Using thermo-mechanical models of subduction to constrain effective mantle viscosity

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To cite this version:

Fanny Garel, Catherine Thoraval, Andrea Tommasi, Sylvie Demouchy, D. Rhodri Davies. Using

thermo-mechanical models of subduction to constrain effective mantle viscosity. Earth and Planetary

Science Letters, Elsevier, 2020, 539, pp.116243. �10.1016/j.epsl.2020.116243�. �hal-02733023�

Author copy of the final (peer-reviewed) version

Article published in Earth and Planetary Science Letters in March 2020

Please, reference to:

Garel, F., Thoraval, C., Tommasi, A. Demouchy, S., Davies, D.R. (2020)

Using thermo-mechanical models of subduction to constrain effective

mantle rheology. Earth Planet. Sci. Lett., 539: 116243,

doi:10.1016/j.epsl.2020.116243

Using

thermo-mechanical

models

of

subduction

to

constrain

effective

mantle

viscosity

Fanny Garel

a

,

∗

,

Catherine Thoraval

a

,

Andréa Tommasi

a

,

Sylvie Demouchy

a

,

D. Rhodri Davies

b

a_{GéosciencesMontpellier,Univ.Montpellier,CNRS,Montpellier,France}

b_{ResearchSchoolofEarthSciences,TheAustralianNationalUniversity,Canberra,Australia}

a

b

s

t

r

a

c

t

Keywords: olivine dislocationcreep subductiondynamics mantleviscosity rheologyparameterization

thermo-mechanicalnumericalmodeling

Mantle convection and plate dynamics transfer and deform solid material on scales of hundreds to thousands of km. However, viscoplastic deformation ofrocks arises from motions ofdefects at sub-crystal scale, such as vacancies or dislocations. In this study, results from numerical experiments of dislocation dynamics in olivine for temperatures and stresses relevant for both lithospheric and asthenospheric mantle (800–1700 K and 50–500 MPa; Gouriet et al., 2019) are used to derive three sigmoid parameterizations (erf,tanh,algebraic), which express stress evolution as afunction of temperature and strain rate. The three parameterizations fitwell the results of dislocation dynamics models and may be easily incorporatedinto geodynamical models.Here, they are used inan upper mantlethermo-mechanicalmodel ofsubduction,inassociationwithdiffusioncreepand pseudo-brittle flow laws. Simulations using different dislocation creep parameterizations exhibit distinct dynamics, from unrealistically fast-sinking slabs in the erf case to very slowly-sinking slabs in the algebraic

case. Thesedifferencescould not have been predicted apriorifrom comparison with experimentally determinedmechanicaldata,sincetheyprincipallyarisefromfeedbacksbetweenslabsinkingvelocity, temperature, drag,and buoyancy, whichare controlledbythe strain ratedependence ofthe effective asthenosphere viscosity. Comparisonof model predictionsto geophysicalobservations and to upper-mantle viscosityinferred fromglacial isostaticadjustment showsthat the tanh parameterizationbest fits bothcrystal-scale andEarth-scale constraints. However, the parameterizationof diffusioncreep is alsoimportantforsubductionbulkdynamicssinceitsetstheviscosityofslowlydeformingdomainsin the convectingmantle.Withinthe rangeofuncertaintiesofexperimentaldata and,most importantly, oftheactualrheologicalparametersprevailingintheuppermantle(e.g.grainsize,chemistry),viscosity enablingrealisticmantlepropertiesandplatedynamicsmaybereproducedbyseveralcombinationsof parameterizationsfordifferentdeformationmechanisms.Derivingmantlerheologycannotthereforerely solelyontheextrapolationofsemi-empiricalflowlaws.Thepresentstudyshowsthatthermo-mechanical modelsofplateandmantledynamicscanbeusedtoconstraintheeffectiverheologyofEarth’smantle inthepresenceofmultipledeformationmechanisms.

1. Introduction

Mantle and lithosphere dynamics is a multi-scale process, in whichthemotionof defectsata sub-crystalscaleinduces defor-mationatscales exceedingthousandsofkilometers. Earth’s litho-spheredeforms elasticallyatshort time-scales,frommilliseconds forseismic stress release (e.g. Melosh and Fleitout, 1982) to the severalthousandyearsofglacialloadingandunloading(e.g.Watts,

*

Correspondingauthor.

E-mailaddress:fanny.garel@umontpellier.com(F. Garel).

2001). Earthquake distribution in oceanic plates away from

sub-ductionzonesindicatesthatbrittlefailureisactiveatlowpressure and temperature conditions (

<

1 GPa,

<

600 K; McKenzie et al.,

2005). On the time-scales of million years, viscous deformation

overhundredsofkilometersisrevealedbyseismictomography im-ages showingdeformedsubducted slabs inthe deepmantle (e.g. Kárason andVanDer Hilst,2000). Thisdeformation isessentially accommodatedbyviscoplasticprocessesallowingtheflowof man-tlerocksathighpressure.

Laboratoryexperimentshavelonginvestigatedthedeformation ofolivine,themostabundantuppermantlemineral(

∼

60%in vol-ume), to constrain its rheological properties, i.e. how it deforms

Fig. 1. a) Resultsfromdislocationdynamicssimulations(Gourietetal.,2019)withstressasafunctionofstrainratefortemperaturesrangingfrom800to1700 K.b) Sigmoid functions(erf,tanh,algebraic)usedasfittingfunctions.Theyreproduceboththerapidincreaseathighstrainratesandtheasymptoticbehaviortowardszeroatlowstresses andlowstrainrates.Thedatasetisrestrainedtostresseslowerorequalto500 MPa.Giventheshapeofasigmoidfunction,theparameterizationsshouldnotbeextrapolated athighstrainrateandhighstresses(shadeddomainonFig.1b).(Forinterpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

underanappliedstress.Athightemperatures,ductiledeformation throughcreepprocessespredominates.Theseprocessesinvolvethe motionofcrystaldefects.Athighstressesorstrainrates,the mo-tionoflinedefectsacrossthecrystalpredominates,characterizing the dislocation creep regime (see review by Ashby and Verrall,

1978; Hirth and Kohlstedt, 2003). At lower stresses and higher

temperatures,thediffusionofpoint defects–vacancies –plays a majorrole. Thisdiffusioncreepregime isgrain-size sensitive. Ex-perimentaldatafordislocationcreepinolivinepointtoavariation inmechanicalbehaviorasafunctionofstressand/ortemperature. The high-temperature data are classically adjusted using a semi-empirical flow law in which strain rates depend on stress to a powern (power-lawrheology; Baietal., 1991;Hirth and

Kohlst-edt,2003),whereas thelow-temperature dataisbetter described

byarheologicalparameterizationinwhichstrainratesdepend ex-ponentially on stress (e.g.Evans andGoetze,1979; Demouchy et

al.,2013).

