Technical Note: A new SIze REsolved Aerosol Model (SIREAM) · 2017. 2. 3. · CEREA, Joint Research Laboratory, Ecole Nationale des Ponts et Chauss´ ees / EDF R&D, France´ Received:

Technical Note: A new SIze REsolved Aerosol Model

(SIREAM)

E. Debry, K. Fahey, K. Sartelet, B. Sportisse, M. Tombette

To cite this version:

E. Debry, K. Fahey, K. Sartelet, B. Sportisse, M. Tombette. Technical Note: A new SIze RE-solved Aerosol Model (SIREAM). Atmospheric Chemistry and Physics, European GeosciencesUnion, 2007, 7 (6), pp.1547.

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Atmos. Chem. Phys., 7, 1537–1547, 2007www.atmos-chem-phys.net/7/1537/2007/© Author(s) 2007. This work is licensedunder a Creative Commons License.

AtmosphericChemistry

and Physics

Technical Note: A new SIze REsolved Aerosol Model (SIREAM)

E. Debry, K. Fahey, K. Sartelet, B. Sportisse, and M. Tombette

CEREA, Joint Research Laboratory,École Nationale des Ponts et Chaussées / EDF R&D, France

Received: 9 November 2006 – Published in Atmos. Chem. Phys. Discuss.: 22 November 2006Revised: 26 February 2007 – Accepted: 2 March 2007 – Published: 21 March 2007

Abstract. We briefly present in this short paper a newSIze REsolved Aerosol Model (SIREAM) which simu-lates the evolution of atmospheric aerosol by solving theGeneral Dynamic Equation (GDE). SIREAM segregatesthe aerosol size distribution into sections and solves theGDE by splitting coagulation and condensation/evaporation-nucleation. A quasi-stationary sectional approach is usedto describe the size distribution change due to condensa-tion/evaporation, and a hybrid equilibrium/dynamical mass-transfer method has been developed to lower the computa-tional burden. SIREAM uses the same physical parameter-izations as those used in the Modal Aerosol Model, MAMSartelet et al. (2006). It is hosted in the modeling systemPOLYPHEMUS Mallet et al., 2007, but can be linked to anyother three-dimensional Chemistry-Transport Model.

1 Introduction

Atmospheric particulate matter (PM) has been negativelylinked to a number of undesirable phenomena ranging fromvisibility reduction to adverse health effects. It also has astrong influence on the earth’s energy balance Seinfeld andPandis (1998). As a result, many governing bodies, es-pecially in North America and Europe, have imposed in-creasingly stringent standards for PM. As an exemple the1999/30/CE European Directive has imposed a daily PM10concentration limit of 50µg/m3 since January 2005.

Atmospheric aerosol is a complex mixture of inorganicand organic components, with composition varying over thesize range of a few nanometers to several micrometers. Theseparticles can be emitted directly from various anthropogenicand biogenic sources or can be formed in the atmosphere byorganic or inorganic precursor gases.

Given the complexity of PM, its negative effects, andthe desire to control atmospheric PM concentrations, mod-els that accurately describe the important processes thataffect the aerosol size/composition distribution are there-fore crucial. Three-dimensional Chemistry-Transport Mod-

Correspondence to: B. Sportisse([email protected])

els (CTMs) provide the necessary tools to develop not onlya better understanding of the formation and the distributionof PM but also sound strategies to control it. Historically,CTMs focused on ozone formation or acid deposition and didnot include a detailed treatment of aerosol. Several modelshave been developed that include a very thorough treatmentof aerosol processes Gong et al. (2003); Adams and Sein-feld (2002); Spracklen et al. (2005), but there are still manylimitations Seigneur (2001).

In detailed models that seek to describe the time and spa-tial evolution of atmospheric PM, it is necessary to includethose processes described in the General Dynamic Equa-tion for aerosols (condensation/evaporation, coagulation, nu-cleation, inorganic and organic thermodynamics). Theseand additional processes like heterogeneous reactions at theaerosol surface, mass transfer between aerosol and clouddroplets, and aqueous-phase chemistry inside cloud dropletsrepresent some of the most important mechanisms for alter-ing the aerosol size/composition distribution.

Among the aerosol models, one usually distinguishes be-tweenmodal modelsWhitby and McMurry (1997) andsizeresolved or sectional models Gelbard et al. (1980). We referfor instance to the modal model of Binkowski and Roselle(2003) and the sectional model of Zhang et al. (2004) fora description of state-of-the-science aerosol models, hostedby the Chemistry-Transport Model, CMAQ Byun and Schere(2004).

Here we present the development of a new SIze RE-solved Aerosol Model (SIREAM). SIREAM is the size-resolved alternative to the modal model, MAM ModalAerosol Model, Sartelet et al. (2006). Both models use thesame physical parameterizations through the library for at-mospheric physics and chemistry ATMODATA Mallet andSportisse (2005). Both have a modular approach and relyon different model configurations. They are hosted in themodeling system POLYPHEMUS Mallet et al., 2007 and usedin several global, regional and local eulerian applications. Adetailed description of SIREAM and MAM can be found inSportisse et al. (2006) (available at http://www.enpc.fr/cerea/polyphemus). A key feature of SIREAM is its modular de-sign, as opposed to an all-in-one model. SIREAM can be

Published by Copernicus GmbH on behalf of the European Geosciences Union.

http://www.enpc.fr/cerea/polyphemushttp://www.enpc.fr/cerea/polyphemus

1538 E. Debry et al.: A new SIze REsolved Aerosol Model

used in many configurations and is intended for ensemblemodeling (similar to Mallet and Sportisse (2006)).

