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Home , Langmuir circulation

IMACS Mathematics and Computers in Simulation

Planktonic interactions

and chaotic advection

in Langmuir circulation

1;2 2 2

M A Bees I Mezic and J McGlade

1

Mathematics Institute

University of Warwick Coventry CV AL UK

2

Ecosystems Analysis and Management Group

Department of Biological Sciences

University of Warwick Coventry CV AL UK

Page heading Planktonic interactions and chaotic advection

Key words mixing resonant retention zones patchiness

Langmuir circulation chaotic advection physical forcing

March

Abstract

The role of unsteady laminar ows for planktonic communities is investigated Langmuir

circulation is used as a typical mediumscale structure to illustrate mechanisms for the

generation of plankton patches Two b ehaviours are evident chaotic regions that help to

spread plankton and lo cally coherent regions that do not mix with the chaotic regions and

which p ersist for long p erio ds of time The interaction of p opulations of phytoplankton with

zo oplankton is discussed taking into account the variations in plankton

Intro duction

Recent studies of plankton dynamics eg Solow Steele Pinelalloul Gallager et al

identify plankton patchiness as a ma jor cause of the discrepancies b etween predictions from bulkaveraged

mo dels and eld measurements In particular lo cal aggregations of planktonic sp ecies have b een observed

that are entrained in larger scale o ceanic and limnetic circulations Gallager et al Wiafe Frid

Owen Haury Wieb e These patches of phytoplankton can p ersist for long p erio ds

of time irresp ective of the apparently strong mixing pro cesses characteristic of turbulent ows

In this pap er a mechanism for the spread and aggregation of advected buoyant plankton is presented

that employs a deterministic nearintegrable ow eld We do not include turbulence or any sto chastic

forcing in the description but its p ossible eects are discussed and will b e included in later publica

tions We cho ose Langmuir circulation in a stably stratied uid a typical mediumscale structure to

demonstrate our approach in detail

Langmuir circulations Langmuir o ccur as a result of a balance b etween a destabilizing wind

induced gradient and a stabilizing temp erature driven vertical density stratication Craik

Leib ovich They typically manifest themselves as long roll structures of width b etween m and

m commonly identied from the debris at regions of surface convergence which are called windrows

Barstow Rather than studying plankton transp ort in Langmuir circulation in the eld we cho ose

to simulate a mathematical ow eld that can b e directly controlled and insert plankton into the ow

in order to investigate their passive b ehaviour We insist that the plankton can do no more than swim

up or down eectively mo delled as buoyancy In this way we are able to study the eect of the ow on

the plankton without getting distracted by other physical or biological factors

A successful physical mo del of Langmuir circulation was derived by Craik Leib ovich This

was subsequently used by Moroz Leib ovich to generate a reduceddimensional system that

adequately describ es key features of the phenomenon The reduced system was applied to investigate

the bifurcation structure of Langmuir circulation employing metho ds from dynamical systems Both

steady and unsteady ow regimes were observed We shall consider the same reduced mo del in our

description only with an additional timedep endent p erturbation The particular p erturbation that we

will use whilst b eing physically realizable is somewhat arbitrary and similar results can b e obtained

with other p erturbations However we wish to emphasize our qualitative arguments and interpretations

with quantitative results

Chaotic advection is a fashionable mechanism for the deterministic transp ort of scalars that is inde

p endent of molecular diusion Jones Young Hydon Camassa Wiggins Unsteady

or quasisteady ows can exhibit nonintegrable dynamics that can induce transp ort of advected scalars

across what would normally b e imp enetrable barriers or equivalently streamsurfaces for steady inte

grable ows Arnold Small timedep endent p erturbations to steady ows can break up these

barriers and pro duce an eective diusivity for purely advected scalars The resulting lob e dynamics

Wiggins for hetero clinic connections may describ e imp ortant mixing pro cesses and diusion in

