This study investigates the effects of cloud condensation nuclei (CCN) and ice nuclei (IN) on ice formation in mixed-phase clouds using an adiabatic parcel model. The simulations with lower updrafts represent mixed-phase stratiform clouds, and those with higher updrafts represent deep convective clouds. Increasing CCN concentration can lead to more ice crystals with smaller sizes, but does not affect the height of freezing nucleation at lower updrafts. At higher updrafts, ice number concentration is more sensitive to CCN, and CCN effect on ice nucleation height is also more noticeable. This study also investigates the effects of IN properties (number concentration, size, contact angle) on ice formation assuming the immersion freezing mechanism. At very low IN number concentration, ice nucleation is dominated by homogeneous freezing and ice number concentration is relatively high. As IN number concentration is increased, ice number concentration decreases because both heterogeneous and homogeneous freezing processes occur. A mixed-phase layer extends up to the height for homogeneous freezing and there is a competition between the two freezing processes. As IN number concentration is further increased, ice number concentration increases because heterogeneous freezing dominates ice nucleation. IN number concentration needs to be higher for heterogeneous freezing to dominate ice nucleation at higher updrafts and lower at lower updrafts. Increasing IN size and contact coefficient also increases the contribution of heterogeneous freezing to ice nucleation. Finally, spheroids are used to represent columnar and plate-like ice crystals. CCN and IN effects are not dependent on the assumption of ice crystal shapes.
 Aerosol effects on clouds and climate have been one of the focuses for climate research in the past years. The complicated interactions of aerosol particles and clouds are not well understood and cause the largest uncertainties in climate forcing estimation [e.g., Cox, 1971; Knight and Heymsfield, 1983; Curry, 1995; Solomon et al., 2007; Stevens and Feingold, 2009; Morrison, 2012]. Cloud particle number concentrations, sizes, phases, and geometries are major factors that are related to cloud radiative properties [Stephens et al., 1990]. In ice-containing clouds, ice particle number and size can affect cloud albedo just as liquid-phase clouds do. Ice water content can also change the infrared radiation. The height of freezing nucleation and the height of glaciation can affect the vertical distribution of latent heat, and possibly affect cloud dynamics [Rosenfeld et al., 2008]. As ice-containing clouds appear widely and frequently all over the world [Hobbs and Deepak, 1981], the effect of aerosols as cloud condensation nuclei (CCN) and ice nuclei (IN) on these clouds needs to be understood.
 It has been suggested that the suppression of warm rain by increased CCN causes most of the condensates to ascend, freeze, and release the latent heat of freezing aloft [Williams et al., 2002; Andreae et al., 2004]. This can result in an invigoration of the convective clouds and additional rainfall [e.g., Koren et al., 2005, 2010; Tao et al., 2007; Rosenfeld et al., 2008]. The enhanced precipitation can then lead to a stronger cooling due to the evaporation or melting of precipitating particles in the lower troposphere, hence stronger downdrafts and convergence in the lower layer and more vigorous precipitation [Tao et al., 2007; Phillips et al., 2007]. Observations have shown evidence for the aerosol invigoration effect on convective clouds [Yuan et al., 2011].
 However, modeling studies also showed that the response of deep convection to CCN can be opposite to the invigoration effect or nonmonotonic. Morrison and Grabowski  suggested that convection can be weakened in polluted cloud ensembles. This is because more ice from the freezing of more droplets in the polluted case at cloud top will lead to more radiative heating and stabilize the troposphere. It has also been shown that aerosol effect on convective precipitation depends on the environmental conditions [Khain et al., 2005; Tao et al., 2007; Cui et al., 2011]. For example, in a continental environment, increasing CCN can lead to a later onset of precipitation, weaker evaporative cooling below clouds, and therefore weaker precipitation [Tao et al., 2007]. Evaporative cooling is a key process in determining whether higher CCN reduces or enhances precipitation [Tao et al., 2007]. Cui et al.  showed that the accumulated precipitation and the riming rate increase with aerosol in tropical marine convective clouds, but are nonmonotonic in continental convective clouds. Aerosol effects on deep convective clouds also depend on vertical wind shear [Fan et al., 2009]. Increasing aerosols can suppress the convection under strong wind shear and invigorate the convection under weak wind shear.
 In addition to the studies of CCN effects on convective clouds, there are also investigations about CCN effects on stratiform mixed-phase clouds, especially arctic mixed-phase clouds. Early investigation confirmed that arctic stratiform mixed-phase clouds are very susceptible to CCN [Hobbs and Rangno, 1998]. Aircraft observations show that smaller droplets and fewer precipitating ice crystals in polluted conditions can suppress the riming process in arctic mixed-phase clouds [Lance et al., 2011]. Similarly, pollution aerosols can suppress the riming process in orographic clouds [Xue et al., 2012]. However, there is evidence that the change of ice and liquid water content (LWC) evolution with the increase of CCN concentration is insignificant in arctic mixed-phase clouds [Ervens et al., 2011].
