Statistical steady state in turbulent droplet condensation

Abstract

Motivated by systems in which droplets grow and shrink in a turbulence-driven supersaturation field, we investigate the problem of turbulent condensation in a general manner. Using direct numerical simulations we show that the turbulent fluctuations of the supersaturation field offer different conditions for the growth of droplets which evolve in time due to turbulent transport and mixing. Based on that, we propose a Lagrangian stochastic model for condensation and evaporation of small droplets in turbulent flows. It consists of a set of stochastic integro-differential equations for the joint evolution of the squared radius and the supersaturation along the droplet trajectories. The model has two parameters fixed by the total amount of water and the thermodynamic properties, as well as the Lagrangian integral timescale of the turbulent supersaturation. The model reproduces very well the droplet size distributions obtained from direct numerical simulations and their time evolution. A noticeable result is that, after a stage where the squared radius simply diffuses, the system converges exponentially fast to a statistical steady state independent of the initial conditions. The main mechanism involved in this convergence is a loss of memory induced by a significant number of droplets undergoing a complete evaporation before growing again. The statistical steady state is characterised by an exponential tail in the droplet mass distribution. These results reconcile those of earlier numerical studies, once these various regimes are considered.Comment: 24 pages, 12 figure

Search for Microphysical Signatures of Stochastic Condensation in Marine Boundary Layer Clouds Using Airborne Digital Holography

“…Various mechanisms such as entrainment with environmental air (Baker et al, 1980;Paluch & Knight, 1986;Telford & Chai, 1980;Warner, 1969), stochastic condensation (Khvorostyanov & Curry, 1999;Mazin & Smirnoff, 1969), and particle-turbulence interactions (Grabowski & Wang, 2013;Shaw, 2003) have been considered possible mechanisms to explain this broadening. Recently, numerical studies (Field et al, 2014;Grabowski & Abade, 2017;Paoli & Shariff, 2009;Sardina et al, 2015;Siewert et al, 2017) and observations of supersaturation variability (Ditas et al, 2012;Siebert & Shaw, 2017) have reignited the debate that stochastic condensation may play an important role in broadening of the size distribution. These build on the work by Cooper (1989) showing that stochastic condensation may still produce broadening despite the Journal of Geophysical Research: Atmospheres 10.1029/2018JD029033 limitations pointed out by Bartlett and Jonas (1972).…”

Cloud droplet size distribution broadening during diffusional growth: ripening amplified by deactivation and reactivation

“…2b. s qs , the environmental supersaturation in quasi-steady state, is inversely proportional to the integral of the mean droplet size r and the droplet number concentration (n), s qs ∝ (rn) −1 (e.g., Squires, 1952;Politovich and Cooper, 1988;Korolev and Mazin, 2003;Lamb and Verlinde, 2011). Here the decrease in n due to droplet deactivation is much greater than the change of r; therefore, s qs will increase with decreasing n. This means that larger droplets grow even faster in the updraft region, and smaller droplets evaporate even faster in the downdraft region -beyond the solute effect alone.…”

“…A qualitative description of this mechanism is that some "lucky" cloud droplets experience relatively larger supersaturation or stay a relatively longer time in the cloud compared with the other cloud droplets; therefore they can grow larger in size and broaden the CDSD. Recent theoretical and experimental studies support this mechanism and provide ways to quantify the resulting width of the droplet size distribution (e.g., McGraw and Liu, 2006;Sardina et al, 2015;Chandrakar et al, 2016;Grabowski and Abade, 2017;Siewert et al, 2017). Turbulence can also modulate the condensational growth of cloud droplets through mixing and entrainment (e.g., Lasher-Trapp et al, 2005;Cooper et al, 2013;Korolev et al, 2013;Yang et al, 2016).…”

Droplet size distributions in turbulent clouds: experimental evaluation of theoretical distributions

“…Physical processes affecting the cloud DSD include the curvature and solute effects (and associated mechanisms such as spectral ripening and competing activation of cloud condensation nuclei of varying size, solubility, etc.) (Johnson, 1982;Korolev, 1995;Wood et al 2002;Yang et al 2018), turbulent fluctuations (Cooper, 1989;Paoli and Shariff, 2009;Sardina et al 2015;Chandrakar et al 2016;Siewert et al 2017), entrainment and mixing (Baker et al 1980;Lehmann et al 2009;Yum et al 2015), microphysical variability (Cooper, 1989;Desai et al 2018), internal mixing of parcels of different growth history (Hudson and Yum, 1997;Lasher-Trapp et al 2005), and droplet collision-coalescence (Pruppacher and Klett, 2010;Glienke et al 2017;Witte et al 2017). Cloud droplet growth prior to onset of collision-coalescence occurs through water vapour condensation, and therefore the central variable affecting the DSD is the water vapour supersaturation, including both its mean value and its variability (Ditas et al 2012;Siebert and Shaw, 2017).…”

“…Several different theoretical models have been proposed to characterize the shape of cloud DSDs formed through the condensation process. Two approaches that have received considerable recent attention are systems-theory derivations based on the principle of maximum entropy (Liu and Hallett, 1998;Liu et al 2002;Yano et al 2016;Wu and McFarquhar, 2018) and derivations based on a Langevin equation representation of stochastic condensation (McGraw and Liu, 2006;Paoli and Shariff, 2009;Chandrakar et al 2016;Siewert et al 2017;Abade et al 2018;Saito et al 2019). By and large, however, these theoretical models have not been subjected to systematic experimental evaluation to validate the proposed functional forms for the DSD.…”