FRACTAL RIVER BASINS

Transcription

1 FRACTAL RIVER BASINS CHANCE AND SELF-ORGANIZATION Ignacio Rodriguez-Iturbe Texas A & M University Andrea Rinaldo University of Padua, Italy CAMBRIDGE UNIVERSITY PRESS

2 Contents Foreword Preface page xiii xv 1 A View of River Basins Introduction River Basin Geomorphology: A Brief Review Ordering of the Channel Network Drainage Density and the Hillslope Scale Relation of Area to Length Relation of Area to Discharge Relation between Magnitude and Area Stream Channel Geometry The Width Function The Three-Dimensional Structure of River Basins River Basins from Digital Elevation Models Slope-Area Scaling Empirical Evidence Where Do Channels Begin? Experimental Fluvial Geomorphology Statistical Models of Network Evolution Introduction Random-Walk Drainage Basin Models The Random Topology Model Limitations of Statistical Models Deterministic Models of Drainage Network Development Introduction Models Based on Junction Angle Adjustments Models of Erosion and the Evolution of River Networks A Process-Response Model of Catchment and Network Development 77 VII

3 viii ^^^^^ ^^^^^^ Contents Detachment-Limited Basin Evolution Limitations of Deterministic Models Lattice Models 95 Fractal Characteristics of River Basins Introduction Fractals and Fractal Dimensions The Box-Counting Dimension The Cluster Dimension or Mass Dimension The Correlation Dimension Self-Similarity and Power Laws Self-Similarity in River Basins Horton's Laws and the Fractal Structure of Drainage Networks Peano's River Basin Power Law Scaling in River Basins Scaling of Slopes Scaling of Contributing Areas, Discharge, and Energy Self-Similarity of Topographic Contours Self-Affinity in River Basins Brownian Motion and Fractional Brownian Motion Power Spectrum and Correlation Structure of Fractional Brownian Motion Characterization of Self-Affine Records Self-Affine Characteristics of Topographic Transects Self-Affine Characteristics of Width Functions Other Self-Affine Characterizations Self-Affine Scaling of Watercourses Self-Affine Scaling of Basin Boundaries Transects, Contours, Watercourses, and Mountain Ridges as Parts of the Basin Landscape Hack's Law, the Self-Affinity of Basin Boundaries, and the Power Law of Contributing Areas Does Hack's Law Imply Elongation? Power Law of Contributing Areas, Hack's Relationship, and the Self-Affinity of Basin Boundaries Hack's Law and the Probability Distribution of Stream Lengths to the Divide Generalized Scaling Laws for River Networks Scaling of Areas Scaling of Lengths 190

4 Multifractal Characteristics of River Basins Introduction Peano's Basin and the Binomial Multiplicative Process Multifractal Spectra Multifractal Spectra of Width Functions Multiscaling and Multifractality Other Multifractal Descriptors Multifractal Topographies Fractal versus Multifractal Descriptors Generalized Variogram Analysis Random Cascades Canonical Random Cascades Conservative Random Cascades and Width Functions 247 Optimal Channel Networks: Minimum Energy and Fractal Structures Introduction The Connectivity Issue Principles of Energy Expenditure in Drainage Networks Energy Expenditure and Optimal Network Configurations Stationary Dendritic Patterns in a Potential Force Field Scaling Implications of Optimal Energy Expenditure Optimal Channel Networks Geomorphologic Properties of OCNs Fractal Characteristics of OCNs Multifractal Characteristics of OCNs Multiscaling in OCNs Fractals in Nature: Least Energy Dissipation Structures? On Feasible Optimality OCNs, Hillslope, and Channel Processes On the Interaction of Shape and Size Are River Basins OCNs? Hack's Relation and OCNs Renormalization Groups for OCNs OCNs with Open Boundary Conditions Disorder-Dominated OCNs Thermodynamics of OCNs Space-Time Dynamics of Optimal Networks Exact Solutions for Global Minima and Feasible Optimality 347 Contents ix

5 Contents 5 Self-Organized Fractal River Networks Introduction Self-Organized Criticality SOC Systems in Geophysics On Forest Fires, Turbulence, and Life at the Edge Sandpile Models and Abelian Groups Fractals and Self-Organized Criticality Self-Organized Fractal Channel Networks Optimality of Self-Organized River Networks River Models and Temporal Fluctuations Fractal SOC Landscapes Renormalization Groups for SOC Landscapes Thermodynamics of Fractal Networks Self-Organized Networks and Feasible Optimality On Landscape Self-Organization Introduction Slope Evolution Processes and Hillslope Models The Effects of Nonlinearity The Effects of a Driving Noise Landscape Self-Organization On Heterogeneity Fractal and Multifractal Descriptors of Landscapes Geomorphologic Signatures of Varying Climate Geomorphologic Hydrologic Response Introduction Travel Time Formulation of Transport Geomorphologic Unit Hydrograph Travel Time Distributions in Channel Links Geomorphologic Dispersion Hortonian Networks Width Function Formulation of the GIUH Can One Gauge the Shape of a Basin? Estimation of Basin Shapefromthe Width Function Geomorphologic Hydrologic Response On the Spatial Organization of Soil Moisture Fields Introduction The Effect of Aggregation on the Statistics of the Soil Moisture Field 518 References 525 Index 540