Lithos 109 (2009) 47–60

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Thermal evolution of cratonic roots

Chloe Michaut a,1, Claude Jaupart a, Jean-Claude Mareschal b,⁎a Institut de Physique du Globe de Paris, Franceb Geotop-UQAM-McGill, Montréal, Canada

⁎ Corresponding author.E-mail address: mareschal.jean-claude@uqam.ca (J.-

1 Now at Yale University, United States.

0024-4937/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.lithos.2008.05.008

A B S T R A C T

A R T I C L E I N F O

Article history:

Cratons stabilized in the Ar Received 8 April 2008Accepted 15 May 2008Available online 5 June 2008

Keywords:Heat flowThermal evolutionHeat productionArchean cratonsConvective instabilityDelamination

chean because their mantle roots remained strong despite temperatures in theconvecting mantle and amounts of radiogenic elements in the crust and in the lithospheric mantle that werelarger than today. Thermal evolutionary models are developed using constraints from heat flow and heatproduction data in Archean provinces. The large time-scale of diffusive heat transport implies that thelithospheric mantle can remain thermally decoupled from the crust for as long as 1 Gyr. Heat production inthe lithospheric mantle is a key variable in determining thermal conditions that permit stabilization of thecrust and the preservation of a thick cratonic mantle root.Archean provinces are currently characterized by low heat flow, with an average of 41 mW m−2 less than theglobal continental average (56 mW m−2). The range of regionally averaged heat flow values in Archeanprovinces (18–54 mW m−2) is narrower than in Proterozoic and Paleozoic terranes. However, at the end ofthe Archean, when crustal heat production was double the present value, surface heat flow would havevaried over a range (≈45–90 mW m−2) at least as wide as that presently observed in Paleozoic provinces. Thehigh crustal heat production during the Archean is not sufficient to account for elevated lower crustaltemperatures recorded in some metamorphic assemblages without some additional heat input, or withoutthe crust being thicker, or the vertical distribution of radioelements being different from today's.Thermal models and the observed correlation between the degree of upper crustal enrichment and heatproduction values indicate that crustal melting and differentiation were largely driven by heating due to in-situ radiogenic heat production. Even for values of the surface heat flow higher than average in cratons, thecrust can be stabilized before 2.5 Ga if the radioelements are confined to shallow crustal layers. For presentsurface heat flow of 40–45 mW m−2, calculations indicate that, prior to differentiation, the lowermost crustwas near the solidus.Present heat production in the mantle root is constrained by estimates of the mantle heat flow. Furtherconstraints can be obtained by modeling the past thermal regime of the root when heat production washigher. If heat generation is high and/or if the root is thick, the lower lithosphere must have cooled morerapidly than the convecting mantle, which implies that the temperature gradient was inverted at the base ofthe lithosphere and a weak mechanical layer formed in the middle of the root, precluding its survival. A lowtemperature gradient at the base of the root leads to the development of convective instabilities and possibleremoval of the lowermost part.Following its isolation from the convecting mantle and stabilization, thick continental lithosphere adjustsslowly to in-situ heat production and basal heat supply. We show that, due to in-situ heat production,temperatures in the lithospheric mantle may rise above their initial values inherited from the process of rootformation due to in-situ radiogenic heat production.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Cratons that have been stable since the end of the Archean (2.5 Ga),are underlaid by thick cold roots to depth in excess of 250 km.Following the work of Jordan (1975), much research has beenconducted to understand their composition and how they formed

C. Mareschal).

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and stabilized (Pollack, 1986; Abbott, 1991; Griffin et al., 2003; Lee,2006). The presence of thick lithospheric roots beneath the cratonsraises two related questions. How did cratonic lithosphere form in thefirst place? Available evidence indicates that lithospheric roots formedat about the same time the overlying crust stabilized (Richardsonet al., 1984; Boyd et al., 1985; Moser et al., 2001). The second questionis how was this root preserved through geological time? Or did itthicken thermally? The fact that the lithospheric roots are cold impliesthat they are dense and stiff. The high viscosity of the root keeps itfrom being deformed and may help preserving it; on the other hand,

48 C. Michaut et al. / Lithos 109 (2009) 47–60

its density makes it gravitationally unstable. Jordan (1981) thusargued that, for it to survive, the cold mantle lithosphere must becompositionally buoyant. Without this compositional buoyancy, thecratons could not maintain their present elevation above sea level.

Two end member models have been proposed for the formation ofthe lithospheric root: one class of models calls for the formation of theroot in a plate tectonics framework by subduction related processes(Helmstaedt and Schulze, 1989; Abbott, 1991; Bostock, 1998; Kelemenet al., 1998); the alternative model invokes plume related activity(Pollack, 1986; Griffin et al., 2003). Two stage evolutions have alsobeen invoked. For example, Griffin et al. (2003) suggested theimpingement of a mantle plume at the base of a preexisting rootpossibly formed by subduction related processes. The other alter-native is a root made of mantle plume material which gets refertilizedby metasomatic fluids and/or melts generated in subduction zones.

Regardless of the mechanism of root emplacement, the litho-spheric mantle will initially be far from thermal equilibrium. Ifemplaced by subduction processes, the root will be colder thansteady state. The temperature in a root formed from the residue of aplume will follow the plume adiabat and be hotter than steady state.When considering the thermal evolution of the sub-continentallithosphere, we must keep in perspective of the different timeconstants involved: (1) For a thick lithosphere, the thermal relaxationtime is on the order of 1 Gyr; (2) radiogenic heat production has ahalf-life on the order of 2.5 Gyr; and (3) the Earth's mantle cools at arate of about 40–60 KGyr−1. Therefore, the lithosphere could not havebeen close to thermal steady state when it stabilized, and it is nottoday in steady state with the present heat production. In order toestimate the thermal regime of the cratons, we need to know theradiogenic heat production in the crust and lithospheric mantle, andto specify the initial thermal structure and thermal conditions at thebase of the lithosphere. The basal boundary condition depends on theinteraction between the lithosphere and the convecting mantle. Theinitial structure depends on the mechanism of lithospheric rootformation.

Thermal models for the formation of the cratonic roots must alsobe consistent with two seemingly inconsistent geological observa-tions: (1) In some cratons, crustal temperatures were elevated duringthe Archean (Davis et al., 2003), but (2) the rootwas sufficiently cold toremain stable and to allow the preservation of Archean features in themantle (Bostock, 1998; Davis et al., 2003). This inconsistency can bereconciled with thermal evolution models depending on the initialand boundary conditions.

The heat flow data in cratons are now sufficient to provide robustestimates of the crustal heat production and theMoho heat flow. Sincethe review byMorgan (1985), new heat flow and heat production datahave been obtained in several Archean cratons (Kaapvaal, Superior,Dharwar) (Jones,1987,1988,1992; Roy and Rao, 2000;Mareschal et al.,2000; Perry et al., 2006a, and references therein). Lithospherictemperature profiles inferred from heat flow data are in goodagreement with inferences from geothermobarometry on xenoliths(e.g. Russell and Kopylova, 1999; Rudnick and Nyblade, 1999) andseismic velocity profiles (e.g., Poupinet et al., 2003; Shapiro andRitzwoller, 2004). Combined with geothermobarometry data frommantle xenoliths and seismic velocity profiles, heat flow data alsoprovide constraints on heat production in the lithospheric mantle andon the heat flow from the convecting mantle. Heat production in thelithospheric mantle can be constrained much further by consideringthe rundown of the radioelements in thermal evolutionmodels for thelithospheric root. The interpretation of heat flow data and theirimplication for the distribution of radiogenic heat production in thecrust, the heat flow from the mantle and the calculation oflithospheric geotherms have been discussed in numerous papers(Jaupart et al., 1998; Jaupart and Mareschal, 1999; Mareschal andJaupart, 2004; Michaut and Jaupart, 2004; Mareschal and Jaupart,2005; McKenzie et al., 2005; Michaut and Jaupart, 2007).

