1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 13:
THE QUASI-STEADY APPROXIMATION
The conservation equations governing 1D morphodynamics can be summarized as
H UH

0
t
x
Conservation of flow mass
U
U
H

U2
U
 g
 g  Cf
t
x
x
x
H
Conservation of flow momentum
qt

(1   p )
t
x
Conservation bed sediment
(sample form using total bed
material load)
For many applications in morphodynamics, however, it is possible to neglect the
time derivatives in the first two equations, retaining it only in the Exner equation of
conservation of bed sediment. That is, the flow over the bed can be approximated
as quasi-steady. This result, first shown by de Vries (1965), is often implicitly used
1
in morphodynamic calculations without justification. A demonstration follows.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-DIMENSIONALIZATION USING A REFERENCE STATE
The essence of morphodynamics is in the interaction between the flow and the
bed. The flow changes the bed, which in turn changes the flow.
Consider a reference mobile-bed equilibrium state with constant flow velocity Uo
flow depth Ho, bed slope So and total volume bed material transport rate per unit
width qto. For the sake of simplicity the bed friction coefficient Cf is assumed to be
constant. The analysis easily generalizes, however, to the case of varying friction
coefficient. The application of momentum balance to the equilibrium flow imposes
the conditions
o  u  So x
, Cf Uo2  gHoSo
where u is the value of  at x = 0. In a problem of morphodynamic evolution, the
flow and bed can be expected to deviate from this base state. In general, then,
  u  So x  d
The following non-dimensionalizations are introduced:
~
~
H  HoH , U  UoU ,
d  Ho ~
d ,
qt  qto~
qt ,
x  Ho ~
x,
t
Ho ~
t
Uo
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-DIMENSIONALIZATION USING A CHARACTERISTIC HYDRAULIC
TIME SCALE
Note that the non-dimensionalization of time involves the “hydraulic” time scale
Ho/Uo, which physically corresponds to the time required for the flow to move a
distance equal to one depth in the downstream direction. Substituting the nondimensional variables into the balance equations yields the results
~
~~
H UH
0
~  ~
x
t
~
~
~2 
 ~ ~
 U
d 
U ~ U
 2  H
 Fro
 ~  C f 1  ~ 
~  U ~
~



x
t
H 
 x x 

~
d
~
qt
~  - ~
x
t
where

qto
1
(1  p ) UoHo
, Fro 
Uo
gHo
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HOW LARGE IS 
Bed porosity p is typically in the range 0.25 ~ 0.45 for beds of noncohesive
sediment. The parameter

qto
qto
1
1

(1  p ) UoHo (1  p ) qwo
thus scales the ratio of the volume transport of solids to the volume transport of
water by a river. For the great majority of cases of interest this ratio is exceedingly
small, even during floods.
A case in point is the Minnesota River near Mankato, Minnesota, a medium-sized
sand-bed stream. Some sample calculations follow.
Minnesota River at the Wilmarth
Power Plant just downstream of
Mankato, Minnesota, USA. Flow is
from left to right.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HOW LARGE IS  contd.
Given below are 13 grain size distributions for the bed material of the Minnesota River
at Mankato, along with an average of all 13. The fraction of sediment finer than 0.062
microns in the bed is negligible; such material can be treated as wash load.
Bed Grain Size Distributions, Minnesota River at Mankato
100
90
GSD1
GSD2
GSD3
GSD4
GSD5
GSD6
GSD7
GSD8
GSD9
GSD10
GSD11
GSD12
GSD13
Average
Percent Finer
80
70
60
50
40
30
20
10
0
0.01
0.1
From http://www.usgs.gov/
1
D (mm)
10
100
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HOW LARGE IS  contd.
During the period 1967-1995 the highest measured suspended load concentration was
2850 mg/liter, or C = 2850/2.65*1x10-6 = 0.001075; the discharge Q was 340 m3/s, so
the volume total suspended load (bed material load + wash load) Qsbw = 0.366 m3/s.
Suspended Sediment Concentration Minnesota River
Mankato
1
0.1
C
0.01
C = 1E-05(Q)0.388
0.001
0.0001
0.00001
0.000001
1
From http://www.usgs.gov/
10
100
Q (m3/s)
1000
10000
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HOW LARGE IS  contd.
At a discharge of 340 m3/s, about 79.5% of the suspended load is wash load, giving a
suspended bed material load Qs of 0.075 m3/s. Estimating the bedload Qb as about
15% of the total bed material load, an estimate for the highest value of Qt of 0.088 m3/s
is obtained.
Percent of Suspended Load Finer than 0.062 mm
100
Fload<62 = [-0.0069(Q) + 81.9]/100
90
Percent Finer
80
70
60
50
40
30
20
10
0
0
200
400
From http://www.usgs.gov/
600
800
1000
Q (m3/s)
1200
1400
1600
1800
2000
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HOW LARGE IS  contd.
Assuming a value of bed porosity p of 0.35, then, an estimate of the very high
end of the value of  that might be attained by the Minnesota River near
Mankato is
qto
qto
1
1
1  Qt 





