The simulations are performed with the climate version of the non-hydrostatic COSMO model . More specifically, we use COSMO-crCLIM, a version of COSMO that is able to run on hybrid CPU-GPU architectures . Hereafter, we refer to COSMO-crCLIM as COSMO for simplicity. The simulations are conducted following a two-step, one-way nesting approach with a horizontal grid spacing of 12 km for the first nest and 2.2 km for the second nest (Fig. a). The simulations are driven by the ERA5 reanalysis with a boundary updating interval of 1 h. Both domains are discretized with 60 terrain-following hybrid vertical levels, where vertical spacing ranges from 20 m near the surface to about 1.2 km at the model top are located at around 23.5 km above mean sea level.

From the parameterization packages, we apply a single-moment bulk microphysics scheme with prognostic cloud water, cloud ice, graupel, rain, and snow , and a radiation scheme with a δ-two-stream approach . For the outer 12 km domain, the convection scheme is turned on for shallow convection and switched off for deep and mid-level convection following . For the inner 2.2 km domain, the convection parameterization scheme is switched off entirely to resolve the convection processes explicitly as far as feasible.

The COSMO model is run with the HAILCAST module, a diagnostic hail growth model that predicts the size of hailstones falling to the ground. It was originally a 1D coupled cloud and hail model developed by and further improved by and . The HAILCAST version used in this study is adopted from WRF-HAILCAST . If the grid column or any adjacent grid columns has a maximum updraft exceeding 10 m s-1 between the previous and current model time steps, the updraft duration field is incremented by one model time step. This field is used to track the convective cells and limit the maximum updraft time in the hail model. HAILCAST activates when the updraft velocity is larger than 10 m s-1 and the updraft duration is more than 15 min. At that point, the vertical profile at the given model time step is passed to HAILCAST, which then calculates the evolution of five hail embryos selected based on microphysical considerations. Two embryos of 5 and 7.5 mm in diameter are initialized at -8 ∘C level, and three embryos of 5, 7.5, and 10 mm in diameter are initialized at -13 ∘C. HAILCAST is activated every 5 min in the inner COSMO 2.2 km domain. The hourly maximum hailstone diameter among the five prescribed hail embryos is stored at hourly output intervals, providing information on hail swaths and the maximum expected hail size over an hour.

The lightning potential index (LPI, J kg-1) is a measure of the potential for charge generation and separation that leads to lightning flashes in thunderstorms . It considers the separation region of clouds within the main charging zone (0 to -20 ∘C), where the contribution of non-inductive mechanisms is the most efficient. Non-inductive mechanisms refer to the rebounding collisions between cloud ice crystals and graupel particles under the presence of supercooled liquid water . We use the updated LPI version after : 1LPI=f1f21H-20∘C-H0∘C∫H0∘CH-20∘Cεw2g(w)dz, with 2ε=2(qLqF)0.5qL+qF, and 3qL=qc+qr,4qF=qg(qiqg)0.5qi+qg+(qsqg)0.5qs+qg, where qc, qr, qi, qs, and qg are the mixing ratios of cloud water, rain water, cloud ice, snow, and graupel, respectively. g(w) is a boolean function equal to 1 when vertical velocity w≥ 0.5 m s-1, and 0 otherwise. ε is a dimensionless number that has a value between 0 and 1, and it scales the cloud updrafts and reaches the maximum when the vertically averaged mixing ratios of liquid (qL) and combined ice (qF) species are equal. Thus, the LPI is non-zero when liquid water and ice species co-exist in the grid boxes with updraft velocity above 0.5 m s-1, a threshold that identifies the growth phase of the thunderstorm. However, this chosen threshold generates many LPI signals. To overcome this issue two Boolean functions f1 and f2 are included to filter out weak and noisy LPI signals caused by isolated single-grid-column updrafts (f1) and to filter out false LPI signals in strong orographic gravity wave clouds (f2) following . f1 is TRUE if more than 50 % of grid boxes in a surrounding area of 10 × 10 km2 have an updraft larger than (or equal to) a threshold wmax. The threshold wmax is somehow arbitrary (see ) and depends on the grid spacing used. In our application, we have set it to 2 m s-1, which showed a reasonable distribution of LPI. However, this threshold is slightly different from 1.1 m s-1 used by with a grid spacing of 2.8 km. Note that these values are much lower than what is observed in the real world due to the simulated wider convective clouds and weaker updrafts, given that the 2.2 km grid spacing is high but not to a level that matches reality. f2 is TRUE if a column integrated buoyancy (see Eq. 16 in ) in a surrounding area of 20 × 20 km2 is larger than (or equal to) -1500 J kg-2. As for wmax, this threshold is also arbitrary, but in this case, we did not do any additional test and simply used the one recommended by . Thus, for more detail on these functions and choices, please see . LPI is calculated every 15 min in the COSMO 2.2 km simulations, and it is saved as an hourly maximum.

