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

Authors


Abstract

[1] Empirical power law expressions of the form NCCN = Csk have been used in cloud physics for over 4 decades to relate the number of cloud condensation nuclei (CCN) and droplets formed on them to cloud supersaturation, s. The deficiencies of this parameterization are that the parameters C and k are usually constants taken from the empirical data and not directly related to the CCN microphysical properties and this parameterization predicts unbounded droplet concentrations at high s. The activation power law was derived in several works from the power law Junge-type aerosol size spectra and parameters C and k were related to the indices of the power law, but this still does 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 based upon the lognormal aerosol size spectra that yield finite NCCN at large s, but does not explain the activation power law, which is still traditionally used in the interpretation of CCN observations, in many cloud models and some large-scale models. Thus the relation between the lognormal and power law parameterizations is unclear, and it is desirable to establish a bridge between them. In this paper, algebraic and power law equivalents are found 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 the power law expression for cloud drop activation from basic thermodynamic principles (Köhler theory). In the new power law formulation, the index k and coefficient C are obtained as continuous analytical functions of s and expressed directly via parameters of aerosol lognormal size spectra (modal radii, dispersions) and physicochemical properties. This approach allows reconciliation of this modified power law and the lognormal parameterizations and their equivalence is shown by the quantitative comparison of these models as applied to several examples. The advantages of this new power law relationship include bounded Nd at high s, quantitative explanation of the experimental data on the k-index and possibility to express k(s) and C(s) directly via the aerosol microphysics. The modified power law provides a framework for using the wealth of data on the C, k parameters accumulated over the past decades, both in the framework of the power law and the lognormal parameterizations.

1. Introduction

[2] The power law and lognormal parameterizations of aerosol size spectra and supersaturation activity spectra are widely used in studies of aerosol optical and radiative properties, cloud physics, and climate research. These two kinds of parameterizations exist in parallel, implicitly compete, but their relation is unclear. One of the most important applications of the power and lognormal laws is parameterization of the concentrations of cloud drops Nd and cloud condensation nuclei NCCN (CCN), on which the drops form. They influence cloud optical and radiative properties as well as the rate of precipitation formation. Precise evaluation of NCCN is required, in particular, for correct estimation of anthropogenic aerosol effects on cloud albedo [Twomey, 1977] and precipitation [Albrecht, 1989].

[3] The most commonly used parameterization of the concentration of cloud drops or cloud condensation nuclei NCCN on which the drops activate is a power law by the supersaturation s reached in a cloud parcel is

equation image

which is referred to in the literature as the integral or cumulative CCN activity spectrum. Numerous studies have provided a wealth of data on the parameters k and C for various geographical regions [see, e.g., Hegg and Hobbs, 1992; Pruppacher and Klett, 1997, Table 9.1, hereafter referred to as PK97]. Parameterizations of the type (1) were derived by Squires [1958] and Twomey [1959], using several assumptions and similar power law for the differential CCN activity spectrum, ϕs(s)

equation image

Equations (1) and (2) have been used for several decades in many cloud models with empirical values of k and C that are assumed to be constant for a given air mass [PK97] and are usually constant during model runs.

[4] To explain the empirical dependencies (1) and (2), models were developed of partially soluble CCN with the size spectra of Junge-type power law f(r) ∼ rμ (total aerosol concentration Nar(μ−1)) and the index k was expressed as a function of μ. Jiusto and Lala [1981] found a linear relation

equation image

which implies k = 2 for a typical Junge index μ = 4, while the experimental values of k compiled in that work varied over 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 for ϕs(s) and expressed k as a function of μ and CCN soluble fraction. Khvorostyanov and Curry [1999a, 1999b, hereafter referred to as KC99a, KC99b] derived power laws for the size spectra of the wet and interstitial aerosol, for the activity spectra ϕs(s), NCCN(s) and for the Angstrom wavelength index of extinction coefficient, and found linear relations among these indices expressed in terms of the index μ and aerosol soluble fraction.

[5] A deficiency of (1) is that it overestimates the droplet concentration at large s and predicts values of droplet concentration that exceeds total aerosol concentration; this occurs because of the functional form of the power law and use of a single value of k. Many field and laboratory measurements have shown that a more realistic NCCN(s) spectrum in log-log coordinates is not linear as it would be with k = const, but has a concave curvature, i.e., the index k decreases 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 introducing the C-k space and constructing nomograms in this space [Braham, 1976]; (2) by constructing ϕs(s) that rapidly decreases at high s [Cohard et al., 1998, 2000] or integral empirical spectra NCCN(s) [Ji and Shaw, 1998] that yield finite drop concentrations at large s; (3) by considering the decrease in the indices μ, k with decreasing aerosol size [KC99a]; (4) by using a lognormal CCN size spectrum instead of the power law, which yields concave spectra [von der 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 the drop concentration are recently being incorporated into climate 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 chamber experiments. The relation between the power law and newer lognormal parameterization is unclear. In this paper, a modified power law is derived and its equivalence to the lognormal parameterizations is established. In section 2, a model of mixed (partially soluble) dry CCN is developed with parameterization of the soluble fraction as a function of the dry nucleus radius and a power law Junge-type representation of the lognormal size spectra is derived. To allow a more accurate estimate of aerosol contribution into the cloud optical properties and to the Twomey effect, a lognormal spectrum of the wet interstitial aerosol is found using Köhler theory. In section 3, a new algebraic representation of the lognormal spectra is derived. The differential CCN activity spectra as a modification of (2) are derived in section 4 in both the lognormal and algebraic forms. In section 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 modification of (1), expressing the parameters C and k as continuous algebraic functions of supersaturation and parameters of aerosol microstructure and physicochemical properties. The advantages of this new power law relationship include drop concentration bounded by the total aerosol concentration at high supersaturations, quantitative explanation of the experimental data on the k-index, and the possibility to express k(s) and C(s) directly via the aerosol microphysics. This formulation allows reconciliation of this modified power law and the lognormal parameterizations, which is illustrated with several examples. Summary and conclusions are given in section 6.

