GEOPHYSICALRESEARCHLETTERS,VOL. 20, NO. 2, PAGES101-104,JANUARY22, 1993
CATASTROPHIC OVERTURN OF THE EARTH'S MANTLE DRIVEN BY MULTIPLE PHASE CHANGES
AND INTERNAL
HEAT GENERATION
Stuart A. Weinstein
Department
of Geological
Sciences,
TheUniversity
of Michigan
•.
Theeffectsof phasechanges
andstrong
internal governingequationsfor heat and momentumconservation
heatgeneration
may combineto bring aboutbrief, but
extremely
intenseepisodes
of rapidthermalconvection
in the
are
•T
Earth'smantle. Numerical calculations using realistic
thermodynamic
propertiesfor the exothermicOlivine --+
Spinel and endothermic Spinel ---> Perovskite +
Magnesiowustite
phasetransitionssuggestthe transition
regionof the Earth'smantlemay act as a capacitorfor
subducting
slabs.Slabmaterialaccumulates
in the transition
region
untila threshold
levelof thermalbuoyancy
isreached,
andis thenrapidly dischargedinto the lower mantle.In my
calculations,
this occursas a catastrophic
burstof convection
lasting~10 Myr, with elevatedheat transferrateslasting
~100 Myr. Such episodesmay be analogsto superplume
activitywhich has been hypothesizedto give rise to an
intenseepisodeof intra-platevolcanismand stabilizethe
geodynamo
againstreversals.The topographyon the two
phase
change
boundaries
is foundto be negatively
correlated,
withthecorrelationbecomingmore negativeduringperiods
of rapidconvection.Furthermore,the resultsof this study
suggest
the transitionregionof the Earth'smantlecouldbe
~250K cooler on average,and consequentlymore viscous,
thanthesurrounding
mantle.
Introduction
The effects of multiple phase transitionson thermal
convection
in the mantleis an importantdynamicalproblem
thathasbeenreceivingmuchattention[Liu et al., 1991,Zhao
•t +J(T,T)
=V2T+ R
(1)
dF1+•Rp2
_,•'.
dF2)
3T (2)
V2ooy
=Ra(l+
• TI*
'•
T2
•xx
v2q,= o5,
(3)
ED2
gApl.2D
3
Ra=
PøgtxATD3
R=
•KAT Rpl,2=
•cg
•
(4)
whereT is the temperature,
ß is the streamfunction,
R is the
dimensionless
heatsource,
COy
is thevorticity,ri is the
reducedpressure,Ra is the Rayleighnumberbasedon the
imposed
temperature
difference
AT, Rpl,2 are the phase
changeRayleighnumbers,
F 1 andF2 arethephasefunctions
determining
thefraction
of denser
phase
andTl* and)t2*are
the dimensionless
Clapeyronslopes.The other symbolsD, g,
ct, •c, g, Po, APa,2,K, and •; are the layer thickness,
acceleration of gravity, thermal expansion coefficient,
thermaldiffusivity, dynamicviscosity,density at the top of
the layer, phase transition density changes, thermal
conductivity and internal heat generation rate per unit
volume. The subscripts 1,2 denote the exothermic and
endothermic
transitions,
respectively.
The phasefunctionsare
calculated using the method of Richter [1973] with a
transition half-width constant of 43 km.
All of the fluid boundaries are assumed to be stress-free
et al., 1992, Honda et al., 1992, Pelfier and Solhelm, 1992,
andimpermeable.In addition,thetop andbottomboundaries
are isothermaland the sidewallsare insulating.The boundary
Steinbach
andYuen, 1992].Thesestudiesshowhowmultiple
conditions are:
transitions
filter ascending
plumes,[Zhaoeta!., 1992],affect
secondaryinstabilities arising from an internal thermal
boundarylayer [Liu et al., 1991], control the thermal
spectrum
of the transitionzone [Peltier and Solheim, 1992,
• = coy=T=0
[z= 1]
(5)
ß = O)y= O, T=i
[z = 0]
(6)
[x = 0,•1
(7)
OT
Zhaoet al., 1992] andinfluencethe largescalestructureof
=• = 0
T'= COy
8x
convection[Pelfier and Solheim, 1992, Steinbachand Yuen,
1992,Zhaoet al., 1992].Thisstudyfocuses
onthecombined Equations(1)-(3) are solvedwith respectto the boundary
effectsof phasechangesand internalheat generation; conditions(5)-(7) usingthemethodof finite differenceson a
togethertheseeffectsproduceshortperiodsof vigorous uniformmesh.Thereare225 grid pointsin the verticalgiving
12.8 km resolution when scaled to the thickness of the
convection
whichcouldhaveplayeda fundamental
role in
Earthhistory.
