Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-02-11T17:21:08.266Z Has data issue: false hasContentIssue false
  • English
  • Français

Longitudinal profile of channels cut by springs

Published online by Cambridge University Press:  13 December 2010

O. DEVAUCHELLE ,
A. P. PETROFF ,
A. E. LOBKOVSKY  and
D. H. ROTHMAN
O. DEVAUCHELLE*
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
A. P. PETROFF
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
A. E. LOBKOVSKY
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
D. H. ROTHMAN
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
*
†Email address for correspondence: devauche@mit.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose a simple theory for the longitudinal profile of channels incised by groundwater flow. The aquifer surrounding the stream is represented in two dimensions through Darcy's law and the Dupuit approximation. The model is based on the assumption that, everywhere in the stream, the shear stress exerted on the sediment by the flow is close to the minimal intensity required to displace a sand grain. Because of the coupling of the stream discharge with the water table elevation in the neighbourhood of the channel head, the stream elevation decreases as the distance from the stream's tip with an exponent of 2/3. Field measurements of steephead ravines in the Florida Panhandle conform well to this prediction.

JFM classification

Interfacial Flows (free surface): Fingering instability Geophysical and Geological Flows: River dynamics Geophysical and Geological Flows: Sediment transport
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 38 - 47
Copyright
Copyright © Cambridge University Press 2010

