An Integrated Hillslope and Channel Evolution Model as an...

An Integrated Hillslope and Channel Evolution Model as anInvestigation and Prediction Tool

Year 2 annual report

DACA88-95-R-0020

Prepared by

Greg Tucker

Nicole Gasparini

Stephen Lancaster

Rafael Bras, PI

Department of Civil and Environmental Engineering

Massachusetts Institute of Technology

Cambridge, MA

August, 1997

An Integrated Hillslope and Channel Evolution Model as anInvestigation and Prediction Tool

Year 2 Annual Report

Introduction ............................................................................................................................... 1-1Overview of Model Components ............................................................................................. 2-1

Grid Elements ................................................................................................................. 2-3Storm Size, Duration, and Frequency ............................................................................. 2-5Flow Routing and Runoff ............................................................................................... 2-6Bedrock Weathering (Regolith Production) ................................................................... 2-7Hillslope Sediment Transport: Continuous Processes .................................................... 2-7Hillslope Sediment Transport: Mass Movement ............................................................ 2-8Stream Erosion and Deposition ...................................................................................... 2-8Grain-Size Sorting ........................................................................................................ 2-10Lateral Stream Channel Erosion (Meandering) ............................................................ 2-10Vegetation .....................................................................................................................2-10

Modeling Landscape Evolution Using an Adaptive Irregular Simulation Mesh ................ 3-1Grid Elements and Data Structures ................................................................................. 3-3Drainage Networks on a Triangulated Irregular Mesh ................................................... 3-7Numerical Algorithms .................................................................................................... 3-8Variable Resolution for Modeling River Meandering .................................................. 3-10

A Stochastic Approach to Modeling Drainage Basin Evolution ........................................... 4-1Model Description .......................................................................................................... 4-2Sensitivity of Erosion Rate to Rainfall Variability ......................................................... 4-5Morphologic Consequences of Rainfall Variability: Numerical Example ..................... 4-9

Dynamics of Vegetation and Runoff Erosion ......................................................................... 5-1Model of Vegetation and Erosion ................................................................................... 5-2Characteristic Form Profiles ........................................................................................... 5-6Numerical Examples ..................................................................................................... 5-11

Understanding the Interactions of Multiple Grain Sizes ...................................................... 6-1Meandering: A Simple Model ................................................................................................. A-1

CHAPTER 1 Introduction

rob-uc-tent-

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iteult-er-

eis atate

Understanding the dynamics of landscape evolution is a challenging plem, for two reasons. First, the processes involved are inherently destrtive, and therefore the geologic record of landscape development is offragmentary. Second, the sculpture of terrain involves a fascinating bucomplex set of interacting nonlinear processes. While challenging, however, the problem is not intractable. Information on landscape history istill preserved in the form of topography itself, and often also in the formassociated sedimentary deposits such as alluvial valley fills. And despthe complexity of geomorphic processes and their interactions, the resant landforms often exhibit an underlying similarity associated with diffent geologic and climatic settings.

One of the important challenges in modern geomorphic research is toinvestigate landscape dynamics by developing simulation models thatincorporate the process interactions we observe in nature. This reportdescribes progress to date on the development of one such model. ThChannel-Hillslope Integrated Landscape Development model (CHILD) descendant of earlier modeling efforts at MIT and at the Pennsylvania SUniversity, and it incorporates many of the important processes not

Channel Hillslope Integrated Landscape Development Model 1-1

Introduction

1-2

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included in previous models. Its primary near-term purpose is to serve as a quative tool for investigating the impact of Quaternary-scale climate change on hillslope, channel, and floodplain evolution in non-glaciated drainage basins. This rfocuses on the theory and implementation of the model itself, and on its applicato several theoretical problems that are closely related to the problem of geomoresponses to environmental change.

The CHILD model is designed to simulate the evolution of fluvially-dominated lascapes formed chiefly by physical erosion (thus, it does not include glacial erosikarst development, for example). It simulates the interaction of two general typprocess: “fluvial” processes, a category which encompasses erosion or depositrunoff cascading across the landscape (including slope wash and channel anderosion), and “hillslope” processes, which includes weathering, soil creep, and slope transport processes. The computer code is written to maximize flexibility modularity, so that the user is free to formulate the problem in a way that is suito the problem at hand, rather than being dictated by the software itself. To thisthe model provides a number of different options for activating or deactivating dferent processes and/or different formulations of the same process.

The various components in the model are diagrammed in Figure 1-1. Those shogray are planned components that have not yet been fully implemented as of thwriting. In some cases, prototypes of process modules have been developed wthe framework of the fixed-grid GOLEM model but not yet adapted to the variabmesh data structure used in CHILD. Some of the important improvements overvious models include:

• Adaptive, irregular mesh: the model uses an irregular finite-difference griddinmethod based on the model of Braun and Sambridge (1997) to represent thescape surface. Use of an irregular simulation mesh allows for the incorporatiolateral stream erosion and makes it possible to represent different parts of thlandscape at different spatial resolutions. It also eliminates some of the griddartifacts associated with fixed-grid methods, and opens up the possibility of pling the model with three-dimensional tectonic deformation models (e.g., stslip or thrust faulting). Implementation of the irregular mesh is described in Cter 3.

Channel Hillslope Integrated Landscape Development Model

LANDSCAPE STATEVARIABLES:

Elevation z(x,y,t)Drainage area A(x,y,t)Surface runoff Q(x,y,t)Regolith thickness C(x,y,t)Percent gravel D%(x,y,t)Vegetation cover V(x,y,t)

RUNOFFAND FLOWROUTING

CLIMATEMODULE

STORMGENERATOR

FLUVIALEROSION &

DEPOSITION

SELECTIVEGRAIN SIZETRANSPORT

LATERALEROSION

(MEANDERING)

MESHUPDATER

(move, add,delete points)

VEGETATION

WEATHERING

HILLSLOPETRANSPORT

(soil creep,landsliding, etc.)

FIGURE 1-1. Schematic illustration of the state variables and process modules in the CHILDmodel. Boxes shown in gray are still under development.


Introduction

1-4

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pro-) areandeen

ed

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of the-iptione sec- differ-cuseseoryuid-

• Stochastic rainfall forcing: Previous models have always modeled sediment port by using a single “effective” rainfall or runoff rate that represents a geomphic average. CHILD relaxes that assumption by providing the option ofstochastic rainfall input. The stochastic rainfall model is described further inChapter 4.

• Stream meandering: this is the first landscape evolution model in which the cesses of vertical stream erosion and lateral channel migration (meanderingcoupled. Meandering is simulated using the model developed by Lancaster others (in prep) (a copy of which appears in Appendix A). The interface betwCHILD and the meander model is currently under development; it is discussbriefly in Chapter 3.

• Multiple sediment sizes: A model of mixed-size sediment transport has beendeveloped within the framework of the GOLEM model, and is currently beingadapted for incorporation into CHILD. The multi-size model is discussed inChapter 6.

• Vegetation: CHILD provides a module to simulate the interaction of vegetatiogrowth and runoff erosion. This module simulates the effect of vegetation onface resistance to overland flow, vegetation removal due to runoff erosion, anvegetation regrowth during interstorm periods. The vegetation model is discusion further in Chapter 5.

The CHILD code is written in C++ in order to exploit the advantages of an objeoriented programming language.

This report is divided into two parts. The first part describes the present versionthe model and its implementation. Chapter 2 presents an overview of the basicory and a summary of the different model components. Chapter 3 gives a descrof the algorithms and data structures used to implement the irregular mesh. Thond part of the report consists of a three short chapters that address the role ofent processes incorporated in the model. Essentially, each of these chapters foon the role of one process, and includes both a description of the underlying thand a sensitivity analysis. These chapters are built around the following set of ging questions:


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ro-

differ-

hat fieldt pre- they

• What is the role of rainfall variability in controlling drainage basin morphologand evolution? (Chapter 4)

• What is the nature of the interaction between vegetation growth and runoff esion in catchment evolution? (Chapter 5)

• How do interactions between hillslope and channel processes and between ent branches of a drainage network influence spatial variations in grain size?(Chapter 6)

We do not expect the model to provide definitive answers to these questions. Wthe model can do, however, is provide insights that can be further tested throughstudies. The sensitivity analyses described in these chapters are also importanrequisites to calibrating and applying the model to a specific field site, becauseprovide a basis for understanding the model’s behavior under idealized cases.


Introduction

1-6

CHAPTER 2 Overview of ModelComponents

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The CHILD model simulates landscape evolution by tracking the passaof water and sediment across an irregular lattice of points that represethe landscape surface. Figure 2-1 shows a typical simulation, highlightthe drainage networks that form naturally when converging flow excavavalleys and leads to further flow convergence. The model tracks four bstate variables that determine the depth of erosion or deposition at eacpoint during a given iteration: elevation, slope, drainage area, and surfrunoff (Figure 1-1). In addition, at the user’s option the model can trackseveral additional state variables, including regolith thickness (depth tobedrock), fractional vegetation cover, percent gravel in the active surfasediment layer, and the number, thickness, and composition of sedimelayers previously deposited at a given point.1

In generic mathematical terms, the model can be expressed by the foling equation:

1. At this writing, the module for selective transport of different sediment sizes is currently undevelopment, as noted below.


