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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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morphology
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example
connections
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R
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n
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)
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Q
Q
X probability
Qchannel forming
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Q flow discharge m3/s
Q = Au
A ≈ Wh →
Q = Whu
H depth
bankfull
u flow velocity
Area (bankfull)
Width
wetted
Perimeter
0
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0
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AA?,A=
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flood
Q
low flow
t
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water surface elevation (base
level) is raised at t = 0 by e.g.
installation of a dam
sediment supply
remains constant
at qsa
η
antecedent equilibrium bed
profile established with load qsa
before raising base level
About Froude: subcritical and supercritical flow
• slow
• downstream control
• fast
• no downstr. control
• Fr < 1
• Fr > 1
"
bed shear stress (and sediment mobility)
τ = ρgR sin (S ) ≈ ρgRS
τ
τ* =
[(ρ s − ρ )gD50 ]
Flow shear stress on the bed (Newton)
Shields number:
Sediment-entraining ‘force’ vs.
sediment-detraining ‘force’
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q = αt τ −τ
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∗ nt
c
, τ >τ
∗
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qt
q =
[( ρ s − ρ ) / ρ s ]gD D
∗
t
∗
c
4
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Shields mobility number
10
Suspended transport
Bed load transport
1
RIJN
ALLIER
0,1
GRENSMAAS
Shields criterion
0,01
sand
silt
0.1
gravel
1
10
Grain size/diameter (mm)
100
4
'
♣ Allier
♦ Meuse
♥ Rhine
♠ Volga
Braiding, often Fr~1
Stream
power
(Van den Berg, 1995)
♣
♥
♠
♦
Meandering, often Fr<<1
Grain size
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transport
gradient
λ=poros
η=bed level
qb=transp
t=time
x=location
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parameter
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discharge
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slope i
water depth h1
water grain flow
depth thickness
h0
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sediment bed
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Long Profile of the Amazon River
3000
2500
η (m)
2000
1500
1000
500
0
-7000
-6000
-5000
-4000
-3000
x (km)
-2000
-1000
0
D
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The Kosi River flows into a zone of rapid
subsidence. Subsidence forces a
streamwise decline in the sediment load in
a similar way to sea level rise. Note how
the river width decreases noticeably in the
downstream direction.
https://zulu.ssc.nasa.gov/mrsid/mrsid.pl
7 !
5%
%
N
:
Ultimate water surface
Initial water surface
Ultimate bed
Initial bed
η
transient bed profile
(prograding delta)
%
0
%%
.
'
D
% /'
%
final equilibrium bed profile in
balance with load qt > qta
transient aggradational profile
sediment supply
increases from qta
to qt at t = 0
η
antecedent equilibrium bed profile
established with load qta
$8
%%
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%
Bed evolution
160
0 yr
5 yr
10 yr
15 yr
20 yr
25 yr
Ultimate
140
100
80
60
40
degradation
20
0
0
2000
4000
6000
8000
Bed evolution
10000
Distance in m
90
80
aggradation
0 yr
20 yr
40 yr
60 yr
80 yr
100 yr
Ultimate
70
Elevation in m
Elevation in m
120
60
50
40
30
20
10
0
0
2000
4000
6000
Distance in m
8000
10000
%
Bed evolution (+ Water Surface at End of Run)
25
Elevation in m
20
bed 0 yr
bed 20 yr
bed 40 yr
bed 60 yr
bed 80 yr
bed 100 yr
bed 120 yr
ws 120 yr
15
10
5
0
-5
0
10000
20000
30000
Distance in m
40000
50000
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Calculation of River Bed Elevation Variation with Normal Flow Assumption
(Qf)
(Inter)
(B)
(D)
(lamp)
Calculation of ambient river conditions (before imposed change)
Assumed parameters
Q
70 m^3/s
Flood discharge
If
0.03
Intermittency
The colored boxes:
B
25 m
Channel Width
indicate the parameters you must specify.
D
30 mm
Grain Size
The rest are computed for you.
λp
0.35
Bed Porosity
(kc)
kc
(S)
S
75 mm
0.008
Roughness Height
If bedforms are absent, set kc = ks, where ks = nk D and nk is an order-one factor (e.g. 3).
Ambient Bed Slope
Otherwise set kc = an appropriate value including the effects of bedforms.
Computed parameters at ambient conditions
H
0.875553 m
Flow depth (at flood)
τ*
0.141503
Shields number (at flood)
q*
0.232414
Einstein number (at flood)
qt
0.004859 m^2/s
Volume sediment transport rate per unit width (at flood)
Gt
3.05E+05 tons/a
Ambient annual sediment transport rate in tons per annum (averaged over entire year)
Calculation of ultimate conditions imposed by a modified rate of sediment input
Gtf
7.00E+05 tons/a
Imposed annual sediment transport rate fed in from upstream (which must all be carried during floods)
qtf
0.011161 m^2/s
Upstream imposed volume sediment transport rate per unit width (at flood)
τult∗
0.211523
Ultimate equilibrium Shields number (at flood)
Sult
0.014207
Ultimate slope to which the bed must aggrade
Hult
0.736984 m
Ultimate flow depth (at flood)
Click the button to perform a calculation
Calculation of time evolution toward this ultimate state
L
qt,g
∆x
∆t
10000
0.011161
1.67E+02
0.01
m
m^2/s
m
year
length of reach
Ntoprint
sediment feed rate (during floods) at ghost node
Nprint
spatial step
M
αu
time step
Duration of calculation
200
5
60
0.5
10
Number of time steps to printout
Number of printouts
Intervals
Here 1 = full upwind, 0.5 = central difference
years
8
,