CHAPTER 30: BEDROCK INCISION DUE TO WEAR. This chapter was written by Phairot Chatanantavet and Gary Parker. A slot canyon in the southwestern United States resulting from bedrock incision. FROM m&n’s TO A MORE PHYSICALLY-BASED MODEL OF BEDROCK INCISION.
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BEDROCK INCISION DUE TO WEAR
This chapter was written by Phairot Chatanantavet and Gary Parker.
A slot canyon in the southwestern United States resulting from bedrock incision
In Chapter 16: Morphodynamics of Bedrock-alluvial Transitions, it is assumed that the bedrock platform is fixed in time and is not free to undergo incision. This is generally true at the scale of adjustment of alluvial streams, but is not true in longer geomorphic time.
In Chapter 29: Knickpoint Migration in Bedrock Streams, a formulation for the morphodynamics of bedrock streams was developed using the following incision law:
This relation has provided useful results, but does not adequately express the physics of the incisional process. Recently Parker (2004) has developed a model that incorporates three mechanisms: a) wear caused as bedload particles strike bedrock (Sklar and Dietrich, 2004), plucking, by which chunks of fractured bedrock are torqued out of the bed by the flow and broken up, and macroabrasion, by which these chunks are further broken up as bedload particles strike them (Whipple et al., 2000).
Here a model of incision due to wear based on Sklar and Dietrich (2004) is developed.
OVERVIEW OF BEDROCK INCISION INCISION
As noted in the previous slide, aspects of bedrock rivers were introduced in Chapter 16 and 29. As described in Chapter 16, a bedrock river has patches of bed that are not covered by alluvium, and where bedrock is exposed.
There are many ways to cause a river to incise into its own bedrock. In this chapter, only the process of wear (abrasion) is considered (e.g. Sklar and Dietrich, 2004). That is, the bedrock is gradually worn away as bedrock particles strike regions of the bed where bedrock is exposed.
A bedrock river in Kentucky (tributary of Wilson Creek) with a partial alluvial covering.
Image courtesy A. Parola.
A bedrock river in Japan. Image courtesy H. Ikeda.
BEDROCK INCISIONAL ZONE INCISION
Floor of subsiding graben
Uplifting, incising zone
Incisional zone and alluvial fan in Tarim Basin, China.
Bedrock incision does not need to, but can be strongly driven by uplift. The
Above example shows incision in an uplifting mountain zone, with the resulting
sediment deposited in an adjacent subsiding graben.
HILLSLOPE DIFFUSION AND LANDSLIDING INCISION
Oregon Coast Range USA, Image courtesy Bill Dietrich
As the channel cuts down in response to uplift, it causes the adjacent hillslopes
to erode by hillslope diffusion or landsliding.
The model for incision driven by wear presented here is similar to that given in Sklar and Dietrich (2004). Wear or abrasion is the process by which stones colliding with the bed grind away the bedrock to sand or silt.
Wear is treated in terms of relations of the same status as those used to predict gravel abrasion in rivers (e.g. Parker, 1991). The stones that do the wear are assumed to have a characteristic size Dw.
Let q(x) denote the volume transport rate of sediment in the stream per unit width (L2/T) during the storm events that drive abrasion. Let the fraction of this load that consists of particles coarse enough to do the wear be . The volume transport rate per unit width qwear of sediment coarse enough to wear the bedrock is then given as
For simplicity, might be set equal to the fraction of the load that is gravel or coarser. A more sophisticated formulation might use a discriminator such as the ratio of shear velocity to fall velocity, as in Sklar and Dietrich (2004). Here is taken to be a prescribed constant.
Consider the case of saltating bedload particles. Let Esaltw denote the volume rate at which saltating wear particles bounce off the bed per unit bed area [L/T] and Lsaltw denote the characteristic saltation length of wear particles [L]. It follows from simple continuity that
The mean number of bed strikes by wear particles per unit bed area per unit time is equal to Esaltw/Vw, where Vw denotes the volume of a wear particle. It is assumed that with each collision a fraction r of the particle volume is ground off the bed (and a commensurate, but not necessary equal amount is ground off the wear particle). The rate of bed incision vIw due to wear is then given as (number of strikes per unit bed area) x (volume removed per unit strike), or
It is found that
Here the parameter w has dimensions [1/L], and has exactly the same status as the abrasion coefficients used to study downstream fining by abrasion in rivers.
This parameter could be treated as a constant. In so far as Lsaltw depends on flow conditions and r depends on rock type and perhaps the strength of the collision, w might be expected to vary somewhat with flow and lithology.
