DEPARTMENT OF PHYSICS ATMOSPHERIC,OCEANIC AND PLANETARY PHYSICS A Fast Stratospheric Aerosol Microphysical Model (SAMM): H SO -H O Aerosol Development and Validation S. N. Tripathi Department of Civil Engineering, Indian Institute of Technology Kanpur, India X. P. Vancassel, R. G. Grainger Atmospheric, Oceanic, and Planetary Physics, Clarendon Laboratory, University of Oxford, U.K. H. L. Rogers Centre for Atmospheric Science, Department of Chemistry, University of Cambridge, U.K. AOPP Memorandum 2004.1 27 May 2004 University of Oxford
Transcript
DEPARTMENT OF PHYSICS
ATMOSPHERIC,OCEANICAND PLANETARY PHYSICS
A FastStratosphericAerosolMicrophysical Model(SAMM): H � SO� -H � O AerosolDevelopment and
Validation
S.N. TripathiDepartment of Civil Engineering, Indian Institute of Technology Kanpur, India
X. P. Vancassel,R. G. GraingerAtmospheric, Oceanic, and Planetary Physics, Clarendon Laboratory, University of Oxford, U.K.
H. L. RogersCentre for Atmospheric Science, Department of Chemistry, University of Cambridge, U.K.
AOPPMemorandum2004.1
27May 2004
Universityof Oxford
Abstract
A Fast Stratospheric Aerosol Microphysical Model (SAMM) hasbeendeveloped to studyaerosol behaviour in the lower stratosphere. This modelsimulateshomogeneousbinary nucle-ation, condensational growth, coagulation, andsedimentation of sulphuric acid-waterparticlesin order to predict the composition andsize-distribution of stratospheric aerosols. The princi-pal advantageof SAMM is that it is non-iterative, i.e. computing time is reducedby findingsemi-implicit solutionsto aerosol processes. Condensation andcoagulation aresolvedusing theoperator-split method. Hencethe effect of coagulation is determinedin a single iteration. Thesemi-implicit solution for coagulation agreeswell with Smoluchowski’s solution for a constantcoagulation kernel. Similarly, starting from the fundamentalgrowth equation, a solution forcondensational growth is derived that doesnot require iteration. The solution conserves massexactly, andis unconditionally stable. In SAMM, homogeneousnucleation andcondensation arecoupled in a mannerthatallowsrealistic competition betweenthe two processesfor the limitedamount of vapour. With geometrically relatedsizebins(around40 binsfor sulphuric acid-waterparticles in therange0.3nm to 1.5 � m) andan1800s time-step, SAMM takesabout3 minutesCPUon a 1.4 GHz computer to calculate thebackground stratospheric aerosol sizedistribution.HenceSAMM allows aerosol processesto be includedin global modelsfor a relatively smallcomputational expense.SAMM hasbeenusedto simulatebackgroundstratosphericaerosolsandvolcanically disturbedaerosol andhasbeenshown to be in good agreementwith observationsandother modelling studies.
1 INTRODUCTION
1 Introduction
Stratosphericaerosolscanaffecttheglobalclimatesystemin avarietyof ways.Theseaerosolsplayasignificantrolein theEarth’sradiativebalance(Lacisetal.,1992)andtheattenuationof UV radiation(Michaelangelietal.,1992).Also, they provideasurfacefor heterogeneouschemicalreactions:theseareimportantfor ozonelossin themiddleatmosphere(Solomonetal.,1986;HofmannandSolomon,1989; Rodriguezet al., 1991). The magnitudeof theseeffects is significantlyenhancedwhenthebackgroundaerosollayer is perturbedby strongvolcaniceruptions,or possiblyby high-speedciviltransportaircraft(Tie et al., 1994;Weisenteinet al., 1997;Bekki andPyle,1992;Pitari etal., 1993).
Stratosphericaerosolsaremainly composedof supercooledsulphuric acid droplets(SteeleandHamill, 1981). They mayalsocontainsmallamountsof othercomponentssuchasammonium sul-phate(Bigg, 1975;CadleandKiang, 1977). The sulphuric acid fraction of the droplets,which isin the rangeof 50-80%, is a strongfunctionof the relative humidity andthe ambienttemperature(SteeleandHamill, 1981),andthereforevarieswith latitudeandseason.
Severalmodelshave beendevelopedin orderto understandtherole of stratosphericaerosolsintheatmosphericsystem.A one-dimensionalmodelwasdevelopedby Turcoet al. (1979)andlaterrefinedby Toonet al. (1988)andZhaoet al. (1995).Anotherclassof aerosolmicrophysical model(0D models)includingaircraftplumedynamicshasbeendevelopedin thelasttenyears(e.g.Brownet al., 1996; Yu andTurco 1997; Karcher, 1998). Recently, several studieshave beenperformedwhereaerosolmicrophysicshasbeenaddedto the existing 2D and3D models. Thesestudiesareconcernedwith the impact of either aircraft emissions(e.g. Bekki and Pyle, 1992; Pitari et al.,1993)or volcaniceruptions(Tie et al., 1994;Bekki andPyle,1994;Weistensteinet al., 1997)on thebackgroundstratospheric aerosoldistribution.
Theonly 3D studysimulatingtheformationanddevelopmentof stratosphericaerosolsis thatre-portedby Timmreck(2001).In thatpaper, theemphasiswason theevolution andseasonalvariationof stratosphericaerosolsusingtheHamburg climatemodelECHAM4. Thereis a needfor 3D simu-lationsof theeffectsof aircraftemissionson backgroundstratosphericaerosols,includingradiativeandchemistryeffects. Global studiesare importantbecauseof the complex interactionsbetweenradiation,chemistryandaerosolmicrophysics.Furthermore,severalauthors(e.g.Pitari et al., 1993)have pointedout the role of transportwhensimulating the effectsof aircraft sulphuremissionsonbackgroundstratospheric aerosols.
