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MODELS OF SMALL-SCALE PATCHINESS

Dennis McGillicuddy, Woods Hole OceanographicInstitution, Bigelow 209b, Mail Stop d11, Woods Hole,MA 02543, USA

Copyright ^ 2001 Academic Press

doi:10.1006/rwos.2001.0405

Introduction

Patchiness is perhaps the most salient characteristicof plankton populations in the ocean. The scale ofthis heterogeneity spans many orders of magnitudein its spatial extent, ranging from planetary down tomicroscale (Figure 1). It has been argued thatpatchiness plays a fundamental role in the function-ing of marine ecosystems, insofar as the mean con-ditions may not reSect the environment to whichorganisms are adapted. For example, the fact thatsome abundant predators cannot thrive on the meanconcentration of their prey in the ocean implies thatthey are somehow capable of exploiting small-scalepatches of prey whose concentrations are much lar-ger than the mean. Understanding the nature of thispatchiness is thus one of the major challenges ofoceanographic ecology.

The patchiness problem is fundamentally one ofphysical}biological}chemical interactions. This in-terconnection arises from three basic sources: (1)ocean currents continually redistribute dissolved andsuspended constituents by advection; (2) space}timeSuctuations in the Sows themselves impact biolo-gical and chemical processes; and (3) organisms arecapable of directed motion through the water. Thistripartite linkage poses a difRcult challenge tounderstanding oceanic ecosystems: differentiationbetween the three sources of variability requiresaccurate assessment of property distributions inspace and time, in addition to detailed knowledge oforganismal repertoires and the processes by whichambient conditions control the rates of biologicaland chemical reactions.

Various methods of observing the ocean tend tolie parallel to the axes of the space/time domain inwhich these physical}biological}chemical interac-tions take place (Figure 2). Given that a purelyobservational approach to the patchiness problem isnot tractable with Rnite resources, the coupling ofmodels with observations offers an alternativewhich provides a context for synthesis of sparsedata with articulations of fundamental principlesassumed to govern functionality of the system. In

a sense, models can be used to Rll the gaps in thespace/time domain shown in Figure 2, yieldinga framework for exploring the controls on spatiallyand temporally intermittent processes.

The following discussion highlights only a few ofthe multitude of models which have yielded insightinto the dynamics of plankton patchiness. Exampleshave been chosen to provide a sampling of scaleswhich can be referred to as ‘small’ } that is, smallerthan the planetary scale shown in Figure 1A. Inaddition, the article attempts to furnish some expo-sure to the diversity of modeling approaches whichcan be brought to bear on this problem. These rangefrom abstract theoretical models intended toelucidate speciRc processes, to complex numericalformulations which can be used to actually simulateobserved distributions in detail.

Formulation of the Coupled Problem

A general form of the coupled problem can bewritten as a three-dimensional advection-diffusion-reaction equation for the concentration Ci of anyparticular organism of interest:

LCi

Lt# + ) (vCi) ! + ) (K+Ci)

hij hgigj hggiggj

local rate of changeadvection diffuion

" Rihij [1]

biological sources/sinks

where the vector v represents the Suid velocity plusany biologically induced transport through thewater (e.g., sinking, swimming), and K the turbulentdiffusivity. The advection term is often written sim-ply as v )+Ci because the ocean is an essentiallyincompressible Suid (i.e., + ) v"0). The ‘reactionterm’ Ri on the right-hand side represents the sour-ces and sinks due to biological activity.

In essence, this model is a quantitative statementof the conservation of mass for a scalar variable ina Suid medium. The advective and diffusive termssimply represent the redistribution of materialcaused by motion. In the absence of any motion,eqn [1] reduces to an ordinary differential equationdescribing the biological and/or chemical dynamics.The reader is referred to the review by Donaghayand Osborn for a detailed derivation of the advec-tion-diffusion-reaction equation, including explicit

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MODELS OF SMALL-SCALE PATCHINESS 1

VVCrMDrScanrPadmarRWOS 0405

treatment of the Reynolds decomposition for biolo-gical and chemical scalars (see Further Reading).

Any number of advection-diffusion-reaction equa-tions can be posed simultaneously to represent a setof interacting state variables Ci in a coupled model.For example, an ecosystem model including nutri-ents, phytoplankton, and zooplankton (an ‘NPZ’model) could be formulated with C1"N, C2"Pand C3"Z. The biological dynamics linking thesethree together could include nutrient uptake, pri-mary production, grazing, and remineralization.Ri would then represent not only birth and mortal-ity, but terms which depend on interactions betweenthe several model components.

