Labs WMT Child Exercises

Landscape Evolution Modeling with CHILD

Gregory E. Tucker, University of Colorado Boulder, and Stephen T. Lancaster, Oregon State University
These short course notes prepared for SIESD 2012: Future Earth: Interaction of Climate and Earth-surface Processes,
University of Minnesota, Minneapolis, Minnesota, USA, August 2012. Notes and exercises updated for WMT by Stephanie
Higgins, University of Colorado Boulder, August 2014.

Before beginning these exercises, download the CHILD visualization tools here

1. Overview
The learning goals of this exercise are:

To gain a clearer understanding of how a typical landscape evolution model (LEM) solves the governing equations that represent geomorphic processes.
To gain hands-on experience actually using a LEM.
To understand how continuity of mass is maintained by a typical LEM, and some of the limitations that arise.
To appreciate some of the ways in which climate and hydrology can be represented in a LEM, and some of the simplifications involved.
To appreciate that working with LEMs involves choosing a level of simplification in the governing physics that is appropriate to the problem at hand.
To get a sense for how and why soil creep produces convex hillslopes.
To appreciate the concepts of transient versus steady topography.
To acquire a feel for the similarity and difference between detachment-limited and transport-limited modes of fluvial erosion.
To understand the connection between fluvial physics and slope-area plots.
To appreciate that LEMs (1) are able to reproduce (and therefore, at least potentially, explain) common forms in fluvially carved landscapes, (2) can enhance our insight into dynamics via visualization and experimentation, but (3) leave open many important questions regarding long-term process physics.
To develop a sense of ``best practices in using landscape evolution models.

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Landscape Evolution Modeling with CHILD

LANDSCAPE EVOLUTION MODELING WITH CHILD

GREGORY E. TUCKER
UNIVERSITY OF COLORADO

STEPHEN T. LANCASTER
OREGON STATE UNIVERSITY

Date: Short Course notes prepared for SIESD 2012: Future Earth: Interaction of Climate and Earth-surface Processes, University of Minnesota, Minneapolis, Minnesota, USA, August 2012.

1.  Overview
2.  Introduction to LEMs
   2.1.  Brief History
   2.2.  Brief Overview of Models and their Uses
3.  Continuity of Mass and Discretization
4.  Gravitational Hillslope Transport
   4.1.  Linear Diffusion
  Exercise 1: Getting Set Up with CHILD
  Exercise 2: Hillslope Diffusion and Parabolic Slopes with CHILD
   4.2.  Nonlinear Diffusion
  Exercise 3: Nonlinear Diffusion and Planar Slopes
   4.3.  Remarks
5.  Rainfall, Runoff, and Drainage Networks
   5.1.  Methods Based on Drainage Area
  Exercise 4: Flow Over Noisy, Inclined Topography
   5.2.  Shallow-Water Equations
   5.3.  Cellular Automata
   5.4.  Depressions in the Terrain
   5.5.  Precipitation and Discharge
  Exercise 5: Visualizing a Poisson Storm Sequence
   5.6.  Remarks
6.  Hydraulic Geometry
7.  Erosion and Transport by Running Water
   7.1.  Detachment-Limited Models
  Exercise 6: Detachment-Limited Hills and Mountains
  Exercise 7: Zooming in to the Hillslopes
  Exercise 8: Knickzones and Transient Response
   7.2.  Transport-Limited Models
  Exercise 9: A Pile of Fine Sand
  Exercise 10: A Pile of Cobbles
   7.3.  Hybrid Model: Combining Detachment and Transport
  Exercise 11: Erosion and Deposition, Together at Last
   7.4.  Other Sediment-Flux-Dependent Fluvial Models
8.  Multiple Grain Sizes
9.  Exotica
10.  Forecasting or Speculation?
11.  Ten Commandments of Landscape Evolution Modeling
References

1. Overview

Figure 1: Schematic diagram of CHILD model’s representation of the landscape: hexagonal Voronoi cells, nodes (at centers of cells) connected by the edges of the Delaunay triangulation, vegetated cell surfaces, channelized cells, and soil and sediment layers above bedrock.

The learning goals of this exercise are:

To gain a clearer understanding of how a typical landscape evolution model (LEM) solves the governing equations that represent geomorphic processes.
To gain hands-on experience actually using a LEM.
To understand how continuity of mass is maintained by a typical LEM, and some of the limitations that arise.
To appreciate some of the ways in which climate and hydrology can be represented in a LEM, and some of the simplifications involved.
To appreciate that working with LEMs involves choosing a level of simplification in the governing physics that is appropriate to the problem at hand.
To get a sense for how and why soil creep produces convex hillslopes.
To appreciate the concepts of transient versus steady topography.
To acquire a feel for the similarity and difference between detachment-limited and transport-limited modes of fluvial erosion.
To understand the connection between fluvial physics and slope-area plots.
To appreciate that LEMs (1) are able to reproduce (and therefore, at least potentially, explain) common forms in fluvially carved landscapes, (2) can enhance our insight into dynamics via visualization and experimentation, but (3) leave open many important questions regarding long-term process physics.
To develop a sense “best practice” in using landscape evolution models.

