ICON is a non-hydrostatic compressible atmospheric model, which uses a triangular icosahedral mesh . ICON is developed and maintained jointly by the Deutscher Wetterdienst (DWD), the Max Planck Institute for Meteorology (MPI-M), the German Climate Computing Center (DKRZ), and the Karlsruhe Institute of Technology (KIT). The operational NWP system at DWD consists of a global ICON model, currently at 13 km grid spacing, with a European two-way nest at 6.5 km grid spacing (called ICON-EU), and the regional ICON-D2 with approximately 2 km grid spacing over central Europe . ART is a component of ICON that enables treatment of atmospheric chemistry and aerosols . In the following, we apply ICON-ART in a regional configuration similar to the operational ICON-D2, which we call ICON-D2-ART. The model domain has 542 040 cells in each of the 65 model levels. ICON uses a vertically stretched grid. For ICON-D2 the vertical grid spacing in the lowest levels is smaller than 100 m, but near the tropopause it is approximately 500 m seep. 121. The domain top is at 22 km height. In this study, we use ART only to simulate mineral dust. The aerosol model uses a modal distribution with a two-moment formulation . Mineral dust is represented by three log-normal modes with standard deviations of 1.7, 1.6, and 1.5 for modes dustA, dustB, and dustC, respectively. While the standard deviations are kept constant, the median diameters are variable depending on the simulated mass and number concentration. Chemical aging of dust is not taken into account in this study, although ICON-ART is in principle able to treat coated aerosol particles . DWD and KIT maintain a pre-operational global dust forecasting system based on ICON-ART with a grid of approximately 40 km with 90 vertical levels and a two-way nest with 20 km grid spacing (ICON-EU-ART). In this study we use boundary conditions for ICON-D2-ART from the ICON-EU-ART nest of the analysis cycle of this global mineral dust forecasting system.

The model physics of ICON-D2-ART is largely based on the physical parameterizations of COSMO-DE as described in , but in ICON-D2-ART the two-moment mixed-phased cloud microphysics of as described in is used. The most important recent change to the two-moment microphysics is that the PDA08 ice nucleation scheme has been replaced with a parameterization of the ice nucleation active surface site (INAS) density . The INAS approach greatly simplifies the coupling of the ice microphysics to the predicted mineral dust modes because only the total surface area of dust is needed as an input to the ice nucleation parameterization in addition to the supersaturation. In the setup of ICON-D2-ART used in the current study, mineral dust is not depleted by ice nucleation. Therefore we use an additional tracer variable to track activated ice nuclei as described in to avoid an overestimation of heterogenous ice nucleation. Homogeneous ice nucleation of liquid aerosols is parameterized based on .

Radiative fluxes in ICON are calculated using the ecRad radiation scheme . ICON makes use of the aerosol climatology. In the present study, ICON-D2-ART with prognostic mineral dust applies the monthly climatological values for the sulfate, organic and black carbon, and sea salt aerosol modes. Furthermore, we have reduced the sulfate, organic carbon, and black carbon aerosol optical depth compared to the original Tegen climatology to take into account the reduction in anthropogenic aerosol sources in Europe in the last few decades. For the prognostic dust, the radiative transfer parameters are calculated online depending on the optical properties and the size distributions . For resolved clouds the effective radius is calculated consistent with the microphysical assumptions of the two-moment scheme. Ice optical properties of are used. For effective radii larger than 100 µm, which is outside the tables currently provided by ecRad, the effective radius is rescaled with the assumption qi/reff= const., where qi is the ice mass fraction and r the ice effective radius. As an offline diagnostic the RTTOV forward operator is used to simulate Meteosat Second Generation (MSG) observations of the spinning enhanced visible and infrared imager (SEVIRI). RTTOV is called with model-consistent cloud information including the effective radii from the two-moment microphysics. For the visible channel at 0.6 µm, the method for fast satellite image synthesis (MFASIS) operator is applied as part of RTTOV.

