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1

Chapter 12

The Channel-Hillslope Integrated Landscape

Development Model (CHILD)

Gregory E. Tucker1, Stephen T. Lancaster2, Nicole M. Gasparini, and Rafael L.

Bras

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology,

Cambridge, MA USA

1. INTRODUCTION

Numerical models of complex Earth systems serve two important

purposes. First, they embody quantitative hypotheses about those systems

and thus help researchers develop insight and generate testable predictions.

Second, in a more pragmatic context, numerical models are often called

upon as quantitative decision-support tools. In geomorphology,

mathematical and numerical models provide a crucial link between small-

scale, measurable processes and their long-term geomorphic implications. In

recent years, several models have been developed that simulate the structure

and evolution of three-dimensional fluvial terrain as a consequence of

different process “laws” (e.g., Willgoose et al., 1991a; Beaumont et al.,

1992; Chase, 1992; Anderson, 1994; Howard, 1994; Tucker and Slingerland,

1994; Moglen and Bras, 1995). By providing the much-needed connection

between measurable processes and the dynamics of long-term landscape

evolution that these processes drive, mathematical landscape models have

posed challenging new hypotheses and have provided the guiding impetus

1 Now at: School of Geography and the Environment, University of Oxford, Mansfield Road,

Oxford OX1 3TB United Kingdom, email: greg.tucker@geog.ox.ac.uk

2 Now at: Department of Geosciences, Oregon State University, Corvallis, OR

2 Chapter 12

behind new quantitative field studies and DEM-based analyses of terrain

(e.g., Snyder et al., 2000). The current generation of models, however, shares

a number of important limitations. Most models rely on a highly simplified

representation of drainage basin hydrology, treating climate through a simple

“perpetual runoff” formulation. The role of sediment sorting and size-

dependent transport dynamics has been ignored in most studies of drainage

basin development, despite its importance for understanding the interaction

between terrain erosion and sedimentary basin deposition (e.g., Gasparini et

al., 1999; Robinson and Slingerland, 1998). Furthermore, with the exception

of the pioneering work of Braun and Sambridge (1997), the present

generation of models is inherently two-dimensional, describing the dynamics

of surface evolution solely in terms of vertical movements without regard to

lateral displacement by tectonic or erosional processes.

Our aim in this paper is to present an overview of the Channel-Hillslope

Integrated Landscape Development (CHILD) model, a new geomorphic

modeling system that overcomes many of the limitations of the previous

generation of models and provides a general and extensible computational

framework for exploring research questions related to landscape evolution.

We focus here on reviewing the underlying theory and illustrating the

capabilities of the model through a series of examples. Discussion of the

technical details of implementation is given by Tucker et al. (1999, 2000)

and Lancaster (1998). We begin by briefly reviewing previous work in

landscape evolution modeling. We then discuss the theory and capabilities of

the modeling system, and present a series of examples that highlight those

capabilities and yield some useful insights into landscape dynamics.

2. BACKGROUND

The first quantitative geomorphic process models began to appear in the

1960s, stimulated by the combination of an intellectual shift toward

investigating the mechanics of erosion and sedimentation processes, and the

appearance of digital computers. The earliest models were one-dimensional

slope simulations developed to explore basic concepts in hillslope profile

development (e.g., Culling, 1960; Scheidegger, 1961; Ahnert, 1970; Kirkby,

1971; Luke, 1972; Gossman, 1976). These studies helped to quantify and

formalize some of the concepts of hillslope process and form enunciated by

early workers such as Gilbert (1877) and Penck (1921). Similar one-

dimensional (1D) approaches have more recently been used to examine the

evolution of stream profiles (e.g., Snow and Slingerland, 1987) and alluvial

deposystems (e.g., Paola et al., 1992; Robinson and Slingerland, 1998).

There are clear limitations of the 1D approach for understanding terrain

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

3

morphology, however, and these limitations prompted early efforts to extend

erosion models to two dimensions, though still with a focus primarily on

hillslope morphology (Ahnert, 1976; Armstrong, 1976; Kirkby, 1986).

Driven in part by technological advances, there has been a flowering of

landscape evolution models during the past decade. Many of these models

have focused on watershed evolution and dynamics (Willgoose et al., 1991a;

Howard, 1994; Moglen and Bras, 1995; Coulthard et al., 1997; Tucker and

Slingerland, 1997; Densmore et al., 1998). Although spatial scale is often not

specified, these modeling studies have generally focused on the formation of

hillslope-valley topography within small- to moderate-sized drainage basins

(on the order of several square kilometers or smaller). In parallel with these

developments in watershed geomorphology, a number of researchers have

attempted to model the evolution of terrain on the scale of a mountain range

or larger (e.g., Koons, 1989; Beaumont et al., 1992; Lifton and Chase, 1992;

Anderson, 1994; Tucker and Slingerland, 1994, 1996; Braun and Sambridge,

1997). In these applications, computational limitations dictate the use of a

coarse spatial discretization in which individual grid cells are much larger

than the scale of an individual hillslope, making it impossible to address

explicitly the role of hillslope dynamics, and raising the issue of “upscaling”

as a need in large-scale geomorphic models (Howard et al., 1994). A third

category of models includes cellular statistical-physical models that employ

simple rule sets to address the origin and nature of scaling properties

observed in river networks and terrain (e.g., Chase, 1992; Rigon et al., 1994;

Rodriguez-Iturbe and Rinaldo, 1997). Finally, a number of two-dimensional

models of hillslope-scale soil erosion and rill development have been

developed to study and predict patterns of slope erosion and drainage pattern

initiation (e.g., Smith and Merchant, 1995; Favis-Mortlock, 1998; Mitas and

Mitasova, 1998).

Despite significant progress in theory and model development over the

past decade, the current generation of physically based models suffers from

several limitations: (1) temporal variability in rainfall and runoff has been

largely ignored (cf. Tucker and Bras, 2000); (2) with a few exceptions (e.g.,

Ijjasz-Vasquez et al., 1992; Tucker and Bras, 1998), runoff is usually treated

as spatially uniform (Hortonian) across the landscape, despite the well

known importance of variable source-area runoff generation in humid

regions; (3) lateral erosion by channels has been ignored in the context of

drainage basin evolution; (4) most models use a fixed and uniform spatial

discretization in which only vertical movements of the terrain surface are

allowed (for an exception, see Braun and Sambridge, 1997); (5) the role of

heterogeneous sediment and sorting dynamics is usually ignored for

simplicity, despite their potential impacts on stream profile shape (e.g.,

4 Chapter 12

Snow and Slingerland, 1987; Sinha and Parker, 1996; Robinson and

Slingerland, 1998) and drainage basin structure (Gasparini et al., 1999); and

(6) few efforts have been made to examine the coupling between erosional

and depositional systems (e.g., Johnson and Beaumont, 1995; Tucker and

Slingerland, 1996; Densmore et al., 1998).

3. MODEL FORMULATION

3.1 Overview

The CHILD model simulates the evolution of a topographic surface and

its subjacent stratigraphy under a set of driving erosion and sedimentation

processes and with a prescribed set of initial and boundary conditions (Fig.

1). Designed to serve as a computational framework for investigating a wide

range of problems in catchment geomorphology, CHILD is both a model, in

the sense that it comprises a set of hypotheses about how nature works, and a

software tool, in the sense that it provides a simulation environment for

exploring the consequences of different hypotheses, parameters, and

boundary conditions. Here we will use the term “model” to refer collectively

to the software and the assumptions and hypotheses embedded within it.

