On the dynamics of long-lived plume conduits in the convecting mantle

214

Earth and Planetary Science Letters, 103 (1991) 214 227

Elsevier Science Publishers B.V., Amsterdam [cL]

On the dynamics of long-lived plume conduits in the convecting mantle R o s s W . G r i f f i t h s a n d I a n H. C a m p b e l l Research School of Earth Sciences, Australian National University, GPO Box 4, Canberra. A.C.T. 2601, A ustralia

Received May 21, 1990; revision accepted November 20, 1990

ABSTRACT Long-lived plumes, responsible for mantle hotspots and associated chains of volcanism, will be deflected from the vertical as they penetrate through a mantle which is overturning on much larger length scales in response to surface cooling, Recent experiments have shown that plume inclination leads to entrainment of surrounding material as a result of coupling between conduction of heat and laminar stirring. We develop an analysis for entrainment and its effects on the structure and dynamics of continuing, steady conduits in a mantle undergoing horizontal shear. It is found that the stirring of surrounding mantle with plume source material can result in major modifications of the diameter, temperature and mass flux in plumes. This model is a departure from the axisymmetric vertical pipe models of plumes and predicts that the temperature in a defected plume will decrease with height, thus reducing the plume temperature in the upper mantle while allowing a sizeable temperature difference at the source. The cooling of plume conduits, after they are deflected by plate motion, can lead to a lower maximum temperature and lower MgO contents of basalts along some linear hotspot tracks compared with picrites from their associated flood basalt provinces. The effects of entrainment depend primarily on the buoyancy (or heat) flux carried by the plume, with greater cooling for smaller fluxes. Strong continuing plumes such as Hawaii, Easter Island, Macdonald, Tahiti and Reunion are predicted to produce picrites having MgO contents comparable to the picrites found in continental flood basalt provinces. Weak plumes such as Crozet, Bouvet and Juan de Fuca will give only cooler melts. Through entrainment long-lived plumes can sample the whole depth of the mantle from the source to the surface, and stirring within the plume leads to compositional zonation which may contribute to compositional heterogeneity in hotspot melts. We also predict that plumes originating at the core-mantle boundary and having buoyancy fluxes less than about 3 × 103 N s i will not give surface hotspots, in good agreement with the lower limit of hotspot buoyancy fluxes inferred from seafloor topography.

1. Introduction If m a n t l e hotspots are p r o d u c e d by b u o y a n t plumes rising from deep in the m a n t l e [1,2] it follows that the relative m o t i o n of lithospheric plates a n d hotspots, indicated by the age progression of volcanism along hotspot tracks, implies a vertical gradient of horizontal velocity in the m a n tle. At least some of the plumes are sufficiently long-lived to m a i n t a i n active hotspots for 10 s years. Irrespective of the depth d i s t r i b u t i o n of the shear, the long-lived plumes m u s t be deflected from their vertical ascent, the degree of i n c l i n a t i o n reflecting both the shear stress a n d the d e p t h of the source [3,4]. Once inclined the p l u m e c o n d u i t itself will ascend by virtue of its b u o y a n c y a n d displace the overlying mantle. L a b o r a t o r y experi0012-821x/91/$03.50

© 1991 - Elsevier Science Publishers B.V.

m e n t s [4] have shown that the s u b s e q u e n t path of the p l u m e is given b y a s u p e r p o s i t i o n of the horizontal advection b y the s u r r o u n d i n g s a n d the b u o y a n c y - d r i v e n vertical ascent through the surr o u n d i n g s (Fig. 1). T h e same experiments also revealed that the b u o y a n c y - d r i v e n c o m p o n e n t of the c o n d u i t m o t i o n leads to a n o n - a x i s y m m e t r i c recirculation in vertical planes within the p l u m e (Fig. 2). This circulation is s u p e r i m p o s e d o n the more rapid, u p w a r d axial flow of hot material along the c o n d u i t [4]. A n established vertical p l u m e is expected to consist of a c h a n n e l c o n t a i n i n g hot, low-viscosity material t h r o u g h which the b u o y a n t material can How w i t h o u t having to displace the cooler, more viscous s u r r o u n d i n g s . A x i s y m m e t r i c flow in a purely vertical p l u m e c o n d u i t was analysed b y

O N T H E D Y N A M I C S O F L O N G - L I V E D P L U M E C O N D U I T S IN T H E C O N V E C T I N G M A N T L E

occurs as a result of conduction of heat into a thin boundary layer around the sphere. The boundary layer is heated, expands and adds to the buoyant volume. It is also stirred within the plume so that a large temperature gradient is maintained around the perimeter of the plume, thereby assuring continued entrainment. The rate of entrainment and its effects in spherical plume heads are well modelled by a similarity solution involving an experimentally evaluated constant. In the case of longlived plume conduits deflected by large-scale shear conduction and stirring occur in an approximately cylindrical geometry which again may lead to cooling and an increase in the plume mass flux with height. The dyed source fluid is seen to split into two parallel strips encased in a sheath of heated entrained mantle (Fig. 2). The interaction between thermal plumes and their surroundings must be correctly modelled if we are to understand the behaviour of plumes in

Loper and Stacey [5], who showed that upward flow in that case is concentrated near the axis of the plume where the temperature is greatest and the viscosity much reduced. A broad thermal halo extends to much larger radii, its width determined by a balance between radial conduction of heat and a very slow inflow at all depths. This halo effectively insulates the steady plume and little heat is lost from the source material as it ascends. For deflected plumes, on the other hand, laboratory experiments [4,6,7] show that nonaxisymmetric flow leads to entrainment of surrounding fluid into the interior of the plume. Major effects on the distribution of a tracer in the plume are observed when the plume is driven by thermal buoyancy [7]. This feature of the flow is common to isolated spherical volumes of hot fluid [8-10] and to the spherical heads of newly initiated plumes [11] (Fig. 1). For these spherical cases it has been demonstrated that entrainment

plate

215

plate velocity =.

