Deep convective clouds are an important component of the global hydrological cycle and radiative budget (e.g. Lohmann et al., 2016; Massie et al., 2002). The anvil cirrus cloud they produce can persist in the atmosphere for several hours to a few days and therefore impact outgoing radiation long after the deep convection has decayed (Luo and Rossow, 2004). However, accurately representing the spatial and temporal complexity of large convective systems and therefore convectively generated cirrus presents extensive challenges for atmospheric modelling (Prein et al., 2015).

Deep convective cloud systems extend vertically from the boundary layer to the tropopause and can have a horizontal radius of over 1000 km. They are dynamic and powerful systems with updraught speeds of up to 50 ms-1 (Frank, 1977; Musil et al., 1986; Xu et al., 2001). In addition, a multitude of different thermodynamic and microphysical conditions can exist within the same system. There is also a scarcity of measurements of these climatically important clouds, particularly profile measurements within the convective core (Fan et al., 2016), and thus a scarcity of data with which to validate representations of deep convective clouds in models. The myriad of competing microphysical processes operating within deep convective clouds, along with the difficulty in validating model simulations against observations, cause the simulation of deep convective clouds to be subject to a large number of parametric and structural uncertainties (Johnson et al., 2015; Wellmann et al., 2018). In particular, mixed-phase microphysics presents a challenge for cloud modelling because it is critical for deep convective cloud properties and poorly understood (Prein et al., 2015).

One of the largest uncertainties in quantifying aerosol–cloud interactions and the resultant climate impacts is the amount of, and balance between, liquid and ice in mixed-phase clouds. In particular, the representation of microphysical processes affecting cloud phase in tropical convection contributes substantial uncertainty to the simulated climate response to global warming in climate models (Medeiros et al., 2008; Stevens and Bony, 2013). The representation of the amount of ice within deep convective clouds is also important for the representation of the amount and intensity of precipitation, the prediction of which is one of the most socially and economically important roles of numerical weather forecasting (Arakawa, 2004; Prein et al., 2015).

Within the mixed-phase region of deep convective clouds, i.e. the region between 0 and ∼ -38 ∘C where both liquid and ice can coexist, we hypothesise that three factors controlling ice production strongly influence the partitioning of condensate into cloud liquid and ice, which are listed as follows. i.

The limiting number concentration of ice-nucleating particles (INPs: aerosol particles with the ability to initiate the freezing of cloud droplets at mixed-phase temperatures) at the top of the mixed-phase cloud regime (in this work, this is the INP number concentration at -38 ∘C). In an aerosol population made up entirely of dust particles, all with ice-nucleating potential, the limiting number concentration of INP would equate to the number concentration of dust.

The limiting number concentration of INPs in the atmosphere is extremely variable and depends on several interacting factors. For example, Saharan dust is an efficient INP at temperatures below -15 ∘C and the largest component by mass of the global aerosol budget (Tang et al., 2016; Textor et al., 2006). The export of this atmospherically important INP, across the Atlantic Ocean, varies hugely depending on factors such as season (Ridley et al., 2012); desert soil moisture (Laurent et al., 2008); local wind speed (Grini et al., 2005; Laurent et al., 2008); and the occurrence and intensity of convection, wet removal, and dry deposition (Bou Karam et al., 2014; Marsham et al., 2011; Provod et al., 2016), both in source (Heinold et al., 2013) and transport regions (Sauter et al., 2019; Twohy and Twohy, 2015). As a result of variations in dust emission and transport, summertime INP number concentrations in the Saharan Air Layer can vary by up to 4 orders of magnitude at -33 ∘C (Boose et al., 2016). Variations in INP number concentrations can impact cloud properties and cloud radiative forcing (e.g. Shi and Liu, 2019; Solomon et al., 2018). However, the reported effect of changes to INP number concentrations on cloud properties can be non-linear, counterintuitive, or conflicting depending on the environmental conditions, magnitude of the tested perturbation, or study methodology (Deng et al., 2018; Fan et al., 2010a, b; Gibbons et al., 2018; Hawker et al., 2021; van den Heever et al., 2006; Phillips et al., 2005, 2007).

