MORPHODYNAMICS OF BEDROCK-ALLUVIAL TRANSITIONS

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CHAPTER 16 MORPHODYNAMICS OF BEDROCK-ALLUVIAL
TRANSITIONS
An alluvial river has a bed that is completely
covered with sediment that the river can move
freely during flood flow. A bedrock river has
patches of bed that are not covered by alluvium,
where bedrock is exposed. In some bedrock rivers
the bed is almost completely bare of sediment.
This is, however, not the usual case. In most
cases of interest there is a mixture of patches
covered by alluvium and patches where bedrock is
exposed.
A bedrock river in Kentucky (tributary of Wilson
Creek) with a partial alluvial covering. Image
courtesy A. Parola.
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THE CONCEPT OF TRANSPORT CAPACITY
Equilibrium bedrock streams transport alluvium
under below-capacity conditions, whereas alluvial
streams transport sediment under at-capacity
conditions. These concepts can be explained as
follows. If the sediment supply of an alluvial
river is increased, the bed can be expected to
aggrade toward a new, steeper slope capable of
carrying the extra sediment. A bedrock stream,
on the other hand, may experience no aggradation
when sediment supply is increased. Instead, the
stream responds by reducing the fraction of the
bed covered by bedrock and increasing the
fraction covered by alluvium. Only when the bed
is completely covered with alluvium can the river
respond to increased sediment supply by aggrading.
Big Box Creek, USA, a bedrock river with a
stepped profile. Image courtesy E. Wohl.
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QUANTIFICATION OF TRANSPORT CAPACITY
The concept of a mobile-bed equilibrium state was
outlined in Chapter 14. In the case of the
Chezy resistance relation and the sample sediment
transport relation introduced in that chapter,
the governing equations of this equilibrium state
take the following forms
Now let grain size D, sediment submerged specific
gravity R, resistance coefficient Cf, critical
Shields number ?c and the parameters g, ?t and
nt be given. The relations specify two equations
in the following four parameters depth H, bed
slope S, water discharge per unit width qw and
volume total bed material sediment discharge per
unit width qt. Consider a stream with given
values of water discharge per unit width qw and
bed slope S. The capacity transport qt is that
computed from the above equation.
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BELOW-CAPACITY CONDITIONS
Now suppose that for given values of qw and S,
the actual sediment supply qts is less that the
value qt associated with mobile-bed equilibrium,
i.e.
An alluvial stream would degrade to a lower slope
S that would allow the above equation to be
satisfied with qts. A bedrock stream, however,
cannot degrade. So in the event that for given
values of qw and S the sediment supply rate qts
is less than the equilibrium mobile-bed value qt,
the river responds by exposing bedrock on its bed
instead of degrading. As qts is further reduced
the river responds by increasing the fraction of
the bed over which bedrock is exposed (Sklar and
Dietrich, 1998). The river so adjusts itself to
transport sediment at the rate qts which is below
its capacity qt for the given values of qw and S.
This allows a below-capacity equilibrium. In
the event that the actual sediment supply qts is
greater than the capacity transport rate qt at
the given slope S , the river will aggrade to a
new, higher slope in consonance with qts that
satisfies the above equation. There is
no above-capacity equilibrium.
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A SAMPLE CALCULATION
The following values are assumed in the sediment
transport relation below ?t 3.97, nt 1.5,
?c 0.0495 (Wong and Parker, submitted,
modification of Meyer-Peter and Müller), R
1.65, g 9.81 m2/s and Cf 0.01.
Consider a river with D 20 mm, flood Qw 90
m3/s and width B 30 m. The flood value of qw
Qw/B 3 m2/s. For any slope S, then, the
capacity value of qt can be computed from the
above relation. Assume that a bedrock river is
just barely completely covered with alluvium at
the slope S. How will the river respond if
sediment supply qts is reduced or increased? The
following two slides illustrate that the river
will aggrade to a new mobile-bed equilibrium when
qts gt qt. When qts lt qt, the river cannot
degrade due to the presence of bedrock, and
instead reaches a below-capacity equilibrium with
exposed bedrock.
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A SAMPLE CALCULATION contd.
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A SAMPLE CALCULATION contd.
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ILLUSTRATION OF BELOW-CAPACITY TRANSPORT OF 7 MM
GRAVEL OVER A BEDROCK BED
The video clip is from the Ph.D. research of
Phairot Chatanantavet.
rte-bookbelowcaptrans.mpg to run without
relinking, download to same folder as PowerPoint
presentations.
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BEDROCK-ALLUVIAL TRANSITIONS THE FALL LINE
The southeastern coastal plain of the United
States is characterized by a feature called the
Fall Line. Upstream (westward) of this line
the streams are in bedrock. Downstream
(eastward) of this line they are in alluvium. It
is of interest to speculate how the position of
the fall line might respond to changing sea
level.
