GUIDE WAVE ANALYSIS AND FORECASTING - WMO

Figure 3.7 —
Structure of spectral energy growth.
The upper curves show the components
S nl, S in, and S ds. The lower
curves are the frequency spectrum
E(f), and the total growth curve S
Energy density (E) Rate of growth (S)
growth at a given frequency is dominated by the nonlinear
wave-wave interaction. As a wind sea develops (or
as we move out along a fetch) the peak frequency
decreases. A given frequency, f e, will first be well below
a peak frequency, resulting in a small amount of growth
from the wind forcing, some non-linear interactions, and
a little dissipation. As the peak becomes lower and
approaches f e, the energy at f e comes under the influence
of a large input from non-linear interactions. This can be
seen in Figures 3.5 and 3.7 in the large positive region of
S or S nl just below the peak. As the peak falls below f e
this input reverses, and an equilibrium is reached
(known as the saturation state). Figure 3.6 illustrates the
development along a fetch of the energy density at such
a given frequency f e.
Although the non-linear theory can be expressed,
as in Equation 3.4, the evaluation is a problem. The
integral in Equation 3.4 requires a great deal of
computer time, and it is not practical to include it in
this form in operational wave models. Some wave
models use the similarity of spectral shape, which is a
manifestation of this process, to derive an algorithm so
that the integral calculation can be bypassed. Having
established the total energy in the wind-sea spectrum,
these models will force it into a pre-defined spectral
shape. Alternatively, it is now possible to use integration
techniques and simplifications which allow a
reasonable approximation to the integral to be evaluated
(see the discrete interaction approximation (DIA)
of Hasselmann and Hasselmann, 1981, Hasselmann
and Hasselmann, 1985, and Hasselmann et. al., 1985;
or the two-scale approximation (TSA) of Resio et al.,
1992). These efficient computations of the non-linear
WAVE GENERATION AND DECAY 41
S n1
S
S in
E (f)
S ds
transfer integral made it possible to develop third
generation wave models which compute the non-linear
source term explicitly without a prescribed shape for
the wind-sea spectrum.
Resonant weakly non-linear wave-wave interactions
are only one facet of the non-linear problem.
When the slopes of the waves become steeper, and the
non-linearities stronger, modellers are forced to resort to
weaker theories and empirical forms to represent
processes such as wave breaking. These aspects have
been mentioned in Section 3.4.
3.6 General notes on application
The overall source term is S = S in + S ds + S nl. Ignoring
the directional characteristics (i.e. looking only at the
frequency dependence), we can construct a diagram for
S such as Figure 3.7. This gives us an idea of the relative
importance of the various processes at different frequencies.
For example, we can see that the non-linear transfer
is the dominant growth agent at frequencies near the
spectral peak. Also, for the mid-frequency range (from
the peak to about twice the peak frequency) the growth
is dominated by the direct input from the atmosphere.
The non-linear term relocates this energy mostly to the
lower frequency range. The dissipation term, so far as is
known, operates primarily on the mid- and highfrequency
ranges.
The development of a frequency spectrum along a
fetch is illustrated in Figure 3.8 with a set of spectra
measured during the JONSWAP experiment. The downshift
in peak frequency and the overshoot effect at each
frequency are evident.
f
f