The experimental data used as a basis for the parameterizations include the following ice nucleating particle types: bacteria, pollen, feldspar, illite, and kaolinite. Murray et al. (2011) investigated kaolinite KGa-1b and Atkinson et al. (2013) K-feldspar. Illite NX was studied by Broadley et al. (2012) and recently by Hiranuma et al. (2015). The latter publication summarized the results from 17 experimental techniques. Wex et al. (2015) include the results from seven experimental techniques using Snomax^® as a proxy for bacteria; previous data for pollen (Diehl et al., 2002; v. Blohn et al., 2005) were newly evaluated. As a parameter for immersion freezing, the number of active sites per unit mass nm was selected as this is derived directly from the particle mass concentration in the drops during the experiments. Some references give as an outcome from their measurements the surface density of active sites per unit particle surface ns; however, this parameter was derived afterwards by using a specific particle surface.

For kaolinite KGa-1b, K-feldspar, and tree and grass pollen, an exponential increase of nm with temperature T was found, which is described by nm=exp⁡(aimm+bimmTs), where nm is in g-1, aimm and bimm are particle-related constants, Ts=T0-T, and T0= 0 ∘C, with T in ∘C. The constants for all particle types are listed in Table 1. Given in Table 1 are also the parameters Tini and Tlim, representing the onset of immersion freezing during experiments and the lowest temperature that was investigated in the experiments, respectively.

Based on the best fit of the data of Murray et al. (2011), the constants in Eq. (2) were derived for kaolinite KGa-1b by using an average specific particle surface area of 11.8 m2 g-1 as given by Murray et al. (2011); the result is shown in Fig. 1 as an orange line. The solid part of the line represents the range that is validated by measurements of Murray et al. (2011) while the dotted part shows an extrapolation towards higher temperatures. The value of Tini is based on earlier measurements of Pitter and Pruppacher (1973). The same was performed for K-feldspar by using the best fit to the experimental data given by Atkinson et al. (2013), which were changed into Eq. (2) by using the specific particle surface area of 3.2 m2 g-1 from Atkinson et al. (2013). The parameters are listed in Table 1, the result of Eq. (2) is shown in Fig. 1 as a red line. Regarding pollen, previous data from Diehl et al. (2002) and v. Blohn et al. (2005) were evaluated. From the frozen fractions of drops, the drop volume, and the mass of pollen in the drops, the numbers of active sites nm as functions of temperature were calculated according to (e.g., Murray et al., 2011) nm=-ln⁡( 1-fice(T))cpollenVdrop, where fice(T) is the fraction of frozen drop at temperature T, cpollen the pollen concentration per drop, and Vdrop the drop volume. Data for tree and grass pollen were summarized leading to two parameterizations for tree and grass pollen as given in Eq. (2). The constants are listed in Table 1 and the results of Eq. (2) are shown in Fig. 1 as light green lines. The solid parts of the lines represent the ranges, which are validated by measurements, while the dotted parts show extrapolations towards lower temperatures.

Regarding illite NX and Snomax^®, parameterizations were suggested by Broadley et al. (2012), Hiranuma et al. (2015), and Wex et al. (2015). In order to derive analogue descriptions, as in the cases of the other particle types, these parameterizations were replaced by new ones following Eq. (2). Regarding illite NX, the parameterization was derived from the data reported by Hiranuma et al. (2015). The fit as given by Hiranuma et al. (2015) was changed into an expression for nm by using the specific particle surface area of 124.4 m2 g-1 of the illite NX sample (Hiranuma et al., 2015). From the nonlinear curve (solid blue line in Fig. 2) a linear regression line was derived, which is included in Fig. 1 (blue solid line). For Snomax^®, it was concluded in Wex et al. (2015) that the average nm values (given in Fig. 2 as open green symbols) are very well represented by the Hartmann et al. (2013) parameterization (solid green line in Fig. 2). One can notice a linear increase in the temperature range down to approximately -9 ∘C and a further progress on a maximum value of 1.4 × 1012 g-1. Thus, a linear regression curve was derived for the increase of nm until the maximum value was reached; it is included in Fig. 2 as a solid green line.

It can be noted from Fig. 1 that bacteria act at the highest temperatures starting not far below 0 ∘C. Pollen lie in the range of the mineral dust particles. The differences between the mineral dust types are significant, with feldspar being the most efficient one and kaolinite the least efficient one.

