Transcript Slide 1

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 18:
MOBILE AND STATIC ARMOR IN GRAVEL-BED STREAMS
Whereas sand-bed rivers often show dunes
on the surface of their beds, gravel-bed
streams often show a surface armor layer.
That is, the surface layer is coarser than the
substrate. In addition, the surface layer is
usually coarser than the mean annual load of
transported gravel (e.g. Lisle, 1995).
The surface of even an equilibrium gravelbed stream must be coarser than the gravel
load because larger material is somewhat
harder to move than finer material. The river
renders itself able to transport the coarse half
of its gravel load at the same rate as its finer
half by overrepresenting coarse material on
its surface, where it is available for transport.
Bed sediment of the River Wharfe, U.K., showing a pronounced
surface armor. Photo courtesy D. Powell.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MOBILE AND STATIC ARMOR
The principle of mobile-bed armor is
explained in Parker and Klingeman (1982)
and Parker and Toro-Escobar (2002).
Most gravel-bed streams display a mobile
armor. That is, the surface has coarsened to
the point necessary to move the grain size
distribution of the mean annual gravel load
through without bed degradation or
aggradation. In the case of extremely high
gravel transport rates, no armor is necessary
to enable the coarse half of the gravel load to
move through at the same rate as the fine
half (e.g. Powell et al., 2001). A mobile-bed
armor gives way to a static armor as the
sediment supply tends toward zero.
Bed sediment of the unarmored Nahal Eshtemoa, a wadi in Israel
subject to severe flash floods with intense gravel transport.
Photo courtesy D. Powell.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MECHANISM OF MOBILE ARMOR
The mechanism of armoring can be explained with e.g. the transport equation of
Powell, Reid and Laronne (2001), which can be cast in the following form.
u3 
1
1  
qbi  11.2 Fi
Rg  i 
4.5
 D 
, i  s50  i 
 Ds50 
0.26
s50
u2

s50  
, s50 
, sc 50  0.03
sc 50
RgD s50
Recall that here s50* denotes the Shields number based on the surface median
size Ds50.
The functional form that drives armoring (and its disappearance at sufficiently high
flows) is the term [1-(1/i)]4.5 in the above relation.
Now consider the function

1
K(i )   1  
i 

plotted on the next slide.
4.5
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MECHANISM OF MOBILE ARMOR contd.
1.0E+00
1.0E-01
1.0E-02
Note that the bedload transport rate is a multiple of K(i),
which is a steeply-increasing function of i for values of i that
are not much greater than 1 (just above the threshold of
function), but becomes nearly horizontal for value of I that
are large compared to 1 (far above the threshold of motion.
1.0E-03
K(i)
1.0E-04
1.0E-05
1.0E-06
In the next slide it is shown that this feature of the function
biases the bedload to be finer than the surface material (or
surface material to be coarser than the bedload) at conditions
not far above the threshold of motion. By the same token, at
conditions far above the threshold of motion the bedload and
surface grain size distributions become nearly identical.
1.0E-07
1.0E-08
1.0E-09
1.0E-10
1.0E-11
1
10
100
i
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MECHANISM OF MOBILE ARMOR contd.
Now consider a mixture of only three grain sizes, D1 = 0.5 Ds50, D2 = Ds50 and D3 =
2 Ds50.
A condition fairly typical of bankfull flows in many perennial gravel-bed streams is
characterized by the value s50 = 1.5, i.e. 50% above the threshold of motion for
the surface median size. The value s50 = 8, on the other hand, corresponds to a
condition far above the threshold of motion for the surface median size. Using the
relations
 D 
i  s50  i 
 Ds50 
0.26

1
, Ki  K(i )   1  
i 

4.5
the following values are obtained:
finer
1
s50
1.5
8
2
1.80
9.58
coarser
3
1.50
1.25
8.00
6.68
finer
K1
K2
2.57E-02 7.13E-03
6.09E-01 5.48E-01
coarser
K3
7.43E-04
4.82E-01
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MECHANISM OF MOBILE ARMOR contd.
Note that the value of i is largest for the finest grain and smallest for the coarsest
grain for both values of s50.
finer
1
s50
1.5
8
2
1.80
9.58
coarser
3
1.50
1.25
8.00
6.68
finer
K1
K2
2.57E-02 7.13E-03
6.09E-01 5.48E-01
coarser
K3
7.43E-04
4.82E-01
Now the bedload transport equation can be written in the form

