GEOPHYSICAL
RESEARCH
LETTERS, VOL. 16, NO. 7, PAGES617-620, JULY 1989
INFLUENCE OF HEATING MODE ON THREE-DIMENSIONAL
MANTLE CONVECTION
D. Bercovici, G. Schubert
Department
of EarthandSpaceSciences,
University
of California,
LosAngeles
G. A. Glatzmaier
EarthandSpaceSciences
Division,Los AlamosNationalLaboratory
Abstract. Numerical models of three-dimensional, thermal conthe meltingtemperature
of ironimplya muchhighercore-mantle
boundary
temperature
thanpreviously
believed,andhencea possivectionin highly viscoussphericalshellswith differentcombinationsof internal and basalheatingconsistentlyhave upwelling
blyevengreater
component
of basalheating.Giventheuncertainty
concentrations
in the form of cylindricalplumesand downwelling in the heat source for mantle convection, we seek to determine,
of three-dimensional
convection
in
in planarsheets.As the proportionof internalheatingincreases, throughnumericalsimulations
shell,whetherthecharacter
of theconvective
flowitself
thenumberof upwellingplumesincreases,
anddownwelling
sheets a spherical
bearssomesignatureof the heatingmode.
becomemorevigorousand time-dependent.
With any amountof
basalheating,the entireconvectivepattern,duringits evolution,
is anchoredto the upwellingplumes. As the proportionof inNumerical
Model
ternal heating increases,the heat flow carried by the upwelling
plumesremainsa large fractionof the basalheat flow. DownWe solve the three-dimensional, infinite Prandtl number, anelaswellingsheetscarryonly a minorfraction(approximately
30%) of
tic equationsof mass,momentumand energyconservation
in a
the basal heat flow (even when the shell is entirely heatedfrom
self-gravitating
sphericalshellwith an innerto outerradiusratioof
below),but theyadvectalmostall of the internallygenerated
heat.
0.55, characteristicof the Earth's whole mantle (Glatzmaier, 1988;
The relativelylargenumberof plumesin the Earth'smantle(inBercoviciet al., 1989a,b). The top and bottomboundariesof the
ferred from hotspots),the possibilitythat downwellingslabsare
shellare impermeable
and shearstressfree andtheupperboundary
vigorousenoughto penetrate
thelowermantle,andthe smallfracof the shell is isothermal.With basalheating,the bottomboundary
tion of terrestrialsurfaceheat flow carriedby plumesall suggest is also isothermal;when the shell is only heatedfrom within, the
that the mantleis predominantly
heatedfrom within.
lower boundaryis insulated.
We solvefor perturbations
to a hydrostatic,adiabaticreference
Introduction
state based on the Murnaghanequationwhich prescribesa linear dependence
of bulk moduluson pressure(Murnaghan,1951;
Convectivecirculationin the Earth's mantle is distinctlythreeGlatzmaier,
1988).The dissipation
number/)•' (theratioof shell
dimensionaland time dependent.No strongerevidenceof this
thickness
d to adiabatic
temperature
scaleheight)is notconstant
existsthan in the very surfaceexpressionof mantle convection,
in
this
formulation;
its
radial
average/)i
= !'•(•bo!/•t,.,i,)
isapi.e., platetectonicswith its diverse,migratingplateboundaries
and
proximately
0.5(where
•b,!and•! ol,arethebottom
andtoptemwide distributionof hotspots. Indeed, the combinationof threeperaturesof the adiabaticreferencestate)as in the Earth'swhole
dimensionality
with sphericityin numericalsimulationsof mantle
convectioncan accountfor some major featuresof plate tectonics: for a wide range of heatingmodesand a simple Newtonian
rheology,planarsheetsof downwellingand concentrated
cylindrical plumesof upwellingare the basicforms of verticaltransport
(Bercoviciet el., 1989a,b;Glatzmaier,1988; Baumgardner,1988),
in agreement
with theoccurrence
of descending
slabsandplumesin
the mantle. However, the lack of any activesheet-likeupwellingin
the numericalsimulationscorrelateswith evidencesuggestingthat
mid-oceanridgesare passivefeatures(Lachenbruch,
1976; Silver
et el., 1988; Bercovici et el., 1989b).