This paper is structured as follows. The model formula-tion and main parameterizations included in SIREAM aredescribed in Sect. 2. The numerical algorithms used for solv-ing the GDE are given in Sect. 3. A specific focus is devotedto condensation/evaporation, which is by far the most chal-lenging issue.

2 Model formulation

In this section we focus on aerosol dynamics, i.e. on thenucleation, condensation/evaporation, and coagulation pro-cesses. In addition, we briefly describe some processes thatare strongly related to aerosols (heterogeneous reactions atthe aerosol surface, mass transfer between the aerosols andthe cloud droplets and aqueous-phase chemistry in clouddroplets). We also include the parameterizations for Semi-Volatile Organic Compounds (SVOCs).

In order to deal with different parameterizations and toavoid the development of an “all-in-one” model, the parame-terizations have been implemented as functions of the libraryATMODATA (Mallet and Sportisse (2005)), a package for at-mospheric physics. As such, they can be used by other mod-els.

2.1 Composition

The particles are assumed to beinternally mixed, i.e., thatthere is a unique chemical composition for a given size. Eachaerosol may be composed of the following components :

– liquid water;

– inert species : mineral dust, elemental carbon and, insome applications, heavy trace metals (lead, cadmium)or radionuclides bound to aerosols;

– inorganic species : Na+, SO2−4 , NH4+, NO−3 and Cl

−;

– organic species : one species forPrimary OrganicAerosol (POA), 8 species for Secondary OrganicAerosol (see below for more details).

A typical version of the model (trace metals or radionuclidesare not included) tracks the evolution of 17 chemical speciesfor a given size bin (1+2+5+1+8). These species (externalspecies) should be distinguished from the species that are ac-tually inside one aerosol in different forms (ionic, dissolved,solid). Letne be the number of external species.

The internal composition for inorganic species is deter-mined by thermodynamic equilibrium, solved by ISOR-ROPIA V.1.7 Nenes et al. (1998). Water is assumedto quickly reach equilibrium between the gas and aerosolphases. Its concentration is given by the thermodynamicmodel (through the Zdanovskii-Stokes-Robinson relation).

The organic composition is given by the SORGAM modelSchell et al. (2001b) which we detail in Sect. 2.2.5.

Hereafter, the particle massm refers to thedry mass. Inorder to reduce the wide range of magnitude over the particlesize distribution and to better represent small particles, theparticle distribution is described with respect to the logarith-mic massx = ln m Wexler et al. (1994); Meng et al. (1998);Gaydos et al. (2003).

The particles are described by a number distribution,n(x, t) (in m−3), and by the mass distributions for species Xi ,{qi(x, t)}i=1,ne (in µg.m−3). The mass distributions satisfy∑i=ne

i=1 qi = m n. We also define the massmi(x, t) =qi (x,t)n(x,t)

of species Xi in the particle of logarithmic massx. It satisfies∑i=ne

i=1 mi(x, t) = ex .

2.2 Processes and parameterizations for the GDE

2.2.1 Nucleation

The formation of the smallest particles is given by the ag-gregation of gaseous molecules leading to thermodynami-cally stableclusters. The mechanism is poorly known andmost models assume homogeneous binary nucleation of sul-fate and water to be the major mechanism in the formationof new particles. Binary schemes tend to underpredict nucle-ation rates in comparison with observed values. Korhonenet al. (2003) has indicated that for the conditions typical inthe lower troposphere ternary nucleation of sulfate, ammo-nium and water may be the only relevant mechanism.

SIREAM offers two options for nucleation: the H2O-H2O4 binary nucleation scheme ofVehkamki et al. (2002)and the H2O-H2O4-NH3 ternary nucleation scheme of Na-pari et al. (2002).

The output is a nucleation rate,J0, a nucleation diameter,and chemical composition for the nucleated particles. Thenew particles are added to the smallest bin.

2.2.2 Coagulation

Atmospheric particles may collide with one another due totheir Brownian motion or due to other forces (e.g., hydro-dynamic, electrical or gravitational). SIREAM includes adescription of Brownian coagulation, the dominant mecha-nism in the atmosphere. There may be a limited effect onthe particle mass distribution and this process is usually ne-glected Zhang et al. (2004). However coagulation may havesubstantial impact on the number size distribution for ultra-fine particles.

The coagulation kernelK(x, y) (in unit of volume per unitof time) describes the rate of coagulation between two par-ticles of dry logarithmic massesx andy. K has differentexpressions depending on the relevant regime Seinfeld andPandis (1998).

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E. Debry et al.: A new SIze REsolved Aerosol Model 1539

2.2.3 Condensation/evaporation

Some gas-phase species with a low saturation vapor pressuremay condense on existing particles while some species in theparticle phase may evaporate. The mass transfer is governedby the gradient between the gas-phase concentration and theconcentration at the surface of the particle. The mass flux forvolatile species Xi between the gas phase and one particle oflogarithmic massx is computed by:

dmi

dt= Ii = 2πDgi dpfFS(Kni , αi)

(

cgi − c

si (x, t)

)

(1)

dp is the particle wet diameter (see Sect. 2.2.6 for the relationto mass).Dgi andc

gi are the molecular diffusivity in the air

and the gas-phase concentration of species Xi , respectively.The concentrationcsi at the particle surface is assumed to beat local thermodynamic equilibrium with the particle compo-sition:

csi (x, t) = η(dp) ceqi (q1(x, t), . . . , qne (x, t), RH, T ) (2)

T is the temperature andRH is the relative humidity.

