The nested UM (Unified Model) from the UK Met Office was coupled with a double-moment cloud microphysics scheme CASIM (Cloud AeroSol Interacting Microphysics) . We used UM version 13.0 with the RAL (Regional Atmosphere and Land) 3.2 configuration . The boundary conditions of the nested domains were provided hourly by a global model (N216) with the GAL (Global Atmosphere and Land) 6.1 configuration . The cloud parameterisation in the nested model was from the bimodal cloud scheme . This approach uses subgrid turbulent kinetic energy to diagnose the subgrid humidity distribution and hence cloudy fraction. The cloud cover was derived using cloud fractions in a column with the maximum-random overlap assumption. The radiative processes were from SOCRATES (Suite Of Community Radiative Transfer codes based on Edwards and Slingo) , and mass mixing ratio and number concentration of ice hydrometeors were both considered when calculating the single-scattering albedo of ice in SOCRATES . The parameterisation of boundary layer processes in our model is from .

The selected CAO event occurred on 24 October 2022 over the Labrador Sea, which coincided with two flights (C322 and C323) during the M-Phase aircraft campaign . Figure  shows the tracks of both flights, the corresponding nested model domains, and the analysis domains. Flight C322 measured regions dominated by stratocumulus (Sc) clouds, while Flight C323 focused on regions dominated by cumulus (Cu) clouds.

Both nested model domains contain 500×500 gridpoints with a horizontal grid spacing of 1.5 km, centred at 60° N, 60.5° W for capturing the early stage and Sc region and 56° N, 51° W for the Cu region. has demonstrated that this grid spacing is capable of reproducing the general cloud features of a CAO event. The nested model domains were designed to ensure the model analysis domains remained sufficiently distant from the domain boundaries, allowing cloud fields to fully develop when they leave the ice edge and before entering the model analysis domains. At this point, the model has already spun up from the boundaries. The regional model has 90 vertical levels capped at 40 km with 16 model levels below 1 km and 28 model levels below 3 km. The model is set to the atmosphere-only configuration with prescribed sea surface temperature (SST) and fixed surface fluxes. The mean SST in the Sc region is 277.5 K and is 280.5 K in the Cu region.

The simulations were initialized at 00:00 UTC and performed for 24 h, with the model output from the first 12 h excluded from the analysis due to model spin-up.

Six parameters were investigated in this study: Nd (cloud droplet number concentration), NINP-15 (INP concentration at -15 °C), EHM (efficiency of the Hallett–Mossop process), EBR (leading coefficient controlling the efficiency of the ice-ice collisional breakup process), EDS (efficiency of the droplet shattering process), and mpof (mixed-phase overlap factor). All the parameters are within the CASIM scheme. A description of each perturbed parameter and the selection of their perturbed ranges are introduced in this section. The combinations of the six parameter values were designed with the maximin Latin hypercube sampling method. The single OAT test was performed for every parameter before running the PPEs to investigate the effects of perturbing each parameter individually around the control simulation (not shown).

Figure  shows the five hydrometeors and the main processes in CASIM with relevant processes for the perturbed parameters in this work highlighted. Not all processes in CASIM are shown here (e.g. self-collection, sedimentation, evaporation and sublimation); see for a complete description of CASIM implementation in the UM. To keep the consistency of symbols, we used the same symbol for mass tendency (P) and number tendency (N) in the description of parameters below. In CASIM, cloud ice is represented as a spherical particle (m∝D3, where m is mass and D is diameter of a hydrometeor) and snow is represented as aggregate particles (m∝D2) based on findings from . Details for terminal fall speed, mass-dimension and shape of the particle size distribution of each hydrometeor in CASIM can be found in Table A1 in and the calculation of the ventilation coefficient is presented in Eq. (A23) in .

Gaussian Process (GP) emulation is a non-parametric supervised machine learning method that learns the relationship between model input parameters and outputs of interest . GP emulators were constructed on the output from the PPE simulations and acted as a statistical “surrogate” model for the computationally-expensive numerical model. The GP emulators enabled efficient exploration of the entire parameter space with dense sampling, allowing for construction of multi-dimensional response surfaces and a variance-based sensitivity analysis to assess the importance of the perturbed parameters to the cloud variables. Detailed methodologies and explanations on building GP emulators for climate and atmospheric models can be found in and .

