the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
New straightforward formulae for the settling speed of prolate spheroids in the atmosphere: theoretical background and implementation in AerSett v2.0.2
Abstract. We propose two direct expressions to calculate the settling speed of atmospheric particles with prolate spheroidal shapes under the hypothesis of horizontal and vertical orientation. The first formulation is extremely simple and based on theoretical arguments only. The second method, valid for particles with massmedian diameter up to 1000 µm, is based on recent heuristic drag expressions based on CFD numeric simulations. We show that these two formulations show equivalent results within 2 % for d_{eq} ≤ 100 µm, and within 10 % for particles with d_{eq} ≤ 500 µm falling with a horizontal orientation, showing that the first, more simple method is suitable for virtually all atmospheric aerosols, provided their shape can be adequately described as a prolate spheroid. Finally, in order to facilitate the use of our results in chemistrytransport models, we provide an implementation of the first of these methods in AerSett v2.0.2, a module written in Fortran.
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RC1: 'Comment on egusphere20232637', Carlos Alvarez Zambrano, 17 Jan 2024
Review of New straightforward formulae for the settling speed of prolate spheroids in the atmosphere: theoretical background and implementation in AerSett v2.0.2. by Sylvain Mailler et al.
Summary:
In this paper, the authors deduced two equations for calculating the settling velocity of atmospheric particles with elongated spheroidal shapes, considering both horizontal and vertical orientations. The first formulation relies solely on theoretical reasoning. The second method is based on drag expressions derived from numerical simulations using computational fluid dynamics (CFD). Their findings indicate that these two formulations yield comparable results, with a deviation, based on the mean particle diameter, within 2% and 10% for particles falling horizontally. The authors also implemented their formulations into a Fortranbased model to calculate dust transport.
Overall Evaluation:
The manuscript is wellwritten, and the authors have done a great job deducing the equations and providing explanations for the reasoning behind them. However, certain sections of the paper, including those related to the formulation deduction, could benefit from additional explanations and discussion. With the incorporation of extra clarifications and/or inclusion of details, in my opinion, this manuscript will ultimately make a good contribution to the atmospheric dust transport community. Below, I include some questions and comments that could enhance the quality of this paper.
 I recommend that the authors provide a brief description of AerSett v2.0.2 in the Introduction section, as not everyone may be familiar with this module previously published by (almost) the same authors.
 Line 67: I suggest changing the expression "might be tricky" to a more formal expression, such as "pose challenges."
 Line 69: It would be advisable to include the definition of the aspect ratio, even though it is defined later in the document.
 Abstract and Line 85: It is not clear if the authors implemented both formulations as mentioned in Line 85, or if they used the equation obtained from the first approach, as stated in the Abstract.
 Equation 10: Define x in the D(x).
 Equation 11: Is v_inf the settling velocity for prolate spheroidshaped particles? If so, what is the main difference with U^(\lambda, phi)?
 Section 2.3: Why is the slip correction factor needed? Is the correction being applied to the whole range of particle sizes? To determine the applicability of the slipcorrection factor, the Knudsen number (Kn), the ratio of the mean free path to the particle diameter, needs to be observed. Depending on the calculated value of Kn, the correction may be relevant or not. However, the mean free path depends on the pressure, density, and dynamic viscosity of the air. This raises a question for the authors: do the calculations include variations in these air parameters, or was only a constant pressure considered? I recommend that the authors explore in detail the impact and applicability of the slip correction and include in the paper a discussion of for what particle sizes and/or air pressures the correction is important.
 Equation 29: Define u in F_cg(u).
 Line 195: The authors state that Eq. 31 provides an accuracy better than 2.5%. However, it is not clear what was the reference used to calculate/compare the results of this equation.
 Conclusions: I suggest that the authors expand the discussion of the limitations of this formulation. They can explore, for example: i) how other orientation values would change their findings. Although the authors stated that particles tend to fall horizontally, it is also known that during the particle lifespan, they change their orientation. ii) Are there any ideas on how to incorporate porosity into each particle for this new formulation?
Citation: https://doi.org/10.5194/egusphere20232637RC1 
AC1: 'Answer to Reviewer Comment by Carlos Alvarez Zambrano on gusphere20232637', Sylvain Mailler, 25 Jan 2024
We are grateful to Reviewer Carlos Alvarez Zambrano for his careful reading of our manuscript and his insightful questions and suggestions. This comment suggests some clarifications as well as the addition of missing discussion elements. We feel that these suggestions are useful and will help improve the quality of the Manuscript. Please find our answers to this comment in the form of a pdf document (attached).

