Cloud-Top Entrainment Instability through Small-Scale Mixing and Its Parameterization in Numerical Models

Abstract

In a recent paper, Kuo and Schubert demonstrated the lack of observational support for the relevance of the criterion for cloud-top entrainment instability proposed by Randall and by Deardorff. Here we derive a new criterion, based on a model of the instability as resulting from the energy released close to cloud top, by Mixing between saturated boundary-layer air and unsaturated air from above the capping inversion. The condition is derived by considering the net conversion from potential to kinetic energy in a system consisting of two layers of fluid straddling cloud-top, when a small amount of mixing occurs between these layers. This contrasts with previous analyses, which only considered the change in buoyancy of the cloud layer when unsaturated air is mixed into it. In its most general form, this new criterion depends on the ratio of the depths of the layers involved in the mixing. It is argued that, for a self-sustaining instability, there must be a net release of kinetic energy on the same depth and time scales as the entrainment process itself. There are two plausible ways in which this requirement may be satisfied. Either one takes the depths of the layers involved in the mixing to each be comparable to the vertical scale of the entrainment process, which is typically of order tens of meters or less, or alternatively, one must allow for the efficiency with which energy released by mixing through a much deeper lower layer becomes available to initiate further entrainment. In both cases the same criterion for instability results. This criterion is much more restrictive than that proposed by Randall and by Deardorff; furthermore, the observational data is then consistent with the predictions of the current theory. Further analysis provides estimates of the turbulent fluxes associated with cloud-top entrainment instability. This analysis effectively constitutes an energetically consistent turbulence closure for models of boundary layers with cloud. The implications for such numerical models are discussed. Comparisons are also made with other possible criteria for cloud-top entrainment instability which have recently been suggested.