Geoscience Reference
In-Depth Information
Figure 4.16
Energy spectrum measured
off Vancouver Island by Grant
et al. (1962). The straight line has a
slope of
10
4
=
3
. Reproduced courtesy
Cambridge University Press.
5
10
2
10
0
Slope -5/3
10
-2
10
-4
10
-6
10
-2
10
-1
10
0
10
1
k
1
=
4
.This
is the universal equilibrium form of the spectrum predicted by Kolmogorov. It
was first shown to apply to turbulence in the ocean by a famous series of
measurements by Grant and co-workers in the early 1960s (Grant et al., 1962)
who used hot film anemometers to measure turbulent fluctuations of velocity in a
tidally energetic channel off the west coast of Canada. The measurements were
made from an anchored research vessel so the data constituted time series of
velocity fluctuations from which the frequency spectrum can be determined.
In order to transform from a frequency spectrum determined at a fixed point into
a wavenumber spectrum, the Taylor hypothesis was used.
The wavenumber spectrum resulting from these measurements is shown in
Fig. 4.16
, along with a fit to k
5/3
extending over two decades of k and a fall-off in
the spectrum for wave numbers above k
n
) is a function of the non-dimensional variable k
n
¼
k n
3
=
where F(k
e
1. Many other spectra observed in the
atmosphere and ocean have been found to fit a k
5/3
form over a wide range of scales,
indicating the presence of an inertial subrange and seeming to confirm Kolmogorov's
hypothesis about the nature of the cascade. Some doubts, however, have been raised
about the accuracy of the fit to k
5/3
(e.g. Long,
2003
) and about the finding that, in
the presence of weak stratification, the motion at high wave numbers is not fully
isotropic as it should be in the Kolmogorov model. Nevertheless, the Kolmogorov
theory stands as a key reference model in investigations of turbulence, and spectral
fitting to the k
5/3
form is widely used in the estimation of dissipation.
In the shelf seas, where dissipation in the tidal flow may exceed 1 W m
2
n
∼
in
10
5
Wkg
1
so, with the kinematic viscosity n
10
6
m
2
s
1
¼
100 metres depth, e
the microscale
0.6 mm. Even at much lower dissipation rates
typical of low energy stratified conditions, where e
n
may be as small as
10
8
Wkg
1
, the microscale is
only
∼
3 mm. The inertial subrange of turbulence extends from these millimetre
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