V////////////////////////////////////////////////////////////////////////////////////////////////////////////////////•d I

starting plume i

steady state

source layer Fig. 1. A schematic diagram of two stages in the development of a mantle plume fed by a constant source flux: ( a ) the initial ascent of a large plume head followed by a nearly vertical feeder conduit; (b) the final steady state, inclined conduit. Black indicates hot, uncontaminated source material, stippled regions indicate cooler, stirred plume material. The arrows indicate the ascent of the conduit which leads to entrainment. The steady long-rived plume (b) is everywhere fixed relative to the source but is left behind by the plume head which effectively becomes attached to the moving lithosphere. Once deflected the plume conduit entrains surrounding mantle.

216

R.W. G R I F F I T H S A N D I.H. C A M P B E L l .

(a)

plumes to long-lived plumes, and allowing us to extrapolate results of the laboratory observations of Richards and Griffiths [4,7] to mantle conditions. Since the results will be dependent in detail on the viscosity distribution and structure of large-scale flow in the mantle, neither of which are fully prescribed, our aim is to demonstrate principles and, in particular, to point out the potential consequences of entrainment. 2. The model

Consider plumes emanating from an anomalously hot and unstable boundary layer deep in the mantle. The horizontal spacing of plumes, the mass flux into the base of each plume and the temperature of the material entering the plumes will depend on the details of the flow within the boundary layer source region. The flux is that carried into the plume by a convergent horizontal flow of heated, softened material in that part of the boundary layer having smallest viscosity [12,13], However, the behaviour of the plumes as they ascend through the mantle is otherwise insensitive to the source region. It is therefore sufficient for our purposes to model the flow as if a localised source of hot fluid is turned on at a time t = t 0. For simplicity and clarity we let both the volume and heat fluxes from the source be constant in time. It is assumed that the mantle, except for heated material in the plume and its source layer, is of uniform potential density 0~, potential temperature T~¢ and kinematic viscosity u~. If the source volume flux is Q and the source temperature is ~ , the plume heat flux becomes Fig. 2. Diagrams showing (a) the form of the non-axisymmetric recirculation (in a plane normal to the plume axis) resulting from horizontal deflection of plume conduits, and (b) the consequent distribution of source material in the conduit far above the source as determined from laboratory experiments.

where c is the specific heat and A ~ = 7~ -- T~ is the temperature anomaly. Taking the simplest form of the thermal expansion, #/~

the mantle, their depth of origin and the causes of compositional heterogeneity in hotspot magmas. However, a complete analysis of the flow within and around deflected plumes would be extremely difficult. Here we present an approximate solution obtained by modelling the essential features of the steady flow, thus extending the discussion of Griffiths and Campbell [11] from the heads of starting

(1)

Q H = pcQA7~,

= 1 - a(T-

T~)

(2)

the buoyancy flux is QB = gp~o~A ~ Q = gp~e~Qu/PC ,

(3)

where g is the gravitational acceleration. We take the case of a plume rising through a horizontal shear flow in the mantle and assume that the plume flow and the driving forces of the shear (convective overturn of the mantle or plate

ON THE DYNAMICS

OF LONG-LIVED

PLUME CONDUITS

IN T H E C O N V E C T 1 N G

motion) have reached a steady state. Conservation of heat then demands that the heat flux QH, which is integrated across the plume, must be independent of height along the conduit. Similarly, in the Boussinesq approximation, the buoyancy flux (3) must be independent of height. The temperature anomaly and volume flux, on the other hand, are functions of the height from the source. In order to evaluate these quantities it is assumed that azimuthal circulation resulting from the buoyancy-driven ascent of the conduit largely expels thermal (but not velocity) gradients from the interior of the plume to a thin boundary layer. Hence the temperature need only be represented by a square, or "top-hat", profile such that the heat flux QH = ocATq, where AT and q are the temperature anomaly and volume flux at each height. Thence (4)

qA T = QA 72 = QH/OC.

Assuming parallel axisymmetric flow driven by buoyancy in a circular pipe of diameter d, tilted at an angle `5 from the vertical, the volume flux q is given by q = rr(g cos ` 5 ) a A T d 4 / 1 2 8 v ,

(5)

where the viscosity v in the plume is much less than that outside the conduit and, consistent with the temperature profile, is assumed constant over any cross-section (again as a result of recirculation within the conduit). The viscosity does, of course, remain a function of temperature and will increase if the fluid ascending in the conduit cools with height. As in pipe flow, the velocity component parallel to the conduit axis and leading to the flux (5) is parabolic. The maximum velocity occurs on the axis, whereas the assumptions of a large viscosity contrast and square temperature profile imply that the component of flow parallel to the axis effectively vanishes at the edge of the hot conduit. From (4) and (5) the flux, diameter and temperature anomaly are related according to q = (rr~g

cos

`sQH/128pc)l/2d2v -1/2

a = (Ocrrag cos ` 5 / 1 2 8 Q H ) - l / n v l / 4 A T

(6a) 1/2.

(6b)

The tilted conduit will rise vertically by displacing the overlying mantle. This velocity is given by

[41 U = 0.135go~ATd2/v~,

(7)

217

MANTLE

where the only variable is the buoyancy cross-section A T d 2. We note that (6) implies A T d 2 - v 1/2. Hence U - v l / 2 / v ~ . Thus the Stokes speed of ascent (7) will be independent of distance along the conduit if the viscosities both inside and outside the plume are constant. In that case an entraining continuous plume in a simple shear will follow a parabolic path, as found experimentally for compositionally buoyant plumes having negligible entrainment [4]. The rate of entrainment of surrounding mantle material into a conduit as a result of conductive heating is estimated in a similar way as that into a diapir [8], but in this case is given by e ~ b28U sin `5,

(8)

where e is now the volume rate of entrainment per unit length of the conduit, 8 is the thermal boundary layer thickness given by 6 ,-~ b l ( ~ d / U

sin `5)1/2

(9)

and ~ is the thermal diffusivity. Combining (8) and (9): e = b~b2(~dU sin 4}) '/2.

