We use the Community Atmosphere Model version 6 (CAM6), which is the atmosphere component of the Community Earth System Model version CESM-2.0 . The CAM6 model uses a two-moment microphysics scheme for stratiform clouds, with liquid, ice, rain, and snow hydrometeors calculated as prognostic variables, allowing CAM6 to explicitly represent the aerosol indirect effect .

We leverage a perturbed parameter ensemble (PPE) hosted in CAM6 . A PPE is a large set of simulations based on the structure of a single ESM (e.g., CAM6) with a different combination of parameter values to examine parameter uncertainty . The CAM6 PPE is fully described in . CAM6 is run at the standard resolution of 1.25°×0.9375° resolution. Briefly, 262 model simulations (i.e., 262 parameter combinations) of CAM6 sample 45 sources of uncertainty in the parameterizations for cloud, precipitation, convection, boundary layer, and aerosol processes. The 45 parameters are simultaneously perturbed using Latin Hypercube within the plausible range of realistic values based on expert-elicitation. We examine 203 ensemble members out of 262 integrated. The remaining 59 members were excluded based on criterion: (1) the linear regression slope of Nd to CCN in log space (dln⁡Nd/dln⁡CCN) is less than 0; (2) the correlation coefficient between Nd and CCN is less than 0.3. The two criteria are used to exclude PPE members that are too far outside the observational constraint behaviors (i.e., the Southern Ocean field campaign measurements analyzed in Fig. 14 in ). Following , we also exclude PPE members that simulate too much ice in tropics, which is inconsistent with satellite observations .

Two scenarios are simulated and each of them use the same parameter combinations – consistent with previous studies . First, 2-year global simulations saved at monthly-mean are completed for pre-industrial (PI) and present-day (PD) emissions. PI and PD aerosol emission scenarios are integrated from 2019 to 2020 so anthropogenic perturbations in Nd can be calculated over global coverage by taking the difference between PI and PD. The atmosphere is nudged to horizontal winds and temperature and sea surface temperature and sea ice fraction are prescribed from observations. Wind and temperature fields are nudged to the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA2) reanalysis with 24 h relaxation time. MERRA2 output is interpolated to CAM6 vertical resolution with standard 32 vertical levels from the surface to 3 hPa following . Previous studies have shown the CAM6 PPE produces a wide range of perturbations in cloud microphysics (e.g., ΔNd, PD-PI) and cloud macrophyiscs (ΔLWPPD-PI). In this study, we focus on diagnosing the parametric effects on cloud microphysical responses to anthropogenic aerosols from different parameter combinations using the PPE.

In addition to the two-year integrations of the PPE used to calculate anthropogenic perturbations in Nd, the PPE is integrated over short periods consistent with the Southern Ocean Clouds, Radiation, Aerosol, Transport Experimental Study (SOCRATES) field campaign based from Hobart, Tasmania (Fig. ). The SOCRATES campaign occurred over the midlatitude Southern Ocean (SO) during austral summer and was dominated by a series of frontal systems, postfrontal stratocumulus decks, and cyclonic activity typical of the storm track region . Model outputs are saved along flight tracks over SOCRATES and is sampled at 1 min resolution following . It applies atmospheric nudging to horizontal winds and temperature, consistent with global simulation with PI and PD aerosol emissions scenarios, but nudged to the period of January–March 2018 when the aircraft observations were conducted. The behavior of the default parameter configuration in CAM6 has been characterized using this approach in , , , .

Previous studies have shown that the CAM6 PPE, configured with 2-year global simulations, produces a wide spread in present-day (PD) cloud microphysical (Nd) and macrophysical (LWP) properties. The mean-state PD values have been shown to fall within the observational range derived from satellite remote sensing . Additionally, CAM6 simulations along flight tracks using the default parameter configuration reproduce many features of in-situ observations, including cloud phase, cloud location, and boundary layer structure . These results give us confidence that at least some members of the nudged PPE simulations provide a physically plausible baseline in terms of cloud microphysical and macrophysical properties. In this study, we focus specifically on microphysical properties.

