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 Assume : wide rectangular cross section

FORMULATION OF THE PROBLEM Governing equations (Continuity of the liquid phase ) (Conservation of Momentum) (Continuity of the solid phase )

The morphodynamic response of the channel to small initial perturbations of bed elevation Seek solution Basic uniform stateSmall perturbation Make the formulation dimensionless

Linearize governing equations and reduce Parameters

Seek solutions for perturbations in the form of normal modes Dispersion relationship THREE MODES

Typically  ~ O ( ), hence expand [O  [O   

Hydrodynamic modes: F 0 < 2 they are both stable Hydrodynamic modes: F 0 > 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): F 0 > 1 both modes migrate downstream F 0 < 1 one mode migrates upstream

Morphodynamic mode: F 0 = 0.5 : Downstream migration F 0 = 1 : No Migration F 0 = 2.5 : upstream Migration Invariably stable for any Froude number Long waves (k <<1) are Weakly damped and nearly non migrating

Upstream morphodynamic influence in supercritical flows: Fully non linear numerical solution (Siviglia, 2005) F 0 = 2.4  = Short perturbation: Propagation is very fast!

Downstream morphodynamic influence in subcritical flows: Fully non linear numerical solution (Siviglia, 2005) F 0 = 0.51  = 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) Propagation Is very slow! Damping is very weak! F 0 = 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  *, d s ) -  : average Shields stress of the mean flow - d s 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=0 Numerical simulations (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)    d s = 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 = (----) perturbation amplitude = The forced development of bars lead to an equilibrium amplitude

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 evolution of 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 experiment H-2 of Fujita & Muramoto (1985)

Numerical simulation of the laboratory experiment H-2 of Fujita & Muramoto (1985)  = 10 (  c =7)    d s = t=480’ t=290’ t=240’ t=145’

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 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 solutions Dominant DOWNSTREAM INFLUENCE MORPHODYNAMIC INFLUENCE: The 4 characteristic exponents Dominant UPSTREAM INFLUENCE 3 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. Tubino Department 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 small random 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 (F 0 =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 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 ? Downstream influence Local effect of curvature Only one characteristic exponent 1 =2  C f0 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 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 F o > 1 C< 0 Fo < 1 C = 0 Fo = 1 C → 0 as k→ 0 C → -  /(F ) as 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 u m (s) 4 Characteristic exponents mj