ICON does come with a diagnostic sub-grid cloud-cover scheme, which is similar in spirit to the schemes of and , and provides cloud fraction and sub-grid liquid and ice water content for the radiation calculation. This sub-grid cloud-cover scheme is usually not explicitly coupled to the aerosol information provided by ICON-ART; i.e., aerosol–cloud interaction happens only on the grid scale but is neglected for sub-grid clouds. A parameterization, which changes this, is described in the following.

The sub-grid parameterization of dusty cirrus is based on the concept of a fundamental mixing instability of cold moist clean air with a drier air mass containing ice-nucleating particles (INPs). During the Saharan dust events, and especially as part of a dust-infused baroclinic storm (DIBS), the dynamical lifting can lead to a stratification with moist clean air in the upper troposphere located above drier dusty air below. The clean moist air has very few INPs, whereas the dusty air has INPs but lacks the moisture to form a cloud. At the interface between the two layers, heterogeneous ice nucleation can occur if the air mass has cooled to a sufficiently low temperature. This can be achieved, e.g., through lifting of the air layers by ascending motion underneath . Small amounts of mineral dust, which are mixed into the moist air above the rather dry dust-carrying layer, can initiate the formation of an ice cloud at the interface between the two layers (see Fig. ). Longwave cooling at this thin cirrus layer can further destabilize the layer and leads to turbulent mixing. This will entrain more dust from below into the moist layer and further thicken the cirrus cloud layer. At some point, the convective overturning becomes strong enough that even homogeneous nucleation may become relevant e.g.,. We favor the hypothesis of a mixing instability driven by heterogeneous nucleation over the original hypothesis of longwave cooling at the dust layer postulated earlier because it explains more naturally why the dusty cirrus cloud forms above rather than in the Saharan dust layer. Longwave cooling at the dust layer could play a role in the formation of the very first cirrus cloud at the moist–dusty interface, but on the other hand, the moist layer above reduces the longwave effect of the dust below. The longwave cooling at the dusty layer can contribute to the destabilization of the background profile, but lifting of the air masses, e.g., in a warm conveyor belt, leads by itself to a near-neutral or at least weakly stable stratification in the upper troposphere e.g.,. Given a weakly stable stratification, the longwave cooling at the thin ice cloud that forms at the interface between the moist clean air and the drier dusty air is sufficient to further destabilize the layer and will eventually lead to the formation of the dusty cirrus deck. Hence, it remains unclear whether the radiative effect of the dust is important in this process. In our hypothesis, the main role of the mineral dust is to provide INPs for the cloud formation, whereas the cloud itself causes the longwave cooling that leads to the turbulent mixing of the layers and the development of the shallow convective cirrus.

To characterize the amount of dust in the Saharan dust layer, we use the mass concentration of mineral dust cmode with mode∈{dustA, dustB, dustC}. That sufficient amounts of mineral dust reach the upper troposphere is the most important predictor in our parameterization. The moisture is quantified by the ice saturation ratio sice=pv/psat,ice, where pv is the vapor pressure, and psat,ice is the saturation vapor pressure over ice. As a measure of atmospheric stability we use the temperature lapse rate 1γk=∂T∂zk≈Tk-Tk+1Δz.

Note that ICON uses top-down indices; i.e., level k+1 is below level k. Dusty cirrus occurs in model level k if the following conditions are fulfilled: 2Tk<240K,3c^dust,k=max⁡j=k+1k+NcdustB,j+2cdustC,j>cdust∗,4s^ice,k=max⁡j=kk+Nsice,j>sice∗,5γ^k=min⁡j=kk+1γj<γ∗, with empirically determined thresholds cdust∗ = 50 µgkg-3, sice∗ = 0.7, and γ∗ = -6.5 Kkm-1 and with N=4 corresponding to a vertical depth of approximately 1500 m. The non-locality of this parameterization corresponds to the convective overturning and the mixing instability described above.

Note that cdust∗ does not include the smallest dust mode dustA, and the largest mode dustC has double the weight of dustB. This choice helps to avoid false alarms in the prediction of the dusty cirrus. The fact that the larger dust modes, dustB and dustC, are better predictors for the occurrence of a dusty cirrus than dustA is consistent with the increased ability of large mineral dust particles to act as INPs, whereas smaller particles are less relevant for the formation of ice clouds by heterogeneous nucleation .

