Turbulence is intermittent by nature. Even relatively low (by ocean standards) Reynolds number simulations of homogeneous, isotropic turbulence show strongly coherent turbulent regions embedded in a highly disordered background [ She et al., 1990]. (The Reynolds number is a measure of the tendency for turbulence in a fluid, the ratio of the magnitude of the non-linear terms in the governing equation to the viscous term, calculated as a typical velocity times a typical length divided by the viscosity.) Most of the measurements of ocean turbulence to date have been made by vertically-profiling devices. The data obtained represent vertical single-line passages through turbulent patches of unknown horizontal extent at unknown stages in their evolution. If we see a patch on one cast, the interval between casts is necessarily so long even with rapidly-profiling devices in the upper ocean (some minutes at best) that we cannot tell whether the next cast samples the same patch or whether another one has taken its place by advection or by re-creation. (To follow a patch in time we would need to sample it non-destructively at a rate much greater than N.) The problem is greater for thermocline or deeper measurements, where the time between samples is much greater. Measurements using horizontally-profiling devices, such as towed bodies or submarines, have similar limitations. With the means available now, we cannot produce exact images of turbulence; we can only produce data that must be analyzed in statistical terms.
Gurvich and Yaglom [1967] predicted that
positive-definite, uncorrelated turbulence variables should be
lognormally distributed. Many of us have examined the statistics
of turbulence data (
and 
, for example) in various
data sets and have found populations that had lognormal statistics,
though we have found as many that did not. It was clear to most that
the conditions for lognormality required consideration of a flow regime
in which the flow did not depart from similarity. Yamazaki and
Lueck [1990] provided a much-needed demonstration of the conditions
of applicability of the lognormal probability density function
for volume-averaged dissipation rates: that 
must
be statistically homogeneous within the sample and that the averaging
scale must be small compared to the patch size but large compared to
the Kolmogoroff scale. From horizontal measurements in the
seasonal thermocline, they found that the shortest averaging scale
that produced a lognormal distribution was three Kolmogoroff scales.
This was corroborated by Wijesekera et al. [1993] from
vertical turbulence measurements of mixing patches in the thermocline
over steep topography.
In the Gregg et al. [1993] examination of the statistics
of 
and shear
from a subset of Gregg's
[1989] thermocline data, they found that 
, when averaged
over 10m and scaled by N
, was lognormally distributed. We infer
that turbulence produced by one statistically-steady source, in their
case the internal wave field, is distributed lognormally. A
lognormal distribution should not be expected when turbulence production
is due to more than one source, or to a changing source.
The authors believe that the calculations of Gibson [1991, for example] based on a universal lognormality are inconsistent with the above results.
One type of information routinely gathered in laboratory studies
of turbulence that has received little attention in the ocean is
horizontal correlation. This is understandable in terms of difficulty
and expense. A recent comparison of a three and a half-day time series
of 
at the equator obtained by two different groups on
research vessels separated by 1--11 km, gave a result which was
encouraging from the technical viewpoint and lent some insight into
the lateral variability of turbulence [ Moum et al., 1995].
Statistics of 
from hundreds of profiles showed no
systematic bias in the estimates between the two groups. Since
the measurements were made independently by two separate groups from
two different ships using very different instrumentation and
procedures, this result increases our faith in our measurement
techniques. The fact that these measurements did show occasional
large differences (several factors of 10) over periods of several
hours shows that great natural variability can be expected in averages
taken over periods of hours, even over distances of only a few
kilometers, at least at the equator.
Although we may now have a better idea of the statistics of turbulence in the thermocline, we must be wary of the variability to be expected.