Metamodeling of Droplet Activation for Global Climate Models
1. Introduction
Interactions between aerosol and clouds yield one of the largest sources of uncertainty in understanding climate and future climate change on regional and global scales (Boucher et al. 2013). Within Earth’s atmosphere, homogeneous liquid water droplet formation is not thermodynamically favorable (Pruppacher and Klett 1997); instead, the pathway to nucleating cloud droplets is aided by the presence of ambient aerosol, a subset of which possess physical and chemical characteristics that allow them to serve as cloud condensation nuclei (CCN). These CCN provide a linkage between the physiochemical processes of atmospheric particles and cloud microphysics.
Changes in the background aerosol population can directly affect the properties of a nascent cloud droplet population. For instance, holding liquid water content constant, an increase in the number of CCN would tend to increase the total cloud droplet number concentration (the “Twomey” effect) while necessarily reducing the average size of the droplets (Twomey 1974). Such a change could enhance a cloud’s albedo, an effect that could be further amplified through microphysical feedbacks since smaller droplets impede the production of drizzle and thus lengthen cloud lifetime (Albrecht 1989). Mechanisms whereby aerosol influence the properties of clouds (and ultimately climate) are generally known as “aerosol indirect effects” (Haywood and Boucher 2000; Lohmann and Feichter 2005) and provide a path for changes in the ambient aerosol to produce cascading effects up to progressively larger scales of atmospheric motion (e.g., Wang 2005; Ekman et al. 2011; Morrison et al. 2011; Tao et al. 2011; Fan et al. 2012; Altaratz et al. 2014).
Aerosol indirect effects can either warm or cool the climate, but they all fundamentally depend on a subset of the ambient aerosols that function as CCN. The theory describing the dependency of cloud droplet nucleation (also known as aerosol activation) on CCN availability and ambient aerosol has been rigorously developed using adiabatic and entraining parcel theory (Seinfeld and Pandis 2006; Pruppacher and Klett 1997) and depends on details of the heterogeneous chemical composition, number, size distribution(s), and mixing state of the background aerosol (Mcfiggans et al. 2006) as well as local meteorology (Morales and Nenes 2010). Under polluted conditions, effects relating to chemical composition could produce a climatic effect as large as the basic Twomey effect (Nenes et al. 2002; Lance et al. 2004).
The development of activation parameterizations was pioneered by Twomey (1959) and Squires and Twomey (1961), who derived a relationship between the number of activated particles and the environmental supersaturation based on an aerosol size distribution approximated by a power law. Ghan et al. (2011) presented a thorough overview of subsequent developments over the past five decades and an intercomparison of several modern parameterizations. However, there is still an active effort to improve these parameterizations, as they are increasingly called upon to mediate between ever more complex aerosol models and the climate models to which they are coupled. For instance, the parameterization initially developed by Nenes and Seinfeld (2003) has seen continuous development, including modifications to handle condensation onto insoluble but wettable particles using adsorption activation theory (Kumar et al. 2009), environmental entrainment (Barahona and Nenes 2007), and numerical improvement of the population-splitting technique (Barahona et al. 2010; Morales Betancourt and Nenes 2014). Similarly, Ghan et al. (2011) extended the parameterization of Abdul-Razzak and Ghan (2000) to account for nonunity values of the accommodation coefficient 
Additionally, following the original integral/geometric approach by Twomey (1959), analytical representations of supersaturation evolution from adiabatic parcel theory have been progressively generalized to relate aerosol distributions to activation kinetics (Cohard et al. 1998; Khvorostyanov and Curry 2006, 2008; Shipway and Abel 2010; Shipway 2015). Although fundamentally analytical parameterizations, schemes of this class typically must rely on expensive numerical operations, such as in the evaluation of hypergeometric functions and iterative loops.
While most of these recent efforts toward improving activation parameterizations have focused on building highly generalized, “physically based” tools, there is still an application for other parameterization approaches. Saleeby and Cotton (2004) parameterized droplet nucleation for a cloud-resolving model, the Regional Atmospheric Modeling System (RAMS), by constructing a four-dimensional lookup table based on temperature, vertical velocity, aerosol number concentration, and the median radius of a lognormal aerosol mode with assumed chemical composition. Ward et al. (2010) added a fifth dimension representing chemical composition via aerosol hygroscopicity [following κ–Köhler theory (Petters and Kreidenweis 2007)] to the lookup table and later generalized this dimension to aerosol soluble fraction (Saleeby and van den Heever 2013). Constructing lookup tables of detailed parcel model results can be considered a form of model emulation combining a cache of known results and local polynomial (linear) approximation.
