Upper-tropospheric deep convective outflows during an event on 10–11 June 2019 over central Europe are analysed in ensembles of the operational Icosahedral Nonhydrostatic (ICON) numerical weather prediction model. Both a parameterised and an explicit representation of deep convective systems is studied. Near-linear response of deep convective outflow strength to net latent heating is found for parameterised convection, while different but physically coherent patterns of outflow variability are found in convection-permitting simulations at 1 km horizontal grid spacing. We investigate if the conceptual model for outflow strength proposed in our previous idealised large-eddy simulation (LES) study is able to explain the variation in outflow strength in a real-case scenario. Convective organisation and aggregation induce a non-linear increase in the magnitude of deep convective outflows with increasing net latent heating in convection-permitting simulations, consistent with the conceptual model. However, in contrast to expectations from the conceptual model, a dependence of the outflow strength on the dimensionality of convective overturning (two-dimensional versus three-dimensional) cannot be fully corroborated from the real-case simulations.

Our results strongly suggest that the interactions between gravity waves emitted by heating in individual deep convective elements within larger organised convective systems are of prime importance for the representation of divergent outflow strength from organised convection in numerical models.

It is well known that the process of (deep) convective organisation and clustering is an important actor in physics and dynamics of the Earth's atmosphere

Top view of flow resulting from latent heating in convective updrafts (orange shading), as occurring in the upper troposphere. Left: a point source of heating with radially diverging flow. Centre: a line source of heating associated with squall lines leads to laterally diverging flow (blue arrows), but in the longitudinal direction, individual heating patterns along the squall line compensate; the diverging winds away from individual cells converge (red arrows) and neutralise divergent winds along the convective line. Right (top): complex situation with several updrafts in irregular positions relative to each other, (bottom) which leads to a more complex pattern of divergence and convergence zones as outflows collide, simplified in the schematic with an oval orientation of those zones. Reality will be even more complicated. The conceptual understanding is based on

Comparing different representations of deep convection (i.e. LES, convection-permitting simulations and deep convection parameterisations) is important as forecast products are increasingly based on high-resolution simulations, while global ensembles of weather and climate simulations are currently treating deep convection as a parameterised process

In this work, the state-of-the-art Icosahedral Nonhydrostatic (ICON) numerical weather prediction (NWP) model

Divergent and convergent flows can be interpreted as results of gravity wave emissions at the location of convective heating

Work fundamental for the interpretation of the conceptual model has been done in the late 1980s and 1990s

A continuous stream of upward-moving parcels in a convective system results in continuously generated perturbations, leading to gravity wave adjustment within the convective system and in the surrounding atmosphere. The upper branch of the flow following such an adjustment mechanism in the plane perpendicular to a quasi-two-dimensional convective system (e.g. a squall line) is the divergent outflow from deep convection, which has been investigated in

From the perspective of

Most of the above-mentioned works focus on short timescales of several hours but usually a fraction of a day. On timescales beyond about 6–10 h, it is thought that the distinction between instantaneous and integrated divergence response is important. The instantaneous divergence is produced by gravity waves, which while propagating on longer timescales (or, equivalently, in a system that rotates faster) are affected by system rotation

The variability in the instantaneous rate of upper-tropospheric divergence is governed by the combination of either quasi-two-dimensional or quasi-three-dimensional vertical overturning (or a mixture of those) and, herewith connected, the morphology of organised convective systems. In the current study, we examine whether variability in instantaneous outflow strength from realistic convective systems in NWP may be explained by the same concepts.

One of the mechanisms that can organise convection is actively driven by gravity wave dynamics

As gravity waves simultaneously impact the spatial distribution of convective updrafts and downdrafts and are generated by heating (cooling) signals produced in updrafts and downdrafts, complicated mutual interactions can occur

Other mechanisms, like vertical wind shear, cold pool propagation and (related) moisture convergence, also impact the organisation and clustering of convective systems. These mechanisms may interact with the gravity wave dynamics that (co-)organise the convection. In this work it is not of relevance which mechanisms cause convective organisation and aggregation, but it is important to be aware that those factors interact. A comprehensive review of convective organisation and relevant mechanisms is provided by

Furthermore, convective momentum transport (CMT) may modify upper-tropospheric flow perturbations induced by deep convection

The following hypotheses are investigated here:

The geometry of a convective system is statistically related to the local divergent outflow strength, where updrafts approximately in line produce comparatively less divergent outflow than those that resemble a point source of heating at given heating rates (as in Fig.

