Quantifying Aerosol Mixing State With Entropy And .

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Open AccessAtmosphericChemistryand PhysicsAtmos. Chem. Phys., 13, 11423–11439, 194/acp-13-11423-2013 Author(s) 2013. CC Attribution 3.0 License.Quantifying aerosol mixing state with entropy anddiversity measuresN. Riemer1 and M. West21 Department2 Departmentof Atmospheric Sciences, University of Illinois at Urbana-Champaign, IL, USAof Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, IL, USACorrespondence to: N. Riemer (nriemer@illinois.edu)Received: 12 May 2013 – Published in Atmos. Chem. Phys. Discuss.: 12 June 2013Revised: 14 September 2013 – Accepted: 30 September 2013 – Published: 25 November 2013Abstract. This paper presents the first quantitative metricfor aerosol population mixing state, defined as the distribution of per-particle chemical species composition. This newmetric, the mixing state index χ, is an affine ratio of theaverage per-particle species diversity Dα and the bulk population species diversity Dγ , both of which are based oninformation-theoretic entropy measures. The mixing state index χ enables the first rigorous definition of the spectrumof mixing states from so-called external mixture to internal mixture, which is significant for aerosol climate impacts,including aerosol optical properties and cloud condensationnuclei activity. We illustrate the usefulness of this new mixing state framework with model results from the stochastic particle-resolved model PartMC-MOSAIC. These resultsdemonstrate how the mixing state metrics evolve with timefor several archetypal cases, each of which isolates a specificprocess such as coagulation, emission, or condensation. Further, we present an analysis of the mixing state evolution fora complex urban plume case, for which these processes occursimultaneously. We additionally derive theoretical propertiesof the mixing state index and present a family of generalizedmixing state indexes that vary in the importance assigned tolow-mass-fraction species.1IntroductionOur quantitative understanding of the aerosol impact on climate still has large gaps and hence introduces large uncertainties in climate predictions (IPCC, 2007). One of the challenges is the inherently multi-scale nature of the problem:the macro-scale impacts of aerosol particles are governedby processes that occur on the particle-scale, and these microscale processes are difficult to represent in large-scalemodels (Ghan and Schwartz, 2007).An important quantity in this context is the so-called mixing state of the aerosol population, which we define as thedistribution of the per-particle chemical species compositions. Recent observations made in the laboratory and in thefield using single-particle measurement techniques have revealed that the mixing states of ambient aerosol populationsare complex. Even freshly emitted particles can have complex compositions by the time they enter the atmosphere. Forexample, the mixing state of particles originating from vehicle engines depends strongly on fuel type and operatingconditions (Toner et al., 2006). The initial particle composition is further modified in the atmosphere as a result of agingprocesses including coagulation, condensation of secondaryaerosol species, and heterogeneous reactions (Weingartneret al., 1997).While the extent to which mixing state needs to be represented in models is still an open research question, thereis evidence that mixing state matters for adequately modeling aerosol properties such as optical properties (Jacobson,2001; Chung and Seinfeld, 2005; Zaveri et al., 2010), cloudcondensation nuclei activity (Zaveri et al., 2010), and wet removal (Koch et al., 2009; Stier et al., 2006; Liu et al., 2012).Therefore, in recent years, efforts have been made to represent mixing state in models to some extent. This is the casefor models on the regional scale (Riemer et al., 2003) as wellas on the global scale (Jacobson, 2002; Stier et al., 2005;Bauer et al., 2008; Wilson et al., 2001).In discussions about mixing state, the terms “external mixture” and “internal mixture” are frequently used to describePublished by Copernicus Publications on behalf of the European Geosciences Union.