η(dp) = exp(

4σvpRT dp

)

is a correction for the Kelvin effect,

with σ the surface tension,R the gas constant andvp theparticle molar volume. In practice,ceqi is computed by there-verse mode of a thermodynamics package like ISORROPIANenes et al. (1998) in the case of kinetic mass transfer.

The Fuchs-Sutugin function,fFS , describes the non-continuous effects (Dahneke (1983)). It depends on theKnudsen number of species Xi , Kni =

2λidp

(with λi the airmean free path), and on the accommodation coefficientαi(default value is 0.5):

fFS(Kni , αi) =1 + Kni

1 + 2Kni (1 + Kni )/αi(3)

When particles are in a liquid state, the condensation of anacidic component may free hydrogen ions and the conden-sation of a basic component may consume hydrogen ions.Thus the condensation/evaporation (c/e hereafter) processmay have an effect on the particlepH . The hydrogen ionflux induced by mass transfer is:

JH+ = 2JH2O4 + JHCl + JHNO3 − JNH3 (4)

with Ji the molar flux in species Xi . ThepH evolution dueto c/e can be very stiff and cause instabilities, due to the verysmall quantitynH+ of hydrogen ions inside the particle. Thehydrogen ion flux is then limited to a given fractionA of thehydrogen ion concentration following Pilinis et al. (2000) :|JH+ | ≤ AnH+ , whereA is usually chosen arbitrarily be-tween 0.01 and 0.1. A is a numerical parameter that has nophysical meaning and does not influence the final state ofmass transfer. It just modifies the numerical path to reachthis state. We refer to Pilinis et al. (2000) for a deeper under-standing.

2.2.4 Inorganic thermodynamics

There are a range of packages available to solve thermody-namics for inorganic species Zhang et al. (1998). ISOR-ROPIA Nenes et al. (1998) was shown to be a computation-ally efficient model that is also numerically accurate and sta-ble and provides both aclosed mode (for global equilibrium,a.k.a. forward mode) andopen mode (for local equilibriumand kinetic mass transfer, a.k.a.reverse mode). Particles canbe solid, liquid, both or in a metastable state, where particlesare always in aqueous solution.

Moreover, the inclusion of sea salt (NaCl) in the computa-tion of thermodynamics is also an option in SIREAM.

When the particles are solid, fluxes of inorganic speciesare governed by gas/solid reactions at the particle surface.In this case, thermodynamic models are not able to computegas equilibrium concentrations. For solid particle, SIREAMcalculations are based on the solutions proposed in Piliniset al. (2000).

2.2.5 Secondary Organic Aerosols

The oxidation of VOCs leads to species (SVOCs) that haveincreasingly complicated chemical functions, high polariza-tions, and lower saturation vapor pressure.

There are many uncertainties surrounding the formation ofsecondary organic aerosol. Due to the lack of knowledge andthe sheer number and complexity of organic species, mostchemical reaction schemes for organics are very crude repre-sentations of the “true” mechanism. These typically includethe lumping of “representative” organic species and highlysimplified reaction mechanisms.

The default gas-phase chemical mechanism for SIREAMis RACM Stockwell et al. (1997). Notice that the gas-phasemechanism and the related SVOCs are parameterized andcan be easily modified.

The low volatility SOA precursors and the partitioning be-tween the gas and particle phases are based on the empiri-cal SORGAM model (Schell et al. (2001a); Schell (2000)).Eight SOA classes are taken into account (4 anthropogenicand 4 biogenic). Anthropogenic species include two fromaromatic precursors (ARO 1 and ARO 2), one from higheralkanes (OLE 1) and one from higher alkenes (ALK 1).The biogenic species represent two classes fromα-pinene(API 1 and API 2) and two from limonene (LIM 1 andLIM 2) degradation. Some oxidation reactions of the formVOC + Ox → P where Ox is OH, O3, or NO3 have beenmodified to VOC+ Ox → P+ α1 P1 + α2 P2 with P1 andP2 representing SVOCs among the eight classes. Updatedvalues of these parameters have also been defined in otherversions of the mechanism (not reported here).

The partitioning between the gas phase and the particlephase is performed in the following way. LetnOM be thenumber of organic species in the particle mixture (this in-

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cludes primary and secondary species) which are assumed toconstitute anideal mixture:

(qi)g = γi (xi)a qsati (5)

For species Xi , qsati is the saturation mass concentration in apure mixture,(xi)a is the molar fraction in the organic mix-ture andγi is the activity coefficient in the organic mixture(a default value of 1 is assumed).(xi)a is computed through:

(xi)a =

(qi)a

MiqOM

MOM

=

(qi)a

Mij=nOM∑

j=1

(qj )a

Mj+

(qPOA)a

MPOA

(6)

qOM is the total concentration of organic matter (primary andsecondary) in the particle phase. The molar massMi of com-ponenti is expressed inµg/mol (in the same unit as the massconcentrationsqi); MOM is the average molar mass for or-ganic matter inµg/mol. POA stands for the primary organicmatter, assumed not to evaporate.

qsati is computed from the saturation vapor pressure with

qsati =MiRT

psati . A similar way to proceed is to define the

partitioning coefficientKi = (qi )aqOM (qi )g (in m3/µg). Ki can

be computed from the thermodynamic conditions and the sat-uration vapor pressure through:

Ki =RT

psati γi(MOM)(7)

The saturation vapor pressurepsati (T ) is given by theClausius-Clapeyron law:

psati (T ) = psati (298K) exp

(

−1Hvap

R(

1

T−

1

298)

)

(8)

with 1Hvap the vaporization enthalpy (in the default version,a constant value 156 kJ/mol).