2. Correspondence Between the Lognormal and Power Law Size Spectra

2.1. Lognormal Size Spectra of the Dry and Wet Aerosol

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

equation image

where Na is the aerosol number concentration, σd is the dispersion of the dry spectrum, and rd0 is the mean geometric radius related to the modal radius rm as

equation image

As the relative humidity H exceeds the threshold of deliquescence Hth of the soluble fraction, the hygroscopic growth of the particles begins and initially dry CCN convert into wet “haze particles.” To derive the size spectrum of the wet aerosol, we need relations between the dry radius rd and a corresponding radius of a wet particle rw. These relations can be obtained using the Köhler equation for supersaturation s = (ρv − ρvs)/ρvs that can be written for the dilute solution particles as [PK97]:

equation image
equation image

Here ρv, ρvs, and ρw are the densities of vapor, saturated vapor, and water, H is the ambient relative humidity, Ak is the Kelvin curvature parameter, Mw is the molecular weight of water, ζsa is the surface tension at the solution-air interface, R is the universal gas constant, T is the temperature (in degrees Kelvin). The parameter B describes effects of the soluble fraction, ν is the number of ions in solution, Φs is the osmotic potential, ms and Ms are the mass and 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 soluble fraction and the parameter B, following Levin and Sedunov [1966], Sedunov [1974]:

equation image

where the parameters b and β depend on the chemical composition and physical properties of the soluble part of an aerosol particle. The parameter β describes the soluble fraction of an aerosol particle. Generally, the solubility decreases with increasing particle size [e.g., Laktionov, 1972; Sedunov, 1974; PK97], so that β decreases with increasing rd and may vary from 0.5, when soluble fraction is proportional to the volume, to −1, when soluble fraction is independent of the radius. This variation describes the change of the dominant mechanisms of accumulation of soluble fraction with growing particle size. A detailed analysis of these mechanisms is given, e.g., in the work of PK97 and Seinfeld and Pandis [1998] and is beyond the scope of this paper. We consider in more detail two particular cases.

[9] 1. β = 0.5. The value β = 0.5 corresponds to the case B = brd3, when soluble fraction is distributed within particle volume, is proportional to it, and is usually implicitly assumed in most parameterizations of CCN deliquescence and 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 of KC99a that with β = 0.5, the quantity b is a dimensionless parameter:

equation image

where ρs is the density of the soluble fraction, ɛv is its volume fraction.

[10] 2. β = 0. Then B = brd2, i.e., mass of soluble fraction is proportional to the surface area where it is accumulated as a film or shell. The particle volume fraction was parameterized in the work of KC99a as ɛv(rd) = ɛv0(rd1/rd), where rd1 is some scaling parameter and ɛv0 is the reference soluble fraction (dimensionless). We then obtain for b

equation image

For this case, b has the dimension of length and is proportional to the scaling radius rd1. If the thickness l0 of the soluble shell is much smaller than the radius rd of a spherical insoluble core, l0rd, then approximately ms ≈ 4πρsrd2l0, ɛv(rd) = 3l0/rd, and in (9b) ɛv0 = 1, rd1 = 3l0, and

equation image

The parameterizations (9b) and (9c) might be used, in particular, for surface-active substances that are accumulated near the particle surface [e.g., Rissman et al., 2004] or for soluble shell coatings of particles with insoluble cores such as dust when heterogeneous chemical reactions on the particle 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-proportional soluble components (β = 0.5) consisting of NaCl and ammonium sulphate evaluated from (9a) are determined as follows, using data obtained from PK97. If the soluble material is NaCl (Ms = 58.5, ρs = 2.16 g cm−3, ν = 2, Φs ≈ 1 for dilute solutions), then b = 1.33 for ɛv = 1 (fully soluble nuclei), and b = 0.26 for ɛv = 0.20. If the soluble material is ammonium sulphate (Ms = 132, ρs = 1.77 g cm−3, ν = 3, Φs ≈ 0.767 for molality of 0.1, so that νΦs ≈ 2.3), then b = 0.55 for fully soluble nuclei (ɛv = 1), and b = 0.25 for ɛv = 0.45. In the calculations below, we consider mostly the case β = 0.5 (volume-proportional) and use the value b = 0.25, which may correspond to the dry aerosol containing ∼50 % ammonium sulphate, as found by Hänel [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 β = 0 with an appropriate model of the soluble shell (some estimates are given below for the case β = 0), to any other ɛv, or for any other chemical composition using appropriate values of Ms and ρs. The soluble aerosol fraction often decreases with increasing radius [PK97], and this feature can be described by choosing an appropriate superposition of the aerosol size spectra with the two soluble fractions: volume- proportional (β = 0.5 for smaller fraction) and surface-proportional (β = 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 rw radii. Various solutions for the humidity dependence of rw(H) at subsaturation and supersaturation were found in algebraic and trigonometric forms using the Kohler's equation (6) [e.g., Levin and Sedunov, 1966; Sedunov, 1974; Hänel, 1976; Fitzgerald, 1975; Smirnov, 1978; Fitzgerald et al., 1982; Khvorostyanov and Curry, 1999a; Cohard et al., 2000]. In this work, we consider only the wet interstitial aerosol, and humidity transformations at subsaturation will be considered in a separate paper.

