Severe hailstorms have the potential to damage buildings and crops. However, important processes for the prediction of hailstorms are insufficiently represented in operational weather forecast models. Therefore, our goal is to identify model input parameters describing environmental conditions and cloud microphysics, such as the vertical wind shear and strength of ice multiplication, which lead to large uncertainties in the prediction of deep convective clouds and precipitation. We conduct a comprehensive sensitivity analysis simulating deep convective clouds in an idealized setup of a cloud-resolving model. We use statistical emulation and variance-based sensitivity analysis to enable a Monte Carlo sampling of the model outputs across the multi-dimensional parameter space. The results show that the model dynamical and microphysical properties are sensitive to both the environmental and microphysical uncertainties in the model. The microphysical parameters lead to larger uncertainties in the output of integrated hydrometeor mass contents and precipitation variables. In particular, the uncertainty in the fall velocities of graupel and hail account for more than 65 % of the variance of all considered precipitation variables and for 30 %–90 % of the variance of the integrated hydrometeor mass contents. In contrast, variations in the environmental parameters – the range of which is limited to represent model uncertainty – mainly affect the vertical profiles of the diabatic heating rates.

Due to the large damage potential associated with severe convective storms, the forecast of deep convective clouds should be as accurate as possible. Thus, numerous studies have been published on simulating deep convective clouds. These have investigated how environmental parameters like wind shear

In

In addition to thermodynamic profiles and environmental conditions determining the formation and structure of deep convective clouds, microphysical parameterizations also have been shown to play a role.

Additional relevant parameters are the size distributions and the fall speeds of hydrometeors.

The development of deep convective clouds is sensitive to both environmental conditions and model parameters, but these sensitivities are usually examined separately. A few studies, including

In general, the approach usually applied for the analysis of the sensitivity of the model output to changing input parameters is to vary a chosen parameter in a given range while other parameters are kept constant. This so-called

Here, we focus on the warm bubble as the trigger mechanism, as it is frequently used in idealized studies, but we extend the set of uncertain input parameters to include not only environmental conditions but also microphysical parameters. Consequently, we compare the impact of environmental conditions and microphysics
to quantify the individual contributions of the various parameters to the forecast uncertainty of precipitation-related quantities including hail. We also consider the vertical profiles of the diabatic heating rates in our analysis. This analysis and the choice of output variables are based on the results of the first author's PhD thesis

A general description of the model setup and the input parameters is given in Sect.

For the simulations in this study, the limited-area numerical weather prediction (NWP) model COSMO (Consortium for Small-Scale Modeling)

We have taken a staged approach to our analysis of the effects of uncertain inputs on model output uncertainty for COSMO. We first explored the effects of the environmental conditions (Sect.

The input parameters of interest in this study are assigned to either describe environmental conditions, microphysics, or both, where the parameter ranges relate to observations and model uncertainty. Regarding the environmental conditions, CCN concentration, INP concentration, wind shear, vertical temperature profile, and characteristics of the warm bubble, in terms of temperature perturbation and horizontal radius, are perturbed.
An overview of these parameters and their respective ranges is given in Table

Overview of the uncertain input parameters and their ranges regarding environmental conditions (Setup 1). The parameters marked by

CCN, essential for the formation of cloud droplets, affect the dynamics and microphysics of the clouds

According to several observational and modelling studies, directional shear is most important for the organization of convection

Depending on the choice of

The vertical profile of the potential temperature is implemented according to

It is based on the near-surface potential temperature

The warm bubble is characterized by a temperature perturbation

As the wind shear and the temperature are part of the operational forecast, their parameter ranges are the only ones that can be related to typical forecast errors. The ranges of the remaining parameters cover a wide variety of atmospheric conditions, since there is no information from a forecast. These specifications are identical to those of the sensitivity analysis related to typical forecast errors in

The microphysical parameters analysed in Setup 2 (S2) are the fall velocities of rain, graupel, and hail; the strength of the ice multiplication; and the shape parameter of the size distribution of cloud droplets. In addition, the CCN and INP concentrations are included in this set of input parameters.
Table

Overview of the uncertain input parameters and their ranges regarding cloud microphysics (Setup 2). The parameters marked by

The fall velocities of the precipitating hydrometeors rain, graupel, and hail are implemented in the model following mainly empirical equations based on measurements that describe the relation between the size or other characteristics of the particles and their fall velocities

