|
|
| Line 23: |
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| <li>To develop a sense of ``best practices'' in using landscape evolution models.</li> | | <li>To develop a sense of ``best practices'' in using landscape evolution models.</li> |
| </ul> | | </ul> |
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| <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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| <head><title>Landscape Evolution Modeling with CHILD</title>
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| <meta name="date" content="2014-08-28 15:01:00">
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| </head><body
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| >
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| <div class="maketitle">
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| <h2 class="titleHead">
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| LANDSCAPE EVOLUTION MODELING WITH CHILD
| |
| </h2><div class="authors"><span class="author" >
| |
| <span
| |
| class="cmr-10">GREGORY E.</span><span
| |
| class="cmr-10"> TUCKER</span><br
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| class="newline" /><span
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| class="cmti-10">UNIVERSITY OF COLORADO</span><br
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| class="newline" /><br
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| class="newline" /><span
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| class="cmr-10">STEPHEN T.</span><span
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| class="cmr-10"> LANCASTER</span><br
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| class="newline" /><span
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| class="cmti-10">OREGON</span>
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| <span
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| class="cmti-10">STATE UNIVERSITY</span>
| |
| </span></div>
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| <div class="submaketitle">
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| <div class="date" >
| |
| <!--l. 29--><p class="noindent" ><span
| |
| class="cmti-12">Date</span>: Short Course notes prepared for SIESD 2012: Future Earth: Interaction of Climate and
| |
| Earth-surface Processes, University of Minnesota, Minneapolis, Minnesota, USA, August
| |
| 2012.</div></div></div>
| |
| <h3 class="sectionHead"><a
| |
| id="x1-1000"></a></h3>
| |
| <div class="tableofcontents"><span class="sectionToc" ><a
| |
| href="#x1-1000" id="QQ2-1-1"></a></span><br /><span class="sectionToc" > 1.  <a
| |
| href="#x1-20001" id="QQ2-1-2">Overview</a></span><br /><span class="sectionToc" > 2.  <a
| |
| href="#x1-30002" id="QQ2-1-4">Introduction to
| |
| LEMs</a></span><br /><span class="subsectionToc" >   2.1.  <a
| |
| href="#x1-40002.1" id="QQ2-1-5">Brief History</a></span><br /><span class="subsectionToc" >   2.2.  <a
| |
| href="#x1-50002.2" id="QQ2-1-6">Brief Overview of Models and
| |
| their Uses</a></span><br /><span class="sectionToc" > 3.  <a
| |
| href="#x1-60003" id="QQ2-1-8">Continuity of Mass and Discretization</a></span><br /><span class="sectionToc" > 4.  <a
| |
| href="#x1-70004" id="QQ2-1-10">Gravitational
| |
| Hillslope Transport</a></span><br /><span class="subsectionToc" >   4.1.  <a
| |
| href="#x1-80004.1" id="QQ2-1-11">Linear Diffusion</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-90004.1" id="QQ2-1-12"><span
| |
| class="cmbxti-10x-x-120">Exercise 1: Getting</span>
| |
| <span
| |
| class="cmbxti-10x-x-120">Set Up with CHILD</span></a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-100004.1" id="QQ2-1-13"><span
| |
| class="cmti-12">Exercise 2: Hillslope Diffusion and</span>
| |
| <span
| |
| class="cmti-12">Parabolic Slopes with CHILD</span></a></span><br /><span class="subsectionToc" >   4.2.  <a
| |
| href="#x1-110004.2" id="QQ2-1-14">Nonlinear Diffusion</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-120004.2" id="QQ2-1-15"><span
| |
| class="cmti-12">Exercise 3:</span>
| |
| <span
| |
| class="cmti-12">Nonlinear Diffusion and Planar Slopes</span></a></span><br /><span class="subsectionToc" >   4.3.  <a
| |
| href="#x1-130004.3" id="QQ2-1-16">Remarks</a></span><br /><span class="sectionToc" > 5.  <a
| |
| href="#x1-140005" id="QQ2-1-17">Rainfall,
| |
| Runoff, and Drainage Networks</a></span><br /><span class="subsectionToc" >   5.1.  <a
| |
| href="#x1-150005.1" id="QQ2-1-18">Methods Based on
| |
| Drainage Area</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-160005.1" id="QQ2-1-20"><span
| |
| class="cmti-12">Exercise 4: Flow Over Noisy, Inclined</span>
| |
| <span
| |
| class="cmti-12">Topography</span></a></span><br /><span class="subsectionToc" >   5.2.  <a
| |
| href="#x1-170005.2" id="QQ2-1-21">Shallow-Water Equations</a></span><br /><span class="subsectionToc" >   5.3.  <a
| |
| href="#x1-180005.3" id="QQ2-1-23">Cellular
| |
| Automata</a></span><br /><span class="subsectionToc" >   5.4.  <a
| |
| href="#x1-190005.4" id="QQ2-1-24">Depressions in the Terrain</a></span><br /><span class="subsectionToc" >   5.5.  <a
| |
| href="#x1-200005.5" id="QQ2-1-25">Precipitation
| |
| and Discharge</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-210005.5" id="QQ2-1-26"><span
| |
| class="cmti-12">Exercise 5: Visualizing a Poisson Storm</span>
| |
| <span
| |
| class="cmti-12">Sequence</span></a></span><br /><span class="subsectionToc" >   5.6.  <a
| |
| href="#x1-220005.6" id="QQ2-1-27">Remarks</a></span><br /><span class="sectionToc" > 6.  <a
| |
| href="#x1-230006" id="QQ2-1-28">Hydraulic Geometry</a></span><br /><span class="sectionToc" > 7.  <a
| |
| href="#x1-240007" id="QQ2-1-29">Erosion
| |
| and Transport by Running Water</a></span><br /><span class="subsectionToc" >   7.1.  <a
| |
| href="#x1-250007.1" id="QQ2-1-30">Detachment-Limited
| |
| Models</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-260007.1" id="QQ2-1-31"><span
| |
| class="cmti-12">Exercise 6: Detachment-Limited Hills and Mountains</span></a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-270007.1" id="QQ2-1-32"><span
| |
| class="cmti-12">Exercise</span>
| |
| <span
| |
| class="cmti-12">7: Zooming in to the Hillslopes</span></a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-280007.1" id="QQ2-1-33"><span
| |
| class="cmti-12">Exercise 8: Knickzones and Transient</span>
| |
| <span
| |
| class="cmti-12">Response</span></a></span><br /><span class="subsectionToc" >   7.2.  <a
| |
| href="#x1-290007.2" id="QQ2-1-34">Transport-Limited Models</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-300007.2" id="QQ2-1-35"><span
| |
| class="cmti-12">Exercise 9: A Pile of Fine</span>
| |
| <span
| |
| class="cmti-12">Sand</span></a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-310007.2" id="QQ2-1-36"><span
| |
| class="cmti-12">Exercise 10: A Pile of Cobbles</span></a></span><br /><span class="subsectionToc" >   7.3.  <a
| |
| href="#x1-320007.3" id="QQ2-1-37">Hybrid Model: Combining
| |
| Detachment and Transport</a></span><br /><span class="subsectionToc" >  <a
| |
| href="#x1-330007.3" id="QQ2-1-38"><span
| |
| class="cmti-12">Exercise 11: Erosion and Deposition, Together at</span>
| |
| <span
| |
| class="cmti-12">Last</span></a></span><br /><span class="subsectionToc" >   7.4.  <a
| |
| href="#x1-340007.4" id="QQ2-1-39">Other Sediment-Flux-Dependent Fluvial Models</a></span><br /><span class="sectionToc" > 8.  <a
| |
| href="#x1-350008" id="QQ2-1-40">Multiple
| |
| Grain Sizes</a></span><br /><span class="sectionToc" > 9.  <a
| |
| href="#x1-360009" id="QQ2-1-41">Exotica</a></span><br /><span class="sectionToc" > 10.  <a
| |
| href="#x1-3700010" id="QQ2-1-42">Forecasting or Speculation?</a></span><br /><span class="sectionToc" > 11.  <a
| |
| href="#x1-3800011" id="QQ2-1-43">Ten
| |
|
| |
| Commandments of Landscape Evolution Modeling</a></span><br /><span class="sectionToc" ><a
| |
| href="#x1-3900011" id="QQ2-1-44">References</a></span><br />
| |
| </div>
| |
| <h3 class="sectionHead"><span class="titlemark">1. </span> <a
| |
| id="x1-20001"></a>Overview</h3>
| |
| <!--l. 38--><p class="noindent" ><span class="floatingfigure-r" style="width:234.8775pt"><img
| |
| src="child_exercises_nced_aug20120x.png" alt="PIC" class="graphics"><!--tex4ht:graphics
| |
| name="child_exercises_nced_aug20120x.png" src="mesh_schematic.pdf"
| |
| -->
| |
| <br /><span class="caption"><span class="id">Figure 1</span>: Schematic
| |
| diagram of CHILD model’s representation
| |
| of the landscape: hexagonal Voronoi cells,
| |
| nodes (at centers of cells) connected by the
| |
| edges of the Delaunay triangulation, vegetated
| |
| cell surfaces, channelized cells, and soil and
| |
| sediment layers above bedrock.</span><br /> </span>
| |
| <!--l. 46--><p class="noindent" >The learning goals of this exercise are:
| |
| <ul class="itemize1">
| |
| <li class="itemize">To gain a clearer understanding of
| |
| how a typical landscape evolution
| |
| model (LEM) solves the governing
| |
| equations that represent geomorphic
| |
| processes.
| |
| </li>
| |
|
| |
| <li class="itemize">To gain hands-on experience actually
| |
| using a LEM.
| |
| </li>
| |
| <li class="itemize">To understand how continuity of mass
| |
| is maintained by a typical LEM, and
| |
| some of the limitations that arise.
| |
| </li>
| |
| <li class="itemize">To appreciate some of the ways in
| |
| which climate and hydrology can be
| |
| represented in a LEM, and some of
| |
| the simplifications involved.
| |
| </li>
| |
| <li class="itemize">To appreciate that working with LEMs involves choosing a level of simplification in
| |
| the governing physics that is appropriate to the problem at hand.
| |
| </li>
| |
| <li class="itemize">To get a sense for how and why soil creep produces convex hillslopes.
| |
| </li>
| |
| <li class="itemize">To appreciate the concepts of transient versus steady topography.
| |
| </li>
| |
| <li class="itemize">To acquire a feel for the similarity and difference between detachment-limited and
| |
| transport-limited modes of fluvial erosion.
| |
| </li>
| |
| <li class="itemize">To understand the connection between fluvial physics and slope-area plots.
| |
| </li>
| |
| <li class="itemize">To appreciate that LEMs (1) are able to reproduce (and therefore, at least potentially,
| |
| explain) common forms in fluvially carved landscapes, (2) can enhance our insight into
| |
| dynamics via visualization and experimentation, but (3) leave open many important
| |
| questions regarding long-term process physics.
| |
| </li>
| |
| <li class="itemize">To develop a sense “best practice” in using landscape evolution models.</li></ul>
| |
| <h3 class="sectionHead"><span class="titlemark">2. </span> <a
| |
| id="x1-30002"></a>Introduction to LEMs</h3>
| |
| <!--l. 63--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">2.1. </span> <a
| |
| id="x1-40002.1"></a><span
| |
| class="cmbx-12">Brief History.</span></span>
| |
|
| |
| <!--l. 65--><p class="noindent" >G.K. Gilbert, a member of the Powell Expedition, produced “word pictures” of landscape
| |
| evolution that still provide insight (<a
| |
| href="#Xgilbert1877report"><span
| |
| class="cmti-12">Gilbert</span></a>, <a
| |
| href="#Xgilbert1877report">1877</a>). For example, consider his “Law of Divides”
| |
| (<a
| |
| href="#Xgilbert1877report"><span
| |
| class="cmti-12">Gilbert</span></a>, <a
| |
| href="#Xgilbert1877report">1877</a>):
| |
| <!--l. 69--><p class="noindent" >
| |
| <!--l. 71--><p class="noindent" ><span
| |
| class="cmr-10x-x-109">We have seen that the declivity over which water flows bears an inverse relation</span>
| |
| <span
| |
| class="cmr-10x-x-109">to the quantity of water. If we follow a stream from its mouth upward and pass</span>
| |
| <span
| |
| class="cmr-10x-x-109">successively the mouths of its tributaries, we find its volume gradually less and less</span>
| |
| <span
| |
| class="cmr-10x-x-109">and its grade steeper and steeper, until finally at its head we reach the steepest</span>
| |
| <span
| |
| class="cmr-10x-x-109">grade of all. If we draw the profile of the river on paper, we produce a curve concave</span>
| |
| <span
| |
| class="cmr-10x-x-109">upward and with the greatest curvature at the upper end. The same law applies</span>
| |
| <span
| |
| class="cmr-10x-x-109">to every tributary and even to the slopes over which the freshly fallen rain flows</span>
| |
| <span
| |
| class="cmr-10x-x-109">in a sheet before it is gathered into rills. The nearer the water-shed or divide the</span>
| |
| <span
| |
| class="cmr-10x-x-109">steeper the slope; the farther away the less the slope.</span>
| |
| <!--l. 83--><p class="noindent" ><span
| |
| class="cmr-10x-x-109">It is in accordance with this law that mountains are steepest at their crests. The</span>
| |
| <span
| |
| class="cmr-10x-x-109">profile of a mountain if taken along drainage lines is concave outward...; and this is</span>
| |
| <span
| |
| class="cmr-10x-x-109">purely a matter of sculpture, the uplifts from which mountains are carved rarely if</span>
| |
| <span
| |
| class="cmr-10x-x-109">ever assuming this form.</span>
| |
| <!--l. 90--><p class="noindent" >Flash forward to the 1960’s, and we find the emergence of the first one-dimensional profile models.
| |
| <a
| |
| href="#Xculling1963soil"><span
| |
| class="cmti-12">Culling</span></a> (<a
| |
| href="#Xculling1963soil">1963</a>), for example, used the diffusion equation to describe the relaxation of escarpments
| |
| over time.
| |
| <!--l. 94--><p class="noindent" >Models became more sophisticated in the early 1970’s. Frank Ahnert and Mike Kirkby, among others,
| |
| began to develop computer models of slope profile development and included not only diffusive soil
| |
| creep but also fluvial downcutting as well as weathering (<a
| |
| href="#Xahnert1971general"><span
| |
| class="cmti-12">Ahnert</span></a>, <a
| |
| href="#Xahnert1971general">1971</a>; <a
| |
| href="#Xkirkby1971hillslope"><span
| |
| class="cmti-12">Kirkby</span></a>, <a
| |
| href="#Xkirkby1971hillslope">1971</a>).
| |
| Meanwhile, Alan Howard developed a simulation model of channel network evolution
| |
| (<a
| |
| href="#Xhoward1971simulation"><span
| |
| class="cmti-12">Howard</span></a>, <a
| |
| href="#Xhoward1971simulation">1971</a>).
| |
| <!--l. 102--><p class="noindent" >The mid-1970’s saw the first emergence of fully two-dimensional (and even quasi-three-dimensional)
| |
| landscape evolution models, perhaps most noteworthy that of <a
| |
| href="#Xahnert1976"><span
| |
| class="cmti-12">Ahnert</span></a> (<a
| |
| href="#Xahnert1976">1976</a>). Geomorphologists
| |
| would have to wait nearly 15 years for models to surpass the level of sophistication found in this
| |
| early model.
| |
| <!--l. 108--><p class="noindent" >During that time, computers would become much more powerful and able to model full
| |
| landscapes. The late 1980’s through the mid-1990’s saw the beginning of the “modern era”
| |
| of landscape evolution models, and today there are many model codes with as many
| |
| applications, scales, and objectives, ranging from soil erosion to continental collision (Table
| |
| 1).
| |
| <!--l. 115--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">2.2. </span> <a
| |
| id="x1-50002.2"></a><span
| |
| class="cmbx-12">Brief Overview of Models and their Uses.</span></span>
| |
| <!--l. 117--><p class="noindent" >Some examples of landscape evolution models (LEMs) are shown in Table 1. LEMs have been
| |
|
| |
| developed to represent, for example, coupled erosion-deposition systems, meandering, Mars
| |
| cratering, forecasting of mine-spoil degradation, and estimation of erosion risk to buried
| |
| hazardous waste. These models provide powerful tools, but their process ingredients are
| |
| generally provisional and subject to testing. For this reason, it is important to have
| |
| continuing cross-talk between modeling and observations—after all, that’s how science
| |
| works.