This paperdescribesa StratosphericAerosol Microphysical Model (SAMM) that is a micro-physicalbox modelof sulphuric acid-wateraerosols.SAMM hasbeendevelopedfor inclusion inglobalmodels.While eachof thestratosphericmicrophysicalmodelsdescribedearlierwassuccess-ful at reproducingtheobservedbackgroundaerosolsizedistributionparameters,they wereall basedon iterative methods. Computerprocessingtime and memoryuseare importantissuesin globalsimulations of atmosphericprocesses:Keepingthis in mind, non-iterative solutions to growth andcoagulationequationsareusedin SAMM. Thesearebasedon techniquesdevelopedby Jacobson(1997,1999,2002)andJacobsonandTurco(1995).Jacobson(2002)hassuccessfullyappliedthemto troposphericaerosols,and this is the first time the techniqueshave beenextendedto simulatestratosphericaerosols.
wherethesubscriptsnuc,coag,cond,sed,anddiff standfor homogeneousnucleationof sulphuricacid-water, coagulationof particles,condensationof sulphuric acid/waterontoparticles,sedimenta-tion of particles,anddiffusion,respectively. Following Turcoetal. (1979)andTimmreckandGraf’s(2000)assessmentsof therelative importanceof theseprocesses,SAMM includes:
- homogeneousheteromolecularnucleationof H F SOG /H F O,- condensationandevaporationof H F SOG andH F O,- coagulationamongtheparticles,- sedimentation removal. Figure1 shows a schematicof thevariousprocessesusedin SAMM.
It canbe seenthat all the sulphuricacid gasis availablefor nucleationat the beginning of simula-tion. Oncenew particlesareformed,they undergo sulphuricacidcondensation. As particlegrowthoccurs,all particlesarebroughtto thermodynamic equilibrium with respectto watervapourpres-sure.Condensationandcoagulationarecoupledin anoperator-split manner. Numberconcentrationsdeterminedfrom thecondensation solution areusedto initialisevaluesbeforecoagulation.Finally,largerparticlesareremovedasthey undergoasedimentationprocessdueto gravity effects.
tributedin thesebinsasafunctionof thenumberof sulphuricacidmoleculesJLK they contain(Sorokinet al., 2001).
2.2 Condensation of Sulphuric Acid
If thevapourpressureof H F SOG is in excessof theequilibriumvapourpressurethencondensationwill occur. In thelower stratosphere,wheretheatmosphereis generallysupersaturatedwith respectto sulphuricacid,condensationoccurs,whereasin theupperstratospheresulphuricacid moleculesevaporatefrom the particles. Jacobson(1997,1999)givesan equationdescribingcondensationalgrowth of H F SOG ontoparticlesin thesizebin M as:��N K1O P��� !RQ K1O PTS�UVPVW XYP[Z]\[K1O PTS�UVPAX ? 9_^K1O PTS�U[Pa` (3)
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2 MODEL DESCRIPTION 2.2 Condensationof SulphuricAcid
In theseandthefollowing equations,thesubscriptsb , c and c�dfegc refer respectively to thesizebinb considered,the time c andtheprevioustime chdiegc at which calculationsarecarriedout. On theleft handside, jlk1m n is theparticlemoleconcentrationof H o SOp (molem q�r ). We canalsowrite it ass k1m ntukvxw , where
s k5m n is the particlenumberconcentration(m q�r ) and w is Avogadro’s number. Onthe right handsideof the equation,y�k5m n q�z n is the transferrate(sq�{ ) betweenthe gasphaseandallparticlesin the bin i. We alsouse |Cn , the ambientH o SOp vapourmole concentration(mole m q�r ),which is expressedas}�v�~�� , where} is thesulphuricacidvapourpressure(in Pa), ~ is theuniversalgasconstant(J moleq�{ K q�{ ) and � is theair temperaturein Kelvin. Finally, |��T�_�k1m n q�z n is thesaturationvapourmoleconcentrationof condensingH o SOp above a flat surfacehaving thesamecompositionasthe droplet in bin b (molesm q�r ). It is alsodefinedas } �T�_�"�*� kTv�~�� , where } �T�_� is the saturationvapourpressureabovea flat surfaceof puresulphuric acid,determinedfrom Ayerset al. (1980)andcorrectedfor low temperaturesby KulmalaandLaaksonen(1990),and� k is thesulphuric acidactivitycorrespondingto thecomposition of particlesin bin b . Themostrigorousmethodfor evaluatingtheactivity coefficientof asolutionincludingionic soluteshasbeengiven by Clegg(1995)andCleggetal. (1998).However, it is computationally slow andonly valid upto asulphuric acidmolefractionof0.42.More recentlyNoppeletal. (2002),havecomparedthethermodynamicmodelof Clegg(1995)andClegget al. (1998)with theliquid phaseactivity modelof Zeleznik(1991).They foundthattheresultingsulphuricacidhydratesdistribution andnucleationrates,aspredictedfrom thetwo models,do not differ significantly. Hencewe have usedthecomputationally fastactivity coefficientsgivenby Zeleznik(1991)in the SAMM. Finally, ��k1m n q�z n is the saturationratio of sulphuric acid above acurvedsurfaceof puresulphuricacid.This ratio is definedby theKelvin equation:
�[k5m n q�z n,���l��������� kA���Lm k� ka~����&� (4)
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2.2 Condensationof SulphuricAcid 2 MODEL DESCRIPTION
Here, ×�� is thediffusion coefficient of H � SO¶ molecules(m� s��¤ ), ¹°ê � is theKnudsennumber,Û�� is the ventilation factor, and ëÑ� is the sticking coefficient of sulphuricacid gasmolecules.Theexpressionfor ×ì� is givenby (Davis, 1983):
is thediameterof a sulphuricacidmolecule,commonlytakenas Ó Ð���� Ü� ��¤ à m, ô ø ú û ü ý isits molecularweight(kg mole��¤ ), ô À is themassof air permole(kg mole��¤ ) and ð À is thedensityofair (kg m � £ ). TheKnudsennumber, ¹°ê � , with respectto particlesize � � , is given by
¹°ê �V��x� ª (11)
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2 MODEL DESCRIPTION 2.2 Condensationof SulphuricAcid
wherethe meanfree pathof sulphuric acid gasis given by �� ������������������! (FuchsandSutugin,1971).Here, �"� � ��� is thethermalvelocity of H # SO$ molecules(m s%'& ).
