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A model of pycnocline thickness
modified by the rheological properties of
phytoplankton exopolymeric substances
IAN R. JENKINSON 1,2* AND JUN SUN 1
1
7 NANHAI ROAD, QINGDAO 266071,
19320 LA ROCHE CANILLAC, FRANCE
CHINESE ACADEMY OF SCIENCES, INSTITUTE OF OCEANOLOGY,
ET DE RECHERCHE OCÉANOGRAPHIQUES, LAVERGNE,
2
PEOPLE’S REPUBLIC OF CHINA AND AGENCE DE CONSEIL
*CORRESPONDING AUTHOR: [email protected]
Received May 11, 2010; accepted in principle July 1, 2010; accepted for publication July 19, 2010
Corresponding editor: William K.W. Li
We model how phytoplankton-produced exopolymeric substances (EPS) may
change pycnocline thickness dz through control by Richardson number Ri. Shear
stress tFORCED is imposed across the pycnocline, giving a shear rate du/dz, where
du is cross-pycnocline velocity difference, modulated by viscosity h. In natural
waters, viscosity is composed of two components. The first is Newtonian, perfectly
dispersed viscosity due to water and salts. The second is non-Newtonian, and
depends on phytoplankton abundance raised to an exponent between þ1.0 and
þ1.5. It is also generally shear-thinning, depending on (du/dz)P, where P is negative. In suspensions of microbial aggregates, viscosity depends also on (length
scale)d. Published measurements of EPS rheology in a culture of Karenia mikimotoi
are input. These measurements were made at 0.5 mm, so they can be scaled to
du/dz if d is known for this EPS over the appropriate range of length scales. The
model shows that du/dz is very sensitive to d. At values of d , 20.2 or 20.3,
however, high concentrations of K. mikimotoi (,31 million L21) have no
significant effect on dz at ambient values (.1 cm). Future investigations of
pycnocline dynamics should include measurements of rheological properties, and
particularly d.
KEYWORDS: rheology; phytoplankton; pycnocline; thin layer; exopolymeric substance; biogeochemistry
I N T RO D U C T I O N
Effects of the physics on the biology
Small-scale turbulence and shear rates are important
determinants of the productivity, biomass and composition of bacterio- and phytoplankton communities
(Iversen et al., 2010), cell division rates in dinoflagellates
(Berdalet, 1992; Juhl and Latz, 2002) and grazing rates
of and by nanoplankton (Jenkinson and Wyatt, 1992;
Jenkinson, 1995; Shimeta et al., 1995). This may be
particularly true of the communities associated with
pycnoclines, which often appear as if aggregated vertically into highly concentrated thin layers (TLs). TLs of
dense phytoplankton often occur associated with density
discontinuities, or pycnoclines, in coastal seas (Bjørnsen
et al., 1993; Dekshenieks et al., 2001; Alldredge et al.,
2002; Gentien et al., 2007; GEOHAB, 2008), oceans
and lakes. This suggests that pycnoclines are often part
of the cause of the occurrence and biodynamics of
these TLs.
doi:10.1093/plankt/fbq099, available online at www.plankt.oxfordjournals.org. Advance Access publication August 18, 2010
# The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please email: [email protected]
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Effects of the biology on the physics
In natural waters, viscosity consists of the sum of two
components. Newtonian, perfectly dispersed viscosity is
contributed by the water and salts (Jenkinson and Sun,
2010). Non-Newtonian viscosity is contributed mainly
by dissolved and flocculated exopolymeric substances
(EPS), related to shear rate by a mainly negative exponent P (Jenkinson, 1986, 1993a, b) as well as to the
plankton concentration raised to the power of another
exponent, roughly between þ1 and þ1.5 (Jenkinson
and Biddanda, 1995). Measurements made on other
microbe-rich suspensions (Spinosa and Lotito, 2003)
have shown EPS that viscosity is related to length scale
by a third, negative exponent d (Jenkinson et al., 2007),
but this has yet to be investigated in natural waters.
Dense phytoplankton is often associated with increased
viscosity as well as elasticity of the seawater or lake water
(Jenkinson and Sun, 2010, and references therein;
Seuront et al., 2010). The question thus arises whether
the phytoplankton in TLs may alter pycnocline characteristics through the rheological effects of their secreted
EPS. The present paper is limited to investigating this
possible “Biology to Physics” causality. Since the degree
of stratification clearly influences the phytoplankton in
TLs (“Physics to Biology” causality), the potential exists
for a feed-back loop, in which the phytoplankton,
through modification of the rheological properties of the
water by secreted EPS might cause changes in the
dynamics of pycnocline formation and maintenance.
