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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 OCE´ANOGRAPHIQUES, LAVERGNE,

2 PEOPLE’S REPUBLIC OF CHINA AND AGENCE DE CONSEIL

*CORRESPONDING AUTHOR: [email protected]

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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Received May 11, 2010; accepted in principle July 1, 2010; accepted for publication July 19, 2010

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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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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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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-Va¨isa¨la¨) 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

tW ty tFORCED

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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½m s1 

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tLo

a

No Not directly Not directly

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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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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).

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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 g˙ . G0 was found to be between 10 and 50% of excess viscous modulus, G00E ¼ hE . g˙. (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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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,

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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

We take the appropriate value of g_ to be the shear rate vertically through the pycnocline (du/dz), so that 1:85  10

Kmik

1:3

½Pa

ð21Þ

 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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½Pa ð18Þ

5

ð20Þ

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

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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. 3. Examples of output from model. (A) For Kmik ¼ 0 cells mL21, tFORCED ¼ 1024 Pa, t and tW (with tFORCED) versus g˙ (marked as g dot). Since t ¼ tW, the two lines are superimposed. After iteration, the values of t and g˙ 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 g˙ (as g dot). (D) as for (B) but with the same inputs as (C).

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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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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).

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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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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’Alcala`, 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

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Grindley, J. R. and Taylor, F. J. R. (1971) Factors affecting plankton blooms in False Bay. Trans. R. Soc. S. Afr., 39, 201– 210.

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) High-resolution numerical modeling of wave-supported gravitydriven mudflows. J. Geophys. Res. Oceans, 114, C05014. Iversen, K. R., Primicerio, R., Larsen, A et al. (2010) Effects of small-scale turbulence on lower trophic levels under different nutrient conditions. J. Plankton Res., 32, 197– 208. Jenkinson, I. R. (1986) Oceanographic implications of non-Newtonian properties found in phytoplankton cultures. Nature, 323, 435– 437. Jenkinson, I. R. (1993a) Viscosity and elasticity of Gyrodinium cf. aureolum and Noctiluca scintillans exudates in relation to mortality of fish and damping of turbulence. In: Smayda, T. and Shimizu, Y. (eds), Toxic Phytoplankton Blooms in the Sea. Elsevier, Amsterdam, pp. 757–762.

FUNDING

Jenkinson, I. R. (1993b) Bulk-phase viscoelastic properties of seawater. Oceanol. Acta, 16, 317–334. Jenkinson, I. R. (1995) A review of two recent predation-rate models: the dome-shaped relationship between feeding rate and shear rate appears universal. ICES J. Mar. Sci., 52, 605– 610. Jenkinson, I. R. (2004a) Effects of phytoplankton on turbulence from small to large Reynolds numbers: A model with fractally intermittent turbulent dissipation and fractally distributed excess shear-thinning viscosity in fractally aggregated phytoplankton exopolymer. Geophys. Res. Abstr., 6, 05825. Jenkinson, I. R. (2004b) Effects of phytoplankton on ocean turbulence: Observations using Particle Tracking Velocimetry (PTV). Geophys. Res. Abstr., 6, 06557. Poster: http://assoc.orange.fr/acro/egu1.pdf.

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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 Science Foundation of China (40776093) also to J.S.

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