The previous Eq. (1), which gives the freezing rate of the supercooled drops, had to be replaced by a similar expression that couples the number of active sites and the mass of insoluble particles in the drops. According to the singular description of heterogeneous freezing (Vali, 1971; Broadley et al., 2012), the frozen fraction of drops fice is given by fice(T)=Nf(T)Nliq=1-exp⁡(-nm(T)mpid), where Nf(T) is the number of frozen drops at temperature T, Nliq the number of liquid drops, mpid the mass of particles immersed in the drops, and nm(T) the number of active sites per unit mass at temperature T, which is related to the cumulative nucleus spectrum K(T) per unit mass per unit temperature: nm(T)=∫T0TK(T)dT when lowering the temperature from T0=0 ∘C to T. From Eqs. (4) and (5) an expression for the change of the number of frozen drops ΔNf per temperature interval ΔT can be derived (Connolly et al., 2009): ΔNf=Nliq(1-exp⁡(-K(T)mpidΔT), where Nliq is the number of supercooled liquid drops and mpid the mass of particles immersed in the drop. Thus, Eq. (1) can be replaced by dNfdt=Nliq1-exp⁡(-K(T)mpiddT)dt.

As the aerosol particles are internally mixed, one can assume that only a fraction of the insoluble mass per drop consists of ice nucleating material. Thus, the mass mpid was reduced by a factor FINP, so that only this mass fraction accounts for possible numbers of active sites nm and Eq. (7) was modified to dNfdt=Nliq1-exp⁡(-K(T)mpidFINPdT)dt.

A similar treatment was applied in Diehl and Wurzler (2010). From Eqs. (2) and (5) it follows that K(T)=dnm(T)dT=bimm exp⁡(aimm+bimmTs).

In Eq. (9) the freezing point depression due to the content of soluble material in the drops is considered; for details see Diehl and Wurzler (2004). In the model simulations, immersion freezing starts at the particle-related temperature Tini, and at temperatures below Tlim it is assumed that the numbers of active sites stay constant. The sizes of possibly ice nucleating particles are restricted; i.e., dust particles must be larger than 0.1 µm in diameter and bacteria are limited to their typical diameters of 0.3 to 2 µm (Matthias-Maser and Jaenicke, 1995). Pollen are large particles of 10 µm at least (Straka, 1975); however, for the present simulations a lower limit of 2 µm was selected to allow for at least some freezing by pollen (see Sect. 3, initial dry particle number size distributions).

During the model simulations the content of insoluble particles per drop varies by several orders of magnitude, which is due to the sizes of the condensation particles, the uptake of particles by the drops via impaction scavenging, and the drop collisions followed by coalescence. Thus, not all drops of the same sizes freeze at certain temperatures, which describes the situation in real clouds (Diehl and Wurzler, 2004).

Measurements, which give a direct correlation between particle size and contact freezing efficiency, are not yet available for a wider range of ice nucleating particle types. Therefore, for the present parameterization different size classes were examined. Hoffmann et al. (2013a, b), and N. Hoffmann and A. Kiselev (personal communication, 2014) performed contact freezing experiments using an electrodynamical balance with monodisperse particles of 150 to 750 nm diameter. The results showed rather low median freezing temperatures T50 (the temperature where 50 % of an observed drop population freeze), e.g., -34 ∘C for illite NX, -32.5 ∘C for kaolinite Fluka, and less than -25 ∘C for Snomax^®, in comparison to measurements with polydisperse particle samples.

Those experiments with polydisperse particle samples were performed at the UCLA vertical wind tunnel by Levin and Yankofsky (1983) and Pitter and Pruppacher (1973). The latter studied kaolinite and montmorillonite with particle diameters between 0.1 and 10 µm with a mode between 1 and 2 µm and measured median freezing temperatures of -12 and -8 ∘C, respectively. Levin and Yankofsky found a median freezing temperature of -4.5 ∘C for bacteria. Diehl et al. (2012) investigated polydisperse mineral particles with supercooled drops suspended in an acoustic levitator. Median freezing temperatures were -11.5 ∘C for illite NX and -8.7 ∘C for montmorillonite K10. The latter agrees very well with the value found by Pitter and Pruppacher (1973) within the measurement error (1 K). Extrapolating the data of Hoffmann et al. (2013b) for illite NX towards larger particle sizes as shown in Fig. 3 indicates that a median freezing temperature of -11.5 ∘C could be affected by dust particles with sizes between 2.3 and 3.0 µm in diameter (Diehl et al., 2012). Those particles are part of the polydisperse particle spectrum of illite NX (Hiranuma et al., 2015). Thus, one might conclude that the median freezing temperatures determined for polydisperse particle samples are affected by larger particles present in the size spectrum. This is probably the case also for previous findings measured with polydisperse particle samples.