The temperature dependence of INP number concentration depends, amongst other factors, on the aerosol type providing INP in a given scenario. A large number of aerosol types have the ability to act as INPs, including mineral dust (Atkinson et al., 2013; Niemand et al., 2012; Price et al., 2018; Welti et al., 2018), organic material in sea spray (McCluskey et al., 2018; Wilson et al., 2015), bacteria (Šantl-Temkiv et al., 2015), and pollen (Diehl et al., 2002). INPs comprised of marine organics emitted with sea spray tend to have a shallower temperature dependence than INP comprised of mineral dusts. This means marine organic INPs tend to have a higher ice-nucleating ability than mineral dust INPs at warm temperatures but lower ice-nucleating ability at colder temperatures (Atkinson et al., 2013; DeMott et al., 2016; McCluskey et al., 2018; Niemand et al., 2012; Vergara-Temprado et al., 2017; Wilson et al., 2015). In numerical weather and climate models, the temperature dependence of INP number concentration can be described by the slope of the INP parameterisation (i.e. d(log⁡10NINP)/dT, as described in Hawker et al., 2021). The INP parameterisation slope depends on aerosol type (DeMott et al., 2010; Harrison et al., 2016, 2019) and any ageing the aerosol has been subjected to (Boose et al., 2016; Brooks et al., 2014) as well as particle properties yet to be fully understood, such as surface morphology (Holden et al., 2019). The INP slope of any one aerosol population (composed of different INP types) is extremely uncertain and difficult to accurately predict without specific measurements. Variation in ice nucleation active site densities (ns) even of materials of similar mineralogy can span several orders of magnitude at any one temperature (Atkinson et al., 2013; Harrison et al., 2016, 2019). The factors governing active site location and densities are not fully understood but are theorised to be related to features on an INP surface such as surface pits (Holden et al., 2019), hydrophilic sites (Freedman, 2015), or lattice mismatches (Kulkarni et al., 2015). Variations in the temperature dependence of INP number concentration can affect the cloud development and the altitude at which cloud glaciation occurs, as was noted by Takeishi and Storelvmo (2018). This difference in glaciation altitude has been shown to cause differences in hail amount, intensity, and size (Liu et al., 2018); anvil ice crystal number concentration (Nice) (Takeishi and Storelvmo, 2018); and radiative forcing (Hawker et al., 2021) of convective clouds.

Observational campaigns have long documented the existence of ice crystals at concentrations vastly exceeding the concentration of INP in clouds with relatively warm cloud top temperatures, indicating the presence of SIP mechanisms (e.g. Crawford et al., 2012; Field et al., 2017; Huang et al., 2017; Ladino et al., 2017; Lasher-Trapp et al., 2016). SIP can occur via processes such as rime splintering (i.e. the Hallett–Mossop process), droplet shattering, collision fragmentation, and sublimation fragmentation (Field et al., 2017; Korolev et al., 2020; Korolev and Leisner, 2020). The most well-studied SIP mechanism is the Hallett–Mossop process by which small ice splinters are produced during the riming of liquid drops onto existing frozen hydrometeors (Crawford et al., 2012; Field et al., 2017; Hallett and Mossop, 1974; Ladino et al., 2017; Phillips et al., 2007). However, even the Hallett–Mossop process is relatively poorly defined and its importance disputed. A recent laboratory study failed to observe rime splintering in conditions designed to stimulate the Hallett–Mossop process (Emersic and Connolly, 2017), and some recent literature suggests that previous observations of ice crystal number concentration attributed to the Hallett–Mossop process may have been indicative of other secondary ice formation mechanisms (Korolev et al., 2020). Nevertheless, as it is the only SIP mechanism that is currently represented in most numerical weather prediction (NWP) models, we focus on the uncertainty associated with the Hallett–Mossop process in this study.

