Buoyant melting instabilities beneath extending lithosphere: 1. …eps.berkeley.edu/~hernlund/Hernlund_Melt1_JGR_2008.pdf · 2010. 10. 21. · Buoyant melting instabilities beneath

Buoyant melting instabilities beneath extending lithosphere:

1. Numerical models

John W. Hernlund,1,2 Paul J. Tackley,1,3 and David J. Stevenson4

Received 18 November 2006; revised 18 October 2007; accepted 14 December 2007; published 8 April 2008.

[1] Buoyant decompression melting instabilities in regions of partially molten uppermantle have been proposed to be an important process that might account for somecharacteristics of intraplate volcanism on Earth and other terrestrial planets. The instabilityis driven by variations in the melting rate within a partially molten layer whenever arelative decrease in density accompanies decompression melting of ascending mantle.Here, the development of buoyant decompression melting instabilities in a plane layer ofpassively upwelling and partially melting mantle beneath diffusely extending lithosphereis studied using numerical convection models covering a wide range of physicalparameters. We find that the occurrence and nature of these instabilities in such a scenariois strongly affected by the rate of extension and melt percolation, as well as depthdistribution of solid density variations arising from melt depletion. In some cases,instabilities do not occur during extension, but only develop after extension has slowed orstopped completely. This behavior creates two pulses of magma generation due topassive upwelling accompanying extension followed by the subsequent instability and isfavored by a faster rate of extension, higher mantle viscosity, higher rate of meltpercolation, and smaller amount of solid residuum depletion-derived buoyancy. Largerdegrees of solid density changes accompanying melt depletion can enhance theinstability of partially molten mantle during extension but decrease the cumulativevolume of generated melt. This kind of behavior modifies the conventional expectationof spatially and temporally correlated volcanism and extension andmay lend insight into theobserved increase in localized volcanic activity following Miocene Basin and Rangeextension in the western United States.

Citation: Hernlund, J. W., P. J. Tackley, and D. J. Stevenson (2008), Buoyant melting instabilities beneath extending lithosphere:

1. Numerical models, J. Geophys. Res., 113, B04405, doi:10.1029/2006JB004862.

1. Introduction

[2] If a portion of Earth’s upper mantle crosses thesolidus during ascent and undergoes decompression melt-ing, in most circumstances its density will decrease. Thesmaller density is due to the larger molar volume of anyretained melt in the matrix as well as changes in composi-tion and/or phase abundance in the residual solid rock. Solong as the upward percolation of melt through the solidmatrix is not instantaneous, a small fraction of melt will beretained which, along with solid residuum density changes,is available to drive buoyant flow in the partially moltenregion. Because the buoyancy of partially melted mantlerelative to any adjacent unmelted (or less melted) mantle

increases as both its upward displacement and velocityincreases, a positive feedback can arise that causes internalinstability (i.e., runaway convective overturn) of a partiallymolten region (see Figure 1). This ‘‘Rayleigh-Taylor-like’’melting instability [Stevenson, 1988] may carry importantimplications for the dynamics of the mantle asthenosphereand the production of melt in Earth’s mantle. This processhas more recently been termed a ‘‘buoyant decompressionmelting instability’’ [Raddick et al., 2002], and has beensuggested to be a potentially important source of intraplatevolcanism on Earth as well as other terrestrial planets[Tackley and Stevenson, 1993; Raddick et al., 2002].[3] The simplest example of a Rayleigh-Taylor instability

is the classical scenario where one fluid is overlain byanother fluid of greater density. In this gravitationallyunstable arrangement, the smallest perturbation will leadto growing corrugations in the interface between the twofluids as they undergo gravity-driven exchange at a ratecontrolled by their viscosities and relative buoyancy. Anasthenospheric instability of this type was suggested byAnderson and Sammis [1970] whenever lower-density par-tially molten layers in Earth’s mantle is overlain by higherdensity lithosphere. Such mechanisms have been proposed

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1Department of Earth and Space Sciences, University of California, LosAngeles, California, USA.

2Now at Department of Earth and Ocean Sciences, University of BritishColumbia, Vancouver, British Columbia, Canada.

3Now at Institute for Geophysics, ETH-Zurich, Zurich, Switzerland.4Division of Geological and Planetary Sciences, California Institute of

Technology, Pasadena, California, USA.

Copyright 2008 by the American Geophysical Union.0148-0227/08/2006JB004862$09.00

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to account for the inferred length scales of buoyant meltdiapirism beneath mid-ocean ridges [Whitehead et al., 1984;Crane, 1985] and arc volcanoes [Lingenfelter and Schubert,1974]. Complex interactions between Rayleigh-Taylor two-phase instabilities under the additional influence of meltpercolation have also been analyzed by Ricard et al. [2001].However, the buoyant decompression melting instabilitydiffers from the classical scenario in several ways. This isbecause it does not directly involve the mantle lithosphere(which may be too viscous to flow in response to theavailable forces) and because the buoyancy which drivesthe flow develops as a direct consequence of variations inthe rate of melting within the layer rather than unstabledisplacement of the entire layer (although external influen-ces may also play an important role). Additional complica-tions include the fact that the generation of buoyancy withina partially molten layer may be asymmetric, in that onlyupwellings undergo active melting while adjacentdownwellings remain unchanged if they are initiallyunmelted or contain very little or no melt [Tackley andStevenson, 1993]. The primary similarities between thedecompressionmelting instability and the classical Rayleigh-Taylor scenario are its inherently limited duration and appar-ent unconditional instability [Tackley and Stevenson, 1993;Raddick et al., 2002].[4] There also exist distinctions between the buoyant

decompression melting instability and the enhanced insta-bility caused by interaction between thermal convection andpartial melting. Although important interactions of this kindhave been studied in the context of mid-ocean ridges [Scottand Stevenson, 1989; Buck and Su, 1993], and mantleplumes [Ribe and Christensen, 1999], the buoyant decom-pression melting instability is different because it is derivedfrom an inherent internal instability of a ‘‘two-phase’’ (i.e.,melt + solid) region. This instability need not be anenhancement of larger-scale convection, but rather is aprocess which could develop independently in the absenceof significant lateral temperature variations in the mantle.

However, the existence of any sort of background thermalor chemical heterogeneity would likely serve as a nucleationpoint for instability, causing activity associated with thisphenomenon to be preferentially correlated with changes indeep-rooted lithospheric structure [Raddick et al., 2002].[5] The basic requirements for the occurrence of a buoyant

melting instability are that a layer of mantle be brought to thesolidus temperature for a ‘‘sufficient’’ period of time,adiabatic upwelling leads to further melting, and suchmelting results in a cumulative decrease in density inupwelling regions relative to downwelling regions. This isalso subject to the more fundamental requirement that thelayer of mantle not be too depleted that it cannot producemore melt. There are numerous processes by which aportion of mantle may be brought to the solidus tempera-ture, such as upwelling of deep mantle that has risen underits own buoyancy (e.g., a mantle plume head or its wake) orunder the influence of tectonic activity (e.g., beneathextending continental lithosphere or mid-ocean ridges). Aregion of mantle brought to the solidus does not immedi-ately cool, but may remain at or near this temperature untilthermal conduction or some other process modifies its state.The timescale for conductive cooling of a partially meltedregion several tens of km in dimension is typically on theorder of several million years. So long as the instabilitymanifests itself on shorter timescales, a partially meltedregion may be susceptible to the occurrence of buoyantmelting instabilities.[6] Basaltic melt extracted from a typical ‘‘dry’’ upper

mantle peridotite can be up to about 15% less dense than thesource rock, although this value varies widely with pressureand composition [e.g., Stolper and Walker, 1980]. Becauseit is a partial melt, its actual contribution to bulk densityvariations is proportional to the volume fraction of retainedmelt f, which is expected to be small (of order 1%) in theupper mantle. The total contribution of melt to densityvariations is therefore on the order of 0.15%. Densitychanges in the solid residuum following melt extraction

Figure 1. Illustration showing the cause of a runaway instability within a partially molten layer. Upwardmotion is associated with increased rates of melting relative to downward motion. The density is reducedas a consequence of melting so that upwellings become more buoyant, move upward at a faster rate, andgenerate more melt. This positive feedback can lead to convective overturn of the layer and mightpotentially generate significant amounts of melt.

