This work investigates the multi-fractal nature of a turbulent urban atmosphere using high-resolution atmospheric data. Meteorological and concentration measurements of passive and reactive pollutants collected over a 3-year period in a sub-urban high-Reynolds number atmospheric field were analyzed. Scaling laws characterizing the self-similarity and thereby depicting the multi-fractal nature are determined by calculating the singularity spectra, where a range of Hölder exponents, , are estimated. In doing so, the complexity of the urban atmosphere entailing different stability regimes was addressed. Using the Monin-Obukhov length () as a marker of atmospheric stability and thereby an indication of the magnitude of anisotropy, we find where and how self-similarity is manifested relative to the different regimes and we estimate corresponding appropriate scaling laws. We find that the wind speed obeys the law suggested by Kolmogorov only when the atmosphere lies within the stable regime as defined by Monin-Obukhov theory. Specifically, when the ratio of the atmospheric boundary layer height () over is greater than 15, and at the same time, the ratio of the height above ground of the wind measurements () over is higher than 3 (i.e., in stable regime), then the singularity spectra of wind speed time series indicate that the dominant Hölder exponent, , coincides with Kolmogorov’s second hypothesis. On the contrary under unstable regimes in the atmosphere where the anisotropy is approached, different scaling laws are estimated. In detail, when , the dominant Hölder exponent, , of the singularity spectra of the wind speed time series is either negative or close to zero, which is an indication of an impulse-like singularity, that is associated with rapid changes. For the ambient temperature and air quality measurements such as of carbon monoxide and particulate matter concentrations, it was found that they obey different laws, which are related with the long-term correlation of their data fluctuation.
REFERENCES
1
E.
Bacry
, A.
Arneodo
, U.
Frisch
, Y.
Gagne
, and E.
Hopfinger
, “Wavelet analysis of fully developed turbulence data and measurement of scaling exponents,” in Turbulence and Coherent Structures (Springer, 1991), pp. 203–215.2
E.
Bacry
, J.-F.
Muzy
, and A.
Arneodo
, “Singularity spectrum of fractal signals from wavelet analysis: Exact results
,” J. Stat. Phys.
70
(3–4
), 635
–674
(1993
). 3
K. E.
Bassler
, G. H.
Gunaratne
, and J. L.
McCauley
, “Markov processes, hurst exponents, and nonlinear diffusion equations
,” Physica A
369
(2
), 343
–353
(2006
). 4
F.
Böttcher
, J.
Peinke
et al., “Small and large scale fluctuations in atmospheric wind speeds
,” Stoch. Environ. Res. Risk. Assess.
21
(3
), 299
–308
(2007
). 5
Z.
Chen
, P. C.
Ivanov
, K.
Hu
, and H. E.
Stanley
, “Effect of nonstationarities on detrended fluctuation analysis
,” Phys. Rev. E
65
(4
), 041107
(2002
). 6
J.
Ching
, G.
Mills
, B.
Bechtel
, L.
See
, J.
Feddema
, X.
Wang
, C.
Ren
, O.
Brousse
, A.
Martilli
, M.
Neophytou
et al., “Wudapt: An urban weather, climate, and environmental modeling infrastructure for the anthropocene
,” Bull. Am. Meteorol. Soc.
99
(9
), 1907
–1924
(2018
). 7
D. A.
Dickey
and W. A.
Fuller
, “Distribution of the estimators for autoregressive time series with a unit root
,” J. Am. Stat. Assoc.
74
(366a
), 427
–431
(1979
). 8
J. F.
Eichner
, E.
Koscielny-Bunde
, A.
Bunde
, S.
Havlin
, and H.-J.
Schellnhuber
, “Power-law persistence and trends in the atmosphere: A detailed study of long temperature records
,” Phys. Rev. E
68
(4
), 046133
(2003
). 9
P.
Embrechts
and M.
Maejima
, Selfsimilar Processes, Princeton Series in Applied Mathematics (Princeton University Press, 2002) ISBN: 9780691096278.10
F.
Esposti
, M.
Ferrario
, and M.
Gabriella Signorini
, “A blind method for the estimation of the hurst exponent in time series: Theory and application
,” Chaos
18
(3
), 033126
(2008
). 11
M.
Farge
and E.
Guyon
, A Philosophical and Historical Journey Through Mixing and Fully-developed Turbulence
(Springer US
, Boston, MA
, 1999
), pp. 11
–36
.12
T.
Feng
, Z.
Fu
, X.
Deng
, and J.
