Chronologically, one of my first interests was modeling the evolution of channel networks according to principles of minimal energy dissipation and self-organization by critical states. These two types of models proved to be capable of reproducing the two- and three-dimensional statistical characteristics of channel networks and natural basins, as well as the fractal and multifractal characteristics. This work, born with the intent of identifying a minimum set of characteristic dynamic elements in the evolution a hydrographic basin, has always been carried out in parallel to the refinement of measurement and analysis techniques of topographic data [J3].

The concept of optimality of a hydrographic basin was introduced in [J3, J4, J5]. In these works, the three postulates of optimal channel networks (OCNs) are stated and developed, proving how such principles can have quantitative effects on the morphology of river networks, particularly affecting the structure of slopes and of contributing areas, the geometry of the channels, and the characteristic velocity of the peak flow of a basin. All of these results explain numerous empirical laws and are still the basis of measurement campaigns.

In [J4], that which was postulated in [J3, J5] was verified by numeric simulation. It should be noted that the minimization of dissipated energy generates fractal forms that reproduce the quantitative characteristics of real basin. In [J6] the concept of optimality is further refined by introducing the hillslope contribution and presenting some case studies. In [J6], more tools are introduced for the qualitative comparison between the numeric models and the natural data. In [J8, J9, A6] a model of the evolution of river basins is presented that is based on the concepts of self-organization by critical states. This model proved to be equivalent to optimization model of [J3-J6]. In [J13] the impact of climatic variability on the morphology of the fluvial landscape is simulated, so offering an interpretative framework for some fluvial forms that can be found in nature.

Subsequently, the concept of optimality was refined observing that real basins do not have the configuration that would give an absolute minimum of dissipated energy, but rather that of states of local minimum that are dynamically accessible. From here the concept of feasible optimality was derived [A10, J18, J19]. It was also demonstrated that the states of absolute minimum, dynamically unreachable, have statistical properties that are not realistic, while accessible minimum states have the desired statistical characteristics. A relevant characteristic of the space-time dynamics of hydrographic networks is that they can be described by means of a parameter that can be linked to temperature [J16]. It is therefore possible to define the thermodynamics of the river networks. As with the thermodynamics of other physical systems, the relevant quantities are energy (dissipated in unit time), entropy, and the temperature.

It should be noted that the space-time evolution of river networks happens with an intermittent behavior similar to the concept of point equilibrium proposed for the evolu- tion of the biological species. It was also demonstrated that the temporal dynamics of river networks is coupled with the spatial activity at all scales and that natural networks, therefore, evolve according to conditions of minimum dissipation of energy but in the presence of a great variety of possible dynamic states.

In [J10] an accurate analysis of the fractal and multifractal properties of optimal river networks was carried out. The note [J18] is a review article, sent to the Annual Review of Earth and Planetary Sciences, that treats the aforementioned topics. In [J20] the results of a theorem on network topology that relates the sum of the contributing areas with the contributing areas themselves and hypothesizes that these quantities are analogous to the ratio of metabolic rhythm and mass of living beings. The two quantities are linked an exponential law with an exponent that was proved to be Hack’s exponent.

This work, born with the intent of identifying a minimum set of characteristic dynamic elements in the evolution of a hydrographic basin, has always been carried out in parallel to the refinement of measurement and analysis techniques of topographic data [s2,eb3]. Recently, this field of study has produced a work [J26] where the morphometric statistics of tributaries of natural rivers and OCNs are studied. These are related to the characteristics of peak flows and they have ecological implications such as, for example, the velocity of diffusion of waterborne diseases and the diffusion of species along the river network. In Rigon’s work, the morphological relations between the different parts of fluvial basins have been analyzed with ever more refined numeric instruments, to the point of creating a series of GIS methods known as the Horton Machine [eb3].

