This work presents a new physically based parameterization of cirrus cloud formation for use in large-scale models which is robust, computationally efficient, and links chemical effects (e.g., water activity and water vapor deposition effects) with ice formation via homogenous freezing. The parameterization formulation is based on ascending parcel theory and provides expressions for the ice crystal size distribution and the crystal number concentration, explicitly considering the effects of aerosol size and number, updraft velocity, and deposition coefficient. The parameterization is evaluated against a detailed numerical cirrus cloud parcel model (developed during this study), the equations of which are solved using a novel Lagrangian particle-tracking scheme. Over a broad range of cirrus-forming conditions, the parameterization reproduces the results of the parcel model within a factor of 2 and with an average relative error of −15%. If numerical model simulations are used to constrain the parameterization, error further decreases to 1 ± 28%.
 The effect of aerosols on clouds and climate is one of the major uncertainties in anthropogenic climate change assessment and prediction [Intergovernmental Panel on Climate Change (IPCC), 2007]. Cirrus is one of the most poorly understood cloud types, yet they can strongly impact climate. Cirrus are thought to have a net warming effect because of their low emission temperatures and small thickness [Liou, 1986]. They also play a role in regulating the ocean temperature [Ramanathan and Collins, 1991] and maintaining the water vapor budget of the upper troposphere and lower stratosphere [Hartmann et al., 2001]. Concerns have been raised on the effect of aircraft emissions [Penner et al., 1999; Minnis, 2004; Stuber et al., 2006; IPCC, 2007] and long-range transport of pollution [Fridlind et al., 2004] changing the properties of upper tropospheric clouds, that is, cirrus and anvils, placing this type of clouds in the potentially warming components of the climate system.
 Cirrus clouds form by the homogenous freezing of liquid droplets, by heterogeneous nucleation of ice on ice nuclei, and the subsequent growth of ice crystals [Pruppacher and Klett, 1997]. This process is influenced by the physicochemical properties of the aerosol particles (i.e., size distribution, composition, water solubility, surface tension, and shape), as well as by the thermodynamical state (i.e., relative humidity, pressure, and temperature) of their surroundings. Dynamic variability (i.e., fluctuations in updraft velocity) also impact the formation of cirrus clouds, potentially enhancing the concentration of small crystals [Lin et al., 1998; Kärcher and Ström, 2003; Hoyle et al., 2005].
 The formation of cirrus clouds is modeled by solving the mass and energy balances in an ascending (cooling) cloud parcel [e.g., Pruppacher and Klett, 1997]. Although models solve the same equations (described in section 2.1), assumptions about aerosol size and composition, J calculation, deposition coefficient, and numerical integration procedure strongly impact simulations. This was illustrated during the phase I of the Cirrus Parcel Model Comparison Project [Lin et al., 2002]; for identical initial conditions, seven state-of-the-art models showed variations in the calculation of ice crystal concentration, Nc, (for pure homogeneous freezing cases) up to a factor of 25, which translates to a factor of two difference in the infrared absorption coefficient. Monier et al.  showed that 3 orders of magnitude difference in the value of J, which is typical among models at temperatures above −45°C, will account only for about a factor of two variation in Nc calculation. The remaining variability in Nc results from the numerical scheme used in the integration, the calculation of the water activity inside the liquid droplets at the moment of freezing, and the value of the water vapor deposition coefficient.
 Introducing ice formation microphysics in large-scale simulations requires a physically based link between the ice crystal size distribution, the precursor aerosol, and the parcel thermodynamic state. Empirical correlations derived from observations are available [i.e., Koenig, 1972]; their applicability, however, for the broad range of cirrus formation presented in a GCM simulation is questionable. Numerical simulations have been used to produce prognostic parameterizations for cirrus formation [Sassen and Benson, 2000; Liu and Penner, 2005], which relate Nc to updraft velocity and temperature (the Liu and Penner  parameterization also accounts for the dependency of Nc on the precursor aerosol concentration, and was recently incorporated into the NCAR Community Atmospheric Model (CAM3) [Liu et al., 2007]). Although based on theory, these parameterizations are constrained to the values of parameters (i.e., deposition coefficient, aerosol composition and characteristics) used during the parcel model simulations (the uncertainty of which is quite large). Kärcher and Lohmann [2002b, 2002a] introduced a physically based parameterization solving analytically the parcel model equations. In their approach, a “freezing timescale” is used (related to the cooling rate of the parcel) to obtain an approximated crystal size distribution at the peak saturation ratio through a function describing the temporal shape of the freezing pulse. This function, along with the freezing timescale, should be prescribed (the freezing pulse shape and freezing timescale may also change with the composition and size of the aerosol particles). An analytical fit of the freezing timescale based on Koop et al.  data was provided by Ren and Mackenzie . The parameterization of Kärcher and Lohmann [2002a, 2002b] has been applied in GCM simulations [Lohmann and Kärcher, 2002] and extended to include heterogeneous nucleation and multiple particle types [Kärcher et al., 2006]. All parameterizations developed to date provide limited information on the ice crystal size distribution, which is required for computing the radiative properties of cirrus clouds [Liou, 1986].
