Oceanography and Marine Biology: an Annual Review 1996, 34, 71-107 ) A. D. Ansell, R. N. ...

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OceanographyandMarineBiology:anAnnualReview1996,34, 71-107
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A. D. Ansell, R. N. Gibsonand MargaretBarnes, Editors
VCLPress
NUTRIENT FLUXES TO PLANKTONIC OSMOTROPHS
IN THE PRESENCE OF FLUID MOTION
L. KARP-BOSS, E. BOSS & P. A. JUMARS
University ofWashington, School ofOceanography,
Box357940, Seattle, WA 98195-7940, USA
Abstract Wepresentsolutionsfor nutrienttransfertoosmotrophsinthe full rangeofflow regimesfor
whichsolutionshavebeenpublished,andweextendsomeofthosesolutionstonewparameterdomainsand
flow environments. These regimes includestagnantwater;steady, unitormflow arisingfrom swimmingor
sinking; steadyshearflows; and fluctuating shearfrom dissipationofturbulence, aswell as the combined
effectsofturbulence-inducedshearandswimmingorsinking.Solutionsfornutrientfluxescannotbecarried
overfromoneflow regimetoanother. Inallcases, however, masstransferincreaseswithcellsizeandwith
flow velocity. Cell shapebecomesparticularly importantathighflow velocities. Forsteady, uniformflow
arisingfromsinkingorswimming,we find asymptoticanalyticandnumericalsolutionsfrom theengineer
ingliteraturesuperiortothoseinmorecommonusewithinoceanography.Theseengineeringsolutionssug
gest flow effects an order of magnitude smaller than commonly supposed. A cell radius near 20Ilm is
neededbeforeswimmingorsinkingcanbeexpectedtoincreasethefluxofnutrients,suchasnitrateorphos
phate, substantially (by ~50%) over the stagnant-watercase. We find sound asymptotic solutions for the
caseoflinearshearandsupplementthemwithnumericalsolutionsofourowntocovertherangeofcellsizes
andshearratesofinterestforphytoplankton.Weextendthemfurthertocoverviscousshearsfromdissipat
ing turbulencefor cells smallerthan the Kolmogorov scale (orderof1-6mm in the ocean). Ouranalysis
suggests turbulence effects an orderof magnitude greater than previously postulated, with a cell size of
60~lmneededtoexperiencesubstantialgain.Cellrotation,whetherinducedbythepropulsionmechanismin
swimmingorpassivelybyshearacrossthecellperimeter,willreducetherateofnutrienttransferrelativeto
a non-rotating cell unless the axis of rotation parallels the direction of flow. Although in calm water
dinoflagellatesbyswimmingareableto increase nutrientuptake, instrongturbulence theymay notbeable
tomaintainarotational axis parallel to the directionofswimmingorthe directionofshear, resulting ina
relative reductioninflux. Conversely, largechainsofdiatomsandfilamentouscyanobacteriathatspanthe
radiusofthesmallestvorticesarebestabletotake advantageofturbulence. Despitethesedeductionsfrom
a diversity ofanalytic and numerical solutions, unequivocal data to test the contribution ofadvection to
nutrientacquisitionbyphytoplanktonarescarce- owing, inlarge part, to the inability tovisualize, record
andthusmimicfluid motions inthevicinitiesofcellsinnaturalflows.
Introduction
Effects of small-scale fluid motions on diffusional fluxes of nutrients to bacteria and
phytoplanktonhavebeenreviewedseveraltimes(Gavis 1976,Roberts 1981, Lazier& Mann
1989, Mann & Lazier 1991, Ki0rboe 1993). It is thus worth asking why it might be worth
while to visit them yet again. In the process of a review focused much more closely on
bacteria, we (Jumars et al. 1993) became aware ofseveral inconsistencies in the published
treatments of solute fluxes to osmotrophs. Among the accumulated problems we encoun
tered, one solution (Munk& Riley 1952) is oftenusedfor sinking cells while another (Berg
& Purcell 1977)isoftenusedfor swimmingcells, despite the fact that both invoke the iden
tical governing equation and boundary conditions and thus can neither apply to disparate
71
L. KARP-BOSS, E. BOSS & P. A. JUMARS
processes nor both be correct. Unfortunately, we perpetuated several other inconsistencies.