Nevertheless,extrapolation ofexperimentally determined flow lawsto nature is hindered by the fact that the strain raterange inthe deformation experiments(

>

10−6 _s−1_{) is atleast 6orders}

ofmagnitudefasterthanthestrainratesexpectedinthe convect-ing mantle (

∼

10−20–10−12 s−1). Moreover, steady state is never achievedin the low temperatures experiments(Demouchy etal.,

2013; Thieme et al., 2018). Recently, the increased efficiency of

numericalcalculus allowedthedevelopmentofnumerical simula-tionsinwhichthedislocation dynamicsinolivine,includingboth glideandclimb,isexplicitlymodeled(Boiolietal.,2015).Forthe firsttime,thesemodelscalculatedtheinteractions between dislo-cationglideandclimbinolivineatbothlaboratoryandgeological strain rates, avoidingextrapolation ofthe empirically-determined parameters.Recently,Gourietetal.(2019) continuedthistask, es-tablishingthatthechangeinmechanicalbehaviorassociatedwith the “power-law breakdown”, which in these models occurs for applied stressesgreater than 150 MPa, especiallyat low temper-atures,isnotduetoa changefroma climb-controlledtoa glide-controlledcreep.Theyshowedacontinuityofdynamicalprocesses. Strain is produced almost entirely by glide, but strain rates are controlled by climb. Variations in stress and temperature within arangerelevant forboththelithospheric andthe asthenospheric mantle(50–500 MPaand800–1700 K)continuouslyimpactthe in-teractionsbetweenclimbandglide.Themechanical behaviorcan thenbedescribedbyaunifiedsemi-empiricalflowlaw(Gourietet

al.,2019).

Constitutiveequationsofrocksrheologyderivedfrom deforma-tion experiments usually expressthe strain rateasa function of the applied stress. However, it is often more convenient in

nu-merical models of geodynamics to express stress as a function of strain rate and temperature. The flow laws implemented in these modelsare parameterizations, partlybased on deformation and diffusion experiments on minerals and rocks. The constants in these equations (e.g. activation volumes and energies)are ei-ther derived directly from experimental data (e.g. van Hunen et

al., 2000; Crameriand Kaus,2010) orare assigned adhoc values

(e.g.Tackley,1996).However, eveninthefirstcase,thereare un-certainties inthe actualvaluesof mostrheologicalparameters in the mantle (pre-exponential constant, activation volume and en-ergy forboth dislocation anddiffusioncreep)aswell asinother importantparameters,such asgrain size,hydrogenconcentration, oxygenfugacity,etc.Moreover,numericalmodelsimplicitly extrap-olatethecrystal- orpolycrystalrheologicalpropertiestothescale of modelspatial resolution(usually larger than severalhundreds ofmeters,e.g.Daviesetal.,2011).

In this study, we derive several parameterizations based on the results of the dislocation dynamics simulations, in addition to thefirst one proposed inGourietetal.(2019) and implement thesedislocationcreepparameterizations,alongwithclassicalflow lawsfordiffusioncreep andpseudo-brittleyielding,in a thermo-mechanical model of upper mantle subduction. We analyze the effects of therheological parameterizations on thefeedbacks be-tween slabsinking velocity,temperature, drag, andbuoyancyand comparethesubduction dynamicspredictedbythedifferent sim-ulationstoobservationalconstraintsonlarge-scalemantle dynam-ics. This approach, which is complementary to inversion of rhe-ological parameters fromsurface observations (e.g.Baumann and Kaus, 2015), showshow thermo-mechanical models of plateand mantledynamicsmayfurtherhelpconstrainingmantlerheology.

2. Methods

2.1. Parameterizingdislocationcreepinolivine

We search for a mathematical formulation expressing stress solely as a function of strain rate and temperature and ensur-ingmechanicalconsistency(strainratesshouldtendtozerowhen stress tends to zero). The parameterization should fit the most recentnumericaldeformationsimulationsforforsteritesingle crys-tals obtained from dislocation dynamics models (Gouriet et al.,

2019).Thedataset,showninFig.1a,correspondsto49strainrate

valuesobtainedfrom150numericalsimulationsperformedat am-bientpressure,forsevenfixedapplieddifferentialstresses(i.e.the differencebetweenmaximalandminimal appliedstress)between 50and500 MPa(uniaxialloading)andforseventemperatures

be-Fig. 2. Thethreesigmoidfunctions(Eq.(1) and(2))withcoefficientsfromTableS1(SupplementaryMaterial)aredisplayedfortheseventemperaturescorrespondingtothe resultsofdislocation-dynamicscalculations(Gourietetal.,2019,colored markersasinFig.1)inadifferentialstressσdvs.uniaxialstrainrateε˙graph.Theinsertshowsa

zoomathightemperaturesandhighstrainrates,inarangerelevantforadeformingasthenosphericmantle,withdashedcurvesdisplayingthepower-lawfit(Eq. (3))for thefivehighesttemperaturesandwithparametersfromGourietetal.(2019),seesection2.1fordetails.

tween800 Kand1700 K.Theshapeofthemathematicalfunctions usedfortheparameterizationisconstrainedby: (i) anasymptotic behaviortowardszerostressatzerostrainrate;and(ii) therapid increaseofstressabovea criticalstrain rateshownbytheresults from Gouriet et al. (2019), the latter being steeper as tempera-tureincreases. Sigmoid functionsmeet these criteria. We choose three differentsigmoid functions: the errorfunction erf (already presented in Gouriet et al., 2019), the hyperbolic tangent func-tiontanh,andanalgebraic function f

(

x

)

= (

x

/

√

1

+

x2

₎

_.Theyrange

from

−

1 to1,aresymmetricalrelativeto0,anddifferonlyatthe transitionbetweentheasymptoticpartandthesteepincrease,as theyexhibitdistinctcurvatures(Fig.1b).

Differential stress

σ

d at each temperature is expressed as a

function of strain rate

ε

˙

(shortening parallel to the direction of theappliedcompression),following

σ

d

=

A0



1

+

SF



A1



log₁₀

(

ε)

˙

−

A2



(1) withSF thesigmoid function,and A0, A1, A2 beingthree

coeffi-cientsdependentontemperature.TheformulationofEq. (1) shifts theasymptoticlimitto0asrequired.Inafirststep,thethree co-efficients A0, A1, A2 are calculated independently to bestmatch

thedataateachtemperatureusinganonlinearleast-squaressolver using a trust-region-reflective algorithm based on the interior-reflectiveNewton method(herethe“lsqcurvefit” MATLABsolver). Ina secondstep, thetemperature-dependence ofeach coefficient isexpressedasafirst-order( A0,A1)orsecond-order( A2)

polyno-mial functionin orderto bestfit thetemperature-dependencyof deformation:

A0

(

T

)

=

a0

+

b0T

A1

(

T

)

=

a1

+

b1T (2)

A2

(

T

)

=

a2

+

b2T

+

c2T2

withT thetemperaturein K.

Finally,the sevenpolynomial coefficientsa0,b0,a1,b1,a2,b2,

c2 aredeterminedforthewholedatasetsimultaneouslywiththe

“lsqcurvefit” MATLAB solver (with initial guesses from the fitted

polynomialfunctions, followedby iterativeleast-squaredfits). Ad-ditionaltestshavebeenperformedtoensuretherobustnessofthe solutionwhenexcludingeitherthehighestorthelowest tempera-turedata.