11424how different chemical species are distributed over the particle population. An external mixture consists of particlesthat each contain only one pure species (which may be different for different particles), whereas an internal mixturedescribes a particle population where different species arepresent within one particle. If all particles consist of the samespecies mixture and the relative abundances are identical, theterm “fully internal mixture” is commonly used.While these terms may be appropriate for idealized cases,observational evidence shows that ambient aerosol populations rarely fall in these two simple categories. In this paperwe present the first quantitative measure of aerosol mixingstate, the mixing state index χ, based on diversity measuresderived from the information-theoretic entropy of the chemical species distribution among particles.The measurement of species diversity and distribution using information-theoretic entropy measures has a long history in many scientific fields. In ecology, the study of animaland plant species diversity within an environment dates backto Good (1953) and MacArthur (1955), but rose to prominence with the work of Whittaker (1960, 1965, 1972). Whittaker proposed measuring species diversity by the speciesrichness (number of species), the Shannon entropy, and theSimpson index, which are now referred to as generalized diversities of order 0, 1, and 2, respectively (see Appendix Afor details).Whittaker also introduced the fundamental concepts of alpha, beta, and gamma diversity, where alpha diversity Dαmeasures the average species diversity within a local area,beta diversity Dβ measures the diversity between local areas,and Dγ measures the overall species diversity within the environment, given as the product of alpha and beta diversities.In the context of aerosols, we regard alpha diversity as measuring the average species diversity within a single particle,beta diversity as quantifying diversity between particles, andgamma diversity as describing the overall diversity in bulkpopulation (see Section 2 for details). From these measureswe construct the mixing state index χ as an affine ratio ofalpha and gamma diversity.The ecology literature in the 1960s and 1970s containsmuch work on species diversity and distribution, althoughthere was also significant confusion about the underlying mathematical framework (Hurlbert, 1971; Hill, 1973).Within the last decade, the profusion of diversity measures have been largely categorized (Tuomisto, 2013, 2012,2010), although disagreement in the literature is still present(Tuomisto, 2011; Gorelick, 2011; Jurasinski and Koch, 2011;Moreno and Rodríguez, 2011). Despite the current controversies, certain principles are now well-established, such asthe use of the effective number of species as the fundamentally correct way to measure diversity (Hill, 1973; Jost, 2006;Chao et al., 2008, 2010; Jost et al., 2010).As well as the basic mathematical framework of measuring diversity, there has also been much effort on understanding its ecological impacts, with a particular interest inAtmos. Chem. Phys., 13, 11423–11439, 2013N. Riemer and M. West: Quantifying aerosol mixing statethe relationship between ecosystem stability and diversity(MacArthur, 1955; Goodman, 1975; McCann, 2000; Ives andCarpenter, 2007). Other important research questions includethe sources of diversity (Tsimring et al., 1996; De’ath, 2012),extensions of diversity to include a concept of species distance Chao et al. (2010); Leinster and Cobbold (2012); Feoli(2012); Scheiner (2012), and techniques for measuring diversity (Chao and Shen, 2003; Schmera and Podani, 2013;Gotelli and Chao, 2013), despite the well-known difficultiesin estimating entropy in an unbiased fashion (Harris, 1975;Paninski, 2003). Beyond ecology, the study of diversity isalso important in economics (Garrison and Paulson, 1973;Hannah and Kay, 1977; Attaran and Zwick, 1989; Maliziaand Ke, 1993; Drucker, 2013), immunology (Tsimring et al.,1996), neuroscience (Panzeri and Treves, 1996; Strong et al.,1998), and genetics (Innan et al., 1999; Rosenberg et al.,2002; Falush et al., 2007).This paper is organized as follows. In