Are turbulence effects on droplet collision–coalescence a key to understanding observed rain formation in clouds?

Significance

The flow in most clouds is turbulent, and the effects of cloud–turbulence interactions are challenging to represent in atmospheric models. It has been hypothesized that impacts of turbulence on drop coalescence strongly influence rain formation. State-of-the-art numerical simulations with Lagrangian particle-based microphysics model and observations of drop size distributions in cumulus congestus clouds from the NASA CAMP²Ex campaign are combined to provide evidence of critical impacts of turbulence on rain formation. Turbulent coalescence substantially enhances rain initiation and must be included in the model to accurately simulate the tail of drop size distributions, especially near cloud base. Large aerosols “giant” cloud condensation nuclei (“giant CCN”) have little impact on rain formation in cumulus clouds when turbulence effects are considered.

Abstract

Rain formation is a critical factor governing the lifecycle and radiative forcing of clouds and therefore it is a key element of weather and climate. Cloud microphysics–turbulence interactions occur across a wide range of scales and are challenging to represent in atmospheric models with limited resolution. Based on past experiments and idealized numerical simulations, it has been postulated that cloud turbulence accelerates rain formation by enhancing drop collision–coalescence. We provide substantial evidence for significant impacts of turbulence on the evolution of cloud droplet size distributions and rain formation by comparing high-resolution observations of cumulus congestus clouds with state-of-the-art large-eddy simulations coupled with a Lagrangian particle-based microphysics scheme. Turbulent coalescence must be included in the model to accurately represent the observed drop size distributions, especially for drizzle drop sizes at lower heights in the cloud. Turbulence causes earlier rain formation and greater rain accumulation compared to simulations with gravitational coalescence only. The observed rain size distribution tail just above cloud base follows a power law scaling that deviates from theoretical scalings considering either a purely gravitation collision kernel or a turbulent kernel neglecting droplet inertial effects, providing additional evidence for turbulent coalescence in clouds. In contrast, large aerosols acting as cloud condensation nuclei (“giant CCN”) do not significantly impact rain formation owing to their long timescale to reach equilibrium wet size relative to the lifetime of rising cumulus thermals. Overall, turbulent drop coalescence exerts a dominant influence on rain initiation in warm cumulus clouds, with limited impacts of giant CCN.

Precipitation development is a critical process affecting cloud dynamics and lifetime, the hydrological cycle, atmospheric radiation, and aerosol properties. Cloud droplets grow through condensation of water vapor on the aerosol surface initially. Once they grow sufficiently large, they start colliding and coalescing to form rain drops under the influence of gravity. However, even after numerous studies using observations and process models, understanding of precipitation development in warm (liquid-only) clouds remains limited. In particular, how cloud droplets grow through the “rain formation bottleneck” between the condensation-dominated growth regime for small droplets (less than $\sim$25 $\mathrm{\mu }$m radius) and the gravitational collision–coalescence regime for larger drops is uncertain. The formation of embryo rain drops in the bottleneck size range depends strongly on the cloud droplet size distribution and its variability. Various factors that shape drop size distributions, like turbulent motion and aerosol properties, increase the complexity of the problem under different environmental conditions.

The flow in most clouds is turbulent, driving fluctuations down to $\mathcal{O}$ (1 mm) dissipation scales and impacting how droplets interact with one another and with their environment. Turbulent variability in water vapor and temperature fields impacts cloud droplet formation and drives variability in droplet size through condensation growth (1, 2). Indirectly, this variability in droplet size (and concentration) can potentially impact the collision–coalescence process (hereafter “coalescence”) and rain formation. Turbulent velocity fluctuations can also directly influence coalescence by 1) enhancing droplet spatial variability through preferential clustering owing to droplet inertia (characterized by radial distribution functions) (e.g., refs. 3–5); 2) driving changes in the relative velocity differences between droplet pairs (e.g., refs. 4 and 6–8); and 3) modifying the collision efficiency of droplet pairs by influencing the hydrodynamic field around droplets (e.g., refs. 9–14). These factors have been hypothesized to impact the rain formation process significantly. But do we observe such an impact of turbulence in atmospheric clouds? This study directly evaluates the hypothesized impacts of turbulence on rain formation using observations of cumulus congestus clouds during the cloud, aerosol and monsoon processes philippines experiment (CAMP²Ex, 15).

