SCI. MAR., 61 (Supl. 1): 25-45
SCIENTIA MARINA
1997
LECTURES ON PLANKTON AND TURBULENCE. C. MARRASÉ, E. SAIZ and J.M. REDONDO (eds.)
“Theories” and techniques for observing turbulence
in the ocean euphotic zone*
ANN E. GARGETT
Institute of Ocean Sciences, P.O. Box 6000, Sidney, B.C. V8L 4B2, Canada.
SUMMARY: With growing recognition of the important roles which turbulence plays in the functioning of marine food
webs, interest in the tools and techniques of measuring turbulence in the ocean has spread from the physical to the biological oceanographic community, for which this paper is intended. The subject of ocean turbulence and its measurement is
introduced, with emphasis on the euphotic zone of both deep ocean and coastal environments. A discussion of important
characteristics of turbulence and the various means by which turbulence may affect components of the biological system is
followed by a simplified outline of the mathematical means used to describe scales of variability produced by turbulent
fields. Existing and developing techniques for field measurements of turbulence variables are described, with discussion of
the “theories” which are often necessary to transform those variables which we can measure into those we actually wish to
know.
Key words: Turbulence, mixing.
INTRODUCTION
Turbulence occurring in the upper sunlit layer of
the oceans is increasingly recognized as critical to
the functioning of the marine food web. Since different properties of a turbulent field are important to
different aspects of the embedded biology, these lecture notes must encompass a broad description of
various turbulent properties and our present abilities
to observe them. I will first mention some defining
general characteristics of three-dimensional turbulence and indicate various ways in which these characteristics may influence marine life systems, then
describe the turbulence environments of the euphotic zone in both offshore and coastal areas. A simplified outline of the mathematical means used to
*Received December 10, 1995. Accepted May 30, 1996.
describe the scales of variance generated by turbulence precedes description of the sensors and associated “theories” which have been used to quantify the
smallest (dissipation) scales of turbulence over the
past 20 years. Finally, I will consider current developments in oceanic observations which promise to
provide new and broader views of the turbulent environment of plankton.
Clearly, such broad scope forces radical abbreviation of all the above topics, so this article provides only starting points, frequently via references to specific sections in two texts, “A First
Course in Turbulence” (Tennekes and Lumley,
1972), and Turner (1973) “Buoyancy Effects in
Fluids”. In keeping with the tutorial rather than
research focus of this volume, other references
are not exhaustive, but hopefully will provide the
student with entry points to the literature in specific areas.
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 25
TURBULENCE CHARACTERISTICS
AND THEIR INFLUENCES ON MARINE
ECOSYSTEMS
General characteristics of three-dimensional
turbulence
Turbulence is not a property of a fluid, but of its
state of motion. Turbulent flows are highly individualistic, depending strongly on such features as flow
geometry, stability properties of the fluid, etc; nevertheless, turbulent flows share a number of common characteristics.
(1) Threshold: Turbulence is a threshold phenomenon. If U is a mean speed and L a characteristic length scale in a flow of fluid of kinematic viscosity ν, the flow becomes turbulent only when the
value of the Reynolds number Re ≡ UL/ν exceeds
some critical threshold value Rec, which must be
determined for specific cases (for example, see the
various photos in Van Dyke (1982) of the changes in
the flow around a circular cylinder as Re increases).
(2) Randomness: Turbulent motion is unpredictably variable in instantaneous speed and direction, and is usually characterized by values of averaged quantities.
(3) 3-dimensionality: The turbulent fields
which are the topic of these notes are those with
strong 3-dimensionality (excluding the 2-dimensional geostrophic turbulence covered elsewhere in
this volume).
(4) Dissipative and diffusive: A flow that has
become turbulent has rates of energy dissipation and
property diffusion which are many times the molecular values associated with laminar flows.
supplied from the deep-ocean reservoir. Because
nutrient gradients are usually quite large across the
base of the surface mixed layer (SML), as shown
schematically in Fig. 1, the occurrence and intensity of turbulence at the layer base is a limiting
process for primary productivity. Understanding
the process(es) and rate of nutrient resupply
requires measurement of turbulent fluxes, i. e. the
vertical diffusive transport of nutrients associated
with the (sporadic) occurrence of turbulent flow at
the top of the nutricline.
Importance of turbulence to the marine
food web
Turbulence in the euphotic zone of the ocean
affects the marine food web in a variety of ways, a
few of which will be used here to illustrate particular features of a turbulent field which are required
for specific applications.
Nutrient supply
Phytoplankton requires both light and nutrients
for continuing growth. When light is not limiting,
growth of the phytoplankton population can rapidly exhaust dissolved inorganic nutrients in the
euphotic zone if such nutrients are not being re26 A.E. GARGETT
FIG. 1. – Schematic of nutrient concentrations (proportional to
intensity of shading) in the upper layer of the Sargasso Sea, superimposed on a profile of temperature (T) and small-scale shear
(∂u/∂z) obtained from a free-fall microscale profiler (redrawn from
Gargett et al. 1979). Nutrient concentrations are assumed to be low
within a weakly stratified surface layer (above ~ 100m), but to rise
rapidly below. The turbulent activity revealed by the “burst” of
shear signal between 100 and 110m will cause enhanced transport
of nutrients up into the euphotic zone.
Light regime
FIG. 2 – Measurements of P = time-integrated average oxygen production rate of a laboratory phytoplankton culture as a function of I
= light intensity for three different incubation times (redrawn, with
permission, from Marra 1978). Photoinhibition, the decrease of P at
high I, becomes evident only for incubation times of order 2 hours
or longer. Such results suggest that net productivity is a function of
the time history of light intensity, and can thus be affected by the
largest energy-containing eddies of a turbulent field, acting to
advect plankters in the vertically variable light field of the nearsurface ocean.
FIG. 3 – Schematic (redrawn, with permission, from Davis et al.
1991) illustrating possible competing effects of turbulence on the
growth rate of a planktonic predator in an ocean surface mixing layer
(SML) where turbulent intensity (proportional to some power of the
wind speed) is greatest near the surface. If growth rate increases with
both predator/prey encounter rate and prey patchiness (assumed to
be some inverse function of turbulent intensity), net growth rates
may differ with depth. In the lower SML, the (small) increase in turbulence intensity associated with higher winds does not increase
encounter rates sufficiently to compensate for the decrease in prey
patchiness; thus net growth rate decreases in high winds. If in contrast, a large increase in encounter rates in the stronger turbulence of
the upper SML more than makes up for the homogenization of the
prey population during high winds, the upper SML
growth rate will exhibit a double-maximum structure, as shown.
The dependence of phytoplankton productivity
upon ambient light intensity is both nonlinear and
time-dependent. As one increases the average light
intensity at which a laboratory culture is grown, its
productivity first increases, then eventually decreases again due to photoinhibition. However, as seen in
Fig. 2 from Marra (1978), the onset of photoinhibition has an associated time constant; a culture grown
at 1500 µE m-2 s-1 for only 12.5 minutes does not
exhibit the photoinhibition which is apparent if the
incubation period is increased to 2 hours. This timedependence of photoinhibition, coupled with the
strongly surface-intensified light field of the upper
ocean, means that the net productivity of a phytoplankter will depend on the time scale with which it
is moved vertically by the fluctuating motion fields
of the upper ocean. For this research area, we
require information on the time and space scales of
the largest energy-containing eddies of the turbulent
field.
