GEOPHYSICAL RESEARCH LETTERS, VOL. 24, NO. 2, PAGES 201-204, JANUARY 15, 1997
The clusteringof rising diapirs and plume heads
AmandaKelly and DavidBercovici
•
Departmentof Geology& Geophysics,
SOEST, Universityof Hawaii, Honolulu
Abstract. Buoyantdiapirsor plumeheadsin the General Rayleigh-Taylor Experiment
Earth's mantle are typically modelledas a solitary bodWe conducted
severalbasicRayleigh-Taylor
experiies,ascending
verticallyand uninfluenced
by neighbor- ments in a cylindrical tank with a 30.75cm outer diameter, 29.8cm inner diameter, and 30.5cm height. The
ing diapirs. However,laboratory experilnentsand basic
theory suggestthat two or more risingdiapirs can becomeattracted to eachother, therebyinfluencingtheir
ascenttrajectory such that they cluster and possibly
coalesce.Clusteringis controlledby the horizontaland
vertical separationbetweendiapirs, as well as their relative sizes. This phenomenahas implications for the
originsof large igneousprovincesas well as the Earth's
long-wavelength
hotspotdistribution.
Introduction
Mantle plumeshave been invokedto explain a wide
variety of geologicaleventssuchas hot spot volcanism,
flood basalts,and continentalrifting [seeWhite and
McKenzie, 1995]. Mantle plumespossiblyinitiate as
large diapirs or plume headsin the D" layer above the
core-mantleboundary. After the diapirs separatefrom
D" it is assumedthat they continueto rise alonga vertical line as essentiallysolitary bodies. Any deflection
from vertical ascent is presumedto arise from largescalemantle flow associatedwith plate tectonicmotions
[e.g.,Skilbeckand Whitehead,1978;Olsonand Singer,
1985;Richardsand Griffiths,1988].Howeverin this paper we presentlaboratoryexperimentsand basictheory
whichsuggestthat risingdiapirscaninteract,attracting
eachother to form diapir clusters,and possiblymerge
into larger diapirs. The clusteringand coalescence
of
diapirs naturally has implicationsfor many geological
phenomena,mostnotablyfor the originof floodbasalts,
and the globaldistributionof hotspots.
Laboratory Experiments
tank wasfilled to a height of 28.8cmwith LSI Specialty
Products Liquidose No. 444 corn syrup with density
of 1460kg/m
3 and a viscosity
of 607,329.8cP
(where
1cP=10-spas). Overthe Liquidose
syrupwasadded
a 5mm layer of Karo light corn syrup (dyed purple to
enhance
its visibility)with a densityof 1320kg/m
3 and
a viscosity of 3839cP. The tank was eqUilibrated for
12 hours, sealed and turned upside down. Approximately 11 min thereafter, the Karo syrup layer developed dome-shapedundulations10-25mm in width. Af-
ter another15mintheundulations
separated
fromthe
sourcearea and beganto rise as distinct diapirs. Approximately 10 min later, the attraction between two of
the diapirsbecameapparent(Figure1). Oneof the diapirs was slightly larger and rosefaster than the other,
eventuallybecoming
the leadingdiapir. The trailing
diapir becamemore oblate as its ascentpath was de-
flectedtowardthe leadingdiapir. in time the lowerdiapir moved to a trajectory almost directly underneath
the leading diapir which also becamedistorted; after
another 3 min the two diapirsmerged.
Controlled Experiments
Controlled experiments were conductedin a rectan-
gulartank (53cmlongby 40.5cmhighby 21.5cmwide)
filledto a heightof 33 cm with Karo syrup. Mounted
above the tank were two dispensingburets beneath
whichwerefixed42cmlongglasstubeswith endscurved
into a J-shape; the inner diameter of the tubes was
0.8cm and the upward-curvedend of each tube was 7cm
long.The buretswerepartiallyfilledwitha mixtureof
The phenomenaof diapir attraction occursnaturally
in simpleRayleigh-Taylorexperimentswhereina heavy
viscousfluid overlies a thinner, less viscous and less
densefluid. To isolateand examinethis phenomenawe
alsoperformedexperimentswherethe distancebetween
the plumescould be controlled.
aAuthor to whom correspondence
shouldbe addressed.
40% (by volume)water and 60% Karo Syrupwith a
densityof 1080kg/m
s and a viscosity
of 103.2cP;fluid
flux was controlled by stopcoCkson the buret-nozzles.
