structure and dynamics of sheared mantle plumes

Structure and dynamics of sheared mantle plumes

Ross C. Kerr and Catherine MeriauxResearch School of Earth Sciences, Australian National University, Mills Road, Canberra, ACT 0200, Australia([email protected])

[1] An extensive series of laboratory experiments is used to investigate the behavior of sheared thermalplumes. The plumes are generated by heating a small circular plate on the base of a cylindrical tank filledwith viscous fluid and then sheared by rotating a horizontal lid at the fluid surface. The motion of passivetracers in the plumes is visualized by the release of several dye streams on the hot plate. We systematicallyexamine the dependence of the convective flow on four dimensionless numbers: a velocity ratio, aRayleigh number, the viscosity ratio, and an aspect ratio. We identify and delineate two transitions in theconvective behavior: from a regime where the plume can spread upstream against the shear to a regimewhere the entire plume is advected downstream, and from a regime of negligible cross-stream circulationto a regime with significant cross-stream circulation and thermal entrainment. Our analysis of the steadyprofiles of the plumes shows that they initially rise with a constant vertical rise velocity. This rise velocitydepends on the buoyancy flux and ambient viscosity but is almost independent of the centerline plumeviscosity, which suggests that most of the thermal plume has a viscosity that is much closer to the ambientviscosity than the centerline viscosity. As the plumes approach the lid, they decelerate as the viscous dragon them steadily increases. The lateral spreading of the plumes under the lid is found to be well describedby similarity solutions derived for the spreading of compositional plumes on a rigid surface, if the effectiveviscosity of the thermal plumes is taken to be the ambient viscosity rather than the centerline viscosity. Asimilar theoretical model is found to roughly predict the upstream spreading of thermal plumes at lowshear, but it breaks down at moderate to high shear, where the entire plumes are advected downstream.When our results are applied to the Earth, we find that mantle plumes are mostly divided into only twoflow regimes in the upper mantle: plumes under slow moving plates experience upstream flow andnegligible cross-stream circulation, while plumes under faster moving plates (including all Pacific plumes)experience significant cross-stream circulation and are advected downstream. We also demonstrate thatgeochemical heterogeneities in a plume’s source region will result in an azimuthally zoned plume and in anasymmetric geographical distribution of geochemical heterogeneities in the erupted hot spot basalts, as isseen in the Hawaiian, Galapagos, Marquesas, and Tahiti/Society island chains. For individual mantleplumes, we determine their diameter and vertical rise velocity as well as the extent of upstream spreadingand the rate of lateral spreading under the lithosphere.

Components: 16,471 words, 30 figures, 14 tables.

Keywords: dynamics; mantle plume; shear; thermal plumes.

Index Terms: 8120 Tectonophysics: Dynamics of lithosphere and mantle—general; 8121 Tectonophysics: Dynamics,

convection currents and mantle plumes.

Received 28 April 2004; Revised 15 July 2004; Accepted 23 August 2004; Published 15 December 2004.

Kerr, R. C., and C. Meriaux (2004), Structure and dynamics of sheared mantle plumes, Geochem. Geophys. Geosyst., 5,

Q12009, doi:10.1029/2004GC000749.

G3G3GeochemistryGeophysics

Geosystems

Published by AGU and the Geochemical Society

AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES

GeochemistryGeophysics

Geosystems

Article

Volume 5, Number 12

15 December 2004

Q12009, doi:10.1029/2004GC000749

ISSN: 1525-2027

Copyright 2004 by the American Geophysical Union 1 of 42

1. Introduction

[2] Convection in the Earth’s mantle is driven bythree important buoyancy sources [cf. Davies,1999; Schubert et al., 2001]. The first (and dom-inant) source arises from the surface cooling of theEarth, which results in the sinking of cold plates ofoceanic lithosphere at subduction zones. The sec-ond source arises from internal heating of themantle due to the decay of radioactive elements.The third buoyancy source arises from heat con-ducted into the mantle through the core-mantleboundary, which leads to hot, rising plumes [Staceyand Loper, 1983]. These plumes are believed to beresponsible for intraplate volcanism [Condie,2001], with flood basalts being produced fromthe partial melting of plume heads, and linearvolcanic island chains being produced from thepartial melting of plume tails [Morgan, 1971, 1972,1981; Richards et al., 1989].

[3] As mantle plumes rise through the mantle, theyare sheared by large scale, plate-driven convection[e.g., Steinberger and O’Connell, 1998]. The be-havior of sheared mantle plume tails was initiallystudied using laboratory experiments with shearedcompositional plumes [Skilbeck and Whitehead,1978; Whitehead, 1982; Richards and Griffiths,1988; Griffiths and Richards, 1989]. However, thefirst few laboratory experiments with sheared ther-mal plumes [Richards and Griffiths, 1989] showeddramatically different behavior, with strong ther-mal entrainment and no gravitational instability ofstrongly tilted plume tails. Since Richards andGriffiths [1989], the only other experimentalinvestigation of sheared thermal plumes wasby Kincaid et al. [1995], who examined theinteraction of a mid-ocean ridge with an off-axisthermal plume head. In this paper we reportan extensive series of laboratory experimentsthat systematically investigate the behavior ofsheared thermal plume tails as a function of thegoverning dimensionless numbers. We find anumber of convective flow regimes, and wedelineate the transition boundaries between thesevarious regimes. We also use the experiments toquantify the deflection of a sheared thermalplume tail, which we find differs significantlyfrom that of a sheared compositional plume tail[Richards and Griffiths, 1988]. We then analyzeboth the upstream and lateral spreading ofthermal plumes under a moving plate, whichare also found to differ significantly from simpletheoretical models for the spreading of compo-sitional plumes.

[4] An important aim of our experiments is toinvestigate for the first time the stirring ofgeochemical heterogeneities as they rise insheared mantle plume tails, so that geographicalvariations in the geochemistry of ocean islandbasalts [e.g., DePaolo et al., 2001; Harpp andWhite, 2001] can be used to infer the distributionof geochemical heterogeneities in mantle plumesand in their source regions at the base of themantle. To examine this stirring, we introducestreams of dye into our thermal plumes, andobserve the motion of the dye in the sourceregion, in the rising plumes, and in the spreadingunder the moving plate.

[5] In section 2, we introduce the relevant fluiddynamical parameters and define the dimensionlessnumbers needed to describe the shearing of thermalplumes. Our experimental apparatus and proce-dures are then given in section 3. In section 4,we present the qualitative observations of severalsets of experiments that systematically explore theeffect of each dimensionless number on the con-vective behavior of the plumes. In section 5, wequantify the diameters and vertical rise velocitiesof thermal plumes, as well as their upstream andlateral spreading under the moving plate. We alsodelineate the transition boundaries between thevarious convective flow regimes. Our results arethen applied in section 6 to describe the motion ofgeochemical heterogeneities in sheared mantleplumes, and to predict the convective flow regime,diameter, vertical velocity, and sublithosphericspreading of individual mantle plumes. Our con-clusions are outlined in section 7.

2. Dimensionless Numbers

[6] Consider the shearing of the viscous thermalplume shown schematically in Figure 1. The fluidhas depth H, density ro, viscosity mo, coefficient ofthermal expansion a, thermal conductivity k, spe-cific heat cp and thermal diffusivity k = k/rocp. Theacceleration due to gravity is g, and the fluid isassumed to be incompressible. The fluid is heatedfrom below from a circular plate of radius of a,with a total heat flow rate Q, which produces athermal plume with a buoyancy flux B = gaQ/cpthat rises through the fluid. The fluid is alsosheared by a rigid upper boundary that moves withuniform horizontal velocity U, which tilts the risingthermal plume.

[7] The convective flow is described mathemati-cally by the continuity equation, the Navier-Stokes

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equations and the thermal advection-diffusionequation, together with the appropriate boundaryconditions [e.g., see Tritton, 1977, section 14].When these equations and boundary conditionsare nondimensionalized, the system is found tobe described by four dimensionless numbers: theRayleigh number

RaQ ¼ agQa2

pk2cpmo¼ Ba2

pk2mo; ð1Þ

the Prandtl number

Pr ¼ morok

; ð2Þ

the aspect ratio

A ¼ H

a; ð3Þ

and a velocity ratio

U ¼ U

wip

; ð4Þ

where wpi = (B/8pmo)

1/2 is the centerline velocity ofan isoviscous thermal plume (see Appendix A). Wenote that the velocity ratio U = (8p/Bn)

1/2, whereBn = B/(moU

2) is the buoyancy number used byFeighner and Richards [1995] and Kincaid et al.[1995]. Similarly, the Peclet number used byKincaid et al. [1995] is given by Pe = UH/k =2�3/2UARaQ

1/2.

[8] In a fluid with a temperature-dependent viscos-ity, there is a fifth dimensionless number: a vis-cosity ratio

M ¼ momi

; ð5Þ

where the centerline viscosity of the thermalplume, mi, is a function of the centerline tempera-ture Tp.

3. Laboratory Experiments

3.1. Apparatus

[9] The experiments were conducted in a cylindri-cal, Plexiglas tank with a radius of 300 mm(Figure 2). A circular electrical heating pad ofradius a = 50 mm was screwed onto the base ofthe tank, with the center of the pad 150 mm from thecenter of the tank. The padwas overlaid by a circularcopper plate, 3 mm thick, to ensure that the heating

Figure 1. Idealized experiment. A thermal plume isinitiated over a region of surface area pa2 which isheated at a rate Q, in a fluid layer of depth H whoseupper surface is moving with horizontal velocity U. Thefluid is Newtonian with density ro, viscosity mo,coefficient of thermal expansion a, thermal conductivityk, and heat capacity cp. The acceleration due to gravityis g.

Figure 2. Sketches of the experimental apparatus. Notshown are a side camera, a DC power supply used toelectrically heat the plate, and an electric motor andpulley system used to rotate the lid. (a) Tubes positionedto release dyed fluid on the upstream and downstreamedges of the plate. (b) Alternative positioning of thetubes to release dyed fluid on the inner and outer edgesof the plate.


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was relatively uniformly distributed. The heat flowrate Q was varied, from 10 to 60 W, to generatethermal plumes with different buoyancy fluxes.

[10] Two tubes filled with dyed fluid were set intothe tank such that the ends of the tubes lay about10 mm inside the edges of the plate at diamet-rically opposite positions. The tank fluid waseither glycerol or glucose syrup, whose relevantphysical properties are listed in Table 1. Mea-surements (with an Anton Paar digital densitom-eter) confirmed that the effect of the dye on thefluid densities was negligible in comparison tothe imposed thermal density differences in ourexperiments.

[11] A Plexiglas lid was inserted in the fluid at aheight H of either 120 mm or 240 mm. The lid wasrotated, at a range of velocities, by an electricmotor connected by belt and pulley systems. Rel-ative to the sense of rotation, two sets of positionsof the dye tubes were used. In the first set, the endsof the tubes lay at the ‘‘upstream’’ and ‘‘down-stream’’ edges of the plate (Figure 2a). In thesecond set, the ends of the tubes lay on the ‘‘inner’’and ‘‘outer’’ edges of the plate (Figure 2b).

3.2. Procedure

[12] The experiments were started by turning onthe electrical heating. We then waited a fewminutes for the thermal boundary layer to develop,and then used the shadowgraph technique toobserve a transient phase where one or moreplume heads rose through the fluid. After about10 minutes, a stable plume tail was established, andthe motor rotating the lid was turned on. After theshear had diffused through the depth of fluid,which took about a minute, the dyed fluid wasreleased (at a flow rate, about 6 ml/s, sufficient tomark most of the thermal boundary layer on the hotplate).

[13] The motion of the dye streamlines wasrecorded throughout the experiments by takingphotographs with three cameras giving a top view,a front view, and a side view from the downstreamdirection. The side and front views were taken withthe cameras either at middepth or at the height ofthe lid (except for the front view in Figure 6awhere the camera was tilted upward from about theheight of the hot plate), and with illuminatedscreens behind the tank. To reduce optical distor-tion in the front and side views, the cylindrical tankwas placed inside a 700 mm square tank, whichwas filled with deaerated water. Measurementsshowed that the remaining horizontal optical dis-tortion was less than 2% up to a distance of 150 mmfrom the center of the tank.

4. Qualitative Experimental Results

[14] The parameters varied in the experiments werethe lid velocity U, the fluid depth H, the heat flowrate Q, the fluid used, and the ambient temperatureof the fluid To (see Tables 1, 2, and 4). Theresulting values of the Rayleigh number RaQ,Prandtl number Pr, aspect ratio A, velocity ratio

Table 1. Physical Properties of the Experimental Fluids

Experiment

Variable 1–15 16–21 22–30 Units

Fluid Glycerol Glucose GlycerolAmbient temperature To 22 22 5 �CAmbient viscosity mo 0.83 1.59 5.0 Pa sThermal expansion coefficient a 6.1 � 10�4 3.6 � 10�4 6.1 � 10�4 K�1

Density ro 1260 1350 1260 kg m�3

Specific heat cp 2300 2100 2300 J kg�1 K�1

Thermal conductivity k 0.30 0.34 0.30 W m�1 K�1

Thermal diffusivity k 1.0 � 10�7 1.2 � 10�7 1.0 � 10�7 m2 s�1

Table 2. Lid Velocity

Experiment U, mm s�1

1 02, 3 0.234, 5, 22, 23 0.386, 7, 26, 27 0.938, 9, 28, 29 2.2510, 11 0.4612, 13 0.7614, 15 1.8616, 17 0.4518, 19 1.1020, 21 2.6624, 25, 30 0.60



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U and viscosity ratio M are listed in Table 3. Theviscosity ratio M was determined from the fluidviscosity mi at the measured temperature Tp at thecenter of the hot plate (Table 4).

