Download The Thermal Evolution of an Earth with Strong Subduction Zones

GEOPHYSICAL RESEARCH LETTERS, VOL. 26, NO. 19, PAGES 3041-3044, OCTOBER 1, 1999
The thermal evolution of an earth with strong
subduction
zones
Clinton P. Conrad
Bradford H. Hger
Department of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology
Abstract.
It is commonly supposed that plate tectonic
rates are controlled by the temperature-dependent viscosity
of Earth's deep interior. If this were so, a small decrease in
mantle temperature would lead to a large decreasein global
heat transport. This negative feedback mechanism would
prevent mantle temperatures from changing rapidly with
time. We propose alternatively that convection is primarily
resisted by the bending of oceanic lithosphere at subduction zones. Becauselithospheric strength should not depend
strongly on interior mantle temperature, this relationship
decreasesthe sensitivity of heat flow to changesin interior
mantle viscosity, and thus permits more rapid temperature
changes there. The bending resistance is large enough to
limit heat flow rates for effective viscositiesof the lithosphere
cosity fluid at plate boundaries. These weak zones diminish the role of the lithosphere in controlling plate velocities.
Becauseinterior mantle viscosity remains as the controlling
parameter, mantle temperatures are again stabilized.
The geometry of Earth's subduction zones requires cold,
strong, lithosphere to deform as it bends and descendsinto
the mantle. Conrad and Hayer [1999]showthat this bend-
ing deformation providesat least as much resistanceto plate
motions as viscous deformation of the underlying mantle.
Thus, the rheologicalproperties of the bending lithosphere
should play an important role in determining plate velocities. If this is the case, the negative feedback mechanism
that stabilizes temperatures is disrupted while plate motions
and subduction are maintained. In this study, we include the
greaterthan about 1023Pa s, and increases
with the cube bending resistancein studies of parameterized convection to
of plate thickness. As a result, processesthat affect plate
show how plate bending at subduction zones could play an
thickness, such as small-scale convection or subduction iniimportant role in Earth's thermal evolution.
tiation, could profoundly influence Earth's thermal history.
Parameterized
Introduction
Convection
Models
The relationshipbetweenconvectiveheat transport and
interior temperatureis typicallyinvestigatedin the context
of Rayleigh-Bernard
convection,
wherea constanttemperature boundaryconditionis imposedat the baseof the system. For this scenario,the dimensionless
heat flux is given
Plate velocities are typically thought to depend on the viscosity of the mantle's deep interior, which depends strongly
on temperature. In this view, as the earth cools, mantle viscosity increases, which forces slower convection and
less efficient heat transport. This slows the mantle's cool-
by the Nusselt number, Nu, which is the ratio of the heat
ing rate and stabilizesits internal temperature[e.g., Davies, flow due to convection to that due to conduction alone. For
Nu is oftenrelatedto the Rayleighnum1980; Tozer, 1972]. Mantle temperaturescan changemore steadyconvection,
berof thesystem,Ra,•, by a powerlaw [e.g.,Davies,1980]:
rapidly with time if convection is influenced by the lithosphere, which is colder, and therefore stiffer, than the underlying mantle. In simple convectioncalculations that in-
Nu Ra• whereRa,•= Pgc•TDa
clude temperature-dependent
viscosity,Christensen[1985]
found surface heat flow to depend more on the viscosity of
the cold thermal boundary layer than on that of the under-
where p, g, c•, and n are the density,gravitationalacceleration, thermal expansivityand diffusivity, and T is the
temperature differenceacrossthe system'sdepth, D. For a
lying mantle. Becauselithospheric viscositydependson surface temperatures, which are thought to have changed little
sincethe Archean, Christensen's
[1985]calculationspredict system with constant viscosityr/,• and constant basal temnearly constant heat flow. The decouplingof heat flow from perature,boundarylayertheorygives/3- 1/3 [e.g.,Turcotte
temperature prevents heat lossrates from slowing as the in- and Oxburyh,1967]. For internal heating,both the definiterior cools. This leads to greater present-day cooling rates tion of N u and the valueof/3 predictedby boundarylayer
in thermal evolution models that reproduce current mantle
theorydiffer[e.g.,O'ConnellandHayer,1980].
temperatures[Christensen,1985].
scribe convection beneath a "stagnant lid," where only a
The strength of the negative feedbackmechanismthat
stabilizesmantle temperaturesdependson/3. As the earth
cools,Ra,• decreasesbecauser/,• dependson temperature.
