Wave Statistics and Space-time Extremes via Stereo Imaging

We present an analysis of the space-time dynamics of oceanic sea states exploiting stereo imaging techniques. In particular, a novel Wave Acquisition Stereo System (WASS) has been developed and deployed at the oceanographic tower Acqua Alta in the Northern Adriatic Sea, off the Venice coast in Italy. The analysis of WASS video measurements yields accurate estimates of the oceanic sea state dynamics, the associated directional spectra and wave surface statistics that agree well with theoretical models. Finally, we show that a space-time extreme, defined as the expected largest surface wave height over an area, is considerably larger than the maximum crest observed in time at a point, in agreement with theoretical predictions.

Understanding long-term hunting statistics: the case of Spain (1972-2007)

Hunting is assuming a growing role in the current European forestry and agroforestry landscape. However, consistent statistical sources that provide quantitative information for policy-making, planning and management of game resources are often lacking. In addition, in many instances statistical information can be used without sufficient evaluation or criticism. Recently, the European Commission has declared the importance of high quality hunting statistics and the need to set up a common scheme in Europe for their collection, interpretation and proper use. This work aims to contribute to this current debate on hunting statistics in Europe by exploring data from the last 35 years of Spanish hunting statistics. The analysis focuses on the three major pillars underpinning hunting activity: hunters, hunting grounds and game animals. First, the study aims to provide a better understanding of official hunting statistics for use by researchers, game managers and other potential users. Second, the study highlights the major strengths and weaknesses of the statistical information that was collected. The results of the analysis indicate that official hunting statistics can be incomplete, dispersed and not always homogeneous over a long period of time. This is an issue of which one should be aware when using official hunting data for scientific or technical work. To improve statistical deficiencies associated with hunting data in Spain, our main suggestion is the adoption of a common protocol on data collection to which different regions agree. This protocol should be in accordance with future European hunting statistics and based on robust and well-informed data collection methods. Also it should expand the range of biological, ecological and economic concepts currently included to take account of the profound transformations experienced by the hunting sector in recent years. As much as possible, any future changes in the selection of hunting statistics should allow for comparisons between new variables with the previous ones.

Optimal fits of diffusion constants from single-time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.

First passages in bounded domains: When is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ1/(τ1+τ2), which is a measure for how similar the first passage times τ1 and τ2 are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Comment on the existence of a long range correlation in the geomagnetic disturbance storm time (Dst) index

Very recently (Banerjee et al. in Astrophys. Space, doi:1007/s10509-011-0836-1, 2011) the statistics of geomagnetic Disturbance storm (Dst) index have been addressed, and the conclusion from this analysis suggests that the underlying dynamical process can be modeled as a fractional Brownian motion with persistent long-range correlations. In this comment we expose several misconceptions and flaws in the statistical analysis of that work. On the basis of these arguments, the former conclusion should be revisited.

Space-time Measurements of Oceanic Sea States

Stereo video techniques are effective for estimating the space–time wave dynamics over an area of the ocean. Indeed, a stereo camera view allows retrieval of both spatial and temporal data whose statistical content is richer than that of time series data retrieved from point wave probes. We present an application of the Wave Acquisition Stereo System (WASS) for the analysis of offshore video measurements of gravity waves in the Northern Adriatic Sea and near the southern seashore of the Crimean peninsula, in the Black Sea. We use classical epipolar techniques to reconstruct the sea surface from the stereo pairs sequentially in time, viz. a sequence of spatial snapshots. We also present a variational approach that exploits the entire data image set providing a global space–time imaging of the sea surface, viz. simultaneous reconstruction of several spatial snapshots of the surface in order to guarantee continuity of the sea surface both in space and time. Analysis of the WASS measurements show that the sea surface can be accurately estimated in space and time together, yielding associated directional spectra and wave statistics at a point in time that agrees well with probabilistic models. In particular, WASS stereo imaging is able to capture typical features of the wave surface, especially the crest-to-trough asymmetry due to second order nonlinearities, and the observed shape of large waves are fairly described by theoretical models based on the theory of quasi-determinism (Boccotti, 2000). Further, we investigate space–time extremes of the observed stationary sea states, viz. the largest surface wave heights expected over a given area during the sea state duration. The WASS analysis provides the first experimental proof that a space–time extreme is generally larger than that observed in time via point measurements, in agreement with the predictions based on stochastic theories for global maxima of Gaussian fields.