The ice saturation threshold is rather low with sice∗=0.7. We assume that the process starts at scales considerably smaller than the resolved scales of the model. Fluctuation due to gravity waves or shear-induced clear-air turbulence can locally initiate the cirrus cloud formation at the air mass interface. Once an initial cirrus layer has formed, this triggers the microphysical mixing instability as described above.

The dusty cirrus is further characterized by a cloud fraction of 1, C=1; a maximum ice water content of IWCdusty∗ = 50 mgm-3; and an ice particle number density of 500 × 10-3 m-3. For the ice water content of the dusty cirrus in a model level k we apply a linear tapering with 6IWCdusty,k=IWCdusty∗max⁡min⁡1.0,c^dust,k-cdust∗cdust∘-cdust∗,0.1×max⁡min⁡1.0,s^ice,k-sice∗sice∘-sice∗,0.1, with cdust∘=70 µgkg-3 and sice∘=1.0.

According to our hypothesis and consistent with observations, convective overturning on scales smaller than 10 km is a crucial ingredient for the formation of the dusty cirrus cloud deck. Given the horizontal grid spacing of 2 km, with an effective resolution larger than 10 km, and a vertical grid spacing of approximately 500 m in the upper troposphere, it is reasonable that ICON-D2-ART is not able to explicitly simulate the chain of processes that leads to the formation of the dusty cirrus. It seems therefore appropriate to describe the dusty cirrus by a parameterization as part of the sub-grid cloud scheme as formulated above.

On 15 March 2022, 00:00 UTC, the center of storm Celia is located west of the Strait of Gibraltar. In the mid- to upper-troposphere it is associated with a deep trough that reaches equatorward as far as the western Sahara. Located south of a quasi-stationary anticyclone over Europe, the trough decouples from the westerly flow and forms a cutoff on 16 March 2022. On the eastern flank of the trough in a region of quasi-geostrophic forcing for ascent, large amounts of Saharan dust are transported poleward towards Spain and lifted from the lower into the mid to upper troposphere. Over the next 2 d, the dusty air mass moves further eastward over France and Germany with a dust aerosol optical depth (AOD) exceeding 0.8 in the ICON-D2-ART simulation (Fig. ).

The dusty air mass is associated with an extended cirrus cloud deck, which covers most of France and Switzerland on 16 March, 12:00 UTC, as can be seen in the Meteosat SEVIRI visible image shown in Fig. a (left). The infrared brightness temperature from SEVIRI reaches 215 K over large parts of the cirrus cloud deck (Fig. b; left). The ACI simulation of ICON-D2-ART shows clear-sky conditions over most of France and Switzerland and mid-level clouds with brightness temperatures larger than 240 K in the northern part of France (Fig. a and b; center). This simulation fails to simulate the cirrus associated with the Saharan dust, although the simulated dust AOD matches the spatial extension of the cirrus cloud. This changes with the sub-grid dusty cirrus parameterization in the ACI–dusty simulation, which enables ICON-D2-ART to simulate the cirrus cloud deck that is associated with the Saharan dust (Fig. a; right). The infrared brightness temperature is somewhat too low for 16 March, 12:00 UTC, but the spatial extent agrees well with the SEVIRI observations (Fig. b; right cf. left). On 17 March, 00:00 UTC, when cirrus clouds extend over the Netherlands, the North Sea, Denmark, and most of Germany, we find a good agreement between SEVIRI and the ACI–dusty simulation (Fig. c). The purely grid-scale ACI simulation is not able to simulate the cirrus clouds and shows only low- and mid-level clouds.

An overpass of the Aqua satellite operated by NASA provides CERES radiative flux data for 16 March at 12:30 UTC as shown in Fig. . Throughout this study the CERES SSF level-2 edition 4A instantaneous fluxes are used . The cirrus cloud deck over France leads to reduced outgoing longwave flux and a large reflected shortwave flux at the top of the atmosphere (TOA). The solar irradiance at the surface is greatly reduced to values below 200 Wm-2 under the cirrus cloud deck. Consequently, the ACI simulation without the cirrus cloud deck overestimates the shortwave surface flux and the outgoing longwave radiation (OLR) and underestimates the reflected shortwave at TOA. The errors in the solar irradiance exceed a factor of 2 over large areas in the ACI simulation. In the ACI–dusty simulation with the sub-grid dusty cirrus scheme, the cirrus cloud deck is properly represented, and all three radiative fluxes are reasonably consistent with the CERES observations.

