Analysis and Modelling of Small-Scale Turbulence

The analysis and modelling of small-scale turbulence in the atmosphere play a significant role in improving our understanding of cloud processes, thereby contributing to the development of better parameterization of climate models. Advancement in our understanding of turbulence can be fueled from a more in-depth study of small-scale turbulence, which is the subject of this thesis. Within this thesis, small scales are understood as turbulent structures affected by viscosity as well as scales from the highwavenumber part of the inertial range which are of O(0.1m−1m) typically neglected in numerical simulations of atmospheric turbulence. This work is divided into two parts. In the first part, various approaches to estimate the turbulence kinetic energy (TKE) dissipation rate , from one-dimensional (1D) intersections that resemble experimental series, are tested using direct numerical simulation (DNS) of the stratocumulus cloudtop mixing layer and free convective boundary layer. Results of these estimates are compared with “true” DNS values of in buoyant and inhomogeneous atmospheric flows. This research focuses on recently proposed methods of the TKE dissipation-rate retrievals based on signal’s zero crossings and on recovering the missing part of the spectrum. The methods are tested on fully resolved turbulence fields and compared to standard retrievals from power spectra and structure functions. Anisotropy of turbulence due to buoyancy is shown to influence retrievals based on the vertical velocity component. TKE dissipation-rate estimates from the number of crossings correspond well to spectral estimates. As far as the recovery of the missing part of the spectrum is concerned, different models for the dissipation spectra was investigated, and the best one is chosen for further study. Results were improved when the Taylors’ microscale was used in the iterative method, instead of the Liepmann scale based on the number of signal’s zero crossings. This also allowed for the characterization of external intermittency by the Taylor-to-Liepmann scale ratio. It was shown that the new methods of TKE dissipation-rate retrieval from 1D series provide a valuable complement to standard approaches. The second part of this study addresses the reconstruction of sub-grid scales in large eddy simulation (LES) of turbulent flows in stratocumulus cloud-top. The approach is based on the fractality assumption of the turbulent velocity field. The fractal model reconstructs sub-grid velocity fields from known filtered values on LES grid, using fractal interpolation, proposed by Scotti and Meneveau [Physica D 127, 198–232 1999]. The characteristics of the reconstructed signal depend on the stretching parameter d, which is related to the fractal dimension of the signal. In many previous studies, the stretching parameter values were assumed to be constant in space and time. To improve the fractal interpolation approach, the stretching parameter variability is accounted for. The local stretching parameter is calculated from DNS data with an algorithm proposed by Mazel and Hayes [IEEE Trans. Signal Process 40(7), 1724–1734, 1992], and its probability density function (PDF) is determined. It is found that the PDFs of d have a universal form when the velocity field is filtered to wave-numbers within the inertial range. The inertial-range PDFs of d in DNS and LES of stratocumulus cloud-top and experimental airborne data from physics of stratocumulus top (POST) research campaign were compared in order to investigate its Reynolds number (Re) dependence. Next, fractal reconstruction of the subgrid velocity is performed and energy spectra and statistics of velocity increments are compared with DNS data. It is assumed that the stretching parameter d is a random variable with the prescribed PDF. Moreover, the autocorrelation of d in time is examined. It was discovered that d decorrelates with the characteristic timescale of the order of the Kolmogorov’s time scale and hence can be chosen randomly after each time step in LES. This follows from the fact that the time steps used in LES are typically considerably larger than Kolmogorov’s timescale. The implemented fractal model gives good agreement with the DNS and physics of stratocumulus cloud (POST) airborne data in terms of their spectra and PDFs of velocity increments. The error in mass conservation is smaller compared to the use of constant values of d. In conclusion, possible applications of the fractal model were addressed. A priori LES test shows that the fractal model can reconstruct the resolved stresses and residual kinetic energy. Also, based on the preliminary test, the fractal model can improve LES velocity fields used in the Lagrangian tracking of droplets for the simulation of cloud microphysics. Both parts of the thesis are based on the assumptions of scale self-similarity of Kolmogorov and local isotropy, which may not be satisfied in real atmospheric conditions. Since the standard methods for TKE dissipation rate retrieval are derived from these assumptions, the level of discrepancy is investigated by comparing the actual value of from DNS with estimates from these methods. Also, in the case of the modelling of small (subgrid) scales, the improved fractal model relies on scale-similarity. Range of scales, in which this assumption is sufficiently satisfied (i.e. inertial range scales) is reconstructed. Statistical tools from the Kolmogorov’s similarity hypotheses are used to assess the performance of the improved fractal model.