Time Evolution of the Fractal Dimension in Turbulent Plumes

Turbulent mixing is a very important issue in the study of geophysical phenomena because most fluxes arising in geophysics fluids are turbulent. We study turbulent mixing due to convection using a laboratory experimental model with two miscible fluids of different density with an initial top heavy density distribution. The fluids that form the initial unstable stratification are miscible and the turbulence will produce molecular mixing. The denser fluid comes into the lighter fluid layer and it generates several forced plumes which are gravitationally unstable. As the turbulent plumes develop, the denser fluid comes into contact with the lighter fluid layer and the mixing process grows. Their development is caused by the lateral interaction between these plumes at the complex fractal surface between the dense and light fluids

Fractal Behaviour and Irregularity of Wetting Fronts in Heterogeneous Porous Media

The wetting front is the zone where water invades and advances into an initially dry porous material and it plays a crucial role in solute transport through the unsaturated zone. Water is an essential part of the physiological process of all plants. Through water, necessary minerals are moved from the roots to the parts of the plants that require them. Water moves chemicals from one part of the plant to another. It is also required for photosynthesis, for metabolism and for transpiration. The leaching of chemicals by wetting fronts is influenced by two major factors, namely: the irregularity of the fronts and heterogeneity in the distribution of chemicals, both of which have been described by using fractal techniques. Soil structure can significantly modify infiltration rates and flow pathways in soils. Relations between features of soil structure and features of infiltration could be elucidated from the velocities and the structure of wetting fronts. When rainwater falls onto soil, it doesn?t just pool on surfaces. Water ?or another fluid- acts differently on porous surfaces. If the surface is permeable (porous) it seeps down through layers of soil, filling that layer to capacity. Once that layer is filled, it moves down into the next layer. In sandy soil, water moves quickly, while it moves much slower through clay soil. The movement of water through soil layers is called the the wetting front. Our research concerns the motion of a liquid into an initially dry porous medium. Our work presents a theoretical framework for studying the physical interplay between a stationary wetting front of fractal dimension D with different porous materials. The aim was to model the mass geometry interplay by using the fractal dimension D of a stationary wetting front. The plane corresponding to the image is divided in several squares (the minimum correspond to the pixel size) of size length ". We acknowledge the help of Prof. M. García Velarde and the facilities offered by the Pluri-Disciplinary Institute of the Complutense University of Madrid. We also acknowledge the help of European Community under project Multi-scale complex fluid flows and interfacial phenomena (PITN-GA-2008-214919). Thanks are also due to ERCOFTAC (PELNoT, SIG 14)

Fractal scaling of apparent soil moisture estimated from vertical planes of Vertisol pit images

Image analysis could be a useful tool for investigating the spatial patterns of apparent soil moisture at multiple resolutions. The objectives of the present work were (i) to define apparent soil moisture patterns from vertical planes of Vertisol pit images and (ii) to describe the scaling of apparent soil moisture distribution using fractal parameters. Twelve soil pits (0.70 m long × 0.60 m width × 0.30 m depth) were excavated on a bare Mazic Pellic Vertisol. Six of them were excavated in April/2011 and six pits were established in May/2011 after 3 days of a moderate rainfall event. Digital photographs were taken from each Vertisol pit using a Kodak™ digital camera. The mean image size was 1600 × 945 pixels with one physical pixel ≈373 μm of the photographed soil pit. Each soil image was analyzed using two fractal scaling exponents, box counting (capacity) dimension (DBC) and interface fractal dimension (Di), and three prefractal scaling coefficients, the total number of boxes intercepting the foreground pattern at a unit scale (A), fractal lacunarity at the unit scale (Λ1) and Shannon entropy at the unit scale (S1). All the scaling parameters identified significant differences between both sets of spatial patterns. Fractal lacunarity was the best discriminator between apparent soil moisture patterns. Soil image interpretation with fractal exponents and prefractal coefficients can be incorporated within a site-specific agriculture toolbox. While fractal exponents convey information on space filling characteristics of the pattern, prefractal coefficients represent the investigated soil property as seen through a higher resolution microscope. In spite of some computational and practical limitations, image analysis of apparent soil moisture patterns could be used in connection with traditional soil moisture sampling, which always renders punctual estimates

A Program for Fractal and Multifractal Analysis of Two-Dimensional Binary Images. Computer Algorithms versus Mathematical Theory.