Althoughsheet-likedownwellingand cylindricalplume-likeupwelling occurover a wide rangeof heatingmodes,convectivepatternsdependon the styleof heating.The strengthand numberof
upwellingplumes, the structureof sinkingsheetsand the nature
of time dependence
all changewith heatingmode. Theseeffects
of heatingmodehaveparticularsignificance
for the Earth'smantle which is both heatedfrom below by the hotter outer core and
from within by radiogenicheat sourcesand secularcooling. While
thermalhistoriesindicatethat the mantle is 70% to 80% internally
heated(Schubertet el., 1980; Davies, 1980), recenthigh pressure
diamond-anviland shockexperiments(Williams et el., 1987) on
mantle.
TheGriineisen
parameter
') = •!(!•)/d(!•)
(where
•
and• are,respectively,
theadiabatic
reference
statedensity
and
temperature)is constantat unity as is approximatelytrue in the
Earth'sinterior.Specificheatat constant
pressure
el,, thermalconductivity!c,dynamic
viscosity
•], andrateof internal
heatinge (in
units of power per unit mass)are all constant.
The conservation
equationsare solvedby a spectraltransform,
Chebyshev-collocation
method(Glatzmaier,1984). Dependentvariablesare expandedin termsof sphericalharmonics(for latitudinal
and longitudinaldependences)
and Chebyshevpolynomials(for radial dependence). Accuracy of the spatial representationis assuredby a drop of four ordersof magnitudeor more with decreasing wavelengthin the power spectraof the dependentvariables.
Time integrationis performedvia finite differences:linear terms
are treatedimplicitly with a Crank-Nicolsonschemeand nonlinear termsare treatedexplicitly with an Adams-Bashforthmethod.
Time stepsare constrainedto satisfythe Courantcondition.
Four different modes of heating are considered. Cases I and
II are for the sphericalshell heated only from below and only
from within, respectively. CasesIII and IV are 50% and 80%
heatedfrom within, respectively;the proportionof internalheating
is 1 -- (q)hot/(q)tol,
where(q) is thetotalheatflow(inunits
Copyright 1989 by the AmericanGeophysical Union.
of power)through
a bo,ndaryof lheshell.TheRayleighnumber
Paper number 89GLOO806.
cases
is ]•It, • 100•Itcr (wheref•ttc•,is the•tt for theonsetof
]•'tt (a nondimensional
measureof convectivevigor) of all four
convection),
aboutan orderof magnitude
belowthelikelyrangeof
OO94-8276/89/89GL-OO806503. OO
617
618
Bercovici et al.: Three-Dimensional Mantle Convection
/•a for the Earth's whole mantle. Three additional nondimensional
CaseIII (50% heatingfrom within)is similarin planformto case
parameters
are
thesame
forallfour
cases;
they
are4"rGISbotd
- I (Figure1). However,theradialvelocitiesof the upwellingcylin!tbot
dricalplumesanddownwellingplanarsheetsarecomparable.With
internalheating,the thermalboundarylayershaveroughlyequal
1.26,
cp'l•)øt
0.12,
and
]it
=
3.5,
where
.qbol
is
the
gravity
gbo!
d _
at the bottomof the shell(dueto the core)and Ii t is the slope
of the linear relation for bulk modulusas a functionof pressure.
The
amount
ofsuperadiabaticity
iscontrolled
by
temperaturedrops,hencethe buoyanciesof upwellingand downwellingregionsare comparable.As in caseI, upwellingsconserve
their plume-likestructureas they traversethe shell. However, the
downwelling
sheets,
whileundergoing
somedispersion
astheyap-
proachthe bottomboundary,maintaintheir coherencemore than
(whereZ•Tsaisthesuperadiabatic
temperature
drop)whichequals in caseI (Figure la).