If/3 > 0, this decreasesNu, and slowsthe mantle's rate of
temperaturechange.Conversely,knowledgeof the mantle's
coolingrate allowsus to distinguishamongconvectionmod-
small portion of the upper thermal boundary layer participates in downwelling beneath nearly stationary surface
elsthat yielddifferentvaluesof/3 [e.g.,Christensen,
1985].
The earth's presentrate of secularcoolingis obtained
and plate motions by including "weak zones" of low vis-
rent mantle heat production to total heat flow. The lat-
The notion that the viscosity of the lithosphere limits
mantle heat flow has received some criticism, particularly
becauseChristensen's
[1985]convectionmodelsdo not produce realistic subduction zones. Instead, these models de-
plates. Gurnis [1989] producedmore realistic subduction from estimates of the Urey ratio, which is the ratio of cur-
Copyright1999by theAmericanGeophysical
Union.
ter has been estimatedat about 39 x 10•2 W, after the
contributionfrom radioactivity in the continentalcrust,
5 x 10x2W, is subtracted
fromthe worldwidetotal heatflow
Papernumber1999GL005397.
[e.g.,O'ConnellandHayer,1980;$clateret al., 1980;Stein,
0094-8276/99/1999GL005397505.00
1995]. The rate of internalheatingcan be estimatedfrom
3041
3042
CONRAD
AND HAGER:
THERMAL
EVOLUTION
the abundances of the heat producing elements, U, Th, and
K, in the mantle. Their relative abundances are relatively
WITH
STRONG
SUBDUCTION
Hager
[1999]
as•v•d= 3rlmv}(L/D
+ 2). Conrad
andHager
2(h•/R)a forthe
[1999]
also
derive
theexpression
•[d _ 2rllVp
well constrainedto a ratio of about 1:4:10000[e.g., Jochem viscousdissipationwithin the bendingsubductingslab and
eta!., 1983], but their absoluteabundanceis more uncer- notethat •[d could
be aslargeas,or evenmuchgreater
tain. Geochemical models of the primitive mantle yield 21
ppb for the present abundance of U, giving a current man-
than,•Vmd.
Herer/l is the effective
viscosity
of thelithosphereand R is the radius of curvatureof the subducting
tle heat productionof 15 x 1012W [Jochemeta!., 1983]. plate.Setting
•pe _ •Vmd
+ •[d andsolving
forVpyields:
For comparison, if the mantle were solely composed of the
materialsampledby mid-oceanridges(MORB), with a con-
Vp---3rim
(LIDq-2)q-2r/l
(hs/R)
3
centration of 3 ppb for U, the mantle heat production would
be2.4x 1012W [Kellogg
eta!., 1999].Theseestimates,
when
compared with the mantle heat flow, yield a Urey ratio of
0.4 for models using a primitive mantle and 0.06 for a mantle composedof MORB source. Thus, 60ø-/0
or more of the
current mantle heat flow may represent secular cooling.
A thermal history for the earth is obtained by assuming
a heating history from estimates of the Urey ratio and in-
(3)
Wenowsolve
forplatethickness
h• = 2v/nL/vp using(3):
.lls t•l:l
l --8¾/•L/ls
(4)
where we have defined a dimensionless"lithospheric" Ray-
tegrating heat flow rates from (1) backwardin time from leighnumberRat = pgaTR•/(n•t) that is a measure
of
Earth's present thermal state. For /3 = 1/3, Christensen convectiveinstability resisted by the bending lithosphere.
[1985]findsthat the mantle coolsfrom a molten state to its
It is cle• from (4) that as Rat approaches
8•L/l•,
the
present one in significantly less than the expected 4.5 billion
years if more than about 15% of current mantle heat flow is
due to secular cooling. Thus, to allow Urey ratios less than
0.85, the feedback mechanism that stabilizes mantle temperatures must be diminished, which implies a decrease in /3.
If/3 < 0.1, Christensen[1985]obtainsplausiblethermal his-
t•ckness h• becomes infinite. This occ•s because a plate
slowedby lithospheric bending will thicken due to increased
cooling, and thus slow even f•ther because a thicker plate
is more diffic•t to bend. This runaway process leads to
in•itely thick plates, which are not plausible for the e•th,
so we impose a maximum plate thickness hm. Thus, if h•
tories for Urey ratios between 0.3 and 0.6. These values are
consistent with the above estimates of the Urey ratio, but,
in (4) is l•ger than hm, we obtain an expression
for N by
settingh• = hm in (3) and applyingthe restting expression
as describedabove, Christensen's[1985]convectionmodels for Vpto (2). In terms of Ram and Rat, this yields:
that yield/3 < 0.1 do not produce realistic plate motions or
subducting slabs. In what follows, we introduce an analysis
based on plate bending in subduction zones that produces
small/3 while still allowing realistic plate and slab behavior.