 This study presents a new physically based parameterization for ice formation from homogeneous freezing. The parameterization unravels much of the stochastic nature of the cirrus formation process by linking crystal size with the freezing probability, and explicitly considers the effects the deposition coefficient and aerosol size and number, on Nc. With this approach, the requirement of prescribed parameters is relaxed and the size distribution, peak saturation ratio, and ice crystal concentration can be computed. The parameterization is then evaluated against a detailed numerical parcel model (also developed here), which solves the model equations using a novel Lagrangian particle-tracking scheme.
2. Numerical Cirrus Parcel Model
 Homogenous freezing of liquid aerosol droplets is a stochastic process resulting from spontaneous fluctuations of temperature and density within the supercooled liquid phase [Pruppacher and Klett, 1997]. Therefore, only the fraction of frozen particles at some time can be computed (rather than the exact moment of freezing). At anytime during the freezing process, particles of all sizes have a finite probability of freezing; this implies that droplets of the same size and composition will freeze at different times, so freezing of a perfectly monodisperse droplet population will result in a polydisperse crystal population. This conceptual model can be extended to a polydisperse droplet population; each aerosol precursor “class” will form an ice crystal distribution with its own composition and characteristics, which if superimposed, will represent the overall ice distribution. In the following sections, the formulation of a detailed numerical model, taking into account these considerations, is presented. The equations of the model share similar characteristics with those proposed by many authors [Pruppacher and Klett, 1997; Lin et al., 2002, and references therein] as the ascending parcel framework is used for their development.
2.1. Formulation of Equations
 The equations that describe the evolution of ice saturation ratio, Si (defined as the ratio of water vapor pressure to equilibrium vapor pressure over ice), and temperature, T, in an adiabatic parcel, with no initial liquid water present, are [Pruppacher and Klett, 1997]
where ΔHs is the latent heat of sublimation of water, g is the acceleration of gravity, cp is the heat capacity of air, pio is the ice saturation vapor pressure at T [Murphy and Koop, 2005], p is the ambient pressure, V is the updraft velocity, Mw and Ma are the molar masses of water and air, respectively, and R is the universal gas constant. For simplicity, radiative cooling effects have been neglected in equation (2), although in principle they can be readily included. By definition, the ice mixing ratio in the parcel, wi, is given by
where ρi and ρa are the ice and air densities, respectively. Dc is the volume-equivalent diameter of an ice particle (assuming spherical shape), Do is the wet diameter of the freezing liquid aerosol, nc(Dc, Do) = is the ice crystal number distribution function, Nc(Do) is the number density of ice crystals in the parcel formed at Do; Do,min, and Do,max are the limits of the droplet size distribution, and Dc,min andDc,max are the limits of the ice crystal size distribution. Taking the time derivative of equation (3) we obtain
where the term Dc3 was neglected as instantaneous nucleation does not substantially deplete water vapor from the cloudy parcel. The growth term in equation (4) is given by [Pruppacher and Klett, 1997]
where ka is the thermal conductivity of air, Dv is the water vapor diffusion coefficient from the gas to ice phase, Si,eq is the equilibrium ice saturation ratio, and αd is the water vapor deposition coefficient on ice.
 The crystal size distribution, nc(Dc, Do) is calculated by solving the condensation equation [Seinfeld and Pandis, 1998]
subject to the boundary and initial conditions (neglecting any change of volume upon freezing),
where no(Do, t) is the liquid aerosol size distribution function, ψ(Do, t) is the nucleation function (which describes the number concentration of droplets frozen per unit of time), and Pf(Do, t) is the cumulative probability of freezing, given by [Pruppacher and Klett, 1997]
where J is the homogeneous nucleation rate coefficient, being the number of ice germs formed per unit volume of liquid per unit of time [Pruppacher and Klett, 1997].
Equation (7) is a simplified version of the continuous general dynamic equation applied to the ice crystal population [Gelbard and Seinfeld, 1980; Seinfeld and Pandis, 1998], where the nucleation term has been set as a boundary condition to facilitate its solution. This can be done since the size of the ice particles equals the size of the precursor aerosol only at the moment of freezing.
 The evolution of the liquid droplets size distribution, no(Do, t), is calculated using an equation similar to equation (7),
The first term of the right-hand side of equation (11) represents the growth of aerosol liquid particles by condensation of water vapor, and the second term the removal of liquid particles by freezing. Boundary and initial conditions for equation (11) are simply the initial aerosol size distribution and the condition of no particles at zero diameter.
