the energetics of melting fertile heterogeneities within the...

Volume 12, Number 10Article

N21 October 2011

Q0AC16, doi:10.1029/2011GC003834

ISSN: 1525‐2027

The energetics of melting fertile heterogeneitieswithin the depleted mantle

Richard F. KatzDepartment of Earth Sciences, University of Oxford, South Parks Road, Oxford OX1 3AN, UK([email protected])

John F. RudgeInstitute of Theoretical Geophysics, Bullard Laboratories, University of Cambridge, Madingley Road,Cambridge CB3 0EZ, UK ([email protected])

[1] To explore the consequences of mantle heterogeneity for primary melt production, we develop amathematical model of energy conservation for an upwelling, melting body of recycled oceanic crustembedded in the depleted upper mantle. We consider the end‐member geometric cases of spherical blobsand tabular veins. The model predicts that thermal diffusion into the heterogeneity can cause a factor‐of‐two increase in the degree of melting for bodies with minimum dimension smaller than ∼1 km, yieldingmelt fractions between 50 and 80%. The role of diffusion is quantified by an appropriately defined Pecletnumber, which represents the balance of diffusion‐driven and adiabatic melting. At intermediate Pecletnumber, we show that melting a heterogeneity can cool the ambient mantle by up to ∼20 K (spherical)or ∼60 K (tabular) within a distance of two times the characteristic size of the body. At small Pecletnumber, where heterogeneities are expected to be in thermal equilibrium with the ambient mantle, wecalculate the energetic effect of pyroxenite melting on the surrounding peridotite; we find that each 5%of recycled oceanic crust diminishes the peridotite degree of melting by 1–2%. Injection of the magma fromhighly molten bodies of recycled oceanic crust into a melting region of depleted upper mantle may nucleatereactive‐dissolution channels that remain chemically isolated from the surrounding peridotite.

Components: 11,900 words, 9 figures, 1 table.

Keywords: eclogite; heat conduction; magma; mantle heterogeneity; partial melting; pyroxenite.

Index Terms: 1011 Geochemistry: Thermodynamics (0766, 3611, 8411); 1037 Geochemistry: Magma genesis and partialmelting (3619); 5134 Physical Properties of Rocks: Thermal properties.

Received 12 August 2011; Revised 16 September 2011; Accepted 18 September 2011; Published 21 October 2011.

Katz, R. F., and J. F. Rudge (2011), The energetics of melting fertile heterogeneities within the depleted mantle, Geochem.Geophys. Geosyst., 12, Q0AC16, doi:10.1029/2011GC003834.

Theme: Geochemical Heterogeneities in Oceanic Island Basalt and Mid-ocean RidgeBasalt Sources: Implications For Melting Processes and Mantle Dynamics

Guest Editors: C. Beier and P. Asimow

Copyright 2011 by the American Geophysical Union 1 of 22

http://dx.doi.org/10.1029/2011GC003834

1. Introduction

[2] It is well established that the mantle is chemi-cally heterogeneous on length‐scales smaller thanthe 100 km typical size of the melting regionbeneath a plate‐boundary or hot spot volcano [e.g.,Hofmann, 1997]. Despite this, most studies ofmelting at plate tectonic boundaries and hot spotsstill treat the mantle as a homogeneous source.Such models, irrespective of their sophistication,probably miss crucial aspects of basalt petrogene-sis; they may therefore lead to incorrect inversionsof geochemical data for mantle properties andprocesses. Treatment of melting of a lithologicallyheterogeneous mantle remains largely qualitative,lacking the rigor necessary for quantitative geo-chemical modeling. The present work seeks toaddress this deficiency.

[3] A range of studies have inferred that hetero-geneities are formed from previously subductedoceanic and continental crust and lithosphere [e.g.,Hofmann and White, 1982; Willbold and Stracke,2010] that has been stirred into the mantle overgeologic time. This stirring continuously reduces thecharacteristic size of heterogeneities [Hoffman andMcKenzie, 1985; Allègre and Turcotte, 1986], andincreases the rate of homogenization by solid‐statediffusion. The surviving heterogeneities are thosethat remain larger than the decimeter scale overwhich diffusion would erase chemical variations onthe billion‐year time‐scale of mantle convection[Hofmann and Hart, 1978]. Many of these have apyroxene‐rich lithology that may contain garnet,spinel, and other accessory phases, but has little or noolivine. In this paper, we do not attempt to distinguishbetween the pyroxenites, garnet pyroxenites, andeclogites that are included in this range of lithologies;rather we generalize them to a class of mantle het-erogeneity composed of fertile, recycled crustalrocks. We do not directly address other classes ofmantle heterogeneity such as recycled sediments.

[4] Within the class considered here, the size‐distribution and concentration of heterogeneitiesthroughout the mantle remains the subject of con-troversy. Geochemical measurements and modelsput constraints on the proportion of enrichedmaterials in the source regions of mid‐ocean ridges.Hirschmann and Stolper [1996] used a variety oftrace element and isotopic measurements to estimatethat the MORB‐source contains 3–6% garnetpyroxenite. Melting experiments by Pertermannand Hirschmann [2003a] more accurately deter-mined the productivity of upwelling pyroxenite,

and lead them to suggest 2–3% pyroxenite, thoughthey note that this result is sensitive to the assumedfertility of the pyroxenite. Sobolev et al. [2007]studied trace element and forsterite content in oliv-ine phenocrysts from MORB, OIB, komatiites, andwithin‐plate lavas and suggested that variations inMn and Ni content, among other indicators, call forabout 5% recycled oceanic crust in theMORB sourceand 20% in the OIB source. This is approximatelyconsistent with estimates by Ito andMahoney [2005],determined by modeling the melting process for rid-ges and plumes, and seeking the range of startingcompositions that can explain both OIB and MORBgeochemical and isotopic systematics. Each of thesestudies used a different approach to quantify thecontribution of recycled oceanic crust, yet theirresults are similar; it seems reasonable to expect thatenriched heterogeneities compose less than one fifthof the mantle that melts at most ridges and hot spots.

[5] Constraining the sizes and shapes of mantleheterogeneities has proven more difficult thandetermining their relative proportion. Seismicscattering techniques described by Helffrich [2006]provide support for blobs of mineralogically dis-tinct heterogeneities with a characteristic scale of] 10 km, distributed uniformly throughout themantle. Kogiso et al. [2004] argued for a size of^ 1 m, which they calculated based on the obser-vational constraint of radiogenic osmium sig-natures in rocks recovered from mid‐ocean ridgesand hot spots. Yasuda and Fujii [1998] noted thatthe negative buoyancy of eclogite blobs means thatblobs of diameter 40 km or larger cannot ascendthrough the upper mantle, though the density thatthey assumed for eclogite may have been too large[Pertermann and Hirschmann, 2003b]. Others haveargued for heterogeneity of equal magnitude at allscales based on observed geographical variation inthe isotopic ratios of lavas [e.g., Gurnis, 1986].

[6] Since fertile heterogeneities are a small fractionof mantle volume, and are apparently dispersed atsizes smaller than 10 km, we study their meltingsystematics using an idealized model of an isolatedheterogeneity. Such a heterogeneity would initiallybe in thermal equilibrium with the surroundingmantle, because the rate of thermal diffusion is fastrelative to that of convective overturning. Becauseof its composition, it would begin decompressionmelting at higher pressures than the ambient, moredepleted mantle. This melting requires energy toconvert solid to liquid, and it diminishes the tem-perature of the blob relative to the surroundingmantle. The associated temperature gradient drivesheat flow into the blob, and increases the melting

GeochemistryGeophysicsGeosystems G3G3 KATZ AND RUDGE: MELTING FERTILE MANTLE HETEROGENEITIES 10.1029/2011GC003834

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rate, while cooling the surrounding mantle. What isthe magnitude of this effect? This question was firstaddressed by Sleep [1984], who developed modelsof heat conduction into a fertile, melting sphere andtabular body.

[7] Sleep [1984] considered a fertile heterogeneityundergoing decompression melting at an upwellingrate of 3 cm/yr and calculated the size it must havesuch that the characteristic thermal‐diffusion timeis approximately equal to its duration in the meltingcolumn, obtaining a value of 5 km. Moreover, herecognized that heterogeneities much smaller thanthis size would melt in thermal equilibrium with thesurrounding mantle, while those much larger wouldmelt in thermal isolation. Sleep [1984] then deriveda mathematical model of partial melting of suchheterogeneities based on diffusive heat flow andparameterized thermodynamic properties. Analysisof this model lead him to conclude that thermaldiffusion could cause an enhancement of meltingby up to a factor of seven over the thermally iso-lated case. This extremely large enhancementcontrasts with the results that we present below.

