Journal of Geophysical Research: Atmospheres

A computationally efficient Multicomponent Equilibrium Solver for Aerosols (MESA)

Authors


Abstract

[1] Development and application of a new Multicomponent Equilibrium Solver for Aerosols (MESA) is described for systems containing H+, NH4+, Na+, Ca2+, SO42−, HSO4, NO3, and Cl ions. The equilibrium solution is obtained by integrating a set of pseudo-transient ordinary differential equations describing the precipitation and dissolution reactions for all the possible salts to steady state. A comprehensive temperature dependent mutual deliquescence relative humidity (MDRH) parameterization is developed for all the possible salt mixtures, thereby eliminating the need for a rigorous numerical solution when ambient RH is less than MDRH(T). The solver is unconditionally stable, mass conserving, and shows robust convergence. Performance of MESA was evaluated against the Web-based AIM Model III, which served as a benchmark for accuracy, and the EQUISOLV II solver for speed. Important differences in the convergence and thermodynamic errors in MESA and EQUISOLV II are discussed. The average ratios of speeds of MESA over EQUISOLV II ranged between 1.4 and 5.8, with minimum and maximum ratios of 0.6 and 17, respectively. Because MESA directly diagnoses MDRH, it is significantly more efficient when RH < MDRH. MESA's superior performance is partially due to its “hard-wired” code for the present system as opposed to EQUISOLV II, which has a more generalized structure for solving any number and type of reactions at temperatures down to 190 K. These considerations suggest that MESA is highly attractive for use in 3-D aerosol/air-quality models for lower tropospheric applications (T > 240 K) in which both accuracy and computational efficiency are critical.

1. Introduction

[2] It is well known that the phase state of an atmospheric aerosol particle plays an important role in governing its composition, water content, size, and direct radiative forcing [Wexler and Seinfeld, 1991; Seinfeld and Pandis, 1997; Tang, 1997; Martin et al., 2004]. At thermodynamic equilibrium, a particle made up of a single salt will be solid if the ambient relative humidity (RH) is below its deliquescence RH (DRH) and will be completely dissolved (liquid) at RH ≥ DRH. Once completely deliquesced, the solution droplet will continue to grow by water vapor condensation as RH further increases. Conversely, as RH decreases, water vapor will evaporate and the solution droplet will continue to shrink. Numerous laboratory experiments have also shown that a completely deliquesced particle can remain in a metastable liquid solution that is supersaturated with the salt until crystallization finally takes place at relative humidities that are much below the DRH [Tang et al., 1995]. While the DRH for a given salt depends only on temperature, the crystallization RH (CRH) is a complex function of homogeneous nucleation of the solid phase, and is sensitive to solid impurities in the solution [Lightstone et al., 2000; Onasch et al., 2000].

[3] Tropospheric particles typically contain a mixture of electrolytes of ammonium, sodium, calcium, sulfate, nitrate, chloride, etc. Hygroscopic growth of such mixed-salt particles also behaves in a similar manner as described above except that it exhibits a complex multistage deliquescence behavior instead of a single deliquescence point. This is illustrated in Figure 1 with the example of an equimolar mixture of Na2SO4 and NaCl [Tang, 1997]. As single salts, NaCl and Na2SO4 deliquesce at ∼75.3% and ∼87% RH, respectively, at 298.15 K. However, as a mixture, the solid particle partially deliquesces first at ∼74.2% RH when NaCl is completely dissolved while some solid Na2SO4 remains in equilibrium with the aqueous phase. As RH further increases, more Na2SO4 continues to dissolve into the aqueous phase until it is completely deliquesced at ∼80% RH. As the RH is now decreased, the liquid particle remains in a metastable state until ∼52.5% RH when it spontaneously crystallizes. The first deliquescence point is called the mutual deliquescence RH (MDRH) and the point when the particle is completely dissolved is called the complete deliquescence RH (CDRH). The point where a completely deliquesced particle finally crystallizes during efflorescence (evaporation) is called the mutual crystallization RH (MCRH).

Figure 1.

Mass growth factor as a function of RH during growth and evaporation of a particle containing an equimolar mixture of NaCl and Na2SO4 at 298.15 K [Tang, 1997]. Also shown are mutual crystallization RH (MCRH); mutual deliquescence RH (MDRH); and complete deliquescence RH (CDRH). Dotted and dashed lines are used for pure NaCl and Na2SO4 curves, respectively, while solid line is used for the equimolar mixture.

[4] At a given temperature, it has been shown that MDRH of a salt mixture corresponds to the eutonic point of the mixture, and is lower than the minimum DRH of each salt. Moreover, it depends only on which salts are present in the dry state, but not their relative amounts [Wexler and Seinfeld, 1991; Tang and Munkelwitz, 1993; Potukuchi and Wexler, 1995a, 1995b]. On the other hand, each subsequent partial deliquescence points during the multistage growth as well as the CDRH depend on the relative amounts of the salts present. For example, Tang et al. [1978] experimentally showed that particles having different initial compositions in wt% of NaCl and KCl, respectively, of 80-20, 64-36, and 20-80 exhibited the first deliquescence point (i.e., MDRH) at 73.8% RH, which is lower than the respective individual DRHs of 75.3% and 84.3%. However, the CDRH of each mixture was different, and ranged between 73.8% and 84.3% RH depending on the relative amounts of each salt.

1.1 Current Thermodynamic Equilibrium Models

[5] Accurately predicting the multistage deliquescence points and the associated solid-liquid partitioning is a challenging task. In the recent years, numerous thermodynamic models have been developed that attempt to solve this problem using different numerical techniques – SCAPE2 [Kim et al., 1993a, 1993b; Kim and Seinfeld, 1995; Meng et al., 1995], ISORROPIA [Nenes et al., 1998], EQUISOLV II [Jacobson et al., 1996; Jacobson, 1999], GFEMN [Ansari and Pandis, 1999b], and AIM [Clegg et al., 1998a, 1998b; Wexler and Clegg, 2002].

[6] SCAPE2 and ISORROPIA classify a given multicomponent problem into one of the several sub-domains based on the relative composition where only certain components/reactions are assumed to occur. The relevant equilibrium equations in a chosen sub-domain are then solved with the bisection method in an iterative manner. However, SCAPE2 does not predict MDRH as a function of composition. Instead, it assumes that the particle first deliquesces at the lowest DRH of all the individual salts present in a given mixture. While ISORROPIA predicts MDRH as a function of composition and temperature, it assumes a “mutual deliquescence region” between MDRH and the lowest DRH (denoted as RHwet) of all the individual salts present in the mixture. Moreover, for any RH between MDRH and RHwet, the aerosol water content and dissolution of different salts in ISORROPIA are approximated using a weighted mean of the “dry” state (MDRH) and the “wet” state (RHwet). Thus, it is evident that the composition and water content of mixed phase systems predicted by SCAPE2 and ISORROPIA will not be at true thermodynamic equilibrium as dictated by the various solid-liquid reactions, and therefore may distort the composition and size of the particle. However, it should be noted that these assumptions and approximations were made to increase the computational efficiency of the models.

[7] On the other hand, EQUISOLV II employs a more rigorous approach to solve the non-linear system of equilibrium equations without any such simplifying assumptions, and therefore adequately predicts the multistage deliquescence behavior of mixed-electrolyte systems. The EQUISOLV II code consists of a numerical scheme that requires the solution of one equilibrium equation at a time either by an analytical technique or by iteration, and all equations are iterated many times until convergence to solve the entire system.

[8] GFEMN and AIM models are rigorous thermodynamic models that directly minimize the total Gibbs free energy of the system using an iterative numerical technique to obtain the equilibrium solution. This approach eliminates many of the assumptions used in SCAPE2 and ISORROPIA such as divided RH and composition domains where only certain reactions are assumed to occur and the various simplifications related to DRH. Also, GFEMN and AIM both use the highly accurate Pitzer-Simonson-Clegg (PSC) [Pitzer and Simonson, 1986; Clegg et al., 1992, 1998a, 1998b] module for estimating the multicomponent activity coefficients of aqueous electrolytes, and are therefore regarded as the most accurate thermodynamic models available in the literature. Ansari and Pandis [1999a] conducted a comparison of GFEMN with SCAPE2 and ISORROPIA and found that the latter two indeed have difficulty in reproducing the complex multistage deliquescence behavior of mixed-electrolyte systems. A detailed comparative review of EQUISOLV II, SCAPE2, and AIM along with two other models is also available in Zhang et al. [2000], and a brief comparison of all the above models and the techniques they use for estimating multicomponent activity coefficients and for predicting solid-liquid partitioning is summarized here in Table 1.