Overview of Model Components

en-w-

,

(EQ 1)

wherez is surface elevation,t is time, and terms in brackets [] denote optional depdencies. The first term,U(), represents tectonic uplift (or equivalently, baselevel loering), which may vary in time and space. The second term,F(), represents runofferosion, and is a function of the runoff rate,Q, surface slope in the direction of flowS, and optionally also of sediment flux,Qs, regolith thickness,C, vegetation cover,V,and percent gravel in the substrate,D. No distinction is made between erosion byoverland flow and by fully channelized flow. The surface runoff rate,Q, is a function

FIGURE 2-1. Drainage basin simulated using the CHILD model. (a) Perspective view oflandscape. (For graphical display purposes, the mesh has been interpolated onto a regulargrid. Visualization is from the SG3D module of GRASS.) (b) Plan view of drainage networkand irregular mesh. The catchment outline is that of the Forsyth Creek watershed, FortRiley, Kansas, and represents an area of about 11.5 km2.

(a) (b)

t∂∂z

U x y t, ,[ ]( ) F Q S Qs C V D, , ,[ ], ,( )– H z( )–=


Grid Elements

od- runoff

mpact,sg

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eled. formelow.

iefid 2-

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of drainage area,A, and optionally of slope,S (which can influence the likelihood ofsurface saturation and therefore of runoff production). Surface runoff can be meled using a series of random storm events, or it can be modeled as a constantrate that represents an average effective geomorphic event. The third term,H(), rep-resents sediment transport by hillslope processes such as soil creep, raindrop iand landsliding. In the present version of the model, hillslope transport dependsolely on local topography; however, future versions may incorporate landslidinand its dependence on soil pore pressure.

To compute the effects of these different processes, the model iterates throughseries of discrete storm events and interstorm periods. These storm events canmodeled stochastically, with the intensity and duration of each chosen at randofrom a distribution, or they may be uniform, in which case the storm properties resent a geomorphically “average” event (the assumption made in most previoumodels [e.g., Ahnert, 1976; Kirkby, 1987; Willgoose et al., 1991; Howard, 1994Tucker and Slingerland, 1994; Moglen and Bras, 1995]).

The model provides flexibility in the way that each of the process terms are modImplementation of the various processes within the model is diagrammed in theof a flow chart in Figure 2-2. The component process models are summarized bFurther discussion of many of these components is provided in later chapters.

Grid Elements

The irregular grid used in the model is described in detail in Chapter 3, but a broverview now will help to clarify some of the concepts that follow. The model grconsists of a set of pointsN that are connected to form a mesh of triangles (Figure3). Elevation, drainage area, and other state variables are computed at the poirather than within the triangles (in other words, the model uses a finite-differencrather than a finite-element approach). Points (or nodes) are connected using tDelaunay triangulation, which is the unique set of triangles that connect a givenof points in such a way that a circle passing through the three points in any triawill contain no other points. Each nodeNi is associated with a surrounding VoronoRegion (or Voronoi Cell). The Voronoi Region for a nodeNi is the region within



INITIALIZE

UPDATEMESH1

GENERATENEW STORM2

COMPUTEEDGE SLOPES

DRAINAGEDIRECTIONS

DRAINAGEAREA3

SURFACEFLOW3

FIND OUTLETSFOR CLOSED

DEPRESSIONS4

ADD NODES TOMEANDERINGCHANNELS5

STREAM EROSION/DEPOSITION

DURING STORM(INCL. EROSION OF

VEG. COVER)

CHANNELMEANDERING6

INTERSTORMHILLSLOPE

TRANSPORT

BASELEVELCHANGE

OUTPUT7

DONE?

END

FIGURE 2-2. Flow chartshowing the sequence ofcomputations in themodel.

Notes:

1. Only on first iterationor if mesh haschanged during theprevious iteration.

2. If option for stochasticstorms is selected;otherwise, meanstorm properties areused.

3. If at least one flowdirection haschanged, or if themesh has beenupdated.

4. If the option for lake-filling is selected; oth-erwise, flow entering aclosed depression isassumed to evapo-rate.

5. If the meanderingoption is selected andhigher resolution isneeded.

6. If the meanderingoption is selected.

7. Only at selected out-put intervals.

VEGETATIONREGROWTH

yes

no


Storm Size, Duration, and Frequency

.acent

with a

g then. Inghouts (see

ing

which any arbitrary pointQ would be closer toNi than to any other node on the gridThe boundaries between Voronoi Cells are lines of equal distance between adjgrid points. Each Voronoi Cell has surface areaAv.

Storm Size, Duration, and Frequency

Each model iteration represents a storm event. Each storm event is associatedrainfall (or runoff) intensity,P, a duration,Td, and an interstorm period before thenext event,Ti. These parameters may be chosen at random for each storm usinmodel of Eagleson (1978), or they may be held constant throughout a simulatioeither case, storms are approximated as having constant rainfall intensity throutheir duration, and the same assumption is applied to the resulting hydrographbelow).

For variable storms, the storm properties are chosen at random from the followdistributions:

Rainfall (runoff) intensity (EQ 2)

Storm duration (EQ 3)

Points

Edge

Voronoi Cell

FIGURE 2-3. Schematic illustration of model grid components.

(nodes)

f P( ) 1P--- P

P---–

exp=

f Td( ) 1

Td

------Td

Td

------–

exp=



pec-uring

itse. If atheran

ys.

’s

n-s-

a-w, the

Interstorm period (EQ 4)

where are mean storm intensity, duration, and interstorm period, restively. If stochastic storm generation is not used, the mean values are applied deach iteration (in which case the duration of each iteration is ).

Flow Routing and Runoff

Surface flow collected at a point on the grid is routed downslope toward one ofadjacent neighbor nodes, following the edge that has the steepest downhill sloppit occurs on the grid, with no downhill route away from a given node, water is eiassumed to evaporate at that point, or a lake-filling algorithm is invoked to find outlet for the closed depression (see Chapter 3).

The local contribution from rainfall at a node can be computed in one of two waIn the first method, the rainfall associated with a nodeNi is equal to the effective run-off rate times the node’s Voronoi area,Av. Alternatively, flow can be collected oneach triangular element and routed downslope toward the lowest of the trianglethree vertices.

The drainage area,A, for a node is the sum of the area of all Voronoi cells (or triagles, if the second method is used) that contribute flow to that node. Surface dicharge is computed from drainage area in one of the following ways:

(1) Hortonian (infiltration-excess) runoff: Runoff production (rainfall minus infiltrtion) is assumed to be uniform across the landscape. Assuming steady-state flosurface discharge at any point is equal to

, (EQ 5)

whereIc is infiltration capacity (Q = 0 if P < Ic).

f Ti( ) 1

Ti

-----Ti

Ti

-----–

exp=

P Ts andTi, ,

Ts Ti+

Q P Ic–( )A=


Bedrock Weathering (Regolith Production)

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r 3.

(2) Saturation-excess runoff: An option is currently under development for coming direct runoff from saturated areas, with saturation dependent on position inlandscape, groundwater flux rate, and elapsed time since the last storm.

Bedrock Weathering (Regolith Production)

The model allows for two basic types of material, bedrock and regolith. The terregolith is used to represent any disaggregated sediment material, and includecolluvium, and stream sediment. Conversion of bedrock to regolith by weatheriprocesses is assumed to take place at the bedrock-regolith contact. The rate oversion, expressed in units of depth per unit time, depends on the thickness of regolith mantle,C, according to the exponential function

, (EQ 6)

wheret is time,kw is the rate of bedrock-to-regolith conversion whenC=0, andC0 isa parameter that controls the rate of decrease in weathering rate with increasinregolith thickness. Note that for purposes of computational efficiency, equation (only computed when the model is run in “bedrock-alluvial” mode (see “Stream sion and Deposition” on page 8).

Hillslope Sediment Transport: Continuous Processes

Sediment transport by “continuous” hillslope processes such as soil creep anddrop impact is modeled using the well-known geomorphic diffusion equation (eCulling, 1960),

. (EQ 7)

Numerical solution of equation (7) on an irregular mesh is discussed in Chapte

t∂∂C

weathering

kwCC0------–

exp=

t∂∂z

creep

kdx2

2

∂∂ z

y2

2

∂∂ z+

=



nd local

repre-

ded

cess

-

Hillslope Sediment Transport: Mass Movement

A module for landsliding presently exists within the GOLEM model, and will beincorporated into CHILD in a future update.

Stream Erosion and Deposition

The model distinguishes between detachment of material from a stream bed atransport of the detached material. The maximum detachment rate depends onslope and discharge according to

, (EQ 8)

whereDb is the detachment (erosion) rate,θcb is a threshold, andkb, kt, mb, nb, andpb are parameters. Note that with suitably chosen parameters, equation (8) cansent excess shear stress, with (τ = bed shear stress,τcb = criticalshear stress for erosion).

