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Atmos. Chem. Phys., 13, 11423–11439, 2013www.atmos-chem-phys.net/13/11423/2013/doi:10.5194/acp-13-11423-2013© Author(s) 2013. CC Attribution 3.0 License.

Atmospheric Chemistry

and PhysicsO

pen Access

Quantifying aerosol mixing state with entropy anddiversity measures

N. Riemer1 and M. West2

1Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, IL, USA2Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, IL, USA

Correspondence to:N. Riemer ([email protected])

Received: 12 May 2013 – Published in Atmos. Chem. Phys. Discuss.: 12 June 2013Revised: 14 September 2013 – Accepted: 30 September 2013 – Published: 25 November 2013

Abstract. This paper presents the first quantitative metricfor aerosol population mixing state, defined as the distribu-tion of per-particle chemical species composition. This newmetric, the mixing state indexχ , is an affine ratio of theaverage per-particle species diversityDα and the bulk pop-ulation species diversityDγ , both of which are based oninformation-theoretic entropy measures. The mixing state in-dex χ enables the first rigorous definition of the spectrumof mixing states from so-called external mixture to inter-nal mixture, which is significant for aerosol climate impacts,including aerosol optical properties and cloud condensationnuclei activity. We illustrate the usefulness of this new mix-ing state framework with model results from the stochas-tic 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. Fur-ther, 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.

1 Introduction

Our quantitative understanding of the aerosol impact on cli-mate still has large gaps and hence introduces large uncer-tainties in climate predictions (IPCC, 2007). One of the chal-lenges is the inherently multi-scale nature of the problem:the macro-scale impacts of aerosol particles are governed

by processes that occur on the particle-scale, and these mi-croscale processes are difficult to represent in large-scalemodels (Ghan and Schwartz, 2007).

An important quantity in this context is the so-calledmix-ing stateof the aerosol population, which we define as thedistribution of the per-particle chemical species composi-tions. Recent observations made in the laboratory and in thefield using single-particle measurement techniques have re-vealed that the mixing states of ambient aerosol populationsare complex. Even freshly emitted particles can have com-plex compositions by the time they enter the atmosphere. Forexample, the mixing state of particles originating from ve-hicle engines depends strongly on fuel type and operatingconditions (Toner et al., 2006). The initial particle composi-tion is further modified in the atmosphere as a result ofagingprocessesincluding coagulation, condensation of secondaryaerosol species, and heterogeneous reactions (Weingartneret al., 1997).

While the extent to which mixing state needs to be rep-resented in models is still an open research question, thereis evidence that mixing state matters for adequately model-ing 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 re-moval (Koch et al., 2009; Stier et al., 2006; Liu et al., 2012).Therefore, in recent years, efforts have been made to repre-sent 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 mix-ture” and “internal mixture” are frequently used to describe

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

11424 N. Riemer and M. West: Quantifying aerosol mixing state

how different chemical species are distributed over the par-ticle population. An external mixture consists of particlesthat each contain only one pure species (which may be dif-ferent 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 popula-tions rarely fall in these two simple categories. In this paperwe present the first quantitative measure of aerosol mixingstate, themixing state indexχ , based on diversity measuresderived from the information-theoretic entropy of the chem-ical species distribution among particles.

The measurement of species diversity and distribution us-ing information-theoretic entropy measures has a long his-tory 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 promi-nence with the work ofWhittaker(1960, 1965, 1972). Whit-taker proposed measuring species diversity by the speciesrichness (number of species), the Shannon entropy, and theSimpson index, which are now referred to as generalized di-versities of order 0, 1, and 2, respectively (see AppendixAfor details).

Whittaker also introduced the fundamental concepts of al-pha, beta, and gamma diversity, where alpha diversityDα

measures the average species diversity within a local area,beta diversityDβ measures the diversity between local areas,andDγ measures the overall species diversity within the en-vironment, given as the product of alpha and beta diversities.In the context of aerosols, we regard alpha diversity as mea-suring 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 Section2 for details). From these measureswe construct the mixing state indexχ 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 underly-ing mathematical framework (Hurlbert, 1971; Hill , 1973).Within the last decade, the profusion of diversity mea-sures 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 contro-versies, certain principles are now well-established, such asthe use of the effective number of species as the fundamen-tally 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 mea-suring diversity, there has also been much effort on under-standing its ecological impacts, with a particular interest in

the 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 dis-tanceChao et al.(2010); Leinster and Cobbold(2012); Feoli(2012); Scheiner(2012), and techniques for measuring di-versity (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 mix-ing state indexχ . This section also contains examples of di-versity and mixing state and a summary of the properties ofthese measures. Section3 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, in-cluding emissions, dilution, coagulation, and gas-to-particleconversion. A more complex urban plume simulation is thenconsidered in Sect.4, for which the above processes occursimultaneously. AppendixA presents a generalization of thediversity and mixing state measures to ascribe different lev-els of importance to low-mass-fraction species, while Ap-pendixB contains mathematical proofs for the results sum-marized in Sect.2.

2 Entropy, diversity, and mixing state index

We consider a population ofN aerosol particles, each con-sisting of some amounts ofA distinct aerosol species. Themass of speciesa in particlei is denotedµa

i , for i = 1, . . . ,N

anda = 1, . . . ,A. From this basic description of the aerosolparticles we can construct all other masses and mass frac-tions, as detailed in Table1. Using the distribution of aerosolspecies within the aerosol particles and within the popula-tion, we can now define mixing entropies, species diversi-ties, and the mixing state index, as shown in Table2. 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 entropyHi or diversityDi of a single particlei mea-sures how uniformly distributed the constituent species arewithin the particle. This ranges from the minimum value

Atmos. Chem. Phys., 13, 11423–11439, 2013 www.atmos-chem-phys.net/13/11423/2013/

N. Riemer and M. West: Quantifying aerosol mixing state 11425

Table 1.Aerosol mass and mass fraction definitions and notations.The number of particles in the population isN , and the number ofspecies isA.

Quantity Meaning

µai mass of speciesa in particlei

µi =

A∑a=1

µai total mass of particlei

µa=

N∑i=1

µai total mass of speciesa in population

µ =

N∑i=1

µi total mass of population

pai =

µai

µimass fraction of speciesa in particlei

pi =µi

µmass fraction of particlei in population

pa=

µa

µmass fraction of speciesa in population

(Hi = 0, Di = 1) when the particle is a single pure species,to the maximum value (Hi = lnA, Di = A) when the parti-cle is composed of equal amounts of allA species. As shownin Fig. 1, the diversityDi of a particle measures the effec-tive 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 num-ber of species somewhat less than 3.