As the degrees of freedom and number of parameters describing a given aerosol population in a model increase, the burden of saving enough known points to interpolate through the parameter space via lookup table to some reasonable accuracy increases algebraically. For instance, the CESM features a modal aerosol population with three predefined, internally mixed lognormal modes, each with a fixed geometric standard deviation (Liu et al. 2012). Each mode is uniquely described by two moments (total number and total mass concentration) and the chemical composition of the mode by a single prognostic hygroscopicity term. Thus, the entire aerosol population has 
The goal of this study is to apply a surrogate modeling or emulation technique commonly used in the uncertainty quantification literature to a detailed parcel model capable of describing aerosol activation; this yields an efficient parameterization optimized for the high-dimensional parameter space affecting droplet nucleation in coupled aerosol–climate model. In essence, employing the derived emulator as an activation parameterization would be akin to directly coupling a detailed parcel model to a global model. Such a parameterization would be directly physically based but rely on fewer assumptions that affect the condensational growth of aerosol into CCN. However, it would also incorporate the efficiency of a lookup table, since the emulator would be designed to require a scarce amount of cached information and to be computationally cheap to evaluate. In this way, the emulator would improve upon the framework of a lookup table and be extensible to a very high-dimensional parameter space and thus be compatible with aerosol–climate models of increasing complexity.
Additionally, this study aims to better understand which parameters and inputs into droplet nucleation calculations are key to determining the resulting activated number concentration. Morales Betancourt and Nenes (2014) supplemented traditional error metrics by computing local sensitivities of the number concentration activated to key aerosol size distribution parameters using a detailed parcel model and comparing them to the adjoint of their parameterization. This study applies a related approach instead using global sensitivity analysis, which is suitable for identifying how uncertainty in inputs and parameters contributes to uncertainty in a model response. This analysis has not previously been applied to droplet activation and can provide additional metrics for evaluating parameterizations and their potential biases.
Section 2 describes the parcel model and the probabilistic collocation method (PCM) used to build its emulator. Section 3 presents results from applying the PCM to build a parcel model emulator designed to simulate the activation of a single lognormal aerosol mode under a wide variety of background environments and compares the new emulator to existing activation parameterizations. Section 4 motivates an extension of the technique to an emulator suitable of mediating aerosol activation in a coupled aerosol–climate model.
2. Methodology
a. Parcel model
Adiabatic parcel models are commonly used to study droplet activation and its sensitivity to factors such as environmental conditions and ambient aerosol properties. For this work, a novel parcel model based on previous studies (Leaitch et al. 1986; Nenes et al. 2001; Seinfeld and Pandis 2006) was designed and implemented to accommodate diverse, chemically heterogeneous, and polydisperse aerosol populations. The model simulates droplet growth on the initial aerosol population due to condensation within a constant-speed adiabatic updraft.
where the parameter set 
To simulate droplet activation, the parcel model first computes an equilibrium wet-size distribution from the given initial aerosol population and initial environmental temperature, pressure, and relative humidity. Then, a set of conservation equations that describe the evolution of the parcel temperature, supersaturation, liquid/vapor water content, and pressure are integrated forward in time using a solver suitable for stiff systems [variable-coefficient ordinary differential equation solver (VODE); Brown et al. (1989)]. The complete system of equations and further details on the parcel model can be found in appendix A.
b. Polynomial chaos expansion
We construct an emulator of the parcel model in order to assess droplet activation by applying the probabilistic collocation method (PCM; Tatang et al. 1997). The PCM maps a set of input parameters to an output from the parcel model by building a response surface using a polynomial chaos expansion. The polynomial that results from this process is a computationally efficient, high-fidelity reproduction of the detailed parcel model simulation. Although often used for conducting global sensitivity analyses (Pan et al. 1997; Calbó et al. 1998; Mayer et al. 2000; Lucas and Prinn 2005; Anttila and Kerminen 2007) chaos expansion-based emulators have also been used to build deterministic parameterizations (Cohen and Prinn 2011). To apply and build the chaos expansions discussed here, the open-source Design Analysis Kit for Optimization and Terascale Applications (DAKOTA; Adams et al. 2014), version 6.1, was used, which automates the sampling of the PCM collocation points and the computation of the coefficients of the polynomial chaos expansion given a user-generated interface to a numerical model (the parcel model described in section 2a) and a description of the inputs and outputs to and from that interface.