The mass concentration of a gas at local equilibrium withthe particle mixture is given by Eq. (5). The global equi-librium between a gas and the particle mixture is given byEq. (5) and mass conservation for speciesXi :

(qi)a + (qi)g = (qi)tot (9)

with (qj )tot representing the total mass concentration (forboth phases) to be partitioned. This with Eq. (6) leads toa system ofnOM algebraic equations of second degree:

− ai ((qi)a)2 + bi(qi)a + ci = 0 (10)

where the coefficients depend on concentrations{(qj )a}j 6=ithroughai = 1Mi , bi =

qsatiMi

− 6i , ci = qsati 6i and

6i =j=nOM∑

j=1,j 6=i

(qj )a

Mj+

(qPOA)a

MPOA.

The resulting system is solved by an iterative approachwith a fixed point algorithm. Each second degree equationis solved in an exact way: the only positive root is computedfor each equation of type (10).

2.2.6 Wet diameter

Parameterizations of coagulation, condensation/evaporation,dry deposition and wet scavenging depend on the particle“wet” diameterdp. Two methods have been implementedin SIREAM to compute it, one based on thermodynamics,another on the Gerber’s Formula.

The thermodynamic method consists in using the particleinternal composition{mi} provided by the thermodynamicmodel ISORROPIA. Many of aerosol models use a constantspecific particle massρp Wexler et al. (1994); Pilinis and

Seinfeld (1988) supposed to satisfyρpπd3p

6 =∑ne

i=1 mi . InSIREAM, following Jacobson (2002), the particle volume is

split into a solid part and a liquid part:πd3p

6 = Vliq + Vsol. Aseach solid represents one single phase, the total solid particlevolume is the sum of each solid volume :Vsol =

∑

is

misρ∗is

,

with ρ∗is the specific mass of pure componentXis . The liquidparticle phase is a concentrated mixing of inorganic species,whose volume is a non linear function of its inorganic in-ternal composition :Vliq =

∑

ilVilnil whereVil is the par-

tial molar volume of ionic or dissolved speciesXil andnil isthe molar quantity inXil . Due to some molecular processeswithin the mixture (e.g. volume exclusion), the partial molarvolume is a function of the internal composition. However,we assume thatVil ≃

Miρ∗il

whereMi andρ∗il are the molar

mass of Xi and the specific mass of a pure liquid solutionof X i , respectively. This method is well suited for condensa-tion/evaporation for which thermodynamic computation can-not be avoided.

For other processes (coagulation, dry deposition and scav-enging) the particle “wet” diameter is computed through afaster method, the Gerber’s Formula (Gerber (1985)). Thisone is a parameterization of the “wet” radius as a function ofthe dry one :

rw =[

C1(rd)C2

C3(rd)C4 − logRH+ (rd)3

] 13

(11)

whererw andrd are respectively the wet and dry particle ra-dius in centimeters,RH is the atmospheric relative humiditywithin [0, 1]. Coefficients(Ci)i=1,4 depend on the particletype (urban, rural or marine). TheC3 coefficient is tempera-ture dependent (T) through the Kelvin effect:

C3(T ) = C3[1 + C5(298− T )] (12)

We have actually modified the coefficients given by Gerberthrough a minimization method so that the Gerber’s Formulagive results as close as possible to the “wet” diameters givenby the thermodynamic method Sportisse et al. (2006):

C1 = 0.4989, C2 = 3.0262, C3=0.5372 10−12

C4 = −1.3711, C5=0.3942 10−02 (13)

The choice of which method to use (thermodynamics orGerber’s Formula) is up to the user.



2.2.7 Logarithmic formulation for the GDE

On the basis of the parameterizations described above, theevolution of the number and mass distributions is governedby the GDE :

∂n

∂t(x, t) =

∫ x̃

x0

K(y, z)n(y, t)n(z, t) dy

−n(x, t)∫ ∞

x0

K(x, y)n(y, t) dy

−∂(H0n)

∂x(14)

∂qi

∂t(x, t) =

∫ x̃

x0

K(y, z)[qi(y, t)n(z, t) + n(y, t)qi(z, t)] dy

−qi(x, t)∫ ∞

x0

K(x, y)n(y, t) dy

−∂(H0qi)

∂x+ (Iin)(x, t) (15)

H0 = I0m (in s−1) is the logarithmic growth rate. The nucle-

ation threshold isx0= ln m0. Moreover,x̃ = ln(ex −ex0) andz = ln(ex − ey) in the above formula.

At the nucleation threshold, the nucleation rate determinesthe boundary condition :

(H0n)(x0, t) = J0(t) , (H0qi)(x0, t) = mi(x0, t)J0(t) (16)

The evolution of the gaseous concentration for the semi-volatile species Xi is given by:

dcgi

dt(t) = −mi(x0, t)J0(t) −

∫ ∞

x0

(Iin)(x, t) dx (17)

or by mass conservation :cgi (t) +∫ ∞x0

qi(x, t) dx = Ki .

2.3 Other processes related to aerosols

The following processes are not directly part of SIREAM.As such, the core of SIREAM (the parameterizations and thealgorithms for the GDE) is independent. As for SOA, otherparameterizations can be used. For the sake of completeness,we have chosen to include a brief description of the defaultcurrent parameterizations.