Extending the single-particle diversityDi to an entire pop-ulation of particles gives three different measures of popula-tion diversity.Alpha diversityDα measures the average per-particle diversity in the population,beta diversityDβ mea-sures the inter-particle diversity, andgamma diversityDγ

measures the bulk population diversity. The bulk populationdiversity (Dγ ) is the product of diversity on the per-particlelevel (Dα) and diversity between the particles (Dβ ), giving

Dα︸︷︷︸per-particle

diversity

× Dβ︸︷︷︸inter-particle

diversity

= Dγ .︸︷︷︸bulk population

diversity

(1)

Alpha diversityDα measures the average per-particle effec-tive 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 pop-ulation, ranging from 1 if the entire population contains justone species, to a maximum when there are equal bulk massfractions of all species. Beta diversityDβ is defined by anaffine ratio of gamma to alpha diversity, so it measures inter-particle 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. Table3 sum-

N. Riemer and M. West: Quantifying Aerosol Mixing State 3

Di = 1 Di = 1.4 Di = 1.9 Di = 2 Di = 2.5 Di = 3

1 2 3

Fig. 3: Particle diversities Di of representative particles. Theparticle diversity measures the effective number of specieswithin a particle, so a pure single-species particle has Di = 1and a particle consisting of 2 or 3 species in even proportionwill have Di = 2 or Di = 3, respectively. A particle with un-equal amounts of 2 species will have an effective number ofspecies somewhat less than 2, while a particle with unequalamounts of 3 species will have effective species below 3, andpossibly even below 2 if the distribution is very unequal.

(MacArthur, 1955; Goodman, 1975; McCann, 2000; Ives andCarpenter, 2007). Other important research questions include120

the sources of diversity (Tsimring et al., 1996; De’ath, 2012),extensions of diversity to include a concept of species dis-tance Chao et al. (2010); Leinster and Cobbold (2012); Feoli(2012); Scheiner (2012), and techniques for measuring di-versity (Chao and Shen, 2003; Schmera and Podani, 2013;125

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; Malizia130

and 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 Section 2 we define135

the well-established entropy and diversity measures, adaptedto the aerosol context, and use these to define our new mix-ing state index �. This section also contains examples of di-versity and mixing state and a summary of the properties ofthese measures. Section 3 presents a suite of simulations for140

archetypal 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, in-cluding emissions, dilution, coagulation, and gas-to-particle145

conversion. A more complex urban plume simulation is thenconsidered in Section 4, for which the above processes oc-cur simultaneously. Appendix A presents a generalization ofthe diversity and mixing state measures to ascribe differentlevels of importance to low-mass-fraction species, while Ap-150

pendix B contains mathematical proofs for the results sum-marized in Section 2.

0 1 2 3 4 51.0

1.5

2.0

2.5

3.0

ord.-q

div

.qD

i

0 1 2 3 4 5order parameter q

0 1 2 3 4 5

Fig. 4: Generalized per-particle diversity qDi of order q forvarying q, shown for three different particles (inset squareplots). Left: a particle with equal amounts of two species.Center: a particle with two species in unequal amounts.Right: a particle with three species in unequal amounts. Theorder q controls the importance of species with small massfraction. When q = 0 all species are taken to be equallypresent, irrespective of mass fraction, so 0Di is simply thenumber of species present in the particle. When q = 1 thegeneralized diversity is equal to the regular diversity definedfrom the Shannon entropy in Section 2. When q = 2 the gen-eralized diversity 2Di is the inverse of the Simpson index�i =

PAa=1

(pai )2 for particle i (Simpson, 1949), often used

in the form of the Gini-Simpson index 1��i (Peet, 1974;Jost, 2006).

2 Entropy, diversity, and mixing stateindex

We consider a population of N aerosol particles, each con-155

sisting of some amounts of A distinct aerosol species. Themass of species a in particle i is denoted µa

i , for i = 1, . . . ,Nand a = 1, . . . ,A. From this basic description of the aerosolparticles we can construct all other masses and mass frac-tions, as detailed in Table ??. Using the distribution of160

aerosol species within the aerosol particles and within thepopulation, we can now define mixing entropies, species di-versities, and the mixing state index, as shown in Table ??.Note that entropy and diversity are equivalent concepts, andthat either could be taken as fundamental. We retain both in165

this paper to enable connections with the historical and cur-rent literature.

The entropy Hi or diversity Di of a single particle i mea-sures how uniformly distributed the constituent species arewithin the particle. This ranges from the minimum value170

(Hi = 0, Di = 1) when the particle is a single pure species,to the maximum value (Hi = lnA, Di = A) when the parti-cle is composed of equal amounts of all A species. As shownin Figure 3, the diversity Di of a particle measures the effec-tive number of equally distributed species in the particle. If175

the 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 num-ber of species somewhat less than 3.

Fig. 1. Particle diversitiesDi of representative particles. The par-ticle diversity measures the effective number of species within aparticle, so a pure single-species particle hasDi = 1 and a particleconsisting of 2 or 3 species in even proportion will haveDi = 2 orDi = 3, respectively. A particle with unequal amounts of 2 specieswill have an effective number of species somewhat less than 2,while a particle with unequal amounts of 3 species will have effec-tive species below 3, and possibly even below 2 if the distribution isvery unequal.

marizes the conditions under which the diversity measuresattain their maximum and minimum values.

The two population diversitiesDα (per-particle) andDγ

(bulk) can be combined to give the singlemixing state indexχ , which measures the homogeneity or heterogeneity of thepopulation. It ranges fromχ = 0 when all particles are pure (afully externally mixedpopulation) toχ = 1 when all particleshave identical mass fractions (a fullyinternally mixedpopu-lation). For example, a population with a mixing state indexof χ = 0.3 (equivalently,χ = 30 %) can be interpreted as be-ing 30 % internally mixed, and thus 70 % externally mixed.

Examples for different population diversities and mixingstates are shown graphically in Fig.2. Because the popula-tion diversityDγ cannot be less than the per-particle diver-sity Dα, only a triangular region is accessible on the mixingstate diagram. Representative populations and their diversi-ties are indicated on this diagram, as listed in Table4.

The diversity measures and mixing state index behave incharacteristic ways when the particle population undergoescoagulation or when two particle populations are mixed, asis the case when particles are emitted into a pre-existing pop-ulation. This is summarized in Table5. The population mix-ing results (Table5 and Theorem3) show that the diversitiesand entropies are intensive quantities. For example, doublingthe size of particlei leavesHi unchanged, and doubling thepopulation leavesHα unchanged. Extensive versions of thesequantities can be defined by mass-weighting, so that the totalmass-extensive entropy isH =

∑i µiHi , for example.

3 Single-process studies

Having established the key quantities to characterize mixingstate, and having explored their properties and their physicalinterpretation, we illustrate in this section their behavior witha suite of simulation scenarios. The cases presented in thissection are “single-process” simulations. They are designedto isolate the impacts of emission, coagulation, and conden-sation on the aerosol mixing state, and exemplify how each

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Table 2. Definitions of aerosol mixing entropies, particle diversities, and mixing state index. In these definitions we take 0ln0= 0 and00

= 1.

Quantity Name Units Range Meaning

Hi =

A∑a=1

−pai lnpa

i mixing entropy ofparticlei

– 0 to lnA Shannon entropy of species dis-tribution within particlei

Hα =

N∑i=1

piHi average particlemixing entropy

– 0 to lnA average Shannon entropy perparticle

Hγ =

A∑a=1

−pa lnpa population bulkmixing entropy

– 0 to lnA Shannon entropy of species dis-tribution within population

Di = eHi =

A∏a=1

(pai )−pa

i particle diversityof particlei

effectivespecies

1 toA effective number of species inparticlei

Dα = eHα =

N∏i=1

(Di)pi average particle

(alpha) speciesdiversity

effectivespecies

1 toA average effective number ofspecies in each particle

Dγ = eHγ =

A∏a=1

(pa)−pabulk population(gamma) speciesdiversity

effectivespecies

1 toA effective number of species inthe population

Dβ =Dγ

Dαinter-particle(beta) diversity

– 1 toA amount of population species di-versity due to inter-particle di-versity

χ =Dα − 1

Dγ − 1mixing state index – 0 to 100 % degree to which population is

externally mixed (χ = 0) versusinternally mixed (χ = 100 %)

Table 3.Conditions under which the maximum and minimum diversity values are reached. See Fig.2 for a graphical representation of thisinformation, and see Theorem1 for precise statements.