where 
A practical consideration in applying the PCM to a particular problem is what subset of the 
choose parameter sets with roots closest to the origin (Sudret 2008) and
cross validate the regression result using
in an iterative, greedy fashion with the potential to yield sparse solutions with some coefficients 
c. Emulation of parcel model
The PCM was applied to emulate the activation of a single, lognormal aerosol mode embedded in a constant-speed adiabatic updraft as simulated by the parcel model described in section 2a. Specifically, the model was used to predict the maximum supersaturation 
The roots of these Legendre polynomials can be inverted using Eq. (5) to determine values in the original, physical parameter space to use in sampling the parcel model.
The bounds for the physical parameters supplied to the PCM were chosen in order to characterize activation near cloud base (Table 1). The logarithm of several variables (aerosol number concentration, aerosol geometric mean radius, and updraft velocity) is used because the supersaturation maximum computed by the parcel model is sensitive to changes in these parameters over several orders of magnitude. Updraft velocity is permitted to vary between 0.01 and 10.0 m s−1; over this range (which covers a spectrum from weakly convecting, stratiform clouds to strong, deeply convecting ones) and the range of aerosol number concentration (which includes clean and very polluted regimes), activated fraction can range from virtually nothing to complete activation of the entire aerosol population. The lower bound of updraft speeds considered here is less than the minimal value allowed in many climate models (Golaz et al. 2011). The aerosol mode geometric mean radii 
Input parameters and bounds used in computing chaos expansion for emulating droplet activation from a single, lognormal aerosol mode embedded in a constant-speed updraft.
Many parameterizations of droplet nucleation diagnose activation directly by applying equilibrium Köhler theory. To do this, the maximum supersaturation achieved by a cloudy parcel is used as a threshold; particles with a Köhler-predicted critical supersaturation lower than this maximum environmental supersaturation are considered to be activated. However, physically, for a droplet to activate it must grow beyond a critical size corresponding to this critical supersaturation. Because of kinetic limitations on droplet growth, this may not be realized for droplets growing on very large CCN (Nenes et al. 2001). Ghan et al. (2011) suggests particles with radius larger than 0.1 μm or those whose critical supersaturations are close to the environmental maximum supersaturation are likely to suffer from this effect. By directly considering a detailed parcel model, the emulators constructed here consider kinetic limitations on droplet growth and their feedback on the evolving parcel supersaturation. In existing, physically based parameterizations in the literature, an instantaneous growth-rate assumption must be applied. This assumption causes parameterizations to underpredict supersaturation maximum because instantaneous growth will tend to condense water from the vapor phase too quickly and release surplus latent heat, which suppresses the increase of the supersaturation.
Because the computed supersaturation maximum in a parcel model activation simulation can also vary over several orders of magnitude, we use 
For the eight-parameter input space governing single-mode activation considered here, the third- and fourth-order chaos expansions produced by the PCM have 165 and 495 terms, respectively. The number of terms is equivalent to the number of coefficients one must store in order to reuse a given chaos expansion. This small memory footprint affords chaos expansions a huge advantage over similar parameterizations based on detailed lookup tables. An isotropic lookup table with M parameters and n sample points for each parameter would require 
where 
d. Global sensitivity analysis
We supplement the assessment of our new droplet activation emulator by calculating a set of global sensitivity metrics not previously applied to this problem. The method deployed here is a variance-based decomposition, which seeks to assign uncertainty in a model response to uncertainty in both individual model input parameters and their interactions with one another. Two different quantities, called Sobol’ indices (Sobol’ 2001), are produced by this method: main (Si) and total (Ti) effect indices. Sobol’ indices can be used to rank the relative importance of model inputs in influencing its response and for identifying potential inputs that are unimportant and also candidates to be held fixed without grossly biasing the accuracy of a model emulator (Sobol’ 2001).