Langmuir circulation This phenomenon is thought to play a signicant role in causing planktonic mixing

andor patchiness eg Stommel retention zones Stommel This mechanism is likely to work

in addition to other patchforming mechanisms such as reactionswimmingdiusion Davis et al

Bees Bees Hill and reactionadvectiondiusion Spiegel Zaleski Malchow

but tends to work on shorter time scales hours instead of tens of days see Malchow

This study aims to clarify the relationship b etween plankton interactions such as growth predation

swimming and diusive pro cesses and the aggregation and transp ort of ecosystems due to physical

forcing In particular the lowdimensional system of Moroz Leib ovich is used to investigate

the role of chaotic advection in Langmuir circulation and how it couples with strategies and lifecycles of

individual plankton By simulating the tra jectories of buoyant particles we can investigate the disp ersion

of plankton in a typical unsteady ow eld paying particular attention to the qualitative eect on the

overall crosssectional structure of planktonic communities This enables us to discuss in a relevant

language the implications on the planktonic fo o dweb taking into account the diversity of foraging and

evasion strategies In particular simulation aids in the discussion of the following questions

How do es chaotic advection help to disp erse phytoplankton and do patches emerge and p ersist as

noted in the eld studies

How do zo oplankton minimize their foraging time or eort and hence maximize their growth in

an unsteady ow eld Should predators do as the prey do or is there a b etter strategy to nd

fo o d in unsteady ows

Do es plankton patchiness p ersist What are the eects of variations in light temp erature and

forcing frequency on Langmuir circulations and what consequences do these have for plankton

patchiness and their dynamics

Section describ es the key mathematical features of the mo del the results of the simulations are

presented in Section and we interpret the results in Section

Mo del foundations

Here we make use of the reduced dimensional system of Moroz Leib ovich to form a simple

T

velo city eld which adequately describ es basic Langmuir circulation The uid velo city u u v w

is given by

u z C t cos z B t cos l y

v At cos z sin l y

and

w Atl sin z cos l y

This naturally satises the incompressibility condition r u l is the asp ect ratio of circulation

depth ie depth of the thermo cline D to circulation width Figure shows the co ordinate system

and p ortrays the ow describ ed by the equations for which typical eld measurements Leib ovich

Barstow have b een added Both eld observations see Leib ovich and theory Craik

Leib ovich Moroz Leib ovich nd that there is a critical windsp eed ab ove which Langmuir

1

circulation is evident The critical windsp eed is most often found to b e approximately m s eg Faller

Wo o dco ck Walther For regions of parameter space for which steady states are stable

the steady states are given by

A

B

2 2

l A

and

AB l

C

where A is a parameter that represents the sp eed of the circulation and is a function of the viscosity

thermal diusivity and the Stokes drift gradient as detailed in Moroz Leib ovich all of which

T

are dicult to estimate or measure For our purp oses A may b e regarded as a constant determined

directly from observations In order for us to do this we require the dimensional variables indicated by

hats which are given by

y z y z D

2

u D

T



v w v w u u

D

T

and

2

D

t t

T

2

is the stress acting in the x direction applied to the surface of a mass of uid having density where u



u is the water friction velo city determined by the applied wind stress A typical value of is

 T

2 1

cm s see Leib ovich Simple calculus reveals that u for z unless l is unphysically

very small

The ab ove description of a velo city eld is from what is commonly called a Eulerian p ointofview

where the whole system is observed with resp ect to a xed lo cation In some instances it is b enecial to

follow a single blob of uid or a particle through the ow and track its path This is a Lagrangian

p ersp ective and is straightforward for velo city elds for which we study the dynamical system

dy dz dx

uy z v y z and w y z

dt dt dt

and follow the tra jectory of the particle

The velo city eld is indep endent of x and the crosssectional velo cities can b e written in terms of a

streamfunction as

v and w

z y

where

A sin z sin l y

In this sense the ow is Hamiltonian and integrable with Hamiltonian where all the streamlines are

closed and can b e found explicitly see Fig Clearly in the absence of diusion an advected particle

will remain on the closed orbit that it started on To follow an individual particle with p ositive or

negative buoyancy V we amend the ab ove equations by putting

s

dz

w y z V z

s dt

We make sure that V go es rapidly to zero at the upp er and lower b oundaries by insisting

s

   

