aerosol size spectra and ccn activity spectra: reconciling

Aerosol size spectra and CCN activity spectra: Reconciling the

lognormal, algebraic, and power laws

Vitaly I. Khvorostyanov1 and Judith A. Curry2

Received 27 July 2005; revised 2 November 2005; accepted 6 February 2006; published 17 June 2006.

[1] Empirical power law expressions of the form NCCN = Csk have been used in cloudphysics for over 4 decades to relate the number of cloud condensation nuclei (CCN) anddroplets formed on them to cloud supersaturation, s. The deficiencies of thisparameterization are that the parameters C and k are usually constants taken from theempirical data and not directly related to the CCN microphysical properties andthis parameterization predicts unbounded droplet concentrations at high s. The activationpower law was derived in several works from the power law Junge-type aerosol sizespectra and parameters C and k were related to the indices of the power law, but this stilldoes not allow to describe observed decrease of the k-indices with s and limited NCCN.Recently, new parameterizations for cloud drop activation have been developed basedupon the lognormal aerosol size spectra that yield finite NCCN at large s, but does notexplain the activation power law, which is still traditionally used in the interpretation ofCCN observations, in many cloud models and some large-scale models. Thus the relationbetween the lognormal and power law parameterizations is unclear, and it is desirableto establish a bridge between them. In this paper, algebraic and power law equivalents arefound for the lognormal size spectra of partially soluble dry and wet interstitial aerosol,and for the differential and integral CCN activity spectra. This allows derivation of thepower law expression for cloud drop activation from basic thermodynamic principles(Kohler theory). In the new power law formulation, the index k and coefficient C areobtained as continuous analytical functions of s and expressed directly via parameters ofaerosol lognormal size spectra (modal radii, dispersions) and physicochemical properties.This approach allows reconciliation of this modified power law and the lognormalparameterizations and their equivalence is shown by the quantitative comparison of thesemodels as applied to several examples. The advantages of this new power law relationshipinclude bounded Nd at high s, quantitative explanation of the experimental data on thek-index and possibility to express k(s) and C(s) directly via the aerosol microphysics. Themodified power law provides a framework for using the wealth of data on the C, kparameters accumulated over the past decades, both in the framework of the power lawand the lognormal parameterizations.

Citation: Khvorostyanov, V. I., and J. A. Curry (2006), Aerosol size spectra and CCN activity spectra: Reconciling the lognormal,

algebraic, and power laws, J. Geophys. Res., 111, D12202, doi:10.1029/2005JD006532.

1. Introduction

[2] The power law and lognormal parameterizations ofaerosol size spectra and supersaturation activity spectra arewidely used in studies of aerosol optical and radiativeproperties, cloud physics, and climate research. Thesetwo kinds of parameterizations exist in parallel, implicitlycompete, but their relation is unclear. One of the mostimportant applications of the power and lognormal laws is

parameterization of the concentrations of cloud drops Nd

and cloud condensation nuclei NCCN (CCN), on which thedrops form. They influence cloud optical and radiativeproperties as well as the rate of precipitation formation.Precise evaluation of NCCN is required, in particular, forcorrect estimation of anthropogenic aerosol effects oncloud albedo [Twomey, 1977] and precipitation [Albrecht,1989].[3] The most commonly used parameterization of the

concentration of cloud drops or cloud condensation nucleiNCCN on which the drops activate is a power law by thesupersaturation s reached in a cloud parcel is

NCCN sð Þ ¼ Csk ; ð1Þ

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, D12202, doi:10.1029/2005JD006532, 2006

1Central Aerological Observatory, Dolgoprudny, Moscow Region,Russia.

2School of Earth and Atmospheric Sciences, Georgia Institute ofTechnology, Atlanta, Georgia, USA.

Copyright 2006 by the American Geophysical Union.0148-0227/06/2005JD006532$09.00

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which is referred to in the literature as the integral orcumulative CCN activity spectrum. Numerous studies haveprovided a wealth of data on the parameters k and C forvarious geographical regions [see, e.g., Hegg and Hobbs,1992; Pruppacher and Klett, 1997, Table 9.1, hereafterreferred to as PK97]. Parameterizations of the type (1) werederived by Squires [1958] and Twomey [1959], usingseveral assumptions and similar power law for thedifferential CCN activity spectrum, js(s)

js sð Þ ¼ dNCCN=ds ¼ Cksk�1: ð2Þ

Equations (1) and (2) have been used for several decades inmany cloud models with empirical values of k and C thatare assumed to be constant for a given air mass [PK97] andare usually constant during model runs.[4] To explain the empirical dependencies (1) and (2),

models were developed of partially soluble CCN with thesize spectra of Junge-type power law f(r) � r�m (totalaerosol concentration Na � r�(m�1)) and the index k wasexpressed as a function of m. Jiusto and Lala [1981] found alinear relation

k ¼ 2 m� 1ð Þ=3; ð3Þ

which implies k = 2 for a typical Junge index m = 4, whilethe experimental values of k compiled in that work variedover the range 0.2–4. Levin and Sedunov [1966], Sedunov[1974], Smirnov [1978], and Cohard et al. [1998, 2000]derived more general power law or algebraic equations forjs(s) and expressed k as a function of m and CCN solublefraction. Khvorostyanov and Curry [1999a, 1999b, hereafterreferred to as KC99a, KC99b] derived power laws for thesize spectra of the wet and interstitial aerosol, for theactivity spectra js(s), NCCN(s) and for the Angstromwavelength index of extinction coefficient, and found linearrelations among these indices expressed in terms of theindex m and aerosol soluble fraction.[5] A deficiency of (1) is that it overestimates the droplet

concentration at large s and predicts values of dropletconcentration that exceeds total aerosol concentration; thisoccurs because of the functional form of the power law anduse of a single value of k. Many field and laboratorymeasurements have shown that a more realistic NCCN(s)spectrum in log-log coordinates is not linear as it would bewith k = const, but has a concave curvature, i.e., the index kdecreases with increasing s [e.g., Jiusto and Lala, 1981;Hudson, 1984; Yum and Hudson, 2001]. This deficiency in(1) has been corrected in various ways: (1) by introducingthe C-k space and constructing nomograms in this space[Braham, 1976]; (2) by constructing js(s) that rapidlydecreases at high s [Cohard et al., 1998, 2000] or integralempirical spectra NCCN(s) [Ji and Shaw, 1998] that yieldfinite drop concentrations at large s; (3) by considering thedecrease in the indices m, k with decreasing aerosol size[KC99a]; (4) by using a lognormal CCN size spectruminstead of the power law, which yields concave spectra [vonder Emde and Wacker, 1993; Ghan et al., 1993, 1995, 1997;Feingold et al., 1994; Abdul-Razzak et al., 1998; Abdul-Razzak and Ghan, 2000; Nenes and Seinfeld, 2003;Rissman et al., 2004; Fountoukis and Nenes, 2005]. On

the basis of these studies, the prognostic equations for thedrop concentration are recently being incorporated intoclimate models [Ghan et al., 1997; Lohmann et al., 1999].[6] However, the power laws (1), (2) are still used in

many cloud models and in analyses of field and chamberexperiments. The relation between the power law and newerlognormal parameterization is unclear. In this paper, amodified power law is derived and its equivalence to thelognormal parameterizations is established. In section 2, amodel of mixed (partially soluble) dry CCN is developedwith parameterization of the soluble fraction as a function ofthe dry nucleus radius and a power law Junge-type repre-sentation of the lognormal size spectra is derived. To allow amore accurate estimate of aerosol contribution into thecloud optical properties and to the Twomey effect, alognormal spectrum of the wet interstitial aerosol is foundusing Kohler theory. In section 3, a new algebraic repre-sentation of the lognormal spectra is derived. The differen-tial CCN activity spectra as a modification of (2) are derivedin section 4 in both the lognormal and algebraic forms. Insection 5, the cumulative CCN spectra as a modification of(1) are derived in both lognormal and algebraic forms.Finally, a modified power law is derived as a modificationof (1), expressing the parameters C and k as continuousalgebraic functions of supersaturation and parameters ofaerosol microstructure and physicochemical properties. Theadvantages of this new power law relationship include dropconcentration bounded by the total aerosol concentration athigh supersaturations, quantitative explanation of the exper-imental data on the k-index, and the possibility to expressk(s) and C(s) directly via the aerosol microphysics. Thisformulation allows reconciliation of this modified powerlaw and the lognormal parameterizations, which is illustratedwith several examples. Summary and conclusions are givenin section 6.