Quantity Minimum value Maximum value

Di 1 when particlei is pure A when particlei has all mass fractions equalDα 1 when all particles are pure Dγ when all particles have identical mass frac-

tionsDβ 1 when all particles have identical mass frac-

tionsA when all particles are pure and the bulk

mass fractions are all equalDγ Dα when all particles have identical mass frac-

tionsA when all bulk mass fractions are equal

χ 0 % when all particles are pure 100 % when all particles have identical mass frac-tions

process impacts the quantitiesDα, Dγ andχ . Expanding onthis, in Sect.4 we analyze a more complex urban plume casewith emission, dilution, coagulation and condensation occur-ring simultaneously.

We used the particle-resolved model PartMC-MOSAIC(Particle Monte Carlo Model for Simulating Aerosol Inter-actions and Chemistry) (Riemer et al., 2009; Zaveri et al.,2008) for this study (PartMC version 2.2.0). This stochas-tic particle-resolved model explicitly resolves the composi-tion of individual aerosol particles in a population of differ-

ent particle types in a Lagrangian air parcel. PartMC simu-lates particle emissions, dilution with the background, andBrownian coagulation stochastically by generating a realiza-tion of a Poisson process. Gas- and aerosol-phase chemistryare treated deterministically by coupling with the MOSAICchemistry code. The governing model equations and the nu-merical algorithms are described in detail inRiemer 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 excluded



Table 4.Representative particle populations shown on Fig.2, with the average per-particle diversityDα , the bulk population diversityDγ ,and the mixing state indexχ listed for each population.

PopulationPer-part.div. Dα

Bulk div.Dγ

Mix. stateindexχ

Description

51 1 1 undefined all particles identical and just one bulk species

52 3 3 100 %all particles identical (fully internally mixed)with identical bulk fractions

53 1 3 0 %all particles pure (1 effective species per par-ticle, fully externally mixed) but identical bulkfractions (3 effective bulk species)

54 1 1.89 0 %all particles pure (1 effective species per par-ticle, fully externally mixed) but less than twoeffective bulk species

55

P5

2.37 3 68 %each particle has less than three effectivespecies (unequal fractions) but bulk fractionsare identical (3 effective bulk species)

56 1.89 1.89 100 %all particles are identical (fully internallymixed) but less than 2 effective bulk species

57 1.35 2.37 26 % generic state with partial mixing

Table 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 , andχ stay constant when all particles haveidentical mass fractions. See Theorems2 and3 for precise statements.

Quantity Change due to coagulation Combination of populations5X and5Y into 5Z

Dα increases (or constant) min(DXα ,DY

α ) ≤ DZα ≤ max(DX

α ,DYα )

Dβ decreases (or constant) min(DXβ ,DY

β ) ≤ DZβ

Dγ constant min(DXγ ,DY

γ ) ≤ DZγ

χ increases (or constant) χZ≤ max(χX,χY )

aerosol water from calculating total particle masses of par-ticles (i.e., we use dry mass to define the mass fractions inTable1). Note that from the information on per-particle com-position, it is straightforward to calculate per-particle prop-erties, 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., 2011).

3.1 Single-process case descriptions

The following model setup applies to the cases listed in Ta-ble 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 weight-ing function in the sense ofDeVille et al. (2011). The tem-perature 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 initial



Table 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 initial Aerosol emissions Chem. Coag.Dα χ

1 1 monodisperse mode73 % SO4, 27 % NH4, Di = 1.8

monodisperse100 % BC,Di = 1

No No decr. decr.

2 2 monodisperse modes(1) 50 % SO4, 50 % OIN,Di = 2(2) 50 % BC, 50 % OC,Di = 2

monodisperse50 % OC, 50 % BC,Di = 2

No No const. incr.

3 1 monodisperse mode100 % BC,Di = 1

monodisperse73 % SO4, 27 % NH4, Di = 1.8

No No incr. incr.

4 2 monodisperse modes(1) 100 % BC,Di = 1(2) 100 % SO4, Di = 1

No No Yes incr. incr.

5 1 monodisperse mode100 % BC,Di = 1

No Yes No incr. 100 %

6 1 monodisperse mode37 % BC, 16 % SO4, 16 % NH4, 32 % NO3,Di = 3.7

No Yes No decr. 100 %

7 2 monodisperse modes(1) 65 % SO4, 24 % NH4, 10 % BC,Di = 2.35(2) 7 % SO4, 3 % NH4, 90 % BC,Di = 1.5

No Yes No incr. incr.

8 2 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

No Yes No incr. decr.

total number concentration ofNtot = 3× 107 m−3 and by aninitial diameter ofD = 0.1 µm.

For the cases that included particle emissions (Cases 1,2, and 3), the diameter of the emitted particles wasD =

0.1 µm. The emitted particle flux wasE = 5× 107 m−2 s−1

for Case 1,E = 1.6× 1010 m−2 s−1 for Case 2, andE =

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, re-spectively. Photolysis rates were constant, corresponding toa solar zenith angle of 0◦. Coagulation was not simulated ex-cept for Case 4.

The results from the single-process studies are summa-rized in Figs.3, 4, and 5. The left column of Figs.3 and4 shows time series of the per-particle species diversity dis-tribution, n(t,Di). The right column in these figures showsthe time series of the average per-particle diversityDα, thepopulation diversityDγ , and the corresponding mixing stateindexχ . Each simulation is also depicted in the mixing statediagram in Fig.5. The next sections discuss the main featuresof these results.

3.2 Emission 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 coagulation


Fig. 24: Diversity and mixing state evolution for BC-containing particles in the urban plume case. (a) Distributionof per-particle diversity Di as a function of time. (b) Timeseries of average particle diversity D↵, population diversityD� , and the mixing state index �.

Fig. 25: Diversity and mixing state evolution for all particlesin the urban plume case. (a) Distribution of per-particle di-versity Di as a function of time. (b) Time series of averageparticle diversity D↵, population diversity D� , and the mix-ing state index �.

⇧1

⇧2⇧3

⇧4

⇧5

⇧6

⇧7

D↵

1 31

3

pure

parti

cles

identi

calpa

rticles

equal bulk amounts

D�

�=

0%

�=

25%

�=

50%

�=

75%

�=

100%

Fig. 26: Mixing state diagram to illustrate the relationship be-tween per-particle diversity D↵, bulk diversity D� , and mix-ing state index � for representative aerosol populations, aslisted in Table ??. See Section 2 and Table ?? for more de-tails.

Fig. 2. Mixing state diagram to illustrate the relationship betweenper-particle diversityDα , bulk diversityDγ , and mixing state indexχ for representative aerosol populations, as listed in Table4. SeeSection2 and Table3 for more details.

and condensation were not simulated. Depending on the rel-ative magnitudes of the per-particle diversities of the emittedversus the pre-exisiting particles, emissions can have differ-ent impacts on the aerosol mixing state.



