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On the heat flow variation from Archean craton to Proterozoic mobile belts










to Proterozoic
A. Len?rdic


Department of Earth and Space Sciences,University of California, Los Angeles

Abstract. Heat flow data from a number of continentalshield regionsshow a trend of relatively low to high valuesfrom Archcancratonsto borderingProterozoicmobile belts. Of the two end-member explanations for this trend, low heat production in Archcan crust or a relatively thick cratonic lithosphere,the latter has come to be generally preferred. Such an explanation assumes strong, one-to-one correspondence a between the mantle component of surface heat flow and local lithospheric thickness. This assumptionhas been well validated below oceansand its applicability to continentallithospherehas gone largely unchallenged. It is, however,not fully valid. This is demonstratedthrough numerical models that allow continentsto form over a convectingmantle. Model continentsconsistof a core region of thickened crust and mantle residuum and a peripheral region of thick crust, analogsto a craton and a mobile belt, respectively. Despite a thicker thermal lithosphere in the core relative to the periphery, the equilibrium surfaceheat flux acrossa continent showslittle variation. The finite thermal conductivity of buoyant continental material is at the heart of this behavior as it allows continents to enforce a spatially near-constant heat flux condition on the mantle below. Such a condition is associatedwith a weak correspondence between mantle heat flow and lithosphericthicknessdefinedthermally or mechanically.This general result, together with specificmodeling results applied to heat flow data, suggests that variable lithospheric thicknessis most likely not the primary causeof heat flow variations near Archcan cratons,leavingdifferingdegrees crustal heat productionas the more likely of

Available continental heat flow data indicate that less

strongcorrespondence valid for the entire Earth. This is

heat flows out of many Archcan cratons than out of

the Proterozoicregionsthat border them [Ballard et
al., 1987; Jones, 1987, 1988; N$tbladeet al., 1990; N?t-

bladeand Pollack, 1993a]. Two end-member models
have been advanced to isolate the principal factor be-

hind this observation, with proponents both realizing of
that the Earth is likely not an end-member. The first attributes the majority of the observedheat flow varia-

tion to a relatively low concentration heat-producing of

elements Archcancrust [Morgan,1985],the second in
attributes the variation principally to a relatively thick

Archcan lithosphere[Ballard and Pollack, 1987; N?tbladeandPollack,1993b].This latter explanation relies argument, that low heat flow out of Archcan cratons
on the assumptionthat there is a strong, one-to-oneremantle heat loss,with a thicker thermal lithosphereimplying a lower mantle heat flux. The wealth of mantle convection models that show this correspondence and

beliefno doubt madethe explanation that relativelylow cratonic heat flow was due to a thick lithosphereseem quite reasonable, main unanswered the question being how a thick lithospherewas stabilized, with the tectosphere idea of Jordan[1978]providinga viable answer. I haveno thoughtson challenging idea that a chemthe ically light mantle melt-residuumlayer, a byproduct of crustalformation, can help to stabilizea relatively thick lithosphere cratonicregions in [Jordan,1978]. In fact, taking it as given, I intend to challenge assumption the that a locally thick continentallithospherenecessarily impliesrelatively low mantle heat loss. Showingthe assumptionfalse for continentallithosphereat, or near, thermal equilibrium,I will then suggest, a negative via
relative to bordering Proterozoic mobile belts results

in lationship between locallithospheric?;?ickness local principally from differences crustal radioactivity. and

the fact that data confirmit for oceanic regions [ParResultsfrom near-surface thermal/chemical boundsonsand Sclater,1977]has lead to the beliefthat this ary layer convection models form the base of this secCopyright 1997bytheAmerican Geophysical Union.
Papernumber 96JB02849.
0148-0227/97/96 JB-02849509.00

tion. The two-dimensionalmodels consistof coupled conservation equationsfor mass, energy, momentum, and chemicalcomposition.The equationsare solvednumericallyusinga finite elementformulation.Initially, a







near-surface chemical boundary layer (CBL) is embed- mantle, respectively,and a middle step value representded within the upperthermalboundary layer(TBL) of ing residuum. Boundary conditionsfor the modelsare a chemically dense, thermallyconvecting layer(ini- reflectingsidewalls;free slip horizontalboundarieswith fluid
tial thermal fields are obtained by solving model equations with only the chemically dense layer present and are taken to steady state or, for casesin which a steady state is not achieved, the initial thermal field calculaa fixed temperature and compositionvalue of 0 on the upper boundary and i on the lower. Full model equations, assumptions,and results from similar models to thosepresentedherein are givenby ?enardic and Kaula [1996];discussions numericalsolution procedures of are tionsare run out to several convective overturn times). and Kaula Light CBL material is a model analog to crust and given by King et al. [1990]and ?enardic residuum, the chemicallydenselayer an analog to the [?993]. Figure 1 showsresultsfrom a prototype model. Free bulk mantle, and the surfaceTBL an analog to the thermal lithosphere. Chemical density is tracked through a parametersare Rs, the bottom heatedthermal Rayleigh compositionfunction initially assigned uniquevaluesfor number which determines the vigor of thermal conveccrust, residuum, and bulk mantle; that is, the function tion in the mantle; Rat/Rs, the ratio of the chemical is a two-dimensional double step function with top and to the thermal Rayleigh number which determinesthe bottom values of 0 and i representing crust and bulk strengthof chemical,relative to thermal, densityvaria-








"' ,.

