The increase in computing power and recent model developments allow for the use of global kilometer-scale weather and climate models for routine forecasts. At these scales, deep convective processes can be partially resolved explicitly by the model dynamics. Next to horizontal resolution, other aspects such as the applied numerical methods, the use of the hydrostatic approximation, and time step size are factors that might influence a model's ability to resolve deep convective processes.

In order to improve our understanding of the role of these factors, a model intercomparison between the nonhydrostatic COSMO model and the hydrostatic Integrated Forecast System (IFS) from ECMWF has been conducted. Both models have been run with different spatial and temporal resolutions in order to simulate 2 summer days over Europe with strong convection. The results are analyzed with a focus on vertical wind speed and precipitation.

Results show that even at around 3 km horizontal grid spacing the effect of the hydrostatic approximation seems to be negligible. However, time step proves to be an important factor for deep convective processes, with a reduced time step generally allowing for higher updraft velocities and thus more energy in vertical velocity spectra, in particular for shorter wavelengths. A shorter time step is also causing an earlier onset and peak of the diurnal cycle. Furthermore, the amount of horizontal diffusion plays a crucial role for deep convection with more diffusion generally leading to larger convective cells and higher precipitation intensities. The study also shows that for both models the parameterization of deep convection leads to lower updraft and precipitation intensities and biases in the diurnal cycle with a precipitation peak which is too early.

The Earth's atmosphere is home to processes ranging from scales as large as the planet itself, such as the trade winds, down to scales of angstroms (

Atmospheric models with a grid spacing of around 4 km and smaller have been considered to at least partially resolve deep convection

Nonetheless, deep convection is not yet fully resolved with grid spacings of 1–4 km. To fully resolve deep convection, one would require a grid spacing of around 250 m or less

Another subject that is often associated with weather and climate models at the kilometer scale is the use of the hydrostatic approximation in the governing equations. The hydrostatic approximation assumes the vertical accelerations to be small compared to the buoyancy force. This is normally the case when the horizontal length scale of the flow is much larger than the vertical length scale. With the hydrostatic approximation, vertical velocity can be derived from the continuity equation and thus becomes a diagnostic variable. The resulting system of equations is simpler and usually computationally less expensive to solve, which makes it an attractive option for models as long as the hydrostatic approximation is still suitable. For example, the nonhydrostatic version of the Integrated Forecast System (IFS) model from the
European Centre for Medium-Range Weather Forecasts (ECMWF) is about 80 % more expensive than the corresponding hydrostatic version at a grid spacing of around 9 km

There is not really a consensus in the scientific community about the horizontal resolution at which the hydrostatic approximation is no longer suitable. For example,

Several studies also primarily looked at the vertical velocities of hydrostatic and nonhydrostatic models at different resolutions. A maybe counterintuitive behavior of the hydrostatic regime is the development of too high vertical wind velocities at resolutions where the hydrostatic assumption is no longer valid. This is due to the fact that the vertical wind velocity is directly diagnosed from the horizontal velocities, and there is no nonhydrostatic process limiting the vertical mass flux. Simulations of a squall line with horizontal grid spacings reaching from 20 to 1 km by

Not only the system of equations but also the applied numerical methods are important when it comes to understanding the model behavior. The two models used for this study are very different in this regard: while the hydrostatic IFS model is a spectral model with a semi-Lagrangian semi-implicit scheme, the nonhydrostatic COSMO model is a Eulerian model with a split explicit scheme in the horizontal and a implicit scheme in the vertical dimension. These differences in design have direct implications on the conditions for numerical stability and the associated time step of the models. Thanks to the semi-Lagrangian treatment of advection, the time step in IFS is not limited by the Courant–Friedrichs–Lewy (CFL) condition

Compared to the many studies addressing spatial resolution in atmospheric models, the sensitivity to temporal resolution has received relatively little attention. Several studies identified time step as a very important factor when it comes to precipitation patterns

Deep convection is a dynamical process that is often happening very locally, involving only a few grid points in kilometer-scale models. The dynamics and concentration of moist variables at such scales are largely affected by diffusion. Diffusion may serve many purposes, such as eliminating numerical noise, increasing model stability, absorbing vertically propagating gravity waves at the model top, or also emulating cumulative effects of unresolved subgrid-scale processes

In order to improve our understanding about the role of some of the aforementioned factors in the representation of deep convection, we here present a model intercomparison between COSMO and IFS, addressing the following key questions:

What are the main differences between COSMO and IFS in the representation of deep convective precipitation? How do the precipitation patterns, precipitation intensities, and the diurnal cycle of precipitation look like at different resolutions, and how do they compare with observations?

