In Cascade, it is assumed that the local shoreline evolves with respect to the regional

shoreline, yielding the following transport equation:

(5)

determined, *y *can be obtained directly by solving Eq. 5 in combination with the sand

volume conservation equation,

∂*Q*

∂*y*

+ *D * = *q*( *x*, *t *)

(6)

∂*x*

∂*t*

where *D *= depth of transport, *t *= time, and *q *= a source (sink) term varying in time and

space. The regional trend *y*r may be determined trough spatial filtering using a window

size appropriate for the features that should be resolved.

In Cascade, sediment may be added or taken away from the coastal area through

sources or sinks, respectively, creating a shoreline response. Common sediment sources

are: cliff erosion, dune erosion, beach nourishment, wind-blown sand, river-transported

sediment, and onshore transport of material from deeper water. Common sediment sinks

are: wind-blown sand, dredging, barrier-island wash-over, and offshore transport of

material to deeper water. Thus, certain processes may act both as a source and a sink for

the shoreline depending on the particular conditions. Four different types of boundary

conditions have been formulated for Cascade: (1) no transport (*Q*=0), (2) no shoreline

change (∂*Q*/∂*x*=0), (3) bypassing, and (4) bypassing and inlet sediment storage and transfer

(for the two latter boundary conditions, *Q *is derived in submodels, discussed below).

The transport rate *Q *must be calculated at a large number of points in space and for

many time steps. Thus, the wave properties at the break point are computed a large number

of times, and it is of great value to have an efficient algorithm to do this. Assuming input

wave conditions in deep water, the wave properties at breaking are obtained by

simultaneously solving the energy flux conservation equation and Snell's law, both

equations taken from deep water to the break point. The two equations are written,

(7)

sin θo sin θb

=

(8)

where *H *= wave height, *C*g = group speed, *C *= phase speed, θ = wave angle (with respect

to the local shoreline orientation), and *o *and *b *denote deep water and the break point,

respectively. The two equations are coupled and are solved through an iterative procedure.

Introducing expressions for the various wave quantities valid for deep and shallow water,

and substituting the unknown angle from Snell's law into the energy flux conservation

Larson, Kraus, and Hanson

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