SCI. MAR., 61 (Supl. 1): 141-158
SCIENTIA MARINA
1997
LECTURES ON PLANKTON AND TURBULENCE, C. MARRASÉ, E. SAIZ and J.M. REDONDO (eds.)
Small-scale turbulence, marine snow
formation, and planktivorous feeding*
THOMAS KIØRBOE
Danish Institute for Fisheries Research, Charlottenlund Castle. DK-2920 Charlottenlund, Denmark.
SUMMARY: This paper examines how turbulence influences two very basic properties of planktonic ecosystems, namely
trophic interactions and vertical flux of particulate material. It starts with a simple account of classical particle encounter
theory which forms the basis of the substance of both problems. Turbulent fluid motion will bring suspended particles to
collide, and the basic equations describing the collision rate as a function of dissipation rate and particle size, concentration
and motility will be presented. The classical (coagulation) theory is then applied to marine snow formation in the ocean: colliding suspended particles may stick together and form mm-cm sized aggregates (marine snow). These aggregates are
believed to account for the vertical flux of matter in the ocean. Aggregation of microscopic phytoplankton cells is a special
case. Examples from laboratory and field experiments are used to demonstrate how phytoplankton cells may coagulate, how
their stickiness may be measured, how coagulation determines the sedimentation of particulate matter in the ocean, and how
it may control the population dynamics of phytoplankton. Subsequently the collision equations are used to describe how
planktivorous predators encounter prey in turbulent environments, and the equations are modified to take predator and prey
behaviour into account. Simple equations that describe prey encounter rates for cruising predators, suspension feeders,
ambush feeders, and pause-travel predators in calm and turbulent water are derived. The influence of fluid motion on postencounter prey capture (pursuit success) is examined. Experimental results on various copepod and larval fish predators will
be used to illustrate the theory. Finally, the significance of size and behaviour is discussed. It is shown that turbulence is
potentially very important for prey encounter in mm-cm sized planktonic predators, while it is unimportant for most larger
and smaller ones.
Key words: Turbulence, phytoplankton aggregation, copepods, fish larvae, feeding.
INTRODUCTION
Ocean turbulence has implications for a number of
fundamental biological processes in the plankton
(Fig. 1). On the larger scale, turbulence may dissipate
patches of elevated food concentration, hence affecting food availability to planktivorous predators. It
may cause entrainment of deep, nutrient rich water
across the pycnocline into the euphotic zone, thereby
making inorganic nutrients available to phytoplankton populations in the upper mixed layer. Finally, it
*Received: November 27, 1995. Accepted: March 14, 1996.
may increase vertical mixing within this layer, causing vertical excursions of phytoplankton cells whereby these become exposed to a variable light climate.
On the micro-scale, which is the topic of this chapter,
turbulence may increase transport of nutrients
towards the surface of phytoplankton cells, thus
increasing their nutrient uptake rate. The effect of this
is normally considered insignificant for small, spherical cells (e.g. Lazier and Mann, 1989). Small-scale
turbulence may increase also the collision rate
between suspended particles and, thus, facilitate their
aggregation (McCave, 1984) and it may increase the
contact rate between planktivorous predators and
their prey (Rothschild and Osborn, 1988). This chapMARINE SNOW AND PLANKTIVOROUS FEEDING 141
the ambient turbulent fluid shear. It is a classical
problem to derive expressions for β. In this account
I shall combine physical and biological approaches
to elaborate on the basic equation 1 to examine (i)
the formation of marine snow aggregates by physical coagulation and its consequences to vertical
material fluxes and to the population dynamics of
diatoms, and (ii) prey encounter and feeding rates in
planktonic predators. In each of the following two
main sections, one dealing with each of these two
topics, I shall first provide a theoretical analysis of
the problem, subsequently present examples and
evidence from laboratory and field studies, and
finally briefly discuss some of the implications. A
list of the notation used is given in Table 1.
This article is not a review and no attempt has
been made to cover the entire literature. The present
lecture notes are based mainly on previously published material, and most of the ideas presented here
stem from Rothschild and Osborn (1988), Jackson
(1990), Kiørboe et al. (1990) and Kiørboe and Saiz
(1995) as well as the literature on which those
papers are based. Throughout this presentation simplicity has been given priority over a more complete
(and complex) description.
FIG. 1. – Implications to biological processes in the plankton of
small-scale and larger scale turbulence. N, P, Z, and F symbolise
inorganic nutrients, phytoplankton, zooplankton, and fish.
ter particularly examines the effects of small-scale
turbulence on (i) particle aggregation, especially
aggregation of phytoplankton cells, and its implications for vertical flux in the ocean and for the dynamics of phytoplankton populations, and (ii) planktivorous feeding. The core of both of these issues is that
particles need come into contact (collide) for something to happen. Contact or encounter with prey is a
prerequisite for planktivorous feeding; and collisions
between suspended particles (such as phytoplankton
cells) is a prerequisite for particles to combine into
aggregates. Small-scale turbulence increases the rate
at which this happens.
The encounter rate, E (# of encounters per unit
time and unit volume), between two types of suspended particles, occurring at concentrations Ca and
Cb, can be written as:
E = βCaCb
Notation
Name
Dimensions
r
Particle radius, encounter
radius
v
Swimming, sinking or feeding
current velocity
f
Pause frequency for a
pause-travel predator
T-1
τ
Pause duration for a
pause-travel predator
T
ω
Tumbling frequency for
random walk
T-1
α
Stickiness
β
Encounter rate kernel
L3T-1
D
Diffusion coefficient
L2T-1
ε
Dissipation rate
L2T-3
γ
Sub-Kolmogorov scale
shear rate
η
Kolmogorov length scale
L
ν
Kinematic viscosity
2
χ
Contact efficiency
(1)
where β is the encounter rate kernel. β has dimensions of volume per unit time (L3T-1) and is a function of the size and motility of the particles and of
142 T. KIØRBOE
TABLE 1. – Notation used
L
LT-1
Dimensionless
T-1
L T-1
Dimensionless
FORMATION OF MARINE SNOW
BY PHYSICAL COAGULATION
Some background
In the ocean there is a constant flux of organic
particles from the euphotic surface layer towards the
bottom. This flux includes live, intact phytoplankton
cells. Observed sinking velocities of, e.g., phytoplankton cells, however, frequently exceed the sinking velocity predicted by Stokes’ law. For example,
Stokes’ settling velocities of say 10 µm diameter
phytoplankton cells is on the order of < 1 m d-1,
while there are many reports of observed settling
velocities exceeding 100 m d-1. The reason for this is
that most of the vertical flux in the ocean is due to
sinking of mm-cm sized particle-aggregates with
enhanced Stokes’ settling velocities; such aggregates are known as ‘marine snow’. Because marine
snow aggregates are extremely fragile they cannot
be sampled by traditional means (nets, bottles), and
they were only discovered by divers in the fifties
and rediscovered in the sixties. Subsequent studies
have demonstrated that marine snow aggregates
occur abundantly in the ocean - and appear to be the
main vehicle for vertical flux (Alldredge and Silver,
1988). They may consist of all kinds of particles;
some are mainly composed of phytoplankton cells,
while others are composed mainly of detritus and
inorganic particles. There are several ways particle
aggregates can be formed; we shall particularly
examine the formation of phytoplankton aggregates
by physical coagulation.
