[18] Vautard R, Ghil M. Singular spectrum analysis in nonlinear dynamics, with applications o paleoclimatic time series, Physica D 1989;35(3): 395-424, doi:10.1016/0167-2789(89)90077-8. [19] Grassberger P, Procaccia I. Characterization of strange attractors, Physical Review

[20] Grassberger P, Procaccia I. Measuring the strangeness of strange attractors, Physica D

[21] Higuchi T. Approach to an irregular time series on the basis of the fractal theory,

[22] Higuchi T. Relationship between the fractal dimension and the power law index for a ime series: A numerical investigation, Physica D 1990;46(2): 254-264, doi:10.1016/0167-

[23] Donner R, Witt A. Characterisation of long-term climate change by dimension estimates of multivariate palaeoclimate proxy data. Nonlinear Processes in Geophysics 2006;13(5):

[24] Donner R, Witt A. Temporary dimensions of multivariate data from paleoclimate records - A novel measure for dynamic characterization of long-term climate change. nternational Journal of Bifurcation and Chaos 2007;17(10): 3685-9,

[25] Donner R. Advanced Methods for Analysing and Modelling of Multivariate

[27] Preisendorfer RW. Principal Component Analysis in Meteorology and Oceanography.

Palaeoclimatic Time Series. PhD thesis. University of Potsdam; 2005. [26] Joliffe JT. Principal Component Analysis. New York: Springer; 1986.

Physical Review Letters 1980;45(9): 712-716, doi:10.1103/PhysRevLett.45.712.

Physica D 1986;20(2-3): 217-236, doi:10.1016/0167-2789(86)90031-X.

Letters 1983;50(5): 346-349, doi:10.1103/PhysRevLett.50.346.

Physica D 1988;31(2): 277-283, doi:10.1016/0167-2789(88)90081-4.

1983;9(1-2): 189-208, doi:10.1016/0167-2789(83)90298-1.

1983;97(6): 227-230, doi:10.1016/0375-9601(83)90753-3.

3403-3411, doi:10.1103/PhysRevA.45.3403.

2789(90)90039-R.

485-97, doi: 10.5194/npg-13-485-2006.

doi:10.1142/S0218127407019573.

Amsterdam: Elsevier; 1988.

## **Author details**

Reik V. Donner *Research Domain IV – Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Potsdam, Germany*

## **Acknowledgement**

This work has been supported by the Leibniz Society (project ECONS).

## **7. References**

[10] Donges JF, Donner RV, Rehfeld K, Marwan N, Trauth MH, Kurths J. Identification of dynamical transitions in marine palaeoclimate records by recurrence network analysis. Nonlinear Processes in Geophysics 2011;18(5): 545-62, doi:10.5194/npg-18-545-2011.

22 Fractal Analysis and Chaos in Geosciences

even complex systems sciences in general.

10.1103/RevModPhys.65.1331

Analysis. Singapore: World Scientific; 2000.

University Press; 1997.

Scientific; 1999.

*Research Domain IV – Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Potsdam, Germany*

This work has been supported by the Leibniz Society (project ECONS).

Physical Science Basis. Cambridge: Cambridge University Press; 2007.

Modern Physics 1985;57(3): 617-56, doi:10.1103/RevModPhys.57.617

[4] Tong, H. Nonlinear Time Series and Chaos. Singapore: World Scientific; 1993. [5] Abarbanel, HDI. Analysis of Observed Chaotic Data. New York: Springer; 1996.

[1] Intergovernmental Panel on Climate Change (IPCC). Climate Change 2007: The

[2] Eckmann J-P, Ruelle D. Ergodic theory of chaos and strange attractors. Reviews of

[3] Abarbanel, HDI, Brown R, Sidorowich JJ, Tsimring LS. The analysis of observed chaotic data in physical systems. Reviews of Modern Physics 1993;65(4): 1331-92, doi:

[6] Kantz H, Schreiber T. Nonlinear Time Series Analysis. Cambridge: Cambridge

[7] Diks C. Nonlinear Time Series Analysis: Methods and Applications. Singapore: World

[8] Galka A. Topics in Nonlinear Time Series Analysis – With Implications for EEG

[9] Sprott JC. Chaos and Time Series Analysis. Oxford: Oxford University Press; 2003.

**Author details**

**Acknowledgement**

Reik V. Donner

**7. References**

properties of various statistical characteristics in space or time that can be found in many complex systems, fractal dimensions have demonstrated their great potential to quantitatively distinguish between time series obtained under different conditions or at different locations, thus contributing to a classification of behaviors based on nonlinear dynamical properties. However, as it has been demonstrated both empirically and numerically, established concepts of fractal dimensions reach their fundamental limits when being applied to relatively short and noisy geoscientific time series, e.g. climate records. As potential alternatives providing measures with comparable meaning, but different conceptual foundations, two promising approaches based on the evaluation of serial correlations and complex network theory have been discussed. Although both concepts still need to systematically prove their capabilities and require further methodological improvements as highlighted in this chapter, they constitute promising new research avenues for future problems in climate change research, other fields of geosciences, and

	- [28] Kramer MA. Nonlinear principal component analysis using autoassociative neural networks. American Institute for Chemical Engineering Journal 1991;37(2): 233-43, doi: 10.1002/aic.690370209.

Complexity Concepts and Non-Integer Dimensions in Climate and Paleoclimate Research 25

[46] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU. Complex networks: Structure and dynamics, Physics Reports 2006;424(4-5): 175-308, doi:10.106/j.physrep.2005.10.009. [47] Costa LF, Rodrigues FA, Travieso G, Villas Boas PR. Characterization of complex networks : A survey of measurements, Advances in Physics 2007 ;56(1) : 167-242,

[48] Kenett DY, Tumminello M, Madi A, Gur-Gershgoren G, Mantegna RN, Ben-Jacob E. Dominating clasp of the financial sector revealed by partial correlation analysis of the

[50] Tsonis AA, Roebber PJ. The architecture of the climate network, Physica A 2004;333:

[51] Yamasaki K, Gozolchiani A, Havlin S. Climate networks around the globe are significantly affected by El Nino, Physical Review Letters 2008;100(22): 228501,

[52] Donges JF, Zou Y, Marwan N, Kurths J. The backbone of the climate network,

[53] Donges JF, Schultz HCH, Marwan N, Zou Y, Kurths J. Investigating the topology of nteracting networks: Theory and application to coupled climate subnetworks, European

[54] Donner RV, Small M, Donges JF, Marwan N, Zou Y, Xiang R, Kurths J. Recurrencebased time series analysis by means of complex network methods, International Journal

[56] Lacasa L, Luque B, Ballesteros F, Luque J, Nuno JC. From time series to complex networks: The visibility graph, Proceedings of the National Academy of Sciences

[57] Zhang J, Small M. Complex network from pseudoperiodic time series: Topology versus dynamics, Physical Review Letters 2006;96(23): 238701, doi: 10.1103/

[58] Yang Y, Yang H. Complex network-based time series analysis, Physica A 2008 ;387(5-

[59] Xu X, Zhang J, Small M. Superfamily phenomena and motifs of networks induced from ime series, Proceedings of the National Academy of Sciences 2008;105(50): 19601-19605,

[60] Marwan N, Donges JF, Zou Y, Donner RV, Kurths J. Complex network approach for recurrence analysis of time series, Physics Letters A 2009;373(46): 4246-4254,

[61] Donner RV, Zou Y, Donges JF, Marwan N, Kurths J. Recurrence networks – a novel paradigm for nonlinear time series analysis, New Journal of Physics 2010;12(3): 033025,

of Bifurcation and Chaos 2011;21(4): 1019-1046, doi:10.1142/S0218127411029021. [55] Nicolis G, Garcia Cantu A, Nicolis C. Dynamical aspects of interaction networks, nternational Journal of Bifurcation and Chaos 2005;15(11): 3467-3480,

Europhysics Letters 2009;87(4): 48007, doi:10.1209/0295-5075/87/48007.

Physical Journal B 2011;84(4): 635-651, doi:10.1140/epjb/e2011-10975-8.

stock market, PLoS ONE 2010;5(12): e15032, doi:10.1371/journal.pone.0015032. [49] Zhou C, Zemanova L, Zamora G, Hilgetag CC, Kurths J. Hierarchical organization unveiled by functional connectivity in complex brain networks, Physical Review Letters

2006;97(23): 238103, doi:10.1103/PhysRevLett.97.238103.

2008;105(13): 4972-4975, doi:10.1073/pnas.0709247105.

6) : 1381-1386, doi :10.1016/j.physa.2007.10.055.

doi :10.1080/00018730601170527.

497-504, doi:10.1016/j.physa.2003.10.045.

doi:10.1103/PhysRevLett.100.228501.

doi:10.1142/S0218127405014167.

PhysRevLett.96.238701.

doi:10.1073/pnas.0806082105.

doi:10.1016/j.physleta.2009.09.042.

doi:10.1088/1367-2630/12/3/033025.

[46] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU. Complex networks: Structure and dynamics, Physics Reports 2006;424(4-5): 175-308, doi:10.106/j.physrep.2005.10.009.

24 Fractal Analysis and Chaos in Geosciences

10.1002/aic.690370209.

science.290.5500.2319.

2001.

[28] Kramer MA. Nonlinear principal component analysis using autoassociative neural networks. American Institute for Chemical Engineering Journal 1991;37(2): 233-43, doi:

[29] Tenenbaum J, de Silva V, Langford JC. A global geometric framework for nonlinear diomensionality reduction. Science 2000;290(5500): 2319-23, doi: 10.1126/

[30] Hyvärinen A, Karhunen J, Oja E. Independent Component Analysis. New York: Wiley;

[31] Ciliberti S, Nicolaenko B. Estimating the number of degrees of freedom in spatially extended

[32] Pomeau Y. Measurement of the information density in turbulence. Comptes Rendus de

[33] Kaneko K. Spatiotemporal chaos in one-dimensional and two-dimensional coupled map attices. Physica D 1989;37(1-3): 60-82, doi: 10.1016/0167-2789(89)90117-6. [34] Mayer-Kress G, Kaneko K. Spatiotemporal chaos and noise. Journal of Statistical

[35] Zoldi SM, Greenside HM. Karhunen-Loeve decomposition of extensive chaos. Physical

[36] Zoldi SM, Liu J, Bajaj KMS, Greenside HM, Ahlers G. Extensive scaling and nonuniformity of the Karhunen-Loeve decomposition for the spiral-defect chaos state.

[37] Meixner M, Zoldi SM, Bose S, Schöll E. Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system. Physical Review E 2000;61(2): 1382-

[38] Varela H, Beta C, Bonnefort A, Krischer K. Transitions in electrochemical turbulence. Physical Review Letters 2005;94(17): 174104, doi: 10.1103/PhysRevLett.94.174104. [39] Xie X, Zhao X, Fang Y, Cao Z, He G. Normalized linear variance decay dimension density and its application of dynamical complexity detection in physiological (fMRI) ime series. Physics Letters A 2011;375(17): 1789-95, doi: 10.1016/j.physleta.2011.03.003. [40] Donner R. Spatial Correlations of River Runoffs in a Catchment. In: Kropp J, Schellnhuber HJ. (eds.) In Extremis - Disruptive Events and Trends in Climate and

Hydrology. Berlin: Springer; 2011. p286-313. doi:10.1007/978-3-642-14863-7\_14. [41] Donner R. Multivariate analysis of spatially heterogeneous phase synchronisation in complex systems: Application to self-organised control of material flows in networks.

European Physical Journal B 2008;63(3): 349-61, doi:10.1140/epjb/e2008-00151-8. [42] Toonen C, Lappe D, Donner RV, Scholz-Reiter B. Impact of Machine-Driven Capacity Constellations on Performance and Dynamics of Job-Shop Systems. In: El Maraghy HA. ed.) Enabling Manufacturing Competitiveness and Economic Sustainability. Berlin:

[43] Grenander U, Szegö G. Toeplitz Forms and Their Applications. Berkeley: University of

[44] Albert R, Barabasi AL. Statistical mechanics of complex networks, Reviews of Modern

[45] Newman MEJ. The structure and function of complex networks, SIAM Review

Springer; 2011. p611-6, doi:0.1007/978-3-642-23860-4\_100.

Physics 2002;74(1): 47-97, doi:10.1103/RevModPhys.74.47.

2003;45(2): 167-256, doi:10.1137/S003614450342480.

Review Letters 1997;78(9): 1687-90, doi: 10.1103/PhysRevLett.78.1687.

Physical Review E 1998;58(6): 6903-6. doi: 10.1103/PhysRevE.58.R6903.

systems. Europhysics Letters 1991;14(4): 303-8, doi: 10.1209/0295-5075/ 14/4/003.

'Academie des Sciences. Series II 1985;300(7): 239-41.

Physics 1989;54(5-6): 1489-508, doi:10.1007/BF01044730.

85, doi: 10.1103/PhysRevE.61.1382.

California Press; 1958.

[62] Donner RV, Zou Y, Donges JF, Marwan N, Kurths J. Ambiguities in recurrence-based complex network representations of time series, Physical Review E 2010;81(1): 015101(R), doi:10.1103/PhysRevE.81.015101.