Plate Bending and Mantle Heat Flow
We now study limiting casesfor weak and stronglithosphere.
Weak Bending Lithosphere
The mantle losesmuch of its heat through the formation
of oceanic lithosphere. The total heat flow from an oceanic
plate, expressedhere as N, a dimensionlessnumber analogous to a Nusselt number, is given by:
(1]2)
4D
20(Vp)
-- 47h
( 41shinRamRal
)(1/2)
N- a/2LD
3(L/D+2)Ra7Ramh/D
If lithosphere is weak or does not bend sharply, Rat is
large.If Rat >> 8x/'-•L/l•, applying(4) to (2) yields:
(
16/•)(l/a)
Ill=
1023
Pa
s
L=2500
km/
N-- ]:•"m
37r2
L(-•/Dq-2)
(6)
(2)
where vp is the plate velocity, L is the ridge to trench plate
103
length,
andh• = 2v/nL/vpistheplatethickness
atthetime
of subduction[e.g., Turcotteand Schubert,1982, p. 280-3].
hm=
100km
It has been suggestedthat oceanicplates only thicken for 80
Ma, the age at which seafloorflattening is observedto begin
[e.g., Sclater eta!., 1980]. If additionalheat transport for
older plates limits their thickness to some maximum value
hm < h•, then N shouldbe larger than that givenby (2).
For Earth, the additional heat flow is lessthan 10%, and is
important only for the oldestplates [Sclatereta!., 1980].
•: 102
/
•
L=7500km
The velocity of a plate can be estimated by considering
the energy budget of a single convectingcell [e.g., Conrad and Hager, 1999; Richter, 1984; Turcotte and Oxburgh,
1967]. Convectionis driven by the potential energyrelease
of a descending
slab,givenby •pe: pgc•Tvpl•h•/x/• [Conrad and Hager, 1999] where l• is the effectiveslab length.
Slab reheating is accompanied by cooling of the surrounding mantle, so a slab's net buoyancy change as it descends
should be negligible. Thus, l• is determined by the depth
of convective circulation, and can be taken as a constant.
Potential energy release is typically balanced by viscousdissipation within the shearing mantle, given by Conrad and
10•
106
........
•
.
.
_
lOs
9
Ra
m
Figure 1.
A log-logplot of N vs. Ram for four dif-
ferent plate lengths, L, using the "preferred" values of
r/t = 102aPa s and hm: 100km. The slopes,/3, are the
exponentof the N ,-..,Ra•mrelation.
CONRAD AND HAGER: THERMAL EVOLUTION WITH STRONG SUBDUCTION
whichappliesif Ram is sufficientlylargethat h, < hm. In
this case,convection
is adequatelydescribed
by boundary
layertheory,giving/3• 1/3. If, however,Ramis sufficiently
small that h• > hm, plates reach their maximum thickness
t03[
a)
rl;=i0•z3
Pa-s
.......
L=-••-00
km•
] am=50
km
beforesubductingand (5) yields:
3rr3/2LD
(LID+2))(1/2)(7)
N=(Ram
41shin
3043
[
•
• 102|
•L=5000
/
]
km
•7500km
In this scenario,plate velocitiesare slowedby resistance
in the mantle until the plate thicknesssaturates. Because
slabthickness
is constant,fasterplatesareno longerslowed
becausetheir slabsare thinner. As a result, N is more sensitiveto changes
in Ram than it is for simpleboundarylayer
101•••........................
theory,asshownby the power-law
exponentof/3 = 1/2.
Strong Bending Lithosphere
103
A strong resistance to bending occurs if Rat is less than
or about 8x/rL/l•, in whichcaseplatesalwaysreachtheir
maximumthickness,so (5) applies.If Ram is smallso that
the denominatorof (5) is dominatedby Rat, then (5) reduces
to (7). Otherwise,if Ram is large,(5) becomes:
21,D
2)(1/2)
Ratrr3
/2Lh---7•
•
• 102
(8)
t03
a)B/=t022
Pas
•
hm=100
km
101
••
106
107
t08
109
t01ø
Ra
In
Figure 3. N vs. Ramcurvesfor r/t = 1023Pa s and (a)
hm = 50 km or (b) hm = 200 km. To examinea full range
• 102
./'•••
Plate
Lengths:
•///
L=5000
km
/'//
L=7500
km
x\%
L=
t0000
km
101
103
b)rll=t024Pas
of plate thicknesses,hm, compare to Figure 1.
Herethebending
dissipation
(I)[d, overwhelms
thedissipationin theunderlying
mantle(I)Vm
d, andthuscontrols
plate
velocities.This causesN to be sensitiveonly to Rat. If Rat
is independent
of changes
in Ram,/3 ,.• 0 in (1) is implied.
