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, please download the CHILD visualization tools.
Overview
The learning goals of these exercises are:
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.
Fig. 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.
Introduction to LEMs
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).
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.
Model
Example reference
Notes
SIBERIA
Willgoose, Bras, and Rodriguez-Iturbe (1991)
Transport-limited;
Channel activator function
DRAINAL
Beaumont, Fullsack, and Hamilton (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, Kirkby, and Macklin (1996)
Cellular automaton algorithm
for 2D flow field
ZSCAPE
Densmore, Ellis, and Anderson (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, Veldkamp, and Bouma (2002)
Multiple flow directions
APERO/CIDRE
Carretier and Lucazeau (2005)
Single or multiple
flow directions
Table 1: Partial list of numerical landscape models published between 1991 and 2005.
Continuity of Mass and Discretization
A typical mass continuity equation for a column of soil or rock is:
\frac{\partial n}{\partial t} = B - \nabla \vec{q}_s     (1)
where η is the elevation of the land surface [L]; 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 q⃗s is sediment flux per unit width [L2/T]. (The letters in square brackets indicate the dimensions of each variable; L stands for length, T for time, and M for mass.) 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.)
This statement can be expressed mathematically, for cell i, as follows:
\frac{d\eta_i}{dt} = B + \frac{1}{\Lambda_i} \sum_{j=1}^N q_{sj} \lambda_j     (2)
where Λ i is the horizontal surface area of cell i; N is the number of faces surrounding cell i; qsj is the unit flux across face j; and λj 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), Gregory E Tucker, Lancaster, Gasparini, Bras, et al. (2001).)
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, Howard, and Wiberg (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:
\frac{d\eta_i}{dt} \approx \frac{\eta_i(t+\Delta t) - \eta_i(t)}{\Delta t}     (3)
\eta_i(t+\Delta t) = \eta_i(t) + U\Delta t + \Delta t \frac{1}{\Lambda_i} \sum_{j=1}^N q_{sj} \lambda_j      (4)
The main disadvantage of this approach is that very small time steps are typically needed in order to ensure numerical stability. (CHILD uses a variant of this that seeks the largest possible stable value of Δ t 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).
 
Fig. 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.
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.
Linear Diffusion
For relatively gentle, soil-mantled slopes, there is reasonably strong support for a transport law of the form:
q⃗s =  − 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 $\vec{q_s}$ 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:
S_{ik} = \frac{\eta_k - \eta_i}{L_{ik}}     (6)
where Lik is the distance between nodes. On a raster grid, Lik = Δ x is simply the grid spacing. The sediment flux per unit width is then
q_{sik} \simeq D \frac{\eta_k - \eta_i}{L_{ik}}     (7)
where qsik is the volume flux per unit width from node k to node i (if negative, sediment flows from i to k), and Lik is the distance between nodes. To compute the total sediment flux through face j, we simply multiply the unit flux by the width of face j, which we denote λij (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 ensure that CHILD can be run and visualized. You should already have an account on WMT and an account on beach. Note that your login credentials may not be the same for WMT as for beach. You should also have downloaded the CHILD visualization files. Note that Matlab is required to run the visualization scripts.
>> If you do not have an account on beach, request one here.
>> If you do not have an account on WMT, create one here.
>> If you have not downloaded the visualization tools, download them and unzip them.
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 unzipped visualization scripts. You can also add a folder to your path by selecting File- > Set Path... from the menu.
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
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:
\vec{q}_s = \frac{-D \nabla z}{1-|\nabla z/S_c|^2}     (10)
Exercise 3: Nonlinear Diffusion and Planar Slopes
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, Hayes, and Grant 2003).
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
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).
For the Voronoi cell matrix that CHILD and CASCADE use, the simplest routing procedure is a generalization of D8 (Fig. 1). Each cell i has Ni 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 Ni 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.
Fig. 3: Flow accumulation by D8, or single flow directions, and multiple flow directions (Schauble et al., 2008)
Exercise 4: Flow Over Noisy, Inclined Topography
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 = PA     (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.