Thestickingcoefficient ()� isanuncertainparameterbecausenoexperimentshavebeenconductedunderstratosphericconditions. Laboratoryexperimentsundernormalconditions(VanDingenenandRaes,1991)havequantifiedastickingcoefficient in therangeof *�+,*.-�/102()�)03*�+,*54�6 with anaveragevalueof 0.04.Clementetal. (1996)from theoreticalconsiderationsdeterminedastickingcoefficient(7�8 :9 . We have usedthis last assumption andmadea sensitivity studyto seethe impactof thechoiceof sucha value(seesection3). To correctthe increasedrateof vapourandenergy transferto theupstreamsurfaceof a largeparticle( ; 5< m), a ventilation factorfor vapour( =�� ) needsto beincludedin theexpressionfor aneffectivediffusion coefficient. Thesizesof particlesencounteredinstratosphericsimulationsaresmallandtherefore=>�? @9 in thiscase.
Equations3 and7 togetherrepresentA1BDCE9 ordinarydifferentialequations. Most of themicro-physicalmodelsto dateuseiterative schemes(suchasdifferentialequationsolvers)to solve thesegrowth equations.Theseiterative schemestake considerablecomputer time. However, our primaryconcernhereis to develop a fast microphysical model that is computationally the leastdemand-ing whencoupledwith largemodels.We usethenon-iterative schemeproposedby Jacobson(1997,1999)to solvetheseequations. Assumingthatthefinal concentrationcanbeintegratedfrom equation7, wegetanexplicit expression:F �HG IJ F �HG I %LK I!CNM1OQP��HG I %LK ISRQTUI?VXWY�HG I %LK IZT\[H]_^�HG I %LK I�`ba (12)
wherethefinal H # SO$ gasconcentration,TcI , is currentlyunknown. Fromthemass-balanceequationwe canwrite: TUI�Cedgfh�ji & F �HG IJ kTUI %LK I!Cedgfh�li & F �HG I %LK I�+ (13)
Substituting 12 into 13andsolving for final gasconcentrationgivesTUIS TUI %LK I!CNMmO!n dgf�ji & P��HG I %LK IZWY�HG I %LK IZT [o]_^�HG I %LK I9pCNMmO n d f�ji & P��HG I %LK I + (14)
Substituting Equation12 into Equation14givesF �HG IS F �HG I %LK I!C M1OQP��HG I %LK I9pCXM1O n dgf�li & P��HG I %LK Irqts a (15)
wheres ETUI %LK I!CNMmO dufh�ji & P��HG I %LK IvWY�HG I %LK IZT [o]_^�wG I %LK I VxWY�HG I %LK IvT [o]_^�HG I %LK Izy 9pCNMmO dufh�ji & P��HG I %LK I|{ (16)
Two limits, however, areusedto containthe artificially growing solutions,assuggestedby Ja-cobson(1997).Theseare TUI} ~��H� y TUI a T�{ (17)F �HG I} ~��"� y F �HG I a *5{ (18)
whereC is thetotalsulphuric acidconcentration(liquid+gas,in molem %L� ). A third limit derivedfromJacobson(2002)hasbeenaddedto the previous onesin order to avoid sulphuricacid evaporationexceedingthetotalamountof acidincludedin a particle.
whereall thesymbolsusedhave alreadybeendescribedexcept �� which is thedensityof a particlein thesizebin ® (kg m ¯L° ) determinedfrom Vehkamaki et al. (2002),and � � � � (kg mole'± ) whichis themolecularmassof water. In thecaseof H ² SO³ , SteeleandHamill (1981)demonstratedthat,understratosphericconditions,the water-vapourpressureover the H ² SO³ -H ² O droplet is equaltothe ambientpartial pressureof water, i.e. the sulphuricacid weight fraction of a solution dropletis principally a function of water vapourconcentration. If environmentalconditions change(forexamplethetemperature)sodoesthewatersaturationvapourpressure,andthis resultsin a changeof sulphuricacidweight fractionanddropletsize. However, thesolution dropletalwaysremainsinequilibriumwith respectto atmosphericwatervapour(SteeleandHamill, 1981).Therateof collisionof watermoleculeson a stratosphericparticleis muchhigherthanthatof gaseoussulphuricacid,soparticlesareassumedto reachwaterequilibrium instantaneously(foundby solvingEquation19).
2.4 Homogeneous Heteromolecular Nucleation
Theprocessof new particleformation(homogeneousnucleation)becomesimportantonly in areasoflow temperatureandhighambientsulphuricacidconcentration(Yue,1981;YueandDeepak,1982).The modelusesthe homogeneousnucleationtheoryto describenew particleformation. Althoughclassicaltheoryhasnever beendemonstratedto be accuratelyableto matchobserved ratesof nu-cleation,it hasbeena useful tool to predict trendsof new particleformationin the uppertropicaltroposphereandlowerstratosphere(Brocketal., 1995).Indeed,aftermajorvolcaniceruptions,suchasMt. Pinatubo,98% of theobservedaerosolis volatile (e.g. Deshleret al., 1992)indicatingthathomogeneousnucleationis themostimportantprocessfor stratosphericaerosolformation.Further-more,the large-scalefeaturesof the stratosphericaerosollayer arenot very sensitive to the rateofnucleationbut arecontrolledmoreby coagulationandgrowth processes(Weisensteinet al., 1997).This justifiesour choiceof modellingstratosphericaerosolsasconsistingsolelyof H ² SO³ -H ² O liq-uid droplets.Here,webriefly describetheclassicalnucleationtheory. For amoredetaileddescriptionthereaderis referred to, for instance,Kulmalaetal. (1998)andNoppeletal. (2002).
