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Hurst-Kolmogorov dynamics in hydrometeorological processes and in the microscale of turbulence

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Abstract

The high complexity and uncertainty of atmospheric dynamics has been long identified through the observation and analysis of hydroclimatic processes such as temperature, dew-point, humidity, atmospheric wind, precipitation, atmospheric pressure, river discharge and stage etc. Particularly, all these processes seem to exhibit high unpredictability due to the clustering of events, a behaviour first identified in Nature by H.E. Hurst in 1951 while working at the River Nile, although its mathematical description is attributed to A. N. Kolmogorov who developed it while studying turbulence in 1940. To give credits to both scientists this behaviour and dynamics is called Hurst-Kolmogorov (HK). In order to properly study the clustering of events as well as the stochastic behaviour of hydroclimatic processes in general we would require numerous of measurements in annual scale. Unfortunately, large lengths of high quality annual data are hardly available in observations of hydroclimatic processes. However, the microscopic processes driving and generating the hydroclimatic ones are governed by turbulent state. By studying turbulent phenomena in situ we may be able to understand certain aspects of the related macroscopic processes in field. Certain strong advantages of studying microscopic turbulent processes in situ is the recording of very long time series, the high resolution of records and the controlled environment of the laboratory. The analysis of these time series offers the opportunity of better comprehending, control and comparison of the two scientific methods through the deterministic and stochastic approach. In this thesis, we explore and further advance the second-order stochastic framework for the empirical as well as theoretical estimation of the marginal characteristic and dependence structure of a process (from small to extreme behaviour in time and state). Also, we develop and apply explicit and implicit algorithms for stochastic synthesis of mathematical processes as well as stochastic prediction of physical processes. Moreover, we analyze several turbulent processes and we estimate the Hurst parameter (H >> 0.5 for all cases) and the drop of variance with scale based on experiments in turbulent jets held at the laboratory. Additionally, we propose a stochastic model for the behaviour of a process from the micro to the macro scale that results from the maximization of entropy for both the marginal distribution and the dependence structure. Finally, we apply this model to microscale turbulent processes, as well as hydroclimatic ones extracted from thousands of stations around the globe including countless of data. The most important innovation of this thesis is that, to the Author’s knowledge, a unique framework (through modelling of common expression of both the marginal density distribution function and the second-order dependence structure) is presented that can include the simulation of the discretization effect, the statistical bias, certain aspects of the turbulent intermittent (or else fractal) behaviour (at the microscale of the dependence structure) and the long-term behaviour (at the macroscale of the dependence structure), the extreme events (at the left and right tail of the marginal distribution), as well as applications to 13 turbulent and hydroclimatic processes including experimentation and global analyses of surface stations (overall, several billions of observations). A summary of the major innovations of the thesis are: (a) the further development, and extensive application to numerous processes, of the classical second-order stochastic framework including innovative approaches to account for intermittency, discretization effects and statistical bias; (b) the further development of stochastic generation schemes such as the Sum of Autoregressive (SAR) models, e.g. AR(1) or ARMA(1,1), the Symmetric-Moving-Average (SMA) scheme in many dimensions (that can generate any process second-order dependence structure, approximate any marginal distribution to the desired level of accuracy and simulate certain aspects of the intermittent behaviour) and an explicit and implicit (pseudo) cyclo-stationary (pCSAR and pCSMA) schemes for simulating the deterministic periodicities of a process such as seasonal and diurnal; and (c) the introduction and application of an extended stochastic model (with an innovative identical expression of a four-parameter marginal distribution density function and correlation structure, i.e. g(x;C)=λ/[(1+|x/a+b|^c )]^d, with C=[λ,a,b,c,d]), that encloses a large variety of distributions (ranging from Gaussian to powered-exponential and Pareto) as well as dependence structures (such as white noise, Markov and HK), and is in agreement (in this form or through more simplified versions) with an interestingly large variety of turbulent (such as horizontal and vertical thermal jet of positively buoyancy processes using laser-induced-fluorescence techniques as well as grid-turbulence generated within a wind-tunnel), geostatistical (such as 2d rock formations), and hydroclimatic processes (such as temperature, atmospheric wind, dew-point and thus, humidity, precipitation, atmospheric pressure, river discharges and solar radiation, in a global scale, as well as a very long time series of river stage, and wave height and period). Amazingly, all examined physical processes (overall 13) exhibited long-range dependence and in particular, most (if treated properly within a robust physical and statistical framework, e.g. by adjusting the process for sampling errors as well as discretization and bias effects) with a mean long-term persistence parameter equal to H ≈ 5/6 (as in the case of isotropic grid-turbulence), and (for the processes examined in the microscale such atmospheric wind, surface temperature and dew-point, in a global scale, and a long duration discharge time series and storm event in terms of precipitation and wind) a powered-exponential behaviour with a fractal parameter close to M ≈ 1/3 (as in the case of isotropic grid-turbulence).
