| 27 Apr 2026
Stochastic homogeneous freezing of supercooled droplets in mixed-phase clouds in particle-based microphysics framework
Abstract. Homogeneous freezing of supercooled cloud droplets is an important process governing ice formation during the transition from mixed-phase clouds to pure ice-phase clouds in the upper troposphere. In this study, we implement a stochastic representation of homogeneous freezing in the particle-based aerosol–cloud microphysics model PySDM, treating freezing as a Poisson process dependent on droplet volume, time step, and the nucleation rate. We compare two parameterisations of the nucleation rate: a temperature-dependent formulation and a saturation-dependent formulation. Using an idealised adiabatically ascending air-parcel framework, we investigate the distribution of freezing temperatures and the resulting ice number concentrations across ensembles that vary updraft speed, cloud condensation nuclei number concentration, droplet size distribution, and the number of super-particles. Simulations are performed both with and without vapour deposition on ice, enabling assessment of the role of the Wegener-Bergeron-Findeisen process. We find that homogeneous freezing occurs over a broad temperature range rather than at a single threshold, with freezing temperatures strongly controlled by cooling rate and droplet size. When vapour deposition on ice is active, early stochastic freezing events dominate the evolution of the frozen droplet fraction and substantially reduce the fraction of droplets that ultimately freeze. The two nucleation-rate formulations produce similar behaviour near water saturation but diverge significantly when supersaturation or subsaturation with respect to water develops, leading to pronounced differences in freezing temperatures and ice number concentrations. Our results highlight the importance of stochastic freezing formulations and nucleation-rate choice for representing cloud glaciation in models.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Geoscientific Model Development.
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The overall quality of the preprint by Lüttmer et al. is high with good focus, detail, and scientific impact. The authors revisit the accuracy of current models in assuming a homogeneous temperature cutoff for the formation of ice in the atmosphere and instead they use Monte Carlo simulations of droplets freezing under different conditions in the atmosphere, including updraft velocity, super particle number, and CCN concentration. They consider two sets of simulations with and without vapor deposition and two parameterizations for Jhom. This work provides a valuable contribution to the field in untangling the importance of homogeneous nucleation albeit under relatively idealistic assumptions. The following specific comments and technical corrections should be considered before publication.
Specific comments.
line 100 and 174. The authors claim that Jhom should not depend on the ambient vapor pressure because the ice embryo is not in contact with it. However, an alternative explanation could explain this dependence. If the liquid water is in a subsaturated environment, then the activity of the liquid water changes correspondingly. This change in the liquid water could then have an impact on the homogeneous nucleation rate (similar to how a solute would reduce water activity), so that a change in ambient supersaturation ultimately affects Jhom in an indirect manner through the water activity. This interpretation aligns with the authors' Fig. 1b, where a water saturation < 1 leads to reduced Jhom, while water saturation > 1 leads to higher Jhom. At subsaturated Sw conditions, there may be a driving force for the liquid water molecules to evaporate that competes with Jhom. At supersaturated Sw conditions, the driving force reverses, such that there could be condensation to enhance Jhom. Thus, the authors should provide additional justification of their stated hypothesis or consider alternative explanations.
line 138. The authors claim that the KP00 parameterization shows that Jhom is "independent of the amount of solute". To my knowledge, KP00 showed that Jhom is directly independent of the type of solute, not the amount. It could be better to instead state that Jhom depends on water activity and that water activity decreases with the addition of more solute.
The organization of the manuscript could be improved slightly. For example, the first two paragraphs of the Results before 3.1 appear to be detailed descriptions of the parameters used, which could be better placed in a new subsection at the end of the Methods. The end of the Methods section could conclude with an overview, such as a table, of all the simulations performed and ensembles studied to help the reader obtain an overall impression of which variables were studied in which subsection of the Results. Similarly, the beginning of Section 3.3 could be moved to the Methods section to define sigma prior to its use in the Results.
Technical corrections.
line 63. typo 'an' to 'a' stochastic process
line 110. the notation in Equation 1 and the text could be more consistent. The text defines Pfrz(Δt), but Equation 1 uses the notation Pfrz(T), and the sentence afterwards states that Pfrz does not depend on the time interval, so the use of Pfrz(Δt) could instead be replaced by Pfrz alone or Pfrz(T).
line 189. 'capacity' may be better as 'capacitance'
line 196. 'were,' can be deleted
line 222. typo 'a' to 'an' adiabatic parcel model
Figure 2 (a) and (e) are missing x-axis labels. In the caption, the number of simulations performed could be stated. Was it just one simulation each for JHOM-T and -DWA? The reader could be directed here to section 3.2.1 for the discussion that one simulation is sufficient given nsd = 1000.
line 274. typo ΔT to Δt for time
line 287. in both panels (d) and (h), the blue line is solid. Should the red line be solid in panel (h)?
line 443. missing reference in parentheses