1D MORPHODYNAMICS OF MOUNTAIN RIVERS: UNIFORM SEDIMENT

1
1D MORPHODYNAMICS OF MOUNTAIN RIVERS UNIFORM
SEDIMENT
Morphodynamics is the study of the variation of
morphology in response to net erosion or
deposition of sediment. The images below
illustrate a reach of the Mad River, California
which has undergone bed lowering (degradation) in
response to gravel mining.
2
AGGRADATION AND DEGRADATION
A river reach aggrades (bed elevation increases)
when it is supplied more sediment than it
exports. A river reach degrades (bed elevation
decreases) when it exports more sediment than it
is supplied.
Degraded reach of the Uria River, Venezuela after
the Vargas disaster, 1999. Cour. J. Lopez.
The Ok Tedi in Papua New Guinea had aggraded some
5 m near this bridge in response to mine disposal.
3
RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING
CAUSED BY AN EARTHQUAKE
View in November, 1999, shortly after the
earthquake caused a sharp 3 m elevation drop at a
fault.
View in May, 2000 after aggradation and
degradation have smoothed out the elevation drop.
The above images of the Deresuyu River, Turkey,
are courtesy of Patrick Lawrence and François
Métivier (Lawrence, 2003)
4
RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING
CAUSED BY AN EARTHQUAKE contd.
Inferred initial profile immediately after
faulting in November, 1999
Profile in May, 2001
Upstream degradation (bed level lowering) and
downstream aggradation (bed level increase) are
realized as the river responds to the knickpoint
created by the earthquake (Lawrence, 2003).
5
BEDLOAD TRANSPORT
The morphodynamics of mountain rivers is
controlled by the differential transport of
gravel moving as bedload. Bedload particles
slide, roll, or saltate just above the bed, as
opposed to suspended particles, which can be
wafted high in the water column by turbulence.
Bedload transport of uniform 7-mm gravel in a
flume is illustrated below. The video clip is
from the experiments of Miguel Wong.
6
BELOW-CAPACITY BEDLOAD TRANSPORT
The video clip is from the experiments of Phairot
Chatanantavet.
7
EXNER EQUATION FOR THE CONSERVATION OF BED
SEDIMENT
In 1D morphodynamics bed elevation variation is
considered in the absence of width variation, and
local bed features such as bars are not
specifically modeled. 1D morphodynamics
describes the time variation of the longitudinal
profile of river bed elevation in response to net
sediment deposition or erosion. The first step
in characterizing 1D morphodynamics is the
derivation of the Exner (19??) equation of bed
sediment conservation. The parameters defined
below are used in the derivation.

• qb volume bedload transport rate per unit width
  L2T-1
• ?s sediment density ML3/T
• gb ?sqb mass bedload transport rate per unit
  width ML-1T-1
• bed elevation L
• ?p porosity of sediment in bed deposit 1
• (volume fraction of bed sample that is holes
  rather than sediment 0.25 0.55 for
  noncohesive material)
• x streamwise coordinate L
• t time T
• B channel width L

8
CONSERVATION OF BED SEDIMENT
?/?t(gravel mass in control volume of bed) mass
gravel inflow rate mass gravel outflow rate
or thus
This corresponds to the original form derived by
Exner.
The control volume has a unit width normal to x
9
BACKGROUND AND ASSUMPTIONS FOR 1D MORPHODYNAMICS

• Change in channel bed level (aggradation or
  degradation) can occur in response to
• increase or decrease in upstream sediment
  supply
• change in hydrologic regime (water diversion or
  climate change)
• change in river slope (e.g. channel
  straightening)
• increased or decreased sediment supply from
  tributaries
• sudden inputs of sediment from debris flows or
  landslides
• faulting due to earthquakes or other tectonic
  effects such as tilting along the reach,
• and
• changing base level at the downstream end of
  the reach of interest.

Here base level loosely means a controlling
elevation at the downstream end of the reach of
interest. It means water surface elevation if
the river flows into a lake or the ocean, or a
downstream bed elevation controlled by e.g.
tectonic uplift or subsidence at a point where
the river is not flowing into standing water.
Base level of this reach of the Eau Claire river,
Wisconsin, USA is controlled by a reservoir, Lake
Altoona
10
THE EQUILIBRIUM STATE contd.
