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Dpt. of Civil and Environmental Engineering
University of Trento (Italy)
Channel competition in tidal flatsChannel competition in tidal flats
Marco Toffolon & Ilaria Todeschini
1/12
Channel competition in tidal flats
Tidal channels on a tidal flat, Coos Bay, Oregon
Tidal channels on a tidal flat, Venice Lagoon
2/12
TWO POSSIBLE DIFFERENT APPROACHES
1. THE PROBLEM OF THE INITIAL FORMATION OF TIDAL CHANNELS IN A SALT MARSH
2. THE PROBLEM OF THE STABILITY OF ALREADY DEVELOPED CHANNELS WITH RESPECT TO A PERTURBATION OF THEIR STATE
Channel competition in tidal flats
a. one single channel (Fagherazzi and Furbish, 2001)
b. a network of channels (D’Alpaos et al.,2005)
c. initiation of tidal channels
VERY SIMPLIFIED APPROACH
3/12
FORMULATION OF THE PROBLEM
THE SYSTEM IS CONSTITUTED ONLY BY TWO ELEMENTS
B1 B2 B3
Xx
y LTexternal basin
Di
Bi
z
y
Channel competition in tidal flats
SIMPLE CONCEPTUAL MODEL:
1. TIDAL FLAT
2. CHANNELS
We suppose that the free surface level varies in the flat without drying
4/12
HYPOTHESIS:
3
5
iisiii DJkBQ
)(tQt
hS
n
iit QQ
1
The contribution of the flat to the total discharge is neglected
the total discharge depends only on the planimetric surface S (quasi-static model)
)( critii E
t
Simplified EROSION LAW typical of cohesive
sediments
The discharge in each channel is estimated using the usual equilibrium relationship
Tidally averaged
(e.g. Fagherazzi and Furbish)
Channel competition in tidal flats
Only the altimetric evolution is considered, while the variation of the width is neglected
iii JgD
5/12
STABILITY WITH CONSTANT ENERGY SLOPE
If we assume a constant J along the transversal direction (e.g. Fagherazzi and Furbish, 2001)
Channel competition in tidal flats
n
iiit DJQ
1
3
5
J
Perturbation analysis starting from the equilibrium configuration:
) 1( 0 iii dDD
) 1( 0 iii jJJ
d1, d2 perturbations of the two water depths
UNKNOWNS: 21
22111
1
3
10
dd
dt
d
21
22112
2
3
10
ddd
t
d
INSTABILITY except for the case d01=d02
In the case of TWO CHANNELS:
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2time
d1d2
d10 = 1
d20 = 0 (only the first channel is perturbed)
PER
TU
RB
ATIO
N O
F TH
E
FLO
W D
EPTH
channel 1
channel 2
6/12
BASIC IDEA: Every channel drains a portion of the tidal flat
“COMPETENCY AREA”
STABILITY WITH VARIABLE ENERGY SLOPE
Channel competition in tidal flats
QQQ - 02
QQQ 01
22
3
72
d
dd
t
d
11
3
72
d
dd
t
d
TWO COUPLED MODELS:
1. TRANSVERSAL DRAINAGE AND WATERSHED DELIMITATION
2. LONGITUDINAL WAVE PROPAGATION ALONG THE CHANNELS
to establish a relationship between Q and the free surface elevation in each channel, h1 and h2
to determine h1 and h2 as functions of d1 and d2
0Q
Q
in the case of TWO INITIALLY IDENTICAL channels Q0 is the same
7/12
1. WATERSHED DELIMITATION
Poisson equation
(Rinaldo et al. 1999)t
h
Dh m
m
ff
22
where:
2
)(),( 21 hh
xthm
Channel competition in tidal flats
h1 L
D0(f)
Q1z
h2
mean sea levely
Q2
External Basin
Tidal flat
Ch
ann
el 1
x
Ch
ann
el 2
X
y
f is the friction factor
t
h
Dy
hm
f
ff
2
)(02
2
2
hypothesis:
Rough salt marsh
(from Lawrence et al., 2004)
•the longitudinal fluxes in the salt marsh are neglected
•the tidal oscillation is small with respect to the average depth D0(f) in the flat Dm= D0(f) +hm D0(f)
8/12
L
hhDtxq
f
f
)(
),( 21
2)(0
In a tidal cycle in the generic section x:
T
dttxqT
xq0
),(1
)(
the absolute value is due to the fact that these are periodic functions
The variation of the total discharge at the mouth:
X
dxxqQ0
)(
Channel competition in tidal flats
1. WATERSHED DELIMITATION
h1 L
D0(f)
Q1z
h2
mean sea levely
Q2
External Basin
Tidal flat
Cha
nnel
1
x
Cha
nnel
2
X
y
9/12
2. WAVE PROPAGATION IN THE CHANNELS
The free surface elevation in ith channel:
following DRONKERS (1964))cos()exp(),( xktxkaxth widiii
Channel competition in tidal flats
SYMMETRIC CONFIGURATION WITH TWO INITIALLY IDENTICAL CHANNELS
2 )(
28
2
21
2)(00 X
AddL
DaQ
f
f
if we substitute (d1-d2) from the previous expression and we integrate in time to have the average value in a tidal cycle
if we integrate in the longitudinal direction to have the difference of the total discharge
)exp( )(8)( 021
2)(00 xkAxdd
LT
Daxq d
f
f
Linearizing, we obtain:
ii Hdhh 0 )(),( 21
2)(0 dd
L
DHtxq
f
f
L
hhDtxq
f
f
)(
),( 21
2)(0
10/12
Substituting the expression for Q into the differential equation system:
Channel competition in tidal flats
11
3
72
d
dd
t
d
22
3
72
d
dd
t
d
becomes21
1 2)3
7(2
d
ddd
t
d
212 )
3
72(2
d
ddd
t
d
EIGENVALUES:
1. (-7/3)
2. (4-7/3)positive for 12
7χχ c
always negative
Two channel separated by a distance L > Lc can be considered INDEPENDENT
Two channel separated by a distance L < Lc influence each other and tend to form an unstable system
INSTABILITY if Xc
DLLf
fc0
)(0
1
7
24
2
0
0
2L
D
L
XT f
gf
where
11/12
Channel competition in tidal flats
Xc
DLf
fc0
)(0
1
7
24
THRESHOLD VALUE OF THE DISTANCE L
Tidal flatCha
nne
l 1
x
X Cha
nne
l 2
y
LL > LC STABLE
L < LC UNSTABLE
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3time
d1d2
L L c UNSTABLE
L>Lc STABLE
PER
TU
RB
ATIO
N O
F TH
E F
LOW
DEPTH channel 1
channel 2
Physical interpretation:
Increasing L,
Q decreases
Q0 increases
Mutual influence (Q/ Q0) DECREASES
0
250
500
750
1000
1250
0 50 100 150 200X [m]
Lc [
m]
(a)(b)
12/12
Channel competition in tidal flats
Xc
DLf
fc0
)(0
1
7
24
THRESHOLD DISTANCE FOR CHANNEL STABILITY Tidal flatCha
nnel
1
x
X Cha
nnel
2
y
L
Shallow and rough flat and channels
Deep and smooth flat and channels
Lc is quite large:
do channels in nature usually influence each other?
Limitations of the model:
• the longitudinal flux has been neglected
• the wave propagation theory ignores finite-length effects
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