 Insoluble aerosols can serve as IN and greatly affect cloud microphysics. For example, ice can form in altocumulus clouds under temperatures as high as −5.2°C to −8.8°C when dust was present in the cloud layer [Sassen et al., 2003]. In arctic mixed-phase clouds, it has been observed that IN concentration and ice crystal number concentration are positively correlated [Jackson et al., 2012; McFarquhar et al., 2011]. IN also has a strong effect on liquid water content in arctic mixed-phase clouds [Morrison et al., 2005]. However, simulations using an adiabatic parcel model showed that IN concentration generally does not have a strong effect on the ratio of LWC to ice water content (IWC) in arctic mixed-phase clouds [Ervens et al., 2011]. In an earlier modeling study on stratiform mixed-phase clouds, it was found that increasing IN concentration caused more ice crystals to form and a more rapid Bergeron process, leading to larger precipitation fluxes, weaker convection, and lower cloud fraction [Harrington and Olsson, 2001]. However, Seifert et al.  showed that CCN and IN concentrations have little effect on surface precipitation for mixed-phase clouds, because clouds are a buffered system of microphysics and macrophysics [Stevens and Feingold, 2009], but that CCN and IN may have a significant effect on radiation and near-surface temperature. Similarly, for deep convective clouds, additional IN will increase ice concentration, decrease ice sizes, but lead to a small effect on precipitation [Lebo and Seinfeld, 2011]. In cirrus clouds, IN and the associated heterogeneous freezing process is also important [Karcher and Lohmann, 2003]. The presence of IN may lead to fewer ice particles and larger sizes, hence stronger sedimentation and reduction in IWC, compared to homogeneous freezing [Hendricks et al., 2011]. In cirrus clouds where heterogeneous and homogeneous freezing processes compete, IN and the associated heterogeneous freezing can strongly affect the subsequent homogeneous freezing, leading to a strong reduction in ice crystal concentration [Spichtinger and Cziczo, 2010]. Recently, results from an observational study for orographic clouds showed that dust particles from the Sahara serve as IN and play an important role in orographic precipitation processes over the United States [Creamean et al., 2013].
 Many atmospheric species have been proven to serve as efficient IN, such as mineral dust, volcanic ash, soot, and several organic species [e.g., Gorbunov et al., 2001; Diehl et al., 2002; Zuberi et al., 2002; DeMott et al., 2003a, 2003b; Abbatt et al., 2006; Beaver et al., 2006; Möhler et al., 2007; Zobrist et al., 2007; Murray et al., 2010; Niedermeier et al., 2010; Baustian et al., 2012]. Mineral dust has been observed to be the dominant residue in the cirrus ice that formed during a dust transport event from the Sahara [Cziczo et al., 2004]. A field campaign at the high alpine station also found an enrichment of black carbon (BC) mass in residual small ice crystals [Cozic et al., 2008]. INSPECT-II (the second ice nuclei spectroscopy campaign) proved that IN has a much higher percentage of mineral dust/fly ash and metallic particle compared to ambient particles [Richardson et al., 2007]. In wintertime cumulus over the UK, mineral dust particles play an important role in ice nucleation at relatively high temperature (−8°C) [Crawford et al., 2012]. However, a recent laboratory study pointed out that both uncoated and coated soot particles are unlikely to significantly affect ice nucleation [Friedman et al., 2011].
 Not only does aerosol composition determine whether it is an efficient IN, but aerosol size and contact angle also determine its ice nucleation activity. Dust particles with larger sizes are reported to have a better efficiency for ice nucleation compared to particles with smaller sizes [Welti et al., 2012]. Heterogeneous ice nucleation is found to be strongly dependent on the surface area available for nucleation based on laboratory measurements [Wheeler and Bertram, 2012; Hoose and Möhler, 2012; Murray et al., 2012]. Higher freezing temperatures for immersion freezing were also observed when larger surface areas of IN were present within the drops [Broadley et al., 2012]. It is found that the freezing rate is particularly sensitive to the contact angle, which is considered as an interface parameter for IN [e.g., Fletcher, 1958; Khvorostyanov et al., 2006]. Although the contact angle is not a basic physical property of aerosol particles, it can be used to describe properties of the ice-nucleating surface and the ice nucleation activity of aerosol particles. Note that except for the contact angle, there are also different ways to describe ice nucleation activity, as summarized in Wheeler and Bertram  and Hoose and Möhler . Effective contact angles for several IN species are presented in the literature [Archuleta et al., 2005; Chen et al., 2008; Kulkarni and Dobbie, 2010; Wang and Knopf, 2011], while laboratory data also suggest that a single contact angle for one type of IN is inadequate [Marcolli et al., 2007; Luond et al., 2010; Niedermeier et al., 2010; Murray et al., 2011; Broadley et al., 2012; Wheeler and Bertram, 2012]. The above investigations showed that effective IN have a wide range of sizes and contact angles. Therefore, it is important to study the effect of IN properties (concentration, size, and contact angle) on ice formation.
 Current modeling studies on ice freezing nucleation deal with freezing processes as either: (1) a singular, time-independent process, and (2) a stochastic, time-dependent process. The former calculates the ice number concentration with instant temperature or supersaturation with respect to ice [e.g., Fletcher, 1969; Meyers et al., 1992; Cotton and Field, 2002; Morrison et al., 2008; Phillips et al., 2008; de Boer et al., 2010; Demott et al., 2010]. The number concentration of ice particles is reported as an integrated number over the investigated temperature and/or supersaturation range. The latter is based on the classical nucleation theory [e.g., Fukuta and Schaller, 1982; Khvorostyanov and Curry, 2000, 2004, 2005; Shaw et al., 2005; Curry and Khvorostyanov, 2012], where the number concentration of newly formed ice crystals depends on time for a certain temperature or supersaturation. However, some experimental data showed that the time dependence in the classical nucleation theory can be ignored in parameterizations [e.g., Broadley et al., 2012; Welti et al., 2012]. Other studies combine the deterministic and stochastic approaches, and some results match the data well [Niedermeier et al., 2011; Broadley et al., 2012; Field et al., 2012; Ervens and Feingold, 2012; Kulkarni et al., 2012]. The main disadvantage of reconciling the stochastic and deterministic descriptions of ice nucleation is that more free parameters are included.
