Version 3.7.1 of the WRF () is used in this study. The model was run with a 50 km (low resolution; LR), a 16.6 km (medium resolution; MR), and a 3.3 km (high resolution; HR) horizontal resolution in a domain displayed in Fig. . It is forced by the Global Forecast System (GFS) model as initial and boundary conditions. Temperature, humidity, geopotential, and velocity components are nudged towards GFS analysis data with a Newtonian-type method using a relaxation coefficient of 5×10-5 s-1 as recommended by, for example, , .

The microphysical scheme used is the scheme, which explicitly calculates the number concentrations of aerosols. The latter are represented in a simplified way according to their capacity to nucleate cloud water (“water-friendly” aerosols; WFAs) or ice water (“ice-friendly” aerosols; IFAs). Aerosol number concentrations are initialized and forced at domain boundaries by a climatology based on Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. While no surface emissions are applied to IFAs, surface emission fluxes are applied to WFAs in order to approximately equilibrate the loss of WFAs due to scavenging and nucleation. The radiation scheme is RRTMG (rapid radiative transfer model for general circulation models; ) and uses the cloud water droplets, ice, and snow effective radii of the microphysical scheme to resolve the radiative transfer equations. Another climatology of aerosols from is used in this radiative scheme and therefore is not affected by any changes in the microphysical aerosol climatology, which enables us to perform sensitivity experiments on the indirect effects of aerosols with a fixed aerosol direct effect. The scheme is used to parameterize convection. The microphysical effects of aerosols are not taken into account explicitly in this parameterization, although they can affect convection indirectly through modifications in the temperature or moisture profiles.

This configuration is the same as in , to which the reader is referred for additional detail.

The model was run to make two extreme simulations in terms of WFA and IFA microphysical concentrations. Both simulations start on 1 April 2013 (after 1 month of spin-up) and end on 17 September 2013. A very high aerosol emission level (1.75×107 kg s-1 for the whole domain) is applied in the first simulation, referred to as MAX or polluted simulation, and a very low aerosol emission level (1.75×10-4 kg s-1 for the whole domain) is applied for the other simulation, referred to as MIN or pristine simulation. Although these emission rates are extreme, maximal and minimal values permitted by the microphysics scheme reduce the range of variation of the number of WFAs (NWFAs) to between ∼10 and ∼10000 cm-3, and of the number of IFAs (NIFAs) to between 0.005 and 10 000 cm-3. Therefore these latter extreme emission rates ensure that both the NIFAs and the NWFAs in the MIN (and MAX) simulation remain close to their minimal (and maximal) permitted values, which correspond to a factor of 2×106 for the NIFAs and a factor of 103 for the NWFAs between the MAX and the MIN simulations. Such high differences in aerosol concentration between the two simulations ensure that aerosol indirect effects are strong enough to emerge from the potential noise between the MAX and the MIN simulations. On the flip side, extreme values of aerosol concentrations reach the bounds of permitted values in the microphysical scheme, suggesting that for these ranges of concentrations, the uncertainties associated with the parameterizations of microphysical processes may be more pronounced.

Another set of MIN and MAX simulations has been performed at a resolution at which convection is resolved (3.3 km) and in a smaller domain (HR domain) as seen in Fig. . An intermediate set of simulations was used to perform one-way nesting between the LR and the HR simulations, ensuring that the LR simulations force the HR simulations at their boundaries. These intermediate simulations were done at a resolution of 16.6 km in an intermediate domain (MR; see Fig. ) and with the same configuration as the LR simulations. Under these conditions, each grid cell of the LR domain corresponds to exactly 15×15 grid cells of the HR domain. The HR simulations have been performed without activating any convection scheme since horizontal resolution (3.3 km) is sufficient to resolve convection processes, which is the only difference in model configuration between the LR simulations and the HR simulations. While the microphysical effects of aerosol on convective clouds were not taken into account due to the use of a convection scheme insensitive to aerosol concentrations, the whole set of indirect effects is represented in the HR simulations, including small-scale and large-scale processes.

