Towards a Solution of the Closure Problem for Non-Homogeneous and Anisotropic Turbulence: Non-Gaussian Closures for Convective Atmospheric Boundary Layer Turbulence Based on an Analytically Solvable Multivariate Delta-PDF Model.

The atmospheric convective boundary layer (CBL) plays an important role in the energy and momentum budget of the Earth system and is of importance in simulations for the weather forecast, the present-day climate and future climate projections. Currently the physical processes in the CBL are not or only partly resolved, therefore need to be parameterised. The turbulent mixing in the CBL is driven by small-scale eddies near the surface and in the inversion, and by large plumes in the well-mixed middle part. To properly describe the CBL turbulence an accurate parameterisation of the effect of both small scale turbulence as well as the plumes together with their statistical properties (distribution in amplitudes and sizes) is of primary importance.

An analytically tractable one-point turbulence closure model based on an assumed multivariate Delta-PDF approach is developed. The Delta-PDF model is suited to represent the effect of coherent structures - populations of plumes in the most transparent way. It is an extension of the model of Gryanik and Hartmann [1] (GH02), and additionally includes a term for background turbulence. We show that the closure model has an exact solution, all higher order moments (HOM) can be determined analytically. The HOMs are algebraic functions of the following irreducible moments: variances, bivariate correlation coefficients, skewnesses and trivariate correlation coefficients, and of only one fourth order moment (FOM) which is the correlation coefficient in four variables, the velocity components and the temperature. In particular, new closures include the familiar monovariate kurtosis-skewness relationships and generalise these to the multivariate case.

The a priory testing of the closures predicted by the new Delta-PDF model shows a good agreement with data from the aircraft campaign ARTIST and from LES and DNS simulations. The accuracy of presenting the vertical profiles of HOMs is of the same order or better then that of the LES and DNS. The data sets reveal that the model is universially applicable to all stratification conditions, including the unstably stratified surface layer, the well-mixed zone and the stably stratified air in the inversion and aloft. There is also no obvious reason why the approach should not work for neutral conditions.

The solution provides a proof of the extended universality hypothesis of GH02 which is a refinement of the Millionshchikov hypothesis (quasi-normality of FOM). This refined hypothesis states that CBL turbulence can be considered as a result of a linear interpolation between the quasi-Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations for a wide range of conditions, including even deep-ocean convection and convection in stars (see e.g. [2-5]), several questions remained unexplained. These are now answered by the new solutions including the physical reasons of the universality of the functional form of the HOMs, the explanation of the scatter of the values of the ``universal" coefficients and the source of the magic of the linear interpolation. Moreover, the model presents new alternatives to the Millionshchikov hypothesis. We also discuss the reduction of our general fourth-order closure model to a hierarchy of more simple third- and second-order closure models for practical use in weather prediction and climate models.

Our main conclusion is that the new multivariate Delta-PDF model is able to describe the HOM statistics and closures of dry CBL conditions properly up to FOM. However, the accuracies of the approximation decrease with increasing order of the moment, so that the shape of the PDF (related with sub-plume and inter-plume fluctuations) becomes an important characteristic. Finally, the
insight gained from this simple model could provide a basis for equivalent studies in more advanced models of CBL turbulence.

References:
[1] Gryanik V.M. and J. Hartmann, 2002: J. Atmos. Sci., 59, 2729. Gryanik, V.M., J. Hartmann, S. Raasch and M. Schröter, 2005: J. Atmos. Sci., 62, 2632.
[2] Losch M., 2004: Geophys. Res. Lett., 31, L23301, doi:10.1029/2004GL021412.
[3] Kupka F. and F. Robinson 2007: Mon. Not. Roy. Astron. Soc. 374, 305, 79.
[4] Lenschow, D.H., M. Lothon, S.D. Mayor, P.P. Sullivan and G. Canut, 2011: Boundary-Layer Meteorol. 143, 107.
[5] Waggy S., A. Hsieh and S. Biringen, 2016: Geophys. Astrophys. Fluid Dyn. doi: 10.1080/03091929.2016.1196202.