2. Correspondence Between the Lognormal andPower Law Size Spectra

2.1. Lognormal Size Spectra of the Dry and WetAerosol

[7] We consider a polydisperse ensemble of mixed aero-sol particles consisting of soluble and insoluble fractions.The lognormal size spectrum of dry aerosol fd(rd) by the dryradii rd can be presented in the form:

fd rdð Þ ¼ Naffiffiffiffiffiffi2p

pln sdð Þrd

exp � ln2 rd=rd0ð Þ2 ln2 sd

� �; ð4Þ

where Na is the aerosol number concentration, sd is thedispersion of the dry spectrum, and rd0 is the meangeometric radius related to the modal radius rm as

rm ¼ rd0 exp � ln2 sd� �

: ð5Þ

As the relative humidity H exceeds the threshold ofdeliquescence Hth of the soluble fraction, the hygroscopicgrowth of the particles begins and initially dry CCN convertinto wet ‘‘haze particles.’’ To derive the size spectrum of thewet aerosol, we need relations between the dry radius rd anda corresponding radius of a wet particle rw. These relations

D12202 KHVOROSTYANOVAND CURRY: CCN ACTIVITY SPECTRA

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can be obtained using the Kohler equation for super-saturation s = (rv � rvs)/rvs that can be written for the dilutesolution particles as [PK97]:

s ¼ H � 1 ¼ Ak

r� B

r3: ð6Þ

Ak ¼2MwVsaRTrw

; B ¼ 3nFsmsMw

4pMsrw: ð7Þ

Here rv, rvs, and rw are the densities of vapor, saturatedvapor, and water, H is the ambient relative humidity, Ak isthe Kelvin curvature parameter, Mw is the molecular weightof water, zsa is the surface tension at the solution-airinterface, R is the universal gas constant, T is thetemperature (in degrees Kelvin). The parameter B describeseffects of the soluble fraction, n is the number of ions insolution, Fs is the osmotic potential, ms and Ms are the massand molecular weight of the soluble fraction.[8] The radius rw can be related to rd as in the work of

KC99a using a convenient parameterization of the solublefraction and the parameter B, following Levin and Sedunov[1966], Sedunov [1974]:

B ¼ br2 1þbð Þd ð8Þ

where the parameters b and b depend on the chemicalcomposition and physical properties of the soluble part of anaerosol particle. The parameter b describes the solublefraction of an aerosol particle. Generally, the solubilitydecreases with increasing particle size [e.g., Laktionov,1972; Sedunov, 1974; PK97], so that b decreases withincreasing rd and may vary from 0.5, when soluble fractionis proportional to the volume, to �1, when soluble fractionis independent of the radius. This variation describes thechange of the dominant mechanisms of accumulation ofsoluble fraction with growing particle size. A detailedanalysis of these mechanisms is given, e.g., in the work ofPK97 and Seinfeld and Pandis [1998] and is beyond thescope of this paper. We consider in more detail twoparticular cases.[9] 1. b = 0.5. The value b = 0.5 corresponds to the case

B = brd3, when soluble fraction is distributed within particle

volume, is proportional to it, and is usually implicitlyassumed in most parameterizations of CCN deliquescenceand drop activation [e.g., von der Emde and Wacker, 1993;Ghan et al., 1993, 1995; Abdul-Razzak et al., 1998; Abdul-Razzak and Ghan, 2000]. It was found in the work ofKC99a that with b = 0.5, the quantity b is a dimensionlessparameter:

b ¼ nFsð Þevrsrw

Mw

Ms

; ð9aÞ

where rs is the density of the soluble fraction, ev is itsvolume fraction.[10] 2. b = 0. Then B = brd

2, i.e., mass of soluble fractionis proportional to the surface area where it is accumulated asa film or shell. The particle volume fraction was parame-terized in the work of KC99a as ev(rd) = ev0(rd1/rd), where

rd1 is some scaling parameter and ev0 is the reference solublefraction (dimensionless). We then obtain for b

b ¼ rd1ev0 nFsð Þ rsrw

Mw

Ms

: ð9bÞ

For this case, b has the dimension of length and isproportional to the scaling radius rd1. If the thickness l0 ofthe soluble shell is much smaller than the radius rd of aspherical insoluble core, l0 � rd,, then approximately ms 4prsrd

2l0, ev(rd) = 3l0/rd, and in (9b) ev0 = 1, rd1 = 3l0, and

b ¼ 3l0 nFsð Þ rsrw

Mw

Ms

: ð9cÞ

The parameterizations (9b) and (9c) might be used, inparticular, for surface-active substances that are accumu-lated near the particle surface [e.g., Rissman et al., 2004] orfor soluble shell coatings of particles with insoluble coressuch as dust when heterogeneous chemical reactions on theparticle surface may lead to formation of such coatings [e.g.,Laktionov, 1972; Levin et al., 1996; Sassen et al., 2003;Bauer and Koch, 2005].[11] The values of b for aerosols with volume-propor-

tional soluble components (b = 0.5) consisting of NaCl andammonium sulphate evaluated from (9a) are determined asfollows, using data obtained from PK97. If the solublematerial is NaCl (Ms = 58.5, rs = 2.16 g cm�3, n = 2,Fs 1 for dilute solutions), then b = 1.33 for ev = 1(fully soluble nuclei), and b = 0.26 for ev = 0.20. If thesoluble material is ammonium sulphate (Ms = 132, rs =1.77 g cm�3, n = 3, Fs 0.767 for molality of 0.1, so thatnFs 2.3), then b = 0.55 for fully soluble nuclei (ev = 1),and b = 0.25 for ev = 0.45. In the calculations below, weconsider mostly the case b = 0.5 (volume-proportional) anduse the value b = 0.25, which may correspond to the dryaerosol containing �50 % ammonium sulphate, as found byHanel [1976] in measurements of continental and marine(Atlantic) aerosols, or 20% NaCl, as was assumed by Junge[1963]. All of the results below can be recalculated for b = 0with an appropriate model of the soluble shell (someestimates are given below for the case b = 0), to any otherev, or for any other chemical composition using appropriatevalues of Ms and rs. The soluble aerosol fraction oftendecreases with increasing radius [PK97], and this featurecan be described by choosing an appropriate superpositionof the aerosol size spectra with the two soluble fractions:volume- proportional (b = 0.5 for smaller fraction) andsurface-proportional (b = 0 for larger fraction).[12] The size spectrum of the wet aerosol fw(rw) can be

obtained using the relations between the dry rd and wet rwradii. Various solutions for the humidity dependence ofrw(H) at subsaturation and supersaturation were found inalgebraic and trigonometric forms using the Kohler’s equa-tion (6) [e.g., Levin and Sedunov, 1966; Sedunov, 1974;Hanel, 1976; Fitzgerald, 1975; Smirnov, 1978; Fitzgerald etal., 1982; Khvorostyanov and Curry, 1999a; Cohard et al.,2000]. In this work, we consider only the wet interstitialaerosol, and humidity transformations at subsaturation willbe considered in a separate paper.


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[13] The supersaturation in clouds is small (s 10�4–10�2), the left-hand side in (6) tends to zero and the radiusof a wet particle can be found from the quadratic equation:

rw H ffi 1ð Þ ¼ B

Ak

� �1=2

¼ b1=2r1þbd

A1=2k

: ð10Þ

[14] This expression can be used at the stage of cloudformation (CCN activation into the drops) or for interstitialunactivated cloud aerosol. Now the size spectrum fw(rw) canbe found from the conservation equation

fw rwð Þdrw ¼ fd rdð Þdrd : ð11Þ

Using (10)–(11), we obtain that the dry distribution (4)transforms into the wet spectrum

fw rwð Þ ¼ Naffiffiffiffiffiffi2p

pln swð Þrw

exp � ln2 rw=rw0ð Þ2 ln2 sw

� �: ð12Þ

The wet mean geometric radii rw0 and dispersions sw arerelated to the corresponding rd0 and sd of the dry aerosol as

rw0 ¼ r1þbð Þd0 b=Akð Þ1=2; sw ¼ s 1þbð Þ

d ; H ffi 1; ð13Þ

[15] Thus the shape of thewet interstitial size spectrum (12)is the same lognormal as for the dry aerosol (4) but the valuesof rw0 and sw are different. The size spectra of the interstitialaerosol at s > 0 are limited by the boundary radius rb = 2Ak/3s[Levin and Sedunov, 1966; Sedunov, 1974].

2.2. Approximation of the Lognormal Size Spectra bythe Junge Power Law

[16] A correspondence between the lognormal and powerlaw size spectra can be established using the same methodas in the work of Khvorostyanov and Curry [2002, 2005]

for a continuous power law representation of the fallvelocities. Suppose f(r) is smooth and has a smooth deriv-ative f 0(r). Then they can be presented at a point r as thepower law functions

f rð Þ ¼ cf r�m; f 0 rð Þ ¼ �mf rð Þ=r ð14Þ

Solving (14) for m and cf we obtain:

m rð Þ ¼ � rf 0 rð Þf rð Þ ; cf rð Þ ¼ f rð Þrm ð15Þ

The power index m and cf are here the functions of radius. Ifaerosol size spectrum f(r) is described by the dry fd(rd) orwet fw(rw) lognormal distributions (4), (12), substitution offd,w and f 0d,w into (15) yields the power index for the dry andwet aerosol spectra

md;w rd;w� �

¼ 1þln rd;w=rd;w0� �ln2 sd;w

; ð16Þ

where the indices ‘‘d, w’’ denote dry or wet aerosol, and rw0,sw0 are defined by (14). Thus the lognormal dry and wetspectra (4), (12) can be presented at every point rd, rw, in thepower law form (14) with the index md,w (16).