AC1: 'Answer to Reviewer Comment by Carlos Alvarez Zambrano on gusphere20232637', Sylvain Mailler, 25 Jan 2024
We are grateful to Reviewer Carlos Alvarez Zambrano for his careful reading of our manuscript and his insightful questions and suggestions. This comment suggests some clarifications as well as the addition of missing discussion elements. We feel that these suggestions are useful and will help improve the quality of the Manuscript. Please find our answers to this comment in the form of a pdf document (attached).

RC2: 'Reply on AC1', Carlos Alvarez Zambrano, 05 Feb 2024
Dear Authors,
Thank you for providing additional clarifications. As a final suggestion, I would like to emphasize my earlier recommendation outlined in Comment 7 regarding the Slipcorrection factor. To reiterate, I propose that the authors thoroughly investigate the impact and relevance of the slip correction, incorporating a detailed discussion in the paper regarding the particle sizes and/or air pressures for which the correction holds significance.
While the authors have responded well to the raised question in the modified version of the manuscript, it appears that they suggest referring to another paper, Mailler et al., 2023. While this reference is appropriate, the revised publication should include a concise summary of the key findings from Mailler et al., 2023. This addition would be crucial as the cited paper focuses on sphericalshaped particles, necessitating discussions to explore variances, limitations, and applicability.
Best regards,
Dr. Carlos A. Alvarez
Citation: https://doi.org/10.5194/egusphere20232637RC2

RC2: 'Reply on AC1', Carlos Alvarez Zambrano, 05 Feb 2024

RC3: 'Comment on egusphere20232637', Anonymous Referee #2, 12 Mar 2024
Mailler et al: Settling speed of prolate spheroids in the atmosphere
The authors revisit two published results (Mallios et al 2020 and Sanjeevi et al 2022) for the drag coefficient on prolate spheroids in vertical and horizontal orientations settling in an incompressible Newtonian fluid. I would be happy to recommend the manuscript it the authors address my concerns.
Major:
First of all, I am confused whether why the authors need to explicitly calculate the velocity if the drag coefficients are already calculated as functions of two governing dimensionless quantities: Reynolds numbers and aspect ratios. Usually in simulations, equations of motion are made dimensionless and these drag coefficients can then be directly used and there is usually no need of calculating velocities. If the authors can include a justification, that would be better for the readers. Usually in fluid dynamics literature, analyses and calculations are performed in dimensionless forms and as a result one can make use of the drag coefficients themselves and one does not need to worry about the dependence of nondimensional particle velocity and its variation with d_eq. The goal of making governing variables dimensionless is to encode information in a compact form which can then be easily used in calculations. I do not think such an effort to explicitly calculate the “steady state” particle velocity as a function of d_eq is needed in the first place if one solves dimensionless equations of motion.
Instead of presenting their results a functions of d_eq, I urge the authors to first plot the drag coefficients as functions of Re for different \lambda using the results of Mallios et al 2020 and Sanjeevi et al 2022 simultaneously in a single plot. This would clearly showcase the ranges of Re these results can respectively be used and the range for which they are consistent. This eliminates the need to worry about exact values of d_eq.
What are the values of \rho, \rho_P, and \mu did the authors use for their calculations? Does \rho and \mu correspond to values of air? I think these results should equally be valid even in the case when prolate particles are settling in liquids given their Re are in the range when the expressions given by Mallios et al 2020 and Sanjeevi et al 2022 are valid. Why do then the authors focus only on atmosphere?
In addition to Re and \lambda, in the case when Re > 1 Stokes number also becomes important. The authors should comment on why they ignored it. They should also include a discussion addressing the time evolution of particle speed in the cases when particles have a finite Re.
It is well known that prolate particles (spheroidal particles in general see Ardekani et al IJMF 87 2016 PP1634) rotate as they settle under gravity and attain a steady state orientation such that their broadside is horizontal if their R is lower than a critical value. At R above this critical value, they undergo orientation instability such that they do not have a steady a steady state orientation. In such a case, how useful are the assumptions about \phi = 0 and \pi/2?
It is well known that the earth’s atmosphere has density and viscosity stratification (see Magnaudet and Mercier Annual Review of Fluid Mechanics 2020; More and Ardekani Annual Review of Fluid Mechanics 2023). In such a case how useful are these calculations? Shouldn’t one need to include effects of stratification and time dependence then?
Aerosols can also mean liquid droplets suspended in air. The authors should clearly state that by aerosol they strictly focus on solid particles in air as in the case of liquid droplets, the surface boundary conditions are different and the calculations presented are not valid.
Minor:
Eq 15 should have \mu^2 in the denominator. This definition of R is equivalent to something called Archimedes number
Line 233: typo should be “expression”

