gas/aerosol partitioning: 1. a computationally efficient...

Gas/aerosol partitioning:

1. A computationally efficient model

Swen Metzger,1 Frank Dentener,2 Spyros Pandis,3 and Jos Lelieveld1

Received 2 July 2001; revised 24 January 2002; accepted 2 February 2002; published 29 August 2002.

[1] A computationally efficient model to calculate gas/aerosol partitioning of semivolatileinorganic aerosol components has been developed for use in global atmospheric chemistryand climate models. We introduce an approximate method for the activity coefficientcalculation that directly relates aerosol activity coefficients to the ambient relativehumidity, assuming chemical equilibrium. We demonstrate that this method provides analternative for the computationally expensive iterative activity coefficient calculationmethods presently used in thermodynamic gas/aerosol models. The gain of our method isthat the entire system of the gas/aerosol equilibrium partitioning can be solvednoniteratively, a substantial advantage in global modeling. We show that our equilibriumsimplified aerosol model (EQSAM) yields results similar to those of current state-of-the-art equilibrium models. INDEX TERMS: 1610 Global Change: Atmosphere (0315, 0325); 5704

Planetology: Fluid Planets: Atmospheres—composition and chemistry; 0305 Atmospheric Composition and

Structure: Aerosols and particles (0345, 4801); KEYWORDS: inorganic aerosols, aerosol composition, aerosol

water, thermodynamic equilibrium, gas/aerosol partitioning parameterization

1. Introduction

[2] Atmospheric aerosols are usually mixtures of manycomponents, partly composed of inorganic acids (e.g.,H2SO4, HNO3), their salts (e.g., (NH4)2SO4, NH4NO3),and water [e.g., Charlson et al., 1978; Heintzenberg,1989]. Because many compounds are highly hygroscopic,aerosol associated water often exceeds the dry aerosolmass [e.g., Pilinis et al., 1989], especially in the humidlower troposphere. Both the dry aerosol mass and theaerosol associated water are important in climate changescenario simulations [e.g., Charlson et al., 1987; Charlsonand Wigley, 1994; Charlson and Heintzenberg, 1995;Pilinis et al., 1995; Intergovernmental Panel on ClimateChange (IPCC), 1996]. Nevertheless, multicomponentaerosol concentrations are not yet routinely calculatedwithin global atmospheric chemistry or climate models.The reason is that simulations of these aerosol particles,especially those including semivolatile components, re-quire complex and computationally expensive thermody-namic calculations. For instance, the aerosol-associatedwater depends on the composition of the particles, whichis determined by the gas/liquid/solid partitioning, which isin turn strongly dependent on temperature and relativehumidity.[3] In the past two decades much effort has been devoted

to the development of methods for the calculation of aerosolproperties that are difficult to measure. These properties

include the aerosol phase composition (i.e., solid or liquid)and the aerosol-associated water mass. Most attention hasfocused on the inorganic aerosol compounds that are oftenpredominant, such as sulfate, ammonium, nitrate and aero-sol water. These compounds partition between the liquid-solid aerosol phases and the gas phase of aerosol precursorgases such as HNO3 and NH3. Therefore numerous ther-modynamic models have been developed [Bassett andSeinfeld, 1983, 1984; Saxena et al., 1986; Binkowski,1991; Pilinis and Seinfeld, 1987; Wexler and Seinfeld,1991; Kim et al., 1993a, 1993b; Kim and Seinfeld, 1995;Meng and Seinfeld, 1996; Nenes et al., 1998; Clegg et al.,1998a, 1998b; Jacobson et al., 1996; Jacobson, 1999], andseveral gas/aerosol dynamic models [Meng et al., 1998; Sunand Wexler, 1998; Pilinis et al., 2000]. While the firstgeneration models, to which we refer in the following asequilibrium models (EQMs), assumed thermodynamic equi-librium between the gas/liquid/solid aerosol phases, neglect-ing mass transfer between the liquid/solid aerosol phasesand the gas phase, the second generation dynamical modelscalculate mass transfer explicitly, e.g., by incorporating anEQM [Pilinis et al., 2000]. For a comparative review, seeZhang et al. [1999a, 1999b].[4] Most of these models were developed for incorpo-

ration into urban and regional scale air quality models.Numerical EQM schemes generally solve the system ofnonlinear gas/aerosol equilibrium equations using differenttypes of iterative schemes. The number of iterations neededto solve the equilibrium equations strongly depends on theaerosol composition and the meteorological conditions. Forinstance, if some aerosol compounds are predicted to besolid, additional iterations are needed to calculate both theliquid-solid phase transitions, and the partitioning betweenliquid/solid aerosol phases and the gas phase. Even in therelatively simple case of pure aqueous phase particles, the

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D16, 4312, 10.1029/2001JD001102, 2002

1Max-Planck-Institute for Chemistry, Mainz, Germany.2Institute for Environment and Sustainability, Joint Research Centre,

Ispra, Italy.3Carnegie Mellon University, Pittsburgh, Pennsylvania, USA.

Copyright 2002 by the American Geophysical Union.0148-0227/02/2001JD001102$09.00

ACH 16 - 1

composition is calculated iteratively. The reason is thatcommon calculation methods of the aerosol compositionrequire information about the solute activity, which includesthe aerosol-associated water, and which determines thevapor pressure above the aerosol.[5] To reduce the computational costs of the gas/aerosol

partitioning calculations, different approaches have beenproposed as an alternative to the iterative calculations.These include the use of neural networks [Potukuchi andWexler, 1997] to obtain the equilibrium partial pressuresof the semivolatile aerosol components, the use of pre-calculated sets of activity coefficients which are the crucialpart of the iteration procedure in an EQM (ISORROPIA,see Nenes et al. [1998]), and polynomial fits of pre-calculated activity coefficients [Metzger et al., 1999].However, if incorporated into a global model, even theoptimized EQM ISORROPIA accounts for most of thetotal computational burden [Adams et al., 1999, 2001].Therefore, we have developed a computationally efficientgas/aerosol partitioning model specifically for global mod-eling (EQSAM: Equilibrium Simplified Aerosol Model).The approach used for EQSAM is based on the relation-ship between activity coefficients and the relative humidity[Metzger et al., 1999]. This relationship allows parameter-ization of the relevant nonideal solution properties, whichis sufficiently accurate for global modeling [Metzger,2000]. Although EQSAM is solely based on parameter-izations, the thermodynamic framework is based on thesame assumptions used by other EQMs; for a moredetailed discussion, we refer to Metzger [2000] (availableat http://www.library.uu.nl/digiarchief/dip/diss/1930853/inhoud.htm).[6] For a general discussion of the aerosol thermodynam-

ics and the EQMs, we refer to Denbigh [1981], Seinfeld andPandis [1998], Wexler and Potukuchi [1998], and thereferences therein; for a discussion of the interaction ofaerosols with clouds, see, e.g., Pruppacher and Klett [1997].[7] EQSAM is introduced in section 2. In section 3,

EQSAM is evaluated against other thermodynamic mod-els presently in use in urban and regional chemicaltransport models. This model comparison includes box-model calculations and global offline calculations, as wellas online calculations. For the latter part, EQSAM andthe state-off-the-art EQM ISORROPIA will be applied toglobal atmospheric chemistry modeling, with both rou-tines fully coupled to an atmospheric chemistry-transportmodel. The results will be discussed in section 4. Weextend the online model study in the accompanying paperof Metzger et al. [2002] (hereinafter referred to asM2002) to investigate and discuss the gas/aerosol parti-tioning on a global scale (using EQSAM). There, we alsofocus on the relevance of the gas/aerosol partitioning forglobal modeling, and on the estimation of the associateduncertainties, including a comparison with ground-basedmeasurements.

2. Model Description

[8] The basic concept of EQSAM is that the activities ofatmospheric aerosols in equilibrium with the ambient air aregoverned by the relative humidity (RH). Since the wateractivity is fixed by RH, the solute activity is, for a given

aerosol composition, a function of RH; the molality dependson the water mass, which solely depends on RH (for a givensolute). This is also approximately true for activity coeffi-cients of salt solutes of binary and multicomponent solu-tions. The latter is a direct consequence of the ZSR-relation(see section 2.1). Consequently, activity coefficients can bedirectly derived from specific functions (see section 2.5).Using the ‘‘domain structure’’ (see section 2.2), and takinginto account that gas/aerosol equilibrium is only valid forcertain domains where sulfate is completely neutralized, wecan noniteratively calculate the aerosol composition, includ-ing aerosol-associated water. A schematic description ofEQSAM is given in Figure 1; the most important aspectswill be discussed in the following.