2.3. Examples of the Dry and Wet Aerosol SizeSpectra and Power Indices

[17] Figure 1 shows in log-log coordinates a lognormaldry aerosol size spectrum fd(rd) with modal radius rm = 0.03mm and dispersion sd = 2, close to Whitby [1978] for theaccumulation mode, and concentration Na = 200 cm�3. ItsJunge-type power law approximations shows the radiusdependence of the power index m: it changes from m =�1.27 at r1 = 0.016 mm < rm, to m = 0 at the modal radiusr2 = rm, and m is positive at all r > rm; in particular, m = 2.5 atr3 = 0.1 mm and m = 5.0 at r4 = 0.33 mm. Recall that thedimensional analysis of the coagulation equation and itsasymptotic solutions yield the inverse power laws with theindex m = 2.5 for rd � 0.1 mm due to the dominant effects ofBrownian coagulation, and m = 4.5 for rd > 1 mm due toprevailing sedimentation [PK97, chap. 11]. The lognormalspectrum with the described parameters is in general agree-ment with these power law solutions. Thus the lognormalsize spectrum can be considered as the superposition of thepower laws with the index from (16).[18] The transition of the dry lognormal spectrum to the

limit H = 1 (curve with open circles) is fairly smooth,although there is a distinct decrease of the slopes. Thisresembles a corresponding effect for the power law spectra,when the transition to H = 1 is accompanied by a decreaseof the Junge power index, e.g., m = 4 at H < 1 converts intom = 3 at H ! 1 over a very narrow humidity range [e.g.,Sedunov, 1974; Fitzgerald, 1975; Smirnov, 1978; KC99a].[19] The dependence of the effective power indices cal-

culated for lognormal distributions with various modal radiiand dispersions is shown for the dry aerosol in Figure 2,which illustrates the following general features of (16). Theindices (the slopes of the spectra) increase with radii for allr. The indices m increase with decreasing dispersions andwith increasing rd0 of the dry lognormal spectrum. The

Figure 1. Lognormal dry (solid curve) and interstitial(open circles) aerosol size spectra with the parameters:modal radius rm = 0.03 mm, sd,w = 2, Na = 200 cm�3, andthe Junge-type power law approximations to the dryspectrum at four points: before modal radius, r1 =0.016 mm, negative index m = �1.27; modal radius r2 =0.03 mm, m = 0; r3 = 0.1 mm, m = 2.5; r4 = 0.33 mm, m = 5.0).


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calculated m intersect the average Junge’s curve m = 4 in therange of radii 0.06 mm to 0.7 mm, i.e., mostly in theaccumulation mode. Since the values chosen here for rd0and sd are realistic (close to Whitby’s multimodal spectra),this may explain how a superposition of several lognormalspectra may lead to an ‘‘effective average’’ m = 4, observedby Junge. The growth of the indices above 4 for larger r isin agreement with the coagulation theory that predictssteeper slopes, m = 19/4, of the power laws for greater radiir � 5 mm [PK97]. The described relation between thelognormal and power law size spectra might be useful foranalysis and parameterizations of the measured spectra andof solutions to the coagulation equation.

3. Algebraic Approximation of the LognormalDistribution

[20] A disadvantage of the lognormal size spectra is thedifficulty of evaluating analytical asymptotics andmoments of this distribution. A convenient algebraicequivalent of the lognormal distribution can be obtainedusing two representations of the smoothed (bell-shaped)Dirac’s delta-function d(x). The first one was taken as[Korn and Korn, 1968]

d1ðx;aÞ ¼affiffiffip

p expð�a2x2Þ; ð17Þ

where a determines the width of the function. Anotherrepresentation as an algebraic function of exp(x/a) is givenby Levich [1969]. We transformed it to the following form:

d2 x;að Þ ¼ affiffiffip

p 1

ch2 2ax=ffiffiffip

pð Þ; ð18Þ

[21] It can be shown that both functions d1(x,a) and d2(x,a)are normalized to unity, tend to the Dirac’s delta-functionwhen a! 1 and exactly coincide in this limit:

lima!1

d1 x;að Þ ¼ lima!1

d2 x;að Þ ¼ d xð Þ: ð19Þ

Thus we can use the following equality:

affiffiffip

p exp �a2x2� �

affiffiffip

p 1

ch2 2ax=ffiffiffip

pð Þð20Þ

A comparison shows that both functions are alreadysufficiently close at values a � 1, which is illustrated inFigure 3a, where these functions divided by a/

ffiffiffip

pare

plotted, the difference generally does not exceed 3–5%. Theapproximate equality of these smoothed delta functions isused further for an algebraic representation of the aerosolsize and CCN activity spectra.[22] Integration of (20) from 0 to x yields another useful

relation:

erf axð Þ tanh 2ax=ffiffiffip

p� �; ð21Þ

where erf(z) is the Gaussian function of errors:

erf zð Þ ¼ 2ffiffiffip

pZz

0

e�x2dx: ð21aÞ

Equation (21) with a = 1 was found by Ghan et al. [1993],who showed its accuracy, which is also illustrated inFigure 3b for two values of a. It is known that integration ofthe Dirac’s delta-function yields Heaviside’s step-function.Comparison of Figures 3a and 3b shows that the same isvalid for the correspondent smoothed functions, therefore

Figure 2. Effective power indices calculated fromequation (16) for lognormal distributions with modal radiiand dispersions indicated in the legend; the constant valuem = 4 is given for comparison.

Figure 3. (a) Comparison of the smoothed Dirac delta-functions: exp(�a2x2), and its approximation1/ch2(2ax/

ffiffiffip

p),

for two values a = 1 and 3. The difference generallydoes not exceed 3–5%. (b) Comparison of thesmoothed Heaviside step-functions, erf(ax), andtanh(2ax/

ffiffiffip

p).


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the derivation above shows that the relation (21) is anapproximate equality of the two representations of thesmoothed Heaviside step-functions. Equation (21) is used insection 5 for the proof of equivalence of the lognormal andalgebraic representations of the drop concentration and k-index.[23] If we introduce in (20) the new variables x = ln (r/r0),

a = (ffiffiffi2

pln s)�1, then 2ax/

ffiffiffip

p= ln(r/r0)

k0/2, where k0 is aparameter introduced by Ghan et al. [1993]:

k0 ¼4ffiffiffiffiffiffi

2pp

ln s: ð22Þ

Substituting these variables into (20), we obtain

1ffiffiffiffiffiffi2p

plns

exp � ln2 r=r0ð Þ2 ln2 s

� � k0 r=r0ð Þk0

r=r0ð Þk0þ1h i2 : ð23Þ

[24] The expression on the left-hand side is the lognormaldistribution dN/dlnr with mean geometric radius r0 anddispersion s, and the right-hand side is its algebraic equiv-alent, which is another representation of the smoothed deltafunction. When the dispersion tends to its lower limit s !1, then a ! 1, and according to (19), both distributionstend to the delta function, i.e., become infinitely narrow(monodisperse) at s = 1. For s > 1, they representdistributions of the finite width. Using the equality (23),the dry (4) and wet (12) lognormal aerosol size spectra canbe rewritten in algebraic form as

fd rdð Þ ¼ kd0Na

rd0

rd=rd0ð Þkd0�1

1þ rd=rd0ð Þkd0h i2 ; ð24Þ

fw rwð Þ ¼ kw0Na

rw0

rw=rw0ð Þkw0�1

1þ rw=rw0ð Þkw0h i2 ; ð25Þ

where rd0 and sd are the geometric radius and dispersion ofthe dry spectrum, rw0 and sw are corresponding quantitiesfor the wet spectrum defined in (13), and the indices k0d andk0w are

kd0 ¼4ffiffiffiffiffiffi

2pp

ln sd; kw0 ¼

4ffiffiffiffiffiffi2p

plnsw

: ð26Þ

A comparison of the algebraic (24) and lognormal (4)aerosol size spectra calculated for the two pairs of spectrawith the same parameters is shown in Figure 4. One can seethat algebraic and lognormal spectra are in a goodagreement in the region from the maxima and down bytwo to three orders of magnitude. A discrepancy is seen inthe region of the tails, but this region provides a smallcontribution into the number density.[25] These algebraic representations of the size spectra

can be useful in applications. Evaluation of their analyticaland asymptotic properties is simpler than those of thelognormal distributions, and various moments over the sizespectra can be expressed via elementary functions ratherthan via erf as in the case with the lognormal distributions.Being a good approximation to the lognormal distributionbut simpler in use, the algebraic functions (24), (25) can beused for approximation of the aerosol, droplet, and crystalsize spectra as a supplement or an alternative to thetraditionally used power law, lognormal, and gamma dis-tributions. Some applications are illustrated in the nextsections.