FIG. 2. – Schematic of physical particle collision mechanisms: differential settling, turbulence and Brownian motion.
ters due to differential settling, because all particles
are of the same size and density and settle with the
same velocity; and we can ignore encounters due to
Brownian motion because this is insignificant for
particles > 1 µm (McCave, 1984). Thus, in this case
(Table 2, eq. 2)
Theory
Consider a monospecific suspension of phytoplankton cells. Here Ca = Cb = C and eq. 1 then simplifies to:
E = βC2
(2)
There are several physical mechanisms that may
bring suspended particles to collide; viz. Brownian
motion, differential settling, and turbulent fluid
shear (Fig. 2). Kernels have been derived to quantify each of these processes (Table 2). In the engineering literature it is frequently assumed that the β
that goes into eqs. 1 and 2 is the sum of βs of the relevant processes (e.g. O’Melia and Tiller, 1993). For
a monospecific suspension of phytoplankton cells,
however, we can (at least initially) ignore encoun-
TABLE 2. – Physical encounter rate kernels for suspended spherical
particles with radii ra and rb, diffusion coefficients Da and Db, and
sinking velocities va and vb. ε is the energy dissipation rate and ν is
the kinematic viscosity. The Kolmogorov length scale, η = (ν3/ε)0.25.
Note that the magnitudes of the lead coefficients are uncertain, and
varies among authors. Based on Jackson (1990) and Delichatsios
and Probstein (1975).
Mechanism
Brownian motion
Differential settling
Turbulent shear (at scales <<
Kolmogorov scale)
Turbulent shear (at scales >>
Kolmogorov scale)
Encounter rate kernel (L3T-1)
βab
4π(Da+Db)(ra+rb)
π(ra+ rb)2 |va-vb|
1.3γ(ra+rb)3
(where γ = (ε/ν)0.5)
1.37π(ra+rb)2(ε(ra+rb))0.33
MARINE SNOW AND PLANKTIVOROUS FEEDING 143
E = 1.3γ(r1+r1)3C2 = 10.4γr13C2
(3)
Particles (or phytoplankton cells) that collide
may adhere upon collision provided they are
‘sticky’. Assume that the probability of adhesion
upon collision is given by the stickiness coefficient
α. The concentration of single (unaggregated) cells
(C1) suspended in a turbulent shear field will then
decline due to aggregation according to:
dC1/dt = -αE = -10.4αγr13C12
(4)
As coagulation proceeds aggregates consisting of
2, 3, 4,... cells begin to form, and collisions between
single cells and small aggregates, and between
aggregates of various sizes will occur. The bookkeeping of all possible collisions can be accomplished by infinitely many coupled differential equations of the same principal form as equation (3).
These are not easily managed (however, see Jackson
and Lochman 1992) but it can be shown that if one
considers only the initial process, a good approximation is (Kiørboe et al., 1990):
Ct = C0e-α(7.8φγ/π)t
(5)
where C0 is the initial concentration of cells, Ct is the
total concentration of particles (single cells + aggregates consisting of 2, 3, 4,... cells) at time t, and φ is
the volume-concentration of cells (=4/3 πr13C0). It
can likewise be shown that the average solid volume
of particles (including aggregates) will increase
according to:
Vt = V0eα(7.8φγ/π)t
(6)
where V0 and Vt are the average volume of particles
at time 0 and t, respectively. Thus, theory predicts
that initially particle concentration will decline
exponentially and average particle volume will
increase exponentially due to coagulation. This
growth in particle size due to aggregation will lead
to enhanced Stokes’ settling velocities and, hence, to
increased vertical flux of phytoplankton.
While these considerations present the basic
ideas of aggregate formation by physical coagulation and describe some of the fundamental properties, the above simple equations do not contain all
the complexity of the real world. As aggregates of
various sizes are formed differential settling as a
collision mechanism becomes important, as does
loss of particles from the system due to settling (e.g.
144 T. KIØRBOE
FIG. 3. – Qualitative demonstration that phytoplankton cells may aggregate when suspended in a laminar shear field. A suspension of diatoms,
Skeletonema costatum (equivalent spherical diameter ca. 5 µm), were
sheared in a Couette device at a rate of 30 s-1, and the temporal development of the particle size distribution was monitored with a laser diffraction instrument during 52 min. Aggregates were gradually
formed during the incubation period. (Kiørboe, unpublished).
FIG. 4. – Quantitative demonstration that phytoplankton cells may aggregate in a turbulent environment as predicted by coagulation theory.
Diatoms, Phaeodactylum tricornutum, were suspended (33 ppm) in a beaker with an oscillating grid generating a turbulent dissipation rate,
ε, of 25 cm2s-3 (equivalent shear rate, γ, of 50 s-1). The total concentration of particles (single cells + aggregates of all sizes) declines exponentially, and average particle volume increases exponentially, as predicted by eqs. 5 and 6. The stickiness, α, of the cells can be estimated
from the slope of the exponential decline (-0.029 min-1) (or increase) as 0.12 (from eq. 5). Data from Kiørboe et al. (1990).
Jackson, 1990; Jackson and Lochmann, 1992). A
further complication is that collisions between
unlike sized particles are restricted by hydrodynamics (e.g. Hill, 1992; Stolzenbach and Elimelech,
1994). Finally, aggregates are fractal objects (e.g. Li
and Logan, 1995); i.e., they are porous, and their
encased volume is larger than their solid volume,
which, of course, has implications for their collision
rate with other particles and their further aggregation. A treatment of these topics is beyond the scope
of this chapter (and the capability of the present
author); interested readers are referred to the above
quoted papers and references therein.
Evidence
Laboratory observations
While aggregate formation by physical coagulation has been demonstrated in many systems, e.g.,
for particles in sewage treatment plants, the question
remains whether phytoplankton cells are sticky and
can aggregate as predicted by coagulation theory.
Figs. 3 and 4 are qualitative and quantitative demonstrations, respectively, that this is in fact the case in
laboratory experiments. Phytoplankton cells
(diatoms) were suspended in a shear field generated
in either a Couette device or by an oscillating grid.