[8] In the present paper we take a similar approachto Sleep [1984]—we consider thermal diffusioninto a fertile heterogeneity of spherical or tabularshape—but we provide a more comprehensivestudy and obtain significantly different results. Ourapproach is more rigorous in that we derive anequation for conservation of energy that explicitlycouples melting with thermal diffusion. In contrastto Sleep [1984], we obtain full analytical solutionsin terms of non‐dimensional parameters, and usethem to elucidate the controls on and limits ofpartial melting of heterogeneities. Furthermore, onempirical [Pertermann and Hirschmann, 2003a]and theoretical grounds [Hirschmann et al., 1999],we consider the nonlinear relationship betweenmelt fraction and temperature, where Sleep [1984]assumed linearity. For the nonlinear case, we relyon numerical solutions of the governing equations;these are documented and validated in Appendix B.Our calculations are based on parameter valuesfrom published experiments on G2 pyroxenite[Pertermann and Hirschmann, 2003a], which has acomposition thought to approximate recycled oce-anic crust. We compare the results of these calcu-lations with those obtained by Sleep [1984],who used different and probably unrealistic para-meters for pyroxenite melting (for example, Sleep[1984] assumed a near‐solidus isobaric productiv-ity ∂F/∂T∣P that is a factor of five to twenty timeslarger than the value obtained empirically byPertermann and Hirschmann [2003a]).

[9] Our work is also related to that of PhippsMorgan [2001], who considered narrow, multilay-ered tabular veins of fertile pyroxenite withindepleted peridotite, and computed melting behaviorfor a variety of compositional scenarios. Heassumed thermal equilibrium between all the layersin the model. Similar models were developed byHirschmann and Stolper [1996] and Stolper andAsimow [2007]. And whereas these authors’ workfocused on the effects of melting a compositionalmélange with variable solidus (and solidus slope)between solid phases, here we are interested inaccounting for the finite time‐scale of thermalequilibration. Our model is not restricted to any oneset of pyroxenite melting parameters, though in thispaper we limit consideration to a single examplethat we hope is representative.

[10] The relative simplicity of the physical model isbased on a number of key assumptions. The mostimportant of these is that heterogeneities are iso-lated from one another, such that their thermaldisturbances do not interact. We further assumethat the characteristic size of a heterogeneity issmall relative to the vertical distance between thedepth at which it reaches its solidus and the depthof the ambient mantle solidus. This latter assump-tion allows us to approximate the lithostatic pres-sure as being constant within the heterogeneity. Weassume that heterogeneities upwell at the samespeed as the ambient mantle; other authors haveconsidered the possibility that chemically dense,unmolten, recycled material upwells at a slowerspeed due to its negative buoyancy [Yasuda andFujii, 1998; Pertermann and Hirschmann, 2003b].Finally, we assume that melt does not segregatefrom the host rock, which is the thermodynamicequivalent of assuming batch melting. If theambient mantle remains unmolten as an embeddedheterogeneity partially melts, magma may be heldwithin the heterogeneity by a permeability barrier.Alternatively, it may chemically react with thesurrounding rock [Yaxley and Green, 1998; Kogisoet al., 2004], propagate through it by diking, oropen the pores through surface‐energy driven flow[Riley and Kohlstedt, 1991]. Even a leaky perme-ability barrier around a fertile enclave would sig-nificantly restrict magmatic segregation if the scaleof the sphere is smaller than the compaction length[Spiegelman, 1993]. For the case of a tabular vein,these arguments are clearly tenuous; we thereforejustify the assumption on grounds of mathematicalconvenience, and flag this as an issue to be remediedin future work. These assumptions enable us to for-mulate a mathematical model that admits analytical


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solution in some cases, and high‐accuracy, efficientnumerical solutions in others. We can thus provide athorough exploration of the model behavior, anddraw conclusions that will inform more detailedwork in which assumptions are relaxed.

[11] Although this paper focuses on the energeticsof pyroxenite melting, the results also have geo-chemical implications. It is well‐known that garnetis an important phase in recycled oceanic crust, andthat it has a unique set of affinities for trace ele-ments [e.g., Stracke et al., 1999; Pertermann et al.,2004]. The trace element budget of melts derivedfrom pyroxenite is therefore sensitive to the pres-ence of residual garnet; the persistence of garnet inthe residue depends on pressure, but it also dependson the degree of melting. The work presented hereprovides a framework for evaluating the degree ofmelting of pyroxenite enclaves within the depleted,peridotite upper mantle; it also highlights the dif-ficulties in inverting geochemical data for thephysical characteristics of mantle heterogeneity byshowing that the size and shape of heterogeneitieshas a significant influence on their melting behavior.Perhaps a larger difficulty is that the pathways ofmelt transport from deep‐melting, fertile hetero-geneities to the surface are poorly constrained, and itis increasingly understood that the details of melttransport have a crucial impact on observed geo-chemical patterns [e.g., Spiegelman and Kelemen,2003; Liang et al., 2011].

[12] The next section describes the idealizedphysical scenario to be analyzed, and lays out amathematical formulation of the problem. TheResults section then presents non‐dimensionaloutput from analytical and numerical solutions. TheDiscussion section examines these results in more

detail and provides a subset of them in dimensionalform. We summarize our findings briefly at theend, and give the details of analytical and numer-ical methods in two appendices. Throughout thispaper, the focus is on the slightly more complicatedcase of a spherical heterogeneity, however in theDiscussion section, we compare our results forspherical and tabular bodies.

2. The Mathematical Model

[13] Figure 1 is a schematic representation of themodel set‐up. We consider two end‐member cases:a spherical blob of radius R and an infinite tabularvein of half‐width R. Both are made of a uniform,fertile lithology, and are embedded within thedepleted upper mantle, upwelling at speed W > 0.We develop the mathematical model using the caseof the spherical blob and present the case of atabular vein in Section 2.3.

[14] We assume that the blob is small enough thatwe can neglect vertical variation of all propertieswithin it. This assumption is awkward for the tab-ular vein, but not unreasonable if we consider thattemperature gradients (and hence heat flow) normalto the vein will greatly exceed those within it. Asthe heterogeneity ascends, it reaches a pressurep0 = rgz0 where its temperature is equal to thesolidus temperature T0; we label this moment t = 0(r is the density of all materials under consider-ation; g is the acceleration due to gravity; both areassumed constant). The ambient mantle is alsoupwelling with speed W, and it reaches its solidustemperature T1 at a shallower pressure p1 = rgz1(and hence at a later time t = t1). We investigate thetime interval between 0 and t1, when the blob is

Figure 1. Schematic diagram of the upwelling column with idealized fertile blob and fertile tabular vein. The blob isa sphere in three dimensions, while the tabular vein is an infinite sheet. The fertile material crosses its solidus at z = z0and t = 0; far‐field ambient mantle crosses its solidus at z = z1 and t = t1.


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partially molten but the ambient mantle is entirelysolid.

[15] Motivated by the experimental results ofPertermann and Hirschmann [2003a], the solidusand liquidus temperatures of the blob are taken todepend on pressure only:

Ts pð Þ ¼ T0 þ ��1 p� p0ð Þ; ð1aÞ

Tl pð Þ ¼ T0 þ ��1 p� p0ð Þ þDT ; ð1bÞ

where DT is a constant and g is the Clapeyronslope. Because of the uniform upward motion, thepressure experienced by the blob changes with timeaccording to

p tð Þ ¼ p0 � �gWt: ð2Þ

[16] The coordinate system is fixed to the center ofthe spherical heterogeneity, which has a radius R.The goal is to determine the temperature field T(r, t)outside the blob as it melts, and to calculate theaverage temperature and melting rate within theblob. Magmatic segregation can be neglected ifthe blob is much smaller than the compactionlength within it. This will be true for smaller valuesof R, but since we are concerned only with theaveraged melting properties of the blob, we expectthat mass redistribution within the blob would haveonly a small effect on our results.

[17] With the above assumptions, conservation ofenergy is

@H

@tþ �gW ¼ kr2T ; ð3Þ

where rgW is the rate of change of potential energyand

dH ¼ �L dF þ �cp dT þ 1� �Tð Þ dp ð4Þ

represents an infinitesimal change in bulk enthalpyin terms of its contributing parts. F here is thevolume fraction of melt, and the remaining sym-bols are defined in Table 1. Combining (2), (3), and(4) gives

�L@F

@tþ �cp

@T

@t� �T

@p

@t¼ kr2T : ð5Þ

[18] We will non‐dimensionalize with the follow-ing scales

x½ � ¼ R; t½ � ¼ Dp

�gW; ð6Þ

and define the non‐dimensional temperature andpressure as

� ¼ �

DpT � T0ð Þ; P ¼ p� p0

Dp; ð7Þ

whereDp = p0 − p1 and the temperature scale is thechange in the solidus temperature from p0 to p1.Using these scales and linearizing the adiabaticgradient about T = T0 gives

S @F

@tþ @�

@tþA ¼ 1

Per2�; ð8Þ

where the Peclet number is

Pe ¼ �R�a

¼ �gWR2

�Dp;

