## 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 *N*_{d} and cloud condensation nuclei *N*_{CCN} (CCN), on which the drops form. They influence cloud optical and radiative properties as well as the rate of precipitation formation. Precise evaluation of *N*_{CCN} 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 *N*_{CCN} on which the drops activate is a power law by the supersaturation *s* reached in a cloud parcel is

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, ϕ_{s}(*s*)

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*) ∼ *r*^{−μ} (total aerosol concentration *N*_{a} ∼ *r*^{−(μ−1)}) and the index *k* was expressed as a function of μ. *Jiusto and Lala* [1981] found a linear relation

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 ϕ_{s}(*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 ϕ_{s}(*s*), *N*_{CCN}(*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 *N*_{CCN}(*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 ϕ_{s}(*s*) that rapidly decreases at high *s* [*Cohard et al.*, 1998, 2000] or integral empirical spectra *N*_{CCN}(*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.