In SG experiments G4 and G4K, SO2 is injected at the Equator (0∘ longitude) throughout the altitude range from 18 to 25 km with a Gaussian distribution centered at 21.5 km. The OH oxidation of SO2 starts the production of supercooled H2O-H2SO4 particles; the size distribution of these particles is calculated in an aerosol microphysics module with a sectional approach, starting from gas–particle interaction processes (nucleation, H2SO4 condensation and H2O growth) and then including aerosol particle coagulation. Removal processes are included via gravitational settling across the tropopause and evaporation in the upper stratosphere ().

In the troposphere, the ULAQ-CCM model includes sulfate production from dimethyl sulfide (DMS) and SO2 emissions, with gas phase and aqueous/ice SO2 oxidation (by OH and H2O2, O3, respectively) to produce SO4 (; ). The tropospheric and stratospheric SOx budget in the ULAQ-CCM (for unperturbed background conditions) was recently discussed in , with a focus on the role of non-explosive volcanic sulfur emissions, and in , in connection with the SG.

Aerosol extinction, optical thickness, single scattering albedo and surface area density are calculated online at all model grid points every hour. This allows the interactive calculation of up/down diffuse radiation and absorption of solar near-infrared and planetary radiation by SG aerosols, with explicit full coupling of the aerosol, chemistry and radiation modules in the ULAQ-CCM model. This justifies the “composition–climate” name for this coupled model, which is more general than the usual “chemistry-climate” model denomination.

The ability of ULAQ-CCM to produce the correct confinement of sulfate aerosols in the tropical stratosphere has already been documented in the literature. This was carried out by a comparison with SAGE II data following the Pinatubo eruption or looking at the SG conditions during the period following the eruption (see ; ).

The formation of UT ice particles may take place via heterogeneous and homogeneous freezing mechanisms. In the latter case, the ULAQ-CCM model adopts the approach initially described in , which assumes ice crystals are only formed via the homogeneous freezing of solution droplets as a function of local UT temperatures and updraft velocities; this approach also includes the effects of a variable aerosol size distribution. These updraft velocities are obtained as the sum of a dominant term related to the TKE and a much smaller contribution from the large-scale tropospheric circulation (). Typical vertical velocity net values are on the order of 10–20 cm s-1 (see Sect. 3.1) and allow the formation of thin cirrus.

For the ice supersaturation ratio, we adopt a simplified probabilistic approach, starting from the knowledge of climatological frequencies of the UT relative humidity (RHICE), from which a mean value and a standard deviation can be calculated, assuming a normal distribution. Local ice supersaturation conditions (RHICE>100 %) are a result of turbulent ascent and can be found in the UT, in the vertical layer below the tropopause (where turbulent updraft conditions may be found) and above an altitude where T<238 K (i.e., the assumed threshold for the spontaneous freezing of solution droplets). Here, the conditions for ice formation are met and we may calculate the probability that RHICE>1.5 (PHOM). This represents the assumed threshold for the activation of homogeneous freezing (in our model this threshold does not depend on local temperature or water activity conditions), which is considerably higher with respect to the threshold for heterogeneous freezing (RHICE>∼1.3) (). This represents the probability that an ice particle could be formed via heterogeneous freezing (PHET) on a preexisting population of ice condensation nuclei (NIN), typically mineral dust or black carbon (BC) particles transported from the surface.

The size distribution and number density of ice particles formed via heterogeneous freezing (nHET) is calculated utilizing the formulation of and the ULAQ microphysical scheme adopted for polar stratospheric ice particle formation (). NIN is the sum of grid point model-predicted concentrations of mineral dust (DU) and BC aerosols (NDU and NBC, respectively) and is used as the population of available condensation nuclei, with PHET being the probability that RHICE>1.3 at any model grid point. The problem in this case is the actual availability of solid ice nuclei. A low fraction of activated IN is suggested in the literature (fDU=1 % for mineral dust and fBC=0.25 % for BC) because the large majority of IN will be rapidly coated by sulfate (). The number density nHET is then obtained as nHET=fBCNBC+fDUNDUPHET.

The specification of the active ice fraction for both mineral dust and BC represents the major source of uncertainty for UT ice particle formation via heterogeneous freezing. Considering the above assumptions, homogeneous freezing normally dominates ice particle formation in the ULAQ-CCM model, with respect to the heterogeneous freezing mechanism. Whilst this may not be considered a general conclusion, it is assumed to be valid in all thermodynamics conditions and any local atmospheric composition; this has been shown, for instance, in , where a predominance of heterogeneous freezing over homogeneous was found. In general, the freezing mechanism that dominates in the atmosphere is still very uncertain.