While convective systems aggregate, grow upscale and organise, the precipitation rate tends to increase, but the ratio between the instantaneous mass divergence rate and precipitation rate decreases on average (compare Fig.

Variability in CMT does not alter the typical (i.e. mean) ratio between the instantaneous mass divergence rate and precipitation rate, as found by

Therefore, we firstly investigate if sub-linear increases in the instantaneous mass divergence rate occur at increasing precipitation rates (corresponding to latent heating rates) in convection-permitting and parameterised convection ICON ensembles of a single event. The event is exemplary and will demonstrate whether the methodology is useful, as well as indicate first conclusions on whether the conceptual understanding is likely correct and represents dynamical feedbacks of convective aggregation in state-of-the-art NWP at mid-latitudes. Another aim is to investigate whether patterns resembling line and point sources may be separated, using our proposed methodology. If both of the leading aspects of storm morphology, resulting from line and point sources of heating on the one hand and convective organisation and clustering on the other hand, are connected to the instantaneous divergence variability, simulation setups are able to represent gravity wave interactions and the impact of storm morphology on instantaneous upper-level divergence patterns. Supposedly this is possible at 1 km grid spacing but not at 13 km resolution, when convection is parameterised.

In Sect.

Afterwards, we reflect on the results and their coherence in the discussion in Sect.

The organised convection over central Europe on 10 and 11 June is notorious for the Munich Hail Storm

Equivalent potential temperature at 600 hPa (blue–white–red), isotachs at 250 hPa (30 to 60

After all, several systems with mesoscale convective activity developed over Germany and the Alps during the afternoon and evening, which were co-located within the parameterised deep convection ensemble. Similarly, convection was relatively active in convection-permitting simulations over southern Germany in the (late) afternoon of 10 June (e.g. Fig.

This study investigates numerical simulations with ICON 2.6

global simulations, with a nest over Europe (“PAR”)

convection-permitting simulations over southern Germany (“PER”) using the local area mode (LAM).

Simulation settings within the three domains.

Ensembles have been used with the aim to sample an unspecific form of background convective variability within a similar large-scale flow configuration. To further sample the variability of the PAR setup, additional experiments with adjustments following

The results presented are mostly focused on the comparison of the PAR and PER ensembles and on the PER ensemble itself.

For a fair comparison, the divergence in convection-permitting simulations is low-pass filtered, whereby the variability in the wind field at scales up to 45 km is removed using a discrete cosine transform. Thereby, the convection-permitting and parameterised convection simulations obtain roughly the same effective resolution in the divergence field. Therefore the divergent outflows are well intercomparable, and there is no problem of small-scale divergence patterns in convection-permitting simulations (lacking in the parameterised configuration). The filtering step assures that the box integrations that we carry out are applied to datasets with very similar truncation scales.

Extraction of properties of individual convective systems (shape, area, etc.) can be achieved in the PER simulations. On the contrary, parameterised treatment does not lend itself very well to such an extraction procedure because it assumes that a statistically averaged effect of convection over larger scales exists and is represented

In order to single out the expected outflow regime (two-dimensional-like or three-dimensional-like), the dataset with properties of convective systems must be able to describe the degree of convective clustering, orientation, and the relative state of elongation of convective systems in time and space. These factors have been found to determine the relative (instantaneous) magnitude of outflow from deep convection

initiation of a moving box to track each convective system in a simulation;

ellipse fitting (blue box nos. 1–4 in Fig.

validation and ordering of obtained ellipse parameters (blue box nos. 5–6 in Fig.

matching between ellipse parameters and moving-box diagnostics, as obtained from a specific convective system and that specific simulation (brown arrow and first brown box at the bottom of Fig.

final check of the matched records (second brown arrow at the bottom of Fig.

Processing of the raw ICON PER simulation data to obtain the dataset used in the analysis in Sect.

The box (step in red) is used for integration of precipitation and divergence over a horizontal subspace that is constant in time (with respect to the moving-box centre). The convective systems propagate with relatively constant velocity north- or northeastward, and only one to three systems have been tracked in each simulation (see also Fig.