1, 13.9, 2.4, and 14.9 for casesI- IV respectively. The relative
The convectivepatternof caseIII evolvesin the samemanner
/x7'.•a which
equals as that of caseI (Figure lb). Upwellingplumesagain tend to
coalesce
andreducetheirnumber.Evenwith internalheating,the
1, 0, 0.14, and 0.07 in casesI- IV, respectively.
existence
of strongupwellingplumescontrolsthe time-dependent
CasesI - III have been partially discussedin Bercoviciet al.
natureof the convectiveplanform;plumesanchorthe rest of the
(1989b). Here we presentnew resultsfor a heatingmodepossibly
characteristic
of the Earth, and describethe advectiveheattransport patternto their own slow drift. However, the minimum number
of plumesachievedis greaterthanfor caseI; caseIII finallyhas
and changesin convectivepatternswith depthand time.
four separated
plumeswhilecaseI hasthreeplumes,two of which
are very closeto one another.The intensityand coherenceof the
Three-Dimensional
Planform and Pattern Evolution
downwellingsheetsin caseIII are greaterthanin caseI. Henceit
is more difficultfor the upwellingregionsto breachthe sheetsas
they attemptto fuse.
The convectivepattern of case I (heating only from below)
The convectivepatternof caseIV (80% heatingfrom within)
is characterizedby narrow cylindricalplumesof upwelling surroundedby a weaker (i.e., lower velocity and less concentrated) resemblesthat of caseII, exceptthat the upwellingregionsare
morewell defined(Figure2). Downwellingis againin theform of
interconnected
networkof downwellingsheets(seeCovertophalf).
The upwelling plumesmaintain their concentrationand structure sheetsandcylinders,whileupwellingtakesplacein many(on the
orderof ten) smallcylindricalplumes.As in caseIll, upwelling
throughoutthe entire shell. However, the weakerdownwellingis
and downwellingvelocitiesare comparable
and the downwelling
most sheet-likein the upper half of the shell; in the lower half of
sheetsmaintaintheir integrityas theyapproachthe bottomof the
the shell, the sheetsdisintegrateas they divergeinto the bottom
shell(Figure2a). Downwellingsheetsin caseIV are relatively
boundarylayer (Cover top left).
short-livedand displaya very chaoticevolution. Althoughit is
Althoughas many as six upwellingplumesoccurearly in the
difficultto followthepatternevolutionin Figure2b, moredetailed
calculationof caseI, the solutionslowly evolvesto a statewith
analysisof theresultsshowsa continuous
formationanddispersal
one major plume (in the easternhemisphere)separatedfrom two
of
downwelling
sheets.
While
the
upwelling
plumesgrow and
smallerplumes(Covertop fight). The reductionin plumenumber
diminishin vigor, they do not migratesignificantly;even with
occursby fusion of plumesat their bases.Once this final stateis
reachedthe two smallerplumesin thewesternhemisphere
continue a small amountof basalheating,the convectivepatternremains
anchoredto theupwellingplumes.
to alternatelyapproachandrecedefrom oneanother.The temporal essentially
amounts
ofheating
are
controlled
by/xTs,+pl),•l•d2/k
evolutionof thesolutionis controlledby thedriftof theplumes;i.e.,
the convectivepatternis anchoredto the strongupwellingplumes.
Calculations
similarto caseI butfor a Boussinesq
fluid(Bercovici
et al., 1989a) yield regularpolyhedralpatternswith six and four
plumesthatare stableat leastup to ]?a = 100f•ocr. The time
dependence
andlackof anypolyhedralsymmetryin caseI mayeitherbe dueto a difference
in initialconditions
fromtheBoussinesq
casesor to the additionof compressibility.The anelasticconservation equationscontainaddednonlinearitiesthat may break the
regularpolyhedralsymmetryof the Boussinesq
equations.Compressibilityalsoweakensthe downwellingsheetsbecauseadiabatic
and viscousheatingswarm downwellingregionsand reducetheir
negativebuoyancy;this may facilitatethe coalescence
of plumes
(becausethe downwelling sheetsare easierto breach).
