G. SEMINARA Dipartimento di Ingegneria Ambientale, Università di Genova, Italy

Coworkers: M. Colombini, B. Federici, M. Guala,S. Lanzoni, N. Siviglia, L. Solari, M. Tubino, D. Zardi, G. Zolezzi,

Morphodynamic Influence and related issues

Major issue

In what directions does morphodynamic influence (i.e. perturbations of bottom topography and/or river alignement)

propagate in a river reach ?

Plan of the talk

1. The settled case of 1-D perturbations.

2. The case of 2-D perturbations : free bars.

3. The case of 2-D perturbations : forced bars

Can a bend (or other geometrical constraint) affect bed topography upstream ?

4. Plan form perturbations in meandering rivers:

Can meander evolution be upstream influenced?

1.THE SETTLED CASE OF 1-

D PERTURBATIONS

Sketch of the channel and notations

Independent Variables : x , t Unknown functions : Y , , U

Y

Y0

x

Y

b

Assume : wide rectangular cross section

0

x

UY

t

Y

FORMULATION OF THE PROBLEM

Governing equations

(Continuity of the liquid phase )

02

2

YC

U

x

hg

x

UU

t

U

(Conservation of Momentum)

0 1

x

Q

tp s

(Continuity of the solid phase )

The morphodynamic response of the channel to small initial perturbations of bed elevation

Seek solution

1111100000 ,,,, ,,,,,,,, UQYhUQYhUQYh sss

Basic uniform state Small perturbation

Make the formulation dimensionless

002

02

0

1

0

1

0

1 , , , , UYC

tT

YC

xX

Y

hh

Y

Yy

U

Uu

Linearize governing equations and reduce

031

123

3

20

2

2

20

2

2

3

3

x

u

Fx

u

x

u

Ftu

x

u

tt

u

Parameters

discharge fluid and solidbetween Ratio 1

number Froude

00

0

202

0

YU

Qb

gY

UF

so

Seek solutions for perturbations in the form of normal modes

01

13122

0

32

0

223

Fik

Fkikiiki

ctxikfu exp

wavenumberL

k 2

rategrowthkc

wavespeedkc

ii

rr

Dispersion relationship

THREE MODES

3,2,1 10 jjj

0

1

1

03

20

2

02

20

2

01

F

kikik

F

kikik

Typically ~ O (10-3 - 10-4 ), hence expand

[O

[O

2

20

42

20

2

20

3

13

12

11

119

113

...................

...................

Fkk

Fikk

F

ki

Hydrodynamic modes:F0 < 2 they are both stable

Hydrodynamic modes:F0 > 2 one mode is unstable (roll waves)

In the long wave limit (k small) both modes migrate downstream

In the short (inertial) wave limit (k large): F0 > 1 both modes migrate downstreamF0 < 1 one mode migrates upstream

Morphodynamic mode:

F0 = 0.5 : Downstream migration

F0 = 1 : No Migration

F0 = 2.5 : upstream Migration

Invariably stable for any Froude number

Long waves (k <<1) areWeakly damped and nearly non migrating

Upstream morphodynamic influence in supercritical flows:Fully non linear numerical solution (Siviglia, 2005)

F0 = 2.4 = 0.028

Short perturbation:Propagation is very fast!

Downstream morphodynamic influence in subcritical flows:Fully non linear numerical solution (Siviglia, 2005)

F0 = 0.51 = 0.001

Short perturbation:- Propagation is still fast but less so as is smaller!- Non linear effects generate fronts

Growthrate and wavespeed tend to vanish as k → 0Fully non linear numerical solution (Siviglia, 2005)

PropagationIs very slow!

Damping is very weak!

F0 = 0.51 = 0.001

2. 2-D PERTURBATIONS:

free bars

Free bars

arise spontaneously

whenever bed topography is

unstable to 2-D perturbations

of spatial scale of the order of channel width

Under what conditions do they form?

In the case of alternate bars we find:

c = c * , ds )

- : average Shields stress of the mean flow - ds relative roughness of the mean flow

Incipient conditions for bar formation determined by classical linear stability analysis:

Bars form provided the width to depth ratio of the channel exceeds critical value c

(e.g. Blondeaux and Seminara, JFM, 1985)

Multiple row bars form for higher values of

A MORE DELICATE PROBLEM RELATED TO THE ISSUE OF MORPHODYNAMIC INFLUENCE: How does bar growth occur?