u3 
 Fi K i
qbi   11.2
Rg 

When s50 = 1.5, the values of Ki are strongly dependent on grain size Di, such
that K1/K3 = 34.6. At such a condition, then, the finer sizes will be overrepresented
in the bedload compared to the surface (underrepresented in the surface
compared to the bedload). The result is a mobile armor.
When s50 = 8, the values of Ki are weakly dependent on the grain size Di, such
that K1/K3 = 1.26. At such a condition, the grain size distributions of the bedload
and the surface material will not differ much, and only weak mobile armor is
present.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPUTATION OF MOBILE AND STATIC ARMOR
In principle the computation of equilibrium mobile-bed armor is a direct calculation
(Parker and Sutherland, 1990). Let the bedload transport rate qT and fractions in
the bedload pbi be specified. A knowledge of pbi allows computation of the
geometric mean size Dlg and arithmetic standard deviation l of the load. The
bedload transport relation of Parker (1990), for example, can be written in the form
2


D
u

i


W
,
, s 
3
 D RgD

(u ) Fi
sg
 sg

Rgq bT pbi
where W*( ) denotes a function. After some rearrangement,
Fi 
Rgq bT
 Di

u
(u ) W 
,
, s 
 Dsg RgD sg

3

2

pbi
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPUTATION OF MOBILE AND STATIC ARMOR contd.
Letting i = ln2(Di) and recalling that
Dsg  2
s
N
s   iFi
,
i1
N
,    i  s  Fi
2
2
s
i1
and taking the 0th, 1st and 2nd moments of the equation below,
Rgq bT
Fi 
pbi
2


u
3
 Di

(u ) W 
,
, s 
 Dsg RgD sg

three equations for the three unknowns u*, Dsg and s are obtained;
1
Rgq bT
3
(u )
N

i1
pbi
n2 (Dsg ) 
2
 Di

u

W 
,
, s 
 Dsg RgD sg


 
2
s
Rgq bT
(u )3
i1
(u )3
ipbi
N

i1
2
 Di

u


W
,
, s 
 Dsg RgD sg


(i  s )2 pbi
N

Rgq bT
2
 Di

u

W 
,
, s 
 Dsg RgD sg


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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPUTATION OF MOBILE AND STATIC ARMOR contd.
The solution for u*, Dsg and s is obtained iteratively (e.g. using a Newton-Raphson
scheme). Once this is done the surface fractions are obtained directly from the
relation
Rgq bT
Fi 
pbi
2


D
u
(u )3 W   i ,
, s 
 Dsg RgD sg

It can be verified from e.g. the Parker (1990) relation that the armor becomes
washed out as the Shields number based on the geometric mean size of the
sediment feed becomes large:
Fi  pbi
2
u

as lg 

RgD lg
On the other hand, the mobile-bed armor approaches a constant static armor as
2
u