Rivers are different in many ways from laboratory
flumes. It nevertheless helps to conceptualize
rivers in terms of a long, straight, wide,
rectangular flume with high sidewalls (no
floodplain), constant width and a bed covered
with alluvium. Such a river has a simple
mobile-bed equilibrium (graded) state at which
flow depth H, bed slope S, water discharge per
unit width qw and bed material load per unit
width qt remain constant in time t and in the
streamwise direction x. A recirculating flume
(with both water and sediment recirculated) at
equilibrium is illustrated below.
11
THE EQUILIBRIUM STATE contd.
The hydraulics of the equilibrium state are those
of normal flow. Here the case of a plane bed (no
bedforms) is considered as an example. The bed
consists of uniform material with size D. The
governing equations are (see lecture on
hydraulics)
Momentum conservation
Water conservation
Friction relations where kc is a composite
bed roughness which may include the effect of
bedforms (if present).
Generic transport relation of the form of
Meyer-Peter and Müller for total bed material
load where ?t and nt are dimensionless constants
12
THE EQUILIBRIUM STATE contd.
In the case of the Chezy resistance relation, the
equations governing the normal state reduce to
(Slide 26 of the lecture on hydraulics)
In the case of the Manning-Stickler resistance
relation with an exponent of 1/6, the equations
governing the normal state reduce to (Slide 30 of
the lecture on hydraulics, with kc ? ks and ?g ?
?r)
Let D, ks and R be given. In either case above,
there are two equations for four parameters at
equilibrium water discharge per unit width qw,
volume sediment discharge per unit width qt, bed
slope S and flow depth H. If any two of the set
(qw, qb, S and H) are specified, the other two
can be computed. In a sediment-feed flume, qw
and qb are set, and equilibrium S and H can be
computed from either of the above pair. In a
recirculating flume, qw and H are set (total
water mass in flume is conserved), and qb and S
can be computed.
13
SIMPLIFICATIONS

• The concepts of aggradation and degradation are
  best illustrated by using simplified relations
  for hydraulic resistance and sediment transport.
  Here the following simplifications are made in
  addition to the assumptions of constant width and
  the absence of a floodplain
• The case of a Manning-Strickler formulation with
  constant roughness ks is considered
• Bed material is taken to be uniform with size D
• Only the portion of boundary shear stress due to
  skin friction is available to transport sediment
• The Exner equation of sediment conservation is
  based on a computation of bedload, which is
  computed via the generic equation
• where ?s ? 1 is a constant to convert
  total boundary shear stress to that due to skin
  friction (if necessary). For example, to recover
  the corrected version of Meyer-Peter and Müller
  (1948) relation of Wong (2003) gravel transport,
  set ?t 3.97 , nt 1.5, ?c 0.0495 and ?s
  1. A setting ?s 0.75 implies that 75
• of the total resistance is skin friction and 25
  is form drag.

14
SIMPLIFICATIONS contd.
5. The full flood hydrograph or flow duration
curve of discharge variation is replaced by a
flood intermittency factor If, so that the river
is assumed to be at low flow (and not
transporting significant amounts of sediment) for
time fraction 1 If, and is in flood at constant
discharge Q, and thus constant discharge per unit
width qw Q/B for time fraction If (Paola et
al., 1992). The implied hydrograph takes the
conceptual form below
In the long term, then, the relation between
actual time t and time that the river has been in
flood tf is given as Let the value of the bed
material load at flood flow qb be computed in
m2/s. Then the total mean annual sediment load
Gt in million tons per year is given as
15
AN ISSUE OF NOTATION
A numerical method and program for computing the
1D morphodynamics of rivers using the normal flow
approximation is introduced in the succeeding
slides. The code was originally written for a
generic river (gravel-bed or sand-bed), for which
the total volume bed material load per unit width
is denoted as qt. Here this code is applied to
mountain rivers, so wherever qt appears, the user
of this lecture material should make the
transformation
16
AGGRADATION AND DEGRADATION AS TRANSIENT
RESPONSES TO IMPOSED DISEQIUILBRIUM CONDITIONS
Aggradation or degradation of a river reach can
be considered to be a response to disequilibrium
conditions, by which the river tries to reach a
new equilibrium. For example, if a river reach
has attained an equilibrium with a given sediment
supply from upstream, and that sediment supply is
suddenly increased at t 0, the river can be
expected to aggrade toward a new equilibrium.
