Coherent vortices, Lagrangian particles and the marine ecosystem

Claudia PasqueroISAC-CNR, Torino, Italy

Annalisa BraccoThe Abdus Salam ICTP, Trieste, Italy

Antonello ProvenzaleISAC-CNR, Torino, and CIMA, Savona, Italy

Coherent vortices characterize the dynamics of large and mesoscale geophysical flows, and affectmaterial transport in several different ways. Here we show that vortices are one of the actors inthe marine ecosystem, and discuss some of their effects on plankton dynamics. More generally, theresults reported here can help addressing the problem of the interplay between coherent vortices andthe dynamics of reactive tracers.

1 INTRODUCTION

Large and mesoscale geophysical flows in theocean and the atmosphere are turbulent, and arecharacterized by the dominance of the Earth’srotation. In such conditions, the vorticity fluc-tuations generated by the flow are just smallperturbations to the planetary vorticity. To us,however, who participate in the spinning mo-tion of the planet, these small fluctuations ap-pear in their full power as the flow patterns thatcharacterize the dynamics of the ocean and theatmosphere.

In the case of stable stratification, as hap-pens in large portions of the ocean and in thestratosphere, the simultaneous presence of rota-tion and stratification makes the flow stronglyanisotropic, favouring movements on the lo-cal horizontal plane and inhibiting vertical fluidmotions.

This strong anisotropy makes geophysicalturbulence very different from homogeneousand isotropic three-dimensional turbulence. Infact, a better model - albeit a very simplifiedone - of large-scale geophysical turbulence is

two-dimensional turbulence, obtained when theflow is assumed to be independent of the verti-cal coordinate. Two-dimensional turbulence is asimplified version of the more general dynamicsdescribed by the quasi-geostrophic approxima-tion, obtained from the full equations of motionin the case of stably-stratified, rapidly-rotatingflows that take place in a thin fluid layer. Here,thin means that the horizontal scale of motion,L, is much larger than the vertical scale, H , ashappens for large-scale flows in the ocean andthe atmosphere. Rapidly-rotating, on the otherhand, means that the Rossby number, U/fL, is(much) smaller than one. Here, U is a horizontalvelocity scale and f = 2Ωsinφ where Ω is theEarth rotation rate and φ is latitude (Pedlosky1987, Salmon 1998).

Rapidly-rotating flows have special char-acteristics, one of which is that they oftenself-organise into coherent vortices. This phe-nomenon reminds of a phase-separation mech-anism where the energy and the vorticity of thesystem concentrate in long-lived, localized vor-tical structures that occupy only a small frac-

1

tion of the fluid domain. The background tur-bulence between the vortices has much lowerenergy and vorticity levels, and it is feeded byvortex filamentation processes.

The coherent vortices that characterize thedynamics of rapidly-rotating flows have impor-tant effects on the transport and dispersion ofpassively advected Lagrangian particles, as dis-cussed by several authors in the last fifteenyears (see e.g. Provenzale 1999 for a review).In the case of reactive tracers, vortices do alsoplay an important role. A particularly interest-ing instance of the interaction between coher-ent vortices and reactive tracers is found in ma-rine ecosystems, where vortices affect planktondynamics in several ways. In this contributionwe shall focus on the interplay between vortexdynamics and plankton distributions, using theidealized model of two-dimensional turbulence.

2 VORTICES IN THE MARINE ECOSYS-TEM

Phytoplankton is a crucial component of themarine ecosystem, both for its role as the baseof the trophic web and for its contribution tothe global carbon cycle. At present, much at-tention is devoted to the interaction between theocean ecosystem and climate, as well as to theeffects of climate variability on plankton distri-butions in the ocean. For these reasons, quanti-tative modelling of biogeochemical cycles andof the interplay between the ocean circulationand plankton dynamics is an important area ofresearch (see, e.g., Sarmiento 1992, Gruber andSarmiento 2004).

Phytoplankton growth in the ocean is limitedby the availability of light and of dissolved min-eral nutrients, see e.g. the book of Mann andLazier (1996) for an introduction to biological-physical interactions in the ocean. The high ab-sorpion rate of light in water confines photosyn-thetic biological organisms to live in the surfaceeuphotic layer, that typically extends in the up-per 100 meters of the water column. This layeris usually depleted in nutrients, that are con-sumed by phytoplankton and converted into or-ganic matter (primary production). Deeper wa-ters, on the other hand, are abundant in nu-trients, due to gravitational sinking of detri-tus generated by the biological activity in theeuphotic layer. Nutrients are then supplied tothe euphotic layer by several processes, such

as winter convection, eddy pumping, diapycnalmixing, isopycnal mixing, and submesoscalefeatures in frontal dynamics. It is believed thatmixing (isopycnal or diapycnal) does not pro-vide by itself a significant contribution to theobserved vertical flux of nutrients (Siegel et al1999).

In the last two decades, some attention hasbeen payed to the effect of mesoscale ed-dies and sub-mesoscale features on the marineecosystem. An important issue concerns the po-tential effect of mesoscale vortices on primaryproductivity in the ocean (and, consequently,on the carbon cycle). The eddy pumping mech-anism (Falkowski et al 1991, McGillicuddyand Robinson 1997, McGillicuddy et al 1998,Siegel et al 1999) is based on the fact that isopy-cnals are lifted upwards, towards the surface,in cyclonic eddies (in geostrophic equilibrium,cyclones have low pressure and negative sealevel anomaly in their core). This mechanismcan thus bring up nutrients from the deeper wa-ters. In this view, cyclonic eddies act as nutrientpumps for the marine ecosystem. The fact thatthe nutrient supply is concentrated in small in-dividual regions with eddy sizes rather than ina large uniform region has been shown by Mar-tin et al (2002) to significantly affect numericalestimates of primary productivity in the ocean.

In the presence of a meridional density gradi-ent, however, raised isopycnals within cyclonesare not necessarily associated with upward ver-tical transport (Levy 2003). Moreover, poten-tial vorticity analysis (Williams and Follows2003, Levy 2003) suggests that the formation ofa cyclone is associated with vortex stretching,that leads to downwelling. Similarly, the resultsfrom the quasi-geostrophic ecosystem model ofSmith et al (1996), indicates that during springblooms peaks in phytoplankton production oc-cur in deep mixed layer regions where verticalturbulent mixing entrain nutrient-rich deeperwater.

