Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 23:
TRANSPORTATIONAL CYCLIC STEPS
Froude-supercritical normal flow over an erodible bed is inherently unstable. In
many cases this instability is expressed in terms of antidunes, which tend to be
ephemeral features that
form, collapse and reform.
jumps
In some cases, however,
antidunes give way to a
much more stable cousin,
i.e. cyclic steps. Such
features occur in trains,
with each step locked in
place by a hydraulic jump
upstream and downstream.
Cyclic steps are one of the
few periodic 1D bedforms
that can be explained at
first order using the shallow
Trains of cyclic steps in a coastal
water equations.
outflow channel on a beach in Calais,
France. Image courtesy H. Capart.
flow
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTRODUCTION TO TRANSPORTATIONAL CYCLIC STEPS
The first clear description of transportational cyclic steps is due to Winterwerp et al.
(1992), who described their formation as water drains out of a polder on the verge of
being closed, so forming steps in loose sand. Winterwerp et al. also developed a
formulation for flow and sediment
transport over these steps that has
many of the elements of a
morphodynamic model. The name
jumps
“cyclic steps” was proposed by Taki and
Parker (in press) and Sun and Parker
(in press), who developed the first
complete first-order morphodynamic
model for cyclic steps.
Trains of cyclic steps in an outflow
channel of a polder in the Netherlands
that is in the process of being closed.
Image courtesy M. de Groot.
The cyclic steps in the image to the left
are “transportational” in that they result
from the differential transport of
sediment over an alluvial bed, as
described in the next slide. Cyclic steps
can also form in a purely erosional 2
setting.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMICS OF EQUILIBRIUM TRANSPORTATIONAL STEPS
Consider the train of bedforms illustrated below. The bedforms have wavelength L,
wave height  and upstream migration speed c. Each bedform is bounded
upstream and downstream by a hydraulic jump. The dominant mode of sediment
c
transport is suspension. The
slow flow just downstream of
the jump causes net
deposition, and the swift flow
just upstream of the jump

L
H
causes net erosion.
c
net
deposition
net
erosion
As a result, it is possible for the
train to migrate upstream
without changing form, and
with no net bed aggradation or
degradation when averaged
over one wavelength.
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM TRANSPORTATIONAL STEPS IN THE LABORATORY
Taki and Parker (in press) illustrated the formation of equilibrium transportational
cyclic steps in the laboratory. Each of the squares denotes 1 cm2.
supercritical
subcritical
hydraulic jump
zone reworked by
upstream-migrating steps
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM TRANSPORTATIONAL STEPS IN THE LABORATORY contd.
In a sufficiently long flume, Taki and Parker (in press) were able to form equilibrium
trains of up to four successive steps. As each step migrates into the headbox of the
flume, a new one forms at the downstream end.
one
two
three
four
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TRANSPORTATIONAL CYCLIC STEPS CAN ALSO FORM IN ALLUVIUM UNDER
NET-AGGRADATIONAL AND NET-DEGRADATIONAL CONDITIONS.
The video clip below documents cyclic steps on an experimental alluvial fan-delta
undergoing first degradation due to base level fall, and then aggradation due to base
level rise. The video has been speeded up by a factor of 360. Double-click to view.
Video clip courtesy N. Strong, B. Sheets, W. Kim and C. Paola (Strong et al., in press)
The steps are manifested
as trains of upstreammigrating hydraulic
jumps.
6
rte-bookstepsfan.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CYCLIC STEPS CAN ALSO FORM IN A PURELY EROSIONAL SETTING
Reid (1989) has documented trains of erosional steps in discontinuous gullies. Similar
purely erosional steps have been observed in bedrock (Wohl and Grodek, 1994)).
Parker and Izumi (2000) have developed a theory for purely erosional cyclic steps.
Train of erosional steps in a gully in
California. Adapted from Reid (1989).
Train of bedrock steps
on Gough Island, South
7
Atlantic Ocean.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EROSIONAL CYCLIC STEPS
The hydraulic jumps of erosional cyclic steps can eventually evolve into a train of
plunge pools (Brooks, xxxx). Double-click on the video (right) to see steps form in a
simulated bedrock made with silica flour and kaolinite.
Train of plunge pools in a
bedrock stream in
California. Image
courtesy of M. Neumann.
rte-bookbedrocksteps.mpg: to run without relinking, download to 8
same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EROSIONAL CYCLIC STEPS IN A MODEL DRAINAGE BASIN
Hasbargen and Paola (2000) have performed experiments on the formation of
drainage basins due to rainfall on a sediment consisting of a mix of silica flour and
kaolinite. Base level was lowered at a constant rate during the experiments. Erosional
cyclic steps were often found to form spontaneously during the experiments. Doubleclick to view the video (courtesy L. Hasbargen and C. Paola).
The video is speeded up
considerably. The steps
are mainfested as
upstream-migrating
headcuts. The steps
formed even though base
level was lowered a
constant rate.
9
rte-bookdrainagebasin.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONCEPT: THE HYDRAULIC JUMP
The morphodynamics of cyclic steps is intimately tied to the mechanics of hydraulic
jumps. The Froude number Fr = U2/(gH), where H denotes depth and U denotes
depth-averaged flow velocity, was introduced in Chapter 5. In open channel flow, the
transition from swift, shallow supercritical flow (Fr > 1) to tranquil, deep subcritical
flow requires passage through the shock known as a hydraulic jump. A hydraulic
jump in a flume is illustrated below.
flow
supercritical
subcritical
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONCEPT: THE HYDRAULIC JUMP contd.
Spillways are often designed so as to dissipate energy by means of a hydraulic
jump. The images below are from the upper St. Anthony Falls on the Mississippi
River, Minneapolis, Minnesota USA.
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP
The illustrated channel below carries constant water discharge Q in a wide,
rectangular channel of constant width B. The flow makes the transition from
supercritical to subcritical through a hydraulic jump.
hydraulic jump
Fr < 1
Fr > 1
Let the values of H and U just upstream of the hydraulic jump be denoted as Hu and
Uu, and the corresponding values just downstream of the jump be denoted as Hd and
Ud. Momentum balance in the illustrated control volume is considered (width B out
of the page).
control volume
Q
Hu
Hd
Q
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP contd.
The control volume is assumed to be short enough to allow for the neglect of bed
friction. The momentum discharge Qm and pressure force Fp were introduced in
Chapter 5;
Qm  U2HB
, Fp 
1
gH2B
2
Applying balance of streamwise momentum to a steady flow in the control volume
below, the following relation is obtained.

