Part C | 27.1
600 Part C Coastal Design
ing. Note that outside the breaker zone, where the lefthand side of the above equation is zero, the energy flux
method will reduce to a linear shoaling model. This
wave transformation model has been adopted by a number of large-scale morphology change simulators, which
require a good estimate of the breaker location and resulting longshore currents for large horizontal areas.
However, this approach has been largely displaced by
more recent numerical tools, which can better include
directionality, dissipation, and nonlinear effects. Some
of these tools are discussed in the following sections.
27.1.2 Spectral Modeling: Phase-Averaged
and Phase-Resolving Approaches
In the open ocean, wind wave generation and propagation are typically described using phase-averaged
spectral models. These spectral models determine spatial and temporal changes in the wave spectrum, or
the wave energy as a function of frequency and direction. A spectral energy balance is derived, accounting
for wave growth, propagation, and dissipation based
on wind energy input, whitecapping, and bottom friction. Examples of such models are WISWAVE (wave
information study wave model) [27.8], WAVEWATCH
III [27.9], and WAM (wave model) [27.10]. These models are highly developed for open ocean waves, but do
not account properly for coastal effects such as shallow water wave–wave interactions and depth-induced
breaking [27.11]. They output a directional spectrum,
which can then be employed as boundary conditions
in a coastal zone model to simulate nearshore propagation. For example, WAM could be coupled with SWAN
(simulating waves nearshore) [27.12], a coastal spectral
model, to estimate the spectral evolution from deep to
shallow water [27.11]. However, even coastal spectral
models only crudely approximate dynamic nearshore
phenomena, due to the inherent approximations such
as phase-averaging, weak nonlinear effects, and no
diffraction.
In its most general form, the spectral wave action
density balance equation is given by
D w .N/
Dt
D
S
!
;
where N is the time-dependent wave action density, or
energy density divided by frequency, defined for an individual frequency; the total derivative D w .N/=Dt is
taken in the direction of wave motion, and S represents
any external source or sink acting on the conservation volume for which the total derivative is taken. S
by definition includes forcings such as those of wind
and viscous dissipation, such as whitecapping, wave
breaking, and bottom friction. If the nonlinear advection terms are neglected from the total derivative, then
S must also include nonlinear energy transfers among
the various frequencies. Expanding the above balance
equation, and including the terms of coastal importance,
we have
@N
@t
C .linear energy transfers/
D
S bottom friction
!
C
S breaking
!
C
S whitecap
!
S wind input
!
.nonlinear energy transfers/ ;
where the mechanism of each source/sink S term is
given in its subscript, and the nonlinear energy transfer
term represents energy transfer from one frequency to
a different frequency. For example, the bottom friction
sink is most commonly expressed in the Joint North Sea
Wave Project (JONSWAP) form [27.13]
S bottom friction D DC b G .f / E ;
where C b is a bottom friction coefficient dependent on
local wave and bottom conditions, G is the functional
form of the sink and is dependent on the frequency, and
E is the local energy density. G has the physically reasonable form of being large at low frequencies (high
dissipation in shallow water) and approaches zero (no
dissipation) in the deep water region. The breaking dissipation source function is restricted by breaker-index
uncertainties similarly as discussed with the linear modeling in the previous section. However, due to the ability
of these coastal spectral models to include nonlinearity
and approximate dissipation, they have been shown to
be excellent predictors of wave height transformation
for a wide range of physical configurations. It is reasonably stated that, if the user is not interested in detailed
wave-by-wave breaking estimation, runup, infragravity waves, three-dimensional (3-D) flow profiles, and
turbulent dynamics such as rip currents and coastal eddies, then coastal spectral models are the most efficient
and accurate option available. For a further discussion
and a nice review of state-of-the-art spectral modeling
methods, the reader is directed to [27.14]. Figure 27.1
shows a result from the SWAN model [27.12], with an
improved numerical scheme for the geographical propagation terms [27.15], applied to the area surrounding
Santa Rosa and San Miguel Islands in the Southern California Bight. The extreme refraction around the islands
served as a stringent test of the model; the single-peaked
incident wave spectrum (Fig. 27.1c) has split into two
discrete spectra at the location of the measurement station between the two islands.
