A one-equation model is used to describe the production, advection, diffusion,
and dissipation of the turbulent kinetic energy produced by wave breaking:
3/ 2
l ∂u ∂v
2
2
2
k 3/ 2
- uη ⋅ ∇k + σ∇ ⋅ ∇ (νt k ) + B
=
+
- CD
(18)
t
kt
CD ∂z ∂z z =η
lt
The waves are assumed to start breaking when the horizontal component of the
orbital velocity at the free surface, uη, exceeds the phase velocity of the waves,
C. The parameter B is introduced to ensure that production of turbulence occurs
after the waves break, i.e.,
|uη| < C
0
=
(19)
B
|uη| ≥ C
1
The phase velocity is determined from the linear dispersion relation (Equa-
tion 10) using the average zero-crossing period of the incident wave train. While
this approach leads to some waves in an irregular wave train breaking at slightly
different locations in the model than they would in nature, it was found to be
more stable than trying to estimate a time-dependent phase velocity using
The empirical constants CD and σ have been chosen as 0.02 and 0.2 respec-
tively. The turbulent length scale, lt, remains the only free parameter in the turbu-
lence model and is determined from comparisons of numerical model results with
experimental data. Recommended values are the significant wave height (lt =
Hmo) for irregular waves, and the wave height (lt = H) for regular waves.
Bottom Friction
The bottom boundary layer in wave fields is typically confined to a tiny
region above the seabed, unlike river and tidal flows where it extends all the way
up to the free surface. There is, thus, very little wave energy attenuation due to
bottom friction over typical wave propagation distances of O (1km) used in
Boussinesq-type models. The bottom friction factor, however, plays a more
important role in wave transformation close to the shoreline and nearshore circu-
lation patterns. The effect of energy dissipation due to a turbulent boundary layer
at the seabed has been modeled by adding a bottom shear stress term to the right-
hand side of the momentum equation (Equation 5 or 8):
1
=
-
fwuα uα
Fbfriction
(20)
h+η
in terms of the Chezy coefficient, Cf, used in tidal flows by replacing fw with
2
g/Cf .
10
Chapter 2 Theoretical Background
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