the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 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.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-2217', Anonymous Referee #1, 14 Jul 2026
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RC2: 'Comment on egusphere-2026-2217', Anonymous Referee #2, 27 Jul 2026
The presented manuscript has a well-thought structure and a fairly good scientific content. The implemented stochastic freezing method represents an important and valuable step towards an accurate description of the homogeneous freezing process although the final conclusions are somewhat limited by the absence of pre-existing ice in the idealized simulations. The step-by-step approach with incrementing complexity between the three presented ensemble sets is well thought and also the decision to vary always one variable at a time which keeps results and dependencies traceable in a simple manner. The following comments should be regarded before publication. If confirmed by the authors that there was a bug in the PySDM formula for the thermal conductivity correction used for the ICE-DEP simulations, it should be fixed and at least the high-CCN simulations with DEP-ICE be re-run before publication and the corresponding n_ccn ensemble plots re-rendered. Usually this would be clearly a recommendation for major revisions as part of the simulations need to be redone but in this case the conclusions might not change much or at all and therefore I'll leave it to the editor to decide if minor revisions could be okay as well.
Major Comments
In section 2.2 the Eq.10 for the kinetic regime correction of the thermal conductivity is missing a factor of 4 in the right part of the denominator, it should be 4*K_T instead of just K_T there (compare e.g. with Eq. 8.32 of Lamb & Verlinde / Eq. 5.2.13b of Khvorostyanov & Curry). Unfortunately this factor of 4 seems to be also a absent in the PySDM code as well of the zenodo archive, meaning that there was a bug in the DEP-ICE part of the simulation code (under PySDM/physics/diffusion_ice_kinetics/standard.py at the bottom, although I could not understand which K_T kinetic effects correction the condensation uses, it should be the same as for DEP-ICE). This bug leads to an overestimation of the thermal conductivity for equivalent radii around 2µm and smaller: a short juptyer-notebook-based analysis with copy-pasted formulas (see supplement) gave me an overestimation of K_T* around 12% at r_i=2µm, 23% at 1µm and 47% at 0.5µm for the conditions of the simulations. Nevertheless the growth parameter G_v for ice deposition (you call it Howell-factor) doesn't seem to be too sensitive to the thermal conductivity for these conditions: I get a bug-introduced overestimation of G_v of just about 0.5% at r_i=2µm, 0.9% at 1µm and 1.6% at 0.5µm which seems almost negligible but should be considered enough to re-run the high-CCN simulations which lead to the smallest droplets/crystals.Minor Comments
In the title is says "in mixed-phase clouds" which might be a bit misleading since pre-existing ice is absent at the start of the homogeneous freezing process (it's a purely liquid-phase cloud). Something like "idealized developing cirrus clouds" could be clearer here.
The title should include the word "idealized" to clarify the idealized nature of the simulations.There is mixed spelling of vapour/vapor throughout the manuscript, it should be consistent and in line with other regional spelling choices (like idealised, discretise, emphasise).
Fig.1 in (a) and (b) the black lines representing JHOM-T seem both to coincide with each other but the red line in (a) and the 1.00 red line in (b) should also do so. It seems unexplainable that the black and red line in (a) diverge for < 235K but they don't diverge in (b) for S_w=1.00. It is stated that the temperature range below 235K is rarely relevant for the studied events which is okay but the lines should show the same thing in (a) and (b).
Do the CCN concentration numbers in the manuscript and figures get always downscaled to the air density of the simulations (so around -40%) in general? I guess this is implied in L.247 but it would be a lot clearer to mention this explicitly there and add an "(STP)" note to the y-axis descriptions of the CCN number concentrations plots.
Shorter/Technical Comments
1.
L.26-27 "liquid-origin pathway" appears twice in the comparison
L.55 as WRF has a whole range of different options for microphysical models e.g. mentioning "P3" instead of "WRF" would be more specific, the given reference is refering to P3 anyway2.
L.113 there's a typo in the exponent sign of the J_hom value, should be positive
L.196 nanometer size range is probably not the best word here (there are no droplets 1-10nm in size and the mean droplet radii in the later figures are in the range of 0.5-10µm), sub-micrometer size range would be more precise
L.188 Eq.6 is missing a pi factor (although the pi factor seems to be present in the code of the zenodo archive and no introduced bug here)
L.222 should be "an" adiabatic parcel3.
Fig.2 in (h) the solid and dashed of the blue and red lines should be switched (JHOM-DWA is active, JHOM-T not)
L.289 it seems slightly confusing that the reference value is mentioned as JHOM-T, with JHOM-DWA being higher than the reference, the relative error should be positive then, not negative. Negative relative errors are fine and maybe also better readable in the plots but then the reference (to compare against) should be JHOM-DWA.
L.300 & L.371 leave out the word "solid" as only the black line is solid and the other two are dashed/dotted.
L.322/323 The strong updrafts of supercells can reach easily over 30m/s, sometimes even 50m/s or more (e.g. Marinescu et al., 2020, MWR). In this context, calling "very strong updrafts of 5 and 10m/s" representative of the conditions in deep convetive systems might also be a bit of an overstatement. Maybe calling it just "strong updrafts" instead of "very strong updrafts" would be an easy midway here.
L.403 hard to tell if it increases actually exponentially, as w is on a logarithmic y-axis, so it could be also linear to w.
L.420 should probably be "narrower range" rather than now being "broader range" if refering to the histograms of JHOM-T
L.443 missing reference at the end of the line
L.450 ambigous reference to "the literature", at least a reference should be given
L.491 refering to Fig. 5 (g) and (h) which don't exist in the manuscript. This should probably be Fig. S5 (a) and (b)4.
L.519 in section 2. the droplet volume was named uppercase V (or V_l), here it's called lowercase v, the variable names should be consistent.
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- 1
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