The paper [J41] is partially a review of old results, that were not collected before, and were overlooked by people because they did not appear in Rodriguez-Iturbe and Rinaldo 1997 book. It includes however some new set of simulation were injection of rainfall is assigned with certain distributions (with given correlation structure) producing differentiated power laws for discharge and contributing areas. Clearly a result to further explore.

References

In English:

[J3] - Rodriguez-Iturbe, I. , A. Rinaldo, R. Rigon, R. L. Bras, A. Marani and E.J. Ijjasz-Vasquez, Energy dissipation, runoff production, and the 3-dimensional structure of river basin, Water Resources Research, (28)4, 1095-1103, 1992.

[J4] - Rinaldo, A., I. Rodriguez-Iturbe, R. Rigon, R.L. Bras, E. J. Ijjasz-Vasquez e A. Marani, Minimum energy and fractal structures of drainage networks, Water Resources Research, (28), 2183, 1992.

[J5] - Rodriguez-Iturbe, I. , A. Rinaldo, R. Rigon, R. L. Bras, A. Marani and E.J. Ijjasz-Vasquez, Fractal structure as least energy patterns: The case of river networks, Geophysical Res. Letters, (19)9, 889-892, 1992.

[J6] - Rigon R., A. Rinaldo, I. Rodriguez-Iturbe, R. L. Bras and E. Ijjasz-Vasquez, Optimal channel networks: a framework for the study of river basin morphology, Water Resources Research, 29(6), 1635-1646, 1993.

[J7] - Ijjasz-Vasquez, E., R.L. Bras, I. Rodriguez-Iturbe, A. Rinaldo and R. Rigon, Are river networks OCN?, Advances in Water Resources, 16, 69-79, 1993.

[J8] - Rinaldo, A., I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez, and R.L. Bras, Self organized fractal river networks, Physical Review Letters,70(6), 822-26, 1993.

[J9] - Rigon, R., A. Rinaldo and I. Rodriguez-Iturbe, On landscape self-organization, Journal of Geophysical Research, 99(B6), 11971-11993,1994.

[J10] - Rodriguez-Iturbe, I., M. Marani, R. Rigon and A. Rinaldo, Self-organized river basin landscapes: fractal and multifractal characteristics,Water Resources Research, 30(12), 3531-3539,1994.

[J16] - Rinaldo, A. Maritan, A. Flammini, F. Colaiori, R. Rigon, I. Rodriguez-Iturbe and J. R. Banavar, Thermodynamics of fractal networks, Physical Review Letters, 76(18), 3364-3367, 1996.

[J18] Rinaldo, A., I. Rodriguez-Iturbe and R. Rigon, Channel Networks, Annual Review of Earth and Planetary Sciences, 26, 289-327, 1998

[J27] - Convertino, M, Rigon, R; Maritan, A; I. Rodriguez-Iturbe and Rinaldo, A, Probabilistic structure of the distance between tributaries of given size in river networks, Water Resour. Res., Vol. 43, No. 11, W11418, doi:10.1029/2007WR006176, 2007

[J41] -Rinaldo, A., Rigon R., Banavar, J., Maritan, A. and Rodriguez-Iturbe, I., Evolution and selection of river networks: Statics, dynamics, and complexity, PNAS 2014

In Italian:

[A04]- Rigon, R., Il clima è scritto nella forma del reticolo idrografico?, Rapporti e studi della commissione di studio dei provvedimenti per la conservazione e la difesa della cittá di Venezia, Tomo CLI, Classe di Scienze ff. mm. e nn., 1-21, 1992.

[A06] - Rigon, R. - Principi di auto-organizzazione nella dinamica evolutiva delle reti idrografiche, Tesi di Dottorato, Università degli Studi di Genova, Firenze, Padova, Trento, 1994

[A12] - Rigon, R., Che cosa guida i processi morfologici nei bacini fluviali? Reti ottime di canali e la legge di Hack, Atti XXVI Convegno di Idraulica e Costruzioni Idrauliche, Vol II, 121, 1998

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