GUIDE WAVE ANALYSIS AND FORECASTING - WMO

88
dissipation in such a bore can be determined analytically
and, by assuming a certain random distribution
of the wave heights in the surf zone, the total rate of
dissipation can be estimated. This model (e.g. Battjes
and Janssen, 1978) has been very successful in predicting
the decrease of the significant wave height in the
surf zone.
The effect of triad interactions is not as obvious
but, when wave records are analysed to show the spectral
evolution of the waves in the surf zone, it turns out
that, in the case of mild breaking, a second (highfrequency)
peak evolves in the spectrum. In that case
the waves transport energy to higher frequencies with
little associated dissipation. This second peak is not
necessarily the second harmonic of the peak frequency.
In cases with more severe breaking, the transport to
higher frequencies seems to be balanced by dissipation
at those frequencies as no second peak evolves (while
the low frequency part of the spectrum continues to
decay). In either case numerical experiments have
shown that these effects can be obtained by assuming
that the dissipation is proportional to the spectral
energy density itself.
The evolution of the waves in the surf zone is
extremely non-linear and the notion of a spectral model
may seem far-fetched. However, Battjes et al. (1993)
have shown that a spectral triad model (Madsen and
Sorensen, 1993), supplemented with the dissipation
model of Battjes and Janssen (1978) and the assumption
that the dissipation is proportional to the energy
density, does produce excellent results in laboratory
conditions, even in high-intensity breaking conditions
(Figure 7.5). The source term for breaking in the spectral
energy balance, based on Battjes and Janssen
(1978), is then:
S
breaking
( )
2
QbωHmEωθ ,
( ωθ , )= – α
,
8π
E
total
(7.14)
in which α is an empirical coefficient of the order one,
–ω is the mean wave frequency, Q b is the fraction of
breaking waves determined with
1–
Qb
Etotal
= – 8 , 2
(7.15)
lnQb
Hm Hm is the maximum possible wave height (determined as
a fixed fraction of the local depth) and Etotal is the total
wave energy.
The triad interactions indicated above have not (yet)
been cast in a formulation that can be used in a spectral
energy balance. Abreu et al. (1992) have made an attempt
for the non-dispersive part of the spectrum. Another more
general spectral formulation is due to Madsen and
Sorensen (1993) but it requires phase information.
GUIDE TO WAVE ANALYSIS AND FORECASTING
7.8 Currents, set-up and set-down
Waves propagate energy and momentum towards the
coast and the processes of refraction, diffraction, generation
and dissipation cause a horizontal variation in this
transport. This variation is obvious in the variation of the
significant wave height. The corresponding variation in
momentum transport is less obvious. Its main manifestations
are gradients in the mean sea surface and the
generation of wave-driven currents. In deep water these
effects are usually not noticeable, but in shallow water
the effects are larger, particularly in the surf zone. This
notion of spatially varying momentum transport in a
wave field (called “radiation stress”; strictly speaking,
restricted to the horizontal transport of horizontal
momentum) has been introduced by Longuet-Higgins
and Stewart (1962) with related pioneering work by
Dorrestein (1962).
To introduce the subject, consider a normally incident
harmonic wave propagating towards a beach with
straight and parallel depth contours. In this case the
waves induce only a gradient in the mean sea surface
(i.e. no currents):
(7.16)
where h is the still-water depth, – η is the mean surface
elevation above still-water level and Mxx is the shoreward
transport (i.e. in x-direction) of the shoreward component
of horizontal momentum:
Mxx = (2 β – 0.5) E , (7.17)
where E = ρwgH2 /8 and β = 1 dη
1 d Mxx
= – ,
dx
ρwg( h+
η)
dx
/2 + kh/sinh(2kh).
Outside the surf zone the momentum transport
tends to increase slightly with decreasing depth (due to
shoaling). The result is a slight lowering of the mean
water level (set-down). Inside the surf zone the dissipation
is strong and the momentum transport decreases
rapidly with decreasing depth, causing the mean sea
surface to slope upward towards the shore (set-up). The
maximum set-up (near the water line) is typically 15– 20
per cent of the incident RMS wave height, Hrms. In general waves cause not only a shoreward transport
of shoreward momentum Mxx but also a shoreward
transport of long-shore momentum Mxy. Energy dissipation
in the surf zone causes a shoreward decrease of Mxy which
manifests itself as input of long-shore momentum into the
mean flow. That input may be interpreted as a force that
drives a long-shore current. For a numerical model and
observations, reference is made to Visser (1984). In arbitrary
situations the wave-induced set-up and currents can
be calculated with a two-dimensional current model driven
by the wave-induced radiation stress gradients (e.g.
Dingemans et al., 1986).