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Given these results, we suggest a possible new representation for vertical mixing (a variant of the CLUBBE scheme) that assumes an underlying joint probability density function for fluctuations of momentum and tracers. With this assumption only seven turbulence equations must be predicted in total. While for a standard higher order turbulence closure needs 15 equations for second order and a closure for higher orders. This formulation can also be used to derive an explicit equation for boundary layer entrainment and is amenable to any coordinate system. Initial results where LES data is used in the proposed scheme to reconstruct turbulent fluxes are encouraging.
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