We examine in-situ airborne observations taken from SOCRATES as our observational constraint . The importance of the Southern Ocean (SO) to understanding the global anthropogenic contribution to Nd has been shown in several previous studies . The National Science Foundation Gulfstream-V (GV) aircraft was deployed during January–March 2018 for SOCRATES. There were 15 flights sampling data from 42 to 62° S with aerosol and cloud properties sampled at 1 Hz frequency. The GV was equipped with a variety of sensors and instruments. In this work, Nd from the cloud droplet probe (CDP) and aerosol number concentrations from the ultra-high sensitivity aerosol spectrometer (UHSAS) are examined. We focus on accumulation mode aerosols, with diameters ranging from 0.1 to 1 µm, reported as UHSAS100 in this paper following . Accumulation mode aerosol usually accounts for most of the surface area of aerosols and is a good estimate of the CCN concentration for stratocumulus updraft velocities .

With a focus on low-level, liquid cloud, we restrict the aircraft measurements of aerosol and cloud to be below 2 km. As in previous studies , in-situ aircraft aerosol measurements are discarded when the liquid water content (LWC) from the CDP exceeds 0.001 g m⁻³, along with the subsequent 10 s after cloud detection. This is to avoid measurement contamination from cloud . In-cloud Nd measurements are restricted to regions where the LWC from the CDP is greater than a threshold (0.1 g m⁻³) following . Because the observations of aerosols and in-cloud Nd that are considered valid for use are taken at different locations, direct comparison is challenging due to inconsistencies in spatial and temporal coverage. To make comparisons between Nd and aerosol observations, we bin the aircraft measurements by 2 min in duration and 50 m in altitude so that aerosols and Nd can be compared in the same bin. Only bins with at least ten 1 Hz flight observations are considered valid composites for use. Median values of aerosol concentration and Nd are computed for each bin for observations from each flight following . The instrument limitation inevitably forces us to look either at small clouds or cloud edges, where both the measurements of aerosol and cloud are valid for use. This has minimal impact on our comparison between models and observations as we colocate model output with observations as detailed in Sect. .

In this study, we focus exclusively on low-level, liquid clouds simulated by the stratiform (large-scale) cloud microphysics scheme (MG2) in CAM6, as CAM6’s convective scheme does not include prognostic microphysical variables such as Nd, which is a key quantity in our analysis. As such, all Nd values analyzed in this study originate from the stratiform cloud scheme. Furthermore, we limit our comparison with aircraft observations to altitudes below 2 km, corresponding to the marine boundary layer and excluding a large potion of clouds formed by deep convection . The majority of simulated Nd in CAM6 is also concentrated below 2 km . The convective scheme, while it may be triggered during postfrontal cloud conditions, does not contribute to Nd in CAM6. The convective scheme can contribute to precipitation, while this is beyond the scope of analysis in the present study.

The default configuration of CAM6 has been extensively evaluated in and and has been shown to be able to reproduce many features consistent with in-situ observations in . Here, we examine a PPE that is hosted in the same model evaluated in previous studies . The CAM6 model parameterization and the prior distribution of parameters (i.e., 217 sets of parameter combinations) in the PPE produce simulated aerosol and cloud properties that we can compare with observations to evaluate how process representation impacts aerosol-cloud interactions. Here, we focus on microphysical quantities that are available from in-situ measurements but hard to observe from spaceborne remote sensing.

Nd is directly available from both CAM6 and in-situ measurements from the CDP. CAM6 in-cloud Nd is calculated as Nd divided by liquid cloud fraction (when cloud fraction ≤ 1 %, we set Nd=0). This cloud fraction threshold is smaller than the one used in as we found it retains more flight composites but does not significantly change the results of our analysis (Fig. S1 in the Supplement).

CCN is a subset of aerosols that can be activated to cloud droplets at a given supersaturation. CAM6 outputs CCN at a set of fixed supersaturations. Here, we look at supersaturation at 0.2 %. It is found that observed CCN at 0.2 % supersaturation (CCN02) has an one-to-one relationship with accumulation mode aerosol (e.g., UHSAS100) measured over SOCRATES . Following previous work , we use UHSAS100 as a proxy to CCN02 over SOCRATES as UHSAS100 lies very close to the one-to-one line with CCN02. This supersaturation level is shown to be representative of marine low-level stratocumulus .

