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REVIEW 4 major objections 6 minor 41 references

Drop size distribution from laboratory experiments based on single-drop fragmentation and comparison with aerial in-situ measurements

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Laboratory drop fragmentation, modelled with a two-parameter gamma distribution, recovers the Marshall-Palmer raindrop-size law and matches in-situ aerial data.

desk verdict New lab DSD data and a plausible rain-model application, but Eq. (14) is currently an assertion until the characteristic sizes and gamma parameters are reported. read the letter →

arxiv 2501.16686 v2 pith:FF3BSB3F submitted 2025-01-28 physics.flu-dyn

Shubham Chakraborty , Someshwar Sanjay Ade , Lalsingh Devsoth , Lakshmana D. Chandrala , Thara Prabhakaran , Omar K. Matar , Kirti Chandra Sahu This is my paper · ORCID
classification physics.flu-dyn
keywords raindropsizedistributiondropbreakupbaggammaMarshall-Palmerrelationdigitalin-lineholographyWebernumberaerialin-situmeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Raindrop size distributions determine how weather radar echoes are interpreted, since reflectivity scales with the sixth power of drop diameter. This paper asks whether the distribution observed in steady rain can be produced from the physics of a single drop breaking up in an upward airstream. It reports laboratory experiments on drops of 3.6 mm diameter in Weber-number regimes that produce bag, bag-stamen, and dual-bag breakup, together with a two-parameter gamma model whose parameters come from eleven characteristic fragment sizes. The model reproduces the measured distributions and yields $d_{\rm avg}^{-1} = 40R^{-0.21}$, close to the Marshall-Palmer relation for steady rain ($d_{\rm avg}^{-1} = 41R^{-0.21}$), and it aligns with airborne in-situ measurements from the CAIPEEX campaign.

What carries the argument

The central object is the two-parameter gamma distribution $P(x = d/d_{\rm avg}) = x^{\alpha-1}e^{-x/\beta}/(\beta^{\alpha}\Gamma(\alpha))$, with shape $\alpha = (\bar{x}/s)^2$ and rate $\beta = s^2/\bar{x}$, where $\bar{x}$ and $s$ are the mean and standard deviation of eleven characteristic fragment sizes. Four sizes come from bag rupture (bag thickness, receding-rim thickness, the Rayleigh-Plateau ligament size, and satellite droplets), four from rim fragmentation (rim-thickness size, receding-rim collision size, and two satellite corrections), and three from node breakup at minimum, mean, and maximum node volume fractions. This replaces the single-parameter gamma model in which $\alpha = \beta$, and it turns the previously assumed average drop diameter into a computed output. The distribution is then fed through the rainfall-rate integral (Eq. 2) to reach the Marshall-Palmer-type law.

What would settle it

Measure the full child-drop size distribution for the three Weber numbers (9.38, 16.9, 18.9) with the same holography setup, recompute the eleven characteristic sizes from the cited formulas, and check whether the gamma distribution with $\alpha=(\bar{x}/s)^2$ and $\beta=s^2/\bar{x}$ reproduces the measured tail at $d/d_{\rm avg}\approx 7.3$ within a few percent; if the tail deviates by more than the claimed 7\%, the Marshall-Palmer match is not established.

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Extended reading notes

Core claim

The central claim is that all breakup modes contribute characteristic sizes that, combined in a two-parameter gamma distribution, predict the full child-drop size distribution, and that this distribution matches both laboratory measurements and the standard steady-rain relation. Computing the integral in Eq. (2) with the model gives $d_{\rm avg}^{-1} = 40R^{-0.21}$, so the average drop diameter tracks rainfall rate with the same exponent (0.21) as Marshall-Palmer and a coefficient of 40 versus 41. At large normalized sizes ($d/d_{\rm avg} = 7.3$), the model deviates from Marshall-Palmer by 6.6--7.6\%, while the single-parameter gamma model deviates by 24--27\%. The same distribution is then compared with in-situ DSD measurements from three research flights, with which it shows satisfactory alignment.

Load-bearing premise

A two-parameter gamma distribution whose shape and rate are fixed by the mean and standard deviation of eleven theoretically estimated characteristic sizes captures the full measured child-drop distribution, including the large-drop tail used for the Marshall-Palmer comparison.

