An analytical study of cumulus onset

Abstract

A slab model of the well-mixed daytime boundary layer is combined with a surface-layer parametrization based on Monin-Obukhov similarity to investigate analytically basic aspects of cumulus onset. Cumulus clouds are defined to occur when the equilibrium height of surface-layer air equals or exceeds its lifting condensation level (LCL). For a linearized version of the model, a closed solution of onset time is derived as a function of atmospheric and surface parameters. It is found that the effect of the various parameters on the timing of cumulus onset can be encompassed by two non-dimensional ‘cumulus’ numbers, which describe the relationship between differential variations of equilibrium height and LCL, one for the mixed layer and one for the surface layer. the most important parameter, appearing in both these numbers, is the ratio between the potential-temperature gradient above the mixed layer and the dry-adiabatic lapse rate of dew-point depression (∽ 0.008 K m−1). Results from both the linearized and the nonlinear model indicate that cumulus onset is delayed by increasing the Bowen ratio in cases of moderate-to-high stability, whereas the opposite is found for less stable conditions. Qualitative aspects of the model results are discussed with reference to observed relations between cumulus formation and surface characteristics.