Based on the results of the sensitivity analysis for hydrometeor and precipitation variables in S1 and S2, where the sets of environmental conditions and the cloud microphysics parameters are treated separately (Fig. 5 of

We identify the parameters leading to the uncertainty in each model output via a variance-based approach, which is a global sensitivity analysis meaning that all of the multi-dimensional parameter space is sampled

First, a set of uncertain input parameters including their respective ranges has to be defined. Depending on the number of input parameters, a choice of input combinations of the parameters is selected within the parameter uncertainty space. As the emulator is required to predict the model output equally well across the

The extension of a Gaussian distribution to an infinite number of variables is referred to as a Gaussian process

Variance-based sensitivity analysis aims to decompose output variance into contributions from the uncertain input parameters. These include both contributions from each individual parameter and contributions from interactions of the parameters. The decomposition of the variance

In the analysis, we consider several output variables for which emulators are derived as described above. These output variables, including vertically integrated hydrometeor mass contents, precipitation, diabatic heating rates, and the size distribution of surface hail, will be described in more detail in this section. The results of the sensitivity analysis are shown for variations of the microphysical parameters only (S2). Similar analyses for variations of the environmental conditions (S1) have been discussed in

The output variables of the model have to be reduced to zero dimensions in order to be represented by the emulators. We are not only interested in the variables that are linked to severe weather at the surface (as precipitation maxima and hail) but also in the in-cloud processes causing them and therefore in the microphysical properties of the cloud. To reduce the dimensionality of the output, the composition of the cloud is described by the vertically integrated mass content of each hydrometeor class that includes cloud water, hail, ice, snow, graupel, and rain.
The spatial and temporal mean is taken for the considered vertically integrated hydrometeor mass contents (all in kg m

The set of considered precipitation variables include the amount of hail at the ground per an output interval of 15 min, the precipitation rate of hail and the total precipitation rate (all in kg m

The results of the variance-based sensitivity analysis are shown as a bar plot in Fig.

Bar plot of the main effect for vertically integrated hydrometeor mass contents

Figure

The output uncertainties of the considered precipitation variables are all dominated by contributions from the CCN concentration (13 %–47 %) and the fall velocity of hail, modified by the scaling factor

Deep convective clouds usually cover a large area and thus are able to influence the surrounding atmosphere. Furthermore, diabatic processes cause a redistribution of energy such as heating due to condensation and freezing or cooling due to evaporation and melting. To examine how the simulated storm impacts the temperature profile, we interpret the vertical profiles of the diabatic heating rates.

In order to obtain statistically robust results and to minimize the effect of single extreme events, emulators are used to generate 10 000 realizations of the vertical profiles of the heating rates covering the whole parameter space. Subsequently, the mean and standard deviation are calculated over all profiles together. Using this method, we are able to link changes of the total heating rate to the individual hydrometeor classes. Furthermore, the standard deviation is a measure of how much the heating rates react to variations of the input parameters. Figure

Close to the ground the total heating rate is negative because of the cooling caused by the evaporation of rain. As there is a strong increase of the heating due to the formation of cloud water, the total heating rate becomes positive above a height of about 1.3 km and reaches its maximum of 5.7 K h

Corresponding to the heating by the formation of ice between 7 and 10 km, there are large contributions to the output uncertainty from the INP concentration in this height. Above, the output uncertainty of the total heating rate is dominated by the CCN concentration and the fall velocity of graupel. This is probably linked to the indirect effect of CCN and riming efficiency on the amount of supercooled water transported to the homogeneous freezing level. Furthermore, graupel is produced at these levels in our model as a result of the freezing of rain drops, and the graupel fall speed factor thus impacts the gravitational sink of the (small) graupel particles present at these altitudes.

The size distribution of hailstones reaching the ground is of interest regarding the damage potential of hail events. For the size distributions of hydrometeors, a generalized

Input values representing both lower and higher values of the parameter ranges used to analyse the size distribution of hail. Parameters marked with

The size distribution of surface hail is simulated using the emulators for all possible combinations of the high and low input parameter values for each setup (128 combinations in S2; 64 combinations in S1 and S3). The aim of this approach is to attribute the minimum and maximum hail size distributions to specific parameter combinations. Figure

The distributions in the two groups with either very low or very high number concentrations share common features regarding the combination of the input parameters. The lowest number concentrations of hail (over the entire size distribution) are found for regimes with a low value of the fall velocity of hail and a high value for the strength of the ice multiplication. These distributions show maximum number concentrations of 0.06–0.15 mm

The corresponding plot of the main effect (Fig.

