  Cloud droplets containing ice-nucleating particles (INPs) may freeze at
temperatures above than the homogeneous freezing threshold. This process,
referred to as immersion freezing, is one of the major modulator of
aerosol-cloud interactions in the Earth's atmosphere that impacts cloud
radiative properties, and thus, climate. In modeling studies, immersion
freezing is often implemented using either so-called ``singular'' or
``time-dependent'' parameterizations. In this paper, we juxtapose both
approaches and discuss them in the context of probabilistic particle-based
cloud microphysics modeling, referred to as the super-particle or super-droplet
techniques. These techniques work by populating the particle-attribute phase
space with large numbers of computational particles and tracking the evolution
of particle attributes. We outline the congruence of the singular and
time-dependent approaches with Poissonian process description and point to
limitations pertinent to the singular approach. Simulations with both box and
two-dimensional kinematic models focus on the following aspects of immersion
freezing modeling. First, we contrast how both parameterizations respond to
different idealized ambient cooling rate profiles. Second, we quantify how the
polydispersity of the immersed surface spectrum impacts macroscopic features of
the system. Third, we illustrate the implications of applying the singular
model in simulations with flow regimes relevant to ambient cloud conditions
rather than to the cloud-chamber experiments on which these parameterizations
are built upon. We discuss the critical role of the attribute-space sampling
strategy for particle-based model simulations in modeling heterogeneous ice
nucleation which is contingent on the presence of relatively sparse immersed
INPs.