2. Introduction to LEMs

2.1. Brief History.

G.K. Gilbert, a member of the Powell Expedition, produced “word pictures” of landscape evolution that still provide insight (Gilbert, 1877). For example, consider his “Law of Divides” (Gilbert, 1877):

We have seen that the declivity over which water flows bears an inverse relation to the quantity of water. If we follow a stream from its mouth upward and pass successively the mouths of its tributaries, we find its volume gradually less and less and its grade steeper and steeper, until finally at its head we reach the steepest grade of all. If we draw the profile of the river on paper, we produce a curve concave upward and with the greatest curvature at the upper end. The same law applies to every tributary and even to the slopes over which the freshly fallen rain flows in a sheet before it is gathered into rills. The nearer the water-shed or divide the steeper the slope; the farther away the less the slope.

It is in accordance with this law that mountains are steepest at their crests. The profile of a mountain if taken along drainage lines is concave outward...; and this is purely a matter of sculpture, the uplifts from which mountains are carved rarely if ever assuming this form.

Flash forward to the 1960’s, and we find the emergence of the first one-dimensional profile models. Culling (1963), for example, used the diffusion equation to describe the relaxation of escarpments over time.

Models became more sophisticated in the early 1970’s. Frank Ahnert and Mike Kirkby, among others, began to develop computer models of slope profile development and included not only diffusive soil creep but also fluvial downcutting as well as weathering (Ahnert, 1971; Kirkby, 1971). Meanwhile, Alan Howard developed a simulation model of channel network evolution (Howard, 1971).

The mid-1970’s saw the first emergence of fully two-dimensional (and even quasi-three-dimensional) landscape evolution models, perhaps most noteworthy that of Ahnert (1976). Geomorphologists would have to wait nearly 15 years for models to surpass the level of sophistication found in this early model.

During that time, computers would become much more powerful and able to model full landscapes. The late 1980’s through the mid-1990’s saw the beginning of the “modern era” of landscape evolution models, and today there are many model codes with as many applications, scales, and objectives, ranging from soil erosion to continental collision (Table 1).

2.2. Brief Overview of Models and their Uses.

Some examples of landscape evolution models (LEMs) are shown in Table 1. LEMs have been developed to represent, for example, coupled erosion-deposition systems, meandering, Mars cratering, forecasting of mine-spoil degradation, and estimation of erosion risk to buried hazardous waste. These models provide powerful tools, but their process ingredients are generally provisional and subject to testing. For this reason, it is important to have continuing cross-talk between modeling and observations—after all, that’s how science works.

In this exercise, we provide an overview of how a LEM works, including how terrain and water flow are represented numerically, and how various processes are computed.

Table 1: Partial list of numerical landscape models published between 1991 and 2005.


Model	Example reference	Notes

SIBERIA	Willgoose et al. (1991)	Transport-limited;
		Channel activator function
DRAINAL	Beaumont et al. (1992)	“Undercapacity” concept
GILBERT	Chase (1992)	Precipiton
DELIM/MARSSIM	Howard (1994)	Detachment-limited;
		Nonlinear diffusion
GOLEM	Tucker and Slingerland (1994)	Regolith generation;
		Threshold landsliding
CASCADE	Braun and Sambridge (1997)	Irregular discretization
CAESAR	Coulthard et al. (1996)	Cellular automaton algorithm
		for 2D flow field
ZSCAPE	Densmore et al. (1998)	Stochastic bedrock
		landsliding algorithm
CHILD	Tucker and Bras (2000)	Stochastic rainfall
EROS	Crave and Davy (2001)	Modified precipiton
TISC	Garcia-Castellanos (2002)	Thrust stacking
LAPSUS	Schoorl et al. (2002)	Multiple flow directions
APERO/CIDRE	Carretier and Lucazeau (2005)	Single or multiple
		flow directions

3. Continuity of Mass and Discretization

A typical mass continuity equation for a column of soil or rock is:

∂η ∂t-= B - ∇ ⃗qs

(1)

where η is the elevation of the land surface [L]1; t is time; B [L/T] represents the vertical motion of the rocks and soil relative to baselevel (due, for example, to tectonic uplift or subsidence, sea-level change, or erosion along the boundary of the system); and _s is sediment flux per unit width [L²/T]. This is one of several variations; for discussion of others, see Tucker and Hancock (2010). Some models, for example, distinguish between a regolith layer and the bedrock underneath (Fig. 1). Note that this type of mass continuity equation applies only to terrain that has one and only one surface point for each coordinate; it would not apply to a vertical cliff or an overhang.

A LEM computes η(x,y,t) given (1) a set of process rules, (2) initial conditions, and (3) boundary conditions. One thing all LEMs have in common is that they divide the terrain into discrete elements. Often these are square elements, but sometimes they are irregular polygons (as in the case of CASCADE and CHILD; Fig. 1). For a discrete parcel (or “cell”) of land, continuity of mass enforced by the following equation (in words):

Time rate of change of mass in element = mass rate in at boundaries - mass rate out at boundaries + inputs or outputs from above or below (tectonics, dust deposition, etc.)

Figure 2: Schematic diagram of CHILD mesh with illustration of calculation of volumetric fluxes between cells. Dashed lines indicate cells and their faces, solid circles are nodes, and solid lines show the edges between nodes.