The net effect of individual eddies on the ver-tical fluxes of nutrients is thus currently underdebate. Correspondingly, estimates of the meaneffect of mesoscale structures on primary pro-ductivity have large uncertainties. Smith et al(1996) found no significant effect of eddies onspatially averaged primary production in theirmodel, although the spatial distribution of pri-mary productivity was affected by the presence

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of the eddies. On the other hand, Oschlies andGarcon (1998) reported on eddy-induced en-hancement of primary productivity in a modelof the North Atlantic Ocean, and Siegel et al(1999) suggested that half of the nutrient sup-ply in the Sargasso Sea can be accounted for bycyclonic eddy pumping. These results have re-cently been questioned by Martin and Pondaven(2003), who report on smaller values for thevertical nitrate flux due to eddy pumping. Fi-nally, modulations of the mixed layer depth bymesoscale processes interfere with primary pro-duction by changing the exposure time of phy-toplankton to light (Smith et al 1996, Levy et al1999).

Submesoscale processes have also been sug-gested to be an important source of nutrientsin the surface layer. Mesoscale eddies are as-sociated with filaments and other submesoscalestructures characterized by large vorticity gra-dients and strong vertical velocity, which cangenerate a significant nutrient supply (Martinand Richards 2001, Levy, Klein and Treguier2001, Levy 2003). Similarly, vertical velocitiesat ocean fronts are much larger than those asso-ciated with eddy pumping, and they can playan important role in determining the nutrientsupply and the variability of plankton distribu-tions (Mahadevan and Archer 2000). For study-ing some of these submesoscale processes andthe role of convective events, integration of thenon-hydrostatic equations is required (Mahade-van et al 1996, Mahadevan and Archer 1998).

A different mechanism is related to the factthat eddies act as horizontal barriers to trans-port. In the presence of a front (e.g., the GulfStream), eddies that are born by pinching offa meander are an effective way of transportingwater across the front. The water carried withinthe eddy has biogeochemical properties that aredifferent from those of the surrounding waters,and it does not mix with the outer water for longtimes. In this view, eddies are horizontal carri-ers, independent of their effect on vertical nu-trient transport. The horizontal velocity field in-duced by the eddies has also been suggested toplay an important role in determining the spatialdistribution of phyto- and zooplankton (Abra-ham 1998, Mahadevan and Campbell 2002).

In the remainder of this contribution, weshall consider some of the issues related tothe interplay between vortex dynamics and ma-

rine ecosystem functioning, using the simpli-fied model of two-dimensional turbulence. Inthe next section we introduce the equations oftwo-dimensional turbulence and discuss someof the effects of coherent vortices on the dynam-ics of passively advected tracers. In section 4 weintroduce the equations for the dynamics of a re-active tracer such as plankton, and we discusssome simple ecosystem models. In section 5we introduce the semi-Lagrangian method thatwe use to integrate the dynamics of the reac-tive tracers, and we briefly reconsider the prob-lem of plankton patchiness addressed by Abra-ham (1998). In section 6 we consider the prob-lem of how vortices affect primary productivityin the ocean, extending the work of Martin etal (2002). Finally, in section 7 we discuss therole that vortices play in segregating differentspecies of competing phytoplankton, allowingfor the survival of the temporarily unfavouredspecies (Bracco et al 2000a).

3 PASSIVE TRACERSIN VORTEX-DOMINATED FLOWS

Coherent vortices induce specific signatures onparticle transport in geophysical flows (Elh-maidi et al 1993, Weiss et al 1998, Proven-zale 1999, Bracco et al 2000b,c, Pasquero etal 2001). In the rest of this contribution weshall use two-dimensional (2D) turbulence asa model system for the dynamics of vortex-dominated, stably-stratified geophysical flows.This model is by no means perfect, complete,and perhaps not even satisfactory, but it doesshare some of the properties of rotating geo-physical turbulence: an inverse cascade of en-ergy, and the formation of long-lived vortices.A better model is based on the continuously-stratified quasigeostrophic equations, which in-clude the basic phenomenon of baroclinic insta-bility (which is absent in 2D turbulence). An-other interesting model is based on the primi-tive equations, and a still better one comes fromconsidering the full 3D, stratified, convectiveequations.

For the sake of simplicity, however, in thiscontribution we stay with 2D turbulence, as de-scribed by the model equation

∂∇2ψ

∂t+[

ψ,∇2ψ]

= D+ F (1)

where ψ is the streamfunction and ∇2ψ is the

3

vertical component of relative vorticity. Thesymbol [·, ·] is the two-dimensional horizontalJacobian, [ψ,∇2ψ] = ∂xψ∂y∇

2ψ− ∂x∇2ψ∂yψ.

The horizontal fluid velocity, u = (u, v), isgiven by u = −∂yψ and v = ∂xψ, and D and Fare dissipation and forcing terms, respectively.The motion takes place on a horizontal planewith coordinates (x, y), there is no dependenceon the vertical coordinate, z, and the horizon-tal velocity field is non-divergent. Numericalsimulations of this model system (McWilliams1984,1990, Bracco et al 2000d), and controlledlaboratory experiments (Hopfinger and van Hei-jst 1993, Longhetto et al 2002), show that an ini-tially random vorticity field self-organizes intoa collection of vortices that live for times thatare much longer than the typical eddy turnovertime of the flow.

The velocity fields described by eq.(1) can beused to advect a passive tracer field, C(x, y, t),and/or an ensemble of passive point-like La-grangian particles that represent fluid elements.The dynamics of a passive tracer field is de-scribed by the (Eulerian) advection-diffusionequation,

∂C

∂t+ [ψ,C] = DC (2)

where DC is a diffusion term. The dynamics ofpassively advected Lagrangian particles is de-scribed by the (Lagrangian) equations of mo-tion

dX

dt= −

(

∂ψ

∂y

)

(X,Y )

,dY

dt=

(

∂ψ

∂x

)

(X,Y )

(3)where (X(t), Y (t)) is the particle position attime t. Formally, eq.(3) represent a (non au-tonomous) one-degree-of-freedom Hamiltoniansystem.