Sect. 2 we definethe well-established entropy and diversity measures, adaptedto the aerosol context, and use these to define our new mixing state index χ. This section also contains examples of diversity and mixing state and a summary of the properties ofthese measures. Section 3 presents a suite of simulations forarchetypal cases using the stochastic particle-resolved modelPartMC-MOSAIC (Riemer et al., 2009; Zaveri et al., 2008).These simulations show how the diversity and mixing statemeasures evolve under common atmospheric processes, including emissions, dilution, coagulation, and gas-to-particleconversion. A more complex urban plume simulation is thenconsidered in Sect. 4, for which the above processes occursimultaneously. Appendix A presents a generalization of thediversity and mixing state measures to ascribe different levels of importance to low-mass-fraction species, while Appendix B contains mathematical proofs for the results summarized in Sect. 2.2Entropy, diversity, and mixing state indexWe consider a population of N aerosol particles, each consisting of some amounts of A distinct aerosol species. Themass of species a in particle i is denoted µai , for i 1, . . . , Nand a 1, . . . , A. From this basic description of the aerosolparticles we can construct all other masses and mass fractions, as detailed in Table 1. Using the distribution of aerosolspecies within the aerosol particles and within the population, we can now define mixing entropies, species diversities, and the mixing state index, as shown in Table 2. Notethat entropy and diversity are equivalent concepts, and thateither could be taken as fundamental. We retain both in thispaper to enable connections with the historical and currentliterature.The entropy Hi or diversity Di of a single particle i measures how uniformly distributed the constituent species arewithin the particle. This ranges from the minimum valuewww.atmos-chem-phys.net/13/11423/2013/

N. Riemer and M. West: Quantifying Aerosol MixingState11425Table 1. Aerosol mass and mass fraction definitions and notations.The number of particles in the population is N, and the number ofspecies is A.Di 1 Di 1.4 Di 1.9 Di 2QuantityMeaningµaimass of species a in particle iµi µa AXa 1NX1µaitotal mass of particle iµaitotal mass of species a in populationµ µii 1µaipia µµiipi µµaap µtotal mass of populationper-particlediversity mass fraction of particle i in populationmass fraction of species a in populationDβ {z}inter-particlediversity Dγ . {z}120125130135(1)bulk populationdiversityAlpha diversity Dα measures the average per-particle effective number of species in the population, and ranges from1 when all particles are pure (each composed of just onespecies, not necessarily all the same), to a maximum whenall particles have identical mass fractions. Gamma diversityDγ measures the effective number of species in the bulk population, ranging from 1 if the entire population contains justone species, to a maximum when there are equal bulk massfractions of all species. Beta diversity Dβ is defined by anaffine ratio of gamma to alpha diversity, so it measures interparticle diversity and ranges from 1 when all particles haveidentical mass fractions, to a maximum when every particleis pure but the bulk mass fractions are all equal. Table 3 sumwww.atmos-chem-phys.net/13/11423/2013/3possibly even below 2 if the distribution is very unequal.mass fraction of species a in particle i(Hi 0, Di 1) when the particle is a single pure species,to the maximum value (Hi ln A, Di A) when the particle is composed of equal amounts of all A species. As shownin Fig. 1, the diversity Di of a particle measures the effective number of equally distributed species in the particle. Ifthe particle is composed of equal amounts of 3 species thenthe number of effective species is 3, for example, while 3species unequally distributed will result in an effective number of species somewhat less than 3.Extending the single-particle diversity Di to an entire population of particles gives three different measures of population diversity. Alpha diversity Dα measures the average perparticle diversity in the population, beta diversity Dβ measures the