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

equation image

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

equation image

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

equation image

The wet mean geometric radii rw0 and dispersions σw are related to the corresponding rd0 and σd of the dry aerosol as

equation image

[15] Thus the shape of the wet interstitial size spectrum (12) is the same lognormal as for the dry aerosol (4) but the values of rw0 and σw are different. The size spectra of the interstitial aerosol 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 by the Junge Power Law

[16] A correspondence between the lognormal and power law size spectra can be established using the same method as in the work of Khvorostyanov and Curry [2002, 2005] for a continuous power law representation of the fall velocities. Suppose f(r) is smooth and has a smooth derivative f′(r). Then they can be presented at a point r as the power law functions

equation image

Solving (14) for μ and cf we obtain:

equation image

The power index μ and cf are here the functions of radius. If aerosol size spectrum f(r) is described by the dry fd(rd) or wet fw(rw) lognormal distributions (4), (12), substitution of fd,w and fd,w into (15) yields the power index for the dry and wet aerosol spectra

equation image

where the indices “d, w” denote dry or wet aerosol, and rw0, σw0 are defined by (14). Thus the lognormal dry and wet spectra (4), (12) can be presented at every point rd, rw, in the power law form (14) with the index μd,w(16).

2.3. Examples of the Dry and Wet Aerosol Size Spectra and Power Indices

[17] Figure 1 shows in log-log coordinates a lognormal dry aerosol size spectrum fd(rd) with modal radius rm = 0.03 μm and dispersion σd = 2, close to Whitby [1978] for the accumulation mode, and concentration Na = 200 cm−3. Its Junge-type power law approximations shows the radius dependence of the power index μ: it changes from μ = −1.27 at r1 = 0.016 μm < rm, to μ = 0 at the modal radius r2 = rm, and μ is positive at all r > rm; in particular, μ = 2.5 at r3 = 0.1 μm and μ = 5.0 at r4 = 0.33 μm. Recall that the dimensional analysis of the coagulation equation and its asymptotic solutions yield the inverse power laws with the index μ = 2.5 for rd ∼ 0.1 μm due to the dominant effects of Brownian coagulation, and μ = 4.5 for rd > 1 μm due to prevailing sedimentation [PK97, chap. 11]. The lognormal spectrum with the described parameters is in general agreement with these power law solutions. Thus the lognormal size spectrum can be considered as the superposition of the power laws with the index from (16).

Figure 1.

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

[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. This resembles a corresponding effect for the power law spectra, when the transition to H = 1 is accompanied by a decrease of the Junge power index, e.g., μ = 4 at H < 1 converts into μ = 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 calculated for lognormal distributions with various modal radii and dispersions is shown for the dry aerosol in Figure 2, which illustrates the following general features of (16). The indices (the slopes of the spectra) increase with radii for all r. The indices μ increase with decreasing dispersions and with increasing rd0 of the dry lognormal spectrum. The calculated μ intersect the average Junge's curve μ = 4 in the range of radii 0.06 μm to 0.7 μm, i.e., mostly in the accumulation mode. Since the values chosen here for rd0 and σd are realistic (close to Whitby's multimodal spectra), this may explain how a superposition of several lognormal spectra may lead to an “effective average” μ = 4, observed by Junge. The growth of the indices above 4 for larger r is in agreement with the coagulation theory that predicts steeper slopes, μ = 19/4, of the power laws for greater radii r ≥ 5 μm [PK97]. The described relation between the lognormal and power law size spectra might be useful for analysis and parameterizations of the measured spectra and of solutions to the coagulation equation.

Figure 2.

Effective power indices calculated from equation (16) for lognormal distributions with modal radii and dispersions indicated in the legend; the constant value μ = 4 is given for comparison.

3. Algebraic Approximation of the Lognormal Distribution

[20] A disadvantage of the lognormal size spectra is the difficulty of evaluating analytical asymptotics and moments of this distribution. A convenient algebraic equivalent of the lognormal distribution can be obtained using two representations of the smoothed (bell-shaped) Dirac's delta-function δ(x). The first one was taken as [Korn and Korn, 1968]

equation image

where α determines the width of the function. Another representation as an algebraic function of exp(x/α) is given by Levich [1969]. We transformed it to the following form:

equation image

[21] It can be shown that both functions δ1(x, α) and δ2(x, α) are normalized to unity, tend to the Dirac's delta-function when α → ∞ and exactly coincide in this limit:

equation image

Thus we can use the following equality:

equation image

A comparison shows that both functions are already sufficiently close at values α ∼ 1, which is illustrated in Figure 3a, where these functions divided by α/equation image are plotted, the difference generally does not exceed 3–5%. The approximate equality of these smoothed delta functions is used further for an algebraic representation of the aerosol size and CCN activity spectra.

Figure 3.

(a) Comparison of the smoothed Dirac delta-functions: exp(−α2x2), and its approximation 1/ch2(2αx/equation image), for two values α = 1 and 3. The difference generally does not exceed 3–5%. (b) Comparison of the smoothed Heaviside step-functions, erf(αx), and tanh(2αx/equation image).

[22] Integration of (20) from 0 to x yields another useful relation:

equation image

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

equation image

Equation (21) with α = 1 was found by Ghan et al. [1993], who showed its accuracy, which is also illustrated in Figure 3b for two values of α. It is known that integration of the Dirac's delta-function yields Heaviside's step-function. Comparison of Figures 3a and 3b shows that the same is valid for the correspondent smoothed functions, therefore the derivation above shows that the relation (21) is an approximate equality of the two representations of the smoothed Heaviside step-functions. Equation (21) is used in section 5 for the proof of equivalence of the lognormal and algebraic representations of the drop concentration and k-index.