Furthermore, in waters of the Eastern Antarctic in
summer, Seuront et al. (Seuront et al., 2010) measured the
viscosity of seawater using a measuring gap of 0.23 mm
at constant but unquantified shear stress. They found
that in subsurface water (depth 10 m), the biological
component of viscosity correlated well with bacterial
abundance but not with chlorophyll a (Chl a), microautotroph abundance or microheterotroph abundance. In
the deep chlorophyll maximum (DCM), however, the
biological viscosity correlated strongly with Chl a, microautotroph abundance and microheterotroph abundance,
but not with bacterial abundance. Moreover, the taxa
whose abundance correlated the most with biological
viscosity in the DCM were the dinoflagellates Alexandrium
tamarense and Prorocentrum sp. The methods used by
Seuront et al., however, did not allow biological viscosity
to be quantified in terms of shear stress or, as in all
natural-water studies to date, length scale.
A parallel “Biology-to-Physics” causality is the effect
of differential solar heating caused by layers of phytoplankton on thermal stratification structures (Grindley
and Taylor, 1971; Lewis et al., 1983; Edwards et al.,
2004; Murtugudde et al., 2002). Although this
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mechanism might sometimes act in synergy with the
rheological mechanism, it is not subject of this paper.
Aims of this paper
We begin this investigation by modeling the possible
effects of increased viscosity on pycnocline thickness.
Published values are used for the viscosity and elasticity
of seawater (Jenkinson, 1993b; Jenkinson and Biddanda,
1995) and phytoplankton cultures (Jenkinson, 1986,
1993a). Recent findings that the effects of length scale
on rheological properties of heterogeneous suspensions
of microbial flocs can be modeled using a power law
(Jenkinson and Wyatt, 2008) are applied to a naturalwater situation for the first time. The aim of using this
model is to help identify appropriate targets for future
research on TL and pycnocline dynamics.
In this paper, the model is presented, some results for
seemingly appropriate values of pycnocline parameters,
and densities of the harmful algal bloom (HAB) dinoflagellate Karenia mikimotoi, that occurs widespread, often
forming TLs in relation to pycnoclines (Bjørnsen et al.,
1993). Rheological measurements in cultures and seawater have been measured at a length scale only of
0.5 mm, and it is shown that whether rheological effects
can modify the values of pycnocline thickness that actually occur, will depend critically on the length-scale
exponent d. Since this model has so far no field data
with which to validate it, we then discuss appropriate
targets for future investigation.
In brief, the aim of this model is not to predict the
dynamics of TLs or pycnoclines. It is to rather to
suggest how the dynamics in TLs or pycnoclines may
be sensitive to modification by algal EPS, particularly
those of Karenia mikimotoi.
We have tried to keep rheological terminology in the
present paper to a minimum, but some terms are
explained in Table I. For a broader view of thalassorheology (the rheology of the sea and other natural
waters), however, the reader is encouraged to consult
Jenkinson and Sun (Jenkinson and Sun, 2010), and to
peruse any good textbook about Rheology.
Model set-up and examples
We build the model with a standard set of parameters,
ready to change each parameter separately for numerical experiments (Table I). This is a 1D three-layer equilibrium model of a pycnocline (Fig. 1) in which
Richardson number Ri determines the thickness.
There are three layers: an upper and a lower mixed
layer, with a stratified pycnocline between them. The
mixed layers are turbulent, with a turbulent eddy
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Table I: Parameters used in the simulation, with standard values where appropriate
Symbol
Meaning
Standard value
Units
Change investigated?