In this paper we present a tool to carry out the multifractal analysis of binary, two-dimensional images through the calculation of the Rényi D(q) dimensions and associated statistical regressions. The estimation of a (mono)fractal dimension corresponds to the special case where the moment order is q = 0.

Fractal scaling of apparent soil moisture estimated from vertical planesof Vertisol pit images

Image analysis could be a useful tool for investigating the spatial patterns of apparent soil moisture at multiple resolutions. The objectives of the present work were (i) to define apparent soil moisture patterns from vertical planes of Vertisol pit images and (ii) to describe the scaling of apparent soil moisture distribution using fractal parameters. Twelve soil pits (0.70 m long × 0.60 m width × 0.30 m depth) were excavated on a bare Mazic Pellic Vertisol. Six of them were excavated in April/2011 and six pits were established in May/2011 after 3 days of a moderate rainfall event. Digital photographs were taken from each Vertisol pit using a Kodak? digital camera. The mean image size was 1600 × 945 pixels with one physical pixel ?373 ?m of the photographed soil pit. Each soil image was analyzed using two fractal scaling exponents, box counting (capacity) dimension (DBC) and interface fractal dimension (Di), and three prefractal scaling coefficients, the total number of boxes intercepting the foreground pattern at a unit scale (A), fractal lacunarity at the unit scale (?1) and Shannon entropy at the unit scale (S1). All the scaling parameters identified significant differences between both sets of spatial patterns. Fractal lacunarity was the best discriminator between apparent soil moisture patterns. Soil image interpretation with fractal exponents and prefractal coefficients can be incorporated within a site-specific agriculture toolbox. While fractal exponents convey information on space filling characteristics of the pattern, prefractal coefficients represent the investigated soil property as seen through a higher resolution microscope. In spite of some computational and practical limitations, image analysis of apparent soil moisture patterns could be used in connection with traditional soil moisture sampling, which always renders punctual estimates.

Characterization of rock joints by fractal analysis

From a physical perspective, a joint experiences fracturing processes that affect the rock at both microscopic and macroscopic levels. The result is a behaviour that follows a fractal structure. In the first place, for saw-tooth roughness profiles, the use of the triadic Koch curve appears to be adequate and by means of known correlations the JRC parameter is obtained from the angle measured on the basis of the height and length of the roughnesses. Therefore, JRC remains related to the geometric pattern that defines roughness by fractal analysis. In the second place, to characterise the geometry of irregularities with softened profiles, consequently, is proposed a characterisation of the fractal dimension of the joints with a circumference arc generator that is dependent on an average contact angle with regard to the mid-plane. The correlation between the JRC and the fractal dimension of the model is established with a defined statistical ratio.

Polymer interaction with macroscopic carbon nanotube fibres and fabrication of nanostructured composites