Understanding mechanisms of rain formation is important because weather and climate simulations are sensitive to how it is parameterized. For example, model climate sensitivity is sensitive to the representation of rain formation in cumulus clouds (e.g., ref. 16). Uncertainties in rain formation also reduce confidence in model estimates of aerosol impacts on clouds and their radiative forcing (e.g., refs. 17 and 18). Weather and climate models rely on microphysics parameterizations that are typically developed from detailed process models (e.g., large-eddy simulations with size-resolved cloud microphysical schemes). However, these detailed models cannot resolve microscale interactions of cloud droplets with their turbulent environment and instead use theoretical or empirically derived schemes based on laboratory experiments in nonturbulent environments (e.g., refs. 19 and 20). Commonly used droplet coalescence rate formulations (“collision kernels”) in benchmark atmospheric models ignore the effects of turbulence. Instead, they calculate the collision rate based only on the differential gravitational fall speed between drops of different sizes. However, as discussed above, laboratory experiments, theory, and direct-numerical simulations clearly point to the potential for significant impacts of turbulence on droplet dynamics and coalescence (see refs. 21–23 for detailed reviews of past literature).

A different mechanism—activation of large cloud condensation nuclei (“giant CCN” or GCCN)—has long been hypothesized to promote rain formation in clouds (e.g., refs. 24–27). GCCN can produce drops with an equilibrium size through condensation alone that is sufficient to initiate rapid collision coalescence growth in clouds. However, they require a long time (tens of minutes for large GCCN) to reach equilibrium drop size due to kinetic limitations on growth (28). In cumulus updrafts, with a lifetime of individual rising thermal circulations of 10 to 20 min (29), there may be insufficient time for these particles to reach large enough sizes to significantly impact drop coalescence (27). Nonetheless, it is important to consider GCCN when evaluating the impact of turbulence on droplet coalescence in cumulus clouds.

Using a state-of-the-art Lagrangian particle-based microphysics scheme (“superdroplet method,” SDM) (30), we performed high-resolution (50 m horizontal and vertical grid spacing) large-eddy simulations (LES) of a CAMP²Ex precipitating cumulus congestus case (Sep 7, 2019). Thermodynamic, wind, and aerosol observations during CAMP²Ex provide a well-constrained base state for these simulations. Fig. 1 demonstrates the realistic three-dimensional structure and cloud water mixing ratio along a vertical cross-section of the simulated cloud at time $=$ 7800 s. The cloud macroscale structure (e.g. cloud base and top heights) is similar to the observed congestus clouds, and the maximum vertical velocity of vertical cross-sections through the cloud center at different times is $\sim$9 to 12 m s⁻¹, similar to the observed maximum vertical velocity of 9.1 m s⁻¹.

Fig. 1.

(A) Isosurfaces of the simulated cloud water field (0.1 g kg⁻¹) from time $=$ 7,800 s and (B) corresponding x-z cross-section through the domain center showing cloud water mixing ratios from the CONTROL case.

Model results are compared to observed droplet size distributions (DSDs) obtained by combining different high-resolution cloud probes (31). We conducted multiple sensitivity simulations (listed in Table 1) to investigate the impacts of turbulence and GCCN on DSDs and precipitation formation. A “piggybacking” framework (32, 33), which allows us to isolate sensitivities to microphysical changes (here, GCCN and turbulent coalescence) from feedbacks between the microphysics and flow dynamics, is used for these simulations. In this framework, the simulations feature exactly the same flow evolution, and the simulated clouds have the same macroscale structure (see Material and Methods and SI Appendix for details).

Table 1.

List of simulation cases

Case	Turbulent collision kernel	Aerosols
1. CONTROL	No	Bimodal
2. TURBULENT	Yes	Bimodal
3. 10xGCCN	No	Bimodal with 10 times higher GCCN concentration
4. NO-GCCN	No	Unimodal
5. TURBULENT-NO-GCCN	Yes	Unimodal
6. TURBULENT-10xGCCN	Yes	Bimodal with 10 times higher GCCN concentration

Results

Drop Size Distribution Observations.

As a reference for this investigation, drops measured at different altitudes in a cumulus congestus cloud field (classified based on temperature: warmest near cloud base and coldest near the top) are used to create average DSDs (blue lines in Fig. 2).

Fig. 2.

