Boudreau & McBirney 1997.pdf

JOURNAL OF PETROLOGY VOLUME 38 NUMBER 8 PAGES 1003–1020 1997

The Skaergaard Layered Series.Part III. Non-dynamic Layering

ALAN E. BOUDREAU1∗ AND ALEXANDER R. McBIRNEY2

1DEPT. OF GEOLOGY, DUKE UNIVERSITY, DURHAM, NC 27708-0227, USA2DEPT. OF GEOLOGICAL SCIENCES, UNIVERSITY OF OREGON, EUGENE, OR 97403, USA

RECEIVED MAY 6, 1996 REVISED TYPESCRIPT ACCEPTED MARCH 18, 1997

Layering in the Skaergaard Intrusion has been divided into two INTRODUCTIONgeneral types, one produced by magmatic flow and another by Recent studies of the structural and textural features ofprocesses resulting from variations of rates of nucleation and crys- the Skaergaard Intrusion (McBirney & Nicolas, 1997)tallization, and, in the case of the Layered Series, by compaction- have distinguished two broad types of layering, onerelated processes. Modal variations caused by shifts of cotectic produced by the dynamic effects of magmatic flow andproportions produce thick layers which, in the Layered and Upper another by processes that operate in situ such as variedBorder Series, are diffuse and normally lack strong foliation and nucleation and growth of crystals, recrystallization, or bylineation. In the Marginal Border Series, the layers are thinner and compaction-related mechanisms. We refer to layeringsharper; possibly because the rate of accumulation was slower. formed by these latter processes as ‘non-dynamic’ toOscillatory nucleation may have played a role in producing fine- emphasize that it is not the result of fluid dynamicscale cyclic layers, but it was less important than solution and processes. Although most layering combines elements ofreprecipitation during slow cooling and Ostwald ripening. Evidence more than one process, the contribution of each mech-for compaction is found in deformed plagioclase laths and a relative anism can usually be recognized from its distinctive formdeficiency of incompatible elements in rocks formed on the floor. and setting.Layering related to compaction becomes sharper with increasing The distinguishing features of dynamic magmatic lay-height in the Layered Series until it suddenly disappears above the ering have been described in Part II of this series (McBir-trough horizon near the base of Upper Zone b. Mechanical sorting ney & Nicolas, 1997). They are best seen near the steepduring compaction may have produced crude layering, but if it did margins of the Layered Series in what Wager & Brownthe evidence has long since been destroyed by the superimposed effects (1968) referred to as the ‘cross-bedded zone’ where theof solution and reprecipitation when interstitial liquid rose through layering is disrupted by slumping and channeling andthe overlying crystals and re-equilibrated with them. Numerical the rocks have a marked foliation and lineation.simulations illustrate how small differences of surface energy caused Layering not associated with magmatic flow takes aby variations of grain size, textural dependence of solubility, and variety of forms, but we can distinguish two generalpressure solution can cause segregation of minerals into layers during end-members, each of which has a distinct origin andsolution and reprecipitation. characteristics. The first results from varied rates of

nucleation and crystal growth; it can be seen throughoutthe intrusion. The second is associated with compactionand is confined to rocks formed on the floor. Althoughthese mechanisms are closely associated and tend toreinforce one another, each has its own distinctive char-acteristics.KEY WORDS: compaction; layering; metasomatism; pressure solution

∗Corresponding author. Telephone: (919) 684-5646. Fax: (919) 684-5833. e-mail: [email protected] Oxford University Press 1997

JOURNAL OF PETROLOGY VOLUME 38 NUMBER 8 AUGUST 1997

(Boudreau, 1987, 1994, 1995). Small initial differencesGENERAL FEATURES OF NON-of grain size or modal proportions are accentuated and

DYNAMIC LAYERING repeated by cyclical solution and crystal growth underLayering produced by variations of conditions similar to those of Ostwald ripening. None ofintensive parameters the examples we have found in the Skaergaard Intrusion

measures more than a few meters in vertical or horizontalThe basic mechanism Wager (1961) credited for well-extent, but in larger bodies, such as the Stillwater Com-layered rocks that formed on the steep walls was alsoplex, they are much more extensive (Boudreau, 1987;responsible for the indistinct layering in the interior ofNaslund & McBirney, 1996).the Layered Series and nearly all the layering in rocks

that crystallized under the roof. It is thought to bethe result of transitory excursions about the cotecticproportions of precipitating minerals (Harker, 1909;

Layering associated with compactionWager, 1959; Maaløe, 1978). These variations may haveAt the other extreme of non-conventional layering is abeen brought on by any of a variety of events includingmore conspicuous variety seen throughout most of theconvective overturn, invasions of new magmas, con-interior of the Layered Series. It consists of sharplytamination with country rocks, gain or loss of volatiles,defined mafic and felsic layers, a few centimeters orand any other factor affecting intensive parameters, suchdecimeters thick and separated by thicker intervals ofas the composition, temperature, or oxygen fugacity ofhomogeneous gabbro of widely differing thickness. Wethe magma (Hort et al., 1993; Naslund & McBirney, 1996).will not describe this layering in detail, for it has been[A mechanism based on double-diffusive convection wasthe subject of several detailed studies (Wager & Deer,thought to be an important effect of this kind (McBirney1939; Wager & Brown, 1968; McBirney & Noyes, 1979;& Noyes, 1979), but closer examination of its theoreticalIrvine, 1987; Conrad & Naslund, 1989). We attributebasis (McBirney, 1985) raised doubts as to its importancethis layering to the mineral segregation that accompaniesin natural magmas.]or is significantly enhanced by compaction. Although weIn the case of rocks formed on the floor or underdiscuss several mechanisms that can give rise to this typethe roof, these layers tend to be diffuse and, thoughof layering, its development is broadly analogous to thatconspicuous when viewed from a distance, may be farof metamorphic banding in the sense that it is the resultfrom apparent at close range. The most conspicuousof solution and reprecipitation of minerals in responseexample is the Triple Group (Fig. 1), a set of three felsicto the stresses of compaction.layers near the top of Middle Zone. From almost any