Feeding
Rothschild and Osborn (1988) first assessed a
potential increase in encounter rate between planktonic predators and their prey due to turbulence-generated shear at small scales. Since then, numerous
laboratory studies have sought to quantify the effect
of turbulence on the feeding success of zooplankton
and larval fish, using the dissipation rate of turbulent
kinetic energy ε, to quantify the shear on scales
comparable to the organisms involved. However
various studies suggest that in the ocean, increased
encounter rates may be entwined with other turbulence-associated effects on the food web, such as
changes in the coagulation processes by which
marine snow is formed and destroyed (Jackson
1990, Kiørboe, 1997), or in the patchiness of the
prey species. Fig. 3 (after the numerical model
results of Davis et al. 1991) illustrates the effect of
this latter interaction on the net growth rate of a
predator operating in an upper ocean environment
where turbulence intensity is greatest near the surface. Growth rate is assumed to increase with turbulent intensity (due to increased encounter rates) and
with the degree of prey patchiness. As shown, the
net growth rate will not everywhere be simply related to turbulence intensity if prey patchiness itself
decreases with increasing turbulent intensity (Owen
1989). An understanding of the interaction between
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 27
prey patchiness and turbulent intensity will probably
require information on both the large-eddy structures and the horizontal diffusive fluxes of the turbulent field.
Since both characteristics and challenges vary considerably between deep-sea and coastal regions,
these environments will be considered separately.
The offshore ocean
THE TURBULENCE ENVIRONMENT
OF THE EUPHOTIC ZONE
Before addressing questions of observational
ability, it is necessary to gain some idea of the
underlying characteristics of the turbulent environment of the euphotic zones of the ocean, and the
observational challenges associated with them.
Far from land, the upper euphotic zone of (icefree) oceans is most strongly influenced by the overlying atmosphere. Wind-driven waves produce turbulent mixing both directly, when they break, and
indirectly by generation of Langmuir circulations;
sheared currents driven by surface wind stress
become unstable and generate turbulence in the interior; wind-induced inertial wave motions produce
FIG. 4 – Time series of parameters related to forcing by (a) wind (U103) and (b) buoyancy (I) of the turbulence field of the upper ocean in the
northeast Pacific (adapted from Denman and Gargett 1988). The “mountain and valley” response of temperature (c) measured at thermistors
between 2.5 and 25m depth (arrowed) show that daytime radiation is sufficient to stratify the upper euphotic zone during most days, despite
periods of strong winds. The exception (May 15th) is the one day when very low winds produced little turbulent diffusion, leaving the surface
heat flux trapped at depths less than 2.5m. This example illustrates that turbulence in the offshore upper ocean involves a delicate balance
between wind and buoyancy forcings, which are known to have significant variation on diurnal, storm and annual time scales.
28 A.E. GARGETT
sporadic shear instability at the base of the surface
mixing layer (a useful general reference here is
Phillips 1977). In addition to wind forcing, the upper
ocean exchanges heat and fresh water with the
atmosphere, exchanges which can be either stabilizing or destabilizing, depending upon the net effect
on surface water density. Wind stress forcing varies
on the period of storms, themselves modulated
annually, while buoyancy forcing has strong diurnal
and seasonal variability. The often subtle interactions between stress and buoyancy forcings makes
turbulence in the SML extremely variable in time
(spatial variability is generally less pronounced in
offshore regions, although of course exceptions may
be found). Fig. 4, adapted from Denman and Gargett
(1988), illustrates this point with records, taken over
several days in the northeast Pacific, of (a) the cube
of the 10m wind speed U103 , proportional to the
power of wind driving, and (b) the intensity of photosynthetically available radiation I, proportional to
the (stratifying) heat flux into the ocean during daylight hours. The bottom panel (c) is a composite of
time series registered at thermistors spaced throughout the upper 200m of the ocean: records from thermistors in the upper 25m are emphasized by arrows.
The “valleys” in these records show that temperature becomes nearly uniform in the upper 25m at
night, when convection driven by surface heat loss
mixes this part of the water column. However
despite frequently intense winds, daytime surface
heating was usually sufficient to cause the upper
25m to stratify, as shown by the “mountains” caused
by spreading apart of temperatures during the day.
The single exception to this statement is May 15th,
when the SML below 2.5m (the depth of the shallowest thermistor) remained relatively well mixed
FIG. 5 – Turbulence in coastal areas is often spatially inhomogeneous and site-specific, as seen in this transect of a front
formed by horizontal convergence of two tidal streams (large arrows) meeting at a “T-junction” (upper cartoon). The vertical velocity field measured by a specialized Doppler profiler (Gargett 1994) across such a front shows the strong downward flow at the front which generates vigorous turbulence downstream. Turbulence generated regularly by such tidal
flows is associated with values of vertical eddy size, vertical transport of nutrients, and ε which are often 100 - 1000 times
those typical of the euphotic zone in the offshore ocean. Ordinary microscale profiling (at (a) and (b)) severely undersamples such vigorous turbulent flows.
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 29
throughout the day. Contrary to expectations based
on the simplistic assumption that strong winds=strong
turbulence=well mixed surface layer, this was not
the day with the strongest wind forcing (May 14th),
but the day with the lowest wind forcing. Why? The
comparison between May 14th and 15th, both days of
reduced radiation (due to fog) but very different
wind forcing, indicates that diffusion associated
with wind-driven turbulence distributes the surface
heat flux in the vertical, thus contributes to stratification. On May 15th, in the absence of wind-generated turbulence, the heating remained trapped near
the surface, so the euphotic zone did not stratify
below 2.5m. This example clearly demonstrates that
a comprehensive understanding of the strongly
time-dependent turbulent environment of the offshore upper ocean requires significant meteorological information as well as oceanic measurements.
strength of local buoyancy forcing. Here as in most
coastal regions, freshwater runoff is an important
and often poorly determined source of near-surface
buoyancy.
A final complication of coastal euphotic zones is
that the effects of turbulence may be non-local. To
appreciate this problem, consider the geographic
area shown in Fig. 6a, where the shaded area is a
highly productive coastal shelf off Vancouver
Island. When Crawford (1991) estimated the contributions of various processes to the supply of inorganic nutrients to the euphotic zone (see the
schematic of Fig. 6b), he concluded that the largest
contribution came by advection in the surface outflow from Juan de Fuca Strait. (The coastal waters
inside Vancouver Island act as a giant estuary, with
The coastal ocean
To the time variability of the offshore euphotic
zone, associated with varying atmospheric forcing,
the coastal ocean adds time variability associated
with tides and spatial variability associated with the
mechanisms by which tidal flows generate turbulence. The time scales over which tidal flows vary
are well-known; a reasonably thorough investigation of the associated turbulence characteristics
should include observations over the major modulation cycles, minimally the spring-neap cycle but
preferably also the annual cycle. The real problem in
many coastal environments however is finding those
regions where energy is removed from the mean
tidal flow. It is increasingly being recognized that
such locations are spatially localized, frequently
associated with site-specific characteristics such as
bottom topography or channel morphology. Examples of turbulence-generating structures are convergent tidal fronts which form at “T-junctions”, sites
where two tidal channels meet as shown in the upper
cartoon of Fig. 5. During a transect of such a front
along the major channel (y-axis), the vertical velocity field w (lower panel, observed with a specialized
acoustic Doppler current profiler, Gargett 1994)
exhibits very large downwards values associated
with the horizontal convergence of the two tidal
streams, a structure which generates fluctuating turbulent velocities far downstream. While a front
forms in this location on most ebb tides, the strength
of the associated turbulent velocities depends not
only upon available tidal energy but also upon the
30 A.E. GARGETT
FIG. 6 – Due to the strength of horizontal advective flows in the
coastal ocean, effects of turbulence may be non-local. (a) Map
showing (shaded) the highly productive shelf off southern Vancouver Island. Sites of strong localized turbulent mixing which have
been identified in waters inside Vancouver Island are marked with
dots: locations of the front (F) of Fig.5 and the tidal narrows (N) of
Fig.18 are marked). (b) Crawford’s (1991) estimates of the contribution to the nutrient supply of the shelf euphotic zone in summer:
local tidal (t) and wind (w) mixing are insignificant terms in a balance dominated by advection in the surface layer Juan de Fuca outflow. Nutrients are supplied to this outflow by strong turbulent mixing at sites (such as those noted in (a) above) which are spatially far
removed from the shelf itself.
