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Chapter 5
−5/3 Kolmogorov Turbulent Behaviour and
Intermittent Sustainable Energies
Rudy Calif, François G. Schmitt and
O. Durán Medina
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/106341
Provisional chapter
−5/3 Kolmogorov Turbulent Behaviour
and Intermittent Sustainable Energies
Rudy Calif, François G. Schmitt and
O. Durán Medina
Additional information is available at the end of the chapter
Abstract
The massive integration of sustainable energies into electrical grids (non-interconnected
or connected) is a major problem due to their stochastic character revealed by strong
fluctuations at all scales. In this paper, the scaling behaviour or power law correlations
and the nature of scaling behaviour of sustainable resource data such as flow velocity,
atmospheric wind speed, solar global solar radiation and sustainable energy such as,
wind power output, are highlighted. For the first time, Fourier power spectral densities
are estimated for each dataset. We show that the power spectrum densities obtained are
close to the 5/3 Kolmogorov spectrum. Furthermore, the multifractal and intermittent
properties of sustainable resource and energy data have been revealed by the concavity
of the scaling exponent function. The proposed analysis frame allows a full description
of fluctuations of processes considered. A good knowledge of the dynamic of fluctua-
tions is crucial to management of the integration of sustainable energies into a grid.
Keywords: turbulence, kolmogorov spectrum, intermittency, multifractality
1. Introduction
The installed capacity for energy from solar farms, wind farms and marine energy systems is
constantly increasing in response to worldwide interest in low-emissions power sources and a
desire to decrease the dependence on petroleum. The variability and unpredictability of this
kind of resources over short time scales remains a major problem, as its penetration of this
energy into the electric grid is limited. Hence, a good knowledge of renewable resource
© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and eproduction in any medium, provided the original work is properly cited.
© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
variations and intermittency is of real practical importance in managing the electrical network
integrating this kind of energy.
Figure 1 illustrates examples of temporal increments of atmospheric wind speed and global
solar radiation for a time scales Δv¼5 min. We can observe the existence of intermittent
bursts. Following, the disciplinary field, the concept of intermittency can be defined differ-
ently [1, 2]. In the wind and solar energies fields, the concept of intermittency is often defined
as the variability [2]. In turbulence field, Batchelor and Townsend have observed the inter-
mittency experimentally for the first time in 1949 [3] and formalized in the multifractal
framework after the seminal works of Kolmogorov [4]. The meaning of intermittency can
change according to the authors. Frisch defines an intermittent signal if “it displays activity
during only a fraction time, which decreases with the scale under consideration”. According
to Pope, a motion “sometimes turbulent and sometimes non-turbulent”characterizes an
intermittent flow. In the engineering field, the intermittency is considered as a transition
between a laminar and turbulent flows [1].
Here, the concept of intermittency in the fully developed turbulence framework is used, with
with the help of multifractal analysis. This allows a better description of a stochastic signal at
all scales and all intensities.
Multifractal analysis techniques have encountered an amount success through several disci-
plinary fields, such as, for instance, turbulence [5–8], finance [9–11], physiology [12], rainfall
[13, 14] and geophysics [15, 16].
In this chapter, the intermittent properties of renewable resources data (wind speed, solar
radiation and flow velocity data) and sustainable energy data (power output data from WECS
and marine energy systems) are investigated using a classical multifractal analysis method,
structure functions analysis.
The structure of this chapter is as follows. Section 2 describes briefly the fully developed
turbulence framework. Section 3 presents the results analysis.
Figure 1. Examples of temporal increments of atmospheric wind speed Δvand the global solar radiation Δg. These
sequences show intermittent bursts.
Sustainable Energy - Technological Issues, Applications and Case Studies96
2. Fully developed turbulence framework
2.1. Richardson’s cascade and Kolmogorov theory
The intuitive scheme of Richardson has largely inspired numerous authors in the turbulence
field. Richardson provided a poetic form of energetic cascade [17] this is represented by a
schematic illustration of Kolmogorov-Obhukov given in (Figure 2):
“Big whirls have little whirls that feed on their velocity,
And little whirls have lesser whirls
And so on to viscosity in the molecular sense”
The mathematical formalization of this scheme is given in 1940s by Kolmogorov who postu-
lated the local-similarity hypothesis, i.e. small-scale turbulence is homogeneous and statisti-
cally isotropic in the inertial sub-range and hypothesized that velocity fluctuations Δvbetween
two points separated by a distance rdepend only on the average dissipation rate ε. This trans-
lates into the following expression for the squared fluctuations S2ðrÞ¼ΔvrÞ2¼ðvxþr−vxÞ2[4]:
S2ðrÞ≈ε2=
3r2=
3(1)
This has been generalized, considering the structure functions for moments of order q>0of
the absolute spatial velocity increments as follows [18]:
SqðrÞ≈εq=
3rq=
3(2)
This leads to the famous K41 linear law (when there is no intermittency):
ζðqÞ¼q
3(3)
where ζðqÞis the scaling exponent of the structure functions:
Figure 2. A schematic illustration of Kolmogorov-Obhukov spectrum that a −5/3 slope, based on Richardson’s cascade
concepts.