(z +1)d z d

e V V e

s

where the relaxation length d is very small and V is a constant buoyancy By dening V y we see

b

again that the system is Hamiltonian and incompressible for the main b o dy of the ow where the buoyancy

cuto terms are insignicant Flows in the are unlikely to b e as p erfect as the ab ove description

suggests To investigate the eects of a small timedep endent variation we consider small amplitude

uctuations of the ow p erp endicular to the direction of the wind such that the ow is unchanged

at z Figure indicates the typ e of p erturbation that is considered This is obtained by the

transformation

y y g tz

where g t is the timep erio dic forcing function given by

g t sin t

It is hop ed that more eld data will b ecome available in order to validate this typ e of p erio dic p erturbation

or suggest another one Theoretical studies at least indicate the p otential for complex b ehaviour in

Langmuir circulation including the p ossibility of secondary Hopf bifurcations see discussion To rst

order in the streamfunction b ecomes

A sin z sin l y Ag tl z sin z cos l y

which denes a very similar ow to the full streamfunction and hence we will use the ab ove truncation

throughout the remainder of the pap er The ab ove system is similar to the even oscillatory instability

for RayleighBenard convection as studied by Camassa Wiggins and many of the techniques that

they employ can b e used here eg Melnikov theory can b e used to investigate the spread of plankton

However imp ortant distinctions are made here in the application and interpretation of the results

Results

For the case of steady state circulation the crosssectional streamlines for neutrally buoyant particles are

p ortrayed in Fig All streamlines are closed and in the absence of any diusive pro cesses particles

stay on streamlines for all time

The crosssectional streamlines for p ositively buoyant particles are displayed in Fig where it is

clear that there are two qualitatively dierent b ehaviours for particles some particles are trapp ed in

closed orbits at some distance b elow the surface whereas others accumulate at the p oint or line when

also considering the longitudinal ow of convergence of streamlines at the uid surface There is a

clear b oundary b etween these two regions of varying b ehaviour The set of closed orbits form what is

called a Stommel retention zone after Stommel In the absence of any other transp ort or

diusive pro cesses buoyant particles that b egin at the surface cannot submerge due to the upward vertical

comp onent of the buoyant particle tra jectories Fig and hence will not enter the Stommel retention

zone no matter how fast the uid ow is in the Langmuir circulation in contrast to the conclusions of

Wo o dco ck

We now consider the streamfunction given by Eq for an unsteady ow eld To simplify the

analysis we calculate the Poincare section we record the p osition of an advected particle at times

t n where n is an integer ie at times t when g t A go o d measure of how

much the timedep endent p erturbation aects the tra jectory of a particle is to calculate the particles

escap e time or the value of n for which a particle rst leaves the Langmuir cell that it started in

Figures to display the escap e time as a function of the particles initial condition within a Langmuir

cell In contrast to the steady velo city eld particles are not constrained to a closed orbit and can

wander through the whole space There are no clear b oundaries b etween regions of varying escap e times