Table 1. Dimensional and Non‐Dimensional Parameters

Parameter Value or Range Units Comment

a 0.25 – 1 ‐ Coefficient in melting relation (11)cp 1200 J kg−1K−1 Specific heat capacityg 10 m s−2 Gravitational accelerationk = rcp� J K−1m−1s−1 Thermal conductivityL 4 × 105 J kg−1 Latent heat of meltingR 10−2 – 103 km Characteristic size of heterogeneityT0 1623 K Reference melting temperatureW 10−1 – 103 cm a−1 Upwelling ratea 3 × 10−5 K−1 Thermal expansivityg 8.3 × 106 Pa K−1 Clausius‐Clapeyron slope (120 K/GPa)Dp 1.7 × 109 Pa Pressure interval (110–60 km depth)DT 250 K T‐difference between liquidus and solidus� 10−6 m2s−1 Thermal diffusivityr 3300 kg m−3 DensityA 0.1 ‐ Adiabatic parameterD� 1.2 ‐ Melting‐temperature intervalPe ‐ Peclet numberS 1.7 ‐ Stefan number


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the Stefan number is

S ¼ L�

cpDp;

and the adiabatic parameter is

A ¼ ��T0�cp

:

All symbols in equation (8) represent non‐dimensional quantities. The Peclet number is theratio of the time‐scale tR = R2/� for diffusion ofheat across the blob to the time‐scale ta =Dp/(rgW)for advection of the blob from p0 to p1. It isthe principal control parameter in the problem; forPe → 0 the diffusive heat transport dominates thethermal budget and the blob is in thermal equilib-riumwith the surroundingmantle, while for Pe→∞,diffusion is negligible relative to advective trans-port, and the blob melts adiabatically.

[19] The adiabatic parameter is the linearized rateof energy consumption by adiabatic expansion. Weneglect the term associated with the adiabaticparameter in what follows. In A5 we show thatincorporating this term gives rise to the usual adi-abatic temperature gradient, and introduces a factorof (1 − A) into the melt productivity.

[20] With the above scaling, the equations for thesolidus and liquidus (1) become

�s ¼ P ¼ �t; ð9aÞ

�l ¼ P þD� ¼ �t þD�; ð9bÞ

where D� = gDT/Dp is the non‐dimensional,constant temperature offset between solidus andliquidus. The non‐dimensional pressure is given byP(t) = − t with 0 ≤ t ≤ 1.

[21] We can put rough constraints on all of thematerial parameters, as given in Table 1, but cannotprescribe the radius of the blob or its upwellingrate. However, these latter two parameters arecombined in the Peclet number, so we need explorethe variation in only a single dimensionlessparameter. Note that a value ofD� larger than unityindicates that the temperature difference betweenthe solidus and the liquidus of the blob is largerthan the temperature difference between its solidusfor non‐dimensional pressures P = 0 and P = −1.This means that for D� > 1 and t ≤ 1, the blobcannot melt to F = 1. In the current work we do notconsider the case of a fully molten heterogeneity,which can only occur for t ≤ 1 if D� ≤ 1.

2.1. Within the Blob

[22] Since we are interested in melting of a fertileblob of recycled material, we first formulate equa-tions that capture its melting properties.

2.1.1. Simplified Melting Relations for Fertile,Recycled Oceanic Crust

[23] When the temperature within the blob is abovethe solidus, a dimensionless normalized tempera-ture is

Q ¼ �� sD�

; when �s � � � �l: ð10Þ

[24] Pertermann and Hirschmann [2003a] found,for the anhydrous pyroxenite composition G2(N.B. Pertermann and Hirschmann [2003a] use theterm pyroxenite to describe all pyroxene‐rich,olivine‐poor mantle heterogeneities), that the rela-tionship between the non‐dimensional normalizedtemperature Q and the degree of melting can berepresented as

F ¼ aQþ 1� að ÞQ2; ð11Þ

where 0 ≤ a ≤ 1. Their data was best fit for a ≈ 1/4.Their pyroxenite composition, chosen to be similarto typical oceanic crust, cannot represent all flavorsof mantle heterogeneity. It is, instead, a well‐characterized and important example. We assert thatthe melting consequences of other compositions canbe investigated within the mathematical frameworkestablished below, by repeating our calculationswith modified values for material constants. Forexample, one could model the presence of volatileelements in the pyroxenite by modifying para-meters T0, p0 and a to produce a “tail” of low‐Fmelting at high pressure [Hirschmann et al., 1999].

2.1.2. The Degree of Melting of the Blob

[25] To simplify the analysis, we now assume ahomogeneous distribution of temperature and meltfraction within the blob, and define these as �B andFB respectively. This approximation will hold atsmall Peclet numbers; in Section 3.3 we relax thisconstraint and consider radially variable blobs.Proceeding with the averaged quantities �B and FB,we can integrate equation (8) over the non‐dimensional volume of the blob V = 4p/3. Thisgives

S @FB

@tþ @�B

@t¼ 3

4�

1

Pe

ZSrrr� � dS; ð12Þ


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where we have used the divergence theorem toconvert the volume integral into a surface integral.We can evaluate this integral in spherical coordi-nates as the spherically symmetric gradient in theradial direction times the non‐dimensional surfacearea of the blob 4p. We then integrate with respectto time to obtain

SFB þ �B ¼ 3

Pe

Z t

0

@�

@r

��1;�

d�: ð13Þ

[26] Both �B and FB can be expressed in terms ofQB by using (10) and (11). Equation (13) becomes

D�þ aSð ÞQB þ 1� að ÞSQ2B ¼ t þ 3

Pe

Z t

0

@�

@r

��1;�

d�; ð14Þ

which relates the homologous temperature insidethe blob to the integrated heat flux into the blobfrom the ambient mantle. The above equation canbe solved for QB; the result is

QB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4LfB

p � 1

2L; ð15Þ

where

L ¼ 1� að ÞSD�þ aS ; fB ¼ 1

aS þD�t þ 3

Pe

Z t

0

@�

@r

��1;�

d�

!:

ð16Þ

In the linear limit where a → 1 (L → 0),equations (15) and (16) simplify to

QB ¼ fB ¼ 1

S þD�t þ 3

Pe

Z t

0

@�

@r

��1;�

d�

!: ð17Þ

This linear case can be treated analytically; detailsare provided in A.

2.2. Outside the Blob

[27] Outside the blob, the mantle is below its soli-dus and hence F = 0 . Equation (8) becomes

@�

@t¼ 1

Pe

1

r2@

@rr2@�

@r

� �; ð18Þ

where we have chosen spherical coordinates andused the symmetry of the problem to discard terms.

[28] Equation (18) has boundary conditions

� 1; tð Þ ¼ �t þQBD�; ð19aÞ

� ∞; tð Þ ¼ 0; ð19bÞ

where QB is given by (15) and (16). The first ofthese boundary conditions represents the continuity

of temperature at the surface of the blob, and thesecond represents the constant far‐field temperatureof the ambient mantle. The initial condition isuniform temperature,

� r; 0ð Þ ¼ 0: ð20Þ

2.3. A Tabular Heterogeneity

[29] The governing equations are simplified slightlyif we consider a tabular heterogeneity of half‐widthR and infinite extent; equations (14) and (18) arereplaced by

D�þ aSð ÞQB þ 1� að ÞSQ2B ¼ t þ 1

Pe

Z t

0

@�

@x

��1;�

d�; ð21Þ

@�

@t¼ 1

Pe

@2�

@x2; ð22Þ

where we have taken the x‐axis in the directionnormal to the tabular body. The boundary conditions(19) are unchanged, except for use of the modifiedsolution for QB. As with the spherical blob, weneglect magmatic segregation for the tabular body.A model similar to this was considered by Sleep[1984]. In the Results section, we limit our atten-tion to solutions for the spherical blob. The tabularvein is reintroduced in the Discussion section, wherewe examine the consequences of blob shape onmelting behavior, for these two end‐member cases.

[30] For a spherical (or tabular) blob, the solution isobtained by solving the system of equations (14)and (18) (or (21) and (22)), with (16), (19), and(20) on the domain 1 ≤ (r, x) < ∞ and 0 ≤ t ≤ 1 forgiven values of a, Pe, S, and D�.

[31] In the end‐member cases of melting for infiniteand zero Peclet number corresponding, respec-tively, to purely adiabatic and purely isothermalmelting, it is not necessary to solve for the tem-perature structure outside the blob [Sleep, 1984].Figure 2 shows a schematic representation of thetemperature–pressure path taken by the blob inthese two cases, and illustrates the thermodynamicphase diagram (for a = 1). Solutions for interme-diate Peclet numbers follow paths between the end‐member curves, but must be obtained throughanalytical or numerical solutions to the governingequations. These are presented in the next section.

3. Results

[32] In this section we present results from bothanalytical and numerical solutions of the governing


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equations for a spherical blob. We fix all thedimensionless parameters except the Peclet num-ber, which captures the variation in both upwellingrate and blob size. Solutions are presented for arange of Peclet numbers.