The study of particle motion in vortex-dominated 2D turbulent flows has shown thatvortices trap particles for very long times, andinhibit particle exchanges between the inside ofthe vortex and the surrounding background tur-bulence (e.g., Elhmaidi et al 1993). This hasimportant effects on the dynamics of advectedtracers, such as the lack of dilution of a pol-luting tracer deployed in a vortex, and its bulktransport over long distances by the vortex mo-tion. The limited mixing between the inside andthe outside of coherent vortices has interesting

conceptual implications, such as the fact thatergodicity of Lagrangian statistics is achievedonly on times that are longer than the vor-tex lifetime (see Weiss and McWilliams 1991,Weiss et al 1998 for the case of point-vortexsystems).

Coherent vortices have important effects onLagrangian statistics as well. In particular, 2Dvortices exert their influence over long dis-tances, inducing non-Gaussian velocity distri-butions in the background turbulence (Braccoet al 2000b,c, 2003). The presence of coherentvortices makes particle dispersion more com-plicated, and the standard description based onthe Langevin equation (or Ornstein-Uhlenbeckprocess) is no longer sufficient (Pasquero et al2001). Novel parameterizations of dispersionin vortex-dominated flows have been devel-oped, see e.g. Pasquero et al (2001), Reynolds(2002). This is an active area of research, wherecomplete agreement on the best strategy to beadopted has not been reached yet.

In the rest of this contribution we shall fo-cus on the dynamics of reactive tracers, that is,tracers that react between themselves withoutaffecting the advecting flow. For these tracers,the standard advection-diffusion (2) becomesan advection-reaction-diffusion equation. Re-cently, reactive tracers have sometimes beencalled, with a slight abuse of terminology, ac-tive tracers. In fact, we believe that it is better toreserve the term “active” for tracers that do feedback on the advecting flow, such as temperaturein turbulent convection. Our reactive tracers arestill passive from the point of view of the ad-vecting flow.

The work that we discuss below is numer-ical, and it is based on the integration of themodel equations (1) complemented by appro-priate equations for the reactive tracers. The di-mensionless Eulerian equations of motion (1)are integrated on the doubly-periodic domain(2π,2π) by a pseudo-spectral method and athird-order Adams-Bashforth time integration.Dissipation is provided by the sum of a hyper-viscosity term, Dp = (−1)p−1νp∇

2p∇2ψ (New-tonian viscosity corresponds to p = 1), and bya hypoviscosity term Dq = (−1)q−1∇−2q∇2ψ(Eckman friction corresponds to q = 0). Forc-ing is obtained by keeping the power spectrumfixed at a given wavenumber kF , and allowingthe Fourier phase at kF to evolve dynamically.

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In this way, a statistically stationary state can beachieved. To obtain a large Reynolds number,we fix forcing and dissipation to be as small aspossible, consistent with the numerical resolu-tion. This choice of the forcing allows for vortexformation, but it breaks spatial correlations onscales larger than about 1/kF . As a result, co-herent vortices have sizes in the range betweenthe dissipation scale and the forcing scale, butno larger than the latter.

4 DYNAMICS OF REACTIVE TRACERSA typical advection-reaction-diffusion equationfor a single reactive tracer, P (x, y, t), is writtenas

∂P

∂t+ [ψ,P ] = f(P ) + κP∇

2P (4)

where f(P ) is the reaction term and we haveused a Fickian diffusion term, κP∇

2P . Thistype of equation has been studied intensively inthe absence of an advection term; in this case,we speak of a reaction-diffusion equation.

In the context of marine ecosystem dynamics,the advected reactive tracers represent the abun-dance - or concentration - of selected speciesor aggregates of species, or of other fields ofbiological significance such as nutrients or de-tritus. For example, in the simple equation (4)the variable P can be taken to represent phy-toplankton abundance. Classic ecological ex-amples of reaction-diffusion equations are theKISS model, where f(P ) = αP , where α is apositive constant measuring the growth rate ofP (Kierstead and Slobodkin 1953), developedas a model for red tide outbreaks, and the Fisherequation, where f(P ) = αP (1− P/K) and Kis a positive constant measuring the carrying ca-pacity of the system (Fisher 1937). In the homo-geneous version of the KISS model, the popu-lation grows exponentially without bound (i.e.,it follows a Malthusian growth). The homoge-neous version of the Fisher equation is the cel-ebrated equation for logistic growth. Here, theabundance of the species under study saturatesat the carrying capacity of the system, P = K.

When more than one tracer is present, asystem of coupled advection-reaction-diffusionequations must be adopted. Several introduc-tions to predator-prey models exist, here we re-fer to the book by Kot (2001) for further details.For a predator-prey system, such as the one de-scribing the coupled dynamics of phytoplank-

ton, P , and zooplankton, Z, we can write

∂P

∂t+[ψ,P ] = αP (1−

P

K)−g(P )Z+κP∇

2P

(5)∂Z

∂t+ [ψ,Z] = −µZ + γg(P )Z + κZ∇

2Z .

(6)In the above equations, the term αP (1− P/K)models the logistic growth of the prey popula-tion, and the term −µZ the exponential mortal-ity of a predator population in the absence ofprey. The term g(P )Z models the interactionbetween predator and prey and γ is a positiveconstant that measures the conversion rate fromprey to predator biomass. The interaction termbetween prey and predator is assumed to be lin-ear in the abundance of predators (more preda-tors, more prey is collected), while the func-tion g(P ) models the functional response of theprey. Three standard choices for g(P ) in genericpredator-prey systems have been described byHolling (1959a,b, 1965, 1966). The first two area mass-action term (Holling type I function),g = g1(P ) = δP where δ is a positive constant,and the so-called Holling type II function (alsocalled Michaelis-Menten or Monod kinetics inthe context of chemostat models, see e.g. Kot,2001)

g = g2(P ) = βP

kP + P(7)

where kP and β are other positive constants (inthe limit for P kP , g2 → g1 with δ = β/kP ).The homogeneous version of the predator-preyequations (5,6) becomes the Lotka-Volterramodel with logistic prey growth for the Hollingtype I functional response, and the Rosenzweig-MacArthur model for an Holling type-II formfor g. In general, a type-II functional responsemodels the fact that the predator cannot eatan infinite amount of prey, and that saturationoccurs in the predation efficiency (for a givenabundance of predators) when the amount ofprey grows.

A third type of functional response is the so-called Holling type III function,

g = g3(P ) =aεP 2

a+ εP 2, (8)

where a and ε are positive constants. This func-tional form is appropriate for predators that

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have a low foraging efficiency at low prey den-sity. There are of course many discussions onwhat is the best choice of the functional re-sponse in different conditions and for differ-ent species, which we skip here. Suffices to saythat the choice is not completely harmless: forexample, a predator-prey system with a typeIII functional response becomes analogous toan excitable medium with relaxation oscilla-tions (Truscott and Brindley 1994), so differentchoices of the functional form can lead to dif-ferent types of dynamical behavior.