Can we detect significant differences due to the use of the hydrostatic and nonhydrostatic dynamics in IFS and COSMO, respectively?

What is the effect of the time step on deep convection in the different models? Are there any disadvantages to using a large time step for simulations with explicit treatment of deep convection?

How does horizontal diffusion affect deep convection? Do the two models show differences in the representation of deep convection that can be accounted for by differences in horizontal diffusion?

The Consortium for Small-scale Modeling (COSMO) model

The Integrated Forecasting System (IFS) is the model used by the European Centre for Medium-Range Weather Forecasts (ECMWF) for its daily data assimilation and subsequent global forecasts. It is a hydrostatic model but can also be run using a nonhydrostatic extension which originally has been developed for the ARPEGE/Aladin models

The simulations cover 2 d from 29 May 2018 00:00 UTC to 30 May 2018 with heavy thunderstorms over Europe. Both models use 1 d lead time (28 May) and are initialized with ECMWF operational analysis data at a horizontal grid spacing of

Model description and setup.

For this experiment, COSMO has been run for the same case as above but with a varying amount of explicit diffusion from a monotonic fourth-order linear scheme with orographic limiter. This will give us some idea about the influence of diffusion on the model results and might explain some characteristic differences between IFS and COSMO. In COSMO, fourth-order diffusion is applied on model levels by introducing an additional operator at the right-hand side of the prognostic equation, similar to the following:

Three datasets are used for the evaluation of the model results: IMERG, RADKLIM, and IDAWEB. Comparing model results with observational data is a difficult undertaking. Next to the differences in spatial sampling (i.e., point measurement vs. grid cell averages), observations also suffer from several deficiencies (see below); therefore, different observational datasets often provide substantially different results, which is also the case in this study. Thus, observations should only be taken as a point of reference and not the absolute truth.

The Integrated Multi-satellitE Retrievals for GPM (IMERG) dataset

RADKLIM (Radarklimatologie) is a radar-derived and gauge-adjusted precipitation product from the German Weather Service (DWD, Deutscher Wetterdienst) that works on a

Radar-based estimates of rainfall allow for a high resolution in space and time, but they are also associated with some uncertainties. Sources of errors include cluttering from other objects, attenuation, variability of the relation between reflectivity and rainfall rate (

An intercomparison between RADOLAN, which is very similar to RADKLIM, and IMERG can be found in

Like satellite-based or radar-based products, also rain gauge observations involve uncertainties. They suffer from various errors such as evaporation, splashing, and most importantly wind effects which usually result in a low bias. The mean undercatch for Switzerland in summer is estimated to be 7 % with exposed stations having roughly twice the bias as well-protected sites

We start by showing an example of the spatial precipitation distribution. Figure

Accumulated hourly precipitation between 17:00 and 18:00 UTC on 29 May 2018 over the European domain. The top left panel shows precipitation data obtained from the multi-satellite product IMERG which are provided on a 0.1

Accumulated hourly precipitation interpolated to the RADKLIM domain over Germany. While IMERG and the results from the model runs show precipitation from 17:00 to 18:00 UTC on 29 May 2018, the corresponding RADKLIM interval is 16:50 to 17:50 UTC (see Sect.

When looking at the observations only, RADKLIM and IMERG agree well on the location of the precipitation. There are, however, visible differences in intensity and spatial extent. While some of these differences might come from the different measurement and processing techniques, differences will also be caused by the much higher spatial resolution of RADKLIM.

The cumulative frequencies of hourly precipitation within the European domain for all 48 h are depicted in Fig.

(a) It could be that the original time step in IFS is too large to properly represent deep convective processes associated with such high vertical wind velocities (see also Sect.