nutrient-rich deep water entering Juan de Fuca Strait
at depth, underneath a fresher surface layer moving
seaward). In the strongly turbulent flows that occur
in certain interior locations (dots in Fig. 6a mark
some which have been identified), nutrients are
“pumped” up into this surface layer, where they are
advected out onto the continental shelf, providing a
major part of the inorganic nutrient supply which
supports the marine ecosystem in the shaded area.
The significance of this result is that no amount of
measurement on the shelf would reveal the turbulence actually responsible for a major portion of the
nutrient supply to the shelf. This problem of nonlocal effects of turbulence must always be kept in
mind when working in the strongly advective flows
characteristic of many coastal regions.
SCALES OF TURBULENCE
Oceanic flow fields ut can be partitioned into a
mean U and a fluctuating or turbulent component u
if an averaging operator < > can be defined so that
ut = U + u , where <ut> = U and <u> = 0 ;
this partition is known as the Reynolds decomposition (see Tennekes and Lumley, 1972, Sec.2.1). Frequently the operator < > is defined as a time average
and the mean field U = U(z) is assumed to be a function only of the vertical coordinate z (taken positive
upwards). In the schematic of the ocean surface
mixing layer shown in Fig. 7, typical scales of the
mean field are U ~ Uo and L ~ H, where Uo is a typical magnitude of U (perhaps its maximum value)
and H is the depth of the mixing layer. The scales of
the fluctuating field are a typical speed u (often a
root-mean-square speed u ~ <u · u>1/2 ) and length l
characteristic of the largest eddies.1
The turbulent kinetic energy (TKE) spectrum
A component u(x) of the instantaneous turbulent
velocity, measured as a function of horizontal distance at some depth in the SML, might look like the
uppermost trace seen in Fig. 8a. A common mathematical means of describing the spatial scales pre1
In this text, use of the symbol “~“ implies only that a suggested
relationship between variables is dimensionally correct: replacement of “~“ by an equality sign requires specification of a nondimensional constant value, normally obtained by referring to experimental material on the flow in question.
FIG. 7 – Schematic of the upper mixing layer of the ocean, showing
velocity and length scales associated with a Reynolds decomposition
of the flow into a (time-) mean component U(z), characterized by
velocity scale Uo and length scale H, and a fluctuating or turbulent
component {u,l}. The horizontal dashed line is the mean depth of
the mixing layer, the convoluted interface an instantaneous position.
sent in such a record is the Fourier transform, a
method of decomposing the signal u(x) into a finite
sum of sinusoids:
u(x) = ∑ ck ei (kx+α k ) =
= ∑ ( a kc o s ( k x ) + b ks i n ( k x ) )
(1)
where ck = 1/2(ak2 + bk2)1/2 and k = 2π/l is the radian
wavenumber associated with the (here horizontal)
wavelength l. Because wavenumber k and wavelength l are inversely related, low wavenumbers are
associated with large scales, high wavenumbers with
small scales. The amplitude ck and phase angle αk (or
alternatively the coefficients ak and bk) are chosen so
that the various sinusoids depicted in the lower panel
of Fig. 8a sum to the signal u(x). Then the turbulent
kinetic energy (per unit mass) E, proportional to the
square of the turbulent velocity, can be written as
E ~ u2 ~ ∑ uk2 ~ ∑Ek ~ ∑ φE ∆k
(2)
where Ek is called the energy spectrum. If ∆k is the
wavenumber resolution of the Fourier transform, the
kinetic energy spectral density function
φE ≡
Ek
∆k
(3)
is defined so that turbulent kinetic energy E is the
area (shaded in Fig. 8b) under the curve of φE as a
function of wavenumber k. The shape of φE, in particular the location of its maximum, tells us which
scales contain the turbulent kinetic energy. The
schematic of Fig. 8b, drawn using experimental
information on a wide variety of turbulent flows,
indicates that E is concentrated at wavenumbers
associated with the large-eddy scale l, near the scale
H of the mean flow. Much smaller and much larger
scales contain very little energy.
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 31
FIG. 8 – (a) Sketch of the Fourier decomposition (lower panel) of the measurement of a turbulent velocity component
u(x) as a function of horizontal distance (upper panel): for details, see p. 31. (b) Sketch of the power spectral density φE
of turbulent kinetic energy E as a function of horizontal wavenumber k. E is proportional to the area under this curve.
E is supplied (S(Uo)) from the mean flow mostly at low k ~ kH (large scales, comparable to H, the scale of the mean
flow), and dissipated (ε(ν)) at high k ~ kν, corresponding to scales small enough that molecular viscosity ν becomes
important. (c) Sketch of the power spectral density of turbulent shear, φε ~ ν k2 φE. The turbulent kinetic energy dissipation rate ε is proportional to the area under this curve, which is small both at low wavenumbers, where k is small, and
at high wavenumbers, where φE is small.
The turbulent energy “cascade”
and the Kolmogoroff length scale
A conceptual equation for the time rate of change
of turbulent kinetic energy E in an unstratified fluid
is:
∂E = S - ε
(4)
∂t
here S denotes the rate at which the turbulent field
receives energy from the mean flow field or external
forcing, and ε denotes the loss due to viscous (intermolecular) forces which occurs in all fluids at a rate
proportional to the fluid kinematic viscosity ν (see
Tennekes and Lumley, 1972, Sec.3.2). A common
assumption is that the turbulent flow is “stationary”
or steady-state, i. e. that E does not vary with time;
for this to be true, the source(s) of TKE must balance the sink, i. e. S = ε for a stationary flow 2. Energy is supplied mainly at scales of variability near
those of the mean flow, schematically S(Uo) in Fig.
32 A.E. GARGETT
8b. Viscous losses, depicted as ε(ν) in Fig. 8b, occur
only at very small scales of motion, where viscous
drag forces convert velocity fluctuations to heat
(intermolecular motion). The intermediate scales are
generally assumed to be part of an energy “cascade”,
first hypothesized by Kolmogoroff (1941), in which
energy is moved progressively from larger to smaller scales: there is no energy supply or loss directly
to scales within this “cascade” range, merely a transfer of energy (small arrows in Fig. 8b) which ends at
the smallest scale of velocity variance, that associated with wavenumber kν in Fig. 8b, where viscosity
smooths out the turbulent fluctuations.