| |
| <!--l. 126--><p class="noindent" >In this exercise, we provide an overview of how a LEM works, including how terrain and water flow
| |
| are represented numerically, and how various processes are computed.
| |
| <div class="center"
| |
| >
| |
| <!--l. 130--><p class="noindent" >
| |
| <!--l. 132--><p class="noindent" ><a
| |
| id="x1-50011"></a><a
| |
| id="x1-60032"></a><a
| |
| id="x1-150013"></a><hr class="float"><div class="float"
| |
| >
| |
| <br /> <div class="caption"
| |
| ><span class="id">Table 1: </span><span
| |
| class="content">Partial list of numerical landscape models published between 1991 and 2005.</span></div><!--tex4ht:label?: x1-50011 -->
| |
| <div class="tabular"> <table id="TBL-1" class="tabular"
| |
| cellspacing="0" cellpadding="0"
| |
| ><colgroup id="TBL-1-1g"><col
| |
| id="TBL-1-1"><col
| |
| id="TBL-1-2"><col
| |
| id="TBL-1-3"></colgroup><tr
| |
| class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-1-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-1-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">Model </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-2"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Example reference </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Notes </span></td>
| |
| </tr><tr
| |
| class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-2-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-2-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">SIBERIA </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-2"
| |
| class="td11"> <a
| |
| href="#Xwillgoose1991coupled"><span
| |
| class="cmti-10x-x-109">Willgoose et</span><span
| |
| class="cmti-10x-x-109"> al.</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xwillgoose1991coupled"><span
| |
| class="cmr-10x-x-109">1991</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Transport-limited; </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-3-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-3-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-3-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-3-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Channel activator function </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-4-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-4-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">DRAINAL </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-4-2"
| |
| class="td11"> <a
| |
| href="#Xbeaumont1992erosional"><span
| |
| class="cmti-10x-x-109">Beaumont et</span><span
| |
| class="cmti-10x-x-109"> al.</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xbeaumont1992erosional"><span
| |
| class="cmr-10x-x-109">1992</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-4-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">“Undercapacity” concept </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-5-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-5-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">GILBERT </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-5-2"
| |
| class="td11"> <a
| |
| href="#Xchase1992fluvial"><span
| |
| class="cmti-10x-x-109">Chase</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xchase1992fluvial"><span
| |
| class="cmr-10x-x-109">1992</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-5-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Precipiton </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-6-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-6-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">DELIM/MARSSIM</span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-6-2"
| |
| class="td11"> <a
| |
| href="#Xhoward1994detachment"><span
| |
| class="cmti-10x-x-109">Howard</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xhoward1994detachment"><span
| |
| class="cmr-10x-x-109">1994</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-6-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Detachment-limited; </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-7-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-7-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-7-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-7-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Nonlinear diffusion </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-8-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-8-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">GOLEM </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-8-2"
| |
| class="td11"><a
| |
| href="#Xtucker1994erosional"><span
| |
| class="cmti-10x-x-109">Tucker and Slingerland</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xtucker1994erosional"><span
| |
| class="cmr-10x-x-109">1994</span></a><span
| |
| class="cmr-10x-x-109">)</span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-8-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Regolith generation; </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-9-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-9-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-9-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-9-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Threshold landsliding </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-10-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-10-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">CASCADE </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-10-2"
| |
| class="td11"> <a
| |
| href="#Xbraun1997modelling"><span
| |
| class="cmti-10x-x-109">Braun and Sambridge</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xbraun1997modelling"><span
| |
| class="cmr-10x-x-109">1997</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-10-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Irregular discretization </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-11-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-11-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">CAESAR </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-11-2"
| |
| class="td11"> <a
| |
| href="#Xcoulthard1996cellular"><span
| |
| class="cmti-10x-x-109">Coulthard et</span><span
| |
| class="cmti-10x-x-109"> al.</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xcoulthard1996cellular"><span
| |
| class="cmr-10x-x-109">1996</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-11-3"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">Cellular automaton algorithm</span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-12-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-12-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-12-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-12-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">for 2D flow field </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-13-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-13-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">ZSCAPE </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-13-2"
| |
| class="td11"> <a
| |
| href="#Xdensmore1998landsliding"><span
| |
| class="cmti-10x-x-109">Densmore et</span><span
| |
| class="cmti-10x-x-109"> al.</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xdensmore1998landsliding"><span
| |
| class="cmr-10x-x-109">1998</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-13-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Stochastic bedrock </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-14-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-14-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-14-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-14-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">landsliding algorithm </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-15-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-15-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">CHILD </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-15-2"
| |
| class="td11"> <a
| |
| href="#Xtucker2000stochastic"><span
| |
| class="cmti-10x-x-109">Tucker and Bras</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xtucker2000stochastic"><span
| |
| class="cmr-10x-x-109">2000</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-15-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Stochastic rainfall </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-16-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-16-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">EROS </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-16-2"
| |
| class="td11"> <a
| |
| href="#Xcrave2001stochastic"><span
| |
| class="cmti-10x-x-109">Crave and Davy</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xcrave2001stochastic"><span
| |
| class="cmr-10x-x-109">2001</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-16-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Modified precipiton </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-17-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-17-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">TISC </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-17-2"
| |
| class="td11"> <a
| |
| href="#Xgarcia2002interplay"><span
| |
| class="cmti-10x-x-109">Garcia-Castellanos</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xgarcia2002interplay"><span
| |
| class="cmr-10x-x-109">2002</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-17-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Thrust stacking </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-18-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-18-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">LAPSUS </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-18-2"
| |
| class="td11"> <a
| |
| href="#Xschoorl2002modeling"><span
| |
| class="cmti-10x-x-109">Schoorl et</span><span
| |
| class="cmti-10x-x-109"> al.</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xschoorl2002modeling"><span
| |
| class="cmr-10x-x-109">2002</span></a><span
| |
| class="cmr-10x-x-109">) </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-18-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Multiple flow directions </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-19-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-19-1"
| |
| class="td11"><span
| |
| class="cmr-10x-x-109">APERO/CIDRE </span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-19-2"
| |
| class="td11"><a
| |
| href="#Xcarretier2005does"><span
| |
| class="cmti-10x-x-109">Carretier and Lucazeau</span></a><span
| |
| class="cmr-10x-x-109"> (</span><a
| |
| href="#Xcarretier2005does"><span
| |
| class="cmr-10x-x-109">2005</span></a><span
| |
| class="cmr-10x-x-109">)</span></td><td style="white-space:nowrap; text-align:center;" id="TBL-1-19-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">Single or multiple </span></td>
| |
| </tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-20-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-20-1"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-20-2"
| |
| class="td11"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-20-3"
| |
| class="td11"> <span
| |
| class="cmr-10x-x-109">flow directions </span></td>
| |
| </tr><tr
| |
| class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
| |
| style="vertical-align:baseline;" id="TBL-1-21-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-21-1"
| |
| class="td11"> </td></tr></table></div></div><hr class="endfloat" />
| |
| </div>
| |
| <h3 class="sectionHead"><span class="titlemark">3. </span> <a
| |
| id="x1-60003"></a>Continuity of Mass and Discretization</h3>
| |
| <!--l. 167--><p class="noindent" >A typical mass continuity equation for a column of soil or rock is:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-6001r1"></a>
| |
|
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20121x.png" alt="∂η
| |
| ∂t-= B - ∇ ⃗qs
| |
| " class="math-display" ></center></td><td class="equation-label">(1)</td></tr></table>
| |
| <!--l. 170--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">η </span>is the elevation of the land surface
| |
| [L]<span class="footnote-mark"><a
| |
| href="child_exercises_nced_aug20122.html#fn1x0">1</a></span><a
| |
| id="x1-6002f1"></a>; <span
| |
| class="cmmi-12">t</span>
| |
| is time; <span
| |
| class="cmmi-12">B </span>[L/T] represents the vertical motion of the rocks and soil relative to baselevel (due, for
| |
| example, to tectonic uplift or subsidence, sea-level change, or erosion along the boundary
| |
| of the system); and <img
| |
| src="child_exercises_nced_aug20122x.png" alt="⃗q" class="vec" > <sub><span
| |
| class="cmmi-8">s</span></sub> is sediment flux per unit width [L<sup><span
| |
| class="cmr-8">2</span></sup>/T]. This is one of several
| |
| variations; for discussion of others, see <a
| |
| href="#Xtucker2010modelling"><span
| |
| class="cmti-12">Tucker and Hancock</span></a> (<a
| |
| href="#Xtucker2010modelling">2010</a>). Some models, for
| |
| example, distinguish between a regolith layer and the bedrock underneath (Fig. <a
| |
| href="#x1-20011">1<!--tex4ht:ref: fig:schem --></a>). Note
| |
| that this type of mass continuity equation applies only to terrain that has one and
| |
| only one surface point for each coordinate; it would not apply to a vertical cliff or an
| |
| overhang.
| |
| <!--l. 173--><p class="noindent" >A LEM computes <span
| |
| class="cmmi-12">η</span>(<span
| |
| class="cmmi-12">x,y,t</span>) given (1) a set of process rules, (2) initial conditions, and
| |
| (3) boundary conditions. One thing all LEMs have in common is that they divide the
| |
| terrain into discrete elements. Often these are square elements, but sometimes they are
| |
| irregular polygons (as in the case of CASCADE and CHILD; Fig. <a
| |
| href="#x1-20011">1<!--tex4ht:ref: fig:schem --></a>). For a discrete
| |
| parcel (or “cell”) of land, continuity of mass enforced by the following equation (in
| |
| words):
| |
| <!--l. 177--><p class="noindent" ><span
| |
| class="cmti-12">Time rate of change of mass in element = mass rate in at boundaries - mass rate out at boundaries</span>
| |
| <span
| |
| class="cmti-12">+ inputs or outputs from above or below (tectonics, dust deposition, etc.)</span>
| |
|
| |
| <!--l. 179--><p class="noindent" ><span class="floatingfigure-r" style="width:171.0pt"><img
| |
| src="child_exercises_nced_aug20123x.png" alt="PIC" class="graphics"><!--tex4ht:graphics
| |
| name="child_exercises_nced_aug20123x.png" src="child_mesh_schem.pdf"
| |
| -->
| |
| <br /><span class="caption"><span class="id">Figure 2</span>: Schematic diagram of
| |
| CHILD mesh with illustration of
| |
| calculation of volumetric
| |
| fluxes between cells. Dashed lines
| |
| indicate cells and their faces, solid
| |
| circles are nodes, and solid lines
| |
| show the edges between nodes.</span><br /> </span>
| |
| <!--l. 187--><p class="noindent" >This statement can be expressed mathematically, for cell <span
| |
| class="cmmi-12">i</span>, as follows:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-6004r2"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20124x.png" alt="dη 1 ∑N
| |
| --i= B + --- qsjλj
| |
| dt Λi j=1
| |
| " class="math-display" ></center></td><td class="equation-label">(2)</td></tr></table>
| |
| <!--l. 191--><p class="nopar" >
| |
| where Λ<sub><span
| |
| class="cmmi-8">i</span></sub> is the horizontal surface area of cell <span
| |
| class="cmmi-12">i</span>; <span
| |
| class="cmmi-12">N </span>is the number of faces surrounding cell <span
| |
| class="cmmi-12">i</span>; <span
| |
| class="cmmi-12">q</span><sub><span
| |
| class="cmmi-8">sj</span></sub> is
| |
| the unit flux across face <span
| |
| class="cmmi-12">j</span>; and <span
| |
| class="cmmi-12">λ</span><sub><span
| |
| class="cmmi-8">j</span></sub> is the length of face <span
| |
| class="cmmi-12">j </span>(Fig. <a
| |
| href="#x1-60032">2<!--tex4ht:ref: fig:cmesh --></a>). (Note that, for the sake of
| |
| simplicity, we are using volume rather than mass flux; this is ok as long as the mass density of the
| |
| material is unchanging). Equation (<a
| |
| href="#x1-6004r2">2<!--tex4ht:ref: eq:finvol --></a>) expresses what is known as a <span
| |
| class="cmti-12">finite-volume </span>method because
| |
| it is based on computing fluxes in and out along the boundaries of a finite volume of
| |
| space.
| |
| <!--l. 194--><p class="noindent" ><span
| |
| class="cmti-12">Some terminology: a </span><span
| |
| class="cmbx-12">cell </span><span
| |
| class="cmti-12">is a patch of ground with boundaries called </span><span
| |
| class="cmbx-12">faces</span><span
| |
| class="cmti-12">. A </span><span
| |
| class="cmbx-12">node </span><span
| |
| class="cmti-12">is the point</span>
| |
| <span
| |
| class="cmti-12">inside a cell at which we track elevation (and other properties). On a raster grid, each cell is square</span>
| |
|
| |
| <span
| |
| class="cmti-12">and each node lies at the center of a cell. On the irregular mesh used by CASCADE and CHILD,</span>
| |
| <span
| |
| class="cmti-12">the </span><span
| |
| class="cmbx-12">cell </span><span
| |
| class="cmti-12">is the area of land that is closer to that particular node than to any other node in the</span>
| |
| <span
| |
| class="cmti-12">mesh. (It is a mathematical entity known as a </span><span
| |
| class="cmbx-12">Voronoi cell </span><span
| |
| class="cmti-12">or </span><span
| |
| class="cmbx-12">Thiessen polygon</span><span
| |
| class="cmti-12">; for more, see</span>
| |
| <a
| |
| href="#Xbraun1997modelling"><span
| |
| class="cmti-12">Braun and Sambridge</span></a><span
| |
| class="cmti-12"> (</span><a
| |
| href="#Xbraun1997modelling"><span
| |
| class="cmti-12">1997</span></a><span
| |
| class="cmti-12">), </span><a
| |
| href="#Xtucker2001object"><span
| |
| class="cmti-12">Tucker et</span><span
| |
| class="cmti-12"> al.</span></a><span
| |
| class="cmti-12"> (</span><a
| |
| href="#Xtucker2001object"><span
| |
| class="cmti-12">2001a</span></a><span
| |
| class="cmti-12">).)</span>
| |
| <!--l. 196--><p class="noindent" >Equation <a
| |
| href="#x1-6004r2">2<!--tex4ht:ref: eq:finvol --></a> gives us the time derivatives for the elevation of every node on the grid. How do we
| |
| solve for the new elevations at time <span
| |
| class="cmmi-12">t</span>? There are many ways to do this, including matrix-based
| |
| implicit solvers (see for example <a
| |
| href="#Xfagherazzi2002implicit"><span
| |
| class="cmti-12">Fagherazzi et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xfagherazzi2002implicit">2002</a>); <a
| |
| href="#Xperron2011numerical"><span
| |
| class="cmti-12">Perron</span></a> (<a
| |
| href="#Xperron2011numerical">2011</a>)). We won’t get into the
| |
| details of numerical solutions (at least not yet), but for now note that the simplest solution is the
| |
| forward-difference approximation: <div class="eqnarray">
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20125x.png" alt=" dη η (t + Δt ) - η(t)
| |
| --i ≈ -i------------i-- (3)
| |
| dt Δt
| |
| 1 ∑N
| |
| ηi(t + Δt ) = ηi(t) + U Δt + Δt--- qsjλj (4)
| |
| Λi j=1
| |
| " class="math-display" ></center>
| |
| </div>The main disadvantage of this approach is that very small time steps are typically needed in
| |
| order to ensure numerical stability. (CHILD uses a variant of this that seeks the largest
| |
| possible stable value of Δ<span
| |
| class="cmmi-12">t </span>at each iteration). A good discussion of numerical stability,
| |
| accuracy, and alternative methods for diffusion-like problems can be found in <a
| |
| href="#Xpress2007numerical"><span
| |
| class="cmti-12">Press</span>
| |
| <span
| |
| class="cmti-12">et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xpress2007numerical">2007</a>).
| |
| <h3 class="sectionHead"><span class="titlemark">4. </span> <a
| |
| id="x1-70004"></a>Gravitational Hillslope Transport</h3>
| |
| <!--l. 207--><p class="noindent" >Geomorphologists often distinguish between hillslope and channel processes. It’s a useful
| |
| distinction, although one has to bear in mind that the transition is not always abrupt,
| |
| and even where it is abrupt, it is commonly either discontinuous or highly dynamic or
| |
| both.
