THE NEAR-SURFACE LAYER OF THE OCEAN
1/ 4
5
u
x
u
x
E k
F k Q
HQ
K
(3.21)
where u
E is the longitudinal (in the x direction) velocity spectrum, x
k is the
wavenumber in the x direction (k x = 2Sf /U 0 by Taylor’s hypothesis (3.8)),
u
F is a universal function of its non-dimensional argument x
k Q
K , and
3/ 4
1/ 4
Q
K Q H
is the Kolmogorov internal scale of turbulence. In the inertia
interval (
1
x
k Q
K ), equation (3.21) reduces to
3
/
5
3
/
2
1
)
(
x
x
u
k
k
E
H
D
,
where dimensionless constant
5
.
0
1 |
D
.
Several interpolation formulas of the universal function u
F can be found
in the literature on turbulence (Novikov, 1961; Hinze, 1975; Oakey, 1982;
Moum et al., 1995). Here we will use the form of function u
F as empirically
determined by Nasmyth (1970; c.f. Oakey, 1982), which has been used in
many studies of oceanic turbulence.
The theoretical spectrum of turbulence and its fit to a measured velocity
spectrum using the Stewart and Grant (1962) techniques are shown in Figure
3-10. The measured spectrum is taken as a frequency spectrum; then, it is
converted into the wavenumber spectrum using Taylor’s hypothesis and
transfer functions for anti-alias filter and spatial averaging (Figure 3-9b).
The theoretical spectrum in Figure 3-10 corresponds to H =
6
1.7 10
u
W kg
-1 .
The large deviation from the theoretical turbulence spectrum on the left
(Figure 3-10, upper subplot) is due to the surface wave and ship pitching
disturbances, which is consistent with the results of Stewart and Grant
(1962). There is also a slight difference between the experimental and
theoretical spectra in the wavenumber range from 20 m
-1 to 120 m
-1 . This is
presumably an effect of the rapid flow distortion produced by the pressure
wave in front of the moving ship. Recently, Fornwalt et al. (2002) modeled
this effect numerically and found that the rapid flow distortion results in the
net production of TKE concentrated at relatively small scale, which thus
affects the velocity spectrum primarily at high wave numbers. Note that the
observed deviation might also be introduced by the correction factor for the
probe spatial resolution that is known with only 20% accuracy (Figure
3-9b). Similar to the disturbance from surface waves, this deviation is not
expected to affect the dissipation rate estimate made with the spectrum
fitting techniques of Stewart and Grant (1962).
The uncertainty of the Hestimation due to spectral scatter is small in this
example because confidence intervals are small. The spectral scatter,
however, is not the only source of error in the dissipation rate estimation.
Other errors are introduced by the uncertainty of the instrument towing
speed and probe calibration. As we analyze only the data that satisfy
Taylor’s hypothesis of frozen turbulence, the fluctuation of the towing speed
168
1/ 4
5
u
x
u
x
E k
F k Q
HQ
K
(3.21)
where u
E is the longitudinal (in the x direction) velocity spectrum, x
k is the
wavenumber in the x direction (k x = 2Sf /U 0 by Taylor’s hypothesis (3.8)),
u
F is a universal function of its non-dimensional argument x
k Q
K , and
3/ 4
1/ 4
Q
K Q H
is the Kolmogorov internal scale of turbulence. In the inertia
interval (
1
x
k Q
K ), equation (3.21) reduces to
3
/
5
3
/
2
1
)
(
x
x
u
k
k
E
H
D
,
where dimensionless constant
5
.
0
1 |
D
.
Several interpolation formulas of the universal function u
F can be found
in the literature on turbulence (Novikov, 1961; Hinze, 1975; Oakey, 1982;
Moum et al., 1995). Here we will use the form of function u
F as empirically
determined by Nasmyth (1970; c.f. Oakey, 1982), which has been used in
many studies of oceanic turbulence.
The theoretical spectrum of turbulence and its fit to a measured velocity
spectrum using the Stewart and Grant (1962) techniques are shown in Figure
3-10. The measured spectrum is taken as a frequency spectrum; then, it is
converted into the wavenumber spectrum using Taylor’s hypothesis and
transfer functions for anti-alias filter and spatial averaging (Figure 3-9b).
The theoretical spectrum in Figure 3-10 corresponds to H =
6
1.7 10
u
W kg
-1 .
The large deviation from the theoretical turbulence spectrum on the left
(Figure 3-10, upper subplot) is due to the surface wave and ship pitching
disturbances, which is consistent with the results of Stewart and Grant
(1962). There is also a slight difference between the experimental and
theoretical spectra in the wavenumber range from 20 m
-1 to 120 m
-1 . This is
presumably an effect of the rapid flow distortion produced by the pressure
wave in front of the moving ship. Recently, Fornwalt et al. (2002) modeled
this effect numerically and found that the rapid flow distortion results in the
net production of TKE concentrated at relatively small scale, which thus
affects the velocity spectrum primarily at high wave numbers. Note that the
observed deviation might also be introduced by the correction factor for the
probe spatial resolution that is known with only 20% accuracy (Figure
3-9b). Similar to the disturbance from surface waves, this deviation is not
expected to affect the dissipation rate estimate made with the spectrum
fitting techniques of Stewart and Grant (1962).
The uncertainty of the Hestimation due to spectral scatter is small in this
example because confidence intervals are small. The spectral scatter,
however, is not the only source of error in the dissipation rate estimation.
Other errors are introduced by the uncertainty of the instrument towing
speed and probe calibration. As we analyze only the data that satisfy
Taylor’s hypothesis of frozen turbulence, the fluctuation of the towing speed
168