inter-particle diversity, and gamma diversity Dγmeasures the bulk population diversity. The bulk populationdiversity (Dγ ) is the product of diversity on the per-particlelevel (Dα ) and diversity between the particles (Dβ ), givingDα {z}Di 3Fig.Di Dofi representativeparticles.The f heeffectivenumberofspecieswithinaparticle diversity measures the effective number of speciesparticle,soapuresingle-speciesparticlehasD 1andaparticleiwithin a particle, so a pure single-species particle has Di 1consisting of 2 or 3 species in even proportion will have Di 2 orand a particle consisting of 2 or 3 species in even proportionDi 3, respectively. A particle with unequal amounts of 2 specieswillhavehaveanDieffective 2 or Drespectively.A particlei 3, ofwillnumberspecies somewhatless withthan un2,equala amountsof 2unequalspeciesamountswill haveeffectivenumberwhileparticle withof 3anspecieswill haveeffec-ofspeciessomewhatlesspossiblythan 2,evenwhilea particlewith unequaltivespeciesbelow 3, andbelow2 if the distributionisamountsof 3 species will have effective species below 3, andveryunequal.i 1NX2Di 2.5140145150marizes the conditions under which the diversity measuresattain their maximum and minimum values.The two population diversities Dα (per-particle) and Dγ(bulk) can be combined to give the single mixing state indexχ, which measures the homogeneity or heterogeneity of thepopulation.It 1955;rangesGoodman,from χ 01975;whenMcCann,all particlesare opulation)χ 1 whenall uestionsincludehaveidenticalfractions(a fullyinternallymixed poputhe sourcesof massdiversity(Tsimringet al.,1996; De’ath,2012),lation).For example,a populationa mixingstate indexextensionsof diversityto includewitha conceptof speciesdisofχ Chao0.3 (equivalently, 30 %) andcan Cobboldbe interpretedasFeolibetanceet al. (2010);χLeinster(2012);ing30 % Scheinerinternally(2012),mixed, andand thus70 % externallymixed.di(2012);techniquesfor measuringExamplesforanddifferentdiversitiesand mixingversity(ChaoShen, population2003; Schmeraand Podani,2013;Because thedifficultiespopulastatesareandshownin Fig.GotelliChao,graphically2013), despitethe2.well-knowntiondiversity Dthan theper-particleγ cannotin estimatingentropyin bean lessunbiasedfashion(Harris,diver1975;sityDα , onlya triangularis accessiblemixingis 155Paninski,2003).Beyondregionecology,the study onof thediversitystateandPaulson,their diversialso diagram.importantRepresentativein economics populations(Garrison and1973;tiesare indicatedthis diagram,listedin TableHannahand Kay,on1977;Attaran asandZwick,1989;4. MaliziaThemeasures2013),and mixingstate indexbehaveinandKe,diversity1993; Drucker,immunology(Tsimringet al.,characteristicways whenthe particlepopulationundergoes1996), neuroscience(Panzeriand Treves,1996; Stronget al.,coagulationwhen twoparticlepopulationsare mixed,1998), and orgenetics(Innanet al.,1999; Rosenberget asal., 160is2002;the casewhenare emitted into a pre-existing popFalushet particlesal., 2007).ulation.Table 5.InThepopulationmixThis Thispaperisissummarizedorganized asinfollows.Section2 we defineing(Table 5 andTheoremshow thatthe diversitiesthe resultswell-establishedentropyand ample,to theaerosolarecontext,andquantities.use these Forto defineour doublingnew mixtheparticle. Thisi leavesHi onalso containsexamplesdi- 165populationHαstateunchanged.Extensiveof theseofversity andleavesmixingand a summaryofversionsthe propertiesquantitiescan be Sectiondefined bymass-weighting,that the totalthese measures.3 presentsforP a suite ofsosimulationsmass-extensiveis Hstochastic i µi particle-resolvedHi , for example. modelarchetypal casesentropyusing thePartMC-MOSAIC (Riemer et al., 2009; Zaveri et al., 2008).These simulations show how the diversity and mixing state 1703measuresSingle-processstudiesevolve undercommon atmospheric processes, including emissions, dilution, coagulation, and gas-to-particleHavingestablishedkey quantitiesto characterizeconversion.A morethecomplexurban plumesimulationmixingis calconsidered in Section 4, for which the abovetheirprocessesocinterpretation,we illustratein thissection theirbehavior withof 175cur simultaneously.AppendixA presentsa generalizationathesuiteof simulationscenarios.The casestopresentedin thisdiversityand mixingstate measuresascribe .Theyaredesignedlevels of importance to low-mass-fraction species, while AptoisolateBtheimpactsof hematicalfor xemplifyhoweachmarized in Section 2.Atmos. Chem. Phys., 13, 11423–11439, 2013ord.-q div. q DiN. Riemer and M. West: Quantifying aerosol mixing state3. rafrom teralizePi in theJost, 22EiWe cosistingmass oand apartictions,aerosopopulversitiNote tthat eithis parent liThesureswithin(Hi to thecle isin Figtive nthe pathe nuspecieber of

