pith. machine review for the scientific record. sign in

arxiv: 2604.10564 · v1 · submitted 2026-04-12 · ⚛️ physics.ao-ph · physics.flu-dyn

Recognition: unknown

Direct Lagrangian tracking simulation of droplet growth in vertically-developing turbulent cloud

Masaya Iwashima, Ryo Onishi

Pith reviewed 2026-05-10 15:46 UTC · model grok-4.3

classification ⚛️ physics.ao-ph physics.flu-dyn
keywords cloud microphysicsturbulencecollision-coalescenceLagrangian trackingDNSprecipitationdroplet growth
0
0 comments X

The pith

Turbulence speeds droplet collisions in clouds, causing earlier precipitation with larger first raindrops at the ground.

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

The paper builds a DNS model that tracks every droplet's full life cycle inside a tall vertical slab stretching from ground to cloud top. It runs the same cloud development twice, once with added wind fluctuations and once without, to isolate turbulence effects on activation, condensation, collisions, and fallout. The runs show turbulence first boosts same-size droplet mergers in the middle layer while an updraft is active, then shifts to large droplets sweeping up small ones lower down. This shortens the time until rain reaches the ground and increases the size of the very first surface drop. The vertical setup reveals height-dependent statistics that box-style simulations miss.

Core claim

Turbulence promotes collision-coalescence: early autoconversions between similar-sized droplets occur in the middle layer during the updraft phase, followed by accretions of small droplets by larger ones in middle and lower layers, so precipitation arrives earlier at the ground and the first raindrop is larger than in the corresponding non-turbulent run.

What carries the argument

Vertically-elongated quasi-1D domain with Lagrangian particle tracking of all warm-rain processes plus an embedded homogeneous isotropic turbulence field.

If this is right

  • Turbulence shortens the time from cloud formation to surface rain.
  • Autoconversion dominates early growth in the middle cloud levels while updrafts persist.
  • Accretion takes over later and lower, where larger drops have already formed.
  • Box-domain DNS without vertical structure or turbulence underestimates rain formation speed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If real cloud turbulence is stronger or more organized than the homogeneous field used here, the speedup of precipitation could be even greater.
  • Weather and climate models that ignore turbulence-enhanced collisions will tend to delay predicted rain onset.
  • The same vertical tracking method could test how changes in updraft strength or aerosol loading alter the turbulence effect.

Load-bearing premise

The added uniform random turbulence field stands in for the real, spatially varying motions that arise from the cloud's own updraft and developing structure.

What would settle it

Measurements or simulations that find identical precipitation onset times and identical sizes of the first ground-level raindrops in turbulent versus non-turbulent conditions would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.10564 by Masaya Iwashima, Ryo Onishi.

Figure 1
Figure 1. Figure 1: Schematic of vertically-elongated quasi-1D computational domain. Liquid droplets at t = 1000 s are visualized. 2015; Saito and Gotoh, 2018; Chen et al., 2018). Although the periodic box domain is numerically convenient, it is not phys￾ically realistic: real atmospheric clouds have vertical structure in terms of, for example, supersaturation ratio or droplet size distribution. However, the periodic box doma… view at source ↗
Figure 2
Figure 2. Figure 2: A snapshot of homogeneous isotropic turbulence (HIT) is repeated in the vertical direction to fill the vertically-elongated domain. In the figure of HIT, the colored isosurfaces and arrows represent the Q-criterion isosurfaces and the velocity vectors on the x − z plane, respectively. LCS with vertically-elongated quasi-1D computational domain is based on Kunishima and Onishi (2018) [PITH_FULL_IMAGE:figur… view at source ↗
Figure 3
Figure 3. Figure 3: The collision frequency [m−3 s −1 ] between droplets with dp < 80 µm that result in droplets with dp ≥ 80 µm. 300 400 500 600 700 800 900 Time [s] 0 20 40 60 80 100 A v erage particle diam eter [ m] LAM case, z = 700 750 m (lower layer) LAM case, z = 1200 1250 m (middle layer) LAM case, z = 1700 1750 m (upper layer) TURB case, z = 700 750 m (lower layer) TURB case, z = 1200 1250 m (middle layer) TURB case,… view at source ↗
Figure 4
Figure 4. Figure 4: Temporal evolutions of average diameter (mass-weighted average) at three representative altitudes (lower layer: z = 700−750 m, middle layer: z = 1200 − 1250 m, and upper layer: z = 1700 − 1750 m). In addition, the average diameter ratio of the colliding droplets with the threshold diameter of 80 µm was calculated from 300 s to 600 s. The average diameter ratio was 1.88 in LAM-case, while it was 1.66 in TUR… view at source ↗
Figure 5
Figure 5. Figure 5: Temporal evolutions of droplet size distribution (DSD) at three altitude ranges in the cloud (lower layer: z = 700−750 m, middle layer: z = 1200 − 1250 m, and upper layer: z = 1700 − 1750 m). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The diameter of surface-reaching raindrops and the time when they reach the ground. 0 50000 100000 150000 Nmemb 0 1 × 10 9 2 × 10 9 3 × 10 9 4 × 10 9 Vsr [m3 ] TURB-case LAM-case Vsr = Nmemb 2.52×10 14 : TURB-case Vsr = Nmemb 2.36×10 14 : LAM-case [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The volume of surface-reaching raindrops Vsr [m3 ] against the number of constituent particles Nmemb. These results indicate that promoted collisions caused earlier precipitation onset and larger initial surface-reaching raindrops with the presence of turbulence. In the Lagrangian tracking simulation, the growth history of each precipitation particle is individually tracked [PITH_FULL_IMAGE:figures/full_f… view at source ↗
read the original abstract