The transport capacity for detached sediment material (referred to hereafter as“regolith,” a term which includes both sediment weathered from bedrock and erofrom the channel bed) is

, (EQ 9)

whereCs is transport capacity,W is channel width, andkf, kt, mf, nf, andpf areparameters. As with equation (8), equation (9) can be expressed in terms of exbed shear stress using suitably-chosen values forkf, mf, andnf. In the present versionof the model, channel width is computed using the empirical relationship

(the constantkcw is assumed to be absorbed into the transport coefficientkf).

Db kb ktQmbS

nb θcb–( )pb

=

Db kb τ τcb–( )pb=

Cs Wkf ktQmf S

nf θc–( )pf

=

W kcwQmcw=


Stream Erosion and Deposition

d (9),es sep-

ntach-

erly-esentSeidlckert has

hetheirf

ow.

ment-helf (1)golithacity

Two end-member cases and one intermediate case arise from equations (8) anand special subroutines are provided in the model to handle each of these casarately:

Detachment-limited:If the sediment transport capacity is everywhere greater thathe sediment flux, the rate of stream erosion is simply equal to the maximum dement rate,

, (EQ 10)

wherezb represents elevation of the channel bed above a datum within the unding rock column. This formulation has been used in a number of studies to reprbedrock channel erosion (or more generally, detachment-limited erosion) (e.g., and Dietrich, 1992; Anderson, 1994; Howard et al., 1994; Seidl et al., 1994; Tuand Slingerland, 1994; Moglen and Bras, 1995; Humphrey and Heller, 1995). Ithe practical advantage of being simple and efficient to integrate numerically.

Transport-limited: If sufficient sediment is always available for transport and/or tbed material is easily detached, streams can be assumed to be everywhere at carrying capacity. Under this condition, continuity of mass gives the local rate oerosion or deposition as

, (EQ 11)

whereρs is sediment bulk density and is a vector oriented in the direction of fl

These two different cases can be invoked in the model by selecting the “detachlimited” or “transport-limited” stream erosion option, respectively. Alternatively, tmodel may be run in “bedrock-alluvial” mode. This represents the most generacase. In bedrock-alluvial mode, the rate of erosion is computed as the lesser othe excess sediment carrying capacity or (2) the maximum detachment rate. Reis assumed to have an effectively infinite detachment capacity; detachment caponly becomes a limiting factor when bedrock is exposed in a channel.

t∂∂zb Dc–=

t∂∂zb 1

ρs-----

x∂∂Cs–=

x



d.devel-l

r 6.

besion on

ss as aeer-er et

ff ero-

tionver.

Grain-Size Sorting

A model of transport and sorting of multiple sediment sizes has been developewithin the framework of the GOLEM fixed-grid landscape evolution model by NGasparini. The sorting model uses a two-phase sand-gravel transport formula oped by P. Wilcock at Johns Hopkins University. The multi-size transport modeincludes a set of routines for recording the depth and composition of previouslydeposited sediment layers. The multi-size model is described further in ChapteAdaptation of the multi-size model to the CHILD framework is currently inprogress.

Lateral Stream Channel Erosion (Meandering)

An interface is currently under development that will allow the CHILD model to coupled with a 2D model of river meandering in order to compute later river eroand floodplain widening. The meander model, written by S. Lancaster, is basedtopographic steering of channel flow. The model parameterizes bank shear strefunction of the cross-channel transfer of fluid momentum due to topographic sting by point bars. The meander model is described in greater detail by Lancastal. (Appendix A).

Vegetation

Vegetation is modeled in terms of a percent vegetation cover on the surface,V. Vege-tation increases the threshold shear stress that must be exceeded before runosion can occur, according to

, (EQ 12)

whereτcs is the critical shear stress for an unvegetated surface (primarily a funcof grain size) andτcv is the added critical shear stress under 100% vegetation co

τc V( ) τcs Vτcv+=


Vegetation

r

vege-

During storms, vegetation cover is eroded at a rate proportional to excess sheastress,

, (EQ 13)

wherekvd is a constant that controls the rate of vegetation removal. The rate of tation regrowth during interstorm periods is modeled as a linear function of theamount of vegetation cover present,

. (EQ 14)

The vegetation module is discussed further in Chapter 5.

tddV

(erosion) kvdV τ τc–( )η–=

tddV

(growth) kvg 1 V–( )=


CHAPTER 3 Modeling LandscapeEvolution Using anAdaptive IrregularSimulation Mesh

-.singal92;

all,ingeralatial

any5tionand of aork

st

Continuing advances in computing technology have made three-dimensional modeling an attractive tool for investigating landscape evolutionMost landscape evolution models represent three-dimensional terrain ua regular matrix of points, the same representation that is used in digitelevation models (e.g., Ahnert, 1976; Kirkby, 1987; Beaumont et al., 19Willgoose et al., 1991; Chase, 1992; Slingerland et al., 1993; Howard,1994; Rigon et al., 1994; Tucker and Slingerland, 1994; Sinclair and B1996). Although significant insights have been gained from models usthis type of regular spatial discretization, the technique suffers from sevdrawbacks: (1) landform elements must be represented at a uniform spresolution, which in practice means the highest resolution required by feature or process of interest; (2) drainage directions are restricted to 4degree increments (though for watershed-scale applications, this limitamay be reduced by using multiple-flow algorithms (e.g., Costa-Cabral Burgess, 1994; Tarboton, 1997)); (3) under certain circumstances, useregular grid introduces anisotropy that can lead to bias in drainage netwpatterns; and (4) use of a fixed grid makes it difficult or impossible tomodel processes with a significant horizontal component, such as thrupropagation or river meandering.


Modeling Landscape Evolution Using an Adaptive Irregular Simulation Mesh

nt ofhedodnship

lrrentm ero-

Ns)ren-ctheirof92; Sam- a tri-

gulard

odellse-

tivesh,ional

The last of these constraints is especially significant. Although we conventiallyspeak of “uplift,” most crustal deformation processes involve a significant amouhorizontal translation. Previous coupled models have either incorporated only tvertical component of deformation (e.g., Tucker and Slingerland, 1996; Kooi anBeaumont, 1996) or have represented lateral translation by simply offsetting twfixed grids (e.g., Anderson, 1994). Coupled models of deformation, erosion, ansedimentation promise to yield important insights into such issues as the relatiobetween deformation and the stratigraphic record, but such models ultimatelyrequire the ability to model deformation in three dimensions. Similarly, erosionaprocesses often have a significant horizontal component that is neglected in cumodels. One of the most important horizontal erosion processes is lateral streasion, which by widening a valley can signicantly alter the depositional geometrywithin a floodplain over geologic time.

One alternative to regular grids is the use of triangulated irregular networks (TIfor representing topographic surfaces. Triangulated irregular networks, which aoften based on the so-called Delaunay triangulation, are commonly used for costructing finite element meshes and for representing surfaces within geographiinformation systems. Techniques for constructing Delaunay triangulations and corresponding Voronoi (or Thiessen) diagrams are well established in the field computational geometry (e.g., Guibas and Stolfi, 1985; Sloan, 1987; Knuth, 19Sugihara and Iri, 1994; Sambridge et al., 1995; Du, 1996). Recently, Braun andbridge (1997) adapted an existing large-scale landscape model to operate withangular irregular spatial discretization that overcomes many of the limitationsassociated with regular grids. They demonstrated that use of a triangulated irrenetwork has the advantages of eliminating anisotropy in drainage directions anallowing for a variable-resolution representation of topography.

In this chapter, we describe the irregular mesh technique used in the CHILD mof hillslope and channel evolution. The mechanics of the model are described ewhere (see Chapters 2 and 4-7); here we focus on implementation of the adapmesh. We describe an efficient data structure for implementing the irregular meand briefly discuss how the technique makes it possible to model three-dimensvalley formation by stream erosion.


Grid Elements and Data Structures

ra-

leva- com-ses a

ronoi

ydrol-rd


The irregular mesh used by the CHILD model is illustrated in Figure 3-1. Topogphy is represented in the model by a set of nodesN that are connected to form amesh of triangles using the Delaunay triangulation ofN. The Delaunay triangulationis a unique set of triangles that connect a set of points in such a way that a circpassing through the three points of any triangle will contain no other points. Eletion, drainage area, and other state variables in the model (see Figure 1-1) areputed at the nodes rather than within the triangles; in other words, the model ufinite-difference rather than a finite-element approach. Each nodeNi is associatedwith a Voronoi Region (or Voronoi cell, also known as a Thiessen polygon). TheVoronoi Region for a nodeNi is the region within which any arbitrary pointQ wouldbe closer toNi than to any other node on the mesh. The boundaries between VoCells are lines of equal distance between adjacent grid points.