In particular, we were not adequately aware of the diversity of field and laboratory flow
regimes for which solutionshad beengivenorofthe fact thatcompetingsolutionshad been
publishedfor someregimes. Theprincipalpurposeofourpresentreview, therefore, is tolay
outthe diversityofflow regimes plainly and topoint outthe best oftheknown solutions for
each specific case. We focus on those physical characteristics ofthe cell and its flow envi
ronment that affect nutrient flux.
Asecondary purpose is to specify, as accurately as can be done from the existing litera
ture, the size of organism for which relative fluid motion begins to become important in
nutrient flux. It has been firmly established that, for diffusion coefficients typical ofsmall
molecules in water (of order 1O-5cm2s-I), 1-~lm long, freely-suspended bacteria cannot
experience any substantial flux enhancement from the fluid shear about them (Berg &
Purcell 1977, Lazier& Mann 1989, Logan& Dettmer 1989). Utilizationofslowlydiffusing
molecules, larger "cell" sizeorattachmenttolargerparticles, however, canresultinsignifi
cant benefit from relative fluid motion (Roberts 1981, Logan& Hunt 1987, 1988, Lazier&
Mann 1989, Logan& Dettmer 1989, Confer& Logan 1991). The rangesofsizes and diffu
sion coefficients for which such enhancements can become important have been given
implicitly or explicitly in several treatments (Logan & Hunt 1987, 1988, Lazier & Mann
1989), but have sometimes been based on equations from inappropriate flow regimes. As
with many other real-world biological-physical interactions of interest, these substantial
enhancements of fluxes begin in an uncomfonable middle range of parameters for which
analytic solutions typically are not available. Our systematic exploration identifies flow
regimes and parametervalues for whichfurther numerical modellingand laboratory experi
mentationwould beprofitable.
The role offluid motioninthe nutritionofplanktonicautotrophsandosmotrophs is simi
lar to its role inproblems ofheatand mass transfer long studiedbycivil and chemical engi
neers. Since the governing equations andboundary conditions for the heat-transferproblem
andthe dilute-solutionmass-transferproblemare identical, heat-transfertheory isapplicable
directly to the study ofnutrient fluxes to planktonic osmotrophs. Most ofthe solutions we
presentthus comefrom theengineering literature; we use, however, oceanographic(Appen
dix I) rather than engineering convention for notation. In keeping with practice in physics
and engineering, we use square brackets to denote primary dimensions oflength [L], mass
[M] and time lTl. Toavoidconfusion, we avoidthe term "convection" andlabel any organ
ized fluid motion as "advection". In this review diffusion and moleculardiffusion are syn
onymous. Several inconsistencies have occurred because some authors used radii while
others used diameters as characteristic dimensions; for consistency, we give characteristic
cell sizes as radii. We work from stagnant fluids, through simple, steady shear flows to the
three-dimensional, rotational shear flows characteristic ofdissipating turbulence. We warn
fluid dynamicists that we use the word "particle" exlusively for a solid object of finite di
mensions suspendedin the flow and notsynonymously witha small parcel ofwater.
