Proc. Natl. Acad. Sci. USAVol. 90, pp. 11914-11918, December 1993Geology

Equilibration during mantle melting: A fractal tree modelSTANLEY R. HARTDepartment of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, MA 02543

Contributed by Stanley R. Hart, August 30, 1993

ABSTRACT Many basalts from oceanic islands, ridges,and arcs show strong trace element evidence for melting atgreat depths, where garnet is a stable phase in mantle peri-dotites. If partial melts ascend to the surface by porous(intergranular) flow processes, the high-pressure garnet sig-nature will be obliterated by diffusive reequilibration at shal-lower depths in the mantle. Spiegelman and Kenyon [Spiegel-man, M. & Kenyon, P. (1992) Earth Planet Sci. Left. 109,611-620] argued that partial melts must therefore be focusedinto a coarser transport network, for high-speed delivery to thesurface. Numerous natural network systems, such as rivers andthe human vascular and bronchial systems, have fractal struc-tures that are optimal for minimizing energy expenditureduring material transport. I show here that a fractal magma"tree" with these optimal properties provides a network inwhich magma rapidly loses diffusive chemical "contact" withits host matrix. In this fractal network, magma conduitscombine by twos, with the radius and flow velocities scaling as(2)"/3, where n is the generation number. For reasonable valuesof volume diffusivities, viscosities, and aspect ratios, melts wiliexperience only limited diffusive reequilibration once they havetraveled some hundreds ofmeters from their source. Melts thusrepresent rather local mantle domains, and there is littleproblem in delivering melts with deep (<100 km) geochemicalsignatures to the surface.

I. IntroductionDuring adiabatic upwelling of peridotitic mantle under oce-anic ridges and island hot spots, melting is initiated at depthsof50-100 km when the mantle material crosses the peridotitesolidus. Commonly, the depth of this melting is large enoughfor garnet to be a stable phase in the peridotite (1-3). Thisincipient melting occurs along mineral grain boundaries andis believed to lead to an interconnected network of microme-ter-sized melt tubules (4, 5). Important fractionation of ele-ments between melt and solid takes place during this process.This fractionation is usually deemed to occur under condi-tions oflocal equilibrium (6). Ultimately, this melt is collectedor aggregated into magma chambers at much shallowerdepths, and some fraction is erupted as basaltic volcanicrock. The melt transport system at these shallower depths ismeter-scale dikes and sills. The extent to which the melt inthese meter-scale features undergoes any equilibration withthe wall rocks is unknown. However, it has become quiteclear over the past few years that erupted ridge melts stillretain recognizable chemical signatures of their original high-pressure formation, proving that shallow-level reequilibra-tion with mantle host rock was incomplete (3, 7, 8).The basic question I will address here is the nature of the

transition of the melt transport system from an intergranularporous network at great depth to a meter-scale crack systemat shallow depth. I will address the issue of where and howthe melts "lose touch" with their host rocks during upwardmigration. I argue that if the melt transport system is modeled

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as a fractal tree, then melts reach high enough migrationvelocities, within hundreds of meters of their origin, toescape to the surface without further diffusive reequilibrationbetween melt and host rock.H. Equilibrium During Melt Generation

Hofmann and Hart (6) gave quantitative arguments to showthat isotopic and trace element equilibrium between melt andcrystalline phases was highly likely during mantle melting. Iwill revisit this question here, in view of the fact that muchnew information has been published since 1978 concerningelemental diffusion rates in silicates, melt generation rates,and melt topologies.

Spiegelman and Kenyon (9) addressed this issue using amodel network of parallel tubes and defining equilibrationtimes in terms of intertube diffusion times. Here I will modelthe problem instead as a cylinder of melt of variable radius,which is small compared with the grain size (distance be-tween tubes); equilibration times are defined in terms ofhowlong it takes for the melt to reach approximate partitioningequilibrium with the host. This time may be quite differentfrom the length of time it requires for diffusion betweenadjacent melt tubes, as the solid volume around the tube thatneeds to be diffusely accessed is dependent on the crystal/melt partition coefficient for a given element. Compatibleelements that prefer to be in the solid phases will have shorterequilibration times because only a small volume around themelt tubule needs to be "milked."

