Fate of turbulence-driven continuous growth of cloud droplet size variance by condensation under realistic activation, entrainment, and collision coupling

Sardina, Picano, Brandt, and Caballero showed by a stochastic model and direct numerical simulation that in homogeneous isotropic turbulence the standard deviation of the squared cloud droplet radius grows continuously as $t^{1/2}$, with a rate linear in the Taylor-scale Reynolds number and independent of the mean turbulent dissipation, so that turbulence-induced supersaturation fluctuations alone can produce realistically broad droplet size spectra within about 20 minutes. The result was derived for an idealized closed, homogeneous, non-activating, collisionless cloud with one-way droplet coupling, and the authors conjectured that real-cloud inhomogeneity and entrainment would further enhance the broadening. The open problem is to determine, within one quantitative and validated framework, whether and at what rate this variance growth survives when cloud condensation nucleus activation, entrainment of dry air and spatial inhomogeneity, collision-coalescence, and droplet feedback act simultaneously, and whether a generalized stochastic-condensation closure — a probability-density-function (PDF) or Langevin model with estimated parameters — can predict observed droplet size spectra in real clouds.

Background

Warm clouds form on aerosol particles that act as cloud condensation nuclei (CCN): when the ambient supersaturation $s$ (the fractional excess of water vapor pressure over its equilibrium value at the local temperature) exceeds a size- and composition-dependent critical value, a particle activates and grows as a liquid droplet. An activated droplet grows by diffusion of water vapor (condensation), a process whose rate is inversely proportional to the droplet radius, which tends to narrow the size distribution, while growth to raindrop size requires collision and coalescence between droplets, which is effective only once the size distribution is broad enough. The resulting condensation-collision bottleneck between roughly 10 and $50\,\mu\mathrm{m}$ is a central obstacle to warm rain formation. The classical Twomey equation governs the supersaturation field: $s$ is produced by cooling of rising air, transported by turbulence, and relaxed to equilibrium by condensational uptake at the droplet surfaces, with a characteristic phase relaxation time $\tau_s$ set by the droplet concentration and mean radius. Sardina, Picano, Brandt, and Caballero (Phys. Rev. Lett. 115, 184501, 2015) studied how turbulence-driven supersaturation fluctuations broaden the size distribution in homogeneous isotropic turbulence (HIT, an idealized turbulent state with no preferred direction and statistical uniformity in space), characterized by the Taylor-scale Reynolds number $\mathrm{Re}_{\lambda}$, which measures the separation between the large energy-containing scales and the small dissipative scales. Using a stochastic (Langevin) version of the Twomey equation, valid when the phase relaxation time is much shorter than the turbulence integral time $T_0$, together with direct numerical simulation (DNS, resolving all turbulent scales) and large-eddy simulation, they showed that the standard deviation of the squared droplet radius fluctuations, $\sigma_{R^2}$, grows as $t^{1/2}$ (equivalently the variance of $R^2$ grows linearly in time), with a growth rate proportional to the scale separation $\mathrm{Re}_{\lambda}$ and to the turbulence integral scales, but independent of the mean energy dissipation rate; at cumulus-like values $\mathrm{Re}_{\lambda}\sim10^4$ the predicted broadening matches observations within about 20 minutes. At small $\mathrm{Re}_{\lambda}$, viscous effects on the supersaturation field cause the inviscid stochastic model to overestimate the broadening, so it provides an upper bound. The analysis deliberately excluded CCN nucleation and activation, droplet collisions, entrainment of dry environmental air, spatial inhomogeneity, mean updrafts, and the dynamical feedback of droplets on the flow, and the authors conjectured, without demonstration, that entrainment and fluctuating temperature fields in real clouds would lead to even larger broadening. Subsequent work developed the pieces separately: a mean updraft and finite activation size reduce the turbulent broadening (Sardina et al., J. Atmos. Sci. 75, 451, 2018); in a closed system with finite total water the $R^2$ diffusion is transient and the droplet population relaxes to a statistical steady state (Siewert, Bec, Krstulovic, J. Fluid Mech. 810, 254, 2017), and a Fokker-Planck description (the probability-density transport equation governing a stochastic process) with Köhler-theory activation effects (Köhler theory gives the equilibrium vapor pressure over a curved solution droplet and thereby sets the critical activation size) yields steady-state spectra consistent with laboratory cloud-chamber measurements (Saito, Gotoh, Watanabe, J. Meteorol. Soc. Japan 97, 867, 2019); combined condensation-plus-collision DNS shows that fluctuation-induced broadening enhances collisional growth (Chen et al., Atmos. Chem. Phys. 18, 7251, 2018; Li et al., J. Atmos. Sci. 77, 337, 2020); entraining parcel and cloud-edge studies address entrained-CCN activation and Lagrangian (measured along individual droplet trajectories) supersaturation fluctuations at the cloud boundary (Abade et al., J. Atmos. Sci. 75, 3365, 2018; Fries et al., Phys. Rev. Lett. 131, 254201, 2023); scaled-up DNS and comparisons of Eulerian bin microphysics with Lagrangian superdroplet simulations (a superdroplet is a Monte Carlo computational particle representing many identical physical droplets) establish the Reynolds-number and domain-size validity range of the stochastic model (Thomas et al., Atmos. Chem. Phys. 20, 9087, 2020; Grabowski and Thomas, Atmos. Chem. Phys. 21, 4059, 2021); and field holography finds signatures consistent with stochastic condensation in marine boundary-layer clouds (Desai et al., J. Geophys. Res. Atmos. 124, 2739, 2019). No published study combines all of these processes in one validated framework or tests the enhancement conjecture at cloud scale.

Problem Statement

Determine, within a single quantitative framework validated against simulation or observation, the fate of turbulence-driven condensational broadening of the cloud droplet size distribution when the processes excluded by Sardina et al. (2015) act simultaneously — spatial inhomogeneity and entrainment of dry air with fluctuating temperature fields, nucleation and activation of cloud condensation nuclei, collision-coalescence, and droplet feedback on the flow. Concretely: (i) does the standard deviation of the squared droplet radius continue to grow, transiently or indefinitely, and at what rate relative to the inviscid prediction $\sigma_{R^2}\propto t^{1/2}$ with growth rate linear in $\mathrm{Re}_{\lambda}$, once these couplings are present; (ii) is the conjecture that entrainment and inhomogeneity enhance the broadening correct, and by how much; (iii) once these couplings are present, over what parameter range does the inviscid Langevin model remain a valid upper bound, given that viscous small-scale effects already make it overestimate broadening at low $\mathrm{Re}_{\lambda}$ in homogeneous turbulence; and (iv) can a generalized stochastic-condensation (probability-density-function/Langevin) closure incorporating activation, collision, and entrainment reproduce observed droplet size spectra in real clouds, with its closure parameters (such as the velocity-supersaturation correlation constant $C_{ws}$ and the Lagrangian supersaturation correlation time) estimated from laboratory or field data? These four parts are facets of the single objective stated above and are to be addressed jointly within the one validated framework; no subset of them alone resolves the task.

References

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