Theclassicalnucleationtheorydescribestheformationof a new phasefrom a motherphasethatis becomingunstable.The formationof new particlesis assumedto result from the formationofstableclustersthatwill be ableto grow spontaneouslyafter they reacha critical size. This criticalsizerepresents,in thecaseof homomolecularnucleationinvolving only onecondensingvapour, thesizeor thenumberof moleculesproviding themaximum of theGibbsfreeenergy change,resultingfrom the phasetransition. This maximumis thencalledthe free energy of formationof a particle.In the caseof a binarynucleation(herewaterandsulphuricacid), theGibbsfree energy changeisa functionof thenumberof sulphuric acidandwatermoleculesandis representedby a surface.Tobecomestable,a clusterwill have to follow thepathof leastenergy on this surface,thusleadingtotheso-calledsaddlepoint (Reiss,1950).Findingthesaddlepointusingiterativemethodsis compu-tationallydemanding.This is why we usedtheparameterizationof Kulmala et al. (1998),updated
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2 MODEL DESCRIPTION 2.5 Coagulation
by Vehkamaki et al. (2002). This schemeprovidesthenucleationrate ´'µ�¶�·�¸!¹ªºLµ�¶�¸ (numberof newparticlesformedpersecondandpercubicmeter)in thewater/sulphuricacidmixture,andthecriticalclustercomposition (total numberof moleculesandsulphuricacidmolefraction). This parameteri-zationis valid at temperaturesbetween190.15K and305.15K, relative humiditiesbetween0.01%and100%,andsulphuricacidconcentrationfrom »¼.½ to »¼¿¾�¾ cmºLÀ . This rangeof relative humiditycoversthevariationfound in theglobalstratosphericrelative humidity. Theseparameterizedequa-tionsreducethecomputingtime by a factorof 500comparedto non-parameterizednucleationratecalculations.
Sincein thepresentmicrophysicalmodelnucleationandgrowth arecoupled,thenucleationrateis first convertedtoamasstransferratebetweengasandparticle,andis thenaddedto thegrowthmasstransferrate(definedin section2.2) in thesizebin Á . More precisely, when Â�ÃoºLÄYÃÆÅÈÇYÉHÊ ÃoºLÄ?Ã�Â\ËHÌ_ÍÉHÊ ÃoºLÄ?Ãand ´.µ�¶�·�¸!¹ªºLµ!¶_¸NÅȼ thenthenucleationrateis convertedto a gas-phasetransferrateas(Jacobson,2002) ÎLÏÑÐ�ÒÉHÊ ÃoºLÄ?ÃSÓ ´.µ ¶ ·�¸ ¹ ºLµ ¶ ¸pÔ µ!¶�·�¸!¹ÖÕ!ÉZ×ÉØ µ ¶ ·ª¸ ¹ Ù »ÂUÃoºLÄ?ÃYÚXÇYÉHÊ ÃoºLÄ?ÃZ ËoÌ_ÍÉHÊ ÃoºLÄYÃ�Û¥Ü (20)
where Ô µ ¶ ·ª¸ ¹ is thedensityof puresulphuricacid (kg m ºLÀ ), Õ!É is thevolumeof a single nucleatedparticle (mÀ ), and ×ÝÉ is the volume fraction of sulphuricacid in a nucleateddroplet. The mass-transferratecalculatedfrom equation20 is thenaddedto thegrowth masstransferrate,calculatedfrom equation8, for theappropriatesizebin whereall thenucleatedparticlesareplaced.
After solving equations14-18 with the new transferrates(growth + nucleation),the numberconcentrationof new H Þ SO½ -H Þ O particlesformeddueto homogeneousnucleationin the bin Á iscalculatedas ß ÉHÊ Ã Ó ß ÉHÊ ÃoºLÄYÃ'àâáäã"åSævç Ü ¼.è Ü (21)
where ç Ó ævéÖÉHÊ ÃYÚxéêÉHÊ ÃoºLÄ?Ã|è Ø µ ¶ ·�¸ ¹Ô µ�¶�·�¸!¹ÖÕ!ÉZ×ÉÎ ÏÑÐ�ÒÉHÊ ÃoºLÄ?ÃÎ Íwë�ÍÉHÊ ÃoºLÄ?à (22)
and
Î Íwë�ÍÉHÊ ÃoºLÄ?Ã Ó Î ÏÑÐ�ÒÉHÊ ÃoºLÄ?à à ÎLì�í ë|îÍwïÉHÊ ÃoºLÄ?à .2.5 Coagulation
Aerosolcoagulationis importantbecauseit altersthe sizedistribution andthe composition of par-ticles,primarily thosesmallerthanonemicron in diameter. Sincemostof thebackgroundaerosolsfound in the stratosphereareof this size, it is necessaryto includecoagulationwithin the model.Atmosphericparticlescollideasa resultof Brownianmotion, differencesin fall velocities,turbulentmotionandinter-particleforces. Differentapproacheshave beendevelopedin the pastto simulatecoagulation,dependingupontheneedandavailability of computerresources.Sincethemodelhastobeincorporatedinto a 3D globalchemicaltransportmodel,wehaveconcentratedonasemi-implicitschemewhich is computationally fast. Aerosolcoagulation, in the discreteform, is given by (e.g.Friedlander, 2000) ð ß�ñ Ê Ãð'ò Ó »ó ñ º ¾ôõQö ¾L÷ õ Ê ñ º õ ß�ñ º õ Ê Ã ß õ Ê ÃuÚ ßøñ Ê Ãúùôõªö ¾�÷ ñ Ê õ ß õ Ê Ã Ü (23)
where ÷ ñ Ê õ is thecoagulationkernelof two colliding particles(m À sº ¾ ). Usingsemiimplicit finitedifferencing,we obtaina generalformulafor a volume-conservingsolution of uniform composition
� �û� . The generalisedBrownian coagulationkernel,
� � ÿ � , is taken from Fuchs’(1964)interpolation formula� � ÿ � � L1M #ON � � N � & #QPSR� � PTR� &U 8 @ U ;U 8 @ U ; @�VXW5Y8 @ W<Y;[Z 6]\ Y � ^ V`_ba8 @c_ba;dZV U 8 @ U ; Z VeV 0 a8fZ Y @�V 0 a;�Z Y Z 6]\ Y ' (26)
wheretheparticlediffusioncoefficient P R� is givenby
P R� � ( A�gihkjl M N �]m1n ' (27)
wheretheCunningham-slip factor, h%j , is definedashkj � � � �po � � �1q`r L7s � * q L r I ��t"u vxwzy5{ |~} � q (28)
The Knudsennumber�po � ��� n9� N � is determinedfrom the meanfree pathof air molecules,� n ,
which is givenby � n � r m1n3���=n9��n ' (29)
where m�n is thedynamicair viscosity(kg m � � s� � ) calculatedfrom List (1984):
m�nk� �1qX�7����� * ��w�� L �dl�qe�dlg � ��r *!� � gr s l�q]��l �E�Y > (30)�=n is theair densityand �7n is thethermalvelocityof air molecules(m s� � ) calculatedfrom��n �)� � (7� g%� M�� n (31)
where ¦� is theparticlethermalvelocity(m s�c� ) . Othercoagulationkernels(gravitationalcollection,turbulentinertialmotion andturbulentshearmotions)arenotincludedbecausethey aremuchsmallercomparedto theBrowniankernelin typicalstratosphericconditions.