[15] In this section, we systematically describe howthe motion of the dye streamlines (shown inFigures 3–18) depends on the velocity ratio U ,the Rayleigh number RaQ, the viscosity ratio M,and the aspect ratio A. We note that variations inthe Prandtl number Pr in our experiments are notdynamically significant, since the large viscositiesof the fluids used ensured that momentum wasdiffused on timescales (t = roH

2/mo = 14–84 s) thatwere small compared to the timescales of ourexperiments.

4.1. Effect of Velocity Ratio UU[16] In experiments 1–9, we varied U while keep-ing RaQ = 2.4 � 106, A = 4.8 and M = 3.1 (seeTable 3).

4.1.1. No Shear

[17] In experiment 1, the lid was stationary, sothere was no shear in the fluid. After the blueand red dyed fluid was released on the edges of thehot plate, the dye streams thinned laterally intowedges as they converged radially toward thecenter of the plate (Figure 3c). There, the dyestreams were bent upward and became thin verticalsheets (Figure 3b). The intensity of the dyesdecreased from the edge toward the center of theplume (Figure 3a), which reflected horizontal gra-dients of temperature, viscosity and vertical veloc-ity in the plume. Fluid parcels remained separate

entities in the upward laminar flow but they werestretched by the velocity gradients. As the plumeapproached the lid, the upward flow was replacedby radial outflow as a gravity current. The dyesheets were bent outward and then spread radiallywhile broadening laterally (Figure 3c). The pale,hotter, faster flowing fluid from the plume centernow overlay the darker, cooler, slower flowingfluid from the margins of the plume (Figure 3a).

4.1.2. Very Low Shear

[18] In the next two experiments, a small lidvelocity was imposed on the fluid, with the dyetubes placed in the upstream/downstream orienta-tion (see Figure 2a) in experiment 2, and in theinner/outer orientation (see Figure 2b) in experi-ment 3. Near the hot plate, the behavior of all thedye streams was indistinguishable from that seen inthe axisymmetric plume described in section 4.1.1.However, in the upper half of the fluid, the im-posed shear led to a noticeable tilting of the plume(Figure 4a), and the paths of the dye streams werevery different from the axisymmetric flow in sec-tion 4.1.1.

[19] For the dye stream from the downstream edgeof the plate (i.e., the red dye stream in Figures 4a,4b, and 4c), the flow remained relatively straight-forward. It was tilted backward by the shear as itapproached the lid, and was then bent outward toleave the pale, hotter fluid overlying the darker,cooler fluid. The fluid was then dragged down-stream at a constant velocity by the lid andremained a relatively thin sheet, in marked contrastto the radial spreading and broadening seen in theaxisymmetric plume in section 4.1.1.

[20] As the two dye streams from the inner andouter edges of the plate (i.e., the red and blue dyestreams in Figures 4d, 4e, and 4f) approached thelid, they were first pushed sideward by the shear(Figure 4d), and then rotated to leave the pale,hotter fluid overlying the darker, cooler fluid(Figure 4e). Finally the dye streams were pushedapart as the buoyant plume flattened verticallyagainst the lid while spreading out horizontally at

Table 3. Dimensionless Numbers of the Experiments

Experiment RaQ Pr A U M

1 2.4 � 106 6400 4.8 0 3.12 and 3 2.4 � 106 6400 4.8 0.21 3.14 and 5 2.4 � 106 6400 4.8 0.35 3.16 and 7 2.4 � 106 6400 4.8 0.85 3.18 and 9 2.4 � 106 6400 4.8 2.05 3.110 and 11 9.6 � 106 6400 4.8 0.21 11.112 and 13 9.6 � 106 6400 4.8 0.35 11.114 and 15 9.6 � 106 6400 4.8 0.85 11.116 and 17 2.3 � 106 9800 4.8 0.35 11.518 and 19 2.3 � 106 9800 4.8 0.85 11.520 and 21 2.3 � 106 9800 4.8 2.05 11.522 and 23 2.4 � 106 39000 4.8 0.35 5624 and 25 2.4 � 106 39000 4.8 0.55 5626 and 27 2.4 � 106 39000 4.8 0.85 5628 and 29 2.4 � 106 39000 4.8 2.05 5630 2.4 � 106 39000 2.4 0.55 56

Table 4. Measurements of Plume Parameters

ExperimentQ,W

Tp,�C

mi,Pa s

wp,mm s�1

rp,mm

1–9 10 36 0.27 3.0 8.5–10.310–15 40 56 0.075 6.8 11.516–21 40 59 0.14 5.9 9.722–30 60 55 0.090 6.4 7.0–8.8



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right angles to the direction of motion of the lid(Figure 4f ). This radial spreading caused the dyesheets to be curved around the laterally spreadingplume, as can be clearly seen at the end of the reddye stream in Figure 4d.

[21] For the dye stream from the upstream edge ofthe plate (i.e., the blue dye stream in Figures 4a,4b, and 4c), the flow was more complicated. It wasfirst tilted forward by the shear as it approached thelid (Figure 4a), before it then bifurcated dramati-cally. The pale, hotter fluid rose close to the lid andwas then dragged downstream over the top of thered downstream dye, while the darker, cooler fluidbent to initially flow about 45 mm upstream beforeit then flowed down and around to mark the edgesof the gravity current under the lid (Figures 4aand 4c).

4.1.3. Low Shear

[22] In experiments 4 and 5, the lid velocitywas increased by a factor of 1.7 more than insection 4.1.2, which resulted in a greater tilting ofthe plume (e.g., Figure 5a). The increased shearmade little difference to the behavior of the dyestream from the downstream edge of the plate (thered dye stream in Figures 5a, 5b, and 5c), but it hada major effect on the upstream dye (the blue dyestream in Figures 5a, 5b, and 5c). At this (almostcritical) shear, the upstream dye was only able toflow about 3 mm upstream (see Figure 4a) beforebeing dragged downstream, where it spread later-ally over the narrow, vertical sheet of downstreamdye (Figures 5b and 5c).

[23] For the two dye streams from the inner andouter edges of the plate (i.e., the red and blue dyestreams in Figures 5d, 5e, and 5f), the flow wasqualitatively similar to that seen at lower lid speeds(see section 4.1.2). The only changes were quan-titative: greater tilting of the sheets during their risein the plume (Figure 5d), and reduced radialspreading as a function of distance downstream(Figure 5f ).

4.1.4. Moderate Shear

[24] In experiments 6 and 7, the lid velocity wasincreased by a factor of 2.4 more than insection 4.1.3. This increased shear resulted in a

Figure 3. Experiment 1: no shear. (a) Front view,9 min after dyed glycerol was released, showing thepaths of the two blue and red streams of glycerol.(b) Side view, 3 min after dye release, showing that thestreams of dye are sheets that remain in the same verticalplane during ascent. (c) Top view, 5 min 30 after dyerelease, showing the narrowing wedges of dye on thehot plate from initial release point to the point of plumelift up and, farther up, the radial spreading andbroadening under the lid.



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more pronounced tilt of the plume (Figures 6a and6d), and reduced radial spreading of the dyestreams from the inner and outer edges of the plateas a function of distance downstream (Figure 6f ).

[25] As the sheet of upstream dye (the blue dyestream in Figures 6a, 6b, and 6c) rose in the plume,it slowly thinned in the vertical and spread hori-zontally, so that it was virtually a horizontal sheetby the time it reached the lid (Figure 6a). Theupstream dye then spread laterally under the lid,but remained above the narrow vertical sheet of

downstream dye (Figures 6b and 6c) as both dyestreams were dragged downstream.

4.1.5. High Shear

[26] In experiments 8 and 9, the lid velocity wasfurther increased, by a factor of 2.4 more thanin section 4.1.4. This increased shear resultedin a significant tilt of the plume at all depths(Figures 7a and 7d). It also affected the thermalboundary layer on the hot plate, causing the lift-off point of the plume to be displaced somewhat

Figure 4. Experiments 2 and 3: very low shear. (a) Front view, 12 min after dyed glycerol was released on theupstream and downstream edges of the plate. (b) Side view, 10 min after upstream/downstream dye release. (c) Topview, 29 min after upstream/downstream dye release. (d) Front view, 20 min after dyed glycerol was released on theinner and outer edges of the plate. (e) Side view, 20 min after inner/outer dye release. (f ) Top view, 20 min after inner/outer dye release.



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downstream of the center of the plate (Figures 7dand 7f ). The paths of all the dye streams werealso found to be significantly different to thoseseen in section 4.1.4.

[27] As the upstream dye (the blue dye stream inFigures 7a, 7b, and 7c) rose in the tilted plume,it thinned in the vertical and spread horizontally,before wrapping itself around the sheet of reddownstream dye. Meanwhile, the downstreamdye sheet was initially stretched in the vertical

(Figure 7a), before it then became thinner as itrose to the top of the plume (Figure 7b), as mostof the upstream dye was rotated to the sides ofthe plume (Figure 7c). This cross-stream circula-tion was also seen in dye streams from the innerand outer edges of the plate (Figures 7d, 7e, and7f ), which were rotated downward and inward asthe plume rose and spread under the rapidlymoving lid. These observations are similar toprevious observations of cross-stream circulationand thermal entrainment in tilted thermal plumes

Figure 5. Experiments 4 and 5: low shear. (a) Front view, 6 min 30 after dyed glycerol was released on theupstream and downstream edges of the plate. (b) Side view, 7 min 30 after upstream/downstream dye release. (c) Topview, 14 min 30 after upstream/downstream dye release. (d) Front view, 23 min after dyed glycerol was released onthe inner and outer edges of the plate. (e) Side view, 11 min after inner/outer dye release. (f ) Top view, 23 min afterinner/outer dye release.



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from a continuous ‘‘point source’’ [cf. Richardsand Griffiths, 1989, Figure 2].

4.2. Effect of Rayleigh Number RaQ

[28] To explore the effect of Rayleigh number,we conducted two sets of experiments in whichRaQ and U were varied, while maintaining ap-proximately constant values of A = 4.8 and M 11 (see Tables 1–4). The first set (experiments10–15) used glycerol (see Figures 8–10), and

had a Rayleigh number of 9.6 � 106, while thesecond set (experiments 16–21) used glucosesyrup (see Figures 11–13), and had a Rayleighnumber of 2.3 � 106. From these experiments,the effect of a factor of 4 difference in RaQ, for fixed

U , can be examined by comparing the dye streams ofexperiments 12–13 (Figure 9) with those of experi-ments 16–17 (Figure 11), and those of experiments14–15 (Figure 10) with those of experiments 18–19(Figure 12). However, it is very important to realizethat varying RaQ while imposing fixed U involves

Figure 6. Experiments 6 and 7: moderate shear. (a) Front view, 16 min after dyed glycerol was released on theupstream and downstream edges of the plate. (b) Side view, from a camera raised to the height of the lid, 8 min afterupstream/downstream dye release. (c) Top view, 17 min after upstream/downstream dye release. (d) Front view,16 min 40 after dyed glycerol was released on the inner and outer edges of the plate. (e) Side view, 9 min after inner/outer dye release. (f ) Top view, 16 min 40 after inner/outer dye release.



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applying larger lid velocities at larger Rayleighnumbers, since U / (RaQ)

�1/2 (see equations (4)and (A8)).

[29] The factor of 4 increase in RaQ (with asimultaneous doubling in U (see Tables 2 and 3))had three main effects. First, it increased cross-stream circulation in the plumes, which resulted inmore of the upstream dye being rotated and pushedto the sides of the plume as it spread under the lid.Second, it reduced lateral spreading of the gravitycurrent under the lid. Third, it decreased, from

22.5 mm (Figure 11c) to 12.5 mm (Figure 9c),the distance that upstream dye was able to flowupstream under the lid.

4.3. Effect of Viscosity Ratio MMMMM[30] To determine the effect of the viscosity ratio,experiments 2–9 and 16–21 were combined with afurther set of experiments (22–29), which wereconducted in a cold room (see Figures 14–17).The result of an order of magnitude increase in theviscosity ratio M, at almost constant values of U,

Figure 7. Experiments 8 and 9: high shear. (a) Front view, 6 min 15 after dyed glycerol was released on theupstream and downstream edges of the plate. (b) Side view, from a camera raised to the height of the lid, 6 min afterupstream/downstream dye release. (c) Top view, 6 min after upstream/downstream dye release. (d) Front view, 9 minafter dyed glycerol was released on the inner and outer edges of the plate. (e) Side view, 6 min 30 after inner/outerdye release. (f ) Top view, 9 min 30 after inner/outer dye release.



