MOBILE AND STATIC ARMOR IN GRAVEL-BED STREAMS

1
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 gravel-bed 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.
2
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.
3
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. 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 plotted on the next
slide.
4
MECHANISM OF MOBILE ARMOR contd.
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. 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.
5
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 the following values are
obtained
6
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. Now the bedload
transport equation can be written in the form
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.
7
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 where W( )
denotes a function. After some rearrangement,
8
COMPUTATION OF MOBILE AND STATIC ARMOR contd.
Letting ?i ln2(Di) and recalling that and
taking the 0th, 1st and 2nd moments of the
equation below, three equations for the three
unknowns u, Dsg and ?s are obtained
9
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 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 On the other
hand, the mobile-bed armor approaches a constant
static armor as
10
ALTERNATIVE COMPUTATION OF MOBILE AND STATIC ARMOR
An alternative way to compute armor is with the
code of the Excel workbook RTe-bookAgDegNormGravMi
xPW.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.
11
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.
12
qbTo 1x10-2 m2/s After 120 years
qbTf 1x10-2 m2/s After 120 years
13
qbTo 3x10-3 m2/s After 240 years
qbTf 3x10-3 m2/s After 240 years
14
qbTo 1x10-3 m2/s After 240 years
qbTf 1x10-3 m2/s After 240 years
15
qbTo 3x10-4 m2/s After 480 years
qbTf 3x10-4 m2/s After 480 years
16
qbTo 1x10-4 m2/s After 480 years
qbTf 1x10-4 m2/s After 480 years
17
qbTo 1x10-5 m2/s After 960 years
qbTf 1x10-5 m2/s After 960 years
18
qbTo 1x10-6 m2/s After 7680 years
qbTf 1x10-6 m2/s After 7680 years
19
qbTo 1x10-8 m2/s After 15360 years
qbTf 1x10-8 m2/s After 15360 years
20
qbTo 1x10-2 m2/s After 120 years
qbTf 1x10-2 m2/s After 120 years
21
qbTo 3x10-3 m2/s After 240 years
qbTf 3x10-3 m2/s After 240 years
22
qbTo 1x10-3 m2/s After 240 years
qbTf 1x10-3 m2/s After 240 years
23
qbTo 3x10-4 m2/s After 480 years
qbTf 3x10-4 m2/s After 480 years
24
qbTo 1x10-4 m2/s After 480 years
qbTf 1x10-4 m2/s After 480 years
25
qbTo 1x10-5 m2/s After 960 years
qbTf 1x10-5 m2/s After 960 years
26
qbTo 1x10-6 m2/s After 7680 years
qbTf 1x10-6 m2/s After 7680 years
27
qbTo 1x10-8 m2/s After 15360 years
qbTf 1x10-8 m2/s After 15360 years
28
qbTf
qbTf
29
qbTf
30
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,
doi10.1029/2001WR000669. 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.