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... In this work, similarities are found to exist in both (a) the marginal structure, through the mixed Hurst-Kolmogorov (e.g., Pareto-Burr-Feller for positively defined processes) probability density distribution functions, which, depending on the selected shape and scale parameters, is applicable to all the examined processes from (truncated) nearly Gaussian distributions to heavy-tail Pareto ones; and (b) the second-order dependence structure, through a Hurst-Kolmogorov generalized model, which is also expanded to include the observed curved behavior at the intermediate scales. These similarities can be well described within the framework of the maximum entropy and the Hurst-Kolmogorov dynamics (see definitions in [14][15][16][17][18]), and can be implemented through the method of moments (e.g., explicit models), through nonlinear transformation (e.g., copulas), or disaggregation (e.g., downscaling and pulse models) schemes, as summarized for the field of Hydrology [19,20] and beyond [3,4,[21][22][23][24][25][26][27][28][29][30][31], preserving both the marginal and dependence structures for a vast range of scales, including double periodic and intermittent behaviors (see the discussion in [18], and references therein). ...
... In this work, similarities are found to exist in both (a) the marginal structure, through the mixed Hurst-Kolmogorov (e.g., Pareto-Burr-Feller for positively defined processes) probability density distribution functions, which, depending on the selected shape and scale parameters, is applicable to all the examined processes from (truncated) nearly Gaussian distributions to heavy-tail Pareto ones; and (b) the second-order dependence structure, through a Hurst-Kolmogorov generalized model, which is also expanded to include the observed curved behavior at the intermediate scales. These similarities can be well described within the framework of the maximum entropy and the Hurst-Kolmogorov dynamics (see definitions in [14][15][16][17][18]), and can be implemented through the method of moments (e.g., explicit models), through nonlinear transformation (e.g., copulas), or disaggregation (e.g., downscaling and pulse models) schemes, as summarized for the field of Hydrology [19,20] and beyond [3,4,[21][22][23][24][25][26][27][28][29][30][31], preserving both the marginal and dependence structures for a vast range of scales, including double periodic and intermittent behaviors (see the discussion in [18], and references therein). ...