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RaQ and A, can then be seen by carefully compar-ing: Figures 5, 11, and 14; Figures 6, 12, and 16; andFigures 7, 13, and 17. One effect is stronger cross-stream circulation in the plumes, which resulted inmore of the upstream dye being rotated and pushedto the sides of the gravity current as it spread underthe lid, while the downstream dye rose to the top ofthe current. Another effect, seen when the upstreamdye streams in Figures 5c, 11c, and 14c are com-pared, is a systematic increase in the upstreamspreading of the plume fluid (see Figure 19a). Theincrease in M also slightly decreases tilting of the

plume, but it has little effect on lateral spreading ofthe plume fluid under the lid.

[31] Using our observations from experiments 2-9and 16-29, it is also possible to delineate theboundaries in (U, M) space between several con-vective regimes, for A = 4.8 and RaQ 2.4 � 106.First, there is a boundary (see Figure 19a) thatseparates upstream spreading of the plume fromsolely downstream advection of the plume. Sec-ond, there is a boundary (see Figure 19b) thatseparates regions where there was negligible

Figure 8. Experiments 10 and 11: high Rayleigh number and very low shear. (a) Front view, 10 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 5 min after upstream/downstream dye release. (c) Top view, 11 min after upstream/downstream dyerelease. (d) Front view, 4 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Side view,8 min after inner/outer dye release. (f ) Top view, 12 min after inner/outer dye release.



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cross-stream circulation from regions where signif-icant cross-stream circulation was observed.

4.4. Effect of Aspect Ratio AAAAAA[32] To examine the effect of aspect ratio, weperformed experiment 30 (Figure 18), which hadthe same heat flow rate and shear as experiment24 (Figures 15a, 15b, and 15c) but half the fluiddepth, which approximately halved the rise timeof the thermal plume. In both experiments, thevelocity ratio was found to be very close to the

critical value that separated upstream spreadingfrom downstream advection, which suggests thatthis transition is almost independent of the aspectratio.

4.5. Plume Instabilities

[33] In the course of our experimental program,two types of plume instability were observed. Thefirst type appeared in experiments with very highvelocity ratios (e.g., U � 2.05 at RaQ = 2.4 � 106,and U � 0.85 at RaQ = 9.6 � 106). In these

Figure 9. Experiments 12 and 13: high Rayleigh number and low shear. (a) Front view, 8 min after dyed glycerolwas released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to the height ofthe lid, 9 min after upstream/downstream dye release. (c) Top view, 14 min after upstream/downstream dye release.(d) Front view, 10 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Side view, 8 minafter inner/outer dye release. (f ) Top view, 12 min after inner/outer dye release.



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experiments, the very high shear pushed the ther-mal plume to the edge of the hot plate, producingan instability of the thermal boundary layer on thehot plate that lead to the formation of new thermalplumes.

[34] The second type of instability was seen insome preliminary experiments with a very largeviscosity ratio and inner/outer dye release. In thiscase, a varicose instability gradually developed inthe tilted thermal plume, which was probably trig-gered by the symmetrical disturbance created by the

dye release. The varicose instability appeared to besimilar to the varicose instability of compositionalplumes described by Huppert et al. [1986], and itoccurred at internal and external plume Reynoldsnumbers (Rei = ro

4/5B3/5g�2/5Dr�2/5mi

�1 = 1.6 andReo = ro

4/5B3/5g�2/5Dr�2/5mo

�1 = 0.029, where Dr isthe density difference between the two fluids)fairly close to varicose regime boundary foundby Huppert et al. [1986]. However, the instabil-ity was not wanted in the current study, becauseit is not expected to occur in the mantle (whereRei � 10�15 for mantle plumes). Fortunately,

Figure 10. Experiments 14 and 15: high Rayleigh number and moderate shear. (a) Front view, 4 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 6 min after upstream/downstream dye release. (c) Top view, 7 min after upstream/downstream dyerelease. (d) Front view, 2 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Side view,6 min after inner/outer dye release. (f ) Top view, 4 min after inner/outer dye release.



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careful experimentation showed that we coulddelay or eliminate the instability, by waiting asufficiently long time that the thermal plume wasvery stable before the lid was rotated.

[35] In experiments with tilted compositionalplumes, Whitehead [1982] found that stronglytilted plumes can break up into discrete diapirs(see also run A1 in Figure 7 of Richards andGriffiths [1988]). He found that the critical tiltangle required for instability increased with de-creasing plume Reynolds number (see Whitehead

[1982, Figure 5], where we note that the plumeReynolds number used is equal to 0.25 Rei

5/4),and that sufficiently small plumes were ‘‘perpet-ually passively stretched’’ and remained stable.In stark contrast, this gravitational instabilitywas never observed in our experimental program(which had thermal plumes with internal Reynoldsnumbers Rei ranging from 0.42 to 1.8), evenat the very large tilt angles seen near the lid inFigures 7, 13, and 17. Similarly, the instability wasnever observed in the thermal plume experimentsof Richards and Griffiths [1989]. The marked

Figure 11. Experiments 16 and 17: low shear and large viscosity ratio. (a) Front view, 12 min after dyed glucosewas released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to the height ofthe lid, 12 min after upstream/downstream dye release. (c) Top view, 14 min after upstream/downstream dye release.(d) Front view, 10 min 30 after dyed glucose was released on the inner and outer edges of the plate. (e) Side view, 12min after inner/outer dye release. (f ) Top view, 10 min 30 after inner/outer dye release.



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difference in stability between tilted compositionalplumes and tilted thermal plumes is probably dueto diffusion, which is much greater in the thermalplume experiments because the thermal diffusivityis about four orders of magnitude larger thanthe compositional diffusivity. Little diffusion isseen in the compositional plume experiments,but it is substantial in the thermal plume experi-ments, where it leads to broad gradients in vis-cosity and axial velocity inside the plumes, andto significant thermal entrainment into tiltedplumes, which are likely to inhibit the growth of

gravitational instabilities [Richards and Griffiths,1989].

4.6. Summary of Qualitative Observations

[36] When all the experiments (Figures 3–18) arecombined, the following general observations canbe made about the shearing of viscous thermalplumes:

[37] 1. In the plume’s source region, distinct par-cels of fluid are focused into narrow vertical sheets(see Figures 3b, 4b–17b, and 4d–17d).

Figure 12. Experiments 18 and 19: moderate shear and large viscosity ratio. (a) Front view, 7 min after dyedglucose was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 9 min after upstream/downstream dye release. (c) Top view, 9 min after upstream/downstream dyerelease. (d) Front view, 6 min after dyed glucose was released on the inner and outer edges of the plate. (e) Side view,6 min after inner/outer dye release. (f ) Top view, 7 min after inner/outer dye release.



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[38] 2. In the rising plume, fluid parcels experi-ence vertical laminar stretching due to radialvariations in temperature, viscosity, and verticalvelocity.

[39] 3. Shear in the ambient fluid results in tiltingof the plumes, and in the plume fluid beingdragged downstream as it spreads out as a gravitycurrent under the lid.

[40] 4. Axisymmetry is lost in sheared plumes, butleft/right symmetry about the direction of shearremains. For example, in Figures 3f–17f, the red

fluid parcels from the ‘‘inner’’ edge of the platealways remain on that side, and are not stirred tothe other side of the plume.

[41] 5. As the shear ratio is increased, there is atransition from some upstream spreading under thelid, to solely downstream advection of plume fluid.Upstream spreading increases slightly with increas-ing viscosity ratio, and with decreasing Rayleighnumber.

[42] 6. As the shear ratio is further increased, thereis another transition as cross-stream circulation

Figure 13. Experiments 20 and 21: high shear and large viscosity ratio. (a) Front view, 6 min after dyed glucosewas released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to the height ofthe lid, 6 min after upstream/downstream dye release. (c) Top view, 6 min after upstream/downstream dye release.(d) Front view, 7 min 30 after dyed glucose was released on the inner and outer edges of the plate. (e) Side view, 4min after inner/outer dye release. (f ) Top view, 6 min after inner/outer dye release.



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becomes apparent in the plume. Cross-stream cir-culation increases with increasing viscosity ratioand Rayleigh number.

[43] 7. Tilting of plume fluid under the lidincreases greatly with increasing shear ratio, butdecreases slightly with increasing viscosity ratio.

[44] 8. Lateral spreading of plume fluid under thelid decreases greatly with increasing shear ratio andincreasing Rayleigh number, but is insensitive tothe viscosity ratio.

[45] 9. The behavior of plume fluid from theupstream side of the source region is very sensitiveto the convective flow regime (Figure 19):

[46] . In the upstream spreading regime, the up-stream fluid parcels bifurcate, with the hotterupstream fluid being dragged downstream over athin vertical sheet of fluid from the downstreamside of the source, while the cooler upstream fluidflows upstream and around to the edges of thespreading gravity current under the lid.

Figure 14. Experiments 22 and 23: low shear and very large viscosity ratio. (a) Front view, 14 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 15 min 30 after upstream/downstream dye release. (c) Top view, 22 min after upstream/downstreamdye release. (d) Front view, 15 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Sideview, 17 min after inner/outer dye release. (f ) Top view, 14 min after inner/outer dye release.



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[47] . In the intermediate flow regime of down-stream advection with negligible cross-stream cir-culation, all the upstream source fluid is draggeddownstream where it spreads out over the verticalsheet of downstream source fluid.

[48] . In the regime of significant cross-streamcirculation, the upstream fluid is rotated to the edgesof the plume, while the downstream fluid risesthrough the center of the plume to its upper surface.

[49] 10. The thermal plumes were stable even atthe very large tilt angles. This behavior is prob-

ably due to thermal diffusion, which results inbroad gradients in viscosity and axial velocityinside the plumes, as well as significant thermalentrainment.

5. Quantitative Experimental Results

[50] In this section, we use our experimental obser-vations (Figures 3–18) to quantify many aspects ofthe rise and shearing of thermal plumes, and theirspreading under the rotating lid.

Figure 15. Experiments 24 and 25: a critical shear and very large viscosity ratio. (a) Front view, 19 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 17 min 30 after upstream/downstream dye release. (c) Top view, 19 min after upstream/downstreamdye release. (d) Front view, 11 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Sideview, 11 min after inner/outer dye release. (f ) Top view, 10 min after inner/outer dye release.



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5.1. Thermal Plume Parameters

[51] During the experiments, a number of plumeparameters were measured (see Table 4). First, thetemperature Tp was measured using a thermistor onthe hot plate in the center of the thermal plume.Second, the plume radius rp was measured from theseparation of the outer edges of the dye streams atmiddepth in Figures 3a, 4a–17a, and 4e–17e,although we note that it is unclear what range oftemperatures are marked by these dye streamlines.Third, the centerline rise velocity wp was measured

from videos of the rise of small blobs of dyed fluidinjected into the base of unsheared plumes. Theseblobs were found to rise with a constant velocity(Figure 20), which suggests that the centerlinetemperature and viscosity are also constant in thesethermal plumes, which can be explained by notingthat the fluid depths (H = 120 or 240 mm) aremuch less than the theoretically predicted scaleheights (zscale � a RaT

1/3 5–7 m) for thermaldiffusion within the plumes [Olson et al., 1993,equation (28)].

Figure 16. Experiments 26 and 27: moderate shear and very large viscosity ratio. (a) Front view, 19 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 16 min 30 after upstream/downstream dye release. (c) Top view, 21 min after upstream/downstreamdye release. (d) Front view, 8 min after dyed glycerol was released on the inner and outer edges of the plate. (e) Sideview, 9 min after inner/outer dye release. (f ) Top view, 8 min after inner/outer dye release.



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[52] In Table 5, we present theoretical estimates ofthe plume parameters from two simple models. Thefirst model is the isoviscous model of Olson et al.[1993], summarized in Appendix A, which givespredictions of the centerline velocity wp

i , radiallengthscale dp and plate temperature Tp

i = To + DT.The second model is a Poiseuille pipe flow model[cf. Skilbeck and Whitehead, 1978; Whitehead,1982; Griffiths and Campbell, 1991a; Steinberger,2000], which predicts a rise velocity

wp* ¼ B

2pmi

� �1=2

ð6Þ

and a plume radius

rp* ¼ 8Bmipg2Dr2

� �1=4

; ð7Þ

where Dr = roa(Tp � To). We find that theisoviscous model underestimates wp and over-estimates rp 2dp (i.e., taking the edge of theGaussian temperature distribution to be at about 2lengthscales from the center), while the Poiseuillepipe flow model overestimates wp and under-estimates rp. These discrepancies are probably dueto neither model giving an adequate description of

Figure 17. Experiments 28 and 29: high shear and very large viscosity ratio. (a) Front view, 8 min after dyedglycerol was released on the upstream and downstream edges of the plate. (b) Side view, from a camera raised to theheight of the lid, 7 min after upstream/downstream dye release. (c) Top view, 7 min after upstream/downstream dyerelease. (d) Front view, 7 min 30 after dyed glycerol was released on the inner and outer edges of the plate. (e) Sideview, 6 min 30 after inner/outer dye release. (f ) Top view, 6 min after inner/outer dye release.



