1. Potential mechanism for the initial formation of rhythmic coastline features M.van der Vegt, H.M. Schuttelaars and H.E. de Swart Institute for Marine and Atmospheric research Utrecht, Utrecht University, The Netherlands “ Tidal motion can cause coastline variations with wave lengths of a couple of kilometers “

2. Mesoscale rhythmic coastlines Shoreline sand waves: planimetric variations of coastline  1-10 km C  100m/yr A  10-100 m Red colors: protruding Blue colors: retreating (adapted from Ruessink&Jeuken,2002) Protruding or retreating coast

3. Motivation Waves and tides are potentially important Tides not considered so far Can tidal motion cause initial formation of rhythmic coastline features? What are the underlying physical mechanisms? Research questions

4. Physical model Geometry: Assumptions: - Near shore zone has constant width -Sediment transport q only in near shore zone -Sediment transport determined by velocities at transition line 20 m 5 m 10 km 500 m q Side view Top view Coastline Transition line Inner shelf Tide Near shore zone

5. Physical model 2-DH Shallow water equations, no diffusion, rigid lid approximation. Only on inner shelf. Boundary conditions: at x=x t and at x  Width-integrated sediment transport for the near shore zone: T m >>T  Coastline evolution is slow and tidally averaging is allowed Alongshore variations of sediment transport causes changes of shoreline position. No change of bathymetry.

6. Basic state and perturbations The model allows for a basic state with uniform alongshore conditions Note that basic state velocity has vorticity: Basic state is perturbed by perturbation of coastline Solutions cyclic in y and t. Model calculates initial complex growth rate Coast V(x,t) Sea 1 m/s Near Shore zone

7. Results: Growth rate curve

8. Residual flow on inner shelf Tidal residual circulations cells Vorticity dynamics? Growth determined by depth dependent friction Coriolis force migration Without Coriolis effectsWith Coriolis effects

9. Mechanism behind model results: vorticity dynamics Basic state velocity:V Basic state vorticity: Perturbed velocity: u,v Perturbed vorticity: Unstable Stable

10. Battle between the fluxes Stabilizing fluxes in cross-shore, destabilizing in alongshore direction Convergence cross-shore flux ~ width of inner shelf Convergence alongshore flux ~ wave length Positive growth rate for < width inner shelf

11. Conclusions Tidal motion can initially form rhythmic coastlines Length scale in the range of shoreline sand waves Steeper profile Critical wave length shorter Mechanism can be understood in terms of vorticity fluxes. Cross-shore Alongshore fluxes No damping mechanism With diffusion a preferred mode will occur Inclusion of wave actiondiffusion Model parameterization Future work

12. Stability analysis Consider perturbations (with alongshore structure) of coastline. This also perturbs the flow. After linearising, the model allows for solutions which are cyclic in y and t: Combination of u and v results in vorticity equation, which can be rewritten into one equation for u- component. Chebyshev approximation in x, Galerkin in t Eigenvalue problem K=alongshore wave number

13. Mesoscale rhythmic coastlines Barrier Islands along the Dutch and German coast Tidal range increasing Length scale 10 – 20 km Length scale decreasing Tidal range increasing Length scale barrier islands determined by tidal motion? Sha, 1989

14. Scaling Timescale T=1/  10 4 s Velocity: U=1m/s Surface elevation: z =U  L/g  0.1 Horizontal length scale: L=length of inner shelf=10 4 m Vertical length scale H s =10m, H  =20m After scaling: -Froude number is small  Rigid lid, local description. -Diffusion is small - T m >>T  coastline evolution is slow and tidally averaging is allowed

15. Mechanism behind model results: vorticity dynamics Basic state velocity:VBasic state vorticity: Perturbed velocity: u,vPerturbed vorticity: 0 0 0 u

16. Mechanism behind model results: vorticity dynamics Basic state velocity:VBasic state vorticity: Perturbed velocity: u,vPerturbed vorticity: v00

17. Why is the bottom profile so important for the results? Change in bottom profile has a strong effect on cross-shore vorticity flux. plays the crucial role, and thereby

18. Sensitivity to model parameters Sensitivity to steepness of bottom profile: Steeper profile smaller pref Migration speed strongly influenced by residual flow in basic state