The coagulationequationsfor particlesof different composition (for example water-sulphuricacid)canbewritten in asimilarmannerto Equation24:µf¶K·¸¶º¹ » � µc¶K·¼¶º¹ » ��½ » ¤¾�¿ À ¶Á� ��à À ¶ �c��  ��Ä � ¹ Á ¹ ¶KÅ � ¹ Á µ � · � ¹ »®· Á ¹ » ��½ »�ÆÇ ¤�¾�¿cÀ�È!ÉÁ" � Ç ¨ Ä ¶[¹ Á ¹ ¶ § ÅT¶º¹ Á · Á ¹ » ��½ » Ê (34)
Browniancoagulationkernelsobtainedfrom equation26 at 216K areplottedin figure2. Fromthis plot, it is clearthat thesmallestvalueof thekernelsoccurswhenbothparticlesareof thesamesize. The coagulationkernelrisesrapidly whenthe ratio of the particlesradii divergesfrom unity.This increaseis dueto the interplaybetweensizeandvelocity of two colliding particles. A largeparticlehasasmallvelocitybut it providesalargesurfaceareacomparedto averyfastsmallparticle,thustheresultis alargecoagulationrate.In theeventof two smallparticlescolliding,thecoagulationkernelis smallbecauseof their smallcross-sectionalarea.
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2.6 Sedimentation 3 SELECTIONOF MODEL PARAMETERS
2.6 Sedimentation
Thesedimentationmechanismis responsiblefor thedescentof particlesfromonelayertoanotherandthusit affectsthesizedistribution of stratosphericparticlesundertheinfluenceof gravity. Assumingthat all the particlesarespherical,sedimentationvelocity, ËcÌÎÍ®ÏÐ , is given as(PruppacherandKlett,1997) Ë ÌÎÍ®ÏÐ Ñ±Ò Ó®Ô Ð!Õ Ô=Ö[×�ØkÙzÚ�ÛÐ[ÜÝ1Þ Ö ß (35)
whereÜ is thegravitationalconstantin m sà Û .3 Selection of Model Parameters
Computerprocessingmemoryis animportantissuein globalsimulations.Thepresentmicrophysicalmodelis intendedto beincludedin a 3D globalchemicalmodel(SLIMCAT) to simulate theeffectsof aircraftemissionsonbackgroundaerosols.Keepingthis in mind,wehaveperformedseveralsen-sitivity studiesin orderto seetheeffectsof thechoiceof thenumberof sizebinsusedonthepredictedaerosolsizedistribution features.The upperandlower sizelimits adoptedfor SAMM areroughly1.5 á m and0.3nmrespectively. Thisuppersizelimit is basedonthefactthatin typicalstratosphericconditionsvery few water-sulphuricacid particlesbecomelarger thanthis size,and thosethat dorapidlysediment.Thelowersizelimit is setby thesizeof thenewly formedparticlesestimatedfromclassicalnucleationtheory. Whenusingalargernumberof sizebins,theaerosoldistribution is betterresolved.But theintegrationtimestepis alsoa very importantparameter. Thesmallerthetimestep,themoreaccuratetheresultsthatareexpected.Weperformedabenchmarkcalculationusingroughly300sizebins(against44, for example,in TimmreckandGraph’s runs,2000)anda relatively smalltimestepof integration(600s,whichis 3 timessmallerthantypicaldynamicaltimestepsusedin 3Dmodels).Suchacalculationprovidesaccurateresultsbut is computationallyslow anddemanding. Inorderto find thebestcompromisebetweenaccuracy andcomputing time required,we performedasensitivity study:thiswasdoneby changingthenumberof binsandtimesteps.Results(surfaceareadensity, particlenumberconcentration,volume density)have thenbeencomparedto our referencecase,the benchmarkrun. Figure3 (top) plots the backgroundaerosolsurfaceareadensity(SAD,surfaceper unit volume of air) asa function of the numberof sizebins and the integrationtime-stepfor conditionstypical of analtitudeof 20 km. ThecalculatedSAD lie between0.5â 10 à�ã and6.5â 10à�ã cmÛ cmà�ä (0.5and0.65 á mÛ cmà�ä ). Thesevaluesarecomparableto thoserecommendedfor heterogeneouschemicalreactionsof atmosphericimportance.Tie et al. (1994)reporteda valueof 0.6 á mÛ cmà�ä for aerosolSAD at 20 km from their 2D modelcalculations. It canbe seenthatlittle accuracy is gainedby decreasingthemodeltimestepbelow about2000s.
Thereforeall thestandardmodelrunsusedan1800s time-stepwhich is indicatedon thefigure(dashedline). It canalsobeseenthat thenumberof sizebinsdoesnot have a pronouncedeffect onthecalculatedaerosolSAD, sincethemodelresultsdiffer by only 25%using10or 300bins.Figure3 (bottompanel)plotsthevariationof particlesurfacedensitywith numberof sizebinsfor an1800s time-step.Thesurfaceareadensityreachesits asymptotic valuewhenthenumberof binschosenis greaterthan åçædè1è . Fromthisplot, wecanestimatethedifferenceof aerosolSAD betweenthe40binscaseandthebenchmark(300bins)to bearound5%.
Figure3: Top: Aerosolsurfaceareadensity(mé m ê�ë ) plottedasa function of the numberof sizebinsusedandtheintegrationtime-step.Bottom:Aerosolsurfaceareadensityvsnumberof sizebinsfor an integration time-stepof 1800s. Initial parametersadoptedare: temperature= 216K, watervapourpressure= 3.68mb,sulphuric acidconcentration= 0.048 ì g m ê�ë .
