An Extended Eddy‐Diffusivity Mass‐Flux Scheme for Unified Representation of Subgrid‐Scale Turbulence and Convection

Abstract

Large‐scale weather forecasting and climate models are beginning to reach horizontal resolutions of kilometers, at which common assumptions made in existing parameterization schemes of subgrid‐scale turbulence and convection—such as that they adjust instantaneously to changes in resolved‐scale dynamics—cease to be justifiable. Additionally, the common practice of representing boundary‐layer turbulence, shallow convection, and deep convection by discontinuously different parameterizations schemes, each with its own set of parameters, has contributed to the proliferation of adjustable parameters in large‐scale models. Here we lay the theoretical foundations for an extended eddy‐diffusivity mass‐flux (EDMF) scheme that has explicit time‐dependence and memory of subgrid‐scale variables and is designed to represent all subgrid‐scale turbulence and convection, from boundary layer dynamics to deep convection, in a unified manner. Coherent up and downdrafts in the scheme are represented as prognostic plumes that interact with their environment and potentially with each other through entrainment and detrainment. The more isotropic turbulence in their environment is represented through diffusive fluxes, with diffusivities obtained from a turbulence kinetic energy budget that consistently partitions turbulence kinetic energy between plumes and environment. The cross‐sectional area of up and downdrafts satisfies a prognostic continuity equation, which allows the plumes to cover variable and arbitrarily large fractions of a large‐scale grid box and to have life cycles governed by their own internal dynamics. Relatively simple preliminary proposals for closure parameters are presented and are shown to lead to a successful simulation of shallow convection, including a time‐dependent life cycle.

Parameterization of Entrainment Rate and Mass Flux in Continental Cumulus Clouds: Inference From Large Eddy Simulation

“…In a bulk plume approach, an ensemble of clouds is contributing to the transport of heat and moisture rather than an individual cloud. The LES model has been proven to be an extremely useful tool to diagnose bulk entrainment rate in cumulus clouds using a simple entraining plume model (Betts, ; de Rooy & Siebesma, ; Lu et al, ; Tan et al, ) with moist conserved variables: where ϕ is a moist conserved parameter, for example, moist static energy temperature T h . The parameter ε refers to the fractional entrainment rate.…”

Numerical solution of the conditionally averaged equations for representing net mass flux due to convection

“…This builds on traditional mass‐flux schemes by solving equations for mass, temperature, and momentum both inside and outside the plumes, and so net mass transfer by convection can be represented. Our numerical method would also be suitable for the similar extended EDMF scheme (Tan et al ) and would allow net mass transport by convection. Transfers of mass between the fluids are proposed for two purposes. The conditionally averaged equations with a single pressure are ill‐posed, so transfer terms are formulated to represent drag and mass transfer between fluids to ensure that the different fluids do not diverge. Transfer terms are proposed so that well‐resolved convection is transferred to the buoyant fluid and stable or sinking air is transferred to the stable fluid.…”

A two‐fluid single‐column model of the dry, shear‐free, convective boundary layer

“…The conceptual similarity of the multi‐fluid and EDMF approaches is further emphasized by the recent extension of the EDMF approach by Tan et al . () to include prognostic equations for updraught properties; their model is then essentially a two‐fluid anelastic model.…”