On a broad scale, these layers become sharper andpart of the intrusion they are seen extending for hundredsmore numerous with increasing height in the Layeredof meters across almost the entire width of the intrusion,Series until they suddenly disappear above the troughbut on an outcrop scale they are so indistinct that theyhorizon near the base of Upper Zone b. On a local scale,easily pass unnoticed.however, they are particularly well developed in theWe find no systematic spacing of these layers; theirvicinity of blocks that fell from the roof and disturbeddistribution seems totally random. Modal variations arethe mush of crystals on the floor. Because this sharp,normally gradational on a scale of tens or hundreds ofintermittent layering is normally more mafic at the basecentimeters, but in the Marginal Border Series individualand felsic at the top and has a superficial resemblancelayers are thinner and much sharper, possibly becauseto sedimentary deposits laid down by turbidity currents,the sequence accumulated more slowly and is relativelyit was once thought to be the result of crystal-ladencompressed. Similarly, grading from mafic to felsic min-magma descending from the walls and sweeping acrosserals is much more pronounced in the Marginal Borderthe floor. The objections to this explanation are nowSeries where the outer, wallward side of an individualwell known. No gaps, unconformities, or other evidencelayer is normally more mafic. Except in the specialhave been found to support the notion that sections ofcase of crescumulate textures produced by constitutionalthe walls supplied the large masses of crystals formingsupercooling close to the contact, fabrics are essentiallyextensive layers in interior parts of the floor. Similar layersisotropic.are equally well developed in many larger intrusions, suchAn unusual variety of this diffuse layering, found in aas the Bushveld and Stillwater Complexes, where thefew local parts of the Layered Series, is cyclical on awalls were less steep and turbidity currents could scarcelyscale of one or two centimeters. It is confined to smallhave had sufficient energy to flow for tens or hundredsareas, mainly in Upper Zone a (Fig. 2a), and is alsoof kilometers across the floor. Although they may haveobserved in a nearby rhyolitic dike (Fig. 2b). The rhythmica well-developed foliation (Brothers, 1964), the rocksspacing was first ascribed to oscillatory nucleation (McBir-rarely have a strong lineated fabric in which elongatedney & Noyes, 1979), but we now assign more importance

to competitive growth of crystals during slow cooling minerals have a preferred orientation within the plane

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BOUDREAU AND McBIRNEY NON-DYNAMIC LAYERING OF THE SKAERGAARD INTRUSION

Fig. 1. Triple Group at the top of Middle Zone is clearly visible from a distance (a) but is scarcely noticeable at close range (b). In places theplagioclase-rich units have fine-scale rhythmic layers.

of foliation or layering. This is not to say that currents Perhaps the most curious type of layering is seen in aset of trough-like structures near the boundary betweenmoving across the floor did not influence the orientation

of mineral grains, but that they did not deposit them by Upper Zones a and b. Wager & Deer (1939) originallyproposed that these features resulted from turbidityconventional sedimentary processes.

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Fig. 2. (a) Fine-scale cyclic layering near the top of Upper Zone a. Layering of this kind is restricted to areas of limited extent, mainly in theupper half of the Layered Series. (b) and (c) Fine concentric layers in a rhyolitic dike ~500 m beyond the eastern margin of the Skaergaard

Intrusion (McBirney et al., 1990).

currents sweeping across the floor, an explanation that STRUCTURES AND TEXTURESIrvine (1987) has elaborated in detail. We see several

PRODUCED BY COMPACTIONobjections to this explanation. The most obvious is theAlthough ‘filter-pressing’ has been considered a potentialtotal absence of any of the features normally associatedmechanism of differentiation since it was first proposedwith high-energy, sedimentary deposits. The troughs areby Bowen (1928), it is only in recent years that compactionconcordant synforms with sides that become steeperhas been given serious consideration. This neglect is dueupward in narrowing stacks of layers. The cross-bedding,mainly to early impressions that its effectiveness wouldscour and fill, and lateral migration of channels so typicalbe severely limited by the small density difference betweenof sedimentary deposits laid down by agrading dis-crystals and iron-rich interstitial liquids and by the in-tributory streams are conspicuously absent. Taylor &ferred low permeability resisting the force of this weakForester (1979) and Sonnenthal (1992) have commenteddensity contrast (Sparks et al., 1985; Morse, 1986). How-on isotopic and trace-element relations they find difficult

to reconcile with the turbidity current hypothesis. ever, these objections have not considered reactions

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between liquid and the crystals through which it rises,nor the growth of large crystals at the expense of smallones during crystal coarsening, both of which can aug-ment permeability manyfold. Numerical simulations(Sonnenthal, work in progress) show that, when thesefactors are taken into account, the process is effectiveeven in thick sills and ponded lavas. This conclusionfinds support in the facility with which liquids ooze intofractures in partly crystallized lavas and shallow sillswhere the pressure differences driving flow are very small.

Although rarely obvious in the field, the effects ofcompaction have been recognized in a growing numberof studies of ponded lava flows (Helz, 1987; Helz et al.,1989; Philpotts et al., 1996) and thick sills (Shirley, 1987).In the case of the Skaergaard Intrusion, it appears tohave been pervasive throughout most of the LayeredSeries, where it was a primary factor in the compositionalevolution of the magma (McBirney, 1995). The evidencefor this is both chemical and textural.

Fig. 3. Concentrations of Ba, a representative incompatible elementin the principal series of the Skaergaard Intrusion.

Geochemical effectsThe geometric configuration of the intrusion provides a

Petrographic and structural effectsconvenient way of identifying the relative importance ofOur co-worker, Robert Hunter, has drawn our attentionany petrologic mechanism driven by gravity. By com-to several significant petrographic features of igneousparing the compositions of rocks that crystallized sim-‘cumulates’ that have strongly influenced our view ofultaneously on the floor, walls, and under its roof, onehow these rocks form (Hunter, 1987; McBirney & Hunter,can relate their differences to the geometrical orientation1995). Planar fabrics that have long been taken as ain which they formed. On doing this, one sees thatnatural consequence of sedimentation and compactionthe roof series has consistently larger concentrations ofmust now be viewed with caution. As Higgins (1991)incompatible elements than equivalent units on the floorpointed out, foliation, in itself, is not a reliable criterionor walls (Fig. 3). Long thought to be the result of con-for mechanical rotation of plagioclase by compactiontamination, isotopic evidence shows that the greateralone. This conclusion has been reinforced by geo-abundance of lithophile elements, such as Ba, Rb, Zr,chemical evidence that the strong foliation to whichand Nd, in the Upper Border Series could not haveWager & Deer (1939) gave the name ‘igneous lamination’come from assimilated Archean gneiss, for the strontiumhas little if any correlation with the amount of interstitialisotopic ratios of these same rocks are, on average,liquid the rocks retained (McBirney & Hunter, 1995).even less radiogenic than those of the Layered SeriesQuantitative petrofabric and compositional analyses of(McBirney, 1995).rocks from the Stillwater Complex have shown thatThe relatively depleted character of rocks formed ondevelopment of these fabrics is a function not only ofthe floor is more logically attributed to a gravitationalcompaction but also of a number of other factors in-process, either compaction or convective fractionation,cluding interaction with exsolved fluids (Meurer & Bou-or possibly both. Density relations make convective frac-dreau, 1997).tionation less likely, at least during the early stages of

