Parameterization of homogeneous ice nucleation for cloud and climate models based on classical nucleation theory

A new analytical parameterization of homogeneous ice nucleation is developed based on extended classical nucleation theory including new equations for the critical radii of the ice germs, free energies and nucleation rates as simultaneous functions of temperature and water saturation ratio. By representing these quantities as separable products of the analytical functions of temperature and supersaturation, analytical solutions are found for the integral-differential supersaturation equation and concentration of nucleated crystals. Parcel model simulations are used to illustrate the general behavior of various nucleation properties under various conditions, for justifications of the further key analytical simplifications, and for verification of the resulting parameterization. The final parameterization is based upon the values of the supersaturation that determines the current or maximum concentrations of the nucleated ice crystals. The crystal concentration is analytically expressed as a function of time and can be used for parameterization of homogeneous ice nucleation both in the models with small time steps and for substep parameterization in the models with large time steps. The crystal concentration is expressed analytically via the error functions or elementary functions and depends only on the fundamental atmospheric parameters and parameters of classical nucleation theory. The diffusion and kinetic limits of the new parameterization agree with previous semi-empirical parameterizations.


Introduction
Homogeneous freezing of haze particles and cloud droplets plays an important role in crystal formation in cirrus, orographic, deep convective clouds and other clouds under low temperatures.Development of parameterizations of homogeneous ice nucleation suitable for cloud and climate models has been underway for the past several decades.These parameterizations have been mostly semi-empirical, based on heuristic relations for various properties of ice nucleation: nucleation rates, critical humidities, nucleated crystal concentrations, etc.These parameterizations have been developed using parcel model simulations and either experimental data or some relations of classical nucleation theory or alternative nucleation theories.
These parameterizations can be separated into two general types.The first type provides equations for the instantaneous characteristics of the nucleation process at any given intermediate time of nucleation.The second type considers the entire nucleation process as a sub-step process (taking less than one time step in a model) and derives equations for the final characteristics of the nucleation process after the nucleation has ceased: crystal concentrations, radii, masses.
Parameterizations of the first type.One of the most important characteristics of freezing is the nucleation rate, J hom , the number of ice embryos formed per unit volume per unit time.Heymsfield and Miloshevich (1993) used results from the statistical molecular model of Eadie (1971) and fitted J hom,0 for pure water with a power law expression.
J hom,0 (T ) = 10 −X(T c ) , X(T ) = Published by Copernicus Publications on behalf of the European Geosciences Union.
V. I. Khvorostyanov and J. A. Curry: Homogeneous ice nucleation for cloud and climate models with J hom , 0 in cm −3 s −1 , T c is temperature in degrees Celsius, and A 0,HM = 606.3952,A 1,HM = 52.6611,A 2,HM = 1.7439,A 3,HM = 0.0265, A 4,HM = 1.536 × 10 −4 .Experimental data show that the freezing rates of haze particles are smaller than given by this equation, since they are depressed by the presence of solute.Sassen andDodd (1988, 1989) suggested describing this depression of the nucleation rate by introducing an effective freezing temperature where T is temperature in degrees Kelvin, T m and T f are the depressions of the melting and freezing temperatures respectively.Then the nucleation freezing rate J f,hom of the haze particles could be calculated with Eq. ( 1) but with T * instead of T .The value λ SD = 1.7 was chosen in Sassen and Dodd (1988) as an average over the experimental data by Rasmussen (1982) on the relationship between depressions of the nucleation and melting temperatures for a number of salts.It was clarified later that the coefficient 1.7 is not universal, and can vary over the range 1.4-2.4 and may reach 3-5 for some organic substances, depending on the chemical composition and concentration of a solute (Martin, 2000;Chen et al., 2000;Lin et al., 2002;DeMott, 2002;Koop and Zobrist, 2009).DeMott et al. (1994) suggested a parameterization of T m for ammonium sulfate as a function of molality M. Molality was evaluated in terms of the equilibrium particle diameter, which was calculated using Köhler's (1936) equation and the freezing point depression was calculated with Eq. (2a).De-Mott et al. (1994) used Eqs.(1), (2a) and their parameterization of M to calculate the frozen fraction F hf of the haze particles at various T and water saturation ratios S w .Having calculated F hf at various T and S w and assuming an exponential size spectrum of haze particles, DeMott et al. (1994) suggested a fit for the concentration of nucleated crystals as an integral of F hf over the haze size spectrum.This scheme reproduced the experimental data on ice nucleation of haze particles and was suitable for use in cloud models.
An important characteristic of homogeneous ice nucleation is the critical humidity or the critical water saturation ratio S hom w,cr .Sassen andDodd (1988, 1989) and Heymsfield and Miloshevich (1995) parameterized S hom w,cr as polynomial fits by the temperature.Sassen and Benson (2000) generalized these equations to account for wind shear.Koop et al. (1998) and Bertram et al. (2000), based on their measurements of the freezing temperatures T f of aqueous solutions droplets of sulfuric acid and ammonium sulfate, parameterized T f as polynomial functions of the solution concentration.Using these data and thermodynamic model of Clegg et al. (1998), these authors developed parameterizations of the critical humidities, water activity and freezing point depression as the polynomial functions of the water vapor pressure.Koop et al. (2000) suggested a parameterization of J hom,f similar to Heymsfield and Miloshevich (1993) for pure water, but accounted for solute effects parameterized with polynomial fits of a w = a w − a i w , where a w is the water activity in the liquid solution and a i w is the activity of water in solution in equilibrium with ice.Koop et al. (2000) assumed that in equilibrium a w is equal to the environmental water saturation ratio S w , and a i w was parameterized as an exponential function of the chemical potentials of water in pure ice and pure liquid water, respectively.
Many of these empirically based dependencies can be described with classical nucleation theory (CNT) for homogeneous and heterogeneous ice nucleation (Frenkel, 1946;Dufour and Defay, 1963;Defay et al., 1966;Pruppacher and Klett, 1997, hereafter PK97;Seinfeld, and Pandis, 1998;Kashchiev, 2000).CNT was extended further in a number of works as reviewed in Laaksonen et al. (1995), Mishima and Stanley (1998), Ice Physics (1999), Slezov and Schmelzer (1999).Subsequent extensions of CNT were performed by Khvorostyanov andSassen (1998a, 2002, hereafter KS98a, KS02), by Khvorostyanov and Curry (2000, 2004a, b, 2005, 2009a, hereafter KC00, KC04a,b, KC05, KC09a) and Curry and Khvorostyanov (2012, hereafter CK12).Analytical expressions for the critical radii r cr of ice germs, critical energies F cr , and nucleation rates J nuc derived in these works describe the dependence of these quantities not only on the temperature T as in CNT, but also the dependencies on water saturation ratio S w , finite radius of freezing particles, external pressure and some other factors.In particular, KS98a showed that the concentrations of nucleated crystals calculated with this extended CNT were very close to those in the semi-empirical scheme by DeMott et al. (1994).The expressions for r cr , F cr , J nuc for solution particles in KS98a and KC00 depended on water saturation ratio S w , but dependence on chemical composition vanished in the derivation.Thus, these expressions predicted that nucleation characteristics are a colligative property that do not depend on chemical nature of solute substance.This was confirmed by Koop et al. (2000) from an analysis of experimental data on freezing temperatures of various substances.It was shown in KC04a,b that the relation between the freezing and melting point depressions analyzed in Sassen andDodd (1988, 1989) can be derived from the extended CNT.