In fact the b oundaries are fractal in nature The dark regions in Figs to indicate that the particle

never leaves the Langmuir cell and in this pap er we call these regions retention zones By varying the

frequency of the unsteady p erturbation the structure of the escap e space changes dramatically Increasing

the frequency from Fig to Fig reduces the size of the central retention zone

but intro duces extra resonant retention zones In particular the resonance retention zone is clearly

visible orbiting the central retention zone These resonant retention zones are advected with the ow

but remain separate from the surrounding uid In Fig we plot the crosssectional p ositions of a

uniform grid of neutrally buoyant particles after forcing oscillations n allowing particles to

escap e from the Langmuir cell No particles were allowed to enter the Langmuir cell from elsewhere but

a similar picture was obtained when particles were allowed to reenter the Langmuir cell There are clear

regions which exhibit coherent b ehaviour and have b een left b ehind after approximately half the initial

numb er of particles have escap ed The coherent b ehaviour is even more evident when we also consider the

longitudinal displacement as a function of the particles initial conditions as shown in Fig Particles

in the retention zones on average all travel at the same longitudinal sp eed whereas particles in the

mixed or chaotic regions are widely disp ersed in the longitudinal direction All the gures clearly show

the stretching and folding that o ccurs in the coherent regions Increasing the forcing frequency further to

Fig we see that the central retention zone decreases in size again but higher order resonant

retention zones are pro duced well dened and retention zones orbit the central retention zone

The resonant retention zones p ersist for as long as the physical forcing remains constant The escap e

times for p ositively buoyant particles ie V are displayed in Figs and in which we keep the

s

same forcing frequency as in Fig ie The gures share characteristics from b oth Figs

and but it is evident that the total area of the retention zones in Figs and are less than the area

of the corresp onding Stommel retention zones for the steady ow Again resonant retention zones are

pro duced that circulate the central retention zone

Interpretation and Discussion

As p ointed out previously we could apply the Melnikov theory to this problem and extract a measure

of the size of the coherent regions Wiggins It is also p ossible to calculate an estimate of the

hetero clinic lob es and hence quantify the transp ort and eective diusivity of plankton due to the forcing

Camassa Wiggins Because of the rather arbitrary nature of the timedep endent p erturbation

we refrain from p erforming such analysis in this pap er and concentrate on the qualitative interpretation

There are clear regions where phytoplankton tend to stay in the same patches as can b e seen in Figs

to and regions where much mixing is evident b oth in the crosssectional space and longitudinally

Fig The transp ort of particles in the chaotic regions is p otentially much quicker than for molecular

diusion alone as particles a small distance apart can move to adjacent Langmuir cells after only one

p erio d Regions where particle tra jectories can deviate quickly are clearly seen particularly close to

the b oundaries due to the structure of the hetero clinic connection This is in contrast to particles that

start in the coherent regions in which the structure is clear The eect on a patch of plankton of a

similar scale to the Langmuir cell is to quickly remove plankton in the chaotic region from the system

and leave b ehind a skeleton of plankton in the coherent regions The coherent regions are dierent for

particles of varying buoyancy see Figs and In general increasing the magnitude of the buoyancy

of the particles decreases the size and changes the lo cation of the coherent regions Therefore dierent

sp ecies of phytoplankton with dissimilar buoyancy will form patches in a variety of lo cations and hence

zo oplankton must b e able to adapt to lo cate and follow dierent prey eciently It is also likely that

phytoplankton can adjust their buoyancy over a short time scale in resp onse to the available light source

Mo ore Villareal and hence plankton could escap e a coherent lightdecient patch by changing

their buoyancy This could result in the organisms lo cal environment eectively changing into a chaotic

region thus allowing the plankton to harness the chaotic regions transp ort and mixing dynamics

The result of resonance b etween two typ es of physical forcing is to provide additional retention zones

which are advected with the ow and do not diminish in size The retention zones consist of coherent

motion and are isolated in regions of chaotic b ehaviour Longitudinal transp ort of the retention zones

is also coherent and the phytoplankton are given the opp ortunity to b e surrounded by organisms of the

same sp ecies Given sucient time this may enable the phytoplankton to mate sexually and form high

lo cal concentrations Clearly the growth rate of phytoplankton will b e limited dep ending on their time

averaged depth dep endent light source and hence some patches may b e more p opulated than others