0 6 12 18 24

time t / h

1.01.21.41.61.82.0

par

t.d

iv.Di

Cas

e4

(g)

0 6 12 18 24

time t / h

1.01.21.41.61.82.0

div

ersi

tyDα,D

γ

Dγ

Dα

χ

1.0

1.5

2.0

2.5

3.0

par

t.d

iv.Di

Case

3

(e)

1.0

1.5

2.0

2.5

3.0

div

ersi

tyDα,D

γ

Dα

Dγχ

1.0

2.0

3.0

4.0

5.0

part

.d

iv.Di

Cas

e2

(c)

1.0

2.0

3.0

4.0

5.0

div

ersi

tyDα,D

γ

Dα

Dγ

χ

1.0

1.5

2.0

2.5

3.0

part

.d

iv.Di

Cas

e1

(a)

1.0

1.5

2.0

2.5

3.0

div

ersi

tyDα,D

γ

Dα

Dγ

χ

107108109101010111012

n(t,D

i)/

m−

3

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1011

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3

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020406080100

mix

.st

ateχ

/%

(h)

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10

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ateχ

/%

(f)

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60

80

100

mix

.st

ateχ

/%

(d)

20

40

60

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100

mix

.st

ateχ

/%

(b)

Fig. 3.Diversity and mixing state evolution for archetypal cases. Left column: Distributions of per-particle diversityDi as a function of time.Right column: time series of average particle diversityDα , population diversityDγ , and the mixing state indexχ . Note that the left axisshowsDα andDγ , and the right axis showsχ . The rows correspond to Cases 1 to 4 as defined in Table6.

– Case 1: we considered an initial particle populationthat contained ammonium sulfate (Di = 1.8 effectivespecies) combined with emissions of pure BC parti-cles (Di = 1 effective species). This process is shownin Fig. 3a with the number concentration of the am-monium sulfate particles remaining constant, and thenumber concentration of the emitted BC particles in-creasing over time. Due to the emission of particleswith lower Di than the initial population, the averageper-particle diversityDα decreased (Fig.3b). On theother hand, adding particles of a different species thanthe initial particles increased the population species di-versity Dγ . This results in a decreasing mixing stateindex χ and is consistent with the particles becom-ing on average more simple, and the population moreinhomogeneous. In this particular case the popula-tion 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 of

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 consisted ofmode-2 particles. Since theDi-values of all particleswere identical, we only observe one line in Fig.3c, andthe average per-particle species diversityDα in Fig.3dwas constant with time. However, since the emittedparticles had the same composition as one of the ini-tial modes,Dγ decreased in this simulation, hence themixing state indexχ increased from 33 % to 90 % in-ternally mixed. This is an example of a process wherethe average diversity on a particle-level did not change,but on a population-level diversity decreased. The bluedashed line in Fig.5 shows this process on the mixingstate diagram.

– Case 3: we considered an initial particle population ofpure BC particles (Di = 1 effective species) combinedwith emissions of particles containing ammonium sul-fate (Di = 1.8 effective species) (Fig.3e). This caserepresents the opposite of Case 1. The emission of



0 6 12 18 24

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5

(a)

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DαDγ

χ

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120

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.st

ateχ

/%

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ateχ

/%

(b)

Fig. 4.Diversity and mixing state evolution for archetypal cases. Left column: distributions of per-particle diversityDi as a function of time.Right column: time series of average particle diversityDα , population diversityDγ , and the mixing state indexχ . Note that the left axisshowsDα andDγ , and the right axis showsχ . The rows correspond to Cases 5 to 8 as defined in Table6.

mixed particles with a higher per-particle diversitythan that of the initial particles increasedDα, as theparticles became more diverse on average. At the sametime population diversityDγ also increased. As a re-sult the mixing state indexχ increased, indicating thatthe population became more homogeneous (Fig.3e).The cyan line in Fig.5 shows this process on the mix-ing state diagram.

3.3 Coagulation case (Case 4)

Case 4 explores the impact of coagulation. Emissions andcondensation were not simulated. We considered an initialparticle population that contained a subpopulation of pureBC particles and another subpopulation of pure SO4, givingDi = 1 effective species for all particles at the start of thesimulation. Coagulation of the particles produced mixed par-ticles with 1≤ Di ≤ 2, as Fig.3g shows. The largest possiblevalue forDi was 2 effective species, resulting from coagula-tion events that led to equal amounts of SO4 and BC in theparticles. Values ofDi smaller than 2 developed as a resultof multiple coagulation events, when one species dominated

the composition of the constituent particles. Since coagula-tion produced mixed particles,Dα increased, indicating thatparticles became more complex on average. In contrast, asstated in Theorem2, the population diversityDγ remainedconstant, as shown in Fig.3h. As a resultχ increased from0 % to about 75 % internally mixed, indicating that the pop-ulation became more homogeneous. The red line in Fig.5shows this process on the mixing state diagram. Since in thisscenario the bulk amounts of SO4 and BC were equal, thisline traces the upper edge of the triangle in the mixing statediagram.

3.4 Condensation cases (Cases 5–8)

Cases 5–8 explore the impact of condensation. In these casesparticle emissions and coagulation were not simulated. Sincewe only prescribed gas phase emissions of NO and NH3, onlyammonium nitrate formed as a secondary species. Similarto the emission cases, the condensation cases illustrate thatthe same process (here condensation) can lead to differentoutcomes in terms of mixing state, depending on the con-ditions of the scenario. As we will demonstrate below, on a



Table 7. Initial, background, and emission aerosol populations for the urban plume case, giving the number concentrationNa or area rate ofemissionEa as appropriate. The aerosol population size distributions are log-normal and defined by the geometric mean diameterDg andthe geometric standard deviationσg.

Initial/background Na / cm−3 Dg / µm σg Composition by mass Di

Aitken mode 1800 0.02 1.45 50 % (NH4)2SO4 + 50 % SOA 2.7Accumulation model 1500 0.116 1.65 50 % (NH4)2SO4 + 50 % SOA 2.7

Emission Ea / m−2 s−1 Dg / µm σg Composition by mass Di

Meat cooking 9× 106 0.086 1.91 100 % POA 1Diesel vehicles 1.6× 108 0.05 1.74 30 % POA + 70 % BC 1.8Gasoline vehicles 5× 107 0.05 1.74 80 % POA + 20 % BC 1.7

0.0 0.2 0.4 0.6 0.8 1.0

(Dα − 1)/(A− 1)

0.0

0.2

0.4

0.6

0.8

1.0

(Dγ−

1)/(A−

1)

1

2

3

4

5

6

7

8

Fig. 5. Mixing state diagram showing the normalized populationspecies diversityDγ versus the normalized average particle speciesdiversityDα for all single-process cases. The number labels refer tothe cases defined in Table6 and shown in Figs.3 and4.

population level, condensation can produce either more ho-mogeneous or less homogeneous populations, and on a par-ticle level, it can produce either less diverse or more diverseparticles.

– Case 5: we considered an initial monodisperse par-ticle population of pure BC particles, henceDi wasinitially 1 (Fig. 4a). Over the course of the simula-tion, secondary ammonium nitrate formed on the par-ticles, with the same amount on each particle. There-fore Dα increased and was at all times equal toDγ .This resulted in a constant valueχ = 1, as shown inFig. 4b. While the population was always 100 % inter-nally mixed, the increase ofDα andDγ can be inter-preted as an increase in diversity both on a per-particle

level and on a population level. This process is repre-sented by the black solid line in the mixing state dia-gram (Fig.5).