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
http://dx.doi.org/10.5772/106341
97
SqðrÞ≈rζðqÞ(4)
This leads to the following expression for the power spectrum of velocity fluctuations in the
Fourier space:
EðkÞ≈k−5=
3(5)
where kis the wave number.
In 1949, the experimental works of Batchelor and Townsend [3] highlighted the nonlinearity of
the scaling exponent ζðqÞcontrary to the K41 prediction. This nonlinearity indicates the
intermittent character of the dissipation energy, caused by the inhomogeneity and anisotropy
of the turbulent flow. To take intermittency into account, many theoretical formulations have
been provided for a quantitative description of cascade processes and fitting the scaling
exponent function ζðqÞ:The log-normal model was the first prediction describing the intermit-
tency of the fully turbulence [18]:
ζðqÞ¼q
3þμ
18 ð3q−q2Þ(6)
where μis the intermittency parameter. Thereafter, others models have been proposed. The
most used are given in Section 2.3.
2.2. A description of scale invariance and multifractal framework
2.2.1. Self-similarity and scale invariance
The idea of describing natural phenomena by the study of statistical scaling laws is not recent
[19]. Self-similarity has been widely observed in nature: self-similarity concept being the simplest
form of scale invariance. A process xðtÞis self-similar if these statistical properties remains
unchanged with the process aHxðt=aÞobtained by simultaneously dilating the time axis by a
factor a>0, and the amplitude axis by a factor a−H.His called the self-similarity or Hurst
parameter. This parameter provides information on the variability degree of process. A primitive
model of self-similar signals is the fractional Brownian motion (fBm) BHðtÞ[20] for illustration,
(Figure 3) shows a protion of flow velocity u dilated in the box, exibithing the statistical self-
similarity features of flow velocity signal considered in this study.
The Fourier spectral density EðfÞof scale invariance or self-similar processes follows a power
law obtained over a range of frequency f:
EðfÞ∼f−β(7)
where βis the spectral exponent. According to some authors [19, 21, 22], it defines the degree
of stationary of the signal:
•β<1, the process is stationary
•β>1, the process is no stationary
•1<β<3, the process is no stationary with increments stationary.
Sustainable Energy - Technological Issues, Applications and Case Studies98
2.2.2. Multifractal framework
The mathematical multifractal framework was appeared with the cascade multiplicative emer-
gence in order to consider the intermittency of the energy dissipation in Turbulence.
Multiscaling concept allows the statistical description of stochastic signals for the modelling of
physical systems, using multifractal technique analyses.
If xðtÞis a stochastic signal function of time, his scaling behaviour is highlighted when the time
absolute time increments jΔxj¼jxðtþτÞ−xðtÞj, more precisely, the structure functions of order
qrespect the following relationship [5]:
SqðτÞ¼ðjΔxjÞ≈τζðqÞ(8)
where τis a time lag and ζis the scaling exponent function. The full ðq,ζðqÞÞ curve for integer
and non-integer qmoments provides a full characterization of signal considered at all scales
and at all intensities. The parameter ζð2Þ¼β−1 relates the second order moment to the β
Fourier power spectrum scaling exponent. The parameter H¼ζð1Þis the Hurst exponent with
0<H<1. This parameter defines the degree of roughness or smoothness of a measured
signal: more H is, the more the signal is smooth. The values of the ζðqÞfunction are estimated
from the slope of the SqðτÞversus τin a log-log representation for all moments q. Concerning
the scaling behaviour, the scaling exponent function is useful to characterize the statistics of a
stochastic process. For a linear scaling function of the form qH, the signal is said to be
monofractal; Brownian motion is described by H¼1=2, fractional Brownian motion is
described by 0 <H<1, and homogeneous non-intermittent turbulence is described by
H¼1=3. While for a nonlinear scaling exponent function, the signal is said to be multifractal.