 The goal of this study is to investigate ice formation affected by aerosol properties, such as soluble aerosol concentration, IN concentration, IN size, and contact angle, using an adiabatic cloud parcel model with the stochastic, time-dependent approach. We focus on the effects of aerosols on ice particle number concentration, ice particle size, IWC, the height of freezing nucleation, the thickness of the mixed-phase layer, and the height of glaciation in mixed-phase clouds. Section 2 discusses the theory of ice nucleation and growth. Section 3 introduces the model used in this study. Section 4 presents the results. The first part of the results is focused on the effects of CCN on ice nucleation. The second part of the results is related to the effects of IN properties on ice cloud formation. The last part of the results shows whether the assumption of ice crystal shapes affects the cloud response to CCN and IN. Spheroids are used to represent columnar and plate-like ice crystals. Section 5 presents the conclusion and discussion.
2 Classical Theory of Ice Nucleation and Diffusional Growth
 In general, two fundamental nucleation processes exist to form ice in the atmosphere: homogeneous and heterogeneous ice nucleation mechanisms. There are four modes in heterogeneous ice nucleation: deposition, contact freezing, condensation freezing, and immersion freezing [Lamb and Verlinde, 2011, pp 312]. The accurate contribution of each nucleation mode to ice concentration is not clear from model studies [e.g., Harrington and Olsson, 2001; Morrison et al., 2005; Marcolli et al., 2007]. Recent investigations suggest that immersion freezing is the main nucleation mode for ice [e.g., Diehl and Wurzler, 2004; Zobrist et al., 2006; Popovicheva et al., 2008; Vali, 2008; Crosier et al., 2011] and has more than 60% contribution to ice formation in arctic mixed-phase clouds [de Boer et al., 2010, 2011].
 The homogeneous freezing nucleation rate is determined by the following equation [Pruppacher and Klett, 1997, pp 207]:
where Δg≠ is the activation energy for the diffusion of water molecules across the water-ice boundary; R is the universal gas constant; ΔFg is the critical energy of ice germ formation; ρi and ρw are the densities of ice and water, respectively; k and h are the Boltzmann and Planck constants; σi/w is the surface tension at the ice and water interface; Nc is the number of molecules in contact with a unit area of ice; T is the droplet temperature, which can be assumed to be equal to the environmental temperature. We choose to use a revised equation from Khvorostyanov and Sassen  (T ≤ −30°C) and Pruppacher and Klett  (T > −30°C) to describe Δg≠ in this paper:
where T is in the unit of °C. In the classical theory, ΔFg is related to the critical radius rg of a spherical ice germ. It can be derived from the equilibrium condition at the ice-water interface as [Khvorostyanov and Sassen, 1998]
 The expression of rg is based on Khvorostyanov and Sassen :
where T0 is 273.15 K. is an “effective” melting heat, which is a simple fit for the melting heat introduced by Khvorostyanov and Sassen  and provides good agreement for the freezing nucleation rate in a wide range of temperature. It is expressed as
with the unit of cal g−1 and T in the unit of °C. We assume that droplets are pure water when freezing occurs. The solute has little effect because cloud droplets are larger than 5 µm and the solution is extremely dilute at the time of freezing. Khvorostyanov and Sassen  compared their parameterization of homogeneous freezing with other parameterizations [e.g., Sassen and Dodd, 1989; Heymsfield and Sabin, 1989; DeMott et al., 1994; Jensen et al., 1994], and found that their parameterization was reasonable.
 The heterogeneous freezing nucleation rate is expressed as [Khvorostyanov and Curry, 2000]:
assuming that the IN, ice germ, and droplet are all spherical for simplicity and their radii are represented as rN, rg, and rd, respectively. ΔFg,S is the critical energy of ice germ formed on IN and can be simply written as [Khvorostyanov and Curry, 2000]:
where f(m,x) is the geometric factor, m is the contact coefficient (cosine of contact angle), x is the ratio of rN to rg. We also use equation (4) to calculate the critical radius of an ice germ rg, which is simplified from Khvorostyanov and Curry  for heterogeneous freezing. The parameter cI,S is the concentration of water molecules adsorbed on a unit area of a surface. The value of cI,S varies from different measurements, but the variation does not have significant effect on freezing nucleation rate [Fletcher, 1962]. We choose cI,S = 1015 cm−2, following Ervens et al. . The chosen parameterization of heterogeneous freezing [Khvorostyanov and Curry, 2000] has been described in more detail in Khvorostyanov and Curry , and used in parcel model studies [Khvorostyanov and Curry, 2005; Eidhammer et al., 2009] and cloud resolving model studies [Khvorostyanov et al., 2006; Morrison and Pinto, 2006; Sassen and Khvorostyanov, 2007]. It was seen in Khvorostyanov and Curry  that their simulated ice nucleation rate had a good agreement with observations [e.g., Ohtake et al., 1982; Curry et al., 1990; Pinto et al., 2001]. The heterogeneous nucleation rate as a function of supercooling is plotted in Figure 1. It can be seen that nucleation rate is sensitive to supercooling at a wide temperature range.
 For both freezing mechanisms, the frozen number concentration of cloud droplets in time step Δt can be described as [DeMott et al., 1994; Khvorostyanov and Curry, 2000]:
where Φ stands for the freezing nucleation frequency (rate of ice crystal formation) [s−1]. In homogeneous freezing, Φ is the multiplication of the freezing nucleation rate and the volume of a droplet; in heterogeneous freezing, Φ is simply the freezing nucleation rate. rd is the radius of cloud droplets, and n(rd) is droplet size distribution. Note that only the fraction of droplets that contain IN are available for heterogeneous freezing.