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the viscosity structure of the plume; that is, theisoviscous model ignores the lower viscosities inthe interior of the plume, while the pipe flow modeloverestimates this effect in assuming that all of theplume interior has the measured centerline tem-perature and viscosity.

[53] We also note the isoviscous model under-estimates the measured values of Tp, which mayreflect a fundamental difference between the

model and our experiments: the isoviscous modelof Olson et al. [1993] has a uniform imposedplate temperature, while our experiments have auniform imposed heat flux, which leads to platetemperatures that systematically increase from

Figure 18. Experiment 30: small aspect ratio, verylarge viscosity ratio, and a critical shear. (a) Front view,13 min after dyed glycerol was released on the upstreamand downstream edges of the plate. (b) Side view,14 min after upstream/downstream dye release. (c) Topview, 15 min after upstream/downstream dye release.

Figure 19. Observed convective regimes as a functionof the viscosity ratio M and the velocity ratio U (forexperiments with A = 4.8 and RaQ 2.4 � 10 6).(a) The dashed line separates experiments in which therewas some upstream spreading of plume (solid circles)from those where there was solely downstream advec-tion of the plume (diamonds). The numbers beside thesolid circles give the observed upstream spreadingdistance (in mm). (b) Dashed line that separatesexperiments in which there was negligible cross-streamcirculation (black squares) from those in whichsignificant cross-stream circulation of the upstreamdye was observed (empty circles).



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low values at the edge of the plate to high valuesat its center.

5.2. Thermal Plume Profiles

[54] From the front views of the experiments withthe dye tubes in the upstream/downstream posi-tions (Figures 4a–18a), measurements were madeof the horizontal displacements d of the centerlinesof the sheared thermal plumes as a function ofheight z. As the plumes were being rotated in acircular tank at a radius R = 150 mm, the horizontaldisplacements were greater than the apparent dis-placements da as seen from the front views, butthey were easily determined using the expression

d ¼ R sin�1 da

R

� �: ð8Þ

[55] For compositional plumes in which there isnegligible diffusion of buoyancy, Richards andGriffiths [1988] found that steady plumes tilted ina linear shear had a parabolic trajectory given by

d zð Þ ¼ Uz2

2vsH; ð9Þ

where the vertical rise of the plume was governedby a modified Stokes velocity

vs ¼ ksgDr r2pmo

ð10Þ

and the constant ks = 0.54 ± 0.02. For the thermalplumes produced in our experiments, we can testthe simple parabolic description in equation (9) byplotting d/H against z2/H2 to see if a straight lineresults. This test is shown for 10 experiments inFigure 21, in which horizontal displacementsrelative to the center of the plumes are shownfrom a height 45 mm above the hot plate to avoiddifferences in the lift-off behavior of the plumes.For z2/H2 up to about 0.4–0.5 (i.e., for z/H up toabout 0.6–0.7), the plumes were found to beaccurately described by equation (9), and linearleast squares fits to the data (e.g., Figure 22) wereused to determine the plume rise velocities vs. Therise velocities were then combined with values ofrp* (see equation (7) and Table 5) to determine theStokes constants ks, which are listed in Table 6.The Stokes constants are found to be independentof U and RaQ. However, there is a strongdependence on the viscosity ratio M, shown inFigure 23, which is accurately represented by theexpression

ks ¼ 0:30M0:55 0:05: ð11Þ

The plume rise velocity is therefore given by

vs ¼ 0:308B

pmo

� �1=2

M0:05 0:05 ¼ 2:4wip M

0:05 0:05; ð12Þ

a result which indicates that the rise velocity of athermal plume does not have a significant depen-dence on the centerline viscosity mi. This resultprobably reflects the fact that most of the plume hasa viscosity that is much closer to the ambientviscosity mo than the centerline viscosity mi.

[56] Figures 21 and 22 also show that the simpleparabolic description in equation (9) breaks downwhen z/H is greater than about 0.6–0.7, whichreflects a slow down of the thermal plumes asthey approach the overlying rigid lid. This de-celeration, which is due to increasing viscousdrag on the plumes, has been previously studiedfor rigid spheres [see Happel and Brenner, 1965,pp. 329–331] and fluid spheres [Griffiths and

Figure 20. Measurements of the height z as a functionof time t of a small blob of dyed fluid injected into thecenter of an unsheared thermal plume of the type used inexperiments 1–9. The line is a linear least squares fitwhich gives a centerline rise velocity wp = 3.0 mm s�1.

Table 5. Theoretical Estimates of Plume Parameters

Experimentwp

i,mm s�1

dp,mm

Tpi,

�Cw*p,

mm s�1r*p,mm

1–9 1.1 9.3 27 3.8 6.310–15 2.2 7.9 38 14.6 4.216–21 1.3 9.4 42 8.7 5.322–30 1.1 9.3 38 16.3 4.0



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Campbell, 1991b] as they approach either a rigidboundary or a free surface. In an attempt toquantify the deceleration of our thermal plumes,fourth-order polynomial expressions for d(z) werefit to the data for 8 experiments, which werethen differentiated to obtain expressions for the

Figure 22. Two straight lines showing linear leastsquares fits to the tilting data for experiments 26 and 28.The slope of these lines is then used to infer the Stokesconstant ks for each experiment.

Figure 21. Measurements of the tilting of thermal plumes. The horizontal axis gives the horizontal displacements drelative to the center of the plume at a height 45 mm above the hot plate, and are nondimensionalized by the fluiddepth H. The vertical axis gives the square of the dimensionless height in the tank (i.e., z2/H2). (a) Experiments 6 and8. (b) Experiments 12 and 14. (c) Experiments 18 and 20. (d) Experiments 22, 24, 26, and 28.

Table 6. Stokes Constant ks Determined From theProfiles of the Thermal Plumes

Experiment ks

6 0.548 0.5412 1.1614 1.1118 1.1120 1.2822 2.8824 2.9426 2.2528 2.66



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plume rise velocities vs(z) [cf. Richards andGriffiths, 1988]:

vs zð Þ ¼ Uz

H

dd

dz

� ��1

: ð13Þ

The rise velocities in the upper half of the tank areshown in Figure 24, where the velocities have beennondimensionalized by the velocity vs

m at middepth(at z = H/2). Despite some scatter in the velocitycurves, the figure clearly shows the strong effect ofthe rigid lid, which has halved the rise velocity of theplumes by the time they are about 30 mm from thelid (i.e., by z/H = 0.875). The data show no cleardependence on either the diameter or the tilt of theplumes, and can be roughly fit by the power law:

vs zð Þvms

¼ H � z

H=2

� �0:64 0:09

; ð14Þ

where 0 < H � z < H/2, which satisfies theconstraint that vs(H) = 0.

5.3. Lateral Spreading of Thermal PlumesUnder the Moving Lid

[57] Using the top views of the experiments(Figures 4c–17c and 4f–17f ), we measured theapparent plume width W as a function of the arclength s measured downstream of the hot plate. Forupstream/downstream dye release (Figure 2a) theupstream dye indicates both edges of the spreadingplume (e.g., Figure 11c), while for inner/outer dyerelease (Figure 2b) each dye stream indicates oneedge of the plume (e.g., Figure 5f ). The plumewidths W were only measured for intermediatevelocity ratios (U = 0.35 � 0.85), because lowvelocity ratios produced very wide plumes (e.g.,

Figure 4) whose spreading was probably affectedby the finite size of our experimental tank, whilehigh velocity ratios produced strongly tilted plumes(e.g., Figure 7) where there is little spreadingduring one revolution of the lid. The arc lengthss = Rmq were determined along a circle that best fitthe centerline between the plume edges, where Rm

was the measured radius of the best-fit circle and qwas the radial angle (which varied from 0�–120�to 0�–220� in different experiments).

[58] The nondimensionalized plume widths of 9experiments are shown in Figure 25, where theyare compared with a similarity solution (see equa-tions (B6)–(B7) of Appendix B), valid at largedistances downstream, for the spreading of a com-positional plume in which there is negligible dif-fusion of buoyancy:

W

Li¼ 2C

s

Li

� �1=5

; ð15Þ

where the constant C = 3.70. . . and the lengthscaleLi is given by

Li ¼B3

96p3mig2Dr2 U 4

� �1=4

: ð16Þ

From Figure 25, several observations can be made.First, the initial spreading is generally more rapid,

Figure 23. The Stokes constant ks as a function of theviscosity ratio M. The straight line shows a leastsquares fit to the data: ks = 0.30 M0.55±0.05, where theerror bar shown is two standard deviations.

Figure 24. Curves showing the plume rise velocityvs in the upper half of the tank in eight experiments,as a function of the dimensionless distance from thelid (H � z)/H. The velocities have been nondimen-sionalized by the velocity vs

m at middepth. The datacan be roughly fit by a power law expression, givenin equation (14) and shown by the dashed curve.



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but at large distances downstream the spreadingappears to gradually approach the 1/5 power law inequation (15), in a manner that is typical of theslow approach to a similarity solution that is onlyasymptotically valid (e.g., see the fixed-volumerelease experiments presented in Figures 6a and 6cof Lister and Kerr [1989]). Second, the depen-dence on U in the similarity solution is supportedby the consistent asymptotic spreading of threepairs of experiments that differed only in theirvelocity ratios (experiments 5 and 7 in Figure 25a,experiments 12 and 14 in Figure 25b, andexperiments 17 and 19 in Figure 25c). Third, theplume widths indicated by upstream dye release arelarger than those indicated by inner/outer dyerelease.

[59] In Figure 26, the spreading data are orga-nized into two groups: the experiments withupstream/downstream dye release in Figure 26a,and the experiments with inner/outer dye releasein Figure 26b. In both these figures, a systematiceffect with increasing viscosity ratio can be seen:in Figure 26a, the asymptotic spreading of ex-periment 26 lies below that of experiments 12and 14, while in Figure 26b, the asymptoticspreading of experiments 17 and 19 lies belowthat of experiments 5 and 7. This variation withviscosity ratio implies that the plume spreadinghas a weaker dependence on mi than is predicted byequations (15)–(16). To determine whether thespreading data shows any dependence on mi, theexperimental results were rescaled (see Figure 27)

Figure 25. Dimensionless plume width W/Li as a function of dimensionless downstream arc length s/Li, where Li isgiven by equation (16). The solid line shows the scaling law given in equation (15). The experimental data have beenseparated into the four groups listed in Table 4, which correspond to different values of RaQ andM. (a) Experiments 5and 7. (b) Experiments 12, 13, and 14. (c) Experiments 17, 18, and 19. (d) Experiment 26.



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with mi replaced by mo in equations (15)–(16),giving

W

Lo¼ 2C

s

Lo

� �1=5

; ð17Þ

where

Lo ¼B3

96p3mog2Dr2 U 4

� �1=4

: ð18Þ

In both Figure 27a and Figure 27b, the collapse ofthe data is much better: in particular, the asymptoticspreading of experiments 12 and 14 is consistentwith that of experiment 26 (Figure 27a), and theasymptotic spreading of experiments 5 and 7 isconsistent with that of experiments 17 and 19

(Figure 27b). We can therefore conclude that thespreading of a thermal plume, like the Stokesvelocity of a thermal plume (see equation (12)), doesnot have a significant dependence on the centerlineviscosity mi, which again probably reflects the factthat most of the plume has a viscosity that is muchcloser to the ambient viscosity mo than the centerlineviscosity mi. We can also conclude that the constant2C in equation (21) has a value of about 10 ± 1 forour upstream dye release experiments (Figure 27a),and about 5.5 ± 0.5 for our lateral dye releaseexperiments (Figure 27b).

5.4. Upstream Spreading of ThermalPlumes

[60] Using the front and top views of the experi-ments with upstream/downstream dye release (Fig-

Figure 26. Dimensionless plume width W/Li as afunction of dimensionless downstream arc length s/Li,where Li is given by equation (16). The solid line showsthe scaling law given in equation (15). The experimentaldata have been separated according to the orientation ofthe dye tubes (see Figure 2). (a) Upstream/downstreamdye release: experiments 12, 14, 18, and 26. (b) Inner/outer dye release: experiments 5, 7, 13, 17, and 19.

Figure 27. Dimensionless plume width W/Lo as afunction of dimensionless downstream arc length s/Lo,where Lo is given by equation (18). The solid line showsequation (17). The experimental data have beenseparated according to the orientation of the dye tubes(see Figure 2). (a) Upstream/downstream dye release:experiments 14, 16, 20, and 28. (b) Inner/outer dyerelease: experiments 5, 7, 15, 19, and 21.



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ures 4a–8a and 4c–18c), we measured the distanceLs that the blue upstream dye was able to spreadupstream before being dragged downstream by themoving lid. The upstream spreading is substantialat very small velocity ratios, but at moderate tohigh velocity ratios, all the dye is advected down-stream and there is no upstream spreading (seeFigure 19a). The spreading distances are given inTable 7, where they are compared with the dis-tances predicted by a simple theoretical model forthe upstream spreading of a compositional plume.The theoretical model is presented in Appendix C,and has two possible flow regimes, depending onthe viscosities of the two fluids. For our experi-ments, the distance Ls to the upstream stagnationpoint is given by the second regime (equation (C6))in experiment 28, and by the first regime (equation(C3)) in all the other experiments. The calculatedtimes ts for the gravity currents to reach theirupstream stagnation points are all less than 90 s,so all the experiments had sufficient time to reachsteady-state.