To make comparisons between the modeled and observed Nd and aerosol properties, model data are colocated to observations by linearly interpolating to temporal and spatial locations from the 2 min × 50 m observational composites following . Our comparison between observations and models follows two strategies. First, model outputs (CCN and Nd) are confronted with in-situ observations for collocated bins (flight track composites) along flight tracks for each simulation ensemble. This method allows for the evaluation of simulated CCN, Nd, and the inferred efficiency of aerosol activation (dNddCCN) relative to observations within individual PPE members. The results are discussed in Sect. . Second, campaign-means of CCN and Nd are calculated for each PPE ensemble and compared with campaign-means of CCN and Nd calculated from in-situ observations. In this approach, we evaluate aerosol activation efficiency across the CAM6 PPE members (run with different parameter sets) and use campaign-means of in-situ aerosol properties and Nd to constrain the CAM6 PPE (Sect. ).

The intention of taking the campaign-mean is to reduce random error by averaging over a large number of samples. However, there remains potential sources of systematic error. One possible source of systematic error is from differences in sampling between the observations and the model (e.g. if the pilot only flew through clear air and avoided cloud). Sampling during airborne campaigns may have some systematic sampling biases as discussed in . Output from CAM6 is representative of an average within the grid box of the model, whereas flight patterns in a similar-sized domain may not be sampling randomly (e.g. focusing on convective cores). We believe that this is a minimal concern for SOCRATES. The SOCRATES flight pattern was designed to focus on cold sectors of cyclones and synoptically uplifted aerosol layers, but followed a random sampling pattern in those large-scale features . In addition to any systematic errors from sampling strategy, instrument error inherent in the CDP introduces additional uncertainties in the measurements of Nd. CDP measures cloud droplets within a specific size range (i.e., 2 to 50 µm in diameter). It has limitations regarding droplets that fall outside its designed size range. Coincidence errors may occur when multiple droplets pass through the sensor's detection volume but is counted as a single droplet. The impact of observational uncertainty on the model constraints is examined in in Sect. .

Another potential source of systematic uncertainty may arise from the use of UHSAS100 as a proxy to CCN02 over SOCRATES. While a near one-to-one relationship between UHSAS100 and CCN02 has been reported for the SOCRATES campaign , the campaign-mean ratio of CCN02 to UHSAS100 is approximately 1.08 (±0.3), based on the median and interquartile range of the CCN02 : UHSAS100 ratio uncertainty shown in their Fig. S2. This suggests that UHSAS100 may underestimate CCN02 by 8 % on average. Moreover, the activation diameter for SO aerosol is typically below 100 nm at 0.2 % supersaturation, and likely closer to 80 nm for the aerosol population sampled during SOCRATES . This suggests that USHS100 may introduce an even greater underestimation of CCN02 compared to UHSAS100. To reflect the potential offset between UHSAS100 and CCN02, we conducted sensitivity tests by increasing the observed “CCN” by 8 % and 40 %, representing the lower and upper bounds of the CCN02 to N100 ratio uncertainty, to examine how this affects our results (Sect. ).

Having discussed uncertainty in the observations, we can turn our attention to uncertainty in the representation of processes in models. While the PPE samples a large number of possible representations of the underlying physics, it is still quite sparse . To systematically explore parametric uncertainty across the PPE, we build emulators (surrogate models) for the campaign-mean Nd and CCN using Gaussian Process (GP) regression . Emulators are trained by using the 45 perturbed parameters as inputs and simulation outputs (e.g., campaign-mean Nd and CCN) using a subset of the PPE ensemble as training data. Emulators are trained on the sample of different process representations in the CAM6 PPE data (Fig. S2). The creation and validation of the emulators follows previous literature . With the GP emulators, we sample 1 million model realizations of Nd and CCN (e.g., model variants) with 1 million different combinations of parameter values sampled uniformly across 45 dimensional parameter space.

Model variants are ruled out when they are observationally implausible based on a implausibility measure 4I(x)=|M-O||Error(M)|+|Error(O)|>1 where M is the emulator campaign-mean and O is the observed campaign-mean . Error(M) and Error(O) denote the deviation from the emulator campaign-mean and observation campaign-mean, respectively. Error(M) comes from emulator uncertainty and the variance Var(M) in the emulator estimate is directly calculated from GP regression. Error(M) is estimated as ±1.96×Var(M). The number of 1.96 is chosen as ±1.96×Var(M) covers approximately 95 % confidence bounds of the emulator uncertainty. Estimating observational uncertainty Error(O) as fractional value is commonly used in observational constraints on models . We discuss observational uncertainty in terms of a fractional error fobs. Finally, we write the implausibility metric I(x) where we account for 95 % uncertainty in the emulator and an arbitrary observational uncertainty as 5I(x)=|M-O||Var(M)⋅1.96|+|O⋅fobs|>1 Model variants are excluded when I(x) exceeds 1. An illustration of our constraint process is summarized in Fig. S3.