The long-term evolution and dynamics of thick continentallithosphere depend on a host of processes and physical properties.Most of our knowledge stems from present-day observations and,more importantly, from continents that, by definition, have survivedthe vagaries of mantle convection and instability processes. In thispaper, we focus on thermal structure and how it may have evolvedwith time. We do not restrict the analysis to the present-daycharacteristics and consider wider ranges of physical properties thanthose of the present-day Archean cratons. We thus discuss potentiallyunstable situations leading to destruction or modification of thicklithospheric roots and address the conditions which led to, or allowed,the survival of Archean cratonic lithosphere. For the same reason, wetreat independently properties that are in fact coupled, such as heatproduction and water content in the lithospheric mantle. One way toillustrate the problem of lithosphere stabilization is provided byArchean diamonds, which have been found in the Witwatersrandsedimentary basin, South Africa, and in volcaniclastic rocks of theWawa region of the Superior Province, Canada (Poujol et al., 2003;Stachel et al., 2006). The Wawa diamonds demonstrate thatcontinental lithosphere beneath the Superior Province was depletedand thick before it finally stabilized. They also show, however, that theSuperior lithosphere was not stable yet at the time of the diamond-carrying eruptions, all the more as the very process of diamonddelivery to the surface involved some form of lithospheric thinning ordetachment (Stachel et al., 2006). Thus, onemust consider at least twosteps in the evolution of a continent: the formation of a thick depletedlithospheric root and its early thermal relaxation as it adjusts tointernal heat production and heat supply from below.

In this paper, we shall review the data on heat production in thelithosphere and examine their implication for the thermal evolution ofthe cratonic roots. We shall show that the differentiation of radio-elements is a necessary and, perhaps the most important condition,both for the stabilization of the crust and for the long time survival ofthe cratonic lithosphere. We shall also show the importance of thelong thermal relaxation time of a thick lithosphere in preserving itsmantle root after it is emplaced. Present-day data allow constraints onthermal transients in the past. In particular, we shall study howcontinental roots evolve thermally once they have been isolated fromthe convective mantle.We shall show that theymay undergo an initialheating phase leading to in-situ melting.

2. Stabilization and destabilization processes

2.1. Lithospheric structure

The formation of thick lithosphere involves three different physicaleffects. One is the strong temperature dependence of viscosity, suchthat the upper part of the thermal boundary layer remains stablebecause of its low temperature and large viscosity. This upper layercan be called the rheological layer. Theory, laboratory experiments andnumerical simulations have provided robust scaling laws for thetemperature difference across the active part of the thermal boundarylayer, called the convective boundary layer (Davaille and Jaupart,1993;Solomatov, 1995). This temperature difference, noted ΔTν depends onthe rheological law for the material. Another stabilization factor ischemical depletion due to melt extraction or to reaction with apercolating melt phase, such that the residue is buoyant with respectto undepleted mantle. A third factor is dehydration, which makes theresidue stronger and more viscous than the starting material (Hirthand Kohlstedt, 1996). Because chemical depletion and dehydration areboth achieved by the same process involving a melt phase, they can belumped together.

The continental thermal boundary layer may be split into twodifferent units, with an upper part made of depleted and dehydratedmaterial, called the chemical lid, overlying a lower layer ofundepleted mantle (Fig. 1). This definition relies on composition

Fig. 1. Top: schematic vertical temperature profile through the continental thermalboundary layer. The lithosphere is defined as a rigid layer of thickness L, such that heatfed from the basal convective boundary layer is transported by conduction. Bottom: twopossibilities for the chemical lid made of depleted, strong and buoyant mantle. If thechemical lid is thinner than the rigid layer, it is underlain by a layer of undepletedmantle and the temperature difference across the convective boundary layer (ΔTT) isequal to the rheological temperature difference, ΔTnu, which corresponds to themaximum viscosity contrast for instability. If, on the other hand, the chemical lid isidentical to the rigid layer, it may be thicker than the rheological layer and thetemperature difference across the convective boundary layer, ΔTT, is necessarily lessthan or equal to, the rheological temperature difference, ΔTnu.

49C. Michaut et al. / Lithos 109 (2009) 47–60

alone and one must also consider rheological differences and heattransport mechanisms. From the latter standpoint, one mustdistinguish between the “rheological” upper layer where heat getstransported by conduction and the convective boundary layer. Thetwo types of definition may not coincide and it is useful to evaluatewhy. For this purpose, we call ΔTT the temperature difference acrossthe convective boundary layer, which may or may not be equal to therheological temperature difference, ΔTν. The respective magnitudesof these two temperature differences depend on the lithospherearchitecture. One possibility is that the chemical lid extends over alarger thickness than the rheological layer, implying that ΔTT issmaller than ΔTν. In this case, the chemical lid determines thecharacteristics of sub-lithospheric convection, as studied by Sleep(2003) and Lee et al. (2005). By definition, ΔTT cannot be larger thanΔTν. Thus, if the chemical lid is thinner than the rheological layer,ΔTT=ΔTν, and there may be a sub-layer of undepleted mantle belowthe chemical lid that is not involved in sub-lithospheric convection.In that other case, continental lithosphere thickens following thechemical depletion event.

Which one of these two cases applies to Archean cratons can bedetermined from the respective magnitudes of ΔTν and ΔTT. Both canbe calculated, the former from well-established scaling laws and thelatter from xenolith data, surface heat flow and knowledge of the heatproduction distribution in the crust and in the lithospheric mantle.One key point is that ΔTT may have changed due to modifications of

the distribution of radioactive elements in the continental column, tothermal relaxation following the initial process of root formation, andto secular cooling of the Earth.

2.2. Thermal structure

Following isolation from the well-mixed convecting mantle, thicklithosphere undergoes an early thermal transient as it adjusts tointernal heat sources and heat input from the convecting mantle.Depending on the initial thermal structure, thickness and the amountof heat sources, this initial transient may in principle follow heating orcooling trends. Consider for example a hot initial lithosphere, such asthat which would form out of a mantle plume: it would clearly bebuoyant with respect to the underlying mantle because it would beboth hotter than surrounding mantle and chemically depleted, butcould become negatively buoyant and unstable as it cools down.

It is useful to determine the magnitude of temperature variationsand differences that are relevant in order to evaluate the significanceof the various effects involved. Since the Archean, the Earth has cooleddown by about 150 K (Abbott et al., 1994). Simultaneously, thelithosphere has also been cooling down because its radioactivesources decay and may get redistributed during orogenic or magmaticevents. Clearly, one must ascertain whether such effects implychanges of lithospheric temperatures that are larger or smaller thanthe amount of secular cooling. An alternative method to evaluate thetemperature changes that are relevant is to compare thermal andcompositional buoyancy effects. Various methods indicate that thedensity contrast between depleted lithospheric mantle and unde-pleted convecting mantle below is about 45 kg m−3 today (Doin et al.,1996). For the large thermal expansion coefficient of mantle materialat pressures and temperatures that prevail in the lower part of thelithosphere, such a density contrast is counterbalanced by atemperature difference of 300 K (Schutt and Lesher, 2006).

The present-day thermal structure of continental lithosphere iswell-constrained by xenolith geotherms as well as heat flow data.These data require that the lithosphere is heated frombelowandhencethat ΔTT is positive. This is not necessarily true at all times, however.The well-mixed convecting mantle has been cooling at a rate of about50 K Gyr−1 (Abbott et al., 1994; Jaupart et al., 2007a). Because of heatproduction, however, temperatures at the base of the lithosphere mayhave decreasedmore rapidly depending on U and Th concentrations inthe crust and the lithospheric mantle. This raises the possibility thatthe sub-lithosphericmantlemay have been hotter than the convectingmantle in the Archean. We shall argue against that possibility, anddeduce constraints on heat production and lithosphere thickness.

2.3. Melting and dehydration

Lithospheric roots may be modified through melting after theirinitial separation from the convecting mantle. This may be caused byexternal perturbations such as heating by mantle plumes andhydration from a subducting slab, or by internal processes, such asheating due to in-situ radiogenic heat production. Here, we shall focuson heating due to heat production because it can be calculated directlywithout invoking an external event. At present, Archean continentalroots still contain substantial amounts of water (Peslier and Luhr,2006). The source of komatiitic magmas, which is relevant forcontinents in the Archean, may have had as much as 1 wt.% dissolvedH2O (Grove and Parman, 2004).