Averaged drop size distributions at different altitudes (A: near cloud base to E: near cloud top) from observations in cumulus congestus clouds during CAMP²Ex (blue) and corresponding LES outputs from the CONTROL simulation at time $=$ 7,500 s using the Lagrangian microphysics scheme with gravitational drop coalescence only (black).

Averaging is over segments in cloudy updrafts with measured liquid water content $\ge$ 0.1 g m⁻³; similar averaging is applied to obtain the model-simulated DSDs. All of the DSDs were collected from nearly similar locations (see SI Appendix, Fig. S2 showing flight track and sampling locations). Note that averaging DSDs over congestus cloud updraft improves robustness but removes in-cloud variability. This might affect the DSD shape to some extent, especially at upper cloud levels, but is not expected to have a major impact on the DSD tail near the cloud base, which is a focus here. This is supported by SI Appendix, Fig. S3, which shows individual 1 Hz DSD samples (large-drop tail) used to create the average DSD near cloud base within the cumulus updraft core. All of the DSD samples have a similar shape with some variability in total number concentration. The observed average DSD near cloud base is unimodal, relatively narrow, and has an apparently power-law tail approaching drizzle drop sizes (with a maximum diameter of 205 $\mathrm{\mu }$m). This power law drizzle tail is a robust feature of the observed DSDs near cloud base, considering the close similarity of distribution shape among the individual DSD samples (SI Appendix, Fig. S3). The DSD near cloud base was unlikely to have been contaminated by large drops falling from above as evidenced by the lack of a secondary mode of larger drops. In this case, vertical shear of the mean horizontal wind likely pushed trajectories of most falling drops away from the cloud base updrafts, as demonstrated by our modeling results (SI Appendix, Figs. S16 and Fig. 2). Thus, the occurrence of drizzle-size drops in updrafts near cloud base indicates rapid drop growth through the bottleneck between the condensation and gravitational coalescence regimes. With an increase in altitude, the DSDs become broader toward both smaller (i.e., the mode radius becomes larger while small drops extend to a similar minimum size) and larger sizes. More rain drops with larger sizes (up to 3,000 $\mathrm{\mu }$m, i.e., 3 mm) appear at higher altitudes. At cloud temperatures $\le$ 14 °C, there is a second mode and distinct DSD shoulder consisting of drops with diameter $D>100$ $\mathrm{\mu }$m. The droplet number concentration is also significantly higher near cloud base than at higher altitudes (almost three times larger than near cloud top).

This DSD evolution suggests the growth of cloud droplets by collecting other droplets as they ascended in updrafts above cloud base, with some reaching rain drop sizes at higher altitudes. Some of the reduction in droplet number concentration was also from the expansion and dilution of the cloud volume through entrainment and mixing. DSD broadening toward smaller sizes with increasing height was likely from the activation of new cloud droplets on aerosol particles entrained into the cloud from the environment (34, 35). A decrease in drop size from evaporation owing to entrainment and mixing of cloudy and dry environmental air may have also contributed to this DSD broadening (36).

LES with Gravitational Droplet Coalescence.

To answer our primary science question—is gravitational coalescence alone, which is commonly assumed in detailed process models, sufficient to represent rain formation in cumulus clouds?—we analyzed DSDs from the LES using a gravitational collision kernel and aerosol distribution derived from observations (SI Appendix, Fig. S2), which we refer to as the CONTROL simulation. These DSDs (black lines in Fig. 2) are analyzed at the same altitudes as the observed DSDs. This simulation captures the observed cloud droplet mode reasonably well but fails to reproduce the drizzle and rain drop size tail at lower altitudes (higher temperatures). The large size tail is significantly suppressed relative to the observed DSDs near the cloud base (at the 21 and 18 °C levels). At higher altitudes, the secondary rain size mode and DSD shoulder appear in the modeled distributions consistent with observed DSDs. Although the simulation captures the general shape of the observed distributions better at upper cloud levels (12 and 8.5 °C), the rain size tail is suppressed relative to the observations. Over time, the simulated DSD tail at rain sizes becomes closer to the observations at upper levels, but the difference at lower levels persists (SI Appendix, Fig. S6). Note that the small droplet tail ($D<4$ $\mathrm{\mu }$m) has a higher concentration and deviates from the observations at all levels and times. This is likely due primarily to measurement limitations (degradation of detection efficiency for small droplets, coincidence count, counting uncertainties, the detection efficiency of small drops, etc.) (37). Aerosol variability could also cause deviations between simulated and observed small droplet concentrations since the simulations use a horizontally uniform aerosol vertical profile. However, this is not expected to impact the primary conclusions as extremely small drops are unimportant for collision–coalescence. Moreover, the cloud droplet mode diameter near the cloud base and DSD width toward larger sizes, which are the key for collision–coalescence, match well between the simulations and observations. Drop recirculation and growth is another mechanism that could produce large drops near cloud base. However, SDM considers this explicitly as it follows Lagrangian trajectories of particles. Thus, drop recirculation together with gravitational coalescence cannot explain the drizzle size tail of the observed DSD near cloud base.

