Aerosol size spectra and CCN activity spectra: Reconciling the lognormal, algebraic, and power laws

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, oi: /2005jd006532, 2006 Aerosol size spectra an CCN activity spectra: Reconciling the lognormal, algebraic, an power laws Vitaly I. Khvorostyanov 1 an Juith A. Curry 2 Receive 27 July 2005; revise 2 November 2005; accepte 6 February 2006; publishe 17 June [1] Empirical power law expressions of the form N CCN = Cs k have been use in clou physics for over 4 ecaes to relate the number of clou conensation nuclei (CCN) an roplets forme on them to clou supersaturation, s. The eficiencies of this parameterization are that the parameters C an k are usually constants taken from the empirical ata an not irectly relate to the CCN microphysical properties an this parameterization preicts unboune roplet concentrations at high s. The activation power law was erive in several works from the power law Junge-type aerosol size spectra an parameters C an k were relate to the inices of the power law, but this still oes not allow to escribe observe ecrease of the k-inices with s an limite N CCN. Recently, new parameterizations for clou rop activation have been evelope base upon the lognormal aerosol size spectra that yiel finite N CCN at large s, but oes not explain the activation power law, which is still traitionally use in the interpretation of CCN observations, in many clou moels an some large-scale moels. Thus the relation between the lognormal an power law parameterizations is unclear, an it is esirable to establish a brige between them. In this paper, algebraic an power law equivalents are foun for the lognormal size spectra of partially soluble ry an wet interstitial aerosol, an for the ifferential an integral CCN activity spectra. This allows erivation of the power law expression for clou rop activation from basic thermoynamic principles (Köhler theory). In the new power law formulation, the inex k an coefficient C are obtaine as continuous analytical functions of s an expresse irectly via parameters of aerosol lognormal size spectra (moal raii, ispersions) an physicochemical properties. This approach allows reconciliation of this moifie power law an the lognormal parameterizations an their equivalence is shown by the quantitative comparison of these moels as applie to several examples. The avantages of this new power law relationship inclue boune N at high s, quantitative explanation of the experimental ata on the k-inex an possibility to express k(s) an C(s) irectly via the aerosol microphysics. The moifie power law provies a framework for using the wealth of ata on the C, k parameters accumulate over the past ecaes, both in the framework of the power law an the lognormal parameterizations. Citation: Khvorostyanov, V. I., an J. A. Curry (2006), Aerosol size spectra an CCN activity spectra: Reconciling the lognormal, algebraic, an power laws, J. Geophys. Res., 111,, oi: /2005jd Introuction [2] The power law an lognormal parameterizations of aerosol size spectra an supersaturation activity spectra are wiely use in stuies of aerosol optical an raiative properties, clou physics, an climate research. These two kins of parameterizations exist in parallel, implicitly compete, but their relation is unclear. One of the most important applications of the power an lognormal laws is 1 Central Aerological Observatory, Dolgopruny, Moscow Region, Russia. 2 School of Earth an Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia, USA. Copyright 2006 by the American Geophysical Union /06/2005JD006532$09.00 parameterization of the concentrations of clou rops N an clou conensation nuclei N CCN (CCN), on which the rops form. They influence clou optical an raiative properties as well as the rate of precipitation formation. Precise evaluation of N CCN is require, in particular, for correct estimation of anthropogenic aerosol effects on clou albeo [Twomey, 1977] an precipitation [Albrecht, 1989]. [3] The most commonly use parameterization of the concentration of clou rops or clou conensation nuclei N CCN on which the rops activate is a power law by the supersaturation s reache in a clou parcel is N CCN ðþ¼cs s k ; ð1þ 1of15

2 which is referre to in the literature as the integral or cumulative CCN activity spectrum. Numerous stuies have provie a wealth of ata on the parameters k an C for various geographical regions [see, e.g., Hegg an Hobbs, 1992; Pruppacher an Klett, 1997, Table 9.1, hereafter referre to as PK97]. Parameterizations of the type (1) were erive by Squires [1958] an Twomey [1959], using several assumptions an similar power law for the ifferential CCN activity spectrum, j s (s) j s ðþ¼n s CCN =s ¼ Cks k 1 : Equations (1) an (2) have been use for several ecaes in many clou moels with empirical values of k an C that are assume to be constant for a given air mass [PK97] an are usually constant uring moel runs. [4] To explain the empirical epenencies (1) an (2), moels were evelope of partially soluble CCN with the size spectra of Junge-type power law f(r) r m (total aerosol concentration N a r (m 1) ) an the inex k was expresse as a function of m. Jiusto an Lala [1981] foun a linear relation ð2þ k ¼ 2ðm 1Þ=3; ð3þ which implies k = 2 for a typical Junge inex m = 4, while the experimental values of k compile in that work varie over the range Levin an Seunov [1966], Seunov [1974], Smirnov [1978], an Cohar et al. [1998, 2000] erive more general power law or algebraic equations for j s (s) an expresse k as a function of m an CCN soluble fraction. Khvorostyanov an Curry [1999a, 1999b, hereafter referre to as KC99a, KC99b] erive power laws for the size spectra of the wet an interstitial aerosol, for the activity spectra j s (s), N CCN (s) an for the Angstrom wavelength inex of extinction coefficient, an foun linear relations among these inices expresse in terms of the inex m an aerosol soluble fraction. [5] A eficiency of (1) is that it overestimates the roplet concentration at large s an preicts values of roplet concentration that excees total aerosol concentration; this occurs because of the functional form of the power law an use of a single value of k. Many fiel an laboratory measurements have shown that a more realistic N CCN (s) spectrum in log-log coorinates is not linear as it woul be with k = const, but has a concave curvature, i.e., the inex k ecreases with increasing s [e.g., Jiusto an Lala, 1981; Huson, 1984; Yum an Huson, 2001]. This eficiency in (1) has been correcte in various ways: (1) by