Effective drift velocities and effective diffusivities of swimming microorganisms in external flows

J. Math. Biol. (1999) 39: 172}192

E4ective drift velocities and e4ective di4usivities ofswimming microorganisms in external 6ows

A. N. Yannacopoulos1, G. Rowlands2

1School of Mathematics and Statistics, University of Birmingham, Edgbaston,Birmingham B15 2TT, UK. e-mail: [email protected] Department, University of Warwick, Coventry CV4 7AL, UK

Received: 4 January 1999

Abstract. In this note we calculate the e!ective drift velocities and thee!ective di!usivities of swimming microorganisms in an external #ow"eld. It is shown that if the ambient velocity "eld is incompressible thenthe e!ects of reorientation of the cells can under certain circumstancesenhance the e!ective drift velocity along the preferred direction.

Key words: Gyrotaxis } E!ective transport } homogenization

1. Introduction

In this note we study the motion of swimming microorganisms in anexternal #ow "eld. The microorganisms are considered to be gyrotacticbacteria or algae. Typically, these microorganisms are bottom heavyand in the absence of external #ow "elds they will tend to swimupwards along a preferred direction denoted by the unit vector k.These organisms are considered to be spheroidally shaped cells, and asa result of their shape a viscous torque is exerted on them if they movein a shear #ow. The combined e!ect of these two mechanisms willresult to a reorientation of the cells as they move through the #uid, insuch a way that their swimming direction is no longer along k. Thisphenomenon is called gyrotaxis. The cells are also considered to havetheir own swimming capacity, along the direction given by the gyro-tactic mechanism, as a result of beating #agellae.

Gyrotaxis has been studied extensively by Pedley, Kessler, Hill andcoworkers in a number of papers [8}10]. These authors have construc-ted models on how the cells reorientate, based on results of Je!ery on

the viscous torque exerted on an ellipsoidal body by a shear #ow,which seem to capture in a very satisfactory manner the basic phenom-enology of gyrotaxis. In particular they show that as long as the#uid #ow in the vicinity of the cell remains essentially laminar(that is describable by the theory of Je!ery) then the spatial andtemporal distribution of cells in the #uid can be described by a localadvection}di!usion equation. Both the advection and di!usion coe$-cients are in general spatially varying, and expressions for these quant-ities are given in terms of the local #uid velocity in [9]. Importantly it isto be noted that though the imposed external velocity "eld is incom-pressible, the velocity "eld of the cells is compressible.

In many situations of biological interest, for example in the oceanor in great lakes, the #uid velocity is a periodic function of space and/ortime. When one is interested in the behaviour of cell transport overspatial/time scales which involve a number of the basic periods, it isnatural to try to describe the system in terms of macroscopic equationswhich in some way describe the average behaviour over a number ofperiods. The aim of the present paper is to show that this averagedbehaviour can still be described by an advection}di!usion equationwith constant transport coe$cients. Furthermore these coe$cients arerelated to suitably averaged quantities involving the spatially varyingtransport coe$cients appearing in the work of Pedley and Kessler [9].These constant transport coe$cients will be called e+ective di+usivities.In order to obtain these large scale results we shall use a multiple scalesperturbation theory approach. This approach has been used heavily inhomogenisation theory and o!ers an extremely clear and relativelystraightforward way to obtain results. Since, as pointed out above, thevelocity "eld of the cells is compressible the homogenisation methodused must be applicable to such #ows. The homogenisation method weuse is an extension of a homogenization technique, recently publishedby Vergassola and Avellaneda [12]. There are of course other methodsof homogenisation [2] but as far as we are aware the only workinvolving compressibility e!ects is that of Vergassola and Avellaneda.

We must emphasize that we study the motion of cells in an external#uid #ow and neglect any bioconvection e!ects that is convectionpatterns created in the #uid by the motion of the cells, as an e!ect of thedensity di!erence of the cells and the #uid. That is we assume thatthe e!ect of the cells on the #uid #ow can be neglected. The rationalebehind this assumption is that the forces on the #uid, driving theexternal #ow (e.g. temperature gradients, wind induced stress etc)are stronger than the body force imposed on the #uid as a result of thedensity di!erence of the cells and the #uid. This is a reasonableassumption for large scale #ows. Finally we wish to emphasize that

Swimming microorganisms in external #ows 173

the results here can be generalised in other cases where there is motionalong a preferred direction in the absence of external #ow, possiblyoriginating from chemotaxis or phototaxis, and reorientation becauseof the e!ects of viscous torque. In this paper we will always refer to thispreferred direction as the vertical direction.