The origin of this strong foliation in the Skaergaard iscrystallization when the differentiating liquid increasedstill unclear, but we suspect that rotation of grains canin density and would have been heavier than the over-be aided by differential solution and reprecipitation oflying, less-differentiated magma. These same relationscrystals with different crystallographic orientations withprobably account for the relative depletion of the Mar-respect to the principal stresses. Two distinct mechanismsginal Border Series where heavy, differentiated liquidscontribute to the development of foliation during com-are thought to have flowed down the wall to pond onpaction, one mechanical rotation and the other selectivethe floor. At a later stage after iron enrichment passedpressure solution and recrystallization (Meurer & Bou-its peak and concentrations of silica and volatiles greatlydreau, 1997). In the first, crystals with an initially randomreduced the density of the residual liquid, extraction oforientation (Fig. 4a) are rotated toward a weak planarthese late liquids by convective fractionation would have

augmented the effects of compaction. orientation of their long axis (Fig. 4b). This effect is then

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plagioclase laths, that are unequivocal signs of anisotropicstrain. Unfortunately, evidence of this kind is far lesscommon in large intrusions that cooled slowly enoughto permit extensive recrystallization and annealing. In-dividual crystals are weakly deformed, if at all, whenthey grow in the presence of interstitial liquids that takeup much of the strain caused by differential stresses, andeven when grain-to-grain contacts make crystals selfsupporting, dissolution and reprecipitation can ac-commodate a large measure of deformation.

Lineation, although it is an important consequenceof magmatic flow, does not normally develop duringcompaction unless there is a component of simple shear.In the interior regions of the Skaergaard Layered Serieslineation is weak or undetectable, except where founderedblocks have caused local deformation or where irregularcompaction has led to strong warping of the layers(McBirney & Nicolas, 1997). With this exception, foliationwithout lineation is more likely to reflect pure shearand compaction than the simple shear expected frommagmatic flow.

LAYERING RELATED TOCOMPACTIONDemonstrating that the Skaergaard gabbros have under-gone compaction is one thing; proving that this com-paction resulted in modal layering is quite another. Theevidence is largely circumstantial and stems not onlyfrom the inadequacies of the alternative explanationsbut also from an improved understanding of magmaticcrystallization. We know of only two mechanisms thatFig. 4. Schematic diagram illustrating the formation of igneous lam-

ination during compaction. In this diagram the number, sizes and have been seriously proposed to account for compaction-shapes of crystals remain unchanged during mechanical rotation (a to related layering, one through mechanical sorting and theb), whereas only the total cross-sectional area is constant in the transition

other through solution and reprecipitation.from b to c.

accentuated by solution of stressed corners, edges andMechanical segregation during compactionsmall crystal faces, and by equivalent growth that favors

large, sub-horizontal faces (Fig. 4c). More than half a century ago, Coats (1936) observedthat crystals of differing sizes and densities tend to sortWe find (Park & McBirney, work in progress) that the

large (010) faces of plagioclase are normally more stable themselves in crude layers as they settle from a suspensionand compact under the force of gravity. His simplethan the smaller (001) faces and that the boundaries

between the two tend to lie close to the plane of foliation. experiments were largely ignored, possibly because theyhad no satisfactory theoretical basis. Apart from a fewFrom this we infer that plagioclase crystals with long

axes at steep angles to the horizontal are less stable and studies of industrial materials that show that particlescan be segregated by upward-infiltrating liquids (e.g.tend to dissolve and contribute to growth of other grains

with stable faces normal to the direction of maximum Font, 1990), no adequate explanation has been givenfor the layering produced during compaction. We arestress (Fig. 5). This is supported by observations from the

Stillwater Complex, where the aspect ratio of plagioclase convinced, however, that the phenomenon is real, for ourexperiments have fully verified all of Coats’ observations.shows a strong positive correlation with a quantitative

measure of foliation (Meurer & Boudreau, 1997). To do this, we used natural minerals in a size rangeof 0·1–0·5 mm and bromoform diluted with acetone toThis interpretation is also in accord with other signs

of mechanical deformation, such as bent or broken a density slightly less than that of the lightest mineral. A

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Fig. 5. Distorted and broken laths of plagioclase are a conspicuous result of compaction of the layering. Twinning tends to be much moreintense where the crystals have been deformed. Vertical thin sections were cut from an oriented sample of Middle Zone. The top of the

photograph is toward the top of the specimen. Width of field is 5 mm.

50:50 mixture of plagioclase and pyroxene was placed and liquid; (3) proportions of the crystal species; (4) grainsize and shape; (5) proportions of liquid and solids; (6)in a 250 ml mixing cylinder with a somewhat smaller

volume of bromoform, brought into suspension by vig- viscosity of the liquid; (7) flow velocity of the liquid. Thelast parameter is not independent of the others. To assessorous shaking, and allowed to settle. Little if any se-

gregation was observed during the initial stage of settling, these various effects properly, one should conduct a seriesof experiments in which each factor could be variedbut as the bed of crystals continued to compact, irregular

aggregates of plagioclase began to form. Even though independently. To date, however, our efforts to do thishave had only limited success, owing mainly to thethe crystals formed a self-supporting framework, they

continued to compact, reducing the pore space and problem of finding materials with appropriate physicalproperties. At this stage, we can only offer a few broaddriving out interstitial liquids. The rising liquid seems to

entrain the lighter crystals of plagioclase, carrying some generalizations.Alternating layers develop best when the density con-to the top of the bed but segregating others into crude

layers (Fig. 6). trast is small and the proportion of liquid is less thanthat of the crystals. In mixtures in which the densityAs Coats noted, the effectiveness of separation is a

function of several factors, including: (1) density contrast contrast between liquid and crystals is very large or theproportion of liquid is greater than that of the crystals,of the crystal species; (2) density contrast of the solids

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the extensive recrystallization that subsequently affectedmore slowly cooled bodies. Lacking any firmer evidence,one can only speculate on the importance of this unusualtype of mechanical sorting.

Solution and recrystallizationInterstitial liquids escaping from a compacting pile ofcrystals rise to levels where temperature and cotecticrelations differ from those deeper in the section. Theliquids are therefore reactive and must re-equilibrate,both thermally and chemically, with the new environmentthrough which they rise. Under conditions of slow crys-tallization this equilibration can proceed, even when thedifferences of chemical potential are small. In additionto temperature differences, several factors affect the rel-ative free energy of crystals and may alter the fabric andmodal proportions of the original liquidus assemblage.

Just as the textures of igneous rocks are governed bythe physical and chemical properties of the liquid fromwhich they crystallize, liquids in equilibrium with crystalshave local differences that stem from three factors: (1)differential pressure solution; (2) differences of grain size;(3) the affinities of like and unlike crystals.

