The numerical simulations were conducted with version 2.6.6 of the ICOsahedral Non-hydrostatic (ICON) model. This fully compressible model employs an unstructured triangular grid with C-type staggering, derived from the successive refinement of a spherical icosahedron . It supports both global and limited-area modes with grid-nesting capabilities. Since February 2021, the German Weather Service (DWD) has been using the convection-permitting ICON-D2 configuration, which features a horizontal grid spacing of 2.1 km, for operational forecasts over central Europe. The model domain (Fig. ) and the horizontal and vertical resolutions used in this study align with those of the operational ICON-D2 setup. Instead of the operationally used single-moment scheme, we use a double-moment bulk microphysics scheme to study aerosol effects on mixed-phase clouds and precipitation. This scheme predicts the mass and number concentrations of cloud water, rain water, ice, snow, graupel, and hail. It uses pre-calculated activation ratios stored in look-up tables for the CCN activation of aerosol particles. To prevent generating very high ice number concentrations by CCN activation at low temperatures and immediate homogeneous freezing of the activated cloud droplets, CCN activation was restricted to temperatures above -38 °C as in . Additional model settings are detailed in Table .

In this study, we construct an ensemble entirely based on microphysical uncertainties, hereafter referred to as MPHYS-ENS. We do not consider initial and boundary condition uncertainties as all our simulations use the same initial and boundary data, i.e. 7 km ICON-EU analyses. The following four microphysical uncertainties are applied:

CCN uncertainty

To investigate the impact of different aerosol and CCN concentrations on precipitation, numerical simulations were conducted with 3 different number densities of condensation nuclei (NCN) representing low (NCN = 100 cm-3), high (NCN = 1700 cm-3), and very high concentrations (NCN = 3200 cm-3). Typical conditions in central Europe are represented by the high (i.e., continental) aerosol assumption . However, there is considerable uncertainty regarding aerosol number concentrations, as there are few in situ observations or routine measurements of aerosols in three-dimensional space . As aerosols can be highly variable in central Europe (see e.g., Copernicus Atmosphere Monitoring Service (CAMS) forecasted aerosols), assuming different CCN concentrations seems to be a suitable perturbation parameter for the model. Furthermore, previous studies have demonstrated a relatively large impact of CCN on domain-averaged total precipitation amounts e.g., and it could even be crucial in determining whether a supercell storm forms in the model or not .

Table  summarizes the parameters and their uncertainty ranges that we apply for generating the microphysical ensemble. Using three different CCN concentrations, 3 INP concentrations, 3 scalings of the graupel sedimentation velocity, and 4 different shape parameter values, we end up with a large ensemble size of 108 members. The simultaneous perturbation of these parameters allows us to investigate whether and how they interact synergistically and what collective effects they have on simulated convection and convective-scale predictability.

Since February 2021, the convection-permitting ensemble prediction system ICON-D2-EPS has been used operationally at the DWD as successor of the COSMO-D2-EPS and thereby continuing the sequence of operational ensembles on the convective scale, which started in May 2012 with the COSMO-DE-EPS. Forecast variability in the global ICON-EPS and ICON-EU-EPS systems is achieved through 40-member initial condition (IC) perturbations, generated via ensemble data assimilation with a 3 h assimilation cycle. Currently, the operational ICON-D2-EPS consists of 20 members. Initial condition uncertainty is considered by selecting the first 20 members of the hourly Kilometer-Scale Ensemble Data Assimilation KENDA , which is a Local Ensemble Transform Kalman Filter. In contrast, the perturbation of boundary conditions is implemented using forecast members of ICON-EU-EPS from the run 3 h before the ICON-D2-EPS initialization. Additionally, variability is introduced through ensemble physics perturbations, where a pre-defined fixed selection of tuning parameter perturbations is randomly assigned to each ensemble member at each new forecast start and remains consistent throughout the forecast lead time. In a statistical sense, considering many forecast starts, each of the affected parameters is perturbed in 50 % of the members in each forecast run. The physics perturbation comprises 19 parameters in total, which cover, e.g., the low-level wake drag constant, the critical Froude number in the subgrid-scale orography scheme, the entrainment parameter in the convection scheme, or a minimum vertical diffusion for heat, moisture, and momentum. However, these perturbations do not cover parameters and properties of the double-moment physics scheme, which is employed in MPHYS-ENS.

