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

Abstract. 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 = C₀ε^1/3s^1/3/A_k, where A_k = K_St (R²_k − R²_k−1) is the Stokes differential settling speed and s is the mean inter-drop separation. The kernel takes the factored form

K (R_k, R_k−1; ε) = π (R_k + R_k−1)² |∆v_tᵉᶠᶠ | Eᵉᶠᶠ , Eᵉᶠᶠ = E_lam (R_k, R_k−1) · η_E,

with an effective approach velocity |∆v_tᵉᶠᶠ| = A_k (1 + ξ ) and a turbulent efficiency enhancement η_E (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) ε^1/3 turbulent kinetic-theory scaling, and the monotonic ordering η_E,2 > η_E,3 > η_E,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 α_kᵉᶠᶠ = ξ / [3(1 + ξ )] + 2 − 2 / (N_k + 1) that interpolates analytically between the gravity-only and turbulence-only limits, together with closed-form parametric trajectories s_k (θ) = s₀ sinⁿ θ with n = 2 / (p_kᵉᶠᶠ + 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.