Table 1. Comparison of Different Equilibrium Models and the Methods They Use to Solve Solid-Liquid Partitioning
Model NameaSystem SolvedMulticomponent Activity Coefficient MethodbMDRH Calculation MethodSolid-Liquid Partitioning Method
SCAPE2gas-solid-liq: NH4-Na-Ca-Mg-K-NO3-SO4-Cl-CO3Choice of Bromley, KM, and Pitzer at 298.15 KIndividual salt DRH as a function of temperaturecIterative bisection method with temperature-dependent equilibrium constants
ISORROPIAgas-solid-liq: NH4-Na-NO3-SO4-ClBromley at 298.15 KParameterized as a function of composition and temperatureApproximation using a weighted mean of dry and wet solutionsd
EQUISOLV IIgas-solid-liq: NH4-Na-Ca-Mg-K-NO3-SO4-Cl-CO3BromleyeN/AfAnalytical equilibrium iteration and mass flux iteration techniques with temperature-dependent equilibrium constants
GFEMNgas-solid-liq: NH4-Na-NO3-SO4-ClPSC at 298.15 KN/AfIterative Gibbs free energy minimization with temperature-dependent chemical potentials.
AIM (Model III)gas-solid-liq: NH4-Na-NO3-SO4-ClPSC at 298.15 KN/AfIterative Gibbs free energy minimization at 298.15 K
MESA (this work)solid-liq: NH4-Na-Ca-NO3-SO4-ClChoice of PSC, MTEM, KM, and Bromley at 298.15 KParameterized as a function of composition and temperatureModified pseudo-transient continuation (PTC) technique with temperature-dependent equilibrium constants

[9] Numerical techniques based on the direct minimization of Gibbs free energy are highly accurate, but also computationally very expensive and therefore not suitable for inclusion in 3-D chemical transport models [Ansari and Pandis, 1999b; Wexler and Clegg, 2002]. EQUISOLV II is much more efficient. However, it involves up to three levels of nested iteration loops. For instance, individual equilibrium equations are first iterated (level-3), and when one equation is converged the updated concentrations are used as inputs into subsequent equations (level-2). When the local convergence criterion is met, the water content and activity coefficients are updated, and the level-2 and level-3 iterations are repeated several times (level-1) until the entire system of equations is converged [Jacobson, 1999]. While it is possible to solve the second-order equilibrium equations analytically (e.g., NaCl(s)equation image Na(aq)+ + Cl(aq)), level-3 iterations are still required to solve higher order equilibrium reactions (e.g., Na2SO4(s)equation image 2Na(aq)+ + SO4(aq)2−) that give rise to third- or higher-degree polynomials. EQUISOLV II requires less than 30 level-1 iterations with the analytical method for typical aerosol composition and ambient RH and temperature conditions; however, the total number of iterations for a given solid-liquid partitioning problem (equal to the product of the number of iterations for each iteration level) is expected to be quite large, especially when third- and higher-order equilibrium reactions are present. The advantage, though, is that the code is generalized to allow the solution to any number and type of equilibrium equations. EQUISOLV II also contains temperature dependence of activity coefficients for the highly sulfate-rich systems containing H+, NH4+, SO42−, HSO4, NO3, and Cl ions valid at low RH and temperatures found in the upper troposphere and stratosphere. However, temperature dependent parameterizations of binary activity coefficients for all the salts that can form under sulfate-poor conditions (e.g., salts containing Na, Ca, Mg, and K) are either not available or applicable at low RHs, or have not been verified due to lack of experimental data.

1.2. Scope

[10] Here, we present a new multicomponent equilibrium solver for aerosols (MESA) based on the pseudo-transient continuation technique for solving solid-liquid equilibria. The algorithm is generally applicable to any number of salts and solid-liquid equilibrium reactions of any order, and involves only one iteration loop. MESA is designed for inclusion in 3-D Eulerian models, and is embedded in a new dynamic gas-aerosol model MOSAIC (Model for Simulating Aerosol Interactions and Chemistry), which will be described elsewhere. The scope of this paper is limited to MESA and the aerosol-phase solid-liquid equilibrium problem. Numerical algorithms for dynamic gas-aerosol partitioning will be described in the future publication on MOSAIC.

[11] The MESA algorithm can be used with different activity coefficient models. Zaveri et al. [2005] recently developed a new method, MTEM (Multicomponent Taylor Expansion Method), for estimating activity coefficients of electrolytes in multicomponent atmospheric aerosols. MTEM was applied to aerosol systems containing H+, NH4+, Na+, Ca2+, SO42−, HSO4, NO3, Cl, and H2O, and was evaluated and contrasted in a stand-alone mode against other multicomponent activity coefficient method such as the comprehensive PSC model and the widely-used Kusik and Meissner [1978] (KM) and Bromley [1973] mixing rules.

[12] In this paper, we evaluate the performance of MESA using MTEM for several multicomponent systems commonly found in tropospheric aerosols. The Web-based AIM Model III (http://mae.ucdavis.edu/∼wexler/aim.html) is used as a benchmark to evaluate the accuracy of MESA-MTEM. Furthermore, since EQUISOLV II is the only other rigorous equilibrium solver that is efficient enough for use in 3-D models, we use its both non-vectorized and vectorized versions to assess the relative computational speed of MESA-MTEM for the test cases considered in this study.

2. Theoretical Development

2.1. Solid-Liquid Phase Equilibrium

[13] The total Gibbs free energy (G) of the system should be at a minimum to attain chemical equilibrium in a closed multiphase system [Denbigh, 1981]. Equilibrium composition can be obtained by minimizing G directly [e.g., Ansari and Pandis, 1999b; Wexler and Clegg, 2002], or indirectly by solving equilibrium relations for each phase and chemical reaction [e.g., Kim et al., 1993a, 1993b; Jacobson et al., 1996; Jacobson, 1999]. In this work we adopt the latter approach of solving a set of equilibrium equations described by

equation image

where ai is the activity of species i, and vij is the stoichiometric coefficient of the ith species in the jth reaction, such that it is negative for reactants and positive for products. Kjsl (T) is the solid-liquid equilibrium constant of jth reaction at temperature T, which is expressed as

equation image

where μi0 (T) is the standard chemical potential of species i, and R is the universal gas constant. The standard chemical potential at a given temperature can be estimated from [Wexler and Seinfeld, 1991]

equation image

where ΔGf,0o, ΔHf,0o, and Cp,0o are the standard state Gibbs free energy of formation, enthalpy of formation, and specific heat capacity at 298.15 K and 100 kPa pressure. T0 is the reference temperature also equal to 298.15 K. Note that Cp,0o is assumed to be constant over the desired temperature range (T to T0). Combining equations (2) and (3) we get:

equation image

where the enthalpy of the hydration reaction is ΔHr,0o = equation imagevij ΔHf,0o and ΔCp,0o = equation imagevijCp,0o. ΔHf,0o and Cp,0o values for all the salts and aqueous ions considered in this work are listed in Table 2, and the associated equilibrium reactions along with their equilibrium constant parameters (required in equation (4)) are listed in Table 3. Since CaSO4 and CaCO3 are sparingly soluble [Sheikholeslami and Ong, 2003], they are treated as solids over the entire RH range in the model, and therefore not explicitly listed in these tables.

Table 2. Standard State Thermodynamic Parameters for Various Salts and Aqueous Ionsa
SpeciesΔHf,0o, kJ mol−1Cp,0o, J mol−1 K−1
  • a

    All values refer to standard conditions of 298.15 K and 100 kPa pressure, and are taken from Lide [2003] unless indicated otherwise.

  • b

    Wexler and Seinfeld [1991].

  • c

    The standard state specific heat capacities are apparent molar hear capacities at infinite dilution (equal to standard partial molar heat capacities).

  • d

    Criss and Millero [1999].

Solid Phase
(NH4)2SO4−1180.9187.5
NH4NO3−365.6139.3
NH4Cl−314.484.1
Na2SO4−1387.1128.2
NaNO3−467.992.9
NaCl−411.250.5
Ca(NO3)2−938.2149.4
CaCl2−795.872.59
(NH4)3H(SO4)2b−2207.0315.0
NH4HSO4b−1027.0127.5
NaHSO4b−1125.585.0
 
Aqueous Phasec
H+00
NH4+−132.579.9
Na+−240.146.4
Ca2+−542.8−26.7d
HSO4−887.3−84.0
SO42−−909.3−293.0
NO3−207.4−86.6
Cl−167.2−136.4
Table 3. List of Solid-Liquid Equilibrium Reactions Solved in MESA
No.Solid-Liquid Equilibrium ReactionEquilibrium Constant,aKEsl (T)
Ksl (T0) bequation imageequation imageUnits
  • a

    Equilibrium constant at any temperature, T, is calculated from Ksl(T) = Ksl(T0) exp equation image, where the reference temperature T0 = 298.15 K.

  • b

    The values of equilibrium constants at 298.15 K are from Clegg et al. [1998b] unless otherwise indicated.

  • c

    Clegg et al. [1992].

  • d

    Kim and Seinfeld [1995].

  • e

    Adjusted to reproduce the deliquescence points predicted by AIM at 298.15 K.