(10)

The volume flux along the conduit consists of the source flux Q~ plus a contribution from entrainment. Hence d q / d X = e,

(11)

where X is the distance along the conduit. Our attempts to evaluate the constant bib 2 in (10) from experiments with sheared conduits similar to those reported by Richards and Griffiths [4] give values with large uncertainties as a result of difficulties in measuring variations in the small conduit diameter. On the other hand, the corresponding constant for spherical diapirs (as distinct from the quasi-cylindrical conduit) has been carefully determined from the entrainment-induced enlargement of the (relatively large) diameters of diapirs in a large number of experiments [8]. This value lies within the bounds of uncertainty for the conduit measurements. At the same time, we see no reason for the value to differ markedly between the spherical and cylindrical cases. We therefore set bib 2 ~ 4, keeping in mind that the uncertainty is about a factor of two.

218

R.W. GRIFF1THS A N D I.H. CAMPBELL

Using relations (4), (5), (7) and (10) the differential equation (11) becomes

d q / d X = blb2(0.135x sin ~,)'/2(128/rr cos q,)3/s X

(agQH/Pcv~)I/8(v/voo)3/~q '/4, (12)

where variables dependent upon the distance X are 0, v and q. The inclination ~a of conduits depends intimately on the vertical structure and magnitude of horizontal motions in the mantle [4] and may even change sign with depth. Given a lack of knowledge of the mantle motions, however, we choose first to solve a relatively simple and instructive case and find solutions for conduits having a prescribed and constant angle of tilt. In this case, if the viscosity v within the conduit is also constant, relations (4), (5), (7) and (12) lead to simple power-law solutions for q, d and AT as functions of X. Applying the boundary condition q = 0 (and d = 0) at X = 0 the solution can be put into the form

and larger angles of tilt. However, this solution does not take into account variations of the angle ,/, with height, a dependence of q~ on the buoyancy flux, or an increase in viscosity with height as material within the plume ascends and cools. Viscosity variation with temperature within the plume can be important because this viscosity controls the axial mass flux along the conduit. For a given rate of entrainment and cooling an increase in viscosity will lead to additional enlargement of the conduit diameter in order to pass the imposed buoyancy flux. This enlargement will in turn effect subsequent entrainment. We can explore the magnitude of the effects of viscosity variation by choosing a temperature dependence of the form: p, = 7e 1~/7

(14)

q / Q = (X/Xs) 4/3

where the values of "/ and fi are determined by the molecular properties of the fluid of interest. In order to predict q, AT and v it is then necessary to resort to numerical solution of (12) with (4) and (14). Plume diameter and Stokes ascent speed are available from (6) and (7).

d/d --- (x/x

3. Solutions for mantle plumes

,~T/,,.~77 s

)273

= (h/As)

4/3,

(13a)

where d~ is the plume diameter at the source and

b = 0.23(bib2)

1 cos3/aq, sin i/2q,

Rp = gaQA ~d2/K2v~

(13b)

The parameter R p in (13b) is a Rayleigh number, which can be written as R p = QBd2/I¢.21x ~ and which describes the strength of the driving buoyancy force compared to the retarding influences of conduction and viscosity. The distance X~ is the position of the source with respect to a virtual source at X = 0. It is also the length scale over which the plume properties are modified by entrainment. The plume transport at a distance h from the source is simply q = Q(1 + h/Xs) 4/3. This simple solution illustrates the way in which entrainment leads to enlargement of the plume diameter, an increasing volume flux and a cooling of the plume with height. Clearly, entrainment has

larger effects in plumes with smaller buoyancy fluxes

3.1. The conditions Although some aspects of the mantle flow are poorly known, it is indicative to calculate plume behaviour in the mantle using approximate values of material properties and simple models incorporating horizontal shear. For properties in the deep mantle we use: x = 10 -6 m 2 s 1 a = 2 x 10 ~5 K -1, p ~ = 4 × 1 0 3 kg m -3 and c = 7 5 0 J kg 1 K -1. We use two viscosities: ff~ = 10 21 Pa s (or u~ = 2.5 × 1017 m 2 s 1) which models the modern upper mantle and 1022 Pa s which provides a reasonable average over the whole mantle [14]. Two source parameters are required: the source temperature anomaly and volume flux. Both of these are discussed by Griffiths and Campbell [11] in relation to the initial unsteady stage of plume ascent. Briefly, the potential temperatures of the source region required to produce picritic melts in flood basalt provinces can be taken as a lower bound on the potential temperature of the mantle in the boundary layer from which the plume originates. Potential temperatures of 1500°C for high-MgO picrites and 1280°C for normal mantle