The paper is structured as follows. In the next section we givea brief account of the model of Pedley and Kessler giving the &local'advection di!usion equation the concentration of the cells satisfy. InSect. 3 we present the results of the homogenization theory and give theresults for the e!ective drift velocities and the e!ective di!usivities in anexternal #ow. In general these quantities can only be calculated numer-ically but in Sect. 4 we present some results in the small Peclet limit(small velocities, large di!usion) which can be obtained analytically. Inthat section we give analytical expressions for the e!ective drift vel-ocities and the e!ective di!usivities for swimming cells in a generalambient incompressible #ow, and their dependence on the character-istics of the cell. These results are valid in the small Peclet number limit,and show the e!ect of gyrotaxis on the drift velocity in the verticaldirection. This e!ect is shown to depend on the geometrical character-istics of the cells and the details of the #ow. The results show theenhancement of e!ective vertical drift velocity of elongated cells whichswim in #ows with su$ciently rich spatial structure. This can beunderstood intuitively from the consideration that the elongated cellswill use the strain of the #ow to move towards regions with highvelocities along the vertical direction. However, cells close to sphericalwill experience a reduced average vertical drift, compared to their driftin the case where they were swimming upwards with a constant speedand direction. Cells close to spherical will not interact strongly with thestrain of the #ow. Furthermore as a result of gyrotaxis, the cells acquiree!ective drift velocities in other directions apart from the vertical. Thisleads to a reduced e!ective drift in the vertical direction compared tothe one experienced with purely upswimming cells. These changes inthe e!ective drift velocity along a preferred direction could haveimportant consequences on the feeding habits of microorganismswhich rely on spatial transport in a particular direction to move toregions of high nutrient concentration or light intensities [4]. Thesee!ects could furthermore a!ect the behaviour of species at a higherlevel of the food chain [11]. In the same section we give small Pecletnumber analytical expressions on the e!ective di!usivities.

In Sect. 5 we generalise the methods proposed, for the case wherethe #ow is bounded in the vertical direction. Such cases are of interestin marine population dynamics, in cases such as the advection of algaein shallow water or in Langmuir circulations. In Sect. 6 we give some

174 A. N. Yannacopoulos, G. Rowlands

concrete results for two examples and in Sect. 7 we conclude and givesuggestions for further study.

We feel the results given in this note, will enhance the understand-ing of mean motion of swimmers in external #ow "elds. They have beenused to develop some intuition on the basic movement of cells in #uid#ows, something that is not easy to obtain from purely numericalsimulations. The class of #uid #ows considered in this paper naturallyappear in nature. One can consider as examples internal waves formedby density strati"cation in the ocean or Langmuir circulations formedby wind #ow over lakes or the ocean. Both of these types of #ows areknown to have important consequences on plankton ecology [11, 1, 4].The macroscopic transport equations derived in this paper could formthe basis of more detailed models which involve inter-species interac-tion. The requirement is simply that the biological interaction timescale is slower than the time scale of the advection. This point isdiscussed more fully in Sect. 3.

2. The local di4usion+advection equation

This section draws heavily on the work of Pedley and Kessler [9]. Herewe just present their basic results for convenience so as to make thepresent paper self contained.

The cells are considered to be gyrotactic and their swimmingdirection p (a unit vector) satis"es the following di!erential equation

p5 "B~1[k!(k . p)p]#X]p#2a0p .E . (I!pp)

where B (called the gyrotactic parameter) is the time scale for reorienta-tion of a cell by the gravitational torque against the resistive viscoustorque (given in terms of the characteristics of the cells and theviscosity of the #uid), X is the vorticity of the external #ow "eld, k is theprefered swimming direction of the cells in still #uid, hereafter to betaken as z, E is the rate of strain tensor of the external #ow and a

0is

a geometric characteristic of the cells, showing how elongated the cellis. The value a

0"0 corresponds to spherical cells while the limiting

value a0"1 corresponds to rods. Dyadic notation has been used.

The cells are considered to swim randomly in all directions in theabsence of all torques, viscous or gravitational. Their swimming direc-tion is given by a probability distribution f (p) which is isotropic. In thepresence of torques (gyrotaxis) there will be a bias in this distributionwhich must satisfy the Fokker}Planck equation in the form

LfLt#$ ' (p5 f )"D

u$2f


where $ is the gradient operator in p-space and Du

corresponds toa di!usion coe$cient in direction space. Pedley and Kessler solved thisequation for the probability distribution f using perturbation methodsin the case where f has negligible or slow time dependence and whenthe ambient #ow can be considered weak. This condition is quanti"edby the introduction of a dimensionless quantity e"Bu where u isa scale for the amplitude of the vorticity of the ambient #ow.

Using this perturbative solution they show that the concentrationof the cells n(r, r) satis"es the advection-di!usion equation

LnLt

#$ ' ((u#<sSpT)n)"$ ' (D$n)

whereSp

1T"eJ

1u

2!2ea

0J4e13

Sp2T"!eJ

1u

1!2ea

0J4e23

Sp3T"K

1!3ea

0K

4e33

and the di!usion tensor D"(Dij) is given by

D11

"

K1

j#e

32

a0K

5e33!e

12

a0J6(e

11!e

22)

D12

"!ea0J6e12"D

21

D13

"e(J2!K

1J1)u

2!2ea

0(J

5!K

1J4)e

13"D

31

D22

"

K1

j#e

32

a0K

5e33#e

12

a0J6(e

11!e

22)

D23

"!e(J2!K

1J1)u

1!2ea

0(J

5!K

1J4)e

23"D

32

D33

"K2!3ea

0(K

5!2K

1K

4)e

33.

In the equations above, j is a dimensionless parameter giving therelative strength of gyrotaxis and randomness j"(BD

u)~1, e

ijare the

components of the rate of strain tensor of the ambient #oweij"1/2(Lu

i/Lx

j#Lu

j/Lx

i), <

sis the amplitude of the swimming velo-

city of the cells, and Ki, J

i, i"1, 2 , 6 are numbers depending on

j and can take both positive and negative values.We now assume that the external velocity "eld is incompressible

and periodic. We write it as a complex Fourier series in the form

uj"+

k

A(j)k

exp(ik .r), j"1, 2, 3

where k"(k1, k

2, k

3) is an integer vector, k . r"k

1x#k

2y#k

3z.

