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 $k-\varepsilon$ algebraic second-moment closure by relaxing the assumption on the equilibrium of the temperature variance $\overline{\theta’^2}$. With this additional transport equation for the temperature variance, we obtain a $k - \varepsilon - \overline{\theta’^2}$ model (the “$k \varepsilon t$” 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 $k \varepsilon t$ 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 $k - \varepsilon$ 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.