GUIDE WAVE ANALYSIS AND FORECASTING - WMO

only once, since it is usual to store this hindcast and
progressively update it as part of each model run.
In some ocean areas in the northern hemisphere,
there is sufficient density of observations to make a
direct field analysis of the parameters observed (such as
significant wave height or period). Most of the available
data are those provided visually from ships; these data
are of variable quality (see Chapter 8). Such fields are
not wholly satisfactory for the initialization of computer
models. Most computer models use spectral representations
of the wave field, and it is difficult to reconstruct a
full spectral distribution from a height, period and direction.
However, the possibility of good quality wave data
from satellite-borne sensors with good ocean coverage,
has prompted attempts to find methods of assimilating
these data into wave models. The first tested methods
have been those using significant wave heights measured
by radar altimeters on the GEOSAT and ERS-1 satellites.
It has been shown that a positive impact on the
wave model results can be achieved by assimilating
these data into the wave models (e.g. see Lionello et al.,
1992, and Breivik and Reistad, 1992). Methods to assimilate
spectral information, e.g. wave spectra derived from
SAR (Synthetic Aperture Radar) images, are also under
development. More details on wave data assimilation
projects are given in Section 8.6.
5.4.2 Wind
Perhaps the most important element in wave modelling
is the motion of the atmosphere above the sea surface.
The only input of energy to the sea surface over the
time-scales we are considering comes from the wind.
Transfer of energy to the wave field is achieved through
the surface stress applied by the wind, which varies
roughly as the square of the wind speed. Thus, an error
in wind specification can lead to a large error in the wave
energy and subsequently in parameters such as significant
wave height.
The atmosphere has a complex interaction with the
wave field, with mean and gust wind speeds, wind
profile, atmospheric stability, influence of the waves
themselves on the atmospheric boundary layer, etc., all
needing consideration. In Chapter 2, the specification of
the winds required for wave modelling is discussed.
For a computer model, a wind history or prognosis
is given by supplying the wind field at a series of timesteps
from an atmospheric model. This takes care of the
problem of wind duration. Similarly, considerations of
fetch are taken care of both by the wind field specification
and by the boundary configuration used in the
propagation scheme. The forecaster using manual
methods must make his own assessment of fetch and
duration.
5.4.3 Input and dissipation
The atmospheric boundary layer is not completely independent
of the wave field. In fact, the input to the wave
field is dominated by a feedback mechanism which
INTRODUCTION TO NUMERICAL WAVE MODELLING 59
depends on the energy in the wave field. The rate at
which energy is fed into the wave field is designated by
Sin. This wind input term, Sin, is generally accepted as
having the form:
Sin = A(f,θ) + B(f,θ) E(f,θ) (5.2)
A(f,θ) is the resonant interaction between waves and
turbulent pressure patterns in the air suggested by
Phillips (1957), whereas the second term on the righthand
side represents the feedback between growing
waves and induced turbulent pressure patterns as
suggested by Miles (1957). In most applications, the
Miles-type term rapidly exceeds the Phillips-type term.
According to Snyder et al. (1981), the Miles term
has the form:
U
Bf,
max , f cos – – f
g θ
⎡ ρ ⎛
⎞ ⎤
a 5
( )= ⎢0
K1 ⎜K2
( θ ψ) 1⎟2π ⎥
⎣ ρw
⎝
⎠ ⎦
(5.3)
where ρ a and ρ w are the densities of air and water,
respectively; K 1 and K 2 are constants; ψ is the direction
of the wind; and U 5 is the wind speed at 5 m (see also
Section 3.2).
Equation 5.3 may be redefined in terms of the friction
speed u * = √(τ/ρ a), where τ is the magnitude of the
wind shear stress. From a physical point of view, scaling
wave growth to u * would be preferable to scaling with
wind speed U z at level z. Komen et al. (1984) have
approximated such a scaling, as illustrated in Equation
3.1, however, lack of wind stress data has precluded
rigorous attempts. U z and u * do not appear to be linearly
related and the drag coefficient, C d, used to determine τ
(τ = ρ aC dU z 2 ), appears to be an increasing function of Uz
(e.g. Wu, l982; Large and Pond, 1981). The scaling is an
important part of wave modelling but far from resolved.
Note that C d also depends on z (from U z) (see for
example Equation 2.14). Recent advances with a quasilinear
theory which includes the effects of growing
waves on the mean air flow have enabled further refinement
of the formulation (Janssen, 1991; Jenkins, 1992;
Komen et al., 1994).
Note also that in the case of a fully developed sea,
as given by the Pierson-Moskowitz spectrum E PM (see
Equation 1.28), it is generally accepted that the dimensionless
energy, ε,
∫
ε = g2 EPM f
4 u *
( ) df
= g2 E total
u * 4
is a universal constant. However, if ε is scaled in terms
of U 10 this saturation limit will vary significantly with
wind speed since C d is a function of U 10. Neither
Equation 5.3 nor the dependence of C d on U z are well
documented in strong wind.
The term S ds describes the rate at which energy is
lost from the wave field. In deep water, this is mainly
through wave breaking (whitecapping). In shallow water