The simulation domain covers the Euro-Mediterranean region as displayed in Fig. . This figure also shows the difference in accumulated convective precipitation over the period of study between the MAX and the MIN simulations. It shows that most of the negative signal is concentrated over land regions where precipitation is more intense in this period of the year . The following analysis of convective precipitation reduction in the MAX simulation is conducted over the HR domain. Indeed, the location of the HR domain was chosen because of the high negative values of convective precipitation differences between the MAX and the MIN simulations in this area and because it is far away from oceanic areas where flux imbalance with the non-coupled oceanic surface may hinder interpretation as discussed in . Because of the short duration of our simulations, temperature at the first vertical grid level (centred around 28 m above the ground; hereafter referred to as surface) and convective precipitation hourly time series were collected for all grid points of the WRF that were inside the HR domain and then concatenated. To avoid snow precipitation, we selected only the events with daily mean temperatures warmer than 5 ∘C.

The method used to scale precipitation with temperature is similar to the one used by . Temperature has a diurnal variation and may be impacted by precipitation events. Since for each precipitation event we want the corresponding temperature that represents the air mass, the daily average temperature is used. We selected hours with strictly positive precipitation amount in both the MIN and MAX time series and placed the pairs of daily mean temperatures and hourly precipitation into eight bins of 5896 samples according to the daily temperatures. In each bin the 50th percentile of daily mean temperature, the 50th percentile of precipitation, and the 95th percentile of precipitation are used for our analysis.

We focus on the contributions of precipitation efficiency, surface water vapour mixing ratio, and maximum vertical wind speed to the difference of convective precipitation scaling with a temperature between the MAX and the MIN simulations. We define precipitation efficiency as the ratio between the total mass of condensate of a column and the effective rate of precipitation that reaches the surface. For the LR simulations, it is calculated using hourly output variables of the WRF and following the parameterization of implemented in the model in which precipitation efficiency is a decreasing function of cloud base height and vertical wind shear. Because the model output frequency is lower than the typical convective characteristic time, we expect large uncertainties. Precipitation efficiency is not explicitly calculated in the HR simulations; we therefore estimated it using the ratio of precipitation divided by the product of maximum vertical wind speed and surface water mixing ratio. For the LR simulations, the maximum vertical wind speed is calculated using the square root of surface-based convective available potential energy (CAPE), which is more representative of convective vertical motions than the resolved vertical velocity. These three variables are computed 1 h before the convective precipitation occurrence to better represent the air inside the updraught of the convective cell rather than the air inside its downdraught.

The contribution of each variable to the change in precipitation between the MAX and MIN simulations is computed for both median and extreme precipitation events, which are defined as follows. Median events are all events in which precipitation is between the 40th and the 60th percentile in at least one of the simulations (MIN or MAX). Extreme events are all events beyond the 90th percentile in at least one of the simulations (MIN or MAX). Median and extreme events are sorted as a function of the corresponding daily mean temperature and placed in eight bins with the same number of events per bin. For median or extreme precipitation, the median daily mean temperature is paired with each of the four variables (precipitation, precipitation efficiency, surface water vapour mixing ratio, and maximum vertical wind speed along the atmospheric column) in the MIN and the MAX simulations.

Figure  displays the 50th (panels a, c) and 95th (panels b, d) percentiles of hourly convective (panels a, b) and total (panels c, d) precipitation as a function of daily mean temperature at the surface for both the LR MIN (magenta) and the LR MAX (blue) simulations. Median total precipitation displays a negative scaling with surface temperature for both LR and HR simulations (Fig. a). Since the temperature range is spread over two seasons, it is likely that changes in large-scale forcings between spring and summer events may explain the decrease in median precipitation with surface temperature. Sub-CC scaling for median total precipitation is consistent with the study of in Australia. On the other hand, median convective precipitation follows a nearly CC scaling in our LR simulations, indicating that, unlike median total precipitation events, convective precipitation events seem to be mostly affected by changes in surface temperature rather than changes in large-scale dynamics.