introucing the C-k space an constructing nomograms in this space [Braham, 1976]; (2) by constructing j s (s) that rapily ecreases at high s [Cohar et al., 1998, 2000] or integral empirical spectra N CCN (s) [Ji an Shaw, 1998] that yiel finite rop concentrations at large s; (3) by consiering the ecrease in the inices m, k with ecreasing aerosol size [KC99a]; (4) by using a lognormal CCN size spectrum instea of the power law, which yiels concave spectra [von er Eme an Wacker, 1993; Ghan et al., 1993, 1995, 1997; Feingol et al., 1994; Abul-Razzak et al., 1998; Abul- Razzak an Ghan, 2000; Nenes an Seinfel, 2003; Rissman et al., 2004; Fountoukis an Nenes, 2005]. On the basis of these stuies, the prognostic equations for the rop concentration are recently being incorporate into climate moels [Ghan et al., 1997; Lohmann et al., 1999]. [6] However, the power laws (1), (2) are still use in many clou moels an in analyses of fiel an chamber experiments. The relation between the power law an newer lognormal parameterization is unclear. In this paper, a moifie power law is erive an its equivalence to the lognormal parameterizations is establishe. In section 2, a moel of mixe (partially soluble) ry CCN is evelope with parameterization of the soluble fraction as a function of the ry nucleus raius an a power law Junge-type representation of the lognormal size spectra is erive. To allow a more accurate estimate of aerosol contribution into the clou optical properties an to the Twomey effect, a lognormal spectrum of the wet interstitial aerosol is foun using Köhler theory. In section 3, a new algebraic representation of the lognormal spectra is erive. The ifferential CCN activity spectra as a moification of (2) are erive in section 4 in both the lognormal an algebraic forms. In section 5, the cumulative CCN spectra as a moification of (1) are erive in both lognormal an algebraic forms. Finally, a moifie power law is erive as a moification of (1), expressing the parameters C an k as continuous algebraic functions of supersaturation an parameters of aerosol microstructure an physicochemical properties. The avantages of this new power law relationship inclue rop concentration boune by the total aerosol concentration at high supersaturations, quantitative explanation of the experimental ata on the k-inex, an the possibility to express k(s) an C(s) irectly via the aerosol microphysics. This formulation allows reconciliation of this moifie power law an the lognormal parameterizations, which is illustrate with several examples. Summary an conclusions are given in section Corresponence Between the Lognormal an Power Law Size Spectra 2.1. Lognormal Size Spectra of the Dry an Wet Aerosol [7] We consier a polyisperse ensemble of mixe aerosol particles consisting of soluble an insoluble fractions. The lognormal size spectrum of ry aerosol f (r ) by the ry raii r can be presente in the form: f ðr N a Þ ¼ pffiffiffiffiffi exp ln2 ðr =r 0 Þ 2pð ln s Þr 2ln 2 ; ð4þ s where N a is the aerosol number concentration, s is the ispersion of the ry spectrum, an r 0 is the mean geometric raius relate to the moal raius r m as r m ¼ r 0 exp ln 2 s : ð5þ As the relative humiity H excees the threshol of eliquescence H th of the soluble fraction, the hygroscopic growth of the particles begins an initially ry CCN convert into wet haze particles. To erive the size spectrum of the wet aerosol, we nee relations between the ry raius r an a corresponing raius of a wet particle r w. These relations 2of15

3 can be obtaine using the Köhler equation for supersaturation s =(r v r vs )/r vs that can be written for the ilute solution particles as [PK97]: s ¼ H 1 ¼ A k r B r 3 : ð6þ A k ¼ 2M wv sa RTr w ; B ¼ 3nF sm s M w 4pM s r w : ð7þ Here r v, r vs, an r w are the ensities of vapor, saturate vapor, an water, H is the ambient relative humiity, A k is the Kelvin curvature parameter, M w is the molecular weight of water, z sa is the surface tension at the solution-air interface, R is the universal gas constant, T is the temperature (in egrees Kelvin). The parameter B escribes effects of the soluble fraction, n is the number of ions in solution, F s is the osmotic potential, m s an M s are the mass an molecular weight of the soluble fraction. [8] The raius r w can be relate to r as in the work of KC99a using a convenient parameterization of the soluble fraction an the parameter B, following Levin an Seunov [1966], Seunov [1974]: B ¼ br 21þb ð where the parameters b an b epen on the chemical composition an physical properties of the soluble part of an aerosol particle. The parameter b escribes the soluble fraction of an aerosol particle. Generally, the solubility ecreases with increasing particle size [e.g., Laktionov, 1972; Seunov, 1974; PK97], so that b ecreases with increasing r an may vary from 0.5, when soluble fraction is proportional to the volume, to 1, when soluble fraction is inepenent of the raius. This variation escribes the change of the ominant mechanisms of accumulation of soluble fraction with growing particle size. A etaile analysis of these mechanisms is given, e.g., in the work of PK97 an Seinfel an Panis [1998] an is beyon the scope of this paper. We consier in more etail two particular cases. [9] 1. b = 0.5. The value b = 0.5 correspons to the case B = br 3, when soluble fraction is istribute within particle volume, is proportional to it, an is usually implicitly assume in most parameterizations of CCN eliquescence an rop activation [e.g., von er Eme an Wacker, 1993; Ghan et al., 1993, 1995; Abul-Razzak et al., 1998; Abul- Razzak an Ghan, 2000]. It was foun in the work of KC99a that with b = 0.5, the quantity b is a imensionless parameter: r b ¼ ðnf s Þe s M w v ; r w M s Þ ð8þ ð9aþ where r s is the ensity of the soluble fraction, e v is its volume fraction. [10] 2. b = 0. Then B = br 2, i.e., mass of soluble fraction is proportional to the surface area where it is accumulate as a film or shell. The particle volume fraction was parameterize in the work of KC99a as e v (r )=e v0 (r 1 /r ), where r 1 is some scaling parameter an e v0 is the reference soluble fraction (imensionless). We then obtain for b b ¼ r 1 e v0 ðnf s Þ r s M w : r w M s ð9bþ For this case, b has the imension of length an is proportional to the scaling raius r 1. If the thickness l 0 of the soluble shell is much smaller than the raius r of a spherical insoluble core, l 0 r,, then approximately m s 4pr s r 2 l 0, e v (r )=3l 0 /r, an in (9b) e v0 =1,r 1 =3l 0, an b ¼ 3l 0 ðnf s Þ r s M w : r w M s ð9cþ The parameterizations (9b) an (9c) might be use, in particular, for surface-active substances that are accumulate near the particle surface [e.g., Rissman et al., 2004] or for soluble shell coatings of particles with insoluble cores such as ust when heterogeneous chemical reactions on the particle surface may lea to formation of such coatings [e.g., Laktionov, 1972; Levin et al., 1996; Sassen et al., 2003; Bauer an Koch, 2005]. [11] The values of b for aerosols with volume-proportional soluble components (b = 0.5) consisting of NaCl an ammonium sulphate evaluate from (9a) are etermine as follows, using ata obtaine from PK97. If the soluble material is NaCl (M s = 58.5, r s = 2.16 g cm 3, n =2, F s 1 for ilute solutions), then b = 1.33 for e v = 1 (fully soluble nuclei), an b = 0.26 for e v = If the soluble material is ammonium sulphate (M s = 132, r s = 1.77 g cm 3, n =3,F s for molality of 0.1, so that nf s 2.3), then b = 0.55 for fully soluble nuclei (e v = 1), an b = 0.25 for e v = In the calculations below, we consier mostly the case b = 0.5 (volume-proportional) an use the value b = 0.25, which may correspon to the ry aerosol containing 50 % ammonium sulphate, as foun by Hänel [1976] in measurements of continental an marine (Atlantic) aerosols, or 20% NaCl, as was assume by Junge [1963]. All of the results below can be recalculate for b =0 with an appropriate moel of the soluble shell (some estimates are given below for the case b = 0), to any other e v, or for any other chemical composition using appropriate values of M s an r s. The soluble aerosol fraction often ecreases with increasing raius [PK97], an this feature can be escribe by choosing an appropriate superposition of the aerosol size spectra with the two soluble fractions: volume- proportional (b = 0.5 for smaller fraction) an surface-proportional (b = 0 for larger fraction). [12] The size spectrum of the wet aerosol f w (r w ) can be obtaine using the relations between the ry r an wet r w raii. Various solutions for the humiity epenence of r w (H) at subsaturation an supersaturation were foun in algebraic an trigonometric forms using the Kohler s equation (6) [e.g., Levin an Seunov, 1966; Seunov, 1974; Hänel, 1976; Fitzgeral, 1975; Smirnov, 1978; Fitzgeral et al., 1982; Khvorostyanov an Curry, 1999a; Cohar et al., 2000]. In this work, we consier only the wet interstitial aerosol, an humiity transformations at subsaturation will be consiere in a separate paper. 3of15

4 for a continuous power law representation of the fall velocities. Suppose f(r) is smooth an has a smooth erivative f 0 (r). Then they can be presente at a point r as the power law functions fðþ¼c r f r m ; f 0 ðþ¼ mf r ðþ=r r ð14þ Solving (14) for m an c f we obtain: mðþ¼ r rf 0 ðþ r fðþ r ; c f ðþ¼f r ðþr r m ð15þ Figure 1. Lognormal ry (soli curve) an interstitial (open circles) aerosol size spectra with the parameters: moal raius r m = 0.03 mm, s,w =2,N a = 200 cm 3, an the Junge-type power law approximations to the ry spectrum at four points: before moal raius, r 1 = mm, negative inex m = 1.27; moal raius r 2 = 0.03 mm, m =0;r 3 = 0.1 mm, m = 2.5; r 4 = 0.33 mm, m = 5.0). [13] The supersaturation in clous is small (s ), the left-han sie in (6) tens to zero an the raius of a wet particle can be foun from the quaratic equation: r w ðh ffi 1Þ ¼ B 1=2 ¼ b1=2 r 1þb A k A 1=2 k : ð10þ [14] This expression can be use at the stage of clou formation (CCN activation into the rops) or for interstitial unactivate clou aerosol. Now the size spectrum f w (r w ) can be foun from the conservation equation f w ðr w Þr w ¼ f ðr Þr : ð11þ Using (10) (11), we obtain that the ry istribution (4) transforms into the wet spectrum f w ðr w Þ ¼ N a pffiffiffiffiffi exp ln2 ðr w =r w0 Þ 2pð ln sw Þr w 2ln 2 : ð12þ s w The wet mean geometric raii r w0 an ispersions s w are relate to the corresponing r 0 an s of the ry aerosol as r w0 ¼ r ð1þbþ 0 ðb=a k Þ 1=2 ; s w ¼ s ð1þbþ ; H ffi 1; ð13þ [15] Thus the shape of the wet interstitial size spectrum (12) is the same lognormal as for the ry aerosol (4) but the values of r w0 an s w are ifferent. The size spectra of the interstitial aerosol at s > 0 are limite by the bounary raius r b =2A k /3s [Levin an Seunov, 1966; Seunov, 1974] Approximation of the Lognormal Size Spectra by the Junge Power Law [16] A corresponence between the lognormal an power law size spectra can be establishe using the same metho as in the work of Khvorostyanov an Curry [2002, 2005] The power inex m an c f are here the functions of raius. If aerosol size spectrum f(r) is escribe by the ry f (r )or wet f w (r w ) lognormal istributions (4), (12), substitution of f,w an f 0,w into (15) yiels the power inex for the ry an wet aerosol spectra m ;w r ;w ln r ;w =r ;w0 ¼ 1 þ ln 2 ; ð16þ s ;w where the inices, w enote ry or wet aerosol, an r w0, s w0 are efine by (14). Thus the lognormal ry an wet spectra (4), (12) can be presente at every point r, r w, in the power law form (14) with the inex m,w (16) Examples of the Dry an Wet Aerosol Size Spectra an Power Inices [17] Figure 1 shows in log-log coorinates a lognormal ry aerosol size spectrum f (r ) with moal raius r m = 0.03 mm an ispersion s = 2, close to Whitby [1978] for the accumulation moe, an concentration N a = 200 cm 3. Its Junge-type power law approximations shows the raius epenence of the power inex m: it changes from m = 1.27 at r 1 = mm <r m,tom = 0 at the moal raius r 2 = r m, an m is positive at all r > r m ; in particular, m = 2.5 at r 3 = 0.1 mm an m = 5.0 at r 4 = 0.33 mm. Recall that the imensional analysis of the coagulation equation an its asymptotic solutions yiel the inverse power laws with the inex m = 2.5 for r 0.1 mm ue to the ominant effects of Brownian coagulation, an m = 4.5 for r >1mm ue to prevailing seimentation [PK97, chap. 11]. The lognormal spectrum with the escribe parameters is in general agreement with these power law solutions. Thus the lognormal size spectrum can be consiere as the superposition of the power laws with the inex from (16). [18] The transition of the ry lognormal spectrum to the limit H = 1 (curve with open circles) is fairly smooth, although there is a istinct ecrease of the slopes. This resembles a corresponing effect for the power law spectra, when the transition to H = 1 is accompanie by a ecrease of the Junge power inex, e.g., m =4atH < 1 converts into m =3atH! 1 over a very narrow humiity range [e.g., Seunov, 1974; Fitzgeral, 1975; Smirnov, 1978; KC99a]. [19] The epenence of the effective power inices calculate for lognormal istributions with various moal raii an ispersions is shown for the ry aerosol in Figure 2, which illustrates the following general features of (16). The inices (the slopes of the spectra) increase with raii for all r. The inices m increase with ecreasing ispersions an with increasing r 0 of the ry lognormal spectrum. The 4of15