The process modules in CHILD are summarized graphically in Figure 2.

Processes incorporated in the model include: (1) climate forcing via a

sequence of discrete storm events with durations, intensities, and inter-

arrival times that may be either random or constant; (2) generation of runoff

by infiltration-excess or saturation-excess mechanisms; (3) downslope

routing of water and sediment using a steepest-descent method; (4)

detachment (erosion) of sediment or bedrock by channelized surface runoff

in rills or stream channels; (5) water-borne downslope transport of detached

sediment; (6) transport of sediment by soil creep and related processes on

hillslopes; (7) meandering of large stream channels; (8) overbank

sedimentation on floodplains; and (9) tectonic deformation. Note that not all

of these processes need to be, or even should be, considered in any particular

application. The point of including a number of different processes is to

allow one to investigate different types of geomorphic system under

different space and time scales, using a common modeling framework that

handles the basic spatial and temporal simulation framework.

In addition to these process modules, CHILD includes capabilities for

managing the spatial simulation framework. The use of an adaptive, irregular

spatial discretization adds several useful capabilities (Braun and Sambridge,

1997; Tucker et al., 2000), including the ability to vary spatial resolution and

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

5

to incorporate the horizontal components of erosion processes (e.g., stream

channel migration) and tectonic motions (e.g., strike-slip displacement). In

addition, the model can simulate depositional history and stratigraphy by

tracking and updating “layers” of deposited material underlying each point

in the landscape, thereby making it possible to model coupled erosion-

deposition systems such as mountain drainage basins and their associated

alluvial fans (e.g., Ellis et al., 1999).

3.2 Continuity of Mass and Topographic Change

Changes in ground surface height, z(x,y), are described by the continuity

of mass equation for a terrain surface, which is expressed in terms of the

divergence of the sediment flux qs (dimensions of bulk volume rate per unit

width):

),,( tyxU

t

z+−∇=

∂

∂s

q

where z is surface height, t is time, and U is a source term that represents

baselevel change or tectonic uplift. The first term on the right-hand side

embodies several different sediment transport and erosion terms and can take

on a number of different forms depending on the assumptions made about

process mechanics. The formulations of the transport and erosion terms and

the numerical solution to (1) are described below.

3.3 Spatial Framework

In order to avoid the limitations associated with grid-based models, the

terrain surface may be discretized as a set of points (nodes) in any arbitrary

configuration. These nodes are connected to form a triangulated irregular

mesh (Figs. 1, 3) (Braun and Sambridge, 1997; Tucker et al., 2000). The

mesh is constructed using the Delaunay triangulation, which is the

(generally) unique set of triangles having the property that a circle passing

through the three nodes of any triangle will contain no other nodes (e.g., Du,

1996). The use of an irregular spatial framework offers several significant

advantages: (1) the model resolution can vary in space in order to represent

certain landscape features, such as floodplains or regions of complex terrain,

at a locally high level of detail (e.g., Fig. 1); (2) adaptive remeshing can be

used to adjust spatial resolution dynamically in response to changes in the

nature or rates of processes occurring at a particular location (e.g., Braun and

Sambridge, 1997; Tucker et al., 2001; and examples below); (3) nodes can

(1)

6 Chapter 12

be moved horizontally as well as vertically, making it possible to simulate

lateral and surface-normal, as opposed to purely vertical, erosion (as, for

example, in the cases of meandering channels and cliff retreat); (4) nodes

can be added to simulate lateral accretion of, for example, point bars in

meandering streams or accretionary wedges at active margins; and (5) the

terrain can be coupled with 3D kinematic or dynamic models of tectonic

deformation in order to simulate interactions between crustal deformation

(e.g., shortening, fold growth) and topographic change. The data structures

used to implement the triangular mesh are described by Tucker et al. (2001).

The Delaunay framework lends itself to a numerical solution of the

continuity equation (Eq 1) using finite-volume methods. Each node (vertex)

in the triangulation, Ni, is associated with a Voronoi (or Thiessen) polygon

of surface area Λi (Fig. 3), in which the polygon edges are perpendicular

bisectors of the edges connecting the node to its neighbors (e.g., Du, 1996;

Guibas and Stolfi, 1986). Thus, the Delaunay triangulation defines the

connectivity between adjacent nodes, while the associated Voronoi diagram

defines the surface area associated with each node as well as the width of the

interface between each pair of adjacent nodes (Fig. 3B). In CHILD, each

Voronoi polygon is treated as a finite-volume cell. Continuity of mass for

each node is written using an ordinary differential equation:

∑

=

Λ

=i

M

j

Sji

i

iQ

dt

dz

1

1,

where zi is the average surface height of node i, Mi is the number of neighbor

nodes connected to node i, and QSji is the total bulk volumetric sediment flux

from node j to node i (negative if the net flux is from i to j). Note that by this

method it is only possible to describe the average rate of erosion or

deposition within a given Voronoi polygon. As described below, the method

used to integrate the flux terms depends on whether the flux is two-

dimensional (e.g., for diffusive sediment transport or kinematic-wave

overland flow routing) or one-dimensional (for streamwise water and

sediment routing). For discussion of the implementation, application, and

advantages of irregular discretization in landscape models, the reader is

referred to Braun and Sambridge (1997) and Tucker et al. (2001).

3.4 Temporal Framework

One of the challenges in modeling terrain evolution lies in addressing the

great disparity between the time scales of topographic change (e.g., years to

geologic epochs) and the time scales of storms and floods (e.g., minutes to

days). Most previous models of drainage basin evolution have dealt with this

(2)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

7

disparity by simply assuming a constant average climatic input (e.g., a

steady rainfall rate or a “geomorphically effective” runoff coefficient). This

approach, while computationally efficient, has three drawbacks: (1) it

ignores the influence of intrinsic climate variability on rates of erosion and

sedimentation (e.g., Tucker and Bras, 2000); (2) it fails to account for the

stochastic dynamics that arise when a spectrum of events of varying

magnitude and frequency acts in the presence of geomorphic or hydrologic

thresholds; and (3) the approach typically relies on a poorly calibrated “cli-

mate coefficient” that cannot be directly related to measured climate data.

In order to surmount these limitations, and to address the role of event

magnitude and frequency in drainage basin evolution, CHILD uses a

stochastic method to represent rainfall variability. The method is described

in detail by Tucker and Bras (2000), and is only briefly outlined here. In

solving the continuity equation, the model iterates through a series of

alternating storms and interstorm periods, based on the Poisson rainfall

model developed by Eagleson (1978). Each storm event is associated with a

constant rainfall intensity, P, a duration, Tr, and an inter-arrival “waiting

time”, Tb (Fig. 4). For each storm, these three attributes are chosen at

random from exponential probability distributions, the parameters for which

can be readily derived from hourly rainfall data (Eagleson, 1978; Hawk,

1992). Alternatively, storm intensity, duration, and frequency may be kept

constant, in which case the approach reduces to the “effective rainfall

intensity” approximation (Tucker and Slingerland, 1997). In either case,

storms are approximated as having constant intensity throughout their

duration, and the same assumption is also applied to the resulting

hydrographs. Runoff-driven transport and erosion processes (described

below) are computed only during storm events. Other processes, including

diffusive creep transport and tectonic deformation, are assumed to occur

continuously, and are updated at the end of each interstorm period (Fig. 5).