A GP emulator consists of a mean function and a covariance function, also called the kernel. To obtain a good GP emulator of each variable of interest we tested four commonly-used stationary kernels: the Radial Basis Function (RBF) kernel and three Matérn kernels with different smoothness parameters: the Matérn12 kernel (0.5), the Matérn32 kernel (1.5), and the Matérn52 kernel (2.5), implemented in the GP Jax Python library for Gaussian processes . When the smoothness parameter becomes infinite in a Matérn kernel, it becomes equivalent to the RBF kernel. For each variable, we conducted cross-validation by dividing the 66 PPE members into 11 sets with 60 training and 6 validation data points per set. The cross-validation was used here to maximise the use of our limited data due to the high computational cost of running the numerical simulations. Emulator performance was assessed with graphical diagnostics including individual standardized error plots and Q-Q plots following the guidance from , as well as the RMSE (root mean squared error) and R2 (coefficient of determination). The kernel selection was mainly based on the RMSE values, with the Q-Q plots and R2 for supporting the assessment. Once the best-performing kernel was selected for each variable, we trained the GP emulator with the best kernel on all 66 PPE members and tested it with the corresponding testing set. Appendix C describes and discusses the emulator performances from the kernel selection and testing steps, as well as the dependency of the RMSE on the different numbers of training points.

Before analysing emulated response surfaces of domain-mean outputs in the next section, we first illustrate the strength of perturbing parameters in a PPE design and use TOA albedo here as an example in Fig. . Note that the emulators (Sect. 3.2) define the relationship between input parameters and mean output variables (e.g. domain-mean albedo) and cannot describe the specific cloud patterns shown here, hence why we start by showing the results of some PPE members. We selected two parameters to demonstrate the PPE results: Nd and NINP-15 for the Sc region, and Nd and mpof for the Cu region. These parameter pairs were chosen only for illustration because, as we show in Sect. 3.2, they have the strongest joint effects on cloud albedo in each region.

Figure  shows the maps of TOA albedo in 25 of the 66 PPE simulations over the two regions. The intervals between each PPE output are not equal in Fig.  because of the Latin hypercube design method used for sampling the parameter space, and the subplots shown in Fig.  are only ordered using the values of the selected parameters with the values of other four parameters ignored.

In both regions, the selected parameters jointly affect the TOA albedo. For example, in the Sc region (Fig. a), low Nd and high NINP-15 (top left subplot) result in the lowest TOA albedo, while high Nd and low NINP-15 (bottom right subplot) result in the highest TOA albedo. With PPE simulations, we can see clear joint effects of parameters on cloud properties. However, the number of simulations is too small to identify whether the joint effects are linear (combined effects) that could be predicted from combinations of OAT simulations, or non-linear (interactive effects) . See Appendix E for a simple illustration of linear and non-linear joint effects between two parameters. Using the emulators enables a much denser sampling of the parameter space and, as we show in Sect. 3.2, exposure of non-linear effects (interactions) of combinations of parameters that are not apparent in Fig. .

The PPE members provide considerable information about cloud behaviour across the 6-D parameter space, but the data sparseness (just 66 points in 6-D space) limits our ability to visualise joint parameter effects or to perform statistical analysis. Therefore, we built emulators of temporally and spatially averaged cloud properties, which enable 6-D continuous response surfaces to be created. The time of averaging for variables (except the cloud top temperature) is from 12:00 to 19:00 UTC, and the domains of averaging are shown as analysis domains in Fig. . Cloud top temperature (CTT) is derived from the air temperature at the highest model level in a column with total cloud water content ≥10-6 kgkg-1. The model output is taken at 14:00 UTC for the Sc region and 19:00 UTC for the Cu region, corresponding to the times of the M-Phase C322 and C323 flights respectively. An evaluation of modelled air temperature against aircraft measurements can be found in . The response surfaces were generated by predicting one million output values from the emulators across the parameter space.

Figure  shows the distributions of TOA albedo, cloud cover, all-sky LWP (liquid water path), all-sky IWP (ice water path), all-sky TWP (total water path), and CTT from one million combinations of parameters drawn from the GP emulators. By perturbing all 6 parameters in the Sc region, the TOA albedo ranges from 0.43 to 0.54 (inter-percentile range from 5 % to 95 %), cloud cover from 0.84 to 0.91, LWP from 74.5 to 149.5 gm-2, IWP from 0.0 to 38.1 gm-2, TWP from 97.6 to 148.4 gm-2 and CTT from -12.3 to -11.2 °C. In the Cu region, the TOA albedo ranges from 0.37 to 0.44, cloud cover from 0.78 to 0.84, LWP from 47.9 to 98.4 gm-2, IWP from 71.3 to 186.1 gm-2, TWP from 147.8 to 244.4 gm-2 and CTT from -11.0 to -9.3 °C.