[23] If we introduce in (20) the new variables x = ln (r/r0), α = (equation image ln σ)−1, then 2αx/equation image = ln(r/r0)k0/2, where k0 is a parameter introduced by Ghan et al. [1993]:

equation image

Substituting these variables into (20), we obtain

equation image

[24] The expression on the left-hand side is the lognormal distribution dN/dlnr with mean geometric radius r0 and dispersion σ, and the right-hand side is its algebraic equivalent, which is another representation of the smoothed delta function. When the dispersion tends to its lower limit σ → 1, then α → ∞, and according to (19), both distributions tend to the delta function, i.e., become infinitely narrow (monodisperse) at σ = 1. For σ > 1, they represent distributions of the finite width. Using the equality (23), the dry (4) and wet (12) lognormal aerosol size spectra can be rewritten in algebraic form as

equation image
equation image

where rd0 and σd are the geometric radius and dispersion of the dry spectrum, rw0 and σw are corresponding quantities for the wet spectrum defined in (13), and the indices k0d and k0w are

equation image

A comparison of the algebraic (24) and lognormal (4) aerosol size spectra calculated for the two pairs of spectra with the same parameters is shown in Figure 4. One can see that algebraic and lognormal spectra are in a good agreement in the region from the maxima and down by two to three orders of magnitude. A discrepancy is seen in the region of the tails, but this region provides a small contribution into the number density.

Figure 4.

Comparison of the algebraic and lognormal aerosol size spectra for the two pairs of parameters: rm = 0.1 μm, σd = 2.15 (solid and open circles for the algebraic and lognormal, respectively), and rm = 0.0.43 μm, σd = 1.5 (solid and open triangles).

[25] These algebraic representations of the size spectra can be useful in applications. Evaluation of their analytical and asymptotic properties is simpler than those of the lognormal distributions, and various moments over the size spectra can be expressed via elementary functions rather than via erf as in the case with the lognormal distributions. Being a good approximation to the lognormal distribution but simpler in use, the algebraic functions (24), (25) can be used for approximation of the aerosol, droplet, and crystal size spectra as a supplement or an alternative to the traditionally used power law, lognormal, and gamma distributions. Some applications are illustrated in the next sections.

4. Differential CCN Supersaturation Activity Spectrum

4.1. Critical Supersaturations and Radii

[26] To derive the supersaturation activity spectrum, we need relations among the dry radius rd, corresponding critical water supersaturation scr, and the critical nucleus radius rcr. The values rcr and scr for CCN activation can be found as usually using Köhler's equation (6) from the condition of maximum ds/dr = 0:

equation image
equation image

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

equation image

where

equation image

[27] Figure 5 depicts the critical supersaturations, scr, and critical radii, rcr, calculated with (27), (28) as functions of rd with four values of b (0.55, 0.275, 0.135, 0.07), corresponding to various soluble fractions of ammonium sulfate (100, 50, 25, and 12.5%) indicated in the legend. The values of scr required for activation of a given CCN increase with decreasing rd from 0.1–0.3% at rd = 0.1 μm (a typical cloud case) to 3–10% at rd = 0.01 μm (as can be found in a cloud chamber) and reach 70–200% at rd = 0.001 μm. The values of rcr increase with rd and the ratio rcr/rd reaches 5–13 at rd = 0.1 μm and 13–40 at rd = 1 μm. The 8-fold decrease in the soluble fraction causes scr to increase and rcr to decrease by a factor of ∼3–5.

Figure 5.

(a) Critical supersaturation, scr, and (b) critical radius, rcr, calculated as functions of the dry radius rd with the four values of b corresponding to various soluble fractions of ammonium sulfate indicated in the legend.

[28] These features are important for understanding and quantitative description of the separation between activated and unactivated CCN fractions. Figure 6 shows four lognormal size spectra of the dry aerosol with various modal radii and dispersions along with the corresponding critical radii and supersaturations calculated with β = 0.5, b = 0.55 (50% soluble fraction of ammonium sulfate). One can see that if the maximum supersaturation does not exceed ∼0.1 %, the right branch of the dry spectrum with rm = 0.1 μm can be activated down to the mode, and the left branch (rd < rm) remains unactivated. For the spectrum with rm = 0.03 μm, activation of the right half down to rm requires a supersaturation of 0.6%. Such a supersaturation is typical of some natural clouds. Activation of the left branches with rd < 0.01–0.03 μm requires higher supersaturations and may be typical of the CCN measurements in cloud chambers where s ∼ 5–20% can be reached [e.g., Jiusto and Lala, 1981; Hudson, 1984; Yum and Hudson, 2001].

Figure 6.

Lognormal aerosol size spectra for rm = 0.1 μm, σd = 2 (solid circles), rm = 0.1 μm, σd = 1.5 (crosses), rm = 0.03 μm, σd = 2 (rhombs), and rm = 0.03 μm, σd = 1.5 (triangles), plotted as the functions of dry radii and of corresponding critical radii and critical supersaturations. Critical supersaturations scr corresponding to the maxima at 0.1 μm and 0.03 μm are shown near the curves.

4.2. Lognormal Differential CCN Activity Spectrum

[29] The CCN differential supersaturation activity spectrum ϕs(s) can be obtained now using the conservation law in differential form for the concentration:

equation image

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

equation image

where

equation image

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

equation image

where we introduced the mean geometric supersaturation s0 and the supersaturation dispersion σs:

equation image
equation image

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

equation image

The supersaturation activity spectrum (33) has a maximum at sm and the region s = sm ± σs gives the maximum contribution into the droplet concentration.