The above relation is valid only to the extent that all wear particles collide with exposed bedrock. If wear particles partially cover the bed, the wear rate should be commensurately reduced. This effect can be quantified in terms of the ratio qwear/qwearc, where qwearc denotes the capacity transport rate of wear particles. Let po denote the areal fraction of surface bedrock that is not covered with sediment. In general po can be expected to approach unity as qwear/qwearc 0, and approach zero as qwear/qwearc 1.
COVER FACTOR FOR INCISION BY WEAR INCISION
A “cover function” of the following type is
proposed by Sklar and Dietrich (2004);
Wear particles striking other wear particles do not wear the bed
Note that vIw drops to zero when q becomes equal to qwearc, downstream of which a completely alluvial gravel-bed stream is found. That is, the above formulation can describe the end point of the incisional zone as well as the incision rate.
The image on the left shows an inerodible concrete- bed flume with grooves at St. Anthony Falls Laboratory, University of Minnesota, USA. The design of the grooves is based on Piccaninny Creek, Australia (Wohl, 1998). Experiments are underway to investigate the value and dependence of the exponent no in the cover function. The picture below shows a top view of a sample experimental run with the ratio qwear/qwearc = 0.64; also channel slope = 2.0%, Froude number ~ 1.3, and Shields number * ~ 0.11. The size of the gravel is 7 mm. Note that the bed is not completely covered with gravel.
The parameter qwearc can be quantified in terms of standard bedload transport relations. A generalized relation of the form of Meyer-Peter and Müller (1948), for example, takes the form
where g, , and R are given, b denotes bed shear stress, c denotes a dimensionless critical Shields number, T is a dimensionless constant and nT is a dimensionless exponent. For example, in the implementation of Fernandez Luque and van Beek (1976), T = 5.7, nT = 1.5 and c is between 0.03 and 0.045.
As outlined in Chapter 5, the standard formulation for boundary shear stress places it proportional to the square of the flow velocity U = qf/H where qf denotes the flow discharge per unit width and H denotes flow depth. More precisely,
where Cf is a friction coefficient, which here is assumed to be constant
For the steep slopes of bedrock streams, the normal flow approximation, according to which the downstream pull of gravity just balances the resistive force at the bed, should apply, so that momentum balance takes the form (Chapter 5)
The bedload transport rate of wear
material is then evaluated as
The concept of below-capacity conditions is reviewed in Chapter 16. Briefly described here, an alluvial stream that is too steep relative to its sediment supply rate of wear material qwear would degrade to a lower slope S that would allow the above equation to transport wear material at the rate qwear. A bedrock stream, however, cannot degrade (without bedrock incision). So if for given values of qf and S it turns out that the sediment supply rate qwear is less than the equilibrium mobile-bed value qwearc, the river responds by exposing bedrock on its bed instead of degrading. As qwear is further reduced the river responds by increasing the fraction of the bed over which bedrock is exposed (Sklar and Dietrich, 1998). The bedrock river so adjusts itself to transport wear sediment at a rate qwear which is below its capacity qwearc for the given values of qf and S.
Now let i denote the precipitation rate (L/T), Bc(x) denote channel width, and A(x) denote the drainage basin area upstream of the point at distance x from a virtual origin near the headwater of the main-stem stream . Assuming no storage of water in the basin, the balance for water flow is
The parameter [L] is a surrogate for down-channel distance x. It will appear naturally in the model. Also, note that hydrology now enters into the model through the rainfall rate.
The capacity bedload transport rate of effective tools for wear is then given as
A routing model is necessary to determine the volume sediment transport rate per unit width q, and thus qwear. The equation of sediment conservation on a bedrock reach can be written as
where qh denotes the volume of sediment per unit stream length per unit time entering the channel from the hillslopes (either directly or through the intermediary of tributaries). Several models can be postulated for qh depending on hillslope dynamics. For simplicity, it is assumed that the watershed consists of easily-weathered rocks that are rapidly uplifted, so that bed lowering by channel incision results in hillslope lowering at the same rate. In this case
Note that in the latter equation, vI is the total incision rate and not just that due to wear. Note that the latter equation is just an example that must later be generalized to forms including e.g. hillslope diffusion, hillslope relaxation due to landslides driven by e.g. earthquakes or saturation in the absence of uplift, etc.
The above two equations lead to
The above relation can be used in the case of weak deviation from steady-state incision. In the case of steady-state incision in response to spatially uniform (piston-style) uplift, it reduces to qBc = vIA, or thus
Note that the parameter naturally arises from the formulation.
To obtain an approximate treatment of the case of deviation from steady-state incision in response to piston-style uplift, it is useful to postulate the structure relation
In general, b = 0.02 and nb ~ 0.3 to 0.5 (Montgomery and Gran, 2001; Whipple, 2004).