In each experiment, the diapirs were formed bY com-
pletely filling the J-tubeswith the plume mixture and
then stopping the sourcefeed. The diapirs inflated at
the mouth of each J-tube to about 1.3 cm diameter af-
ter which they separatedfrom their tube and proceeded
to rise with a total ascenttime of approximately 10 seconds. Multiple experimentswere run for a range of
separationdistancesbetweenthe j-tubes. No signifi-
Copyright1997by theAmericanGeophysical
Union.
cant interaction betweendiapirs was noticed until the
J-tubes were 3.4cm apart or less. Four runs were conducted with the 3.4cm separation. In two runs which
Papernumber96GL03904
0094-8534/97/96GL-03904505.00
201
202
KELLY AND BERCOVICI:
CLUSTERING
cm-?•!,•..
00:31.
:11 i;::':iii('i
'•:i•....
:•i•
'.................
•z•:•.:.•.•.•.:.:.•.•.•:•:•.•...?
.......
:;...
OF RISING PLUME HEADS
cendinglow-viscosityspheres. The majority of work
on the interaction betweenrising diapirs or bubblesis
mostlynumerical[e.g.,Wijngaarden,1993;Mangaand
Stone,1993].Herewedevisea simpleapproximate
theory of how two diapirs interact through the dipole flow
and pressurefields that they establishin the surround-
5
ing viscousmedium[seealsoMangaand Stone,1993].
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........
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................
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Figure 1. Sequentialphotographsof two clusteringdiapirs in a basic Rayleigh-Taylorexperiment. Scaleis
shownon the left of the top frame;times (sincestart
of experiment)are indicatedin the upperleft cornerof
each frame. See text for completediscussionof experiments.
showedstrongdiapir attraction,onediapir alwayslead
the other by a significantdistance(Figure2). In one.
run, the two diapirs coalescedbeforereachingthe surface of the tank. In the two runs which showed no no-
ticeableinteraction, neither diapir lead the other as the
diapirsformedsimultaneously
and ascendedat nearly
the same rate.
Discussion of Experiments
Our experimentssuggestthat the verticalseparation
betweendiapirs is one of the primary criteria for clustering. For the diapirsto showany attraction,one of
the two neededto be slightly higher than the other,
by at least I cm. Total separationbetweendiapirsalso
controlledattraction; diapirsdisplayedlittle interaction
whenseparatedby more than 3.4 cm. In the RayleighTaylor experiments,attractionwasapparentonly when Figure 2. Sequentialphotographsof two clustering
the leadingdiapir formedand, approximatey5 minutes diapirsin a controlledexperiment.The diapirsare genlater, a secondtrailing diapir formednearby.
eratedby feedingfluid throughtwo separatetubes,with
adjustable separation,into the base of a tank of more
viscousfluid. Scaleis shownon the left of the top frame;
Theory
times (sincestart of experiment)are indicatedin the
In this section we theoretically examine diapir clus- upper left corner of each frame. See text for complete
tering by consideringthe interactionbetweentwo as- discussionof experiments.
KELLY
AND BERCOVICI:
CLUSTERING
OF RISING
PLUME
HEADS
203
Our twodiapirshavebuoyantdensityanomalies
(rel- pel and Brenner,1965;Mangaand Stone,1993];there
ativeto the surrounding
medium)of Ap, andhaveradii is, however,no equivalentmethodfor invisciddiapirs.)
of al and a2. The surroundingmedium is infinite, in- Thus, our approximateforceFz is only valid aslongas
compressibleand has a viscosityof p, which is much
larger than the viscositiesof the two diapirs. We define the x-z plane to contain the direction of gravity
along the z axis and the line connectingthe two diapirs. We assumethat the diapirsmaintain their spherical symmetry,thus there are no forcesto pull them out
of the x-z plane. The positionsand velocitiesof each
the viscousnormal stresseson diapir I due to its own
ascentare much larger than thosedue to diapir 2's induced dipole flow. Therefore,not only must diapir 1
be very far awayfrom diapir 2 (sothat diapir 2 maintains a dipolefield), but diapir I must not be so small
comparedto diapir 2 that it movestoo slowly.In brief,
the theory assumesthat the diapirsare not only widely
diapir are thereforerl - (Xl, 0, Zl), r2 - (x2,0, z•) separatedbut also are not of greatly disparatesizes.