| |
| <!--l. 209--><p class="noindent" >Alternatively, one can also distinguish between processes that are driven nearly exclusively by
| |
| gravitational processes, and those that involve a fluid phase (normally water or ice). This
| |
| distinction too has a gray zone: landslides are gravitational phenomena but often triggered by fluid
| |
| pore pressure, while debris flows are surges of mixed fluid and solid. Nonetheless, we
| |
| will start with a consideration of one form of gravitational transport on hillslopes: soil
| |
| creep.
| |
|
| |
| <!--l. 211--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">4.1. </span> <a
| |
| id="x1-80004.1"></a><span
| |
| class="cmbx-12">Linear Diffusion.</span></span>
| |
| <!--l. 213--><p class="noindent" >For relatively gentle, soil-mantled slopes, there is reasonably strong support for a transport law of
| |
| the form:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-8001r5"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20126x.png" alt="⃗qs = - D ∇ η
| |
| " class="math-display" ></center></td><td class="equation-label">(5)</td></tr></table>
| |
| <!--l. 216--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">D </span>is a transport coefficient with dimensions of L<sup><span
| |
| class="cmr-8">2</span></sup>T<sup><span
| |
| class="cmsy-8">-</span><span
| |
| class="cmr-8">1</span></sup>. Using the finite-volume method
| |
| outlined in equation <a
| |
| href="#x1-6004r2">2<!--tex4ht:ref: eq:finvol --></a>, we want to calculate <img
| |
| src="child_exercises_nced_aug20127x.png" alt="⃗qs" class="vec" > at each of the cell faces. Suppose node <span
| |
| class="cmmi-12">i </span>and node <span
| |
| class="cmmi-12">k</span>
| |
| are neighboring nodes that share a common face (we’ll call this face <span
| |
| class="cmmi-12">j</span>). We approximate the
| |
| gradient between nodes <span
| |
| class="cmmi-12">i </span>and <span
| |
| class="cmmi-12">k </span>as:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-8002r6"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20128x.png" alt=" ηk - ηi
| |
| Sik = -------
| |
| Lik
| |
| " class="math-display" ></center></td><td class="equation-label">(6)</td></tr></table>
| |
| <!--l. 220--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">L</span><sub><span
| |
| class="cmmi-8">ik</span></sub> is the distance between nodes. On a raster grid, <span
| |
| class="cmmi-12">L</span><sub><span
| |
| class="cmmi-8">ik</span></sub> = Δ<span
| |
| class="cmmi-12">x </span>is simply the grid spacing.
| |
| The sediment flux per unit width is then
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-8003r7"></a>
| |
|
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug20129x.png" alt=" ηk---ηi
| |
| qsik ≃ D Lik
| |
| " class="math-display" ></center></td><td class="equation-label">(7)</td></tr></table>
| |
| <!--l. 224--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">q</span><sub><span
| |
| class="cmmi-8">sik</span></sub> is the volume flux per unit width from node <span
| |
| class="cmmi-12">k </span>to node <span
| |
| class="cmmi-12">i </span>(if negative, sediment flows from
| |
| <span
| |
| class="cmmi-12">i </span>to <span
| |
| class="cmmi-12">k</span>), and <span
| |
| class="cmmi-12">L</span><sub><span
| |
| class="cmmi-8">ik</span></sub> is the distance between nodes. On a raster grid, <span
| |
| class="cmmi-12">L</span><sub><span
| |
| class="cmmi-8">ik</span></sub> = Δ<span
| |
| class="cmmi-12">x </span>is simply the grid
| |
| spacing. To compute the total sediment flux through face <span
| |
| class="cmmi-12">j</span>, we simply multiply the
| |
| unit flux by the width of face <span
| |
| class="cmmi-12">j</span>, which we denote <span
| |
| class="cmmi-12">λ</span><sub><span
| |
| class="cmmi-8">ij</span></sub> (read as “the <span
| |
| class="cmmi-12">j</span>-th face of cell
| |
| <span
| |
| class="cmmi-12">i</span>”):
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-8004r8"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201210x.png" alt="Qsik = qsikλij
| |
| " class="math-display" ></center></td><td class="equation-label">(8)</td></tr></table>
| |
| <!--l. 228--><p class="nopar" >
| |
| <!--l. 230--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-90004.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 1: Getting Set Up with CHILD</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 232--><p class="noindent" >
| |
| <!--l. 235--><p class="noindent" ><span
| |
| class="cmss-12">Our first exercise is simply to (1) get the model, input files, documentation, and</span>
| |
| <span
| |
| class="cmss-12">visualization tools and (2) run the executable file to make sure it is installed and</span>
| |
| <span
| |
| class="cmss-12">working correctly. In some cases, it might be necessary to create a new executable</span>
| |
| <span
| |
| class="cmss-12">file from the source code.</span>
| |
|
| |
| <!--l. 241--><p class="noindent" ><span
| |
| class="cmss-12">For SIESD 2012, the package will already have been installed on the computers</span>
| |
| <span
| |
| class="cmss-12">in the lab. Look for it in the folder: </span><span
| |
| class="cmtt-12">C:</span><span
| |
| class="cmsy-10x-x-120">\</span><span
| |
| class="cmtt-12">child</span><span
| |
| class="cmsy-10x-x-120">\</span><span
| |
| class="cmtt-12">ChildExercises</span><span
| |
| class="cmss-12">.</span>
| |
| ______________
| |
| <!--l. 245--><p class="noindent" ><span
| |
| class="cmbxti-10x-x-120">If you are working on your own computer:</span>
| |
| <!--l. 247--><p class="noindent" ><span
| |
| class="cmssi-12">If you are working on your own computer, you will need to download a copy of the</span>
| |
| <span
| |
| class="cmssi-12">latest CHILD release from the Community Surface Dynamics Modeling System</span>
| |
| <span
| |
| class="cmssi-12">(CSDMS) web site:</span>
| |
| <!--l. 249--><p class="noindent" ><a
| |
| href="http://csdms.colorado.edu" class="url" ><span
| |
| class="cmitt-10x-x-120">http://csdms.colorado.edu</span></a>
| |
| <!--l. 251--><p class="noindent" ><span
| |
| class="cmssi-12">Once you have downloaded and unwrapped the package, locate the users’ guide</span>
| |
| <span
| |
| class="cmssi-12">and follow the instructions to compile the model on your particular platform. You</span>
| |
| <span
| |
| class="cmssi-12">will need to use either a UNIX shell or the Command window under Windows.</span>
| |
| <span
| |
| class="cmssi-12">On a mac, use the </span><span
| |
| class="cmbx-12">Terminal </span><span
| |
| class="cmssi-12">application. On a windows machine, use either</span>
| |
| <span
| |
| class="cmssi-12">a UNIX emulator shell such as cygwin on a PC, or the command window. In a</span>
| |
| <span
| |
| class="cmssi-12">UNIX shell, to change folders (“directories” in UNIX-speak), use </span><span
| |
| class="cmtt-12">cd </span><span
| |
| class="cmssi-12">followed by</span>
| |
| <span
| |
| class="cmssi-12">the folder name. A single period represents the current working directory; two</span>
| |
| <span
| |
| class="cmssi-12">periods represent the next directory up. For example, the command </span><span
| |
| class="cmtt-12">cd .. </span><span
| |
| class="cmssi-12">takes</span>
| |
| <span
| |
| class="cmssi-12">you one level up. To get a list of files in a directory, use </span><span
| |
| class="cmtt-12">ls</span><span
| |
| class="cmssi-12">. For Command prompt</span>
| |
| <span
| |
| class="cmssi-12">under windows, use </span><span
| |
| class="cmtt-12">dir </span><span
| |
| class="cmssi-12">instead of </span><span
| |
| class="cmtt-12">ls </span><span
| |
| class="cmssi-12">and backslashes instead of forward slashes.</span>
| |
| <!--l. 261--><p class="noindent" ><span
| |
| class="cmss-12">Start up Command Window. In the command window, type </span><span
| |
| class="cmtt-12">child</span><span
| |
| class="cmss-12">. You should see</span>
| |
| <span
| |
| class="cmss-12">something like the following:</span>
| |
|
| |
| <div class="verbatim" id="verbatim-1">
| |
| Usage: child [options] <input file>
| |
|  <br /> --help: display this help message.
| |
|  <br /> --no-check: disable CheckMeshConsistency().
| |
|  <br /> --silent-mode: silent mode.
| |
|  <br /> --version: display version.</div>
| |
| <!--l. 269--><p class="nopar" >
| |
| <!--l. 271--><p class="noindent" ><span
| |
| class="cmss-12">While we’re at it, let’s get ready to visualize the output. Start Matlab. The first thing we</span>
| |
| <span
| |
| class="cmss-12">will do is tell Matlab where to look for the plotting programs that we will use. At the</span>
| |
| <span
| |
| class="cmss-12">Matlab command prompt type:</span>
| |
| <!--l. 273--><p class="noindent" ><span
| |
| class="cmtt-12">path( path, ’</span><span
| |
| class="cmssi-12">childFolderLocation</span><span
| |
| class="cmsy-10x-x-120">\</span><span
| |
| class="cmtt-12">ChildExercises</span><span
| |
| class="cmsy-10x-x-120">\</span><span
| |
| class="cmtt-12">MatlabScripts’</span>
| |
| <span
| |
| class="cmtt-12">)</span>
| |
| <!--l. 275--><p class="noindent" ><span
| |
| class="cmss-12">For </span><span
| |
| class="cmssi-12">childFolderLocation</span><span
| |
| class="cmss-12">, use the path name of the folder that contains the CHILD</span>
| |
| <span
| |
| class="cmss-12">package. You can also add a folder to your path by selecting </span><span
| |
| class="cmssi-12">File-</span><span
| |
| class="cmmi-12">></span><span
| |
| class="cmssi-12">Set Path... </span><span
| |
| class="cmss-12">from the</span>
| |
| <span
| |
| class="cmss-12">menu.</span>
| |
| <!--l. 278--><p class="noindent" ><span
| |
| class="cmss-12">In Matlab, navigate the current folder to the location of the example input file</span>
| |
| <span
| |
| class="cmtt-12">hillslope1.in </span><span
| |
| class="cmss-12">(which should end in: </span><span
| |
| class="cmtt-12">ChildExercises</span><span
| |
| class="cmsy-10x-x-120">\</span><span
| |
| class="cmtt-12">Hillslope1</span><span
| |
| class="cmss-12">).</span>
| |
| <!--l. 280--><p class="noindent" ><span
| |
| class="cmss-12">Note that the “package” also includes some documentation that you may find</span>
| |
| <span
| |
| class="cmss-12">useful: the </span><span
| |
| class="cmtt-12">ChildExercises </span><span
| |
| class="cmss-12">folder contains an earlier version of this document,</span>
| |
| <span
| |
| class="cmss-12">and the </span><span
| |
| class="cmtt-12">Doc </span><span
| |
| class="cmss-12">folder contains the Users’ Guide (</span><span
| |
| class="cmtt-12">child</span><span
| |
| class="cmtt-12">_users</span><span
| |
| class="cmtt-12">_guide.pdf</span><span
| |
| class="cmss-12">). The</span>
| |
| <span
| |
| class="cmss-12">guide covers the nuts and bolts of the model in much greater detail than these</span>
| |
| <span
| |
| class="cmss-12">exercises and includes a full list of input parameters.</span>
| |
| <!--l. 287--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-100004.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 2: Hillslope Diffusion and Parabolic Slopes with CHILD</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 288--><p class="noindent" >
| |
| <!--l. 291--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-10002x1"><span
| |
| class="cmss-10x-x-109">In your terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">ChildExercises</span><span
| |
| class="cmsy-10x-x-109">\</span><span
| |
| class="cmtt-10x-x-109">Hillslope1 </span><span
| |
| class="cmss-10x-x-109">folder.</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-10004x2"><span
| |
| class="cmss-10x-x-109">To run the example, in your terminal window type:</span>
| |
| <!--l. 297--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child hillslope1.in</span>
| |
|
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-10006x3"><span
| |
| class="cmss-10x-x-109">A series of numbers will flash by on the screen. These numbers represent time</span>
| |
| <span
| |
| class="cmss-10x-x-109">intervals in years. The 2-million-year run takes about 20 seconds on a 2GHz Intel</span>
| |
| <span
| |
| class="cmss-10x-x-109">Mac. When it finishes, return to Matlab and type:</span>
| |
| <!--l. 303--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">m = cmovie( ’hillslope1’, 21, 200, 200, 100, 50 );</span>
| |
| <!--l. 305--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">(This command says “generate a 21-frame movie from the run ‘hillslope1’ with</span>
| |
| <span
| |
| class="cmss-10x-x-109">the x-, y- and z- axes set to 200, 200 and 100 m, respectively, and with the color</span>
| |
| <span
| |
| class="cmss-10x-x-109">range representing 0 to 50 m elevation).</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-10008x4"><span
| |
| class="cmss-10x-x-109">To replay the movie, type </span><span
| |
| class="cmtt-10x-x-109">movie(m)</span><span
| |
| class="cmss-10x-x-109">.</span>
| |
| </li></ol>
| |
| <!--l. 310--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">(Windows note: we found that under Vista and Windows 7, the movie figure gets erased after</span>
| |
| <span
| |
| class="cmss-10x-x-109">display; slightly re-sizing the figure window seems to fix this).</span>
| |
| <!--l. 313--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The analytical solution to elevation as a function of cross-ridge distance </span><span
| |
| class="cmmi-10x-x-109">y </span><span
| |
| class="cmss-10x-x-109">is:</span>
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-10009r9"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201211x.png" alt=" U ( )
| |
| z(y) = --- L2 - (y - y0)2
| |
| 2D
| |
| " class="math-display" ></center></td><td class="equation-label"><span
| |
| class="cmr-10x-x-109">(9)</span></td></tr></table>
| |
| <!--l. 316--><p class="nopar" >
| |
| <span
| |
| class="cmss-10x-x-109">where </span><span
| |
| class="cmmi-10x-x-109">L </span><span
| |
| class="cmss-10x-x-109">is the half-width of the ridge (100 m in this case) and </span><span
| |
| class="cmmi-10x-x-109">y</span><sub><span
| |
| class="cmr-8">0</span></sub> <span
| |
| class="cmss-10x-x-109">is the position of the ridge</span>
| |
| <span
| |
| class="cmss-10x-x-109">crest (also 100 m). The effective uplift rate </span><span
| |
| class="cmmi-10x-x-109">U</span><span
| |
| class="cmss-10x-x-109">, represented in the input file by the parameter</span>
| |
| <span
| |
| class="cmtt-10x-x-109">UPRATE</span><span
| |
| class="cmss-10x-x-109">, is </span><span
| |
| class="cmr-10x-x-109">10</span><sup><span
| |
| class="cmsy-8">-</span><span
| |
| class="cmr-8">4</span></sup> <span
| |
| class="cmss-10x-x-109">m/yr. The diffusivity coefficient </span><span
| |
| class="cmmi-10x-x-109">D</span><span
| |
| class="cmss-10x-x-109">, represented in the input file by parameter</span>
| |
| <span
| |
| class="cmtt-10x-x-109">KD</span><span
| |
| class="cmss-10x-x-109">, is 0.01 m</span><sup><span
| |
| class="cmr-8">2</span></sup><span
| |
| class="cmss-10x-x-109">/yr. Next, we’ll make a plot that compares the computed and analytical</span>
| |
| <span
| |
| class="cmss-10x-x-109">solutions.</span>
| |
| <!--l. 320--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Enter the following in Matlab:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">ya = 0:200; </span><span
| |
| class="cmssi-10x-x-109">% This is our x-coordinate</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">U = 0.0001; D = 0.01; y0 = 100; L = 100;</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">za = (U/(2*D))*(L</span><img
| |
| src="child_exercises_nced_aug201212x.png" alt="^ " class="circ" ><span
| |
| class="cmtt-10x-x-109">2-(ya-y0).</span><img
| |
| src="child_exercises_nced_aug201213x.png" alt="^ " class="circ" ><span
| |
| class="cmtt-10x-x-109">2);</span>
| |
|
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2), plot( ya, za ), hold on</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">xyz = creadxyz( ’hillslope1’, 21 ); </span><span
| |
| class="cmssi-10x-x-109">% Reads node coords, time 21</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">plot( xyz(:,2), xyz(:,3), ’r.’ ), hold off</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">legend( ’Analytical solution’, ’CHILD Nodes’ )</span></li></ul>
| |
| <!--l. 337--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Diffusion theory predicts that equilibrium height varies linearly with </span><span
| |
| class="cmmi-10x-x-109">U</span><span
| |
| class="cmss-10x-x-109">, inversely with</span>
| |
| <span
| |
| class="cmmi-10x-x-109">D</span><span
| |
| class="cmss-10x-x-109">, and as the square of </span><span
| |
| class="cmmi-10x-x-109">L</span><span
| |
| class="cmss-10x-x-109">. Make a copy of </span><span
| |
| class="cmtt-10x-x-109">hillslope1.in </span><span
| |
| class="cmss-10x-x-109">and open the copy in</span>
| |
| <span
| |
| class="cmss-10x-x-109">a text editor. Change one of these three parameters. To change </span><span
| |
| class="cmmi-10x-x-109">U</span><span
| |
| class="cmss-10x-x-109">, edit the number</span>
| |
| <span
| |
| class="cmss-10x-x-109">below the line that starts with </span><span
| |
| class="cmtt-10x-x-109">UPRATE</span><span
| |
| class="cmss-10x-x-109">. Similarly, to change </span><span
| |
| class="cmmi-10x-x-109">D</span><span
| |
| class="cmss-10x-x-109">, edit the value of</span>
| |
| <span
| |
| class="cmss-10x-x-109">parameter </span><span
| |
| class="cmtt-10x-x-109">KD</span><span
| |
| class="cmss-10x-x-109">. If you want to try a different ridge width </span><span
| |
| class="cmmi-10x-x-109">L</span><span
| |
| class="cmss-10x-x-109">, change both </span><span
| |
| class="cmtt-10x-x-109">Y</span><span
| |
| class="cmtt-10x-x-109">_GRID</span><span
| |
| class="cmtt-10x-x-109">_SIZE</span>
| |
| <span
| |
| class="cmss-10x-x-109">and </span><span
| |
| class="cmtt-10x-x-109">GRID</span><span
| |
| class="cmtt-10x-x-109">_SPACING </span><span
| |
| class="cmss-10x-x-109">by the same proportion (changing </span><span
| |
| class="cmtt-10x-x-109">GRID</span><span
| |
| class="cmtt-10x-x-109">_SPACING </span><span
| |
| class="cmss-10x-x-109">will ensure that</span>
| |
| <span
| |
| class="cmss-10x-x-109">you keep the same number of model nodes). Re-run CHILD with your modified input</span>
| |
| <span
| |
| class="cmss-10x-x-109">file and see what happens.</span>
| |
| <!--l. 341--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">4.2. </span> <a
| |
| id="x1-110004.2"></a><span
| |
| class="cmbx-12">Nonlinear Diffusion.</span></span>
| |
| <!--l. 343--><p class="noindent" >As we found in our study of hillslope transport processes, the simple slope-linear transport law
| |
| works poorly for slopes that are not “small” relative to the angle of repose for sediment and rock.