Growth and Diffusion ^ the‘KISS’ Model

Some of the earliest models used to investigateplankton patchiness dealt with the competing effectsof growth and diffusion. In the early 1950s, modelsdeveloped independently by Kierstead (KI) andSlobodkin (S) and Skellam (S) } the so-called ‘KISS’model } were formulated as a one-dimensional dif-fusion equation with exponential population growthand constant diffusivity:

LCLt

!KL2CLx2"aC [2]

Note that this model is a reduced form of eqn [1]. Itis a mathematical statement that the tendency fororganisms to accumulate through reproduction iscounterbalanced by the tendency of the environmentto disperse them through turbulent diffusion. Seek-ing solutions which vanish at x"0 and x"L(thereby deRning a characteristic patch size of di-mension L), with initial concentration C(x, 0)"f(x), one can solve for a critical patch sizeL"n(K/a)1

2 in which growth and dispersal are inperfect balance. For a speciRed growth rate a anddiffusivity K, patches smaller than L will be elimi-nated by diffusion, while those that are larger willresult in blooms. Although highly idealized in itstreatment of both physical transport and biologicaldynamics, this model illuminates a very importantaspect of the role of diffusion in plankton patchi-ness. In addition, it led to a very speciRc theoreticalprediction of the initial conditions required to starta plankton bloom, which Slobodkin subsequentlyapplied to the problem of harmful algal blooms onthe west Florida shelf.

Homogeneous Isotropic Turbulence

The physical regime to which the preceding modelbest applies is one in which the statistics of theturbulence responsible for diffusive transport is spa-tially uniform (homogeneous) and has no preferreddirection (isotropic). Turbulence of this type mayoccur locally in parts of the ocean in circumstanceswhere active mixing is taking place, such as ina wind-driven surface mixing layer. Such motionsmight produce plankton distributions such as thoseshown in Figure 1E.

The nature of homogeneous isotropic turbulencewas characterized by Kolmogoroff in the early1940s. He suggested that the scale of the largesteddies in the Sow was set by the nature of theexternal forcing. These large eddies transfer energyto smaller eddies down through the inertial sub-range in what is known as the turbulent cascade.This cascade continues to the Kolmogoroff micro-scale, at which viscous forces dissipate the energyinto heat. This elegant physical model inspired thefollowing poem attributed to L.F. Richardson:

Big whorls make little whorlswhich feed on their velocity;little whorls make smaller whorls,and so on to viscosity...

Based on dimensional considerations, Kol-mogoroff proposed an energy spectrum E of theform

E(k)"Ae23k~5

3

where k is the wavenumber, e is the dissipation rateof turbulent kinetic energy, and A is a dimensionlessconstant. This theoretical prediction was later borneout by measurements, which conRrmed the ‘minusRve-thirds’ dependence of energy content onwavenumber.

In the early 1970s, Platt published a startling setof measurements which suggested that for scalesbetween 10 and 103 m the variance spectrum ofchlorophyll in the Gulf of St Lawrence showed thesame !5/3 slope. On the basis of this similarity tothe Kolmogoroff spectrum, he argued that on thesescales, phytoplankton were simply passive tracers ofthe turbulent motions. These Rndings led to a burg-eoning Reld of spectral modeling and analysis ofplankton patchiness. Studies by Denman, Powell,Fasham, and others sought to formulate more uni-Red theories of physical}biological interactions us-ing this general approach. For example, Denmanand Platt extended a model for the scalar variance

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2 MODELS OF SMALL-SCALE PATCHINESS


spectrum to include a uniform growth rate. Theirtheoretical analysis suggested a breakpoint in thespectrum at a critical wavenumber kc (Figure 3),which they estimated to be in the order of 1 km~1 inthe upper ocean. For wavenumbers lower than kc,phytoplankton growth tends to dominate the effectsof turbulent diffusion, resulting in a k~1 depend-ence. In the higher wavenumber region, turbulentmotions overcome biological effects, leading tospectral slopes of !2 to !3. Efforts to includemore biological realism in theories of this type havecontinued to produce interesting results, althoughPowell and others have cautioned that spectral char-acteristics may not be sufRcient in and of themselvesto resolve the underlying physical}biological interac-tions controlling plankton patchiness in the ocean.

Vertical Structure

Perhaps the most ubiquitous aspect of plankton dis-tributions which makes them anisotropic is theirvertical structure. Organisms stratify themselves ina multitude of ways, for any number of differentpurposes (e.g., to exploit a limiting resource, toavoid predation, to facilitate reproduction). Forexample, consider the subsurface maximum which ischaracteristic of the chlorophyll distribution inmany parts of the world ocean (Figure 4) The deepchlorophyll maximum (DCM) is typically situatedbelow the nutrient-depleted surface layer, where nu-trient concentrations begin to increase with depth.Generally this is interpreted to be the result of jointresource limitation: the DCM resides where nutri-ents are abundant and there is sufRcient light forphotosynthesis. However, this maximum in chloro-phyll does not necessarily imply a maximum inphytoplankton biomass. For example, in the nutri-ent-impoverished surface waters of the open ocean,much of the phytoplankton standing stock is sus-tained by nutrients which are rapidly recycled; thusrelatively high biomass is maintained by low ambi-ent nutrient concentrations. In such situations, theDCM often turns out to be a pigment maximum,but not a biomass maximum. The mechanism re-sponsible for the DCM in this case is photoadapta-tion, the process by which phytoplankton alter theirpigment content according to the ambient light envi-ronment. By manufacturing more chlorophyll percell, phytoplankton populations in this type ofDCM are able to capture photons more effectivelyin a low-light environment.

