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Mantle convectionEncyclopedia of Solid Earth Geophysics, Harsh Gupta (ed.), Springer

David BercoviciProfessor & ChairDepartment of Geology & GeophsyicsYale UniversityNew Haven, Connecticut 06520-8109, USAE-mail: [email protected]: (203) 432-3168Fax: (203) 432-3134

December 26, 2010

Encyclopedia of Solid Earth Geophysics Mantle Convection, David Bercovici

MANTLE CONVECTION

Synonyms

Mantle dynamics. Mantle circulation.

Definition

Mantle convection:Thermal convection in theterrestrial planetary mantles, the rocky layer be-tween crust and core, in which hot material rises,cold material sinks and the induced flow governsplate tectonic and volcanic activity, as well aschemical segregation and cooling of the entireplanet.

Mantle convection

Introduction and History

All planetary bodies retain some heat from theirearly formation but are inexorably cooling tospace. Planetary surfaces are therefore coldrelative to their hotter interiors, and thus un-dergo thermal convection wherein cold materialis dense and sinks while hot material is light andrises (liquid water near freezing being one of therare exceptions to this process). Planetary at-mospheres, oceans, rocky mantles and metallicliquid cores convect and are subject to uniquepatterns of circulation in each domain. Silicatemantles however tend to be the most massiveand sluggish part of terrestrial planets and there-fore govern how planetary interiors evolve andcool to space. (See Fig 1.)

The theory of mantle convection was origi-nally developed to understand the thermal his-tory of the Earth and to provide a driving mech-anism for Alfred Wegener’s theory of Continen-tal Drift in the 1930s [seeSchubert et al., 2001;Bercovici, 2007]. Interest in mantle convectionwaned for decades as Wegener’s theory was crit-icized and apparently discredited. However, theaccumulation of sea-floor sounding data duringWar World II and refinement of paleomagen-

tic techniques paved the way for the discoveryof sea-floor spreading [Hess, 1962; Vine andMatthews, 1963] and the birth of the grand uni-fying theory of Plate Tectonics in the 1960s;this consequently revived interest in mantle con-vection as the driving mechanism for plate mo-tions [Runcorn, 1962a,b] as well as non-plate-tectonic volcanism such as the possible Hawai-ian plume [Morgan, 1971]. The success of man-tle convection theory in explaining plate veloc-ities, sea-floor subsidence, volcanism, gravityanomalies, etc., lead to its further application toother terrestrial planets such as Venus and Mars,which also sustained unique forms of mantleconvection, evident from volcanic activity.

Basics of thermal or free convection

Rayleigh-Benard ConvectionThe simplest form of thermal convection is

referred to as Benard convection named afterthe French experimentalist Henri Benard whoin 1900 performed the first systematic experi-ments on convection in thin layers of oil (sper-macetti) and recognized both the onset of con-vection from a static conductive state and theregular patterns formed in a convecting layer[Benard, 1900, 1901]. Fifteen years later, theBritish theoretical physicist and mathematicianLord Rayleigh (William John Strutt), attemptedto explain Benard’s results for the onset of con-vection [Strutt, John William (Lord Rayleigh),1916] – the delay in communication betweenthem being caused by the First World War.However, the mismatch between theory and ex-periment was profound, and not resolved untilthe late 1950s [Pearson, 1958] when it was in-ferred that Benard’s experiments were stronglyinfluenced by surface tension or Marangoni ef-fects not included in Rayleigh’s theory (al-though Benard himself was aware of these ef-fects). Because Rayleigh’s work provided theframework for nearly all thermal convection the-ory to follow, the simple Benard convective sys-

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Figure 1: Graphic renditions of cut aways of Earth’s structure showing crust, mantle and core (left) andof the convecting mantle (right). The relevant dimensions are that the Earth’s average radius is 6371km;the depth of the base of the oceanic crust is about 7km and continental crust about 35km; the base of thelithosphere varies from 0 at mid-ocean ridges to about 100kmnear subduction zones; the base of the uppermantle is at 410km depth, the Transition Zone sits between 410km and 660km depths; the depth of the baseof the mantle (the core-mantle boundary) is 2890km; and the inner core-out core boundary is at a depth of5150km. Left frame adapted fromLamb and Sington[1998]. Right frame, provenance unknown.

tem is also referred to as Rayleigh-Benard con-vection.

Although Benard’s experiments were in ametal cavity, Rayleigh-Benard convection actu-ally refers to Rayleigh’s idealized model of athin fluid layer infinite in all horizontal direc-tion such that the only intrinsic length scale inthe system is the layer thickness. The Rayleigh-Benard system is heated uniformly on the bot-tom by a heat reservoir held at a fixed temper-ature (i.e., the bottom boundary is everywhereisothermal) and the top is likewise held at afixed colder temperature by another reservoir(see Figure 2). If the layer were not fluid, heatwould flow from the hot boundary to the coldone by thermal conduction. But since the fluidnear the hotter base is (typically) less dense thanthe fluid near the colder surface, the layer isgravitationally unstable, i.e., less dense materialunderlies more dense material. To release gravi-

tationaly potential energy and go to a minimumenergy state, the layer is induced to turn over.

Convective onset and the Rayleigh numberWhile the fluid in a Rayleigh-Benard layer

might begravitationallyunstable, it is not nec-essarilyconvectivelyunstable. Convective over-turn of the layer is forced by heating but resistedor damped in two unique ways. Clearly the ther-mal buoyancy (proportional to density contrasttimes gravity) of a hot fluid parcel rising fromthe bottom surface through colder surroundingsacts to drive convective overturn. However, vis-cous drag acts to slow down this parcel, andthermal condcution, or diffusion, acts to erase itshot anomaly (i.e., it loses heat to its colder sur-roundings). Thus while the fluid layer might begravitationally unstable, hot parcels rising mightmove too slowly against viscous drag before be-ing erased by thermal diffusion. Similar argu-

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0.00.20.40.60.81.0

Z

0 1 2 3 4 5 6 7 8

XFigure 2: Rayleigh-Benard convection: initially conducting layer (top) and numerical simulation of con-vection (bottom).

ments can be made for cold material sinkingfrom the top surface through warmer surround-ings. The competition between forcing by ther-mal buoyancy, and damping by viscosity andthermal diffusion, is characterized in dimension-less ratio called the Rayleigh number

Ra =ρgα∆Td3

µκ(1)

whereρ is fluid density,g is gravity,α is ther-mal expansivity (units of ofK−1),∆T is the dif-ference in temperature between the bottom andtop surfaces,d is the layer thickness,µ is fluidviscosity (units ofPa s) andκ is fluid thermaldiffusivity (units ofm2s−1).

Even though∆T > 0 (i.e., heating is from be-low and causes gravitational instability),Ra stillmust exceed a certain value, called the criticalRayleigh numberRac for convection to occur.ForRa < Rac the layer is stable and transportsheat by conduction; forRa > Rac the layer willbe convectively unstable and transport heat morerapidly via convection. See Figure 3.

Although Rac varies depending on the me-chanical nature of the horizontal boundaries(whether rigid or a free surface) it is typicallyof order 1000. This value is easily understoodby considering the fate of a hot (or cold) parcel

of sizea and temperature anomaly∆T . Dimen-sional analysis readily shows that the typical as-cent rate of the parcel isρgα∆Ta2/µ (with unitsof m/s). However the rate that heat diffuses outof the parcel isκ/a (smaller parcels lose heatfaster). The critical state occurs when these tworates are equal; i.e., if the buoyant ascent ratejust exceeds the diffusion rate, the parcel shouldrise without being erased, but if the ascent rateis less than the diffusion rate it will not rise veryfar before being lost. Therefore the critical stateoccurs ifρgα∆Ta3/(µκ) ≈ 1. Scaling purelyby orders of magnitude, a small parcel of fluidcan be assumed to be of order 10 times smallerthan the entire layer; thus assuminga ≈ d/10leads to a critical condition for onset of convec-tion of ρgα∆Td3/(µκ) ≈ 1000.