5 A SEMI-LAGRANGIAN APPROACHTO MARINE ECOSYSTEM DYNAMICS

In general, advection-reaction-diffusion equa-tions for biogeochemical tracers are quite dif-ficult to solve, both analytically and numer-ically. The concentration fields described bythe equations are (and must stay) positive def-inite and display sharp gradients that need tobe maintained in the numerical integration. Inthe past, comparisons between different numer-ical advection schemes have shown that the nu-trient supply in the euphotic layer is signifi-cantly affected by the numerics used (Oeschliesand Garcon 1999, Levy, Estublier and Madec2001). New production sensitivity to the advec-tion schemes is comparable to uncertainties inthe estimation of biological parameters (Levy,Estublier and Madec 2001), and is thus a majorcontributor to the errors in the results of biogeo-chemical models. Numerical errors are usuallyidentified as non-physical diffusion, dispersion,and non-monotonicity related to the appearanceof under- and over-shoots in presence of sharpgradients.

Due to the difficulties with the Eulerian for-mulation, we adopt here a semi-Lagrangian ap-proach to the numerical integration of the bio-geochemical equations. This method is an ex-tension of that used by Abraham (1998) in astudy of phytoplankton patchiness (see Steele,1978, for an early discussion of spatial vari-ability of plankton distributions). The advect-ing fluid velocity comes from the Eulerian in-tegration of the momentum equations, such aseq. (1). However, one does not try to solve theadvection-reaction-diffusion equations in theirEulerian form, but rather computes the reac-tions in a large number of independent fluidparcels, whose motion is described by the La-

grangian equations (3). Each parcel represents agiven water volume (usually taken to have a sizecomparable with the grid spacing used to inte-grate the Eulerian momentum equations), and itis assumed to have homogeneous properties. (Inreality, water properties on scales smaller thanthe grid resolution are not homogeneous; to de-scribe the effects of inhomogeneities it is nec-essary to include a sub-grid scale parameteriza-tion such as a closure scheme for the turbulentdynamics and a specific representation of the re-active components of the system. This latter is-sue is largely unexplored).

Biogeochemical reactions occur within eachparcel, and do not depend on the concentra-tion of biogeochemical fields in the neighbour-ing particles. This allows for the formation ofsharp gradients, in case the system is prone tothis behavior. When a concentration field is re-quired, the distribution of Lagrangian particlesis gridded and/or interpolated onto a regulargrid and a tracer field is obtained. In principle,this method does not require any diffusion ofbiological tracers, although diffusion of the bio-logical components can be accounted for by in-troducing mixing among nearby water parcels:consider two water parcels, i and j, at distancerij from each other, with concentration of - say- phytoplankton Pi and Pj , respectively. Mixingis introduced by assuming that there is a fluxof the biological component towards the waterparcel with lower concentration. At each timestep in the integration, we use the mixing lawPmixed

i = Pi + h(rij)(Pj − Pi) for any couplei, j of water parcels that are closer to each otherthan a given threshold. The weight function thatwe use is h(r) ∝ exp(−r2/r2

0), where r0 mea-sures the scale of mixing.

Of course, the semi-Lagrangian method re-quires integrating the motion of a large num-ber of fluid parcels, and it works best when theadvecting flow is non-divergent. In this case,an initially homogeneous particle distributionremains so. As implemented here, the semi-Lagrangian approach requires that the biolog-ical components do not swim. These assump-tions are appropriate for plankton at mesoscaleand at large scale, where the horizontal size ofthe advected parcels is of the order of at least afew hundred meters. On these scales, both phy-toplankton and zooplankton are passively ad-vected on the horizontal. Vertical migration of

6

zooplankton is a different matter, and it canchange the picture drawn here. Inclusion of ver-tical motions of zooplankton is left to futurework.

To illustrate the method, we first reconsiderthe system studied by Abraham (1998), using2D turbulence as the advecting field (in the orig-inal work Abraham used a synthetic velocityfield generated by a random distribution of cir-cular eddies). We integrate equation (1) withforcing at the wavenumber kF = 10 and resolu-tion 5122 grid points. In dimensional units, thesquare simulation domain has linear size 256km and resolution 0.5 km. The turbulent field ischaracterized by an Eulerian decorrelation timeTE = 3.6 days and r.m.s. velocity fluctuationσ = 4 cm/s, values in the appropriate range formid-ocean conditions.

The biological model is indeed very crudeand it has three components: the carrying capac-ity of the system, K (which is herein assumedto vary in space and time), the phytoplanktonconcentration, P , and the zooplankton concen-tration, Z. The dimensionless equations that de-scribe ecosystem dynamics within a water vol-ume are:

dKdt

= −ω(K −K0)

dPdt

= P (1− PK

)− PZ

dZdt

= P (t− τ)Z(t− τ)− µZ2

(9)

where (X,Y ) is the particle position at time t,ω is the relaxation rate of the carrying capacityK to a reference carrying capacity K0, functionof space but not of time and determined by thedistribution of limiting factors such as nutrientsand light, τ is the time lag required by the zoo-plankton to grow and become an efficient preda-tor, and µ is the coefficient of (quadratic) zoo-plankton mortality. Time is in units of the phy-toplankton growth rate.

In the following, we compare results ob-tained by the semi-Lagrangian approach withthose provided by the integration of the Eule-rian advection-reaction-diffusion equations,

∂K

∂t+ u · ∇K = −ω(K −K0) + κK∇

2K

∂P

∂t+ u · ∇P = P (1−

P

K)− PZ + κP∇

2P

∂Z

∂t+u ·∇Z = P (t− τ)Z(t− τ)−µZ2 +κZ∇

2Z .

The reference carrying capacity is fixed asK0(x, y) = [1− cos y]/2, where the square two-dimensional domain has non-dimensional size2π. For the Eulerian numerical integration, thediffusion coefficients cannot be set to zero,and the tracer fields obtained from the integra-tion of the advection-reaction-diffusion equa-tions are smoother than those obtained with theLagrangian parcel method.