The calculated mass mixing ratio of ice formed in the ULAQ-CCM model through both freezing mechanisms is shown in Fig. a, c for two pressure layers, 150–200 and 350–400 hPa, where the ice formation is greater in the tropics and mid–high latitudes, respectively. These calculations are compared against the MERRA-2 (; ) and ERA5 reanalyses (), and are all averaged over the same decade (2003–2012). For the upper layer (150–200 hPa), we also show the MLS satellite retrieval in Fig. S1 in the Supplement (), which compares very closely to the ERA5 reanalysis. Tropical ice formation shows a strong land–ocean asymmetry due to significantly higher PHOM and PHET values over land. For both pressure layers, the magnitude and spatial distribution of the ice mass mixing ratio are comparable between the ULAQ-CCM and MERRA. Regarding the datasets used to compare against our model results, note that there is a large spread amongst retrievals (such as MODIS or CALIPSO) and amongst reanalyses (; ). In particular, MERRA-2 appears to be at the lower end of the spectrum with regards to some quantities, such as ice water path. Considering that the dataset only considers non-precipitating ice (), this quantity might be closer to the one simulated in our model and thus would allow for a more correct comparison.

While the probability of homogeneous ice formation is defined as above, the number density and size of the ice particles formed this way is determined by the local temperatures and vertical velocities, in addition to the competing ice formation mechanism – heterogeneous freezing. The lower the temperature, the faster the nucleation rate; thus, more ice crystals can be formed. Conversely, higher vertical velocities increase the saturation ratio, leading to more ice crystals being formed before water deposition on ice crystals reduces supersaturation below the threshold. The spatial distribution of the cirrus ice optical depth (OD) in the model is calculated as follows: τice=Δz∑i∑jQextπrij2nij(r), where the extinction efficiency coefficient Qext∼2 at all visible wavelengths for ice particle sizes is on the order of 5–50 µm; i is an index for the vertical layers, and the sum is over all the vertical layers in the UT; j is an index for the particle size bins, and the sum is over the whole size distribution; rij is the particle radius at the ith layer and jth bin; and nij is the corresponding ice number density.

Equation () can easily be applied to the model, and the results are shown in Fig. a. An evaluation can again be made using the ice mixing ratio from MERRA-2 and ERA5 (shown in Fig. b, c for two specific pressure layers), and the ULAQ-CCM values of the ice particle effective radius. With these two quantities we indirectly derived τice at every horizontal grid point in Eq. (2), using the hydrostatic equation: τice=Qext32Δpg1ρice∑iχiri, where the sum is, again, over all the vertical layers (constant Δp=50 hPa), g is the acceleration of gravity, ρice is the ice bulk density, ri is the ULAQ-CCM effective radius at the ith layer, and χi is the MERRA-2 and ERA5 ice mass mixing ratio at the ith layer. Through this process, we obtain the optical depth values in Fig. b, c. The ODs are comparable in terms of spatial distribution, with the highest values in the tropics over land. However, the absolute values in the ULAQ-CCM model are significantly smaller over the tropics. The reason for this is that updraft velocities result in a relatively narrow interval (w<30 cm s-1) when only calculated as a function of TKE (as in the ULAQ-CCM), while thick cirrus formation takes place from strong (and less frequent) convective events (w<100 cm s-1). This detrained ice that originates in deep convection is not included in our model formulation.

In Fig. , we show the model-predicted fraction of ice formed through heterogeneous freezing in terms of optical depth (Fig. a) and zonally averaged extinction (Fig. b). In both panels, we see that a large part of the ice particles formed via heterogeneous freezing is located in the tropical band at lower altitudes, where a higher concentration of mineral dust and BC ice nuclei can be transported from the surface. In these regions, the fraction of ice formed this way can be as much as 80 % of the total.

In Fig. a, b we show the model calculated vertical profiles of the ice particle number density averaged over the tropics (Fig. a) and the extratropics (Fig. b), superimposed with the time variability produced by changing conditions of vertical velocity, temperature, and PHOM and PHET. The ice number density maxima are located at different altitudes in the two latitude bands, close to 13 km in the tropics and 8 km elsewhere. This is clearly expected from the latitudinal variability of the tropopause height.