3.1. Linear Melting Approximation

[33] When a = 1, the boundary condition describedby (17) and (19a) is linear and an analytical solutioncan be obtained using Laplace transforms (seesection A1). These are plotted in Figure 3 for dif-ferent values of the Peclet number. Figure 3a shows

the degree of melting of the blob as a function oftime. The range of curves is bounded by dashedlines for solutions at asymptotic values of Pe.When Pe → 0, the blob is in thermal equilibriumwith its surroundings. In this case, the degree ofmelting is determined entirely by the Clapeyronslope and the relationship between temperature anddegree of melting in (11). Blobs of radius O(1 km)or smaller would behave according to the small‐Peclet limit. When Pe → ∞, the blob is a closedthermodynamic system, exchanging no heat withthe surrounding mantle. In this case, the degree ofmelting is controlled by the decreasing solidus

Figure 2. Thermodynamic analysis of the melting of the blob in the two extreme cases of complete thermal isolation(Pe = ∞) and complete thermal equilibration (Pe = 0). Diagram based on Sleep [1984, Figure 5]. Orange and greenlines represent temperature‐pressure paths for these extremes. Linear solidus and liquidus lines are assumed (red andblue lines), the dependence of melt fraction on homologous temperature is taken to be linear (i.e., a = 1), and weneglect the adiabatic temperature gradient (i.e., A = 0). Degrees of melting are depicted by dotted contours.

Figure 3. Analytical solution for equations (17)–(20) with a = 1 and other parameters as in Table 1. Curves arelabeled with log10 Pe. (a) The mean degree of melting within the blob as a function of time, for different values of Pe.The red dashed curves are the limiting cases of Pe → 0 and Pe → ∞. (b) The dimensionless temperature disturbance∣�(r, 1)∣ caused by diffusion into the blob as a function of dimensionless radius at the dimensionless time t = 1.


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temperature and the latent‐heat cost of melting.Blobs of radius O(100 km) or larger would behaveaccording to the large‐Peclet limit. Further dimen-sional considerations are deferred to the discussionsection, below.

[34] For finite values of Pe, the ambient thermalstate, �(r, 0) = 0, is altered by diffusion of heat intothe relatively cool blob. The magnitude of thedisturbance is plotted in Figure 3b for t = 1. Theamplitude of the near‐field disturbance grows withincreasing Pe, while the decay length of the dis-turbance decreases. In other words, when diffusionis unimportant (Pe → ∞), temperature differencesfrom the background are large, but the radius of thethermal halo is small. When diffusion is important(Pe → 0), the opposite is true. In the latter case,where diffusive equilibrium is reached in infinites-imal time, we can make the quasi‐steady approxi-mation, setting ∂�/∂t ≈ 0 in (18) to obtain

0 � 1

r2@

@rr2@�

@r

� �: ð23Þ

Inspection of this equation shows that solutionsmust take the form �(r, t) = �B(t)/r. This predictsthat at small Peclet number, the thermal disturbanceclose to the blob should scale as r−1 (Figure 3).

3.2. Nonlinear Melting

[35] Melting experiments on pyroxenite byPertermann and Hirschmann [2003a] are best fitwith a ≈ 1/4, so we now examine solutions for thatcase. Since equation (14) is then nonlinear, wemust rely on numerical methods to solve the

problem (see Appendix B). Figure 4 shows theresults of a suite of calculations for different valuesof Pe, with a = 1/4. As with the case of a = 1, thecurves for FB(t) are bounded above and below byasymptotes for Pe → 0 and Pe → ∞, respectively.The latter case, representing adiabatic melting,results in a temperature evolution given by

QB t; Pe ! ∞ð Þ ¼ 1

2L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4Lt

D�þ aS

r� 1

!; ð24Þ

which can be obtained from (15) and (16) byneglecting the term describing conduction of heatinto the blob. The Pe → 0 case represents meltingin thermal equilibrium with the surrounding man-tle, and hence at the fixed temperature � = 0. Using(9a) and the definition of Q in (10) we obtain

QB t; Pe ! 0ð Þ ¼ t

D�: ð25Þ

Equations (24) and (25) can be substituted into themelting relationship (11); the results are plotted inFigure 4 as red, dashed lines.

[36] Figure 4b shows that the perturbation to theambient temperature field surrounding the blob isnot significantly different from the linear case. Forsmall Peclet numbers, the temperature perturbationfalls off as r−1, in accordance with a quasi‐steadyapproximation (23) of the diffusion equation.

3.3. Radially Variable Melting Withinthe Blob

[37] At small Peclet number, diffusion efficientlytransports heat to the center of the blob, and neutralizes

Figure 4. Numerical solutions for equations (15), (16), and (18)–(20) with a = 1/4 and other parameters as inTable 1. Each curve is labeled with log10 Pe. Details as in Figure 3.


9 of 22

any temperature gradients. In contrast, when thePeclet number is O(1), the time‐scale for diffusionof heat across the blob is comparable to the time forvertical advection through the domain. In this case,we expect a significant temperature gradient acrossthe radius of the blob, with warmer temperatures atthe edge and cooler temperatures at the center. Forlarger Peclet numbers, diffusive heat transport isinefficient, and the temperature gradient does notpenetrate far into the blob, but rather yields highertemperatures and melt fractions in a rim at the edgeof the domain. To capture the behavior for Pe ^ 1,we relax the assumption of homogeneous propertiesFB(t) and �B(t) within the blob, and instead calculateradially variable properties F(r, t) and �(r, t). This isthe approach taken by Sleep [1984], although hiscalculations used 1‐dimensional Cartesian ratherthan spherical geometry. In this section we consideronly the linear melting relation, a = 1; based on thesimilarity of Figures 3 and 4, we infer that com-parable behavior would be observed for the non-linear case. For a = 1, the Laplace transform method

can be used, with a numerical scheme to invert thetransform (details and references in section A2).

[38] Figure 5 shows the results of calculations for aradially variable blob. Figure 5a illustrates thediscussion in the preceding paragraph about thestructure of the blob at Peclet numbers around orabove unity. The degree of melting shows a radialdependence at Pe ≈ 1 and this gradient steepens andlocalizes with increasing Pe. Figure 5b shows theevolution of the radially resolved degree of meltingwith time, for Pe = 1. From an early state that isnearly homogeneous in F, the diffusive heat fluxpenetrates progressively further into the blob withtime, as shown in Figure 5d, and establishes atemperature gradient that spans the radius of theblob. Figure 5c compares the final temperaturedistribution within and outside the blob for a rangeof Peclet numbers. Section 4.1 includes a quanti-tative comparison of the radially resolved modeland the radially averaged model, showing that

Figure 5. Radial profiles of melt fraction F(r, t) and temperature �(r, t) through the blob with a = 1 (compare withFigure 6 of Sleep [1984]). (a) F(r, t) at t = 1 for different values of Pe. Red, dashed lines are the asymptotic valuesof F(r < 1, t) for Pe → 0, ∞. (b) F(r, t) at Pe = 1 for different values of t. (c) �(r, t) at t = 1 for different values ofPe. (d) �(r, t) at Pe = 1 for different values of t.


10 of 22

differences are relatively small, but are maximizedfor 1 ] Pe ] 10.

[39] We obtain a parallel set of results (not shown)for the case of the tabular vein, with moderate butsystematic differences from those above. Part of thediscussion below is a comparison of results forspherical and tabular models.

4. Discussion

[40] In this section we make a detailed examinationof results for spherical and tabular heterogeneities.Before doing that, however, we compare thebehavior of the radially averaged spherical model(Figures 3 and 4) with the radially resolvedspherical model (Figure 5). Most of this Discussionsection compares results as a function of the Pecletnumber, but at the end, we interpret the resultsin terms of dimensional heterogeneity size andupwelling rate. At that point, we also provide aquantitative comparison with the results obtainedby Sleep [1984].

4.1. Model Behavior as a Function of PecletNumber

[41] Section 3.3 showed that at intermediate andlarge Peclet numbers, the spherical blob can havesignificant radial structure. What error do we make,then, in assuming a homogeneous distribution ofmelting a priori? Figure 6 shows that in terms ofmean quantities, this difference is rather small (the

radially resolved model has been averaged over thespherical blob to produce an a posteriori mean). Ingeneral, the figure shows that the assumption of ahomogeneous blob leads to a larger melt produc-tion and larger temperature contrast. This isbecause imposing homogeneity within the blob isequivalent to requiring that diffusion is infinitelyfast there, leading to perfect redistribution ofenergy. Without this assumption, the rim of thesphere warms more rapidly than the core, andinsulates it from inward diffusion of heat. Thedifference is largest at intermediate Peclet numbersbecause there, as shown in Figure 5, the gradient inF and � spans much or all of the spherical blobradius.

[42] The maximum difference for the mean degreeof melting between the two models is about 5%,and indicates that a blob at Pe ≈ 1 may melt to∼60% rather than ∼65% (for a = 1). In the presentidealized context, this difference may be consid-ered insignificant. Similar models that assumeprogressive removal of melt from the blob (e.g.,fractional melting), or that are concerned with thechemistry of individual “packets” of melt producedwithin the blob, might find the assumption ofhomogeneity to be problematic, especially for largePeclet numbers.