Summarizing, the operational set-up differs from MPHYS-ENS in the coverage of sources of forecast uncertainties, i.e., the ICON-D2-EPS has initial and boundary conditions perturbed, leading to forecast variability right from the forecast start, already without physics perturbations. However, the MPHYS-ENS covers more comprehensively the aspects of model error in microphysical processes by perturbing parameters of the double-moment microphysics, which are not available in ICON-D2-EPS. Moreover, MPHYS-ENS covers all combinations of its microphysics parameter perturbations in its 108 members. Apart from the perturbation approach, the MPHYS-ENS configuration at KIT (see Sect. ) uses the same model domain and grid, the same time step, and mostly the same physical parameterizations. Whereas the operational configuration uses a tuned version of the convection scheme that bridges the gap between parameterized and resolved convection (so-called grayzone-tuned version), our configuration uses a scheme for shallow convection only (see Table ). In addition, heterogeneous ice nucleation is parameterized operationally using the scheme.

Even though there is already valuable information about the potential of MPHYS-ENS to quantify forecast uncertainty only by its perturbed microphysics in comparison to the operational ICON-D2-EPS, we would like to allow for a comparison focusing better on the significance of microphysical processes by using an EPS, which emphasizes the differences between MPHYS-ENS and ICON-D2-EPS mostly in the aspect of physics perturbations. For this purpose, we set up the ensemble ICON-D2-2MOM, which uses no initial and boundary condition perturbations but the double-moment microphysics like the MPHYS-ENS, and perturbs physics identical to the operational ICON-D2-EPS without perturbing the parameters of MPHYS-ENS. ICON-D2-2MOM has 20 members like ICON-D2-EPS and will be compared to the full 108 as well as 20-member subsets of MPHYS-ENS in Sect. .

Table  summarizes the ensemble strategies used for the investigated cases.

In this study, we investigate convective-scale predictability with numerical simulations of 4 d from the Swabian MOSES 2021 and 2023 field campaigns. Both campaigns were dedicated to hydro-meteorological extremes, i.e., convective storms with strong winds, heavy rainfall, hail, and large-scale heat waves. The observation area was located in southwestern Germany and parts of neighboring countries where previous campaigns on deep convection above complex terrain took place e.g.. We simulate one day from 2021 (23 June) and 3 d from 2023: 8 June, 22 June, and 15 July. The 4 cases analyzed here represent a variety of convective scenarios in central Europe. They include single cells, multicells, supercells, and larger convective clusters in weather regimes with different strengths of synoptic forcing. We therefore believe that the case selection is representative of summertime convective situations in that area. Two of these cases (23 June 2021, 22 June 2023) were also examined with 1 km grid spacing under pseudo-global warming conditions by .

We start the analysis of the microphysical ensemble by investigating the spatial variability of 24 h accumulated precipitation amounts. Accumulated precipitation amounts exhibit a large spatial variability of the individual location and intensity of the convective cells (not shown). To assess this spatial variability quantitatively, we computed the median and the interquartile range (IQR, representing the spread between the 25th and 75th percentiles) of the ensemble for each analyzed day (Fig. ). The characteristics of the different convective scenarios on these days are generally well simulated by the ICON model, i.e., the location of precipitation and sizes of convective areas generally agree with the observed precipitation (see radar-derived total precipitation in Fig. ). The median panels indicate coherent precipitation patterns with very high values up to 100 mm for almost all cases, indicating the strong convective activity simulated by the model. The IQR maps in the lower row reveal a substantial spatial variability across the domain, highlighting regions of enhanced forecast uncertainty. High IQR values exceeding 30 mm in some locations indicate considerable disagreement among ensemble members. Regions of high IQR are, however, not located at topographic boundaries or regions with different land-use, which reflects the fact that surface heterogeneities do not control the spread in the microphysical ensemble. This spatial heterogeneity is particularly pronounced in cases 1–3, for which the synoptic-scale forcing was low to moderate. Case 4 with stronger synoptic-scale forcing and weaker rain intensities generally shows a smaller, but still large, ensemble spread. The large spread in the spatial precipitation distribution solely introduced by microphysical perturbations is remarkable and nicely exemplifies the heteroscedastic nature of precipitation forecasts, linking high values of mean precipitation to high forecast uncertainty and hence variability between ensemble members. All this underlines the importance of probabilistic forecasts in capturing the full range of possible precipitation outcomes and the importance of accounting for as many of the relevant sources of uncertainty in the ensemble generation, particularly during high-impact weather events.