Note that the model imposes no special restrictions on time scale, aside

from the fact that it is designed for periods longer than the duration of a

single storm. For simulations involving terrain evolution over thousands to

millions of years (e.g., Tucker and Slingerland, 1997), however, it becomes

computationally intractable to simulate individual storms. For many

applications this problem can be overcome by simply amplifying the storm

and interstorm durations. As long as the ratio Tr/Tb remains the same, the

underlying frequency distributions are preserved. Perturbations in climate

can also be simulated by changing the parameters of the three frequency

distributions (Fig. 4).

8 Chapter 12

3.4.1 Stochastic Rainfall: Example

Figure 6 illustrates the behavior of the model under stochastic rainfall

forcing, in a case where a high erosion threshold (see below) lends the

system a high sensitivity to extreme events. The initial condition consists of

a 30 degree slope upon which are superimposed small random perturbations

in the elevation of each node. Erosion of the slope in response to a random

series of rainfall and runoff events (Fig. 6A) is highly episodic (Fig. 6B). In

this example, a gully forms early on in response to a series of large-

magnitude and relatively long-duration storms (Figs. 6C and E). The gully

develops in an area where the topography of the initial surface leads to local

flow convergence. The reduction in gradient along the gully effectively

stabilizes the system, so that later events have little or no impact. Subsequent

mass movement by slope-driven diffusive creep (see below) leads to gradual

healing of the scar (Figs. 6D and F).

3.5 Surface Hydrology and Runoff Generation

Surface runoff collected at each node on the mesh is routed downslope

toward one of its adjacent neighbor nodes, following the edge that has the

steepest downhill slope (Fig. 3). If a closed depression occurs on the mesh,

water can either be assumed to evaporate at that point, or alternatively a

lake-filling algorithm can be invoked to find an outlet for the closed

depression (Tucker et al., 2000).

The local contribution to runoff at a node is equal to the effective runoff

rate (defined below) multiplied by the node’s Voronoi area, Λ. The drainage

area, A, at a node is the sum of the area of all Voronoi cells that contribute

flow to that node. Total surface discharge can be computed from drainage

area using one of three methods. The first two assume that runoff generation

is spatially uniform, while the third represents variable-source area runoff

generation.

3.5.1 Hortonian (infiltration-excess) runoff

Runoff production (rainfall rate minus infiltration rate) is assumed to be

uniform across the landscape. Assuming steady-state flow, the surface

discharge at any point is equal to

cc IPAIPQ >−= ,)( ,

where Ic is infiltration capacity (L/T).

(4)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

9

3.5.2 Excess storage capacity runoff

Under this approach, the soil, canopy, and surface are collectively

assumed to have a finite and spatially uniform capacity to absorb rainfall.

Any rainfall exceeding this storage depth will contribute to runoff according

to

srr

r

srr DPT,

T

DPT

R>

−

=

where R is local runoff rate (L/T), Dsr is the soil-canopy-surface retention

depth (L), and the resulting discharge at any point is Q = RA. Note that

equation (4) describes a runoff rate that is constant throughout a storm and

equal to the total volume of excess rainfall divided by the storm duration.

Note also that R = 0 if Dsr > TrP.

3.5.3 Saturation-excess runoff

With this option, a modified form of O’Loughlin’s (1986)

topographically based method is used to partition rainfall between overland

and shallow subsurface flow. The capacity for shallow subsurface flow per

unit contour length (qsub) is assumed to depend on local slope (S) and soil

transmissivity (T, dimensions of L2/T),

TS

w

Q

q== sub

sub

where contour length is represented by the width of adjoining Voronoi cell

edges, w. The surface flow component is equal to the total discharge minus

the amount that travels in the subsurface,

TSwPATSwPAQ >−= , .

Here, Q represents surface discharge resulting from a combination of

saturation-excess overland flow and return flow. Note that this method

assumes hydrologic steady state for both surface and subsurface flows, and

thus is most applicable to prolonged storm events and/or highly permeable

shallow soils.

(5)

(5)

(6)

10 Chapter 12

3.5.4 Example

The mechanism of runoff production can impact both terrain morphology

and dynamic responses to changing climate, land-use, or tectonism. For

example, theoretical studies have shown that the mode of runoff production

can have a significant impact on terrain morphology, drainage density, and

the scaling of drainage density with relief and climate (Kirkby, 1987; Ijjasz-

Vasquez et al., 1992; Tucker and Bras, 1998). Figure 7 compares two

simulated drainage basins formed under Hortonian and saturation-excess

runoff production, respectively. All other parameters in the two simulations

are identical. In the saturation case, runoff is rarely generated on hillslopes.

As a result, hillslopes are steep and highly convex (reflecting the dominance

of diffusive creep-type sediment transport; see below). The difference is

reflected in slope-area plots for the two simulated basins. In the case of the

saturation-dominated basin, the hillslope-channel break is well described by

the line of saturation for the mean-intensity storm (Fig. 7D).

3.6 Hillslope Mass Transport

Sediment transport by “continuous” hillslope processes such as soil creep

is modeled using the well-known hillslope diffusion equation (e.g., Culling,

1960; McKean et al., 1993),

zkzk

t

z

dd

2

creep

)( ∇=∇−−∇=

∂

∂,

where kd is a transport coefficient with dimensions of L2/T. Numerical

solution to equation (7) is obtained using a finite-volume approach (Tucker

et al., 2000). The net mass flux for a node is taken as the sum of the mass

fluxes through each face of its Voronoi polygon (Eq (2)). For each pair of

adjacent nodes, the gradient across their shared Voronoi polygon edge is

approximated as the gradient between the nodes themselves. The total flux

between each pair of nodes is thus equal to the topographic gradient between

them multiplied by the width of their shared Voronoi edge, so that Eq (7) is

approximated numerically as

∑

=

Λ

−=

∂

∂i

M

j

ijij

i

di wS

k

t

z

1

creep

,

(7)

(8)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

11

where Sij = (zi - zj)/λij is the downslope gradient from node i to node j, λij is

the distance between i and j (i.e., the length of the triangle edge connecting

them), and wij is the width of the shared Voronoi polygon face (Fig. 3B).

For steeper gradients, a nonlinear form of Eq (7) is arguably more

appropriate to describe the effects of accelerated creep and intermittent

landsliding (e.g., Anderson and Humphrey, 1990; Roering et al., 1999).

Although this type of nonlinear rate law is not presently coded in CHILD, its

incorporation would be straightforward.

Note also that equation (7) is intended to model creep-type processes

rather than wash erosion. Instead, wash and channel erosion are effectively

lumped together under the same formulation, as described below. This

approach has the obvious disadvantage that wash is effectively treated as a

form of rill erosion in which rills have the same hydraulic geometry (i.e.,

width, depth, and roughness properties) as larger channels, with all the

attendant limitations this implies. On the other hand, lumping rill and

channel erosion in a single “runoff erosion” category has the advantage of

simplicity: no extra parameters are needed to differentiate between hillslopes

and channels (as is the case, for example, in the model of Willgoose et al.,

1991a), which emerge solely as a result of process competition (Kirby, 1994;

Tucker and Bras, 1998). Thus, while we acknowledge a need for a more

rigorous sub-model for wash erosion in the future, the treatment of wash as a

general form of channel erosion seems justified given the aims of the model

and the present uncertainty regarding the dynamics of channel initiation.