[30] Equations (4) and (33)(36) show the following. The lognormal size spectrum of the dry aerosol with the parameters described above is equivalent to the lognormal CCN activity spectrum by supersaturations ϕs(s); that is, the shape is preserved under transformations between the radius and supersaturation variables. This feature is similar to that found in the work of Abdul-Razzak et al. [1998] and Fountoukis and Nenes [2005]. The difference of our model with these works is in that they assume soluble fraction proportional to the volume (β = 0.5), while our model for ϕs(s) generalizes this expression and allows variable soluble fraction. For β = 0.5 (homogeneously mixed in volume), (35) for σs shows that the argument in the exponent (33) can be rewritten as (−1/2)[(2/3)ln(s/s0)/lnσd]2, which coincides with the cited works. For soluble fraction proportional to the surface, β = 0, as can be for the dust particles covered by the soluble film, the mean geometric supersaturation s0rd0−1, and not ∼rd0−3/2, as in the case β = 0.5; the logarithm lnσs is 50 % smaller with β = 0, the argument becomes (−1/2)[ln(s/s0)/lnσd]2 and ϕs(s) becomes much narrower. This means that the maximum in the activation spectrum is reached at another sm and drop activation occurs over narrower s-range with β = 0.

4.3. Algebraic Differential CCN Spectrum

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

equation image

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

equation image

Here

equation image
equation image
equation image

(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) using the similarity relation (23), which again illustrates the equivalence of the lognormal and algebraic distributions.

[32] The first term, image on the right-hand side of (38) represents Twomey's [1959] power law (2). In contrast to the C0 and ks0 based on fits to experimental data, here the analytical dependence is derived from the Köhler's theory and algebraic approximation of the lognormal functions, and parameters C0, ks0 are expressed directly via aerosol parameters. This first term describes drop activation at small supersaturations ss0 (as in the clouds with weak updrafts and in the “haze chambers”) and grows with s as image The second term in parenthesis decreases at large ss0 as image overwhelms the growth of the first term, and ensures the asymptotic decrease ϕs(s) ∼ image Thus the term in parenthesis serves effectively as a correction to the Twomey law and prevents an unlimited growth of concentration of activated drops at large supersaturations.

[33] The entire CCN spectrum (38) resembles the corresponding equation from Cohard et al. [1998, 2000]:

equation image

where the first term is also the Twomey's differential power law, and the term in parenthesis was added by the authors as an empirical correction with the two parameters η, λ that were varied and chosen by fitting the experimental data. Thus the algebraic representation of ϕs(s) (38) found here establishes a bridge between the Twomey's power law with its corrections as in the work of Cohard et al. [1998, 2000] and the lognormal parameterizations of drop activation as in the work of Ghan et al. [1993, 1995], Abdul-Razzak et al. [1998], Abdul-Razzak and Ghan [2000], and Fountoukis and Nenes [2005].

[34] Shown in Figure 7 is an example of the CCN differential activity spectrum ϕs(s) (38) calculated with b = 0.25 (CCN with ∼50 % of ammonium sulfate or ∼20 % of NaCl) and four combinations of modal radii and dispersions. The area beneath each curve represents the drop concentration Ndr that could be activated at any given supersaturation s, so that the maximum of ϕs(s) indicates the region of s where its increase leads to the most effective activation. A remarkable feature of this figure is a substantial sensitivity to the variations of rm, σd. The maximum for the maritime-type spectrum with rm = 0.1 μm, σd = 2 lies at very small s ∼ 0.01 %. When σd decreases to 1.5 or rm decreases to 0.03 (closer to the continental spectrum), the maximum of ϕs(s) shifts to s ∼ 0.04–0.1 %. With rm = 0.03 μm, σd = 1.5, the region s ∼ 0.01–0.1% becomes relatively inactive and the maximum shifts to 0.4%. Thus even moderate narrowing of the dry spectra or decrease in the modal radius may require much greater vertical velocities for activation of the drops with the same concentrations.

Figure 7.

CCN differential activity spectrum ϕs(s) calculated with b = 0.25 and four combinations of modal radii rm and dispersions σd of the dry aerosol spectra indicated in the legend.

[35] A comparison of the algebraic (37) and lognormal (33) differential activity spectra in Figure 8 shows their general good agreement. The discrepancy increases in the both tails, but their contributions into the number density is small as will be illustrated below. Thus the algebraic functions (37), (38) approximate the lognormal spectrum (33) with good accuracy.

Figure 8.

Comparison of the algebraic (alg.) and lognormal (log.) differential CCN activity spectra ϕs(s). A pair of digits in the parentheses in the legend denotes modal radius rm in μm and dispersion σd of the corresponding dry aerosol spectrum.

5. Droplet Concentration and Modified Power Law for Drops Activation

5.1. Droplet Concentration Based on the Lognormal and Algebraic CCN Spectra

[36] The concentration of CCN or activated drops can be obtained by integration of the differential activity spectrum ϕs(s) (33) over the supersaturations to the maximum value s reached in a cloudy parcel:

equation image

where s0 and σs are defined by (34), (35). Equation (41) is similar to the corresponding expressions derived by von der Emde 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 was discussed for the differential activation spectrum in previous section, in variable soluble fraction β, which is discussed below.

[37] Droplet or CCN concentration in algebraic form can be obtained now in two equivalent ways. The first way is integration of the algebraic CCN spectrum ϕs(s) (37) or (38) over s:

equation image

where C0 and η0 are defined in (39b) and (39c). The second way is as in the work of Ghan et al. [1993] by substituting (21) for the equality of erf and tanh into (41), which also yields (42). The derivation of (41) and (42) along with equivalence (20), (21) emphasizes again the fact that differential CCN distributions by critical radii or supersaturations are smoothed Dirac delta functions, while their integrals, droplet concentrations, are smoothed Heaviside step functions.

[38] According to (42), the asymptotic of NCCN(s) at ss0 is the Twomey power law NCCN(s) = image The term in parenthesis is a correction, its asymptote at ss0 is image it compensates the growth of the first term and prevents unlimited drop concentration. The asymptotic limit of drop concentration is NCCN(s) → Na at large s, and hence the number of activated drops is limited by the total aerosol concentration.