In this paper, we vary the observational uncertainty by varying fobs under two conditions: (1) with emulator uncertainty and (2) without emulator uncertainty, to characterize the impact of different sources of uncertainty on our ability to constrain the response of Nd to anthropogenic aerosol. We discuss the impact of different values of fobs on the model constraint process in Sect. . Equation () is a simplified implausibility metric as in , . Here, we only consider observational uncertainty and emulator uncertainty in the comparison between 1 million model variants with observations. Spatial-temporal representation uncertainty and model structural uncertainty are also important as discussed in . We set the spatial-temporal representation uncertainty to 0 in Eq. () as we collocated the model outputs to flight track locations in 2 min × 50 m composites. The characterization of model structural uncertainty is conceptually ambiguous to quantify and is not considered in this work.

We conducted sensitivity tests on the observationally plausible 2.5–97.5th percentile range of ΔNd, PD-PI to the emulator and observational uncertainties. The observationally plausible 2.5–97.5th percentile range of ΔNd, PD-PI was calculated with varying presumed observation uncertainties. To reduce noise from imperfect emulators, we conduct another set of sensitivity tests with emulator uncertainties set to 0 (Error(M) = 0) in Eq. () in the sensitivity test. The “constraint” is quantified as the reduction in the observational plausible range relative to the prior range predicted from the 1 million model variants. The relative reduction in range is calculated using 6constraint=1-Posterior: ΔNd,(PD-PI),obs, 97.5-ΔNd,(PD-PI),obs, 2.5Prior: ΔNd,(PD-PI), 97.5-ΔNd,(PD-PI), 2.5 Where the subscript “obs” denotes the sources of observations that are used for constraints. The posterior range refers to the range of observationally plausible ΔNd, PD-PI at 2.5–97.5th percentiles. The prior range refers the 2.5–97.5th percentile range of ΔNd, PD-PI derived from the original 1-million-member sample. Mathematically, the range of constraint in Eq. () should vary between 0 to 1. The greater the magnitude, the better constraints we can achieve.

In addition to the comparison between aircraft measurements and model outputs saved along flight tracks, we examine the simulated global oceanic mean Nd and confront it with observations. Observations of global oceanic Nd are derived from the Moderate Resolution Imaging Spectroradiometer (MODIS). MODIS is a passive radiometer onboard NASA’s Terra and Aqua satellites. Nd is calculated from MODIS retrievals of effective radius (re) and optical depth (τ) assuming an adiabatic cloud . MODIS Nd is calculated for daily means for the period 2003–2015 and is gridded to 1 by 1° resolution as in . During winter, high-latitude regions (e.g., Arctic, Antarctic) have greater solar zenith angle (SZA), resulting in lower reflected solar radiation, making retrievals of cloud properties (e.g., re and τ) less reliable. MODIS Nd is unavailable during wintertime high latitude regions. To ensure consistency in the comparison between MODIS Nd and the model data, Nd data from months and latitudes where MODIS retrievals are unavailable are removed from the ESM dataset. In addition to global oceanic mean Nd from MODIS, we also examine a box region from MODIS with latitude range of 65–42° S and longitude range of 132–165° E, which covers the SOCRATES campaign. MODIS Nd is computed in this box region and compared with campaign-mean Nd from SOCRATES in-situ measurements.

We examine relationships between CCN and Nd across flight composites (50 m × 2 min bin median) within individual PPE members. The number of flight composites valid for CCN-Nd comparisons from SOCRATES in-situ observations is 44 (Fig. : red dots). This number is smaller than the results in as we choose a lower altitude level for analysis with a focus on warm liquid cloud. The number of colocated flight composites valid for CCN-Nd comparisons for each PPE ensemble member (Fig. : black dots) is less than that from observations (red dots) since some flight composites simulates near-zero Nd and are excluded from our analysis. This might be due to the coarse vertical resolution of CAM6 and linear interpolation cannot fully capture the Nd variability in the vertical. Despite the limitations, PPE ensemble members simulate CCN and Nd flight composites that are comparable with observations.

