Multiscale Asymptotics Analysis for the Mesoscale Dynamics of Cloud-Topped Boundary Layers

Abstract

This paper presents the derivation of a model that captures the coupling between the dynamic and thermodynamic processes of a cloud topped boundary layer on the mesoscales using a formal multiscale asymptotic approach. The derived equations show how the anomalies in the heat, moisture, and mass budgets in the boundary layer affect boundary layer motions, and how these motions can organize and amplify (or damp) such anomalies. The thermodynamics equations are similar to those that have been suggested in mixed layer studies, that is, the evolution of the thermodynamics variables depends upon the surface heat and moisture fluxes, cloud top radiative cooling rate, temperature and moisture jumps across the capping inversion. However, these equations are coupled to the dynamics equation through the entrainment rate at the top of the cloud deck. The entrainment rate is parameterised from results obtained in laboratory experiments and clearly shows the dependence upon the velocity perturbation which in turn strongly depends upon the horizontal gradient of the thermodynamics variables. The derived entrainment rate is applicable when the thermal jump at cloud-top is sufficiently weak and the velocity jump is of the order of the velocity perturbation. The mathematical properties and physical characteristics of the system of equations will be explored in future papers.