We developed a new explicit cloud microphysical model, based on direct numerical simulation (DNS) with Lagrangian particle tracking. The model employs a vertically-elongated quasi-1D computational domain extending from the ground to the cloud top to explicitly capture the vertical structure of clouds. This allows us to simulate the all warm-cloud microphysical processes, including activation, condensation growth, collision-coalescence growth, and sedimentation. A homogeneous isotropic turbulence field is incorporated into this domain to explicitly resolve the turbulent wind fluctuations. Cloud microphysics simulations with and without turbulent wind fluctuations were performed to clarify the impact of turbulence on droplet growth. We obtained new insights into the altitude- and time-dependent microphysical statistics, which cannot be obtained through conventional DNS researches for a cubic box domain with periodic boundaries. The comparison have shown that turbulence promoted the collision-coalescence growth of droplets. During the early developing stage, where the updraft was present, turbulence promoted the collisions between droplets with similar sizes (autoconversions) in the middle layer of the cloud. In later stage, relatively large droplets produced by autoconversions actively collected smaller droplets (accretions) in the middle and lower layers. The onset of precipitation at the ground occurred earlier and the first raindrop at the ground was larger in turbulence case than that in non-turbulence case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. This paper introduces a DNS model with Lagrangian particle tracking for warm-cloud microphysics in a vertically elongated domain from ground to cloud top. It adds a homogeneous isotropic turbulence field and compares runs with and without turbulence to isolate effects on activation, condensation, collision-coalescence, and sedimentation. The central claim is that turbulence promotes autoconversion in the middle layer during the early updraft stage and accretion in middle/lower layers later, producing earlier surface precipitation onset and a larger first raindrop.

Significance. The vertically elongated domain is a clear strength, enabling explicit capture of vertical structure, updraft interactions, and altitude-dependent statistics unavailable in periodic-box DNS. The direct with/without comparison cleanly isolates turbulence effects on specific growth stages. If the turbulence representation and numerics are shown to be adequate, the work would provide useful mechanistic insights into turbulence-enhanced droplet growth.

major comments (3)
  1. [Methods] Methods section on turbulence implementation: The homogeneous isotropic field is superimposed on a vertically developing flow containing an updraft; no quantitative comparison is given to observed cloud dissipation rates, anisotropy, or vertical correlation lengths at droplet scales. This directly affects whether the reported layer-specific autoconversion and accretion enhancements can be attributed to realistic turbulence rather than the forcing choice.
  2. [Results] Results on collision statistics: No grid-resolution convergence tests or sensitivity to Lagrangian collision kernel parameters are presented, despite the fact that collision rates depend on resolved velocity gradients and interpolation. Without these, the claimed differences between turbulent and non-turbulent cases lack demonstrated numerical robustness.
  3. [Results] Results, early and late stages: Collision-rate differences are reported without error bars, ensemble variability, or statistical significance tests. This undermines the quantitative statements on promotion of autoconversion versus accretion and the timing of surface precipitation.
minor comments (3)
  1. [Abstract] Abstract: grammatical error ('The comparison have shown' should be 'The comparison has shown').
  2. [Results] Figures and text: the precise altitude ranges defining 'middle' and 'lower' layers should be stated explicitly when discussing layer-specific statistics.
  3. [Introduction] Introduction: additional citations to existing Lagrangian cloud DNS studies would better situate the vertically elongated domain approach.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for their thorough and constructive review of our manuscript. We address each of the major comments in detail below, indicating the revisions we plan to make to strengthen the paper.