Models have been developed which can produceboth aspects of the DCM. For example, consider thenutrient, phytoplankton, zooplankton, detritus(NPZD) type of model (Figure 5) which simulates

the Sows of nitrogen in a planktonic ecosystem. Thevarious biological transformations (such as nutrientuptake, primary production, grazing excretion, etc.)are represented mathematically by functional rela-tionships which depend on the model state variablesand parameters which must be determined empi-rically. Doney et al. coupled such a system to aone-dimensional physical model of the upper ocean(Figure 6). Essentially, the vertical velocity (w) anddiffusivity Relds from the physical model are used todrive a set of four coupled advection-diffusion-reaction equations (one for each ecosystem statevariable) which represent a subset of the fullthree-dimensional eqn [1]:

LCi

Lt#w

LCi

Lz!

LLzAK

LCi

Lz B"Ri [3]

The Ri terms represent the ecosystem interactionterms schematized in Figure 5. Using a diagnosticphotoadaptive relationship to predict chlorophyllfrom phytoplankton nitrogen and the ambient lightand nutrient Relds, such a model captures the over-all character of the DCM observed at the BermudaAtlantic Time-series Study (BATS) site (Figure 6).

Broad-scale vertical patchiness (on the scale of theseasonal thermocline) such as the DCM is accom-panied by much Rner structure. The special volumeof Oceanography on ‘Thin layers’ provides an excel-lent overview of this subject, documenting small-scale vertical structure in planktonic populations ofmany different types. One particularly strikingexample comes from high-resolution Suorescencemeasurements (Figure 7A). Such proRles often showstrong peaks in very narrow depth intervals, whichpresumably result from thin layers of phytoplan-kton. A mechanism for the production of this layer-ing was identiRed in a modeling study by P.J.S.Franks, in which he investigated the impact of near-inertial wave motion on the ambient horizontal andvertical patchiness which exists at scales much lar-ger than the thin layers of interest. Near-inertialwaves are a particularly energetic component in theinternal wave spectrum of the ocean. Their horizon-tal velocities can be described by:

u"U0cos(mz!ut) v"U0sin(mz!ut) [4]

where U0 is a characteristic velocity scale, m is thevertical wavenumber, and u the frequency of thewave. This kinematic model prescribes that thevelocity vector rotates clockwise in time andcounterclockwise with depth; its phase velocity isdownward, and group velocity upward. In his

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words, ‘the motion is similar to a stack of pancakes,each rotating in its own plane, and each slightly outof phase with the one below’. Franks used thisvelocity Reld to perturb an initial distribution ofphytoplankton in which a Gaussian vertical distri-bution (of scale p) varied sinusoidally in both x andy directions with wavenumber kP . Neglecting theeffects of growth and mixing, and assuming thatphytoplankton are advected passively with the Sow,eqn [1] reduces to:

LCLt

#uLCLx

#vLCLy

"0 [5]

Plugging the velocity Relds [4] into this equation,the initial phytoplankton distribution can be integ-rated forward in time. This model demonstrates thestriking result that such motions can generate verti-cal structure which is much Rner than that presentin the initial condition (Figure 7B). Analysis of thesimulations revealed that the mechanism at workhere is simple and elegant: vertical shear can trans-late horizontal patchiness into thin layers by stretch-ing and tilting the initial patch onto its side (Figure7C).

Mesoscale Processes: The InternalWeather of the Sea

Just as the atmosphere has weather patterns thatprofoundly affect the plants and animals that live onthe surface of the earth, the ocean also has its ownset of environmental Suctuations which exert funda-mental control over the organisms living within it.The currents, fronts, and eddies that comprise theoceanic mesoscale, sometimes referred to as the ‘in-ternal weather of the sea’, are highly energetic fea-tures of ocean circulation. Driven both directly andindirectly by wind and buoyancy forcing, their char-acteristic scales range from tens to hundreds of kilo-meters with durations of weeks to months. Theirspace scales are thus smaller and timescales longerthan their counterparts in atmospheric weather, butthe dynamics of the two systems are in many waysanalogous. Impacts of these motions on surfaceocean chlorophyll distributions are clearly visible insatellite imagery (Figure 1B).