Fig. 6. Crude layers produced by segregation of plagioclase andpyroxene during compaction of a slurry suspended in dilute bromoform. Pressure solution

Because the surface energy of a crystal increases withstress, points where stress is concentrated tend to dissolve,the light and heavy crystals are able to separate completelywhereas those under smaller stress grow. The effect ofand form two layers with the light mineral overlying thepressure differs from one mineral species to another. Inheavy.a mixture of two minerals, the more pressure-sensitiveAlthough the forces responsible for this sorting arephase has a greater free energy when the proportions ofpoorly understood, they seem to be related to some formthat mineral are large than when they are small andof self-organization of particles according to their dragstress is taken up by grains of a more resistant mineral.coefficients in a viscous fluid. The fact that the mechanismPressure solution has long been considered an importantseems to operate only within a restricted range of con-factor in metamorphic rocks (Fyfe, 1976), but Dick &ditions may explain why it is not more common. UntilSinton (1979) seem to have been the first to suggest thatmore has been learned about the phenomenon, we canit could produce layering in igneous rocks. They proposedonly speculate on its importance, but we can point to a fewthat some of the layering in the ultramafic rocks ofpossible cases of what could be interpreted as ‘Coatsianophiolites developed when olivine and pyroxene werelayering’. All known examples are in sills that have asegregated into separate layers of dunite and pyroxenitebasal zone of coarse mafic minerals that were carried inin a zone of strong tectonic deformation close to the basesuspension at the time of intrusion and settled to theof the crust. Because olivine dissolves and reprecipitatesfloor as a single mass. Bruce Marsh has shown us pla-more readily than pyroxene, individual crystals of pyr-gioclase-rich lenses in the Yorkhaven Sill that closelyoxene in olivine-rich rocks bear a disproportionatelyresemble the crude layers produced in our experiments,greater share of the total stress than do crystals ofand has supplied photographs of similar layers in thepyroxene in an olivine-poor rock. Thus, they tend toFerrar Dolerites of Antarctica (Fig. 7; B. D. Marsh,dissolve in regions where they are less abundant andpersonal communication, 1995). In both places, the pla-reprecipitate where they are modally more important.gioclase-rich lenses developed in a coarse bronzite-richAs the relative sizes of grains are reduced by pressuremass that was brought in as a dense suspension andsolution, the chemical potential difference is further in-underwent gravitational compaction on the floor.creased by the size-dependent difference of surface en-The scarcity of this type of layering in larger intrusions

may be due to the slower accumulation of crystals or to ergy. Thus modal and grain-size differences are

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Fig. 7. Example of ‘Coatsian’ layering in a dolerite sill in Antarctica (photo courtesy of B. D. Marsh).

enhanced, and an initially weak inhomogeneity can de- those rocks in which the mineral is modally dominantand probably explains a similar correlation observed invelop into layers that are increasingly monomineralic.

This mechanism could be equally effective under either graded layers of the Skaergaard Middle Zone (Conrad& Naslund, 1989). Crystal aging can occur in a non-simple shear, as Dick and Sinton proposed, or under

pure shear associated with compaction. compacting assemblage, but would aid any other com-The stability of crystals is also a function of the crys- paction-driven effects.

tallographic face on which stress is concentrated. As we The tendency for large crystals to grow at the expenseobserved in an earlier section, the large 010 faces of of smaller ones is a function of the size-dependent con-tabular plagioclase crystals appear to be more resistant centration variations defined by the Gibbs–Thomsonthan other smaller faces. When contacts bear stress, the equation (see Table 1 for explanation of symbols):latter tend to dissolve whereas the former will grow andincrease the proportion of grains oriented with 010

C=C xexpA 2rRTrB. (1)normal to the direction of maximum stress. We credit

much of the foliation in plagioclase-rich rocks to thiseffect. A small difference of surface energy gives larger grains

a competitive advantage, so that small initial variationsowing to some random effect are magnified and can lead

Grain size to cyclic layering. Examples of such layering have beendescribed in detail and successfully modeled by computerOwing to their greater volume-to-surface ratio, the freesimulations (Boudreau, 1987, 1994, 1995; McBirney etenergy of small crystals (and hence their solubility) isal., 1990).greater than that of larger ones. This difference is the

driving force by which Ostwald ripening leads to ageneral coarsening of grain size as large crystals grow atthe expense of smaller ones. Crystal size distributions of

Affinity of like crystalsslowly crystallized rocks, which typically show a pro-The magnitude of surface free energy of a given mineralnounced paucity of smaller grain sizes, are one line ofis not a unique value but depends on the nature of theevidence that their minerals have undergone modificationcrystal’s surroundings. In general, one would expect theby the aging process (Waters & Boudreau, 1996). Thesurface energy of a crystal to increase as the surroundingpositive correlation between grain size and modal abund-material, whether it be a silicate liquid or other crystallineance of olivine and chromite seen in olivine chromititesphases, becomes less compatible with the surface of theof the Stillwater Complex ( Jackson, 1961) was attributed

by Boudreau (1995) to more rapid aging of minerals in crystal structure. For example, McLean (1957; in Spry,

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Table 1: List of symbols and units

A, B component A or B, respectively

C liquid concentration of component A (mol/cm3)

Cx equilibrium liquid concentration for a crystal of infinite (large) radius

C0 equilibrium liquid concentration of component A for the case in which

mineral a has 0% contact with other a mineral grains (mol/cm3)

C100 equilibrium liquid concentration of component A for the case in which

mineral a has 100% contact with other a mineral grains (mol/cm3)

C(W) concentration of component A in a liquid that is in equilibrium with a rock of

mode/texture characteristics W

H ratio of mass of mineral a to that of mineral b (non-dimensional)

Q source term for component A (mol/cm3/s)

R gas constant (J/mol/K)

T temperature (K)

V liquid velocity (cm/s)

a, b denotes mineral a or b, respectively

f liquid fraction (nondimensional)

fa fraction of solids composed of mineral a

n crystal number density (per cm3)

q crystal growth constant (cm/s)

r radius (cm)

s supersaturation (non-dimensional)

t time (s)

z distance (cm)

K maximum mode/texture-dependent concentration variation (mol/cm3)

U fraction crystallized (non-dimensional)

W variable describing the rock mode/texture

q crystal density (mol/cm3)

r surface tension (J/cm2)

′ non-dimensional value of a variable

¯ (overbar) characteristic quantity of a variable

1969) noted that the surface energy of copper may vary result in zones or ‘protolayers’ in which, for example,a–a contacts are slightly more abundant than in theby as much as two orders of magnitude, and is generally

lower when in contact with other copper phases (crystals immediately surrounding rock (e.g. ‘Coatsian’ layers).Because of this small difference, crystals of a in this zoneor Cu liquid). This is also supported by the tendency for

monomineralic strings of minerals to be among the last will have a lower solubility and will grow at the expenseof crystals in the surrounding rock. Similarly, for a two-material to melt during fusion (Philpotts & Carroll, 1996).