2.4. Entrainment of lithospheric material by convection

Lithospheric material may get entrained by convection currentsinto the convecting mantle, implying the gradual erosion ofcontinental roots. This may be achieved at subduction zones, wherethe descending slab acts to drag downwards material at the edge of

50 C. Michaut et al. / Lithos 109 (2009) 47–60

continents (Morency et al., 2002; Lenardic et al., 2003). Currentmodels indicate that a moderate intrinsic viscosity contrast betweenthe lithosphere and the underlying mantle, due to dehydration forexample, can limit the efficacy of this process. Such models do notaccount for island arc formation and onemay note that subduction hasled to the lateral accretion of volcanic arc material against cratoniccores and hence have in fact led to continental growth. Furthermore,subduction can only operate at the edge of a landmass and hence mayleave cratonic cores unaffected. Another erosion mechanism isconvection in the unstable layer at the base of the lithospheric root,where downgoing plumes can entrain small slivers of buoyantdepleted material. Once again, a moderate intrinsic viscosity contrastis sufficient to limit this mechanism.

2.5. Instability of the chemical lid

The chemical lid is cooled from above and hence may be unstable.In contrast to the previous processes, an instability would involvelithospheric material over a large depth extent and over the wholewidth of the continent. Stability against convective overturn in avariable viscosity material depends on a suitably defined Rayleighnumber which must be less than a critical value Rc:

Ra ¼ gαΔT�d3

κ�bVRc ð1Þ

where g is the acceleration of gravity, α is the coefficient of thermalexpansion, κ is thermal diffusivity and νb is the viscosity at the base ofthe lithosphere. Instability cannot affect the most viscous parts of thelithosphere and will be restricted to a lower region where viscosityvariations do not exceed about two orders of magnitude (Davaille andJaupart, 1993). Thus, in, Eq. (1), ΔTν is the “rheological” temperaturedifference across the potentially unstable part of the lithosphere and dis the thickness over which the temperature difference is ΔTν. Here,we are investigating the possibility that part of the chemical lid is infact unstable, and hence that the continental thermal boundary layermay in fact be made of two unstable layers: the chemical lid itself andthe convective boundary layer below the rheological layer. Because ofthe differences in density and viscosity between the convectiveboundary layer and the chemical lid, the two types of instabilitiesoccur on different time-scales and length-scales, with fast overturn inthe former and a slowly-developing instability in the latter. Thispeculiar behaviour has been illustrated by laboratory experiments(Jaupart et al., 2007b).

For our present purposes, it is worthwhile to rewrite the instabilitycriterion (1) using the geothermal gradient at the base of thelithosphere, noted G, which is such that ΔTν=Gd:

Ra ¼ gαΔT4� G

−3

κ�b¼ Rc: ð2Þ

One should note that basal gradient G depends on lithospherethickness and radiogenic heat production in both crust and mantleroot, and hence depends on the local geological and magmatic historyof each province. Thus, G must be determined locally on each cratonand calculated backward in time.

A complete stability analysis of the isoviscous case has beenperformed recently (Jaupart et al., 2007b). These authors considered abuoyant layer with uniform viscosity overlying a fluid with a differentviscosity. This layer is cooled from the top, as in the Earth. Jaupart et al.(2007b) solved the stability problem and compared theoretical predic-tions to laboratory experiments. They showed that the critical Rayleighnumber which separates stable and unstable states is a strong functionof the buoyancy ratio, which is defined as follows:

B ¼ Δρc

ΔρTð3Þ

where Δρc is the compositional density contrast between thebuoyant root and the underlying mantle and ΔρT the negativebuoyancy due to cooling. This study led to two surprising results.One is that the viscosity ratio between the upper and lower layers(lithosphere and asthenosphere) plays a very small role as long as itis larger than 1, which is relevant to cratonic roots. The otherunexpected result is that the critical Rayleigh number (Rac) is verysmall when the buoyancy ratio is less than about 0.3: it may be assmall as 0.28 when the buoyancy ratio tends to zero. The relevanceof this result is best appreciated when compared to commonstability arguments for thermal boundary layers (e.g. Parsons andMcKenzie (1978) rely on values of about 103 for the critical Rayleighnumber). The critical Rayleigh number increases sharply with B andtends to a constant value of about 1200 for BN0.5. This sharpvariation of Rac indicates that stabilization may be achieved bychanging the buoyancy ratio with all other parameters and proper-ties constant.

One concludes from Eq. (1) that lithosphere stability dependsstrongly on the basal temperature gradient and that instability isfavored when this gradient is small. In a steady-state thermal regime,how small this gradient is can be determined from geothermobaro-metry onmantle xenoliths of from the basal heat flow (which may bederived from the surface heat flow and heat production in the crustand mantle root). Here, we shall not be concerned with specificpredictions of the Rayleigh number for the lithosphere, which wouldinvolve many parameters, but will discuss the physical controls onthe value of the basal gradient and how it evolves with time.Assuming for simplicity that there is no heat production in themantle root and that temperatures are in steady state, the basalgradient is equal to:

G ¼ Tb−TmL−zc

ð4Þ

where Tb is the basal temperature and Tm the Moho temperature.Thus, internal crustal differentiation leading to the production of anenriched upper crust, which acts to decrease Tm and hence to increaseG, may play a role in stabilizing the mantle root against convectiveoverturn. If the Moho temperature stays constant, the above equationdictates that the basal gradient decreases with time as the Earth coolsdown which would trigger instability. We shall see that this is notlikely to be the case because of radiogenic heat production in themantle root. Another factor associated with secular cooling is theimplied increase of viscosity. The local Rayleigh number for thelithosphere depends on the basal viscosity, which depends in turn ontemperature in the underlying convective mantle. Thus, secularcooling of the mantle stabilizes roots because it increases theirviscosities.

3. Thermal structure of continental lithosphere: radioactiveheating in crust and mantle root

All physical processes affecting continents depend on the thermalstructure of the lithosphere, which may vary with time due to severalprocesses that are the focus of this paper. Calculating temperatureprofiles in the lithosphere requires the knowledge of the verticaldistribution of the radioelements. As will be shown below, the basicrule is that, for fixed surface heat flow, the deeper the heat sources thehigher the temperature. The relationship is not linear and it is worthpointing out that the effect on temperature increases as the square ofthe depth of the heat sources. Vertical differentiation in the crust andthroughout the lithosphere is the key controlling parameter onthermal stability of the crust. In this section, we present simplecalculations of the steady-state temperature effects due to crustal andlithospheric mantle heat sources to illustrate that differentiation ofthe radiogenic elements is crucial to stabilize the crust and the

Table 1Crustal heat production and differentiation index in different Precambrian provincesworldwide

Province NA bAN±σA Ac DI References

μW m−3 μW m−3

CanadaSlave 20 2.1±1.0 1.00±0.18 2.1±0.5 (1)Superior 64 0.76±0.8 0.63±0.10 1.2±0.1 (2)Trans Hudson Orogen 44 0.73±0.5 0.55±0.10 1.1±0.2 (3)Lynn Lake Belt 9 0.72±0.5 0.35±0.1 2 (3)Flin-Flon Snow Lake Belt 15 0.32±0.2 0.62±0.1 0.5 (3)Grenville 0.8† 0.65±0.12 1.3±0.2 (4)Appalachians 50 2.6±1.9 1.05±0.11 2.5±0.2 (4)

South AfricaKaapvaal granite domes (Archean) 7 2.3±0.8 0.5 4.6 (5)Namaqa belt (Proterozoic) 10 2.3 1.1 2 (6)

AustraliaWestern Shield (Archean) 12 2.6±2.1 0.9 3 (7)Central Shield (Proterozoic) 38 3.6±1.9 1.5 2.4 (7)

FennoscandiaEastern Finland (Archean) 100 1.2±0.4 0.6 2.0 (8)

Mean surface heat production±standard deviation (bAN±σA), Ac mean crustal heatproduction, and DI differentiation index.† Area-weighted averageReferences:(1) Mareschal (2004) and references therein;(2) Perry (2006a,b) and references therein;(3) Mareschal (2005) and references therein;(4) Mareschal (2000) and references therein;(5) Jones (1988);(6) Jones (1987);(7) Cull (1991);(8) Kukkonen and Lahtinen (2001).

