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For such a truncated power law spectrum, the surface variance is given by
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Similarly, the total slope variance, or the mean-square slope, is obtained from
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Thus, the variance of a power law spectrum ocean is dominated by its low frequency components, since they comprise the highest amplitude portions of the spectrum, and the slope variance is determined by the width of the spectrum. It is noted that this slope variance dependence emphasizes the need for a cutoff wavelength in the composite surface model, because the slope variance of such a surface could be made very large by extending the spectrum into very short wavelengths beyond the Bragg scattering region.
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Pierson-Moskowitz Spectrum
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The Pierson-Moskowitz ocean spectrum [Pierson and Moskowitz, 1964J is frequently applied in oceanography for a fully developed wind-generated sea. This model is derived based on the assumption that the wind has been constant for an adequate duration and fetch (the effective distance over which the wind transfers energy to the ocean). As a function of wavenumber, gravity, and wind speed, the form of Pierson-Moskowitz spectrum is
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where ao = 0.0081, (3 = 0.74, 9 is the gravitational acceleration equal to 9.81 m/sec 2 , and U19 .5 is the wind speed, in m/sec, at the altitude of 19.5 m above the mean sea level [Thorsos, 1990]. The subscript PM denotes the Pierson-Moskowitz spectrum. In (4.8.9), if>(k, ) is the directional factor, which can in general be a function of wavenumber k. A cos2 azimuthal dependence has often been assumed with the use of the Pierson-Moskowitz spectrum [Thorsos, 1990J:
if>(k, ) = cos
The peak of the Pierson-Moskowitz spectrum occurs at the wavenumber
-4m1s . 8m1s
~ 10-'
~ 10-1
::: 10-
10" 10-3
k (rad/m)
Figure 4.8.1 Amplitude of the Pierson-Moskowitz spectrum for along-wind direction and wind speeds U19.5 = 4, 8, and 20 m/sec.
For the Pierson-Moskowitz model, the surface variance is given by (4.8.11) 4 9 which is constant depending on the gravitational acceleration and the wind speed. The mean-square slope of this spectrum diverges if integrating k3 WPM(k, </J) over 0 ~ k < 00. If we let the lower cutoff wavenumber be kdl' then the mean-square slope becomes
dk k
121f d</J WPM(k, </J) = aU4 {J19;}
2 SpM=
k d1
31 21f d</JWPM(k,</J)=-TEi ( -k 2(Jg2) (4.8.12) ao U4
0 dl 19.5
where Ei ( ) is the exponential integral [Gradshteyn and Ryzhik, 1980; Thorsos, 1990]. The Pierson-Moskowitz spectra for along-wind direction (</J = 0) are illustrated in Fig. 4.8.1 for three wind speeds U19 . 5 = 4, 8, and 20m/sec. As shown in Fig. 4.8.1, much of the wave energy is located at smaller wavenumbers. Note the changes in the low-frequency cutoff predicted by the spectrum, the spectrum amplitude increases significantly with the wind speed, and the peak shifts toward smaller wavenumbers (Le., longer spatial wavelengths), indicating a much larger surface 1mS height at the higher wind speeds. The spectra at large wavenumbers are essentially unchanged by the wind speed, and the spectral slope is -4, demonstrating the k- 4 power law dependence.
Ocean Surface
Durden-Vesecky Spectrum
This ocean spectrum model was proposed by Durden and Vesecky [1985] by empirically fitting model parameters to radar backscattering data based on an assumed two-scale rough surface scattering model. The Durden-Vesecky spectrum is given by
WDv(k, 4
2"k 4 <i> Dv(k,
p (ex ( -
k> k J
where k j = 2m- I . In (4.8.13) ao is a constant, originally set equal to 0.004 [Durden and Vesecky, 1985], but doubling the value to 0.008 has been used [Yueh et al. 1994, 1997] to achieve a better fit to passive ocean measurements. Other parameters are (3 = 0.74, b = 1.25, a = 0.225, 9* = 9 +,k 2 , 9 = 9.81 m/sec 2 is acceleration due to gravity, and, = 7.25 x 1O- 5 m 3 jsec2 is the ratio of surface tension to water density. The subscript DV denotes the DurdenVesecky spectrum. u* is the wind friction velocity at the ocean surface in m/sec, U19 . 5 is the wind speed, in m/sec, at a height of 19.5m. The windspeed Uh in mjsat h meters above the ocean surface is related to the wind friction velocity u* by
u* Uh = -log 0.4 0.0000684/u*
+ 0.00428u* -
(4.8.14 )
Thus for a measured Uh at a height h, u* is first determined by using (4.8.14). In (4.8.13), the directional factor cPDv(k, ) is