References

REFERENCES

Abrams, D. M., Lobkovsky, A. E., Petroff, A. P., Straub, K. M., McElroy, B., Mohrig, D. C., Kudrolli, A. & Rothman, D. H. 2009 Growth laws for channel networks incised by groundwater flow. Nature Geosci. 2, 193–196.CrossRefGoogle Scholar
Bear, J. 1988 Dynamics of Fluids in Porous Media. Dover.Google Scholar
Chanson, H. 2004 The Hydraulics of Open Channel Flow: An Introduction, 2nd edn. Elsevier Butterworth-Heinemann.Google Scholar
Churchill, R. V. & Brown, J. W. 1984 Complex Variables and Applications. McGraw-Hill.Google Scholar
Dade, W. B. 2000 Grain size, sediment transport and alluvial channel pattern. Geomorphology 35 (1–2), 119–126.CrossRefGoogle Scholar
Derrida, B. & Hakim, V. 1992 Needle models of Laplacian growth. Phys. Rev. A 45 (12), 8759–8765.Google Scholar
Devauchelle, O., Malverti, L., Lajeunesse, É., Lagrée, P. Y., Josserand, C. & Thu-Lam, K. D. N. 2009 Stability of bedforms in laminar flows with free surface: from bars to ripples. J. Fluid Mech. 642, 329–348.CrossRefGoogle Scholar
Dunne, T. 1980 Formation And Controls Of Channel Networks. Prog. Phys. Geogr. 4 (2), 211–239.CrossRefGoogle Scholar
Dupuit, J. 1863 Études théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables, 2nd edn. Dunod.Google Scholar
Fourrière, A. 2009 Morphodynamique des rivières: sélection de la largeur, rides et dunes. PhD thesis, Université Paris Diderot, Paris, France.Google Scholar
Fowler, A. C., Kopteva, N. & Oakley, C. 2007 The formation of river channels. SIAM J. Appl. Math. 67 (4), 1016–1040.CrossRefGoogle Scholar
Fox, G. A., Chu-Agor, M. L. M. & Wilson, G. V. 2007 Erosion of noncohesive sediment by ground water seepage: lysimeter experiments and stability modeling. Soil Sci. Soc. Am. J. 71 (6), 1822–1830.CrossRefGoogle Scholar
Gilbert, G. K. 1877 Report on the Geology of the Henry Mountains. Government Printing Office.CrossRefGoogle Scholar
Henderson, F. M. 1961 Stability of alluvial channels. J. Hydraul. Div. ASCE 87, 109–138.CrossRefGoogle Scholar
Higgins, C. G. 1982 Drainage systems developed by sapping on Earth and Mars. Geology 10 (3), 147–152.2.0.CO;2>CrossRefGoogle Scholar
Howard, A. D. 1988 Groundwater sapping experiments and modeling. In Sapping Features of the Colorado Plateau: A Comparative Planetary Geology Field Guide (ed. Howard, A. D., Kochel, R. C. & Holt, H. E.), pp. 71–83. NASA.Google Scholar
Howard, A. D. & McLane, C. F. III 1988 Erosion of cohesionless sediment by groundwater seepage. Water Resour. Res. 24 (10), 1659–1674.Google Scholar
Katul, G., Wiberg, P., Albertson, J. & Hornberger, G. 2002 A mixing layer theory for flow resistance in shallow streams. Water Resour. Res. 38 (11), 1250.CrossRefGoogle Scholar
Kochel, R. C. & Piper, J. F. 1986 Morphology of large valleys on Hawaii: evidence for groundwater sapping and comparisons with Martian valleys. J. Geophys. Res. 91 (B13), E175–E192.CrossRefGoogle Scholar
Laity, J. E. & Malin, M. C. 1985 Sapping processes and the development of theater-headed valley networks on the Colorado Plateau. Geol. Soc. Am. Bull. 96 (2), 203–217.Google Scholar
Lamb, M. P., Dietrich, W. E., Aciego, S. M., DePaolo, D. J. & Manga, M. 2008 Formation of Box Canyon, Idaho, by megaflood: implications for seepage erosion on Earth and Mars. Science 320 (5879), 1067–1070.CrossRefGoogle ScholarPubMed
Lamb, M. P., Howard, A. D., Johnson, J., Whipple, K. X., Dietrich, W. E. & Perron, J. T. 2006 Can springs cut canyons into rock? J. Geophys. Res 111, E07002.CrossRefGoogle Scholar
Lane, E. W. 1955 Design of stable channels. Trans. Am. Soc. Civil Eng. 120, 1234–1260.CrossRefGoogle Scholar
Leopold, L. B. & Wolman, M. G. 1957 River channel patterns: braided, meandering, and straight. Geol. Surv. Prof. Pap. 282 (B), 39–85.Google Scholar
Lobkovsky, A. E., Orpe, A. V., Molloy, R., Kudrolli, A. & Rothman, D. H. 2008 Erosion of a granular bed driven by laminar fluid flow. J. Fluid Mech. 605, 47–58.CrossRefGoogle Scholar
Paola, C., Heller, P. L. & Angevine, C. L. 1992 The large-scale dynamics of grain-size variation in alluvial basins. Part 1. Theory. Basin Res. 4, 73–90.CrossRefGoogle Scholar
Parker, G. 1978 Self-formed straight rivers with equilibrium banks and mobile bed. Part 2. The gravel river. J. Fluid Mech. 89 (1), 127–146.CrossRefGoogle Scholar
Perron, J. T., Kirchner, J. W. & Dietrich, W. E. 2009 Formation of evenly spaced ridges and valleys. Nature 460 (7254), 502–505.Google Scholar
Petroff, A. P., Devauchelle, O., Abrams, D., Lobkovsky, A., Kudrolli, A. & Rothman, D. H. 2010 Physical origin of amphitheater shaped valley heads. (preprint). arXiv:1011.2782v1.Google Scholar
Rice, S. P. & Church, M. 2001 Longitudinal profiles in simple alluvial systems. Water Resour. Res. 37 (2), 417–426.Google Scholar
Savenije, H. H. G. 2003 The width of a bankfull channel; Lacey's formula explained. J. Hydrol. 276 (1–4), 176–183.CrossRefGoogle Scholar
Schmidt, W. 1985 Alum Bluff, Liberty County, Florida. Open File Rep. 9. Florida Geological Survey.CrossRefGoogle Scholar
Schumm, S. A., Boyd, K. F., Wolff, C. G. & Spitz, W. J. 1995 A ground-water sapping landscape in the Florida Panhandle. Geomorphology 12 (4), 281–297.CrossRefGoogle Scholar
Sinha, S. K. & Parker, G. 1996 Causes of concavity in longitudinal profiles of rivers. Water Resour. Res. 32 (5), 1417–1428.CrossRefGoogle Scholar
Sklar, L. S. & Dietrich, W. E. 2008 Implications of the saltation-abrasion bedrock incision model for steady-state river longitudinal profile relief and concavity. Earth Surf. Process. Landf. 33 (7), 1129–1151.CrossRefGoogle Scholar
Snow, R. S. & Slingerland, R. L. 1987 Mathematical modeling of graded river profiles. J. Geol. 95 (1), 15–33.CrossRefGoogle Scholar
Whipple, K. X. 2001 Fluvial landscape response time: how plausible is steady-state denudation? Am. J. Sci. 301 (4–5), 313–325.CrossRefGoogle Scholar
Wolman, M. G. & Miller, J. P. 1960 Magnitude and frequency of forces in geomorphic processes. J. Geol. 68 (1), 54–74.CrossRefGoogle Scholar