2.1. General Assumptions

[9] The main assumptions basic to EQSAM, as to otherEQMs, are that aerosols are internally mixed and obeythermodynamic gas/aerosol equilibrium. Both assumptionsare expected to be accurate under most atmospheric con-ditions considering the time steps used by global chemicaltransport models. For instance, Meng and Seinfeld [1996]have shown that the time required for gas/aerosol equilibra-tion depends primarily on the aerosol size and less on thetemperature; small particles have a larger surface-to-volumeratio, therefore equilibrating faster than larger ones. Inaddition, the equilibration times of relevant aerosol specieshave been shown to be of the order of minutes [Khlystov,1998; Dassios and Pandis, 1999]. These timescales aremuch shorter than the time steps of global atmosphere/chemistry models calculating chemistry and transport pro-cesses (typically 0.5–2 hours). The equilibrium approachcan therefore be assumed adequate for global modeling.[10] The equilibrium assumption further implies that the

water activity (aw) of an aqueous aerosol particle is equal tothe ambient relative humidity (RH), i.e., aw = RH; RHdefined on the fraction scale [0–1] [Bassett and Seinfeld,1983]. This is valid for atmospheric applications, since theambient relative humidity is not influenced by the smallwater uptake of aerosol particles. From aw = RH it followsthat the aerosol molality of a single solute or a mixture ofsolutes changes with ambient relative humidity. To obtainthe molality of a mixture of solutes, the total water contentof the aerosols is calculated in EQMs based upon a mixingrule that is widely used in atmospheric aerosol modeling,i.e., the so-called ZSR-relation named after Zdanovskii[1948] and Stokes and Robinson [1966]. This semiempiricalrelation simply assumes that a mixture of single-solutesolutions has the same water activity as each single-solutesolution, because aw = RH [Chen et al., 1973]. The ZSR-relation has been shown to be an excellent approximationfor most atmospheric applications [Cohen et al., 1987a,1987b; Kim et al., 1993b].

2.2. Concentration Domains

[11] To increase the computational efficiency of EQSAMthe total number of equilibrium reactions is minimized bymaking use of concentration domains, as described byNenes et al. [1998]. Each of these domains contains fewerspecies than the entire set of possible aerosol compositions.The domains are based on additional assumptions: Becausesulfuric acid has a very low vapor pressure, it is assumed

ACH 16 - 2 METZGER ET AL.: GAS/AEROSOL PARTITIONING 1

Figure 1. Sequential overview of the EQSAM structure. Note that the parameterizations in row 5 arediscussed by Metzger [2000]. The parameters are given in Table 2; f = 0.81 for all species.

METZGER ET AL.: GAS/AEROSOL PARTITIONING 1 ACH 16 - 3

that it resides completely in the aerosol phase. Dependingon mole ratios of the input concentrations (expressed inmolar units) of, e.g., total ammonium to sulfate, sulfate isneutralized either completely or partially, so that eachdomain represents a certain aerosol type (total denotes thesum of the gas and aerosol concentration). In EQSAMcurrently 3 main domains are used (Figure 1, point 1; theprefix t denotes total).[12] The subdivision of the input concentration into

concentration domains according to the mole ratios thusyields the number of reactions, order and domain specificparameterizations (see section 2.5). Additionally, each con-centration domain is further divided into several subdo-mains, according to the regime of deliquescence relativehumidity of the corresponding salt compound.

2.3. Relative Humidity of Deliquescence

[13] Whether gas/aerosol equilibrium is calculated be-tween the gas phase and a pure solid, or ions in an aqueoussolution, depends on the deliquescence behavior of theconsidered aerosol compound. Certain salts, such as ammo-nium sulfate or ammonium nitrate, deliquesce if the relativehumidity reaches a threshold value; below that these saltsmay be crystalline. However, certain aerosol mixtures (e.g.,solutions containing sulfuric acid) do not deliquesce; ratherthey remain aqueous regardless the ambient relative humid-ity. As in most EQMs, the deliquescence of various saltcompounds is determined in EQSAM in the correspondingsubdomains. The deliquescence of salt aerosols depends onthe ambient RH and temperature. The deliquescencebehavior has been investigated for single salt solutionsby, for instance, Wexler and Seinfeld [1991], and formultiple-salt solutions by, e.g., Tang and Munkelwitz[1993].[14] For modest atmospheric temperature variations, T -

To, with T the ambient temperature and To the standardtemperature, the relative humidity of deliquescence (RHD)and the mutual deliquescence relative humidity (MDRH) of

multicomponent salt particles can be expressed as [Neneset al., 1998]

ln½MDRHðTÞ=MDRHðToÞ�¼�Mw ��iðmsolute;i �Lsolute;iÞ=ðR � 1000Þ�ð1=T� 1=ToÞ; ð1Þ

where MDRH (To) is known (usually at To = 298.15K). Mw

denotes the molar mass of water, msolute,i the molality [molesolute/kg water] of electrolyte i that deliquesces, and Lsolute,i

the latent heat of fusion of salt i (RHD = MDRH, if i = 1). Ris the universal gas constant.[15] In EQSAM we use the RHD/MDRH values of

ISORROPIA according to the aerosol system considered,i.e., the ammonium-sulfate-nitrate-water system (values aregiven by Nenes et al. [1998]). At a relative humidity abovethe deliquescence point, the electrolyte is present in theaqueous phase. If the RH is above the RHD of all speciesconsidered, no solid crystalline phase of water-solublecompounds is present. Liquid/solid phase partitioning isthus determined in EQSAM according to the aerosol com-position (domain) and the corresponding RHD values forbinary and mixed salt solutions (Figure 1, point 2). Chem-ical equilibrium of individual salts is subsequently calcu-lated for the solid (or liquid) phase, if the RH is below (orabove) the temperature dependent RHD values.

2.4. Chemical Equilibrium

[16] The mass and composition of aerosols, includingvolatile species in the gas and aerosol phases, is calculatedin EQSAM, as within all EQMs, from chemical equilibrium[Stelson and Seinfeld, 1982; Bassett and Seinfeld, 1983,1984; Saxena et al., 1986; Pilinis and Seinfeld, 1987]. Forpartitioning between the gas/liquid/solid aerosol phases,chemical equilibrium is determined by the temperaturedependent equilibrium constant, K(T), where K is calcu-lated from [e.g., Denbigh, 1981]

K ¼ �i avii : ð2Þ

Figure 1. (continued)


Subscript i denote an ion pair, ai the activity, and ni thestoichiometric coefficient. The temperature dependence andthe K values used in EQSAM (Figure 1, point 3) are givenby Nenes et al. [1998].[17] To account for the effect of nonideal behavior at high

ionic strength on equilibrium, K is expressed in terms ofactivities ai. For the gas phase, the activity is defined as ai =pi/p, with pi the partial pressure of compound i at atmosphericpressure p. All vapors considered (e.g., water vapor, ammo-nia and nitric acid) are at sufficiently low partial pressures, sothat they are assumed to behave as a mixture of ideal gases.For the crystalline phase, the compound is assumed to bepure and the activity is unity, ai = 1. In solid solutions(supersaturated solutions, where individual compounds cancrystallize) the activity of the compound may be less thanunity, but such cases are not considered. For the aqueousphase, the activity is defined as the product of molality mi andactivity coefficient gi of compound i, i.e., ai = mi

.gi.

2.5. Activity Coefficients

[18] To solve the differential equilibrium equations andgas/aerosol partitioning noniteratively, we approximate theactivity coefficients by utilizing a relationship betweenactivity coefficients of atmospheric aerosols and the ambi-ent relative humidity in equilibrium [Metzger, 2000]. Thelatter work presents a more complete discussion, toodetailed for the present paper, of the thermodynamic back-ground of the functional relationship, on which the follow-ing parameterizations are based.[19] For atmospheric aerosols in thermodynamic equili-

brium with ambient air an increase of the aerosol concen-tration, e.g., ammonium nitrate, is always associated withthe condensation of a certain amount of water vapor so thatthe solution molality remains constant, if aw = RH. For theaqueous phase in turn a decrease of solute water due toevaporation or precipitation (into the solid phase) causes acertain amount of water to evaporate to maintain thesolution molality of the aerosol constant.[20] Thus, an equilibrium reaction that takes place in the

aqueous phase causes, for a given RH, a change of the molenumber of aerosol water. The same applies if the electrolytedissociates. Taking the change of aerosol water associatedwith a given equilibrium reaction into account, the relation-ship between aerosol activity coefficients and RH can bedescribed by the following empirical equation, derived bytrial and error, for the mean activity coefficient of the saltsolute, gi

± [Metzger, 2000],

g�i ¼ ½RHN=ð1000=N � ð1� RHÞ þ NÞ�1=x: ð3Þ

RH denotes the fractional relative humidity [0–1], and Nand x domain specific parameters. For a theoreticalinterpretation of equation (3) see Appendix A.[21] In equation (3) the domain specific parameter N

(Figure 1) accounts for the number of water moleculesinvolved in the dissolution of the salt solute of a binary ormixed solution, while x accounts for the charge and stoi-chiometric coefficient of the dissociated electrolyte. Valuesfor x are given in Table 1; parameters for the parameterizationof single solutes (ss) are given in Table 2. For instance, theactivity coefficients of ammonium nitrate, letovicite, ammo-nium sulfate, or calcium sulfate can be obtained withequation (3), by using the parameters x = 4 for NH4NO3,x = 2.5 for (NH4)3H(SO4)2, x = 2 for (NH4)2SO4, and x = 1for CaSO4, with N the number of the domain: For neutralizedsalt solutions (domain 2) N = 2, for partly neutralizedsolutions containing NH4HSO4 (domain 3) N = 3, and fornonneutralized solutions containing sulfuric acid (domain 4)N = 4. Values for x and N were chosen to yield an optimal fitwith the results of ISORROPIA and of the EQMs presentedin the following. Note that the activity coefficients of otherelectrolytes with similar type and charge used in variousEQMs (listed in Table 2) can be obtained by using theappropriate parameters for x and N [Metzger, 2000]. Inaddition, activity coefficients can be obtained from eachother, if equation (3) is used. Furthermore, equation (3) isnot restricted to binary solutions. The aerosol activity andequilibrium composition of binary and mixed solutions cantherefore be calculated noniteratively using equation (3).This is possible because aerosol activity coefficients ofmixed solutions do not differ significantly from those ofbinary solutions if the ZSR-relation is used to calculate theaerosol water content (aw = RH). It should be noted thatequation (3) is only valid for particles larger than about 0.1micrometer since the Kelvin-term is neglected (for aw=RH);the complete equation reads: aw + 2s � Vm/(R�T�r) = RH[e.g.,Warneck, 1988]. Note further that equation (3) does notinclude a temperature dependency; the latter is includedthrough equilibrium (see below).