3.7 Water Erosion and Sediment Transport

At each node, the local rate of water erosion is equal to the lesser of (1)

the detachment capacity, or (2) the excess sediment transport capacity. Both

of these are represented as power functions of slope and discharge, and they

are assumed to be mutually independent. Deposition occurs where sediment

flux exceeds transport capacity (for example, due to a downstream reduction

in gradient). The maximum detachment capacity depends on local slope and

discharge according to

b

b

b

p

cb

n

m

tbc S

W

Q

kkD

−

=

θ

,

where Dc is the maximum detachment (erosion) capacity (L/T), W is channel

width, θcb is a threshold for particle detachment (e.g., critical shear stress),

and kb, kt, mb, nb, and pb are parameters. Note that with suitably chosen

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12 Chapter 12

parameters, equation (9) can represent either excess shear stress (i.e., τ = bed

shear stress, τcb = critical shear stress for detachment) or excess stream

power (Whipple and Tucker, 1999). The shear stress formulation is similar

to that used in the drainage basin evolution models of Howard (1994) and

Tucker and Slingerland (1997), as well as a number of soil erosion models

(e.g., Foster and Meyer, 1972; Mitas and Mitasova, 1998). If the Manning

equation is used to model roughness, mb=0.6, nb=0.7, and kt=ρgn3/5, where ρ

is water density, g is gravitational acceleration, and n is Manning’s

roughness coefficient. If the Chezy equation is used, mb=2/3, nb=2/3, and

kt=ρgC-1/2, where C is the Chezy roughness coefficient (for derivations, see

Tucker and Slingerland, 1997; Whipple and Tucker, 1999).

Channel width is computed empirically, using the well-known scaling

relationships between channel width and discharge (Leopold and Maddock,

1958; Leopold et al., 1964):

b

s

bwbbb QkWQQWW

ω

ω

== ,)/(

where Wb is bankfull channel width, Qb is a characteristic discharge (such as

bankfull or mean annual), kw is bankfull width per unit scaled discharge, and

ωb and ωs are the downstream and at-a-station scaling exponents,

respectively. Although these laws were developed for alluvial streams, they

appear to be applicable to other fluvial systems (e.g., Ibbitt, 1997) such as

steep mountain channels (Snyder et al., 2000).

The transport capacity for detached sediment material of a single grain

size is based on a generalization of common bedload and total-load sediment

transport formulas, which are typically expressed as a function of excess

shear stress or stream power (e.g., Yang, 1996). For steady, uniform flow in

a wide channel,

f

f

f

p

c

n

m

tfs S

W

Q

kWkC

−

=

θ

,

where Cs is transport capacity (L3/T) and kf, kt, mf, nf, and pf are parameters.

As with equation (9), equation (11) can be expressed in terms of excess

shear stress or stream power using suitably chosen values for kt, mf, and nf.

For transport of multiple sediment size-fractions, an alternative approach

based on the method of Wilcock (1997, 1998) is used (this is described

below).

Three end-member cases can arise from equations (9) and (11):

detachment-limited behavior, transport-limited behavior, and mixed-channel

behavior:

(10)

(11)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

13

1. Detachment-limited: If the sediment transport capacity is everywhere

much larger than the rate of sediment supply, the rate of water erosion is

simply equal to the maximum detachment rate,

c

bD

t

z−=

∂

∂,

where zb represents elevation of the channel bed above a datum within the

underlying rock column. This type of formulation has been used in a number

of studies to represent bedrock channel erosion (or more generally, detach-

ment-limited erosion of cohesive, cemented, or non-granular materials) (e.g.,

Howard and Kerby, 1983; Seidl and Dietrich, 1992; Anderson, 1994;

Howard et al., 1994; Moglen and Bras, 1995; Sinclair and Ball, 1996; Stock

and Montgomery, 1999; Whipple and Tucker, 1999; Snyder et al., 2000). An

important assumption is that the sediment flux has no direct control on the

rate of incision, as long as there is sufficient capacity to transport the eroded

material (cf. Sklar and Dietrich, 1998). Note that this case has the practical

advantage of being efficient to solve numerically. Though widely used,

however, the accuracy of this approximation for long-term stream profile

development remains to be evaluated.

2. Transport-limited: If sufficient sediment is always available for

transport and/or the bed material is easily detached (i.e., high kb), streams are

assumed to be everywhere at their carrying capacity. Under this condition,

continuity of mass gives the local rate of erosion or deposition as

x∂

∂

−

−=

∂

∂WC

t

zsb /

)1( 1

ν

,

where ν is bed sediment porosity (usually absorbed into the transport coeffi-

cient kf) and x is a vector oriented in the direction of flow. Transport-limited

behavior has been assumed in a number of models (e.g., Snow and Slinger-

land, 1987; Willgoose et al., 1991a; Tucker and Bras, 1998; Gasparini et al.,

1999), though its applicability to bedrock streams has been questioned (e.g.,

Howard et al., 1994).

3. Mixed-channel systems: The detachment and transport formulas imply

a third category of behavior that arises under conditions of (1) active erosion

into resistant material (e.g., bedrock) and (2) high sediment supply. Under

these conditions, active detachment of bed material must occur (by

definition), but the sediment supply rates are sufficiently high that the local

rate of incision is limited by the excess transport capacity (e.g., Tucker and

Slingerland, 1996). Stream channels falling into this category might be

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14 Chapter 12

expected to have (on average) a partial cover of alluvium over bedrock; we

thus refer to streams falling into this category as mixed-channel systems

(Howard, 1998). Under certain conditions, the transition point between one

type of behavior (e.g. detachment-limited) and another (e.g. mixed) can be

computed analytically. Mixed channel behavior is discussed in greater depth

by Whipple and Tucker (in review).

3.7.1 Example

In the special case of a constant rate of surface lowering, equations (12)

and (13) both imply a power-law relationship between channel gradient and

contributing area (Willgoose et al., 1991b; Howard, 1994; Whipple and

Tucker, 1999), which is consistent with river basin data (e.g., Hack, 1957;

Flint, 1974). Figure 8 shows an example of such scaling for two simulated

landscapes. The straight lines indicate the trend that would occur under

purely transport-limited conditions (solid line; e.g., Willgoose et al., 1991b)

and under purely detachment-limited conditions (dashed line; e.g., Howard,

1994). Theoretical considerations suggest that longitudinal profile concavity,

which is indicated by the slope of the lines on Figure 8, should generally be

lower in transport-limited alluvial channels (Howard, 1994). The intersection

of the two lines indicates the point at which the gradient required to transport

eroded sediment becomes equal to the gradient required to detach particles.

Upstream of this point, channel gradient is dictated by detachment capacity;

downstream, the channel falls into the “mixed” category in which active

incision occurs but the gradient is controlled by sediment supply. Under

constant runoff, the transition point is abrupt (Fig. 8A), but in the more

realistic case of variable flows, the transition is gradational and spread over

two or more orders of magnitude in drainage area (Fig. 8B). This result

implies that such detachment-to-transport transitions, even if they do exist,

would be very difficult to identify on the basis of morphology alone (this of

course excludes channel-type transitions that are forced by tectonic or other

controls).