| |
| The next example explores what happens to our ridge when we (1) increase the relative uplift rate,
| |
| and (2) use the nonlinear diffusion transport law:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-11001r10"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201214x.png" alt="⃗qs = ---- D-∇z----
| |
| 1 - |∇z ∕Sc |2
| |
| " class="math-display" ></center></td><td class="equation-label">(10)</td></tr></table>
| |
| <!--l. 346--><p class="nopar" >
| |
|
| |
| <!--l. 348--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-120004.2"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 3: Nonlinear Diffusion and Planar Slopes</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 350--><p class="noindent" >
| |
| <!--l. 353--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-12002x1"><span
| |
| class="cmss-10x-x-109">Navigate to the </span><span
| |
| class="cmtt-10x-x-109">Hillslope2 </span><span
| |
| class="cmss-10x-x-109">folder</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-12004x2"><span
| |
| class="cmss-10x-x-109">Run CHILD: </span><span
| |
| class="cmtt-10x-x-109">child hillslope2.in</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-12006x3"><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Hillslope2 </span><span
| |
| class="cmss-10x-x-109">folder</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-12008x4"><span
| |
| class="cmss-10x-x-109">When the 70,000-year run (</span><span
| |
| class="cmsy-10x-x-109">~</span><span
| |
| class="cmss-10x-x-109">1 minute on a 2GHz mac) finishes, type in Matlab:</span>
| |
| <!--l. 363--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">m = cmovie( ’hillslope2’, 21, 200, 200, 100, 70 );</span></li></ol>
| |
| <!--l. 365--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">If we had used linear diffusion, the equilibrium slope gradient along the edges of the</span>
| |
| <span
| |
| class="cmss-10x-x-109">ridge would be </span><span
| |
| class="cmmi-10x-x-109">S </span><span
| |
| class="cmr-10x-x-109">= </span><span
| |
| class="cmmi-10x-x-109">UL∕D </span><span
| |
| class="cmr-10x-x-109">= (0</span><span
| |
| class="cmmi-10x-x-109">.</span><span
| |
| class="cmr-10x-x-109">001)(100)</span><span
| |
| class="cmmi-10x-x-109">∕</span><span
| |
| class="cmr-10x-x-109">(0</span><span
| |
| class="cmmi-10x-x-109">.</span><span
| |
| class="cmr-10x-x-109">01) = 10 </span><span
| |
| class="cmss-10x-x-109">m/m, or about 84</span><sup><span
| |
| class="cmsy-8">∘</span></sup><span
| |
| class="cmss-10x-x-109">! Instead,</span>
| |
| <span
| |
| class="cmss-10x-x-109">the actual computed gradient is close to the threshold limit of 0.7. Notice too how</span>
| |
| <span
| |
| class="cmss-10x-x-109">the model solution speed slowed down as the run went on. This reflects the need for</span>
| |
| <span
| |
| class="cmss-10x-x-109">especially small time steps when the slopes are close to the threshold angle.</span>
| |
| <!--l. 369--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">4.3. </span> <a
| |
| id="x1-130004.3"></a><span
| |
| class="cmbx-12">Remarks.</span></span>
| |
| <!--l. 371--><p class="noindent" >There is a lot more to mass movement than what is encoded in these simple diffusion-like transport
| |
| laws. Some models include stochastic landsliding algorithms (e.g., CASCADE, ZSCAPE). Some
| |
| impose threshold slopes (e.g., GOLEM). One spinoff version of CHILD even includes debris-flow
| |
| generation and routing (<a
| |
| href="#Xlancaster2003effects"><span
| |
| class="cmti-12">Lancaster et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xlancaster2003effects">2003</a>).
| |
| <h3 class="sectionHead"><span class="titlemark">5. </span> <a
| |
| id="x1-140005"></a>Rainfall, Runoff, and Drainage Networks</h3>
| |
| <!--l. 375--><p class="noindent" >In order to calculate erosion, sediment transport, and deposition by running water, a model needs
| |
| to know how much surface water is flowing through each cell in the model. Usually, the
| |
| erosion/transport equations require either the total discharge, <span
| |
| class="cmmi-12">Q </span>[L<sup><span
| |
| class="cmr-8">3</span></sup>/T], the discharge per unit
| |
| channel width, <span
| |
| class="cmmi-12">q </span>[L<sup><span
| |
| class="cmr-8">2</span></sup>/T], or the flow depth, <span
| |
| class="cmmi-12">H</span>.
| |
| <!--l. 377--><p class="noindent" >There are three main alternative methods for modeling the flow of water across the
| |
| landscape:
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-14002x1">Methods based on contributing drainage area
| |
|
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-14004x2">Numerical solutions to the 2D, vertically integrated and time-averaged Navier-Stokes
| |
| equations
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-14006x3">Cellular automaton methods</li></ol>
| |
| <!--l. 384--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.1. </span> <a
| |
| id="x1-150005.1"></a><span
| |
| class="cmbx-12">Methods Based on Drainage Area.</span></span>
| |
| <!--l. 386--><p class="noindent" ><span
| |
| class="cmti-12">Drainage area</span>, <span
| |
| class="cmmi-12">A</span>, is the horizontally projected area of land that contributes flow to a
| |
| particular channel cross-section or to unit length of contour on a hillslope. For a numerical
| |
| landscape model that uses discrete cells, <span
| |
| class="cmmi-12">A </span>is defined as the area that contributes flow to a
| |
| particular cell. When topography is represented as a raster grid, the most common
| |
| method for computing drainage area is the <span
| |
| class="cmti-12">D8 method</span>. Each cell is assigned a flow
| |
| direction toward one of its 8 surrounding neighbors. An algorithm is then used to trace
| |
| flow paths downstream and add up the number of cells that contribute flow each cell
| |
| (Fig. <a
| |
| href="#x1-150013">3<!--tex4ht:ref: fig:d8mfd --></a>).
| |
| <!--l. 388--><p class="noindent" ><span class="floatingfigure-r" style="width:296.30743pt"><img
| |
| src="child_exercises_nced_aug201215x.png" alt="PIC" class="graphics"><!--tex4ht:graphics
| |
| name="child_exercises_nced_aug201215x.png" src="Schauble_EtAl_flow_dirs.pdf"
| |
| -->
| |
| <br /><span class="caption"><span class="id">Figure 3</span>: Flow accumulation by
| |
| D8, or single flow directions, and multiple flow directions
| |
| (<a
| |
| href="#Xschauble2008gis"><span
| |
| class="cmti-12">Schauble et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xschauble2008gis">2008</a>).</span><br /> </span>
| |
| <!--l. 396--><p class="noindent" >For the Voronoi cell matrix that CHILD and CASCADE use, the simplest routing procedure is a
| |
|
| |
| generalization of D8 (Figure <a
| |
| href="#x1-20011">1<!--tex4ht:ref: fig:schem --></a>). Each cell <span
| |
| class="cmmi-12">i </span>has <span
| |
| class="cmmi-12">N</span><sub><span
| |
| class="cmmi-8">i</span></sub> neighbors. As we noted earlier, the
| |
| slope from cell <span
| |
| class="cmmi-12">i </span>to neighbor cell <span
| |
| class="cmmi-12">k </span>is defined as the elevation difference between the
| |
| nodes divided by the horizontal distance between them (Fig. <a
| |
| href="#x1-60032">2<!--tex4ht:ref: fig:cmesh --></a>). Thus, one can define a
| |
| slope for every <span
| |
| class="cmti-12">edge </span>that connects each pair of nodes. There is a slope value for each
| |
| of the <span
| |
| class="cmmi-12">N</span><sub><span
| |
| class="cmmi-8">i</span></sub> neighbors of node <span
| |
| class="cmmi-12">i</span>. The flow direction is assigned as the steepest of these
| |
| slopes.
| |
| <!--l. 398--><p class="noindent" >Single-direction flow algorithms have advantages and disadvantages. Some models use a <span
| |
| class="cmti-12">multiple</span>
| |
| <span
| |
| class="cmti-12">flow direction </span>approach to represent the divergence of flow on relatively gentle slopes or
| |
| divergent landforms (Fig. <a
| |
| href="#x1-150013">3<!--tex4ht:ref: fig:d8mfd --></a>). This is most appropriate for models that operate on a grid
| |
| resolution significantly smaller than the length of a hillslope. When grid cells are relatively
| |
| large, conceptually each cell contains a primary channel, narrower than the cell, that is
| |
| tracked.
| |
| <!--l. 400--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-160005.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 4: Flow Over Noisy, Inclined Topography</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 402--><p class="noindent" >
| |
| <!--l. 405--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-16002x1"><span
| |
| class="cmss-10x-x-109">In the terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Network1 </span><span
| |
| class="cmss-10x-x-109">folder and run the input file by</span>
| |
| <span
| |
| class="cmss-10x-x-109">typing:</span>
| |
| <!--l. 409--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child network1.in</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-16004x2"><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Network1 </span><span
| |
| class="cmss-10x-x-109">folder</span></li></ol>
| |
| <!--l. 414--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">colormap pink</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">a = cread( ’network1.area’, 1 );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">ctrisurf( ’network1’, 1, a );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">view( 0, 90 ), shading interp, axis equal</span></li></ul>
| |
| <!--l. 429--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The networks are formed because of noise (</span><span
| |
| class="cmsy-10x-x-109">±</span><span
| |
| class="cmr-10x-x-109">1 </span><span
| |
| class="cmss-10x-x-109">m in this case) in the initial surface,</span>
| |
| <span
| |
| class="cmss-10x-x-109">which causes flow to converge in some places.</span>
| |
|
| |
| <!--l. 433--><p class="noindent" >The simplest method for computing discharge from drainage area is to simply assume (1) all rain
| |
| runs off, and (2) rain lasts long enough that the entire drainage network is in hydrologic steady
| |
| state. In this case, and if precipitation rate <span
| |
| class="cmmi-12">P </span>is uniform,
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-16005r11"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201216x.png" alt="Q = P A
| |
| " class="math-display" ></center></td><td class="equation-label">(11)</td></tr></table>
| |
| <!--l. 436--><p class="nopar" >
| |
| A number of landscape modeling studies have used this assumption, on the basis of its
| |
| simplicity, even though it tends to make hydrologists faint! The simplicity is indeed a
| |
| virtue, but one needs to be extremely careful in using this equation, for at least three
| |
| reasons. First, obviously <span
| |
| class="cmmi-12">Q </span>varies substantially over time in response to changing seasons,
| |
| floods, droughts, etc. We will return to this issue shortly. Second, there is probably no
| |
| drainage basin on earth, bigger than a hectare or so, from which <span
| |
| class="cmti-12">all </span>precipitation runs off.
| |
| Typically, evapotranspiration returns more than half of incoming precipitation to the
| |
| atmosphere. Third, hydrologic steady state is rare and tends to occur only in small
| |
| basins, though it may be a reasonable approximation for mean annual discharge in some
| |
| basins.
| |
| <!--l. 439--><p class="noindent" >River discharge, whether defined as mean annual, bankfull, mean peak, or some other way, often
| |
| shows a power-law-like correlation with drainage area. Some models take advantage of this fact by
| |
| computing discharge using an empirical approach:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-16006r12"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201217x.png" alt="Q = bAc
| |
| " class="math-display" ></center></td><td class="equation-label">(12)</td></tr></table>
| |
|
| |
| <!--l. 442--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">c </span>typically ranges from 0.5-1 and <span
| |
| class="cmmi-12">b </span>is a runoff coefficient with awkward units that represents
| |
| a long-term “effective” precipitation regime.
| |
| <!--l. 445--><p class="noindent" >CHILD’s default method for computing discharge during a storm takes runoff at each cell
| |
| to be the difference between storm rainfall intensity <span
| |
| class="cmmi-12">P </span>and soil infiltration capacity
| |
| <span
| |
| class="cmmi-12">I</span>:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-16007r13"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201218x.png" alt="Q = (P - I)A
| |
| " class="math-display" ></center></td><td class="equation-label">(13)</td></tr></table>
| |
| <!--l. 448--><p class="nopar" >
| |
| which of course is taken to be zero when <span
| |
| class="cmmi-12">P < I</span>.