Table 1. Parameters for Equation (3) for Major Inorganic Aerosol Compounds as

Considered in Various EQMs (e.g., ISORROPIA and SCAPE)a

x Species

4 NH4NO3, NaNO3, KNO3, NH4HSO4, NaHSO4, KHSO4, NACl, KCl, HCl2.5 (NH4)3H(SO4)2 = (NH4)2SO4*NH4HSO4 (letovicite)2 (NH4)2SO4, CaCl2, MgCl2, K2SO4

1 CaSO4, MgSO4

aParameter N of equation (3) is given in Figure 1 (point 1). For neutralized salt solutions(domain 2), N = 2; for partly neutralized solutions containing NH4HSO4 (domain 3), N = 3;and for nonneutralized solutions containing sulfuric acid (domain 4), N = 4.

Table 2. Parameters for the Parameterization of Single Solute (ss)

Molalities (Figure 1)a

Species (i) Mi vivi zi

NH4NO3 80 4 1NH4HSO4 115 4 1(NH4)2SO4 132 27 0.75(NH4)3H(SO4) 247 9 1H2SO4 98 4 0.5aThe parameters are discussed in more detail by Metzger [2000].


[22] We cannot a-priori predict if equation 3 would alsohold for other aerosol systems, for which we did not haveexisting EQM results available. However, the gi

±- RHrelationship (equation (3)) allows us to reproduce activitycoefficients of various neutralized salt solutions includingammonium, sulfate, nitrate, sea salt and mineral dustcomponents relative to various state-of-the-art EQMs. TheseEQMs calculate gi

± based on the widely used activitycoefficient calculation methods of Bromley [1973], Pitzerand Mayorga [1973], and Kusik and Meissner [1978]. TheEQMs to which we refer in the following (section 3), i.e.,MARS [Saxena et al., 1986; Binkowski, 1991] and SEQUI-LIB [Pilinis and Seinfeld, 1987] use the Pitzer method forthe calculation of binary activity coefficients, and theBromley method for the calculation of multicomponentactivity coefficients. SCAPE [Kim et al., 1993a, 1993b;Kim and Seinfeld, 1995] and SCAPE2 [Meng et al., 1998]have an option to use either one of the three methods, whileISORROPIA [Nenes et al., 1998] makes use of all threemethods depending on the aerosol composition. A compar-ison of these methods is presented by Kim et al. [1993a,1993b], Kim and Seinfeld [1995], in addition to Saxena andPeterson [1981]. Note that most of the methods that predictthe activity coefficients of a multicomponent solution areempirical or semiempirical, and typically use the activitycoefficients of single-electrolyte solutions of the same ionicstrength. In EQSAM, we use equation (3) to obtain theactivity coefficients needed in EQSAM (Figure 1, point 4).In section 3, we discuss the results of a model comparisonof EQSAM with various other EQMs presently is use,including a parameterized version of the state-of-the-artEQM ISORROPIA. For the latter we substitute the originaland iterative activity coefficient calculation method byequation (3), which yield a noniterative alternative forlarge-scale applications.[23] The types of activity coefficients are shown in

Figure 2, plotted as a function of relative humidity. The

black lines represent the gi± of SCAPE (using the Pitzer-

method), the gray lines are obtained using equation (3).Generally, the activity coefficient departs for a given RHfurther from unity as the charge carried by the speciesincreases. From top to bottom the types of activity coef-ficients represent NH4NO3, (NH4)3H(SO4)2, (NH4)2SO4 andCaSO4. The gi

± - RH relationship (Figure 2) shows that themean activity coefficients of salt solutes approaches one asthe fractional relative humidity approaches one, owing to theincreasing dilution with increasing RH, which is caused bythe equilibrium growth due to water uptake.[24] The activity coefficients calculated with equation (3)

are within the range of the activity coefficients given byvarious EQMs. Note that this is also true for cases where therelative humidity is not the only variable (temperature andinput concentration are kept constant in Figure 2). The reasonis that temperature variations and a concentration range donot significantly influence the aerosol activity coefficients fora given RH in thermodynamic equilibrium, assuming aw =RH. Under this assumption, the water uptake is proportionalto the total amount of dissolved matter, while the temperatureaffects the gi

± most noticeably due to changes in aerosolcomposition, i.e., when evaporation or crystallization occurs(when RH equals RHD). However, evaporation or crystal-lization also changes the total amount of dissolved matter,which is accounted for by the use of different domains.[25] The appropriate activity coefficients can therefore be

obtained by using a domain specific parameter N, whichaccounts for all salt solutes under consideration. Neverthe-less, equation (3) breaks down for a sulfuric acid solution (atleast compared to uncertain results of other EQMs). Fortu-nately, for aerosol equilibrium calculations we can neglectthe activity coefficients of strongly acidic aerosols sincevolatile species, such as ammonium nitrate, cannot, or, areat least not assumed to be present in this case. Furthermore,the amount of water associated with the aerosol is also notaffected; the amount of water solely depends on the amount

Figure 2. Relationship between activity coefficients of atmospheric aerosols and the ambient relativehumidity at thermodynamic equilibrium for different types of activity coefficients; black lines areobtained from a reference model; gray lines are obtained from equation (3).


and type of dissolve matter, which remains in this caseaqueous independent of the activity (it is sufficient to treata sulfuric acid solution as nonvolatile and noncrystalline).

2.6. Aerosol Composition

[26] Gas/aerosol partitioning only occurs for volatile andsemivolatile aerosol compounds. For the ammonium-sul-fate-nitrate-water aerosol system it is only relevant to theammonium nitrate salt. However, according to the assump-tion made by using domains, this salt can only form ifsurplus ammonia is available, which is the case if sulfate iscompletely neutralized (domain 2 in Figure 1). All otherammonium salts and nonneutralized sulfate (sulfuric acid)are treated as nonvolatile.[27] In addition, sulfuric acid is assumed to remain in the

aqueous phase regardless of its solute activity and activitycoefficients. Thus, these assumptions allow simplificationswith respect to the explicit determination of the varioussulfate states, i.e., H2SO4, HSO4

�, SO42�. To increase the

computational efficiency we directly determine the sulfatestates in EQSAM (denoted as � at point 1 in Figure 1),based on the ammonium to sulfate mole ratio (domain). Theuse of domains thus yields the number and order of theequilibrium reactions, and the stoichiometric coefficient ofsulfate. The latter is needed to determine the neutralizationby ammonia, and the aerosol water mass associated withammonium sulfate, ammonium bisulfate, or sulfuric acid.[28] If sufficient ammonia is available to neutralize all

sulfate (domain 2), the residual amount of ammonia mightneutralize nitric acid to form ammonium nitrate. Dependingon the relative humidity and temperature, it is determined inthe subdomain whether the RH exceeds the RHD ofammonium nitrate (Figure 1, point 2), or, if hysteresis isconsidered, the history of the aerosol compounds is takeninto account as well (see M2002 for the definition and thetime dependency). Then, either gas-liquid, or gas-solidequilibrium is calculated, if the partial pressure product ofgaseous ammonia and nitric acid exceeds the value given bythe corresponding temperature dependent equilibrium con-stant (Figure 1, point 3). For the aqueous phase, the activitycoefficient and the single solute (ss) molalities are calcu-lated with parameterizations, i.e., equation (3) within point4 and point 5 of Figure 1, respectively, so that the equili-brium equations can be solved directly. From the totalamount of aerosol nitrate, the residual amount of the nitricacid can be determined. The residual amount of the nitricacid and ammonia finally yield the gas phase concentra-tions, while the sum of ammonium, sulfate and nitrate yieldthe total particulate matter (Figure 1, point 6). For theaqueous phase, the amount of water associated with theaerosol is determined (Figure 1, point 7, Metzger [2000]).[29] Note that because of the gi

± -RH relationship (equa-tion (3)), single solute molalities and aerosol water can beparameterized; the single solute molalities are presentlyobtained from tabulated measurements in EQMs. The singlesolute molalities are expressed in terms of relative humidity(Figure 1, point 5). They are used to calculate the watermass of the aerosol upon substitution of the expression formolality in the definition of aerosol water by using the ZSR-relation (Figure 1, point 7). For illustration and furtherdiscussion of these empirical parameterizations we refer toMetzger [2000].