Complexity Concepts and Non-Integer Dimensions in Climate and Paleoclimate Research 27

[79] Donges JF, Donner RV, Trauth MH, Marwan N, Schellnhuber HJ, Kurths J. Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution, Proceedings of the National Academy of Sciences 2011;108(51): 20422-20427,

[80] Hays JD, Imbrie J, Shackleton NJ. Variations in the Earth's orbit: Pacemaker of the ice

[81] Buck CE, Millard AD, editors. Tools for Constructing Chronologies. London: Springer; 2004. [82] Parnell AC, Buck CE, Doan TK. A review of statistical chronology models for highresolution, proxy-based Holocene palaeoenvironmental reconstruction, Quaternary

[84] Ghil M, Allen MR, Dettinger MD, Ide K, Kondrashov D, Mann ME, Robertson AW, Saunders A, Tian Y, Varadi F, Yiou P. Advanced spectral methods for climate time

[85] Babu P, Stoica P. Spectral analysis of nonuniformly sampled data – a review, Digital

[86] Rehfeld K, Marwan N, Heitzig J, Kurths J. Comparison of correlation analysis echniques for irregularly sampled time series, Nonlinear Processes in Geophysics 2011;18(3): 389-

[87] Lomb NR. Least-squares frequency analysis of unequally spaced data Astrophysics

[88] Scargle J. Studies in astronomical time series analysis. II. Stattistical aspects of spectral analysis of unevenly spaced data, Astrophysical Journal 1982;263: 835-853,

[89] Scargle J. Studies in astronomical time series analysis. III. Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data,

[90] Mudelsee M. Climate Time Series Analysis: Classical Statistical and Bootstrap Methods.

[91] Priestley MB. Non-linear and non-stationary time series analysis. London: Academic

[93] Holschneider, M. Wavelets: An Analysis Tool. Oxford: Oxford University Press; 1995. [94] Percival DB, Walden AT. Wavelet Methods for Time Series Analysis. Cambridge:

[95] Foster G. Wavelets for period analysis of unevenly sampled time series, Astronomical

[96] Andronov IL. Method of running parabolae: Spectral and statistical properties of the smoothing function, Astronomy and Astrophysics Supplement Series 1997;125: 207-217,

[97] Andronov IL. Wavelet analysis of time series by the least-squares method with supplementary weights, Kinematics and Physics of Celestial Bodies 1998;14(6): 374-392. [98] Andronov IL. Wavelet analysis of the irregular spaced time series. In: Priezzhev VB,

Spiridonov VP (eds.) Self-Similar Systems. Dubna: JINR; 1999. p57-70.

Science Reviews 2011;30(21-22): 2948-2960, doi:10.1016/j.quascirev.2011.07.024. [83] Priestley MB. Spectral Analysis and Time Series. London: Academic Press; 1981.

series, Reviews of Geophysics 2002;40(1), 1003, doi:10.1029/2000RG000092.

Signal Processing 2009;20(2): 359-378, doi:10.1016/j.dsp.2009.06.019.

and Space Science 1976;39: 447-462, doi:10.1007/BF00648343.

Astrophysical Journal 1989;343): 874-887, doi:10.1086/167757.

[92] Daubechies, I. Ten Lectures on Wavelets. Philadelphia: SIAM; 1992.

ages, Science 1976;194(4270): 1121-1132, doi:10.1126/science.194.4270.1121.

doi:10.1073/PNAS.1117052108.

404, doi:10.5194/npg-18-389-2011.

doi:10.1086/160554.

Press; 1988.

Dordrecht: Springer; 2010.

doi:10.1051/aas:1997217.

Cambridge University Press; 2000.

ournal 1996;112(4): 1709-1729, doi:10.1086/118137.

[79] Donges JF, Donner RV, Trauth MH, Marwan N, Schellnhuber HJ, Kurths J. Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution, Proceedings of the National Academy of Sciences 2011;108(51): 20422-20427, doi:10.1073/PNAS.1117052108.

26 Fractal Analysis and Chaos in Geosciences

015101(R), doi:10.1103/PhysRevE.81.015101.

doi:10.1103/PhysRevE.79.066303.

doi:10.1080/02693799408902004.

2001;28(1): 103-121, doi:10.1068/b2684.

doi:10.1103/PhysRevE.66.016121.

doi:10.1103/PhysRevE.85.046105.

653-672, doi:10.1140/epjb/e2011-10899-1.

Chaos 2010;20(4): 043130, doi:10.1063/1.3523304.

2009;86(3): 30001, doi:10.1209/0295-5075/86/30001.

2009;373(42): 3822-3826, doi:10.1016/j.physleta.2009.08.041.

Letters 2012;98(4): 48001, doi:10.1209/0295-5075/98/48001.

doi:10.1145/359156.359164.

8493(94)90002-7.

[62] Donner RV, Zou Y, Donges JF, Marwan N, Kurths J. Ambiguities in recurrence-based complex network representations of time series, Physical Review E 2010;81(1):

[63] Gao Z, Jin N. Flow-pattern identification and nonlinear dynamics of gas-liquid twophase flow in complex networks, Physical Review E 2009;79(6): 066303,

[64] Lozano-Perez T, Wesley MA. An algorithm for planning collision-free paths among polyhedral obstacles, Communications of the ACM 1979;22(10): 560-570,

[65] de Floriani L, Marzano P, Puppo E. Line-of-sight communication on terrain models, nternational Journal of Geographical Information Systems 1994;8(4): 329-342,

[66] Nagy G. Terrain visibility, Computers & Graphics 1994;18(6): 763-773, doi:10.1016/0097-

[67] Turner A, Doxa M, O'Sullivan D, Penn A. From isovists to visibility graphs: A methodology for the analysis of architectural space, Environment and Planning B

[68] Lacasa L, Luque B, Luque J, Nuno JC. The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion, Europhysics Letters

[69] Ni XH, Jiang ZQ, Zhou WX. Degree distributions of the visibility graphs mapped from ractional Brownian motions and multifractal random walks, Physics Letters A

[70] Gneiting T, Schlather M. Stochastic models that separate fractal dimension and the Hurst effect, SIAM Review 2004;46(2): 269-282, doi:10.1137/S0036144501394387. [71] Donner RV, Donges JF. Visibility graph analysis of geophysical time series: Potentials and possible pitfalls, Acta Geophysica 2012;60(3): 589-623, doi:10.2478/s11600-012-0032-x. [72] Telesca L, Lovallo M. Analysis of seismic sequences by using the method of visibility graph, Europhysics Letters 2012;97(5): 50002, doi:10.1209/0295-5075/97/50002. [73] Zou Y, Heitzig J, Donner RV, Donges JF, Farmer JD, Meucci R, Euzzor S, Marwan N, Kurths J. Power-laws in recurrence networks from dynamical systems, Europhysics

[74] Dall J, Christensen M. Random geometric graphs, Physical Review E 2002;66(1): 016121,

[75] Donges JF, Heitzig J, Donner RV, Kurths J. Analytical framework for recurrence network analysis of time series, Physical Review E 2012;85(4): 046105,

[76] Donner RV, Heitzig J, Donges JF, Zou Y, Marwan N, Kurths J. The geometry of chaotic dynamics – a complex network perspective, European Physical Journal B 2011;84(4):

[77] Zou Y, Donner RV, Donges JF, Marwan N, Kurths J. Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods,

[78] Zou Y, Donner RV, Kurths J. Geometric and dynamic perspectives on phase-coherent

and noncoherent chaos, Chaos 2012 ;22(1) : 013115, doi :10.1063/1.3677367.

[99] Schumann AY. Waveletanalyse von Sedimentdaten unter Einbeziehung von Alters-Tiefen-Modellen (in German). Diploma Thesis. University of Potsdam; 2004.

**Chapter 2**

© 2012 Maqueda et al., licensee InTech. This is an open access chapter 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.

© 2012 Maqueda et al., licensee InTech. This is a paper 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.

**Analysis of Fractal Dimension of**

Additional information is available at the end of the chapter

these flows are related with fractal geometry.

http://dx.doi.org/10.5772/51876

**1. Introduction**

**the Wind Speed and Its Relationships**

**with Turbulent and Stability Parameters**

Manuel Tijera, Gregorio Maqueda, Carlos Yagüe and José L. Cano

The atmospheric fluxes in the boundary layer at large Reynolds numbers are assumed to be a superposition of periodic perturbations and non-periodic behavior that can obey an irregular state and variable motion that is referred as turbulence. Turbulence can be observed in time series of meteorological variables (wind velocity, for example). The analysis of these series presents a self-similarity structure, [1]. So, the wind velocity can be seen as a fractal magnitude. Fractal Dimension (FD) is an artifice that shows in some way the complexity of the time series and the variability degree of the physical magnitude including. Fractal dimensions will is correlated with the characteristic parameters of the turbulence. The non-integer values of the FD are assigned to time series which exhibit a self– similarity geometry and which show that structure on all length scales. In general, turbulent flows allowed us to recognize the coexistence of structure and randomness, they are a set of solution that are not unique or depend sensitively on initial conditions [2]. The structures of

A structure or time series is said to be self–similar if it can be broken down into arbitrarily small pieces, each of which is a small replica of the entire structure. There is a way measure this degree of complexity by means of FD. The concept of dimension is no easy to understand probably to determine what dimension means and which properties have been one of the big challenges in Mathematics. In addition, mathematicians have come up with some tens of different notions of dimension: topological dimension, Hausdorf dimension, fractal dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more [3, 4]. They are all related. We focus in the fractal dimension of these series by means of dividing its structure onto a grid with size L, and counting the number of grid boxes which contain some of the structure of series, N. This number will be depending on the size L. To

**Chapter 2**

## **Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters**

Manuel Tijera, Gregorio Maqueda, Carlos Yagüe and José L. Cano

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51876

## **1. Introduction**

28 Fractal Analysis and Chaos in Geosciences

doi:10.5194/npg-12-345-2005.

372, doi:10.1046/j.1365-8711.2002.05396.x.

doi:10.1086/304206

doi:10.1038/311529a0.

doi:10.1038/323609a0.

10.1007/VF00207399.

10.1007/BF02083118

313, doi:10.1007/BF00874493.

Berlin: Springer; 1994. p377-387.

German). PhD thesis. University of Kiel; 1995.

http://www.jstor.org/stable/27375.

[99] Schumann AY. Waveletanalyse von Sedimentdaten unter Einbeziehung von Alters-Tiefen-Modellen (in German). Diploma Thesis. University of Potsdam; 2004.

[100] Witt A, Schumann AY. Holocene climate variability on millennial scales recorded in Greenland ice cores, Nonlinear Processes in Geophysics 2005;12(3): 345-352,

[101] Frick P, Baliunas SL, Galyagin D, Sokoloff D, Soon W. Wavelet analysis of stellar hronomspheric activity variations, Astrophysical Journal 1997;483: 426-434,

[102] Frick P, Grossmann A, Tchamitchian P. Wavelet analysis of signals with gaps, Journal

[103] Otazu X, Ribo M, Peracaula M, Paredes JM, Nunez J. Detection of superimposed ignals using wavelets, Monthly Notices of the Royal Astronomical Society 2002;333(2), 365-

[104] Otazu X, Ribo M, Paredes JM, Peracaula M, Nunez J. Multiresolution approach for period determination on unevenly sampled data, Monthly Notices of the Royal

[106] Nicolis C, Nicolis G. Reconstruction of the dynamics of the climatic system from timeeries data, Proceedings of the National Academy of Sciences 1986;83(3): 536-540,

[107] Grassberger P. Do climatic attractors exist? Nature 1986;323(6089): 609-612,

[109] Fluegeman Jr RH, Snow RS. Fractal analysis of long-range paleoclimatic data: Oxygen sotope record of Pacific core V28-239, Pure and Applied Geophysics 1989;131(1-2): 307-

[110] Schulz M, Mudelsee M, Wolf-Welling TCW. Fractal analyses of Pleistocene marine oxygen isotope records. In: Kruhl JH (ed.) Fractals and Dynamic Systems in Geoscience.

[111] Mudelsee M, Stattegger K. Application of the Grassberger-Procaccia algorithm to the

[112] Mudelsee M, Stattegger K. Plio-/Pleistocene climate modeling based on oxygen isotope ime series from deep-sea sediment cores: The Grassberger-Procaccia algorithm and haotic climate systems, Mathematical Geology 1994;26(7): 799-815, doi:

[113] Mudelsee M. Entwicklung neuer statistischer Analysemethoden für Zeitreihen mariner, stabiler Isotopen: die Evolution des globalen Plio-/Pleistozänen Klimas (in

[114] Gibson JF, Farmer JD, Casdagli M, Eubank S. An analytic approach to practical state space reconstruction, Physica D 1992;57(1-2): 1-30, doi:10.1016/0167-2789(92)90085-2.

and Dynamic Systems in Geoscience. Berlin: Springer; 1994. p390-413.

*18O* record from ODP site 659: Selected methodical aspects. In: Kruhl JH (ed.) Fractals

Grassberger-Procaccia algorithm, Climate Dynamics 1989;4(1): 45-55, doi:

*18O* records by the

Astronomical Society 2004;351(1), 215-219, doi:10.1111/j.1365-2966.2004.07774.x. [105] Nicolis C, Nicolis G. Is there a climatic attractor? Nature 1984;311(5986): 529-532,

of Mathematical Physics 1998;39(8): 4091-4107, doi:10.1063/1.532485.

[108] Maasch KA. Calculating climate attractor dimension from

The atmospheric fluxes in the boundary layer at large Reynolds numbers are assumed to be a superposition of periodic perturbations and non-periodic behavior that can obey an irregular state and variable motion that is referred as turbulence. Turbulence can be observed in time series of meteorological variables (wind velocity, for example). The analysis of these series presents a self-similarity structure, [1]. So, the wind velocity can be seen as a fractal magnitude. Fractal Dimension (FD) is an artifice that shows in some way the complexity of the time series and the variability degree of the physical magnitude including. Fractal dimensions will is correlated with the characteristic parameters of the turbulence. The non-integer values of the FD are assigned to time series which exhibit a self– similarity geometry and which show that structure on all length scales. In general, turbulent flows allowed us to recognize the coexistence of structure and randomness, they are a set of solution that are not unique or depend sensitively on initial conditions [2]. The structures of these flows are related with fractal geometry.