We focus on phytoplankton, because it spans the full size range ofinterest (from a few
micrometresto afewmillimetres) and comprisesawidediversityofbody, chainandcolony
shapes. Our analysis also applies, however, to metazoan larvae and otherfree-living plank
tonic osmotrophs. To save space we refer to the individual as a cell and leave implicit the
applicability to multicellularorganisms ofcomparable sizes. Wealsoleave implicitthe con
verse problem ofsolute leakage from a cell, although it is plain (Jumars et al. 1993, their
Fig. 1)thatour results applyequally well there. The principal differenceis asignchange in
theconcentrationgradientandhenceinthefluxdirection; magnitudesoffloweffectsonsuch
72
NUTRIENT FLUXES TO PLANKTONIC OSMOTROPHS
fluxes are identical. Wealsocautionthatsolutionsfrequently givenfor apointsourceorsink
cannot adequatelydescribe theproblemofsolute leakagefrom oruptake byacell; animpor
tant aspectoffluid motion is the divergence ofstreamlines around cells.
Mass transfer to cells in stagnant water
Non-motile cells:pure diffusion
Rate of nutrient delivery into a planktonic cell at steady state proceeds at the rate of the
slower oftwo processes in the series, nutrient delivery to the cell membrane and transport
across the membrane. If the former is much faster, no nutrient-depleted layer develops
around the cell. If the latter is faster, however, nutrient concentration at the cell surface
drops below its far-field level, thereby driving a faster delivery rate by diffusion.
The concentrationdistribution ofanutrient aboutaplanktonic cell ineither case is given
by the diffusionequation (Crank 1975),
ac =V(DVC) (Equ. 1)
at '
where D is the diffusion coefficient [L2 T-1l and C is the concentration of the nutrient of
interest. The del operator (V) indicates partial derivatives in all three spatial dimensions
(e.g. in Cartesian co-ordinates alox+ol8y+olaz). In a heterogeneous environment like the
ocean, Dvaries in time andspace (due to temperature and salinity). Since we are interested
in processes at the microscale, D for a given nutrient is assumed to be constant. Thus,
V(DVC) = DV2C.
At steady state (aClat =0) the solution ofEquation 1for a sphere becomes:
(Equ. 2)
where '0 is the cell radius, , is the radial distance from the centre ofthe cell, C= is the far
field concentrationand Cois concentrationatthe cell surface. Whenuptakecapacityinto the
cell exceedsdiffusional supplyrate, Equation2indicatesthatthereexists anutrient-depleted
region in the vicinity ofthe cell (Koch 1971, Jumars 1993). For convenience, hereafter we
refer to this depleted regionas the diffusional boundarylayer and define it morespecifically
as the region in which C::;90% ofthe ambientconcentration. By this definition it extends to
about nine cell radii away from the cell surface (Fig. 1herein; Koch 1971, Jumars 1993).
The outer limit is arbitrary, since approach tobackground concentration is asymptotic. The
rate at which nutrients will betransported across this diffusional boundary layercan beesti
mated from the characteristic diffusion timescale (tD)'
(Equ.3)
where L is a characteristic length scale (e.g. the thickness ofthe layer). For a cell with a
c
radius of 10/lm and adiffusional shell 90/lm thick, it will take c. 8s for a molecule such as
nitrate or phosphate (D ofthe order 1O-5cm2s-l) to cross this layer.
73
L. KARP-BOSS, E. BOSS& P. A. JUMARS
goo
Concentration Fields Around Spherical Cells,
C]]
--0.9
..---- 0.5
.............. 0.1
PureDiffusion
(noflow)
Pe=O Sh=1
UniformFlow
(swimmingor
sinkingin
o
stagnantwater;
flowfrom -Xto +X)
·1
Pe=1 Sh=1.21
·2
·2 ·1
.~.~
ShearFlow
(purestrain;
toward originin
YandZ; away
from origin in X)
Pe=1 Sh=1.44
-2 ~::::~::::~:::..<:::::
-2 ·1 0 2
)(fro
Figure 1 Concentration distributions ofa given nutrient around cells (black circles) in three
different flow regimes: stagnant water (pure diffusion), uniform flow (i.e. cell swimming or
sinking instagnantwater; the cell is movingfrom right to leftsoflow is from left toright) and
shearflow (uniaxialextensional flow whereE = (2,0,0; 0, -1, 0; 0,0, -1). The thirddimen
sion can be visualized by rotation about thex axis. The left panel illustrates the concentration
field atdistancesupto 10cellradii fromthecentreofthecell.Therightpanelisa "blow-up"of
theconcentrationfield nearthecellsurface, uptoadistanceof2 radiifromthecentreofthecell.