Therefore, as an approximation, data obtained from polydisperse particle samples were used in the present parameterizations for particles with diameters mostly larger than 1 µm, and at least larger than 0.7 µm. Data obtained from monodisperse particle samples were taken for ranges around the respective particle sizes.

The experimental data used as basis for the parameterizations include the following ice nucleating particle types: bacteria, feldspar, montmorillonite, illite, and kaolinite. In most cases, a linear correlation between the frozen fraction of drops and the temperature was found. From the experimental data, regression lines were calculated, which are shown in Fig. 4 for various particle types (marked by different colors) and particle sizes (marked by different line styles).

Particle sizes of bacteria were restricted to their typical sizes with diameters between 0.3 and 2 µm (Matthias-Maser and Jaenicke, 1995). Measurements of N. Hoffmann and A. Kiselev (personal communication, 2014) were performed with monodisperse Snomax^® particles of 0.32 and 0.55 µm diameter. These data were used for size ranges from 0.3 to 0.5 µm and from 0.5 to 0.7 µm. For particles between 0.7 and 2 µm the data of Levin and Yankofsky (1983) were taken. The results are given in Fig. 4 as green lines.

For mineral dust particles, a lower size limit of 0.1 µm in diameter was assumed. Illite NX particles were investigated by Hoffmann et al. (2013b) with 0.15, 0.32, 0.55, and 0.75 µm particles. These data were taken for size ranges from 0.1 to 0.2 µm, 0.2 to 0.4 µm, 0.4 to 0.6 µm, and 0.6 to 0.8 µm. For larger particles, data from Diehl et al. (2012) were used. The results are shown in Fig. 4 as blue lines. K-feldspar was investigated by N. Hoffmann and A. Kiselev (personal communication, 2014) with 0.32 and 0.55 µm particles and by K. Diehl and S. K. Mitra (personal communication, 2014) with polydisperse particles. Here three size ranges were defined, from 0.1 to 0.4, 0.4 to 0.8 µm, and larger than 0.8 µm. The data are marked in Fig. 4 as red lines. For kaolinite and montmorillonite, two size ranges were also specified, from 0.1 and 1 µm and larger than 1 µm. The data for the larger particles sizes were both taken from Pitter and Pruppacher (1973; polydisperse particle samples). Results from Hoffmann et al. (2013a) were used for the smaller size range of kaolinite. Under the assumption that the differences between larger and smaller ice nucleating particles (INP) are similar for kaolinite and montmorillonite the data for the smaller size range of montmorillonite were obtained by a parallel shifting analogue to kaolinite. In Fig. 4, results for kaolinite are given in orange, results for montmorillonite in cyan.

From Fig. 4 it can be noted that particles in the larger size ranges affect freezing already at temperatures around -10 ∘C, while smaller particles become active in a temperature range around -25 ∘C. Bacteria act at the highest temperatures, and kaolinite and illite at the lowest. For all particle types and sizes except bacteria smaller than 0.7 µm, the frozen fraction of drops increases linearly with temperature T (given in ∘C) according to Diehl et al. (2006): NfNliq=aconT+bcon, where Nf is the number of frozen drops, Nliq the number of liquid drops colliding with inactivated particles at temperature T, and the constants acon and bcon. Note that in Eq. (10) the frozen fraction is limited to values between 0 and 1. In Diehl et al. (2006), the constants were given for several particle types independent of their sizes, while in the present parameterization the constants acon and bcon are size resolved. They are listed in Tables 2 and 3. In the case of small bacteria, the equation to calculate the frozen fraction of drops with respect to the temperature T (in ∘C) has the form NfNliq=acon+b1,conT+b2,conT2+b3,conT3.

The size-resolved constants acon, b1,con, b2,con, and b3,con are given in Table 2.

The description of contact freezing includes the following conditions: (1) inactivated particles have to be present, and (2) particles and supercooled drops have to collide with each other. Furthermore, the sizes of the particles allowed as activating ice nucleating particles are restricted as for deposition freezing (i.e., dust particles > 0.1 µm in diameter, bacteria 0.3 to 2 µm in diameter).