mantle. The equilibrium positions (when there is a
Model
conductive temperature profile) of the exothermic and
endothermicphase transitionscorrespondto dimensional
Our model consistsof a 2-D, isoviscous,Newtonian,
depthsof 280 km and 460 km respectively.The initial
incompressible
fluidlayerheatedfrombelowandinternally, temperature distribution consists of upper and lower
with two isochemicalphasechangesrepresenting
the boundarylayersandhas a meandimensionless
temperature
exothermic
Olivine --> Spinelandendothermic
Spinel--+ of 0.5. In order to avoid a long transient period, R is set
Perovskite
+ (Mg,Fe)Ophase
transitions.
Thedimensionlessinitially to 30 and then lowered to 20 after the average
dimensionless
temperaturereached0.6.
Table
Copyright
1993
bytheAmerican
Geophysical
Union.
1 contains
both
the
values
used
for
the
dimensionless
parametersin equations(!)-(3) and the values
of the constantsthat were used in calculating them. The
values for the C!apeyron slopes and transition density
changesusedhere are givenby [Ito andTakahashi,1989, Ito
Papernumber93GL00044
0094-8534/93/93
GL-00044503.00
101
Wainstein:
Catastrophic
Overturn
InTheEarth's
Mantle
102
Table
1' Parameter
descends
thehot,lowerpartof thefluid layermovesupwards
Values
through
theendothermic
transition.
Thisresults
in a rapidrise
Parameters
Values
in thebasalandsurfaceaveragedNusseltnumbersby a factor
of-5 (Figure2b), andreducesthe transitionzonethickness
Ra
1.0 x 107
(Figure3). Afterthecoldfluid hasdescended
through
the
Rp1
Rp2
1.67x 107
2.0x 107
lowerlayerandspreadacrossthe baseof the box (Figures
4d-4f), the pattern of convection again becomes
R
20
•tl*
Y2*
.106
-.09
O:
2.0 x 10-5K -1
AT
2750 K
Po
'Y1
T2
3300kgm-3
3.6MPaK-1
-3.0MPaK-1
Apl/p o
Ap2/Po
.09
.]1
!l
4.2x 102! Pas
œ
K
1.9x10 -8W m-3
1.0 x 10-6 m'3 s-1
g
9.8ms'2
D
2.88 x 106 m
predominantlylayered. The horizontally averaged
temperature
profiles(< T >) nowreflectthe abundance
of
coldfluid on the bottomof the fluid layer and hot fluid above
the endothermic transition (Figures 4e,4f). This thermal
distributionexcites the formation of instabilities in the lower
andupperthermalboundary
layers(Figure40 maintainin
highheattransfer
rates.Theelevated
heattransfer
ratesdecay
10. I
......
i , , , i , , ,
u.i
et al., 1990,Ringwood,1991,Anderson,1987]. The other
constants
aregivenvaluesappropriate
for themantle.
10 '5
, ,
Results and Discussion
,
........
DIMENSIONLESS
TIME
x 100
Fig. 1. Log of volume kinetic energytime series.The kinetic
The kineticenergy,volume,basalandsurface
averaged energyis normalized
by Ra2. A dimensionless
timeinterva
Nusseltnumber,and averagetransitiondepthtime series of .001 corresponds
to-260 Myr.