[39] Equation (42) is similar to the corresponding equation from Ghan et al. [1993], but is expressed via supersaturation dispersion instead of the modal radius as in that work and is generalized here for the variable soluble fraction β, which causes the following difference. As follows from definitions of ks0(39a), the power index ks0 differs by the factor (1 + β) from the corresponding expression kgh = 4/(equation image ln σd) in the work of Ghan et al. [1993], who assumed soluble fraction proportional to the volume, i.e., in our terms β = 0.5. With this value, ks0 = (2/3)kgh, and the asymptotic of NCCN for small s is NCCNimageimage which coincides with the expression in the work of Ghan et al. [1993]. However, for β = 0 (surface-proportional soluble fraction), the index ks0 is 1.5 times higher than with β = 0.5, and, as mentioned before, the functions NCCN(s) increase (droplets activate) over much narrower interval of s.

[40] If the dry aerosol size spectrum is multimodal with I modes (e.g., I = 3 in a three-modal distributions [Whitby, 1978]), then it can be written as a superposition of I lognormal fractions (4) with Nai, σdi, and rd0i being the number concentration, dispersion, and mean geometric radius of the ith fraction. The mean geometric supersaturations s0i, dispersions σsi, and the other parameters for each fraction are then defined by the same equations as described above for a single fraction. Then the algebraic dry and wet size spectra, the differential and integral activity spectra are obtained for each fraction as described above and the corresponding multimodal spectra are superpositions of the I fractions. The multimodal CCN activity spectrum in algebraic form is derived from (42):

equation image

Equation (43) for β = 0.5 is similar to the corresponding equation from Ghan et al. [1995] where Nd(s) is expressed in terms of critical radii, but expresses it here directly via supersaturation and generalizes for the other β. The parameters βi can be different for each fraction, e.g., βi = 0.5 may correspond to the accumulation mode, and βi = 0 may correspond to the coarse mode consisting of the dust particles 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 for parameterization of the results of CCN measurements and of drop activation in many cloud models. An analytical function for the k- index can be found now (similar to the μ-index (16)) using the method from Khvorostyanov and Curry [2002, 2005] for the power law representation of the continuous functions. Using (1) and its derivative N′CCN(s) (2) and solving these two equations for k(s), we obtain:

equation image

Equation (44) allows evaluation of k(s) with known differential ϕs(s) and cumulative NCCN(s) CCN spectra. Substituting here (33) for ϕs(s) and (41) for NCCN(s), we obtain k(s) in terms of lognormal functions:

equation image

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

equation image

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

equation image

where the parameters ks0i, s0i, Nai are defined as before but for each ith mode.

[42] Equations (45)(47) allow evaluation of k(s) for any s with given σs and s0 that are expressed via rd, σd, b, and β. That is, knowledge of the size spectra and composition of the dry or wet CCN allows direct evaluation of k(s) over the entire s-range. This solves the problem outlined in Jiusto and Lala [1981], Yum and Hudson [2001] and many other studies: various values of the k-index at various s. Equations (45)(47) provide a continuous representation of k over the entire s-range. Note that the k-index is not constant for a given air mass or aerosol type but depends on a given s, at which it is measured. Equation (46) shows that for ss0, the asymptotic value k(s) = ks0. For ss0, the index decreases with s as k(s) = image and tends to zero. This behavior is in agreement with the experimental data 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)

equation image

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

equation image

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

equation image

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

equation image

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

equation image

Equation (51) is similar to (1), however, now with the parameters depending on s as just described. Hence k(s) and C(s) are now expressed directly via the parameters of the lognormal spectrum (Na, rd0, σd) and its physicochemical properties 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)-index evaluated from (47) and the coefficient C(s)

equation image

where χi = ks0ik(s), and ks0i, C0i, s0i, Nai are defined for each ith mode. The coefficient Cpc(s) with supersaturation in percent is defined by (52). The number of modes I is determined by the nature of aerosol and is the same as used for multimodal size spectra with the lognormal approach in the work of Ghan et al. [1995], Abdul-Razzak and Ghan [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 particular, the C-k space constructed by Braham [1976] based on experimental data follows now from the equations for k(s) and C(s) with use of any relation of maximum s to vertical velocity. The decrease of the k-slopes with increasing s analyzed by Jiusto and Lala [1981], Yum and Hudson [2001], Cohard et al. [1998, 2000], and many others is predicted by (45)(47), as well as its dependence on the nature of aerosol. The proportionality of coefficient C to the aerosol concentration explains substantially greater values of C observed in continental than in maritime air masses as compiled by Twomey and Wojciechowski [1969] and Hegg and Hobbs [1992] and used by cloud modelers.

5.3. Supersaturation Dependence 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 in Figures 9a and 9b and is compared with the experimental data from Jiusto and Lala [1981], who collected the data from several chambers and to the field data collected by Yum and Hudson [2001] with a continuous flow diffusion chamber (CFD) at two altitudes in the Arctic in May 1998 during the First International Regional Experiment-Arctic Cloud Experiment (FIRE-ACE). It is seen that the k-index is not constant as assumed in many cloud models. The calculated k-indices reach the largest values of 1.5–4.2 at s ≤ 0.01 %, where it tends to the asymptotic ks0(39a) and increases with decreasing dispersion of the dry CCN spectrum since ks0 ∼ (lnσd)−1. As s increases, k(s) is initially almost 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%. The k-indices increase with decreasing modal radii and dispersions. Agreement of calculated k(s) with haze chamber data is reached at σd = 1.3, rd = 0.085 μm, and with CFD at σd = 1.5, rd = 0.043 μm. Agreement with the Arctic field data by Yum and Hudson [2001] is reached with rm = 0.045 μm, σd = 2.15 in the layer 960–1010 mb and rm = 0.043 μm, σd = 1.9 in the layer 560–660 mb.

