```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 1D PERTURBATIONS
Sketch of the channel and notations
Assume : wide rectangular cross section
Q, U
Y
Y0
Qs
h
x
Y

b
Independent Variables : x , t
Unknown functions : Y , h , U
FORMULATION
OF
THE PROBLEM
Governing equations
Y UY

0
t
x
(Continuity of the liquid phase )
U
U
h U 2
U
g  2 0
t
x
x C Y
(Conservation of Momentum)
h Qs
1  p  
0
t
x
(Continuity of the solid phase )
The morphodynamic response of the channel to
small initial perturbations of bed elevation
Seek solution
h,h , Y , Qs ,U   h0 ,h0 , Y0 , Qs 0 ,U 0    h1 ,h1 , Y1 , Qs1,U1 
Basic uniform state
Small perturbation
Make the formulation dimensionless
U1
Y1
h1
x
t
u
, y
, h
, X 2
, T 2
U0
Y0
Y0
C Y0
C Y0 U 0
Linearize governing equations and reduce
2
3




u
  u  
1 u
u
 u
 2 2   u   1  2  2  3      2 3  0
3
t
t  x  t  F0  x
x  F0 x
3
2
Parameters
2
U
F02  0
gY0
Froude number
b Qso

 1
U 0Y0
Ratio between solid and fluid discharge
Seek solutions for perturbations in the form of normal modes
u  f exp ik x  ct 
2
k
 wavenumber
L
cr  kr  wavespeed
ci  ki  growth rate
Dispersion relationship

1 
2
3 
i  2 ik  1  i ik 3     k 1  2   ik 2  0
F0
 F0 

3
2
THREE MODES
Typically  ~ O (10-3 - 10-4 ), hence expand
 j  1,2,3
  0 j   1 j
[O0]
k2
01  k  i  ik  1  2
F0
k2
02  k  i  ik  1  2
F0
03  0
[O]
11  ...................
12  ...................

1 

3k  ik 1  2 
3
F0 
k

 i 2
2
F0


1
9k 2  k 4 1  2 
F0 

2
13
Hydrodynamic modes:
F0 < 2 they are both stable
Hydrodynamic modes:
F0 > 2 one mode is unstable
(roll waves)
In the short (inertial) wave limit (k large):
F0 > 1 both modes migrate downstream
F0 < 1 one mode migrates upstream
In the long wave limit
(k small) both modes
migrate downstream
Morphodynamic mode:
Invariably stable for any
Froude number
F0 = 0.5 : Downstream migration
F0 = 1 : No
Migration
Long waves (k <<1) are
Weakly damped and
nearly non migrating
F0 = 2.5 : upstream
Migration
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 → 0
Fully non linear numerical solution (Siviglia, 2005)
F0 = 0.51
Damping is
very weak!
 = 0.001
Propagation
Is very slow!
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?
Incipient conditions for bar formation determined by
classical linear stability analysis:
Bars form provided the width to depth ratio of the channel b exceeds critical value bc
(e.g. Blondeaux and Seminara, JFM, 1985)
In the case of alternate bars we find:
b c = b c  t * , ds )
- t* : average Shields stress of the mean flow
- ds relative roughness of the mean flow
Multiple row bars form for higher values of b
A MORE DELICATE PROBLEM RELATED TO THE ISSUE OF
MORPHODYNAMIC INFLUENCE: How does bar growth occur?
For values of b larger than bc
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 versus Convective instability
(Briggs, 1964 and Bers,1975 in plasma physics,
Huerre & Monkewitz, 1990 in hydrodynamics)
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)
Numerical simulations (Federici and Colombini, 2003)
If initial perturbation of bed topography is not persistent
localized initial perturbation of bed topography
3-DSpatially
view
t=0
s
t=500
n
Initial perturbation of bed topography randomly distributed in space
t=500
t=1000
t=1500
Because of the convective nature of bar instability:
Need persistent small perturbation of bed topography in the initial crosssection of the channel
b = 8 (bc=5.6) t*  0.057 ds= 0.053
t=500
t=650
only if the domain is sufficiently long!!!