d
g
G’
G00
G00E
K
Kmik
L
LDIA
LCOUETTE
N2
P
Q
Ri
Ricr
du
dz
dzz
dr
h
hE
hW
g_
ðg_ Þ2
ðg_ 2 Þ
r
t
tE
tHi
Exponent of a rheological property as a function of length scale L
Acceleration due to gravity
Elastic modulus, the elastic stress resisting deformation/g
Viscous modulus, hg_ , the viscous stress resisting deformation/g_
Excess viscous modulus, that due to EPS, hE g_
The inequality factor defined in (9)
Concentration of phytoplankton dinoflagellate Karenia mikimotoi
Length scale of interest, or of measurement
Length scale corresponding to diameter of a viscometer flow-pipe
Length scale corresponding to measuring gap in a Couette rheometer
Square of the buoyancy (Brunt-Väisälä) frequency
a
Exponent of hE as a function of g_
b
Exponent of hE as a function of Kmik
Gradient Richardson number
Critical gradient Richardson number
Velocity difference between upper and lower mixed layers
Vertical distance across pycnocline (pycnocline thickness)
dz calculated from equation (4)
Density difference between top and bottom of pycnocline
Viscosity
Non- Newtonian “Excess” viscosity due to plankton-produced EPS
Newtonian “Solvent” viscosity due to water and salts
Shear rate, in our case across the pycnocline
Square of the arithmetic mean shear rate in a volume –time bin
Arithmetic mean of the shear rate squared in the same volume –time bin
Mean density of the water
Shear stress
“Excess” component of t due to shearing of EPS
Component of tE corresponding to first element of sum in left-hand side of
equations (16), (17), (19)
Component of tE corresponding to second element of sum in left-hand side
of equations (6), (17), (19)
“Solution” component of t due to shearing of water and salts
Yield stress (in this case, of sewage sludge)
Shear stress vertically across pycnocline
20.2
10
No
No
No
In turbulence, 7.5
4.4
No
No
5 1024
No
Yes
1.3
No
0.25
No
No
No
1.0
No
No
1 1023
No
No
No
1 103
No
No
No
None
m s22
Pa
Pa
Pa
None
cells mL21
m
m
m
(rad s21)2
None
None
None
None
m s21
m
m
kg m23
Pa s
Pa s
Pa s
s21
s22
s22
kg m23
Pa
Pa
Pa
No
Pa
No
No
1 1024
Pa
Pa
Pa
tLo
tW
ty
tFORCED
a
No
Not directly
Not directly
Yes
No
No
No
Yes
No
No
Yes
a
The variable P is not used directly, but is shown as 20.75 or 20.18, depending on the value of g_ . See equation (12).
The variable Q is not used explicitly, but shown as “1.3” in exponents of Kmik in equations (15), (16), (18), (19). Standard value taken from Jenkinson
and Biddanda (1995).
c
Bar notation: A horizontal bar over a quantity denotes the mean of that quantity, in this particular case in a 4D volume –time bin. The quantity to be
averaged is that covered vertically by the bar. ðg_ Þ2 and ðg_ 2 Þ thus mean, respectively, the square of the mean of all g_ values in the bin, and the mean of
all the values of g_ 2 in the bin.
b
viscosity Az of 1021 to 102 Pa s in the vertical plane
(Massel, 1999), several orders of magnitude greater
than the viscosity in the pycnocline, which comprises
only molecular viscosity h. There exists a difference in
density dr between the upper and the lower mixed
layers, which we vary around a standard value of
1 kg m23, and this produces a uniform vertical gradient
dr/dz in the pycnocline, where dz is the thickness of
the pycnocline. The mean density of the water r in all
layers is set to 103 kg m23, and we ignore the 0.1% or
so variation induced by dr.
Initially, we consider the pycnocline without phytoplankton or its associated EPS. Eddies in this layer are
damped by the stratification, so the only resistance to
shearing is that due to molecular viscosity due to water
and salt, hW, that is set at 1023 Pa s. Because of the
large difference between the viscosity in the mixed
layers Az, and that in the pycnocline, [h], shearing is
neglected in the mixed layers, and assumed to take
place only in the pycnocline. Vertical shear rate in the
pycnocline,
g_ ¼
t
hW
½s1 ð1Þ
where g is shear (or deformation) [m m21 ¼ dimensionless] and the dot on g_ signifies a differential with time,
giving shear rate (or deformation rate) [s21], and is the
same as dg/dt. t is the shear stress.
The velocity difference between the top and base of
the pycnocline,
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@u ¼ @zg_
½m s1 ð2Þ
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Fig. 1. One-dimensional three-layer equilibrium model of a pycnocline in which the thickness is determined by Richardson number Ri. It is
inspired by the pycnocline model described in words by Miles and Howard (1964), reproduced graphically by Woods (1968).
Pycnocline stability is governed by the gradient
Richardson number (Goldstein, 1931; Turner, 1972),
Ri ¼
gðdr=dzÞ
rðdu=dzÞ2
½dimensionless
ð3Þ
where g is the acceleration due to gravity.