(1)(NH4)2SO4(s)equation image 2NH4(aq)+ + SO4(aq)2−1.04−2.6638.57mol3 kg−3
(2)NH4NO3(s)equation image NH4(aq)+ + NO3(aq)12.21−10.3617.56mol2 kg−2
(3)NH4Cl(s)equation image NH4(aq)+ + Cl(aq)17.37−6.0316.92mol2 kg−2
(4)Na2SO4(s)equation image 2Na(aq)+ + SO4(aq)2−0.490.9839.75mol3 kg−3
(5)NaNO3(s)equation image Na(aq)+ + NO3(aq)11.95−8.2216.01mol2 kg−2
(6)NaCl(s)equation image Na(aq)+ + Cl(aq)38.28−1.5716.90mol2 kg−2
(7)Ca(NO3)2(s)equation image Ca(aq)2+ + 2NO3(aq)4.31 × 105c7.8342.01mol3 kg−3
(8)CaCl2(s)equation image Ca(aq)2+ + 2Cl(aq)7.97 × 1011d32.8444.79mol3 kg−3
(9)(NH4)3H(SO4)2(s)equation image 3NH4(aq)+ + HSO4(aq) + SO4(aq)2−11.8e−5.1954.40mol6 kg−6
(10)NH4 HSO4(s)equation image NH4(aq)+ + HSO4(aq)117.0e−2.8715.83mol3 kg−3
(11)NaHSO4(s)equation image Na(aq)+ + HSO4(aq)313.0e0.8014.79mol3 kg−3
(12)HSO4(aq)equation image H(aq)+ + SO4(aq)2−1.058.8525.14mol kg−1

2.2. Pseudo-Transient Formulation of Phase Equilibrium

[14] As discussed earlier, the system of non-linear algebraic equations given by equation (1) can be solved iteratively to yield the equilibrium solution [e.g., Jacobson et al., 1996; Jacobson, 1999]. Such techniques usually involve multiple nested iteration loops and tend to be computationally expensive. Pseudo-transient continuation (PTC) methods offer a powerful alternative to this approach. PTC is an iterative technique for finding roots of some implicit non-linear equilibrium problems which can be formulated as ordinary (or partial) differential equations (ODE) [Coffey et al., 2003]. The equilibrium solution is obtained by integrating the ODEs to their steady-state solutions. The iterations therefore mimic a temporal integration scheme with an adaptive time step, although temporal accuracy is not the objective in this technique. The classical PTC technique uses a time-stepping scheme which starts with the Forward Euler method and then switches over to Newton's method as the iteration gets closer to steady-state [Coffey et al., 2003]. Such techniques are widely used in aerodynamics [Venkatakrishna, 1989], magneto-hydrodynamics [Knoll and McHugh, 1998], reacting flow [Smooke et al., 1989], and structural analysis [Kant and Patel, 1990] modeling communities.

[15] Here we will first cast the solid-liquid phase equilibrium problem as a pseudo-transient system, and then apply a modified form of the classical PTC technique that uses only the Forward Euler method to solve the resulting set of non-linear ODEs. Let Cs,i and Cl,i denote the concentrations of a generic species (ions) i in the solid and liquid phases (mol cm−3 air) of a particle, respectively. Therefore, at any instant the total particulate concentration (mol cm−3 air) of i is Ca,i = Cs,i + Cl,i. Now, consider a salt E containing the cation M and anion X, and let vM and vX be the number of moles of M and X, respectively, per mole of E. The transient dissolution and precipitation reactions of E in a multicomponent aqueous solution can be written as:

equation image

where kd and kp are the rate constants for the dissolution and precipitation reactions, respectively. The rate of change of moles of salt E with time θ in the solid phase can be expressed as the difference between the precipitation and dissolution rates:

equation image

where ns,E is the concentration of salt E in the solid phase in mol cm−3 (air), W is the aerosol water content in kg cm−3 (air). The molal-scale activity of E in the liquid phase, al,E, is expressed as

equation image

where mM and mX are the cation and anion molalities, respectively; γE is the mean ionic activity coefficient of the salt E in a given multicomponent aqueous solution; and vE = vM + vX. The solid-phase activity of all the salts, as,E, is unity by definition. At equilibrium, the rate of dissolution and precipitation are equal, and the equilibrium constant can be expressed as KEsl = kd/kp. While the absolute values of kd and kp are usually unknown, only their relative values are sufficient for this algorithm. By arbitrarily setting kp = 1/(KEslW) and kd = kpKEsl, equation (6) becomes:

equation image

where ΨE = al,E/KEsl is the saturation ratio. A saturation ratio equal to unity means that the salt is in equilibrium with its solid phase, a value greater than unity means that it is super-saturated, while a value less than unity means it is sub-saturated with respect to its solid phase. A Heaviside function, HE, is introduced in equation (8) to force the derivative to zero when there exists a tendency to dissolve a salt that is absent in the solid-phase, and is defined as

equation image
equation image

The goal is to integrate the initial-value ODEs described by equation (8) to steady-state and thereby obtain the desired equilibrium solid-liquid partitioning for a given overall particle composition. We will use the Forward Euler method with an adaptive time-stepping scheme. The discretized form of equation (8) is written as

equation image

where q is the time index, Js,Eq is the flux of salt E at q, and hEq is the time step for that salt. Normally, in the Forward Euler method a common time step that is much smaller than the smallest of all the individual ODE time-scales is specified for integrating the entire system of ODEs. However, the objective of the ODE solver used here is not to produce an accurate transient solution, but rather to reach the final equilibrium state in as few time steps (or iterations) as possible. We therefore allow each ODE to take an independent and adaptive time step based on its own time-scale. This approach substantially reduces the number of iterations to reach equilibrium, especially when the time scales of individual ODEs differ significantly (i.e., stiff ODEs). In this approach the time variable θ loses its physical significance; and q should be viewed as an iteration index here rather than the time index.

[16] All the quantities at the current iteration index q are known (starting with known initial conditions), and the new solid-phase moles of salt E at index (q + 1) can be integrated from equation (11) as

equation image

The new solid-phase moles of individual ions, Cs,iq+1, can be computed from the new salt moles ns,Eq+1, and the new liquid-phase moles of all the ions, Cl,iq+1, can then be computed from mass balance as

equation image

Multicomponent activity coefficients and aerosol water content are updated every iteration step to calculate the new liquid phase salt activities. These steps are repeated until the system reaches equilibrium. The entire MESA algorithm is summarized with a flowchart in Figure 2. The following sub-sections describe various parts of the MESA algorithm including initial conditions, convergence criteria, and the algorithms for calculating time steps.

Figure 2.

Flowchart of the MESA algorithm.

2.3. Initial Conditions

[17] Since CaSO4 and CaCO3 are sparingly soluble they are treated as solids over the entire RH range in the present model. Thus, for a given mixture of various cations and anions, the number of moles of solid CaSO4 and CaCO3 are first calculated from the available moles of calcium, sulfate, and carbonate. The integration is then initiated by assuming all the remaining soluble material in the aerosol is completely dissolved at the given RH, assuming the water activity aw = RH. If all the initial salt fluxes Js,E0 = 0, i.e. ΨE0 < 1, then it means that the system is sub-saturated at the given RH, and no precipitation will occur. In this situation the system is already at equilibrium, and no further steps are necessary. However, if any one or more Js,E0 > 0, then precipitation of one or more salts is possible.

2.4. Convergence Criteria

[18] Two independent criteria must be used to check for convergence of the numerical solution at every iteration step. One criterion is for the overall equilibrium condition and the other for the total soluble mass. Relative driving force as defined below is used to diagnose the equilibrium condition for each salt present in the particle.

equation image

The relative driving force for a salt is a measure of its departure from equilibrium, and its value varies between −1 and 1. A value of zero indicates that the solid phase of the salt (if it exists) is in equilibrium with the aqueous phase while values 1 or −1 means that the salt is farthest away from equilibrium. In practice, the numerical solution is assumed to have converged when

equation image

where RTOLeqbm is a user-specified relative equilibrium tolerance, set at 0.01 in the present study.

[19] When a completely solid aerosol is predicted at a given RH, the absolute relative driving forces for all the soluble salts present may never be less than the specified value of RTOLeqbm. A mass convergence criterion must therefore be applied independent of the equilibrium criterion at every iteration step to check if all the soluble material has been precipitated. Thus, the aerosol may be assumed to be completely solid if

equation image

where Mi is the molecular weight of the ion i and RTOLmass is the relative mass tolerance, set at 0.99.

2.5. Adaptive Time-Stepping Scheme

[20] Stability, robust convergence, and computational efficiency of the solver critically depend on the time-stepping scheme used. The time-step, hEq, for each salt depends on the precipitation and dissolution time scales τp,Eq and τd,Eq, respectively, which are defined as:

equation image

Physically, the time scales are based on the precipitation and dissolution of ions. But because a particular ion may precipitate to or dissolve from several salts, determining time steps using ion time scales becomes complicated. Instead, it is more convenient to use the time scales based on salts. However, while the moles of all the salts in the solid phase are directly known at any given iteration q, estimating the moles of individual salts in the liquid phase is not as straightforward. Because electrolytes in the aqueous phase exist as completely or partially dissociated ions, many different combinations of salts and acids (electrolytes) are possible for a given ionic composition. Nevertheless, a unique electrolyte composition in the liquid phase is needed for the purpose of estimating the precipitation time scale as well as for calculation of aerosol water content using the widely-used ZSR mixing rule [Zdanovskii, 1948; Stokes and Robinson, 1966]. The algorithms used in MESA for apportioning liquid-phase ions to electrolytes are based on Zaveri et al. [2005]. The algorithm for sulfate-rich systems is modified slightly to ensure apportionment of ions to all the possible electrolytes in each of the sulfate-rich sub-domains defined in Table 2 of Zaveri et al. [2005].

[21] During the precipitation process (JEq > 0) the time-step hEq must be less than τp,Eq to yield a stable (positive) solution for the liquid-phase ions that constitute salt E. Thus, the time-step is allowed to be a fraction, αE, of this time-scale where 0 < αE < 1:

equation image

On the other hand, during the dissolution process (JEq < 0) the time-step hEq depends on τd,Eq for stability and on τp,Eq for robust convergence. Therefore the time-step is allowed to be a fraction of the minimum of the two time-scales as

equation image

With these theoretical developments the problem reduces to choosing judicious values for αE that will produce a robust and computationally efficient solution.

[22] In a stiff system of ODE such as the present non-linear system, a large value of αE ∼ 0.5 would result in an oscillating solution and have difficulty converging. On the other hand, a small value of αE ∼ 0.01 would produce a smooth but computationally expensive solution. This is illustrated here for a 5:1 molar mixture of (NH4)2SO4 and NH4NO3 whose mutual deliquescence relative humidity (MDRH) point at 298.15 K is ∼60% RH.