Sudret (2008) derives alternative equations that further clarify the meanings of these terms in the context of studying model emulators. Critically, these terms can be expressed as multidimensional integrals over model input parameters that can be approximated via Monte Carlo or other sampling techniques, although it is very computationally expensive to do so. We adopt the column swap-out sampling method of Weirs et al. (2012), which combines Latin hypercube sampling with perturbed combinations of parameters to efficiently approximate 
Although the sampling procedure can be repeated for the emulators derived via polynomial chaos expansion, the orthogonality of the basis terms that constitute each expansion lends itself to a more direct computation of 
3. Results
a. Evaluation of emulators
To assess the performance of the emulator, two sets of n = 10 000 samples were drawn using maximum Latin hypercube sampling from the parameter space defined in Table 1. This randomized design ensured that representative, equal numbers of samples were drawn from across the multidimensional parameter space. In the first set, variables whose logarithms were used to build the emulator were sampled in logarithmic space; in the second set, these variables were transformed back to their original values (e.g., from 
Figure 1 illustrates the performance of a fourth-order expansion whose coefficients were derived using ordinary least squares. The large range of initial temperatures, pressures, aerosol populations, and updraft speeds sampled here leads to a very large range of supersaturation maxima achieved by the ascending parcel. Weaker updraft speeds are generally associated with lower maximum supersaturations and corresponding to lower aerosol activated fraction; the opposite is true when strong updrafts are present, although there are some cases where a strong updraft activates a small fraction of aerosol. This typically occurs when initial aerosol size distribution is shifted toward larger radii and under polluted conditions with aerosol number concentrations greater than 3000 cm−3. However, over the large parameter space sampled, the chaos expansion accurately reproduces the parcel model’s determination of 
One–one plot comparing (a) predicted supersaturation maximum and (b) diagnosed equilibrium droplet activated fraction between parcel model and a polynomial chaos expansion of order 
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0223.1
For parameter sets leading to a large activated fraction of 0.8–1.0, the relative error of the chaos expansion (compared to the parcel model) rarely exceeds 5% and on average (for all activated fractions) is 5.7%. While the mean relative error in each activated fraction decile is close to 0, the standard deviation in the relative error tends to increase for the lower ones; the standard deviation in relative error decreases from 19.7% for activated fractions in the range 0.1–0.2 to 3.5% for those in the range 0.8–0.9. This suggests that there is a nonlinear component in the mapping from the input parameter space to the emulated maximum supersaturation and diagnosed droplet number concentration that is prevalent in the weak droplet activation regime; the predicted activated fraction is more sensitive to small changes in the input parameters in this regime than in others.
Increasing the order of the chaos expansion tends to improve the accuracy of the predicted 
Summary statistics for supersaturation maxima predicted by chaos expansions derived in this study compared to detailed parcel model calculations. The chaos expansions are organized by the method used to derive their coefficients and the expansion order in the two leftmost columns. (left to right) The reported statistics are the normalized root-mean-square error (NRMSE), coefficient of determination r2, mean relative error (MRE), and standard deviation of the mean relative error (MRE std dev).
These same statistics, computed for the diagnosed droplet number concentration given the predicted supersaturation maximum, are summarized in Table 3. Here, the trend is similar to before; increasing the order of the expansion tends to improve the accuracy of the diagnosed number concentration in terms of mean relative error and also tends to decrease spread around that value. Third-order expansions tend to produce more accurate results with respect to the mean relative error, but this is overshadowed by the fact that there is far more variance in their predicted values as indicated by the standard deviation of their relative errors, which are almost twice as large as those of the higher-order expansions.
b. Comparison with other parameterizations
We compare the performance of the chaos expansion-based emulators to two existing parameterizations from the literature. The scheme by Abdul-Razzak and Ghan (2000) (ARG)—which is widely used in global models—utilizes a psuedoanalytical solution to an integro-differential equation derived from the adiabatic parcel system with embedded aerosol growing via condensation. This is in contrast to the scheme by Morales Betancourt and Nenes (2014) (MBN), which is based on an iterative scheme to separate the aerosol population into subsets whose growth is inertially limited or not and uses this information to derive a maximum supersaturation for a given parcel system. The MBN scheme is generally more expensive to evaluate than the ARG scheme because of its iterative nature but is often more accurate owing to its consideration of the potentially important effect of large albeit unactivated aerosol particles (Simpson et al. 2014). In contrast with these schemes, our emulators simulate the activation process based on the explicit numerical solution obtained from a detailed parcel model, which is similar to the one used to build and evaluate the MBN scheme (e.g., Nenes and Seinfeld 2003).