(b) By halving the time step, any subgrid-scale parameterization scheme will be called twice as often, which may affect precipitation. Notably, parameterizations such as cloud microphysics, shallow convection, or vertical mixing could experience time step sensitivity which could affect convective processes.

(c) One possibility that has been investigated was the sensitivity of the interpolation error in the semi-Lagrangian scheme to time step. In semi-Lagrangian schemes, the accumulation of errors is also a function of the time step and the error of the spatial interpolation procedure

Presumably, the time step dependency in IFS stems from a combination of (a), (b), and (c). But getting a better insight into the respective role of these factors would require further studies. For example, to quantify the effect of (a), one could perform a convergence analysis with IFS by changing the time step for the dynamics while keeping the time step for the subgrid-scale parameterizations constant. Similarly, to quantify the effect of (b), the time step for the subgrid-scale parameterizations could be changed while keeping the time step for the dynamics constant. However, this is beyond the scope of this work.

Cumulative frequency of accumulated hourly precipitation over the European domain for the whole 48 h period. Panel

Figure

Same as Fig.

Several studies have already shown that parameterized deep convection leads to a premature diurnal cycle in COSMO

For COSMO,

Diurnal cycle of precipitation over land in the European domain from 29 May 2018 01:00 UTC to 31 May 2018 00:00 UTC. The plot shows accumulated hourly precipitation, meaning that at 29 May 2018 01:00 UTC it shows the precipitation accumulated from 29 May 2018 00:00 UTC to 29 May 2018 01:00 UTC. Panel

Total precipitation during the 2 d has been analyzed for four domains: the whole European domain, the land part of the European domain, the RADKLIM domain, and the IDAWEB stations. The results are summarized in Table

Mean precipitation per 48 h in the different domains. The bold numbers represent the reference values of the respective domain that have been used to calculate the differences in percent.

For the whole European domain, all COSMO runs show clearly less precipitation than IMERG. And while all COSMO simulations with explicit deep convection produce about the same amount of precipitation, the one with parameterized deep convection is clearly an outlier with even less precipitation. The IFS runs with explicit deep convection show about the same amount of precipitation as IMERG. Also here, the run with parameterized deep convection shows significantly less precipitation than the explicit ones, similar to the results of the global simulations with IFS by

If one looks only at the precipitation over land, COSMO is much closer to IMERG, while the values from IFS are clearly larger. The effect of parameterized deep convection for both models is the same as for the whole European domain but even more distinct as the larger part of deep convection is happening over land. Moreover, IFS shows a clear sensitivity to time step with the amount of precipitation increasing with decreasing time step.

One of the properties of hydrostatic systems is supposed to be the overestimation of convective precipitation amount and area compared to nonhydrostatic systems

Total precipitation in the RADKLIM domain and at the IDAWEB stations has to be interpreted cautiously as both domains are rather small and the simulations cover only 48 h. But the numbers support the findings from the European domain in the sense that IFS seems to overestimate precipitation while COSMO generally underestimates it. Also, the precipitation-reducing effect of parameterized deep convection is visible for both domains.

Figure

Cumulative frequency of vertical wind velocity on the 500 hPa (upper row) and 850 hPa pressure levels (lower row) with the same layout as in Fig.

The profound impact of deep convection parameterization on the vertical motions in the atmosphere can be seen on the two panels on the left-hand side of Fig.

Both models show some sensitivity to horizontal resolution and the updraft velocities at 500 hPa are also comparable between the respective horizontal resolutions. The time step sensitivity seems to be more pronounced in IFS at both levels, which we interpret as resulting from the larger vertical motion in combination with a large time step. Nevertheless, the updraft velocities for the 4.5 and 2.9 km runs of the hydrostatic IFS are similar to those of the nonhydrostatic COSMO runs with 4.4 and 2.2 km grid spacing, respectively. Hence, the presumption that the vertical velocities could become unrealistically high due to the violation of the hydrostatic assumption at these resolutions cannot be confirmed. It is not clear to what extent the large time step of IFS influences these results, but results from