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are an inevitable consequence of incongruent fractionalmelting, and is attributed to two effects: (1) a change inchemistry of individual mineral phases comprising the rockdue to preferential partitioning of some of its dissolvedcomponents into the melt and (2) a change in the volumetricabundance of different crystalline phases, each of which hasa different density. The relative importance of these twomechanisms is sensitive to solid-solid phase changes, suchas the spinel-garnet transition occurring at �70–80 kmdepth in Earth’s mantle, because garnet is a dense phase andis preferentially consumed by melting. Density changes alsooccur because the Fe/Mg ratio of depleted rock is oftensmaller than unmelted rock. The calculations of Oxburghand Parmentier [1977] yielded a density reduction of about1.5% for 15% fractional melting of peridotite in the garnetstability field, and about half this amount in the spinelstability field, similar to estimates reported by Jordan[1979]. Recent estimates by Schutt and Lesher [2006] usingimproved experimental data found a 1.2% density reductionin the garnet stability field after 20% melt extraction andmuch smaller density changes for melt extraction in thespinel stability field. Thus the degree to which melt extrac-tion in the spinel stability field affects solid density may besignificantly smaller than previously thought. Thermalbuoyancy may also become important in a partially meltedlayer if temperature variations are not strongly buffered bylatent heat effects associated with melting. Temperaturechanges density by aDT, where a is the thermal expansivityand DT is the magnitude of horizontal temperature varia-tions. Assuming a typical value of a = 3 � 10�5 K�1 andDT = 50 K yields a 0.15% density change. Thus insummary, a temperature increase of 50 K yields the samedensity variation as 1% increase in melt fraction, or about1.5% melt extracted from garnet peridotite. However, thisratio is also generally variable because depletion also affectsthe thermal expansivity of mantle rocks [Schutt and Lesher,2006].[7] The full physics of the buoyant decompression melt-

ing instability is complex because it involves up to threesources of buoyancy for the bulk circulation, each of whichis governed by different transport processes. Temperaturevariations are subject to thermal diffusion, and additionallycan be affected by the absorption (or release) of latent heatwhenever melting (or freezing) occurs. Solid compositionand phase variations arise from partial melting and segre-gation, and unlike temperature variations do not diffuseaway at a significant rate over the relevant spatiotemporalscales. Partial melt is more complicated, because it mayundergo gravity-driven percolation through the pore spacesit occupies in the solid matrix, which can accelerate itsremoval up and out of regions where it is produced.Although percolation is not a strictly diffusive process, itmay still act as a dampening agent in this kind of instabilityby removing a source of buoyancy and weakening thestrength of any unstable circulation driven by lateral varia-tions in partial melt fractions. A straightforward analysis ofthe effects of percolation is also complicated by the fact thatthe rate of melt migration is itself dependent upon theamount of melt present in the rock as well as a variety ofpotentially complex microstructural processes.[8] Tackley and Stevenson [1993] first studied this kind of

instability using convection models including the effects of

melt buoyancy, a simple univariant melting model, andpercolation of the melt according to Darcy’s law. Theyignored the effects of thermal and melt depletion buoyancyin order to isolate the role of melt retention buoyancy andpercolation. Beginning with an initial condition where themantle finds itself at the solidus temperature over a layer ofthickness D, the duration and volume of melt generated bythe instability were found to depend strongly (and non-linearly) upon the rate of melt percolation. When scaled togeologically relevant ranges of parameters, characteristicvelocities of order 3 cm a�1 and durations of order severalmillion years were obtained from their numerical results.The spacing between upwellings was found to be about2–3 X the thickness of the partially molten layer, whichdefines a characteristic length scale for the occurrence ofvolcanic activity which might arise as a consequence ofthese kinds of instabilities.[9] Jha et al. [1994] considered the occurrence of this

kind of instability beneath the axis of mid-ocean ridgesusing melt retention and depletion buoyancy, followingearlier work that only considered melt depletion buoyancy[Parmentier and Morgan, 1990]. Such instabilities cangenerate three-dimensional structure in an otherwise two-dimensional ridge setting. These studies demonstrated thatthe instabilities only occur beneath a ridge for slower platespreading rates, perhaps explaining why slow spreadingridges exhibit more along axis variability than fast spread-ing ridges [Parmentier and Morgan, 1990], though not at alllength scales [Barnouin-Jha et al., 1997] unless the influ-ence of variable viscosity is taken into account [Choblet andParmentier, 2001]. The influence of melt buoyancy wasfound to be strongly controlled by the grain size, whichmodulates the rate of melt percolation and hence theretained fraction of melt in the mantle. The study ofBarnouin-Jha et al. [1997] also considered buoyancy aris-ing from temperature variations beneath a ridge, in additionto melt retention and solid depletion effects, and found avariety of behavior for instabilities occurring beneath andaway from the ridge axis, and suggested that the occurrenceof these different behaviors must be controlled by some typeof effective ‘‘Rayleigh number.’’[10] Schmeling [2000] considered the rate of propagation

of this type of instability across an initially uniform partiallymolten asthenospheric layer. An experimentally constrainedpower law rheology was adopted, and partially moltenregions were assigned a ‘‘wet’’ rheology. The physics ofthe two-phase (melt + solid) flow was treated using a‘‘compaction Boussinesq approximation’’ that was designedto account for the effects of matrix stresses on meltpercolation while still maintaining some simple features ofthe standard Boussinesq approximation. The rate of propa-gation of the instability was found to be on the order ofseveral centimeters per year, and was compared to themotion of hot spots. Circulation was largely confined tothe partially molten layer, most likely because it wasassigned a lower viscosity than other portions of the modeldomain.[11] Raddick et al. [2002] completed the most recent

systematic study of the instability, treating the effects ofthermal, solid depletion, and melt buoyancy along withviscosity variations induced by melting in a plane layer ofasthenosphere. Percolation was not explicitly modeled,


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rather a threshold criterion was used instead where the meltwas considered to escape instantly after reaching a criticalvalue. They found that the duration and amount of meltgenerated by this kind of instability can be strongly affectedby preexisting melt depletion buoyancy. Another signaturecharacteristic of this instability was found to be a durationthat is inversely proportional to the volume of melt pro-duced, a feature that is apparently related to solid depletionbuoyancy effects. Additionally, the spacing between upwel-lings was found to be slightly sensitive to the relativeimportance of thermal and compositional buoyancy, withspacing becoming smaller with increasing solid depletion-derived buoyancy.[12] The goal of the present study is to consider the

occurrence of these kinds of instabilities beneath diffuselyextending lithosphere, an important and common geologicalprocess on Earth in which partial melting of the mantle iscommonly attributed to simple passive upwelling of themantle. Numerical models are used to study the develop-ment of buoyant decompression melting instabilities under avariety of extension rates, mantle viscosities, rates of meltpercolation, and effects of solid depletion-related buoyancy.We then discuss a possible application of the wide variety ofobserved behaviors to the complex spatial and temporalrelationships between volcanism and extension in the west-ern U.S. Basin and Range province. The companion paperby Hernlund et al. [2008] (hereafter referred to as HST)employs a linear analysis in order to interpret these numericalresults and to obtain a better understanding of the basicprocesses leading to the onset of buoyant decompressionmelting instabilities. Further applications of this theory tomelting instabilities beneath mid-ocean ridges are alsodiscussed in HST.[13] The setting for the models performed in this study is