To quantify the errors in the radiative fluxes, we make use of all CERES data from Aqua and Terra overpasses over the ICON-D2-ART domain during the 5 d period. These are a total of 35 overpasses but only 17 during daytime with non-zero shortwave data. The CERES data are processed in 20 min time windows with radiative flux output from ICON-D2-ART every 20 min. Based on this data, the biases and mean absolute errors (MAEs) of the ICON-D2-ART simulations can be calculated using each single CERES SSF footprint of 25 km diameter. For each CERES footprint, 64 ICON triangles around its center are averaged. Here and in the following the bias is identical to the mean error (ME). Figure  shows the result stratified by clear-sky/clean, clear-sky/dusty, cloudy/clean, and cloudy/dusty conditions. Clear sky is defined as both observation and model, having a cloud cover smaller than 5 %. As observed cloud cover, the MODIS cloud cover as provided with the CERES SSF data is used. Clean and dusty pixels are defined solely based on ICON-D2-ART because MODIS does not provide an AOD for cloudy pixels. The dust AOD threshold for dusty pixels is set to 0.1. Due to the explicit representation of the direct aerosol–radiation effect, we would expect that all three ICON-D2-ART simulations with prognostic mineral dust can improve the radiative fluxes in clear-sky/dusty conditions but show little changes in the clear-sky/clean situations. This is confirmed for the shortwave surface flux in Fig. c, but only a minor improvement is seen in the TOA fluxes. For cloudy/dusty pixels the ARI and ACI simulations show an improvement over the control run, but this is most likely also due to the direct aerosol effect because no significant difference exists between ARI and ACI. In fact, ACI with grid-scale aerosol–cloud interaction is slightly worse than the ARI simulation. The dusty cirrus parameterization used in the ACI–dusty simulation greatly improves all three radiative fluxes for cloudy/dusty pixels and also in the all-sky statistics. Especially the biases are dramatically reduced for ACI–dusty. The MAE is for all simulations rather high in cloudy situations because it suffers from a double-penalty problem and the fact that the clouds are far from perfect on small scales, but the reduction in MAE in the ACI–dusty simulation is nevertheless a great improvement over the other three simulations.

To gain some understanding of the vertical structure of this dusty cirrus event, Fig.  shows a comparison of the radiosonde measurement at Payerne, Switzerland, on 16 March, 12:00 UTC, with the corresponding profiles from ICON-D2-ART. The skew-T log-p diagram for the ACI simulation (Fig. a) shows a good agreement for the temperature profile and also for dew point temperature, except for a layer between 250 and 375 hPa in which the dew point temperature of the ACI simulation is 10 K too low. Initially, we thought that the lack of moisture in the layer, in which the dusty cirrus is observed, is the reason that the ACI simulation fails to predict the cirrus cloud deck. All attempts to track down this error, for example, in the initial or boundary conditions, proved to be futile, though. Interestingly, the ACI–dusty simulation corrects this error in the dew point temperature and shows a good agreement for the whole profile. This is even more remarkable as the dusty cirrus scheme itself does not directly modify the (grid-scale) moisture profile. As described in the previous section, it is a sub-grid cloud scheme that affects only the radiative fluxes. Hence, the only possible explanation is that the longwave cooling at the parameterized sub-grid dusty cirrus changes the dynamics and the moisture transport in the model in such a way that it becomes much more consistent with the observations.