This statement can be expressed mathematically, for cell i, as follows:

dη 1 ∑N --i= B + --- qsjλj dt Λi j=1

(2)

where Λ_i is the horizontal surface area of cell i; N is the number of faces surrounding cell i; q_sj is the unit flux across face j; and λ_j is the length of face j (Fig. 2). (Note that, for the sake of simplicity, we are using volume rather than mass flux; this is ok as long as the mass density of the material is unchanging). Equation (2) expresses what is known as a finite-volume method because it is based on computing fluxes in and out along the boundaries of a finite volume of space.

Some terminology: a cell is a patch of ground with boundaries called faces. A node is the point inside a cell at which we track elevation (and other properties). On a raster grid, each cell is square and each node lies at the center of a cell. On the irregular mesh used by CASCADE and CHILD, the cell is the area of land that is closer to that particular node than to any other node in the mesh. (It is a mathematical entity known as a Voronoi cell or Thiessen polygon; for more, see Braun and Sambridge (1997), Tucker et al. (2001a).)

Equation 2 gives us the time derivatives for the elevation of every node on the grid. How do we solve for the new elevations at time t? There are many ways to do this, including matrix-based implicit solvers (see for example Fagherazzi et al. (2002); Perron (2011)). We won’t get into the details of numerical solutions (at least not yet), but for now note that the simplest solution is the forward-difference approximation:

dη η (t + Δt ) - η(t) --i ≈ -i------------i-- (3) dt Δt 1 ∑N ηi(t + Δt ) = ηi(t) + U Δt + Δt--- qsjλj (4) Λi j=1

The main disadvantage of this approach is that very small time steps are typically needed in order to ensure numerical stability. (CHILD uses a variant of this that seeks the largest possible stable value of Δt at each iteration). A good discussion of numerical stability, accuracy, and alternative methods for diffusion-like problems can be found in Press et al. (2007).

4. Gravitational Hillslope Transport

Geomorphologists often distinguish between hillslope and channel processes. It’s a useful distinction, although one has to bear in mind that the transition is not always abrupt, and even where it is abrupt, it is commonly either discontinuous or highly dynamic or both.

Alternatively, one can also distinguish between processes that are driven nearly exclusively by gravitational processes, and those that involve a fluid phase (normally water or ice). This distinction too has a gray zone: landslides are gravitational phenomena but often triggered by fluid pore pressure, while debris flows are surges of mixed fluid and solid. Nonetheless, we will start with a consideration of one form of gravitational transport on hillslopes: soil creep.

4.1. Linear Diffusion.

For relatively gentle, soil-mantled slopes, there is reasonably strong support for a transport law of the form:

⃗qs = - D ∇ η

(5)

where D is a transport coefficient with dimensions of L²T^-1. Using the finite-volume method outlined in equation 2, we want to calculate ⃗qs at each of the cell faces. Suppose node i and node k are neighboring nodes that share a common face (we’ll call this face j). We approximate the gradient between nodes i and k as:

ηk - ηi Sik = ------- Lik

(6)

where L_ik is the distance between nodes. On a raster grid, L_ik = Δx is simply the grid spacing. The sediment flux per unit width is then

ηk---ηi qsik ≃ D Lik

(7)

where q_sik is the volume flux per unit width from node k to node i (if negative, sediment flows from i to k), and L_ik is the distance between nodes. On a raster grid, L_ik = Δx is simply the grid spacing. To compute the total sediment flux through face j, we simply multiply the unit flux by the width of face j, which we denote λ_ij (read as “the j-th face of cell i”):

Qsik = qsikλij

(8)

Exercise 1: Getting Set Up with CHILD.

Our first exercise is simply to (1) get the model, input files, documentation, and visualization tools and (2) run the executable file to make sure it is installed and working correctly. In some cases, it might be necessary to create a new executable file from the source code.

For SIESD 2012, the package will already have been installed on the computers in the lab. Look for it in the folder: C:\child\ChildExercises. ______________

If you are working on your own computer:

If you are working on your own computer, you will need to download a copy of the latest CHILD release from the Community Surface Dynamics Modeling System (CSDMS) web site:

http://csdms.colorado.edu

Once you have downloaded and unwrapped the package, locate the users’ guide and follow the instructions to compile the model on your particular platform. You will need to use either a UNIX shell or the Command window under Windows. On a mac, use the Terminal application. On a windows machine, use either a UNIX emulator shell such as cygwin on a PC, or the command window. In a UNIX shell, to change folders (“directories” in UNIX-speak), use cd followed by the folder name. A single period represents the current working directory; two periods represent the next directory up. For example, the command cd .. takes you one level up. To get a list of files in a directory, use ls. For Command prompt under windows, use dir instead of ls and backslashes instead of forward slashes.

Start up Command Window. In the command window, type child. You should see something like the following:

Usage: child [options] <input file>
--help: display this help message.
--no-check: disable CheckMeshConsistency().
--silent-mode: silent mode.
--version: display version.

While we’re at it, let’s get ready to visualize the output. Start Matlab. The first thing we will do is tell Matlab where to look for the plotting programs that we will use. At the Matlab command prompt type:

path( path, ’childFolderLocation\ChildExercises\MatlabScripts’ )

For childFolderLocation, use the path name of the folder that contains the CHILD package. You can also add a folder to your path by selecting File->Set Path... from the menu.

