), stood out as the best instability predictor across Europe, the U.S., and its subregions. This universal skill in both continents suggests that it may better represent what is relevant for hail formation, i.e., large buoyancy high in the storm.

A similar comparison of the skill of shear-related parameters was carried out. In contrast to instability parameters, little variation in skill was found between different geographical regions. Overall, the Effective Most Unstable Shear (EFF_MU_BS) was the best shear parameter, outperforming bulk wind shear between 0 and 6 km AGL (BS_0–6 km). EFF_MU_BS is the bulk wind shear between the height of the most unstable parcel origin and halfway to its Equilibrium Level, similar to Thompson et al. (2007) but without assuming an effective-inflow layer.

d. Final models and their validation

The predictor parameters for the lightning, hail ≥ 2 cm and hail ≥ 5 cm models are shown in Table 3.

Table 3. Selected predictor parameters for the regression models. The acronyms are explained in the text.

Table 3. Selected predictor parameters for the regression models. The acronyms are explained in the text.

Model	Lightning	Hail ≥ 2 cm	Hail ≥ 5 cm
Predictors	MU_LI (K) RH_500–850hPa (%) 1h Acc. Conv. Precip. (kg m-2) MU MIXR (g kg-1) Land-sea Mask	MU500_CAPE-10° (J kg-1) EFF_MU_BS (m s-1) ML_MIXR (g kg-1) 0° height (m)	MU500_CAPE-10° (J kg-1) EFF_MU_BS (m s-1) ML_MIXR (g kg-1) ML_LCL height (m)

The base 2-dimensional model for lightning was developed using the Most Unstable Lifted Index (MU_LI) defined as by Galway (1956) and the average relative humidity between 500 and 850 hPa (RH_500–850hPa). MU_LI and RH_500–850hPa represent two crucial ingredients for convective initiation: instability and mid-level moisture, as shown by Rädler et al. (2019) using ERA-Interim. The model with CAPE, instead of MU_LI, performed worse (deviance explained 28.0% against 30.1% of the MU_LI model) and was found to underestimate lightning across mountainous regions whereas the MU_LI model suffers less from this bias (e.g., across the Atlas Mountains and the Spanish Plateau in Europe). While CAPE by itself can only quantify positive buoyancy, LI is a continuous quantifier regardless of stability (Púčik et al. 2017; Pilguj et al. 2022). This may make LI more suitable in a logistic model which needs to represent accurately whether low amounts of instability are or are not sufficient to support lightning.

Convective precipitation added the most skill (deviance explained: 32.5%, BIC 3.05 ∙ 10⁶) to this initial 2-dimensional model (deviance explained: 30.1%, BIC 3.16 ∙ 10⁶), arguably because it distinguishes situations with convective inhibition despite sufficient instability and moisture. It is a common parameter in modelling studies using reanalyses and climate projections (Trapp et al. 2009; Tippett et al. 2012; Romps et al. 2014; Allen and Tippett 2015; Púčik et al. 2017; Tippett et al. 2019; Taszarek et al. 2021a). The fourth parameter is mixing ratio for the most unstable parcel (MU_MIXR). The probability of lightning is bimodally distributed as a function of MU_MIXR (not shown): given sufficient instability and mid-level moisture, lightning probability increases with increasing MU_MIXR up to 12–15 g/kg, depending on the amount of instability (higher for more unstable environments), and then decreases again. Possibly, the probability of storm electrification is suppressed by excessive water loading in very moist environments. The inclusion of MU_MIXR reduced the frequency of lightning across the Gulf of Mexico (not shown). Finally, a land-sea-mask was added to reflect the observation that lightning is less common over the sea than over land, which may be attributed to other parameters not explored here, e.g., aerosol load or LCL height (Williams and Stanfill 2002; Romps et al. 2018).

The hail ≥ 2 cm and hail ≥ 5 cm model each have four predictors and share three of them, i.e., MU500_CAPE-10°, EFF_MU_BS, and ML_MIXR. The fourth predictor giving most additional skill differs between the two models. For hail ≥ 2 cm, it is the height of the 0° isotherm while, for hail ≥ 5 cm, it is the height of the Lifting Condensation Level for the mixed layer (0–500 m AGL) parcel (ML_LCL). ML_LCL lends the additional skill to the model for hail ≥ 5 cm because for the same amount of instability, high cloud bases are associated with stronger and wider updrafts which favor the development of very large hailstones (McCaul and Weisman 2001, Mulholland et al. 2021). On the other hand, for smaller hailstones, which are more sensitive to melting (Rasmussen and Heymsfield 1987; Kumjian et al. 2019), the height of the wet-bulb zero degree level is more important.