To analyze how precipitation amounts respond to the perturbations introduced in this study, we calculated the total precipitation integrated over the evaluation domain shown in Fig. . The percentage deviations of each of the 108 ensemble members generally show large numbers of -15 % and +12 % (Fig. ) compared to the corresponding reference run. When taking the respective minimum precipitation amount as a reference, the percentage increases to the maxima are considerable: We find a 18 %, 17 %, 33 %, and 21 % increase on cases 1–4, respectively. As already reported for other summertime convective events in previous studies , we find a mostly negative aerosol effect (i.e., a monotonic decrease between low and very high CCN) on total precipitation amounts for the first 3 cases (Fig. a–c). For case 4 (15 July 2023), however, we find a mostly systematic positive aerosol effect on total precipitation amounts. The precipitation reduction with increasing CCN concentrations is most probably related to a reduced warm-rain process (autoconversion and accretion) for larger aerosol loads, where smaller droplet sizes result in a weaker collision–coalescence process. The increase for the last case could be due to either warm-phase i.e., condensational or water-phase invigoration, e.g., or cold-phase invigoration e.g.,. This will be investigated later on by the analysis of microphysical process rates. On days with a dominating negative aerosol effect, the largest precipitation amounts are simulated for low CCN concentration and increased graupel sedimentation. In contrast, the largest precipitation amounts for the day with a positive aerosol effect are simulated mostly for reduced graupel sedimentation and very high CCN concentration. Only for the reference shape parameter of 1, the maximum precipitation amount is simulated for increased graupel sedimentation for this case study. Possibly, the mixed-phase riming growth of ice hydrometeors is also impacted by aerosol-induced changes to the cloud droplet size distribution and the different assumptions of the shape parameter. The efficiency with which cloud droplets contribute to the riming growth of ice hydrometeors can be substantially reduced if the droplet size is reduced, consequently inhibiting ice growth . The influence of a higher number of smaller cloud droplets on riming, either enhancing or hindering it, depends on both the droplet size and their concentration. In a high aerosol scenario, if the droplet sizes are not reduced substantially enough to offset the greater number of droplets contributing positively to the collection rate between ice hydrometeors and cloud droplets, then greater riming and ice production can result e.g.,. At very high aerosol concentrations, however, a tipping point towards reduced riming may be reached e.g.. A higher graupel fall velocity is expected to enhance the riming growth of ice hydrometeors, affecting total precipitation either in solid or melted form. If the size of cloud droplets is reduced either by higher CCN concentrations or larger shape parameters, the increased riming growth due to higher graupel fall velocities may be overcompensated by the reduced riming efficiency associated with smaller cloud droplets. On the other hand, the increased fall velocity could also reduce the time during which riming can be active and thus reduce ice growth. In our study, the riming of graupel and hail with cloud droplets is two orders of magnitude higher than for snow and ice and therefore dominates the cloud-droplet riming growth of ice hydrometeors. For all cases, we find a systematic decrease in riming with increasing CCN concentrations. However, increasing the shape parameter usually leads to a systematic increase in riming: under clean conditions, riming increases by 14 %–28 %, while under polluted conditions, the increase is slightly lower, at between 7 %–15 %. The narrowing of the cloud droplet size distribution reduces the number of larger and smaller cloud droplets, but the higher occurrence of mid-sized cloud droplets (see Fig. ) seems to compensate for the reduction at larger droplet sizes.

