Dpt. of Civil and Environmental Engineering University of Trento (Italy) Channel competition in...

Dpt. of Civil and Environmental Engineering

University of Trento (Italy)

Channel competition in tidal flatsChannel competition in tidal flats

Marco Toffolon & Ilaria Todeschini

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Channel competition in tidal flats

Tidal channels on a tidal flat, Coos Bay, Oregon

Tidal channels on a tidal flat, Venice Lagoon

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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


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

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FORMULATION OF THE PROBLEM

THE SYSTEM IS CONSTITUTED ONLY BY TWO ELEMENTS

B1 B2 B3

Xx

y LTexternal basin

Di

Bi

z

y


SIMPLE CONCEPTUAL MODEL:

1. TIDAL FLAT

2. CHANNELS

We suppose that the free surface level varies in the flat without drying

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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)


Only the altimetric evolution is considered, while the variation of the width is neglected

iii JgD

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STABILITY WITH CONSTANT ENERGY SLOPE

If we assume a constant J along the transversal direction (e.g. Fagherazzi and Furbish, 2001)


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

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BASIC IDEA: Every channel drains a portion of the tidal flat

“COMPETENCY AREA”

STABILITY WITH VARIABLE ENERGY SLOPE


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

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1. WATERSHED DELIMITATION

Poisson equation

(Rinaldo et al. 1999)t

h

Dh m

m

ff

22

where:

2

)(),( 21 hh

xthm


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)

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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

)(


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

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2. WAVE PROPAGATION IN THE CHANNELS

The free surface elevation in ith channel:

following DRONKERS (1964))cos()exp(),( xktxkaxth widiii


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

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Substituting the expression for Q into the differential equation system:


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

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Xc

DLf

fc0

)(0

1

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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)

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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

Dpt. of Civil and Environmental Engineering University of Trento (Italy) Channel competition in...

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