(τ m - τ c )

*qB *= α

*U* ,

(6)

ρ*g*

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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