For values of larger than c

any perturbation

e.g. located at some cross section of the channel

leads to sand wave which migrates

with amplitude growing in space and time

But:• does the perturbation spread both upstream and downstream?

• does it eventually reach an equilibrium (possibly periodic) state?

Absolute instability

an impulse response propagates for large times at all points in the flow

Convective instability

an impulse response decays to zero for large times at all points in the flow: disturbances

are convected away as they amplify

Is bar instability convective or absolute ? (Federici and Seminara, JFM, 2003)

Absolute versus Convective instability(Briggs, 1964 and Bers,1975 in plasma physics, Huerre & Monkewitz, 1990 in hydrodynamics)

Spatially localized initial perturbation of bed topography

Initial perturbation of bed topography randomly distributed in space

t=1500

t=1000

t=500

t=500

t=0

NNumericumericalal ssimulaimulattionions (Federici and Colombini, 2003)s (Federici and Colombini, 2003)

If initial perturbation of bed topography is not persistent

s

n

3-D view

Because of the convective nature of bar instability:

Need persistent small perturbation of bed topography in the initial cross-section of the channel

t=500

t=650

t=750

= 8 (c=5.6) 0.057 ds= 0.053

Forced development of bars leads to equilibrium amplitude

only if the domain is sufficiently long!!!

does influence the spatial position at which equilibrium amplitude is reached

Varying the amplitude of the initial perturbation

does not influence the equilibrium amplitude

(— ) perturbation amplitude = 0.001

(----) perturbation amplitude = 0.002

if monochromatic, it influences the equilibrium amplitude

does not influence the spatial position at which the equilibrium amplitude is reached

The frequency of the perturbation at initial cross-section

(—) perturbation frequency = 7.9*10-4

(----) perturbation frequency = 5.7*10-4

forcing a discrete spectrum containing (10) 20 harmonics of equal amplitudes

with frequencies obtained from the linear dispersion relationship in the unstable

range.

linearly most unstable

Temporal evolutionof bar wavenumber

Temporal evolution of bar wavespeed

bars hardly develop uniformly along the

whole reach

upstream and downstream bars have decreasing

heights

Wave group

a more developed bar always forms

The tail of the wave group remains in the upstream reach

All bars migrating downstream amplify

Was an equilibrium amplitude reached?

Numerical simulation of the laboratory Numerical simulation of the laboratory eexperiment H-2 xperiment H-2 of Fujita & Muramoto (1985)of Fujita & Muramoto (1985)

Numerical simulation of the laboratory Numerical simulation of the laboratory eexperiment H-2 xperiment H-2 of Fujita & Muramoto (1985)of Fujita & Muramoto (1985)

= 10 (c=7) 0.064 ds =

0.047

t=480’

t=290’

t=240’

t=145’

Hence:Hence:

A persistent perturbation is needed to reproduce the mechanism of formation and development of bars in straight channels correctly.

Bars evolve spatially and reach an equilibrium amplitude that is independent of the amplitude and frequency of the initial perturbation

Bars lengthen and slow down as they grow in amplitude

The distance from the initial cross section where equilibrium amplitude is

reached does depend on the intensity of the initial perturbation.

Length of the channel in laboratory experiments must be large enough for equilibrium conditions to be reached :

Uncertainty on significance of values of bar amplitude, wavelength and wavespeed reported by different authors

2-D informations are propagated downstream

3.

Forced Bars

3.

Forced Bars

Forced bars

arise as a response of bed topography

i) to variations of channel geometry, e.g. channel curvature

ii) to perturbed boundary conditions

Fundamental question: does the presence of a bend affect bed

topography In the downstream and/or upstream reach ?

(Struiksma et al., 1985, J. Hydr. Res.Zolezzi and Seminara, 2001, J. Fluid Mech.

Zolezzi et al., 2005, J. Fluid Mech.)

NOTATIONS

ASSUMPTIONS

Curvature ratio

Width ratio

• slowly varying approach

• wide channel

Curvature s

sC

)(

FORMULATION (Zolezzi and Seminara, 2001)

characteristic exponents

integration constants

The exact solution of the linear problem of fluvial morphodynamics

Upstream-downstream influence

Local effect of curvature

Required to fit Boundary condtns.