[A Couette device consists of two cylinders, either
or both of them rotating, thus generating well
defined laminar shear in the annular gap between
cylinders; because these phytoplankton cells (5-10
µm) are very much smaller than the Kolmogorov
scale, they will experience laminar shear in a turbulent environment, and turbulence at this scale can
thus be mimicked by laminar shear in the laboratory]. As the suspension of cells is sheared, aggregates
are formed (Fig. 3), and the change in particle concentration and average particle volume proceeds
exactly as predicted by eqs. 5 and 6, i.e., exponential
decline and increase, respectively (Fig. 4). Because
both the volume-concentration of suspended cells
(φ) and the fluid shear rate (γ) are known in such an
experiment, the only unknown in eq. 5 is the stickiness, α, of the cells; α can therefore be determined
from the slope of the exponential decline. In the present case a stickiness of 0.12 can be estimated; i.e.,
12% of the collisions result in the cells sticking
together.
By this type of approach the stickiness of several
species of phytoplankton has now been measured in
laboratory experiments with monocultures (Kiørboe
et al., 1990; Kiørboe and Hansen, 1993; Drapeau et
al., 1994). The pattern that emerges is that most
diatom species appear to be sticky while the other
groups hitherto examined are not. However, the stickiness of diatoms varies considerably both between
and within species. For example, the stickiness of the
diatom Skeletonema costatum, which has been particularly well studied, may vary between 0 and 1. The
mechanism of sticking is unknown; it has been
hypothesized that the thin capsule of mucus that covMARINE SNOW AND PLANKTIVOROUS FEEDING 145
ers many diatoms may act as a biological glue (e.g.
Kiørboe and Hansen, 1993). This is supported by the
observation that bacterial exoenzymes, which digest
mucus, may inhibit aggregation (Smith et al., 1995),
and consistent with the observation in Skeletonema
costatum and other diatoms, that stickiness declines
as cultures age and the cells become overgrown with
bacteria (Kiørboe et al., 1990; Dam and Drapeau,
1995). Other diatoms may aggregate when sheared,
even though their cell surface is not by itself sticky,
e.g. Chaetoceros affinis. This diatom excretes organic material that form 1-100 µm sized sticky mucus
particles; these particles may collide with cells and,
thus, make them sticky (Kiørboe and Hansen, 1993;
Passow et al., 1994).
Mesocosm and field observations
While there are several observations of phytoplankton aggregates and aggregation in the ocean
(e.g. Alldredge and Gotschalk, 1989; Riebesell,
1991b; Tiselius and Kuylenstierna, 1996) there are
only very few field studies that have attempted to
quantitatively compare observed aggregate formation in the sea with predictions based on coagulation
theory.
Riebesell (1991a) quantified the occurrence of
phytoplankton aggregates during a diatom bloom in
the North Sea and found that the volume concentration of aggregates increased disproportionally with
the concentration of phytoplankton. Aggregate concentration was low and almost constant during the
initial phase of the bloom but increased dramatically when the phytoplankton exceeded a certain concentration. This observation is consistent with coagulation theory, because aggregation rate is a power
function of cell concentration (cf. eq. 2). A similar
observation can be inferred from data collected by
Kiørboe et al. (1994) during a diatom bloom in a
shallow fjord. Their data show that the sedimentation rate of the phytoplankton scaled with the phytoplankton concentration raised to a power > 1 (Fig.
5), again consistent with the general prediction of
coagulation theory, since coagulation is a higher
order process.
Kiørboe et al. (1994) also more directly attempted to test predictions of coagulation theory in their
fjord system, albeit in a simple way. They noted that
aggregation rate is a function of φ (volume concentration of particles), γ (the turbulent shear rate) and
α (the stickiness of the particles) (cf. equation 5).
They monitored these parameters during a 24-d peri146 T. KIØRBOE
FIG. 5. – The sedimentation rate of phytoplankton as a function of the
concentration of suspended phytoplankton during a diatom bloom in
a shallow, Danish fjord. Sedimentation rates were measured
by sediment traps. Based on data in Kiørboe et al. (1994).
od while a diatom bloom was developing. Based on
these measurements and on coagulation theory they
constructed a semiquantitative predictor of aggregation rate. The temporal variation in the predictor
mimicked fairly well the temporal variation in (i)
observed settling velocity of phytoplankton (an indirect measure of aggregation rate) and (ii) the vertical volume-flux of aggregates (Fig. 6). The fair correspondence suggests that aggregate formation by
physical coagulation could actually account for
important properties of aggregation and sedimentation of the phytoplankton in this system.
The hitherto most complete attempt to test predictions of coagulation theory in seminatural algal
systems was reported by Alldredge and Jackson
(eds, 1995). The experiment was conducted in a 1
m3 mesocosm system inoculated with natural phytoplankton populations and stirred to simulate turbulence. A diatom bloom developed in the tank and
intense phytoplankton aggregation was observed.
Almost all relevant parameters were monitored during the course of the diatom bloom, including particle size spectra and fractal dimensions (porosity) of
the particles, occurrence of phytoplankton aggregates, algal growth rate, stickiness of the particles,
chemical properties of particles and solutes, etc. The
temporal development of the particle size spectrum
was then modelled utilizing coagulation theory and
compared to that observed. The conclusion of the
exercise was that coagulation theory was useful in
that live in gelatinous houses that are renewed at a
high rate (several times per day); phytoplankton and
other particles become aggregated at the surface of
these discarded house. Such aggregates may contain
phytoplankton at concentrations 3 orders of magnitude higher than the ambient water. Although coagulation due to differential settling velocities of houses and phytoplankton cells has been suggested as a
mechanism by which these aggregates are formed,
recent work suggests that this is not an important
mechanism, and that other mechanisms are quantitatively much more important (Hansen et al. 1996).
Implications
FIG. 6. – Field test of predictions based on coagulation theory.
During a 3.5 week period a diatom bloom was monitored; particle
concentration, particle stickiness and shear rates were measured at
2-3 d intervals and were combined into a predictor of aggregation
rate (a). The predictor shows the same temporal variation as the
observed variation in phytoplankton sinking velocity (c) and measured sedimentation of aggregates (b). From Kiørboe et al. (1994).
predicting the developing particle size spectra and,
thus, that physical coagulation could explain phytoplankton aggregation in this seminatural system. It
was also evident, however, that adjustments or
improvements of the theory are necessary to accommodate the complexity of such systems.
The above examples demonstrate that phytoplankton aggregates can be formed by physical
coagulation, and that coagulation may be important
for the vertical flux of phytoplankton (and other particles) in the ocean. For the sake of completeness it
should be pointed out, however, that (phytoplankton-) aggregates may also form by processes other
than coagulation. One example of this is aggregates
build up around discarded larvacean houses.
