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

1/15

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

2/15

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)


3/15

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


4/15

FORMULATION OF THE PROBLEM

D* = water depthB* = channel widthh* = bottom elevationa* = tidal amplitudeT0

* = tidal period


•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

5/15

SEDIMENT FLUX qs*

ENGELUND & HANSEN FORMULA (1967)

FRICTIONAL TERM

**2h

**

DgC

UUj


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

6/15

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:


Sea

B0 B

Lb

x

B=B +(B -B ) exp(-x/L )b0

eL

7/15

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


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

8/15

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


9/15

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


0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1x

chan

nel w

idth

time

10/15

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


-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 !

11/15

-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


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?

12/15


)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)

13/15

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


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

14/15

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)


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

15/15


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

Dpt. of Civil and Environmental Engineering University of Trento (Italy) Long term evolution of...

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