600 Part C Coastal Design
ing. Note that outside the breaker zone, where the lefthand side of the above equation is zero, the energy flux
method will reduce to a linear shoaling model. This
wave transformation model has been adopted by a number of large-scale morphology change simulators, which
require a good estimate of the breaker location and resulting longshore currents for large horizontal areas.
However, this approach has been largely displaced by
more recent numerical tools, which can better include
directionality, dissipation, and nonlinear effects. Some
of these tools are discussed in the following sections.
27.1.2 Spectral Modeling: Phase-Averaged
and Phase-Resolving Approaches
In the open ocean, wind wave generation and propagation are typically described using phase-averaged
spectral models. These spectral models determine spatial and temporal changes in the wave spectrum, or
the wave energy as a function of frequency and direction. A spectral energy balance is derived, accounting
for wave growth, propagation, and dissipation based
on wind energy input, whitecapping, and bottom friction. Examples of such models are WISWAVE (wave
information study wave model) [27.8], WAVEWATCH
III [27.9], and WAM (wave model) [27.10]. These models are highly developed for open ocean waves, but do
not account properly for coastal effects such as shallow water wave–wave interactions and depth-induced
breaking [27.11]. They output a directional spectrum,
which can then be employed as boundary conditions
in a coastal zone model to simulate nearshore propagation. For example, WAM could be coupled with SWAN
(simulating waves nearshore) [27.12], a coastal spectral
model, to estimate the spectral evolution from deep to
shallow water [27.11]. However, even coastal spectral
models only crudely approximate dynamic nearshore
phenomena, due to the inherent approximations such
as phase-averaging, weak nonlinear effects, and no
diffraction.
In its most general form, the spectral wave action
density balance equation is given by
D w .N/
Dt
D
S
!
;
where N is the time-dependent wave action density, or
energy density divided by frequency, defined for an individual frequency; the total derivative D w .N/=Dt is
taken in the direction of wave motion, and S represents
any external source or sink acting on the conservation volume for which the total derivative is taken. S
by definition includes forcings such as those of wind
and viscous dissipation, such as whitecapping, wave
breaking, and bottom friction. If the nonlinear advection terms are neglected from the total derivative, then
S must also include nonlinear energy transfers among
the various frequencies. Expanding the above balance
equation, and including the terms of coastal importance,
we have
@N
@t
C .linear energy transfers/
D
S bottom friction
!
C
S breaking
!
C
S whitecap
!
S wind input
!
.nonlinear energy transfers/ ;
where the mechanism of each source/sink S term is
given in its subscript, and the nonlinear energy transfer
term represents energy transfer from one frequency to
a different frequency. For example, the bottom friction
sink is most commonly expressed in the Joint North Sea
Wave Project (JONSWAP) form [27.13]
S bottom friction D DC b G .f / E ;
where C b is a bottom friction coefficient dependent on
local wave and bottom conditions, G is the functional
form of the sink and is dependent on the frequency, and
E is the local energy density. G has the physically reasonable form of being large at low frequencies (high
dissipation in shallow water) and approaches zero (no
dissipation) in the deep water region. The breaking dissipation source function is restricted by breaker-index
uncertainties similarly as discussed with the linear modeling in the previous section. However, due to the ability
of these coastal spectral models to include nonlinearity
and approximate dissipation, they have been shown to
be excellent predictors of wave height transformation
for a wide range of physical configurations. It is reasonably stated that, if the user is not interested in detailed
wave-by-wave breaking estimation, runup, infragravity waves, three-dimensional (3-D) flow profiles, and
turbulent dynamics such as rip currents and coastal eddies, then coastal spectral models are the most efficient
and accurate option available. For a further discussion
and a nice review of state-of-the-art spectral modeling
methods, the reader is directed to [27.14]. Figure 27.1
shows a result from the SWAN model [27.12], with an
improved numerical scheme for the geographical propagation terms [27.15], applied to the area surrounding
Santa Rosa and San Miguel Islands in the Southern California Bight. The extreme refraction around the islands
served as a stringent test of the model; the single-peaked
incident wave spectrum (Fig. 27.1c) has split into two
discrete spectra at the location of the measurement station between the two islands.