In addition to the individual uncertainties in INP number concentration, INP temperature dependence, and SIP rates, these three factors can also interact causing non-linear or counterintuitive changes in cloud properties, further motivating the exploration of their combined effects here. For example, intermediate INP number concentrations and intermediate Hallett–Mossop ice production rates have been found to produce higher cloud ice crystal number concentrations (ICNC) than high INP number concentrations or high SIP rates alone due to non-linear interactions between the two freezing mechanisms whereby very high heterogeneous freezing rates affect the availability and efficiency of secondary ice production, and vice versa (Crawford et al., 2012; Sullivan et al., 2017).

We investigate the individual and interacting effects of the INP number concentration, the temperature dependence of INP number concentration across the full spectrum of mixed-phase temperatures, and the Hallett–Mossop ice production efficiency on the micro- and macro-physical properties of an idealised deep convective cloud by conducting a large ensemble of idealised simulations using Latin hypercube sampling to select the input parameter combinations and using statistical emulation where appropriate to analyse the ensemble output. We quantify the importance of the three uncertain input parameters and their interactions with one another for the anvil properties of the simulated deep convective cloud. Our methodology proves to be a powerful tool for analysing and understanding the behaviour of complex systems (Johnson et al., 2015; Lee et al., 2011; Marshall et al., 2019; Wellmann et al., 2018) because Latin hypercube sampling enables dense sampling over a defined parameter uncertainty space, allowing more extensive coverage of the defined parameter space than a traditional one-at-a-time test, and statistical emulation allows the production of detailed response surfaces of system behaviours across the three dimensions.

This paper is structured as follows: Sect. 2 describes the idealised cloud model and the simulation set-up, as well as the methods used in our analysis. In Sect. 3, we examine the role of the uncertain input parameters in determining the ice crystal number concentration, ice crystal size, and the cloud fraction of the simulated deep convective anvil cirrus. In Sect. 4, we detail the limitations of our study. Section 5 summarises the main findings and implications of this study.

The role of λINP in determining the ice particle production by the Hallett–Mossop process highlights the importance of the interaction of INP number concentrations with SIP mechanisms. The Hallett–Mossop process is the only SIP mechanism included in simulations in this work, but other SIP mechanisms have been identified in convective clouds (Field et al., 2017; Korolev et al., 2020). We recommend that the effect of these other SIP mechanisms, including those occurring at temperatures below -10 ∘C such as droplet shattering (Field et al., 2017; Lauber et al., 2018) on deep convective clouds, be tested in similar studies in the future.

The simulated cloud is a single idealised case, and as such the results cannot be directly extrapolated to more realistic convective cloud cases, where less idealised triggering mechanisms are at play (Wellmann et al., 2018, 2020) and different clouds in the population can interact (Hawker et al., 2021). It was not feasible to conduct the necessary number of simulations required to study the impact of three uncertain input parameters on a larger more complicated cloud field due to time and cost restrictions. However, the results presented here provide an interesting stepping stone to understanding the interacting effects of INP number concentrations, INP efficiency, and SIP on deep convective anvil properties. We recommend similar studies be undertaken with more realistic cases, including with less idealised triggering mechanisms, in the future.

The chosen uncertain input parameters are just three of a multitude of microphysical parameters that contribute uncertainty to convective cloud processes which should be considered in future work. For example, uncertainty in ice crystal number and mass concentrations were strongly affected by assumed ice crystal shape in simulations of a continental deep convective cloud simulated using the 3D MAC3 model (Johnson et al., 2015). Uncertainties in the riming, sedimentation, and aggregation rates of snow and graupel particles should also be addressed in the future. Uncertainty in environmental conditions that may affect the cloud properties, for example, the size and temperature perturbation value of the warm bubble initiating our deep convective cloud, or the potential temperature profile (Wellmann et al., 2018), have also not been addressed here. Uncertainties in the initial conditions of our simulations have also not been tested and should be explored in the future (Miltenberger et al., 2018a; Miltenberger and Field, 2020). Additionally, MONC–CASIM is configured to be a two-moment scheme in this work and uses multiple ice categories with fixed parameters for bulk physical properties. The representation of ice properties using a continuous spectrum of physical properties could be explored in the future.