51C. Michaut et al. / Lithos 109 (2009) 47–60

lithospheric mantle. For our present purposes, one maywrite the totaltemperature difference across the continental thermal boundary layeras the sum of four components:

ΔT ¼ ΔTb þ ΔTT ð5Þ

¼ ΔTc þ ΔTm þ ΔTQ þ ΔTT ð6Þ

where ΔTT has already been defined and ΔTb is the temperaturedifference across the stable layer above the convective boundary layer.The latter can be split into three components due to heat production inthe crust (ΔTc), heat production in the lithospheric mantle (ΔTm) andheat supply from the convective boundary layer (ΔTQ). One has:

ΔTQ ¼ QbLk

ð7Þ

where Qb is the convective heat flux at the base of rheological layer,L the rheological layer thickness and k thermal conductivity hereassumed constant. ΔT is determined by the temperature of the well-mixed mantle. Thus, it is known today and may be determined inthe past using various methods (Abbott et al., 1994; Jaupart et al.,2007a). The key point is that ΔTc and ΔTm decrease with time due toradioactive decay in a conduction regime, whereas ΔT also decreaseswith time, but in a convective regime. Thus, they may not decreaseat the same rate, with important consequences for lithosphericstructure and behaviour.

3.1. Crustal component

In steady state, the temperature at Moho can be broken down intotwo components:

TM ¼ QMzck

þ ΔTc ð8Þ

where QM is the Moho heat flow and zc is the crustal thickness. Forundifferentiated crust, the latter is:

ΔTc ¼ Acz2c2k

ð9Þ

where Ac is the heat production. Because lattice thermal conductivitydecreases with temperature, including the temperature dependenceof k increases Tm.

The above result overestimates crustal temperatures because itrelies on the assumption of a homogeneous crust. In reality,continental crust is stratified in radiogenic heat production, suchthat the upper crust is enriched with respect to the lower crust. It isnot possible to determine the vertical distribution of radioelements inthe crust for lack of sufficient data. To get around this problem, Perryet al. (2006a,b) noted that one can obtain constraints on the totalamount of heat generated by radioactivity in the crust. Once this totalis fixed, a two-layer crustal model allows temperature calculationsthat are sufficiently accurate for most purposes: more complicatedcrustal models involving more layers or a continuous verticaldistribution of radioactivity lead to temperature changes of less than40 K in most circumstances. In such conditions, one needs four inputsfor a crustal thermal model. Heat production in the upper crust isdeduced from surface measurements. Crustal thickness zc and surfaceheat flow Qs can be determined by standard techniques. The Mohoheat flow Qm can be estimated using various geophysical, geochemicaland petrological constraints (Jaupart and Mareschal, 1999). One maythen determine the average crustal heat production Ac from heat flowvalues at the surface and at the Moho:

Ac ¼ Qs−Qm

zc: ð10Þ

The ratio of the surface heat production As to the average crustalheat production Ac defines the differentiation index DI (Perry et al.,2006a,b):

DI ¼ As

Ac: ð11Þ

The differentiation index is usually N1, unless depleted surface layershave been emplaced tectonically or deposited over a more radiogenicbasement. Table 1 gives estimates of crustal heat production anddifferentiation index for different Precambrian provinces worldwide.

For a stratified crustmodel,we consider two layers. h is the thicknessof the enrichedupper layer andAs=DI×Ac is its heat production.Wemayfirst assume for the sake of example that the lower layer has zero heatproduction, an extreme case of differentiation. In this case:

h ¼ zcDI

ð12Þ

and the crustal component of mantle temperature is:

ΔTc ¼ 1DI

Acz2c2k

ð13Þ

which illustrates the impact of crustal differentiation on the Mohotemperature. The prediction of the homogeneous model is recoveredfor DI=1 and one can see clearly that increasing DI decreases ΔTc. Amore general two-layer model has non-zero heat production in thelower layer and an upper layer which is thinner than zc/DI. In this case:

ΔTc ¼ DIAch2

2kþ Ach z2c−h2

� �2k

1−hDI

zc

� �: ð14Þ

Note that this equation is only valid if DIh /zcb1 (or, equivalently,hbzc /DI), which corresponds to the condition that heat production inthe lower layer is larger than zero. For a homogeneous crust with

Table 2Heat production estimates for the sub-continental mantle lithosphere

Sample Heat production rate Reference

μW m−3

For comparisonBulk silicate earth 0.014–0.020 (1)

Peridotite samplesOff-craton, massif peridotite 0.018 (2)Off-craton, spinel peridotite 0.033 (2)

Average values for xenolith suites worldwideOn craton, all 0.093 (2)On craton, kimberlite hosted 0.104 (2)On craton, non-kimberlite hosted 0.028 (2)

Jericho kimberlite, Slave Province, CanadaLow T xenoliths 0.15–0.27 (3)High T xenoliths 0.096–0.461 (3)

Inversion of (P,T) arraySlave Province b0.025 (3), (4)Kaapvaal craton b0.03 (4)

References:(1) Palme and O'Neil (2003).(2) Rudnick et al. (1998).(3) Russell et al. (2001).(4) Michaut et al. (2007).

52 C. Michaut et al. / Lithos 109 (2009) 47–60

Ac=0.65 μW m−3 and k=3 W m−1 K−1, crustal heat sources contribute26 mW m−2 to the surface heat flow and 140 K to the temperature atMoho and in the mantle. For a differentiated crust with DI=2.5 andh=10 km, the contribution of crustal heat production to Mohotemperature is only 80 K. These numbers were double at 2.5 Gawhen heat production was double the present. More accuratecalculations relying on realistic crustal models have been developedfor the Superior Province and compared to other geophysical data byPerry et al. (2006a).

Fig. 2 illustrates how the temperature at Moho at 2.5 Ga dependson the differentiation index for different values of the surface heatflow referred to present values.

3.2. Mantle component

Heat production in the lithospheric mantle is often neglectedwhen estimating the geotherm. It is too small to cause sufficientcurvature of the temperature profiles and be resolved from xenolithsgeothermobarometry. Heat production measurements in mantlexenoliths have suggested a high value (0.09 mW m−3) (Rudnicket al., 1998). This value is too high because the inferred geotherm doesnot intersect the mantle adiabat. Whether the xenolith values arerepresentative of the lithospheric mantle over long time intervals hasthus been questioned. Values around 0.02 μW m−3 have beendetermined from large orogenic peridotite exposures and have beenadopted for cratonic roots (Rudnick et al., 1998; Russell et al., 2001).Michaut et al. (2007) have obtained a range 0.0–0.04 μW m−3 asconsistent with geothermobarometry in xenoliths from the Kaapvaaland Superior cratons. Table 2 gives some of the estimates of heatproduction in the continental lithospheric mantle from variousstudies.

The steady-state temperature at depth z due to heat production Am

in the lithospheric mantle is:

Tm zð Þ ¼ Am L−zcð Þzk

−Am z−zcð Þ2

2kð15Þ

where L is the thickness of the lithosphere. For L=240 km and Am=0.02 μW m−3, the lithospheric mantle contributes 4 mW m−2 to thesurface heat flow, and for k=3Wm−1 K−1, it contributes ΔTm=185 K tothe temperature at the base of the lithosphere (more than the crust).Although the present heat production in the lithospheric mantle doesnot contribute much to the heat flow, it affects mantle temperatures.

Fig. 2. Temperature at the base of the crust at 2.55 Ga as a function of present surfaceheat flow for different values of the differentiation index. Mantle heat flow value is15 mW m−2 at present and at the end of the Archean.

The effect on mantle temperature and heat flow was higher at 2.5 Ga,when the heat production was double the present. Because of therundown of the radioelements, the lowermost lithosphere could becooling at a rateN50 K Gyr−1.

4. Heat flow and heat production in Archean cratons

4.1. Observations

We shall only summarize the main results of heat flow studies instable cratons.