the simplest possible scenario for bringing a layer of mantleabove the solidus temperature without introducing spuriousperturbations or external influences: passive decompressionmelting of a plane layer of mantle as it upwells in responseto regional (diffuse) lithospheric extension (Figure 2). Incontrast to the study of these instabilities beneath mid-ocean

ridges [Parmentier and Morgan, 1990; Jha et al., 1994;Barnouin-Jha et al., 1997], the basic setting for the insta-bility is essentially one-dimensional rather than two-dimensional, which allows us to isolate different behaviorsof the instability from confounding influences and otherfactors that may arise in more complex settings. There areadditional motivations for studying buoyant decompressionmelting instabilities beneath diffusely extending lithosphere.First, the generation of mantle melt in regions undergoingdiffuse lithospheric extension represents an importantsource of volcanism away from plate boundaries. Anyinstability of this kind occurring in such a setting willinevitably have an impact upon the volumes, timing, andlength scales of volcanism observed at the surface whichcannot be accounted for by simpler one-dimensional models[e.g., McKenzie and Bickle, 1988]. Second, the occurrenceof this kind of instability beneath the Basin and Rangeprovince of the western United States has previously beenproposed on the basis of geological and geophysical obser-vations [Humphreys and Dueker, 1994a, 1994b], and thebasic setting for the model studied here is similar toconditions that are expected to have accompanied the lateCenozoic history of this region.[14] An important feature of this study is the incorpora-

tion of a self-consistent treatment of melt percolation, and itis again found that the rate of this process strongly modu-lates the dynamics of this kind of instability. It will also beshown that for certain ranges of mantle viscosity, extensionand percolation rates, buoyant decompression melting insta-bilities are suppressed during extension and only begin todevelop after extension has slowed or stopped. This behavioris found to be sensitive to both the strength and depthdistribution of melt depletion buoyancy. The possibility foran increased rate of localized volcanism following thecessation of extension is predicted to be a distinctivediagnostic characteristic of these kinds of instabilities thatmay be difficult to produce by any other mechanism. Anincrease in localized volcanism following extension hasbeen reported in several late Cenozoic volcanic fields inthe Basin and Range province [Gans and Bohrson, 1998;Stockli et al., 2002], and these kinds of instabilities might

Figure 2. Schematic illustration of passive decompression melting in a plane layer of mantleaccompanying distributed lithospheric extension.


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therefore be an important dynamical process that hascontributed to the complex geologic history of this region.

2. Mathematical Model

[15] Numerous models have emerged to describe partialmelting and two-phase (melt and solid) flow in themantle [e.g., McKenzie, 1984; Scott and Stevenson,1989; Spiegelman, 1993; Schmeling, 2000; Bercovici etal., 2001]. The situation can either be described byconstructing momentum equations governing the melt andsolid individually, or alternatively by taking both an averageof these equations to obtain a mixture momentum equationand a weighted difference of the two momentum equationsto yield an ‘‘action-reaction’’ equation for the separation offluid and solid. Here, the latter strategy is adopted because itallows for a simple implementation in the numerical rou-tines to be used in this study. The model described here issimilar to the one used by Tackley and Stevenson [1993],with the inclusion of solid depletion similar to that ofRaddick et al. [2002]. Although the model we adopt isadmittedly simple, it will become clear that it produces avariety of interesting behavior that needs to be thoroughlyunderstood before implementing more complex models.

2.1. Momentum Equations

[16] The kinematics of the melt and solid mixture can bedescribed by assigning to the solid an average matrixvelocity vs, and to the melt an average velocity throughthe pore spaces vm. The flow separation is defined as thedifference between these two velocities vm � vs, and thebulk (or average/barycentric) velocity of melt and solid is v =fvm + (1� f) vs, where f is the volume fraction of melt. Thesimplest model for flow separation is percolation driven bythe lithostatic pressure gradient according to Darcy’s law:

uz ¼ f vm � vsð Þ ¼ k fð Þ rs � rmð Þgmm

z; ð1Þ

where u is the ‘‘Darcy velocity’’ (volume flux of meltthrough the solid matrix), z is an upward directed unitvector, rs is the density of the solid, rm is the melt density, gis the acceleration of gravity, and mm is the melt viscosity.For small melt fractions and laminar percolation throughquasi-cylindrical pore spaces, the Darcy permeability k(f) isapproximately k = k0f

2, where k0 is a constant that dependson the shape and size of solid grains [e.g., Turcotte andSchubert, 2002, chapter 9]. We note, however, that theparticular form of the permeability relation appropriate forEarth’s mantle is not well constrained. Equation (1) is an‘‘action-reaction’’ equation representing the difference inmomentum between melt and solid. It assumes that for thelength scales of interest in this study, the vertical (lithostatic)pressure gradient is much larger (of order several GPa) thanany lateral dynamic pressure gradients arising from mantleflow (of order of several megapascals), and thus meltseparation only occurs in the direction of gravity. Aposteriori comparison of dynamical pressure gradients fromthe numerical models reveals that the dynamic pressuregradient is typically at least 2 orders of magnitude smallerthan the lithostatic pressure gradient for the presentscenario.

[17] The momentum equation for the mixture is written interms of the bulk velocity, which is taken to be incompress-ible for simplicity,

rrrr � vv ¼ 0: ð2Þ

Assuming the Boussinesq and infinite Prandtl numberapproximations, the bulk force balance is given by

2rrrr � mrrrrvvð Þ þ rrrr� mrrrr� vvð Þ � rrrrP þ rg ¼ 0; ð3Þ

where r is the gradient vector, m is the viscosity of themixture, and P is the pressure. Equations (1) and (3)implicitly assume that (1) pressure differences between fluidand solid arising from surface tension effects are negligible,(2) f is small (i.e., of order 1%), (3) mm is small incomparison to the solid viscosity ms, and (4) matrixcompaction due to melt removal occurs instantaneouslyand without resistance. For the bulk density, r, a linearizedequation of state is used to measure the response tovariations in temperature T, melt fraction f, and degree ofsolid depletion z (defined below), after which the mixturemomentum equation becomes

2rrrr � mrrrrvvð Þ þ rrrr � mrrrr � vvð Þ � rrrrp

¼ gz@r@T

T þ @r@f

f þ @r@z

z� �

; ð4Þ

where p is the nonhydrostatic pressure. The derivatives @r/@T, @r/@f, and @r/@z are taken to be constant parameters inthis study.

2.2. Transport Equations

[18] The conservation of melt is given by [e.g.,McKenzie,1984]

@f@t

þrrrr � vmfð Þ ¼ rsrm

_m; ð5Þ

or alternatively in terms of the bulk and Darcy velocities,

@f@t

þ v � rrrrf ¼ � @ u 1� fð Þ½ �@z

þ rsrm

_m; ð6Þ

where _m is melt production rate. For the conservation ofenergy,

@T

@tþ v � rrrrT ¼ kr2T � L

cp_m; ð7Þ

where k is the thermal diffusivity, L is the latent heat ofmelting, and cp is the specific heat. In writing equation (7),it is assumed that differences in k, cp, etc., between melt andsolid are negligible, and that temperature is advected withthe bulk velocity. The solid depletion, z, is defined as thecumulative volume fraction of melt extracted from the solid,and because it must move with the solid velocity, it isgoverned by

@z@t

þrrrr � vszð Þ ¼ rsrm

_m; ð8Þ


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or in terms of the bulk and Darcy velocities,

@z@t

þ v � rrrrz ¼ @ uzð Þ@z

þ rsrm

_m: ð9Þ

The meaning of z can be understood by noting that in theabsence of percolation u ! 0, and the conservationequations for melt (equation (5)) and depletion (equation(8)) are exactly the same.