[61] When Table 7 is examined, it is found that thetheoretical model gives a rough prediction of Lswhere the thermal plume is only slightly tilted(�10�), which is the case for experiments 2 and10 (where the velocity ratio is very small). Wherethe plume tilt is somewhat larger (�10�–20�), themodel overestimates the upstream spreading byabout a factor of 2, as seen for experiments 12,16 and 22 (where the velocity ratio is small and theviscosity ratio is large to very large). Finally, forthe remaining experiments, where the plume tiltbecomes large near the lid (>20�), the model failsto predict that there is a regime of no upstreamspreading at moderate to high velocity ratio. This

failure presumably reflects a breakdown in themodel assumption that ‘‘details of the supply ofmaterial by the plume conduit’’ can be ignored.

5.5. Convective Flow Regimes

[62] Using the observations presented in Table 7 andFigure 19a, the velocity ratio Ua required for noupstream flow (i.e., downstream advection) can beroughly quantified. From the dashed line that marksthe regime boundary in Figure 19a, we find that

Ua ¼ 0:33M0:13 ð19Þ

when A = 4.8 and RaQ = 2.4 � 106. Next weobserve that experiments 24 and 30 were both veryclose to the critical velocity ratio, despite the factorof 2 difference in A, which shows that Ua is almostindependent of A. Lastly, from experiments 12 and16, it is seen that a factor of 4 increase in RaQ hasreduced the upstream spreading from 22.5 mm to12.5 mm. We can therefore infer that a furtherfactor of 4 increase in RaQ should eliminateupstream spreading, so that Ua = 0.35 at RaQ 4 � 10 7 and M 11.3, and hence that

Ua ¼ 1:26M0:13 Ra�0:091Q : ð20Þ

[63] The velocity ratio Uc that marks the transitionfrom negligible to significant cross-stream circula-tion can be evaluated similarly, but less completely.First, from the approximate regime boundaryshown in Figure 19b, we obtain

Uc ¼ 1:83M�0:20 ð21Þ

when A = 4.8 and RaQ = 2.4 � 106. Next, whenFigures 11, 13, and 14 are compared, it is seen thatthe circulation in Figure 11 is roughly intermediatebetween that observed in Figures 13 and 14. Afactor of 4 increase in RaQ is therefore about half aseffective as a factor of 2.4 change in U, from whichwe can infer that Uc is roughly proportional toRaQ

�0.316, and hence that

Uc ¼ 189M�0:20 Ra�0:316Q ð22Þ

when A = 4.8. Experimental constraints preventedus from determining Uc at small aspect ratios, butwe anticipate that Uc increases as A decreases.

6. Application to Mantle Plumes

[64] In this section, we examine the application ofour experiments to thermal plumes in the mantle. At

Table 7. Upstream Spreading Distances

ExperimentMeasuredLs, mm

Ls FromTheoretical Model, mm

2 45 424 3 266 0 108 0 410 85 5312 12.5 3114 0 1316 22.5 3918 0 1620 0 722 40 6824 0 4726 0 2728 0 930 1 47



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the outset, we acknowledge that this discussion issimplistic and at best approximate, as our experi-ments do not model many features of mantle con-vection, including the free-slip boundary conditionat the base of the mantle, large vertical variations inmantle viscosity, a non-Newtonian rheology, pres-sure-induced phase changes, partial melting, sub-duction zones, and plume-ridge interactions.

6.1. Dimensionless Numbers for MantlePlumes

[65] To examine the dynamical similarity betweenour experiments and mantle plumes, we need toevaluate the 5 dimensionless numbers: RaQ, Pr, A,

U and M. We begin by listing in Table 8 sometypical parameter values for mantle plumes [cf.Schubert et al., 2001, Table 11.3]. In the table, wegive viscosity values appropriate for the uppermantle, which are significantly less than the ambi-ent viscosity of the lower mantle (which is of order1022 Pa s) [Lambeck and Johnson, 1998]. From theparameter values in Table 8, we obtain Pr = 1.5 �1023 and M = 102.

[66] The other three dimensionless numbers arespecific to each mantle plume, and depend on theplume buoyancy flux B (or the plume heat flowrate Q = Bcp/ga), the radius a of the plume’scatchment area and the velocity U of the plateoverriding the plume (Table 9). The anomalousmass fluxes B/g associated with mantle plumeshave been estimated (with levels of reliability thatrange from ‘‘good’’ to ‘‘poor’’) for about 40 hotspots from the rate of topographic swell formation[cf. Davies, 1988; Sleep, 1990; Steinberger, 2000].The anomalous mass fluxes range from 0.2 Mg/s to6.2 Mg/s (Table 9).

[67] To estimate a, we first assume that all theplumes come from the core-mantle boundary,

which has a total heat flow QT of about 3.5 TW[Stacey, 1992; Davies, 1999] through a surface areaS = 4pRc

2, where Rc = 3485 km is the core radius.For each plume, the catchment area pa2 is thengiven by QS/QT, so that a = 2Rc(Q/QT)

1/2. Thevalues of a are found to range from 340 km to1890 km (Table 9).

[68] In Table 9, we also list the plate velocity foreach plume [Davies, 1988; Sleep, 1990; Schubert etal., 2001; Koppers et al., 2001; Wang and Wang,2001]. The values of U range from 5 mm/yr to121 mm/yr. (It should be noted however that, forplumes rising under relatively stationary mid-oceanridges (e.g., Iceland), the effective value of U maybe much less than the overlying plate velocities.)

[69] The resulting values of RaQ, A and U arelisted in Table 9 and plotted in Figure 28. Thevalues of RaQ are found to range from 3 � 105 to3 � 108, the values of A range from 1.5 to 8.5, andthe values of U range from about 0.2 to 3.1.

[70] In Table 10, the dimensionless numbers ofmantle plumes are compared with those in ourlaboratory experiments. The range of values issimilar, except for a difference in Prandtl numbersthat is not dynamically significant (since bothvalues are large enough to ensure that momentumis rapidly diffused in both convecting systems). Wetherefore expect that mantle plumes are character-ized by convective flows similar to those observedin our experiments. We also note the complicationthat the mantle viscosity mo increases significantlywith depth [Lambeck and Johnson, 1998], whichresults in a decrease in the Rayleigh number(RaQ / mo

�1) and an increase in the velocity ratio(U/ mo

1/2) for a mantle plume.

6.2. Diameters of Mantle Plumes

[71] From a comparison of the data given Tables 3and 4, it is seen that the isoviscous lengthscale dpprovides a very approximate estimate of the appar-ent radii rp of the thermal plumes in our experi-ments. If we apply this empirical result to mantleplumes, we can make rough predictions of theplume radius by evaluating dp. The resulting valuesof dp are listed in Table 9: they range from about 80to 200 km, which implies plume diameters in theupper mantle of about 160 to 400 km. In the lowermantle, where the viscosity is up to a factor of 100greater (see Table 11), the plume diameters arepredicted to be a factor of 1001/8 greater (i.e., about280 to 710 km). This range in plume diameters isin broad agreement with recent observations of

Table 8. Mantle Parameters

Variable Value Units

Depth H 2900 kmThermal expansioncoefficient

a 3 � 10�5 K�1

Thermal anomaly DT 200 KDensity ro 3400 kg m�3

Specific heat cp 1250 J kg�1 K�1

Viscosity mo 4 � 1020 Pa sPlume viscosity mi 4 � 1018 Pa sThermal conductivity k 3.3 W m�1 K�1

Diffusivity k 8 � 10�7 m2 s�1

Gravity g 9.8 m s�2



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mantle plumes from seismic tomography [Li et al.,2000; Montelli et al., 2004].

[72] For the Hawaiian plume, Ribe and Christensen[1994] found that the axial topography of the swellwas best fit by a numerical model with an imposedsource temperature anomaly that had a Gaussianprofile with a lengthscale of 91 km. If we take theedge of this Gaussian temperature profile to be atabout 2 lengthscales from the center, then the plumeradius is consistent with our rough prediction of200 km in Table 9.

6.3. Shearing of Mantle Plumes

[73] In an important recent advance, Steinbergerand O’Connell [1998] and Steinberger [2000,

2002] have used a number of numerical modelsto help understand the shearing of mantle plumesin the Earth’s mantle. In their models, the risevelocity of the mantle plumes was evaluated usingthe modified Stokes velocity found by Richardsand Griffiths [1988] for sheared compositionalplumes (i.e., equation (10)). Steinberger [2000]used the Poiseuille pipe flow model (equation(7)) to determine the plume radius, giving

vs ¼0:54

mo

8Bmip

� �1=2

: ð23Þ

On the basis of equation (23), model 1 ofSteinberger [2000] assumed that the plume

Table 9. Parameter Values and Dimensionless Numbers for Mantle Plumes

Hot Spot B/g, Mg s�1 a, km dp, km U, mm yr�1 Lo, km RaQ A U

Ascension 0.9 720 120 40 49 5.7 � 106 4.0 1.35Azores 1.1 800 130 15 151 8.5 � 106 3.6 0.46Bermuda 1.1 900 130 15 151 8.5 � 106 3.6 0.46Bouvet 0.4 480 100 17 62 1.1 � 106 6.0 0.86Bowie 0.8 680 120 50 36 4.5 � 106 4.3 1.79Canary 1.0 760 120 15 140 7.0 � 106 3.8 0.48Cape Verde 1.6 960 140 18 166 1.8 � 107 3.0 0.46Caroline 1.6 960 140 121 25 1.8 � 107 3.0 3.07Cobb 0.3 420 90 51 17 6.3 � 105 7.0 2.99Crozet 0.5 540 100 16 78 1.8 � 106 5.4 0.73Darfur 0.4 480 100 10 106 1.1 � 106 6.0 0.51Discovery 0.4 480 100 13 81 1.1 � 106 6.0 0.66East Africa 0.6 590 110 12 120 2.5 � 106 4.9 0.50East Australia 0.9 720 120 63 31 5.7 � 106 4.0 2.13Easter 3.3 1380 170 105 49 7.7 � 107 2.1 1.86Ethiopia 1.0 760 120 7 300 7.0 � 106 3.8 0.22Fernando 0.9 720 120 32 61 5.7 � 106 4.0 1.08Galapagos 1.0 760 120 60 35 7.0 � 106 3.8 1.93Great Meteor 0.4 480 100 5 212 1.1 � 106 6.0 0.25Hawaii 6.2 1890 200 96 86 2.7 � 108 1.5 1.24Hoggar 0.4 480 100 5 212 1.1 � 106 6.0 0.25Iceland 1.4 900 140 20 135 1.4 � 107 3.2 0.54Juan Fernandez 1.7 990 140 65 48 2.0 � 107 2.9 1.60Kerguelen 0.2 340 80 6 105 2.8 � 105 8.5 0.43Lord Howe 0.9 720 120 63 31 5.7 � 106 4.0 2.13Louisville 3.0 1320 160 90 53 6.3 � 107 2.2 1.67MacDonald 3.9 1500 170 105 56 1.1 � 108 1.9 1.71Marquesas 4.6 1630 180 105 63 1.5 � 108 1.8 1.57Martin Vaz 0.8 680 120 30 59 4.5 � 106 4.3 1.08Pitcairn 1.7 990 140 100 31 2.0 � 107 2.9 2.46Reunion 0.9 720 120 15 130 5.7 � 106 4.0 0.51St Helena 0.3 420 90 10 85 6.3 � 105 7.0 0.59Samoa 1.6 960 140 72 42 1.8 � 107 3.0 1.83San Felix 2.3 1150 150 60 66 3.7 � 107 2.5 1.27Tahiti 5.8 1830 190 105 75 2.4 � 108 1.6 1.40Tasmanid 0.9 720 120 63 31 5.7 � 106 4.0 2.13Tibesti 0.3 420 90 5 171 6.3 � 105 7.0 0.29Tristan 0.5 540 100 12 104 1.8 � 106 5.4 0.54Vema 0.4 480 100 12 88 1.1 � 106 6.0 0.61Yellowstone 1.5 930 140 35 81 1.6 � 107 3.1 0.92



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viscosity mi had a constant value (of 5.4 � 1019 Pas) throughout the mantle, giving

vs ¼ vrB

Br

� �1=2 mrmo

; ð24Þ

where Br/g = 1 Mg/s, mr = 1021 Pa s, vr = 2 cm/yr,and the mantle viscosity mo was a function of depth[see Steinberger, 2000, Figure 1]. In models 2 and3 of Steinberger [2000], equation (24) was

assumed where mo � mr, but where mo � mr, theplume viscosity mi was taken to be proportional tothe mantle viscosity mo, giving

vs ¼ vrB

Br

� �1=2 mrmo

� �1=2

ð25Þ

when the constant viscosity ratio M = 18.5. Inintroducing models 2 and 3, Steinberger [2000]explained that equation (25) was not applied wheremo � mr ‘‘in order to achieve a higher rising speedand prevent conduits from becoming stronglytilted.’’

