Riccardo Rigon

Energy Dissipation, Runoff Production, and the Three-Dimensional Structure of River Basins

Three principles of optimal energy expenditure are used to derive the most important structural characteristics observed in drainage networks: (1) the principle of minimum energy expenditure in any link of the network, (2) the principle of equal energy expenditure per unit area of channel anywhere in the network, and (3) the principle of minimum total energy expenditure in the network as a whole. Their joint application results in a unified picture of the most important empirical facts which have been observed in the dynamics of the network and its three-dimensional structure. They also link the process of runoff production in the basin with the characteristics of the network.

On the spatial organization of soil moisture fields

We examine the apparent disorder which seems to characterize the spatial structure of soil moisture by analyzing large-scale experimental data. Specifically, we address the statistical structure of soil moisture fields under different scales of observation and find unexpected results. The variance of soil moisture follows a power law decay as function of the area at which the process is observed. The spatial correlation remains unchanged with the scale of observation and follows a power law decay typical of scaling processes. Soil moisture also shows clear scaling properties on its spatial clustering patterns. A well-defined organization of statistical character is found to exist in soil moisture patterns linking a large range of scales through which the process manifests itself and impacts other processes. We suggest that such scaling properties are crucial to our current understanding and modeling of the dynamics of soil moisture in space and time.

GEOtop: A Distributed Hydrological Model with Coupled Water and Energy Budgets

Abstract This paper describes a new distributed hydrological model, called GEOtop. The model accommodates very complex topography and, besides the water balance, unlike most other hydrological models, integrates all the terms in the surface energy balance equation. GEOtop uses a discretization of the landscape based on digital elevation data. These digital elevation data are preprocessed to allow modeling of the effect of topography on the radiation incident on the surface, both shortwave (including shadowing) and longwave (accounting for the sky view factor). For saturated and unsaturated subsurface flow, GEOtop makes use of a numerical solution of the 3D Richards’ equation in order to properly model, besides the lateral flow, the vertical structure of water content and the suction dynamics. These characteristics are deemed necessary for consistently modeling hillslope processes, initiation of landslides, snowmelt processes, and ecohydrological phenomena as well as discharges during floods and interstorm...

MINIMUM ENERGY AND FRACTAL STRUCTURES OF DRAINAGE NETWORKS

This paper explores the similarities of digital elevation maps (DEMs) of natural river basins and optimal channel network (OCN) configurations obtained minimizing the total rate of energy expenditure in the system as a whole and in its parts. Striking similarities are observed for natural and optimal networks in their fractal aggregation structure and in certain multifractal structures found to be characteristic of river basins. Our results suggest, upon critical assessment of the reliability of the identification of the attractor of the underlying dynamics implied by our optimality concepts, that fractal structures are indeed possibly a product of least energy dissipation. Power laws emerging in the description of the distribution of aggregated quantities from both DEMs and OCNs suggest a link with the framework of self-organized criticality in the dynamics of natural channel network formation. Also, the geomorphological description of OCNs reveals surprising analogies with well-known empirical or experimental results. A comparison of Peanos basins with OCNs suggests that nature seems to reject the type of strict self-similarity exhibited by Peanos construct in favor of different shapes implying statistical self-similarity not only because of chance acting through random conditions but also because of necessity as reflected by least energy expenditure considerations.

On Hack's Law

Hacks law is reviewed, emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics. The relation between Hacks law and the internal structure of river basins is investigated experimentally through digital elevation models. It is found that Hacks exponent, elongation, and some relevant fractal characters are closely related. The self-affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to Hacks law. An explanation for Hacks law is derived from scaling arguments. From the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of Hacks law.