AC2: 'Reply to Anonymous Referee #2', Sylvain Mailler, 22 Mar 2024
We are grateful to Anonymous Reviewer 2 for their careful reading of our manuscript and their insightful questions and suggestions. Their comment seem to be written from a fluid mechanicist point of view, which makes it a particularly useful apport to this discussion, since the manuscript was prepared from an atmospheric physics point of view. We are particularly grateful to the Reviewer to bring to our attention that what we had called the “pseudoReynolds number” in the present study and in Mailler et al. (2023) is already known as the Archimedes number. This information will permit us to alleviate and clarify the redaction of our manuscript.

AC2: 'Reply to Anonymous Referee #2', Sylvain Mailler, 22 Mar 2024
Status: closed

RC1: 'Comment on egusphere20232637', Carlos Alvarez Zambrano, 17 Jan 2024
Review of New straightforward formulae for the settling speed of prolate spheroids in the atmosphere: theoretical background and implementation in AerSett v2.0.2. by Sylvain Mailler et al.
Summary:
In this paper, the authors deduced two equations for calculating the settling velocity of atmospheric particles with elongated spheroidal shapes, considering both horizontal and vertical orientations. The first formulation relies solely on theoretical reasoning. The second method is based on drag expressions derived from numerical simulations using computational fluid dynamics (CFD). Their findings indicate that these two formulations yield comparable results, with a deviation, based on the mean particle diameter, within 2% and 10% for particles falling horizontally. The authors also implemented their formulations into a Fortranbased model to calculate dust transport.
Overall Evaluation:
The manuscript is wellwritten, and the authors have done a great job deducing the equations and providing explanations for the reasoning behind them. However, certain sections of the paper, including those related to the formulation deduction, could benefit from additional explanations and discussion. With the incorporation of extra clarifications and/or inclusion of details, in my opinion, this manuscript will ultimately make a good contribution to the atmospheric dust transport community. Below, I include some questions and comments that could enhance the quality of this paper.
 I recommend that the authors provide a brief description of AerSett v2.0.2 in the Introduction section, as not everyone may be familiar with this module previously published by (almost) the same authors.
 Line 67: I suggest changing the expression "might be tricky" to a more formal expression, such as "pose challenges."
 Line 69: It would be advisable to include the definition of the aspect ratio, even though it is defined later in the document.
 Abstract and Line 85: It is not clear if the authors implemented both formulations as mentioned in Line 85, or if they used the equation obtained from the first approach, as stated in the Abstract.
 Equation 10: Define x in the D(x).
 Equation 11: Is v_inf the settling velocity for prolate spheroidshaped particles? If so, what is the main difference with U^(\lambda, phi)?
