In Cascade, it is assumed that the local shoreline evolves with respect to the regional
shoreline, yielding the following transport equation:
Q = Qo sin 2 ( αbr + αb - arctan ( ∂y / ∂x ))
(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 yr 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).
Breaking Wave Properties
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,
H o2Cgo cos θo = H b2Cgb cos θb
(7)
sin θo sin θb
=
(8)
Co
Cb
where H = wave height, Cg = 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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