Metocean Extreme and Operating Conditions 3.2 Overview of WWC Processes 51
Part A | 3.2
be excited by long period variations in wind speeds.
Knowledge of the wind spectrum is required in order
to calculate the response. Again, the standard engineering wind spectrum is that given by NORSOK [3.4]. It
is
S.f ; z/ D
.320 m
2 s
1
/.U.10/=U ref /
2
.z=z ref /
0:45
1 C Q
f 0:468
3:561
;
(3.17)
where
Q
f D .172 s/f
 z
z ref
à 2=3  U.10/
U ref
à 0:75
:
(3.18)
The reference elevation above the mean sea surface z ref
is 10 m, and the reference wind speed U ref is 10 m s
1 .
The drag of wind on the sea surface produces waves
and currents, so accurate knowledge of the drag as
a function of wind speed is important for modeling
waves and currents. The drag coefficient depends on
atmospheric stability, but in the high winds that interest us, the equations for neutral conditions usually
apply. The wind stress is equal to u
2
o where u
is the so-called friction velocity given by (3.6) and o
is the air density. The friction velocity is dependent
on the drag coefficient, C d . There are many formulations for C d but one of the more popular is from Large
and Pond [3.8] as shown in (3.8) and (3.9). For years,
many metocean experts used (3.9) well above the maximum 25 m s
1 suggested by Large and Pond [3.8],
but more recent hurricane measurements by Powell
et al. [3.10] showed that the drag coefficient starts to
level off around 30 m s
1 . They conjecture that high
wind speeds create a layer of sea foam and bubbles at
the sea surface thus dropping the effective roughness
of the sea. This reasoning was supported by the laboratory experiments by Donelan et al. [3.11]. Powell [3.12]
provided additional support from field measurements.
Frolov [3.13] showed that capping the drag coefficient
at 0.0022 was essential to model currents measured in
Hurricane Katrina in the Gulf of Mexico.
3.2.2 Waves
Waves grow because of the input of momentum from
the wind, but knowledge of the exact mechanism by
which this momentum is transferred has remained
elusive. The fundamental mechanism, first proposed
by Miles [3.14], seems to be a resonance interaction
between wave-induced pressure fluctuations and the
waves. As the waves propagate, they are modified by
nonlinear interactions between different frequencies,
frictional dissipation and wave breaking. A fuller discussion of wave generation and modeling is given in
Sect. 3.4.
Ocean waves are a complex and irregular function
of space and time. This complexity is best understood
by considering the sea surface to be the superposition
of many cosine waves, as shown in Fig. 3.1. Each of the
cosine waves is characterized by a period T and an amplitude a. The height of a cosine wave H D 2a. Later we
will see that this relation is not true for real waves. The
wave frequency f D 1=T is the inverse of the wave period. The wave length L between two crests is given by
L D gT
2
=2 in deep water. The phase speed or celerity is given by c D L=T. A more detailed discussion of
wave kinematics and dynamics is given in Chap. 2.
A wave record measured at a point can be analyzed
into its component cosine waves using the Fourier transform. This transform gives the amplitude and phase of
each component. It includes all of the information and
irregularity of the original record. This is too much
detail for most purposes because an individual wave
record is a single realization of a random process. We
would usually prefer to know the distribution of wave
energy with frequency in the underlying process. If F.f /
is the Fourier transform of the wave record, its power
Surface elavation, η (m)
T = 3.76 s, H = 0.91 m
Superposed waves
1
0
–1
1
0
–1
T = 5.04 s, H = 1.83 m
1
0
–1
T = 6.63 s, H = 2.17 m
1
0
–1
T = 8.69 s, H = 4.18 m
T = 13.03 s, H = 2.38 m
2
1
0
–1
–2
0
1 0
2 0
3 0
4 0
5 0
t (s)
5
4
3
2
1
0
–1
–2
–3
–4
Fig. 3.1 Superposition of cosine waves to make regular waves
Part A | 3.2
be excited by long period variations in wind speeds.
