A Theoretical Model of Pattern Formation in Coral Reefs · continuum models and thus have largely...

A Theoretical Model of PatternFormation in Coral Reefs

Susannah Mistr and David Bercovici*

Department of Geology and Geophysics, School of Ocean and Earth Science and Technology, University of Hawaii,Honolulu, Hawaii 96822, USA

ABSTRACTWe present a mathematical model of the growth ofcoral subject to unidirectional ocean currents usingconcepts of porous-media flow and nonlinear dy-namics in chemical systems. Linear stability analysisof the system of equations predicts that the growthof solid (coral) structures will be aligned perpendic-ular to flow, propagating against flow direction.Length scales of spacing between structures are se-lected based on chemical reaction and flow rates. Inthe fully nonlinear system, autocatalysis in thechemical reaction accelerates growth. Numerical

analysis reveals that the nonlinear growth createssharp fronts of high solid fraction that, as predictedby the linear stability, advance against the predom-inating flow direction. The findings of regularlyspaced growth areas oriented perpendicular to floware qualitatively supported on both a colonial and aregional reef scale.

Key words: coral reefs; pattern formation; self-organization; colonial organisms.

INTRODUCTION

Coral reefs are composed of small, calcifying coralpolyps that together build complex architectures.That these architectures can be different for colo-nies of the same species in different environmentalconditions suggests that quantifiable, nongeneticchemical and physical controls are important in theorganization of the system. Although the complexinteractions controlling the chemistry of coral cal-cificaltion are difficult to model, it is possible tomodel some aspects of growth using fundamentalprinciples of physics and chemistry. Important fac-tors in the growth of coral include light, whichinfluences the rate of calcification, and flow, whichinfluences the delivery and uptake of the nutrientsessential for growth.

Herein we consider the simplest examples of

coral colonies and reef systems subject to one-di-mensional nutrient-transporting water currentsand construct a theoretical model of coral growth.We simplify the growth process to a hypotheticalchemical reaction between a limiting nutrient sup-plied by the water column and solid material man-ifest as a porous skeleton. The chemical reaction,represented mathematically using the law of massaction, produces additional solid matrix. The self-enhancing growth of the solid matrix is said to beautocatalytic and reflects the need for an existingorganism in order to sustain growth. The density ofsolid matrix influences the flow of the nutrient-bearing current by changing the constriction and/ortortuosity of fluid pathways or by inducing subcriti-cal turbulence by means of enhanced surfaceroughness. Overall, we assume that an increase inmatrix density impedes flow, creating a nonlinearfeedback mechanism between matrix growth andnutrient supply. To consider how natural patternsmay arise, we consider the behavior of perturba-tions in an initially uniform model system.

Alan Turing was one of the first to describe atheory for the onset of biological form and to con-sider theoretically the dynamics of spatially varying

Received 1 October 2001; accepted 22 May 2002.*Corresponding author’s current address: Department of Geology and Geo-physics, Yale University, PO Box 208109, New Haven, Connecticut06520-8109, USA; email: [email protected] address for S. Mistr: College of Medicine, Medical University ofSouth Carolina, 96 Jonathan Lucas Street, PO Box 250617, Charleston,South Carolina 29425, USA

Ecosystems (2003) 6: 61–74DOI: 10.1007/s10021-002-0199-0 ECOSYSTEMS

© 2003 Springer-Verlag

61

nonlinear chemical systems (Turing 1952). The keypoint in Turing’s work relevant to many aspects ofpattern formation is that asymmetry could be am-plified by means of competing chemical reactions,and, as mitigated by diffusion, the asymmetry couldbe spatially distributed. By considering autocata-lytic, or self-enhancing, chemical species Turing wasable to show how steep gradients could result from asmall perturbation to uniform concentrations.

Patterns in coral reefs range from the small-scalearchitecture of a particular colony to large-scale reefdistribution patterns. Because of the many complexfactors that can contribute to the development ofthese patterns, examples for theoretical study aretaken from environments where one particular en-vironmental parameter appears to play a distinctlysignificant role. In particular, the influence of oceancurrents appears to affect coral reef orientation overmany scales. On the colony scale, certain coral col-onies orient as a series of plates perpendicular to thepredominating direction of flow (Figure 1). On the

regional scale, reef islands orient in series (Figure 2)along the predominating direction of large-scalecurrents (Figure 3).

Well-developed coral reefs are thought to bepresent in otherwise low-productivity waters due tothe tight recycling of nutrients between symbioticzooxanthellae within the tissues of the coral organ-ism and the coral organisms themselves (Lewis1973). Classically, these photosynthetic zooxan-thellae provide oxygen and food for the coral or-ganism through photosynthesis, while the coral inturn provides essential nutrients to the zooxanthel-lae through metabolic waste (Muscatine 1973). Cal-cification can be positively correlated with light ex-posure, indeed calcification rates can be about threetimes faster in light than in dark (Kawaguti andSakumoto 1948; Baker and Weber 1975a, b; seealso Gattuso and others 1999). It is therefore widelysuspected that the photosynthesis in zooxanthellaecontributes to increased rates of calcification. For abasic understanding of how photosynthesis mayenhance calcification, we consider the overall reac-tions of both.

For photosynthesis, carbon dioxide (CO2) andwater (H2O) are used to produce glucose and oxy-gen:

CO2 � H2OO¡sunlight

C6H12O6 � O2 (1)

For calcification, calcium and bicarbonate ions pre-

Figure 1. Aggregations of common reef coral Agariciatenuifolia, Carrie Bow Cay, Belize. The colony consists ofa series of upright plates oriented perpendicular to flow.The sand channel is parallel to the flow direction (afterHelmuth and others 1997).

Figure 2. Bunker Reef group, Great Barrier Reef, Aus-tralia (after Jupp and others 1985).