Stressvariation asa functionofstrainrateandtemperatureis fullydescribedbythesevencoefficientsinEq. (2).Thebestfitting coefficients (see Fig. 2) for the three different sigmoid functions (more than 98% ofvariance explainedforall three parameteriza-tions) are givenin Supplementary Material Table S1.For stresses rangingfrom150–500 MPa,whichcorrespondtothesteepestpart of thecurves,the three parameterizations are almost exactly su-perimposed. On the other hand, at stresses below 150 MPa, the curves,andhencethe predictionsof thethree parameterizations, differasaconsequenceofthedifferentcurvatures(Fig.1b).For in-stance,thestresspredictedforastrainrateof10−15_s−1 _{at1400 K}

is 1, 4, and 18 MPa for the erf, tanh and algebraic parameteri-zations, respectively. Theexperimental data at50 MPa arebetter adjustedbytheerf andtanh parameterizations.Thefitforallthree parameterizationsbecomespoorerasbothstress andtemperature decrease,in particularbelow 150 MPaand 1100 K.Yet,the con-ditionsatwhichthefitisdegradedareassociatedwithveryslow strainrates(

<

10−16_s−1_{).Notethattheparameterizationsarenot}

constrained by DD simulations at stresses higher than 500 MPa: usingthembeyondthisvalueisanextrapolation.

The unified creep parameterizationof Eq. (1) is a continuous function oftemperatureandstrain rate, applicablefordislocation creep at both low and hightemperatures. It is a useful alterna-tivetothedoubleparameterizationofplasticitydepending onthe temperature or stress range, which have been used in previous geodynamicalmodels(e.g.Nevesetal.,2008).

Forcomparison,thecurvescorrespondingtoa“classical”, high-temperaturepower-lawparameterizationfordislocationcreep

˙

ε

=

APL

σ

dnexp



−

Edisl R T



(3) withpre-exponentialfactorAPL

=

5

.

2710−29Pa-n.s−1,n

=

4

.

5,and

activation energy Edisl

=

443 kJ.mol−1 (Gouriet et al., 2019) are

shownintheinsertofFig.2.TheexpressionofEq. (3) alsopredicts an asymptoticbehavior towards null strain rates at null stresses

Fig. 3. Subductionmodelset-up(10,000×660 kmbox).Theinitialagesofthesubductingandoverridingplatesatthetrenchare40 Myr(seeadditionalthermalanddensity profilesinSupplementaryFig. S1).Theweaklayeris7.5-kmthick.Theinitialslabtipdepthis194 km,andtheinitialslabwidthis96 km.Seetextforfurtherdetailson materialproperties.

andfits wellthedislocationdynamicsdataatstresseslowerthan 150 MPa. The tanh parameterization is the closest to the power lawone atstressesbelow50 MPa.Since thedislocation dynamic simulationswereperformedatambientpressure,Eq. (1) doesnot feature any pressure-dependency, nor does Eq. (3), e.g. there is noactivation volume.Depth-dependency ofrheological flow laws arises from the increase of both pressure and temperature with depth,whichareexpectedtoatleastpartlycompensateeachother (see more details in sections 2.3and 4.3). Hence, potential tem-peraturesT areusedinEq. (1) tocalculaterheologyforthewhole mantledomaininoursimulationsatmantlepressures.

2.2. Set-upofsubductionmodel

The three parameterizations for dislocation creep in olivine (Eq. (1)–(2))areimplementedina2-Dnumerical thermo-mechani-cal model of subduction to quantify how they influence the dy-namicsinasubductionsetting.ThemodelgeometryfollowsGarel et al. (2014), with cold subducting and overriding plates at the top of a 660-km deep (i.e. restricted to the upper mantle) and 10,000-km wide domain (Fig. 3). The trench corresponds to the interfacebetweenthetwo platesandisfree tomoveinresponse to their dynamics.It is initially located inthe middleof the do-main (X

=

5

,

000 km). The initial plate ages are zero at the left andrightcornerridges.Theyincreaselinearlytowardsthetrench, whereboththesubductingandtheoverridingplateshavean age of 40 Myr. The initial temperature profile follows the half-space cooling model (Supplementary Fig. S1). The surface and bottom thermalboundariesareisothermalatTs

=

273 KandTm

=

1573 K,

respectively.Thesidewallsare insulating(zeroheat-flux). Thetop boundary is a free surface; all other boundaries are free-slip. A hangingslabtip(withabendingradiusof250 km,extendingdown to194 kmdepth)isprescribedtoinitiatefreesubduction.

“Mantle”propertiesare ascribedtomostofthesimulation do-main, exceptfora 7.5-km thick low-viscositylayer located along thetop ofthesubducting plate, whichallows formechanical de-couplingbetweenthetwo plates.Bothmantleandweakmaterial havedensitiesinarangerelevantformantleperidotite,with

ρ

s of

3300 kg/m3_atT

s.Thedensitydecreaseswithincreasing

tempera-tureT ,following

ρ

=

ρ

s



1

−

α(

T

−

Ts

)



(4) with

α

thecoefficientofthermalexpansion(3

×

10−5K−1).

The standard equations describing the conservation of mass, momentum andenergy foran incompressibleStokes fluid under theBoussinesqapproximationaresolvedwithintheFluidity com-putationalmodelingframework(e.g.Daviesetal.,2011;Krameret

al., 2012). This finite-element,control-volume codeis built upon

adaptive,unstructureddiscretizations (e.g.Daviesetal., 2007), al-lowingto achieve a minimum element size of400 m in regions ofdynamicalsignificance(in thepresentmodels,suchmeshsizes are observed for instance in the decoupling layer), with coarser resolutionelsewhere. Thelocation of thedecoupling layer,atthe

interface betweenthesubductingandoverridingplates,istracked using avolume fractionapproach;its evolution isdescribed bya linear advection equation. In order to avoid excessive numerical diffusionfromtheweaklayerintoneighboringregions,theformer isdiscretizedonthecontrolvolumemeshusingtheminimally dif-fusive HyperC face-value scheme (see Wilson (2009) for further details). Weverified that the 7.5-kmthick weaklayer is well re-solved,withnearidenticalresultsobtainedforsimulationswitha smallerminimumelementsize.

2.3. Compositeviscosityanddeformationpartitioning

Tosolvethemomentumequation,therheologicalflowlawsare implemented in Fluidityusing a viscosityformulation. The effec-tive viscosity

μ

eff oftheincompressible, isotropicmaterialcanbe

derivedfromrheologicallawsusing(Gerya,2010)

μ

eff

=

σ

II

2

ε

˙

II

(5) with

σ

II the second invariant ( J2) ofthe deviatoric stress tensor

and

ε

˙

II the second invariant of the strain ratetensor defined in

two-dimensionas

˙

ε

i j

=

1 2



∂

vi

∂

xj

+

∂

vj

∂

xi



(6) The viscosity is modeled as controlled by three deformation mechanisms:diffusioncreep,dislocationcreep,andpseudo-brittle mechanism (noted with the subscript “disl”, “diff ” and “br” re-spectively in thefollowing). The present approachconsiders that deformation ispartitionedbetweenthe threecoexisting deforma-tionmechanisms(i.e.theindividualstrainratesaddup;cf.chapter 3inFrostandAshby,1982):

ε

˙

tot

= ˙

ε

diff

+ ˙

ε

disl

+ ˙

ε

br.Accordingly,the

correspondingbulkcompositeviscosity,

μ

,iscalculatedthrougha pseudo-harmonicmean: 1

μ

=

1

μ

diff

+

1

μ

disl

+

1

μ

br (7) with

μ

diff,

μ

disl,

μ

br the viscosities associated with each

defor-mation mechanism, as detailed below. Note that the composite viscositycorrespondstothelowestviscosityoftheindividual pro-cesses when differences between them are large. In the present models, the composite viscosity is boundedby upper andlower valuesof1018 _and1025_{Pa.s.Thereisnocompositionaldifference}

betweenlithosphericandasthenosphericmantlematerial.The lim-its between slab/plate and surrounding asthenosphere arise and evolvenaturallyasafunctionofvariationsinthetemperatureand strainratefields.