4. Differential CCN SupersaturationActivity Spectrum

4.1. Critical Supersaturations and Radii

[26] To derive the supersaturation activity spectrum, weneed relations among the dry radius rd, correspondingcritical water supersaturation scr, and the critical nucleusradius rcr. The values rcr and scr for CCN activation can befound as usually using Kohler’s equation (6) from thecondition of maximum ds/dr = 0:

rcr ¼3B

Ak

� �1=2

¼ r1þbd

3b

Ak

� �1=2

; ð27Þ

scr ¼4A3

k

27B

� �1=2

¼ 2

3

Ak

rcr¼ r

� 1þbð Þd

4A3k

27b

� �1=2

¼ c1=k11 r

�1=k1d ; ð28Þ

where we used again the parameterization (8) for B. Theseequations can be reverted to express rd via rcr or scr:

rd rcrð Þ ¼ rk1cr Ak=3bð Þk1=2; rd scrð Þ ¼ c1s�k1cr ð29Þ

where

k1 ¼1

1þ b; c1 ¼

4A3K

27b

� �k1=2

ð30Þ

[27] Figure 5 depicts the critical supersaturations, scr, andcritical radii, rcr, calculated with (27), (28) as functions of rd

Figure 4. Comparison of the algebraic and lognormalaerosol size spectra for the two pairs of parameters: rm =0.1 mm, sd = 2.15 (solid and open circles for the algebraicand lognormal, respectively), and rm = 0.0.43 mm, sd =1.5 (solid and open triangles).


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with four values of b (0.55, 0.275, 0.135, 0.07),corresponding to various soluble fractions of ammoniumsulfate (100, 50, 25, and 12.5%) indicated in the legend.The values of scr required for activation of a given CCNincrease with decreasing rd from 0.1–0.3% at rd = 0.1 mm (atypical cloud case) to 3–10% at rd = 0.01 mm (as can befound in a cloud chamber) and reach 70–200% at rd =0.001 mm. The values of rcr increase with rd and the ratio rcr/rd reaches 5–13 at rd = 0.1 mm and 13–40 at rd = 1 mm. The8-fold decrease in the soluble fraction causes scr to increaseand rcr to decrease by a factor of �3–5.[28] These features are important for understanding and

quantitative description of the separation between activatedand unactivated CCN fractions. Figure 6 shows four lognor-mal size spectra of the dry aerosol with various modal radiiand dispersions along with the corresponding critical radiiand supersaturations calculated with b = 0.5, b = 0.55 (50%soluble fraction of ammonium sulfate). One can see that ifthe maximum supersaturation does not exceed �0.1 %, theright branch of the dry spectrum with rm = 0.1 mm can beactivated down to the mode, and the left branch (rd < rm)

remains unactivated. For the spectrum with rm = 0.03 mm,activation of the right half down to rm requires asupersaturation of 0.6%. Such a supersaturation is typicalof some natural clouds. Activation of the left brancheswith rd < 0.01–0.03 mm requires higher supersaturationsand may be typical of the CCN measurements in cloudchambers where s � 5–20% can be reached [e.g., Jiustoand Lala, 1981; Hudson, 1984; Yum and Hudson, 2001].

4.2. Lognormal Differential CCN Activity Spectrum

[29] The CCN differential supersaturation activity spec-trum js(s) can be obtained now using the conservation lawin differential form for the concentration:

fd rdð Þdrd ¼ �js sð Þds: ð31Þ

Here s is the critical supersaturation required to activate adry particle with radius rd; the minus sign occurs sinceincrease in rd (drd > 0) corresponds to a decrease in s (ds < 0)as indicated by (28). Using (29) and (30), (31) can berewritten in the form:

js sð Þ ¼ �fd rdð Þ drd sð Þds

¼ c2s�k2 fd rd sð Þ½ �; ð32Þ

where

c2 ¼1

1þ b4A3

K

27b

� �1=2 1þbð Þ¼ k1c1; k2 ¼ k1 þ 1 ¼ 2þ b

1þ b: ð32aÞ

Figure 6. Lognormal aerosol size spectra for rm = 0.1 mm,sd = 2 (solid circles), rm = 0.1 mm, sd = 1.5 (crosses), rm =0.03 mm, sd = 2 (rhombs), and rm = 0.03 mm, sd = 1.5(triangles), plotted as the functions of dry radii and ofcorresponding critical radii and critical supersaturations.Critical supersaturations scr corresponding to the maxima at0.1 mm and 0.03 mm are shown near the curves.

Figure 5. (a) Critical supersaturation, scr, and (b) criticalradius, rcr, calculated as functions of the dry radius rd withthe four values of b corresponding to various solublefractions of ammonium sulfate indicated in the legend.


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Substituting the dry lognormal size spectrum (4) into (32),we obtain:

js sð Þ ¼ Naffiffiffiffiffiffi2p

plnssð Þs

exp � ln2 s=s0ð Þ2 ln2 ss

� �; ð33Þ

where we introduced the mean geometric supersaturation s0and the supersaturation dispersion ss:

s0 ¼ rd0=c1ð Þ�1=k1¼ r� 1þbð Þd0

4A3k

27b

� �1=2

; ð34Þ

ss ¼ s1=k1d ¼ s 1þbð Þd : ð35Þ

The modal supersaturation sm is related to s0 similar to thatfor the dry size spectrum (5)

sm ¼ s0 exp � ln2 ss� �

: ð36Þ

The supersaturation activity spectrum (33) has a maximumat sm and the region s = sm ± ss gives the maximumcontribution into the droplet concentration.[30] Equations (4) and (33)–(36) show the following.

The lognormal size spectrum of the dry aerosol with theparameters described above is equivalent to the lognormalCCN activity spectrum by supersaturations js(s); that is, theshape is preserved under transformations between the radiusand supersaturation variables. This feature is similar to thatfound in the work of Abdul-Razzak et al. [1998] andFountoukis and Nenes [2005]. The difference of our modelwith these works is in that they assume soluble fractionproportional to the volume (b = 0.5), while our model forjs(s) generalizes this expression and allows variable solublefraction. For b = 0.5 (homogeneously mixed in volume),(35) for ss shows that the argument in the exponent (33) canbe rewritten as (�1/2)[(2/3)ln(s/s0)/lnsd]

2, which coincideswith the cited works. For soluble fraction proportional to thesurface, b = 0, as can be for the dust particles covered by thesoluble film, the mean geometric supersaturation s0 � rd0

�1,and not �rd0

�3/2, as in the case b = 0.5; the logarithm lnss is

50 % smaller with b = 0, the argument becomes (�1/2)[ln(s/s0)/lnsd]

2 and js(s) becomes much narrower. This meansthat the maximum in the activation spectrum is reached atanother sm and drop activation occurs over narrower s-rangewith b = 0.

4.3. Algebraic Differential CCN Spectrum

[31] An algebraic form of the differential activity spec-trum can be obtained substituting rd(s) from (29) and rd0(s0)from (34) into the conservation equation (32) with fd(rd)from (24):

js sð Þ ¼ ks0Na

s0

s=s0ð Þks0�1

1þ s=s0ð Þks0h i2 ; ð37Þ

or, in a slightly different form that is more convenient forcomparison with the other models

js sð Þ ¼ ks0C0sks0�1 1þ h0s

ks0� ��2

; ð38Þ

Here

ks0 ¼4ffiffiffiffiffiffi

2pp

ln ss¼ 4ffiffiffiffiffiffi

2pp

1þ bð Þ ln sd; ð39aÞ

C0 ¼ Nas�ks00 ¼ Na

27b

4A2k

� �ks0=2

rd0ð Þks0 1þbð Þ; ð39bÞ

h0 ¼ s�ks00 ¼ 27b

4A2k

� �ks0=2

rd0ð Þks0 1þbð Þ: ð39cÞ

(It is shown below that ks0 is the asymptotic value of the k-index in the CCN activity power law at small s). Expression(38) can be also obtained from the lognormal law (33) usingthe similarity relation (23), which again illustrates theequivalence of the lognormal and algebraic distributions.[32] The first term, ks0C0s

ks0�1, on the right-hand side of(38) represents Twomey’s [1959] power law (2). In contrastto the C0 and ks0 based on fits to experimental data, here theanalytical dependence is derived from the Kohler’s theoryand algebraic approximation of the lognormal functions,and parameters C0, ks0 are expressed directly via aerosolparameters. This first term describes drop activation at smallsupersaturations s� s0 (as in the clouds with weak updraftsand in the ‘‘haze chambers’’) and grows with s as sks0�1. Thesecond term in parenthesis decreases at large s � s0 ass�2ks0, overwhelms the growth of the first term, and ensuresthe asymptotic decrease js(s) � s�ks0�1. Thus the term inparenthesis serves effectively as a correction to the Twomeylaw and prevents an unlimited growth of concentration ofactivated drops at large supersaturations.[33] The entire CCN spectrum (38) resembles the

corresponding equation from Cohard et al. [1998, 2000]:

js sð Þ ¼ kCsk�1 1þ hs2� ��l

; ð40Þ

where the first term is also the Twomey’s differential powerlaw, and the term in parenthesis was added by the authors as

Figure 7. CCN differential activity spectrum js(s) calcu-lated with b = 0.25 and four combinations of modal radii rmand dispersions sd of the dry aerosol spectra indicated in thelegend.


