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Such threshold-like dependences are consistent with parameterisations of the breaking
probability (see Chapter 5 ), at least for straightforward wave-development conditions when
the threshold wind speed can be related to the threshold characteristic steepness in the wave
field (see also Section 5.3.4 ).
When using the power of 3 in (3.3) , Stramska & Petelski ( 2003 ) followed the approach
prevailing in the 1990s (e.g. Monahan & Lu , 1990 ; Monahan , 1993 ). While in the earlier
parameterisations of
U 10 ,
W
(3.5)
the power p was obtained by means of fitting the experimental data points, the latter
approach assumes and enforces p
3. The semi-theoretical reasoning behind such an
assumption is the same as that of Wu ( 1979 ) whose conclusion, however, was p
=
75.
Therefore, it is worth briefly outlining the argument, and reviewing it somewhat in the light
of more recent knowledge.
The physical argument states, by definition, that the total energy flux from the wind to
the waves, and therefore the total wave energy dissipation rate in a quasi-stationary case is
approximately:
=
3
.
S ds = τ
U
,
(3.6)
where
τ = ρ a u 2
= ρ a C D U 10
(3.7)
is the wind stress or the momentum flux from the air to the water, U is some characteristic
velocity of energy propagation in the low atmospheric boundary layer, u
is the so-called
friction velocity,
ρ a is the air density, and
u 2
C D =
U 10 ,
(3.8)
is the drag coefficient, a parameter introduced for convenience for converting the U 10
wind into u
. Thus, according to (3.6) - (3.7) , the dissipation is proportional to some wind
speed cubed, but the precise nature of this proportion depends on what the characteristic
speed U is.
If U
=
u
(e.g. Wu , 1979 ; Soloviev et al. , 1988 ; Agrawal et al. , 1992 ; Melville , 1994 ),
then
= ρ a C 3 / 2
S ds = ρ a u 3
D U 10 .
(3.9)
If U
=
U 10 (e.g. Demchenko , 1993 ; Bister & Emanuel , 1998 ), we obtain
S ds = ρ a u 2
U 10 = ρ a C D U 10 .
(3.10)
Obviously, a conclusion on speed U depends on which model of the boundary layer is
employed, and other options are possible between the two extremes mentioned, includ-
ing more complicated characteristic speeds defined by integral properties of the sheared
boundary-layer air flow (e.g. Kudryavtsev et al. , 2001 ).
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