What is this smallest scale? The answer to this
question is a classical derivation of turbulence “theory”, which is frequently nothing more than dimen2
In a stably-stratified fluid, this steady-state equation must be written rather as S = ε + b, where b is the buoyancy work, work done
against gravity in raising heavy over light fluid: it is generally
assumed that b/ε << 1, so that this effect is only a small correction
to the arguments which follow.
sional analysis inspired by a vision of the underlying
physical processes. In this case, based on the vision
of the “cascade” depicted in Fig. 8b, Kolmogoroff
(1941) assumed that only two parameters could
affect kν, the rate ε at which energy was arriving at
dissipation scales, and ν the kinematic viscosity of
the fluid. Using the notation [variable] to denote the
dimensions of the variable, [ε] = [E/t] = L2 / t3, [ν] =
L2 / t , and [kν] = 1/L, so for dimensional consistency alone, we must have
ε
kν ~ ⎛ 3 ⎞
⎝ν ⎠
1
4
ε
= cν ⎛ 3 ⎞
⎝ν ⎠
1
4
(5)
Information from a wide variety of oceanic turbulent flows (Grant et al. 1962, Oakey 1982, Gargett
et al. 1984) shows that the non-dimensional coefficient cν required to replace “~” by an equality is in
fact of order 1, i. e.
ε
kν = ⎛ 3 ⎞
⎝ν ⎠
1
4
,
associated with a length scale lν = 2π/kν .
Note that Kolmogoroff originally derived the
dimensionally correct viscous length scale
⎛ ν3 ⎞
η~ ⎜ ⎟
⎝ ε ⎠
1
4
= cη kν-1 = cη
lν
.
2π
If the non-dimensional constant cη is taken as 1,
the result is a length scale which is a factor of 2π
smaller than the length scale lν determined by reference to actual observations. The difference between
η and lν , the source of some confusion in the biological literature (Lazier and Mann, 1989), is a clear
example of the way in which the purely dimensional arguments so widespread in turbulence “theory”
must always be referenced to observations if one is
interested in specific values of variables, rather than
just functional relationships.
2
⎛ ∂u ⎞
ε ~ ν ⎜ ⎟ ~ ν∑ k2 ck2 ~ ν∑ k2 Ek ~
⎝ ∂x ⎠
~ ν∑ k2 φE∆k ~ ν∑ φε ∆k
(6)
we see that the shear (or dissipation) spectrum φε is
determined by multiplying the energy spectrum by k2.
This results in a shear spectrum (Fig. 8c) which is
small both at small k (because of the k2 factor) and at
large k (due to small φE). Again from measurements,
the maximum shear is found to reside at relatively
small scales3 (high wavenumbers), of order (3-4) lν .
Scalar spectra
While turbulent velocity spectra, at least at small
scales, have revealed an acceptable degree of “universality”, i. e. common properties over a wide variety of flows, universality is much more controversial for the spectra of scalar properties of the fluid.
Some such properties may be “passive” in the sense
that their presence does not significantly affect the
dynamics of the flow: others such as temperature
and salinity (hence, by association, nutrient concentrations) may be “active” scalars, affecting flow
dynamics through their effect on fluid density. Lacking a comprehensive theory, we presently continue
to believe that the finest scale of variance in the field
of a scalar property C is set by its molecular diffusivity DC and related to the Batchelor wavenumber
(Batchelor 1959; Tennekes and Lumley, 1972, Sec.
8.6):
1
1
⎛ ε ⎞ 4 ⎛ ν ⎞ 2
kc = ⎜
(7)
⎟ = ⎜ ⎟ kν
⎝ ν D 2c ⎠
⎝ Dc ⎠
For the ocean environment of plankton, temperature T and salinity S are important scalars; given
typical values of ν, DT and DS:
ν ≅ 1x10-6 m2/s
The spectrum of turbulent shear (dissipation)
DT ≅ 1x10-7 m2/s ≅ ν ⇒ kT ≅ 3 kν
10
To calculate kν, it is necessary to determine ε, the
dissipation rate of E, which is proportional to the
shear of the turbulent velocity field, i. e.
⎛ ∂u ⎞
ε~ν⎜ ⎟
⎝ ∂x ⎠
DS ≅ 1x10-9 m2/s ≅
ν
⇒ kS ≅ 30 kν
1000
2
(Tennekes and Lumley, 1972, Sec.3.2). Taking the
derivative with respect to x of expression (1) for the
Fourier decomposition of u(x) and squaring,
3
All shear spectral peaks lie within this range. However shear
spectral shape does vary slightly depending upon whether a sensor measures u or v, the fluctuating velocity component measured
respectively parallel or perpendicular to the direction of the vector wavenumber k. A hot film probe measures u, an airfoil probe
measures v.
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 33
is large (10-1 cm2/s3), the peak of the velocity shear
(dissipation) spectrum is at scales smaller (wavenumbers larger) than 1 cm; if ε is small (10-4 cm2/s3),
the shear variance lies at larger scales (smaller
wavenumbers). Thus the shear environment at the
scale of a particular biological entity will depend
not only upon its size, but on the strength of turbulence in its environment. On a plot similar to
Fig. 9, Denman and Gargett (1995) show the locations of sizes typical of a zooplankton predator
and a phytoplankton prey, as well as a size range
for marine snow.
Notes on the shear environment of single
plankters
FIG. 9. – Non-dimensional
dissipation spectra of u,v,T, and S as
^
functions of k = k/kν, wavenumber k non-dimensionalized by kν ^=
of kν
(ε/ν3)1/4 . Amplitudes are arbitrary, but the relative^ locations
^
and the non-dimensional Batchelor wavenumbers kT and kS associated with temperature
T and salinity S are accurate. The non-dimen^
sional wavenumber kη = kη/ kν = 2π, associated with the length scale
originally formulated by Kolmogoroff
(1941), is considerably larg^
er than the wavenumber kν = 1 at which observations show that
shear variance is effectively zero. Arrowed locations show the
wavenumbers corresponding to a physical length scale of 1 cm, for
two values of ε. With ε = 10-1 cm2s-3 (a reasonably large value, typical of the upper SML under moderate wind forcing), the shear
maximum is at higher wavenumbers, ie at scales smaller than 1 cm:
with ε = 10-4 cm2s-3 (typical of the lower mixed layer or, sporadically, the stratified region at the mixed layer base), the shear maximum is at lower wavenumbers, ie the shear is contained in scales
mostly larger than 1 cm.
we see that in a turbulent field, scalar variance is
present to much higher wavenumbers (smaller
scales) than velocity variance, a fact that may have
biological implications, for example when considering chemosensory behaviour of individual organisms.
Fig. 9 is a summary schematic4, showing the
relative location of velocity, T and S gradient
^
spectra as a function of k, wavenumber normalized
by kν. Magnitudes are arbitrary, but the relative
^
^
^
locations of the wavenumbers kν, kT, and kS, as
^
well as that (kη) associated with the Kolmogoroff
length scale η, are accurate. Because kν depends
on ε, the location in this plot of the wavenumber
associated with a particular physical length scale
will vary with ε. The location of a 1 cm length
scale is shown by arrows for two values of ε: if ε
4
Because of the large range of wavenumbers involved in depicting
scalar as well as velocity spectra, these are plots as functions of
log(k) rather than k. In order to preserve areas (proportional to dissipation rates), the variable plotted is k3 φ(k), since Σ k2 φ(k) ∆k ~
Σ k3 φ(k) (∆k/k) ~ Σ k3 φ(k)∆(log(k)).
