1. River networks as emergent characteristics of open dissipative systems Kyungrock Paik and Praveen Kumar 4th IAHR Symposium on River, Coastal and Estuarine Morphodynamics Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Acknowledgements National Science Foundation grant no. EAR 02-08009 Dissertation Completion Fellowships from the University of Illinois at Urbana-Champaign

2. River networks exhibit apparent self-similarity, or more broadly fractal Rogue River in Oregon Self-similar topological organization [e.g., Peckham, 1995] What causes this regularity?

3. Is this a result of an optimization process? Optimal channel network (OCN) [Rodriguez-Iturbe et al., 1992b; Rinaldo et al., 1992] Minimum energy expenditure

4. Is this a result of an optimization process?  Figures from Stevens[1974]

5. Is this a result of an optimization process?  Figures from Stevens[1974] “This ‘law-like’ feature seems to emerge as the inevitable result of a dynamic process that minimizes the dissipation of energy”- Mark Buchanan [Nature 419, 787 (24 October 2002) ]

6. However, the mechanism that enables the systems to find these extremal states remains elusive Hypothesis Evolutionary dynamics driven by a flow gradient and subject to proximity constraint, that is, the matter and energy can traverse only through a continuum, in the presence of inherent randomness of media properties, give rise to a tree topological organization.

7.  Grid size: 500m × 500m  Domain: 40401 cells (201 × 201)  Effective cells: 31397 (=7849.25 km 2 ) < Puerto Rico   t =2 years  =0.5   s =2650 kg/m 3  Dynamic equilibrium condition To test the proposed hypothesis, a deterministic numerical model is built

8. Regard river networks as streamlines. Streamlines: orthogonal to the geographic contour D8 method [O'Callaghan and Mark, 1984] for flow path decision Governing eq: Schoklitsch [1934] i e =0.1mm/hr (=876mm/yr)

9. Simulation result Time: 0 years

10. Simulation result Time: 10 years

11. Simulation result Time: 10 2 years

12. Simulation result Time: 10 3 years

13. Simulation result Time: 10 4 years

14. Simulation result Time: 10 5 years 294 Watersheds

15. Power law relationships can be measures of self- similarity Figures from [Maritan et al., 1996] and [Rigon et al., 1996]

16. Power law relationships can be measures of self- similarity P(A d   )     =0.43±0.03 [Rodriguez-Iturbe et al., 1992] x  h h=0.6±0.1 [Hack, 1957] P(L  l)  l -   =0.68±0.24 [Rigon et al., 1996; Crave and Davy, 1997] Figures from [Maritan et al., 1996] and [Rigon et al., 1996]

17. Simulated networks exhibit these power law distributions P(L  l)  l - 

18. Key results are robust regardless of the shape of islands

19. The insights gained here may be extended to explain the formation of other networks Images from [Merrill, 1978], http://www.lightningsafety.noaa.gov/photos.htm,http://www.lightningsafety.noaa.gov/photos.htm [Huber et al., 2000], and [Jun and Hübler, 2005]

20. In summary, evolutionary dynamics driven by a flow gradient and subject to proximity constraint, that is, the matter and energy can traverse only through a continuum, in the presence of inherent randomness of media properties, give rise to a tree topological organization The minimization of energy expenditure is not the cause but a consequent signature Findings may serve as a motif for the formation of other networks. Questions?