Abstract:
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.