Mao
, “A brief description to different multi-fractal behaviors of daily wind speed records over china
,” Phys. Lett. A
373
(45
), 4134
–4141
(2009
). 13
14
R. B.
Govindan
and H.
Kantz
, “Long-term correlations and multifractality in surface wind speed
,” Europhys. Lett.
68
(2
), 184
(2004
). 15
K. P.
Harikrishnan
, R.
Misra
, G.
Ambika
, and R. E.
Amritkar
, “Computing the multifractal spectrum from time series: An algorithmic approach
,” Chaos
19
(4
), 043129
(2009
). 16
D.
Hurst
and J. C.
Vassilicos
, “Scalings and decay of fractal-generated turbulence
,” Phys. Fluids
19
(3
), 035103
(2007
). 17
S.
Jaffard
, “Multifractal formalism for functions part i: Results valid for all functions
,” SIAM J. Math. Anal.
28
(4
), 944
–970
(1997
). 18
S.
Jaffard
, “Multifractal formalism for functions part ii: Self-similar functions
,” SIAM J. Math. Anal.
28
(4
), 971
–998
(1997
). 19
J. W.
Kantelhardt
, S. A.
Zschiegner
, E.
Koscielny-Bunde
, S.
Havlin
, A.
Bunde
, and H.
Eugene Stanley
, “Multifractal detrended fluctuation analysis of nonstationary time series
,” Physica A
316
(1
), 87
–114
(2002
). 20
R. G.
Kavasseri
and R.
Nagarajan
, “A multifractal description of wind speed records
,” Chaos, Solitons Fractals
24
(1
), 165
–173
(2005
). 21
N.
Kitova
, K.
Ivanova
, M.
Ausloos
, T. P.
Ackerman
, and M. A.
Mikhalev
, “Time dependent correlations in marine stratocumulus cloud base height records
,” Int. J. Mod. Phys. C
13
(02
), 217
–227
(2002
). 22
E.
Koscielny-Bunde
, A.
Bunde
, S.
Havlin
, and Y.
Goldreich
, “Analysis of daily temperature fluctuations
,” Physica A
231
(4
), 393
–396
(1996
). 23
E.
Koscielny-Bunde
, A.
Bunde
, S.
Havlin
, H.
Eduardo Roman
, Y.
Goldreich
, and H.-J.
Schellnhuber
, “Indication of a universal persistence law governing atmospheric variability
,” Phys. Rev. Lett.
81
(3
), 729
(1998
). 24
D.
Kwiatkowski
, P. C. B.
Phillips
, P.
Schmidt
, Y.
Shin
et al., “Testing the null hypothesis of stationarity against the alternative of a unit root
,” J. Econom.
54
(1–3
), 159
–178
(1992
). 25
G.
Lin
and Z.
Fu
, “A universal model to characterize different multi-fractal behaviors of daily temperature records over China
,” Physica A
387
(2
), 573
–579
(2008
). 26
J. L.
López
and J. G.
Contreras
, “Performance of multifractal detrended fluctuation analysis on short time series
,” Phys. Rev. E
87
(2
), 022918
(2013
). 27
S.
Lovejoy
and D.
Schertzer
, “Scale invariance, symmetries, fractals, and stochastic simulations of atmospheric phenomena
,” Bull. Am. Meteorol. Soc.
67
(1
), 21
–32
(1986
). 28
S.
Mallat
, A Wavelet Tour of Signal Processing: The Sparse Way
(Academic Press
, 2008
).29
S.
Mallat
and W. L.
Hwang
, “Singularity detection and processing with wavelets
,” IEEE Trans. Inf. Theor.
38
(2
), 617
–643
(1992
). 30
B. B.
Mandelbrot
, A. J.
Fisher
, and L. E.
Calvet
, “A multifractal model of asset returns
,” in Cowles Foundation Discussion Papers
(Cowles Foundation, Yale University
, 1997
).31
C.
Meneveau
and K. R.
Sreenivasan
, “The multifractal spectrum of the dissipation field in turbulent flows
,” Nucl. Phys. B Proc. Suppl.
2
, 49
–76
(1987
). 32
Q.
Minwei
and L.
Shida
, “Scale exponents of atmospheric turbulence under different stratifications
,” Fractals
4
(01
), 45
–48
(1996
). 33
P.
Mouzourides
, A.
Eleftheriou
, A.
Kyprianou
, J.
Ching
, and M. K.-A.
Neophytou
, “Linking local-climate-zones mapping to multi-resolution-analysis to deduce associative relations at intra-urban scales through an example of metropolitan London
,” Urban Climate
30
, 100505
(2019
). 34
P.