momentum  Qm upstream  Qm downstream  Fp upstream  Fp downstream
t
control volume
Q
Hu
Fp
Fp
Qm
Qm
Hd
Q
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP contd.
Momentum balance thus yields the relation:
0  Uu2HuB  U2dHdB 
1
1
gHu2B  gH2dB
2
2
The relation for water conservation is
Q  UHB  qwB
where Q denotes the constant water discharge, B denotes the (constant) width and
qw denotes the (constant) water discharge per unit width. Now denoting  = Hd/Hu
and Fru2 = Uu2/(gHu) = qw2/(gHu3), the above two equations can be reduced to the
form
3  (1 2Fru2 )  2Fru2  (  1)(2    2Fru2 )  0
The only physically realistic roots to the polynomial are  = 1 (no jump) and the
conjugate depth relation for a hydraulic jump:

Hd 1

 1  1  8Fru2
Hu 2

14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP contd.
In so far as a) according to flow continuity qw = UH and b) the Froude number Fr is
given as Fr = U2/(gH) = qw2/(gH3), the conjugate depth relation can be used to find
the relation between the upstream and downstream (conjugate) flow velocities and
Froude numbers:

1
  1 
2



1  8Fr 

Ud  1

  1  1  8Fru2 
Uu  2

Frd
Fru
3.5
1
3
2.5
3 / 2
2
u
Hd/Hu, Frd

Hd 1
 1  1  8Fru2
Hu 2
2
Hd/Hu
Frd
1.5
1
A hydraulic jump thus entails an
increase in depth and decrease
in velocity as the flow makes the
transition from supercritical (Fru >
1) to subcritical (Frd < 1).
0.5
0
1
1.5
2
2.5
Fru
3
3.5
4
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM CYCLIC STEPS AS A PERMANENT PERIODIC WAVE TRAIN
The experiments of Taki and
c
Parker (in press) indicate
that cyclic steps can reach
an equilibrium state at which
Se
a periodic wave train
d
migrates upstream without
changing form. When this
u
state is reached, average
 u - Se x
bed slope Se, wavelength L,
wave height  and wave
migration speed c attain
constant values. In this simplest of cases the morphodynamic problem reduces a
search for a solution of permanent form. That is, the deviation of bed elevation d(x,
t) from the line of equilibrium slope becomes a function of spatial coordinate alone
when transformed to a coordinate system moving upstream with constant speed c.
More precisely,
xm  x  ct ,
d ( x, t )  d ( xm )
tm  t
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PARAMETERS
The analysis given here is based on that of Sun and Parker (in press), which
captures the essential features of the experimental cyclic steps described in Taki and
Parker (in press). The flow is treated as suspension-dominated, and bedload
transport is neglected. The suspension is characterized with a single sediment size.
The following parameters are used in this chapter.
x = streamwise coordinate [L]
t = time [T]
 = bed elevation [L]
H = flow depth [L]
U = depth-averaged flow velocity [L/T]
L = step wavelength [L]
C = depth-averaged susp. sed conc. [1]
c = upstream step migration speed [1]
qw = water discharge per unit width [L2/T]
g = gravitational acceleration [L/T2]
Cf = bed friction coefficient [1]
vs = fall velocity of susp. sed. [L/T]
 = water density [M/L3]
s = sediment density [M/L3]
R = (s/ - 1)
L = step wavelength [L]
c = upstream migration speed of step [L/T] Se = mean equilibrium bed slope [1]
b = CfU2 = bed shear stress [M/L/T2]
p = bed porosity
D = grain size [L]
E = dimensionless rate of entrainment of bed sediment into suspension [1]
ro  cb / C = ratio of near bed concentration of suspended sediment to depth- 17
averaged value [1]
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GOVERNING EQUATIONS
The governing equations for momentum and mass balance of the flow in a channel
of constant width were derived in Chapter 5. They are as follows;
U
U
H

U2
U
 g
g
 Cf
t
x
x
x
H
H UH

0
t
x
The Exner equation of conservation of bed sediment was derived in Chapter 4.