For the Earth’s mantle, the typical averageproperties from which the Rayleigh number isconstructed areρ ≈ 4000kg/m3, g = 10m/s2,α = 3× 10−5K−1, ∆T ≈ 3000K, d = 2900km,µ = 1022Pa s (dominated by the lower man-tle), andκ = 10−6m2/s [see Schubert et al.,2001]. Taken together these lead to a Rayleighnumber of approximately107, which is well be-yond supercritical; although the mantle viscos-ity is extremely high, the mantle is also very hotand very large and hence convecting vigorously.

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0 2 4 6 8 10 12

0

500

1000

1500

2000

2500

3000

3500

4000

criti

cal R

a (d

imen

sion

less

T d

rop

or h

eatin

g)

size (wavelength) of perturbation (T fluctuation)

unstable

most unstablemode

stable

min heating to induce convection

super−critical heating

not enough heating

Figure 3:The critical Rayleigh numberRa for the onset of convection is a function of wavelength or sizeof the thermal perturbation to a static conductive state. For a layer with isothermal and free-slip (shear-stress free) top and bottom boundaries, the relationship isRacrit = (k2 + π2)3/k2 wherek = 2π/λ andλ is wavelength [Chandrasekhar, 1961]. Values ofRa above theRacrit curve are associated with theconductive layer being convectively unstable (perturbations grow), while below the curve the layer is stable(perturbations decay). The minimum in theRacrit curve occurs at the wavelength of the first perturbationto go unstable as heating andRa is increased, often called the most unstable mode.

Thermal boundary layers and the Nusseltnumber

For a Rayleigh numberRa above the criti-cal valueRac, convective circulation will mixthe fluid layer, and the mixing and homogeniza-tion of the fluid will become more effective themore vigorous the convection, i.e., asRa is fur-ther increased. With very largeRa and vig-orous mixing most of the layer is largely uni-form and isothermal. (In fact, if the layer isdeep enough such the pressures are comparableto fluid incompressibility, the fluid layer is notisothermal but adiabatic, wherein even withoutany heating or cooling, the temperature woulddrop with fluid expansion on ascent and increasewith fluid compression on descent.) Most ofthe fluid in the Rayleigh-Benard system is atthe mean temperature between the two bound-ary temperatures. However the temperature still

must drop from the well-mixed warm interior tothe cold temperature at the top, and to the hot-ter temperature at the bottom. The narrow re-gions accomodating these jumps in temperatureare called thermal boundary layers (Figure 4).

Thermal boundary layers are of great impor-tance in thermal (and mantle) convection for tworeasons. First, most of the buoyancy of thesystem is bound up in thermal boundary layerssince these are where most of the cold materialat the top and hot material at the bottom residesand from where hot and cold thermals or cur-rents emanate. Moreover, with regard to con-vection in the mantle itself, the top cold thermalboundary layer is typically associated with theEarth’s lithosphere, the 100km or so thick layerof cold and stiffer mantle rock that is nominallycut up into tectonic plates.

Second, since fluid in these boundary layers

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temperaturehe

ight

convection

conduction

temperaturetemperature

all bottom heated

heig

ht

add internalheating

Figure 4: Sketch of temperature profiles, showing how convective mixing homongenizes the conductivemean temperature into a nearly isothermal state (if the fuidis incompressible) with thermal boundary layersconnecting it to the cold surface and hot base (top frame). With no internal heating the interior meantemperature is the average of the top and bottom temperatures; the effect of adding internal heating (bottomframes) is to increase the interior mean temperature and thus change the relative size and temperature dropacross the top and bottom thermal boundary layers.

is near a horizontal boundary, most of the fluidmotion is horizontal and thus heat is only trans-ported vertically across these thin layers by con-duction; but since the layers are very thin suchconductive transport is rapid. Indeed, the en-tire cycle of heat transport occurs by heat con-ducted in rapidly through the bottom boundarylayer, after which hot fluid in this layer will, invarious spots, become unstable and rise to forma convective upwelling that carries heat out ofthe boundary layer rapidly across most of thefluid layer and deposits it at the upper bound-

ary, where the heat is then transported out ofthe system by conduction across the top bound-ary layer. The eventual heat flow (power out-put per unit area) out of the well mixed layeris essentiallyk∆T/δ wherek is thermal con-ductivity (units ofW K−1m−1), ∆T/2 is thetemperature drop from the well mixed interior tothe surface and we defineδ/2 is the thickness ofthe boundary layer. By comparison, the thermalconduction across a static non-convecting layeris k∆T/d. The ratio of heat flow in the con-vectively well mixed layer to the purely conduc-

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tive layer is thusd/δ, which is called the NusseltnumberNu (named after the German engineer,Wilhelm Nusselt 1882-1957). The relation be-tweenNu and convective vigor parameterizedby Ra is important for understanding how con-vection transports heat and cools off bodies in-cluding planets. Convective heat transport is of-ten written asNu(k∆T/d) and in consideringthis relationHoward [1966] argued that vigor-ous convective heat transport across the bulk ofthe well-mixed fluid layer is so fast it is not therate limiting factor in releasing heat (only con-duction across the thermal boundary layer is),and thus heatflow should be independent of fluiddepthd; this implies that sinceRa ∼ d3 thenNu ∼ Ra1/3, which yields a convective heat-flow Nu(k∆T/d) that is independent ofd. Ingeneral, since the fluid is conductive forRa ≤Rac, one often writes thatNu = (Ra/Rac)

1/3

(althoughNu = 1 for Ra < Rac), whichis a reasonably accurate relationship born outby simple experiments and computer modeling.This relationship also implies that the ratio ofthermal boundary width to fluid layer depth isδ/d ∼ Ra−1/3, which shows that the bound-ary layers become increasingly thin as convec-tive mixing of the layer becomes more vigorous.

The relation ofδ ∼ Ra−1/3 applies to thehorizontally averaged boundary layer thickness.However, boundary layers change with time ordistance from their first formation, e.g., wherean upwelling impinges on the top boundary. Asthe fluid in the boundary layer moves from anupwelling to a downwelling it cools and theboundary layer thickens as more material coolsnext to the cold surface. The thickening de-pends on the thermal diffusivityκ (with unitsof m2/s) and the residence time or aget nearthe cold boundary (i.e., time since leaving theupwelling). Simple dimensional considerationsshow that the boundary layer thickness goes as√κt; this corresponds to the well known

√age

law for subsidence of ocean sea floor with agesince formation at mid-ocean ridges, implyingthat sea-floor gets deeper because of the cooling

and thickening lithosphere.

Patterns of convection, structure of up-wellings and downwellings: plumes, andslabs

When convection occurs upwellings anddownwellings will be separated horizontally bysome optimal distance. If they are too close toeach other they can induce too much viscousdrag on each other and/or lose heat rapidly toeach other; if they are too far apart they must rolltoo much mass between them. The separationdistance is also determined by heat transport inthe thermal boundary layer between upwellingsand downwelling. When hot upwelling fluidreaches the surface it spreads laterally into thethermal boundary layer. As it travels horizon-tally it cools to the surface and eventually getscold and sinks into the downwelling; thus theupwelling-downwelling separation is also deter-mined by the distance it takes for fluid to cooloff and become heavy enough to sink.

The upwelling-downwelling separation dis-tance or convection cell size is predicted by con-vective stability theory to approximately equalto the layer depthd (a bit larger at the onset ofconvection but identicallyd asRa becomes verylarge); i.e., the cell that is either least stable andthus most likely to convect – or equivalently thecell that optimizes gravitational potential energy(and thus heat) release – is usually the cell thatis as wide as it is deep.

