1. Initial Formation of Estuarine Sections Henk Schuttelaars a,b, George Schramkowski a and Huib de Swart a a Institute for Marine and Atmospheric Research, Utrecht University b Delft University of Technology

2. Contents Introduction Model Formulation Instability Mechanisms Numerical Experiments Conclusions + Future Research

3. Tidal Embayments: Introduction Semi-enclosed bodies of water Connected to the open sea Driven by tides Examples: Frisian Inlet System Western Scheldt Inlets East Coast of the US

4. Marine Part of the Western Scheldt

5. Amplitude Hor. Vel. Phase Hor. Vel.

6. Salinity Distribution in the Western Scheldt

7. Properties Length: 160 km Depth: 20 m - 50 m - decreasing Complex Pattern of Channels and Shoals Main Channels: 3-1 Dredging Cyclic Behaviour Fractal Patterns Tidal Range: Vl 3.2 m Ant 4.0 m decreasing M2/M4 ~ 0.1 (at entrance) Phase difference (approx. 1 tidal wave length)

8. (From Jeuken, 2000)

9. Research Questions Can Estuarine Sections be modelled as Free Instabilities Can the Physical Mechanisms be understood How do these results depend on Physical Parameters

10. Model Formulation Idealized Models: Water Motion Sediment Transport Bed Evolution } Short Time Scale Long Time Scale Averaging

11. Model Equations and Assumptions Depth Averaged Shallow Water Equations Only Bed Erodible Noncohesive Material Suspended Load Transport Sediment Balance: holebar Fine Sand

12. Geometry Side View: Top View:

13. Linear Stability Analysis Find a (one dimensional) equilibrium solution h eq (x). This equilibrium h eq (x) is usually not stable w.r.t. small perturbations with a 2D structure: h = h eq (x) + h’(x,y, t ) The perturbation h’ can be found by solving an eigen value problem. The resulting eigenfunction reads h’ mn = e wt f m (x) cos(l n y) If Re( w ) > 0 : unstable bedform Re( w ) < 0 : stable bedform If Im( w ) = 0 : migrating bedforms

14. Instability Mechanisms Net ( tidally averaged) Sediment Transport: Advective Transport: Diffusive Transport:   F adv = x + y ~ (A/H) 2   F diff = -  xx -  yy ~  L 2

15. Diffusive Mechanism

16. Advective Mechanism

17. Numerical Experiments Short Embayment: Long Embayment: L=20 km, H=10 m, A=1.75m, B=5km.  = 25 m 2 s -1 Focus on influence of frictional strength L=60 km, H=10 m, A=1.75m, B=5km.  = 25 m 2 s -1, weak friction Focus on local/blobal modes

18. Short Embayment  Realistic Friction  Advectively Dom.  Unstable Mode  Local Mode  Weak Friction  Diffusively Dom.  Stable Mode  Global Mode Bed ProfileFluxes

19. Long Embayment  Global Mode  Diffusively Dom.  Stable Mode  Global Mode  Local Mode  Advectively Dom.  Unstable Mode  Local Mode Bed Profile Flux

20. Conclusions Two Types of Modes: Very Sensitive for Frictional Strength diffusively dominated: advectively dominated:  Scale with L  Non-migrating  Scale with B, U/ s  Migrating and Non-Migrating

21. Future Research Are Estuarine Sections Free Instabilities? What Determines the Position of the Advective Instabilities in the Estuary? Why are Advective and Diffusive Divergences of Fluxes (Always) Out of Phase?  Diffusive Instabilities?  Strongly Nonlinear Advective Instabilities?