Please find my review attached below. 

This paper extends the theoretical study of homogeneous ice nucleation by Dolapchiev et. al. (2023). The results may well be worthy of publication, but serious presentation issues in the current manuscript need to be addressed before this can be determined. In short, the paper is not yet ready to be properly reviewed. I would like to encourage all the authors to revise and clarify the text so that the referees can focus on the scientific content. Some suggestions are provided below. Generally, in the whole manuscript, the punctuation (commas), grammar (verbs should appear in all propositions), symbols (definition, consistent use) and equations (homogeneity) should be checked.

I include a few preliminary comments on the methods and results but, as explained above, the paper will need to be reassessed once the writing has been clarified.

General comments

I/ Legibility of the figures: please include labels to all figures (a, b...) so that they can be referenced in the caption and text. The very dense scatter plots in figures 5 and 6 are not very informative due to the extreme overlap of the symbols. Perhaps consider adding median and quantiles (as, e.g., in the scatter plots of Kramer et al., 2016, their figure 2). Figures 9, 10 and 11 would be clearer with an x-axis for the pdfs.

II/ To develop a parameterization based on a limited set of numerical simulations of nucleation, it is indeed important to assess the degree of realism of the initial conditions and forcing used to construct this dataset. The authors touch that point, but should elaborate further on it.

1) Regarding the forcing, Figure 3 and the associated discussion should be made more quantitative and the choices should be explained. I suggest

a) explaining the choices made to build the forcing dataset: why is the background updraft not always included? are tropical regions excluded (f=10^-4 s^-1) ?

b) extending the altitude range (currently 9 to 14 km as specified on line 165) to about 18 km in the tropics to include the climatically important TTL cirrus, which this parameterization should also represent

c) including a normalization for the first three panels in FIg. 3 (with units). In particular, clarify what is shown in the 3rd panel and whether it relates to the power spectral density of w

d) including more references to observational studies, for instance Podglajen et al. (2016) or Kohler et al. (2023). The first one is dedicated to observations of w which the authors are using to drive their simulations. The second contains comparisons with models, and some information on geographic variability. From a quick look (to be confirmed by the authors), it seems that the dataset of the authors underestimates vertical wind variability.

e) considering assessing other quantities than momentum fluxes, which do not directly affect ice microphysics. Note that the fact that a model can simulate quite well some GW-induced quantities (momentum fluxes) does not necessarily mean that others (vertical wind) are well represented (see, e.g., Podglajen et al, 2020).

2) A more thorough description of the simulations' initial conditions is required. How exactly were ni and mi selected ? On lines 137-139, it is written "Other conditions are varied randomly using a uniform distribution in physically meaningful ranges motivated from the paper (Krämer et al. (2016)): we choose 10 −4 kg −1 < n init (t = 0) < 10 7 kg −1 , and 10 −16 kg < m mean (t = 0) < 10 −12 kg"

What is meant here by uniform distribution? An arguably wise choice would be a uniform distribution in log(Ni) (to be consistent with Kramer et al.), but this sentence means a uniform distribution in Ni, i.e. oversampling large log(Ni) (see, e.g., the distributions in Jensen et al., 2013). Also, the lower end of your Ni values would be smaller than 10^-11 cm-3, which is well below the range shown in Kramer et al. (2016). Please explain.

Specific comments:

l 16: for citations, here and elsewhere in this context, please use \citep{} instead of (\cite{} or \citet{})

l 17: "meaning the" -> "meaning THAT the"

l22-23: unclear, please clarify

l 26: "lower" than what ? should it be "low"

l 28: "for the description of " by "to describe"

l 33: "is extended" by "has extended"

l 60: "pressure" and temperatures

Equations 1-3: it would be helpful to clarify which parameters depend on T (Sc ? J ?)

n , q are not defined, but ni, qi are introduced above. The homogeneity of all terms in equation 2 should be checked.

l 68:"Extner"-> "Exner"

l 75: "valid during long periods of the nucleation" -> "valid over the duration of the nucleation"

l91: for clarity, you could define F(t) as a function of the Exner function tendency

lines 95 and 119: couldn't you directly include the crystal's mass parameter m_0 instead of these D* and D** which I didn't find elsewhere in the manuscript

l 129: m0 appears here for the first time, but has not been defined before (the reader can guess it is q0/n0)

l 142-144: "This parameterisation is based on WKB theory (Bölöni et al. (2021)) and implemented using Lagrangian ray volumes, which are considered as carriers of the GW fields’ wave-action density.": consider moving the reference to Bölöni et al. (2021) at the end of the sentence - here it could be interpreted as WKB theory is due to Bölöni et al.

l 169: the chosen reference is not the best suited here; perhaps use VanZandt (1982) or one of the older references which proposed this form of the GW spectrum for the atmosphere

l 173: it seems to me that this may be an understatement and the vertical wind is actually underestimated compared to available observations

l 193-199: the extension is not shown- does it need to be mentioned?

l 207: I find it difficult to assess the agreement with this figure.

l 261: missing "are"

l 273: higher -> larger

l 343: consider referring to Karcher et al. (2024) about turbulence and cirrus

Figure 9: clarify the caption. Please explain why the number of horizontal bars varies.

In the bibliography section, some references are missing the doi/url

References:

Jensen, E. J., R. P. Lawson, J. W. Bergman, L. Pfister, T. P. Bui, and C. G. Schmitt (2013), Physical processes controlling ice concentrations in synoptically forced, midlatitude cirrus, J. Geophys. Res. Atmos., 118, 5348–5360, doi:10.1002/jgrd.50421

Kärcher, B., and Coauthors, 2024: Effects of Turbulence on Upper-Tropospheric Ice Supersaturation. J. Atmos. Sci., 81, 1589–1604, https://doi.org/10.1175/JAS-D-23-0217.1

Köhler, L., Green, B., & Stephan, C. C. (2023). Comparing Loon superpressure balloon observations of gravity waves in the tropics with global storm-resolving models. Journal of Geophysical Research: Atmospheres, 128, e2023JD038549. https://doi.org/10.1029/2023JD038549

Podglajen, A., Hertzog, A., Plougonven, R., & Legras, B. (2016). Lagrangian temperature and vertical velocity fluctuations due to gravity waves in the lower stratosphere. Geophysical Research Letters, 43, 3543–3553. https://doi.org/10.1002/2016GL068148

Podglajen, A., Hertzog, A., Plougonven, R., & Legras, B. (2020). Lagrangian gravity wave spectra in the lower stratosphere of current (re) analyses. Atmospheric Chemistry and Physics, 20, 9331–9350. https://doi.org/10.5194/acp-20-9331-2020

VanZandt, T.E. (1982), A universal spectrum of buoyancy waves in the atmosphere. Geophys. Res. Lett., 9: 575-578. https://doi.org/10.1029/GL009i005p00575