[43] How does melting vary as a function of Pecletnumber? Figure 7 provides an answer in terms ofthe mean degree of melting (Figure 7a) and tem-perature (Figure 7b) inside the heterogeneity. Evi-dently, for Peclet numbers smaller than ∼10−1,

Figure 6. Difference between the radially averaged FB(t) and the radially resolved F(t) solutions for a = 1 at timet = 1. For comparison, the radially resolved solution has been averaged over the blob. The temperature differencebetween the two models (not shown) is a factor of D� larger than the difference in degree of melting.


11 of 22

diffusion of heat into the spherical blob (black lines& symbols) dominates the melting budget andkeeps the temperature of the blob equal to theambient temperature. For Peclet numbers largerthan ∼102, diffusion of heat makes a negligiblecontribution to the overall budget and blob tem-peratures approach their adiabatic limit (althoughFigure 5a shows that the diffusive heat flux may beimportant in a narrow rim at the edge of the blob).In the intermediate‐Peclet regime, diffusion makesa significant but not dominant contribution, and thefinal degree of melting and blob temperature aresensitive to the value of Pe.

[44] The tabular vein shows a qualitatively similarbehavior (blue lines and symbols in Figure 7). Atvery large Peclet numbers, where diffusion makesan insignificant contribution to melting, the spheri-cal blob and the tabular vein melt to equal extents; atvery small Peclet numbers, where diffusion makes asubstantial contribution to powering melting, thetwo also agree. Differences appear between theseextremes, with the tabular vein exhibiting lessextensive melting at any intermediate value of Pe.This can be understood in terms of the symmetry ofeach blob shape. In the spherical case, a circularpatch on the surface of the heterogeneity draws adiffusive heat flow from a (truncated) cone ofmantle, its volume increasing with the cube of dis-tance from the blob. In the tabular case, a circularpatch on the surface of the heterogeneity draws heatonly from a cylinder of mantle, with a volumeproportional to linear distance from the vein.

[45] Figure 7 also shows that the nonlinearity of thetemperature–melting function (11) has a simple anduniform effect on the final degree of melting andthe final temperature. Lower values of a requirehigher temperature to reach a given FB. With otherparameters held constant, decreasing a givessmaller FB as a function of time, which means lessconversion of sensible to latent heat, and thereforea smaller blob‐temperature difference from ambi-ent. Since it is this temperature difference thatdrives thermal diffusion and further melting, thediffusive flow of energy into the blob decreaseswith decreasing a.

[46] Despite reductions to maximum FB fromnonlinear effects, geometry of the heterogeneity,and adiabatic decompression (which gives a ∼10%reduction), melting at low Peclet numbers yieldsdegrees of partial melting that are well in excess of50%. And while less fertile compositions than theones considered here would generate smaller FB,our prediction should hold if the G2 pyroxenitecomposition ofPertermann andHirschmann [2003a]is representative of recycled oceanic crust. Fur-thermore, Figure 7a shows that even for the non-linear case, we can approximate the maximumincrease in degree of melting due to thermal dif-fusion by 1/l where

1

¼ S þD�

D�¼ Lþ cpDT

cpDT; ð26Þ

which is independent of the temperature differenceg/Dp between the solidii of recycled oceanic crust

Figure 7. (a) Summary of degree of melting and (b) dimensionless temperature at t = 1 for three values of a,computed for a range in Peclet number spanning the transition between asymptotic values. Black lines and symbolscorrespond to the spherical‐blob model; blue lines and symbols correspond to the tabular‐vein model. Dotted linesshow asymptotic values as labeled. Note that in interpreting Pe for the tabular vein, R is the half‐width, as shownin Figure 1.


12 of 22

and depleted upper mantle (recall that L is latentheat, cp is specific heat capacity, and DT is theliquidus–solidus temperature interval of pyroxe-nite). This equation is equivalent to equation (11)of Sleep [1984].

[47] How broad is the effect of melting within theheterogeneity on the mantle temperature around it?Figure 8 addresses this question for a range ofPeclet numbers for both the spherical blob (blacklines and symbols) and the tabular vein (blue lines).At large values of Pe, diffusion is highly localizednear the heterogeneity, causing large differencesfrom the ambient temperature in a narrow region.Moving toward smaller values of Pe, diffusionbecomes more efficient, and heat flows into theheterogeneity from a broad region. In the sphericalgeometry, the volume of this region increases withthe cube of distance from the blob; when a largevolume of ambient mantle contributes heat to theblob, the ambient temperature change associatedwith that contribution is small. These two tendenciesare reflected by the trends in Figure 8, followingthe black curves from right to left. At Pe ≈ 1, whereadvective and diffusive heat transport are roughlyin balance for the spherical heterogeneity, we finda thermal perturbation that is relatively large inboth amplitude and extent. In Cartesian geometry,

volume increases linearly with distance from thetabular vein, and hence for a given inward heatflow, the thermal halo reaches greater distances.

4.2. Dependence on Dimensional Sizeand Upwelling Rate

[48] To conclude the discussion we reintroducedimensional parameters and consider, indepen-dently, the effect of changing the characteristic sizeR and the upwelling speed W. Although theseresults could be deduced from earlier, dimension-less plots, they are presented in Figure 9 for clarity.To produce this figure, we have assumed values ofmaterial parameters as given in Table 1. Figure 9ashows contours of degree of melting at t = 1 fora = 1 (analytical, black lines) and a = 1/4 (numerical,red lines) for a spherical heterogeneity. Degree ofmelting is smallest at the top‐right of the figure, forlarge blobs that upwell rapidly and melt adiabati-cally, and largest at the bottom‐left of the figure, forsmall blobs that upwell slowly and melt in thermalequilibrium with their surroundings. Figure 9bshows the dimensional temperature perturbationat a distance 1.5 R from the center of the blob. Forlarge blobs that upwell rapidly, loss of heat fromthe ambient mantle by diffusion into the bloboccurs in a region narrowly confined around the

Figure 8. The non‐dimensional size of the diffusively cooled halo around the spherical blob (black) and the tabularvein (blue) at t = 1, as a function of the Peclet number. Lines are calculated with the analytical solution for a = 1; eachline represents the radius at which �(r, 1) reaches a specified value (see legend for values). Points are derived fromspherical‐blob simulations with a = 1/4; their close correspondence with the lines indicates the associated value of�(r, 1).


13 of 22

edge of the blob, hence the temperature at r = 1.5 Ris unaffected. For small blobs that upwell slowly,the heat required to maintain thermal equilibriumwith the ambient mantle is small, and it is extractedover a very large volume around the blob, includ-ing r = 1.5 R. The case of a tabular vein is shownby contours of degree of melting and temperatureperturbation in Figures 9c and 9d. The transition todiffusion‐dominated melting is shifted to lowerPeclet numbers, meaning that tabular veins must benarrower or upwelling more slowly to overcometheir geometry and reach the same FB as sphericalblobs. Finally, note that contours in all four panelshave a slope of −2, which is consistent with thedependence of Peclet number on R and W (notedifferent x‐scales in the two panels in Figure 9).

[49] Results presented here are qualitatively con-sistent with those obtained by Sleep [1984]. Wherehis scaling analysis predicted a transition betweenthermally isolated and thermally equilibrated at acharacteristic size of about five kilometers for anupwelling rate of 3 cm/yr, Figure 9c suggests thatfor a tabular vein, the transition occurs at a slightlysmaller size of about one kilometer. This differenceis insignificant given the model assumptions. Moresubstantial differences can be noted in the amountof melting and the enhancement factors obtained bySleep [1984] compared to our work. His modelconsidered only the linearized melting relationshipa = 1, and used an isobaric productivity of 0.02 K−1

corresponding to a temperature difference betweenthe liquidus and solidus ofDT = 50 K. This is about

Figure 9. Contour plots of dimensional quantities as a function of characteristic heterogeneity size R in km andupwelling rate W in cm/yr. Other dimensional parameters as in Table 1. (a and c) Degree of melting for the spher-ical blob and tabular vein, respectively. Black contours are computed with the analytical solution and a = 1; redcontours are computed with the numerical simulation and a = 1/4. Degree of melting is maximal in the bottom leftcorner and minimal in the top right corner. Contour spacing is linear and the maximum (minimum) contour values are1% less (more) than the appropriate asymptotic values. Asymptotic values for FB are obtained using equations (24)and (25) with equation (11). For the nonlinear melting solution, only the maximum and minimum contours are shown.(b and d) The temperature perturbation at a distance 1.5 R from the center of the spherical blob or tabular vein,respectively. The largest absolute perturbation is ∼20°C (spherical) and ∼56°C (tabular) at this radius. Only the lineara = 1 solution is plotted for temperature. As in previous figures, the results here exclude the background adiabaticgradient (see A5 for details).