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an empirical correction with the two parameters h, l thatwere varied and chosen by fitting the experimental data.Thus the algebraic representation of js(s) (38) found hereestablishes a bridge between the Twomey’s power law withits corrections as in the work of Cohard et al. [1998, 2000]and the lognormal parameterizations of drop activation as inthe work of Ghan et al. [1993, 1995], Abdul-Razzak et al.[1998], Abdul-Razzak and Ghan [2000], and Fountoukisand Nenes [2005].[34] Shown in Figure 7 is an example of the CCN

differential activity spectrum js(s) (38) calculated withb = 0.25 (CCN with �50 % of ammonium sulfate or�20 % of NaCl) and four combinations of modal radiiand dispersions. The area beneath each curve represents thedrop concentration Ndr that could be activated at any givensupersaturation s, so that the maximum of js(s) indicates theregion of s where its increase leads to the most effectiveactivation. A remarkable feature of this figure is a substan-tial sensitivity to the variations of rm, sd. The maximum forthe maritime-type spectrum with rm = 0.1 mm, sd = 2 lies atvery small s � 0.01 %. When sd decreases to 1.5 or rmdecreases to 0.03 (closer to the continental spectrum),the maximum of js(s) shifts to s � 0.04–0.1 %. Withrm = 0.03 mm, sd = 1.5, the region s � 0.01–0.1% becomesrelatively inactive and the maximum shifts to 0.4%. Thuseven moderate narrowing of the dry spectra or decrease inthe modal radius may require much greater vertical velocitiesfor activation of the drops with the same concentrations.[35] A comparison of the algebraic (37) and lognormal

(33) differential activity spectra in Figure 8 shows theirgeneral good agreement. The discrepancy increases in theboth tails, but their contributions into the number density issmall as will be illustrated below. Thus the algebraicfunctions (37), (38) approximate the lognormal spectrum(33) with good accuracy.

5. Droplet Concentration and Modified PowerLaw for Drops Activation

5.1. Droplet Concentration Based on the Lognormaland Algebraic CCN Spectra

[36] The concentration of CCN or activated drops can beobtained by integration of the differential activity spectrum

js(s) (33) over the supersaturations to the maximum value sreached in a cloudy parcel:

Nd sð Þ ¼ NCCN sð Þ ¼Zs

0

js s0ð Þds0 ¼ Na

21þ erf

1ffiffiffi2

p ln s=s0ð Þln ss

� �� ;

ð41Þ

where s0 and ss are defined by (34), (35). Equation (41) issimilar to the corresponding expressions derived by von derEmde and Wacker [1993], Ghan et al. [1993, 1995], Abdul-Razzak et al. [1998], Abdul-Razzak and Ghan [2000],Fountoukis and Nenes [2005] that also arrive at erf function.A generalization of these works in (41) is the same as wasdiscussed for the differential activation spectrum in previoussection, in variable soluble fraction b, which is discussedbelow.[37] Droplet or CCN concentration in algebraic form can

be obtained now in two equivalent ways. The first way isintegration of the algebraic CCN spectrum js(s) (37) or (38)over s:

NCCN sð Þ ¼ Na

s=s0ð Þks0

1þ s=s0ð Þks0h i ¼ C0s

ks0 1þ h0sks0

��1; ð42Þ

where C0 and h0 are defined in (39b) and (39c). The secondway is as in the work of Ghan et al. [1993] by substituting(21) for the equality of erf and tanh into (41), which alsoyields (42). The derivation of (41) and (42) along withequivalence (20), (21) emphasizes again the fact thatdifferential CCN distributions by critical radii or super-saturations are smoothed Dirac delta functions, while theirintegrals, droplet concentrations, are smoothed Heavisidestep functions.[38] According to (42), the asymptotic of NCCN(s) at s �

s0 is the Twomey power law NCCN(s) = C0sks0. The term in

parenthesis is a correction, its asymptote at s � s0 is s�ks0, it

compensates the growth of the first term and preventsunlimited drop concentration. The asymptotic limit of dropconcentration is NCCN(s) ! Na at large s, and hence thenumber of activated drops is limited by the total aerosolconcentration.[39] Equation (42) is similar to the corresponding equa-

tion from Ghan et al. [1993], but is expressed via super-saturation dispersion instead of the modal radius as in thatwork and is generalized here for the variable solublefraction b, which causes the following difference. Asfollows from definitions of ks0 (39a), the power index ks0differs by the factor (1 + b) from the correspondingexpression kgh = 4/(

ffiffiffiffiffiffi2p

pln sd) in the work of Ghan et al.

[1993], who assumed soluble fraction proportionalto the volume, i.e., in our terms b = 0.5. With this value,ks0 = (2/3)kgh, and the asymptotic of NCCN for small s isNCCN � sks0 � s(2/3)kgh, which coincides with the expressionin the work of Ghan et al. [1993]. However, for b = 0(surface-proportional soluble fraction), the index ks0 is 1.5times higher than with b = 0.5, and, as mentioned before,the functions NCCN(s) increase (droplets activate) over muchnarrower interval of s.

Figure 8. Comparison of the algebraic (alg.) and lognor-mal (log.) differential CCN activity spectra js(s). A pair ofdigits in the parentheses in the legend denotes modal radiusrm in mm and dispersion sd of the corresponding dry aerosolspectrum.


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[40] If the dry aerosol size spectrum is multimodal with Imodes (e.g., I = 3 in a three-modal distributions [Whitby,1978]), then it can be written as a superposition of Ilognormal fractions (4) with Nai, sdi, and rd0i being thenumber concentration, dispersion, and mean geometricradius of the ith fraction. The mean geometric supersatura-tions s0i, dispersions ssi, and the other parameters for eachfraction are then defined by the same equations as describedabove for a single fraction. Then the algebraic dry and wetsize spectra, the differential and integral activity spectra areobtained for each fraction as described above and thecorresponding multimodal spectra are superpositions ofthe I fractions. The multimodal CCN activity spectrum inalgebraic form is derived from (42):

Nd sð Þ ¼XI

i¼1

Nai

s=s0ið Þks0i

1þ s=s0ið Þks0ih i ¼ XI

i¼1

C0isks0i 1þ h0is

ks0i ��1

:

ð43Þ

Equation (43) for b = 0.5 is similar to the correspondingequation from Ghan et al. [1995] where Nd(s) is expressedin terms of critical radii, but expresses it here directly viasupersaturation and generalizes for the other b. Theparameters bi can be different for each fraction, e.g., bi =0.5 may correspond to the accumulation mode, and bi = 0may correspond to the coarse mode consisting of the dustparticles coated with the soluble film.

5.2. Revived Power Law for the Drop Concentration,Power Index k(s), and Coefficient C(s)

[41] The power law (1), NCCN = Csk, is often used forparameterization of the results of CCN measurements and ofdrop activation in many cloud models. An analytical func-tion for the k- index can be found now (similar to them-index (16)) using the method from Khvorostyanov andCurry [2002, 2005] for the power law representation ofthe continuous functions. Using (1) and its derivativeN0

CCN(s) (2) and solving these two equations for k(s), weobtain:

k sð Þ ¼ sN 0CCN sð Þ

NCCN sð Þ ¼ sjs sð Þ

NCCN sð Þ ð44Þ

Equation (44) allows evaluation of k(s) with knowndifferential js(s) and cumulative NCCN(s) CCN spectra.Substituting here (33) for js(s) and (41) for NCCN(s), weobtain k(s) in terms of lognormal functions:

k sð Þ ¼ffiffiffi2

p

r1

lnssexp � ln2 s=s0ð Þ

2 ln2 ss

� �1þ erf

ln s=s0ðffiffiffi2

plnss

� �� 1

:

ð45Þ

Here s0 and ss are defined by (34), (35). The algebraic formof the index k(s) is obtained by substitution of (37) for js(s)and (42) for NCCN(s) into (44):

k sð Þ ¼ ks0 1þ s=s0ð Þks0h i�1

¼ ks0 1þ sks0rks0 1þbð Þd0

27b

4A3k

� �ks0=2" #�1

:

ð46Þ

If the aerosol size spectrum is multimodal and is given by asuperposition of I algebraic distributions, then the k-index isderived from (44), (43) by summation over I modes:

k sð Þ ¼XI

i¼1

ks0iNai s=s0ið Þks0i

1þ s=s0ið Þks0ih i2

�XI

i¼1

Nai

s=s0ið Þks0i

1þ s=s0ið Þks0ih i

8<:

9=;

