Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks
An analytical theory is developed that obtains Horton laws for six hydraulic–geometric (H–G) variables (stream discharge <i>Q</i>, width <i>W</i>, depth <i>D</i>, velocity <i>U</i>, slope <i>S</i>, and friction <i>n'</i>) i...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2014-09-01
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| Series: | Nonlinear Processes in Geophysics |
| Online Access: | http://www.nonlin-processes-geophys.net/21/1007/2014/npg-21-1007-2014.pdf |
| _version_ | 1818590036616019968 |
|---|---|
| author | V. K. Gupta O. J. Mesa |
| author_facet | V. K. Gupta O. J. Mesa |
| author_sort | V. K. Gupta |
| collection | DOAJ |
| description | An analytical theory
is developed that obtains Horton laws for six hydraulic–geometric (H–G)
variables (stream discharge <i>Q</i>, width <i>W</i>, depth <i>D</i>, velocity <i>U</i>, slope
<i>S</i>, and friction <i>n'</i>) in self-similar Tokunaga networks in the limit of a
large network order. The theory uses several disjoint theoretical concepts
like Horton laws of stream numbers and areas as asymptotic relations in
Tokunaga networks, dimensional analysis, the Buckingham Pi theorem,
asymptotic self-similarity of the first kind, or SS-1, and asymptotic
self-similarity of the second kind, or SS-2. A self-contained review of these
concepts, with examples, is given as "methods". The H–G data sets in
channel networks from three published studies and one unpublished study are
summarized to test theoretical predictions. The theory builds on six
independent <i>dimensionless river-basin numbers</i>. A mass conservation
equation in terms of Horton bifurcation and discharge ratios in Tokunaga
networks is derived. Assuming that the H–G variables are homogeneous and
self-similar functions of stream discharge, it is shown that the functions
are of a power law form. SS-1 is applied to predict the Horton laws for
width, depth and velocity as asymptotic relationships. Exponents of width and
the Reynolds number are predicted and tested against three field data sets.
One basin shows deviations from theoretical predictions. Tentatively assuming
that SS-1 is valid for slope, depth and velocity, corresponding Horton laws
and the H–G exponents are derived. Our predictions of the exponents are the
same as those previously predicted for the optimal channel network (OCN)
model. In direct contrast to our work, the OCN model does not consider Horton
laws for the H–G variables, and uses optimality assumptions. The predicted
exponents deviate substantially from the values obtained from three field
studies, which suggests that H–G in networks does not obey SS-1. It fails
because slope, a dimensionless river-basin number, goes to 0 as network order
increases, but, it cannot be eliminated from the asymptotic limit. Therefore,
a generalization of SS-1, based on SS-2, is considered. It introduces two
anomalous scaling exponents as free parameters, which enables us to show the
existence of Horton laws for channel depth, velocity, slope and Manning
friction. These two exponents are not predicted here. Instead, we used the
observed exponents of depth and slope to predict the Manning friction
exponent and to test it against field exponents from three studies. The same
basin mentioned above shows some deviation from the theoretical prediction. A
physical reason for this deviation is given, which identifies an important
topic for research. Finally, we briefly sketch how the two anomalous scaling
exponents could be estimated from the transport of suspended sediment load
and the bed load. Statistical variability in the Horton laws for the H–G
variables is also discussed. Both are important open problems for future
research. |
| first_indexed | 2024-12-16T09:50:09Z |
| format | Article |
| id | doaj.art-ee8223688507442882237af2dadd35c9 |
| institution | Directory Open Access Journal |
| issn | 1023-5809 1607-7946 |
| language | English |
| last_indexed | 2024-12-16T09:50:09Z |
| publishDate | 2014-09-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Nonlinear Processes in Geophysics |
| spelling | doaj.art-ee8223688507442882237af2dadd35c92022-12-21T22:36:05ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462014-09-012151007102510.5194/npg-21-1007-2014Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networksV. K. Gupta0O. J. Mesa1Department of Civil, Environmental and Architectural Engineering and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado, USADepartamento de Geociencias y Medio Ambiente, Universidad Nacional de Colombia, Medellín, ColombiaAn analytical theory is developed that obtains Horton laws for six hydraulic–geometric (H–G) variables (stream discharge <i>Q</i>, width <i>W</i>, depth <i>D</i>, velocity <i>U</i>, slope <i>S</i>, and friction <i>n'</i>) in self-similar Tokunaga networks in the limit of a large network order. The theory uses several disjoint theoretical concepts like Horton laws of stream numbers and areas as asymptotic relations in Tokunaga networks, dimensional analysis, the Buckingham Pi theorem, asymptotic self-similarity of the first kind, or SS-1, and asymptotic self-similarity of the second kind, or SS-2. A self-contained review of these concepts, with examples, is given as "methods". The H–G data sets in channel networks from three published studies and one unpublished study are summarized to test theoretical predictions. The theory builds on six independent <i>dimensionless river-basin numbers</i>. A mass conservation equation in terms of Horton bifurcation and discharge ratios in Tokunaga networks is derived. Assuming that the H–G variables are homogeneous and self-similar functions of stream discharge, it is shown that the functions are of a power law form. SS-1 is applied to predict the Horton laws for width, depth and velocity as asymptotic relationships. Exponents of width and the Reynolds number are predicted and tested against three field data sets. One basin shows deviations from theoretical predictions. Tentatively assuming that SS-1 is valid for slope, depth and velocity, corresponding Horton laws and the H–G exponents are derived. Our predictions of the exponents are the same as those previously predicted for the optimal channel network (OCN) model. In direct contrast to our work, the OCN model does not consider Horton laws for the H–G variables, and uses optimality assumptions. The predicted exponents deviate substantially from the values obtained from three field studies, which suggests that H–G in networks does not obey SS-1. It fails because slope, a dimensionless river-basin number, goes to 0 as network order increases, but, it cannot be eliminated from the asymptotic limit. Therefore, a generalization of SS-1, based on SS-2, is considered. It introduces two anomalous scaling exponents as free parameters, which enables us to show the existence of Horton laws for channel depth, velocity, slope and Manning friction. These two exponents are not predicted here. Instead, we used the observed exponents of depth and slope to predict the Manning friction exponent and to test it against field exponents from three studies. The same basin mentioned above shows some deviation from the theoretical prediction. A physical reason for this deviation is given, which identifies an important topic for research. Finally, we briefly sketch how the two anomalous scaling exponents could be estimated from the transport of suspended sediment load and the bed load. Statistical variability in the Horton laws for the H–G variables is also discussed. Both are important open problems for future research.http://www.nonlin-processes-geophys.net/21/1007/2014/npg-21-1007-2014.pdf |
| spellingShingle | V. K. Gupta O. J. Mesa Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks Nonlinear Processes in Geophysics |
| title | Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks |
| title_full | Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks |
| title_fullStr | Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks |
| title_full_unstemmed | Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks |
| title_short | Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks |
| title_sort | horton laws for hydraulic geometric variables and their scaling exponents in self similar tokunaga river networks |
| url | http://www.nonlin-processes-geophys.net/21/1007/2014/npg-21-1007-2014.pdf |
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