 The depositional growth of ice crystals can be described as:
where mp is the mass of ice particles; Si is the saturation ratio of water vapor over ice; Ls is the specific latent heat of sublimation; Mw is the molecular weight of water; ei is the saturation vapor pressure over plane ice surface; is the effective diffusion coefficient of water vapor; is the effective thermal conductivity of air [Pruppacher and Klett, 1997, pp 547]; C is the capacitance of ice crystals which is determined by crystal geometry. In this study, we first assume that the crystals are spherical and that C is equal to ice radius.
 The nonspherical shape of an ice crystal introduces complicated boundary conditions, which impose great difficulty in solving the Laplace equation for vapor distribution. Therefore, the geometry of ice crystals needs to be idealized. Jayaweera and Cottis  have shown that spheroids are good analogs for simple columnar and plate-like ice crystals. Chen and Lamb  described ice crystals as spheroids instead of spheres. Spheroidal particles can act much differently from spherical particles through the nonlinear feedbacks between mass growth and habit evolution [e.g., Fukuta and Takahashi, 1999; Sheridan et al., 2009; Sulia and Harrington, 2011]. It has been shown that detailed consideration of temperature-dependent habit growth can significantly affect the glaciation timescales [Harrington et al., 2009].
 The growth rate and habits of ice crystals are specified in a self-consistent way as in Chen and Lamb . Under a certain supercooling, the growth ratio of c axis (perpendicular to the basal face) and a axis (perpendicular to the prism face) can be represented as:
where the inherent growth ratio Γ(T) is the ratio of the deposition coefficient for the basal face to that for the prism face. Experimental data of Γ(T) between 0 and −30°C are used in Chen and Lamb  and this study. Below −30°C, Γ(T) is assumed to be a constant and has the same value as at −30°C. φ(c,a) is the ratio of the vapor density gradient along the c and a axes, and can be simplified as the aspect ratio c/a. Thus, the ice axes have different growth rates and then change from a sphere to a spheroid. The capacitance C of a spheroid can be described as
for oblates, where
for prolates, where
3 Model Description
 This study is conducted with an adiabatic parcel model [Feingold and Heymsfield, 1992] with the stochastic, time-dependent approach for ice nucleation. The original version of the model is designed to simulate warm cloud processes and has been applied in several studies [e.g., Feingold et al., 1998; Feingold and Kreidenweis, 2000; Xue and Feingold, 2004; Ervens et al., 2005; Koehler et al., 2006]. Eidhammer et al.  and Ervens et al.  added ice processes in this parcel model to investigate ice formation. In this study, ice nucleation and depositional growth are also added to the adiabatic parcel model to study the effect of aerosols on homogeneous and heterogeneous ice freezing in mixed-phase clouds.
 The soluble aerosols that act as CCN are composed of ammonium sulfate and have a logarithmic size distribution over the radius range 0.01–1 µm, with a median radius of 0.1 µm, a geometric standard deviation of 1.4, and number mixing ratios (Na) of 40, 100, and 1000 mg−1 to represent clean to polluted conditions. The water activity of ammonium sulfate solution is calculated based on Tang and Munkelwitz . The activation and condensational growth of liquid-phase cloud droplets are simulated with 20 discrete Lagrangian bins. Collision-coalescence and sedimentation are not considered in this study.
 The insoluble aerosols that act as IN are externally mixed with ammonium sulfate. When investigating the effect of IN properties on ice nucleation, the mixing ratio of ammonium sulfate Na is fixed at 100 mg−1, corresponding to a droplet number mixing ratio (Nd) of 89 mg−1 for the parcel model settings in this study at an updraft of 0.5 m s−1. The relative IN number concentration NIN/Nd is varied from 0.00001 to 0.1, i.e., 0.001% to 10% in percentage, and about 0.00089 to 8.9 cm−3 in absolute number concentration, to represent the range of IN number concentration in the atmosphere based on measurements. It has been found that IN concentration was less than 0.05 cm−3 at −24°C and 104% relative humidity with respect to water [DeMott et al., 2003a]. Lower concentration of IN (0.0004 – 0.006 cm−3) has been measured in lee wave clouds at a temperature range from −24 to −34°C and relative humidity from 100.5% to 103.8% with respect to water [Field et al., 2012]. There are also high IN concentration (0.5 – 1 or even > 1 cm−3) measured at −37°C and 86% relative humidity with respect to water [DeMott et al., 2003a; Sassen et al., 2003]. In this study, the highest IN concentration of 8.9 cm−3 (NIN/Nd = 10%) is used as an upper limit to test the IN effect. IN radius is varied from 200 to 1000 nm, and IN contact coefficient is varied from 0.35 to 0.6 [Eidhammer et al., 2009; Chen et al., 2008; Khvorostyanov and Curry, 2004].
 The ice nucleation rate is calculated based on equations (1) and (6). The heterogeneous ice nucleation is assumed to be through the immersion freezing mechanism. We use to parameterize the nucleation rates for both homogeneous and heterogeneous freezing. The threshold for freezing nucleation, that is, the threshold for the model to turn on the ice diffusional growth process, is that the freezing nucleation frequency is larger than 0.001 s−1 for the biggest droplet size bin. Because the droplet spectrum is relatively narrow at the stage of freezing nucleation, the newly formed ice crystals from different droplet size bins have similar size. We therefore add one size bin to represent the newly formed ice crystals in each time step. The concentration of the newly formed ice crystals is calculated based on equation (8). The mass of the newly formed ice bin is the total mass of newly formed ice divided by ice number. Instead of using a hybrid Lagrangian/Eulerian framework [Cooper et al., 1997; Eidhammer et al., 2009], we simulate ice crystal growth explicitly with Lagrangian bins. The size of each ice bin due to vapor diffusional growth is simulated based on equation (9). The collection process of ice crystals is not included in this study. We assume that the ice crystals are spherical in sections 4.1 and 4.2. The geometric effect is also considered in this parcel model, using the method by Chen and Lamb , as will be presented in section 4.3.