�1

; ð47Þ

where the parameters ks0i, s0i, Nai are defined as before butfor each ith mode.[42] Equations (45)–(47) allow evaluation of k(s) for any

s with given ss and s0 that are expressed via rd, sd, b, and b.That is, knowledge of the size spectra and composition ofthe dry or wet CCN allows direct evaluation of k(s) over theentire s-range. This solves the problem outlined in Jiustoand Lala [1981], Yum and Hudson [2001] and many otherstudies: various values of the k-index at various s.Equations (45)–(47) provide a continuous representationof k over the entire s-range. Note that the k-index is notconstant for a given air mass or aerosol type but depends ona given s, at which it is measured. Equation (46) shows thatfor s � s0, the asymptotic value k(s) = ks0. For s � s0, theindex decreases with s as k(s) = ks0(s/s0)

�ks0 and tends tozero. This behavior is in agreement with the experimentaldata from Jiusto and Lala [1981], Yum and Hudson [2001],and others.[43] The coefficient C(s) can be now calculated using

(42) for NCCN(s) and (46) for k(s)

C sð Þ ¼ NCCN sð Þs�k sð Þ ¼ C0sc sð Þ 1þ s=s0ð Þks0

h i�1

; ð48Þ

where C0 is defined in (39b). The power index c(s) of s in(48) is

c sð Þ ¼ ks0 � k sð Þ ¼ ks0s=s0ð Þks0

1þ s=s0ð Þks0: ð49Þ

Now we can write the CCN (or droplet) concentration in theform of a modified power law

NCCN sð Þ ¼ C sð Þsk sð Þ; s in share of unit: ð50Þ

This equation is usually used with supersaturation s inpercent, related to ssu in share of unit as s = 10�2 ssu.Substitution of this relation into (50) yields a moreconventional form

NCCN sð Þ ¼ Cpc sð Þsk sð Þ; s in percent: ð51Þ

In (51), both k(s) and C(s) depend on the ratio s/s0 and thusdo not depend on units; k(s) is calculated again from (46)but with s and s0 in percent, and the coefficient

Cpc spc� �

¼ 10�2k sð ÞC sð Þ: ð52Þ


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Equation (51) is similar to (1), however, now with theparameters depending on s as just described. Hence k(s) andC(s) are now expressed directly via the parameters of thelognormal spectrum (Na, rd0, sd) and its physicochemicalproperties via simple parameterization of b in (9a)–(9c).[44] Finally, the power law for the multimodal aerosol

types is described again by (50), (51) with the k(s)-indexevaluated from (47) and the coefficient C(s)

C sð Þ ¼XI

i¼1

C0isci sð Þ 1þ s=s0ið Þks0i

h i�1

; ð53Þ

where ci = ks0i � k(s), and ks0i, C0i, s0i, Nai are definedfor each ith mode. The coefficient Cpc(s) with super-saturation in percent is defined by (52). The number of modesI is determined by the nature of aerosol and is the sameas used for multimodal size spectra with the lognormalapproach in thework ofGhan et al. [1995],Abdul-Razzak andGhan [2000], and Fountoukis and Nenes [2005].[45] This analytical s-dependence of C and k in (50), (51)

provides a theoretical basis for many previous findings and

improvements of the drop activation power law. In partic-ular, the C-k space constructed by Braham [1976] based onexperimental data follows now from the equations for k(s)and C(s) with use of any relation of maximum s to verticalvelocity. The decrease of the k-slopes with increasing sanalyzed by Jiusto and Lala [1981], Yum and Hudson[2001], Cohard et al. [1998, 2000], and many others ispredicted by (45)–(47), as well as its dependence on thenature of aerosol. The proportionality of coefficient C to theaerosol concentration explains substantially greater valuesof C observed in continental than in maritime air masses ascompiled by Twomey and Wojciechowski [1969] and Heggand Hobbs [1992] and used by cloud modelers.

5.3. SupersaturationDependence of k(s), C(s), and Nd(s)

[46] The equivalent power index k(s) calculated with (46)and a monomodal lognormal size spectrum is shown inFigures 9a and 9b and is compared with the experimentaldata from Jiusto and Lala [1981], who collected the datafrom several chambers and to the field data collected byYum and Hudson [2001] with a continuous flow diffusionchamber (CFD) at two altitudes in the Arctic in May 1998during the First International Regional Experiment-ArcticCloud Experiment (FIRE-ACE). It is seen that the k-index isnot constant as assumed in many cloud models. Thecalculated k-indices reach the largest values of 1.5–4.2 ats 0.01 %, where it tends to the asymptotic ks0 (39a) andincreases with decreasing dispersion of the dry CCN spec-trum since ks0 � (lnsd)

�1. As s increases, k(s) is initiallyalmost constant, then decreases slowly up to s � 0.03–0.05% and then decreases more rapidly in the region s �0.05–0.12% to values less than 0.1–0.2 at s > 1%. Thek-indices increase with decreasing modal radii and disper-sions. Agreement of calculated k(s) with haze chamber datais reached at sd = 1.3, rd = 0.085 mm, and with CFD at sd =1.5, rd = 0.043 mm. Agreement with the Arctic field data byYum and Hudson [2001] is reached with rm = 0.045 mm,sd = 2.15 in the layer 960–1010 mb and rm = 0.043 mm,sd = 1.9 in the layer 560–660 mb.[47] Thus this model explains and describes quantitatively

the observed decrease in the k-index with increasing super-saturation reported previously by many investigators (e.g.,Twomey and Wojciechowski, [1969], Braham [1976], Jiustoand Lala [1981]; see review in PK97) and relates it to theaerosol size spectra. A more detailed analysis of experi-mental data with these equations should be based onsimultaneous use of the measured multimodal aerosol sizespectra and CCN supersaturation activity spectra.[48] In contrast to most models where both k and C are

constant and to some other models where the k-indexdepends on s, but the coefficient C is constant [e.g., Cohardet al., 1998, 2000], the values of ‘‘effective’’ C(s), Cpc(s) inthis model also depend on supersaturation. The spectralbehavior of Cpc(s) calculated from (52) with different rm, sdand the same Na = 100 cm�3 is shown in Figure 10. For s�s0, the coefficients tend to the asymptotic limit Cpc(0) =10�2ks0C0 and reach values of 103–105 cm�3 caused bythe dependence C0 � Ak

�2ks0rd0ks0(1+b) with Ak � 10�7 cm and

rd0� 3� 10�6 to 10�5 cm. This dependence causes increasein Cpc with decreasing sd (increasing ks0) and with increas-ing rd0 seen at small s in Figure 10; i.e., the narrower spectraand the larger modal radius, the faster drop activation at

Figure 9. Supersaturation dependence of the k-indexcalculated for the lognormal size spectrum of dry CCNwith various modal radii rm and dispersions sd indicated inthe legend and comparison: (a) with the experimental datafrom Jiusto and Lala [1981] from four chambers (hazechamber; continuous flow diffusion chamber, CFD; andstatic diffusion chamber, SDC), and (b) with the datacollected in FIRE-ACE in 1998 [Yum and Hudson, 2001,hereinafter referred to as YH01] at the heights 960–1010 mband 560–660 mb.


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small s. The values of Cpc(s) decrease with increasing s by1–3 orders of magnitude and tend to Na at spc > 0.1–0.7%,since the asymptotics at large s is Cpc(spc) = 10�2k(s)Naspc

�k(s),and k(s) ! 0 at large s, hence all Cpc ! Na at spc � 0.5% asseen in Figure 10. Thus NCCN(s) = Cpcspc

k(s) � 10�2k(s)Na andtends to the limiting value Na, which ensures finite values ofNa in contrast to the models with constant k, C.[49] A comparison of the concentrations NCCN(s) calcu-

lated using the power law (51) and erf function (41) showsin Figure 11 their good agreement and illustrates goodaccuracy of the algebraic distributions and renewed powerlaw as compared to the direct integration of the lognormaldistributions. The curves NCCN(s) calculated with the newerpower law are not linear in log-log coordinates as it is with(1) and constant C0, k, but are concave in agreement withthe measurements [e.g., Jiusto and Lala, 1981; Hudson,

1984; Yum and Hudson, 2001] and other models [e.g., Ghanet al., 1993, 1995; Feingold et al., 1994; Cohard et al.,1998, 2000]. The reason for this concavity is quite clearfrom Figures 6–8 for the size and activity spectra: the rateof increase in concentration of activated drops dNd(s)/ds isgreatest at small supersaturations in the left branch of js(s)(Figures 7 and 8), or in the right branch of fd(rd) at rd > rm(Figure 6), then dNd(s)/ds decreases with growing s, espe-cially at s > 1%, i.e., in the left branch of the small dry CCNwith rd < 0.03 mm (Figure 6). Effectively, some ‘‘saturation’’with respect to supersaturation occurs, causing k(s), C(s)and dNd(s)/ds to decrease with s and leading to finite Nd(s)at large s.[50] Finally, Figure 12 shows NCCN(s) calculated with the

lognormal size spectra and compared to the correspondingfield data obtained by Yum and Hudson [2001] in the FIRE-ACE campaign in May 1998 in the Arctic. The parametersrm, sd, and Na were varied slightly in the calculations,although we constrained the calculations to the measured k-indices (Figure 10b), so that the values of the parametershad to provide an agreement for both k(s) and NCCN(s).Good agreement over the entire range of supersaturationswas found with the values indicated in the legend. Thevalues of modal radii and dispersions are typical for theaccumulation mode. This comparison shows that this modelof the power law can reasonably reproduce the experimentaldata on CCN activity spectra and concentrations of activateddrops and can be used for parameterization of drop activa-tion in cloud models.