Larvaceans are small (mm-cm) planktonic animals
One immediate implication of phytoplankton
coagulation is, of course, to enhance the vertical flux
of phytoplankton to the seafloor. Marine snow
aggregates, whether composed mainly of phytoplankton or of other kinds of organic and inorganic
particles, are also believed to be important to microbial processes in the pelagic environment, since they
provide habitats for microorganisms. Aggregates
typically house a rich microbial flora and fauna, and
microbial activity is high. At times a high proportion
of the microbial biomass and activity in the pelagic
zone, in fact, may be located within marine snow
aggregates. The interior of marine snow aggregates
may become anoxic and, thus, be important sites of
methane production in otherwise well oxygenated
water (Shanks and Reeder, 1993).
Besides these implications, phytoplankton aggregation also may have important consequences for
the dynamics of phytoplankton populations, and
may help us understand the development (and fate)
of phytoplankton blooms in the ocean. Jackson
(1990) and Jackson and Lochmann (1992) combined phytoplankton growth kinetics and coagulation theory in a model of the dynamics of a monospecific phytoplankton bloom. In its simplest form,
where only collisions between individual cells are
considered (even when large aggregates have
formed, this is still the quantitatively most important
process), Jackson (1990) showed that
dC1/dt = µC1 - αβC12,
(7)
where µ is the net phytoplankton growth rate in the
absence of coagulation, and the other parameters are
as described above. The first term on the right side
of the equation is the cell gain due to growth and the
second term (compare with eq. 4) is the cell loss due
MARINE SNOW AND PLANKTIVOROUS FEEDING 147
to coagulation (and subsequent sedimentation). This
model predicts a sigmoid population growth curve
for the phytoplankton, even when light and nutrient
supply to the phytoplankton is constant, and the
growth rate, therefore, constant. Kiørboe et al.
(1994) examined exactly such a situation, and
demonstrated that the populations of 5 species of
diatoms showed sigmoid growth. Because the cell
gain increases linearly with cell concentration while
the cell loss scales with the concentration squared,
coagulation becomes relatively more important as
the population increases. There will be a cell concentration (CCr) at which growth is balanced by
coagulation, and cell concentration is constant.
Putting dC1/dt = 0 and inserting the expression for β
(Table 2) in eq. 7 yields
CCr = 0.096µ(αγr3)-1
(8)
This simple model actually predicts fairly accurately the diatom equilibrium concentrations found
by Kiørboe et al. (1994) (Fig. 7).
FIG. 8. – Comparison of observed (closed symbols) and modelled
(open symbols) (equation 9) population dynamics of two diatom
species in a shallow, Danish fjord. From Hansen et al. (1995)
This approach ignores the fact that species may
interact; i.e., that interspecific aggregation may
occur. Kiørboe et al. (1994) expanded Jackson’s
simple model to take interspecific aggregation into
account. Due to coagulation the concentration of the
i’th species will change according to:
dCi/dt = µiCi -Ci∑αijCjχijβij
FIG. 7. – Observed diatom equilibrium concentrations versus that
predicted by combining phytoplankton growth dynamics with coagulation theory. Data obtained during a spring diatom bloom in a
shallow, Danish fjord. From Kiørboe et al. (1994).
148 T. KIØRBOE
(9)
where the subscripts refer to species and χ is the contact efficiency. The contact efficiency is the probability that two particles in close proximity come into
direct contact. χ is close to unity for like-sized particles (thus ignored when considering monospecific
algal blooms), but declines rapidly with increasing
difference in size between approaching particles (see
Hill, 1992 for a formulation of χ). The encounter
rate kernel, β, is the sum of the kernels for turbulent
shear and differential settling. Because phytoplankters of different species are differently sized, differential settling needs be taken into account. Hansen et
al. (1995) used this model to simulate temporal
development of phytoplankton populations in the
study by Kiørboe et al. (1994), using estimates of
ambient shear, field measurements of µi, sizes of the
seeding populations, and estimates of stickiness
based on lab experiments as input (Fig. 8).
Obviously, predicted and observed cell concentrations during the development of the bloom are similar. The conclusion is, that coagulation may control
phytoplankton population dynamics.
ENCOUNTER AND FEEDING RATES IN
PLANKTONIC PREDATORS
Some background
There has been a long lasting interest among
plankton ecologists to quantify feeding rates of
planktivorous organisms. Knowledge of such rates
(and their variability) is a prerequisite for understanding trophic interactions in the pelagic environment and the functioning of plankton food webs.
Most of the experimental work hitherto conducted to
achieve this goal has been carried out in stagnant
water and with the concentration of food organisms
as the only proxy of food availability. Likewise,
encounter rate models of planktivorous feeding have
considered only predator-prey velocity differences
caused by motility of the organisms. However,
recent theory (Rothschild and Osborn, 1988) and
simulations (MacKenzie and Leggett, 1991) have
suggested that microscale turbulence may significantly enhance the contact rate between planktonic
predators and their prey by increasing velocity difference between prey and predator. This finding
may potentially solve a problem in fisheries
oceanography: most laboratory studies have found
that planktivorous fish larvae require concentrations
of food (e.g. copepods) that are orders of magnitude
higher than those concentrations that the larvae
would (on average) encounter in their nursery areas.
In any case, if the effect of turbulence is substantial
then our present comprehension of planktivorous
feeding, which is based mainly on calm water experiments and considerations, needs be revised.
The predation process can be divided into several components: (1) prey encounter, (2) pursuit and
attack, and (3) capture. Micro-scale turbulence
potentially affects both prey encounter and post
encounter processes. We shall first examine the
encounter process. This problem is equivalent to the
problem addressed above, and we shall use the basic
equation as a starting point of the analysis.
Theory
If we interpret the two ‘particles’ in equation 1 (a
and b) as predator and prey, then E is the number of
predator-prey encounters per unit time and volume.
The prey encounter rate per predator, e, is then (from
eq. 1):
e = E/Ca = βCb
(10)
where Ca and Cb are the concentrations of predator
and prey, respectively. In this formulation the
encounter rate kernel, β, has a clear biological interpretation. It is the volume searched for prey per unit
time or, provided all encountered prey are captured,
the clearance rate. As noted above β is a function of
both the size and motility of the ‘particles’, and of
the ambient turbulent fluid shear. We shall in the following assume that the particles are spherical, and
we will model the predator-particle as the predators
reactive sphere, i.e. a sphere with an encounter
radius equal to the distance at which the predator
can perceive and react to prey. When considering
the diatom coagulation problem above we could
ignore particle self-motility. Obviously, this is not
the case when considering swimming predators (and
prey).
Several processes can cause velocity differences
between predator and prey and, thus, contribute to
prey encounter rates. We shall here distinguish
between physical processes and behavioral
processes. One can derive expressions for
encounter rate kernels for both of these groups of
processes. As in the coagulation literature we shall
assume that the kernel that goes into equation 10 is
the sum of the relevant kernels (typically one physical and one behavioral kernel). This assumption
introduces a small error. There are more exact ways
of combining kernels than just adding them; interested readers are referred to Rothschild and Osborn
(1988) and Evans (1989).
