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Earth and Planetary Science Letters 160 (1998) 289–296

Edge-driven convection 

Scott D. King a, , Don L. Anderson b

a  Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN, USA b Seismological Laboratory, California Institute of Technology, Pasadena, CA, USA

Received 15 January 1998; revised version received 21 April 1998; accepted 28 April 1998

Abstract

We consider a series of simple calculations with a step-function change in thickness of the lithosphere and imposed, far-field boundary conditions to illustrate the influence of the lithosphere on mantle flow. We consider the effect of aspect ratio and far-field boundary conditions on the small-scale flow driven by a discontinuity in the thickness of the lithosphere. In an isothermal mantle, with no other outside influences, the basic small-scale flow aligns with the lithosphere such that there is a downwelling at the lithospheric discontinuity (edge-driven flow); however, the pattern of the small-scale flow is strongly dependent on the large-scale thermal structure of a much broader area of the upper mantle. Long-wavelength temperature anomalies in the upper mantle can overwhelm edge-driven flow on a short timescale; however, convective motions work to homogenize these anomalies on the order of 100 million years while cratonic roots can remain stable for longer time periods. A systematic study of the effect of the boundary conditions and aspect ratio of the domain shows that small-scale, and large-scale flows are driven by the lithosphere. Edge-driven flow produces velocities on the order of 20 mm yr. This is comparable to calculations by others and we can expect an increase in this rate as the mantle viscosity is

decreased.  1998 Elsevier Science B.V. All rights reserved. Keywords: mantle; convection; flood basalts; mantle plumes

pointed out that thermal conditions at the edge of a

1. Introduction

continent drive a strong, unsteady flow.



The lithosphere of the Earth is composed of nu merous provinces with different ages, thicknesses,

and compositions. These provinces impose heterogeneous thermal and mechanical boundary conditions

at the top of the mantle lithosphere system. While some variations, such as the thickness of the oceanic vary smoothly, except at fracture zones, lithosphere, there may be large discontinuities at other locations, such as the edges of Archean cratons. Elder [1]

 Corresponding author. Tel.:  1 (765) 494-3696; Fax:   4961210; E-mail: [email protected]

1 (765)

Laboratory (described by Elder) gave a strongly pulsating upflow of hot fluid near the edge of the continent. These variations in surface boundary con ditions are seldom considered in mantle convection studies.  Several studies have suggested that discontinuities in lithospheric thickness drive small-scale convective that are responsible for observed surface tinstabilities opography anomalies and or tectonic features. Vogt [2] proposed that variation in lithospheric thickness

at a continent–ocean lithospheric boundary creates

a small-scale convective instability with a down-

experiments

0012-821X/98/$19.00  1998 Elsevier Science B.V. All rights reserved.  PII S0012-821X(98)00089-2

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welling that remains fixed at the boundary between the continental and oceanic lithosphere. This downwelling has a corresponding upwelling about 600 km

from the edge of the continent. This mechanism is used to explain the Bermuda Rise, an elongated swell

approximately 1500 km long and 500–1000 km wide that is up to 1 km shallower than normal seafloor of that age, and the Appalachian–Labrador Rise, a belt of elevated topography approximately 500 km wide that extends from the southern Appalachians to the ! "Labrador Highlands. While the Bermuda Rise has been attributed to a mantle plume, there are nu merous problems reconciling the observations with

a plume model [2], leading to suggestions that the plume is highly episodic and moves with respect to other hotspots [3] or perhaps travels with the North "American plate [4]. It is easier to visualize an instability generated by the lithosphere traveling with a plate, than one generated at great depth that travels through the mantle along with the North American plate. The plume hypothesis attributes high rates of magmatism to high mantle temperatures. Plume models require potential temperatures of approxi mately 300ºC above the background mantle temper ature to sufficient melt for flood basalt provinces Hot plume material passively rises and melts as [5]. the lithosphere is stretched or thinned. Some of the problems with current plume models are discussed in Cordery et al. [5]. In the convective partial melting process [6,7], increased melting is accomplished by convecting more material through the melting zone. Keen and Boutilier [7] showed that reasonable parameters could provide 3.5 times more melt than the passive upwelling plume model, about the same as a # 300º  C temperature rise. The role of small-scale convection in rift en$vironments has been addressed by several authors % e.g. [7–9]). In these models, small-scale convec(tion is driven by horizontal temperature gradients that develop as the lithosphere is thinned. Mutter et al. [6] suggested that small-scale convection may " e the mechanism that generated the large quantitbies of magma needed to explain the thick igneous crust observed seismically at the edge of the North continent. Mutter [10] pointed out that tAmerican he thickness of volcanic margins thins away from the continent boundary and not with distance from