– Case 6: here we initialized each particle with a mix-ture of BC, ammonium sulfate and ammonium ni-trate, so the particles started out withDi = 3.7 effec-tive species (Fig.4c). The formation of secondary am-monium nitrate led to a decrease ofDα, again withχ = 1 at all times for a 100 % internally mixed pop-ulation (Fig.4d). The decrease ofDα andDγ can beinterpreted as a decrease in the complexity of the par-ticles. This is consistent with the particle compositionbecoming more dominated by the condensing species.This process is represented by the black dashed line inthe mixing state diagram (Fig.5).

– Case 7: we initialized the population with twomonodisperse modes. One consisted predominantly ofBC with some ammonium sulfate (Di = 1.5 effec-tive species). The other was mainly ammonium sul-fate with a small amount of BC (Di = 2.35 effectivespecies). The condensation of ammonium nitrate onall particles led to increasingDi for each subpopu-lation (Fig. 4e). Ammonium nitrate condensed on allparticles, hence the overall population became morehomogeneous, indicated by increasingχ from 50 % toabout 75 % internally mixed (Fig.4f). This process isrepresented by the green solid line in the mixing statediagram (Fig.5).

– Case 8: we initialized the population with twomonodisperse modes. They both consisted predomi-nantly of BC, but differed in their composition of inor-ganic species (see Table6) and contained 1.4 and 1.6effective species, respectively. This case was designedso that differences would occur in the ammonium ni-trate formation on the two modes based on differencesin aerosol water content (Fig.4g). The result was thatthe two subpopulations diverged from each other incomposition. WhileDα increased,Dγ increased even



0 12 24 36 48time / h

0

2

4

6

8

10

mas

sco

nc.

/µ

gm−

3

SO4

NO3

POA

0

2

4

6

8

mas

sco

nc.

/µ

gm−

3

BC NH4

SOA

Fig. 6.Evolution of bulk aerosol species for the urban plume case.

faster, henceχ decreased (Fig.4h) from 90 % to 78 %internally mixed. In this case ammonium nitrate con-densed preferentially on one of the two subpopula-tions, hence condensation caused the overall popula-tion to be more inhomogeneous. This process is rep-resented by the green dashed line in the mixing statediagram (Fig.5).

4 Complex urban plume simulation

4.1 Overview of urban plume case

In this section we discuss the case of a more complex urbanplume scenario. The details of this scenario are described inZaveri et al.(2010) and Ching et al.(2012). We assumedthat a Lagrangian air parcel containing background air wasadvected within the mixed layer across a large urban area.The start of the simulation was at 06:00 LT in the morn-ing. During the first 12 h of simulation, while the air parceltraveled over the urban area, we prescribed continuous gasemissions NOx, SO2, CO, and volatile organic compounds(VOCs), as well as emissions of three different particle types,which originated from gasoline engines, diesel engines, andmeat-cooking activities. The specifics of the particle emis-sions and the initial particle distributions, here initialized aslog-normal distributions, are listed in Table7. We slightlymodified two details of the original urban plume case pre-sented inZaveri et al.(2010). The initial and backgroundparticles of the urban plume case inZaveri et al.(2010) con-tained small amounts of BC. Here we changed this, so thatthese particle types only contained ammonium sulfate andSOA, while BC is exclusively associated with particle emis-sions. We also set the initial concentration of HCl to zero.

Both of these modifications simplify the discussion in thissection.

Unlike the single-process cases presented in Sect.3, thisurban plume case included diurnal variations of the mete-orological variables (temperature, relative humidity, mixingheight and solar zenith angle). Dilution with background airoccured during the entire simulation period, and gas andaerosol chemistry as well as coagulation amongst the par-ticles modified the aerosol population further. For referencewe show the bulk time series of the aerosol species in Fig.6.The BC and POA mass concentration increased during theemission phase, and decreased thereafter due to dilution withthe background. The time series of the secondary aerosolspecies sulfate and SOA were determined by the interplaybetween loss by dilution and photochemical production. Theammonium nitrate mass concentration depended on the gasconcentrations of its precursors, HNO3 and NH3. When thetwo gas precursors were abundant during the emission phase,ammonium nitrate formed rapidly. After emissions and pho-tochemistry ceased, HNO3 and NH3 decreased due to dilu-tion, and the ammonium nitrate evaporated.

4.2 Evolution of mixing state for urban plume case

To analyze the mixing state evolution for the urban plumecase, we graph the same quantities as for the single-processcases. We show two versions of this analysis, first we only in-clude the subpopulation of BC-containing particles in Fig.7,and then we include the whole population in Fig.8.

Focusing on the BC-containing particles, Fig.7a showsthat the particle diversity valuesDi covered a wide rangeat any point in time during the simulation, and this rangechanged over the course of the simulation. To explain this,we refer to Table7, which lists the initial particle diversityvalues of the different particle types. Particles originatingfrom diesel engine emissions and gasoline engine emissionsentered the simulation withDi = 1.8 effective species andDi = 1.7 effective species, respectively. However, coagula-tion and, more importantly, condensation altered these ini-tial values quickly. Given the particular mix of gas precursoremissions, the number of secondary species in this simula-tion was 8, and adding the primary species BC and POA, thetotal number of species is 10, which is the maximum num-ber of effective species for this simulation. Indeed, as shownin Fig. 7a, many particles during the first 12 h of simulationacquired diversity values of up to 9 effective species. At thesame time, due to fresh emissions, particles with lower parti-cle diversity values were replenished, which maintained thespread ofDi values during the emission period. During theemission phase we also observe BC-containing particles withDi values lower than their initial value of 1.7 or 1.8 effectivespecies. These arose due to coagulation with meat cookingaerosol particles withDi = 1 (see Table7). After the emis-sion period, and especially on the second day, the majorityof particles resided in a narrow range ofDi values between 6



0 12 24 36 48

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107

108

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i)/

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40

50

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.st

ateχ

/%

Fig. 7.Diversity and mixing state evolution for BC-containing particles in the urban plume case. (a) Distribution of per-particle diversityDi

as a function of time. (b) Time series of average particle diversityDα , population diversityDγ , and the mixing state indexχ .

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(b)

1061071081091010

n(t,D

i)/

m−

340

50

60

70

80

90

mix

.st

ateχ

/%

Fig. 8.Diversity and mixing state evolution for all particles in the urban plume case. (a) Distribution of per-particle diversityDi as a functionof time. (b) Time series of average particle diversityDα , population diversityDγ , and the mixing state indexχ .

1 2 3 4 5 6 7 8 9 10Average particle diversity Dα

1

2

3

4

5

6

7

8

9

10

Bu

lkp

opu

lati

ond

iver

sityDγ

χ = 50% χ = 75% χ = 100%

Fig. 9.Mixing state diagram for urban plume case showing the pop-ulation species diversityDγ versus the average particle species di-versityDα for the urban plume case presented in Fig.8 (all particlesincluded).

and 7 effective species. This is consistent with our notion of“aging”, which results in a less diverse population. Howeverit is interesting to note that at all times a range of particlesstill existed with lower number of effective species.