The physical processes that can cause velocity
differences between predator and prey are exactly
the same as those considered above for the coagulation problem and the physical encounter rate kernels
are identical to those in Table 2. Again Brownian
motion is insignificant for particles (organisms)
larger than 1 µm. Also, we shall consider differential
settling a behavioral process (and model it slightly
differently from that in Table 2). We are then left
with the kernels for turbulent shear. While coagulation between small (say 10 µm) phytoplankton cells
occur at a spatial scale much smaller than the
Kolmogorov length scale, many planktonic predators have sizes or reactive distances very close to the
Kolmogorov scale (η = (ν3/ε)0.25 ~ 0.1 cm for typical
dissipation rates in coastal surface waters). While
physical encounter rate kernels have been derived
for particles much larger and much smaller than the
Kolmogorov scale, theory for encounter rate of particles similar in size to the Kolmogorov scale does
not exist. However, Hill et al. (1992) have shown
MARINE SNOW AND PLANKTIVOROUS FEEDING 149
TABLE 3. – Behavioral encounter rate kernels for predators with a
spherical reactive sphere and spherical prey particles. ra and rb are
predator reactive distance and radius of prey, respectively, va and
vb are predator and prey swimming or sinking velocities, ωi is the
average ‘tumbling frequency’ for organisms with a random walk
type of motility pattern, and f is pause frequency for pause-travel
predators.
Behaviour
Predator swimming
(Cruise predator)
Encounter rate kernel (L3T-1)
βab
π(ra+rb)2va
Both predator and prey
swimming
π(ra+rb)2(vb2+3va2)/3va
(for vb < va)
Predator and prey have
random walk
4π(Da+Db)(ra+rb)
where Di = vi2/3ωi
Passive sinking
(ambush)
Pause-travel predator
π(ra+rb)2 |va-vb|
4/3fπ(ra+rb)3
experimentally that supra-Kolmogorov scale theory
applies at, or even somewhat below, the
Kolmogorov scale. While this appears to be controversial and not generally accepted, we shall make
this assumption in the following.
There has been some confusion in the literature
as to the appropriate scale to be used when calculating the velocity difference due to turbulence. Here
we assume that the predators reaction distance (+ the
radius of the prey) is the correct scale (cf. Table 2);
see Kiørboe and MacKenzie (1995) for a discussion
of the issue.
The behavioral processes that will cause velocity differences between predator and prey and,
hence, lead to encounters, have to do mainly with
the motility of the organisms. Different motility
patterns and hunting strategies have been described
in the literature, and behavioral kernels for these
various behaviours have been derived (Table 3). For
example, the kernel for a cruising predator is the
well known π(ra+rb)2va (Fig. 9). If the prey is also
moving the kernel becomes modified (Table 3).
Other types of behaviours and motility patterns are
common. For example, many copepods generate
feeding currents; this behaviour can be described by
the kernel for a cruise predator, except that the relevant velocity, va, is the velocity of the feeding current (relative to the predator) at one reaction distance, rather than the swimming velocity of the
copepod. Many planktonic predators have a random
walk type of motility pattern. Several ciliates, for
example, swim for short periods, then tumble and
continue swimming in a new random direction.
150 T. KIØRBOE
FIG. 9. – Schematic explanation of the behavioral kernel for a cruising predator. Consider a predator swimming with velocity va and
having a spherical visual field with (reactive) radius ra searching for
spherical prey particles with radius rb. Per unit time the predator will
encounter all prey particles in the volume given by π(ra+rb)2va,
which is then the behavioral encounter rate kernel.
Such a motility pattern can be quantified by a diffusion coefficient, Da = va2/3ω, where va is the swimming velocity, and ω is the tumbling frequency. The
encounter rate kernel for this type of motility has
been given in Table 3. Other common strategies
among planktonic predators belong to the ambush
type of hunting behaviours. Some predators hang
quietly in the water while slowly sinking and
searching for prey (e.g., many copepods), while
others swim short distances between pauses, where
they search the perceptive sphere for prey (many
larval fish, for example). These ambush type behaviours have been termed ‘passive sinking’ and
‘pause-travel’ strategies, and the corresponding
kernels have been given in Table 3.
The equations for β in Tables 2 and 3 may overpredict encounter rates in organisms in which prey
capture depends on direct interception with the prey
particle (e.g., many protozoans); i.e., when the
encounter sphere is a solid body, not a volume of
water. This is because direct contact is restricted by
hydrodynamics. While the contact efficiency, i.e.,
the probability that two particles in close proximity
come into direct contact, is close to unity for likesized particles, it declines rapidly with differences
in particle sizes (e.g. Hill, 1992). These considerations are particularly relevant to protozoans, but
will not be dealt with here any further. Interested
readers are referred to Shimeta (1993) and Shimeta
et al. (1995).
A simple example may illustrate the application
of the kernels in Tables 2 and 3. To model the prey
encounter rate in calm water one just has to insert
the relevant behavioral kernel in eq. 10. To model
prey encounter in a turbulent environment, β in eq.
10 is assumed to be the sum of the relevant behavioral and physical kernels, β= βbehaviour+βturbulence. For
example, assume that the cruising predator in Fig. 9
is a herring larva with a reaction distance (ra) of 1.5
cm and a swimming velocity (va) of 1 cm s-1. If we
ignore the motility and size of the prey (copepod
nauplii), because they are both very small, then in
calm water β = βbehaviour = πx1.52x1 = 7.1 cm3 s-1 (~
25 l h-1). Assume now that the larva experiences a
turbulent environment characterized by a dissipation
rate ε = 10-2 cm2 s-3 (typical for coastal surface
waters). Then, β = βbehaviour+βturbulence = 7.1 +
1.37x1.52x(10-2x1.5)0.33 = 7.9 cm3 s-1 (~ 28 l h-1).
Thus, in this imaginary example, turbulence of this
magnitude will increase the volume searched for
prey by only 3/25 = 12%. In a similar manner kernels for turbulence and behaviour (Tables 2 and 3)
can be combined to examine the effect of turbulence
on prey encounter rates for predators with various
behaviours and behavioral parameters, and at various intensities of turbulence.
In the above example it was assumed that the
behaviour of the predator was unaffected by turbulence. This is not necessarily the case. Thus, there
are several examples that predators change swimming speeds, reactive distances and/or time allocated to searching for food as a function of turbulence
(e.g., Costello et al., 1990; Saiz, 1994). If the
changes in the relevant behavioral parameters (i.e.,
in the above example: time spent searching for food,
swimming velocity and reactive distance) can be
quantified, it is a simple task to account for it in the
model.
Evidence
Observational evidence is accumulating, that
small-scale turbulence in fact enhances prey
encounter and feeding rates in planktivorous predators. Thus, positive effects of turbulence have been
demonstrated in both laboratory and field experiments for protozoa (Shimeta et al., 1995; Peters and
Gross, 1994), copepods (Saiz et al., 1992; Marrasé
et al., 1990; Saiz and Kiørboe, 1995; Kiørboe, 1993)
and fish larvae (Sundby and Fossum 1990; Sundby
et al., 1994, MacKenzie and Kiørboe, 1995).
