Analysis of radiation-induced cooling and growth of mist and cloud droplets

Abstract

Radiative heat transfer from a dilute, optically thin assembly of water droplets in air to a remote heat sink (e.g., cloud tops to outer space) is analyzed, including droplet-vapor conduction, vapor-phase mass transfer, and liquid-vapor phase change. Predictions are compared with a recent laboratory study and show improvement over preliminary modeling efforts [Brewster et al. 2020]. Results highlight: (i) the validity of the quasi-steady vapor-supersaturation assumption, (ii) the importance of unintended droplet solution effects, and (iii) the effects of internal (isothermal) equilibration of droplet-surface supersaturations vs. external (radiative cooling) equilibration on droplet growth. This analysis produces simple formulas for the time scales for condensational growth and for cooling or thermal equilibration of radiatively cooled, emitting/absorbing, conducting, and condensing droplets under near-saturation conditions. Both time scales—growth and cooling—are of the order of minutes. The growth time scale is a strong function of temperature and the cooling time scale is a strong function of droplet volume fraction. The growth time scale decreases, i.e., the process goes faster, for warmer droplets, e.g., with simultaneous conductive heating; it also decreases as droplet diameter decreases, indicating that radiatively-induced growth can happen even for small (< 1 µm diameter) droplets. The cooling time scale, a modification of the corresponding time scale for non-volatile (non-condensing) droplets, varies inversely with droplet volume fraction. The cooling time is analogous to the classical result for lumped-capacitance (low Biot-number), convective-conductive cooling. A dimensionless factor is identified that represents the fractional increase in radiative cooling time due to condensation, which arises from latent energy supplied by condensation that would have been drawn out of the vapor internal energy for non-condensing droplets. A simple thermophysical formula is presented for estimating this factor, which exhibits significant temperature dependence (primarily through water-vapor mole fraction) and varies from 67% at 272 K to 230% at 293 K, 1 bar. A conduction-radiation parameter is also identified that characterizes the importance of conductive/convective heating by surrounding air, when that mode competes with radiative cooling, and quantifies how closely droplets radiating to a remote sink at a fixed temperature can approach that temperature, given enough time. These results bring to light an important distinction between the cooling and droplet-growth effects of droplet radiation and the dependence of these effects on competition between radiative cooling and conductive/convective heating. When conductive heating is negligible, significant droplet growth and cooling both occur, within minutes; with conductive heating present, the cooling effect is diminished while the growth effect is enhanced. These findings have implications for modeling cloud microphysics, both with respect to liquid droplets overcoming the condensation-coalescence barrier in “warm” (non-freezing) clouds and with respect to droplets freezing in “cold” clouds.