read point-by-point responses

  1. Referee: [Methods] Methods section on turbulence implementation: The homogeneous isotropic field is superimposed on a vertically developing flow containing an updraft; no quantitative comparison is given to observed cloud dissipation rates, anisotropy, or vertical correlation lengths at droplet scales. This directly affects whether the reported layer-specific autoconversion and accretion enhancements can be attributed to realistic turbulence rather than the forcing choice.

    Authors: Our study is designed as an idealized numerical experiment to isolate the impact of turbulence on droplet growth processes by direct comparison of otherwise identical simulations with and without the superimposed HIT field. The turbulence intensity and scales were selected to be representative of typical warm cloud conditions based on established literature values for the dissipation rate and integral scales. We agree that additional context would be beneficial. In the revised manuscript, we will expand the Methods section to provide a more detailed rationale for the chosen turbulence parameters, including comparisons to typical observed values from the literature (e.g., dissipation rates in the range of 10^{-4} to 10^{-2} m^2 s^{-3}), and explicitly discuss the assumptions and limitations of superimposing isotropic turbulence on a developing updraft flow. This will help clarify that the enhancements are due to the presence of turbulence rather than specific forcing details. revision: partial

  2. Referee: [Results] Results on collision statistics: No grid-resolution convergence tests or sensitivity to Lagrangian collision kernel parameters are presented, despite the fact that collision rates depend on resolved velocity gradients and interpolation. Without these, the claimed differences between turbulent and non-turbulent cases lack demonstrated numerical robustness.

    Authors: We acknowledge this as a valid concern for ensuring the robustness of the collision statistics. The spatial resolution in our DNS was chosen to resolve the smallest turbulent scales (Kolmogorov length) relevant to droplet interactions, consistent with previous validated studies. However, explicit convergence tests were not included. In the revised manuscript, we will add convergence tests by comparing results at the current resolution with a higher-resolution run for a subset of the simulation period, focusing on collision rates and droplet size distributions. We will also include a sensitivity analysis to the Lagrangian particle interpolation scheme and collision kernel parameters to demonstrate that the differences between the turbulent and non-turbulent cases are robust. revision: yes

  3. Referee: [Results] Results, early and late stages: Collision-rate differences are reported without error bars, ensemble variability, or statistical significance tests. This undermines the quantitative statements on promotion of autoconversion versus accretion and the timing of surface precipitation.

    Authors: We appreciate the referee's emphasis on statistical presentation. Given the substantial computational expense associated with DNS in a vertically elongated domain, performing multiple ensemble realizations was not feasible within the scope of this study. The reported results are from single, well-resolved deterministic simulations. In the revision, we will add a discussion in the Results section addressing potential sources of variability, such as by examining statistics over different horizontal subdomains or time windows to provide informal estimates of uncertainty. We will also moderate the language around quantitative claims to focus on the qualitative and mechanistic differences observed, while noting the limitations regarding formal error bars and significance testing. This approach maintains the integrity of the findings without overclaiming statistical precision. revision: partial

standing simulated objections not resolved
  • Conducting full ensemble simulations to provide error bars and statistical significance tests, due to the prohibitive computational cost of the DNS setup.

Circularity Check

0 steps flagged

No significant circularity; results from direct comparative simulations

full rationale

The paper reports results from explicit DNS Lagrangian simulations of warm-cloud microphysics in a vertically-elongated domain, with one run including an added homogeneous isotropic turbulence field and the other without. All reported effects (enhanced autoconversion early, accretion later, earlier surface precipitation) are direct numerical outputs of the differential comparison between these two explicitly constructed cases. No fitted parameters are renamed as predictions, no equations reduce to self-definition, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The derivation chain is therefore self-contained and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The simulation rests on standard numerical methods for DNS and Lagrangian tracking plus the modeling choice of homogeneous isotropic turbulence superimposed on a developing cloud.