Figure 9.

Supersaturation dependence of the k-index calculated for the lognormal size spectrum of dry CCN with various modal radii rm and dispersions σd indicated in the legend and comparison: (a) with the experimental data from Jiusto and Lala [1981] from four chambers (haze chamber; continuous flow diffusion chamber, CFD; and static diffusion chamber, SDC), and (b) with the data collected in FIRE-ACE in 1998 [Yum and Hudson, 2001, hereinafter referred to as YH01] at the heights 960–1010 mb and 560–660 mb.

[47] Thus this model explains and describes quantitatively the observed decrease in the k-index with increasing supersaturation reported previously by many investigators (e.g., Twomey and Wojciechowski, [1969], Braham [1976], Jiusto and Lala [1981]; see review in PK97) and relates it to the aerosol size spectra. A more detailed analysis of experimental data with these equations should be based on simultaneous use of the measured multimodal aerosol size spectra 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-index depends on s, but the coefficient C is constant [e.g., Cohard et al., 1998, 2000], the values of “effective” C(s), Cpc(s) in this model also depend on supersaturation. The spectral behavior of Cpc(s) calculated from (52) with different rm, σd and the same Na = 100 cm−3 is shown in Figure 10. For ss0, the coefficients tend to the asymptotic limit Cpc(0) = image and reach values of 103–105 cm−3 caused by the dependence C0image with Ak ∼ 10−7 cm and rd0 ∼ 3 × 10−6 to 10−5 cm. This dependence causes increase in Cpc with decreasing σd (increasing ks0) and with increasing rd0 seen at small s in Figure 10; i.e., the narrower spectra and the larger modal radius, the faster drop activation at small s. The values of Cpc(s) decrease with increasing s by 1–3 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)Naspck(s), and k(s) → 0 at large s, hence all CpcNa at spc ≥ 0.5% as seen in Figure 10. Thus NCCN(s) = Cpcspck(s) ∼ 10−2k(s)Na and tends to the limiting value Na, which ensures finite values of Na in contrast to the models with constant k, C.

Figure 10.

Equivalent coefficient Cpc(s) in the power law for NCCN calculated for the lognormal size spectrum of dry CCN with modal radii rm and dispersions σd indicated in the legend and other parameters described in the text. For the convenience of comparison, all curves are calculated with Na = 102 cm−3.

[49] A comparison of the concentrations NCCN(s) calculated using the power law (51) and erf function (41) shows in Figure 11 their good agreement and illustrates good accuracy of the algebraic distributions and renewed power law as compared to the direct integration of the lognormal distributions. The curves NCCN(s) calculated with the newer power law are not linear in log-log coordinates as it is with (1) and constant C0, k, but are concave in agreement with the measurements [e.g., Jiusto and Lala, 1981; Hudson, 1984; Yum and Hudson, 2001] and other models [e.g., Ghan et al., 1993, 1995; Feingold et al., 1994; Cohard et al., 1998, 2000]. The reason for this concavity is quite clear from Figures 68 for the size and activity spectra: the rate of increase in concentration of activated drops dNd(s)/ds is greatest at small supersaturations in the left branch of ϕs(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, especially at s > 1%, i.e., in the left branch of the small dry CCN with rd < 0.03 μm (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.

Figure 11.

Comparison of the accumulated CCN spectra NCCN(s) calculated using power law equation (51) (solid symbols) and using (41) with erf function (open symbols). The values of the corresponding modal radii rm and dispersions σd of the dry aerosol spectra are indicated in the legend. For convenience of comparison, NCCN is calculated with the dry aerosol concentrations of 250, 200, 150, and 100 cm−3 from the upper to lower curves.

[50] Finally, Figure 12 shows NCCN(s) calculated with the lognormal size spectra and compared to the corresponding field data obtained by Yum and Hudson [2001] in the FIRE-ACE campaign in May 1998 in the Arctic. The parameters rm, σd, 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 parameters had to provide an agreement for both k(s) and NCCN(s). Good agreement over the entire range of supersaturations was found with the values indicated in the legend. The values of modal radii and dispersions are typical for the accumulation mode. This comparison shows that this model of the power law can reasonably reproduce the experimental data on CCN activity spectra and concentrations of activated drops and can be used for parameterization of drop activation in cloud models.

Figure 12.

Supersaturation dependence of the CCN integral activity spectrum NCCN(s) calculated with the new power law (51) for the lognormal size spectra with rm, σd and Na (in cm−3) indicated in the legend (solid rhombs and circles), compared to the corresponding field data obtained by Yum and Hudson [2001] in FIRE-ACE campaign in May 1998 in the Arctic (open rhombs and circles).

6. Summary and Conclusions

[51] A model of the lognormal size spectrum of mixed (partially soluble) dry CCN with parameterization of the soluble fraction as a function of the dry nucleus radius is used to consider the processes of humidity transformation of the dry size spectrum into the spectrum of the wet interstitial aerosol and drop activation. Using the expressions for the radius of a solution particle as a function of relative humidity, it was shown that a lognormal size spectrum of the dry aerosol transforms into the lognormal spectra of the cloud interstitial aerosol at supersaturation. The mean geometric radii and dispersions of these spectra are expressed via the corresponding parameters of the dry aerosol. A quasi-power law Junge-type representation of the lognormal spectra was derived, and corresponding power indices and coefficients are expressed via the corresponding mean geometric sizes and dispersions of the dry and interstitial aerosol. Using various representations of the smoothed Dirac delta function, an algebraic equivalent was found for the lognormal dry and interstitial aerosol size spectra. These algebraic representations allow simplification of analytical estimations and evaluation of various moments of the spectra such as condensation rate, etc.