Real time passenger information systems and quality of bus services

One of the main problems in urban areas is the steady growth in car ownership and traffic levels. Therefore, the challenge of sustainability is focused on a shift of the demand for mobility from cars to collective means of transport. For this end, buses are a key element of the public transport systems. In this respect Real Time Passenger Information (RTPI) systems help citizens change their travel behaviour towards more sustainable transport modes. This paper provides an assessment methodology which evaluates how RTPI systems improve the quality of bus services in two European cities, Madrid and Bremerhaven. In the case of Madrid, bus punctuality has increased by 3%. Regarding the travellers perception, Madrid raised its quality of service by 6% while Bremerhaven increased by 13%. On the other hand, the users ́ perception of Public Transport (PT) image increased by 14%.

Inter-operator reliability of live football match statistics from OPTA Sportsdata

The aim of the study was to evaluate the inter-operator reliability of OPTA Client System which is used to collect live football match statistics by OPTA Sportsdata Company. Two groups of experienced operators were required to analyze a Spanish league match independently. Results showed that team events coded by independent operators reached a very good agreement (kappa values were 0.92 and 0.94) and average difference of event time was 0.06±0.04 s. The reliability of goalkeeper actions was also at high level, kappa values were 0.92 and 0.86. The high intra-class correlation coefficients (ranged from 0.88 to 1.00) and low standardized typical errors (varied from 0.00 to 0.37) of different match actions and indicators of individual outfield players showed a high level of inter-operator reliability as well. These results suggest that the OPTA Client System is reliable to be used to collect live football match statistics by well trained operators.