Figure 4 illustrates the scaling behaviour of the ζðqÞfunction for instance a monofractal and
multifractal processes. Furthermore, the concavity of ζðqÞfunction gives an indication on the
intermittency degree of process considered: the more concave the curve is, the more intermit-
tent the process [5, 22].
Figure 3. A portion of flow velocity u dilated in the box. This shows the statistical self-similarity features of flow velocity u.
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
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2.3. Some multifractal models
Several models have been proposed to fit the scaling exponent function ζðqÞsince in the
literature, for instance, the “black and white”model [23], the log-normal model [18] and the
log-stable model [22].
The “black and white”model proposed by Frisch et al. in 1978 is the simplest model [23]:
ζðqÞ¼qH −μðq−1Þ(9)
where H is the Hurst exponent and μthe intermittency parameter.
The classical lognormal model of the form:
ζðqÞ¼qH −μ
2ðq2−qÞ(10)
The log-stable or log-Lévy model proposed by Schertzer and Lovejoy in 1987 [22]:
ζðqÞ¼qH −C1
ðα−1Þðqα−qÞ(11)
where H is the Hurst exponent. The parameter C1is the fractal co-dimension measuring the
mean intermittency: the larger C1, the more the signal is intermittent. Furthermore, 0 <C1<d
with d the dimension space (here d¼1). The multifractal Lévy parameter 0 <α<2 inquires
on the degree of multifractality i.e., how fast the inhomogeneity increases with the order of the
Figure 4. Examples of scaling exponent functions ζðqÞfor a monofractal and a multifractal processes. The scaling
exponent functions represented are linear and nonlinear (concave), respectively, for monofractal and multifractal pro-
cesses.
Sustainable Energy - Technological Issues, Applications and Case Studies100
moments. Furthermore, α¼0 corresponds to the monofractal case and α¼2 corresponds to
the multifractal log-normal case.
In this chapter, we consider the log-normal model that provides a reasonable fit for the scaling
exponent of data considered. In [24], the log-stable is considered for the global solar radiation
data.
3. Results
In this chapter, we present analysis results from multiple time series sampled at different
sampling rates and at different places. The atmospheric wind speed uwas measured with a
sampling frequency of 20 Hz during 40 h, on the wind energy site production of Petit-Canal in
Guadeloupe an island located at 16°15’N latitude and 60°30’W longitude. The wind power
output Pwas measured at the same place, with a sampling frequency of 1 Hz over a one-year
period. A 10 MW wind farm delivers this wind power output. The global solar radiation
measurements Gwas collected with a sampling frequency of 1 Hz over a one-year period, at
the University site of Pointe-à-Pitre in Guadeloupe. The flow velocity measurements were
generated from the facilities of the wave and current flume tank of IFREMER (French Research
Institute for Exploitation of the Sea) in Boulogne-sur-mer (North of France). The data are
collected with a sampling frequency of 100 Hz. Figure 5 illustrates extract of signals consid-
ered. All the signals fluctuate over a large range scales showing the intermittent nature of
sustainable resources and energy considered in this study.
Figure 5. Examples of extract of signal considered: (a) flow velocity u, (b) atmospheric wind speed, (c) global solar
radiation G, (d) normalized wind power output delivered by a wind farm. All the signals display strong fluctuations at
all scales.
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
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3.1. Fourier analysis of sustainable energy data and −5=3 Kolmogorov spectrum
The Fourier power spectral density separates and measures the amount of variability occurring
in different frequency bands. In this section, the Fourier power spectral densities are estimated
for our database in order to detect scale invariance. For a scale invariant signal, the following
scaling power law is obtained over a range of frequency f:
EðfÞ≈f−β(12)
where βis the exponent spectral.
Figure 6 shows the Fourier power spectral densities of databases described above, compared
to the −5=3 Kolmogorov spectrum (red straight line), log-log representation. The spectra
computed follow a power law of the f−βwith βclose to 5=3. As expected, the atmospheric
wind and the flow velocity spectra demonstrate a scaling behaviour for the respective frequen-
cies from about f¼0:1−10 Hz and f¼0:1−50 Hz with β¼1:67 close to the 5/3 Kolmogorov
value [4, 25]. This is consistent with the values obtained for the inertial range in previous
studies [26–28]. The wind power output spectrum displays a power law with β¼1:68 close
to the 5/3 Kolmogorov value, for frequencies from about f¼10−4to 0:5 Hz. In 2007, Apt has
shown that the wind power output from a wind turbine, follows a Kolmogorov spectrum over
more than four orders of magnitude in frequency [29]. In [30], we show the wind power output
spectrum with an exponent spectral close to the 5/3 value, which is observed for particular
conditions.