2.2. Numerical Solution of Parcel Model Equations
Equations (1)–(11) are solved numerically using a Lagrangian particle-tracking scheme; this uses a particle-tracking grid for the ice crystal population (the growth of groups of ice crystals is followed after freezing) coupled to a moving grid scheme (the liquid aerosol population is divided into bins the size of which is changing with time), for the liquid aerosol population (Figure 1). For each time step, the number of frozen aerosol particles is calculated using equation (9) and placed in a node of the particle-tracking grid, after which their growth is subsequently followed. This group of ice crystals represents a particular solution of equation (7) for which all particles freeze at the same time, and have the same size and composition. Since a particular solution of equation (7) can be obtained for each time step and droplet size, the general solution of equation (7) is obtained from the superposition of all generated ice crystal populations during the freezing process; wi can then be calculated and equations (1)–(4) readily solved. To describe the evolution of no(Do, t), a moving grid is employed, where frozen particles are removed from each liquid drop size bin in each time step.
 The particle tracking scheme allows ice particles to grow in each node of the particle-tracking grid (no sorting after each time step is needed); thus, the effect of numerical diffusion on the simulation is minimized without losing information about ice crystal compostion. The discretization of equation (7) transforms the partial differential equation into a system of ordinary differential equations, each of which represents the growth of a monodisperse ice crystal population. Thus, simple integration schemes can be used without compromising solution accuracy, although a large number of points may be required (the total number of nodes in the particle-tracking grid is the number of time steps times the number of nodes of the liquid aerosol moving grid). However, the particle-tracking grid size can be substantially reduced by grouping the newly frozen particles in a fewer number of sizes [i.e., Khvorostyanov and Curry, 2005],
where o′ is the assumed size of the frozen particles. If all aerosol particles freeze at the same size, the integral in equation (12) is evaluated over the entire size spectrum of the liquid aerosol population. A further reduction in the size of the particle-tracking grid is achieved by considering that the freezing process occurs after a water vapor saturation threshold is reached [Sassen and Benson, 2000; Kärcher and Lohmann, 2002b]. The initial time step size in the model is set to 2 V−1 s, and reduced to 0.05 V−1 s (with V in m s−1) when ice nucleation starts (J > 104 m−3 s−1) and growth of ice particles needs to be accounted for.
2.3. Baseline Simulations
 The formulation of the parcel model was tested using the baseline protocols of Lin et al. . Pure ice bulk properties were used to calculate the growth terms (equations (5) and (6)). Do was assumed to be the equilibrium size at Si, given by Köhler theory [Pruppacher and Klett, 1997], and solved iteratively using reported solution bulk density and surface tension data [Tabazadeh and Jensen, 1997; Myhre et al., 1998]. This assumption may bias the results of the parcel model simulations at low T and high V [Lin et al., 2002]. Alternatively, the aerosol size can be calculated using explicit growth kinetics, although the water vapor uptake coefficient from the vapor to the liquid phase is uncertain [Lin et al., 2002] (recent measurements indicate a value between 0.4 and 0.7 [Gershenzon et al., 2004]). Although any model can be employed for J, the Koop et al.  parameterization was used owing to its simplicity and its widely accepted accuracy for a broad range of atmospheric conditions [i.e., Abbatt et al., 2006]. The dry aerosol population was assumed to be pure H2SO4, lognormally distributed with geometric mean diameter, Dg,dry = 40 nm, geometric dispersion, σg,dry = 2.3, and total number concentration, No = 200 cm−3. The runs were performed using 20 size bins for the liquid aerosol; the newly frozen particles were grouped into 4 size classes, producing a grid between 1500 and 2000 nodes; numerical results showed that little accuracy was gained by using a finer grid (not shown). Runs of the parcel model using a regular PC (2.2 GHz processor speed and 1 GB of RAM), usually took between 5 and 12 min.
Figure 2 shows results of the performed simulations for the protocols of Lin et al.  and αd = 1. The value of αd is still uncertain and may impact Nc [Lin et al., 2002]. Simulations using αd = 0.1 (not shown) produced Nc (cm−3) of 0.20, 2.87, 24.06, for the cases Ch004, Ch020 and Ch100 (using the nomenclature of Lin et al. ) respectively, and 0.043, 0.535, and 5.98 for the cases Wh004, Wh020 and Wh100, respectively. Results from the INCA campaign summarized by Gayet et al.  indicate Nc ∼ 1.71 cm−3 for T between −43 and −53°C, and Nc ∼ 0.78 cm−3 for T between −53 and −63°C (V was mainly below 1 m s−1, at conditions favorable for homogeneous freezing) [Haag et al., 2003b]. These values are consistent with a low value for αd (around 0.1) which is supported by independent studies [Gierens et al., 2003; Hoyle et al., 2005; Khvorostyanov et al., 2006; Monier et al., 2006]. However, direct comparison of the parcel model with experimental results is a rather simplistic view of the cirrus formation process, and overlooks other important effects (i.e., variation in aerosol characteristics, V and T fluctuations [Kärcher and Ström, 2003; Kärcher and Koop, 2005]). Theoretical calculations and direct experimental observations have reported αd values from 0.03 to 1 at temperatures ranging from 20 to 263 K [i.e., Haynes et al., 1992; Wood et al., 2001]. Owing to these considerations, αd is explicitly introduced in the parameterization.
