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\begin{document}

\title{Derivation and implementation of a non-gradient term to improve the
oceanic convection representation within the k-e \selectlanguage{english}parameterization\selectlanguage{english}}



\author[1]{Alexandre Legay}%
\author[1]{Bruno Deremble}%
\author[2]{Hans Burchard}%
\affil[1]{Institut des Géosciences de l'Environnement}%
\affil[2]{Leibniz Institute for Baltic Sea Research Warnemünde}%


\vspace{-1em}



  
  \date{May 25, 2024}


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\begin{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-gradient component in their expression in addition to the
usual downgradient part. To do so, we extend the
\$k-\textbackslash{}varepsilon\$ algebraic second-moment closure by
relaxing the assumption on the equilibrium of the temperature variance
\$\textbackslash{}overline\{\textbackslash{}theta'\^{}2\}\$. With this
additional transport equation for the temperature variance, we obtain a
\$k - \textbackslash{}varepsilon -
\textbackslash{}overline\{\textbackslash{}theta'\^{}2\}\$ model (the
``\$k \textbackslash{}varepsilon t\$'' model) which includes a
non-gradient term for the temperature flux. We validate this new model
against Large Eddy Simulations (LES) in both wind-forced and
buoyancy-driven regimes. In both cases, we find that the vertical
profile of temperature is well captured by the \$k
\textbackslash{}varepsilon t\$ model. Particularly, for the
buoyancy-driven regime, the non-gradient term increases the portion of
the mixed layer that is stably stratified. This is an improvement since
this portion is too small with the \$k - \textbackslash{}varepsilon\$
parameterization. Finally, a comparison of the non-gradient term with
the KPP non-local term gives insights for refining the KPP's ad hoc
shape polynomial.%
\end{abstract}%



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