14 of 22

a factor of five smaller than the liquidus–solidusdifference obtained empirically by Pertermann andHirschmann [2003a]; furthermore, near‐solidusproductivity for G2 pyroxenite is about 0.001 K−1,a factor of 20 smaller than the value used bySleep [1984]. This difference, as prescribed byequation (26), explains the very large enhancementfactors (3–7 × adiabatic) that he obtained. Despitethis enhancement to melting, the melt fractionsreported by Sleep [1984] are small relative to thoseobtained here. This is because he computed meltingcurves for only 5 to 12 km of mantle ascent, whichagain is small relative to the value of ∼50 kmbetween the melting onset‐depth of pyroxeniteand peridotite, as estimated by Pertermann andHirschmann [2003a] and used here.

5. Summary and Implications

[50] In this paper we have presented new theory forthe melting of fertile mantle heterogeneities, andshown that a properly formulated Peclet numbermeasures the importance of diffusion‐driven versusadiabatic melting. Our results also show that asimplified theory that considers a uniformly melt-ing spherical blob, rather than one that captures thegradient in F and T with radius, accurately modelsthe gross behavior of the system, deviating only forPeclet numbers near unity. For intermediate valuesof upwelling rate, our model predicts that uni-formly melting spherical blobs ] 5 km in radiushave substantial melt enhancement by diffusion;for tabular veins, this transition occurs at a smallersize of ] 1 km. These conclusions confirm andextend the scaling analysis of Sleep [1984].

[51] Under the assumptions and parameter choiceslisted above, we have demonstrated that diffusionof heat into an upwelling, fertile heterogeneity canlead to an increase in the degree of partial meltingby a factor of two, generating extents of melting of50–80% beneath the bottom of the ambient meltingregion. Furthermore, we have shown that thethermal anomaly imprinted on the ambient mantleis confined to within about two times the charac-teristic size of the heterogeneity, and ranges downto about −60 K. This temperature difference issmall relative to the absolute temperature of themantle, but significant when compared with thetemperature drop due to decompression meltingbeneath a ridge. As such, melting of the ambientmantle will be suppressed in the neighborhood of aheterogeneity, with the onset of ambient meltingoccurring at shallower depths. The expected changein the overall degree of ambient mantle melting due

to this effect is on the order of 1–2% for a pyrox-enite fraction of 5% [Phipps Morgan, 2001].

[52] A detailed consideration of the geochemicalimplications of variable melting of heterogeneitiesbased on their size and shape is beyond the scopeof the present paper. It might be argued, forexample, that larger heterogeneities, which melt tolesser extents, preserve residual garnet to shallowerdepth, and hence may impart a greater garnet sig-nature than smaller heterogeneities. Such argu-ments are based on the details of the partitioncoefficients, the mineral mode of recycled oceaniccrust, and the rate and style of melt segregationfrom pyroxenitic heterogeneities [e.g., Prytulakand Elliott, 2009]. Moreover, most geochemicalmodels of the contribution of magma from recycledoceanic crust indicate that to preserve a distinctivegeochemical signature, such melts must ascendrapidly, in chemical isolation from the ambientmantle. This is thought to occur either by hydro-fracture and propagation of dikes, or by reactiveflow and transport though high‐flux dunite chan-nels [e.g., Kelemen et al., 1995; Lundstrom et al.,2000; Spiegelman and Kelemen, 2003; Elliott andSpiegelman, 2003; Kogiso et al., 2004].

[53] In this context, we emphasize that because therate of reactive melting is proportional to the ver-tical magmatic flux, a local excess of melt suppliedto the melting region from below can inducereactive channelization [Hewitt, 2010; Liang et al.,2010]. This prediction, considered in light of thelarge extents of melting for recycled crust belowthe base of ambient‐mantle melting regime, sup-ports the hypothesis by Lundstrom et al. [2000]that melt released from a fertile heterogeneitycould induce chemically isolated, channelizedmelt transport. Furthermore, it is possible thatthe pyroxenite‐derived magmatic flux through achannel would reduce the local solidus tempera-ture, cool the channel by consumption of latentheat, and give rise to a diffusive flux of heat intothe channel (as we predict to occur around theheterogeneity). A cool diffusion‐halo around adunite channel will suppress adjacent melting ofmantle peridotite, further isolating the magma as itis transported.

Appendix A: Analytical SolutionsA1. Homogeneous Meltingof a Spherical Blob

[54] When the degree of melting is linearlydependent on the homologous temperature (a = 1


15 of 22

in (11)), analytical methods can be used to obtainsolutions. For the homogeneous blob, the problemis to solve for �(r, t) satisfying (18),

@�

@t¼ 1

Pe

1

r2@

@rr2@�

@r

� �; ðA1Þ

with boundary conditions

� SD�

þ 1þ SD�

� �@�

@t1; tð Þ ¼ 3

Pe

@�

@r1; tð Þ; ðA2Þ

� ∞; tð Þ ¼ 0; ðA3Þ

and initial condition

� r; 0ð Þ ¼ 0: ðA4Þ

The first of the two boundary conditions representsthe heat balance between the heat flowing into theblob due to the temperature gradient outside, theheat used to melt the blob (latent heat), and the heatused to raise the temperature of the blob (sensibleheat). The expression (A2) follows directly from(10), (11), and (12) when a = 1 (and also followsfrom partial differentiation with respect to t of theintegral boundary condition described by (17) and(19a)). The second boundary condition states thatthe far‐field temperature is constant (i.e., neglect-ing adiabatic decompression effects: see A5 fordiscussion of these effects). The total degree ofmelting of the blob can be obtained from (9a), (10),and (11) as

FB tð Þ ¼ � 1; tð Þ þ t

D�: ðA5Þ

[55] In order to simplify later algebra, it is helpfulrescale time and temperature by the Peclet numberand introduce a parameter l as

�′ ¼ �

Pe; t′ ¼ t

Pe; ¼ D�

S þD�: ðA6Þ

With this new scaling, the problem becomes

@�′

@t′¼ 1

r2@

@rr2@�′

@r

� �; ðA7Þ

with boundary conditions and initial condition

@�′

@t′1; t′ð Þ ¼ �1þ þ 3

@�′

@r1; t′ð Þ; ðA8Þ

�′ ∞; t′ð Þ ¼ 0; ðA9Þ

�′ r; 0ð Þ ¼ 0; ðA10Þ

and

FB t′ð Þ ¼ Pe

D��′ 1; t′ð Þ þ t′ð Þ: ðA11Þ

For the remainder of this appendix, we will dropthe primes and use the rescaled variables.

[56] Laplace transform solution: The governingpartial differential equation (A7) can be simplifiedby introducing a new variable u(r, t) as

� r; tð Þ ¼ u r; tð Þr

; ðA12Þ

to give

@u

@t¼ @2u

@r2ðA13Þ

with boundary conditions and initial condition

@u

@t1; tð Þ ¼ �1þ þ 3

@u

@r1; tð Þ � u 1; tð Þ

� �; ðA14Þ

u r; tð Þ=r ! 0 as r ! ∞; ðA15Þ

u r; 0ð Þ ¼ 0: ðA16Þ

[57] Introduce the Laplace transform in time as

~u r; sð Þ ¼Z ∞

0u r; tð Þe�st dt: ðA17Þ

The transformed problem is then

s~u ¼ @2~u

@r2; ðA18Þ


s~u 1; sð Þ ¼ �1þ

sþ 3

@~u

@r1; sð Þ � ~u 1; sð Þ

� �; ðA19Þ

~u r; sð Þ=r ! 0 as r ! ∞: ðA20Þ

(A18) and (A20) imply

~u r; sð Þ ¼ A sð Þe�ffiffis

pr�1ð Þ; ðA21Þ

for some function A(s) to be determined. (A19)then becomes

sA sð Þ ¼ �1þ

sþ 3 � ffiffi

sp

A sð Þ � A sð Þ� � ðA22Þ

which gives A(s) as

A sð Þ ¼ �1þ

s sþ 3ffiffis

p þ 3ð Þ : ðA23Þ


16 of 22

Hence we have

~� r; sð Þ ¼ � 1ð Þe�ffiffis

pr�1ð Þ

rs sþ 3ffiffis

p þ 3ð Þ : ðA24Þ

To find �(r, t) we need to find the inverse Laplacetransform of the above function. This can beobtained by factoring the denominator, splittinginto partial fractions, and performing the inverseLaplace transform term‐by‐term. Writing

1

s sþ 3ffiffis

p þ 3ð Þ �1

sffiffis

p þ að Þ ffiffis

p þ bð Þ ðA25Þ

where

a ¼ 3þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi92 � 12

p

2; b ¼ 3�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi92 � 12

p

2;

ðA26Þ

the inverse Laplace transform of ~�(r, s) is

� r; tð Þ ¼ � 1

r

1

aberfc ð Þ þ e�2

a � b

"

� 1

aw i þ ia

ffiffit

p� �� 1

bw i þ ib

ffiffit

p� �� #; ðA27Þ

where h = (r − 1)/(2ffiffit

p) and w(z) = e−z

2

erfc(−iz) isthe Faddeeva function. (A27) is the analyticalsolution to the linear homogeneous blob problem,and can be calculated rapidly with the aid of effi-cient routines for calculating the Faddeeva function[Weideman, 1994]. A solution similar to the abovewas recently obtained by Oliver [2008] for a relatedproblem of spherical heat generation and conduc-tion. The corresponding degree of melting is givenby (A11),

FB tð Þ ¼ Pe

D�

�t þ � 1ð Þ

� 1

abþ 1

a � b

1

aw ia

ffiffit

p� �� 1

bw ib

ffiffit

p� �� ;

ðA28Þand is plotted in Figure 3.