34 A.E. GARGETT
In an influential paper, Lazier and Mann (1989)
suggested that over distances smaller than lν, the
motion field due to turbulence consists of an approximately linear shear with magnitude determined
by ε:
du
ε
= csh ⎛ ⎞
⎝ ν⎠
dz
1
2
(8)
where the coefficient csh is of order 1/2. Although
this relationship may provide an estimate of the
magnitude of a small-scale linear shear associated
with a turbulent field, Lazier and Mann also point
out that turbulent shear varies randomly in direction and that shear magnitude is a distributed variable, i. e. one characterized as much by its variability as its mean value. To these complications
must be added the fact that turbulent shear is not
evenly distributed in space. Indeed another characteristic of turbulence is that of “intermittency”,
basically the observation that mean square shear
(and variance of any scalars) in a volume of fluid is
made up of some very large values in a small fraction of the volume and many small values in the
rest (the subject of intermittency is treated extensively elsewhere in these notes). While integrating
a shear spectrum is an appropriate means of determining the volume-average value of mean square
shear, it provides no information on its spatial distribution within the volume. The question of how
to relate a measured value of ε (necessarily an
averaged quantity) to the spatial distribution of
small-scale shear and the length of time over which
directionality is maintained needs further consideration if we seek to understand the instantaneous
shear field surrounding individual plankters.
FIELD MEASUREMENTS OF TURBULENCE
PROPERTIES
As discussed in Sec.2, different questions with
respect to the functioning of the marine food web
require information on different aspects of turbulence. Thus we would like to be able to measure
A. fluxes
B. large eddy scales: any 2 of the set
{u, l, and te}, as related by u ~ l/te
C. dissipation rates, particularly ε,
associated with turbulence in the euphotic zone of
the ocean. What has been possible to measure has
been much more limited, for various reasons. Sensors which work in a laboratory setting often do not
perform in the ocean environment (heated film
probes very quickly become fouled with material of
biological origin, laser systems use too much power
or have been physically too large), and there is
always the problem of a suitable platform for measurements of turbulence at sea, where velocities
associated with platform motion can easily swamp
the turbulent signal. Our knowledge of the properties listed above has thus been (and continues to be)
a strong function of the instruments available at the
time. Turbulence instrumentation and techniques
have evolved steadily over the roughly thirty years
since the pioneering measurements of Grant et al.
(1962), and are most simply classified by decade.
many profiles needed to produce meaningful averages in the often intermittent turbulence characteristic of the ocean environment. A second problem is
that thermistor encounters with the biological material in the ocean can produce both transitory
“spikes” in temperature records, which must be
identified and edited before analysis, and/or permanent change in thermistor calibration. However the
major disadvantage of the χT measurement is that it
is not a direct measurement of any of the features
(A, B or C above) which we wish to know. Instead
some “theory” is needed to proceed.
The first “theory” was that of Osborn and Cox
(1972) who suggested a means of deriving <ρw>,
the vertical turbulent flux of density, via a simplification to the equation for fluctuation density variance <ρ2> (the original Osborn and Cox (1972) formulation is a somewhat complicated version in
terms of entropy: for a simpler derivation, see Gargett (1993)):
<ρw>
(9)
in which (in addition to the Reynolds decomposition) it has been assumed that the turbulence is
steady-state and spatially homogeneous. If it is then
assumed that the turbulent density flux can be parameterized as an eddy diffusivity Kρ for density times
the vertical gradient of mean density ∂ ρ /∂z (see
Tennekes and Lumley, 1972, Sec.2.4), i. e.
<ρw> = -Kρ
The 1970s: Scalar dissipation rate, χT
then
In the early 1970s, Osborn and Cox (1972) pioneered the use of “fast” thermistors on freely falling
vehicles to resolve the temperature spectrum to high
wavenumber, hence determine the temperature dissipation rate χT ~ DT∑ k2 φT . While measurement of
χT formed the basis for the first decade of ocean
microscale observations, it has several difficulties.
There are two practical disadvantages. First,
“fast” thermistors are not very fast, so to resolve the
T gradient spectrum (to kT, see Fig. 9) in strong turbulence, it is necessary to move the sensor rather
slowly through the water (recall the relation F k = ω
between radian wavenumber k and frequency ω for
a probe moving at speed F; since a thermistor has a
fixed upper frequency limit to its response, the higher the value of k which must be resolved, the smaller F must be). The disadvantage of a slow fall speed
is merely the length of time required to gather the
∂ρ
= -χρ
∂z
Kρ=
∂ρ
,
∂z
(10)
χρ
⎛ ∂ρ ⎞
⎜
⎟
⎝ ∂z ⎠
2
–
In fact, we measure instead KT = χT /(∂T/∂z)2 and
use a final assumption that Kρ = KT to complete an
estimate of the density flux from the measurement
of χT and mean gradients. The accuracy of the
results of this “theory” is of course dependent upon
the unknown degree to which all the underlying
assumptions are met.
A second “theory” involving the interpretation of
χT was the suggestion by Dillon and Caldwell
(1980) that measurement of the cut-off wavenumber
ko of the temperature gradient spectrum could be
used to deduce ε, if it is assumed that ko is a constant
times the Batchelor scale for temperature; i. e.
ko = cT kT = cT (ε/νDT2)1/4 can be solved for ε if cT is
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 35
known. The accuracy of this result unfortunately
depends upon the degree of universality of the shape
and magnitude of the temperature spectrum, which
is increasingly being questioned (Gargett 1985,
Miller and Dimotakis 1991).
The 1980s: Turbulent kinetic energy dissipation
rate ε
In the mid-1970s, Osborn (1974) introduced a
new ocean turbulence sensor, the airfoil probe. The
sensing element is a tiny bimetallic strip (essentially a phonograph cartridge) which produces an electrical signal when bent. This element is mounted at
the end of a narrow tube and waterproofed in an airfoil-shaped tip, made of a soft epoxy which allows
it to bend under the cross-forces associated with a
velocity field v(z) normal to its length, shown
schematically in Fig. 10. The electrical signal produced is proportional to v(z), but is often differentiated before recording5, providing a signal proportional to velocity gradient or shear; hence the alternate name “shear probe”. By the beginning of the
1980s, the airfoil probe had been proven in the field
against a hot-film technique (Gargett et al. 1984)
and laboratory measurements of its high wavenumber response had been carried out against a laserDoppler velocimeter (Ninnis 1984). This high
wavenumber resolution is a function only of the
physical size of the airfoil, so does not degrade with
increase of mean speed F of the probe through the
water. Since the gain of the probe increases as F2, a
larger value of F also means larger signal levels.
Thus successful airfoil probe profiles of ε are taken
more quickly than thermistor measurements of χT.
However there remain some problems with the airfoil probe technique. Impacts with individual
plankters cause “spikes” in the velocity record
which must be edited before ε is calculated. More
serious is contamination of the shear record by
vibrations of the vehicle which carries the probe,
limiting use of airfoils mostly to free-fall profilers
and fall speeds less than ~1 m/s. In this configuration however, airfoil probes have been used extensively in many deep-ocean and coastal applications
during the last 20 years.
5
Because the probe is usually mounted on a free-fall body which
moves horizontally with the flow on vertical scales larger than its
own length (usually around 2m), there is no relative velocity signal
to be sensed by the airfoil on larger scales. Thus with no loss of
available signal, differentiation can be used to emphasize the smallscale velocity gradients which contribute to ε.
36 A.E. GARGETT
FIG. 10. – Sketch of airfoil (shear) probe operation. When the probe
moves through v(z), a variable velocity normal to its axis, side
forces deflect the soft epoxy tip, bending a sensing element (essentially a phonograph cartridge) which produces an electrical signal
proportional to v(z).
Properly calibrated and operated, airfoil probes provide direct measurements of ε (Item C above), a major
advance over the previous indirect technique via χT.
