A framework is provided for scaling and scale issues in hydrology. The first section gives some basic definitions. This is important as researchers do not seem to have agreed on the meaning of concepts such as scale or upscaling. ‘Process scale’, ‘observation scale’ and ‘modelling (working) scale’ require different definitions. The second section discusses heterogeneity and variability in catchments and touches on the implications of randomness and organization for scaling. The third section addresses the linkages across scales from a modelling point of view. It is argued that upscaling typically consists of two steps: distributing and aggregating. Conversely, downscaling involves disaggregation and singling out. Different approaches are discussed for linking state variables, parameters, inputs and conceptualizations across scales. This section also deals with distributed parameter models, which are one way of linking conceptualizations across scales. The fourth section addresses the linkages across scales from a more holistic perspective dealing with dimensional analysis and similarity concepts. The main difference to the modelling point of view is that dimensional analysis and similarity concepts deal with complex processes in a much simpler fashion. Examples of dimensional analysis, similarity analysis and functional normalization in catchment hydrology are given. This section also briefly discusses fractals, which are a popular tool for quantifying variability across scales. The fifth section focuses on one particular aspect of this holistic view, discussing stream network analysis. The paper concludes with identifying key issues and gives some directions for future research.

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Scale issues in hydrological modelling: A review

A drainage basin simulation model introduced here incorporates creep and threshold slumping and both detachment- and transport-limited fluvial processes. Fluvial erosion of natural slopes and headwater channels is argued to be dominantly detachment-limited. Such slopes undergo nearly parallel retreat and replacement with alluvial surfaces under fixed base level, in contrast with gradual slope decline for transport-limited conditions. The arrangement of divides and valleys is sensitive to initial conditions, although average morphology is insensitive. Dissected, initially flat surfaces in which downstream concavity is slight exhibit nearly parallel drainage, compared to very wandering main valleys when concavity is great. Steady state is reached after a cumulative base level drop approximately 3 times the final relief. Simulated valley systems are similar to those predicted by a previous model of optimal drainage basins. A critical value of slope divergence normalized by average slope gradient is a useful criterion for defining the valley network.

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A detachment-limited model of drainage basin evolution

From the Publisher:
Why do similar patterns and forms appear in nature in settings that seem to bear no relation to one another? The Windblown ripples of desert sand follow a sinuous course that resembles the stripes of a zebra or a marine fish In the trellis-like shells of microscopic sea creatures we see the same geometry as in the bubble walls of foam, Forks of lightning mirror the branches of a river network or a tree
This book explains why these are not coincidences Nature commonly weaves its tapestry without any master plan or blueprint Instead, these designs build themselves by self-organization The interactions between the component parts -- whether they be grains of sand, molecules or living cells -- give rise to spontaneous patterns that are at the same time complex and beautiful Many of these patterns are universal, recurring again and again in the natural order: spirals, spots, stripes, branches, honeycombs Philip Ball conducts a profusely illustrated tour of this gallery, and reveals the secrets of how nature's patterns are made

The Self-Made Tapestry: Pattern Formation in Nature

Fifty years of hyporheic zone research have shown the important role played by the hyporheic zone as an interface between groundwater and surface waters. However, it is only in the last two decades that what began as an empirical science has become a mechanistic science devoted to modeling studies of the complex fluid dynamical and biogeochemical mechanisms occurring in the hyporheic zone. These efforts have led to the picture of surface-subsurface water interactions as regulators of the form and function of fluvial ecosystems. Rather than being isolated systems, surface water bodies continuously interact with the subsurface. Exploration of hyporheic zone processes has led to a new appreciation of their wide reaching consequences for water quality and stream ecology. Modern research aims toward a unified approach, in which processes occurring in the hyporheic zone are key elements for the appreciation, management, and restoration of the whole river environment. In this unifying context, this review summarizes results from modeling studies and field observations about flow and transport processes in the hyporheic zone and describes the theories proposed in hydrology and fluid dynamics developed to quantitatively model and predict the hyporheic transport of water, heat, and dissolved and suspended compounds from sediment grain scale up to the watershed scale. The implications of these processes for stream biogeochemistry and ecology are also discussed.

Hyporheic flow and transport processes: Mechanisms, models, and biogeochemical implications

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Supplement to "Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications"

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.

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Energy dissipation, runoff production, and the three-dimensional structure of river basins

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 Peano's basins with OCNs suggests that nature seems to reject the type of strict self-similarity exhibited by Peano's 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.

Minimum energy and fractal structures of drainage 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, Horton's bifurcation ratio R(B) tends, for high values of Strahler's order OMEGA of the basin, to the area ratio R(A), and Horton's 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 (Melton's law, Hack's relation, Moon's 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 Peano's construct (for which R(B) = R(A) = 4, R(L) = 2) closes the paper.

A Note on Fractal Channel 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.

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

A mathematical model based on conditional probability distributions is proposed for solute yield response to rainfall impulses at the basin scale. Solutes enter the mass balance because of sorption processes between fixed phases and the carrier flow. The present approach addresses the individual role and the mutual connection of the hydrologic response and of macroscopic sorption kinetics controlling basin scale release processes. Mass response functions from geomorphology and statistical mechanics are derived and used to analyze experimental records. A synthesis of nonlinear effects in the mechanism of chemical supply to the carrier and the derivation of concentration distributions are also attempted. -from Author

Alessandro Marani

Papers

Energy dissipation, runoff production, and the three-dimensional structure of river basins

Minimum energy and fractal structures of drainage networks

A Note on Fractal Channel Networks

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

Basin scale model of solute transport