A structure or time series is said to be self–similar if it can be broken down into arbitrarily small pieces, each of which is a small replica of the entire structure. There is a way measure this degree of complexity by means of FD. The concept of dimension is no easy to understand probably to determine what dimension means and which properties have been one of the big challenges in Mathematics. In addition, mathematicians have come up with some tens of different notions of dimension: topological dimension, Hausdorf dimension, fractal dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more [3, 4]. They are all related. We focus in the fractal dimension of these series by means of dividing its structure onto a grid with size L, and counting the number of grid boxes which contain some of the structure of series, N. This number will be depending on the size L. To

© 2012 Maqueda et al., licensee InTech. This is an open access chapter 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. © 2012 Maqueda et al., licensee InTech. This is a paper 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.

obtain the FD it is needed to represent in a diagram the logarithm of N against the logarithm of the reciprocal of L (log (N(L)) versus log (1/L)). Then, the slope of the better linear fit or the linear regression between them corresponds to the searched fractal dimension or box-counting dimension. This dimension proposes a systematic measurement of degree of similarity or complexity of wind series. Various works have approached the problem of calculating the dimension associated with time series [5, 6].

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 31

The data analysed was recorded in the experimental campaign SABLES-98 at the Research Centre for the Lower Atmosphere (CIBA in the Spanish acronyms), located in Valladolid province (Spain). SABLES-98 was an extensive campaign of measures in the PBL with a large number of participant teams and took place from 10th to 27th September 1998 [9]. The experimental site is located around 30km NW from Valladolid city, in the northern Iberian Plateau, on a region known as Montes Torozos, which forms a high plain of nearly 200 km2 elevated above the plateau. The surrounding terrain is quite flat and homogeneous, consisting mainly on different crop fields and some scattered bushes. The Duero river flows along the SE border of the high plain, and two small river valleys, which may act as drainage channels in stable conditions, extends from the lower SW

The main instrumentation available at CIBA is installed on the 100m meteorological tower, and includes sonic and cup anemometers, platinum resistance thermometers, wind vanes, humidity sensors, etc. Other instruments are spread in minor masts and ground based in order observes the main meteorological variables of the PBL. In this work we study five minute series from the sonic anemometers (20 Hz) installed at 5.8 (~ 6), 13.5 (~13) and 32m were used to evaluate the fractal dimension and turbulent parameters. These series have been obtained once we have carried out the necessary transformation to get the mean wind series in short intervals, namely 5 minutes, to ensure the consistent

We focus this study in a period of eight consecutive days (from 14 to 21 September 1998) in which have been analyzed every records of the velocity fluctuations. The synoptic conditions were controlled by a high pressure system which produces thermal convection during the daily hours and from moderate to strong stable stratification during the nights. The evolution of wind speed, Bulk Richardson number (Rib), Turbulent Kinetic Energy (TKE), friction velocity (*u\** ), potential temperature difference between 32m and 5.8m, and the temperature difference near surface but in an deeper layer (50-0.22 m, named invT50- 0.22, in reference to thermal inversion), with fractal dimension of velocity *u* and *w* component fluctuations at three levels above the ground (5.8 m, 13.5 m, 32 m) are

The turbulent flows show very high irregularity for wind velocity time series. They present a self–similarity structure, it is to say, that for different scales the structure of the variables remains similar, as it is shown in Fig. 1. This property of the turbulent flows is related with the Fractal Dimension, since the irregularity is a common characteristic. Their non-integer values can help us analyze how the irregularity of the sign is, as well as of its geometry. As the bigger the values of FD, the more irregularity and random is the

**2. Data**

region of the plateau.

properties of turbulence [10].

analysed.

flow.

**3. Methodology**

In this Chapter it is going to study the Fractal Dimension (FD) of *u* and *w* fluctuations of time series of velocity recorded close to the ground, it is to say, into the low Planetary Boundary Layer, more specifically into the Surface Layer, closer to the ground and where the dynamic influence is predominant. The main aim is to seek physical quantities included in turbulent flows that correlate better with FD. Nevertheless, thermal, dynamical and combined parameters will be explored to complete the Chapter. The central idea is that the kind of stability controls the turbulent fluxes and it is showing in the values of the FD also. Differences of potential temperatures between two levels make the Surface Layer or the Low PBL unstable or stable depending on its sign. Shears in the wind produce dynamical instability making it easier the mix of properties, physical variables and mass. In neutral conditions mixture is completed and turbulence is well developed. The values of Fractal Dimension must be in accordance with the turbulent level.

It has been calculated the fractal dimension, *d* (Kolmogorov capacity or box-counting dimension) of the time series of the velocity component fluctuations *u'* and *w'* ( *u' = u - U*, *w' = w - W*) with *U* the horizontal mean velocity, both in the physical space (velocity-time) [2]. It has been studied the time evolution of the fractal dimension of the *u'* and *w'* components (horizontal and vertical) of wind velocity series during several days and three levels above the ground (5.8 m, 13.5 m, 32 m).

This study is focused on the simplest boundary layer kind, over a flat surface. So, we could assume that the flow to be horizontally homogeneous. Its statistical properties are independent of horizontal position; they vary only with height and time [1, 7, 8]. The experimental data have been taken in a flat terrain with short uniform vegetation. It allows us to take on this approximation of horizontal homogeneity and on this context we focus this study on variation of fractal dimension of the horizontal and vertical components of the velocity of flow turbulent in the diurnal cycle versus to a variety of turbulent parameters: difference of potential temperatures in the layers 50-0.22 m and 32 – 5.8 m, Turbulent Kinetic Energy, friction velocity and Bulk Richardson number.

It has been observed that there is a possible correlation between the fractal dimension and different turbulent parameters, both from dynamical and thermal origin: turbulent kinetic energy, friction velocity, difference of temperature between the extreme of the layer studied close to the surface (∆T50-0.22m). Finally, it has been analysed the behaviour of fractal dimension versus stability evaluated from the Richardson number.

The knowledge of turbulence and its relationships with fractal dimension and some turbulent parameters within the Planetary Boundary Layer (PBL) can help us understand how the atmosphere works.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 31

## **2. Data**

30 Fractal Analysis and Chaos in Geosciences

dimension associated with time series [5, 6].

Dimension must be in accordance with the turbulent level.

levels above the ground (5.8 m, 13.5 m, 32 m).

Energy, friction velocity and Bulk Richardson number.

how the atmosphere works.

dimension versus stability evaluated from the Richardson number.

obtain the FD it is needed to represent in a diagram the logarithm of N against the logarithm of the reciprocal of L (log (N(L)) versus log (1/L)). Then, the slope of the better linear fit or the linear regression between them corresponds to the searched fractal dimension or box-counting dimension. This dimension proposes a systematic measurement of degree of similarity or complexity of wind series. Various works have approached the problem of calculating the

In this Chapter it is going to study the Fractal Dimension (FD) of *u* and *w* fluctuations of time series of velocity recorded close to the ground, it is to say, into the low Planetary Boundary Layer, more specifically into the Surface Layer, closer to the ground and where the dynamic influence is predominant. The main aim is to seek physical quantities included in turbulent flows that correlate better with FD. Nevertheless, thermal, dynamical and combined parameters will be explored to complete the Chapter. The central idea is that the kind of stability controls the turbulent fluxes and it is showing in the values of the FD also. Differences of potential temperatures between two levels make the Surface Layer or the Low PBL unstable or stable depending on its sign. Shears in the wind produce dynamical instability making it easier the mix of properties, physical variables and mass. In neutral conditions mixture is completed and turbulence is well developed. The values of Fractal

It has been calculated the fractal dimension, *d* (Kolmogorov capacity or box-counting dimension) of the time series of the velocity component fluctuations *u'* and *w'* ( *u' = u - U*, *w' = w - W*) with *U* the horizontal mean velocity, both in the physical space (velocity-time) [2]. It has been studied the time evolution of the fractal dimension of the *u'* and *w'* components (horizontal and vertical) of wind velocity series during several days and three

This study is focused on the simplest boundary layer kind, over a flat surface. So, we could assume that the flow to be horizontally homogeneous. Its statistical properties are independent of horizontal position; they vary only with height and time [1, 7, 8]. The experimental data have been taken in a flat terrain with short uniform vegetation. It allows us to take on this approximation of horizontal homogeneity and on this context we focus this study on variation of fractal dimension of the horizontal and vertical components of the velocity of flow turbulent in the diurnal cycle versus to a variety of turbulent parameters: difference of potential temperatures in the layers 50-0.22 m and 32 – 5.8 m, Turbulent Kinetic

It has been observed that there is a possible correlation between the fractal dimension and different turbulent parameters, both from dynamical and thermal origin: turbulent kinetic energy, friction velocity, difference of temperature between the extreme of the layer studied close to the surface (∆T50-0.22m). Finally, it has been analysed the behaviour of fractal

The knowledge of turbulence and its relationships with fractal dimension and some turbulent parameters within the Planetary Boundary Layer (PBL) can help us understand The data analysed was recorded in the experimental campaign SABLES-98 at the Research Centre for the Lower Atmosphere (CIBA in the Spanish acronyms), located in Valladolid province (Spain). SABLES-98 was an extensive campaign of measures in the PBL with a large number of participant teams and took place from 10th to 27th September 1998 [9]. The experimental site is located around 30km NW from Valladolid city, in the northern Iberian Plateau, on a region known as Montes Torozos, which forms a high plain of nearly 200 km2 elevated above the plateau. The surrounding terrain is quite flat and homogeneous, consisting mainly on different crop fields and some scattered bushes. The Duero river flows along the SE border of the high plain, and two small river valleys, which may act as drainage channels in stable conditions, extends from the lower SW region of the plateau.

The main instrumentation available at CIBA is installed on the 100m meteorological tower, and includes sonic and cup anemometers, platinum resistance thermometers, wind vanes, humidity sensors, etc. Other instruments are spread in minor masts and ground based in order observes the main meteorological variables of the PBL. In this work we study five minute series from the sonic anemometers (20 Hz) installed at 5.8 (~ 6), 13.5 (~13) and 32m were used to evaluate the fractal dimension and turbulent parameters. These series have been obtained once we have carried out the necessary transformation to get the mean wind series in short intervals, namely 5 minutes, to ensure the consistent properties of turbulence [10].

We focus this study in a period of eight consecutive days (from 14 to 21 September 1998) in which have been analyzed every records of the velocity fluctuations. The synoptic conditions were controlled by a high pressure system which produces thermal convection during the daily hours and from moderate to strong stable stratification during the nights. The evolution of wind speed, Bulk Richardson number (Rib), Turbulent Kinetic Energy (TKE), friction velocity (*u\** ), potential temperature difference between 32m and 5.8m, and the temperature difference near surface but in an deeper layer (50-0.22 m, named invT50- 0.22, in reference to thermal inversion), with fractal dimension of velocity *u* and *w* component fluctuations at three levels above the ground (5.8 m, 13.5 m, 32 m) are analysed.

## **3. Methodology**

The turbulent flows show very high irregularity for wind velocity time series. They present a self–similarity structure, it is to say, that for different scales the structure of the variables remains similar, as it is shown in Fig. 1. This property of the turbulent flows is related with the Fractal Dimension, since the irregularity is a common characteristic. Their non-integer values can help us analyze how the irregularity of the sign is, as well as of its geometry. As the bigger the values of FD, the more irregularity and random is the flow.

**Figure 1.** a) Variation of wind speed versus time from 14 to 21 September 1998, at 5.8 m and 32 m. b) If we zoom red window, in figure c), it is observed that the structure of this flow is similar over all scales of magnification.

In this section we describe the methodology applied to calculate the Fractal Dimension *d* (Kolmogorov capacity or box-counting dimension) [11].The more precise definition of the fractal dimension are in Hausdorff's work, become later known as Hausdorff dimension [12]. This dimension is not practical in the sense that it is very difficult to compute even in elementary examples and nearly impossible to estimate in practical applications. The boxcounting dimension simplifies this problem, being an approximation of the Hausdorff dimension and is calculated approximately by

$$d = \lim\_{L \to 0} \frac{\log N(L)}{\log \left(\frac{1}{L}\right)} \tag{1}$$

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 33

**Figure 2.** Example of linear regression between number of boxes not empty and length side of the box.

The Richardson number is a parameter that includes both dynamic and thermal effects to measure the degree of stratification stability in the low Atmosphere. The static stability parameter, � � �� ⁄ ����� ⁄ �����, only takes in mind the buoyancy, i.e., the thermal effects. Nevertheless, a ratio between *s* and squared wind shear���� ��� ⁄ ��, gives a nondimensional product more appropriate for to calculate the stability. This ratio is known as gradient Richardson number. For the most practical cases a needed numerical approximation will be

In the low Planetary Boundary Layer the atmosphere responds to changes in stratification stability brought about by the heating and cooling of the ground. We search the behaviour or relation between the Fractal Dimension and a parameter to establish a proper measure of stability in the surface layer. One of the most widely used indicator of stability close to the ground in atmospheric studies is the Bulk Richardson number RiB, a nondimensional parameter representing the ration of the rate of production or destruction of turbulence by buoyancy to that by wind shear strain is caused by mechanical forces in the

> <sup>2</sup> ( ) *<sup>B</sup> g z*

where *g* is the gravity acceleration and �̅ the average potential temperature a the reference

stratification, negative for unstable stratification and approximate to zero for neutral

stratification, [10, 14, 15]. The way to calculate this number is following:

is referred to as the buoyancy parameter. RiB is positive for stable

(4)

*<sup>u</sup> Ri*

The slope (*d*) is the fractal dimension, *d* =1.21 ± 0.02 for the example case.

introduced below.

atmosphere:

level, the term *<sup>g</sup>*

*N(L)* is the number of the boxes of side *L* necessary to cover the different points that have been registered in the physical space (velocity-time) [13]. As *L* 0 then *N(L)* increases, *N* meets the following relation:

$$N(L) \cong kL^{-d} \tag{2}$$

$$
\log N(L) = \log k - d \log L \tag{3}
$$

By means of least – square fitting of representation of log *N(L)* versus log *L*, it has been obtained of the straight line regression given by equation (3), as is shown in Fig 2. The fractal dimension *d* will come given by the slope of this equation.