Case II (heatingonly from within) has a horizontalconvective
patterndominatedby one very long downwellingsheet(in the westernhemisphere)andseveralsmallcylindricaldownwellingsembedded in a broadweak backgroundof upwelling(Coverbottomhalf).
There is no well definedupwelling structurebecausethere is no
lower thermal boundarylayer from which narrow instabilitiescan
arise. Unlike caseI, downwellingzonesin caseII maintaintheir
structureand concentration
as they approachthe bottomboundary
(Cover bottom left).
While the evolutionof the convectivepatternin caseI is con-
trolledby the drift andfusionof upwellingplumes,the patternof
caseII evolveswith no apparentend state(Cover bottomfight).
Althoughthe long downwellingsheetdriftsand bends,it is very
long lived (persistingthroughoutthe entiresimulation).The downwelling plumesare, by comparison,transientfeatures.
Heat Flow
Any amountof basalheatingresultsin cylindricalplumesof upwelling. However,do theplumesonly advectheatenteringthrough
the bottomof the shell or do they also carry heat generatedinternally? The best measureof a plume's heat flow is its advective
heat flow near the middle of the shell (Glatzmaier, 1988). The net
advectiveheat flow acrossa sphericalsurfaceat some radius ro
for compressible
flow is
(1)
wherettr is theradialvelocity,S't is thenonspherically
symmetric
partof the specificentropy,0 is colatitude,
and O is longitude;
St is usedinstead
of thetotalentropybecause,
by conservation
of mass,the net advectionof any sphericallysymmetricquantity
througha concentricsphericalsurfaceis zero. An estimateof the
heatflow carriedupwardsby a plumecanbe madeby integrating
/3•orS' overthehorizontal
cross-sectional
areaoftheplume
at
•'o= 0.77't'top
(i.e.,at a radiusmidwaythrough
theshell).
The last time frame of caseI (Cover top fight) has threeupwelling plumes(the middle plume has a minor plume-likeappendage)The totalheatflow (in unitsof power)advectedby these
plumesis approximately70% of the total heat flow throughthe
shell.The final stateof caseIII hasfour upwellingplumes(Figure
lb). The total advectiveheat flow carriedby all of theseplumes
Bercoviciet al.: Three-DimensionalMantle Convection
619
R/R,op
0.89
180
0.77o•\
0.77
0.66
0.66
o•,,,
,"
61 Oa
84
Fig. 1. Contoursof radial velocity tt.'on a sphericalsurfaceat a)
differentdepthsfor one time, and b) differentlimesfor onedepth(at
r/l'tol,= 0.77)forcase
III (inwhich
theshell
is50%heated
from
within). The imagesare in an equal area projection.Colorsrepresentequalintervalsof velocity. Red contoursarefor •t: in the range
•>
I,,'1,,,,.,.
> ,,' > I
yellow
isfor I
_ •t: •>0;
green
isfortt..'=0;cyan
(light
blue)
isfor0
blueisfor--5]u,[,,,,.,.;> It, > -I,,'1,,,,,.,..A, coZors
appear
in
I
I
theseframesbecausethe upwelling and downwellingvelocitiesare
comparable.
midwaythroughthe shellis approximately
80% of the total heat
flow into the base of the shell, or 40% of the total heat flow out of
the shell.CaseIV hasa broaddistribution
of relativelyweakupwellingplumes;thereareabout10 plumesat anyonetime. At the
Oa
Fig. 2. Same as Figure 1, but for case IV (in which the shell is
80% heated from within). As in Figure 1, all colors appear in
theseframesbecausethe upwelling and downwellingvelocitiesare
comparable.