Upon the neglect of bedload transport, it takes the form
deposition

(1   p )
 v s ( c b  E)
t
entrainment
where cb denotes the near-bed volume concentration of suspended sediment. The
equation of conservation of sediment in suspension was also derived in Chapter 4.
Neglecting bedload transport, it takes the form
entrainment
 H
UCH

 v s E  c b 
 0 c dz  

t 
x
deposition
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION OF THE RELATIONS FOR SEDIMENT CONSERVATION
The following assumptions are introduced to close the equations of conservation of
bed sediment and sediment in suspension;
cb  r0C
H

0
cdz   c CH
where ro and c are (for most cases of interest) order-one parameters. The first of
these assumptions was introduced in Chapter 21. The material of Chapter 10
(Slides 21-25 and the condition E = cb) can be used to derive quasi-equilibrium forms
for ro and c;
vs


u
 1  (1  ) /    n 30 H
 k
 b  (1  b ) / b 
c

ro  
 H

n11 

 kc 


 
 d
 




1
vs
u
 H  1  (1  ) /  
n11   
b (1   ) /  
k
c 
b
b


c 
vs
1  (1   ) /   u

H


n
30
b  (1  b ) / b   k c
d

 d

In the present analysis ro and c are approximated as constants. It is easily
demonstrated that ro and c tend toward unity as vs/u* approaches 0.
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION contd
With the assumptions of the previous slide, the following approximate shallow-water
forms are used to treat the case of cyclic steps:
U
U
H

U2
U
 g
g
 Cf
t
x
x
x
H
H UH

0
t
x

 v s (roC  E)
t
CH UCH
c

 v s E  ro C
t
x
(1   p )
In the case of the experiments of Taki and Parker (in press), the suspensions
were sufficiently vigorous that the assumptions ro = c = 1 could be applied.
These assumptions are also used here. The analysis easily generalizes,
however, to the range for which ro and c are not equal to unity.
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MOBILE-BED EQUILIBRIUM IN THE ABSENCE OF STEPS
An equilibrium solution can always be found for the case of a constant flow over a
constant slope in the absence of steps. In the range where steps form, this
equilibrium without steps is unstable, and devolves to a new equilibrium with steps.
Having said this, however, it is useful to characterize the equilibrium solution without
steps before proceeding with the analysis. Let Sn, Hn, Un, Cn, En and qsn denote
equilibrium values of S, H, U, C, E and qs = UHC in the absence of steps. Where qw
denotes constant water discharge per unit width, Cfeed denotes the constant volume
feed concentration of sediment and Cf denotes the bed friction coefficient (here
assumed to be constant for simplicity), the governing equations reduce to the
following equilibrium forms:
UnHn  q w
qsn = qw Cfeed
C f Un2  gHnSn
En  Cn  C feed
Hn
qsn  q w C feed
Sn
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FRICTION COEFFICIENT IN THE ABSENCE OF STEPS
Taki and Parker (in press) used data from their experiments pertaining to equilibrium
without steps to evaluate Cf and E in the absence of steps. They used three grades
of silica flour, with nominal sizes of 19, 45 and 120 m. Within the scatter of the
data, the friction coefficient was found to be roughly constant, as shown below. Note
that D denotes grain size.
1
Cf
0.1
19 micron
45 micron
120 micron
0.01
0.001
0.0001
10
100
hd/D
H/D
1000
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT ENTRAINMENT IN THE ABSENCE OF STEPS
In the experiments of Taki and Parker (in press), the flow was only a few mm thick,
and no existing relation for the entrainment rate E could be used. With this in mind,
they used data for the case of equilibrium without steps to determine a relation of the
general form of van Rijn (1984); where  denotes the kinematic viscosity of water,
RgD D
 t   t (Re p ) , Re p 

