A probabilistic description of entrainment instability for cloud-topped boundary-layer models

Cloud-top radiative and evaporative cooling effects on entrainment fluxes are essential to turbulence generation and growth of cloud-topped boundary layers. While the cloud radiative cooling effects are currently not well understood, their interaction with the cloud-top evaporation further complicates efforts to parametrize the phenomena due to the random behaviour of the turbulent mixing at the cloud-top interface. Here we focus on the cloud-top mixing and treat the typical turbulent mixing eddy in a statistical manner to relate the mean evaporative cooling rate to the distribution of effective mixing fraction (defined as the fraction of air coming from the free atmosphere that enhances the entrainment flux). Existing observations suggest that this effective mixing fraction can be parsimoniously parametrized with beta distributions, the shape parameters of which control the locations of stability lines in the Cloud-Top Entrainment Instability (CTEI) diagrams used to discriminate between stable and unstable clouds. The probabilistic description of the cloud-top mixing process allows us to coherently reinterpret various forms of CTEI criteria and generalize them to form new cloud-top entrainment schemes. We expect that such schemes will help improve cloud dynamic models by embedding realistic distributions of the effective mixing fraction.