[23] At 70% RH, NH4NO3 is completely deliquesced while (NH4)2SO4 is expected to be partially deliquesced with NH4+ and SO42− ions in equilibrium with its solid phase. Figure 3 shows the behavior of the relative driving force of (NH4)2SO4 as a function of iteration step for different values of α. A fixed value of α = 0.5 produced large oscillations in Φ about the zero-line before converging after 59 iterations. A lower value of α = 0.2 produced fewer and smaller oscillations in Φ, and the solution converged after 21 iterations. On the other hand, fixed values of α = 0.05 and 0.01 produced increasingly smoother decay in Φ, but required 62 and 283 iterations for convergence, respectively. However, by setting αEq = ∣ΦEq∣ a smooth solution is obtained that converges in only 8 iterations. Initiating the integration with a large value of α (∼0.6 in this case) results in a rapid decrease in ∣Φ∣, which feeds back into the next iteration step with a lower value of α. This dampens the oscillations while drastically reducing the number iterations required for convergence.

Figure 3.

Relative driving force for (NH4)2SO4 as a function of number of iterations with different choices of image values. The aerosol composition is [(NH4)2SO4]:[NH4NO3] = 5:1 at RH = 70% and T = 298.15 K.

[24] The same salt mixture at 59% RH and 298.15 K is expected to be completely solid. In the present algorithm both the salts are expected to precipitate until the total solid mass fraction exceeds ɛmass. Figure 4 shows the behavior of image image and solid mass fraction for different fixed values of α as well as for αEq equal to the respective ∣ΦEq∣. In this case, a value of α = 0.5 did not produce any oscillations in Φ for both salts, and required only 10 iterations for solid mass fraction to exceed ɛmass as opposed to 27 and 111 iterations for α = 0.2 and 0.05, respectively. However, interesting behavior is observed when αEq = ∣ΦEq∣. It can be seen that the solid mass fraction increased exponentially from 0.0 to ∼0.88 in the first five iterations, but abruptly slowed down to a rather gradual linear increase since then. It required an additional 52 iterations before the solid mass convergence criterion was met. This behavior can be explained by the sharp drop in the driving force for both salts to ∼0.033, which remained nearly constant after the first 5 iterations. The gradual subsequent increase in the solid mass was, thus, due to a relatively much smaller value of α that also remained nearly constant after the 5th iteration. However, at 50% RH, which is much lower than the MDRH, the same system converged in only 13 iterations (not shown here) for αEq = ∣ΦEq∣. Thus, difficulty in mass convergence is expected for any salt mixtures when the ambient RH is just below the mutual deliquescence point and the saturation ratios of the salts are only slightly greater than unity. In practice, a more aggressive time-stepping scheme is used based on the rate at which ∣ΦEq∣ changes with αEq as follows:

equation image
equation image
equation image

where equation imageE is the derivative of ∣ΦE∣ with respect to αE, and is calculated from two consecutive iterations as:

equation image

Thus, if equation imageEq < 0 then ∣ΦE∣ has decreased since the previous iteration, and the solver takes a more aggressive time step in the current iteration as expressed in equation (20). On the other hand, if equation imageEq > 0, then ∣ΦE∣ has increased since the previous iteration instead of decreasing, and therefore the solver takes a conservative time step as shown in equation (21). However, if (ΦEq × ΦEq−1) < 0 then the relative driving force has changed its sign, indicating an oscillatory behavior in the solution. In this case the solver takes a more conservative time step as given in equation (22). Note that the maximum value of αEq is limited to less than unity so that the solver always remains positive definite to conserve species mass.

Figure 4.

Relative driving forces for (NH4)2SO4, NH4NO3, and solid mass fraction as a function of number of iterations with different choices of αE values. The aerosol composition is [(NH4)2SO4]:[NH4NO3] = 5:1 at RH = 59% and T = 298.15 K.

[25] Now, although using a large fixed value of α may sometimes result in an efficient solution when RH < MDRH, it does not always guarantee robust convergence, especially for more complex mixtures. However, if the MDRH for a given aerosol composition can somehow be known a priori, then the problem of solving the phase equilibrium can be eliminated altogether for RH < MDRH. Fortunately, the MDRH for a mixed-salt system at a given temperature depends only on which salts are present when the system is completely solid. In other words, the knowledge of their relative amounts is not necessary. Equations for temperature dependence of DRH and MDRH for a few salts and a small set of mixtures have been derived based on latent heat of fusion or integral heat of solution [Wexler and Seinfeld, 1991; Tang and Munkelwitz, 1993; Nenes et al., 1998]. However, algorithms for determining unique dry salt mixtures from a given set of ionic concentrations and temperature dependent MDRH equations for all the possible salt mixtures for the ions considered in this study are not available. An alternative approach based on temperature dependent equilibrium constants for estimating MDRH(T) for any given ionic aerosol composition (for the ions considered here) is described below.

2.6. MDRH(T) Parameterization

[26] To be able to parameterize MDRH for a salt mixture as a function of T we must first be able to determine the “dry-state” salt composition from the given amounts of various cations and anions, i.e., which salts may exist if the particle is completely solid at some very low RH. Two electrolyte composition domains are possible in the H+ - NH4+ - Na+ - Ca2+ - SO42− -HSO4 -NO3 - Cl - H2O system as defined by the sulfate ratio

equation image

where Ca,SULF is number of moles of sulfate (S(VI)). Particles that do not contain any sulfate or which have a sulfate ratio Xa,T ≥ 2 fall under the sulfate-poor domain while particles that have a sulfate ratio Xa,T < 2 fall under the sulfate-rich domain. The MESA solver was used to examine the equilibrium salt formation characteristics for several different cation-anion systems, and it was found that simple algorithms could be developed for determining the dry-state salt compositions in sulfate-poor and sulfate-rich systems as shown in Tables 4 and 5, respectively.

Table 4. Dry-State Salt Formation in Sulfate-Poor Domain
No.SaltsOrder of Formation of Salt Moles, ns,E
(1)CaSO4image = min image Ca,SULF)
(2)Ca(NO3)2image = min imageimage equation imageimage
(3)CaCl2image = min imageimageimage equation imageimage
(4)CaCO3image = imageimageimageimage
(5)Na2SO4image = min (equation imageimage Ca,SULFimage
(6)NaNO3image = min imageimage imageimage
(7)NaClns,NaCl = min imageimageimage imageimage
(8)(NH4)2SO4image = min (equation imageimage [Ca,SULFimageimage SO4])
(9)NH4NO3image = min imageimage imageimageimage
(10)NH4Climage = min imageimageimage imageimagens,NaCl])
Table 5. Dry-State Salt Formation in Sulfate-Rich Domain a,b
Salt ESalt Moles, ns,E
Xa,T = 11 < Xa,T < Xa,NaXa,NaXa,T < 2
1 ≤ image < equation imageequation imageimage < 2
  • a

    The formulae in this table are used after first forming solid CaSO4 as image = min image Ca,SULF). The various sulfate ratios XaT, Xa,Na, and image are then calculated using the remaining SULF moles, as Xa,T = image + image Xa,Na = 1 + equation image image and image = image (2Ca,SULFimageimage

  • b

    x = Ca,SULFimage

Na2SO40imagex)equation imageimageequation imageimage
NaHSO4image(2ximage00
(NH4)2SO4000(equation imageimage − 3ximage
(NH4)3H(SO4)200(equation imageimagex)(2x + imageimage
NH4HSO4imageimage(3(xequation imageimage + image0

[27] In sulfate-poor systems, the hierarchy in Table 4 dictates that all the calcium salts are allowed to form first, then sodium, and then ammonium. Furthermore, for a given cation, sulfate salts are allowed to form first, followed by nitrate, and then chloride. For example, consider a mixture of NH4+, Na+, NO3, and Cl ions such that the system is electrically neutral. It can be seen that four different salts are possible, viz., NaCl, NH4NO3, NaNO3, and NH4Cl. However, according to the order of formation in Table 4 either a mixture of NaNO3, NaCl, and NH4Cl or NaNO3, NH4NO3, and NH4Cl may form at equilibrium depending on the relative amounts of the individual ions. Similarly, large mixtures such as Na2SO4-NaCl-NaNO3-(NH4)2SO4-NH4NO3-NH4Cl do not exist at equilibrium, but several submixtures are possible as shown in Table 4. This is consistent with the thermodynamic analysis of solid-solid equilibrium by Wexler and Potukuchi [1998], who showed that reacting solid salts are in equilibrium when one of them has been completely depleted in favor of formation of the other compounds to yield a system that has the minimum Gibbs free energy.

[28] In sulfate-rich systems the salt formation hierarchy is more complicated and depends on additional sub-domains as shown in Table 5. As before, CaSO4 is allowed to form first so that all the calcium moles are exhausted. The sulfate ratio is recalculated using the remaining sulfate moles, and if sodium is present and Xa,T > 1, then a preference is given to Na2SO4 formation, after which other salts such as (NH4)2SO4, (NH4)3H(SO4)2, NH4HSO4, and NaHSO4 are formed based on the relative amounts of NH4+ and Na+, mass-balance, and electro-neutrality constraints. For Xa,T = 1 only NaHSO4 and NH4HSO4 are possible, and the mixture never becomes completely solid at any RH if Xa,T < 1 due to the presence of liquid H2SO4.

