Nonlinear Dynamics in Geosciences pp 121-151 | Cite as

# Towards a Nonlinear Geophysical Theory of Floods in River Networks: An Overview of 20 Years of Progress

## Abstract

Key results in the last 20 years have established the theoretical and observational foundations for developing a new nonlinear geophysical theory of floods in river basins. This theory, henceforth called the scaling theory, has the explicit goal to link the physics of runoff generating processes with spatial power-law statistical relations between floods and drainage areas across multiple scales of space and time. Published results have shown that the spatial power law statistical relations emerge asymptotically from conservation equations and physical processes as drainage area goes to infinity. These results have led to a key hypothesis that the physical basis of power laws in floods has its origin in the self-similarity (self-affinity) of channel networks. Research within the last 20 years has also shown that self-similarity is the basis for the widely observed fractal structure and Horton relations in river networks. Observed power laws in floods span a broad range of spatial scales and multiple time scales that range from hours of individual flood events to annual time scale of flood frequencies. They serve as the foundation for developing a new diagnostic framework to test different assumptions governing spatial variability in physical processes that arise in predicting power law statistical relations between floods and drainage areas. The structure of the diagnostic framework is illustrated using two examples. Contemporary relevance and future applications of the scaling theory come from predicting floods in the context of a significant warming of the Earth’s climate, which is altering local, regional, continental and global balances of water and energy. This anthropogenic perturbation to the planetary hydro-climate precludes making flood predictions at gauged and ungauged sites based on regional statistics from historical stream flow data, which is a well-established practice in hydrologic engineering.

### References

- Band, L. (1986) Topographic partition of watersheds with digital elevation models. Water Resour. Res. 22, 15-24.Google Scholar
- Barenblatt, G. I. (1996)
*Scaling, Self-Similarity and Intermediate Asymptotics*. Cambridge Texts in Applied Mathematics, 14, Cambridge, UK.Google Scholar - Beven, K. (2003)
*Rainfall-Runoff Modelling: the Primer*. John Wiley and Sons Ltd.Google Scholar - Birnir, B., Smith, T.R., and Merchant, G.E. (2001) The scaling of fluvial landscapes. Comp. Geosci. 27, 1189-1216.CrossRefGoogle Scholar
- Booij M. J. (2005) Impact of climate change on river flooding assessed with different spatial model resolutions. J. Hydrol. 303, 176–198.CrossRefGoogle Scholar
- Budyko, M.I., (1974)
*Climate and Life*, (D. H. Miller (Ed.), English Edition), Academic Press, New York.Google Scholar - Burd, G.A., Waymire, E. and Winn, R.D. (2000) A self-similar invariance of critical binary Galton-Watson trees. Bernoulli. 6, 1-21.CrossRefGoogle Scholar
- Cathcart, J. (2001) The effects of scale and storm severity on the linearity of watershed response revealed through the regional L-moment analysis of annual peak flows, Ph.D. dissertation, University of British Colombia, Vancouver, Canada.Google Scholar
- Choudhury, B.J. (1999) Evaluation of an empirical equation for annual evaporation using field observations and results from a biophysical model. J. Hydrol. 216, 99-110.CrossRefGoogle Scholar
- Corradini, C., Govindaraju, R.S. and Morbidelli, R. (2002) Simplified modeling of areal average infiltration at the hillslope scale. Hydrol. Process. 16, 1757-1770.CrossRefGoogle Scholar
- Cox, D. R., and Isham, V. (1998) Stochastic spatial-temporal models for rain. In: O. E. Barndorff-Nielsen, V. K. Gupta, V. Perez-Abreu, and E. C. Waymire (Eds.),