[58] The expressions in (A27) and (A28) aresomewhat cumbersome to work with when study-ing the asymptotic behaviors of the solution. It iseasier to study the asymptotic behavior of theLaplace transform solution in s, and then relate theasymptotics in s to the asymptotics in t (see A5).For this purpose, note that ~FB(s) is given bythe expression

~FB sð Þ ¼ Pe

D�

1

s2þ ~� 1; sð Þ

� �¼ Pe

D�

1

s2þ A sð Þ

� �: ðA29Þ

A2. Radially Variable Melting of a SphericalBlob

[59] If we do not assume that the blob is homoge-neous, and instead allow it to have a radial tem-perature profile due to the conduction of heatthrough the blob, then we must solve the heatconservation equation both inside and outside theblob. Using the rescaled variables of (A26), thegoverning equations are

@�

@t¼ �1þ þ

r2@

@rr2@�

@r

� �; 0 � r < 1; ðA30Þ

@�

@t¼ 1

r2@

@rr2@�

@r

� �; r > 1: ðA31Þ

On the surface of the blob, both the temperatureand the heat flux must be continuous, i.e.,

� r; tð Þ; @�@r

r; tð Þ continuous on r ¼ 1; ðA32Þ

and as before the initial condition is

� r; 0ð Þ ¼ 0; ðA33Þ

and the boundary condition in the far‐field is

� ∞; tð Þ ¼ 0: ðA34Þ

[60] Laplace transform solution: As before, thegoverning partial differential equations can besimplified by writing

� r; tð Þ ¼ u r; tð Þr

ðA35Þ

to obtain

@u

@t¼ � 1ð Þr þ

@2u

@r2; 0 � r < 1; ðA36Þ

@u

@t¼ @2u

@r2; r > 1; ðA37Þ

with boundary and initial conditions

u r; tð Þ; @u

@rr; tð Þ continuous on r ¼ 1; ðA38Þ

u r; tð Þ=r f inite as r ! 0; ðA39Þ

u r; tð Þ=r ! 0 as r ! ∞; ðA40Þ

u r; 0ð Þ ¼ 0: ðA41Þ


17 of 22

[61] The Laplace transformed problem is

s~u ¼ � 1ð Þrs

þ @2~u

@r2; 0 � r < 1; ðA42Þ

s~u ¼ @2~u

@r2; r > 1; ðA43Þ


~u r; sð Þ; @~u

@rr; sð Þ continuous on r ¼ 1; ðA44Þ

~u r; sð Þ=r f inite as r ! 0; ðA45Þ

~u r; sð Þ=r ! 0 as r ! ∞: ðA46Þ

The governing equations (A42) and (A43) can beintegrated using the boundary conditions (A45) and(A46) to give

~u r; sð Þ ¼� 1ð Þrs2

þ B sð Þ sinhffiffiffis

rr

� �; 0 � r < 1;

C sð Þe�ffiffis

pr�1ð Þ; r > 1:

8><>:

ðA47Þ

The two functions B(s) and C(s) are determined bythe continuity requirements of (A44)

B sð Þ ¼ � � 1

s21þ ffiffi

spffiffi

s

pcosh

ffiffis

p þ ffiffis

psinh

ffiffis

p !

; ðA48Þ

C sð Þ ¼ � 1

s2

ffiffis

pcosh

ffiffis

p � sinhffiffis

pffiffis

pcosh

ffiffis

p þ ffiffis

psinh

ffiffis

p !

: ðA49Þ

The solution for ~�(r, s) is thus

~� r; sð Þ ¼� 1ð Þs2

þ B sð Þr

sinh

ffiffiffis

rr

� �; 0 � r < 1;

C sð Þr

e�ffiffis

pr�1ð Þ; r > 1:

8>><>>:

ðA50Þ

To find �(r, t) we must obtain the inverse Laplacetransform of the above function. Unfortunately,there does not appear to be a simple analyticalinverse of (A50). However, the inverse can becalculated numerically using efficient routines fornumerical inverse Laplace transforms [de Hooget al., 1982; K. Hollenbeck, INVLAP.M: A matlabfunction for numerical inversion of Laplace trans-forms by the de Hoog algorithm, 1998, http://www.isva.dtu.dk/staff/karl/invlap.htm, hereinafter referredto as Hollenbeck, matlab function, 1998].

[62] The degree of melting within the blob is givenby

F r; tð Þ ¼ Pe

D�t þ � r; tð Þð Þ; ðA51Þ

and hence using (A50) we have

~F r; sð Þ ¼ Pe

D�

1

s2þ ~� r; sð Þ

� �¼ Pe

D�

s2þ B sð Þ

rsinh

ffiffiffis

rr

� �:

ðA52ÞThe Laplace transform of the mean degree ofmelting F(t) is thus

~F sð Þ ¼ 3

Z 1

0

~F r; sð Þr2 dr

¼ Pe

D�

s2þ 3B sð Þ

s

ffiffiffis

rcosh

ffiffiffis

r� sinh

ffiffiffis

r� ��

¼ Pe

D�

1

s2� 3

s1þ ffiffi

sp� �

C sð Þ� �

: ðA53Þ

F(t) can be obtained by finding the inverse Laplacetransform of the above function. This was donenumerically using the routines of Hollenbeck(matlab function, 1998) to produce the profilesplotted in Figure 5.

A3. Melting of a Tabular Vein

[63] In this section we briefly derive the tabularequivalents of the solutions given in A1 and A2.The general method of solution for a tabulargeometry is identical to that for a spherical geom-etry. The only change that needs to be made is thatthe Laplacian operator is now given by

r2 ¼ @2

@x2ðA54Þ

where 0 < x < 1 is inside the sheet, and x > 1 isoutside to sheet. Symmetry is assumed about theorigin, so that

@�

@x¼ 0 on x ¼ 0: ðA55Þ

A3.1. Homogeneous Melting

[64] The problem is

@�

@t¼ @2�

@x2ðA56Þ

with boundary conditions and initial condition@�

@t1; tð Þ ¼ �1þ þ

@�

@x1; tð Þ; ðA57Þ

� ∞; tð Þ ¼ 0; ðA58Þ

� x; 0ð Þ ¼ 0: ðA59Þ


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The Laplace transformed problem is

s~� ¼ @2 ~�

@x2; ðA60Þ


s~� 1; sð Þ ¼ �1þ

sþ

@ ~�

@x1; sð Þ; ðA61Þ

~� ∞; tð Þ ¼ 0; ðA62Þ

and solution

~� x; sð Þ ¼ � 1ð Þe�ffiffis

px�1ð Þ

s3=2ffiffis

p þ ð Þ : ðA63Þ

The inverse Laplace transform of this is

� x; tð Þ ¼ � 1

2e�2 w i þ i

ffiffit

p� �þ 2ffiffit

pffiffiffi�

p� ��

� 1þ 2ffiffit

p

� �erfc ð Þ

; ðA64Þ

where h = (x − 1)/(2ffiffit

p) and w(z) is the Faddeeva

function. The above solution can also be foundin the study by Carslaw and Jaeger [1959] (theirequation (12) in section 12.4). The correspondingdegree of melting is given by

FB tð Þ ¼ Pe

D�t þ � 1

2w i

ffiffit

p� �þ 2ffiffit

pffiffiffi�

p � 1

� � �: ðA65Þ

A3.2. Laterally Variable Melting

[65] The governing equations are

@�

@t¼ �1þ þ @2�

@x2; 0 � x < 1; ðA66Þ

@�

@t¼ @2�

@x2: x > 1: ðA67Þ

The Laplace transformed problem is

s~� ¼ � 1

sþ

@2 ~�

@x2; 0 � x < 1; ðA68Þ

s~� ¼ @2 ~�

@x2; x > 1: ðA69Þ

with solution

~� x; sð Þ ¼� 1

s2þ B sð Þ cosh

ffiffiffis

rx

� �; 0 � x < 1;

C sð Þe�ffiffis

px�1ð Þ; x > 1:

8><>: ðA70Þ

The two functions B(s) and C(s) are determined bycontinuity as

B sð Þ ¼ �� 1

s21

coshffiffis

p þ 1ffiffi

p sinhffiffis

p !

; ðA71Þ

C sð Þ ¼ � 1

s2

1ffiffi

p sinhffiffis

pcosh

ffiffis

p þ 1ffiffi

p sinhffiffis

p !