2.7. Model Summary

[30] Figure 1 gives an example of the structure of EQSAMand can be summarized as follows: Row 1 defines thedivision in domains according to the mole ratio of totalammonia (t denotes total = sum of ammonia and ammonium)and total sulfate (sum of sulfuric acid and sulfate), for whichcertain parameters are defined for the parameterizations; i.e.,the domain number N in column 1 and the stoichiometriccoefficient � of ammonium in column 4 (both row 1). Theconcentration of free ammonia (superscript F) is determined,i.e., the amount of ammonia which is available after theneutralization of total sulfate by using � depending on thedomain. For a given domain (here domain 2, where sulfate isneutralized), gas/aerosol partitioning can occur. Dependingon the subdomain (determination in row 2), i.e., whether therelative humidity (RH) is below or above the temperaturedependent relative humidity of deliquescence (RHD) of agiven aerosol composition (domain), partitioning betweenthe gas and solid or liquid aerosol phase is considered; row2–9 is therefore divided into a left and a right column,respectively. Note that the RHD is considered for single saltcompounds (e.g., ammonium nitrate), while the mutualdeliquescence relative humidity (MDRH) is used for mixedsalts (e.g., a mixture of ammonium sulfate and ammonium bi-sulfate). According to the domain and subdomain the appro-priate temperature dependent equilibrium constant is used tocalculate either gas-solid or gas-liquid equilibrium (row 3).For the former case, i.e., RH<RHD, the activity coefficient is1 and, e.g., solid ammonium nitrate (subscript s) is inequilibrium with free ammonia and total nitric acid (t denotesthe sum of nitrate and nitric acid). For the case RH>RHD, theactivity coefficient is directly derived from equation (3) (row4). Additionally, the solution molalities are derived fromparameterizations (row 5), which depend for a given RH onlyon the type of (salt) solute (since the amount of water andsolute are proportional). The equilibrium concentration arethen determined in row 6: for the gas/aerosol partitioning ofammonium nitrate, gaseous nitric acid is determined as theresidual after neutralization of total nitric acid by the con-centration of ammonium (in equilibrium with nitrate); like-wise, the amount of residual gaseous ammonia is determinedas the difference between free ammonia and nitrate, while thedetermination of sulfuric acid additionally depends on �.Subsequently, the solid or aqueous concentrations of ammo-nium and sulfate are calculated, as well as the dry or aqueousparticulate matter (PM) and the total PM (t denotes sum ofsolid and aqueous PM; the subscript s and aq denotes solidand aqueous, respectively). The amount of aerosol water (W)and the approximate radius increase (rinc), caused by thewater uptake, are determined in row 7 (for the wet case). Theproton concentration (H+) is calculated in row 8 and the ionicstrength (I) of the solution in row 9. Row 10 defines thediagnostic output for both the solid and aqueous case. Alldiagnostic output will be discussed in the next section.

3. Equilibrium Model Comparison

[31] In this section we compare EQSAM with otherEQMs in use. We present results of a model comparisonfor different modeling tasks, using various equilibriummodels presented in section 2.5 (i.e., MARS, SEQUILIB,SCAPE, SCAPE2, ISORROPIA). Since SCAPE has the


option to choose different activity coefficient calculationmethods (ACCM), i.e., the Pitzer Method (PM), Kusik-Meissner Method (KM), or Bromley Method (BM) (intro-duced in section 2.5), we will use these for the modelcomparison to investigate the differences caused by differ-ent ACCMs. The different SCAPE models will be termedaccording to the ACCM used: SCAPEa-PM (SCAPE),SCAPEb-KM (SCAPE2), SCAPEb-BM (SCAPE2).Unfortunately not all versions were numerically stableunder all conditions, therefore only combinations have beenused which yield reliable results. The comparison includes aparameterized version of ISORROPIA, denoted by thesuffix ‘P’. In ISORROPIA-P we have replaced the originalACCM by the activity coefficient - RH relationship used inEQSAM, i.e., equation (3) as described in section 2.5.[32] The comparison is performed on three levels: (1) In

box model calculations, all EQMs are applied to an artificialthough realistic set of input data, using fixed concentrationsand temperature, so that the results (differences) can bestudied as a function of RH. (2) In global offline calcula-tions, all EQMs are applied to global chemistry fields (10�7.5� resolution) as produced with an atmospheric/chemistrytransport model (TM3) and ECMWF meteorology usingmonthly mean surface values for January 1997; this allows astudy of the differences for a wide range of realisticatmospheric conditions, i.e., temperature, relative humidityand various aerosol precursor concentrations. (3) In globalonline calculations, EQSAM and ISORROPIA have beenincorporated into the atmospheric/chemistry transport modelTM3 to calculate the gas/aerosol partitioning online, i.e.,interactively with the other chemistry simulations, account-ing for feedback between the gas and aqueous phase(including cloud phase chemistry), transport and depositionprocesses (as described by M2002).

3.1. Box Model Calculations

[33] To explore the errors introduced by using EQSAMrelative to the more comprehensive EQMs in use, we firstcompare the models for three cases of selected inputconcentrations of the ammonium/sulfate/nitrate/water-sys-tem, which corresponds to three different concentrationdomains, i.e., ammonium/sulfate ratios (section 2.2). Theresults of the gas/liquid/solid equilibrium partitioning willbe shown as function of RH (10–95%). The input con-centrations [mmol/m3] have been fixed to the following:tSO4 = 0.6 / 0.3 / 0.1 (for domain = 4 / 3 / 2), tNH3 = 0.4,and tNO3 = 0.1. The prefix t denotes total gas and aerosolconcentration; the temperature has been fixed to T = 20�C.[34] In the light of global modeling we focus here on the

gas/aerosol partitioning of nitrate and ammonium, thedegree of neutralization, the total (nonwater) particulatematter, the aerosol associated water mass, and the approxi-mated radius increase due to water uptake (Figure 1).[35] The results of this model comparison are shown in

Figure 3 as a function of RH (10–95%) for each domain.The comparison shows that nitrate and ammonium partition-ing is predicted for all domains by all models and all ACCMversions with differences smaller than a few percent. Whilenitric acid partitions almost completely into nitrate only atrelative high humidities above 90% for the case of surplusammonia (domain 2), ammonia remains in the aerosol phaseindependently from RH if sulfate is not completely neutral-

ized by ammonium (domain 3 and 4). Only for domain 2,ammonia partitioning occurs by RH dependent ammoniumnitrate formation. Note that EQSAM follows the referencemodel ISORROPIA very well, which is presently the mostsophisticated model in terms of accuracy and computationalperformance. MARS, on the other hand, is the simplest of allequilibrium models and shows the largest deviations fromother models predictions. Remarkably, the parameterizedversion of ISORROPIA (ISORROPIA-P) shows also (asEQSAM) only small deviations from the original version.Note that the iterative structure of the ISORROPIA-P waskept unchanged to minimize errors associated with otheraspects than the ACCM.[36] According to Figure 3 the relatively strongest differ-

ences between all EQMs occur for the sulfate neutral case(domain 2). Nevertheless, differences are overall rather smalland EQSAM follows closely ISORROPIA for all cases. Notethat the uptake of aerosol water is strongest for the ‘sulfatevery rich case’ (domain 4), because nonneutralized sulfuricacid exists in the aerosol phase. This causes water uptake at arelative humidity even below 10%, since sulfuric acid isalways associated with water. For the other cases (domain 2and 3) water uptake takes place only if the relative humidityexceeds the lowest deliquescence relative humidity of thecorresponding salt solution, or the lower limit of the mutualdeliquescence relative humidity range. Although wateruptake is strong for domain 4, the radius increase due towater uptake is not. The reason is that the radius increase isproportional to the ratio of total particulate matter (includingaerosol water) and the total dry particulate matter (approx-imately to a power of 1/3 to obtain the radius from mass).Because the aerosol water mass is proportional to the dryparticulate matter, the main differences in water uptake andradius increase are confined to the relative humidity rangewhere solid particles can form. However, this is also thehumidity range where the main differences between allmodels occur. The aqueous phase is determined for the gas/liquid/solid equilibrium calculations by the relative humidityof deliquescence (section 2.3), for which different assump-tions are used for different EQMs. For instance, ISORROPIAuses temperature dependent mutual deliquescence ranges forsalt mixtures, which are also used in EQSAM (according tothe aerosol system considered). All other models use RHDvalues only for individual salts. RHD values for individualsalts generally lead to the formation of solid particles at ahigher RH. Additionally, MARS and SEQUILIB do notaccount for the temperature dependence of RHDs.[37] We thus note that the difference in deliquescence

humidity used in various EQMs is the main cause of devia-tions associated with the prediction of aerosol water. To amuch lesser extent, differences in the prediction of the soluteconcentration (at a given RH near the RHD) are responsiblefor the fact that some EQMs predict solids while others donot. Similarly, any failure in predicting solids due to differ-ences in the activity coefficient calculation or the numericalstability in reaching convergence in iterative processes isnegligible compared to the failure caused by differences inthe RHDs. This is demonstrated by the comparison ofISORROPIA-P with the other EQMs (Figure 3).[38] Because the main differences in the gas/aerosol

partitioning calculations are associated with the assumptionson the deliquescence relative humidity, another source of


