STA TURBULENCE
Wavelets and Turbulence
Papers
Katul, G. and Vidakovic, B. (1996). IDENTIFICATION OF LOW-DIMENSIONAL ENERGY CONTAINING/FLUX TRANSPORTING EDDY MOTION IN THE ATMOSPHERIC SURFACE LAYER USING WAVELET THRESHOLDING METHODS
Abstract: The partitioning of turbulent perturbations into
a ``low-dimensional" part responsible for much of the turbulent energy and fluxes
and a ``high-dimensional" passive part that contributes little to turbulent energy
and transport dynamics is investigated using atmospheric surface layer (ASL)
measurements.
It is shown that such a partitioning scheme can be
achieved by transforming the ASL measurements into a domain that
concentrates the ``low-dimensional" part into few coefficients and thus
permits a global threshold of the remaining coefficients. In this
transformation-thresholding approach, Fourier rank reduction (FRR), and
orthonormal wavelet and wavelet packet methods are considered.
The efficiencies of these three thresholding methods to extract the
events responsible for much of the heat and momentum turbulent fluxes are
compared for a wide range of atmospheric stability conditions. The
inter-comparisons are performed in four ways: (i) compression ratios,
(ii) energy conservation, (iii) turbulent flux conservation, and (iv)
fine-scale filtering via departures from Kolmogorov's {\bf K41} power-laws.
For orthonormal wavelet and wavelet packets analysis, wavelet functions
with varying time-frequency localization properties are also considered.
Our study showed that wavelet and wavelet packet Lorentz thresholding can
also achieve high compression ratios (98\%) with minimal
loss in energy (3\% loss) and fluxes (4\%). However, these compression
ratios and energy and flux conservation measures are comparable to the
linear Fourier rank reduction method if a Lorentz threshold function is
applied to the later. Finally, it is demonstrated that orthonormal wavelet
and wavelet packets thresholding are insensitive to the analyzing wavelet.
Katul, G. and Vidakovic, B. (1995). The partitioning of attached and detached eddy motion in the atmospheric surface layer using Lorentz wavelets; in Press, Boundary Layer Meteorology.
Abstract: Townsend's attached eddy hypothesis states that the
turbulent structure in the constant stress layer can be decomposed
into attached and detached eddy motion. This paper
proposes and tests a methodology for
separating the attached and detached eddy motion from time series
measurements of velocity and temperature.
The proposed methodology is based on the time-frequency
localization and filtering capabilities of the orthonormal wavelet
transforms. Using a relative entropy statistical measure, the
optimal wavelet basis is identified first. The turbulence time
series measurements are then transformed into the wavelet domain
where the contribution of specific events in the time-frequency
domain are identified. The filtering scheme utilizes a recently
constructed Lorentz thresholding methodology that successfully
eliminated all wavelet coefficients associated with the detached
eddy motion. While this filtering scheme lacks the compression
efficiency of the classical Donoho and Johnstone's universal
thresholding model, it conserves the higher-order statistics and
important turbulence interactions related to the Reynolds stresses.
Following the filtering scheme, the attached eddy motion time
series is re-constructed by an inverse wavelet transform of the
non-zero wavelet coefficients. The proposed partitioning
methodology for attached and detached eddy motion is tested using
$56 Hz$ triaxial sonic anemometer velocity and temperature
measurements above a uniform dry lake bed in Owens valley,
California, for a wide range of atmospheric stability conditions.
Validation that the wavelet filtered time series represented the
attached eddy motion is also discussed in the context of
conservation of turbulence energy and surface fluxes.
Katul, G., Parlange, M., and Chu, C. R. (1995).
Analysis of land surface heat fluxes using the orthonormal wavelet
approach. Water Resources Research, Vol. 31, 2743-2749.
Abstract: Heat fluxes under unstable atmospheric conditions are measured
and analyzed using orthonormal wavelet expansions. Both wavelet and Fourier
power spectra display a -1 power law that is derived from dimensional
arguments for latent and sensible heat flux. The wavelet expansion is
used to investigate the spatial structure of the heat fluxes for scales that
exhibit a - 1 power law. Dimensionless statistical measures, which provide
a space-scale representation of the flux power spectrum, are developed and
applied to the sensible and latent heat flux measurements. Deviations from
Gaussian statistics were observed over many scales that are comparable to
the flux integral length scale. It was found that the extreme flux events
(positive and negative) in the heat flux signals contribute directly to the
energy and spatial structure of the -1 power law. We also used the wavelet
transform to identify turbulent scales directly responsible for the large
tails observed in the horizontal gradient probability de nsity function of
both heat fluxes.