17
NORMAL FLOW FORMULATION OF MORPHODYNAMICS
GOVERNING EQUATIONS
In this chapter the flow is calculated by
approximating it with the normal flow
formulation, even if the profile itself is in
disequilibrium. The approximation is of loose
validity in most cases of interest. It is
particularly justifiable in the case of mountain
rivers, as shown in the lecture on hydraulics
Using the Exner formulation for sediment
conservation and the Manning-Strickler
formulation for flow resistance, the
morphodynamic problem has the following character
In the above relations t denotes real time (as
opposed to flood time) and the intermittency
factor If accounts for the fact that the river is
only occasionally in flood (and thus
morphologically active).
18
THE NORMAL FLOW MORPHODYNAMIC FORMULATION AS A
NONLINEAR DIFFUSION PROBLEM
The previous formulation can be rewritten
as where ?d is a kinematic diffusivity of
sediment (dimensions of L2/T) given by the
relation
The top equation is a (nonlinear) diffusion
equation. In the bottom equation, it is seen
that ?d is dependent on S - ??/?x, so that the
diffusion formulation is nonlinear. The problem
is second-order in x and first order in t, so
that one initial condition and two boundary
conditions are required for solution.
19
INITIAL AND BOUNDARY CONDITIONS
The reach over which morphodynamic evolution is
to be described must have a finite length L.
Here it extends from x 0 to x L.
The initial condition is that of a specified bed
profile The simplest example of this is a
profile with specified initial downstream
elevation ?Id at x L and constant initial slope
SI The upstream boundary condition can be
specified in terms of given sediment supply, or
feed rate qtf, which may vary in time The
simplest case is that of a constant value of
sediment feed. The downstream boundary condition
can be one of prescribed base level in terms of
bed elevation Again the simplest case is a
constant value, e.g. ?d 0.
20
NOTES ON THE DOWNSTREAM BOUNDARY CONDITION

• In principle the best place to locate the
  downstream boundary condition is at a bedrock
  exposure, as illustrated below. In most alluvial
  streams, however, such points may not be
  available. Three alternatives are possible
• Set the boundary condition at a point so far
  downstream that no effect of e.g. changed
  sediment feed rate is felt during the time span
  of interest
• Set the boundary condition where the river joins
  a much larger river or
• Set the boundary condition at a point of known
  water surface elevation, such as a lake.

Alluvial Kaiya River, Papua New Guinea, and
downstream bedrock exposure
Bedrock makes a good downstream b.c.
21
DISCRETIZATION FOR NUMERICAL SOLUTION
The morphodynamic problem is nonlinear and
requires a numerical solution. This may be done
by dividing the domain from x 0 to x L into M
subreaches bounded by M 1 nodes. The step
length ?x is then given as L/M. Sediment is fed
in at an extra ghost node one step upstream of
the first node.
Bed slope can be computed by the relations to the
right. Once the slope Si is computed the
sediment transport rate qt,i can be computed at
every node. At the ghost node, qt,g qtf.
22
DISCRETIZATION OF THE EXNER EQUATION
Let ?t denote the time step. Then the Exner
equation discretizes to
where
and au is an upwinding coefficient. In a pure
upwinding scheme, au 1. In a central
difference scheme, au 0.5. A central
difference scheme generally works well when the
normal flow formulation is used. At the ghost
node, qt,g qtf. In computing ?qt,i/?x at i
1, the node at i 1 ( 0) is the ghost node. At
node M1, the Exner equation is not implemented
because bed elevation is specified as ?M1 ?d.
23
INTRODUCTION TO RTe-bookAgDegNormal.xls
The basic program in Visual Basic for
Applications is contained in Module 1, and is run
from worksheet Calculator. The program is
designed to compute a) an ambient mobile-bed
equilibrium, and b) the response of a reach to
changed sediment input rate at the upstream end
of the reach starting from t 0. The first set
of required input includes flood discharge Q,
intermittency If, channel (bankfull) width B,
grain size D, bed porosity ?p, composite
roughness height kc and ambient bed slope S
(before increase in sediment supply). Here
composite roughness height is meant to include
the effect of bedforms. For mountain streams in
the absence of form drag, it is appropriate to
set kc equal to ks nkD, where nk is in the
range 3 4. When form drag is present it is
appropriate to increase nk to somewhat larger
values ( 5 or 6). Various parameters of the
ambient flow, including the ambient annual bed
material transport rate Gt in tons per year, are
then computed directly on worksheet Calculator.
