Energetically Consistent Eddy-Diffusivity Mass-Flux Convective Schemes: 1. Theory and Models

Abstract

This paper presents a self-contained derivation, from first principles, of a convective vertical mixing scheme based on the Eddy-Diffusivity Mass-Flux (EDMF) approach. This type of closure separates vertical turbulent fluxes into two components: an eddy-diffusivity (ED) which accounts for local small-scale mixing in a nearly isotropic environment, and a mass-flux (MF) transport term, which represents the non-local transport driven by vertically coherent plumes. Using the multi-fluid averaging underlying the MF concept, we review consistent energy budgets between resolved and subgrid scales for seawater and dry atmosphere, in anelastic and Boussinesq frameworks. We demonstrate that when using an EDMF scheme, closed energy budgets can be recovered if: (a) bulk production terms of turbulent kinetic energy (TKE) by shear buoyancy include MF contributions; (b) boundary conditions are consistent with EDMF, to avoid spurious energy fluxes at the boundary. Furthermore, we show that lateral mixing, due to either entrainment or detrainment induces a net production of TKE via the shear term, with enhanced production under increased horizontal drag. We also provide constraints on boundary conditions to ensure mathematical consistency. Throughout the theoretical development, we maintain transparency regarding underlying assumptions. In a companion paper (Perrot and Lemarié (2024, https://hal.science/hal-04666049); hereafter Part II) we assess the validity of these hypotheses, and analyze the sensitivity of the scheme to modeling choices against Large Eddy Simulations (LES) and observational data on oceanic convection. Part II also details an energy-conserving discretization and quantifies energy biases in inconsistent formulations.