In warm clouds, rain formation begins when micron-sized droplets grow by collision–coalescence, and small-scale turbulence can strongly accelerate this growth. Quantitative prediction requires both a faithful geometric collision kernel and accurate collision efficiencies — the fraction of geometric encounters in which two droplets actually touch, set by their near-field hydrodynamic interaction. Krueger and Kerstein (2018) developed ClusColl, an economical triplet-map Lagrangian simulation method whose turbulence enhancement of coalescence growth matches DNS for dissipation rates up to about 100 cm² s⁻³, but whose evaluation at higher intensities is blocked: the available efficiency inputs — Hall (1980) gravitational tables multiplied by the Wang and Grabowski (2009) turbulent enhancement factor — disagree with the DNS-derived efficiencies of Chen, Yau and Bartello (2018) by factors of up to 4–9 for collector radii of 20–25 μm at collected-to-collector radius ratios near unity. The open problem is to obtain internally consistent, validated collision efficiencies covering the parameter space needed for coalescence-growth simulations and thereby settle how strongly turbulence enhances coalescence growth and rain initiation at dissipation rates of 200 cm² s⁻³ and above, together with the associated questions raised by the source paper of representing Reynolds-number/intermittency effects and large-Stokes-number inertial response within economical simulation frameworks and embedding them in cloud-resolving models.

Background

In warm (ice-free) clouds, precipitation begins when cloud droplets with radii of roughly 5–20 μm collide and coalesce. Because the droplet size distribution must broaden rapidly for rain to initiate within observed cloud lifetimes, the rate of this collision–coalescence growth is a central quantity in cloud microphysics. The collision rate of a droplet pair is conventionally factorized into a geometric collision kernel — the rate at which turbulent motions and differential sedimentation bring two droplets into contact, ignoring their mutual hydrodynamic interaction — and a collision efficiency EE, the fraction of geometric encounters in which the droplets actually touch. The efficiency is below unity because each droplet induces a disturbance flow in the surrounding air that deflects the other droplet around it, with viscous lubrication effects in the thin air gap between the droplet surfaces becoming important just before contact.

Small-scale turbulence enhances the geometric kernel through mechanisms quantified since the classical Saffman–Turner shear result: strain and shear at scales down to the Kolmogorov scale (the smallest turbulent scales, with length η\eta and time τK\tau_K) increase relative droplet velocities, and finite-inertia droplets cluster preferentially in high-strain regions, an effect measured by the radial distribution function (RDF), the local enhancement of the pair number density at a given separation. Droplet inertia is measured by the Stokes number St=τp/τK\mathrm{St}=\tau_p/\tau_K, the ratio of the droplet viscous response time τp\tau_p to the Kolmogorov time; clustering is strongest for St\mathrm{St} of order unity. Turbulence also modifies the collision efficiency itself. The turbulent kinetic energy dissipation rate ϵ\epsilon ranges from a few cm2 s−3\mathrm{cm^2\,s^{-3}} in stratiform clouds to several hundred or more in vigorous cumulus.

Direct numerical simulation (DNS), which resolves all turbulent scales while tracking individual droplets, is the reference method, but its computational cost prohibits the large ensembles and long simulated times needed for growth studies. Krueger and Kerstein (2018) therefore built ClusColl, an economical Lagrangian method in the linear-eddy tradition: turbulent advection is represented by random triplet maps — rearrangements of a line segment that compress the middle third and stretch the flanking thirds, mimicking a turbulent eddy — applied one coordinate at a time so that droplet positions evolve in three dimensions. A droplet slip factor SS introduces inertial response, related empirically to the Stokes number by S=c St/(1+a St2)S=c\,\mathrm{St}/(1+a\,\mathrm{St}^2) with coefficients calibrated against DNS clustering exponents and collision kernels; the map size l=20ηl=20\eta and map frequency λp=0.464 (ϵ/ν)1/2\lambda_p=0.464\,(\epsilon/\nu)^{1/2} tie the model to physical scales. With collision detection and sedimentation added, ClusColl reproduces published DNS turbulent geometric kernels for droplet radii of 10–60 μm and ϵ\epsilon from 50 to 1535 cm2 s−3\mathrm{cm^2\,s^{-3}} reasonably well, and its turbulence-induced enhancement of coalescence growth matches hybrid DNS (a method combining a DNS turbulent flow field with an analytical superposition of the droplets' disturbance flows) for ϵ≤100 cm2 s−3\epsilon\le 100\ \mathrm{cm^2\,s^{-3}}.

The bottleneck is the collision-efficiency input. ClusColl does not resolve droplet disturbance flows and must be supplied with efficiencies, and the available sources disagree: the classical gravitational efficiencies tabulated by Hall (1980), multiplied by the turbulent enhancement factor of Wang and Grabowski (2009) obtained from hybrid DNS, overestimate the DNS-derived efficiencies of Chen, Yau and Bartello (2018) by factors of up to 4–9 for collector radii R=20R=20–25 μm at collected-to-collector radius ratios r/Rr/R near 0.95 at high ϵ\epsilon, and also overestimate gravitational efficiencies at large sizes. Consequently, the source paper states that for ϵ≥200 cm2 s−3\epsilon\ge 200\ \mathrm{cm^2\,s^{-3}} the lack of accurate collision efficiencies precludes definitive quantitative evaluation of ClusColl's coalescence growth. The same paper leaves open the related questions: how to obtain efficiency data consistently covering the full range of collector radii and dissipation rates encountered in growth simulations, or how to include hydrodynamic disturbance flows in the map model; how to represent Reynolds-number and intermittency effects that the present triplet-map formulation (whose parameters depend only on ϵ\epsilon and the air viscosity ν\nu) cannot capture; how to extend the empirical slip–Stokes relation beyond its small-St\mathrm{St} calibration regime; and how to embed the method in three-dimensional large-eddy simulations (which resolve only the large turbulent eddies) or cloud-resolving models at acceptable computational cost.

Problem Statement

Make the quantitative evaluation of turbulence-enhanced droplet collision–coalescence growth at high dissipation rates (ϵ≥200 cm2 s−3\epsilon\ge 200\ \mathrm{cm^2\,s^{-3}}) well-posed by removing the collision-efficiency input uncertainty that currently dominates it. Concretely: (1) obtain collision efficiencies for droplet pairs — gravitational and turbulence-modified, accounting for near-field hydrodynamic interactions — that are internally consistent across sources and validated over the parameter space needed for coalescence-growth simulations (droplet radii spanning roughly 10–60 μm, collected-to-collector radius ratios r/Rr/R approaching unity, and dissipation rates up to the vigorous-cloud values of order 10210^2–103 cm2 s−310^3\ \mathrm{cm^2\,s^{-3}}); and (2) determine definitively how strongly turbulence enhances coalescence growth and rain initiation at ϵ≥200 cm2 s−3\epsilon\ge 200\ \mathrm{cm^2\,s^{-3}}, closing the outcome spread that the current factor-4–9 efficiency discrepancy produces. The source paper also raises associated model-level questions for economical frameworks such as ClusColl — representing any residual Reynolds-number and intermittency dependence of collision statistics absent from the present triplet-map formulation, extending the empirical slip–Stokes relation S=c St/(1+a St2)S=c\,\mathrm{St}/(1+a\,\mathrm{St}^2) beyond its small-St\mathrm{St} calibration regime, and achieving computationally feasible embedding in three-dimensional large-eddy or cloud-resolving simulations — as part of the same broader agenda, rather than as additional parts of (1) and (2).

References

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