We evaluate daily accumulated total precipitation against IMERG using an object-based verification method. For a fair comparison, both observations and model outputs are interpolated to the same grid spacing of 12 km (Fig. b). We use the SAL (structure-amplitude-location) method proposed by to evaluate the model performance. The A component is calculated as the normalized difference between the domain-averaged observed and simulated fields. A positive (negative) A component indicates an overestimation (underestimation) by the model. The L component considers the displacement of the center of mass between the observed and simulated fields, as well as the weighted average distance between individual objects and the mass center of the total field. Lower L values indicate a more accurate placement of the simulated field. The S component accounts for the size and shape of the objects. Positive (negative) S values suggest a more widespread (peaked) simulated field. The computation requires the identification of precipitation objects within the analysis domain, separately for the observed and simulated fields. An object is defined as the grid points above the threshold of 1/15 of the maximum value of precipitation within the domain as suggested in . As a result, the influence of interpolation on the result is rather small.

For each case, we have estimated whether the atmospheric instability was generated by local conditions or synoptic atmospheric processes. This classification depends on the convection adjustment scale τ, which is derived using the precipitation rate P (kg m-2 s-1) and CAPE according to the following equation : 5τ∼CAPEdCAPE/dt∼12cpLvρT0gCAPEP, where the change of CAPE due to the release of latent heat dCAPE /dt is estimated from precipitation rate P. Reference values of density (ρ= 1.292 kg m-3), temperature (T0= 273.15 K), specific heat of air at constant pressure (cp), latent heat of vaporization (Lv), and acceleration due to gravity (g) are taken. τ considers the timescale within which CAPE is removed by convection. For the calculation of τ, we use the hourly domain-averaged CAPE and total precipitation from ERA5 (same domain as in Fig. ) and calculate the daily maximum τ for each case, as it must be calculated over a region large enough to smooth the variability from individual clouds. suggests that if τ is shorter than 12 h, the atmospheric instability is governed by the synoptic conditions, and the event is then classified as strong synoptic forcing. A larger τ (> 12 h), however, indicates that the convection is driven by high local CAPE values, in which case the event is classified as weak synoptic forcing. We should note that the threshold between weak and strong synoptic forcing varies in the literature (see, e.g., ), and should thus not be taken strictly, especially for the cases close to it. Here, we just use it as an indication of prevailing conditions.

In this study, we simulate and analyze eight cases of observed severe convective storms covering different synoptic situations (Fig. ). The following cases are selected according to the severity of their impacts (e.g., the size of hailstones, number of lightning strikes, and cost of damages):

23 July 2009. Severe hailstorms occurred over eastern and central Switzerland and caused damage to buildings amounting to around CHF 261 million in Switzerland . The weather over Central Europe was dominated by a southwesterly flow and large temperature contrasts (Fig. a). Strong lifting associated with a cold front resulted in severe thunderstorms, leading to several long hail swaths that can be seen in radar observations (Fig. a).

In this section, we assess how COSMO, with a 2.2 km grid spacing, performs in simulating total precipitation, hail, and lightning. To do so we look into the model performance with SAL diagrams (explained in Sect. 2.5) shown in Fig. , and spatial distribution of total precipitation, hail, and lightning obtained from the model and observations for all eight cases shown in Figs. –.

The SAL diagram of daily accumulated total precipitation is shown in Fig. . The amplitude (i.e., intensity) of precipitation is overestimated for 1 June 2013 and 17 May 2018 and underestimated for 18 June 2013 and 8 July 2017. The structure component is captured relatively well for most of the cases except for 2 cases – 18 June 2013 and 25 June 2017 – for which the precipitation objects are too small and peaked compared to observations. For the case of 17 May 2018, the simulated precipitation is more scattered (Fig. k, o). Finally, the location component is particularly large in two cases – 8 July 2017 and 18 June 2013. The bias shown for the 8 July 2017 case is partially due to the southerly shift of the convective system (Fig. i, m). On 18 June 2013, COSMO fails to simulate precipitation over eastern France and overestimates peak precipitation over the Black Forest (Fig. c, g), which results in a large location error together with the largest negative bias for amplitude and structure components.