| |
| <!--l. 451--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.2. </span> <a
| |
| id="x1-170005.2"></a><span
| |
| class="cmbx-12">Shallow-Water Equations.</span></span>
| |
| <!--l. 453--><p class="noindent" >Some landscape models are designed to address relatively small-scale problems such as
| |
| channel initiation, inundation of alluvial fan surfaces, channel flood flow, etc. In such
| |
| cases, the convergence and divergence of water in response to evolving topography is an
| |
| important component of the problem, and is not adequately captured by the simple
| |
| routing schemes described above. Instead, a tempting tool of choice is some form of
| |
| the <span
| |
| class="cmti-12">shallow-water equations</span>, which are the vertically integrated form of the general
| |
| (time-averaged) viscous fluid-flow equations. One form of the full shallow-water equations is:
| |
| <div class="eqnarray">
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201219x.png" alt=" ( )
| |
| ∂-η = i - ∂qx-+ ∂qy- (14)
| |
| ∂t ∂x ∂y
| |
| ∂q ∂q u ∂q u ∂h ∂η τ
| |
| --x-+ --x--+ ---y- + gh ---+ gh ---+ -bx-= 0 (15)
| |
| ∂t ∂x ∂y ∂x ∂x ρ
| |
| ∂qy- ∂qyv- ∂qxv- ∂h- ∂η- τby-
| |
| ∂t + ∂y + ∂x + gh ∂y + gh ∂y + ρ = 0 (16)
| |
| " class="math-display" ></center>
| |
| </div>These equations express continuity of mass, x-directed momentum, and y-directed momentum,
| |
| respectively. They are challenging and computationally expensive to integrate numerically in their
| |
| full form. However, there are several approximate forms that are commonly used, including the
| |
| non-accelerating flow form (in which convective accelerations are assumed negligible) and the
| |
| kinematic-wave equations (in which gravitational and friction forces are assumed to dominate). An
| |
| example of use of the shallow-water equations in a landform evolution model can be found in the
| |
| work of T.R. Smith and colleagues (Fig. <a
| |
| href="#x1-170024">4<!--tex4ht:ref: fig:shalflow --></a>). Various forms of the shallow-water equations can often
| |
| be found in hydrologic models, and sometimes in soil-erosion models (e.g., <a
| |
| href="#Xmitas1998distributed"><span
| |
| class="cmti-12">Mitas and</span>
| |
| <span
| |
| class="cmti-12">Mitasova</span></a>, <a
| |
| href="#Xmitas1998distributed">1998</a>).
| |
| <!--l. 470--><p class="noindent" ><hr class="figure"><div class="figure"
| |
| >
| |
|
| |
| <a
| |
| id="x1-170024"></a>
| |
|
| |
| <!--l. 472--><p class="noindent" ><img
| |
| src="child_exercises_nced_aug201220x.png" alt="PIC" class="graphics"><!--tex4ht:graphics
| |
| name="child_exercises_nced_aug201220x.png" src="Smith_Merchant_shallow_flow.pdf"
| |
| -->
| |
| <br /> <div class="caption"
| |
| ><span class="id">Figure 4: </span><span
| |
| class="content">Simulated water surface elevations and flow depth (<a
| |
| href="#Xbirnir2001scaling"><span
| |
| class="cmti-12">Birnir et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xbirnir2001scaling">2001</a>).</span></div><!--tex4ht:label?: x1-170024 -->
| |
|
| |
| <!--l. 476--><p class="noindent" ></div><hr class="endfigure">
| |
| <!--l. 478--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.3. </span> <a
| |
| id="x1-180005.3"></a><span
| |
| class="cmbx-12">Cellular Automata.</span></span>
| |
| <!--l. 480--><p class="noindent" >Some models use cellular automaton methods to calculate flow over a cellular topography. These
| |
| include:
| |
| <ul class="itemize1">
| |
| <li class="itemize"><a
| |
| href="#Xchase1992fluvial"><span
| |
| class="cmti-12">Chase</span></a> (<a
| |
| href="#Xchase1992fluvial">1992</a>) precipiton algorithm
| |
| </li>
| |
| <li class="itemize"><a
| |
| href="#Xcrave2001stochastic"><span
| |
| class="cmti-12">Crave and Davy</span></a> (<a
| |
| href="#Xcrave2001stochastic">2001</a>) modified precipiton algorithm
| |
| </li>
| |
| <li class="itemize"><a
| |
| href="#Xmurray1994cellular"><span
| |
| class="cmti-12">Murray and Paola</span></a> (<a
| |
| href="#Xmurray1994cellular">1994</a>) multiple-flow-direction river-flow algorithm
| |
| </li>
| |
| <li class="itemize"><a
| |
| href="#Xcoulthard1996cellular"><span
| |
| class="cmti-12">Coulthard et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xcoulthard1996cellular">1996</a>) generalization of Murray-Paola for 2D flow (CAESAR model)</li></ul>
| |
| <!--l. 488--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.4. </span> <a
| |
| id="x1-190005.4"></a><span
| |
| class="cmbx-12">Depressions in the Terrain.</span></span>
| |
| <!--l. 490--><p class="noindent" >What happens when flow enters a topographic depression? In the real world, three possibilities:
| |
| complete evaporation/infiltration, formation of a lake with overflow, or formation of a closed lake.
| |
| CHILD can be set either to have water in “pits” evaporate, or to use a lake-fill algorithm to route
| |
| water through depressions in the terrain (with no evaporation).
| |
| <!--l. 492--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.5. </span> <a
| |
| id="x1-200005.5"></a><span
| |
| class="cmbx-12">Precipitation and Discharge.</span></span>
| |
| <!--l. 494--><p class="noindent" >Water supply to the channel network varies dramatically in both time and space, but there is a big
| |
| gap in time scale between, on the one hand, storms and floods and, on the other hand, topographic
| |
| evolution. Many landscape evolution models have therefore used the “effective discharge”
| |
| concept, or the idea that there is some value of discharge that represents the cumulative
| |
| geomorphic effect of the natural sequence of storms and floods. <a
| |
| href="#Xwillgoose1991coupled"><span
| |
| class="cmti-12">Willgoose et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xwillgoose1991coupled">1991</a>)
| |
| used mean peak discharge, but <a
| |
| href="#Xhuang2006evaluation"><span
| |
| class="cmti-12">Huang and Niemann</span></a> (<a
| |
| href="#Xhuang2006evaluation">2006</a>) recognized that the return
| |
| period of effective discharge events is not necessarily the same at different times and
| |
| places.
| |
| <!--l. 506--><p class="noindent" >Basically, landscape models tend to use one of four methods:
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-20002x1">Steady flow with uniform precipitation or a specified runoff coefficient (effective
| |
| discharge concept)
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-20004x2">Steady flow with nonuniform precipitation or runoff (e.g., orographic precipitation)
| |
|
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-20006x3">Stochastic-in-time, spatially uniform runoff generation
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-20008x4">“Short storms” model (<a
| |
| href="#Xsolyom2004effect"><span
| |
| class="cmti-12">S</span><span
| |
| class="cmti-12">Ûlyom and Tucker</span></a>, <a
| |
| href="#Xsolyom2004effect">2004</a>)</li></ol>
| |
| <!--l. 513--><p class="noindent" >We will not examine all of these in detail. Instead, we will take a brief look at the Poisson
| |
| rectangular pulse model implemented in CHILD.
| |
| <!--l. 515--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-210005.5"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 5: Visualizing a Poisson Storm Sequence</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 517--><p class="noindent" >
| |
| <!--l. 520--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-21002x1"><span
| |
| class="cmss-10x-x-109">In the terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Rainfall1 </span><span
| |
| class="cmss-10x-x-109">folder and run the input file</span>
| |
| <span
| |
| class="cmss-10x-x-109">by typing:</span>
| |
| <!--l. 524--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child rainfall1.in</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-21004x2"><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Rainfall1 </span><span
| |
| class="cmss-10x-x-109">folder</span></li></ol>
| |
| <!--l. 528--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf, cstormplot( ’rainfall1’ );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2), clf, cstormplot( ’rainfall1’, 10 );</span></li></ul>
| |
| <!--l. 536--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The first plot shows a 1-year simulated storm sequence; the second shows just the first</span>
| |
| <span
| |
| class="cmss-10x-x-109">10 storms.</span>
| |
| <!--l. 541--><p class="noindent" >The motivation for using a stochastic flow model is (1) that nature <span
| |
| class="cmti-12">is </span>effectively stochastic, and (2)
| |
| variability matters when the erosion or transport rate is a nonlinear function of flow. For more on
| |
| this, see <a
| |
| href="#Xtucker2000stochastic"><span
| |
| class="cmti-12">Tucker and Bras</span></a> (<a
| |
| href="#Xtucker2000stochastic">2000</a>); <a
| |
| href="#Xsnyder2003importance"><span
| |
| class="cmti-12">Snyder et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xsnyder2003importance">2003</a>); <a
| |
| href="#Xtucker2004drainage"><span
| |
| class="cmti-12">Tucker</span></a> (<a
| |
| href="#Xtucker2004drainage">2004</a>), and <a
| |
| href="#Xdibiase2011influence"><span
| |
| class="cmti-12">DiBiase and</span>
| |
| <span
| |
| class="cmti-12">Whipple</span></a> (<a
| |
| href="#Xdibiase2011influence">2011</a>).
| |
| <!--l. 548--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">5.6. </span> <a
| |
| id="x1-220005.6"></a><span
| |
| class="cmbx-12">Remarks.</span></span>
| |
| <!--l. 550--><p class="noindent" >Landscape evolution models can be, and have been, used to study climate impacts on erosion,
| |
| topography, and mountain building. But be careful—climate and hydrology amount to much more
| |
| than a “sprinkler over the landscape.”
| |
|
| |
| <h3 class="sectionHead"><span class="titlemark">6. </span> <a
| |
| id="x1-230006"></a>Hydraulic Geometry</h3>
| |
| <!--l. 554--><p class="noindent" >Channel size, shape, and roughness control delivery of hydraulic force to the bed and banks. Most
| |
| landscape models either implicitly assume constant width (practical but dangerous) or use the
| |
| empirical relation <span
| |
| class="cmmi-12">W </span>= <span
| |
| class="cmmi-12">K</span><sub><span
| |
| class="cmmi-8">w</span></sub><span
| |
| class="cmmi-12">Q</span><sup><span
| |
| class="cmmi-8">b</span></sup>, where <span
| |
| class="cmmi-12">b </span><span
| |
| class="cmsy-10x-x-120">≈ </span>0<span
| |
| class="cmmi-12">.</span>5. Models with time-varying discharge must also
| |
| specify how width varies at a point along the channel as <span
| |
| class="cmmi-12">Q </span>rises and falls. Width-discharge scaling
| |
| is practical but incomplete, because channels may narrow or widen downstream in concert
| |
| with variations in incision rate, as observed in Italy (<a
| |
| href="#Xwhittaker2007bedrock"><span
| |
| class="cmti-12">Whittaker et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xwhittaker2007bedrock">2007</a>), Nepal
| |
| (<a
| |
| href="#Xlave2001fluvial"><span
| |
| class="cmti-12">Lav</span><span
| |
| class="cmti-12">È and Avouac</span></a>, <a
| |
| href="#Xlave2001fluvial">2001</a>), New Zealand (<a
| |
| href="#Xamos2007channel"><span
| |
| class="cmti-12">Amos and Burbank</span></a>, <a
| |
| href="#Xamos2007channel">2007</a>), Taiwan (<a
| |
| href="#Xyanites2010incision"><span
| |
| class="cmti-12">Yanites</span>
| |
| <span
| |
| class="cmti-12">et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xyanites2010incision">2010</a>), and California (<a
| |
| href="#Xduvall2004tectonic"><span
| |
| class="cmti-12">Duvall et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xduvall2004tectonic">2004</a>). Some models have begun to explore these
| |
| sensitivities (<a
| |
| href="#Xwobus2006self"><span
| |
| class="cmti-12">Wobus et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xwobus2006self">2006</a>, <a
| |
| href="#Xwobus2008modeling">2008</a>; <a
| |
| href="#Xattal2008modeling"><span
| |
| class="cmti-12">Attal et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xattal2008modeling">2008</a>; <a
| |
| href="#Xturowski2009response"><span
| |
| class="cmti-12">Turowski et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xturowski2009response">2009</a>; <a
| |
| href="#Xyanites2010controls"><span
| |
| class="cmti-12">Yanites and</span>
| |
| <span
| |
| class="cmti-12">Tucker</span></a>, <a
| |
| href="#Xyanites2010controls">2010</a>), but full treatment of the channel geometry adjustment problem is a frontier
| |
| area.
| |
| <h3 class="sectionHead"><span class="titlemark">7. </span> <a
| |
| id="x1-240007"></a>Erosion and Transport by Running Water</h3>
| |
| <!--l. 569--><p class="noindent" >There are several competing models for erosion by channelized flow. Detachment-limited models
| |
| assume that eroded material leaves the system without significant re-deposition and that
| |
| lowering of channels is limited by the ability of the stream to detach material from the bed
| |
| (<a
| |
| href="#Xhoward1994detachment"><span
| |
| class="cmti-12">Howard</span></a>, <a
| |
| href="#Xhoward1994detachment">1994</a>; <a
| |
| href="#Xwhipple1999dynamics"><span
| |
| class="cmti-12">Whipple and Tucker</span></a>, <a
| |
| href="#Xwhipple1999dynamics">1999</a>). Transport-limited models assume plentiful supply of
| |
| loose sediment and that lowering of channels is limited by the stream’s capacity to transport
| |
| sediment (<a
| |
| href="#Xwillgoose1991coupled"><span
| |
| class="cmti-12">Willgoose et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xwillgoose1991coupled">1991</a>; <a
| |
| href="#Xwhipple2002implications"><span
| |
| class="cmti-12">Whipple and Tucker</span></a>, <a
| |
| href="#Xwhipple2002implications">2002</a>). In simple hybrid models, lowering
| |
| may be limited either by excess transport capacity or by detachment rate, depending
| |
| on local sediment supply and substrate resistance (<a
| |
| href="#Xtucker2001channel"><span
| |
| class="cmti-12">Tucker et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xtucker2001channel">2001b</a>; <a
| |
| href="#Xwhipple2002implications"><span
| |
| class="cmti-12">Whipple and</span>
| |
| <span
| |
| class="cmti-12">Tucker</span></a>, <a
| |
| href="#Xwhipple2002implications">2002</a>). With the undercapacity concept, detachment rate depends on surplus transport
| |
| capacity (<a
| |
| href="#Xbeaumont1992erosional"><span
| |
| class="cmti-12">Beaumont et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xbeaumont1992erosional">1992</a>). In the saltation-abrasion model, detachment is driven by
| |
| grain impacts and limited by sediment shielding (<a
| |
| href="#Xgasparini2007predictions"><span
| |
| class="cmti-12">Gasparini et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xgasparini2007predictions">2007</a>; <a
| |
| href="#Xwhipple2002implications"><span
| |
| class="cmti-12">Whipple and</span>
| |
| <span
| |
| class="cmti-12">Tucker</span></a>, <a
| |
| href="#Xwhipple2002implications">2002</a>).
| |
| <!--l. 583--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">7.1. </span> <a
| |
| id="x1-250007.1"></a><span
| |
| class="cmbx-12">Detachment-Limited Models.</span></span>
| |
| <!--l. 585--><p class="noindent" >On a cohesive or rock bed with a discontinuous or absent cover of loose sediment, detachment of
| |
| particles from the bed may be driven primarily by hydraulic lift and drag (“plucking”). Most
| |
| models assume that the rate of detachment (or more generally the capacity for detachment)
| |
| depends on excess bed shear stress:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-25001r17"></a>
| |
|
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201221x.png" alt=" p p p
| |
| Dc = Kb (τ - τc)b , or alternatively, Dc = Kb (τ b - τcb)
| |
| " class="math-display" ></center></td><td class="equation-label">(17)</td></tr></table>
| |
| <!--l. 589--><p class="nopar" >
| |
| where <span
| |
| class="cmmi-12">τ </span>is local bed shear stress, <span
| |
| class="cmmi-12">τ</span><sub><span
| |
| class="cmmi-8">c</span></sub> is a threshold stress below which detachment is ineffective, <span
| |
| class="cmmi-12">K</span><sub><span
| |
| class="cmmi-8">b</span></sub>
| |
| is a constant, and <span
| |
| class="cmmi-12">p</span><sub><span
| |
| class="cmmi-8">b</span></sub> is an exponent.
| |
| <!--l. 592--><p class="noindent" >Bed shear stress fluctuates in space and time, but is often treated using the cross-sectional average,
| |
| which in turn is based on a force balance between gravity and friction.
| |
| <!--l. 594--><p class="noindent" >Some models assume that the detachment rate depends on stream power per unit width,
| |
| <span
| |
| class="cmmi-12">ω </span>= <span
| |
| class="cmmi-12">ρg</span>(<span
| |
| class="cmmi-12">Q∕W</span>)<span
| |
| class="cmmi-12">S</span>:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-25002r18"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201222x.png" alt=" ( Q )pb
| |
| Dc = Kb --S - Φc
| |
| W
| |
| " class="math-display" ></center></td><td class="equation-label">(18)</td></tr></table>
| |
| <!--l. 598--><p class="nopar" >
| |
| where Φ<sub><span
| |
| class="cmmi-8">c</span></sub> is, again, a threshold below which detachment is ineffective. Stream power per unit width
| |
| turns out to be proportional to <span
| |
| class="cmmi-12">τ</span><sup><span
| |
| class="cmr-8">3</span><span
| |
| class="cmmi-8">∕</span><span
| |
| class="cmr-8">2</span></sup>, so the two erosion formulas are closely related (<a
| |
| href="#Xwhipple1999dynamics"><span
| |
| class="cmti-12">Whipple and</span>
| |
| <span
| |
| class="cmti-12">Tucker</span></a>, <a
| |
| href="#Xwhipple1999dynamics">1999</a>). In the following example, we will use the unit stream power formula with
| |
| Φ<sub><span
| |
| class="cmmi-8">c</span></sub> = 0.