24
INTRODUCTION TO RTe-bookAgDegNormal.xls contd.
The next required input is the annual average bed
material feed rate Gtf imposed after t gt 0. If
this is the same as the ambient rate Gt then
nothing should happen if Gtf gt Gt then the bed
should aggrade, and if Gtf lt Gt then it should
degrade. The final set of input includes the
reach length L, the number of intervals M into
which the reach is divided (so that ?x L/M),
the time step ?t, the upwinding coefficient au
(use 0.5 for a central difference scheme), and
two parameters controlling output, the number of
time steps to printout Ntoprint and the number of
printouts (in addition to the initial ambient
state) Nprint. The downstream bed elevation ?d
is automatically set equal to zero in the
program. Auxiliary parameters, including ?r
(coefficient in Manning-Strickler), ?t and nt
(coefficient and exponent in load relation), ?c
(critical Shields stress), ?s (fraction of
boundary shear stress that is skin friction) and
R (sediment submerged specific gravity) are
specified in the worksheet Auxiliary Parameters.
25
INTRODUCTION TO RTe-bookAgDegNormal.xls contd.
The parameter ?s estimating the fraction of
boundary shear stress that is skin friction,
should either be set equal to 1 or estimated
using the techniques of Chapter 9. In any given
case it will be necessary to play with the
parameters M (which sets ?x) and ?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 Do a
Calculation from the worksheet Calculator.
Output for bed elevation is given in terms of
numbers in worksheet ResultsofCalc and in terms
of plots in worksheet PlottheData The
formulation is given in more detail in the
worksheet Formulation, which is also available
as a stand-alone document, Rte-bookAgDegNormalFor
mul.doc.
26
MODULE 1 Sub Main
This is the master subroutine that controls the
Visual Basic program.
Sub Main() Clear_Old_Output
Get_Auxiliary_Data Get_Data
Compute_Ambient_and_Final_Equilibria
Set_Initial_Bed_and_time Send_Output j
0 For j 1 To Nprint For w 1 To
Ntoprint Find_Slope_and_Load
Find_New_eta Next w
More_Output Next j End Sub
27
MODULE 1 Sub Set_Initial_Bed_and_time
This subroutine sets the initial ambient bed
profile.
Sub Set_Initial_Bed_and_time() For i 1
To N 1 x(i) dx (i - 1)
eta(i) Sa L - Sa dx (i - 1)
Next i time 0 End Sub
28
MODULE 1 Sub Find_Slope_and_Load
This subroutine computes the load at every node.
Sub Find_Slope_and_Load() Dim i As
Integer Dim taux As Double Dim qstarx As
Double Dim Hx As Double Sl(1) (eta(1)
- eta(2)) / dx Sl(M 1) (eta(M) -
eta(M 1)) / dx For i 2 To M
Sl(i) (eta(i - 1) - eta(i 1)) / (2 dx)
Next i For i 1 To M 1
Hx ((Qf 2) (kc (1 / 3)) / (alr 2) / (B
2) / g / Sl(i)) (3 / 10) taux
Hx Sl(i) / Rr / D If fis taux lt
tausc Then qstarx 0
Else qstarx alt (fis taux -
tausc) nt End If qt(i)
((Rr g D) 0.5) D qstarx Next
i End Sub
29
MODULE 1 Sub Find_New_eta
This subroutine implements the Exner equation to
find the bed one time step later.
Sub Find_New_eta() Dim i As Integer
Dim qtback As Double Dim qtit As Double Dim
qtfrnt As Double Dim qtdif As Double For
i 1 To M If i 1 Then
qtback qqtf Else
qtback qt(i - 1) End If
qtit qt(i) qtfrnt qt(i 1)
qtdif au (qtback - qtit) (1 - au)
(qtit - qtfrnt) eta(i) eta(i) dt
/ (1 - lamp) / dx qtdif Inter Next i
time time dt End Sub
30
A SAMPLE COMPUTATION
The ambient sediment transport rate is 305,000
tons/year. At time t 0 this is increased to
700,000 tons per year. The bed must aggrade in
response.