When Ri becomes ,0.25, turbulence is generated
within the pycnocline, which entrains water in from one
or both mixed layers, thus increasing pycnocline thickness (Turner, 1972). In this equilibrium model, we fix a
critical Richardson number Ricr at 0.25. Substituting
Ricr for Ri in equation (3) and rearranging,
ðduÞ
g:dr
½m
ð4Þ
In simple, steady shear flow of liquid of uniform,
Newtonian viscosity, the dissipation rate per unit mass
of mechanical energy into heat
1¼
hg_ 2
r
ðg_ Þ2 ¼ ðg_ 2 Þ ½s2 ð7Þ
The bar notation is explained in the footnote of
Table I.
When shear rate shows variation, however, as in turbulence,
ð8Þ
ðg_ Þ2 ¼ kðg_ 2 Þ ½s2 where k is an inequality factor .1. From field observations, Kitaigorodskii (Kitaigorodskii, 1984) suggested
that a suitable relationship is
h
½W kg1 ð9Þ
r
Since for a uniformly shearing fluid, 1 ¼ g_ 2 h=r, a
value of k . 1, as in equation (9), is due to the fact that
turbulence contains a range of g_ values.
The value of 7.5 in equation (9) is thus the inequality
factor k in equation (8), being positively related to turbulence variance or intermittence (Gibson, 1991), but
in the almost laminar-flow pycnocline, k will only
slightly exceed 1.
The mean value of g_ is scale dependent and published values represent those in 4D volume – time “bins”
corresponding to the measurements made by standard
methods of Kitaigorodskii and other ocean-turbulence
workers. A hot-wire probe is used to measure water velocity with both time and space resolution smaller than
the respective Kolmogorov (i.e. turbulent) time
and length scales (e.g. Osborn and Lueck, 1985).
1 ¼ g_ 2 7:5
2
dz ¼ Ricr r
time and space,
½W kg1 ¼ m2 s3 ð5Þ
and the shear stress,
t ¼ hg_ ½Pa ¼ kg m s1 ð6Þ
Turbulent shearing, however, varies in time and space.
In natural water bodies, it generally takes a distribution
of values close to lognormal (Yamazaki and Lueck,
1990). When shear rate is uniform over a given range of
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Neither e nor h are actually measured in such investigations. e is calculated from values of g_ derived from
closely spaced velocity measurements, while h is
derived from temperature, salinity and hydrographic
pressure using tables such as those in Miyake and
Koizumi (Miyake and Koizumi, 1948) and Stanley and
Batten (Stanley and Batten, 1969), respectively.
In the oceans, e is considered by Mann and Lazier
(Mann and Lazier, 2006) to vary typically from 1026 to
1029 W kg21, although Van Leer and Rooth (Van Leer
and Rooth, 1975) found the values of g_ in deep thermoclines as low as about 0.0001 s21. Assuming a value
of 1023 Pa s for h, this gives a corresponding value of
7.5 10214 W kg21 for e , and for shear stress t down
to 1027 Pa, but e would have been higher if biorheological thickening had been increasing these tiny shear
stresses. “Biorheological thickening” means increasing
the viscosity or elasticity of the medium by means of
biologically produced substances. Such substances can
be termed thickening agents.
To the water in the TL, we now add phytoplankton
and its associated EPS, in this case K. mikimotoi (syn.:
Gyrodinium aureolum). Jenkinson (Jenkinson, 1993a)
measured the rheological properties of a culture of
4.4 cells mL21 of K. mikimotoi over a range of g_ from
0.0021 to 8 s21. These results are reproduced in Fig. 2.
From measurements of this figure, it may be deduced
that the excess viscosity (due to EPS) varied with g_ thus,
h ¼ hW þ hE ½Pa s
hW ¼ 1:00 103 ½Pa s
4 0:75
hE ¼ 1:27 10 g_
½Pa s
ð10Þ
ð11Þ
Fig. 2. Rheological properties of a culture of 4.4 cells mL21 of Karenia
mikimotoi (syn: Gyrodinium cf. aureolum). Excess viscosity hE and elastic
modulus G0 are plotted against shear rate ġ. G0 was found to be
between 10 and 50% of excess viscous modulus, G00E ¼ hE . ġ.
(Slightly redrawn from Jenkinson, 1993a.)
excess part due to EPS. So from equation (1),
tW ¼ hW g_ ;
tE ¼ 1:27 104 g_ þ0:25 þ 3:16 104 g_ þ0:82
½Pa
ð12Þ
where the subscripts W and E stand, respectively, for the
Newtonian component of a quantity contributed by
water and salt, and the excess (generally
non-Newtonian component) contributed by EPS.