Comparison against IMERG and high-resolution RhiresD observations reveals that COSMO can capture the main spatial distribution of daily accumulated total precipitation (Fig. ). The best performance is seen for the case of 23 July 2009 (Figs.  and a, e), characterized by stronger synoptic forcing and convection ahead of the cold front. The worst performance is seen in Figs.  and c and g for the case of 18 June 2013. The inability of the model to simulate this event properly is attributed to the local processes involved. The event is associated with weak synoptic forcing, with the largest convective timescale of all cases (Table ). Thus, due to its more chaotic nature, this event has small predictability. We should note, however, that the SAL components of precipitation are computed against IMERG, which has a much coarser resolution than the model. Therefore, some of the biases can be attributed to the rather smooth precipitation distribution (larger precipitation objects of lower intensity) shown by the observations (Fig. ). It is also interesting to note that none of the precipitation observations used, neither IMERG nor RhiresD, captured the record-breaking hourly precipitation amount of 82 mm as observed at the rain gauge station in Lugano (southern Switzerland) on 25 June 2017. However, such a high precipitation intensity is simulated by the model, even though it is slightly misplaced.

To evaluate the hail produced by COSMO HAILCAST, we first compare the model output against radar-based observations available over Switzerland and its surrounding areas (Fig. a–h). We first show the simulation against the POH data in terms of the hail footprint and coverage, but the comparison against MESHS data looks qualitatively similar. In general, the occurrence of hail is simulated well, but the placement and coverage are not captured very well in some cases (e.g., 8 July 2017). The case of 1 June 2013 with heavy rain but no hail over Switzerland is well reproduced, even though a very small number of grid cells produced small hail (Fig. b). Among the best-simulated cases, the same as for total precipitation, we can again consider the case of 23 July 2009 characterized by stronger synoptic forcing and elongated hail swaths reproduced by the model (Fig. a). The record-breaking rainfall event of 25 June 2017 also produced hail south of the Alps, as observed by radar and crowd-sourced reports and over the Adriatic region, including the continental part of Croatia as observed by hailpads (Fig. d, i, l). The widespread occurrence of hail in this case is reproduced by the model, even though the spatial extent is overestimated. The case of 17 May 2018 is characterized by hail recorded on the Istrian peninsula in Croatia and was well reproduced by the model. However, the model produced very light and scattered hail over both Adriatic and Alpine regions where it was not observed (Fig. g, n). Another case with poorer model performance is 18 June 2013, when the model overestimates the spatial extent of hail swaths, especially over the Black Forest (Fig. c). Overall, we can see that the performance in the simulation of hail aligns with the performance in the simulation of total precipitation. However, we should note many difficulties in comparing model output with available hail observations. As described earlier, MESHS data provides information on hailstone diameters above 20 mm only, which can potentially lead to underestimating the hail frequency and spatial coverage. POH only provides the probability of hail but does not indicate whether or not hail has occurred. Furthermore, both MESHS and POH detect hail at higher altitudes and not on the ground, and melting may influence hailstones . Last but not least, even though the hailpad network consists of many hailpad stations, many areas are not well covered and, thus, are prone to miss recording very localized events such as thunderstorms and hail associated with them.

To further explore the performance of the COSMO HAILCAST, we compare the simulated hail against available observations for different cases as listed in Table . We compared the area affected by hail using COSMO HAILCAST against MESHS over the radar-covered area (Fig. ), frequency of simulated hail diameter against crowd-sourced data over Switzerland (hailstones larger than coffee beans – 5 mm – are shown; Fig. a–c), and hailpad observations over the hailpad-covered area (Fig. d–f). When compared to MESHS, the results show a large difference between the observed and simulated area. We see that, while MESHS observation only estimates hail diameter above 20 mm, the model produces hail diameter below 20 mm.

According to , considering the 23 and 32 mm reports, MESHS tends to exceed the crowd-sourced reported hailstone size by 10–15 mm on average. Yet, compared to crowd-sourced data over Switzerland (Fig. a–c), the model shows a reasonable hail size distribution, although it still tends to overestimate small hailstones and underestimate large hailstones. Moreover, when compared to hailpad observations over Croatia (Fig. d–f), the model exhibits a distribution that closely aligns with the observed data, particularly for the two cases of 25 June 2017 and 17 May 2018. For the case of 24 July 2017, an event relatively well captured by the model, the simulated hailstones above 15 mm are underestimated. Therefore, the comparisons against radar-based MESHS should be considered with caution. They show that there is a need to improve radar-based hail algorithms, although comparisons with other observations show that COSMO in general tends to overestimate small hailstones and underestimate large ones.