Di = 1 Di = 1.4 Di = 1.9 Di = 2 Di = 2.5 Di = 3

1 2 3

Fig. 3: Particle diversities Di of representative particles. Theparticle diversity measures the effective number of specieswithin a particle, so a pure single-species particle has Di = 1and a particle consisting of 2 or 3 species in even proportionwill have Di = 2 or Di = 3, respectively. A particle with un-equal amounts of 2 species will have an effective number ofspecies somewhat less than 2, while a particle with unequalamounts of 3 species will have effective species below 3, andpossibly even below 2 if the distribution is very unequal.

(MacArthur, 1955; Goodman, 1975; McCann, 2000; Ives andCarpenter, 2007). Other important research questions include120

the sources of diversity (Tsimring et al., 1996; De’ath, 2012),extensions of diversity to include a concept of species dis-tance Chao et al. (2010); Leinster and Cobbold (2012); Feoli(2012); Scheiner (2012), and techniques for measuring di-versity (Chao and Shen, 2003; Schmera and Podani, 2013;125

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; Malizia130

and 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 Section 2 we define135

the well-established entropy and diversity measures, adaptedto the aerosol context, and use these to define our new mix-ing state index �. This section also contains examples of di-versity and mixing state and a summary of the properties ofthese measures. Section 3 presents a suite of simulations for140

archetypal 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, in-cluding emissions, dilution, coagulation, and gas-to-particle145

conversion. A more complex urban plume simulation is thenconsidered in Section 4, for which the above processes oc-cur simultaneously. Appendix A presents a generalization ofthe diversity and mixing state measures to ascribe differentlevels of importance to low-mass-fraction species, while Ap-150

pendix B contains mathematical proofs for the results sum-marized in Section 2.

0 1 2 3 4 51.0

1.5

2.0

2.5

3.0

ord.-q

div

.qD

i

0 1 2 3 4 5order parameter q

0 1 2 3 4 5

Fig. 4: Generalized per-particle diversity qDi of order q forvarying q, shown for three different particles (inset squareplots). Left: a particle with equal amounts of two species.Center: a particle with two species in unequal amounts.Right: a particle with three species in unequal amounts. Theorder q controls the importance of species with small massfraction. When q = 0 all species are taken to be equallypresent, irrespective of mass fraction, so 0Di is simply thenumber of species present in the particle. When q = 1 thegeneralized diversity is equal to the regular diversity definedfrom the Shannon entropy in Section 2. When q = 2 the gen-eralized diversity 2Di is the inverse of the Simpson index�i =

PAa=1

(pai )2 for particle i (Simpson, 1949), often used

in the form of the Gini-Simpson index 1��i (Peet, 1974;Jost, 2006).

2 Entropy, diversity, and mixing stateindex

We consider a population of N aerosol particles, each con-155

sisting of some amounts of A distinct aerosol species. Themass of species a in particle i is denoted µa

i , for i = 1, . . . ,Nand a = 1, . . . ,A. From this basic description of the aerosolparticles we can construct all other masses and mass frac-tions, as detailed in Table ??. Using the distribution of160

aerosol species within the aerosol particles and within thepopulation, we can now define mixing entropies, species di-versities, and the mixing state index, as shown in Table ??.Note that entropy and diversity are equivalent concepts, andthat either could be taken as fundamental. We retain both in165

this paper to enable connections with the historical and cur-rent literature.

The entropy Hi or diversity Di of a single particle i mea-sures how uniformly distributed the constituent species arewithin the particle. This ranges from the minimum value170

(Hi = 0, Di = 1) when the particle is a single pure species,to the maximum value (Hi = lnA, Di = A) when the parti-cle is composed of equal amounts of all A species. As shownin Figure 3, the diversity Di of a particle measures the effec-tive number of equally distributed species in the particle. If175

the 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 num-ber of species somewhat less than 3.

Fig. 10.Generalized per-particle diversityqDi of orderq for vary-ing q, shown for three different particles (inset square plots). Left:a particle with equal amounts of two species. Center: a particle withtwo species in unequal amounts. Right: a particle with three speciesin unequal amounts. The orderq controls the importance of specieswith small mass fraction. Whenq = 0 all species are taken to beequally present, irrespective of mass fraction, so0Di is simply thenumber of species present in the particle. Whenq = 1 the gener-alized diversity is equal to the regular diversity defined from theShannon entropy in Sect.2. Whenq = 2 the generalized diversity2Di is the inverse of theSimpson indexλi =

∑Aa=1(pa

i)2 for parti-

cle i (Simpson, 1949), often used in the form of the Gini-Simpsonindex 1− λi (Peet, 1974; Jost, 2006).

Figure7b shows the corresponding evolution ofDα, Dγ ,andχ . The average particle diversityDα displayed a rapidincrease during the condensation period of the first 6 h ofsimulation, consistent with the BC-containing particles be-coming more complex in composition as condensation andcoagulation were at work. After this,Dα stayed essentially



constant, with small modulations induced by the diurnal cy-cle of evaporation and condensation of secondary material.The importance of condensation is also reflected by the in-crease in the bulk population diversityDγ . The mixing stateindexχ decreased from 70 % to 50 % internally mixed dur-ing the first two hours because of the effect of meat cook-ing aerosol emissions; some of these particles coagulatedwith the BC-containing particles, making the subpopulationof BC-containing particles more heterogeneous. During theperiod of secondary aerosol formation on the first day (t = 2–11 h), χ increased to 75 % internally mixed, which meansthat the population of BC-containing particles became morehomogeneous during that time. After another day of process-ing the aerosol in the air parcel, without fresh emissions, thesimulation period ended with the aerosol population being82 % internally mixed.

The same analysis performed for all particles reveals aqualitatively similar pattern for the distributions ofDi andthe time series ofDα, Dγ , andχ , with some differences inthe details, as shown in Fig.8. For example, during the first12 h, when emissions were present, the minimum value ofthe per-particle diversities wasDi = 1 effective species, dueto the meat cooking particle emissions (Fig.8a). The red lineat Di = 1.8 in Fig.8a can be attributed to background parti-cles from which SOA components had evaporated once theyentered the air parcel, hence ammonium sulfate particles re-mained. The values forχ from 2 h of simulation onwardswas lower when considering the whole population, comparedto χ of the subpopulation of BC-containing particles. Thismakes sense, since including all particle types allows fora more heterogeneous population. Figure9 shows the mix-ing state diagram that corresponds to the urban plume caseshown in Fig.8. At the end of the two-day simulation periodthe whole population was 75 % internally mixed.

5 Conclusions

With the advent of sophisticated measurement techniqueson a single-particle level, a wealth of information about thecomposition of an aerosol population has become available.The observations show that, on a particle level, aerosols arecomplex mixtures of many species, and different particletypes can coexist within one population. This reflects the par-ticles’ sources as well as their history during the transport inthe atmosphere. To describe this distribution of per-particlecompositions the term “mixing state” has been coined; how-ever, so far this term has not been rigorously defined and noconcept existed to quantify it.

This paper, for the first time, presents a framework forquantifying the mixing state of aerosol particle popula-tions. In developing this framework we borrowed the ideaof entropy-derived “diversity parameters” from other disci-plines, allowing us to define the per-particle species diversity

Di , the average particle diversityDα, the population diver-sity Dγ , and the mixing state indexχ .

The average particle diversityDα is a measure for thenumber of effective species on a per-particle level, whileDγ

quantifies the number of effective species of the bulk popu-lation. An affine ratio of the two, represented by the mixingstate indexχ , measures how close the population is to anexternal or internal mixture.

