Dally and Osiecki (1994) generalized the wave energy balance equation
for the roller introduced by Dally and Brown (1995) to obliquely incident
waves,
d 1
A
A
PD +
ρRC 2 cos2 α = ρR gβ D
(31)
dx 2
T
T
where
PD = loss in organized wave energy flux through wave
breaking (obtained from Equation 13)
ρR = density of the roller
C = roller speed (taken to be proportional to the wave speed,
that is, C = βRCr, where βR is a coefficient)
α = wave angle
A
roller cross-sectional area,
T = wave period
βD = dissipation coefficient (about 0.1)
By defining the period-averaged mass flux (mR = ρrA/T), Equation 31 can be
solved conveniently for this quantity yielding:
d 1
2
PD +
mRCr cos α = gβ D mR
2
(32)
dx 2
where βR = 1.0 was assumed, and T = Tr is employed in the definition of mR.
The momentum flux in the roller is then obtained as MR = mRCr in the
direction of wave propagation. The additional terms in the longshore and
cross-shore momentum equations due to the roller are MRl = mRCr
sin (α) cos (α) and MRc = mRCr cos2(α), respectively, bearing in mind that
these are tensor quantities as are the radiation stresses.
Here it is assumed that Equation 32 can describe the transfer of energy
from the organized wave motion to the roller and the eventual dissipation
also for a situation where a current is present. However, the equation should
be solved by inserting the relative wave properties. It is not obvious that the
dissipation coefficient would be the same if a current is present, but this
assumption will be made here. The roller model proposed by Dally and
Brown (1995) was implemented in NMLong-CW, and test simulations were
carried out to assess the functioning of the roller model on the computed
mean water level and longshore current.
Numerical Implementation
The numerical implementation to calculate the cross-shore wave height
distribution in NMLong-CW follows that of Kraus and Larson (1991), who
employed an explicit finite-difference solution scheme for a staggered grid.
The discretization of the wave action flux conservation equation followed the
approach in NMLong of discretizing the wave energy flux conservation
equation. Calculations start from the most seaward grid point, where the
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Chapter 3 Wave Model
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