Image of the southeastern coastal plain of the
United States from NASA https//zulu.ssc.nasa.gov/
mrsid/mrsid.pl
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EQUILIBRIUM STATE WITH BEDROCK-ALLUVIAL TRANSITION

A bedrock channel has constant slope Sb and
carries flood discharge per unit width qw.
Sediment with size D is fed in at the upstream
end at rate qts. The at-capacity slope S
consonant with qst, qw and D (as computed, for
example, from the transport relation of Slide 5)
is less than Sb. Base level is maintained at
some elevation at the downstream end this level
is higher than the elevation of the bedrock
basement there. An equilibrium bedrock-alluvial
transition must occur. To find it, draw a
straight line with slope S and intercept at the
point of base level maintenance, and extend it
upstream until it intersects the bedrock profile.
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DYNAMICS OF THE MIGRATION OF BEDROCK-ALLUVIAL
TRANSITIONS
Bedrock-alluvial transitions can migrate upstream
or downstream due to the effects of e.g. changing
sediment supply from upstream or changing base
level downstream. The figure below shows a case
where the alluvial region is (for whatever
reason) aggrading, resulting in an upstream
migration of the bedrock-alluvial transition.
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CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL
TRANSITION
The elevation profile of the bedrock basement is
denoted as ?base(x) it is assumed to be
unchanging in time. The elevation profile of the
alluvial zone is denoted as ?(x, t) it can
change in time due to aggradation or degradation.
The position of the bedrock-alluvial transition
is denoted as x sba(t). It is a function of
time because the position of the transition can
change in time. In order for the bedrock channel
to join continuously with the alluvial channel,
the following condition must hold
or
Now take the derivative with respect to time of
both sides of the equation. For example,
where S -??/?x denotes the alluvial bed slope
and dsba/dt denotes the speed of
migration of the bedrock-alluvial transition.
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CONTINUITY CONDITION AT THE BEDROCK-ALLUVIAL
TRANSITION contd.
Taking the derivative of both sides of the
relation results in where Sb -??base/?x
the slope of the bedrock channel. Reducing,
the following cute little relation is obtained
(Parker and Muto, 2003). Now since x sba
denotes a bedrock-alluvial transition, it can
always be expected that the bedrock slope Sb
exceeds the alluvial slope S there. So the
continuity condition says simply If the bed
aggrades, the transition moves upstream and if
the bed degrades the transition moves downstream.
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MOVING-BOUNDARY FORMULATION FOR RIVER
MORPHODYNAMICS WITH A BEDROCK-ALLUVIAL TRANSITION
The downstream end of the reach is located at the
constant value x sd, where base level is
maintained. The bedrock-alluvial transition is
located at x sba(t) lt sd. The goal is to
describe the morphodynamics of the evolution of
the stream so as to obtain both the change in the
alluvial profile ?(x,t) as a function of time and
the trajectory sba(t) of the transition as a
function of time. To this end we introduce the
coordinate transformation Note that the
bedrock-alluvial transition is located at
, and the downstream end of the reach is
located at . Using the chain rule,
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TRANSFORMATION OF THE EXNER EQUATION TO
MOVING-BOUNDARY COORDINATES
Evaluating the derivatives,
Transforming the Exner equation of sediment
continuity to the moving-boundary coordinate
system results in the form
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TRANSFORMATION OF THE CONTINUITY CONDITION TO
MOVING-BOUNDARY COORDINATES
Now from Slide 12 and the moving-boundary
coordinate transformation.
However slope is given as
Between these two relations,
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CHARACTER OF THE MORPHODYNAMIC PROBLEM
There is one more variable to solve than before,
i.e. the speed of the moving boundary, but
there is one more equation as well. Further
reducing the continuity condition with Exner,
or thus
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DISCRETIZATON FOR NUMERICAL SOLUTION
The domain from to (x sba
to x sd) is discretized into M intervals
bounded by M1 nodes. The node i 1 denotes the
bedrock-alluvial transition and the node i M1
denotes the point where base level is maintained.
The sediment feed rate qtf during floods is
specified at the node i 1 no ghost node is
needed in this formulation. The Exner equation
discretizes to
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DISCRETIZATION FOR NUMERICAL SOLUTION
As noted in the previous slide, the Exner
equation discretizes to The derivatives
discretize to the forms Derivatives need not
be evaluated at i M1 because bed elevation
?M1 is held constant. The shock condition
discretizes to
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INTRODUCTION TO RTe-bookBedrockAlluvialTrans.xls
The treatment of bedrock-alluvial transitions is
implemented in RTe-bookBedrockAlluvialTrans.xls.