5 Thus we can use the following equality: a p ffiffiffi exp a 2 x 2 a 1 p ffiffiffi p p ch 2 p ð2ax= ffiffiffi pþ ð20þ Figure 2. Effective power inices calculate from equation (16) for lognormal istributions with moal raii an ispersions inicate in the legen; the constant value m = 4 is given for comparison. calculate m intersect the average Junge s curve m = 4 in the range of raii 0.06 mm to 0.7 mm, i.e., mostly in the accumulation moe. Since the values chosen here for r 0 an s are realistic (close to Whitby s multimoal spectra), this may explain how a superposition of several lognormal spectra may lea to an effective average m = 4, observe by Junge. The growth of the inices above 4 for larger r is in agreement with the coagulation theory that preicts steeper slopes, m = 19/4, of the power laws for greater raii r 5 mm [PK97]. The escribe relation between the lognormal an power law size spectra might be useful for analysis an parameterizations of the measure spectra an of solutions to the coagulation equation. A comparison shows that both functions are alreay sufficiently close at values a 1, which is illustrate pffiffiffi in Figure 3a, where these functions ivie by a/ p are plotte, the ifference generally oes not excee 3 5%. The approximate equality of these smoothe elta functions is use further for an algebraic representation of the aerosol size an CCN activity spectra. [22] Integration of (20) from 0 to x yiels another useful relation: p erf ðaxþ tanh 2ax= ffiffiffi p ; ð21þ where erf(z) is the Gaussian function of errors: erf ðþ¼ z p 2 ffiffiffi p Z z 0 e x2 x: ð21aþ Equation (21) with a = 1 was foun by Ghan et al. [1993], who showe its accuracy, which is also illustrate in Figure 3b for two values of a. It is known that integration of the Dirac s elta-function yiels Heavisie s step-function. Comparison of Figures 3a an 3b shows that the same is vali for the corresponent smoothe functions, therefore 3. Algebraic Approximation of the Lognormal Distribution [20] A isavantage of the lognormal size spectra is the ifficulty of evaluating analytical asymptotics an moments of this istribution. A convenient algebraic equivalent of the lognormal istribution can be obtaine using two representations of the smoothe (bell-shape) Dirac s elta-function (x). The first one was taken as [Korn an Korn, 1968] 1 ðx; aþ ¼p a ffiffiffi expð a 2 x 2 Þ; ð17þ p where a etermines the with of the function. Another representation as an algebraic function of exp(x/a) is given by Levich [1969]. We transforme it to the following form: 2 ðx; a Þ ¼ a 1 pffiffiffi p ch 2 p ð2ax= ffiffiffi pþ ; ð18þ [21] It can be shown that both functions 1 (x, a) an 2 (x, a) are normalize to unity, ten to the Dirac s elta-function when a!1an exactly coincie in this limit: lim 1ðx; aþ ¼ lim 2ðx; aþ ¼ ðþ: x a!1 a!1 ð19þ Figure 3. (a) Comparison of the smoothe Dirac eltafunctions: exp( a 2 x 2 ), an itsapproximation 1/ch 2 p ffiffiffi (2ax/ p ), for two values a = 1 an 3. The ifference generally oes not excee 3 5%. (b) Comparison of the smoothep ffiffiffiheavisie step-functions, erf(ax), an tanh(2ax/ p ). 5of15

6 where r 0 an s are the geometric raius an ispersion of the ry spectrum, r w0 an s w are corresponing quantities for the wet spectrum efine in (13), an the inices k 0 an k 0w are 4 4 k 0 ¼ pffiffiffiffiffi ; k w0 ¼ pffiffiffiffiffi : ð26þ 2p ln s 2p ln sw Figure 4. Comparison of the algebraic an lognormal aerosol size spectra for the two pairs of parameters: r m = 0.1 mm, s = 2.15 (soli an open circles for the algebraic an lognormal, respectively), an r m = mm, s = 1.5 (soli an open triangles). the erivation above shows that the relation (21) is an approximate equality of the two representations of the smoothe Heavisie step-functions. Equation (21) is use in section 5 for the proof of equivalence of the lognormal an algebraic representations of the rop concentration an k- inex. [23] pif ffiffiffi we introuce in (20) the new variables x =ln(r/r 0 ), a =( 2 ln s) 1 p ffiffiffi, then 2ax/ p = ln(r/r0 ) k0/2, where k 0 is a parameter introuce by Ghan et al. [1993]: 4 k 0 ¼ pffiffiffiffiffi : ð22þ 2p ln s Substituting these variables into (20), we obtain 1 pffiffiffiffiffi exp ln2 ðr=r 0 Þ 2p ln s 2ln 2 k 0ðr=r 0 Þ k0 h i s 2 : ð23þ ðr=r 0 Þ k0 þ1 [24] The expression on the left-han sie is the lognormal istribution N/lnr with mean geometric raius r 0 an ispersion s, an the right-han sie is its algebraic equivalent, which is another representation of the smoothe elta function. When the ispersion tens to its lower limit s! 1, then a!1, an accoring to (19), both istributions ten to the elta function, i.e., become infinitely narrow (monoisperse) at s = 1. For s > 1, they represent istributions of the finite with. Using the equality (23), the ry (4) an wet (12) lognormal aerosol size spectra can be rewritten in algebraic form as f ðr Þ ¼ k 0N a r 0 f w ðr w Þ ¼ k w0n a r w0 h ðr =r 0 i 2 ; ð24þ 1 þ ðr =r 0 Þ k0 Þ k0 1 h ðr w =r w0 i 2 ; ð25þ 1 þ ðr w =r w0 Þ kw0 Þ kw0 1 A comparison of the algebraic (24) an lognormal (4) aerosol size spectra calculate for the two pairs of spectra with the same parameters is shown in Figure 4. One can see that algebraic an lognormal spectra are in a goo agreement in the region from the maxima an own by two to three orers of magnitue. A iscrepancy is seen in the region of the tails, but this region provies a small contribution into the number ensity. [25] These algebraic representations of the size spectra can be useful in applications. Evaluation of their analytical an asymptotic properties is simpler than those of the lognormal istributions, an various moments over the size spectra can be expresse via elementary functions rather than via erf as in the case with the lognormal istributions. Being a goo approximation to the lognormal istribution but simpler in use, the algebraic functions (24), (25) can be use for approximation of the aerosol, roplet, an crystal size spectra as a supplement or an alternative to the traitionally use power law, lognormal, an gamma istributions. Some applications are illustrate in the next sections. 4. Differential CCN Supersaturation Activity Spectrum 4.1. Critical Supersaturations an Raii [26] To erive the supersaturation activity spectrum, we nee relations among the ry raius r, corresponing critical water supersaturation s cr, an the critical nucleus raius r cr. The values r cr an s cr for CCN activation can be foun as usually using Köhler s equation (6) from the conition of maximum s/r = 0: 1=2 s cr ¼ 4A3 k ¼ 2 A k ¼ r 27B 3 r cr r cr ¼ 3B 1=2 ¼ r 1þb 3b 1=2 ; ð27þ A k A k ð 1þb Þ 4A 3 1=2 k ¼ c 1=k1 27b 1 r 1=k1 ; ð28þ where we use again the parameterization (8) for B. These equations can be reverte to express r via r cr or s cr : where r ðr cr Þ ¼ r k1 cr A k=3b ð Þ k1=2 ; r ðs cr k 1 ¼ 1 k1=2 1 þ b ; c 1 ¼ 4A3 K 27b Þ ¼ c 1 s k1 cr ð29þ ð30þ [27] Figure 5 epicts the critical supersaturations, s cr, an critical raii, r cr, calculate with (27), (28) as functions of r 6of15