For each value of g_ , this culture thus showed three
components of viscosity: first hW, the Newtonian aquatic
component due to water and dissolved salt, constant
across all shear rates and equivalent to a “solvent viscosity”; secondly, the first term on the right-hand side of
equation (12), a polymeric component giving viscosity
related to shear rate g_ by a power of -0.75; thirdly, the
second term on the right-hand side of equation (12), a
polymeric component similarly related to g_ by a power
of 20.18. These two polymeric components might represent two different polymers present, or else a single
polymer with different behavior at different shear rates.
Like viscosity, shear stress can be decomposed into
the Newtonian part due to water and salts and an
t ¼ h g_ ½Pa
ð13Þ
Multiplying both sides of equation (12) by g_ gives the
excess shear stress for this culture
4 0:18
þ 3:16 10 g_
tE ¼ hE g_ ;
ð14Þ
Excess viscosity is generally proportional to a power of
the concentration of the causative agent, whether this is
a dissolved polymer (Ross-Murphy and Shatwell, 1993)
or phytoplankton. For phytoplankton, Jenkinson and
Biddanda (Jenkinson and Biddanda, 1995) found the
power to be about þ1.3. We assume the same powerlaw relationship between tE and K. mikimotoi concentration Kmik,
Kmik 1:3 þ0:25
tE ¼ ð1:27 104 Þ:
:g_
þ ð3:16 104 Þ:
4:4
Kmik 1:3 þ0:82
g_
Þ ½Pa
ð15Þ
4:4
where the terms 4.4 come from the concentration (nos.
mL21) in the culture whose rheological properties have
been used as the basis for this model (Fig. 2). Equation
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resistance to shearing by the water phase (water þ salt),
(15) simplifies to
tE ¼ 1:85 105 Kmik 1:3 g_ þ0:25 þ 4:6
105 Kmik 1:3 g_ þ0:82
½Pa
tW ¼ h W
ð16Þ
Spinosa and Lotito (Spinosa and Lotito, 2003) investigated the yield stress, or resistance to flow just enough
to stop flow ty in sewage sludge in tubes of different
diameters. Their data show that, for different sludge
concentrations,
du
dz
ty LDIA
d
½Pa
½Pa
ð18Þ
We take the appropriate value of g_ to be the shear rate
vertically through the pycnocline (du/dz), so that
5
1:85 10
Kmik
1:3
ð20Þ
½Pa
ð21Þ
ð17Þ
where LDIA is the length scale of tube diameter, and d is
an exponent in this particular case close to 22. The
particular value of d might be related to the fractal
dimension of the aggregates’ shape (Mari and Kiørboe,
1996), size spectrum distribution or both. Like sewage
sludge aggregates, marine aggregates and transparent
exopolymeric particles (TEP) are quasi-fractally hierarchical organic aggregates rich in EPS (Mari and
Kiørboe, 1996). However, in places where the rheologically active polymer in the sea is more diffuse, then d is
likely closer to zero, but the effect of length scale on
seawater rheological properties has not yet been investigated. The rheological data in Jenkinson (Jenkinson,
1993a) were all obtained using Couette flow with a
measuring gap LCOUETTE of 0.5 mm. Therefore, to
model the effect of d (for simplicity assumed to be the
same for all values of g_ ) on excess viscosity, we take the
appropriate in situ length scale to be dz, applying a
“correction” relative to the value of L equal to (dz/
LCOUETTE)d thus:
tE ¼ ð1:85 105 Kmik 1:3 g_ þ0:25 þ 4:6
d
dz
5
1:3 þ0:82
10 Kmik g_
Þ
LCOUETTE
½Pa
where hW is independent of both shear rate and length
scale and is assumed to be 0.001 Pa s. We have taken
respective values for dr and r to be 1 and
1000 kg m23. Combining equations (6) and (10),
t ¼ tW þ tE
tE ¼
j
þ0:25
du
þ4:6 105
dz
d
þ0:82 ! du
dz
Kmik
½Pa
dz
LCOUETTE
1:3
ð19Þ
tE thus depends on Kmik, du/dz, dz and d. The
To run the model, let us first consider an example of no
plankton (Kmik ¼ 0 cells mL21) and a forced shear stress
across the pycnocline tFORCED ¼ 0.0001 Pa. Here the
value of d makes no difference to the outcome. The
computation proceeds in two steps.
In step 1, values are put in for a forced shear stress
tFORCED and a value is guessed for dz. We solve for t
¼ tFORCED and read off the corresponding value for
pycnocline shear rate du/dz¼ (Fig. 3A).
In step 2, dz is now calculated in two ways, first from
equation (2), using a range of values for du, and secondly from equation (4) (where dz is called dzz), using
the same range of values for du. Here when dz ¼ dzz,
dz is read off and a new value for du (Fig. 3B and D).