Another set of variables for the same sounding at Payerne is shown in Fig. c and d. Here we focus on the vertical profiles of the ice saturation ratio and the dust concentration. Although standard radiosondes do not measure cloudiness or ice water content, the ice saturation ratio allows us to identify the location of the dusty cirrus in the observations. Due to the high IWC, the dusty cirrus layer is close to ice saturation, i.e., sice=1. The ACI simulation underestimates the ice saturation ratio in the dusty cirrus layer and is consequently not able to simulate any cloud formation. The sub-grid dusty cirrus parameterization represents the dusty cirrus as a cloud cover of 1 (dashed green line in Fig. d). To make the connection with the SEVIRI data, the infrared brightness temperature from SEVIRI and the RTTOV forward calculation is depicted by a small blue cloud symbol (darker colors represent the observations). For optically thick clouds this can be identified with the cloud-top pressure. First, this sounding confirms the concept of moist air located over drier dusty air; i.e., the dusty cirrus initially forms above, not in the dust layer. Second, it shows that the dusty cirrus layer of ACI–dusty is consistent with the observed profile, and the cloud top is consistent with SEVIRI. Third, it is remarkable that the simulated ice saturation ratio increases and reaches almost 1 in the dusty cirrus layer, although the dusty cirrus parameterization does not directly change the water vapor. That the increase in moisture has dynamical causes is consistent with the fact that the dust profiles change significantly as well when the dusty cirrus parameterization is employed. Compared to the ACI simulation the ACI–dusty case has a stronger vertical transport of mineral dust, and the dust mixes more efficiently with the moist layers above. This is consistent with our conceptual model but was not necessarily expected from this very simple sub-grid parameterization that does not explicitly modify the vertical transport of dust or moisture.

This behavior is also confirmed by the radiosonde profile at Lindenberg (Germany) from 17 March, 05:00 UTC (Fig. ). The observed sounding shows as a cirrus cloud layer between 200 and 300 hPa at ice saturation, similar to the Payerne sounding. The maximum of the dust layer is here at 300 hPa and no longer separated from the cloud layer. Hence, an observation like this could be interpreted as if the cloud has formed in the dust layer. The comparison with the earlier sounding at Payerne suggests, though, that the Lindenberg sounding is characteristic of the mature state of the dusty cirrus when the mixing of the moist layer and the dusty layer has already happened.

To substantiate the concept of moist clean air above drier dusty air prior to the dusty cirrus formation, we calculate 5 d kinematic backward trajectories with the Lagrangian analysis tool Lagranto;. The underlying data are 3-hourly ERA5 reanalysis on model levels and a regular 0.5∘ × 0.5∘ latitude–longitude grid. The backward trajectories are started from an equidistant grid of 25 km spacing surrounding Payerne (46–49.5∘ N, 6–10∘ E) and every 50 hPa between 450 and 200 hPa. The levels between 450 and 350 hPa represent the dust layer, and levels between 300 and 200 hPa represent the moist layer above (cf. Fig. c and d). The trajectory analysis reveals that the bulk of air parcels in the moist layer reaching Payerne on 16 March, 12:00 UTC, ascended almost 5 d earlier over the Caribbean and were transported eastward with the midlatitude jet (Fig. a). Humidity in this layer was anomalously high, reflected by values of specific humidity in the upper 10 % of the climatological distribution in the month of March based on ERA5 (not shown). The characteristic U-shape of the trajectories over the eastern North Atlantic shows their transport around the upper-level trough associated with storm Celia. As already indicated in the synoptic description, the air masses were lifted from 450 to 250 hPa ahead of the upper-level trough 36 to 18 h (between 15 March, 00:00 to 18:00 UTC) before arriving over Payerne (Fig. a and triangles in Fig. a showing air parcel locations 24 h prior to arriving over Payerne). We note that this lifting of the moist layer also occurred spatially in the region of initial dusty cirrus formation over France. Since the potential temperature remained nearly constant during this ascent, we conclude that the lifting was mostly dry adiabatic and – as discussed in the following – initiated through the injection of the dusty layer underneath via an ascending air stream. Concerning the dry dusty layer, the majority of air parcels ascended from the lower troposphere over North Africa during the 5 d period (Fig. b). Several air parcels start below 800 hPa and reach above 400 hPa a few days later, reflecting ascent in a warm conveyor belt WCB; in the warm sector of the forming cyclone. Only a smaller fraction, probably reminiscent of the moist layer, already ascended over the Caribbean. About 36 to 24 h prior to reaching the region of Payerne, the air parcels ascended on average from 600 to above 400 hPa, accompanied by latent heat release, reflected by an increase in potential temperature (Fig. b). This coincides with the lifting of the moist layer above and suggests that WCB ascent also lifted the air column above, corroborating the notion of two distinct air layers and the formation of a cirrus cloud deck through lifting and thereby adiabatic cooling and the injection of ice-nucleating particles into the moist layer see. Our main conclusion is therefore that the lifting of an anomalously upper-tropospheric moist air layer originating from the Caribbean through a synoptic-scale ascending WCB airstream underneath emerging from the Sahara and the associated injection of dust particles as ice-nucleating particles at the interface of the moist and dusty layers have created the conditions for the formation of the dusty cirrus cloud deck. Although a climatological investigation is beyond the scope of this study, we hypothesize that the complex interaction of the various components (upper-level moist layer, synoptic forcing, dust transport in WCB) makes dusty cirrus events so rare.

