A two-fluid single-column model of turbulent shallow convection. Part I: Turbulence equations in the multi-fluid framework

Abstract

The multi-fluid equations are derived from the compressible Euler equations (or any of the usual approximate equation sets used in meteorology) by conditional filtering. They have the potential to provide the basis for an improved representation of cumulus convection and its coupling to the boundary layer and larger scale flow in numerical models. The present article derives the prognostic equations for subfilter-scale turbulent second moments in the multi-fluid framework, along with certain systematic simplifications of them, thus providing a multi-fluid analogue of the well-known Mellor and Yamada hierarchy of turbulence closures. As well as enabling a more accurate calculation of subfilter-scale fluxes and the effects of subfilter-scale variability on cloud fraction, liquid water, and buoyancy, the second moment information can be used to obtain a more accurate parameterization of entrainment and detrainment. A subset of the turbulence equations derived here is employed in the two-fluid single-column model described in Part~II and applied to the simulation of shallow cumulus cases in Part~III.