Turbulence Effects Versus “giant CCN” Impacts on Rain Initiation.

Cumulus congestus clouds are highly turbulent due to substantial buoyant production of turbulence (see tables 1 and 2 in ref. 38). The mean value of the turbulent kinetic energy dissipation rate is estimated to be $\sim$5 $\times {10}^{-4}$ m² s⁻³ and $3\times {10}^{-3}$ m² s⁻³ from CAMP²Ex observations near the cloud base and at upper cloud levels, respectively. These are close to the average dissipation rate of $6.1\times {10}^{-4}$ m² s⁻³ and $2.3\times {10}^{-3}$ m² s⁻³ in simulated cloud updraft (w $>$ 0.1 m s⁻¹) at 1.05 km and 4 km heights, respectively. Are turbulence effects on drop coalescence a missing piece of the rain drop formation puzzle? To address this question, we implemented a turbulent collision kernel (39, 40) in the Lagrangian microphysics scheme and simulated the case with the same cloud dynamics as the CONTROL simulation using the piggybacking method.

Note that when the turbulent effects on coalescence are included in the model, gravitational coalescence still occurs. Particularly for larger drop sizes, gravitational coalescence is dominant due to the significant fall-speed differences. Overall, results from this simulation (TURBULENT) show a fairly close match to the observed DSDs at all cloud levels (Fig. 3). In particular, the TURBULENT simulation captures the observed drizzle drop tail near the cloud base and at lower cloud levels overall (21 to 14 °C), where the gravitational collision kernel alone gave poor results. Thus, the impact of turbulence on drop coalescence appears to be a critical factor in shaping drop size distributions and rain formation. This result is robust across different simulation times (SI Appendix, Fig. S11) and model grid spacings (SI Appendix, Figs. S14 and S15). It is also consistent across CONTROL and TURBULENT simulations with different flow realizations of the cloud (SI Appendix, Figs. S12 and S13).

Fig. 3.

Same as Fig. 2 but for LES outputs from the TURBULENT simulation with a turbulent drop coalescence kernel.

To examine how GCCN impact rain initiation for this case, and to test whether increased concentrations of GCCN could also explain the mismatch between the observed DSDs and those from the CONTROL simulation with gravitational coalescence only, we performed two additional simulations (again with the same flow evolution using piggybacking). In the first simulation, the large aerosol mode corresponding to GCCN was removed (NO-GCCN). In the second test, the concentration of GCCN in this large mode was enhanced by a factor of 10 (10xGCCN) relative to the observed GCCN concentration. Both of these simulations applied the gravitational collision kernel only (no turbulent coalescence). The simulated DSDs from 10xGCCN (Fig. 4) have higher concentrations of rain drops relative to CONTROL, and the deviation from the observed DSD tails at higher altitudes (at the 12 and 8.5 °C levels) decreases. However, there is still a large discrepancy between the modeled and observed DSDs at lower levels (21 to 18 °C) that persists over time even with the very high GCCN concentration. Conversely, removing GCCN entirely (NO-GCCN) increases the mismatch between the modeled and observed DSDs relative to the CONTROL simulation, mainly at upper cloud levels (SI Appendix, Figs. S7 and S10). The bias at lower levels (21 to 18 °C) is only slightly increased when GCCN are neglected entirely, and both CONTROL and NO-GCCN greatly underpredict the concentration of drizzle-sized drops there. The limited impact of GCCN is because of the long time required to reach equilibrium drop size for large aerosols, consistent with the idea mentioned in the introduction.

Fig. 4.

Same as Fig. 2 but for LES outputs from the 10xGCCN simulation with gravitational coalescence only and 10 times higher concentration of GCCN.