E  t [b  th ]n
In the above relations
b

b 
RgD
b  Cf U2  gHS
Cf U2t
 
 th (Re p )
RgD

th
and Ut denotes a critical velocity for the entrainment of bed sediment into
suspension.
The above relations reduce to the form
 
E  t 
 2
th
 U

 Ut


  1


2
n
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EVALUATION OF THE SEDIMENT ENTRAINMENT RELATION
The scatter in the data was sufficient to allow the fitting of a relation with exponents
in the range 1.5  n  2.5. The fit using the exponent of 2.0 is shown below.
100
n = 2.0
E/
E/
 tt
10
19 micron
45 micron
120 micron
power
1
0.1
0.01
0.1
1
*-th*
10
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EVALUATION OF THE SEDIMENT ENTRAINMENT RELATION contd
The fits of t and th* versus Rep to the data of Taki and Parker (in press) are given
below for the cases n = 1.5, 2.0 and 2.5. The reference “Mantz” pertains to
experiments due Mantz (1977) on the threshold of motion for fine particles.
0.1
1
th* (data)
th* (Mantz)
t
0.01
0.1
th*
t (data)
n = 1.5
n = 1.8
n = 2.0
0.001
0.01
0.1
1
Rep
10
25
1D SEDIMENT
TRANSPORT
MORPHODYNAMICS
National
Center for
Earth-surface Dynamics
with
applications
to
Our vision: Integrated, predictive
science of the processes
RIVERS AND TURBIDITY CURRENTS
and interactions
that 2004
shape the Earth’s surface
© Gary Parker November,
CALCULATION OF MOBILE-BED EQUILIBRIUM WITHOUT STEPS
In many of the experiments of Taki and Parker (in press), steps formed only after the
establishment of a mobile-bed equilibrium without steps. Such an equilibrium is
shown below.
flow
26
The flume is 10 cm wide and either 2 or 4 m long. The flow is only a few mm deep.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATION OF MOBILE-BED EQUILIBRIUM WITHOUT STEPS contd
The results and relations used in the previous slides can be used to “predict” the
mobile-bed equilibrium state without steps. Let D, R, , qw, Cfeed and Cf be specified,
and let n be chosen (e.g. n = 2.0). Rep is computed from the first three of these
parameters, and t and th* are then computed from the diagram of Slide 25. The
parameter Uth is computed from the relation
Cf U2t
 th
RgD
Since according to Slide 21 E = Cfeed for this state, flow velocity Un in the absence of
steps is computed from the relation
n
2


2 U 
E  C feed   t th   n   1
 Ut 

The flow depth Hn and bed slope Sn in the absence of steps are computed from
qw  UnHn
, CfUn2  gHnSn
The latter relation can be rearranged to the form
where Frn denotes the Froude
Un2
q2w
2
2
Cf Frn  Sn , Frn 

