Dpt. of Civil and Environmental Engineering University of Trento (Italy) Long term evolution of...
Transcript of Dpt. of Civil and Environmental Engineering University of Trento (Italy) Long term evolution of...
Dpt. of Civil and Environmental Engineering
University of Trento (Italy)
Long term evolution of self-formed estuarine channels
Ilaria Todeschini, Marco Toffolon and Marco Tubino
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TIDE-DOMINATED ESTUARIES
(Thames, Bristol Channel,
Columbia River, etc.)
Delaware Bay
Bristol Channel
Long-term evolution of self-formed estuarine channels
Sea
B0 B
L b
x
B=B +(B -B ) exp(-x/L )b0
eL
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Scheldt
Potomac
Le = 77 km
Lb = 54 km
Le = 184 km
Lb = 54 km
Thames
Le = 95 km
Lb = 25 km
(data from Lanzoni and Seminara, 1998)
Long-term evolution of self-formed estuarine channels
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Which reasons could explain this
FUNNEL- SHAPE ?
SEA ACTION
SEA LEVEL
COASTAL UPLIFT
INTERNAL DYNAMICS
SEA ACTION
SEA LEVEL
COASTAL UPLIFT
BIDIMENSIONAL PROCESSES
ONEDIMENSIONAL PROCESSESetc …
LONG-TERM EQUILIBRIUM CONFIGURATION
•Schuttelaars & de Swart 2000
•Lanzoni & Seminara 2002
• …
Equilibrium bed profile
HERE
A simplified model to try to explain the shape of the estuary
Long-term evolution of self-formed estuarine channels
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FORMULATION OF THE PROBLEM
D* = water depthB* = channel widthh* = bottom elevationa* = tidal amplitudeT0
* = tidal period
Long-term evolution of self-formed estuarine channels
•RECTANGULAR CROSS SECTION AREA
•INTERTIDAL AREAS ARE NEGLECTED
Dt
B
x
)(Bq
t
ηp)(1
*s
0j gΩx
HgΩ)
Ω
Q(
xt
Q 2
ONE-DIMENSIONAL MODEL
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SEDIMENT FLUX qs*
ENGELUND & HANSEN FORMULA (1967)
FRICTIONAL TERM
**2h
**
DgC
UUj
Long-term evolution of self-formed estuarine channels
0t
Ω
x
Q
Continuity equation
Momentum equation
Exner equation
B*0
L*
B*
b
x*
SEAWARD BOUNDARY
•H(t) = sin(2t) (M2)
•qs = qs equilibrium
BOUNDARY CONDITIONS
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HYDRODYNAMICS: finite differences MacCormack + TVD filter to avoid oscillations (second order accurate both in space and in time)
EXNER EQUATION: finite differences First-order upwind
since Tbed >>T0 Hydrodynamic problem decoupled from the morphodynamic one
NUMERICAL SCHEME
LANDWARD BOUNDARY
•Q = 0
•Q = Qriver
qs=qs equilibrium
2 CASES:
Long-term evolution of self-formed estuarine channels
Sea
B0 B
Lb
x
B=B +(B -B ) exp(-x/L )b0
eL
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FIXED BANKS (Lanzoni & Seminara, 2002;
Todeschini et al, 2003)
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
after 2 years
after 10 years
after 50 years
after 100 years
equilibrium
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
after 2 yearsafter 10 yearsafter 20 yearsafter 50 yearsafter 100 yearsafter 130 yearsequilibrium
Convergent channel
Long-term evolution of self-formed estuarine channels
LENGTH OF THE DOMAIN
EQUILIBRIUM LENGTH
LONG-TERM BED EVOLUTION
0
100
200
300
400
500
0 200 400 600 800 1000
Length of the domain [km]
Equ
ilibr
ium
leng
th [k
m]
DEGREE OF CONVERGENCE
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from: http://sposerver.nos.noaa.gov/bathy/finddata.htm (National Ocean Service, USA)
TIDE-DOMINATED ESTUARIES
Delaware Bay Columbia River
D* water depth
B* width
*bottom elevation
Long-term evolution of self-formed estuarine channels
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A MODEL FOR WIDTH CHANGE