Editorial extensions

If this is right

  • The model yields $d_{\rm avg}^{-1}=40R^{-0.21}$, effectively recovering Marshall-Palmer's $41R^{-0.21}$ for steady rain.
  • At the large-drop end ($d/d_{\rm avg}\approx 7.3$), the model stays within about 7\% of Marshall-Palmer across all three breakup modes, whereas the single-parameter model deviates by 24--27\%.
  • The same two-parameter distribution describes bag, bag-stamen, and dual-bag breakup, suggesting the eleven characteristic sizes capture the breakup physics relevant to rain.
  • In-situ drop spectra from the three CAIPEEX flights fall within the envelope of the laboratory and theoretical distributions, supporting laboratory-scale experiments for rainfall modelling.
  • Radar-relevant quantities such as the sixth moment of the drop size distribution can be computed from the model rather than assumed as input.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the eleven characteristic sizes can be predicted across the Weber-number range rather than measured case by case, the model could extend to unsteady or intense rain without in-situ calibration.
  • The comparison with flight data is visual and qualitative; a quantitative, altitude-resolved test against the CAIPEEX spectra would be a sharper check of the model's tail behaviour.
  • The model's fidelity at large $d/d_{\rm avg}$ could be tested directly by computing the sixth moment $\int d^6 P(d)\,dd$ from the holography data and comparing it with radar reflectivity--rain-rate relations.
  • A natural extension is to test whether the same gamma parameters also describe DSDs altered by collisional breakup or mixed-phase microphysics, processes the paper itself lists as unaddressed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper reports laboratory experiments on the fragmentation of single water drops in an upward airstream at three Weber numbers (We = 9.38, 16.9, and 18.9), corresponding to bag, bag-stamen, and dual-bag breakup morphologies. The child-drop size distributions are measured using shadowgraphy and machine-learning-assisted digital in-line holography. The authors then construct a two-parameter gamma distribution model, Eq. (4), whose parameters are derived from the mean and standard deviation of eleven characteristic breakup sizes taken from the bag, rim, and node fragmentation modes. They claim that this model reproduces the experimental distributions, and that evaluating the integral in Eq. (2) yields the Marshall-Palmer-like relation d_avg^{-1} = 40 R^{-0.21} (Eq. (14)). The paper also compares the resulting drop size distributions with in-situ raindrop measurements from the CAIPEEX research flights. The central claims are that a lab-derived, breakup-based model can predict the steady-rain drop size distribution and that the model matches aerial observations.

Significance. If the central derivation and comparisons are correct, the paper offers a potentially important bridge between single-drop fragmentation physics and operational rainfall parameterization: a model that estimates davg from measurable breakup mechanisms without treating it as an unknown input. The experimental data set, which combines controlled lab breakup with in-situ aircraft measurements, is a valuable resource, and the use of holography with a U-Net segmentation pipeline is a methodological strength. However, the main quantitative result, Eq. (14), is not verifiable from the preprint as written, and the parameterization in Eq. (4) contains a dimensional ambiguity. The significance of the paper in its current form is therefore conditional on a revised and fully documented derivation.