Figure 3. Comparison of equilibrium calculations of EQSAM with various EQMs in use (ISORROPIA,SEQUILIB, MARS, SCAPEa-PM, SCAPEb-KM, SCAPEb-BM), including a parameterized EQM(ISORROPIA-P). Results are shown for domains 2–4 as a function of relative humidity (10–95%).Figure 3a shows the percentage for nitrate- (top) and ammonium- (middle) partitioning, and the degree ofneutralization (bottom), and Figure 3b shows the total particulate matter (top), the aerosol-associatedwater (middle), and the radius increase due to water uptake (bottom).


uncertainty needs to be discussed and quantified, i.e., thehysteresis effect of aerosols. Many salt solutes exhibithysteresis, which is that they do not crystallize at the sameRH at which they deliquesce. Instead, these compoundsremain in a metastable supersaturated aqueous phase, rather

than forming solids once they have been wet. Especially forhygroscopic particles at midlatitudes or marine environ-ments, where the relative humidity generally exceeds thedeliquescence points of the salts of interests (at least atnight), the hysteresis effect of aerosols can become impor-



tant. Therefore we have quantified in Figure 4 the differencesfor the aerosol water associated with metastable aerosols(gas/liquid partitioning), in addition to the aerosol watershown in Figure 3, where the aerosol water is associated withthe full gas/liquid/solid partitioning. However, in Figure 4only the results of EQSAM and ISORROPIA are included;all other EQMs do not account for metastable aerosols. Themain result is that differences between both models arenegligible compared to the differences in the total amountof aerosol water associated with either the gas/liquid/solid orthe gas/liquid partitioning. Furthermore, also the absolutedifference between both models is acceptably small forglobal modeling applications (with respect to other modeluncertainties, e.g., those associated with the parameteriza-tion of the boundary layer; see discussion in M2002). Notethat the maximum values in Figure 4 are the same as beforebut they have been truncated to highlight the differences thatonly occur at water masses below 50 mg/m3 (due to the effectof deliquescence).

3.2. Global Offline Calculations

[39] In this section we extend the previous comparison toa wide range of realistic atmospheric conditions so that wecan quantify the relative errors for different aerosol concen-trations and temperatures.[40] We now apply all EQMs to global fields of a CTM

(TM3) and meteorological fields from ECMWF, focusingon monthly mean values for January 1997 at the surface.The global chemistry fields were calculated on a horizontal10� 7.5� grid, the TM3 version used is described byHouweling et al. [1998] and Lelieveld and Dentener [2000],and in more detail by M2002.[41] Figure 5 shows the results of the model comparison

for the gas/liquid/solid equilibrium calculations. We focuson the mole ratio of aerosol nitrate and sulfate (upper

panels), the aerosol radius increase owing to water uptake(middle panels), and the nitrate partitioning (lower panels).Each set is shown for a rural location (10�E, 50�N) inGermany (left), a regional average over Europe (middle),and the global average (right); for January 1997 at surfacelevel. The EQMs are shown along the x axis, i.e., from left toright EQSAM, ISORROPIA, SEQUILIB, MARS, SCAPEa-PM, SCAPEb-KM, SCAPEb-BM, ISORROPIA-P (01Z to08Z, respectively). The main results are that the deviationsassociated with aerosol predictions of EQSAM are smallrelative to ISORROPIA and other EQMs. The same is truefor the parameterized version of ISORROPIA (02Z vs. 08Z).The effect on the gas/aerosol partitioning is rather small forall models, and in particular smaller than the differencesassociated with different EQMs. Accordingly, the two differ-ent iterative ACCMs applied to the SCAPEb EQM yieldsconsistent results, indicated by the comparison of the resultsof SCAPEb-KM (06Z) and SCAPEb-BM (07Z). In addition,the comparison of the results for the location in Germanywith the results of the regional and global average furthershows that the differences between EQSAM and ISORRO-PIA are smallest for the global average. This suggests thatthe results of both models are also in good agreement (oreven better) for locations outside of Europe, where inparticular semivolatile ammonium nitrate is less predomi-nant. However, the differences for the single grid box arealso small; about a per mille for the nitrate/sulfate mole ratioand the nitrate partitioning, and a few percent for the radiusincrease. The absolute values for this location furthermoreindicate that nitric acid is predicted to partition almostcompletely into the aerosol phase under northern hemi-spheric winter conditions, so that the aerosol nitrate con-centrations are twice as high as the sulfate concentrations.On the regional scale, the mean nitrate concentration iscomparable to the sulfate concentration, which is associated

Figure 4. Comparison of equilibrium calculations shown in Figure 3 (middle right panel) supplementedfor EQSAM and ISORROPIA by the results of the aerosol associated water based on metastable aerosols(indicated by triangles and squares, respectively), and based on solid aerosols (indicated by solid lineswith triangles and squares, respectively). Note that the maximum values have been truncated since theyremained unchanged.


with considerable nitrate partitioning of about 70% into theaerosols over Europe. Although these values are much lessfor the global average, approximately 20% for the nitratepartitioning and about 33% for the nitrate/sulfate mole ratio,they indicate that gas/aerosol partitioning is important on aglobal scale.[42] To investigate these differences in greater detail, we

have plotted in Figure 6 the global patterns (upper panel) aspredicted by EQSAM (color scale) and ISORROPIA (con-tour) for the total particulate matter; the middle panel showsthe corresponding differences of EQSAM-ISORROPIA,and the lower panel the scatterplot, including the grid boxarea-weighted (left) and nonweighted (right) correlationcoefficients for January 1997 at the surface. Note that thegrid box area-weighted correlation coefficient accounts forthe decrease of the grid box area with increasing latitude. Ifboth differ, this indicates that they include nonzero values atnorthern latitudes, e.g., that gas/aerosol partitioning takesplace; the absolute difference additionally indicates therelative importance of the northern latitudes. In accord withthe previous findings, differences are overall rather small,and limited to locations with rather extreme atmosphericconditions, i.e., a high temperature and low relative humid-ity. Under these conditions hygroscopic aerosols favor thesolid phase. The prediction of crystalline particles howeverstrongly depends on the assumptions about the deliques-cence points, as mentioned before. Uncertainties associatedwith the deliquescence points explain most of the differ-ences, which are basically limited to the occurrence ofsolids. However, for these dry cases the solute activity isusually high and likely to exceed the range for which theactivity coefficient calculation methods are tested (i.e., 30Mfor the Pitzer method). The results of ISORROPIA and theother EQMs may also be questioned for these conditions.[43] Since the differences are mostly associated with the

calculations of solids, and because of the hysteresis effect ofaerosols, it is probably appropriate to limit the gas/aerosolcalculations in global modeling to metastable aerosols,rather than to include the calculation of solids withoutconsidering hysteresis, unless hysteresis is explicitly calcu-lated (to include solids alone seems insufficient according tothe discussion above). For this reason, Adams et al. [1999]also followed this approach, focusing on metastable aerosolcalculations with a general circulation model (GCM).Adams et al. [1999] nevertheless devote roughly threequarters of the total processor time to the thermodynamicequilibrium calculations by ISORROPIA. This would evenhave been much more if they would have accounted for thefull gas/liquid/solid partitioning calculations.3.2.1. CPU Time[44] Table 3 shows the CPU time used by all EQMs

applied at two different platforms. The parameterizedversion of ISORROPIA is included, although the iterativestructure was not adapted to the activity coefficient

parameterizations and therefore not optimized with respectto the CPU time consumption (to minimize errors notrelated to this modification as mentioned before). Never-theless, ISORROPIA-P also shows a gain in computingtime, which is caused by the fact that fewer iterations areneeded to reach convergence. However, the largest gain incomputational performance is achieved with EQSAM. Thecomputational burden is about 1–2 orders of magnitudeless compared to the nonparameterized EQMs. The largegain in CPU-time results from the noniterative calculationof the gas/liquid/solid aerosol partitioning with EQSAM;this model version is not even numerically optimized forthis particular application. Note that the average CPU-timeby iterative models strongly depends on the atmosphericconditions, i.e., the number of cases where solids must becalculated. For the same reason the average CPU-timedepends on the assumptions made on the gas/aerosol parti-tioning (metastable or gas/liquid/solid aerosols). Moreover,also the hardware architecture on which the computations areperformed can have a strong impact on the computationalburden, as shown by the results in Table 3; the left side showsthe computing times for the calculations performed on apersonal computer (PowerBook G3) with a 250MHz CPUrunning under Mac OS 9, and the right side for the samecalculations performed on a supercomputer infrastructure ofa CRAY-C916/121024. The columns show the computa-tional burden, i.e., CPU seconds for 8640 grid boxes (36 24 horizontal grids, 10 vertical levels). From the comparisonof the right and left columns of Table 3 it becomes apparentthat the EQMs in use are not efficient for the presentgeneration of supercomputers with vector architectures, suchas the CRAY-C916. EQMs are purely scalar and too complexfor vectorization. Global modeling however largely takesplace on supercomputers because of the large amount of dataneeded and produced. The use of vector architecturesstrongly limits the application of EQMs. Although all EQMsin use were developed for incorporation in larger air-qualitymodels, they are not numerically efficient because the gas/aerosol partitioning is solved iteratively. This limitation willremain also for the next generation supercomputers, whetherthey are based on scalar or vector architectures. Because purescalar models generally perform poorly on a vector com-puter, it is not very surprising that even a personal computeroutperforms a supercomputer (by a factor of 4 or so for scalarmodels), which is true even for the most efficiently opti-mized EQM (ISORROPIA). The parameterized modelEQSAM, on the other hand, performs better on a vectormachine, i.e., by a factor of 3 or so for the version includingsolids, and up to 3 orders of magnitudes for the reducedversion excluding solids as compared to ISORROPIA (dueto vectorization).3.2.2. CPU Burden of a 1 Year Simulation[45] For a one year integration on a coarse grid resolution