For each box and time step the following variables are calculated: firstly, the strength of convective momentum transport (CMT) is computed to determine whether and how this acceleration (deceleration) affects the upper-tropospheric divergence. The estimate of CMT is based on the cross-correlation products of flow deviation vectors (

The moving boxes are initiated, and they then track the systems independently of the ellipse-fitting procedure because merging of ellipses occurs frequently in the ellipse dataset. In the case of a merging event, ellipse parameters will weakly vary in time, but the spatial integration mask of the moving box should not change accordingly. If ellipse parameters vary strongly, the ellipses cannot be validated. The signal of the instantaneous divergence rate and precipitation rate within a box should predominantly be affected by the main, central convective system within the box and only be weakly affected by small/shallower neighbouring cells that develop from time to time around some of the systems.

Ellipse fitting and verification are used to quantify the geometry of convective systems, in line with

Subsequently, after integrating the instantaneous precipitation rate, divergence rate and convective momentum transport spatially (red arrows and boxes), additional validation measures check the distance between an ellipse centre (set to be

Finally, the ellipse characteristics of the ellipses contained within each box (elongation

The ellipse dataset fulfilling all conditions of quality control contains 456 records, in which the time evolution of 22 of a total of 28 convective systems is represented (following the validation procedure). This dataset is the basic dataset for the assessment in Sects.

Computationally feasible NWP resolutions require the application of a parameterisation to represent deep convection, although mostly just in current global models. Global convection-permitting simulations have only recently been utilised for research purposes

The philosophy behind the representation of deep convection is and has been generally different between parameterised convection and convection-permitting simulation approaches. For simulations with parameterised deep convection, the following is done:

Convective cells are not advected with the background flow but have their full life cycle within a cell; there is a split between larger-scale, explicitly represented dynamics and the parameterisations of sub-grid-scale motion (including deep convection) in each grid cell

An equilibrium assumption is done

The temporal resolution of the full life cycle of convection within an individual cell is either represented within a full time step (typically in climate models) or the adjustment process and reduction in CAPE take place over several consecutive time steps.

Even though there are small differences between the assumptions applicable to different convection schemes

Furthermore, the LAM domain is small (400 by 500 km), whereas the parameterised convection simulations cover most parts of Europe with a grid spacing of 13 km. A typical (mesoscale) convective system is easily contained within a box of several to tens of grid cells in each horizontal direction for ICON PAR, which means the system starts to get resolved if it grows sufficiently large

The track of one of the two convective systems in ensemble member 14 of the PER simulations is illustrated in Fig.

Precipitation rate (mm h

A total of 2 h later, two matches are found again: one very near the box centre and one to the north of the centre but within the box. The larger central one matches through the distance rule, but the northern one gets rejected.

At 18:30 UTC, an elongated convective system develops in association with the earlier central system (14:30, 16:30 UTC). Still sitting close to the box centre, it is the only ellipse within the box.

In Fig.

However, the northeastern system matches at just one instance: the last time step. This match is only valid for the larger dataset with relaxed conditions. This illustrates how convective (re-)initiation and small displacements can affect the ellipse parameters. Corresponding jumps in the evolution of ellipse parameters are filtered out. The wobbly interval is indicated by the dark purple rectangle at 17:30–18:00 UTC. Most ellipses in this interval are rejected due to wobbly ellipse parameters, but some are retained during the interval. A temporary shrinking in the axis lengths is seen (without consequent rejection in the validation) due to the stability criteria and interpolation from any prior and successive records within an hour. Another jump within the time window is seen in the offset parameters (Fig.

The evolution of the upper-tropospheric divergence, CMT and precipitation rate over the moving box can be found in the Supplement (Fig. S3). Around 13:00 UTC no records of the system are validated: the validation criteria have not been fulfilled (solid green outline in Fig.

Between 14:00 and 15:00 UTC two convective systems have been matched with the box (Fig.

The distance between the centre of an ellipse and the associated convective box centre is a maximum of 9 grid cell distances (20 km) for the strict dataset of 456 records (purple line versus solid pink rectangle in Fig.

Example of time evolution of ellipse parameters in the dataset for the same convective system as shown in Fig.

The representation and variability of instantaneous convective outflow rates in ICON PER and ICON PAR ensembles are compared here. In particular, the mean mass divergence rate over moving boxes and in the corresponding areas of persistent thunderstorm activity is investigated. First, the spatial–temporal characteristics of instantaneous divergent outflow rates are broadly assessed for the selected systems in both ICON PER and ICON PAR. This provides a basis for the quantitative intercomparison of the divergent outflow rates between both configurations, for which we condition on the precipitation rate (equivalent to net latent heating rate). Case-related information on the convective organisation and plausible assumptions on the outflow characteristics are used to further characterise the dataset.