Time-resolved evolution of coherent structures in turbulent channels

El objetivo de esta tesis es estudiar la dinámica de la capa logarítmica de flujos turbulentos de pared. En concreto, proponemos un nuevo modelo estructural utilizando diferentes tipos de estructuras coherentes: sweeps, eyecciones, grupos de vorticidad y streaks. La herramienta utilizada es la simulación numérica directa de canales turbulentos. Desde los primeros trabajos de Theodorsen (1952), las estructuras coherentes han jugado un papel fundamental para entender la organización y dinámica de los flujos turbulentos. A día de hoy, datos procedentes de simulaciones numéricas directas obtenidas en instantes no contiguos permiten estudiar las propiedades fundamentales de las estructuras coherentes tridimensionales desde un punto de vista estadístico. Sin embargo, la dinámica no puede ser entendida en detalle utilizando sólo instantes aislados en el tiempo, sino que es necesario seguir de forma continua las estructuras. Aunque existen algunos estudios sobre la evolución temporal de las estructuras más pequeñas a números de Reynolds moderados, por ejemplo Robinson (1991), todavía no se ha realizado un estudio completo a altos números de Reynolds y para todas las escalas presentes de la capa logarítmica. El objetivo de esta tesis es llevar a cabo dicho análisis. Los problemas más interesantes los encontramos en la región logarítmica, donde residen las cascadas de vorticidad, energía y momento. Existen varios modelos que intentan explicar la organización de los flujos turbulentos en dicha región. Uno de los más extendidos fue propuesto por Adrian et al. (2000) a través de observaciones experimentales y considerando como elemento fundamental paquetes de vórtices con forma de horquilla que actúan de forma cooperativa para generar rampas de bajo momento. Un modelo alternativo fué ideado por del Álamo & Jiménez (2006) utilizando datos numéricos. Basado también en grupos de vorticidad, planteaba un escenario mucho más desorganizado y con estructuras sin forma de horquilla. Aunque los dos modelos son cinemáticamente similares, no lo son desde el punto de vista dinámico, en concreto en lo que se refiere a la importancia que juega la pared en la creación y vida de las estructuras. Otro punto importante aún sin resolver se refiere al modelo de cascada turbulenta propuesto por Kolmogorov (1941b), y su relación con estructuras coherentes medibles en el flujo. Para dar respuesta a las preguntas anteriores, hemos desarrollado un nuevo método que permite seguir estructuras coherentes en el tiempo y lo hemos aplicado a simulaciones numéricas de canales turbulentos con números de Reynolds lo suficientemente altos como para tener un rango de escalas no trivial y con dominios computacionales lo suficientemente grandes como para representar de forma correcta la dinámica de la capa logarítmica. Nuestros esfuerzos se han desarrollado en cuatro pasos. En primer lugar, hemos realizado una campaña de simulaciones numéricas directas a diferentes números de Reynolds y tamaños de cajas para evaluar el efecto del dominio computacional en las estadísticas de primer orden y el espectro. A partir de los resultados obtenidos, hemos concluido que simulaciones con cajas de longitud 2vr y ancho vr veces la semi-altura del canal son lo suficientemente grandes para reproducir correctamente las interacciones entre estructuras coherentes de la capa logarítmica y el resto de escalas. Estas simulaciones son utilizadas como punto de partida en los siguientes análisis. En segundo lugar, las estructuras coherentes correspondientes a regiones con esfuerzos de Reynolds tangenciales intensos (Qs) en un canal turbulento han sido estudiadas extendiendo a tres dimensiones el análisis de cuadrantes, con especial énfasis en la capa logarítmica y la región exterior. Las estructuras coherentes han sido identificadas como regiones contiguas del espacio donde los esfuerzos de Reynolds tangenciales son más intensos que un cierto nivel. Los resultados muestran que los Qs separados de la pared están orientados de forma isótropa y su contribución neta al esfuerzo de Reynolds medio es nula. La mayor contribución la realiza una familia de estructuras de mayor tamaño y autosemejantes cuya parte inferior está muy cerca de la pared (ligada a la pared), con una geometría compleja y dimensión fractal « 2. Estas estructuras tienen una forma similar a una ‘esponja de placas’, en comparación con los grupos de vorticidad que tienen forma de ‘esponja de cuerdas’. Aunque el número de objetos decae al alejarnos de la pared, la fracción de esfuerzos de Reynolds que contienen es independiente de su altura, y gran parte reside en unas pocas estructuras que se extienden más allá del centro del canal, como en las grandes estructuras propuestas por otros autores. Las estructuras dominantes en la capa logarítmica son parejas de sweeps y eyecciones uno al lado del otro y con grupos de vorticidad asociados que comparten las dimensiones y esfuerzos con los remolinos ligados a la pared propuestos por Townsend. En tercer lugar, hemos estudiado la evolución temporal de Qs y grupos de vorticidad usando las simulaciones numéricas directas presentadas anteriormente hasta números de Reynolds ReT = 4200 (Reynolds de fricción). Las estructuras fueron identificadas siguiendo el proceso descrito en el párrafo anterior y después seguidas en el tiempo. A