The presence of inactivated particles is always the case during the air parcel ascent because of entrainment; i.e., new inactivated particles are continuously mixed in at the edges of the simulated cloud. However, in the presently employed air parcel model the particles are in equilibrium with respect to the water vapor in their environment and, thus, they take up some water due to their size and soluble fraction. As introduced in Diehl et al. (2006) the dryness of a potential ice nucleating particle is defined by the assumption that the water mass should be smaller than half of the dry particle mass.

The second condition is considered by a collision kernel K calculated for supercooled drops and particles (Kerkweg et al., 2003; for more details see Diehl et al., 2006): K=EcollV∞,drop-V∞,ap⋅πrdrop-r∞2, where V∞,drop and V∞,ap are the terminal velocities of the drop and the particle, respectively, and rdrop and rap the drop and particle radii, respectively. The collision kernel shows the highest values for collisions between large drops and particles (Diehl et al., 2006); i.e., contact freezing is most efficient when large supercooled drops and particles are present. If during the model simulations inactivated particles collide with supercooled drops, the number of frozen drops is calculated. It is assumed that only a fraction FINP of the aerosol particles is able to act as ice nucleating particles. Only drops that collide with those INP are allowed to freeze, which is included in the following modified Eqs. (13) and (14): Nf=FINPNliq(aconT+bcon),Nf=FINPNliq(acon+b1,conT+b2,conT2+b3,conT3).

This is under the requirement that these equations are based on measurements with one drop–particle collision per freezing event so that the fraction of frozen drops in Eqs. (13) and (14) can be set equal to the freezing probability (Ladino Moreno et al., 2013). This requirement is fully achieved in the experiments of Hoffmann et al. (2013a, b), and N. Hoffmann and A. Kiselev (personal communication, 2014). During the experiments of Pitter and Pruppacher (1973), Levin and Yankofsky (1983), and Diehl et al. (2012) the number of collisions per freezing event is not documented. Single supercooled drops were freely levitated (in a wind tunnel or an acoustic levitator) while one burst of INP was blown on it. Therefore, as the particles collided almost simultaneously with the supercooled drop, one could assume that in case the drop froze this was triggered by the first collision.

The experimental data used as basis include the following ice nucleating particle types: bacteria, feldspar, illite, and Saharan and Asian dust. The measurements were performed with INP counters FRIDGE (FRankfurt Immersion and Deposition freezinG Experiment) or with a continuous flow diffusion chamber (CFDC). Data for Asian and Saharan dust were taken from measurements with the FRIDGE-TAU (FRankfurt Ice-nuclei Deposition freezinG Experiment, the Tel Aviv University version) (K. Ardon-Dryer and Z. Levin, personal communication, 2012), data for illite NX and Snomax^® from INUIT FRIDGE experiments (Hiranuma et al., 2015; A. Danielczok and H. Bingemer, personal communication, 2014; Weber, 2014), and data for illite IMt1 and K-feldspar from CFDC measurements (Yakobi-Hancock et al., 2013). The Snomax^® data were considered as representative for bacteria in the model simulations.

K. Ardon-Dryer and Z. Levin (personal communication, 2012) measured the activation of Saharan and Asian dust particles at different ice supersaturations and temperatures and observed an increase of the activated particles with supersaturation but no temperature dependence in the observed temperature range between -15 and -20 ∘C. Figure 5 shows the data of K. Ardon-Dryer and Z. Levin (personal communication, 2012) as activated fraction of particles as a function of ice supersaturation. From all data, mean values were calculated and regression lines were derived. These are included in Fig. 6 as cyan and pink solid lines.

The INUIT FRIDGE measurements were performed with illite NX and Snomax^® in a temperature range from -10 to -25 ∘C. A significant temperature dependence was not observed and, therefore, the activated particle fraction was derived as a function of ice supersaturation only. As ice saturation is dependent on temperature, an implicit dependence of deposition freezing on temperature is already incorporated. Regression lines were calculated based on the average values. They are given in Fig. 6 as blue (illite NX) and green (Snomax^®) solid lines. It can be noticed that illite NX is much more efficient than the Saharan and Asian dust particles, which are characterized by a mixed composition. Asian dust mostly consists of quartz (Möhler et al., 2006), which acts in the deposition mode at lower temperatures than illite (Zimmermann et al., 2008). Also Saharan dust contains a quartz fraction of nearly one-third (Möhler et al., 2006). As expected the activated fractions of Snomax^® particles are the highest. Measurements with a CFDC were performed at -40 ∘C with illite IMt1 and K-feldspar particles (Yakobi-Hancock et al., 2013). Regression lines were derived from the data and are given in Fig. 6 as blue (illite IMt1) and red (K-feldspar) broken lines, respectively.