However, only few experiments have been designed
to quantitatively test the predictions of the models
described above.
The study of the feeding of the copepod Acartia
tonsa in calm and turbulent environments by Saiz
and Kiørboe (1995) may serve as one such example.
Acartia tonsa has two different feeding modes (Fig.
10), which can each be triggered by different types
of food (see Jonsson and Tiselius 1990 for a detailed
account of the behaviour). When offered a suspen-
FIG. 10. – The two feeding modes of the copepod Acartia tonsa. In the ambush feeding mode the
copepod sinks slowly through the water with the antennae extended. Motile prey are perceived by
hydromechanical receptors on the antennae. In the suspension feeding mode a feeding current is generated. Prey particles that pass through the hatched volume may be captured. Drawn by E. Saiz and
based on Jonsson and Tiselius (1990).
MARINE SNOW AND PLANKTIVOROUS FEEDING 151
sion of ciliates or other motile organisms as food the
copepod adopts an ambush type of ‘passive sinking’
strategy: it hangs quietly in the water with the antennae extending from the body, sinking slowly (0.069
cm s-1) while scanning for prey. Swimming ciliates
generate a hydromechanical signal which is perceived by the copepod by mechanoreceptory hairs
on the extending antennae. Once a ciliate has been
perceived, the copepod reorients towards the ciliate,
jumps towards it and attempts to capture the ciliate.
Ciliates can be perceived at a distance of about 1
mm from the antennae, which themselves extend
about 1 mm from the body; thus, ra = 0.1 cm. With
va = 0.069 cm s-1 and ignoring the size (rb = 3x10-3
cm) and sinking velocity (vb = 2x10-3 cm s-1; Fenchel
and Jonsson, 1988) of the ciliate (Strombidium sulcatum) the behavioral kernel for ‘passive sinking’
(Table 3) becomes 193 ml d-1. This is close to the
average clearance rate actually observed in calm
water, 182 ± 15 ml d-1 (Fig. 11a). Clearance rates in
turbulent environments can be modelled by adding
the kernel for turbulence (above Kolmogorov scale)
(Table 2) to that for ‘passive sinking’, and both predicted and observed variation in clearance as a function of turbulent dissipation rates are shown in Fig.
11a. Obviously, the effect of turbulence is significant. For example, at a dissipation rate of 10-2 cm2
s-3, predicted clearance is enhanced fourfold over
that in calm water. It also appears from Fig. 11a that
the model provides a good prediction of observed
clearance rates at low and moderately high dissipation rates. At (unrealistic) high dissipation rates,
however, observed clearance declines while predicted clearance continues to increase, and the model,
thus, fails to describe the feeding of Acartia tonsa in
such energetic environments. We shall return to this
discrepancy later.
When offered a suspension of diatoms, Acartia
tonsa adopts a different feeding strategy: suspension
feeding (Fig. 10). It establishes a feeding current and
diatoms are captured within the volume of water that
passes within reach of the food collecting
appendages (the 2nd maxillae). These measure
about 0.02 cm; i.e., ra = 0.02 cm. The feeding current
accelerates as it approaches the copepod and reaches a velocity of ca. 0.8 cm s-1 at the tip of the 2nd
maxillae. With this information we can model the
clearance rate in calm and turbulent environments
(Fig. 11b). Both predicted and observed effects of
turbulence are very small. This is very different
from the situation in the ambush feeding mode. The
difference arises because of the very different mag152 T. KIØRBOE
FIG. 11. – Comparison of observed and predicted clearance rate as
a function of turbulent dissipation rate in the copepod Acartia tonsa
as an ambush predator (a) and as a suspension feeder (b). In both
cases were >η encounter rate kernels employed; this is not warranted in the suspension feeding mode, where ra = 0.02 cm << η. Using
the <η kernel in the latter case renders the predicted clearance rate
practically constant over the range of dissipation rates considered.
nitude of the velocity difference between predator
and prey caused by the behaviour relative to the
velocity difference due to ambient turbulent shear.
However, here again the simple model provides predictions that are consistent with observations.
An example of the effect of turbulence for prey
encounter rates in a pause-travel predator was provided by MacKenzie and Kiørboe (1995). By videorecording the feeding behaviour of larval cod these
authors quantified the reaction distance (ra), pause
frequency (f) and pause duration (τ) of the larvae as
well as the rate at which they encountered prey
(copepod nauplii) in calm (ε = 0) and turbulent (ε ~
10-3 cm2s-3) environments. By combining the kernels
for turbulence and pause-travel feeding they predicted that larvae in this size range (5.2-6.1 mm length)
should increase their encounter rate with prey 2.23.2-fold; they observed a 2.2-4.7-fold increase. In
this comparison observed changes in behavioral
parameters (pause frequency and pause duration) in
turbulent as compared to calm water were taken into
account. Thus, apparently the simple model was
able to predict the magnitude of the increase in prey
encounter rate.
Yet another example may be provided by studies of protozoan feeding in calm and turbulent environments. Cruising and suspension feeding ciliates
and flagellates normally swim or produce feeding
currents that are by far too fast for ambient shear to
significantly influence prey encounter rates (see
below), and accordingly clearance rates measured
in calm and turbulent environments do not appear
to vary in most species (Shimeta et al., 1995; Peters
and Gross, 1994). However, some protozoans, like
heliozoans, do not move and appear to depend
either on the motility of the prey or on ambient
fluid motion to encounter prey (Shimeta and
Jumars, 1991). Shimeta et al. (1995) measured
clearance rates of the helioflagellate Ciliophrys
marina offered nonmotile bacteria both in still
water and in a laminar shear field (generated in a
Couette device). C. marina is functionally a heliozoan (Fig. 12); although it does posses a flagellum
this is normally not functional and the organism is
immobile when feeding (Fenchel, 1986). Bacterial
prey are captured as they intercept with the sticky
pseudopodia. Assuming that in still water prey
encounters depend only on the Brownian motion of
the bacteria, one can calculate the still water
encounter rate kernel as 4πDbra (cf. Table 2, assuming Da ~ rb ~0). Brownian diffusivity of bacteria can
be estimated as Db = 4×10-9 cm2 s-1 (Shimeta and
Jumars, 1991). It is difficult to assign an exact
effective capture radius to C. marina; obviously it
must be larger than the radius of the spherical cell
body (~ 4 µm) and smaller than the longest
pseudopodia (~30 µm; see Fig. 12). Shimeta et al.
(1995) measured a still water clearance rate of ca.