&

a

hotspot and, in some cases, there is no nearby hotspot. This has led to suggestions that plume material rises to near the surface and then travels hundreds, even thousands, of kilometers laterally in order to explain the occurrence of some large igneous provinces where there is no obvious hotspot chain [11]. Elder [1] showed that edge-driven con$vection is intrinsically unsteady and pulsating, a characteristic of some large igneous provinces. It is

also intrinsically transient, particularly in a newly opening ocean. ' In a previous paper [12], we described a mechanism relying on the variability of near-surface ther mal and mechanical conditions that is capable of explaining the correlation of large igneous provinces

and the boundaries of Archean cratons. This mech anism is based on the discontinuity in the thickness of the lithosphere at the boundary between older Archean craton and younger continental lithosphere. We concentrate on a continent–ocean bound ary where the discontinuity is pre-existing, rather than being formed gradually by the rifting process. In this paper, we use the term lithosphere in the traditional sense, e.g. the strong outer shell of the Earth. The lithosphere can be strong because of low temper ature or low volatile content. The thermal boundary  is about twice the thickness of lithosphere and layer the lower part of the thermal boundary layer is weak.  The previous paper [12] has generated confusion because the pattern of flow from the calculations is opposite to intuition and other results [7]. A critical

assumption in King and Anderson [12] is that the mantle under the Archean craton is warmer than the mantle under the thinner boundary. Continental insulation and lack of subducted material cooling the (upper mantle in a region are possible mechanisms or generating such a thermal anomaly. Because of fthe  interaction of long-wavelength thermal anomalies in the upper mantle and a small-scale thermal

anomaly at the craton boundary, the small-scale flow

at the discontinuity drives a flow in the opposite di)rection compared with what would be expected from the analysis of the lithospheric discontinuity in an * isothermal fluid. It is important to consider models where the small-scale flow is a part of a larger dynamical system because the upper mantle is certainly not

an isothermal system, as generally assumed in the

S.D. King, D.L. Anderson / Earth and Planetary Science Letters 160 (1998) 289–296

plume +

hypothesis and most edge-driven models. While it is clear that the upper boundary layer of the Earth is heterogeneous, leading to discontinuous $variations in thermal and mechanical properties at the surface, the small-scale flow driven by variations "in the lithosphere and the larger-scale flow driven by density variations in the upper mantle interact. For example, if the upper mantle under the region of thick lithosphere (i.e., craton) is hotter than average, the pattern of the small-scale instability at the lithospheric discontinuity is reversed compared with the * intuitive solution based only on the lithospheric discontinuity. That is, fluid flows from beneath the thick toward the thinner lithosphere, producing a weak upwelling at the lithospheric discontinuity. A temperature anomaly of only 30ºC under the thick lithosphere reverses the small-scale flow pattern. Some studies of the formation and break-up of supercontinents show that the mantle may be hotter than

average under large, thick continents [13]. Continental insulation, or isolation, is most effective under large, long-lived supercontinents [14]. In general, th " e large-scale flow of the mantle will most likely be directed downwards under thick cratons and upwards under thin oceanic lithosphere, the opposite to the initial conditions assumed in our previous paper. In this paper we simplify the problem so that the essence of the edge-driven flow is revealed, and

also investigate the interaction of a large-scale flow  driven by temperature anomalies in the upper mantle with the edge-driven flow.