Following Moresi and Solomatov (1998), brittle failure atlow pressureisapproximatedthroughthepseudo-brittleviscosity

μ

br

μ

br

=

τ

y

2

ε

˙

II

(8) where

τ

y istheyieldstrengthderivedfromByerlee’slaw:

τ

y

=

min

(τ

0

+

fcP

,τ

y,max

)

(9)

with

τ

0 thecohesionatthesurface(2 MPa), fc thefriction

coeffi-cient(0

.

2), P isthelithostaticpressure( P

=

ρ

sgz where

ρ

s isthe

referencedensity for mantle rocks, g the acceleration of gravity andz thedepth),and

τ

y,max amaximumyield strength(10 GPa).

Thebrittlestrengthofrockdiffersunderextensiveorcompressive stresses,butinthepresentmodel,thepseudo-brittleviscosity

μ

br

onlydependsonthestrain-ratesecondinvariant

ε

˙

II.

Theviscosityinthediffusioncreepregimeiscalculatedas:

μ

diff

=

A−_diff1exp



Ediff

+

P Vdiff

R

(

T

+ δ

T

)



,

(10)

with R the gas constant.The activation energy Ediff andthe

ac-tivation volume Vdiff are set to 410 kJ.mol−1 and 4 cm3.mol−1,

respectively.The formulationof Eq. (10) accountsfora pressure-dependency, hence we also consider the depth-dependency of temperature by adding a

δ

T to the Boussinesq temperature so-lution,calculatedfroman adiabatic gradientof 0.5 K/km. Adiff is

10−7 _{Pa.s,whichcorresponds to aconstant grain sized}

_∼

_{3 mm,}

calculatedwitha pre-exponentialfactorof1.510−15Pa−1.m3 for diffusioncreepindryolivineinHirthandKohlstedt(2003).These valuesyield aviscosity foran asthenosphericlayer deforming by diffusion creep between 1020 and 1021 Pa.s (see Supplementary Fig. S2). Note that the depth-dependency of the diffusion creep viscosityaccountsforbothincreasinglithostaticpressureP and in-creasingtemperature(T

+δ

T )withincreasingdepthinthemantle. SupplementaryFig. S2 illustratesthat the twoeffects counterbal-anceeachother:diffusioncreepviscositycalculatedfromEq. (10) isalmostdepth-independentintheconvectiveuppermantle.

The viscosity associated with deformation by diffusion (

μ

diff)

ordislocation creep (

μ

disl) is calculated using Eq. (5). For

dislo-cationcreep, we implementEq. (1) directly intoEq. (5) equating the differential stress and the uniaxial strain rate to the second invariantsofdeviatoricstressandstrainratetensors,respectively. This is a first-order approximation, which considers the rock as an isotropic material with a scalar (and not a tensor) viscosity. Moreover,deformation experiments (Eq. (1))are often conducted underuniaxial loading (whereas stress is multidirectional in our 2-Dmodels), andthedifferential stress(Eq. (1))is differentfrom the deviatoric stress (Eq. (5)) since the minimal applied stress is different from the mean stress (e.g. Schmalholz et al., 2019). Therelationship betweendifferentialstress (Eq. (1)) andthe sec-ondinvariant ofthe deviatoricstress tensor(Eq. (5)) isgiven by Schmalholzet al.(2019) in 2-D:

σ

d

=

2

σ

II; andby Gerya (2010)

in 3-D:

σ

d

=

√

3

σ

II

≈

1

.

7

σ

II. The relationship between uniaxial

strain rate to the second invariant of the strain rate tensor is

˙

ε

=

2

/

√

3

ε

˙

II

≈

1

.

2

ε

˙

II (Gerya (2010)). Implementing these

correc-tions would lead to slightly smaller (by less a factor 2) mean viscosityforallthreeparameterizations,whereasthediscrepancies betweenthemaremuchlarger(Fig.4).

Theweaklayerdeformsonlyby“yielding”withareduced fric-tioncoefficientof0.02anditsviscosityiscappedat1020Pa.s.The weaklayerpropertiesrevertto“mantle”propertiesbelow200 km depth.Thesimulationisfirstrunforashorttime(

<

0.3 Myr)with onlydiffusioncreepand“yielding”rheologiestogeneratenon-zero strainrateandvelocity fieldsthat is identicalforall simulations. Thedislocationcreeprheology(eithertanh,erf oralgebraic param-eterizationinEq. (1))is thenintroduced inthecalculationofthe bulkviscosity.

The partitioning of the deformation between the different mechanisms is calculated using the individual strain rates asso-ciatedtoeachdeformationmechanism:

˙

ε

y

=

μ

μ

y

˙

ε

tot

Fig. 4. Verticalprofilesofthedislocationcreepviscosityundera40-Myrold litho-sphericplate (temperatureprofileshown inSupplementary Fig. S1), assuminga constant strain rate of10−15 _s−1_{. The viscosity μ}

disl is either calculated from

combinationsofEq. (1),(2) and(5) (sigmoid functionserf,tanhoralgebraic,see section2.1)orfrompower-lawexpressioninEq. (13).Theparameterization ofthe power-lawcasesissimilartotheoneusedinGareletal.(2014),withanactivation energyEdislof540 kJ.mol−1,anexponentn of3.5,apre-exponentialfactor APLof

4.410−17_Pa-n_s−1_{andaadditionaltemperaturetermδT calculatedusingan}

adia-baticgradientof0.5 K/km.Theactivationvolumeiseither0(blackcurve),6(red curve)or12 cm3_.mol−1_{(bluecurve).Alsoshownisacasewithnoadditional}

tem-peratureterm,i.e.adiabaticgradientof0 K/km,andnullactivationvolume(green curve).