Is the combination of GCCN and turbulent coalescence a key for producing the close match of the TURBULENT simulation with the observed DSDs? We addressed this question by eliminating the large aerosol mode but keeping the turbulent collision kernel in another sensitivity run (TURBULENT-NO-GCCN), again using the piggybacking method to constrain the cloud flow field. The results (SI Appendix, Fig. S8) still show a reasonably good match between simulated and observed DSDs. Thus, the hypothesis that the combination of GCCN and turbulent coalescence is a key to shaping the DSDs is not supported. One might also ask a related question—are the results with turbulent droplet coalescence sensitive to the concentration of GCCN? We investigated this question by performing another simulation with the turbulent kernel but increasing the concentration of the large aerosol mode by a factor of 10 (TURBULENT-10xGCCN). The results (SI Appendix, Fig. S9) are still consistent with the other two runs with turbulent coalescence (TURBULENT and TURBULENT-NO-GCCN), and there is a similarly close match between the simulated and observed DSDs. Overall, these results suggest that GCCN are not essential to bridge the rain formation bottleneck if turbulent coalescence is active.

Theoretical Consideration

We performed a theoretical analysis of the observed DSDs that provides SI Appendix evidence of the impact of turbulence on drop coalescence. This also constrains the scaling form of the turbulent kernel. The turbulent collection kernel may be expressed in a general form as $K_{\mathit{ij}}=2\pi {\eta }_{\text{col}}{\eta }_{\text{coa}}{(r_i+r_j)}^2⟨|v_r(a=R)|⟩g_{\mathit{ij}}(a=R)$, where ${\eta }_{\text{col}}$ and ${\eta }_{\text{coa}}$ are the turbulent collision and coalescence efficiencies (the latter usually assumed to be unity), respectively, of drop pairs with radii $r_i$ and $r_j$. Here, $a$ is the radial coordinate (distance in the radial direction), and $⟨|v_r(a=R)|⟩$ and $g_{\mathit{ij}}(a=R)$ are the radial relative velocity and the radial distribution function characterizing preferential concentrations at a separation distance $R\equiv r_i+r_j$. Various turbulent collision kernels have been proposed in recent years based on direct-numerical simulation (e.g., refs. 8, 14, and 41–43), trajectory models (44, 45), and theoretical studies (39) following the pioneering work of Saffman and Turner (6).

Following the approach presented in refs. 46 and 47, motivated by the wave turbulence theory, a quasi-steady-state power-law solution of the Smoluchowski coalescence equation is derived and compared with the observations. The Smoluchowski coalescence equation (48) for a continuum drop size distribution $n(\mathbf{x},t,v)$, as a function of spatial coordinate $\mathbf{x}$, time $t$, and drop volume ($v\equiv \pi D^3/6$), can be expressed as (46):

$$\begin{array}{cc}& \frac{\partial n}{\partial t}+(\mathbf{u}·\mathrm{\nabla })n\hfill \\ \hfill & =\frac{1}{2}{\int }_0^{\infty }{\int }_0^{\infty }K(v_i,v_j)n_i{n}_j\delta (v-v_i-v_j)d{v}_{i}d{v}_j\hfill \\ \hfill & -\frac{1}{2}{\int }_0^{\infty }{\int }_0^{\infty }K(v,v_j)n{n}_j\delta (v_i-v-v_j)d{v}_{i}d{v}_j\hfill \\ \hfill & -\frac{1}{2}{\int }_0^{\infty }{\int }_0^{\infty }K(v,v_i)n{n}_i\delta (v_j-v-v_i)d{v}_{i}d{v}_j+J-S,\hfill \end{array}$$