It is thus apparent that the solubility of a mineral depends component liquid, the local abundance of a–a contactsmeans that mineral b is relatively less stable than in thenot only on grain size and stress but also on the nature

of its contact with surrounding phases. Specifically, it is surroundings, where it is slightly more abundant andhence will dissolve as components of b migrate out intoprobable that certain precipitating mineral grains have

lower solubility against ‘like’ mineral grains than against the surrounding rock. The result is that the initial texturalirregularity will not only continue to grow with time but‘unlike’ ones. As with crystal aging, this phenomenon

can occur regardless of whether the system is undergoing will accelerate as the number of a–a contacts increasesand the number of b–b contacts decreases. Furthermore,compaction, as it is driven by interfacial energy effects

alone. this mechanism is self-propagating as regions dominatedby a–a contacts cause the surrounding regions to beMineral segregation can occur in the following manner.

Cotectic crystallization of two minerals a and b will dominated by b–b contacts and vice versa. This growthof textural irregularities is similar to that proposed bynominally result in random variations of the distribution

of a and b with no preferred arrangement of a–a, a–b, Ortoleva and others for formation of metamorphic band-ing (e.g. Ortoleva et al., 1987).and b–b contacts. However, minor irregularities may

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In layered intrusions, some of the most striking layered B) which crystallizes two minerals a and b. For a two-component system, any change in the liquid con-mineral assemblages involve spinels (chromite or

magnetite). The most extreme examples consist of al- centrations of component A will result in an equal butopposite change in the liquid concentration of componentternating layers of almost pure spinel and silicate minerals

(Fig. 8). Textural evidence indicates that initial modal B. Hence, one can relate all changes of the liquid solutionto that of component A alone, as is done in the followingvariations may have been enhanced by surface energy

differences at spinel–spinel contacts (i.e. contacts in- formulations (i.e. the quantity C refers to the con-centration of component A). It is further assumed thatvolving an isotropic mineral) and spinel–silicate contacts.

The relative ease by which chromite grains nucleate both minerals are in contact with an interstitial liquid,that the liquid is moving through the system undertogether and even 1ink together to form extensive chains

during growth (e.g. Hunter, 1987) suggests that spinel– the influence of a mechanism such as compaction orconvection, and that mass transport by advection isspinel contacts have relatively lower surface energies

than other types. Because of the lower surface energy, dominant over that by diffusion and hence diffusionaltransport is ignored (but see the scaling discussion, below).spinel–spinel contacts would have lower equilibrium

liquid concentrations than would sites of spinel–silicate At any location, the change in liquid concentration ofcomponent A is equal to its net rate of flux of liquid intomineral contacts. This would result in spinel grains at

mixed contacts dissolving in favor of grains in which or out of the location, plus the rate at which it is producedor destroyed within the system by reaction with the solidspinel–spinel contacts predominate. Over time, this might

lead to a nearly complete segregation of spinel from assemblage. For a one-dimensional transport, one hassilicate minerals.

∂C∂t=−V

∂C∂z−Q. (2)

The term Q is a source term defined by the gain or lossCombined effects of solution component A by growth or dissolution ofIt is important to note that the effects of these three factors minerals a and b (assumed to be spherical with average(differential pressure solution, grain size and affinity of radius r):like crystals) tend to reinforce one another, so that anysmall initial inhomogeneity is enhanced as crystals coar-

Q=∂C∂U∂U∂r∂r∂t

(3)sen, the relative abundance of one of the minerals in-creases, and more grains of the dominant mineral are in

=−naqa 4p(ra)2∂ra

∂t+nbqb4p(rb)2

∂rb

∂t.direct contact with one another. In this process, the

intergranular liquid plays a crucial role. If the liquid doesAssuming a first-order reaction mechanism, the rate ofnot move and mass transport is solely by diffusion withinprecipitation or dissolution of crystals can be expressedthe liquid, the scale of the effects is on the order ofascentimeters only (Boudreau, 1994, 1995). In the presence

of a moving fluid, however, the efficiency and scale ofmass transfer is potentially much greater. Thus, liquid ∂ra

∂t=

qa

qa

[C−C(w)] (4)expelled by compaction and rising through the crystalmush surmounts the limitations of diffusion and increases andthe vertical dimensions and intensity of the layering.

∂rb

∂t=−

qb

qb

[C−C(w)] (5)

where q is the crystal growth constant and q is the densityQUANTITATIVE MODEL OF LAYERof the crystalline phase. C(W) is the solution concentration

GROWTH in equilibrium with a rock with modal/textural char-Presented here is a very general model for separation of acteristics W. That is, it is assumed that all the mechanistictwo crystals in a two-component system into modally mineralogical/textural phenomena that affect local equi-segregated layers. It is much simplified from the more librium liquid concentrations as described previously candetailed treatment of the general phenomenon of self- be summarized in a general phenomenological expressionorganization in geologic systems discussed by Ortoleva for C(W). In detail, W is a function of mineral modes as(1994), but illustrates the pertinent mechanisms of interest well as grain size, shape and orientation. As a firstas they might occur in a large layered intrusion. approximation, C(W) is taken to be a simple function of

Following a similar derivation by Boudreau (1994), we local mineral mode, which is itself a function of grainsize, r, and the crystal number density, n, such thatconsider a two-component system (components A and

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Fig. 8. (a) Layers of chromitite alternating with anorthosite at the Dwars River in the Bushveld Complex of South Africa. (b) Detail of coarse,plagioclase-rich segregation or residual clot within a magnetite layer.

C(W)=C100+(1−fa)K (6) completely surrounded by a. (It is noted that the liquidcan be crystallizing at the eutectic but that in any smallregion there may be no a–b contacts.) K is a constantwhereequivalent to the maximum concentration differencebetween an assemblage where mineral a is surrounded

fa=C na(ra)3

na(ra)3+nb(rb)3D (7)entirely by a and one where a is surrounded entirely byb. The value of K can be small if one is consideringsolubility variations arising from surface energy effectsand K is a constant. The quantity C 100 is the equilibrium

liquid concentration for the situation where mineral a is alone, but could be substantial for the case where crystals

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of one phase are experiencing most of the strain duringcompaction.

A qualitative view of the situation is shown schem-atically in a two-crystal, two-component system, wherethe location of the eutectic is assumed to be a functionof the local mode/texture as shown in Fig. 9. In thisfigure, the eutectic for the case where a is surroundedcompletely by a is at a lower concentration (i.e. at C 100)than is the case for which a is surrounded by b (i.e. atC 0). It should be noted that we are assuming that theeutectic does not change its location with respect totemperature, although a temperature dependence couldbe readily added (Boudreau, 1994). Also, the effect ofthe rock mode/texture term W as defined in equations(6) and (7) includes grain size and number but notorientation of the mineral grains as would be required ifone were not considering spherical grains.