11426N. Riemer and M. West: Quantifying aerosol mixing stateTable 2. Definitions of aerosol mixing entropies, particle diversities, and mixing state index. In these definitions we take 0 ln 0 0 and00 1.QuantityAXHi Hα NameUnitsRangeMeaning pia ln piamixing entropy ofparticle i–0 to ln AShannon entropy of species distribution within particle ipi Hiaverage particlemixing entropy–0 to ln Aaverage Shannon entropy perparticle pa ln papopulation bulkmixing entropy–0 to ln AShannon entropy of species distribution within populationparticle diversityof particle ieffectivespecies1 to Aeffective number of species inparticle iaverage particle(alpha) speciesdiversityeffectivespecies1 to Aaverage effective number ofspecies in each particlebulk population(gamma) speciesdiversityeffectivespecies1 to Aeffective number of species inthe populationinter-particle(beta) diversity–1 to Aamount of population species diversity due to inter-particle diversitymixing state index–0 to 100 %degree to which population isexternally mixed (χ 0) versusinternally mixed (χ 100 %)a 1NXHγ i 1AXa 1Di eHi AYa 1NYDα eHα a(pia ) pi(Di )pii 1Dγ eHγ AY(pa ) pa 1Dβ χ DγDαDα 1Dγ 1aTable 3. Conditions under which the maximum and minimum diversity values are reached. See Fig. 2 for a graphical representation of thisinformation, and see Theorem 1 for precise statements.QuantityMinimum valueMaximum valueDiDα11when particle i is purewhen all particles are pureADγDβ1ADγDαχ0%when all particles have identical mass fractionswhen all particles have identical mass fractionswhen all particles are pureprocess impacts the quantities Dα , Dγ and χ. Expanding onthis, in Sect. 4 we analyze a more complex urban plume casewith emission, dilution, coagulation and condensation occurring simultaneously.We used the particle-resolved model PartMC-MOSAIC(Particle Monte Carlo Model for Simulating Aerosol Interactions and Chemistry) (Riemer et al., 2009; Zaveri et al.,2008) for this study (PartMC version 2.2.0). This stochastic particle-resolved model explicitly resolves the composition of individual aerosol particles in a population of differ-Atmos. Chem. Phys., 13, 11423–11439, 2013A100 %when particle i has all mass fractions equalwhen all particles have identical mass fractionswhen all particles are pure and the bulkmass fractions are all equalwhen all bulk mass fractions are equalwhen all particles have identical mass fractionsent particle types in a Lagrangian air parcel. PartMC simulates particle emissions, dilution with the background, andBrownian coagulation stochastically by generating a realization of a Poisson process. Gas- and aerosol-phase chemistryare treated deterministically by coupling with the MOSAICchemistry code. The governing model equations and the numerical algorithms are described in detail in Riemer et al.(2009). Since the model tracks the per-particle compositionas the population evolves over time, we can calculate themixing state quantities as detailed in Sect. 2. We excludedwww.atmos-chem-phys.net/13/11423/2013/

N. Riemer and M. West: Quantifying aerosol mixing state11427Table 4. Representative particle populations shown on Fig. 2, with the average per-particle diversity Dα , the bulk population diversity Dγ ,and the mixing state index χ listed for each population.Per-part.div. DαBulk div.DγMix. stateindex χDescription5111undefinedall particles identical and just one bulk species5233100 %all particles identical (fully internally mixed)with identical bulk fractions53130%all particles pure (1 effective species per particle, fully externally mixed) but identical bulkfractions (3 effective bulk species)5411.890%all particles pure (1 effective species per particle, fully externally