Viewing a convecting layer from above, theupwelling and downwellings may be separatedin various patterns, such as two-dimensionalrolls (sheets of upwelling rolling over into sheetsof downwelling, and each cell counter-rotatingwith its neighboring cell). In the Rayleigh-Benard layer, which is infinite horizontally, noone location of the layer should be different thanany other one so ideally the pattern should bea regular repeating tile; as there are only somany polygons that can form a repeating tile,the patterns usually involve convection cells inthe shapes of rolls (already mentioned), squares,

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hexagons and triangles (Figure 5). Of coursenon-ideality and irregularities can occur due totiny imperfections for example in the boundariesleading to irregular patterns.

In many instances the 3-D pattern of convec-tion, especially in fluids where hot fluid is lessviscous than cold fluid (as is true in many ma-terials, including the mantle) the upwelling is inthe form of a cylindrical plume at the center of acanopy of sheet-like downwellings, much like afountain.

Plumes and Slabs in the mantle: Simpleview

The common occurence of sheet-like down-wellings and columnar upwellings in simpleconvection is crudely applicable to mantle con-vection. Subducting slabs are where tectonicplates sink into the mantle; these are analagousto the cold sheet-like downwellings seen in 3-D convection, although much more complicatedby rheology as discussed below. Deep hot nar-row upwelling plumes are infered to explainanomalous intraplate volcanism as in Hawaii, aswell as the fixity of these volcanic hotspots rela-tive to each other (which suggests deep anchor-ing in the mantle). These mantle plumes are os-tensibly analogous to the pipe-like upwelling insimple 3-D convection, but again more compli-cated by unique mantle properties. (See “MantlePlumes” essay by Farnetani and Hofmann.)

While mid-ocean ridges or spreading centerstransport material vertically to surface, they arevery narrow features and for the most part in-volve shallow upwelling (inferred from theirweak gravity anomalies that suggest shallowisostatic compensation, i.e., they are floating ona shallow buoyant root). Ridges are likely bestexplained as being pulled and rifted apart pas-sively from a distant force (ostensibly slabs)rather than involving a deep convective up-welling that pries them open.

Energy sources for mantle convection andEarth’s thermo-chemical history

Heatflow coming out of the Earth is measuredby heat-flow gauges (measuring conductivity ofrocks first and then thermal gradients in bore-holes) both in continents and oceans [seeTur-cotte and Schubert, 1982]. The total heat flow-ing out from beneath the Earth’s surface is ap-proximately 46TW (46 trillion Watts) [Jaupartet al., 2007], which is in fact not a large num-ber given the surface area of the Earth, and isactually tens of thousands of times smaller thanthe heat absorbed from the Sun. Nevertheless itrepresents the source of power driving dynamicsinside our planet, including mantle convection(and hence tectonic, volcanological and othergeological activity) as well as core cooling andflow.

The source of the Earth’s internal heat is acombination of primordial heat, i.e., left overfrom accretion (gravitational potential energyfrom formation and collisions), and heat gener-ated by unstable radioactive isotopes, especiallythe isotopes of uranium (238U), thorium (232Th)and potassium (40K), although the 40K half-life is much shorter than the others and so gen-erated a large heat pulse primarily in the earlyEarth. Because continental crust is originallyformed by partial melting and chemical segrega-tion of early mantle material (indeed the chem-ical separation allows another energy source inthe release of gravitational potential energy, butother than early core formation this is a rela-tively minor contribution), these radioactive ele-ments tend to be concentrated in crust (i.e., meltmore readily dissolves these elements than doessolid rock, so they partition toward the melt).Thus the crust itself produces a significant frac-tion of the net heat output through the surface;removing the crustal component leaves approx-imately 31TW emanating from the mantle andcore [Schubert et al., 2001;Jaupart et al., 2007].

The relative contributions of primordial cool-ing and radiogenic heating to the mantle (and

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Figure 5: Patterns of convection from laboratory experiments in rectangular tank byWhite [1988] (left),and numerical simulations in a spherical shell byZhong et al.[2000].

core) heat output remains an active area of de-bate even today and leads to various quandaries.The most direct estimate of the concentraionof radiogenic sources (U, Th, K) is by look-ing at the concentration in chondritic meteorites,which come for the solar systems’ main aster-oid belt and because they have been largely un-altered for 4.5Gyrs (i.e., unmelted) are thoughtto be the same as the original building blocksof the terrestrial planets. The chondritic con-centrations of U, Th and K would allow for ra-diogenic heating to contribute 50% or less ofthe total heat output [Korenaga, 2008]; this isoften called the Urey ratio, i.e., the radiogenicheat output to the total output. With radio-genic heating this small, the only way the man-tle could be as hot as it is today – while alsotransporting heat as it does presently – is if itwere very hot and nearly molten in the geologi-cally recent past (a few hundred million years);this is geologically untenable since petrologicaland geochemical analysis demonstrate the pres-ence of solid rock and even liquid water in theearly Earth [seeHalliday, 2001], which there-fore ceased to be molten not long after it’s for-mation 4.5 billions years ago. This paradox hasled some researchers to assume that the radio-genic sources are super-chondritic, i.e., to al-

low heat to be produced continously through-out the past rather than by rapid cooling froma molten state. Alternatively, researchers havesought ways to keep chondritic concentrationsof U, Th and K by arguing that heat-transport inthe past was different than it is today; for exam-ple higher temperatures in the past might haveallowed for more melting and thus more buoyantcrust and/or stronger dehydrated lithosphere thatkept the top part of the convecting mantle slug-gish or immobile hence bottling up primordialheat for much later release [Korenaga, 2008].

Finally, the release of both radiogenic and pri-mordial heat, again termed collectively “inter-nal heating” (i.e., heat coming from the bulkof the fluid) means that the idealized Rayleigh-Benard model of convection, where heating isonly along the base, is inaccurate. The effect ofinternal heating in addition to “basal heating” isrelatively straightforward to understand. Whilein the Rayleigh-Benard model the temperatureof the well mixed interior is at the average ofthe two boundary temperatures, the addition ofinternal heating acts to heat up the well mixedinterior to a higher mean temperature. This putsthe interior temperature closer to the hot bottomtemperature, but further from the cold top tem-perature; the effect is to create a larger tempera-

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ture drop across the top thermal boundary layerthan across the bottom one; in essence, the topboundary layer must conduct out heat injectedthrough the bottom plus heat generated or re-leased from the interior (Fig 4). These very dif-ferent boundary layers tend to cause more neg-atively buoyant and stronger downwelling cur-rents and smaller and weaker upwelling cur-rents; this effect seems to be borne out in theEarth by the presence of many large cold slabsdriving plate tectonic motion with large ther-mal anomalies (of order 500K) relative to fewerupwelling plumes with weaker (200K) thermalanomalies (although this is still somewhat a mat-ter of debate).

Effects of mantle properties

Mantle rheologyThe entire mantle of the Earth is potentially a

convecting and over-turning fluid. However, themantle is almost entirely solid (with some smallportions of melting near the surface delineatedby volcanism, and possibly much smaller areasof melting at depth) and thus flows extremelyslowly. (Indeed, the term fluid does not sug-gest a liquid state but refers to how a mediumdeforms, as opposed to elastic or brittle defor-mation; these do not necessarily correlate withstates of matter, i.e., gas, liquid, solid, each ofwhich can display, for example, either fluid flowor elastic behavior depending on the nature ofthe deformation; e.g., atmospheric sound wavesare elastic behavior in gas.)