62 S. Mistr and D. Bercovici

cipitate calcium carbonate and evolve carbon diox-ide (Ware and others 1991, from Stumm and Mor-gan 1981):

Ca�� 2HCO3� 3 CaCO3 � CO2 � H2O (2)

The calcification reaction produces carbon dioxide,which can be used in photosynthesis. This removalof carbon dioxide by photosynthesis provides amechanism by which the calcification reaction inEq. (2) can be driven toward the right; it has beenproposed that this mechanism explains enhancedcalcification (Goreau 1963). Although this is a via-ble mechanism, the details of calcium supply andbicarbonate usage are poorly constrained; there-fore, few conclusions can be drawn as to the signif-icance of these reactions alone in enabling organis-mal calcification.

For example, complicating the simple photosyn-thesis-driven calcification hypothesis, some speciesof coral possess an organic matrix on the outside ofcells lining the skeleton that contains calcium-bind-ing substances (Isa and Okazaki 1987). The organicmatrix provides a way to lower the energy for pre-cipitation and thus drive calcification (Borowitzka1987). Additionally, there is evidence that the in-corporation of calcium into the coral skeleton is a

cellularly controlled process that requires metabolicenergy for the pumping of materials into or out ofthe cell (see Gattuso and others 1999). Finally, theintroduction of a mineral inhibitor can hold calcifi-cation to 1% of normal incorporation of calciuminto the skeleton without affecting rates of photo-synthesis (see Gattuso and others 1999).

The combination of these observations suggeststhat photosynthesis-driven calcification involvesmore complex chemical schemes than Eqs. (1) and(2); however, since the aim of this paper is to con-sider the relationship of growth and shape of coralcolonies in response to differing environmental fac-tors, it is not entirely necessary to understand theexplicit mechanism of calcification. What we mustconsider is that skeletogenesis is an active processthat occurs in the presence of the organism, whichmust be supplied with some form of food or energy.Because corals are known to feed on zooplanktonand remove food fragments from the water column(Yonge 1930; Porter 1976), we devise very simpli-fied “calcification” reactions where preexisting cal-cium carbonate “reacts” with some limiting nutrientto produce more calcium carbonate, representingcoral growth. The nutrient is supplied at the sea sur-face and diffuses or settles downward. We then con-sider the effect of currents circulating through the reefby imposing a unidirectional flow on the system.

An interesting characteristic of coral reef mor-phology is that there is often a strong pattern ofvertical zonation along the reef (see Jackson 1991).The absorption of light as it penetrates through thewater column and the change in flow regime aresuspected to contribute to the change of dominatingform with depth (Baker and Weber 1975a, b; Grausand Macintyre 1989). Although the difference indominating forms with depth is often related to achange in the dominating coral species and is thusmore of an evolutionary problem, there are in-stances where corals of the same species exhibitmorphological plasticity in accordance with flowregime. Using an example of Pocillopora damicor-nis, we see that as flow increases, morphologychanges from more space-filling knobby forms inhigh-flow regimes to more delicately branchingforms in low-flow regimes (Figure 4). Since thesame species can exhibit different growth formsunder different flow regimes, it is evident that theambient environmental conditions profoundly in-fluence morphology. The effect of water and nutri-ent flux in controlling biological pattern formationis obviously important in other environments aswell, such as vegetation in semiarid areas (HilleRis-Lambers and others 2001)

It is our goal to understand how the calcification

Figure 3. Southwest Pacific Circulation (after Maxwell1968). “X” denotes the location of the Bunker ReefGroup.

Theoretical Model of Coral Reefs 63

rate and the flow regime affect the growth of coralreefs. We hypothesize that colonies and reefs de-velop their complex structures as they obstruct flowand intercept available nutrients, either by en-hanced surface roughness or by increased complex-ity of the path for the flow field. Complex feedbacksamong turbulent flow, nutrient supply, and calcifi-cation likely operate to generate the elaboratethree-dimensional colony architectures. Thesecomplex interactions are unwieldy in theoreticalcontinuum models and thus have largely been ap-proached using idealized lattice Boltzman or cellu-lar-automata models of aggregation to explain theincrease of space-filling forms with increased flowregime (Preece and Johnson 1993; Kaandorp andothers 1996).

Here we attempt to provide a simple, physicallybased model using basic carbonate chemistry inconjunction with a porous-flow model to describethe overall decrease in flow rate that accompaniesthe increase in coral cover that occurs as corals takenutrients from the water column. We representincreasing coral cover in the face of nutrient supplyby the autocatalytic construction of a porous matrixinfiltrated with fluid subject to a one-dimensionalpressure gradient. The nutrient, supplied from thesurface, diffuses through and is carried by the fluid.We arrive at the model by considering the conser-vation of mass for the fluid, dissolved nutrient, andthe solid. We then analyze the model to explore thedynamic interaction of nutrient supply, solid andnutrient diffusion rates, flow rates, and chemicalreaction rates and to find cases where small depar-tures from the uniform system are amplified andpatterning phenomena arise.

THEORETICAL MODEL

Although it is a simplification, considering that tis-sue, and thus growth, occurs only on the outersurface of the coral, we employ a porous-medium

formulation to model our skeletal framework. Mak-ing use of a porous matrix allows us to describe flowof fluid by Darcy’s law in porous materials. Aha-ronov and others (1995) used a porous-mediumformulation to address feedbacks between dissolu-tion of host matrix and fluid infiltration for under-standing melt-focusing beneath midocean ridges.Their model considered the interaction of dissolu-tion, changing porosity, and enhanced fluid infiltra-tion and revealed that larger channels develop atthe expense of smaller channels, thus providing amechanism for focusing melt beneath the ridge.The basic premise of their work is that changingporosity can affect flow velocity. This idea will beretained in our model; however, rather than con-sidering a positive feedback between infiltrationand dissolution, we examine the interaction of theautocatalytic growth of the matrix and the advec-tive supply of a limiting nutrient.