6. Summary and Conclusions

[51] A model of the lognormal size spectrum of mixed(partially soluble) dry CCN with parameterization of thesoluble fraction as a function of the dry nucleus radius isused to consider the processes of humidity transformation ofthe dry size spectrum into the spectrum of the wet interstitialaerosol and drop activation. Using the expressions for theradius of a solution particle as a function of relativehumidity, it was shown that a lognormal size spectrum of

Figure 10. Equivalent coefficient Cpc(s) in the power lawfor NCCN calculated for the lognormal size spectrum of dryCCN with modal radii rm and dispersions sd indicated in thelegend and other parameters described in the text. For theconvenience of comparison, all curves are calculated withNa = 102 cm�3.

Figure 11. Comparison of the accumulated CCN spectraNCCN(s) calculated using power law equation (51) (solidsymbols) and using (41) with erf function (open symbols).The values of the corresponding modal radii rm anddispersions sd of the dry aerosol spectra are indicated inthe legend. For convenience of comparison, NCCN iscalculated with the dry aerosol concentrations of 250,200, 150, and 100 cm�3 from the upper to lower curves.

Figure 12. Supersaturation dependence of the CCNintegral activity spectrum NCCN(s) calculated with the newpower law (51) for the lognormal size spectra with rm, sdand Na (in cm�3) indicated in the legend (solid rhombs andcircles), compared to the corresponding field data obtainedby Yum and Hudson [2001] in FIRE-ACE campaign in May1998 in the Arctic (open rhombs and circles).


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the dry aerosol transforms into the lognormal spectra of thecloud interstitial aerosol at supersaturation. The mean geo-metric radii and dispersions of these spectra are expressedvia the corresponding parameters of the dry aerosol. Aquasi-power law Junge-type representation of the lognormalspectra was derived, and corresponding power indices andcoefficients are expressed via the corresponding meangeometric sizes and dispersions of the dry and interstitialaerosol. Using various representations of the smoothedDirac delta function, an algebraic equivalent was foundfor the lognormal dry and interstitial aerosol size spectra.These algebraic representations allow simplification ofanalytical estimations and evaluation of various momentsof the spectra such as condensation rate, etc.[52] On the basis of Kohler theory, the differential CCN

activity spectrum is expressed in lognormal form for varioussoluble fractions. This spectrum coincides with the previousparameterizations from Abdul-Razzak et al. [1998] andFountoukis and Nenes [2005] for soluble fraction propor-tional to the particle volume and generalizes them for othersoluble fractions (e.g., soluble fraction proportional to sur-face area as can be for the insoluble dust particles coated witha thin soluble shell). The algebraic equivalent of the lognor-mal differential CCN activity spectra is found, js(s) =ks0C0s

ks0�1(1 + h0sks0)�2. This is a generalization and correc-

tion of the Twomey’s differential activity spectrum: the firstterm is Twomey’s power law, and the term in parenthesesensures finite droplet concentrations at high supersaturations.The functional form of this activity spectra is similar to theempirical correction by Cohard et al. [1998, 2000] ofTwomey’s power law but differs from these works by theanalytical dependence that leads to the simple algebraicexpressions for the droplet concentration, and the threeparameters C0, ks0, h0 of this spectrum are directly expressedvia the size parameters and composition of the dry aerosol.[53] The cumulative CCN activity spectrum, NCCN(s) is

found in lognormal form, which coincides with the previousparameterizations by von der Emde and Wacker [1993],Ghan et al. [1993, 1995], Abdul-Razzak et al. [1998], andFountoukis and Nenes [2005] for the soluble fraction pro-portional to the particle volume (homogeneously mixedparticles). We have generalized the previous lognormalparameterizations for other soluble fractions, in particular,for the soluble fraction proportional to the particle surface, as,e.g., for dust or other insoluble particles with soluble coatingson the surface [Levin et al., 1996; Bauer and Koch, 2005]. Itis shown that the algebraic form of the cumulative CCNspectrum,NCCN(s), as found in the work ofGhan et al. [1993]from the lognormal spectrum using the equivalence of erf andtanh, can be obtained also by integration of the algebraicdifferential spectrum js(s). This establishes a bridge andillustrates the relation among the numerous approaches basedon the differential power law or algebraic CCN spectra, andthose that derive NCCN(s) by direct integration of lognormalaerosol size spectra using the Kohler relation between thecritical radius and supersaturation.[54] A modified power law is derived for the drop

activation, NCCN(s) = C(s)sk(s). The modified power lawyields drop concentration limited by the total aerosolnumber, in contrast to the previous power law with unlim-ited NCCN(s) at high supersaturations. Algebraic formulaeare found for C(s), and k(s) as continuous functions of

supersaturation s, which correctly describe their decreasewith s, i.e., predict the observed concave NCCN(s) spectra.The new formulation expresses C(s) and k(s) via parametersof the dry aerosol spectrum (the modal radius, dispersion,and aerosol concentration) and the physicochemical prop-erties of the aerosol and the distribution of the solublefraction in the aerosol particles. The derivation of the newpower law for the CCN activity spectrum clearly shows itsequivalence and correspondence to the lognormal parame-terizations. The modified power law permits using thewealth of the data on the C, k parameters accumulated overthis time. Additionally, the C, k parameters can be directlycalculated based on the aerosol microstructure and recalcu-lated from s = 1% to any s. The C-k space suggested byBraham [1976] based on experimental data can now beobtained from calculations using aerosol microstructure asthe input.[55] The relationships derived here for CCN concentra-

tion and activity spectra can be easily incorporatedinto cloud and climate models. Relative to the lognormalparameterization that is included in climate models [Ghan etal., 1997; Lohmann et al., 1999; Nenes and Seinfeld, 2003],the parameterizations presented here include an additionalelement of necessary physics, related to the treatment of thesoluble fraction and how the aerosol particles are mixed. Anadditional advantage of the new parameterizations is theiralgebraic and computational simplicity, allowing for simpleanalytical determination of derivatives and higher ordermoments. Further, the power law and algebraic representa-tions are easily superimposed to represent multimodalspectra.

Notation

Ak the Kelvin curvature parameter.B = brd

2(1+b) the nucleus activity.b the parameter defined in (9a), (9b), (9c)

that describes the soluble fraction of anaerosol particle.

C the coefficient in the power law forintegral CCN activity spectrum.

C0 the coefficient in the differential powerlaw CCN activity spectrum (38).

c1, c2 the coefficients defined in (30) and(32a).

cf the coefficient in the Junge-type sizespectrum.

f(r) the general notation for the aerosol sizespectrum.

fd(rd), fw(rw) the size distribution functions of the dryand wet aerosol.

H, Hth the relative humidity and the thresholdof deliquescence of the soluble fraction.

k the index in the power law for integralCCN activity spectrum.

k0 the parameter of algebraic size spectra.k0d and k0w the parameters of the dry and wet

algebraic size spectra.k1, k2 the indices defined in (30) and (32a).

ks0 the power index of the algebraic super-saturation spectrum.


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l0 the thickness of the soluble shell on aninsoluble aerosol particle.

Mw and Ms the molecular weights of water andaerosol soluble fraction.

ms the mass of the soluble fraction of anaerosol particle.

Na the aerosol number concentration.NCCN the concentration of the cloud conden-

sation nuclei (CCN).Nd the droplet number concentration.R the universal gas constant.

rb = 2Ak/3s the boundary radius of interstitial CCN.rcr the CCN critical radius of drop activa-

tion.rd the radius of a dry aerosol particle.rd0 the mean geometric radius of the dry

aerosol spectrum.rd1 the scaling parameter in (9b).rm the modal radius of the aerosol size

spectrum.rw(H) the radius of a wet particle.rw0 the wet mean geometric radius.

s = (rv � rvs)/rvs the supersaturation.s0 the mean geometric supersaturation.sm the modal supersaturation.T the temperature (in degrees Kelvin).a the parameter of width in the smoothed

Dirac delta function.b the power index defined in (8) that

describes the soluble fraction of anaerosol particle.

c the power index of s in the coefficientC(s) in (48).

d(x) Dirac’s delta function.ev the volume fraction of the soluble

fraction.ev0 the reference soluble fraction.Fs the osmotic potential.

js(s) differential CCN activity spectrum.h the parameter of the differential activity

spectrum in (39c).m the power index of Junge-type power

law spectrum � r�m.n the number of ions in solution.

rv, rvs and rw the densities of vapor, saturated vaporand water.

rs the density of the soluble fraction.sd the dispersion of the dry spectrum.ss the supersaturation dispersion.sw the dispersion of the wet aerosol size

spectrum.zsa the surface tension at the solution-air

interface.

