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Title: 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 » 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.« less

Authors:
 [1];  [2];  [3]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Univ. of Northern Iowa, Cedar Falls, IA (United States)
  3. 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.
@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 = {2021},
month = {11}
}

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