Furthermore, the equivalence of the solution and pressure effects discussed in Kanno and Angell (1977) and in Koop et al. (2000) based on the experimental data was derived in Khvorostyanov and Curry (2004a) from the extended CNT in a simple quantitative form where p is the external pressure, R is the universal gas constant, ρ i and ρ w are the densities of ice and water, M w is the molecular weight of water.This equation relates S w (or equivalent molality) and p, and shows that a decrease in S w (increase in solution molality) is equivalent to an increase in p, with proportionality determined by the function Q p that depends on the densities and temperature.The proportionality is p ∼ −T lnS w with constant densities, although they in turn depend on p, T .The value of Q p in Eq. (2b) calculated in KC04a is very large, Q p ∼ 10 4 atm at T ∼ 273 K and increases with decreasing T , so that a saturation ratio S w = 0.9 (ln S w ≈ −0.1) at T ∼ 273 K is equivalent to a high external pressure of ∼ 10 3 atm.It was shown in KC04 that Eq. ( 2b) allows a simple quantitative description of the solution-pressure equivalence in the depression of the melting and freezing points experimentally derived in Kanno and Angell (1977) and is in a good agreement with the laboratory measurements at high pressures.These comparisons show that many empirical functional dependencies of nucleation and parameterizations can be derived from CNT.
The densities of ice and water, ρ i and ρ w , and other thermodynamic parameters of water and ice at low temperatures and high pressures can be calculated from the equations of state for water and ice or can be obtained from standard tables recommendable from the new International Thermodynamic Equation Of Seawater 2010 (TEOS-10) (e.g., Jeffery and Austin, 1997;Feistel and Wagner, 2006;IOC, SCOR, and IAPSO, 2010;McDougall et al., 2010;Holten et al., 2011Holten et al., , 2012;;Feistel, 2012;IAPWS, 2012).Reviews of the recently refined equations of state for water and ice, the recent developments of the nucleation theory and their applications for ice nucleation and deliquescence-efflorescence phenomena are given in Hellmuth et al., 2012a, b).
Parameterizations of homogeneous freezing of the second type as a sub-step process in the models include more intermediate steps and assumptions.Such parameterizations are also semi-empirical, and as examples we describe the parameterizations developed by Kärcher and Lohmann (2002a, b) and Ren and MacKenzie (2005).The methods used in these parameterizations are similar to the method developed by Twomey (1959) for drop activation.The basis of these parameterizations is the equation for ice saturation ratio S i .The sink term in this equation, the deposition rate in an ensemble of the crystals, is defined as the time integral of the number density of aerosol particles dn c (t 0 )/dt 0 that freeze within the time interval between t 0 and t 0 + dt 0 , with monodisperse or polydisperse model of the haze particles.To solve this nonlinear system of equations, the authors introduce several additional hypotheses.Following Ford (1998a, b), a hypothesis on the exponential time behavior of the nucleation rate R f,hom = dn c /dt 0 was introduced where τ nuc is a characteristic time scale of the nucleation event, unknown for now, which has to be determined.Integration of Eq. ( 3) by t yields An additional heuristic hypothesis was introduced for the timescale of the nucleation event τ nuc by Kärcher and Lohmann (2002a, b) relating it to the temperature change rate dT /dt, The unknown parameter c τ was parameterized in Kärcher and Lohmann (2002a) as a function of temperature, and was replaced with a constant value c τ = 50 in Kärcher and Lohmann (2002b).Ren and MacKenzie (2005) arrived at a simpler expression, τ −1 nuc ≈ c τ (T )(dT /dt), where c τ was approximated by the temperature polynomial.A further hypothesis was that the ice saturation ratio S i changes only slightly around its critical value S i,cr during the nucleation event, and it can be assumed that S i (t) ≈ S i,cr (T ).An additional assumption is that diffusional growth of the nucleated crystals is described by the equations for the diffusion growth regime with kinetic corrections.And finally, they assume that homogeneous ice nucleation stops when S i reaches a maximum, dS i /dt = 0 at S i,cr .
With these assumptions, Kärcher and Lohmann (2002b) and Ren and MacKenzie (2005) found analytical solutions for R f,hom (t 0 ) and the concentrations of the nucleated crystals N c , and studied several limiting cases.In particular, they found for the diffusion growth regime, N c ∼ w 3/2 , and , where w is the vertical velocity and ρ is is the saturated vapor density over ice.For the kinetic crystal growth, Ren and MacKenzie (2005) found that N c ∼ w for the large particles, and is for small particles.Barahona and Nenes (2008) developed a similar substep parameterization of homogeneous ice nucleation, using Twomey's (1959) upper limit approximation for ice supersaturation, and a representation for the nucleation rate similar to that from Khvorostyanov and Curry (2004b) They used the temperature dependence for b τ (T ) from Koop et al. (2000), made several auxiliary simplifications and arrived at a parameterization that required an iterative numerical solution.All the parameterizations described above used parcel models for tuning the parameters of the final paramaterization equations.
In the studies reviewed above, it was assumed that stable hexagonal ice Ih nucleates in supercooled water or solution droplets.The thermodynamic parameters associated with ice Ih were therefore used when interpreting the data.Evidence V. I. Khvorostyanov and J. A. Curry: Homogeneous ice nucleation for cloud and climate models was provided recently that metastable cubic ice Ic may form first in some cases at low temperatures, especially at T <200 K, with subsequent relaxation to the stable ice Ih (e.g., Murray et al., 2005Murray et al., , 2010;;Murray and Bertram, 2006;Malkin et al., 2012).However, uncertainty remains in the general conceptual picture of this sequence of the processes, and a wide spread in the current data on the thermodynamic parameters for Ic, so that the nucleation rates for Ic estimated with CNT may vary by many orders of magnitude (e.g., Murray et al., 2010).We therefore assume in this work, as in most of the others, that hexagonal ice Ih nucleates in droplets and use the corresponding parameters for Ih.Calculations for Ic or any other type of ice can be done using the same equations derived in this work with corresponding changes of the thermodynamic parameters: the surface tension, melting heat, saturated vapor pressure, etc.
We have shown above that many (or most) parameterizations of ice nucleation of the first type can be derived from CNT.A question arises as to whether the more complicated parameterizations of the second type (integral) can be also derived from the CNT.This paper addresses homogeneous freezing of deliquescent haze particles and water drops.The new analytical parameterization developed here is based directly on extended classical nucleation theory with minimum auxiliary hypotheses and simplifications.Parcel model simulations are used in Sect. 2 to illustrate the general behavior of various nucleation properties under various conditions, for justification of key analytical simplifications, and for their verification.The new analytical solutions are derived in Sect.3, and the diffusion and kinetic limits are determined.It is shown that the new analytical dependencies agree with the previous parameterizations and can be expressed in terms of the primary parameters of modified classical theory.
2 Kinetics of homogeneous ice nucleation simulated with a parcel model

Parcel model
The parcel model used here was described in Khvorostyanov and Curry (2005, hereafter KC05).The parcel model is a zero-dimensional or Lagrangian model of an adiabatic rising air parcel that cools, causing nucleation and growth of the drops and crystals.All variables depend only on time t.