Mesoscale phenomenologies accommodate a di-verse set of physical}biological interactions whichinSuence the distribution and variability of planktonpopulations in the sea. These complex yet highlyorganized Sows continually deform and rearrangethe hydrographic structure of the near-surface re-gion in which plankton reside. In the most general

terms, the impact of these motions on the biota istwofold: not only do they stir organism distribu-tions, they can also modulate the rates of biologicalprocesses. Common manifestations of the latter areassociated with vertical transports which can affectthe availability of both nutrients and light tophytoplankton, and thereby the rate of primary pro-duction. The dynamics of mesoscale and submeso-scale Sows are replete with mechanisms that canproduce vertical motions.

Some of the Rrst investigations of these effectsfocused on mesoscale jets. Their internal mechanicsare such that changes in curvature give rise tohorizontal divergences which lead to very intensevertical velocities along the Sanks of the meandersystems (Figure 8). J.D. Woods was one of the Rrstto suggest that these submesoscale upwellings anddownwellings would have a strong impact on upperocean plankton distributions (see his article con-tained in the volume edited by Rothschild; seeFurther Reading). Subsequent modeling studies haveinvestigated these effects by incorporating plank-tonic ecosystems of the type shown in Figure 5 intothree-dimensional dynamical models of meanderingjets. Results suggest that upwelling in the Sank ofa meander can stimulate the growth of phytoplan-kton (Figure 9). Simulated plankton Relds are quitecomplex owing to the fact that Suid parcels arerapidly advected in between regions of upwellingand downwelling. Clearly, this complicated convo-lution of physical transport and biological responsecan generate strong heterogeneity in plankton distri-butions.

What are the implications of mesoscale patchi-ness? Do these Suctuations average out to zero, orare they important in determining the mean charac-teristics of the system? In the Sargasso Sea, it ap-pears that mesoscale eddies are a primarymechanism by which nutrients are transported tothe upper ocean. Numerical simulations were usedto suggest that upwelling due to eddy formation andintensiRcation causes intermittent Suxes of nitrateinto the euphotic zone (Figure 10A). The mecha-nism can be conceptualized by considering a densitysurface with mean depth coincident with the base ofthe euphotic zone (Figure 10B). This surface is per-turbed vertically by the formation, evolution, anddestruction of mesoscale features. Shoaling densitysurfaces lift nutrients into the euphotic zone whichare rapidly utilized by the biota. Deepening densitysurfaces serve to push nutrient-depleted water out ofthe well-illuminated surface layers. The asymmetriclight Reld thus rectiRes vertical displacements ofboth directions into a net upward transport of nutri-ents, which is presumably balanced by a commen-

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surate Sux of sinking particulate material. Severaldifferent lines of evidence suggest that eddy-drivennutrient Sux represents a large portion of the annualnitrogen budget in the Sargasso Sea. Thus, in thisinstance, plankton patchiness appears to be an es-sential characteristic which drives the mean proper-ties of the system.

Coastal Processes

Of course, the internal weather of the sea is notlimited to the eddies and jets of the open ocean.Coastal regions contain a similar set of phenomena,in addition to a suite of processes in which thepresence of a land boundary plays a key role. A ca-nonical example of such a process is coastal upwell-ing, in which the surface layer is forced offshorewhen the wind blows from a particular direction.This triggers upwelling of deep water to replace thedisplaced surface water. The biological ramiRcationsof this were explored in the mid-1970s by Wrob-lewski with one of the Rrst coupled physical}biolo-gical models to include spatial variability explicitly.ConRguring a two-dimensional advection-diffusion-reaction model in vertical plane cutting across theOregon shelf, he studied the response of an NPZD-type ecosystem model to transient wind forcing. His‘strong upwelling’ case provided a dramatic demon-stration of mesoscale patch formation (Figure 11).Deep, nutrient-rich waters from the bottom bound-ary layer drawn up toward the surface stimulatea large increase in primary production whichis restricted to within 10 km of the coast. The phy-toplankton distribution reSects the localizedenhancement of production, in addition to advectivetransport of the resultant biogenic material. Notethat the highest concentrations of phytoplankton aredisplaced from the peak in primary production, ow-ing to the offshore transport in the near-surfacelayers.

Although Wroblewski’s model was able to cap-ture some of the most basic elements of the bio-logical response to coastal upwelling, its two-dimensional formulation precluded representationof alongshore variations which can sometimes be asdramatic as those in the cross-shore direction. Thecomplex set of interacting jets, eddies, and Rlamentscharacteristic of such environments (as in Figure 1C)have been the subject of a number of three-dimen-sional modeling investigations. For example,Moisan et al. incorporated a food web and bio-optical model into simulations of the CoastalTransition Zone off California. This model showedhow coastal Rlaments can produce a complex biolo-gical response through modulation of the ambient

light and nutrient Relds (Figure 12). The simulationssuggested that signiRcant cross-shelf transport ofcarbon can occur in episodic pulses when Rlamentsmeander offshore. These dynamics illustrate the tre-mendous complexity of the processes which link thecoastal ocean with the deep sea.