3.8 Extension to Multiple Grain Sizes

Size-selective erosion, transport, and deposition are important as agents

that control the texture of alluvial deposits. Textural properties of ancient

strata are important not only for the information they reveal about the

geologic past (e.g., Paola et al., 1992; Robinson and Slingerland, 1998), but

also for their control on the movement and storage of water and hydrocarbon

resources (e.g., Koltermann and Gorelick, 1992). Perhaps surprisingly,

recent work has also shown that the dynamics of size-selective erosion and

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

15

transport can have a significant impact on drainage basin architecture and

evolution (Gasparini, 1998; Gasparini et al., 1999). Size-selective sediment

transport and armoring can also exert important controls on the erosional

history of artificial landforms such as mine tailing heaps, and are therefore

important for engineering applications (Willgoose and Riley, 1998).

To model size-selective erosion and deposition, CHILD uses a two-

fraction (sand and gravel) approach based on the bedload entrainment and

transport functions developed by Wilcock (1997, 1998). The rock or

sediment column underlying each node in the model contains a mixture of

sand and gravel sediment fractions. An active layer of depth Lact defines the

depth over which sediment near the surface is well mixed and accessible for

active erosion and deposition (Gasparini, 1998; Gasparini et al., 1999). The

transport capacities of the two size fractions are given by

54

51

1

1

.

cg

.

gw

sg g)s(

fC

q

τ

τ

−

ρ

τ

−

=

54

51

1

1

.

cs

.

sw

ss g)s(

fC

q

τ

τ

−

ρ

τ

−

=

where qsg and qss are the transport rates of gravel and sand, respectively

(kg/ms), Cw is a dimensionless constant equal to 11.2, fg and fs are the

fractions of gravel and sand in the bed, ρ is water density, s is the ratio of

sediment and water density, g is gravitational acceleration, τ is bed shear

stress, and τcg and τcs are the critical shear stresses needed to entrain gravel

and sand, respectively.

Wilcock (1997; 1998) analyzed the relative mobility of sand and gravel

fractions in gravel-sand mixtures, and found that the initiation of motion

threshold for both fractions approaches a constant (and minimum) value for

mixtures containing more than about 40% sand. The threshold of motion

criterion criteria used in CHILD’s gravel-sand transport module is based on

a piecewise linear fit to the data of Wilcock (1997) (Gasparini et al., 1999).

The active layer represents the depth over which active particle exchange

takes place. For modeling instantaneous transport, the active layer is

typically defined on the basis of grain diameter. For modeling average

transport rates over the duration of one or many floods, this definition is

inappropriate, because local scour and the movement of bars and bedforms

(14)

(15)

16 Chapter 12

allow the flow to access significantly more near-bed sediment than simply

the uppermost one or two grain diameters. Paola and Seal (1995) suggested

that bankfull channel depth might be an appropriate choice for active layer

thickness for calculating long-term average transport rates. However, in the

absence of data on what controls the “effective mixing depth” over a given

time period, we adopt here the simple approach of using an active layer of

constant thickness. Sensitivity experiments by Gasparini (1998), which show

little variation in equilibrium texture patterns with varying active layer

depth, provide some justification for this approach, though we acknowledge

a need for deeper understanding of this issue.

Detachment of cohesive or intact sediment is assumed to be size-

independent and governed by Eq (9). When the multiple grain-size option is

used, detached material is assumed to break down into a user-specified

proportion of gravel and sand, which is then subject to differential

entrainment and transport according to Eqs (14) and (15).

3.9 Deposition and Stratigraphy

There has been an increasing recognition of the importance of coupling

between erosional and depositional systems (e.g., Humphrey and Heller,

1995; Johnson and Beaumont, 1995; Tucker and Slingerland, 1996;

Densmore et al., 1998). An important goal behind developing CHILD has

been to create a system that can be used to investigate these interactions and

their role in shaping the terrestrial sedimentary record. For this reason,

CHILD includes a “layering” module that records depositional stratigraphy.

Each node in the model is underlain by a column of material divided into

a series of layers of variable thickness and properties. Physical attributes

associated with each layer include the relative sand and gravel fractions (if

applicable), the median grain size of each sediment fraction, and the material

detachability coefficient, kb. These properties are assumed to be

homogeneous within a given layer. The time of most recent deposition is

also stored for each layer, so that chronostratigraphy can be simulated.

Finally, each layer also records the amount of time it has spent exposed at

the surface, which is useful for identifying periods of quiescence and may be

applicable to modeling exposure-age patterns in conjunction with

cosmogenic isotope studies.

The active layer depth is fixed in time and space. When material is

eroded from the surface, the active layer is replenished with material from

the layer below. The active layer texture and time of surface exposure are

then updated as a weighted average between the current properties of the

active layer and those of the layer below. Bedrock is never mixed into

sediment layers. When there are no sediment layers below the surface, the

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

17

active layer is depleted until no sediment remains and the channel is on

bedrock. During deposition, material from the active layer is moved into the

layer below before material is deposited into the active layer, so that the

active layer depth remains constant. The layers below the active layer have a

maximum depth; when this depth will be exceeded due to deposition, a new

layer is created.

3.9.1 Example

Fault-bounded mountain ranges and alluvial fans in regions of tectonic

extension are classic examples of close coupling between erosional and

depositional systems (e.g., Leeder and Jackson, 1993; Ellis et al., 1999).

Alluvial fan stratigraphy is shaped by a combination of forces, including

extrinsic factors such as tectonic uplift/subsidence and climate change, and

intrinsic factors related to the dynamics and geometry of sediment erosion,

transport, and deposition. Numerical modeling of these systems can be used

to evaluate the feasibility of conceptual models, to explore their sensitivity to

external controls, and to suggest new hypotheses regarding the stratigraphic

and geomorphic signatures of tectonic and climatic change.

Figure 9 shows a simple example of a simulated mountain range bounded

by an alluvial fan complex. In this example, we have chosen a simple

experimental design in which a block consisting of a cohesionless sand-

gravel mixture rises vertically at a constant rate relative to an adjacent

(fixed) basin surface and its associated baselevel. As one might expect, the

simulation shows a set of alluvial fans that prograde across the basin surface

(Fig. 9). A “wave” of sand-rich sediment progrades ahead of the advancing

fan toes (Figs. 9A, B). Interestingly, size-selective transport occurs not only

within the fan complex but also within the source terrain. Initially, finer

material is removed from the surface of the rising block, leaving behind a

coarsened layer of surface sediment that rims the headward-encroaching

drainages. Thus, the grain-size patterns within the fan complex are

influenced in part by sorting within the source terrain. Whether this effect

occurs in nature must depend on the regolith thickness; while bare rock

slopes offer little opportunity for grain-size fractionation, such fractionation

has been observed to occur on soil-mantled, wash-dominated slopes (e.g.,

Abrahams and Parsons, 1991).

Note that in this example no attempt is made to simulate either

downstream flow branching or sheetflow; rather, flow is effectively spread

across the fan surface through time as channels shift in response to

depositional patterns. A transverse section across the fan complex (Fig. 10)

reveals that the main fan bodies are, perhaps counter-intuitively, slightly

18 Chapter 12

finer than the inter-fan areas. This behavior would have implications for

fluid reservoir modeling, as it implies that in some cases inter-fan areas may

have locally higher hydraulic conductivity.