Let us concentrate on the carrying capacityfield, K, as the other components behave sim-ilarly. Figure 1 shows the two fields of carry-ing capacity, obtained respectively by the Eu-lerian and the Lagrangian methods. In the La-grangian method we used one particle for eachEulerian grid point. Of course, during the inte-gration Lagrangian particles move and do notstay on grid points, but their distribution re-mains homogeneous on scales larger than thegrid spacing. The fields produced by the Eu-lerian and the Lagrangian methods are verysimilar to each other, confirming that the La-grangian particle approach is a good alterna-tive to the integration of the Eulerian advection-reaction-diffusion equations. The power spec-tra of the fields produced by the two methods,shown in fig. 2, are very similar at large scales.At small scales, the spectrum obtained with theLagrangian method displays saturation at a con-stant value, due to the Poisson white noise gen-erated by the random fluctuations in the particlepositions at small scales. By contrast, dissipa-tion in the Eulerian integration forces the Eule-rian spectrum to decay rapidly at small scales.The total variance, however, is approximatelythe same in the two cases, as the contribution ofthe small scales is almost insignificant.

To summarize, in the semi-Lagrangianmethod the advection part is taken care of by thewater parcels that carry, during their motion, thebiogeochemical components. The reaction partis then easily integrated in terms of ordinarydifferential equations within each water parcel.In standard Eulerian biological models, mostof the integration time is spent in computingthe advective and diffusive fluxes (Oschlies andGarcon, 1999, say that this account for morethan 90% of the total integration time). In theLagrangian scheme, these fluxes are no longerneeded, and the heaviest part of the integration

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Figure 1: Snapshots of the carrying capacityfield, K, obtained from (a) integration of theadvection-reaction-diffusion equations; (b) thesemi-Lagrangian method. Units on the axes arein grid points.

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10 100

Figure 2: Power spectrum of the carrying capac-ity field, K, obtained from the semi-Lagrangianmethod (solid line) and from integration of theadvection-reaction-diffusion Eulerian equations(dashed line).

is the interpolation of the velocity field requiredto integrate the trajectories of the water parcels.(With our code, the Eulerian method is almost50% slower than the Lagrangian one. We usecubic spline interpolation for the Lagrangianmethod, and the finite-difference scheme dis-cussed by Smolarkiewicz and Margolin, 1998,for the Eulerian advection of biological tracers).

The integration of the biological reactions re-quires a negligible amount of time, and it canbe performed off-line. The semi-Lagrangianmethod is perfectly suited for integrating dif-ferent ecosystem models in the same advectingflow and for studying the effects of different for-mulations of the ecosystem dynamics.

6 COHERENT VORTICESAND PRIMARY PRODUCTIVITY

The occurrence of intense nutrient fluxes insmall regions associated with mesoscale vor-tices, rather than in large homogeneous areas,can significantly increase the amount of pri-mary production (Martin et al 2002). Here, wereconsider this issue, following a similar ap-proach, and discuss the differences that arisewhen the dynamics of nutrient supply to the sur-

8

face layer is modified. Instead of the integrationof the Eulerian reaction equations, as done byMartin et al (2002), here we adopt the semi-Lagrangian method discussed in section 5.

The plankton ecosystem model that is usedhere is a simplification of that adopted byMartin et al (2002). It includes three compo-nents, that represent nutrient, N , phytoplank-ton, P , and zooplankton, Z. The dynamics ofthe ecosystem is described by the reaction equa-tions

dNdt

= ΦN − β NkN +N

P+

+µN

(

(1− γ) aεP 2

a+εP 2Z + µPP + µZZ2)

dPdt

= β NkN+N

P − aεP 2

a+εP 2Z − µPP

dZdt

= γ aεP 2

a+εP 2Z − µZZ2 .

(10)The terms on the right hand side of the equa-

tion for the nutrient represent respectively ver-tical nitrate supply from deep water, conversionto organic matter through phytoplankton activ-ity, and regeneration of the dead organic mat-ter into nutrients. The phytoplankton dynam-ics is regulated by production, dependendingon available nutrients through a Holling type-II functional response, by a Holling type IIIgrazing by zooplankton, and by linear mortal-ity. Finally, zooplankton grows when phyto-plankton is present (γ is the assimilation effi-ciency of the zooplankton), and has a quadraticmortality term used to close the system andparameterize the effects of higher trophic lev-els. The specific form of the terms used inthis model is quite standard in marine ecosys-tem modelling (Oschlies and Garcon 1999).The term µN is smaller than one and it repre-sents the fact that not all biological substanceis immediately available as nutrient: the frac-tion (1− µN) is lost by sinking to deeper wa-ters. Note, also, that nutrient enters this modelby affecting the growth rate of phytoplankton.This is different from the formulation of Abra-ham (1998), discussed in the previous section,who modelled the spatio-temporal dynamics ofthe carrying capacity instead of the nutrient.(The carrying capacity determines the equilib-rium value of the phytoplankton abundance in-stead of the growth rate). In addition, no timelag for zooplankton growth is included in thecurrent model. Since the formulation adopted

here is two-dimensional in the horizontal and novertical structure of the fields is allowed, verti-cal upwelling has to be parameterized. The pa-rameter values used in the model are listed intable 1.

Let us now explore further the dynamics ofthe nutrient. Nutrient is brought up to the sur-face from deep water via (isopycnal and di-apycnal) turbulent mixing and upwelling. Whenthese processes are sufficiently intense, it is rea-sonable to assume that the surface water be-comes saturated in nutrients (with respect tothe nutrient content in deep water) and furthermixing does not change the concentration ofavailable nutrients. Only when nutrient is re-moved (by phytoplankton growth and/or hori-zontal dispersion), vertical mixing becomes ef-fective again. This situation can be representedby a relaxation flux, where the concentration ofnutrient at the surface relaxes to a value that de-pends on the nutrient content in the deep reser-voir. The nutrient supply can in this case bewritten in restoring form,

ΦN = −s(x, y) (N −N0) , (11)

where N0 is the (constant) nutrient content indeep waters and s is the (spatially varying) re-laxation rate of the nutrient, which is large inregions of strong vertical mixing and small inregions of weak vertical mixing. This form canalso be interpreted as the finite-difference ap-proximation to a vertical turbulent advectiveterm that acts between two layers with nutrientconcentration N and N0, and it is the standardformulation used for chemostat models whenthe reservoir has infinite capacity (Kot 2001).