The performance of the models can be quantified by the Area Under the ROC Curve (AUC) score. The lightning (EU: 0.943 and U.S: 0.931), hail ≥ 2 cm (EU: 0.983, U.S: 0.964) and hail ≥ 5 cm (EU: 0.989, U.S: 0.979) models are all highly skillful. A further investigation of the robustness of the models was carried out by training the models on even year data across each region and, subsequently, generating predictions for odd years. The performance of the models was similar to the full year models (lightning EU: 0.939, lightning U.S: 0.926; hail ≥ 2 cm EU: 0.977, hail ≥ 2 cm US: 0.961; hail ≥ 5 cm EU: 0.981, hail ≥ 5 cm U.S: 0.973). These findings suggest that the AR-CHaMo models are robust. The performance of the 4-dimensional conditional hail models was also compared with that of a logistic model based on SHP (Table 4). The AR-CHaMo hail outperformed SHP for both hail sizes and regions showing that it discriminates hail and non-hail environments better than SHP.

Table 4. AUC scores for hail ≥ 2 cm and hail ≥ 5 cm models given convective initiation. The performance of AR-CHaMo is compared with that of a logistic model based on SHP.

Table 4. AUC scores for hail ≥ 2 cm and hail ≥ 5 cm models given convective initiation. The performance of AR-CHaMo is compared with that of a logistic model based on SHP.

	Europe		U.S.
	AR-CHaMo	SHP	AR-CHaMo	SHP
Hail ≥ 2 cm	0.778	0.764	0.764	0.739
Hail ≥ 5 cm	0.894	0.865	0.878	0.819

4. Modelled lightning and its evolution since 1950

a. Mean distribution

Modelled annual and seasonal lightning distributions from AR-CHaMo are compared to lightning observations across Europe and the U.S. by constraining the modelled data to the period for which lightning observations are available, i.e., 2008-2020 (2010-2020) across Europe (U.S.).

In Europe, the annual mean modelled distribution reflects the observed spatial patterns well (Figure 3). Both the model and the observations highlight maxima across mountainous areas, especially the Alps and the Caucasus. Although coastal areas also stand out as local maxima in the model output, lightning is underestimated across the Balkans and the southern Mediterranean coasts (especially Turkey). The seasonal cycles match well: in winter, the occurrence of convective storms is limited to the south-eastern Mediterranean. During spring, the storm activity picks up over the continent, more rapidly across Eastern Europe and Turkey where a seasonal maximum is present. Lightning is most common during the summer over the continent where a band of lightning activity stretches from the Pyrenees across Central Europe to Russia. Although the model accurately reproduces such spatial feature, it underestimates the activity across parts of south-eastern Europe. Finally, in autumn the peak in the activity shifts to the relatively warm waters of the southern Mediterranean and the surrounding coastal regions where the model underestimates the activity compared to the observations.

Figure 3. Mean annual and seasonal number of hours with lightning from model and observations. Data refers to the period 2008-2020 across Europe and to the period 2010-2020 across the U.S.

Figure 3. Mean annual and seasonal number of hours with lightning from model and observations. Data refers to the period 2008-2020 across Europe and to the period 2010-2020 across the U.S.

Across the U.S., a maximum in lightning occurrence is modelled across southern Louisiana and Florida, part of a large region with high thunderstorm activity that extends from the Southeast U.S. into the Central Plains. The spatial distribution of lightning is less influenced by orography than in Europe but does play a role in the Southwest, where local maxima correspond to mountain ranges. Along the Gulf of Mexico coast, a sharp land-sea contrast exists with less lightning over sea than over the surrounding land. The model underestimates lightning across the Southeast, Florida, and the mountains of the Southwest. The modelled seasonal cycle reflects the observed one rather well: in winter, the occurrence of convective storms is limited to the Southeast while in Spring it peaks over the southern Great Plains. The occurrence is underestimated during summer when the observed occurrence peaks higher over Florida and the Southwest. A decrease in occurrence is modelled and observed during Autumn.

The modelled and observed seasonal cycle can be compared quantitatively by averaging over the training region (Figure 4). In Europe, the number of hours of lightning matches the observations almost perfectly in all months apart from May, June, and July when the model slightly underestimates the lightning activity (Figure 4a). In the U.S., lightning activity is somewhat underestimated by the model throughout the year (Figure 4d), but the seasonal cycle is modelled well quantitatively.

