4.4 Influence of Mesoscale Effects on Wind Wave Evolution
139
The equation of spectral energy density evolution can be written in its
simplest form. The linear term connected with the wind wave energy input
according to the Miles theory is taken only into account on the right-hand
side of the equation:
{)
8t S(a) = 1 S(a) ,
(4.61)
where 1 is the wave energy increment. This can be written in the simplified
form as:
1
(
Pa ( U*
)
)
-;;. = max 0.2 Pw 28~ - 1 , 0 .
(4.62)
It should be noted that the value 1 cannot be negative: 1 > 0 with
28U*/c > 1.0.
The friction velocity U* can be written in the form of its mean value (J*
and the fluctuations oU*:
(4.63)
After substituting (4.63) into (4.62), the wave energy increment can be presented as its mean value and fluctuations:
(4.64)
where b « 1.
Since 1 is always positive, the mean value of (b/1) is not equal to zero.
Thus, the mechanism of wind wave energy input within the approximation
( 4.62) acts as a "rectifier" in an electric circuit. The wind gusts increase the
intensity of wave energy input for 28U*/c > 1.0, whereas in the opposite case,
it converts to zero.
Using previous equations the following stochastic equation can be considered:
(4.65)
When the correlation time of a random process is small, the equation of
wave spectrum evolution (S), averaged over an ensemble of states, can be
written in the following form:
8 (S)
---at = ('? + b/1) (S) .
(4.66)
Assuming that the statistics of friction velocity fluctuations is described
by the Gaussian distribution:
(4.67)