#### **4. Stability of stratification and turbulence**

The origin of turbulence cannot be easily determined, but it is know that both the dynamic and the thermal effects contribute strongly to turbulence by producing a breakdown of streamlined flow in a previously nonturbulent movement. The dynamical effects are represented by wind shear production and the thermal ones make differences of density in the fluid giving rise to hydrostatic phenomena and buoyance.

of magnification.

dimension and is calculated approximately by

meets the following relation:

� � ��� ���

fractal dimension *d* will come given by the slope of this equation.

the fluid giving rise to hydrostatic phenomena and buoyance.

**4. Stability of stratification and turbulence**

**Figure 1.** a) Variation of wind speed versus time from 14 to 21 September 1998, at 5.8 m and 32 m. b) If we zoom red window, in figure c), it is observed that the structure of this flow is similar over all scales

In this section we describe the methodology applied to calculate the Fractal Dimension *d* (Kolmogorov capacity or box-counting dimension) [11].The more precise definition of the fractal dimension are in Hausdorff's work, become later known as Hausdorff dimension [12]. This dimension is not practical in the sense that it is very difficult to compute even in elementary examples and nearly impossible to estimate in practical applications. The boxcounting dimension simplifies this problem, being an approximation of the Hausdorff

> ��� �(�) ��� �� �

*N(L)* is the number of the boxes of side *L* necessary to cover the different points that have been registered in the physical space (velocity-time) [13]. As *L* 0 then *N(L)* increases, *N*

By means of least – square fitting of representation of log *N(L)* versus log *L*, it has been obtained of the straight line regression given by equation (3), as is shown in Fig 2. The

The origin of turbulence cannot be easily determined, but it is know that both the dynamic and the thermal effects contribute strongly to turbulence by producing a breakdown of streamlined flow in a previously nonturbulent movement. The dynamical effects are represented by wind shear production and the thermal ones make differences of density in

� (1)

�(�) � ���� (2)

��� �(�) � ��� � � � ��� � (3)

**Figure 2.** Example of linear regression between number of boxes not empty and length side of the box. The slope (*d*) is the fractal dimension, *d* =1.21 ± 0.02 for the example case.

The Richardson number is a parameter that includes both dynamic and thermal effects to measure the degree of stratification stability in the low Atmosphere. The static stability parameter, � � �� ⁄ ����� ⁄ �����, only takes in mind the buoyancy, i.e., the thermal effects. Nevertheless, a ratio between *s* and squared wind shear���� ��� ⁄ ��, gives a nondimensional product more appropriate for to calculate the stability. This ratio is known as gradient Richardson number. For the most practical cases a needed numerical approximation will be introduced below.

In the low Planetary Boundary Layer the atmosphere responds to changes in stratification stability brought about by the heating and cooling of the ground. We search the behaviour or relation between the Fractal Dimension and a parameter to establish a proper measure of stability in the surface layer. One of the most widely used indicator of stability close to the ground in atmospheric studies is the Bulk Richardson number RiB, a nondimensional parameter representing the ration of the rate of production or destruction of turbulence by buoyancy to that by wind shear strain is caused by mechanical forces in the atmosphere:

$$\mathbf{R}\dot{\mathbf{r}}\_B = \frac{\underline{g}}{\overline{\theta}} \frac{\Lambda \overline{\theta} \Delta \underline{\omega}}{\left(\Delta \overline{\hat{u}}\right)^2} \tag{4}$$

where *g* is the gravity acceleration and �̅ the average potential temperature a the reference level, the term *<sup>g</sup>* is referred to as the buoyancy parameter. RiB is positive for stable stratification, negative for unstable stratification and approximate to zero for neutral stratification, [10, 14, 15]. The way to calculate this number is following:

1. Calculate the mean potential temperatures at height *z* = 32 m, and the close surface *z* = 5.8 m, namely 32 and 5.8 respectively. Being 32 5.8 .

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 35

the nocturnal cooling that produces inversion. During the noon the potential temperatures decrease slightly with height. It is corresponding to instable conditions in the surface layer and we have a possible mixture of both mechanical and convective turbulence [16]. However, the evolution of potential temperature from minimum value until maximum is identical for the two levels in every day studied. It is the consequence of neutral situation

**Figure 3.** The variation of the Fractal Dimension (FD) *versus* time present an analogous behavior for the *u'* and *w'* component fluctuations ((a) and (b), respectively) at the 3 levels, showing the influence of the

In the Figure 4, it is clear to observe the diurnal cycle, observing a strong thermal inversion during the nights that is starting after noon and increasing uniformly to reach the maximum value. The main differences between potential temperatures are observed under inversion

In the Fig. 5, it is shown the fractal dimension at three heights and the difference of potential temperature between 50 and 0.22 m. In this study we will name it as thermal inversion because positive values correspond to actually inversion, in terms of potential temperature (invT50-0.22). The layer 0.22 to 50 m is more extensive than the stratum defined by the levels of the sonic anemometers covering a deep part of the PBL and likely the whole Surface Layer. Besides the instruments to measure the invT are others different of the sonic anemometers which can to give a temperature by the speed of the sound measured also in they. This has

It is observed that fractal dimension correlates in opposed way with the difference of potential temperatures in the layer between 50 and 0.22 m. The more stable conditions are coincident with the bigger values of fractal dimension. In strong thermal inversions the fractal dimensions are lower. Although this good correlation is observed in the figure, we can justify it by mean of a least–squares fitting between the data in the different temporal

condition as a result of the separation among layers because the strong stratification.

with efficient mix during the morning.

diurnal cycle.

an evident benefit in the results.

intervals corresponding to the variables analyzed.

2. Obtain *<sup>z</sup> u* the module mean wind velocity at height z = 32 m and z = 5.8 m, denoted <sup>32</sup> *u* and 5.8 *u* respectively, where 32 5.8 *uu u* .

Once obtained the values of , *u* and *z* by means of the Eq. 4 calculate the Bulk Richardson number in a layer 32m to 5.8m. The mean properties of the flow in this layer as the wind speed and temperature experience their sharpest gradients. In order to know the influence of the kind of stratification over de fractal structure of the flow in the PBL, we going to analyse the behaviour of FD and its possible changes versus the parameter RiB, as the better parameter of stability obtained from the available data.

## **5. Results**

In this section we present the variation of the Fractal Dimension of the *u',* horizontal, and *w',* vertical, components of the velocity fluctuations along the time at the three heights of the study: 5.8 m, 13.5 m and 32 m. We observe that these variations in the three heights are similar. The daily cycle during the period of study is clearly shown in Fig. 3. No significant different values are observed in levels, but a light increasing seems outstanding in diurnal time at the lower level (red line corresponds to 5.8m above the ground). Two components, *u'* and *w'*, present no differences in the time evolution.

As is shown in the Fig. 3, it is observed that the variation interval values of the fractal dimension range between 1.30 and nearly to 1.00. During the diurnal hours the fractal dimension is bigger than at night. A subtle question that concerns us is to what owes this. A possible explanation is that fractal dimension is related with atmospheric stability and, by the same reason, with the turbulence. It is well known that intensity of the turbulence grows as solar radiation increase, producing instability close the ground. In other hand, it is observed that Fractal Dimension is lightly inferior for stable stratification. We shall come back to this matter forward. In the nights a strong stability atmospheric usually exists, so the fractal dimension is usually smaller than during the diurnal hours.

## **5.1. Potential temperature and fractal dimension**

This section concerns with the exam of the relation between the potential temperature differences and the Fractal Dimension. Potential temperature is a very useful variable in the Planetary Boundary Layer that can be replace the observed temperature in the vertical thermal structure, since an air parcel rises or goes down adiabatically at potential temperature constant. A vertical profile of potential temperature uniform represents a neutral stratification or it is called as adiabatic atmosphere.

Next, we show the variation of potential temperature at heights *z* = 32 m and *z* = 5.8 m along the time in Fig. 4. The features shown in this figure need some comment. The potential temperature at height *z* = 5.8 m is bigger than to 32 m in the nights. It is a characteristic of

Once obtained the values of

**5. Results**

 and 5.8

<sup>32</sup> *u* and 5.8 *u* respectively, where 32 5.8 *uu u* .

the better parameter of stability obtained from the available data.

fractal dimension is usually smaller than during the diurnal hours.

**5.1. Potential temperature and fractal dimension**

neutral stratification or it is called as adiabatic atmosphere.

and *w'*, present no differences in the time evolution.

5.8 m, namely 32

1. Calculate the mean potential temperatures at height *z* = 32 m, and the close surface *z* =

2. Obtain *<sup>z</sup> u* the module mean wind velocity at height z = 32 m and z = 5.8 m, denoted

Richardson number in a layer 32m to 5.8m. The mean properties of the flow in this layer as the wind speed and temperature experience their sharpest gradients. In order to know the influence of the kind of stratification over de fractal structure of the flow in the PBL, we going to analyse the behaviour of FD and its possible changes versus the parameter RiB, as

In this section we present the variation of the Fractal Dimension of the *u',* horizontal, and *w',* vertical, components of the velocity fluctuations along the time at the three heights of the study: 5.8 m, 13.5 m and 32 m. We observe that these variations in the three heights are similar. The daily cycle during the period of study is clearly shown in Fig. 3. No significant different values are observed in levels, but a light increasing seems outstanding in diurnal time at the lower level (red line corresponds to 5.8m above the ground). Two components, *u'*

As is shown in the Fig. 3, it is observed that the variation interval values of the fractal dimension range between 1.30 and nearly to 1.00. During the diurnal hours the fractal dimension is bigger than at night. A subtle question that concerns us is to what owes this. A possible explanation is that fractal dimension is related with atmospheric stability and, by the same reason, with the turbulence. It is well known that intensity of the turbulence grows as solar radiation increase, producing instability close the ground. In other hand, it is observed that Fractal Dimension is lightly inferior for stable stratification. We shall come back to this matter forward. In the nights a strong stability atmospheric usually exists, so the

This section concerns with the exam of the relation between the potential temperature differences and the Fractal Dimension. Potential temperature is a very useful variable in the Planetary Boundary Layer that can be replace the observed temperature in the vertical thermal structure, since an air parcel rises or goes down adiabatically at potential temperature constant. A vertical profile of potential temperature uniform represents a

Next, we show the variation of potential temperature at heights *z* = 32 m and *z* = 5.8 m along the time in Fig. 4. The features shown in this figure need some comment. The potential temperature at height *z* = 5.8 m is bigger than to 32 m in the nights. It is a characteristic of

respectively. Being 32 5.8

 .

, *u* and *z* by means of the Eq. 4 calculate the Bulk

the nocturnal cooling that produces inversion. During the noon the potential temperatures decrease slightly with height. It is corresponding to instable conditions in the surface layer and we have a possible mixture of both mechanical and convective turbulence [16]. However, the evolution of potential temperature from minimum value until maximum is identical for the two levels in every day studied. It is the consequence of neutral situation with efficient mix during the morning.

**Figure 3.** The variation of the Fractal Dimension (FD) *versus* time present an analogous behavior for the *u'* and *w'* component fluctuations ((a) and (b), respectively) at the 3 levels, showing the influence of the diurnal cycle.

In the Figure 4, it is clear to observe the diurnal cycle, observing a strong thermal inversion during the nights that is starting after noon and increasing uniformly to reach the maximum value. The main differences between potential temperatures are observed under inversion condition as a result of the separation among layers because the strong stratification.

In the Fig. 5, it is shown the fractal dimension at three heights and the difference of potential temperature between 50 and 0.22 m. In this study we will name it as thermal inversion because positive values correspond to actually inversion, in terms of potential temperature (invT50-0.22). The layer 0.22 to 50 m is more extensive than the stratum defined by the levels of the sonic anemometers covering a deep part of the PBL and likely the whole Surface Layer. Besides the instruments to measure the invT are others different of the sonic anemometers which can to give a temperature by the speed of the sound measured also in they. This has an evident benefit in the results.

It is observed that fractal dimension correlates in opposed way with the difference of potential temperatures in the layer between 50 and 0.22 m. The more stable conditions are coincident with the bigger values of fractal dimension. In strong thermal inversions the fractal dimensions are lower. Although this good correlation is observed in the figure, we can justify it by mean of a least–squares fitting between the data in the different temporal intervals corresponding to the variables analyzed.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 37

period study. These results are obtained for fluctuations of *w*' component of velocity at height z = 5.8 m. Similar results are obtained for other two heights *z* =13 m and *z* = 32 m, also for fluctuations of *u'* component, along of the mean wind direction at three heights

In order to extent the study to all days of the experimental campaign it has been analysed the relation between the Fractal Dimension and the difference of potential temperatures using averages of FD in defined intervals of invT. Figure 7 presents the cloud of the points for all values measured in 5 minutes interval for FD and InvT with the average and the error bars based on standard deviation. Both *u'* as *w'* have similar behaviour respect the variation of invT. In these cases FD is calculate for the lowest level, i. e. 5.8 m (6m labelled in the figure). An analogous shape of the cloud of points and identical results can be obtained if the potential temperature differences when the stratum 32-5.8m has been used. It is not

**Figure 6.** As it is observed the fit is very good in the temporal intervals with r satisfactory, it is showed the linear regression between the fractal dimension and la difference of the temperature between two

It is observed that in two components a similar behaviour. Fractal Dimension have bigger values in unstable and near neutral conditions (negative and close to zero invT), reaching fractal dimension average near 1.15 value for the interval from -1 to 1 K of temperature differences. For strong inversion (> 1 K) the fractal dimension becomes smaller, around 1.05. The presented figures and values can be extent to FD of *u'* and *w'* fluctuations at others two

Finally, the good relation between FD and invT has been valued by mean a linear regression shown in Figure 8, where it has been used the average value of the scatter plot in Figure 7. It is notable the negative slope (- 0.029 –in the corresponding units -) of the regression and the

levels in the periods of 24 h and 18-06h ( (a) and (b) respectively).

heights z = 13 m and z = 32 m with similar results.

high correlation coefficient, R = -0.93.

studied.

shown in this Chapter because repetitive.