In order to effectively decouple the NINP-38 and λINP, for the purposes of simulation, NINP-38 was fixed to be equal to the coarse-mode dust number concentration (in reality NINP-38 would be subset of dust and other aerosols with ice-nucleating abilities), and the calculation of the intercept of the parameterisation at 0 ∘C from NINP-38 and λINP ensures that INP number concentrations decline constantly between -38 and 0 ∘C. This was necessary because a key assumption of the variance-based sensitivity analysis shown in Figs. 7 and 11 is that the uncertain input parameters are independent of one another. However, decoupling the λINP and NINP-38 in this way means that the plateauing of the INP number concentrations at low temperatures and high aerosol concentrations, as can be seen in the Niemand et al. (2012) parameterisation in Fig. 4a, is not represented in the simulations in this study. Therefore, the effect of a high NINP-38 and steep λINP on the cloud properties may be larger in reality than was found in this idealised study because of this design feature. This feature of our experiment design also means that combinations of INP number concentration and INP parameterisation slope causing unrealistically low INP concentrations at temperatures above -35 ∘C are very common in our sampling design. To compensate for this we conducted 22 additional simulations for use in the emulator design and 6 additional validation simulations in the realistic region of parameter space (red and black lines in Fig. 5d). Our variance-based sensitivity analysis is conducted over all simulated parameter space, including the unrealistic space shown in Fig. 5.

It should also be borne in mind that INP concentration spectra often do not follow a simple logarithmic relationship. Instead, different INP species can contribute to the overall INP number concentrations in different temperature regimes, and different INPs tend to have different temperature dependencies. For example, mineral dust tends to have a steep slope and dominates INP populations below around -15 ∘C, whereas biological material can enhance the INP concentration in the Hallett–Mossop regime (e.g. O'Sullivan et al., 2018). Hence, the INP concentration spectra in real clouds can be much more complex than those used in this model. The work presented here highlights the importance of improving our capabilities to represent the complexities of INP number concentrations at mixed-phase temperatures.

The ice crystal properties of the convectively generated anvil are analysed and the implications for anvil lifetime and radiative effect hypothesised. However, the short length of our simulations due to computational limitations means we do not examine the full life cycle of the generated anvil. Conducting similar simulations covering a longer time period would address this limitation and is recommended for the future.

We quantify the impact of varying the limiting INP number concentration (NINP-38), the INP parameterisation slope (λINP), and the Hallett–Mossop splinter production efficiency (HM-eff) on the anvil properties of an idealised deep convective cloud. A schematic of the main effects identified in this study is shown in Fig. 16. Overall, we find that both λINP and NINP-38 play a role in determining the anvil cloud properties, with the HM-eff being relatively unimportant in determining the anvil cloud properties. Despite this, we find that the interaction of λINP with HM-eff is important for determining the resultant amount of ice particle production by the Hallett–Mossop process, which in turn has large effects on the cloud development.

Anvil ICNC is strongly reduced at high-NINP-38 values, with the reduction being more pronounced at shallow-λINP values. Conversely, anvil ice crystal size is increased at shallow-λINP values, with the enhancement being more pronounced at high-NINP-38 values. This is because the lower the altitude of heterogeneous freezing, the more cloud droplets are consumed by riming and depositional growth and the lower the number of droplets that are available for either homogeneous or cold temperature heterogeneous freezing. A shallow λINP reduces the number of cloud droplets reaching the top of the mixed-phase regime, and a high NINP-38 reduces the number of cloud droplets reaching the homogeneous freezing regime. Consequently, the anvil consists of a smaller number of large heterogeneously frozen crystals in a high-NINP-38 and shallow-λINP scenario. The ice crystals transported from the heterogeneous freezing regime to the anvil in this case are larger than those that would be frozen heterogeneously at very cold temperatures or homogeneously. Anvil cloud fraction within the time period studies is enhanced at shallow-λINP values, and this enhancement is lower at high-NINP-38 values. This is because a shallow λINP induces an invigoration effect (inferred from higher cloud updraught speeds) due to an increase in Hallett–Mossop ice production and enhanced glaciation at mixed-phase temperatures, leading to larger condensate mass divergence in the upper troposphere and a more extensive anvil. The anvil is smaller in a high-NINP-38 scenario due to the reduced ICNC and increased ice crystal size discussed above, which serves to reduce cloud lifetime.