mixed) but less than twoeffective bulk species552.37368 %each particle has less than three effectivespecies (unequal fractions) but bulk fractionsare identical (3 effective bulk species)561.891.89100 %all particles are identical (fully internallymixed) but less than 2 effective bulk species571.352.3726 %generic state with partial mixingPopulationP5Table 5. Change in population diversities and mixing state index due to change in the particle population. For population combinations thesuperscript indicates the population for which a quantity is evaluated. For coagulation, Dα , Dβ , and χ stay constant when all particles haveidentical mass fractions. See Theorems 2 and 3 for precise statements.QuantityChange due to coagulationCombination of populations 5X and 5Y into 5ZDαDβDγχincreases (or constant)decreases (or constant)constantincreases (or constant)min(DαX , DαY ) DαZ max(DαX , DαY )min(DβX , DβY ) DβZmin(DγX , DγY ) DγZχ Z max(χ X , χ Y )aerosol water from calculating total particle masses of particles (i.e., we use dry mass to define the mass fractions inTable 1). Note that from the information on per-particle composition, it is straightforward to calculate per-particle properties, such as hygroscopicity (Riemer et al., 2010; Zaveriet al., 2010; Ching et al., 2012; Tian et al., 2013), opticalproperties (Zaveri et al., 2010), or particle reactivity (Kaiseret al., gle-process case descriptionsThe following model setup applies to the cases listed in Table 6. The simulation time was 24 h, and 105 computationalparticles were used to initialize the simulations. To simplifythe interpretation of the results, we applied a flat weighting function in the sense of DeVille et al. (2011). The temperature was 288.15 K, the pressure was 105 Pa, the mixingheight of the box was 300 m, and the relative humidity (RH)was 0.7. Dilution with background air was not simulated.Each initial monodisperse mode was defined by an initialAtmos. Chem. Phys., 13, 11423–11439, 2013

11428N. Riemer and M. West: Quantifying aerosol mixing stateTable 6. List of single-process case studies. Column “Chem.” indicates if gas and aerosol phase chemistry were simulated, and column“Coag.” indicates if coagulation was simulated.Case Aerosol initialN. Riemer and M. West:Chem.QuantifyingAerosolAerosol r.decr.100 % BC, Di 1monodisperseNoNoconst.incr.50 % OC, 50 % BC, Di 2Fig. 24: Diversity and mixing state evolution for BCcontaining particles in theNourban plumeDistributionmonodisperseNocase. (a)incr.incr.73 % SO4 ,of27per-particle% NH4 , Didiversity 1.8 Di as a function of time. (b) Timeseries of average particle Nodiversity DYes , populationNoincr. diversityincr.D , and the mixing state index .incr.100 %NoYesNodecr.100 %3.2Emission cases (Cases 1, 2, and 3)Cases 1, 2, and 3 explore the impact of particle emissions intoa pre-existing aerosol population. In these cases coagulationAtmos. Chem. Phys., 13, 11423–11439, 2013 5 3 2D ical 10pa 0%rticles%equal bulk amounts3nttotal number concentration of Ntot 3 107 m 3 and by aninitial diameter of D 0.1 µm.For the cases that included particle emissions (Cases 1,2, and 3), the diameter of the emitted particles was D 0.1 µm. The emitted particle flux was E 5 107 m 2 s 1for Case 1, E 1.6 1010 m 2 s 1 for Case 2, and E 1.6 108 m 2 s 1 for Case 3. For the cases that includedchemistry (Cases 5, 6, 7, and 8), the initial conditions forthe gas phase were 50 ppb O3 , 4 ppb NH3 and 1 ppb HNO3 .The gas phase emissions of NO and NH3 were prescribed ata constant rate of 60 nmol m 2 s 1 and 9 nmol m 2 s 1 , respectively. Photolysis rates were constant, corresponding toa solar zenith angle of 0 . Coagulation was not simulated except for Case 4.The results from the single-process studies are summarized in Figs. 3, 4, and 5. The left column of Figs. 3 and4 shows time series of the per-particle species diversity distribution, n(t, Di ). The right column in these figures showsthe time series of the average per-particle diversity Dα , thepopulation diversity Dγ , and the corresponding mixing stateindex χ . Each simulation is also depicted in the mixing statediagram in Fig. 5. The next sections discuss the main featuresof these results.75NoFig. 