˙

ε

diff

=

μ

μ

diff

˙

ε

tot (11)

˙

ε

disl

=

μ

μ

disl

˙

ε

tot

Weverifythatthetotalstrainrateequalsthesumofallindividual strain rates calculated in Eq. (11), consistently with the expres-sion of the composite viscosity in Eq. (7). The individual strain ratesarenotusedintheresolutionoftheconservationequations, which refers only to the bulk composite viscosity and the total strain rate. Theexpressions of Eq. (11) are solelyusedto discuss deformationpartitioning. Weconsiderthatonemechanism domi-nateswhenitsviscosityismuchlowerthantheothertwo(

<

20%), whichcorrespondstoastrainratemuchhigher(

>

80%)thanthese associated with the two other mechanisms. Equivalent deforma-tionforall threemechanisms isnot observedinoursimulations. We quantifythedeformation partitioningbetweenthe two coex-istingcreepmechanismsasthepercentageofdislocationcreep in totaldeformationpdisl:

pdisl

=

˙

ε

disl

˙

ε

diff

+ ˙

ε

disl (12) 3. Results 3.1. Subductiondynamics

Snapshots of the temperature, strain rate, and viscosity fields (top,middleandbottompanels,respectively)foreachofthethree dislocation creep parameterizations: erf, tanh, and algebraic (left, center and right columns) at the point when the 1300 K slab isothermreaches400 kmdeptharepresentedinFig.5.Thispoint inthesimulationischaracterizedbyarapidsinkingoftheslabin theuppermantle,beforethe descentishampered bythebottom boundaryat660 km. Thethreeparameterizations leadtodistinct subductiondynamics,withtheslabreaching400 kmdepthin0.7, 5.0, and 11.9 Myr, for the erf, tanh, and algebraic parameteriza-tions, respectively. A quantitative comparison of key subduction diagnostic features is provided is Table 1. Although the average temperature inthe asthenosphere is the samein all three cases,

Fig. 5. Snapshotsofthetemperature(top)strainrate(middle)andviscosity(bottom)fieldsinadomainsurroundingthesubductingslabwhenthe1300 Kisothermreaches 400 kmdepthforsimulationsusingthethreedifferentdislocationcreepparameterizations:erf (left),tanh (center)andalgebraic (right).Thedashedgreenlinesshowthe crosssectionalongwhichprofilesareextractedinFig.6.Theisothermsaredrawn100 Kapart(whitelines),theblacklineisthe1300 Kisotherm.

Table 1

Quantitativecomparisonsbetweenthesimulationswitheitherofthethreedislocationcreepparameterizations (erf,tanh,andalgebraic)whentheslab(isotherm1300 K)reaches400 kmdepth,correspondingtoFigs.5,6and

7. Thediffusivelengthiscalculatedwithathermaldiffusivityof10−6_m2_.s−1_.

ERF TANH ALGEBRAIC

Time(Myr) 0.7 5.0 11.9

Associateddiffusivelength(km) 5 13 19 Widthofslabat350 km

(distance between 1500-K isotherms)

113 126 141 Minimumslabtemperatureat350 kmdepth(K) 586 854 1034 Subductingplatevelocity(cm/yr) 223 5.1 1.5 Meanasthenosphereviscosity(Pa.s)below200 km

depth(averagedfortemperatures>1500 K)

8.0 1018 _{2.2 10}20 _{3.8 10}20

Slabmeanviscositybelow200 kmdepth(T<1300 K) 4.5 1024 _{3.2 10}24 _{2.4 10}24

thethermalstructureofthecoldslabvaries:slabwidthand ther-malstructure areinfluenced by thermaldiffusion,the magnitude of which depends on the elapsed time and, hence, on the slab sinking velocity. As a consequence, by the time the fast-sinking erf slabreaches400 kmdepth,itsthermalstructurehasonlybeen slightlymodified,whereas thetemperaturecontrastbetweenslab andasthenospherehasbeensmoothedinbothtanh and algebraic parameterizations. The strain rate field also reflects the distinct subduction dynamics: in the erf simulation, large strain rates in theasthenospherefarawayfromtheslab(about10−13_s−1_)arise

in response to the strong viscous drag induced by the rapidly sinking slab. Strain rates in the asthenosphere are

∼

100 times smallerinthetanh andalgebraic simulations,thelatterexhibiting

the smalleststrain rates.Asthenosphereviscosities (bottompanel of Fig.5) are in thesame orderof magnitudefortanh and alge-braic cases (

∼

1020 _{Pa.s), but significantly lower for the erf case}

(

<

1019_Pa.s).

Horizontal profilesoftemperature, strain rate,andviscosityat 350 km depth for the three simulations at the times shown on Fig.5 arepresentedinFig.6.The profilesare locatedsothat the cross-section is far away from both the slab tip andthe base of the surface lithospheric plate. The slab is visible on the thermal profiles ascolder temperaturesincomparisonto thesurrounding asthenosphere, whichhasan averagetemperatureof1573 K.The asymmetricaltemperatureprofileintheerf caseisinheritedfrom theinitialsurfaceplatethermalstructure,asopposedtothe

sym-Fig. 6. Temperature(top),strainrate(middle)andviscosity(bottom)horizontal pro-filesacross the slabat 350 kmdepth whenthe slabisotherm 1300 Kreaches 400 kmdepth.Theprofilesaredrawnbetween X=4800 kmand X=5500 km (dashedlinesinFig.5)forthethreesimulations(erf,tanh andalgebraic),withfilled circlemarkerslocatedintheasthenosphereclosetotheslabasshowninFig.5.

metrical thermal profiles in the tanh and algebraic cases, which havebeensmoothedby thermaldiffusion(Fig. 6,top panel).The coldesttemperatureintheslabisrelatedtothetimeelapsedsince subductionstarted.Itis584 Kfortheerf slabthathassubducted to400 kmdepthin0.7 Myrold, 854 Kforthetanh slabthathas subducted to 400 km depth in 5 Myr, and1034 K for the alge-braic slabthathassubductedto400 kmdepthin12 Myr.Thecold slabcoreisthe leastdeformedregion (Fig.6, middlepanel).The contrastinstrain rateandviscositybetweentheslabandthe sur-roundingasthenospheredependsstronglyontheparameterization andisnotsolelyafunctionofspatialvariationsintemperature(cf. topandmiddle/bottompanelsinFig.6).Asalreadyhighlightedin Fig.5,asthenosphericstrainratesintheerfcase(upto10−12_s−1₎

areup to threeorders ofmagnitudehigherthanin thetanh and algebraic cases(10−16–10−15 s−1). The erf simulationpresents a thick undeformedcold slab core(characterized by strain ratesof 10−18 _s−1_{), and a fast deforming asthenosphere (strain rates of}

10−13 s−1). A region inthe mantle wedge stronglydeformed by viscousdragisvisibleontherightsideoftheslabinFigs.5and6 forallsimulations.Strainratesinthislayerarethehighestinthe erf simulationandthelowest inthealgebraic case,thelatter ex-hibitingtheweakestcontrastsinstrainrateandviscositybetween slabandasthenosphere.Theviscosityprofilesfurtherillustratethe combined influence of the strain rateand temperature. The dis-crepancybetweenasthenosphericviscosities(

∼

1018–19Pa.sinthe erf caseand

∼

1020-21_{Pa.sintheothercases)reflectsthevariation}

instrainratesbetweenthethreesimulations.