Katul, G., Finkelstein, P., Clarke, J., and Ellestad, T. (1995).
An Investigation of the conditional sampling method used to estimate
fluxes of active, reactive, and passive scalars.
To appear in: Journal of Applied Meteorology, 1996.
Abstract: The conditional sampling flux measurement technique was
evaluated for four scalars (temperature, water vapor, ozone, and
carbon dioxide) by comparison with direct eddy correlation measurements
at two sites. The empirical constant \beta relating the turbulent flux
to the accumulated concentration difference between updrafts and downdrafts
was computed from 10 Hz turbulence measurements. Comparison between the
simulated relaxed eddy accumulation flux formulation and the eddy
correlation measurements allowed the direct determination of \beta for all
four scalars. The \beta models previously proposed overpredicted the
measured \beta by about 8-10%. It was found that a mean \beta=0.58
reproduced the eddy correlation measurements dependent of the scalar
type being analyzed, roughness and atmospheric stability conditions,
in agreement with previous studies. The role of energy-containing eddy
motion in the deviations between the measured and predicted \beta is
considered using orthonormal wavelet expansion in conjunction with a
wavelet shrinkage approach. It was demonstrated that the energy-containing
large eddy motion contributed to a reduction in \beta when compared to the
predicted \beta. Finally, the deadband vertical velocity effects were also
considered and found to reduce \beta exponentially, in agreement with
other studies.
Szilagyi, J., G.G. Katul, M.B. Parlange, J.D. Albertson, and A.T. Cahill,
(1996). The local effect of intermittency of the inertial subrange energy
of the atmospheric surface layer,
Boundary Layer Meteorology, 1996.
Abstract: Orthonormal wavelet expansions are applied to atmospheric surface
layer velocity measurements. The effect of intermittent events on the
energy spectrum of the inertial subrange is investigated through the
analysis of the wavelet coefficients. The local nature of the orthonormal
wavelet transform in physical space makes it possible to identify a
relationship between the inertial subrange slope of local wavelet spectrum
and simpler indicator (e.g. local variance of the signal) of local
intermittency buildup. The slope of the local wavelet energy spectrum
in the inertial subrange is shown to be sensitive to the presence of
intermittent events. During well developed intermittent events (coherent
structures), the slope of the energy spectrum is somewhat steeper than -5/3,
while in less active regions, the slope is found to be flatter than - 5/3.
When the slopes of local wavelet spectra are ensemble averaged, a slope
of - 5/3 is recovered in the inertial subrange.
Katul, G., Albertson, J., Chu, C., and Parlange, M. (1994).
Intermittency in atmospheric surface turbulence:
The orthonormal wavelet approach,
In Wavelets in Geophysics, ed. E. Foufoula, Academic
Press, 81-105.
Abstract: Orthonormal wavelet expansions are applied
to atmospheric surface layer velocity measurements that exhibited
about three decades of inertial subrange energy spectrum. A direct
relation between the nth order structure function and the wavelet
coefficients is derived for intermittency investigations. This
relation is used to analyze power-law deviations from the
classical Kolmogorov theory in the inertial subrange. The local
nature of the orthonormal wavelet transform in physical space aided
the identification of events contributing to inertial subrange
intermittency buildup. By suppressing these events, intermittency
effects on the statistical structure of the inertial subrange is
eliminated. The suppression of intermittency on the nth order
structure function is carried out via a conditional wavelet sampling
scheme. The conditional sampling scheme relies on an indicator function
that identifies the contribution of large dissipation events in the
wavelet space-scale domain. The conditioned wavelet statistics
reproduce the Kolmogorov scaling in the inertial subrange and resulted
in a zero intermittency factor. A relation between Kolmogorov's theory
and Gaussian statistics is also investigated. Intermittency resulted
in non-Gaussian statistics for the inertial subrange scales.
Katul, G., and Parlange M. (1994). On the active role of
temperature in surface layer turbulence, Journal of
Atmospheric Science, 51, 2181-2195.
Abstract: Orthonormal wavelet expansions were derived and applied
to atmospheric surface layer turbulence measurements of temperature
and vapor concentration under unstable and stable atmospheric stability
conditions. These expansions were used to investigate both the
statistical and spectral structure of turbulence simultaneously in
space and scale using two tracers: temperature and specific humidity.
