The turbulent length scale problem in cloud resolving models

The turbulent length scale (a.k.a. mixing length) represents the size of the large energy-containing eddies in a turbulent flow. In cloud resolving models (CRMs) this term is usually needed to parameterize the sub-grid scale (SGS) dissipation rate as well as eddy diffusivity. Oftentimes, the length scale is set equal to the vertical grid spacing, but because turbulence can vary significantly in time and space, this is typically not a good approximation. Furthermore, there is a general lack of guidance as to what the proper mixing length should be and CRMs tend to exhibit moderate to high sensitivity to the specification of this value (Moeng and Randall 1984).

This study aims to address what the proper turbulent length scale is for coarse-grid CRMs and, based on these results, to provide a general formulation of this term that is independent of the model grid size. In addition, the formulation for the new length scale should be unified, which is important for CRMs that are implemented as “superparameterizations” within General Circulation Models (GCMs). Utilizing several LES benchmark simulations we are able to diagnose the proper length scale for a variety of boundary layer regimes and cloud types, in order to find similarity and to aide in the formulation process. Due to the large domain sizes used in our LES simulations, we are able to compute the appropriate length scale for grid sizes ranging from those typical of CRMs to GCMs (800 m to 204.8 km).

Our findings for the appropriate turbulent length scale differs from popular formulations used in some coarse-grid CRMs (Blackadar 1964; Bougeault and Andre 1986). In addition, model results utilizing this new length scale are greatly improved compared to simulations utilizing a specified length scale, or ones based on traditionally used formulations. This work presents our new formulation and the resulting improved representation of turbulence in coarse-grid CRMs.