(τ m - τ c )
qB = α
U ,
(6)
ρg
in which α = empirical coefficient of order unity, τ = c ρ U U is the time-
m
f
averaged bottom shear stress, τ c is the critical shear for sediment motion, and
ρ is the density of water. In the presence of waves, the quadratic dependence
of the shear stress τ m is expressed by the Nishimura (1988) approximation.
Field observation indicates that the rapidly flowing water through a breach
will remove material both by direct shear on the sides of the opening and by
notching of the side, causing collapse of the material above the notch. This
complex process is simply represented as a fraction of the total transport at the
bottom as,
,
qS = β qB
(7)
where the value β = 0.7 is typically assigned.
2.3. Numerical Solution Method
Eqs. (3) and (4) are solved by a trapezoidal finite-difference method, and for
field conditions a time step of 10 sec gave good results. Longer time steps, for
example, 60 sec, are possible. However, in some situations, physically
generated transients can be generated following rapid changes in water level and
opening of a breach. A 10-sec time step was found to control these transients.
After the velocity is obtained at time step n, transport rates are calculated and
substituted into explicit finite-difference forms of Eqs. (1) and (2). The solution
then proceeds forward.
2.4. Layered Barrier Island
The original model (Kraus 2003) developed solutions for a rectangular barrier
island (Fig. 1). Although a rectangle is a reasonable first approximation, barrier
islands, especially those prone to breaching, have a pyramidal or curved cross
section. In a numerical solution, such a shape can be represented by a series of
stacked rectangles to give a layered barrier island shape (Fig. 2). As the breach
deepens, new layers are opened in the model, giving a new length L and surface
area on the sides for calculating sediment transport. A similar extension of the
model allows representation of island width to represent a common feature of
narrowing of a barrier island where it may be more vulnerable to breaching.
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