It is clear that HAILCAST underestimates the frequency of larger hail sizes, i.e., does not produce many hailstones larger than 30 mm. As noted by , the hail size strongly depends on the initial hail size embryo – the larger the initial embryo, the larger the output hail size. However, the size also depends on the model micro-physics, the strength of the updrafts that hail has to overcome to fall to the surface, and the initial temperature level. For example, if updrafts are weaker, larger hail falls down faster and does not have enough time to grow further, while smaller hail has more time to grow but does not reach sizes above 20 mm. In a parallel study, in which the same eight cases are simulated with the WRF model , larger hailstones are obtained with the WRF model than with COSMO. This result indicates that the simulated hail size strongly depends upon the model formulation.

Next, we turn our attention to the evaluation of lightning. A comparison of lightning patterns between the model and LINET observations for each of the cases is shown in Fig. . In addition, to overcome difficulties related to different variables represented by observations and model (lightning flashes versus lightning potential), we calculate and display a coverage bias in the figure (red number). It is defined as the ratio of the number of gridpoints with lightning in the model and observations, respectively. Note that the coverage bias does not provide any information on the overlap of simulated and modeled lightning, but this is qualitatively assessed from the spatial representation. Overall, the model using LPI diagnostics is able to capture the lightning patterns for each case, although it tends to slightly overestimate the spatial patterns of the signal (as for total precipitation and hail). The largest overestimation of spatial patterns, and, thus, the coverage bias, is found in the case of 1 June 2013, when very little lightning was observed over the Adriatic and no lightning over the Alpine region. However, the model diagnostics produced lightning over the eastern Alps, which coincides with the area of very intense precipitation. The case of 1 June 2013 is the case without hail over the Alpine region, which was successfully reproduced by the model. Differences in representing hail and lightning can be related to different updraft thresholds used by LPI and HAILCAST, which is lower for LPI: 0.5 m s-1 for LPI (Sect. ) versus 10 m-1 for HAILCAST (Sect. ). The smallest coverage bias is obtained for the case of 24 July 2017 and 30 May 2018, even though there is a slight shift between the observations and the model. We should also note that both of these cases are characterized by weaker synoptic forcing and more locally driven convection, which is well reproduced by the model. The largest underestimation of the spatial coverage of lightning is found in the case of 25 June 2017. A large part of this bias is visible over the Adriatic Sea – the area over which the model fails in reproducing total precipitation as well.

Overall and not surprisingly, we can see that the performance of both hail and lightning diagnostics strongly depends on simulated total precipitation, since both hail and lightning diagnostics depend on the same ingredients as precipitation.

A central element of the simulation strategy is the use of ERA5 lateral boundary conditions, with the initialization taking place at 12:00 UTC on the day before the event to account for the spin-up of the storms. The simulation is thus guided along the reanalysis, and the predictability in our simulations is much higher than in a numerical weather prediction (NWP) forecast. The strategy is ideal for testing diagnostic tools that require adequate synoptic forcing. Despite the enhanced predictability due to the use of ERA5 lateral boundaries, there is some remaining internal variability. To test the effect of model internal variability on our results, we conducted a small ensemble of simulations for three of the eight cases, by shifting the initialization by +6 and -6 h. The ensemble simulations are initialized at 06:00, 12:00, and 18:00 UTC on the day before the storms occurred. Consideration is given to the whole modeling chain with nested simulations at 12 and 2 km resolutions.

Results show that even for localized deep convective storms, the predictability of precipitation and hail is overall quite high (Fig. ). However, there are significant differences in detail, due to the chaotic nature of the nonlinear flow evolution. For example, in the case of July 2009 (top two rows of Fig. ), there are considerable differences in the length and location of the hail swaths. Likewise, in the case of 18 June 2013, precipitation is simulated over the Black Forest when initialized at 12:00 UTC but not when initialized at 06:00 and 18:00 UTC. Similarly, in the case of 30 May 2018, there are pronounced differences in the precipitation fields with concomitant differences in hail. Overall, however, the internal variability is rather small, and, hence, the simulations confirm the suitability of the selected modeling strategy for assessing the performance of the modeling approach for case studies of severe convection. A comparison of the cases shown in Fig.  suggests that synoptically driven convective storms have a higher predictability.

To further investigate the environmental conditions and the mechanisms that are favorable for the development of thunderstorms over the Alpine-Adriatic region we present a more detailed analysis of three specific cases, which affected different areas under different synoptic situations.