In Matlab, navigate the current folder to the location of the example input file hillslope1.in (which should end in: ChildExercises\Hillslope1).

Note that the “package” also includes some documentation that you may find useful: the ChildExercises folder contains an earlier version of this document, and the Doc folder contains the Users’ Guide (child_users_guide.pdf). The guide covers the nuts and bolts of the model in much greater detail than these exercises and includes a full list of input parameters.

Exercise 2: Hillslope Diffusion and Parabolic Slopes with CHILD.

In your terminal window, navigate to the ChildExercises\Hillslope1 folder.
To run the example, in your terminal window type:
child hillslope1.in
A series of numbers will flash by on the screen. These numbers represent time intervals in years. The 2-million-year run takes about 20 seconds on a 2GHz Intel Mac. When it finishes, return to Matlab and type:
m = cmovie( ’hillslope1’, 21, 200, 200, 100, 50 );
(This command says “generate a 21-frame movie from the run ‘hillslope1’ with the x-, y- and z- axes set to 200, 200 and 100 m, respectively, and with the color range representing 0 to 50 m elevation).
To replay the movie, type movie(m).

(Windows note: we found that under Vista and Windows 7, the movie figure gets erased after display; slightly re-sizing the figure window seems to fix this).

The analytical solution to elevation as a function of cross-ridge distance y is:

U ( ) z(y) = --- L2 - (y - y0)2 2D

(9)

where L is the half-width of the ridge (100 m in this case) and y₀ is the position of the ridge crest (also 100 m). The effective uplift rate U, represented in the input file by the parameter UPRATE, is 10^-4 m/yr. The diffusivity coefficient D, represented in the input file by parameter KD, is 0.01 m²/yr. Next, we’ll make a plot that compares the computed and analytical solutions.

Enter the following in Matlab:

ya = 0:200; % This is our x-coordinate
U = 0.0001; D = 0.01; y0 = 100; L = 100;
za = (U/(2*D))*(L2-(ya-y0).2);
figure(2), plot( ya, za ), hold on
xyz = creadxyz( ’hillslope1’, 21 ); % Reads node coords, time 21
plot( xyz(:,2), xyz(:,3), ’r.’ ), hold off
legend( ’Analytical solution’, ’CHILD Nodes’ )

Diffusion theory predicts that equilibrium height varies linearly with U, inversely with D, and as the square of L. Make a copy of hillslope1.in and open the copy in a text editor. Change one of these three parameters. To change U, edit the number below the line that starts with UPRATE. Similarly, to change D, edit the value of parameter KD. If you want to try a different ridge width L, change both Y_GRID_SIZE and GRID_SPACING by the same proportion (changing GRID_SPACING will ensure that you keep the same number of model nodes). Re-run CHILD with your modified input file and see what happens.

4.2. Nonlinear Diffusion.

As we found in our study of hillslope transport processes, the simple slope-linear transport law works poorly for slopes that are not “small” relative to the angle of repose for sediment and rock. The next example explores what happens to our ridge when we (1) increase the relative uplift rate, and (2) use the nonlinear diffusion transport law:

⃗qs = ---- D-∇z---- 1 - |∇z ∕Sc |2

(10)

Exercise 3: Nonlinear Diffusion and Planar Slopes.

Navigate to the Hillslope2 folder
Run CHILD: child hillslope2.in
In Matlab, navigate to the Hillslope2 folder
When the 70,000-year run (~1 minute on a 2GHz mac) finishes, type in Matlab:
m = cmovie( ’hillslope2’, 21, 200, 200, 100, 70 );

If we had used linear diffusion, the equilibrium slope gradient along the edges of the ridge would be S = UL∕D = (0.001)(100)∕(0.01) = 10 m/m, or about 84^∘! Instead, the actual computed gradient is close to the threshold limit of 0.7. Notice too how the model solution speed slowed down as the run went on. This reflects the need for especially small time steps when the slopes are close to the threshold angle.

4.3. Remarks.

There is a lot more to mass movement than what is encoded in these simple diffusion-like transport laws. Some models include stochastic landsliding algorithms (e.g., CASCADE, ZSCAPE). Some impose threshold slopes (e.g., GOLEM). One spinoff version of CHILD even includes debris-flow generation and routing (Lancaster et al., 2003).

5. Rainfall, Runoff, and Drainage Networks

In order to calculate erosion, sediment transport, and deposition by running water, a model needs to know how much surface water is flowing through each cell in the model. Usually, the erosion/transport equations require either the total discharge, Q [L³/T], the discharge per unit channel width, q [L²/T], or the flow depth, H.

There are three main alternative methods for modeling the flow of water across the landscape:

Methods based on contributing drainage area
Numerical solutions to the 2D, vertically integrated and time-averaged Navier-Stokes equations
Cellular automaton methods

5.1. Methods Based on Drainage Area.

Drainage area, A, is the horizontally projected area of land that contributes flow to a particular channel cross-section or to unit length of contour on a hillslope. For a numerical landscape model that uses discrete cells, A is defined as the area that contributes flow to a particular cell. When topography is represented as a raster grid, the most common method for computing drainage area is the D8 method. Each cell is assigned a flow direction toward one of its 8 surrounding neighbors. An algorithm is then used to trace flow paths downstream and add up the number of cells that contribute flow each cell (Fig. 3).