Concentration values for the case of pure diffusion were obtained by solving Equation 1 at
steady stateanalytically. Values for the two other caseswere obtained by solving Equation 11
numerically for Pe= 1tosteady state (Appendix 11). In theabsenceoffluid motiona diffusive
boundarylayer(C;:5;0.9CJextendstoabout9cellradiifromthecellsurface. Uniformandshear
flows distorttheboundarylayerandsteepentheconcentrationgradientincertainregions. Since
transportisdominatedbythethinnestregionsofthediffusionalboundarylayer,cellsinuniform
orshear flow willexperienceenhancedfluxes comparedto non-motilecells instagnantwater.
The diffusional flux ofnutrients to a spherical cell (QD) depends on the diffusion coeffi
cient, the cell radius ('0) and the concentration gradient between the cell surface and the
ambient water,
(Equ.4)
There are onlytwomeans bywhichthe cell canincrease theflux to itselfwithoutrelative
fluid motion: increasing,oandlowering Co(Jumarsetal. 1993). There are, however, limits
74
NUTRIENT FLUXES TO PLANKTONIC OSMOTROPHS
on the increase achievable. As cell size increases, demand for nutrients increases more
rapidly (proportional approximately to r 2 cf. Roberts 1981) than does the diffusive flux.
0
In addition, for a given cell size, it is impossible to enhance the flux beyond its value for
Co = 0, when every molecule that reaches the cell surface is absorbed immediately.
Assuming delivery to a spherical cell by molecular diffusion and active uptake of
Michaelis-Mentenform acrossthe cell membrane, Pasciak& Gavis (1974) introducedacri
terion for recognizing transport limitation. They defined the parameter P to be the ratio
betweendiffusional delivery and maximal uptake rates,
=
p 14.47rroDKm
(Equ.5)
V. '
max
where V is the maximal uptake rate [mol ceU-' T-'] andK is the concentrationat which
lnlU m
the uptake rate equals IhV ' WhenPis small, the cellhas the capacity to absorb nutrients
max
muchfaster thanthe rate atwhichnutrientscandiffuse towards the cell, and transport to the
cell membrane becomes limiting. When Pis large, the maximal uptake rate is much slower
than diffusion, and for such an organism uptake rate will not be controlled by its physical
environment. Pasciak& Gavis (1974) applied this criterionto phytoplankton, for which val
ues ofKin and V were known from laboratory experiments, and thereby established that
mat
forsomephytoplankton(mostlylargecells)transportlimitationcanbe significant. Ithas also
beensuggestedthat undercertainconditionsuptake ofcarbondioxide (Riebesell etal. 1993)
and iron (Morel et al. 1991) by phytoplankton and uptake of phosphate by cyanobacteria
(Mierle 1985) may become transport limited.
Determining whether uptake plays a role in limitation is clearly important but is outside
the scope ofour review. For all subsequent analyses, we assume that the cell is a perfect
absorber, i.e. thatcell-surfacenutrientconcentration(Co)equalszero. Wedosofor two rea
sons. One istoprovideanupperboundonthe importanceoffluid dynamiceffects: ifthere is
no significanteffectoffluid motiononuptake for aperfectabsorber, then therecanbe none
for an imperfectone. The moreobvious reason is for compactness. Ifkinetics ofuptake are
known and suspected to be important, they can be included in the problem in the manner
shown by Pasciak & Gavis (1974, 1975).