través de la interseción geométrica de estructuras pertenecientes a instantes de tiempo contiguos, hemos creado gratos de conexiones temporales entre todos los objetos y, a partir de ahí, definido ramas primarias y secundarias, de tal forma que cada rama representa la evolución temporal de una estructura coherente. Una vez que las evoluciones están adecuadamente organizadas, proporcionan toda la información necesaria para caracterizar la historia de las estructuras desde su nacimiento hasta su muerte. Los resultados muestran que las estructuras nacen a todas las distancias de la pared, pero con mayor probabilidad cerca de ella, donde la cortadura es más intensa. La mayoría mantienen tamaños pequeños y no viven mucho tiempo, sin embargo, existe una familia de estructuras que crecen lo suficiente como para ligarse a la pared y extenderse a lo largo de la capa logarítmica convirtiéndose en las estructuras observas anteriormente y descritas por Townsend. Estas estructuras son geométricamente autosemejantes con tiempos de vida proporcionales a su tamaño. La mayoría alcanzan tamaños por encima de la escala de Corrsin, y por ello, su dinámica está controlada por la cortadura media. Los resultados también muestran que las eyecciones se alejan de la pared con velocidad media uT (velocidad de fricción) y su base se liga a la pared muy rápidamente al inicio de sus vidas. Por el contrario, los sweeps se mueven hacia la pared con velocidad -uT y se ligan a ella más tarde. En ambos casos, los objetos permanecen ligados a la pared durante 2/3 de sus vidas. En la dirección de la corriente, las estructuras se desplazan a velocidades cercanas a la convección media del flujo y son deformadas por la cortadura. Finalmente, hemos interpretado la cascada turbulenta, no sólo como una forma conceptual de organizar el flujo, sino como un proceso físico en el cual las estructuras coherentes se unen y se rompen. El volumen de una estructura cambia de forma suave, cuando no se une ni rompe, o lo hace de forma repentina en caso contrario. Los procesos de unión y rotura pueden entenderse como una cascada directa (roturas) o inversa (uniones), siguiendo el concepto de cascada de remolinos ideado por Richardson (1920) y Obukhov (1941). El análisis de los datos muestra que las estructuras con tamaños menores a 30η (unidades de Kolmogorov) nunca se unen ni rompen, es decir, no experimentan el proceso de cascada. Por el contrario, aquellas mayores a 100η siempre se rompen o unen al menos una vez en su vida. En estos casos, el volumen total ganado y perdido es una fracción importante del volumen medio de la estructura implicada, con una tendencia ligeramente mayor a romperse (cascada directa) que a unirse (cascade inversa). La mayor parte de interacciones entre ramas se debe a roturas o uniones de fragmentos muy pequeños en la escala de Kolmogorov con estructuras más grandes, aunque el efecto de fragmentos de mayor tamaño no es despreciable. También hemos encontrado que las roturas tienen a ocurrir al final de la vida de la estructura y las uniones al principio. Aunque los resultados para la cascada directa e inversa no son idénticos, son muy simétricos, lo que sugiere un alto grado de reversibilidad en el proceso de cascada. ABSTRACT The purpose of the present thesis is to study the dynamics of the logarithmic layer of wall-bounded turbulent flows. Specifically, to propose a new structural model based on four different coherent structures: sweeps, ejections, clusters of vortices and velocity streaks. The tool used is the direct numerical simulation of time-resolved turbulent channels. Since the first work by Theodorsen (1952), coherent structures have played an important role in the understanding of turbulence organization and its dynamics. Nowadays, data from individual snapshots of direct numerical simulations allow to study the threedimensional statistical properties of those objects, but their dynamics can only be fully understood by tracking them in time. Although the temporal evolution has already been studied for small structures at moderate Reynolds numbers, e.g., Robinson (1991), a temporal analysis of three-dimensional structures spanning from the smallest to the largest scales across the logarithmic layer has yet to be performed and is the goal of the present thesis. The most interesting problems lie in the logarithmic region, which is the seat of cascades of vorticity, energy, and momentum. Different models involving coherent structures have been proposed to represent the organization of wall-bounded turbulent flows in the logarithmic layer. One of the most extended ones was conceived by Adrian et al. (2000) and built on packets of hairpins that grow from the wall and work cooperatively to gen- ´ erate low-momentum ramps. A different view was presented by del Alamo & Jim´enez (2006), who extracted coherent vortical structures from DNSs and proposed a less organized scenario. Although the two models are kinematically fairly similar, they have important dynamical differences, mostly regarding the relevance of the wall. Another open question is whether such a model can be used to explain the cascade process proposed by Kolmogorov (1941b) in terms of coherent structures. The challenge would be to identify coherent structures undergoing a turbulent cascade that can be quantified. To gain a better insight into the previous questions, we have developed