2.3. Melting Model

[19] In this study, density changes in the residual solid areconsidered to arise primarily as a consequence of thepreferential consumption of distinct phases during melting[Schutt and Lesher, 2006]. Hence a simple eutectic meltingmodel is adopted, with melt being generated by the reactionbetween two immiscible solid phases A and B. The bulkcomposition of undepleted material (i.e., z = 0) is taken tolie between pure A and the eutectic composition E (Figure 3)such that melting preferentially consumes phase B. Nonzerodegrees of depletion (i.e., z0 > 0) are therefore associatedwith a higher relative abundance of phase A. Phase B is ageneric proxy for a dense solid phase (e.g., garnet), whosepreferential consumption by eutectic melting causes thebulk density of the solid to decrease (i.e., rA < rB) as aconsequence of its decreased volume abundance followingmelt extraction. For simplicity, the total fraction of meltextracted in the model is considered to be less than thefraction for which B would be entirely consumed (i.e., Bout),and therefore the temperature does not rise above theeutectic, nor does the composition of the melt depart from

the eutectic composition E. The eutectic temperature Ts isassumed to be linear in depth h,

Ts ¼ Ts0 þdTs

dhh; ð10Þ

where Ts0 is the eutectic at the surface (h = 0) and dTs/dh isthe rate of change with depth. Because the bulk flow isincompressible in the models considered here, the eutecticslope appearing in equation (10) is defined relative to thepotential, rather than absolute, temperature scale. Under theabove assumptions, the exact details of the binary eutecticsystem need not be specified, since the effects of meltingupon solid density, melt depletion, and temperature arealready specified by the linearized equation of state andlatent heat. More realistic melting models in the context ofthis instability might be worth considering in the future,however, this is beyond the scope of the present study.[20] Melting can also be described in the context of

adiabatic upwelling, where the rate at which mass is con-verted from solid to liquid, _m, is proportional to theupwelling velocity. At the eutectic temperature, upwellingproduces a rate of melting given by

_m ¼ rmrs

F z � vð Þ; ð11Þ

where F is the volume fraction of melt produced per unitdistance of adiabatic upwelling. Given the above assump-tions, the adiabatic melt productivity is related to latent heatby combining equation (11) with equation (7) to give

dTs

dh¼ LF

cp: ð12Þ

Although F is in reality a complicated function of pressureand composition [e.g., Asimow et al., 1997], it is treated as aconstant here for simplicity. It should be noted that our goalin this study is not to reproduce the precise details of mantlemelting, but rather to formulate and understand its effects inas simple a way as possible and lay the groundwork forfuture studies of the instability.

2.4. Nondimensionalization

[21] It is useful to nondimensionalize the governingequations and all quantities relative to a depth scale D, athermal diffusion time scale D2/k, a temperature scale DT, areference viscosity m0 and density r0. Indicating nondimen-sionalized quantities with a prime, the equations of motionbecome

rrrr0 � v0 ¼ 0; ð13Þ

2rrrr0 � m0rrrr0v0ð Þ þ rrrr0 � m0rrrr0 � v0ð Þ � rrrr0p0

¼ �z RaT 0 þ Rmfþ Rdzð Þ; ð14Þ

u0z ¼ f v0m � v0s� �

¼ Rm

Mrf2z; ð15Þ

Figure 3. Illustration of the simple eutectic melting modelassumed in this study. Melting is considered to take placebetween two solid phases, A and B, with a ‘‘bulk’’composition lying between the eutectic E and phase A.Phase B is considered to be more dense than phase A; thuspreferential consumption of B by the eutectic melt results ina bulk density decrease in the residual solid.


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@T 0

@t0þ v0 � rrrr0T 0 ¼ r02T 0 � L0 _m0; ð16Þ

@f@t0

þ v0 � rrrr0f ¼ � @u

@z0þ _m0; ð17Þ

@z@t0

þ v0 � rrrr0z ¼ @ uzð Þ@z0

þ _m0; ð18Þ

where

Ra ¼ r0gaDTD3

km0

; ð19Þ

Rm ¼ � gD3

km0

@r@f

; ð20Þ

Rd ¼ � gD3

km0

@r@z

; ð21Þ

Mr ¼ mmD2

m0k0: ð22Þ

Ra, Rm, and Rd are the respective ‘‘Rayleigh numbers’’ fortemperature, melt, and solid depletion derived buoyancy.The quantity Mr is called a ‘‘melt retention’’ number, sinceit is inversely proportional to the rate of melt percolation. Inthe following, the primes on nondimensionalized quantitieswill be dropped for brevity, and unless stated otherwisevariables are nondimensional.

2.5. Numerical Model

[22] Numerical models are performed using the codeSTAG3D [Tackley, 1996]. The governing equations (13)–

(18)) are solved using a finite volume technique in a two-dimensional Cartesian domain of aspect ratio two. A griddensity of 32 in the vertical and 64 in the horizontaldirection is used for most calculations, and cases run atdouble this grid density have also been performed to ensurethat the resolution is sufficient. The cases are run in twodimensions with the horizontal axis coinciding with thedirection of extension, which allows us to explore a largeparameter space at relatively small computational expense.[23] A Newtonian temperature-dependent viscosity is

adopted which follows a simple Arrhenius relationship ofthe form

m ¼ exp Ea

1

T þ Tsurf� 1

1þ Tsurf

� �� ; ð23Þ

where Ea is the nondimensional activation energy (i.e., thedimensional value in kJ mol�1 normalized by the gasconstant R and DT) and Tsurf is the nondimensionaltemperature of the surface (i.e., 273 K normalized byDT). This rheology gives a strong temperature dependenceof the viscosity so that a highly viscous ‘‘lithosphere’’ arisesnaturally as a consequence of its cooler temperatures, andcauses the circulation due to buoyant convection to beconfined to the partially molten zone and underlyingmantle. The nondimensional viscosity is equal to unity inthe underlying mantle, where the temperature is T = 1, andis truncated at a maximum value of m = 103, which is morethan sufficient to suppress convective motions in theoverlying lithosphere.[24] The initial condition is a geotherm described by a

subeutectic error function profile, but which is taken to bevery close to the eutectic temperature so that only a smallamount of extension is required to induce a partially moltenlayer of material (Figure 4). In order to isolate the effects ofpassive upwelling upon the solid depletion profile, initiallyz = 0 everywhere. Random perturbations of 1% are added tothe temperature, which are transformed into perturbations inmelt and depletion after the temperature exceeds the eutec-tic. Extension is imposed as a kinematic boundary conditionby setting the velocities at the edges of the model domain tomatch a pure shear flow pattern of the form

v ¼ vxxþ vzz ¼ _e x� 1ð Þxþ 1� zð Þz½ �; ð24Þ

where x is the unit vector directed along the coordinate x, vxis the horizontal velocity, vz is the upward velocity, _e is thestrain rate, and x = 1 is the x coordinate of the middle of thedomain. The pattern of flow that develops in the absence ofinternal buoyancy forces is plotted in Figure 5. This type ofboundary condition assumes that extension is uniformlydistributed with depth, which is the simplest possiblearrangement and removes the possibility for any potentiallycomplicating effects of background lateral shear flow in theasthenosphere.[25] The temperature at the top of the domain is set to T =

0, while the temperature and depletion are set to T = 1 and z =0 at the bottom of the model domain to ensure that fertilematerial of uniform temperature is advected into the modelfrom below. The run durations are shorter than the timerequired to develop a lower thermal boundary layer, so that

Figure 4. Initial conditions for temperature (solid line)and the position of the eutectic (dashed line) used in thenumerical models. For the choice of parameters used here,the initial temperature is given by erf(depth/0.21). Thedepth of the spinel-garnet transition used in the ‘‘mixedphase’’ cases (see section 3 for details) is indicated by adotted line.