Figure 4. Comparison between modelled and observed seasonal cycles for lightning (a,d), hail ≥ 2 cm (b,e), hail ≥ 5 cm (c,f) across Europe and U.S. training regions.

Figure 4. Comparison between modelled and observed seasonal cycles for lightning (a,d), hail ≥ 2 cm (b,e), hail ≥ 5 cm (c,f) across Europe and U.S. training regions.

b. Long-term trends

AR-CHaMo suggests that the occurrence of thunderstorm environments has significantly increased across most of Europe during the past 72 years (Figure 5). The strongest absolute upward trends are found over the Alpine and Caucasus Mountain ranges with an increase up to 5 hours of lightning more per decade. Scandinavia stands out as the region with the largest relative increases with 2 more hours per decade while the annual mean is 20–25 hours. These increases in lightning occur throughout the year, but especially in the summer. In a belt from Finland to Turkey, small, statistically insignificant changes are modelled. Across parts of Russia, AR-CHaMo indicates a significant decrease in thunderstorms. Some significant trends are also observed in areas with very low absolute values, which are displayed in white. These findings are consistent with Taszarek et al. (2021a), however, their analysis was limited to the period 1979–2021. For direct comparison, figures for the same period are included in Appendix B.

Figure 5. Annual and seasonal trends in lightning hours across Europe and the U.S. between 1950 and 2021. Values are expressed as the change in the number of hours per decade. Trends significant at a 95th percentile level are hatched.

Figure 5. Annual and seasonal trends in lightning hours across Europe and the U.S. between 1950 and 2021. Values are expressed as the change in the number of hours per decade. Trends significant at a 95th percentile level are hatched.

The strongest positive trends in the U.S. occur in the southern states, specifically Florida and the Texas-Louisiana coasts. Although not as strong, upward trends can also be found in the Midwest and across southern Canada, mostly during summer. Conversely, a region of significant negative trends is located across the Colorado Plateau and the Great Basin.

When comparing the trends in Figure 5 with those found by Taszarek et al. (2021b), there are a number of similarities and differences. We do not find any strong negative trends across the southern Great Plains. There are two possible reasons for this: the difference in the definition of favorable thunderstorm environments and the difference of period for which the trends were calculated with Taszarek et al. (2021b)’s analysis starting in 1979. Indeed, when constraining the trend analysis with AR-CHaMo to the period 1979–2021 (Appendix B) similar patterns to Taszarek et al. (2021b) result, i.e., negative trends across most of the southern Great Plains and the Southwest. Nonetheless, some differences between the two approaches remain, especially across the Gulf Coast, probably because of the different lightning proxies used, i.e., AR-CHaMo versus those of Taszarek et al. (2021b).

5. Modelled hail ≥ 2 cm and hail ≥ 5 cm and their evolution since 1950

a. Europe

Hail ≥ 2 cm is most frequent in the Po Valley in northern Italy, averaging to around one event per year (per grid point) while south-western France, eastern and northern Spain also exhibit a relatively high frequency of occurrence (Figure 6). The modelled spatial distribution in Italy accurately reflects that of observed data with a maximum in the northeast (Figure 1b, c), but the maxima in France and Spain are not confirmed by ESWD reports, likely because of the lower reporting rate in those countries (Groenemeijer et al. 2017; Taszarek et al. 2020a). Nonetheless, the spatial distribution across these regions corresponds with that of local hail climatologies based on hail pad network data (Vinet 2001; Merino et al. 2014). Here, the model correctly simulates hail maxima outside of the region where it was trained.

Figure 6. Mean modelled annual and seasonal number of hail hours (hail ≥ 2 cm and hail ≥ 5 cm) between 1950 and 2021 across Europe. Annual and seasonal trends are also shown and expressed as the change in the number of hail hours per decade. Trends significant at a 95th percentile level are hatched.

Figure 6. Mean modelled annual and seasonal number of hail hours (hail ≥ 2 cm and hail ≥ 5 cm) between 1950 and 2021 across Europe. Annual and seasonal trends are also shown and expressed as the change in the number of hail hours per decade. Trends significant at a 95th percentile level are hatched.

Compared to the earlier hail hazard maps from Rädler et al. (2019) based on ERA-Interim, local patterns are much better resolved, especially near complex orography, such as the local maxima across Switzerland and Southern Germany, where the model output is consistent with high resolution climatologies based on radar and observed data (Junghänel et al. 2016; Schrooer et al. 2019). The Atlas region in northern Africa has the highest modelled hail occurrence, which is consistent with the climatology from Punge et al. (2017) that was based on overshooting top detections. The lack of ground-truth observations from this region makes it impossible to verify this prediction but reports of hail ≥ 5 cm are known from this region. For instance, gargantuan hail (Kumjian et al. 2020) with a verified diameter of 17 cm occurred in Libya on the 27th of October 2020 and was reported to the ESWD.