[29] Based on these salt formation hierarchies a total of 63 unique salt mixtures are possible. As mentioned earlier, equations for MDRH(T) have been derived for a few salts and their mixtures in the past by applying the Clausius-Clapeyron equation to the phase transformation. However, it is difficult to extend this technique to all the 63 salt mixtures considered in this study. Instead, an alternative approach is used here in which MDRH points are found from the equilibrium mass growth factors (MGF) predicted by MESA as a function of RH and T. In these calculations only the equilibrium constants in MESA were allowed to vary in with temperature, while the multicomponent activity coefficients were computed with the comprehensive PSC model at 298.15 K. Of the 63 unique salt mixtures possible, 20 mixtures containing calcium nitrate and/or calcium chloride were found to have MDRHs ∼10% RH or less over the temperature range of 240 to 320 K. This is consistent with the findings of Tang and Fung [1997] and Cohen et al. [1987] who observed very low DRHs for Ca(NO3)2 and CaCl2, respectively. Tang and Fung also noted that a completely deliquesced Ca(NO3)2 particle could easily remain supersaturated down to less than 1% RH, and does not dry out unless evaporated in vacuum. Similarly, Cohen et al. found that CaCl2.4H2O particles were non-crystalline at 10% RH, and had to be dried at high temperatures (50 to 90°C) before they lost more water. They also observed that these particles did not display a clear deliquescence point, but simply absorbed water gradually as the RH was increased. These results suggest that Ca(NO3)2 and CaCl2 are completely dissolved in mixed atmospheric particles even at very low RH. Therefore, MDRHs for all the 20 salt mixtures containing Ca(NO3)2 and/or CaCl2 are assumed to be equal to 10% in MESA for simplicity and computational efficiency.

[30] The predicted MDRH vs T data for the remaining 43 salt mixtures were fit to cubic polynomials in T, and are listed in Table 6. The accuracy of this approach could be verified for a limited number of salt systems for which temperature dependent MDRH data are available. Figure 5 shows plots of the predicted DRHs for six individual salts against the Web-based AIM model II output [Wexler and Clegg, 2002]. Figure 6 shows comparisons of predicted MDRHs for eight different salt mixtures against the available data from Tang and Munkelwitz [1993, 1994] or the AIM Model II output. It is seen that the DRH(T) and MDRH(T) predictions agree quite well over a wide range of temperature (240 to 310 K) considering the fact that the multicomponent activity coefficients were estimated at 298.15 K. Thus, in the MESA algorithm, the dry-state salt mixture for a given particle composition is first determined using Table 4 or Table 5. MDRH(T) is then computed using the appropriate polynomial given in Table 6. If the ambient RH < MDRH(T), then the system is assumed to be completely solid, otherwise the ODE solver subroutine is called to compute the equilibrium solid-liquid partitioning. It is worth noting here that Tables 4 and 5 are not used to determine the actual solid-phase salt moles that may exist in equilibrium with the aqueous phase. The equilibrium solid-liquid partitioning is always determined by solving the set of ODEs to steady-state.

Figure 5.

Comparison of DRH predicted by MESA-PSC for individual salts with available data from Wexler [2003] or Web-based AIM model II output [Wexler and Clegg, 2002].

Figure 6.

Comparison of MDRH predicted by MESA-PSC for selected salt mixtures against temperature-dependent Web-based AIM model II output [Wexler and Clegg, 2002] or experimental data from Tang and Munkelwitz [1993, 1994] (see text).

Table 6. Polynomial Fits of MDRH (%) as a Function of Temperature (K)a
No.Sulfate-Poor SystemsbMDRH (%) at 298.15 KD0D1D2D3
  • a

    MDRH(T) = D0 + D1T + D2T2 + D3T3. Read coefficients in the format 1.23(−4) as 1.23 × 10−4.

  • b

    SS = Na2SO4, SB = NaHSO4, SN = NaNO3, SC = NaCl, AS = (NH4)2SO4, LV = (NH4)3H(SO4)2, AB = NH4HSO4, AN = NH4NO3, AC = NH4Cl.

(1)AC (NH4Cl)77.6−58.0032.0311−8.2812(−3)1.0045(−5)
(2)AN (NH4NO3)61.21039.14−11.47854.7703(−2)−6.7767(−5)
(3)AS ((NH4)2SO4)80.0115.8370.49188−4.2281(−3)7.2927(−6)
(4)SC (NaCl)75.3253.242−1.43003.7276(−3)−3.1304(−6)
(5)SN (NaNO3)73.9−372.4315.3956−1.9804(−2)2.2566(−5)
(6)SS (Na2SO4)87.1286.127−1.67084.4314(−3)−3.5776(−6)
(7)AN + AC53.31761.18−19.29817.5677(−2)−1.0117(−4)
(8)AS + AC71.5122.1070.42969−3.9283(−3)6.4327(−6)
(9)AS + AN59.12424.64−26.54030.10163−1.3154(−4)
(10)AS + AN + AC51.52912.08−31.88940.12119−1.5565(−4)
(11)SC + AC68.8172.260−0.511004.2724(−4)4.1280(−7)
(12)SN + AC52.71596.19−16.37956.0281(−2)−7.6161(−5)
(13)SN + AN49.51916.07−20.85598.1140(−2)−1.0795(−4)
(14)SN + AN + AC41.41467.17−16.01176.3505(−2)−8.6672(−5)
(15)SN + SC67.3158.447−0.628172.0144(−3)−3.1304(−6)
(16)SN + SC + AC52.91115.89−11.76944.5577(−2)−6.0578(−5)
(17)SS + AC68.2269.543−1.32002.5924(−3)−1.4448(−6)
(18)SS + AN48.52841.34−31.18890.11881−1.5301(−4)
(19)SS + AN + AC41.82199.37−24.11939.2932(−2)−1.2177(−4)
(20)SS + AS71.5395.005−2.52116.1393(−3)−4.4376(−6)
(21)SS + AS + AC65.7386.515−2.46326.1393(−3)−4.9880(−6)
(22)SS + AS + AN48.83101.54−34.19980.13012−1.6687(−4)
(23)SS + AS + AN + AC42.12307.58−25.43149.8065(−2)−1.2830(−4)
(24)SS + SC74.5291.831−1.82895.0531(−3)−4.5752(−6)
(25)SS + SC + AC67.4188.391−0.631356.2281(−4)4.4720(−7)
(26)SS + SN72.7−167.1252.9698−1.0637(−2)1.1318(−5)
(27)SS + SN + AC52.21516.78−15.79235.8942(−2)−7.5301(−5)
(28)SS + SN + AN47.21739.96−19.06587.4550(−2)−9.9430(−5)
(29)SS + SN + AN + AC39.92152.11−23.75009.2257(−2)−1.2195(−4)
(30)SS + SN + SC66.8221.998−1.31134.4061(−3)−5.8824(−6)
(31)SS + SN + SC + AC52.21205.65−12.71354.8804(−2)−6.4190(−5)
No.Sulfate-Rich SystemsbMDRH (%) at 298.15 KD0D1D2D3
(32)AB (NH4HSO4)36.6−493.6196.7471−2.6955(−2)3.4512(−5)
(33)LV ((NH4)3H(SO4)2)69.353.37871.0137−5.8875(−3)8.9439(−6)
(34)SB (NaHSO4)39.2206.619−1.34273.1977(−3)−1.9360(−6)
(35)AB + LV36.6−493.6196.7471−2.6955(−2)3.4512(−5)
(36)LV + AS69.353.37871.0137−5.8875(−3)8.9439(−6)
(37)SS + SB39.2206.619−1.34273.1977(−3)−1.9360(−6)
(38)SS + LV63.041.76191.3039−7.6479(−3)1.1785(−5)
(39)SS + AS + LV63.041.76191.3039−7.6479(−3)1.1785(−5)
(40)SS + AB30.8−369.7145.5129−2.3020(−2)3.0303(−5)
(41)SS + LV + AB30.8−369.7145.5129−2.3020(−2)3.0303(−5)
(42)SB + AB35.0−162.8092.3993−9.3362(−3)1.1785(−5)
(43)SS + SB + AB30.1−735.4298.8855−3.3488(−2)4.1246(−5)

3. Results and Discussion

[31] It is ideally desirable to evaluate the accuracy and computational speed of MESA relative to various other models/solvers available in the literature, but a fair comparison between different models is not a straightforward task. One must not only ensure, to the extent possible, that all the models are set up to solve the same set of equilibrium reactions and species, but also consider the tradeoff between accuracy and speed to draw meaningful conclusions from the intercomparison exercise. For example, models such as GFEMN and AIM were developed for the highest accuracy possible without worrying about computational costs, while ISORROPIA and SCAPE were developed using several simplifying assumptions for high efficiency, but at the cost of accuracy. On the other hand, EQUISOLV II is a highly generalized equilibrium solver that can rigorously solve any number of equilibrium reactions for any number of species, and at the same time, it is designed to be efficient enough for use in 3-D models. Zhang et al. [2000] have performed accuracy and speed comparisons for some of these models.

[32] In this paper we limit the accuracy and speed comparisons between MESA and EQUISOLV II, with the Web-based AIM Model III serving as a benchmark for accuracy only. To make the models similar in terms of species and reactions, precipitation of only single salts was considered in the AIM Model III runs while all the double salts and salt hydrates were excluded. Similarly, EQUISOLV II was configured such that it solved only those reactions that are included in MESA. Performances of the models were evaluated using several representative salt mixtures commonly found in tropospheric aerosols. Table 7 displays the ionic compositions for the 16 test cases considered in this exercise, and Table 8 displays the dry-state electrolyte compositions for these cases as computed using the algorithms described in subsection 2.6. Test cases 1–10 consist of different sulfate-poor salt mixtures involving NH4+, Na+, Ca2+, SO42−, NO3, and Cl ions, while cases 11–16 consist of sulfate-rich salt mixtures involving H+, NH4+, Na+, SO42−, and HSO4 ions. MESA calculations were carried out using the MTEM multicomponent activity coefficient method recently developed Zaveri et al. [2005].