number at mobile-bed equilibrium
3
gHn gHn
without steps.
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RANGE OF INTEREST; EQUILIBRIUM AT THRESHOLD STATE
Taki and Parker (in press) found that cyclic steps would form only when the mobilebed equilibrium flow without steps was supercritical. The range of interest is, then,
2
2
U
q
Frn2  n  w3  1
gHn gHn
It will also prove of interest here to examine the equilibrium state at the threshold of
motion that prevails for the same values of D, R, , qw and Cf as in the case of
mobile-bed equilibrium, but with vanishing values of Cfeed and qsn. Let Ht, St, Ut and
Frt be the values of H, U, S and Froude number Fr at the threshold of motion.
qsn = 0
Ht
St
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM AT THRESHOLD STATE contd
The threshold state is of interest for the following reason. When steps form, they are
accompanied by hydraulic jumps. It is reasonable to assume that the bed shear
stress b just after the hydraulic jump is reduced to the threshold value for the
entrainment of sediment.
Recall that D, R, , qw and Cf take the same values as in the case of mobile-bed
equilibrium. The method of computation of th* and Ut has already been outlined in
Slide 27. The values of Ht, St and Froude number Frt are computed from the
relations
2
qw  HtUt
, Cf U2t  gHtSt
, Frt2 
qw
gH3t
 Cf Frt2  St
In so far as it is speculated that
threshold conditions are realized just
after the hydraulic jump (where by
definition the flow must be subcritical,
the range of interest is then
Threshold of motion attained here?
Fr  1  Fr
2
t
2
n
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FORMULATION IN THE PRESENCE OF STEPS
The following simplifying assumptions are made to treat the case for which cyclic
steps are present.
a) The friction coefficient Cf takes the same value as it does at mobile-bed
equilibrium without steps.
b) The entrainment relation evaluated at mobile-bed equilibrium without steps can
be generalized to the the case for which steps are present, so that where U
denotes the local flow velocity, the local entrainment rate is given as

 2  U
E   t  th 
 Ut
 



2

 1

n
The appropriate balance relations are
taken to be those of Slide 20 (but with
ro = c = 1).
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM STEPS: TRANSFORMATION TO MOVING COORDINATES
The solution sought here is one of permanent form: the steps migrate upstream at
constant speed c without changing form, and without net aggradation or degradation
averaged over one wavelength. Let Se denote the equilibrium average slope in the
presence of steps, and let bed elevation  be given as
  u  Se x  d ( x, t )
where u denotes an upstream bed elevation at x = 0 and d denotes the deviatoric
bed elevation associated with the steps.
c
Se
d
u
 u - Se x
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM STEPS: TRANFORMATION TO MOVING COORDINATES contd.
The balance equations of Slide 20 are reduced with the assumptions ro = c = 1 and
the relation of the previous slide to obtain the forms
d
U
U
H
U2
U
 g
 gSe  g
 Cf
t
x
x
x
H

(1   p ) d  v s (C  E)
t
H UH

0
t
x
CH UCH

 v s E  C 
t
x
c
They are then transformed to coordinates
moving upstream with speed c as
xm  x  ct ,
Se
tm  t
d
u
 u - Se x
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EQUILIBRIUM STEPS: TRANSFORMATION TO MOVING COORDINATES contd.
The results of the transformation are
d
U
U
H
U2
 (U  c )
 g
 gSe  g
 Cf
t m
xm
xm
xm
H
 
 
(1  p ) d  c d   v s (C  E)
x m 
 t m
c
H (U  c )H

0
t m
x m
CH (U  c )CH

 v s E  C
t m
x m
For a solution of permanent form, all time
derivatives /tm must vanish. In addition, in
the experiments of Taki and Parker (in press),
the following condition was satisfied:
Se
d
u
 u - Se x
c
 1
U
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESULTS OF THE TRANSFORMATION
After the coordinate transformation, neglect of the time derivatives and application of
the condition c/U << 1, the following relations are obtained.
dd
dU
dH
U2
U
 g
 gSe  g
 Cf
dxm
dxm
dxm
H
d(UH)
 0  UH  qw
dxm
dd
(1  p )c
 v s (C  E )
cxm
dC
qw
 v s E  C
dxm
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BOUNDARY CONDITIONS
Recall that the wavelength of a step is denoted as L. According to the figure below,
the boundary conditions on deviatoric bed elevation are
d x
m 0
0
d x
m L
0
d
xm = 0
xm = L
xm
The flow at xm = 0 is just after a hydraulic jump where it is assumed that threshold
conditions are attained, and the flow at xm = L is just before a hydraulic jump. As a
result, the boundary conditions on U are
U x 0  Ut
Threshold velocity
1
U x L
 1  8Fr 2  1
t
 U


 t
2


Conjugate velocity
See Slide 15
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BOUNDARY CONDITIONS contd
In order for the train of steps to be in equilibrium with the sediment supply, the
transport rate of suspended sediment averaged over the steps must be equal to the
feed rate. Thus the transport rate qs = UHC = qwC averaged over one wavelength
must equal the feed rate qwCfeed:
L
1 L
1
qs dx m  qw  Cdx , qw C feed