obtainedfrom a long integrationof equations(1)-(3) are
shownin Figures1-3. Thesetime seriesstartapproximately
ß
'
'
i
'
ß
i
ß
ß
'
i
'
'
'
i
,
'
'
I 03
,--2.5overturnsafter R was lowered to 20. The kinetic energy
and Nusseltnumbertime seriesare dominatedby a few large
peakswhichoccurduringbriefepisodes
of extremeheatand
LLI
mass transfer when the fluid above the endothermic transition
tTI
•;1
plummets
throughthe transition
destroying
a predominantly
layeredconvection
pattern.The kineticenergyof thefluid
layer variesby nearly four ordersof magnitudeand the
volume averagedNusseltnumbervariesby more thantwo
ordersof magnitude.The peakvaluesof thevolumeaveraged
02
z
LU 01
(/)1
z
Nusselt number exceed 500 with a maximum of 1230 and are
an orderof magnitudegreaterthan theircounterparts
in the
basal and surface averagedNusselt number.During these
brief episodes,a tremendousamountof potentialenergyis
releasedin a shortburstwhichthenslowlydissipates
through
the boundarylayers.This accountsfor thediscrepancy
in size
betweenthe peak volume averagedand surfaceand basal
averagedNusseltnumbers[GrossmanandLohse,1991].
The episode
whichoccursat t=0,009will beexamined
in
detail and is shown in a sequenceof temperatureand
streamfunctionfields (Figure 4). During the first two time
instants(Figures4a, 4b) coldfluid accumulates
from drifting
instabilitiesoff the top boundary-layeron the left sideof the
layer abovethe endothermic
transition.Eventuallyenough
buoyancyexists to overcomethe stabilizingeffect of the
endothermic transition. When this occurs, a substantial
fraction of the cold fluid above the endothermic transition
I 0ø
0.7
0.88
1.06
1.24
DIMENSIONLESS
1.42
TIME
1.6
x 100
90 , ,,',. , --Surface
,v,,.u I_1
80
,,m,
7o
---Basal
,v,.,u I/
• 5o
n4o
•3o
z
I"" ,"' .;",:,/',
10
0.7
0.88
1.06
DIMENSIoNLESS
1.24
TIME
1.42
x
1.6
100
catastrophically
descends
to thebottomof thelayerat speeds
exceeding50 cm/yr (Figure4c). The descentthroughthe
Fig. 2. (a) Log of volumeaveragedNusseltnumbertime
lower layer requires about ,-,10 Myr. As the cold fluid
number time series.
series.
(b)Linearplotof basalandsurface
averaged
Nusse
Weinstein:CatastrophicOverturnIn The Earth'sMantle
350F
400l-
•¾-m
-
•'450
-
'-'500F F'-"-Exøthern•i•
'Transiiion
!-.
u.I550•6001-I'""Endøthermic
. •. Transition
- ß]a
/t
....
,V'-.-".
65O
F"-"'
'"J
.........
7001
0.7
0.925
1.15
DIMENSIONLESS
:' -
1.375
1.6
TIME x 100
Fig.3.Timeseries
of average
phase
transition
depths.
The
phase
transition
depth
isthedepth
atwhich
F1,2=.5.
103
by Hondaet al., [1992, 1993] and the 3-D spherical
calculations
of Tackleyet al., [1992].While theeffectsof
compressibility
(e.g.,viscousdissipation,
latentheat)may
effecttheseepisodes,
it is clearlynotresponsible
for them.
Thegreaterintensity
of thecatastrophic
eventsfoundin this
studymaybe dueto theeffectsof smallaspect-ratio,
the
presence
of the exothermic
phasetransitionor dueto the
effectsof greaterinternalheatgeneration.
For thetimeintervalusedin Figures1-3, the transition
zone width thickness varies between 138 km and 255 km,
withanaverage
of 177km.Themeandepthsare449.5km
and 626.5 km for the exothermic and endothermic transitions.
Present
estimates
of depths
of 410km and660 km for these
transitions
in theEarth are bracketedby valuesobtainedin
thecourseof the calculation.
Duringthe episodesof rapid
convection,the transitionzonethicknessdecreases
rapidly,
bynearly100km,andslowlyincreases
thereafter.