where subscripts “$i$” and “$j$” denote the two drop classes in a collision pair, $K$ is the collection kernel as a function of drop volume that is analogous to $K_{\mathit{ij}}(r_i,r_j)$ following the relation between drop radius and volume, $\mathbf{u}$ is the velocity field, $\delta$ is the Dirac delta function, $J$ is the source, and $S$ is the sink of droplets (other than from coalescence). For a volume extending horizontally across the cloud with a small vertical extent just above the cloud base (as in the 21 °C leg), we assume that the vertical fluxes into and out of the volume, the source through activation, and the sink through deactivation are approximately in balance to produce a quasi-steady-state volume-mean DSD. This assumption is well supported by the model which shows the horizontally averaged cloud base DSD is nearly steady in time over the cloud lifetime. In this case, the left side of the above equation and the source and sink terms will approach zero for drop sizes away from the source and sink size ranges (e.g., less than $\sim$20 $\mathrm{\mu }$m owing to impacts of activation and condensation on small drops, or larger than several hundred microns in the large drop regime strongly influenced by gravitational sedimentation). Following an analogy between wave turbulence theory and drop coalescence, Horvai et al. (46) and Saito et al. (47) derived steady-state solutions for the two extreme scenarios, assuming the gravitational coalescence kernel ($K_{\mathit{ij}}\propto {(r_i+r_j)}^2|r_i^2-r_j^2|$) and the Saffman–Turner turbulent collision kernel ($K_{\mathit{ij}}\propto {(r_i+r_j)}^3$), respectively, the latter of which neglects droplet inertia and gravity (6). They obtained power-law solutions using the Zakharov transformation for these two kernels: $n(D)\propto D^{-9/2}$ (gravitational) and $n(D)\propto D^{-4}$ (Saffman–Turner). Using a similar approach (see details in SI Appendix), we obtained a solution for a generalized collection kernel with the scaling $K_{\mathit{ij}}\sim C{(r_i+r_j)}^2{(r_i\pm r_j)}^{\lambda }$, where $C$ is a proportionality constant. The corresponding solution for a quasi-steady-state drop size distribution scales as $n(D)\sim D^{-(7+\lambda )/2}$. This expression leads to the same solutions as (46) and (47) for $\lambda =2$ (gravitational) and $\lambda =1$ (Saffman–Turner), respectively.

Fig. 5 compares the above steady-state distribution scalings for the gravitational (46) and Saffman–Turner turbulent (47) kernels with the observed mean DSD just above the cloud base. The observed DSD tail is close to a power-law scaling, consistent with the theory, but does not follow either scaling relationship. This suggests neither the purely gravitational kernel nor the Saffman–Turner turbulent kernel (without droplet inertial or gravitational effects) is sufficient to explain the observed DSD near cloud base, which exhibits a scaling close to $D^{-11/2}$ and thus $\lambda \approx 4$. Relating the form of the turbulent collection kernel given in Results above to the generalized collection kernel above, $K_{\mathit{ij}}=2\pi {\eta }_{\text{col}}{\eta }_{\text{coa}}{(r_i+r_j)}^2⟨|v_r(R)|⟩g_{\mathit{ij}}(R)\approx C{(r_i+r_j)}^2{(r_i\pm r_j)}^{\lambda }$, $\lambda =4$ implies that together the radial distribution function $g_{\mathit{ij}}(R)$ times the mean radial relative velocity function $⟨|v_r(R)|⟩$ (inertial effects in both are neglected in the Saffman–Turner kernel) should approximately scale as $D^4$. The local fluid acceleration and gravitational acceleration terms in the relative velocity function both approximately scale as $D^2$ for small drops, implying that $⟨|v_r(R)|⟩\sim D^2$. Therefore, $\lambda =4$ implies that the average radial distribution function at a separation distance $R$ also scales approximately as $g_{\mathit{ij}}(R)\sim D^2$ in the small drizzle size range. Note that the collision efficiency ${\eta }_{\mathit{col}}$ also depends weakly on drop sizes, so that the radial distribution function scaling may differ slightly from $\sim$D². This scaling of the radial distribution function at separation distance $R$ is similar to the trend presented in earlier studies for particle Stokes number $<$1 (e.g., see figure 17 in refs. 39, 49, and 50). As inferred from this theoretical analysis, turbulence effects on both the radial relative velocity difference and radial distribution function scaling are crucial in shaping the drizzle-sized tail of the DSDs. However, according to the theoretical study and model from ref. 39, the radial distribution function contributes more to turbulent enhancement of the collision kernel than the other effects and is more sensitive to turbulence intensity and scales.

Fig. 5.

Comparison of drop size distribution near the cloud base (T$=$ 21 °C) with the theoretical scaling relationships from refs. 46 and 47.

Because the simulated DSDs using the turbulent kernel compare well with the observations, the simulated mean DSD near cloud base also follows an approximately $D^{-11/2}$ scaling. Note that the simulations using only the gravitational kernel have negligible coalescence near cloud base and thus produce a much steeper slope of the DSD tail. At higher altitudes in these simulations, when gravitational collection becomes significant, the slope is indeed closer to $D^{-9/2}$. Similarly, the observed DSDs transition from a slope closer to $D^{-11/2}$ near the cloud base to a shallower slope for drops with D $>$ 100 $\mathrm{\mu }$m (evident for the 14 °C cloud level). This transition represents a shift from turbulence-dominated coalescence lower in the cloud to gravitational-dominated coalescence at higher altitudes. Finally, the theoretical analysis supports the idea that the structure of the turbulent collision kernel is the crucial part rather than the parameters of the kernel. Without this analysis, it could be argued that poorly calibrated parameters in either the turbulent or gravitational kernel could be solely responsible for the data mismatch between the CONTROL simulation with gravitational collection only and the observations.