... Therefore, based on the scientific boost, the climacogram (and not the other two metrics) was found to be adequate for the identification and model building of a stochastic process. Since then, interest in the scale domain and the climacogram estimator has increased, and the climacogram has been implemented in education material [49], and has been used to identify the LTP behaviour in various scientific studies, such as 2D precipitation fields [50], multidimensional spatiotemporal domain [51], paleoclimatic temperature [52] and precipitation [53,54], Bayesian statistical models of rainfall and temperature [55], higher-order moments of skewness and kurtosis vs. scale in grid turbulence [26], annual precipitation [56], water demand [57], daily river flows [58], precipitation and temperature for a bivariate drought analysis [59], wind and solar energy [60], water-energy nexus [61], solar radiation [62], wave height and period [63], daily streamflow [64], and monthly temperature and precipitation ( [65,66]), annual streamflow ( [30,66]), ecosystem variability [67], 2D rock formations [68], urban streamflows [69], global temperature and wind of resolution spanning 10 orders of magnitude from ms to several decades [70], disaggregation schemes from daily to hourly rainfall and runoff [71], hourly wind and daily precipitation [26], fine scale precipitation [3,22,[72][73][74][75][76][77][78], fine scale wind [26,70], 2D landscapes [79,80], flood risk assessment [81], bridge scour [82], art [83][84][85], spatiotemporal evolution of clustering [86], alternative statistical moments (e.g., L-moments, K-moments; [3,87]), comparison of multi-source data [88], daily extreme global temperature [89], hourly offshore and coastal wind for energy assessment [90], and in many other applications [3,18]. ...
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To seek stochastic analogies in key processes related to the hydrological cycle, an extended collection of several billions of data values from hundred thousands of worldwide stations is used in this work. The examined processes are the near-surface hourly temperature, dew point, relative humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow and precipitation. Through the use of robust stochastic metrics such as the K-moments and a secondorder climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic similarities exist in both the marginal structure, in terms of the first four moments, and in the secondorder dependence structure. Stochastic similarities are also detected among the examined processes, forming a specific hierarchy among their marginal and dependence structures, similar to the one in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can be also justified by the principle of maximum entropy. Therefore, they allow for the development of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics, with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence at large scales, and a transient behavior at intermediate scales.
... A variety of processes exhibit Long-Term Persistence (LTP) behaviour such as temperature, humidity, surface wind, precipitation, atmospheric pressure, river discharges etc (Dimitriadis, 2017). Particularly, all these processes are characterized by high unpredictability due to the clustering of events. ...
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Living organisms pass through life seeking prosperity in a materialistic world. There are different meanings of prosperity. Some people think that it is measured in money, others relate it to pleasure and life satisfaction, while others link it to spirituality. However, it could be argued that the basic human needs related to the Water, Energy and Food (WEF) compose a nexus not only necessary for the survival of humans, but able to explain their prosperity as well. Unfortunately, decision-making in modern world is largely driven by economic aspects and monetarist policies. Koutsoyiannis (personal communication) notes that water, energy and food are not derived by money; rather money and economic growth derives from the availability and the access to water, energy and food. In this thesis, we study critical issues of prosperity rationally, using publicly available data, historical evidences and stochastic tools. The studied issues are based on the WEF nexus but extend to various other societal, environmental and cultural aspects of human life in societies, ranging from social stratification and urban clustering, to the aesthetic quality of surrounding environment.
... Particularly, to account for the double periodicity, heavy-tail distribution, and long-term persistence, evident in hydrometeorological processes (Dimitriadis, 2017), the stochastic algorithm proposed by Dimitriadis and Koutsoyiannis (2015) is applied in offshore wind speed. An outline of the appropriate steps to be followed during the stochastic generation scheme is presented in Figure 5 ...