 Section 2.3: Why is the slip correction factor needed? Is the correction being applied to the whole range of particle sizes? To determine the applicability of the slipcorrection factor, the Knudsen number (Kn), the ratio of the mean free path to the particle diameter, needs to be observed. Depending on the calculated value of Kn, the correction may be relevant or not. However, the mean free path depends on the pressure, density, and dynamic viscosity of the air. This raises a question for the authors: do the calculations include variations in these air parameters, or was only a constant pressure considered? I recommend that the authors explore in detail the impact and applicability of the slip correction and include in the paper a discussion of for what particle sizes and/or air pressures the correction is important.
 Equation 29: Define u in F_cg(u).
 Line 195: The authors state that Eq. 31 provides an accuracy better than 2.5%. However, it is not clear what was the reference used to calculate/compare the results of this equation.
 Conclusions: I suggest that the authors expand the discussion of the limitations of this formulation. They can explore, for example: i) how other orientation values would change their findings. Although the authors stated that particles tend to fall horizontally, it is also known that during the particle lifespan, they change their orientation. ii) Are there any ideas on how to incorporate porosity into each particle for this new formulation?
Citation: https://doi.org/10.5194/egusphere20232637RC1 
AC1: 'Answer to Reviewer Comment by Carlos Alvarez Zambrano on gusphere20232637', Sylvain Mailler, 25 Jan 2024
We are grateful to Reviewer Carlos Alvarez Zambrano for his careful reading of our manuscript and his insightful questions and suggestions. This comment suggests some clarifications as well as the addition of missing discussion elements. We feel that these suggestions are useful and will help improve the quality of the Manuscript. Please find our answers to this comment in the form of a pdf document (attached).

AC1: 'Answer to Reviewer Comment by Carlos Alvarez Zambrano on gusphere20232637', Sylvain Mailler, 25 Jan 2024
We are grateful to Reviewer Carlos Alvarez Zambrano for his careful reading of our manuscript and his insightful questions and suggestions. This comment suggests some clarifications as well as the addition of missing discussion elements. We feel that these suggestions are useful and will help improve the quality of the Manuscript. Please find our answers to this comment in the form of a pdf document (attached).

RC2: 'Reply on AC1', Carlos Alvarez Zambrano, 05 Feb 2024
Dear Authors,
Thank you for providing additional clarifications. As a final suggestion, I would like to emphasize my earlier recommendation outlined in Comment 7 regarding the Slipcorrection factor. To reiterate, I propose that the authors thoroughly investigate the impact and relevance of the slip correction, incorporating a detailed discussion in the paper regarding the particle sizes and/or air pressures for which the correction holds significance.
While the authors have responded well to the raised question in the modified version of the manuscript, it appears that they suggest referring to another paper, Mailler et al., 2023. While this reference is appropriate, the revised publication should include a concise summary of the key findings from Mailler et al., 2023. This addition would be crucial as the cited paper focuses on sphericalshaped particles, necessitating discussions to explore variances, limitations, and applicability.
Best regards,
Dr. Carlos A. Alvarez
Citation: https://doi.org/10.5194/egusphere20232637RC2