**Figure 4.** Time series of potential temperature at 32 and 5.8 m along the complete period of study.

**Figure 5.** Fractal dimension of component fluctuations u' and w' and the thermal inversion in layer between 0.22 and 50 m versus time at three heights: z=5.8 m, z=13 m and z=32 m

As it is observed in the Fig. 6, we show the linear regression in two temporal intervals corresponding to the first 24 hours -from 06 UTC of first day to 06 UTC of the second dayand at night interval from 18 UTC to 06 UTC. The values obtained of the correlation coefficient are very good, in the first case R = -0.925, in the second one R = -0.707. It is observed a better correlation during 24 hours than the night hours because the variation of the invT and FD are wider along all day. The fractal dimension is bigger during the day time than during the night, outstanding a strong inversion at night hours. We could achieve similar results in others temporal intervals of the two variables analysed in the period study. These results are obtained for fluctuations of *w*' component of velocity at height z = 5.8 m. Similar results are obtained for other two heights *z* =13 m and *z* = 32 m, also for fluctuations of *u'* component, along of the mean wind direction at three heights studied.

36 Fractal Analysis and Chaos in Geosciences

**Figure 4.** Time series of potential temperature at 32 and 5.8 m along the complete period of study.

**Figure 5.** Fractal dimension of component fluctuations u' and w' and the thermal inversion in layer

As it is observed in the Fig. 6, we show the linear regression in two temporal intervals corresponding to the first 24 hours -from 06 UTC of first day to 06 UTC of the second dayand at night interval from 18 UTC to 06 UTC. The values obtained of the correlation coefficient are very good, in the first case R = -0.925, in the second one R = -0.707. It is observed a better correlation during 24 hours than the night hours because the variation of the invT and FD are wider along all day. The fractal dimension is bigger during the day time than during the night, outstanding a strong inversion at night hours. We could achieve similar results in others temporal intervals of the two variables analysed in the

between 0.22 and 50 m versus time at three heights: z=5.8 m, z=13 m and z=32 m

In order to extent the study to all days of the experimental campaign it has been analysed the relation between the Fractal Dimension and the difference of potential temperatures using averages of FD in defined intervals of invT. Figure 7 presents the cloud of the points for all values measured in 5 minutes interval for FD and InvT with the average and the error bars based on standard deviation. Both *u'* as *w'* have similar behaviour respect the variation of invT. In these cases FD is calculate for the lowest level, i. e. 5.8 m (6m labelled in the figure). An analogous shape of the cloud of points and identical results can be obtained if the potential temperature differences when the stratum 32-5.8m has been used. It is not shown in this Chapter because repetitive.

**Figure 6.** As it is observed the fit is very good in the temporal intervals with r satisfactory, it is showed the linear regression between the fractal dimension and la difference of the temperature between two levels in the periods of 24 h and 18-06h ( (a) and (b) respectively).

It is observed that in two components a similar behaviour. Fractal Dimension have bigger values in unstable and near neutral conditions (negative and close to zero invT), reaching fractal dimension average near 1.15 value for the interval from -1 to 1 K of temperature differences. For strong inversion (> 1 K) the fractal dimension becomes smaller, around 1.05. The presented figures and values can be extent to FD of *u'* and *w'* fluctuations at others two heights z = 13 m and z = 32 m with similar results.

Finally, the good relation between FD and invT has been valued by mean a linear regression shown in Figure 8, where it has been used the average value of the scatter plot in Figure 7. It is notable the negative slope (- 0.029 –in the corresponding units -) of the regression and the high correlation coefficient, R = -0.93.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 39

would be interesting to relate the TKE to different kind of atmospheric stability. The budgets of TKE in the instable, stable and neutral conditions near the surface are next

In the stable case, the production of the TKE by shear is not sufficient to balance the dissipation energy at all levels and production by buoyancy is not happening. So, turbulence decay and TKE decrease along the time. In neutral conditions near the surface in agreement with the observations suggest that exists a near balance between shear production and viscous dissipation [14, 17]. In this case, a strong shear wind is normally present in the low atmosphere. In the unstable layer, convective case, vertical gradients of *w* are bigger than the vertical gradients of *u* and *v*, the TKE production is mainly due to buoyancy and transport from other levels. Turbulence is supported by thermal effects in the

**Figure 9.** Variation of Fractal dimension and Energy kinetic turbulent versus time for *u* (upper-left panel) and *w* (lower-left) component of wind velocity at height *z* = 5.8 m. Scattered diagram of the

As results it has found that an strong correlation between the dynamic magnitude, the Turbulent Kinetic Energy, and the Fractal Dimension exists. Figs. 9 and 10, in the left panels, show the evolution of the FD and TKE along the time for all days studied at two levels (5.8m Fig 9 and 13m in Fig 10). Diurnal cycle is also observed like in the study of invT in previous sections. This first result is in agreement with the normal variation of the turbulence between the day time and night time. Values of TKE are similar at two levels analyzed which may be interpreted as

fractal dimension and TKE with averages values and error bars (right panels).

summarized.

case of instability [17].

**Figure 7.** Scatter diagram of FD versus InvT50-0.22.It is observed a similar behavior of the fractal dimension with the inversion of temperature, which is evaluated by difference of potential temperature at 50 – 0.22m layer. Average points and error bars are shown for a better understanding.

**Figure 8.** Linear regression of the FD against invT, using the average values from the cloud of points in Fig 7. Negative slope and the high correlation coefficient show a good correlation.

#### **5.2. Turbulent Kinetic Energy and fractal dimension**

The Turbulent Kinetic Energy (TKE) correspond to the quantity of energy associated to the movement of the turbulent flow and it is evaluated from the variances of the components of ݒ ,തതതᇱଶതݑ :velocity തതതᇱଶത and ത ݓതതᇱଶതത . TKE is given by Eq. 5, in terms of energy per unit mass [15].

$$TKE = \frac{1}{2} \left( \overline{\mu^{\cdot 2}} + \overline{\upsilon^{\cdot 2}} + \overline{\upsilon^{\cdot 2}} \right) \tag{5}$$

In this Section we will examine the relationship between TKE and Fractal Dimension of the fluctuations *u'* and *w'* . Before incoming to study with certain details of this relationship, it would be interesting to relate the TKE to different kind of atmospheric stability. The budgets of TKE in the instable, stable and neutral conditions near the surface are next summarized.

38 Fractal Analysis and Chaos in Geosciences

**Figure 7.** Scatter diagram of FD versus InvT50-0.22.It is observed a similar behavior of the fractal

at 50 – 0.22m layer. Average points and error bars are shown for a better understanding.

dimension with the inversion of temperature, which is evaluated by difference of potential temperature

**Figure 8.** Linear regression of the FD against invT, using the average values from the cloud of points in

The Turbulent Kinetic Energy (TKE) correspond to the quantity of energy associated to the movement of the turbulent flow and it is evaluated from the variances of the components of

In this Section we will examine the relationship between TKE and Fractal Dimension of the

2

 <sup>1</sup> 22 2 '' '

ݓതതᇱଶതത . TKE is given by Eq. 5, in terms of energy per unit mass [15].

. Before incoming to study with certain details of this relationship, it

*TKE u v w* (5)

Fig 7. Negative slope and the high correlation coefficient show a good correlation.

**5.2. Turbulent Kinetic Energy and fractal dimension**

ݒ ,തതതᇱଶതݑ :velocity

fluctuations *u'* and *w'*

തതതᇱଶത and ത

In the stable case, the production of the TKE by shear is not sufficient to balance the dissipation energy at all levels and production by buoyancy is not happening. So, turbulence decay and TKE decrease along the time. In neutral conditions near the surface in agreement with the observations suggest that exists a near balance between shear production and viscous dissipation [14, 17]. In this case, a strong shear wind is normally present in the low atmosphere. In the unstable layer, convective case, vertical gradients of *w* are bigger than the vertical gradients of *u* and *v*, the TKE production is mainly due to buoyancy and transport from other levels. Turbulence is supported by thermal effects in the case of instability [17].

**Figure 9.** Variation of Fractal dimension and Energy kinetic turbulent versus time for *u* (upper-left panel) and *w* (lower-left) component of wind velocity at height *z* = 5.8 m. Scattered diagram of the fractal dimension and TKE with averages values and error bars (right panels).

As results it has found that an strong correlation between the dynamic magnitude, the Turbulent Kinetic Energy, and the Fractal Dimension exists. Figs. 9 and 10, in the left panels, show the evolution of the FD and TKE along the time for all days studied at two levels (5.8m Fig 9 and 13m in Fig 10). Diurnal cycle is also observed like in the study of invT in previous sections. This first result is in agreement with the normal variation of the turbulence between the day time and night time. Values of TKE are similar at two levels analyzed which may be interpreted as

turbulence is development into the stratum in a uniform way. Right panels, in the Figs. 9 and 10, is showing the scatter plot of FD and TKE values for all data used. Averaged of FD for arbitrary intervals of TKE together its error bars are also represented. It is observed a continuous increasing of the Fractal Dimension when the TKE grows, both for the *u'* and *w'* fluctuations. The maxima values of the kinetic energy are 2 m2 s-2, in average, at heights *z* =5.8 m, *z* =13 m and *z* = 32m (do not shown) corresponds to maxima values of mean FD. Differences may be observed for FD among the figures. For *u'* components at 5.8 m present the highest values of FD (around 1.20 ± 0.05) while in the remainder cases *w'* in two levels and *u'* at 13m the maximum of FD are nearly to 1.15. The relation studied does not seem linear, but FD increases very fast for small values of TKE and it has a light increasing of FD beyond of 1 m2/s2.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 41

In the atmosphere the relevant turbulent velocity scale is the surface friction velocity u\*

Friction velocity is also a turbulent parameter that measures the shearing stress into the

important dynamic property of the vertical structure of the lower atmosphere. Friction velocity can be obtained from vertical profile of average horizontal component of the wind

**Figure 11.** Linear regression between fractal dimension *w* component and TKE at height z=5.8 m. It is

**Figure 12.** Average points according defined interval of TKE of Fractal Dimension of vertical

component fluctuations at 13 m level. Coefficient of correlation, *R* = 0.8872, is shown together the right

shown the correlation coefficient in the two temporal intervals analyzed.

\* *u uw vw* '' ''

1/4 2 2

(6)

�. It is considered constant in whole Surface Layer. So, it is an

**5.3. Friction velocity and fractal dimension**

but also by mean of Eq. 6, based in turbulent fluxes.

surface layer, ���� � ��∗

of regression.

which includes the vertical momentum flux related by Eq. 6.

In order to investigate the correlation between FD and TKE, we were tried a linear regression for two different time interval, at first, from 06 UTC to 24 UTC of the initial day, including the period of day time and early night. In this record the variation of FD is maximum. Figure 11a presents the scatter plot and the linear regression obtaining an R2 coefficient of 0.698 that can be to consider satisfactory in spite of the shape of the cloud of points. The second period valued corresponds to the third day of the experiments from 48 UTC to 60 UTC, i.e., from midnight to noon. The scatter plot (Fig. 11b) presents similarity with the previous one, a cumulated group closes to the zero of TKE and there is spread points of higher values of FD for bigger TKE. The determination coefficient �� = 0.689 seems similar to the other period. Analogous results are obtained for *u'* component.

**Figure 10.** Variation of Fractal dimension and Energy kinetic turbulent versus time for u and w component of wind velocity at height z= 13 m. Diagram of fractal dimension and TKE with error bars

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 41

#### **5.3. Friction velocity and fractal dimension**

40 Fractal Analysis and Chaos in Geosciences

TKE and it has a light increasing of FD beyond of 1 m2/s2.

turbulence is development into the stratum in a uniform way. Right panels, in the Figs. 9 and 10, is showing the scatter plot of FD and TKE values for all data used. Averaged of FD for arbitrary intervals of TKE together its error bars are also represented. It is observed a continuous increasing of the Fractal Dimension when the TKE grows, both for the *u'* and *w'* fluctuations. The maxima values of the kinetic energy are 2 m2 s-2, in average, at heights *z* =5.8 m, *z* =13 m and *z* = 32m (do not shown) corresponds to maxima values of mean FD. Differences may be observed for FD among the figures. For *u'* components at 5.8 m present the highest values of FD (around 1.20 ± 0.05) while in the remainder cases *w'* in two levels and *u'* at 13m the maximum of FD are nearly to 1.15. The relation studied does not seem linear, but FD increases very fast for small values of

In order to investigate the correlation between FD and TKE, we were tried a linear regression for two different time interval, at first, from 06 UTC to 24 UTC of the initial day, including the period of day time and early night. In this record the variation of FD is maximum. Figure 11a presents the scatter plot and the linear regression obtaining an R2 coefficient of 0.698 that can be to consider satisfactory in spite of the shape of the cloud of points. The second period valued corresponds to the third day of the experiments from 48 UTC to 60 UTC, i.e., from midnight to noon. The scatter plot (Fig. 11b) presents similarity with the previous one, a cumulated group closes to the zero of TKE and there is spread points of higher values of FD for bigger TKE. The determination coefficient �� = 0.689

seems similar to the other period. Analogous results are obtained for *u'* component.

**Figure 10.** Variation of Fractal dimension and Energy kinetic turbulent versus time for u and w component of wind velocity at height z= 13 m. Diagram of fractal dimension and TKE with error bars In the atmosphere the relevant turbulent velocity scale is the surface friction velocity u\* which includes the vertical momentum flux related by Eq. 6.

$$\mu\_\* = \left[ \left( \overline{-\mu^\* w^\*} \right)^2 + \left( \overline{-v^\* w^\*} \right)^2 \right]^{1/4} \tag{6}$$

Friction velocity is also a turbulent parameter that measures the shearing stress into the surface layer, ���� � ��∗ �. It is considered constant in whole Surface Layer. So, it is an important dynamic property of the vertical structure of the lower atmosphere. Friction velocity can be obtained from vertical profile of average horizontal component of the wind but also by mean of Eq. 6, based in turbulent fluxes.