| |
| <!--l. 602--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-260007.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 6: Detachment-Limited Hills and Mountains</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 604--><p class="noindent" >
| |
|
| |
| <!--l. 607--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-26002x1"><span
| |
| class="cmss-10x-x-109">In the terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Dlim </span><span
| |
| class="cmss-10x-x-109">folder and run the input file by</span>
| |
| <span
| |
| class="cmss-10x-x-109">typing:</span>
| |
| <!--l. 612--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child dlim.in</span>
| |
| <!--l. 614--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The 3 m.y.</span><span
| |
| class="cmss-10x-x-109"> run should take about 20 seconds.</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-26004x2"><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Dlim </span><span
| |
| class="cmss-10x-x-109">folder</span></li></ol>
| |
| <!--l. 619--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf, colormap jet</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’dlim’, 31, 3e4, 3e4, 1e3, 500 );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">csa( ’dlim’, 31 ); </span><span
| |
| class="cmssi-10x-x-109">% Shows slope-area graph</span></li></ul>
| |
| <!--l. 632--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Notice that the landscape has come close to a state of equilibrium between erosion and</span>
| |
| <span
| |
| class="cmss-10x-x-109">relative uplift. The resulting terrain has about 200 m of relief over a 30 km half-width</span>
| |
| <span
| |
| class="cmss-10x-x-109">mountain range—more Appalachian than Himalayan. Notice that the log-log</span>
| |
| <span
| |
| class="cmss-10x-x-109">slope-area graph shows a straight line, indicating a power-law relationship. This is</span>
| |
| <span
| |
| class="cmss-10x-x-109">exactly to be expected, and we can predict the plot slope and intercept analytically.</span>
| |
| <span
| |
| class="cmss-10x-x-109">Finally, note the points on the upper left of the graph. These “first order” cells, at</span>
| |
| <span
| |
| class="cmss-10x-x-109">about 2500 m</span><sup><span
| |
| class="cmr-8">2</span></sup> <span
| |
| class="cmss-10x-x-109">contributing area, have slopes less than 10%. They represent</span>
| |
| <span
| |
| class="cmss-10x-x-109">embedded channels, not hillslopes, which are too small to resolve at this grid</span>
| |
| <span
| |
| class="cmss-10x-x-109">spacing.</span>
| |
| <!--l. 645--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Now, what happens when we increase the relative uplift rate?</span>
| |
| <!--l. 647--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-26006x1"><span
| |
| class="cmss-10x-x-109">Run the </span><span
| |
| class="cmtt-10x-x-109">dlimC1.in </span><span
| |
| class="cmss-10x-x-109">input file by typing:</span>
| |
| <!--l. 651--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child dlimC1.in</span>
| |
| <!--l. 653--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">This run starts off where the previous one ended, but with a 10</span><span
| |
| class="cmsy-10x-x-109">× </span><span
| |
| class="cmss-10x-x-109">higher rate of</span>
| |
| <span
| |
| class="cmss-10x-x-109">relative uplift.</span></li></ol>
| |
| <!--l. 656--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1)</span>
| |
|
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’dlimC1’, 31, 3e4, 3e4, 1e4, 5000 ); </span><span
| |
| class="cmssi-10x-x-109">% 10</span><span
| |
| class="cmsy-10x-x-109">× </span><span
| |
| class="cmssi-10x-x-109">vertical scale</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2)</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">hold on, csa( ’dlimC1’, 31, ’r.’ ); hold off</span></li></ul>
| |
| <!--l. 669--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Because we are using a slope-linear detachment law, a 10</span><span
| |
| class="cmsy-10x-x-109">× </span><span
| |
| class="cmss-10x-x-109">increase in relative uplift rate leads</span>
| |
| <span
| |
| class="cmss-10x-x-109">to a 10</span><span
| |
| class="cmsy-10x-x-109">× </span><span
| |
| class="cmss-10x-x-109">increase in relief. Notice that the points have shifted upward by a factor of 10 on the</span>
| |
| <span
| |
| class="cmss-10x-x-109">slope-area graph.</span>
| |
| <!--l. 671--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">We still do not see any hillslopes, because the scale of landscape dissection is too fine</span>
| |
| <span
| |
| class="cmss-10x-x-109">for the model to resolve.</span>
| |
| <!--l. 676--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-270007.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 7: Zooming in to the Hillslopes</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 678--><p class="noindent" >
| |
| <!--l. 681--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Next, we will “zoom in” by repeating the </span><span
| |
| class="cmtt-10x-x-109">dlim </span><span
| |
| class="cmss-10x-x-109">run but with a twenty-fold decrease in</span>
| |
| <span
| |
| class="cmss-10x-x-109">domain size and model cell size.</span>
| |
| <!--l. 684--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-27002x1"><span
| |
| class="cmss-10x-x-109">Run the </span><span
| |
| class="cmtt-10x-x-109">dlim</span><span
| |
| class="cmtt-10x-x-109">_small.in </span><span
| |
| class="cmss-10x-x-109">input file by typing:</span>
| |
| <!--l. 688--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child dlim</span><span
| |
| class="cmtt-10x-x-109">_small.in</span>
| |
| <!--l. 690--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">This run is identical to </span><span
| |
| class="cmtt-10x-x-109">dlim </span><span
| |
| class="cmss-10x-x-109">but with a domain of 1.5 by 1.5km and </span><span
| |
| class="cmsy-10x-x-109">~</span><span
| |
| class="cmss-10x-x-109">25m wide</span>
| |
| <span
| |
| class="cmss-10x-x-109">cells, instead of 30x30km and </span><span
| |
| class="cmsy-10x-x-109">~</span><span
| |
| class="cmss-10x-x-109">500m cells.</span></li></ol>
| |
| <!--l. 694--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1)</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’dlim</span><span
| |
| class="cmtt-10x-x-109">_small’, 31, 1.5e3, 1.5e3, 500, 200 );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2)</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">hold on, csa( ’dlim</span><span
| |
| class="cmtt-10x-x-109">_small’, 31, ’g.’ ); hold off</span></li></ul>
| |
| <!--l. 708--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Note how the hillslopes become evident in the topography. In the slope-area plot,</span>
| |
| <span
| |
| class="cmss-10x-x-109">the points seem to continue the trend of the coarser-scale run, but somewhat shifted</span>
| |
| <span
| |
| class="cmss-10x-x-109">upward. Can you guess why they are shifted upward? (The answer is subtle, and lies</span>
| |
|
| |
| <span
| |
| class="cmss-10x-x-109">hidden in </span><span
| |
| class="cmtt-10x-x-109">dlim</span><span
| |
| class="cmtt-10x-x-109">_small2.in</span><span
| |
| class="cmss-10x-x-109">).</span>
| |
| <!--l. 716--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-280007.1"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 8: Knickzones and Transient Response</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 718--><p class="noindent" >
| |
| <!--l. 721--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">For the next exercise, we return to our earlier </span><span
| |
| class="cmtt-10x-x-109">dlimC1 </span><span
| |
| class="cmss-10x-x-109">run and plot a representative</span>
| |
| <span
| |
| class="cmss-10x-x-109">stream profile at different times, to look at how the profile responds to the increased</span>
| |
| <span
| |
| class="cmss-10x-x-109">rate of relative uplift.</span>
| |
| <!--l. 723--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">[d,h,x,y] = cstrmproseries( ’dlimC1’, 10, 15000, 29000 );</span>
| |
| <!--l. 731--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">This command traces the stream profile starting from </span><span
| |
| class="cmmi-10x-x-109">x </span><span
| |
| class="cmr-10x-x-109">= 15 </span><span
| |
| class="cmss-10x-x-109">km, </span><span
| |
| class="cmmi-10x-x-109">y </span><span
| |
| class="cmr-10x-x-109">= 29 </span><span
| |
| class="cmss-10x-x-109">km. It</span>
| |
| <span
| |
| class="cmss-10x-x-109">will plot the first 10 profiles.</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2), clf, plot( x, y )</span>
| |
| <!--l. 737--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">This shows the horizontal trace of the stream course.</span></li></ul>
| |
| <!--l. 741--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">During the period of transient response, the stream profile shows a pronounced</span>
| |
| <span
| |
| class="cmss-10x-x-109">convexity, or knickzone, along the profile. The knickzone marches upstream through</span>
| |
| <span
| |
| class="cmss-10x-x-109">time. This pattern is characteristic of the “stream power” erosion law, which is actually</span>
| |
| <span
| |
| class="cmss-10x-x-109">a form of wave equation.</span>
| |
| <!--l. 750--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">7.2. </span> <a
| |
| id="x1-290007.2"></a><span
| |
| class="cmbx-12">Transport-Limited Models.</span></span>
| |
| <!--l. 752--><p class="noindent" >We next explore the dynamics of landscapes and networks with transport-limited models. One
| |
| caution as we do so: we will assume that channel width is independent of grain size, slope,
| |
| etc.
| |
| <!--l. 754--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-300007.2"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 9: A Pile of Fine Sand</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 756--><p class="noindent" >
| |
| <!--l. 759--><p class="noindent" >
| |
|
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-30002x1"><span
| |
| class="cmss-10x-x-109">In the terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Tlim </span><span
| |
| class="cmss-10x-x-109">folder and run:</span>
| |
| <!--l. 763--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child tlim1.in</span>
| |
| <!--l. 765--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The 1 m.y.</span><span
| |
| class="cmss-10x-x-109"> run should take about 2 minutes.</span>
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-30004x2"><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Tlim </span><span
| |
| class="cmss-10x-x-109">folder</span></li></ol>
| |
| <!--l. 770--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’tlim1’, 21, 3e4, 3e4, 40, 10 );</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">csa( ’tlim1’, 21 ); axis([1e-1 1e3 1e-4 1e-3])</span></li></ul>
| |
| <!--l. 784--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In this run, we are effectively assuming that 0.1 mm sand moves as bed-load, according</span>
| |
| <span
| |
| class="cmss-10x-x-109">to a Meyer-Peter and Mueller-like transport formula. The landscape takes on an</span>
| |
| <span
| |
| class="cmss-10x-x-109">effectively uniform and very shallow gradient, on the order of </span><span
| |
| class="cmr-10x-x-109">3 </span><span
| |
| class="cmsy-10x-x-109">× </span><span
| |
| class="cmr-10x-x-109">10</span><sup><span
| |
| class="cmsy-8">-</span><span
| |
| class="cmr-8">4</span></sup><span
| |
| class="cmss-10x-x-109">.</span>
| |
| <!--l. 789--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-310007.2"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 10: A Pile of Cobbles</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 791--><p class="noindent" >
| |
| <!--l. 794--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Now let’s try the same experiment with 5cm cobbles.</span>
| |
| <!--l. 796--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-31002x1"><span
| |
| class="cmss-10x-x-109">Run:</span>
| |
| <!--l. 800--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child tlim2.in</span>
| |
| <!--l. 802--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The 3 m.y.</span><span
| |
| class="cmss-10x-x-109"> run should take about 2-3 minutes.</span></li></ol>
| |
| <!--l. 805--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’tlim2’, 31, 3e4, 3e4, 1000, 300 );</span>
| |
|
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(2)</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">hold on, csa( ’tlim2’, 31, ’r.’ ); hold off</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">axis([1e-1 1e3 1e-4 1e-1])</span></li></ul>
| |
| <!--l. 821--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Lesson: grain size matters!</span>
| |
| <!--l. 823--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">But let’s remember the caveat that channel width matters too, and we haven’t taken</span>
| |
| <span
| |
| class="cmss-10x-x-109">that into account with these simple runs. Also, Nicole Gasparini’s work (</span><a
| |
| href="#Xgasparini1999downstream"><span
| |
| class="cmssi-10x-x-109">Gasparini</span>
| |
| <span
| |
| class="cmssi-10x-x-109">et</span><span
| |
| class="cmssi-10x-x-109"> al.</span></a><span
| |
| class="cmss-10x-x-109">,</span><span
| |
| class="cmss-10x-x-109"> </span><a
| |
| href="#Xgasparini1999downstream"><span
| |
| class="cmss-10x-x-109">1999</span></a><span
| |
| class="cmss-10x-x-109">,</span><span
| |
| class="cmss-10x-x-109"> </span><a
| |
| href="#Xgasparini2004network"><span
| |
| class="cmss-10x-x-109">2004</span></a><span
| |
| class="cmss-10x-x-109">) tells us that channel concavity is less sensitive to grain size when</span>
| |
| <span
| |
| class="cmss-10x-x-109">there is a mixture of sizes available to the river.</span>
| |
| <!--l. 831--><p class="noindent" >
| |
| <!--l. 833--><p class="noindent" ><span
| |
| class="cmbx-10x-x-109">Optional exercise: </span><span
| |
| class="cmss-10x-x-109">Make a copy of </span><span
| |
| class="cmtt-10x-x-109">tlim2.in </span><span
| |
| class="cmss-10x-x-109">and configure it to re-start from</span>
| |
| <span
| |
| class="cmtt-10x-x-109">tlim2 </span><span
| |
| class="cmss-10x-x-109">but with a higher uplift rate. Use the Matlab script </span><span
| |
| class="cmtt-10x-x-109">cstrmproseries </span><span
| |
| class="cmss-10x-x-109">to</span>
| |
| <span
| |
| class="cmss-10x-x-109">plot fluvial profiles undergoing transient response. How do these compare with the</span>
| |
| <span
| |
| class="cmss-10x-x-109">detachment-limited model?</span>
| |
| <!--l. 840--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">7.3. </span> <a
| |
| id="x1-320007.3"></a><span
| |
| class="cmbx-12">Hybrid Model: Combining Detachment and Transport.</span></span>
| |
| <!--l. 842--><p class="noindent" >Next, we’ll look at a more complex situation with simultaneous erosion and sedimentation, and
| |
| simultaneous detachment-limited and transport-limited behavior. In this case, we use a fluvial
| |
| model in which erosion rate can be limited either by transport capacity or by detachment capacity,
| |
| depending on their relative magnitudes:
| |
| <table
| |
| class="equation"><tr><td><a
| |
| id="x1-32001r19"></a>
| |
| <center class="math-display" >
| |
| <img
| |
| src="child_exercises_nced_aug201223x.png" alt=" { ∑N ∑N
| |
| Qc---ji=1Qsij if Qc---j=i1Qsij< D
| |
| Ei = Λi Λi c
| |
| Dc otherwise
| |
| " class="math-display" ></center></td><td class="equation-label">(19)</td></tr></table>
| |
|
| |
| <!--l. 849--><p class="nopar" >
| |
| <!--l. 851--><p class="noindent" ><span class="subsectionHead"><a
| |
| id="x1-330007.3"></a><span
| |
| class="cmbxti-10x-x-120">Exercise 11: Erosion and Deposition, Together at Last</span><span
| |
| class="cmbx-12">.</span></span>
| |
| <!--l. 853--><p class="noindent" >
| |
| <!--l. 856--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-33002x1"><span
| |
| class="cmss-10x-x-109">In the terminal window, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Hybrid </span><span
| |
| class="cmss-10x-x-109">folder and run:</span>
| |
| <!--l. 860--><p class="noindent" ><span
| |
| class="cmtt-10x-x-109">child erodep1.in</span>
| |
| <!--l. 862--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">The 1 m.y.</span><span
| |
| class="cmss-10x-x-109"> run should take about 5 minutes (but of course you can peek at earlier</span>
| |
| <span
| |
| class="cmss-10x-x-109">time steps while the run is going, by reducing the number of frames in your movie).</span></li></ol>
| |
| <!--l. 868--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">In Matlab, navigate to the </span><span
| |
| class="cmtt-10x-x-109">Hybrid </span><span
| |
| class="cmss-10x-x-109">folder and type:</span>
| |
| <ul class="itemize1">
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">figure(1), clf</span>
| |
| </li>
| |
| <li class="itemize"><span
| |
| class="cmtt-10x-x-109">cmovie( ’erodep1’, 21, 6e4, 6e4, 4000 );</span></li></ul>
| |
| <!--l. 878--><p class="noindent" ><span
| |
| class="cmss-10x-x-109">Here we have a block rising at 1 mm/yr and an adjacent block subsiding at 0.25 mm/yr.</span>
| |
| <span
| |
| class="cmss-10x-x-109">Uplift and subsidence shut down after 500 ky. The subsiding block forms a large lake</span>
| |
| <span
| |
| class="cmss-10x-x-109">that gradually fills in with fan-deltas.</span>
| |
| <!--l. 886--><p class="noindent" ><span class="subsectionHead"><span class="titlemark">7.4. </span> <a
| |
| id="x1-340007.4"></a><span
| |
| class="cmbx-12">Other Sediment-Flux-Dependent Fluvial Models.</span></span>
| |
| <!--l. 888--><p class="noindent" >We won’t take the time to address some of the other models, including
| |
| <ul class="itemize1">
| |
| <li class="itemize">“Under-capacity” models (detachment rate depends on degree to which sediment flux
| |
| falls below transport capacity), and
| |
| </li>
| |
| <li class="itemize">Saltation-abrasion models (detachment rate driven by particle impacts, and limited
| |
| by alluvial shielding of bed)</li></ul>
| |
| <!--l. 895--><p class="noindent" ><a
| |
| href="#Xgasparini2007predictions"><span
| |
| class="cmti-12">Gasparini et</span><span
| |
| class="cmti-12"> al.</span></a> (<a
| |
| href="#Xgasparini2007predictions">2007</a>) explore the behavior of these models with CHILD simulations.
