Turbulent rivers
Author:
Björn Birnir
Journal:
Quart. Appl. Math. 66 (2008), 565-594
MSC (2000):
Primary 35Q30, 35R60, 60H15, 76D06
DOI:
https://doi.org/10.1090/S0033-569X-08-01123-8
Published electronically:
June 5, 2008
MathSciNet review:
2445529
Full-text PDF Free Access
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Additional Information
Abstract: The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one-dimensional turbulence is established. The turbulent solutions are not smooth but Hölder continuous with exponent $3/4$. The scaling of the solutions’ second structure (or width) function gives rise to Hack’s law (1957), stating that the length of the main river, in mature river basins, scales with the area of the basin $l \sim A^{h}$, $h = 0.568$ being Hack’s exponent.
References
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- J.K. Weissel, L.F. Pratson, and A. Malinverno, The length-scaling of topography, Journal of Geophysical Research 99 (1994), 13997–14012.
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References
- R. Betchov and W. O. Criminale, Stability of parallel flows, Academic Press, New York, 1967.
- B. Birnir, Turbulence of uniform flow, Proceedings of the conference Probability, Geometry and Integrable Systems, MSRI Dec. 2005, MSRI Publication Series nr. 55, Cambridge University Press (2008), Available at: http://www.math.ucsb.edu/˜birnir/papers.
- B. Birnir, Uniqueness, an invariant measure and Kolmogorov’s scaling for the stochastic Navier-Stokes equation, Preprint (2007), Available at: http://www.math.ucsb.edu/˜birnir/papers.
- B. Birnir, J. Hernández, and T. R. Smith, The stochastic theory of fluvial landsurfaces, J. Nonlinear Science 17 (2007), 13–57, DOI:10.1007/s00332-005-0688-3. MR 2281138 (2007k:86002)
- B. Birnir, Keith Mertens, Vakhtang Putkaradze, and Peter Vorobieff, Meandering of fluid streams on acrylic surface driven by external noise, To appear in Journ. Fluid Mech. (2008).
- B. Birnir, T.R. Smith, and G. Merchant, The Scaling of Fluvial Landscapes, Computers and Geoscience 27 (2001), 1189–1216.
- R. F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Springer, New York, 1995. MR 1351248 (96i:93001)
- P. S. Dodds and D. Rothman, Geometry of river networks. I. Scaling, fluctuations and deviations, Phys. Rev. E 63 (2000), 016115.
- ---, Geometry of river networks. II. Distributions of component size and number, Phys. Rev. E 63 (2000), 016116.
- ---, Geometry of river networks. III. Characterization of component connectivity, Phys. Rev. E 63 (2000), 016117.
- ---, Scaling, universality and geomorphology, Annu. Rev. Earth Planet. Sci. 28 (2000), 571–610.
- ---, Unified view of scaling laws for river networks, Phys. Rev. E 59 (2000), no. 5, 4865.
- S. F. Edwards and D. R. Wilkinson, The surface statistics of granular aggregate, Proc. Roy. Soc. London. Series A (1982). MR 661715 (83m:82032)
- U. Frisch, Turbulence, Cambridge Univ. Press, Cambridge, 1995. MR 1428905 (98e:76002)
- D.M. Gray, Interrelationships of watershed characteristics, Journal of Geophysics Research 66 (1961), no. 4, 1215–1223.
- J. Hack, Studies of longitudinal stream profiles in Virginia and Maryland, U.S. Geological Survey Professional Paper 294-B (1957).
- R. E. Horton, Erosional development of streams and their drainage basins: A hydrophysical approach to quantitative morphology, Geol. Soc. Am. Bull. 56 (1945), 275–370.
- T. Kato, Perturbation theory for linear operators, Springer, New York, 1976. MR 0407617 (53:11389)
- S. L. Dingman, Fluvial hydrology, W. H. Freeman and Company, New York, 1984.
- J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 3, 193–248. MR 1555394
- E. Levi, The science of water, ASCE Press, New York, 1995.
- A. Maritan, A. Rinaldo, R. Rigon, A. Giacometti, and I. Rodriguez-Iturbe, Scaling laws for river networks, Phys. Rev. E 53 (1996), 1510.
- H. P. McKean, Turbulence without pressure: Existence of the invariant measure, Methods and Applications of Analysis 9 (2002), no. 3, 463–468. MR 2032342 (2005d:76023)
- A. S. Momin and A. M. Yaglom, Statistical fluid mechanics, vol. 1, MIT Press, Cambridge, MA, 1971.
- ---, Statistical fluid mechanics, vol. 2, MIT Press, Cambridge, MA, 1975.
- J. E. Mueller, Re-evaluation of the relationship of master streams and drainage basins: Reply, Geo. Soc. Amer. Bull. 84 (1973), 3127–3130.
- B. Oksendal, Stochastic differential equations, Springer, New York, 1998. MR 1619188 (99c:60119)
- L. Onsager, Statistical hydrodynamics, Nuovo Cimento. 6 (1945), no. 2, 279–287. MR 0036116 (12:60f)
- G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge UK, 1992. MR 1207136 (95g:60073)
- ---, Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge UK, 1996. MR 1417491 (97k:60165)
- I. Rodriguez-Iturbe and A. Rinaldo, Fractal river basins: Chance and self-organization, Cambridge University Press, Cambridge UK, 1997.
- T.R. Smith, G.E. Merchant, and B. Birnir, Transient attractors: Towards a theory of the graded stream for alluvial and bedrock channels, Computers and Geosciences 26 (2000), no. 5, 531–541.
- T.R. Smith, B. Birnir, and G.E. Merchant, Towards an elementary theory of drainage basin evolution: I. The theoretical basis, Computers and Geoscience 23 (1997), no. 8, 811–822.
- T.R. Smith, G.E. Merchant, and B. Birnir, Towards an elementary theory of drainage basin evolution: II. A computational evaluation, Computers and Geoscience 23 (1997), no. 8, 823–849.
- J.K. Weissel, L.F. Pratson, and A. Malinverno, The length-scaling of topography, Journal of Geophysical Research 99 (1994), 13997–14012.
- E. Welsh, B. Birnir, and A. Bertozzi, Shocks in the evolution of an eroding channel, AMRX Appl. Math. Research eXpress 2006 (2007), 1–27. MR 2278493 (2007h:86014)
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Additional Information
Björn Birnir
Affiliation:
Center for Complex and Nonlinear Science, and Department of Mathematics, University of California, Santa Barbara
Email:
birnir@math.ucsb.edu
Received by editor(s):
May 15, 2007
Published electronically:
June 5, 2008
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.