major comments (4)
  1. [Section IV, Eq. (14)] The computation of the integral in Eq. (2) that leads to Eq. (14) is not reported. The eleven characteristic sizes, the resulting mean and standard deviation, the values of alpha and beta for each Weber number, and the numerical value of the integral are all absent. The coefficient 40 in Eq. (14) therefore cannot be checked. The manuscript must specify how the three Weber-number cases (We = 9.38, 16.9, and 18.9), which have distinct gamma parameters, are combined into a single power law with fixed constants; if one representative case is used, that choice must be justified.
  2. [Section IV, Eq. (4)] The definitions alpha = (xbar/s)^2 and beta = s^2/xbar are dimensionally inconsistent as written. The text states that xbar and s are the mean and standard deviation of the eleven characteristic sizes, which are measured in micrometers, but the variable x = d/davg in Eq. (4) is dimensionless. If xbar and s are meant to be normalized by davg, that normalization is not stated. Moreover, because davg is set equal to the arithmetic mean of the characteristic sizes, the normalized mean is identically 1, which reduces the gamma distribution to a single-parameter form; this reduction needs to be acknowledged and the subsequent parameter counting made explicit.
  3. [Section IV, Eqs. (2) and (14)] The value n0 = 0.08 cm^-4 is taken from the Marshall-Palmer empirical fit and inserted into Eq. (2). Since Eq. (14) is then compared with the Marshall-Palmer relation, the comparison is partly anchored by the chosen n0. The authors should either derive n0 from the fragmentation model or clearly label it as an external input, and they should report the sensitivity of Eq. (14) to a realistic range of n0 values.
  4. [Section V, Figure 6] The conversion of the CAIPEEX in-situ droplet concentration data C(d) into the normalized distribution P(x) plotted in Figure 6 is not described. In particular, the authors do not state how davg is estimated from the measured size-resolved concentrations or how the normalization is performed. Without this information, the claimed 'satisfactory alignment' between the laboratory model and the aerial data cannot be quantitatively assessed.
minor comments (6)
  1. [Section IV] The phrase 'we compute the integral R x^{7/2} P(x) x. in Eq. (2)' is malformed; it should read 'the integral \int x^{7/2} P(x) dx in Eq. (2)'.
  2. [Section II and Figure 3 caption] The initial droplet diameter is stated as 3.6 ± 0.08 mm in the main text but as 3.6 ± 0.06 mm in the caption of Figure 3; please reconcile the uncertainty.
  3. [Section VI] The sentence fragment '7 also emphasized the importance...' should be rephrased, for example as 'Villermaux and Bossa also emphasized the importance...'.
  4. [Section II] The description 'n0 is the average spatial density of drops (measured in cm^-4)' is ambiguous; n0 is the intercept of the number concentration per unit size interval per unit volume, and the text should use standard terminology.
  5. [Section IV] The statement 'All characteristic sizes are measured in µm' should be accompanied by explicit unit conversions, since davg in Eq. (14) is in centimetres and R in mm/hr.
  6. [Section V, Figure 6] The shaded envelope in Figure 6 is not defined in the caption; specify whether it covers the full range of scattered data points, a percentile interval, or the combined range of all in-situ and laboratory data.

Circularity Check

1 steps flagged · score 6.0 of 10

Eq. (14)'s exponent 0.21 is inherited from Marshall-Palmer/Villermaux-Bossa, while true inversion of Eq. (2) gives 2/9; the headline agreement is partly an input, not an output.

  1. fitted input called prediction [Section IV, Eq. (14); Section I, Marshall-Palmer parameters]
    "Marshall and Palmer 5 found that for steady rainfall, n0 = 0.08 cm−4, Λ = 41, and a = 0.21. ... d−1 avg = 48.5R−2/9 (i.e. Λ = 48.5 and a = 2/9). ... we compute the integral ... in Eq. (2), leading to the relationship ... d−1 avg = 40R−0.21, such that Λ = 40 and a = 0.21. This relationship closely matches the one proposed by Marshall and Palmer for steady rain5"

    Eq. (2) has the form R = C d_avg^(9/2) I with I a constant from the normalized P(x); inverting gives d_avg^(-1) = C' R^(-2/9) for any normalized P(x). The exponent 0.21 reported in Eq. (14) is therefore not an output of the authors' gamma model: the actual computed exponent is 2/9 ≈ 0.222. The paper instead reports a = 0.21, exactly the Marshall-Palmer input quoted earlier, and it feeds n0 = 0.08 cm^-4 from Marshall-Palmer into the same Eq. (2). The central 'prediction' d_avg^(-1) = 40 R^(-0.21) thus reduces to the Marshall-Palmer parameters used as inputs; the exponent is forced and the prefactor is anchored by the chosen n0.

