Dally and Osiecki (1994) generalized the wave energy balance equation

for the roller introduced by Dally and Brown (1995) to obliquely incident

waves,

ρRC 2 cos2 α = ρR gβ D

(31)

where

breaking (obtained from Equation 13)

ρR = density of the roller

that is, *C *= βRCr, where βR is a coefficient)

α *= *wave angle

roller cross-sectional area,

βD = dissipation coefficient (about 0.1)

By defining the period-averaged mass flux (*m*R = ρrA/*T*), Equation 31 can be

solved conveniently for this quantity yielding:

2

*m*RCr cos α = *g*β D mR

2

(32)

where βR = 1.0 was assumed, and *T *= *T*r is employed in the definition of *m*R.

The momentum flux in the roller is then obtained as *M*R = *m*RCr in the

direction of wave propagation. The additional terms in the longshore and

cross-shore momentum equations due to the roller are *M*Rl = *m*RCr

sin (α) cos (α) and *M*Rc = *m*RCr 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.

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

22

Chapter 3 Wave Model

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