I assume a melt cylinder of radius r instantly emplaced insolid host. The diffusion coefficient of element i in the meltis assumed to be infinite and that in the host is Di (in reality,diffusion rates in silicate melts are only 104-106 times fasterthan in silicate solids; ref. 10). Based on the equivalentheat-flow problem (11), for the melt to reach 83% of itsultimate equilibrium value requires

diffusion time = tD = D(K) .5 [1]

where K is the solid/melt partition coefficient (ratio of heatcapacities in the thermal analogue). The 83% equilibrationpoint was chosen simply to lend simplicity to Eq. 1; the power1.5 is a close approximation to a more complicated functionalform for K (appropriate only for 83% equilibration). For K =1, Eq. 1 reduces to the conventional diffusion rule-of-thumb,r2 = Dt. Eq. 1 shows that incompatible elements (K << 1) willrequire much longer equilibration times than compatibleelements (K > 1); every factor of 10 decrease in K requiresan increase of a factor of =30 in time. Note that this diffusionmodel holds true only as long as the diffusion "volumes" ofadjacent tubes do not overlap.To assess the extent of equilibration during melting, I will

compare the above diffusion time with the time required togrow the tubule by melt generation. Production of a givenmelt fraction 4 requires

CR(S - a)tm0= L [2]

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where R is the upwelling rate of the mantle's heat capacity Cand enthalpy of fusion L; S and a are the change in temper-ature with pressure of the solidus and adiabat, respectively;and tm is the elapsed melting time.To relate the melt fraction to a tubule size, I use the results

from the theoretical treatment ofvon Bargen and Waff (5). Tofirst order, the cross-sectional area Cch of a melt tubule at itsmidpoint is related to the grain size, d, and melt fraction 4 by

Cch -0.05d2 . [3]

This is valid for small melt fractions (<2%) and for melt/solidwetting angles of =50° (the channel area would increase by afactor of =1.3 if the wetting angle were as small as 200). Themelt tubules are of course not circular in cross section butrather are curvilinear triangular sections. By estimating aradius r for a cylinder of comparable surface area to thesemelt tubules, Eq. 3 becomes

[4]

From Eqs. 2 and 4, the growth time for a melt tubule of givenradius is then

59r2Ltm d2CR(S -a)' [5]

Fig. 1 shows a compilation of diffusion data for a variety ofelements in various peridotite phases. The overall range ishuge, but if we ignore He, Si, and 0 in olivine, the range attemperatures appropriate to the 30-kb solidus is from 1 x10-11 cm2/sec to 2 x 10-9 cm2/sec. While this data is far froma comprehensive coverage of the most important elementsand phases involved in mantle melting, I believe it providesa reasonable representation of diffusion in mantle phases.The heat capacity ofperidotite (0.3 cal/gm.K, 1 cal = 4.184

J) and the slope of the peridotite adiabat (0.4°C/km) are fromrefs. 26-28; these values are unlikely to be in error by morethan 20-30%. The enthalpy of fusion of peridotite has re-cently been put on a firm footing by Fukuyama (29), whodetermined L for the melting of a natural peridotite to give arealistic basaltic composition melt; his measured value of 162cal/g at 1260°C has been corrected to a 30-kb pressure valueof 185 cal/g, according to his discussion and that ofHess (30).