Can One Gauge the Shape of a Basin

This paper investigates the effects of geometrical factors characterizing the shape of a river basin on the features of its hydrologic response. In particular, we wonder if by measuring the hydrologic response (i.e., gauging) the salient geomorphic features of the basin can be recovered. We argue that the basic structure of the channel network tends, in ideal conditions, to yield some universal characters of the width function W(x) defining the relative proportion of a contributing area at a distance x from the outlet. W(x) exhibits low-frequency features, which are geometry-dominated, and high-frequency features determined by recurrent aggregation patterns. It is suggested that given the shape of the basin one can indeed forecast in a rational manner the main characters of the hydrologic response which are imprinted in reproducible width functions. However, the inverse problem (i.e., the determination of the shape from the measure of the hydrologic response) is less solidly defined because of the possible loss of irretrievable information induced by the dynamics of runoff processes. Therefore the question posed in the title cannot be solved in general, although many elements for a general theory are seemingly established.

Optimal Channel Networks - a Framework for the Study of River Basin Morphology

Optimal channel networks (OCNs) are dendritic structures obtained by minimizing the local and global rates of energy dissipation in a continuously fed (in space and time) plane aggregation system reminiscent, and based on the properties, of the planform of three-dimensional natural drainage networks. Geomorphological and fractal properties of OCNs are known from earlier studies by the authors. This paper explores further the structures derived by optimization of energy dissipation rates. Optimality of subnetworks and of basin shapes is investigated as a by-product of competition for drainage. A new perspective on the possible prediction of the width function of a basin network, and hence of its hydrologic response, is obtained by exploiting OCN techniques, requiring only the definition of the outer boundaries of the basin. The interplay between hillslope processes and the development of drainage networks is addressed, aiming at the relative role and the mutual interrelations of geology and optimal organization in the structure of mature river basins. Also addressed is the issue of multiscaling and multifractality in the spatial organization of the network. It is concluded that OCN approaches provide a comprehensive framework for the study of the morphology of geophysical structures.

A Note on Fractal Channel Networks

This paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response. A mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry. In view of the connectivity conjecture, Hortons bifurcation ratio R(B) tends, for high values of Strahlers order OMEGA of the basin, to the area ratio R(A), and Hortons length ratio R(L) equals, in the limit, the single-order contributing area ratio R(a). The relevance of these arguments is examined by reference to data from real basins. Well-known empirical results from the geomorphological literature (Meltons law, Hacks relation, Moons conjecture) are viewed as a consequence of connectivity. It is found that in Hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1/R(B). The application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph, in particular generalizing its width function formulation. A quantitative example of multifractal hydrologic response of idealized networks based on Peanos construct (for which R(B) = R(A) = 4, R(L) = 2) closes the paper.

On landscape self-organization

A new quantitative characterization of landscape-forming processes in the general framework of self-organized criticality and of fractal analyses is proposed. The coupled processes considered are threshold-independent hillslope evolutions and threshold-dependent fluvial transport phenomena. From a body of experimental and theoretical evidence we argue that geomorphological thresholds, principles of minimum energy expenditure and concepts of self-organized criticality are of crucial importance for the understanding of the basic general mechanisms which govern landscape evolution. This paper considerably extends both the theoretical framework and the empirical evidence for a recently developed theory which incorporates the above general principles. The modeling of landscape evolution by principles of self-organization is accomplished through the introduction of diffusion processes operating mainly on the hillslopes and the coupling of these processes with the fluvial evolution of the network previously studied through principles of self-organized criticality. The effects of spatial variability of surface erodibility are investigated under the general framework of random space functions with a correlation structure. Finally, a fractal analysis of the characteristics of the resulting landscape is performed and compared with recent results from real landforms to suggest the relationship of landscape fractal dimensions with the underlying landscape-forming processes.

Fractal Structures as Least Energy Patterns - the Case of River Networks

Natural drainage networks, like river basins, exhibit fractal and multifractal properties. This work shows that these properties evolve from arbitrary initial conditions by minimizing the local and global rates of energy expenditure in the system. This suggests that many fractal structures may arise as a natural consequence of least energy dissipation requirements.