The dynamics in this parcel model is parameterized by prescription of a vertical velocity w constant in time.The primary thermodynamic equations are the prognostic equations for supersaturation and temperature.This system of equations includes terms that describe the phase transitions and is closed using the two kinetic equations for the drop and ice crystal size distribution functions that account for nucleation, condensation and deposition, and two equations for the droplets and crystals growth rates.Similar to the methodology adopted for the Cirrus Parcel Model Comparison Project (CPMCP, Lin et al., 2002), we exclude from consideration coagulation among the droplets and aggregation between the droplets and crystals, sedimentation, entrainment, turbulent exchange, etc. to isolate the effects directly related to nucleation processes.The system of equations comprising the parcel model is described below.The heat balance is calculated using the equation for the temperature T in a wet adiabatic process: where γ a is the dry adiabatic lapse rate, L e , L s and L m are the latent heats of condensation, deposition and melting (cal g −1 ), c p is the specific heat capacity, ρ a is the air density, I con , I dep and I fr are the rates of condensation, deposition, and freezing (g cm −3 s −1 ).
Both water and ice supersaturation govern ice nucleation kinetics: water supersaturation determines the nucleation process, and growth of ice particles is determined by ice supersaturation.We consider the equations for fractional water and ice supersaturations, s w = (ρ v − ρ ws )/ρ ws , and s i = (ρ v − ρ is )/ρ is , where ρ v is the environmental water vapor density, ρ ws and ρ is are the densities of vapor saturated over water and ice, respectively.In a rising air parcel, supersaturation is governed by two competing processes: supersaturation generation by cooling in an updraft and supersaturation absorption by the crystals in the vapor deposition process.This process can be described by the supersaturation equations that account for homogeneous ice nucleation (KC05, Sect.2a therein): Here 12 and 2 are the psychrometric corrections associated with the latent heat release at condensation derived in KC05, I dep is the deposition integral that describes the vapor flux onto the crystals, and where M w and M a are the molecular weights of water and air, R a is the gas constant of air.The vapor flux I dep to the crystals is the integral of the mass growth rate over the crystal size spectrum.We consider ice nucleation in haze particles at water subsaturation, formation of water drops is not considered in this work and the term I con is absent in Eqs.(8a, b).However, both s w and s i are required for further consideration since ice nucleation is governed by s w , and crystal growth is governed by s i .We assume that crystal size can be characterized by an effective radius r c , then I dep is expressed via crystal growth rate (dr c /dt) where f c (r c , t 0 ) is the size distribution function of the crystals nucleated at a time t 0 , and r c (t, t 0 ) denotes the radius at time t of a crystal nucleated at time t 0 .We use (dr c /dt) in the form similar to Fuchs (1959) and Sedunov (1974) where D v is the water vapor diffusion coefficient, ξ dep is the kinetic correction to the radius growth rate, V w is the thermal speed of water vapor molecules, R is the universal gas constant, and α d is the deposition coefficient.This equation for dr c /dt accounts for the kinetic correction ξ dep .Substitution of Eq. ( 12) into Eq.( 11) yields The radius r c (t, t 0 ) at time t of a crystal nucleated at time t 0 is evaluated by integrating Eq. ( 12) with constant ρ i , ρ is during the relatively short time of integration, where r c0 = r i (t 0 ) is the initial crystal radius at the activation time t 0 , and y i (t) is the integral ice supersaturation defined as Ice nucleation via haze freezing depends simultaneously on T and s w , and we can consider the integrand in Eq. ( 14) for I dep using a kinetic equation for the crystal size spectrum and introducing two activity spectra, by supersaturation φ s (T , s w ) and by temperature φ T (T , s w ) where the Dirac delta function δ(r c − r c (t 0 )) describes nucleation of a crystal with radius r c (t 0 ) and ψ fc denotes the righthand side that will be specified in the finite difference approximation as described below.Equation ( 17) can be viewed as a generalization of the known relation for the drop activation on the CCN, where usually only the supersaturation activity spectrum is accounted for (e.g., Twomey, 1959;Sedunov, 1974;Khvorostyanov andCurry, 2008, 2009b;Ghan et al., 2011;Tao et al., 2012).We could consider each of these spectra in Eq. ( 17) separately, and this will be done in Sect. 3.3,Eqs. (52b,c), but a simpler and faster way is to use an equivalent equation for concentration conservation where R f,hom = dN c (t)/dt (cm −3 s −1 ) is the polydisperse homogeneous freezing nucleation rate describing effects of both T and s w on freezing defined below.
The probability of freezing of a haze particle or a drop with radius r a and volume v(r a ) during the time interval from t 0 to t is where J f,hom is the homogeneous nucleation rate (cm −3 s −1 ) considered in Sect.3.2, Eq. ( 36).The crystal concentration N c in a polydisperse aerosol with uniform size and surface properties can be calculated by integrating the probability of freezing P f,hom of an individual haze or cloud droplet over the size spectrum f (r a ) of aerosol or droplets normalized to the aerosol or drop concentration N a : The polydisperse nucleation rate R f,hom can be calculated as where v(r a ) is the volume of a freezing particle with radius r a , J f,hom is the homogeneous nucleation rate that is calculated from the extension of the classical nucleation theory (CNT) as developed by the authors and employed here (see Sect. 3.2).It is expressed via the activation and critical energies of an ice germ freezing that depend simultaneously on the temperature and water saturation ratio.Substituting the conservation law for the nucleated crystals f c (r c )dr c = R f,hom (t 0 )dt 0 from Eq. (18a) into Eq.( 14) for I dep and using Eq. ( 15) we obtain where we introduced the effective radius r c,ef (t, t 0 ), which is the first multiplier in the integrand in Eq. ( 14) . Substituting Eq. ( 19) into Eq.( 8b) and using Eq. ( 16) for y i (t), we obtain an equation for integral ice supersaturation where Substitution of R f,hom from Eq. (18d) into Eq.( 22) yields Substitution of Eq. ( 23) into Eq.( 21) and using the relation This equation describes evolution of integral ice supersaturation.It is analogous to Twomey's (1959) and Sedunov's (1974) supersaturation equations for the drop activation, but includes a more complicated description of crystal nucleation.The first term on the RHS describes supersaturation generation by cooling action of updrafts, and the second term accounts for its depletion by the newly nucleated and growing crystals.We consider in this section homogeneous ice nucleation at cold temperatures and not very vigorous updrafts when the haze solution particles freeze at water subsaturation, so that drops do not form.The crystal nucleation term in Eq. ( 17) can be calculated in the finite difference scheme as where N c,fr is the number concentration of the crystals nucleated via homogeneous freezing in a time step t and calculated with Eq. (18c) using equations for the nucleation rate J f,hom (Eq.36 here) and r c denotes the first size step by the crystal radii (0.1-0.2 µm).The crystal size spectrum includes 30 radius intervals: 10 steps by 0.1-1 µm and the next 20 steps increasing logarithmically to 100-350 µm.This division allows coverage of both small and large size ranges without loosing accuracy.