'width i,,, i , , 1

0 ,(b)









Figure 1. Results froma model with Ra- 105,Rac/Ra - 2.0, ICD - 0.03d, IRD - 0.08d,

and C(Res 0.5.(a)Model )evolution displayed byplotting isotherms composition over shade
plots(sixequally spaced isotherms thehot 2/3 of thetemperature span field);arrows delineate upwelling downwelllng and thermal in the mantle jets layer.(b) Surface heatfluxevolution. (c)
Blow up of the top third of the modeldomain(15 equallyspaced isotherms spanthe coldhalf

of the temperature field). Timeis nondimensionalized by g2/d,where is thermal g diffusivity
and d is the systemdepth. Heat flux is nondimensionalized KAT/d, whereK is thermal by conductivityand AT is temperaturedrop across system. An 80 x 80 finite elementmesh the
with enhancedvertical resolutionacrossthe upper thermal boundarylayer is employedfor this model and all subsequent unit aspect ratio models.







20=] ?o

' ? ' ' I ....

I , , , , I , , , , I , , , , I ....

I ?




Figure a. Results froma ? x 1 aspect ratiomodel with Ra- 10 RamiRa- 2.0, IGD - 0.04d, ?,

IRD - 0.1d, G(Res) 0.7.Nondimensional hea? profiles shown isotherm and surface flux are over
and compositionshadeplo?s of ?he ?ype used in Figure la. A 200 x S0 finite elemen? mesh wi?h enhanced vertical resolution across?he upper ?hermal boundary layer is employed. ?he ?op

portion of ?he images s?re?ched is verticallyfor easeof visuMi?a?ion(?he vertical exaggeration across ?op 0.2d of ?he domainis a factor of 2.5). ?he initial ?hermalfield for ?his model ?he
contained ?hree equal dimensionconvectioncells.

comparedto modelswith smaller continentsthat span initially passiveupflow which deflectsthe lower thermal less extreme mantle temperature gradientsbelow, one boundarylayer leadingto the growth of a hot upwelling sees that this is not the case. Notice also that in all plume belowthe accumulation.This plume inducessigthe models the ratio of averageoceanicto continental nificant lateral surface heat flux variations only after it heat flow is quite large, again likely unrealisticly so for has pushed aside near surface chemical material. In many of them. This is due to the fact that, for these time the left portion of the model no longer servesas models, thermal convectionin the mantle is driven en- an analog to a continental region as both model crust tirely by bottom heating. Later models that allow for and residuumare sweptaside(the issueof long-lived internal heating lead to lessexaggeratedaverageocean dynamic upflows below model mobile belts or model
to continent heat flow variations.

cratons will be addressed moredetail below). in

The spatially near-constant heat flow above model Figure 3 showsthat the reflectingboundary conditions imposed on model continentsin Figures 1 and 2 continentsin Figures 1-3 resultsfrom the finite thermal are not responsiblefor model heat flux behavior. The conductivity of near-surface chemical material and its chemical accumulation that forms in the center of the positive buoyancy. The latter condition allows chemdomain in Figure 3 is not forced to have a reflecting ical accumulations to remain mechanically isolated to symmetry nor are the thermal patterns below it. Varia- a large degree from thermal convection in the mantions in thermal boundary layer thicknessare enhanced tle below (they are deformedby convective motions, by the secondarydownwellinginstabilities triggered at but theydo not participate full convective in overturn). the edges the accumulation of (secondary that, un- The former condition, combinedwith the latter, allows in like the downwellingcentered below the accumulation, temperature perturbations generated in the mantle to theydonot formcell-defining downflows). Formation of penetrate the base of chemical accumulations and not these edgeinstabilities leads to small local hot patches be rapidly diffused away. As a result, the lateral tembelow the central accumulation,that is, significantlat- perature variation generated at the base of a chemical eral temperature variations are induced below the ac- accumulation, by heat lossfrom the mantle, can come cumulation. Despite this, surface heat flux variations to parallelthat in the mantlejust below(Figure lc). remain mild. The chemical accumulation at the left of This tends to establish a laterally constant flux of mandomain induces a flow reversal below itself. The instatle heat into the base of a model continent. The degree bility triggered at the edgeof this accumulationdrives to which this occursdependson the thermal resistance

of a continent relative to that

The resis-

20 J'
? ,-,?

' ' ' ' I , , , I , , ,

of the mantle.