1x10-7 ml h-1; equating this value with the above
kernel yields an estimate of effective encounter
radius, ra= 5.5 µm. Brownian motion is, thus, sufficient to account for clearance in still water. At a
laminar shear rate of, e.g., 1 s-1, which is equivalent
to the small-scale laminar shear occurring at a dissipation rate of 10-2 cm2 s-3, they observed that the
FIG. 12. – The helioflagellate Ciliophrys marina with bacteria
attached to the pseudopodia. Scale bar: 10 µm. Reproduced from
Fenchel (1986) with permission.
clearance rate increased to about 3 × 10-7 ml h-1, i.e.,
a three-fold increase. The kernel for turbulent shear
(< Kolmogorov scale), assuming ra= 5.5 µm, predicts a clearance rate of ca. 8 × 10-6 ml h-1, which is
somewhat higher than observed. However, even
though the present simple models are insufficient to
accurately predict the effect of turbulence in these
protozoans, the fact remains that measurable
effects occur in those protozoans for which a significant effect is predicted, and that no effects are
found in most fast-swimming protozoans in which
an insignificant effect is predicted (see also
Shimeta et al., 1995 and below).
Implications
The examples above already illustrate that the
effect of micro-scale turbulence on prey encounter
rates differ among predators and depend, among
other things, on the behavioral type of the predator,
on the velocity difference between predator and prey
due to the motility of the predator (and prey), and on
the spatial scale of the predator-prey interaction.
These insights can be quantified by comparing the
kernels for behaviour and turbulence. One can
define a critical dissipation rate (εCr) at which the
encounter rate due to ambient fluid shear is equal to
the encounter rate due to the behaviour of the predator; i.e., the dissipation rate at which
βturbulence(εCr) = βbehaviour
(11)
MARINE SNOW AND PLANKTIVOROUS FEEDING 153
Thus, at dissipation rates exceeding εCr, turbulence is more important than behaviour for prey
encounter; and vice versa. In the foregoing we have
identified two different kernels for turbulence, and
we have defined four different behavioral types and
their associated kernels (Table 2 and 3). This yields
eight different combinations of behaviour and turbulence. For simplicity we shall in the following
ignore the motility and size of the prey, because
these are often (but not always) small compared to
the motility and behaviour of the predator. Let us
first examine the random walk type of predator
motility pattern at scales > Kolmogorov scale.
Inserting the relevant kernels form Tables 2 and 3
into eq. 11 yields:
1.37πra2(εCrra)0.33= 4πDara
which simplifies to
εCr = 24.9Da3ra-4
and likewise for scales smaller than the Kolmogorov
scale:
εCr = 93Da2ra-4ν
where we recall that ν is the kinematic viscosity (~
10-2 cm2s-1). In a similar fashion critical dissipation
rates can be derived for the other behavioral types
(Table 4). Note that the formula for the critical dissipation rate is the same for cruising predators, passively sinking predators, and for predators that generate a feeding current, but that the interpretation of
the velocity (va) varies (swimming velocity, sinking
velocity, and velocity of the feeding current at one
reaction distance away from the predator, respecTABLE 4. – Critical dissipation rates for predators with different foraging strategies operating at scales above and below the
Kolmogorov length scale (η). The lead coefficients may vary (up to
an order of magnitude) depending on the magnitude of the lead
coefficient in the governing equations, and on the way kernels have
been combined. Here we have combined kernels as suggested by
Evans (1989). Parameters as in Tables 2 and 3.
Predator behaviour\
Spatial scale
Cruise predator
passive sinking
Suspension feeder
Random walk
Pause-travel predator
154 T. KIØRBOE
ra>η
ra<η
0.14va3ra-1
1.7va2ra-2ν
3 -4
a a
25D r
2 -3
a
0.33r τ
2 -4
a a
93D r ν
10τ-2ν
FIG. 13. – Critical dissipation rate as a function of body length. A.
Critical dissipation rate calculated for cruise predators employing
the relationship between swimming velocity and body length of
Peters et al. (1994) and using the expressions for εCr given in Table
4 for scales above and below the Kolmogorov length scale (η). In
calculating critical dissipation rates it was assumed that the
encounter radius is equal to half a body length. The typical variation
in the magnitude of the Kolmogorov length scale in the ocean is
also indicated. Note that dissipation rates exceeding 10-1 cm2s-3 are
rare in the ocean. B. Critical dissipation rates for small cruising and
suspension feeding ciliated metazoans and protozoa (left side of the
graph - open circles) and for fish larvae (right side - different symbols for different species) employing different predation strategies
(closed symbols: pause-travel predators; open symbols: cruising
predators). Data on swimming and feeding current velocities for the
protozoans and small metazoans are from Hansen et al. (in press),
and critical dissipation rates were calculated assuming that the
encounter radius is half a body length. Critical dissipation rates for
fish larvae are taken from Kiørboe and MacKenzie (1995); these
were calculated employing >η theory.
tively). Note also that the magnitude of the lead
coefficients in these expressions may vary (up to
one order of magnitude) depending on the assumed
magnitude of the lead coefficients in the governing
equations, on the way that kernels are combined,
and on modifications of the kernels relevant to
details in the behaviour of specific organisms. The
form of the expressions, however, are invariant with
these variations in assumptions. In Table 4, kernels
have been combined in the way suggested by Evans
(1989); see also Kiørboe and MacKenzie (1995).
The form of the equations for critical dissipation
rate suggest that this is minimum at scales near the
Kolmogorov length scale (η), at least for cruising
predators (Fig. 13a,b). In other words, turbulence
appears to be potentially most important for planktivorous predators with an encounter radius close to
η. To see this, consider first predators smaller than
η; for these, εCr scales with the ratio of swimming
velocity to encounter radius squared, (va/ra)2 (Table
4). Swimming or feeding current velocities in flagellates, ciliates and other small ciliated cruising
and suspension feeding predators (e.g. rotifers)
depends only weakly on size (Peters et al., 1994.)
and appear to scale with size raised to exponents
considerably less than one (e.g., 0.24, Sommer,
1988; 0.6, Hansen et al., in press). For these predators the encounter radius is typically simply the
radius of the (assumed spherical) body and it, thus,
follows that (va/ra)2 and, hence, εCr decrease with
size. For predators with encounter radius > η the
critical dissipation rate scale with va3/ra (Table 4).
For such predators (e.g. fish larvae) both swimming
velocity and reactive distance appear to increase
almost proportionally to body length (e.g. Blaxter,
1986; Miller et al., 1988). Thus, va3/ra and εCr
increases with size and it follows that εCr is minimum for predators with encounter radius near η.