However they provide no information on the largeeddy scales (Item B), since this information is lost in
the cascade of turbulent kinetic energy to the dissipation scales which are measured. Turbulent fluxes (Item
A) must again be derived by use of “theory”, here that
of Osborn (1980) who suggested simplifying the
steady-state equation for turbulent kinetic energy per
unit mass E (see Tennekes and Lumley 1972, Sec.3.2)
by dropping both the pressure/velocity correlation term
and the term containing triple correlations among turbulent velocities and their gradients (because we cannot measure either term) to yield a balance
∂E
∂U i
+ <uiw>
= – ε – b,
∂t
∂z
in which E is supplied by Reynolds stresses <uiw>
working against the vertical gradient of mean velocity ∂Ui /∂z , and dissipated by viscosity (ε) and by
work against buoyancy forces in a stably stratified
fluid (b ≡ gρo-1<ρw> , see Turner, 1973, Sec.5.1.2).
At steady-state (∂E/∂t = 0), this can be rewritten as
S=ε+b
(11)
where S ≡ - <uiw> (∂Ui /∂z) > 0. Defining the flux
Richardson number Rf as
Rf =
b
b
=
S ε +b
(12)
(Tennekes and Lumley 1972, Sec.3.4), eqn. (11) can
be rewritten as
b
= ε+ b or <ρw> = Γ ρo g-1 ε
Rf
where
Rf
b
=
1- R f ε
With the eddy diffusion assumption of eqn. (10 ),
Kρ can then be determined as
Rf ε
ε
Kρ =
=Γ 2
2
1- R f N
N
,
FIG. 11. – Flux coefficients as functions of gradient Richardson
Number Ri, from a variety of laboratory measurements. The curve
is drawn through values of Γtotal ≡ b/εtotal collated by Linden (1979).
Filled squares scattered around the value 0.25 are McEwan’s (1983)
measurements of Γ ≡ b/εmix , where εmix = εtotal - εν , εν being the energy dissipated in sidewall viscous boundary layers without causing
vertical mixing of density. See text for discussion of the
differences between these two measures.
Γ=
where
∂ρ
∂z
is the Brunt-Väisälä or buoyancy frequency (Tennekes and Lumley 1972, Sec.3.4).
An actual value for Kρ is usually determined by
assuming that Rf ≅ constant ≅ 0.2, hence Γ ≅ 0.25 .
Although this procedure is the basis of most of
the estimates of vertical eddy diffusivity, hence turbulent fluxes, made in the past decade, it is important to realize that such estimates are only as good as
the assumptions built into the above derivation. Two
main questions must be kept in mind.
N 2 ≡ −gρo−1
Is Rf really constant?
The basis which presently exists for an assumption of near-constant Γ, equivalent to near-constant Rf, is shown in Fig. 11, redrawn from McEwan
(1983)6. The curve (roughly fitting observations compiled by Linden (1979) from a range of laboratory
experiments) shows non-constant Γ, with an apparent
maximum at an intermediate value of Ri, the overall
6
Note that McEwan identified b/ε = “ Rf”, whereas in the usual definition (eqn. (12)), the flux Richardson number Rf = b/(b+ε); McEwan’s “ Rf” = Γ in the notation employed here.
gradient Richardson number of the mean flow (see
Turner, 1973, Sec. 1.4). McEwan (1983) argued that
this result is an artifact of the small scale of most laboratory experiments, which causes a large amount
(εν) of turbulent kinetic energy dissipation to occur
through the action of molecular viscosity in sidewall
boundary layers where it is not associated with any
vertical density transport. Thus εtotal = εν + εmix where
εmix is the amount of dissipation which is associated
with density transport. In cases where Ri is relatively
large and density transport is associated with instabilities in a small fraction of the stratified interior of the
fluid (as was the case in McEwan’s laboratory flow
and is characteristic of the ocean interior) εν >> εmix,
hence Γtotal ≡ b/εtotal << Γ ≡ b/εmix. McEwan (1983)
corrected his laboratory results by calculating and
removing εν, and reported the scattered but much
more constant values of Γ shown as filled squares in
Fig. 11; these values are the basis for the assumption
of constant Γ ≅ 0.25, hence Rf ≅ 0.2 . Constancy of Rf
continues to be an open question, however, as very
recent ocean observations suggest that Rf may be a
more complicated function of flow morphology (Gargett and Moum 1995) and/or water column T/S structure (Ruddick et al. 1996). This latter result introduces the second question associated with estimates
of diffusivities from dissipation rates ...
Are vertical eddy diffusivities the same for all
water properties?
Strong continuous turbulence is likely to mix all
water properties equally thoroughly. However turTURBULENCE IN THE OCEAN EUPHOTIC ZONE 37
bulence in the ocean, even in the near-surface layer
which is directly forced by the atmosphere, is often
weak and/or sporadic. In addition, the fact that the
density of ocean water depends upon both T and S
allows the existence of “double diffusive” instabilities (see Turner, 1973, Chapter 8) which are characterized by different effective diffusivities for the two
stratifying properties. The salt-fingering (SF) instability and the double-diffusive layering (DDL) instability, shown schematically in the upper two sets of
panels in Fig. 12, occur when one of the two stratifying components (S for SF, T for DDL) has a vertical gradient which by itself would cause static instability; the water column is stabilized by the vertical
distribution of the second component. Mean property distributions (panels (i)) favouring SF are typical
of the large subtropical gyres of the world ocean,
while those allowing DDL are widespread in Arctic
and Antarctic regions.
Underlying both double diffusive instabilities is
the factor of roughly 100 difference in the molecular
diffusivities for T and S. Consider the example of
salt fingers (upper row of Fig. 12), where the mean
state shown in panel (i) is a hot (H), salty (S) upper
layer lying above a colder (C), fresher (F) layer.
Suppose that a small perturbation moves “blobs” of
fluid across the interface between the layers, as
shown schematically in panel(ii). Because the mean
state of the fluid is stable, these “blobs” will tend to
fall back towards their original positions; however
before they can do so, they very rapidly exchange
heat (T) with their surroundings, without exchanging any significant amount of salt (because of the
much smaller molecular diffusivity of S). The result
(panel(iii)) is a “blob” above (below) the interface
which is lighter (heavier) than its surroundings,
hence tends to rise (fall). These tendencies produce
the long thin vertical columns of fluid, alternately
rising and falling, which are called salt fingers.
Because the heat transfer takes place close to the
interface, while the finger structures move the salt
anomalies much farther in the vertical, salt fingers
are characterized by more effective vertical transfer
of S than T, denoted by a diffusivity ratio d ≡ KS/KT
> 1. Following the same kind of arguments in the
middle set of panels for DDL, we see that the
“blobs” left after heat exchange with the surroundings tend to fall back to their stable positions, returning with salt content nearly unchanged but having
transferred heat (which results in layering in a continuously stratified fluid, as the released heat causes
convection over a vertical scale dependent on the
38 A.E. GARGETT
FIG. 12. – Schematics of the salt-fingering (SF: upper panels) and
diffusive layering (DDL: centre panels) types of double-diffusive
instabilities, and of a possible differential diffusion (DD: lower panels) mechanism of “ordinary” turbulence. In panels (ii), wavy
arrows denote the direction of (rapid) heat transfer into or out of displaced fluid parcels: in panels (iii) straight arrows show the direction of the vertical motion of the fluid parcels which results from
their modified buoyancy. The SF mechanism transfers salt more
efficiently than temperature, resulting in a ratio of effective diffusivities d ≡ KT/KS > 1. Both the DDL and DD mechanisms are
associated with d < 1. For details, see text.
original stratification). Thus the DDL instability is
characterized by more efficient vertical transfer of T
than S, hence d < 1. Although not “turbulent” in the
usual sense of the word, both double-diffusive
processes result in vertical diffusivities for both
components which are greater than molecular values, hence must be considered in a more general
interpretation of “turbulent transfers”.