On 2–3 March 2021, the synoptic situation is similar to the case of 16 March 2022, with a trough in 500 hPa extending from the British Isles over Spain to Morocco. The southerly flow at the eastward flank of the trough advects mineral dust directly from the Saharan desert and North Africa to Europe. The event is not as massive as the 16 March 2022 event, but dust AODs simulated by ICON-D2-ART do exceed 0.5 over southern France on 3 March 2021, 12:00 UTC (Fig. ), and high dust AOD extends along a northeast line into Switzerland and southern Germany.

The MSG–SEVIRI images for 3 March 2021, 12:00 UTC, show a cirrus cloud band associated with the Saharan dust event, which agrees well with the simulated dust plume of ICON-D2-ART (Fig. ). As in the previous case, the ACI simulation fails to predict the dusty cirrus and shows clear-sky conditions instead (including the snow-covered Alps). The ACI–dusty simulation does have the cirrus cloud band, but the boundaries of the cloud deck are too sharp, and the infrared brightness temperature is too homogeneous compared to the SEVIRI observations. These are probably deficiencies of the simple diagnostic parameterization that can not capture any transient behavior during the formation and decay of the dusty cirrus. Nevertheless, the sub-grid parameterization clearly improves the simulation of this dusty cirrus event by ICON-D2-ART. These improvements are also seen in the comparison to the CERES SSF data from the Aqua overpass of 3 March 2022, 11:55 UTC (Fig. ). Whereas the ACI simulation completely misses the radiative signature of the dusty cirrus cloud deck, the ACI–dusty matches the CERES data remarkably well (the shortwave TOA flux is shown in Fig. S12 in the Supplement). For the broadband fluxes the tapering of the IWC of Eq. () seems to work better than for the RTTOV forward simulation of the MSG–SEVIRI channels. Both simulations have problems to properly represent the low clouds over the North Sea and England, but this is most likely not related to Saharan dust. For the validation of the vertical distribution of cloud occurrence, we can also make use of ceilometer data for this event. Results show a significant improvement of the modeled vertical cloud fraction for the ACI–dusty simulation compared to the other simulations (see Supplement and Fig. S3 for details).

The good agreement of ACI–dusty with the CERES SSF fluxes is also confirmed by the scores for all Aqua and Terra overpasses from 1–5 March 2021 shown in Fig. . For this time period all three ICON-D2-ART simulations with prognostic dust show a clear improvement in bias and MAE of the shortwave flux at the surface for clear-sky/dusty pixels (for scores of reflected shortwave at TOA see Fig. S5). For cloudy/dusty pixels the ACI–dusty simulation drastically improves the representation of the radiative fluxes. For all-sky conditions the bias of the outgoing longwave radiation is reduced almost to 0 for ACI–dusty, but significant biases remain for the shortwave fluxes due to issues with low clouds in ICON during this period. Validation of the case with radiosondes shows very similar results as in the case of 15–20 March 2022 (Figs. S14–S16).