Revenaugh
andJordan[1991] foundnegativelinear
correlationcoefficients between travel times through the
tobackground
levelsover-100Myr.Based
onthesurface transitionzoneand lower mantle (r=-0.65), betweentravel
andbasal
Nusselt
number
timeseries
duringthetimeinterval timesthrough
thetransition
zoneanduppermantle(r=-0.78)
displayed
inFigures
1-3,radiogenic
heating
contributes
40%- anda positivecorrelation
between
traveltimesthrough
the
60%of thesurfaceheatflow.
uppermantleandtraveltimesthroughthe lowermantle
Intermittent
layeredconvection,
in whicha layered (r=0.35).Thesecorrelations
areprimarilydueto topography
convection
patternis disruptedby occasionalstrong on the 410 km and 670 km discontinuitiesand are consistent
downwellings
wasfirstdescribed
by Machetel
andWeber with the behaviorexpectedfor phasetransitionswith
[1991].
Theepisodes
foundin thisstudy
arequalitatively Clapeyron
slopesof differentsign. Figure5 showsthe
similar
to,butclearlymoreintense,
thanthosefoundin the variation of the linear correlation coefficient of the
spherical
axisymmetric
calculations
ofMachetel
andWeber topography
onthetwophasetransitions
andkineticenergy
[1991],
andthose
found
recently
inthe3-DCartesian
studies over a time interval centeredon a catastrophicepisode
CATASTROPHIC
t=0.00883
MANTLE
OVERTt•RN
t=0.00928
t=0.00932
occurring
at 0.0141dimensionless
time. The correlation
coefficients
are alwaysnegativeandrangein valuefrom
-0.40 to -0.88. When the kinetic energy is large, the
correlation
coefficientis morenegative.This relationship
exists
duringperiods
of rapidconvection,
because
thereis a
largefluidfluxthrough
bothphase
transitions.
At othertimes
whenthereisa significant
amount
of layering,
thecorrelation
is muchweaker.Perhaps
a globaldetermination
of r for the
<T>
410 km and 670 km discontinuities can be used as a
barometer
forplumeactivityin themantle.
,
Recently,
LarsonandOlson[1991]hypothesized
that
superplume
activityduringtheCretaceous
stabilized
the
.,
t=0.00966
t=0.00936
t=0.00988
10 -2
<T>
z -0.7
0
10-3I'd
10 -4
Fig.4. Sequence
of temperature
andstreamfunction
fields
0 -o.5
spanning
the time interval0.00883- 0.00988.In the
I
P
temperature
fields,blueandredrepresent
coldandhotfluid
-0.4 .... •..... ,.I,,,, .... , .... 10 '5
respectively.
In thestreamfields,
blueandredindicate
fluid
1.3
1.35
1.4
1.45
1.5
1.55
moving
counter-clockwise
respectively.
Thehistogram
to the
DIMENSIONLESS
TIME X 100
fightof thetemperature
fieldsrepresents
thehorizontally
thevariation
of thelinearcorrelation
averaged
temperature
distribution.
Theblackandwhitelines Fig.5. Plotshowing
are the .2 and .8 contours of the F]
respectively.
andF2 functions
coefficient
(R)and
kinetic
energy
inatime
window
centered
on theeventat .0141 dimensionless
time.
Weinstein:
Catastrophic
Overturn
InTheEarth's
Mantle
104
References
Anderson,D.L., Thermally inducedphasechanges,lateral
heterogeneity
of themantle,
continental
rootsanddeep
E 720 ...........
:
slabanomalies,
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Grossman S., and Lohse D., Fourier-Weierstrass Mode
"'r'
1440
......................
:
-'
C::l
2160 ..................
i
2880
. ,
-,
1.018
1.041
...........
1.064
DIMENSIONLESS
....
1.087
1.110
TIME
1.133
x 100
Fig. 6. Sequenceof horizontallyaveragedtemperature
profilesspanning
the timeinterval.01018- .01330.The
horizontal bar indicates a variation in T of 1.0.
Analysis
for ThermallyDrivenTurbulence,
Ph.•s5_•.
•,
67,445-448, 1991
HondaS., Balachandar
S., Yuen D. A. and ReutelerI).,
Dynamicsof Three-Dimensionalmantle Convection
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•
in press,1992
HondaS., Balachandar
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Three-Dimensional Onstabilities of Mantle Convection
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Ito E., Akaogi M., Topor L., and Navrotsky,A., Negative
geodynamo
againstreversals
andwasresponsible
for the
pressure-temperature
slopes for reactionsforming
Cretaceous
superchron.