2.3.1 Mass transfer and aqueous-phase chemistry for clouddroplets

For cells with a liquid water content exceeding a criticalvalue (the default value is 0.05 g/m3), the grid cell is as-sumed to contain a cloud and the aqueous-phase module iscalled instead of the SIREAM module. A part of the parti-cle distribution is activated for particles that exceed a criticaldry diameter the default value isdactiv = 0.7µm Straderet al. (1998). The microphysical processes that govern the

evolution of cloud droplets are parameterized and not explic-itly described. Cloud droplets form on activated particles andevaporate instantaneously (during one numerical timestep) inorder to take into account the impact of aqueous-phase chem-istry for the activated part of the particle distribution Fahey(2003); Fahey and Pandis (2001).

The activated particle fraction is then incorporated into thecloud droplet distribution. The VSRM model can simulatea size-resolved droplet distribution, but we use only a bulkapproach in order to decrease the computational. In this casethe average droplet diameter is fixed at 20µm. The chemicalcomposition of the cloud droplet is then given by the acti-vated particle fraction.

Aqueous-phase chemistry and mass transfer between thegaseous phase and the cloud droplets (bulk solution) are thensolved. The aqueous-phase model is based on the chem-ical mechanism developed at Carnegie Mellon UniversityStrader et al. (1998). It contains 18 gas-phase species and28 aqueous-phase species. Aqueous-phase chemistry is mod-eled by a chemical mechanism of 99 chemical reactions and17 equilibria (for ionic dissociation).

Mass transfer is solved dynamically only for “slow”1

species, while “fast” species are assumed to be described byHenry’s equilibrium, we refer to Sportisse et al. (2006) for adetailed list of species and their status.

The radical chemistry is not taken into account. The com-putation of H+ is made with the electroneutrality relationwritten asfelectroneutrality(H+) = 0. This nonlinear algebraicequation is solved with the bisection method. If no conver-gence occurs, we take a default valuepH=4.5 as observeddroplet pH often ranges between 4.0 en 5.0 Seinfeld and Pan-dis (1998); Pruppacher and Klett (1998).

After one timestep, the new mass generated from aqueouschemistry is redistributed onto aerosol bins that were acti-vated. To do so the initial aerosol distribution is assumedto have a bimodal shape (log-normal distributions) that givesweighting factors for each aerosol bin. Median diameter andvariance for each mode are respectively 0.4µm and 1.8 forfirst one, 2.5µm and 2.15 for second one. The tests in Fahey(2003) illustrate the low impact of the choice made for thisassumption.

We use a splitting method, the gas-phase chemistry beingsolved elsewhere (in the gas-phase module of the Chemistry-Transport Model). Aqueous-phase chemistry and mass trans-fer are solved with VODE Brown et al. (1989).

2.3.2 Heterogeneous reactions

The heterogeneous reactions at the surface of condensed mat-ter (particles and cloud or fog droplets) may significantly im-pact gas-phase photochemistry and particles. This process issolved together with gas-phase chemistry. Following Jacob

1“slow” and “fast” refer to the time for given species to reachequilibrium



(2000), these processes are described by the first-order reac-tions:

HO2PM→ 0.5 H2O2

NO2PM→ 0.5 HONO+ 0.5 HNO3

NO3PM→ HNO3

N2O5PM,clouds−→ 2 HNO3

The heterogeneous reactions for HO2, NO2 and NO3 at thesurface of cloud droplets are assumed to be taken into ac-count in the aqueous-phase model and are considered sepa-rately.

The first-order kinetic rate is computed for gas-phase

species Xi with ki =(

a

Dgi

+ 4c̄gi γ

)−1Sa wherea is the par-

ticle radius,c̄gi the thermal velocity in the air,γ the reactionprobability andSa the available surface for condensed matterper air volume.

γ strongly depends on the chemical composition andon the particle size. We have decided to keep the varia-tion ranges from Jacob (2000) for these parameters in or-der to evaluate the resulting uncertainties:γHO2 ∈ [0.1-1], γNO2 ∈ [10−6-10−3], γNO3 ∈ [2.10−4-10−2] andγN2O5 ∈ [0.01-1]. The default values are the lowest val-ues. For numerical stability requirements, these reactions arecoupled to the gas-phase mechanism.

3 Numerical simulation

3.1 Numerical strategy

On the basis of a comprehensive benchmark of algorithmsDebry (2004), the numerical strategy relies on methods thatensure stability with a low CPU cost. First, we use a split-ting approach for coagulation and condensation/evaporation.Second, the discretization is performed with sectional meth-ods which remain stable even with a few discretizationpoints, contrary to spectral methods Sandu and Borden(2003); Debry and Sportisse (2005b). Third, condensa-tion/evaporation is solved with a Lagrangian method, thequasistationary method of Jacobson Jacobson (1997) is em-ployed to reduce the numerical diffusion associated with Eu-lerian schemes in the case of a small number of discretizationpoints (typically the case in 3D models).

The splitting sequence goes from the slowest pro-cess to the fastest one (first coagulation and thencondensation/evaporation-nucleation). The nucleation pro-cess is not a numerical issue and is solved simultaneouslywith condensation/evaporation. In the following, we presentthe numerical algorithm used for each process.