2. Comparison with King and Anderson (1995) [12]

+

*

We present a series of models where we vary the initial thermal temperature field within the mantle with a long-wavelength anomaly of the form T , - x . /z 021

4 5  pert sin 687:9 /z ; cos =@? - x A 2B

13 0

%

(1)

#

where ‘pert’ is the magnitude of the anomaly. In fig. 3c of Ref. [12], ‘pert’ was 0.1. These calculations performed in a two by one box with free-slip $velocity boundary conditions on the top, bottom and sides using the two-dimensional, Cartesian, finiteelement convection code ConMan [15]. The depth of the box is scaled to be 600 km. The lithosphere

are

%

on

291

the left-hand side of the box is 200 km thick (0.333) and on the right-hand side is 50 km (0.0833) thick. The viscosity of the lithosphere is 1000 times the background viscosity and, the viscosity structure does not evolve with the flow. The thermal and chemical properties of the plates are identical to the " fluid, except for the viscosities. Because background these calculations do not evolve far from the initial condition, we do not consider the assumption of a spatially fixed viscosity structure to significantly affect our conclusions; however, this viscosity structure inhibits the formation of instabilities at the " *bottom of the upper thermal boundary layer as seen in experiments with a temperature-dependent fluid [16]. Our view is that the cratonic root is stabilized " by compositional buoyancy, not by rheology; however, we greatly simplify our model by choosing a stiff cratonic root that does not evolve with time so that we can isolate, and study, the edge-driven flow. The fact that the thickest, coldest lithosphere occurs "under Archean cratons suggests that it has been stable for some time, probably due to a combination of neutral buoyancy and cold, dry rheology. A grid of 120 by 60 uniformly spaced elements is used, * identical to the grid used in Ref. [12]. The temper ature boundary condition along the top is T C 40 D 40

and along the bottom we impose T E 1 F 40. The H G Rayleigh number in these calculations is 106 and all other parameters are identical to those in Ref. [12]. In Fig. 1 we present the isotherms and velocities

at approximately 10 million years after the start of the calculation for three models where we vary the 4initial mantle thermal anomaly using values of 0.1, 0.01 and 0.001 for ‘pert’ in Eq. 1. If we assume an

average adiabatic mantle temperature of 1400ºC, this corresponds to a peak-to-peak temperature anomaly of 280, 28, and 2.8ºC, respectively. The flow in I Fig. 1A is identical to that of fig. 3c from Ref. [12]. The results from Fig. 1C qualitatively agree with the )results of Keen and Boutilier [7]. From Fig. 1, we observe that the flow pattern discussed in the King and J Anderson model is strongly dependent on the size of th e mantle thermal anomaly. If the anomaly is 10% of the background temperature (Fig. 1A), the buoy ancy from the long-wavelength temperature drives a long-wavelength flow pattern. When the long-wavetemperature anomaly is 1% of the background tlength emperature (Fig. 1B), the long-wavelength flow pat-

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Fig. 1. Edge-driven convection with an initial temperature anomaly given by Eq. 1. (A) pert K 0L 1, (B) pert M Q The maximum velocities are 30, 4, and 2.5 mmR S yr, respectively. All other parameters are described in the text.

0N 01, (C) pert

O

0P 001.

S.D. King, D.L. Anderson / Earth and Planetary Science Letters 160 (1998) 289–296

tern

still dominates; however, the edge-driven flow be seen. When the long-wavelength temperature anomaly is 0.1% of the background temperature % (Fig. 1C), the edge-driven flow dominates the flow pattern; however, the long-wavelength flow is still $visible. The peak velocities in each of these models

are 30, 4, and 2.5 mmT Uyr, respectively.  These calculations from Fig. 1 are time dependent. Because the initial temperature distribution within the mantle evolves with time, the long-term V Wflow field will differ from the instantaneous flow fields shown in Fig. 1. As the flow patterns evolve with time, the long-wavelength flow weakens and the V flow pattern assumes a form like that in Fig. 1C for

all three calculations. It takes longer for this to happen for the 10% internal mantle temperature pertur" bation (Fig. 1A) than the 1% perturbation (Fig. 1B). If we examine the flow just after the beginning of each calculation, the flow patterns for all three calculations are almost identical, although the magnitudes of the velocities differ. Thus, even for the smallest internal mantle temperature perturbation, there is a finite period of time where the long-wavelength flow nates over the edge-driven flow, but eventually tdomi he edge-driven flow does dominate the flow pattern. + We can examine the time evolution of the calculatio ns by examining the time-series of the average $velocity of the flow (Fig. 2). The average velocity of the calculation with the 10% (solid line) initial thermal perturbation differs significantly from the calculations with a 0.1% (short dash) and 1% % (dashed line) initial thermal perturbation. For com-

parison,

can

2. Average velocity time series from the three calculations X Fig. in Fig. 1: pert Y 0Z 1 (solid), pert [ 0\ 01 (dashed), pert ] 0^ 001 (short dash), pert _ 0` 0 (dotted).