25: Diversity and mixing state evolution for all particlesYesNoincr.incr.in the urban plume case. (a) Distribution of per-particle diversity Di as a function of time. (b) Time series of averageparticle diversity D , population diversity D , and the mixYesNoincr.decr.ing state index . 4 6ideNo 8No0%7Yes 56No%5 254 0%3pure particles21 monodisperse mode73 % SO4 , 27 % NH4 , Di 1.82 monodisperse modes(1) 50 % SO4 , 50 % OIN, Di 2(2) 50 % BC, 50 % OC, Di 21 monodisperse mode100 % BC, Di 12 monodisperse modes(1) 100 % BC, Di 1(2) 100 % SO4 , Di 11 monodisperse mode100 % BC, Di 11 monodisperse mode37 % BC, 16 % SO4 , 16 % NH4 , 32 % NO3 ,Di 3.72 monodisperse modes(1) 65 % SO4 , 24 % NH4 , 10 % BC, Di 2.35(2) 7 % SO4 , 3 % NH4 , 90 % BC, Di 1.52 monodisperse modes(1) 92 % BC, 2 % NH4 , 6 % NO3 , Di 1.4(2) 89.3 % BC, 7.2 % SO4 , 2.8 % NH4 , 0.3 %NO3 , Di 1.6 71 113D Fig. 2. Mixing state diagram to illustrate the relationship betweenper-particle diversity Dα , bulk diversity Dγ , and mixing state indexFig. 26: Mixing state diagram to illustrate the relationship beχ for representative aerosol populations, as listed in Table 4. Seetween per-particle diversity D , bulk diversity D , and mixSection 2 and Table 3 for moredetails.ing state index for representative aerosol populations, aslisted in Table ?. See Section 2 and Table ? for more details.and condensation were not simulated. Depending on the relative magnitudes of the per-particle diversities of the emittedversus the pre-exisiting particles, emissions can have different impacts on the aerosol mixing state.www.atmos-chem-phys.net/13/11423/2013/

4.010133.02.010121.0n(t, Di ) / m 3(c)5.010112.01.510101.0part. div. DiCase 4101210111010109108710061218time t / .03.0(g) Dα , Dγ2.5601.0diversity Dα , Dγ3.080Dα1.5n(t, Di ) / m 100806040200χDα061218time t / hmix. state χ / %1.0Dγ2.5100mix. state χ / %1.5(b)χmix. state χ / %10103.0mix. state χ / %10112.0diversity Dα , Dγ2.5n(t, Di ) / m 3(a)3.0diversity Dα , Dγ11429n(t, Di ) / m 3part. div. Dipart. div. Dipart. div. DiCase 3Case 2Case 1N. Riemer and M. West: Quantifying aerosol mixing state24Fig. 3. Diversity and mixing state evolution for archetypal cases. Left column: Distributions of per-particle diversity Di as a function of time.Right column: time series of average particle diversity Dα , population diversity Dγ , and the mixing state index χ . Note that the left axisshows Dα and Dγ , and the right axis shows χ . The rows correspond to Cases 1 to 4 as defined in Table 6.– Case 1: we considered an initial particle populationthat contained ammonium sulfate (Di 1.8 effectivespecies) combined with emissions of pure BC particles (Di 1 effective species). This process is shownin Fig. 3a with the number concentration of the ammonium sulfate particles remaining constant, and thenumber concentration of the emitted BC particles increasing over time. Due to the emission of particleswith lower Di than the initial population, the averageper-particle diversity Dα decreased (Fig. 3b). On theother hand, adding particles of a different species thanthe initial particles increased the population species diversity Dγ . This results in a decreasing mixing stateindex χ and is consistent with the particles becoming on average more simple, and the population moreinhomogeneous. In this particular case the population evolved from 100 % internally mixed (χ 1) to30 % internally mixed (χ 0.3). The blue solid line inFig. 5 shows this process on the mixing state diagram.– Case 2: we prescribed an initial particle population oftwo monodisperse modes, with mode 1 consisting ofwww.atmos-chem-phys.net/13/11423/2013/mineral dust (model species OIN, “other inorganics”)and SO4 , and mode 2 consisting of BC and OC. Sinceall particles contained two species in equal amounts,Di 2 for all particles. The emissions consist

tion diversity. Alpha diversity Dα measures the average per-particle diversity in the population, beta diversity Dβ mea-sures the inter-particle diversity, and gamma diversity Dγ measures the bulk population diversity. The bulk population diversity (Dγ) is the product of diversity on the per-particle

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