Drainage area A can be written in the function of down-channel distance x in terms of Hack’s law (Hack, 1957).
the following relation is obtained after some work:
MODIFICATION OF EXNER INCISION
If the river is assumed to be morphologically active only intermittently (during floods), the Exner equation becomes
where vI denotes the instantaneous incision rate during a flood (rather than the long-term average value used in Chapter 29) and
= uplift rate
p= porosity of bedrock ( ~ 0)
I = intermittency of large flood events (fraction of time)
In the case of a more general hydrologic model
where Ik = fraction of time the flood flow is in the kth flow range
Uplift is not really continuous, but it is treated as such here for simplicity.
FLOW DURATION CURVE INCISION
Q = flow discharge, PQ = fraction of time exceeded
The fractions Ik can be extracted from a flow duration curve such as the example given above.
Summary of the previous results
The sediment transport rate and
the incision rate talk to each other.
The incision rate at a point is a function of all incision upstream.
To solve this equation, introduce the new variable
x = 0 INCISION
x = xb
GOVERNING EQUATION ANDUPSTREAM BOUNDARY CONDITION
This equation is a first-order ordinary differential equation (ODE). After obtaining one boundary condition, it can be solved numerically, i.e. by the Runge-Kutta method.
It is assumed that the channel begins
at x = xb, upstream of which is a debris
flow dominated zone (x = 0 to xb).
at the channel head (x = xb) is
where again the subscript
“b” denotes the channel head
UPSTREAM BOUNDARY CONDITION contd. INCISION
which is the boundary condition for the first order ODE below.
Note that qwearcb denotes the value of qwearc at = b.
INCISION = b
MAKING THE PARAMETER DIMENSIONLESS
To solve the O.D.E. numerically, is first recast into a dimensionless parametervarying from 0 to 1. Where L denotes the value of at the downstream end of the basin, where x = L,
Thus the ODE becomes
This is solved numerically to obtain Wd, i.e. by the Runge-Kutta method with the previously derived boundary condition.
If bed elevation is held constant at the downstream end, the downstream boundary condition on the Exner equation becomes
Not too difficult to model in any program
INPUT: Initial values , Wdb, step size h, and M(=1/h)
OUTPUT: Approximation Wdn+1 to the solution
where n=0,1, … M-1
For n=0, 1, …, M-1 do:
subject to the b.c.
i = N+1
NUMERICAL MODEL: DISCRETIZATION
INTRODUCTION TO O.D.E.RTe-bookBedrockIncisionWear.xls
The program computes the time evolution of the long profile of a bedrock river with incision due to wear (abrasion). The output also includes plots of sediment transport a, slope, incision rate vI and areal fraction of bed exposure po as they vary in time.
A generalized relation of the form of Meyer-Peter and Müller (1948) relation is used to compute bedload transport capacity. Resistance is specified in terms of a constant Chezy coefficient Cz. The flow is calculated using the normal flow (local equilibrium) approximation. The drainage area is computed by using Hack's law and the river has varying width by the relation
The basic input parameters are nb, i, I, bw, Dw, Cz, a, Sinit, xb, L, N, dt, and au.
The auxiliary parameters include gt, nt, tc*, R, lp, no, Kh, nh, Kb, nb, and h.
Note that the value of the initial slope Sinit must be sufficiently high so that the lowest value of sediment transport, which is at the headwater, exceeds zero.
INTRODUCTION TO O.D.E.RTe-bookBedrockIncisionWear.xls contd.
An estimat of the minimum initial slope (Sinit) for each set of inputs is also shown at the bottom of the page “Calculator.” This estimation is calculated by fitting a line to the lower bound of a band given in Sklar and Dietrich (1998) dividing alluvial coarse bed streams from bedrock streams. The relation so obtained is
where drainage basin area A is measured in km2.
Figure from Sklar and Dietrich (1998)
INTRODUCTION TO O.D.E.RTe-bookBedrockIncisionWear.xls contd.
The final set of input includes the reach length L, the number of intervals N into which the reach is divided (so that x = L/N), the time step t, the upwinding coefficient u , and two parameters controlling output; the number of time steps to printout Ntoprint and the number of printouts Nprint. A value of u = 0.25 is recommended for stability in this program.
The basic program in Visual Basic for Applications is contained in Module 1, and is run from worksheet “Calculator”.
In any given case it is necessary try various values of the parameter N (which sets x) and the time step t in order to obtain good results. For any given x, it is appropriate to find the largest value of t that does not lead to numerical instability.