293

the dotted-line shows the average velocity

of the flow in the case where there is an isothermal

a

mantle. This shows the evolution of the edge-driven flow. The dotted line and the short-dashed line (0.1% perturbation) almost overlay for the entire length of the calculation. The long-dashed line (1% perturbation) starts out differently and merges by 40 million Uyears. The solid line (10%) approached the others " by about 60 million years; however, the spike in ve*locity at 40 million years indicates a boundary layer instability that dominates the flow pattern. At this point, applying these calculations to the Earth becomes unrealistic as the edge effects of the box and  ns of the 2D flow come into play because tlimitatio he large-scale flow is confined to the dimensions of the box and this creates an instability in the lithosphere over time. The unstable thinner lithosphere sinks along the side of the box. The resulting geoid

and topographic anomalies over the sinking lithosphere are significantly larger than any observed

anomalies.

3. Additional results



The thermal anomaly used in the calculations in Fig. 1 is limited by the size of the box % (the thermal anomaly is the maximum wavelength in this domain). The average width of a continent is significantly longer than the 1200 km length in our calculation. We address this problem by increasing the domain of the computation. We studied a four-by-one box and an eight-by-one box, where the thickness of the lithospheres was the same as the original problem and the lithospheric discontinuity remaining in the center of the box. We increased the wavelength of the sine perturbation so that the ther mal anomaly remains centered under the left-hand side of the box. The results, not shown, confirm that the aspect ratio of the domain has no discernible effect on the flow pattern. The flow patterns ( from the wider boxes are identical to the pattern in Fig. 1B. + hile it is clear that temperature anomalies in theWmantle have an impact edge-driven flow, this is only can true if the mantle under the continental (or craton) boundary is warmer than the mantle under the oceanic boundary. A more important

shown

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b Fig. 3. Flow field for edge-driven models with imposed Couette side-wall boundary conditions. (A) 10 mmc S yr to the left; (B) 10 mmd S yr e to the right. All other parameters are identical to the calculations in Fig. 1C. consideration is that the

lithosphere is moving, not While the degree of coupling between the lithosphere and mantle is unknown, it is un that the entire upper mantle is moving with ltikely he lithospheric plate. We examine the effect of shear " between the mantle and lithosphere on this smallscale flow by imposing Couette velocity boundary conditions on the left- and right-hand edges (Fig. 3). f Plate-scale shear flow between the lithosphere and "mantle does not overwhelm the edge-driven instability if the mantle moves from the craton toward th e ocean relative to the lithosphere (Fig. 3A), but it does if the mantle moves from the ocean toward the craton and the imposed shear velocity exceeds 30 mmg Uyr (Fig. 3B).

stationary.

4. Discussion Having demonstrated that the flow driven by longtemperature anomalies can overwhelm the flow driven by the edge of the lithospheric discontinuity, even for anomalies as small as 30ºC, it * is natural to ask: “why isn’t this pattern observed everywhere?” The assumption behind the initial temperature field is that cratons are over warmer than

average mantle. While this might be true at times of supercontinental breakup, it has been suggested that continents are currently moving toward colder than

average mantle [17]. In addition, the leading edge of

a moving continent is always pushing cold mantle under itself, except when it overrides a ridge. Thus,

wavelength

S.D. King, D.L. Anderson / Earth and Planetary Science Letters 160 (1998) 289–296