Evidence of the occurrence of both types of double-diffusive instability in the ocean has accumulated over the years (Schmitt, 1994) provides a recent
overview of both theory and observations). Fig. 13
shows records of fluctuating temperature T′ measured from a horizontal tow in the subtropical North
Pacific, contrasting the ragged appearance of a typical turbulence signal (TT) with the limited amplitudes and length-scales typical of a field of salt fingers (SF) sampled normal to their long (vertical)
dimension. In other suitable areas, small-scale structures appear to be more complex (Schmitt et al.
1987), but the ratio of T to S fluxes still suggest the
action of double diffusion in the SF sense. Evidence
for the DDL instability is found in observations of
strongly layered structures, such as those seen in Fig.
14, which have been observed in many regions of the
Arctic (Perkin and Lewis, 1984) and Antarctic
oceans (Muench et al. 1990) where the mean fields
are suitable. While there remains little doubt that
FIG. 13. – Contrast between the small-scale temperature signals (T′) of salt fingers (SF) and turbulence (TT), reported by
Gargett and Schmitt (1982) from constant-depth towed measurements in the subtropical North Pacific.
double diffusive processes occur in the ocean, we are
unfortunately far from being able to parameterize
their net effects on vertical transfers of T and S.
Finally, it is also necessary to consider the possibility (Gargett, 1988) that even “ordinary” turbulence may be characterized by differential transfer
rates of T and S, as shown schematically in the bottom panels of Fig. 12. Here the mean distributions of
both T and S are stabilizing (as for example in the
subarctic North Pacific in summer). If turbulence is
sporadic, displaced “blobs” of fluid exchanging heat
more quickly than salt by molecular processes (the
schematic shows a particular case when both
“blobs” come to the same T, denoted as W for
“warm”) may tend to fall back to their original positions with S content nearly unchanged; such a
process would be associated with d < 1. This
process, denoted DD for differential diffusion, is not
double-diffusive in nature, since neither T nor S has
a destabilizing mean gradient.
The 1990s: Large-eddy characteristics
FIG. 14. – Layered structure observed in the Arctic Ocean where
mean T/S structure is favourable to DDL, the diffusive layering
double-diffusive instability: from Perkin and Lewis (1984).
During the mid-1980s, as observation of ε with
airfoil probes was becoming relatively routine
(though by no means easy, and certainly not inexpensive), attention moved towards direct observation of features of the large-eddy structures of ocean
turbulence. Reasons for this were many. First, measurements at the dissipation end of the energy “cascade” provide no information on the scales which
receive the energy, i. e. no direct information on the
actual mechanisms responsible for generating turbulence under different conditions of flow and/or forcing. Second, as discussed in pp. 26-28, some applications require estimates of length and time scales of
the large eddies. Third, the number of assumptions
involved in deducing fluxes from either dissipationscale variable is uncomfortably large. Finally, measurement of large-eddy characteristics raises the
possibility of providing information on all of the
fundamental turbulence properties required for
items A through C above, as follows:
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 39
(1) any 2 of the large-eddy scales {u, l, te} provide advective space/time scales;
(2) the scaling relation
ε~
u3
u2
~
l
te
(14)
(see Tennekes and Lumley 1972, Sec.1.5) may provide an estimate of ε, to a constant factor which
must be determined;
(3) the scaling relation
K~ul
(15)
(see Tennekes and Lumley 1972, Sec.1.5) may provide an estimate of the turbulent eddy diffusivity for
scalars, again within a factor which must be determined.
The word “may” is used advisedly in association
with relations (14) and (15), since these remain
areas of active research; (15) is particularly uncertain due to the probable influence of the molecular
diffusivity of the scalar. However the possibility of
achieving even order of magnitude estimates of ε
and K from the measurement of large-eddy characteristics is exciting. This last section looks briefly at
some new instruments and techniques for measuring
large-eddy characteristics, as well as their extensions to estimates of ε via (14).
Neutrally buoyant floats
Fig. 15 is an example of the diurnal evolution of
large-eddy scales in a convectively forced SML, as
revealed by the vertical excursions of neutrally
buoyant floats (E. D’Asaro and G. Dairiki, personal
communication; redrawn with permission). Each
light line is the depth/time history of a single float;
one such track (actually made up from three separate
float tracks) is emphasized to depict a possible depth
history of a non-motile plankter. Vertical excursions
are reduced during the day as solar heating stabilizes
the near-surface water column. At night, the surface
ocean is cooled by heat loss to the atmosphere, forcing convective motion with vertical scale which
increases with time. Note the large variability in
FIG. 15. – Displacements of an acoustically-tracked neutrally-buoyant float in a convectively forced ocean surface layer
(modified with permission from E. D’Asaro and G. Dairiki, Univ. of Washington). The diurnal cycle of surface heat
loss/gain has a dramatic influence on the vertical scale of the turbulent eddies which advect the non-motile part of the
marine food web in the strongly surface-intensified light field of the ocean.
40 A.E. GARGETT
FIG. 16. – (a) An acoustic drifter for observing large-eddy structures in the near-surface ocean (courtesy of Dr. D. Farmer, Institute of Ocean
Sciences, Canada). (b) A vertical narrow-beam sonar observes clouds of bubbles (dark shading), generated by breaking surface waves and
swept into the interior by convergences associated with Langmuir circulations. Doppler processing of the same signal provides estimates of w,
the vertical velocities associated with this particular form of large-eddy structure. (Redrawn with permission from Farmer and Polonichko
(1995)). (c) A sweeping side-scan sonar on the same instrument reveals the horizontal spatial structure of Langmuir cells (the dark bands are
regions of high acoustic return from bubbles concentrated in the surface convergences). This measurement demonstrates the familiar elongation of cells in the direction of the wind, but also the substantial 3-dimensionality associated with the circulations. (Figure courtesy of
M. Trevorrow and D. Farmer, Institute of Ocean Sciences, Canada).
depths achieved by individual floats at any particular time of day, a result of the day-to-day variation
of the surface forcing.
Acoustics
Over the past decade, high frequency acoustic
backscatter has been widely used to provide at least
rudimentary information on distribution of zooplankton biomass. More recently, both backscatter
amplitude and, increasingly, Doppler velocity measurements are being used to determine the characteristics of large-eddies in the ocean.
Fig. 16a is an example of the kind of instrumentation now being used to study turbulent processes
in the near-surface ocean. A variety of acoustic
sounders, isolated from surface wave motions by a
long elastic cable between submerged and surface
buoys, look upwards towards the sea surface. A nar-
row-beam sonar directed straight upwards measures
the backscatter amplitude from clouds of microbubbles formed by breaking waves and pulled
downwards in the convergences associated with
Langmuir cells (a particular kind of large-eddy
structure formed when wind blows in a fixed direction, Plueddemann et al. 1996). The same signal can
be processed for Doppler-shift, resulting in the estimates of along-beam (here vertical) velocities as a
function of range (depth) shown superimposed on
the bubble clouds of Fig. 16b. Side-scan (widebeam) sonars on the same buoy provide 2-dimensional images of the distribution of bubble clouds
just below the sea-surface, as shown in Fig. 16c.
While the Langmuir convergences (associated with
large backscatter from high bubble density, coded
black) tend to lie approximately in the downwind
direction, it is clear that such structures contain a
high degree of 3-dimensionality.