Figures  and show the joint effects of each combination of two parameters with the other parameter effects averaged for TOA albedo and all-sky LWP, which are two key properties for cloud-phase feedback, respectively over the two selected regions. These are known as “paired projections”. Paired projections for cloud cover, all-sky IWP, all-sky TWP, and CTT are shown in Appendix F. The values shown on the paired projection plots are calculated by averaging over the other four dimensions, and therefore describe the mean response of each variable to each paired parameter set.

Using the paired-projections, we are able to qualitatively identify key parameters controlling the cloud properties over the parameter space in this CAO event. In the Sc region, Nd and NINP-15 exert the strongest effects on TOA albedo and LWP. All parameters from SIP processes and mpof only show weak effects across the projections due to the liquid-dominated nature of these clouds, demonstrating limited effects in the Sc region. In the Cu region, Nd maintains its strong influence on TOA albedo and LWP, while the second strongest effect is now from varying mpof. The perturbation of mpof now results in a stronger effect on cloud properties compared to in the Sc region and to NINP-15 in the Cu region. A reduction in mpof leads to less interactions of liquid and ice particles, which weakens the removal of liquid water through accretion. The increasing effect from mpof in the Cu region compared to in the Sc region comes from an increasing ice formation in Cu clouds when the cloud top deepens and more mixing between ice and liquid hydrometeors. The effects from mpof in the Sc region are also limited by a large liquid cloud fraction with little ice cloud fraction, as the extent to which liquid and ice interacts in a model grid box is not just determined by mpof but also cloud fraction.

All three SIP parameters have a relatively weak effect on TOA albedo and LWP in both Sc and Cu regions in this case, even though the cloud temperature covers most of the active temperature range of the HM process. Note that a low sensitivity does not imply that SIP is unimportant, only that the cloud properties are fairly insensitive to its precise value. Although the influences from varying SIP processes on albedo and LWP are small in both regions, the effect of each parameter is specific to this case study and is not representative of all mixed-phase CAO clouds. The HM process has a much stronger effect on IWP and TWP in the Cu region (Appendix F), but such influence is weakly transferred to the radiative properties (TOA albedo).

When ice formation processes are stronger in the Cu region (e.g. with a higher NINP-15 or EHM), the reduction in cloud liquid is smaller than the increase in cloud ice, leading to a net increase of TWP (Fig. ). This is because the increase in IWP results not only from the direct conversion of liquid to ice, but also from stronger vapour deposition onto ice than onto liquid at the same temperature. In addition, a higher ice number concentration can reduce snow autoconversion efficiency and decrease the size and fall speed of ice hydrometeors, thereby reducing cloud-water removal through frozen precipitation and increasing IWP and TWP. This effect is more pronounced in the Cu region, where precipitation is substantially stronger than in the Sc region. Analysing the relevant process rates would provide further insight into these mechanisms, but this is beyond the scope of the present study. We therefore recommend that future work investigate the joint effects of these processes on TWP responses across different cloud regimes and CAO conditions.

The influences of NINP-15 and mpof on TOA albedo are weaker compared to their influences on LWP in both regions. This is because perturbing Nd also exerts a strong Twomey effect on the liquid-dominated Sc clouds (over 80 % of the Sc cloud water mass is liquid in the control simulation). As a result, the effect of perturbing NINP-15 and mpof on TOA albedo becomes smaller than their effects on LWP. There are also non-monotonic responses of LWP to the perturbation of NINP-15 at lower NINP-15 values in the Sc region. An increase in NINP-15 does not lead to a monotonic decrease in LWP; instead, LWP initially increases with increasing NINP-15, followed by a decrease at higher NINP-15. This may be because the removal of liquid through warm rain processes is first reduced when increasing NINP-15 as the amount of liquid is reduced, hence leading to a higher LWP; but with further increases in INPs, the removal of liquid is then dominated by the mixed-phase processes rather than the warm rain processes, hence leading to a lower LWP eventually. One can also see that, as Nd increases, the warm rain processes reduce due to less rain autoconversion, and the value of NINP-15 required to let mixed-phase processes start dominating the water removal process (where the increasing LWP becomes decreasing LWP with increases in INP) decreases. To further test this, one can compare different model response surfaces between Nd and NINP-15 with and without the warm rain and cold rain processes.