V

the

most likely effect of any coherent, large-scale flow driven by temperature anomalies in the upper mantle would be to drive circulation in the same direction as the edge-driven mechanism. An important point to bear in mind is that any edge-driven flow will be occurring in a heterogeneous upper mantle, while most analyses of lithospheric, or edge-, or bottom-driven instabilities assume an isothermal layer below the lithosphere. It is possible that at some locations, temperature anoma lies in the upper mantle drive local flow that overwhelms edge-driven effects, while in other areas the edge-driven flows dominate. With our parameters, we obtained small-scale flow velocities of 20 mmh Uyr, of the order of previous calculations. Keen

and Boutilier [7] obtained velocities as high as 18 mmi Uyr even for plate separation velocities of 1 mmj Uyr. The edge-driven flow velocities and melt productivity depend mainly on mantle viscosity and the lateral temperature contrast and less so on the plate separation velocities [7]. In our previous paper, we showed that the edge-driven mechanism is capable of delivering sufficient melt to explain the presence of large igneous provinces [12]. To esti mate the region of partial melt, we used a simple parameterized melting formulation where the pressure and temperature at a given point are compared with an analytic function representing the solidus

and liquidus as a function of pressure and temperature for garnet peridotite and the fraction of melt is calculated by fitting a polynomial to the experimental data [18]. This formulation gives a rough estimate of the region of melting; however, it ignores the between the latent heat of melting and the tfeedback emperature field and, it does not take into account the volatile content of the underlying mantle on the melting curves. We have argued that the mantle material that is drawn into the melting region may have & high concentrations of volatiles. Because the effect of volatiles is to reduce the solidus and liquidus temperatures, our melting results represent a lower " bound. Even so, these estimates show that sufficient melt can be generated with edge-driven flow. Other calculations show that melt volumes are enhanced by small-scale convection driven by thermal gradients

at the edges of lithospheric discontinuities and that large melt volumes can be produced without high mantle temperatures [7].

V

+

295

We conclude that both large-scale flow and ‘edge’ flow driven by variations in lithospheric architecture can be comparable to, and in some cases exceed, flow driven by large-scale density anomalies in the mantle. Most investigations assume that large ig"neous provinces, swells and hotspots are all formed by deep mantle structures like plumes, while we sugkgest that asymmetries in the lithosphere may fix the large-scale pattern of mantle flow and small-scale flow like the edge drive flow discussed here. The l Bermuda rise has been suggested to be a result of such small-scale flow at the boundary of the North J American continent and the old Atlantic Ocean [2]. Some of the eastern Atlantic swells are at similar distances from Europe and Africa and may be similarly explained. In addition, these swells were )ridge-centered prior to westward migration of the mid-Atlantic ridge, and may have been induced by passive plate divergence. Plume models for large igneous provinces require large excess temperatures; however, by feeding more material through the melting zone at continental margins than can be accomplished by the plume hypotheses, which involves passive upwelling, large melt volumes can be generated without excessive temperatures. Large-scale V w may be controlled by the distribution of cratflo ons, as well as slabs and ridges. These all represent $variations in the upper boundary conditions, effects that are often unaccounted for in convection models

applied to plumes and rifts. + We have shown, as have others, that horizontal temperature gradients set up by lithospheric architecture can produce small-scale flow velocities comparable to plate scale velocities. The most common types of abrupt boundaries are at fracture zones

and at suture zones where different age lithospheres & have been juxtaposed. In addition, long-wavelength kgradients are set up between oceans and cratons. + We envision the following scenario for the gener ation of continental flood basalts and coast-paral lel oceanic swells. Continents are generally under compression. The subsurface convection caused by  lateral temperature gradients focus upwellings under th in lithosphere. All continental flood basalts occur on the edges of cratons. We infer that this is where the lithosphere is most likely to fail when placed under extension and where melts will be focused, draining uphill. The extension phase may have to do