 0.0004  1
(1  p ) UoHo (1  p ) qwo (1  p )  Q w max
The Minnesota River is by no means atypical of rivers. The largest values of 
attained in the great majority of rivers is much less than unity. The exceptions
include streams with slopes so high that the flows are transitional to debris
flows, streams carrying lahars, or heavily sediment laden flow from regions
recently covered with volcanic ash, and many streams in the Yellow River
Basin of China.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOT ALL FLOWS SATISFY THE CONDITION  << 1
Double-click on the image to see a debris flow in Japan. The volume (mass) of sediment
carried by debris flows is of the same order of magnitude as the volume (mass) of water carried
by such flows. The quasi-steady approximation breaks down for such flows. Video courtesy
Paul Heller.
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rte-bookjapandebflow.mpg: to run without from relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HYDRAULIC TIME SCALES
Over short, or “hydraulic” times scales t ~ Ho/Uo, then, when  << 1 the governing
equations approximate to
~
~~
H UH
0
~  ~
x
t
~
~
~
~2 
~





U ~ U

H
U
 Fro2  ~  ~d   C f 1  ~ 
~  U ~
 x x 

x
t
H 



~
qt
~

0
~  - ~
x
t
That is, the bed can be treated as unchanging for computations over “hydraulic”
time scales, even though sediment is in motion. This is because the condition  <<
1 implies lots of water flows through but very little sediment, so that the bed does
not have time to change in response to the flow.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMIC TIME SCALES
A dimensionless morphodynamic time t* can be defined as
U
~
t   t   o t
Ho
An order-one change t* corresponds to a change in dimensioned time
t ~
H
1 Ho
 o
 Uo
Uo
i.e. much longer than the characteristic “hydraulic” time. The governing equations
thus become
~
~
~
~2 
~
~~
~





U ~ U

H
U
H UH
   U ~  Fro2  ~  ~d   C f 1  ~ 
   ~ 0
 x x 

t
x
t
x
H 



~
qt
~

- ~

t
x
That is, when the time scales of interest are of “morphodynamic” scale, the flow
can be treated as quasi-steady even though the bed is evolving, and thus
changing the flow.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE DIMENSIONED EQUATIONS WITH THE QUASI-STEADY APPROXIMATION
According to the quasi-steady approximation, the bed changes so slowly compared
to the characteristic response time of the flow that the flow can be approximated
as responding immediately. The dimensioned equations thus reduce to the
following forms:
UH
0
x
 UH  qw  const
(1   p )
q

- t
t
x
U
H

U2
U
 g
 g  Cf
x
x
x
H
The quasi-steady approximation greatly simplifies morphodynamic calculations.
There are, however, reasons not to use it. These include
a) Cases of rapidly varying hydrographs, when it is desired to characterize the
sediment transport over the entire hydrograph;
b) Cases when one wishes to capture the effect of a flood wave (with a high water
surface slope on the upstream side of the wave and a low water surface slope
on the downstream side) on sediment transport; and
c) Cases when the flow makes transitions between subcritical and supercritical
flow, in which case a shock-capturing method capable of automatically
12
locating hydraulic jumps is required.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 13
de Vries, M. 1965. Considerations about non-steady bed-load transport in open channels.
Proceedings, 11th Congress, International Association for Hydraulic Research, Leningrad:
381-388.
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