3 exponentially DECAYING solutionsDominant DOWNSTREAM INFLUENCE

MORPHODYNAMIC INFLUENCE:

The 4 characteristic exponents

Dominant UPSTREAM INFLUENCE3 exponentially GROWING solutions

R

+

-

U-FLUME EXPERIMENTS

Validation of the theory of upstream overdeepening

• Reproduce sub- and super- resonant conditions

• Measure temporal bed evolution

• Obtain steady bed topography by time-averaging Laboratory of D.I.A.M. - University of Genova

PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUPERRESONANT CONDITIONS

(Zolezzi, Guala & Seminara, 2005, JFM)

PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUPERRESONANT CONDITIONS

(Zolezzi et al., 2005, JFM)

PATTERN OF STEADY COMPONENT OF BED TOPOGRAPHY IN U- CHANNELS UNDER SUBRESONANT CONDITIONS

(Zolezzi et al., 2005, JFM)

BIFURCATION AS A PLANIMETRIC DISCONTINUITY

Under super-resonant conditions (> R) upstream influence

INITIAL STAGE: MIGRATING BARS

FINAL STAGE: STEADY BARS

W. Bertoldi, A. Pasetto, L. Zanoni, M. TubinoDepartment of Civil and Environmental Engineering, Universtiy of Trento, Italy

4.

Downstream and upstream influence

In the plan form evolution

Of meandering channels

Fundamental questions: Is bend instability convective or absolute ? In what directions do wavegroups migrate?(Lanzoni, Federici and Seminara , 2005)

planform configurations after several neck cut offs

Plan form response of initially straight channel to smallrandom perturbations: Free boundary conditions

Subresonant : instability is convective and meander groups migrate downstream

Superresonant : instability is convective and meander groups migrate upstream

=15, *= 0.3, dune covered bed

Conclusion and main message

The direction of propagation of 1-D morphodynamic information Changes as the critical value of the Froude number (F0=1) is crossed

Role of the Froude number

somewhat taken by the aspect ratio of the channel when the propagation of 2-D morphodynamic information is considered

Superresonant channels display features quite differentfrom those of subresonant channels

Field verification of this framework urgently needed : a challenge for geomorphologists ?

The end

Why does the “standard model” (Ikeda & al., 1981)not predict upstream influence ?

Downstream influenceLocal effect of curvature

•Only one characteristic exponent 1=2 Cf0

•Only downstream influence

STEADY BED TOPOGRAPHY IN U- CHANNELS : SUPERRESONANT BED PROFILES AVERAGED AT THE INNER

AND OUTER BANKS (Zolezzi et al., 2005, JFM)

Run U2

STEADY BED TOPOGRAPHY IN U- CHANNELS : SUPERRESONANT BED PROFILES AVERAGED AT THE INNER

AND OUTER BANKS (Zolezzi et al., 2005, JFM)

Run U3

STEADY BED TOPOGRAPHY IN U - CHANNELS : SUBRESONANT BED PROFILES AVERAGED AT THE INNER

AND OUTER BANKS (Zolezzi et al., 2005, JFM)

Run D2

R

R

Subresonant meanders migrate downstream while superresonant meanders migrate upstream

Bend instability:Linear theory (Blondeaux &Seminara, 1985)

-Bend instability selects near resonant wavenumber

Planimetric response of initially straight channel to smallrandom perturbations: Periodic boundary conditions

Subresonant

Superresonant

Planimetric response of initially straight channel to smallrandom perturbations: Free boundary conditions

Subresonant : instability is convective and meander groups migrate downstream

Superresonant : instability is convective and meander groups migrate upstream

Highly Superresonant : instability is absolute and meander groups migrate upstream

The third (morphodynamic) mode

2

20

2

20

20

213

119

11

Re

Fk

F

F

k

kc

C > 0 Fo > 1C< 0 Fo < 1C = 0 Fo = 1

C → 0 as k→ 0C → - /(F0

2 -1) as k → ∞

2

20

2

20

2

13

119

3Im

Fk

F

k

i) Growth rate is negative for any value of Fo and k

ii) Damping tends to vanish for very

long waves.

MORPHODYNAMIC INFLUENCE:LINEAR THEORY

Fourier expansion in n

IV order ordinary problem for um(s)

4 Characteristic exponents mj