Field studies also rep ort the presence of coherent patches on a variety of scales that are not destroyed by

an otherwise turbulent velo city eld Beneld et al Wiafe Frid Lenz et al Wishner

et al Visman et al Owen Haury Wieb e Denman Herman

The upshot of the dierence in buoyancy b etween phytoplankton and purely advected particles is that

their retention zones dier ie the tra jectories of the p ositively buoyant phytoplankton that get trapp ed

in the retention zones of Fig will intersect the tra jectories of neutrally buoyant particles in the chaotic

regions of Fig This enables the phytoplankton to have a fresh mixed supply of essential nutrients

Zo oplankton are able to change their swimming characteristics in resp onse to the availability of prey

and their vulnerability to higher predators eg Munk Davis et al Hunter Thomas

Dep ending on the zo oplanktons feeding characteristics living in a chaotic region may either b e b enecial

or detrimental Saiz Kiorb o e For a lter feeder that do esnt swim and waits for prey to come

close the chaotic region has the eect of increasing the zo oplankters encounter rates with phytoplankton

However for a zo oplankter that actively pursues prey the diverging ow tra jectories could prove dicult

to traverse thus decreasing its capacity for successful capture Also there may b e less prey in chaotic

regions as lo cal patches may diuse quicker To leave the chaotic regions the zo oplankter must change

its eective buoyancy ie swimming up or down until the prey are encountered in the coherent region

similar to Davies et al but transp ort of zo oplankton can o ccur at much greater sp eeds making

use of the ow than for swimming alone It would obviously b e b enecial for zo oplankton to remain

in this region until the fo o dsource has run out or if they get chased by larger predators and therefore

they should match the buoyancy of the phytoplankton Thus zo oplankton may make use of the chaotic

regions to move swiftly and economically b etween phytoplankton coherent regions and to evade predators

Yen Strickler To do this they could make use of the light intensity and adjust their buoyancy

accordingly Analyses of eld data suggest that zo oplankton move into phytoplankton patches as so on

as they form then move on b efore getting eaten themselves

In op en o ceans and lakes it is likely that there will b e signicant variations in windsp eed illumination

and forcing frequency It is the timescales of these variations that are imp ortant for the construction

and maintenance of the resonant retention zones One can imagine a smo oth transition b etween Figs

to for which certain retention zones would p ersist throughout the pro cess but others would exist for

relatively short p erio ds of time

Variations of light and wind lead to a path in the parameter space of Moroz Leib ovich

and present the p ossibility of oscillatory solutions as well as steady solutions Cox et al a b

demonstrate the existence of a whole range of solutions which can coexist such as steady oscillatory and

multiple oscillatory states They nd however that the only robustly observable motions that should

b e anticipated are travelling waves and steady states Also they rep ort the existence of transversally

drifting Langmuir circulations in eld studies and indicate that stable stratication causes the most

unstable linear mo des to to rotate from the wind direction although this is only a minor aect What

we conclude from these studies is that a whole host of instabilities arise from the mo del irresp ective of

other arbitrary external forcing It is exp ected that some of these oscillatory mo des will exhibit similar

b ehaviour to the system describ ed in this pap er

If the Langmuir circulation slows down and stops it will leave b ehind patches that may diuse relatively

slowly due to turbulence but this could b e balanced by the active aggregation of plankton Davis et al

Therefore the patches may p ersist in the absence of Langmuir circulation and may even b e

entrained in the ow if the circulation restarts Some patches will b e p ositioned within the coherent

regions and others not and so some patches will remain and others will quickly disapp ear This may

indeed b enet one sp ecies with a particular buoyancy and not another but this would dep end explicitly

on the physical conditions

The eect of other diusive pro cesses on the unsteady ow such as turbulence may do no more than

blur the edges of the retention zones enabling particles to diuse in and out However it has b een shown

by Davis et al that by including the eects of the planktons swimming b ehaviour see also Pedley

