GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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Here, σ is used to denote the particular frequency associated
with wavenumber, k, and the property of the
medium, Ψ, which in this context will be the bottom
depth and/or currents. Characteristic curves are then
obtained by integration of
dx
δσ
= c
(5.6)
g =
dt
δk
and in an ocean with steady currents these curves need
only be obtained once. Examples including refraction
due to bottom topography are shown in Figures 5.2 and
7.1. For more details on ray theory see LeBlond and
Mysak (1978).
Thus, starting from the required point of interest,
rays or characteristics are calculated to the boundary of
the area considered necessary to obtain reliable wave
energy at the selected point. Since we are considering
the history of a particular wave frequency, our reference
frame moves with the component and all we need
consider is the source function along the rays, i.e.
δE
δt
= S.
(5.7)
Rays are calculated according to the required directional
resolution at the point of interest; along each ray,
Equation 5.7 may be solved either for each frequency
separately or for the total energy. In the former approach
S nl is not considered at all. In the latter case interactions
in frequency domain are included but the directions are
uncoupled.
The ray approach has been extensively used
in models where wind sea and swell are treated separately.
In such cases, swell is propagated along
the rays subjected only to frictional damping and
geometric spread. Interactions with the wind sea may
take place where the peak frequency of the Pierson-
Moskowitz spectrum (≈ 0.13g/U 10) is less than the swell
frequency.
5.4.6 Directional relaxation and wind-sea/swell
interaction
Many of the differences between numerical wave models
result from the way in which they cater for the weakly nonlinear
wave-wave interactions (S nl). The differences are
particularly noticeable in the case of non-homogeneous
and/or non-stationary wind fields. When the wind direction
changes, existing wind sea becomes partly swell and a new
wind sea develops. The time evolution of these components
results in a relaxation of the wave field towards a new
steady state that eventually approaches a fully developed
sea in the new wind direction.
Three mechanisms contribute to the directional
relaxation:
(a) Energy input by the wind to the new wind sea;
(b) Attenuation of the swell; and
(c) Weak non-linear interactions, resulting in energy
transfer from swell to wind sea.
GUIDE TO WAVE ANALYSIS AND FORECASTING
The way these mechanisms are modelled may yield
significant deviations between models. The third mechanism
appears to be dominant in this respect.
5.4.7 Depth
Water depth can considerably affect the properties of
waves and how we model them. We know that waves
feel the sea-bed and are changed significantly by it at
depths less than about one-quarter of the deep-water
wavelength (see also Section 1.2.5). In a sea with a wide
spectrum, the longer waves may be influenced by the
depth without much effect on the short waves.
One major effect of depth is on the propagation
characteristics. Waves are slowed down and, if the seabed
is not flat, may be refracted. Also, the non-linear
interactions tend to be enhanced and, of course, there are
more dissipative processes involved through interaction
with the sea-bed. The framework we are describing for
wave modelling is broad enough in its concept to be able
to cope with depth-related effects without drastic alterations
to the form of the model outlined in Figure 5.1. In
Chapter 7 the effect of shallow water will be discussed
in more detail.
5.4.8 Effects of boundaries, coastlines and
islands
With the exception of global models, most existing wave
models have an open ocean boundary. Wave energy may
then enter the modelled area. The best solution is to
obtain boundary data from a model operating over a
larger area, e.g. a global model. If there is no knowledge
of the wave energy entering the model area, a possible
boundary condition is to let the energy be zero at the
boundaries at all times. Another solution may be to
specify zero flux of energy through the boundary. In
either case it will be difficult to get a true representation
of distantly generated swell. The area should therefore
be sufficiently large to catch all significant swell that
affects the region of interest.
In operational models, with grid resolution in the
range 25–400 km, it is difficult to get a true representation
of coastlines and islands. A coarse resolution will
strongly affect the shadowing effects of islands and
capes. To obtain a faithful representation of the sea state
near such a feature we need to take special precautions.
One solution may be to use a finer grid for certain areas,
a so-called “nested” model, where results from the
coarse grid are used as boundary input to the fine grid. It
may also be necessary to increase the directional resolution
so as to model limited depth and shadowing effects
better. Another way may be to evaluate the effects of the
topographic feature for affected wave directions at a
certain number of grid points and tabulate these as
“fudge factors” in the model.
5.5 Model classes
Wave models compute the wave spectrum by numerical
integration of Equation 5.1 over a geographical region.