The global solar radiation spectrum shows also a power law behaviour with β¼1:66 close to
the 5/3 Kolmogorov value for frequencies from about f¼0:7·10−4to 0:07 Hz. This scale
invariance is indirectly linked to scale invariance of cloud field transported by atmospheric
turbulence. In [31], a power law is also observed for the spectrum of cloud radiances obtained
Figure 6. The Fourier power spectral densities for each dataset, compared with the −5/3 Kolmogorov spectrum.
Sustainable Energy - Technological Issues, Applications and Case Studies102
from ground-based photography: the exponent spectral β¼1:67 is observed for clouds over
ocean.
In summary, the spectra of sustainable data considered in this study, display power law
behaviour with an exponent spectral close to the 5/3 Kolmogorov value. The slight difference
with the exact 5/3 value is usually caused by intermittency effects [5, 22].
Furthermore, the Fourier power spectrum is a second order statistic providing information on
medium level fluctuations, and consequently, its slope is not sufficient to fully describe a
scaling process. Multifractal analysis is a natural generalization to fully study the scaling
behaviour of a nonlinear phenomenon using, for example, the qth order structure functions.
3.2. Multifractal analysis of sustainable energy data
In order to qualify the nature of scaling behaviour (monofractal or multifractal), a multifractal
analysis using qth order structure functions is applied to sustainable energy data to determine
the scaling exponents ζðqÞ. For each dataset, the structure functions are computed on the
temporal increments Δxas defined above. The details concerning the scale range of τand q
are given in the following references [24, 32–34]. As shown in [24, 32–34], the straight lines of
structure functions indicate that the scaling of the relationship is well respected. Consequently,
the scaling exponents ζðqÞare extracted from the slopes of the straight lines using a linear
regression. Figure 7 represents the scaling exponents ζðqÞcorresponding to each dataset
compared with a model proposed by Kolmogorov, the linear model K41, ζðqÞ¼q=3. We can
see that the scaling exponents ζðqÞobtained are nonlinear and concave. This highlights the
Figure 7. The scaling exponent functions ζðqÞfor each dataset compared with the linear non-intermittent K41 model.
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
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103
multifractal and intermittent character of considered sustainable data here. Furthermore, the
degree of concavity gives an indication on the degree of intermittency: the more concave the
scaling exponent curve is, the more intermittent the process. We recall that the intermittency
parameter can be estimated by μ¼2ζð1Þ−ζð2Þwith 0 <μ<1. Table 1 draws up some param-
eters for each dataset: H the Hurst exponent, ζð2Þ, and the intermittency parameter μ.
As shown in Figure 6 and indicated in Table 1, the global solar radiation G is the most
intermittent.
4. Conclusion
This work highlights the intermittency and the scale invariance properties of flow velocity u,
atmospheric wind speed v, wind power output P and global solar radiation G data, at all
intensities and at all scales, in the fully developed turbulence framework.
We have shown for all datasets over the period encountered:
•The presence of a scaling regime or power law correlation of the form f−βover a broad
range of time scales, in the Fourier space. The exponent spectral βis close to the exact 5/3
Kolmogorov value for all the datasets.
•The nature of the scaling behaviour for each dataset is determined using qth order struc-
ture functions analysis. The nonlinearity and the concavity of the scaling exponent func-
tions ζðqÞobtained reveal the intermittent and the multifractal properties of datasets
considered in this manuscript. This could result from the complex interaction of the turbu-
lent atmospheric and the energy converter systems such as, for example, wind turbine.
With the increase in sustainable energies, a good knowledge of their nonstationary and
intermittent properties is crucial. The fully developed turbulence framework is a relevant
frame to analysis stochastic processes such as those considered in this manuscript. It allows
providing a sharp description of fluctuations of processes at all scales and at intensities. The
Hurst and the intermittency parameters can be used in stochastic simulations based on
multifractal cascade model, as performed in [33]. Here, with a dynamical modelling of
fluctuations sustainable energy considered, the interest could be, for instance, to test the
stability evaluation of electricity grid.