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no boundary layer or associated instabilities in the formof upwelling plumes from below occur in the time span ofinterest. Models with an effective dimensional depth of670 km have also been performed for a variety of cases toensure that the side and lower boundary conditions haveno significant effect on the results reported here.[26] The imposed extensional strain rate is taken to be

time-dependent ( _e = _e(t)), and after a prescribed amount oftime it tends to zero according to

_e ¼_e02

1� tanh5 t � tsð Þ

ts

� �� ; ð25Þ

where _e0 is a reference strain rate and ts is a stopping timewhich is set to produce a given amount of cumulativestretch b according to ts = _e0

�1 ln b (e.g., b = 2 when thesurface is expanded by a factor of 2). The factor of 5appearing in the argument of the hyperbolic tangent isarbitrarily chosen so that it produces a gradual and smoothtransition to a zero rate of extension.[27] Melt production is treated by a simple energy bal-

ance that converts excess temperature into melt fraction viathe latent heat in order to satisfy the condition that thetemperature remains capped by the eutectic. Freezing isstraightforwardly accounted for in this procedure. Becausef ! 0 in the ‘‘lithosphere’’ above the partially molten layer,u ! 0 and the melt fraction in the topmost cell of a partiallymolten column exhibits runaway growth. A threshold cri-terion for melt ‘‘eruption’’ was used by Raddick et al.[2002], where melt was considered to erupt after attaininga critical fraction fc, thus avoiding this kind of runawayaccumulation of melt. However, it will be important in thisstudy to allow the equations to directly govern the meltfraction. Thus we adopt a new kind of filter that removes theexcess melt accumulating at the top of a partially moltenlayer, and mimics the transition to rapid melt transport

through dikes or cracks in the lithosphere. The filter is verysimple, and is applied by first finding the uppermost cell inthe model domain containing partial melt within everyvertical column. A fraction of melt in this uppermost cellin excess of the fraction in the underlying cell is thenremoved and added to a total cumulative amount of melt‘‘erupted’’ from the mantle. We note, however, that inreality only a fraction of this ‘‘erupted’’ melt might reachthe surface and it therefore represents an upper bound uponany actual expected volcanic eruption rate.

3. Results

[28] Several thousand cases have been investigated, andthe results of these runs reveal a variety of interestingbehaviors. The parameter values used in the suite ofcalculations are listed in Table 1. The most uncertainphysical parameters are the reference viscosity of the bulkmixture m0, the reference Darcy permeability k0, and themelt viscosity mm, which are accordingly varied within areasonable range of values. The reference strain rate _e0 isalso varied (between 1 � 10�15 sec�1 and 10�14 sec�1) inorder to explore the effects of different extension rates uponthe characteristics of the instability. The dimensional dura-tion of extension resulting from these extension rates rangesbetween roughly 2 and 20 Ma for a stretch factor of b = 2.The contribution of solid depletion to density variations isconsidered in three distinct scenarios: (1) one in which @lnr/@z = 0% throughout the entire model domain (hereaftertermed the ‘‘spinel stability case’’), (2) one in which @ lnr/@z = 6% throughout the entire model domain (hereaftertermed the ‘‘garnet stability case’’), and (3) one in which aspinel-garnet transition is included with @ lnr/@z = 6% atdepths greater than 80 km and @ lnr/@z = 0% at shallowerdepths (hereafter termed the ‘‘mixed phase case’’). Al-though this is admittedly oversimplified, it will serve to

Figure 5. Boundary conditions for the x and z components of velocity (black arrows) and the pattern ofpure shear that develops inside the domain in the absence of buoyancy forces (grey arrows). In thenumerical models, the internal velocity field (grey arrows) is obtained in accordance with themathematical model described in the text. The x and z coordinates are measured in units of the domaindepth D.


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illustrate that the behavior of the instabilities under theinfluence of solid depletion buoyancy depends not onlyupon the degree of density changes, but also upon thelocation of density changes within the partially moltenlayer. Because the ratio ko/mm (hereafter termed the ‘‘Darcycoefficient’’) appears in the melt retention number Mr as asingle parameter, the exploration of parameters leads tovariations in three independent quantities in each scenariofor the solid depletion, while all other parameters are takento be fixed.[29] Three distinct types of behavior are found for the

timing of the onset of instability. For the higher range ofviscosities and melt percolation rates considered, no signif-icant circulation occurs during or after extension, and the

instability is very weak or essentially nonexistent. For thespinel stability cases, a mantle viscosity m0 less than 10

20 Pas is required for any significant instabilities to occur, whileviscosities less than about 1019.5 Pa s are required for theoccurrence of instabilities in either the garnet stability ormixed phase cases. For the lower range of referenceviscosities (m0), Darcy coefficients (k0/mm), and strain rates( _e0), the instability sets in almost immediately after exten-sion and passive upwelling lead to the development of a thinpartially molten layer. This kind of timing will be referred toas ‘‘synextensional instability,’’ since the layer becomesunstable while extension is ongoing. For an intermediaterange of reference viscosities, Darcy coefficients, and strainrates, on the other hand, instabilities do not develop duringthe time period of extension, but only begin to develop afterthe rate of extension slows significantly. However, this‘‘postextensional instability’’ behavior is only observed inthe spinel stability and mixed phase cases, and does notoccur in any of the garnet stability cases. The ‘‘synexten-sional’’ versus ‘‘postextensional’’ instability distinction isshown in a series of time slices for two different spinelstability cases in Figure 6. Figure 7 shows time slices fortwo garnet stability cases: one in which no significantinstability occurs and another in which synextensionalinstability occurs.[30] The effect of various parameter values upon the

occurrence of synextensional and postextensional behaviorsis shown in Figure 8. The criterion used to make thisdistinction is the clear manifestation of instability in thelayer prior to (synextensional) or after (postextensional) therate of extension begins to tail off significantly (for times ofabout jt � tsj < ts/5). Interestingly, both the spinel stabilityand mixed phase cases exhibit synextensional or postexten-sional behavior for the same range of parameters. In all

Table 1. Parameter Values Considered in This Study

Quantity Dimensional Values

r0 3300 kg m�3

g 10 m s�2

@r/@f �500 kg m�3

@r/@z 0–200 kg m�3

b 2_e0 10�15–10�14 s�1

a 3 � 10�5 K�1

DT 1500 Km0 1019–1021 Pa sk 10�6 m2 s�1

D 300 kmEa 500 kJ mol�1

Ts0 1400 KdTs/dh 3.5 K km�1

F 5 � 10�3 km�1

k0/mm 1010–1012 Pa s m�12

Figure 6. Plots of temperature, melt fraction, and deple-tion during and after extension for two spinel stability cases,showing the distinction between (top) postextensionalinstability and (bottom) synextensional instability.

Figure 7. Plots of temperature, melt fraction, and deple-tion during and after extension for two garnet stability cases.(top) No significant instability. (bottom) Synextensionalinstability.


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cases considered, synextensional instability is favored by alower reference viscosity, lower rate of extension, andsmaller Darcy coefficient (k0/mm). The range of parametersover which the occurrence of postextensional instabilitiesdevelop becomes more narrow as the rate of extensiondecreases, so that for very slow extension only synexten-sional or weak/no instabilities should occur. For higherstrain rates than those considered here, only mantle vis-cosities that are smaller than 1019 Pa s will be capable ofpermitting the occurrence of synextensional instabilities. Ahigh rate of melt percolation, which rapidly drains thelayer of melt, suppresses the development of instabilityboth before and after extension in the spinel stability andmixed phase cases.[31] A reference mantle viscosity less than 1020 Pa s

appears to be necessary for the instabilities to developsignificantly before growth of the overlying thermal bound-ary layer by conduction cools the partially molten layer. Atmuch later times in the models this thermal boundary layeritself becomes unstable and ‘‘delaminates’;’ however, this isa fundamentally different phenomenon than the decompres-sion melting instability. The contribution of any laterthermally driven instabilities to quantities of interest istherefore removed. However, it is important to distinguishthis effect from the postextensional instability behavior seenin this study when it produces melt at later times, sinceadditional pulses of melting can also occur that could beconfused with a postextensional instability. In practice, thetimescales involved in the growth and subsequent ‘‘delami-nation’’ of a portion of the thermal boundary layer (on theorder of several tens of millions of years) are much longerthan the timescales over which the decompression meltinginstabilities take place (on the order of a few million years),and as a consequence the two phenomena can easily bedistinguished based upon their relative timing.[32] The synextensional versus postextensional distinc-

