Derivation of an energetically consistent Eddy-Diffusivity Mass-Flux scheme for oceanic convection

During shallow and deep oceanic convection, eddy-diffusivity parameterizations are known to fail since advection by non-local plumes is the primary source of turbulent transport. Integration of the Mass-Flux concept to correct Eddy-Diffusivity approaches has long been studied and applied for parameterizing convection in atmospheric models, and very recently in ocean models. This closure involves breaking down vertical turbulent fluxes into two components: (1) a diffusion term that addresses local small-scale mixing in an near isotropic environment, which intensity is typically scaling with turbulent kinetic energy (TKE) (2) a mass-flux transport term, that accounts for the non-local transport due to vertically coherent plumes within the environment. We expose an energetically consistent coupling of Eddy-Diffusivity Mass-Flux (EDMF) schemes with TKE schemes, in order to model oceanic convection. To achieve such a goal, we reexamine PDE-based derivations from first principles relying on multi-fluid averaging techniques. This approach offers several key advantages. Firstly, it allows to establish fully consistent local and global energy budgets
between resolved and subgrid scales, effectively rectifying energy biases present in prior EDMF schemes. Notably, it facilitates a clear separation of convective and turbulent small-scale energy reservoirs. This is a significant departure from traditional schemes used in ocean modeling to account for non-local effects (e.g. KPP). During the theoretical development we maintain transparency regarding underlying assumptions and systematically assess their validity in the light of LES data. When compared to existing oceanic schemes, our model demonstrates performance in reproducing mean fields as well as higher-order moments such as TKE, vertical fluxes, and turbulent transport of TKE. It is validated against Large Eddy Simulation (LES) and observational data of oceanic convection.