• The mean heat flow in Archean provinces is lower than in youngerprovinces, but this difference must be qualified. Pollack et al. (1993)have reported that the global average of heat flow values for Archean“tectonothermal age” is 51 mW m−2 while Nyblade and Pollack(1993) report a global Archean average of 41 mW m−2, comparedwith a mean continental heat flow of 61 mW m−2. This globalcontinental average includes the contribution of the young andactive tectonic provinces where the transient thermal regime isassociated with surface heat flow exceeding 80 mW m−2. Excludingthe very young provinces, the difference between Archean andyounger provinces is small. For example, between the stableprovinces of the Canadian Shield, there is no variation of the meanheat flow with age (e.g., Perry et al., 2006a).

• Within each craton, the small horizontal scale of the heat flowvariations requires that they originate in the crust (Mareschal andJaupart, 2004). Within each craton, variations in the heat flow fromthe mantle are b3 mW m−2. Different methods have been used tocalculate the heat flow at the base of the crust in the cratons. Theyinvariably lead to estimates in the range 11–18 mW m−2. As seen inSection 3.1, these variations in heat flow have little effect on mantletemperatures.

• Heat flow varies between different cratons (Table 3). Thesedifferences may in part be due to small (≤3 mW m−2) differencesin the heat flow from the mantle.

• Heat flow varies on regional scale within cratons (Table 4). Regions(200×200 km)with very low heat flow (e.g. ≤18mWm−2) are foundin several cratons. This confirms that the heat flow at the base of thecrust in these cratons must be b18 mW m−2. Regional average heat

Table 3Mean heat flow and heat production in different Archean provinces in the world

bQN σQ NQ bAN σA NA

mW m−2 μW m−3

Slave 50 10 2 2.3 1 20Superior (N2.5 Ga) 42±2 12 57 0.95±0.15 1. 44Superior (excluding Abitibi) 45±2.4 12 31 1.4±0.26 1.2 22Abitibi 37±1 7 26 0.41±0.07 0.33 22

Western Australia 39±2.1 7 27 2.7 2.5 13Yilgarn Block 39±1.5 7 23 2.7 2.5 13Pilbara Block 43±1.5 3 4

South AfricaKaapvaal granite domes 33±0.8 2 7 2.3 0.8 7Witwatersrand 45±1.8 6 10(Witwatersrand† 51±0.7 6 81)Lesotho 61±5.3 16 9

Siberian Shield 38±1.2 14 143Baltic Shield 34±1.5 12 60Ukraine 36±2.4 8 12 0.9 0.2 7

bQN mean heat flow±one standard error, σQ standard deviation on the distribution, NQ

number of sites, bAN mean heat generation±standard error, σA standard deviation, NA

number of heat production values. Each value is based on many samples.† Including heat flow estimates from bottom hole temperature.

53C. Michaut et al. / Lithos 109 (2009) 47–60

flow values can be as high as 54 mW m−2. If a geotherm could bedirectly inferred from regional surface heat flow as proposed byPollack and Chapman (1977), variations in the mantle roottemperatures and thicknesses would be very large, much largerthan suggested by seismic velocity variations or geothermobaro-metry on xenoliths. Neither age, nor surface heat flow are sufficientto determine the temperature profile of the lithosphere (Mareschaland Jaupart, 2004).

• The heat generation in the lithospheric mantle is poorly constrainedby the data. Heat production measurements on mantle xenolithsyield values that are too high (0.09 μW m−3) to be plausible.

• The average heat production in theArchean crust is 0.55–0.75 μWm−3

(Jaupart and Mareschal, 2003). Crustal heat production varies muchbetween different Archean provinces because of their distinctmagmatic and tectonic evolutions. At the time of crustal stabilization,some Archean cratons had much higher heat production thanProterozoic and Paleozoic provinces.

• The bulk silicate earth (BSE) has an average heat production ofb0.02 μW m−3 (McDonough and Sun, 1995). The average heatproduction of Archean crust is much larger than this, implying thatits radioactive elements have been extracted from a volume of theBSE that is many times larger than the volume of the mantle root.Thus, the continental crust/mantle root system cannot be consid-ered as a closed system and is best viewed as open to melt and/ormetasomatic fluids carrying heat producing elements (Carlson et al.,2005).

Table 1 lists values of the differentiation index (defined in Eq. (11)above) for several Archean and Proterozoic provinces. There is a well-marked correlation between DI and the crustal heat production (Perryet al., 2006a), which suggests that crustal differentiation is largelydriven by in-situ heat production, i.e. that it is weakly sensitive to the

Table 4Regional variations of the heat flow in different cratons

Minimum Maximum

mW~m−2

Superior 22 48Australia 34 54Baltic 15 39Siberia 18 46

Minimum and maximum values obtained by averaging over 200 km×200 kmwindows.

deeper thermal regime. Thick lithosphere effectively isolates the crustfrom mantle perturbations and it would take large changes of heatsupply at the base of the lithosphere to generate high crustaltemperatures. Furthermore, after root formation, it takes a very longtime for the crust to equilibrate thermally with a thick mantle root.The initial stages of crustal evolution are decoupled from tempera-tures deep in the root. We shall return to these points below.

4.2. Archean crustal geotherms

High temperature metamorphic conditions have been reported indifferent Archean cratons. For instance, in the Slave Province, Canada,temperatures in the range 650–800 °C have been reported forpressures 0.9–1.1 GPa during the neo-Archean (2.55 Ga) (Thompson,1989; Davis et al., 2003). Present crustal heat production in the Slave ishigher than in other Archean cratons (Thompson et al., 1995;Mareschal et al., 2004). At 2.55 Ga, when crustal heat productionwas double the present, surface heat flowwas ≈85mWm−2. Althoughthe present heat production is high in the Slave craton (Mareschalet al., 2004), the crust is highly differentiated with DI≈3 (Table 5).

We have calculated how the temperatures in the lower crustdepend on surface heat flow and crustal differentiation for the SlaveProvince at 2.5 Ga for differentiated or undifferentiated crust. Over therange of crustal temperatures, the thermal conductivity can vary by asmuch as 50%. In order to calculate a geotherm, we have thus used thefollowing law for the thermal conductivity of the crust (Durham et al.,1987):

k ¼ 2:26−618:241

Tþ k0

255:576T

−0:30247� �

ð16Þ

where k is thermal conductivity (in W m−1 K−1), T is the absolutetemperature and k0 is the thermal conductivity at the surface (forT=273 K). The values used are based on measurements on samplesfrom the Superior Province. Measurements on lower crustal granuliteshave yielded values that fall within the same range.

Fig. 3 compares the present temperature profile for the SlaveProvince with the profiles at 2.5 Ga for differentiated and non-differentiated crust, with temperature dependent thermal conductiv-ity. In steady state for a differentiated crust, the temperature at Mohodepth remains ≤750 °C while it exceeds 900 °C for a non-differentiated crust. Several authors have explained the elevatedtemperatures by invoking additional heat input following delamina-tion of the lithospheric mantle root or underplating of hot magmas atthe base of the crust. Alternatively, if the crust is not differentiated,elevated temperatures are reached in the lower crust without anyadditional heat input (Mareschal and Jaupart, 2005) (Fig. 3). The pointis not to argue against heat input from the mantle, but to argue thatcrustal differentiationwas a necessary condition for crustal stability inArchean provinces with higher than average heat flow (Sandiford andMcLaren, 2002).

To avoid lower crustal melting, temperatures must be below≈800 °C. The stability of the crust depends thus on two conditions: the

Table 5Heat production models for the Slave Province

H A0 A0×H Q0 A2.6 A2.6×H Q2.6

km μW m−3 mW m−2 mW m−2 μW m−3 mW m−2 mW m−2

Slave(Lac de Gras)

10 1.7 17 3.4 3410 1.2 12 2.4 2420 0.4 8 0.8 16

50 87 (94)

H: layer thickness, A0: present heat production, A2.6 Archean heat production, presentsurface heat flow, Q0, Archean surface heat flow, Q2.6 (with 20 mW m−2 mantle heatflow).

Fig. 5. Evolution of the lithospheric geotherm with time, when the basal temperaturedecreases with time at a rate of ≈40 K Gyr−1. Present-day values of crustal and mantleheat production are 0.8 and 0.04 μW m−3 respectively.