[56] Acknowledgments. This research has been supported by theDOE Atmospheric Radiation Program. Input from Hugh Morrison isappreciated. We are grateful to reviewer for careful reading the manuscriptand useful remarks that allowed to improve the text. Jody Norman isthanked for help in preparing the manuscript.

ReferencesAbdul-Razzak, H., and S. J. Ghan (2000), A parameterization of aerosolactivation: 2. Multiple aerosol types, J. Geophys. Res., 105, 6837–6844.

Abdul-Razzak, H., S. J. Ghan, and C. Rivera-Carpio (1998), A parameter-ization of aerosol activation: 1. Single aerosol type, J. Geophys. Res.,103, 6123–6131.

Albrecht, B. (1989), Aerosols, cloud microphysics and fractional cloudi-ness, Science, 245, 1227–1230.

Bauer, S. E., and D. Koch (2005), Impact of heterogeneous sulfate forma-tion at mineral dust surfaces on aerosol loads and radiative forcing in theGISS GCM, J. Geophys. Res., 110, D17202, doi:10.1029/2005JD005870.

Braham, R. R. (1976), CCN spectra in c-k space, J. Atmos. Sci., 33, 343–346.

Cohard, J.-M., J.-P. Pinty, and C. Bedos (1998), Extending Twomey’sanalytical estimate of nucleated cloud droplet concentrations from CCNspectra, J. Atmos. Sci., 55, 3348–3357.

Cohard, J.-M., J.-P. Pinty, and K. Suhre (2000), On the parameterization ofactivation spectra from cloud condensation nuclei microphysical proper-ties, J. Geophys. Res., 105(D9), 11,753–11,766.

Feingold, G., B. Stevens, W. R. Cotton, and R. L. Walko (1994), Anexplicit cloud microphysics/LES model designed to simulate the Twomeyeffect, Atmos. Res., 33, 207–233.

Fitzgerald, J. W. (1975), Approximation formulas for the equilibriumsize of an aerosol particle as function of its dry size and composi-tion and ambient relative humidity, J. Appl. Meteorol., 14, 1044–1049.

Fitzgerald, J. W., W. A. Hoppel, and M. A. Vietty (1982), The size andscattering coefficient of urban aerosol particles at Washington D.C. as afunction of relative humidity, J. Atmos. Sci., 39, 1838–1852.

Fountoukis, C., and A. Nenes (2005), Continued development of a clouddroplet formation parameterization for global climate models, J. Geo-phys. Res., 110, D11212, doi:10.1029/2004JD005591.

Ghan, S., C. Chuang, and J. Penner (1993), A parameterization of clouddroplet nucleation. part 1, Single aerosol species, Atmos. Res., 30, 197–222.

Ghan, S., C. Chuang, R. Easter, and J. Penner (1995), A parameterization ofcloud droplet nucleation. 2, Multiple aerosol types, Atmos. Res., 36, 39–54.

Ghan, S., L. Leung, R. Easter, and H. Abdul-Razzak (1997), Prediction ofcloud droplet number in a general circulation model, J. Geophys. Res.,102, 777–794.

Hanel, G. (1976), The properties of atmospheric aerosol particles as func-tions of the relative humidity at thermodynamic equilibrium with thesurrounding moist air, Adv. Geophys., 19, 73–188.

Hegg, D. A., and P. V. Hobbs (1992), Cloud condensation nuclei in themarine atmosphere: A review, in Nucleation and Atmospheric Aerosols,edited by N. Fukuta and P. E. Wagner, pp. 181–192, A. Deepak, Hamp-ton. Va.

Hudson, J. G. (1984), Cloud condensation nuclei measurements withinclouds, J. Clim. Appl. Meteorol., 23, 42–51.

Ji, Q., and G. E. Shaw (1998), On supersaturation spectrum and size dis-tribution of cloud condensation nuclei, Geophys. Res. Lett., 25, 1903–1906.

Jiusto, J. E., and G. G. Lala (1981), CCN-supersaturation spectra slopes (k),J. Rech. Atmos., 15, 303–311.

Junge, C. E. (1963), Air Chemistry and Radioactivity, 424 pp., Elsevier,New York.

Khvorostyanov, V. I., and J. A. Curry (1999a), A simple analytical model ofaerosol properties with account for hygroscopic growth: 1. Equilibriumsize spectra and CCN activity spectra, J. Geophys. Res., 104, 2163–2174.

Khvorostyanov, V. I., and J. A. Curry (1999b), A simple analytical model ofaerosol properties with account for hygroscopic growth: 2. Scattering andabsorption coefficients, J. Geophys. Res., 104, 2175–2184.

Khvorostyanov, V. I., and J. A. Curry (2002), Terminal velocities of dro-plets and crystals: power laws with continuous parameters over the sizespectrum, J. Atmos. Sci., 59, 1872–1884.

Khvorostyanov, V. I., and J. A. Curry (2005), Fall velocities of hydrome-teors in the atmosphere: Refinements to a continuous analytical powerlaw, J. Atmos. Sci., 62, 4343–4357.

Korn, G. A., and T. M. Korn (1968), Mathematical Handbook for Scientistsand Engineers, 831 pp., McGraw-Hill, New York.

Laktionov, A. G. (1972), Fraction of soluble in water substances in theparticles of atmospheric aerosol, Izv. Acad. Sci. USSR, Atmos. OceanicPhys., Engl. Transl., 8, 389–395.

Levich, V. G. (1969), The Course of Theoretical Physics, vol. 1, 910 pp.,Nauka, Moscow.

Levin, L. M., and Y. S. Sedunov (1966), A theoretical model of condensa-tion nuclei. The mechanism of cloud formation in clouds, J. Rech. At-mos., 2, 2–3.

Levin, Z., E. Ganor, and V. Gladstein (1996), The effects of desert particlescoated with sulfate on rain formation in the eastern Mediterranean,J. Appl. Meteorol., 35, 1511–1523.


14 of 15

D12202

Lohmann, U., J. Feichter, C. C. Chuang, and J. E. Penner (1999), Predictingthe number of cloud droplets in the ECHAM GCM, J. Geophys. Res.,104, 9169–9198.

Nenes, A., and J. H. Seinfeld (2003), Parameterization of cloud dropletformation in global climate models, J. Geophys. Res., 108(D14), 4415,doi:10.1029/2002JD002911.

Pruppacher, H. R., and J. D. Klett (1997), Microphysics of Clouds andPrecipitation, 2nd ed., 954 pp., Springer, New York.

Rissman, T. A., A. Nenes, and J. H. Seinfeld (2004), Chemical amplifica-tion (or dampening) of the Twomey effect: Conditions derived fromdroplet activation theory, J. Atmos. Sci., 61, 919–930.

Sassen, K. P. J. DeMott, J. M. Prospero, and M. R. Poellot (2003), Saharandust storms and indirect aerosol effects on clouds: CRYSTAL-FACEresults, Geophys. Res. Lett., 30(12), 1633, doi:10.1029/2003GL017371.

Sedunov, Y. S. (1974), Physics of Drop Formation in the Atmosphere,translated from Russian by D. Lederman, 234 pp., John Wiley, Hoboken,N. J.

Seinfeld, J. H., and S. N. Pandis (1998), Atmospheric Chemistry and Phy-sics, 1326 pp., John Wiley, Hoboken, N. J.

Smirnov, V. I. (1978), On the equilibrium sizes and size spectra of aerosolparticles in a humid atmosphere, Izv. Acad. Sci. USSR, Atmos. OceanicPhys., 14, 1102–1106.

Squires, P. (1958), The microstructure and colloidal stability of warmclouds, II, The causes of the variations of microstructure, Tellus, 10,262–271.

Twomey, S. (1959), The nuclei of natural cloud formation. II. The super-saturation in natural clouds and the variation of cloud droplet concentra-tion, Pure Appl. Geophys., 43, 243–249.

Twomey, S. (1977), Atmospheric Aerosols, 302 pp., Elsevier, New York.Twomey, S., and T. A. Wojciechowski (1969), Observations of the geo-graphical variations of cloud nuclei, J. Atmos. Sci., 26, 684–696,1969.

von der Emde, K., and U. Wacker (1993), Comments on the relationshipbetween aserosol size spectra, equilibrium drop size spectra and CCNspectra, Contrib. Atmos. Phys., 66, 157–162.

Whitby, K. (1978), The physical characteristics of sulfur aerosols, Atmos.Environ., 12, 135–159.

Yum, S. S., and J. G. Hudson (2001), Vertical distributions of cloud con-densation nuclei spectra over the springtime Arctic Ocean, J. Geophys.Res., 106, 15,045–15,052.

��J. A. Curry, School of Earth and Atmospheric Sciences, Georgia Institute

of Technology, 311 Ferst Drive, Atlanta, GA 30332-0340, USA.([email protected])V. I. Khvorostyanov, Central Aerological Observatory, 3, Pervomayskaya

str., Dolgoprudny, Moscow Region, 141700, Russia. ([email protected])


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aerosol size spectra and ccn activity spectra: reconciling

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