(10� 7.5�) with 19 vertical levels and a 2 hours time step

Figure 5. (opposite) Comparison of equilibrium calculations with EQSAM and various EQMs in use, including aparameterized EQM. Each point of the x axis represents a model, i.e., from left to right EQSAM, ISORROPIA, SEQUILIB,MARS, SCAPEa-PM, SCAPEb-KM, SCAPEb-BM, ISORROPIA-P (01Z to 08Z, respectively). Shown are the mole ratioof aerosol nitrate and sulfate (upper panels), the aerosol radius increase due to water uptake (middle panels), and the nitratepartitioning [%] (lower panels); for a rural location in Germany (left), a regional average over Europe (middle), and theglobal average (right); for January 1997 at the surface.


Figure 6. Comparison of EQSAM with ISORROPIA for the total particulate matter. The upper panelsshow the global pattern (color scale EQSAM, contour ISORROPIA), the middle panels show thecorresponding difference between EQSAM and ISORROPIA, and the lower panels show the scatterplot,including the regionally weighted and unweighted correlation coefficients; for January 1997 at thesurface.


for the chemistry/equilibrium calculations (gas/liquid/solidpartitioning), which already yields a total number of calls tothe equilibrium routine of the order of many millions(71.90208E+06), the computational burden for EQSAMon the CRAY would be about 43 min. Correspondingly,139 hours would be needed for ISORROPIA, and of about 6hours for the parameterized version (ISORROPIA-P). Notethat these numbers refer to the lower limit. The actualintegration by an iterative model usually consumes morecomputing time due to the dependence of the solid calcu-lations on the atmospheric condition as mentioned before(these numbers were based on monthly mean values forJanuary 1997). However, the same calculations with theEQM SCAPEb would add up to 400 hours CPU-time,therefore, ISORROPIA is already regarded as computation-ally efficient (and accurate). Nevertheless, our comparisonshows that for global applications alternatives are desiredand feasible (e.g., EQSAM), despite increasing computingpower. Especially for climate calculations, where hundredsof years of simulations must be performed, parameteriza-tions are needed. Even if the gas/aerosol calculations arelimited to metastable aerosols the computational burden forISORROPIA remains too high, i.e., of about 22.5 hours peryear. This is a factor 10 more compared to the entirecomputing time consumption of the atmospheric/chemistrytransport model for this resolution. TM3, for example, usesabout 650 CPU seconds per month or �2 hours per year,without thermodynamic gas/aerosol calculations. Thereduced version of EQSAM, without the option for solidcalculation, only adds 3.5 CPU seconds per month.

3.3. Global Online Calculations

[46] In the previous subsections we have shown that theresults of the aerosol calculations based on EQSAM arewell within the range of the results of various EQMs in use.To extend the comparison to global modeling and to test thestability of the previous results by integration over time andvarious altitudes, we further compare the results of EQSAMand ISORROPIA in so-called online calculations. Thus bothequilibrium models were incorporated into the atmospheric/

chemistry transport model TM3 (introduced in section 3.2and described in more detail in M2002). Contrary to Adamset al. [1999], who have coupled ISORROPIA to a GCM byusing prescribed HNO3-fields from a CTM, we calculate thegas/aerosol equilibrium of the ammonium/sulfate/nitrate/water-system online with the gas-phase and cloud chem-istry. This therefore includes feedback of nitrate and ammo-nium partitioning with the gas-phase chemistry. The onlinecalculations are performed on a 10� 7.5� horizontalresolution, with 19 vertical levels and a 2 hour time stepfor gas- and aerosol phase chemistry. Following Adams etal. [1999] we only consider metastable aerosols.[47] Since the largest uncertainties, which are associated

with thermodynamic gas/aerosol calculations, arise athigher temperatures, we focus on a comparison of EQSAMand ISORROPIA for July 1997. Figure 7 shows themonthly mean total direct aerosol radiative forcing at thetop of the atmosphere, based on the vertical integral ofammonium, sulfate, nitrate and water. The upper panelshows a comparison of the global aerosol patterns (shadedEQSAM, contour ISORROPIA), the middle panel showsthe corresponding difference (EQSAM – ISORROPIA),and the lower panel shows the scatter, including the region-ally weighted and unweighted correlation coefficients.Figure 8 additionally shows scatterplots (similar to Figure7, lower panel) for the total dry aerosol mass, the aerosolassociated water mass (burdens); aerosol sulfate [mgS],nitrate [mgN], and ammonium [mgN] (burdens); the residualgases nitric acid [mgN] and ammonia [mgN] (burdens); thecolumn aerosol radiative forcing of sulfate and nitrate;including the regionally weighted correlation coefficients.Figure 9 shows the time evolution of the aerosol radiativeforcing of transported nitrate for a rural location in Germany(top), a regional average over Europe (middle), and theglobal average (bottom). In addition to the results ofEQSAM, the differences between these results and thoseobtained by ISORROPIA (accordingly coupled to TM3) areincluded in each panel (EQSAM - ISORROPIA).[48] The radiative forcing is calculated online with the

method of Van Dorland et al. [1997] in TM3. However, werefrain from a detailed discussion on the climate forcing byaerosols. Rather we use the aerosol forcing to quantifyrelative differences associated with the aerosol calculations.The forcing provides an integrated difference, which isbased on various aerosol properties (column particulatematter of aerosol ammonium, sulfate, nitrate, water). Theseproperties interact with meteorological and geophysicalparameters such as cloud cover, solar constant, solar-zenithangle, i.e., effective daylight and albedo. Since the directradiative forcing includes the uncertainties that are associ-ated with the calculation of aerosol water, and because it ismost important for climate modeling, we use it as aparameter in our online comparison.[49] In general, relative errors introduced by the use of

the computationally efficient gas/aerosol model EQSAMare also small for the online equilibrium calculations,compared to the reference model ISORROPIA. This isparticularly true for the monthly mean total direct aerosolradiative forcing, which is shown in Figure 7. While thecontour lines (values of ISORROPIA) nicely enclose thecolor-coded areas (values of EQSAM), the absolute differ-ences are also small; differences are generally less than

Table 3. CPU Times

CPU Seconds (Burdena)

Mac Pb-G3 (250 MHz) Cray-C916

Gas/Liquid/Solid Aerosol PartitioningEQSAM 0.94 0.31ISORROPIA 14 60SEQUILIB 105 64MARS 8 12SCAPEa-PM 52 49SCAPEb-KM 40 181SCAPEb-BM 38 172ISORROPIA-P 1.5 2.7

Metastable AerosolsEQSAM 0.93 0.28EQSAMb 0.88 0.0048ISORROPIAc 3.1 9.7ISORROPIA-P 1.7 3.3aNumber of gridboxes = 8640 (values are rounded).bReduced version of EQSAM (code without solids for better vectoriza-

tion).cDouble precision was omitted on the CRAY for ISORROPIA

(ISORROPIA-P) to achieve better performance.


Figure 7. Online equilibrium calculations (metastable aerosols). Comparison of EQSAM withISORROPIA, showing the monthly mean total direct aerosol radiative forcing for July 1997. The upperpanel shows the global pattern (color scale EQSAM, contour ISORROPIA), the middle panel shows thecorresponding difference between EQSAM and ISORROPIA, and the lower panel shows the scatterplot,including the regionally weighted and unweighted correlation coefficients.


10%, with a low scatter and a high correlation coefficient(nearly one). Furthermore, Figure 8 demonstrates that differ-ences are also small for other aerosol properties. Therelatively largest uncertainty is associated with the calcu-lation of aerosol nitrate (Figure 8), as a result of the gas/aerosol partitioning. In contrast, differences are smallest foraerosol sulfate, because sulfate is treated as nonvolatile withthe consequence that total sulfate [mgS] remains unchangedby the gas/aerosol calculations. Only the absolute sulfatemass can be affected by the equilibrium calculations (due todifferences in the neutralization by ammonium), which cansubsequently result in erroneous gaseous ammonia predic-tion and therefore of aerosol nitrate as well as the totalaerosol mass and the aerosol associated water mass (which

depend on the absolute mass). Nevertheless, the agreementis rather good also for these latter aerosol properties.[50] Note that the deviations are largest in July. In winter

the overall temperature is lower and the relative humidityhigher, which favors the formation of ammonium nitrate.Ammonium nitrate is then more stable, with HNO3 oftencompletely neutralized: all nitric acid simply remains in theaerosol phase in form of nitrate, without gas/aerosol parti-tioning (also see the discussion of Figure 2b in M2002).[51] To focus on the largest uncertainties, which are

associated with the calculation of aerosol nitrate, we showin Figure 9 the time variability of the direct radiative forcingof nitrate, based on EQSAM, and the differences relative toISORROPIA. The differences are largest for the rural

Figure 8. Online equilibrium calculations (metastable aerosols). EQSAM versus ISORROPIA for July1997 (monthly mean). Scatterplots are shown for the total dry aerosol mass, the aerosol associated watermass (burdens); the aerosol sulfate [mgS], nitrate [mgN], and ammonium [mgN] (burdens); the residualgases nitric acid [mgN], and ammonia [mgN] (burdens); the column aerosol radiative forcing of sulfate,and nitrate; including the regionally weighted correlation coefficients.