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:
\frac{\partial \eta}{\partial t} = i - \left( \frac{\partial q_x}{\partial x}
+ \frac{\partial q_y}{\partial y} \right )      (14)
\frac{\partial q_x}{\partial t} + \frac{\partial q_x u}{\partial x}
+ \frac{\partial q_y u}{\partial y}
+ g h \frac{\partial h}{\partial x}
+ g h \frac{\partial \eta}{\partial x}
+ \frac{\tau_{bx}}{\rho} = 0      (15)
\frac{\partial q_y}{\partial t} + \frac{\partial q_y v}{\partial y}
+ \frac{\partial q_x v}{\partial x}
+ g h \frac{\partial h}{\partial y}
+ g h \frac{\partial \eta}{\partial y}
+ \frac{\tau_{by}}{\rho} = 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).
Fig. 4: Simulated water surface elevations and flow depth (Birnir et al., 2001)
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, Kirkby, and Macklin (1996) generalization of Murray-Paola for 2D flow (CAESAR model)
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).
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, Bras, and Rodriguez-Iturbe (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 (Sólyom 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
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).
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.”
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 = KwQb, 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 (Lavé and Avouac 2001), New Zealand (Amos and Burbank 2007), Taiwan (Yanites et al. 2010), and California (Duvall, Kirby, and Burbank 2004). Some models have begun to explore these sensitivities (Wobus, Tucker, and Anderson 2006; Wobus et al. 2008; Attal et al. 2008; Turowski, Lague, and Hovius 2009; Yanites and Tucker 2010), but full treatment of the channel geometry adjustment problem is a frontier area.
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, Bras, and Rodriguez-Iturbe 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 (Gregory E Tucker, Lancaster, Gasparini, and Bras 2001; Whipple and Tucker 2002). With the undercapacity concept, detachment rate depends on surplus transport capacity (Beaumont, Fullsack, and Hamilton 1992). In the saltation-abrasion model, detachment is driven by grain impacts and limited by sediment shielding (Gasparini, Whipple, and Bras, 2007; Whipple and Tucker, 2002.
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:
Dc = Kb(τ − τc)pb, or alternatively, Dc = Kb(τpb − τcpb)     (17)
where τ is local bed shear stress, τc is a threshold stress below which detachment is ineffective, Kb is a constant, and pb 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:
D_c = K_b \left( \frac{Q}{W}S - \Phi_c \right)^{p_b}     (19)
where Φ c is, again, a threshold below which detachment is ineffective. Stream power per unit width turns out to be proportional to τ3 / 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
Exercise 7: Zooming in to the Hillslopes
Exercise 8: Knickzones and Transient Response
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
Exercise 10: A Pile of Cobbles
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:
E_i =
\begin{cases}
\frac{Q_c - \sum_{j=1}^{N_i} Q_{sij}}{\Lambda_i} & \text{if $\frac{Q_c - \sum_{j=1}^{N_i} Q_{sij}}{\Lambda_i} < D_c$} \\
D_c & \text{otherwise}
\end{cases}     (19)
Exercise 11: Erosion and Deposition, Together at Last
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, Whipple, and Bras (2007) explore the behavior of these models with CHILD simulations.
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
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, Bras, and Tucker 2004; Istanbulluoglu and Bras 2005) and trees (Lancaster, Hayes, and Grant 2003)
Alternate forms of mass wasting, including landslides and debris flows (Densmore, Ellis, and Anderson 1998; Lancaster, Hayes, and Grant 2003; Istanbulluoglu et al. 2005)
Knickpoints, hanging valleys, and plunge pools (Flores-Cervantes, Istanbulluoglu, and Bras 2006; Crosby et al. 2007)
Glaciation (Herman and Braun 2006; Herman, Braun, and Dunlap 2007; Herman and Braun 2008)
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.
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).
References
Ahnert, F. 1976. “Brief Description of a Comprehensive Three-Dimensional Process-Response Model of Landform Development.” Zeitschrift F&Uuml;R Geomorfologie, Supplementband 25: 29–49.
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