Without loss of generality we take the velocity "eld to have periodicity


2n (which can be changed by a proper rescaling of r). The incompressi-bility property is ensured by the additional property of the Fourieramplitudes

k1A(1)

k1#k

2A(2)

k2#k

3A(3)

k3"0

and the fact that the velocities are real functions of space is ensured bythe property

(A(j)k

)*"A(j)~k

, j"1, 2, 3

where the star denotes the complex conjugate. We further assumewithout loss of generality that the velocity "eld has vanishing averagecomponents. For a velocity "eld of this form, according to the Pedleyand Kessler model we have that the Fourier components of the velocity"eld the cells will experience are given by

AM (1)k"A(1)

k#i<

se ((J

1!a

0J4)k

3A(1)

k!(J

1#a

0J4)k

1A(3)

k)

AM (2)k"A(2)

k#i<

se ((J

1!a

0J4)k

3A(2)

k!(J

1#a

0J4)k

2A(3)

k)

AM (3)k"A(3)

k#<

sK

1dk,0

!3iea0<sK

4k3A(3)

k

where we notice that the component of the velocity "eld in the verticaldirection will have an average value of<

sK

1. Similarly we can calculate

the Fourier amplitudes of the components of the di!usion tensor.These will be

DM (11)k

"

K1

jdk,0

!ieAJ6a0

2k1A(1)

k!

J6a0

2k2A(2)

k!

3K5a0

2k3A(3)

k BDM (12)

k"!ie

a0J6

2(k

1A(2)

k#k

2A(1)

k)

DM (13)k

"ie((J2!K

1J1!a

0J5#a

0K

1J4)k

3A(1)

k

!(J2!K

1J1#a

0J5!a

0K

1J4)k

1A(3)

k)

DM (22)k

"

K1

jdk,0

#ieAa0J6

2k1A(1)

k!

a0J6

2k2A(2)

k#

3a0K

52

k3A(3)

k BDM (23)

k"!ie((J

2!K

1J1#a

0J5!K

1J4a0)k

2A(3)

k

!(J2!K

1J1!a

0J5#a

0K

1J4)k

3A(2)

k)

DM (33)k

"K2dk,0

!3iea0(K

5!2K

1K

4)k

3A(3)

k.

The di!usion tensor is a symmetric tensor. We see that only thediagonal components of the di!usion tensor have non-zero averagevalues.

Having presented the local di!usion advection the cells will satisfyin a weak ambient velocity "eld we are ready to present some hom-ogenization results.


3. Average transport coe7cients

In this section we present a method of "nding the average transportcoe$cients for the cells, which will be valid for large enough temporaland spatial scales. The method is an extension of some recent results ofVergassola and Avelaneda [12] who studied scalar transport in com-pressible #ows, but did not take into account the e!ects of a spatiallyvarying di!usion tensor. Since the inclusion of a spatially varyingdi!usion tensor modi"es the average drift velocity we give a briefoutline of this calculation.

3.1. Average drift velocities

We de"ne fast and slow spatial and temporal scales. The fast arex1, x

2, x

3, t and the slow are X

1"e

1x1, X

2"e

1x2, X

3"e

1x3

and¹"e

1t. This scaling is chosen because we want to study advective

e!ects. Here e1

is essentially the ratio of the wavelength (spatialperiodicity) of the velocity "eld to a length which characterises thespatial variable of the macroscopic behaviour. Expanding the timederivatives as L/Lt"L/Lt#e

1L/L¹ and the spatial derivatives likewise

and assuming that n can be expanded as n0#e

1n1#) ) ) (where it is

a function of both slow and fast variables which are considered inde-pendent) we can expand the advection di!usion equation in powersof e

1. Matching these powers we get to leading order, and to "rst order

respectively

¸n0"0 (1)

¸n1"!

Ln0

L¹!

LLX

j

(uNjn0)#

LLX

jADjk

Ln0

LxkB#

LLx

jADjk

Ln0

LXkB (2)

where ¸ is the parabolic operator

¸ f"LLt

f#L

Lxj

(uNj

f )!L

LxjADjk

LLx

k

fBwhere repeated indices denote summation over them, and uN

iis the i-th

component of velocity "eld of the cells, uNi"u

i#<

sSp

iT. The coe$-

cients of this operator are periodic functions of the fast variables (notnecessarily of time).

It is important to note that the velocity "eld of the cells is no longerincompressible. This implies that the solution of the zeroth orderequation (1) will not be a constant, with respect to the fast variables, as


would be the case for an incompressible velocity "eld but will havesome dependence on them, re#ecting the possibility of high and lowconcentration regions of cells as a result of e!ective compressibility. Inanalogy to the case where the di!usion coe$cients are constant (see[12]) we write the solution of the zeroth order equation in the formn0"N

0(X

1, X

2, X

3, ¹ )m (x

1, x

2, x

3, t) where it is readily seen that

m will have to satisfy the auxiliary equation

¸m"0 (3)

with periodic boundary conditions. The solution of this equation willbe normalised in such as way as to satisfy the condition

SmT"1

where S.T denotes averaging over the fast variables.The next order equation (2) has exactly the same form as the zeroth

order one but is non-homogeneous. Because of the periodic boundaryconditions the solvability condition for this equation will be that theaverage of the right hand side over the fast spatial variables and overtime will have to vanish. Substituting the form of the solution forn0

into (2) we obtain

¸n1"!m

LNL¹

!uNjm

LN0

LXj

#

LLX

jADjk

LmLx

k

N0B#

LLx

jADjk

mLN

0LX

kB .