The Saharan dust case of 4–8 May 2022 differs from the two previous cases in that the dust transport is more to the east, and dust is advected from North Africa to Italy and the larger Alpine region. In the ICON-D2-ART domain, only the southeastern part is affected by high dust AOD, which is rather persistent over several days (Fig. ). The dust event is associated with extended cirrus clouds, but in this case, all four simulations do have some cirrus cloud coverage over the Alps. In ACI–dusty the cirrus clouds are thicker and have a lower IR brightness temperature compared to ACI (Fig. ). Compared to the SEVIRI observations ACI–dusty significantly overestimates the cirrus cloud deck, and ACI underestimates the spatial extent but matches the IR brightness temperature better than ACI–dusty. Hence, in this case, the dusty cirrus parameterization intensifies the cirrus deck in ICON-D2-ART but overestimates the impact of the dust. This might be due to the simple diagnostic nature of the scheme that is unable to represent the microphysical aging and dissipation of the cirrus cloud. An Aqua overpass for 6 May, 12:55 UTC, confirms the behavior seen with SEVIRI and is shown in Fig. S13.

In the statistical evaluation of all CERES overpasses from 4–8 May 2022 shown in Fig. , the ACI and ARI simulations are clearly superior to ACI–dusty (for reflected shortwave at TOA see Fig. S8). Hence, the dusty cirrus parameterization leads to an unrealistic intensification of the cirrus formation during this dust episode. The result is a significant negative bias in the OLR and increased biases in the shortwave fluxes compared to the other ICON simulations. This suggests that the dusty cirrus parameterization is too simple to describe all dust events and the related cirrus clouds correctly. Given the simplicity of the parameterization, this is not surprising, though. The two main processes that are missing are, first, the transient behavior of the cirrus cloud depth during formation and decay of the cirrus deck and, second, the microphysical processes in the cirrus cloud, like aggregation and sedimentation. The lack of microphysical processes contributes to a prolonged lifetime and could cause the overestimation that we see in this case.

So far we have focused on case studies of Saharan dust events, which show some observational evidence for dusty cirrus occurrence. For any practical application of the dusty cirrus parameterization, it is essential that it does not deteriorate the forecasts during dust events without dusty cirrus formation; i.e., the false alarm rate for dusty cirrus needs to be small. To check this, we have performed simulations for three more Saharan dust events without observational evidence for dusty cirrus (see Table ).

Figure  shows the bias and mean absolute errors (MAEs) for the radiative fluxes validated by CERES SSF data (for reflected shortwave at TOA see Fig. S10). For the solar irradiance at the surface, we see a clear improvement in clear/dusty conditions dominated by the direct radiation effect of mineral dust. Hence, the improvement is already seen in ARI without further gain from ACI or ACI–dusty. The impact of mineral dust in clear/clean conditions is small, as expected. For cloudy/dusty conditions we find an improvement by ARI (because this includes partially cloudy pixels) and further improvements in MAE from ACI–dusty. The mean bias is the smallest for ARI and ACI and becomes negative for ACI–dusty. But a close inspection reveals that the small mean bias for ARI and ACI is an error compensation between cases in early spring, which have a positive bias in those simulations, and late spring, which have a negative bias. This change in bias behavior is related to the occurrence of deep convection in May/June, which contributes to a negative bias due to the overestimation of convective anvils in ICON-D2. Hence, the small bias of ARI and ACI should not be interpreted as a general forecast improvement. Similar arguments apply to OLR and reflected shortwave radiation at TOA. For both TOA fluxes we find the lowest MAE for ACI–dusty, but the mean bias becomes negative for OLR and increases for reflected shortwave radiation. Again, this is most likely related to an overestimation of deep convective anvils in ICON-D2, which leads to an error compensation in ARI and ACI. Hence, from this validation with CERES SSF level-2 data we find a small overall improvement from ACI–dusty compared to the other simulations, but further work would be necessary to, first, reduce the false alarms of the dusty cirrus parameterization and, second, improve the representation of convective anvils in ICON-D2.

The retrieval of solar irradiance at the surface from satellite data can lead to considerable uncertainties for cloudy pixels . To further support our results we have performed an additional validation with the 27 pyranometers of DWD's radiation station network in Germany. Due to the rather short time periods of 5 d, we refrain from a decomposition in clear-sky and cloudy conditions for the pyranometer data. As for the validation using CERES data, dusty conditions are defined based on ICON-D2-ART dust AOD exceeding 0.1. Figure  shows the results for the two major dusty cirrus episodes and for all six dust episodes investigated in this study. The biases and MAEs compare well with the results against CERES in Figs. , , and . This not only confirms our findings but also confirms that CERES level-2 instantaneous fluxes can indeed be used for this kind of episode-based analysis. The pyranometer results for the other dust episodes are provided in the Supplement (Fig. S2).