tion in behavior can also be seen in the time series of thetotal cumulative erupted volume of melt. Figures 9 and 10show the volume of erupted melt over time for threedifferent strain rates and for different values of the Darcycoefficient. Extension alone produces a linearly increasing

volume of melt that saturates and begins to level off asextension slows. Synextensional instabilities are character-ized by a growth in erupted volume of melt prior to theslowing of extension, and the total amount of erupted meltis usually larger in these cases than in postextensional cases.Postextensional instabilities, on the other hand, yield erup-ted volumes that initially follow extension-induced eruptionrates and only after production begins to plateau does asecond pulse in increased erupta begin to develop. Thespinel stability cases illustrate that the propensity for syn-extensional instability is correlated with the total volumeand rate of erupted melt, with postextensional affinityassociated with a greater time lag in the onset of instability.The garnet stability cases, on the other hand, yield a muchsmaller variation in erupted volume with time and do notexhibit a clear signal heralding the onset of instabilitysimilar to those seen in the time series for the spinel stabilitycases.[33] The cumulative total erupted volume of melt, termed

the ‘‘total erupta,’’ is plotted in Figure 11 for the samevalues of the strain rate. The total erupta usually increases asboth k0/mm and m0 decrease, while the spinel stability casesgenerated significantly more melt than either the garnetstability or mixed phase cases. In some cases, the totalerupta is also smaller for the smallest values of the Darcycoefficient, a phenomenon also observed in the study ofTackley and Stevenson [1993] for higher values of Mr. Themixed phase cases exhibit a total erupta similar to that foundin the garnet stability cases, but slightly smaller in volume.In either case where solid depletion buoyancy is present, thedownward entrainment of the depleted material into theunderlying mantle is suppressed.

4. Discussion

[34] The numerical modeling results presented abovereveal a range of interesting behavior for the occurrenceof the instability, its timing relative to extension, the effectsof solid depletion buoyancy, and a strong influence ofpercolation. All of the features seen in this study carryobservational consequences for the spatiotemporal charac-

Figure 8. Dependence of synextensional (solid circles) versus postextensional (open circles) timing ofinstabilities upon the reference mantle viscosity m0 and the Darcy coefficient k0/mm for three differentreference strain rates _e0 in both the spinel stability and mixed phase cases and for a stretch factor of b = 2.Cases where only weak or no instabilities developed (crosses) are also indicated.


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teristics of volcanism in regions that have undergone diffuselithospheric extension. Here we discuss these issues ingreater detail.

4.1. Synextensional Versus Postextensional Instability

[35] The lack of instabilities during extension for some ofthe cases suggest that the instability in this scenario is notstrictly analogous to a Rayleigh-Taylor instability, since thiskind of terminology carries the implication of uncondition-ally unstable behavior. A better analogy might be a Rayleigh-Benard type of behavior, where some critical value orparameter must be exceeded in order for instabilities to occur.However, as the occurrence of instabilities depend upon thestrain rate, it appears that any such ‘‘Rayleigh number’’describing the onset of convection must itself depend uponthe rate of extension. The most direct consequence of anincreased extension rate is a higher rate of passive upwellingand melt production in the partially molten layer, which isaccompanied by a larger fraction of retained melt. While it

seems counterintuitive that a larger amount of retained meltsuppresses the onset of instability, it is shown in HST that thisarises due to the presence of an increased rate of percolationin the basic state, with which the growth of any instabilitymust compete in order to sustain lateral variations in meltfraction.[36] The results also demonstrated that the timing of

instability is highly sensitive to the depth distribution ofdensity changes accompanying solid depletion. In particu-lar, an absence of large density changes at depths smallerthan 80 km allowed for postextensional behavior, while auniform strong density change everywhere promoted syn-extensional behavior. Thus the conclusion of Schutt andLesher [2006] that density changes at shallower depths aresmall is important in assessing whether or not postexten-sional instability behavior can actually occur in Earth’smantle.[37] If instability occurs during extension, the distribution

of mantle melt production will be very different than the

Figure 9. Time series of the cumulative erupted volume of melt in spinel stability cases with m0 = 1019

Pa s and b = 2 for three different strain rates, where the values are normalized by the volume of melterupted by extension alone. Solid curves are cases having k0/mm = 10�9 m2 Pa s�1, dashed curves k0/mm =10�9.5 m2 Pa s�1, dotted curves k0/mm = 10�10 m2 Pa s�1, dot-dashed curves k0/mm = 10�10.5 m2 Pa s�1,and double-dot-dashed curves k0/mm = 10�11 m2 Pa s�1.


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case where instability only sets in after extension hasstopped. In all cases where a significant synextensionalinstability was observed in the numerical models, onlyupwelling regions exhibited a nonzero fraction of meltwhile the downwellings were frozen. (The exceptions werecases where a weak or no significant instability occurred atall, resulting in no significant gain in melt productionrelative to extension alone.) The lack of melt in downwel-ling portions of the circulation reflects the fact that the flowwhich arose due to the instability (excluding the ‘‘no-instability’’ cases) was greater in magnitude than the passiveupwelling velocity due to imposed extension. Thus there isa net downward motion of material in the downwellingregions even though extension has the effect of imposing anet upward motion upon the entire partially melting layer.[38] Because melt is produced only in upwelling portions

of an unstable partially melting layer, magma delivered tothe crust by this mechanism will tend to be localized aboveregions that are correlated with upwelling asthenospheric

flow arising from the instability. The consequences for thespatial distribution of magmatism (and possibly volcanismif melt reaches the surface) in the crust is shown schemat-ically in Figure 12. The cumulative melt production forpostextensional instabilities will be characterized by thedevelopment of a distribution of magmatism consistent withextension alone (which is uniform in the cases consideredhere), followed by localized delivery of melt above upwell-ing portions of the instability. Synextensional instability, onthe other hand, will exhibit very little magmatism in areasoverlying downwelling currents, so that the majority of meltproduction will be focused in patches above upwellings.Thus the different distributions of magmatism (or possiblyvolcanism) in the crust predicted for synextensional versuspostextensional instabilities may present a good diagnosticfor the characteristics of any asthenospheric instabilitiesoccurring beneath extensional provinces. However, this isnot necessarily a unique diagnostic. A potential complica-tion is the uncertainty of the initial condition in any part of

Figure 10. Time series of the cumulative erupted volume of melt in garnet stability cases with m0 = 1019

Pa s and b = 2 for three different strain rates, where the values are normalized by the volume of melterupted by extension alone. Solid curves are cases having k0/mm = 10�9 m2 Pa s�1, dashed curves k0/mm =10�9.5 m2 Pa s�1, dotted curves k0/mm = 10�10 m2 Pa s�1, dot-dashed curves k0/mm = 10�10.5 m2 Pa s, anddouble-dot-dashed curves k0/mm = 10�11 m2 Pa s�1.


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the upper mantle prior to extension. For example, someregions of the upper mantle could be cooler or more highlydepleted relative to adjacent material prior to extension, sothat a decreased propensity for melt production alreadyexists in some locations at the outset. It is important tonote that this sort of preexisting lateral heterogeneity willalso influence the behavior of the instability because onlysome portions of the upwelling asthenosphere are capable ofproducing melt. Such an arrangement would be expected toinfluence the pattern of any upwelling and downwellingflow, and might promote the occurrence of synextensionalinstabilities.