Using particle-resolved simulations to illustrate the evo-lution of the mixing state metrics for selected test cases re-vealed the following results. Coagulation always increasesthe degree of internal mixture. The impact of emission andcondensation on mixing state is not as straightforward, and inSect.3 we showed examples of scenarios whereχ increased,decreased or stayed constant as a result of emissions or con-densation. However, in the case of emissions, perhaps themost intuitive scenario is where “fresh” particles (lowDi) areemitted into an “aged” population (highDi) which decreasesthe degree of internal mixture and is reflected by decreasingχ . Similarly, in the case of condensation, the most intuitivecase is where the same species condenses on all particles ofan initially low-χ population, increasing the mixing state in-dex, consistent with our notion of “aging” as a process thatincreases the degree of internal mixing.

We expect that the mixing state indexχ will prove usefulin communicating, discussing, and categorizing the aerosolmixing state of both observed and modeled aerosol popu-lations. This, in turn, will facilitate answering the key re-search questions: (1) what is the mixing state at emissionand how does it evolve in the atmosphere; (2) what is theimpact of mixing state on climate-related and health-relevantaerosol properties; and (3) to what extent do models need toaccount for mixing state to answer these questions? In thiscontext, it would be very useful to obtain quantitative infor-mation on per-particle composition from field observationsand laboratory experiments, together with measurements ofapplication-relevant bulk properties.

Appendix A

Generalized entropy and generalized diversity

The mixing entropy can be generalized to give more or lessimportance to species with small mass fractions. This gen-eralization was originally due toHavrda and Charvát(1967)and then was independently rediscovered at least three times:in information theory (Daróczy, 1970; Aczél and Daróczy,1975), in ecology (Patil and Taillie, 1979, 1982), and inphysics (Tsallis, 1988, 2009). In the physics literature thisgeneralized entropy is frequently called the Tsallis entropyor the HCDT (Havrda–Charvát-Daróczy–Tsallis) entropy.



The per-particlegeneralized entropy of orderq ≥ 0 is de-notedqHi and leads to generalized average mixing entropyqHα and generalized population bulk entropyqHγ . These aredefined by

qHi =

|{a : pa

i > 0,a = 1, . . . ,A}| − 1 if q = 0

−∑A

a=1pai lnpa

i if q = 11

q−1(1−∑A

a=1(pai )q) if q /∈ {0,1}

(A1)

qHα =

N∑i=1

piqHi (A2)

qHγ =

|{a : pa > 0,a = 1, . . . ,A}| − 1 if q = 0

−∑A

a=1pa lnpa if q = 11

q−1(1−∑A

a=1(pa)q) if q /∈ {0,1}.

(A3)

Note that forq = 0 the definition of0Hi is one less than thenumber of species present in particlei. The special cases ofq = 0 andq = 1 are such that the generalized entropy is con-tinuous inq. To convert a generalized entropy of orderq to acorresponding generalized diversity, the conversion functionqf is defined by

qf (qH) =

{e

qH if q = 1(1− (q − 1)qH

)1/(1−q) if q 6= 1(A4)

domqf =

{[0,∞) if q ≤ 1

[0, 1q−1) if q > 1,

(A5)

where the domain ofqf is as given.Thegeneralized diversities of orderq are now defined by

qDi =qf (qHi) =

|{a : pa

i> 0,a = 1, . . . ,A}| if q = 0∏A

a=1(pai)−pa

i if q = 1(∑Aa=1(pa

i)q

)1/(1−q)if q /∈ {0,1}

(A6)

qDα =qf (qHα) =

∏N

i=1(Di)pi if q = 1(∑N

i=1pi(qDi)

1−q)1/(1−q)

if q 6= 1

(A7)

qDγ =qf (qHγ ) =

|{a : pa > 0,a = 1, . . . ,A}| if q = 0∏A

a=1(pa)−paif q = 1(∑A

a=1(pa)q)1/(1−q)

if q /∈ {0,1}

(A8)

qDβ =

qDγ

qDα. (A9)

For q = 0 the generalized diversity0Di is simply the num-ber of species present in particlei. To see that the gener-alized diversities are well-defined, we observe thatqHi ismaximized forpa

i =1A

(Theorem1), so forq > 1 we have

qHi ≤1

q−1(1−A1−q) < 1q−1, which is within the domain of

qf . For largeq the generalized diversity takes the limitingvalue of

limq→∞

qDi =1

pmaxi

, (A10)

wherepmaxi = maxa pa

i is the largest mass fraction of anyspecies in particlei. This is the exponential of themin-entropy∞Hi = − lnmaxa pa

i of particlei.From the generalized diversities, we can now define the

generalized mixing state index of orderq by

qχ =

qDα − 1qDγ − 1

. (A11)

The per-particle generalized diversity of orderq is illus-trated in Fig.10, and the population-level generalized diver-sitiesqDα andqDγ behave similarly, with low-mass-fractionspecies ranging from equally important atq = 0 to entirelyirrelevant asq → ∞.

The entropy and diversity measures defined in Sect.2are theq = 1 case of the generalized entropy and diversity.An alternative derivation of the generalized diversity can begiven using theRényi entropy of orderq (Rényi, 1961, 1977),which is given by lnqD. This latter definition was origi-nally used byHill (1973) to define the generalized diversi-ties, for which reason these diversities are sometimes calledHill numbersin the ecology literature.

Appendix B

Mathematical proofs

In this section we give precise statements and proofs for theproperties of the entropies, diversities, and mixing state indexlisted in Tables3 and5. While many of these facts are clas-sical (MacKay, 2003; Tsallis, 2009), we restate them herein the aerosol particle setting for clarity, as well as adding re-sults particular to aerosol particle populations and the mixingstate indexχ . The provided proofs aim to capture the essen-tial properties that lead to the results, rather than being ex-haustive derivations. Although Tables3 and5 only summa-rize the properties for order parameterq = 1, the results holdfor all orders as indicated below, withq = 0 typically havingonly sufficient conditions rather the necessary and sufficientconditions for positive orders.

The key observations that lead to the properties below are(1) the functionqHi(p

1i , . . . ,p

Ai ) is concave (strictly con-

cave forq > 0); and (2) the mapqf is strictly increasingand convex on its domain (strictly convex forq > 0). Bothof these facts follow immediately from second-order deriva-tive tests (Boyd and Vandenberghe, 2004, Sect. 3.1.4).

We use the notationµi = (µ1i , . . . ,µ

Ai ) for the mass vector

describing particlei, and we regard a particle population5



as a multiset (Knuth, 1998, p. 473) of particle mass vectors. Amultiset is, roughly speaking, a set where identical elementscan appear multiple times, and for which the set differenceoperator\ and union operator] have been appropriately ex-tended.

Theorem 1. The diversitiesqDi , qDα, andqDγ all lie in theinterval [1,A] and qDα ≤

qDγ . Furthermore, forq > 0, theextreme values satisfy:qDi = 1 if and only if particle i ispure (consists of a single species);qDi = A if and only ifparticlei contains all species in equal mass fractions;qDα =

1 if and only if all particles are pure;qDγ = A if and only ifall species are present with equal bulk mass fractions; andqDα = Dγ if and only if all particles have identical speciesmass fractions.

Proof. Here we show the results for the entropies and thisimplies the corresponding results for the diversities becauseqf is strictly increasing on its domain.