It is also evident from Fig.  that the CCN concentration and the graupel sedimentation impact the results most. To quantify that, we computed the mean total precipitation deviation for varying each uncertain parameter while keeping the other parameters constant (Fig. ). The graupel sedimentation has the largest impact on precipitation changes, followed by the CCN concentration in the first three cases, whereas the CCN concentration dominates precipitation change on the last day on which the positive aerosol effect on total precipitation amount was simulated. The INP concentration has the smallest impact on precipitation totals for all cases. The shape parameter has a stronger impact on the results than the INP concentration in all cases. On the last day (15 July 2023), it is even the parameter with the second strongest influence.

The analysis of aerosol effects on total precipitation amounts showed a mostly negative impact for cases 1–3, whereas a positive impact exists for case 4. Rain, on the other hand, shows an increase with increasing CCN concentration in 3 of the cases (not shown). Therefore, precipitation via the cold phase, like hail and graupel, must be considered for the prevailing negative aerosol effect on total precipitation. To further quantify this effect, we computed the relative contributions of rain, graupel, and hail (no snow on the ground) to the domain-averaged total precipitation amount for each of the 108 ensemble members. As expected, rain has the highest contribution, ranging from 87 % to 98 %. The maximum graupel contribution is almost identical for all days (2 %–3 %). For case 4 with a positive aerosol effect, the maximum contribution of hail to total precipitation is only 3 %, whereas it is much larger for the other cases (7 %, 9 %, 10 %). On average for the entire ensemble, hail contributes only 1.4 % to total precipitation on case 4, whereas the other days have contributions between 2.8 % and 4.8 %. We therefore hypothesize that the positive aerosol effect could be related to the relative role of cold-rain processes, which will be analyzed later in Sect. .

To investigate the reasons for the overall strong effects on precipitation totals, we now analyze the daily cycle of 30 min precipitation rates to assess whether the model simulates longer/shorter lifetimes and increased/reduced rain rates. Convection mostly starts simultaneously when averaging over the comparatively large evaluation domain (Fig. ). Only for cases 1 and 2 do the runs with low CCN concentrations have a 30–60 min earlier onset of precipitation compared to the more polluted runs. The later onset can be explained by a weaker collision–coalescence process due to smaller droplet sizes in polluted conditions lifetime effect;. After convective precipitation has started, however, the rate at which precipitation then increases can differ. For example, on 8 June 2023, the runs assuming low CCN concentration have larger rain intensities than the runs with high and very high CCN concentration. This also holds for case 1 (23 June 2021), whereas the last two cases show similar increasing rain intensities after convection starts to form. Only case 3 shows the highest rain intensities for the runs with low CCN concentrations, all remaining cases have their maxima at high or very high CCN concentrations. This is particularly true in case 4, where the precipitation rates for a polluted atmosphere are always the highest. Cases 1 and 2 demonstrate the importance of analyzing a longer period for precipitation totals: Although the maximum rain intensities are simulated for high and very high CCN concentrations, the longer times with higher rain intensities before and after the respective maximum control the total precipitation amount as a negative aerosol effect on total precipitation mostly characterized these cases. The effect of different graupel sedimentation velocities is small for cases 1, 3, and 4. For case 2 (8 June 2023) with prevailing small and isolated showers with shorter lifetimes, however, there is a clear tendency for higher maximum rain intensities with increased INP concentrations. The higher the graupel sedimentation, the earlier the maximum rainfall intensities occur (RED at 17:30 UTC, REF at 17:00 UTC, INC between 15:30 and 16:30 UTC). In brief, these results indicate that the timing of convection initiation is, on average, not sensitive to the varied microphysical parameters in this study and that different total precipitation amounts are caused by varying rain intensities throughout the simulation period and/or the number of convective cells.

Except for case 2, the temporal evolution in Fig.  shows that the convective events are not yet complete and precipitation rates are still declining at the end of the 24 h integration period. To assess the effects of the incomplete simulation of the entire convective event during the night, we repeated all 108 simulations with a longer integration time of 30 h for case 1. We find that the precipitation characteristics (deviation of total precipitation from the reference run, integrated warm- and cold-rain processes) do not change. Thus, the findings regarding the sensitivity to the microphysical perturbations are considered to be robust and not influenced by the fact that the decaying stage of convective activity is not fully covered in all days when using an integration time of 24 h.