Regarding convective precipitation extremes, a nearly CC scaling appears in the LR simulation. Using in situ measurements in Switzerland, found a scaling of 8.9 % ∘C-1 of hourly convective precipitation as a function of daily mean temperature. Lower but similar slopes are obtained in our study with a value of 6.1 % ∘C-1 for the LR MIN simulation and a value of 8.6 % ∘C-1 in the LR MAX simulation. and showed that the scaling between total extreme precipitation and daily mean temperature could be super-CC because of the distribution of convective and stratiform precipitation with respect to daily mean temperature. Convective precipitation is generally more intense and occurs at higher temperatures. Supposing that both convective and stratiform precipitation follow CC scaling, they argued that total precipitation will display a super-CC scaling for temperatures corresponding to the transition between stratiform and convective precipitation. Such an effect does not appear in our study since we can observe a slight sub-CC scaling for total extreme precipitation. The scaling of total extreme precipitation is therefore different from the hook shape found in the study in the Mediterranean area. As expected , precipitation extremes are increased in the HR simulations with respect to the LR simulations. However, the slopes of the HR simulations are rather similar to the slopes of total precipitation in the LR simulations.

Differences between the MAX and the MIN simulations are similar for both extremes and medians in the HR and LR simulations. We find that convective precipitation is reduced in the MAX simulation, but only at low temperatures. This temperature dependency slightly changes the scaling between the MAX and the MIN simulations, with higher slopes in the MAX simulation (around 8.5 % ∘C-1 in LR) compared to the MIN simulation (around 6.2 % ∘C-1 in LR). The fact that indirect effects of aerosols are weaker at high temperatures is probably due to the lower occurrence of clouds under these conditions. Figure  shows the mean COD calculated as in , as a function of daily mean temperature for both the MIN and MAX simulations for low and high resolutions. It shows a decrease in COD with temperatures in all of the simulations. When averaging over hours with a strictly positive COD, we found that the optical thickness of clouds is relatively constant over the temperature range (not shown), confirming that the decrease in COD with temperature is mostly due to a decrease in the occurrence of clouds with temperature. This tendency maximizes the indirect effects of aerosols at low temperatures and minimizes them at high temperatures. In their study of the impact of the microphysical scheme on the scaling of precipitation extremes with temperature, have also shown that the main effect occurs at low temperatures. They attributed the change in slope at low temperatures to a difference in the parameterization of hydrometeor fall speed. They argued that slower hydrometeor fall speed decreases precipitation efficiency through enhanced evaporation and detrainment and through reduced precipitation rate. In our case, convective precipitation is diagnosed with the same convective scheme in the LR MAX and LR MIN simulations, which takes into account neither aerosol concentration nor rain fall speed. Such a microphysical effect is therefore impossible for the LR simulations. In the HR simulations, however, convective precipitation is diagnosed by the microphysical scheme, which does represent the terminal fall speed of hydrometeors that may be different depending on the aerosol concentration, as found in other studies . This effect is one among others that possibly acts to change the precipitation efficiency between the HR MAX and the HR MIN simulations. The next section is dedicated to evaluating how strong these changes in precipitation efficiency are relative to the background radiative effect between the MIN and the MAX simulations.

To analyse the reduction in convective precipitation at low temperatures, we consider that precipitation can be approximately described by the following equation : 2Pr∝ϵ×Q×W, with ε corresponding to the precipitation efficiency, Q the water vapour mixing ratio at the surface, and W the maximum vertical wind speed. This description is mostly valid for convective precipitation which results from a parcel that raises from the surface . Assuming the small changes in precipitation that we observe between the MAX and the MIN simulations, one can write 3PrMAX-PrMINPrMIN≈ϵMAX-ϵMINϵMIN+QMAX-QMINQMIN+WMAX-WMINWMIN. Figure  displays relative changes in convective precipitation vertical wind speed, precipitation efficiency, and the surface water vapour mixing ratio between the LR MAX and LR MIN simulations for median and extreme precipitation. As expected from Fig. , the decrease in convective precipitation in the MAX simulation with respect to the MIN simulation tends to be weaker with increasing temperature, from -25 % at 10 ∘C to almost 0 % at 22 ∘C. Among the three factors that may impact the precipitation intensity, the vertical velocity seems to explain much of the reduction in convective precipitation. Indeed, among the 25 % of precipitation reduction at low temperatures, around 15 % is attributable to the weakening of vertical velocity in the MAX simulation. It is also striking in Fig.  that the variations in the difference of vertical velocity and of convective precipitation with temperature are perfectly similar, with stronger reductions for low temperatures than for higher ones, while both precipitation efficiency and the surface water vapour mixing ratio display insignificant or erratic variations with temperature. Indeed, the high variations in precipitation efficiency differences with temperature for precipitation extremes may not reflect a physical process but only the difficulty in retrieving precipitation efficiency from hourly outputs.