Fixing these values for C, a and L, Eq. 6 becomes

tD 9 x 10-' d2R(S - 0.4)tm D(K)15 1 [7]

where d is in cm, R in cm/year, S in °C/km, andD in cm2/sec.For a "most likely" case scenario for subridge melting, I

choose an upwelling rate R = 3 cm/year, a grain size d of 0.3cm, a solidus slope S of 4°/km, a partition coefficient K of

The ratio of the diffusion time from Eq. 1 and the tubulegrowth time from Eq. 5 gives a type of "Peclet" number withwhich to assess melt equilibration:

tD d2CR(S - a)tm 59LD(K)15

1o-8

[6]

For tD/tm ratios < 1, equilibration between melt and solid willbe >83%. From Eq. 6, equilibration is favored, as might beexpected, by small grain size, slow upwelling rates, lowsolidus slopes, large fusion enthalpies, large partition coef-ficients, and high diffusion coefficients.

Table 1 gives a list of the parameters and ranges of valuesappropriate to the mantle. The solidus of peridotite dependson pressure and bulk composition. If melting starts in thegarnet stability field as argued by Salters and Hart (3), thisrequires pressures in excess of 25-30 kbar (1 kbar = 100MPa), and at these pressures the solidus of fertile-to-mildlydepleted peridotite lies at temperatures of 1450-1500°C (12).The slope of the solidus averaged over the 20- to 30-kbinterval is close to 4°C/km; however, there may be a largevariation in slope depending on whether or not a cusp existsat the spinel facies -* garnet facies transition. Takahashi (13)and Takahashi and Kushiro (14) claim a cusp at this boundaryin two different peridotites; in contrast, Hirose and Kushiro(12) show that a linear 4°C/km solidus is consistent with theirnew melting data. The solidus slopes at the cusp in theTakahashi (13) data range from 1.5°C/km to 10°C/km. I willtake this range as the maximum likely range in S.

Table 1. Equilibration model parametersParameter Value(s)

d, grain size, cm 0.1-1.0C, peridotite heat capacity, cal/g°K 0.3R, mantle upwelling rate, cm/year 1-5000S, solidus slope, IC/km 1.5-10a, adiabat slope, °C/km 0.4L, enthalpy of fusion, cal/gram 185D, diffusion coefficient, cm2/sec* 1 x 10-11-2 x 10-9K, partition coefficientt 0.001-10*At 15000C.tSolid concentration/melt concentration.

10-9

lo-lo0a)

E0Cz

-1110

-1210

-1310

1600 1500Temperature, 'C

1400 1300 1200

5.5 6.0 6.51 04/K

FIG. 1. Arrhenius plot for experimentally measured diffusioncoefficients D of various elements in olivine, diopside, garnet, andspinel. Data are from refs. 15-23. Not shown are the data of Houlieret al. (24) for Si in olivine, which has a very low D at 1500'C ("2 x10-16 cm2/sec). Sr diffusion is shown for both a natural diopside anda synthetic diopside at 20 kb. Fo is pure forsterite; the spinel isMgAI204; Di CATS is interdiffusion between diopside and Ca-Tschermak's (19); the oxygen diffusion was in a natural Fo92 olivine.Comparable data for Ca, Fe, Mg, Mn, and Al in olivine is also citedby Jurewicz and Watson (25).

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0.03, and a diffusion coefficient D of 1x 10-10 cm2/sec. FromEq. 7, the "Peclet" number for this scenario is =2x 10-5; thisis well into the "equilibrated" domain and shows that meltingin spreading ridge situations will clearly take place underconditions of local equilibrium.We might ask if there are any conceivable conditions under

which disequilibrium melting might occur. For a worst-casescenario, I will model a very robust mantle plume. Fluiddynamic modeling of mantle plumes shows that upwellingvelocities may reach 50 m/year as an upper limit for non-Newtonian plumes with large buoyancy fluxes (31). Pushingall other parameters to the limits indicated in Table 1 (d = 1cm, S = 10°/km, D = 1x10-11 cm2/sec, K = 0.01) gives a"Peclet" number (from Eq. 7) of 43; this is mildly into thedisequilibrium domain and shows that mantle melting is likelyto always represent equilibrium except for extreme situationsof very rapid plume upwelling for highly incompatible ele-ments of lower-limit diffusion rates. Even for this worst-casescenario, changing just the partition coefficient from 0.01 to0.1 will bring the "Peclet" number down to order unity (i.e.,functional equilibrium).Im. Equilibrium During Melt TransportA number of authors have studied various aspects of chem-ical equilibrium and fractionation during melt transport in themantle (9, 32-38). Most of these studies dealt with meltmigration in either simple one-dimensional systems or inhomogeneous porous networks. As recognized by Spiegel-man and Kenyon (9) and emphasized above, equilibration ingrain-scale porous networks is so rapid that melts wouldcontinuously reequilibrate as they migrated to the surface.Since most melts in fact preserve deep geochemical signa-tures, some kind offocusing of melts into high-speed conduitsis required.While one can simply advocate a coarser vein or channel

