A computationally efficient description of heterogeneous freezing: A simplified version of the Soccer ball model


Correspondence to:

D. Niedermeier,



In a recent study, the Soccer ball model (SBM) was introduced for modeling and/or parameterizing heterogeneous ice nucleation processes. The model applies classical nucleation theory. It allows for a consistent description of both apparently singular and stochastic ice nucleation behavior, by distributing contact angles over the nucleation sites of a particle population assuming a Gaussian probability density function. The original SBM utilizes the Monte Carlo technique, which hampers its usage in atmospheric models, as fairly time-consuming calculations must be performed to obtain statistically significant results. Thus, we have developed a simplified and computationally more efficient version of the SBM. We successfully used the new SBM to parameterize experimental nucleation data of, e.g., bacterial ice nucleation. Both SBMs give identical results; however, the new model is computationally less expensive as confirmed by cloud parcel simulations. Therefore, it is a suitable tool for describing heterogeneous ice nucleation processes in atmospheric models.

1 Introduction

Heterogeneous ice nucleation, which is a primary pathway to form ice in the atmosphere, has a large influence on both weather and climate [Murray et al., 2012]. Within the last decades, several techniques have been suggested (summarized in, e.g., Hoose and Möhler [2012] and Murray et al. [2012]) for the description of heterogeneous ice nucleation in cloud models.

On the one hand, various empirical [e.g., Niedermeier et al., 2010; Murray et al., 2011] and theory-based parameterizations [e.g., Khvorostyanov and Curry, 2000, 2005; Hoose et al., 2010] have been developed which base on the “stochastic approach” implying time to be an important parameter for the ice nucleation process. On the other hand, there are empirical parameterizations based on the “singular approach” (summarized in Hoose and Möhler [2012]) assuming instantaneous (i.e., time independent) ice nucleation on so called “active sites” at specific temperatures. Especially within the last decade, new descriptions have been developed and applied, which combine the key features of both approaches [e.g., Vali, 1994; Marcolli et al., 2007; Murray et al., 2012; Welti et al., 2012]. The underlying concept can be outlined as particles possessing active sites, each site being characterized by a given nucleation rate coefficient (i.e., contact angle).

To contrast the performance of the two approaches, Eidhammer et al. [2009] and Ervens and Feingold [2012] performed parcel model simulations considering various parameterization schemes (singular/stochastic, single/multiple contact angles). It was found that due to the inconsistencies between the two approaches, and the different assumptions underlying the schemes, the application of these schemes over a wide range of atmospherically-relevant conditions could lead to differences in predicted cloud related parameters such as ice crystal number concentration, ice water content, cloud life time, etc. [Ervens and Feingold, 2012].

In order to provide a consistent description of the heterogeneous ice nucleation process as well as to shed light upon the somewhat bewildering range of interpretations of stochastic and singular ice nucleation, we developed the concept of the Soccer ball model (SBM) [Niedermeier et al., 2011]. The SBM applies contact angle based classical nucleation theory and assumes a Gaussian probability density function (PDF) for the distribution of contact angles over the nucleation sites of a particle distribution. The original SBM utilizes the Monte Carlo technique. Therefore, to gain statistically significant results, a large number of particles and/or a large number of independent nucleation events need to be considered. This makes the original SBM numerically expensive and not well-suited for atmospheric modeling applications.

Therefore, we have developed a simplified and computationally much more efficient version of the SBM which is introduced in the framework of the present paper. Within this new version, the mean ice nucleation behavior of a particle population is directly determined in terms of the mean frozen droplet fraction. In order to prove the applicability of the new SBM for describing and parameterizing heterogeneous ice nucleation processes, it will be first compared to the original version of the SBM. Then we exemplarily apply the new SBM for parameterizing laboratory data concerning the ice nucleating behavior of SNOMAXTM [Hartmann et al., 2013] recently gained at LACIS (Leipzig Aerosol Cloud Interaction Simulator) [Hartmann et al., 2011]). The determined parameterization will then be used to predict the ice nucleation behavior of SNOMAXTM/P. syringae bacteria determined in other laboratory studies. Finally, in the framework of a cloud parcel model, we compare ice particle number predictions based on the new and original versions of the SBM and a simplified assumption of stochastic freezing (i.e., applying a single contact angle).