RC2: 'Reply on AC1', Carlos Alvarez Zambrano, 05 Feb 2024

RC3: 'Comment on egusphere20232637', Anonymous Referee #2, 12 Mar 2024
Mailler et al: Settling speed of prolate spheroids in the atmosphere
The authors revisit two published results (Mallios et al 2020 and Sanjeevi et al 2022) for the drag coefficient on prolate spheroids in vertical and horizontal orientations settling in an incompressible Newtonian fluid. I would be happy to recommend the manuscript it the authors address my concerns.
Major:
First of all, I am confused whether why the authors need to explicitly calculate the velocity if the drag coefficients are already calculated as functions of two governing dimensionless quantities: Reynolds numbers and aspect ratios. Usually in simulations, equations of motion are made dimensionless and these drag coefficients can then be directly used and there is usually no need of calculating velocities. If the authors can include a justification, that would be better for the readers. Usually in fluid dynamics literature, analyses and calculations are performed in dimensionless forms and as a result one can make use of the drag coefficients themselves and one does not need to worry about the dependence of nondimensional particle velocity and its variation with d_eq. The goal of making governing variables dimensionless is to encode information in a compact form which can then be easily used in calculations. I do not think such an effort to explicitly calculate the “steady state” particle velocity as a function of d_eq is needed in the first place if one solves dimensionless equations of motion.
Instead of presenting their results a functions of d_eq, I urge the authors to first plot the drag coefficients as functions of Re for different \lambda using the results of Mallios et al 2020 and Sanjeevi et al 2022 simultaneously in a single plot. This would clearly showcase the ranges of Re these results can respectively be used and the range for which they are consistent. This eliminates the need to worry about exact values of d_eq.
What are the values of \rho, \rho_P, and \mu did the authors use for their calculations? Does \rho and \mu correspond to values of air? I think these results should equally be valid even in the case when prolate particles are settling in liquids given their Re are in the range when the expressions given by Mallios et al 2020 and Sanjeevi et al 2022 are valid. Why do then the authors focus only on atmosphere?
In addition to Re and \lambda, in the case when Re > 1 Stokes number also becomes important. The authors should comment on why they ignored it. They should also include a discussion addressing the time evolution of particle speed in the cases when particles have a finite Re.
It is well known that prolate particles (spheroidal particles in general see Ardekani et al IJMF 87 2016 PP1634) rotate as they settle under gravity and attain a steady state orientation such that their broadside is horizontal if their R is lower than a critical value. At R above this critical value, they undergo orientation instability such that they do not have a steady a steady state orientation. In such a case, how useful are the assumptions about \phi = 0 and \pi/2?
It is well known that the earth’s atmosphere has density and viscosity stratification (see Magnaudet and Mercier Annual Review of Fluid Mechanics 2020; More and Ardekani Annual Review of Fluid Mechanics 2023). In such a case how useful are these calculations? Shouldn’t one need to include effects of stratification and time dependence then?
Aerosols can also mean liquid droplets suspended in air. The authors should clearly state that by aerosol they strictly focus on solid particles in air as in the case of liquid droplets, the surface boundary conditions are different and the calculations presented are not valid.
Minor:
Eq 15 should have \mu^2 in the denominator. This definition of R is equivalent to something called Archimedes number
Line 233: typo should be “expression”

AC2: 'Reply to Anonymous Referee #2', Sylvain Mailler, 22 Mar 2024
We are grateful to Anonymous Reviewer 2 for their careful reading of our manuscript and their insightful questions and suggestions. Their comment seem to be written from a fluid mechanicist point of view, which makes it a particularly useful apport to this discussion, since the manuscript was prepared from an atmospheric physics point of view. We are particularly grateful to the Reviewer to bring to our attention that what we had called the “pseudoReynolds number” in the present study and in Mailler et al. (2023) is already known as the Archimedes number. This information will permit us to alleviate and clarify the redaction of our manuscript.

AC2: 'Reply to Anonymous Referee #2', Sylvain Mailler, 22 Mar 2024
Model code and software
AerSett v2.0.2 S. Mailler, L. Menut, A. Cholakian, and R. Pennel https://doi.org/10.5281/zenodo.10261378
Interactive computing environment
FreeFall v2.1 S. Mailler https://doi.org/10.5281/zenodo.10254770
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