GUIDE WAVE ANALYSIS AND FORECASTING - WMO

The models may differ in several respects, e.g.: the representation
of the spectrum, the assumed forms of S in
and S ds, the representation of S nl and whether the integration
is carried out in natural characteristic coordinates
along individual rays or in terms of a discretized advection
operator in a grid-point system common to all wave
components.
The most difficult term to model is the non-linear
source term, S nl, and it is in its specification that the
differences between the categories are to be found.
In the energy-balance Equation 5.1, the interactive
term S nl couples the components. Models based on discrete
spectral components with a non-linear term which
is formulated in terms of several (if not all) components
are called coupled discrete (CD) models. In such models
estimates of all components are needed just to be able to
compute the evolution of any one component.
Computations for these models are often time
consuming, and some modellers prefer to dispense with
the coupling term, and include the weak non-linear interactions
implicitly in their formulation for S in + S ds. Such
models are decoupled propagation (DP) models. Each
component can then be calculated independently.
Advanced models in this class may include a simple
parametric form for S nl, but they are nevertheless distinguished
by the pre-eminence of S in and S ds in the
source term.
The third type of model uses the evidence that
spectra of growing seas are shaped by the non-linear
interactions to conform to a self-similar spectrum (e.g.
JONSWAP — see Section 1.3.9). The spectral shape is
characterized by some small number of parameters and
the energy-balance equation can then be written in these
terms. This gives an evolution equation for each of a
small number of parameters rather than one for each of a
large number of components. However, this parametric
representation is only valid for the self-similar form of
the wind-sea spectrum, and waves outside the generating
area (swell) require special treatment. This is usually
achieved by interfacing the parametric model for wind
sea with a decoupled propagation model for “swell”
through a set of algorithms by which wind-sea energy
and swell energy are interchanged, hence the naming of
this class as coupled hybrid (CH) models.
Table 6.2 in Chapter 6 gives details and references
for a variety of numerical wave models. A thorough
description and discussion of the model classes can be
found in SWAMP Group (1985).
5.5.1 Decoupled propagation (DP) models
Models of this class generally represent the wave
spectrum as a two-dimensional discretized array of
frequency-direction cells in which each cell or component
propagates at its appropriate group velocity along
its own ray path. The components are grown according
to a source function of the form
S = A + BE(f,θ) .
INTRODUCTION TO NUMERICAL WAVE MODELLING 63
As non-linear energy transfer is basically neglected, the
factors A and B are usually empirically determined.
Each component is grown independently of all the
other components up to a saturation limit, which is also
independent of the other spectral components and is
represented by a universal equilibrium distribution. If
non-linear coupling is considered at all, it is parameterized
in a simple way, e.g. by one or two spectral
parameters. The saturation limit may be given by the
energy of a fully developed sea, often represented by the
Pierson-Moskowitz spectrum (see Section 1.3.9). Let the
fully developed sea spectrum be given by E∞ . A modification
of Sin may then appear as
E
S =
t
⎛ E ⎞
E
= A ⎜ ⎟ BE
⎝ E ⎠
E
+
δ
δ
⎡
2 ⎤
2 ⎡
⎢
⎥
⎛ ⎞ ⎤
1– ⎢1–
⎢
⎥ ⎜ ⎟ ⎥
∞ ⎝ ⎠
⎣
⎦ ⎣
⎢ ∞ ⎦
⎥
(Pierson et al., 1966; Lazanoff and Stevenson, 1975), or
E
E
S = = ( A + BE)
t
E
⎛
δ
⎞
⎜1–
⎟
δ
⎝ ⎠
(Ewing, 1971). It is also possible to use the Phillips’
saturation range,
2
αg
– 5
E
4 f
2π
∞ = ( )
as the saturation limit (Cavaleri and Rizzoli, 1981).
The introduction of a saturation limit also works as
an implicit representation of wave-energy dissipation,
except for dissipation due to bottom friction and dissipation
of swell. None of these effects is specific to DP
models and may vary from model to model.
For strictly decoupled models, and for only weakly
coupled models, the differential time and space scales dt
and ds are related through the group velocity c g for a
wave component, ds = c gdt. From this it follows that for
DP models the laws for fetch-limited waves under
uniform stationary wind conditions are immediately
translated into the corresponding duration-limited
growth laws by replacing the fetch, X, with c gt for each
wave component.
Another feature of DP models that can be traced to
the decoupling of wave components is that the spectrum
generally develops a finer structure both in frequency
and direction than coupled models, which continually
redistribute energy and smooth the spectrum.
5.5.2 Coupled hybrid (CH) models
The independent evolution of individual wave components
is effectively prevented by the non-linear energy
transfer. Unless the wind field is strongly non-uniform,
the non-linear transfer is sufficiently rapid relative
to advection and other source functions that a
∞
2