GUIDE WAVE ANALYSIS AND FORECASTING - WMO

3.1 Introduction
This chapter gives an overview of the processes involved
in wave generation and decay. An indication of how
these processes are formulated is given.
Wave forecasting is the process of estimating how
waves evolve as changing wind fields act on the surface
of the ocean. To understand this we need to identify the
processes affecting the energy of the waves. In simple
terms, wave energy at a given location is changed
through advection (rate of energy propagated into and
away from the location), the wave energy gains from the
external environment and wave energy losses due to
dissipation. In wave modelling, the usual approach is to
represent these influences as a wave energy conservation
equation, as presented in Chapter 5 (Equation 5.1), and
then to solve it.
The sources of wave energy (gains and losses) are
identified as three major processes, the external gains
(S in), the dissipative loss (S ds) and the shifting of energy
within the spectrum due to weakly non-linear wavewave
interactions (S nl). A description of these terms as
well as of propagation are presented in this chapter.
3.2 Wind-wave growth
The only input of energy to the sea surface over the timescales
we are considering comes from the wind. Transfer
of energy to the wave field is achieved through the surface
stress applied by the wind and this varies roughly as the
square of the wind speed. Thus, as already noted in
Section 2.1, an error in wind specification can lead to a
large error in the wave energy and subsequently in parameters
such as significant wave height.
After the onset of a wind over a calm ocean there
are thought to be two main stages in the growth of wind
waves. First, the small pressure fluctuations associated
with turbulence in the airflow above the water are sufficient
to induce small perturbations on the sea surface
and to support a subsequent linear growth as the
wavelets move in resonance with the pressure fluctuations.
This mechanism is called the Phillips’ resonance
(see Phillips, 1957). Formulations can be found in
Barnett (1968) and Ewing (1971). However, this mechanism
is only significant early in the growth of waves on
a calm sea.
Most of the development commences when the
wavelets have grown to a sufficient size to start affecting
the flow of air above them. The wind now pushes and
drags the waves with a vigour which depends on the size
of the waves themselves.
CHAPTER 3
WAVE GENERATION AND DECAY
A. K. Magnusson with M. Reistad: editors
This growth is usually explained by what is called a
shear flow instability: the airflow sucking at the crests
and pushing on the troughs (or just forward of them). A
useful theory has been presented by Miles (1957). The
rate of this growth is exponential as it depends on the
existing state of the sea. This is usually described in
terms of the components of the wave energy-density
spectrum (see Section 1.3.7).
From the formulation of Miles (1960):
or
k
2π
ftµ
E f,
θ
P k, f e – 1
2
4πf
ρ µ g
( ) = ( )( )
( ) =
S f θ
in ,
where E(f,θ) is a frequency-direction component, k is the
wavenumber, P(k,f) is the spectrum of wave-induced
turbulence, µ is a coupling coefficient to be defined, g is
gravitational acceleration and ρ w is the water density.
It has been noted that the rates of growth predicted by
Miles are much smaller than observed growth rates
from laboratory and field studies. Based on a field experiment,
Snyder and Cox (1966) proposed a simple form:
µ = ρ a
ρ w
where c and θ are the phase speed and the direction,
respectively, of the component being generated, ψ and u
are the direction and speed of the wind and ρ a is the air
density.
Measurements made in the Bight of Abaco in the
Bahamas in 1974 enabled Snyder et al. (1981) to
propose a revision which can be expressed by:
Sin( f ,θ ) = E( f,θ)
⎡
⎣
max 0,K1 2πf ρa ⎢
ρw The height at which the wind speed was specified in the
original work was 5 m. Since application of this to other
situations can be affected by the structure of the lower
part of the atmospheric boundary layer (see Section 2.4),
it may be better to express the wind input in terms of the
friction velocity u * with magnitude:
u * =
w
( ) = ( )
δE f,
θ
δt
⎡u
cos( θ – ψ
⎤
)–1
⎣⎢ c ⎦⎥ ,
τ
ρ a
2πµ
f E f,
θ ,
= u C d
where τ is the magnitude of the wind stress and C d is the
drag coefficient.
⎛
⎝
U 5
c
cos( θ – ψ
⎞ ⎤
)–1
⎠ ⎥.
⎦