The particle mass distribution is discretized intonb bins[xj , xj+1]. We define the integrated quantities over the bin

j for the number distribution and the mass distributions forspecies Xi :

N j (t) =∫ xj+1

xjn(x, t) dx , Q

ji =

∫ xj+1

xjqi(x, t) dx (18)

m̃ji =

Qji

Njis the average mass per particle inside binj for

species Xi .We use a Method of Lines by first performing size dis-

cretization and then time integration. After discretization, theresulting system of Ordinary Differential Equations (ODEs)has the generic form:

dc

dt= f (c, t) (19)

where the state vectorc is specific for each process.cn is thenumerical approximation ofc(tn) at timetn, with a timestep1tn = tn+1 − tn. A second-order solver is specified foreach case with a first-order approximationc̃n+1. The vari-able timestep1tn is adjusted by:

1tn+1 = 1tn

√

εr‖cn+1‖2‖c̃n+1 − cn+1‖2

(20)

whereεr is a user parameter, usually between 0.01 and 0.5.The higherεr is, the faster1tn increases.‖.‖2 is the Eu-clidean norm.

3.2 Size discretization

3.2.1 Coagulation

Coagulation is solved by the so-calledsize binning method(Jacobson et al. (1994)). Equations (14) and (15) are inte-grated over each bin, which gives:

dNk

dt(t) =

1

2

k∑

j1=1

k∑

j2=1f kj1j2Kj1j2N

j1N j2 − Nknb∑

j=1KkjN

j

dQki

dt(t) =

k∑

j1=1

k∑

j2=1f kj1j2Kj1j2Q

j1i N

j2 − Qkinb∑

j=1KkjN

j(21)

Kj1j2 is an approximation of the coagulation kernel betweenbinsj1 andj2.

The key point is to compute the partition coefficientsf kj1j2that represent the fraction of particle combinations betweenbins j1 andj2 falling into bin k. As these coefficients onlydepend on the chosen discretization, they can be computedin a preprocessed step. The computation depends on theassumed shape of continuous densities inside each bin forthe closure scheme, see Debry and Sportisse (2005a). InSIREAM, we use a closure scheme similar to Fernndez-Dazet al. (2000).



3.2.2 Condensation/evaporation-nucleation

Lagrangian formulation Let x̄j (t) be the logarithmic massof one particle at timet whose initial value corresponds topoint xj of the fixed discretization. The time evolution ofx̄j (t) is given by the equation of the characteristic curve :

dx̄j

dt(t) = H0(x̄j , t) , x̄j (0) = xj (22)

One crucial issue is to ensure that the characteristic curvesdo not cross themselves. If this happens the Lagrangian for-mulation is no longer valid. In real cases we have no proofthat this does not happen, even though we have not seen sucha situation up to now.

Provided that the characteristic curves do not cross, wecan define integrated quantitiesN j and Qji for each La-

grangian bin[x̄j , x̄j+1] : N j (t) =∫ x̄j+1

x̄jn(x, t) dx and

Qji =

∫ x̄j+1

x̄jqi(x, t) dx.

Mass conservation can be easily written in the form :cgi (t) +

∑nbj=1 Q

ji (t) = Ki .

The time derivation of integrated quantities leads to theequations:

dN j

dt= 0 ,

dQji

dt= N j Ĩ ji (23)

Ĩji is an approximation of the mass transfer rate for species

Xi in bin j :

Ĩji = 2πDid

jpf (K

jni , αi)

︸︷︷︸

aji

(

Ki −nb∑

k=1Qki − η

j (ceqi )

j

)

(24)

with ηj = e4σvp

RT djp . (ceqi )

j is computed at̃mji .For the nucleation process, the first boundx1 is assumed

to correspond to the nucleation threshold, so that the La-grangian bound̄x1 does not satisfy (22) but:

dx̄1

dt= j (t) , x̄1(0) = x1 (25)

where j (t) is the growth law of the first bound due tonucleation and given by the nucleation parameterization. Theequations for the first Lagrangian bin therefore are written as:

dN1

dt= J0(t) ,

dQ1i

dt= N1Ĩ1i + mi(x1, t)J0(t) (26)

where[m1(x1, t), . . . , mne (x1, t)] is the chemical composi-tion of the nucleated particles, also given by the nucleationprocess.

The Lagrangian formulation consists in solving Eqs. (22),(23) and (26). In the next section we detail the variousnumerical strategies to perform the time integration , whichis by far the most challenging point in particle simulation.

Interpolation of Lagrangian boundaries One has to solvethe equations for the characteristic curves in order to knowthe boundaries of each bin. Notice that the c/e equations forboundaries are similar to those for integrated quantities. In-deed, forj = 1, . . . , nb and x̃j = ln(m̃j ), one gets fromEq. (23):

dx̃j

dt= H̃ j0 , H̃

j

0 =Ĩ

j

0

m̃j, (27)

In practice, in order to reduce the computational burden, onetries to avoid solving boundary equations. An alternative isto interpolate the bin boundaries from integrated quantities.

First method Koo et al. (2003) consists of utilizing the ge-ometric mean of two adjacent bin :

for j = 2, . . . , nb , m̄j (t) =√

m̃j−1(t)m̃j (t) (28)

This algorithm would have a physical meaning if Eqs. (22)and (27) were conserving formula (28), which is not the case.We have therefore developed another algorithm.

Equations (22) and (27) are similar and thereforex̃j andx̄j evolve in the same proportion given byλj (t) (j ≥ 2):

λj (t) =x̄j (t) − x̃j−1(t)x̃j (t) − x̃j−1(t)

(29)

λj (0) is known becausēxj (0) = xj . The time integrationover[0, t] of Eqs. (22) and (27) gives forj ≥ 1:

x̄j (t) = xj + 1x̄j , 1x̄j =∫ t

0H

j

0 (t′) dt ′

x̃j (t) = x̃j (0) + 1x̃j , 1x̃j =∫ t

0H̃

j

0 (t′) dt ′ (30)

The variation of each boundarȳxj is then computed fromthat of its two adjacent bins̃xj−1 andx̃j :

1x̄j ≃ (1 − λj (0))1x̃j−1 + λj (0)1x̃j (31)

where one assumes thatλj remains constant.