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3 SELECTIONOF MODEL PARAMETERS
Figure4: Top: Percentagedifferencein total particlevolumedensityrelative to benchmarkvalue.Thebenchmarkvalueis thebestestimateof particledistribution using306sizebinsand600s inte-grationtimestep.Middle: Sameastheplot at thetopexceptfor aerosolsurfaceareadensity. Bottom:Sameasthetoponeexceptfor numberconcentration.
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3 SELECTIONOF MODEL PARAMETERS
Figure5: Rootmeansquareerrorasa functionof numberof sizebinsandintegrationtime-stepfor asizedistribution relative to thebenchmarksizedistribution.
In figure 4 (top) we plot the percentagedifferencefrom thebenchmarkrun of the total particlevolumedensityobtainedusingdifferentnumbersof binsandintegration time-steps.
First, thepercentagedifferencedoesn’t dependon the numberof binsor the time stepused,aswe find thesamevalueusing20 or 200binsat thesametime step(e.g.0.0014around2000s). Thesecondpoint is thatthepercentagedifferenceis verysmall(lessthan0.5%)andshows how accuratethe numericalschemeis as it conserves volume. Theseobservations indicatethat the differenceobserved betweenthe benchmarkrun and other calculationsis not significant. Figure 4 (middlepanel),shows the percentagedifference(from thebenchmark)of theaerosolSAD asa functionofthe numberof sizebins and the integration time-stepused. The percentagedifferencein surfaceareadensityincreasesas the numberof bins decreases,which is as expected,but remainsfairlysmall(maximum of í)î1ï7ð of difference).Figure4 (bottompanel)shows thepercentagedifferencein particle numberconcentration( ñ7ò�ó moment)plotted as a function of numberof size bins andintegrationtime-step.We seethatpercentagedifferenceincreasesasnumberof sizebinsdecreases.This clearlyindicatesthatusingcoarseresolutionbinsresultsin only asmallerrorwhencalculatingtotal numberconcentrations,surfaceandvolumedensities.Our mainconcernremainstheability togetanaccuratevaluefor theSAD, asheterogeneouschemistrystronglydependson it. Hence,thesetestsaresatisfactorybut wehaveto keepin mindthatthetail of theaerosolsizedistribution couldbeartificially broadenedwhenusingasmallnumberof bins.
Figure 5 shows this result more explicitly, giving the ( ð ) root meansquare(rms) differencebetweenthecalculatedsizedistribution at20km andthebenchmarkdistribution. For an1800stimestep,the% rmsdifferenceis ashighas40% whenthenumberof binsis about40.
A sensitivity studywasalsoperformedto seetheeffect of theH ô SOõ molecules’stickingcoeffi-
13
4 VALIDATION
cient ( öø÷ ) on themodelledsizedistribution. Changingöù÷ from 0.01to 1 did not make a significantdifferenceto themodelledsizedistribution. Hence,we usea fixedvalueof 1 for öú÷ in SAMM. Thesamevaluehasbeenusedto describethestickingcoefficient betweencolliding particles(i.e. theircollisionefficiency), assulphuricacid-waterparticlesarelikely to attachveryefficiently.
A time stepof 1800s and a bin numberof approximately40 (dependingon the geometricalincreaseused)wasadoptedfor SAMM. Basedon this evaluation,we expectcomputationalerrors û0.005%, 5 % and40% in thevolume,surfaceareaandnumberdensitiesrespectively.
4 Validation
The coagulationschemeandits efficiency asa functionof the numberof sizebinsusedhave beeninvestigated.Figure6 comparesthesemi-implicit solution after12hoursof simulationto theSmolu-chowski’sanalyticalsolution givenbyü¸ýºþ ÿ������ ü ÿ���� �1ü¼ÿ�� ÷ þ ������� ý������������ �1ü¼ÿ�� ÷ þ ������� ý ���"! (36)
.Figure6 showsthatthesolution from thesemi-implicit numericalapproachmatchesveryclosely
theanalyticalsolutionwhenthenumberof binsusedis large. It confirmsalsothatthetail of thesizedistribution is dramaticallymodifiedwhenusingasmallnumberof bins.
Figure7 (top) shows theevolution of particles’sizespectrafor coagulationalone,usinga timestepof 600s. Figure7 (bottom)plotstheevolution of particlesizedistribution dueto all processeshappeningtogether. Comparingfigures7 top andbottom,onecannotethatdueto growth, particlesarebecominglarger andthesizespectrumis shifting moretowardsthe largerparticles. It canalsobe seenthat large particlesareremovedandthesizedistribution shifts towardssmallersizeswhensedimentationis included. An importantfinding is that resultsappearedto be very sensitive to thesedimentationschemeused. In particular, whenusinga single box model, whenmasshasto beconserved, the questionof replacementof the sulphuricacid lost by sedimentationis important.Replacingit by gaseoussulphuricacidcanleadto aburstof nucleation(TimmreckandGraf,2000).In reality, the lossof massby sedimentationis balancedby falling particlesfrom upperlayersandalsoby advectionor evenconvection.In thepresentstudy, transportphenomenahavebeenneglected.Hence,all the sulphuricacid lost by sedimentation hasbeenreplacedby the samemassin the gasphase.
We have performeda numberof validationstudieswith this modelandcomparedthecalculatedsize-distributionparameterswith someobservedvalues,ananalyticalsolutionandabenchmarkrun.Thesestudiesareimportantat thispointandhaveservedto testtheability of themodelto reproducetheparametersof stratosphericaerosolsbeforecouplingit to globalmodels.
Severalmeasurementsof stratosphericaerosolsizedistributionexist, from in situ(opticalparticlecounter)to remotesensing(e.g. satellite,lidar). To retrieve particlesizedistribution, remotesens-ing techniquesrely on additional information from theoreticalmodels;hencewe choseto comparethe modelledaerosolpopulation parameterswith only in situ observations andpreviousmodellingstudies.
Figure6: Comparisonof semi-implicit resultswith Smoluchowski’s analyticalsolution for differentnumbersof sizebins.
15
4 VALIDATION
Figure7: Top: Evolution of particlesizedistributionundergoingBrowniancoagulationonlyatdiffer-enttimes.Bottompanel:Evolutionof nucleatedparticlesizedistribution by condensationalgrowth,coagulationandsedimentationat differentsimulation times,1 year, 2 yearsand3 years(1y, 2y, 3y).