The parameter sets used in section 3a were also used to compute droplet activation with the ARG and MBN parameterizations. The relative errors between the supersaturation maximum and droplet number predicted by these schemes and the chaos expansions compared to the parcel model are illustrated in Fig. 2. Both the ARG and MBN parameterizations are more accurate than the second- and third-order chaos expansions. The ARG scheme tends to underpredict the maximum supersaturation, which is consistent with previous investigations into its performance (Abdul-Razzak and Ghan 2000; Ghan et al. 2011; Simpson et al. 2014). This tends to produce a bias toward underprediction of droplet number. The MBN scheme tends to yield more accurate predictions of both maximum supersaturation and droplet number. However, both schemes are outperformed by the fourth- and fifth-order chaos expansions, both on average and in terms of the variance of the predictions; for instance, the OLS-derived fourth- and fifth-order expansions yield relative error in predicted 
Box plots illustrating mean relative error between (a) supersaturation max and (b) droplet number concentration predicted by chaos expansions and parameterizations vs detailed parcel model. The chaos expansions have been grouped by expansion order (x axis) and method for computing their coefficients (OLS, LARS, and LASSO; hue). “ARG” refers to the scheme of Abdul-Razzak and Ghan (2000); “MBN” refers to the scheme of Morales Betancourt and Nenes (2014).
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0223.1
As a consequence of tending to slightly underpredict 
In addition to producing low mean relative error in predicted 
Sensitivity of (a)–(d) parameterized and simulated maximum supersaturation and (e)–(h) activated number fraction to changes in mode number concentration, mode geometric mean radius, mode hygroscopicity, and updraft speed with all other parameters held fixed at the values 
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0223.1
The chaos expansions, as well as both the ARG and MBN schemes, capture these subtleties of activation dynamics as well as the detailed parcel model. More importantly, the expansions reproduce the sensitivity of activation to updraft speed (Figs. 3g,h), which is an important factor controlling 
Although all the chaos expansion results in Fig. 3 appear biased high compared to the parcel model, this bias does not hold true in general. Note that the ARG scheme is biased low in this particular analysis; this is generally just an artifact of fixing seven of the eight parameters and analyzing one-dimensional transects. Fixing the nonvarying parameters at different values tends to shift the bias positive or negative in a nonsystematic way. Critically, the choice of values for these parameters does not affect the chaos expansion’s ability to reproduce the sensitivity to the varying parameter, which lends confidence that the expansions accurately reproduce the behavior of the parcel model.
Since Fig. 3 highlights the fact that different schemes potentially perform better in different parts of the parameter space governing droplet activation, we stratified the sampling results based on level of pollution and updraft-speed strength and computed activated fraction relative error statistics in each of these bins as shown in Fig. 4. All of the schemes are accurate in clean and lightly polluted conditions (with aerosol number concentration 
Mean relative error in activated fraction for (a) fourth-order OLS-derived chaos expansion, (b) ARG, and (c) MBN schemes relative to detailed parcel model. Updraft speeds are light (10–50 cm s−1), moderate (0.5–2.0 m s−1), and strong (2.0–10.0 m s−1); pollution levels are clean (10–250 cm−3), light (250–1000 cm−3), moderate (1000–2500 cm−3), and heavy (2500–10 000 cm−3). Error bars denote 95% confidence interval on mean relative error from in-bin samples; samples with 
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0223.1
The fourth-order OLS-derived chaos expansion is plotted in Fig. 4a as a representative example of the chaos expansions, and it retains its accuracy across the pollution level–updraft strength spectrum. The combination of light updrafts and heavy pollution tends to produce the largest underprediction in activated droplet number, ranging from 10% to 30% for the ARG and MBN schemes. Unsurprisingly, relative error in activated fraction tends to be least sensitive to increasing aerosol number concentration in the strong updraft regime. In this case, the vigorous updraft produces strong adiabatic cooling that overwhelms latent heat release from condensation as the droplets in the parcel grow, contributing to a rapid and large 
It should be noted that parts of the parameter space we considered in applying the PCM, evaluating its output, and comparing to existing parameterizations may not be typical of real atmospheric cases. We chose a large parameter space in order to derive the most general emulator possible for this particular single-mode aerosol case. In the real world, there should be some correlation between the ambient temperatures, pressures, and updraft speeds used when diagnosing aerosol activation, while we sample these factors as if they were independent from one another. The output from applying the parcel model to nonrealistic activation scenarios could tend to inflate the computed relative errors and exacerbate differences between the parcel model and the existing, physically based parameterizations.