? ' '



- time=O.008282

tance to heat lossout of the mantle dependsinverselyon its thermal diffusion coefficientand directly on the local thicknessof its upper thermal boundary layer, that is, the layer across which internal heat is transferred from the mantle to its surroundingsby thermal diffusion. Similarly, the thermal resistanceof a model continent dependson its local thicknessand its thermal diffusion coefficient. In fluid dynamical literature dealing with the manner of heat exchangebetweena thermally convectinglayer and its surroundings, ratio of these the

?v-?-? 16 -q'---- ?, \ ? ?


average from time 0 .004911-0.008282








.. .





resistances termedthe Blot number[e.g.,Sparrow is et al., 1964]. If the resistance a continentis infinite of relativeto that of the mantle (Blot numberof zero),
then in thermal equilibrium a constant heat flux condi-


tion will holdat its base; the converse true (infinite if is Blot number),then a constant temperature condition
will hold. The ratio of resistances in the models of this



paper is of the order of unity acrossthe extent of model c continents at, or near, thermal equilibrium. Although o this leads to a formally imperfect thermal condition on Ill the mantle below model continents, the local condition p is, relative to heat flux variationsin other regions,one of spatially near-constant heat flux. That the finite conductivity of continental material played a crucial role in time=O.008282 modelheat flow behaviorwasconfirmed throughmodels that varied the thermal diffusivity of buoyant continen- Figure 4. Resultsfrom a variableviscosity model with
tal material relative to that of bulk mantle.

Ra = 5 x 106,Rac/Ra - 2.0, H - 4.0, ICD - 0.02d,

Models so far have assumedconstant viscosity;however, the general conclusion drawn from them, that the relationship between heat flow and lithospheric thickness should be relatively weak below continents near thermal equilibrium, doesnot hinge on this simplifica-

IRD - 0.06d, C(Res-- 0.7. (top)Nondimenand )
sional surface heatflux profiles, (middle)logof the viscosityover the top 0.2d of the model,and (bottom) isotherms overa composition shade plot (eightequally spacedisotherms span the hot 2/3 of the temperature field).

tion. The argument given in the previousparagraph
for what is at the heart of model heat flow behavior

already hints at this, and it can be shown more quantitatively via models that incorporatetemperature- and composition-dependentviscosity. For such models the nondimensionalviscosityof pure mantle, p,?, and pure crustal, pc, material is given by

Figure 4 showsthe results from a unit aspect ratio, temperature- and composition-dependent viscosity model. Material parameters are A - 1.5888 and

B -- 1.3815,which givesa viscosity drop of 104 from mantle at T -- 0 to mantle at T -- 1, and Visr -- 0.1,
which allows crust to be 10 times weaker than mantle

- xp((z z0) + -

and/or residuumat equivalenttemperature. In addition to variable viscosity, the model also allows for a


+z0) -

uniformdistribution internalheatsources heatraof (a
tio H which is the thermal Rayleighnumberdefinedfor internal heating divided by that definedfor basal heating, parameterizes proportionof heatingdue to inthe

where T is nondimensional temperature, To is a temperature offset set to a value of 0.15, A and B are material constants, and Visr is a viscosityratio lessthan 1; that is, crust is assumedto be weaker than mantle at equivalent temperature. For the purpose of viscosity determination, residuum is treated as pure mantle. That is, residuum is no strongeror weaker than mantle

ternalsources). This allows thermalmantleconvection to be drivenby both bottomandinternalheating(coupled with top cooling, course).The modelassumes of
no enhanced concentration heat-producing of elements
in the crust relative to the mantle. This is also true of

at equivalent temperature but, like mantle,is Visr -z

subsequent mixedheatingmodels.The modelof Figure

strongerthan crust at equivalenttemperature. The two 4 doesnot reacha steadystate. Secondary downwelling temperature-dependent functions above are combined instabilities are repeatedlygenerated the edgeof the at to form a temperature- and composition-dependent vis- modelcontinent.As with the high-Ra modelof Figure

cosity function[Lenafdic Kaula, 1993]. and

2, the instabilitiescirculatewithin the domain filling





0 ] i. i i i i t

surface heat flux

isothermsover comp

log v?scos?ty

Unstretched Composition Field

Stretched Composition Field

Figure 5. Results froma variable viscosity,x 1 aspect 6 ratiomodel with Ra - 5 x 10 H - 4.0, a,

RamiRa 2.0,ICD - 0.02d, - 0.06d, C(Res 0.7.A 360 80element is IRD and )x mesh
employed with enhancedvertical resolutionacrossthe upper thermal boundary layer. The top triplet of plots showsfrom top to bottom the nondimensional surfaceheat flux, isothermsover a

composition shade plot (eightequally spaced isotherms spanthe hot 2/3 of the temperature field and the top of the domainis stretched vertically visualization), a shade for and plot of the logof
the viscosity with the top stretched vertically. The two compositionfields at the bottom of the figure show the effectsof the vertical stretchingthat is usedfor easeof visualization.