Figure 3: Flow accumulation by D8, or single flow directions, and multiple flow directions (Schauble et al., 2008).

For the Voronoi cell matrix that CHILD and CASCADE use, the simplest routing procedure is a generalization of D8 (Figure 1). Each cell i has N_i neighbors. As we noted earlier, the slope from cell i to neighbor cell k is defined as the elevation difference between the nodes divided by the horizontal distance between them (Fig. 2). Thus, one can define a slope for every edge that connects each pair of nodes. There is a slope value for each of the N_i neighbors of node i. The flow direction is assigned as the steepest of these slopes.

Single-direction flow algorithms have advantages and disadvantages. Some models use a multiple flow direction approach to represent the divergence of flow on relatively gentle slopes or divergent landforms (Fig. 3). This is most appropriate for models that operate on a grid resolution significantly smaller than the length of a hillslope. When grid cells are relatively large, conceptually each cell contains a primary channel, narrower than the cell, that is tracked.

Exercise 4: Flow Over Noisy, Inclined Topography.

In the terminal window, navigate to the Network1 folder and run the input file by typing:
child network1.in
In Matlab, navigate to the Network1 folder

In Matlab, type:

figure(1), clf
colormap pink
a = cread( ’network1.area’, 1 );
ctrisurf( ’network1’, 1, a );
view( 0, 90 ), shading interp, axis equal

The networks are formed because of noise (±1 m in this case) in the initial surface, which causes flow to converge in some places.

The simplest method for computing discharge from drainage area is to simply assume (1) all rain runs off, and (2) rain lasts long enough that the entire drainage network is in hydrologic steady state. In this case, and if precipitation rate P is uniform,

Q = P A

(11)

A number of landscape modeling studies have used this assumption, on the basis of its simplicity, even though it tends to make hydrologists faint! The simplicity is indeed a virtue, but one needs to be extremely careful in using this equation, for at least three reasons. First, obviously Q varies substantially over time in response to changing seasons, floods, droughts, etc. We will return to this issue shortly. Second, there is probably no drainage basin on earth, bigger than a hectare or so, from which all precipitation runs off. Typically, evapotranspiration returns more than half of incoming precipitation to the atmosphere. Third, hydrologic steady state is rare and tends to occur only in small basins, though it may be a reasonable approximation for mean annual discharge in some basins.

River discharge, whether defined as mean annual, bankfull, mean peak, or some other way, often shows a power-law-like correlation with drainage area. Some models take advantage of this fact by computing discharge using an empirical approach:

Q = bAc

(12)

where c typically ranges from 0.5-1 and b is a runoff coefficient with awkward units that represents a long-term “effective” precipitation regime.

CHILD’s default method for computing discharge during a storm takes runoff at each cell to be the difference between storm rainfall intensity P and soil infiltration capacity I:

Q = (P - I)A

(13)

which of course is taken to be zero when P < I.

5.2. Shallow-Water Equations.

Some landscape models are designed to address relatively small-scale problems such as channel initiation, inundation of alluvial fan surfaces, channel flood flow, etc. In such cases, the convergence and divergence of water in response to evolving topography is an important component of the problem, and is not adequately captured by the simple routing schemes described above. Instead, a tempting tool of choice is some form of the shallow-water equations, which are the vertically integrated form of the general (time-averaged) viscous fluid-flow equations. One form of the full shallow-water equations is:

( ) ∂-η = i - ∂qx-+ ∂qy- (14) ∂t ∂x ∂y ∂q ∂q u ∂q u ∂h ∂η τ --x-+ --x--+ ---y- + gh ---+ gh ---+ -bx-= 0 (15) ∂t ∂x ∂y ∂x ∂x ρ ∂qy- ∂qyv- ∂qxv- ∂h- ∂η- τby- ∂t + ∂y + ∂x + gh ∂y + gh ∂y + ρ = 0 (16)

These equations express continuity of mass, x-directed momentum, and y-directed momentum, respectively. They are challenging and computationally expensive to integrate numerically in their full form. However, there are several approximate forms that are commonly used, including the non-accelerating flow form (in which convective accelerations are assumed negligible) and the kinematic-wave equations (in which gravitational and friction forces are assumed to dominate). An example of use of the shallow-water equations in a landform evolution model can be found in the work of T.R. Smith and colleagues (Fig. 4). Various forms of the shallow-water equations can often be found in hydrologic models, and sometimes in soil-erosion models (e.g., Mitas and Mitasova, 1998).

Figure 4: Simulated water surface elevations and flow depth (Birnir et al., 2001).

5.3. Cellular Automata.

Some models use cellular automaton methods to calculate flow over a cellular topography. These include:

Chase (1992) precipiton algorithm
Crave and Davy (2001) modified precipiton algorithm
Murray and Paola (1994) multiple-flow-direction river-flow algorithm
Coulthard et al. (1996) generalization of Murray-Paola for 2D flow (CAESAR model)

5.4. Depressions in the Terrain.

What happens when flow enters a topographic depression? In the real world, three possibilities: complete evaporation/infiltration, formation of a lake with overflow, or formation of a closed lake. CHILD can be set either to have water in “pits” evaporate, or to use a lake-fill algorithm to route water through depressions in the terrain (with no evaporation).

