Three-dimensional numerical simulations of crustal deformation and subcontinental mantle convection

EPSL Earth and Planetary Science Letters 150 (1997) 233-243

Three-dimensional numerical simulations of crustal deformation and subcontinental mantle convection

L.-N. Moresi a**, A. Lenardic ’

a Research School of Earth Sciences, Australian National University. Canberra ACT 0200. Australia ’ Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095, USA

Received 2 January 1997; revised 8 May 1997; accepted 8 May 1997

Abstract

3-D simulations of mantle convection allowing for continental crust are explored to study the effects of crnstal thickening

on lithosphere stability and of continents on large-scale mantle flow. Simulations begin with a crustal layer within the upper

thermal boundary layer of a mantle convection roll in a 1 X 1 X 1 Cartesian domain. Convective stresses cause crust to thicken above a sheet-like mantle downwelling. For mild convective vigor an initial crustal thickness variation is required to

induce 3-D lithospheric instability below the zone of crustal convergence. The amplitude of the required variation decreases with increasing convective vigor. Morphologically, instability is manifest in the formation of drip-like thermals that exist within the large-scale roll associated with initial crustal thickening. A strong surface signature of the drips is their ability to

cause deviations from local Airy compensation of topography. After the initial thickening phase, the crustal accumulation that forms serves as a model analog to a continent. Its presence leads to mantle flow patterns distinctly different from the

steady-state roll that results in its absence. Large lateral thermal gradients are generated at its edge allowing this region to be

the initiation site for continued small-scale thermal instabilities. Eventually these instabilities induce a restructuring of large-scale mantle flow, with the roll pattern being replaced by a square cell. Although preliminary and idealized, the

simulations do show the fluid dynamical plausibility behind the idea that significant mantle variations can be generated

along the strike of a largely 2-D mountain chain by the formation of the chain itself. The ability of a model continent to

cause a change in fundamental convective planform also suggests that the effects of continental crust on mantle convection may be low-order despite the seemingly trivial volume of crust relative to mantle. 0 1997 Elsevier Science B.V.

Keywords: mantle; isostasy: continents; Mohorovicic discontinuity; orogeny; gravity anomalies; tectonics

1. Introduction

Both seismic tomography [l] and theoretical mod-

eling [2] suggest that the dominant mode of mantle

convection is characterised by cold sinking sheets

that are, by consensus, associated with subducted slabs [3]. That the mode of local subcontinental

mantle flow might differ from that associated with broad sinking slabs was suggested some time ago based on the geologic pattern of ridges and domes

* Corresponding author. Present address: CSIRO Exploration & Mining, P.O. Box 437, Nedlands 6009, Western Australia. Tel.: +61 8 9389 8421. Fax: f61 8 0389 1906. E-mail: [email protected]

observed in Africa [4,5] and, later, in Australia [6]. The general idea was that the features suggested a

honeycomb pattern of subcontinental convection with

relatively short distances between upflows and

0012-821X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOOl2-821X(97)00093-9

234 L.-N. Moresi. A. Lenardic/Earth and Planetary Science Letters 150 (1997) 233-243

downflows. Downflow morphology was inferred to be sheet-like, with a smaller length scale than that associated with oceanic plates 171, or to be more drip-like [8].

More direct evidence into the morphology of subcontinental mantle convection comes from seismic mapping below terrestrial mountain ranges. Such mapping has revealed large variations in mantle structure that have been interpreted as evidence of small-scale convection. Example ranges covered are: the European Alps [9]; the Sierra Nevada of the Western U.S. [lo]; the Transverse Ranges of Califor- nia [ 111; the Rif and Betic belts of Morocco and Spain [12]; and the Tien Shan of central Asia 1131. For these ranges the inferred pattern of local mantle flow often appears more drip-like than sheet-like and, in many cases, it does not follow the general trend of surface deformation. In the Tien Shari, for example, the dominant surface trend is range perpendicular compression associated with India-Asia collision; however, seismic studies suggest a subcontinental convection pattern with a component of range-parallel flow [13]. Further, the linear nature of the range, with mild variation in geologic structure along strike, belies pronounced variations in seismi- cally mapped mantle structure along its strike [14]. These observations suggest that the 2-D character of the mountain belt may not be reflected in the mantle flow below [151.

