Warm rain in shallow cumulus clouds forms when cloud droplets roughly 10 µm in radius grow by binary collision–coalescence into raindrops. Small-scale turbulence can accelerate this growth by enhancing droplet collisions through three factors: the radial relative velocity at contact wrw_r, the radial distribution function g12g_{12} (preferential concentration), and a turbulence-induced modification of the collision efficiency ηE\eta_E. Seifert, Nuijens and Stevens (2010) embedded these factors — taken from direct numerical simulations (DNS) at Taylor-microscale Reynolds number Reλ≈72Re_\lambda\approx72 with a linear extrapolation in the dissipation rate ϵ\epsilon — into a bulk autoconversion parametrization and showed, in large-eddy simulations (LES) of precipitating shallow convection, significantly enhanced rain formation. They explicitly flagged the central uncertainty: the kernel's turbulence modifications rest on DNS far below the Reλ∼103Re_\lambda\sim10^3–10410^4 and high ϵ\epsilon of real cloud turbulence. The problem is to determine the joint dependence of wrw_r, g12g_{12}, and ηE\eta_E on ReλRe_\lambda and ϵ\epsilon over cloud-relevant ranges, validate it against independent data, and use it to re-quantify the turbulence enhancement of autoconversion in shallow convection. Later work extended DNS to Reλ≈1140Re_\lambda\approx1140 (monodisperse statistics) and Reλ≈418Re_\lambda\approx418 (polydisperse kernels), produced two kernel families whose autoconversion-enhancement predictions diverge by up to a factor of about 3 at cloud ReλRe_\lambda, and began in-situ holographic clustering measurements — but cloud-like kernel data and direct pair-statistics validation are still missing, so the core remains open.

Background

In warm (liquid) clouds, droplets nucleate on aerosol particles and grow by vapor condensation to radii of about 10–20 µm, after which further growth to raindrop sizes proceeds by binary collisions and merging (coalescence). In cloud models this collisional stage is described by the stochastic collection equation (SCE), an integro-differential equation for the droplet size distribution whose key input is the collection kernel K(r1,r2)K(r_1,r_2): the rate at which droplets of radii r1r_1 and r2r_2 collide and coalesce. For droplets settling in still air, KK is the gravitational kernel Kgrav=π(r1+r2)2∣v(r1)−v(r2)∣EcollK_{\mathrm{grav}}=\pi(r_1+r_2)^2|v(r_1)-v(r_2)|E_{\mathrm{coll}}, where v(r)v(r) is the terminal fall speed and EcollE_{\mathrm{coll}} the collision efficiency (the fraction of geometrically swept collisions that actually occur, reduced by hydrodynamic deflection of the droplets around each other). Gravitational collection is inefficient for droplets smaller than about 20 µm, leaving a historical 'size gap' between condensational growth and efficient coalescence — one motivation for studying turbulent collision enhancement.

Small-scale air turbulence modifies the kernel in three distinct ways, conventionally written as K=2π(r1+r2)2 wr g12 Ecoll ηEK=2\pi(r_1+r_2)^2\,w_r\,g_{12}\,E_{\mathrm{coll}}\,\eta_E. First, turbulent velocity fluctuations increase the radial relative velocity at contact, wrw_r (the Saffman–Turner effect). Second, inertial droplets preferentially concentrate in regions of the small-scale flow, raising the pair number density at contact; this is quantified by the radial distribution function g12g_{12}. Third, turbulence perturbs the hydrodynamic interaction between approaching droplets, modifying the collision efficiency by a factor ηE\eta_E. The system is characterized by the Stokes number St=τp/τkSt=\tau_p/\tau_k (droplet relaxation time over Kolmogorov time; clustering peaks near St≈1St\approx1, i.e. droplets of roughly 20–100 µm), the turbulent energy dissipation rate ϵ\epsilon (in clouds typically 1010–10310^3 cm² s⁻³, locally higher near vigorous cloud tops), and the Taylor-microscale Reynolds number Reλ=urmsλT/νRe_\lambda=u_{\mathrm{rms}}\lambda_T/\nu, which measures the scale separation of the turbulence and reaches 10310^3–10410^4 in cumulus clouds — far beyond what direct numerical simulation (DNS, which resolves all turbulent scales down to the Kolmogorov scale) can currently reach (roughly 10210^2–10310^3).