Note that the overall improvement seen with the dusty cirrus parameterization may become smaller for longer time periods and with more non-dusty cirrus episodes. Due to the fact that the dusty cirrus is a rare event, false alarms and double-penalty issues make it a challenging forecasting problem. The double-penalty problem occurs due to predictability limits on scales smaller than 100 km, when the verification is done on the cloud scale. Hence, further investigations and improvements of the scheme would be necessary before it can be generally applied.

More insight into aerosol–cloud–radiative effects can be gained from an analysis of the errors on individual CERES footprints. Figure  shows the (model-obs) errors as a function of the mineral dust AOD simulated by ICON-D2-ART. Hence, this analysis gives an indication of whether the modeled radiative flux has the correct sensitivity to mineral dust AOD. The control ICON-D2 simulation shows large positive errors for high dust AOD. This corresponds to the lack of direct and indirect aerosol effects during the Saharan dust episodes. A linear regression gives a slope of 336.8 Wm-2 for the sensitivity ΔSWDsfc/AOD. Taking into account the direct aerosol–radiative effect in ARI reduces these errors significantly and reduces the slope to 187.5 Wm-2, but large errors remain especially for high AOD, which correspond to the occurrence of aerosol–cloud interaction. Considering grid-scale aerosol–cloud effects due to mineral dust acting as INPs has only a marginal impact on ICON-D2-ART and gives only a reduction in the slope to 175.6 Wm-2. Only the newly developed sub-grid dusty cirrus parameterization is able to correct the large errors in solar irradiance for large AODs, which obviously correspond to the pixels below the extended cirrus clouds decks. Interestingly, the slope ΔSWDsfc/AOD becomes almost 0; actually it is already slightly negative with a value of -12.1 Wm-2. Similar behavior is found for OLR (see Fig. S11), but the impact of the direct aerosol is much weaker with a decrease in slope from 84.4 Wm-2 for control to 75.3 Wm-2 for ARI. The improvement from ACI is again marginal with a slope of 71.3 Wm-2, whereas the dust cirrus parameterization removes the dependency on mineral dust AOD, resulting in a slope of -4.9 Wm-2.

The fact that on the one hand grid-scale ACI has only a negligible impact and, on the other hand, the dusty cirrus parameterization fully removes the aerosol sensitivity is remarkable. It shows that for the six Saharan dust episodes of the current study, dusty cirrus formation is the dominant aerosol–cloud–radiative effect of mineral dust over Europe. This is somewhat unexpected because enhanced mineral dust concentrations during Saharan dust episodes should lead to more INPs being available for ice formation. It is often argued that high INP numbers increase the ice particle number concentration and subsequently increase ice water content and ice water path e.g.,. Our simulations suggest that this sensitivity is very weak in ICON-D2-ART. A possible physical explanation is the so-called freezing–relaxation feedback, which is usually described for homogeneous ice nucleation , but which does in principle also apply to heterogeneous ice nucleation. Freezing–relaxation means that within an active updraft the ice supersaturation increases until enough ice particles are present to efficiently deplete the supersaturation. This establishes a nonlinear feedback or buffering mechanism, which can dramatically reduce the sensitivity of the ice particle number concentration to INPs. Hence, a low number of INPs will lead to larger ice supersaturation but not necessarily to a significant decrease in ice particle number concentration. Similarly, a high number of INPs, as during a Saharan dust event, will reduce the local supersaturations within the updrafts but has little effect on the actual cloud properties like ice particle number, ice water content, or ice effective radius. This behavior is consistent with the recent findings of , who showed that aerosol–cloud effects due to volcanic aerosol perturbations in low-level liquid clouds are dominated by changes in cloud cover rather than by cloud brightening. Similarly, in our case the formation of the dusty cirrus and the corresponding increase in cloud cover dominate over changes in cloud optical properties due to mineral dust.