It was found that at small wavenumbers, both temperature and specific
humidity Fourier and wavelet spectra exhibit a -1 power law behavior
consistent with other atmospheric boundary layer experiments. The mean
values of the energy spectrum obtained from the wavelet analysis are
in agreement with the classical Fourier counterparts. The wavelet
flatness factors (values up to 10) indicate strong deviation from Gaussian
statistics in space for the temperature fluctuations as the wavenumber
increases. In contrast, the spatial wavelet flatness factor for the
specific humidity exhibits near Gaussian statistics (values up to 4)
for all wavenumbers. The wavelet skewness in space indicates that the
specific humidity attains a near isotropic state with increasing
wavenumber for both stability condi tions. Unlike the specific humidity,
the temperature wavelet skewness in space did not decay with increasing
wavenumber indicating the presence of large eddy anisotropy in space.
Land surface heating/cooling inhomogeneity appears to affect the local
structure of turbulence and therefore, at small scales, temperature
behaves as an active scalar when compared to specific humidity.
The active role of temperature was also analyzed within the framework
of Bolgiano' s spectral theory. Deviations from Bolgiano's theory for
the temperature spectrum were observed at all wavenumbers, with measured
energy power law behaviour of |1.2| which is less than the theoretical
value of |7/5|. Conditional wavelet analysis was developed and used to
investigate the nature of these deviations from Bolgiano's scaling law
for the temperature measurements. It was found that by suppressing
energy-containing and intermittent events, Bolgiano's scaling law for
the temperature spectrum held under stable stability conditions. The
effect of different wavelet basis functions on the statistical and
spectral description of atmospheric turbulence was also considered.
Katul, G., and Parlange M. (1995).
The spatial structure of turbulence at production wavenumbers:
The Orthonormal wavelet,
Boundary Layer Meteorology 75, 81-108.
Abstract: Orthonormal wavelet expansions are applied to surface layer
measurements of vertical wind speed under various atmospheric stability
conditions. The orthonormal wavele t transform allows for the unfolding
of these measurements into space and scale simultaneously to reveal the
large intermittent behavior in space for the turbulent production wavenumbers.
Both Fourier and wavelet power spectra indicated the existence of a -1
power law for the vertical velocity production wavenumbers. The -1 power
law in the turbulent production range was also derived from surface layer
similarity theory. A dimensionless skewness structure function is applied
to the wavelet decomposed vertical velocity field to trace the destruction
of the shear or buoyancy induced asymmetry under various stability
conditions. The structure skewness funct ion revealed shear or buoyancy
induced eddy asymmetry dependence on stability at each scale within the -1
power-law wavenumber range with more isotropy during propagation from smaller
to larger wavenumbers. The asymmetry of these events at the turbulent
production wavenumbers appeared very localized in space as well as in
scale and could be described within the context of a simple idealized
eddy-overturning model. It is demonstrated that the wavelet transform is
suitable for such analysis.
Katul, G., Parlange, M., and Chu, C. (1994).
Intermittency, local isotropy, and non-Gaussian statistics in
atmospheric surface layer turbulence
Physics of Fluids, 6, 2480-2492.
Abstract: Orthonormal wavelet expansions are applied to atmospheric
surface layer velocity and temperature measurements above a uniform
bare soil surface that exhibit a long inertial subrange energy spectrum.
In order to investigate intermittency effects on Kolmogorov's theory,
a direct relation between the nth order structure function and the wavelet
coefficients is derived. This relation is used to examine deviations from
the classical Kolmogorov theory for velocity and temperature in the inertial
subrange. The local nature of the orthonormal wavelet transform in physical
space aided the identification of events directly contributing to
intermittency buildup at inertial subrange scales. These events occur at
edges of large eddies and contaminate the Kolmogorov inertial subrange
scaling. By suppressing these events, the statistical structure of
the inertial subrange for the velocity and temperature, as described by
Kolmogorov's theory, is recovered. The suppression of intermittency on
the nth order structure function is carried out via a conditional wavelet
sampling scheme. The conditioned wavelet statistics reproduced the
Kolmogorov scaling (up to n=6) in the inertial subrange and result in a zero
intermittency factor. The conditional wavelet statistics for the mixed
velocity temperature structure functions is also presented. It was found
that the conditional wavelet statistics for these mixed moments result in
thermal intermittency parameter consistent with other laboratory and field
measurements. The relationship between Kolmogorov's theory and near Gaussian
statistics for velocity and temperature gradients is also considered.
NSF Grant Proposal Abstract
Abstract of NSF 95 DMS-626159 proposal: Vidakovic and Katul.