Redistribution on a fixed size gridUsing a Lagrangian ap-proach for condensation/evaporation requires the redistribu-tion or projection of number and mass concentrations ontothe fixed size grid required by a 3D model or for coagula-tion.

Let N and(Qi)nei=1 be the integrated quantities of one La-

grangian bin after condensation/evaporation. We assume thatthis Lagrangian bin is covered by two adjacent fixed bins la-belledj andj + 1.

The redistribution algorithm must be conservative for thenumber and mass distribution of speciesXi :

N = N j + N j+1 , Qi = Qji + Qj+1i (32)

Two algorithms have been developed:



1. If x̄lo andx̄hi are the boundaries of the Lagrangian binafter condensation/evaporation, the redistribution is per-formed as follows for the number distribution and themass distribution of speciesXi :

N j =x̄

jhi − x̄lo

x̄hi − x̄loN , Q

ji =

x̄jhi − x̄lo

x̄hi − x̄loQi

N j+1 =x̄hi − x̄j+1lox̄hi − x̄lo

N , Qj+1i =

x̄hi − x̄j+1lox̄hi − x̄lo

Qi (33)

The number and aerosol mass are redistributed in equalproportions, depending upon the part of each fixed binscovered by the Lagrangian one. Nevertheless this im-plies that the average masses of fixed bins be equal tothe Lagrangian bin one (Q/N ) and may fall out of fixedbin boundaries.

2. Another approach consists in conserving the averagemass. Letm̃ = Q/N be the averaged Lagrangian bin,m̃j andm̃j+1 be respectively the centered mass of binsj andj+1. The number distribution and the mass distri-bution of speciesXi is redistributed ensuring conserva-tion of relationsQj = m̃jN j andQj+1 = m̃j+1N j+1,which together with (32) lead to the algorithm:

N j =1 − m̃

m̃j+1

1 − m̃jm̃j+1

N , Qji =

m̃j+1m̃

− 1m̃j+1

m̃j− 1

Qi

N j+1 =1 − m̃

m̃j

1 − m̃j+1m̃j

N , Qj+1i =

1 − m̃jm̃

1 − m̃jm̃j+1

Qi (34)

This algorithm comes to the fitting method developedby Jacobson in Jacobson (1997).

The first method takes advantage of the more sophisticatedcomputation of bound sections but does not conserve aver-age mass. The second method conserves average mass butmay increase numerical diffusivity due to the lack of boundsection information.

Both schemes are available in SIREAM.

3.3 Time integration

3.3.1 Coagulation

As coagulation is not a stiff process, we solve it by the secondorder explicit scheme ETR (Explicit Trapezoidal Rule) withthe sequence:

c̃n+1 = cn + 1tf (cn, tn)

cn+1 = cn +1t

2

(

f (cn, tn) + f (c̃n+1, tn+1))

(35)

with c = (N1, . . . , Nnb , Q11, . . . ,Qnb1 , . . . ,Q

1ne

, . . . ,Qnbne ).

3.3.2 Condensation/evaporation

Here,c = (Q11, . . . ,Q1ne , . . . ,Qnb1 , . . . ,Q

nbne )

T . nc = ne ×nb is the dimension ofc.

SIREAM offers three methods for solving conden-sation/evaporation: a fully dynamic method that treatsdynamic mass transfer for each bin, a bulk equilibriumapproach, and a hybrid approach that combines the twoprevious approaches.

Fully dynamic method Due to the wide range of timescalesrelated to mass transfer, the system is stiff and implicit al-gorithms have to be used. The second-order Rosenbrockscheme Verwer et al. (1999); Djouad et al. (2002), ROS2,is applied for the time integration :

cn+1 = cn +1tn

2(3k1 + k2)

[I − γ1tnJ (f )]k1 = f (cn, tn)[I − γ1tnJ (f )]k2 = f (c̃n+1, tn+1) − 2k1 (36)

wherec̃n+1 = cn + 1tnk1 andγ = 1 + 1√2.This scheme requires the computation of the Jacobian ma-

trix of f (a matrixnc ×nc) defined by[J (f )]kl = ∂fk

∂cl. f k is

thekth component of functionf andcl is thelth componentof c.

Let us writek = (i − 1)nb + j andl = (i′ − 1)nb + j ′wherei andi′ label the semi-volatile species whilej andj ′

label the bins. The(kl)th element of the Jacobian matrix maythen be written as

∂f k

∂cl=

∂Iji

∂Qj ′

i′

(37)

The derivation off k may be split into one linear part, dueto mass conservation, and one non-linear part related to thecoefficientaji , to the Kelvin effectη

j , and to the gas equi-librium concentration(ceqi )

j . The linear part is analyticallyderived :(

∂f k

∂cl

)

lin= −aji N

j ′ (38)

The non-linear part has to be differentiated by numericalmethods, like the finite difference method :(

∂f k

∂cl

)

non−lin=

f k(. . . , cl(1 + εjac), . . .) − f k(. . . , cl, . . .)clεjac

(39)

whereεjac is generally close to 10−8. During the numeri-cal computation, the linear part is arbitrarily kept constant toavoid deriving it twice.

A default option, advocated for 3D applications, is toapproximate the Jacobian matrix by its diagonal. Themotivation here is to reduce the CPU time.