ON THE DYNAMICS OF LONG-L[VED PLUME CONDUITS IN THE CONVECTING MANTLE

[15] imply an anomaly of ATs>~ 200°C. In addition, if the source is a thermal boundary layer at the core-mantle boundary (CMB), estimates of around 800°C for the core-to-mantle temperature difference [12] (although unreliable) give an upper bound on the plume anomaly. Hence we have investigated the implications of source temperature anomalies in the range 200 ° ~ AT~ ~< 800°C. In order to find the volume flux Q from (3) it is necessary to know the source temperature anomaly and the plume buoyancy flux QB- Davies [16] and Sleep [17] have evaluated the buoyancy fluxes for a large number of currently "active" hotspots from the rate of creation of seafloor topography. They obtain values in the range 0.3 ~< Qg/g <~8 M g s l. These become 3 × 1 0 3 ~ < Q B ~ 8 × 1 0 4 N s 1. The hotspot population shows a broad spectrum of fluxes, with a median value of 1 × 10 4 N s 1. The corresponding volume fluxes are approximately 20 ~< Q ~< 500 m 3 s-1 if AT~ = 200°C or 5 ~< Q ~< 125 m 3 S-1 if AT., = 800°C. From (1) the largest buoyancy flux, that inferred for Hawaii, also corresponds to a heat flux Q~ = 3 × 1011 W, whereas the average identifiable plume carries 3 × 10 ~° W. In the following we consider buoyancy fluxes in the range 103-105 N s 1. As an example with a constant viscosity inside the plume we take QB = 1 0 4 N s 1 and /1~ = 1022 P a s ( v ~ = 2 . 5 × 1 0 1 ~ m 2 s 1) along with the arbi_ trary values v/v~ ~ 10 -3, d~ = 50 km and ~ = 20 °. From (13) the length X~ = 2.3 × 1 0 4 km and the corresponding increase in plume volume flux due to entrainment is only 6% at a distance 1000 km above the source, or 21% at 3000 km. These values are halved if the ambient viscosity is 1021 Pa s. This calculation therefore suggests that entrainment may have only small effects on the physical characteristics of plumes. The importance of this amount of entrainment in determining the characteristic trace element or isotopic ratios of plumes depends upon whether the plume source or the entrained material has the larger concentration of those components. On the other hand, a 21% decrease in temperature anomaly can have a large effect on plume viscosity and this can, in turn, alter entrainment. Hence viscosity variations must be allowed for. The implications of a range of values for fl and 3' in (14) were explored, but the results were found to be insensitive to the precise values chosen.

103

-~ ~ 102

219

I "1 .... I Jg Cr Ca

I Re

i

.... P

i: Ha.

EAT=100°C

~5

t0 1

103

104 buoyancyflux (N/s)

105

Fig. 3. The diameter of conduits carrying a given buoyancy flux for selected local temperature differences. This calculation is for vertical conduits. However, the diameter increases only as (cos~) i/4 when the plume is tilted by an angle e~. Viscosity in the plume is estimated from (14). Temperature anomalies of order 100 200°C in the melt zone beneath ocean island chains imply plume diameters of 100 300 km near that depth before melting. Anomalies of order 500°C responsible for Archaean komatiites [18] imply diameters of 30 70 km if buoyancy fluxes were comparable. We also indicate the strengths of several plumes as estimated by Sleep [17]: Ha = Hawaii, P = Pitcairn, Re = Reunion, = Canary, Cr = Crozet, J F = Juan de Fuca.

Ca

Hence we present results for fi = 4 × 10 4 K and "I' = 6.52 × 109 Pa s, a combination which gives an ambient viscosity (at 1280°C) of 1021 Pa s. We choose this ambient viscosity because the greatest deflection of conduits is expected to occur in the upper mantle, where shear stresses are likely to be greatest. It should also give a conservative estimate for entrainment. The conduit diameter (6b) is now a function of the angle of inclination, the buoyancy flux and the temperature anomaly. Diameters for the case ~ = 0 are shown in Fig. 3. These are relevant at the source but are also representative of conduits subtending small angles to the vertical as they ascend through a sheared mantle. Thus modern plumes feeding ocean island chains and having a potential temperature of, say, 1400°C ( A T = 100 °) at the top of the mantle are predicted (given the present choice of/~ and y) to have diameters of 200-300 km, depending on Q~. At the other extreme there may have been Archaean plumes for which a value of A ~ ~ 500°C may be more appropriate [181, and these would have had diameters of 30-70 km near their source.

220

R.W. G R I F F I T H S AND I.H. CAMPBELL

In order to illustrate the effects of entrainment on these plumes we have selected two idealised models: the semi-dynamic case of a straight conduit prescribed to have a constant inclination to the vertical; and the fully dynamic case of a conduit with a steady state trajectory determined by a simple mantle shear (a horizontal velocity decreasing linearly with depth). The first case is of greatest use in illustrating the fundamental effects of entrainment; the second involves additional complications due to the variation of tilt with height, the inclination itself being influenced by the effects of entrainment. 3.2. Plumes with f i x e d inclination

Results for the case of a plume with constant tilt but temperature dependent viscosity are shown in Fig. 4. Entrainment in this case is much greater than that calculated above with a constant plume viscosity. This is because the viscosity increase on cooling leads to a larger conduit diameter (in order to accommodate the buoyancy flux), a larger conduit ascent speed, and greater surface area

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4 3 1000

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0

~

~

~

J

10

20

30

40

50

60

angle of tilt (deg) Fig. 4. The local mass f l u x in a model plume c o n d u i t relative to t h e s o u r c e m a s s flux a t h e i g h t s 1000 a n d 3000 k m f r o m t h e s o u r c e for a b u o y a n c y f l u x QB = 1 0 4 ture

anomaly

ATe= 300°C

and

N s a

1 a source tempera-

temperature-dependent

v i s c o s i t y . A l m o s t i d e n t i c a l r e s u l t s a r e f o u n d f o r AT~ = 2 0 0 a n d 400°C. The temperature-dependence

o f v i s c o s i t y l e a d s t o en-

h a n c e d e n t r a i n m e n t , w h i c h c a n b e l a r g e e v e n 1000 k m f r o m t h e source.