In this CAO case, CTT is strongly controlled by Nd in both regions, with stronger effects from NINP-15 and mpof in the Cu region. SIP processes have limited influences on CTT in both regions. A reduction in Nd or an increase in NINP-15 leads to a higher CTT, and hence a shallower vertical development of clouds. Both the reduction of Nd and the increase of NINP-15 can lead to an enhancement of precipitation, as shown in our previous work for this CAO case. This response of CTT to the change of precipitation is consistent with previous work from on CAO clouds, which show that enhancement of precipitation can stabilise the boundary layer during CAO event and lead to less vertical development of clouds. The limited cloud vertical development from a high NINP-15 can also prevent the effects of converting liquid to ice, as the INP concentration is temperature-dependent.

For most of the paired projections, non-linear joint effects (interactions) of the parameters on the cloud properties can be identified. Here we use the projections between Nd and NINP-15 in the Sc region and the ones between Nd and mpof in the Cu region as examples. Parameter interactions are apparent when the sensitivity of a cloud property to a parameter perturbation depends on the value of another parameter – i.e. the number of contours across for the same change of a parameter differs when the value of another parameter changes. Using LWP in the Sc region as an example, by perturbing Nd from low to high values and keeping NINP-15 low (less than 0.01 L-1), the LWP ranges from approximately 90 to 140 gm-2; if the NINP-15 is high (more than 1 L-1), the LWP ranges approximately from 70 to 95 gm-2. In the Cu region, the LWP ranges from 70 to 90 gm-2 for low to high Nd values when mpof is low (less than 0.2), but from 50 to 60 gm-2 when mpof is high (more than 0.6).

Another example of parameter interactions is that the local sensitivity of a parameter can vary across the parameter space. In the Sc region, the LWP is insensitive to NINP-15 when Nd is lower than approximately 20 cm-3 and NINP-15 less than 0.1 L-1, while it becomes very sensitive to NINP-15 when Nd is higher than approximately 50 cm-3. The relative importances of SIP processes and mpof in the Sc region also depend on the values of Nd and NINP-15, despite the perturbations of SIP parameters and mpof having limited effect on cloud radiative properties in this region. When Nd is higher than about 40 cm-3 or NINP-15 lower than 0.01 L-1, the influence on LWP is strongly enhanced by SIP processes and mpof. It is worth noting that the effects of SIP processes and mpof on LWP are still quite small, so although they strongly affect the change in LWP in that part of the parameter space, their importance over the whole parameter space is small compared to Nd and NINP-15. Such interdependency between the parameters shows that a single OAT test is insufficient to draw conclusions about the importance of a parameter or a process, and the results from the OAT test are highly dependent on the default values of the parameters in the control simulation.

Figures 6–8 also show that different combinations of parameter values may result in the same value of cloud properties, which is called equifinality . For example, the mode TOA albedo in the Sc region is 0.54, meaning a large number of model variants with different parameter values produce the same albedo of 0.54. The equifinality is also visible in the filled contour plots in the paired projections shown in Figs.  and – each contour colour defines parameter combinations that produce equifinal model output. Equifinality becomes important when model output is compared to observations, because model output from different combinations of parameters will agree equally well with observations. Perturbing each parameter individually to match observations may fall into the problem of equifinality because it would result in a single and overly confident estimate of a parameter value rather than a range. Here we use TOA albedo as an example. The observed, domain-mean TOA albedo from CERES onboard the Aqua satellite is around 0.48 in the Sc region and around 0.39 for the Cu region at 17:00 UTC. There are more than 70 000 model variants out of the 1 million model variants that produce this value of TOA albedo (±0.005) in the Sc region, and more than 90 000 model variants in the Cu region. There would also be many more equifinal variants when accounting for observation uncertainty. Such equifinality problem suggests that using a single variable to constrain the parameter space is not sufficient, and multiple variables for different cloud properties are needed. A key question is whether there exists a single part of parameter space for which all the variables agree well with observations, which will be shown in our subsequent paper comparing the PPE simulations shown in this work with multiple observations (in-situ and satellite).