Wang, Ayala and co-workers derived from DNS the parametrizations of wrw_r and g12g_{12} by Ayala et al. (2008) and the collision-efficiency enhancement ηE\eta_E by Wang et al. (2008), the latter at Reλ=72.4Re_\lambda=72.4 with values interpolated linearly in ϵ\epsilon. Seifert, Nuijens and Stevens (2010) fitted the resulting 'Ayala–Wang' kernel into the kernel parameter kcck_{\mathrm{cc}} of a two-moment bulk autoconversion scheme (a compact parametrization used in weather and climate models, predicting the conversion rate of cloud water to rainwater from prognosed liquid water content and droplet number), calibrating the fit against several thousand SCE bin-model simulations over ϵ∈[50,400]\epsilon\in[50,400] cm² s⁻³ at the fixed kernel-data Reynolds number Reλ=72.4Re_\lambda=72.4, and implemented it in the UCLA large-eddy simulation (LES, a cloud-resolving technique in which the largest turbulent eddies are resolved and subgrid scales are parametrized) of precipitating shallow cumulus convection. The turbulence-enhanced coalescence produced markedly more rain. The authors themselves stated the key limitations: the kernel modifications derive from DNS at low ReλRe_\lambda and limited ϵ\epsilon, so their extrapolation to cloud-like ReλRe_\lambda and to the high local dissipation rates near vigorous cloud tops is unvalidated; the LES at O(100 m)O(100\ \mathrm{m}) grid spacing with a subgrid dissipation estimate cannot resolve the fine-scale cloud microstructure where collisions and dissipation interact; and the bulk fit assumes linearity in ϵ\epsilon. A competing 'Onishi' kernel family was derived by Onishi, Matsuda and Takahashi (2015) from Lagrangian-tracking simulations of droplet growth in turbulence and re-derived by Onishi and Seifert (2016) from new direct numerical simulations extending to Reλ≈1140Re_\lambda\approx1140; the two families agree near Reλ≈66Re_\lambda\approx66 but their stochastic-collection-equation predictions diverge by up to a factor of about 3 at cloud-typical ReλRe_\lambda (Onishi and Seifert 2016), the Onishi family predicting weaker autoconversion but stronger accretion enhancement. Reviews by Devenish et al. (2012) and Grabowski and Wang (2013) confirmed the Reynolds-number dependence of the turbulent kernel as a central open problem of warm-rain physics.

Problem Statement

Determine the joint dependence of the turbulent collision–coalescence kernel on the Taylor-microscale Reynolds number ReλRe_\lambda and the turbulent energy dissipation rate ϵ\epsilon over cloud-relevant ranges, and thereby establish a validated quantification of the turbulence enhancement of warm-rain autoconversion in precipitating shallow convection. Concretely, the problem — as posed by Seifert, Nuijens and Stevens (2010) and sharpened by the follow-up literature — comprises jointly serving components: (1) derive, from DNS or very-high-resolution LES (or laboratory measurement) extended to ReλRe_\lambda well beyond the O(102)O(10^2) of the original kernel data toward the cloud-like 10310^3–10410^4, and to the high local ϵ\epsilon found near vigorous cloud tops (including possible nonlinearity in ϵ\epsilon beyond the fitted ranges), robust parametrizations of the kernel enhancement factors — the radial relative velocity wrw_r, the radial distribution function g12g_{12}, and the collision-efficiency modification ηE\eta_E — with quantified uncertainty over the stated cloud-relevant (Reλ,ϵ)(Re_\lambda,\epsilon) ranges, adjudicating between the qualitatively conflicting ReλRe_\lambda trends of the Ayala–Wang and Onishi kernel families, whose autoconversion-enhancement predictions differ by up to a factor of about 3 at cloud-typical ReλRe_\lambda; (2) validate the turbulent-kernel predictions against the best available independent in-situ data — in particular the shallow-cumulus holographic droplet-clustering measurements of Thiede et al. (2026) and the turbulent wind-tunnel droplet size distributions of Frey et al. (2026) — and propagate the residual uncertainty this validation leaves on the kernel enhancement factors over the stated cloud-relevant (Reλ,ϵ)(Re_\lambda,\epsilon) ranges; (3) repeat the bin-to-bulk calibration of the autoconversion (and accretion) closure on the best kernel data available, quantifying its sensitivity to the Ayala–Wang versus Onishi choice; and (4) establish the resolution convergence of LES responses, including the collocation of high-ϵ\epsilon regions with droplet-growth hotspots, to determine whether the simulated turbulence enhancements persist as finer-scale cloud structure is resolved, and to quantify the residual subgrid collision–dissipation limitation and its uncertainty.

References

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