Implementation of the quasi-normal scale elimination theory in models of ABLs

Recently, we have developed a quasi-normal spectral theory (QNSE) of turbulence which is based upon the procedure of successive small-scale modes elimination. This procedure allows one to analytically derive scale-dependent expressions for horizontal and vertical eddy viscosities and eddy diffusivities as well as the dispersion relation for internal waves in the presence of turbulence, various one-dimensional spectra and other important characteristics of turbulence. The elimination of all scales yields models of the Reynolds stress closure type, i.e., models of the RANS (Reynolds averaged, Navier-Stokes equations-based) family. Since the derivation of the QNSE-based models relies on a rigorous procedure in Fourier space rather than the closure assumptions in the physical space typical of the Reynolds stress closures, the QNSE-based models are devoid of the problems typical of the Reynolds stress closure-based RANS models. From this viewpoint, the QNSE-based models present a viable alternative to the Reynolds stress closure-based models. We report further progress in application of the QNSE-based models to simulate ABLs. We have derived $K-\epsilon$ and $K-l$ models of stably and unstably stratified atmospheric boundary layers and tested them against data (here, $K$ is the turbulence kinetic energy, $l$ is the turbulence macroscale, and $\epsilon$ is the rate of turbulent dissipation). Although $K-l$ models are simpler than $K-\epsilon$ models, we demonstrate that the performance of both models is comparable in simulations of stable ABLs and the diurnal cycle. We have also implemented the QNSE-based $K-l$ model in the numerical weather prediction system HIRLAM (High Resolution Limited Area Model). Currently, for parameterization of turbulent mixing, HIRLAM employs a $K-l$ model which requires a prognostic equation for the turbulence kinetic energy. The original stability functions used in HIRLAM were replaced by those derived within the QNSE theory; these functions are used for calculation of the vertical turbulent mixing coefficients, $K_M$ and $K_H$. For preliminary testing and validation of the $K-l$ model with new stability functions, a 1-D variant of the HIRLAM has been employed. The results of simulations with 1-D HIRLAM were compared with the observational datasets from CASES-99 and Sodankyla station (polar Finland). The simulated profiles of wind, temperature and turbulence dissipation (when available) are in good agreement with the observational data.

For further validation and verification of the new $K-l$ model, the 3-D HIRLAM model (variant 6.4.1) was used for +48h forecasts during January 2005, total 120 forecasts. These datasets are used for statistical analysis and calculation the skills of the forecasts. The results show an improvement in bias and rms of 2m temperature and 2m relative humidity and significant improvement in bias of mean sea level pressure, $P_{msl}$.