1. Morphodynamic Equilibria in Tidal Embayments with Decreasing Cross-Section Henk Schuttelaars The Institute for Marine and Atmospheric research Utrecht (IMAU) Utrecht University

2. Contents Introduction Model Formulation Numerical Experiments Comparison with Observations 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. Research Questions Do Long-Term Equilibria Exist Are They Unique and Stable } PARA- METERS ?

9. Research Questions Do Long-Term Equilibria Exist Are They Unique and Stable Can We Explain/Model/Predict Cyclic Bar behaviour Formation of Main Channels Splitting of Channels Formation of Thresholds } PARA- METERS ?

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. The Equations Continuity: Momentum: Concentration: Bed Evolution: The Boundary Conditions X=0:X=L:

13. Sediment Flux

14. Geometry W E R W Embayment X=0X=L Side View: Top View:

15. Parameter Continuation Short Embayment: Analytical Solution constantly sloping bed spatially uniform hor and vert velocities spatially uniform bed stress For other parameters, continue this known solution

16. Parameter Continuation Short Embayment: Analytical Solution constantly sloping bed uniform hor and vert velocities spatially uniform bed stress net sediment import after channel reduction For other parameters, continue this known solution

17. The Experiments Reducing W R Different Types of Equilibria Comparison with data Experiment 1: Experiment 2:

18. The Experiments Reducing W R Local Lorentz Linearization ( L ~ 110 km << L MAX ) Different Types of Equilibria Comparison with data Global Lorentz Linearization (L ~ 100 km ~ L MAX ) Experiment 1: Experiment 2:

19. Maximum Embayment Length Figure A1. The dimensionless frictional length scale Lf =Lg of the M2 tide versus the dimensionless length of the em-bayment. Here Lg is the frictionless tidal wavelength and Lf has been computed using (A2) with = 0 : 6. The water depths Hf and Hmax are determined by the characteristics of the morphodynamic equilibrium. The intersection of these curves with the solid curve determines the maximum embayment length Lmax.

20. Width Variation Bed Profile SeaLand110 km W R /W E M2 Phase Hor Vel SeaLand110 km W R /W E W R /W E ~ 1 Bed Weakly Concave Travelling Wave M 4 extern negligible W R /W E ~ 0.5 Bed Very Deep Standing Wave M 4 extern amplified Decreasing W R 0 -10 -17 0 -10 -27 Depth in meters

21. 110 kmSeaLand Amplitude of M 4 extern Hor Vel W R /W E

22. Frictionally Dominated Equilibrium

23. “External” Equilibrium Plate 2. Contour plots of the dominant contributions to the net sediment flux (defined in (11)) for the morphodynamic equilibrium that owes its existence to the presence of external overtides. Dependent variables are the dimensionless coordinate x and the length scale ratio L=Lg. Here Figure 2a shows us Cs, Figure 2b shows us ^ Cs, Figure 2c shows u 2c C 2c, and Figure 2d shows ^ u 2c C 2c. The scaling for the fluxes is identical to that in Figure 5. Here = 0 :1 5, other parameter values are as in Table 3.

24. Comparison With Field Data (1) WidthWidth-Averaged Depth

25. Comparison With Field Data (2) Sea Land

26. Conclusions Two Types of Equilibria: Idealized Model Reproduces Global Characteristics Quite Well Multiple Equilibria Maximum Embayment Length externally driven friction-related

27. Future Research Closer Comparison With Process-Based Models and Observations Introduce Two Dimensional Perturbations

28. Multiple Equilibria