over which the conductive boundary layer develops. As illustrated in Fig. 4, entrainment increases with the angle of inclination. The angles of tilt of probable relevance to mantle plumes are poorly constrained. However, the lateral displacement of most plumes between source and surface cannot be greater than about 300 kin, otherwise hotspots would not appear to remain nearly stationary relative to each other immediately after changes in surface plate velocities. The small radius of curvature ( < 200 km) of the bend in the H a w a i i a n - E m p e r o r chain confirms this conclusion [6]. Hence the mean tilt of a plume originating at the CMB is likely to be no greater than 10 °. In this case the tilt is likely to be greatest in the upper mantle [4], where inclination has had a chance to accumulate and where horizontal return flow may be concentrated. Angles of 30 ° may be possible if the mantle shear is such that most of the horizontal displacement occurs between depths of 100 and 700 km. Even larger angles may occur if shear is confined to a shallow low-viscosity layer. However, depth averaged angles of tilt will be smaller, with a maximum (of about 30 °) in the case where a plume originates at the mid-mantle transition zone. Selecting a 20 ° angle of tilt we compute the effects of entrainment on plume mass flux, diameter and temperature anomaly for a range of buoyancy fluxes (Fig. 5). The most obvious result is that stronger plumes entrain less as a fraction of their source flux. Plumes having fluxes as large as that inferred for Hawaii ( - 1 0 5 N s 1), are not greatly altered by stirring, although even these are predicted to be diluted by around 30%, a result that could be important in chemical terms. Plumes having fluxes of order 103 N s - t are predicted to be so strongly diluted and cooled (by a factor greater than 10: 1) that they would be unlikely to cause melting in the upper mantle, where the temperature anomaly they introduce would be less than 10°C. This result is almost independent of the source temperature. There is a very sharp cutoff dividing buoyancy fluxes that can carry thermal anomalies of order 100°C to the top of the mantle from those that cannot. This cut-off lies between 3000 and 5000 N s -t, a value which is in close agreement with the weakest buoyancy flux inferred for observed hotspot tracks [17]. It varies little with the choice

()N ]'HE DYNAMICS

OF LONG-LIVED

PLUME CONDUITS

IN T H E C O N V E C T 1 N G

of rheology and angle. For example, a mantle viscosity of 1022 Pa s leads to only slightly greater entrainment effects. The largest uncertainty in the predictions arises from the factor-of-two uncertainty in the empirical value for bib 2 in (10): halving the value decreases the predicted rate of entrainment by close to two. Buoyancy fluxes (a)

10

. . . . . . .

,

221

MANTLE

greater than 2000-3000 N s ~ originating from the CMB are then able to produce the required hotspots at the top of the mantle. For plumes arising from the 670 km transition zone the cutoff flux is smaller, but still of order 103 N s 1. (We say more about the behaviour of weak plumes later.) (b) 1000

. . . . . . . .

. . . . . . . .

i

. . . . . . .

200 °

u~

IO0 oi E

/~'~',`

~(400/° X X

(lO00km)

8

"":::::"!!? """........ .....11% ]03

~ ~

104

1003

0 s

buoyancy flux (N/s) (c)

. . . . . . . . . . . . . .

o

I

600 (at source)

104

105

buoyancy flux (N/s)

ool 800

iI f

d.

600 o,.,, 400

2O0

0

104

10 s

buoyancy flux (N/s) Fig. 5. The properties of a plume conduit at the top of the mantle for a fixed (20 °) angle of tilt: (a) local mass flux compared with the source flux; (b) conduit diameter; (c) local temperature anomaly. A mantle viscosity of 102a Pa s was used and a slightly greater entrainment is found for 1022 Pa s. Curves are labelled with the source temperature anomaly A~. The properties of the plume are shown at 1000 km from the source (dotted curves) and at 3000 km from the source (solid curves). Note the logarithmic ordinate scale in (a) and the very rapid variation of entrainment with buoyancy flux at small fluxes, which implies a sharp cutoff to those fluxes likely to produce a surface hotspot.

222

The d i a m e t e r s of conduits (Fig. 5b) reflect the degree of dilution a n d cooling. F o r the largest source fluxes the p l u m e d i a m e t e r at the top of the m a n t l e is a r o u n d 50% greater than at the source. F o r example, our e s t i m a t e d d i a m e t e r for the H a w a i i a n plume, assuming QB = 8 × 10 4 N s [17], A ~ = 300°C a n d a source at the c o r e - m a n t l e b o u n d a r y , is 180 k m at its source. This increases to 270 km at the surface, where the c a l c u l a t e d t e m p e r a t u r e has d e c r e a s e d to A T ~ 230°C. (An i n d e p e n d e n t estimate by M c K e n z i e [19] of the near-surface d i a m e t e r is 240 km.) F o r the smallest of the fluxes inferred by Sleep [17], a n d the s a m e source t e m p e r a t u r e a n o m a l y of 300°C, the d i a m e ter increases from 80 k m at the source to 600 k m (~T--~ 40°C) at the surface. This dilution of weak p l u m e s m a y be o v e r - e s t i m a t e d b y our selection of a fixed inclination t h r o u g h o u t the d e p t h of the mantle. However, it serves to d e m o n s t r a t e a p o t e n t i a l l y large role for e n t r a i n m e n t in influencing the dynamics, t e m p e r a t u r e a n d c o m p o s i t i o n of ascending plumes.

R.W.

10

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GRIFFITHS

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105

buoyancy flux (N/s)

Fig. 6. Local mass fluxes compared with the source flux for tilted conduits in a simple shear flow driven by a surface plate speed of 0.05 m/yr. Results are shown for heights 3000 km (heavy lines) and 1000 km (light lines) from the source. Dilutions of the source material are similar to those found for a constant angle of tilt (Fig. 4). Again, the results imply a cut-off in the strength of plumes expected to be detectable at the surface.