7 remains unactivate. For the spectrum with r m = 0.03 mm, activation of the right half own to r m requires a supersaturation of 0.6%. Such a supersaturation is typical of some natural clous. Activation of the left branches with r < mm requires higher supersaturations an may be typical of the CCN measurements in clou chambers where s 5 20% can be reache [e.g., Jiusto an Lala, 1981; Huson, 1984; Yum an Huson, 2001] Lognormal Differential CCN Activity Spectrum [29] The CCN ifferential supersaturation activity spectrum j s (s) can be obtaine now using the conservation law in ifferential form for the concentration: f ðr Þr ¼ j s ðþs: s ð31þ Here s is the critical supersaturation require to activate a ry particle with raius r ; the minus sign occurs since increase in r (r > 0) correspons to a ecrease in s (s <0) as inicate by (28). Using (29) an (30), (31) can be rewritten in the form: where j s ðþ¼ f s ðr Þ r ðþ s s ¼ c 2 s k2 f ½r ðþ s Š; ð32þ c 2 ¼ 1 4A 3 1=2ð1þb Þ K ¼ k 1 c 1 ; k 2 ¼ k 1 þ 1 ¼ 2 þ b 1 þ b 27b 1 þ b : ð32aþ Figure 5. (a) Critical supersaturation, s cr, an (b) critical raius, r cr, calculate as functions of the ry raius r with the four values of b corresponing to various soluble fractions of ammonium sulfate inicate in the legen. with four values of b (0.55, 0.275, 0.135, 0.07), corresponing to various soluble fractions of ammonium sulfate (100, 50, 25, an 12.5%) inicate in the legen. The values of s cr require for activation of a given CCN increase with ecreasing r from % at r = 0.1 mm(a typical clou case) to 3 10% at r = 0.01 mm (as can be foun in a clou chamber) an reach % at r = mm. The values of r cr increase with r an the ratio r cr / r reaches 5 13 at r = 0.1 mm an at r =1mm. The 8-fol ecrease in the soluble fraction causes s cr to increase an r cr to ecrease by a factor of 3 5. [28] These features are important for unerstaning an quantitative escription of the separation between activate an unactivate CCN fractions. Figure 6 shows four lognormal size spectra of the ry aerosol with various moal raii an ispersions along with the corresponing critical raii an supersaturations calculate with b = 0.5, b = 0.55 (50% soluble fraction of ammonium sulfate). One can see that if the maximum supersaturation oes not excee 0.1 %, the right branch of the ry spectrum with r m = 0.1 mm can be activate own to the moe, an the left branch (r < r m ) Figure 6. Lognormal aerosol size spectra for r m = 0.1 mm, s = 2 (soli circles), r m = 0.1 mm, s = 1.5 (crosses), r m = 0.03 mm, s = 2 (rhombs), an r m = 0.03 mm, s = 1.5 (triangles), plotte as the functions of ry raii an of corresponing critical raii an critical supersaturations. Critical supersaturations s cr corresponing to the maxima at 0.1 mm an 0.03 mm are shown near the curves. 7of15

8 50 % smaller with b = 0, the argument becomes ( 1/2)[ln(s/ s 0 )/lns ] 2 an j s (s) becomes much narrower. This means that the maximum in the activation spectrum is reache at another s m an rop activation occurs over narrower s-range with b = Algebraic Differential CCN Spectrum [31] An algebraic form of the ifferential activity spectrum can be obtaine substituting r (s) from (29) an r 0 (s 0 ) from (34) into the conservation equation (32) with f (r ) from (24): Figure 7. CCN ifferential activity spectrum j s (s) calculate with b = 0.25 an four combinations of moal raii r m an ispersions s of the ry aerosol spectra inicate in the legen. j s ðþ¼ s k s0n a ðs=s 0 h i s 2 ; ð37þ 0 1 þ ðs=s 0 Þ ks0 Þ ks0 1 or, in a slightly ifferent form that is more convenient for comparison with the other moels Substituting the ry lognormal size spectrum (4) into (32), we obtain: N a j s ðþ¼ s pffiffiffiffiffi 2pð ln ss Þs exp ln2 ðs=s 0 Þ 2ln 2 ; ð33þ s s where we introuce the mean geometric supersaturation s 0 an the supersaturation ispersion s s : s 0 ¼ ðr 0 =c 1 Þ 1=k1 ¼ r ð 1þb Þ 0 4A 3 1=2 k ; ð34þ 27b Here j s ðþ¼k s s0 C 0 s ks0 1 2; 1 þ h 0 s ks0 ð38þ 4 4 k s0 ¼ pffiffiffiffiffi ¼ pffiffiffiffiffi ; ð39aþ 2p ln ss 2p ð1 þ bþln s 27b ks0=2 C 0 ¼ N a s ks0 0 ¼ N a 4A 2 ðr 0 Þ k h 0 ¼ s ks0 0 ¼ 27b ks0=2 4A 2 ðr 0 Þ k ð Þ ; ð39bþ ks0 1þb ð Þ : ð39cþ ks0 1þb s s ¼ s 1=k1 ¼ s ð1þbþ : ð35þ The moal supersaturation s m is relate to s 0 similar to that for the ry size spectrum (5) s m ¼ s 0 exp ln 2 s s : ð36þ The supersaturation activity spectrum (33) has a maximum at s m an the region s = s m ± s s gives the maximum contribution into the roplet concentration. [30] Equations (4) an (33) (36) show the following. The lognormal size spectrum of the ry aerosol with the parameters escribe above is equivalent to the lognormal CCN activity spectrum by supersaturations j s (s); that is, the shape is preserve uner transformations between the raius an supersaturation variables. This feature is similar to that foun in the work of Abul-Razzak et al. [1998] an Fountoukis an Nenes [2005]. The ifference of our moel with these works is in that they assume soluble fraction proportional to the volume (b = 0.5), while our moel for j s (s) generalizes this expression an allows variable soluble fraction. For b = 0.5 (homogeneously mixe in volume), (35) for s s shows that the argument in the exponent (33) can be rewritten as ( 1/2)[(2/3)ln(s/s 0 )/lns ] 2, which coincies with the cite works. For soluble fraction proportional to the surface, b = 0, as can be for the ust particles covere by the soluble film, the mean geometric supersaturation s 0 r 1 0, an not r 3/2 0, as in the case b = 0.5; the logarithm lns s is (It is shown below that k s0 is the asymptotic value of the k- inex in the CCN activity power law at small s). Expression (38) can be also obtaine from the lognormal law (33) using the similarity relation (23), which