Simulation results
The design of the simulations generally follows the protocol of the Cirrus Parcel Model Comparison Project (CPMCP; Lin et al., 2002).To simulate the ice crystal nucleation process, the parcel model was run for 1 h with most initial data specified following the CPMCP and varying some parameters to estimate the sensitivity of the results.We describe the results for three values of the vertical velocity, w = 4, 20, and 100 cm s −1 , two values of the initial temperature, T 0 = −40 • C and −60 • C, and two values of the aerosol concentration, N a = 200 cm −3 , and with increased N a = 500 cm −3 .The initial humidities were chosen as RHW 0 = 90 % for T 0 = −40 • C and RHW 0 = 78 % for T 0 = −60 • C. The initial pressure p 0 was specified to be 340 hPa.The parcel model includes the option of isolating specific ice crystal nucleation modes.Here we consider only the homogeneous freezing of deliquescent haze particles, excluding the other modes (heterogeneous freezing, deposition, contact, immersion).Integration over the haze size spectrum was performed using a lognormal size spectrum of soluble haze particles with the mean radius of 0.02 µm and dispersion σ s = 2.5.The time steps were 0.01-0.2s in the main program, but the time step can be divided further, if necessary, in the nucleation or condensation subroutines to meet stability conditions.The accuracy of the calculations was controlled by comparing the total number of crystals nucleated with those obtained by integration over the size spectrum of the grown crystals at the end of a parcel run.If the error exceeded 5 % (especially at low temperatures), the time and radius steps were varied and several additional runs were performed until the error became less than 5 %.Figures 1 and 2 illustrate the effect of the vertical velocity (w = 4 and 20 cm s −1 ) on the kinetics of homogeneous freezing at T 0 = −40 • C and N a = 200 cm −3 .It is seen that the nucleation process has two branches with increasing and  decreasing supersaturations.At the ascending branch, the first term on the right-hand side of Eq. ( 21) or Eq. ( 24) with supersaturation generation dominates; therefore the relative humidity and supersaturation increase from the initial values to the maximum values reached at the time t max .At the descending branch, RHW, s w , and s i decrease due to domination of the second term on the RHS of Eq. ( 21) or Eq. ( 24) with supersaturation depletion.Due to cooling in the parcel, RHW increases in the ascending branch and reaches at w = 4 cm s −1 a maximum of 97.7 % at t ∼ 35 min, then begins to decrease (Fig. 1a).
The critical or threshold water and ice supersaturations s w,cr and s i,cr can be defined as the points where the nucleation rates become significant and the crystal concentrations    reach some threshold values, e.g., N c ∼ 10 −3 l −1 (a more rigorous quantitative description is given in Khvorostyanov and Curry, 2009a).The water and ice supersaturation pass in the ascending branch the first critical values of s w,cr1 = −4.2% and s i,cr1 = 42 % at about t ≈ 22 min, reach maxima of −2.45 % and 46 % respectively at t = 33.67 min, then de-crease in the descending branch to the second critical values reached at about t = 40 min (Fig. 2a).Note that the change in ice supersaturation s i = s i,max − s i,cr1 ≈ 4 %, or s i /s i,max is less than 10 %.Thus it can be assumed that nucleation occurs at almost constant ice supersaturation.
Noticeable ice nucleation with w = 4 cm s −1 begins after the first critical point s w,cr1 at t ≈ 22 min (Fig. 1d, e, f).At the time of maximum RHW and s w , the crystal critical radius and energy reach minima of 1.36×10 −7 cm and 1.38×10 −12 erg respectively (Fig. 1b, c), while the nucleation rate per particle (J f,hom r 3 h , with r h = 0.11 µm) and the polydisperse nucleation rate R f,hom reach maxima of 4.90 × 10 −6 s −1 and 4.93 × 10 −4 cm −3 s −1 (Fig. 1d, e).The values of r cr and F cr are substantially greater, while J f,hom r 3 h and R f,hom are smaller at the later times, although the temperature continues to decrease.This illustrates an important key role of humidity in ice nucleation.
In contrast to drop activation, the ice nucleation process continues after t m along the descending branch until the point when the second critical values s w,cr and s i,cr are reached (this process has been mostly disregarded in previous parameterizations of ice nucleation.)The entire nucleation process takes 15-20 min with w = 4 cm s −1 , and the final crystal concentration is 66 l −1 (Fig. 1f).The crystal mean radius grows to 43 µm by t = 1 h, the ice water content (IWC) increases to 0.044 g m −3 and the supersaturation relaxation time τ fc decreases from more than 3 h at the beginning of nucleation to 17 min by the end of simulation.This indicates that deposition of the vapor is not instantaneous but a significant amount of vapor is deposited over a period of hours.
For quantitative illustration, it is convenient to introduce the two quantities, vapor excess, M v , and the relative amount, or percentage of condensed ice, Fr con , These quantities characterize the mass of uncondensed ice and the fraction of condensed ice.In a bulk model with instantaneous condensation and deposition, M v = 0, and Fr con = 100 %, but it is not so in this microphysical model with explicit calculation of supersaturation.Fig. 2f shows that the vapor excess is greater or comparable to IWC and the fraction of condensed ice is less than 50 % during 30 min.This means that optical thickness and emissivity of cirrus clouds at the initial stages of their formation are significantly smaller than predicted in a bulk model.The corresponding curves for the case with w = 20 cm s −1 (solid circles in Figs. 1 and 2) show much faster nucleation, about 5 min.The other features of the nucleation process are qualitatively similar, with some quantitative differences.The minimum critical radius and energy are somewhat smaller, the nucleation rates increase by almost two orders of magnitude, and the final crystal concentration increases to 649 l −1 , almost 10 times greater than with w = 4 cm s −1 .Because of more numerous crystals and their competition for vapor, the mean crystal radius is smaller than with w = 4 cm s −1 , but the relaxation time τ fc is also smaller with a minimum of 2.6 min.The deposition is faster with w = 20 cm s −1 , but the vapor excess and fraction of condensed ice are still smaller for 15-20 min that would be in a bulk model with instantaneous deposition (Fig. 2e, f).
A comparison of the results with N a = 200 cm −3 and 500 cm −3 at T 0 = −40 • C, w = 4 cm s −1 is shown in Figs. 3  and 4; all other parameters are as before.This comparison shows that a significant increase in N a causes very weak effect on nucleation kinetics and all the resulting quantities.Nucleation with higher N a begins and ceases a little earlier, and the resulting crystal concentration is 68.6 l −1 vs. 66 l −1 with N a = 200 cm −3 ; that is, an increase 2.5 times in N a causes and increase of only 4 % in N c .This remarkable insensitivity to the initial concentration of deliquescent freezing aerosol indicates a kind of "saturation" with respect to N a at values of N a much smaller than these values typical for the upper troposphere.
The fraction of nucleated haze particles (the ratio N c /N a ), is tiny (66 l −1 )/(200 000 l −1 ) = 3.3 × 10 −4 , which is much smaller than the typical fraction of CCN activated into the drops, ∼ 0.3-0.7.This very small fraction of freezing solution particles is explained by the following factors: (a) very strong negative feedback by the water supersaturation: even a small decrease in s w causes a significant decrease in the nucleation rate J f,hom ,; and (b) much faster crystal growth at high ice supersaturation than drop growth at small water supersaturation.