As confirmation, Ri, calculated from equation (3),
equals Ricr always, as it turns out, at the same value of
du as when dz ¼ dzz. In the present case, t equals tW
since in the absence of phytoplankton and EPS, tE is
zero.
Iteration now proceeds by repeating step 1, putting in
the value of dz obtained in step 2, and so on, until the
values remain unchanged. In this case dz ¼ 4.0 m,
du ¼ 0.4 m s21, g_ , or du/dz, ¼ 0.1 s21, h ¼ hW ¼
1023 Pa s and the buoyancy frequency N2 ¼ 2.5 23
10 rad s21.
Now let us add phytoplankton EPS to the model
with Kmik ¼ 4.4 cells mL21 and d ¼ 0 (rheological properties independent of length scale) and, as before,
tFORCED ¼ 0.0001 Pa. The results are shown in Fig. 3C
and D. In Fig. 3C, while the contributions, tHi, tLo, tE,
tW and t are each shown, the co-ordinate of t g_ is
retained for input to step 2. (tHi corresponds to the first
term in the sum on the right-hand side of equation (15),
and tLo to the second term, while tW, tE and t are all
defined in equation (14). After iteration, this simulation
with phytoplankton EPS gives dz ¼ 50.0 m, du ¼
1.43 m s21, g_ , or du/dz, ¼ 0.029 s21, h ¼ 3.5 24
1023 Pa s and N2 ¼ 2.0 10 rad s21.
The code is available at http://assoc.pagespro-orange.
fr/acro/TL/SupInfo.html.
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Fig. 3. Examples of output from model. (A) For Kmik ¼ 0 cells mL21, tFORCED ¼ 1024 Pa, t and tW (with tFORCED) versus ġ (marked as g dot).
Since t ¼ tW, the two lines are superimposed. After iteration, the values of t and ġ are read off. (B) As for (A), but solution of dz and Ri as
functions of du. (C) As for (A) but with Kmik ¼ 4.4 cells mL21, tFORCED ¼ 1024 Pa and d ¼ 0.tHi (dotted line), tLo (dashed line), tE (dash-dotted
line), t (upward curving thin continuous line), tW (straight thin continuous line) and tFORCED (horizontal line) versus ġ (as g dot). (D) as for (B)
but with the same inputs as (C).
R E S U LT S
In our model pycnocline, the Richardson number Ri
controls its thickness. With no extra viscosity from phytoplankton EPS, Fig. 4A illustrates that dz shows a
linear relationship with dr and is proportional to the
square of tFORCED.
At our standard value for tFORCED of 0.0001 Pa, and
20.2 for d, Fig. 4B shows how K. mikimotoi concentration
Kmik progressively increases dz, by over three times when
Kmik ¼ 10 cells mL21. The two lower curves, for Kmik ¼
0.1 and 0.316 cells mL21, cannot be distinguished, and
even the curve for Kmik ¼ 1 cell mL21 is barely distinguishable. The relative effect is higher at lower values
of tFORCED (Supplementary Table SI, http://assoc.
orange.fr/acro/TL/091011SupTab1.xls). Figure 4B also
shows that the effect of Kmik on dz decreases as dz itself
increases. This is because the negative value of d lessens
the functional value of viscosity h as pycnocline thickness
increases.
In summary, Fig. 4B shows that for our standard
value of d of 20.2, a value of 107 cells L21 would
increase pycnocline thickness by a factor of 3 – 4 over
the thickness for no cells. For our standard concentration of K. mikimotoi, 4.4 million cells L21, Fig. 4C
shows that a value of d equal to 20.2 roughly doubles
pycnocline thickness compared to the case when there
is no K. mikimotoi, while a value of d equal to 0 thickens
the pycnocline by over an order of magnitude.
Figure 4C shows the effect of d on the relationship
between dz and dr, in relation to that when Kmik ¼ 0.
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Fig. 4. Some examples of model output of pycnocline thickness dz plotted against density difference across the pycnocline. Standard values of
parameters (Table I) are used except where stated to be otherwise (A) no phytoplankton. Values of shear stress across pycnocline tFORCED from
lower to upper curves: 0.001, 0.000316, 0.0001, 0.0000316, 0.00001 Pa. (B) Karenia mikimotoi concentrations, shown as Kmik, from lower to upper
curves 0.1, 0.316, 1, 3.16, 10 cells mL21; the curves for Kmik equal to 0.1 and 0.316 cannot be distinguished, and serve also to illustrate the case
for no phytoplankton. (C) Different values of the exponent d defining the relationship between viscosity and length scale. For the lowest curve,
the case is given for Kmik ¼ 0, and for the other curves, the standard value of Kmik 4.4 cells mL21 is used, with d values of 20.1, 20.2, 20.3,
20.4. Note the different vertical scales on the graphs, and that the lowest curve of (B) and (C) corresponds to the middle one of (A).