a novel method to track coherent structures in time, and used it to characterize the temporal evolutions of eddies in turbulent channels with Reynolds numbers high enough to include a non-trivial range of length scales, and computational domains sufficiently long and wide to reproduce correctly the dynamics of the logarithmic layer. Our efforts have followed four steps. First, we have conducted a campaign of direct numerical simulations of turbulent channels at different Reynolds numbers and box sizes, and assessed the effect of the computational domain in the one-point statistics and spectra. From the results, we have concluded that computational domains with streamwise and spanwise sizes 2vr and vr times the half-height of the channel, respectively, are large enough to accurately capture the dynamical interactions between structures in the logarithmic layer and the rest of the scales. These simulations are used in the subsequent chapters. Second, the three-dimensional structures of intense tangential Reynolds stress in plane turbulent channels (Qs) have been studied by extending the classical quadrant analysis to three dimensions, with emphasis on the logarithmic and outer layers. The eddies are identified as connected regions of intense tangential Reynolds stress. Qs are then classified according to their streamwise and wall-normal fluctuating velocities as inward interactions, outward interactions, sweeps and ejections. It is found that wall-detached Qs are isotropically oriented background stress fluctuations, common to most turbulent flows, and do not contribute to the mean stress. Most of the stress is carried by a selfsimilar family of larger wall-attached Qs, increasingly complex away from the wall, with fractal dimensions « 2. They have shapes similar to ‘sponges of flakes’, while vortex clusters resemble ‘sponges of strings’. Although their number decays away from the wall, the fraction of the stress that they carry is independent of their heights, and a substantial part resides in a few objects extending beyond the centerline, reminiscent of the very large scale motions of several authors. The predominant logarithmic-layer structures are sideby- side pairs of sweeps and ejections, with an associated vortex cluster, and dimensions and stresses similar to Townsend’s conjectured wall-attached eddies. Third, the temporal evolution of Qs and vortex clusters are studied using time-resolved DNS data up to ReT = 4200 (friction Reynolds number). The eddies are identified following the procedure presented above, and then tracked in time. From the geometric intersection of structures in consecutive fields, we have built temporal connection graphs of all the objects, and defined main and secondary branches in a way that each branch represents the temporal evolution of one coherent structure. Once these evolutions are properly organized, they provide the necessary information to characterize eddies from birth to death. The results show that the eddies are born at all distances from the wall, although with higher probability near it, where the shear is strongest. Most of them stay small and do not last for long times. However, there is a family of eddies that become large enough to attach to the wall while they reach into the logarithmic layer, and become the wall-attached structures previously observed in instantaneous flow fields. They are geometrically self-similar, with sizes and lifetimes proportional to their distance from the wall. Most of them achieve lengths well above the Corrsin’ scale, and hence, their dynamics are controlled by the mean shear. Eddies associated with ejections move away from the wall with an average velocity uT (friction velocity), and their base attaches very fast at the beginning of their lives. Conversely, sweeps move towards the wall at -uT, and attach later. In both cases, they remain attached for 2/3 of their lives. In the streamwise direction, eddies are advected and deformed by the local mean velocity. Finally, we interpret the turbulent cascade not only as a way to conceptualize the flow, but as an actual physical process in which coherent structures merge and split. The volume of an eddy can change either smoothly, when they are not merging or splitting, or through sudden changes. The processes of merging and splitting can be thought of as a direct (when splitting) or an inverse (when merging) cascade, following the ideas envisioned by Richardson (1920) and Obukhov (1941). It is observed that there is a minimum length of 30η (Kolmogorov units) above which mergers and splits begin to be important. Moreover, all eddies above 100η split and merge at least once in their lives. In those cases, the total volume gained and lost is a substantial fraction of the average volume of the structure involved, with slightly more splits (direct cascade) than mergers. Most branch interactions are found to be the shedding or absorption of Kolmogorov-scale fragments by larger structures, but more balanced splits or mergers spanning a wide range of scales are also found to be important. The results show that splits are more probable at the end of the life of the eddy, while mergers take place at the beginning of the life. Although the results for the direct and the inverse cascades are not identical, they are found to be very symmetric, which suggests a high degree of reversibility of the cascade process.