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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]. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :.:: .<• .......... •:• •::•:•:.•:•:•!•.-_-'::.. .................. . .--------:•:•-.-.:•:•:•:•:•:•:• .............................. •:•:•:•:•:•:•:•::2:•-,•:•:•:•:..•:::•::•..*•j•,•:•.•.;g•;•::• ........ ::•,•:...,• •':'•'":'•ii'""'"'"'"••!•;•:'-"•'"":•'•"• •¾'"•'•'•'•' '":••••••!• .'" - :•::.''" ............... •:••.•`•`.• `•. •:•ii;•!•i•..•``•..•:•••••:••:• • .: •'•'"" .............. •-•i:i•:.:.?'" ------'-'-"•'"'"'"'"'•:'=•:'""•'••••i"-'"'•"'-•":•:' - ................................. •'•:..•.........•••::-'.".-'-'•:•:-..."• ......................... "•'"•"•'•'"•••"'• •'"'"'""-"'"-"•'"'••"":••••:•• .•;•.•.A<•..•,..•:• ......... •................ .< ................. •..•<•:.::• • •••••<..•?, ................ .,.•.•.:.:.:.• 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, œ•6, 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.)