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13 pages, 1255 KiB

Open AccessArticle

A Joint Multifractal Approach to Solar Wind Turbulence

by Giuseppe Consolini and Paola De Michelis

Fractal Fract. 2023, 7(10), 748; https://doi.org/10.3390/fractalfract7100748 - 11 Oct 2023

Cited by 1 | Viewed by 914

Previous studies have shown that solar wind, a plasma medium with turbulent dynamics, exhibits anomalous scaling features, i.e., intermittency, in the inertial domain. This intermittent nature has primarily been investigated through the study of the scaling features of the structure functions of single [...] Read more.

Previous studies have shown that solar wind, a plasma medium with turbulent dynamics, exhibits anomalous scaling features, i.e., intermittency, in the inertial domain. This intermittent nature has primarily been investigated through the study of the scaling features of the structure functions of single quantities. We use a novel approach based on joint multifractal analysis in this study to simultaneously investigate the scaling characteristics of both the magnetic field and the plasma velocity in solar wind turbulence. Specifically, we focus on the joint multifractal behavior of magnetic and velocity field fluctuations in both fast and slow solar wind streams observed by the ESA-Ulysses satellite, with the goal of identifying any differences in their joint multifractal characteristics. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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21 pages, 5865 KiB

Open AccessArticle

Fractal Derivatives and Singularity Analysis of Frequency—Depth Clusters of Earthquakes along Converging Plate Boundaries

by Qiuming Cheng

Fractal Fract. 2023, 7(10), 721; https://doi.org/10.3390/fractalfract7100721 - 30 Sep 2023

Cited by 3 | Viewed by 1098

Abstract

Fractional calculus (FC) has recently received increasing attention due to its applications in many fields involving complex and nonlinear systems. However, one of the key challenges in using FC to deal with fractal or multifractal phenomena is how to relate functions to geometries [...] Read more.

Fractional calculus (FC) has recently received increasing attention due to its applications in many fields involving complex and nonlinear systems. However, one of the key challenges in using FC to deal with fractal or multifractal phenomena is how to relate functions to geometries with fractal dimensions. The current paper demonstrates how fractal calculus can be used to represent physical properties such as density defined on fractal geometries that no longer have the Lebesgue additive properties required for ordinary calculus. First, it introduces the recently proposed concept of fractal density, that is, densities defined on fractals and multifractals, and then shows how fractal calculus can be used to describe fractal densities. Finally, the singularity analysis based on fractal density calculation is used to analyze the depth clustering distribution of seismic frequencies around the Moho transition zone in the subduction zone of the Pacific plates and the Tethys collision zones. The results show that three solutions (linear, log-linear, and double log-linear) of a unified differential equation can describe the decay rate of the fractal density of depth clusters at the number (frequencies) of earthquakes. The spatial distribution of the three groups of earthquakes is further divided according to the three attenuation relationships. From north latitude to south latitude, from the North Pacific subduction zone to the Tethys collision zone, and then to the South Pacific subduction zone, the attenuation relationships of the earthquake depth distribution are generally from a linear, to log-linear, to double log-linear pattern. This provides insight into the nonlinearity of the depth distribution of earthquake swarms. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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15 pages, 1707 KiB

Open AccessArticle

Lagrangian Particle Dispersion in a Poor Man’s Magnetohydrodynamic Turbulence Model

by Tommaso Alberti and Vincenzo Carbone

Fractal Fract. 2023, 7(9), 662; https://doi.org/10.3390/fractalfract7090662 - 31 Aug 2023

Viewed by 677

Abstract

Lagrangian dispersion of fluid particle pairs refers to the study of how individual fluid particles disperse and move in a fluid flow, providing insights to understand transport phenomena in various environments, from laminar to turbulent conditions. Here, we explore this phenomenon in synthetic [...] Read more.

Lagrangian dispersion of fluid particle pairs refers to the study of how individual fluid particles disperse and move in a fluid flow, providing insights to understand transport phenomena in various environments, from laminar to turbulent conditions. Here, we explore this phenomenon in synthetic velocity and magnetic fields generated through a reduced-order model of the magnetohydrodynamic equations, which is able to mimic both a laminar and a turbulent environment. In the case of laminar conditions, we find that the average square distance between particle pairs increases linearly with time, implying a dispersion pattern similar to Brownian motion at all time steps. On the other hand, under turbulent conditions, surprisingly enough we observe a Richardson scaling, indicating a super-ballistic dispersion pattern, which aligns with the expected scaling properties for a turbulent environment. Additionally, our study reveals that the magnetic field plays an organizing role. Lastly, we explore a purely hydrodynamic case without magnetic field effects, showing that, even in a turbulent environment, the behavior remains Brownian-like, highlighting the crucial role of the magnetic field in generating the Richardson scaling observed in our model. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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9 pages, 466 KiB

Open AccessArticle

Scaling Analysis of Time-Reversal Asymmetries in Fully Developed Turbulence

by François G. Schmitt

Fractal Fract. 2023, 7(8), 630; https://doi.org/10.3390/fractalfract7080630 - 18 Aug 2023

Viewed by 865

Abstract

In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized [...] Read more.