"master" cellcausing time variationsin continental heat flow profiles.Spatial heat flow variationsacross model a continentat any given time, however,remain mild relative to those acrossthe continent free region. This, as with previousmodels,despitelarge variationsin the thicknessof the upper thermal boundary layer below a continent, in the thicknessof the chemicalboundary layer below a continent, and, unlike previousmodels, variationsin the integratedstrengthacross continent. a That is, largevariationsin the thickness a modelanaof log to the thermal, chemical, and rheologic lithosphere below a continent lead to relatively mild variations in
surface heat flow above a model continent.

tinent has an upwellingplume centeredbelowit, while the othertwo continents not overlay do activeupwelling
plumes; local relatively hot patches are presentbelow

them, however.All threeare associated with strong lateral variationsin the thickness a model lithosphere of definedthermally,chemically, rheologically, only or but the continentwith an active,long-lived upwelling plume
below it showsstrong lateral variations in surfaceheat

flux. Continuedmodelevolutionleadsto a mergingof the two left-mostcontinents (Figure6). As the continents coalesce, the upwellingplume that was between

them initially decays strength(Figure6, top). It in
is not, however, extinguished,and eventually its up-

We movenow to a large aspectratio, high-Ra, mixed welling velocity rises again(Figure6, bottom).As with heating, temperature-and composition-dependent vis- the right-mostcontinent,the presence an active,relaof cositymodel that represents most complexclassof tively long-livedupwellingplume below a modelmobile the models explored herein(Figure 5). Viscosity parame- belt doesallow for significantheat flow variationsabove ters are A - 1.5888,B -- 1.3815,and Visr - 0.01 (a a model continent. model with Visr = 0.1 for the same A and B values, Figure 7 comparesthe model of Figure 6 to a model and modelswith A and B set to give maximum possible that is identical in every way except for the fact that viscosity dropsof 10 and 105across mantlewith residuumis not present.In both modelsa portionof the s the Visr - 0.01, gaveresultsthat werenot qualitatively dif- temperature drop potentially available to drive thermal ferent in terms of the specificheat flux behaviorfocused convectionin the unstablemantle layer is absorbed by on in this paper). Three continents formedinitially in the convectivelystable, chemicallylight, near-surface the model of Figure 5; one centeredat a distanceof ld material. This actsto lowerthe effective Rayleighnumin from the left margin, one centeredroughly 3d from the ber driving thermal convection the mantle. The lowthermal Rayleighnumberis greater left margin, and one centered5d from the left margin. eringof the effective At the time frame shownin Figure 5 the right-mostcon- for the model with both crust and residuumpresent,






! i i , i i i i i i I i i i i i i i i i I i i i i i i i i i



_ ........




- .-?j?.,? .



?:?:??. ..........

-, ;...?-'.'-!?:? ?.



!:?!?.i-'.-?::?::??? ....

:.:?:?"?"..? ....

Figure 6. Continuedevolutionof the model in Figure 5. Nondimensional surfaceheat flux
profilesare shownabove isothermsover compositionshadeplots aL three different nondimensional

which is why dynamic thermal upwelling and down-

seenby comparingthe heat flow abovethe two left-most
continents in the crust and residuum model to the heat

wellingjets are thinnerfor the crustonly model (recall that thermal boundary layers thin as the thermal

flow abovecontinents the crust only model. Simiin larly, the greater variationsin upper thermal boundfectivethermal Rayleighnumberof the crustonly model ary layer thickness and the greater'variationsin conalso causes upwellingplumeswithin the model domain tinental rheologicstructure, both of which are associLomigratein space a greaterdegree to than they do for ated with a morelaterallyvariablechemical boundary the crustand residuum model(that a highereffective layer,alsodo not havea greateffect increasing on latRayleighnumbershouldcauseplumesLobe moretran- eral heat flow variations above model continents. What

Rayleigh number drivingflowincreases). higher The ef-

sient in exact spatial position and more Lime variable

does have relatively a large effect long-lived is upwelling
plumes that can be stabilized below continents in the
crust and residuum model due to the lower effective

in strengthis not surprising, iL has beenwell estabas lished in purely thermal convectionmodeling in large aspect ratio domains [e.g.,Larsenet al., 1995]). Comparing the two modelsof Figure 7 one can seethat the

thermal Rayleigh number associated with said model.
This reinforces the inference that in order for models

magnitude of the heal flow rise acrossmodel continents of the type discussed thus far to be able to account for

is comparable all continents for the right-most significant for save continental flowvariations active heat an upcontinent the crustand residuum in model,the onlyone wellingplume must lie below a model mobilebelt. The
with an active, long-livedplume below it. That is, the largerlateral variationsin the upperchemical boundary layer associated with the crust and residuummodel, as opposed the crust only model, do not havea great to effect on increasingthe potential for large lateral heal
flow variations above a model continent. This can be

plume must also be situated below the mobile belt for

the time it takesthermalperturbations generated it by to diffuse throughthe crustand be felt at the surface; this is physically equivalent placinga constraint to on the time a driftingcontinent mustremainabove spaa
tially fixed plume.







terozoic crustif observed flowisto bematched heat (recall that the model itself assumes no relative enrichment

surface heat flux

of HPEs for the crust). Whetheronefeelsthis reasonabledepends the crustalcomposition on modeladopted. Those with relatively high to intermediateabundances of HPEs in the crust [e.g., Christensen and Mooney, 1995;Rudnickand Fountain, 1995]can account the for difference,while those with relatively low abundances [e.g.,TaylorandMcLennan, 1985] implydifficulties for
the model of Figure 8. The main result from modelsin which dynamic upwelling plumes may underlay model mobile belts is encapsulatedin Figure 9. The model used in Figure 9 is the same as that of Figures 5 and 6. As with the model of Figure 8, this model suggests that large varia-

log viscosity




t. ,






, I , , , I , , , I , , , I , , , I , , , I., lOO


model times (Gyr)
---1.6 -2.5

surface heat flux
E 8O








!,,,s9enns comp._ over ?:: ['"? ''"--??77/-;:;----,-?-' -i;::: ';--*-:????::"
logviscosity fim?=0.001964




..... 4.7





' I ' ' ' I ' ' ' I ' ' "! ' ' ' I ' ' ' I '

Figure 7. A comparisonof results from the model of Figure 5 and a similar model that lacks residuum. Nondimensional surface heat flux profiles, isotherms over composition shade plots, and shade plots of the log of the viscosity,are shownfor both modelsat nearly







DistanceFrom CratonicMargin(kin)
, I , , , I , , , I , , , I , , , I , , , I ?

equivalent nondimensional evolution times. (top) Crust only and (bottom) crustand residuum models.