The model system is considered to take place in alayer of thickness H that rests atop the sea floor butabove which no coral grows (Figure 5). The coral reefmass is considered to be a porous, immobile, and solidmatrix that reacts chemically with dissolved nutrientto form more solid matrix. The nutrient itself is sup-plied by a uniform source from the overlying watercolumn and by flux of the nutrient-bearing water,forced to flow horizontally through the solid matrixby an imposed pressure gradient. The solid matrix hasmolar concentration � (that is, moles in a unit volumecontaining solid, fluid, and nutrient); the nutrient hasmolar concentration �. Both values are given in unitsof mol/m3 to facilitate the provisions of a stoichiomet-ric relation for their chemical reaction (see Appendix1). The pressure whose gradient drives flow is de-noted by P and has units of Pa. All three dependentvariables (�, �, and P) change in space and time andobey equations of mass conservation and fluid flow(Darcy’s law); however, these equations are verticallyaveraged in z so that they can be posed in terms ofhorizontal position (x, y) only, as well as time t (seeAppendix 1).

The growth, loss, and transport of the nutrient isdetermined by its advective flux with waterthrough the solid matrix, its diffusion through thewater, and the mass lost through chemical reactionwith the solid matrix; when all dependent and in-dependent variables are nondimensionalized by thenatural mass, length, time, and molar scales of thesystem (see Appendix 1), this relation is expressed as:

��

�t� �h � ��1 � ��2��hP � h

2� � �s � � � ��2

(3)

Figure 4. Three specimens of the coral Pocillopora dami-cornis from sites increasingly sheltered from water move-ment. Specimen a is most exposed to flow; specimen c ismost protected from flow. After Kaandorp and others1996 (from Veron and Pichan 1976).


where h is the horizontal gradient operator, �s isthe uniform supply of nutrient through the topsurface of the layer, and � is a chemical reactionparameter (that is, the dimensionless number thatcontains the reaction rate; see Appendix 1). Theterms on the left side of Eq. (3) are (from left toright) the growth rate of nutrient density and theadvective nutrient flux carried by Darcy flow that isitself forced by a pressure gradient hP through amatrix with permeability proportional to (1 � �)2

(such that when the solid density � � 1, the matrixis impermeable). The terms on the right side repre-sent horizontal diffusion of nutrient (h

2�), verticaldiffusion relative to the overlying source concentra-tion (�s � �, such that if � �s nutrient diffusesdown into the layer, and if � � �s it diffuses up outof the layer), and mass loss due to chemical reaction(term proportional to �). A coefficient of diffusivityfor � does not explicitly appear in Eq. (3) because itis absorbed into the nondimensionalizing time scale(see Appendix 1).

The equation for change of solid matrix concen-tration is controlled only by mass addition throughchemical reaction with nutrients and by lossthrough minor diffusion; such solid diffusion is usedto represent both slight dissolution of coral reefmass and—perhaps more importantly—erosionand damage near the perimeter of coral reef con-centrations, which are here represented as steepgradients in �. This leads to the following equation(again dimensionless):

��

�t� �h

2� � �� 2 (4)

where � is the solid matrix diffusivity (scaled by thenutrient diffusivity) and will be referred to as “thediffusion parameter” (see Appendix 1 for more de-tails). The terms in Eq. (4) show growth of solidconcentration (left side), diffusion both horizontallyand vertically (terms proportional to �), and massaddition from chemical reactions with the nutrient(term proportional to �).

Lastly, the mass conservation of water flowingthrough the matrix leads to an equation for pres-sure itself:

h2P �

2�h� � �hP

�1 � ��

1

�1 � ��2

��

�t(5)

where spatial adjustments in pressure (left side ofthe equation) are controlled by flow through con-strictions in the matrix—for example, a “nozzle”effect (first term on right side) and a growth of thematrix that would squeeze fluid out of pores (lastterm on the right side).

Eqs. (3), (4), and (5) are the dimensionless gov-erning equations that we use to analyze the stabilityof our autocatalytic, nutrient-limited porous ma-trix. The complete derivation of these equationsand details about nondimensional parameters andnondimensionalizing scales are discussed in Appen-dix 1.

ANALYSIS OF THEORETICAL MODEL

Stability AnalysisPhilosophy. One of the most powerful tools for

elucidating self-organization and pattern formationin natural systems is linear stability analysis. Mostsystems have a simple equilibrium state that is es-sentially uniform and patternless. However, thisequilibrium state might be unstable, so that distur-bances or perturbations would grow and force thesystem to a different state with some inherent pat-tern. A ball atop a hill is a simple zero-dimensionalexample: Undisturbed, the ball can be balanced andremains at rest; but with a sufficient disturbance(for example, wind, earthquake, a light kick), theball will roll off the hill and not stop until it reachessome other equilibrium state (for example, a valleydeep enough to capture it).

The classic paradigm of instability and patternformation in physics is that of thermal convection,whereby a fluid layer is uniformly heated along itsbase and cooled along its top. This system has astatic and patternless thermally conductive statewhereby heat is transported from the hot base to acooler surface by transmission of molecular vibra-tions, or phonons. But since fluid near the hot baseis less dense than fluid near the colder top, it isgravitationally unstable. Thus, perturbations or dis-turbances such as slight fluid motion can jostle thesystem and initiate overturn such that light fluidmoves to the top and heavy fluid moves to thebottom; with the continuous heat supply throughthe bottom and heat loss through the top, the mo-tion persists as convective circulation. The pertur-bation of a given length scale that can cause thelayer to turn over most efficiently, or for the leastamount of heat input, is generally the one that willgrow the fastest and dominate the pattern of con-vection (that is, it will determine the length scale ofthe convection cells).