With a procedure similar to the one described above for the ice OD, we may derive a first-order approximation of the ice number density from the MERRA-2 and ERA5 ice mass mixing ratio and ULAQ-CCM radii. Similar to Eq. (), for the ice number density (ni) at each vertical layer we obtain the following expression: ni=34π1ρice1r3χi.

The results from Eq. () (circles in Fig. a, b) show that while the model and the indirectly derived values from the reanalyses agree in terms of the general vertical distribution and localization of the maxima in the extratropics, the ULAQ-CCM tends to have smaller number densities in the tropics in the 10–13 km layer. Again, this should not be surprising in light of the fact that we are focusing on a specific type of cirrus cloud particle.

Figure c shows the model-calculated values of PHOM, as a 2-D zonally averaged distribution. Using these PHOM values, it is possible to scale a ni value measured in the midlatitude airborne campaign of during a young cirrus formation, to derive an average climatological value to be considered consistent with our modeling approach. measured a midlatitude ice concentration value of n=0.3 cm-3 in a young cirrus cloud at T=220 K and p=320 hPa. If we scale this result with our corresponding PHOM=12±3 %, a “climatological-mean” value n=0.025±0.005 cm-3 is obtained, which is close to our model prediction value of 0.031±0.008 cm-3 (Fig. b).

Relevant aerosol and ice quantities calculated in the ULAQ-CCM model are summarized in Table  and compared with available satellite observations. The first two rows in Table 2 compare the ULAQ-CCM results for stratospheric sulfate optical depth (OD) and the tropical effective radius (reff) against SAGE-II and AVHRR satellite observations (; ), under post-Pinatubo conditions (). This is undertaken to highlight the realistic representation of the gas–particle conversion and aerosol microphysics processes in the model, along with the aerosol large-scale transport in the lower stratosphere in the case of a major tropical volcanic eruption, which may be used as a proxy for SG with an equatorial SO2 injection. A comparison of the aerosol effective radii under volcanic and background conditions (see rows 2 and 3 in Table ) clearly shows the effects of the sulfuric acid condensation on the size extension of the aerosol accumulation mode and how this is represented in the model.

The bottom 5 rows in Table  compare the global budget calculations for tropospheric ice particles with values obtained from the MERRA-2 and ERA5 reanalyses (ice mass mixing ratio) and ULAQ-CCM effective radius (compared in row 7 with the ice effective radius as retrieved by MODIS). The simultaneous use of these two products (reanalysis values for ice mass mixing ratio and model calculated radius) allows for an indirect calculation of the ice optical depth (row 8 of Table 2), as previously discussed. The ULAQ-CCM OD underestimation is mostly related to the lower values of the ice mass mixing ratio in the largest portion of the upper troposphere (see row 5 of Table ) and may be, in part, explained with the inclusion of a relatively narrow interval for updraft velocities (w<30 cm s-1).

The values are given separately for the ice formed through homogeneous and heterogeneous freezing.

The ice extinction anomalies of G4-Base that are calculated in the ULAQ-CCM are negative in the whole UT (Fig. a, b) due to the decreasing number density of the particles caused by the reduced vertical velocities in the SG dynamical conditions (see Figs. –). Although the UT cooling in G4 tends to partially offset the effects of the updraft decrease on the ice particle number density, the overall impact is of a general decrease of the UT ice extinction and is even more pronounced than in G4K where the tropospheric cooling is not taken into account. In the latter case, however, the particle effective radius is larger than in G4, as discussed above for Fig. . These size distribution changes affect not only ice extinction, but also the shortwave and longwave radiative responses per unit optical depth (see ahead Sect. 3.2.2).

Following the procedure described in Sect. 2.2 (see Eq. ), an evaluation of the model calculated ice extinction profiles is attempted (Fig. c, d). This is made using indirectly derived values from the MERRA-2 and ERA5 ice mass mixing ratio and the ULAQ-CCM effective radius, as in Eq. () below. Here, χext,i is the ice extinction at the ith vertical layer and ρatm,i is the atmospheric mass density at the same vertical layer: χext,i=Qext32ρatm,iρiceχir

The ULAQ-CCM tropical underestimation of the ice extinction below 13 km is consistent with that of the ice number density and is partly justified by the specific assumptions made on cirrus cloud formation in the model, as pointed out in the discussion of Fig. .