Figure 9. Online equilibrium calculations (metastable aerosols). Time series for July 1997 of thecolumn aerosol radiative forcing of nitrate for a rural location in Germany (top), a regional average overEurope (middle), and the global average (bottom). The results are obtained by EQSAM (black lines). Inaddition, the differences between these results and those obtained by using ISORROPIA are included ineach panel and marked in light gray (EQSAM - ISORROPIA). Note that the differences are very smalland therefore difficult to distinguish from the minimum values.


location in Germany, although they are overall quite small.Note that the differences are negligible and therefore diffi-cult to distinguish from the minimum values that occur atnight, therefore being small as well.[52] As mentioned above, the main source of uncertain-

ties can be attributed to differences in the nitrate partition-ing, which in turn is strongest in summer for this particularlocation. In general, the relative difference between EQSAMand ISORROPIA decreases as the area increases over whichthe averaging is performed, reflecting the nonuniform spa-tial distribution of the aerosol load. The strongest forcing ofthese anthropogenic species occurs mainly over land, i.e.,over Europe, northeast America and Asia (Figure 7). A moredetailed discussion of the space-time variability is presentedin M2002.[53] The online comparison thus demonstrates that the

relative errors associated with the computationally efficientgas/aerosol equilibrium calculations are practically negli-gible for online calculations. Particularly in the light ofother model uncertainties the results of the online modelcomparison of our computationally efficient gas/aerosolmodel EQSAM, as compared to the EQM ISORROPIA,are satisfactory for global modeling applications (see alsothe discussion in M2002).

4. Discussion and Conclusions

[54] We have introduced a simplified approach to calcu-late the gas/aerosol partitioning on a global scale. Ourapproach is based on the activity coefficient - RH relation-ship, which enables the solution of the nonlinear algebraicequilibrium equations noniteratively. Based on our approachwe present a computationally efficient gas/aerosol model,EQSAM (Equilibrium Simplified Aerosol Model). TheEQSAM thermodynamic framework is similar to the currentstate-of-the-art gas/aerosol models, but uses parameteriza-tions for the calculation of the aerosol activity, includingactivity coefficients and water. These nonideal solutionproperties are usually derived iteratively in present EQMsby using time consuming numerical schemes. We haveshown that the activity coefficient - RH relationship canbe used to approximate the equilibrium calculations. Severalstudies have previously addressed the importance of aerosolhygroscopic growth [e.g., Low, 1969a, 1969b; Winkler,1973, 1988; Hanel, 1976; Tang, 1980; Tang and Munkel-witz, 1994], but none of these were aimed at the develop-ment of a gas/aerosol model for global modeling.Nevertheless, there are studies that have explored the issueof the ammonium-sulfate-nitrate aerosol climate forcing ona global scale, for instance, those of Van Dorland et al.[1997] and Adams et al. [1999, 2001]. The former work,however, did not include an explicit determination of theaerosol composition, while in the latter an EQM wascoupled to a climate model by using offline calculatedchemistry fields of a CTM. We extend these studies byproviding a method that can easily be used for the inter-active (online) determination of the aerosol compositionwithin global chemistry-transport and climate models.Although our method is based on equilibrium, for globalmodeling this assumption is reasonable. The equilibrationtimes of relevant aerosol particles are of the order ofminutes and thus much shorter than the typical time steps

of 0.5–2 hours applied in CTM or GCMs to calculatechemistry and transport.[55] To demonstrate that our simplified approach used for

EQSAM yields consistent results with other EQMs, wehave presented an equilibrium model comparison, includinga state-of-the-art EQM (ISORROPIA) and a parameterizedversion of ISORROPIA. For the latter we have applied theparameterization used in EQSAM for the activity coefficientcalculation, by replacing the original iterative activity coef-ficient calculation method by our noniterative one, i.e.,equation (3). The model comparison shows that the mostimportant aerosol properties for global modeling, such asthe total particulate matter and the aerosol associated watermass, can be sufficiently accurately reproduced with respectto other model uncertainties. The results of the box-modelcomparison further showed that the relative errors (e.g., thedifferences between EQSAM and ISORROPIA) are smallcompared to the assumptions on the aerosol state. Especiallythe calculations of aerosol associated water, important forthe aerosol radiative forcing simulations in climate studies,show only small differences, being much smaller than forother model assumptions. The main differences betweenvarious EQMs are associated with the determination of theaqueous phase. The reason is that this strongly depends onthe assumptions about deliquescence behavior of varioussalt compounds and salt mixtures. For instance, the deli-quescence humidity of multicomponent salts is consideredin ISORROPIA and EQSAM, while many EQMs onlyconsider the deliquescence humidity of single salt com-pounds.[56] We emphasize that the application of any EQM in

global atmospheric studies is associated with considerableuncertainties. One should keep in mind that these models(including EQSAM) are based on simplifying assumptions,including thermodynamical equilibrium, which implies thatthe water activity of the aerosol equals the ambient relativehumidity. The so-called ZSR relation is a consequence,simply assuming that the total aerosol associated water isthe sum of the water fractions of all single-solute solutions.Furthermore, multicomponent aerosols are assumed to beinternally mixed. To introduce and demonstrate the activitycoefficient - RH relation, we only addressed submicronparticles (bulk approach). The above assumptions are validfor inorganic salt compounds if the aerosol modeling islimited to the ammonium-sulfate-nitrate-water system.These assumptions are, however, no longer valid if particlessuch as sea-salt or mineral dust are considered. Especiallythe latter particles often provide a pre-existing solid corethat favors condensation; heterogeneous nucleation is gen-erally thermodynamically favored. They therefore can diur-nally redistribute (semi-) volatile compounds such asammonium nitrate from smaller particles to larger ones(those including a solid core) as the relative humiditydecreases [Wexler and Seinfeld, 1990; Wexler and Potuku-chi, 1998]. The equilibrium approach might fail for coarsemode particles, such as sea salt and mineral dust. Theequilibration times are considerable longer for larger par-ticles (because of the smaller surface-to-volume ratio),possibly exceeding the timescales over which transportand chemistry are calculated. Especially in remote and lesspolluted regions, or regions exposed to cold conditions, theequilibrium assumption may not be satisfied. In these


locations, nucleation might become important as well,which is not considered in EQMs. However, such aerosoldynamical processes can be included separately in theoverall aerosol module of which the EQM is one compo-nent.[57] Finally, the limitations of our approach should be

evaluated in the light of assumptions in atmospheric chem-istry and climate models (CTM or GCM). The efforts toachieve high accuracy for the gas/aerosol calculationsshould balance with the efforts to describe other processesin the CTM or GCM. In fact, other process descriptions maydominate the uncertainties, or affect the assumptions onwhich the state-of-the-art model is based so that they maynot hold under all circumstances.[58] For instance, we show in M2002 that the uncertain-

ties in predicting the variability of the planetary boundaryaffect the aerosol concentrations much more strongly thanthe uncertainties associated with the activity coefficientparameterization.

Appendix A: Theoretical Interpretationof Equation (3)

[59] For the interested reader we present here a theoreticalinterpretation of our empirically derived equation (3).Although we are able to theoretically derive in the followingthe relationship between activity coefficients and the rela-tive humidity, we are not able to clearly determine therequired parameters. We can only make an interpretation ofthese parameters in the light of the models used forvalidation; state-off-the-art EQMs such as ISORROPIAare subsequently needed to fit these parameters. However,one should realize that these models are based on simplify-ing assumptions. In particular, the methods used to predictactivity coefficients are empirical or semiempirical; refer-ences are given in section 2.5.[60] Equilibrium condition:�G= 0, G�U+ p �V�T � S.

A differential change yields:

d G ¼ d Uþ p � d Vþ V � d p� T � d S� S � d T: ðA1Þ

Eliminating dU with the Gibbs-relation (dT = 0 and dp = 0)gives:

d GjT;p ¼ �i mi � d ni ¼ 0: ðA2Þ

mi and ni are the chemical potential and number of moles ofcompound i, respectively.[61] For one chemical reaction, the amount of each

component is ni = nio + ni � e, where ni

o is the initialamount of each compound; ni is their stoichiometric coef-ficients and e the reaction coordinate.[62] Taking the derivative, dn i = ni � de, when substituted

into equation (A2) gives

�i ni � mi ¼ 0: ðA3Þ

The chemical potentials mi of each compound can beexpressed in terms of their activities ai, i.e., mi = mi

o + R � T �ln ai. mi

o is the chemical potential at a standard state, which

is equal to the partial Gibbs free energy gio, R is the

universal gas constant. Substitution into equation (A3) gives

�ini � gio þ R � T � �ini � ln ai ¼ 0 ðA4Þ

Equation (A4) yields the equilibrium constant K (section2.4) of a given equilibrium reaction at a given temperature.Neglecting the temperature dependency, i.e., upon rearran-ging and exponentiation and expressing the sum of thelogarithm as their product K ¼ �ia

nii .