5.5. Precipitation and Discharge.

Water supply to the channel network varies dramatically in both time and space, but there is a big gap in time scale between, on the one hand, storms and floods and, on the other hand, topographic evolution. Many landscape evolution models have therefore used the “effective discharge” concept, or the idea that there is some value of discharge that represents the cumulative geomorphic effect of the natural sequence of storms and floods. Willgoose et al. (1991) used mean peak discharge, but Huang and Niemann (2006) recognized that the return period of effective discharge events is not necessarily the same at different times and places.

Basically, landscape models tend to use one of four methods:

Steady flow with uniform precipitation or a specified runoff coefficient (effective discharge concept)
Steady flow with nonuniform precipitation or runoff (e.g., orographic precipitation)
Stochastic-in-time, spatially uniform runoff generation
“Short storms” model (SÛlyom and Tucker, 2004)

We will not examine all of these in detail. Instead, we will take a brief look at the Poisson rectangular pulse model implemented in CHILD.

Exercise 5: Visualizing a Poisson Storm Sequence.

In the terminal window, navigate to the Rainfall1 folder and run the input file by typing:
child rainfall1.in
In Matlab, navigate to the Rainfall1 folder

In Matlab, type:

figure(1), clf, cstormplot( ’rainfall1’ );
figure(2), clf, cstormplot( ’rainfall1’, 10 );

The first plot shows a 1-year simulated storm sequence; the second shows just the first 10 storms.

The motivation for using a stochastic flow model is (1) that nature is effectively stochastic, and (2) variability matters when the erosion or transport rate is a nonlinear function of flow. For more on this, see Tucker and Bras (2000); Snyder et al. (2003); Tucker (2004), and DiBiase and Whipple (2011).

5.6. Remarks.

Landscape evolution models can be, and have been, used to study climate impacts on erosion, topography, and mountain building. But be careful—climate and hydrology amount to much more than a “sprinkler over the landscape.”

6. Hydraulic Geometry

Channel size, shape, and roughness control delivery of hydraulic force to the bed and banks. Most landscape models either implicitly assume constant width (practical but dangerous) or use the empirical relation W = K_wQ^b, where b ≈ 0.5. Models with time-varying discharge must also specify how width varies at a point along the channel as Q rises and falls. Width-discharge scaling is practical but incomplete, because channels may narrow or widen downstream in concert with variations in incision rate, as observed in Italy (Whittaker et al., 2007), Nepal (LavÈ and Avouac, 2001), New Zealand (Amos and Burbank, 2007), Taiwan (Yanites et al., 2010), and California (Duvall et al., 2004). Some models have begun to explore these sensitivities (Wobus et al., 2006, 2008; Attal et al., 2008; Turowski et al., 2009; Yanites and Tucker, 2010), but full treatment of the channel geometry adjustment problem is a frontier area.

7. Erosion and Transport by Running Water

There are several competing models for erosion by channelized flow. Detachment-limited models assume that eroded material leaves the system without significant re-deposition and that lowering of channels is limited by the ability of the stream to detach material from the bed (Howard, 1994; Whipple and Tucker, 1999). Transport-limited models assume plentiful supply of loose sediment and that lowering of channels is limited by the stream’s capacity to transport sediment (Willgoose et al., 1991; Whipple and Tucker, 2002). In simple hybrid models, lowering may be limited either by excess transport capacity or by detachment rate, depending on local sediment supply and substrate resistance (Tucker et al., 2001b; Whipple and Tucker, 2002). With the undercapacity concept, detachment rate depends on surplus transport capacity (Beaumont et al., 1992). In the saltation-abrasion model, detachment is driven by grain impacts and limited by sediment shielding (Gasparini et al., 2007; Whipple and Tucker, 2002).

7.1. Detachment-Limited Models.

On a cohesive or rock bed with a discontinuous or absent cover of loose sediment, detachment of particles from the bed may be driven primarily by hydraulic lift and drag (“plucking”). Most models assume that the rate of detachment (or more generally the capacity for detachment) depends on excess bed shear stress:

p p p Dc = Kb (τ - τc)b , or alternatively, Dc = Kb (τ b - τcb)

(17)

where τ is local bed shear stress, τ_c is a threshold stress below which detachment is ineffective, K_b is a constant, and p_b is an exponent.

Bed shear stress fluctuates in space and time, but is often treated using the cross-sectional average, which in turn is based on a force balance between gravity and friction.

Some models assume that the detachment rate depends on stream power per unit width, ω = ρg(Q∕W)S:

( Q )pb Dc = Kb --S - Φc W

(18)

where Φ_c is, again, a threshold below which detachment is ineffective. Stream power per unit width turns out to be proportional to τ^3∕2, so the two erosion formulas are closely related (Whipple and Tucker, 1999). In the following example, we will use the unit stream power formula with Φ_c = 0.

Exercise 6: Detachment-Limited Hills and Mountains.