A different situation arises when the verti-cal flux of nutrient is associated with an indi-vidual upwelling event. In this case the nutrientincome into the euphotic layer is independentof the actual concentration in the surface layer.Even though phytoplankton consumes nutrientat a rate that is faster than the upwelling rate, noextra supply can occur. Such a situation is betterrepresented by a fixed-flux supply of nutrient.The fixed-flux form for the nutrient supply is

ΦN = Φ0(x, y) . (12)

Note that vertical upwelling is allowed to varyin space, but it is held constant in time. In par-ticular, we shall consider instances where there

9

β = 0.66 day−1 kN = 0.5 mmol N m−3

ε = 1.0 (mmol N m−3)−2 day−1 µN = 0.2γ = 0.75 µP = 0.03 day−1

a = 2.0 day−1 µZ = 0.2 (mmol N m−3)−1 day−1

s = sp = 0.00648 day−1 in nutrient-poor regions N0 = 8.0 mmol N m−3

s = sa = 0.648 day−1 in nutrient-rich regions

Table 1: List of parameters used in the ecosystem model.

is a well-defined region of significant upwellingwhile the rest of the domain is considered asambient water with small vertical nutrient flux.

The choice between the two formulationsof nutrient supply depends on the intensity ofthe mixing between surface and deep waters,compared to the removal rate of nutrient fromthe surface layer. Whenever the time scale ofthe vertical mixing (due to turbulence, eddy-induced fluxes, or upwelling) is shorter than theremoval timescale, the restoring nutrient supply(11) is appropriate. In the opposite situation, thenutrient supply is limited by the intensity of theupwelling, and the fixed-flux condition (12) isa better choice. Presumably, both cases are en-countered in the ocean.

The model ecosystem equations are solved,using the semi-Lagrangian method, for eachwater parcel moving in a forced and dissipated,statistically stationary 2D turbulent field, as dis-cussed in sections 3 and 5. The turbulent fieldused here is forced at wavenumber k = 40 andhas a resolution of 5122 grid points. Assumingthat the forcing scale corresponds to the typi-cal size of an eddy, of about 25 km, then thedomain size becomes 1000 km in dimensionalunits and the resolution is about 2 km. The tur-bulent velocity field has mean eddy turnovertime TE = 2.8 days.

In all the numerical simulations discussed inthis section, the turbulent velocity field is thesame. What changes from one simulation to an-other is the type of nutrient supply, and the spa-tial distribution of the regions where the nu-trient is supplied. The restoring-flux case (RF)and the fixed-flux case (FF) are each run in twodifferent situations, characterized by a differentspatial distribution of the region where strongupwelling takes place (which we call active re-

gion). While the total area of the active regionwith strong upwelling is kept constant at the12% of the domain area, the intense upwellingis respectively confined to a circular patch atthe center of the domain (CP), or to a numberof small patches correlated with eddy structures(EP). The two types of active regions are shownin figure 3. For the EP case, the figure is a snap-shot at a particular time, and the active regionsare dynamically moving and changing with theflow.

In the EP case, the intense nutrient flux isassumed to take place in vortex cores and inannular regions around the vortices. These ac-tive patches are defined using the value of theOkubo-Weiss parameter, Q = s2 − ω2 wheres2 is squared strain and ω2 is squared vortic-ity (Okubo 1970, Weiss 1991). This parame-ter assumes strongly negative values in vortexcores, where vorticity dominates, and stronglypositive values in the annular regions around thevortices, where vorticity gradients are large andstrain dominates. In the rest of the domain, theOkubo-Weiss parameter oscillates around zero.The nutrient flux is defined to take place in re-gions for which |Q|>Q0, where Q0 is a thresh-old fixed by the requirement that the total areacovered by the active regions is 12%± 0.5% atany time.

When there is no advecting velocity field andno diffusion, the system reduces to a large num-ber (5122) of independent, point-like (homo-geneous) ecosystem models described by thesystem of ordinary differential equations (10).Each of these systems is labelled by the (fixed)spatial position of the corresponding fluid par-cel and it is characterized by a specific valueof the nutrient flux. For the parameter valuesadopted here, each of these systems tends to

10

(a)

(b)

Figure 3: White regions indicate where the in-tense nutrient flux takes place. Panel (a) refersto the case with a single region of strong up-welling (CP), while panel (b) shows a snapshotof the case where the active regions are corre-lated with the dynamical structure of the flowand upwelling is found in and around eddies(EP). The total area of the regions with strongupwelling is approximately the same in the twocases.

a steady state, (N ∗, P ∗,Z∗), determined by thevalue of the nutrient input. Note that primaryproduction in the steady state is larger than thenutrient flux, as a consequence of the fact thatpart of the organic nitrogen content is regener-ated into nutrients (such as ammonium).

To perform the simulations, all fluid parcelsare initialized at their steady state solution. Tur-bulent advection is then turned on and the evo-lution of the system is followed for 300 modeldays. This allows for describing both transientbehavior and asymptotic states of the spatially-extended ecosystem.

We have chosen to calculate themean primary production, defined asPP = 〈βNP/(kn + N)〉, where the angu-lar brackets indicate average over the wholedomain, and the ratio between primary pro-duction and nutrient upwelling flux, PP/ΦN ,as an indicator of the efficiency of the biolog-ical model to convert inorganic into organicmatter. To distinguish the primary productionoriginated by the two different nutrient sources,a quantity called “new production” is usuallyintroduced. This is defined as the primaryproduction associated with the newly upwellednitrate. In a steady state, new productionand the flux of nitrate into the euphotic zonecoincide. However, the two quantities can berather different during transients. This suggeststhat the common practice of taking upwellednitrate as a proxy for new production, althoughreasonable when the fluxes are integrated overlong times, can give biased results when usedon short time scales.

The horizontal stirring induced by the tur-bulent velocity field displaces parcels of waterthat are rich in nutrient and planktonic life intoareas of limited upwelling, where the supplyis not sufficient to sustain the biological activ-ity at the level present in the parcel. Viceversa,parcels with poor nutrient content and limitedplanktonic abundance can be displaced into ac-tive regions where the newly available nitratecan stimulate a plankton bloom. This effect de-pends on the intensity of turbulent stirring. Theexchange rate of water parcels between activeand inactive regions depends on the flow char-acteristics and on the amount of parcels that areclose to regions with strong gradients in the bio-geochemical properties of water. ExperimentsCP and EP are designed to illustrate the effects

11

of a variation in the stirring rate on the bio-logical activity when the extension of the re-gion with strong biogeochemical gradients ischanged from small (CP) to large (EP), whilethe total upwelling area is kept constant.