0
L 0
L
Upon reduction, then,
1 L
Cdx m  Cfeed

0
L
d
xm = 0
xm = L
36
xm
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION AND NON-DIMENSIONALIZATION
The following equations of Slide 34
dd
dU
dH
U2
U
 g
 gSe  g
 Cf
dxm
dxm
dxm
H
, UH  qw
reduce to a form of the backwater equation involving flow velocity;
dd
U3
gSe  g
 Cf
dU
dx m
qw

dx m
U(1  Fr 2 )
U3
, Fr 
gqw
2
The above equation and the relations of sediment conservation of Slide 34, i.e.
(1  p )c
dd
dC
 v s (C  E) , qw
 v s (E  C)
dxm
dxm
are made dimensionless using the following transformations;

qw 
q 
U  UtU , H 
H , d  w  ,
Ht
Ut
xm 
qw 1 
x , C  Cfeed 
Ut St
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-DIMENSIONALIZATION contd
When placed in non-dimensional form and reduced with the relation CfFrt2 = St of
Slide 29, the following relations result:
 
d

 S    U3
r
dU
dx 

 
dx
Frt2U  U2
,



 d
d
  (E  ) , c   (E  )
dx
dx
where dimensionless wave speed, sediment fall velocity and bed slope and
normalized entrainment rate are given as
 (1 p )
c
c
Ut Cfeed
v
,  s
StUt
S
, Sr  e
St

E
, E
Cfeed

In addition, dimensionless wavelength L is given as

U
L  St t L
qw
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-DIMENSIONALIZATION contd
The dimensionless forms for the boundary conditions are

 x 0  0

U
x 0
1

 x L  0

U
x L


L

L   dx
 1  8Fr 2  1
t




2


1
0

The normalized entrainment rate E can be reduced with the forms on Slides 27 and
30 to the form

 (U2  1)n
E  2
(Un  1)n

Un
, Un 
Ut
and Un is the flow velocity at the mobile-bed equilibrium state discussed in Slides 26
and 27.
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION OF DIMENSIONLESS FORMS
The normalized entrainment rate
2
 (U  1)n
E  2
(Un  1)n

U
, Un  n
Ut
can be further reduced with the definitions
q2w
Un3
Fr 

3
gHn gqw
2
n
U3t
q2w
, Fr 

3
gHt gqw
2
t
to the form

E
2
(U  1)n
 Fr 
 n 
 Frt 
4/3

 1

n
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION OF DIMENSIONLESS FORMS contd
Substituting


 d
c   (E   )
dx
into
results in the form
Between
 
3
 S  d

U

r
dU
dx 

 
dx
Frt2U  U2

3
 S 
 (E  )  U
r
dU
c  
 
dx
Frt2U  U2



 d
d
c   (E   )
  (E  ) and
dx
dx
it is found that

d  1 d
  
dx c d x
Applying the boundary conditions
 1
   ( x 0  ) ,
c
 1
      const
c


 x 0  0 ,  x L  0 yields the results
 x L   x 0
Suspended sediment
concentration is continuous
through the hydraulic jump.
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
FINAL FORM OF THE PROBLEM
Solve


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2
where

E
,

d
  (E  )
dx
2
(U  1)n
 Fr  4 / 3 
 n   1
 Frt 

n
subject to the boundary conditions

U
x 0

1 , U

x L
 1  8Fr 2  1
t




2


1
,
 x L   x 0


L

, L   dx
The problem consists of two first-order ordinary differential equations
subject to four boundary conditions.
0
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHARACTER OF THE PROBLEM: SINGULARITY
The problem defines an overspecified system of ordinary differential equations. That
is;

a) There are 2 first-order differential equations in U and ,

b) plus the 4 parameters Frt, Frn, Sr and c ,
c) Subject to 4 boundary conditions.


Thus if any two of Frt, Frn, Sr and c are specified, U ,  and the other two parameters
can be computed.

Note, however, that the equation for U becomes singular when the denominator on
the right-hand side vanishes.