The solid-state viscous flow of the mantle isa complex process and there are various mech-anisms that permit such ”irrecoverable” defor-mation (one of the definitions of viscous flow).A survey of mantle deformation mechanisms or”rheology” is beyond the scope of this essay[seeRanalli, 1995;Karato, 2008]. However, inbrief the two primary deformation mechanismsare called diffusion and dislocation creep. Inany solid-state creep mechanism, mobility de-

pends on the statistical-mechanical probabilityof a molecule in a crystal lattice leaving thepotential well of its lattice site; the potential-well itself is defined by electrostatic or chemi-cal bonds inhibiting escape, and Pauli-exclusionpressure preventing molecules squeezing tooclosely to each other. Thus mobility depends onthe Boltzman distribution measuring the prob-abilty of having sufficient energy to overcomethe lattice potential well barrier, which is oftencalled the activation energy (or allowing insteadfor pressure variations the activation enthalpy).This probability depends on the (Arrhenius) fac-tor e−Ea/RT whereEa is the activation energy(J/mol),R is the gas constant (J/K/mol) andTis temperature;RT represents the thermal exci-tation energy of the molecule in the well. AsTgoes to infinity the probability of escaping thewell goes to1, while asT goes to 0 the proba-bility of escape goes to 0.

Stress imposed on the medium effectivelychanges the shape of the potential well, such thatcompressive stress steepens the walls of the well(squeezing molecules closer makes coloumb at-traction in the chemical bonds stronger) whiletension lowers the walls (separating moleculesweakens the bonds); thus the probability of es-cape is preferred in the direction of tensionand away from compression, thus allowing themedium to stretch in the tensile direction bysolid-state diffusion of molecules.

Simple diffusive creep works much in thisway where differential stress (i.e., non-uniformstress) causes slow diffusion of molecules to al-low the entire substance to deform accordingly.However, such deformation occurs at the grainor mineral level inside a rock, with either diffu-sion through the grains or along the grain bound-aries; thus the response depends significantly ongrainsize.

Dislocation creep is more complicated. A dis-location can be in the form of a truncated row ofmolecules in a crystal lattice; shortening of thecrystal under compression perpendicular to therow can be accomplished by simply removing

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that row. Thus differential compressive stresswould act to force molecules to diffuse out ofthat dislocated row into other parts of the lattice.However, stress not only governs the preferen-tial diffusion of molecules (as in diffusion creep)but also the geometry of the dislocations (theirspacing and directions), hence the multiple ac-tions of stress are compounded into a nonlinearresponse.

The viscosities for diffusion and dislocationcreep mechanisms can be written as

µ =

BameEa

RT for diffusion creep

Aσ1−neE′

a

RT for dislocation creep(2)

whereA andB are proportionality constants,ais grainsize,σ is stress (in fact since stress isa tensorσ is the scalar second invariant of thestress tensor), andm andn are exponents, typ-ically both equal to 3. It should be emphasizedthat diffusion and dislocation creep occur inde-pendently of each other depending on stress andgrainsize: for high stress and large grains dislo-cation creep dominates; for low stress and smallgrains diffusion creep dominates.

Dislocation creep allows for moderate soften-ing as stress increases; diffusion creep poten-tially allows for significant softening if stresscan reduce grainsize, although mechanisms toallow this are controversial still (see section be-low on generating plates), and significant hard-ening via grain growth by standard coarseningof the material (i.e., what happens to all grainedmaterials under the action of grain-surface en-ergy reduction).

The strongest rheological effect is clearly thatof temperature; the temperature-dependence ofviscosity allows for many orders of magnitudevariations in viscosity. For example, while thisrheological effect allows subducting slabs tokeep their strength and integrity to great depthsas they sink, it would make hot upwelling man-tle plumes more fluid and, if they have a con-duit structure, the plume flow would be rela-tively rapid, of order 100cm/yr or more. How-

ever, this effect is most profound in the cold topthermal boundary layer or lithosphere. If viscos-ity is strongly temperature dependent, as it is inEarth, the lithosphere can become so stiff thatit becomes immobile; in this case convectionin the mantle would proceed beneath the litho-sphere, which in turn would act like a rigid lidto the mantle [Solomatov, 1995]. If mantle rhe-ology obeyed only diffusion or dislocation creeplaws, then the lithosphere should be locked andimmobile and there should be no plate tectonics.While this scenario might be relevant for Venusand Mars (and Moon and Mercury) which haveno plate tectonics, obviously it is missing a vitalingredient to allow plate tectonics on Earth. Thisparadox underlies the fundamental question andmystery about why Earth has plate tectonics atall and how it is generated on our planet but notothers in our solar system [Bercovici, 2003].

Compressibility, Melting and Solid phasechanges

Pressures deep inside the Earth’s mantle areso large they are sizable fractions of rock in-compressibility or bulk modulus (e.g., mantlepressures reach 140GPa, or 1.4 million atmo-spheres, while bulk modulus – which has thesame units as pressure – are typically a few toseveral 100GPa). As downwelling mantle mate-rial travels from near the surface to the base ofthe mantle its density and temperature increasedue to compression, called ”adiabatic compres-sion and heating” (and likewise upwelling ma-terial undergoes ”adiabatic decompression andcooling”), although these increases are not large(of order several degress celsius). The com-pression and decompression of circulating ma-terial establishes a weak adiabatic temperatureand density increase with depth, which has aslight stabilizing effect on convection; however,because the mantle is so viscous the thermalanomalies needed to get it to move – and inparticular the temperature variations across thethermal boundary layers – are so large (of or-der several 100 to 1000 degrees celsius) that the

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adiabatic variations are small in comparison.Where compressibility and pressure play a

dual important role is in phase changes. First,as hot upwelling mantle material approachesthe surface it actually travels along a graduallycooling adiabatic temperature profile. The up-welling does eventually melt when it gets nearthe surface but not because it gets hotter. Melt-ing occurs because the melting temperatureTm

drops with decreasing pressure faster than theupwelling adiabat (in essence, decreasing con-fining pressure makes it easier for molecules tomobilize into a melt); thus at a certain (usuallyshallow depth of a few 10s of km to 100km) theupwelling mantle crosses the melting tempera-ture from solid to liquid phase and undergoesmelting; however, the mantle is not a single puresubstance so in fact only partially melts. Such”pressure release” melting is a shallow processbut is vital for chemical segregation of the man-tle and development of oceanic and continentalcrust. In particular, melting is sequential in thatthe most easily melted material (usually moresilica rich material with lower melting temper-ature) melts first, freezes last and is typicallychemically less dense, and thus comes to the sur-face as lighter crust eventually gathering, aftermore weathering and reactions into continentalcrust. (Continental crustal rocks like sandstoneand granite have typical densities of2300kg/m3

and2700kg/m3, respectively.) The more refrac-tory (harder to melt, silica poor and heavier ma-terial) melts last, freezes first and either stays inthe mantle or lithosphere or sits in the heavybasaltic oceanic crust. (Oceanic crustal rockslike basalt have densities of3000kg/m3, whilemantle peridotites at near-surface pressure havedensities of around3400kg/m3.)

Extreme pressures with depth can also over-come a mineral’s elastic resistance to com-pression and cause solid-solid phase changeswhere the minerals change their crystalographicstructure to a more compact and incompress-ible state (but of course their chemistry remainsthe same). Such mineralogical phase changes

have been observed in laboratory experiments inolivine, which is the major component mineralof the upper mantle (at about 60% by weight,the remainder being mostly pyroxene at shal-low depths, and garnet at slightly greater depth);moreover, the pressures at which they are pre-dicted to occur have been verified seismolog-ically, wherein the seismic wave speeds anddensity undergo a jump at the predicted pres-sures. The first major phase change to occurwith depth is from olivine to the same mate-rial with a wadsleyite structure, at 410km depth.Wadsleyite changes slightly to a similar ring-woodite structure at 510km depth. The largestphase change occurs from ringwoodite to per-ovskite/magnesiowustite at 660km depth.