c. Global sensitivity analysis
Total Sobol’ indices 
Total Sobol’ indices corresponding to the prediction of 
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0223.1
Indices derived using the chaos expansions very closely approximate those derived by sampling the full parcel model. This is expected since they simply provide an alternative framework for calculating the indices from the parcel model. However, the differences between the ARG and MBN schemes and the parcel model highlight the potential for biases in these schemes due to oversensitivity to particular model parameters. For instance, the ARG scheme is less sensitive to variations in 
Maximum supersaturations produced by the MBN scheme are more sensitive to the geometric mean size of the aerosol than those from the parcel model or ARG scheme and generally less sensitive to the strength of the updraft speed. The relative importance of each term for both the ARG and MBN scheme generally agrees with the estimates from the parcel model and chaos expansions, although the ARG scheme is most sensitive to updraft speed rather than geometric mean aerosol size. By means of those parameters’ higher 
The chaos expansions help rank the relative importance of each input parameter with the potential for important interactions between terms. The leading eight terms ranked by main Sobol’ index computed using the OLS-derived chaos expansions are summarized in Table 4. For all four (second- through fifth-order) chaos expansions summarized, all the terms with rank greater than eight were an order of magnitude less important than those in the table. With the exception of the second-order scheme, the rankings of the top eight terms do not change relative order, and the main terms dominate the higher-order ones. The only higher-order terms that contribute grossly to the variance in 
Input parameters and combinations ranked by main Sobol’ index for the OLS-based chaos expansions.
d. Computational efficiency of chaos expansions
As detailed in appendix B, evaluating the emulator produced by the chaos expansion requires two sets of straightforward floating-point operations. The first set requires the projection of the input parameters into the vector space spanned by the basis polynomials of the chaos expansion, which can then be used to evaluate the basis polynomials up to the required order. The remaining operations simply multiply these intermediate evaluations together and sum them to evaluate the full expansion. In general, this procedure should lie in between the ARG and MBN schemes in terms of computational complexity. The ARG scheme relies on straightforward floating-point operations to derive an estimate for 
On average, the second- and fifth-order OLS-derived chaos expansion was 10–17 times faster than the MBN scheme given the same single-mode aerosol population. The exact speedup depended on the background velocity; for weak updraft speeds, the performance of the MBN scheme fared better, although it became much worse for updrafts where V < 2 m s−1. The ARG scheme was consistently 1–3 times faster than those same chaos expansions. Since the pathway for evaluating either the ARG or chaos expansion schemes do not change depending on the input parameters, their performance was the same regardless of what inputs were provided.
4. Summary and conclusions
An efficient parameterization of droplet activation for a single aerosol model under a wide variety of different physico-chemical properties and thermodynamic conditions was developed via statistical emulation of a detailed parcel model using polynomial chaos expansion. The emulators predict the maximum supersaturation achieved by a parcel, which is then used to diagnose activated droplet number using Köhler theory in a similar framework to existing activation parameterizations. The fourth- and fifth-order chaos expansions derived from the detailed parcel model are more accurate on average than two commonly used, physically based parameterizations from the literature (Abdul-Razzak and Ghan 2000; Morales Betancourt and Nenes 2014). Additionally, the chaos expansions are all at least 10 times faster to evaluate than the MBN scheme and only about twice as expensive as the ARG scheme. A simple algorithm was suggested for evaluating a chaos expansion that requires a minimal amount of data about the expansion (such as the basis polynomials and the coefficients of the expansion terms) to be saved; in this way, the chaos expansions offer a method for extending lookup tables to very high dimensionalities without suffering from exponentially rising storage costs.
Based on the large set of aerosol properties and thermodynamic conditions we sampled in order to derive and evaluate the chaos expansions, we observed that our emulators particularly outperform the existing schemes in conditions where a light updraft and heavy aerosol pollution (with respect to number concentration) are present. Because the ultimate goal of an activation parameterization is to couple the aerosol physics and chemistry to the cloud microphysics of a global-scale model, this deficiency in the existing parameterizations could be particularly important. Few global models have aerosol–cloud microphysics connections in their deep convection parameterizations, but many source potential cloud droplet formation based on a detailed aerosol activation calculation for their shallow convection and stratiform cloud microphysics schemes. These schemes sometimes artificially restrict the lowest possible updraft speed available for estimating droplet activation, but as a consequence they ensure that weak updrafts make up a large portion of the activation conditions considered during a model run. In regions of the world with heavy anthropogenic aerosol pollution—such as southern and eastern Asia—this provides a recipe for systematically underpredicting droplet number and potentially impacting either a global model’s simulated aerosol indirect effect on climate or the modeled aerosol–cloud interaction’s sensitivity to changes in anthropogenic aerosol emissions.