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2
The problem cannot be solved without overcoming this singularity.
43
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NATURE OF THE SINGULARITY
Fr > 1
Fr < 1


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2
Fr = 1


 
x  x1

x0
 
x L

x
Now at the singularity

U  Frt2 / 3
or thus

U3
2 3
Fr 
 Frt U  1
gqw
2
Image courtesy J. White
Just after the hydraulic jump, the
flow is subcritical. As the flow
progresses from left to right, it
must pass through a Froude
number Fr of unity at some point
and become supercritical. The
equation blows up as the flow
passes from subcritical to 44
supercritical.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REMOVAL OF THE SINGULARITY: L’HOPITAL’S RULE
The denominator of the right-hand side of the equation


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2
 
 
2 / 3
vanishes at x  x1 where U  U1  Frt .
Thus in order to avoid singular behavior the numerator of the right-hand side must
also vanish, yielding the relation


c
1  E1  (U13  Sr )
  
Here the subscript “1” denotes
the this relation with

 the value at x  x1. Reducing
the relation given above for U1 and the evaluation of E of Slide 40, it is found that
1   x  x 
1

c
2

(
Fr
 Sr )
t
n


 1

(Frt4 / 3  1)n
 Fr  4 / 3
 n 
 Frt 
45
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004

REMOVAL OF THE SINGULARITY: L’HOPITAL’S RULE contd

Now at x  x1 the equation in question takes the form

dU
0


dx x  x
0
1
 
Applying L’hopital’s rule at x  x1 ,
 3 
d 
 
  Sr  (E  )  U  

c
  x  x 1
 dx 
dU


  3 
dx x  x
d

2
1
 dx Frt U  U   


x  x1


dU
Now denoting U1   , after some algebra it is found that
dx x
1



2 / 3
4 / 3
n 1


2
2
4 / 3
n
Fr
(
Fr

1
)
d
E
t
t
3Frt (U1 )  (3Frt
 Eu1 )U1 where EU 1   
n
4/3
c
dU x


 Frn 
1
2

  1

 (Frt  Sr )  0
 Frt 
 46
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION PROCEDURE
The parameter  must be specified in advance.
1. Specify values of Frn and Frt.

2. Guess solutions for Sr and
c
 .
3. Define the variable xr  x  x1



4. Integrate the following two equations upstream to the point xr  Lu where U  1
(threshold condition).


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2
,

d
  (E  )
dx

The conditions on these equations at xr  0 are
2






dU
  (3Fr  4 / 3   E ) dU
3Frt2  
U   Frt2 / 3 ,
t
u1
 dx r  
 dx r
x r 0
c
x r 0 


 x  x 
1

c
2

(
Fr
 Sr )
t
n


 1





x r 0 
 (Frt2  Sr )  0
(Frt4 / 3  1)n
 Fr  4 / 3
 n 
 Frt 
47
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION PROCEDURE contd
5. Integrate the same two equations i.e.


 S   (E  )  U3
r
dU
c  
 
dx
Frt2U  U2

d
,
  (E  )
dx
 

x
subject to the same conditions at xr  0, downstream to r  Ld , where
1
2
 1  8Fr  1 

t

U  
x r L d


2



6. For every guessed pair (c, Sr ) the dimensionless wavelength is then given as
  

L  Lu  Ld  L(c,Sr )

7. The resulting solutions for Uand  take the forms
  

  
U  U(x; c, Sr ) ,   (x; c, Sr ) , x  xr  Lu
48
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOLUTION PROCEDURE contd
8. The remaining two boundary conditions are
 x L   x 0
or


L

, L   dx
0
 



F1(c, Sr )  [L(c, Sr ); c, Sr ]  (0; c, Sr )  0
 
 
L ( c , Sr )

 

F2 (c, Sr )  L(c, Sr )  
( x; c, Sr )dx  0
0

For specified values of Frt and Frn, the correct values of c and Sr are the ones
that satisfy the above equations. These values must be found iteratively. Both
the bisection and Newton-Raphson iterative techniques have been found to yield
solutions.
In the material presented subsequently the bisection method has been used.
Details are presented in Sun and Parker (in press).
49
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTED AND OBSERVED REGION OF FORMATION OF STEPS, 19 m
SEDIMENT
The solid lines enclose the
region within which cyclic
steps are predicted by Sun
and Parker (in press). The
points (M or S) are
observed values from Taki
and Parker. Here
“M” denotes the fact that a
train of multiple steps was
observed, whereas “S”
denotes the fact that a
single step was observed.
50
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTED AND OBSERVED REGION OF FORMATION OF STEPS, 45 m
SEDIMENT
The solid lines enclose the
region within which cyclic
steps are predicted by Sun
and Parker (in press). The
points (M or S) are
observed values from Taki
and Parker. Here
“M” denotes the fact that a
train of multiple steps was
observed, whereas “S”
denotes the fact that a
single step was observed.
51
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTED AND OBSERVED REGION OF FORMATION OF STEPS, 120 m
SEDIMENT
The solid lines enclose the
region within which cyclic
steps are predicted by Sun
and Parker (in press). The
points (M or S) are
observed values from Taki
and Parker. Here
“M” denotes the fact that a
train of multiple steps was
observed, whereas “S”
denotes the fact that a
single step was observed.
52
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004

