where
D = derivative
R = coordinate in the direction of the wave ray
n = coordinate normal to the wave orthogonal
The governing equation for steady-state conservation of spectral wave action
along a wave ray is given by (Jonsson 1990):
∂ C a C ga cos ( - α )E (ω a , α)
S
=Σ
(9)
(C ga ) i
ωr
ωr
∂xi
where
E = wave energy density divided by (ρw g), where ρw is density of
water
S = energy source and sink terms
Refraction and shoaling
Refraction and shoaling are implemented in STWAVE by applying the
conservation of wave action along backward traced wave rays. Rays are traced
in a piecewise manner, from one grid column to the next. The two-dimensional
wave spectra are set as input along the first grid column (the offshore boundary).
For a point on the second grid column, the spectrum is calculated by back tracing
a ray for each frequency and direction component of the spectrum. The ray
direction, μ, is determined by Equation 7. Only ray directions propagating
toward the shore (-87.5 to +87.5 deg) are included. Energy propagating toward
the offshore is neglected.
The wave ray is traced back to the previous grid column, and the length of
the ray segment DR is calculated. Derivatives of depth and current components
normal to the wave orthogonal are estimated (based on the orthogonal direction
at column 2) and substituted into Equation 8 to calculate the wave orthogonal
direction at column 1. Then, the wave number, wave and group celerities, and
ray angle in the previous column are calculated. The energy is calculated as a
weighted average of energy between the two adjacent grid points in the column
and the direction bins. The energy density is corrected by a factor that is the ratio
of the 5-deg standard angle band width to the width of the back-traced band to
account for the different angle increment in the back-traced ray. The shoaled and
refracted wave energy in column 2 is then calculated from the conservation of
wave action along a ray (Equation 9).
In a strong opposing current (e.g., ebb currents at an entrance), waves may be
blocked by the current. Blocking occurs if there is no solution to the dispersion
equation (Equation 2). Or, to state it another way, blocking occurs if the relative
8
Chapter 2 Governing Equations and Numerical Discretization
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