Jacquot and Sartelet. “Numerical investigations on the modelling of ultrafine particles in SSH-aerosol-v1.3a: size resolution and redistribution”

General comments:

This paper presents the development of analytical equations and a numerical algorithm for Lagrangian calculation of the coagulation term in aerosol distribution dynamics, specifically for calculating the partitioning of coagulated aerosol (number and mass) between discrete size bins in a manner consistent with the Lagrangian formulation of other aerosol dynamics processes.  The approach was developed with the purpose of reducing numerical dispersion during aerosol dynamics simulation caused by redistribution due to changing between Lagrangian and Eulerian formulations. In this study, the new approach was applied within a Eulerian chemical transport model to a real-world case study, with three different resolution particle size distribution schemes. Impacts on accuracy (compared with measurements) and numerical dispersion were investigated. The main results suggest that the impact of the new formulation/algorithm on numerical diffusion is small compared with the impact of size resolution itself (and compared with the overall error in representing measurements). Overall, I find the work to be interesting and publication of the new more-internally-consistent algorithm is likely to be useful to CTM model developers.  However, the impact on the field may be small due to the findings (of small effects); hence this may be better as a technical note.  Additionally, some of the explanation needs improved clarity and some corrections are required, as discussed below.

Specific Comments:

The importance and impact of this work may be limited for a few reasons, making me question whether this would be better as a technical note than a full paper.

• The findings indicate that numerical dispersion is less sensitive to the new coagulation formulation/algorithm than to the resolution of the aerosol size discretization. The differences are also much smaller than the simulation errors (determined by comparison to measurements).
• The authors state that the new formulation is basically repeating a derivation of Debry and Sportisee (2007) but correcting a mistake from that paper. This seems like a focus for a technical correction rather than for a novel contribution.
• The authors further conclude that the new formulation/algorithm may be most useful for low size resolution simulations (9 bins), due to the computational costs of the new Lagrangian formulation/algorithm. Given this, it is not clear why one should invest (computationally) in the new formulation versus just investing in a higher resolution size distribution. A discussion of the relative computational costs should be provided.

Tables 1 and 2 show comparison of model results to measurements for the mixed Eulerian/Lagrangian algorithm (1), not the fully Lagrangian algorithm (2). The text indicates this is because statistics “are very similar”.  Because the fully Lagrangian algorithm is the focus of this paper, the statistics resulting from using that algorithm should be shown (and perhaps those for the mixed algorithm put in an appendix for comparison)

Additional checking of the derivations is needed (by someone whose work involves similar derivations).  I did not have the time to go through the equations in detail to fully understand them, but I did compare them to equations in the existing literature and was left with some questions.  First, Appendix A lists the general aerosol dynamics equations for evolution of the number and mass distributions; a citation for this classical formulation is listed as Gelbard et al. (1980) [L91]. Although the number density equations seem generally consistent with other sources (Seinfeld and Pandis, 2006 textbook), I did not find these formulations in the Gelbard reference, so the citation seems to need correction.  More importantly, it looks like there may be a typo or error in the mass distribution equation (A4) as the units don’t seem to work out.  Specifically, the last term in the equation (I_s*n) appears to be missing some kind of multiplicative density term (mass of species s per volume of species s?).  Without that, there is no mass, so it is unclear how that term contributes to a mass density. 

Additional explanation of some terms is also needed.  For example, what is the meaning of the blackboard bold 1 symbol in Equations 3 and 4? Perhaps this is a well-known symbol in this subfield, but I didn’t recognize it.  If this work is to reach an audience that isn’t excessively narrow, non-standard mathematical symbols should be defined.  

Table 2 caption.  The units of mass concentration appear to be mislabeled here as #.cm^-3, which are units for a number concentration, not a mass concentration.  The units need to be corrected (or clarified).

Figure 2.  The figure labelling should be improved.  In addition to labelling the sites based on whether they measure number or mass concentration, sites should also be labelled using the network names provided in the text (LHVP, SIRTA AIRPARIF).  Additionally, labeling the location of Paris and the meaning of the polygon’s boundaries would be helpful.

Technical corrections:

Figure 3.  The exponent (4) is missing from the color scales and the units of number concentration are also missing (except for the exponent).

L20 typo: should be “focused”

L42 “formerly equivalent to ...’ I don’t understand this.  Should it say "formally"?

L86. Missing word “be” before “a partitioning”

L107 Typo: “substraction”

L186.  Should refer to Figure 6, not Figure 3.

Jacquot and Sartelet describes a semi-Lagrangian approach for representing the evolution of the aerosol size distribution through coagulation. They implemented this approach in the SSH-aerosol model and quantified the impact on simulated aerosol number concentrations over Greater Paris by coupling the updated ssh-aerosol scheme with a 3D chemical transport model. In these simulations, they quantified the sensitivity of the simulations to the resolution of the size distribution.

While the technical issues with this paper (described below) may be fixable, I am not convinced that the study addresses a critical research gap, and the technical approach is rather weak. The paper is also poorly written with many grammatical errors. Further, I do not understand why this paper is being considered for the special issue on particle-based methods; they describe an algorithm for a sectional model. I do not see how this is a particle-based method.

• The paper introduces an algorithm designed to improve simulations of the aerosol size distribution, but they do not include any benchmarking for this algorithm. They show differences between their approach and a traditional sectional modeling approach, and they also quantify the impact of these differences for varying resolution of particle sizes. However, they do not show how their approach compares to analytical solutions, quantify performance against a benchmark model, or even show convergence of their solutions. They use the 25-bin sectional approach as their benchmark, but they do not show any verification of the 25-bin model. They evaluate their findings with observations, but they do not include any verification of their new algorithm against a benchmark model, which would be more appropriate.
• The paper describes their algorithm as “Lagrangian”, but they project the size distribution onto a fixed grid at every time step. As there are several truly Lagrangian aerosol and models in the literature (e.g., Shima et al., 2009 and Riemer et al., 2009), I found this characterization misleading. I think it would be more accurately described as “semi-Lagrangian”. I do not understand why this approach is being described in a special issue on particle-based methods.
• The authors state that the numerical results are more sensitive to the resolution of the size distribution than the incorporation of their new algorithm. Even with substantial revisions, the impact of this paper as it is currently framed seems limited. Perhaps it would be better to reformulate the paper to focus on the impact of the size resolution, rather than advocating for a new algorithm that has a relatively small impact on the simulation results.
• Since the authors state that the impact of their semi-Lagrangian scheme has the greatest impact in their low-resolution simulation of 9 bins, I wonder if the proposed algorithm may be more relevant for extremely low-resolution sectional schemes (e.g., 4 bins). WRF-Chem, for example, is often run with 4 sections.
• The references in the introduction should be double-checked. For one thing, the aerosol chemistry model is “MOSAIC” not “MOSAIC”, and the correct citation is “Zaveri, R. A., Easter, R. C., Fast, J. D., & Peters, L. K. (2008). Model for simulating aerosol interactions and chemistry (MOSAIC). Journal of Geophysical Research: Atmospheres, 113(D13).” I noticed this error in the references because the model was misspelled, but I suggest double-checking to be sure the correct paper is referenced.
• In my opinion, Algorithms 1 and 2 would be better described in paragraph form.
• I suggest adding the units into the headings of Table 1 to improve readability.
• This paper contains many grammatical errors and typos. Aside from overt errors, the phrasing is often strange and unclear. The technical editing that would be required to bring this paper to a publishable form is beyond the responsibility of a peer reviewer. I strongly recommend sending this paper to a technical editor before resubmission.