Statistical emulation and variance-based sensitivity analysis allow us to identify complex interdependencies between input and output variables. We find that the interaction between λINP and NINP-38 account for up to 30 % of the variation in values of anvil ICNC and anvil ice crystal size. The emulator surfaces help us to see variation in the importance of input parameters depending on the value of other uncertain inputs that would not otherwise be apparent in one-at-a-time tests. In particular, we identify several important interdependencies between different freezing mechanisms. For example, at high NINP-38, λINP is important in determining the anvil ICNC and homogeneous freezing rates because the shallower the λINP, the fewer cloud droplets available to be frozen homogeneously or heterogeneously at the top of the mixed-phase regime. However, at lower NINP-38 values, λINP is nearly inconsequential to homogeneous freezing rates and anvil ICNC because the NINP-38 is low enough that a shallow λINP does not substantially affect the number of droplets reaching upper cloud levels. Furthermore, the dominant effect of λINP on many cloud properties is in part attributed to the fact that high-λINP values provide seed crystals for the Hallett–Mossop process, vastly increasing the number concentration of ice crystals between 5 and 7.5 km and subsequently the cloud riming and deposition rates.

The amount of Hallett–Mossop ice particle production is determined by both HM-eff and λINP, with a λINP above ∼ -0.3 ∘C-1 causing a jump of about an order of magnitude in Hallett–Mossop ice particle production while the effect of HM-eff is linear. A regime shift to a cloud with extensive glaciation at warm temperatures, stronger convective updraughts, larger condensate mass divergence in the upper troposphere, and a more extensive anvil occurs for λINP values between -0.3 and -0.1 ∘C-1, with the exact value of the transition depending on the NINP-38 values (the transition occurs at steeper λINP values if the NINP-38 is high). This regime change is driven by a shallow λINP (particularly when combined with a high NINP-38) forming more ice crystals in the Hallett–Mossop temperature regime and thus seeding a stronger ice particle production by the Hallett–Mossop process. We find a strong enhancement in Hallett–Mossop ice particle production occurs at an INP number concentration anywhere between 10-4 to 1 L-1 depending on the HM-eff. This indicates that the threshold INP number concentration needed to cause substantial SIP varies depending on the strength of the SIP mechanism in question. Whether the interaction of INP with other SIP mechanisms that operate over different temperature ranges has a similar effect on SIP production rates should be explored in the future.

Tropical cirrus can typically persist, and therefore affect radiation, in the atmosphere for days after the convective cloud that formed them has decayed (e.g. Luo and Rossow, 2004). While the simulations were not long enough to study the full life cycle of the anvil, we can infer possible implications of the uncertain input parameters on anvil lifetime based on the anvil ice crystal properties. An anvil with more numerous, smaller crystals will persist longer in the atmosphere than one with fewer, larger crystals. High-NINP-38 values lead to both fewer and larger anvil ice crystals and a slightly reduced cloud fraction, implying that deep convection generated in an environment with a high concentration of ice-nucleating aerosol, such as in high dust loading, may have a shorter lifetime than those generated in lower NINP-38 environments. At shallow-λINP values, the anvil ice crystals are larger, indicating that a cloud anvil formed in the presence of an INP population with high efficiency at warmer temperatures, e.g. marine organics (Wilson et al., 2015), may have a shorter lifetime than that formed in the presence of an INP population with a steeper temperature dependence. However, this effect is compensated for by an invigoration effect (inferred from higher in-cloud vertical velocities) driven by higher rates of mixed-phase ice formation in simulations with shallow-λINP values, which leads to larger condensate mass divergence in the upper troposphere and a larger anvil in the timescale simulated in this study. Future studies should cover the entire life cycle of the generated anvil cirrus in order to quantify the resultant lifetime of the convective anvil due to compensation between these two effects of λINP and NINP-38.