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Planning an offshore wind project is considered as a highly complex and multivariable task since it involves a large number of parameters, controversial objectives and constraints to be considered. During the pre-feasibility and pre-planning stages for offshore wind farm site-prospecting, the current manual and sequential design approaches are not always sufficient to guarantee optimal solutions because inherent interactions and trade-offs are most of the times disregarded. Most of the already existing wind energy design tools are specifically built either for onshore environments or for specific offshore activities; hence most of them ignore many relevant key design aspects extended in both space and time. In addition, with the rapid evolution of the Geographic Information Systems (GIS) during the last two decades, numerous research studies, spatial modelling and spatial optimization approaches in the field of the Renewable Energy Sources (RES) gained attention. Highlighting the promising results occurred, considering the planning and designing procedures of such projects, in the near future, geospatial technology with its numerous services and fields can effectively be utilized for timely analysis and future planning assessments. Considering the aforementioned challenges, this Ph.D. thesis proposes the development of a set of tools, as a Spatial Decision Support System (SDSS) entitled SpOWNED-Opt (Spatial Optimization for Offshore WiNd Energy Development), in order to model, map, evaluate and identify continuous space for future OWF siting, towards the mathematical programming approach, based on GIS data structures and algorithms. Thus, the proposed tool can be defined as a more integrated GIS-based framework for the pre-feasibility assessment as also for parts of the Front and Engineering Stage of the Design (FEED) for offshore wind farm site-prospecting procedures in the North and Central Aegean Sea in Greece. In particular, the SpOWNED-Opt approach proposes a multi-level methodological framework for integrating different spatial modelling tools separated at four stages of development. The first stage consists of all preparative steps considering data acquisition pre-processing along with the screening analysis module, based on the Maritime Spatial Planning (MSP) guidelines and the national legislative regulations. Vector and raster data 10 are used expressing existing potential conflicts among human activities combined with socio-economic and environmental factors affecting the selection procedures. The second stage is linked to the cost assessment modules for the capital, operation and maintenance and decommissioning expenses (CAPEX, O&M and DECEX) approximation. An extensive review of all sub-cost components is carried out in order to formulate analytical expressions embedded in the SDSS. Moreover, graph-based optimization techniques are applied, based on Least Cost Path (LCP) algorithms upon raster surfaces in order to extract distance-based costs (transmission lines, installation, decommissioning and O&M costs). The third stage focuses on the energy yield estimation and wind power output variability based on the UERRA Regional Reanalysis data. Different probabilistic models (Weibull, Burr Type II and XII, Gen. Gamma), reanalysis data errors quantification, wind speed intermittent characteristics and the second-order dependence structure are examined, analyzed and modelled in order to stochastically generate wind power output time series that are served as inputs to the last stage of the SDSS. The final module refers to a multi-objective integer non-linear programming (INLP) algorithm; as a unified framework that allows exploring in a rigorous and systematic mode numerous alternatives for offshore wind farm site-prospecting. The economic viability and the performance of the proposed wind farms are assessed along with the optimality of the different scenarios, from which the best ones are finally identified and mapped. The novelty of this research lies both on the integrated nature of the SDSS and on the models used in the spatial modelling field. A critical advantage of the SDSS is that it addresses existing gaps on OWFs siting and overall, in RES location-allocation issues, by: i) introducing a holistic, step-by-step, spatial modelling framework, ii) providing a long-term planning approach, iii) implemented in a user-friendly graphical user interface (GUI), giving the opportunity to national and local authorities and stakeholders to delineate systematic assessment strategies in order to succeed an effective and sustainable renewable energy sources penetration.
... [18][19][20][21] With this aim, complex network analysis combines graph theory and statistical physics have been exploited to study different fluid flows, such as two-phase flows, 22,23 turbulent flows, [24][25][26][27] biology, 28,29 and atmospheric and oceanic flows. [30][31][32][33][34][35] To the best of the authors' knowledge, no study has been conducted to apply complex network analysis into magnetohydrodynamic turbulent flows. ...
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... The mathematical field of Stochastics has been introduced as an alternative to deterministic approaches and is used to model the so-called random, i.e., complex, unexplained or unpredictable, fluctuations observed in (but not limited to) non-linear geophysical processes [45][46][47]. Stochastics helps develop a unified perception for natural phenomena and expel dichotomies like random vs. deterministic. Under the viewpoint of stochastics, there is no such thing as a virus of randomness that infects some phenomena to make them random, leaving other phenomena unaffected. ...
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... [5,6,7,8] In this research a stochastic computational tool based in climacogram, called 2D-C is used to analyze art paintings. [9,10,11,12,13] (white=1, black=0) Benchmark of image analysis; (a) White noise; (b) Image with clustering; (c) Art painting; the lower row depicts the average brightness in the upper one [13]. ...
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