full rationale

The paper's shape-level comparison in Fig. 5 is a genuine, externally anchored exercise: a two-parameter gamma distribution with parameters derived from characteristic breakup sizes is compared with measured P(x), and this does not reduce to fitting the target DSD. However, the central quantitative rainfall claim, Eq. (14), is not an independent prediction. Eq. (2), taken from Villermaux and Bossa, already imposes d_avg^(-1) proportional to R^(-2/9) for any normalized P(x). The authors state 2/9 for the Villermaux-Bossa result but then report a = 0.21 after evaluating their integral; 0.21 is the Marshall-Palmer exponent, not 2/9. The prefactor 40 is similarly computed using n0 = 0.08 cm^-4 taken from Marshall-Palmer, so the 'close agreement' of the prefactor with Marshall-Palmer's Λ = 41 is partly anchored by that input. In addition, the eleven characteristic sizes, the resulting α and β values, and the integral I are never reported, and Eq. (4)'s parameter definitions are dimensionally inconsistent as written (if x̄ and s are in µm, β has units of length and P(x) is not a properly normalized density in dimensionless x). These omissions prevent independent verification of the derived Λ = 40 but are verifiability concerns rather than the core circularity. The central exponent reduction warrants a score of 6: one or more headline predictions reduce by construction to inputs, while there remains independent content in the shape comparison and experimental data.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central coefficient in Eq (14) depends on n0 from Marshall-Palmer and on node-volume fractions from Jackiw-Ashgriz, both imported rather than derived. The paper does not report the measured characteristic sizes or alpha and beta, so additional hidden dependencies exist. No new physical entities are postulated.

free parameters (2)
  • n0, spatial density of drops = 0.08 cm^-4
    Taken from Marshall-Palmer and inserted into Eq (2); this external empirical input sets the coefficient 40 in Eq (14) and is not derived in the paper.
  • Vn, node volume fractions = 0.2, 0.4, 1.0
    Values from Jackiw and Ashgriz (2022) used to produce three node-breakup characteristic sizes; they affect the mean and standard deviation and are not measured in this paper.
assumptions (5)
  • domain assumption The child-drop number size distribution follows a two-parameter gamma distribution.
    Eq (4) postulates the gamma form; this is a phenomenological assumption from prior work, not derived in this paper.
  • domain assumption Eleven characteristic sizes from bag, rim, and node breakup are sufficient to determine the full drop size distribution.
    The model collapses all breakup outcomes onto the mean and standard deviation of eleven characteristic sizes, with no validation that this captures the large-drop tail.
  • domain assumption Villermaux-Bossa Eq (2) correctly relates rainfall rate to average droplet diameter.
    The paper takes Eq (2) as the bridge to rainfall rate; this fixes the exponent structure and introduces n0 as an input.
  • domain assumption A drop in a counter-current airstream is analogous to a raindrop falling in quiescent air.
    The meteorological interpretation relies on this analogy, cited from refs 19-20 in the introduction.
  • domain assumption CIP and PIP in-situ measurements represent raindrop size distributions at altitude.
    The comparison treats airborne particle measurements as raindrops, although the paper later admits possible snowflake contamination at larger sizes.

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Pith. "Pith review of Drop size distribution from laboratory experiments based on single-drop fragmentation and comparison with aerial in-situ measurements." pith.science (2026). https://pith.science/paper/FF3BSB3F

@misc{pith2026250116686,
  author       = {Pith},
  title        = {Pith review of: Drop size distribution from laboratory experiments based on single-drop fragmentation and comparison with aerial in-situ measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FF3BSB3F}},
  note         = {Machine review of arXiv:2501.16686}
}
read the original abstract

Laboratory experiments and theoretical modelling are conducted to determine the raindrop size distribution (DSD) resulting from distinct fragmentation processes under various upward airstreams. Since weather radar echoes are proportional to the sixth power of the average droplet diameter, understanding the fragmentation mechanisms that lead to different breakup sizes is crucial for accurate rainfall predictions. We utilize a two-parameter gamma distribution for theoretical modelling and estimate the average droplet diameter from the theoretically obtained characteristic sizes, often treated as assumed input parameters for different rain conditions in rainfall modelling. Our experimental and theoretical findings demonstrate a close agreement with the DSD predicted by the Marshall and Palmer relationship for steady rain conditions. Additionally, in situ DSD measurements at different altitudes were obtained through research flights equipped with advanced sensors, further validating our rainfall model. This study underscores the effectiveness of laboratory-scale experiments and the critical importance of accurately characterizing DSD to enhance rainfall predictions.

Figures

Figures reproduced from arXiv: 2501.16686 by the authors.

Figure 1
Figure 1. FIG. 1. Updraft (upward movement) of warm air from the surface [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic diagram of the experimental setup (side view) used in in-line holography to obtain the drop size distribution. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Temporal evolution of the droplet breakup dynamics for (a) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Temporal evolution of the droplet size distribution, illustrating droplet counts [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Drop size distribution for different breakup phenomena. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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