[74] However, when the two plume velocity mod-els above are compared with typical mantle param-eters given by Schubert et al. [2001] (and listed inTable 8), we see that the value of mi implicitlyassumed in equation (24) is somewhat large, whilethe value ofM implicitly assumed in equation (25)is somewhat small. For more reasonable values,mi = 1019 Pa s in equation (24) and M = 102 inequation (25), the velocity scale in both equationsdecreases to vr = 0.86 cm/yr. As a consequence, theplume rise velocity in all three models is decreasedby a factor of 2.3, which in turn increases thesteady-state plume tilts in the upper mantle of allthree models by a factor of 2.3, giving very largetilts for many mantle plumes.

[75] The origin of the above tilting problems lies inthe use of a rise velocity found for compositionalplumes (equation (10)). In section 5.2, we foundthat this equation is not appropriate for the thermalplumes; instead, the rise velocity is given byequation (12) and is almost independent of theplume viscosity mi. For M = 102, equation (12) canbe written in the same form as equation (25),except that the velocity scale vr now has a valueof 6.0 cm/yr. Equation (12) therefore predicts thatthe plume rise velocity is a factor of 3.0 greaterthan assumed by Steinberger [2000] where mo � mr

Figure 28. (a) The range of Rayleigh numbers RaQand velocity ratios U for the mantle plumes listed inTable 9. (b) The range of aspect ratios A and velocityratios U for the mantle plumes listed in Table 9.

Table 10. Comparison of the Dimensionless Numbersfor the Experiments and for Mantle Plumes

Dimensionless Number Experiments Mantle Plumes

RaQ 106–107 3 � 105–3 � 108

A 2.4–4.8 1.5–7.0U 0.2–2 0.2–3.1M 3.1–56 102

Pr 6400–39000 1.5 � 1023

Table 11. Correction Factors, Predicted Using Equa-tion (12), for the Plume Rise Velocity in MantleConvection Models of Steinberger [2000]

DepthRange, km

MantleViscosity, Pa s Model 1

Models 2and 3

0–100 6 � 1021 7.3 3.0100–400 4 � 1020 1.9 1.9400–670 1021 3.0 3.0670–870 1022 9.5 3.0870–1070 2 � 1022 13.4 3.01070–1270 3 � 1022 16.4 3.01270–2800 4 � 1022 19.0 3.0



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in models 2 and 3 (see Table 11). Similarly, whereSteinberger [2000] used equation (24), equation(12) predicts that the plume rise velocity is greaterby a factor, 3(mr/mo)

1/2, that ranges from 1.9 to 19(see Table 11).

[76] The correction factors listed in Table 11 showthat the corrections required for models 2 and 3 ofSteinberger [2000] are smaller than those requiredfor model 1 (which greatly underestimates theplume rise velocities in the lower mantle). How-ever, to fully assess the accuracy of models 2 and3, the deceleration of the thermal plumes as theyapproach the rigid overlying lithosphere needs tobe considered. For the thermal plumes in ourexperiments, this effect is roughly given by equa-tion (14) (Figure 24). When equations (12) and(14) are combined, a revised velocity correctionfactor is obtained for models 2 and 3, which isshown as a function of depth in Figure 29. We findthat the plume rise velocities in models 2 and 3 ofSteinberger [2000] are overestimated in the upperhalf of the upper mantle, but are underestimated inthe lower half of the upper mantle (by a factorincreasing from 1.35 to 1.87) and in all the lowermantle (by a factor increasing from 1.87 to 3.0). Asa consequence, we suggest that models 2 and 3 ofSteinberger [2000] underestimate the lateral dis-placement of plumes in the upper half of the uppermantle, but overestimate the lateral displacement inthe lower half of the upper mantle and in all thelower mantle. For most mantle plumes, the neteffect of these displacement corrections will be areduction in the distance from the hot spot to the

predicted plume location at the core-mantle bound-ary [cf. Steinberger, 2000, Table 2]. In addition, areduction is expected in the maximum tilt achievedduring the lifetime of most mantle plumes [cf.Steinberger, 2000, Table 2]. We note, however,that the maximum tilt does not appear to be animportant dynamical parameter for mantle plumes,as our experiments (and those of Richards andGriffiths [1989]) show no sign of gravitationalinstability in strongly tilted thermal plumes (seesection 4.5).

[77] In the particular case of the Hawaiian mantleplume, equations (12) and (14) predict a rise timethrough the mantle of only about 100 million years.This rise time is less than or equal to the estimatedage of this hot spot [Steinberger, 2000, 2002;Tarduno et al., 2003], so this plume might havebeen able to reach a steady state shape, if a changein mantle motion and/or plate motion had notoccurred about 47 million years ago [Tarduno etal., 2003], and if the buoyancy flux of this plumehad not varied with time [Davies, 1992].

6.4. Flow Regimes of Mantle Plumes

[78] In section 5.5, we determined an empiricalexpression (equation (20)) from our experimentsfor the velocity ratio Ua required for downstreamadvection. For mantle plumes where M = 102, thisexpression becomes

Ua ¼ 2:29Ra�0:091Q : ð26Þ

This equation is plotted in Figure 30a, togetherwith upper mantle values of U and RaQ fromTable 9. Examination of the figure shows thatsubstantial upstream flow is predicted at the 5 hotspots that lie on plates with very small velocities(5–7 mm/yr): Ethiopia, Great Meteor, Hoggar,Kerguelen, and Tibesti. In contrast, downstreamadvection is predicted at hot spots on plates thathave large velocities, including all the Pacific hotspots, a conclusion that is reasonably consistentwith recent 3D numerical models of the interactionof Hawaiian plume with the Pacific plate [e.g.,Ribe and Christensen, 1994, 1999; Zhong andWatts, 2002].

[79] In section 5.5, we also found an expression(equation (22)) for the velocity ratio Uc that marksthe transition from negligible to significant cross-stream circulation. For mantle plumes where M =102, this expression becomes

Uc ¼ 75:4Ra�0:32Q ð27Þ

Figure 29. The velocity correction factor, as afunction of depth, for mantle convection models 2 and3 of Steinberger [2000]. The correction factor combinesthe plume rise velocity in equation (12) and thedeceleration due to the rigid lid in equation (14).



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when A = 4.8. This equation is plotted in Figure30b, together with upper mantle values of U andRaQ from Table 9. Cross-stream circulation andthermal entrainment is predicted for plumesexposed to high plate velocities, including all thePacific plumes. For plumes subject to very smallplate velocities, negligible cross-stream circulationis predicted in the upper mantle. However, if theseplumes are long-lived, they may experience weakcross-stream circulation in the lower mantle, wherethe very high viscosities produce slow plume risevelocities, enabling the tilting of plumes exposed tolarge scale circulation driven by density hetero-geneities [cf. Steinberger, 2000].

6.5. Upstream Spreading of Mantle PlumesUnder the Lithosphere

[80] In section 5.4, we found that the theoreticalmodel in Appendix C gave a rough prediction ofthe upstream spreading distance Ls in our experi-ments at very small velocity ratios, where the

thermal plume is only slightly tilted. In Table 12,the theoretical model (equation (C3)) is thereforeused to estimate the distances to the upstreamstagnation point for the 5 hot spots with very smallplate velocities: Ethiopia, Great Meteor, Hoggar,Kerguelen, and Tibesti. Table 12 also gives theestimated time ts for the gravity currents to flowupstream to the stagnation point, together withestimates of the ages of some of these hot spots[see Steinberger, 2000, Table 1]. In the cases ofKerguelen and Tibesti, their ages are larger than ts,so these plumes should have reached their up-stream stagnation points (unless their plate veloc-ities have decreased during their lifetime). Thepredicted values of Ls are found to be quite large:480 km for Kerguelen and 770 km for Tibesti. Inthe case of Hoggar, its age (20 Myr) is muchsmaller than ts, so the upstream spreading of thisplume has not reached steady-state. Instead, itsupstream spreading is given by its age and equation(C4), and is estimated to be about 430 km. Lastly,for Ethiopia and Great Meteor, their ages are notknown [cf. Steinberger, 2000], so it is not clearwhether these plumes have reached their upstreamstagnation points.

[81] In Table 13, we give the model predictions ofLs and ts for the 9 other hot spots in Figure 30a that

Figure 30. (a) The velocity ratio Ua required fordownstream advection, given by equation (26), togetherwith the mantle plume data from Figure 28a. (b) Thevelocity ratio Uc required for cross-stream circulation,given by equation (27), together with the mantle plumedata from Figure 28a.

Table 13. Upstream Spreading of Mantle Plumes WithSmall Plate Velocities

Hot Spot Age, Myr

Ls FromTheoreticalModel, km

ts FromTheoreticalModel, Myr

Azores 100 680 23Bermuda – 680 23Canary 65 640 21Cape Verde 20 750 21Dafur 140 480 24East Africa 40 540 23Reunion 67 590 20St Helena 100 390 19Tristan 125 470 20Vema 40 400 17

Table 12. Upstream Spreading of Mantle Plumes WithVery Small Plate Velocities

Hot Spot Age, Myr

Ls FromTheoreticalModel, km

ts FromTheoreticalModel, Myr

Ethiopia – 1360 97Great Meteor – 960 96Hoggar 20 960 96Kerguelen 117 480 40Tibesti 80 770 77



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lie to the left of the predicted regime boundary,where some upstream spreading is suggested. Thehot spot ages are comparable to or greater than ts,so these plumes should have reached their up-stream stagnation points. The estimated values ofLs range from 390–730 km. However, becausethese 9 plumes lie very close to the regime bound-ary in Figure 30a, our experimental results insection 5.4 suggest that these values of Ls and tsare likely to be significant overestimates. Thisconclusion is consistent with geoid and oceandepth observations presented by Sleep [1990,Figures 10, 11, and 13], who inferred that thedistance to the upstream stagnation point is onlyabout 500 km for the Bermuda hot spot, about400 km for the Cape Verde hot spot, and about285 km for the Reunion hot spot.

6.6. Lateral Spreading of Mantle PlumesUnder the Lithosphere

[82] In section 5.3, we found that the lateralspreading of our thermal plumes was best de-scribed by equations (17) and (18), where theconstant 2C had a value of about 10 ± 1 forupstream dye release and about 5.5 ± 0.5 forlateral dye release. For mantle plumes, the esti-mated values of Lo range from 17 to 301 km(see Table 9). However, as noted in section 6.5,spreading from the Hoggar hot spot has notreached steady-state, and the same may be truefor the Ethiopia and Great Meteor hot spots. Ifthese 3 hot spots are ignored, the values of Lorange from 17 to 171 km.

[83] For the Hawaiian plume, Lo is equal to 86 km,so at a distance s downstream of 500 km,the predicted plume width W ranges from 670–1220 km (for the two empirical values of 2Cgiven above). This prediction is in reasonableagreement with the observed swell width of about1000 km [e.g.,Wessel, 1993; Ribe and Christensen,1994, 1999]. We also note that our experimentalresult that W mostly depends on the ambientviscosity mo, rather than the centerline viscositymi, is in agreement with the recent numericalmodeling of the Hawaiian plume by van Hunenand Zhong [2003]. As a consequence, there is noneed to appeal to melt extraction increasing theviscosity of the mantle plume [Phipps Morgan etal., 1995] in order to explain the width of theHawaiian swell.

[84] In Table 14, plume widths are also predictedfor 3 other hot spots (Bermuda, Cape Verde,Reunion), which are also found to be in reasonable

agreement with the observed swell widths [cf.Sleep, 1990, Figures 10, 11, 13].

6.7. Motion of GeochemicalHeterogeneities in Sheared Mantle Plumes

[85] As outlined in section 1, an important moti-vation of this experimental study was to examinethe behavior of geochemical heterogeneities insheared mantle plume tails. These heterogeneitiesare up to two billion years old [Hofmann andWhite, 1982], and are expected to exist with awide range of shapes and scales [cf. Davies, 2002,Figure 8a].

[86] At the base of the mantle, we expect thatgeochemical heterogeneities will be drawn down-ward into the thermal boundary layer at the core-mantle boundary, before flowing laterally and thenrising in mantle plumes [cf. Stacey and Loper,1983; Olson et al., 1993]. In section 6.1, we haveshown that mantle plumes have quite large catch-ment areas, with radii a that range from 340 km to1890 km. We therefore expect that mantle plumesare able to simultaneously sample heterogeneitiesthat are widely separated in the mantle.

6.7.1. Fluid Dynamical Predictions

[87] From observations of the dye streamlines inour experiments (see section 4.6), we can make anumber of predictions about the motion of thesegeochemical heterogeneities as they rise in shearedmantle plumes:

[88] 1. Our experiments show that small hetero-geneities are focused into thin vertical sheets in aplume’s source region (see section 4.1.1), resultingin mantle plumes that are zoned azimuthally, ratherthan concentrically as proposed by Hauri et al.[1994].