| |
|
| |
| <h3 class="sectionHead"><span class="titlemark">8. </span> <a
| |
| id="x1-350008"></a>Multiple Grain Sizes</h3>
| |
| <!--l. 899--><p class="noindent" >Although we won’t explore the effects of including multiple grain sizes of sediment in transport,
| |
| grain size introduces some interesting issues, including:
| |
| <ul class="itemize1">
| |
| <li class="itemize">Bed armoring and its impact on transport rates
| |
| </li>
| |
| <li class="itemize">Downstream fining
| |
| </li>
| |
| <li class="itemize">Abrasion and lithologic controls</li></ul>
| |
| <h3 class="sectionHead"><span class="titlemark">9. </span> <a
| |
| id="x1-360009"></a>Exotica</h3>
| |
| <!--l. 909--><p class="noindent" >Landscape evolution models include more than diffusion and stream-power models:
| |
| <ul class="itemize1">
| |
| <li class="itemize">Stream meandering in the context of landscape evolution and valley stratigraphy
| |
| (<a
| |
| href="#Xclevis2006simple"><span
| |
| class="cmti-12">Clevis et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xclevis2006simple">2006</a>, a,b).
| |
| </li>
| |
| <li class="itemize">Vegetation,
| |
| including both grass (<a
| |
| href="#Xcollins2004modeling"><span
| |
| class="cmti-12">Collins et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xcollins2004modeling">2004</a>; <a
| |
| href="#Xistanbulluoglu2005vegetation"><span
| |
| class="cmti-12">Istanbulluoglu and Bras</span></a>, <a
| |
| href="#Xistanbulluoglu2005vegetation">2005</a>) and trees
| |
| (<a
| |
| href="#Xlancaster2003effects"><span
| |
| class="cmti-12">Lancaster et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xlancaster2003effects">2003</a>)
| |
| </li>
| |
| <li class="itemize">Alternate forms of mass wasting, including landslides and debris flows (<a
| |
| href="#Xdensmore1998landsliding"><span
| |
| class="cmti-12">Densmore</span>
| |
| <span
| |
| class="cmti-12">et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xdensmore1998landsliding">1998</a>; <a
| |
| href="#Xlancaster2003effects"><span
| |
| class="cmti-12">Lancaster et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xlancaster2003effects">2003</a>; <a
| |
| href="#Xistanbulluoglu2005implications"><span
| |
| class="cmti-12">Istanbulluoglu et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xistanbulluoglu2005implications">2005</a>)
| |
| </li>
| |
| <li class="itemize">Knickpoints, hanging valleys, and plunge pools (<a
| |
| href="#Xflores2006development"><span
| |
| class="cmti-12">Flores-Cervantes</span>
| |
| <span
| |
| class="cmti-12">et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xflores2006development">2006</a>; <a
| |
| href="#Xcrosby2007formation"><span
| |
| class="cmti-12">Crosby et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xcrosby2007formation">2007</a>)
| |
| </li>
| |
| <li class="itemize">Glaciation (<a
| |
| href="#Xherman2006fluvial"><span
| |
| class="cmti-12">Herman and Braun</span></a>, <a
| |
| href="#Xherman2006fluvial">2006</a>; <a
| |
| href="#Xherman2007tectonomorphic"><span
| |
| class="cmti-12">Herman et</span><span
| |
| class="cmti-12"> al.</span></a>, <a
| |
| href="#Xherman2007tectonomorphic">2007</a>; <a
| |
| href="#Xherman2008evolution"><span
| |
| class="cmti-12">Herman and</span>
| |
| <span
| |
| class="cmti-12">Braun</span></a>, <a
| |
| href="#Xherman2008evolution">2008</a>)</li></ul>
| |
| <h3 class="sectionHead"><span class="titlemark">10. </span> <a
| |
| id="x1-3700010"></a>Forecasting or Speculation?</h3>
| |
| <!--l. 924--><p class="noindent" >Some mathematical models in the physical sciences have such firm foundations that they can be
| |
| relied upon to forecast the behavior of the natural world. For example, laws of motion of objects in
| |
| a vacuum are absolutely reliable (as long as their speed is much less than that of light). The
| |
| same can be said for numerical solutions to these equations, provided the solution is
| |
| reasonably accurate. For these kinds of model, the verb “to model” means to calculate
| |
|
| |
| with high reliability what would happen under a particular set of initial and boundary
| |
| conditions.
| |
| <!--l. 926--><p class="noindent" >At the other end of the spectrum, we have mathematical models that are essentially tentative
| |
| hypotheses. Such models are often based on intuition about a physical system, and represent a sort
| |
| of educated guess about the quantitative relationships between things. For example, when
| |
| <a
| |
| href="#Xahnert1976"><span
| |
| class="cmti-12">Ahnert</span></a> (<a
| |
| href="#Xahnert1976">1976</a>) presented his inverse-exponential equation for regolith generation from
| |
| bedrock, he was essentially expressing a conceptual hypothesis in mathematical terms. For
| |
| these models-as-hypotheses, the phrase “to model” means to perform a quantitative
| |
| “what if” experiment, asking the question: what kinds of pattern would I see if my
| |
| hypothesis were correct? Comparing the prediction with observations provides a test of the
| |
| hypothesis.
| |
| <!--l. 928--><p class="noindent" >One can find many models that fall between these extremes. There are models that are based on
| |
| well-known physics, but which are forced to use approximations of unknown accuracy in
| |
| order to solve the governing equations. For example, climate models typically use simple
| |
| parameterization schemes to represent convective mass and energy transport. Then too there are
| |
| models that combine basic physical principles with elements of intuition, empiricism, and
| |
| approximation. Arguably, many sediment-transport laws fall into this category: they are
| |
| based on firm mechanical foundations (the force balance on a sediment grain) but also
| |
| rely on strong approximations of factors like grain geometry, local flow velocity, and so
| |
| on.
| |
| <!--l. 930--><p class="noindent" >By now, it should be obvious that landscape evolution models also fall somewhere between
| |
| the end-member cases of “model as truth” and “model as speculative hypothesis.” As
| |
| we have seen throughout this course, there is a varying degree of experimental and
| |
| observational support for the individual transport, weathering and erosion laws that go into a
| |
| typical landscape model. In that sense, then, these models amount to more than just
| |
| speculation. But equally there is still an element of speculation behind many of the
| |
| process laws used in landscape models. Also, the process laws and algorithms represent a
| |
| significant amount of upscaling in space and (especially) time. For example, the use
| |
| of a steady precipitation rate as a proxy for the natural sequence of flows in a river
| |
| channel represents a major approximation. For these reasons, we believe that three of
| |
| the most important frontiers in landscape evolution research are (1) continuing to test
| |
| individual process laws in the field and lab, (2) testing whole-landscape models using natural
| |
| experiments, and (3) using mathematics, computation and experiments to study how the rates
| |
| of various processes scale upward in time and space, and how these can be effectively
| |
| parameterized.
| |
| <h3 class="sectionHead"><span class="titlemark">11. </span> <a
| |
| id="x1-3800011"></a>Ten Commandments of Landscape Evolution Modeling</h3>
| |
| <!--l. 936--><p class="noindent" >
| |
| <ol class="enumerate1" >
| |
| <li
| |
| class="enumerate" id="x1-38002x1">Thou shalt not use a model without understanding the ingredients therein.
| |
|
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38004x2">Be thou ever mindful of uncertainty.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38006x3">Thou shalt use thy model to develop insight.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38008x4">Thou shalt take delight when thy model surprises thee.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38010x5">Thou shalt kick thy model hard, that it may notice thee (an injunction borrowed
| |
| gratefully from the 10 Climate Modeling Commandments).
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38012x6">Thou shalt diagnose the reasons for thy model’s behavior.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38014x7">Thou shalt conduct sensitivity experiments and “play around.”
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38016x8">Thou shalt use thy model to discover the necessary and sufficient conditions needed
| |
| to explain thy target problem.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38018x9">If thou darest use a model to calculate what happened in your field area in the past,
| |
| thou shalt find a way to test and calibrate it first.
| |
| </li>
| |
| <li
| |
| class="enumerate" id="x1-38020x10">If thou darest to predict future erosion, thou shalt heed the previous commandment
| |
| ten times over (but thou mightest point out to skeptics that a process-based prediction
| |
| is usually better than one based on pure guesswork, provided that commandment #2
| |
| is obeyed).</li></ol>
| |
| <!--l. 949--><p class="noindent" >
| |
|
| |
| <h3 class="sectionHead"><a
| |
| id="x1-3900011"></a>References</h3>
| |
| <!--l. 1--><p class="noindent" >
| |
| <div class="thebibliography">
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xahnert1971general"></a><span class="bibsp">   </span></span>Ahnert, F.
| |
| (1971), Brief description of a comprehensive three-dimensional process-response model of
| |
| landform development, <span
| |
| class="cmti-12">Zeitschrift fur Geomorfologie, Supplementband</span>, <span
| |
| class="cmti-12">25</span>, 29–49.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xahnert1976"></a><span class="bibsp">   </span></span>Ahnert, F. (1976), Brief description of a comprehensive
| |
| three-dimensional process-response model of landform development, <span
| |
| class="cmti-12">Zeitschrift f&uuml;r</span>
| |
| <span
| |
| class="cmti-12">Geomorfologie, Supplementband</span>, <span
| |
| class="cmti-12">25</span>, 29–49.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xamos2007channel"></a><span class="bibsp">   </span></span>Amos, C. B., and D. W. Burbank (2007), Channel width response to differential uplift,
| |
| <span
| |
| class="cmti-12">Journal of Geophysical Research: Earth Surface (2003–2012)</span>, <span
| |
| class="cmti-12">112</span>(F2).
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xattal2008modeling"></a><span class="bibsp">   </span></span>Attal, M., G. E. Tucker, A. C. Whittaker, P. A. Cowie, and G. P. Roberts (2008),
| |
| Modeling fluvial incision and transient landscape evolution: Influence of dynamic channel
| |
| adjustment, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">113</span>, F03,013.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xbeaumont1992erosional"></a><span class="bibsp">   </span></span>Beaumont, C., P. Fullsack, and J. Hamilton (1992), Erosional control of active
| |
| compressional orogens, <span
| |
| class="cmti-12">Thrust tectonics</span>, <span
| |
| class="cmti-12">99</span>, 1–18.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xbirnir2001scaling"></a><span class="bibsp">   </span></span>Birnir, B., T. R. Smith, and G. E. Merchant (2001), The scaling of fluvial landscapes,
| |
| <span
| |
| class="cmti-12">Computers & geosciences</span>, <span
| |
| class="cmti-12">27</span>(10), 1189–1216.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xbraun1997modelling"></a><span class="bibsp">   </span></span>Braun, J., and M. Sambridge (1997), Modelling landscape evolution on geological time
| |
| scales: a new method based on irregular spatial discretization, <span
| |
| class="cmti-12">Basin Research</span>, <span
| |
| class="cmti-12">9</span>, 27–52.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xcarretier2005does"></a><span class="bibsp">   </span></span>Carretier, S., and F. Lucazeau (2005), How does alluvial sedimentation at range fronts
| |
| modify the erosional dynamics of mountain catchments?, <span
| |
| class="cmti-12">Basin Research</span>, <span
| |
| class="cmti-12">17</span>(3), 361–381.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xchase1992fluvial"></a><span class="bibsp">   </span></span>Chase, C. G. (1992), Fluvial landsculpting and the fractal dimension of topography,
| |
| <span
| |
| class="cmti-12">Geomorphology</span>, <span
| |
| class="cmti-12">5</span>, 39–57.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
|
| |
| <a
| |
| id="Xclevis2006simple"></a><span class="bibsp">   </span></span>Clevis, Q., G. Tucker, S. Lancaster, A. Desitter, N. Gasparini, and G. Lock (2006),
| |
| Geoarchaeological simulation of meandering river deposits and settlement distributions: a
| |
| three-dimensional approach, <span
| |
| class="cmti-12">Computers and Geosciences</span>, <span
| |
| class="cmti-12">21</span>(8), 843–874.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xclevis2006geoarchaeological"></a><span class="bibsp">   </span></span>Clevis, Q., G. E. Tucker, G. Lock, S. T. Lancaster, N. Gasparini, A. Desitter, and
| |
| R. L. Bras (2006), Geoarchaeological simulation of meandering river deposits and
| |
| settlement distributions: A three-dimensional approach, <span
| |
| class="cmti-12">Geoarchaeology</span>, <span
| |
| class="cmti-12">21</span>(8), 843–874,
| |
| doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.1002/gea.20142<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xcollins2004modeling"></a><span class="bibsp">   </span></span>Collins, D., R. Bras, and G. Tucker (2004), Modeling the effects of vegetation-erosion
| |
| coupling on landscape evolution, <span
| |
| class="cmti-12">Journal of Geophysical Research—Earth Surface</span>,
| |
| <span
| |
| class="cmti-12">109</span>(F3), doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.1029/2003JF000028<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xcoulthard1996cellular"></a><span class="bibsp">   </span></span>Coulthard, T., M. Kirkby, and M. Macklin (1996), A cellular automaton landscape
| |
| evolution model, in <span
| |
| class="cmti-12">Proceedings of the First International Conference on GeoComputation</span>,
| |
| vol. 1, pp. 248–81.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xcrave2001stochastic"></a><span class="bibsp">   </span></span>Crave, A., and P. Davy (2001), A stochastic ’precipiton’ model for simulating
| |
| erosion/sedimentation dynamics, <span
| |
| class="cmti-12">Computers and Geosciences</span>, <span
| |
| class="cmti-12">27</span>, 815–827.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xcrosby2007formation"></a><span class="bibsp">   </span></span>Crosby, B. T., K. X. Whipple, N. M. Gasparini, and C. W. Wobus (2007), Formation
| |
| of fluvial hanging valleys: Theory and simulation, <span
| |
| class="cmti-12">JOURNAL OF GEOPHYSICAL</span>
| |
| <span
| |
| class="cmti-12">RESEARCH-EARTH SURFACE</span>, <span
| |
| class="cmti-12">112</span>(F3), doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.1029/2006JF000566<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xculling1963soil"></a><span class="bibsp">   </span></span>Culling, W. (1963), Soil creep and the development of hillside slopes, <span
| |
| class="cmti-12">The Journal of</span>
| |
| <span
| |
| class="cmti-12">Geology</span>, pp. 127–161.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xdensmore1998landsliding"></a><span class="bibsp">   </span></span>Densmore, A. L., M. A. Ellis, and R. S. Anderson (1998), Landsliding and the evolution
| |
| of normal-fault-bounded mountains, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">103</span>, 15,203–15,219.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xdibiase2011influence"></a><span class="bibsp">   </span></span>DiBiase, R., and K. Whipple (2011), The influence of erosion thresholds and runoff
| |
| variability on the relationships among topography, climate, and erosion rate, <span
| |
| class="cmti-12">Journal of</span>
| |
| <span
| |
| class="cmti-12">Geophysical Research</span>, <span
| |
| class="cmti-12">116</span>(F4), F04,036.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xduvall2004tectonic"></a><span class="bibsp">   </span></span>Duvall, A., E. Kirby, and D. Burbank (2004), Tectonic and lithologic controls on bedrock
| |
|
| |
| channel profiles and processes in coastal california, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>,
| |
| <span
| |
| class="cmti-12">109</span>(F3), F03,002.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xfagherazzi2002implicit"></a><span class="bibsp">   </span></span>Fagherazzi, S., A. Howard, and P. Wiberg (2002), An implicit finite difference method
| |
| for drainage basin evolution, <span
| |
| class="cmti-12">Water Resources Research</span>, <span
| |
| class="cmti-12">38</span>(7), 21.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xflores2006development"></a><span class="bibsp">   </span></span>Flores-Cervantes, H., E. Istanbulluoglu, and R. L. Bras (2006), Development of gullies
| |
| on the landscape: A model of headcut retreat resulting from plunge pool erosion, <span
| |
| class="cmti-12">Journal</span>
| |
| <span
| |
| class="cmti-12">of Geophysical Research</span>, <span
| |
| class="cmti-12">111</span>, F01,010.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xgarcia2002interplay"></a><span class="bibsp">   </span></span>Garcia-Castellanos, D. (2002), Interplay between lithospheric flexure and river transport
| |
| in foreland basins, <span
| |
| class="cmti-12">Basin Research</span>, <span
| |
| class="cmti-12">14</span>(2), 89–104.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xgasparini1999downstream"></a><span class="bibsp">   </span></span>Gasparini, N., G. Tucker, and R. Bras (1999), Downstream fining through selective
| |
| particle sorting in an equilibrium drainage network, <span
| |
| class="cmti-12">Geology</span>, <span
| |
| class="cmti-12">27</span>(12), 1079.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xgasparini2004network"></a><span class="bibsp">   </span></span>Gasparini, N., G. Tucker, and R. Bras (2004), Network-scale dynamics of grain-size
| |
| sorting: Implications for downstream fining, stream-profile concavity, and drainage basin
| |
| morphology, <span
| |
| class="cmti-12">EARTH SURFACE PROCESSES AND LANDFORMS</span>, <span
| |
| class="cmti-12">29</span>(4), 401–421,
| |
| doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.1002/esp.1031<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xgasparini2007predictions"></a><span class="bibsp">   </span></span>Gasparini, N. M., K. X. Whipple, and R. L. Bras (2007), Predictions of steady state
| |
| and transient landscape morphology using sediment-flux-dependent river incision models,
| |
| <span
| |
| class="cmti-12">JOURNAL OF GEOPHYSICAL RESEARCH-EARTH SURFACE</span>, <span
| |
| class="cmti-12">112</span>(F3), doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.