Hζð2Þμ
Flow velocity u 0.34 0.66 0.02
Atmospheric wind speed v 0.35 0.68 0.02
Wind power output P 0.38 0.70 0.06
Global solar radiation G 0.43 0.61 0.25
Table 1. Hurst exponent H¼ζð1Þ,ζð2Þlinked to the exponent spectral by ζð2Þ¼β−1 and the intermittency parameter μ
estimated for each dataset.
Sustainable Energy - Technological Issues, Applications and Case Studies104
Author details
Rudy Calif
1
*, François G. Schmitt
2
and O. Durán Medina
2
*Address all correspondence to: rcalif@univ-ag.fr
1 Laboratory in Geosciences and Energies, University of Antilles, France
2 Laboratory in Geosciences and Oceanology, CNRS & University of Lille, Wimereux, France
References
[1] Seuront L and Schmitt FG: Intermittency. In Baumert H, Simpson J and Sündermann J
editors. Marine Turbulences: Theories, Observations and Models. Cambridge, Royaume-
Uni Press; 2005. pp. 66–78
[2] Tarroja B, Mueller F and Samuelsen S: Solar power variability and spatial diversification:
implications from an electric grid load balancing perspective. International Journal of
Energy Research. 2013; 37: 1002–1016. doi:10.1002/er.2903
[3] Batchelor GK and Townsend AA: The nature of turbulent motion at large wave-numbers.
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering
Sciences. 1949; 199: 238–255. doi:10.1098/rspa.1949.0136
[4] Kolmogorov AN: The local structure of turbulence in incompressible viscous fluid for
very large Reynolds numbers. Doklady Akademii Nauk SSSR. 1941; 30: 301–305
[5] Frisch U: Turbulence: the legacy of AN Kolmogorov. Cambridge, Royaume-Uni Press;
1995. 312 p. ISBN 978–0521457132
[6] Sreenivasan KR and Antonia RA: The phenomenology of small-scale turbulence. Annual
Review of Fluid Mechanics. 1997; 29: 435–472. doi:10.1146/annurev.fluid.29.1.435
[7] Bottcher F, Barth S, Peinke J: Small and large scale fluctuations in atmospheric wind
speeds, Stochastic Environmental Research Risk, 2007; 21: 299–308. doi:10.1007/s00477-
006-0065-2
[8] Lohse D, Xia KQ: Small-scale properties of turbulent Rayleigh-Bénard convection.
Annual Review of Fluid Mechanics 2010; 42: 335–364. doi:10.1146/annurev.fluid.010908.
165152
[9] Ghashghaie S, Breymann W, Peinke J, Talkner P, Dodge Y: Turbulent cascades in foreign
exchange markets. Nature. 1996; 381: 767–770. doi:10.1038/381767a0
[10] Schmitt FG, Schertzer D, Lovejoy S: Multifractal analysis of foreign exchange data.
Applied Stochastic Models and Data Analysis. 1999; 15:29–53. doi:10.1002/(SICI)1099-
0747(199903)15:1<29::AID-ASM357>3.0.CO;2-Z
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
http://dx.doi.org/10.5772/106341
105
[11] Calvet L, Fisher A: Multifractality in asset returns: theory and evidence. Review Econom-
ics and Statistics. 2002; 84: 381–406. doi:10.1162/003465302320259420
[12] Ivanov P, Bunde A, Amaral L, Havlin S, Fritsch-Yelle L, Baevsky R, Stanley H, Goldberger
A: Sleep-wake differences in scaling behavior of the human heartbeat: Analysis of terres-
trial and long-term space flight data. Europhysics Letters. 1999; 48: 594–600
[13] Schertzer D and Lovejoy S: Physical modeling and analysis of rain and clouds by aniso-
tropic scaling multiplicative processes. Journal of Geophysical Research. 1987; 92: 9693–
9714. doi:10.1029/JD092iD08p09693
[14] Venugopal V, Roux SG, Foufoula-Georgiou E, Arnéodo A: Scaling behavior of high
resolution temporal rainfall: new insights from a wavelet-based cumulant analysis. Phys-
ics Letters A. 2006; 348: 335–345. doi:10.1016/j.physleta.2005.08.064
[15] Kantelhardt J, Koscielny-Bunde E, Rybski D, Braun P, Bunde A, Havlin S: Long-term
persistence and multifractality of precipitation and river runoff. Journal of Geophysical
Research. 2006; 111:1–13. doi:10.1029/2005JD005881
[16] Mauas PJD, Flamenco E, Buccino AP: Solar forcing of the stream flow of a continental
scale South American river. Physical Review Letters. 2008; 101: 168501. doi:10.1103/
PhysRevLett.101.168501
[17] Richardson LF: Weather Prediction by Numerical Processes, 2nd ed. Cambridge,
Royaume-Uni Press; 1922. 262 p. ISBN 978-0-521-68044-8
[18] Yaglom AM: The influence of fluctuations in energy dissipation on the shape of turbu-
lence characteristics in the inertial interval. Soviet Physics Doklady. 1966; 11:26p.