4.2. Melt Buoyancy and Production

[39] The cumulative ‘‘erupted’’ volume of melt was foundto be sensitive to both the viscosity m0 and Darcy coefficientk0/mm in all cases where a significant instability developed.The simultaneous balance between melt percolation, pro-duction, and buoyant circulation is central to understandinghow these physical parameters affect the total volume ofmelt. A simple model may be constructed to show how thecombination of these factors gives rise to the strong influ-ence of m0 and k0/mm upon the total erupta observed in thenumerical models.[40] First, consider upwelling in a layer at the eutectic

with bulk nondimensional velocity vc0 (primes will again be

used to denote nondimensionalized quantities in this sec-tion). Here, vc

0 will refer to the magnitude of the buoyantcirculation rather than the rate of passive upwelling. Asimple balance between melt production and percolation(ignoring the effects of compaction and advection) yields anapproximate characteristic melt fraction

fc �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMrF 0v0cRm

r; ð26Þ

where the dimensional length scale D is taken to be thethickness of the partially molten layer. Here, fc is intendedto be only a representative value of f rather than a fullsolution. The more general case is presented in HST [seealso Schmeling, 2000].[41] Now consider a stationary bulk circulation within a

partially molten layer with an aspect ratio of unity. In thiscase the vertical velocity field can be roughly approximatedby vz

0 = vc0 sin(px) sin (pz), where vc

0 is now identified withthe characteristic amplitude of the circulation. The flow isdriven by a ‘‘rectified’’ buoyancy force arising from partialmelt that can be crudely approximated as Rmfc sin (px) sin(pz)/2, where the factor of 2 is included to account for thelack of melt in downwellings. A simple momentum balancein the absence of thermal or depletion buoyancy forces thengives

v0c �Rmfc

8p2: ð27Þ

This is the characteristic diapiric rise rate for upwelling ofrock containing a fraction of melt fc.[42] Combining equations (26) and (27) yields an ap-

proximate value for the fraction of melt supported bybuoyant circulation:

fc �MrF 0

8p2; ð28Þ

fc does not depend upon Rm because melt buoyancy drivesbulk circulation (and hence production) as well aspercolation of melt out of the layer, with the net result thatthe two effects cancel one another. Another expression for

Figure 11. Total erupted volume of melt normalized by the volume of melt erupted by extension alonefor a strain rate of _e0 = 5 � 10�15 s�1. Each series is labeled by its value of k0/mm, which is given in unitsof m2 Pa s�1.


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the velocity amplitude can be obtained that only depends onthe parameters Rm, Mr, and F0,

v0c �RmMrF 0

64p4: ð29Þ

The proportionalities expressed by equations (28) and (29)are in good agreement with the modeling results of Tackleyand Stevenson [1993], which only considered partial meltbuoyancy as a driving force for the instability.[43] In terms of the dimensional velocity (vc = vc

0k/D),equation (29) leads to the proportionality

vc /DrMgFD

5

m2o k0=mmð Þ ; ð30Þ

where DrM = rs � rm. The dimensional velocity is seen toscale inversely to the square of the viscosity. This nonlinearbehavior can be understood by noting that the strength ofthe buoyancy source (/ Rmfc) itself depends on thecirculation rate vc, with higher rates supporting a propor-tionally greater fraction of melt in upwellings, andconsequently receiving the benefit of extra buoyancy.Estimates for the importance of melt buoyancy in this kindof system must therefore take into account the fact that theretained fraction of melt is itself proportional to thecirculation.

[44] This nonlinear behavior for the role of melt buoyancycan help explain the strong sensitivity of erupted melt uponviscosity and the Darcy coefficient. For example, the rate ofmelt production due to buoyant circulation in the layer, _Q, isnecessarily proportional to vcF, so that,

_Q / DrMgF2D5

m2o k0=mmð Þ : ð31Þ

Defining an effective duration of the circulation tc (whichneed not be constant for all parameter values) such that thecolumn height of melt H erupted at the surface is H = _QtcD:

H / DrMgtcF2D6

m2o k0=mmð Þ : ð32Þ

The proportionality relation in (32) captures the observeddependence of erupted melt upon k0/mm as well as the strongdependence upon the reference viscosity. It is worth notingthat this kind of dependence would not be present if thevalue of fc were imposed as a constant, since it would nolonger be coupled to the characteristic velocity of thecirculation. It is also apparent that the dependence of Hupon F is not linear, and therefore caution should be takenin extrapolating any results presented in this study todifferent values of F.

4.3. Effects of a Buoyant Depleted Layer

[45] Unlike partial melt, depletion is a cumulative prop-erty that generally increases with the degree of upwarddisplacement (instead of upward velocity) of material in thepartially molten layer. Therefore the effects of buoyant meltdepletion of the solid residuum are inherently different thanthe type of behavior described above for melt buoyancy. Inthe garnet stability cases, a buoyant depleted layer formedas a consequence of passive upwelling following extension,and the volume of melt produced by the instability wasgenerally less than in the spinel stability cases. However, thecumulative amount of erupted melt still depends upon k0/mm

even in these cases. This behavior makes sense if oneconsiders that lower values of viscosity and rates of meltpercolation can nevertheless enhance the ability for meltbuoyancy to push depleted material aside and for downwel-lings to entrain some of this material downward, asdescribed by Jha et al. [1994].[46] This resistance to downward entrainment of the

depleted layer in the mixed phase and garnet stability casesis a likely explanation for the reduced total volumes oferupted melt measured in these scenarios because it opposesthe tendency for deeper (and longer duration) circulation. Itis somewhat surprising that the garnet stability cases pro-duced slightly more erupta than the mixed phase cases,since the latter scenario is intermediate between the end-member garnet and spinel stability cases. The reason for theslight gain in melt productivity in the garnet stability casesrelative to the mixed phase cases might be attributable to thefact that the former always exhibited synextensionalbehavior, while the latter exhibited postextensional insta-bility for certain ranges of parameters.

Figure 12. Schematic illustration of the differing distribu-tions of mantle melt production depending upon whethersynextensional (solid line) or postextensional (dashed line)instabilities occur. The flat background distribution in bothcases is due to extension in the absence of instabilities,while the peaks are correlated with locations lying aboveupwellings portions of the circulation resulting frominstability. A synextensional instability occurs early duringextension, suppressing the production of melt in locationsabove downwelling portions of the resulting circulation,while a postextensional instability allows extension toproduce a greater amount of melt prior to forming localizedpatches of magma production.


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4.4. Western United States Basin and Range Volcanism

[47] The suppression of buoyant decompression meltinginstabilities by extension is a surprising characteristic of thistype of instability. This kind of behavior can lead todistinctive observational consequences if these instabilitiesoccur in regions undergoing lithospheric extension becausethere is no other straightforward dynamical process whichwould produce a similar spatiotemporal pattern of volca-nism without appealing to ad hoc mechanisms. While thereare many extensional provinces on Earth, none offers bettergeological exposure and geophysical coverage than theBasin and Range province of the western United States.In this region, the relationship between volcanism andextension have long been a subject of debate, and the resultsfound in this study might offer some insights into the natureof recent volcanic activity in this province.[48] During the late Cenozoic (mid-Miocene), much of

the western United States underwent large-scale diffuselithospheric extension, with elongation of the crust mostlyaccommodated by a fault-block mode of deformation that islargely responsible for the present-day ‘‘horst and graben’’physiography of the Basin and Range province. This ‘‘Basinand Range’’ episode of extension followed earlier episodesof localized high-magnitude extension which is character-ized by unroofing of the upper crust along detachment faults[e.g., Axen et al., 1993]. Thermochronological constraints inthe Great Basin region indicate that the duration of mid-Miocene extension was relatively short (on the order of amillion years) and largely contemporaneous [Stockli et al.,2002]. The Cenozoic volcanic history of this region is alsotemporally bimodal, with ubiquitous episodes of ignimbrite