A number of algorithms and data structures for representing topography and hogy on a fixed grid have been developed, and are generally fairly straightforwa

0

1000

2000

3000

4000

5000

0

200

400

FIGURE 3-1. Isometric view of a simulatedcatchment, showing irregular mesh.



atarors,et-

iable with-namic that

e isnsists. The

(e.g., Jenson and Domingue, 1988; Tarboton et al., 1991). Designing efficient dstructures for an irregular triangulated mesh is more complicated. Unlike regulamatrices, where each node is connected to either four or eight adjacent neighbthe number of neighbors connected to a given node in a triangulated irregular nwork may be arbitrarily large. Ideally, a data structure should represent this varconnectivity in a way that (1) provides rapid access to adjacent mesh elementsout demanding excessive storage space, and (2) is flexible enough to handle dychanges in the mesh itself. The CHILD model uses a “DualEdge” data structureprovides an efficient way to satisfy these requirements. The DualEdge structuradapted from the QuadEdge data structure of Guibas and Stolfi (1985), and coof three geometric elements: nodes, triangles, and directed edges (Figure 3-2)data structure is summarized in Figure 3-3.

Voronoi CellA

BC

DA

BC

D

a

b

A.edg = ABAB.nextedg = ACAB.vvertex = a

B.edg = BABA.nextedg = BDBA.vvertex = b

a

b

(a) (b)

FIGURE 3-2. Illustration of the dual-edge data structure, showing triangular lattice (black)and corresponding Voronoi diagram (gray). (a) Directed edge AB, counterclockwisenextedgAC, and right-hand Voronoi vertex a. (b) Complementary directed edge BA,counterclockwisenextedg BD, and right-hand Voronoi vertexb.



samenednd-

one thatrep-g

mme-

i ver-iatedd-

nd aro-

hei

Directed Edges

Each edge of a triangle is associated with two directed edges, which share theendpoints but point in opposite directions (Figure 3-2). Directed edges are defiby their origin and destination nodes; two directed edges that share the same epoints are termed complementary edges. In addition to the origin and destinatinodes, each directed edge data object includes a reference to the directed edglies immediately counter-clockwise relative to its origin node (Figure 3-2). This resentation makes it possible to rapidly access all of the edges and neighborinnodes connected to a given node.

Each directed edge object also includes the coordinates of the Voronoi vertex idiately clockwise. A Voronoi vertex is defined as the intersection point of threeVoronoi cells (Figure 3-2). In general, each triangle is associated with a Voronotex; each directed edge object includes a reference to the Voronoi vertex assocwith the triangle on its right-hand side (clockwise). Pseudo-code for the DirecteEdge data structure is shown in Figure 3-3.

Nodes

Each node object includes x, y, z coordinates, the number of adjacent nodes, apointer to one of its directed edges (along with other information relevant to hydlogic routing and physical parameters, not shown) (Figure 3-3). The format of tNode and DirectedEdge objects makes it possible to efficiently find the Voronopolygon associated with a node, using the following algorithm:

FindVoronoiPolygon( Node thenode )BEGIN

XYPoint voronoi_polygon(1..thenode.nnbrs)current_edge = thenode.edgFOR i=1,thenode.nnbrs DO

voronoi_polygon(i).x := current_edge.vvertex_xvoronoi_polygon(i).y := current_edge.vvertex_ycurrent_edge := current_edge.nextedg

ENDEND



igh-

uch a

ofre 3-rtex

Similarly, a list of neighboring nodes can be obtained by accessingcurrent_edge.dest for each edge originating at a given node.

Triangles

Triangle objects include pointers to the three nodes in the triangle, the three neboring triangles, and the three directed edges that are oriented clockwise withrespect to the triangle. The nodes and neighboring triangles are numbered in sway that thenth neighboring triangle lies opposite thenth vertex (see Figure 3-4).Each triangle is associated with a Voronoi vertex. This vertex is the intersectionthe three Voronoi cells associated with each of the triangle’s vertices (e.g., Figu2). Because the three edges are clockwise-oriented, their right-hand Voronoi ve

Class Nodex // x-coordinatey // y-coordinatez // z-coordinate (elevation)edg // first connected edgennbrs // number of neighboring nodes

Class DirectedEdgeorg // origin nodedest // destination nodenextedg // next directed edge counterclockwisevvertex_x // x-coordinate of right-hand Voronoi vertexvvertex_y // y-coordinate of right-hand Voronoi vertex

Class Trianglep(3) // Vertex nodest(3) // Adjacent triangles (t(1) is opposite p(1), etc.)e(3) // Clockwise-oriented directed edges

FIGURE 3-3. Pseudo-code summary of the DualEdge data structure, showing the datamembers belonging to Node, DirectedEdge, and Triangle objects.


Drainage Networks on a Triangulated Irregular Mesh

cesrian-ases

con-ds of

sneigh-westcondf

an

a-akeally

is the Voronoi vertex associated with the triangle itself. In general, Voronoi vertican be found by locating the intersection of the perpendicular bisectors of the tgle’s edges (note that there exist special cases in which this is not true; these cmust be detected and handled differently).

Drainage Networks on a Triangulated Irregular Mesh

The surface discharge at each node is computed as a function of the upstreamtributing area (see Chapters 2 and 4). The model provides two alternative methofinding the upstream contributing area at a node. In both methods, each node iassigned a drainage direction along the steepest path (edge) toward one of its boring nodes. In the first method, the area of each triangle is assigned to the loof the triangle’s three nodes and to all nodes downstream of that point. In the semethod, the contributing area at a nodei is equal to the sum of the Voronoi areas oall nodes that flow toi (includingi itself).

In some cases a node may form a local depression, with no neighbors lower thitself. This case can be handled in one of two ways in the model. The simplestmethod assumes that all water entering a “sink” evaporates at that point. Alterntively, an outlet can be found for each sink using the “Lake Fill” algorithm. The LFill algorithm starts by creating a stack of contiguous flooded nodes, which initi

P0

P1P2

T0

T1T2

e1 e0

e2

FIGURE 3-4. Illustration of numbering of triangle nodes, adjacent triangles, and clockwiseedges in Triangle data objects.



iter-nointd that the Fig- and

an typi-anmires atinu-

lly

the

ux is

contains just the sink itself. The perimeter of the flooded region (“lake”) is then atively searched to identify the lowest node along the perimeter. If this node cadrain downhill to a location other than the lake itself, it is flagged as the outlet pfor all nodes on the stack. If not, it is added to the stack. If a node is encountereis part of a pre-existing lake (one initiated at a different sink), it is also added tostack. An example of a lake computed using the Lake Fill algorithm is shown inure 3-5. The algorithm is robust enough to handle any arbitrary initial condition,is useful for modeling a rising baselevel or the damming of water and sedimentbehind an uplifting block.

For a mesh with numerous sinks, the lake-filling algorithm is probably slower ththe “cascade” algorithm of Braun and Sambridge (1997). However, in the morecal case of a few isolated sinks, the lake-filling algorithm is likely to be faster ththe cascade algorithm. The number of iterations needed by the lake-fill algorithdepends on the number of flooded points, whereas the cascade algorithm requnumber of iterations equal to the maximum number of segments along any conous stream regardless of the number or depth of sinks.

Numerical Algorithms

Fluvial erosion and deposition at each node in the mesh is computed numericafrom

(EQ 1)

wherezi is the elevation at nodei, t is time,n is the number of nodes that flowdirectly toi, Qs is sediment flux, andAvi is the Voronoi area of nodei (the deposi-tional area associated with each node is always its Voronoi area, regardless ofmethod used to compute contributing drainage area). The sediment outfluxQsidepends on discharge, slope, and (possibly) sediment influxΣQsj, as discussed inChapter 2. (Note that this equation only applies to the case in which sediment fl

td

dzi

Qsj

j 1=

n

∑

Qsi–

Avi-----------------------------------=


Numerical Algorithms

ge (1)

ing

oped

tracked; in the “detachment-limited” end-member case, the erosion ratedz/dt isassumed to be independent ofQs, and is computed directly as a function of discharand slope). The system of ordinary differential equations described by equationis solved using a predicted-corrector method (e.g., Acton, 1970).

The hillslope diffusion equation (Chapter 2, equation (7)) is solved in the followway. The downslope sediment flux per unit slope width is given by

, (EQ 2)

where the vector symbol denotes orientation in the downslope direction. The slwidth between two adjacent nodes is approximated by the length of their shareVoronoi cell edge,Λij . Erosion or deposition at a node due to diffusion is thenapproximated numerically by

FIGURE 3-5. Example of lake formation in the model. The green asterisks denote lakenodes. The lake has formed in response to a “digital dam” that was created by artificiallyraising the elevation of nodes near the catchment outlet. The lake outlet is indicated by thethick blue line at the right-hand edge.

qs kd x∂∂z

=



s-

can bee (or

ions

nder-irregu-ee-

rou-rea;ata

ove,odelsng

(EQ 3)

whereSij is the slope from nodei to nodej, defined as positive downwards. This sytem of equations is solved using a simple forward-difference method. Note thatbecause diffusive mass exchange takes place along mesh edges, the equationsolved efficiently by first computing the mass exchange along each physical edgequivalently, across each Voronoi polygon face), then updating the node elevataccordingly.