The global sensitivity analysis framed on the input parameter set used to derive the new chaos expansion emulators provides an additional, new check on the performance of existing activation schemes compared to the detailed parcel model. The breakdown of main Sobol’ indices calculated using the chaos expansions provides insight into the importance of interactions between the dominant first-order terms (updraft speed, number concentration, and geometric mean size of the aerosol distribution), which are further summarized by the total Sobol’ indices derived for the parcel model and both ARG and MBN schemes. The oversensitivity of the MBN scheme to the geometric mean radius and the undersensitivity of the ARG scheme to the updraft speed contribute to their disagreement with the parcel model across the range of pollution levels and updraft speeds studied here. Further, such sensitivity analyses could shed additional light on the potential biases of activation schemes and could provide useful metrics for evaluating the improvement of parameterizations more generally than simple ones based on relative or absolute error alone.
Critically, the framework from which the chaos expansions reported here are derived is extensible to the case where a complex, multispecies/multimodal aerosol population is tracked by a global model; in that case, the number of parameters describing the aerosol size distribution and chemical composition simply increases. Future work will derive chaos expansions emulating activation for a multimodal aerosol distribution specific to a particular global aerosol–climate model. Additionally, physical processes not considered here can also be introduced into the chaos expansion framework. For instance, entrainment can be incorporated into the parcel model following Seinfeld and Pandis (2006) and Barahona and Nenes (2007). Subgrid-scale variability in updraft speeds due to the coarse resolution of global model grids and the distribution of these updrafts can be represented either by a characteristic value (Morales and Nenes 2010) or by numerical integration over a distribution (Lohmann et al. 1999; Golaz et al. 2011). In the latter case, many activation calculations must be performed, incurring a large computational cost. However, the entire integration over a spectrum of droplet speeds could be parameterized in the chaos expansion framework, greatly reducing the cost of this calculation and potentially improving the accuracy of diagnosed cloud droplet number.
As the complexity of global aerosol–climate models increases with respect to the number of aerosol modes and species tracked by the model, there is a pressing need to understand how biases in activation calculations across the high-dimensional parameter spaces defining the aerosol–climate model affect cloud properties and ultimately impact modeled climate. This work highlights a novel way to build efficient, accurate activation schemes for this purpose akin to customized lookup tables, which cannot themselves extend to cover the necessary parameters. Employing such schemes should help improve simulated cloud microphysical properties and constrain modeled aerosol indirect effects on climate.
Acknowledgments
The work in this study was supported by the National Science Foundation Graduate Research Fellowship Program under NSF Grant 1122374, NSF Grant AGS-1339264, the National Research Foundation Singapore through the Singapore–MIT Alliance for Research and Technology and the interdisciplinary research group of the Center for Environmental Sensing and Modeling, and the U.S. Department of Energy, Office of Science (DE-FG02-94ER61937). We thank Steve Ghan (PNNL) and Athanasios Nenes (Georgia Tech) for reference implementations of their activation parameterizations.
APPENDIX A
Parcel Model Description
The adiabatic cloud parcel model implemented for this study follows the basic equations of Pruppacher and Klett (1997) and adopts the framework used by Nenes et al. (2001) to account for kinetic limitations on droplet growth. Fundamentally, the model integrates a system of coupled ordinary differential equations that describe the thermodynamic evolution of an adiabatically lifted, nonentraining parcel. In all the simulations described here, we use the variable-coefficient ordinary differential equation solver (VODE; Brown et al. 1989) to integrate the system forward in time.
where 
and then computing 
where 
APPENDIX B
Chaos Expansion Emulator Evaluation
The PCM as applied here produces two outputs: a P-length vector of coefficients α comprising real values and a 
Evaluating the chaos expansion involves two parts. First, the input parameters must be projected to conform to the space supported by the PDFs associated with each basis polynomial type, producing a set of parameters 
In general, the only computationally complex part of the chaos expansion evaluation algorithm is the projection from 
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