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”

Aim of this talk: to model and understand the observed dynamical behaviour of large scale alternating bars

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)

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

Model approachUse a finite number of spatial patterns obtained from a linear stabilityanalysis to describe the finite amplitude bed behaviour:

h’ = Amn(t) cos (m kcx) cos (ny/B)m=0

M

n=0

N

Growth curves

kc

• kc: wavenr. of critical mode channel length c

c= ~ 60 km

• B ~ 5 km.• N,M: truncation numbers

kc

Model approachUse a finite number of spatial patterns obtained from a linear stabilityanalysis to describe the finite amplitude bed behaviour:

h’ = Amn(t) cos (m kcx) cos (ny/B)m=0

M

n=0

N

Growth curves

m=1,n=1

m=1,n=2

m=2,n=1

0

00

00

LcB

B

B Lc

Lc

Model approachUse a finite number of spatial patterns obtained from a linear stabilityanalysis to describe the finite amplitude bed behaviour:

h’ = Amn(t) cos (m kcx) cos (ny/B)m=0

M

n=0

N

Insert expansion in complete nonlinear equations.

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/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