 The vertical velocity of the parcel varies from 0.1 to 10 m s−1. The simulations with lower updrafts can represent mixed-phase stratified clouds, and those with higher updrafts can represent deep convective clouds. The initial air temperature is −10°C, the pressure is 680 mb, and the relative humidity is 85%. The parcel is placed at 4 km initially and lifted up to 8 km (about −43°C). The thermal accommodation coefficient is 0.96 for liquid and 0.93 for solid growth in the model. The mass accommodation coefficient is 1.0 for water and 0.7 for spherical ice [Pruppacher and Klett, 1997, pp 164–165].
4 Results and Discussion
4.1 Effect of CCN Concentration on Ice Nucleation
 Figure 2 shows cloud profiles under different CCN concentrations for homogeneous freezing at an updraft of 0.5 m s−1. Ice number concentration, ice effective radius, and IWC increase rapidly with height between 7 km and 7.2 km (−34.7 to −36°C). Meanwhile droplet number concentration, droplet effective radius, and LWC decrease dramatically in this layer. This indicates that once homogeneous freezing starts, Bergeron process occurs rapidly, and cloud glaciation is fast. Above 7.2 km, ice number concentration becomes constant because the collection process of ice crystals is not considered in this study. Liquid water totally evaporates. Ice effective radius and IWC increase as a result of vapor deposition.
 The ice number concentration in the clean case (Na = 40 mg−1) is 16 mg−1 (approximately 16 cm−3) while in the polluted case (Na = 1000 mg−1) is 31 mg−1, nearly 90% higher than the clean case. The ice effective radius at 8 km is around 33 µm in the clean case and 27 µm in the polluted case. This suggests that more CCN can increase the number but decrease the size of ice crystals. In the liquid phase, the polluted case has much higher droplet number concentration and smaller droplet effective radius (783 mg−1, and 8 µm right before glaciation) compared to the clean case (36 mg−1, and 22 µm right before glaciation). It is also noted that increasing CCN does not affect LWC and IWC in the adiabatic parcel model simulations, consistent with the results of Ervens et al. .
 Despite of the smaller droplet sizes, the polluted case still produces more ice crystals. The reason is that ice crystal number concentration in homogeneous freezing is mainly determined by: (1) the freezing nucleation frequency, which is dependent on the freezing nucleation rate and droplet volume, and (2) droplet number concentration, as shown in equation (8). The freezing nucleation rate is very sensitive to supercooling. A small increase in supercooling (i.e., a small increase in height) can cause a big increase in the nucleation rate. The nucleation frequency is therefore dominated by the supercooling. The decrease of droplet volume due to more CCN has minor effects on nucleation frequency. The polluted case therefore has a similar freezing nucleation frequency compared to the clean case. The much higher droplet number concentration in the polluted case then leads to higher ice number concentration.
 Figure 3 shows the effect of CCN on homogeneous freezing at an updraft of 10 m s−1. Compared with the lower updraft simulations (Figure 2), all the CCN cases at the higher updraft have more ice crystals and smaller ice sizes. This is because, at lower updraft, the unfrozen droplets have enough time to evaporate during the Bergeron process as the parcel slowly moves from 7 to 7.2 km, and cannot survive to height higher than 7.2 km. At higher updraft, the unfrozen droplets can be transported higher than 7.2 km in a short time, survive from evaporation, and then freeze, leading to higher number concentration of ice crystals. It is seen from Figure 3 that the liquid phase can survive to 7.5 km. CCN effect can still be seen at higher updraft: more CCN lead to more liquid-phase droplets with smaller sizes, and more ice crystals with smaller sizes. More importantly, ice number concentration is more sensitive to CCN at the higher updraft. For example, ice number concentration increases from 36 mg−1 to 800 mg−1 as Na increases from 40 to 1000 mg−1 (Figure 3a).
 Because CCN can affect droplet number and size, hence affect ice number and size, we test if CCN has effects on the height of freezing nucleation and the height of glaciation, which can potentially affect the vertical distribution of latent heat. We define the height of freezing nucleation as the height where IWC begins to be higher than 0.1 g kg−1. The height of glaciation can be defined as the height where LWC begins to be lower than 0.1 g kg−1. We focus on the vertical distributions of the liquid-phase mass and the ice-phase mass. This can provide information on whether CCN affects the vertical distribution of latent heat and cloud dynamics.
 Figure 4 is a summary for the height of freezing nucleation and the height of glaciation at different CCN concentrations and different vertical velocities for homogeneous freezing. CCN concentration does not significantly raise the height of freezing nucleation at lower updrafts. It was discussed in Figure 2 that the freezing nucleation rate is very sensitive to supercooling (i.e., height). Therefore, the height of freezing nucleation is dominated by the supercooling, not by droplet sizes. However, at higher updrafts, CCN effect on the height of freezing nucleation is stronger. Increasing CCN from 40 mg−1 to 1000 mg−1 can increase the nucleation height by about 100 m. This is because at higher updrafts, the cloud parcel is lifted through a layer more quickly, and the small droplets in the polluted case may not freeze fast enough. The height of freezing nucleation in the polluted case is therefore higher than that in the clean case. Similarly, the height of glaciation is more affected by CCN at higher updrafts. For w = 5 and 10 m s−1, the polluted case generally has higher glaciation height than the clean case. However, for w = 0.5 m s−1, the polluted case has slightly lower glaciation height than the clean case. This is because liquid droplets are converted to ice particles not only through the freezing process, but also through the Bergeron process. In the polluted case, cloud droplets are smaller and droplet evaporation is faster. Therefore, it is easier to convert liquid droplets to ice particles, resulting in a lower glaciation height. This effect can also be seen in the w = 5 m s−1 simulations: the glaciation height becomes lower as aerosol concentration increases from 500 to 1000 mg−1. For w = 10 m s−1, droplet evaporation effect is not important, and the glaciation height would therefore increase with aerosol concentration, similar to the increase of nucleation height with aerosol concentration as discussed above. Note that the layer between the height of nucleation and the height of glaciation can be seen as the mixed-phase layer. It is very thin for homogeneous freezing.