*Stochastic Methods in Hydrology: Rainfall, landforms and floods*. Adv. Ser. Stat. Sci. Appl. Prob. 7, World Scientific, pp. 1-24.Google Scholar - de Vries, H., Becker, T. and Eckhardt, B. (1994) Power law distribution of discharge in ideal networks. Water Resour. Res. 30, 3541-3544.CrossRefGoogle Scholar
- Dodds, P.S. and Rothman, D.H. (2000) Geometry of river networks. I. Scaling, fluctuations, and deviations. Phys.Rev. E. 63, 16115-1-16115-13.Google Scholar
- Dooge J. C. I. (1997) Scale problems in hydrology. In: N. Buras (Ed.),
*Reflections on hydrology Science and Practice,*American Geophysical Union, pp. 85-143.Google Scholar - Duffy, C. J. (1996) A two-state integral-balance model for soil moisture and groundwater dynamics in complex terrain, Water Resour. Res. 32, 2421-2434.CrossRefGoogle Scholar
- Eagleson, P. S. (1972) Dynamics of flood frequency. Water Resour. Res. 8, 878-898.Google Scholar
- Eaton, B., Church, M. and Ham, D. (2002) Scaling and regionalization of flood flows in British Columbia, Canada, Hydrol. Process. 16, 3245-3263.Google Scholar
- Feder, J. (1988)
*Fractals*. Plenum Press, New York.Google Scholar - Foufoula-Georgiou, E. (1998) On scaling theories of space-time rainfall. Some recent results and open problems. In: O. E. Barndorff-Nielsen, V. K. Gupta, V. Perez-Abreu, and E. C. Waymire (Eds.),
*Stochastic Methods in Hydrology: Rainfall, landforms and floods*. Adv. Series Stat. Sci. and Appl. Prob. 7, World Scientific, pp. 25-72.Google Scholar - Furey, P.R. and Gupta, V.K. (2005) Effects of excess rainfall on the temporal variability of observed peak discharge power laws, Adv. Water Resour. 28, 1240-1253.CrossRefGoogle Scholar
- Goodrich, D.C., Lane, L.J., Shillito, R.M. and Miller, S. (1997) Linearity of basin response as a function of scale in a semiarid watershed. Water Resour. Res. 33, 2951-2965.CrossRefGoogle Scholar
- Griffiths, G.A. (2003) Downstream hydraulic geometry and hydraulic similitude, Water Resour. Res. 39, doi: 10.1029/2002WR001485.Google Scholar
- Gupta, V. K. (2004) Emergence of statistical scaling in floods on channel networks from complex runoff dynamics. Chaos, Solitons and Fractals. 19, 357-365.CrossRefGoogle Scholar
- Gupta, V. K., Castro, S. and Over, T.M. (1996) On scaling exponents of spatial peak flows from rainfall and river network geometry. J. Hydrol. 187, 81-104.CrossRefGoogle Scholar
- Gupta, V. K. and Dawdy, D. (1995) Physical interpretation of regional variations in the scaling exponents in flood quantiles. Hydrol. Proc. 9, 347-361.CrossRefGoogle Scholar
- Gupta, V.K. and Mesa, O.J. (1988) Runoff generation and hydrologic response via channel network geomorphology: Recent progress and open problems. J. Hydrol. 102, 3-28.CrossRefGoogle Scholar
- Gupta, V. K., Mesa, O.J. and Dawdy, D.R. (1994) Multiscaling theory of flood peaks: Regional quantile analysis. Water Resour. Res. 30, 3405-3421.CrossRefGoogle Scholar
- Gupta V. K. and Waymire, E. (1983) On the formulation of an analytical approach to hydrologic response and similarity at the basin scale. J. Hydrol. 65, 95-123.CrossRefGoogle Scholar
- Gupta, V. K. and Waymire, E. (1990) Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res. 95, 1999-2009.Google Scholar
- Gupta, V. K. and Waymire, E. (1998a) Spatial variability and scale invariance in hydrologic regionalization. In: G. Sposito (Ed.),
*Scale Dependence and Scale Invariance in Hydrology*, Cambridge University Press, pp. 88-135.Google Scholar - Gupta, V.K. and Waymire, E. (1998b) Some mathematical aspects of rainfall, landforms and floods. In: O. E. Barndorff-Nielsen, V. K. Gupta, V. Perez-Abreu, and E. C. Waymire (Eds.),
*Stochastic Methods in Hydrology: Rainfall, landforms and floods*. Adv. Series Stat. Sci. and Appl. Prob. 7, World Scientific, pp. 129-171.Google Scholar - Gupta, V.K., Waymire, E. and Wang, C.T. (1980) A representation of an instantaneous unit hydrograph from geomorphology. Water Resour. Res. 16, 855-862.Google Scholar
- Hack, J.T. (1957) Studies of longitudinal stream profiles in Virginia and Maryland.