Fig. 3. Temperature profiles for the Slave Province now, and for a differentiated andundifferentiated crust at 2.55 Ga demonstrating the effect of crustal differentiation oncrustal temperatures.

54 C. Michaut et al. / Lithos 109 (2009) 47–60

surface heat flow and the differentiation index. Fig. 4 summarizes thevalue of present crustal heat flow and differentiation index thatpermitted crustal stabilization at 2.5 Ga.

5. Secular cooling of the lithosphere

5.1. Archean lithospheric geotherms

Presently the crustal and the mantle heat generation componentscontribute 250–420 K to the temperature at the base of thelithosphere. This contribution was double in the Archean, i.e., thetemperatures in the lowermost lithosphere have decreased by at least

Fig. 4. Conditions for lower crustal melting at 2.5Ga. Reaching temperature higher than800°C depends on the differentiation index as well as heat flow. Heat flow Q0 refers tothe present heat flow. For these calculations, mantle heat flow is kept constant and thecrustal component is double the present. Average surface heat flow and differentiationindex estimated in different provinces of the Canadian Shield are displayed as triangle.

250 K since the end of the Archean, i.e. at a rate of about 100 K Gyr−1.This must be compared with the secular cooling of the mantle below.If the mantle cools more slowly that the lithosphere (i.e. at arateb100 K Gyr−1), it is possible that the lithosphere was hotterthan the mantle below and that the geotherms were inverted in thelowermost lithosphere. Whether this indeed occurred depends on thepresent heat generation rate in the lithosphere Am and lithosphericthickness L as well as the cooling rate of the whole mantle. Theargument has been greatly simplified because we did not consider thethermal relaxation time for the lithosphere. Including the relaxationtime, Michaut and Jaupart (2004, 2007) have shown that continentalgeotherms take a different shape and cannot be in equilibrium withthe instantaneous heat sources. One must therefore account forsecular cooling in thermal calculations.

Fig. 5 shows how lithospheric geotherms have evolved throughtime for a specific set of parameters. Calculations are done byspecifying how the basal temperature decreases and by running thethermal model backward in time. The heat production values that arequoted correspond to present-day values, which facilitates compar-isons with measurements. For the example of Fig. 5, the present-daygeotherm is not in equilibriumwith today's heat sources and developssignificant curvature. Comparison between such a transient geothermand xenolith (P,T) data is discussed in Michaut et al. (2007). As we gofurther back in time, the effect of radiogenic heating is to reducesignificantly the temperature gradient in the lowermost lithosphere.For ages older than 2 Ga, the geotherm turns within the lithosphere,such that the basal heat flux is negative, i.e. such that the lithosphereloses heat to the underlying mantle.

For such “turning” geotherms, strength has a marked minimumwithin the lithosphere, as shown in the next section. The low strengthhorizon persists over a length of time that scales with the diffusiontime-scale for thick lithosphere, which is typically as large as 1 Gyr.This is much longer than the tectonic deformation time-scale (1015 s,or about 30Myr), implying that lithospherewith a “turning” geothermwould lose its lower part through delamination. Furthermore, turninggeotherms are characterized by high temperatures at mid-depthswithin the root, andmay cross the hydrous solidus and the carbonatitefield depending on the volatile content of lithospheric material.

“Turning” lithospheric geotherms probably lead to melting anddepletion of the root in radioactive elements, and to delamination andthinning of the root. Both processes would operate until thelithosphere is thin and depleted enough to be stable. Thus it maywell be that continental roots have achieved stability after the initialextraction process that isolated them from the convecting mantle andwithout any further perturbation from the mantle. Michaut andJaupart (2007) have defined the range of lithospheric thicknesses andheat generation that do not lead to inverted geotherms. Fig. 6 shows

Fig. 6. Range of lithospheric mantle heat generation and lithospheric thickness that leadto inverted temperature gradients at the base of the lithosphere. Qb is the heat flux atthe base of the lithosphere and Ac0 is the present-day average crustal heat production.Tb is the temperature at the base of the thermal boundary layer, which decreases withtime due to secular cooling of the Earth. For the function chosen here (where t0 standsfor today, the secular cooling rate is 40 K Gyr−1, which is consistent with theobservations (Abbott et al., 1994; Jaupart et al., 2007a).

Table 6Creep parameters for lithospheric materials used in calculating the strength of thelithosphere (Carter and Tsenn, 1989; Ranalli, 1995; Lee et al., 2005)

A (MPa−n s−1) n H (kJ mol−1)

Upper crust (dry granite) 1.0×10−7 3.2 144Lower crust (mafic granulites) 1.4×104 4.2 445Mantle (dry dunite) 4.85×104 3.5 535Mantle (wet) 4.89 106 3.5 515

The activation volume is assumed to vary linearly from 15×10−6 at surface pressure to7.5×10−6 m3 mol−1 at 8 GPa.

Fig. 7. Yield strength envelope (YSE) of the lithosphere at different times (present, 1, 2and 3 Ga) for a dry mantle. The envelope is calculated for the temperature profiles in 5and the rheological parameters given in Table 6.

55C. Michaut et al. / Lithos 109 (2009) 47–60

the range of values of Am and L that lead to standard or invertedgeotherms for present crustal heat production Ac=0.8 μW m−3 and amantle secular cooling rate of 40 K Gyr−1. If heat generation in themantle lithosphere is N0.03 μW m−3, an inverted geotherm wouldhave occurred for lithosphere thicknessN280 km (Michaut andJaupart, 2007). These calculations with a constant lithospherethickness are presented to illustrate that the cratonic root could nothave been stable if heat production is high (N0.03 μW m−3) in thelithospheric mantle. One way out of this conundrum is to considerthat the lithosphere thickened with time. However, a thinner andhotter lithosphere is not in the diamond stability field, which may bein conflict with the presence of Archean age inclusions in diamonds(Richardson et al., 1984; Boyd et al., 1985).

5.2. Mechanical strength

Thick continents are generated by the stacking of lithosphericsegments or by underplating processes which leave specific structuresand interfaces in the crust and mantle roots. Strong deformationwould obliterate such structures. Furthermore, the lithosphere mayget thinned due to shear imparted by the convective mantle. Thesephenomena depend on stresses that can be generated by convectionand on the mechanical strength of the lithosphere. The former cannotbe estimated with confidence, but the latter can be.

Crustal rocks usually deform by power law creep (Ranalli, 1995):

σ ¼:e1=n

A1=n exp E þ PV⁎ð Þ=nRTð Þ ð17Þ

where ε̇ is the strain rate, σ the deviatoric stress, A and n are constantscharacteristic of the material, E is the activation energy, V⁎ theactivation volume, R the gas constant, P is pressure, and T thethermodynamic temperature. We compared the strength withrheological parameters for dry or wet mantle and a wet crust(Table 6). In particular, the rheology of lower crustal samples fromthe Superior Province, including the Pikwitonei and Kapuskasinggranulites, has been studied in the laboratory (Wilks and Carter,1990).At low temperatures, very large stress is required to maintain steady-state creep, and deformation occurs by frictional sliding on randomlyoriented fractures, leading to a linear increase in deviatoric stress withdepth known as Byerlee's Law (Byerlee, 1978; Brace and Kohlstedt,

1980). The shear stress τ to overcome friction is proportional to thestress normal to the plane of fracture:

jτj ¼ fσn ð18Þwhere f is the coefficient of friction, and σn the effective normal stress(i.e. lithostatic less the fluid pore pressure, usually assumed to behydrostatic). The coefficient of friction was determined to be 0.85 forσnb200 MPa. For horizontal compression, where the maximumprincipal stress is horizontal and the minimum is vertical, and thedip of thrust faults is ≈30°, the deviatoric stress is:

δσ ¼ 1:3σn σnb200 MPa ð19Þ

δσ ¼ 0:85σn þ 90 σnN200 MPa: ð20Þ

A lower deviatoric stress is needed in extension than in compres-sion. The minimum stress needed to maintain a given deformationrate (typically 10−15 s−1) either by frictional sliding or steady-statecreep defines the yield strength envelope. Yield strength envelopeshave been calculated for the lithosphere at different times for a wetand for a dry mantle rheology for the lithospheric parameters used inFig. 5, which, as argued before, do not allow stability. Figs. 7 and 8demonstrate that bothwetness and temperature have a large effect onthe strength. There is a two order of magnitude difference in strengthbetween wet and dry mantle at any time. Another point is that thestrength of a “hot” Archean mantle is two order less than today'smantle. The stresses acting in the lithospheric mantle are poorlyknown. Bokelmann and Silver (2002) suggest that the stress at thebase of the cratonic lithosphere is between 0.7 and 2.5 MPa. For thesake of the argument, we shall arbitrarily fix the stability threshold ata value of 1 MPa. For a dry rheology, the strength of today's mantle

Fig. 8. Yield strength envelope as in Fig. 7 for a wet mantle.