In the next step we analyse the impact of the input parameters on the uncertainty of the output variables of hydrometeor mass contents and precipitation by comparing the results for the three different setups with changes of (1) environmental conditions only, (2) microphysical parameters only, and (3) both environmental conditions and microphysical parameters (S1–S3, see Sect.

To compare the main effects of the three emulator studies, the results are combined in a bubble plot (Fig.

Bubble chart of the contributions from all input parameters of the different emulator studies to the output uncertainty of cloud and precipitation variables. The main effects of all input parameters given on the

The CCN (100 to 4000 cm

The contributions from the INP concentration variations are mostly larger in S1 than in S2 for both integrated hydrometeor mass contents and precipitation. The main effects in S3 are a combination of S1 and S2, but the results are closer to those of S2 than to those of S1. Thus, the main effect of the INP concentration is smaller if other microphysical parameters are used as an input, possibly because other ice phase processes (secondary ice formation or riming) can suppress the sensitivity of a cloud to primary ice formation.

The behaviour of the wind shear is quite consistent for the considered output variables. Its contribution is in general small, except if the integrated rain water content is the target output variable. It is always larger in S1 than in S3, meaning that the wind shear has a larger impact on the output uncertainty if only the environmental conditions are varied. Similarly, the (already small) impact of

The main effect of the fall velocity of graupel is larger for the cloud variables than for precipitation. Furthermore, in most of the cases the fall velocity of graupel has a similar effect on the output uncertainty in S3, such that

When looking at the hydrometeor mass contents, the contribution from the fall velocity of hail to the output uncertainty is negligible except for the integrated hail and rain contents. However, it is the largest contributor to the uncertainty of the precipitation variables, presumably reflecting that hail itself and melted hail constitute a major part of the total precipitation. Here, its impact is larger in S3 compared to S2 for all variables so that its importance expands when also environmental conditions are involved.

The other input parameters (

In summary, we find that the uncertainty of the integrated hydrometeor mass contents and the precipitation mainly emerges from the uncertainty of the microphysics, in particular from the fall velocity of graupel for the hydrometeor mass contents and from the fall velocity of hail for precipitation. The contributions from the parameters characterizing the environmental conditions are rather small in S3.

In the literature, the focus of sensitivity studies is mainly on the effect of CCN concentrations on clouds, but there are also studies examining the effect of other parameters such as wind shear, temperature perturbation, or shape parameter of the cloud droplet size distribution. For example,

The impact of CAPE on deep convection is analysed by

In this study, the diagnostics of diabatic heating rates are implemented similar to

Vertical profiles of the mean total diabatic heating rate

There is diabatic cooling of about

Up to 4 km above the ground, the profiles of the mean heating rates are almost identical for the three considered setups. Also the standard deviations are small and almost negligible, which means that near the ground the total heating rate is rather insensitive to changes of the input parameters, both environmental conditions and microphysical parameters.
However, above 4 km the profiles of S1 and S2 deviate from each other. The maximum of the total heating rate reached in S1 is slightly higher, and the standard deviation enlarges to approximately 1 K h

Condensation of cloud water, which is a substantial contributor to the total heating rate in the lower and middle troposphere, is parameterized via a saturation adjustment scheme in our model. Nevertheless, it yields a large contribution to output uncertainty of the diabatic heating in all three setups. This effect might be even larger if a time-dependent treatment of condensation was used.

In this section, we analyse the impact of variations of environmental conditions and microphysical parameters on the size distribution of surface hail. As described in Sect.

Size distributions of hail at

For S1, the size distribution with the lowest number concentration (dashed blue line) has its maximum of

The maximum of the size distribution with low number concentrations of S2 (dashed red line) is only a fourth of the concentration of S1, while for the distributions with the highest number concentration (continuous red line) it is almost twice the amount. Hence, the spread of all distributions is larger.

