1. Nonlinear channel-shoal dynamics in long tidal embayments
H.M. Schuttelaars1,2, G.P. Schramkowski1,3 and H.E. de Swart1
“Finite amplitude behaviour of large scale alternating bars
can be understood and modelled”

2. Aim of this talk: to model and understand the
Observations on large scale alternating bars:
length scales ~ 20 km.
environments with strong tides
fine sand
Previous studies: Seminara & Tubino (1998), Hibma et al. (2002)
Aim of this talk: to model and understand the
observed dynamical behaviour of large scale
alternating bars

3. Model setup • idealised model • straight channel • only bed erodible
• depth-averaged SW eqns
• suspended load transport
• uniform M2 tidal forcing,
velocity scale ~ 1 m/s
Length scales << channel length, tidal wavelength
Typically: • channel width B
• horizontal tidal excursion length ~ 7 km

4. Model approach h’ = S S Amn(t) cos (m kcx) cos (npy/B)
Use a finite number of spatial patterns obtained from a linear stability
analysis to describe the finite amplitude bed behaviour: M N
h’ = S S Amn(t) cos (m kcx) cos (npy/B)
m=0
n=0
kc: wavenr. of critical mode
channel length Lc
Lc= ~ 60 km
B ~ 5 km.
N,M: truncation numbers 2p kc kc
Growth curves

5. Model approach h’ = S S Amn(t) cos (m kcx) cos (npy/B)
Use a finite number of spatial patterns obtained from a linear stability
analysis to describe the finite amplitude bed behaviour: M N
h’ = S S Amn(t) cos (m kcx) cos (npy/B)
m=0
n=0 B
m=1,n=1 Lc B
m=1,n=2 Lc B
m=2,n=1
Growth curves Lc

6. Model approach h’ = S S Amn(t) cos (m kcx) cos (npy/B)
Use a finite number of spatial patterns obtained from a linear stability
analysis to describe the finite amplitude bed behaviour: M N
h’ = S S Amn(t) cos (m kcx) cos (npy/B)
m=0
n=0
Insert expansion in complete nonlinear equations.
equations describing the behaviour of Amn(t):
steady state solutions
cyclic behaviour

7. Example: channel width ~ 3.5 km.
• multiple steady state
solns:
• trivial soln.
• nontrivial soln.
• no steady equilibrium
soln for r/sH >
periodic soln.

8. Steady state solution (r~0.0213)
Periodic solution (r~0.0214)

9. Sensitivity study: variation of bed friction and channel width
R<rcr(B): horizontal bed
B<3.6 km: stable static solns. exist
B>3.6 km: no static solns time-dependency
Small region of multiple stable steady states

10. Conclusions Present work
existence of finite amplitude alternating bars explicitly
demonstrated
qualitative behaviour depends on channel width and
strength of bed friction
saturation mechanism: importance of destabilizing
sediment fluxes decreases relative to bedslope effects
Present work
explore towards realistic values of bed friction
further identification of physical processes
comparison with more complex models