2 Comparison Between Original and New Soccer Ball Models

2.1 Mathematical Description of the New Soccer Ball Model

Within both the original and the new SBM, we consider a particle population of identically sized, spherical particles, with each particle being immersed in a droplet and able to act as ice nucleus (IN). The fraction of frozen droplets, i.e., the number of frozen droplets divided by the sum of frozen and unfrozen droplets, at a given temperature and time, under these conditions, is directly related to the probability of ice nucleation on the particle surfaces, i.e., the heterogeneous ice nucleation rates of the particles. To calculate these nucleation rates, the surface area of a particle is divided into a number of surface sites nsite, which is identical for all particles. Each surface site is assigned a specific contact angle, θ, based on a Gaussian PDF. This distribution p(θ) is characterized by a mean contact angle μθ and a standard deviation σθ:

display math(1)

In the original model, the freezing probability Pfr of the droplets was determined by means of Monte Carlo simulations. Thereto, the cumulative distribution function of the contact angle distribution was discretized between 0 and π in, e.g., 1800 bins. Uniformly distributed random numbers n∈ [0,1] were applied to sample from the discretized cumulative distribution function and assign specific contact angles to the considered surface sites (see Niedermeier et al. [2011] for details).

In the new version of the SBM, the use of random numbers for contact angle association is avoided. For the derivation of the new SBM, we first assume that all particles have homogeneous surfaces, i.e., nsite=1 and each particle has the probability to feature a single contact angle according to equation (1) with σθ>0. This probability is identical for all particles. Then, the probability of a single droplet to be unfrozen, Punfr, at a given temperature T and time t can be determined by:

display math(2)

jhet represents the nucleation rate coefficient (the equation and necessary parameterizations for its calculation are given in, e.g., Zobrist et al. [2007]), ssite is the surface area of one surface site and t the nucleation time. Since nsite=1 then ssite=Sp, where Sp is the particle surface area. The freezing probability is given by:

display math(3)

The first term in equation ((2)) is similar to the α-pdf model of Marcolli et al. [2007] and Welti et al. [2012]. However, a second and a third term are introduced in equation ((2)) because the Gaussian distribution is a continuous probability distribution function, i.e., it provides contact angles outside the possible interval [0,π]. To account for this, the contribution of the PDF for providing contact angles smaller than 0 rad is directly attributed to θ=0 rad in the nucleation rate coefficient and its contribution of contact angles larger than π is directly attributed to θ=π rad. This needs to be considered, as otherwise Pfr is overestimated. This effect is the more pronounced the larger σθ and the closer μθ is located at the integration limits (0 to π) so that the contact angle distribution contributes beyond these integration limits.

Since the contact angle distribution is identical for all particles of the population, the freezing probability of a single droplet is identical to the freezing probability of the complete droplet population. Therefore, Pfr(T,μθσθ,t) can be directly related to the frozen fraction fice(T,μθ,σθ,t), i.e., fice(T,μθ,σθ,t)=Pfr(T,μθ,σθ,t).

In the next step of the model derivation, we now allow nsite>1, but still nsite is identical for all particles and here ssite=Sp/nsite. However, ssite could also be set to a fixed value independent of the surface area of the particle. This is the extension to the α-pdf model (This extension seems reasonable as recent computer simulations of ice nucleation at the molecular level have shown that ice nucleation on kaolinite particles is preferred at defects like trenches [e.g., Croteau et al., 2010] which could be numerous on the particle surfaces). For each surface site, the same PDF (equation (1)) is valid, but each individual nucleation site on the particle surfaces is treated independently. As independent probabilities are multiplicative, Pfr(T,μθ,σθ,nsite,t) can be calculated as:

display math(4)

Once again, the overall freezing probability of a single droplet is identical to the freezing probability of the droplet population because all particles are statistically identical, i.e., fice(T,μθ,σθ,nsite,t)=Pfr(T,μθσθ,nsite,t) is still valid.

2.2 Model Calculations for the Original and New Versions

In Figure 1, frozen fractions as a function of temperature are presented for a nucleation time t=1 s and the surface area Sp of a spherical 300 nm diameter particle. The frozen fractions were determined with the original as well as with the new version of the SBM. For all calculations, model parameters were chosen identically to the ones used by Niedermeier et al. [2011]. For the calculations with the original SBM, a set of Ndrop= 1000 droplets was chosen. Model runs for each parameter setting, using this original model, were performed thousand times (Nexecute= 1000). That means 1,000,000 droplets are regarded in total (Ntotal=Ndrop×Nexecute= 1,000,000). With the new model, which assumes all particles to have the same probability to feature a certain contact angle, only one single calculation per parameter set was performed.

Figure 1.

Frozen fraction fice as a function of temperature for given Soccer ball settings (μθ=1.0 rad): solid (nsite=1), dotted (nsite=10), and dashed (nsite=100) lines in both panels represent results determined with the new SBM (different σθ values are indicated by different colors). (a) The squares represent the resulting mean frozen fractions determined with the original SBM (Ndrop= 1000) for a given parameter set due to averaging over Nexecute= 1000 model runs. (b) The crosses represent frozen fractions determined with the original SBM (Ndrop= 20) performing only a single model run (Nexecute= 1) for each parameter set.