3. Parameterization of Ice Nucleation and Growth
3.1. Parameterization of nc(Dc, Do)
 The ultimate goal of this study is to develop an approximate analytical solution of equations (5)–(12) to predict number and size of ice crystals as a function of cloud formation conditions. For this, a link should be established between ice particle size and their probability of freezing at the time of nucleation, so that nc(Dc, Do) can be defined at each instant during the freezing pulse. nc(Dc, Do) is determined for a given Si profile by tracing back the growth of a group of ice crystals particles of size Dc down to Do (Figure 3). In the following derivation we assume that most of the crystals are nucleated before maximum saturation ratio, Si,max, is reached (the implications of this assumption are discussed in section 4). We start by writing a solution of equation (7) in the form
where So′ is a value of Si < Si,max at which the ice crystals were formed and Pf(So′) represents the current fraction of crystals of size Dc, that come from liquid aerosol particles of size Do. o(Do) is the average no(Do) during the freezing interval, and is taken to be constant since Nc is usually much less than No [i.e., Lin et al., 2002] and freezing occurs over a very narrow Si range [Kärcher and Lohmann, 2002b]. Since in a monotonically increasing Si, field Pf(So′) decreases with increasing Dc (as explained below), a negative sign is introduced in equation (13).
where α = − , β = . Before the nucleated ice crystals substantially impact saturation (known as “free growth,” Figure 3), ≈ 0, and the integration of equation (14) from So′ to Si,max gives
where the approximation ln(x) ≈ x − 1 has been used. Equation (15) is similar to the “upper bound” expression derived by Twomey  for liquid water clouds. Numerical simulations (section 4) support that “free growth” holds up to Si values very close Si,max; for Si → Si,max however, equation (15) may underestimate t − to and its effect is discussed in section 4.
 By definition, t − to should equal the time for growth of the ice particles from Do to Dc (Figure 3), which is found by integration of equation (5),
where Si has been assumed constant. This is supported by parcel model simulations that suggest that nucleation occurs in a very narrow Si range (i.e., Figure 5). The calculation of Pf(So′) using equation (9) requires the knowledge of J as an explicit function of Si; this can be further simplified given that nucleation occurs on a very narrow interval of saturation so that J(Si) can be approximated with J(Si,max) [i.e., Khvorostyanov and Curry, 2004],
k(T) is obtained by fitting the Koop et al.  data for J between 108 and 1022 m−3 s−1,
where vo = , and the approximation 1 − exp ≈ voJ(t)dt has been used.
 The lower integration limit, So, in equation (19) represents the beginning of the freezing pulse, assumed to be where Pf(Do, So) < 10−6Pf(Do, Si,max), that is, exp[−k(T)(Si,max − So′)] exp[−k(T)(Si,max − So)]. The integration is not very sensitive to the latter assumption, as most of the ice crystals form for Si close to Si,max [i.e., Kärcher and Lohmann, 2002a]. Combining equations (15) and (16) to find Si,max − So′, and replacing into equation (19) gives
where μ = , and Γ(Dc, Do) = (Dc2 − Do2) + Γ2(Dc − Do). The ice number distribution function at Si,max is obtained after computing from equation (20) and substituting into equation (13),
Since ice particles attain large sizes after freezing, the spectrum of Dc values spans over several orders of magnitude [i.e., DeMott et al., 1994; Monier et al., 2006]. Typically variation in Do is much smaller and is further reduced because Pf is significant only for a fraction of the liquid droplets (generally those with larger sizes and low solute concentrations [Pruppacher and Klett, 1997]). With this, the value of exp[−μΓ(Dc, Do)] will be dominated by variation in Dc, so that exp[−μΓ(Dc, Do)] can be approximated with exp[−μΓ(Dc, o)]. Obtaining the crystal size distribution then is done by integration of equation (21) over the contribution from each droplet size class,
which for a lognormal distribution of o(Do) gives
where o = o3 and σg is the geometric dispersion of the droplet size distribution. Equation (23) is the final expression used for the ice crystal size distribution at Si,max. Equations (21)–(23) demonstrate the probabilistic character of ice nucleation: at any time particles of all sizes have finite freezing probabilities; that is, the population of ice crystals of a given size Dc results from the freezing of droplets with different sizes at different times. As a result, the freezing of a monodisperse aerosol size population produces a polydisperse ice crystal population. Since Si increases monotonically with time before reaching Si,max, the number of crystals generally increases as Dc decreases. For a given droplet size, Pf(Do, Si) will increase with time so that the number of newly formed crystals will increase. These crystals in turn will have less time to grow before Si reaches Si,max; in other words, the most recently formed crystals will have the largest probability of freezing, Pf(Dc = o, Si,max). The maximum in the distribution may be shifted to Dc > o if the timescale for the growth of the newly formed particles is larger than the timescale of change of probability (section 4).