Motile cells: swimming and sinking
Theory
Swimming and sinking are two mechanisms by which a phytoplankter can induce relative
fluid motion, steepenits surrounding solute gradientsandhence increasethe fluxes ofnutri
ents to itself. The nature of the flow in the vicinity of the cell is determined by its body
Reynolds number (Re). This Reynolds number describes the relative importance ofinertial
forces comparedto viscous forces,
=
Re ULe
(Equ.6)
v '
where Uis thecharacteristicvelocity, v isthe kinematicviscosity [L2T-'] andL isthechar-
c
acteristic length scale, inour case the cell radius (r )' Since typical values ofRe for plank-
o
75
L. KARP-BOSS, E. BOSS& P. A. JUMARS
Table1 ReynoldsandPeeletnumbersforswimmingandsinkingphytoplankton.
Swimming Sinking
Cellradius speed speed
__
Organism (~lm) (I-lms-I) (I-lms·l) Re _. Pe .S_our-ce_.
Prorocelltrummarial-Iabouriae 6 171 0 Ix10-3 1 Kamykowsk•_iet_al._(19••9•M•2••_).
Gyrodinium(Iorsum 16.5 330 0 5.4xlO-3 5.4 Kamykowskietal. (1992)
Dillophysisllcuta 33 500 0 1.6x102 16.5 Sommer(1988)
CoccolithushlLrleyi 5to6.5 0 3to 15 1.5x10-5to 1.5xlO·2to Smayda(1970)
9.7x10-5 1J.7xlO2
Coscinodiscus wailesii 70 0 81 to 347 5.7xlO,3to 5.7to24 Smayda(l1J70)
2.4xlO,2
tonic micro-organisms are well below unity (Table I), theirenvironments are dominated by
viscous forces, and inertial forces can be neglected.
In the presence offluid motion the concentration distribution of the nutrient of interest
around an organism is givenby
(Equ.7)
where U is the three-dimensional velocity field. As a first approximation, we use Stokes'
solution for uniform, creeping flow past a stationary, rigid sphere (Leal 1992, Ch. 4) to
describe the flow field around cells swimmingorsinking instagnant water. Topermit ana
lytic determinationofthe velocity field and concentrationdistributionin the neighbourhood
ofthe cell, we assumesphericalshapeandsufficientspacingbetweencellstopreclude inter
actions among them.
The boundary conditions applied to Equation7are:
c =0at'0' and (Equ. 8)
C = Coo for, ~oo. (Equ.9)
These boundaryconditions setan upperboundfor the calculatedflux. Theflux ofagiven
nutrient (Q) to the cell is determined by the integrated concentration gradient normal to the
cell surface:
Q = -DfnV'CdA
(Equ. 10)
A
where A is the cell surface and n is an inward unit vectornormal to it.
Much ofthe information and insight about key parameters that determine the concentra
tion distribution in the presence offluid motion can be obtained via nondimensionalization,
= =
without theneedtosolvethe differentialequation. Defining U* UIU; C* C/CooandX*,
y*, Z* =x/'O' ylro' z/ro' the nondimensional form ofEquation7 atsteady state becomes
(Equ. 11)
where each asteriskdenotes anondimensional variable. Correspondingboundaryconditions
are:
76
NUTRIENT FLUXES TO PLANKTONIC OSMOTROPHS
C* =0 at r*= 1 (Equ. 12)
C* = 1as r*~ (Equ. 13)
00,
for r* the nondimensional radial distance from the centre ofthe cell (r*= rlro)and Pe, the
dimensionless Peelet number. The Peclet number indicates the effectiveness of advective
transportcompared withdiffusive transport throughthe fluid overthe specified length scale:
Pe == Uro , (Equ. 14)
D
where Uis the characteristicvelocity (swimmingandsinkingvelocities inthe caseofmotile
cells). Diffusion dominates transportwhen Pe < 1, and advection dominates when Pe > 1.