3.2.Partitioningofdeformationbetweendislocationanddiffusioncreep Thepartitioning ofdeformation betweenthedifferent mecha-nismsconsideredhere (diffusionor dislocationcreep, or pseudo-brittleyielding–seepartitioningdefinitioninsection2.3)is pre-sented in Fig. 7, in a similar way as in Cizkova et al. (2007), Garel et al. (2014) and Bessat et al. (2020), even if the latter

only highlight the mechanism yielding the minimum viscosity ratherthan theirrelative importance.Pseudo-brittleyielding only dominates near the surface and at the subduction interface. For the erf parameterization, deformation occurs almost everywhere by dislocation creep only, except for the cold core of the slab, which is at the maximum cut-off viscosity. In contrast, for the algebraic case, deformation in most of the asthenosphere man-tle occurs dominantly through diffusion creep; dislocation creep only predominates within and around the cold slab. In the tanh case, both dislocation anddiffusion creepcontribute to deforma-tion in mostof theasthenosphere anddislocation creep prevails withinandaroundtheslab,aswellasinmostofthe lithosphere-asthenosphereboundary.ComparingFigs.5and7allowsusto re-late thedifferencesinasthenosphericviscositybetweenthe three parameterizationstochangesinthedominantlyactivated deforma-tion mechanism:the low viscosity inerf simulation results from the strain rate dependence of dislocation creep, which leads to weakeninginpresenceofhighstrainrates,whereas thehigh vis-cosityinthealgebraic simulationreflectsthelackofstrainrate de-pendence ina medium deformingdominantlyby diffusioncreep. However,thesameasthenosphericviscositycanarisefrom differ-entcombinationsofrheologicalparameterizations:similar viscosi-ties (

∼

1020 _{Pa.s) are produced by activation of both creep}

pro-cesses intanh simulationandby dominantdiffusioncreepin the algebraic case.

Thestrainrateandtemperatureconditionsatwhichstrainrate is equally partitioned betweendislocation and diffusion creep at 350 kmdepthforeachofthethreeparameterizationsaredepicted in Fig. 8. Notethat thispartitioning dependson the chosen dif-fusion creep parameters (e.g. grain size). The evolution of strain rateasafunctionoftemperature(Fig.8)highlightsthesensitivity ofdeformationpartitioningtobothdiffusionanddislocationcreep parameterizationsintheconvectingasthenosphere,whereasFig.2 showedonly the dislocation creep sensitivity.For the rheological parameterizations used in this study, diffusion creep prevails at lowstrainrates,anddislocationcreepprevailsathighstrainrates. At low temperature, the three dislocation parameterizations pre-dict similar behaviors. However, the predictions diverge for tem-peratures higherthan 1100 K (i.e.atconditions prevailingin the asthenosphere andatthe baseofthelithosphericplates).For ex-ample,at1550 Kand350 kmdepth,thestrainraterequiredforan equalpartitioningbetweendiffusioncreepanddislocationcreepis 2*10−18 _s−1 _{forthe erf case,9*10}−16 _s−1 _{forthetanh case, and}

10−14s−1 forthealgebraic case.Thus,dependencyoftheeffective viscosityonstrainrate,throughtheactivationofdislocationcreep, appearsatmuchlowerstrain ratesintheerf casethaninthe al-gebraic case. Thisdifference resultssolely fromthemathematical parameterizations(Eq. (1)).

The initial asthenosphere strain rate and temperature condi-tions (3

×

10−16s−1,1573 K)forapoint at350 km depthatthe time when dislocation creep is “switched on” inthe simulations (cf.section 2.2)is indicatedasa blacksquarein Fig.8.Forthese strainrateandtemperatureconditions,theamountofdeformation accommodatedbydislocationcreepis pdisl

=

87,24,or2%forthe

erf, tanh, oralgebraic parameterizations. The partitioning evolves through time, with coloreddots inFigs. 5, 6, and8 locating the conditionsatthetime when theslabreaches400 kmdepth.The differences in deformation partitioning have been enhanced: the contributionofdislocationcreepinthenear-slabasthenosphereis larger in all simulations. The contributionof dislocation creep is larger, both intermsof magnitudeandofvolume ofmaterial, in the erf case (pdisl

=

99%) than in the tanh (pdisl

=

73%) and

al-gebraic (pdisl

=

15%) cases,explaining thecontrasted deformation

maps in Fig. 7.The differencesbetween the three parameteriza-tions, alreadyvisibleinFig. 2,are obviousfromFig.8: eitherfor theinitialorlatertimesinthesimulations,thedeformationoccurs

Fig. 7. Mapsofdeformationmechanismsasafunctionoflocalstrainrateandtemperatureconditions,representingthe deformationpartitioningbetweenthedifferent deformationmechanisms.Mechanismsareindicateddependingontheirrespectivecontributionstototaldeformation(seesection2.3fordetails).Thespatialdomainisthe sameasinFig.5,forthethreesimulations:erf (left),tanh (center)andalgebraic (right).Theminimum(resp.maximum)viscosityof1018_(resp.1025_{)Pa.s,occursinthe}

highlydeformedasthenospherearoundtheslabintheerf case(resp.incoldestandleastdeformedlithospheredomains).

Fig. 8. Strainrateandtemperatureconditionsatwhichthedeformationat350 km depthisequallypartitionedbetweendiffusioncreepanddislocationcreep.Thelines aredrawnforthethreedislocationcreepparameterizations(erf,tanh andalgebraic),

andforapower-lawexpressionofdislocationcreep(Eq. (3) withparametersfrom Gourietetal.,2019).Dislocationcreepprevailsathighstrainrates,whereas diffu-sioncreepprevailsatlowstrainrates.Thefilledcirclesymbolsarelocatedinthe near-slabasthenosphereasshowninFigs.5and7.Theblacksquarerepresentsthe initialconditionsatthatlocation,identicalforallthreesimulations(310−16_s−1_,

1573 K).

mainlybydiffusioncreep inthealgebraic case,mainlyin disloca-tioncreepintheerf case,andbyabalancedmixtureofthetwoin thetanh case,asillustratedinSupplementaryMaterialFig. S3.

Thedeformationpartitioningasafunctionoftemperatureand strain ratefora power lawparameterization ofdislocation creep (Eq. (3))isalsoshowninFig.8.Thetanh parameterizationiscloser tothepowerlawthantothetwoothers.However,stresseshigher than 150 MPa are predicted in the cold slab and surface plates (SupplementaryFig. S4andCizkovaetal.,2007).Intheseregions, the power-law is not suitable to describe dislocation creep (cf. Fig. 2 of Gouriet et al. (2019)). Another illustration of the limi-tation ofthe classical powerlaw formulation of Eq. (3) is shown in Supplementary Fig. S6depicting the variation ofthe apparent stressexponentn asafunctionoftemperatureandstrainratefor thethree parameterizations:n increases withdecreasing temper-ature, with values higherthan 10 at low-temperature values, as predictedforanexponentialflowlawbySchmalholzandFletcher

(2011), and n tends to a value of 1 (i.e. diffusioncreep) athigh temperatureandlowstrainrates.

4. Discussion

4.1. Comparisontolarge-scaleconstraintsonmantleandplate dynamics

Thepredictionsofthepresentsimulationscanbecomparedto a number of differentobservations. First, surface plate velocities reconstructedfrompaleomagneticdataandrecordedthroughGPS are generallylessthan10–15 cm/yr andhavenot oftenexceeded this overthe past200 Ma (e.g.Müller etal., 2008; Larson et al.,

1997). This rules out extremely fast-sinkingslabs (

>

200 cm/yr),

suchasthoseobtainedintheerf case.Second,someslabsseemto penetratedeepintothemantle(

>

1000 km)asimagedbyseismic tomography(e.g. Lietal., 2008). Thisimpliesthatslabs tempera-tureremainssufficientlylowtoinducefasterthanaverageseismic velocities.Suchanobservationisconsistentwiththeslab temper-atures intheerf andtanh cases,butnotwiththeslowandwide slabinthealgebraic simulation,whichalreadyexhibitsacore hot-terthan900 Kat150-kmdepth(Fig.5).