The fact that vertical velocity drives the changes in convective precipitation explains why CC scaling is completely inaccurate for predicting changes in convective precipitation by indirect effects. In fact, even the differences in the surface water vapour mixing ratio between the MAX and MIN simulations do not exactly follow a CC scaling due to increased relative humidity in the MAX simulation: while the CC law prediction is around -4 %, the reduction in the surface water vapour mixing ratio in the MAX simulation is often less significant. One would expect that the sub-CC scaling of surface water vapour mixing ratio differences would result in a sub-CC scaling of convective precipitation differences, but it is actually the reverse (super-CC scaling) because of stronger changes in vertical velocity. The results are similar for both extreme and median precipitation except for precipitation efficiency differences, which display small variations for median precipitation and erratic variations for extreme precipitation, which may not have a physical meaning. Indeed, differences in precipitation efficiency are constrained by the scaling of precipitation (Eq. ) and thus are expected to be negligible compared to the changes in vertical velocity. In our LR simulations, precipitation efficiency is calculated as a decreasing function of cloud base height and vertical wind shear. The increased relative surface humidity in the LR MAX simulation compared to the LR MIN simulation therefore would acts to increase the precipitation efficiency of the polluted simulation. The contribution of the change in the vertical wind shear is less evident, and its high temporal and spatial variability may explain errors in retrieving precipitation efficiency from hourly outputs.

Figure  is the same as Fig.  with the exception that it uses total HR precipitation. Although the differences in vertical velocity and the surface water vapour mixing ratio for median precipitation events have approximately the same behaviour as temperature in the HR simulation with respect to the LR simulation, MAX–MIN differences in the median of total HR precipitation are stronger than the differences of the median of LR convective precipitation. Such positive bias compared to LR convective precipitation differences may be expected since showed that stratiform precipitation is increased in the MAX simulation. In contrast, it was found that hourly extreme precipitation is dominated by convective events at high temperatures . The decomposition of precipitation as a product of a thermodynamics, dynamics, and microphysics term made in the present study is theoretically better adapted to convective precipitation than to stratiform precipitation and thus is not efficient in explaining differences in total median precipitation. In our LR simulations, we found that convective precipitation dominates extreme total precipitation from 10 ∘C (not shown), thus for most of our temperature bins. Therefore, the scaling of precipitation used in the present study (Eq. ) can be used for extreme total precipitation in the HR simulation. Differences in extreme total precipitation in the HR simulation are similar to the differences in extreme convective precipitation in the LR simulation and scale well with the differences in maximum vertical velocities.

In this set of simulations with explicit convection, changes in aerosol concentration may have larger effects on convective precipitation efficiency since aerosols directly interact with convective clouds. Such impact has moreover been stated in many previous studies . The similarity of the precipitation differences with and without parameterized convection suggests that the changes in convective precipitation efficiency are relatively small in the HR simulation. Precipitation efficiency was calculated only indirectly since it is not parameterized for explicitly resolved precipitation. This was done by using the ratio of precipitation to the product of maximum vertical velocity and surface water vapour mixing ratio. Changes in precipitation efficiency between the MAX and the MIN simulations remain small in most of the temperature range. However, one can note a more significant increase in precipitation efficiency (around 15 %) in the warmest temperature bin, which is associated with increased precipitation extremes in the MAX simulation. The exact nature of increased precipitation efficiency only at the highest temperatures is not obvious and requires further investigations which are beyond the scope of the present study. The use of hourly outputs may also not be adapted to analyse these particular extreme events at high temperatures which were shown to be shorter than extreme precipitation events at lower temperatures .