The response surfaces in the previous section show that the sensitivity of cloud properties to a change in a parameter depends strongly on the setting of the other parameters. However, these paired projections neglect variations in the other four parameters and therefore only present the mean response of each model variable to each paired parameter set. Therefore, to understand how much the sensitivities of cloud properties vary within the parameter space, we present the results from multiple OAT tests across the parameter space in Fig. .

We performed a large number OAT tests using the trained GP emulators for each model variable with different starting points. To design the starting points, three values were selected for each parameter: low (25 % of the perturbed range), mid (50 % of the perturbed range), and high (75 % of the perturbed range). For N parameters each with M values, the number of possible parameter combinations are MN. Hence, there were 729 starting points (control sets of parameter values) across the 6-D parameter space in the multiple OAT tests. For each OAT test, the parameters were perturbed to the high-end (95 % of the perturbed range) and the low-end (5 % of the perturbed range) individually. The responses of cloud properties were the change from high-end parameter value output to the low-end parameter value output. The perturbed ranges in the multiple OAT tests were smaller than the perturbed parameter range in the PPE design to avoid the high uncertainties of emulators near boundaries of the parameter space. As for each parameter, we perturbed the value from low-end to high-end without directly including the emulator output from the control simulation, the number of output points from the emulator for each variable is 2×MN-1. Therefore, for each model variable in one domain (one subplot in Fig. ), the total number of investigated model variants is 2916. A compact distribution in Fig.  means that the sensitivity of the cloud property to that parameter is not strongly dependent on the setting of the other parameters.

The magnitudes of cloud responses to individual parameter perturbations are sometimes strongly dependent on the values of other parameters. For example, in the Sc region, the increase in LWP (Fig. e) by perturbing Nd from low to high values ranges from +31 to +65 gm-2, and the decrease in LWP by perturbing NINP-15 from low to high values ranges from -73 to -29 gm-2. In the Cu region, the decrease in LWP (Fig. f) by perturbing mpof from low to high values ranges from -39 to -15 gm-2. Although the projection plots suggest clouds are not very sensitive to variations of SIP processes, the multiple OAT tests here show that in some parameter space, varying SIP processes can have same or even stronger effects on cloud cover, IWP and TWP in the Cu region than Nd and NINP-15. In addition, due to the existence of dominant parameters across the parameter space (e.g. Nd), the effects from SIP processes may be obscured through averaging in the paired projections. The relative importance of each parameter across the parameter space can also be quantitively compared by using the statistics of the sensitivities from the multiple OAT test (e.g. mean or mode), which shows similar results to those from a global sensitivity analysis in Appendix H. Similar to the other variables discussed above, parameter interactions are also identified here for CTT, particularly in the Cu region. Across different parts of the parameter space, an increase in Nd can reduce CTT from about -0.5 °C to nearly -2 °C in the Cu region. An increase in NINP-15 can lead to an increase in CTT ranging from nearly 0 °C to about +1.5 °C. This dependence of individual parameter sensitivities on other parameters indicates that these parameters also jointly affect cloud vertical development in CAO clouds and may therefore influence cloud field evolution, such as the transition from stratocumulus to cumulus. It is therefore important to explore the full parameter space when assessing the effects of aerosols and ice-production processes on cloud evolution during CAO events.

The direction (or sign) of the cloud response can also depend on the control values of the parameters. For example, changes in SIP parameters in this case can cause either increases or decreases in cloud cover (Fig. c and d) depending on the other parameters (primarily Nd and NINP-15 which dominate the sensitivities). In the Cu region, increasing NINP-15 does not consistently lead to a reduction of the TOA albedo (Fig. b). For most OAT tests with NINP-15 perturbations, increasing NINP-15 causes cloud cover to increase (Fig. d), but also leads to a reduction in LWP (Fig. f). These two effects compensate each other for the effect of increasing NINP-15 on TOA albedo. Therefore, the influence from a high NINP-15 on TOA albedo in Cu region is controlled by multiple cloud properties, and these cloud properties can have different responses to changing NINP-15 at different part of the parameter space.

In summary, the sensitivities of modelled cloud properties to each individual perturbed parameter depend on the starting point in the parameter space; this includes both the magnitudes and sign of responses. Results from our multiple OAT tests across the parameter space demonstrate that using a single OAT test is inadequate for determining the magnitude and even sign of the cloud response to model parameters. It is therefore important to investigate the whole parameter space for a more comprehensive understanding of the effects from these parameters.