| |
| 1029/2006JF000567<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xgilbert1877report"></a><span class="bibsp">   </span></span>Gilbert, G. (1877), Report on the geology of the Henry Mountains: US Geog. and Geol,
| |
| <span
| |
| class="cmti-12">Survey, Rocky Mtn. Region</span>, <span
| |
| class="cmti-12">160</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xherman2006fluvial"></a><span class="bibsp">   </span></span>Herman, F., and J. Braun (2006), Fluvial response to horizontal shortening and
| |
| glaciations: a study in the Southern Alps of New Zealand, <span
| |
| class="cmti-12">Journal of Geophysical</span>
| |
| <span
| |
| class="cmti-12">Research-Earth Surface</span>, <span
| |
| class="cmti-12">111</span>(F1), F01,008.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xherman2008evolution"></a><span class="bibsp">   </span></span>Herman, F., and J. Braun (2008), Evolution of the glacial landscape of the Southern
| |
| Alps of New Zealand: Insights from a glacial erosion model, <span
| |
| class="cmti-12">Journal of Geophysical</span>
| |
|
| |
| <span
| |
| class="cmti-12">Research-Earth Surface</span>, <span
| |
| class="cmti-12">113</span>(F2), F02,009.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xherman2007tectonomorphic"></a><span class="bibsp">   </span></span>Herman, F., J. Braun, and W. Dunlap (2007), Tectonomorphic scenarios in the Southern
| |
| Alps of New Zealand, <span
| |
| class="cmti-12">Journal of Geophysical Research-Solid Earth</span>, <span
| |
| class="cmti-12">112</span>(B4), B04,201.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xhoward1971simulation"></a><span class="bibsp">   </span></span>Howard, A. (1971), Simulation model of stream capture, <span
| |
| class="cmti-12">Geological Society of America</span>
| |
| <span
| |
| class="cmti-12">Bulletin</span>, <span
| |
| class="cmti-12">82</span>(5), 1355–1376.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xhoward1994detachment"></a><span class="bibsp">   </span></span>Howard, A. D. (1994), A detachment-limited model of drainage basin evolution, <span
| |
| class="cmti-12">Water</span>
| |
| <span
| |
| class="cmti-12">Resources Research</span>, <span
| |
| class="cmti-12">30</span>(7), 2261–2285.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xhuang2006evaluation"></a><span class="bibsp">   </span></span>Huang, X., and J. Niemann (2006), An evaluation of the geomorphically effective event
| |
| for fluvial processes over long periods, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">111</span>(F3), F03,015.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xistanbulluoglu2005vegetation"></a><span class="bibsp">   </span></span>Istanbulluoglu, E., and R. L. Bras (2005), Vegetation-modulated landscape evolution:
| |
| Effects of vegetation on landscape processes, drainage density, and topography, <span
| |
| class="cmti-12">Journal of</span>
| |
| <span
| |
| class="cmti-12">Geophysical Research</span>, <span
| |
| class="cmti-12">110</span>, F02,012.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xistanbulluoglu2005implications"></a><span class="bibsp">   </span></span>Istanbulluoglu, E., R. L. Bras, H. Flores-Cervantes, and G. E. Tucker (2005),
| |
| Implications of bank failures and fluvial erosion for gully development: Field observations
| |
| and modeling, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">110</span>, F01,014.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xkirkby1971hillslope"></a><span class="bibsp">   </span></span>Kirkby, M. (1971), Hillslope process-response models based on the continuity equation,
| |
| <span
| |
| class="cmti-12">Inst. Br. Geogr. Spec. Publ</span>, <span
| |
| class="cmti-12">3</span>, 15–30.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xlancaster2003effects"></a><span class="bibsp">   </span></span>Lancaster, S., S. Hayes, and G. Grant (2003), Effects of wood on debris flow runout in
| |
| small mountain watersheds, <span
| |
| class="cmti-12">Water Resources Research</span>, <span
| |
| class="cmti-12">39</span>(6), 1168.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xlave2001fluvial"></a><span class="bibsp">   </span></span>LavÈ, J., and J. Avouac (2001), Fluvial incision and tectonic uplift across the Himalayas
| |
| of central Nepal, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">106</span>(B11), 26,561.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xmitas1998distributed"></a><span class="bibsp">   </span></span>Mitas, L., and H. Mitasova (1998), Distributed soil erosion simulation for effective erosion
| |
| prevention, <span
| |
| class="cmti-12">Water Resources Research</span>, <span
| |
| class="cmti-12">34</span>(3), 505–516.
| |
| </p>
| |
|
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xmurray1994cellular"></a><span class="bibsp">   </span></span>Murray, A., and C. Paola (1994), A cellular model of braided rivers, <span
| |
| class="cmti-12">Nature</span>, <span
| |
| class="cmti-12">371</span>(6492),
| |
| 54–57.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xperron2011numerical"></a><span class="bibsp">   </span></span>Perron, J. (2011), Numerical methods for nonlinear hillslope transport laws, <span
| |
| class="cmti-12">Journal of</span>
| |
| <span
| |
| class="cmti-12">Geophysical Research</span>, <span
| |
| class="cmti-12">116</span>(F2), F02,021.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xpress2007numerical"></a><span class="bibsp">   </span></span>Press, W., S. Teukolsky, W. Vetterling, and B. Flannery (2007), <span
| |
| class="cmti-12">Numerical recipes: the</span>
| |
| <span
| |
| class="cmti-12">art of scientific computing</span>, Cambridge Univ Pr.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xschauble2008gis"></a><span class="bibsp">   </span></span>Schauble, H., O. Marinoni, and M. Hinderer (2008), A GIS-based method to calculate
| |
| flow accumulation by considering dams and their specific operation time, <span
| |
| class="cmti-12">Computers and</span>
| |
| <span
| |
| class="cmti-12">Geosciences</span>, <span
| |
| class="cmti-12">34</span>(6), 635–646.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xschoorl2002modeling"></a><span class="bibsp">   </span></span>Schoorl, J., A. Veldkamp, and J. Bouma (2002), Modeling water and soil redistribution
| |
| in a dynamic landscape context, <span
| |
| class="cmti-12">Soil Science Society of America Journal</span>, <span
| |
| class="cmti-12">66</span>(5), 1610.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xsnyder2003importance"></a><span class="bibsp">   </span></span>Snyder, N., K. Whipple, G. Tucker, and D. Merritts (2003), Importance of a stochastic
| |
| distribution of floods and erosion thresholds in the bedrock river incision problem, <span
| |
| class="cmti-12">Journal</span>
| |
| <span
| |
| class="cmti-12">of Geophysical Research</span>, <span
| |
| class="cmti-12">108</span>(B2), 2117.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xsolyom2004effect"></a><span class="bibsp">   </span></span>SÛlyom, P., and G. Tucker (2004), Effect of limited storm duration on landscape
| |
| evolution, drainage basin geometry, and hydrograph shapes, <span
| |
| class="cmti-12">Journal of Geophysical</span>
| |
| <span
| |
| class="cmti-12">Research</span>, <span
| |
| class="cmti-12">109</span>, 13.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker2004drainage"></a><span class="bibsp">   </span></span>Tucker, G. (2004), Drainage basin sensitivity to tectonic and climatic forcing:
| |
| Implications of a stochastic model for the role of entrainment and erosion thresholds, <span
| |
| class="cmti-12">Earth</span>
| |
| <span
| |
| class="cmti-12">Surface Processes and Landforms</span>, <span
| |
| class="cmti-12">29</span>(2), 185–205, doi:<span
| |
| class="cmsy-10x-x-120">{</span>10.1002/esp.1020<span
| |
| class="cmsy-10x-x-120">}</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker2000stochastic"></a><span class="bibsp">   </span></span>Tucker, G. E., and R. L. Bras (2000), A stochastic approach to modeling the role
| |
| of rainfall variability in drainage basin evolution, <span
| |
| class="cmti-12">Water Resources Research</span>, <span
| |
| class="cmti-12">36</span>(7),
| |
| 1953–1964.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker2010modelling"></a><span class="bibsp">   </span></span>Tucker, G. E., and G. R. Hancock (2010), Modelling landscape evolution, <span
| |
| class="cmti-12">Earth Surface</span>
| |
| <span
| |
| class="cmti-12">Processes and Landforms</span>, <span
| |
| class="cmti-12">46</span>, 28–50.
| |
|
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker1994erosional"></a><span class="bibsp">   </span></span>Tucker, G. E., and R. L. Slingerland (1994), Erosional dynamics, flexural isostasy, and
| |
| long-lived escarpments: A numerical modeling study, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">99</span>,
| |
| 12,229–12,243.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker2001object"></a><span class="bibsp">   </span></span>Tucker, G. E., S. T. Lancaster, N. M. Gasparini, R. L. Bras, and S. M. Rybarczyk
| |
| (2001a), An object-oriented framework for hydrologic and geomorphic modeling using
| |
| triangular irregular networks, <span
| |
| class="cmti-12">Computers and Geosciences</span>, <span
| |
| class="cmti-12">27</span>, 959–973.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xtucker2001channel"></a><span class="bibsp">   </span></span>Tucker, G. E., S. T. Lancaster, N. M. Gasparini, and R. L. Bras (2001b), The
| |
| Channel-Hillslope Integrated Landscape Development Model (CHILD), in <span
| |
| class="cmti-12">Landscape</span>
| |
| <span
| |
| class="cmti-12">Erosion and Evolution Modeling</span>, edited by R. S. Harmon and W. W. Doe, pp. 349–388,
| |
| Kluwer Press, Dordrecht.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xturowski2009response"></a><span class="bibsp">   </span></span>Turowski, J. M., D. Lague, and N. Hovius (2009), Response of bedrock channel width
| |
| to tectonic forcing: Insights from a numerical model, theoretical considerations, and
| |
| comparison with field data, <span
| |
| class="cmti-12">Journal of Geophysical Research: Earth Surface (2003–2012)</span>,
| |
| <span
| |
| class="cmti-12">114</span>(F3).
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xwhipple1999dynamics"></a><span class="bibsp">   </span></span>Whipple, K. X., and G. E. Tucker (1999), Dynamics of the stream-power river incision
| |
| model: Implications for height limits of mountain ranges, landscape response timescales,
| |
| and research needs, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">104</span>, 17,661–17,674.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xwhipple2002implications"></a><span class="bibsp">   </span></span>Whipple, K. X., and G. E. Tucker (2002), Implications of sediment-flux-dependent
| |
| river incision models for landscape evolution, <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">107</span>, doi
| |
| 10.1029/2000JB000044.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xwhittaker2007bedrock"></a><span class="bibsp">   </span></span>Whittaker, A., P. Cowie, M. Attal, G. Tucker, and G. Roberts (2007), Bedrock channel
| |
| adjustment to tectonic forcing: Implications for predicting river incision rates, <span
| |
| class="cmti-12">Geology</span>,
| |
| <span
| |
| class="cmti-12">35</span>(2), 103.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xwillgoose1991coupled"></a><span class="bibsp">   </span></span>Willgoose, G., R. L. Bras, and I. Rodriguez-Iturbe (1991), A coupled channel network
| |
| growth and hillslope evolution model, 1, theory, <span
| |
| class="cmti-12">Water Resources Research</span>, <span
| |
| class="cmti-12">27</span>(7),
| |
| 1671–1684.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
|
| |
| <a
| |
| id="Xwobus2006self"></a><span class="bibsp">   </span></span>Wobus, C., G. Tucker, and R. Anderson (2006), Self-formed bedrock channels,
| |
| <span
| |
| class="cmti-12">Geophys.</span><span
| |
| class="cmti-12"> Res.</span><span
| |
| class="cmti-12"> Lett</span>, <span
| |
| class="cmti-12">33</span>, 1–6.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xwobus2008modeling"></a><span class="bibsp">   </span></span>Wobus, C., J. Kean, G. Tucker, and R. Anderson (2008), Modeling the evolution of
| |
| channel shape: Balancing computational efficiency with hydraulic fidelity, <span
| |
| class="cmti-12">Journal of</span>
| |
| <span
| |
| class="cmti-12">Geophysical Research</span>, <span
| |
| class="cmti-12">113</span>.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xyanites2010controls"></a><span class="bibsp">   </span></span>Yanites, B., and G. Tucker (2010), Controls and limits on bedrock channel geometry,
| |
| <span
| |
| class="cmti-12">Journal of Geophysical Research</span>, <span
| |
| class="cmti-12">115</span>(F4), F04,019.
| |
| </p>
| |
| <p class="bibitem" ><span class="biblabel">
| |
| <a
| |
| id="Xyanites2010incision"></a><span class="bibsp">   </span></span>Yanites, B., G. Tucker, K. Mueller, Y. Chen, T. Wilcox, S. Huang, and K. Shi (2010),
| |
| Incision and channel morphology across active structures along the Peikang River, central
| |
| Taiwan: Implications for the importance of channel width, <span
| |
| class="cmti-12">Geological Society of America</span>
| |
| <span
| |
| class="cmti-12">Bulletin</span>, <span
| |
| class="cmti-12">122</span>(7-8), 1192.
| |
| </p>
| |
| </div>
| |
|
| |
| </body></html>
| |
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