Stochastic coalescence in Lagrangian cloud microphysics

Stochasticity in collisional growth of cloud droplets is studied in a box model using the super-droplet method (SDM). The SDM is compared with direct numerical simulations and the master equation. We use the SDM to study fluctuations in autoconversion time and the sol-gel transition. We determine how many computational droplets are necessary to correctly model expected number and standard deviation of autoconversion time. Also, growth rate of lucky droplets is determined and compared with a theoretical prediction. Size of the coalescence cell is found to strongly affect system behavior. In small cells, correlations in droplet sizes and droplet depletion affect evolution of the system. In large cells, unrealistic collisions between rain drops, caused by the assumption that the cell is well-mixed, become important. Maximal size of a volume that is turbulently well-mixed with respect to coalescence is estimated at Vmix = 1.05 · 10−2 cm3. It is argued that larger cells can be considered approximately well-mixed, but only through comparison with fine-grid simulations. In addition, validity of the Smoluchowski equation is tested. Discrepancy between the SDM and the Smoluchowski equation is observed if droplets are initially relatively small. This implies that cloud models that use the Smoluchowski equation might produce rain too soon.

An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

In cloud modeling studies, the time evolution of droplet size distributions due to collision–coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain types of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was seen by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitudes of correlations are significant and must be taken into account when modeling the evolution of a finite volume coalescing system.

An algorithm for the numerical solution of the multivariate master equation for stochastic coalescence

In cloud modeling studies, the time evolution of droplet size distributions due to collision-coalescence events is usually modeled with the kinetic collection equation (KCE) or Smoluchowski coagulation equation. However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitude of correlations are significant, and must be taken into account when modeling the evolution of a finite volume coalescing system.

The breakup of levitating water drops observed with a high speed camera

A wind tunnel was used to interact water drops and were recorded using a high speed camera. Three distinct collisional breakup types were observed and the drop size spectra from each were analysed for comparison with parameterisations constructed by Low and List (1982a). The spectra predicted by the parameterisations did not accurately correlate with the observed breakup distributions for each breakup type when applied to the relatively larger and similarly-sized drop-pairs of size 4–8 mm, comparable to those sometimes observed in nature. We discuss possible reasons for the discrepancies and suggest potential areas for future investigation. A computer programme was subsequently used to solve the stochastic coalescence and breakup equation using the Low and List breakup parameterisation, and the evolving drop spectra for a range of initial conditions were examined. Initial cloud liquid water content was found to be the most influential parameter, whereas initial drop number was found to have relatively little influence. This may have implications when considering the effect of aerosol on cloud evolution, raindrop formation and resulting drop spectra.