The idea that local mantle instabilities beneath mountain belts could be caused by the formation of the belts themselves is well established [161. Less well known is how strongly instability development perturbs mantle flow. Could it be strong enough that the present flow is different to first order from the flow which created the mountain belt to begin with? Is the drip-like character of downflows inferred below some mountain ranges the preferred morphology of subcontinental convection? Further, although it has been argued that small-scale convection has had nontrivial effects on the tectonic evolution of specific mountain ranges, the manner of coupling between subcontinental mantle flow and surface tectonics has gone unmapped, save for inferences based on 2-D modeling [16-191. Given the character of small-scale mantle flow suggested by seismic studies and the inference that it is not fully phased with the large-scale plate flow associated with the formation

of specific ranges, this sort of modeling may not provide sufficient insight.

With this in mind we present preliminary results from 3-D numerical simulations designed to explore the dynamic interaction between continental crust and mantle convection. The initial intent is to explore how crustal thickening above a linear mantle convergence zone could potentially induce local 3-D mantle flow structures and to determine whether such structures could be long lived. These questions are related to the broader issues of the preferred morphology of subcontinental mantle flow and the effects of continents on large-scale mantle circulation. We also present a preliminary exploration of how subcontinental mantle flow might be reflected in surface observables such as gravity and topography. To limit initial scope, the simulations are run at low to moderate degrees of convective vigor and

viscosity is assumed constant. This will hinder direct

application to specific geodynamic problems. How-

ever, given the scarcity of 3-D models that have explored the interaction between mantle convection

and continental crust under any conditions, we feel it

best to begin with the least complicated of models so as to allow for a clear building of physical insight.

2. Models

We consider a 3-D Cartesian model of convective

flow involving thermal and chemical buoyancy

forces. Chemical buoyancy is tracked via a composi-

tion function, C, initially assigned values of one and zero, respectively, for two end-member components

that serve as model analogs for continental crust and mantle. We assume the Boussinesq approximation, infinite Prandtl number, Newtonian viscous rheol-

ogy, thermal convection driven by bottom heating and top cooling, and constant material parameters save for density. Nondimensional equations for con- servation of mass, momentum, energy, and chemical composition, along with a linearized equation of

state, are, respectively:

a,u; = 0

aI2 ui = a, p + RaTiZ - Ra, Ci

c?,T + uicYiT = a:T

(1)

(2)

(3)

L.-N. Moresi. A. Lenardic/ Earth and Planetary Science Letters 150 (1997) 233-243 235

a,c + u,alc = Le-la,?c p= [I --(Y(T- To) + P(C- Cdl where:

(4) (5)

Ra = p. g cw ATd”

/JK

Ra, = PoGACd” ‘

IJK

where u, is the velocity vector; p is pressure; Ra is

the thermal Rayleigh number: T is temperature; i is

the vertical unit vector; Ra, is the compositional

Rayleigh number; Le is the Lewis number, defined

as the ratio of thermal to chemical diffusion; p is density; c~ is the coefficient of thermal expansion; p

is its compositional equivalent; p0 is a reference

density; g is gravitational acceleration; AT is the

system temperature drop; d is the system depth; p is

viscosity; K is thermal diffusivity; and AC is the

difference in the composition function between crust

and mantle.

Equations are solved by a version of the CITCOM

finite element code [20]. CITCOM uses a multigrid scheme for the equations of motion which has the

desirable property of a linear scaling of solution time

with the number of unknowns - crucial when deal-

ing with 3-D problems. Incompressibility is imposed

by a preconditioned Uzawa iteration scheme [21]. Surface stress determination can sometimes be a problem in finite element codes of this variety, so

CITCOM uses a consistent boundary flux method to

compute accurate surface topographies [22]. Exten-

sive benchmarking against 2-D analytic solutions

allowed us to determine the accuracy of the velocity

and surface topography solutions for a given density

of grid points 1231. The mesh used in these calcula-

tions had 32 elements in each horizontal direction

and 40 in the vertical. With some grid refinement in the vertical direction to capture the horizontal bound-

ary layers, we could be confident of a solution accurate to better than 0.1% everywhere. Chemical diffusivity is very low (high Le) but there is an

inherent flaw in all fixed-grid methods which causes some spurious diffusion that leads to a smoothing of sharp features. To minimize this problem, Eq. (4) is treated using a shock-capture algorithm that couples an upwind advection scheme to a nonlinear filter designed to eliminate dispersion errors [24]. The

resulting scheme does retain a small component of

spurious diffusion that spreads an interface into a zone spanning several elements. Numerically, this

allows relatively sharp material transitions to be resolved on a fixed grid; physically, it implies that

the transition from crust to mantle is treated as being

chemically gradational.