The local maxima of the distribution of hail ≥ 5 cm closely resembles that of hail ≥ 2 cm. Northern Italy and eastern Spain have the highest occurrence in Europe with approximately 1 event every 10 years (per grid box) while northern Africa exhibits the highest frequency overall with about 1 event every five years. Furthermore, northern Africa has the highest ratio of hail ≥ 5 cm to hail ≥ 2 cm environments: one in five hail ≥ 2 cm events involves hailstones with a maximum dimension $\ge$5 cm. This is likely because the environmental conditions that characterise the region, namely steep lapse rates and high LCL favor the genesis of large hailstones (Púčik et al. 2015; Taszarek et al. 2020b; Mulholland et al. 2021).

Hail ≥ 2 cm in Europe occurs mostly in summer, except for parts of the Mediterranean region, such as Turkey, where it is more common in Spring. Although a direct pan-European comparison with observations (Figure 1b, c) is not possible, the modelled climatological spatial distribution is in good agreement with previous studies such as that from Punge et al. (2017), Rädler et al. (2019) and various regional and national climatologies. The modelled seasonal cycles for hail ≥ 2 cm and hail $\ge$5 cm were compared with the observation ones across Central Europe (Figure 4b, c). The model agrees with the observations well especially during Spring and the early summer while a tendency to overestimate the activity is present in August, especially for hail ≥ 5 cm.

The strongest upward and significant trends occur across the Po Valley for both hail ≥ 2 cm and hail ≥ 5 cm, especially in summer, which is in line with Rädler et al. (2019) and Taszarek et al. (2021a). Increases of 0.05–0.08 hours per decade per grid box are modelled, corresponding to a relative increase of approximately 8% per decade. Smaller significant increases are found across France, the Benelux and northern Germany, in Switzerland and Austria, across the north-western Balkans into southern Poland and, further east, across Northeast Turkey and Georgia. Finally, significant positive trends are found in northern Africa during spring and autumn while a decrease in hail activity is modelled in summer. Negative and significant trends are present across the Aegean Sea and parts of western Russia. The trend maps of hail ≥ 2 cm and hail ≥ 5 cm are very similar. The results resemble those of Rädler et al. (2019) in that both studies agree on identifying northern Italy as the region with the strongest positive/significant trends in the occurrence of hail ≥ 2 cm and hail ≥ 5 cm. The largest significant increases of favorable severe thunderstorm environments over northern Italy were also found in Taszarek et al. (2021a).

b. United States

Hail ≥ 2 cm and hail ≥ 5 cm occur more frequently in the U.S. than Europe. A high frequency of occurrence (≥ 2 hours per year) is modelled in the Plains across a region extending from southern Texas northward into Nebraska (Figure 7). While modelled hailfall is less frequent further east, it is still relatively common in states like Wisconsin, Illinois, or Louisiana with an average of 1 hour of hail per year, which is comparable to northern Italy. Compared to hail ≥ 2 cm, hail ≥ 5 cm is more confined to the lee of the Rockies, a feature confirmed by hail observations (Figure 1e, f). For both hail size categories, the maximum in occurrence is in the high Great Plains between south-western Nebraska, north-eastern Colorado, and north-western Kansas. The location of the maximum in occurrence corresponds with that in the MRMS-MESH based climatology from Wendt and Jirak (2021). Remarkable local features in the hail occurrence pattern occur across eastern Colorado where a marked south to north gradient is modelled or around Rapid City in South Dakota, where a local maximum in occurrence can be distinguished (Figure 7). Both features also exist in hail observations (Figure 1e, f).

Figure 7. As for Figure 6, but for the U.S. domain.

Figure 7. As for Figure 6, but for the U.S. domain.

Unlike Europe, where most of the hail activity is confined to the summer, hail ≥ 2 cm and hail ≥ 5 cm also occur frequently in spring in the U.S. The model reproduces the observed peak in May followed by a rapid decrease in summer (Figure 4). In spring, hail ≥ 2 cm and hail ≥ 5 cm are most common across the southern Great Plains, in particular in Northeast Mexico, Texas and Oklahoma. During summer, hail activity shifts north to the High Plains into southern Canada (specifically around southern Alberta and Saskatchewan) with a maximum across Western Nebraska and South Dakota. A relatively high occurrence is also found in Northwest Mexico across the Western Sierra Madre Mountains, but this is mostly limited to hail ≤ 5 cm.

