wave group celerity is smaller than the magnitude of the opposing current, so
wave energy cannot propagate against the current. In deep water, blocking
occurs for an opposing current with magnitude greater than one-fourth the
deepwater wave celerity without current (0.25 g Ta/(2π), where Ta is the absolute
wave period). If blocking occurs, the wave energy is dissipated through breaking.
Lai, Long, and Huang (1989) performed laboratory experiments that showed that
wave energy can pass through the linear blocking point through nonlinear energy
transfers to lower frequencies (which are not blocked). These nonlinear energy
transfers are not included in STWAVE.
Diffraction
Diffraction is included in STWAVE in a simple manner through smoothing
of wave energy. The model smoothes energy in a given frequency and direction
band using the following form:
E j (ω a ,α ) = 0.55E j (ω a ,α ) + 0.225[E j+1 (ω a ,α ) + E j-1 (ω a ,α )]
(10)
where E is the energy density in a given frequency and direction band, and the
subscript j indicates the grid row index (alongshore position). Equation 10
provides smoothing of strong gradients in wave height that occur in sheltered
regions, but provides no turning of the waves. This formulation is grid-spacing
dependent, which is a serious weakness. Efforts are ongoing to implement a
more rigorous diffraction representation.
Source/sink terms
Surf-zone wave breaking. The wave-breaking criterion applied in the first
version of STWAVE was a function of the ratio of wave height to water depth:
H momax
= 0.64
(11)
d
where Hmo is the energy-based zero-moment wave height. At a coastal entrance,
where waves steepen because of the wave-current interaction, wave breaking is
enhanced because of the increased steepening. Smith, Resio, and Vincent
(1997) performed laboratory measurements of irregular wave breaking on ebb
currents and found that a breaking relationship in the form of the Miche criterion
(1951) was simple, robust, and accurate:
H momax = 0.1L tanh kd
(12)
(see also Battjes 1982 and Battjes and Janssen 1978). Equation 12 is applied in
version 3.0 of STWAVE as a maximum limit on the zero-moment wave height.
The energy in the spectrum is reduced at each frequency and direction in
proportion to the amount of pre-breaking energy in each frequency and direction
9
Chapter 2 Governing Equations and Numerical Discretization
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