Note that while Reynolds numbers for swimming and sinking planktonic micro-organisms
are alwayssmallerthanunity, Peeletnumbers for large cellscanbe larger(Table 1). Again,
the body Reynolds number ofan organism will determine the flow field in its vicinity while
the Peelet number will determine the concentration field ofnutrients and hence the fluxes.
Since Pe is the only nondimensional parameter arising from the steady-state form of
Equation 7, any desired nondimensional quantity determined by the concentration field,
namely the nondimensional flux, will be dependent solely on Pe. A measure ofthis flux is
the Sherwood number, Sh (its heat-transfer analogue being the Nusselt number, Nu), which
is the ratio between the total flux ofnutrients arriving to the cell surface in the presence of
fluid motion (Q) and the purely diffusional flux (QD): -.
-DfnVCdA
JL
Sh == = _---'-'A:--_---:- (Equ. 15)
QD 41C1iP(Coo - Co)
Shthus indicatesrelativeenhancementofflux duetoadvection. Inthe caseofasphericalcell
in stagnant water Q=Q and Sh= Sho=I
D
In the engineering literature the Sherwood number is commonly defined alternatively in
terms ofthe mass-transfer coefficient, k, [LT-1] as Sh==2kroiD. In this case, Sho=2. The
mass-transfer coefficient is the ratio between the mass flux per unit ofarea and the concen
tration gradient between the boundary of interest and the environment (Cussler 1984,
Rohsenow & Choi 1961). Itcanbe determinedempiricallyand varies with flow regime and
shape ofthe body. This definitionhas beenadopted inoceanographic applications but in the
form Sh==kroiD (Sho=1, e.g. Logan& Hunt 1987, 1988). Thedifferencearises from selec
tionofthediameterversus theradius asthe characteristiclengthscale. Forthesamereason,
and hence by the same factor, definition ofthe Peclet number in the engineering literature
oftendiffers from the oceanographic convention, i.e. in the engineering literature it is often
Pe==2UroID. To reduce confusionamongoceanographicreaders, we haveconvertedall for
mulas extracted from the engineering literature to the oceanographic convention of Equa
tions 14 and 15 (Appendix I). We makethispointand ourparameterizations explicitbecause
factors oftwobasedonchoiceofdiameterversus radiusas the characteristicdimensionhave
caused some mischiefinpastcomparisons.
In order to examine the effect offluid motion on nutrient flux a functional relationship
betweenSh and Pe issought. Firstit is necessary to find asolutionto Equation 11 thatsatis
fies theboundaryconditions 12and 13. Unfortunately, thisproblem is toocomplextopermit
exact solution over the full range of Pe. One way to approach it analytically is to derive
77
L. KARP-BOSS, E. BOSS & P. A. JUMARS
asymptotic solutionsfor limitingvaluesofPeeletnumbers. Asymptotic relationsbetweenSh
and Pe have been derived for the two extreme cases, Pe« 1, when advective effects are
relativelyminorcomparedtopurediffusionandPe» 1, whentheeffectofmoleculardiffu
sionmaybeneglectedeverywhereexceptforathinboundarylayeratthe fluid-cell interface.
The major difference between the two is the nature ofthe dependence ofthe concentration
distributionon flow geometry and hence the nature ofthe dependence ofSh on Pe. Forlow
Peelet number, the concentrationdistribution near the body is governed by moleculardiffu
sion and is therefore relatively insensitive to flow geometry within that region. Cell surface
reliefsmaller thanO.lroin dimension can have no great effect in this regime. Atlarge dis
tance from the body, however, the form of the velocity field determines the concentration
field, sooverallbodyshapeandorientationcanbeinfluential. AtlargePecletnumber, onthe
other hand, the effectofmolecular diffusion is limited to a very thin layer around the cell,
and therefore the concentration distribution will be very sensitive to the flow field in the
vicinity of the cell (Leal 1992). Here, small-scale reliefon the cell surface can have pro
found effect. For this reason, flux will be much more sensitive to fine-scale details ofcell
shape at high Pe than at low.
