4.2 Wind Wave Energy Input
123
- the frequency narrowness of the value B is established as 0.33 ± 0.02 ("the
law of 1/3") and is practically not dependent on the initial spectral form;
- the function of the angular narrowness D( O") reveals a sharp maximum in
the vicinity of the frequency spectral maximum, with its value becoming
smaller in the range of low and high frequencies;
- the maximum of the parameter Dp is established within the range
Dp = 0.9-1.3, depending on the initial angular narrowness Dp(t = 0).
As shown by the numerical computation results, a self-similar spectrum
is established in the process of non-linear evolution. This is revealed as the
establishment of the parameter values B and Dp, and the total form of the
spectrum. It has two main domains: a sharply defined peak and a slowly decreasing high-frequency tail. It should be noted that Kolmogorov's spectrum
S(O") "'0"- 4 · 0 , numerically found by Komatsu & Masuda (1996), is not seen in
all initial conditions. Kolmogorov's spectrum is known to appear in the presence of an energy source located in the low-frequency range, with the energy
sink at the high-frequency range (Zakharov & Zaslavskii, 1982). But, there are
no sources and sinks in our case. Using these numerical results a simplified
intermediate self-similar frequency spectrum approximation is offered (4.28).
This approximation describes the main features of the non-linear spectrum
evolution and provides conservation of the total energy and wave action.
4.2 Wind Wave Energy Input
Miles' model of wind wave energy input. The component Gin of the
wind wave energy input is usually determined with the help of the relation
based on the model of averaged air flow interaction with waves proposed by
Miles (1960). In spite of the fact that this model was proposed in 1957, it
describes accurately enough the mechanism of wind wave energy input. It
is still used nowadays. This mechanism which is specified by using full-scale
observational data (Snyder et al., 1981) can be described as follows:
Gin(O", /3) =max { 0; 0.25al ; ; O" ( a2 U~o cos(/3- /3u)- 1) S(O", /3)} ,
(4.34)
where ulO is the wind speed at the 10m level; /3 - !3u is the angle between the wind speed and the direction of spectral wave component propagation; and a1 and a2 are parameters, assumed to be about 1. It follows
from ( 4.34) that the wind energy is supplied to the wave spectrum range with
a 2(U10/c) cos(/3- /3u) > 1. It should be noted that at lower frequencies, wave
energy transfer takes place only due to the non-linear wave interaction Gnl·
The relation (4.34) can also be expressed with the help of the dynamic
velocity (or friction velocity) u. instead of the wind speed. It is believed to
be more universal with the wind speed ulO in (4.34) being replaced by the
123
- the frequency narrowness of the value B is established as 0.33 ± 0.02 ("the
law of 1/3") and is practically not dependent on the initial spectral form;
- the function of the angular narrowness D( O") reveals a sharp maximum in
the vicinity of the frequency spectral maximum, with its value becoming
smaller in the range of low and high frequencies;
- the maximum of the parameter Dp is established within the range
Dp = 0.9-1.3, depending on the initial angular narrowness Dp(t = 0).
As shown by the numerical computation results, a self-similar spectrum
is established in the process of non-linear evolution. This is revealed as the
establishment of the parameter values B and Dp, and the total form of the
spectrum. It has two main domains: a sharply defined peak and a slowly decreasing high-frequency tail. It should be noted that Kolmogorov's spectrum
S(O") "'0"- 4 · 0 , numerically found by Komatsu & Masuda (1996), is not seen in
all initial conditions. Kolmogorov's spectrum is known to appear in the presence of an energy source located in the low-frequency range, with the energy
sink at the high-frequency range (Zakharov & Zaslavskii, 1982). But, there are
no sources and sinks in our case. Using these numerical results a simplified
intermediate self-similar frequency spectrum approximation is offered (4.28).
This approximation describes the main features of the non-linear spectrum
evolution and provides conservation of the total energy and wave action.
4.2 Wind Wave Energy Input
Miles' model of wind wave energy input. The component Gin of the
wind wave energy input is usually determined with the help of the relation
based on the model of averaged air flow interaction with waves proposed by
Miles (1960). In spite of the fact that this model was proposed in 1957, it
describes accurately enough the mechanism of wind wave energy input. It
is still used nowadays. This mechanism which is specified by using full-scale
observational data (Snyder et al., 1981) can be described as follows:
Gin(O", /3) =max { 0; 0.25al ; ; O" ( a2 U~o cos(/3- /3u)- 1) S(O", /3)} ,
(4.34)
where ulO is the wind speed at the 10m level; /3 - !3u is the angle between the wind speed and the direction of spectral wave component propagation; and a1 and a2 are parameters, assumed to be about 1. It follows
from ( 4.34) that the wind energy is supplied to the wave spectrum range with
a 2(U10/c) cos(/3- /3u) > 1. It should be noted that at lower frequencies, wave
energy transfer takes place only due to the non-linear wave interaction Gnl·
The relation (4.34) can also be expressed with the help of the dynamic
velocity (or friction velocity) u. instead of the wind speed. It is believed to
be more universal with the wind speed ulO in (4.34) being replaced by the