free parameters (2)
  • turbulence intensity
    Amplitude of the added homogeneous isotropic turbulence field; value not stated in abstract but controls the reported collision enhancement.
  • domain aspect ratio and grid spacing
    Vertical elongation and resolution chosen to reach cloud top while resolving droplets; these choices affect sedimentation and collision statistics.
axioms (2)
  • domain assumption Lagrangian particle tracking accurately captures droplet motion and collisions under the resolved flow field
    Invoked when the model is said to explicitly resolve activation, condensation, collision-coalescence and sedimentation.
  • domain assumption Homogeneous isotropic turbulence can be superimposed without altering the mean updraft or cloud development
    Used to create the with-turbulence versus without-turbulence comparison.

pith-pipeline@v0.9.0 · 5539 in / 1472 out tokens · 79619 ms · 2026-05-10T15:46:20.147952+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Ababaei, A., Michel, A., and Rosa, B.: Collision statistics of droplets in turbulence considering lubrication interactions, mobility of interfaces and non-continuum molecular effects, Flow, Turbulence and Combustion, 112, 33–59, https://doi.org/https://doi.org/10.1007/s10494-023- 00450-1,

    work page doi:10.1007/s10494-023-

  2. [2]

    W., and Wang, L

    Ayala, O., Grabowski, W. W., and Wang, L. P.: A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles, J. Comput. Phys., 225, 51–73, https://doi.org/https://doi.org/10.1016/j.jcp.2006.11.016,

    work page doi:10.1016/j.jcp.2006.11.016 2006

  3. [4]

    K., and Bartello, P.: Turbulence effects of collision efficiency and broadening of droplet size distribution in cumulus clouds, J

    Chen, S., Yau, M. K., and Bartello, P.: Turbulence effects of collision efficiency and broadening of droplet size distribution in cumulus clouds, J. Atmos. Sci., 75, 203–217, https://doi.org/https://doi.org/10.1175/JAS-D-17-0123.1,

    work page doi:10.1175/jas-d-17-0123.1

  4. [5]

    N., Vaillancourt, P

    Franklin, C. N., Vaillancourt, P. A., Yau, M. K., and Bartello, P.: Collision rates of cloud droplets in turbulent flow, J. Atmos. Sci., 62, 2451–2466, https://doi.org/https://doi.org/10.1175/JAS3493.1,

    work page doi:10.1175/jas3493.1

  5. [6]

    Grabowski, W. W. and Wang, L. P.: Growth of cloud droplets in a turbulent environment, Annu. Rev. Fluid Mech., 45, 293–324, https://doi.org/https://doi.org/10.1146/annurev-fluid-011212-140750,

    work page doi:10.1146/annurev-fluid-011212-140750

  6. [7]

    W., Morrison, H., Shima, S., Abade, G

    Grabowski, W. W., Morrison, H., Shima, S., Abade, G. C., Dziekan, P., and Pawlowska, H.: Modeling of cloud microphysics: Can we do better?, Bulletin of the American Meteorological Society, 100, 655–672, https://doi.org/https://doi.org/10.1175/BAMS-D-18-0005.1,

    work page doi:10.1175/bams-d-18-0005.1

  7. [8]

    A., Lebo, Z

    Hill, A. A., Lebo, Z. J., Andrejczuk, M., Arabas, S., Dziekan, P., Field, P., Gettelman, A., Hoffmann, F., Pawlowska, H., Onishi, R., and Vie, B.: Toward a numerical benchmark for warm rain processes, J. Atmos. Sci., 80, 1329–1359, https://doi.org/https://doi.org/10.1175/JAS- D-21-0275.1,

    work page doi:10.1175/jas-

  8. [9]

    and Onishi, R.: Direct Lagrangian tracking simulation of droplet growth in vertically developing cloud, Atmos

    Kunishima, Y . and Onishi, R.: Direct Lagrangian tracking simulation of droplet growth in vertically developing cloud, Atmos. Chem. Phys., 18, 16 619–16 630, https://doi.org/https://doi.org/10.5194/acp-18-16619-2018,

    work page doi:10.5194/acp-18-16619-2018 2018

  9. [10]

    K., Khain, A

    Li, X., Tao, W. K., Khain, A. P., Simpson, J., and Johnson, D. E.: Sensitivity of a cloud-resolving model to bulk and explicit bin microphysical schemes. Part I: Comparisons, J. Atmos. Sci., 66, 3–21, https://doi.org/https://doi.org/10.1175/2008JAS2646.1,

    work page doi:10.1175/2008jas2646.1

  10. [11]