and Vl - (u•, 0, Wl), v2 - (u2,0, w2), respectively. Sincethe pressureassociatedwith diapir 2 is assumed
We assumethat each diapir inducesflow and pressure the same as if the diapir were solitary, then
fields in the viscous medium
which differ little from the
dipole fields that would occur in solitary spheresmoving throughan infinite medium;this implicitly assumes
P-
v:2.(r-
Ir- s
(5)
that Jr1- r2[ >> a•, a•. Assuming
creepingflow,the net [Batchelor,1967]wherer is the positionvectorof any
pointin space.The integralin (4) is aroundthe surface
force balance on diapir i is
- 4•ra•pvl
+F•
0-] 4rra•Apg2•
(1)
of a spherecenteredon r•; i.e.,
f0'r••v2-(r
- r2)
sin
OdOd(
F2- -pa:•
Ir- r21
a r-alr•a•
where the first term on the right representsthe buoyancy force of diapir 1, the secondterm is the drag
force on diapir 1 due to its own motion, and F• is the
force due to the flow and pressurefields in the viscous where 0 is colatitudeand • longitudeof a sphericalcomedium induced by the motion of diapir 2. The flow ordinatesystemcenteredat rl. Thus (6) is evaluated
and pressurefieldscausedby diapir 2 are governedby at r = r• + al(cos•bsin0,sin•bsin0,cos0). Defining
Ax = x•-x2, Ay= y•--y2 and Ar= x/Ax 2+Az 2,
0 = -VP-
pV x •
(2)
andexpanding
Ir- r21-a to firstorderin a•/Ar, (6)
becomes
where P is the nonhydrostaticpressure,and w is the
vorticityvectorin the viscousfluid [Batchelor,1967].
Integrationof (2) arounda volumeof the viscousfluid
yieldsthe sum of forcesacting on that volumeby the
surroundingmedium:
a3(otu2+
)
F2- 4rca:•p•
7w2,0,
7u2
+fiw•
(7)
where
Ax 2
1
a- Ar• 3'
Az •
1
fi- Ar• 3'
AxAz
7- Ar2 . (8)
Substitutionof (7) into (1) yieldstwo equationsfor Ul
and Wl; by symmetry,an interchangeof subscripts1
where A is the surfacearea of the volume, h is the unit
normal to this surface;the above forcesare in balance and 2 in theseequationsyields two more equationsfor
accordingto (2). However,if this volumewereoccu- u2 and w2. Combiningall four equations,whileneglectoforder(a•na•-m)2/Ar6 (m- 1 or2),yields
piedby diapir I thenthe forcebalanceof (3) wouldnot ingterms
hold. Much of the imbalance arises because the surface
of diapir I is essentiallyshear-stressfree and thus all
tractionstangentialto its surface,i.e., perpendicularto
h, wouldbe negligible.Eliminationof tractionstangent
to the surfaceof diapir 1, i.e., yh x w, in (3) yieldsan
approximateforce on diapir I due to flow induced by
diapir 2, i.e.,
aja• AzAz.
ui= Ars
Ara Ar2 3 Wj+Wi
(9)
(10)
where(i,j) - (1,2)or (2,1) and W/ - Apga•/(3p).
Equations(9) and (10) showthat the horizontalmotion
one diapir is coupled to the vertical motion of the
F2--/lAPfidS. (4)ofother
diapir. If we nondimensionalize
Ax, Az and Ar
However,as we only use the dipole flow field induced by x/a•a:•,andvelocities
v• andv2 by Apga•a2/(3p),
by diapir 2 as if it were unaffectedby the presenceof then the relative velocityof diapir 1 with respectto
diapir 1, we neglectthe viscousnormal stressesdue to diapir 2 has components
this flowimpingingon and beingdeflectedby diapir
1. (Thesenormalstresseffectscan be approximated
for rigid spheresusingthe methodof reflections[Hap-
dAx
1 - rl AxAz
u•-u•- dt •f5 At5
(11)
204
KELLY AND BERCOVICI: CLUSTERING OF RISING PLUME HEADS
dAz
Wl -- W2 --
where•/-
dr
--
1-•
(3Az•• •+1)
•
3At5Ar
(12)
a•/a2. Thereforethere is no relativemo-
tion betweenthe diapirs if they are of the same size
two regions with a high density of hotspots in the
South Pacificand Africa. Ribe and de Valpine inferred
that if this distribution resulted from an infinitesimal
Rayleigh-Taylor
instabilityin the D" with dominant
mode between œ -
1 and 2, then the mantle in the
D" wouldhaveto be approximately
10• timeslessvis-
(i.e.,r/- 1). ThediaPirs
onlyconverge
oneachother cousthanthe adjacent
mantle.However,
thisassumes
verticallyif dAze/dr < 0 which,bY (12), occursonly that the hotspot distribution at the surfacereflectsthe
if (1- •/)Az > 0 (giventhat Ar >;>i), i.e.,whenthe wavelengthof a linear instability at the core-mantle
largerdiapiris beneaththe smallerdiapir. The diapirs boundary. The clusteringphenomenondiscussedhere
converge
horizontally
if dAx2/dt < 0 which,by (11), showsthat nonlinearinteractions
betweenfully develoccursonlyif (1- •/)Az < 0, i.e., if the smallerdiapiris oped
plume
heads
canstrongly
influence
theirfinaldisbeneaththe largerone. The theory thereforepredicts tribution. That isi if plume conduitsare alsodeflected
two scenarios:
towardeachother as they followtheir clusteringplume
1. If the largerdiapirbeginsabovethe Smallerone,the
vertical separationwill grow, but the two diapirswill
heads,thenthe resultinghotspotswouldconceivably
be
clustered
as well.
be drawn toward the samehorizontal position and thus
will clusterand rise alongthe sameverticaltrack.