3.2. Calculation of Nc at Si,max
 The deposition rate of water vapor upon ice crystals can be approximated by substituting equation (23) into equation (4),
An order of magnitude estimation of parameters based on parcel simulations (not shown) suggest that o ∼ 10−7 m, ∼ 10−5 m and ∼ 1010 m−2. Therefore, the bracketed term in equation (26a) tends to approach in most conditions, that is,
The exponential term in equation (29) approaches unity, which is a result of the assumption made in equation (13) that freezing depletes a negligible amount of aerosol (i.e., o(Do) is constant during the nucleation process). Since J(Si,max) has been eliminated from equation (29), fc is not strongly influenced by small variations in Si,max. Thus, Si,max can be taken as the saturation freezing threshold obtained by solving the parameterization of Koop et al.  for J(Si,max) = 1016 s−1 m−3, which represents an average of J(Si, max) over a wide set of simulations (section 4), and is close to the nucleation rate of pure water at −38°C [Pruppacher and Klett, 1997; Sassen and Benson, 2000]. The total number of crystals would be given by Nc = Nofc; however, such a result is not limited by No. Instead, lifting the assumption made in equation (19), fc can be associated with the solution of the integral in equation (9); that is, fc ≈ voJ(t)dt, so
where represents the number of remaining unfrozen droplets.
3.3. Calculation of
 is required for calculating the ice growth rate (equation (26b)). Two methods are used to calculate it. The first one is based on theoretical arguments (therefore γ = 1 in equation (25)), and assumes as the diameter of the ice crystal at which = 10−6 (i.e., the size above which the number of crystals is below 10−6Nc). With this, equation (20) can be solved for ,
where we have assumed ≫ Do (supported by numerical simulations). Since Γ2 ∝ (equation (6)), the value of would increase as αd decreases. For low values of αd ice crystals grow slowly, and noncontinuum effects limit the condensation rate; when ice become large enough, gas-to-particle mass transfer is in the continuum regime and the crystals grow quickly [Pruppacher and Klett, 1997]. When growth of the newly frozen ice crystals is delayed, water vapor water tends to preferentially condense on preexisting ice crystals. Slow uptake effects became important for αd lower than 0.1, that is, when Γ2 becomes comparable to Γ1 [Lin et al., 2002; Gierens et al., 2003]. Alternatively, can be computed using an empirical fit to numerical simulations obtained with the parcel model,
where V is in m s−1, T is in K, No in cm−3 and Dg,dry in meters. For this case, γ = 1.33 in equation (25). Equation (32) was generated over a broad set of T, p, V, No, Dg,dry, and αd (Table 1, section 4). The T and V dependencies in equation (32) are introduced to adjust the effective growth of the particles correcting for the “free growth” assumption (section 3.1). Variability in the formulation of equation (30) from aerosol property changes is not accounted for. Since no(Do) is determined by the dry aerosol size distribution, a larger Dg,dry will enhance Pf, as the total volume of the liquid aerosol particles is increased. This would produce a longer freezing pulse and increase Nc. The same effect can be achieved by reducing the effective growth of the particles, and explains why Dc,Smax scales with the average volume of the dry aerosol population, that is, ∼(NoDg,dry3)1/3 (equation (32)).
Table 1. Conditions Used in Parameterization Evaluation
V, m s−1
1.7, 2.3, 2.9
3.4. Implementation of the Parameterization
 Application of the parameterization is presented in Figure 4. Inputs are the conditions of cloud formation T, P, V, αd and the aerosol size distribution (i.e., No and Dg,dry); the outputs are Nc and ice crystal size distribution. To apply the parameterization, first is calculated (equation (25)), using computed from either equation (31) or (32), the latter being preferred. Si,max is obtained by solving J(Si,max) = 1016 s−1 m−3 [Koop et al., 2000], for which reported fits can be employed [i.e., Ren and Mackenzie, 2005]. Nc is calculated from equation (30) using fc calculated from equation (29). After fc is calculated, J(Si,max) can be corrected using J(Si,max) = (equation (28)); nc(Dc, Do) is then obtained from equations (21)–(23).
4. Evaluation of the Parameterization
 The parameterization was evaluated against the detailed numerical solution of the parcel model over a wide range of T, V, Dg,dry, No, and αd (Table 1), for a total of 1200 runs that reflect the uncertainty in αd [Lin et al., 2002] and the range of cirrus cloud formation conditions expected in a GCM simulation. The parcel model was run using initial Si = 1.0; initial p was estimated from hydrostatic equilibrium at T using a dry adiabatic lapse rate. The dry aerosol was assumed to follow a lognormal size distribution and be composed of pure H2SO4. “Aged,” “unperturbed,” and “perturbed” aerosol is represented by setting the geometric dispersion of the dry aerosol distribution, σg,dry to 1.7, 2.3, and 2.9, respectively [Seinfeld and Pandis, 1998]. The aerosol was assumed to be deliquesced and in equilibrium with Si in all simulations. Calculated Nc ranged from 0.001 to 100 cm−3 and Si,max varied between 1.4 and 1.6, which agrees with the reported values for cirrus formation by homogeneous freezing [i.e., Heymsfield and Sabin, 1989; Lin et al., 2002].