Estimates of the mean upper mantle viscosity have been derived from a number of independent observations, including glacialisostaticadjustment(e.g.PaulsonandRichards,2009), long-wavelengthgeoidobservationscombinedwithsurfaceplate veloc-ities (e.g. Ricard and Vigny, 1989), constraints from Pacific plate dynamics (e.g. Iaffaldano andLambeck, 2014; Stotz etal., 2018), or melting conditions above mantle plumes (e.g. Thoraval etal.,

2006).Thesedifferentapproachesyieldviscositiesbetween

∼

1020

and

∼

1021 Pa.s,which are incompatible withthe very weak as-thenosphere intheerf case,butconsistentwiththeresultsusing either tanh or algebraic parameterizations (Fig. 4). Moreover, the tanh simulations show that mantle viscosity can locally be as low as1019 Pa.sinhighly deforming regions (e.g.platebaseand slab vicinityinFig.5). Thesevaluesmeettheviscosityneededto activateprocesses such asthesmall-scaleconvectionthat is nec-essary to explain: (i) the surface heat flow in old oceanicplates (e.g. Dumoulin et al., 2005);(ii) high surface heat-flow andthin lithosphere observed in many back-arc regions (e.g. Currie and Hyndman,2006;Daviesetal.,2016)ortoaccountforpost-seismic surfacedeformation(Kleinetal.,2016).

Global seismic anisotropy measurements using surface waves pointtostrongradial andazimuthalanisotropy inthelithosphere and asthenosphere and to a decrease in the intensity of the

anisotropy at depths higher than 250 km (Montagner and Ken-nett,1996).Thisobservationmaybeexplainedbyachangeinthe dominantdislocationslipsystemsofolivine,fromdominant

[

100

]

dislocationtodominantglideof

[

001

]

dislocation withincreasing pressureinan uppermantledeformingdominantlybydislocation creep(Mainprice etal.,2005).However, it isalsoconsistent with adecreaseinthecontributionofdislocationcreepwithincreasing depthinaconvectivemantledeformingbyamixtureofdislocation andgrain-sizesensitiveprocesses,representedhereby the imple-mentation of diffusion creep, asin the tanh simulations (Fig. 7), ifoneconsiders that suchprocesses donot produceolivine crys-tal preferred orientations.Yet thisassumption has been recently challengedbynumericalmodels(Wheeler,2009)andcreep exper-iments(Miyazaki etal., 2013) in whicholivine crystalandshape preferredorientationsdeveloped during deformation by diffusion creepathightemperature(

>

1473 K).

Gourietetal.(2019) modeled the rheology of asingle crystal ofolivine well-oriented for dislocation glide. Olivine is only sta-ble up to 410 km. Hence, using the parameterization of Eq. (1) forboth theuppermantleandthetransitionzoneinour simula-tions isan approximation.We try tolimit its effectby analyzing theresultsonlyup to thetime atwhichtheslab penetratesinto thetransitionzone.Thedynamicsofthesimulationswilllikelybe modifiedbyconsideringamorerealistic,higherstrengthrheology in the transition zone (Ritterbex et al., 2016), but the contrasts betweenthesimulations withdifferentparameterizationswill re-main.

4.2.Physicalfeedbacksinsubductiondynamics

Our results demonstrate that distinct subduction regimes (Fig.5)canarisefromsmalldiscrepanciesbetweenrheological pa-rameterizations(Fig.2).Thegeodynamicalsimulationsthusappear to be an appropriate tool for unraveling the first-order physi-cal processes at stake. Twofeedback loops coupled through slab dynamicsaredescribed inFig.9.Theslabsinkingvelocity is con-trolledbothbythebuoyancyoftheslabitselfandbytheviscosity ofthesurroundingasthenosphere(viscousresistance).Afirst feed-back is relatedto the heating andwidening of the slab through thermaldiffusion, which is a time-dependent process. The resis-tance of asthenosphere to slab penetration increases with slab surfacearea,whichisitselfcontrolledbythetime-integrated ther-maldiffusion.Furthermore,temperaturecontrolstheslabdensity contrastandbuoyancyrelative tothe surroundingasthenosphere. Hence, the slower the slab, the larger the diffusion length, the widerandthehottertheslab,andtheslowerthesinking(e.g. al-gebraic caseinFig.5).

A second feedback arises when the mantleviscosity is strain ratedependent,i.e.whena largepartof thetotaldeformation is accommodated through dislocation creep (erf and tanh cases in Fig. 7). In this case, viscous drag driven by the sinking slab in-duceslargestrain rates intheasthenosphere, whichweakens via dislocationcreep and,inturn,acceleratesslabsinking.This rheo-logicalfeedbackisonlyweakly activeinthe algebraic simulation: intheabsence ofasthenosphere weakening,slowsubduction dy-namicsismainlycontrolledbythethermalfeedbackloop(Fig.9). Incontrast,asthenosphericweakeningdueto dislocation creepis very efficientin the erf case: thermal diffusionis too slow rela-tivelytofastslabsinkingvelocities.Forthetanh parameterization, the rheological feedback is substantial, but does not overwhelm thethermalfeedback, becausethe rangeofasthenosphericstrain rates(10−17_–10−13 _s−1_{, Fig. 5) corresponds to intermediate p}

disl

values(30–70%).

Itisinterestingtocalculateameantimeforaslabtoreachthe transitionzone,whichismorethan3 Myratthevelocitiesinferred fromobservations (

<

15 cm/yr). This minimumtime corresponds

Fig. 9. Feedbackloopsbetweenslabsinkingdynamicsandeither(i) slabthermal structure(left)or(ii) asthenosphereviscosity(right).Theirinteractionsdependon theinterplaybetweenthestrainratedependentandstrainrateindependent vis-cosities,e.g.dislocationcreepanddiffusioncreep,respectively,inthesimulations (seesection4.3fordetails).

toaminimumdiffusivelengthof10 kmusingathermaldiffusivity of10−6 m2/sasinoursimulations.

This suggests that the viscosity in the asthenosphere is ulti-matelyconstrainedfromslabsinkingvelocities,whichshouldlead to balanced thermalanddynamical feedbacks loops(Fig. 9). This rulesoutresultsobtainedinthisstudy forerf (i.e.toorapid sink-ing)andalgebraic parameterizations(i.e.tooslow).

4.3. Constrainingeffectivemantleviscosityusinggeodynamical simulations

Thetanh parameterizationprovidesbothasatisfactory descrip-tion of the dislocation creep process in olivine observed in the dislocation dynamics simulations of Gourietetal. (2019) (Fig.2) andlarge-scale subduction dynamicspredictionsthat fulfill many importantobservationalconstraints(section4.1).Theeffective as-thenosphere viscosity,whichiskeyforplatedynamicsasexplained inFig.9, resultsfromcoexistingdeformationmechanisms.In our study, the mild, steady partitioning of dislocation vs. diffusion creep inthe tanh caseensures that the feedbackbasedon strain ratedependentviscosityisactivatedanddoesnotboltout.