El ensamblado de nanotubos de carbono (CNT) como una fibra macroscópica en la cual están orientados preferentemente paralelos entre sí y al eje de la fibra, ha dado como resultado un nuevo tipo de fibra de altas prestaciones derivadas de la explotación eficiente de las propiedades axiales de los CNTs, y que tiene un gran número de aplicaciones potenciales. Fibras continuas de CNTs se produjeron en el Instituto IMDEA Materiales mediante el proceso de hilado directo durante la reacción de síntesis por deposición química de vapores. Uno de los objetivos de esta tesis es el estudio de la estructura de estas fibras mediante técnicas del estado del arte de difracción de rayos X de sincrotrón y la elaboración de un modelo estructural de dicho material. Mediciones texturales de adsorción de gases, análisis de micrografías de electrones y dispersión de rayos X de ángulo alto y bajo (WAXS/SAXS) indican que el material tiene una estructura mesoporosa con una distribución de tamaño de poros ancha derivada del amplio rango de separaciones entre manojos de CNTs, así como una superficie específica de 170m2/g. Los valores de dimensión fractal obtenidos mediante SAXS y análisis Barrett-Joyner-Halenda (BJH) de mediciones texturales coinciden en 2.4 y 2.5, respectivamente, resaltando el carácter de red de la estructura de dichas fibras. La estructura mesoporosa y tipo hilo de las fibra de CNT es accesible a la infiltración de moléculas externas (líquidos o polímeros). En este trabajo se estudian los cambios en la estructura multiescala de las fibras de CNTs al interactuar con líquidos y polímeros. Los efectos de la densificación en la estructura de fibras secas de CNT son estudiados mediante WAXS/SAXS. El tratamiento de densificación junta los manojos de la fibra (los poros disminuyen de tamaño), resultando en un incremento de la densidad de la fibra. Sin embargo, los dominios estructurales correspondientes a la transferencia de esfuerzo mecánica y carga eléctrica en los nanotubos no son afectados durante este proceso de densificación; como consecuencia no se produce un efecto sustancial en las propiedades mecánicas y eléctricas. Mediciones de SAXS and fibra de CNT antes y después de infiltración de líquidos confirman la penetración de una gran cantidad de líquidos que llena los poros internos de la fibra pero no se intercalan entre capas de nanotubos adyacentes. La infiltración de cadenas poliméricas de bajo peso molecular tiende a expandir los manojos en la fibra e incrementar el ángulo de apertura de los poros. Los resultados de SAXS indican que la estructura interna de la fibra en términos de la organización de las capas de tubos y su orientación no es afectada cuando las muestras consisten en fibras infiltradas con polímeros de alto peso molecular. La cristalización de varios polímeros semicristalinos es acelerada por la presencia de fibras de CNTs alineados y produce el crecimiento de una capa transcristalina normal a la superficie de la fibra. Esto es observado directamente mediante microscopía óptica polarizada, y detectado mediante calorimetría DSC. Las lamelas en la capa transcristalina tienen orientación de la cadena polimérica paralela a la fibra y por lo tanto a los nanotubos, de acuerdo con los patrones de WAXS. Esta orientación preferencial se sugiere como parte de la fuerza impulsora en la nucleación. La nucleación del dominio cristalino polimérico en la superficie de los CNT no es epitaxial. Ocurre sin haber correspondencia entre las estructuras cristalinas del polímero y los nanotubos. Estas observaciones contribuyen a la compresión del fenómeno de nucleación en CNTs y otros nanocarbonos, y sientan las bases para el desarrollo de composites poliméricos de gran escala basados en fibra larga de CNTs alineados. ABSTRACT The assembly of carbon nanotubes into a macroscopic fibre material where they are preferentially aligned parallel to each other and to the fibre axis has resulted in a new class of high-performance fibres, which efficiently exploits the axial properties of the building blocks and has numerous applications. Long, continuous CNT fibres were produced in IMDEA Materials Institute by direct fibre spinning from a chemical vapour deposition reaction. These fibres have a complex hierarchical structure covering multiple length scales. One objective of this thesis is to reveal this structure by means of state-of-the-art techniques such as synchrotron X-ray diffraction, and to build a model to link the fibre structural elements. Texture and gas absorption measurements, using electron microscopy, wide angle and small angle X-ray scattering (WAXS/SAXS), and pore size distribution analysis by Barrett-Joyner-Halenda (BJH), indicate that the material has a mesoporous structure with a wide pore size distribution arising from the range of fibre bundle separation, and a high surface area _170m2/g. Fractal dimension values of 2.4_2.5 obtained from the SAXS and BJH measurements highlight the network structure of the fibre. Mesoporous and yarn-like structure of CNT fibres make them accessible to the infiltration of foreign molecules (liquid or polymer). This work studies multiscale structural changes when CNT fibres interact with liquids and polymers. The effects of densification on the structure of dry CNT fibres were measured by WAXS/SAXS. The densification treatment brings the fibre bundles closer (pores become smaller), leading to an increase in fibre density. However, structural domains made of the load and charge carrying nanotubes are not affected; consequently, it has no substantial effect on mechanical and electrical properties. SAXS measurements on the CNT fibres before and after liquid infiltration imply that most liquids are able to fill the internal pores but not to intercalate between nanotubes. Successful infiltration of low molecular weight polymer chains tends to expand the fibre bundles and increases the pore-opening angle. SAXS results indicate that the inner structure of the fibre, in terms of the nanotube layer arrangement and the fibre alignment, are not largely affected when infiltrated with polymers of relatively high molecular weight. The crystallisation of a variety of semicrystalline polymers is accelerated by the presence of aligned fibres of CNTs and results in the growth of a transcrystalline layer perpendicular to the fibre surface. This can be observed directly under polarised optical microscope, and detected by the exothermic peaks during differential scanning calorimetry. The discussion on the driving forces for the enhanced nucleation points out the preferential chain orientation of polymer lamella with the chain axis parallel to the fibre and thus to the nanotubes, which is confirmed by two-dimensional WAXS patterns. A non-epitaxial polymer crystal growth habit at the CNT-polymer interface is proposed, which is independent of lattice matching between the polymer and nanotubes. These findings contribute to the discussion on polymer nucleation on CNTs and other nanocarbons, and their implication for the development of large polymer composites based on long and aligned fibres of CNTs.