Table 7. Relative Aqueous Phase Ionic Composition in Sulfate-Poor and Sulfate-Rich Test Cases
Test CaseIon Moles
NH4+Na+Ca2+SULFNO3Cl
Sulfate-Poor Domain
(1)5.02.01.0
(2)4.01.01.01.0
(3)3.01.01.0
(4)4.01.01.01.0
(5)1.04.01.01.02.0
(6)4.04.03.01.01.0
(7)2.05.02.02.01.0
(8)2.02.02.0
(9)3.01.02.03.0
(10)3.01.05.0
 
Sulfate-Rich Domain
(11)0.50.21.0
(12)1.03.03.0
(13)4.03.0
(14)4.02.04.0
(15)5.03.0
(16)5.02.04.0
Table 8. Solid-Phase Salt Compositions for the Ionic Compositions Given in Table 7
Test CaseSulfate-Poor Domaina
(NH4)2SO4, moleNH4NO3, moleNH4Cl, moleNaCl, moleNa2SO4, moleNaNO3, moleCa(NO3)2, moleCaCl2, mole
  • a

    Electrolyte compositions in the sulfate-poor domain obey the formulae given in Table 4.

  • b

    Electrolyte compositions in the sulfate-rich domain obey the formulae given in Table 5.

(1)2.01.0
(2)1.01.01.0
(3)1.01.0
(4)1.01.01.0
(5)1.01.01.01.0
(6)1.01.01.02.0
(7)1.01.02.01.0
(8)1.01.0
(9)3.01.0
(10)3.01.0
Test CaseSulfate-Rich Domainb
(NH4)2SO4, mole(NH4)3H(SO4)2, moleNH4HSO4, moleH2SO4, moleNa2SO4, moleNaHSO4, moleXT 
(11)0.50.30.20.7 
(12)1.01.01.01.333 
(13)1.01.01.333 
(14)1.01.01.01.5 
(15)1.01.01.667 
(16)1.01.01.01.75 

3.1. Accuracy

[33] The primary purpose of this sub-section is to demonstrate the overall fidelity of the MESA-MTEM configuration for predicting the complex hygroscopic growth behavior of mixed salt particles, using the Web-based AIM Model III as a benchmark. However, it is first necessary to briefly discuss the different possible sources of errors in MESA and EQUISOLV II before we present the intercomparison results. Predictions of MESA and EQUISOLV II can deviate from AIM Model III due to (1) differences in the degree of convergence for the chosen error tolerances, referred to as “convergence errors”; and (2) differences in the predicted multicomponent activity coefficients, equilibrium constants, and binary molality polynomials, referred to as “thermodynamic errors or deviations.”

3.1.1. Convergence Errors

[34] Convergence errors can be assessed from a self-consistent measure of the actual relative error in a given model. As described in sub-section 2.4, convergence in MESA is based on relative equilibrium criterion given by equation (15), and depends on the user-specified value of the relative equilibrium tolerance parameter RTOLeqbm. Figure 7 shows the actual root mean square (RMS) relative convergence error (%) in mass growth factors (averaged over all the test cases) predicted by MESA for different values of RTOLeqbm. These errors were computed with respect to the “exact” solutions obtained by using an extremely strict value for RTOLeqbm of 1 × 10−10. Furthermore, only those RH ranges were used in error calculations where both the solid and liquid phases co-existed (i.e., completely solid and completely liquid solutions were ignored). The figure shows that the RMS relative error varies linearly as RTOLeqbm on a log-log plot, and that the relative RMS error is ∼0.14% for RTOLeqbm = 0.01. This level of accuracy in MGF is sufficient for most 3-D model applications, and therefore RTOLeqbm = 0.01 is recommended for operational use.

Figure 7.

Root mean square (RMS) relative error in mass growth factor (MGF) as a function of relative equilibrium tolerance used in MESA. The errors shown were computed using results from all the test cases for RHs where both the solid liquid phases were present at equilibrium. The vertical error bars represent one standard deviation.

[35] On the other hand, EQUISOLV II uses a convergence criterion that is based on two successive iterations as:

equation image

where Ciq is the mole concentration (mol cm−3 air) of all ionic, liquid, and solid species including water, and Mi is the molecular weight. Here ETOL is the tolerance in the sum of changes in the mass concentrations of all the species between two successive iterations, and should not be confused with the actual error in the final solution. Also, because this is a lumped convergence criterion, it is possible for the convergence criterion to be met even though relatively minor species may have much larger relative errors. Thus, because the convergence criteria in MESA and EQUISOLV II are fundamentally different, identical tolerance parameter values in the two may yield quite different actual relative convergence errors. For this intercomparison, EQUISOLV II was run with an absolute tolerance of ETOL = 0.01, which produced an actual RMS relative convergence error of ∼3.67% with respect to the “exact” solutions obtained with ETOL = 1 × 10−10. However, this error statistic is skewed because of discrepancies in the MDRH points predicted by EQUISOLV II with ETOL equal to 0.01 and 1 × 10−10. Therefore, selectively excluding the RH points below the lowest MDRH of the two values gives an RMS relative convergence of ∼0.38%. Such discrepancies are eliminated in MESA since it directly diagnoses MDRH(T) for a given aerosol composition using the parameterization described in sub-section 2.6, which is independent of the RTOLeqbm parameter values. In summary, a value of 0.01 for both RTOLeqbm and ETOL produced a factor 25 or 2.5 larger actual RMS relative convergence errors in EQUISOLV II compared to MESA, depending on the range of mixed-phase RH points considered. The RMS convergence error statistics for all the cases are summarized in Table 9.

Table 9. Summary of Convergence Errors in MGF for a Solver Tolerance Value of 0.01 in Both MESA and EQUISOLV II, Relative to a Strict Value of 1 × 10−10
 RMS Relative Convergence Error in MGF (%) With RTOLeqbm = ETOL = 0.01
MESAEQUISOLV II
  • a

    Mixed-phase RH range (MDRH to CDRH), including RH points with discrepancies in MDRH at ETOL = 0.01 compared to ETOL = 1 × 10−10.

  • b

    Mixed-phase RH range (MDRH to CDRH), excluding RH points with discrepancies in MDRH at ETOL = 0.01 compared to ETOL = 1 × 10−10.

Test CaseEntire Mixed-Phase RH RangeEntire Mixed-Phase RH RangeaSelected Mixed-Phase RH Rangeb
(1)0.312.980.43
(2)0.194.790.37
(3)0.1317.150.44
(4)0.0743.250.33
(5)0.143.750.28
(6)0.0891.640.36
(7)0.132.420.43
(8)
(9)0.0650.150.15
(10)0.120.190.19
(11)0.110.130.13
(12)0.052.090.086
(13)0.110.750.68
(14)0.0680.650.65
(15)0.317.530.59
(16)0.287.570.55
Average0.143.670.38

3.1.2. Thermodynamic Errors/Deviations

[36] Having quantified the relative convergence errors, we now proceed to quantify thermodynamic errors in the two models relative to AIM Model III, which is used as a benchmark for accuracy. However, because the thermodynamic parameters used in AIM also have uncertainties, we refer to the errors in MESA and EQUISOLV II relative to AIM as “deviations” rather than “absolute errors.” Figures 8a and 8b show plots of mass growth factors (MGF) predicted by AIM Model III, MESA, and EQUISOLV II as a function of RH for all the 16 test cases. The exact solutions from MESA and EQUISOLV II are shown here corresponding to their respective tolerance parameter values of 1 × 10−10. Since AIM Model III presently does not include calcium, the experimental data of Choi and Chan [2002] are used to evaluate test cases 8, 9, and 10. Excellent visual agreement between the AIM Model III and MESA results demonstrates that MESA-MTEM can accurately solve the complex solid-liquid equilibria under a variety of conditions. The maximum deviation in MDRH was only 2% RH points. This is as expected because the MDRH polynomials were parameterized using the comprehensive PSC activity coefficient model, which is also used in AIM. Also, some of the equilibrium constants used in MESA were adjusted to reproduce the single salt DRH points predicted by AIM (as noted earlier in Table 3).

Figure 8a.

Comparison of mass growth factors (MGF) at 298.15 K as predicted by the Web-based AIM Model III, MESA, and EQUISOLV II for test cases 1–8. Solver tolerances RTOLeqbm and ETOL were set equal to 1 × 10−10 in MESA and EQUISOLV II, respectively. Note that Choi and Chan [2002] data (CC 2002) are used instead of AIM model for calcium containing cases.

Figure 8b.

Same as Figure 8a for test cases 9–16.

[37] Even though EQUISOLV II is not tuned to reproduce AIM results, it is able to capture the complex growth behavior quite well. In most cases it predicted the MDRH within 4% RH points (and exactly in two cases). In two cases, the deviations in the predicted the MDRH were 8–10% RH points (cases 6 and 7), while it was as large as 15% RH point in case 14. Although these deviations occurred at relatively small water contents, such discrepancies in the phase-state of aerosols can potentially affect their interactions with various gas-phase precursors and heterogeneous chemistry, and hence their subsequent growth rates.