The 410km and 660km phase changes are thetwo most remarkable and global phase changesin the mantle, and the region between them iscalled the Transition Zone, since it is wheremost of the mineralological transitions occur,over a relatively narrow region. The mantleabove the Transition Zone is typically identifiedas the Upper Mantle, although in some papersand books Upper Mantle includes the TransitionZone. Below the Transition Zone is the LowerMantle and that is universally agreed upon in theliterature.

The Transition Zone has anomalous proper-ties due to mixing and transitions in mineral or-ganization; for example it is thought to be ableto absorb an order of magnitude more water (perkg) than the mantle both above and below it (al-though this issue is still somewhat controver-sial).

Other phase changes are thought to occur withdepth, although these are less well resolved, andin some instances do not appear to be global.Recently a new phase change has been inferredin the lowest part of the mantle (the bottomfew 100 kilometers), called the perovskite-post-perovskite (or just the post-perovskite) transi-tion [Murakami et al., 2004]. This transition isstill an active area of exploration.

The effect of phase transitions on mantle con-

11


vection has been an active area of research sincethe 1960s. The discovery of the major phasechange at 660km depth coincided with the factthat deep earthquakes along subducting slabs(the Wadati-Benioff Zone) go no deeper than700km. This seemed to imply that subductingslabs, the mantle’s equivalent of cold convectivedownwellings (see below), did not extend intothe lower mantle. Recent seismological studiesusing tomographic techniques to resolve osten-sibly ”cold” and ”hot” areas of the mantle (re-ally seismically fast and slow regions), impliedthat many slabs might stall or pool temporarilyat the 660km boundary but many penetrate intothe lower mantle (Fig 6).

Whether the density jump due to a phasechange could impede vertical flow has beena key question in studying the interaction ofphase changes and convection. In particular, the660km phase change was inferred mineralogi-cally to be ”endothermic” whereby the entropyof the deeper heavier phase (below 660km) in-creases, or more simply latent heat is absorbedon going down through the boundary into thedenser more compact phase (while unusual inmost systems, this is also true of the solid-liquid transition in water). This also meansthat the dependence of the transition temper-ature on pressure has a negative slope; thuscold material impinging on the phase boundarycauses the phase change to deflect in the coldregion to greater pressures; this induces a de-pression in the boundary that acts to buoyantlyrebound upward and oppose the motion of thedescending cold material. Thus the endother-mic boundary at 660km depth possibly impedesflow across that boundary. (However, the ac-tual 660km phase is complex and at higher tem-peratures might become exothermic;Weidnerand Wang[1998].) Computational studies andsimulations of mantle flow across this boundarydemonstrated that downwellings (i.e., slabs) canindeed be impeded and stalled as they impingeon this boundary, but not permanently or glob-ally; i.e., while some are pooling at the bound-

ary others have gathered up enough “weight”to push through the boundary [Tackley et al.,1993;Christensen, 1995]. This picture appearsto be in keeping with the picture from seismol-ogy that while the phase boundary impedes slaband downwelling flow into the mantle, it is notan impermeable boundary and there is in theend significant exchange between the upper andlower mantle and hence whole mantle convec-tion.

However, while the mineralogical, seismo-logical and geodynamical (fluid mechanical) ar-guments imply that there is flow between up-per and lower mantles, there are numerous datafrom geochemical analysis of basaltic lavas im-plying that the mantle is not well stirred on alarge scale, i.e., it is possibly layered with pooror non-existent communication between upperand lower mantle.

Structure of mantle convection and mantlemixing

Upwelling mantle reaching the Earth’s surfaceundergoes melting (see above under pressure-release melting) and this melt reaches the sur-face in two types of volcanic settings: mid-ocean ridges where tectonic plates spread apartand draw mantle up into the opening gap, andocean-islands or hotspots which are anoma-lously productive and localized volcanic fea-tures not necessarily associated with tectonicactivity, Hawaii being the most conspicuoussuch feature. Melts coming from the mantle inthis way are silica poor (relative to more sili-cic rocks such as granite) and largely basaltic;hence these volcanic regions are said to pro-duce Mid-Ocean Ridge Basalts (MORB) andOcean-Island Basalts (OIB), respectively. Thesemelts are in effect messengers from the mantle,and their petrological composition, bulk chem-istry and trace-element chemistry are extensiveareas of research [Hofmann, 1997, 2003;vanKeken et al., 2002;Tackley, 2007]. In the end,while these two basalts nominally come from

12


Figure 6:Tomographic image of slabs beneath Mexico (W. Spakman viavan Keken et al.[2002]) extendinginto the lower mantle (left); and plate velocity vs trench length followingForsyth and Uyeda[1975] showingthe fastest plates are connected to slabs (right). Slabs arenot only cold mantle downwellings but effectiveleydrive plate tectonics

the same mantle, they have distinct features sug-gesting they come from parts of the mantle thathave been isolated from each other for billionsof years. That seismology, mineral-physicsand fluid-dynamics (geodynamics) argue for alargely well-stirred mantle with whole-layer cir-culation creates a dichotomy between geophysi-cal and geochemical observations. This paradoxhas been one of the most fervent areas of debatein mantle dynamics for the last 30-40 years.

Because MORB and OIB are both basalts andthus have similar bulk chemistry, geochemicalmeasurements largely focus on trace elements,in particular incompatible elements, which dis-solve more readily in a rock’s melt phase thanits solid phase; hence during partial melting in-compatible elements partition toward the melt.

Indeed, the trace-element signature of elementssuch as uranium, thorium, helium, have demon-strated that MORB and OIB are very distinct.In particular, MORBs appear to be significantlydepleted in such trace elements relative to OIB.Since such elements tend to be removed by melt-ing, it implies that MORBs come from a re-gion of the mantle that has already been meltedand depleted of trace elements, while OIB comefrom a region of the mantle that has undergonelittle previous melting and depletion. This ob-servation implies that MORBs come from anupper mantle that has been cycled repeatedlythrough the plate tectonic process of mid-oceanridge melting and separation of crust (and traceelements) from mantle, in essence cleaning theMORB source. In contrast, OIB would appear

13


to come from a part of the mantle that has seenlittle of this melt processing, and hence wouldbe isolated presumably at depth from the uppermantle and the plate-tectonic circulation.

There are other geochemical observations thatargue for separated and isolated regions or reser-voirs in the mantle. For example the concen-tration of radioactive daughter isotopes (e.g.,206Pb, which is the final product of the decay of238U) relative to the abundance of that element’sdominant and primordial isotope (e.g.,204Pb)is a metric for reservoir isolation from surfaceprocessing. In particular the relative accumula-tion of daughter products implies that the rockin which they reside has seen little processingor partial melting that would have cleaned outthese elements after they formed. A small rel-ative abundance of daughter isotopes means thesample has been recently processed and cleaned,and thus little time has passed in which to pro-duce new daughter isotopes.

Indeed, many OIBs tend to show distinctlygreater relative abundance of daughter products(e.g., the concentration ratio206Pb/ 204Pb) thando MORB for many isotopic ratios, implyingsome isolation of the OIB source. However,OIBs from various islands are also fairly differ-ent from each other suggesting that reservoirsisolated from the upper mantle (the presumedMORB source) are possibly also isolated fromeach other. Moreover, the OIB isotopic ratioshave some variation, from low MORB-like val-ues to much higher values, which indicates thatthere is some mixing of a ”young” processedMORB-source like mantle and a more primitive,isolated one.