After this comparison, Sect. 6 analyses the upper-tropospheric mass divergence rate versus the precipitation rate and the corresponding ellipse parameters (ICON PER only; this is motivated in the current section), which is a verification of the conceptual understanding presented in the Introduction (e.g. Fig.

The time evolution of the mean horizontal divergence rate over the moving boxes in ICON PER is displayed in Fig.

PAR profiles also reveal a strong divergence maximum directly beneath the tropopause (Figs.

According to Fig.

Results for PAR simulations are shown in Fig.

In Fig.

Figure

The ratio between the instantaneous mass divergence rate and precipitation rate effectively represents the normalised mass divergence rate

Figure

The PAR simulations, illustrated in Fig.

On average, enhanced outflow rates occur in PAR compared to PER at given net latent heating rates. The relationship for unperturbed parameterised deep convection (black, Fig.

In short, Fig.

As there are no systematic patterns of a residual instantaneous outflow rate–net latent heating rate relationship obvious in the ICON PAR simulations, the analysis of such patterns is restricted to the ICON PER configuration in Sect.

This section discusses the representation of instantaneous divergent convective outflow rates in ICON PER, following the conceptual model outlined in the Introduction, and then discusses the role of CMT. The conceptual model in the Introduction suggests that divergent outflow strength depends

linearly on the net latent heating rate (hence, also on the precipitation rate);

on the storm geometry (point or line heating source);

on interactions between outflows from individual convective cells as a result of convective clustering, through outflow collisions.

The elongation of convective systems is quantified by the ratio

Divergence rate–precipitation rate dataset in ICON PER simulations with colours indicating three similarly sized classes of axis ratios. Added are two black lines of constant

At lower precipitation rates below 6 mm h

The ellipse dataset is split into subsets for further investigation. After selecting the subset with the

Furthermore, variability in ellipse orientation

Figures

A reduction in the mass divergence rate may be caused by the collision of individual three-dimensional outflows from individual cells, as induced by convective aggregation. Hence, convective aggregation may reduce divergence rates, relative to isolated convective cells, as more precipitation cells develop within an area. Measures that can indicate the presence of developing and clustering convective systems are ellipse areas and areas of high (

The expected negative correlation of the instantaneous mass divergence per unit precipitation intensity

The robust negative correlation coefficient between

Conditional correlations between the ellipse area and

The analysis suggests the following evolution of the convective characteristics: increasing precipitation intensity forces a linear increase in the instantaneous mass divergence rate in the upper troposphere, initially. However, beyond a certain precipitation intensity the instantaneous mass divergence does not keep up with the initially linear relation anymore. At higher precipitation rates, instantaneous mass divergence tends to grow comparatively slower (i.e. negative feedback). This signal was exemplified by the developing squall-line-like structures in Fig.

In the Supplement (Fig. S6), surface-based and mixed/elevated convection subsets are analysed separately, where a fingerprint of convective aggregation is present, too.

For the larger dataset with less strict matching criteria, the effect of convective momentum transport on the mass divergence rate has been investigated by normalising both quantities with the precipitation rate (

Overview of the correlation structures assessed in Sect.

Overview of the correlation structures conditional on the precipitation rate as assessed in Sect.

Over the 11 bins containing 39–150 samples each, the weighted average of the conditional correlation coefficient is 0.31. The equal-weight average is 0.34. There are exclusively positive correlations with values up to 0.7–0.8 across the range of bins. Given these statistics, the true correlation coefficient probably lies within the interval 0.2–0.5. Therefore, a small fraction of outflow variability in the convective systems can probably be explained by variability in CMT (Fig.

No single data point with upgradient transport occurs within the dataset (Fig.

This study has investigated instantaneous divergent outflow variability from deep convection in ICON, conditional on the precipitation intensity in ensembles with parameterised convection (PAR) and convection-permitting (PER) setups.

PAR simulations show an approximately linear relationship between precipitation rate (a close proxy for vertically integrated latent heating rate) and the instantaneous outflow strength with little spread. Conversely, in PER a non-linear relation between these two quantities is found, accompanied by substantial scatter away from the mean relationship.