When Kmik is the standard value of 4.4 cells mL21 and
d is zero, and over the range of length scales of interest
here, the EPS would be perfectly dispersed. (Water and
its viscosity hW are also perfectly dispersed.) Such
perfect dispersion of EPS would give thickening independent of length scale, and this is why the curve for
d ¼ 0 is parallel to that for Kmik ¼ 0. For negative
values of d, in contrast, as dz increases, hE decreases, so
the curves tend down towards that for Kmik ¼ 0.
DISCUSSION
There have recently appeared complaints of models
that show “failure to fail”, for instance in plankton
trophic models (Franks, 2009), while in the field of
rheology, Woodcock (Woodcock, 2009) bemoans, “computer modeling with many-parameter models is neither
‘theory’ nor ‘experiment’. Moreover, it’s not research,
it[’]s ‘animation’!”. The present model, in contrast,
aims to suggest which parameters are critical to pycnocline dimensions, and under which conditions, and
which are not. Our aim is not to animate, but to help
correctly focus the attention of future research.
Previous work (Jenkinson, 1986, 1993b) modeled, and
attempted to predict, the effect of the shear-thinning
exponent of phytoplankton EPS on isotropic turbulence.
As explained by Jenkinson and Sun (Jenkinson and Sun,
2010), however, a 10-fold increase in Newtonian viscosity (sucrose solution) imposed for calibration of those
experiments did not significantly change the size of
turbulent eddies, possibly because the 3-L container
constrained a harmonic system of eddies (Jenkinson
2004b).
Later work (Jenkinson et al., 2007; Jenkinson and
Wyatt, 2008) pointed out that data presented by Spinosa
and Lotito (Spinosa and Lotito, 2003) indicated that the
yield stress in suspensions of microbial flocs is strongly
length-scale-dependent. This implies that biorheological
thickening of natural waters by EPS is likely to depend
not only on the concentration and the shear-thinning
exponent P of the EPS polymers, but also on their
characteristic length-scale exponent d.
P has already been measured in the sea (Jenkinson
1993b; Jenkinson and Biddanda, 1995) and in phytoplankton cultures (Jenkinson, 1986, 1993a) as mentioned
above, while the effect of P on isotropic turbulence has
also been modeled (Jenkinson, 1986, 1993b, 2004a). The
present model, therefore, has kept a two-component P
constant and investigates the effects of changing d and the
concentration Kmik of a K. mikimotoi bloom, whose rheological properties correspond to a culture measured by
Jenkinson (Jenkinson, 1993a) at a known length scale.
Comparison of Fig. 4B and C shows that for an imposed
shear stress tFORCED, changing d from 0 to 20.2 changes
pycnocline thickness dz by a factor of 10 or more, while a
hefty bloom with Kmik of 10 million cells L21, and
d ¼ 20.2, increases dz only approximately five times
more than with no phytoplankton.
It is important to note that for values of d of 20.4 or
more negative, the effect of blooms on pycnocline thickness is practically negligible at realistic values of dz, say
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1 cm or more, even for blooms of high concentration.
To understand whether EPS thickening may be affecting dz in situ, effort should be given to measuring d in
different blooms and cultures. As mentioned, a value
for d of 0 corresponds to a perfectly dispersed polymer
solution, so by changing the flocculation parameters of
its EPS, a bloom organism may have the potential to
radically change its local environment over a wide
range of scales. Simple changes in molecular properties
may thus be under strong, and perhaps competing,
evolutionary pressures related to niche engineering by
bloom species (Jenkinson and Wyatt, 1992, 1995, 2008;
Wyatt and Ribera d’Alcalà, 2006). Marine snow and
TEP (Alldredge et al., 2002) are common in pycnoclines, and probably form an important component in
the floc-size continuum, thus probably influencing d.
It seems likely that the power-law exponent d would
be closely related to the fractal dimension of particles,
both in the sludge measured by Spinosa and Lotito
(Spinosa and Lotito, 2003), in fluid mud (e.g. Hsu et al.,
2009) and in the particles of EPS (Jenkinson, 2004a)
considered in the present study. Although this relationship requires further investigation, it is not a necessary
input for the present model.