These isotopic ratios are also more easily in-terpreted for refractory daughter products sincethey do not tend to escape the system (alsothe reason they are used to radiometrically daterocks). Volatile products, especially isotopes ofnoble gases such as helium and argon, requiredifferent intepretations since they can readilyescape the mantle and, for helium, escape theEarth. For example, MORB in fact has a high

daughter to primordial isotope ratio4He/ 3Herelative to many OIBs, which is opposite to therefractory ratios involving, for example, leadisotopes. This is often interpreted as result-ing from degassing and loss of primordial he-lium 3He from the upper mantle through platetectonic and mid-ocean ridge processing, andthe subsequent repopulating of helium with itsradiogenic isotope4He (i.e., α-particles frommost large element decay sequences); in con-trast an isolated lower mantle or OIB sourcereservoir would have undergone little loss of pri-mordial helium thus maintaining a smaller iso-topic ratio 4He/ 3He.

The production of the argon isotope40Arfrom the decay of the potassium isotope40K hastwo important arguments relative to mantle lay-ering. First, the total amount of original40K inthe Earth can be roughly estimated from chon-dritic abundances (and other arguments beyondthe scope of this essay). However, the amountof 40Ar it should have produced over the age ofthe Earth is far in excess (by a factor of 2) ofthe 40Ar in the atmosphere, implying that muchof this argon is still buried and isolated. More-over, one can also estimate from the trace el-ement composition of MORBs themselves thatthe MORB source region as it stands now wouldhave been lacking primordial40K and even ifthe entire mantle were composed of this MORBsource region, it would not have been able toproduce even the atmospheric levels of40Ar;this implies that the bulk of original40K wasburied in a layer different and more enrichedthan the MORB-source region, which then pro-duced most of the40Ar, much of which is stillburied in this layer.

An often quoted straw-man argument formantle layering is the heatflow paradox. In thiscase, it is reasoned that if the entire mantle werecomposed of MORB source material with its de-pleted concentration of heat producing elements(U, Th, K), then it would not be able to producethe total heat output through the top of the man-tle (about 30TW). This suggests that the the heat

14


producing elements allowing for the mantle heatoutput must be buried at depth. However, thismakes the assumption that most heat output isfrom radiogenic heating, whereas possibly lessthan half of it is; if most heat output is fromsecular cooling (lost of primordial heat) then theheat-flow paradox argument is questionable. Asimilar but shakier argument is based on the factthat the volcanic flux of the helium isotope4He,which is produced from heat producing radioac-tive decay of U and Th, seems to be very low rel-ative to what would be expected given the heatthat is emanating from the mantle, implying that4He is also buried at depth. Again, if heat ouputis more than 50% from secular cooling and notall from radiogenic heating, then the low fluxof helium is to be expected. Even if that werenot the case, mechanisms for the flux of heliumare not the same as for flux of heat; i.e., heatalways escapes to space eventually by convec-tion, conduction even radiation, whereas heliumonly escapes from the mantle if it passes throughnarrow melting zones, which is not inevitable,and thus it can be hidden and buried almost any-where and not only hidden at great depth.

Finally, the production of continental crustalso argues for an isolated layer in the mantle.Continental crust represents an accumulated his-tory of mantle melting, segregation of lightercomponents and removal of incompatible ele-ments to the surface. If the continental crustwere removed uniformly from a whole mantlemade of primordial ”bulk silicate earth” (i.e., anEarth derived from chondritic material and onlysegregated into mantle and core), then the con-centration of incompatible elements would nothave been reduced enough to produce a mantlemade of MORB source material (i.e., removalof continental crust would not have depleted thewhole mantle enough to make MORB source).However, if the crust were removed from 1/3 to1/2 of the mantle, that resulting depleted por-tion would very closesly match MORB sourcecomposition. This argues that the continentalcrust segregated only from the upper portion of

the mantle, not the whole mantle, and thus thereremains a deeper unsegregated mantle at depth[seevan Keken et al., 2002].

Although there are numerous geochemical ar-guments for a layered mantle with an isolatedand undepleted mantle at depth, they largelyconflict with geophysical evidence for wholemantle convection. Mineral physics experi-ments suggest that the 660km phase changeboundary might provide an impediment to man-tle fow but not an impermeable barrier. Seis-mic tomography consistently shows subductingslabs and apparently cold downwellings extend-ing into the lower mantle [van der Hilst et al.,1997; Grand et al., 1997]. Recent high reso-lution images of a hot upwelling mantle plumebeneath Hawaii [Wolfe et al., 2009] as well asseismic images of other plumes [Montelli et al.,2004] also suggest vertical upward transportacross the 660km boundary. Finally, geodynam-ical arguments against separated layers suggestthat if a lower mantle held most of the man-tles heat producing elements, it would be im-plausibly hot, and by heating up the bottom ofthe upper mantle it would generate much biggermantle plumes than would be observed [Tackley,2002].

The contradiction between geochemical andgeophysical inference of layered vs whole man-tle convection has been and largely remains anunsolved problem. Attempts to reconcile theseobservations have been numerous. A reason-ably popular approach has been to allow that the660km boundary is not a barrier to mantle flow,but that the barrier exists at greater depth. Thereare seismically observable layers at the bottomof the mantle (the D” layer), which could storeenriched material, although these are also so thinthey could possibly overheat (depending on theamount of radioactive heat sources stored there).As a comprosmise, it has been argued thatthe enriched mantle exists in an approximately1000km thick layer at the base of the mantle[Kellogg et al., 1999], although this layer hasnever been seismologically observed (see Fig

15


7). More recently, chemical heterogeneity gath-ered into piles on the core-mantle boundary andbelow upwelling zones [Jellinek and Manga,2002, 2004] has been suggested by convec-tion models [McNamara and Zhong, 2005] withsupport from joint seismology-gravity analyses[e.g.,Ishii and Tromp, 1999].

Other mechanisms for reconciling geochem-ical and geophysical observations, but not in-voking layering, have been recently proposedas well. These mostly involving differentialmelting. For example, one model considers thewhole mantle as a plum-pudding mix of en-riched and volatile (water) enriched plums ina depleted, drier and harder to melt pudding.The pressure-release melting in mantle plumesis stopped at higher pressures at the base of100km thick lithosphere, so this could involvemostly melting of easily melted enriched andvolatile-rich components; melts making it all theway to the surface at ridges would undergo morepressure drop and thus could also melt the de-pleted mantle component, resulting in MORBthat seems depleted relative to OIB [Ito and Ma-honey, 2005a,b]. Another model exploits thefact that Transition Zone minerals seem to beable to absorb water more readily than mate-rial above it (and below it). A Transition-Zonewith a little water will be dry relative to its sol-ubility or water storage capacity. However up-welling material passing through the transitionzone would carry this slightly damp materialinto the upper mantle at the 410km boundary,and since the upper mantle olivine has poor wa-ter solubility it would be closer to saturation andlikely melt. A little melting at 410km depthof the broad upwelling mantle (forced upwardby the downward flux of slabs) passing throughand out of the transition zone would cause itto be stripped of incompatible elements as itflows into the upper mantle leaving a depletedMORB source region; because of the high pres-sures and high compressibility of melt, the par-tial melt that has cleaned this upwelling man-tle would be dense and remain behind, eventu-

ally to be entrained by slab driven downwellingback into the lower mantle. Upwelling mantleplumes on the other hand would go through thetransition-zone too fast to become hydrated andthus would undergo little melting and filteringat 410km depths, leaving largely enriched OIB.This model, called the Transition-Zone WaterFilter [Bercovici and Karato, 2003a; Karatoet al., 2006; Leahy and Bercovici, 2010] (seeFig. 7), predicts that the 410km should be thesite of melting, and this has been born out in var-ious seismological studies [e.g.,Revenaugh andSipkin, 1994; Song et al., 2004; Tauzin et al.,2010]; however, the theory is still controver-sial given poor knowledge of melt propertiesand their solubilities of incompatible elementsat these depths and pressures, so it remains anactive subject of investigation.