[38] In addition to deviations in MDRH points, it would be useful to quantify the relative deviations in MGFs for each model with respect to AIM. To this end, relative deviations in MGF were first computed at individual RHs (at 1% RH intervals), and then averaged over RHs ranging from MDRH to 99%. The first few RH points, wherever applicable, were excluded in due to the discrepancies in MDRH as discussed above. While these deviations were typically small and within acceptable margins in both the models, they were relatively smaller in MESA compared to EQUISOLV II (except in one case). For example, the RMS relative MGF deviations in MESA ranged from 1.7 to 3.3% with an average value of 2.3% over all the cases, while they ranged from 1.9 to 6.5% in EQUISOLV II with an average of 4.3%. Relative deviations in the total water contents, summed over 20 to 99% RH, were also computed for MESA and EQUISOLV II with respect to AIM. This statistic does not take into account the relative deviations at individual RHs as the previous measure does, but rather gives relative deviations in the total water content integrated over a large RH range, and therefore tends to be dominated by deviations in water contents at high RHs (typically above 95%). The values ranged from 0.27 to 4.7% in MESA with an average of 2.46% over all the sixteen cases. Similarly, these deviations typically ranged from 0.15 to 5.1% in EQUISOLV II, except for two cases where the values were ∼12%, with an average of 3.75% over all the cases. The thermodynamic deviation statistics for all the test cases are summarized in Table 10.

Table 10. Summary of Thermodynamic Deviations in MESA and EQUISOLV II Relative to AIM Using a Strict Solver Tolerance Value of 1 × 10−10 in Both MESA and EQUISOLV II
CaseAbsolute MDRH Deviations, % RH PointsRMS Relative Deviation in MGF,a %Relative Deviation in Total Water Content,b %
MESAEQUISOLV IIMESAEQUISOLV IIMESAEQUISOLV II
  • a

    Relative deviations in MGF with respect to AIM were first computed at individual RHs (at 1% RH intervals), and then averaged over RHs ranging from MDRH to 99%, excluding the first few points (where applicable) due to discrepancies in MDRH relative to AIM (shown in the first two columns).

  • b

    Water contents were first summed over 20 to 99%RH (at 1% RH intervals) for each model, and then relative deviations with respect to AIM were computed for the total values. This statistic does not take into account deviations at individual RHs, but rather gives relative deviations in the total water content integrated over a large RH range.

  • c

    Deviations could not be estimated due to lack of calcium salts in AIM.

  • d

    The lower cutoff point was 10% RH since the Web-based AIM model III is restricted to 10% RH.

(1)223.113.244.705.10
(2)041.946.463.1012.80
(3)102.261.923.133.13
(4)112.003.213.643.04
(5)042.462.341.882.42
(6)1101.665.793.0911.34
(7)282.583.012.553.44
(8)cccccc
(9)cccccc
(10)cccccc
(11)1.98d4.54d1.15d0.49d
(12)132.355.630.272.42
(13)133.344.871.512.64
(14)1152.016.361.720.15
(15)202.135.352.620.23
(16)222.263.292.591.59
Average1.174.422.314.312.463.75

[39] The MDRH, MGF, total water content deviations in MESA and EQUISOLV II relative to AIM are likely due to a combination of differences in the various thermodynamic parameters used in these models (e.g., predicted multicomponent activity coefficients, equilibrium constants, and binary molality polynomials). The magnitudes of these deviations are generally consistent with the error analysis presented in Zaveri et al. [2005] for different activity coefficient mixing rules including PSC, MTEM, KM, and Bromley. However, the relative deviations in the EQUISOLV II-Bromley predictions found in this study for a few sulfate-rich cases are somewhat lower than what might be expected with the MESA-Bromley configuration used in the previous study.

3.2. Computational Efficiency

[40] A direct comparison of the computational speeds of MESA and EQUISOLV II is somewhat complicated by the differences in their RMS relative convergence errors for the chosen tolerance values of 0.01 in both the solvers. Second, differences in the general code structure and degree of code generalization can have large effects on a model's computational efficiency. For example, EQUISOLV II uses vectorization around grid cells so that it can simultaneously solve a block of 30 to 500 cells more efficiently than one cell at a time. Third, the EQUISOLV II code is highly generalized, which makes it fairly easy to add more reactions and species directly via input files without having to make any changes to the code itself. However, this convenience comes at the cost of efficiency. On the other hand, the MESA code is “hardwired” for the present set of reactions and species for higher computational efficiency, and would require additional developmental work to add more species and reactions to the code. Despite these fundamental differences, several modifications were carefully made to both the codes to make them as similar as possible in terms of the set of equilibrium reactions and species solved and the general coding style. Also, a non-vectorized single-cell version of EQUISOLV II comparable to MESA was created.

[41] With these caveats, CPU time comparisons between MESA and EQUISOLV II for all the 16 test cases were made on two similar computer platforms using different FORTRAN compilers. The computer at the Pacific Northwest National Laboratory (PNNL) was a dual Intel Xeon 3.0 GHz 32 bit processor system running the Portland Group FORTRAN 77/90 complier, while the computer at Stanford University was an Intel Xeon 3.6 GHz 64 bit processor running the Intel Fortran 77/90 compiler. Both the codes were compiled identically and were run in double precision on each computer. Each test case was run over the entire RH range from 1 to 99% at 1% RH intervals. To obtain statistically significant CPU times, each RH interval was solved 15000 times with MESA and non-vectorized EQUISOLV II, and 100 times with the vectorized version of EQUISOLV II which was set up to solve a block of 350 grid cells. The average CPU times per cell for each case was then calculated by dividing the total CPU time by (15000 X 99) for the MESA and non-vectorized EQUISOLV II codes, and by (350 X 100 X 99) for the vectorized EQUISOLV II code. These results are summarized in Table 11. It can be seen that MESA is considerably faster than EQUISOLV II for ∼94% of the intercomparison runs. The average speedups for MESA with respect to the vectorized and non-vectorized versions of EQUISOLV II were 2.7 to 5.8, respectively.

Table 11. Comparison of Average CPU Times Required by MESA and EQUISOLV II Over the Entire Range of 1 to 99% RHa
CaseAverage CPU Time per Cell (μs) Over the Full Range of 1 to 99% RHa
Platform: Intel Xeon 3.0 GHz at PNNL Compiler: Portland Group Fortran 77/90Platform: Intel Xeon 3.6 GHz at Stanford Compiler: Intel Fortran 77/90
MESA-SbEQII-ScEQII-MdMESA-SEQII-SEQII-M
  • a

    The solver tolerances in MESA and EQUISOLV II were set to 0.01. The RMS relative convergence error was ∼0.14% in MESA, and ∼0.38% and ∼3.7% in EQUISOLV II depending on the RH range considered (see discussion in text and Table 9).

  • b

    MESA-S = single-cell, non-vectorized version of MESA.

  • c

    EQII-S = single-cell, non-vectorized version of EQUISOLV II.

  • d

    EQII-M = multiple-cell, vectorized version of EQUISOLV II with 350 cells.

Sulfate-Poor Test Cases
(1)3.8638.2828.003.0518.6313.20
(2)6.8840.0129.565.1619.6513.94
(3)2.5430.3421.892.0115.1810.96
(4)4.1330.7722.853.1815.1910.96
(5)7.9636.0926.625.5317.6712.69
(6)14.2838.9427.769.9019.0713.20
(7)17.8037.2127.2112.2017.9412.93
(8)2.8214.0711.012.446.704.97
(9)21.1035.4727.0519.3316.3612.35
(10)11.7529.1522.6411.2013.5510.34
 
Sulfate-Rich Test Cases
(11)16.0138.9628.5910.3318.0513.09
(12)13.9940.9629.659.0319.9813.71
(13)10.7444.3831.996.9721.4414.53
(14)18.5946.0132.4111.8322.5714.94
(15)2.5142.6430.691.7821.7614.47
(16)4.1344.2831.352.8022.2814.66

[42] Because MESA diagnoses MDRH directly from a given aerosol composition, it is significantly faster than EQUISOLV II when RH < MDRH. Therefore, to assess the computational efficiency for conditions when both the models are required to solve the same number of equations, another set of runs were made for all the test cases over different RH ranges where both the solid and liquid phases were present (i.e., ignoring RHs where the aerosol is completely solid or completely liquid). The CPU times for these runs are summarized in Table 12. Here too, MESA is found to be appreciably faster than EQUISOLV II for most of the cases, on both the computers. The average speedups for MESA with respect to the vectorized and non-vectorized versions of EQUISOLV II were 1.4 to 2.8, respectively. The minimum, maximum, and average MESA speedups relative to EQUISOLV II for the two computer platforms and two vectorization modes are summarized in Table 13.

Table 12. Comparison of Average CPU Times Required by MESA and EQUISOLV II Over Mixed-Phase RH Ranges Where Both the Solid and Liquid Phases Coexista
Case Average CPU Time per Cell (μs) Over Mixed-Phase RH Ranges Onlya
Platform: Intel Xeon 3.0 GHz at PNNL Compiler: Portland Group Fortran 77/90Platform: Intel Xeon 3.6 GHz at Stanford Compiler: Intel Fortran 77/90
RH Range, %MESA-SbEQII-ScEQII-MdMESA-SEQII-SEQII-M
  • a

    The solver tolerances in MESA and EQUISOLV II were set to 0.01. The RMS relative convergence error was ∼0.14% in MESA, and ∼0.38% and ∼3.7% in EQUISOLV II depending on the RH range considered (see discussion in text and Table 9).

  • b

    MESA-S = single-cell, non-vectorized version of MESA.

  • c

    EQII-S = single-cell, non-vectorized version of EQUISOLV II.

  • d

    EQII-M = multiple-cell, vectorized version of EQUISOLV II with 350 cells.