[52] On the basis of Köhler theory, the differential CCN activity spectrum is expressed in lognormal form for various soluble fractions. This spectrum coincides with the previous parameterizations from Abdul-Razzak et al. [1998] and Fountoukis and Nenes [2005] for soluble fraction proportional to the particle volume and generalizes them for other soluble fractions (e.g., soluble fraction proportional to surface area as can be for the insoluble dust particles coated with a thin soluble shell). The algebraic equivalent of the lognormal differential CCN activity spectra is found, ϕs(s) = image + image This is a generalization and correction of the Twomey's differential activity spectrum: the first term is Twomey's power law, and the term in parentheses ensures finite droplet concentrations at high supersaturations. The functional form of this activity spectra is similar to the empirical correction by Cohard et al. [1998, 2000] of Twomey's power law but differs from these works by the analytical dependence that leads to the simple algebraic expressions for the droplet concentration, and the three parameters C0, ks0, η0 of this spectrum are directly expressed via 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 previous parameterizations by von der Emde and Wacker [1993], Ghan et al. [1993, 1995], Abdul-Razzak et al. [1998], and Fountoukis and Nenes [2005] for the soluble fraction proportional to the particle volume (homogeneously mixed particles). We have generalized the previous lognormal parameterizations 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 coatings on the surface [Levin et al., 1996; Bauer and Koch, 2005]. It is shown that the algebraic form of the cumulative CCN spectrum, NCCN(s), as found in the work of Ghan et al. [1993] from the lognormal spectrum using the equivalence of erf and tanh, can be obtained also by integration of the algebraic differential spectrum ϕs(s). This establishes a bridge and illustrates the relation among the numerous approaches based on the differential power law or algebraic CCN spectra, and those that derive NCCN(s) by direct integration of lognormal aerosol size spectra using the Köhler relation between the critical radius and supersaturation.

[54] A modified power law is derived for the drop activation, NCCN(s) = C(s)sk(s). The modified power law yields drop concentration limited by the total aerosol number, in contrast to the previous power law with unlimited NCCN(s) at high supersaturations. Algebraic formulae are found for C(s), and k(s) as continuous functions of supersaturation s, which correctly describe their decrease with s, i.e., predict the observed concave NCCN(s) spectra. The new formulation expresses C(s) and k(s) via parameters of the dry aerosol spectrum (the modal radius, dispersion, and aerosol concentration) and the physicochemical properties of the aerosol and the distribution of the soluble fraction in the aerosol particles. The derivation of the new power law for the CCN activity spectrum clearly shows its equivalence and correspondence to the lognormal parameterizations. The modified power law permits using the wealth of the data on the C, k parameters accumulated over this time. Additionally, the C, k parameters can be directly calculated based on the aerosol microstructure and recalculated from s = 1% to any s. The C-k space suggested by Braham [1976] based on experimental data can now be obtained from calculations using aerosol microstructure as the input.

[55] The relationships derived here for CCN concentration and activity spectra can be easily incorporated into cloud and climate models. Relative to the lognormal parameterization that is included in climate models [Ghan et al., 1997; Lohmann et al., 1999; Nenes and Seinfeld, 2003], the parameterizations presented here include an additional element of necessary physics, related to the treatment of the soluble fraction and how the aerosol particles are mixed. An additional advantage of the new parameterizations is their algebraic and computational simplicity, allowing for simple analytical determination of derivatives and higher order moments. Further, the power law and algebraic representations are easily superimposed to represent multimodal spectra.

Notation
Ak

the Kelvin curvature parameter.

B = brd2(1+β)

the nucleus activity.

b

the parameter defined in (9a), (9b), (9c) that describes the soluble fraction of an aerosol particle.

C

the coefficient in the power law for integral CCN activity spectrum.

C0

the coefficient in the differential power law CCN activity spectrum (38).

c1, c2

the coefficients defined in (30) and (32a).

cf

the coefficient in the Junge-type size spectrum.

f(r)

the general notation for the aerosol size spectrum.

fd(rd), fw(rw)

the size distribution functions of the dry and wet aerosol.

H, Hth

the relative humidity and the threshold of deliquescence of the soluble fraction.

k

the index in the power law for integral CCN 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 supersaturation spectrum.

l0

the thickness of the soluble shell on an insoluble aerosol particle.

Mw and Ms

the molecular weights of water and aerosol soluble fraction.

ms

the mass of the soluble fraction of an aerosol particle.

Na

the aerosol number concentration.

NCCN

the concentration of the cloud condensation 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 activation.

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 = (ρv − ρvs)/ρvs

the supersaturation.

s0

the mean geometric supersaturation.

sm

the modal supersaturation.

T

the temperature (in degrees Kelvin).

α

the parameter of width in the smoothed Dirac delta function.

β

the power index defined in (8) that describes the soluble fraction of an aerosol particle.

χ

the power index of s in the coefficient C(s) in (48).

δ(x)

Dirac's delta function.

ɛv

the volume fraction of the soluble fraction.

ɛv0

the reference soluble fraction.

Φs

the osmotic potential.

ϕs(s)

differential CCN activity spectrum.

η

the parameter of the differential activity spectrum in (39c).

μ

the power index of Junge-type power law spectrum ∼ rμ.

ν

the number of ions in solution.

ρv, ρvs and ρw

the densities of vapor, saturated vapor and water.

ρs

the density of the soluble fraction.

σd

the dispersion of the dry spectrum.

σs

the supersaturation dispersion.

σw

the dispersion of the wet aerosol size spectrum.

ζsa

the surface tension at the solution-air interface.

Acknowledgments

[56] This research has been supported by the DOE Atmospheric Radiation Program. Input from Hugh Morrison is appreciated. We are grateful to reviewer for careful reading the manuscript and useful remarks that allowed to improve the text. Jody Norman is thanked for help in preparing the manuscript.

Ancillary