Fig. 9. Depth where the mantle strength falls below 1 MPa as a function of timecalculated for the examples of Figs. 7 and 8.

56 C. Michaut et al. / Lithos 109 (2009) 47–60

remains higher than the threshold above 250 km, but it was less than150 km at 2.5 Ga (Fig. 9). The difficulty to preserve the lithosphericmantle root is even greater for a wet rheology because the thresholdstrength is at depth shallower than 200 km today and less than100 km during the Archean. Dehydration was necessary to form astrong lithospheric keel that could have thickened by cooling. Thesecalculations emphasize the effect of a small heat production in themantle on the stability of the mantle root. The value used for thecalculations is within the uncertainty range and less than measure-ments on xenolith samples. Stability for a thick lithospheric rootrequires lower heat generation values than those used for thesecalculations. Another point that will be addressed in the next sectionconcerns the initial conditions when the root became stable.

The total strength of the lithosphere is defined as the integral of theyield stress envelope over the thickness of the lithosphere (Fig. 9). Atpresent, the crustal contribution to the integrated strength isrelatively smaller than that of the mantle (wet or dry). During theArchean, the crustal contribution to the total strength was relativelymore important. For a dry mantle rheology, the strength of the mantleand crust were comparable; for a wet rheology, the mantle wasweaker than the crust.

The geologic record shows that cratons have experienced very littledeformation after they stabilized, suggesting that the stress availableare insufficient to overcome their total strength. The magnitude of thestress available for intraplate deformation is poorly known, but wemay estimate it from examples of localized intracratonic deformationthat took place away from plate boundaries. One such example is theKapuskasing Structural Zone (KSZ), located in the Superior Province ofthe Canadian Shield. In the KSZ, crustal shortening took place withinthe stable craton, with no sign of deformation in the surroundingregions, possibly as a consequence of the Trans Hudsonian orogeny1500 km to the west (Percival, 1994). Perry et al. (2006b) suggestedthat the integrated lithospheric strength was the control parameterthat allowed localized deformation to occur there. The KSZ and theadjacent regions have been extensively sampled for both heat flowand heat production. Complementary information on crustal structureand mantle heat flow is provided by geophysical surveys and mantlexenoliths. These data can be used to determine the thermal structureand the strength of the Superior craton at the time of KSZ uplift anddeformation, and hence the strength which allows tectonic deforma-tion of an Archean craton. In the Superior Province, present litho-spheric strength could vary laterally by more than one order ofmagnitude due to differences in crustal heat production. Far from theKSZ to the east, the lithosphere of the Abitibi sub-province is strongbecause of low heat flow and crustal heat production. On both sides ofthe KSZ, thewestern part of the Abitibi (to the east) and theWawa belt

(to the west) are much weaker (Mareschal et al., 2000). Thus, theregion around Kapuskasing is shown to have been weak before theonset of deformation and uplift, explaining why deformation wouldhave been preferentially concentrated there (Perry et al., 2006b). Atthat time (1.9 Ga), the strength of the lithosphere at Kapuskasing was~1013 N m−1, which corresponds to an average stress≈60 MPa. Bycontrast, the integrated strength in the eastern Abitibi at the sametime was ~1015 N m−1. The variations in lithospheric strength as afunction of age (Fig. 9) suggest that the cratonic lithosphere hasremained strong (under compression) since 2–2.5 Ga. Crustaldeformation such as observed at Kapuskasing was possible onlybecause locally the crust was very radioactive with surface heat flowin excess of 75 mW m−2.

6. Transient lithospheric evolution following root formation

The calculations above have addressed the secular evolution of thelithosphere due solely to the rundown of radioactivity with time.Because of the character of the heat equation, it is impossible tocompletely determine this evolution as the initial conditions have notbeen preserved in the present geotherms (which we know imper-fectly). The discussion in the section above does not account for theinitial thermal transient that followed the root formation. Only afterthe decay of that transient, will the geotherm reach the secular coolingregime.

6.1. Heating of a root by crustal heat production

Steady-state geotherms are adequate to discuss the temperatureregime in the crust, because the thermal equilibration time is shortrelative to the half-life of heat production. This does not hold for thelithospheric root, however. If the root is emplaced cold, itstemperature will rise because of the crustal heat production. Heatingof a thick root, however, occurs on the same time-scale as the decay ofheat producing elements. Crustal heat production decreases substan-tially before the base of the root is heated. Thus the initial temperature“pulse” due to high crustal heat production will never reach the baseof the root. Mareschal and Jaupart (2005) have shown that dependingon lithospheric thickness, maximum temperature at the base of theroot will be reached 1–1.5 Gyr after emplacement and the amplitudeof the temperature pulse is attenuated to 60–70%. This is illustrated byFig. 10 which shows how the temperature at the base of the

Fig. 10. Heating the base of the lithosphere by crustal radioactivity. λ is the decay constantfor crustal radioactivity, τ=L2/κ is the thermal relaxation time for the lithosphere.Temperature is normalized to the initial temperature at the base of the crust.

Fig. 12. Heating of the mantle lithosphere after root emplacement at 3.5 Ga. The initialthermal structure corresponds to the stacking of two thin lithospheres. Ac0 and Am0

stand for the present-day values of heat production in the crust and lithospheric mantle,respectively. In these calculations, heat production decays according to the rundown ofradioactivity. The temperature at the base of the lithosphere is imposed and decreasesat a rate of 40 K Gyr−1.

57C. Michaut et al. / Lithos 109 (2009) 47–60

lithosphere increases due to heating by crustal radioactivity. Thedetails of the model calculation are provided in Appendix A. Fig. 11shows how the heating due to crustal radioactivity propagates withtime in the root. Archean crustal heat production is here assumed tobe 2.2 μW m−3, and thermal conductivity for the crust and mantle is2.5 W m−1 K−1. In these calculations the boundary condition at thebase of the lithosphere is one of constant flux, which explains the longthermal equilibration time. Quasi steady state can be reached morerapidly with a basal temperature boundary condition, as will bediscussed next.

6.2. Initial thermal transients: cooling or heating of a root?

Fig. 12 illustrates how the continental geotherm develops for aspecific choice of parameters starting from an hypothetical tempera-

Fig. 11. Heating the mantle lithosphere by crustal radioactivity. Temperature increase inthe root as a function of time after root emplacement and depth. The crust is 40 km thickand has heat generation rate 2 μW m−3 at 2.7 Ga.

ture profile 3.5 Ga ago. The initial temperature profile is an idealizedrepresentation of the stacking of two lithospheres, corresponding tothe generation of continental roots at subduction zones. The basaltemperature is decreasing at a rate of 40 K Gyr−1 and its present-dayvalue is 1372 °C. Present-day values of heat production in the crustand in the mantle root are 0.8 μWm−3 and 0.04 μWm−3, respectively.With this value of present heat production, mantle material at 3.5 Gawas heated at a rate of ≈70 K per 100 Myr. At 3.5 Ga, radiogenic heatproduction has already heated the lithosphere whereas secularcooling of the mantle has no detectable influence. At 2.5 Ga, thegeotherm is almost turning. After 1 Ga, heat production has decayed

Fig. 13. Cooling of the mantle lithosphere after emplacement of a hot root at 3.5 Ga. Theinitial thermal structure corresponds toahotmantle isentropebelowa thin lithosphere. Thetemperature at the base of the lithosphere is imposed and decreases at a rate of 40 KGyr−1.