However, we acknowledge that subduction dynamics compat-ible with large-scale observables could also arise with different rheologicalparameterizations.Forinstance,diffusioncreep,which seemstocontroltheasthenosphereviscosityandslabsinkingrate (Fig.9),dependsonanassumedgrainsize(e.g.Hirthand

Kohlst-edt,2003).

Furthermore,ourparameterizationofdislocationcreep(Eq. (1)) doesnotfeatureanydepth-dependency,whereaslaboratory defor-mation experiments suggest a dependence on confiningpressure (HirthandKohlstedt,2003).Thisfeatureisaccountedforthrough theterm“ P Vdisl”inthe generalpower-lawexpression,similar to

Eq. (3):

˙

ε

=

APL

σ

_dnexp



−

Edisl

+

P Vdisl R

(

T

+ δ

T

)



(13) withthe depth-dependency oftemperature included inthe term

δ

T asinEq. (10).Verticalprofilesofthedislocationcreepviscosity calculated from Eq. (13) are compared to the sigmoid ones cal-culated from Eq. (1) and Eq. (5) in Fig. 4, for various values of activationvolumesandadiabaticgradients.The higherthe activa-tion volume,the larger therate ofviscosityincrease with depth, whereas the adiabatic increase oftemperature decreases the vis-cosity withdepth.Asfordiffusioncreep (SupplementaryFig. S2), thecombinationofnon-zeroactivationvolumeandthermal gradi-entcanleadtoaquasi-constantviscosityinthemantlebelowthe plates(redcurveonFig.4).Inthisstudy,thepressure-independent rheology derived fromEq. (1) corresponds thus tothe first-order assumption that themean asthenosphere viscositydoes not vary

flow laws.The first one arisesfrom theadjustment ofthe semi-empirical equationsto fit the experimental mechanical data(e.g. Ashby and Verrall, 1978; Bai et al., 1991; Hirth and Kohlstedt,

2003). This affects in particular the pre-exponential factor A in

bothdiffusionanddislocationcreep flowlaws,butalsotheother constants(e.g.activationenergyandactivationvolume)inEq. (10) and (13). There is also a large dispersion in the experimental dataitself,duetodifferencesininitialmaterial(hotpressof fine-powderolivineorsol-gelpure forsterite)andexperimental condi-tions(torsion,axialcompression,temperaturerange),whichcould leadtodifferentassociationsofdeformationprocesses(cf.the dif-ferenceindislocationcreep strengthbetweennaturaldunitesand syntheticolivinepolycrystalsproducedbydifferentmethods–see referencesinDemouchyetal.,2013).

Largeuncertainties alsoarisefromtheextrapolationofthe ex-perimentallyderived parameterizationsover orders of magnitude in strain rateand grain sizes (from microns to millimeters) and fromthe lackof external constraintson the actual valuesof im-portantparameters, such asthe grainsizes inthe asthenospheric mantle.Fordislocationcreep,theformerlimitation hasbeen par-tiallyovercomebytheuseofnumericalmodels,suchasBoioli et al.(2015) andGouriet etal.(2019) to avoidthe extrapolationto geological strain rates.However, thesemodels are still limitedto single crystals. The behavior of a rock, that is a polycrystal, dif-fers:therockwillbestrongerbyafactor2-50, whichdependson theorientation ofolivine crystals,evolving withstrain (e.g.

Tom-masietal., 2000;Mamerietal.,2019).Otherprocessesthanpure

dislocationordiffusioncreep,suchascoupledgrainboundary mi-gration andshear(Mompiou etal., 2009; Cordieretal., 2014) or grain-boundarysliding(e.g.,MaruyamaandHiraga,2017)mayalso playaroleduringthedeformationofapolycrystallineaggregateat depth.

In brief, the enormous experimental investment in the deter-minationofflowlawsfortheuppermantleproducedstrong con-straintsonthedeformationprocesses,ontheequationsdescribing theassociated mechanicalbehavior,andon therangeof physical parameters controlling their activation. However, we are still far fromhaving robust andunique predictions ofits rheology (or of anyotherlayerintheEarth).Thepresentstudyshowsthat geody-namicalmodelsmaybeusedasacomplementarytoolforhandling thequestion.Theycannot discriminatethebestoneamong differ-entparameterizations,buttheycandeterminewhichcombinations of flow laws andtheir parameters produce Earth-like large-scale dynamics,thusconstrainingtheeffectivemantleviscosity.

5. Conclusions

The results of numericalexperiments of dislocation dynamics inolivinehavebeenusedtoderivethreedifferentsigmoid param-eterizations (erf,tanh,algebraic) describing the variation of stress asa function ofstrain rate andtemperature, relevant fora wide rangeofstress andtemperatureconditions prevailinginboth as-thenosphericandlithosphericmantle.Thesedislocation creep pa-rameterizationswere thenimplemented,together withaclassical diffusion creep law, in numerical models of subduction dynam-ics.

While thethree parameterizations fit well the dislocation dy-namicsresults(Fig.2),thetanh parameterizationyieldssubduction

stressesorhighstrainrates.However,mostoftheconvective man-tleinthetanh casedeformsbyamixtureofstrainratedependent (here dislocation creep) andstrain rate independent (here diffu-sioncreep)processesthatmaintainsabalancebetweendynamical and thermalfeedbacks (Fig. 9). The other two parameterizations, erf andalgebraic,canhereberuledoutfromthecomparisonwith large-scale constraints on mantle andplate dynamics,since they are associatedwithtoofastortooslowslabsinking duetoa too largeortoosmallcontributionofdislocationcreeptothetotal de-formation,respectively.

Subduction dynamicsdependsonasthenosphere viscosity. The effective viscosity in the present thermo-mechanical models re-sults from interplay between different deformation mechanisms. The sensitivity to rheological parameterizations is tremendous, yielding distinct dynamics even for deviations within the range of uncertaintyofthe experimental data.However, large-scale ob-servables provide important constraints on plateandmantle dy-namics, which makes the derivation of an effective mantle vis-cosity robust. This viscosity can be reproduced by a variety of combinations of parameterizations of deformation processes. For instance,a unifiedrheology expressionincludingavariablestress exponent n,asproposed byCordier etal.(2012) for lower man-tle MgO and analogue to the apparent n exponent in Fig. S6, is a possible lead to account for strain rate dependent and inde-pendent viscosities at once. We hence recommend testing rheo-logical laws in geodynamicalthermo-mechanical models to eval-uate the interplay and possible feedbacks between different de-formation processes and thermal diffusion; such tests ensuring the large-scale dynamical consistency of the chosen rheological laws.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

Authors thank P. Cordier, P. Carrez and K. Gouriet for fruit-ful discussions. Authors would also like to thank the Applied Modelling and Computation Group (AMCG) at Imperial College London for support with Fluidity. This study was supported by the CNRS-INSU (National Institute of Universe Science) program “TelluS-SYSTER”(2015and2016),andbyfundingfromGéosciences Montpellier. We are grateful to F. Grosbeau, S. Arnal, J. Tack for maintenance and development of the computing cluster at Géo-sciences Montpellier. DRD acknowledges support from the Aus-tralianResearchCouncil,throughFT140101262andDP170100058. The authorsare gratefultoStefanSchmalholzandan anonymous reviewerforconstructivereviews.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi .org /10 .1016 /j .epsl .2020 .116243.

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