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with trench rollback or ridge–trench collision, on the

other side of the plate. Light melts can rise through the

lithosphere when the conditions for failure or hydraulic fracture have been established. Dense picrites are common at the initial stages of flood basalt magmatism. We interpret this as due to transient drainage of newly extended lithosphere. These dense melts are rare under normal conditions. One can calculate that with realistic estimates of the hydrostatic head and the viscosity of magmas that a 1-cm-wide crack can produce flows at the rates observed for m JCFB [19]. If extension continues, as in the North Atlantic, the flow is focused midway between the verging cratons. Iceland would be an example of di this focused flow at a newly opened ocean. As the ocean gets wider the Iceland type plateaus become inactive and the active volcanism follows the oceancentered ridge. Thus, it appears that the ridge has mikgrated away from a hotspot. The swells at Bermuda, Rio Grande, Cape Verde and others in the Atlantic

and Indian oceans are about 1000 km offshore and formed when the oceans were about twice this wide. This scenario is similar to that developed by Vogt [2]. Some CFB formed when cratons were converg* ing (e.g., Deccan, Keweenawen). In these cases an edge- or top-driven flow seems more appropriate than plumes, which attribute CFB to the early stages of continental breakup. n [RV]

[5]

[6]

[7]

[8]

[9]

[10] [11] [12]

[13]

[14]

[15]

References [1] J. Elder, The Bowels of the Earth, Oxford University Press, Oxford, 1976, 222 pp. [2] P.R. Vogt, Bermuda and Appalachian–Labrador rises, Geology 19 (1991) 41–44. [3] R.S. Detrick, P.R. Von Herzen, B. Parsons, D. Sandwell, M. o Dougherty, Heat flow observations on the Bermuda Rise, J. Geophys. Res. 91 (1986) 3701–3723. [4] J.G. Sclater, L. Wixon, The relationship between depth and and heat flow and age in the Western North Atlantic, X age in: P.R. Vogt, B.E. Tucholke (Eds.), The Western North p Atlantic Region, The Geology of North America, Vol. M,

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Geological Society of America, Boulder, CO, 1986, pp. 257–270. M.J. Cordery, G.F. Davies, I.H. Campbell, Genesis of flood q bas eclogite-bearing mantle plumes, J. Geophys. r Res.alts102from (1997) 20179–20197. J.C. Mutter, S.R. Buck, C.M. Zehnder, Convective partial melting, 1: a model for the formation of thick basaltic sequences during the initiation of spreading, J. Geophys. Res. 93 (1988) 1031–1048. Keen, R.R. Boutilier, Lithosphere–asthenosphere ine C.E. teractions below rifts, in: E. Banda et al. (Eds.), Rifted Ocean–Continent Boundaries, Kluwer, Dordrecht, pp. 17– 30, 1995. C.E. Keen, Some important consequences of lithospheric extension, in: M.P. Coward, J.F. Dewey, P.L. Hancock (Eds.), Continental Extensional Dynamics, Geol. Soc. London Spec. Publ. 28 (1985) 67–73. R.W. Buck, Small-scale convection induced by passive riftthe cause for uplift of rift shoulders, Earth. Planet. Sci. s Ling; ett. 77 (1986) 362–372. J.C. Mutter, Margins declassified, news and views, Nature 364 (1993) 393–394. C.J. Ebinger, N.L. Sleep, Africa: one plume goes a long way, Eos 78 (1997) F698. S.D. King, D.L. Anderson, An alternative mechanism of flood basalt formation, Earth Planet. Sci. Lett. 136 (1995) 269–279. M. Gurnis, Large-scale mantle convection and the aggregation and dispersal of supercontinents, Nature 332 (1988) 695–699. P.M. Burgess, M. Gurnis, L. Moresi, Formation of sequences in the cratonic interior of North America by interaction between mantle, eustatic, and stratigraphic processes, Geol. Soc. Am. Bull. 108 (1997) 1515–1535. S.D. King, A. Raefsky, B.H. Hager, ConMan: Vectorizing a finite element code for incompressible two-dimensional convection in the Earth’s mantle, Phys. Earth Planet. Inter. 59 (1990) 195–207. A. Devaille, C. Jaupart, Thermal convection in lava lakes, Geophys. Res. Lett. 20 (1993) 1827–1830. D.L. Anderson, Polet, Depth extent of cratons as inferred from tomographic studies, Geology 23 (1995) 205–208. D. P McKenzie, M.J. Bickle, The volume and composition of melt generated by extension of the lithosphere, J. Petrol. 29 (1988) 625–679. D. Turcotte, Some thermal problems associated with magma migration, J. Volcanol. Geotherm. Res. 10 (1981) 267–278.