 Whether the elevated nucleation height and the associated change in the vertical distribution of latent heat can cause any effect in cloud dynamics as proposed in Rosenfeld et al.  needs further study. Note that this is an idealized simulation where collision-coalescence and sedimentation are not considered. In real clouds, increased CCN can suppress the warm rain process and more liquid water may be transported above the 0°C layer and freeze at an elevated nucleation height, leading to a potentially stronger dynamical effect.
 For heterogeneous freezing, the increase in CCN concentration and the associated decrease in droplet size do not affect ice number concentration, ice effective radius, and the height of freezing nucleation, because heterogeneous freezing nucleation frequency is dependent on IN properties, and not on droplet size, in the parameterization in this study. The ice formation process is the same as long as the IN availability is the same. The ice cloud properties are also not affected by CCN at higher updrafts for heterogeneous freezing (figure not shown).
4.2 Effect of IN Properties on Ice Nucleation
 Heterogeneous freezing nucleation is affected by IN concentration, size, and contact coefficient (i.e., cosine of the contact angle), which complicates the whole process. This section tests the sensitivity of ice cloud properties to IN properties. The cloud parcels are lifted to the homogeneous freezing level eventually (8 km). Figure 5 describes the effect of IN concentration on ice formation at an updraft of 0.5 m s−1. Here we fix the IN radius at 500 nm and contact coefficient at 0.5. Soluble aerosol concentration Na is fixed at 100 mg−1. This means that droplet number concentration Nd is fixed at 89 mg−1 for the parcel model settings at w = 0.5 m s−1. The relative IN concentration NIN/Nd is varied from 0.001% to 10%. Compared with homogeneous freezing (see the black line in Figure 2), all cases here (except for the case with the lowest IN concentration) have lower ice number concentration, bigger ice effective radius, and lower height of freezing nucleation because freezing temperature is higher in the presence of IN.
 Results in Figure 5 show that ice number concentration and effective radius at 8 km have nonmonotonic relationships with IN number concentration. First, when NIN/Nd is less than 0.05%, ice concentration and ice effective radius are not sensitive to IN concentration. The height of freezing nucleation is 7.3 km and the glaciation height is 7.5 km for the 0.001% case, indicating a thin mixed-phase layer. These cloud properties are quantitatively very similar to those in homogeneous freezing (see the black line in Figure 2), which implies that the IN concentration is too low to make noticeable effect on ice nucleation, and that homogeneous freezing is the main mechanism. As NIN/Nd increases from 0.05 to 0.08%, ice concentration decreases, and the NIN/Nd = 0.08% case has the lowest ice concentration because heterogeneous freezing starts to dominate ice nucleation while homogeneous freezing gradually plays a smaller role. Ice concentration is higher than IN concentration due to the contribution from homogeneous freezing in these cases. The freezing nucleation height is lowered to about 5 km and the glaciation height is at about 6–7 km. These cases generally have a much thicker mixed-phase layer because the concentration of ice particles produced from heterogeneous freezing is still low. The liquid droplets evaporate slowly during the Bergeron process and may be carried up to the height for homogeneous freezing. When NIN/Nd is changed from 0.1% to 10%, ice concentration is greatly increased and the size is decreased. Ice concentration is also lower than or equal to IN concentration, which suggests that homogeneous freezing does not contribute to ice nucleation when there are more IN present. The height of freezing nucleation is lowered from 4.8 km to 4.7 km as IN increases (Figure 5c). The height of glaciation is also close to the height of nucleation, indicating that the concentration of ice particles produced from heterogeneous freezing is high enough for Bergeron process to occur quickly.
 We now focus on the cases with NIN/Nd = 0.07% and 0.08% to study the competition between homogeneous and heterogeneous freezing, and the cases with NIN/Nd = 1% and 10% to study the effect of IN concentration on heterogeneous freezing. Figure 6 shows ice spectra for the four cases. It is seen that the NIN/Nd = 0.07% and 0.08% cases have two size modes (Figures 6a–b) but the other two cases only have one size mode (Figures 6c–d). The two size modes are produced by heterogeneous freezing at lower height and homogeneous freezing at higher height, respectively. There is a gap between the two size modes because droplets containing IN have all frozen; hence, no IN are available for further heterogeneous freezing; the temperature is also not low enough for homogeneous freezing to occur. Therefore, there is no ice forming in that size range. For the cases with NIN/Nd = 1% and 10%, ice is produced only by heterogeneous freezing. Increasing IN can progressively lead to more ice and smaller ice size. In the case of NIN/Nd = 10%, ice concentration is very high and no ice can grow to radius larger than 70 µm. For each size mode, ice particles at the larger size end are produced earlier at lower altitude, where the freezing nucleation rate is relatively small and leads to low ice concentration; ice particles at the smaller size end are produced later at higher altitude, where droplet number concentration is low due to freezing and the Bergeron process, leading to low ice concentration. Ice concentration thus has a maximum at the intermediate size.