Thus, in stability analysis, one seeks the charac-teristics (length scale and temporal behavior, suchas growth and propagation) of this dominant per-turbation. The formalism of stability analysis is wellexplained in various texts (see, for example, Chan-drasekhar 1961; Drazin and Reid 1989; Nicolis1995). Here we employ the principles of linear sta-


bility analysis to build a framework for interpretingpattern formation and self-organization in ourmodel coral reef system (see Appendix 2 for fulldetails).

Equilibrium States. Given the cubic nonlinearityof the system of governing equations (in particular,the reaction term proportional to � in Eqs. [3] and[4]), there are three uniform equilibrium states.One such state is the trivial state, wherein there isno solid mass; the two other states involve smallsolid-mass fraction and large solid-mass fraction,respectively. The trivial state is stable and does notlead to any pattern formation, implying that at leastsome minimal amount of initial solid is necessary toinitiate instability and pattern growth; hence, we donot consider this state further. Likewise, the highsolid fraction state is stable under almost all condi-tions and is also not very susceptible to patternformation, implying that little more solid structurecan be built if the solid fraction is near its maximumallowed value; we will therefore not consider thisstate either. (For discussion of these states, see Ap-pendix 2.)

The low solid fraction state is unstable to pertur-bations and leads to the growth of structures. Thisstate is denoted by (�0, �0, P0) where:

�0 �1

2 ��s

�� s

2

�2 �4

�� , �0 � �s � ��0 (6)

and the equilibrium pressure has a constant gradi-ent given dP0/dx � ��, which represents an im-posed background current velocity or flow rate.(Note that in Appendix 2, this equilibrium state isreferred to by �0

� to keep it distinct from the otherequilibrium states; here we drop the “�” super-script.)

Instability of Perturbations. We next address thequestion of perturbations to the equlibrium stategiven by Eq. (6) and consider how their properties(growth rate, spatial structure) are affected by themodel parameters. To maintain a manageable pa-rameter space, we examine only the effect of vary-ing the flow rate � and reaction rate �, and we holdsolid diffusion fixed at � � 10�3 (that is, muchsmaller than nutrient diffusion) and surface nutri-ent supply fixed at �s � 9 � 10�4 (much smallerthan the molar concentration of pure solid bywhich all molar concentrations have been nondi-mensionalized; see Appendix 1).

As shown in Appendix 2, the only perturbationsthat grow are completely uniform in the y (cross-current) direction and have nonuniform structurein the x (along-current) direction. We thereforeonly need to consider the x and t (time) dependence

of our system instead of the x, y, and t dependence.We thus assume that each perturbation to �0, �0,and P0 is a sinusoid in x, with wavenumber k (thatis, wavelength 2�/k) and growth rate s. The growthrate s can be complex; the real part Re(s) impliesgrowth (if positive) or decay (if negative), while theimaginary part Im(s) implies wavelike propagation.(As shown in Appendix 2, there are in fact twopossible growth rates for each k, although only oneallows growth of perturbations; this growth rate isreferred to in Appendix 2 as s�, but for simplicitywe have dropped the “�” subscript here; however,in all the figures, we still refer to s� for consistencywith the figures in the appendixes.)

If we consider an infinite number of perturba-tions with different wavelengths, the one that hasthe largest growth rate max(Re(s)) will dominate thepattern that evolves. The wavelength of this fastestgrowing perturbation is 2�/kmax, and its propaga-tion rate, or wavespeed, is:

cmax � �Im�s�max/kmax (7)

where Im(s)max is the imaginary part of s whose realpart is max(Re(s)).

Figure 6 shows the maximum growth ratemax(Re(s)) along with the corresponding wave-number kmax and wavespeed cmax as functions offlow velocity � and for three reaction rates � � 5,10, and 100. The maximum growth rate increasesmonotonically with both � and � (Figure 6a), al-though for the large reaction rate � � 100, thegrowth rate is nearly constant over �. Because � �100 corresponds to a very fast chemical reaction,there is little influence from current velocity sincethe reaction is nearly instantaneous for any flowrate. On the other hand, for slower reactions, such

Figure 5. Schematic of the model. Fluid infiltrates theporous matrix, and flow is maintained by a pressuregradient in one direction. Limiting nutrient is suppliedfrom the surface, diffuses through the fluid, and reactswith the matrix for the autocatalytic production of addi-tional matrix. The model assumes that the system is con-tinuous in x and y with finite depth z � H.


as � � 5, we see increased growth rates as the flowspeed increases, suggesting that the growth of solidstructures is more sensitive to nutrient supply fromthe current for slower chemical reaction rates. As �3 �, the growth rates for � � 5 approach those for� � 10 and 100 as the growth rate of the dominantperturbation becomes primarily controlled by thenutrient flux from the moving current.

This one-dimensional model also reveals that thesystem develops solid growth maxima at character-istic, finite wavelengths (2�/kmax) over a wide rangeof � and � (Figure 6b). For zero fluid flow, � � 0,spatial variability leads only to diffusive loss andthus decay of perturbations to the solid and nutri-ent fields; therefore, the wavelength or spacing forthe fastest-growing structure is infinite (kmax � 0)for all �—that is, growth is spatially uniform withno fluid motion. However, as the background flowspeed is increased (� � 0), nutrient parcels becomedepleted asymmetrically as upstream reaction af-fects downstream supply; thus, a finite wavelengthor spacing for fastest-growing patterns occurs. How-ever, the wavelength of the fastest-growing struc-tures at first decreases (kmax increases) with increas-ing flow �; then, at a critical �, it increases again(Figure 6b), which suggests competition betweeneffects. In particular, a decrease in wavelength withincreased flow permits sharper pressure gradientswith which to impede flow and more efficientlycapture nutrients; however, at too sharp a wave-length, the growth of structures is limited by diffu-

sion (as occurs in many classic problems of diffu-sion-limited aggregation [DLA]). Beyond thecritical �, diffusion destroys the advantage of havingsharp pressure gradients and thus the system’sdominant length scale becomes determined by therecovery length—that is, the distance traveled by anutrient-depleted parcel of fluid before it recoversits nutrient content via the surface supply �s; thisrecovery distance is obviously influenced by thefluid velocity, as demonstrated in Figure 6b. In thislatter regime (that is, for � greater than the criticalvalue), increased reaction rates (larger �) also causethe spacing of structures to grow (kmax decreases)because the fast reaction depletes the nutrientsmore thoroughly and rapidly, necessitating a longerrecovery.