3.10 Lateral Stream Channel Migration (Meandering)

Owing to the large difference in scale between individual stream

channels and their drainage basins, channels are generally treated as one-

dimensional entities in landscape evolution theory. For many applications,

this choice is entirely appropriate; for others, however, it is problematic

because it neglects the role of floodplains as sediment buffers (e.g., Trimble,

1999). This limitation is particularly severe in analyses of watershed

responses to perturbations (e.g., Tucker and Slingerland, 1997). At the same

time, the morpho-stratigraphic development of floodplains is an important

problem in its own right (e.g., Mackey and Bridge, 1995; Moody et al.,

1999). These issues have motivated the development of a simple “rules

based” model of channel meandering, based on the principle of topographic

steering, which is capable of modeling channel planform evolution on time

scales relevant to valley, floodplain, and stream terrace development

(Lancaster, 1998; Lancaster and Bras, in press).

Lateral channel migration is implemented in CHILD by first identifying

main channel (meandering) nodes on the basis of a drainage area threshold.

Lateral migration of these nodes occurs perpendicular to the downstream

direction, and the rate is proportional to the bank shear stress:

n

ˆ

Eζ

ˆ

weff τ=

where ζ

ˆ

is the migration vector of the outer bank, τw is the bank shear stress

determined by the meandering model of Lancaster and Bras (in press; see

also Lancaster, 1998); n

ˆ

is the unit vector perpendicular to the downstream

direction; and Eeff is the effective bank erodibility, defined by:

0

0

1

0

0

≤=

>

+

−

=

B

B

B

BH

eff

h,E

h,

hH

hP

EE ,

where E0 is the nominal bank erodibility; H is water depth; hB is bank height

above the water surface; and PH is the degree to which the effective bank

erodibility is dependent on bank height, where 10 ≤≤ H

P (Lancaster, 1998).

This bank height dependence directly couples topography and migration

(16)

(17)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

19

rate. Each channel node in the model actually has both right and left bank

erodibilities, and these values are determined from a weighted average of Eeff

values calculated for neighboring nodes falling on either side of the line

perpendicular to the downstream direction (Fig. 11). We write the effective

erodibility at node i of the bank on the e

ˆ

-side as

()

21

1221

dd

dEdE

Ei,effi,eff

e

ˆ

i,eff +

+

=

where e

ˆ

is the unit vector in the direction of either the left ( n

ˆ

) or right

(n

ˆ

−) bank; Eeff,i1 and Eeff,i2 are the effective erodibilities of the bank nodes

with respect to node i; and d1 and d2 are the distances of the bank nodes from

the line parallel to the unit vector, e

ˆ

(Fig. 11).

We use the meandering model of Lancaster (1998) to find τw in (16) as a

function of channel curvature upstream. Movement of a channel node

indicates that the channel centerline has moved, i.e., that one bank has been

eroded while deposition has occurred at the other. As the channel migrates,

existing nodes are deleted from the moving channel’s path, and new nodes

are added in the moving channel’s wake. Node movement and addition

require re-determination of node stratigraphy.

A flow chart in Figure 12 illustrates the implementation of meandering

within the CHILD model. The discretization of meandering channel reaches

is dependent on channel width and is, in general, different from the

discretization of the surrounding landscape. This procedure is described in

more detail in Lancaster (1998).

3.10.1 Examples

An example simulation incorporating the stream meander model is

shown in Figure 13. Here the model is configured to represent an idealized

segment of floodplain, with a large stream (point source of discharge)

entering at the top of the mesh and exiting at the bottom. The hydrology and

initial topography are patterned after Wildcat Creek, a 190 km2 drainage

basin in north-central Kansas. In this example, the mainstream elevation is

forced with a series of cut-fill cycles (representing millennial-scale climate

impacts), while the stream planform is free to migrate laterally. Each point

along the main channel is moveable. Dynamic remeshing is used to ensure

that the mainstream is adequately resolved. Whenever a moving channel

point comes very close to a fixed “bank” point, the latter is removed from

the mesh. To ensure an adequate level of spatial resolution within the

floodplain, a new point is added in the “wake” of a moving channel point

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20 Chapter 12

whenever the moving point has migrated a given distance away from a

previously stored earlier location (which is then updated). The net result is

that the floodplain is modeled at a locally high resolution relative to the

surrounding uplands (Fig. 13).

A similar approach can be used to investigate the development of incised

meanders in bedrock such as those of the Colorado Plateau (e.g., Gardner,

1975) and the Ozark Mountains. Lancaster (1998) modeled the development

of terrain under active uplift, incision, and stream meandering, and found

that coupling between bank height and the rate of cut-bank erosion exerts an

important influence on the resultant topography and channel planforms.

3.11 Floodplains: Overbank Sedimentation

Valley-fill sediments often contain an important record of paleoclimate,

paleo-geomorphology, and prehistory (e.g., Johnson and Logan, 1990). Most

studies of the formation and dynamics of river basins have treated streams as

essentially one-dimensional conduits of mass and energy. Yet valley-fill

sediments are inherently three-dimensional features, and to model their

stratigraphy properly requires an alternative approach. The one-dimensional

approach cannot, for example, resolve important aspects of alluvial

stratigraphy such as the distribution of channel and overbank deposits (e.g.,

Mackey and Bridge, 1995). Motivated by this limitation, CHILD includes

the capability to model overbank sedimentation using a modified form of

Howard’s (1992) floodplain diffusion model. Under this approach, the rate

of overbank sedimentation during a flood varies as a function of distance

from a primary channel and local floodplain topography. Average rates of

floodplain sedimentation are known to decay with distance from the source

channel due to diffusion of turbulent energy. The local rate of sedimentation

is also presumed to depend on the height of the floodplain relative to water

surface height. During a given storm event, the rate of overbank

sedimentation at a given point is

()( )

λ−µ−η= /dzDOB exp

where DOB is the vertical deposition rate (dimensions of L/T), z is local

elevation, d is the distance between the point in question and the nearest

point on the main channel, η is the water surface height at the nearest point

on the main channel, µ is a deposition rate constant (T-1), and λ is a distance-

decay constant. “Main channel” is defined on the basis of a drainage area

threshold; typically, the model would be configured with a large channel fed

in as a boundary condition for this type of application, so that there would be

no ambiguity about what constitutes a primary channel (e.g., Fig. 13). Water

(19)

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

21

surface height is computed as the sum of bed elevation, z, and water depth,

H, using a simple empirical hydraulic geometry approach for H:

b

s

bwbbb QkH,)Q/Q(HH δ

δ==

where Hb is bankfull channel depth, Qb is a characteristic discharge (such as

bankfull or mean annual), kh is bankfull depth per unit scaled discharge, and

δb and δs are the downstream and at-a-station scaling exponents, respectively

(Leopold et al., 1964). Equation (19) is only applied for events in which H >

Hb.

3.11.1 Example

Combining channel meandering and overbank deposition makes it

possible to simulate the development of three-dimensional alluvial

stratigraphic architecture, which has been the goal of a number of different

models (e.g., Howard, 1992; Mackey and Bridge, 1995; Teles et al., 1998).

The fill terraces depicted in Figure 13 are formed during times of rising

baselevel along the main channel. Lateral channel migration etches out the

fills during intervals of cutting. Inset terraces are formed during subsequent

fill episodes (Fig. 13). Among other things, this type of stratigraphic

simulation can provide a basis for developing and testing improved

geostatistical methods for modeling 3D subsurface architecture (e.g., de

Marsily et al., 1998).

4. DISCUSSION: APPLICATIONS AND

LIMITATIONS

All models involve a tradeoff between simplicity and realism. What

makes the CHILD model unique is its ability to examine interactions among

a wide range of processes, in scenarios that range from simple to complex.