[89] 2. Our experiments show that only limitedcross-stream circulation will occur in sheared man-tle plumes, so the identity of geochemical hetero-geneities will not be erased by stirring inside theplume. We therefore predict a geochemically zonedplume and geochemical zonation in the erupted

Table 14. Lateral Spreading of Some Mantle Plumes

Hot Spot Lo, km s, km W Predicted, kmSwell

Width, km

Bermuda 151 680 1060–1920 1600Cape Verde 166 750 1090–1980 1200Hawaii 86 500 670–1220 1000Reunion 130 300 850–1540 1000



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ocean island basalts whenever there are geochem-ical heterogeneities in a mantle plume’s sourceregion.

[90] 3. Our experiments with ‘‘inner’’ and ‘‘outer’’dye release demonstrate that left/right symmetryabout the direction of plate motion will be pre-served in sheared mantle plumes, so geochemicalheterogeneities from one side of a plume’s sourceregion will always remain on that side of theplume, and will never be stirred to the other sideof the plume.

[91] 4. Our experiments demonstrate that the mo-tion of heterogeneities from the upstream side of aplume’s source region is very sensitive to the platevelocity (i.e., the velocity ratio U):[92] . When a mantle plume rises under a stationaryplate, the heterogeneities will spread out radiallyunder the lithosphere, giving an azimuthal geo-graphic distribution that mirrors the azimuthallyentrained distribution of heterogeneities in theplume’s source region (see Figure 3).

[93] . When a mantle plume rises under a slowlymoving plate, the convective regime consists ofupstream flow and negligible cross-stream circula-tion (see section 6.4 and Figure 30). In this case,heterogeneities from the upstream side of theplume’s source region follow two very distinctflow paths: those in the hotter, central part of theplume are dragged downstream where they overlieheterogeneities from the downstream side of theplume’s source region, while those in the cooler,outer part of the plume flow upstream and aroundto the edges of the hot spot swell (see Figure 14).

[94] . When a mantle plume rises under a rapidlymoving plate, the convective regime consists ofdownstream advection, and significant cross-stream circulation and thermal entrainment (seesection 6.4 and Figure 30). The cross-stream cir-culation changes the geographical distribution ofheterogeneities, by causing heterogeneities fromthe upstream side of the plume’s source region tobe rotated toward the edges of the hot spot swell,while heterogeneities from the downstream side ofthe plume’s source region rise through the center ofthe plume to its upper surface (see Figure 17). Thiscross-stream circulation will also affect the tem-perature and melting behavior of heterogeneities,since the upstream heterogeneities will be cooledby their exposure to cold mantle on the outer edgesof the plume, while the downstream heterogene-ities rising in the interior of the plume will remainwarmer and hence are more likely to melt.

6.7.2. Hawaiian Plume

[95] Of all the world’s hot spots, probably the mostintensively studied is the Hawaiian island chain.This mantle plume has the highest buoyancy flux(see Table 9), which results in the largest Rayleighnumber (RaQ = 2.7 � 108) and smallest aspect ratio(A = 1.5). The plume is also under the fastestmoving plate, which results in a moderately largevelocity ratio (U = 1.24). For these dimensionlessparameter values, we predict that the Hawaiianplume will be advected downstream with no up-stream flow (see section 6.4). We also expect thatthe plume will experience cross circulation andthermal entrainment of surrounding mantle (seesection 6.4), although we note that cross circulationis probably limited by the small value of A (seesection 5.5).

[96] The recent volcanism on the Hawaiian islandchain consists of two parallel lines of volcaniccenters, the ‘‘Kea’’ trend and the ‘‘Loa’’ trend,which have systematic geochemical differences intrace element abundances and radiogenic isotoperatios [Frey and Rhodes, 1993; Hauri et al., 1996;Kurz et al., 1996; Lassiter et al., 1996] cannot beexplained by different degrees of partial melting ofa single source. Recently, Abouchami et al. [2000]found four discrete source components in theMauna Kea lavas, which implies ‘‘substantial het-erogeneity within the Hawaiian plume and system-atic tapping of these heterogeneities on the scale ofa single volcano,’’ while DePaolo et al. [2001]report Nd isotopes measurements which suggestthat the Mauna Loa-Hualalai region has ‘‘bloblikeheterogeneity in the plume that has a lengthscale ofcirca 20 km.’’ DePaolo et al. [2001] conclude thatthese geochemical observations preclude the con-centrically zoned plume model proposed by Hauriet al. [1994]. Instead, the observations suggest a‘‘northeast-southwest asymmetry,’’ consistent withour predictions in section 6.7.1, in which the‘‘Kea’’ trend reflects a range of heterogeneities inthe northeast half of the Hawaiian plume’s sourceregion, while the ‘‘Loa’’ trend reflects a range ofheterogeneities in the southwest half of the Hawai-ian plume’s source region.

6.7.3. Galapagos Plume

[97] Another well-studied and geochemically inter-esting hot spot is the Galapagos [e.g., Geist et al.,1988; White et al., 1993; Kurz and Geist, 1999;Harpp and White, 2001; Hoernle et al., 2000;Blichert-Toft and White, 2001]. This plume hasdimensionless parameters (RaQ = 7.0 � 106, A =



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3.8, U = 1.93 and M = 100; see Table 9) that aresimilar to those for the experiments 28 and 29shown in Figure 17 (RaQ = 2.4 � 106, A = 4.8, U =2.05 and M = 56; see Table 3). We thereforeexpect that the motion of the dye streams in Figure17 gives a very good indication of the behavior ofgeochemical heterogeneities in the Galapagosplume. In particular, we predict that the Galapagosplume will be advected downstream with no up-stream flow, and that it will experience substantialcross circulation and thermal entrainment of sur-rounding mantle (see also section 6.4). The pre-dicted thermal entrainment provides a convincingexplanation of the observed ‘‘east facing horseshoepattern’’ of basalts with ‘‘enriched’’ isotope ratiosand trace element abundances, around an easternregion where the basalts have a ‘‘depleted’’ uppermantle signature [Geist et al., 1988; White et al.,1993; Harpp and White, 2001; Blichert-Toft andWhite, 2001], as first proposed by Richards andGriffiths [1989].

[98] The geochemical studies also reveal that thereare at least 3 ‘‘enriched’’ end-members that aregeographically distinct: a northern domain thatincludes the Wolf-Darwin Lineament; a southerndomain that includes Floreana Island, and a centralwestern domain that includes Fernandina Islandand which is more widespread throughout thearchipelago [White et al., 1993; Harpp and White,2001; Blichert-Toft and White, 2001]. In addition, ageochemical study of the Cocos Ridge off the coastof Costa Rica by Hoernle et al. [2000] found thesame 3 enriched sources (northern, central andsouthern domains) as seen in the GalapagosIslands, which shows that ‘‘the asymmetrical spa-tial zonation of the Galapagos hot spot has existedfor 14 m.y.’’ These observations are completelyconsistent with our fluid dynamical predictionsgiven in section 6.7.1. The Galapagos plumeclearly has an azimuthal isotopic zonation, dueto heterogeneities in its catchment area that areonly weakly stretched and stirred during theirrise through the mantle (see Figure 17), as hasbeen suggested by Hoernle et al. [2000] andBlichert-Toft and White [2001]. The northerndomain must reflect heterogeneities in the north-ern half of the Galapagos plume’s source region(see the blue dye in Figure 17f), the southerndomain must reflect heterogeneities in the south-ern half of Galapagos plume’s source region (seethe red dye in Figure 17f ), and the centraldomain with its more widespread signature mustreflect heterogeneities that are more widely dis-tributed throughout the plume’s source region.

6.7.4. Asymmetry at Some Other OceanIsland Chains

[99] The asymmetrical distribution of isotopic val-ues across ocean island chains is not limited to theHawaiian and Galapagos Islands. Desonie et al.[1993] report systematic variations in Sr, Nd, andPb isotopic ratios in basalts from the ‘‘southwest,’’‘‘center,’’ and ‘‘northeast’’ of the Marquesas islandchain. Similarly, Devey et al. [1990] have found alarge increase in maximum Sr isotopic ratio acrossthe Tahiti/Society island chain from southeast tonorthwest. In both cases, these geochemical obser-vations suggest a heterogeneous mantle plume thatis simultaneously sampling a heterogeneous plumecatchment area.

7. Conclusions

[100] In this paper we have used a series of labo-ratory experiments to investigate the behavior ofsheared thermal plumes. We have systematicallyexamined how the plumes depend on four impor-tant dimensionless numbers (a velocity ratio, aRayleigh number, an aspect ratio, and a viscosityratio), using experimental values that are similar tothose appropriate for mantle plumes (Table 10).

[101] Our experiments have discovered two transi-tions in flow behavior: from upstream flow todownstream advection, and from negligible cross-stream circulation to significant cross-stream cir-culation and thermal entrainment. For mantleplumes, we predict that these two transitions nearlyoverlap (Figure 30), so that the population ofmantle plumes is mostly divided into only twoflow regimes in the upper mantle: plumes underslow moving plates experience upstream flow andnegligible cross-stream circulation, while plumesunder faster moving plates (including all Pacificplumes) experience downstream advection andsignificant cross-stream circulation.

[102] In all our experiments, the dye streams arefocused into thin vertical sheets in the plume’ssource region (see section 4.1.1), resulting in plumesthat are zoned azimuthally. In addition, the dyestreams experienced only limited cross-stream cir-culation as they rose, and they preserved left/rightsymmetry about the direction of shear. We thereforepredict that a geochemically zoned plume andgeochemical zonation in the erupted ocean islandbasalts will result whenever there is substantialgeochemical heterogeneity in a mantle plume’ssource region. We also predict that geochemicalheterogeneities from one side of a plume’s source



35 of 42

region will remain on that side of the plume, whichprovides an explanation for the asymmetrical geo-graphical distribution of isotopic values seen inbasalts from the Hawaiian, Galapagos, Marquesas,and Tahiti/Society ocean island chains.

[103] Another observation of our experiments, inagreement with Richards and Griffiths [1989], isthat strongly tilted thermal plume tails (Figures 7,13, and 17) are not prone to the gravitationalinstability seen in similar compositional plumes[Whitehead, 1982; Richards and Griffiths, 1988].This behavior is probably due to the growth ofinstabilities being inhibited by thermal diffusion,which results in broad gradients in viscosity andaxial velocity inside the plumes, as well as signif-icant thermal entrainment. This important conclu-sion for mantle plumes does not seem to be widelyappreciated [cf. Steinberger and O’Connell, 1998;Steinberger, 2000; Condie, 2001, p. 34].

[104] Using the steady profiles of the shearedthermal plumes, we have shown that their initialvertical rise velocity is accurately given by equa-tion (12). This equation depends on the buoyancyflux and ambient viscosity but is almost indepen-dent of the plume viscosity, which suggests thatmost of the plume has a viscosity that is muchcloser to the ambient viscosity mo than the center-line viscosity mi. As the plumes approach the rigidoverlying lid, they are observed to decelerate due toincreasing viscous drag (see Figure 24). Theseresults for thermal plumes can significantly improvenumerical models of the deflection ofmantle plumes[Steinberger and O’Connell, 1998; Steinberger,2000, 2002], which until now have had to rely onexpressions for the vertical rise velocity of compo-sitional plumes (see section 6.3).

[105] We have also shown that the isoviscouslengthscale given by Olson et al. [1993](equation (A9)) is reasonably consistent with theapparent radii of our experimental plumes. Thislengthscale provides rough estimates of the diam-eters of mantle plumes (160 to 400 km in the uppermantle, and 280 to 710 km in the lower mantle)that are in broad agreement with recent observa-tions from seismic tomography [Li et al., 2000;Montelli et al., 2004].

[106] Our experiments have shown that the lateralspreading of thermal plumes under the lid is welldescribed by theoretical models for the lateralspreading of compositional plumes (given in Ap-pendix B) if the centerline viscosity mi is replacedby the ambient viscosity mo (see equations (17) and

(18)), which again suggests that most of the plumehas a viscosity that is much closer to the ambientviscosity mo than the centerline viscosity mi. Equa-tions (17) and (18) are also found to providepredictions in reasonable agreement with the ob-served widths of a number of hot spot swells (seesection 6.6).

[107] Our experiments have also shown that atheoretical model for the upstream spreading ofcompositional plumes (given in Appendix C) ispartly successful in describing the upstreamspreading of thermal plumes under the lid. Atvery small to small velocity ratios, equation (C3)roughly predicts the observed upstream spreading,and can be used to estimate upstream spreadingdistances under slowly moving plates. However, atmoderate to high velocity ratios, the model fails topredict the transition to a regime of no upstreamspreading (i.e., downstream advection of the entireplume).

[108] In the future, we hope that our study willinspire further experimental investigations of thedynamics of mantle plumes. In addition, we hope itwill assist the development of accurate numericalmodels capable of simulating thermal entrainmentinto sheared thermal plumes in a convecting man-tle, once both computer power and numericalmethods have improved sufficiently.

Appendix A: Isoviscous ThermalPlumes

[109] In this appendix, the main parameters thatdescribe the rise of a steady axisymmetric isovis-cous thermal plume are estimated, for the casewhere the total heat flow rate Q from the heatedsurface is specified.

