Zipf's law, Hierarchical Structure, and Shuffling-Cards Model for Urban Development

A new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random phenomena such as cities. Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a rank-size distribution following Zipf's law, and vice versa. The hierarchical structure can be described with a set of exponential functions that are identical in form to Horton-Strahler's laws on rivers and Gutenberg-Richter's laws on earthquake energy. From the exponential models, we can derive four power laws such as Zipf's law indicative of fractals and scaling symmetry. Research on the hierarchy is revealing for us to understand how complex systems are self-organized. A card-shuffling model is built to interpret the relation between Zipf's law and hierarchy of cities. This model can be expanded to explain the general empirical power-law distributions across the individual physical and social sciences, which are hard to be comprehended within the specific scientific domains.Comment: 28 pages, 8 figure

Form, scales and optimality in the basin landscape and its channel network

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 1994.Includes bibliographical references (leaves 252-263).by Ede Jorge Ijjász-Vásquez.Ph.D

Bio-physical controls on tidal network geomorphology

The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the authorLooking over a tidal wetland, the tidal network characterised by its intricate system of bifurcating, blind-ended tidal courses clearly stands out from the overall landscape. This tidal landform exerts a fundamental control on the morphology and ecology within the tidal environment. With today’s recognition of the ecological, economical and societal values provided by tidal wetlands, which has been notably reflected in the development of restoration management strategies across Europe and USA, there is a need to fully understand the nature and development of tidal networks as well as their relationships with associated landforms and biotic components (e.g. vegetation), to eventually guarantee the success of current and future restoration practices. Accordingly, this research aims to bring further insights into the bio-physical controls on the geomorphology of tidal networks. To this end, a combination of remote sensing, modelling and field activities was employed. A geo-spatial analysis was performed at Queen Mary, University of London (UK), to address the variability of tidal network patterns. A series of network scale morphometric variables was extracted using airborne LiDAR data among selected tidal networks across the UK depicting different planview morphologies, and supplemented with the collection of corresponding marsh scale environmental variables from published sources. Multivariate statistics were then performed to characterise the variability of tidal network patterns and identify the inherent environmental controls. The analysis has revealed that every network type can be characterised based upon measures of network size and complexity, with each network pattern depicting proper morphometric aspects. Particularly, the stream Strahler order and the median depth of the network main channel have the highest discriminating weight on the patterns investigated. High correlation between the latter variable and network main channel width has revealed that linear, linear-dendritic and dendritic networks followed a transitional gradient in their aspect ratio approximated by a power law and thus are seen to depict similar erosional processes. To the contrary, meandering networks clearly depart from this relationship, and show particular segregation in their aspect ratios with respect to dendritic networks. Globally, differentiation on network morphometric properties has been linked to environmental conditions specific to the marsh physiographic setting within which a tidal network develops. Conceptually, tidal networks seem to adapt to marsh environmental conditions by adopting suitable morphologies to drain their tidal basin effectively. An eco-geomorphic modelling framework was developed at University of Trento (Italy), to address tidal network morphological development. In line with current theories as well as modelling advances and challenges in the field of tidal network ontogeny, emphasis was thus placed on the investigation of tidal channel formation and evolution in progressive marsh accretional context. Under these environmental conditions, tidal network development can be ascribed to the combination of two channel-forming processes: channel initiation results from bottom incisions in regions where topographic depressions occur; channel elaboration results from differential deposition, contributing to the deepening of the tidal channels relative to the adjacent marsh platform. Further evolutionary stages including channel reduction proceed from the horizontal progradation of the marsh platform which may lead eventually to channel infilling. Moreover, both qualitative and quantitative results allude to an acceleration of the morphological development of the synthetic tidal networks with increasing sediment supply. These different observations thus emphasise the prevalence of depositional processes in shaping tidal channels. In a second stage, the investigation was extended to the role of the initial tidal flat morphology as an inherent control on tidal network development, by considering different scenarios of topographic perturbations, which has revealed its legacy on tidal network morphological features. Modelling experiments have also acknowledged salt marsh macrophytes as a potential control on network evolution depending on their biomass distribution within the tidal frame. However, tidal channel morphodynamcis appears to be sensitive to the way biomass growth is mathematically parameterised in the model. In view of the current challenges in transcribing mathematically such a dynamic process and the relevance of bio-physical interactions in driving salt marsh and tidal network evolution, a field survey was conducted in a temperate salt marsh in the Netherlands, as part of the mobility to UNESCO-IHE (Netherlands) in partnership with University of Antwerp (Belgium), to assess vegetation distribution and productivity in the tidal frame. Particularly, emphasis was placed on extending investigations on the possible presence of relationships involving vegetation properties in different climatic and ecological conditions from those characterising these previously documented relationships. Regression analysis has revealed that biomass growth can be expressed as a linear function of marsh relative elevation, providing therefore direct empirical validation for corresponding assumptions reported in the literature and used in the present modelling framework; surprisingly, that increase did not correlate with an increase in species richness and diversity. Analysis of likely associations between vegetation morphometrics and total standing biomass yielded only a single linear relationship linking the latter variable to stem height. In truth, these observations may bear reconsiderations on the global validity of the assumptions used in the formulation of some eco-geomorphic processes which are applied in the study and prediction of wetland resiliency facing climate change