Another interesting aspect is the disparity in downdraft velocities between IFS and COSMO. The downdraft velocities in IFS are significantly lower for the explicit runs compared to the corresponding COSMO runs. At the same time the probability of having a downdraft is higher in IFS than in COSMO. The lower downdraft velocities in IFS could be related to the hydrostatic formulation of the governing equations, as the results from

Cross section of convective cells over Northern Italy produced by COSMO 2.2 km. The shading represents the vertical wind speed, the green barbs the overall wind direction and velocity, and the blue contour the clouds (cloud water

While kinetic energy spectra are generally not used as a measure of a model's skill, they can be useful in order to determine whether a model is able to reproduce the observed dynamics of the atmosphere

Power spectral density (PSD) plots for horizontal kinetic energy and

So while time step seems to have little influence on the horizontal kinetic energy spectra, it certainly has an influence on the vertical wind spectra, as the lower right panel of Fig.

The amplitude and shape of the power spectral densities of

Figure

The impact of horizontal diffusion on vertical wind velocities is not as prominent as for precipitation, but nevertheless there is a clear pattern. Both updraft and downdraft velocities are reduced with more horizontal diffusion. The relative change is most noticeable for the downdrafts, again with the diffusive configurations showing some similarities to the behavior of IFS in Fig.

Cumulative frequency of hourly precipitation

Figure

This observed increase of convective cell size and heavy precipitation with additional horizontal diffusion is very similar to the results in

One of the most important conclusions from the COSMO diffusion experiments is the evidence that horizontal diffusion in the governing equations does not act to simply smooth the precipitation field (which would weaken and broaden all cells, but not significantly change their number). Rather it appears that diffusion more fundamentally affects the dynamics: with higher diffusion, the available convective available potential energy (CAPE) is consumed by substantially fewer but broader updrafts (Fig.

Vertical wind at 500 hPa (upper row) and accumulated hourly precipitation (lower row) over the Netherlands on 29 May 2018 at 14:00 UTC. The first column shows the values from COSMO 2.2 km without explicit horizontal diffusion, while the middle column shows the results from a simulation with additional diffusion. The right column shows to the values obtained from the IFS 2.9 km simulation with

Power spectral density (PSD) plots for horizontal kinetic energy and

Figure

IFS produces more light precipitation than COSMO in all configurations and generally produces more precipitation. For both models, parameterized deep convection leads to more light precipitation but less medium-to-heavy precipitation. With explicit deep convection, the cumulative frequencies in COSMO are quite constant with regard to horizontal resolution and time step. This is not the case for IFS, which shows an increasing amount of heavy precipitation with increasing resolution. However, the deciding factor for the precipitation frequencies in IFS seems to be the time step. IFS runs with a smaller time step all lead to significantly more heavy precipitation than the respective runs with larger time step. It is not entirely clear how much this behavior is an effect of time step on resolved dynamics or the subgrid-scale parameterizations and their coupling. It is possible that a combination of these factors contribute to this time step sensitivity of precipitation intensities.

The comparison of model results with the three observational datasets IMERG, RADKLIM, and IDAWEB showed that both model's runs with explicit deep convection seem to be in the range of realistic values when it comes to precipitation intensities. In contrast, both runs with parameterized deep convection failed to reproduce the medium-to-heavy precipitation that could be observed during these 2 d and thus also produced significantly less precipitation.

Resolution and time step size also have an effect on the diurnal cycle of precipitation over land. Higher spatial and temporal resolutions seem to lead to an earlier onset and peak of precipitation. While we see a convergence of the diurnal cycle already at 4.4 km grid spacing in COSMO, IFS only shows signs of convergence at the highest resolution with 2.9 km grid spacing, most probably still due to the relatively large time step sizes of 120 and 60 s. Furthermore, this study also reinforces the evidence that parameterized deep convection leads to a much earlier onset and peak in the diurnal cycle. However, besides the two coarsest runs (COSMO 12 km and IFS 9 km) with explicit deep convection, all runs seem to have a too early phase in the diurnal cycle when compared with observations from the multi-satellite product IMERG.