Toward reduction of artifacts in fused images

Most fusion satellite image methodologies at pixel-level introduce false spatial details, i.e.artifacts, in the resulting fusedimages. In many cases, these artifacts appears because image fusion methods do not consider the differences in roughness or textural characteristics between different land covers. They only consider the digital values associated with single pixels. This effect increases as the spatial resolution image increases. To minimize this problem, we propose a new paradigm based on local measurements of the fractal dimension (FD). Fractal dimension maps (FDMs) are generated for each of the source images (panchromatic and each band of the multi-spectral images) with the box-counting algorithm and by applying a windowing process. The average of source image FDMs, previously indexed between 0 and 1, has been used for discrimination of different land covers present in satellite images. This paradigm has been applied through the fusion methodology based on the discrete wavelet transform (DWT), using the à trous algorithm (WAT). Two different scenes registered by optical sensors on board FORMOSAT-2 and IKONOS satellites were used to study the behaviour of the proposed methodology. The implementation of this approach, using the WAT method, allows adapting the fusion process to the roughness and shape of the regions present in the image to be fused. This improves the quality of the fusedimages and their classification results when compared with the original WAT method

Lacunarity of the Spatial Distributions of Soil Types in Europe.

Lacunarity as a means of quantifying textural properties of spatial distributions suggests a classification into three main classes of the most abundant soils that cover 92% of Europe. Soils with a well-defined self-similar structure of the linear class are related to widespread spatial patterns that are nondominant but ubiquitous at continental scale. Fractal techniques have been increasingly and successfully applied to identify and describe spatial patterns in natural sciences. However, objects with the same fractal dimension can show very different optical properties because of their spatial arrangement. This work focuses primary attention on the geometrical structure of the geographical patterns of soils in Europe. We made use of the European Soil Database to estimate lacunarity indexes of the most abundant soils that cover 92% of the surface of Europe and investigated textural properties of their spatial distribution. We observed three main classes corresponding to three different patterns that displayed the graphs of lacunarity functions, that is, linear, convex, and mixed. They correspond respectively to homogeneous or self-similar, heterogeneous or clustered and those in which behavior can change at different ranges of scales. Finally, we discuss the pedological implications of that classification.

Lithologic Control on the Scaling Properties of the First-Order Streams of Drainage Networks: A Monofractal Analysis

The movement of water through the landscape can be investigated at different scales. This study dealt with the interrelation between bedrock lithology and the geometry of the overlying drainage systems. Parameters of fractal analysis, such as fractal dimension and lacunarity, were used to measure and quantify this relationship. The interrelation between bedrock lithology and the geometry of the drainage systems has been widely studied in the last decades. The quantification of this linkage has not yet been clearly established. Several studies have selected river basins or regularly shaped areas as study units, assuming them to be lithologically homogeneous. This study considered irregular distributions of rock types, establishing areas of the soil map (1:25,000) with the same lithologic information as study units. The tectonic stability and the low climatic variability of the study region allowed effective investigation of the lithologic controls on the drainage networks developed on the plutonic rocks, the metamorphic rocks, and the sedimentary materials existing in the study area. To exclude the effect of multiple in- and outflows in the lithologically homogeneous units, we focused this study on the first-order streams of the drainage networks. The geometry of the hydrologic features was quantified through traditional metrics of fluvial geomorphology and scaling parameters of fractal analysis, such as the fractal dimension, the reference density, and the lacunarity. The results demonstrate the scale invariance of both the drainage networks and the set of first-order streams at the study scale and a relationship between scaling in the lithology and the drainage network.

Methodology of image analysis for study of the vertisols moisture content

The main problem to study vertical drainage from the moisture distribution, on a vertisol profile, is searching for suitable methods using these procedures. Our aim was to design a digital image processing methodology and its analysis to characterize the moisture content distribution of a vertisol profile. In this research, twelve soil pits were excavated on a ba re Mazic Pellic Vertisols ix of them in May 13/2011 and the rest in May 19 /2011 after a moderate rainfall event. Digital RGB images were taken from each vertisol pit using a Kodak? camera selecting a size of 1600x945 pixels. Each soil image was processed to homogenized brightness and then a spatial filter with several window sizes was applied to select the optimum one. The RGB image obtained were divided in each matrix color selecting the best thresholds for each one, maximum and minimum, to be applied and get a digital binary pattern. This one was analyzed by estimating two fractal scaling exponents box counting dimension D BC) and interface fractal dimension (D) In addition, three pre-fractal scaling coefficients were determinate at maximum resolution: total number of boxes intercepting the foreground pattern (A), fractal lacunarity (?1) and Shannon entropy S1). For all the images processed the spatial filter 9x9 was the optimum based on entropy, cluster and histogram criteria. Thresholds for each color were selected based on bimodal histograms.