Behavior

The mechanisms for generating plankton patchinessdescribed thus far consist of some combination ofSuid transport and physiological response to thephysical, biological, and chemical environment. Thefact that many planktonic organisms have behavior(interpreted narrowly here as the capability for di-rected motion through the water) facilitates a di-verse array of processes for creating heterogeneity intheir distributions. Such processes pose particularlydifRcult challenges for modeling, in that their effectsare most observable at the level of the population,whereas their dynamics are governed by interactionswhich occur amongst individuals. The latter aspectmakes modeling patchiness of this type particularlyamenable to individual-based models, in contrast tothe concentration-based model described by eqn [1].For example, many species of marine plankton areknown to form dense aggregations, sometimes refer-red to as swarms. Okubo suggested an individual-based model for the maintenance of a swarm of theform:

d2xdt2 "!k

dxdt

!u2x!/(x)#A(t) [6]

where x represents the position of an individual.This model assumes a frictional force on the organ-ism which is proportional to its velocity (with fric-tional coefRcient k), a random force A(t) which iswhite noise of zero mean and variance B, and at-tractive forces. Acceleration resulting from the at-tractive forces is split between periodic (frequencyu) and static (/(x)) components. The key aspect ofthe attractive forces is that they depend on thedistance from the center of the patch. A Fokker-Planck equation can be used to derive a probabilitydensity function:

p(x)"p0expA!u2

2Bx2

!P/(x)

bdxB [7]

where p0 is the density at the center of the swarm.Thus, the macroscopic properties of the system canbe related to the speciRc set of rules governingindividual behavior. Okubo has shown that ob-

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served characteristics of insect swarms compare wellwith theoretical predictions from this model, both interms of the organism velocity autocorrelation andthe frequency distribution of their speeds. Analog-ous comparisons with plankton have proven elusiveowing to the extreme difRculty in making suchmeasurements in marine systems.

The foregoing example illustrates how swarmscan arise out of purely behavioral motion. Yet an-other class of patchiness stems from the joint effectsof behavior and Suid transport. The paper by Flierlet al. is an excellent reference on this general topic(see Further Reading). One of the simplest examplesof this kind of process arises in a population whichis capable of maintaining its depth (either throughswimming or buoyancy effects) in the presence ofconvergent Sow. With no biological sources orsinks, eqn [1] becomes:

LCLt

#v )+HC#C+H ) v!+ ) (K+Ci)"0 [8]

where +H is the vector derivative in the horizontaldirection only. Because vertical Suid motion isexactly compensated by organism behavior (recallthat the vector v represents the sum of physical andbiological velocities), two advective contributionsarise from the term + ) (vC) in eqn [1]: the commonform with the horizontal velocity operating on spa-tial gradients in concentration, plus a source/sinkterm created by the divergence in total velocity(Suid#organism). The latter term provides a mech-anism for accumulation of depth-keeping organismsin areas of Suid convergence. It has been suggestedthat this process is important in a variety of differ-ent oceanic contexts. In the mid-1980s, Olson andBackus argued it could result in a 100-fold increasein the local abundance of a mesopelagic RshBenthosema glaciale in a warm core ring. Franksmodeled a conceptually similar process with asurface-seeking organism in the vicinity ofa propagating front (Figure 13). Simply stated,upward swimming organisms tend to accumulate inareas of downwelling. This mechanism has beensuggested to explain spectacular accumulations ofmotile dinoSagellates at fronts (Figure 1D).

Conclusions

The interaction of planktonic population dynamicswith oceanic circulation can create tremendouslycomplex patterns in the distribution of organisms.Even an ocean at rest could accommodate signiR-cant inhomogeneity through geographic variations

in environmental variables, time-dependent forcing,and organism behavior. Fluid motions tend to amal-gamate all of these effects in addition to introducingyet another source of variability: space}time Suctu-ations in the Sows themselves which impact bio-logical processes. Understanding the mechanismsresponsible for observed variations in planktondistributions is thus an extremely difRcult task.