We demonstrate with the present study that statistical emulation is a powerful tool for visualising and quantifying the relationships between cloud responses and different uncertain parameters (Fig. 8). However, statistical emulation struggles to accurately represent cloud responses where there is a significant regime shift at shallow λINP (Figs. 10 and 14), for example, the anvil cloud fraction. We therefore suggest that emulation be used alongside traditional analysis methods for the further study of the complex processes occurring within deep convective clouds, particularly where sudden transitions or regime shifts are evident or likely. The use of Latin hypercube sampling to capture cloud responses over the full realistic parameter space to multiple uncertain input parameters is very effective, even without undertaking statistical emulation of the simulation data.

The microphysical effects of the variations in INP number concentrations and INP parameterisation slope detailed here build on the results of Hawker et al. (2021) and further our understanding of the role of these two uncertain inputs on deep convection. In both the complex cloud-field simulation of Hawker et al. (2021) and the idealised deep convective cloud presented here, INPs in the warm mixed-phase region enhance Hallett–Mossop ice particle production and increase snow and graupel formation, leading to an invigoration effect, more cloud condensate, and an increased cloud fraction at mixed-phase cloud levels. INPs in the mixed-phase region also reduce homogeneous ice production, leading to reduced overall column-integrated ICNCs in both studies. Conversely, in the Hawker et al. (2021) study, shallow-λINP values led to a reduced cloud fraction above 9 km due to reduced ice particle production by homogeneous freezing. In the deep convective cloud simulated here, a shallow λINP leads to an increased anvil cloud fraction due to an invigoration effect caused by enhanced ice formation and latent heat release in the mixed-phase cloud region. This indicates that the microphysical effects of INPs and the interaction of INPs with the Hallett–Mossop process are relatively consistent between realistic and idealised case studies. However, the consequences of microphysical changes due to INPs on the cloud macro-physical properties such as cloud fraction and outgoing radiation can be different depending on the specific conditions of the simulation and the type of cloud being perturbed. For example, the deep convective cloud simulated here was initiated with a relatively strong warm bubble perhaps predisposing the cloud fraction to be more sensitive to enhancements in an already strong convective updraught strength than the more realistic clouds in the Hawker et al. (2021) study. This study has enhanced our understanding of aspects of cloud microphysical behaviours that was not fully explained in Hawker et al. (2021). For example, in both studies a reduction in homogeneous freezing rates occurs where there are high INP number concentrations at low temperatures, and the results from this study indicate that it is a high NINP-38 rather than a steep λINP that is the main driver of this effect, a distinction that we were unable to make in the previous study. The differences in resolution between the two studies (250 m (here) and 1 km; Hawker et al., 2021) may also cause divergences in the microphysical responses of clouds to perturbations (Varble et al., 2020).

This work highlights the complexity of interactions between mixed-phase ice processes and the challenge of representing them accurately in numerical weather prediction models. Our work indicates that the sensitivity of deep convective cloud properties to mixed-phase ice processes varies depending on ambient ice-nucleating aerosol concentrations (e.g. absolute dust concentrations) as well as the efficiency of the available ice-nucleating aerosol (e.g. whether the INP number concentration consists of dust or marine organic particles). The potential for ice particle production by INPs and SIP to impact anvil cirrus ice properties also presents a challenge for climate models. Climate models do not typically use INP number concentrations to determine ice water path and the resultant outgoing radiation, and this is an important area for future work (Baran et al., 2014; Waliser et al., 2009). The role of the temperature dependence of INP number concentration in determining the observed cloud properties indicates the importance of quantifying the concentration of INP at all mixed-phase temperatures, adding to work by, for example, Hawker et al. (2021), Liu et al. (2018), Shi and Liu (2019), and Takeishi and Storelvmo (2018). Furthermore, the temperature dependence of the INP parameterisation had a substantial effect on Hallett–Mossop ice particle production rates, indicating that heterogeneous freezing can be an important determinant of deep convective cloud properties by affecting SIP mechanisms, even when heterogeneous freezing is not the dominant mechanism of ice formation in the SIP region.