2. If the largerdiapirbeginsbeneaththe smallerone,
the vertical separationwill decreasewhile the horizontal separationincreases
(asthe largerdiapircatchesup
to and pushesasidethe smaller one) Once the larger
diapir passesthe smallerone, the systemis described
Acknowledgments.
Vie wish to thank C. Farnetani,
P. Olson and an anonymous reviewer for helpful comments
and suggestions.This Workwas supportedNSF Grant
EAR93-03402. AK was supported by a ResearchExperi-
encefor Undergraduates(REU) supplementto the above
NSF grant.
by Scenario1 above.(SeealsoWijngaarden[1•31 and References
MangaandStone[1993].)
Batchelor, G.K., An Introduction to Fluid Dynamics, CamAlthoughthismodelis onlyaCCurate
for widelysep- bridge Univ. Press, 1967.
arateddiapirs,it is in keepingwith observations
of the Bercovici,D. and J. Mahoney,DOublefloodbasaltsand
plume head separation at the 660 kilometer discontinulaboratoryexperiments,in particular that 1) there is
ity, Science, •66, 1367-1369, 1994.
little or no clusteringwhen the diapirs are at the same
Happel, J. and H. Brenner,Low ReynoldsNumberHydrolevel (Az -- 0); 2) that the interactionforcesfall off
dynamics,with SpecialApplicationsto Particulate Media,
rapidly with separationdistancebetweendiapirs;and
Prentice Hall, ch. 6, 1965.
3) clustering
largely
occurs
whenthesmaller
trailingdiapir movestowardthe samehorizontalpositionas the
leadingdiapir.
Manga, M. and H,A. Stone, Buoyancy-driven interactions
between two deformable viscous drops, J. Fluid Mech.,
œ56,647-683, 1993.
Olson,P. and H. Singer,Creepingplumes,J. Fluid Mech.,
158, 511-531,1985.
Implicationsand ConcluSions
Flood basalts
Continental flood basalts and ocean-
Ribe N.M. and D.P. de Valpine,The globalhotspotdistribution and instability of D", Geophys. Res. Left., œ1,
1.507-1510, 1994.
ic plateau
basalts
possibly
originate
fromlargeplU•meRichards,M.A. and R.W. Grifliths, Deflectionof plumes
headswhichundergopartial melting upon arriving at
the baseof the lithosphere
[Richardset ai., 1989;c.f.
White and McKenzie,1989, 1995]. However,if rising
plumeheadscanclusterandevencoalesce,
thenthe
by mantle shear flow: Experimental results and a simple
theory, Geophys. J., 9•, 367-376, 1988.
Richards, M.A., R.A. Duncan, and V.E. Courtillot, Flood
basaltsand hot-spot traces:plume headsand tails, Science, ϥ6, 103-107, 1989.
largeplumeheadswhicharriveat the lithosphere
may Skilbeck,J. N. and J. A. Whitehead,Formationof discrete
islandsin linear island chains,Nature, œ7œ,
499-501, 1978.
resultfrom the clusteringand evenmergingof smaller
White, R.S. and D.P. McKenzie,Magmatismat rift zones:
diapirs. Sincethe final plumeheadswouldhavespent
thegeneration
of volcanic
continental
margins
andflood
a portion of their ascentas smallerdiapirs, their net
basalts, J. Geophys. Res., 9•, 7685-7729, 1989.
rise time wouldbe longerand their entrainmentof sur- White, R.S. and D.P. McKenzie, Mantle plumes and flood
basalts, J. Geophys. Res., 100, 17543-17585, 1995.
roundingmantlemoreprofound.Moreover,the apparL. van, The mean rise velocity of pairwiseentoccurence
of double
floodbasaltevents
[BercoviciWijngaarden,
interacting bubbles in liquid, J. Fluid Mech., œ51,55-78,
andMahoney,
i994]mayalsobeexplained
bytheclus- 1993.
tering effect(whereina plumehead drawsa trailing
plumeheadinto its verticalascentpath).
A. Kelly and D. BercoVici, Department of Geology &
Global hotspot distribution The globalhotspot Geophysics,Schoolof Ocean& Earth Science& Technology,
distributionwasfound by Ribe and de Valpine[1994] University of Hawaii, 2525 Correa Road, Honolulu, HI
96822.
to have a very long wavelengthsignature,dominated
by sphericalharmonicdegreeœ - 2 with secondary' (ReceivedAugust15, 1996; revisedOctober28, 1996;
power at œ- 1. This quadrupolarsignaturereflects acceptedNovember16, 1996.)