3.3. A simple shear flow The case of a steady state p l u m e in a simple shear flow d e m o n s t r a t e s the effects of e n t r a i n m e n t on both p l u m e ascent a n d h o r i z o n t a l deflection. W e i m p o s e a shear stress i n d e p e n d e n t of d e p t h , giving a linear decrease of h o r i z o n t a l velocity L, from a m a x i m u m at the overlying surface p l a t e to zero at the source d e p t h [4]. T a k i n g a plate speed of 0.05 m / y r the c o n d u i t m o d e l predicts p l u m e mass fluxes slightly greater than those c o m p u t e d a b o v e for ~ = 20 ° (Fig. 6). These again corres p o n d to dilutions of o r d e r 30% for the strongest p l u m e s and extreme dilutions for fluxes less t h a n 5×103Ns l E n t r a i n i n g c o n d u i t s in a simple shear d o not rise along p a r a b o l i c paths, as they w o u l d if there was no e n t r a i n m e n t or if the viscosity inside an entraining p l u m e was i n d e p e n d e n t of t e m p e r a t u r e (see (7) above). I n s t e a d they take a p a t h that is closer to a straight line. This is because the d i a m e ter and ascent speed of a p l u m e e l e m e n t increase as it ascends higher, thus decreasing the angle s u b t e n d e d to the vertical by the local velocity vector. F o r the same reason, weak c o n d u i t s can, u n d e r some conditions, be deflected less than p l u m e s having larger b u o y a n c y fluxes. However, u n d e r most c o n d i t i o n s that we have investigated

the h o r i z o n t a l defection of p l u m e s continues to increase with d e c r e a s i n g b u o y a n c y flux. T h e overall deflections c o m p u t e d from the m o d e l are c o n s i s t e n t to an o r d e r of m a g n i t u d e with those easily e s t i m a t e d by taking the m e a n characteristics of c o n d u i t s b e t w e e n their source a n d the surface. F o r example, a m o d e r a t e b u o y a n c y flux of 104 N s ~ a n d a t e m p e r a t u r e a n o m a l y of 300°C at the source i m p l y a mean c o n d u i t t e m p e r a t u r e of a p p r o x i m a t e l y 200°C and d i a m e t e r of a p p r o x i m a t e l y 200 km, for which the Stokes ascent speed in the u p p e r m a n t l e is U - - 7 × 10 ~ m s 1 A s s u m i n g a linear t r a j e c t o r y a shear of 0.05 m / y r ( 1 . 6 × 1 0 9 m s i) over a d e p t h of 2800 k m will lead to a deflection of o r d e r 100 km. Because the p a t h is not linear the deflection given by the m o d e l is 3 0 0 - 5 0 0 km, d e p e n d i n g on the a s s u m e d viscosity v a r i a t i o n with t e m p e r a ture. S t r o n g e r p l u m e s are deflected less. They satisfy the c o n s t r a i n t of a deflection less than 200 k m inferred from the r a d i u s of the b e n d in the H a w a i i a n - E m p e r o r s e a m o u n t chain [6] a n d are likely to r e m a i n stable to d i a p i r i c b r e a k u p [4]. W e a k e r p l u m e s are d e f l e c t e d farther. T h e fate of weak p l u m e c o n d u i t s is discussed below. These

ON T H E DYNAMICS OF LONG-LIVED P L U M E C O N D U I T S IN T H E C O N V E C T I N G M A N T L E

results are not greatly altered when a larger mantle viscosity (1022 Pa s) is assumed, or if the gradient of horizontal velocity in a mantle with two viscosity layers is that given by a depth-independent shear stress in place of a constant shear

[41. Although our calculation for the case of a simple shear profile probably over-estimates the effects of entrainment compared with shear profiles having zero stress at the CMB [4], it again shows that dilution is sensitive to the buoyancy flux but not to the source temperature anomaly (Fig. 5c). 4. Discussion

The entraining-conduit model for continuing plumes has a number of other implications for hotspot melt composition and plume source characteristics. These will not depend on the precise form of the mantle shear flow and whether it remains steady, at least not so long as the shear stress penetrates throughout the depth of the upper mantle and horizontal motion beneath plates is not confined to a shallow low-viscosity zone. 4. I Picrites and hotspot tracks Our plume model predicts that entrainment will be greater when the buoyancy flux is smaller, with a relatively weak dependence on the angle of tilt. Furthermore the degree of tilt will depend on both the buoyancy flux and the magnitude of the shear (or plate velocity). Hence we find that the plate velocity will have a very much smaller influence on entrainment than will the flux. If all plumes originate from sources having commensurate potential temperatures, we would expect Hawaii (the strongest of the modern plumes with a buoyancy flux of 8 × 104 N s -1 [17]) to produce the hottest melts. For this plume the model predicts a small cooling and dilution of order 20-30%. Hence the hottest melts may be as hot as any known to be produced by modern hotspots, with MgO contents as high as that of the picrites of continental flood basalt provinces [23]. Other strong plumes such as Easter Island, Marqueses, Pitcairn and Tahiti, which have buoyancy fluxes of approximately 3 × 10 4 N s 1 [17], and Reunion, with a flux of - 2 × 10 4 N s ~, should also produce picrites. Very weak plumes such as Crozet, Kerguelen and Juan de

223

Fuca, which have fluxes of order 5 × 103 N s -1, will give only relatively low-temperature melts. Testing this prediction is difficult for two reasons. First, there is little information available on the maximum MgO content of aphyric samples or the Fo content of olivines, information which would allow a distinction to be drawn between basalts with cumulate olivine and picrites that have crystallised from a high-MgO melt. Second, most samples of hotspot volcanics are taken from the tops of ocean islands or seamounts and represent the most recent volcanic activity, often the alkalic post-erosional stage. Most of the picrites, on the other hand, can be expected to erupt during the main high-temperature phase of activity and to be largely covered by the later lower-temperature tholeiitic and alkalic flows. In spite of these limitations a number of authoritative estimates are available for the maxim u m MgO content of primitive Hawaiian magmas. Most lie within the range 12-15% MgO [25-27]; however, Frey et al [28] have found olivines with cores as Mg-rich a s Fo90.5 implying crystallisation from a m a g m a with an MgO content between 14.4 and 17.0%, and Chen [29] has suggested an MgO content of 16.6% for a primary tholeiite from Haleakala Volcano. More recently picritic basalt, believed by Garcia [30] to have crystallised from magmas with 17-18% MgO, have been dredged from the flanks of Hawaii. Picrites, believed to have crystallised from magmas with 18%, are also recorded from Tahiti but the maxim u m Mg content of the associated olivines is only Fo~8 [31]. Fisk et al. [32] argue for a parent magma to the Reunion basalts with 15-17% MgO. Finally, picritic basalts have recently been dredged from young seamounts above the Pitcairn hotspot [33] but await further study and no estimates are available for the m a x i m u m MgO content of melts from this hotspot. We know of no examples of picrites from ocean islands produced by a plume having a buoyancy flux less than 104 N s 1. We predict that oceanic picrites are to be found by dredging of young seamounts above strong plumes. 4.2 Temperature anomaly at the core-mantle boundary We have argued elsewhere, on the basis of the size of flood basalt provinces, that the strong starting plumes which produce these features must