[63] For atmospheric aerosols in thermodynamic equili-brium with the ambient relative humidity (RH), the follow-ing reactions are of interest, which subsequently lead to therelationship between activity coefficients and RH (as usedin EQSAM).

ðR1Þ ng1 � ðNH3Þg þ ng2 � ðHNO3Þg , ns � ðNH4NO3Þs

ðR2Þ ns � ðNH4NO3Þs þ nsw � ðH2OÞg , n�s � ðNH4NO3Þaqþ nsw � H2Oð Þ

ðR3Þ 1 � ðNH4NO3Þaq þ n�w � ðH2OÞg , nþs � ðNHþ4 Þaq þ n�s

� ðNO�3 Þaq þ n�w � ðH2OÞaq

R1 is determined by K NH4NO3ð Þs, the amount of aerosol waterdepends for R2 on the number of salt molecules and RH(with RH > RHDNH4

NO3), and for R3, additionally onthe number of moles into which each mole of theelectrolyte dissociates. R3 actually depends on thedissociation constant, for ammonium nitrate [see, e.g.,Mozurkewich, 1993]. The stoichiometric coefficients areng1 = ng2 = ns = ns

+ = ns�= 1, ns

± = ns+ + ns

�= 2,while nw

s and nw± are unknown.

[64] For a binary solution of an undissociated electrolyte(R2), equation (A4) yields

vs � gos þ nsw þ gs;ow þ R � T � vs � ln as þ R � T � nsw ln asw ¼ 0; ðA5Þ

where gso and gw

s,o represent the Gibbs free energies at thestandard state temperature T� (and pressure p�) of 1 mole ofsolute and of 1 mole of water, respectively. The activities asand aw

s denote the electrolyte and water, and ns and nws their

stoichiometric coefficients, respectively.[65] Equation (A5) can be rewritten to yield (upon

exponentiation)

avss ¼ as�vws

w � exp½�ðvs � gos þ nsw � gs;ow Þ�=ðR � TÞ�: ðA6Þ

[66] For the formation of a pure compound (then withoutthe terms denoted by the index w) the exponent of equation(A6) would yield the equilibrium constant K, e.g., of pureammonium nitrate (R1). However, if the reaction takesplace in the aqueous phase, additionally a certain numberof moles of water will condense (R2). This requires arelation to exist between the differential changes of themoles of solute and water, i.e., between vs and nw

s . Fur-thermore, in this case the exponent of equation (A6)disappears, i.e., is unity.


[67] The required condition that the exponent of equation(A6) disappears is

vs � gos þ nsw � gs;ow ¼ 0; ðA7aÞ

so that the relation between the stoichiometric coefficients,according to (R2), is

�nsw ¼ vs � gos=gs;ow ¼ vs � Ns

w; ðA7bÞ

or, if the electrolyte is dissociated (R3), i.e.,

�n�w ¼ v�s � g�;os =g�;o

w ¼ v�s � N�w ; ðA7cÞ

with Nw± = vs � Nw

s � k±.[68] The relation between the stoichiometric coefficients

(equation (A7b)) reduces equation (A6) with as = gs � ms to

gs �ms ¼ awsNw

s

: ðA8aÞ

Similarly, for dissociation ðas ) a�v�ss ¼ a

�vsww ¼ a

v�s �N�w

w

�,

according to (R3) and equation (7c), i.e.,

g�s �m�s

� �n�s ¼ asn�s �N�ww ðA8bÞ

Thedefinitionof molality isms=55.51 �ns/nws=55.51�xs/xws ,and for binary solutions xs + xw

s = 1. Here ns and nws are

the number of moles of the solute (undissociated) andwater, respectively; xs, xw

s denote the correspondingmole fractions. They are defined as xs = ns/(ns+nw

s ), andxws = nw

s /(ns + nws ). Taking dissociation into account ns

becomes ns± = ns � ns± and nws becomes nw

± = nws nw

±, so that ms =

55.51 � ns/nws becomes m�v�ss ¼ 55:51 � n�s =v�s =n�w � v�w

� �n�s .Using the relation given by equation (A7c), i.e., m�n�s

s ¼55:51 � n�s =�

n�w � N�wÞ

n�s , or in terms of mole fractions, i.e.,

m�n�ss ¼ 55:51 � x�s =x�w � N�

w

� �n�s . The sum of the molefraction is unity and the mole fraction of the solute can beexpressed by the mole fraction of the aerosol water, i.e., xs

± /xw± = (1�xw

±) � fw± /aw± = (fw± � fw

± � xw±)/aw± = (fw± � aw

±)/aw± =

(fw±/aw

±�1), so that m�v�ss ¼ ½N�

w � 55:51 � ðf�w=aw� � 1Þ�v�s .

Note that we have used the definition of the activities for thesolute and water on the mole fraction scale, i.e., as

± = xs± � fs±

and aw± = xw

± � fw±; where fs± and fw± denote the rational activity

coefficients of the dissociated solute and water, respectively.Substitution of the mean molality m�v�s

s of the dissociatedion-pair into equation (A8b) gives

g�n�ss ¼ aw

n�s �N�w=½N�

w � 55:51 � ðf�w=a�w � 1Þ�n�s : ðA9aÞ

gs± denotes the mean activity coefficient of the dissociated

salt solute. Since activity coefficients are defined to correctfor ion-ion interactions, they depend on the charge densityof the solution. The charge density depends in turn, for agiven solute, on the total charge of the ions zs

±, and theamount of water molecules jvw±j, which are associated withthe formation and dissociation of the electrolyte. Wetherefore express the mean charge density rs

± as rs± = zs

± /jvw±; j with jvw±j the absolute value of vw

±. The mean ioncharge zs

± denotes the charge of each ion, i.e., zs± = (�s

+zs+ +

�s�zs

�) = (vs+ � zs++ vs

�.zs�). Thus, considering dissociation

the activity coefficient of the undissociated salt solute gsbecomes g

�vþs =rþs

s � g�n�s =r�s

s ¼ g�v�s =r

�s

s so that equation (9a)yields

g�s ¼ awr�s �N�

w=½N�w � 55:51 � ðf�w =a�w � 1Þ�r

�s : ðA9bÞ

With (A7c) rs± = zs

±/jvw±j = zs±/(vs

± � Nw±) and the water

activity aw expressed in terms of the fractional relativehumidity (RH), we can finally write the mean activitycoefficient gs

± as a function of RH, i.e.,

g�s ¼ rhz=½N�w � 55:51 � ð1=rh� 1Þ�z=N

�w ; ðA10Þ

with z = zs±/ns

±, and ns± = ns

+ + ns�, Nw

± = ns. Nw.

s k±, Nws = k±, and

z+ = e+e+ and z� = e�

e�; where e+ and e� denote the positive

and negative ion charge, respectively. With the above

definition of z = zs±/vs

± and the ion pair charge zs±, we get

for major inorganic aerosol compounds, given in Table 1 for

various EQMs (e.g., ISORROPIA, SCAPE), the following

values for z : z = 1 = (1�11 + 1�11)/2 for NH4NO3, z = 1.6 =

(4�11 + 1�22)/5 for (NH4)3H(SO4)2, z = 2 = (2�11 + 1�22)/3 for(NH4)2SO4, and z = 4 = (1�22 + 1�22)/2 for CaSO4.[69] If we assume the following values for k±, fw, ns, i.e.,

k± = 2, according to an ±-ion-pair; fw = 1, because g s± varies

(both parameters account for the same solution-non ideal-ity); ns = 1, because of stoichiometry (condensation/evap-oration of water vapor applies to each mole of the saltsolute), we obtain for Nw

± a value of 4. If we further makethe interpretation that the parameter x in equation (3) is x =Nw

±/z, then equation (A10) yields consistent results withequation (3) in the light of the overall general functionaldependency of gs

± on RH.

[70] Acknowledgments. This work was performed at the Instituutvoor Marine en Atmosferisch Onderzoek (IMAU), at the UniversiteitUtrecht, Netherlands. It was funded by the European Community, partlyby a bursary according to Article 6 of the Fixed Contribution Contract forTraining Through Research, ENV4-CT96-5036 (DG 12-ASAL), and partlyby the EU-funded project SINDICATE (Study of the Indirect and DirectClimate Influences of Anthropogenic Trace Gas Emissions). We would liketo acknowledge G. Carmichael and Y. P. Kim for kindly providing thethermodynamic model SCAPE, and we thank J. Seinfeld and M. Steiger forvaluable comments on previous versions of the manuscript.

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��F. Dentener, Institute for Environment and Sustainability, Joint Research

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Mellon University, Pittsburgh, PA 15213, USA. ([email protected])


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