Coupled physical}biological models offer a frame-work for dissection of these manifold contributionsto structure in planktonic populations. Such modelstake many forms in the variety of approaches whichhave been used to study plankton patchiness. Intheoretical investigations, the basic dynamics ofidealized systems are worked out using techniquesfrom applied mathematics and mathematical phys-ics. Process-oriented numerical models offer a con-ceptually similar way to study systems that are toocomplex to be solved analytically. Simulation-oriented models are aimed at reconstructing particu-lar data sets using realistic hydrodynamic forcingpertaining to the space/time domain of interest.Generally speaking, such models tend to be quitecomplex because of the multitude of processeswhich must be included to simulate observationsmade in the natural environment. Of course, thiscomplexity makes diagnosis of the coupled systemmore challenging. Nevertheless, the combination ofmodels and observations provides a unique contextfor the synthesis of necessarily sparse data:space}time continuous representations of the realocean which can be diagnosed term-by-term to re-veal the underlying processes. Formal unionbetween models and observations is beginning tooccur through the emergence of inverse methodsand data assimilation in the Reld of biologicaloceanography. Article no. v v v provides an up-to-date review of this very exciting and rapidlyevolving aspect of coupled physical}biologicalmodeling.

Although the Reld is more than a half-century old,modeling of plankton patchiness is still in its in-fancy. The oceanic environment is replete with phe-nomena of this type which are not yet understood.Fortunately, the Reld is perhaps better poised thanever to address such problems. Recent advances inmeasurement technologies (e.g., high-resolutionacoustical and optical methods, miniaturized biolo-gical and chemical sensors) are beginning to providedirect observations of plankton on the scales atwhich the coupled processes operate. Linkage ofsuch measurements with models is likely to yieldimportant new insights into the mechanisms con-trolling plankton patchiness in the ocean.

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See also

Biogeochemical Data Assimilation (410). CoastalCirculation Models (400). Fish Vertical Migration(20). Fishery Management (459). Fossil Turbulence(138). General Processes in Ocean Circulation(107). Patch Dynamics (290). 205. 346.

Further ReadingDenman KL and Gargett AE (1995) Biological}physical

interactions in the upper ocean: the role of vertical andsmall scale transport processes. Annual Reviews ofFluid Mechanics 27: 225}255.

Donaghay PL and Osborn TR (1997) Toward a theory ofbiological}physical control on harmful algal bloom dy-namics and impacts. Limnology and Oceanography42: 1283}1296.

Flierl GR, Grunbaum D, Levin S and Olson DB (1999)From individuals to aggregations: the interplay be-tween behavior and physics. Journal of TheoreticalBiology 196: 397}454.

Franks PJS (1995) Coupled physical}biological models inoceanography. Reviews of Geophysics (supplement):1177}1187.

Franks PJS (1997) Spatial patterns in dense algal blooms.Limnology and Oceanography 42: 1297}1305.

Levin S, Powell TM and Steele JH (1993) Patch Dynam-ics. Berlin: Springer-Verlag.

Mackas DL, Denman KL and Abbott MR (1985) Plank-ton patchiness: biology in the physical vernacular.Bulletin of Marine Science 37: 652}674.

Mann KH and Lazier JRN (1996) Dynamics of MarineEcosystems: Biological}Physical Interactions in theOceans. Oxford: Blackwell ScientiRc Publications.

Okubo A (1980) Diffusion and Ecological Problems:Mathematical Models. Berlin: Springer-Verlag.

Okubo A (1986) Dynamical aspects of animal grouping:swarms, schools, Socks and herds. Advances in Bi-ophysics 22: 1}94.

Robinson AR, McCarthy JJ and Rothschild BJ (2001) TheSea: Biological}physical Interactions in the Ocean.New York: John Wiley and Sons.

Rothschild BJ (1988) Toward a Theory on Biolo-gical}Physical Interactions in the World Ocean. Dor-drecht: D. Reidel.

Oceanography Society (1998) Oceanography 11(1):Special Issue on Thin Layers. Virginia Beach, VA:Oceanography Society.

Steele JH (1978) Spatial Pattern in Plankton Communi-ties. New York: Plenum Press.

Wroblewski JS and Hofmann EE (1989) U.S. interdisci-plinary modeling studies of coastal-offshore exchangeprocesses: past and future. Progress in Oceanography23: 65}99.

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MODELS OF SMALL-SCALE PATCHINESS


Figure 1 Appears on Previous Page

bFigure 1 Scales of plankton patchiness, ranging from global down to 1 cm. (A-C) satellite-based estimates of surface-layerchlorophyll computed from ocean color measurements. (D) A dense stripe of Noctiluca scintillans, 3 km off the coast of La Jolla. Theboat in the photograph is trailing a line with floats spaced every 20 m. The stripe stretched for at least 20 km parallel to the shore(photograph courtesy of P.J.S. Franks). (E) Surface view of a bloom of Anabaena flos-aquae in Malham Tarn, Scotland. The areashown is approximately 1 m2 (photograph courtesy of G.E. Fogg).