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 41
FIG. 17. – Comparison of ε measurements derived by two independent means. Airfoil probe measurements (heavy line) were taken by a standard microscale profiler (profiles courtesy of Dr. J. Moum, Oregon State University) at the locations marked (a) and (b) in Fig.5. The other
technique (Gargett, 1994) uses large-eddy length and velocity scales measured by a vertical beam Doppler current profiler, interpreted by
eqn.(14): small dots are individual estimates, while the light line is an average over 21 ADCP profiles, taken every 2 s. Such results suggest
that Doppler-based estimates may make possible spatial surveys and/or moored time series measurements of ε, a critical variable in studies
of the interaction between turbulence and zooplankton.
In the coastal ocean, ship-mounted acoustic
Doppler current profilers (ADCPs), modified to
measure vertical velocity (Gargett 1994) offer an
essential ability to survey for significant mixing
structures and, once found, to determine large-eddy
scales from which ε may be estimated from the
relation (14). The unknown constant in (14) may
be determined by comparison with ε values measured by airfoil probes on a standard microscale
profiler. Fig. 17 shows such comparisons for profiles taken at the locations marked (a) and (b) in
Fig. 5. The observed agreement offers the possibility of sampling ε continuously in highly variable
coastal flows which are severely under-sampled by
conventional free-fall profilers. However the very
strength and variability of such flows makes it difficult to decide when the 2 types of profilers ought
to be measuring the same thing (see Gargett 1994).
It is easier to verify the relationship (14) in an offshore ocean environment, where horizontal variability is typically much less and both large-eddy
characteristics and ε are determined by instruments
on the same vehicle. Peters et al. (1995) and Moum
(1996) report such verification using data from
microscale profilers operated in the stratified interior of the ocean; a caution, mentioned by Moum,
is that values of the scaling “constant” must be
expected to vary somewhat, depending on the spe42 A.E. GARGETT
cific measured variables used to define the largeeddy scales.
Before leaving the area of acoustics, it should be
mentioned that commercial ADCPs also provide a
low-resolution measure of backscatter amplitude,
believed to be roughly proportional to zooplankton
biomass for the frequencies at which most ADCPs
operate. An example of the fields of vertical velocity and backscatter amplitude measured during transit of a shallow and narrow tidal channel is shown in
Fig. 18. In the absence of flow, backscatter amplitude would increase towards the channel bottom,
since zooplankton retreat to depth during the daylight hours of these measurements. In this highly
turbulent flow however, the w field (bottom panel)
consists of strong up and down drafts. These often
extend over the full depth of the channel and act to
redistribute zooplankton biomass within the water
column, as evidenced by the “ribbons” of high
backscatter amplitude originating at the bottom.
Despite the limitations involved in interpreting
backscatter amplitude as zooplankton abundance (i.
e. no species decomposition, etc), the simultaneous
measurement of large-eddy characteristics of turbulence and a variable even roughly proportional to
zooplankton biomass has considerable potential in
field studies of the effects of turbulence on predator/prey interactions.
FIG. 18. – Survey of backscatter amplitude A (upper pannel) and vertical velocity w (lower pannel) as measured simultaneously by a Doppler
current profiler during a ship transect of Sansum Narrows, a narrow and shallow tidal passage in the southern coastal waters of British Columbia (marked as N in Fig. 6a). Mean tidal flow causes intense turbulence in the Narrows, generating strong up and down drafts which
scour biological material from its normal daytime position at depth and redistribute it within the water column.
Thorpe scales
The increasing importance of measurements of
ε in biological field programs poses a considerable
problem, given the small number and large cost of
specialized microscale groups. I believe that estimates of ε which would be useful in the biological
context could be provided reasonably cheaply and
easily by measurement of Thorpe scale. Thorpe
(1977) first suggested that an estimate of the vertical overturning length in a stratified fluid could be
obtained by sorting an unstable measured density
profile into a stable configuration. As shown in
Fig. 19a, if measured density at a point is greater
than that at the next deeper point, the two points
are interchanged: in a more complicated measured
profile, this procedure is repeated until an over-all
stable profile results. The Thorpe scale LT for each
point is then the difference between its initial and
final positions, and has been interpreted as the vertical length over which an initially stable density
gradient was overturned. The second step is the
observation that
Lo ≡ (ε/N3)1/2 ~ LT
where Lo is the so-called Ozmidov length scale. Fig.
19b shows some of the original observations of this
relationship, from Dillon (1982): values for ε were
obtained by the indirect temperature spectral technique described in p. 35, and gave a best-fit of Lo =
0.8 LT . Subsequent studies (Crawford 1986; Wesson and Gregg 1994) with direct ε measurements
have found a somewhat higher degree of scatter, eg
those of Peters et al. (1988) shown in Fig. 19c which
exhibit coefficients between 0.5 and 2. Writing
Lo = co LT , and solving for ε yields
ε = co2 N3 LT2
where 0.5 < co < 2, hence 0.25 < co2 < 4 .
Thus an estimate of ε derived from density
measurements at the finescale would only be accurate to about a factor of 16, the difference between
the two ends of the observed range in the “constant” co. However, the variable which is relevant
to the interaction between zooplankton feeding
TURBULENCE IN THE OCEAN EUPHOTIC ZONE 43
FIG. 19. – (a) Schematic of the calculation of Thorpe scale LT . A measured density profile (filled dots) is re-arranged to produce a statically stable density profile: LT is the distance between the initial and final positions of fluid “parcels”. (b) Observations of Dillon (1982)
initially suggested a linear relationship between Thorpe scale LT and the length scale Lo ≡ (ε/N3)1/2 ; here Lo ≅ 0.8 LT, from which ε may be
calculated as ε ≅ 0.6 N3 LT2. (c) Larger data sets have typically shown more scatter; eg Peters et al. (1988) observe Lo ≅ (0.5-2.0) LT in these
measurements from the equatorial Pacific (redrawn with permission). However since kν depends upon the 1/4 power of ε, this spread in
coefficients results in only a factor of 2 inaccuracy in determining the biologically important variable kν.
and turbulence is not ε itself, but the Kolmogoroff
wavenumber
kν ≡ ⎛ ε3 ⎞
⎝ν ⎠
1
4
= co1/2 ⎛ N ⎞
⎝ ν⎠
3
4
LT1/2 ,
where 0.7 < co1/2 < 1.4, which is much more accurate (to about a factor of 2) because ε enters only
to the 1/4 power. Thus it seems likely that measurement of Thorpe scales could produce reasonably accurate estimates for the smallest (average)
shear scale in a turbulent field. A useful system
would cost far less than a full microscale profiler,
in both initial and operating costs, at the same
time providing the more familiar mean profiles of
T and S.
44 A.E. GARGETT
Like any other technique, measurement of Thorpe scale is not without pitfalls - ideally one requires
a freely-falling (for isolation from surface motions)
CTD with matched-response C and T sensors (to
minimize false “overturns” due to salinity spiking),
high data rate (to resolve small overturns), and low
noise (to allow determination of “overturns” in low
density gradients). In addition, work remains to be
done in defining the boundary between the “interior” of a surface mixing layer and the near-surface
stress-driven layer, where measurement of kν via
Thorpe scale will not be valid (but where alternate
scalings are available). Nevertheless, widespread
use of surrogate measurements of kν via Thorpe
scale could yield spatial and temporal coverage of
measurements which is unattainable (at any reason-
able cost) from dedicated microscale profiling. The
associated measurement inaccuracy relative to
direct airfoil probe measurement may be acceptable
in view of the present severe under-sampling of the
turbulent regimes of the ocean in association with
the embedded biological community.
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