Fig. 13a,b illustrates this relation between εCr and
organism size. Peters et al. (1994) compiled literature data on swimming velocities in aquatic organisms ranging from bacteria to whales, and by regression defined a relation to body length (see legend
Fig. 13). Combining this relation with the relevant
expressions for εCr in Table 4, and assuming that the
encounter radius is 0.5 x body-length produces the
pattern in Fig. 13a. Fig. 13b illustrates the same with
data from Hansen et al. (submitted) for small zooplankton and Kiørboe and MacKenzie (1995) for fish
larvae. For cruising predators much smaller than η
(e.g. protozoa) turbulent fluid motion is insignificant
for prey encounters because it is dampened by viscosity; cruise predators much larger than η (e.g. post
larval fish) have typical swimming velocities much
larger than turbulent fluid velocities and, thus, do not
benefit from the latter in encountering prey.
There are, of course, exceptions to this. For
example, some protozoans move very slowly (or not
at all; e.g. heliozoans) and may depend on ambient
fluid shear to encounter prey (e.g. the helioflagellate
above). Likewise, some larger predators move considerably slower than predicted for their size by the
general relationship of Peters et al. (1994); for
example, jellyfish, which may also benefit significantly from turbulence. However, in general, turbulence is potentially most important for mm-cm sized
planktivorous predators.
Turbulence, behaviour, prey perception
and post encounter processes
An analysis of the effects of small-scale turbulence on planktivorous feeding is incomplete without considering the several issues listed in the above
heading. They shall be only briefly discussed in the
following.
As noted above turbulence may interact with
the behaviour of both predator and prey. In its simplest way the predator may change its time budget
in response to turbulence, e.g., the fraction of time
allocated to feeding. For example, the copepod
Acartia tonsa, when in a suspension feeding mode,
changes to more frequent feeding bouts of shorter
duration in turbulent as compared to still water
(Saiz, 1994). Herring larvae allocate less time for
swimming (searching for food) in turbulent than in
calm water (MacKenzie and Kiørboe, 1995). These
kinds of behavioral changes are easily accommodated by the models (if known and quantified), as
also noted above. Costello et al. (1990) reported a
more complex and time dependent behavioral
response in the copepod Centropages hamatus, that
may be less easy to model. Finally, the experimental work of Shimeta et al. (1995) and Peters and
Gross (1994) on protozoan feeding also suggest
that some of these organisms may respond behaviorally to turbulence in an as yet not understood nor
described way.
Turbulence may also interfere with predator
behaviour in more subtle ways. For example, prey
perception may be disturbed by turbulence, particularly in predators that perceive prey by mechanical or chemical cues (vision is unlikely to be affected directly). Kiørboe and Saiz (1995) attempted to
model the effect of turbulence on mechanoreceptory perception of motile prey in Acartia tonsa by
assuming (i) that the signal to noise ratio has to
exceed a critical value for the prey to be detected
MARINE SNOW AND PLANKTIVOROUS FEEDING 155
and that the perception distance is therefore limited by the signal to noise ratio, and (ii) that the signal to noise ratio decreases with increasing dissipation rate and that the perceptive distance, therefore,
declines. Saiz and Kiørboe (1995) provided some
experimental evidence, which also suggests that
the effect is limited at typical and even high oceanic dissipation rates. It does, however, account for
the difference between observed and predicted
clearance rates at high experimental turbulent dissipation rates (Fig. 11).
Chemoreception may be disturbed because turbulence may dissipate patches of odour around prey
organisms. In organisms with a feeding current
chemo-detection depend on the shear of the feeding
current (e.g. Strickler, 1985), which may be eroded
by ambient shear. Kiørboe and Saiz (1995) argued
that in copepods the feeding current shear is considerably larger than ambient shear at typical oceanic
dissipation rates, and therefore unlikely to be significantly disturbed.
Prey that has been encountered should also be
captured (at least from the predators point of view),
and turbulence may interfere with post-encounter
prey capture: perceived prey may be advected out of
the predators reaction sphere by turbulence faster
than the predator can react to it (Granata and Dickey,
1991). This has been modelled by several authors
(MacKenzie et al., 1994; Kiørboe and Saiz, 1995;
Jenkinson, 1995). The risk of loosing an encountered prey this way depends on the predator’s reaction time, the size of its reactive sphere, and of the
ambient shear rate. According to models only the
most slowly reacting predators loose a significant
proportion of prey this way, even at relatively high
turbulent dissipation rates (e.g. Kiørboe and Saiz,
1995), but the models have never been examined
experimentally.
CONCLUDING REMARKS
Small-scale turbulence may affect several fundamental processes in the plankton, such as nutrient
uptake kinetics in phytoplankters, formation of
marine snow aggregates and vertical material fluxes,
population dynamics of phytoplankters, and the
trophodynamics of planktivorous predators. The
foregoing has sketched a theoretical framework for
analyzing several of these processes, and has in several cases demonstrated that these processes in fact
are of quantitative importance in marine planktonic
156 T. KIØRBOE
systems. The analysis presented here is simplistic,
and may require modification (and in some cases
complication) to account for the processes in a complex real environment. However, some robust
results have emerged from the simple analyses presented here and some gaps in our present knowledge
have become evident.
At length scales much smaller than the
Kolmogorov scale (η), the main biological implication of small-scale turbulence is the formation
of marine snow aggregates by physical coagulation, and its effects on the vertical material fluxes
in the ocean. At this length scale trophic interactions are predicted to be influenced only in special cases (heliozoans and radiolarians, for example), although there may be effects that are not
accounted for by present theory (e.g. Peters and
Gross, 1994).
At scales near the Kolmogorov length scale the
main biological implication of small-scale turbulence appears to be its effect on prey encounter rates.
However, this prediction hinges on assumptions
about the ‘structure function’ of turbulence, i.e., the
function that relates encounter velocities to turbulent dissipation rates. As noted earlier, expressions
for encounter velocities and, hence, encounter rate
kernels exist for scales much larger and much smaller than η, but have not been developed for scales
near η. This is particularly crucial because, as
argued above, this is exactly the scale at which turbulence is expected to be most important for planktivorous predators to encounter prey.
In the foregoing it has been assumed that >>-ηtheory also applies at scales close to η. This
assumption is based solely on experiments (and
arguments) of Hill et al. (1992) and this work is
the only one which has addressed the problem. If
one, alternatively, extrapolates <<-η-theory to
scales near η, then estimates of critical dissipation
rates (εCr) will exceed typical ambient levels of turbulence over all spatial scales (see Fig. 13). Thus,
although the prediction that turbulence is potentially most important at length scales near η is still
valid, even at that scale turbulence will significantly enhance prey encounter rates only at turbulent intensities that are in the upper end of what
has been observed in the ocean. An important plea
from biological oceanographers to small-scale
fluid dynamicists would, therefore, be to replicate
the work of Hill (1992), and to examine encounter
velocities at scales near the Kolmogorov length
scale.
ACKNOWLEDGEMENT
Financial support for the work reported here was
provided by the Danish Science Research Council
(grant # 11-0420-1), The National Agency of
Environmental Protection in Denmark (HAV-90, 230), and the U.S. Office of Naval Research
(N00014-93-1-0226).
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