20
Steady, Uniform Flow
10'
(sinking orsteadyswimming)
~/_/pY'
5
Sh
3
~\\'\....... .'
2 ~u~c~.....·· ._.-
ee{g.~..······· _--'
_ _-- -
Eq. 16 ,,~.":':'::'~'~'~ ~ \i'~.'\'0
0.001 0.01 0.1 1 10 100 1000
Pe
Figure2 Sherwoodnumber(Sh)asafunctionofPeeletnumber(Pe) forcellsmovingatacon
stant velocity in stagnant water or cells fixed in a uniform flow (Re« 1). Equation 16 was
derived by Acrivos& Taylor(1962) for Pe« 1and Equation 17 wasderived for Pe» 1by
Acrivos& Goddard(1965). Cliftetal. (1978)suggestedEquation 18asafit to theirnumerical
results; weuse itfortheregionofintermediatePeforwhichanalyticsolutionsarenotavailable.
Berg & Purcell (1977) obtained their relation numerically, but provided no explicit equation.
Forreasonsdetailed inthetextand inAppendix II, we believeBerg& Purcell's(1977) relation
(theirFig. 4) to be inaccurate.
Asymptoticsolutionsfor Equation 11 inthe caseofsteady, uniform, laminarflow andthe
corresponding boundary conditions (12, 13)yield two relations betweenSh and Pe (Fig. 2).
For Pe« 1(Acrivos & Taylor 1962)
(Equ. 16)
»
For Pe 1(Acrivos & Goddard 1965)
78
NUTRIENT FLUXES TO PLANKTONIC OSMOTROPHS
Sh = 0.624SPeJI +0.461 +0(1) , (Equ. 17)
where O(x) is read as "oforderx", meaning~O.Sx but :$;Sx, the precise coefficient varying
with the details of the situation, and o(x) indicates a term much smaller than the term in
brackets. Equations 16 and 17 were derived for the case of heat transfer from a spherical
particle and were given inthe form ofthe Nusselt number (Nu). Fordetailedconsiderations
of the use ofasymptotic techniques in the analysis ofheat-and-mass transfer problems and
for lucid derivations ofthe above solutions we refer the reader to Leal (1992).
Pecletnumbers for many ifnot mostphytoplanktonfall outsidethe regionfor which ana
lyticsolutions are available (Table 1). Numerical solutionsthat include the region ofour in
terest were published by Clift et a1. (1978) and Masliyah & Epstein (1972). The former
suggestedthe following relationship for Sh, valid for all Pe in creeping(Re<0.1), uniform
flow:
(Equ. 18)
So long as very low Re is maintained, the above formula agrees within 0.7% in the region
O.001:$;Pe:$;O.1 (Equation 16) and within 2% of the analytic solution (Equation 17) in the
region 100:$;Pe:$;SOOO.
Applicationofthe theory to swimming andsinking cells
Mostclassesofmarinephytoplanktonaremotile(Sournia1982).Themostthoroughlystudied
motile phytoplankters are the dinoflagellates; their swimming speeds range from SO to
500llms-1(Throndsen1973,Sournia1982).Anon-motilephytoplanktercansinkorrisedueto
densitydifferencesbetweenitselfandthesurroundingfluidandthus alsoexperienceavelocity
relative to its environment. For cells of a given size, measured sinking speeds are usually
slowerthanswimmingspeedsandrangefrom0to34711ms-1 (Smayda 1970,hisTableVII).