The formulation uses the Engelund-Hansen (1967)
total bed material transport relation for
sand-bed streams. (Could you modify the code for
gravel-bed streams?) The downstream end of the
reach is located at the fixed point sd. The
bedrock channel is assumed to have constant slope
Sb. (Could you modify the code to let Sb vary in
space?) Initially the alluvial zone has bed
slope Sinit and length sd, so the
bedrock-alluvial transition is located at sba
0. The water discharge per unit width qw and
volume sediment bed material feed rate per unit
width qtf are specified along with the grain size
D of the bed material and the constant Chezy
resistance coefficient Cz. The flow is computed
using the normal (steady, uniform) flow
approximation. (Could you change it to use a
backwater formulation?). The values of submerged
specific gravity R and bed porosity ?p are set
equal to 1.65 and 0.4, respectively, in Const
statements in the code. They can be changed by
modifying the relevant Const statements.
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A SAMPLE CALCULATION UPSTREAM MIGRATION OF THE
BEDROCK-ALLUVIAL TRANSITION TO A NEW, STABLE
POSITION
The initial bed slope of the alluvial region
Sinit 0.00009 is too low for the specified
water discharge, sediment feed rate (qtf 0.0006
m2/s) and grain size. So the bed must aggrade to
a final equilibrium slope Sfinal, which is less
that the bedrock slope Sb. The result is that
the bedrock-alluvial transition must move
upstream to a new equilibrium from its initial
point at sba 0. The results of the computation
follow.
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A SAMPLE CALCULATION DOWNSTREAM MIGRATION OF THE
BEDROCK-ALLUVIAL TRANSITION TO A NEW, STABLE
POSITION
The initial bed slope of the alluvial region
Sinit 0.0002 is too high for the specified
water discharge, sediment feed rate (qtf 0.0001
m2/s) and grain size. So the bed must degrade to
a final equilibrium slope Sfinal, which is less
that the bedrock slope Sb. The result is that
the bedrock-alluvial transition must move
downstream to a new equilibrium from its initial
point at sba 0. The results of the computation
follow.
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SOME COMMENTS FOR FUTURE WORK

• The simple analysis presented in this chapter
  invites a wide range of extensions. A few are
  suggested below.
• How does sea level change affect the position
  of the fall line on coastal plains of the
  southeastern United States?
• When a dam is placed on a bedrock stream, how
  far upstream does the alluvial-bedrock transition
  created by deltaic deposit behind the dam migrate
  upstream?
• What happens to the position of a
  bedrock-alluvial transition if climate changes,
  resulting in changes in flood sediment supply
  qtf, flood water discharge qw and flood
  intermittency If?
• How could the analysis be generalized to
  gravel-bed streams and sediment mixtures?

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NOTE IN CLOSING BEDROCK INCISION
In the analysis described here it is assumed that
the bedrock platform is fixed in time, and is not
free to undergo incision. This assumption is
correct on the time scale of many alluvial
problems. It is erroneous, however, to assume
that bedrock channels cannot incise their beds
over sufficiently long time scales (e.g. Whipple
et al., 2000). The ability of a river to incise
through bedrock is amply illustrated by the image
below. Bedrock incision is considered in more
detail in Chapters 29 and 30.
Image of a Bolivian river from NASA https//zulu.s
sc.nasa.gov/mrsid/mrsid.pl
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REFERENCES FOR CHAPTER 16
Engelund, F. and E. Hansen, 1967, A Monograph on
Sediment Transport in Alluvial Streams, Technisk
Vorlag, Copenhagen, Denmark. Parker, G. and Muto,
T., 2003, 1D numerical model of delta response to
rising sea level, Proc. 3rd IAHR Symposium,
River, Coastal and Estuarine Morphodynamics,
Barcelona, Spain, 1-5 September. Sklar, L., and
W. E. Dietrich, 1998, River longitudinal profiles
and bedrock incision models Stream power and the
influence of sediment supply, in Rivers Over
Rock Fluvial Processes in Bedrock Channels,
Geophys. Monogr. Ser., vol. 107, edited by K. J.
Tinkler and E. E. Wohl, pp. 237260, AGU,
Washington, D. C. Whipple, K. X., G. S. Hancock,
and R. S. Anderson, 2000, River incision into
bedrock Mechanics and relative efficacy of
plucking, abrasion, and cavitation, Geol. Soc.
Am. Bull., 112, 490503. Wong, M. and Parker,
G., submitted, The bedload transport relation of
Meyer-Peter and Müller overpredicts by a factor
of two, Journal of Hydraulic Engineering,
downloadable at http//cee.uiuc.edu/people/parkerg
/preprints.htm .