The examples herein use simple, idealized scenarios to illustrate these

processes. The model’s design reflects the fundamental recognition that the

characteristics of one part of a river basin are determined in large part by the

characteristics of the basin upstream and, to a lesser degree, downstream.

The inclusion of many process modules and alternative parameterizations

in one model (Fig. 2) is intended to enhance the researcher’s ability to

address very simple and well-posed questions by carefully selecting a subset

of process equations and configuring these with appropriate initial and

boundary conditions. By comparing model behavior under varying levels of

(20)

22 Chapter 12

complexity and/or different process models, the validity and robustness of

different simplifying assumptions can be tested. One can examine, for

example, the consequences of relaxing the common assumption of

homogeneous sediment size (Gasparini et al., 1999), or assess the

appropriateness of using a “characteristic storm” parameter as a surrogate for

time-varying rainfall and flood discharge (Tucker and Bras, 2000).

CHILD has been developed as a framework for modeling changes in

drainage basin terrain over a range of space and time scales. Although there

are no explicit limits to spatial scale, the assumption of hydrologic steady

state during storm events is most valid for relatively small watersheds (less

than perhaps 100km2), in which the time of concentration is shorter than the

duration of a typical storm. Similarly, the assumption of spatially uniform

precipitation rate, infiltration capacity and soil transmissivity is only

appropriate for small watersheds (although one might also wish to make

similar assumptions in simple “what if” studies of large-scale landscape

evolution). At the lower end of spatial scales, the approximation of steady,

uniform unidirectional flow loses validity for the length scales on which

momentum and backwater effects become important (on the order of

decimeters to meters). The assumption of steady rainfall and runoff during

storms also implies that the model is most applicable to time periods much

longer than the duration of a single storm. The upper limit to time scale is

dictated only by performance considerations, and in fact for certain

applications it is feasible to magnify storm and interstorm durations to

enhance computational speed.

Distributed models necessarily involve a tradeoff between speed and

resolution. The CHILD model’s TIN-based framework offers an advantage

in this regard, because it makes it possible to vary spatial resolution as a

function of dominant process or landscape position (Figs. 1 and 13). On the

other hand, the use of variable spatial resolution complicates the inclusion of

“scale-dependent physics” (i.e., equations whose rate constants depend on

spatial scale). This may be a blessing in disguise, for although it makes the

problem of calibration in engineering applications more difficult it also

provides a disincentive to scale-dependent “tuning” of parameters. Use of a

variable-resolution numerical mesh, if handled properly, may also help

resolve certain scaling issues that arise as a result of averaging terrain

properties over an arbitrary and fixed discretization scale. For example, with

an irregular discretization method it becomes possible (at least in principle)

to construct a discretized terrain surface that uses the minimum necessary

number of computational points to accurately represent hillslope gradient at

all points in a region of complex terrain. The TIN framework also opens the

door to bridging the two fundamental and disparate scales in watershed

hydrology, that of the channel and that of the basin as a whole.

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

23

Although it is intended to serve a wide range of applications, the CHILD

model’s roots lie in large-scale drainage basin morphology and evolution.

The form of many of the equations used in CHILD reflects this emphasis.

Thus, the sediment transport equations are based on formulas commonly

used to predict bedload transport rates, and the model at this stage includes

no explicit treatment of suspended or wash load (which are presumably of

lesser importance in controlling stream gradients). Similarly, the model at

present includes no expressions for landsliding or for eolian transport. The

emphasis on physical rather than chemical process renders CHILD

inapplicable in solution-dominated environments (e.g., karst terrain). It

should be emphasized, however, that CHILD is designed with extensibility

in mind, and the modular design of the software reflects this (Tucker et al.,

2001). Recent efforts to adapt CHILD for applications in forestry (Lancaster

et al., 1999) and flood hydrology (Rybarczyk et al., 2000) demonstrate the

utility of constructing modular and extensible numerical modeling systems.

There is no simple answer to the question of how to test and validate a

model such as CHILD because it is in essence not one model but many, each

with different assumptions, aims, and requirements. Ultimately, the basis for

validation or rejection of a model should depend on the nature of the

problem addressed. Nonetheless, it is worth noting that several methods for

evaluating the predictions of landscape evolution theory have been advanced

recently. Statistical approaches have been widely used to examine drainage

network properties (e.g., Rodriguez-Iturbe and Rinaldo, 1997), although

some network statistics suffer from a lack of discriminant ability (e.g.,

Kirchner, 1993). Experimental approaches have also been used (Hancock

and Willgoose, in review). Arguably the most promising tests of landscape

evolution theory come from settings in which paleo-topography (e.g., Stock

and Montgomery, 1998) and driving factors such as uplift rate (e.g., Merritts

and Vincent, 1989; Snyder et al., 2000) are independently known. Much can

be learned by testing the morphologic predictions of landscape evolution

models against observed terrain in regions where some type of equilibrium is

believed to exist (e.g., Willgoose, 1994). The CHILD model, as with other

models based on similar fluvial erosion formulations, can successfully

reproduce theoretically predicted slope-area scaling under conditions of

spatially uniform erosion rate (a form of equilibrium). This simply reflects

the fact that the exponent terms in the fluvial transport and erosion terms can

be chosen such that the model-predicted scaling under either equilibrium

(uplift-erosion balance) or transient decline agrees with observed values

(e.g., Willgoose et al., 1991a; Howard, 1994; Willgoose, 1994; Whipple and

Tucker, 1999; Tucker and Whipple, in review). However, one disadvantage

of testing models on the basis of equilibrium states, aside from the difficulty

24 Chapter 12

in establishing the existence of such states in the first place, is the potential

for equifinality (i.e., different processes may lead to the same outcome, as in

the case of slope-area scaling discussed by Tucker and Whipple, in review).

Landscapes characterized by a transient response to a known perturbation

contain useful information about process dynamics that is often lost in

equilibrium states (Tucker and Whipple, in review; Whipple and Tucker, in

review). Hence, one of the key research needs is to identify transient

landscapes in which knowledge of the nature and timing of the causative

external perturbation, whether of tectonic, climatic, geomorphic, or human

origin, can be obtained. For short-term phenomena such as gully

development, there is a need for detailed monitoring to establish time

sequences of landform development.

5. SUMMARY AND CONCLUSIONS

CHILD is a new computer model of drainage basin evolution that

integrates a wide variety of processes, many of which have not been

included in previous models of drainage basin evolution. The model is

designed to serve as a general-purpose framework for investigating a range

of issues in drainage basin geomorphology, with an emphasis on morpho-

logical development. Some components of the model, such as the treatment

of channel and hillslope erosion, use an approach similar to that of existing

models. The model also includes a number of new features and capabilities

that are designed to foster the development of theoretical geomorphology by

making it possible to investigate in greater detail the feedbacks between

hillslope/channel hydrology and landscape evolution, and to examine

coupling between erosional and depositional systems. The incorporation of

(1) meandering and (2) floodplain deposition, which have not before been

included in models of drainage basin evolution, makes it possible to

investigate the development of alluvial stratigraphy in the drainage basin

context. Other capabilities, which are unique in their combination, include

(3) stochastic storm variability, with an explicit link to climate data; (4) both

detachment- and transport-limited fluvial erosion, with transport of either

single- or dual-size sediment; (5) explicit tracking of subsurface stratigraphy,

including time of deposition, textural properties, and deposit exposure ages;

(6) variable, triangulated discretization and adaptive remeshing, which allow

detailed resolution of particular features and representation of horizontal

surface motion; and (7) infiltration-, storage- and saturation-excess runoff

mechanisms, the last of which provides a direct link between topography and

hydrology. Other capabilities, including a dynamic vegetation component

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

25

(Tucker et al., 1999) and kinematic thrust-fault propagation, are under

development and will be described elsewhere.