network to solve this problem (9), the modeling is thensubject to ad hoc scale parameters. I will take a somewhatdifferent approach and argue that melt migration can bemodeled as a fractal tree network (or more correctly, a fractalroot system). This scheme involves scale-invariant pairwiseconfluence of melt conduits, starting at the finest scale withan intergranular melt tubule network and progressing to muchlarger-scale "thruways." Stevenson (39) has shown theoret-ically that partial melts which are undergoing deformation areunstable with respect to small-scale redistribution and aggre-gation. Grain-scale focusing of melt has also recently beenreported by Daines and Kohlstedt (40) from experimentalperidotite-melt solution experiments. On the largest scale,the fractal tree network will probably be terminated by veinand dike formation during magma fracturing (41-43).

Cortini (44) made the suggestion that magmatic plumbingsystems might be fractal, in analogy with the vascular systemof the human body (45). The human bronchial system is alsofractal (46), and the hydraulics of these systems have longbeen known to operate as minimal resistance or minimalenergy systems (47-49). This principle, known in somecircles as Murray's Law, states that, for a bifurcating system,the cube of the radius of the parent vessel equals the sum ofthe cubes of the radii of the daughter vessels. For a melttransport system, I envision melt conduits that combine bytwos, as sketched in Fig. 2, with conduit radius increasing ateach confluence by the factor 21/3 1.26. Since volume fluxis conserved, the migration velocity in each conduit gener-ation scales in the same way. I assume that matrix compac-tion is rapid enough to keep pace with melt removal from thenetwork (50, 51). Also, while modeled here as discrete "puremelt" tubes, the natural reality may be one of localizedhigh-porosity zones without sharp boundaries.

Radius r

Length I

GENERATIONNUMBER

FIG. 2. Sketch of fractal tree envisioned for melt transport.Radius r increases by 21/3 at each confluence; aspect ratio A isdefined as the length I divided by radius r.

By defining the conduit generation number as n, with theinitial (finest) generation taken as n = 0, then the radius r andvelocity v at any generation level in the system scale as

rn Vnr = n=v(2)n/

ro Vo[8]

which shows that radius and velocity increase by a factor of10 for every 10 generations.While it is reasonable to assert that nature will use a

minimal energy plumbing system of this type, the parameterthat is left without obvious constraint is the conduit length ofeach branch. I define an aspect ratio A as branch length/branch radius. For the human vascular and bronchial systems(52, 53), this aspect ratio is more or less constant at allgeneration levels, with typical values of 15 (vascular) and 7(bronchial). For a natural melt network, where flow is drivenby buoyancy forces (as opposed to muscular forces), thereseems to me to be no obvious a priori way to determine A.Perhaps a full fluid dynamic model will constrain A; other-wise, it may be possible to map melt networks in a fieldsetting by tracing the residual dunite/harzburgite zones invarious ultramafic complexes (41, 54, 55).For the moment, I will acceptA as a free parameter but will

assume it is constant for all generation levels. Becausevelocity, radius, and length then all obey the same scaling(Eq. 8), the time a parcel of melt spends in a given branch ofthe system is constant for all parts of the system. For thecylindrical diffusion model used earlier (see Eq. 1), the extentof equilibration En between melt and matrix in a given branchn is (for K = 1)