Performing the averagings over the fast variables and insisting onperiodicity of n

1on the fast scales, using the divergence form of the

operator ¸ we "nd that the left hand side of this equation vanishes.The solvability condition which simply states that the right hand sideshould vanish as well is equivalent to

LN0

L¹#<

d, j

LN0

LXj

"0

where

<d,j

"SuNjmT!TD

jk

LmLx

kU (4)

and S.T denotes averaging over the fast spatial variables. In obtainingthis result we used the normalisation condition that SmT"1 and thedivergence form (in the fast variables of the last term of the right handside) which makes the last term have a vanishing contribution in thefast scale average.

The solvability condition gives that the evolution of N0

(whichis the average of n

0over the fast scales) is advective with mean


drift velocity given by (4). In the case where the di!usion coe$cientdoes not depend on space the second term vanishes, because of theperiodicity of m, and this result coincides with the result of Vergassolaand Avelaneda.

This equation can be applied to our problem to de"ne e!ective driftvelocities. In most cases the solution of the auxiliary problem isdi$cult and must be obtained numerically. However in the nextsection we will present an analytic result which will be valid in the limitof large di!usion.

3.2. Ewective diwusivities

To obtain large scale di!usive e!ects we need to consider a di!erentscaling and introduce an extra slow time scale of the form ¹

2"e2

1t

while no extra spatial scale is needed. This scaling re#ects thevery nature of di!usion processes, where in general the mean squaredisplacement is a linear function of time. Expanding the di!usionadvection equation in these new spatial and temporal scales we get tozeroth and "rst order the equations (1), (2) and to second order theequation

¸n2"!

Ln0

L¹2

!

Ln1

L¹1

!

LLX

j

(uNjn1)#

LLX

jADjk

Ln1

LxkB

#

LLx

jADjk

Ln1

LXkB#

LLX

jADjk

Ln0

LXkB . (5)

In complete analogy to the analysis given in the previous section, sincewe insist on periodicity of n

2on the fast variables and because of the

divergence form of the operator ¸, the average of the left hand side ofequation (5) over the fast variables will vanish, thus leading to a solva-bility condition: that the average of the right hand side over these scalesshould vanish as well. It should be apparent by now that this solvabil-ity condition can lead to an e!ective transport equation for n valid forthe slow time and spatial scales. Since the right hand side involves n

1,

which is the solution of the eqution obtained by expanding to "rstorder, we need to "nd the general solution of equation (2), in order toget an explicit form of the solvability condition.

Using the advection equation that N0

satis"es, we can see that theright hand side of (2) consists of products of functions of the fastvariables with the gradients of N

0with respect to the slow variables X

j.

Using this observation we can write the general solution for n1

inthe form n

1"s

jLN

0/LX

j#mN

1where s is a vector function depending


*****

1The "rst part is a particular solution of the nonhomogeneous equation while thesecond part is the general solution of the homogeneous.2Of course in the case of periodic time dependence the solutions of the auxiliaryequations will have periodic time dependence as well.

only on the fast variables and N1

depends on the slow variables.1 It iseasily seen that s will be the zero average periodic solution of

¸sj"(<

dj!uN

j)m#D

jk

LmLx

k

#

LLx

k

(Dkjm). (6)

This equation becomes the auxiliary equation obtained by Vergassolaand Avelaneda if the di!usion tensor is constant. Substituting thisansatz for n

1in the RHS of the second order equation and taking into

account that the solvability condition would simply mean that theaverage of the RHS over the fast scales would be zero, we "nd that thesolvability condition is equivalent to

LN0

L¹2

"DEjk

LLX

j

LLX

k

N0,

so that the evolution of N0

at this time scale is di!usive with e!ectivedi!usion coe$cient given by

DEij"SD

ijmT!

12

SuNisj#uN

jsiT#

12 TD

ik

Lsj

Lxk

#Djk

Lsi

LxkU (7)

In the case of constant di!usion tensor the last two terms vanish andthis result agrees with that of Vergassola and Avelaneda. We see thatthe variation of the di!usion tensor with space has an e!ect on theaverage di!usivities. The calculation is similar in spirit as the one forthe drift coe$cient given in Sect. 3.1 so it is not repeated here in detail.

Similarly to the auxiliary equation for m, the auxiliary equation fors is complicated and in most cases it has to be obtained numerically.However here we will solve this auxiliary equation analytically in thelimit of small Peclet number (large di!usion).

