General comments:

In this manuscript, the authors use box-model simulations to estimate ice crystal number concentrations at the end of the jet regime. These are performed for hydrogen combustion and therefore consider only the entrainment of ambient particles (no exhaust emissions). The authors derive conditions under which homogeneous droplet nucleation may compete with heterogeneous droplet formation (on (entrained) particles). They also conclude that the ambient particle size distribution can be treated using a weighted mean approach. Finally, they perform several sensitivity analyses.

Overall, I find this manuscript provides a clear overview of the key considerations for simulating contrail formation from hydrogen exhausts. It is particularly useful to see an assessment of homogeneous versus heterogeneous droplet formation. However, I find that the manuscript would benefit from being restructured, to highlight the main results and differentiate these from observations about the model. Other specific comments are listed below. Therefore, I would recommend publication after the below comments have been suitably addressed.

Specific comments (in form: location-comment):

Overall structure: the conclusions drawn in Sect. 5 of the manuscript are useful and well-defined. I find that there are two main results: (a) the derivation of a conservative boundary for the importance of homogeneous nucleation of water droplets and (b) ambient aerosol properties and/or meteorological conditions that result in asymptotic ice crystal number concentrations.

In addition, the authors show that (c) omission of the coarse-mode population of ambient particles has a negligible impact on model outputs and (d) they can reduce model compute by applying a weighted mean as in Eq. (3). Although the latter results are undeniably useful for reducing the model complexity and associated simulation times, the manuscript does not clearly motivate whether these are as useful to the wider community as (a) and (b). Therefore, I would recommend that either (i) the authors move (c) and (d) to a methodology/model development section and present (a) and (b) as final results or (ii) provide more justification for the wider applicability and limitations of (c) and (d) and retain these in a results section. 

Moreover, I suggest that after restructuring/modifying as above, the main results are presented under a “results” heading rather than “microphysical insights”. Currently, the model development is difficult to disentangle from the results. Accordingly, adopting these changes would more effectively highlight the most applicable results of the study.

Homogeneous droplet nucleation: using information in Figure. 6 and Table. 3, it appears that some simulations may be performed under conditions where homogeneous droplet nucleation competes with heterogeneous droplet formation on entrained ambient particles (i.e., p_a = 400 hPa and T_a = 210 K). However, it is not clear whether homogeneous droplet nucleation is included in the simulations in Sect. 4. Therefore, I would suggest that the authors provide more information on this.

Data availability: I would recommend that the authors consider putting figure data and/or underlying code in a public repository (e.g., Zenodo) rather than only being obtainable from the corresponding author.

Various:

Line 33: this statement would benefit from additional references in support of hydrogen use in aviation. Currently, I find that the reference from Airbus is insufficient to motivate the potential for hydrogen uptake in this sector.

Lines 97 and 100: while the simulations performed by Yu et al., (2024) show the importance of fuel sulfur, the mass of volatile particles is a sum over the masses of all condensable gaseous species, which also include other organic material (Kärcher et al., 2000).

Line 254: this is an interesting observation and would benefit from a fuller explanation. Specifically, why is this the case provided aerosol is entrained and not emitted?

Lines 255-262: It is unclear what the total particle size distribution is being used in these simulations. Are the three modes described using a common total hygroscopicity parameter? If so, presumably it would be possible to represent this trimodal distribution as the linear combination of three monomodal particle size distributions.

Line 359: as some readers will not be familiar with the Koop parameterization, I would suggest that you explain the link between water-supersaturated conditions and the formation of liquid clouds more explicitly.

Line 381: although the temperature dependence is limited for soot-rich exhausts, more recent work (incorporating volatile particles) has shown that ambient temperature regulates contrail ice crystal number concentrations in the soot-poor regime (Yu et al., 2024).

Line 405: this discussion would benefit from additional (supplementary) information on the choice of freezing parameterization adopted in this work.

Figure 8: I would advise clarifying (in both the figure and the main text) that the different colours are being used to represent different co-parameters and that respective ranges are taken from Table. 3.

Minor comments:

Line 13: using the results in Figure. 9, I would advise providing some quantitative information in the abstract to bound this regime (e.g., the information in line 409).

Line 31: I would advise using the term “flight distance” rather than “range”.

Line 73: I would suggest revising the phrasing: “ready to be implemented”.

Lines 77-78 (and Figure 1 caption): I would suggest revising the order of “plume partial water vapor pressure” to “partial pressure of water vapor in the plume”.

Line 80: I would suggest expanding on the significance of “lower calorific value” in this context or removing altogether.

Line 88: I would suggest clarifying that this is only true if the ambient conditions are the same in both cases.

Lines 91-92: I think it is more important to mention that ambient upper-tropospheric temperatures are usually below the Schmidt-Appleman temperature for hydrogen combustion. The comparison with the homogeneous freezing threshold is of secondary importance.

Lines 93-94: I would suggest expanding on why the maximum plume supersaturation is important for the process of contrail formation.

Line 143: I would suggest expanding on the meaning of the term “simple enough”. For instance, does this refer to total computation time?

Line 148: I would suggest revising this sentence for clarity.

Line 175: I would advise defining the meaning of “phase” in this context.

Line 221: I would suggest replacing “spectrum” with “particle size distribution”.

Figure 2: please could you expand (in the text) on why N_{ice, f} shows a slight decrease for large x-values rather than asymptotic behaviour.

Line 274: it would be useful to clarify the units of J_aer(t) in Equation (4).

Line 374: I would suggest introducing a reference for the cutoff at 235 K. Alternatively, this could be enveloped in a supplemental section that outlines your homogeneous ice nucleation parameterization (as described earlier).

Lines 350-354: I would suggest explaining this caveat in more detail as the implication (of the final sentence of this paragraph) is unclear to me.

Technical suggestions (in form: location-comment):

Line 2: “well studied” should be hyphenated.

Line 60: I would suggest replacing “give meteorological condition” with “given set of meteorological conditions”.

Line 63: “globally distributed” should be hyphenated.

Line 76: there is a missing “the” in “(called the plume)”.

Line 77: “Micro-physical” should not be hyphenated.

Lines 77-78 (and Figure 1 caption): I would suggest revising the definition “plume partial water vapor pressure” as “partial pressure of water vapor in the plume”.

Line 121 (and line 210): I would suggest replacing “well” with “highly” and removing the hyphen.

Line 155: I would suggest revising the word order, particularly the use of “used”.

Line 176: “analytically prescribed” should be hyphenated.

Line 178: there is a typographic error in this sentence between “calculated” and “time-resolved”.

Line 205: I would suggest removing “already”.

Line 265: I would suggest rephrasing the “curse of dimensionality”.

References:

Kärcher, B., Turco, R. P., Yu, F., Danilin, M. Y., Weisenstein, D. K., Miake‐Lye, R. C., and Busen, R.: A unified model for ultrafine aircraft particle emissions, J. Geophys. Res., 105, 29379–29386, https://doi.org/10.1029/2000JD900531, 2000.

Yu, F., Kärcher, B., and Anderson, B. E.: Revisiting Contrail Ice Formation: Impact of Primary Soot Particle Sizes and Contribution of Volatile Particles, Environ. Sci. Technol., 58, 17650–17660, https://doi.org/10.1021/acs.est.4c04340, 2024

see the attachment