For S2, the low (dashed red line) and high (continuous red line) hail size distributions are smaller and larger, respectively, than those for S1, leading to a larger spread in the distributions. The fall velocity of hail and the strength of the ice multiplication are the two microphysical parameters that mainly determine the number concentration of surface hail. Low number concentrations are found for a low value of the fall velocity of hail combined with a high value for the strength of the ice multiplication and vice versa.

When both the environmental conditions and the microphysics are perturbed, the lower limit of the size-resolved number concentration of hailstones approximately doubles compared to S1. The distribution with the highest number concentration has similar concentrations to S2. The combination of high INP concentrations and high fall velocities of graupel produce a low number concentration of surface hail, whereas low fall velocities of graupel (presumably resulting in more time for the riming of graupel and growth to hail) and high fall velocities of hail (possibly by leaving less time for melting below the cloud) lead to high number concentrations.

Comparing the results of the different setups, the distribution with the lowest number concentration of S3 is similar to the corresponding distribution of S1. Especially for small diameters the two distributions show similar number concentrations. In contrast, the distribution with the highest number concentration of S3 (continuous green line) resembles the distribution of S2 as high number concentrations are reached that are comparable to S2.
Furthermore, the spread between the distribution with the lowest and the highest number concentration is smaller in S1 and larger in S2 such that the spread of S3 is situated in between. Moreover, the controlling parameters identified in S3 include parameters from both environmental conditions (INP) and microphysics (

In summary, the environmental conditions and the microphysical parameters (with the spread of input parameters chosen in this study) have a comparable impact on the size distribution of surface hail. While the microphysical input parameters mainly determine the maximum number concentration, the environmental conditions substantially influence the minimum number concentration. In general, microphysical input parameters cause a larger spread of the number concentrations of surface hail than the inputs related to environmental conditions.

The results above should not be regarded as definite number concentrations of surface hail, as a bulk model is used here, and several studies note that the representation of hydrometeor sizes is more accurate in bin schemes

In our study, we have investigated how changes in the environmental conditions and cloud microphysics impact deep convection with a focus on the integrated hydrometeor mass contents, precipitation, diabatic heating rates, and the hail size spectrum.

The COSMO model was used to simulate deep convective clouds in an idealized setup, where convection was triggered by an artificial warm bubble. This rather simple setup was required to allow for a large number of simulations in which environmental conditions and microphysical parameters are modified. The straightforward approach for analysing the sensitivity of the model output to changes in the input parameters is to vary a chosen parameter in a given range, while the other parameters are kept constant. However, instead of this one-at-a-time analysis, we employed statistical emulation and variance-based sensitivity analysis where the contributions of the input parameters to the uncertainty of the output are quantified. The emulator approach offers a convenient tool for the identification of relevant parameters without the requirement of running a large number of extensive model simulations. COSMO simulations were used to train the emulators, while the variance-based sensitivity was based on the predictions from the emulators allowing for an identification of not only the impact of each parameter independently but also their interactions which cannot be captured by one-at-a-time analyses. In total, we evaluated three sets of input parameters. First, a set describing environmental conditions such as potential temperature and vertical wind shear was used. Note that the range of variation of these parameters is designed to mimic typical forecast errors and is therefore smaller than in earlier studies, which have encompassed a wider range of possible conditions. The second set of input parameters focused on cloud microphysics consisting of parameters such as the shape parameter of the cloud droplet size distribution or the fall velocity of hydrometeors. The third set combined influential parameters of both environmental conditions and microphysics. For all sets of input parameters, the integrated hydrometeor mass contents, precipitation, size distribution of surface hail, and diabatic heating rates were examined with respect to the output uncertainty or response to variations of the input.

The analysis of the integrated hydrometeor mass contents reveals that the CCN concentration is an important parameter contributing to the output uncertainty if only the environmental conditions are varied, whereas the fall velocity of graupel provides a large contribution if only microphysical parameters are varied. These parameters are crucial for the efficiency of warm- and cold-rain formation, respectively. The decomposition of the output variance given variations of both environmental and microphysical parameters is similar to variations of the microphysical parameters only, implying that regarding the integrated hydrometeor mass contents, the uncertainty in the microphysical parameters is more dominant in causing uncertainty in the output. Similarly, the CCN and INP concentrations are relevant parameters for the uncertainty of the precipitation output when environmental conditions are considered, while the CCN concentration and the fall velocity of hail dominate are relevant when microphysical parameters are analysed. The study combining both sets of input parameters shows a large contribution by the fall velocity of graupel to the output uncertainty of the hydrometeor loads and by the fall velocity of hail to the output uncertainty of the precipitation variables. Consequently, variations of the microphysical parameters are the prevailing source of uncertainty of the integrated hydrometeor mass contents and precipitation compared to variations of the environmental conditions.

