How turbulence accelerates the collision–coalescence growth of micron-sized cloud droplets into raindrops remains incompletely quantified. Direct numerical simulations with a superparticle method (Li et al. 2018) showed that turbulence controls the early (auto-conversion) growth of the droplet-size distribution, produces a power-law tail with slope near −3.7-3.7 between 10 and 40 micrometers, and yields drizzle-sized droplets in roughly 900 s - but only under a unit collision and coalescence efficiency, elevated number density, and Taylor-scale Reynolds numbers far below atmospheric values. The follow-up literature leaves the integrated problem open: condensation–collision coupling has been simulated (at unit efficiency), the superparticle algorithm's fidelity to Poisson collision-time fluctuations has been verified in the quiescent gravity-driven limit, and pairwise turbulent collision efficiencies with hydrodynamic interactions have been computed theoretically and coupled into droplet-population direct numerical simulation with gravity (Michel et al. 2023) - yet no single study carries turbulence-, size-, and settling-state-dependent collision efficiencies through the evolving size distribution across auto-conversion and gravity-dominated growth together with growth-time statistics, quantifies fluctuation-limited growth-time distributions in dilute turbulent clouds, extrapolates collision statistics to atmospheric Reynolds numbers, or explains the observed power-law slope, which 2024 airborne observations confirm deviates from existing theoretical kernels.

Background

In warm (ice-free) clouds, rain forms when micron-sized droplets grow first by vapor condensation and then by collision–coalescence: droplets that collide stick and merge into larger, faster-falling drops. The rate at which the droplet-size distribution broadens through this collisional growth determines how quickly rain begins, and a key open question in cloud physics is the role of air turbulence. The classical reference point is the Saffman–Turner result (Saffman and Turner 1956), which predicts that in turbulent flow the mean collision rate of small, inertialess droplets scales as the square root of the mean turbulent energy-dissipation rate εˉ\bar\varepsilon. Several concepts are needed to state the modern problem. The Kolmogorov scale is the smallest turbulent length scale, where viscous dissipation acts; resolving it is required for direct numerical simulation (DNS) of turbulence. The Taylor-scale Reynolds number Reλ\mathrm{Re}_\lambda measures the width of the turbulent inertial range; atmospheric values are far larger than what DNS can reach (Li et al. 2018 achieved at most Reλ≈158\mathrm{Re}_\lambda\approx 158). The Stokes number St\mathrm{St} is the ratio of a droplet's inertial response time to the smallest turbulent time scale; droplets with St≳1\mathrm{St}\gtrsim 1 detach from the flow, cluster, and collide more often. The collision efficiency is the fraction of geometrically encountered droplet pairs that actually collide once their hydrodynamic interaction (the disturbance flow each droplet induces in the other, plus the lubrication force resisting motion through the thin air film between nearly touching droplets) is accounted for; it is generally below unity. A superparticle (superdroplet) method is a Monte Carlo technique in which each computational particle represents many identical physical droplets, making it feasible to simulate collision–coalescence of the full droplet-size distribution inside a turbulent flow. Li et al. (2018) used such DNS with a superparticle approach and unit collision/coalescence efficiency. Without gravity they found the size distribution depends strongly on εˉ\bar\varepsilon, only weakly on Reλ\mathrm{Re}_\lambda, and exhibits a power law f(r)∝r−3.7f(r)\propto r^{-3.7} between about 10 and 40 micrometers - close to slopes seen for interstellar dust grains and particles in Saturn's rings, suggesting universality. With gravity included, turbulence dominates early growth between similarly sized droplets (auto-conversion, the earliest broadening phase) while gravity takes over above about 20 micrometers, giving a time scale of about 900 s to drizzle size, close to rapid warm-rain formation times. The authors themselves flagged what this leaves open: the unit-efficiency assumption overestimates collision rates; only mean growth was tracked, not fluctuations; the Reynolds numbers are unrepresentative; the power law lacks a theory; and condensation under turbulent supersaturation fluctuations was omitted. Subsequent work has resolved parts of this agenda - condensation–collision coupling was simulated by the same group (Li et al. 2020, still at unit efficiency); the superparticle algorithm's accuracy for Poisson collision-time fluctuations was verified in a quiescent gravity-settling 'lucky droplet' model (Li et al. 2022); and hydrodynamically interacting pairs in turbulence have been treated theoretically and in DNS (Dhanasekaran, Roy and Koch 2021; Michel, Ababaei and Rosa 2023; Patra, Koch and Roy 2026) - but the integrated, source-faithful question of how realistic turbulence-modulated collision efficiencies, stochastic fluctuations, and atmospheric Reynolds numbers jointly shape collisional growth up to rain initiation, and what sets the power-law tail, remains unanswered.

Problem Statement

Quantify, by simulation and/or theory, how air turbulence controls the collisional growth of cloud droplets from the auto-conversion stage to rain initiation once the simplifications of Li et al. (2018) are removed. Concretely: (1) incorporate a realistic, turbulence-modulated collision and coalescence efficiency that depends on droplet sizes and dynamical state (settling rate, local flow) via droplet–droplet aero-hydrodynamics (the coupled aerodynamic disturbance flows and lubrication forces between approaching droplets), and determine how it alters both early growth between similarly sized droplets and late gravity-dominated growth, including the previously predicted 900 s time scale to drizzle-sized droplets; (2) quantify how stochastic fluctuations of collision times in dilute suspensions - where collisions are rare, Poisson-like events - shape the distribution of growth times to rain initiation, beyond the mean behavior; (3) establish whether the weak dependence of the collision statistics on the Taylor-scale Reynolds number Reλ\mathrm{Re}_\lambda (which measures the width of the turbulent inertial range) persists at atmospheric Reynolds numbers, and how an extended inertial range modifies the collision statistics and the prefactor of the εˉ1/2\bar\varepsilon^{1/2} scaling, where εˉ\bar\varepsilon is the mean turbulent energy-dissipation rate; (4) explain theoretically the power-law slope (−3.7-3.7) of the turbulence-generated size distribution in the 10–40 micrometer range and test its universality across parameter ranges and physical systems; and (5) determine the combined condensational–collisional evolution under turbulent supersaturation fluctuations with realistic (non-unit) collision efficiencies. These components jointly serve one objective: a predictive, quantitatively validated account of turbulence-assisted warm-rain initiation. Each of the five components is an independently refutable sub-question and can be resolved on its own. Because (5) applies the realistic efficiency of (1) within the condensation-coupled setting, a result for (5) also addresses the coupled-efficiency part of (1) but not (1)'s isolated reassessment of auto-conversion and gravity-dominated growth.

References

  1. Li, Xiang-Yu; Brandenburg, Axel; Svensson, Gunilla; Haugen, Nils E. L.; Mehlig, Bernhard; Rogachevskii, Igor (2018). 'Effect of turbulence on collisional growth of cloud droplets', Journal of the Atmospheric Sciences 75, 3469–3487. DOI: 10.1175/JAS-D-18-0081.1 (arXiv:1711.10062). https://doi.org/10.1175/JAS-D-18-0081.1
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