 Two main assumptions were introduced in the development of the parameterization: (1) Nc is calculated at Si,max rather than at the end of the freezing pulse, that is, Nc ≈ , and, (2) the newly formed crystals have a negligible impact on Si before Si,max is reached (“free growth”). Figure 5 shows how these assumptions may impact the results of the parameterization for high and low V and T (1–0.04 m s−1, 233–203 K). By using free growth regime to estimate the evolution of Si (Figure 5, dotted black line), Si,max is overestimated by ∼0.5% at low V (Figures 5c and 5d) and ∼2% at high V (Figures 5a and 5b). At high T (Figures 5b and 5d) this overestimation does not impact parameterization performance, as Si,max is low and small overestimations thereof do not significantly influence J(Si,max). However, as T decreases and V increases, Si,max reaches higher values during the parcel ascent; J (which is a very nonlinear function of Si,max), and Pf become very sensitive to small changes in Si,max. As a result, an overestimation in Nc may be expected. Conversely, Figures 5c and 5d show that at low V, underestimates the actual Nc by nearly a factor of 2; since few ice crystals (low Si,max) are formed, it takes a longer time to deplete the available water vapor, resulting in longer freezing pulses. At high V (Figures 5a and 5b), is close to Nc, since Si,max is reached rapidly and a larger number of crystals is formed; this effect will be more important at low T as higher Si,max values are reached.
4.1. Calculation of nc(Dc)
Figure 6 presents the parameterized nc(Dc) (equation (23b)) for two representative cirrus cases, along with obtained from parcel model simulations. Although the effect of the droplet size distribution parameters is explicitly considered in equation (23), it cancels out with its effect on fc (which is a result of assuming constant o(Do)), and a single crystal size distribution is obtained for the three values of σg,dry tested (although small variations in may occur owing to differences in T at Si,max). The latter is not a critical issue, as Nc variations with respect to σg,dry are generally small (i.e., Figure 6). o was approximated with the equilibrium size of Dg,dry at Si,max. nc(Dc) was calculated using the outline in section 3.4 (Figure 4), from which J(Si,max) is corrected using equations (30) and (32) (therefore enforcing Nc ≈ nc(Dc)dDc). Generally, the parameterized nc(Dc) reproduces well the numerical results at Si,max; however, the size of the ice crystals is underestimated (which is a result of the assumption of “free growth” which underestimates t − to, equation (15) and is overestimated.” The adjusted J(Si,max) will then be slightly below the obtained in numerical simulations to satisfy Nc = . Thus, the correction in J(Si,max) produces a reduction in the peak of the ice crystal size distribution. The influence of these factors on the resulting effective radius of the cirrus cloud, and its radiative properties, will require the time integration of nc(Dc), and the comparison with numerical simulations at different stages after cloud formation. Although such a task may be readily achieved, it is out of the focus of this manuscript and will be undertaken in a future study.
4.2. Calculation of Nc
Figure 7 shows the comparison of Nc predicted by the parcel model and the parameterization (equation (30)) using the theoretical calculation of (Figure 7, right; equation (31)) and using the empirical correlation for (Figure 7, left; equation (32)). The effects of assuming “free growth” and the approximation of Nc as are expected to cancel out at moderate T and V. However, at low V and high T (hence low Nc), the parameterization tends to underestimate Nc with respect to the parcel model results; the opposite tendency occurs at very high V (>2 m s−1) and low T (hence, high Nc). The behavior at these T, V extremes is not typical of cirrus formation [i.e., Heymsfield and Sabin, 1989; Gayet et al., 2004], hence not expected to be a source of significant bias in GCM simulations. Overall, using the theoretically calculated gives a parameterized Nc that agrees within a factor of 2 of the parcel model simulations, with a mean relative error about 29% (for all runs in Table 1). When using the empirically calculated (equation (32)), the parameterized Nc is much closer to the numerical parcel model (average relative error 1% ± 28%), as equation (32) allows more flexibility in reproducing the parcel model results, and accounts for the additional variability due to the effect of aerosol size and number.
