124
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
value 28U*. However, the problem of determining U* turns out to be rather
complicated. In the original version of The WAM model (1988) it is assumed
that the dynamical velocity can be determined as:
u* = J(f;;uw ;
{
1.2873 . w- 3
Cv= o.8 . w- 3 + o.65 . w- 4
at ulO : : : : ; 7.5 ms- 1 ;
at Uw > 7.5 ms- 1 .
(4.35)
The drawback of this ratio is that it does not take into account the actual state
of the near-water atmospheric boundary layer determined by the sea surface
roughness and atmospheric stratification. In this connection the model wind
wave energy input, proposed by P. Janssen, was used in the next version of
the WAM model (Komen et al., 1994).
Relation of the friction velocity with the wave surface resistance
coefficient and the wave development stage. The experimental results of investigating the dependence of the friction velocity and the wave
surface resistance coefficient on the wave development stage are in Problems
of Research and Mathematical Modeling of Wind Sea (1995); Huang et al.,
(1986); Komen et al., (1994). Thus, using full-scale data analysis (Problems of
Research and Mathematical Modeling of Wind Sea, 1995) it is shown that the
following dependence between the sea roughness z0 and the non-dimensional
frequency of the wind wave spectral maximum for the deep-water case, excluding the first stage of wind wave development, can be written:
z0 = 0.4 a;;,a.x
(4.36)
The parameters are written in non-dimensional form normalized by the
friction velocity in this relation. The relation ( 4.36) indicates a decrease of
sea surface roughness depending on the wave development stage. Thus, at the
initial stage of wave development the waves are very steep and their speed
is relatively low, creating large resistance to airflow and significant surface
roughness. An intense energy and momentum transfer from the wind flow to
waves appears. The reverse influence of waves on the air flow is significant. It
is revealed in the increase in dynamical velocity and in violation of the selfsimilar regime. The waves become flatter, their phase speed is increased while
the effective roughness is decreased with wave development. Thus, the resistance coefficient is decreased. Correspondingly, the energy and momentum
transfer rates to waves are decreased. The wave increase becomes slower.
If the wind speed profile is assumed to be described by a logarithmic law
with known roughness parameter z0 , it can be written as:
U(z) = -ln -
,
u* ( z)
fi,
zo
(4.37)
where fi, = 0.4 is the Karman constant.
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
value 28U*. However, the problem of determining U* turns out to be rather
complicated. In the original version of The WAM model (1988) it is assumed
that the dynamical velocity can be determined as:
u* = J(f;;uw ;
{
1.2873 . w- 3
Cv= o.8 . w- 3 + o.65 . w- 4
at ulO : : : : ; 7.5 ms- 1 ;
at Uw > 7.5 ms- 1 .
(4.35)
The drawback of this ratio is that it does not take into account the actual state
of the near-water atmospheric boundary layer determined by the sea surface
roughness and atmospheric stratification. In this connection the model wind
wave energy input, proposed by P. Janssen, was used in the next version of
the WAM model (Komen et al., 1994).
Relation of the friction velocity with the wave surface resistance
coefficient and the wave development stage. The experimental results of investigating the dependence of the friction velocity and the wave
surface resistance coefficient on the wave development stage are in Problems
of Research and Mathematical Modeling of Wind Sea (1995); Huang et al.,
(1986); Komen et al., (1994). Thus, using full-scale data analysis (Problems of
Research and Mathematical Modeling of Wind Sea, 1995) it is shown that the
following dependence between the sea roughness z0 and the non-dimensional
frequency of the wind wave spectral maximum for the deep-water case, excluding the first stage of wind wave development, can be written:
z0 = 0.4 a;;,a.x
(4.36)
The parameters are written in non-dimensional form normalized by the
friction velocity in this relation. The relation ( 4.36) indicates a decrease of
sea surface roughness depending on the wave development stage. Thus, at the
initial stage of wave development the waves are very steep and their speed
is relatively low, creating large resistance to airflow and significant surface
roughness. An intense energy and momentum transfer from the wind flow to
waves appears. The reverse influence of waves on the air flow is significant. It
is revealed in the increase in dynamical velocity and in violation of the selfsimilar regime. The waves become flatter, their phase speed is increased while
the effective roughness is decreased with wave development. Thus, the resistance coefficient is decreased. Correspondingly, the energy and momentum
transfer rates to waves are decreased. The wave increase becomes slower.
If the wind speed profile is assumed to be described by a logarithmic law
with known roughness parameter z0 , it can be written as:
U(z) = -ln -
,
u* ( z)
fi,
zo
(4.37)
where fi, = 0.4 is the Karman constant.