**Figure 11.** Linear regression between fractal dimension *w* component and TKE at height z=5.8 m. It is shown the correlation coefficient in the two temporal intervals analyzed.

**Figure 12.** Average points according defined interval of TKE of Fractal Dimension of vertical component fluctuations at 13 m level. Coefficient of correlation, *R* = 0.8872, is shown together the right of regression.

Under neutral conditions an extended model of vertical profiles of wind is the known logarithm profile: � � �∗ � �� � � �� �, where *k* is the Von Karman constant (� � ���� and �� is called the roughness length depending on the terrain. In the case of this study a very high rate of measurement of three components of velocity is available; therefore vertical flux of momentum is utilized here.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 43

**Figure 14.** Linear regression of FD calculated for w' wind component versus friction velocity obtained from vertical fluxes of momentum (Eq. 6). The regression coefficients are presented (*R* = 0.9614).

In this last Sub-section of results we try to investigate the relation between Fractal Dimension and the atmospheric stability in the Planetary Boundary Layer. We utilize the dynamic and thermal parameter denoted Bulk Richardson number, RiB, discussed in Section 4, and presented as a numerical approximation of the gradient Richardson number, Ri. The RiB value enables us to judge the atmospheric stability or, better named, the stability of stratification in the lower atmosphere. When Ri is small with any sign, the air flow is a pure shear flow driven by dynamic forces and we can say that layer is neutrally stratified. If Ri is near zero corresponding to neutrally stable boundary layer, one in which parcels displaced up and down adiabatically maintain exactly the same density as the surrounding air and

When Ri is large the air flow is driven by buoyancy. Positive values of Ri correspond with stable conditions and buoyancy forces keep out vertical displacement and mixing become less active. At the contrary, negatives and large Ri is in instability, where the thermal effects are doing air move vertically and fluctuations of wind components happen. Under instability turbulence increase and mixing of atmospheric properties: momentum, energy

Thus, we go to exam whether the parameter Fractal Dimension can be an appropriated index to classify the atmospheric stratification. As it is already explained the numerical approximation used here is the RiB (Eq. 4). The main difficulty in the use of this parameter is the fact of which is not robust, since small shear in the wind produce values extremes of RiB, both positive and negative. In order to avoid such situations it has been remove of the study cases with RiB < -5 (convective) and RiB > 1.5 (strong stability, hard inversion). The time series of RiB along the complete period of study is drawn in Figure 15a. Can be see how negative values of RiB are in day time and positive or near-zero are far away of the central day.

**5.4. Bulk Richardson number and fractal dimension**

and mass (concentration of the components) are more efficient.

thus experience no net buoyancy forces.

In this section we study as the Fractal Dimension behaves versus the friction velocity. The friction velocity *u\*,* presents maxima at day time closed to noon in every levels studied (5.8 and 13m) and minima at night. This maxima and minima are in concordance with the variation of FD in an analogous way to TKE treated in the last section (see Fig 13 a, c). The scatter plot for FD for *w*' component at the same levels and u\* is showing in Fig 13b and d; it indicates an acceptable correlation between them. FD increase according to growing of u\*. Again, a linear regression has been tried in order to quantify this correlation. Points in Fig 14 correspond to average values of FD of vertical components fluctuations against friction velocity. The positive slope and a good coefficient of correlation (*R* = 0.961) are outstanding results since they improve the ones in the TKE. This trend is observed in the *u'* and *w'* components at three heights studied.

**Figure 13.** a) and c) Fractal dimension of fluctuation of *w* component at 5.8 m and 13 m (blue) and Friction Velocity at the same height (green). b and d) Fractal dimension versus Friction Velocity.

**Figure 14.** Linear regression of FD calculated for w' wind component versus friction velocity obtained from vertical fluxes of momentum (Eq. 6). The regression coefficients are presented (*R* = 0.9614).

#### **5.4. Bulk Richardson number and fractal dimension**

42 Fractal Analysis and Chaos in Geosciences

� �� � � ��

logarithm profile: � � �∗

momentum is utilized here.

components at three heights studied.

Under neutral conditions an extended model of vertical profiles of wind is the known

called the roughness length depending on the terrain. In the case of this study a very high rate of measurement of three components of velocity is available; therefore vertical flux of

In this section we study as the Fractal Dimension behaves versus the friction velocity. The friction velocity *u\*,* presents maxima at day time closed to noon in every levels studied (5.8 and 13m) and minima at night. This maxima and minima are in concordance with the variation of FD in an analogous way to TKE treated in the last section (see Fig 13 a, c). The scatter plot for FD for *w*' component at the same levels and u\* is showing in Fig 13b and d; it indicates an acceptable correlation between them. FD increase according to growing of u\*. Again, a linear regression has been tried in order to quantify this correlation. Points in Fig 14 correspond to average values of FD of vertical components fluctuations against friction velocity. The positive slope and a good coefficient of correlation (*R* = 0.961) are outstanding results since they improve the ones in the TKE. This trend is observed in the *u'* and *w'*

**Figure 13.** a) and c) Fractal dimension of fluctuation of *w* component at 5.8 m and 13 m (blue) and Friction Velocity at the same height (green). b and d) Fractal dimension versus Friction Velocity.

�, where *k* is the Von Karman constant (� � ���� and �� is

In this last Sub-section of results we try to investigate the relation between Fractal Dimension and the atmospheric stability in the Planetary Boundary Layer. We utilize the dynamic and thermal parameter denoted Bulk Richardson number, RiB, discussed in Section 4, and presented as a numerical approximation of the gradient Richardson number, Ri. The RiB value enables us to judge the atmospheric stability or, better named, the stability of stratification in the lower atmosphere. When Ri is small with any sign, the air flow is a pure shear flow driven by dynamic forces and we can say that layer is neutrally stratified. If Ri is near zero corresponding to neutrally stable boundary layer, one in which parcels displaced up and down adiabatically maintain exactly the same density as the surrounding air and thus experience no net buoyancy forces.

When Ri is large the air flow is driven by buoyancy. Positive values of Ri correspond with stable conditions and buoyancy forces keep out vertical displacement and mixing become less active. At the contrary, negatives and large Ri is in instability, where the thermal effects are doing air move vertically and fluctuations of wind components happen. Under instability turbulence increase and mixing of atmospheric properties: momentum, energy and mass (concentration of the components) are more efficient.

Thus, we go to exam whether the parameter Fractal Dimension can be an appropriated index to classify the atmospheric stratification. As it is already explained the numerical approximation used here is the RiB (Eq. 4). The main difficulty in the use of this parameter is the fact of which is not robust, since small shear in the wind produce values extremes of RiB, both positive and negative. In order to avoid such situations it has been remove of the study cases with RiB < -5 (convective) and RiB > 1.5 (strong stability, hard inversion). The time series of RiB along the complete period of study is drawn in Figure 15a. Can be see how negative values of RiB are in day time and positive or near-zero are far away of the central day.

The Fractal Dimension *versus* Bulk Richardson number shows a different behavior depending on the kind of stability as is shown in the Fig. 15b. That is, in strong instability it is observed mean values of the Fractal Dimension almost constant, around 1.15, but increasing when we are going to RiB = 0 (near neutral stratification). In neutral conditions the Fractal Dimension is maximum and decreases quickly for stable conditions. This result agrees with the relationships found for the potential temperature differences in the Fig. 7 and the temperature differences in the layer 50 – 0.22 m (Fig. 5). This behavior is similar at three heights studied.

Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 45

increase, note the negative slope, but the coefficient of correlation is weaker than in dynamic

Maxima values of FD in near-neutral stratification it is supported for the results in the study of TKE or friction velocity, where enhance of dynamical effects gave a rising of the Fractal

This Chapter has treated about the influence of the atmospheric conditions, dynamics and thermal, over the fractal structure of the wind near the ground. The results that have been obtained in the work presented along the Chapter, lead to main following conclusions:

Finally, it can conclude that dynamical origin of the turbulence has a more clear relation

than the thermal origin with the fractal structure of the wind, but both are important.

*Department of Applied Mathematics (Biomathematics), University Complutense of Madrid, Spain*

*Department of Geophysics and Meteorology, University Complutense of Madrid, Spain*

analysis cases (r = 0.751).

Dimension.

**6. Conclusions**

counting dimension concept.

reach its maximum.

Gregorio Maqueda\* and José L. Cano

*University Complutense of Madrid, Spain*

*Department of Earth Physics, Astronomy and Astrophysics II.,*

**Author details**

Manuel Tijera

Carlos Yagüe

Corresponding Author

 \*

inversion in the stratum 0.22 - 50 m.

**Figure 15.** a) Bulk Richardson Number evaluated from the 32 - 5.8 m layer, along the whole period of study, where extreme values have been removed (-5< RiB< 1.5). b) Scatter plot of Fractal Dimension versus Bulk Richardson number; the maxima values of FD are in positive of RiB close to zero, and minima correspond to strong stability.

**Figure 16.**Linear regression of a set points, average FD in RiB intervals for the stable regime at height 13 m.

Behavior of FD according stability must be carry out separately in both kind of stability, as it was say before, instability give near constant values of Fractal Dimension except in quasineutrality, but in stability conditions, variation of FD it is clearly observed. In Fig. 16 it is shown the linear regression of mean values of the set points in stable stratification of the scatter plot of the Fig 13. As a result of that fit, can be conclude FD decrease as stability Analysis of Fractal Dimension of the Wind Speed and Its Relationships with Turbulent and Stability Parameters 45

increase, note the negative slope, but the coefficient of correlation is weaker than in dynamic analysis cases (r = 0.751).

Maxima values of FD in near-neutral stratification it is supported for the results in the study of TKE or friction velocity, where enhance of dynamical effects gave a rising of the Fractal Dimension.

## **6. Conclusions**

44 Fractal Analysis and Chaos in Geosciences

minima correspond to strong stability.

The Fractal Dimension *versus* Bulk Richardson number shows a different behavior depending on the kind of stability as is shown in the Fig. 15b. That is, in strong instability it is observed mean values of the Fractal Dimension almost constant, around 1.15, but increasing when we are going to RiB = 0 (near neutral stratification). In neutral conditions the Fractal Dimension is maximum and decreases quickly for stable conditions. This result agrees with the relationships found for the potential temperature differences in the Fig. 7 and the temperature differences in

**Figure 15.** a) Bulk Richardson Number evaluated from the 32 - 5.8 m layer, along the whole period of study, where extreme values have been removed (-5< RiB< 1.5). b) Scatter plot of Fractal Dimension versus Bulk Richardson number; the maxima values of FD are in positive of RiB close to zero, and

**Figure 16.**Linear regression of a set points, average FD in RiB intervals for the stable regime at height 13 m.

Behavior of FD according stability must be carry out separately in both kind of stability, as it was say before, instability give near constant values of Fractal Dimension except in quasineutrality, but in stability conditions, variation of FD it is clearly observed. In Fig. 16 it is shown the linear regression of mean values of the set points in stable stratification of the scatter plot of the Fig 13. As a result of that fit, can be conclude FD decrease as stability

the layer 50 – 0.22 m (Fig. 5). This behavior is similar at three heights studied.

This Chapter has treated about the influence of the atmospheric conditions, dynamics and thermal, over the fractal structure of the wind near the ground. The results that have been obtained in the work presented along the Chapter, lead to main following conclusions:

Finally, it can conclude that dynamical origin of the turbulence has a more clear relation than the thermal origin with the fractal structure of the wind, but both are important.

## **Author details**

Manuel Tijera *Department of Applied Mathematics (Biomathematics), University Complutense of Madrid, Spain*

Gregorio Maqueda\* and José L. Cano *Department of Earth Physics, Astronomy and Astrophysics II., University Complutense of Madrid, Spain*

Carlos Yagüe *Department of Geophysics and Meteorology, University Complutense of Madrid, Spain*

<sup>\*</sup> Corresponding Author

## **Acknowledgement**

This research has been funded by the Spanish Ministry of Science and Innovation (projects CGL2009-12797-C03-03). The GR35/10 program (supported by Banco Santander and UCM) has also partially financed this work through the Research Group "Micrometeorology and Climate Variability" (No 910437). Thanks to teams participating in SABLES-98 for the facilities with the data.

**Chapter 3**

© 2012 Tanizuka, licensee InTech. This is an open access chapter 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.

© 2012 Tanizuka, licensee InTech. This is a paper 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.

**Evolution of Cosmic System**

Additional information is available at the end of the chapter

What does it mean that the cosmic radio wave flux density varies with the passage of time is an interesting question; the radio wave is of the quasar, a system of galaxy, which is distributed in our universe from a few billions of light years to the distanse close to the big bang age and has been radiating immense electromagnetic energy from it by the synchrotron radiation that we may able to make a measurement of the flux density at micro wave bands with a radio interferometer[3,4]. A group of radio observers and astronomers has been monitoring daily so far over several years extragalactic radio sources (radio galaxies, quasars, etc.) and the monitored data were kindly shared with us who were interested in using for analysis[5]. In a few recent decades, the chaos and fractal theory has been intensively studied and developed in the fields of mathematics, computer numerical analysis, natural sciences and technologies[1], and in same decades, the nonlinear time series analysis methods have been developed intensely based on the newly understood ideas of the theory for

The study in this chapter is motivated by the three factors mentioned above to analyze the time series of the radio wave flux density from the cosmological object, primarily, with one of the nonlinear methods, for finding the dynamics related to the cosmic object, including its information in the flux density variations. We hoped that if we could infer the dynamics and if the result would be found to have any rule changing with the magnitude of the red shift of the object we might have some knowledge concerning to the evolution of our universe. The hope has been prompted us to continue consistently to analyze the time series data. The period of monitor over several years is extremely short compared with the cosmic age, however , the analysis result of the time series data in newly developing methods may give us a new sight viewed from the nonlinear dynamics in the short time scale for the cosmic

Noboru Tanizuka

**1. Introduction**

dynamical system.

http://dx.doi.org/10.5772/51435

analyzing the nonlinear phenonena[2].