16
4 VALIDATION
Figure8: Top: Zonally averagedtemperatureprofile for Laramie,Wyoming taken from ECMWFreanalysisdata.Middle: Sameasthetoppanelexceptfor watervapourpressure.Bottom:Sulphuricacidconcentrationprofile astakenfrom Toonet al. (1979).
In the following experiments,we usedzonallyaveragedtemperatureandrelative humidity pro-files from ECMWF reanalysisdatafor Laramie,Wyomingasgivenin figure8, topandmiddlepanelrespectively, wheremostof the stratosphericaerosolobservationsarebeingcarriedout. Thereisno consensusassuchwith respectto the sulphuric acid gaseousconcentrationin the stratosphere,mostof which is foundin particlephase.Thereareseveralwaysto initialisethemodelfor sulphuricacidgaseousconcentration:for exampleby coupling theaerosolmodelanda chemicalmodelwithsulphurchemistry. However, at eachaltitudewe usedthesulphate concentrationdatafrom Toonetal. (1979),plottedin figure8 (bottom).
Thefigure9 plotsshow theevolution of theaerosolfeatureswith altitude. Thesurfaceareaandthe volumedensities decreasewith altitude as temperaturerisesand sulphuric acid concentrationdecreases.Undersuchconditions,nucleationis thenlessefficientatproducingnew particles,andthecalculatedprofilesobtainedasa functionof altitudewereasexpected.
In figure 10 we plot model-simulatedaerosolmixing ratios (N15, number of particleslargerthan0.15 3 m permassof air) togetherwith observedvaluesfrom HofmannandRosen(1981)andpreviousmodelresults(notethatfor particleslargerthan0.25 3 m,themixing ratioisN25). Whatever
thealtitude,themodelseemsableto reproducethetrendsfor JanuaryandApril. Theaccuracy of theJulyandOctoberresultsis lowerbut remainssatisfying,excluding thecaseof July, at25km. As themixing ratio is a functionof particlenumberdensityandalsoof air density, thesteepincreaseat thisaltitudedoesn’t necessarilymeanthatmoreparticlesareformedathigheraltitudes.Our calculationsindicatethatfewerparticlesareformedat higheraltitudes(asexpectedfrom largertemperaturesandlower sulphuric acid concentrations)but the decreasein the air densityleadsto an increaseof theparticlemixing ratio. For JanuaryandApril, the modelis ableto broadlyreproducethe changeinaerosolmixing ratio with altitude, asobserved by HofmannandRosen(1981)andpredictedfromToon et al. (1979)using the Turco et al. (1979)model. In figure 11 we plot the predictedsizeratio (N15/N25)asa function of altitude. It is clear that modelledvaluesof sizeratio matchwellwith observedvaluestakenfrom HofmannandRosen(1981)exceptfor thecaseof April at 25 km.The main differencesbetweenour calculationsandTimmreck andGraf’s model (2000)probablylie in the methodusedto determinethe gaseoussulphuric acid concentration.We useda profiledeterminedfrom Toon et al. (1979), as Timmreck and Graf (2000) useda 3D model output todeterminethe main initial mixing ratios(watervapour, sulphurdioxide), andalsocalculateda rateof transformationof SO4 into H 4 SO5 . As nucleationschemesaresensitive to the initial amountofsulphuricacid (e.g. Vehkamaki et al., 2002), large differencescould comefrom differentH 4 SO5initialisations.Moreover, discrepanciesbetweenour resultsandmeasurementsmaycomefrom thefact we usedzonally averagedtemperaturefrom ECMWF to get the atmosphericconditions, andnot exactly the experimentalones,in Laramie. Finally, figure 12 illustratesthe behaviour of themixing andthe sizeratio (MR andSR respectively) asa function of the gasphasesulphuric acidconcentration.Whentheinitial amountof acidis high,bothnucleationandthesubsequentgrowth of
18
4 VALIDATION
Figure 10: Top left panel: Numericalvaluesof particle mixing ratio (MR), ( 6 ( 798 0.15 : m)),as a function of altitude for Januaryconditions. Also shown are observed valuestaken from themeasurementsof Hofmannand Rosen(1981) and modelled valuesfrom Toon et al. (1979) andTimmreckandGraf (2000). Thesameresultsarepresentedfor April (top right), July (bottomleft)andOctober(bottomright) conditions.
19
4 VALIDATION
Figure11: Topleft panel:Numericalvaluesof particlesizeratio(SR), ( ; ( <>= 0.15 ? m)/;A@B<C= 0.25? m)), asa functionof altitude for Januaryconditions. Also shown areobservedvaluestaken fromthemeasurementsof HofmannandRosen(1981)andmodelledvaluesfrom Toonet al. (1979)andTimmreckandGraf (2000). Thesameresultsarepresentedfor April (top right), July (bottomleft)andOctober(bottomright) conditions.
20
4 VALIDATION
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Sulfuric acid concentration µg/m3
2
4
6
8
10
12
14
Mix
ing
ratio
(nu
mbe
r/m
g ai
r) a
nd S
ize
ratio
Mixing ratioSize ratio
Figure12: Evolutionof themixing andthesizeratioof particles,asafunctionof theinitial gasphaseH D SOE concentration,at16km.
21
4 VALIDATION
0 6 12 18Time after eruption (m)
0.1
1
10
100
1000
Aer
osol
sur
face
are
a,
µm2 cm
-3
SO2, t=0
=4.95µgm-3
SO2,t=0
=15µgcm-3
Timmreck’s box model resultsObservation from Deshler et al. (1993)
Figure13: Evolutionof themixing andthesizeratioof particles,asafunctionof theinitial gasphaseH F SOG concentration,at16km.
22
6 CONCLUSIONS
particlesareenhanced,producinghighnumberconcentrationsof particleslargerthan0.15 H m (N15).Thesolid line plot showsthistrendfor analtitudeof 16km. As moresulphuricacidis available,N25increasesfasterthanN15, leadingto a total decreaseof thesizeratio (SR),asplottedby thedashedline andasexpected.
5 Simulation of Mt. Pinatubo Volcanic Aerosols
After major volcaniceruptions,large amountsof watervapour, SOI andashparticlesare injectedinto theatmosphere.