For illite NX and Snomax^®, data are available not only for polydisperse particle samples but also for samples with distinct particle sizes (A. Danielczok and H. Bingemer, personal communication, 2014; Weber, 2014). In case of illite, monodisperse particles with diameters of 300 and 500 nm resulted in higher activated fractions for the larger particle sizes, but in both cases the activated fractions were higher than in case of the polydisperse particle samples. In case of Snomax^®, monodisperse particles of 100, 200, 300, and 500 nm were investigated finding again higher activated fractions for the larger particle sizes than in the polydisperse case. As particle-size-resolved measurements are not available for all particle types it was decided to treat deposition freezing size independent in the model. As the activated fractions of the polydisperse particle samples (which are the basis of the present parameterizations) are lower than the ones of monodisperse samples, this would not lead to an overestimation of deposition freezing.

For all particle types, an exponential increase of the activated fraction with the ice supersaturation was found, which is described by NactNtotal=exp⁡(adep+bdepsice), where Nact is the number of activated particles, Ntotal the total particle number, sice the ice supersaturation given in %, and adep and bdep are particle-related constants. The values of adep and bdep are listed in Table 4 for the different particle types. Note that the activated fraction in Eq. (15) is limited to values between 0 and 1.

Although data were available only for ice supersaturation ranges below 25 % (bacteria, illite NX, Saharan and Asian dust) and above 20 % (illite IMt1 and feldspar), respectively, due to the investigated temperature ranges, it is assumed that Eq. (15) is valid for the complete ice supersaturation range. However, according to the onset of deposition freezing in the experiments, lower limits of ice supersaturation and temperature as measured during the FRIDGE experiments were set in the model. That means, deposition freezing of illite IMt1 and feldspar start at the same conditions as illite NX. In the model simulations, the two types of illite were taken as upper and lower limits and referred to as illite 1 and illite 2. The values are given in Table 4.

Conditions for deposition freezing are (1) inactivated particles have to be present as it is required for contact freezing; i.e., the same particles may affect deposition or contact freezing. Additionally, (2) there are two size conditions. First, the inactivated particles have to exceed a critical germ size r∗, depending on temperature and ice supersaturation, which is according to Pruppacher and Klett (2010): r∗=2MWσi,vRTρiceln⁡Sice, where MW is the molecular weight of water, σi,v the surface tension, R the universal gas constant, T the temperature, ρice the density of ice, and Sice the ice saturation ratio. However, this condition is actually redundant as it excludes particles smaller than approximately 0.01 µm and, additionally, size restrictions of the ice nucleating particles were assumed: dust particles larger than 0.1 µm in diameter and bacteria between their typical size range of 0.3 and 2 µm in diameter (Matthias-Maser and Jaenicke, 1995).

In each time step it is checked if temperature and ice supersaturation are above the limit values Tini and sice,ini, if yes it is checked which available inactivated particles exceed the size limits, and from these the activated fraction is calculated. It is assumed that only a part FINP of the available particles is able to act as ice nucleating particles. Therefore, the total number of particles in Eq. (15) is reduced: Nact=FINPNtotalexp⁡(adep+bdepsice).

The activated particles are moved to the ice particle spectrum and grow further by water vapor deposition and they may serve as germs for secondary ice formation; i.e., they may initiate freezing of supercooled drops by collision (Diehl et al., 2006).

With the regional haze particles (larger particles but smaller numbers), less ice was formed in most cases than with the average continental particles (smaller particles but larger number). Less but mostly large drops develop with the regional haze distribution and many but smaller drops with average continental distribution (see Fig. 9). On the other hand, a few still larger drops evolved from the tail of the continental spectrum. The presence of large drops favors immersion and contact freezing (immersion mode: the drops contain more insoluble material; contact mode: the collision kernel between large drops and particles is enhanced); see Tables 5 and 6. This was suggested already by the findings of Diehl et al. (2006) and confirmed by, e.g., Lance et al. (2011), who concluded from their observations that the drop size distribution modulates ice processes in mixed-phase clouds.