Figure 9 Results from a coupled model of the Gulf Stream: thermocline depth (left), phytoplankton concentration (middle), andzooplankton concentration (right). (From Flierl and McGillicuddy, submitted.)

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Temperature (˚C) at 85 mNitrate flux

21 22 23

1% Io

Shoaling isopycnals:________________Nitrate injection stimulates

phytoplankton growth

Deeping isopycnals:_______________No ecosystem response

Euphotic zone

Upwelling

Downwelling

Interaction

(A)

(B)

Figure 10 (A) A simulated eddy-driven nutrient injection event: snapshots of temperature at 85 m (left column, 3C) and nitrate fluxacross the base of the euphotic zone (right column, moles of nitrogen m2 d~1). For convenience, temperature contours from theleft-hand panels are overlayed on the nutrient flux distributions. The area shown here is a 500 km2 domain. (The simulation isdescribed in McGillicuddy DJ and Robinson AR (1997) Eddy induced nutrient supply and new production in the Sargasso Sea.Deep-Sea Research I 44(8): 1427}1450.) (B) A schematic representation of the eddy upwelling mechanism. The solid line depictsthe vertical deflection of an individual isopycnal caused by the presence of two adjacent eddies of opposite sign. The dashed lineindicates how the isopycnal might be subsequently perturbed by interaction of the two eddies. (Reproduced with permission fromMcGillicuddy DJ, Robinson AR, Siegel et al. (1998) Influence of mesoscale eddies on new production in the Sargasso Sea. Nature394: 263}265.

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Figure 12 Modeled distributions of phytoplankton (color shading, mg nitrogen m~3) in the Coastal Transition Zone off California.Instantaneous snapshots in panels (A}C) are separated by time intervals of 10 days. Contour lines indicate the depth of theeuphotic zone, defined as the depth at which photosynthetically available radiation is 1% of its value at the surface. Contours rangefrom 30 to 180 m, 40 to 180 m, and 60 to 180 m in panels (A), (B), and (C) respectively. (Reproduced with permission from Moisanet al. (1996) Modeling nutrient and plankton processes in the California coastal transition zone 2. A three-dimensional physical-bio-optical model. Journal of Geophysical Research 101(C10): 22 677}22691.

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Models104

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Shipboardsurveys

Moorings, time series

Time (days)

Sat

ellit

e im

ager

y

Spa

ce (

km)

101

Figure 2 SpaceItime diagram of the scales resolvable withcurrent observational capabilities. Measurements tend to fallalong the axes; the dashed line running between the ‘shipboardsurvey’ axes reflects the trade-off between spatial coverage andtemporal resolution inherent in seagoing operations of that type.Models can be used to examine portions of the spaceItimecontinuum (shaded area).

k_1

k_2 k

_3

kc Log kLo

g E

b(k)

Figure 3 A theoretical spectrum for the spatial variability ofphytoplankton, Eb(k), as a function of wavenumber, k, displayedon a log-log plot. To the left of the critical wavenumber kc,biological processes dominate, resulting in a k~1 dependence.The high wavenumber region to the right of kc where turbulentmotions dominate, has a dependence. The high wavenumberregion to the right of kc where turbulent motionsdominate, hasa dependence between k~2 and k~3. (Reproduced with per-mission from Denman and Platt (1976). The variance spectrumof phytoplankton in a turbulent ocean. Journal of Marine Re-search 34: 593}601.)

Relative abundance

Light

Chlorophyll

Nutrients

Dep

th

Figure 4 Schematic representation of the deep chlorophyllmaximum in relation to ambient light and nutrient profiles.

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a0405fig0003

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Chl

Grazing nonegestion

Z

DRemineralization

Nutrients

N

Detritus

Zooplankton

Z Mortality

Aggregation

Production

P Mortality('microbial loop')

Grazing egestion

Photoadaptation

Chlorophyll

Phytoplankton

P

Figure 5 A four-compartment planktonic ecosystem model showing the pathways for nitrogen flow. (Reproduced with permissionfrom Doney SC, Glover DM and Najjar RG (1996) A new coupled, one-dimensional biological}physical model for the upper ocean:applications to the JGOFS Bermuda Atlantic Time-series Study (BATS) site. Deep-Sea Research II 43: 591}624.)