As mentioned in Sect. , vertical velocity is calculated as the square root of CAPE. As seen in Fig. , CAPE may be affected by both surface temperature and surface humidity. CAPE is calculated using the entire profile of temperature and relative humidity (RH). Along these lines, we want to quantify the contribution of both the temperature and RH profile changes to the decrease in CAPE in the MAX simulation. For that purpose we have substituted the vertical profile of temperature in the MIN simulation with the vertical profile of temperature from the MAX simulation, and we have calculated two additional CAPEs, i.e. CAPET (and CAPERH) calculated with the temperature profile from the MAX (and MIN) simulation and the relative humidity from the MIN (and MAX) simulation, as represented in Fig. . Using the four CAPEs (CAPEMIN, CAPEMAX, CAPERH, and CAPET) we can compute relative differences (ΔCAPERH,1, ΔCAPERH,2, ΔCAPET,1, ΔCAPET,2, and ΔCAPE; see Fig. ) and thus infer the contribution of temperature and RH vertical profiles to the change in CAPE between the MAX and the MIN simulations.

Figure  shows the total change in CAPE between the MAX and MIN simulations (ΔCAPE), the RH contribution ΔCAPERH=ΔCAPERH,1+ΔCAPERH,22, and the temperature contribution ΔCAPET=ΔCAPET,1+ΔCAPET,22 as a function of daily mean temperature for median and extreme precipitation events. The amount of CAPE is lower in the MAX simulation with respect to the MIN simulation, and ΔCAPE is more negative at low temperatures (-30 %) than at high temperatures (almost 0 %). However one can see that ΔCAPET and ΔCAPERH have opposite signs. Indeed, the RH contribution is positive and decreases from about +40 % at 10 ∘C to about 0 % at 22 ∘C for median precipitation events. The fact that this contribution is positive is not a surprise since we have seen in Fig.  that the surface RH is higher in the MAX simulation. We can see that this apparently weak increase in RH in the MAX simulation has a strong effect on the CAPE at low temperatures. However the main contribution is negative and comes from the differences in vertical temperature profiles: values range between -70 % at low temperatures and -15 % at high temperatures. Moreover, one can see similar variations in ΔCAPE and ΔCAPET with temperature.

Figure  is the same as Fig.  but for the HR simulations and total precipitation. The amount of ΔCAPE is larger in the HR simulation, with values that exceed -50 % for a wide range of low temperatures in both median and extreme precipitation. These large values of ΔCAPE result in small negative differences in maximum vertical wind speed that do not exceed -10 % and are not correlated with total precipitation differences for median total precipitation events (see Fig. ) because of the coexistence of convective and stratiform events. For extreme events, which mostly consist of convective events, the discrepancy between the strong changes in CAPE and the weaker changes in vertical velocities between the HR MAX and the HR MIN simulation may be explained by enhanced release of latent heat at the freezing level caused by increased vertical mass transport of water droplets in polluted conditions, as suggested by previous studies . Indeed, in the present study, CAPE is calculated using a simple formula that does not account for differences in the load of rising parcels as long as their temperature and their relative humidity remain unchanged. According to the theory described in , the aerosol concentrations of our MAX simulation may, however, be too high for invigorating updraughts but would instead weaken them. The reduced changes in vertical velocity between our HR MAX and HR MIN simulations may therefore have another origin. It is also expected that the diffusion efficiency increases when increasing aerosol loading since the resulting increase in cloud drop number would lead to an increase in the total surface area of cloud droplets, enhancing condensation and latent heat release . However, our simulations were done using a saturation adjustment scheme, which excludes the possibility of increased cloud condensation (and its resulting stronger updraught) in the HR MAX simulation due to this process. observed large increases in latent heating in the warm phase of clouds; these increases are thus not related to the theory of convective invigoration exposed in . They attributed these changes to more organized cloud structures that limit dry air intrusions in the core of convective cells. Such an effect may also hold in our simulations since cloud cover and surface relative humidity are increased in the HR MAX simulation compared to the HR MIN simulation (not shown). According to the differences between the variations in CAPE and the variations in vertical velocities, this effect would limit the decrease in precipitation by about 10 % in the HR MAX simulation.