volcanism in the mid to late Tertiary followed by increas-ingly basaltic volcanism from the mid-Miocene to present[e.g., Christiansen and Lipman, 1972].[49] The simplest model for the origin of volcanism

accompanying lithospheric extension is passive upwellingand attendant decompression melting of the underlyingmantle [e.g., McKenzie and Bickle, 1988]. Because magmaproduced in the mantle travels upward into the crustrelatively rapidly, this basic hypothesis predicts that theamount and rate of volcanism should be well correlated(both in space and time) with extension. However, thesepredictions are not borne out by observations in much of thewestern U.S. Basin and Range province. In particular, muchof the Miocene to present basaltic volcanism tends to befocused in several areas (see Figure 13), even though theBasin and Range episode of extension is more uniformlydistributed. Additionally, the temporal relationship betweenvolcanism and extension in the Basin and Range province isnot straightforward [e.g., Axen et al., 1993], and in manylocations an anticorrelation between rates of extension andvolcanism have been reported [Gans and Bohrson, 1998;Stockli et al., 2002]. While the silicic volcanic activitycommonly preceding extension in this region is oftenattributed to delamination and foundering of the shallowlysubducted Farallon plate [e.g., Humphreys, 1995], there isno straightforward mantle process which could explain boththe localization and increase in volcanic activity after thecessation of extension. One possible exception is the fortu-itous arrival of several mantle plumes from the deep mantle.However, a mantle plume origin of volcanism in the westernUnited States (with the exception of the Columbia Riverflood basalts and Snake River Plain/Yellowstone system) isprobably not tenable because the characteristic time pro-gression in ages of volcanic centers expected of thismechanism is not observed. The temporal anticorrelationbetween extension and volcanism is pronounced in theregions surrounding the Colorado plateau, such as the St.George volcanic field and Jemez Lineament [Gans andBohrson, 1998].[50] The strong suppression of volcanism during exten-

sion remains a rather surprising observation. This apparentdilemma is exacerbated by the fact that magma transportthrough crustal dikes and cracks should be enhanced whilethe crust is undergoing extension [e.g., Rubin, 1995]. Gansand Bohrson [1998] proposed that a loss of confiningpressure in the middle crust attending extension might leadto devolatilization and subsequent freezing of midcrustalmagmas, an effect that might be enhanced by the onset ofhydrothermal convection in the upper crust which wouldtransport heat more rapidly toward the surface. This type ofeffect was proposed as a mechanism to impede transmissionof magma to the surface. While reduced rates of magmatransmission through the crust during extension may or maynot explain the relative suppression of volcanism duringextension, these mechanisms alone cannot explain an in-crease in the rate of volcanism following extension, whichrequires a renewed source of mantle melting that cannot begenerated by simple passive decompression melting.[51] The occurrence of anomalously low seismic velocity

anomalies at asthenospheric depths (�60–100 km) beneathseveral Miocene and younger volcanic centers in the westernUnited States ledHumphreys andDueker [1994b] to posit the

Figure 13. Map showing the occurrence of Miocene andyounger volcanic rocks in the western United States, withthe locations of the Columbia River Flood Basalts, SnakeRiver Plain (SRP)/Yellowstone trend, Jemez Lineament,and St. George volcanic field indicated.


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existence of asthenospheric instabilities in the form of small-scale convection rolls, with the axes of upwelling coincidingwith linear trends of volcanism in the Jemez lineament (NewMexico and Arizona), St. George volcanic field (southernUtah), and possibly the Snake River Plain/Yellowstonesystem. They hypothesized that these rolls formed underthe influence of mantle shear accompanying the southwest-ward motion of the North American plate, and were signif-icantly influenced by density variations arising from partialmelt and depletion of the solid residuum. Changes in deep-rooted lithospheric structure along these trends (e.g., theedges of the Colorado plateau) might also explain the spatialdistribution of these fields. Such a mechanism could explainthe spatial distribution of some Basin and Range volcanismand a variety of geophysical observations [Humphreys andDueker, 1994b].[52] The instability behavior offers a potentially useful

way to explain renewed localized volcanism followingBasin and Range extension. The observable effects of thetiming and rate of magma production in the mantle astheno-sphere will, of course, be strongly affected by any suppres-sion of magma transport through the crust. Thus it is notstraightforward to directly map rates of eruption measuredin the present numerical models into expected rates ofvolcanism at the surface for comparison to real data. Thetimescales and volumes of melt generated by instabilitiesseen in the present study and in previous studies [Tackleyand Stevenson, 1993; Raddick et al., 2002] can easilysatisfy a wide range of measured durations in postexten-sional volcanism in the western United States (typically oforder several million years). Rates of extension inferred inseveral volcanic fields of the Basin and Range province[e.g., Gans and Bohrson, 1998; Stockli et al., 2002] may besomewhat higher than those considered in the numericalmodels presented in this study (by up to a factor of abouttwo), enhancing the possibility for postextensional behavior.On the other hand, petrological constraints indicate thatmelting beneath the southern portion of the Great Basinoccurred at depths extending into the garnet peridotitestability field [e.g., Wang et al., 2002], and therefore soliddepletion density changes at shallower depths remains animportant issue.

5. Summary and Conclusion

[53] Buoyant decompression melting instabilities beneathextending lithosphere can have a significant effect upon thegeneration of melt in the underlying mantle and the spatio-temporal characteristics of any consequent magmatism and/or volcanism. These instabilities produce more melt thanextension alone but require a sufficiently small astheno-spheric viscosity (i.e., less than 1020 Pa s) and gains in meltproduction are strongly suppressed by solid depletion buoy-ancy. Suppression of unstable behavior during extension isanother potentially important phenomenon that dependsupon the depth distribution of density changes accompa-nying melt depletion as well as the rate of extension. In acompanion paper, this behavior is assessed in greater detailusing a linear stability analysis.[54] The results of this study also illustrate the importance

of the background processes leading to the formation of a

partially molten layer. Several previous studies of thebuoyant decompression melting instability have oftenbegun with an idealized initial condition in which theprocess of bringing about a partially molten layer hasbeen ignored, and a thick portion of mantle already findsitself at the solidus. The results of this study demonstratethat the process by which a layer of mantle is brought tothe solidus can have dramatic effects upon the character-istics of the instability, and therefore cannot be neglected.This is due in part to the important influence of meltpercolation and the cumulative degree of depletion thelayer obtains upon bringing it to a partially molten state.[55] The explicit treatment of melt percolation also

appears to be critical in assessing the physics and observa-tional consequences of this kind of instability. If a thresholdvalue is used instead where melt is instantly erupted at somecritical fraction, the ability for melt buoyancy to grownonlinearly with the rate of circulation will be suppressedonce it reaches saturation. As a result, many of the featuresof the instability observed in this study would be entirelyabsent. This finding is amplified by the results of the linearstability analysis in the companion paper.

[56] Acknowledgments. Reviews of preliminary versions of thismaterial by Abby Kavner, Paul Roberts, and Gerald Schubert greatlyimproved the quality and flow of the presentation. Reviews by HarroSchmeling, Yanick Ricard, and Justin Revenaugh helped to improve theclarity and substance of this work. We are also grateful for discussions withKen Deuker, Eugene Humphreys, Mark Jellinek, James Kellogg, StephaneLabrosse, William Moore, and Derek Schutt. This work was supported inpart by the NSF CDYEL project and a grant from IGPP Los Alamos.

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��J. W. Hernlund, Department of Earth and Ocean Sciences, University of

British Columbia, 6339 Stores Road, Vancouver, BC, Canada V6T 1Z4.([email protected])D. J. Stevenson, Division of Geological and Planetary Sciences,

California Institute of Technology, 1200 E. California Blvd., Pasadena,CA 91125, USA.P. J. Tackley, Institute for Geophysics, ETH-Zurich, Honggerberg HPP

L13, Schafmattstrasse 30, CH-8093 Zurich, Switzerland.


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