The net result on the ice optical depth (i.e., the vertical integral of ice extinction) is shown in Fig. . In general, a latitude-dependent OD reduction comparable to that found in is present in G4K, while in the G4 case (as expected from the extinction anomalies) a further decrease is calculated mainly in the tropics, even though the UT temperatures are cooler. The effects regarding the temperature and updraft cannot be easily separated, but the colder tropospheric temperatures in G4 with respect to G4K reduce the particle size increase respect to the Base case, producing an additional decrease in the optical depth. The coupled effects of the velocity and temperature anomalies on the ice particle number density and size produce the most relevant impact in our study, pointing out the importance of allowing surface temperatures to respond to the stratospheric aerosol radiative forcing.

The well-tested radiative transfer code online in the ULAQ-CCM (; ; ) has been used to calculate the shortwave and longwave components of the tropopause radiative forcing due to SG aerosols (direct forcing) and to UT ice changes (indirect forcing). As discussed so far, the latter are largely produced by the SG-driven dynamical perturbations on the homogeneous freezing process for ice formation. The ice radiative effects have been calculated using up-to-date wavelength-dependent refractive index available in the literature (; ) and compared against previous results under similar conditions, such as those by . All the radiative calculations shown in this section have been performed offline with the same radiative transfer code as the one present online in the ULAQ-CCM model, in order separate the effects of the single components analyzed.

The results are shown separately for the G4 and G4K experiments, both with respect to the RCP4.5 Base case. Following the previously discussed thinning of the UT ice clouds, a positive SW RF is calculated because of the decreased scattering of the incoming solar radiation by the ice particles. However, such an effect is largely covered by the negative LW RF due to a lessened capacity of the ice particles to trap outgoing planetary radiation; therefore, the obtained net effect on RF is negative, as shown in Table  and Fig. . This indirect negative RF is smaller but still significant when compared to the negative direct net RF due to the SG aerosols (∼30 % of it).

It is interesting to note that the shortwave component of the ice RF is indeed smaller than the longwave component, however, not as much as one could expect from the very different normalized RFs (i.e., forcing per unit OD) at a given particle radius. The reason is that both the SW and LW normalized RFs are decreasing with the increasing particle radius, but the relative changes of these normalized RF components are significantly different between the SW and LW. According to our radiative calculations, the SW normalized values decrease (in magnitude) from -12.1 to -5.7 W m-2 (-53 %) with the ice effective radius increasing from 15 to 40 µm, whereas the instantaneous LW normalized RF remains quasi-constant at an average value of +53 W m-2, with a smooth 3 % decrease over the same radius interval. The resulting SW RF is then controlled not only by the negative OD changes (-0.020 in G4 and -0.012 G4K) but also by the magnitude of the particle radius increase, which is larger in G4K than in G4, and larger in both perturbed cases compared with the Base case (see discussion of Fig. ).

Table  succinctly presents the globally and time-averaged ULAQ-CCM results for the cloud adjustments of the clear-sky RF components due to the SG stratospheric aerosols. The SW and LW cloud adjustments are roughly comparable to the those calculated in (+1.11 and -0.51 W m-2, respectively, calculated at the top of atmosphere for an SG experiment with a 5 Tg-SO2 yr-1 injection). These numbers could be compared with those obtained in the ULAQ-CCM G4K case (although for an 8 Tg-SO2 yr-1 injection), i.e., +1.51 and -0.73 W m-2 for SW and LW, respectively, with a net value of +0.52 W m-2 compared with +0.60 W m-2 in .

In the (more realistic) G4 simulation performed by the ULAQ-CCM model, the SW cloud adjustment is only slightly smaller than in the G4K, while a significantly larger negative LW component is calculated. This results in a net adjustment of +0.52 W m-2 in the G4 case compared with +0.72 W m-2 in the G4K experiment. A latitude-dependent view of these results is presented in Fig. . The black solid line shows the net positive adjustment (SW+LW) due to the mere presence of background clouds, which substantially alter the radiative fluxes (see also ; ; ). These clouds are kept fixed in the ULAQ-CCM model, using climatological values, and thus do not present changes under the G4 scenario. An estimate of the all-sky RF contribution due to SG-driven changes of background clouds is beyond the scope of this study. According to our model calculations, the negative LW is the dominant component of the cloud adjustment due to cirrus ice thinning, and this is particularly true for the more realistic G4 simulation. In this latter case, significantly larger values of the LW adjustment are found over the tropics with respect to G4K, which is consistent with the ice extinction profile changes in Fig. a.

Further information regarding the model calculated RFs is presented in Fig. S3, where we show both the clear-sky latitudinal distribution of the sulfate aerosol RF (Fig. S3a) for G4 and G4K and the LW and SW cloud adjustment due to the presence of background clouds for G4 and G4K (Fig. S3b).