In Figure 1a, the frozen fraction determined with the original (represented through squares) and new SBM (represented through lines) are shown for the different parameter settings. The squares represent the resulting (mean) frozen fractions for a given parameter set determined by averaging over the Nexecute= 1000 model runs. These mean frozen fractions are identical to the ones determined with the new SBM. This clearly shows that old and new versions of the SBM yield identical results if a statistically relevant number of nucleation events is considered in the original SBM.

However, if we now consider a population of Ndrop= 20 and perform only a single model run (Nexecute= 1), the calculated frozen fractions scatter around the mean values determined by the new SBM (see Figure 1b). This apparent discrepancy is caused by a too low number of considered nucleation events so that statistically relevant results cannot be determined. This aspect will be discussed further when applying the old and new SBM versions in a cloud parcel model. In summary, for the description of the mean ice nucleating behavior of a particle population, the new SBM, without loss of accuracy, is simpler to use and reduces the program execution time (the execution time of the original SBM increases by math formula).

3 Parameterization of Laboratory Data

Before applying the new SBM within the framework of a cloud parcel model, the feasibility of the SBM for parameterizing experimental nucleation data needs to be tested. To do so, SBM parameters (number of surface sites nsite, mean μθ and standard deviation σθ of the contact angle distribution) are determined by matching calculated and measured frozen fractions. The SBM is certainly not the first model to apply this concept for parameterizing experimental data.

For example, Marcolli et al. [2007], Welti et al. [2012], and Kulkarni et al. [2012] have shown that only a distribution of contact angles, even for a single IN type (e.g., kaolinite) [Welti et al., 2012], can explain observed freezing behavior. For atmospheric data, Eidhammer et al. [2009] also showed that a single contact angle leads to a too fast freezing and that a contact angle distribution leads to a more realistic temporal evolution of the frozen fraction.

Based on the results of immersion freezing experiments with LACIS, in the supporting information we describe usage of the new SBM for parameterizing the immersion freezing behavior of SnomaxTM and its successful application to model the results gained in two other independent SnomaxTM related ice nucleation studies. Furthermore, we successfully applied the SBM in a recent LACIS study describing the ice nucleating ability of ice nucleating active macromolecules from birch pollen [see Augustin et al., 2013]. This clearly underlines the applicability of the SBM for describing / parameterizing heterogeneous ice nucleation processes.

4 Implementation of Different Freezing Descriptions Into Cloud Parcel Model

4.1 Parcel Model Description

In order to demonstrate the feasibility of the new SBM for atmospheric modeling applications, we implemented it into a cloud parcel model, which already featured different approaches for describing heterogeneous ice nucleation processes [Ervens et al., 2011; Ervens and Feingold, 2012]. Using this model, we assessed differences in predicted Pfr. The following parameters and boundary conditions were applied. The initial aerosol population covers a diameter range of 10nm<Dp<2000nm in eleven size classes, whereas only one size class (Dp = 300 nm) acts as IN. Particle growth by water uptake is tracked on a moving size grid. The air parcel is lifted with a constant updraft velocity (w=20cms−1 for 300 m) starting at a relative humidity with respect to liquid water of 98.5%. Condensation freezing below cloud is excluded and only particles or droplets below a solute concentration threshold of 0.01 M can freeze. We compare three different freezing approaches: (i) single contact angle (θ = 1.0 rad); (ii) the original SBM with 20 randomly chosen θs in each simulation (this approach was previously referred to as θPDF scheme by Ervens and Feingold [2012]), and (iii) the new SBM. For the latter two, we consider two contact angle distributions with nsite=1, μθ=1.0 rad and σθ=0.1 rad, and σθ=0.5 rad, respectively.

Using the original SBM approach, the IN population is treated as an “external mixture” of different particle types that differ in θ and in each model time step, freezing events on such individual particle types removes typically the best IN (smallest θ) from the pool of unfrozen particles. Ice particles originating from different particle types are tracked individually in the model which makes the original SBM scheme computationally very expensive (a computing time of ∼2 h for a single ascent of an air parcel). Such a procedure is of advantage in simulations of several cloud cycles, i.e., when particles are tracked over multiple freezing and melting events. Using the new SBM, after the first freezing event, we apply equations (1) and ((2)) for the interval [θmin,π] in order to take into account the removal of frozen particles with small θ from the unfrozen particle population, and thus a narrowing of the contact angle distribution. θmin is determined by equating the frozen particle fraction to the integrated normalized contact angle distribution, e.g., if Pfr=0.3, for the distribution σθ=0.1 rad (σθ=0.5 rad), math formula(math formula).