The effect of temperature is illustrated in Figs. 5 and  6, where a comparison is made for the cases −40 • C and −60 • C, at w = 4 cm s −1 , and all other parameters as before.The critical and maximum water supersaturations (negative) decrease and ice supersaturations increase with decreasing temperature.Minimum critical radius and energy are comparable at both temperatures, while the nucleation rates grow 4-7 times at lower T .The crystal concentration increases almost 4 times to 242 l −1 at lower T (Fig. 5f), but crystal growth is slower; therefore the mean radius is about 4 times smaller and the fraction of condensed ice is lower by the end of simulation at t = 1 h, and the supersaturation relaxation times are close, ∼ 15-17 min, since increase in crystal concentration is balanced by decrease in the mean radius (Fig. 6).Thus, the amount of condensed ice is again smaller than would be in a bulk model.Some properties of the nucleation rates allow simplification of the nucleation equations.The nucleation rates are very small at all stages of the process, J f,hom r 3 h <10 −5 -10 −4 s −1 and R f,hom < 10 −3 -10 −1 cm −3 s −1 even at their maxima; see Figs. 1d, e, 3d, e, 5d, e. Therefore Eqs.(18b-d) for homogeneous nucleation rate can be substantially simplified since exp The probability P f,hom (r a , t) (Eq.18b) of homogeneous freezing of a haze particle or a drop with radius r a and volume v(r a ) during the time interval from t 0 to t can be simpli- Equation (18c) for the crystal concentration N c,hom in a polydisperse aerosol can be simplified as  The crystal nucleation rate R f,hom (Eq.18d) in a polydisperse aerosol can be simplified and is obtained by differentiating of Eq. ( 29) by t:

Parameterization of homogeneous ice nucleation kinetics
In this section, a new parameterization of homogeneous ice nucleation kinetics is derived, based on extended classical nucleation theory and analytical solutions of the supersaturation equation.

General features of homogeneous ice nucleation kinetics
The general features of homogeneous ice nucleation kinetics are illustrated in more detail in Fig. 7.The symbols t cr,1 and t cr,2 denote the 1st and 2nd times when the critical (threshold) ice supersaturations s i,cr1 and s i,cr2 are reached (marked with ellipses), that is, the start and end of nucleation; t max is the time when maximum ice and water supersaturations, s i,max and s w,max , are reached.Figure 7 shows that homogeneous ice nucleation has features that are both similar and different from drop nucleation.In both cases, supersaturation increases due to cooling by the updraft, but in contrast to drop activation, ice nucleation begins at water subsaturations of a few percent at time t cr,1 , when a critical ice supersaturation s i,cr,1 is reached.The s i (t) and s w (t) curves consist of two branches with increasing and decreasing supersaturations.However, in contrast to the drop activation, nucleation does not cease at t max , when maximum s i,max and s w,max are reached.Only about half of the final crystal concentration has been nucleated by this time (the ellipse in Fig. 7b), and nucleation continues along the branch with decreasing supersaturation to the point t cr,2 , s i,cr2 when s i (t) again intersects the line s i,cr (t).It is seen that an increase in both s w and s i is linear almost to the maximum, and both s w and s i can be well approximated with linear functions.
The basic equations describing kinetics of homogeneous ice nucleation include the integro-differential equations for water and ice supersaturations derived in Sect.2, and the equation for crystal radius growth rate with account for kinetic effects.In addition, we need an equation for homogeneous nucleation rate of haze particles with account for solution effects, an equation for the critical supersaturation s w,cr , and equations for the critical radius and energy of homogeneous nucleation.

Freezing rate
The equation for the critical water supersaturation s w = S w − 1 was derived in Khvorostyanov and Curry (2009a, hereafter KC09a) based on the extension of classical nucleation theory where is the specific melting heat averaged over temperature, R is the universal gas constant, M w is the molecular weight of water, H v,fr and H f,hom are functions of the melting heat, water and ice densities, external pressure and surface tension (KC09a).The last approximate equality in Eq. ( 31) is written neglecting effects of external pressure (small for this case), and for very slow nucleation rates (see KC09a).The corresponding ice saturation ratio S i and supersaturation s i can be obtained using standard relations between s w and s i .
The polydisperse freezing rate R f,hom = dN c (t 0 )/dt 0 can be calculated using classical nucleation theory as described by Eq. (18d).It was illustrated in Figs. 1 and 2 that at typical cooling rates (w), the inner integral in the exponent of Eq. ( 18d) is close to 1. Therefore, Eq. ( 30) can be used as a good approximation for R f,hom : This expression can be further simplified if the depletion of v(r a ) and f a (r a ) are small during freezing, which is usually a good approximation with abundant concentrations of freezing particles where va is the mean aerosol volume averaged over the haze size spectrum In general, N a and va vary with time; however, the fraction of haze particles nucleated into crystals is very small compared to the initial haze population.Therefore, I dep in Eq. ( 23) can be further simplified assuming N a ≈ const, va ≈ const.

Separation of the temperature and supersaturation dependencies
The nucleation rate J f,hom (T , s w ) can be calculated using classical nucleation theory (CNT) (PK97) where ρ w and ρ i are the densities of water and ice, σ is is the surface tension at the solution-ice interface, F act and F cr are the activation and critical energies of an ice germ freezing, N cont is the number of molecules in contact with a unit area of ice surface, k and h are the Boltzmann's and Planck's constants.In CNT, the energy F act is a function of temperature; F cr is a function of the critical germ radius r cr , which is also a function of T (PK97).More general analytical expressions for r cr (T , S w , r d , p) and F cr (T , S w , r d , p) were derived in Khvorostyanov and Sassen (1998a), Khvorostyanov andCurry (2000, 2004a, b).Here we use a somewhat simpler expression from KS98a, KC00, KC04a,b with account for T and S w ,  find a representation of J f,hom with separated T -and S w or s w -dependencies.Here, we express F cr via water supersaturation s w using Eq.(37a) and the relation S w = 1 + s w , then This equation for F cr can be transformed so that the dependencies of T and s w are separated, following KC04b.It s w,max s i,max Fig. 7. General features of homogeneous ice nucleation kinetics (evolution of water and ice supersaturations and crystal concentration) illustrated with a parcel model run with the parameters: initial temperature T c = -40 °C, s w (t = 0) = -0.1 (-10 %), lognormal size spectrum of haze particles with mean geometric radius of 0.02 µm and concentration N a = 200 cm -3 .The symbols t cr,1 and t cr,2 (marked with ellipses) denote the 1st and 2nd times when critical (threshold) ice supersaturations s i,cr1 and s i,cr2 are reached, that is, the start and end of nucleation; t max is the time when maximum ice and water supersaturations s i,max , s w,max are reached; s w,cr denote the curves of critical (threshold) water and ice supersaturations.The point denoted by ellipse on the curve N c (t) with the symbols s i,max , s w,max is the point at t max where maximum s w and s i are reached.The curve N c (t) above this point corresponds to ice nucleation at t max < t < t cr,2 .Fig. 7. General features of homogeneous ice nucleation kinetics (evolution of water and ice supersaturations and crystal concentration) illustrated with a parcel model run with the parameters: initial temperature T c = −40 • C, s w (t = 0) = −0.1 (−10 %), lognormal size spectrum of haze particles with mean geometric radius of 0.02 µm and concentration N a = 200 cm −3 .The symbols t cr,1 and t cr,2 (marked with ellipses) denote the 1st and 2nd times when critical (threshold) ice supersaturations s i,cr1 and s i,cr2 are reached, that is, the start and end of nucleation; t max is the time when maximum ice and water supersaturations s i,max , s w,max are reached; s w,cr denote the curves of critical (threshold) water and ice supersaturations.