In the restoring flux case, the enhanced stir-ring increases the mean flux from deep wa-ters, as seen in Fig. 4a. The enhanced fluxoriginates at active locations when a parcelof water that has low nutrient content is ad-vected over them. To see how this happens,consider two nearby parcels: one is in a re-gion with small nutrient upwelling and char-acterized by a steady-state nutrient concentra-tion N∗

p ; the other is in an active region andhas a steady-state nutrient concentration N ∗

a .In this configuration, the total nitrate flux as-sociated with these two parcels of water is(

sp(N0 −N∗

p ) + sa(N0 −N∗

a ))

, where sp andsa are the relaxation constants for the nutrient-poor and nutrient-rich regions, respectively.Suppose now that, due to advection, the twoparcels switch their position: the parcel withsmall nutrient content gets in a strong upwellingregion and viceversa. In this configuration, thevertical flux is

(

sp(N0 −N∗

a ) + sa(N0 −N∗

p ))

.The net variation of the nutrient flux betweenthe two configurations is (sa − sp)(N

∗

a −N∗

p ),which is proportional to sa − sp. This term ispositive as larger relaxation rates are found inactive regions with strong vertical mixing. Theenhanced nutrient flux is thus due to the asym-metry in the relaxation times between the ac-tive and inactive regions. Note, also, that theexchange rate of water parcels between the twotypes of region directly affects the increased nu-trient flux to the surface, determining larger val-ues of ΦN in the case EP-RF than in the caseCP-RF (Fig. 4a).

Primary productivity follows the increasednutrient flux, although with some delay due tothe time taken by the phytoplankton to grow.This results in a transient with small ratioPP/ΦN , as shown in Fig. 4c, before a statisti-cally steady state is reached. New production isat most as large as primary production (and gen-erally smaller). During the transient, new pro-duction can be as small as 65% of the newlyavailable nitrate. In the final state, the efficiencyof the biological system is about the same asin the no-advection case, indicating that the pri-

(a)

0.15

0.2

0.25

0.3

0 50 100 150 200 250

ΦN

(m

mol

N m

-3 d

-1)

t (d)

(b)

0.2

0.25

0.3

0.35

0 50 100 150 200 250

PP

(m

mol

N m

-3 d

-1)

t (d)

(c)

0.6

0.8

1

1.2

0 50 100 150 200 250

PP

/ Φ

N

t (d)

(d)

1.2

1.3

1.4

0 50 100 150 200 250

PP

/ Φ

N

t (d)

Figure 4: (a) Nitrate flux; (b) Primary produc-tivity; (c) Primary productivity per unit nitrateflux in the restoring nutrient case (RF); (d) Pri-mary productivity per unit nitrate flux in thefixed flux case (FF). Solid lines show the refer-ence case with no advection; dashed lines referto the case with horizontal turbulent advectionand nutrient flux concentrated in a single cir-cular region (CP); dotted lines refer to the casewith horizontal turbulent advection and nutrientflux distributed in small patches associated withthe eddies of the turbulent flow (EP). Time is indays.12

mary production has increased only as a re-sponse to the larger available nitrate, but the dy-namics of the biological system is almost un-changed (the small differences are related to thefact that the ecosystem dynamics is nonlinear).

The results are rather different in the case offixed flux, shown in Fig. 4d. Here, despite thefact that the upwelling rate is constant, primaryproduction during the transient interval can besignificantly increased by stirring, before the fi-nal stationary steady state is reached. Note thatthe duration of the transient depends on the stir-ring rate, and it is shorter when stirring is larger(EP case). This indicates that the system con-sumes more nutrient than in the steady state.The enhanced consumption of nutrient is notdue to newly available nitrate (which is constantin this simulation) but rather to regenerated am-monium. Weak nitrate supply is not sufficientto sustain the biological activity of a plankton-rich parcel of water originated in an active re-gion. The large amount of biomass, however,produces a considerable amount of regeneratednutrient as a result of mortality, and the effect oflow nutrient input is partially counterbalanced.This situation is typical of post-bloom condi-tions and it supports phytoplankton life for alonger period of time, despite the lack of up-welled nitrate. Viceversa, nutrient-poor wateradvected into an active region quickly activatesthe phytoplankton bloom. Here, the asymmetryis related to the different reaction times of theecological model to increased and reduced nu-trient concentration: a bloom starts quickly andit dies slowly. In strong stirring conditions, sev-eral water parcels are in this situation and theprimary production is particularly high (EP).Eventually, a statistically steady state is reachedand the biogeochemical fields become more ho-mogeneous. Homogenization is faster when thestirring rate is larger (EP), explaining the in-verse correlation between transient duration andstirring activity.

7 VORTICAL SHELTERSAs a last topic, we mention the possible role ofvortices in providing a shelter for less-favouredplankton species and allowing their survival forlonger times.

The Principle of Competitive Exclusion(Gause 1934, 1935, Hardin 1960) states that iftwo species are too similar, they cannot coex-

ist in equilibrium: whenever two species com-pete for the same resource, the most favouredwill survive and the less favoured will eventu-ally go locally extinct. By extension, at equilib-rium the number of species competing on thesame resources cannot be larger than the num-ber of resources. Phytoplankton, however, seemto escape this limitation, since a large numberof species that compete for the same resourcesis usually observed. This phenomenon, knownas the paradox of the plankton, was formulatedby Hutchinson about forty years ago (Hutchin-son 1961).

Many solutions to the paradox of the plank-ton have been proposed in the past, includingspatial heterogeneity in the physical or biolog-ical parameters, periodic and chaotic time de-pendence, incomplete mixing and externally-imposed spatial segregation of the compet-ing species, see e.g. the references reported inBracco et al (2000a). In particular, competitionis often avoided by partitioning space and/ortime: the unfavoured species may be segre-gated in a spatial environment forbidden to thestronger species, or the two species might per-form differently at different times (Van Gemer-den 1974). Mesoscale vortices can be one of thecauses of the spatial segregation of unfavouredand favoured competitors, and the shelteringeffect offered by the vortices can allow un-favoured competitors to survive for prolongedperiods of time (Bracco et al 2000a). Here, wepresent some results that confirm that vorticesslow down the decay of the unfavoured species.