[19] Mandelbrot B: The Fractal Geometry of Nature. Macmillan; London, Royaume-Uni 1982.
[20] Manbelbrot BB and Van Ness JW: Fractional Brownian motions, fractional noises and
applications. SIAM Review. 1968; 10(4): 422–437. doi:10.1137/1010093
[21] Marshak A, Davis A, Cahalan R and Wiscombe W: Bounded cascade models as
nonstationary multifractals. Physical Review E. 1994; 49:55–69. doi:0.1103/PhysRevE.49.
55
[22] Schertzer D, Lovejoy S, Schmitt F, Chigirinskaya Y and Marsan D: Multifractal cascade
dynamics and turbulent intermittency. Fractals. 1997; 5(03): 427–471. doi:10.1142/
S0218348X97000371
[23] Frisch U, Sulem PL, Nelkin M: A simple dynamical model of intermittent fully developed
turbulence. Journal of Fluid Mechanics. 1968; 87: 719–736. doi:10.1017/S002211207800
1846
[24] Calif R, Schmit FG, Huang Y and Soubdhan T: Intermittency study of high frequency
global solar radiation sequences under a tropical climate. Solar Energy. 2013; 98: 349–365.
doi:10.1016/j.solener.2013.09.018
Sustainable Energy - Technological Issues, Applications and Case Studies106
[25] Obhukov AM: Energy distribution in the spectrum of turbulent flow. Izvestiya Akademii
Nauk SSSR, Seriya Geologicheskaya. 1941; 5: 453–466.
[26] Schmitt F, Schertzer D, Lovejoy S and Brunet Y: Estimation of universal multifractal
indices for atmospheric turbulent velocity fields. Fractals. 1993; 1: 568–575. doi:10.1142/
S0218348X93000599
[27] Katul G and Chu CR: A theoretical and experimental investigation of energy-containing
scales in the dynamic sublayer of boundary-layer flows. Boundary-Layer Meteorology,
1998; 86: 279–312. doi:10.1023/A:1000657014845
[28] Lauren MK, Menabde M, Seed AW and Austin GL: Characterisation and simulation of
the multiscaling properties of the energy-containing scales of horizontal surface-layer
winds. Boundary-Layer Meteorology. 1999; 90:21–46. doi:10.1023/A:1001749126625
[29] Apt J: The spectrum of power from wind turbines. Journal of Power Sources. 2007; 169:
369–374. doi:10.1016/j.jpowsour.2007.02.77
[30] Calif R, Schmitt FG, Huang Y and Medina O: Hurst exponent, multifractal spectrum and
extreme events versus the installed capacity of wind farms, submitted.
[31] Sachs D, Lovejoy S and Schertzer D: The multifractal scaling of cloud radiances from 1m
to 1km. Fractals. 2005; 10: 253–264. doi:10.1016/j.jhydrol.2005.02.042
[32] Calif R and Schmitt FG: Modeling of atmospheric wind speed sequence using a lognor-
mal continuous stochastic equation. Journal of Wind Engineering and Industrial Aerody-
namics. 2012; 109:1–8. doi:10.1016/j.jweia.2012.06.002
[33] Calif R, Schmitt FG and Huang Y: Multifractal description of wind power fluctuations
using arbitrary order Hilbert spectral analysis. Physica A: Statistical Mechanics and Its
Applications, 2013; 392: 4106–4120. doi:10.1016/j.phisa.2013.04.038
[34] Medina OD, Schmit FG and Calif R: Multiscale Analysis of Wind Velocity, Power Output
and Rotation of a Windmill. Energy Procedia. 2015; 76: 193–199. doi:10.1016/j.egypro.
2015.07.897
−5/3 Kolmogorov Turbulent Behaviour and Intermittent Sustainable Energies
http://dx.doi.org/10.5772/106341
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