Effects of correlated collisions and intermittency on the growth of lucky droplets

TL;DR

This work tackles the warm-cloud size-gap problem by studying how turbulence-driven collisions and intermittency accelerate the growth of lucky droplets that bridge the 15–50 $\mu\mathrm{m}$ gap before rain onset. It combines direct numerical simulations of droplet-laden turbulence with a non-Markovian stochastic framework to quantify memory effects from correlated collisions, and introduces an ensemble toy model that couples droplet growth to fluctuations of the volume-averaged dissipation rate, modeled as a lognormal process. The findings show that short-time collision correlations can modestly speed up early growth, but the dominant acceleration arises from intermittency in the local dissipation rate, which can dramatically shorten the time for lucky droplets to reach larger sizes and bridge the gap. This indicates that dissipation fluctuations, rather than constant collision rates, may be crucial to triggering rain within typical cloud timescales, highlighting the role of intermittency in cloud microphysics and its potential impact on precipitation forecasts.

Abstract

To trigger precipitation, water droplets in warm clouds need to attain a sufficient size. Theoretical estimates based on condensation and gravitational collisions alone fail to explain the observed timescales for the onset of precipitation for a range of droplet sizes. This suggests the involvement of collisional growth mediated by turbulence to resolve the so-called ``size-gap problem''. For the onset of rain, it is sufficient that statistical outliers, coined ``lucky droplets'', cross the size gap. In this study, we explore the influence of turbulence on droplet growth, focusing on correlated collisions and intermittency. Using direct numerical simulations of droplets in turbulent flow, we constrain a non-Markovian stochastic framework that allows us to assess memory effects on the droplet-size distribution arising from correlations between consecutive collisions. Using our framework, we find that correlated collisions accelerate the initial growth of lucky droplets but have sub-leading effect at later stages. Consequently, we neglect correlations from collisions and model an ensemble of cloud parcels representing fluctuations in the volume-averaged dissipation rate. Here, the distribution of droplet sizes in each parcel is described by a linear master equation with a time-dependent collision rate according to the volume-averaged dissipation rate. Our analyses of this toy model show that intermittency can significantly reduce the time required by lucky droplets to cross the size gap.

Figures (10)

• Figure 1: Intermittency in clouds causes strong fluctuations of the normalized dissipation rate $\overline{\varepsilon}_r$ averaged over a volume of extent $r$, here modeled as a lognormally distributed (see text for details). As a result, parameters that depend on the average dissipation rate, such as the Stokes number of droplets of a given size and the Froude number, vary locally across a cloud. We can mimic this in numerical simulations by varying these parameters, see insets, which show typical simulation snapshots for $\epsilon_r=\langle\epsilon\rangle$ (blue frame) and $\epsilon_r=9\langle\epsilon\rangle$ (green frame), where particles are in black, regions of strong vorticity in blue, and regions of strong strain in red.
• Figure 2: Collisions in turbulence are non-Markovian but can be approximated as a superposition of Poisson processes with different timescales. Data from DNS (see SI for parameters) with normalized volume-averaged dissipation rate $\bar{\epsilon}_r=1$ (blue) and $\bar{\epsilon}_r=9$ (green). (Top) Conditional collision rate for droplets of size $n$. The increase at short times implies temporal correlations and violates the memoryless assumption of Poisson processes with a constant rate. (Bottom) The survival probability can be well approximated by a sum of two exponential functions (gray dashed lines). Note that we nondimensionalized with the local Kolmogorov time $\tau_{K,r}$. On the left, this is identical to mean conditions $\tau_K$. On the right, the local and global Kolmogorov times are different and we show the additional axes nondimensionalized by $\tau_K$ in gray.
• Figure 3: (Top) Probability of finding droplets of size $n>2$: We compare our non-Markovian stochastic framework using the empirical fit to $S_n$ to turning off correlations by removing the fast-decreasing exponential and our DNS data of droplet growth. This shows that memory effects can further accelerate growth at short times. In the case of $\epsilon_r=9\langle\epsilon\rangle$, the time $t_{\text{lucky}}(n=2)$ where the one-in-a-million fastest droplets surpass a given size $n=2$ decreases by about 50%. (Bottom) $t_{\text{lucky}}$ for different dissipation rates and as a function of the droplet size: we compare our DNS data to our framework with and without memory effects. The relative effects of memory on $t_{\text{lucky}}$ decrease with the volume-averaged dissipation rate and with increasing droplet size.
• Figure 4: Time-dependent probability distribution of droplet sizes can be approximated by the toy model that integrates \ref{['eq:smol']} with an effective $\lambda_n(\bar{\epsilon}_r)$ parameterized by DNS (data points). (Top) Linear approximation of collision rate as a function of collisions, where faded data points were excluded from the fit due to insufficient statistics. The inset shows slope $a$ and offset $b$ of our linear approximation that are approximated by a linear function and power-law, respectively. (Bottom) Probability distribution at the end time of simulation to benchmark toy model with numerical simulations. The toy model can be easily evaluated for an ensemble drawn from $P(\bar{\epsilon}_r)$ (gray lines) to find that the ensemble mean (black line) has a higher probability for larger droplets.
• Figure 5: (Top) Two realizations of the volume-averaged dissipation rate drawn from the same distribution - the one with the lowest and highest dissipation rate among $10^5$ randomly generated realizations. The corresponding dissipation rate may - as one of the example realizations shows - experience significant spikes. (Bottom) The probability that $n>100$, i.e., a droplet had at least 100 collisions as a function of time within our toy model, assuming a more realistic number density of $200\,\text{cm}^{-3}$. The $10^5$ ensemble realizations are shown in gray, whereas we have the ensemble mean in black. Notice that large outliers (logarithmic $y$-scale) dominate the mean, e.g., the high-dissipation realization (red) towards the end.