Variable Resolution for Modeling River Meandering

Lancaster et al. (Appendix A) describe a physically-based model of stream meaing that represents a meandering stream as a series of (x,y) points. Use of an lar, deformable mesh allows the meander model to be incorporated into the thrdimensional landscape evolution model. An interface between CHILD and themeander model is currently being tested and debugged. The interface includestines to (1) identify meandering channels on the basis of a threshold drainage a(2) collect the nodes within a meandering channel reach and arrange them in dstructures that can be passed to the FORTRAN program meander.f; and (3) madd, and/or delete nodes in response to channel migration. Coupling the two mwill make it possible to investigate the dynamics of valley and floodplain wideniin an eroding landscape.

td

dzi kd

Λi j Sij

j 1=

n

∑Avi

----------------------–=


CHAPTER 4 A Stochastic Approach toModeling Drainage BasinEvolution

tioninearth-c fre-ing geo-h ofcur-ss”r,88)son, vari-ionate noti-lop-sin,

Although it is often modeled as a continuum process, landscape evoluis in fact driven by discrete events. The topography of a typical mountarange, for example, is shaped by a quasi-random sequence of floods, quakes, and landslides, with each process having its own characteristiquency distribution, and with each frequency distribution perhaps varyin time and space as well. The importance of the frequency spectra ofmorphic events is a fundamental problem in geomorphic research. Mucthe previous research on this problem has focussed on defining the rerence interval of the most effective geomorphic event, with “effectivenedefined either on the basis of denudation rate (e.g., Wolman and Mille1960; Andrews, 1980; Webb and Walling, 1982; Ashmore and Day, 19or landform genesis (e.g., Baker, 1977; Harvey, 1977; Wolman and Ger1978). Scant attention has been paid, however, to the question of howability in geomorphic forces impacts the morphology and rate of evolutof landforms. For example, the relative geomorphic significance of climvariability as opposed to mean climate has been widely debated and iswell understood. In this chapter, we address the problem of rainfall varability and its impact on catchment geomorphology. We do so by deveing a stochastic theory for erosion and sedimentation in a drainage ba


A Stochastic Approach to Modeling Drainage Basin Evolution

andom-

us toe

es tonce

s a

el of

val

and exploring the consequences of that theory in the framework of the CHILDmodel. The stochastic model is based on the rainfall model of Eagleson (1978)describes the probability distribution of storm depth, duration, and frequency. Cbining the stochastic rainfall model with the landscape evolution model enablessimulate the long-term geomorphic impact of natural variability in storm size. Wfocus in particular on two questions:

1. What is the predicted sensitivity of long-term average sediment transport ratthe degree of variability in rainfall intensity? To what degree does the importaof variability depend on the presence of thresholds in the landscape, such athreshold for sediment entrainment?

2. What are the morphologic consequences of rainfall variability?

Model Description

Stochastic Rainfall Model

Rainfall is modeled as a series of discrete random storm events, using the modEagleson (1978). Each storm event is treated as having a constant rainfall rateR thatlasts for a durationTd and is separated from the next event by an interstorm interTi. The probability density functions for storm intensity, duration, and interstorminterval are given by

Rainfall (runoff) intensity (EQ 1)

Storm duration (EQ 2)

Interstorm period (EQ 3)

f P( ) 1P--- P

P---–

exp=

f Td( ) 1

Td

------Td

Td

------–

exp=

f Ti( ) 1

Ti

-----Ti

Ti

-----–

exp=


Model Description

enoffr

s.

n of

l or

In order to derive a distribution of runoff ratesf(R), we start by assuming runoff isHortonian and uniform across the landscape. (However, the formulation could bmodified to account for saturation-excess runoff production [Dunne, 1978]). Rurate,R, is defined as precipitation minus losses to infiltration, evaporation, and/ocanopy interception:

, (EQ 4)

whereIc = infiltration capacity andIl = evaporation and canopy interception losseThe derived distribution forR is obtained from

, (EQ 5)

(EQ 6)

The mean runoff rate is therefore

. (EQ 7)

Sediment Transport by Runoff

The instantaneous rate of sediment transport by runoff is modeled as a functioexcess shear stress,

, (EQ 8)

whereQs is the sediment transport rate integrated across the width of a channerill, W is channel width,τ is average bed shear stress,τc is critical shear stress forsediment entrainment,kf is a transport coefficient, andp is an exponent typically onthe order of 1.5-3 for bedload (e.g., Yang, 1996) and higher for suspended load

R P I–= I, I c I l+=

f R( ) f P( )Rd

dP=

f R( )1P--- R I+( )

P-----------------–

R 0;>,exp

0 otherwise.

=

R PIP---–

exp=

Qs kf W τ τc–( )p=



iri-rela-

;

quat-

omet- do

(Whipple et al., in review). Assuming steady, uniform flow and adopting an empcal bed friction relationship (such as the Manning-Strickler or Darcy-Weisbach tion), shear stress can be expressed as a power function of discharge,Q, and slope,S,

, (EQ 9)

with kt, α, andn as parameters. (Derivations are given by Willgoose et al., 1991Howard et al., 1994; Tucker and Slingerland, 1997). Substituting the empiricalwidth-discharge relationship ,

, (EQ 10)

wherem = α(1-ω).

To writeQ in terms of runoff, we use the simple steady-state relationship

, (EQ 11)

whereR is runoff per unit area andA is drainage area. IfR is uniform across thebasin, as we assume in the analysis below, this becomes

. (EQ 12)

This is clearly a simplification, because it neglects hydrograph attenuation by eing hydrograph duration with storm duration. These effects can in principle beaccounted for by relating hydrograph duration and peak attenuation to basin geric parameters such as total stream length, though for the sake of simplicity wenot do so here.

τ ktQW-----

αSn=

W kwQω=

Qs kf kwQω ktkwα– QmSn τc–( )

p=

Q R a( ) adA∫=

Q RA=


Sensitivity of Erosion Rate to Rainfall Variability

turnrived

foreanimes

, is,

ans-

ake

ut its


Fluvial sediment transport can be viewed as a random process in time which inis a function of another random process, runoff. We can combine the models deabove for rainfall and sediment transport to analyze the sensitivity of long-termmean sediment transport rate to the degree of temporal variability in rainfall.

We start by considering only sediment transported during storm events, which most rivers constitutes the bulk of sediment carried. Under this condition, the mannual sediment flux is equal to the mean transport rate produced by a storm tthe mean storm duration times the number of storms per year, or

, (EQ 13)

where is the long-term mean transport rate (mean annual, ifTd is in years),N isthe average number of storms per year,Td is mean storm duration, andQs is themean transport rate produced by a storm.

Note that can be related to mean annual rainfall. Mean annual rainfall,equal to mean storm rainfall times storm frequency times mean storm duration

. (EQ 14)

Combining equations (13) and(14),

. (EQ 15)

All else being equal, equation (15) implies that the long-term mean sediment trport rate should be linearly related to mean annual rainfall.

It remains now to determine the mean storm sediment transport rate, . We mthe following assumptions:

1. Each storm can be approximated as having a constant rainfall rate throughoduration.

Qs⟨ ⟩ NTdQs=

Qs⟨ ⟩

Qs⟨ ⟩ P⟨ ⟩

P⟨ ⟩ PNTd=

Qs⟨ ⟩ P⟨ ⟩P

---------Qs=

Qs



dy

).

nt

unoff

tion

ted

ermnnual

2. At each point in the landscape, runoff (if nonzero) produces a constant, steadischarge equal to the runoff rate times contributing area.

3. Runoff rate per unit area has the exponential distribution given in equation (6

The steady discharge assumption allows us to write the average storm sedimetransport rate as

. (EQ 16)

Substituting the sediment transport formula (equations (10) and (12)) and the rdistribution (equation (6)),

. (EQ 17)

We do not know of an analytical solution to this equation, but an analytical solucan be found for the special caseτc=0, I=0. In that case,

, (EQ 18)

whereγ = mp+ω. The integral term can be solved by the substitution ,which gives

, (EQ 19)

whereΓ() is the gamma function. Combining with equation (15) gives the expecmean annual transport rate:

. (EQ 20)

Assuming thatγ ranges from ~1-2, as is typical, this equation predicts that long-taverage sediment transport rates should generally be more sensitive to mean a

Qs f R( )Qs R( ) Rd0

∞

∫=

Qs

kf kwAω

P------------------ R I+( )

P-----------------–

exp Rω ktkwα– AmRmS

nτc–( )

pRd

0

∞

∫=

Qs

ktk f kw1 α–( )AγSnp

P---------------------------------------- R

P---–

exp Rγ Rd0

∞

∫=

u R P⁄=

Qs ktk f kw1 α–( )AγSnpP

γu–( )exp uγ ud

0

∞

∫ ktk f kw1 α–( )AγSnpP

γ Γ γ 1+( )= =

Qs⟨ ⟩ P⟨ ⟩Pγ 1–

ktk f kw1 α–( )AγSnpΓ γ 1+( )=



rms

lized andould be

s:

rainfall than to mean rainfall intensity, as long as the threshold and infiltration teare negligible.