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originate deep within the lower mantle and almost certainly from the core-mantle boundary [11,23[. According to our hypothesis strong long-lived conduits, especially those that experience little or no tilting, entrain only small amounts of the overlying mantle as they ascend. As a consequence, the plume material retains a potential temperature close to that of the source. The liquidus temperature of melts can be used to assess the temperature of the melt region and the highest liquidus temperature of picrites therefore provides an estimate of the potential temperature of the plume source. The MgO contents of picrites sampled from flood basalts and the tracks of strong plumes normally exhibit maximum values between 18 and 19% and imply potential temperatures of at least 200°C above normal mantle. This temperature, in turn, represents a lower bound on the average temperature of the boundary layer material drawn into the base of the plume. Loper and Stacey [5] and Davies [13] have shown that in a mantle with strongly temperature-dependent viscosity the plume will draw material only from a shallow layer about 15 km thick close to the heat source, where the viscosity is smallest. The material sampled at the base of the plume is therefore only that in a zone of highest temperature at the bottom of the gravitationally unstable part of a thermal boundary layer. Hence the plume temperature is also an estimate of the temperature j u m p across the unstable source layer. Because the above temperature anomaly is a lower bound based on the liquidus temperature of picrites that have been sampled to date, and taking into account some cooling as material ascends in the conduit, we suggest the best estimate for the temperature step across the unstable portion of the source layer is 300 _+ 100°C. This estimate is more reasonable than the 800°C suggested by Stacey and Loper [12], a temperature at which plumes would undergo complete melting at lower mantle pressures and produce picrites or komatiites with MgO contents far in excess of the observed values. If plumes do originate from the c o r e - m a n t l e boundary, and if the interface between the core and the mantle is infinitesimally thin, then our estimate of 300°C for the source temperature anomaly will represent the whole of the temperature difference between the core and the mantle.

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However, the presence of even a very thin layer of compositional gradient, through which heat transport is by conduction alone, will significantly reduce the efficiency of heat transfer. Similarly, the existence of a thick, gravitationally stable, iron-rich layer (possibly identified with the seismic D" layer) as suggested by Knittle and Jeanlot [34] would have an even greater influence. In either case the plume source will be insulated from the core and the core-to-mantle temperature difference will be much greater than the plume temperature anomaly. 4.3 Trace element and isotopic ratios The hottest magmas must be produced by melting of the least contaminated part of a plume. They represent the best geochemical sample of the plume source region (i.e. the thermal boundary layer from which the plume originates). Plumes yielding melts having the highest MgO content are therefore those that have mixed least with their surroundings and, for a given hotspot, the melts having the highest MgO content come from the least contaminated part of the plume. Thus we recommend that geochemists concentrate on picrites when assessing the isotopic and geochemical characteristics of the plume source. Unfortunately trace element data are available for only two ocean island picrites: these come from Reunion and Hawaii. On a trace element normalised abundance diagram (Fig. 7) the Reunion pattern is strongly enriched in the light rare earth elements. This enrichment is slightly less than that for picrites from the Deccan, the flood

100

~ /~

,,,~\ ,~.

~Reunion ~

'I'~'-- Haw aii

g

~o E

0

Rb

Ba

Th

Nb

K

Ca

Ce

Sr

Nd

Sm

Ti

Tb

Y

Yb

Fig. 7. Normalised trace element abundances in picrite samples from Hawaii [35]. Reunion [321 and the Deccan [36]. Mantle normalizing values from Sun and McDonough [37 I.

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basalt provence produced by the Reunion hotspot, especially with respect to highly incompatible elements such as Rb, Ba, Th and K. The Hawaiian picrite, described by Chen and Frey [35] as a tholeiite with 16.6% MgO, has even smaller enrichment of highly incompatible elements. Although no trace element data are available for the Tahiti picrites described by Tracy and Robinson [31], their high TiO 2 and K 2 0 contents suggest that they are enriched in highly incompatible elements. If the incompatible trace element characteristics of the OIB-type mantle end member in plumes can be identified from analysis of picrites, the knowledge will be invaluable in interpreting the trace element and isotopic characteristics of melts produced by lower degrees of partial melting, often from a hybrid source. This is especially so if it is valid to assume that the entrained component in the plume is normal (MORB-type) mantle.