Sulfate-Poor Test Cases
(1)61 – 7714.9467.2148.4911.4733.3623.37
(2)53 – 7128.4297.8972.1020.6447.1533.62
(3)76 – 8217.5263.7144.6513.5232.9023.78
(4)68 – 8120.1044.1031.7615.1622.0715.88
(5)54 – 7628.1493.3967.1119.0244.5132.07
(6)51 – 7533.7993.7165.9223.0747.8433.01
(7)48 – 7829.8963.5346.0820.5531.6822.80
(8)
(9)1 – 6730.1947.0036.0227.4521.3516.07
(10)1 – 6815.9538.2429.7115.1717.3413.01
 
Sulfate-Rich Test Cases
(11)1 – 3044.5652.6237.8827.7624.1316.67
(12)32 – 7330.6254.5339.8819.0626.6018.28
(13)38 – 6730.82103.0780.7019.1849.8933.00
(14)47 – 6743.81136.2295.7627.2674.2348.20
(15)71 – 7519.2070.9350.3412.2434.4724.50
(16)64 – 6940.22102.7872.9525.1051.3135.30
Table 13. Summary of Statistics for MESA Speedups Relative to EQUISOLV IIa
StatisticMESA Speedups Relative to EQUISOLV IIa
Intel Xeon 3.0 GHz Portland Group Fortran 77/90Intel Xeon 3.6 GHz Intel Fortran 77/90
EQII-SbEQII-McEQII-SEQII-M
  • a

    The solver tolerances in MESA and EQUISOLV II were set to 0.01. The RMS relative convergence error was ∼0.14% in MESA, and ∼0.38% and ∼3.7% in EQUISOLV II depending on the RH range considered (see discussion in text and Table 9).

  • b

    EQII-S = single-cell, non-vectorized version of EQUISOLV II.

  • c

    EQII-M = multiple-cell, vectorized version of EQUISOLV II with 350 cells.

Full RH Range
Minimum1.71.30.90.6
Maximum17.012.212.28.1
Average5.84.33.92.7
 
Mixed-Phase RH Range
Minimum1.20.90.80.6
Maximum4.53.32.92.0
Average2.82.02.01.4

[43] Interestingly, the computational speed of EQUISOLV II showed a more pronounced dependence on the computer platform/compiler than MESA. The average speedup for MESA on the Xeon 3.6 GHz processor over the Xeon 3.0 GHz processor was ∼1.4, which is commensurate to the ratio of the processor speeds of 1.2, while the average speedup for EQUISOLV II was ∼2.1 for both the vectorized and non-vectorized versions. The reasons for this behavior are not clear. Also, the average speed up of vectorized over non-vectorized EQUISOLV II in this study was found to be only about 1.4 as opposed to a factor of more than 10 reported in Zhang et al. [2000]. This is partly because the Zhang et al. study used the vectorized code for scalar single-cell calculations by simply reducing the number of cells to unity. For the present study, the single-cell calculations were performed by actually removing the vectorized loops which, however, appeared to increase the computational efficiency by only ∼50%. The reason for the large discrepancy in the speedups due to vectorization between this and Zhang et al. studies are not clear. Furthermore, the vectorized EQUISOLV II code was set up to solve the exact same case in all the 350 grid cells for the intercomparison exercise, while in a real 3-D simulation each grid cell can potentially have different aerosol composition, RH, and temperature. However, because of vectorization, the maximum number of iterations the solver can have for each cell is always fixed, which results in greater convergence errors in some cells but lesser in others. On the other hand, MESA guarantees a low convergence error in all the cells under all conditions.

[44] The above considerations, therefore, suggest that the MESA-MTEM configuration is attractive for use in 3-D aerosol models in which both accuracy and computational efficiency are critical.

4. Summary and Conclusions

[45] Multicomponent aerosol particles are known to exhibit complex multistage deliquescence growth behavior during which both solid and liquid phases co-exist in equilibrium. Accurately and efficiently predicting these phase transitions is one of the many challenging tasks in multicomponent aerosol models. This paper presents the development and application of a new multicomponent equilibrium solver for the aerosol-phase (MESA) for particles containing mixtures of H+, NH4+, Na+, Ca2+, SO42−, HSO4, NO3, and Cl ions, using the newly developed MTEM module for activity coefficients and other thermodynamic data. The algorithm of MESA uses a modified pseudo-transient continuation technique that involves integrating the set of ordinary differential equations describing the precipitation and dissolution reactions for each salt until the system satisfies the equilibrium or mass convergence criteria. An adaptive time-stepping scheme is used with the Forward Euler method such that the resulting solver is unconditionally stable, mass conserving, and computationally efficient. The technique is easy to implement, and is numerically equivalent to iterating all the equilibrium reactions simultaneously with a single iteration loop. Mutual deliquescence relative humidities (MDRH) of all the possible and relevant salt mixtures in present system of ions were parameterized as a function of temperature (valid between 240 to 320 K), thereby eliminating the need for a rigorous numerical solution when ambient RH < MDRH(T).

[46] Performance of MESA-MTEM was evaluated against the Web-based AIM Model III, which served as a benchmark for accuracy, and another computationally efficient and generalized equilibrium solver EQUISOLV II for speed. Excellent agreement between AIM and MESA-MTEM was observed for all sixteen test cases considered in this exercise. The maximum deviation in MDRH for the test cases was only 2% RH points. These results were as expected because MTEM activity coefficient model and MDRH polynomials are parameterized using the comprehensive PSC activity coefficient model, which is also used in AIM. Without tuning to AIM, EQUISOLV II was also able to capture the complex growth behavior quite well, with deviations in MDRH of 0–4% RH points in most of the cases. However, in two cases, it produced MDRH deviations of 8–10% RH points and in one case it was as large as 15%. While these deviations occurred at relatively low water contents, the resulting discrepancies in the phase-state of aerosols can potentially affect their interactions with various gas-phase precursors, and hence their subsequent growth rates.

[47] RMS relative deviations in mass growth factors (MGF) with respect to AIM were also computed for MESA and EQUISOLV II. While these deviations were typically small and within acceptable margins in both the models, they were relatively smaller in MESA (average value of ∼2.3%) compared to EQUISOLV II (average value of ∼4.3%). The relative deviations in total water contents (summed over 20 to 99% RH) ranged from 0.27 to 4.7% in MESA with an average of 2.46 over all the cases. Similarly, these deviations typically ranged from 0.15 to 5.1% in EQUISOLV II, except for two cases where they were ∼12%, with an average value of 3.75% over all the cases. These deviations in MDRHs, MGFs, and total water contents are likely due to differences in thermodynamic parameters such as the predicted multicomponent activity coefficients, equilibrium constants, and binary molality polynomials used in both the models relative to AIM.

[48] A direct comparison of the computational speeds of MESA and EQUISOLV II was complicated by a number of factors including (1) differences in their convergence criteria, (2) differences in their RMS relative convergence errors for the chosen solver tolerance parameters, (3) differences in thermodynamic deviations, (4) differences in the code structure (e.g., vectorized and non-vectorized codes), and (5) the degree of code generalization, such as the ease with which more reactions and species can be added in EQUISOLV II versus the more “hardwired” algorithm in MESA for superior computational efficiency. EQUISOLV II also contains temperature dependence of activity coefficients for the highly sulfate-rich systems valid at low RH and temperatures found in the upper troposphere and stratosphere, although the temperature dependent parameterizations for all the salts that can form under sulfate-poor conditions are either not available or not applicable at low RHs, or have not been verified for accuracy due to lack of experimental data. With these caveats, a brief speed comparison was performed between MESA and EQUISOLV II on two computer platforms using different FORTRAN compilers. With significantly lower RMS convergence errors than EQUISOLV II, MESA was found to be appreciably faster than both the non-vectorized and vectorized versions of EQUISOLV II for ∼94% of the 124 intercomparison runs performed. The average speedups in MESA relative to EQUISOLV II ranged from 1.4 to 5.8 depending on the vectorization mode, computer/compiler used, and the RH range considered. MESA was slightly slower for the remaining ∼6% of runs, with a minimum speedup of ∼0.6, while the maximum speedup obtained was ∼17. Also, because MESA diagnoses MDRH directly from a given aerosol composition, it is significantly faster than EQUISOLV II when RH < MDRH.

[49] Therefore, with regard to the overall accuracy and computational efficiency considerations, the MESA-MTEM configuration appears to be highly attractive for use in 3-D air quality and aerosol models for lower tropospheric applications. These modules are embedded in a new dynamic gas-aerosol model MOSAIC (Model for Simulating Aerosol Interactions and Chemistry) which will be described elsewhere. Work is currently under way to extend the relevant MESA-MTEM model parameters to stratospheric conditions.

Acknowledgments

[50] The authors thank A. S. Wexler (University of California, Davis) for helpful discussions on phase transitions in inorganic salt mixtures and numerical methods for solving equilibrium problems. The authors are grateful to M. Z. Jacobson (Stanford University) for providing the EQUISOLV II codes for intercomparison and the CPU times for MESA and EQUISOLV II runs on a computer at Stanford University, and many useful comments on the draft manuscript. R.A.Z. also thanks S. J. Ghan, J. D. Fast, and C. M. Berkowitz (all at PNNL) for their support throughout this work. Funding for this research was provided by the U.S. Department of Energy (DOE) under the auspices of the Atmospheric Science Program of the Office of Biological and Environmental Research, the NASA Earth Science Enterprise under grant NAGW 3367, and Pacific Northwest National Laboratory (PNNL) Laboratory Directed Research and Development program through the Computational Sciences and Engineering Initiative. Part of the work leading to this paper was performed while R.A.Z. and L.K.P. were at Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830.