En =f(f [9]

wherefis the function given as equation 6 of Jaeger (11).Since t,, = to and rn = ro(2)n/3, then

1101

Because D, to, and ro will be constant for a given system,the ability of melt to maintain diffusive equilibrium with thematrix will decrease very rapidly, (2)-2n/3, as a melt parcelmigrates up the fractal network. To model this quantitativelyonly requires some means of determining to and ro. I do thisby noting that the intergranular melt tubule radius is acontinuous function of the melt fraction (Eq. 4), and the"sstanding" melt fraction in the n = 0 generation is a balancebetween melt generation rate and melt escape rate.The Poiseuille volume flux Pp through a pipe is

11916 Geology: Hart

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Trr4Apg

where r is the pipe radius, Ap is the density differencebetween melt and matrix, and a is the melt viscosity. Thetubule melt volume generation rate, V' from Eqs. 2 and 3 is

0.05d3CR(S - a)V= L [12]

The radius ro of the smallest melt tubules can then bederived as that pipe size that can just carry the flux of meltbeing produced. Equating Eqs. 11 and 12 gives

(0.127d3CR14S a) 0.25

ro= ApgL [13]The time t0 spent in the first (and every other) pipe is

derived from the Poiseuille velocity in the first pipe, from Eq.11, and the length (which is rOA) giving

Ar0 8A,ut0 = tn = =o *Apg [14]

VO roApg

The equilibration of a melt parcel per branch is then, fromEq. 10 and 14,

En =f 3DV L ) [15]

with ro given by Eq. 13.In a real network, one needs to know how long melt has

been flowing in the total system, since the matrix becomesprogressively depleted in a given element as the elementdiffuses into the melt and is carried away. For presentpurposes, I will model the incremental equilibration in eachpipe generation n as if that pipe had already been exposed tomelt for a time = nto. In other words, a melt parcel is trackedthrough successive pipe generations, spending to in eachgeneration and assuming each pipe had already been exposedto melt for a period of time equal to that required to get theparcel to that branch from the starting point.For large n, En will be zero. By summing the argument in

Eq. 15 backwards from large n, one can determine the pointat which melt stops equilibrating with matrix, the total pathlength traveled to that point, the pipe diameter, etc.For L and C and units as in Table 1, Eq. 13 and 15 give

1.96 x 1012DA ,uR \ 1/4En =f (2)2n/3 )tp9R3(S-a3 . [16]

Of the parameters not discussed in Section II, Ap isconstrained between 0.2 and 0.5 g/cm3. The viscosity ,u ofbasalt varies in opposing ways withP and T; along the solidusof peridotite, there is thus some tendency for the P and Teffects to cancel, and the variation of viscosity ofbasalt alongthe peridotite solidus is fairly restricted. Over a depth rangeof 100-40 km, the viscosity of olivine tholeiite is in the rangeof 10-30 poise (56, 57). As discussed earlier, the aspect ratiois not constrained except for the initial (n = 0) generation.From Eq. 13, ro is defined as a function of grain size (amongother things), where dlro is by definition the aspect ratio Aofor this initial generation. For reasonable choices of param-eters in Eq. 13, the aspect ratio Ao varies only from A4000 to10,000 (initial radii are typically 0.3-1.2 ,m).

In lieu of other constraints, I will take this value ofA as an

upper limit to A in higher generations of the fractal network.

Table 2 lists a selection ofparameters for four models chosento embrace likely mantle scenarios; Fig. 3 shows equilibra-tion curves for these four models. The most striking thingabout these curves is the rather limited path length traveledby a melt parcel before it loses "touch" with the solid matrix.I estimate the most likely parameter range to fall betweencurves B and C; once a melt parcel has traveled 60-600meters from its source, it will undergo <10% further equil-ibration during all of its remaining travel to the surface. Evenafter pushing all of the parameters to their extreme limits(curve D), melts will undergo little additional equilibration(<30%) once they have traveled 5km from their source. Notethat the distance traveled given in Fig. 3 is measured along aflow path and probably will not represent a net linear distancebecause of the tortuous nature of a fractal tree path.