31
RESULTS OF SAMPLE COMPUTATION
32
INTERPRETATION
The long profile of a river is a plot of bed
elevation ? versus down-channel distance x. The
long profile of a river is called upward concave
if slope S -??/?x is decreasing in the
streamwise direction otherwise it is called
upward convex. That is, a long profile is upward
concave if
Aggrading reaches often show transient upward
concave profiles. This is because the deposition
of sediment causes the sediment load to decrease
in the downstream direction. The decreased load
can be carried with a decreased Shields number
?, and thus according to the normal-flow
formulation of the present chapter, a decreased
slope
33
INTERPRETATION contd.
The transient long profile of Slide 30 is upward
concave because the river is aggrading toward a
new mobile-bed equilibrium with a higher slope.
Once the new equilibrium is reached, the river
will have a constant slope (vanishing concavity).
This process is outlined in the next slide
(Slide 33), in which all the input parameters are
the same as in Slide 29 except Ntoprint, which is
varied so that the duration of calculation ranges
from 1 year (far from final equilibrium) to 250
years (final equilibrium essentially
reached). Slide 34 shows a case where the
profile degrades to a new mobile-bed equilibrium.
During the transient process of degradation the
long profile of the bed is downward concave, or
upward convex. This is because the erosion which
drives degradation causes the load, and thus the
slope to increase in the downstream direction.
The input conditions for Slide 34 are the same as
those of Slide 29, except that the sediment feed
rate Gtf is dropped to 70,000 tons per year.
This value is well below the ambient value of
305,000 tons per year (see Slide 29), forcing
degradation and transient downward concavity.
In addition, Ntoprint is varied so that the
duration of calculation varies from 1 year to 250
years. Factors such as subsidence or base level
rise can drive equilibrium long profiles which
are upward concave.
34
AGGRADATION TO A NEW MOBILE-BED EQUILIBRIUM
35
DEGRADATION TO A NEW MOBILE-BED EQUILIBRIUM
36
ADJUSTING THE NUMBER M OF SPATIAL INTERVALS AND
THE TIME STEP ?t
The calculation becomes unstable, and the program
crashes if the time step ?t is too long. The
above example resulted in a crash when ?t was
increased from the value of 0.01 years in Slide
29 to 0.05 years. The larger the value M of
spatial intervals is, the smaller is the maximum
value of ?t to avoid numerical instability.
Acceptable values of M and ?t can be found by
trial and error.
37
AN EXTENSION RESPONSE OF AN ALLUVIAL RIVER TO
VERTICAL FAULTING DUE TO AN EARTHQUAKE
The code in RTe-bookAgDegNormal.xls represents a
plain vanilla version of a formulation that is
easily extended to a variety of other cases. The
spreadsheet RTe-bookAgDegNormalFault.xls contains
an extension of the formulation for sudden
vertical faulting of the bed. The bed
downstream of the point x rfL (0 lt rf lt 1) is
suddenly faulted downward by an amount ??f at
time tf. The eventual smearing out of the long
profile is then computed.
38
RESULTS OF SAMPLE CALCULATION WITH FAULTING
39
RESULTS OF SAMPLE CALCULATION WITH FAULTING
contd.
In time the fault is erased by degradation
upstream and aggradation downstream, and a new
mobile-bed equilibrium is reached.
40
REFERENCES
Exner, F. M., 1920, Zur Physik der Dunen,
Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in
German). Exner, F. M., 1925, Uber die
Wechselwirkung zwischen Wasser und Geschiebe in
Flussen, Sitzber. Akad. Wiss Wien, Part IIa, Bd.
134 (in German). Lawrence, P., 2003, Bank Erosion
and Sediment Transport in a Microscale Straight
River, Ph.D. thesis, University of Paris 7
Denis Diderot, 167 p. Meyer-Peter, E. and Müller,
R., 1948, Formulas for Bed-Load Transport,
Proceedings, 2nd Congress, International
Association of Hydraulic Research, Stockholm
39-64. Paola, C., Heller, P. L. Angevine, C.
L., 1992, The large-scale dynamics of grain-size
variation in alluvial basins. I Theory, Basin
Research, 4, 73-90. Wong, M., 2003, Does the
bedload equation of Meyer-Peter and Müller fit
its own data?, Proceedings, 30th Congress,
International Association of Hydraulic Research,
Thessaloniki, J.F.K. Competition Volume 73-80.