200 ? ._
The main result from models in which dynamic upwelling plumes do not underlay model cratons or mobile belts is encapsulatedin Figure 8. The model used in Figure 8 is similar to those of Figures I and 2 except that thermal convectionis driven by both internal and bottom heating. Model heat flow is plotted along
with the observed data trend from the Kalahari cra700 kin

400 300100

' ' I -' ' ' I : ' ' I

- - - 2.5 Gyr
' ' ' I

3.1 Gyr
' ' ' I '


I '

300 K
633 K

967 K 1300 K

ton [Nybladeet al., 1990]. One seesthat a relatively
thick thermal lithosphere in the core of a model continent does not lead to significantvariations in surface
heat flux as one moves from the core to the borderthick cratonic thermal

I'1odelT?me = 3.1 Gyr

Figure 8. Dimensionalized modelresultscompared
with the observed trend in surface heat flow data from

the Kalaharicraton(the trend is redrawn from Nyblade et al. [1990];model parametervaluesare Ra -- 106, lithosphere, but this can not account for the observed H -- 4.0, Rac/Ra -- 2.0, ICD -- 0.02d,IRD - 0.06d,
ing region of significantly thinner lithosphere;that is,
the model accounts for a relative

alongwith modelheat flow at several times;(b, top) depthto the 1300K modelisotherm, gauge model a of lithosphere thickness; bottom)modelchemical (b, and studies[e.g.,Boyd et al., 1985]can be met by models thermal structureplotted in the format of Figure lc. of this type. The final model heat flow above a moModel valuesare dimensionalized assuming - 3000 d bilebeltis? 15mW/m leaving 45mW/m to km, ? -- 10-6 m?'/s, K -- 4 W/m K, AT _--2500K, 2, some 2 be takenup by heat-producing elements (HPEs)in Pro- and a surfacetemperature of 300 K.

heat flow trend. To highlightthis,'severalgeotherms are labeled to show that constraints placed on the thermal structure of cratonic lithosphere by thermobarometry

- 0.7).






60 50 20

data trend


I '




I '




'- '


? I '








0 km






6000 6k


isotherms comp over

1o8 viscosity
70 60 50 40 30

6mc=0.?408 ?
I ....

80? ....

' ....

100 fl, , .... 2 ?h,ea,t , , , ,'-, 0 u,x
0 km 1000 2000






isotherms over comp
........ ....."?"?'::? -:?::"? ...................... ......?............... ...... ? .......... '?d?' ?: ? '?J? ....... '? ?'? ..........'............................................. '


log viscosity

plots show the low end of the observedheat

?igure O. Dimensionalised results from the modelof Figures and 6 compared 5 with the observed trend in surfaceheat flow data from the Kalahari eraton [?blade et al., 1990]. Model values
are dimensionalised per Figure 8. ?he heat fl? as

flow trend, as in Figure 8, alongwith model heat flow across left 1/3 of the domain. ?he the isotherms over composition shadeimagesand the log of the visc?ity imagesspan the left 1/3 and top 1/5 of the modeldomain.?he coldest isotherm plottedcorresponds a temperature to of roughly 1100 K. Increasingviscosity represented progr?sively darker shades is by with warm lower crust having the lowestsystemviscosityand cold mantle the highest.

tions in the thicknessof the thermal lithospherebelow a continent will not necessarily map into equally large
variations in continental heat flow. It also shows that

thus far differ from those of a different

class of models

that were used to suggestthat lithospheric thickness

variations couldbe primarilyresponsible heat flow for

lateral strength variationsin mantle lithospherebelow variationsnear Archcancrotons[N?/blade and Pollack, a continent will alsonot necessarily map into significant 1993b]. These question are best answered tandem s in lateral heat flow variations. Lateral heat flow variations through a reexaminationof modelsof the type employed abovea model continent can comecloseto matching the by N?/bladeand Pollack[1993b]. Thesemodelsdiffer observeddata trend from a representativeeratonic area from thoseof Figures 1-9 in .that local lithosphericthickonly when an active upwellingmantle plume is situated ness variations are imposed.a priori. This is achieved below a model mobile belt. In such a casethe predicted by insertinga block of high:viscosity material at the top variation in the thermal lithosphere thicknessfrom a of a thermally convectinglayer that mimics the mantle. mobile belt above a plume to a eraton is roughly a fac- The block itself mimics continental lithosphere. It has tor of 5. thick and thin portions representingeratonic and modo Two questions can now be asked: (1) how are re- bile belt lithosphere,respectively,and its dimensions suits affected by model eratons being situated above not changein time, that is, it is static. This introduces thermal downflows and (2) why do conclusions drawn the followingfree model parameters: the viscosityof the