Knowledge of the wind spectrum is required in order
to calculate the response. Again, the standard engineering wind spectrum is that given by NORSOK [3.4]. It
is
S.f ; z/ D
.320 m
2 s
1
/.U.10/=U ref /
2
.z=z ref /
0:45
1 C Q
f 0:468
3:561
;
(3.17)
where
Q
f D .172 s/f
 z
z ref
à 2=3  U.10/
U ref
à 0:75
:
(3.18)
The reference elevation above the mean sea surface z ref
is 10 m, and the reference wind speed U ref is 10 m s
1 .
The drag of wind on the sea surface produces waves
and currents, so accurate knowledge of the drag as
a function of wind speed is important for modeling
waves and currents. The drag coefficient depends on
atmospheric stability, but in the high winds that interest us, the equations for neutral conditions usually
apply. The wind stress is equal to u
2
o where u
is the so-called friction velocity given by (3.6) and o
is the air density. The friction velocity is dependent
on the drag coefficient, C d . There are many formulations for C d but one of the more popular is from Large
and Pond [3.8] as shown in (3.8) and (3.9). For years,
many metocean experts used (3.9) well above the maximum 25 m s
1 suggested by Large and Pond [3.8],
but more recent hurricane measurements by Powell
et al. [3.10] showed that the drag coefficient starts to
level off around 30 m s
1 . They conjecture that high
wind speeds create a layer of sea foam and bubbles at
the sea surface thus dropping the effective roughness
of the sea. This reasoning was supported by the laboratory experiments by Donelan et al. [3.11]. Powell [3.12]
provided additional support from field measurements.
Frolov [3.13] showed that capping the drag coefficient
at 0.0022 was essential to model currents measured in
Hurricane Katrina in the Gulf of Mexico.
3.2.2 Waves
Waves grow because of the input of momentum from
the wind, but knowledge of the exact mechanism by
which this momentum is transferred has remained
elusive. The fundamental mechanism, first proposed
by Miles [3.14], seems to be a resonance interaction
between wave-induced pressure fluctuations and the
waves. As the waves propagate, they are modified by
nonlinear interactions between different frequencies,
frictional dissipation and wave breaking. A fuller discussion of wave generation and modeling is given in
Sect. 3.4.
Ocean waves are a complex and irregular function
of space and time. This complexity is best understood
by considering the sea surface to be the superposition
of many cosine waves, as shown in Fig. 3.1. Each of the
cosine waves is characterized by a period T and an amplitude a. The height of a cosine wave H D 2a. Later we
will see that this relation is not true for real waves. The
wave frequency f D 1=T is the inverse of the wave period. The wave length L between two crests is given by
L D gT
2
=2 in deep water. The phase speed or celerity is given by c D L=T. A more detailed discussion of
wave kinematics and dynamics is given in Chap. 2.
A wave record measured at a point can be analyzed
into its component cosine waves using the Fourier transform. This transform gives the amplitude and phase of
each component. It includes all of the information and
irregularity of the original record. This is too much
detail for most purposes because an individual wave
record is a single realization of a random process. We
would usually prefer to know the distribution of wave
energy with frequency in the underlying process. If F.f /
is the Fourier transform of the wave record, its power
Surface elavation, η (m)
T = 3.76 s, H = 0.91 m
Superposed waves
1
0
–1
1
0
–1
T = 5.04 s, H = 1.83 m
1
0
–1
T = 6.63 s, H = 2.17 m
1
0
–1
T = 8.69 s, H = 4.18 m
T = 13.03 s, H = 2.38 m
2
1
0
–1
–2
0
1 0
2 0
3 0
4 0
5 0
t (s)
5
4
3
2
1
0
–1
–2
–3
–4
Fig. 3.1 Superposition of cosine waves to make regular waves