Hybrid resolution Solving the c/e system, even with animplicit scheme, can be computationally inefficient. In or-der to lower the stiffness, hybrid methods for condensa-tion/evaporation have been developed Capaldo et al. (2000).The method consists in partitioning the state vectorc into itsfast components (cf ) and its slow components (cs) respec-tively:

dcs

dt= f s(cs, cf , t) , f f (cs, cf , t) = 0 (40)

The algebraic equation states that the fast part is a functionof the slow part,cf (t)=g(cs(t), t). The time evolution of theslow part is now governed by :

dcs

dt=f s

(

cs, g(cs(t), t), t

)

(41)

As cs gathers particle species and sizes which have a slowc/e characteristic time, stiffness is substantially reduced.

The issue is now to determine whether particle sizes andspecies are “slow” or “fast”. The spectral study of the c/e sys-tem Debry and Sportisse (2006) indicates how to compute acutting diameterdc between “slow” and “fast” species/sizes,such that the partitioning consists of cutting the particle dis-tribution as follows: the smallest bins are at equilibriumwhile the coarsest ones are governed by kinetic mass trans-fer. The cutting diameter can be computed by QSSA criteria,defined by :

QSSAji =

cgi − η

ji (c

eqi )

j

cgi + η

ji (c

eqi )

j(42)

for a given chemical speciesXi and one particle sizej . Thecloser this ratio to unity, the closer the species and the sizeare to equilibrium.

In practice all binsj for which(QSSAji )nei=1 is greater than

one, the user parameterεQSSA (close to unity) will be con-sidered fast and solved by an equilibrium equation. In thefollowing we writejc as the bin corresponding to the cuttingdiameter. Binjc is the largest fast bin and binjc + 1 is thesmallest slow bin.

In SIREAM (to be used in 3D modeling), the defaultoption is a fixed cutting diameter (1.25 or 2.5µm).

The thermodynamic equilibrium between the gas phaseand the fast particle bins is now written for speciesXi as:

Kfi −

jc∑

j=1Q

ji − η

ki c

eqi (Q

k1, . . . ,Q

kne

) = 0 (43)

with Kfi =Ki −∑nb

j=jc+1 Qji the total mass of speciesXi for

fast bins.There are two approaches for solving this equilibrium: the

bulk equilibrium approach and the size-resolved particle ap-proach. For the size-resolved approach, we refer to Jacobson

et al. (1996) (with the use of the fixed point algorithm) and toDebry and Sportisse (2006) (with a minimization procedure).

In SIREAM, the bulk equilibrium has been implemented(Pandis et al. (1993)). It consists in merging all fast binsj ≤ jc into one bin, referred as the “bulk” aerosol phase :

1 ≤ i ≤ ne , Bi =jc∑

j=1Q

ji (44)

The thermodynamic model ISORROPIA is then applied tothe “bulk” aerosol phase(Bi)

nei=1 and one gets equilibrium

“bulk” concentrations(Beqi )nei=1 with theforward mode of the

thermodynamics solver (global equilibrium).The variation from initial to final “bulk” concentrations is

then redistributed among fast bins 1≤ k ≤ jc for speciesXiPandis et al. (1993):

(Qki )eq = Qki + b

ki (B

eqi − Bi) , b

ki =

aki Nk

∑jcj=1 a

ji N

j(45)

This redistribution scheme is exact provided that the particlecomposition is uniform over fast bins and that the variationof the particle diameter can be neglected for fast bins Debryand Sportisse (2006).

Bulk approach It is a special case of the hybrid approachwith the cutting diameterjc = 1 (all bins are at equilibrium).

4 Implementation

The SIREAM module is written entirely in fortran 77.Its external dependencies are the thermodynamic moduleISORROPIA (version 1.7 currently used)Nenes et al.(1998) and the VODE solver from the ODEPACK ordinarydifferential equation package (“double precision” version re-quired). ISORROPIA is not distributed with SIREAM andhas to be retrieved by the user on the ISORROPIA web-site (http://nenes.eas.gatech.edu/ISORROPIA/). The VODEsolver can be retrieved from http://www.llnl.gov/CASC/software.html, but as this solver is in the public domain wealso freely ship it together with SIREAM.

5 Conclusions

We have summarized the main features of the aerosol modelSIREAM (SIze REsolved Aerosol Model). SIREAM sim-ulates the GDE for atmospheric particles and can be easilylinked to a three-dimensional Chemistry-Transport-Model.Moreover, the physical parameterizations used by SIREAMcan be easily modified. They are currently hosted by thelibrary ATMODATA and shared by another aerosol modelMAM, Sartelet et al. (2006).

The next development steps are related to the improvementof the modeling of Secondary Organic Aerosol. The cur-rent parameterization of SOA is limited because it does not


http://nenes.eas.gatech.edu/ISORROPIA/http://www.llnl.gov/CASC/software.htmlhttp://www.llnl.gov/CASC/software.html


take into account the hydrophilic behavior of organic speciesGriffin et al. (2002b,a); Pun et al. (2002). Furthermore newgas precursors such as isoprene and sesquiterpene should beadded.

The modularity of SIREAM will be also strengthened byadding new alternative parameterizations (such as other ther-modynamics models or simplified aqueous-phase chemicalmechanisms) and new numerical algorithms (especially fortime integration of condensation/evaporation).

A further step is also the extension toexternally mixedaerosol.

Acknowledgements. Part of this project has been funded by theFrench Research Program, Primequal-Predit, in the framework ofthe PAM Project (Multiphase Air Pollution). Some of the authors(K. Fahey, K. Sartelet and M. Tombette) have been partially fundedby the Region Ile de France.

Edited by: R. MacKenzie

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