Nonlinear channelshoal dynamics in long tidal embayments H
Nonlinear channel-shoal dynamics in long tidal embayments H. M. Schuttelaars 1, 2, G. P. Schramkowski 1, 3 and H. E. de Swart 1 “Finite amplitude behaviour of large scale alternating bars can be understood and modelled”
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
Model setup • idealised model • straight channel • only bed erodible • depth-averaged SW eqns • suspended load transport • uniform M 2 tidal forcing, velocity scale ~ 1 m/s Length scales << channel length, tidal wavelength Typically: • channel width B • horizontal tidal excursion length ~ 7 km
Model approach 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 = 2 p ~ 60 km kc • B ~ 5 km. • N, M: truncation numbers kc Growth curves
Model approach 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 Growth curves m=1, n=1 0 0 B Lc 0 m=1, n=2 m=2, n=1 Lc
Model approach 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 describing the behaviour of Amn(t): • steady state solutions • cyclic behaviour
Example: channel width ~ 3. 5 km. • multiple steady state solns: • trivial soln. • nontrivial soln. • no steady equilibrium soln for r/s. H > 0. 0213 periodic soln.
Steady state solution (r~0. 0213) Periodic solution (r~0. 0214)
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
Conclusions • 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
- Slides: 10