*US Geological Survey Professional Paper*294-B.Google Scholar - Hobbins, M. T., Ramirez, J., and Brown, T.C. (2004) Trends in pan evaporation and actual evapotranspiration in the conterminous U. S.: Paradoxical or complementary? J. Geophy. Res. 31, L13503, doi: 1029/2004GL019846.Google Scholar
- Ibbitt, R. P., McKerchar, A.I., and Duncan, M.J. (1998) Taieri river data to test channel network and river basin heterogeneity concepts. Water Resour. Res. 34, 2085-2088.CrossRefGoogle Scholar
- IPCC (Intergovernmental Panel on Climate Change) (2001) Third Assessment Report, Climate Change 2001, World Meterol. Assoc. and United Nations Environ. Programme, Geneva.Google Scholar
- Karl, T. R., Knight, R.W., Easterling, D.R. and Quayle, R.G. (1996) Indices of climate change for the United States. Bull. Am. Meteorol. Soc. 77, 179-292.CrossRefGoogle Scholar
- Kean, J.W. and Smith, J.D. (2005) Generation and verification of theoretical rating curves in the Whitewater river basin, Kansas. J. Geophy. Res. 110, F04012, doi:10.1029/2004JF000250.CrossRefGoogle Scholar
- Kirkby, M.J. (1976) Tests of the random network model and its application to basin hydrology. Earth Surf. Proc. Landforms. 1, 97-212.Google Scholar
- Klemes, V. (1978) Physically based stochastic hydrological analysis. Adv. Hydroscience. 11, 285-356.Google Scholar
- Klemes, V. (1983) Conceptualization and scale in hydrology. J. Hydrol. 65, 1-23.CrossRefGoogle Scholar
- Klemes, V. (1989) The improbable probabilities of extreme floods and droughts. In: O. Starosolsky and O. M. Meldev (Eds),
*Hydrology and Disasters.*James & James, London, pp. 43-51.Google Scholar - Klemes, V. (1997) Of carts and horses in hydrologic modeling. J. Hydrol. Eng. 2, 43-49.CrossRefGoogle Scholar
- Koster, R. D. and Suarez, M.J. (1999) A simple framework for examining the interannual variability of land surface moisture fluxes. J. Climate. 12, 1911-1917.CrossRefGoogle Scholar
- Lee, M.T. and Delleur, J.W. (1976) A variable source area model of the rainfall-runoff process based on the watershed stream network. Water Resour. Res. 12, 1029-1035.Google Scholar
- Leopold, L.B. and Langbein, W.B. (1962) The concept of entropy in landscape evolution.
*US Geological Survey Professional Paper*500A.Google Scholar - Leopold, L. B., Wolman, M.G. and Miller, J.P. (1964)
*Fluvial Processes in Geomorphology*, W. H. Freeman, San Francisco.Google Scholar - Leopold, L. B. and Maddock, T. (1953) The hydraulic geometry of stream channels and some physiographic implications.
*US Geological Survey Professional Paper*252.Google Scholar - Leopold, L. B. (1953) Downstream change of velocity in rivers. Am. Jour. Sci. 251, 606-624.CrossRefGoogle Scholar
- Leopold, L. B., and Miller, J.P. (1956) Ephemeral Streams: Hydraulic Factors and Their Relation to Drainage Net.