Our model also shows that fastest-growing per-turbation also propagates into the incoming flow(cmax 0). By supplying more abundant nutrientsupstream, increasing � drives the system towardupstream growth—that is, the structure propagatestoward the upstream accumulation fronts (for ex-ample, the direction a surface travels while being, ineffect, spray-painted). As � affects the nutrient sup-ply, it affects the slowest reaction cases the most—that is, those with small �—since for fast reactions,nutrients stay more ubiquitously depleted. A con-stant propagation value is achieved at a critical flowrate �, since, above a certain flow rate, the nutrientsupply must still be affected by the rate at which itis supplied by nutrients from the surface.

The important features of this linear stabilityanalysis predict that perturbations to our nonlinearsystem will grow, and that the system will be char-acterized by the development of gradients in thesolid and nutrient fields that propagate upstream.

Nonlinear Analysis

For the nonlinear analysis, we build on the resultsfrom the linear stability analysis. We assume thatderivatives in y are zero for Eqs. (3), (4), and (5),thus effectively considering the evolution of theone-dimensional nonlinear system.

We integrate the nonlinear equations using finitedifferences. We take the fully explicit forward-timedifference for time derivatives and a space-centereddifference for second-order derivatives; for first-or-der spatial derivatives, which reflect advection, weuse upwind-differencing (Jain 1984; Press and oth-ers 1992). The Courant condition on the time-dif-ferencing is met by time-stepping at 10% of theCourant time-step (Press and others 1992). For theevaluation of the pressure field, we assume an in-stantaneous response to changed solid and nutrient

Figure 6. Maximum growth rate max(Re(s�)) of leaststable perturbation (a), corresponding wavenumber kmax

(where perturbation wavelength is 2�/k) (b), and corre-sponding propagation velocity cmax � �Im(s�)/k (c) asfunctions of imposed flow velocity (or pressure gradient),�, for three reaction rates, � � 5, � � 10, and � � 100.


fields. We solve for P at each grid point using atridiagonal matrix solver (Press and others 1992).We then use the adjusted pressure field in the nexttime-step integration of the nutrient and solidfields. Convergence tests and comparisons with lin-ear stability analytic solutions suggest that the divi-sion of the domain into 201 grid points providessufficiently precise solutions.

Because the system is open, we expect the nutri-ent and solid fields to obey Eqs. (3), (4), and (5) atthe boundaries. The first derivatives of pressure, P,at the endpoints are approximated by making use ofthe second-order Taylor expansions of the twopoints interior to the boundary, expanded aroundthe boundary itself. For the nutrient � and solid �,derivative expressions are approximated by the de-rivatives of the nearest interior point. Because wecannot specify � and � at the endpoints, we cannotmake use of the Taylor expansion about the bound-aries, as in the case of pressure.

Since there are an infinite number of nonlinearsolutions, here we seek only to extend the results ofthe linear stability analysis into the nonlinear re-gime. Thus, we consider small perturbations to the

uniform steady state, �0 (Eq. [6]), as in the linearstability analysis. We assume the steady state of thesystem is perturbed at 10% of the steady-statevalue, and we integrate the equations over time.We assume that the system selects the scale pre-dicted by the linear stability, and we perturb thesystem at that selected wavelength.

We show the results of the numerical integrationof the nonlinear system for the fastest reaction rate� and for four different values of the pressure gra-dient �, which is defined here, for our finite do-main, as � � (Pl � Pr)/X, where X is the length ofdomain in x and Pl, Pr are the presssures at the leftand right boundaries, respectively (in the cases con-sidered here, Pl � Pr). In these numerical calcula-tions, we consider how the nonlinearities in theautocatalysis affect the shape and propagation of aninfinitesimal perturbation. For � � 100, the fastestreaction parameter examined, we show the devel-opment of solid growth fronts for a range of fourflow parameters (Figure 7). These results showstrong asymmetry between the peaks and troughs,suggesting that the autocatalysis in the chemicalreaction acts to accelerate the growth of the solidmaxima. With respect to the linear system, growthis greatly accelerated (Figure 8).

Examining the behavior of the associated nutri-ent fields for � � 100, � � 50 permits a goodillustration of the system’s behavior (Figure 9).Here we find that the sharply growing solid frontsdeplete the nutrient field and that the downstreamrecovery of the nutrient field controls the spacing ofthe solid fronts. Also, we see that the fronts prop-

Figure 7. Solutions for �, the solid matrix, versus x forthe time integration of the nonlinear system for pertur-bations to the steady state for � � 100. Contours repre-sent different stages in the time evolution; the maximumamplitudes reflect the most recent time integration andare approximately evenly spaced in time. The indicatedpeak and trough values of �, �max, and �min are for thefirst and last contour, and �normalized � (� � �min)/(�max � �min). From top to bottom, solutions for slowflow (� � 4), moderate flow (� � 10), high flow (� � 50),very high flow (� � 100). Perturbations to the steadystate were at the wavelength of the least stable modepredicted by the linear stability analysis at 10% of thesteady-state value �0 � 0.0113; however, the magnitudeof the perturbation for the � � 4 case is 1% of �0.