In the terminal window, navigate to the Dlim folder and run the input file by typing:
child dlim.in
The 3 m.y. run should take about 20 seconds.
In Matlab, navigate to the Dlim folder

In Matlab, type:

figure(1), clf, colormap jet
cmovie( ’dlim’, 31, 3e4, 3e4, 1e3, 500 );
figure(2), clf
csa( ’dlim’, 31 ); % Shows slope-area graph

Notice that the landscape has come close to a state of equilibrium between erosion and relative uplift. The resulting terrain has about 200 m of relief over a 30 km half-width mountain range—more Appalachian than Himalayan. Notice that the log-log slope-area graph shows a straight line, indicating a power-law relationship. This is exactly to be expected, and we can predict the plot slope and intercept analytically. Finally, note the points on the upper left of the graph. These “first order” cells, at about 2500 m² contributing area, have slopes less than 10%. They represent embedded channels, not hillslopes, which are too small to resolve at this grid spacing.

Now, what happens when we increase the relative uplift rate?

Run the dlimC1.in input file by typing:
child dlimC1.in
This run starts off where the previous one ended, but with a 10× higher rate of relative uplift.

In Matlab, type:

figure(1)
cmovie( ’dlimC1’, 31, 3e4, 3e4, 1e4, 5000 ); % 10× vertical scale
figure(2)
hold on, csa( ’dlimC1’, 31, ’r.’ ); hold off

Because we are using a slope-linear detachment law, a 10× increase in relative uplift rate leads to a 10× increase in relief. Notice that the points have shifted upward by a factor of 10 on the slope-area graph.

We still do not see any hillslopes, because the scale of landscape dissection is too fine for the model to resolve.

Exercise 7: Zooming in to the Hillslopes.

Next, we will “zoom in” by repeating the dlim run but with a twenty-fold decrease in domain size and model cell size.

Run the dlim_small.in input file by typing:
child dlim_small.in
This run is identical to dlim but with a domain of 1.5 by 1.5km and ~25m wide cells, instead of 30x30km and ~500m cells.

In Matlab, type:

figure(1)
cmovie( ’dlim_small’, 31, 1.5e3, 1.5e3, 500, 200 );
figure(2)
hold on, csa( ’dlim_small’, 31, ’g.’ ); hold off

Note how the hillslopes become evident in the topography. In the slope-area plot, the points seem to continue the trend of the coarser-scale run, but somewhat shifted upward. Can you guess why they are shifted upward? (The answer is subtle, and lies hidden in dlim_small2.in).

Exercise 8: Knickzones and Transient Response.

For the next exercise, we return to our earlier dlimC1 run and plot a representative stream profile at different times, to look at how the profile responds to the increased rate of relative uplift.

In Matlab, type:

figure(1), clf
[d,h,x,y] = cstrmproseries( ’dlimC1’, 10, 15000, 29000 );
This command traces the stream profile starting from x = 15 km, y = 29 km. It will plot the first 10 profiles.
figure(2), clf, plot( x, y )
This shows the horizontal trace of the stream course.

During the period of transient response, the stream profile shows a pronounced convexity, or knickzone, along the profile. The knickzone marches upstream through time. This pattern is characteristic of the “stream power” erosion law, which is actually a form of wave equation.

7.2. Transport-Limited Models.

We next explore the dynamics of landscapes and networks with transport-limited models. One caution as we do so: we will assume that channel width is independent of grain size, slope, etc.

Exercise 9: A Pile of Fine Sand.

In the terminal window, navigate to the Tlim folder and run:
child tlim1.in
The 1 m.y. run should take about 2 minutes.
In Matlab, navigate to the Tlim folder

In Matlab, type:

figure(1), clf
cmovie( ’tlim1’, 21, 3e4, 3e4, 40, 10 );
figure(2), clf
csa( ’tlim1’, 21 ); axis([1e-1 1e3 1e-4 1e-3])

In this run, we are effectively assuming that 0.1 mm sand moves as bed-load, according to a Meyer-Peter and Mueller-like transport formula. The landscape takes on an effectively uniform and very shallow gradient, on the order of 3 × 10^-4.

Exercise 10: A Pile of Cobbles.

Now let’s try the same experiment with 5cm cobbles.

Run:
child tlim2.in
The 3 m.y. run should take about 2-3 minutes.

In Matlab, type:

figure(1), clf
cmovie( ’tlim2’, 31, 3e4, 3e4, 1000, 300 );
figure(2)
hold on, csa( ’tlim2’, 31, ’r.’ ); hold off
axis([1e-1 1e3 1e-4 1e-1])

Lesson: grain size matters!

But let’s remember the caveat that channel width matters too, and we haven’t taken that into account with these simple runs. Also, Nicole Gasparini’s work (Gasparini et al., 1999, 2004) tells us that channel concavity is less sensitive to grain size when there is a mixture of sizes available to the river.

Optional exercise: Make a copy of tlim2.in and configure it to re-start from tlim2 but with a higher uplift rate. Use the Matlab script cstrmproseries to plot fluvial profiles undergoing transient response. How do these compare with the detachment-limited model?

7.3. Hybrid Model: Combining Detachment and Transport.

Next, we’ll look at a more complex situation with simultaneous erosion and sedimentation, and simultaneous detachment-limited and transport-limited behavior. In this case, we use a fluvial model in which erosion rate can be limited either by transport capacity or by detachment capacity, depending on their relative magnitudes:

{ ∑N ∑N Qc---ji=1Qsij if Qc---j=i1Qsij< D Ei = Λi Λi c Dc otherwise

(19)

Exercise 11: Erosion and Deposition, Together at Last.