Sincedownwelling
sheetshavenegativeu,. and,•", theyalso
carrya portionof the net heal flux; i.e., theycool the shellby
injecting"cold"materialintoit. In casesI, Ill andIV, cylindrical
concentrations
of downwellingindividuallycarrybetweenan order
of magnitude
less(caseI) andanorderof magnitude
more(caseIV)
heatflow thanany individualupwellingplume. One can estimate
theheatflowcarriedby thedownwelling
sheets
by assuming
that
radialheatconduction
is negligible
midwaythroughtheshell;this
assumption
isvalidwhentheheatinputintotheshellis greaterthan
theheatthatcanbeconducted
alongtheadiabat,i.e.,when/'xT, <•
+lSbot,d2/]c
(where/X,T,
istheadiabatic
temperature
drop
84Ga timeframe(bottom
of Figure2b),thetotalheatflowcarried •xT..,,
by the plumesis approximately
60% of the total bottomheatflow
and12%of thetotalheatflowoutof theshell.Theplumesof cases
across
theshellandisapproximately
equalto m • Thor),asis truein
this paper(Glatzmaier,1988; Bercovici,Schubertand Glatzmaier,
sheetscarryapproximately
III andIV carrynomoreheatat I',, = 0.891't,,I
, (i.e.,at a depth
of workin progress).The downwelling
25%of theshellthickness)
thantheydoat I',,= 0.771't,1,,
imply- 30%, 60%, and 90% of the total heat flow out of the shell in cases
In all threecases,theweakbackground
ing thattheyarenotsignificantly
affectedby theinternalheating. I, lII andIV, respectively.
flowcarriesalmostnoheatflux,andin facthaspockets
of negative
heatflow wherecold(hot)materialis beingentrained
by strong
Thesevaluesof plumeheat flow are only approximate
sincethe
estimatesfor plumeheat flow are rough,andall the solutionsare
time dependent.
upwelling(downwelling)currents.In casesIII andIV, with internal
620
Bercovici et al.: Three-Dimensional Mantle Convection
heating,downwellingsheetsplay the major role in the advection
References
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Discussion
and Conclusions
Although thesenumericalmodelslack much of the reality of
the Earth's mantle (e.g., higher Rayleigh numbers,variable viscosily, discontinuitiesfrom phaseand/or compositionalchanges),
their displayof Earth-likefeatures(planardescendingslabs,cylindrical upwelling plumes) is a strongmeasureof the importance
of three-dimensional
flow in sphericalgeometry. Cylindricalupwelling plumes (and by inference,hotspots)are the signatureof
basalheating. Plumescarry about60% to 80% of the heat supplied throughthe bottomboundary.Heat flow carriedby plumes
is thereforerepresentative
of the proportionof basalheating.Estimatesof plume heat flow from convectivelygeneratedtopography
(Davies, 1988) indicate that mantle plumes carry no more than
about 10% of the Earth's total surfaceheat flow. This closelycorrespondsto caseIV, implyingthatthemantleis approximately80%
internallyheated,in agreementwith estimatesfrom thermalhistory
modelling(Schubertet al., 1980). The resultsof this studyalso
suggestthat downwellingslabstransportmost of the heat across
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Acknowledgments.The authorswish to acknowledgehelpful
discussions
with participantsof theLos AlamosMantle Convection
Workshopsponsored
by the Instituteof Geophysicsand Planetary
Physics.This work was supportedby grantNAG5152 from NASA
and a computinggrantfrom the SanDiego Supercomputer
Center,
SDSC. All computations
and graphicswere generatedon a Cray
XMP-48
at SDSC.
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G. A. Glatzmaier, ESS-5 MS-F665, Los Alamos National Laboratory,Los Alamos, NM 87545.
(Received:January31, 1989
Revised:April 19, 1989
Accepted:April 21, 1989)