: ðA72Þ

The Laplace transform of the mean degree ofmelting F(t) is

~F sð Þ ¼Z 1

0

~F x; sð Þ dx

¼ Pe

D�

s2þ B sð Þ

ffiffiffi

s

rsinh

ffiffiffis

r !

¼ Pe

D�

1

s2� C sð Þffiffi

sp

� �: ðA73Þ

A4. Leading Order Asymptotics

[66] All the problems considered in this paper(both spherical/tabular and homogeneous/radiallyvarying) have the same leading order behavior forlarge and small t. This behavior is exactly thatwhich is expected from a simple thermodynamicanalysis of the two extremes of a thermally isolatedblob and a blob in thermal equilibrium with theambient mantle. This leading order behavior hasbeen described by Sleep [1984] and is depicted inFigure 2.

[67] The large‐t and small‐t asymptotic behavior ofFB(t) and F(t) can be obtained directly from theasymptotic behavior of the Laplace transforms~FB(s) and

~F(s) for small s and large s respectively.The leading order asymptotics of ~FB(s) are givenby series expansion of (A29) as

~FB sð Þ �Pe

D�s2þO 1

s5=2

� �; for s 1;

Pe

D�s2þO 1

s

� �; for s 1:

8>><>>: ðA74Þ

The leading order asymptotics of ~F(s) from (A53)are identical. By inverse Laplace transformingterm‐by‐term we obtain the leading order asymp-totics of FB(t) as

FB tð Þ �Pe t

D�þO t3=2

�; for t 1;

Pe t

D�þO 1ð Þ; for t 1:

8>><>>: ðA75Þ


19 of 22

The above expression can be written in dimen-sional units using (2) as

FB tð Þ �p0 � p tð Þ

� DT þ L=cp� �þO t3=2

�; for t R2=�;

p0 � p tð Þ�DT

þO 1ð Þ; for t R2=�;

8>><>>:

ðA76Þ

which agrees with the simple thermodynamicanalysis of Sleep [1984] (his equations 9 and 10respectively, see Figure 2). Higher order asymp-totic expansions for FB(t) can be obtained by con-sidering higher order terms in the series expansionsof (A74). Differences between FB(t) and F(t), andthe tabular and spherical geometries, become evi-dent with the inclusion of higher order terms.

A5. Adiabatic Decompression Effects

[68] Up to this point the effects of adiabaticdecompression have been ignored, as it has beenassumed that the far‐field temperature of the ambi-ent mantle is constant. In fact, the far‐field temper-ature of the ambient mantle will decrease as thepressure decreases as a consequence of adiabaticdecompression. It is straightforward to include thiseffect, at least in a linearized sense. If the tempera-ture differences are small, such that (T0 − T1)/T0 1,the adiabatic decompression term in the energyequation (5) can be approximated as

��T@p

@t� ��T0

@p

@t: ðA77Þ

The non‐dimensional radially varying blob prob-lem is then

@�

@t¼ �A� 1þ þ

r2@

@rr2@�

@r

� �; 0 � r < 1; ðA78Þ

@�

@t¼ �Aþ 1

r2@

@rr2@�

@r

� �; r > 1; ðA79Þ

where A is the adiabatic parameter, defined by

A ¼ �T0�

�cp: ðA80Þ

The boundary conditions on the surface of the blob,and the initial condition are as before. The far‐fieldboundary condition becomes

� ∞; tð Þ ¼ �At; ðA81Þ

reflecting the fact that the far‐field temperaturedrops as the blob ascends. By writing

� r; tð Þ ¼ �At þ 1�Að Þ# r; tð Þ; ðA82Þ

we recover the problem that has already beensolved neglecting adiabatic decompression, i.e.,

@#

@t¼ �1þ þ

r2@

@rr2@#

@r

� �; 0 � r < 1; ðA83Þ

@#

@t¼ 1

r2@

@rr2@#

@r

� �; r > 1; ðA84Þ

# ∞; tð Þ ¼ 0: ðA85Þ

Hence to calculate the solution for a problem whichincludes the adiabatic decompression term, wesimply find the solution without the term, and thenuse (A82). This works for both the homogeneousand the radially varying blob problems, as well asfor the tabular geometry. Since

F r; tð Þ ¼ Pe

D�t þ � r; tð Þð Þ ¼ Pe

D�1�Að Þ t þ # r; tð Þð Þ; ðA86Þ

the melt productivity decreases by a factor of(1 − A) when the adiabatic decompression term isincluded. For example, the values of the dimen-sional degree of melting at the two extremeschanges from that given by (A76) to

FB tð Þ �p0 � p tð ÞDT þ L=cp

1

�� T0

�cp

� �þO t3=2

�; for t R2=�;

p0 � p tð ÞDT

1

�� T0

�cp

� �þO 1ð Þ; for t R2=�:

8>><>>:

ðA87Þ

The first of the above two cases can be recognized asfollowing directly from the usual expression for theproductivity during isentropic decompression melt-ing [e.g., Asimow et al., 1997, equation (3.14)]. Thesecond case is in agreement with expressions for theproductivity assuming complete thermal equilibra-tion [e.g., Phipps Morgan, 2001, equation (19)].

Appendix B: Numerical Solutions

[69] The governing equations with a < 1 are non-linear and must be solved numerically. To do sowe use a semi‐implicit, centered‐difference dis-cretization on a non‐uniform grid. We solve theresulting system of nonlinear algebraic equationswith a Newton‐Krylov (GMRES) scheme and anexplicit LU preconditioner; these are provided bythe Portable, Extensible Toolkit for Scientific


20 of 22

Computation (version 3.1 [Balay et al., 2010; Katzet al., 2007; S. Balay et al., PETSc Web page,2011, http://www.mcs.anl.gov/petsc]). Details ofthe discretization are given in this appendix for thecase of a spherical heterogeneity; correspondingequations for the tabular vein are obtained in asimilar manner. Simulation code is available byemail request to the first author.

[70] The discretization is semi‐implicit in time,

�nþ1 � �n

Dt¼ 1

2Pe

1

r2@

@rr2@�

@r

� �� nþ1

þ 1

r2@

@rr2@�

@r

� �� n" #

;

ðB1Þ

whereDt is the time step, chosen such that tn = nDtfor n 2 [0, Nt − 1]. Superscripts in the semi‐discreteequation (B1) refer to the time step number.

[71] Spatial derivatives are discretized with a cen-tered difference scheme,

1

r2@

@rr2@�

@r

� ��

riþ1þri2

� �2 �iþ1��iriþ1�ri

�h i� riþri�1

2

� �2 �i��i�1ri�ri�1

�h i12 riþ1 � ri�1ð Þ r2i

:

ðB2Þ

Values of the radius are specified at a set of discretepoints i 2 [0, Nr − 1] using

ri ¼ 1þ rmax � 1ð Þ i

Nr � 1

� ��

; ðB3Þ

where x ≥ 1 is a power that determines the relativeconcentration of grid points near the blob. We havefound that x = 2 provides a good balance betweenaccuracy and speed of numerical convergence.

[72] The boundary condition at r → ∞ is astraightforward Dirichlet condition, which weapply at r = rmax 1. The boundary condition atr = 1 is more difficult. For FB ≤ 1, the semi‐implicitdiscretization of this condition is as follows

�n0 ¼ �nDt þD� QnB; ðB4Þ

where QBn is the dimensionless homologous tem-

perature of the blob at the present step, obtainedwith equation (15), which depends on the unknownvalue of fB. The current Newton iterate �f B can becalculated using the current Newton iterate of thesolution vector ��i

n by discrete integration with thetrapezoidal rule,

�f nB ¼ f n�1b þ Dt

D�þ aS 1þ 3

Pe

��n1 þ �n�11 � ��n0 � �n�1

0

2 r1 � r0ð Þ

!:

ðB5Þ

[73] The discrete boundary conditions and diffusionequation are then recast as equations for the ele-ments of the point‐wise Newton residual vector ofthe current iterate �%i. We provide an analyticalJacobian matrix Jij = ∂�%i/∂��j, and the Newtonscheme is iterated until the residual vector satisfiesk�%ik2 < tol. We use a tolerance of 10−10.

[74] Numerical solutions can be compared with theanalytical solution for the linear case, a = 1. Percenterror is computed as

e ¼ k �exact � �numerical k2k �exact k2 � 100; ðB6Þ

where the 2‐norm is calculated over all combina-tions of ri, tn used in the numerical model‐run. Weobtain perfect second order convergence with grid‐spacing for grids up to Nr = 3200 with x = 2; weobtain little improvement in accuracy for Nt ^ 800.For a grid with Nr = 1600 and Nt = 1000 we findthat e = 0.003%.

Acknowledgments

[75] The authors thank C. Langmuir and M. Hirschmann foradvice and comments on the manuscript, T. Becker for his edi-torial efficiency, and S. Weatherley for discussions regardingmelt transport. Katz was supported by a Research CouncilsUK Academic Fellowship and NERC grant NE/H00081X/1.

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the energetics of melting fertile heterogeneities within the...

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