4.2 Parcel Model Results

Figure 2a shows that at T0≃−19°C the single θ approach greatly underestimates Pfr as compared to results from the SBM since the tail of the contact angle distribution with θ<1.0 rad as considered in the SBM versions significantly contributes to Pfr. At lower temperature (T0≃−23°C, Figure 2b), the single θ leads to a rapid increase of the frozen fraction immediately near cloud base. The new SBM predicts a much slower Pfr increase and resulting fraction since particles with θ>μθ composed a significant fraction of the particle population and do not freeze. The comparison to the original SBM shows that predicted Pfr scatter around the mean Pfr from the new SBM. These deviations and the more stepwise increase in Pfr are due to incomplete and random sampling of θ from the contact angle distribution (as already mentioned in section 2). Results of the original SBM scheme show that random θ samples might lead to an underestimation of Pfr in one temperature regime whereas the same θ selection leads to an overestimation in another. The generally more similar evolution of Pfr predicted by the original SBM for the narrower distribution (σθ=0.1 rad) as compared to that as predicted by the new SBM might be explained with the fact that with increasing width of the contact angle distribution the probability of sampling near the mean value becomes lower and thus random deviations might increase. Note that we only show results from two arbitrarily chosen droplet/particle samples here. If a statistically relevant number of droplets would be chosen, the model results from the original SBM would become identical to those of the new SBM, in agreement with findings in Figure 1.

Figure 2.

Frozen fractions as predicted by the parcel model using three different approaches: single contact angle (black) and contact angle distributions (μθ=1.0 rad, nsite=1 and σθ=0.1 rad (red), and σθ=0.5 rad (blue), respectively). Solid-colored lines are for the new SBM; dashed colored lines for the original SBM (different line types denote two random θ selections with 20 θs each). (a) T0=−19°C; (b) T0=−23°C. Results are only shown above cloud base; height denotes the altitude above RH0=98.5%.

Overall, the trends in the parcel model simulations are consistent with those in Figure 1 where the increase in the fozen fraction for the widest distribution (σθ=0.5 rad) exhibits the smallest slope over the temperature range. Over the narrow (average) temperature interval covered by an air parcel during its ascent (here ΔT≃3°C), only a small particle fraction of the population with the wide contact angle distribution will contribute to a significant increase in the frozen fraction. This result is in agreement with large eddy simulation by Kulkarni et al. [2012] who have shown that wide contact angle distributions (they applied the α-pdf model) lead to the lowest ice nucleation onset relative humidity with respect to ice. The major part of the particle population is either immediately frozen near cloud base or do not freeze at all in these temperature and time ranges.

In conclusion, if continuous narrowing of the contact angle distribution is taken into account in the new SBM (i.e., an increase in θmin with time and decreasing temperature), this model gives comparable results to the previous, computationally much more expensive models. It is therefore a highly suitable tool for describing heterogeneous ice nucleation processes in atmospheric modeling applications of bin models, i.e., if the processing of individual IN is not tracked but the target parameters are the total ice particle number concentration and ice water content.

5 Summary and Conclusion

With the original SBM [Niedermeier et al., 2011], a consistent and robust description of the heterogeneous ice nucleation process has become available. However, this model version, as being based on the Monte Carlo method, is numerically too expensive when trying to determine the mean freezing behavior of a droplet population. This deficiency has been overcome with the new version of the SBM presented here. The new SBM allows for the direct calculation of mean frozen droplet fractions. As it avoids the usage of the Monte Carlo technique, it is simpler to use and reduces the program execution time compared to the original model. Ultimately, the SBM parameters can be derived for various IN types (e.g., dust, soot, and biological material) which are routinely included in aerosol/cloud models.

In this study, we successfully used the new SBM to describe and parameterize the ice nucleating ability of SNOMAXTM (see supporting information). This confirms earlier studies [e.g., Marcolli et al., 2007; Welti et al., 2012] which showed that the assumption of a contact angle distribution is useful to represent and parameterize laboratory data on heterogeneous freezing.

Finally, using a cloud parcel model, we showed that the new SBM yields the same average predicted frozen fractions as obtained in the original SBM if a statistically relevant number of nucleation events is considered. In order to account for the fact that the available best IN will be removed first from the pool of unfrozen particles, the contact angle distribution on the remaining unfrozen particles is adjusted by narrowing toward larger contact angles in each model time step. In agreement with Eidhammer et al. [2009], our cloud parcel model simulations confirm that a single contact angle might lead to great underestimates or overestimates and a biased temporal evolution of the frozen fraction, depending on the temperature and contact angle. In conclusion, the new SBM gives comparable results to the previous, computationally much more expensive models and is therefore a highly suitable tool for describing heterogeneous ice nucleation processes in atmospheric modeling applications for a variety of IN.


This work is partly funded by the Federal Ministry of Education and Research (BMBF - project CLOUD 12) and by the German Research Foundation (DFG project WE 4722/1-1, part of the research unit FOR 1525 INUIT). It was financially supported by the EU - FP7 project EUROCHAMP II. B.E. acknowledges support from NOAA's Climate Goal.

The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.