The program is executed by clicking the button “Click to run the program” from the worksheet “Calculator”. Outputs are given in numerical form in worksheet “ResultsofCalc” and in graphical form in four worksheets beginning with the word “Plot”. Some sample calculations are as follows.
The results in the next slide (Slide 32) were generated the following input parameters: uplift rate = 5 mm/yr, initial river bed slope Sinit = 0.006, effective rainfall rate i = 25 mm/hr, flood intermittency = 0.002, wear coefficient bw = 0.0001 m-1, effective size of particles that do the wear Dw = 50 mm, fraction of load consisting of sizes that do the wear = 0.05, bed friction coefficient Cf = 0.01, and value xb at the channel head = 1500 m. The total river length is L 10 km. The total time of calculation is 7200 years. The results produce an autogenic, upstream-propagating knickpoint. Slide 32 explains how such a knickpoint, which is not forced by such factors as base level drop, is formed.
The results in Slide 34 have the same input parameters as in Slide 32 except that the rock is rendered weaker by increasing the wear coefficient w to 0.0002 m-1. The results show that no autogenic knickpoint produced by the model in this case.
Slide 34 shows results for the case of a sudden base level fall. The input parameters are the same as those of Slide 32 except that at in the first year there is a base level fall of 30 m at the downstream end. The results manifest a knickpoint propagating upstream as well but here, but this time allogenically induced by base level fall.
SAMPLE RESULTS WITH WEAR COEFFICIENT O.D.E. = 0.0001 m-1
The process can be briefly explained as follows. Consider the Exner equation of Slide 19. Taking the second derivative in x and assuming a constant uplift rate results in
Now consider the plot of incision rate in the previous slide at year zero. Note that the shape of the curve of the incision rate vIw changes from concave-upward upstream to convex-upward downstream at a point near 4000 m. Thus the term changes from positive to negative near this point. Considering the above equations, a stream with such a shape of the incision curve must gradually form an autogenic knockpoint such that the term has a sign opposite to . This results in an elevation curve that changes from upward convex in the upstream reach to upward concave in the downstream reach. The inflection point sharpens to a knickpoint and migrates upstream. The size of an autogenic knickpoint can vary depending on the input parameters. The next slide shows a case without an autogenic knickpoint. Note that the shape of the curve of the incision rate at the initial year is convex-upward everywhere.
SAMPLE RESULTS WITH WEAR COEFFICIENT O.D.E. = 0.0002 m-1
REFERENCES FOR CHAPTER 30 O.D.E.
Chatanantavet, P. and Parker, G., 2005, Modeling the bedrock river evolution of western Kaua’i, Hawai’i, by a physically-based incision model based on abrasion, River, Coastal and Estuarine Morphodynamics, Taylor and Francis, London, 99-110.
Hack, J.T., 1957, Studies of longitudinal stream profiles in Virginia and Maryland. Prof. Paper 294-B, US Geological Survey, 45-97.
Montgomery, D.R. & Gran, K.B. 2001. Downstream variations in the width of bedrock channels. Water Resources Research, 37, 6, 1841-1846.
Parker, G. 1991. Selective sorting and abrasion of river gravel I: Theory, Jour. of Hydraulic Eng. 117, 2, 131-149.
Parker, G., 2004, Somewhat less random notes on bedrock incision, Internal Memorandum 118, St. Anthony Falls Laboratory, University of Minnesota, 20 p., downloadable at http://cee.uiuc.edu/people/parkerg/reports.htm .
Sklar, L.S. & Dietrich, W.E., 1998, River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply, in River over rock: fluvial processes in bedrock channels. Rivers over Rock, Geophysical Monograph Series, 107, edited by Tinkler, K. and Wohl, E.E., 237 – 260, AGU, Washington D.C.
Sklar, L.S. & Dietrich, W.E, 2004, A mechanistic model for river incision into bedrock by saltating bed load, Water Resources Research, 40, W06301, 21 p.
Whipple, K.X. & Tucker, G.E.,1999, Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs, Jour. of Geophysical Res., 104, B8, 17661-17674.
Whipple, K.X., Hancock, G.S. and Anderson, R.S., 2000, River incision into bedrock: Mechanics and relative efficacy of plucking, abrasion, and cavitation, Geological Society of America Bulletin, 112, 490–503.
Whipple, K.X., 2004, Bedrock rivers and the geomorphology of active orogens, Annual Review Earth and Planetary Sciences, 32, 151-185.
Wohl, E. E., 1998, Bedrock channel morphology in relation to erosional processes, Rivers over Rock, Geophysical Monograph Series, 107, edited by Tinkler, K. and Wohl, E.E., 133 – 151, AGU, Washington D.C.