GUIDE WAVE ANALYSIS AND FORECASTING - WMO

64
quasi-equilibrium spectral distribution is established.
The distributions appear to be of the same shape for a
wide variety of generation conditions and differ only
with respect to the energy and frequency scales. The
quasi-self-similarity was confirmed theoretically by
Hasselmann et al. (1973 and 1976).
Further, a universal relationship appears to exist
between the non-dimensional total energy and frequency
parameters, ε and ν, respectively. The non-dimensional
scalings incorporate g and some wind-speed measure,
e.g. the wind speed at 10 m, U 10, or the friction speed,
u *. Hence ε = E g 2 /u 4 , and ν p = f pg/u, where u = U 10
or u *, and E is the total energy (from the integrated
spectrum).
Since the evolution of the developing wind-sea
spectrum is so strongly controlled by the shape stabilizing
non-linear transfer, it appears reasonable to express
the growth of the wind-sea spectrum in terms of one or a
few parameters, for example ε, the non-dimensional
wave energy. In such a one-parameter, first-order representation,
all other non-dimensional variables (e.g. ν p,
the non-dimensional peak frequency) are uniquely determined,
and hence diagnosed.
Thus, in one extreme the parametric model may
prognose as few as one parameter (e.g. the total spectral
energy), the wind-sea spectrum being diagnosed from
that. For such a model the evolution equation is obtained
by integrating Equation 5.1 over all frequencies and
directions:
δE
δt +∇•cg E ( ) = SE in which c – g is the effective propagation velocity of the
total energy
∫ cg E( f,θ)df
dθ
cg =
∫ E( f,θ)
and S E is the projection of the net source function S onto
the parameter E
SE ∫ S f,θ df dθ
= ( )
c – g is uniquely determined in terms of E by a prescribed
spectral shape and S E must be described as a function of
E and U 10 or u *. This function is usually determined
empirically.
If additional parameters are introduced, e.g. the
peak frequency, f p, the Phillips’ parameter, α, or the
mean propagation direction, – θ, the growth of the windsea
spectrum is expressed by a small set of coupled
transport equations, one for each parameter. A general
method for projecting the transport equation in the
complete (f,θ) representation on to an approximate
parameter space representation is given in Hasselmann et
al. (1976).
In slowly varying and weakly non-uniform wind,
parametric wave models appear to give qualitatively the
same results. The more parameters used, the more varied
are the spectral shapes obtained and, in particular, if the
GUIDE TO WAVE ANALYSIS AND FORECASTING
mean wave direction, – θ, is used, directional lag effects
become noticeable in rapidly turning winds.
The fetch-duration relation for a parametric wave
model will differ from that of a DP model in that a mean
propagation speed takes the place of the group speed for
each frequency band, i.e.
X = Ac – gt ,
where: X = the fetch, t = duration and A is a constant
(typically A = 2/3). Thus, it is not possible to tune the
two types of model for both fetch- and duration-limited
cases.
Once the non-linear energy transfer ceases to
dominate the evolution of the wave spectrum, the parametric
representation breaks down. This is the case for
the low-frequency part of the wave spectrum that is no
longer actively generated by the wind, i.e. the swell part.
The evolution of swell is controlled primarily by advection
and perhaps some weak damping. It is therefore
represented in parametric wave models in the framework
of discrete decoupled propagation. The combination of a
parametric wind-sea model and a decoupled propagation
swell model is termed a coupled hybrid model.
CH models may be expected to encounter problems
when sea and swell interact. Typical transition regimes
arise:
• In decreasing wind speed or when the wind direction
turns, in which cases wind sea is transformed
to swell;
• When swell enters areas where the wind speed is
sufficiently high that the Pierson-Moskowitz peak
frequency fp = 0.13g/U10 is lower than the swell
frequency, in which case the swell suddenly comes
into the active wave growth regime.
These transitions are modelled very simply in CH
models. For turning winds it is common that the wind
sea loses some energy to swell. The loss may be a
continuous function of the rate of change of wind direction
or take place only when the change is above a
certain angle.
When the wind decreases, the CH models generally
transfer frequency bands that travel faster than the wind
to swell. Some models also transfer the energy that
exceeds the appropriate value for fully developed wind
sea into swell.
Swell may be reabsorbed as wind sea when the
wind increases and the wind-sea peak frequency
becomes equal to or less than the swell frequency. Some
CH models only allow reabsorption if the angle between
wind-sea and swell propagation directions fulfil certain
criteria.
Some models allow swell to propagate unaffected
by local winds to destination points. Interaction takes
place only at the destinations. If wind sea exceeds swell
at a point, the swell is completely destroyed. Thus, the
reabsorption of swell into wind sea is non-conservative.
CH models generally use characteristics or rays to
propagate swell.