CCN at 0.2 % supersaturation correlates positively with Nd when comparing matched flight composites along individual flight tracks in the PPE (Fig. ). This is not surprising as we expect CCN at 0.2 % supersaturation to be a reasonable proxy for the aerosol particles that activate to form cloud droplets under typical marine boundary layer updraft conditions, consistent with observations . Hereafter, we refer to the simulated CCN at 0.2 % supersaturation from CAM6 simply as CCN for simplicity. Observations of CCN refer to the observed aerosol concentration with diameters ranging from 0.1 to 1 µm from UHSAS100.

Figure shows a subsample of ensemble members with varying levels of agreement with observations, but a positive correlation between CCN and Nd in log space is found for most of the PPE members (i.e., 224 out of 262). However, the linear regression slope of Nd on CCN is high relative to observations for the majority of PPE members (Fig. S4a). Because Nd is a product of both CCN activating into droplets and precipitation removing drops , a higher CCN-Nd slope in CAM6 does not necessarily indicate a higher simulated aerosol activation efficiency. This diagnostic is broadly telling us that more CCN is required in CAM6 PPE to produce the same amount of Nd through aerosol activation in the presence of coalescence scavenging compared to observations, particularly at low Nd concentration (e.g., Fig. a). Lower Nd over SOCRATES is associated with increased precipitation rate and greater contribution of coalescence scavenging in controlling Nd . The negative correlation between Nd and precipitation rate is also found in the CAM6 PPE (Fig. S5). The high bias in the regression slope of Nd on CCN in CAM6 PPE may indicate a stronger loss in Nd from overestimated coalescence scavenging at low Nd concentration in models. Additionally, the low-biased Nd may also be influenced by an underestimation of subgrid-scale vertical velocity, turbulence intensity, and other dynamical factors that suppress supersaturation and droplet activation. We verify our hypothesis in the discussion of parameter constraints of CAM6 PPE using observations in Sect. .

Most of the PPE members simulate Nd that is low relative to observations, regardless of whether CCN is underestimated (Figs. S4, ). One example is the PPE ensemble of 244 of CAM6 where even though the simulated CCN is relatively close to observations, the Nd is still biased low (Fig. d). This supports the hypothesis in that aerosol biases are not the sole contributors to the low Nd in CAM6, highlighting the importance of other contributing factors.

Although most PPE members exhibit low Nd, there are some members that are close to observations (Figs. S4, c). We next compare the PPE with observations to rule out PPE members that are far away from observations (e.g., Fig. d) and characterize which parameterized processes are important to ACI over SO.

We compare campaign-means of CCN and Nd because averages reduce random errors of aircraft measurements due to instrument noise, atmospheric turbulence, or other transient variations . Unlike random errors, systematic errors such as sensor miscalibration and systematic sampling cannot be reduced by averaging. The model-observation comparison process follows Eq. () as detailed in Sect. .

Observations of CCN and Nd identify and constrain physical processes that are important for ACI (Fig. ). We show the 10 most constrained parameters out of 45 in Fig. . The full list of constrained parameter spaces is shown in Fig. S6. By examining how different parameter values are constrained relative to observables we can try to build an understanding of how different processes drive observables. This also illustrates the problem of equifinality where observed values can be arrived at by combining processes in different ways.

Confronting the PPE with observations of CCN constrains aerosol processes (e.g. sea salt emission) and precipitation processes (e.g. autoconversion, accretion) (Fig. a; the detailed parameter explanation is in Table S1 in the Supplement). The sea salt emission scale factor is constrained to higher values, indicating observations of CCN during SOCRATES are consistent with stronger aerosol production in the CAM6 PPE. This is consistent with a lack of aerosol production in CAM6 .

Constraints on precipitation processes point to the importance of precipitation as an aerosol sink. One of the key parameterization in warm cloud in climate models is the autoconversion, which represents the rate of initial rain formation through collision-coalescence between small cloud droplets.