N!tblade aridPollack[1993b](as with that study,models of the type in Figure 10 suggestthat observedheat
flux variations can be accounted for if a factor of 4 to


5 thickness variation

from cratonic to mobile belt litho-

sphereis employedand if the extent of a craton doesnot significantlyexceedthe thickness the thermal upflow of


The bottom of Figure 10 shows how placingthe "fixed

block"overa hot upwelling (an analogto a manjet tie plume) leadsto largerheat flux variationsthan for
I ' ' ' I ' ' ' I ' ' ' I ' ? ' ;
0.2 0.4

casesin which an identical block is placed over a mantie downflow. The reason for this is related to the




I,? ? ? I ? f ,,I,.,,I ,,,,I,,,, I , ,, ,




If the block is situated


above a thermal upflow, then the way in which the thermal boundary layer of the convectivelyunstable interior fluid thickens acrossthe block is oppositeto the way the block itself thickens. This leads to a pronounced variation in the Biot number as one moves from the craton

'? ? o-I-1

.............. _o._,,_?:..T.?_-..?_o..w"...................
1 ' ' ' ' I .... I .... I ' ' ' ' I .... I" ' ' ' 0.07 0.075 0.08 0.085 0.09 0.095 0

Craton Depth

to the mobile belt. A pronounced Biot number change means a pronouncedchangein the thermal coupling condition between the convectingmantle layer and the continental block, which leadsto enhancedvariations in
surface heat flow. If the block is placed over a thermal
the block thickens same manner across the extent of the block in the leads to milder

Figure 10. Resultsfrom several"fixed block" modan inset plot showingthe fixed dimensionnear sur-

els. (top) Steadystate surface heat flux profiles with downflow, then the thermal boundary layer just below

face block (gray shade)and isotherms from a particular model. In the top graph key, a, bottom heated

as the block itself.


modelwith Ra - 105; b, bottom heatedmodelwith Ra = 10a;m, mixedheating modelwith Ra = 10 and a
H = 4; 3, model with CD = 0.06 and (7L = 0.125; 4, model with CD = 0.08 and (7L = 0.125; 7, model with CD = 0.1 and (7L = 0.125; 6, model with CD = 0.06 and (7L = 0.175. For all models, BD = 0.02, BL = 0.4, and the viscosityof the fixed block is 1000 times that of

Biot number variations and, as a result, milder surface
heat flow variations.

Models of the type in Figure 10 are to my mind less

satisfying than fully dynamic models (Figures1-9) in
that prescribedlithospheric thicknessvariations are adjusted until heat flow observationsare matched. This
leads to a model that is not self-consistent in that a

of how placing the fixed block over a thermal upflow without regard as to whether it would be dynamically leadsto a greater heat flux jump across the block then for cases in which the block is placed over a thermal allowed for by the equations and boundary conditions downflow, otherparameters all being equal(forthisplot that define the model to begin with. In effect, the all models had Ra = 105,(7L = 0.125,BD = 0.02,and model may be overconstrained. This flaw can be al-

the thermallyconvecting interior. (bottom)Indication solution is locally prescribedwithin a modelingdomain

?Z = 0.4).

leviatedby allowing "fixedblock"to deform; the that
is, using the fixed block resultsas initial conditionsfor

fully dynamic models that prescribe higherviscosity a to block material and allow it to freely deformin reblock relative to mantle; the depth and length of the cratonic portion of the lithosphericblock, CD and and the depth and lengthof the mobilebelt portion, BD and B L. The inset in the top of Figure 10 shows iman age of a "fixed block" and temperature contoursfrom one such model. The graph itself showssurfaceheat flux profiles from several models. As with models in Figures 1-9, the relationship betweensurfaceheat flow and upper thermal boundary layer thicknessis weaker in portions of the domain that mimic continents than it is in thosethat mimic oceanicregions.Although weak, there is still a dependenceand, as such, lithospheric

sponseto convective stresses.Figure 11 showsresults from two of these "mildly deformableblock" models.

The initial condition chosen is suchthat a majorityof
the observed heat flux jump across Kalahari craton the
can be accounted for. As the models evolve toward a

moredynamically self-consistent the percentage state,

of the observed heat flux jump that they can account for decreases. the images Figure11 show, As in thisdecreasedoesnot involvelarge near-surface deformation of the modelcratonbut rathera rounding its edges of from below. Thesemodels suggest that for variations in lithosphere thickness be primarilyresponsible obto for thicknessvariations can be adjusted to give variable served heat flow variations across cratons they mustbe
heat flux jumps from a model craton to a mobile belt. A variety of parameter variations were explored for these models, and results were similar to those discussed by

significant a factor 4-5variation), persistent (? of time
in exactstructure(cratonic lithosphere must be effec-

tivelynondeformable its full depth across extent), and,

I,, ? !,,, I,,, I,, , I ,,

, I , , , I








Viscosity 500 Ratio=

lithospheric thickness, examples beingthermobarome-

try studies mantle of xenoliths [e.g., Bo?ld al., 1985], et
which indicate a thick lithospherefor African eratons;