It is worth noting that the results of the e!ective drift and e!ectivedi!usivities also hold in the case where the velocity "eld is periodic intime as well as in space.2

3.3. Population dynamics

To model the e!ect of species interaction we can assume an advection-di!usion equation of the form discussed in Sect. 2 for each species


*****

3We are assuming that the oscillating part of the di!usion tensor is of order d. This isconsistent since the assumption here is that the velocity "eld is weak compared todi!usion and the oscillating part of the di!usion tensor is proportional to the gradientsof the velocity "eld.

whilst their interaction is taken into account by the introduction ofinteraction terms such as bn

1n2

where b is a constant and n1

andn2

refer to the density of each species. For the case where the time rateof change of this interaction is comparable to the di!usion time, theseinteraction terms are of order e2

1. The analysis processes as in Sects. 3.1

and 3.2 but now each di!usion equation is augmented by a nonlinearinteraction term which takes the form bSm2TN

0,1N

0,2. A general non-

linear interaction term f (n1, n

2) is simply replaced by F (N

0,1, N

0,2)"

:dxf (N0,1

m, N0,2

m).

4. E4ective drift velocities and e4ective di4usivities in thesmall Peclet limit

In this section we calculate the e!ective drift velocities and the e!fectivedi!usivities for swimming cells whose local velocities and di!usiontensor are given by the model of Pedley and Kessler. In an attempt toget analytical and general results we limit ourselves to the largedi!usion limit, the opposite limit being highly dependent on the detailsof the velocity "eld and the individual cell trajectories.

4.1. Ewective drift of cells

In order to obtain the e!ective drift velocity of the cells we need toobtain a solution of the auxiliary equation (3). This, as mentionedpreviously, can only be done in general numerically. However, in thelarge di!usion limit we can obtain an analytic result for m. Taking D tobe the largest of the diagonal terms of the di!usion tensor, we rescaletime in the auxiliary equation using q"Dt. De"ning d"1/D to bea small parameter and expanding m"m

0#dm

1we can see that

Hm0"0,

Hm1"!

LLx

j

(m0uNj)#

LLx

iAD@

ik

LLx

k

m0B ,

where the dash in the di!usion tensor denotes the oscillating part,3and H"L/Lq!L2/Lx2

1!L2/Lx2

2!pL2/Lx2

3is the usual heat operator


(see [12]), where p is the ratio of the di!usion coe$cient in the verticalto the di!usion coe$cient in the horizontal direction. These equationshave to be solved using periodic boundary conditions. The solution ofthe "rst equation can be seen to approximate the constant functionm

0"1 for large enough times (we will be interested here in long time

asymptotics, however, the intermediate time results can be obtainedeasily) thus simplifying the RHS of the second equation to simply thedivergence of the velocity "eld of the cells. We thus see that to thisorder the spatial variation of the di!usion coe$cient will not contrib-ute to the drift velocity. This contribution will only become importantwhen we consider second order corrections in d.

From the model of Pedley and Kessler we can easily calculate thatthe divergence of the velocity "eld of the cells is given by

$ ' u6 "L

Lxj

(uNj)"+

k

Ckexp(ik .r)

Ck"e<

s((J

1#a

0J4)k2

1#(J

1#a

0J4)k2

2

#(J1!a

0J4#3a

0K

4)k2

3)A(3)

k,

where we have used the incompressibility of the ambient #ow. We cannow solve for m

1in terms of a Fourier series. If

m1"+

k

Mkexp(ik .r)

we can "nd that the amplitude of the Fourier modes for m1

will be

Mk"!e<

sF(k)G(k)A(3)

k,

F(k)"(k21#k2

2#pk2

3)~1,

G(k)"(J1#a

0J4)k2

1#(J

1#a

0J4)k2

2#(J

1!a

0J4#3a

0K

4)k2

3.

Using the expansion for m, we "nd that an expansion of the driftvelocities in d gives <

dj"SuN

jT#dSm

1uNjT. Since the Pedley and

Kessler model is valid only to "rst order in e we are interested in thee!ective drift velocity to this order. Using the Fourier expansion form

1we "nd that

m1uNj"!e<

s+k,k{

F(k)G(k)A(3)k

A( j)k{

ei(k`k{) >r#O(e2).

From this expression, because of the periodicity in the fast variables itis clear that

Sm1uNjT"!e<

s+

k`k{/0

F(k)G(k)A(3)k

A(j)k{


so that we "nally obtain the "nal result

<dj"<

sK

1dj,3

!ed<s+k

F(k)G(k)A(3)k

A(j)~k

It is interesting to note that the vertical drift velocity becomes

<d3"<

sK

1!ed<

s+k

F(k)G (k) DA(3)k

D2

where G (k) contains the geometrical characteristics of the cells. Forvalues of the geometrical characteristics and j, where the quadraticform G(k) is positive this result assures that gyrotactic cells, will tend toswim in the vertical direction with an average drift which is less thanthe average drift purely upswimming cells would experience, in anyperiodic three dimensional ambient #ow. In addition, the cells due togyrotaxis acquire a mean drift velocity in the other directions, some-thing that would not be true for purely upswimming cells in a zeroaverage ambient #ow. Looking at the table in Pedley and Kessler [9]we see that the term (J

1#a

0J4) will always be positive. We just have to

look at the sign of the term I"J1!a

0J4#3a

0K

4. Below we give an

expression for this term for a number of di!erent values for j:

j I ac

0.3 0.015}34.494a0

0.000431.0 0.14}3.986a

00.035

2.2 0.45}1.45a0

0.313.0 0.6}0.97a

00.618

From these expressions we see that for very small j this expression willbe negative for a

0su$ciently large thus giving the possibility of

enhancement of the average drift velocity in the vertical direction if theambient #ow has high enough modes in x

3. As j increases we "nd that

the term I can be negative for values of a0'a

cand a

cincreases as

j increases. The variation of the critical value acwith j is shown in Fig. 1.

