Review for "Aerosol-cloud interactions: the representation of heterogeneous ice activation in cloud models” by Karcher and Marcolli

This manuscript presents a derivation of "differential active fractions (AFs)” based on the concept of differential spectra (Vali 2019) when budgeting for ice particle number and INPs that are assumed to follow singular time-independent ice nucleation.  This approach contrasts with previous laboratory experiments that have quantified the fraction of activated INPs using AFs that are cumulative in ice supersaturation.  The authors then applied the differential AF concept and their budget equations assuming a lognormal distribution of contrail-processed aircraft soot particles and demonstrated that the differential AF approach results in smaller active fractions, which implies that homogeneous ice nucleation could be underestimated when cumulative AFs are used.

Overall, this is an excellent manuscript that is generally well-written and elegantly presents an important overlooked aspect of ice nucleation that is valuable to the modelling community.  Some aspects of the manuscript could be elaborated and clarified as outlined below.  I would recommend publication of the manuscript once these few minor points have been taken into consideration.

* A more specific discussion on how AFs are relevant to models on various spatial scales on lines 156-160 would be helpful for the reader to understand the feasibility of the approach.

* Why was soot used as the example in Figure 4?  Given its relative low ice-nucleating ability, perhaps dust aerosol particles could be used to better illustrate the effectiveness of differential AFs?  I would also suggest including other examples of INPs and particle size distributions to determine the relative impact of differential AFs compared to cumulative AFs. 

Review for Kaercher and Claudia (2021, ACP)

This study discussed the representation of ice activation in cloud models and identified a problem with the application of cumulative activation fractions when considering the INP/ice particle budget. The authors formulated differential activation fractions that are consistent with the reduction of INP number after activation and demonstrated that the new representation can prevent the INP overestimation. They applied the new formulation with lab-based soot INP measurements and showed using the differential activation fractions indeed prevents the INP overestimation.

The manuscript is concise but very clearly written. The derivation of the formulation is inspiring. This work will improve the INP representation in cloud parameterizations, especially for considering the competition between homogeneous and heterogeneous ice nucleation processes. I recommend publication after some clarifications. Below please find my specific comments.

Title: In my opinion, the title is a little bit too general. A more specific title would be better, e.g., something like “Improving the heterogeneous ice nucleation parameterization using differential activation fractions”?

Line 31-32: Then for immersion freezing measurements that are reported only as a function of temperature, does the INP overestimation problem also exists?

Figure 1 caption (3rd line): “‘no budget’ approach, arrows labeled with φ”. There are three arrows labeled with φ, but it seems only two of them indicate no budget approach?

Page 5, Line 103-105, formula (7): Could you please elaborate how you came up the idea of using such a mathematical form? In other words, why other forms can not conveniently fit cumulative AFs? Does it work for other activation fraction forms other than the one (ns function) reported Ullrich et al. (2017)? Also, maybe consider showing the measured and fitted curves in a figure as appendix? Just to have an idea about how “reasonable” it is.

Page 5, Line 108-110: It seems that the choices of s* and &s (a factor of 3 changes) are a bit arbitrary (or did I miss something important?). If s*=0.352 and &s=0.0175 are used for plotting figure 2, how will the results look like? (I assume &s here is not the grid spacing delta_s).

Page 8, Figure 4: It looks a bit surprising to me that the activation fraction for soot with 400nm size is similar to that for the desert dust particles as shown in Figure 2. Also, do you still need to fit cumulative AFs for this application? If so, how did you choose s* and &s?