again illustrates the equivalence of the lognormal an algebraic istributions. [32] The first term, k s0 C 0 s k s0 1, on the right-han sie of (38) represents Twomey s [1959] power law (2). In contrast to the C 0 an k s0 base on fits to experimental ata, here the analytical epenence is erive from the Köhler s theory an algebraic approximation of the lognormal functions, an parameters C 0, k s0 are expresse irectly via aerosol parameters. This first term escribes rop activation at small supersaturations s s 0 (as in the clous with weak uprafts an in the haze chambers ) an grows with s as s k s0 1. The secon term in parenthesis ecreases at large s s 0 as s 2k s0, overwhelms the growth of the first term, an ensures the asymptotic ecrease j s (s) s k s0 1. Thus the term in parenthesis serves effectively as a correction to the Twomey law an prevents an unlimite growth of concentration of activate rops at large supersaturations. [33] The entire CCN spectrum (38) resembles the corresponing equation from Cohar et al. [1998, 2000]: j s ðþ¼kcs s k 1 1 þ hs 2 l; ð40þ where the first term is also the Twomey s ifferential power law, an the term in parenthesis was ae by the authors as 8of15

9 j s (s) (33) over the supersaturations to the maximum value s reache in a clouy parcel: Z s N ðþ¼n s CCN ðþ¼ s 0 j s ðs 0 Þs 0 ¼ N a þ erf p ln s=s ð 0Þ ffiffi ; 2 ln s s ð41þ Figure 8. Comparison of the algebraic (alg.) an lognormal (log.) ifferential CCN activity spectra j s (s). A pair of igits in the parentheses in the legen enotes moal raius r m in mm an ispersion s of the corresponing ry aerosol spectrum. an empirical correction with the two parameters h, l that were varie an chosen by fitting the experimental ata. Thus the algebraic representation of j s (s) (38) foun here establishes a brige between the Twomey s power law with its corrections as in the work of Cohar et al. [1998, 2000] an the lognormal parameterizations of rop activation as in the work of Ghan et al. [1993, 1995], Abul-Razzak et al. [1998], Abul-Razzak an Ghan [2000], an Fountoukis an Nenes [2005]. [34] Shown in Figure 7 is an example of the CCN ifferential activity spectrum j s (s) (38) calculate with b = 0.25 (CCN with 50 % of ammonium sulfate or 20 % of NaCl) an four combinations of moal raii an ispersions. The area beneath each curve represents the rop concentration N r that coul be activate at any given supersaturation s, so that the maximum of j s (s) inicates the region of s where its increase leas to the most effective activation. A remarkable feature of this figure is a substantial sensitivity to the variations of r m, s. The maximum for the maritime-type spectrum with r m = 0.1 mm, s = 2 lies at very small s 0.01 %. When s ecreases to 1.5 or r m ecreases to 0.03 (closer to the continental spectrum), the maximum of j s (s) shifts to s %. With r m = 0.03 mm, s = 1.5, the region s % becomes relatively inactive an the maximum shifts to 0.4%. Thus even moerate narrowing of the ry spectra or ecrease in the moal raius may require much greater vertical velocities for activation of the rops with the same concentrations. [35] A comparison of the algebraic (37) an lognormal (33) ifferential activity spectra in Figure 8 shows their general goo agreement. The iscrepancy increases in the both tails, but their contributions into the number ensity is small as will be illustrate below. Thus the algebraic functions (37), (38) approximate the lognormal spectrum (33) with goo accuracy. 5. Droplet Concentration an Moifie Power Law for Drops Activation 5.1. Droplet Concentration Base on the Lognormal an Algebraic CCN Spectra [36] The concentration of CCN or activate rops can be obtaine by integration of the ifferential activity spectrum where s 0 an s s are efine by (34), (35). Equation (41) is similar to the corresponing expressions erive by von er Eme an Wacker [1993], Ghan et al. [1993, 1995], Abul- Razzak et al. [1998], Abul-Razzak an Ghan [2000], Fountoukis an Nenes [2005] that also arrive at erf function. A generalization of these works in (41) is the same as was iscusse for the ifferential activation spectrum in previous section, in variable soluble fraction b, which is iscusse below. [37] Droplet or CCN concentration in algebraic form can be obtaine now in two equivalent ways. The first way is integration of the algebraic CCN spectrum j s (s) (37) or (38) over s: ðs=s 0 Þ ks0 1; N CCN ðþ¼n s a h i ¼ C 0 s ks0 1 þ h 0 s ks0 ð42þ 1 þ ðs=s 0 Þ ks0 where C 0 an h 0 are efine in (39b) an (39c). The secon way is as in the work of Ghan et al. [1993] by substituting (21) for the equality of erf an tanh into (41), which also yiels (42). The erivation of (41) an (42) along with equivalence (20), (21) emphasizes again the fact that ifferential CCN istributions by critical raii or supersaturations are smoothe Dirac elta functions, while their integrals, roplet concentrations, are smoothe Heavisie step functions. [38] Accoring to (42), the asymptotic of N CCN (s) ats s 0 is the Twomey power law N CCN (s) =C 0 s k s0. The term in parenthesis is a correction, its asymptote at s s 0 is s k s0,it compensates the growth of the first term an prevents unlimite rop concentration. The asymptotic limit of rop concentration is N CCN (s)! N a at large s, an hence the number of activate rops is limite by the total aerosol concentration. [39] Equation (42) is similar to the corresponing equation from Ghan et al. [1993], but is expresse via supersaturation ispersion instea of the moal raius as in that work an is generalize here for the variable soluble fraction b, which causes the following ifference. As follows from efinitions of k s0 (39a), the power inex k s0 iffers by the factor pffiffiffiffiffi (1 + b) from the corresponing expression k gh = 4/( 2p ln s ) in the work of Ghan et al. [1993], who assume soluble fraction proportional to the volume, i.e., in our terms b = 0.5. With this value, k s0 = (2/3)k gh, an the asymptotic of N CCN for small s is N CCN s k s0 s (2/3)k gh, which coincies with the expression in the work of Ghan et al. [1993]. However, for b =0 (surface-proportional soluble fraction), the inex k s0 is 1.5 times higher than with b = 0.5, an, as mentione before, the functions N CCN (s) increase (roplets activate) over much narrower interval of s. 9of15