The redistribution of heat and moisture due to parameterized deep convection has a distinct effect on the vertical velocities, leading to lower values for the downdrafts and especially the updrafts. From the runs with explicit deep convection, the respective updraft values at the 500 hPa level were quite similar between the nonhydrostatic COSMO and the hydrostatic IFS. This indicates that the hydrostatic approximation at a grid spacing of around 2–3 km still works well and does not lead to too high updraft values. However, the downdraft values in IFS are significantly lower than in COSMO throughout almost all simulations. This could be a characteristic of hydrostatic models

The influence of time step on wind velocities does not seem to be very crucial for the horizontal winds and both models show almost no change in the spectra of horizontal kinetic energy with different time steps. The vertical winds, however, are clearly influenced by the time step. This is visible in changes in spectra and also frequency distributions where a large time step seems to suppress high vertical velocities. The importance of resolving all these high velocities, compared to the significant additional computational costs involved with a smaller time step, is up for debate and probably also depends on application and purpose of the simulation.

Increasing horizontal diffusion in COSMO leads to more medium and heavy precipitation, making the precipitation frequency profile in this range look similar to the ones from IFS. Furthermore, more horizontal diffusion also leads to a reduction of downdraft velocities at the 500 hPa level and thus also makes the vertical velocity profiles of COSMO and IFS more akin. The added diffusion generally leads to fewer convective cells while increasing the horizontal extent of these cells. This could be a reason why the hydrostatic approximation still seems to work quite well even at a grid spacing of around 2–3 km, as the relatively large horizontal width of the cells might prevent them from entering the nonhydrostatic regime where the vertical extent of the buoyant cells becomes larger than the horizontal extent. But while this sensitivity to dissipation certainly would need a more detailed investigation, it seems to explain some of the characteristic differences between COSMO and IFS.

Given the significant structural differences between the two models, it is very difficult to confidently attribute differences in the shown results to specific model properties. While this study is able to give some indications, it also stimulates further research. For example, is the sensitivity in heavy precipitation of IFS with regard to time step size mainly a dynamical effect (with the normal time step being too large to properly resolve the updrafts) or rather an effect from the increased calling frequency of the subgrid-scale parameterization schemes? It would be intriguing to only vary the time step of the dynamics while leaving the time step for the physical parameterizations constant (or the other way around) in order to be able to answer this question. A hypothesis in this study is that we see an increase in heavy precipitation with more horizontal diffusion mainly due to an accumulation effect of the larger convective cells and not necessarily due to heavier instantaneous precipitation. This could be verified by a similar test configuration as for our diffusion experiment but with a focus on instantaneous precipitation rates with a high output frequency. Regarding the validity of the hydrostatic approximation, the current study supports a view that it is still suitable for a grid spacing of around 2–3 km. But is this mainly due to the rather diffusive behavior of IFS with its relatively large convective cells? A study with more focus on convective cell size rather than grid spacing could probably answer this

Model codes developed at ECMWF are the intellectual property of ECMWF and its member states; therefore, the IFS code is not publicly available. Access to a reduced version of the IFS code may be obtained from ECMWF under an OpenIFS license (see

CZ, NPW, and CS designed the experiments. CZ performed the COSMO model simulations and NPW the corresponding IFS simulations. CZ performed the analysis of model output and observations with supervision from CS and NB, as well as technical support from PDD and NPW. NPW, PDD, and CS were strongly involved in the discussion of the results. CZ wrote the paper with input from all other co-authors.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the two anonymous reviewers for their useful comments. We acknowledge PRACE for awarding compute resources for the COSMO simulations on Piz Daint at the Swiss National Supercomputing Centre (CSCS). We also acknowledge the Federal Office for Meteorology and Climatology MeteoSwiss, CSCS, and ETH Zurich for their contributions to the development of the GPU-accelerated version of COSMO. We would like to thank Elmar Weigl and Marcus Paulat (DWD) for their assistance concerning the RADKLIM dataset and Pirmin Kaufmann (MeteoSwiss) for his help regarding the IDAWEB observations. Peter D. Dueben gratefully acknowledges funding from the Royal Society for his University Research Fellowship and the ESIWACE2 project. ESIWACE2 has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement no. 823988.

This paper was edited by Chiel van Heerwaarden and reviewed by two anonymous referees.