 Figure 7 shows profiles of ice concentration, ice effective radius, and saturation ratios with respect to water and ice for the same IN concentrations as in Figure 6. For the NIN/Nd = 0.07% case, the increase of ice concentration below 5 km corresponds to the heterogeneous mode. A mixed-phase layer exists between 5 km and 7 km, where the saturation ratio with respect to ice is very high, and ice particles grow due to Bergeron process (also see Figure 6a). From 7 km to 7.2 km, ice number concentration increases and ice effective radius decreases because of the rapid formation of small ice crystals through homogeneous freezing. The cloud glaciates and no liquid phase exists to maintain the high saturation ratio, so that the saturation ratio rapidly decreases with height. Above 7.2 km, ice concentration is constant but ice effective radius becomes even smaller due to the depositional growth. The reason is that the ice volume concentration is more dominated by the large size mode, and that the ice surface area concentration is relatively more affected by the small size mode. The growth of the numerous small ice particles from homogeneous freezing leads to a faster increase in surface area than in volume, hence a decrease in ice effective radius (ice volume concentration/ice surface area concentration) with height. Similar to the NIN/Nd = 0.07% case, the NIN/Nd = 0.08% case has competition between heterogeneous and homogeneous freezing. Although the ice crystals from the homogeneous mode are too few to be noticeable in ice concentration and to affect ice effective radius, they can make the saturation ratios to decrease rapidly between 7 and 7.2 km. The saturation ratio with respect to ice does not decrease to 1.0 at 8 km, because this case has the minimum number of ice crystals from 7 to 8 km, and hence the minimum consumption of water vapor as ice crystals grow through vapor deposition. In the cases with NIN/Nd = 1% and 10%, ice number concentration increases rapidly with height between 4.7 km and 5 km (−16 to −18°C), as heterogeneous nucleation occurs. Above 5 km, ice number concentration becomes constant because there is no more IN for new ice to form through heterogeneous freezing. Ice effective radius and IWC increase rapidly while the saturation ratio drops rapidly from 5 to 5.5 km (−21°C). The liquid phase totally evaporates through Bergeron process in this layer and cannot reach the height for homogeneous freezing. Another reason for constant ice number is that the collection process of ice crystals is not considered. Ice effective radius and IWC then increase relatively slowly above 5.5 km due to vapor deposition. The mixed-phase layer is therefore thinner and the height of glaciation is lower. Results here indicate that IN number concentration can significantly affect ice number concentration, ice effective radius, the height of freezing nucleation, the thickness of the mixed-phase layer, and the height of glaciation.
 Note that the threshold for freezing nucleation in this study can affect the height of freezing nucleation, the height of glaciation, the ice number concentration, and other cloud properties. For example, by using the current threshold setting (0.001 s−1), heterogeneous ice formation occurs at about −16°C (Figure 1), which corresponds to about 4.7 km (Figure 7). If the threshold for freezing nucleation is set to be 0.00001 s−1, ice forms at about −15°C (Figure 1). If the threshold is changed to 1 s−1, ice forms at about −18°C (Figure 1). Increasing the threshold from 0.001 to 1 s−1 can cause the height of freezing nucleation to be increased by about 0.2 km and ice number concentration to be higher for all CCN conditions in both homogeneous and heterogeneous freezing. However, the setup of this threshold does not affect the sensitivity of cloud properties to CCN. The polluted case still produces more ice crystals compared to the clean case. The threshold also does not affect the sensitivity of cloud properties to IN. The effect of IN can still be noticed when using a different threshold.
 Figure 8 is a summary of IN concentration on ice formation at updrafts of 0.1 and 10 m s−1. It was shown in Figure 5 that, for w = 0.5 m s−1, NIN/Nd needs to be 0.08% for heterogeneous freezing to start dominating ice nucleation. It can be seen from Figure 8 that, when w = 0.1 m s−1, NIN/Nd only needs to be 0.01% (NIN = 0.0087 mg−1) for heterogeneous freezing to dominate over homogeneous freezing. However, when w = 10 m s−1, NIN/Nd needs to be close to 10% (NIN = 8.9 mg−1) for heterogeneous freezing to dominate. Cloud properties also do not change much with IN concentration when NIN/Nd is smaller than 10% at the higher updraft. This is very similar to results for cirrus clouds [Barahona and Nenes, 2009; Eidhammer et al., 2009]. As mentioned earlier in this study, IN concentration ranges from 0.0004 cm−3 to even larger than 1 cm−3 in the atmosphere. This wide range of IN concentration can lead to quite different cloud properties under different updraft velocities.
 As shown in Figure 9, we also test the sensitivity of ice nucleation to IN size and contact angle at w = 0.5 m s−1. The soluble aerosol concentration Na is fixed at 100 mg−1 (i.e., Nd is 89 mg−1), and NIN/Nd is fixed at 0.1% (NIN = 0.089 mg−1). Larger IN and larger contact coefficient (i.e., smaller contact angle) lead to lower ice concentration, larger ice effective radius, and lower height of freezing nucleation, because heterogeneous freezing dominates ice nucleation. The freezing mechanism changes from totally heterogeneous freezing to almost homogeneous freezing as IN size and contact coefficient are decreased. Note that the discontinuity in ice effective radius in Figure 9b for lower contact coefficients (i.e., larger contact angle) is mainly due to the fact that homogeneous freezing occurs and the smaller size mode (see Figures 6a and 6b) contributes to ice particle spectra. When heterogeneous freezing dominates, IN with bigger size and contact coefficient makes the freezing process easier to occur, i.e., heterogeneous freezing occurs at higher temperature, which is consistent with previous studies [Welti et al., 2012]. It is also seen that ice concentration and ice effective radius are more sensitive to contact coefficient than to IN size. For different CCN and IN concentrations, results (not shown here) still suggest that IN size and contact coefficient have strong effects. Similarly, larger IN and larger contact coefficient lead to lower ice concentration, larger ice effective radius, and lower height of freezing nucleation, because heterogeneous freezing dominates over homogeneous freezing.