Figure 8. Values of �, the solid matrix, versus time t forlinear stability solution for perturbations to �0 (dashedline) and for nonlinear solutions (solid line) after 2800time steps. Result is for perturbations to the steady statefor � � 100, � � 100.


agate upstream. For the � � 10 and � � 5 cases theresults are very similar, although growth is muchslower and the asymmetry is not as extensivelyexaggerated by the end of the time integrations(Mistr 1999).

DISCUSSION OF THE MODEL

The main result from the nonlinear analysis is thedevelopment of pronounced nutrient and solidfronts that propagate upstream. Sharp gradients inthe nutrient field develop as the solid depletes theavailable nutrient by reacting autocatalytically tocreate additional solid that can further deplete thenutrient. Growth of the solid due to the autocatal-ysis is greatly accelerated with respect to the linearsystem, and this rapid growth must far exceed dif-fusive loss to allow for the development of the solidfronts. These fronts are most pronounced for in-creased reactions rates and increased flow rates. Wecan conclude from this analysis that without thenonlinearities of the autocatalysis in the reactionchemistry, the development of sharp fronts is notpossible. The self-enhancing behavior of the solidacts to deplete local nutrient concentrationssharply, so that growth occurs most rapidly in theupstream direction, where nutrient supply is great-est. To some extent, the solid structures act as nu-trient filters; thus, it is not surprising that theyaccumulate and absorb nutrients faster on the sup-ply side of the structures. The areas of high solidconcentration act to impede flow, increasing thepotential for nutrient depletion and further growth.

Spacing of the fronts is invariably controlled by thecompetition among (a) the propensity for solidstructures to introduce sharp pressure gradientsthrough obstructions that impede flow and thusmore effectively capture nutrients; (b) the diffusionof sharp gradients in both the solid and (especiallyin the cases considered here) nutrient fields; and (c)both the decay and recovery length scales overwhich the nutrient content in a parcel of fluid is,while being swept downstream, either lost to reac-tions with the solid or replenished by surface sup-ply.

We find that the slowest-reacting systems are themost sensitive to changes in the imposed fluid flow(or pressure gradient) in terms of the growth ratesand selected wavelength for the growing structures(Figure 6a and b), suggesting that slow-growingcorals are most succeptible to environmental plas-ticity. Also, we expect this plasticity to be the mostevident over slow flow regimes. We further findthat for the fastest reactions, as well as the fastestflow rates, long wavelengths dominate (Figure 6b),suggesting that faster-growing systems might ex-hibit broader structures.

As we have mentioned, there are morphologiesof coral reefs that orient perpendicular to flow, withregular spacing of the growth plates parallel to flow(Figure 1). Our theory suggests that this type of reefis organizing to best intercept nutrient supply,whereas nonlinear growth permits the develop-ment of sharp growth fronts in the face of a one-dimensional flow. Additionally, the regular spacingof reef cays off the coast of Queensland parallel tothe main direction of the East Australian Currentleads us to anticipate a similar mechanism for reeforganization (Figure 2). Evidence of upstreampropagation in each of the colonial and large-scalecases is difficult to ascertain, although further in-vestigation of these areas would reveal clues. Ex-amining the paleohistory of Bunker Reef wouldhelp to establish the validity of the hypothesis ofnutrient-controlled growth patterns. Additionally,it would be enlightening to determine whether A.tenuifolia advances into the flow as it grows addi-tional plates. Because the nondimensional scalesinclude hypothetical reaction components, it is notinformative to consider the physical scaling of thespacing.

The model presented here is quite simplified, andthus many possibilities exist for further theoreticalmodeling of reef systems. As information comes tolight regarding the relationship of photosynthesis tocalcification, a simple model of the chemical dynamicsof calcification would be very appropriate.

Figure 9. Normalized solutions for (from top to bottom)�, the solid matrix, and �, the nutrient field, versus x forthe time integration of the nonlinear system for pertur-bations to the steady state for � � 100, � � 50. See Figure7 for information about contours and normalization.


Appendix 1. Derivation of the TheoreticalModel

Let us suppose a system wherein we have a porousmatrix of solid material, � in mol � m�3, infiltratedwith fluid containing dissolved nutrient � inmol � m�3. This dissolved nutrient undergoes a reac-tion with the solid material that generates addi-tional solid material (Figure 5). In other words, thesystem is subject to the autocatalytic production of�. Stated symbolically;

l� � m� ¡k

n� (A1)

with a rate constant, or activity coefficient, k, whereunits depend on the stoichiometric coefficients, l, m,and n, where n � m � l. The stoichiometric coeffi-cients specify the number of mol � m�3 that react andare produced. The law of mass action states that therate of a chemical reaction is proportional to theproduct of the concentrations of the reactants raisedto their stoichiometric coefficients (Bailyn 1994;Nicolis 1995). In an ideal system, this rate would begiven by the product of the number of moles per m3

of the reactants alone, which gives the statisticalprobability of the two species coming into contact;however, in reality, a rate constant, or activity co-efficient, such as k must be used as a proportionalityfactor to account for the gap between the statisticalprobability and the actuality of the reaction occur-ring (Nicolis 1995). To constrain the stoichiometriccoefficients for the hypothetical reaction, we pickedl � 1 for simplicity and investigated the stability ofa system for various m and found that the system isunstable, yielding solid growth only in cases wherem � 2 (Mistr 1999). We chose the lowest-orderunstable case for our model, given by:

� � 2� ¡k

3� (A2)

in which case k’s units are m6 mol�2 s�1. Thus,according to the law of mass action, for rate ofchange of fluid and solid material (with no flow orother sources of mass):

��

�t �reaction

� �k��2 (A3)

��

�t �reaction

� k��2 (A4)

These two expressions provide the chemical sourceand sink terms for our model. We must consider

chemical diffusion and advection of mobile speciesand therefore make the following fundamental as-sumptions. First, we consider that the dissolved spe-cies (nutrients) do not contribute to the volume ofthe fluid infiltrating the porous matrix. Second, thesolid is considered immobile (with respect to advec-tion) although subject to diffusive loss.