The region A is the region in a0!j parameter space where irrespective

of the #ow, gyrotaxis will tend to decrease the e!ective drift of the cells inthe vertical direction. Region B corresponds to the parameter region,where an increase of<

d3is possible because of gyrotaxis for certain #ows.

To sum up, our results show possible enhancement of the verticaldrift velocity of su$ciently elongated cells, in external #ows whichcontain modes of high enough k

3, where as in general cells close to

spherical will experience reduced drifts as an e!ect of gyrotaxis. Itshould be remembered that K

1also depends on j.


Fig. 1. The critical value of a0, a

cas a function of j. Region A corresponds to

parameter values where gyrotaxis leads to decrease in the e!ective vertical velocity,irrespective of the #ow and region B corresponds to parameter values where for certain#ows gyrotaxis may lead to enhancement of the e!ective vertical velocity

4.2. Ewective diwusivities for swimmers in the small Peclet limit

In order to calculate the e!ective di!usivites we need to solve theauxiliary equation (6). This again in principle can be solved onlynumerically but here in an attempt to give asymptotics in the smallPeclet number limit we shall obtain a perturbative solution in the limitof large di!usion (small Peclet number).

Rescaling time in the same manner as in the previous section andexpanding s

j"s(0)

j#ds(1)

jwe can get the following equations for the

di!erent orders:

Hs(0)j"0

Hs(1)j"!

LLx

k

(uNks(0)j

)#(<(0)dj

!uNj)m

0#D@

jk

Lm0

xk

#

Lxk

(D@kj

m0)#

LLx

kAD@

ik

Ls0j

LxiB#2h( j )

Lm1

Lxj


where in the last term h (1)"h(2)"1, h (3)"p and no summation overj is implied there. Taking into account that m

0"1 and the only

periodic solution to the "rst equation with zero average is s(0)j"0 the

second equation simpli"es to

Hs(1)j"(SuN

jT!uN

j)#

LD@kj

xk

#2h( j)Lm

1Lx

j

.

We now solve the second of these equations and get

sj"+

k

X( j)k

exp(ik .r)

where

X(1)k"!F(k)A(1)

k!ie<

sF(k) ((J

1!a

0J4) k

3A(1)

k!(J

1#a

0J4)k

1A(3)

k)

!2ie<sF2(k)G(k)k

1A(3)

k#ieF (k)k

nDM @(n1)

k

X(2)k"!F(k)A(2)

k!ie<

sF(k) ((J

1!a

0J4)k

3A(2)

k!(J

1#a

0J4)k

2A(3)

k)

!2ie<sF2(k)G(k)k

2A(3)

k#ieF (k)k

nDM @(n2)

k

X(3)k

"!F(k)A(3)k#3ieK

4a0F(k) k

3A(3)

k!2iep<

sF2(k)G (k)k

3A(3)

k

#ieF(k)knDM @(n3)

k.

We now substitute this expansion in the equation (7) de"ning thee!ective di!usivities. By separating the constant and the oscillatingpart of the di!usion tensor, and after some algebra we "nd that

DEij"SD

ijT#dSD

ijm

1T!

d2 TAuN i#

LD{ik

LxkBs(1)

j#SU

where by S we denote the symmetric term in i, j. From this stageonwards the calculation of the e!ective di!usivities is straightforwardbut tedious. A sample calculation for DE

ijis given in the appendix. Here

we simply state the result for the diagonal components (keeping onlyterms to "rst order in e since the Pedley-Kessler model is only "rstorder in this parameter)

DEjj"SD

jjT#dF(k) DA( j)

kD2!2ied<

sF(k) (J

1#a

0J4)k

jA(3)

kA(j)

~k

#2ied<sF2(k)G(k) k

jA(3)

kA(j)

~k, j"1, 2

DE33"SD

33T#dF(k) DA(3)

kD2

where summation is taken over all the components of the integervector k. We notice that the second term is always positive andcorresponds to the enhancement of e!ective di!usivity by the incom-pressible ambient #ow which is a well known phenomenon. The next


term can either be positive or negative depending on the geometriccharacteristics of the cells and is in accordance with the generalobservation of Vergassola and Avelaneda that compressible #ow maylead to depletion of e!ective di!usivities. It is interesting to note thatthere is no correction to "rst order in e to D

33.

5. Generalisations

The above results can be generalised to other cases which are possiblyof more interest in applications. Consider the case where the #ow isperiodic in two of the three directions and we only consider a boundeddomain in the third. This would be the case where we have a layer of#uid of "nite depth, which is in"nite in the other two directions eg thesituation of Langmuir cells. This problem has similarities with theproblem of Taylor dispersion which has received much attention andwas studied using a variety of methods e.g. [6, 7, 13]. The novelty of thepresent calculation is in the incorporation of compressibility e!ects inthe velocity "eld.