Finally, liquid velocity will not be constant but willchange as the porosity changes. It is assumed that forsmall changes in total solid fraction, f, then

V=V f

f. (8)

Nondimensional equationsAdditional dimensionless equations equivalent to theabove, derived in the Appendix, are presented below. It

Fig. 9. Schematic representation of the expected shift of the eutecticis assumed that mineral a is the more reactive mineral position in a two-component system as a function of mineral mode. Itphase, and parameters of mineral b are scaled to the is assumed that minerals a and b both have lower free energies when

in contact with like grains than unlike grains. Where mineral a isproperties of mineral a. Thus, the reaction/transportsurrounded by b (i.e. b is modally dominant), b–b contacts are pre-equations (2) and (3) becomedominant and hence a is relatively soluble and b less soluble. Thelocation of the eutectic is then as shown by the continuous lines. Wherea is surrounded mainly by a (i.e. a is modally dominant), then a–a

1b∂s′∂t′=−V ′

∂s′∂z′−na′(ra′)2

∂ra′∂t ′+Hnb′(rb′)2

∂rb′∂t ′

(9)contacts predominate and hence b is relatively soluble and a is relativelyinsoluble. The location of the eutectic is then shifted to the locationshown by the dashed lines. The maximum concentration difference

H=n br b

3qb

n ar a3qa

(10) between rocks rich in a (at C 100, where 100% of the contacts are a–acontacts) and those rich in b (at C 0, where 0% of the contacts are a–acontacts) defines the maximum mode/texture-dependent concentration

The quantity H is the ratio of the characteristic amounts difference term, K.of the two minerals a and b initially present and, for mostcotectic crystallization systems, the ratio is close to one.

The grain growth equations (4) and (5) become V ′=1f ′

(14)

∂ra′∂t ′=[s′−(1−fa)] (11) The mode/texture dependence of the equilibrium

liquid concentration defined by equations (6) and (7) isexpressed through the term fa in equations (11) and (12).∂rb′

∂t ′=−qb′[s′−(1−fa)] (12) The b term in equation (9) is a scaling constant and is

given bywhere

b=4pnaqa(ra)3

K. (15)

qb′=r a

2qaqb

r b2qbqa

. (13)

This scaling constant, b, is simply the ratio of the molesof component A initially present as crystals a (per unitThe liquid velocity equation (8) becomes

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volume) to the maximum molar liquid concentration liquid+crystal mush zone is both thick enough anddifference in component A that can develop by a change persists long enough for the processes that drive self-in rock mode or texture between regions (as illustrated segregation to operate. Even within the Skaergaard, thein Fig. 9). observation that layering becomes more defined with

Finally, the characteristic time and length scales are height is consistent with the interpretation that a thickergiven by crystal mush enhances non-dynamic layer formation.

Assuming characteristic time scales from 1000 to 10 000years, values of the scaling constant, b, would need to

t=ra

qaAqa

KB (16)be in the range of 103 – 104. However, the value of bwill not remain constant over the course of crystallization.andThe scaling constant is initially small during the earlynucleation growth period, when the volume of crystalline

z=t V

b. (17) material is low and solubility differences driven by surface

energy are relatively large. At this liquid-dominant stage,the value of b can be less than one (Boudreau, 1995). Itincreases rapidly, however, as grains become larger and

Estimates of characteristic times and size-driven liquid concentration differences decrease. Itscaling constant, b then decreases as loading by overlying crystals increasesRearranging equation (17), the characteristic time can and crystals begin to deform during compaction. Thus,be expressed as a function of the characteristic length, it is during the early growth phase and then duringscaling constant and interstitial liquid velocity: compaction that non-dynamic layering is most likely to

develop, as these are the times when b is smallest.t=

bz

V. (18) It should be noted that the phenomenon occurs

whether the transport is by advection (as is modeledhere) or by diffusion [as modeled by Boudreau (1994)].Thus, the time it takes for a layer to develop increasesIncluding a diffusion transport term in equation (2) wouldas either the thickness of the layer increases or as thenot change qualitatively the observed segregation, as thescaling constant increases, both of which imply there isdiffusive gradients are such that they also will favora relative increase in the amount of material that must bemineral segregation. Indeed, for slow infiltration rates,transferred between regions to effect mineral segregationthe addition of a diffusional mode of transport along withrelative to the maximum amount of material that can beadvection would lead to more rapid material exchangetransported in a unit volume of liquid. In contrast, thebetween developing layers and hence actually acceleratetime it takes a layer to develop is inversely proportional

to the velocity of the interstitial liquid, as a higher liquid the time scale for layer development. For example, forvelocity speeds up transfer of material between regions. the 1 cm length scale under consideration, the low liquid

From estimates of the cooling time and interstitial velocities at longer characteristic times are such that theliquid velocities one can estimate permissible values of product V L approaches values for diffusion mass trans-the scaling constant, b, and from this the required port appropriate for silicate liquids. For layering thatsupersaturation differences, K, required to effect layer develops on a finer length scale, such as is illustrated information. A plot of characteristic times as a function of Fig. 2, the system can evolve on diffusion length and timevelocity of interstitial liquid and values b ranging from scales and hence could occur without liquid migration.103 to 106 is shown in Fig. 10 for a characteristic length However, for layers with longer characteristic lengthsof 1 cm as a typical length scale for non-dynamic igneous (i.e. thickness), such as the decimeter-scale mafic–felsiclayering. An interstitial liquid velocity of 10–6−10–7 cm/s layers of the Skaergaard Layered series, diffusion aloneis equivalent to 3–30 cm/yr, or in the range of estimated would be insufficient. Compaction-aided advection orvalues for compaction-driven fluid velocities in a typical interstitial liquid convection is required in addition tolarge intrusion (Shirley, 1987; Sonnenthal & McBirney, diffusion to effect the necessary mass transport between1997). Shirley (1987) estimated a minimum compaction regions.time of 200 years for the Muskox intrusion, and this canbe taken as a minimum time scale for non-dynamiclayering to develop.

Numerical model of layer developmentA long characteristic time is consistent with the ob-A one-dimensional numerical model of grain size andservation that well-developed modal segregation layeringtextural evolution with time using a finite-differenceis not common or well developed in small or relativelyanalog of the nondimensional equations (9)–(13) is shownthin intrusions such as the Palisades sill. It is only in

intrusions the size of the Skaergaard or larger that a in Fig. 11. [Because growth of one mineral phase is

1016


Fig. 10. Plot of characteristic times (in seconds) as a function of velocity of intercumulus liquid, for a range of values of the scaling constant, b,all calculated for a characteristic length of 1 cm.

matched by dissolution of the other mineral phase, por-osity changes little and hence velocity variations expressedthrough equation (14) are not considered in the followingsimulations.] In this calculation, the system starts as auniform distribution of a and b mineral grains, all of thesame average size but with a local ‘bump’ in the grainsize of mineral a. For the calculation, the scaling constant,b, is taken to be 104. The calculation follows the evolutionof this initial bump over ten nondimensional space stepsand over nine nondimensional time units. The initialbump of larger grains of mineral a is located at a(nondimensional) distance of 3·3 units: grains of minerala in the peak of this bump are 5% larger than the averageelsewhere. Plotted in the four graphs of Fig. 11 are, fromtop to bottom, the radius of mineral a, the radius ofmineral b, the volume of mineral a as a percent of totalsolids volume (i.e. the modal abundance of mineral a),and the scaled liquid concentration of component A—allplotted as dimensionless quantities. The interstitial liquidis taken as flowing from left to right. The plot showsthe evolution of the profiles for these quantities at thedimensionless times of 0, 3, 6 and 9 time units.