10 [40] If the ry aerosol size spectrum is multimoal with I moes (e.g., I = 3 in a three-moal istributions [Whitby, 1978]), then it can be written as a superposition of I lognormal fractions (4) with N ai, s i, an r 0i being the number concentration, ispersion, an mean geometric raius of the ith fraction. The mean geometric supersaturations s 0i, ispersions s si, an the other parameters for each fraction are then efine by the same equations as escribe above for a single fraction. Then the algebraic ry an wet size spectra, the ifferential an integral activity spectra are obtaine for each fraction as escribe above an the corresponing multimoal spectra are superpositions of the I fractions. The multimoal CCN activity spectrum in algebraic form is erive from (42): N ðþ¼ s XI i¼1 N ai ðs=s 0i Þ ks0i h 1 þ ðs=s 0i Þ ks0i i ¼ XI i¼1 C 0i s ks0i 1 þ h 0i s ks0i 1: ð43þ Equation (43) for b = 0.5 is similar to the corresponing equation from Ghan et al. [1995] where N (s) is expresse in terms of critical raii, but expresses it here irectly via supersaturation an generalizes for the other b. The parameters b i can be ifferent for each fraction, e.g., b i = 0.5 may correspon to the accumulation moe, an b i =0 may correspon to the coarse moe consisting of the ust particles coate with the soluble film Revive Power Law for the Drop Concentration, Power Inex k(s), an Coefficient C(s) [41] The power law (1), N CCN = Cs k, is often use for parameterization of the results of CCN measurements an of rop activation in many clou moels. An analytical function for the k- inex can be foun now (similar to the m-inex (16)) using the metho from Khvorostyanov an Curry [2002, 2005] for the power law representation of the continuous functions. Using (1) an its erivative N 0 CCN(s) (2) an solving these two equations for k(s), we obtain: ks ðþ¼s N CCN 0 ðþ s N CCN ðþ s ¼ s j s ðþ s N CCN ðþ s ð44þ Equation (44) allows evaluation of k(s) with known ifferential j s (s) an cumulative N CCN (s) CCN spectra. Substituting here (33) for j s (s) an (41) for N CCN (s), we obtain k(s) in terms of lognormal functions: ffiffiffi ks ðþ¼ r 2 1 exp ln2 ðs=s 0 Þ p ln s s 2ln 2 s s lnðs=s þ erf pffiffi : 2 ln ss ð45þ Here s 0 an s s are efine by (34), (35). The algebraic form of the inex k(s) is obtaine by substitution of (37) for j s (s) an (42) for N CCN (s) into (44): h i " # 1¼ ks ðþ¼k s0 1 þ ðs=s 0 Þ ks0 ks0 1 þ s ks0 ks0 1þb r ð Þ 27b ks0=2 1 : 0 4A 3 k ð46þ If the aerosol size spectrum is multimoal an is given by a superposition of I algebraic istributions, then the k-inex is erive from (44), (43) by summation over I moes: ks ðþ¼ XI i¼1 8 < : k s0i N ai ðs=s 0i Þ ks0i h i 2 1 þ ðs=s 0i Þ ks0i X I N ai i¼1 ðs=s 0i Þ ks0i h 1 þ ðs=s 0i Þ ks0i 9 = i ; 1 ; ð47þ where the parameters k s0i, s 0i, N ai are efine as before but for each ith moe. [42] Equations (45) (47) allow evaluation of k(s) for any s with given s s an s 0 that are expresse via r, s, b, an b. That is, knowlege of the size spectra an composition of the ry or wet CCN allows irect evaluation of k(s) over the entire s-range. This solves the problem outline in Jiusto an Lala [1981], Yum an Huson [2001] an many other stuies: various values of the k-inex at various s. Equations (45) (47) provie a continuous representation of k over the entire s-range. Note that the k-inex is not constant for a given air mass or aerosol type but epens on a given s, at which it is measure. Equation (46) shows that for s s 0, the asymptotic value k(s) =k s0. For s s 0, the inex ecreases with s as k(s) =k s0 (s/s 0 ) k s0 an tens to zero. This behavior is in agreement with the experimental ata from Jiusto an Lala [1981], Yum an Huson [2001], an others. [43] The coefficient C(s) can be now calculate using (42) for N CCN (s) an (46) for k(s) h i 1; Cs ðþ¼n CCN ðþs s ks ðþ ¼ C 0 s cðþ s 1 þ ðs=s 0 Þ ks0 ð48þ where C 0 is efine in (39b). The power inex c(s) ofs in (48) is ðs=s 0 Þ ks0 cðþ¼k s s0 ks ðþ¼k s0 : ð49þ ks0 1 þ ðs=s 0 Þ Now we can write the CCN (or roplet) concentration in the form of a moifie power law N CCN ðþ¼cs s ðþs ks ðþ ; s in share of unit: ð50þ This equation is usually use with supersaturation s in percent, relate to s su in share of unit as s = 10 2 s su. Substitution of this relation into (50) yiels a more conventional form N CCN ðþ¼c s pc ðþs s ks ðþ ; s in percent: ð51þ In (51), both k(s) an C(s) epen on the ratio s/s 0 an thus o not epen on units; k(s) is calculate again from (46) but with s an s 0 in percent, an the coefficient C pc s pc ¼ 10 2k ðþ s Cs ðþ: ð52þ 10 of 15

11 Figure 9. Supersaturation epenence of the k-inex calculate for the lognormal size spectrum of ry CCN with various moal raii r m an ispersions s inicate in the legen an comparison: (a) with the experimental ata from Jiusto an Lala [1981] from four chambers (haze chamber; continuous flow iffusion chamber, CFD; an static iffusion chamber, SDC), an (b) with the ata collecte in FIRE-ACE in 1998 [Yum an Huson, 2001, hereinafter referre to as YH01] at the heights mb an mb. Equation (51) is similar to (1), however, now with the parameters epening on s as just escribe. Hence k(s) an C(s) are now expresse irectly via the parameters of the lognormal spectrum (N a, r 0, s ) an its physicochemical properties via simple parameterization of b in (9a) (9c). [44] Finally, the power law for the multimoal aerosol types is escribe again by (50), (51) with the k(s)-inex evaluate from (47) an the coefficient C(s) Cs ðþ¼ XI h i C 0i s c i ðþ s 1; 1 þ ðs=s 0i Þ ks0i ð53þ i¼1 where c i = k s0i k(s), an k s0i, C 0i, s 0i, N ai are efine for each ith moe. The coefficient C pc (s) with supersaturation in percent is efine by (52). The number of moes I is etermine by the nature of aerosol an is the same as use for multimoal size spectra with the lognormal approach in the work of Ghan et al. [1995], Abul-Razzak an Ghan [2000], an Fountoukis an Nenes [2005]. [45] This analytical s-epenence of C an k in (50), (51) provies a theoretical basis for many previous finings an improvements of the rop activation power law. In particular, the C-k space constructe by Braham [1976] base on experimental ata follows now from the equations for k(s) an C(s) with use of any relation of maximum s to vertical velocity. The ecrease of the k-slopes with increasing s analyze by Jiusto an Lala [1981], Yum an Huson [2001], Cohar et al. [1998, 2000], an many others is preicte by (45) (47), as well as its epenence on the nature of aerosol. The proportionality of coefficient C to the aerosol concentration explains substantially greater values of C observe in continental than in maritime air masses as compile by Twomey an Wojciechowski [1969] an Hegg an Hobbs [1992] an use by clou moelers Supersaturation Depenence of k(s), C(s), an N (s) [46] The