In this case we have the same local di!usion advection equation forthe cells with periodic boundary conditions in the two directions (x

1and x

2) and no #ux boundary conditions in the third

D3k

LnLx

k

!u3n"0, x

3"0, ¸

In such a situation we need to study average drift velocities anddi!usivities in the periodic direction only (these are the only directionswhere large spatial scales have a meaning). Similarly to the previouscase we de"ne large spatial scales X

1"e

1x1, X

2"e

1x2

and largetemporal scales ¹

1"e

1t and we leave x

3as a local scale only. Expand-

ing in e1

we "nd to the "rst two orders

¸n0"0

¸n1"!

Ln0

L¹!

LLX

l

(uln0)#

LLX

lADlj

Ln0

LxjB#

LLx

jADjl

Ln0

LXlB

where l"1, 2 and j"1, 2, 3 and ¸ is the local advection di!usionoperator but now with no-#ux boundary conditions in x

3. We

take n0"N

0(X

1, X

2, ¹) m (x

1, x

2, x

3, t) and we notice that m solves

the same auxiliary equation as before but with modi"ed boundaryconditions.

The compatibility condition is easily checked to be the same asbefore, that is the average of the right hand side of the "rst order


equation should vanish. This gives an advective equation for N0

andthe e!ective drift is given by

<dl"SuN

lmT!TD

lj

LmLx

jU!SD

3lm DL

0Tx1,x2

The last term is a surface term which is due to the change of boundaryconditions.

The small Peclet number result can still be obtained. Doing a largeD expansion m"m

0#dm

1we see that m

0solves the equation

Hm0"0

Lm0

Lx3

"0, x3"0, ¸

which has the long time solution m0"1. The next order correction

solves

Hm1"!$ ' u

pLm

1Lx

3

"u3, x

3"0, ¸

(where we have expanded the boundary conditions in d"1/D). Thelast equation has non-homogeneous boundary conditions. It is conve-nient to transform it to an equation with homogeneous boundaryconditions by de"ning

m1"mN

1#f (x

1, x

2) x2

3#g(x

1, x

2)x

3

f (x1, x

2)"

u3(x

3"¸)!u

3(x

3"0)

2p¸

g (x1, x

2)"

u3(x

3"0)p

This new function solves

HmN1"!$ ' uN #$2

hfx2

3#$2

hx3#2f

LmN1

Lx3

"0, x3"0, ¸

This problem is well de"ned and can easily be solved using the Fouriercosine series. Having outlined the general method of solution here wewill not pursue the problem any further but leave particular applica-tions to the future.


6. Examples

In this section we illustrate our general results obtained in Sect. 4 byusing some simple examples.

As a "rst example we study the e!ective motion of swimmers in theABC #ow. The ABC #ow is an exact solution of Euler equations inwhich passive tracer dynamics have been studied in detail [5]. TheABC #ow is given by

u1"A sin(z)#C cos(y)

u2"B sin(x)#A cos(z)

u3"B cos(x)#C sin(y)

and is a solution of Euler's equation. Applying the results of theprevious sections for the ABC #ow we "nd that the e!ective driftvelocities will be

<d1"0

<d2"0

<d3"<

sK

1!

ed<s

2(J

1#a

0J4) (B2#C2)

and the e!ective di!usivities (diagonal components)

DE11"D

11#

d2

(C2#A2)

DE22"D

22#

d2

(B2#A2)

DE33"D

33#

d2

(B2#C2)

Since the quantity J1#a

0J4

is positive for all values of a0

we see thatin this #ow the e!ective drift velocity in the vertical direction is reducedby the e!ects of gyrotaxis. This is because the #ow does not containhigh enough z modes to allow for gyrotaxis to enhance the e!ectivevertical drift. Furthermore we see that there is no contribution becauseof the e!ects of gyrotaxis on DE

iito "rst order in e. All such contribution

are of order O(e2).As a second example we will consider the motion of swimmers in the

Arter #ow [3]. The Arter #ow is a perturbative expansion of thesolution of the Rayleigh}Taylor convection equations near a criticalRayleigh number. Convective #ows occur very often in nature and the


Arter #ow is one of the simplest models of the #ow in a convection roll.The velocity "eld is of the form

u1"!sin(x) cos(y) cos(z)#Ak

3sin(x) cos(k

3z)

u2"!cos(x) sin(y) cos(z)#Ak

3sin(x) cos(k

3z)

u3"2cos(x) cos(y) sin(z)!Acos(x) sin(k

3z).

We can easily calculate the e!ective drift in the vertical direction to be

<d3"<

sK

1!

ed<s

6(3J

1#a

0J4#3a

0K

4)

!

ed<sA2

4(1#k2

3)~1(J

1#k2

3J1#a

0J4#3a

0K

4k23!k2

3a0J4)

(for p"1). There are no contributions to this order to the velocities inthe other two directions. It can be seen that since K

4(0, even for

A"0 and cells which are elongated enough there is an enhancement ofthe e!ective drift in the vertical direction. For instance if A"0 then fora0'a

cwhere a

cis a function of j there will be an enhancement of the

e!ective drift in the vertical direction as an e!ect of gyrotaxis. Thedependence of a

con j is given in the following table

j 0.1 1.0 2.2 3.0ac

0.001 0.102 0.685 1.005

It is worth noting that for large j in this particular #ow the e!ectivedrift is always reduced as an e!ect of gyrotaxis, unlike what happensfor smaller values of j.

On the other hand, in this particular #ow, gyrotaxis will not a!ectthe e!ective di!usivities, apart from the usual enhancement which isexpected for the passive tracer, in a divergence free #ow.