The convection-permitting simulations have been utilised to explore hypotheses on the controlling factors in the relationship between convective latent heating–surface precipitation and the upper-level divergence rate derived from idealised studies

The evidence for the clustering hypothesis is strong: the reduction in the ratio

As a third and final hypothesis we investigate the direct impact of convective momentum transport on instantaneous outflow strength. Given a certain precipitation rate, it is suggested by the findings here that flow perturbations induced as a result of convective momentum transport can likely impact the mass divergence rate slightly in ICON PER. However, details of the interactions cannot be derived from this study. Furthermore, note that the indirect impact of convective momentum transport is technically included when convective geometry and clustering are investigated, as convective momentum transport can impact convective organisation and therefore precipitation rates

Our analysis provides insight into the instantaneous divergent outflow variability in ICON for the selected case study and the mechanisms that can govern the variability. The amplitude of instantaneous divergent outflow is proportional to net latent heating rates, which is already known

The dimensionality hypothesis is not strongly supported by our analysis, although some indirect evidence points to an impact of dimensionality on divergent outflow rates. Three suggestions as to why the dimensionality hypothesis is not strongly supported by the statistics are made:

The chosen metric is sub-optimal – it is not able to distinguish nearly two-dimensional (“line source of heating”) and nearly three-dimensional convection (“point source”) well (Fig.

The (elevated) shear profile of this case does not induce the maximum possible variation in dimensionality of the deep convective overturning (from nearly two-dimensional to nearly three-dimensional).

Opposing statistical relations between ellipse parameter estimates (e.g. ellipse elongation

The first possibility is that our metric for system dimensionality, namely the ellipse elongation, does not adequately map the variability in the geometry of small convective systems. In our case study only a few systems develop clear structures. Future assessments involving more extensive datasets of convection-permitting simulations from across the globe and various cases or, alternatively, different algorithms (guidelines by

A second explanation of improper sorting of overturning characteristics could be insufficient variability in the overturning dimensionality in the investigated case study. We essentially may not sample nearly two-dimensional convective overturning, found in well-organised squall lines

Lastly, the estimated ellipse elongation parameter

In summary, the outflow geometry and dimensionality of convective overturn seem to contribute only weakly to outflow variability. Still, the signal associated with squall line development supports the dimensionality argument, consistently with findings by

The impact of CMT on the instantaneous outflow strength may be much more pronounced based on this study than based on the LES study of

Additionally, the ICON PER configuration is arguably most suitable for detecting subtle (reasonable, real-case) CMT impacts on the divergent outflow rates. Conversely, the setup does not allow for an in-depth understanding of the mechanisms behind instantaneous outflow variability due to the complexity of the scenario and the amplitude of the systems in close spatio-temporal proximity.

Overall, it is clear that further research is needed for a basic understanding of the interaction between the characteristics of divergent convective outflow and their relation to convective momentum transport, as well as to further describe the twofold (direct and indirect) role of convective clustering and organisation therein. The different role of CMT for instantaneous upper-level divergence between the LES and NWP study provides little foundation for more than speculation of the mechanisms that may play a role. Similarly, to investigate how the CMT accelerations could mechanistically affect flow predictability is beyond the scope of this work and would require advanced, specifically tailored methods.

Previous studies investigating the predictability of the atmosphere from a dynamical perspective have identified that ensemble spread amplifies strongly in regions of precipitation and convection in particular

An open key question is whether the variability in flow perturbations associated with convective outflows in the upper troposphere is comparably (and reliably) represented in simulations with resolved and parameterised deep convection. The findings of this work suggest that this is only the case when deep convection is explicitly resolved at sufficiently small grid spacing. In the parameterised setup or explicitly resolved setup with coarse-grid spacing (i.e. larger than 10 km), our results suggest that the ensemble is underdispersive in terms of instantaneous outflow variability, with a strong linear correlation between the divergent outflow rate and precipitation rate variability.

This work once more confirms the strong link between precipitation variability and flow variability in an ensemble. This close connection may lead to perturbation growth in a forecast or spread in an ensemble. The downstream propagation of perturbations is not directly addressed here. Nevertheless, the spatial–temporal distribution of instantaneous divergence variability (Fig.