As a metric to quantify forecast error introduced by the microphysical perturbations used in our ensemble, we use the difference total energy DTE;, which is defined as 2DTE(x,t)=12Δu2+Δv2+cpTrΔT2 with the wind components u and v, temperature T, the heat capacity of dry air cp, and a reference temperature Tr = 287 K. Δ denotes the difference between an ensemble member and the reference simulation, x is the 3D position vector, and t is time. At first, we analyze a sequence of DTE maps vertically averaged between 0 and 11 km and integrated over all ensemble members at 2 h intervals, illustrating the tropospheric error growth process over the 24 h simulation period for case 2 (22 June 2023) in Fig. . Initially, high DTE values are confined to localized areas with clouds, but they expand rapidly throughout the domain. By 06:00 UTC, a substantial portion of the model domain is already affected by significant errors, and by 09:00 UTC (not shown), elevated DTE values encompass the entire model domain. This illustrates the rapid and widespread growth of forecast uncertainties, highlighting the challenges in maintaining accurate predictions, even over relatively short periods. Understandably, the largest errors occur in areas with the highest rainfall totals and the greatest rainfall variability. The other cases investigated here behave similarly, but on two of the other days analyzed (not shown), the spread of forecast error, as measured by DTE, remains more localized during the early forecast hours and only begins to affect the entire model domain after approximately 12:00 UTC.

We also computed individual domain-averaged DTE values between every single member of the ensemble and the reference run. The time series plots in Fig.  illustrate the evolution of forecast error throughout the diurnal cycle for the 4 different case studies, with the ensemble members grouped by CCN regime (low, high, and very high) shown as shaded areas. A consistent pattern emerges across all 4 dates: simulations with low CCN concentrations produce the highest DTE values, indicating greater forecast uncertainty or divergence among ensemble members. In contrast, high and very high CCN concentrations lead to reduced DTE growth, suggesting a more constrained or predictable evolution of the atmospheric state. This implies that microphysical processes influenced by aerosol loading, particularly CCN, have a significant impact on predictability. The DTE increase exhibits a strong diurnal signal, with a rapid rise during the afternoon and early evening hours (approximately 12:00–18:00 UTC), followed by a plateau or slight decrease during the night. This pattern closely mirrors the daily cycle of convective precipitation (Fig. ), underscoring the role of convective processes in amplifying forecast errors. As convective activity intensifies, the atmosphere becomes more sensitive to the microphysical uncertainties, particularly under low aerosol conditions, where cloud development is less suppressed for the first 3 cases. Interestingly, case 4 with the positive aerosol–precipitation feedback shows larger DTE values for low CCN concentrations, despite having the lowest rain rates. Overall, the results highlight the importance of aerosol-cloud interactions in ensemble predictability and suggest that lower CCN environments may enhance model divergence during convective regimes, particularly in the presence of strong diurnal forcing.

To investigate the microphysical processes and their response to the perturbations applied in this study, we have expanded the model code to write out a variety of microphysical process rates. To understand which processes dominate and/or exhibit large variability in response to microphysical perturbations, we employ an approach inspired by . For each of the analyzed processes, we computed the vertically-integrated process rates, accumulated over the full integration time of 24 h. Then a domain average was calculated for the evaluation domain given in Fig. . As a second metric, the percentage of occurrence (representing the total duration a process is larger than zero) is used. When displaying the percentage of occurrence against the domain value, a process is considered more relevant the closer it is positioned to the upper-right corner. In this study, we expand the single-model realizations of Arctic clouds from with a 2D histogram computed using all 108 ensemble members. Figure  presents these two-dimensional histograms of accumulated microphysical process rates against their percentage of occurrence for the 4 case study days, derived from the 108-member convection-permitting ensemble. Each panel corresponds to one day and reveals the distribution and variability of key microphysical processes within the simulated weather systems, expressed in relative frequencies of occurrence, so that their sum equals one. The absolute values of homogeneous and heterogeneous ice nucleation are very small compared to the other process rates. Therefore, every point of the histogram is displayed with the same color and symbol to be visible in the figure.

