Labs Landscape Evolution Modeling With Child Part 3

==Landscape Evolution Modeling with CHILD==
Part 3: Erosion and Transport by Running Water, Multiple Grain Sizes, and the Ten Commandments of Landscape Evolution Modeling
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, MN, 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 here: :File:CHILDVisTools.tar.gz.
This material is Part 3 of a three-part introduction to landscape evolution modeling with CHILD in WMT.
Part 1: Labs Landscape Evolution Modeling With Child Part 1
Part 2: Labs Landscape Evolution Modeling With Child Part 2
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
>> In WMT, open the model titled "dlim." Save and submit the job.
>> The 3 m.y. run should take about 20 seconds. Download the results and move them to the folder childFolderLocation∖ChildExercises∖dlim. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
>> In Matlab, navigate to the dlim folder. 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 m^2 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?
>> In WMT, open the model titled "dlimC1." Save and submit the job.
>> Download the results and move them to the folder childFolderLocation∖ChildExercises∖dlim. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
This run starts off where the previous one ended, but with a 10x higher rate of relative uplift.
>> In Matlab, type:
figure(1)
cmovie( 'dlimC1', 31, 3e4, 3e4, 1e4, 5000 ); % 10x vertical scale
figure(2)
hold on, csa( 'dlimC1', 31, 'r.' ); hold off
Because we are using a slope-linear detachment law, a 10x increase in relative uplift rate leads to a 10x 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.
>> In WMT, open the model titled "dlim_small." Save and submit the job.
>> Download the results and move them to the folder childFolderLocation∖ChildExercises∖dlim. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
This run is identical to dlim but with a domain of 1.5 by 1.5km and \sim25m wide cells, instead of 30x30km and \sim500m 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).
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.
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 WMT, open the model titled "tlim1." Save and submit the job.
>> The 1 m.y. run should take about 2 minutes. Download the results and move them to the folder childFolderLocation∖ChildExercises∖tlim. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
>> In Matlab, type:
cd('../tlim')
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\times 10^{-4}.
Exercise 10: A Pile of Cobbles
Now let's try the same experiment with 5cm cobbles.
>> In WMT, open the model titled "tlim2" Save and submit the job.
>> The 3 m.y. run should take about 2-3 minutes. Download the results and move them to the folder childFolderLocation∖ChildExercises∖tlim. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
>> 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; Gasparini et al., 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: Reconfigure tlim2 to have a higher uplift rate. Save the model with a new name, run and download the results. Use the Matlab script cstrmproseries to plot fluvial profiles undergoing transient response. How do these compare with the detachment-limited model?
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
>> In WMT, open the model titled "erodep1.” Save and submit the job.
>> The 1 m.y. run should take about 5 minutes. Download the results and move them to the folder childFolderLocation∖ChildExercises∖hyrbid. For childFolderLocation, use the path to the Child visualization tools package that you downloaded.
>> In Matlab, navigate to the hybrid folder. 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.
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).
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