An Evaluation of Algebraic Turbulence Length Scale Formulations

The most frequently used turbulence parameterizations in numerical weather prediction (NWP) and general circulation (GC) models are turbulence kinetic energy (TKE) schemes. These turbulence schemes are strongly dependent on a key component, the turbulence length scale. The turbulence length scale is used to parameterize the molecular dissipation of TKE and is also required for calculating the turbulence coefficients. Traditionally, the turbulence length scale formulations do not take into account the transfer of TKE across scales, as they are designed for scales above the energy production range of the turbulence spectra. However, with computational power growing, it has become increasingly possible to run numerical models at scales that are within the gray zone of turbulence. At resolutions within this gray zone, the cross-scale transfer of TKE needs to be taken into account in order to accurately represent the turbulence. For this purpose, a turbulence length scale diagnostic was developed. This is achieved by calculating the turbulence length scale from the so-called effective dissipation rate, which is a combination of the cross-scale TKE transfer and the dissipation rate. The effective dissipation rate is estimated from the budget of the TKE using large-eddy simulation (LES) data. A similar approach is used to calculate the turbulence length scale from the budgets of the scalar variances. This study thus makes use of three different turbulence length scale diagnostics based on: the TKE, the variance of the total specific water content, and the variance of the liquid water potential temperature. Using the turbulence length scale diagnostics as a reference, a series of five exisiting algebraic turbulence length scale formulas are evaluated. The objective evaluation is carried out in terms of a local root mean square error and a non-local three-component technique. The algebraic formulations are evaluated for a set of five idealized LES cases, simulated using the MicroHH model. These cases represent different boundary layer conditions. Based on the evaluation, the length scales proposed by Nakanishi and Niino and Honnert et al. are found to be most representative of the turbulence length scale diagnostics.