Because the number of grains is constant, the regionof larger grains centered at a distance of 3·3 units causesa small increase in the modal abundance of a in thisbump, which in turn affects the local equilibrium liquid

Fig. 11. Numerical model of layer development. Plotted against theconcentration. Let us consider first what occurs as thedimensionless distance are, from top to bottom, the nondimensional

interstitial liquid, moving from left to right, begins to radius of mineral a (ra′), the nondimensional radius of mineral b (rb′),encounter the region where the modal abundance of a the volume percent of mineral a as a percentage of total solids volume

(fa), and the scaled nondimensional concentration (s′). Shown is theis beginning to increase. Because of the textural de-evolution of the profile at dimensionless time (t ′) equal to 0, 3, 6 andpendence of the eutectic position, the equilibrium con- 9 time units. (See text for additional discussion.)

centration for component A is lower where mineral a ismodally more abundant. Hence the liquid finds itselfoversaturated in mineral a as it encounters the upstream a. For mineral b it is just the opposite; the liquid finds

itself undersaturated in mineral b as it encounters theside of the bump and begins to precipitate more mineral

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upstream side of the bump and hence b begins to dissolve. from variations of intensive parameters that alter ratesOnce the liquid passes the peak of the bump, however, of nucleation and crystal growth, the layers have diffuse

the situation is the reverse. On the downstream side of boundaries and the minerals have little if any lineation.the bump, the liquid is moving into assemblages that A notable exception is the layering in the Marginal Borderhave progressively more b than a. In this case, the liquid Series, which advanced more slowly and is relativelyis always oversaturated in b but undersaturated in a as compressed. Most of the sharp layering in the interiorit moves to the right of the initial bump, and thus a of the Layered Series is thought to be related in somedissolves whereas b precipitates. The net effect is that a way to compaction and other processes involving porousbecomes more abundant on the upstream side whereas flow of interstitial liquids. Although mechanical se-b becomes more abundant on the downstream side. The gregation may be effective during the initial stages oforiginal peak in grain size of a grows but it also migrates accumulation, differential pressure-solution seems to haveupstream. In addition, the initiation and growth of the been the principal mechanism. Initial modal variationsb peak itself induces a new a peak to form downstream are strongly enhanced and sharpened as the ascendingfrom the first peak, which in turn induces yet other peaks liquid transfers components from one level to another.to form. After nine nondimensional time steps, three a The driving force of segregation is the free-energy differ-peaks and three b peaks have developed. ence resulting from combined effects of grain size, pres-

We note that regions in which a or b are modally sure solution, and the relative affinities of like and unlikedominant tend to become sharply defined from their minerals. Where compaction produces a regionally uni-neighboring regions, as seen in the modal plot of volume form upward percolation of liquid, this segregation leadspercent of mineral a. Also, grains in the individual layers to formation of planar layers. However, focused flowmay be size-graded. That is, the larger grains are at the or non-uniform compaction may cause more irregulardownstream side of each layer (this is better developed structures. The coincidence of the disappearance of ex-in the induced layers than in the layer formed from the tensive strataform layering with the beginning of theinitial bump). If the layering were developed horizontally trough layers in the Upper Zone is consistent with ain response to vertical movement of liquid (as in a change from uniform to focused flow of interstitial liquidcompacting pile of cumulus crystals), then the size grading at this level.would be similar to the size distribution produced byStokes’ law gravitational separation of larger from smallergrains during a crystal settling event. In this case, however,the grain sizes need not be hydraulically equivalent as ACKNOWLEDGEMENTSwould be expected for layering formed by crystal settling.

This work has been supported by grants from the NationalThe scaled concentration profile tends to mirror theScience Foundation to A. E. Boudreau (NSF EAR 92-modal abundance profile for mineral a. This is because17664 and 95-17144). McBirney’s 25 years of work onthe mass of solid material is large as compared with thethe Skaergaard Intrusion would not have been possiblemaximum texture-induced concentration differences andwithout the financial support provided by a series ofcauses the scaled concentration profile to be stronglygrants from the National Science Foundation. Reviewcontrolled at the equilibrium values defined by the localby W. P. Meurer, Bruce Marsh and an anonymousrock mode/texture.reviewer is acknowledged and much appreciated.Finally, on observing the texture and modal variations

evolve, one tends to be taken by the propagation of thepattern. What is perhaps more important, however, isthe fact that minor textural irregularities tend to become

REFERENCESmore sharply defined over time. In a rock composed ofBoudreau, A. E., 1987. Pattern formation during crystallization andinitially weakly defined layers formed by a variety of

the formation of fine-scale layering. In: Parsons, I. (ed.) Origins ofnucleation or mechanical segregation mechanisms, theIgneous Layering. Dordrecht: D. Reidel, pp. 453–471.

processes outlined above will continue to enhance the Boudreau, A. E., 1994. Mineral segregation during crystal aging inmodal and textural contrast between layers. This modal two-crystal, two-component systems. South African Journal of Geology

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Princeton University Press, 331 pp.Brothers, R. N., 1964. Petrofabric analyses of Rhum and SkaergaardCONCLUSIONS

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of Geology 44, 407–419.differs from that caused by magmatic flow. If it results

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Conrad, M. E. & Naslund, H. R., 1989. Modally graded rhythmic Ortoleva, P. J., 1994. Geochemical Self-Organization. New York: OxfordUniversity Press, 411 pp.layering in the Skaergaard Intrusion. Journal of Petrology 30, 251–269.

Ortoleva, P. J., Merino, E., Moore, C. & Chadam, J., 1987. Geo-Dick, H. J. B. & Sinton, J. M., 1979. Compositional layering in alpinechemical self-organization l: reaction–transport feedbacks and mod-peridotites: evidence for pressure solution creep in the mantle. Journal

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melted tholeiitic basalt. Geology 24, 1029–1032.tinuous thickeners. AIChE Journal 36, 3–12.Philpotts, A. R., Carroll, M. & Hill, J. M., 1996. Crystal-mush com-Fyfe, W. S., 1976. Chemical aspects of rock deformation. Philosophical

paction and the origin of pegmatitic segregation sheets in a thickTransactions of the Royal Society of London, Series A 283, 221–228.flood-basalt flow in the Mesozoic Hartford Basin, Connecticut.Harker, A., 1909. Natural History of Igneous Rocks. New York: MacmillanJournal of Petrology 37, 811–836.Company, 384 pp.

Shirley, D. N., 1987. Differentiation and compaction in the PalisadesHelz, R. T., 1987. Differentiation behavior of Kilauea Iki lava lake,Sill. Journal of Petrology 28, 835–865.Kilauea Volcano, Hawaii: an overview of past and current work.