PREDICTED AND OBSERVED VALUES OF c , 19 m SEDIMENT, n = 2
The predictions are in
the same range as the
observed values.
53
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004

PREDICTED AND OBSERVED , 19 m sediment, n = 2

Here  denotes wave
height, or the maximum
value of  . The model
predicts values that are
in the observed range.


54
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004

PREDICTED AND OBSERVED VALUES OF L , 19 m SEDIMENT, n = 2
The model overpredicts
wavelength. The
reasons for this are
unclear, but are probably
related to the many
simplifications in the
analysis.
55
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTED VALUES OF Sf = Se/St, 19 m, n = 2
The model predicts
equilibrium mean bed
slopes slopes that are
well above the bed slope
at the threshold of
motion. Data are not
available for comparison.
56
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PREDICTIONS, 19 m, n = 2
The succeeding
slides show predictions
 
for , U ,  and dimensionless
water
  
surface elevation     H for the points
“a”, “c”, “d”, “g” and “h” in the diagram,
which pertains to 19 m material and n =
2 in the entrainment relation.
57
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004


U

, 
PREDICTIONS, 19 m, n = 2, POINT “a”

xr
58
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004


U

, 
PREDICTIONS, 19 m, n = 2, POINT “c”

xr
59
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004


U

, 
PREDICTIONS, 19 m, n = 2, POINT “d”

xr
60
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004


U

, 
PREDICTIONS, 19 m, n = 2, POINT “e”

xr
61
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004


U

, 
PREDICTIONS, 19 m, n = 2, POINT “h”

xr
62
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SPECULATION
Cyclic steps might also be
produced by turbidity
currents in the submarine
environment. The image
shows the Monterey
Submarine Canyon and
the Monterey Channel
offshore of California,
USA. The region in the
box is expanded in the
next slide.
Image courtesy Monterey Bay
Aquarium Research Institute
63
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ARE THESE SUBMARINE CYCLIC STEPS?
Image of Monterey Submarine Channel and
overflow levee and channel courtesy W Normark
64
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 23
Brooks, P.C., 2001, Experimental study of erosional cyclic steps, M.S. Thesis, University of
Minnesota, Minneapolis, Minnesota.
Hasbargen, L. E., and Paola, C., 2000, Landscape instability in an experimental drainage basin,
Geology, (28)12, 1067-1070.
Parker, G. and Izumi, N., 2000, Purely erosional cyclic and solitary steps created by flow over a
cohesive bed, Journal of Fluid Mechanics, 419, 203-238.
Mantz, P. A., 1977, Incipient transport of fine grains and flakes by fluids – extended Shields
diagram, Journal of Hydraulic Engineering, 103(6), 601-615.
Reid, L. M., 1989, Channel incision by surface runoff in grassland catchments, PhD Thesis,
University of Washington.
Strong, N., Sheets, B., Kim, W. and Paola, C., in press, An experimental study of landscape and
stratigraphic response to eustatic sea level rise, Journal of Sedimentary Research.
Sun, T. and Parker, G., 2005, Transportational cyclic steps created by flow over an erodible bed.
Part 2. Theory and numerical simulation, Journal of Hydraulic Research, in press,
downloadable at http://cee.uiuc.edu/people/parkerg/preprints.htm .
Taki, K. and Parker, G., 2005, Transportational cyclic steps created by flow over an erodible bed.
Part 1. Experiments. Journal of Hydraulic Research, in press, downloadable at
http://cee.uiuc.edu/people/parkerg/preprints.htm .
Winterwerp, J. C., Bakker, W. T., Mastbergen, D. R., and van Rossum, H., 1992,
Hyperconcentrated sand-water mixture flows over erodible bed, Journal of Hydraulic
65
Engineering, 119(11), 1508-1525.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 23 contd
Wohl, E. E. and Grodek, T. 1994), Channel bed-steps along Nahal Yael, Negev desert, Israel,
Geomorphology, 9, 117-126.
66