## **7. References**

## **Chapter 3**

## **Evolution of Cosmic System**

## Noboru Tanizuka

46 Fractal Analysis and Chaos in Geosciences

This research has been funded by the Spanish Ministry of Science and Innovation (projects CGL2009-12797-C03-03). The GR35/10 program (supported by Banco Santander and UCM) has also partially financed this work through the Research Group "Micrometeorology and Climate Variability" (No 910437). Thanks to teams participating in SABLES-98 for the

[2] Falkovich G. and Sreenivasan K. R. (2006). Lessons from Hydrodynamic Turbulence.

[3] Falconer K. J. (1990). *Fractal Geometry Mathematical Foundations and Applications*. John

[6] Abarbanel H. D. I., Rabinovich M. I., and Sushchik M. M. (1993). *Introduction to Nonlinear*

[7] Kolmogorov A. N. (1941a). The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dolk. Akad. Nauk SSSR 30, 299-303 (Reprinted in

[8] Kolmogorov A. N. (1941b). On degeneration (decay) of isotropic turbulence in an

[9] Cuxart J., C. Yagüe, G. Morales, E. Terradellas., J. Orbe, J. Calvo, A. Fernández, M.R. Soler, C. Infante, P. Buenestado, A. Espinalt, H.E. Joergensen, J.M. Rees, J. Vilà, J.M. Redondo, I.R.Cantalapiedra, and Conangla, L. (2000). Stable atmospheric boundary layer experiment in Spain (SABLES 98): A report. Bound.-Layer Meteorol., 96, 337-370. [10] Kaimal, J. C. and Finnigan J. J. (1994). *Atmospheric Boundary Layer Flows. Their Structure*

[11] Vassilicos J. C. and J. C. R. Hunt, (1991). Fractal dimensions and spectra of interfaces

[14] Stull R. B. (1988). *An introduction to Boundary Layer Meteorology*. Kluwer Academic

[15] Arya S. P. (2001). *Introduction to Micrometeorology*. International Geophysics Series,

[16] Lynch A.H and Cassano J.J. (2006). *Atmospheric Dynamics*. John Wiley & Sons, Ltd. 280 pp [17] Garratt, J. R. (1992). *The atmospheric boundary layer*. Cambridge University Press, 316 pp.

[12] Hausdorff F. (1918). Dimension und äusseres Mass. Mathematische Ann. 79 157-179 [13] Tijera, M., Cano J.L., Cano, D., Bolster, B. and Redondo J.M. (2008). Filtered deterministic waves and analysis of the fractal dimension of the components of the

wind velocity. Nuovo Cimento C. Geophysics and Space Physics 31, 653-667

incompressible viscous liquid, Dolk. Akad. Nauk SSSR 31, 538-540.

with application to turbulence. Proc. R. Soc. London Ser. A 435-505.

[4] Peitgen H., Jürgens H. and Saupe D. (2004): *Chaos and Fractals.* Springer-Verlag 971pp. [5] Grassberger P. and Procaccia I. (1983). Characterization of Strange Attractors. Phys. Rev

[1] Frisch, U. (1995). *Turbulence*. Cambridge Univesity Press, England, 296 pp.

**Acknowledgement**

facilities with the data.

Physics Today 59: 43-49.

Wiley & Sons, Ltd. 288 pp

*Dinamic for Physicists*. Word Scientific. 158 pp

*and Measurement*, Oxford University Press. 289 pp

Proc. R. Soc. Lond. A 434, 9-13 (1991))

Lett. 50, 346-349

Publishers.

Academic Press. 420 pp

**7. References**

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51435

## **1. Introduction**

What does it mean that the cosmic radio wave flux density varies with the passage of time is an interesting question; the radio wave is of the quasar, a system of galaxy, which is distributed in our universe from a few billions of light years to the distanse close to the big bang age and has been radiating immense electromagnetic energy from it by the synchrotron radiation that we may able to make a measurement of the flux density at micro wave bands with a radio interferometer[3,4]. A group of radio observers and astronomers has been monitoring daily so far over several years extragalactic radio sources (radio galaxies, quasars, etc.) and the monitored data were kindly shared with us who were interested in using for analysis[5]. In a few recent decades, the chaos and fractal theory has been intensively studied and developed in the fields of mathematics, computer numerical analysis, natural sciences and technologies[1], and in same decades, the nonlinear time series analysis methods have been developed intensely based on the newly understood ideas of the theory for analyzing the nonlinear phenonena[2].

The study in this chapter is motivated by the three factors mentioned above to analyze the time series of the radio wave flux density from the cosmological object, primarily, with one of the nonlinear methods, for finding the dynamics related to the cosmic object, including its information in the flux density variations. We hoped that if we could infer the dynamics and if the result would be found to have any rule changing with the magnitude of the red shift of the object we might have some knowledge concerning to the evolution of our universe. The hope has been prompted us to continue consistently to analyze the time series data. The period of monitor over several years is extremely short compared with the cosmic age, however , the analysis result of the time series data in newly developing methods may give us a new sight viewed from the nonlinear dynamics in the short time scale for the cosmic dynamical system.

© 2012 Tanizuka, licensee InTech. This is an open access chapter 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. © 2012 Tanizuka, licensee InTech. This is a paper 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.

#### **2. Linear and nonlinear systems**

#### **2.1. Linear system**

The result of a linear data computed with Fourier spectral analysis gives a pure periodicity, which means predictability of event(s) for unlimited future. No event of evolution can be expected by a linear data, and a linear system is not a way of our natural world. On the other hand, the result of a pure random data computed with it gives no structure of the periodicity, a continuous spectrum, and means no event of predictability.

#### **2.2. Nonlinear system**

One of discrete nonlinear dynamical systems is given in this section for explanation. The logistic map function is given in Eq.(1) with a difference equation. Given that the time series data is generated from this system by iterating at � = 0�1� . . � �.

$$f(\mathbf{x}) = a\mathbf{x}(1-\mathbf{x});\ \mathbf{x}(t+1) = f(\mathbf{x}(t))\tag{1}$$

Evolution of Cosmic System 49

**Figure 1.** Fourier analysis for time series data of (a) uniform random numbers and (b) logistic map

**Figure 2.** Graphical analysis for (a) random time series data and (b) logistic time series data given in

time. This means that in spite of extremely small error in the chaos deterministic system, the error is extended to a scale of state space in a limited time, that the system is unpredictable.

The parametric dependence is characteristic to a chaos system. The vertical axis on the right diagram in Fig. 3 shows the values of number computed by Eq.(1) at parameter � in the range of �.� ≤ � ≤ 4.0, along the horizontal axis. At a parameter value between 3.82 and 3.83, the behavior of the system drastically changes to periodicity. In the logistic system the parameter is interpreted as the environment for a living thing to survive. For nonlinear systems, a parameter value becomes crucial for the system's behavior. For example, in the logistic dynamics, extinction or evolution of the system depends on the parameter value.

The fact was explained in the Fourier spectrum of the time series in Fig. 1 (b).

system at � � 4.0. The spectrum is shown by the absolute of the Fourier transform.

Fig. 1.

The result of the Fourier analysis for the time series data at parameter � = 4.0 in the function of Eq.(1) is indistinguishable from the result of the random data. The spectra for the random data and logistic time series data are given in Fig.1. The latter comes from the deterministic system, but the analysis result does not disclose the nonlinearity of the data. This is a reason why a help of a new method is necessary for the analysis of data from the natural phenomena which are often complex for us to understand and may include such complexity as above in part. A method to distinguish between the kinds of data is to draw a graph composed of coordinates with time-lagged components, that is, a graphical execution of differentiation for a time series. The result of the graphical execution for the random and logistic time series in Fig. 1 is shown in Fig. 2. As a result, in the figure the two time series are correctly discriminated. The time series generated from Eq.(1) resulted in a quadratic expression with the second dimension of coordinates (see Fig. 2(b)). On the other hand, the random time series resulted in the uncorrelated graph as shown in Fig. 2(a).1

The time series data measured from natural and social systems is not generated in such a mathematical way and so complicated that more intricated process is required to deal with data. This problem will be discussed in later sections. Before we do this, some methods of quantifying fractal data and some characteristics of chaos dynamics (the initial value dependence and the parametric dependence) are exemplified by the logistic system.

Time series data were generated from Eq.(1) at parameter � = �.� with two initial values at �(0) = 0.10000 and at x(0) = 0.10001, and the two time series are drawn as the diagram shown in Fig. 3 (left). The diagram shows that the trajectories of the two time series separate in a few tens of iteration time, which means that a small error was extended to a magnitude of space the time series occupies within the limited time and the prediction is failed over the

<sup>1</sup> It is true in the second dimension of coordinates. If the time series is generated by the function of higher degree, the correlation may be true in the graph at a higher dimension of coordinates.

**2.1. Linear system**

**2.2. Nonlinear system**

**2. Linear and nonlinear systems**

The result of a linear data computed with Fourier spectral analysis gives a pure periodicity, which means predictability of event(s) for unlimited future. No event of evolution can be expected by a linear data, and a linear system is not a way of our natural world. On the other hand, the result of a pure random data computed with it gives no structure of the

One of discrete nonlinear dynamical systems is given in this section for explanation. The logistic map function is given in Eq.(1) with a difference equation. Given that the time series

The result of the Fourier analysis for the time series data at parameter � = 4.0 in the function of Eq.(1) is indistinguishable from the result of the random data. The spectra for the random data and logistic time series data are given in Fig.1. The latter comes from the deterministic system, but the analysis result does not disclose the nonlinearity of the data. This is a reason why a help of a new method is necessary for the analysis of data from the natural phenomena which are often complex for us to understand and may include such complexity as above in part. A method to distinguish between the kinds of data is to draw a graph composed of coordinates with time-lagged components, that is, a graphical execution of differentiation for a time series. The result of the graphical execution for the random and logistic time series in Fig. 1 is shown in Fig. 2. As a result, in the figure the two time series are correctly discriminated. The time series generated from Eq.(1) resulted in a quadratic expression with the second dimension of coordinates (see Fig. 2(b)). On the other hand, the

The time series data measured from natural and social systems is not generated in such a mathematical way and so complicated that more intricated process is required to deal with data. This problem will be discussed in later sections. Before we do this, some methods of quantifying fractal data and some characteristics of chaos dynamics (the initial value

Time series data were generated from Eq.(1) at parameter � = �.� with two initial values at �(0) = 0.10000 and at x(0) = 0.10001, and the two time series are drawn as the diagram shown in Fig. 3 (left). The diagram shows that the trajectories of the two time series separate in a few tens of iteration time, which means that a small error was extended to a magnitude of space the time series occupies within the limited time and the prediction is failed over the

1 It is true in the second dimension of coordinates. If the time series is generated by the function of higher degree, the

correlation may be true in the graph at a higher dimension of coordinates.

dependence and the parametric dependence) are exemplified by the logistic system.

�(�) = ��(1��)� �(��1) = �(�(�)) (1)

periodicity, a continuous spectrum, and means no event of predictability.

random time series resulted in the uncorrelated graph as shown in Fig. 2(a).1

data is generated from this system by iterating at � = 0�1� . . � �.

**Figure 1.** Fourier analysis for time series data of (a) uniform random numbers and (b) logistic map system at � � 4.0. The spectrum is shown by the absolute of the Fourier transform.

**Figure 2.** Graphical analysis for (a) random time series data and (b) logistic time series data given in Fig. 1.

time. This means that in spite of extremely small error in the chaos deterministic system, the error is extended to a scale of state space in a limited time, that the system is unpredictable. The fact was explained in the Fourier spectrum of the time series in Fig. 1 (b).

The parametric dependence is characteristic to a chaos system. The vertical axis on the right diagram in Fig. 3 shows the values of number computed by Eq.(1) at parameter � in the range of �.� ≤ � ≤ 4.0, along the horizontal axis. At a parameter value between 3.82 and 3.83, the behavior of the system drastically changes to periodicity. In the logistic system the parameter is interpreted as the environment for a living thing to survive. For nonlinear systems, a parameter value becomes crucial for the system's behavior. For example, in the logistic dynamics, extinction or evolution of the system depends on the parameter value.

$$\lambda = \lim\_{N \to \infty} \frac{1}{N} \sum\_{t=0}^{N-1} \log |f'(\mathfrak{x}(t))|\,\,\,\,\,\tag{2}$$

$$H(\varepsilon) = -\Sigma\_{l=1}^{M} p\_l \log p\_l, \quad \Sigma\_{l=1}^{M} p\_l = 1 \,, \tag{3}$$

$$D\_0 = \lim\_{\varepsilon \to 0} \frac{\log \mathcal{M}(\varepsilon)}{-\log \varepsilon}, D\_1 = \lim\_{\varepsilon \to 0} \frac{H(\varepsilon)}{-\log \varepsilon}, \ D\_2 = \lim\_{\varepsilon \to 0} \frac{\log \Sigma\_{\varepsilon = 1}^M p\_\varepsilon^2}{\log \varepsilon} \tag{4}$$

$$D\_q = \lim\_{\varepsilon \to 0} \frac{1}{q-1} \frac{\log \sum\_{l=1}^{M} p\_l q}{\log \varepsilon} \tag{5}$$

measured data, even if it is at �=�, because we know little about the system's dynamics, on which we are to study, and in addition, it is impossible to box-count a manifold constructed in a state space with unknown dimension. Fortunately, we have Grassbergar-Procaccia algorithm (GPA) to compute the measured data and substitute the result for the fractal dimension at �=� [2]. The GPA is discussed in section 5.

