Derivation and Implementation of a Non-Local Term to Improve the Oceanic Convection Representation Within the k–ɛ Parameterization

Abstract

The representation of turbulent fluxes during oceanic convective events is important to capture the evolution of the oceanic mixed layer. To improve the accuracy of turbulent fluxes, we examine the possibility of adding a non-local component in their expression in addition to the usual downgradient part. To do so, we extend the $k$–$\epsilon$ algebraic second-moment closure by relaxing the assumption on the equilibrium of the temperature variance $\overline{{\theta }^{\prime 2}}$. With this additional evolution equation for the temperature variance, we obtain a $k$–$\epsilon$–$\overline{{\theta }^{\prime 2}}$ model (the “$k\epsilon t$” model) which includes a non-local term for the temperature flux. We validate this new model against Large Eddy Simulations (LES) in three test cases: free convection (FC), wind-driven mixing, and diurnal cycle (DC). For wind-driven mixing, $k\epsilon t$ is equivalent to $k$–$\epsilon$. However, in the presence of a buoyancy flux (FC and DC), we find that the vertical profile of temperature of the LES is better captured by $k\epsilon t$ than $k$–$\epsilon$. Particularly, the non-local term increases the fraction of the mixed layer that is stably stratified. For FC, this fraction is near 50% for both $k\epsilon t$ and the LES, whereas the $k$–$\epsilon$ value is 20%. We show that this improvement is due to a better representation of the temperature variance in the inner part of the mixed layer. This better representation is mainly caused by the diffusion of temperature variance, which is described by $k\epsilon t$ and not by $k$–$\epsilon$.