Alluvial fan morphology: A self-similar free boundary problem description
Abstract
In this work, we examine approximate geometrically self-similar solutions to a parabolic free boundary value problem applied to alluvial fan surface morphology and growth. Alluvial fans are fan- or cone-shaped sedimentary deposits caused by the rapid deposition of sediment from a canyon discharging onto a flatter plain. Longitudinal, topographic profiles of fans can be readily described by a seemingly time independent dimensionless profile (DeChant et al., 1999). However, because an alluvial fan can be expected to grow over time, it is clear that this “steady” profile is certainly time dependent and can be described using a space-time self-similar solution. In an experimental and theory-based study, Guerit et al. (2014) developed a self-similar (or as they describe it a self-affine) linear solution based upon an approximate first order small parameter expansion solution for a 1-d homogeneous nonlinear diffusion equation. Direct substitution of this result into a linear diffusion equation suggests that this first order expression may not fully satisfy the associated governing equation. In contrast, we develop a more complete solution based upon a modeled approximation for the axi-symmetric formulation such that the associated temporal behavior is consistent with a 1/3 time power-law as described by Reitz and Jerolmack (2014). Themore »
- Authors:
-
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Univ. of Northern Iowa, Cedar Falls, IA (United States)
- Texas A & M Univ., College Station, TX (United States)
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1760430
- Alternate Identifier(s):
- OSTI ID: 1809361
- Report Number(s):
- SAND-2020-14108J
Journal ID: ISSN 0169-555X; 693039
- Grant/Contract Number:
- AC04-94AL85000; NA0003525; AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Geomorphology
- Additional Journal Information:
- Journal Volume: 375; Journal ID: ISSN 0169-555X
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 58 GEOSCIENCES; Alluvial fan; morphology; self-similar; free-boundary model
Citation Formats
DeChant, Lawrence, Pease, Patrick, and Tchakerian, Vatche P. Alluvial fan morphology: A self-similar free boundary problem description. United States: N. p., 2021.
Web. doi:10.1016/j.geomorph.2020.107532.
DeChant, Lawrence, Pease, Patrick, & Tchakerian, Vatche P. Alluvial fan morphology: A self-similar free boundary problem description. United States. https://doi.org/10.1016/j.geomorph.2020.107532
DeChant, Lawrence, Pease, Patrick, and Tchakerian, Vatche P. Fri .
"Alluvial fan morphology: A self-similar free boundary problem description". United States. https://doi.org/10.1016/j.geomorph.2020.107532. https://www.osti.gov/servlets/purl/1760430.
@article{osti_1760430,
title = {Alluvial fan morphology: A self-similar free boundary problem description},
author = {DeChant, Lawrence and Pease, Patrick and Tchakerian, Vatche P.},
abstractNote = {In this work, we examine approximate geometrically self-similar solutions to a parabolic free boundary value problem applied to alluvial fan surface morphology and growth. Alluvial fans are fan- or cone-shaped sedimentary deposits caused by the rapid deposition of sediment from a canyon discharging onto a flatter plain. Longitudinal, topographic profiles of fans can be readily described by a seemingly time independent dimensionless profile (DeChant et al., 1999). However, because an alluvial fan can be expected to grow over time, it is clear that this “steady” profile is certainly time dependent and can be described using a space-time self-similar solution. In an experimental and theory-based study, Guerit et al. (2014) developed a self-similar (or as they describe it a self-affine) linear solution based upon an approximate first order small parameter expansion solution for a 1-d homogeneous nonlinear diffusion equation. Direct substitution of this result into a linear diffusion equation suggests that this first order expression may not fully satisfy the associated governing equation. In contrast, we develop a more complete solution based upon a modeled approximation for the axi-symmetric formulation such that the associated temporal behavior is consistent with a 1/3 time power-law as described by Reitz and Jerolmack (2014). The resulting expression is an exact solution to a linear heat equation. Furthermore, we emphasize that a small parameter is not inherent to the resulting profile result and is not included in our model development. Though developed using rather different approaches, the formal solution developed here is in good agreement with the simple polynomial described by DeChant et al. (1999) suggesting that this self-similar solution is a suitable time dependent representation of alluvial fan longitudinal profile form and improves on earlier work.},
doi = {10.1016/j.geomorph.2020.107532},
journal = {Geomorphology},
number = ,
volume = 375,
place = {United States},
year = {Fri Nov 26 00:00:00 EST 2021},
month = {Fri Nov 26 00:00:00 EST 2021}
}
Works referenced in this record:
Application of a diffusion-erosion model to alluvial channels which degrade due to base-level lowering
journal, September 1988
- Begin, Ze'ev B.
- Earth Surface Processes and Landforms, Vol. 13, Issue 6
Analytical Theory of Erosion
journal, May 1960
- Culling, W. E. H.
- The Journal of Geology, Vol. 68, Issue 3
Braided-Stream Sediment Transport Rates from an Alluvial Fan Diffusivity Model
journal, May 2004
- Chant, L. J. D.
- Environmental and Engineering Geoscience, Vol. 10, Issue 2
Modelling alluvial fan morphology
journal, July 1999
- Chant, Lawrence J. De; Pease, Patrick P.; Tchakerian, Vatche P.
- Earth Surface Processes and Landforms, Vol. 24, Issue 7
Hillslope evolution by diffusive processes: The timescale for equilibrium adjustments
journal, June 1997
- Fernandes, Nelson F.; Dietrich, William E.
- Water Resources Research, Vol. 33, Issue 6
Laboratory alluvial fans in one dimension
journal, August 2014
- Guerit, L.; Métivier, F.; Devauchelle, O.
- Physical Review E, Vol. 90, Issue 2
Correlation and dating of Quaternary alluvial-fan surfaces using scarp diffusion
journal, June 2004
- Hsu, Leslie; Pelletier, Jon D.
- Geomorphology, Vol. 60, Issue 3-4
Erosion-rate determination from foreland basin geometry
journal, January 2007
- Pelletier, Jon D.
- Geology, Vol. 35, Issue 1
The influence of Holocene vegetation changes on topography and erosion rates: a case study at Walnut Gulch Experimental Watershed, Arizona
journal, January 2016
- Pelletier, Jon D.; Nichols, Mary H.; Nearing, Mark A.
- Earth Surface Dynamics, Vol. 4, Issue 2
Longitudinal profiles in simple alluvial systems
journal, February 2001
- Rice, Stephen P.; Church, Michael
- Water Resources Research, Vol. 37, Issue 2
Landform equations fitted to contour maps
journal, September 1965
- Troeh, F. R.
- American Journal of Science, Vol. 263, Issue 7