equivalent power inex k(s) calculate with (46) an a monomoal lognormal size spectrum is shown in Figures 9a an 9b an is compare with the experimental ata from Jiusto an Lala [1981], who collecte the ata from several chambers an to the fiel ata collecte by Yum an Huson [2001] with a continuous flow iffusion chamber (CFD) at two altitues in the Arctic in May 1998 uring the First International Regional Experiment-Arctic Clou Experiment (FIRE-ACE). It is seen that the k-inex is not constant as assume in many clou moels. The calculate k-inices reach the largest values of at s 0.01 %, where it tens to the asymptotic k s0 (39a) an increases with ecreasing ispersion of the ry CCN spectrum since k s0 (lns ) 1.Ass increases, k(s) is initially almost constant, then ecreases slowly up to s % an then ecreases more rapily in the region s % to values less than at s > 1%. The k-inices increase with ecreasing moal raii an ispersions. Agreement of calculate k(s) with haze chamber ata is reache at s = 1.3, r = mm, an with CFD at s = 1.5, r = mm. Agreement with the Arctic fiel ata by Yum an Huson [2001] is reache with r m = mm, s = 2.15 in the layer mb an r m = mm, s = 1.9 in the layer mb. [47] Thus this moel explains an escribes quantitatively the observe ecrease in the k-inex with increasing supersaturation reporte previously by many investigators (e.g., Twomey an Wojciechowski, [1969], Braham [1976], Jiusto an Lala [1981]; see review in PK97) an relates it to the aerosol size spectra. A more etaile analysis of experimental ata with these equations shoul be base on simultaneous use of the measure multimoal aerosol size spectra an CCN supersaturation activity spectra. [48] In contrast to most moels where both k an C are constant an to some other moels where the k-inex epens on s, but the coefficient C is constant [e.g., Cohar et al., 1998, 2000], the values of effective C(s), C pc (s) in this moel also epen on supersaturation. The spectral behavior of C pc (s) calculate from (52) with ifferent r m, s an the same N a = 100 cm 3 is shown in Figure 10. For s s 0, the coefficients ten to the asymptotic limit C pc (0) = 10 2k s0 C 0 an reach values of cm 3 cause by 2k the epenence C 0 A s0 k k r s0 (1+b) 0 with A k 10 7 cm an r to 10 5 cm. This epenence causes increase in C pc with ecreasing s (increasing k s0 ) an with increasing r 0 seen at small s in Figure 10; i.e., the narrower spectra an the larger moal raius, the faster rop activation at 11 of 15

12 Figure 10. Equivalent coefficient C pc (s) in the power law for N CCN calculate for the lognormal size spectrum of ry CCN with moal raii r m an ispersions s inicate in the legen an other parameters escribe in the text. For the convenience of comparison, all curves are calculate with N a =10 2 cm 3. small s. The values of C pc (s) ecrease with increasing s by 1 3 orers of magnitue an ten to N a at s pc > %, since the asymptotics at large s is C pc (s pc )=10 2k(s) N a s k(s) pc, an k(s)! 0 at large s, hence all C pc! N a at s pc 0.5% as seen in Figure 10. Thus N CCN (s) =C pc s k(s) pc 10 2k(s) N a an tens to the limiting value N a, which ensures finite values of N a in contrast to the moels with constant k, C. [49] A comparison of the concentrations N CCN (s) calculate using the power law (51) an erf function (41) shows in Figure 11 their goo agreement an illustrates goo accuracy of the algebraic istributions an renewe power law as compare to the irect integration of the lognormal istributions. The curves N CCN (s) calculate with the newer power law are not linear in log-log coorinates as it is with (1) an constant C 0, k, but are concave in agreement with the measurements [e.g., Jiusto an Lala, 1981; Huson, 1984; Yum an Huson, 2001] an other moels [e.g., Ghan et al., 1993, 1995; Feingol et al., 1994; Cohar et al., 1998, 2000]. The reason for this concavity is quite clear from Figures 6 8 for the size an activity spectra: the rate of increase in concentration of activate rops N (s)/s is greatest at small supersaturations in the left branch of j s (s) (Figures 7 an 8), or in the right branch of f (r )atr > r m (Figure 6), then N (s)/s ecreases with growing s, especially at s > 1%, i.e., in the left branch of the small ry CCN with r < 0.03 mm (Figure 6). Effectively, some saturation with respect to supersaturation occurs, causing k(s), C(s) an N (s)/s to ecrease with s an leaing to finite N (s) at large s. [50] Finally, Figure 12 shows N CCN (s) calculate with the lognormal size spectra an compare to the corresponing fiel ata obtaine by Yum an Huson [2001] in the FIRE- ACE campaign in May 1998 in the Arctic. The parameters r m, s, an N a were varie slightly in the calculations, although we constraine the calculations to the measure k- inices (Figure 10b), so that the values of the parameters ha to provie an agreement for both k(s) an N CCN (s). Goo agreement over the entire range of supersaturations was foun with the values inicate in the legen. The values of moal raii an ispersions are typical for the accumulation moe. This comparison shows that this moel of the power law can reasonably reprouce the experimental ata on CCN activity spectra an concentrations of activate rops an can be use for parameterization of rop activation in clou moels. 6. Summary an Conclusions [51] A moel of the lognormal size spectrum of mixe (partially soluble) ry CCN with parameterization of the soluble fraction as a function of the ry nucleus raius is use to consier the processes of humiity transformation of the ry size spectrum into the spectrum of the wet interstitial aerosol an rop activation. Using the expressions for the raius of a solution particle as a function of relative humiity, it was shown that a lognormal size spectrum of Figure 11. Comparison of the accumulate CCN spectra N CCN (s) calculate using power law equation (51) (soli symbols) an using (41) with erf function (open symbols). The values of the corresponing moal raii r m an ispersions s of the ry aerosol spectra are inicate in the legen. For convenience of comparison, N CCN is calculate with the ry aerosol concentrations of 250, 200, 150, an 100 cm 3 from the upper to lower curves. Figure 12. Supersaturation epenence of the CCN integral activity spectrum N CCN (s) calculate with the new power law (51) for the lognormal size spectra with r m, s an N a (in cm 3 ) inicate in the legen (soli rhombs an circles), compare to the corresponing fiel ata obtaine by Yum an Huson [2001] in FIRE-ACE campaign in May 1998 in the Arctic (open rhombs an circles). 12 of 15