)1(2
2
critu
uk
B
Physically-based erosional law
In literature few contributions can be found, most of them refer to rivers
e.g. Darby & Thorne, 1996
provided the velocity exceeds a threshold value ucrit
PROBLEMS:
estimate of the two parameters
ucrit
k
Darby & Thorne (1996)
smk /10 8
Gabet (1998)
smk /10 11
intermediate value:
smk /10 10
Long-term evolution of self-formed estuarine channels
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1x
chan
nel w
idth
time
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RESULTSBOTTOM BANKS
BOTTOM PROFILE BANKS PROFILE
0.5
1
1.5
2
0 5 10 15 20 25time
chan
nel w
idth
x=0
x=1/3
x=2/3
time
Long-term evolution of self-formed estuarine channels
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
botto
m e
leva
tion
time
-1.5
-1
-0.5
0
0 5 10 15 20 25time
botto
m e
leva
tion
x=0
x=1/3
x=2/3
time
SHORTER TIME SCALE LONGER TIME SCALE
THE BOTTOM EVOLUTION
IS ALMOST THE SAME !THE BANKS PROFILE IS CONCAVE !
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-10
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1x
botto
m e
leva
tion/
tidal
am
plit
ude
6 m8 m 10 m12 m
*0
*
a
η
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1x
chan
nel w
idth
6 m8 m 10 m12 m
Long-term evolution of self-formed estuarine channels
Given the same tidal forcing
Despite the different initial depths at the mouth, the bottom and the banks equilibrium profile are quite similar
IS THE CHOICE OF THE INITIAL DEPTH AT THE MOUTH D0 IMPORTANT?
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Long-term evolution of self-formed estuarine channels
)1(2
2
critu
uk
B
on the other hand its value deeply influences the solution
it’s very difficult to obtain a reliable estimate of this parameter
IS THE CHOICE OF THE CONSTANT K IMPORTANT?
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
1 E-9
1 E-10
1 E-11
(a)
0.5
1.5
2.5
0 0.2 0.4 0.6 0.8 1x
B1 E-10
1 E-11
(b)
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
1 E-9
1 E-10
1 E-11
(a)
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1x
B1 E-9
1 E-10
1 E-11
(b)
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FIXED HORIZONTAL BED
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1x
B
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1x
B
Long-term evolution of self-formed estuarine channels
Irrealistic situation Could other factors induce a funnel- shape geometry?
The banks profile displays a convex shape
(e.g. with an increasing rate of widening seaward)
BANKS PROFILE EVOLUTION EQUILIBRIUM BANKS PROFILE
Moveable bed
Fixed bed
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NON NEGLIGIBLE RIVER DISCHARGE
at the landward boundary
-2
-1.5
-1
-0.5
0
0.5
0 0.2 0.4 0.6 0.8 1x
botto
m e
leva
tion
Q= -0.2
Q= 0
initial bed
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1xch
anne
l wid
th
Q= -0.2
Q= 0
(b)
Long-term evolution of self-formed estuarine channels
CONVEX SHAPE MILD BOTTOM SLOPE
THE RIVER DISCHARGE STRONGLY INFLUENCES THE SOLUTION
with a river discharge
vanishing river discharge vanishing river discharge
with a river discharge
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Long-term evolution of self-formed estuarine channels
comparison between:
• NON NEGLIGIBLE DISCHARGE
• REFLECTIVE BARRIER CONDITION
PHYSICAL INTERPRETATION
-1
-0,5
0
0,5
1
1,5
2
2,5
3
3,5
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1x
qs
RESIDUAL SEDIMENT FLUX
at the beginning of the simulation
at equilibrium
NO RIVER DISCHARGE
WITH RIVER DISCHARGE
at equilibrium