For more information see Gary Parkers e-book 1D
Morphodynamics of Rivers and Turbidity Currents
http//cee.uiuc.edu/people/parkerg/morphodynamics
_e-book.htm
41
1D CONSERVATION OF BED SEDIMENT FOR SIZE
MIXTURES, BEDLOAD ONLY
fi'(z', x, t) fractions at elevation z' in ith
grain size range above datum in bed 1. Note
that over all N grain size ranges qbi(x, t)
volume bedload transport rate of sediment in the
ith grain size range L2/T
Or thus
42
ACTIVE LAYER CONCEPT
The active, exchange or surface layer
approximation (Hirano, 1972) Sediment grains in
active layer extending from ? - La lt z lt ? have
a constant, finite probability per unit time of
being entrained into bedload. Sediment grains
below the active layer have zero probability of
entrainment.
43
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT
Fractions Fi in the active layer have no vertical
structure. Fractions fi in the substrate do not
vary in time.
Thus
where the interfacial exchange fractions fIi
defined as
describe how sediment is exchanged between the
active, or surface layer and the substrate as the
bed aggrades or degrades.
44
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT contd.
Between
and
it is found that
(Parker, 1991).
45
REDUCTION contd.
The total bedload transport rate summed over all
grain sizes qbT and the fraction pbi of bedload
in the ith grain size range can be defined as
The conservation relation can thus also be
written as
Summing over all grain sizes, the following
equation describing the evolution of bed
elevation is obtained
Between the above two relations, the following
equation describing the evolution of the grain
size distribution of the active layer is obtained
46
EXCHANGE FRACTIONS
where 0 ? ? ? 1 (Hoey and Ferguson, 1994
Toro-Escobar et al., 1996). In the above
relations Fi, pbi and fi denote fractions in the
surface layer, bedload and substrate,
respectively. That is The substrate is mined as
the bed degrades. A mixture of surface and
bedload material is transferred to the substrate
as the bed aggrades, making stratigraphy. Stratig
raphy (vertical variation of the grain size
distribution of the substrate) needs to be
stored in memory as bed aggrades in order to
compute subsequent degradation.
47
WHY THE CONCERN WITH SEDIMENT MIXTURES?
Rivers often sort their sediment. An example is
downstream fining many rivers show a tendency
for sediment to become finer in the downstream
direction.
bed slope
elevation
Long profiles showing downstream fining and
gravel-sand transition in the Kinu River, Japan
(Yatsu, 1955)
median bed material grain size
48
WHY THE CONCERN WITH SEDIMENT MIXTURES ? contd.
Downstream fining can also be studied in the
laboratory by forcing aggradation of
heterogeneous sediment in a flume.
Downstream fining of a gravel-sand mixture at St.
Anthony Falls Laboratory, University of Minnesota
(Toro-Escobar et al., 2000)
Many other examples of sediment sorting also
motivate the study of the transport, erosion and
deposition of sediment mixtures.
49
REFERENCES FOR CHAPTER 4
Hirano, M., 1971, On riverbed variation with
armoring, Proceedings, Japan Society of Civil
Engineering, 195 55-65 (in Japanese). Hoey, T.
B., and R. I. Ferguson, 1994, Numerical
simulation of downstream fining by selective
transport in gravel bed rivers Model development
and illustration, Water Resources Research, 30,
2251-2260. Paola, C., P. L. Heller and C. L.
Angevine, 1992, The large-scale dynamics of
grain-size variation in alluvial basins. I
Theory, Basin Research, 4, 73-90. Parker, G.,
1991, Selective sorting and abrasion of river
gravel. I Theory, Journal of Hydraulic
Engineering, 117(2) 131-149. Toro-Escobar, C.
M., G. Parker and C. Paola, 1996, Transfer
function for the deposition of poorly sorted
gravel in response to streambed aggradation,
Journal of Hydraulic Research, 34(1)
35-53. Toro-Escobar, C. M., C. Paola, G. Parker,
P. R. Wilcock, and J. B. Southard, 2000,
Experiments on downstream fining of gravel. II
Wide and sandy runs, Journal of Hydraulic
Engineering, 126(3) 198-208. Yatsu, E., 1955,
On the longitudinal profile of the graded river,
Transactions, American Geophysical Union, 36
655-663.