Across all panels, several processes consistently stand out due to their high mean rates and frequent occurrence, placing them toward the upper-right quadrant of the plots. Notably, condensation and riming emerge as the dominant processes, both in terms of total accumulated mean value and temporal occurrence. The melting of solid hydrometeors to rain droplets also reveals large mean values, indicating the importance of the cold-phase pathways. Their temporal duration, however, is somewhat smaller, ranging between 30 %–47 %. In contrast, processes such as homogeneous and heterogeneous ice nucleation, deposition, sublimation, and rain freeze appear with much lower durations and accumulated rates, clustering in the lower-left regions of the plots. Thus, they contribute less to the overall cloud water or ice budget. In all analyzed cases, the processes with the longest temporal durations are cloud and rain droplet evaporation. As all convective clouds originate from the liquid phase and ice processes only start after condensation and the warm-phase pathway, the high time coverage seems plausible. The percentage difference between the minimum and maximum mean value of rain evaporation (i.e., the left and right limits of the histogram on the abscissa) ranges only between 15 %–28 %. Cloud droplet evaporation, however, shows a much larger response to the microphysical perturbations than rain evaporation (73 %–200 %) with only small variations in their temporal duration. This large variation is mainly caused by the different CCN concentrations, as in polluted conditions, the cloud droplets are more numerous, but smaller in size and therefore more susceptible to evaporation (not shown). Processes like riming and melting also exhibit substantial horizontal elongation, indicating a high sensitivity of their accumulated rates to ensemble perturbations, whereas their temporal durations vary much less. The maximum percentage increase for total riming and melting are 122 % and 165 %, respectively. As for cloud evaporation, the total riming rates mainly depend on the CCN concentration, with a reduction in riming in more polluted conditions. The warm-rain processes autoconversion and accretion also show a large response to the microphysical perturbations: autoconversion varies over two orders of magnitude and accretion increases between 44 %–90 % with an extreme value of 340 %. CCN activation also reveals a change of one order of magnitude. They form 3 clusters depending on the 3 CCN concentrations, where the absolute values are very similar and only a small temporal variation is present, caused by the remaining perturbations (not shown). The freezing of rain is comparatively small in magnitude, but shows an interesting feature for at least 3 of the analyzed cases: The largest mean values occur with the lowest temporal durations. A closer look at this individual process shows that the larger mean values are simulated for more polluted conditions. Probably, the freezing of rain then happens faster and more efficiently than in clean conditions.

In brief, this analysis shows that phase changes involving water and ice (riming, melting) as well as phase changes involving the vapor phase (condensation) dominate the mean rates, whereas the most dominant processes in terms of their duration involve only a phase change from liquid to the vapor phase (cloud and rain droplet evaporation). Overall, the plots highlight which microphysical pathways dominate the cloud and precipitation development, and which are more uncertain or variable in response to ensemble perturbations. This provides insight into both the relative importance and predictability of different microphysical processes in convection-permitting forecasts.