Their modelis comprised
of the
MgSiO3 perovskite from calorimetry, Science,249,
plumemodelof Richards
et al., [1989]anddynamo
theory.
1275-1278, 1990
The basicpremiseis that a plumeheaddetaching
from a
Larson'
R. L. andOlsonP., Mantle plumescontrolmagnetic
thermalboundary
layerat thecore-mantle
boundary
(CMB)
reversalfrequency,Earth Planet. Sci, Lett., 107, 437.
will greatlyincrease
theheatflow at theCMB andstabilize
447, 1991
thegeodynamo
against
reversals.
After10-20Myr theplume Liu M., Yuen D. A. andHondaS., Developmentof Diapiric
headwill reachthe surfacegivingrise to floodbasalts.A
Structures
in the Upper mantleDue to PhaseTransitions,
prediction
of thismodelis thatthesuperchron
beginsbefore
Science,252, 1836-1839, 1991
flood basaltproductioncommences.The catastrophic Machete!P., andWeber P., Intemaittentlayeredconvection
in
episodes
observed
in thecalculation
presented
heremayalso
a modelwith an endothermicphasechangeat 670 krn,
giveriseto superchrons.
In thismodel,heatflowat theCMB
Nature, 350, 55-57, 1991
increasesdue to the cold material which falls on it. Perhaps
Peltier W. R. and Solheim L. P., Mantle Phase Transitions
evenmoreimportant
is thedifference
in timingbetween
the
and Layered Chaotic Convection,Geophys.Res. Lea..,
19, 321-324, 1992
generation
of floodbasaltsandtheonsetof thesuperchron.
Richards M.A., Duncan R.A., and Courtillot V.E., Flood
Figures4b-4cclearlyshowtheinjectionof hotfluidintothe
upperlayer before the cold materialreachesthe bottom.
basalts and hotspot tracks' Plume heads and tails,
Therefore,in thismodelfloodbasaltgeneration
beginsbefore
the onset of the superchron.! suggestthat perhapsa
superchron
indicatesa periodwhereportionsof the upper
mantle becamegravitationallyunstableand precipitateda
transitionfrom two-layer convectionto vigorouswhole
mantle convection.
The generalthermal structurecan be understoodfrom an
examination
of thehorizontallyaveragedtemperature
profiles
obtainedduringa periodof low activity.The profilesshown
in Figure 6 spana periodof time lastingabout1 overturn.
Theprofilesshowa low temperature
zonebetween
thephase
transitions.This featureis alsopresentin the calculations
of
Zhao et al., [1992]. The cold zone develops as cold,
descendingflow is nearly preventedfrom penetratingthe
endothermictransitionand spreadout over the endothermic
transitionboundary.The temperaturecontrastbetweenthe
coldzoneandsurrounding
regionsis ~250 K. Thecoldzone
still exists,to a lesserdegree,even duringepisodesof rapid
convection(Figure4 ).
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RichterF. M., Finite amplitudeconvectionthrougha phase
boundary,Geophys.J. R. Astron. $0C,, 35, 265-276,
1973
RingwoodA. E., Phasetransformations
and their bearingon
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Revenaugh
J. andJordanT. H., MantleLayeringFrom
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Geoohvs. Res. Lett., 1992
_
_
TackleyP.J.,Stevenson
D.J., GlatzmeirG.A., andSchubert
G., Effectsof an endothermic
phasetransition
at 670km
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1992
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Acknow.ledgment. I thank Mike Gumis for his help and
comments and the Pittsburg Supercomputer Center.
Computationalfacilities at Michigan are partly fundedby
NSF EAR-8937164. S. Weinsteinis supportedby a 1991
NSF EarthSciences
Postdoctoral
Research
Fellowship,grant
NSF EAR-9102808.
S. A. Weinstein,Departmentof GeologicalSciences,
TheUniversityof Michigan,Ann Arbor,M148109-1063.
Received:November 24, 1992
Accepted:
December
15,1992