Application
h =tOO km
We now determine how N varies with Ram for a set
of parameter values that characterize the earth. We use
m
PlateLengths'
L=2500
• 102
L=5000
km
km
L=10000
Pm = 3300kg m-3, g = 10 m s-2 ,a=3x10
-s K -1
T = 1200K, n = 10-6 m2 s-1 D = 2500km, R = 200km
[Beyis,1986],and l• = 1000km, and assumethat thesevaluesremainconstantthroughoutEarth history. In doingso,
we assumethat the dynamicaleffect of any changein T,
by at most50ø-/0,
will be overwhelmed
by the accompanying
km
L=7500
to Earth
km
orders of magnitude changein mantle viscosity. Because
these temperature-inducedchangesin rim shouldhave the
largesteffecton the mantle Rayleighnumber, we vary qm
to generate a range of values for Ram.
t01
106
107
108
109
1010
Ra
in
The maximumplate thicknessand the lithosphereviscosity are somewhatdifficult to estimate and may have been
differentin the past. To seehow heat transportis affected
by each of these quantities, we plot N as a function of Ram
Figure 2. N vs. Ram curvesfor hm = 100km and (a) for variouscombinations
of hm and qt (Figures1-3), but
r/l -- 1022Pa s or (b) qt = 1024Pa s. To examinea full showresultsfor "preferred"valuesin Figure 1. We estimate
range of lithospheric viscosities,r/t, compare to Figure 1.
hm = 100 km by assumingplates achievetheir maximum
3044
CONRAD AND HAGER: THERMAL EVOLUTION
thickness after 80 Ma. The effective viscosity of the bend-
WITH STRONG SUBDUCTION
servedsecular cooling of the mantle, expressedby estimates
ing lithosphereis likely the result not only of viscousflow, of Urey ratios near 0.4. The plate bending mechanism is
but also of plastic and brittle deformationassociatedwith
preferable to other models of convection that yield/3 < 0.1
non-Newtonian
creepandfaulting. ConradandHager[1999] becauseit arises naturally from subduction zone geometry,
estimate
thiseffective
viscosity
to beoforderr/t= 102aPa s, and thus produces realistic plate and slab behavior.
a value a few ordersof magnitudestiffer than estimatesfor
the mantle. We vary L between2500 and 10000 km, the approximaterangeof plate lengthson Earth. To calculateN,
We conclude that small values of/3 are only achieved if
bending subducting slabs have an effective viscosity of or-
der 1023Pa s or more. For viscosities
closeto this value,
we firstuse(4) to calculateh•. If h• > hm, we use(5) to only an earth with large, Pacific-sized plates that can grow
sufficiently thick before they subduct will experience a decrease in/3. Because the plate thickness is so influential in
determining plate velocities, any process that affects plate
thicknesscould be an essential aspect of plate tectonics, and
dictedby (7). For largeRam, longplatesshowa slopeof could greatly influence Earth's thermal evolution. Such pro/3 ,-• 0 whileshortplatesgive/3 ,-• 1/3 (Figure1). The di- cessesmay include small-scale convection beneath plates,
vergencebetweenthesetwo behaviorsoccursbecauseshort which may limit the plate thickness through basal erosion,
platesdo not thickensufficientlyfor the bendingresistance or the physical details of subduction initiation, which may
to becomelarge. Long plates, on the other hand, do have control how large, and thus how thick, plates can become.
calculateN, otherwise,we use (2).
The functional dependenceof N on Ram varies differently for differentplate lengths,L- For small Ram, the N
versusRam curvehas a slopeof/3 ,-• 1/2 (Figure 1), aspre-
time to thicken, so their bending resistancebecomeslarge
enoughto limit platevelocities
if Rat < 8x/TL/I,. Thus,for
weaklithosphereleadingto large Rat (Figure 2a), platesof
all lengthsare governed
by/3 ,-• 1/3, aspredictedby (6) for
weakbendingresistance.
If the lithosphere
is strong(Fig-
Acknowledgments.
This work was supported by National
Science Foundation grant 9506427-EAR and by a National Science Foundation Graduate Research Fellowship. We thank M.
Kameyama and an anonymous referee for helpful reviews and P.
Molnar for insightful comments.
Ure2b),platesof all lengths
aredominated
by thebending
resistance,
whichyields/3 ,-• 0, as predictedby (8). Inter- References
mediatebehavioroccursfor r/t = 1023Pa s, for whichonly
Bevis, M., The
longplatesshow/3,-•0 (Figure1).
curvature of Wadati-Benioff zones and the torsional rigidity of subducting plates, Nature, 323, 52-53, 1986.