Evolution of Cosmic System 53

**Figure 6.** Takens' theorem given by a schemetic diagram to infer the system's dynamics from measured

of the system's (source) dynamics at �*-*th dimension. Takens' theory claims that the attractor reconstructed on the embedding space in Fig. 6 is generic embedding under condition � � ��.[2] The attractor in the embedding state space is a theoretical reflection of the manifold, that is, the embedding map is ��M � ��, in which the condition does not need to be satisfied.The source dynamics at the observed system could be inferred in this way. It is generally impossible to solve the function, in a definitive form, of the system's dynamics.

Takens' theorem is summarized in Eq. (7) ~ Eq. (9). ℎ is a map function transforming the

The experimantal expression of Eq.(6), reconstructed by using measured data, corresponds to the theoretical expression of Eq.(8), assuming that a time delay is neglected and the time span is same between both systems. We have the attractor � by analyzing measured data,

It is same to say that Galileo Galilei could find experimentally the gravity on the earth, but could not express it in a deterministic expression as the Newton's equation. The map (ℎ) in Eq.(9) is similar to the gravity in the era. At present, it is only possible to get access to the geometrical manifold with the way given in this section. A discussion on the method how to

It is useful to give attention to noise inevitaby coming into the dynamical system and the

but it is difficult to have a deterministic expression of the manifold M in Eq.(9).

apply the fractal dimension to quantify the geometrical manifold will be given.

observation system. The noise comes into the two systems[2],

�� is measured by �M, with � the observation map and M the manifold

�� M � M � �� M � � � �� M � �� (7)

�(�) � ��(�)� ���(�)�� � � �(����(�))� (8)

ℎ� � � M (9)

�(���) � �(�(�)) � �(�) (10)

�(�) � �(�(�)) � �(�) (11)

time series[2].

The data set ��(�)�����

embedded attractor into the original attractor.

### **4. Dynamical system and time series data**

The purpose of this section is to discuss on the problem: if we measure the time series data from a system of nature how we access to a function of the system with which the data is generated. If the nature was constructed by the mathematics it would have been going well to solve the problem. Unfortunately, the nature, I believe, do not go so easy. We need to solve it by devising the data reconstruction and by applying above nonlinear methods to it .

#### **4.1. Time series data**

The originally measured data is defined as follows: A measurement starts at time �� with sampling rate � and the time series is expressed in the following way, with � natural number

$$\left(\boldsymbol{\chi}(t\_0), \boldsymbol{\chi}(t\_0 + \tau), \dots, \boldsymbol{\chi}(t\_0 + i\tau), \dots\right)$$

If we find a time lag �� proper for the system in three dimensional state space, the time series is reconstructed, for example, to three dimensional vector �� at time �.

$$V\_t = (\mathbf{y}(t), \mathbf{y}(t + \tau'), \mathbf{y}(t + 2\tau')); \ t = 1, 2, \cdots$$

Vector �� is embedded in three dimensional time-lagged state space. As the time goes by, the vector draws a trajectory in the state space. The measured data may geometrically express its functional property in this way. The method can be considered to differentiate the data in the state space in a graphycal way. The dimension of the system is unknown in advance, so an original dimension of the reconstructed vector must be searched by changing it one by one until to find the optimal one which is called the embedding dimension. The dynamics of the observed system have a fractal dimension in the embedding dimension. A manifold is drawn in the embedding dimension in this way from the ofserved data and it is called the attractor of the system given by the observed data. From the manifold we infer the original function, as a mathematical nonlinear equation.

#### **4.2. Embedding theory**

The attractor is the manifold of a dynamical system, from which a physical quantity is continuausly released to be observed and the time series is, as a result, accessed to be analized. The time series is reconstructed in the form of a vector �� at �-th dimension to be embedded in the state space. In the following equation the time lag �� was replaced by �.

$$\mathcal{V}\_t = \{\mathbf{y}(t), \mathbf{y}(t+\tau), \dots, \mathbf{y}(t+(m-1)\tau)\}\tag{6}$$

**4.1. Time series data**

If we find a time lag ��

**4.2. Embedding theory**

dimension at �=� [2]. The GPA is discussed in section 5.

**4. Dynamical system and time series data**

measured data, even if it is at �=�, because we know little about the system's dynamics, on which we are to study, and in addition, it is impossible to box-count a manifold constructed in a state space with unknown dimension. Fortunately, we have Grassbergar-Procaccia algorithm (GPA) to compute the measured data and substitute the result for the fractal

The purpose of this section is to discuss on the problem: if we measure the time series data from a system of nature how we access to a function of the system with which the data is generated. If the nature was constructed by the mathematics it would have been going well to solve the problem. Unfortunately, the nature, I believe, do not go so easy. We need to solve it by devising the data reconstruction and by applying above nonlinear methods to it .

The originally measured data is defined as follows: A measurement starts at time �� with sampling rate � and the time series is expressed in the following way, with � natural number

�(��), �(�� � �),⋯,�(�� � ��), ⋯

), �(� � ���

Vector �� is embedded in three dimensional time-lagged state space. As the time goes by, the vector draws a trajectory in the state space. The measured data may geometrically express its functional property in this way. The method can be considered to differentiate the data in the state space in a graphycal way. The dimension of the system is unknown in advance, so an original dimension of the reconstructed vector must be searched by changing it one by one until to find the optimal one which is called the embedding dimension. The dynamics of the observed system have a fractal dimension in the embedding dimension. A manifold is drawn in the embedding dimension in this way from the ofserved data and it is called the attractor of the system given by the observed data. From the manifold we infer the

The attractor is the manifold of a dynamical system, from which a physical quantity is continuausly released to be observed and the time series is, as a result, accessed to be analized. The time series is reconstructed in the form of a vector �� at �-th dimension to be

is reconstructed, for example, to three dimensional vector �� at time �.

�� = (�(�), �(����

original function, as a mathematical nonlinear equation.

embedded in the state space. In the following equation the time lag ��

proper for the system in three dimensional state space, the time series

))� � = �,�, ⋯

�� = (�(�), �(���), ⋯ , �(� � (� � �)�)) (6)

was replaced by �.

**Figure 6.** Takens' theorem given by a schemetic diagram to infer the system's dynamics from measured time series[2].

The data set ��(�)����� �� is measured by �M, with � the observation map and M the manifold of the system's (source) dynamics at �*-*th dimension. Takens' theory claims that the attractor reconstructed on the embedding space in Fig. 6 is generic embedding under condition � � ��.[2] The attractor in the embedding state space is a theoretical reflection of the manifold, that is, the embedding map is ��M � ��, in which the condition does not need to be satisfied.The source dynamics at the observed system could be inferred in this way. It is generally impossible to solve the function, in a definitive form, of the system's dynamics.

Takens' theorem is summarized in Eq. (7) ~ Eq. (9). ℎ is a map function transforming the embedded attractor into the original attractor.

$$f \colon \mathbb{M} \to \mathbb{M} \quad \Rightarrow \quad g \colon \mathbb{M} \to \mathbb{R} \quad \Rightarrow \quad V \colon \mathbb{M} \to \mathbb{R}^m \tag{7}$$

$$V(\mathbf{x}) = \left( g(\mathbf{x}), g\left( f(\mathbf{x}) \right), \dots, g\left( f^{m-1}(\mathbf{x}) \right) \right) \tag{8}$$

$$h \colon V \to \mathcal{M} \tag{9}$$

The experimantal expression of Eq.(6), reconstructed by using measured data, corresponds to the theoretical expression of Eq.(8), assuming that a time delay is neglected and the time span is same between both systems. We have the attractor � by analyzing measured data, but it is difficult to have a deterministic expression of the manifold M in Eq.(9).

It is same to say that Galileo Galilei could find experimentally the gravity on the earth, but could not express it in a deterministic expression as the Newton's equation. The map (ℎ) in Eq.(9) is similar to the gravity in the era. At present, it is only possible to get access to the geometrical manifold with the way given in this section. A discussion on the method how to apply the fractal dimension to quantify the geometrical manifold will be given.

It is useful to give attention to noise inevitaby coming into the dynamical system and the observation system. The noise comes into the two systems[2],

$$\mathbf{x}(t+1) = f(\mathbf{x}(t)) + \eta(t) \tag{10}$$

$$\mathbf{y}(t) = \mathbf{g}(\mathbf{x}(t)) + \xi(t) \tag{11}$$

where �(�) and �(�) are system's noise and observation noise, respectively. This makes it very difficult to judge observed data that system's function � is a genuine chaos or not, even if the analysis gives a chaotic result, because the chaos trajectory depends severely on the initial condition.

Evolution of Cosmic System 55

��� to eliminate the major error outside

��� , �� and σ the average and the standard

**4.4. Other methods of analysis**

of |�| and � in the frequency range of �

reconstructed time series �(�) � ∑ (�(�) � ��) �

cross-checking the knowledge of the result of source dynamics.

the range of 10 ≤ τ ≤ 1000 days.

**5.1. Monitored cosmic objects**

briefly, and the result was published in [7].

**5. Time series data**

We introduce briefly three methods of analyzing the characteristics of the flux density variation, in which the same data were also computed for reference. The results will cover different aspects of the variation. The first is the spectral index α : The modulus of the fourier spectrum |�|, the intensity of the fluctuations at frequency � is computed over the period of 1024 days; the method of least squares was used for the pairs of the logarithms

��� ≤ �� ≤ �

the frequency range. The power law |�| � ��� was computed for all of the observed data. The second is the Higuchi's fractal dimension � [9]. The absolute change of the flux density at interval � averaged over the period (1024 days) is computed. Given the averaged change �, the relationship ���� stands for the interval 1 ≤ � ≤ 100 days. The detailed algorithm can be reffered in reference [9]. The dimension � expresses a complexity of the variation in the range 1≤�≤�, that is, from liner, ��1, to the plane,�� � �, from fractal dimensional view. The third is the Hurst exponent � [10]. Given the ratio ��σ over the interval τ, with range � a difference from top to bottom levels of the

deviation of �(�), respectively, in the interval τ. The relationship ��� � (���)� stands in

The detailed introductions for the methods be referred in [9-11]2. The result analyzed in these methods will be shown later. The results computed in above principles are useful for

The extragaractic radio sources generate the time series data of the radio wave flux density for us to observe and to analyze their system's dynamics to see a mechanism how the cosmic

Compact extragaractic radio sources had been monitored daily by Waltmann et al. at GBI radio wave observatory over 3000 days from 1979 [5,6]. Waltmann et al were kind to send us the data of 46 extragalactic objects, from which 21 QSOs and 7 BL Lacs were selected for analysis. At the beginning we analyzed the data in the methods of the spectral index, of the Higuchi's fractal dimension and of the Hurst exponent. The methods will be explained

The monitored microwave frequencies were at 2.7 and 8.1GHz; and the red shift (the indicator of cosmological distance) of the monitored objects ranged from 0.15 to 2.22 ( from one billion to ten billions of light years). The name of the objects are shown in Table 1 as

<sup>2</sup> 1�� noise characterized as the power law events in the electronic circuit is in reference [11].

object has been evolved in the cosmological age from a dynamic aspect of view.[3]

�� �

#### **4.3. Correlation dimension**

The attractor is reconstructed in the embedded state space from observed time series. We are interested in the dynamical system of the quasar, the galactic object distributed at cosmological distances (up to ten billions of light years), which releases vast energy by synchrotron radiation and enables us to research some cosmic information by the fluctuations of the microwave flux density. We study the dynamics involving in the time series to know the structure of the dynamics for the system at different cosmological distances. We may be to have some dynamical knowledge of the systems's evolution in the experimental method. The system's information is, in our context, is the manifold for the dynamics of the quasar system at different cosmological distances. The data is the time series of the flux density of the microwaves, 2.7GHz and 8.1GHz, for more than twenty quasars, daily monitored over thousand days. The fractal dimension of the manifold is to be analyzed. The dimensions introduced in the section 3 are difficult to compute with the reconstructed data. Fortunately we have an useful method to compute the correlation dimension developed by Grassberger and Procaccia (GPA) as the substitute of ��. The algorithm for calculation are the correlation sum and the fractal exponent in the following expressions, � the number of points in the embedding state space at �-th dimension.

Given each point �� ��(� � �) in Fig. 5 (not number of the cell), with�� the number of full points and � the diameter of hypersphere, the correlation sum is expressed by [1,2]

$$C^{(m)}(r) = \frac{1}{N} \Sigma\_{k=1}^{N} \frac{1}{N-1} \Sigma\_{f=1}^{N-1} \theta \left[r - \|\nu(f) - \nu(k)\|\right],\tag{12}$$

in which �[�] is the Heavside function, counting a pair of �-th dimensional vector points whose distance is within � over all pairs of points. Eq.(12) counts the probability that any pair of state space points meets within a length of � in �-th dimensional state space. The fractal dimension at �-th dimensional state space is expressed by ( see Eq.(4))

$$D\_2^{(m)} = \lim\_{r \to 0} \frac{\log \mathcal{C}^{(m)}(r)}{\log r} \tag{13}$$

The optimal correlation dimension �� for the attractor embedded in the optimal state space for the time series in quetion is defined as �� (�)������� = �s when the value of �� (�) ceases to increase as the increase of �. This means that at a full embedding dimension, any points in state space of the attractor can not occupy same position by the no-intersection theorem of chaos [1]. The �� is a fractional dimension and the �� is the embedding dimension in which the system works (�� ����� � ��) .

### **4.4. Other methods of analysis**

54 Fractal Analysis and Chaos in Geosciences

**4.3. Correlation dimension**

space at �-th dimension.

�(�)

for the time series in quetion is defined as ��

the system works (�� ����� � ��) .

(�) <sup>=</sup> �

initial condition.