1. On the nature of bend instability Stefano Lanzoni University of Padua, Italy Bianca Federici and Giovanni Seminara University of Genua, Italy

2. Meanders wandering in a flat valley (Alaska -USA) Meanders evolving in a rocky environment (Utah- USA) Tidal meanders within the lagoon of Venice (Italy)

3. Field examples of meander features Flow direction Beaver River Upstream skewed meanders Downstream skewed meanders Flow direction Fly River

4. Coexistence of upstream and downstream skewed meanders Multiple loops White River Flow direction Pembina River

5. Scope of the work Under which conditions is the planimetric development of meandering rivers downstream/upstream controlled? How is downstream/upstream influence related to the nature of bend instability? Which are the implications for the boundary conditions to be applied when simulating the planimetric development of natural rivers?

6. Notations Planform view Sez. A-A

7. Formulation of the problem Dimensionless planimetric evolution equation (Seminara et al., Jfm 2001) Erosion law (Ikeda, Parker & Sawai, Jfm 1981) semilarghezzavelocità media lateral migration speed long term erosion coefficient time longitudinal coordinate depth averaged longitudinal velocity lateral coordinate

8. Flow field (Zolezzi and Seminara, Jfm 2001) Characteristic exponents integration constants channel axis curvature u m = u m ( , C f0,  * ) aspect ratio friction coefficient Shields parameter

9. Dispersion relationship for bend instability (Seminara et al., Jfm 2001) Perturbation Planimetric stability analysis : complex angular frequency : complex phase velocity : complex group velocity  =  ( , C f0,  * )

10. Characteristic of bend instability growth rate dune covered bed plane bed phase speed  r  r  r  response excited at resonance super-resonance: bend migrate upstream sub-resonance: bend migrate downstream

11. Instability classification Absolute instability Convective instability initial impulse perturbation

12. Linear analysis of bend instability (Briggs' criterion, 1964) Absolute instability Convective instability branch point singularities = 0 > 0 the spatial branches of dispersion relationship (  i  given,  r varying) lie in distinct half -planes for large enough values of the temporal growth rate  i

13. Results of linear theory: First scenario  =8,   =0.3, d=0.005, dune covered bed a)  i =[  i ] , b)  i =1.5[  i ] , c)  i =2[  i ]  Convective instability

14.  =25,   =0.7, d=0.005, dune covered bed Results of linear theory: Second scenario a)  i =[  i ]  b)  i =2[  i ]  c)  i =5[  i ]  Absolute instability

15. Linear theory Bend instability is generally convective, but a transition to absolute instability occurs for large values of , dune covered bed and large values of  * The group velocity ∂  r /∂ associated to the wavenumber max characterized by the maximum growth rate changes sign as resonance is crossed moreover, rr

16. Numerical simulations of nonlinear planimetric development    i   = t/E pipi Boundary Conditions:  i = E (U i | n=1 -U i | n=-1 ) U i =U i (  *,d s,c mj ) Free B.C. Periodic B.C. c mj j=1,4 Forced B.C.

17. Numerical results: Free boundary conditions  =8,   =0.3, d=0.005 dune covered bed  =25,   =0.7, d=0.005 dune covered bed Sub-resonant conditions, Convective instability wavegroup migrate downstream Super-resonant conditions, Convective instability wavegroup migrate upstream Super-resonant conditions, Convective instability wavegroup migrate upstream  =15,   =0.3, d=0.005 dune covered bed

18. Numerical results: Periodic boundary conditions Sub-resonant conditions, Convective instability wavegroup migrate downstream Super-resonant conditions, Convective instability wavegroup migrate upstream  =8,   =0.3, d=0.005 dune covered bed  =15,   =0.3, d=0.005 dune covered bed

19. Numerical results: Forced boundary conditions  =30,   =0.1, d=0.01, dune covered bed super-resonant conditions periodic B.C. free B.C. forced B.C.

20. Numerical results: Free boundary conditions incipient cut off configuration  =15,   =0.3, d=0.005 dune covered bed incipient cut off configurationplanform configurations after several neck cut offs The length of straight upstream/downstream reaches continues to increase Cutoff spreads in the direction of morphodynamic influence

21. Conclusions Bend instability is invariably convective Meanders are typically upstream skewed Wave groups travel downstream The upstream reach tends to a straight configuration in absence of a persisting forcing The choice of boundary conditions strongly affects numerical simulations of the planimetric development of alluvial rivers Sub-resonant conditions (  <  r ) Bend instability may be absolute for a dune covered bed and high enough values of the Shields parameter Meanders are typically downstream skewed Wave groups travel downstream The downstream reach tends to a straight configuration in absence of a persisting forcing Super-resonant conditions (  >  r )

22. Open issues S ystematic field observations are needed to further substantiate the morphodynamic upstream influence exhibited by bend instability under super-resonant conditions The role of geological constraints possibly present in nature and their relationships with the features typical of bend instability has to be investigated. Which boundary conditions have to be applied when simulating the planimetric development of alluvial rivers? Further analyses are required to clarify the effects of chute and neck cut off on river meandering.