In addition to the absolute values and the duration of the individual process rates, it is also interesting to examine the relative role of warm-rain and cold-rain processes and if and how they depend on the microphysical perturbations applied in this study. Therefore, Fig.  presents the ratio of cold (vapor deposition and riming; DEP+RIM) to warm (autoconversion and accretion; AC + ACC) rain formation processes for the different sensitivity experiments using the same arrangement of data points as for the 24 h precipitation deviation in Fig. . Across all numerical experiments, this ratio consistently exceeds unity, indicating that precipitation formation is predominantly driven by cold-rain processes in the investigated convective cases. This finding is in agreement with previous studies of convective events simulated with the same model system e.g.. Moreover, the sensitivity experiments reveal that increasing CCN or INP concentrations systematically enhances the dominance of cold-rain processes. Physically, higher CCN concentrations delay warm-rain initiation via suppressed autoconversion and reduced collision-coalescence efficiency. As a result, more cloud water is available to be lofted into the mixed-phase region, where it can participate in cold-rain processes. Similarly, higher INP concentrations promote earlier and more widespread ice formation, accelerating deposition growth and riming. Both mechanisms shift the microphysical balance toward the cold pathway. In contrast, increased graupel sedimentation velocity leads to a reduction in the cold-rain process dominance. This can be attributed to reduced riming rates (not shown): if graupel particles sediment more quickly, the residence time within the mixed-phase region of the cloud is reduced. This highlights the importance of sedimentation feedbacks in modulating the vertical distribution and efficiency of microphysical pathways. In all 3 graupel sedimentation regimes, enlarging the shape parameter does lead to an increased relative role of the processes via the ice phase. This can again be attributed to the reduced autoconversion and accretion processes by the narrowing of the cloud droplet size distribution. The mean ratios found in this study lie in a similar range to previous simulations using the same double-moment scheme . Despite the different aerosol effects on total precipitation amounts found for these 4 cases, the relative roles of the warm- and cold-rain processes behave similarly. We would like to point out that, instead of deposition and riming, melting could be used as a single cold-rain process to calculate the ratio of cold to warm-rain processes. The results are almost identical and exhibit the same systematic behavior (not shown).

Averaged across all 108 data points in the first 3 cases, the cold-to-warm process ratio is 3.5 (case 1), 3.8 (case 2), and 3.4 (case 3). However, the simulation of case 4 (15 July 2023, panel d) stands out with a significantly higher average ratio of 5.9. Notably, this is the only case in which total precipitation increases with increasing CCN concentrations – an effect contrary to the aerosol suppression found for the first 3 cases. The elevated ratio suggests that enhanced cold microphysical activity can, under certain conditions, compensate for or even outweigh the initial suppression of warm-rain processes, leading to a net increase in precipitation. An analysis of the individual contributing processes revealed that in all investigated cases, increasing CCN concentrations lead to a decrease in riming and an increase in deposition. However, in cases where the aerosol has a negative effect on precipitation, the riming decrease is larger than the deposition increase, leading to an overall reduction in the cold-rain processes. Since the warm-rain processes exhibit a greater decrease with increasing aerosol concentration than cold-rain processes, the relative role of cold-rain processes continues to increase. For case 4 with positive aerosol–precipitation feedback, the deposition increase outweighs the decrease in riming, resulting in an increase of cold-rain processes with increasing CCN concentration. Combined with a similar quantitative reduction of the warm-rain processes as in the other cases, the ratio of the cold-to-warm rain processes ends up with larger values, i.e., the relative role of the cold-rain processes in the precipitation formation becomes more important. This connection implies that the cold-to-warm process ratio could potentially serve as a useful diagnostic or threshold to identify regimes where increased aerosol loading leads to precipitation enhancement, rather than suppression. More cases with different environments and convection types need to be investigated to further test this hypothesis.

In an ensemble forecasting system, the spread among ensemble members is crucial for assessing the level of uncertainty in the prediction. To assess the ensemble spread, we calculated the domain-averaged normalized standard deviation Sn at each grid point as follows see, e.g.,: 3Sn(x,y)=1P(x,y)‾1N-1∑i=1NP(x,y)‾-Pi(x,y)2

