EGUsphere

Copernicus Publications

Göttingen, Germany

10.5194/egusphere-2026-2627

A unified closed-form collision kernel for warm-cloud coalescence in the gravity–turbulence coupled regime and the origin of the rain-initiation bottleneck

Chishtie

Farrukh A.

https://orcid.org/0000-0002-6392-6084

¹ ²

Peaceful Society, Science and Innovation Foundation, Vancouver, Canada

Department of Occupational Science and Occupational Therapy, University of British Columbia, Vancouver, Canada

01 07 2026

2026 1 29

2026

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Warm rain forms faster in cloud observations than in standard microphysics models, with the discrepancy concentrated in the 15 to 40 µm droplet-radius size gap where neither condensational growth nor gravitational differential settling is efficient. We present a closed-form unified collision kernel for warm-cloud coales-cence that interpolates continuously between the laminar and turbulent asymptotic regimes through a single dimensionless parameter, the turbulence-to-gravity ratio ξ = C₀ε1/3s1/3/Ak, where Ak = KSt (R2k − R2k−1) is the Stokes differential settling speed and s is the mean inter-drop separation. The kernel takes the factored form K (Rk, Rk−1; ε) = π (Rk + Rk−1)² |∆vtᵉᶠᶠ | Eᵉᶠᶠ , Eᵉᶠᶠ = Elam (Rk, Rk−1) · ηE, with an effective approach velocity |∆vtᵉᶠᶠ| = Ak (1 + ξ ) and a turbulent efficiency enhancement ηE (Stk, ξ ), both derived in closed form from the hierarchical fractional N-body dynamics of Chishtie (2026). Three established results are recovered by construction rather than by fitting: the laminar limit ε → 0 reproduces the Hall (1980) kernel with Pinsky et al. (2001) collision efficiency to within a ±10 % band, the high-ε limit reproduces the classical Saffman–Turner (1956) ε1/3 turbulent kinetic-theory scaling, and the monotonic ordering ηE,2 > ηE,3 > ηE,4 with drop size matches the qualitative signature reported by Wang et al. (2005) and Chen et al. (2018) from DNS. Supporting the kernel construction, we also derive an effective Riesz fractional parameter αkᵉᶠᶠ = ξ / [3(1 + ξ )] + 2 − 2 / (Nk + 1) that interpolates analytically between the gravity-only and turbulence-only limits, together with closed-form parametric trajectories sk (θ) = s₀ sinn θ with n = 2 / (pkᵉᶠᶠ + 1) validated against RK45 integration to relative errors of 10−13. The kernel is directly implementable in bin microphysics and large-eddy simulation super-droplet schemes and carries no free parameters.