## 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 SNOMAX^{TM} [*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 SNOMAX^{TM}/*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).