Sonnenthal, E. L. & McBirney, A. R., 1997. The Skaergaard LayeredIn: Mysen, B. (ed.) Magmatic Processes: Physicochemical Principles. Prince-Series. Part IV: Reaction-transport simulations of foundered blocks.ton, NJ: Princeton University Press, pp. 241–258.Journal of Petrology 38, in press.Helz, R. T., Kirschenbaum, A. & Marinenko, J. W., 1989. Diapiric

Sonnenthal, E. L., 1992. Geochemistry of dendritic anorthosites andtransfer of melt in Kilauea Iki lava lake, Hawaii: a quick, efficientassociated pegmatites in the Skaergaard Intrusion, East Greenland:process of igneous differentiation. Geological Society of America Bulletinevidence for metasomatism by a chlorine-rich fluid. Journal of Vol-101, 57,–591.canology and Geothermal Research 52, 209–230.Higgins, M. D., 1991. The origin of laminated and massive anorthosite,

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oscillatory nucleation and crystal settling in well-mixed magmas.pp.

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and Skaergaard Intrusions: similarities, differences, and origins. In: producing diversity in layered igneous intrusions. Geological MagazineParsons, I. (ed.) Origins of Igneous Layering. Dordrecht: D. Reidel, pp. 96, 75–80.185–246. Wager, L. R., 1961. A note on the origin of ophitic texture in the

Jackson, E. D., 1961. Primary textures and mineral associations in the chilled olivine gabbro of the Skaergaard intrusion. Geological MagazineUltramafic Zone of the Stillwater Complex, Montana. US Geological 98, 353–366.Survey Professional Paper 358, 106 pp. Wager, L. R. & Brown, G. M., 1968. Layered Igneous Rocks. Edinburgh:

Maaløe, S., 1978. The origin of rhythmic layering. Mineralogical Magazine Oliver & Boyd, 538 pp.42, 337–345. Wager, L. R. & Deer, W., 1939. Geological investigations in East

McBirney, A. R., 1985. Further considerations of double-diffusive Greenland, Part III. The petrology of the Skaergaard Intrusion,stratification and layering in the Skaergaard Intrusion. Journal of Kangerdlugssuaq, East Greenland. Meddelelser om Grønland 105, 1–Petrology 26, 993–1001. 352.

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McBirney, A. R. & Hunter, R. H., 1995. The cumulate paradigmreconsidered. Journal of Geology 103, 114–122.

McBirney, A.R. & Nicolas, A., 1997. The Skaergaard layered series:Part II. Dynamic layering. Journal of Petrology 38, 000–000. APPENDIX

McBirney, A. R. & Noyes, R. M., 1979. Crystallization and layering The characteristic and dimensionless forms of the variousof the Skaergaard Intrusion. Journal of Petrology 20, 487–564. values are defined as follows:

McBirney, A. R., White, C. M. & Boudreau, A. E., 1990. Spontaneousdevelopment of concentric layering in a solidified siliceous dike, East ra=rara′ na=nana′Greenland. Earth-Science Reviews 29, 321–330. rb=rbrb′ nb=nbnb′

McLean, D., 1957. Grain Boundaries in Metals. Oxford: Clarendon Press. t=tt ′ z=zz′ V=V V ′ f=f f ′ (A1)Meurer, W. P. & Boudreau, A. E., 1997. Assessing the role of com-

paction and fluid fluxing in the development of igneous foliations – For the radius, crystal number density, and liquid velocity,An example from the Stillwater complex, Montana. Journal of Geology, one can take the typical, or system-averaged, initial valuesIn review. as the characteristic quantities. One can also define a

Morse, S. A., 1986. Convection in aid of adcumulus growth. Journal of dimensionless supersaturation:Petrology 27, 1183–1214.

Naslund, H. R. & McBirney, A. R., 1996. Mechanisms of formation (A2)C=C 100(1+s).of igneous layering. In: Cawthorn, R. G. (ed.) Layered Igneous Intrusions.Amsterdam: Elsevier, pp. 1–44. The characteristic time and length are to be derived

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below. Substitution of the equations (A1), (A2) and (6)qb′=

ra qa qb

r b qb qa

. (A11)into equation (4) and rearrangement givesFor the transport–reaction equation (2), substitution ofequations (3), (A1) and (A2) gives∂ra′

∂t ′=

t qa

r aqa

{C100(1+s)−[C100+(1−fa)K]} (A3)C100

x

∂s∂t ′=−V ′

V C100

z

∂s∂z′−4pqa

na(ra)3

tna′(ra′)2

∂ra′∂t ′

(A12)or

+4pqb

nb(rb)3

tnb′(rb)2

∂rb′∂t ′

.∂ra′∂t ′=

t qa

r aqa

KCC100s

K−(1−fa)D . (A4)

With further substitution of equation (A6), one hasOne can then define a characteristic time as follows: K

x

∂s′∂t ′=−V ′

V Kz

∂s′∂z′−4pqa

na(ra)3

tna′(ra′)2

∂ra′∂t ′

(A13)t=qa ra

qa K(A5)

+4pqb

nb(rb)3

tnb′(rb)2

∂rb′∂t ′

.In addition, one can define a ‘scaled supersaturation’ in

or, on rearranging, one haswhich the supersaturation is scaled to the maximummode/texture-induced concentration difference: ∂s′

∂t ′=−V ′

V t

z

∂s′∂z′−4pqa

na(ra)3

Kna′(ra′)2

∂ra′∂t ′

(A14)s′=sC100

K. (A6)

+4pqb

nb(rb)3

Knb′(rb)2

∂rb′∂t ′

.The non-dimensional crystal growth rate for mineral a

is then given by One can then define the scaling constant, b:

b=4pqana(ra)3

K(A15)∂ra′

∂t ′=[s′−(1−fa)] . (A7)

and furthermore letA similar treatment for mineral b—substitution of equa-tions (A1), (A2) and (6) into equation (5)—gives t V

z=b . (A16)

Then, on substitution of (A15) and (A16) into equation(A14), one arrives at∂rb′

∂t ′=−

t qb

rb qb

{C100(1+s)−[C100+(1−fa)K]} (A8)1b∂s′∂t ′=−V ′

∂s′∂z′−na′(ra′)2

∂ra′∂t ′+Hnb′(rb′)2

∂rb′∂t ′or

where∂rb′∂t ′=−

t qb

rb qb

KCC100s

K−(1−fa)D (A9)

H=qbnb(r b)3

qana(r a)3. (A18)

Finally, for equation (8), substitution of equations (A1)On substitution of the expression for the characteristicgivestime and the scaled supersaturation [equations (A5) and

(A6)], one hasV V ′=

V f

f f ′(A19)

or∂rb′∂t ′=qb′[s′−(1−fa)] (A10)

V ′=1

f ′(A20)

where

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