Discussion and Conclusions

The “rain formation bottleneck” is a critical yet challenging problem that the atmospheric science community has struggled with for several decades. Our study provides strong evidence of the impact of turbulence on drop growth by coalescence through the rain formation bottleneck in cumulus congestus clouds. This evidence is based on a combination of high-resolution observations and detailed LES with a Lagrangian microphysics scheme, supplemented by a theoretical scaling analysis. Multiple model sensitivity tests were also designed and conducted to test a potential competing hypothesis—the impact of GCCN on rain formation. These tests showed that GCCN with gravitational coalescence is insufficient to explain the observed drop size distributions. Moreover, GCCN had only a minor impact on rain formation when turbulence effects on drop coalescence were included in the model.

Past studies (e.g., ref. 51) also suggested that some droplets in cumulus clouds experience growth larger than in an adiabatic cloud parcel due to inhomogeneous mixing with entrained environmental air, which could help produce precipitation embryos at upper cloud levels. However, the current study showed that turbulent effects on droplet collision–coalescence can produce these embryo drops near the cloud base that later enhance drop collection as cloud blobs rise. By comparing simulations with turbulent coalescence and with gravitational coalescence alone, our results suggest that turbulence effects overwhelm the impacts of superadiabatic drop growth on precipitation development.

Impacts of turbulent coalescence on overall rain evolution are seen in Fig. 6. When turbulent effects are included, rain forms much earlier (by 25 to 30 min) and intensifies over time relative to the simulations with only gravitational drop coalescence. The magnitude of the peak rain water mass is more than seven times higher for the simulations with turbulent coalescence compared to those without. This underscores the critical impact on rain formation of a process that is neglected in most atmospheric models, with potential consequences for various feedback mechanisms related to precipitation in weather and climate modeling.

Fig. 6.

Time series of domain integrated rain water mass from the simulations (Table 1) with gravitational and turbulent collision kernels. A threshold drop size of $D=50$ $\mathrm{\mu }$m is used to separate cloud and rain water.

Fig. 6 also shows a small shift to earlier rain formation and greater rain mass when GCCN are increased by an order of magnitude using only gravitation drop coalescence, but with little sensitivity to GCCN using turbulent drop coalescence. This result implies GCCN are a lesser factor or perhaps a slower mechanism in accelerating rain formation compared to turbulent coalescence in cumulus clouds. The lack of sensitivity to GCCN is primarily explained by the limited time available for large aerosol particles in cumulus clouds ($\sim$10 min timescale for individual rising thermals) to reach their equilibrium size, unlike repeated cycling in boundary layer clouds. For example, a 1 $\mathrm{\mu }$m dry aerosol particle takes more than 25 min to reach 25 $\mathrm{\mu }$m if it experiences 0.1% average supersaturation, a size where gravitational coalescence becomes important. This is longer than the typical timescale of rising cumulus thermals. Moreover, the cloud water mixing ratio near cloud top is already large enough to trigger rain formation. Thus, even if GCCN produce large drops near cloud top, this process will not affect the outcome significantly. We also showed that, at lower cloud levels, the impact of GCCN on drop size distributions is minimal, particularly when turbulent coalescence is included in the model. Turbulent effects on drop coalescence could be a lesser factor in other cloud types where turbulent intensity is smaller (e.g., stratocumulus). However, their relative importance in shallow boundary layer clouds needs further investigation, especially relative to the GCCN impact, considering the presence of smaller liquid water contents.

The observations show a $D^{-11/2}$ scaling for the drizzle size tail of the mean DSD near cloud base, a scaling different from that for purely gravitational collision. This provides additional evidence for the role of turbulence on drop coalescence near cloud base where other mechanisms are too weak to produce significant concentrations of drizzle drops. The observed scaling also helps inform the average radial distribution function scaling with drop radius for Stokes number $<$1.