    L., Farley, R

    Lin, Y . L., Farley, R. D., and Orville, H. D.: Bulk parameterization of the snow field in a cloud model, J. Climate Appl. Meteorol., 22, 1065–1092, https://doi.org/https://doi.org/10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2,

    work page doi:10.1175/1520-0450(1983)022 1983

  11. [12]

    K., Shima, S., Lu, C., and Chen, S.: Parameterization and explicit modeling of cloud microphysics: approaches, challenges, and future directions, Adv

    Liu, Y ., Yau, M. K., Shima, S., Lu, C., and Chen, S.: Parameterization and explicit modeling of cloud microphysics: approaches, challenges, and future directions, Adv. Atmos. Sci., 40, 747–790, https://doi.org/https://doi.org/10.1007/s00376-022-2077-3,

    work page doi:10.1007/s00376-022-2077-3 2077

  12. [13]

    Michel, A., Ababaei, A., and Rosa, B.: The influence of gravity and mass loading on the coalescence of aerodynamically interacting droplets in homogeneous isotropic turbulence, J. Atmos. Sci., 80, 2207–2220, https://doi.org/https://doi.org/10.1175/JAS-D-22-0267.1,

    work page doi:10.1175/jas-d-22-0267.1

  13. [14]

    and Takahashi, K.: A warm-bin-cold-bulk hybrid cloud microphysical model, J

    Onishi, R. and Takahashi, K.: A warm-bin-cold-bulk hybrid cloud microphysical model, J. Atmos. Sci., 69, 1474–1497, https://doi.org/https://doi.org/10.1175/JAS-D-11-0166.1,

    work page doi:10.1175/jas-d-11-0166.1

  14. [15]

    Onishi, R., Baba, Y ., and Takahashi, K.: Large-scale forcing with less communication in finite-difference simulations of stationary isotropic turbulence, Journal of Computational Physics, 230, 4088–4099, https://doi.org/https://doi.org/10.1016/j.jcp.2011.02.034,

    work page doi:10.1016/j.jcp.2011.02.034 2011

  15. [16]

    C.: An efficient parallel simulation of interacting inertial particles in homogeneous isotropic turbulence, J

    Onishi, R., Takahashi, K., and Vassilicos, J. C.: An efficient parallel simulation of interacting inertial particles in homogeneous isotropic turbulence, J. Comput. Phys., 242, 809–827, https://doi.org/https://doi.org/10.1016/j.jcp.2013.02.027,

    work page doi:10.1016/j.jcp.2013.02.027 2013

  16. [17]

    15 Onishi, R., Matsuda, K., and Takahashi, K.: Lagrangian tracking simulation of droplet growth in turbulence-turbulence enhancement of autoconversion rate, J. Atmos. Sci., 72, 2591–2607, https://doi.org/https://doi.org/10.1175/JAS-D-14-0292.1,

    work page doi:10.1175/jas-d-14-0292.1

  17. [18]

    Reade, W. C. and Collins, L. R.: A numerical study of the particle size distribution of an aerosol undergoing turbulent coagulation, J. Fluid Mech., 415, 45–64, https://doi.org/https://doi.org/10.1017/S0022112000008521,

    work page doi:10.1017/s0022112000008521

  18. [19]

    Saffman, P. G. and Turner, J. S.: On the collision of drops in turbulent clouds, J. Fluid Mech., 1, 16–30, https://doi.org/https://doi.org/10.1017/S0022112056000020,

    work page doi:10.1017/s0022112056000020

  19. [20]

    and Gotoh, T.: Turbulence and cloud droplets in cumulus clouds, New J

    Saito, I. and Gotoh, T.: Turbulence and cloud droplets in cumulus clouds, New J. Phys., 20, 023 001, https://doi.org/https://doi.org/10.1088/1367-2630/aaa229,

    work page doi:10.1088/1367-2630/aaa229

  20. [21]

    A.: Particle-turbulence interactions in atmospheric clouds, Annu

    Shaw, R. A.: Particle-turbulence interactions in atmospheric clouds, Annu. Rev. Fluid Mech., 35, 183–227, https://doi.org/https://doi.org/10.1146/annurev.fluid.35.101101.161125,

    work page doi:10.1146/annurev.fluid.35.101101.161125

  21. [22]