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Run 6101.30

Run 6113.31Year 20

19

19

20 20

22

21

0

50

100

150

200

250

300

Dep

th (

m)

_ 4 0 4 8 12 16

Time (months)

Temperature

18

0

_ 50

_ 100

_ 150

_ 200

_ 250

Dep

th (

m)

0 60 120 180 240 300 360Time (days)

Chlorophyll (mg m_3)

0.15

0.05

0.25

0.20

0.10

0.10

0.10

0.20

19

18

1920 20

21

0

50

100

150

200

250

300

Dep

th (

m)

_ 4 0 4 8 12 16Time (months)

Temperature

0

50

100

150

200

250

Dep

th (

m)

0 60 120 180 240 300 360Time (days)

BATS Chlorophyll (mg m_3)

0.05

0.15

0.10

0.10

0.050.05

Figure 6 Simulated (left) and observed (right) seasonal cycles of temperature and chlorophyll at the BATS site. (Reproduced withpermission from Doney SC, Glover DM and Najjar RG (1996) A new coupled, one-dimensional biological}physical model for theupper ocean: applications to the JGOFS Bermuda Atlantic Time-series Study (BATS) site. Deep-Sea Research II 43: 591}624.)

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Temperature (˚C)

Drop 8

Phytoplankton

Dep

th (

m)

0

4 5 6

10

9 10 11 12 13 14

20

30

40

0

0 1 2Phytoplankton

t = 0.00 T t = 0.11 Tt = 0.19 T

t = 0.28 Tt = 0.04 T t = 0.42 T

Dep

th (

m)

3 4 5 6

10

20

30

40

50

(A) (B)

u (z)kPH

h

∆ x

∆ zkPV = kPH tanθ

(C)

Figure 7 (A) Observations of thin layer structure in a high-resolution profile of fluorescence (thick line, arbitrary units). The thinline shows the corresponding temperature structure. (Data courtesy of Dr T. Cowles.) (B) Simulated vertical profiles of phytoplan-kton concentration (arbitrary units) as six sequential times. Each profile is offset from the previous by 1 phytoplankton unit. Thetimes are given as fractions of the period of the near-inertial wave used to drive the model. (C) A schematic diagram of the layeringprocess. Vertical shear stretches a vertical column of a property horizontally through an angle h, creating a layer in the verticalprofile. (Reproduced with permission from Franks PJS (1995) Thin layers of phytoplankton: a model of formation by near-inertialwave shear. Deep-Sea Research I 42: 75}91.

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150

100

50

00 50

500 100 cm/s

100 150 200Distance (km)

Dis

tanc

e (k

m)

250

150

100

50

00 50 100 150 200

Distance (km)

Dis

tanc

e (k

m)

250

< _ 9

00

0

>6

>6

>6

< _ 9< _ 9

(A)

(B)

Figure 8 Simulation of a meandering mesoscale jet: (A) velocity on an isopycnal surface with a mean depth of 20 m; (B) verticalvelocity (m d~1) on the same isopycnal surface as in (A). Note the consistent pattern of the vertical motion with respect to thestructure of the jet. (Reproduced with permission from Woods JD (1988) Mesoscale upwelling and primary production. In: RothschildBJ (ed) Toward a Theory on Physical}biological Interactions in the World Ocean. London: Kluwer Academic.)

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U-π Vector plots Elapsed time = 10.0 Days

Dep

th (

m)

Dep

th (

m)

0

50

100

150

200

300

200

100

300

200

100

50 45 40 35 30 25 20 15 10

10

20

30

40

50

5 1

20 15 10 5 0(km)

(km)

Radius 1/2 Dyne

(A)

(B)

(C)

Wind Stress

(mg

N m

_2

d_ 1)

(mg

N m

_2

d_1

)

Dai

ly p

rim

ary

prod

uctio

n

7.210.4

13.616.0

H

7.2

4.0

25

Figure 11 Snapshot from a two-dimensional coupled model of coastal upwelling. (A) Circulation in the transverse plane normal tothe coast (maximum horizontal and vertical velocities are !6.1 and 0.05 cm s~1, respectively); (B) daily gross primary production;(C) phytoplankton distribution (contour interval is 1.6 lg at N l~1. (Reproduced with permission from Wroblewski JS (1977) A modelof plume formation during variable Oregon upwelling. Journal of Marine Research 35(2): 357}394.

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0(A)

(B)

2

4

6

8

10

0

2

4

6

8

10

Dep

th (

m)

Dep

th (

m)

Cross-frontal distance (m)

_ 50 _ 40 _ 30 _ 20 _ 10 100 20 30 40 50

Figure 13 Surface-seeking organisms aggregating ata propagating front. Modeled particle locations (dots, panel (A))and particle streamlines (thin lines, panel (B)) in the cross-frontal flow. The front is centered at x"0, and the coordinatesystem translates to the right with the motion of the front. Flowstreamlines are represented in both panels as bold lines; theydiffer from particle streamlines due to propagation of the front.The shaded area in (B) indicates the region in which cells arefocused into the frontal zone, forming a dense band atx"!20 m. Compare patterns in (A) and (B) with an aerialphotograph of dense bands of Lingulodinium polyedrum popula-tions associated with internal waves about 1 km off the coast ofLa Jolla. (Reproduced with permission from Franks, 1997.)

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