In the deposition mode (Table 7) not the drops but the interstitial particles are relevant. More inactivated particles are present in the regional haze case than in the continental case (see Fig. 9), so that because of the higher competition between the many particles, less ice particles develop by the deposition of water vapor.

Immersion freezing affects the most mixed-phase clouds with ice water fractions of more than 0.5 and even ice clouds with all kinds of investigated INP (Table 5). Pollen are, of course, disadvantaged because of their large size, kaolinite particles are the less efficient mineral dust particles while bacteria and feldspar are the most efficient INP. With bacteria, freezing occurred already with potential fractions of INP FINP as low as 0.001 %. With mineral dust, potential fractions of INP FINP of 0.01 % (feldspar, illite) or 0.1 % (kaolinite) were necessary. Ice formation was sensitive to the type of mineral dust as well as to the potential fraction of INP. For instance, under the same conditions an ice cloud could form with feldspar but a mixed-phase cloud with only a small ice water fraction with kaolinite.

In the contact mode, there is very little ice formation (Table 6). Only liquid clouds formed with the regional haze particle distribution, but in case of the average continental distribution at least some mixed-phase clouds formed with ice water fractions between 0.1 and 0.3. High potential fractions of INP FINP of 10 % were required. Bacteria did not affect ice formation, this probably results from the restricted particle sizes. The type of mineral dust decides whether mixed-phase clouds are formed (feldspar, montmorillonite) or not (illite, kaolinite).

Effective deposition INP are bacteria, feldspar, and illite while the mixed particle samples Saharan and Asian dust form liquid clouds only. Mixed-phase clouds formed with high potential INP fractions FINP between 1 and 10 %. The results for the different mineral dust types are rather similar, but mixed-phase clouds were formed with pure minerals only but not with mixtures (Saharan and Asian dust) that are dominated by quartz (see Sect. 2.3.1).

Ervens et al. (2011), who investigated the impact of immersion and deposition freezing modes on ice formation in mixed-phase clouds, stated that immersion freezing formed less ice than deposition freezing because of lower onset temperatures in the immersion mode. However, this could not be confirmed in the present study. Newer laboratory measurements, which were used as basis of the parameterizations, showed in contrary higher initial temperatures in the immersion mode than in the deposition mode (see Tables 1 and 4 with references therein) and, thus, immersion freezing produced more ice than deposition freezing. This agrees, on the other hand, with the findings of de Boer et al. (2011) and Lance et al. (2011) that liquid-dependent ice nucleation modes are dominant.

In all freezing modes, cases with the lowest temperature difference (ΔT=1.5 K) and the corresponding highest final temperature of -24.5 ∘C showed hardly any ice formation. Exceptions are only cases with bacteria and feldspar in the immersion mode (Table 5). Contact ice formation did not occur with lower ΔT as the temperatures reached during the ascent of the air parcel were not low enough to give the more effective smaller particles the chance to act (Table 6). In the deposition mode, obviously temperatures below -25 ∘C are required to develop mixed-phase clouds (Table 7) as at higher temperatures the ice supersaturation is still too low. On the other hand, the largest ΔT of 3 K also hindered ice formation in contact and deposition modes, which might be affected by the presence of less interstitial particles.

Regarding the cases of immersion freezing listed in Table 5, one has to look at the composition of cloud residuals that was investigated in several field campaigns. For example, Kamphus et al. (2010) measured 8 % minerals in cloud droplet residuals and Hiranuma et al. (2013) found 3 % mineral dust particles in cloud droplet residuals for all particle sizes and particle-size-resolved measurements indicated enhanced fractions for larger particles up to 17 %. In the model simulations mixed-phase clouds were formed already with potential INP fractions between 0.01 and 1 %. Thus, the dominant role of immersion freezing with mineral dust seems to be validated. It is confirmed by the fact that during INUIT field campaigns (Worringen et al., 2015; Schmidt et al., 2015) fractions of mineral dust up to 40 % in ice residuals were observed. Similar numbers between 30 and 40 % are reported by Kamphus et al. (2010) in ice particle residuals.