In the simple model adopted here, twospecies, P1 and P2, compete with regards totheir efficiency at exploiting a common abi-otic resource, N , which is assumed to be con-tinuously pumped into the system. This modeldescribes the competition of two species in achemostat (Kot 2001), and it is written as

dNdt

= −s0(N −N0)−1ρ1

β1NP1

k1+N− 1

ρ2

β2NP2

k2+N

dP1

dt= β1NP1

k1+N− µ1P1

dP2

dt= β2NP2

k2+N− µ2P2 ,

(13)where we have used a Holling type II form fornutrient consumption, as in the previous para-graph. The growth rates, β1 and β2, the pumping(relaxation) rate, s0, the half-saturation values,

13

k1 and k2, the reference nutrient concentration,N0, the mortality rates, µ1 and µ2, and the yieldcoefficients, ρ1 and ρ2, are all positive con-stants. If, for simplicity (but with lack of gener-ality), we assume that the two species have thesame mortality rate and that the pumping rateis equal to the mortality rate, s0 = µ1 = µ2 =µ, we can define B = N/N0 + P1/(N0ρ1) +P2/(N0ρ2) and obtain that

dB

dt= µ(1−B) , (14)

which has solution

B = 1 + [B(t = 0)− 1] exp(−µt) . (15)

Asymptotically (with respect to the pumpingtime scale), we thus have B = 1 and we can ex-press N as a function of P1 and P2. We redefineP ′

1 = P1/(N0ρ1), P ′

2 = P2/(N0ρ2) and obtain,omitting the primes for P1 and P2 (Kot 2001):

dP1

dt= β1(1−P1−P2)P1

k1+1−P1−P2

− µP1

dP2

dt= β2(1−P1−P2)P2

k2+1−P1−P2

− µP2 .

(16)

We then fix k1 = k2, such that the two speciesdiffer only in their maximum growth rate. Ho-mogeneous solutions of this system lead to thesurvival of the species with larger growth rate,and disappearance of the other species (Kot2001).

We next allow for spatial heterogeneity. Wesuppose that the two populations are initiallyseparate and occupy two distinct portions of asquare periodic domain: at time t = 0 all fluidparcels in the region 0 < y < L/2 are occupiedby species 1 while parcels in L/2 < y < L areoccupied entirely by species 2. The fluid ele-ments are then advected by a 2D turbulent ve-locity field, while the plankton species react ac-cording to system (16) and diffuse according tothe gaussian mixing described in the section 5.For this simulation, we fix the rate of mixing at1.5 km/day.

To illustrate the effect of the coherent vor-tices on plankton dynamics, we compare the re-sults of 2D turbulence with a case where wa-ter parcels are displaced according to a stochas-tic Ornstein-Uhlenbeck process (Griffa 1996).This latter represents an unstructured turbu-lent flow with the same kinetic energy and La-grangian decorrelation time of the 2D turbulent

x (Km)

y (K

m)

0 128 256

256

128

0

(a)

x (Km)

y (K

m)

0 128 256

256

128

0

(b)

Figure 5: Snapshot at t = 5 months of the rela-tive concentration of the two planktonic species,P1 and P2, when advection is described by theOrnstein-Uhlenbeck process (panel a) and by2D turbulence (panel b). Black indicates thepresence of the unfavoured species P1 only, andwhite indicates the presence of the favouredspecies P2 only. Parameters of the ecosystemmodel are k1 = k2 = 0.1, µ = 0.04 day−1, β1 =0.1 day−1, β2 = 0.2 day−1. The statistically sta-tionary 2D turbulent field is forced at wavenum-ber kF = 10 and has resolution 5122 grid points.In dimensional coordinates, the simulation usedhere has eddy turnover time TE = 7.2 days,r.m.s. velocity σ = 8 cm/s, and Lagrangiandecorrelation time TL = 25 days.

14

0.01

0.1

0 5 10 15 20

P1

t (month)

Figure 6: Time evolution of the average con-centration of the unfavoured species, P1, in thecase of vortex-dominated 2D turbulence (solidline) and for the Ornstein-Uhlenbeck process(dashed line). Same parameter values as in fig-ure 5.

field. In this case, water parcels are advected byintegrating the stochastic differential equation

dX = Udt

dU = − 1TL

Udt+ σ

T1/2

L

dW

where X = (X,Y ), U = (U,V ) is the La-grangian velocity of the parcel, TL is the La-grangian decorrelation time and σ2 is the veloc-ity variance. The term dW is a random incre-ment, independently extracted from a Gaussiandistribution with zero mean and variance 2 dt.

Snapshots of the relative concentration of theplanktonic species in the two cases, shown inFigure 5, reveal that in vortex-dominated turbu-lent flows strong gradients and almost undilutedconcentrations of different planktonic speciescan survive for long times, while in the unstruc-tured stochastic flow mixing favours the dom-inance of the stronger species. At later times(not shown), for the turbulent case the speciesP1 survives only in the vortex cores, and eventu-ally disappears on time scales long enough to al-low complete mixing of the water (i.e., on timescales longer than the eddy life time). The timeevolution of the average value of the concentra-tion of the unfavoured species, reported in Fig-ure 6, provides a quantitative confirmation ofthe sheltering effects of the vortices. The edges

of the vortices act as transport barriers and limitthe exchanges between the water inside and out-side the coherent structures, permitting the sur-vival of the unfavoured species.

8 CONCLUSIONIn this contribution we have shown how vorticesaffect the dynamics of reactive tracers, and wehave considered the specific example of plank-ton dynamics in the marine ecosystem. We haveintroduced a semi-Lagrangian method that inte-grates the biogeochemical equations in individ-ual fluid parcels advected by an (Eulerian) ve-locity field. Using this approach, we have dis-cussed the interplay of coherent vortices andprimary productivity in the ocean, followingand extending the results of Martin et al (2002),and the sheltering that vortices can offer to tem-porarily unfavoured planktonic species, follow-ing and extending the results of Bracco et al(2000a).

The results discussed here are based onthe adoption of 2D turbulence as a model formesoscale turbulence in the ocean. Although2D turbulence is nothing more than a metaphor,and the ecosystem models discussed hereare extremely simplified, we believe that themain messages provided by the explorationsreported here survive also when more realisticformulations are employed.

We are grateful to the organisers of thesymposium on Shallow Flows for their patienceas well as for their inflexibility in requiring awritten contribution. AP acknowledges Mariafor help with English and computers. This workwas supported by the EU Network on “Stirringand Mixing.”

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