An enhancedSOI concentrationhasa major impacton the stratosphericbackgroundaerosol.Changesin H I O andOH concentrationcanalsoinfluencethe aerosolsizedistribution. Neverthe-less,thereis no agreementasto the amountof watervapourreachingthe stratosphere.Therefore,changesin H I O concentrationsareneglectedwhensimulatingvolcaniceruptions.Theinitial aerosolsizedistribution is assumedto belog-normalfor a totalnumberconcentrationJLK = 10M m NPO , a stan-darddeviation Q = 1.8 anda meanradiusr R 7.10NPS m. Typical stratosphericatmosphericaerosoldistributionscanberepresentedby:JUTWV JXKY T[Z]\^Q0_ `badc�egf hikjml�n oqp Zr\ts Y T�u Y RwvZr\^Q x Izy{ n
(37)
A comparisonbetweenthe calculatedevolution of aerosolsurfaceareaand measurementsbyDeshleret al. (1993)at 20 km height is given in figure 13. We seethat, for initial SOI concen-trationsof 4.95 H g m NPO , simulatedaerosolsurfaceis a reasonablematchcomparedto theobservedvalues.However, for larger initial concentrationsof SOI themodelfails to reproducetheobserveddevelopmentof aerosolsurfacearea.
6 Conclusions
A fastaerosolmicrophysicalmodelfor thestratospherehasbeendeveloped. It simulateshomoge-neousnucleationof sulphuricacid-wateraerosols,condensationalgrowth, coagulationamongparti-clesandsedimentation. Themodelusesa non-iterative schemeto solve final numberconcentrationof particlesundergoing theseprocesses.It is computationally fast: this makesthe model ideal forimporting into 3D globalmodels. Thepresentmodelhasbeenusedto simulatebackgroundstrato-sphericaerosolsizedistribution. The modelled sizedistribution parametersmatchwell within therangeof observedandearlierreportedvalues.Two typesof sensitivity studiesareperformedto seetheeffectsof (a)changesin numberof sizebinsandtimestepand(b) changesin environmentalinputparameterson aerosolsizedistribution. We find that themodelis sensitive to thechoiceof numberof sizebins used. However, changein backgroundaerosolsurfaceareais not significantbetweenhighandlow timeandsizeresolvedexperiments.Themodelledsizedistribution is foundto bemostsensitive to sulphuric acid gaseousconcentration.A simplified caseof Pinatubovolcaniceruptionis usedto simulatethe aerosoltemporaldevelopment. Thesefirst resultsareencouragingandwillbe followedby furtherdevelopment. Themodelthathasbeenconceived,keepingin mind theneedfor computing efficiency, will soonbe includedin SLIMCAT, andwill provide a powerful tool foraircraftemissionsglobalimpactstudies.
23
6 CONCLUSIONS
Notation
a Offsetfor | lineardependenceon temperature.A Avogadronumber.b Slopefor | lineardependenceon temperature.c}]~ � Mole concentrationof H � SO� in aparticlefrom bin i at time t, in molem �P� .C� Mole concentrationof gaseousH � SO� in theatmosphereat time t in molem �P� .C�]���}r~ � Mole concentrationof saturatedH � SO� vapourin theatmosphereattimet, above
aparticlefrom bin � , in molem �P� .Cc Cunninghamslip factor.� } Meaninter-particledistance(centreto surface),for bin � , in m.d Diameterof amoleculeof H � SO� in m.D } , D �} H � SO� andparticlediffusioncoefficient, in m� s��� .f } Volumefractionof H � SO� in acritical clusterplacedin bin i.F} Ventilationfactorfor aparticlein bin � .g Gravitationalconstantin m s� � .� } H � SO� activity in asolution having thesamecompositionasadropletin bin � .k }]~ � Masstransferrates��� of gaseousH � SO� ontoaparticlefrom bin � .k ���z�}]~ � , k ���r�� �]�}]~ � Sameask }]~ � for nucleationandgrowth (for bothsimultaneously, thesuperscript���[�
is used).
k � Boltzmannconstantin JK ��� .K }]~ � Coagulationkernelfor acollisionbetweenparticlesfrom binsi andj, in m � s��� .K Prefactor similar to k.Kn } Knudsennumberfor aparticlein bin � .�
,� � , � �} Respectively air, sulphuric acidandparticlemeanfreepath,in m.� � Dynamicviscosityof air in kg m ��� s��� .
m� , m���¡ z¢�£ Molecularweightof air andsulphuricacidrespectively, in kg mole��� .n} Numberof H � SO� moleculescontainedin aparticlefrom bin � .N, N }]~ � Respectivelyparticlenumberdensityandparticlenumberdensityin bin � , at time
t in m �P� .N ¤ Particletotalnumberdensity, in m �P� .p����¢ Watervapourpressure,in Pa.p�¥�������¢ Watersaturationvapourpressureaboveadroplet,in Pa.p H � SO� vapourpressurein Pa.p�¥��� H � SO� saturationvapourpressureaboveaflat surfaceof pureH � SO� in Pa.
¦®Densityof air, puresulphuricacidandparticlein bin ¯ respectively, in kg m °P± .
r
Radiusof aparticlein bin ¯ , in m.
r ² Meanradiusof thelognormalsizedistribution, in m.R UniversalgasconstantJmole°�³ K °�³ .´ Particle(bin i) surfacetensionin J m °Pµ .s Standarddeviation for a lognormal distribution.S
]¶ ·Saturationratio of H µ SO above a particle (bin i, time t) composedof pureH µ SO .¹ ]¶ º�¶ »Bin partitioning function.
t Time in s.T Temperaturein K.v Wind velocity in m s°�³ .v ¼r½B¾ Sedimentationvelocityof aparticle,in m s°�³ .v
s°�³ .À ² ¶ Volumeof H µ SO permolein a solutionhaving thesamecomposition asa par-ticle from bin ¯ , in m± mole°�³ .
V
Volumeof aparticlefrom bin ¯ , in m ± .
W
Sulphuricacidweightfractionin aparticlefrom bin ¯ .
Acknowledgments
Theauthorsacknowledgefundingof thisworkbyaNERCUTLS-OzoneGrantNER/T/S/2000/01032.Theauthorswouldalsolike to thankH. Vehkamaki for providing hernucleationmodelandfor help-ful discussions.
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