This work suggests that the convective contribution to flow variability can be separated into a component of precipitation variability (i.e. along the

The separation of divergence variability into the above-mentioned components has significance for weather and climate modelling: only the former component seems to be accounted for at coarse grids (

The systematic differences between ICON PAR and ICON PER regarding the relation between surface precipitation and upper-level divergence rates (Fig.

Convective organisation can also affect the vertical extent of mesoscale heating patterns and thereby change the intensity of the local heating's divergent wind forcing, as vertical background stratification changes, although the results in the current study and

It is known that models represent convective organisation imperfectly, especially whenever a parameterisation scheme and coarse-grid spacing are used. Our work suggests that it is important to increase the understanding of convective organisation biases in models. These biases may interact with biases in the precipitation climatology and may even cause compensating errors in NWP. However, these compensating errors may be hidden, unless instantaneous mesoscale mass divergence spread produced at given precipitation rates is specifically included in an analysis. Apart from conditioning on precipitation rates, conditioning on e.g. the diurnal cycle and regional convective characteristics can importantly contribute to improved simulations across resolutions

Upscale growth and clustering of convective systems are found to be key players for the instantaneous magnitude of divergent outflows in practice, which is properly accounted for only by convection-permitting simulations at about 1 km horizontal grid spacing. This is probably because the small-scale gravity waves emitted by individual deep convective elements and their interactions after collisions are only well resolved at this grid spacing. Based on the second and, to a lesser extent, third hypotheses of this work, convective clustering affects dynamics. Divergence rates associated with convective heating increase non-linearly with the heating rate as convective systems grow. Therefore, it is needed to include non-linear increments of divergence rates with increasing intensity of convective systems into error growth studies, assisting these studies to extend all the way from the convective to the planetary scales. Consequently, the conditional convective perspective shaped here can be connected with the

The multivariate exploration of the instantaneous divergent outflow strength of deep moist convection in real-case weather prediction shows that its controlling processes are rather complex and cannot easily be distinguished and assigned to individual mechanisms. However, based on the analysis of a single convective event, major variability of the relationship between the precipitation rate and upper-tropospheric divergent outflow rate is explained by effects that were also present in LES analyses

The outflow is responsible for major ensemble spread in the divergent part of the upper-tropospheric wind during a convective event.

Convection-permitting (1 km horizontal grid spacing) simulations represent the effect of aggregation on instantaneous divergent outflow rates from deep convection, and substantial spread of divergent outflow rates exist at a given net latent heating rate.

Using simulations at coarser resolution probably implies assuming a (near-)linear relationship between the outflow rate and net latent heating rate.

Various indications show that the fingerprint of dimensionality is represented in the variability of the instantaneous convective outflow strength in ICON convection-permitting settings, but a case study comparing squall lines that highly resemble two-dimensional convection with less organised convection is needed to increase the confidence in this finding.

Convective momentum transport seems to weakly affect this outflow strength directly.

To understand convectively induced flow perturbations better, a separation into two components of convective variability is necessary – (1) variability in predicted mesoscale precipitation rates and (2) representation of the residual (conditional) flow perturbations, which depend on the cloud-scale dynamics.

The results of this work strongly suggest that the interactions between gravity waves emitted by heating of individual clouds is likely of prime importance for the representation of instantaneous divergent outflow rates from organised convection, which can successfully be achieved at convection-permitting resolution. Additional case studies are needed to revisit the role of the dimensionality of convective overturning.

List of parameters in the dataset of ellipse records, with their descriptions.

Ellipse parameters associated with two subsets of the full dataset analysed in Sect.

The code used in this work and the output for one PER and one PAR simulation are available in

The supplement related to this article is available online at:

EG (ICON PAR) and PK (ICON PER) carried out the simulations for this work, under the supervision of HT and AM. EG designed the study, developed the ellipse-fitting algorithm, carried out the analysis and wrote the paper with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors would like to acknowledge the computing time granted on the supercomputer MOGON 2 at the Johannes Gutenberg University Mainz (

The research leading to these results has been done within the A1 “Multiscale analysis of the evolution of forecast uncertainty” and B1 “Microphysical uncertainties in hailstorms using statistical emulation and stochastic cloud physics” subprojects of the Transregional Collaborative Research Center SFB/TRR Waves to Weather funded by the German Research Foundation (DFG).

This paper was edited by Juerg Schmidli and reviewed by Michael Whitall and two anonymous referees.