While the curves in Fig. 3 were calculated for a partitioncoefficient K = 1, for incompatible elements (K < 1), thecurves will shift strongly to the left-i.e., melts will "losetouch" even sooner. To a reasonable approximation (for Kranging from 0.01 to 10), the distance at which melts ceaseequilibrating with the rock (E < 10%) scales as (K)0.83 inother words, every factor of 10 variation inK leads to a factorof :7 change in distance. For curves B and C, the point wheremelts stop equilibrating with matrix occurs after some 45-50generations of confluence; the conduit radii at this point arestill quite small (1-4 cm), and the melt velocities are quitehigh (50-100 m/year).IV. CommentaryThe picture that emerges from this fractal model is one inwhich melts are formed as an intergranular network in localequilibrium with the solid matrix. This network feeds a fractalsystem in which small conduits coalesce into ever largerconduits, with melt velocities increasing as the melt migratesupward. Within some hundreds of meters of its source, themelt ceases to undergo any further diffusive exchange withperidotite matrix, effectively "locking-in" a chemical signa-ture. These deep melts will of course mix with shallowermelts, as the shallower melts feed into higher levels of thefractal tree. However, each of the melt parcels in theseaggregate mixtures will represent its own local mantle do-main, probably no more than a few hundreds of meters inscale.Two mechanisms may operate to curtail the efficient

operation of this melt transport system. First, if the shallowconduits get "plugged" or are not effectively open to thesurface, melt can stagnate and undergo reequilibration withperidotite, thereby having its high-pressure chemical signa-ture erased. As mentioned earlier, I believe the buoyancyforces at this point would lead to magma fracturing (41, 42),and rapid transport of melt to the surface would take placealong dikes and veins.

Melt that initially forms in equilibrium with a four-phaseperidotite will be out of equilibrium with that same peridotiteat lower pressure because of the shift in phase equilibriumboundaries with pressure. What I have modeled here is thediffusive processes which attempt to maintain equilibrium.

Table 2. Fractal tree parameters for cases A-D

Parameter* A B C D

D 1 x 10-1 1 x -10 1 x 10-10 2 x 10-9A 1000 1000 5000 10,000,u 10 15 15 30Ap 0.5 0.3 0.3 0.2d 1.0 0.3 0.3 0.2R 20 5 5 1S 10 4 4 2

*Units are as in Table 1 for D, d, R, and S. Other units are A(dimensionless); AL, poise; and Ap, g/cm3.

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Proc. Natl. Acad. Sci. USA 90 (1993)

Cu

U)

.-

03)C

-0

._0~

Uo

ol

Distance traveled, m

As pointed out by Kelemen (54) and Kelemen et al. (55), therewill also be very active solution-redeposition processes oc-curring that act to maintain equilibrium. There will be sometrace element and isotopic redistribution that occurs duringthis solution-redeposition. Depending on the relative massesofmelt/rock that pass through this process, significant chem-ical reequilibration may occur (over and above that which Ihave modeled from a purely diffusive basis).

I appreciate the many stimulating and illuminating discussions Ihave had with P. Kelemen, G. Hirth, H. Dick, and other KeckGeodynamic devotees. I am grateful to K. Helfrich and E. Hauri fortheir careful reviews. National Science Foundation Grant EAR9096194 was also a big help. This is Woods Hole OceanographicInstitution contribution 8483.

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FIG. 3. Model curves showing the extentof diffusive equilibration between melt andmatrix as a function of distance traveled alonga fractal tree. Curves are derived from param-eter values listed in Table 2 (with K = 1) byusing Eq. 16 and the cylindrical diffusion func-tion of Jaeger (11). Curves A and D representparameter choices that give extreme lower andupper bounds to possible mantle melting sce-narios; curves B and C delimit the most likely

i=j bounds. Numbers at the tic marks give conduit5000 radius (in cm) to the left and generation num-

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