In the terminal window, navigate to the Hybrid folder and run:
child erodep1.in
The 1 m.y. run should take about 5 minutes (but of course you can peek at earlier time steps while the run is going, by reducing the number of frames in your movie).

In Matlab, navigate to the Hybrid folder and type:

figure(1), clf
cmovie( ’erodep1’, 21, 6e4, 6e4, 4000 );

Here we have a block rising at 1 mm/yr and an adjacent block subsiding at 0.25 mm/yr. Uplift and subsidence shut down after 500 ky. The subsiding block forms a large lake that gradually fills in with fan-deltas.

7.4. Other Sediment-Flux-Dependent Fluvial Models.

We won’t take the time to address some of the other models, including

“Under-capacity” models (detachment rate depends on degree to which sediment flux falls below transport capacity), and
Saltation-abrasion models (detachment rate driven by particle impacts, and limited by alluvial shielding of bed)

Gasparini et al. (2007) explore the behavior of these models with CHILD simulations.

8. Multiple Grain Sizes

Although we won’t explore the effects of including multiple grain sizes of sediment in transport, grain size introduces some interesting issues, including:

Bed armoring and its impact on transport rates
Downstream fining
Abrasion and lithologic controls

9. Exotica

Landscape evolution models include more than diffusion and stream-power models:

Stream meandering in the context of landscape evolution and valley stratigraphy (Clevis et al., 2006, a,b).
Vegetation, including both grass (Collins et al., 2004; Istanbulluoglu and Bras, 2005) and trees (Lancaster et al., 2003)
Alternate forms of mass wasting, including landslides and debris flows (Densmore et al., 1998; Lancaster et al., 2003; Istanbulluoglu et al., 2005)
Knickpoints, hanging valleys, and plunge pools (Flores-Cervantes et al., 2006; Crosby et al., 2007)
Glaciation (Herman and Braun, 2006; Herman et al., 2007; Herman and Braun, 2008)

10. Forecasting or Speculation?

Some mathematical models in the physical sciences have such firm foundations that they can be relied upon to forecast the behavior of the natural world. For example, laws of motion of objects in a vacuum are absolutely reliable (as long as their speed is much less than that of light). The same can be said for numerical solutions to these equations, provided the solution is reasonably accurate. For these kinds of model, the verb “to model” means to calculate with high reliability what would happen under a particular set of initial and boundary conditions.

At the other end of the spectrum, we have mathematical models that are essentially tentative hypotheses. Such models are often based on intuition about a physical system, and represent a sort of educated guess about the quantitative relationships between things. For example, when Ahnert (1976) presented his inverse-exponential equation for regolith generation from bedrock, he was essentially expressing a conceptual hypothesis in mathematical terms. For these models-as-hypotheses, the phrase “to model” means to perform a quantitative “what if” experiment, asking the question: what kinds of pattern would I see if my hypothesis were correct? Comparing the prediction with observations provides a test of the hypothesis.

One can find many models that fall between these extremes. There are models that are based on well-known physics, but which are forced to use approximations of unknown accuracy in order to solve the governing equations. For example, climate models typically use simple parameterization schemes to represent convective mass and energy transport. Then too there are models that combine basic physical principles with elements of intuition, empiricism, and approximation. Arguably, many sediment-transport laws fall into this category: they are based on firm mechanical foundations (the force balance on a sediment grain) but also rely on strong approximations of factors like grain geometry, local flow velocity, and so on.

By now, it should be obvious that landscape evolution models also fall somewhere between the end-member cases of “model as truth” and “model as speculative hypothesis.” As we have seen throughout this course, there is a varying degree of experimental and observational support for the individual transport, weathering and erosion laws that go into a typical landscape model. In that sense, then, these models amount to more than just speculation. But equally there is still an element of speculation behind many of the process laws used in landscape models. Also, the process laws and algorithms represent a significant amount of upscaling in space and (especially) time. For example, the use of a steady precipitation rate as a proxy for the natural sequence of flows in a river channel represents a major approximation. For these reasons, we believe that three of the most important frontiers in landscape evolution research are (1) continuing to test individual process laws in the field and lab, (2) testing whole-landscape models using natural experiments, and (3) using mathematics, computation and experiments to study how the rates of various processes scale upward in time and space, and how these can be effectively parameterized.

11. Ten Commandments of Landscape Evolution Modeling

Thou shalt not use a model without understanding the ingredients therein.
Be thou ever mindful of uncertainty.
Thou shalt use thy model to develop insight.
Thou shalt take delight when thy model surprises thee.
Thou shalt kick thy model hard, that it may notice thee (an injunction borrowed gratefully from the 10 Climate Modeling Commandments).
Thou shalt diagnose the reasons for thy model’s behavior.
Thou shalt conduct sensitivity experiments and “play around.”
Thou shalt use thy model to discover the necessary and sufficient conditions needed to explain thy target problem.
If thou darest use a model to calculate what happened in your field area in the past, thou shalt find a way to test and calibrate it first.
If thou darest to predict future erosion, thou shalt heed the previous commandment ten times over (but thou mightest point out to skeptics that a process-based prediction is usually better than one based on pure guesswork, provided that commandment #2 is obeyed).

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>> If you have never used WMT, learn how to use it here.
>> Pick SedFlux2D from the list of components in WMT
>> SedFlux2D component is the sole driver of this simulation, so there is no need to assemble its ports.