We can make sense of the relationship between CCN and the autoconversion parameters by looking at how the rate of rain creation through autoconversion works in CAM6. Autoconversion in CAM6 is written as : 7∂qc∂tauto=a⋅qcb⋅Nd-c where ∂qc∂tauto is the rate of generation of rain from cloud water. The autoconversion rate depends on the cloud droplet concentration (Nd [cm⁻³] in Eq. ) and cloud water content (qc [kg kg⁻¹] in Eq ). a, b and -c are uncertain parameters perturbed in the CAM6 PPE (Table S1). They are micro_mg_autocon_fact, micro_mg_autocon_lwp_exp and micro_mg_autocon_nd_exp, respectively. Selecting parts of parameter space that are consistent with observations of CCN leads to lower autoconversion scale factors (a in Eq. ) (less efficient rain production by cloud). The effect of larger exponents on liquid water content (b in Eq. ) on the rain production depends on the relative magnitude of liquid water content qc. Larger b can result in thicker clouds that precipitate more efficiently under conditions of qc>1 kg kg⁻¹. A reversed effect can happen under conditions of qc<1 kg kg⁻¹. The condition of qc>1 kg kg⁻¹ seems unlikely during SOCRATES campaign observations and model simulations . Overall, this results in a lower rain rate across cloud liquid water content values when the scale factor is minimized and the exponent is maximized for the liquid water content in typical stratocumulus clouds (Fig. S7). The shift to lower rain rate with observational constraints in CAM6 PPE indicates that rain rate and the loss of CCN from precipitation scavenging are overestimated for the majority of members in CAM6 PPE.

Observations of Nd from SOCRATES constrain parameters related to aerosol and precipitation process (Fig. b), consistent with the findings in and , that the Nd budget is a function of a source of droplets from CCN and sink from collision-coalescence. The constraint on initial rain formation rate during autoconversion (Eq. ) and the constraint on the strength of precipitation suppression are consistent with CCN observations (Fig. a, b). Broadly, constraints on precipitation formation are consistent with a weaker sink of cloud droplets as well as less cloud droplet removal via precipitation. This supports the hypothesis in Sect.  that rain rate and the loss of Nd from precipitation scavenging are overestimated for the majority of members in CAM6 PPE. We also want to note that the SO is dominated by supercooled liquid cloud , making the glaciation (Bergeron–Findeisen Process: water vapor deposits onto ice crystals) important in this region. This means that the growth of ice crystals might be an important sink for Nd. However, we believe the Nd loss from freezing is minimal when our analysis is restricted to the altitudes below 2 km. This is because a large fraction of snow melts and contributes to rain precipitation at low altitudes (Fig. 2 in ). Mixed-phase and ice cloud processes are important in initiating rain as most rain is derived from ice that has melted to form rain . However, the importance of ice processes is not apparent in our process constraint focused on warmer clouds (Fig. S6). examined ice processes over SOCRATES using observations of detailed aerosol and ice nucleating particle (INP) measurements and models (e.g., CAM6), but the process constraints on ice processes with observations of INPs is beyond the scope of this study.

The parameter constraints from observations of Nd are more stringent than the constraints resulting from using observations of CCN. This is consistent with Nd being the emergent product of aerosol and precipitation processes . In addition to aerosol and precipitation processes, mechanisms important for aerosol activation are also constrained by Nd observations (Fig. b), such as deep convection (e.g., zmconv_capelmt), subgrid velocity (e.g., microp_aero_wsub_scale), and turbulence (e.g., CLUBB: Cloud Layers Unified by Binormals parameters in Table S1), as they play a role in vertical aerosol transport and in generating supersaturation. In particular, microp_aero_wsub_scale is efficiently constrained to higher values, suggesting an underestimated subgrid velocity (i.e., lower updraft speed) that suppresses supersaturation, leading to lower Nd.

Finally, we examine the effect of constraining the PPE using observations of both Nd and CCN. The effect of combining these constraints is similar to the constraint arrived by Nd alone (Fig. c). This is consistent with Nd being an emergent property of both aerosol processes and cloud and precipitation processes.

Observations of CCN and Nd during SOCRATES constrain aerosol, precipitation, and cloud processes. In the next section we examine whether the process constraint from observations of CCN and Nd constrains the response of Nd due to anthropogenic aerosol. Precipitation rate would be a useful constraint on the response of Nd as both observations of CCN and Nd constrain precipitation process as discussed in this section. However, the path to including an observational constraint of cloud base precipitation is somewhat opaque and is not included here. Light precipitation rate at cloud base can be retrieved using radar-lidar techniques , but to provide an apples-to-apples comparison to CAM6 in terms of cloud base precipitation we believe that an instrument simulator is needed .