Forced development of bars leads to equilibrium amplitude
t=750
Varying the amplitude of the initial perturbation
does not influence the equilibrium amplitude
does influence the spatial position at which equilibrium amplitude is
reached
(— ) perturbation amplitude = 0.001
(----) perturbation amplitude = 0.002
The frequency of the perturbation at initial cross-section
does not influence the spatial position at which the equilibrium amplitude is
reached
if monochromatic, it influences the equilibrium amplitude
(—) perturbation frequency = 7.9*10-4
(----) perturbation frequency = 5.7*10-4
linearly most unstable
forcing a discrete spectrum containing (10)
20 harmonics of equal amplitudes with
frequencies obtained from the linear
dispersion relationship in the unstable
range.
Temporal evolution
of bar wavenumber l
Temporal evolution
of bar wavespeed
Numerical simulation of the laboratory experiment H-2
of Fujita & Muramoto (1985)
bars hardly develop
uniformly along the
whole reach
a more developed bar
always forms
upstream and downstream
bars have decreasing heights
Wave group
All bars migrating
downstream amplify
The tail of the wave
group remains in the
upstream reach
Was an equilibrium
amplitude reached?
Numerical simulation of the laboratory experiment H-2
of Fujita & Muramoto (1985)
b = 10 (bc=7) t*  0.064 ds = 0.047
t=145’
t=240’
t=290’
t=480’
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
2-D informations are propagated downstream
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
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.)
FORMULATION (Zolezzi and Seminara, 2001)
NOTATIONS
Curvature
C (s)  

s
Curvature ratio
Width ratio
ASSUMPTIONS
• slowly varying approach
• wide channel
n0  1
b >> 1
The exact solution of the linear problem of
fluvial morphodynamics
characteristic
exponents
Upstream-downstream
influence
integration
constants
Required to fit
Boundary condtns.
Local effect
of curvature
MORPHODYNAMIC INFLUENCE:
The 4 characteristic exponents
Dominant UPSTREAM INFLUENCE
3 exponentially GROWING solutions
+
-
bbR
3 exponentially DECAYING solutions
Dominant DOWNSTREAM INFLUENCE
Validation of the theory of upstream overdeepening
U-FLUME
EXPERIMENTS
• Reproduce sub- and superresonant conditions
• Measure temporal bed evolution
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
W. Bertoldi, A. Pasetto, L. Zanoni, M. Tubino
Department of Civil and Environmental Engineering, Universtiy of Trento, Italy
INITIAL STAGE:
MIGRATING BARS
FINAL STAGE:
Under super-resonant conditions (b > bR)  upstream influence
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)
b=15, t * = 0.3, dune covered bed Plan form response of initially
straight channel to small
random perturbations:
Free boundary conditions
Subresonant : instability is convective and
meander groups migrate downstream
planform configurations after several neck cut offs
Superresonant : instability is convective and
meander groups migrate upstream
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 b
when the propagation of 2-D morphodynamic information is considered
Superresonant channels display features quite different
from 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 ?
Local effect
of curvature
Downstream influence
•Only one characteristic exponent l1=2 b 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
Bend instability:Linear theory
(Blondeaux &Seminara, 1985)
-Bend instability selects
near resonant wavenumber
bR 10
b 25
bR 10
Subresonant meanders
migrate downstream while
superresonant meanders
migrate upstream
b 5
b 25
b 5
Planimetric response of initially straight channel to small
random perturbations: Periodic boundary conditions
Subresonant
Superresonant
Planimetric response of initially straight channel to small
random 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
C > 0 Fo > 1
C< 0 Fo < 1
C = 0 Fo = 1

1 


1

2 

2
F0 
Re 13 
k

c
  2
2
k
F0
1 
2

9  k 1  2 
 F0 
k2
Im13   
F02
C → 0 as
k→ 0
C → -  /(F02 -1) as k → ∞
i) Growth rate is
negative for any
value of Fo and k
3

1 

9  k 2
1

2 

F0 

2
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 lmj
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