*U.S. Geological Survey Professional Paper*282A.Google Scholar - Manabe, S. (1997) Early development in the study of greenhouse warming: The emergence of climate models. Ambio. 26, 47-51.Google Scholar
- Mandelbrot, B. (1982)
*The Fractal Geometry of Nature*. Freeman, San Francisco, USA.Google Scholar - Mantilla, R. (2007) Physical basis of statistical self-similarity in peak flows in random self-similar networks, PhD dissertation, University of Colorado, Boulder.Google Scholar
- Mantilla, R. and Gupta, V.K. (2005) A GIS numerical framework to study the process basis of scaling statistics on river networks. IEEE Geophysical and Remote Sensing Letters. 2, 404-408.CrossRefGoogle Scholar
- Mantilla, R., Gupta, V.K. and Mesa, O.J. (2006) Role of coupled flow dynamics and real network structures on Hortonian scaling of peak flows. J. Hydrol. 322, 155-167.CrossRefGoogle Scholar
- Mantilla, R., Troutman, B.M. and Gupta, V.K. (2007) Statistical scaling of peak flows and hydrograph properties in random self-similar river networks: constant velocity case (preprint).Google Scholar
- McConnell, M. and Gupta, V.K. (2007) A proof of the Horton law of stream numbers for the Tokunaga model of river networks. Fractals (in press).Google Scholar
- McKerchar, A.I., Ibbitt, R.P., Brown, S.L.R. and Duncan, M.J. (1998) Data for Ashley river to test channel network and river basin heterogeneity concepts. Water Resour. Res. 34, 139-142.CrossRefGoogle Scholar
- Meakin, P., Feder, J. and Jossang, T. (1991) Simple statistical models for river networks. Physica A. 176, 409-429.CrossRefGoogle Scholar
- Menabde, M., Seed, A., Harris, D. and Austin, G. (1997) Self-similar random fields and rainfall simulation, J. Geophys. Res. 102, 13509-13515.CrossRefGoogle Scholar
- Menabde, M. and Sivapalan, M. (2001a) Linking space-time variability of river runoff and rainfall fields: a dynamic approach. Adv. Water Resour. 24, 1001-1014.CrossRefGoogle Scholar
- Menabde, M., Veitzer, S.E., Gupta, V.K. and Sivapalan, M. (2001b) Tests of peak flow scaling in simulated self-similar river networks. Adv. Water Resour. 24, 991-999.CrossRefGoogle Scholar
- Mesa, O.J. and Gupta, V.K. (1987) On the main channel length-area relationship for channel networks. Water Resour. Res. 23, 2119-2122.Google Scholar
- Morrison, J. A., and Smith, J. (2001) Scaling properties of flood peaks. Extremes. 4, 5-22.CrossRefGoogle Scholar
- National Research Council (1988) Estimating Probabilities of Extreme Floods: Methods and Recommended Research. National Academy Press, Washington, D.C.Google Scholar
- National Research Council (1991)
*Opportunities in the Hydrologic Sciences.*National Academy Press, Washington D.C.Google Scholar - Newman, W.I., Turcotte, D.L. and Gabrielov, A.M. (1997) Fractal trees with side branching. Fractals. 5, 603-614.CrossRefGoogle Scholar
- Newton, D. W. and Herrin, J. C. (1981) Assessment of Commonly Used Flood Frequency Methods, paper presented at Fall Meeting, Amer. Geophys. Union, San Francisco, CA.Google Scholar
- Nordstrom, K. and Gupta, V.K. (2003) Scaling statistics in a critical, nonlinear physical model of tropical oceanic rainfall. Nonlinear Proc. Geophy. 10, 531-543.Google Scholar