4.3. Comparison Against Other Parameterizations
 The new parameterization is evaluated against several published schemes for different combinations of V, No, Dg,dry, T, and αd. The parameterizations used in this section are those of Liu and Penner  (LP2005), Sassen and Benson  (SB2000), Kärcher and Lohmann [2002a] (KL2002), and Fountoukis and Nenes . SB2000 is based on an empirical fit to numerical simulations relating Nc to T and V. A similar approach is used in LP2005 where an additional dependency on No is included. In both cases, J is calculated through classical nucleation theory (the latter using the effective temperature method [i.e., DeMott et al., 1994]). KL2002 is physically based and employs the freezing timescale and the threshold supersaturation as input parameters. It resolves explicitly the dependency of Nc on T, V, αd, and Do, and uses No as upper limit for Nc. Although the freezing of polydisperse aerosol is discussed in KL2002, not explicit solution is presented; their monodisperse solution is therefore used for comparison. The freezing timescale and supersaturation threshold are calculated using the analytical fits to Koop et al.  data provided by Ren and Mackenzie . o in this case was taken as in equilibrium with the volume-weighted geometric mean diameter of the dry size distribution. By using this definition of o, the best agreement between the parcel model simulations and the results of the KL2002 parameterization was obtained. The three parameterizations were compared to the solution of equation (30) using theoretically calculated Dc,max, equation (31) (although termed “theoretical”, k(T) was derived from an empirical fit to J), and using the empirically adjusted Dc,maxequation (32), (termed “adjusted”). o was calculated as the equilibrium size of Dg,dry at Si,max. All parameterizations are evaluated using T obtained at Si,max from the parcel model simulations. αd was set to 0.1 and 1.0 to test both diffusionally and nondiffusionally limited cases (see section 3.3).
4.3.1. Dependency on V
Figure 8 presents Nc predicted from all parameterizations at To = 213 (Figure 8, left; T between 208.6 and 209.4 K) and 233 K (Figure 8, right; T between 228.8 and 229.2 K), and αd = 0.1 (black line) and 1.0 (gray line). At To = 233 K, all parameterizations agree fairly well when αd = 0.1 and V < 1 m s−1. At higher V, KL2002 and LP2005 predict a larger Nc, whereas SB2000 predicts a lower Nc with respect to the parcel model results and becomes significant when V > 3 m s−1. At these conditions the adjusted parameterization follows the parcel model very well, whereas the theoretical parameterization slightly underpredicts Nc at low V. Runs made using αd = 1 (gray line) showed a good agreement between the parcel model and KL2002, the adjusted and theoretical parameterizations. This is not surprising, as equation (29) bears the same dependency on V and pio, reported by KL2002 in their “fast growth” solution (i.e., Nc ∝ V3/2), and further emphasized by more recent work [Gierens, 2003; Ren and Mackenzie, 2005]. At high fc (i.e., low T, low αd, and high V) the exponential term in equation (30) dampens the effect of V (also because scales with V) and Nc scales almost linearly with V. Results for αd = 1 lie below those of LP2005 and SB2000, who used lower αd values for their numerical simulations (LP2005 used αd = 0.1 and SB2000 used αd = 0.36 [Lin et al., 2002]). At To = 213 K and αd = 0.1 (Figure 8, left, black line), the parameterizations agree only for V below 0.3 m s1 whereas for large V they diverge, with KL2002 giving the largest Nc over the whole V interval. At very high V (>3 m s−1) the adjusted parameterization underpredicts Nc with respect to the parcel model results, which is a result of the exponential term introduced in equation (30). As with To = 233 K, KL2002, the adjusted and theoretical parameterizations agree well with the numerical results when αd = 1, and To = 213 K (Figure 8, left, gray line).
4.3.2. Dependency on No
Figure 9 presents Nc as a function of No for V = 0.2 m s−1 (Figure 9, left) and V = 1.0 m s−1 (Figure 9, right) and To = 213 (black line) and 233 K (gray line); Dg,dry was set to 40 nm and αd to 0.1. In all cases of Figure 9, LP2005 and the adjusted parameterization show the best agreement with the parcel model results. Still, at To = 213 K, V = 1.0 m s−1, and No below 20 cm−3, the adjusted parameterization underpredicts with respect to the numerical results and LP2005 overpredicts. In both cases the difference with the parcel model results is about ±50%, which is not critical as these very low No are atypical of cirrus forming conditions [i.e., Pruppacher and Klett, 1997]. In all cases of Figure 9, KL2002 predicts larger Nc than the parcel model; however, the difference becomes much smaller at large No. SB2000 predicts Nc close to the average of the parcel model results at To = 213 K; when To = 233 K, SB2000 is close to the parcel model results at large No. The theoretical parameterization tends to agree better with the parcel model results at high No. As proposed by LP2005, the parcel model results can be reasonably well expressed in the form Nc = aNob where a and b are functions of T, V, Dg,dry and αd. The dependency of Nc on No generally increases when T and αd decrease and when V increases. For the cases of Figure 9, b lies between 0.19 (To = 233 K, V = 0.2 m s−1) and 0.61 (To = 213 K, V = 1 m s−1). This is consistent with experimental and numerical studies that report a factor of 2 to 4 increase in Nc for a tenfold increase in No [i.e., Heymsfield and Sabin, 1989; Jensen and Toon, 1994; Seifert et al., 2004].