In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized by scaling properties, and we consider here how irreversibility depends on the scale. Indicators of time-reversal symmetry for time series are tested involving triple correlations in a non-symmetric way. These indicators are built so that they are zero for a time-reversal symmetric time series, and a departure from zero is an indicator of irreversibility. We study these indicators applied to two fully developed turbulence time series, from flume tank and wind tunnel databases. It is found that irreversibility occurs in the inertial range and has scaling properties with slopes close to one. A maximum value is found around the injection scale. This confirms that the irreversibility is associated with the turbulent cascade in the inertial range and shows that the irreversibility is maximal at the injection scale, the largest scale of the turbulent cascade. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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28 pages, 5840 KiB

Open AccessArticle

Topological Vortexes, Asymptotic Freedom, and Multifractals

by Alexander Migdal

Fractal Fract. 2023, 7(5), 351; https://doi.org/10.3390/fractalfract7050351 - 25 Apr 2023

Cited by 1 | Viewed by 821

Abstract

In this paper, we study the Kelvinons, which are monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing [...] Read more.

In this paper, we study the Kelvinons, which are monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, which is in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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16 pages, 3981 KiB

Open AccessArticle

Relationships between Reference Evapotranspiration and Meteorological Variables in the Middle Zone of the Guadalquivir River Valley Explained by Multifractal Detrended Cross-Correlation Analysis

by Javier Gómez-Gómez, Ana B. Ariza-Villaverde, Eduardo Gutiérrez de Ravé and Francisco J. Jiménez-Hornero

Fractal Fract. 2023, 7(1), 54; https://doi.org/10.3390/fractalfract7010054 - 01 Jan 2023

Cited by 2 | Viewed by 1590

Abstract

The multifractal relationship between reference evapotranspiration (ET₀), computed by the Penmann-Monteith equation (PM), relative humidity (

R H

) and mean surface temperature (

T_{m e a n}

) was studied in the middle zone of the Guadalquivir River [...] Read more.

The multifractal relationship between reference evapotranspiration (ET₀), computed by the Penmann-Monteith equation (PM), relative humidity (

R H

) and mean surface temperature (

T_{m e a n}

) was studied in the middle zone of the Guadalquivir River Valley (south Spain) in a previous study. This work extends that study to the average wind speed (

U_{2}

) and solar radiation (

S R

), focusing on more recent years. All agro-meteorological variables were analyzed by multifractal detrended cross-correlation analysis (MFCCA) and multifractal detrended fluctuation analysis (MFDFA). The outcomes revealed persistent long-term autocorrelations, with

T_{m e a n}

and

R H

having the highest persistence (

H > 0.75

). More precise results of multifractal properties than in the previous study were obtained for

E T_{0}

,

T_{m e a n}

, and

R H

due to the elimination of trends in the signals. Only medium and large fluctuations in

E T_{0}

showed multifractal cross-correlations with its controlling factors, except for

U_{2}

. Moreover, joint scaling exponents differed from individual exponents. These phenomena contrast with what has been observed in previous cross-correlation studies, revealing that some differences exist in the dynamics of multifractality among the analyzed variables. On the other hand, the

T_{m e a n}

–

E T_{0}

relation showed that extreme events in

E T_{0}

are mainly ruled by high temperature fluctuations, which match conclusions drawn in the previous study. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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29 pages, 68770 KiB

Open AccessArticle

Investigation of the Oriented Structure Characteristics of Shale Using Fractal and Structural Entropy Theory

by Xinhui Xie, Hucheng Deng, Yong Li, Lanxiao Hu, Jinxin Mao and Ruixue Li

Fractal Fract. 2022, 6(12), 734; https://doi.org/10.3390/fractalfract6120734 - 11 Dec 2022

Cited by 3 | Viewed by 1116

Abstract

Unconventional shale reservoirs and typical fine-grained rocks exhibit complicated, oriented features at various scales. Due to the complex geometry, combination and arrangement of grains, as well as the substantial heterogeneity of shale, it is challenging to analyze the oriented structures of shale accurately. [...] Read more.