Here, Pi(x,y) represents the hourly precipitation for ensemble member i, P(x,y)‾ is the ensemble mean hourly precipitation, and N is the total number of ensemble members. Normalizing the spread helps to eliminate variations caused by the diurnal cycle of precipitation. We compare the normalized spread of the microphysical ensemble with the normalized spread of the operational ICON-D2-EPS (Fig. ). In addition, DWD conducted simulations using the same double-moment scheme with the operationally used physical perturbations, but without considering initial and boundary condition uncertainty (ICON-D2-2MOM). In all 4 cases, MPHYS-ENS produces the largest spread, with a pronounced afternoon increase coinciding with the increase in convective activity. This indicates that microphysical perturbations alone can generate substantial variability in convective development when sampled with a sufficiently large number of members. For two of the cases (8 and 22 June 2023), the spread of the operational ensemble is larger than the one from MPHYS in the first 5–7 h. This indicates the larger role of initial condition uncertainty in the first hours after model start, whereas the microphysical ensemble needs some time with cloud development to produce a higher spread. To get a fair intercomparison with the 20-member operational ensemble, we used a bootstrapping technique to randomly pick 20 members of the microphysical ensemble to calculate their normalized spread and repeated that procedure 100 times. When the ensemble size is reduced to 20 members, the bootstrapped subset still lies in a similar range to the operational ICON-D2-EPS for cases 1 and 4, but slightly underestimates the maximum spread for most of the time on cases 2 and 3, even though the temporal evolution remains similar. This loss of amplitude underlines the importance of ensemble size for representing the full range of uncertainty. However, the maximum ICON-D2-EPS spread always lies within the standard deviation of the bootstrapped subset. The ICON-D2-2MOM generally exhibits spread values comparable to or slightly above the 20-member subsample of MPHYS-ENS. This indicates that the microphysical uncertainties introduced in this study can produce a similar large spread as the operationally used physical perturbations in case a double-moment scheme is used. An important finding is the fact that for two of the 4 cases analyzed here, the largest spread in ensembles with 20 members can be gained without considering uncertainties in initial and boundary conditions. Only the two cases in June 2023 benefit from those uncertainties until around midday/early afternoon.

It is also of interest to evaluate the ensemble spread based only on the parameters that have the strongest influence on precipitation amounts. In most of the cases analyzed, these parameters are the CCN concentration and the graupel sedimentation velocity (see Sect. 3.2). Using these parameters, we additionally calculated the ensemble spread using reference values for INP concentrations and shape parameters (9 members) and also averaged over 12 sub-ensembles of 9 members each, in which CCN and GR are perturbed with constant combinations of INP concentrations and shape parameters. Although CCN and graupel sedimentation have the strongest influence on total precipitation amounts, the spread is mostly smaller than in the operational ICON-D2-EPS (not shown) due to the smaller ensemble size. It makes no difference whether only the reference values of INP and NU are used or whether the average of all combinations of constant INP and NU is taken. We therefore conclude that the high-impact parameters represented by a small subset contribute less to the overall spread than a combination with lower-impact parameters using a larger subset.

From a forecast quality perspective, these results imply that (i) large ensemble sizes are essential to avoid underrepresentation of microphysical uncertainty, (ii) randomly picked members produce a similar large spread than the operational 20-member ensemble, and (iii) perturbations in the initial and boundary conditions contribute little (if at all) to the spread of the ensemble, at least for periods with the strongest convective activity. Overall, the comparison highlights that both ensemble design (number of members) and the selection of uncertainty sources are critical for achieving reliable probabilistic forecasts in convective weather regimes.

The present study is based on four 24 h precipitation events. For such a small sample, proper probabilistic scores (e.g., continuous ranked probability score, rank histograms, reliability diagrams) do not allow a statistically robust comparison of MPHYS-ENS and the operational ICON-D2-EPS. Nevertheless, we have computed the Fractions-Skill-Score (FSS) of 24 h total precipitation amounts compared to radar-derived precipitation (not shown). Across all 4 d, the mean values for the selected precipitation thresholds and scales of the FSS is broadly similar between the two ensembles, indicating that MPHYS-ENS is capable of generating a level of skill comparable to that of the operational EPS. Nevertheless, marked day-to-day variability is evident. For the 8 June 2023 case, the scores are uniformly low, reflecting poor agreement with observations, whereas the 22 June 2023 case exhibits consistently higher FSS values, signalling substantially better performance. The 23 June 2021 and 15 July 2023 cases fall in between, with mixed regions of high and low skill. While the ensembles can achieve comparable FSS magnitudes, the skill is highly sensitive to the specific weather situation. Consequently, any quantitative assessment of relative performance would require a much larger sample of cases; the present four-day set primarily serves to demonstrate that the spread produced by MPHYS-ENS is of the same order as that of ICON-D2-EPS, showing the importance and the potential of taking into account microphysical parameterizations in the quantification of model uncertainty in a convective-scale EPS.