Mantle convection and the generation of platetectonics

The oldest problem in mantle convectionThe link between plate tectonics and mantle

convection is one of the oldest and most chal-lenging problems in the history of geodynamics.The original theories of mantle convection putforward by Holmes[1931] were developed inthe context of explaining continental drift as ar-ticulated byWegener[1924]. Although the latertheory of plate tectonics is the grand-unifyingprinciple of geology, it is a kinematic theory inthat it describes surface motions but not theircause. Mantle convection is widely accepted tobe the engine for plate motions since it is a fun-damental mechanism for exploiting the energysources of the Earth’s interior, i.e., loss of pri-mordial and radiogenic heat. It is now generallyregarded that the plates themselves are a featureof mantle convection in that they are the mobileupper thermal boundary layer of convective cellsthat thicken as they cool in their migration awayfrom ridges until they are heavy enough to sinkalong subduction zones.

16


Lower

Mantle

Transition

Zone

Upper

Mantle

Mid-ocean ridge

Subducting SlabPlume

Melt ProductionSlab

Entrainment

Viscous Entrainment

Mantle Velocity Field

Figure 7: Two end-member mantle mixing models. In the abyssal layeredmodel (left) the source forenriched ocean-island basalt (OIB) is in a deep primordial layer, while the source for depleted mid-oceanridge basalt (MORB) is in the upper recycled mantle (afterKellogg et al.[1999]). In differential meltingmodels, such as the water-filter model (right), the MORB and OIB sources undergo different styles ofmelting, but the mantle still undergoes whole-mantle circulation; see text for further explanation (afterBercovici and Karato[2003a];Leahy and Bercovici[2007]).

One of the major accomplishments of mantledynamics theory is that convective fluid veloci-ties, calculated in any number of ways, consis-tently predict the measured scales of plate tec-tonic velocities, i.e., between 1 and 10cm/yr.This was in fact inferred even in the 1930sfrom both gravity and heat-flow measurements[Pekeris, 1935;Hales, 1936] and is well knownby the force balance on sinking slabs [e.g.,Davies and Richards, 1992], as well global heatextraction from mid-mantle cooling by slabs[Bercovici, 2003]. Moreover, plate motions arewell correlated with the presence of slab forc-ing, in particular that tectonic plates with asignificant portion of subduction all have ve-locities roughly an order of magnitude fasterthan plates without substantial subduction zone[Forsyth and Uyeda, 1975]; see Figure 6. Thus,because cold sinking slabs seem to be the ma-jor expression of convection and major driversof plate motions implies that the plates are con-vection.

The plate generation problem

While convective and slab driving forcesfor plate tectonics are important, understandinghow plates self-consistently arise (or why theydo or don’t arise) from planetary convection hasbeen a major goal in geodynamics. Up until theearly 1990s it was believed that since plate-likemotion of the lithosphere was essentially dis-continuous it could not be predicted or repro-duced by fluid dynamical convection theories.However, in the last 15 years or so there hasbeen major progress with fluid dynamical mod-els yielding very plate like motion by incorpo-rating more sophisticated rheological weaken-ing mechanisms, such as brittle/plastic yieldingor damage mechanics.

However, even so, there is still no compre-hensive theory of how the plates and mantleare related, and in particular how plate tectonicsself-consistently arises from a convecting man-tle. Both a clue and frustration is that the Earthappears to be the only known terrestrial planetthat has plate tectonics in addition to liquid wa-ter as well as life, which are all either causative

17


(i.e., necessary conditions for each other) or co-incidental. While our planet supports plate tec-tonics, our ostensible twin, Venus, does not; thisremains a leading-order quandary in Earth sci-ences and to solve it one must understand howand why plate tectonics is generated at all. Ofcourse geoscience has until recently only sam-pled the few planets of our own solar system andthus the data is sparse; with the advent of extra-solar planet discovery, this sparseness should bemitigated and perhaps other planets will be dis-covered with plate like mantle circulation fromwhich we will learn more about how our ownplanet works [e.g.,Valencia et al., 2007].

There are of course various qualitative andphilosophical questions about how and whyplate tectonics forms, evolves and exists, but forthese to be predicted or reproduced in a physi-cal theory, one requires quantifiable questions.In short, what are the metrics of plate gener-ation? Two fundamental features of plate-likemotion exist for instantaneous motions: theseareplateness[Weinstein and Olson, 1992] andtoroidal motion [e.g.,Hager and O’Connell,1979; O’Connell et al., 1991;Bercovici et al.,2000]. Plateness is essentially the extent towhich surface motion and the strength of thelithosphere is like that for nearly rigid blocksseparated by narrow weak boundaries. Toroidalflow is characterized by strike-slip motion andplate spin (Figure 8). Toroidal motion hasno direct driving force in buoyant convection(which drives only vertical and divergent mo-tion), however, it has as much energy in thepresent-day plate tectonic velocity field as thebuoyantly driven motion (called poloidal mo-tion). The globally averaged toroidal motion isdependent on the lithosphere’s reference frame(e.g., hostpot frame), and the field in generalchanges through time [Cadek and Ricard, 1992;Lithgow-Bertelloni et al., 1993]; however, thetoroidal field is a quantifiable and significantfeature of global plate motions. Such measur-able quantities as plateness and toroidal floware important for testing the predictions of plate

generation theories. Both phenomena rely onreasonably strong nonlinear rheological feed-back effects to permit large strength variationsfor high plateness (i.e., rapidly deforming zonesare weak, slowly deforming ones are strong), aswell as coupling of buoyantly driven flow to ver-tical torques that drive toroidal spin and shear[Bercovici, 2003].

Recent progressIn the last decade, plate generation models

have become increasingly sophisticated, in con-cert with further expansion and accessibilityof high-performance computing. Instantaneousplate-like behavior has been achieved with con-vection models employing various forms ofplastic yield and self-weakening criteria. In-corporation of these laws has lead to the pre-diction of reasonable toroidal and poloidal flow[Bercovici, 1995] (Figure 9) and by includ-ing the rheological effects of melting at ridgeshave attained localized passive spreading zones[Tackley, 2000] (Figure 10). Most recentlythese models have been extended to three-dimensional spherical models, creating the firstglobal models of plate generation from mantleconvection [van Heck and Tackley, 2008;Foleyand Becker, 2009] (Figure 11).

However, models that use plastic or instan-taneous self-weakening rheologies only allowweak zones to exist while being deformed, thuscannot correctly model dormant weak zones(e.g., sutures and inactive fracture zones). Rhe-ological mechanisms that allow weakening topersist over time have also been studied. Whilethermal weakening is a well understood mech-anism, thermal anomalies diffuse too fast, andit is highly unlikely that, for example, suturesare thermal remnants. Weakening by hydra-tion or as a secondary phase (i.e, by provid-ing pore pressure) is a strong candidate as well,although the mechanism for ingesting water todepth is problematic, and requires mechanismssuch as cracking enhanced by thermal stresses[Korenaga, 2007]. Damage in the form of mi-

18


Figure 8: The horizontal divergence (which measures spreading and convergence rates) and vertical vor-ticity (which measures angular velocity and horizontal shear) of present day plate motions [fromDumoulinet al., 1998], which are associated with poloidal and toroidal flow, respectively. Poloidal flow is equivalentto basic convective motion driven directly by thermal buoyancy. Although present in Earth’s plate-mantlesystem, toroidal flow does not arise naturally in basic viscous convection, but requires the coupling of con-vective motion with nonlinear rheological effects.

crocracks [Bercovici, 1998] and grain-size re-duction [e.g.,Karato et al., 1980;Braun et al.,1999;Montesi and Hirth, 2003] are also strongcandidate weakening mechanisms because oftheir longevity and evidence in the form of my-lonites [Jin et al., 1998]. The need for shear-localization at significant depth makes grain-size weakening particularly appealing and hasproven to be successful at creating plate-likemantle flows [Bercovici and Ricard, 2005;Lan-duyt et al., 2008;Landuyt and Bercovici, 2009b](Figure 12). However, grainsize reduction and

weakening tend to occur in exclusive areas ofdeformation space (i.e., weakening occurs dur-ing diffusion creep while reduction occurs indislocation creep; seeDe Bresser et al.[2001])and require mixing of mechanisms in physicalor grainsize distribution space [Bercovici andKarato, 2003b;Ricard and Bercovici, 2009].