was found from observations and model simulations that homogeneous freezing of haze droplets in cirrus clouds usually occurs at small water subsaturations of −2 % to −10 %, i.e., s w = −2 × 10 −2 to −10 × 10 −2 , so that |s w | 1 (see e.g., Figs.1-7 here; Sassen and Dodd, 1989;Lin et al., 2002).Since |s w | 1, we can expand the denominator in Eq. (37b) into a power series in s w .The logarithmic term can be transformed as ln where we used a relation ln(1 + G n s w ) ≈ G n s w for |s w | 1 and G n ∼ 0.4-0.6.Substituting this expansion into Eq.(37b), we obtain where That is, F cr,0 is the critical energy for pure water defined by Eq. (37b) but at S w = 1 or s w = 0, i.e., it depends only on temperature but does not depend on supersaturation.For T ∼ −50 • C, G n ∼ 0.5, and κ s ∼ 5, then with s w = −3 × 10 −2 (−3 %), the term κ s s w ∼ −0.15 1.The second order term in expansion by κ s s w in Eq. ( 39) contributes ∼ 3.5 %; therefore, retaining only the first term in Eq. ( 39) is justified.Substitution of Eq. ( 39) into Eq.( 36) yields so that J f,hom can be written such that the s w -dependence is presented in the exponential or power law forms, similar to those derived in KC04b for heterogeneous nucleation.The parameters u s and b hom are where k is the Boltzmann constant, N Av is the Avogadro number, and f,hom is defined by Eq. ( 36) with F cr,0 (T ) from Eq. (40b), i.e., at s w = 0. Thus, J f,hom (T , s w ) is presented in a separable form as a product of the two factors: J (0) f,hom (T , s w = 0) depends on T but does not depend on s w , and the dependence on s w is separated into the exponent in Eqs.(41a), ( 42a).An estimate shows that at cirrus conditions u s ∼ (2-4) × 10 2 1.Since s w <0 in the nucleation process, the value of u s s w is negative.If s w ∼ −(4 to 10) × 10 −2 , at typical nucleation conditions, the value of |u s s w | ≥ 10, and we have an inequality exp(u s s w ) 1.
ssw) that determines this ratio.Calculations for the same conditions as in Fig. 7.It is seen that Jf,hom/J (0) f,hom is very close to exp(usw), which is a good approximation to this ratio.f,hom (T , s w = 0), their ratio J f,hom (T , s w )/J (0) f,hom (T , s w = 0), and exp(u s s w ) that determines this ratio.Calculations for the same conditions as in Fig. 7.It is seen that J f,hom /J ( 0) f,hom is very close to exp(us w ), which is a good approximation to this ratio.
Numerical simulation with the parcel model shows that changes in J (0) f,hom in Eq. ( 41a) are several orders of magnitude smaller than variations in exp(u s s w ).This is illustrated in Fig. 8, which shows that J (0) f,hom (T , s w = 0) ∼ (4-5) × 10 5 cm −3 s −1 and only varies slightly during the nucleation event, while J f,hom (T , s w ) varies (decreases from maximum) by 10 orders of magnitude during nucleation.This is caused by the effect of exp(u s s w ), which reaches a maximum ∼ 10 −5 at t = 34.5 min, the time of maximum of s w .Figure 8 shows that the ratio J f,hom (T , s w )/J (0) f,hom (T , s w = 0) is very close to exp(u s s w ), confirming the validity of the analytical separability of T and s w in Eqs.(41a, b).Further, the primary variations in J f,hom (T , s w ) occur due to variations in s w , while changes due to the temperature are several orders smaller.Therefore, the deposition integral I dep in Eq. ( 35) can be presented in a form that substantially simplifies calculations or introducing the integral J 0i as

Evaluation of nucleation rate and crystal concentration
We seek a solution to the supersaturation equation, similar to that used in the parameterizations of drop activation (e.g., Twomey, 1959;Sedunov, 1974;Khvorostyanov andCurry, 2008, 2009b;Ghan et al., 2012;Tao et al., 2012), as a linear approximation but with the initial critical (threshold) values.
The initial values are zero for drop activation but are equal to some nonzero critical values s w,cr , s i,cr with account for the specifics of ice nucleation as illustrated in the previous figures s w (t)=y w (t)=s w,cr +a 1w t, y w (t)=s w,cr t+(a 1w /2)t 2 .(45) The integral supersaturations y w and y i are written assuming for simplicity that the time is counted from the moment t 0 = t cr when s w,cr and s i,cr are reached, then according to Eq. ( 16) the initial time t 0 = t cr = 0.The parameters a 1w and a 1i can be specified in various ways, which yield the lower and upper limits of the solution similar to drop activation.An approximation that gives a lower bound of the solution can be obtained with a 1w = c 1w w.The difference between the limits is on the order of 10-15 % or smaller, and we for simplicity will consider the approximations a 1w = c 1w w, and a 1i = c 1i w, as prompted by the Eqs.( 8a), (8b), ( 24), and (44), (45).Figures 2 and 7 show that the increase s i = c 1i w(t max − t 0 ) ∼ 0.04 (4 %) during ice nucleation from t 0 to t max is much smaller than the initial critical s i,cr ∼ 0.42 (42 %) or maximum s i,max ∼ 0.46 (46 %).Since s i s i,cr , we can neglect the increase s i of s i in Eq. ( 44) during a nucleation event, which was also neglected by Kärcher and Lohmann (2002a, b), and Ren and MacKenzie (2005).We also assume that s i (t) ≈ const ≈ s i,cr .In contrast, we cannot neglect the term s w = c 1w w(t max − t 0 ) because water supersaturation varies substantially and determines variations in J f,hom (Fig. 7).Thus, assuming again t 0 = t cr = 0, s w (t)=y w (t)=s w,cr +c 1w wt, y w (t)=s w,cr t+(c 1w w/2)t 2 .(46b) Substitution of s w (t) into the separable nucleation rate in Eq. (41a) yields J f,hom (T , s w ) as a function of time in the form f,hom (T cr ) exp(u s s w,cr ) exp(u s c 1w wt), (47a) where u s is defined in (42a).We assume here, based on Fig. 8, that the major time dependence is determined by s w , and the temperature dependence is determined near T cr .Dividing J f,hom (t) by J f,hom (t 0 ) at some initial t 0 , we obtain the time dependence J f,hom (t) of the form For t 0 = t cr , Eq. ( 47b) can be rewritten with Eq. (46b) as Using the relation following from the Clausius-Clapeyron equation where R v is the vapor gas constant and T 0 = 273.15,we express s w in Eq. (47c) via the ice saturation ratio S i = s i + 1 and obtain This expression has the same form as Eq. ( 6) hypothesized by Barahona and Nenes (2008), and their coefficient b τ fitted with empirical data is expressed now from the extended classical nucleation theory as b τ (T ) = u s (T )c iw (T ).Equation (47b) can be also rewritten as where we introduced the characteristic "nucleation time" τ nuc where k is the Boltzmann constant and N Av is the Avogadro number.The temporal dependence of J f,hom (t) as in Eq. ( 48) was hypothesized by Ford (1998a, b), Kärcher and Lohmann (2002a, b) and Ren and MacKenzie (2005) and the time τ nuc was found by fitting to some auxiliary relations Eqs.(3), (5) above.Here, the time dependence of J f,hom (t) and the time τ nuc are derived in terms of the extended classical nucleation theory with the dependence on S w .Equation (49) shows that τ −1 nuc ∼ c 1w w, that is, according to Eq. ( 5), is proportional to (dT /dt), in agreement with Eq. ( 5), the other factors in Eq. (49) determine ∂lnJ hom /∂T and the empirical coefficient c τ in Eq. ( 5).Thus, the approach based on extended CNT confirms the functional forms hypothesized in the previous parameterizations by Ford (1998a, b), Kärcher and Lohmann (2002a, b), Ren and MacKenzie (2005), Barahona and Nenes (2008), and allows to express them via the fundamental thermodynamic parameters reducing the number of hypothesized relations and quantities.