Kessler Hill Hader Bees et al as a biological diusivity the plankton tra jectories

are no longer incompressible and plankton can actively aggregate This may help plankton to accumulate

in areas b enecial to them such as p erhaps the coherent regions and is an obvious extension to the

ab ove work

Acknow ledgments We thank Prof D Rand for useful discussions and supp ort for this work

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diatoms Marine EcologyProgress Series

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windrows up to 6 m critical windspeed approx. 3 m/s 10 cm/s 6 cm/s

CCCCCCC fluidCCCCCCC z = 0 CCCCCCC surfaceCCCCCCC CCCCCCC CCCCCCC CCCCCCC 3 cm/s CCCCCCC CCCCCCC CCCCCCCBBBBBBB CCCCCCCBBBBBBB D = 2 m − 200 CCCCCCCBBBBBBB z = −1 CCCCCCCBBBBBBB *********************************CCCCCCCBBBBBBB *********************************CCCCCCCBBBBBBB *********************************BBBBBBB *********************************deep /lake BBBBBBB *********************************BBBBBBB *********************************BBBBBBB *********************************BBBBBBB *********************************z w BBBBBBB

position x velocity u

y v

Figure Co ordinate system and representation of the steady ow Typical eld measurements as

collated by Leib ovich and Barstow are indicated The windrows consist of oating debris

at regions of surface convergence

regions of convergence or windrows surface

deep ocean/lake

Figure Steady crosssectional streamlines for particles of neutral buoyancy

ε z = 0

z frequency = ω y

z = −1

Figure The timedep endent p erturbation which is sup erimp osed on the steady ow

upper surface

CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC mirror CCCCCCCCCCCCCCCCCCCmirror CCCCCCCCCCCCCCCCCCCimage image CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCStommel CCCCCCCCCCCCCCCCCCCretention CCCCCCCCCCCCCCCCCCCzone

deep ocean/lake

Figure Steady crosssectional streamlines for particles of p ositive buoyancy Suciently near to the

upp er surface all tra jectories have a vertical comp onent which p oints upwards Therefore no p ositively

buoyant particles can move downwards from the upp er surface no matter how strong the Langmuir

circulation The shaded region indicates a region of closed streamlines and is called a Stommel retention zone

Figure Escap e times for neutrally buoyant advected particles whose initial crosssectional p ositions

are plotted The greyscale indicates the numb er of oscillations required b efore the particle leaves the

Langmuir cell where white represents one oscillation and black represents more than oscillations

The circulation sp eed A and the p erturbation amplitude b oth equal and the forcing frequency

equals

Figure Escap e times for neutrally buoyant advected particles whose initial crosssectional p ositions

are plotted The greyscale indicates the numb er of forcing cycles required b efore the particle leaves the

Langmuir cell where white represents one oscillation and black represents more than oscillations

The circulation sp eed A and the p erturbation amplitude b oth equal and the forcing frequency

equals

Figure Escap e times for neutrally buoyant advected particles whose initial crosssectional p ositions

are plotted The circulation sp eed A and the p erturbation amplitude b oth equal and the forcing

frequency equals

Figure Escap e times for p ositively buoyant V advected particles whose initial crosssectional

s

p ositions are plotted The remaining parameters are the same as Fig

Figure Escap e times for p ositively buoyant V advected particles whose initial crosssectional

s

p ositions are plotted The remaining parameters are the same as Fig

Figure Distribution of neutrally buoyant particles after oscillations from a regular grid with a

forcing frequency of Here plankton are allowed to escap e from the region dened by y

Coherent regions of space can b e observed as are regions where particles are quickly removed A

2

with pixels

Figure Longitudinal displacement of particles that start from x at the ab ove crosssectional

lo cation Particles whose initial p osition is contained in the light coloured regions travel a greater lon

2

gitudinal distance A and with pixels The direction of the longitudinal ow is

generally with the wind unless the asp ect ratio l is unrealistically small see text