4.4. A minimum buoyancy flux Having constrained the temperature of the plume source and melt region by the maximum liquidus temperature of melts from the strongest plumes, it is possible to estimate thermal anomalies emplaced by other plumes. The entrainment model predicts that plumes carrying a buoyancy flux of 104 N s-1 (the average of all those collated by Sleep [17]) and having the source anomaly of 200-300°C at the core-mantle boundary will lead to melt temperatures in the range 60-120°C above normal mantle (Figs. 3 and 4). The uncertainty here is largely a result of the uncertainty in the value of the entrainment constant (b~b2). For very weak plumes (QB ~ 5000 N s -1) this same source will produce near-surface temperature anomalies of less than 60°C. An important result is the prediction that longlived plumes having buoyancy fluxes less than about 3000 N s-~ (or heat fluxes less than 10 m W) can deliver only very small thermal anomalies into the upper mantle. They will produce no seafloor swells and little or no volcanism. This prediction is insensitive to assumptions about mantle viscosity, degree of inclination of the plume, and source temperature anomaly. It agrees closely with the smallest observed plume buoyancy flux as inferred from the topography of identified hotspots. The above prediction of a lower bound to the

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buoyancy fluxes that can produce hotspots is based on the calculated plume temperature. The argument is consistent with the extent of horizontal deflection of a conduit as calculated from the present model. Conduits with fluxes as small as 103 N s -I are deflected by over 1000 km. This indicates that they are likely to be unstable to diapirism, at least in the upper mantle. Each diapir will continue to entrain and cool as it ascends. These plumes are therefore characterised by a small thermal signal, are drawn out very extensively and lead to small, well separated diapirs which may or may not reach the lithosphere. 5. Conclusions Recent laboratory observations of deflected plume conduits beneath a moving upper plate show a large amount of stirring between the plume and its surroundings. An analysis of the dynamics of steady thermal plumes bent over by a horizontal shear flow in the mantle enables us, given information about the source depth, source temperature anomaly and buoyancy flux, to predict the properties of a long-lived plume when it reaches the top of the mantle. Deflected conduits can entrain their surroundings through coupling of conduction of heat and non-axisymmetric flow within the conduit [8,11]. Entrainment cools the plume and increases its mass flux. However, the strongest of modern plumes are predicted to undergo only a small amount of cooling. We conclude that these can produce high temperature basalts or picrites. Because these strong plumes also are least contaminated by entrainment the geochemistry of the picrites provides the best sample of the isotopic composition, incompatible trace element ratios and temperature of the thermal boundary layer from which the plume originates. We hope that this study will encourage more attention to picrites than has been the case in the past. Our view of plume conduits is a major departure from models in which plume conduits consist of a very narrow central pipe of high velocity surrounded by a broader thermal halo established by radial conduction [5]. Instead, in the presence of deflection by large-scale convection, we advocate a hot pipe surrounded by a very thin thermal boundary layer and containing material which is

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a p p r o x i m a t e l y h o m o g e n e o u s in t e m p e r a t u r e as a result of stirring a n d c o n d u c t i o n . C o m p o s i t i o n a l differences, on the other hand, are only stirred, not mixed, due to the negligible rate of chemical diffusion. As a consequence, c o m p o s i t i o n a l zonation occurs in plumes where there is a compositional difference between the p l u m e source a n d e n t r a i n e d mantle. The experiments [6,7] indicate that the p l u m e source material becomes largely confined to two separated parallel tubes surr o u n d e d by heated e n t r a i n e d mantle. W e postulate that an effective c o m p o s i t i o n a l mixing m a y occur later in melts derived from the p l u m e if ascent of the p l u m e carries the zoned material through its solidus. The melt c o m p o s i t i o n could then be variable, reflecting the relative volumes extracted from each of the zones, a n d might give rise to some of the chemical a n d isotopic variability detected between hotspot basalts [20-22]. C o n d u i t s can potentially e n t r a i n material from all depths within the m a n t l e above the source, b u t most e n t r a i n m e n t is likely to be from the u p p e r m a n t l e if shear a n d p l u m e i n c l i n a t i o n are greater there. O u r conclusions should not be altered by a degree of u n s t e a d i n e s s in c o n t i n u i n g p l u m e flow resulting from slow changes in the flux delivered from the p l u m e source or from changes in the overlying plate velocities [4,6] a n d in the velocity field of the mantle. T h e a d j u s t m e n t of c o n d u i t s toward the steady state (Richards a n d Griffiths [4]) takes place over the time required for ascent of an element of the plume. F o r m o d e r a t e a n d large b u o y a n c y fluxes this time is less t h a n a transit time for the large scale convection. However, more detailed physical m o d e l l i n g of plumes is needed a n d should consider effects of more realistic large-scale flows, p r e s s u r e - d e p e n d e n t viscosity both inside a n d outside the plume, a n d the d i s t r i b u t i o n of partial melting in the u p p e r reaches of the plume.

Acknowledgements We wish to t h a n k Mark Richards, Geoff Davies a n d Robert Hill for s t i m u l a t i n g discussions a n d for their c o l l a b o r a t i o n in related work.

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31 R.J. Tracy and P. Robinson, Zoned titanium augite in alkali olivine basalt from Tahiti and the nature of titanium substitution in augite, Am. Mineral. 62, 634-645, 1977. 32 M.R. Fisk, B.G.J. Upton, C.E. Ford and W.M. White, Geochemical and experimental study of the genesis of magmas of Reunion Island, Indian Ocean, J. Geophys. Res. 93, 4933-4950, 1988. 33 J.D. Woodhead, personal communication, 1990. 34 E. Knittle and R. Jeanloz, Simulating the core-mantle boundary: an experimental study of high-pressure reactions between silicates and liquid iron, Geophys. Res. Lett. 16, 609 612, 1989. 35 C.-Y. Chen and F.A Frey, Trace element and isotopic geochemistry of lavas from Haleakala volcano, East Maul, Hawaii: implications for the origin of Hawaiian basahs, J. Geophys. Res. 90, 8743-8768, 1985. 36 P. Krishnamurphy and K.G. Cox, Picritic basalts and related lavas from the Deccan Traps of Western India, Contrib. Mineral. Petrol. 62, 53-75, 1977. 37 S.-S. Sun and W.F. McDonough, Chemical and isotopic systematics of oceanic basalts: Implications for mantle coposition and processes, in: Magmatism in Oceanic Basins, A.D. Saunders and MJ. Norry, eds., pp. 313-345, 1989.