Nondimensionalization

To facilitate numerical analysis of equation (19), the equation is nondimensionaas follows. Rainfall and runoff intensity are normalized by mean annual rainfall,shear stress and critical shear stress are normalized by the shear stress that wproduced by the mean annual rainfall (at a given slope and contributing area).

Nondimensional rainfall intensity (EQ 21)

Runoff rate (EQ 22)

Channel width (EQ 23)

Shear stress (EQ 24)

Critical shear stress (EQ 25)

Infiltration rate (EQ 26)

Storm transport rate (EQ 27)

Mean annual transport rate (EQ 28)

Here,Wp refers toW(<P>), and similarly forQp andτp. Parameterm=α(1-ω). Thenondimensionalization reduces the equation to three dimensionless parameterI’ ,τc’, and eitherP’ or R’ (in addition to the exponent terms).

P' P P⟨ ⟩⁄=

R' R P⟨ ⟩⁄=

W' KwQQp------

ϖ=

τ' ττp-----

QQp------

α WWp--------

α– QQp------

α QQp------

αϖ–R'm= = = =

τc'τc

τp-----=

I ' IP⟨ ⟩

---------=

Qs'W τ τc–( )p

Wpτp-------------------------- R'ϖ R'm τc'–( )

p= =

Qs⟨ ⟩ ' 1R'---- I '

I ' R'+--------------–

Qs'exp=



param-he For

plot-

re-

t forllynoff.

Sensitivity to Runoff Variability: Numerical Solutions

In order to analyze the relationship between mean transport rate and the three eters describing rainfall intensity, infiltration capacity, and critical shear stress, tnondimensional form of equation (17) is solved using a Monte Carlo approach.each set of parameters , 10,000 random values ofP’ were chosen and thecorrespondingQs’ was computed for each. The randomQs’ values were then aver-aged and multiplied by to obtain (see equation (28)). The results are ted in Figure 4-1.

Figure 4-1a illustrates how the infiltration parameterI’ influences the relationshipbetween mean sediment discharge and rainfall variability. WithI’=0 andτc’=0,mean sediment discharge increases as the square root of rainfall intensity, as pdicted by equation (20) forγ=ω+mp=3/2. By comparison, equation (20) predicts alinear relationship between mean sediment discharge and mean rainfall, so thalow I’ andτc’, sediment discharge is more sensitive to the total amount of rainfathan to rainfall variability. AsI’ increases, however, sensitivity to rainfall variabilitrises as increasingly larger rainfall events are required to produce significant ru

P' I ' andτc', ,

1 P⁄ Qs⟨ ⟩ '

Tc‘ = 0

Tc‘ = 1

Tc‘ = 2

Tc‘ = 4

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

Pprime

Nor

mal

ized

<Q

s>pr

ime

Monte carlo solution to Qs=f(P)

I = 0

I = 10

I = 100

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

Pprime

Nor

mal

ized

<Q

s>pr

ime

Monte carlo solution to Qs=f(P)

FIGURE 4-1. Plot of normalized mean sediment flux versus the rainfall variabilityparameter , showing how the relationship changes as a function of (a) infiltration rateand (b) critical shear stress . Each plotted point represents an average of 10,000realizations of . To facilitate comparison between the curves, each curve is normalized bythe mean value of . Exponent parameters are .

Qs⟨ ⟩ 'P' I '

τc'Qs'

Qs⟨ ⟩ ' ϖ 0.5 m, 1 3⁄= = and p, 3=

(a) (b)


Morphologic Consequences of Rainfall Variability: Numerical Example

matef infil-

man

abil-ratedlysisdistri-

ite

and/-lueselatiovity

edent

f 2 x,y),is

Which of these curves is most appropriate to natural catchments? A rough estican be obtained from published figures. Dunne (1978) reports measurements otration capacity on the order of 0.2 - 6 cm/hr for midwestern agricultural silt-loasoils, and on the order of 8 cm/hr for vegetated forest soils. Given a typical meannual rainfall of ~1 meter, these values correspond toI’ ~ 20 - 700, implying thatvegetated soils may have the effect of amplifying the importance of rainfall variity. On the other hand, this analysis does not account for direct runoff from satuareas, which would tend to increase the importance of smaller storms. The anaalso does not account for antecedent soil moisture. In natural catchments, the bution of interstorm periods influences the likelihood of precipitation falling onalready-saturated soils, and is therefore also a potentially important variable.

The relationship between mean sediment discharge and rainfall variability is qusensitive to the shear stress thresholdτc’, underscoring the point made by Baker(1977) that extreme events become increasingly important in bedrock channelsor channels bearing very coarse bedload material. Becauseτc’ depends on the meanflow shear stress of the stream in question, it is difficult to judge what typical vamight be. An additional complication is the finding by Parker (1978) that channgeometry in streams with mobile bed and banks tends to adjust such that the rτ/τc ~ Kch = constant. Clearly, however, there exists the possibility for high sensitito climate variability in the case of boulder-bed or bedrock channels.

Morphologic Consequences of Rainfall Variability: NumericalExample

The consequences of rainfall variability for catchment evolution are next explorthrough simulations with the CHILD model. Figure 4-2 shows a synthetic catchmformed by a combination of steady tectonic uplift relative to the outlet at a rate o10-5 m/yr and erosion by a sequence of random storm events. In this simulationmean annual rainfall is 1 meter, mean rainfall intensity is 10 m/yr (~27.4 mm/daand infiltration capacityIc = 10 m/yr. The topography consists of a main valley axflanked by a series of hollows (first-order valleys).



ted totant, ero-tworkad- the

ofship is

ween

Figure 4-3 illustrates what happens when the catchment in Figure 4-2 is subjeca tenfold increase in mean rainfall intensity, with mean annual rainfall held consfor a total duration of 100,000 model years. The increased efficiency of streamsion (predicted by equation (20)) leads to accelerated erosion of the channel ne(Figure 4-3b). Erosion is greatest within the first-order valleys, which extend heward to produce a significant increase in drainage density (Figure 4-3a). Whenmean rainfall intensity later reverts to its original value (Figure 4-4), the patternreverses as the valley network begins to fill in with sediment.

The morphologic changes illustrated in Figures 4-2–4-4 are apparent on a plotlocal slope versus contributing area (Figure 4-5). Initially, the slope-area relationis close to the equilibrium value predicted by equation (20) (the small differencedue to the fact that the simulation has not yet reached complete equilibrium betuplift and erosion) (Figure 4-5a). The increase in rainfall intensity (Figure 4-5b)

0

1000

2000

3000

4000

5000

60002000

0

200

400

FIGURE 4-2. Simulated drainagebasin formed by a combination ofsteady tectonic uplift and storms ofvariable intensity and duration.



0

1000

2000

3000

4000

5000

6000

7000 0

1000

2000

0

200

400

FIGURE 4-3. Effect of an increase inrainfall intensity on the syntheticcatchment shown in Figure 4-2. (a)Topography. (b) Pattern of erosion: red =erosion, green = little or no change, blue(not shown) = deposition or uplift.(Spike-like features in (b) are distortedVoronoi cells created by boundary effectsin the triangulation).



tientslley

ten-t alllllies a ing ontion

mor-

leads to a decrease in gradient along the main valley network as the catchmenadjusts toward a new equilibrium (dashed line). At the same time, however, gradalong the first-order valleys and hillslopes increase in response to headward vaerosion and a consequent steepening of side-slopes.

This numerical example illustrates the predicted relationship between rainfall insity, catchment relief, and drainage density. One implication of the model is thaelse being equal, higher rainfall intensity should be correlated with lower overarelief, higher drainage density, and/or higher sediment flux. The model also imppattern valley erosion and valley infilling, respectively, in response to variationsrainfall intensity, though the nature and pattern of response may vary dependinthe magnitude of erosion thresholds (Tucker and Slingerland, 1997) and vegetacover (Chapter 5). These predictions thus provide a basis for comparison with phologic and sediment flux data.

0

000

000

000

000

FIGURE 4-4. Erosion and deposition pattern following a decrease in mean rainfall intensity(to its original value in the simulation show in Figure 4-2). Initial condition is the topographyin Figure 4-3. Catchment is shown 10,000 model years after decrease in mean rainfallintensity. Blue = net deposition, red = net erosion, and green = little or no change.



rre-mor- ofces.

Climatic factors such as mean rainfall and rainfall variability are often closely colated with vegetation cover, a variable that also can clearly have important geophic consequences. In the next chapter, we develop a model for the interactionvegetation and erosion and use the model to explore some of those consequen

Numerical simulationTheory

103

104

105

106

107

108

10−2

10−1

100

Drainage area (m2)

Slo

pe

(a)

Numerical simulation

Previous equilibrium

New equilibrium

103

104

105

106

107

108

10−2

10−1

100

Drainage area (m2)

Slo

pe

(b)

FIGURE 4-5. Plot of local slope versus contributing area for the numerical simulationsshown in (a) Figure 4-2 and (b) Figure 4-3. Lines show the equilibrium slope-arearelationship calculated by solving equation (20) for slope, under the condition <Qs> = UA(uplift rate times drainage area).