As discussed in the previous section, observations of Nd and CCN constrain the range of possible process representations. In turn, these same processes drive the response of Nd to anthropogenic aerosol. This results in a strong correlation between PD Nd and PI Nd (Fig. ; black line) and by extension the change in Nd between PI and PD (Fig. ; orange line) in the CAM6 PPE. The fractional change in Nd (Δln⁡Nd) is computed by taking the slope of ΔNd, PD-PI to Nd following the definition from . The Δln⁡Nd predicted by the CAM6 PPE (0.23) is greater than the expert elicitation range from (i.e., 0.05 to 0.17) (Fig. ). The emergent relationship between PD Nd (observable) and the ΔNd, PD-PI (unobservable) with a r-value of 0.95 can be used to constrain the likely range of ΔNd, PD-PI if we know the possible range of PD Nd (observable). As suggested in , emergent relationships used for constraints require process-level understanding. We explain the emergent behavior from CAM6 PPE (Fig. ) using a sink-source model of Nd in Sect. .

The uncertainty associated with airborne measurements of aerosol and cloud microphysics is difficult to define as a fixed value because it depends on multiple sources of uncertainties such as: sampling error due to limited spatial and temporal coverage of flight tracks, variability in flight patterns (e.g., altitude, and positioning relative to cloud features), instrument noise due to environmental variability (e..g., turbulence, wind shear). Therefore, assuming a fixed observation uncertainty of ±20% for airborne measurements in Sect.  is just a to provide a baseline amount of uncertainty.

In this section, we perform sensitivity tests on the observationally plausible ΔNd, PD-PI by the varying observational uncertainty under conditions of with emulator uncertainty (i.e., Error(M) = ±1.96×Var(M) in Eq. ) and without emulator uncertainty (i.e., Error(M) = 0×Var(M) in Eq. ). Systematic uncertainty in the observations is assumed to vary from ±5% to ±100% for the campaign-mean. The constraints on ΔNd, PD-PI is calculated as the reduction in the 2.5–97.5 percentile range of ΔNd, PD-PI following Eq. ().

Figure  shows the constraints on ΔNd, PD-PI by observations of CCN (orange line), Nd (blue line) and combining CCN and Nd (green line). Overall, ΔNd, PD-PI is more tightly constrained without emulator uncertainty (Fig. b, d), which makes sense as excluding emulator uncertainties (i.e., Eq. ) allows more model variants to be excluded during model-observation comparison relative to Eq. (). The constraints on ΔNd, PD-PI under two conditions (i.e., with and without emulator uncertainty) both have a positive shift in the 2.5–97.5 percentile range (Fig. c, d), suggesting the constraints are not a result of noise from emulator uncertainties.

Consistent with Fig. , observations of CCN provide nearly no constraint on ΔNd, PD-PI regardless of the uncertainty in CCN (Fig. : orange line). With increasing uncertainties from airborne measurements of Nd, the plausible range of ΔNd, PD-PI is less constrained under both conditions of with and without emulator uncertainty (Fig. : blue line). The weakened constraints on ΔNd, PD-PI with increasing uncertainties in observations is due to the retention of more model variants as plausible during model-observation comparison process (Fig. S3), thereby exacerbating the equifinality problem , for which increasing number of plausible parameter combinations results in broader parameter spaces and thereby reduced constraints on ΔNd, PD-PI (Fig. a, b).

The reduction in ranges (constraints) becomes negligible with observation uncertainty of Nd reaching ±75 % under the condition with emulator uncertainty (Fig. a). The threshold of observational uncertainty of Nd that can achieve constraints is a bit larger under the condition without emulator uncertainty, which is ±85 % (Fig. b), suggesting skillful emulators play an important role in advancing our constraints. Again, we view uncertainty from emulation as an eminently tractable problem compared to developing a better understanding of instrumental systematic uncertainty. Improving emulator uncertainty can be achieved by creating PPEs that can be easily emulated through a more careful selection of perturbed parameters and an increased amount of training data.

Another key insight from Fig.  is that the improvement in constraint when both Nd and CCN are considered emerges at low observational uncertainties for both measurements (Fig. a, b: green line is above the blue). The improved constraint is also found in that adding the number of constraint variables with no structural inadequacies improve the constraints on aerosol forcing when their constraints are consistent across multiple observation types. In our work, although the constraints on ΔNd, PD-PI by observations of CCN is minimal, the constraints on parameter spaces by airborne observations of CCN and Nd are consistent (Fig. ), resulting in the improved constraints on ΔNd, PD-PI. The improvement of the constraints on ΔNd, PD-PI only happens with small uncertainties associated with airborne measurements (Fig. a, b: green line), highlights the importance of accurate airborne measurements of aerosol and cloud properties on constraining ΔNd, PD-PI.