4.3.3. Dependency on Dg,dry
Figure 10 presents Nc as a function of Dg,dry for To = 213 K (Figure 10, left) and To = 233 K (Figure 10, right), and αd = 0.1 (black line) and αd = 1 (gray line); for these simulations V = 1.0 m s−1 and No = 200 cm−3. To apply KL2002, Dg,dry was converted into the volume-weighted mean diameter in equilibrium with Si,max. In all conditions of Figure 10, parcel model results suggest Nc scales almost linearly with Dg,dry, which is also found by the combination of equations (29) and (32). This was also observed in LP2005 and is a result of the increased Pf due to the larger volume of the liquid aerosol particles in equilibrium with the aerosol dry distribution (see section 3.3 and equation (9)). An inverse tendency predicted by KL2002 (Nco−1), and also by the theoretical parameterization (although with a much weaker dependence). In the latter, the effect of increased Pf is not accounted for owing to the assumption of an infinite aerosol source (section 3.1, equation (13)). Although LP2005 does not take into account this dependency, it predicts Nc in agreement with parcel model results at Dg,dry = 80 nm, and αd = 0.1. SB2000 predictions agree with the parcel model results at Dg,dry = 40 nm and To = 213, and at Dg,dry = 120 nm and To = 233, when αd = 0.1. The dependence of Nc with Dg,dry is much stronger in the left than in the right panel of Figure 10, which suggests once more that size effects would be more important at low T and high V. In all cases of Figure 10, the adjusted parameterization closely reproduces the parcel model results. While the comparison of the different parameterizations was carried out over a comprehensive set of conditions, common values of Dg,dry often ranges between 40 and 100 nm (and between 100 and 500 cm−3 for No) [e.g., Heymsfield and Miloshevich, 1995; Gayet et al., 2004]. Figures 8–10 show that the effect of T and V variations on Nc is much stronger than that of Dg,dry and No. The relative importance of each parameter remains to be assessed in global model studies.
5. Summary and Conclusions
 To address the need for improved ice cloud physics in large-scale models, we have developed a physically based parameterization for cirrus cloud formation, which is robust, computationally efficient, and links chemical effects (e.g., water activity and uptake effects) with ice formation via homogenous freezing. This was accomplished by tracing back the growth of ice crystals to their point of freezing, relating their size to freezing probability. Using this approach, an expression for the ice crystal size distribution is derived, the integration of which yields the number concentration.
 The parameterization is evaluated against the predictions of a detailed numerical parcel model developed during this study. The parcel model equations were integrated using a Lagrangian particle-tracking scheme; the evolution of the ice crystal size distribution is described by the superposition of growing of monodisperse crystal populations generated by the freezing of single classes (of same size and composition) of supercooled droplets.
 Two versions of the parameterization are developed and evaluated; one based solely on theoretical arguments, and one with adjustments in the ice crystal growth rate using numerical parcel simulations. When compared against the predictions of the numerical parcel model over a broad set of conditions, the theoretically based parameterization for Nc robustly reproduced the results of the parcel model within a factor of two and with an average relative error of about 29%. When numerical simulations are used to adjust the ice crystal growth rate, the relative error was reduced to 1 ± 28%, which is remarkable given the simplicity of the final expression obtained for Nc, the broad set of conditions tested, and the complexity of the original parcel equations.
 The new parameterization presented in this work offers an analytical and physically based relationship between the parameters affecting cirrus ice formation via homogeneous freezing. The prediction skill of the parameterization is robust across a wide range of parameters (e.g., αd, aerosol characteristics). As shown in Figure 7, the accuracy with which the parameterization reproduces the parcel model results is independent of these parameters. In this regard, only the KL2002 parameterization shares this characteristic; the presented parameterization, however, (1) explicitly links the variables that control the freezing timescale of the particles and (2) successfully reproduces the effect of the aerosol number on Nc.
 The results given are applicable for cirrus formation on predominantly homogenous freezing conditions. Frequently, heterogeneous freezing and competition between multiple particle types can significantly impact cloud formation. Both the numerical model and the parameterization can be readily extended to include these processes, and will be the focus of future work.
ice crystal size distribution function originated at Do.
ice crystal size distribution function.
liquid aerosol size distribution function.
average liquid aerosol size distribution function during freezing.
ice crystal number concentration at Si,max.
ice crystal number concentration.
aerosol number concentration.
ice saturation vapor pressure.
Pf, Pf(Do, t)
cumulative probability of freezing.
ice and air densities, respectively.
universal gas constant.
geometric dispersion of the liquid aerosol size distribution.
geometric dispersion of the dry aerosol size distribution.
water vapor saturation ratio with respect to ice.
equilibrium ice saturation ratio.
maximum ice saturation ratio.
ice saturation ratio at the beginning of freezing.
ice saturation ratio at which an ice crystal was formed.
time of freezing.
volume of a liquid droplet of size Do.
ice mass mixing ratio.
ice nucleation function.
 This study was supported by NASA MAP, NASA EOS-IDS-CACTUS, and a NASA New Investigator Award. We would also like to thank B. Kahn, V. Khvorostyanov, and two anonymous reviewers for comments that substantially improved the manuscript.