CHAPTER 5 Dynamics of Vegetation andRunoff Erosion

olog-laysl

, and

en sta- a

yont-ly,

i-you-

tal

Most models of landscape evolution emphasize physical rather than biical processes, yet the biosphere, and vegetation in particular, clearly pan important role in landscape evolution. Vegetation influences physicaerosion both directly, by increasing surface resistance to wash erosionindirectly, by influencing infiltration, runoff, and evapotranspiration. Forexample, the data of Melton (1958) show an inverse correlation betwedrainage density and humidity, a finding that has been attributed to thebilizing effects of vegetation (e.g., Moglen et al., in press). By imposingsignificant threshold for runoff erosion, vegetation cover may effectivelimpose an upper limit to channel network extent (e.g., Horton, 1945; Mgomery and Dietrich, 1989; Dietrich et al., 1993; Kirkby, 1994). Similarvegetation is widely believed to contribute to the observed relationshipbetween climate and sediment yield. A number of data sets show adecrease in sediment yield with increasing mean annual rainfall in semarid to humid climates, despite the presumed increase in erosive energassociated with more humid climates (Langbein and Schumm, 1958; Dglas, 1967; Wilson, 1973; Moglen et al., in press). Finally, vegetation ispotentially important in governing landscape responses to environmenchange (e.g., Tucker and Slingerland, 1997; Howard, 1996).


Dynamics of Vegetation and Runoff Erosion

iond

eta-mes.vicin-omeis

ention

o-nle

l dif-

es ises,

her-kss on

ls.overandwe

In fully vegetated landscapes, there is a dynamic competition between vegetatgrowth and the disruption of vegetation by runoff erosion. Within well-establishechannels, runoff erosion clearly “wins,” while near drainage divides it is the vegtion that “wins.” The interesting part is what happens in between these two extreThe outcome of the competition between erosion and vegetation growth in the ity of first order channels may have a significant effect on drainage density (in scases, it may be the determining factor), and often also on sediment yield. In thchapter, we develop a simple theory to describe the dynamic interaction betwevegetation and erosion, and use that theory to explore the nature of that interacon “geomorphic” time scales (by which we mean time scales relevant to morphlogic development as opposed to, say, the time scale for significant erosion of aagricultural field). In the first part, we present the theory and explore some simpoutcomes for slope profile development. In the second part, we present severaferent numerical examples using the CHILD model.

Model of Vegetation and Erosion

Conceptual model

In the context of drainage network development, one of the most important issuthe nature of the hillslope-channel transition. In humid or semi-humid landscaphillslopes are typically vegetation-covered, while low-order ephemeral channelscontain sparse, intermittent vegetation (clumps of grass, bushes, etc.), and higorder channels with year-round flow are free of vegetation except along the banand bars. In deriving a model to describe vegetation-erosion dynamics, we focuthe physical interaction of vegetation and runoff erosion within rills and channeThe most important aspects of that interaction are (1) the effect of vegetation con soil/sediment erodibility, (2) the disruption of vegetation due to erosive overlflow, and (3) the rate at which vegetation regrows after being disrupted. Below attempt to quantify these three interacting processes.



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Modeling approach

Vegetation reduces erodibility

There are a number of ways in which the effects of vegetation on surface erodimight be modeled. A simple but physically plausible approach is to assume thaetation increases the both the effective shear stress for runoff erosion and the cshear stress for particle entrainment. Many sediment transport equations haveform

, (EQ 1)

whereqs is sediment transport capacity per unit flow width,τ is fluid shear stress,andτc is a threshold for particle entrainment. By binding the soil with roots and cover (grasses, for example), vegetation effectively increasesτc, an idea consistentwith the threshold channel initiation hypothesis (e.g., Horton, 1945; Willgoose e1990; Montgomery and Dietrich, 1989) and also with Foster’s (1982) soil erosiomodel, which forms part of the WEPP agricultural model (Foster et al., 1995).

Vegetation can be considered in terms of a fractional ground cover,V, which rangesfrom zero to one (or more generally from zero to , whereVmax is the maxi-mum percent cover that can be supported in a given environment). Few data seexist with which to constrain the relationship betweenτc andV (Foster 1982 pro-vides some field-calibrated values; Dietrich et al., 1993 give estimates ofτc at chan-nel heads, based on DEM analysis). In the absence of better information, we asa linear relationship, recognizing that this may be an oversimplification:

, (EQ 2)

whereτcs is the critical shear stress for an unvegetated surface (primarily a funcof grain size) andτcv is the added critical shear stress under 100% vegetation coNote that this approach assumes, via equation (1), that transport capacity depevegetation cover.

We account for changes in the total shear stress due to increased roughness achanges in the fraction of shear stress applied to the soil surface using the app

qs k τ τc–( )p=

Vmax 1≤

τc V( ) τcs Vτcv+=



tal

only,tion

o the

to aed--

ed” onut

discussed by Foster (1982). The total shear stress and effective shear stressτf appliedto the soil are assumed to be related by

, (EQ 3)

with fs/ft being the ratio of the friction factor produced by the soil alone to the tofriction factor (including vegetation cover). The ratioRf=fs/ft is parameterized as afunction ofV, again assuming a linear relationship,

. (EQ 4)

Runoff erosion disrupts vegetation

Surface runoff clearly disrupts vegetation when it becomes strong enough, but recently have there been attempts to model the process quantitatively (Thornes1990; Kirkby, 1995; Kirkby and Cox, 1995). We speculate that the rate of vegetadestruction by rill or channel runoff is proportional to excess shear stress and tfractional vegetation cover remaining:

, (EQ 5)

with kvd being the rate of vegetation loss per unit excess shear stress (or stresspower) at 100% vegetation cover. This has the potential for a self-enhancing feback: removal of vegetation decreasesτc, which in turn increases the rate of vegetation loss (but which is also compensated for by reduction inV). Solutions to (5) aresketched as a function of the dimensionless parameter G =τ/τcv (ratio of appliedshear stress to vegetation-added critical shear stress) in Figure 5-1. The “humpcurves reflect the dual influence of vegetation cover on critical shear stress andvegetation erodibility; when cover is sparse, there is little vegetation to erode, bwhen cover is dense, the erosion potential is reduced due to the increase inτc.

τ f

f s

f t-----τ=

Rf krvV= krv 1≤,

tddV

(erosion) kvdV τ τc–( )η–=



.c

casesenty sup-

tothe

tion con- etela-

Vegetation regrowth

After vegetation is disrupted (for example, during a storm) it will begin to regrowThe rate of regrowth of vegetation biomass Vb is sometimes modeled using a logistigrowth equation, in which the rate of growth approaches zero as the biomassbecomes large, and also approaches zero if the biomass is very small (in whichthere are few or no organisms to reproduce) (e.g., Thornes, 1990). For the prepurposes, however, it is reasonable to assume that there will always be a readply of seeds and colonizing roots near a devegetated area (recall thatV representsvegetation cover within a rill or channel; the surrounding hillsides are assumedhaveV=Vmax). In view of this, a simpler linear approach seems warranted, with rate of regrowth being proportional to the existing cover:

. (EQ 6)

The parameterkvg is the rate of regrowth on a bare surface, and would be a funcof solar radiation, soil moisture, nutrients, and so on. Note that this approach issistent with the crop growth model used in the WEPP agriculture model (Arnoldal., 1995), which assumes a linear rate of biomass growth and an exponential r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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1

% vegetation cover (V)

Non

dim

ensi

onal

veg

etat

ion

dest

ruct

ion

rate

0.5

1

2

FIGURE 5-1. Vegetationdestruction rate as afunction of percentvegetation cover, fordifferent values of G(nondimensional shearstress).

tddV

(growth) kvg 1 V–( )=



EPP

fi-ignif-

, we

racter-eslving

tionship between plant cover and biomass. The cover-biomass relationship in Wis

(EQ 7)

and the biomass growth rate is

, (EQ 8)

which implies that

. (EQ 9)

Equations (8) and (9) say that although the biomass can continue to grow indenitely, it eventually reaches a point where additional biomass growth does not sicantly increase the fractional vegetation cover on the surface.

Combining the terms for vegetation growth and erosion,

. (EQ 10)

In the next section, we explore analytical solutions to this equation. In doing somake the further simplificationsRf=1 (i.e., vegetation does not affect total shearstress),η=1, andτcs=0.

Characteristic Form Profiles

Nondimensionalization

It is interesting to consider what a vegetation profile across a steady-state (chaistic form) hillslope might look like, and how the presence of vegetation influencthe shape of the hillslope. We can obtain a steady-state vegetation profile by so

V 1 Vb B⁄–( )exp–=

td

dVb

tdd

B 1 V–( )ln–( ) k= =

tddV k

B--- 1 V–( )=

tddV

kvg 1 V–( ) kvdV Rf τ τc–( )–=



ysis,

oeta-0),

germ

equation (10) for the casedV/dt=0, and solving equation (1) for the caseqs = Ex,whereE is an erosion rate that is constant along the profile. To facilitate the analwe introduce the following nondimensionalization:

Vegetation growth time scale (EQ 11)

Nondimensional time (EQ 12)

Nondimensional shear stress (EQ 13)

Nondimensional distance along slope profile (EQ 14)

Vegetation number (EQ 15)

The vegetation number represents the efficiency of vegetation growth relative tdestruction; low values represent fast-growing and/or destruction-resistant vegtion, and vice-versa. Introducing these dimensionless numbers into equation (1we have

, (EQ 16)

which is a quadratic equation that can be solved ifdV/dt is constant or zero.