lg 
0
RgD lg
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ALTERNATIVE COMPUTATION OF MOBILE AND STATIC ARMOR
An alternative way to compute armor is with the code of the Excel workbook RTebookAgDegNormGravMixPW.xls. Specified water discharge per unit width qw,
sediment feed rate qbTf and grain size fractions pbf,i of the feed specify a final
equilibrium bed slope S, flow depth H and surface fractions Fi regardless of the
initial conditions.
It thus becomes possible to study
equilibrium mobile-bed armor by allowing the
calculation to run until it converges to
equilibrium. In the succeeding calculations
the sediment feed rate qbTf ( which
eventually becomes equal to the equilibrium
sediment transport rate qbT) is varied from
1x10-8 m2/s to 1x10-2 m2/s, while holding the
following parameters constant: qw = 6 m2/s, If
= 0.05 and L = 20 km. In addition, the size
distribution of the sediment feed is held
constant as given in the table to the right.
D mm
%Finer
256
100
128
95
64
80
32
50
16
25
8
10
4
5
2
0
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ALTERNATIVE COMPUTATION OF MOBILE AND STATIC ARMOR contd.
The input parameters for the highest value of sediment feed rate qbTo of 0.01 m2/s
are given below. The duration of the calculation is longer for smaller feed rates,
because more time is required to approach the final equilibrium.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
200
-2 m22/s
-2
q
=
1x10
qbTo
=
1x10
m /s
bTf
After 120
After
120 years
years
180
Elevation m
160
0 yr
20 yr
40 yr
60 yr
80 yr
100 yr
120 yr
final w.s.
140
120
100
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
140
120
-3 m22/s
q
= 3x10
3x10-3
qbTo
=
m /s
bTf
After
After 240
240 years
years
Elevation m
100
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
qqbTo
= 1x10
1x10-3-3 m
m22/s
/s
bTf =
After
After240
240years
years
Elevation m
100
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
qqbTo
= 3x10
3x10-4-4 m
m22/s
/s
bTf =
After
After480
480years
years
Elevation m
100
0 yr
80 yr
160 yr
240 yr
320 yr
400 yr
480 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
-4
2
qbTo
bTf = 1x10 m /s
After 480 years
Elevation m
100
0 yr
80 yr
160 yr
240 yr
320 yr
400 yr
480 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
-5
2
qbTo
bTf = 1x10 m /s
After 960 years
Elevation m
100
0 yr
160 yr
320 yr
480 yr
640 yr
800 yr
960 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
-6
2
qbTo
bTf = 1x10 m /s
After 7680 years
Elevation m
100
0 yr
1280 yr
2560 yr
3840 yr
5120 yr
6400 yr
7680 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
qqbTf
1x10-8-8m
m22/s/s
bTo==1x10
After
After15360
15360years
years
Elevation m
100
0 yr
2560 yr
5120 yr
7680 yr
10240 yr
12800 yr
15360 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
qqbTf
1x10-2-2m
m22/s/s
bTo==1x10
After
After120
120years
years
Percent finer
80
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
Percent finer
80
70
-3
2
qbTo
bTf = 3x10 m /s
After 240 years
60
50
Ffs
pfeed
Ff
40
30
20
10
0
0.1
1
10
D mm
100
1000
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
qqbTo
= 1x10
1x10-3-3 m
m22/s
/s
bTf =
After
After240
240years
years
Percent finer
80
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
-4
2
qbTo
bTf = 3x10 m /s
After 480 years
Percent finer
80
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
qqbTf
1x10-4-4m
m22/s/s
bTo==1x10
After
After480
480years
years
Percent finer
80
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
Percent finer
80
-5
2
qbTo
bTf = 1x10 m /s
After 960 years
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
Percent finer
80
-6 m22/s
q
= 1x10
1x10-6
qbTo
=
m /s
bTf
After
After 7680
7680 years
years
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Plot of fractions finer in a) substrate (Ffs), b) sediment feed
(pfeed) and c) final surface at node 1 (Ff)
100
90
Percent finer
80
-8
2
qbTo
bTf = 1x10 m /s
After 15360 years
70
60
Ffs
pfeed
Ff
50
40
30
20
10
0
0.1
1
10
D mm
100
1000
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Equilibrium surface geometric mean size Dsg and bed slope
S as functions of sediment feed rate qbTo
bTf
100
mobile armor
static armor
0.01
0.009
Dsg
0.008
0.006
geometric mean size of feed
0.005
0.004
0.003
armor almost
washed out
S
10
1.00E-08
1.00E-07
1.00E-06
1.00E-05
2
qbTf
bTo m /s
1.00E-04
1.00E-03
0.002
0.001
0
1.00E-02
28
S
Dsg mm
0.007
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Equilibrium grain size distributions of sediment feed and
mobile armor at various sediment feed rates qqbTo
bTf
100
nearly unarmored
surface layer
(distribution almost
identical to that of
sediment load)
Percent finer
80
60
Feed
0.01 m2/s
0.003 m2/s
0.001 m2/s
0.0003 m2/s
0.0001 m2/s
0.00001 m2/s
1e-6 m2/s
1e-8 m2/s
40
20
approaching static armor
0
1
10
100
D mm
1000
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 18
Lisle, T. E., 1995, Particle size variations between bed load and bed material in natural gravel
bed channels. Water Resources Research, 31(4), 1107-1118.
Parker, G. and Klingeman, P., 1982, On why gravel-bed streams are paved. G. Parker and P.
Klingeman, Water Resources Research, 18(5), 1409-1423.
Parker, G., 1990, Surface-based bedload transport relation for gravel rivers. Journal of
Hydraulic Research, 28(4): 417-436.
Parker, G. and Sutherland, A. J., 1990, Fluvial Armor. Journal of Hydraulic Research, 28(5).
Parker, G. and Toro-Escobar, C. M., 2002, Equal mobility of gravel in streams: the remains of the
Water Resources Research, 38(11), 1264, doi:10.1029/2001WR000669.
Powell, D. M., Reid, I. and Laronne, J. B., 2001, Evolution of bedload grain-size distribution with
increasing flow strength and the effect of flow duration on the caliber of bedload sediment
yield in ephemeral gravel-bed rivers, Water Resources Research, 37(5), 1463-1474.
30