It follows immediately from the definition thatqHi ≥ 0,and similarly forHα andHγ . The domain of allowable massfractionspa

i is the probability simplex defined bypai ≥ 0 and∑

a pai = 1, which has normal vector(1,1, . . . ,1). Forq > 0,

qHi is a sum of identical functions of each componentpai

and so the gradient ofqHi is in the direction of the normalvector if all pa

i are equal. The KKT conditions are thus sat-isfied for all mass fractions equal (Boyd and Vandenberghe,2004, Sect. 5.5.3), and by strong convexity ofqHi this is aunique maximum. Evaluating atpA

i = 1/A gives the maxi-mumqHi = lnA. Forq = 0 the upper bound is immediate.

The condition forqDγ = A follows from the same argu-ment as above applied to the bulk mass fractionspa . Thecondition for qDα = 1 is due to the fact thatqHα is theweighted arithmetic mean of the particle entropies with pos-itive weights, soqDα = 1 if and only if eachqHi = 0.

The restrictionqHα ≤qHγ is simply Jensen’s inequal-

ity (Boyd and Vandenberghe, 2004, Sect. 3.1.8) for the con-cave entropy function. This can be seen by checking thatpa

=∑

i pipai , and so the weightspi for combiningqHi into

qHα are the same as for combining the per-particle mass frac-tions into the bulk mass fractions. Forq > 0, strict convexityof the entropy implies that Jensen’s inequality is an equalityif and only if the per-particle mass fractions are all identi-cal.

Theorem 2. If population 51 evolves into population52 by coagulation, thenqDα(52) ≥

qDα(51), qDγ (52) =qDγ (51), qDβ(52) ≤

qDβ(51), andqχ(52) ≥qχ(51). For

q > 0, equality in the preceding inequalities occurs if andonly if all coagulating particles have identical mass frac-tions.

Proof. It is sufficient to consider a single coagulation eventand then iterate the result, so without loss of generality weassume that52

= 51\{µi,µj }]{µc}, whereµc = µi +µj .

We observe that the total massµ is preserved by coagu-lation andpc = pi + pj , where the mass fractions are com-

puted with respect to the relevant population51 or 52. Fur-thermore,pcp

ac = pip

ai + pjp

aj so concavity of the entropy

givespcqHc ≥ pi

qHi +pjqHj , with equality forq > 0 if and

only if pai = pa

j by strict convexity. All other per-particleentropies in the populations are identical, sopc = pi + pj

implies qHα(52) ≥qHα(51) with the same condition for

equality.Because the bulk mass fractions are unchanged by coagu-

lation,qHγ (52) =qHγ (51) and the inequalities forqHβ and

χ follow immediately. The fact thatqf is strictly increasingtransfers all results to diversities.

Theorem 3. If populations5X and 5Y are combined togive population 5Z

= 5X] 5Y then min(qDX

α ,qDYα ) ≤

qDZα ≤ max(qDX

α ,qDYα ), min(qDX

β ,qDYβ ) ≤

qDZβ ,

min(qDXγ ,qDY

γ ) ≤qDZ

γ , and qχZ≤

qmax(χX,χY ), wheresuperscripts denote the population.

Proof. As qf is strictly increasing it is sufficient to obtainbounds for entropies, which then imply bounds on the corre-sponding diversities.

Taking the total population mass fractionλ = µY /(µX+

µY ) ∈ (0,1), then for particlesµx ∈ 5X and µy ∈ 5Y themass fractions in the combined population arepZ

x = (1−

λ)pXx andpZ

y = λpYy . ThusqHZ

α is the convex combinationqHZ

α = (1−λ)qHXα +λqH Y

α and soqHZα lies strictly between

qHXα andqH Y

α .The bulk mass fractions satisfypZa

= (1−λ)pXa+λpYa

and so concavity ofqHγ givesqHZγ ≥ (1−λ)qHX

γ +λqH Yγ ≥

min(qHXγ ,qH Y

γ ).

To obtain the lower bound forqHZβ =

qf −1(qDZβ ), we ob-

serve that

qHβ =

qHγ −qHα

1− (q − 1)qHα

. (B1)

Using the expressions above forqHZα andqHZ

γ , this implies

qHZβ ≥

(1− λ)qHXγ + λqH Y

γ − (1− λ)qHXα − λqH Y

α

(1− λ) + λ − (q − 1)(1− λ)qHXα − (q − 1)λqH Y

α

=(1− λ)(qHX

γ −qHX

α ) + λ(qHXγ −

qHXα )

(1− λ)(1− (q − 1)qHXα ) + λ(1− (q − 1)qH Y

α ).

Rearranging (B1) givesqHγ −qHα = (1− (q − 1)qHα)qHβ ,

so

qHZβ ≥

(1− λ)(1− (q − 1)qHXα )qHX

β + λ(1− (q − 1)qH Yα )qH Y

β

(1− λ)(1− (q − 1)qHXα ) + λ(1− (q − 1)qH Y

α )

≥ min(qHXβ ,qH Y

β ).



Finally, to obtain the upper bound forqχZ we again usethe expressions forqHZ

α andqHZγ to compute

qχZ=

exp(qHZα ) − 1

exp(qHZγ ) − 1

(B2)

≤exp(qHX

α )(1−λ) exp(qH Yα )λ − 1

exp(qHXγ )(1−λ) exp(qH Y

γ )λ − 1.

Recallingqχ ≥ 0, rearranging (B2) gives exp(qHα) = (1−qχ) +

qχ exp(qHγ ) ≤qχ exp(qHγ ), so

qχZ≤

(qχX exp(qHXγ ))(1−λ)(qχY exp(qH Y

γ ))λ − 1

exp(qHXγ )(1−λ) exp(qH Y

γ )λ − 1

≤ max(qχX,qχY ),

where we additionally used the boundqχ ≤ 1.

Acknowledgements.This research was supported in part by the USDepartment of Energy’s Atmospheric System Research, an Officeof Science, Office of Biological and Environmental Researchprogram, under Grant No. DE-SC0003921 and in part by USEPA grant 835042. Its contents are solely the responsibility of thegrantee and do not necessarily represent the official views of theUS EPA. Further, US EPA does not endorse the purchase of anycommercial products or services mentioned in the publication.

Edited by: M. Petters

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http://dx.doi.org/10.1029/2006JD007147

http://dx.doi.org/10.1103/PhysRevLett.80.197

http://dx.doi.org/10.5194/acpd-13-16733-2013

http://dx.doi.org/10.1021/es051455x

http://dx.doi.org/10.1007/BF01016429

http://dx.doi.org/10.1103/PhysRevLett.76.4440

http://dx.doi.org/10.1007/s00442-010-1812-0

http://dx.doi.org/10.1007/s00442-011-2128-4

http://dx.doi.org/10.1111/j.1600-0706.2011.19897.x

http://dx.doi.org/10.1111/j.1600-0706.2011.19897.x

http://dx.doi.org/10.1016/B978-0-12-384719-5.00378-6

http://dx.doi.org/10.1016/S1352-2310(97)00023-X

http://dx.doi.org/10.1126/science.147.3655.250

http://dx.doi.org/10.1029/2000JD000198

http://dx.doi.org/10.1029/2007JD008782

http://dx.doi.org/10.1029/2009JD013616

quantifying aerosol mixing state with entropy and ...nriemer/papers/riemer_west_2013.pdf · simpson...

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