Overall, turbulent effects on drop coalescence were found to be critical for DSD evolution and rain initiation in cumulus clouds, particularly because of the strong impact at lower cloud levels. Previous bin microphysics studies showed a relatively modest impact of both turbulence (e.g., ref. 52) and GCCN (e.g., ref. 25) in cumulus simulations. However, several factors hinder the investigation of such effects using bin schemes, particularly their simplified representations of droplet activation and hydrated aerosol growth as well as numerical diffusion that leads to artificial DSD broadening (53). These shortcomings are addressed by Lagrangian microphysics schemes (54, 55), including the scheme used here that was able to well reproduce the observed DSDs at all cloud levels. Overall, this suggests the potential for Lagrangian schemes that include physically based representations of key processes, such as turbulent drop coalescence, to serve as a benchmark for coarser-resolution atmospheric models with bulk microphysics. Past efforts (e.g., ref. 56) to parameterize turbulent coalescence in bulk schemes fit arbitrary functions as a function of turbulence intensity and DSD moments to zero-dimensional box model solutions of the collection equation with assumed initial drop size distribution and without considering realistic flow. However, the coalescence process is fundamentally challenging to represent due to the breakdown of the turbulence scaling law at the grid spacing of larger-scale models ($\sim$10 to 100 km), subgrid-scale cloud variability, and the absence of explicit information on the droplet size spectrum in bulk microphysics schemes. Thus, implementing turbulent coalescence in weather and climate models requires process-level studies like the current one and revisiting the structure of bulk schemes and their coupling with the subgrid-scale cloud schemes.

Materials and Methods

CAMP²Ex Observations.

NASA’s Cloud, Aerosol and Monsoon Processes Philippines Experiment (CAMP²Ex) field campaign was designed to investigate the role of natural and anthropogenic aerosols on precipitation generation in shallow cumulus and cumulus congestus clouds. It was conducted in the Philippines region during August–October 2019 using NASA’s P-3 aircraft and the SPEC Inc. Learjet. The case presented here from September 7, 2019, consisted of a cumulus congestus field in a moderately polluted aerosol environment. (15, 31) discuss the details of this campaign, measurement strategies, instrumentation, and observed conditions. Data presented here are drop size distributions at different altitudes measured using multiple cloud probes: Fast-Forward Scattering Spectrometer, Fast Cloud Droplet Probe, Two-Dimensional Stereo Probe, and the High Volume Precipitation Spectrometer. The aerosol measurements are from the Aerodynamic Particle Sizer and TSI Laser Aerosol Spectrometer onboard NASA’s P-3 aircraft. Additional details on the drop size distribution measurements are provided in SI Appendix, Methods.

LES with Lagrangian Microphysics Scheme.

Cloud Model 1 (57) is used in a LES configuration with the state-of-the-art Lagrangian superdroplet method (30) to simulate the observed CAMP²Ex case. We used the observed sounding and aerosol measurements on Sep 7, 2019, to initialize the model base state. A 50 m isotropic grid length with 128 superparticles on average per grid box are used for the primary LES cases. Additional simulations were performed at 100 m spacing (SI Appendix, Figs. S14 and S15) to check the sensitivity of the results to grid spacing. This cumulus congestus setup and the modeling framework are similar to ref. 35, but for a different case. A cumulus cell in the simulations is initiated using a Gaussian spatial distribution of surface latent and sensible heat fluxes. To compare the various microphysical configurations without complications from feedback with the flow field, we used the piggybacking framework (32, 58) where the Tel Aviv University bin microphysics scheme is coupled to the model dynamics in a two-way feedback and used to drive the flow, and the Lagrangian scheme is coupled to the flow but does not feed back to it. However, additional tests showed that results were qualitatively similar without piggybacking (that is, with the Lagrangian microphysics scheme feeding back to the dynamics). Turbulence effects on droplet coalescence are represented using the turbulent collision kernel from ref. 39 with the turbulent collision efficiency enhancement factor (relative to the Hall gravitational kernel) from (table 1 in ref. 40). See SI Appendix, Methods for additional details.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

CAMP²Ex data are available at https://www-air.larc.nasa.gov/cgi-bin/ArcView/camp2ex (59). CM1 model code can be downloaded from https://www2.mmm.ucar.edu/people/bryan/cm1/ (60). The file size of simulation data is too large to archive, but code modifications, configuration files, name lists, and analysis scripts are available by request to the first author.