- Nordstrom, K., Gupta, V.K. and Chase, T. (2005) Role of the hydrological cycle in regulating the climate of a simple dynamic area fraction model, Nonlinear Proc. Geophy. 12, 741-753.Google Scholar
- Ogden F. L. and Dawdy, D.R. (2003) Peak discharge scaling in small Hortonian watershed. J. Hydrol. Engr. 8, 64-73.CrossRefGoogle Scholar
- Over, T. M. and Gupta, V.K. (1994) Statistical analysis of mesoscale rainfall: Dependence of random cascade generador on the large-scale forcing. J. Appl. Meteor. 33, 1526-1542.CrossRefGoogle Scholar
- Over, T. M. and Gupta, V.K. (1996) A space-time theory of mesoscale rainfall using random cascades. J. Geophys. Res. 101, 26319-26331.CrossRefGoogle Scholar
- Peckham, S. D. (1995) New results for self-similar trees with applications to river networks. Water Resour. Res. 31, 1023-1030.CrossRefGoogle Scholar
- Peckham, S.D. and Gupta, V.K. (1999) A reformulation of Horton’s laws for large river networks in terms of statistical self-similarity. Water Resour. Res. 35, 2763-2777.CrossRefGoogle Scholar
- Poveda G, Vèlez, J.I., Mesa, O.J., Cuartas, A., Barco, J., Mantilla, R.I., Mejìa, J.F., Hoyos, C.D., Ramìrez, J.M., Ceballos, L.I., Zuluaga, M.D., Arias, P.A., Botero, B.A., Montoya, M.I., Giraldo, J.D., and Quevedo, D.I. (2007) Linking long-term water balances and statistical scaling to estimate river flows along the drainage Network of Colombia. J. Hydrol. Eng. 12, 4-13.CrossRefGoogle Scholar
- Reggiani, P, Sivapalan, M., Hassanizadeh, S.M. and Gray, W.G. (2001) Coupled equations for mass and momentum balance in a stream network: theoretical derivation and computational experiments. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 457, 157-189.CrossRefGoogle Scholar
- Richards-Pecou, B. (2002) Scale invariance analysis of channel network width function and possible implications for flood behavior. Hydrol. Sci. J. 47, 387-404.CrossRefGoogle Scholar
- Rinaldo, A. and Rodriguez-Iturbe, I. (1996) Geomorphological theory of the hydrological response. Hydro. Proc. 10, 803-829.CrossRefGoogle Scholar
- Rigon, R., Rinaldo, A., and Rodriguez-Iturbe, I., Bras, R.L., and Ijjasz-Vasquez, E. (1993) Optimal channel networks: A framework for the study of river basin morphology. Water Resour. Res. 29, 1635-46.CrossRefGoogle Scholar
- Robinson, J. S. and Sivapalan, M. (1997) An investigation into the physical causes of scaling and heterogeneity in regional flood frequency. Water Resour. Res. 33, 1045-1060.CrossRefGoogle Scholar
- Rodriguez-Iturbe, I. and Rinaldo, A. (1997)
*Fractal river basins: chance and self-organization*. Cambridge.Google Scholar - Rodriguez-Iturbe, I., Ijjasz-Vasquez, E.J., Bras, R.L., and Tarboton, D.G. (1992) Power law distributions of mass and energy in river basins. Water Resour. Res. 28, 1089-1093.CrossRefGoogle Scholar
- Rodriguez-Iturbe, I. and Valdes, J.B. (1979) The geomorphologic structure of hydrologic response. Water Resour. Res. 15, 1409-1420.CrossRefGoogle Scholar
- Schroeder, M. (1991)
*Fractals, Chaos, and Power Laws*. W.H. Freeman and Com., New York.Google Scholar - Shreve, R.L. (1966) Statistical law of stream numbers. J. Geol. 74, 17-37.Google Scholar
- Shreve, R.L. (1967) Infinite topologically random channel networks. J. Geol. 75, 178-86.CrossRefGoogle Scholar