To implement these capabilities, CHILD includes several process

“modules.” In some cases, these represent alternative models for the same

process (e.g., Hortonian versus saturation-excess runoff generation).

Although the number of parameters in the model is potentially quite large,

the many different capabilities and process equations are in fact developed

with simplicity and flexibility in mind. CHILD’s extensible design facilitates

the process of comparing alternative process models and conducting

sensitivity experiments that address the basic (and important) questions of

“what matters and when.”

Although developed with an emphasis on research applications, CHILD’s

more detailed treatment of hydrology also makes it well suited to potential

applications in land management and erosion prediction. Most soil erosion

models, such as USLE and WEPP, assume a one-dimensional and

unchanging topography. These limitations, though appropriate for estimating

soil loss under rill and interrill erosion, are poorly suited to modeling gully

and channel incision, phenomena in which dynamic modification of

landforms and flow aggregation play a central role.

Acknowledgments

This research was supported by the U.S. Army Corps of Engineers

Construction Engineering Research Laboratories (USACERL), the U.S.

Army Research Office, and the Italian National Research Council. We are

grateful to Lainie Levick for a helpful review, and to William Doe and

Russell Harmon for convening the workshop and GSA Special Session that

led to this volume.

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Coulthard, T.J., Kirkby, M.J., and Macklin, Mark G., 1997, Modelling the impacts of

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FIGURE CAPTIONS

Figure 1. Example terrain simulations produced by CHILD. Thin solid lines are contours

and heavy lines indicate drainage pathways. (A) simulation of gullying on an actual watershed

(unnamed watershed on Fort Carson in the foothills of the Colorado Front Range near

Colorado Springs). Inset shows a segment of the triangular mesh. (B) hypothetical fault-

bounded mountain range. (C) valley and floodplain simulation, illustrating meandering stream

with variable-resolution mesh. (D) rising mountain block with alluvial fans. Scales in (B),

(D), and (D) are nominally 10 km, 1 km, and 2.5 km, respectively.

Figure 2. Components of the CHILD model.

Figure 3. Elements of the irregular computational mesh, showing nodes (solid circles),

triangle edges (black lines), and Voronoi polygons (gray lines). Each Voronoi polygon acts as

a finite volume cell. (A) streamflow is routed downslope from node to node along triangle

edges, following the route of steepest descent. (B) two-dimensional diffusive exchange of

sediment between node N and its neighbors. The diffusive mass flux per unit width between

any two nodes is computed using the gradient between them. Multiplying unit flux by the

width of their shared Voronoi polygon edge gives the total mass exchange rate.

12. The Channel-Hillslope Integrated Landscape Development

Model (CHILD)

31

Figure 4. Schematic illustration of Poisson rectangular pulse rainfall model (after Eagleson,

1978).

Figure 5. Flow chart illustrating the sequence of computations in CHILD.

Figure 6. Example of simulated gully erosion and healing in response to stochastic

variations in rainfall intensity and duration. Here, a gully system forms and begins to heal on

a planar slope (30 degrees, 100 by 100 meters) that is subjected to a series of random storm

events. The landscape is highly sensitive to extreme events, owing to a large threshold for

runoff erosion (τc) and a high soil erodibility coefficient (kb). (A) time series of rainfall events

(durations not shown). (B) mean elevation of the surface through time, highlighting the

episodic nature of denudation. Arrows indicate the times corresponding to plots C and D. (C)

perspective plot of slope immediately after the last gully-forming episode. (D) perspective

plot at the end of the simulation. (E) and (F) show contour plots at these two time slices.

Equation set used in this run is –dz/dt = kb(τ-τc) (kb=1.6 x 10-5 m

2 s kg-1), with τ =

kt(Q/W)2/3S2/3 (Pa) (kt=6.2 x 104 Pa s2/3 m-4/3), τc = 20 Pa, and W = 0.001 Q0.5 with Q in m3/s.

Rainfall parameters are 640.P = mm/hr, 32=

r

T hr, and 148=

b

T hr; hillslope diffusivity

(kd) is 0.01 m2/yr.

Figure 7. The influence of runoff-production mechanism on terrain morphology. (A)

simulated drainage basin under infiltration-excess (Hortonian) runoff production (Eq (3)). (B)

simulated basin under saturation-excess runoff production, using the O’Loughlin (1986)

model (Eq (6)). (C), (D) plots of surface slope versus contributing area for the two cases. The

line in (D) represents the line of saturation for the mean-intensity storm. In these examples

runoff erosion is modeled as SQdt/dz .50

−∝ . Parameters are 92.P = mm/hr, 5=

r

T yr,

95=

b

T yr, kd = 0.01 m2/yr, U = 0.1 mm/yr, and in (B) T=105 m2/yr.

Figure 8. Slope-area plots from two simulations illustrating a downstream transition from

detachment-limited to transport-limited behavior under (A) constant runoff and (B) variable

(stochastic) runoff. Both simulations are in equilibrium with a constant and spatially uniform

rate of baselevel fall. The transport and erosion coefficients are adjusted so that the theoretical

transition point occurs at the same drainage area in both cases. Although fluvial erosion

theory predicts that such a transition should occur in many rivers, the result shown in (B)

implies that transitions may be so smooth as to be undetectable in data.

Figure 9. Example of a simulated mountain-fan system, showing progradation of a set of

alluvial fans in response to block uplift along a vertical fault. The substrate is treated as a

cohesionless sediment pile containing a mixture of sand and gravel sediment fractions.

Shading indicates the relative proportion of sand in the uppermost (active) sediment layer,

with lighter shades indicating higher sand fraction. (A) 20,000 years after onset of uplift; (B)

40,000 years; (C) 100,000 years. Inset in (C) shows the location of cross-section in Figure 10.

Uplift rate is 1 mm/yr, diffusivity is 0.01 m2/yr, and rainfall parameters are 11.P = mm/hr,

3=

r

T yr, and 97=

b

T yr.

Figure 10. Stratigraphic cross-section through the fan complex in Figure 9. Section is taken

parallel to the strike of the range through the center of the fan complex (indicated by a-a’ in

Figure 9C). Black shading indicates less than 60% sand content.

32 Chapter 12

Figure 11. Illustration of right bank ( n

ˆ

e

ˆ

−= ) erodibility determination for node i. Eeff,i1 and

Eeff,i2 are effective erodibilities with respect to node i at adjacent nodes that are distances d1

and d2, respectively, from the line parallel to the unit vector, n

ˆ

−. In the coordinate system

shown, the s-direction is parallel to the flow edge, and the n-direction is perpendicular to the

flow edge. Delaunay triangulation is in thin solid lines, Voronoi diagram is in dashed lines,

and flow edges are in heavy black.

Figure 12. Flow chart showing the implementation of meandering.

Figure 13. Simulation of channel meandering and floodplain development. (A) perspective

view of simulated topography, highlighting stream pattern and development of terraces

(elevations are interpolated to regular grid for plotting purposes). (B) view of triangulated

mesh, showing densification in the area of the floodplain (see text for details).