Mouzourides
, P.
Kumar
, and M. K.-A.
Neophytou
, “Assessment of long-term measurements of particulate matter and gaseous pollutants in south-east mediterranean
,” Atmos. Environ.
107
, 148
–165
(2015
). 35
P.
Mouzourides
, A.
Kyprianou
, M. J.
Brown
, B.
Carissimo
, R.
Choudhary
, and M. K.-A.
Neophytou
, “Searching for the distinctive signature of a city in atmospheric modelling: Could the multi-resolution analysis (mra) provide the dna of a city?
,” Urban Climate
10
, 447
–475
(2014
). 36
P.
Mouzourides
, A.
Kyprianou
, and M. K.-A.
Neophytou
, “A scale-adaptive approach for spatially-varying urban morphology characterization in boundary layer parametrization using multi-resolution analysis
,” Boundary Layer Meteorol.
149
(3
), 455
–481
(2013
). 37
J.-F.
Muzy
, E.
Bacry
, and A.
Arneodo
, “Wavelets and multifractal formalism for singular signals: Application to turbulence data
,” Phys. Rev. Lett.
67
(25
), 3515
(1991
). 38
J.-F.
Muzy
, E.
Bacry
, and A.
Arneodo
, “The multifractal formalism revisited with wavelets
,” Int. J. Bifurcation Chaos
4
(02
), 245
–302
(1994
). 39
M.
Nicollet
, A.
Lemarchand
, and N.
Cavaciuti
, “Detection of atmospheric turbulence by multifractal analysis using wavelets
,” Fractals
12
(02
), 211
–221
(2004
). 40
P.
Oświȩcimka
, J.
Kwapień
, and S.
Drożdż
, “Wavelet versus detrended fluctuation analysis of multifractal structures
,” Phys. Rev. E
74
(1
), 016103
(2006
). 41
42
L. F.
Richardson
, Weather Prediction by Numerical Process
(Cambridge University Press
, 2007
).43
R. H.
Riedi
, Multifractal processes. theory and applications of long range dependence (p. doukhan, g. oppenheim, and ms taqqu, et al.), birkh. auser. (2002).44
K.
Schneider
and M.
Farge
, “Wavelets: Mathematical theory
,” in Encyclopedia of Mathematical Physics, edited by J. P. Françoise, G. Naber, and T. S. Tsun (Elsevier
, 2006
) pp. 426
–438
.45
M. R.
Schroeder
, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise
(Courier Corporation
, 2012
).46
B.
Shi
, B.
Vidakovic
, G. G.
Katul
, and J. D.
Albertson
, “Assessing the effects of atmospheric stability on the fine structure of surface layer turbulence using local and global multiscale approaches
,” Phys. Fluids
17
(5
), 055104
(2005
). 47
H. E.
Stanley
, L. A.
Amaral
, A. L.
Goldberger
, S.
Havlin
, P. C.
Ivanov
and C. K.
Peng
, “Monofractal and multifractal approaches to complex biomedical signals
,” AIP Conf. Proc.
502
(1
), 133
–145
(2000
).48
I.
Stiperski
, M.
Calaf
, and M. W.
Rotach
, “Scaling, anisotropy, and complexity in near-surface atmospheric turbulence
,” J. Geophys. Res. Atmos.
124
(3
), 1428
–1448
(2019
). 49
P.
Talkner
and R. O.
Weber
, “Power spectrum and detrended fluctuation analysis: Application to daily temperatures
,” Phys. Rev. E
62
(1
), 150
(2000
). 50
G. I.
Taylor
, “Statistical theory of turbulence,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (The Royal Society, 1935), Vol. 151, pp. 421–444.51
S. T.
Thoroddsen
and C. W.
Van Atta
, “Experimental evidence supporting kolmogorov’s refined similarity hypothesis
,” Phys. Fluids A Fluid Dyn.
4
(12
), 2592
–2594
(1992
). 52
C. A.
Varotsos
, I.
Melnikova
, M. N.
Efstathiou
, and C.
Tzanis
, “1/f noise in the uv solar spectral irradiance
,” Theor. Appl. Climatol.
111
(3–4
), 641
–648
(2013
). 53
W. W.
Wu-Shyong
, Time Series Analysis: Univariate and Multivariate Methods
, 2nd ed. (Pearson Addison Wesley
, 2006
).54
Y.
Xue
, W.
Pan
, W.-Z.
Lu
, and H.-D.
He
, “Multifractal nature of particulate matters (pms) in hong kong urban air
,” Sci. Total Environ.
532
, 744
–751
(2015
). © 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