[110] We begin with the approximate theoreticalmodel given by Olson et al. [1993], whichdescribes an axisymmetric isoviscous plume in aninfinite Prandtl number fluid above a circularheated surface of radius a and with an imposedtemperature difference DT between the heatedsurface and the overlying fluid. In their model,the Rayleigh number RaT is defined by

RaT ¼ arogDTa3

kmo; ðA1Þ

and the Nusselt number Nu is defined by

Nu ¼ aF

kDT; ðA2Þ



36 of 42

where the heat flux F = Q/(pa2). Their modelassumed that the plume had a parabolic velocityprofile and a Gaussian temperature profile, bothwith radial lengthscale dp, and found that thecenterline plume velocity wp

i is given by

wip ¼

1

4

8

3

� �1=3 kaRa

2=3T ; ðA3Þ

the Nusselt number is given by

Nu ¼ 1

2

8

3

� �2=3

Ra1=3T ; ðA4Þ

and the plume lengthscale

dp ¼ a8

3

� �1=6

Ra�1=6T : ðA5Þ

[111] When the total heat flow Q from the heatedsurface is known, we can define a Rayleigh num-ber RaQ by

RaQ ¼ NuRaT ¼ agroQa2

pkkmo: ðA6Þ

Combining equations (A4) and (A6), RaQ and RaTare related by

RaQ ¼ 1

2

8

3

� �2=3

Ra4=3T ; ðA7Þ

wpi is given by

wip ¼

1

23=2kaRa

1=2Q ¼ B

8pmo

� �1=2

; ðA8Þ

and dp is given by

dp ¼ a32

9

� �1=8

Ra�1=8Q : ðA9Þ

By combining equations (A1), (A6), and (A7), DTcan also be evaluated as

DT ¼ 3

8

� �1=2 kmoarog

� �1=42F

k

� �3=4

: ðA10Þ

Appendix B: Similarity Solutions for theLateral Spreading of CompositionalPlumes

[112] A simple model of the lateral spreading of acompositional plume can be obtained by notingthat, at large distances downstream, the lateral flow

asymptotically approaches that of a fixed-volumeand two-dimensional gravity current that is beingadvected downstream by the moving lid.

[113] When mi � mo, the viscous dissipation occursprimarily in the gravity current. For the release of afixed volume Vof fluid per unit width, the similaritysolution for the length yN of the two-dimensionalcurrent as a function of time t is given by [Pattle,1959; Huppert, 1982]

yN tð Þ ¼ hNgDrV 3

3mi

� �1=5

t1=5; ðB1Þ

where

hN ¼ 510

3

� �1=3

p�1=2 G 5=6ð ÞG 1=3ð Þ

" #3=5

¼ 1:411 . . . ðB2Þ

The current thickness h(y,t) is given by

h y; tð Þ ¼ 3

10

� �1=3

h2=3N

3mi V2

gDr

� �1=5

t�1=5 1� y2

y2N

� �1=3

:

ðB3Þ

When a spreading plume has reached its steady-state shape, the time variable t is related todownstream distance x by t = x/U and V =B/(2gDrU), so the half-width of the spreadingplume is

yN xð Þ ¼ hNB3

24mi g2Dr2 U 4

� �1=5

x1=5 ðB4Þ

and the current depth is

h x; yð Þ ¼ 3

10

� �1=3

h2=3N

3 mi B2

4g3Dr3 U

� �1=5

x�1=5 1� y2

y2N

� �1=3

:

ðB5Þ

Equations (B4)–(B5) are almost identical to asimilarity solution found by Ribe and Christensen[1994]. The only difference is that the half-width in equation (B4) is larger by 22/5 while thedepth in equation (B5) is smaller by the samefactor. This difference is due to an incorrectboundary condition in the model by Olson [Olson,1990; Schubert et al., 2001, section 11.11] thatwas analyzed by Ribe and Christensen [1994] (andalso used by Ribe et al. [1995], Ribe [1996], andRibe and Christensen [1999]): namely, Olson[1990] assumed a no-slip condition at the baseof the current, rather than the continuity ofvelocity and shear stress correctly applied by



37 of 42

Huppert [1982] (and Phipps Morgan et al. [1995]).Equation (B4) can also be written in nondimen-sional form [cf. Ribe and Christensen, 1994,equations (12) and (13)] as

yN

Li¼ C

x

Li

� �1=5

; ðB6Þ

where the constant C = 3.70. . . and the lengthscaleLi is given by

Li ¼B3

96p3mi g2Dr2 U 4

� �1=4

: ðB7Þ

[114] When mi � mo, there is a second flow regimefor viscous gravity currents in which the viscousdissipation occurs primarily in the ambient fluid. Inthis regime, a scaling analysis can be used todetermine the current length yN and the mean currentdepth h as a function of time t [cf. Lister and Kerr,1989; Griffiths and Campbell, 1991b]. The totalbuoyancy force per unit width Fg is given by

Fg � gDr h2; ðB8Þ

while the total viscous drag per unit width Fv isgiven by

Fv � mo UN ; ðB9Þ

where the current velocity

UN � yN

t: ðB10Þ

Conservation of mass for a fixed-volume and two-dimensional gravity current requires that

yN h � V : ðB11Þ

[115] Equating Fg and Fv, and using equations(B10)–(B11), then gives the spreading laws

yN tð Þ � gDrV 2

mo

� �1=3

t1=3 ðB12Þ

and

h tð Þ � mo VgDr

� �1=3

t�1=3: ðB13Þ

In this regime, a spreading plume has a half-width

yN xð Þ � B2

4mo gDrU3

� �1=3

x1=3 ðB14Þ

and a mean depth

h xð Þ � mo B2g2Dr2

� �1=3

x�1=3: ðB15Þ

However, as yN increases, the viscous dissipationinside the current increases until there is atransition to the first flow regime when the aspectratio yN/h equals the viscosity ratio mo/mi, at atransition time tT in (B12)–(B13) given by

tT � m5=2o

gDr m3=2i V 1=2ðB16Þ

or a downstream transition distance xT in (B14)–(B15) given by

xT � m5=2o U3=2

g1=2Dr1=2m3=2i B1=2: ðB17Þ

Appendix C: Theoretical Models for theUpstream Spreading of CompositionalPlumes

[116] A simple model of the upstream spreading ofcompositional plumes can be obtained by makingtwo assumptions proposed by Sleep [1996]: first,that ‘‘the details of the supply of material by theplume conduit are ignored,’’ and, second, that ‘‘thevelocity of material in the asthenosphere is acombination of radial flow away from the plumeand a uniform drag caused by the overlying plate.’’Like the case of lateral spreading described inAppendix B, there are two flow regimes.

[117] When mi � mo, the viscous dissipation occursprimarily inside the axisymmetric gravity current.For the release of a constant volume flux (q =B/Dr) of fluid, Huppert [1982] gives the similaritysolution for the radius r of the axisymmetriccurrent as a function of time t:

r tð Þ ¼ 0:715gDr q3

3mi

� �1=8

t1=2: ðC1Þ

The velocity of the gravity current is

dr

dt¼ 0:715

2

gDr q3

3mi

� �1=8

t�1=2; ðC2Þ

which decreases with time and distance until it isequal to the plate speed U at the upstreamstagnation point, which occurs at a distance

Ls ¼0:715ð Þ2

2

gDr q3

3miU4

� �1=4

¼ 0:194B3

mi g2Dr2 U4

� �1=4

ðC3Þ



38 of 42

upstream from the plume source and after a timets = Ls/2U. The upstream spreading distance Ls istherefore proportional to the lateral spreadinglengthscale Li (i.e., Ls = 1.43 Li), as found bySleep [1996].

[118] When mi � mo, there is a second flow regimefor viscous gravity currents in which the viscousdissipation occurs primarily in the ambient fluid. Inthis regime, a scaling analysis of the buoyancy andviscous forces together with conservation of mass[cf. Lister and Kerr, 1989; Griffiths and Campbell,1991b; Feighner and Richards, 1995] for the fixed-flux and axisymmetric gravity current shows thatthe current radius r is given by

r tð Þ ¼ cgDr q2

mo

� �1=5

t3=5; ðC4Þ

where the constant c has been empirically deter-mined to be 0.79 ± 0.02 [Feighner and Richards,1995]. The velocity of the gravity current is

dr

dt¼ 3c

5

gDr q2

mo

� �1=5

t�2=5; ðC5Þ

which is equal to U at a distance

Ls ¼ c5=23

5

� �3=2gDr q2

moU 3

� �1=2

¼ 0:258B2

mo gDrU3

� �1=2

ðC6Þ

upstream from the plume source and after a time ts =3Ls/5U. However, the viscous dissipation inside thegravity current increases as the radial distanceincreases, until there is a transition from equation(C6) to equation (C3) at a plate speed UT given by

UT ¼ 1:77m1=2i B1=2

moðC7Þ

and a transition distance LT given by

LT ¼ 0:110mo B

1=4

g1=2Dr1=2m3=4i

; ðC8Þ

that is, equation (C6) is valid when U > UT andL < LT, while equation (C3) is valid when U <UT and L > LT.

Notation

a radius of hot plate, mm; radius of mantle

plume’s catchment area, km.

c constant in a flow law for an axisymmetric

gravity current.

cp specific heat, J kg�1 K�1.

d(z) plume horizontal displacement, mm.

da(z) plume apparent displacement, mm.

g gravitational acceleration, m s�2.

h mean current depth, m.

k thermal conductivity, W m�1 K�1.

ks constant in a modified Stokes velocity

expression for a plume.

q volume flux, m3 s�1.

r radius of an axisymmetric gravity current,

m.

rp apparent plume radius, mm.

r*p plume radius predicted by a Poiseuille pipe

flow model, mm.

s arc length, s = Rmq.t time, s.

ts time required for a gravity current to reach

its upstream stagnation point, s.

tT transition time, s.

vs rise velocity of the plume, mm s�1.

vms rise velocity of the plume measured at

middepth (z = 120 mm), mm s�1.

vr rise velocity scale of Steinberger [2000],

vr = 2 cm yr�1.

wp centerline velocity of a thermal plume,

mm s�1.

wpi centerline velocity predicted for an isovis-

cous thermal plume, mm s�1.

wp* centerline velocity predicted by a Poiseuille

pipe flow model, mm s�1.

x downstream distance, m.

xT downstream transition distance, m.

y lateral distance, m.

yN half-width of the spreading plume, m.

z height, mm.

zscale thermal diffusion scale height, m.

B buoyancy flux, B = gaQ/cp, N s�1.

Bn buoyancy number.

Br buoyancy flux scale, N s�1.

C constant in the lateral spreading of a

compositional plume.

F heat flux, F = Q/pa2, W m�2.

Fg total buoyancy force per unit width, kg s�2.

Fv total viscous drag per unit width, kg s�2.

H fluid depth, mm; depth of the mantle, km.

Li lengthscale for lateral spreading, m.

Lo lengthscale for lateral spreading, m.



39 of 42

Ls upstream stagnation distance, mm.

LT transition distance, m.

Nu Nusselt number.

Pe Peclet number.

Pr Prandtl number.

Q heat flow rate, W.

QT total heat flow through the core-mantle

boundary, TW.

R distance from the center of the tank to the

center of the hot plate, mm.

Rc radius of the core, km.

Rm measured radius of the best-fit circle

between the plume edges, mm.

RaQ Rayleigh number defined as a function of Q.

RaT Rayleigh number defined as a function of

DT.

Rei internal plume Reynolds number.

Reo external plume Reynolds number.

S surface area of the core-mantle boundary,

km2.

To ambient temperature, �C.Tp temperature at the center of the hot plate, �C.Tpi plate temperature Tp

i = To + DT in the

isoviscous plume model, �C.U lid velocity mm s�1; plate velocity, mm

yr�1.

UN gravity current velocity, m s�1.

UT transition velocity, m s�1.

V volume of fluid released per unit width, m2.

W apparent plume width, mm.

A aspect ratio.

M viscosity ratio.

U velocity ratio.

Ua velocity ratio required for downstream

advection.

Uc velocity ratio that marks the transition

from negligible to significant cross-stream

circulation.

a thermal expansion coefficient, K�1.

dp plume lengthscale in the isoviscous plume

model, mm.

hN constant in a flow law for a two-dimen-

sional gravity current.

k thermal diffusivity, m2 s�1.

mi plume viscosity, Pa s.

mo ambient viscosity, Pa s.

mr ambient viscosity scale of Steinberger

[2000], mr = 1021 Pa s.

ro density, kg m�3.

t momentum diffusion timescale, s.

q radial angle, radians.

Dr density difference, Dr = aro(Tp � To), kg

m�3.

DT imposed temperature difference in the iso-

viscous plume model, �C.

Acknowledgments

[119] We thank Tony Beasley for expert technical assistance

throughout the long and demanding experimental program and

Ross Griffiths for valuable discussions on many aspects of the

behavior of mantle plumes. Thoughtful reviews from Michael

Gurnis, Neil Ribe, Mark Richards, and Jack Whitehead are

also gratefully acknowledged.

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structure and dynamics of sheared mantle plumes

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