58 C. Michaut et al. / Lithos 109 (2009) 47–60

significantly and the lithosphere is cooling down in the bulk withlittle influence of the initial condition: from then on, the thermalstructure is close to the secular transient regime described in theprevious section. In this model, ‘hydrous’ lithospheric materialgenerated at lower temperatures goes through an initial phase ofheating which may lead to partial melting of the root after a fewhundred million years. If there were enough radioelements andwater in the lithospheric mantle, temperatures may be above thehydrous solidus over a large depth range. In this case, therefore,further melting and dehydration of root material proceeds with noinfluence of the underlying mantle. This occurs at mid-depth in thelithosphere, so that there is no need to thin the root to achievemelting.

For comparison purposes, we have also considered an initialtemperature profile corresponding to the plume model of rootformation (Fig. 13). This profile is obtained by patching an oceanic-type lithosphere with a linear temperature profile in thermalequilibrium with the underlying mantle with an isentropic tem-perature profile for a potential temperature of 1400 °C. Thiscorresponds to a mantle plume not much hotter than the Archeanmantle. Temperature in the root is increased by heat productionuntil a gradient sufficient to transport the heat to the surface isestablished. This heating is independent of the starting plumetemperature. At 3.4 Ga, radiogenic heat production in the root hasgenerated temperatures that are hotter than initial values. It takesabout 500 Myr for root temperatures to drop below the initialvalues. At 2.5 Ga, the temperature profile is close to that of theprevious case, showing further that the geotherm is no longersensitive to the initial condition. For obvious reasons, this case hashigher temperatures than the previous model for as long as 1.5 Gyr.What is significant is that temperatures may rise above their initialvalues after a few 100 Myr depending on the value of heatproduction in the root.

In these calculations, we have fixed the initial thermal structureand the radiogenic heat production values independently of oneanother because they may be determined by different processes. Asexplained above, the amount of radioactive elements in continentalcrust cannot be accounted for by internal differentiation mechanismsconfined to the crust and mantle root system. They probably getintroduced through percolating melt and fluid phases in associationwith subduction. Thus, it may be useful to consider more complicatedtwo-stage models involving for example a thick root made of the solidresidue of mantle plume material which gets “refertilized” bymetasomatic fluids. What these thermal models illustrate is the longtime constants involved in the thermal evolution of thick continentallithosphere and the key role played by radioactive elements in themantle lithosphere.

7. Discussion and conclusion: characteristics of stablecratonic roots

Thermal evolution of continental lithosphere in the Archean issensitive to the initial thermal structure acquired during the process ofroot formation, and boundary conditions that reflect the heatexchanges with the mantle. Below the enriched crust, temperaturesin the mantle root will never be as high as predicted by steady-statethermal calculations with Archean values of crustal heat production.Heat released by radioactive decay in the lithospheric mantle heats upthe root locally and does not get evacuated by the surface heat flux forsome time. Thus, temperatures in the root will increase above theirinitial values. Preservation of a cold and strong root requires a coldinitial temperature profile which is best produced in a subductionenvironment.

Key parameters in all these considerations are the root thicknessand radiogenic heat production in the lithospheric mantle. High(N0.04 μWm−3) heat production in themantle root todaywould imply

that the Archean mantle was weak and a root thicker than 100–150 km could not be preserved. Thick roots are prone to convectiveinstabilities and may well reach temperatures that are higher thanthose of the well-mixed convective mantle. This situation will alsofavor delamination by shearing away of the root. In this sense,therefore, root thickness may well be determined by processes thatare internal to the root and that do not involve perturbations from theconvective mantle. Furthermore, melting may be achieved by in-situradiogenic heat production and the early history of Archeancontinents may have been punctuated by magmatic events that arenot due to mantle plumes or tectonic events.

Acknowledgments

The authors are grateful to Adrian Lenardic, Norman Sleep, andNick Arndt for useful reviews.

Appendix A. Calculation of the transient geotherms

In this section, we shall estimate the time needed for (timedependent) crustal radioactivity to heat up the entire lithosphere. Thetemperature depends on initial conditions, whichwe do not know, butthe time needed to reach a pseudo steady state does not. So we shallconsider the one dimensional heat equation in the crust andlithospheric mantle:

1κATcAt

¼ A2TcAz2

þ H exp −λtð ÞK

0bzbh ðA:1Þ

1κATmAt

¼ A2TmAz2

hbzbL ðA:2Þ

where κ is the thermal diffusivity, K is thermal conductivity, H is thevolumetric heat production. The boundary conditions are Tc=0 at thesurface and ∂T /∂z=0 at the base of the lithosphere. The initialconditions are that the crust is in quasi steady state and that thelithospheric mantle, is in steady statewith a temperature of 0 °C at thesurface:

Tc zð Þ ¼ qm þ Hhð Þzk

−Hz2

2k0bzbh ðA:3Þ

Tm zð Þ ¼ qm z−hð Þk

hbzbL : ðA:4Þ

Heating the lithosphere from below is a distinct problem whichhas been addressed by many studies and it can be calculatedseparately.

The solution is found with the Laplace transform of the equationsabove:

sκTc ¼ d2Tc

dz2þ HK sþ λð Þ 0bzbh ðA:5Þ

sκTm þ ΔT

κ¼ d2Tm

dz2hbzbL ðA:6Þ

where ΔT=(qmh+Hh2 /2) /k and where s is variable of the Laplacetransform defined by:

Ti s; zð Þ ¼Z ∞

0exp −stð ÞTi t; zð Þdt ðA:7Þ

Ti t; zð Þ ¼ 12πi

Z γþi∞

γ−i∞Ti s; zð Þ exp stð Þds ðA:8Þ

with i=c or m, and γ is a real number such that the contour ofintegration lies to the right of all the singularities of the Laplace

59C. Michaut et al. / Lithos 109 (2009) 47–60

transform in the complex s plane. The solution for the mantletemperature is obtained as:

Tm s; zð Þ ¼ HκsK sþ λð Þ

coshffiffiffiffiffiτs

pz−Lð Þ=L

coshffiffiffiffiffiτs

p � coshffiffiffiffiffiτs

p hL

� �−1

� �

þΔTs

1cosh

ffiffiffiffiffiτs

p� �� coshffiffiffiffiffiτs

p hL

� �cosh

ffiffiffiffiffiτs

p z−LL

� �−1

! ðA:9Þ

where τ=L2 /κ. The inverse Laplace transform gives:

Tm z; tð Þ ¼ HL2

λτKexp −λtð Þ cosð

ffiffiffiffiffiffiλτ

pz−Lð Þ=LÞ

cosffiffiffiffiffiffiλτ

p� � � 1− cosffiffiffiffiffiffiλτ

p hL

� �� �

−4HL2

πK

X∞n¼0

−ð Þn2nþ 1ð Þ

1

2nþ 1ð Þ2π2=4−λτ

� 1− cos 2nþ 1ð Þπh2L

� �cos 2nþ 1ð Þ π z−Lð Þ

2L

� �� �

� exp− 2nþ 1ð Þ2π2t

4τ

!

þ2ΔTX∞n¼0

−ð Þn2nþ 1ð Þ cos 2nþ 1ð Þπh

2L

� �cos 2nþ 1ð Þπ h−Lð Þ

2L

� �

� exp− 2nþ 1ð Þ2π2t

4τ

!

ðA:10Þ

The result of such calculations are illustrated by Figs. 10 and 11 thatshow that the lowermost mantle lithosphere never reaches the peaktemperature predicted by steady-state calculations. Fig. 10 shows thetime variation of the crustal radioactivity component of thetemperature at the base of the lithosphere after the emplacement ofthe root. Fig. 11 shows the evolution of the crustal heating componentof temperature in the mantle root. The self-evident conclusion is that,when the half-life of crustal radioactivity is of the same order as thethermal time of the lithosphere, the temperature increase at the baseof the lithosphere is less than the steady state. The temperature at thebase of the lithosphere reaches a maximum after 1–2 Ga, dependingon lithospheric thickness and the peak temperature is ≈70% what onewould infer from steady-state models.

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