U., Thermal evolution models for the earth, J. Geosphere
decreases
(assuming
constant
surface
temperatures).Christensen,
phys. Res., 90, 2995-3007, 1985.
As a result,the lithosphereshouldbecomelessunstable; Conrad, C. P., and B. H. Hager, The effects of plate bending
if small-scale convection erodes the lithospheric base, it
and fault strength at subduction zones on plate dynamics, J.
Geophys. Res., 10J, 17551-17571, 1999.
should become less effective as the mantle cools, increasDavies, G. F., Thermal histories of convective Earth models and
inghm. Thickerplatesproduce
smallerheatflow,solarger constraints on radiogenic heat production in the earth, J. Geohm shouldlead to smallerN (compareN for long plates
phys. Res., 85, 2517-2530, 1980.
in Figures1 and3). Thus,if hmincreases
with decreasingGurnis, M., A reassessment of the heat transport by variable
Ram,/3mustbe greaterthanzero.By comparing
thevalues viscosity convection with plates and lids, Geophys. Res. Lett.,
If the mantle cools,the negativebuoyancyof the litho-
of N for hm = 50 and 100km (Figures3a and 1), we estimate that if hm were to doubledue to a decreaseof Ram by
two ordersof magnitude,
we wouldobtain/3 ,,• 0.15. Such
a largeincrease
in hmseems
unlikely,
so/3should
probably
be somewhat
less than 0.15.
16, 179-182,
1989.
Jochem, K. P., A. W. Hofmann, E. Ito, H. M. Seufert, and W.
M. White, K, U and Th in mid-ocean ridge basalt glassesand
heat production, K/U and K/Rb in the mantle, Nature, 306,
431-436,
1983.
Kellogg, L. H., B. H. Hager, and R. D. van der Hilst, Compositional stratification in the deep mantle, Science, 283, 1881-
Coolermantle temperaturesproducelarger mantle vis1884, 1999.
cosities,
but may alsoincrease
the effectiveviscosity
of the O'Connell, R. J., and B. H. Hager, On the thermal state of the
earth, in Physics of the earth's Interior, ed. by A.M. Dziewonlithosphere.If bendingstresses
controlconvection,
N deski and E. Boschi, pp. 270-317, Soc. Italiana di Fisica, Bologna,
creases
with increasingr/t,leadingto/3 > 0 (compareN for
1980.
longplatesin Figures1 and 2b). An increase
in r/t could Richter, F. M., Regionalized models for the thermal evolution of
alsocausebendingstresses
to suddenlybecomeimportant
the earth, Earth Planet. Sci. Lett., 68, 471-484, 1984.
if Rat becomessmallerthan 8x/TL/I• (comparecurvesfor Sclater, J. G., C. Jaupart, and D. Galson, The heat flow through
and continental crust and the heat loss of the earth,
longplatesin Figures
2aand1). Thus,mantlecooling
could oceanic
Rev. Geophys. and Space Phys., 18, 269-311, 1980.
causebendingslabsto becomemoreeffectiveat resisting Stein, C. A., Heat flow of the earth, in Global Earth Physics,
convection
than the shearingmantle,causing/3to dropfrom
A Handbook of Physical Constants, ed. by T. J. Ahrens, pp.
1/3 to 0 at sometime.
Conclusions
We haveshownthat if plate bendingin subductionzones
is importantin controlling
platevelocities,
heatflowis less
144-158, American Geophysical Union, Washington, 1995.
Tozer, D. C., The present thermal state of the terrestrial planets,
Phys. Earth Planet. Inter., 6, 182-197, 1972.
Turcotte, D. L., and E. R. Oxburgh, Finite amplitude convective
cells and continental drift, J. Fluid Mech., 28, 29-42, 1967.
Turcotte, D. L. and G. Schubert, Geodynamics, John Wiley and
Sons, New York, 1982.
sensitive to interior viscosities and temperatures, so more
rapidchanges
in mantletemperature
arepermitted.Thus, C. P. Conrad, B. H. Hager, Dept. of Earth, Atmospheric, and
platebending
couldbea mechanism
by whichvalues
of/3less Planetary Sciences,Mass. Inst. of Tech., Cambridge, MA 02139.
[email protected])
than0.1canbe obtained
in the relationship
Nu ,,oRaG. As (e-mail: [email protected],
discussed
aboveand by Christensen
[1985],/3mustbe small (ReceivedJune 9, 1999; revisedAugust 9, 1999;
to reconcileparameterized
convection
modelswith the oh- acceptedAugust 11, 1999.)