To make turbulence measurements useful for many oceanographic
purposes, we need a reliable method of estimating the vertical flux of
heat, salt, nutrients and other quantities from the quantities measured.
To simplify our discussion, we'll take heat as our example. Three methods
of estimating heat flux from microstructure measurements are currently
in use, a) the Cox number estimate based on 
, b) the
dissipation method based on 
and c) the direct
eddy-correlation method, which has commonly been used in the
atmosphere [e.g., Lenschow, 1986], but has been employed only
recently in the ocean [ Moum, 1990a; Yamazaki and Osborn,
1993; Fleury and Lueck, 1994]. Which of these methods should be
used? How much confidence can be placed in them?
The Cox number estimate was the original method used in the 70s. It
is derived from the entropy equation [ Osborn and Cox, 1972]. In
its traditional form, isotropy is assumed, and the vertical heat flux Q
is estimated in terms of a vertical diffusivity for heat 
as:
where 
, 
is the mean
vertical gradient and 
is the gradient variance, D is
the molecular diffusivity, and 
is the heat capacity. (The Cox
number is the ratio of the expression for 
defined above to
D.) Usage of this simple form depends on a complete determination of
the variance of temperature fluctuations, which have scales as small as
a few millimeters in the ocean, and on the assumption of isotropy
(numerical simulations, at low Reynolds number admittedly, indicate that
the above isotropic estimate for 
may be twice as large as its
true value [ Itsweire et al., 1993]). Profilers using this method
had to fall very slowly, in order to allow sensors to capture
the temperature variance, or had to rely on questionable
sensor-response corrections.
To allow instruments to fall faster, and to take advantage of
the small-scale shear sensors, which increase sensitivity with fall
speed, the dissipation method came into use. If it is assumed that
mixing takes place with a typical efficiency 
, then the
diffusivity 
[ Osborn, 1980]:
where 
is commonly taken as 0.2. This method suffers of course
from the problems associated with measurement of 
, and
also from the probable variability of 
.
Eddy-correlation measurements are in principle fundamental because the instantaneous, kinematic heat flux is defined in the Reynolds-averaged energy equations as:
This method depends on estimation of 
, which has only recently
been measured from a profiler [ Moum, 1990b] and on a definition
of 
. A problem with current observational methods of making
this estimate is the difficulty of obtaining sufficient data for a
stable estimate of the average value of 
.
Comparisons of the eddy-correlation method and the dissipation
method seem to indicate that 
is not constant. Comparisons in
the main thermocline indicate that 
is only 0.05 there [
Moum, 1994]. Horizontally-profiling comparisons give the same result
[ Yamazaki and Osborn, 1990; Fleury and Lueck, 1994].
But estimates from a tidal channel yielded 
[ Gargett
and Moum, 1995]. 
may depend strongly on the type of
instability causing the turbulence [ Wijesekera et al., 1993].
For example mixing in the main thermocline is probably due to
shear production in which the energy is put initially into the
horizontal velocity and then redistributed among all three components.
On the other hand turbulence in tidal flows may be initiated by
horizontal convergences in the mean flow, forcing vertical
fluctuations first. We might also expect a difference between turbulence
due to breaking high-frequency waves, in which energy is
roughly equipartitioned, and turbulence due to near-inertial shear
(shear associated with internal waves with frequencies near the
local inertial frequency) in which the energy is mostly kinetic.
This difference may explain why larger values of 
were seen on
the Yermak Plateau, where high frequency waves are thought to
predominate [ Wijesekera et al., 1993] than were seen in the
main thermocline, where near-inertial shear dominates [ Hebert
and Moum, 1994].