Use of both plastic/brittle-yielding and dam-age theories of plate generation have been usedto elucidate the planetary dichotomy betweenEarth and Venus and the causal link between cli-mate, liquid water and plate generation. Earth

19


Divergence

-0.94

-0.2125

0.03

0.2725

1

Vorticity

-0.84

-0.1725

0.05

0.2725

0.94

normalized byDmax = 247.68 Gyr

n=3

n=21

n=-1

0.45

0.195

0.11

0.025

-0.23

0.67

0.295

0.17

0.045

-0.33

0.45

0.135

0.03

-0.075

-0.39

(b)(a)

Figure 9: The source-sink model of lithospheric motion uses present day plate motions to drive flow withthe divergent field (upper left) in order to model and recoverthe known vorticity field (lower left) usingnon-Newtonian flow calculations (right column). Simple power-law or pseudo-plastic rheologies (typical ofdislocation creep) do not recover the vorticity field well (top two, upper right), while velocity-weakeningor shear-localizing rheologies (where stress decreases with increased strain-rate) recover it very well (lowerright). FromBercovici et al.[2000] afterBercovici[1995]. American Geophysical Union.

and Venus are ostensible twins but Earth hasplate tectonics and Venus does not. This is usu-ally attributed to lack of water on Venus whichwould otherwise lubricate plate motions; how-ever Earth’s lithosphere is likely to be no morehydrated than Venus’s because of melting dehy-dration at ridges [Hirth and Kohlstedt, 1996].Recent studies have hypothesized that the roleof water in maintaining plate motion is not tolubricate plates, but to be an agent for the car-bon cycle, which thus allows for a temperateclimate on Earth. A cool surface on Earth,according to one hypothesis [Lenardic et al.,2008] causes a larger temperature drop acrossthe lithosphere than would occur on Venus(because of temperature-dependent viscosity),and thus lithospheric buoyant stresses are large

enough on Earth to induce plastic/brittle fail-ure, but not on Venus. An alternative hypoth-esis [Landuyt and Bercovici, 2009a] states thatplate-like motion depends on the competitionbetween damage and healing (where, for exam-ple, if damage is due to grain-size reduction,healing is due to normal grain growth), wherea high damage-to-healing ratio promotes plate-like motion, while a lower ratio yields stagnant-lid behavior. A cooler surface temperature in-hibits healing while a hot surface promotes heal-ing, thus leading to plate like behavior on aplanet like Earth with a temperate climate butnot on a planet like Venus. Both hypotheses em-phasize that water dictates conditions for plategeneration by its modulation of climate and noton direct strength reduction.

20


Figure 10:A simulation of plate generation over mantle convection. The plate rheology is visco-plasticand the viscosity reduction associated with melting is parameterized into the model, leading to exceptionalplate-like behavior and apparent passive spreading (i.e.,narrow spreading centers not associated with anydeep upwelling). The right panels show surfaces of constanttemperature, which here are dominated by colddownwellings; the left panels show the viscosity field (red being high viscosity and blue low viscosity).Different rows show different times in the simulation. After Tackley[2000]. American Geophysical Union.

Figure 11: Three-dimensional spherical shell convection with a plastic-type lithospheric rheology showingplate-like behavior. Left panel shows isothermal surfacesand in particular cold downwellings. The rightpanel shows surface viscosity with velocity vectors super-imposed. Note the passive divergent and rheo-logical weak zone forming mid-way between the two major downwelling regions. Adapted fromFoley andBecker[2009]. American Geophysical Union.

Finally, subduction initiation continues to bean extremely challenging issue in geodynamics

[Stern, 2004;King, 2007]. The strength of thickcold pre-subduction lithosphere is such that it

21


S [min/max= -1/1] G [min/max= -0.00314/0.00254]

Ω [min/max=-1.3/1.3] vh [max vec.length=0.103]

φ [min/max=0.04733,0.05202

A [min/max=0.9849/8.696]

time = 0.639

Figure 12:A simple source-sink model of shallow flow with a two-phase and grainsize-reducing damagemechanism. Damage per se involves transfer of deformational work to the creation of surface energy oninterfaces by void and/or grain boundary generation in the continuum. In the case shown, all damage isfocussed on grainsize reduction. The panel meanings are indicated by symbols whereS is the imposeddivergence rate (i.e. the source-sink field) that drives flow; G is the dilation rate due to void formation;φ is void volume fraction;Ω is vertical vorticity or rate of strike-slip shear;vh is horizontal velocity;andA is the “fineness” or inverse grainsize. This particular calculation shows that fineness-generating,or grainsize reducing, damage is very effective at creatinglocalized fault-like strike-slip zones in vorticityΩ, and solid-body like translation in the velocity fieldvh. Adapted fromBercovici and Ricard[2005].American Geophysical Union.

should never go unstable and sink, at least noton geological time scales (or cosmological oneseither). Thus how and why subduction zonesform remain enigmatic. Mechanisms rangefrom weakening by rifting [Kemp and Steven-son, 1996;Schubert and Zhang, 1997], sedimentloading and water injection [Regenauer-Liebet al., 2001], and re-activation of pre-existingfault-zones [Toth and Gurnis, 1998;Hall et al.,2003], all of which have some observational mo-

tivation, although fault re-activitation might bethe most compelling [e.g.Lebrun et al., 2003].Another unresolved enigma concerns the age ofsubduction zones. The convective picture ofplate motions would have plates subduct whenthey get old, cold and heavy. However, the sea-floor age distribution implies that subductionrate is independent of plate age, such that the ageof plates at subduction zones is distrubted fromage nearly 0 (i.e, subducting ridges), to the old-

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est ages of roughly 200Myrs [seeBecker et al.,2009].

Summary

Mantle convection is the central theory for howthe Earth works, i.e., what drives geological mo-tions as well as the cooling history and chemicalevolution of the planet. The theory is based onthe simple notion that the Earth (and other ter-restrial planets) must cool to space and in so do-ing release heat and gravitational potential en-ergy through thermal convection. Thermal con-vection theory itself is a well established phys-ical theory rooted in classical mechanics andfluid dynamics. Much of the physics of basicconvection goes far in describing circulation andstructure in the Earth’s mantle, for example theflow velocities, establishment of thermal bound-ary layers, and even the prevalence of slab-likedownwellings and plume-like upwellings. How-ever, numerous quandaries and paradoxes per-sist because much remains to be understoodregarding the many complexities of the exoticconvecting “fluid” in the mantle. How the Earthappears to be stirred by deep subducting slabs,but still appear to be unmixed when it comes upat either mid-ocean ridges or ocean-islands, re-mains a major mystery, much of which is due toour incomplete understanding of the processesof melting, chemical segregation and mixing inthe mantle. How and why the Earth’s mantleconvects in the form of plate tectonics at all andunlike other terrestrial planets remains one ofthe biggest questions in geoscience; this prob-lem is undoubtedly related to the mantle’s ex-otic rheology in which flow depends on multipleproperties including temperature, stress, chem-istry and mineral grainsize. Even more than 100years since Kelvin’s controversial dating of theEarth’s age by cooling, in fact the cooling his-tory of the Earth remains poorly understood be-cause of incomplete knowledge of radiogenicheat sources as well as the complex physics ofmantle flow. Thus while mantle convection re-

mains one of the grand unifying physical theo-ries of how the Earth works, many of the majorquestions and mysteries about the mantle remainunsolved and are thus ripe for discovery by fu-ture generations of Earth scientists.

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