Loss of solid mass is accounted for solely by ef-fective diffusion of � representing, for example,disintegration and erosion near the perimeters ofcoral reef concentrations that are represented bysharp gradients in �; thus, its time rate of change is:

��

�t� D�

2� � k��2 (A5)

We relate the velocity of the fluid to the evolvingpressure gradient by Darcy’s law (Bear 1988):

q � �K0�1 � ��2

�P (A6)

where q is the Darcy velocity vector, � is fluidviscosity, � is the volume that one mole of puresolid occupies such that 1 � �� is the matrix poros-ity, and K0(1 � ��)2 is the porosity-dependent per-meability in which K0 is a reference permeability.Conservation of fluid mass (assuming the fluid isincompressible) prescribes that:

�

�t�1 � �� q � 0 (A7)

(see Mistr 1999; also Bercovici and others 2001 andreferences therein), which leads to:

2P �2�

1 � �� P �

��

K0�1 � ��2

��

�t(A8)

With an expression for the velocity of the fluid, wemay then consider the evolution of the nutrientfield. The time rate of change for the nutrient, �, iscontrolled by advection, diffusion, and reactive loss,leading to:

��

�t� � �q�� D�

2� � k��2 (A9)

Our governing equations are (A5), (A6), (A8), and(A9), which interrelate advection of our nutrient togrowth of the solid. We consider the system to becontinuous horizontally, in x and y, with finite ver-tical thickness, H. We supply nutrient to the systemby holding the nutrient density at the top surfaceconstant, � � �s at z � H. Holding the left entranceto the channel (x � 0) at a constant pressure, Pl,and maintaining the right exit at a constant lowerpressure, Pr, we force fluid in the x direction, al-


though the resulting flow with various obstructionsdue to heterogeneity in the matrix can be in anydirection. To simplify the system to a two-dimen-sional model, we integrate the equations over thechannel depth, z, assuming that vertical profiles ofthe solid and the nutrient are parabolic to first orderin order to match the top and bottom boundaryconditions. Thus, the nutrient concentration � isconstrained at the top by the surface nutrient sup-ply �s, and changes parabolically to zero gradient atthe bottom (that is, an effectively insulating bottomboundary through which neither flow nor particlediffusion can occur). The solid concentration � isheld to zero at the top and increases parabolically tozero gradient at the bottom (again, the bottom iseffectively insulating). From these boundary condi-tions we assume that:

�� x, y, z, t� � �s � �� x, y, t� � �s� f� z� (A10)

�� x, y, z, t� � �� x, y, t� f� z� (A11)

where:

f� z� �3

2 �1 �z2

H2� (A12)

and since 1H�

0Hf�z�dz � 1, then �� and �� are the

values of � and �, respectively, averaged over z.Substituting the above functions into Eqs. (A5),(A6), (A8), and (A9), integrating in z, and eliminat-ing q, we find:

��

�t� h � ��

K0�1 � �� 2

�hP�

� D�h2�� D�

3��s � ��

H2 � k�� 2 (A13)

��

�t� D�h

2�� D�

3��

H2 � k�� 2 (A14)

h2P �

2�h�� hP

�1 � ��

��

K0�1 � �� 2

��

�t(A15)

where h is the horizontal gradient. For nonlinearterms, we have assumed that 1

H�

0H fm(z)dz� 1

(where m is any integer), which is generally onlytrue for m � 3; however, more accurate integrationonly introduces factors that are O(1), which can beabsorbed into the system by slightly redefining con-stants such as K0, k, and �. Since the equations aremade nondimensional anyway (see below), thesefactors have no effect on the final governing equa-

tions and merely change the dimensional scales ofthe system by a few tens of percent.

These equations are nondimensionalized accord-ing to natural length, time, mass, and molar scales,

L �H

31/ 2 , T �H2

3D�, M �

�H5

35/ 2D�K0, N �

H3

�33/ 2 ,

respectively, resulting in (dropping the overbars forsimplicity):

��

�t� h � ��1 � ��2�hP � h

2� � �s � � � ��2

(A16)

��

�t� �h

2� � �� 2 (A17)

h2P �

2h� � hP

�1 � ��

1

�1 � ��2

��

�t(A18)

where:

� �D�

D�(A19)

� �kH2

3�2D�(A20)

The first parameter, �, is the ratio of diffusion coef-ficients of the solid to nutrient and will be referredto as “the diffusion parameter.” The second nondi-mensional parameter, which contains k, the rateconstant of the reaction, will be called “the reactionparameter,” �.

Eqs. (A16), (A17), and (A18) are the dimension-less governing equations that we use to analyze thestability of our autocatalytic, nutrient-limited po-rous matrix; these equations are reproduced in themain text as Eqs. (3)–(5).

Appendix 2. Linear Stability Analysis

Here we consider the growth of perturbations to aspatially constant system, characterized by uniformdistribution of solid and nutrient and an imposed,constant pressure gradient. For small perturbations,solutions take the form:

� � �0 � ��1 (A21)

� � �0 � ��1 (A22)

P � P0� x� � �P1 (A23)

where � 1. Substituting these solutions into thetwo-dimensional equations gives rise to the threesteady-state equations O(�0):


�s � �0 � ��0�02 (A24)

��0 � ��0�02 (A25)

d2P0

dx2 � 0 (A26)

the solutions of which are the trivial case:

�00 � 0, �0 � �s (A27)

and the two conjugate roots:

�0� �

1

2 ��s

�� s

2

�2 �4

�� (A28)

where for all solutions �0 � �s � ��0 and the0th-order pressure gradient is dP0/dx � ��, where �is constant.