The contributions are otherwise similar to those of the LR simulations, with mainly a positive contribution of RH and a strongly negative contribution from the temperature vertical profile.

The amount of CAPE is a non-linear function of the temperature and humidity profiles. Therefore, the change ΔCAPET,1 is different from the change ΔCAPET,2. Similarly, the change ΔCAPERH,1 is different from the change ΔCAPERH,2. The quantities ΔCAPET,1 and ΔCAPET,2 (as well as ΔCAPERH,1 and ΔCAPERH,2) delimit a grey area in Fig.  that represents the uncertainty (relative to the non-linearity of CAPE) of the temperature (and RH) contribution. One can see that the effects of CAPE non-linearity are generally lower than the difference between each contribution. Where the grey areas do not intersect, i.e. in almost the entire temperature range for median precipitation, and for the cooler part of the distribution for extreme precipitation, a comparison of ΔCAPET, ΔCAPERH, and ΔCAPE strengthens the interpretation presented above: the negative value of ΔCAPE can be attributed to temperature changes, partially buffered by RH changes.

However the vertical temperature profile can be changed in several ways; e.g. one can only change the vertical gradient of temperature or uniformly reduce the temperature on the vertical. In the first configuration the decrease in CAPE would be purely due to the increase in stability of the environment, whereas in the second configuration the decrease in CAPE would be due to the surface air parcel temperature, more precisely to its reduced release of latent heat due to a reduction in its initial water vapour content.

In this part, the temperature contribution is decomposed into two contributions: one from the vertical gradient of temperature and one from the surface temperature. The amount of CAPE can now be viewed as a function of three variables: the RH profile, the vertical temperature gradient, and the surface temperature. As displayed in Fig. , for a given RH profile (from the MIN or the MAX simulation), we have substituted the vertical temperature gradient (and surface temperature) from the MIN simulation with the vertical temperature gradient (and surface temperature) from the MAX simulation, and we have calculated four additional CAPEs using the four new mixed profiles. By calculating relative differences in CAPE, one can evaluate the contribution of the surface temperature ΔCAPETs=14∑i=1i=4ΔCAPETs,i and of the vertical gradient of temperature ΔCAPE∇zT=14∑i=1i=4ΔCAPE∇zT,i.

Figure  shows ΔCAPE∇zT, ΔCAPETs, and ΔCAPET (as in Fig. ) as a function of daily mean temperature for the LR simulations. The contribution of the vertical gradient of temperature and the contribution of the surface temperature are both negative, indicating not only that the surface temperature is lower in the MAX simulation but also that this cooling is less significant in the higher layers of the troposphere. Both processes tend to reduce the CAPE in the MAX simulation with respect to the MIN simulation. For median precipitation, the reduction in CAPE due to the vertical gradient of temperature (-10 % at high temperatures to -50 % at low temperatures) is more significant than the reduction in CAPE due to the surface temperature (-10 % at high temperatures to -20 % at low temperatures). For extreme precipitation, contributions are similar and range between -20 % at low temperatures to -5 % at high temperatures.

A similar analysis in the HR simulations is displayed in Fig. . The results are very similar to those from the LR simulations with the exception that for extreme precipitation with low temperatures, the temperature gradient contribution is significantly larger than the surface temperature contribution.

The maximum and the minimum values of ΔCAPETs,i (and ΔCAPE∇zT,i) delimit a grey area in Figs.  and that represents the uncertainty related to the CAPE non-linearity. It shows that for both HR and LR simulations, the contributions are clearly different at low temperatures for median precipitation events, whereas the uncertainty ranges tend to overlap at high temperatures. For extreme events, the non-linearity of CAPE does not permit us to distinguish between the two contributions for the entire range of temperatures of the LR simulations. In the HR simulations, the non-linearity uncertainty is also too large at high temperatures to differentiate between the two contributions. However the contribution of the vertical gradient of temperature is significantly weaker than the contribution of the surface temperature at the lowest temperatures of the HR simulations.