Labs Landscape Evolution Modeling With Child Part 2

Landscape Evolution Modeling with CHILD

Part 2: Rainfall, runoff and drainage networks and hydraulic geometry

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 here.

This material is Part 2 of a three-part introduction to landscape evolution modeling with CHILD in WMT.

Part 1: Labs Landscape Evolution Modeling With Child Part 1

Part 3: Labs Landscape Evolution Modeling With Child Part 3

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:

1. Methods based on contributing drainage area

2. Numerical solutions to the 2D, vertically integrated and time-averaged Navier-Stokes equations

3. 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 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.

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 = bA^c     (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:

[math]\displaystyle{ \frac{\partial \eta}{\partial t} = i - \left( \frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} \right ) }[/math]     (14) [math]\displaystyle{ \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 }[/math]     (15) [math]\displaystyle{ \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 }[/math]     (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:

1. Steady flow with uniform precipitation or a specified runoff coefficient (effective discharge concept)

2. Steady flow with nonuniform precipitation or runoff (e.g., orographic precipitation)

3. Stochastic-in-time, spatially uniform runoff generation

4. “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 = 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 (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.