7. Summary

In this paper we have extended and applied the recently publishedresults of Vergassola and Avelaneda, on transport in compressible#ow, to study the e!ective drifts and di!usivities of swimming microor-ganisms in an external incompressible #ow.

The e!ect of the geometric characteristics of cells on the e!ectivedrift velocities and di!usivities is studied in some detail for general#ows and general expressions are obtained for these transport coef-"ents. It is shown that depending on the circumstances gyrotaxis can


cause enhancement or reduction of the vertical drift velocities. Similarresults hold for the e!ective di!usivities. Simple expressions are ob-tained for the e!ective drift velocities and di!usivities in the smallPeclet limit. In particular a critical value for a

0, the geometrical shape

parameter, namely acis de"ned such that for a

0'a

cthe e!ective drift

velocity can be enhanced by gyrotaxis e!ects, whereas the diagonalcomponent of the di!usion tensor in the vertical direction is nota!ected for all a

0. For cells with a

0(a

cthere will be a decrease of the

vertical e!ective velocity, as an e!ect of gyrotaxis, irrespectively of the#ow.

These results will be of use to the scienti"c community interested inmodelling the motion of such swimming microorganisms, and theirinteractions with other species, as our results can be incorporated inlarge scale models of population dynamics.

Finally we have to stress that the opposite limit of large Pecletnumber, (the limit where advection is dominant over di!usion) is a veryinteresting limit which needs to be adressed, and we plan to do so infuture work. Transport in this limit will depend strongly on theparticular class of #ows chosen. In this limit, processes like chaoticadvection will play a dominant role.

Acknowledgements. We wish to thank Prof. J. R. Blake and Dr. S. R. Otto for a criticalreading of the manuscript and useful suggestions. We also wish to thank the referreesfor useful comments.

Appendix

In this appendix we give details of the calculation of the e!ectivedi!usivity DE

33.

To calculate the e!ective di!usivity DE33

we need to calculate theaverage

TAuN 3#LD@

3kLx

kBs(1)

3 Uthat is we need to "nd the average of

I"+k,k{

¸k{N

kei(k`k{) > r

¸k{"A

k{#<

sK

1dk{,0

!3iea0<sK

4k@3A(3)

k{#ik@

nDM {(3n)

k{

Nk"!F(k)A(3)

k#3iea

0<sK4F (k)k

3A(3)

k!2iep<

sF2(k)G(k)k

3A(3)

k

#ieF (k)knDM {(n3)

k


The averaging procedure will only select those terms with k@#k"0and to "rst order in e the average is

SIT"+k

M!F (k) DA(3)k

D2#2iep<sF(k)2G(k) k

3DA(3)

kD2

#ieF (k)kn(DM {(n3)

kA(3)

~k#DM {(n3)

~kA(3)

k)N

where we have taken into account that A(j)~k

"A( j)*k

. Now since thesummation extends from !R to R and because F(k), G(k), DA

kD and

(DM {(n3)k

A(3)~k

#DM {(n3)~k

A(3)k

) are even functions of k we can see that the lasttwo contributions in the summation will vanish thus leading to

SIT"!+k

F(k) DA(3)k

D2#O (e2)

Using that the result given in the main text follows.

References

[1] M. A. Bees, I. Mezic and J. McGlade, Planktonic interactions and chaoticadvection in Langmuir circulation. Mathematics and Computers in Simulation44 (1998) 527}544

[2] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic analysis forperiodic structures. North Holland 1978

[3] A. A. Chernikov and G. Schmidt, Chaotic streamlines in convective cells. Phys.Lett. A 169 (1992) 51}56

[4] K. L. Denman and A. E. Gargett, Biological-physical interactions in the upperocean: the role of vertical and small scale transport processes. Ann. Rev. FluidMech. 27 (1995) 225}255

[5] T. Dombre, U. Frisch, J. M. Green, M. Henon, A. Mehr and A. Soward, Chaoticstreamlines in the ABC #ows. J. Fluid Mech. 167 (1986) 353}391

[6] M. Pagitsas, A. Nadim and H. Brenner, Projection operator analysis of macro-transport processes. J. Chem. Phys. 84 (1986) 2801}2807

[7] M. Pagitsas, A. Nadim and H. Brenner, Multiple scales analysis of macrotrans-port processes. Physica A 135 (1986) 533}550

[8] T. J. Pedley, N. A. Hill and J. O. Kessler, The growth of bioconvection patterns ina uniform suspension of swimming micro-organisms. J. Fluid Mech. 195 (1988)223}237

[9] T. J. Pedley and J. O. Kessler, A new continuum model for suspensions ofgyrotactic micro-organisms. J. Fluid Mech. 212 (1990) 155}182

[10] T. J. Pedley and J. O. Kessler, Hydrodynamic phenomena in suspensions ofswimming microorganisms. Ann. Rev. Fluid Mech. 24 (1992) 313}358

[11] K. W. Mann and J. R. N. Lazier, Dynamics of marine ecosystems: Biological,physical interactions in the oceans. Blackwell 1991

[12] M. Vergassola and M. Avellaneda, Scalar transport in compressible #ow.Physica D 106 (1997) 148}166

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Effective drift velocities and effective diffusivities of swimming microorganisms in external flows