0 ? 65

I : : : I ' I ' I ' ' ' I ' ' ' I ' ' ' I

the lackof diamond-bearing kimberlites mobile in belts, whichsuggests the lithosphere that belowthem is relatively thin; and seismic studies whichalsoindicatea thick eratoniclithosphererelative to that of bordering

mobile belts[e.g.,Clouser Langston, and 1990].These
studies,coupled with the observation low heat loss of
0.2 0.4 0.6 Transit 0.8 Time 1 1.2

from cratons relative to mobile belts, suggesta seem-

ingly tidy explanation localheat flowvariations for in
tectonicallystable continentalregionsthat reliesto a strong degreeon local lithosphericthicknessvariations.
It has, in fact, been argued that interpretations of heat flow variations from cratons to mobile belts that principally rely on variationsin crustal heat productionfail to satisfy independentconstraintson the thermal struc-

ture of the lithosphere [Ballard and Pollack,1987;Nybladeand Pollack,1993a],the progression the arguof ment being(1) if heat flow variations due largely are to variations crustalheat production, in then (2) the

constant, implyingthat (3) the thickness the manof is constant, which(4) is at odds Figure 11. Resultsfrom two "mildly deformable tle lithosphere spatially block"models. For both models,Ra = 106, H - 4, with the results of diamond inclusion and all other stud(TD - 0.1, (7/; = 0.125, BD = 0.02, and B/; = 0.4. ies that suggestthick eratonic lithosphere. The viscosity ratio is that of the blockrelativeto the inThe problem I have with the argument above is step terior. (top) Percentage the heatflux variation of from 3 of its progression. oceanicregionsthe assumption For
the center of a craton to a mobile belt that the models can account for based on data from the Kalahari era-


Transit Time=1.25

mantle component of the heat flow must be spatially

contained in step 3 is valid. The validity relies on two

of ton. (It is assumed that a heat flux jump from 30 to things: first is the low thermal resistance air and wa60 mW/m2 needs be accounted and the model ter relative to rock which maintains a constanttemperato for is dimensionalized such that at time zero the heat flux tureacross Earth'ssurface the (constant terms geoin of above center a model the of eraton 30mW/m2.) This dynamicconcerns) second the fact that oceanic is and is percentage shown is overa timescale whichrepresents crust and residuumparticipate in mantle overturn. This
meansthat the top of the convectivelyunstable portion of the solid Earth corresponds the surfaceof the solid to model.(bottom) Block gray)andisotherms the (in for Earth in oceanicregions. Thus a constanttemperature viscosity ratio=1000 modelat two times. surface boundary condition holdsfor mantle convection below oceans. This condition brings a strong, one-toone correspondence betweenequilibrium heat flux and unless more extreme thickness variations are invoked, the local thicknessof the thermal lithosphere. For conmust alsobe associated with lithosphere situated above tinents this is not the case. The thermal condition at activemantleupfiows. These conditions not, in and the Earth's surface is not what is key to mantle heat can loss below continents; rather, it is the condition at the of themselves, ruled out. be base of the chemically light material that definesa continent. In thermal equilibrium this condition is closer Discussion and Conclusion
to one of constant heat flux than one of constant tem-

the time it takes a fluid particle to travel the length of the modeldomain,basedon the rms velocityof the

between It has long beenheld that the best hopeof using perature. This leads to a weak correspondence continental heat flow data to understand the thermal equilibrium heat flux and local lithosphericthickness.A

structure the lithosphere in regions of is wereit canbe correspondence weakin fully dynamic so models (Figassumed thermalequilibrium beenapproached, ures 1-9) as to suggest that has that underthe assumptions of

i.e.,oldtectonically regions Ozburgh, stable [e.g., 1981]. the modelsa variable thicknesslithospherein equilibGiven that Archeaneratonsand borderingProterozoic rium is most likely not the causeof heat flow variations belts have been stable for long periods, it is not sur- from Archean eratonsto borderingregionsunlessthose prising heatflowdatafromthese that regions been bordering regionsare preferentially situated above achas extensively to inferproperties the deepmantle tive upwelling mantle plumes. For African regionsone used of lithosphere. hasbeen This done conjunction in- could argue that mantle plumes may play a role; howin with dependent studies observations constrain and that local ever, it is hard to argue for this worldwide. It has been






argued that the observedtrend shownin Figure 8 is a
worldwide one with heat flow variations from the North


Americanand Europeancratonsusedas supportIN,/Madearid Pol?ck, 1993a]. Theseregions lack any ev-

Ballard, S., and H.N. Pollack,Diversionof heat flow by Arcbean cratons: model southern A for AfricaandNamibia,

idence for mantle plumes. In fact, seismictomograEarth Planet. Sci. Left., 85, 253-264, 1987. phy showsa broad region of fast anomaliesbelow the Ballard, S., H.N. Pollack,and N.J. Skinner,Terrestrialheat flow in Botswanaand Namibia, J. Geoph!iz. Rez., 9?, North American craton and its surroundings.Fast seismic anomalies extend 300-400 km below the craton and

6291-6300, 1987.