4.3 Ice Crystal Shapes
 This study assumes that ice particles are spherical. However, ice crystals tend to be either columnar or plate-like in real clouds, and have different depositional growth rates than spheres. This could potentially lead to a different speed of the Bergeron process and a different thickness of the mixed-phase layer. Here we investigate whether the assumption of ice crystal shapes affects the response of clouds to CCN and IN. The nonspherical shape of an ice crystal introduces complicated boundary conditions. We therefore use idealized shapes, spheroids, to represent the columnar and plate-like ice crystals in the temperature range in this study, as shown in Figure 10. Ice particles are spherical when they first form and grow from spheres to spheroids through vapor deposition. The inherent growth ratio Γ(T) is smaller than 1 at temperatures from −10 to −20°C and larger than 1 at temperatures below −20°C [Chen and Lamb, 1994]. Therefore, prolate (columnar) ice crystals are produced at the threshold temperature of homogeneous freezing (−34.7°C), while oblate (plate-like) ice crystals are produced at the threshold temperature of heterogeneous freezing (−16°C).
 Figure 11 compares cloud properties at z = 8 km when assuming ice particles are spheroids and spheres. We test if the CCN effect and the IN effect depend on the assumption of ice crystal shapes. An updraft of 0.5 m s−1, fixed IN radius of 500 nm, and contact coefficient of 0.5 are used. Figure 11a shows that in homogeneous freezing, assuming prolate ice crystals nearly produces the same number of ice as assuming spherical ice crystals. CCN effect on ice number concentration is still obvious when assuming prolate shapes. That is to say, CCN effect does not depend on the assumption of ice crystal shapes. Figure 11b shows that, in heterogeneous freezing, assuming oblate ice crystals also produces similar number of ice as assuming spherical ice. It is seen here that IN effect on ice crystal number concentration is also obvious when assuming oblate shapes. In Figures 11c and 11d, we choose the biggest ice bins to show whether the effects of CCN and IN on ice crystal sizes depends on the shapes. For homogeneous freezing, the case with higher CCN produces smaller ice crystals with larger aspect ratio (c/a). For heterogeneous freezing, the case with higher IN concentration has smaller size and higher ratio of a/c. IN effect is therefore not dependent on ice crystal shapes.
5 Conclusions and Discussion
 This study investigates mixed-phase cloud properties affected by aerosol particles (including CCN concentration, IN concentration, IN size, and contact angle). An adiabatic cloud parcel model with a stochastic, time-dependent approach for ice nucleation is used. This study does not consider collision-coalescence, collection, and sedimentation, and therefore the simulations are for idealized nonprecipitating clouds.
 It is shown that, for homogeneous freezing, increasing CCN number concentration can lead to more ice crystals and a slight elevation of the height of freezing nucleation. But the sensitivity of ice cloud properties to CCN number concentration depends on the updraft. Ice number concentration is more sensitive to CCN at higher updrafts. The height of freezing nucleation is also more affected by CCN at higher updrafts. Whether the elevated nucleation height can cause any change in cloud dynamics needs further study. Note that this is an idealized simulation where precipitation process is not considered. In real clouds, increased CCN can suppress the warm rain process and more liquid water may freeze at an elevated nucleation height, leading to a potentially stronger dynamical effects. For heterogeneous freezing, although CCN can change the liquid-phase cloud properties, the ice formation process is the same as long as the IN availability is the same because heterogeneous freezing nucleation rate is dependent on IN size and contact coefficient, and not dependent on droplet size in this study.
 Ice number concentration has a nonmonotonic relationship with IN number concentration, because homogeneous and heterogeneous freezing processes compete and can both contribute to ice formation as cloud parcels are lifted to the homogeneous freezing level. When NIN/Nd is less than 0.05%, the freezing process is dominated by homogeneous nucleation and IN concentration does not have a noticeable effect. When NIN/Nd is higher than 0.1%, the freezing process is dominated by heterogeneous nucleation, and ice number concentration can be increased greatly by IN concentration. For the cases in between (NIN/Nd = 0.07%, 0.08%), IN concentration can affect ice formation through both heterogeneous freezing and the following homogeneous freezing. Ice formation is a result of the competition between heterogeneous and homogeneous freezing though the following mechanism: The number concentration of ice from heterogeneous freezing is low, leading to a slow Bergeron process and a thick mixed-phase layer; As the parcel reaches the temperature for homogeneous nucleation to occur, there are still droplets left to freeze. Therefore, both heterogeneous and homogeneous freezing contribute to ice nucleation, and the ice spectra have two size modes.
 This study also found that, when vertical velocity is low (as in stratiform clouds), not many IN are needed for heterogeneous freezing to dominate or to compete with homogeneous freezing. However, when vertical velocity is high, more IN is needed for heterogeneous freezing to dominate.
 The sensitivity tests show that the ice nucleation changes from totally heterogeneous freezing with big IN size and contact coefficient to almost homogeneous freezing with small IN size and contact coefficient. Ice concentration and ice effective radius are more sensitive to contact coefficient than to IN size.
 We finally investigate whether the assumption of ice crystal shapes affects the cloud response to aerosols. Ice particles are assumed as spheres in this study. Spheres obviously have different depositional growth rates than plates/columns. This could potentially lead to different speeds of the Bergeron process and different thicknesses of the mixed-phase layer. When assuming spheroidal ice crystals, ice number concentration is nearly not changed compared to assuming spherical ice particles. But the CCN effect and the IN effect on clouds are not sensitive to the assumption of ice crystal shapes.
 This study was supported by Chinese 973 Program under project 2013CB955803. We greatly thank Dennis Lamb and Graham Feingold for providing helpful suggestions and comments on this manuscript. We are also very grateful to the three anonymous reviewers and the editor for many suggestions that improved this manuscript.