Toillustratetheeffectofswimmingorsinkingontheconcentrationdistributionofagiven
nutrientwesolvedEquation11 withthecorrespondingboundaryconditions (12, 13)numeri
=
callyforPe 1(Fig. 1;AppendixII). ForbothhighandlowPe, adisproportionateamountof
thetotalfluxtakesthepathofleastresistance. AscanbeseenbyinspectionofEquation10,nu
trient transport will be dominated by the thinnest regions ofthe diffusional boundary layer,
where diffusion can proceed by far the fastest; characteristic diffusion time varies with the
squareofthisthickness(Equation3). Hence, totalfluxarrivingto thecellwillbelargercom
paredwiththepurelydiffusionalfluxdespitethethickeningofthediffusionalboundarylayerat
the"rear"ofthecell.Again,someswimmingmodesmayhelpto"shed"thisthickenedbound
ary layerorto thinthe thinnestregions evenfurther comparedwithsteady, uniform, Stokes'
flow pastasphere. Interestinglyenough, ithasbeenreportedthatchemoreceptorsonthecell
surfaceofthemotilebacteriumEscherichiacoliareclusteredatthepoleofthecellratherthan
being evenly or randomly distributed (Maddock & Shapiro 1993). Such clustering may in
creasetheefficiencyofdetection, especiallyifthereceptor-richpatchislocatedat the "lead
ing" poleofthecell(Parkinson& Blair1993)orwhereaflagellum locallythinstheboundary
layer.Theadvantageisgreatlydecreaseddetectiontime. Berg&Turner(1995)observedthat
tumbles tendto randomize cell orientationofE. coli andthatruns ofeach orientation are of
similarduration. They thereforeconcludedthatpatchesofchemoreceptors atonepole ofthe
79
L. KARP-BOSS, E. BOSS & P. A. JUMARS
celldonotserveasthebacteriu.n's"nose".Clusteringofchemoreceptorsoruptakesitesatthe
polesoratothersitesoflocalboundary-layerthinning,however,maystillbeaneffectivestrat
egyforotherorganismswithdifferentmodesofswimming.
In order to quantify the enhancement offlux due to swimming and sinking, for the full
range of Pe values and cell sizes, we use the solutions for mass transfer in steady, uniform
flow - namely Equations 16-18. The only differences inour proposed solutions for sinking
versus swimming (Fig. 3) arise from differences in the'dependenceofsettling versus swim
mingspeedsoncell sizeandhence the dependenceofPeandShoncell size. Webase swim
ming speeds on the relationship,
(Equ. 19)
Swimming or Sinking
in Stagnant Water
Sh10
1 10 100 1000
Cell radius"o (l.l.m)
Figure 3 Sherwood number (Sh) as a function ofcell radius ('0) for swimming and sinking
cells.Bothswimmingandsinkingvelocitiesarefunctionsofcellsize;Stokes'equationwasused
a,J
tocalculate sinking velocitiesand the relation U= was used tocalculateswimming veloci
ties. Wecautionthatsomecorrelationsbetweencellsizeandswimmingvelocityareveryweak;
we use them onlyfor illustration. Dashed lines representsinking cellsand solid lines represent
swimming cells where: a, sinking cells ~p=0.01; b, swimming cells U=0.16(2'0)0.46
(Kamykowski et al. 1992, R2= 0.32, based on 5 species, ranging in size ('0) from c. 13 to
= ,=
30/lm);c,swimmingcellsU 93(2'0)°·26(Sommer1988, 0.38, basedon19species,ranging
in size ('0) from c. 1 to 40/lm); d, sinking cells ~p=O.I; and, e, swimming cells
U=30.66(2,o>I.I6(Kamykowskietal. 1992,R2=0.79, basedon9species, ranging insize('0)
from c. 8 to 25/lm). To calculate Sh we used equation 16 for Pe<0.01, Equation 18 for
0.01 <Pe<100andEquation 17for Pe> 100. Thediffusioncoefficientforthecalculationsof
Pe was taken to be 1x1O-5cm-2s-l. Thegray segments ofcurvesa, d, and emark regions for
whichswimmingandsinkingvelocitiesextrapolatedfromthegivenrelationsbecomeunrealistic.
80