With pressure specifications, we have introduceda fourth parameter, �, and thus fix two of thenondimensional parameters so as to maintain amanageable parameter space. We assume that thevariation in the diffusion coefficients is minimal,such that the ratio of solid to nutrient diffusion, �,remains constant. We fix � � 0.001, correspondingto nutrient diffusion that is 1000 times faster thansolid diffusion. Since the 0th-order porosity 1 � �0

must be between 0 and 1, then 0 � �0 � 1, thusdemanding that �s/� 1, as well as that �s

2�/(4�2) � 1, to ensure real �0. Therefore, we set �s �0.0009, which, recall, is nondimensionalized by themolar volume of the pure solid. Dimensionally, thisis effectively a surface concentration of around 200mol/m3, assuming solid density for the calcium car-bonate skeleton to be around 2.5 g/cm3. Settingconstant values for two of the parameters, �s and �,permits us to examine changes in the system as thebackground pressure gradient, �, and the reactionparameter, �, are varied.

The trivial case (A27) implies that there is nomatrix present and that the nutrient distributionmatches the surface supply. The two quadraticroots, (A28), represent steady states characterizedby either high (�0

�) or low (�0�) solid density. The

high solid-density steady state evolves when thereis not enough nutrient to react and support growthbeyond that which balances diffusive loss, whereasthe low solid-density state evolves since there is notenough solid to react and sustain growth beyondthat which is lost to diffusion.

The stability of each of the high- and low-densitymatrix steady states, as well as the trivial case, isexamined using equations O(ε1) from substitutionsof solutions (A21), (A22), and (A23) into the two-dimensional Eqs. (3), (4), and (5):

��1

�t� 2�0�1 � �0�P0 � �11

� �1 � �0�2P0 � �1 � �1 � �0�

2�02P1

� 2�1 � �1 � 2��0�0�1 � ��02�1 (A29)

��1

�t� �2�1 � ��1 � 2��0�0�1 � ��0

2�1 (A30)

2P1 �2P0 � �1

�1 � �0��

1

�1 � �0�2

��1

�t(A31)

We assume that solutions to the above equationstake the form:

�P1, �1, �1� � �Pk, �k, �k�eik�r�st (A32)

where s is the growth/decay rate, k � (kx, ky) is thehorizontal wavenumber vector, and r � (x, y) is thehorizontal position vector. Assuming solutions ofthis type implies that for each wavenumber pair (kx,ky) there is an associated growth or decay rate.Since wavenumber is related to wavelength, estab-lishing which wavenumber supports the fastestgrowth allows us to predict the spatial scale overwhich the solid field exhibits preferred growth.

Substituting perturbation solutions (A32) intoEqs. (A29), (A30), and (A31), we find the disper-sion relation:

s2 � s��1 � ��1 � k2� � ��0��0 � 2�0 � �0�0�

� ikx��1 � �0�2 � �1 � k2��1 � k2� � ��0

2

� 2��0�0 � ikx��1 � �0�2��1 � k2� � 2��0�0

� 0 (A33)

where k2 � kx2 � ky

2. Eq. (A33) has two complexroots, s� (where the “�” subscript indicates the signof the radical term in the quadratic formula solu-tion), only one of which, s�, allows growth.

We explore the growth rates Re(s�) versus k �(kx, ky) over a wide range of the nondimensionalparameters for the three steady states and find thatin all cases, the maximum growth rate max(Re(s�))occurs for ky � 0. As an example of this result, weshow the growth rate Re(s�) versus k � (kx, ky) forone reaction parameter � over a range of pressuregradients, � for the steady state �0

� (A28) (FigureA1). We select �0

� because growth is sustained forperturbations to this steady state, whereas growth issustained for neither the trivial case �0

0 nor for thehigh solid-mass concentration case �0

� over thewide range of parameters that were sampled. Herewe see that the fastest growth rate for low solid-mass concentration case �0

� occurs at kx � 0 and ky


� 0. Nonzero kx implies that there is a finite wave-length that characterizes the fastest growth in x,while ky � 0 implies that growth is uniform in y,perpendicular to the imposed pressure gradient.Thus, for further analysis of the dispersion relation,

we set ky � 0 and analyze the system for spatialdependence in x.

We show for the one-dimensional case thatgrowth is not sustained (max(Re(s)) 0) for thetrivial case, and growth is sustained only over very

Figure A1. Two-dimensionaldispersion relation for pertur-bations of the form ei(kxx�kyy)�st

to the equilibrium state �0

given by Eq. (6) with �s �0.0009, � � 0.001, � � 100.Surfaces show the growth rateof the perturbation, Re(s�)(where s� is the one of twopossible solutions for (a) thatallows growth) versus wave-numbers kx and ky for (a) � �5, (b) � � 10; (c) � � 50, (d)� � 100. Note the symmetry inkx and that the least stablemode (location of largestRe(s�)) always occurs at ky � 0.

Figure A2. Growth rateRe(s�) versus wavenumber kfor perturbations to the lowsolid-density quadratic root�0

� (solid curves), to the highsolid-density quadratic root,�0

� (dash–dot curves), and tothe trivial steady state �0

0

(dashed curves). Plots are forincreasing reaction parameter,� from left to right (� � 5,10, 100), and for increasingimposed pressure gradient, �,from top to bottom (� � 5,10, 100).


restricted ranges for the nutrient-limited steadystate �0

� (Figure A2). For the solid-limited steadystate, �0

�, however, growth is sustained over a widerange of parameters. This result does not contradictour findings for sampled parameter ranges for thetwo-dimensional case. We thus examine only thisconsistently unstable root, �0

�, in the main text.

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