Bacteria need to be present as potential INP only as low as 0.001 to 0.01 % to affect mixed-phase clouds, pollen with 1 % potential INP. For pollen these numbers probably do not represent realistic cases; however, the pollen cases were anyway rather artificial as such large particles (> 10 µm) were actually not part of the aerosol particle spectrum. Bacteria concentrations in cloud water are given as average values of, e.g., 1.5 × 109 m-3 (Sattler et al., 2001), 2 × 1010 m-3 (Bauer et al., 2002), and 7 × 104 m-3 (Amato et al., 2005; see also the review paper of Delort et al., 2010). Unfortunately, field measurements that give the number fractions of bacteria or primary biological aerosol particles (PBAP) related to the total concentrations of in-droplet particles are not available so far. An estimation was undertaken based on the results of Bauer et al. (2002). From field measurements in a continental background site, they determined the numbers of bacterial and fungi cells in cloud water and calculated the corresponding amount that would contribute to the amount of organic carbon (OC). For bacteria this contribution was estimated as 0.01 %. Analyses of cloud droplet residuals show, e.g., fractions of 3 % (Twohy and Anderson, 2008) and 9 % (Hiranuma et al., 2013) organic carbon. Based on these numbers, one could assume that 0.0003 to 0.0009 % of the material contained in cloud drops consists of bacteria. This comes near the fraction of potential ice nucleating particles FINP of 0.001 %, which affects mixed-phase clouds via immersion freezing (see Table 5). Joly et al. (2014) investigated the ice nucleation efficiency of cloud water samples and distinguished total and biological INP. They estimated that – assuming that all biological INP were bacteria – in the temperature range between -8 and -12 ∘C 0.6 to 3.1 % of the bacterial cells present in the cloud water samples could have acted as INP. Taking into account these values, a fraction FINP of 0.001 % is probably still an overestimation of realistic bacterial ice nucleating particles in cloud drops.

Thus, one may conclude that the atmospheric immersion freezing with mineral dust particles will play a dominant role, while ice formation via immersion freezing of bacteria might take place in some extreme cases only.

According to the cases listed in Tables 6 and 7, the formation of mixed-phase clouds via contact or deposition freezing requires that 10 % of background aerosols consist of potential ice nucleating particles. Measurements during an INUIT field campaign on the high Alpine site Jungfraujoch (Schmidt et al., 2015) indicate a fraction of 1 % minerals of the background aerosol; however, this value might be enhanced for continental situations in lower altitudes. A closer look was taken on the total number concentrations of ice nucleating particles acting as contact and deposition freezing particles during the model simulations. As shown in Fig. 9 for two different altitudes and corresponding temperatures, interstitial aerosol particles were always present during the ascent of the air parcel. Among these, only particles larger than 0.1 µm in diameter were allowed to act as ice nucleating particles. The total number concentrations of those during the model simulations reached values up to 4 × 105 m-3 for the average continental and up to 2 × 106 m-3 for the regional haze distribution. Thus, fractions of potential ice nucleating particles FINP of 10 % resulted in particle concentrations available for deposition and contact nucleation of at the most 4 × 104 and 2 × 105 m-3, respectively.

In Saharan dust events, Bangert et al. (2012) measured particle number concentrations up to 5 × 107 m-3. In Hande et al. (2015), simulated mineral dust number concentrations are 4 × 105 m-3 on average up to extreme values of 5.8 × 106 m-3. Thus, the numbers of potential INP used in the model simulations do not exceed realistic particle concentrations of mineral dust.

Near-surface concentrations of bacteria range between 1 × 103 and 5 × 105 m-3 depending on ecotypes (Burrows et al., 2009), but these values are certainly not reached in upper cloud regions. On the other hand, DeLeon-Rodriguez et al. (2013) reported from field measurements in low- and high-altitude air masses that bacterial cells represented nearly 20 % of the total particles in the diameter range between 0.25 and 1 µm. This is approximately the size range of potential ice nucleating particles in the present model simulations. For even larger particles in the coarse-mode high fractions of PBAP (primary biological particles) are also reported by Manninen et al. (2014). However, the numbers of potential INP used in the model simulations probably overestimate real bacteria concentrations.

Considering these factors one may conclude that the conditions for atmospheric deposition and contact freezing could be sufficient in some cases to form mixed-phase clouds from primary ice formation by mineral dust particles. In some extreme cases, the formation of mixed-phase clouds might be possible via deposition nucleation on bacteria. On the other hand, the initial particle spectra used for the present model simulations contain small amounts of particles larger than 1 µm (see Fig. 8), which would be able to act as contact ice nucleating particles much more efficiently (see Sect. 2.2.1). Thus, in cases where larger INP are present in or around atmospheric clouds contact freezing might be significantly enhanced.