The linear approximation Eq. (46b) for s w (t) allows description of the time evolution of the nucleation rate R f,hom (t) and crystal concentration N c (t). Substitution of Eq. (47a) into Eq.( 33) yields Integration over time assuming t 0 = t cr = 0 gives N c (t) This is the parameterization for N c (t) that we searched for.The dependencies of N c on s w and T are separated in Eq. ( 52a), this allows to introduce the activity spectra ϕ s (T , s w ) and ϕ T (T , s w ) by s w and T defined in Eqs. ( 17), (18a).Differentiation of Eq. ( 52a) by s w and T yields The activity spectrum ϕ s (T , s w ) characterizes the rate of ice nucleation with increasing humidity and constant temperature, (similar to considered for drop activation), the spectrum ϕ T (T , s w ), vice versa, characterizes the rate of ice nucleation with decreasing temperature and constant humidity.Such processes may occur under natural conditions of cirrus clouds formation with advection of humid air and weak variations of T , or with advection of cold air and weak changes of humidity.Using Eqs.(52b, c), the relative role of variations of the temperature and humidity can be estimated, or these processes can be studied in isolation in a cloud chamber.
The relation between β and t in Eq. (52a) determines the regime of growth of N c with time.For example, at T = −40 • C with u s ∼ 250, c 1w ∼ 10 −5 cm −1 , and w ∼ 10 cm s −1 , an estimate gives β ∼ 2.5 × 10 −2 s −1 and τ nuc = β −1 ∼ 40 s.Thus, for small times, t β −1 ∼ 40 s, yielding a linear growth of N c (t) with time For large times, t β −1 = 40 s, we obtain from Eq. (52a) an exponential time dependence It is interesting to note that Eq. ( 54) for homogeneous nucleation can be presented in the form similar to the empirical The function defined by Eqs. ( 64)-( 67) can be transformed and reduced to the functions more convenient for calculations.Using the recurrent relation for (3/2, x) (Gradshteyn and Ryzhik, 1994, see Appendix A) and the relation between gamma function and error function erf(x), we can transform the gamma function in 1 as Substituting this relation into Eqs.( 65), ( 67) we can rewrite 1 and 3 with use of only erf(x) = (x) and without gamma function, which is more convenient for applications Then the function is expressed with use of only (x) = erf(x): This expression can be further simplified by expressing the transcendent function erf(x) via the elementary function tanh following Ghan et al. (1993) erf Then becomes: +2ξ dep β −1 (e −βt − 1).Now, the deposition integral I dep in Eq. ( 69) is expressed only via the elementary functions.Another transition to the elementary functions can be done using equations for erf(x) given in Ren andMacKenzie (2005, 2007).In the next sections, the solutions of equations for supersaturation and crystal concentration will be expressed via .Although these expressions may look complicated, the analytical representation Eqs. ( 74), ( 76) reduce unavoidable errors caused by finite difference representations and numerical calculations and enables the derivation of simple asymptotic limits of I dep and N c for the diffusion and kinetic regimes of crystal growth as shown below.

Solution of equations for supersaturation and crystal concentration
Substituting Eq. ( 69) for I dep into the integral supersaturation equation Eq. ( 21), multiplying it by (1 + y i ) and using the relation At t = t max with maximum supersaturations s i,max and s w,max , the condition ds i /dt = dy i /dt = 0 is satisfied, thus, the LHS of Eq. ( 77) is zero, which yields exp[u s s w,max (t max )] Now we can rewrite Eq. ( 33) for R f,hom (t) with account for J f,hom from Eq. (41a) as The crystal concentration at the time t is obtained by integrating over t 0 The last equation accounts for the fact that (t max − t cr,1 ) β −1 or β(t max −t cr,1 ) 1 according to Eq. ( 54).Substituting exp[u s s w,max (t max )] from Eq. ( 78) and using the approximate equality s i,max ≈ s i,cr due to small variations of s i during nucleation as discussed above, we obtain finally an analytical parameterization of the concentration of the crystals in homogeneous freezing nucleation: ) after the cease of nucleation at t>t cr,2 .Evaluation of the 2nd stage at t>t max with decreasing supersaturation in principle can be done in a similar way as for t<t max , although it is somewhat more complicated.To simplify the solution, we can use the solutions for t = t max and slightly tune them using the results of the parcel model runs.Their detailed analysis shows that the total N c,tot (t cr,2 ) at t>t cr,2 , when nucleation has ceased, is proportional to N c (t max ); that is, N c,tot can be obtained as Numerical experiments with the parcel model show that K cor ∼ 1.8 to 2.2 (Fig. 7).A more precise fit shows that this coefficient can be chosen as a function of the vertical velocity w as and w sc = 2 m s −1 .Even a simpler choice of the average is K cor ∼ 2, which accounts for about half of the crystals nucleating at decreasing supersaturation at t max <t<t cr,2 , still gives satisfactory results.

Limiting cases
The important asymptotics can be obtained by analysis of the characteristic parameters of the solution Eqs. ( 81), ( 82) with from Eq. ( 74).The parameter λ in Eq. ( 68) can be rewritten in the form Here is a scaling length that characterizes the ratio of the crystal growth rate Eq. ( 12) to the supersaturation generation rate (the first term on the RHS of Eq. 77).Now we present asymptotics of the solution Eq. ( 81) at λ 1 and λ 1.The values of λ and and the physical meaning of the asymptotic limits are analyzed below.

Diffusion growth limit
The values λ 1 in Eq. ( 86) imply small ξ dep and r 0 , and are typical of the diffusion regime of crystal growth with the deposition coefficient α d ∼ 1 or α d >0.1 with not very large w and not very low T .In this case, we can neglect in Eq. ( 74) for all terms with ξ dep and r 0 .Note that erf(λ 1/2 ) → 0 at λ 1 → 0 according to (A29).Using the estimates above, we can assume that βt max 1, use the expansion (A27) for erf[(λ + βt max ) 1/2 ) and neglect the terms with exp(−βt max ).
Then is simplified in this diffusion regime as Substitution of this expression into Eqs.( 81), (82) yields The properties of this solution are discussed below and compared with the other limits.

Kinetic growth, small and large particles limits
The limit λ 1 is seen from Eq. ( 86) to be associated with the kinetic regime with large ξ dep (small α d ) or with large initial particle radius r 0 of freezing particles.It can be studied using the asymptotic property of erf(x) at x 1 (Appendix A, Eq.A27) Expanding in Eq. ( 74) for the functions erf( √ λ) and erf( √ λ + βt max ) with Eq. ( 90), neglecting again the terms with exp(−βt max ) and the terms λ −3/2 compared to λ −1/2 , and collecting the terms of the same order, can be written as This case is divided into 2 subcases: (a) when ξ dep is large (small deposition coefficient α d ) but r 0 is small (small particles limit), that is, ξ dep r 0 ; and (b) when r 0 is large (large particles limit); that is, ξ dep r 0 , which may correspond to both diffusion or kinetic regimes.These limits are considered below.