100-200 km below surroundingcontinentalregionswith Grieve, Variations in effective elastic thickness of the no evidence slowanomalies greaterdepths[GraNd, of at North Americanlithosphere, Nature,3J3, 636-638,1990. 1987, 1994]. Global tomography shows that this obserBoyd,F.R., J.J. Gurney,and S.H. Richardson, Evidence for vation holdsfor many of the world'scratons[St?et al., a 150-200 km thick Arcbean lithosphere from diamond
inclusion thermobarometry, Nature, 315, 387-389,1985. Of course, one can still use models of the type in Christensen, N.I., andW.D. Mooney, Seismic velocity strucFigure 10 to argue that extreme lithosphericthickness ture and composition the continental of crust: A global variations may be able to account for observed heat view, J. Geoph!iz. Rez., ?00, 9761-9788,1995. flow variations. It should be realized that in so doClouser,R.H., and C.A. Langston,Upper mantle structure of southern Africa from P,, waves, Geoph!iz. J. Rez., 95, ing one basesone's argumentson modelsthat are not 17,403-17,415, 1990. fully self-consistent. I find this unsatisfyingand prefer to base arguments on fully dynamic models that Grand, S.P., Tomographic inversionfor shear structure beneath the North Americanplate, J. Geoph!iz. Rez., 9?, if nothing else are self-consistent and do not have any 14,065-14,090, 1987. part of their solution prescribeda priori. Whether one Grand, S.P., Mantle shear structure beneath the Americas agrees with this philosophy or not, models of this paand surrounding oceans,J. GeophFz. Rez., 99, 11,591per can remove a major objection leveled against the 11,621, 1994. hypothesisthat heat flow variations from Archean cra- Guillou-Frottier, J.-C. Mareschal, Jaupart,C. Gariepy, L., C. tons to Proterozoic mobile belts result principally from R. Lapointe, and G. Bienfait, Heat flow variations in the variations in crustal heat production; the objection is GrenvilleProvince,Canada,Earth Planet. Sci. Left., ?36, 447-460, 1995. based on the belief that a spattally near-constantmantle heat flux below continentsrequiresmild lithospheric Jones,M.Q.W., Heat flow and heat productionin the Namaqua mobile belt, South Africa, J. Geoph!iz. Rez., 9?, thickness variations. The demonstration that this is not 6273-6289, 1987. necessarilytrue also removesthe seeminginconsistency Jones,M.Q.W., Heat flow and heat productionin the Witbetween inferences large variationsin the thickness of watersrandbasinand environs and its significance the for of the elastic lithosphere across the North American SouthAfricanshieldgeotherm lithospheric and thickness,

Bechtel, T.D., D.W. Forsyth, V.L. Sharpton,and R.A.F.

craton and surroundings [Bechtel al., 1990]and inet
that mantle heat flow varies little across the


J. Geoph!iz. Rez., 93, 3243-3260,1988.

region[Pir?etet al., 1991;Gsillos-Frottier al., 1995]. et

Jordan,T.H., Composition and development the contiof
nental tectosphere,Nature, g7J, 544-548, 1978.

Given theseconsiderations, am left with a generaland King, S.D., A. Raefsky,and B.H. Hager, ConMan: VecI

a specific conclusion: a strong, (1) one-to-one relationship between mantle heat flow and lithospheric thick-

tortzing a finite element code for incompressible twodimensional convection the Earth'smantle,Ph!iz.Earth in

Dynamics thermalconvection of with Newtontan temperaturedependent viscosity highRayleighnumber,Ph!iz.Earth at Planet. Inter., 89, 9-33, 1995. flow variationsnear cratonsto date, the one relying on Lenafdic, A., and W.M. Kaula, A numerical treatment of a variation in crustal heat production seemsthe more geodynamic viscous flowproblems involving advection the librium is valid to first-order near cratons, then, of the two end-member explanations offered for observedheat

Planet. Inter., 59, 195-207, 1990. ness, defined thermallyand/or mechanically, not be can Larsen, T.B., D.A. Yuen,A.V. Malevsky, J.L. Smedsmo, and assumed be validworldwide; (2) if thermalequito and

of materialinterfaces, Geoph!iz. J. Rez., 98, 8243-8269,

A., thermal/chemical Acknowledgments. This work has been supported by Lenafdic, andW.M. Kaula,Nearsurface boundary layer convection at infinite Prandtl number: NASA under grant NAGW-2997. Computing was carried Two-dimensional numericalexperiments,Geoph!iz. J. out at the San Diego Supercomputer Center. I thank Dale Int., 1?6, 689-711, 1996. Issler, J.-C. Mareschal, P. Morgan, and H. N. Pollack for heat production and the sethoughtful and helpful reviewson earlier versionsof this pa- Morgan,P., Crustal radiogenic per; S. D. King, A. Raefsky, and B. H. Hager for the Conlective survival of ancient continental crust, Proc. Lunar Man finite element code; and NCSA Illinois for visualization Planet. Sci. Conf. 15th, Part 2, J. Geoph!iz. Rez., 90, suppl., C561-C570, 1985. pacl?ges.



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accepted September 1996.) 17,

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