In aquatic systems, biological and physico-chemical
processes, such as diffusion, mixing and encounter, are
governed by water deformation, so viscosity and elasticity need to be measured and known at the scales of
each process under consideration. Measurement over a
range of scales will furthermore allow d to be determined, and to be extrapolated, with caution, to other
length scales. Associated with such measurements of
physical properties, an intense research effort is needed
to visualize water movement, together with videos of
flocculation and break-up of EPS flocs, turbulence and
organism trajectories in 3D over appropriate ranges of
length, time and shear stress. In situ observations and
measurements should be the aim, and laboratory work
only a step on the way towards it.
Finally, this is a proviso concerning the use of the
Richardson number in modeling pycnoclines.
Dekshenieks et al. (Dekshenieks et al., 2001) measured
Richardson number Ri on many occasions in a pycnocline in a shallow fjord, and their Fig. 4F and 7h show
that while a tight lower bound of 0.25 – 0.23 existed for
Ri, no upper bound was apparent. Since Ri is linearly
related to dz, a lower bound thus exists similarly for dz
but no upper bound. Previously, Oakey and Elliott
(Oakey and Elliott, 1982) had worked in Shelf waters off
Nova Scotia. Rather similarly to Dekshenieks et al.
(Dekshenieks et al., 2001), they found (see their Fig. 10)
high Ri, averaging 10, in mixed water below 20 m,
but values of Ri , 0.25 only in the upper 10 m, where
active turbulent events, thought to reduce stratification
and hence Ri, were frequently observed. This implies
that the lower limit of Ri is fairly tightly controlled by
turbulence, but that erosion of the pycnocline at high Ri
is only a weak mechanism, so other causality for thick
pycnoclines, such as their history, may be more important. In the present model, Ri is determined only
indirectly by viscosity, through dz and du. In the past,
the critical value of Ri has been determined empirically,
without explicitly taking viscosity into account, so we do
not know quantitatively the mechanisms behind the
control by Ri. It is important to know, and would be
interesting to investigate, whether viscosity has a direct
effect on Ri. If it does, h would have to appear explicitly
in (3), corresponding to the way in which h would be
controlling dz.
CONCLUSIONS
Based on (1) published values of how viscosity relates to
shear rate in a culture of a Karenia mikimotoi culture and
(2) published values of how yield stress of sewage sludge
varies as a function of length scale, we modeled how
blooms of K. mikimotoi at different concentrations and for
different values of d would affect pycnocline characteristics. It was found that providing d is not more negative
than about 20.2 or 20.3, the phytoplankton could
affect pycnocline thickness several-fold, but below this
value of d, even high concentrations of the plankton
would have no significant effect. This kind of model thus
points out the most critical parameters important for
further investigations. In the present case, investigation of
d, and secondarily P, are required, as well as how hE
depends on the concentrations and physiological states
of the different species of phytoplankton present.
Despite the preliminary calibration of this model
using the published data presented in Fig. 2, future validation is required. We envisage that this will take two
steps. The first step will be measurement of viscosity
and other rheological properties of phytoplankton
blooms and cultures at natural oceanic concentrations
as a function of length scale. The second step will be
investigation of modulation by phytoplankton blooms of
pycnocline dynamics, particularly thickness dz, both in
flume studies and at sea.
S U P P L E M E N TA RY DATA
Supplementary data can be found online at http://
plankt.oxfordjournals.org.
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AC K N OW L E D G E M E N T S
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For many years and particularly at the 2009 GEOHAB
Galway meeting on Modelling Thin Layers, Elisa Berdalet,
the late Patrick Gentien, Dennis McGillicuddy, Tim Wyatt
and Hidekatsu Yamazaki have helped and encouraged this
work on how thalassorheology influences water movement.
We also thank two anonymous referees, as well as two
referees of a former version for their insightful comments.
Hsu, T. -J., Ozdemir, C. E. and Traykovski, P. A. (2009)
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FUNDING
We gratefully acknowledge help by S.C.O.R. for participation by I.R.J. in the GEOHAB Modeling Workshop,
Galway, 2009, while writing of this paper has been supported by a Chinese Academy of Sciences Research
Fellowship for Senior International Scientists
(2009S1-36) to I.R.J., the Knowledge Innovation
Project of The Chinese Academy of Sciences
(KZCX2-YW-QN205) to J.S., the Chinese National
Programme on Key Basic Research Project
2009CB421202 to J.S. and the National Natural
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