Quantifying the relationship between drainage networks at hillslope scale and particle size distribution at pedon scale

Nowadays, translating information about hydrologic and soil properties and processes across scales has emerged as a major theme in soil science and hydrology, and suitable theories for upscaling or downscaling hydrologic and soil information are being looked forward. The recognition of low-order catchments as self-organized systems suggests the existence of a great amount of links at different scales between their elements. The objective of this work was to research in areas of homogeneous bedrock material, the relationship between the hierarchical structure of the drainage networks at hillslope scale and the heterogeneity of the particle-size distribution at pedon scale. One of the most innovative elements in this work is the choice of the parameters to quantify the organization level of the studied features. The fractal dimension has been selected to measure the hierarchical structure of the drainage networks, while the Balanced Entropy Index (BEI) has been the chosen parameter to quantify the heterogeneity of the particle-size distribution from textural data. These parameters have made it possible to establish quantifiable relationships between two features attached to different steps in the scale range. Results suggest that the bedrock lithology of the landscape constrains the architecture of the drainage networks developed on it and the particle soil distribution resulting in the fragmentation processes.

Quantifying the relationship between drainage networks at hillslope scale and particle size distribution at pedon scale

Nowadays, translating information about hydrologic and soil properties and processes across scales has emerged as a major theme in soil science and hydrology, and suitable theories for upscaling or downscaling hydrologic and soil information are being looked forward. The recognition of low-order catchments as self-organized systems suggests the existence of a great amount of links at different scales between their elements. The objective of this work was to research in areas of homogeneous bedrock material, the relationship between the hierarchical structure of the drainage networks at hillslope scale and the heterogeneity of the particle-size distribution at pedon scale. One of the most innovative elements in this work is the choice of the parameters to quantify the organization level of the studied features. The fractal dimension has been selected to measure the hierarchical structure of the drainage networks, while the Balanced Entropy Index (BEI) has been the chosen parameter to quantify the heterogeneity of the particle-size distribution from textural data. These parameters have made it possible to establish quantifiable relationships between two features attached to different steps in the scale range. Results suggest that the bedrock lithology of the landscape constrains the architecture of the drainage networks developed on it and the particle soil distribution resulting in the fragmentation processes.

Computer simulation of the interplay between fractal structures and surrounding heterogeneous multifractal distributions. Applications

In a large number of physical, biological and environmental processes interfaces with high irregular geometry appear separating media (phases) in which the heterogeneity of constituents is present. In this work the quantification of the interplay between irregular structures and surrounding heterogeneous distributions in the plane is made For a geometric set image and a mass distribution (measure) image supported in image, being image, the mass image gives account of the interplay between the geometric structure and the surrounding distribution. A computation method is developed for the estimation and corresponding scaling analysis of image, being image a fractal plane set of Minkowski dimension image and image a multifractal measure produced by random multiplicative cascades. The method is applied to natural and mathematical fractal structures in order to study the influence of both, the irregularity of the geometric structure and the heterogeneity of the distribution, in the scaling of image. Applications to the analysis and modeling of interplay of phases in environmental scenarios are given.

La geometría fractal en los sistemas agroambientales

La formación matemática que se imparte en los diferentes títulos de la ETSI Agrónomos presenta características comunes a otras ingenierías pero también algunos rasgos específicos. Como en otras ingenierías, esta formación está vinculada en parte a la utilización de modelos físicos altamente matematizados. Además, el ingeniero agrónomo debe dar respuesta a desafíos que ponen en juego otras disciplinas, como las ciencias biológicas o las ciencias de la naturaleza. Estas ciencias muchas veces requieren de nuevas teorías matemáticas para alcanzar sus objetivos, lo que representa nuevos retos para las matemáticas tanto en el campo de la docencia como en el de la investigación. A modo de ejemplo, vamos a centrarnos aquí en el uso de una teoría matemática con importantes aplicaciones a los sistemas agroambientales. Esta teoría matemática es la geometría fractal, nacida a principios del siglo pasado.