In the U.S., there are no large areas with positive (significant) trends comparable to those found in Europe. For hail ≥ 2 cm, modest positive statistically significant trends are present in summer across northern Colorado, southern Florida, and southern Canada with the latter two exhibiting the largest relative increases. Weak increases also occurred over western Texas in winter and north-eastern Colorado in spring. A decrease of hail ≥ 2 cm is modelled across the Southeast as also shown by Brimelow et al. (2017), the upper Midwest, the Colorado Plateau, and the Great Basin. In contrast to hail ≥ 2 cm, hail ≥ 5 cm did not decrease significantly across the Southeast and the upper Midwest. These findings contrast with results from Taszarek et al. (2021b) who identified strong positive and significant trends in 99.9th percentile of SHP and the Supercell Composite Index (SCP) and Tang et al. (2019) who find increases in both the LHP and SHP across much of the Central U.S. since 1979. When constraining the analysis to the period starting in 1979, the modelled trends display a more similar pattern to those studies based on SHIP, LHP and SCP (Appendix B), but the match is not perfect, likely because these parameters, do not take storm initiation into account which AR-CHaMo suggest has become less likely across the Rocky Mountains and the southern High Plains (Figure A2).

6. Evolution of hail and lightning in two hail-prone regions: northern Italy and Oklahoma

Figure 8. Modelled timeseries of lightning (c,d), hail ≥ 2 cm (e,f), hail ≥ 5 cm (g,h) across (a) northern Italy (44.5°N – 46.5°N , 7.25°E – 12.25°E) and (b) Central Oklahoma (34°N – 37°N, 260°E – 263.5°E) between 1950 and 2021. Colors indicate the area-averaged percentual departure of each hazard from the long-term (1950-2021) average.

Figure 8. Modelled timeseries of lightning (c,d), hail ≥ 2 cm (e,f), hail ≥ 5 cm (g,h) across (a) northern Italy (44.5°N – 46.5°N , 7.25°E – 12.25°E) and (b) Central Oklahoma (34°N – 37°N, 260°E – 263.5°E) between 1950 and 2021. Colors indicate the area-averaged percentual departure of each hazard from the long-term (1950-2021) average.

Figure 9. Comparison of modelled (blue) and observed (orange) sum of lightning hours per year across northern Italy (44.5°N – 46.5°N, 7.25°E – 12.25°E).

Figure 9. Comparison of modelled (blue) and observed (orange) sum of lightning hours per year across northern Italy (44.5°N – 46.5°N, 7.25°E – 12.25°E).

Figure 10. Seven year moving-average timeseries of modelled lightning (a,d), hail ≥ 2 cm (b,e), hail ≥ 5 cm occurrence (c,f), AMO Index (g) and PDO Index (h). Hazard time series are shown for (a, b, c) northern Italy (44.5°N – 46.5°N , 7.25°E – 12.25°E) and (d, e, f) Central Oklahoma (34°N – 37°N, 260°E – 263.5°E).

Figure 10. Seven year moving-average timeseries of modelled lightning (a,d), hail ≥ 2 cm (b,e), hail ≥ 5 cm occurrence (c,f), AMO Index (g) and PDO Index (h). Hazard time series are shown for (a, b, c) northern Italy (44.5°N – 46.5°N , 7.25°E – 12.25°E) and (d, e, f) Central Oklahoma (34°N – 37°N, 260°E – 263.5°E).

Figure 11. Modelled seasonal cycle of lightning (a,g), hail ≥ 2 cm (b,h), hail ≥ 5 cm (c,i) for three different periods (1950–1973, 1974–1997, 1998–2021 across northern Italy and Central Oklahoma. The mean trends for the three period are also shown for lightning (d,j), hail ≥ 2 cm (e,k) and hail ≥ 5 cm (f,l). Seasonal cycles were smoothed using a B-spline interpolation.

Figure 11. Modelled seasonal cycle of lightning (a,g), hail ≥ 2 cm (b,h), hail ≥ 5 cm (c,i) for three different periods (1950–1973, 1974–1997, 1998–2021 across northern Italy and Central Oklahoma. The mean trends for the three period are also shown for lightning (d,j), hail ≥ 2 cm (e,k) and hail ≥ 5 cm (f,l). Seasonal cycles were smoothed using a B-spline interpolation.

7. Discussion and conclusion

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