    Shima, S., Kusano, K., Kawano, A., Sugiyama, T., and Kawahara, S.: The super-droplet method for the numerical simulation of clouds and precipitation: a particle-based and probabilistic microphysics model coupled with a non-hydrostatic model, Q. J. Roy. Meteorol. Soc., 135, 1307–1320, https://doi.org/https://doi.org/10.1002/qj.441,

    work page doi:10.1002/qj.441

  22. [23]

    Model Dev., 13, 4107–4157, https://doi.org/https://doi.org/10.5194/gmd-13-4107-2020,

    Shima, S., Sato, Y ., Hashimoto, A., and Hisumi, R.: Predicting the morphology of ice particles in deep convection using the super- droplet method: development and evaluation of SCALE-SDM 0.2.5-2.2.0, -2.2.1, and -2.2.2, Geosci. Model Dev., 13, 4107–4157, https://doi.org/https://doi.org/10.5194/gmd-13-4107-2020,

    work page doi:10.5194/gmd-13-4107-2020 2020

  23. [24]

    Shipway, B. J. and Hill, A. A.: Diagnosis of systematic differences between multiple parametrizations of warm rain microphysics using a kinematic framework, Q. J. Roy. Meteorol. Soc., 138, 2196–2211, https://doi.org/https://doi.org/10.1002/qj.1913,

    work page doi:10.1002/qj.1913 1913

  24. [25]

    Squires, K. D. and Eaton, J. K.: Particle response and turbulence modification in isotropic turbulence, Phys. Fluids, 2, 1191–1203, https://doi.org/https://doi.org/10.1063/1.857620,

    work page doi:10.1063/1.857620

  25. [26]

    Fluid Mech., 335, 75–109, https://doi.org/https://doi.org/10.1017/S0022112096004454,

    Direct numerical simulations, J. Fluid Mech., 335, 75–109, https://doi.org/https://doi.org/10.1017/S0022112096004454,

    work page doi:10.1017/s0022112096004454

  26. [27]

    Twomey, S.: The nuclei of natural cloud formation part II: The supersaturation in natural clouds and the variation of cloud droplet concen- tration, Geofisica Pura e Applicata, 43, 243–249, https://doi.org/https://doi.org/10.1007/BF01993560,

    work page doi:10.1007/bf01993560

  27. [28]

    Vaillancourt, P. A. and Yau, M. K.: Review of particle-turbulence interactions and consequences for cloud physics, Bull. Am. Meteorol. Soc., 81, 285–298, https://doi.org/https://doi.org/10.1175/1520-0477(2000)081<0285:ROPIAC>2.3.CO;2,

    work page doi:10.1175/1520-0477(2000)081 2000

  28. [29]

    L., Boyle, J

    Vaillancourt, P. A., Yau, M. K., and Grabowski, W. W.: Microscopic approach to cloud droplet growth by condensation. Part I: Model description and results without turbulence, J. Atmos. Sci., 58, 1945–1964, https://doi.org/https://doi.org/10.1175/1520- 0469(2001)058<1945:MATCDG>2.0.CO;2,

    work page doi:10.1175/1520- 1945

  29. [30]

    Wang, L. P. and Maxey, M. R.: Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence, J. Fluid Mech., 256, 27–68, https://doi.org/https://doi.org/10.1017/S0022112093002708,

    work page doi:10.1017/s0022112093002708

  30. [31]

    P., Wexler, A

    Wang, L. P., Wexler, A. S., and Zhou, Y .: Statistical mechanical descriptions of turbulent coagulation, Phys. Fluids, 10, 2647–2651, https://doi.org/https://doi.org/10.1063/1.869777,

    work page doi:10.1063/1.869777

  31. [32]

    D., and Myers, C.: A Numerical Model of a Hail-Bearing Cloud, J

    Wisner, C., Orville, H. D., and Myers, C.: A Numerical Model of a Hail-Bearing Cloud, J. Atmos. Sci., 29, 1160–1181, https://doi.org/https://doi.org/10.1175/1520-0469(1972)029<1160:ANMOAH>2.0.CO;2,

    work page doi:10.1175/1520-0469(1972)029 1972

  32. [33]

    Woittiez, E. J. P., Jonker, H. J. J., and Portela, L. M.: On the combined effects of turbulence and gravity on droplet collisions in clouds: a numerical study, J. Atmos. Sci., 66, 1926–1943, https://doi.org/https://doi.org/10.1175/2005JAS2669.1,

    work page doi:10.1175/2005jas2669.1 1926