- Sivapalan, M., Takeuchi, K., Franks, S., Gupta, V., Karambiri, H., Lakshmi, V., Liang, X., McDonnell, J., Mendiondo, E., O’Connell, P., Oki, T., Pomeroy, J., Schertzer, D., Uhlenbrook, S. and Zehe, E. (2003) IAHS decade on predictions in ungauged basins (PUB), 2003-2012: Shaping an exciting future for the hydrologic sciences. Hydro. Sci. J. 48, 857-880.CrossRefGoogle Scholar
- Sivapalan, M., Wood, E.F., and Beven, K.J. (1990) On hydrologic similarity, 3, A dimensionless flood frequency model using a generalized geomorphic unit hydrograph and partial area runoff generation. Water Resour. Res. 26, 43-58.CrossRefGoogle Scholar
- Smith, J. (1992) Representation of basin scale in flood peak distributions. Water Resour. Res. 28, 2993-2999.CrossRefGoogle Scholar
- Tokunaga, E. (1966) The composition of drainage networks in Toyohira river basin and valuation of Horton’s first law (in Japanese with English summary). Geophys. Bull. Hokkaido Univ. 15, 1-19.Google Scholar
- Tokunaga, E. (1978) Consideration on the composition of drainage networks and their evolution. Geographical Reports of Tokyo Metropolitan University 13.Google Scholar
- Tokunaga, E. (1984) Ordering of divide segments and law of divide segment numbers. Trans. Jpn. Geomorphol. Union, 5, 71-77.Google Scholar
- Troutman, B.M. (2005) Scaling of flow distance in random self-similar channel networks. Fractals. 13, 265-282.CrossRefGoogle Scholar
- Troutman, B.M. and Karlinger, M.R. (1984) On the expected width function for topologically random channel networks. J. Appl. Prob. 21, 836-849.CrossRefGoogle Scholar
- Troutman, B.M. and Karlinger M.R. (1994) Inference for a generalized Gibbsian distribution on channel networks. Water Resour. Res. 30, 2325-2338.CrossRefGoogle Scholar
- Troutman, B.M. and Karlinger, M.R. (1998) Spatial channel network models in hydrology. In: O. E. Barndorff-Nielsen, V. K. Gupta, V. Perez-Abreu, and E. C. Waymire (Eds.),
*Stochastic Methods in Hydrology: Rainfall, landforms and floods*. Adv. Series Stat. Sci. and Appl. Prob. 7, World Scientific, pp. 85-127.Google Scholar - Troutman, B. M. and Over, T.M. (2001) River flow mass exponents with fractal channel networks and rainfall. Adv. Water Resour. 24, 967-989.CrossRefGoogle Scholar
- Turcotte, D. L. (1997)
*Fractals and Chaos in Geology and Geophysics*, 2nd Edition, Cambridge.Google Scholar - Turcotte, D. L. and Rundle, J.B. (Eds.) (2002) Self-organized complexity in the physical, biological and social sciences, Proc. Natl. Acad. Sci. USA, 99, Suppl. 1, 2463-2465.Google Scholar
- Veitzer, S. A., and Gupta, V.K. (2000) Random self-similar river networks and derivations of Horton-type relations exhibiting statistical simple scaling. Water Resour. Res. 36, 1033-1048.CrossRefGoogle Scholar
- Veitzer, S. A. and Gupta, V.K. (2001) Statistical self-similarity of width function maxima with implications to floods. Adv. Water Resour. 24, 955–965.CrossRefGoogle Scholar
- Veitzer, S.A., Troutman, B.M. and Gupta, V.K. (2003) Power-law tail probabilities of drainage areas in river basins. Phys. Rev. E. 68, 016123.CrossRefGoogle Scholar
- Veneziano, D., Moglen, G.E., Furcolo, P. and Iacobellis, V. (2000) Stochastic model of the width function. Water Resour. Res. 36, 1143-1158.CrossRefGoogle Scholar
- West, G. B. and Brown, J.M. (2004) Life’s universal scaling laws. Physics Today 57, 36-42.CrossRefGoogle Scholar