Wash Profile

The equilibrium vegetation profile depends in part on the rate at which it is beindisrupted by runoff erosion, which may vary along a slope. First we consider thcase of a hillslope profile that is eroded solely by wash (equation (1)), at a uniforateE such thatqs = Ex’. Under that condition, the equilibrium shear stress is

(EQ 17)

Substituting this into equation (16) yields

Tv 1 kvg⁄=

t' t Tv⁄=

τ' τ τ⁄ cv=

x' x L⁄= L, slope length=

Nv

kvdτcv

kvg--------------=

t'ddV

NvV2 1 Nτ'+( )V– 1+=

τ' NE

1p---

x'

1p---

V+= NE, LEkτcv----------=



tress

es

the

ofile,

ofile

es thatert,annot

rmntrolsogra-

. (EQ 18)

Similarly, we can solve for equilibrium slope along the profile using the shear srelationship

. (EQ 19)

Solving for the equilibrium condition,

, (EQ 20)

whereNt = ktLm/τcv. Figure 5-2 depicts solutions to (18) and (20) for different valu

of the dimensionless parametersNv, NE, andNt, assuming typical values ofm=2/3,n=2/3,q=1/p=2/3 (e.g., the Meyer-Peter and Mueller relation) orq=1/p=1/3 (theEinstein-Brown relation). Vegetation cover decreases downslope in response toincreasing shear stress, asymptotically approaching zero asx’ becomes large. Thisvegetation gradient has the effect of increasing the concavity of the hillslope prparticularly when the erosion numberNE is small relative to the other parameters(compare Figure 5-2, top and bottom). The vegetation numberNv controls thedownslope rate of reduction in vegetation cover. WhenNv is large (Figure 5-2, mid-dle), there is an abrupt reduction in vegetation cover in the upper part of the prthat corresponds to a sharp concavity in the topography.

Diffusive Profile

The wash model ignores the effect of creep-related (slope-dependent) processwould tend to produce convex-upward slopes near a drainage divide (e.g., Gilb1909). The profile shape under the action of both creep and wash processes cbe solved analytically, but we can obtain some idea of what a vegetation profilemight look like on a diffusion-dominated slope by solving for the characteristic-foslope as if diffusion were the sole process. Essentially, we assume that wash cothe amount of vegetation but does not contribute significantly to shaping the topphy.

V1

1 NvNEq x'q+

------------------------------= q, 1p---=

τ ktLmx'

mSn=

SNE

q

Nt-------x'q m– 1

Ntx'm 1 NvNEq x'q+( )

-------------------------------------------------+1 n⁄

=



Slope profile

Vegetation profile

Slope profile w/o veg

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Nondimensional distance

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Characteristic form wash profile with vegetation

Slope profile

Vegetation profile


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Characteristic form wash profile with vegetation (Nv=10)

Slope profile

Vegetation profile


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Characteristic form wash profile with vegetation (Ne=10)

Slope profile

Vegetation profile


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Characteristic form wash profile with vegetation (Ne=10, p=3)

Slope profile

Vegetation profile


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Characteristic form wash profile with vegetation (p=3)

Slope profile

Vegetation profile


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Characteristic form wash profile with vegetation (Nv=4, p=3)

FIGURE 5-2. Solutions to equations (18) and (20) for wash-dominated hillslope profiles,showing the effect of vegetation on profile shape.



(23)ege-

mple96),).qui-se and

Assuming that the transport rate by creep is given byqs = kd S, the characteristicform profile is

. (EQ 21)

Substituting into equation (19), the shear stress is given by

. (EQ 22)

Combining with (16) (fordV/dt’=0) gives the vegetation profile

. (EQ 23)

Example solutions to (21) and (23) are shown in Figure 5-3. Note that equationhas two roots. In the example shown in the figure, one of the roots implies full vtation cover along the profile, while the other implies decreasing cover. This exaof bi-stability is similar to the bistable landscape states explored by Howard (19as well as the multiple phase-states in the soil erosion model of Thornes (1990Depending on the initial conditions, the system may evolve toward a different elibrium state. This analytical example of a diffusion-dominated profile is of courrather artificial because it does not incorporate the linkage between slope form

SULkd--------x' Ndx'= =

τ'ktL

mNdn

τcv------------------x'm n+ Ndtx'm n+= =

NvV2 1 NvNdtx'm n++( )V– 1+ 0=

Hillslope profile Vegetation (solution 1)Vegetation (solution 2)

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Non

dim

ensi

onal

ele

vatio

n / V

eget

atio

n co

ver

%

FIGURE 5-3. Solutions to equation(23) for equilibrium vegetation coveron a diffusion-dominated hillslope onwhich vegetation growth is limited bywash erosion.


Numerical Examples

and

atch-ic thegeta- dur- theac-sions duease of

wash erosion. To explore the complete interaction of wash, diffusive transport, vegetation growth, numerical simulations are needed.

Numerical Examples

Figure 5-4 shows the topography and percent vegetation cover in a synthetic cment simulated using the CHILD model. The basin is close to a state of dynamequilibrium between uplift and erosion. Vegetation cover ranges from 100% onhillslopes to very low values in the main channels. (The presence of a small vetion cover within the channels is due to the fact that vegetation always regrowsing interstorm periods, regardless of position within the landscape). In general,downstream transition from full to sparse vegetation cover is a function of two ftors: (1) the threshold imposed by the vegetation itself, which retards runoff eroon upper slopes (as seen in Figure 5-2), and (2) the rounding of hillslope profileto diffusion, which reduces slopes and thus shear stresses (as in the extreme cFigure 5-3). In this particular example, the hillslope diffusivity constantkd is suffi-ciently high that the latter effect is more important.

0

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2000

3000

4000

5000

6000

7000 0

1000

2000

0

200

400

(a)

(b)

FIGURE 5-4. Simulated catchment with variable vegetation cover. (a) Topography. (b)Percent vegetation cover. White = 100% cover, black = 0% cover.



tain-deled

slope

alsopo-ctedcityittlereasetoted) was inn theith a

alleynd

Figure 5-5 illustrates the effect of a hypothetical decrease in the maximum susable vegetation cover. The decrease in sustainable vegetation maximum is moby a tenfold decrease in the threshold parameterτcv. The decrease inτcv is accompa-nied by an increase in the effectiveness of runoff erosion, which leads to an upextension of the sparsely-vegetated tributaries.

Vegetation cover typically influences not only surface resistance to erosion, butthe soil infiltration capacity and hence runoff production. Figure 5-6 shows a hythetical example in which the synthetic catchment shown in Figure 5-4 is subjeto a complete loss in vegetation accompanied by a reduction in infiltration capa(Ic = 0). The figure depicts areas of erosion (red), deposition (not shown), and lor no change (green). The net effect of vegetation loss in this example is an incin the rate of scour along the main channels. This result is somewhat contrary intuition, which would suggest that a loss of vegetation should lead to acceleraerosion on hillslopes, where the initial vegetation cover (and thus the thresholdgreatest (this type of behavior was observed by Tucker and Slingerland (1997)experiments with a constant-threshold model). The explanation appears to lie itopography of the catchment and the nature of the hillslope-valley transition. Wrelatively high diffusivity constantkd, the valley network extent in the simulation islimited by hillslope diffusion rather than by the threshold imposed by vegetationcover. Thus, reducing the threshold does not lead to a rapid expansion of the vnetwork, as it did in the constant-threshold experiments of Tucker and Slingerla

1000

2000

3000

4000

FIGURE 5-5. Short term-effectof a decrease inτcv on percentvegetation cover in the syntheticcatchment shown in Figure 5-4.


Numerical Examples

aycularn-ls thecipateodelene

(1997). An implication of this result is that the nature of catchment responses mdepend to a large extent on the nature of hillslope-valley transitions, and in partion whether that transition is governed by a process transition or by a vegetatioimposed erosion threshold. There is still much to be learned about what controtype of response in the model and, by extension, in natural catchments. We antithat this question can be fruitfully addressed through a program of systematic msensitivity experiments and comparison of the model predictions with the Holocstratigraphic record observed at Fort Riley, Kansas.

0

1000

2000

3000

4000

FIGURE 5-6. Patterns of erosion(red=max. erosion, green=nochange) following a complete lossof vegetation and a correspondingreduction in infiltration capacityin the synthetic catchment shownin Figure 5-4.


6-1

Chapter 6

Understanding the Interactions of Multiple Grain Sizes

A-1

Appendix A

Meandering: A Simple Model

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