Unconventional shale reservoirs and typical fine-grained rocks exhibit complicated, oriented features at various scales. Due to the complex geometry, combination and arrangement of grains, as well as the substantial heterogeneity of shale, it is challenging to analyze the oriented structures of shale accurately. In this study, we propose a model that combines both multifractal and structural entropy theory to determine the oriented structures of shale. First, we perform FE–SEM experiments to specify the microstructural characteristics of shale. Next, the shape, size and orientation parameters of the grains and pores are identified via image processing. Then fractal dimensions of grain flatness, grain alignment and pore orientation are calculated and substituted into the structural entropy equation to obtain the structure-oriented entropy model. Lastly, the proposed model is applied to study the orientation characteristic of the Yan-Chang #7 Shale Formation in Ordos Basin, China. A total of 1470 SEM images of 20 shale samples is analyzed to calculate the structure-oriented entropy (SOE) of Yan-Chang #7 Shale, whose values range from 0.78 to 0.96. The grains exhibit directional arrangement (SOE ≥ 0.85) but are randomly distributed (SOE < 0.85). Calculations of samples with different compositions show that clay and organic matters are two major governing factors for the directivity of shale. The grain alignment pattern diagram analyses reveal three types of orientation structures: fusiform, spider-like and eggette-like. The proposed model can quantitatively evaluate the oriented structure of shale, which helps better understand the intrinsic characteristics of shale and thereby assists the successful exploitation of shale resources. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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13 pages, 2061 KiB

Open AccessArticle

Fractal Clustering as Spatial Variability of Magnetic Anomalies Measurements for Impending Earthquakes and the Thermodynamic Fractal Dimension

by Patricio Venegas-Aravena, Enrique Cordaro and David Laroze

Fractal Fract. 2022, 6(11), 624; https://doi.org/10.3390/fractalfract6110624 - 26 Oct 2022

Cited by 5 | Viewed by 1504

Abstract

Several studies focusing on the anomalies of one specific parameter (such as magnetic, ionospheric, radon release, temperature, geodetic, etc.) before impending earthquakes are constantly challenged because their results can be regarded as noise, false positives or are not related to earthquakes at all. [...] Read more.

Several studies focusing on the anomalies of one specific parameter (such as magnetic, ionospheric, radon release, temperature, geodetic, etc.) before impending earthquakes are constantly challenged because their results can be regarded as noise, false positives or are not related to earthquakes at all. This rise concerns the viability of studying isolated physical phenomena before earthquakes. Nevertheless, it has recently been shown that all of the complexity of these pre-earthquake anomalies rises because they could share the same origin. Particularly, the evolution and concentration of uniaxial stresses within rock samples have shown the generation of fractal crack clustering before the macroscopic failure. As there are studies which considered that the magnetic anomalies are created by lithospheric cracks in the seismo-electromagnetic theory, it is expected that the crack clustering is a spatial feature of magnetic and non-magnetic anomalies measurements in ground, atmospheric and ionospheric environments. This could imply that the rise of multiparametric anomalies at specific locations and times, increases the reliability of impending earthquake detections. That is why this work develops a general theory of fractal-localization of different anomalies within the lithosphere in the framework of the seismo-electromagnetic theory. In addition, a general description of the fractal dimension in terms of scaling entropy change is obtained. This model could be regarded as the basis of future early warning systems for catastrophic earthquakes. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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Review

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21 pages, 3956 KiB

Open AccessReview

Multi-Fractality, Universality and Singularity in Turbulence

by Bérengère Dubrulle

Fractal Fract. 2022, 6(10), 613; https://doi.org/10.3390/fractalfract6100613 - 20 Oct 2022

Cited by 2 | Viewed by 1920

Abstract

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all [...] Read more.

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all turbulent flows. Finer observations however prove that this picture is too simplistic, owing to intermittency of energy dissipation and high velocity derivatives. In this review, we discuss how such intermittency can be explained and fitted into a new picture of turbulence. We first discuss how the concept of multi-fractality (invented by Parisi and Frisch in 1982) enables to generalize the concept of self-similarity in a non-homogeneous environment and recover a universality in turbulence. We further review the local extension of this theory, and show how it enables to probe the most irregular locations of the velocity field, in the sense foreseen by Lars Onsager in 1949. Finally, we discuss how the multi-fractal theory connects to possible singularities, in the real or in the complex plane, as first investigated by Frisch and Morf in 1981. Full article

(This article belongs to the Special Issue Multifractals, Turbulence and Complexity in Geoscience and Space Science)

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