The Probability Density Function Related to Shallow Cumulus Entrainment Rate and Its Influencing Factors in a Large-Eddy Simulation

The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds. Accurately obtaining the entrainment rate (λ) is particularly important for its parameterization within the overall cumulus parameterization scheme. In this study, an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculate λ of cumulus clouds in a large-eddy simulation. The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method, because λ, as calculated from the improved method, falls within the range of λ values obtained from the traditional method using different conserved variables. The probability density functions of λ for all data, different times, and different heights can be well-fitted by a log-normal distribution, which supports the assumed stochastic entrainment process in previous studies. Further analysis demonstrate that the relationship between λ and the vertical velocity is better than other thermodynamic/dynamical properties; thus, the vertical velocity is recommended as the primary influencing factor for the parameterization of λ in the future. The results of this study enhance the theoretical understanding of λ and its influencing factors and shed new light on the development of λ parameterization.