We analysed the variability of the vertical profiles of the diabatic heating rates by using emulators to predict the profiles of 10 000 randomly generated realizations covering the whole parameter space. The mean profiles for the three sets are almost identical, with the exception of a deviation of the set with variations in microphysical parameters in the middle and upper troposphere. The variability is similar for the set with variations of environmental conditions only and the set with combined microphysical and environmental changes. The good agreement between the results of these two sets of input parameters is also confirmed by the component-wise analysis of the heating rates where the contribution from each hydrometeor class to the total heating rate is considered separately. Thus, comparing the impact of environmental conditions and the microphysics on the diabatic heating rates, the effect of the environmental conditions is dominant. This is in contrast to the result of the integrated hydrometeor mass contents and precipitation where the impact of the microphysical parameters is prevalent.

We have assigned two discrete values to each of the input parameters and then used the emulators to predict the hail size distribution for all possible combinations of the input parameters to understand how the surface hail is affected by variations of the environmental conditions and the microphysics. The parameters controlling the size distribution are the CCN concentration, the INP concentration, and the vertical temperature profile for variations of the environmental conditions and the fall velocity of hail and the strength of the ice multiplication for variations of the microphysics. The controlling parameters of the combined input parameters are the INP concentration and the fall velocities of graupel and hail. The range of number concentrations in which the size distributions are found in this combined set is a compromise of the two sets considering the environment and microphysics separately, where the distribution with the lowest number concentration is close to the results for variations of the environmental conditions and the distribution with the highest number concentration is close to the results for variations of the model microphysics. Accordingly, both the environmental conditions and the microphysics affect the size distribution of surface hail comparably.

In conclusion, the aim of this work was to identify the sources of forecast uncertainty and to determine whether the variation of the environmental conditions or the variation of the microphysical parameters leads to larger model output uncertainty. It can be expected that our results (in particular regarding the microphysical parameters) depend to some extent on the microphysics scheme of our model. However, the overarching aim of this study was not to emphasize the impact of a specific parameter but to quantify the relevance of environmental versus microphysical uncertainty in general. We expect that these results are less dependent on the microphysics scheme. In addition, future studies should address how far the results of our idealized simulations are transferable to real cases. For our choices of input parameter ranges, the impact of the environmental conditions versus cloud microphysics depends on the output of interest: the uncertainty in the output of the integrated hydrometeor mass contents and the precipitation is affected more by variations of the microphysics, while variations of the environmental conditions cause more uncertainty in the prediction of the vertical profiles of the diabatic heating rates. Further, a comparable impact of environmental conditions and microphysics on the size distribution of surface hail is found. Therefore, depending on the parameter of interest, the forecast uncertainty could be reduced by either an improved observational network and data assimilation providing a more accurate description of the environmental conditions or a revised microphysics scheme, in particular a revised parameterization of the fall velocity of graupel and hail.

Numerical values represented by the circles in Fig.

Numerical values represented by the circles in Fig.

The processed training datasets and the emulators are published via the open-access institutional repository KITopen (

JJ and KC provided the code for the emulator approach. CW conducted the analysis and wrote the original draft of the paper with contributions from the co-authors. CH conceptualized the project together with MK and BV, and CH edited the revised paper. CH, AB, MK, and BV contributed to the discussion and interpretation of the results.

The authors declare that they have no conflict of interest.

The simulations were performed on the computational resource ForHLR I funded by the Baden-Württemberg Ministry of Science, Research and Arts and DFG. Furthermore, we thank Felix Fundel from the Deutscher Wetterdienst for providing data on the prediction errors of the COSMO model.

This research has been supported by the German Research Foundation through the Transregional Collaborative Research Center SFB/TRR 165 “Waves to Weather” project “Microphysical uncertainties in deep convective clouds and their implications for data assimilation” and the Natural Environment Research Council (grant no. NE/I020059/1).The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Barbara Ervens and reviewed by three anonymous referees.