(a) λ 1, ξ dep r 0 , kinetic regime, small particles limit With these conditions, r 0 can be neglected compared to ξ dep , and Eq. ( 91) for is simplified Thus, in this limit N cm ∼ w 2 , in agreement with Ren and MacKenzie (2005), but all coefficients are expressed now without empirical constants and N cm ∼ ρ −1 is (T ).Note also that the crystal concentration is inversely proportional to the deposition coefficient, N cm ∼ α −1 d ; that is, the smaller α d or the more polluted clouds, the greater nucleated crystal concentration.Gierens et al. (2003) discussed possible reasons for α d as small as 10 −3 ; in these cases, the dependence 1/α d can be significant.This is in agreement with the data from the INCA field experiment (Ovarlez et al., 2002;Ström et al., 2003;Haag et al., 2003;Gayet et al., 2004;Monier et al., 2006) that found greater ice crystal concentrations in cirrus in the more polluted Northern Hemisphere than in the cleaner Southern Hemisphere.This could be caused not only by the heterogeneous ice nucleation mode, but also by a small deposition coefficient in homogeneous nucleation in polluted areas.
(b) Initial r 0 is large and r 0 ξ dep , large particles limit Neglecting ξ dep compared to r 0 , Eq. ( 91) can be further transformed kin,l = The last equality takes into account that λ 1, so the first term in the parentheses is much smaller than the second and can be neglected.Substituting this kin,l into the general solution Eq. ( 81), we obtain That is, the dependence on w is linear, N cm ∼ w.This linear w-dependence is in agreement with predictions in Kärcher and Lohmann (2002a, b) and in Ren and MacKenzie (2005).The term ρ is (T ) is absent; thus the temperature dependence is much weaker than in the previous cases, and is caused by the T -dependence of D v , c 1i , and s i,cr .

Physical interpretation
Two examples of calculations using this new parameterization are shown in Fig. 9 and Fig. 10.The crystal concentrations N c (w) calculated in the diffusion approximation with the new Eqs.( 87)-( 89) and α d = 1 (denoted KC2012) for an air parcel ascending with a vertical velocity w is shown in Fig. 9.The applicability of the diffusion approximation is justified by the small λ ∼ 10 −3 to 0.03 with α d = 1 for all w.It is compared with the parameterizations by Sassen and Benson (2000; SB2000, to w = 1 m s −1 ), Liu and Penner (2005; LP2005), Kärcher and Lohmann (2002;KL2002).Also shown here are the results of several parcel model simulations from Lin et al. (2002) according to the protocols of CPMCP for the three values of w = 4, 20 and 100 cm s −1 .Simulations were performed by Cotton, DeMott, Jensen, Kärcher, Lin, Sassen, and Liu as indicated in Fig. 9 (the models are described in Spice et al., 1999;DeMott et al., 1994;Jensen et al., 1994;Kärcher and Lohmann, 2002a, b;Lin, 1997;Sassen andDodd, 1988, andKhvorostyanov andSassen, 1998a;Liu and Penner, 2005); the results of parcel simulations from Khvorostyanov and Curry (2005) are added (KC2005).This figure shows that the new parameterization KC2012 lies within the spread of the parcel models results, being closer to the lower limit, and to the parcel simulations by Jensen who used a model with spectral microphysics and explicit supersaturation (Jensen et al., 1994).KC2012 is in qualitative agreement with Sassen and Benson (2000) at small w and is especially close to the parameterization by Kärcher and Lohmann (2002a, b), although it was based on a substantially different approach.This supports the validity of the new parameterization based on an extension of the classical nucleation theory and shows that semi-empirical approaches lead to results that can be derived from the extended classical nucleation theory.
Figure 10 shows a comparison of the full solution Eqs. ( 81)-(85) with the diffusion limit Eqs. ( 87)-( 89) at α d = 1 and the kinetic limit Eqs. ( 92)-( 94) at α d = 0.04, 0.01 and 0.001.The diffusion approximation (solid circles) is valid at λ 1, and limited at w ≤ 170 cm s −1 ; the kinetic limit is valid at λ 1 and with α d = 0.04 is limited at w>30 cm s −1 .This figure illustrates good accuracy of the two approximations for corresponding values λ and underscores the important role of the deposition coefficient.With small α d , such as in polluted clouds, the crystal concentrations are substantially higher than with α d = 1 for clean clouds.So, polluted crystalline clouds should have a substantially greater albedo effect and this parameterization provides a quantitative tool for its estimation.
We identify three different regimes of crystal homogeneous nucleation in cold clouds, depending on the cooling time of an air parcel.At small times, t τ nuc (∼ 40 s), the crystal concentrations increase linearly with time and proportional to the concentration of the freezing haze particles N a .At larger times, t τ nuc , but smaller than the time t max of maximum supersaturation in the parcel, N c increases exponentially with time.Crystal concentrations in these two regimes are proportional to the homogeneous nucleation rate and concentration of the aerosol particles.If uplift of an isolated parcel continues so that t>t max and t>t cr2 , the super-saturation reaches and passes a maximum and falls below the threshold value, then a third regime occurs that can be called limiting regime.The dependence on the nucleation rate and haze concentration vanishes in this regime, although concentration of nucleated crystals is much smaller than the concentration of haze particles.
Expressions for the crystal concentration N c in the third limiting regime are very simple, and somewhat surprising.They do not include most of the basic factors present in the original supersaturation equation: neither nucleation rate J hom (T , s w ) nor concentration N a of the haze particle, nor any characteristics of volume or size spectra or chemical composition.The reason why N c does not depend on N a can be explained by the fact that N c is usually on the order of a few or a few tens per liter (rarely, a few hundred), while N a is typically on the order of a few hundred per cubic centimeter.That is, only very small fraction of haze particles freezes, and the dependence of N c on N a vanishes at values of N a much smaller than those available in the upper troposphere studied here.However, if N a is small, N c is limited by N a .
The major factors that govern homogeneous ice nucleation in the third limiting regime are the vertical velocity, w, the temperature, T , and the critical (threshold) saturation ratio s i,cr .The equations for N c derived here show that to first approximation in the diffusion limit, N c ∼ w 3/2 , and N c ∼ ρ −1/2 si (T ), both dependencies are the same as in Kärcher and Lohmann (2002a, b) and in Ren and MacKenzie (2005) in the diffusion growth limit.However, the actual dependence of N c on w and T is more complicated and somewhat different since s −3/2 i,cr also includes dependence on w and T , and the critical supersaturation s i,cr also depends on T and substantially grows toward low T ; the coefficient K i,dif depends on T also via factors D v , c 1w , c 1i , u s .In the kinetic growth or large particle limits, N c can be proportional to w 2 or to w, depending on the initial particle radius, in agreement with the previous semi-empirical parameterizations.
where G n ∼ 0.4-0.6 with relatively weak T -dependence (KC09a).Analytical solution of the supersaturation equation requires some simplifications; in particular, it is desirable to
N c at time t max with maximum supersaturation, i.e., at the end of the 1st stage with growing s i .Some previous parameterizations assumed that N c (t max ) at the time t max of maximum supersaturations is the final crystal concentration.However, as we have seen in Figs.2, 4, 6, 7, during the descending branch at t max <t<t cr,2 , s w (t) decreases but still exceeds s w,cr , therefore nucleation continues after t max until t cr,2 , and N c (t max ) is approximately half the total N c,tot (t cr,2