All simulations are performed in a 1 X 1 X 1 do-

main with reflecting side walls and free slip horizon-

tal surfaces with a fixed nondimensional temperature

of 0 on the upper and 1 on the lower and a no flux

composition condition on both. The initial chemical

configuration is obtained by adding a layer of buoy-

ant ‘crust’ to the upper-most part of the thermal

boundary layer in a convecting ‘mantle’ layer. The

initial thermal state is a steady thermal convection roll obtained by extruding a 2-D, purely thermal

convection calculation. The thermal calculations used

to obtain initial thermal fields were run at the same

Ra as was used for thermal/chemical counterparts.

No perturbations were added to the thermal fields

which, in the absence of a chemically light surface

layer, were stable in time. Using such a simple, and

relatively uninteresting, initial thermal field will al-

low us to track any time-dependent dynamical be-

havior that may ensue directly back to the presence

of crust. The thickness of the crust was locally uniform along the direction of initial surface flow in

the mantle but, for most simulations discussed. a small variable step in thickness was introduced in

half the box perpendicular to this direction. This allows us to test the stability of the flow pattern to

symmetry breaking induced by finite crustal thick-

ness variations. Geologically, such variations repre-

sent pre-existing crustal thickness variations that may

be present in a zone of tectonic convergence; that is,

variations not generated by the convergence itself

but, rather, by past tectonic or magmatic episodes. The ratio of the chemical to thermal Rayleigh number was fixed at 2 for all simulations. This assumes reference crustal and mantle densities of 2800 and

3300 kg/m3, respectively, a thermal diffusivity of

3 X lo-‘, and a temperature drop across the mantle of 2500 K. The latter is a high-end value for upper mantle convection which is also assumed throughout. This buoyancy ratio leads to crustal accumulations that are stable in the sense that they are not remixed into the mantle on a large scale. 2-D models suggest

236 L.-N. Moresi, A. Lenardic/Earth and Planetary Science Letters 150 (1997) 233-243

that, for accumulations to become unstable, the ratio must fall to unity or below, depending on relative crustal volume [2.5]. Those models also show that, provided accumulations are stable, the primary effect of a higher/lower buoyancy ratio is that it leads to a wider/narrower crustal accumulation.

Fig. I shows the evolution of a simulation with Ra = 5 X lo5 and a mild initial step in crustal thickness (initial crustal thickness, relative to system depth, was 0.035 across one half of the box and 0.045 across the other; for a dimensional system depth representative of upper mantle convection this corresponds to a crustal step of less than 7 km). The first column shows an initial evolution phase. Con- vective stresses cause the crust to thicken above the cold sheet-like downwelling in the mantle. Thicken- ing is dominated by perpendicular compression along the strike of the thermal downflow. However, the small but finite step in crustal thickness introduces a finite density gradient along the strike. As there is no critical Rayleigh number for lateral density variations, even the smallest of such variations has the potential to induce instability. In the model of Fig. 1 a lateral instability in the thermal downflow is indeed induced. It manifests itself in the formation of a local drip-like downflow below the region of initially thinner crust. This region of larger than average downflow velocity draws crust toward it and quickly becomes a zone of thicker than average crust. In the absence of a finite step in crustal thickness, pronounced 3-D mantle structure did not develop during the initial crustal thickening phase of evolution.

Column 2 of Fig. 1 shows a second evolution phase characterized by repeated formation of thermal instabilities at the edge of the crustal accumulation. The formation of such instabilities was previously observed in 2-D simulations and was related, in part, to the thermal perturbation exerted on the mantle by a heterogeneous distribution of crust [25]. The crustal accumulation in Fig. 1 does not participate in convective overturn. It thereby forms bounding material for the convectively unstable mantle below. The existence of bounding material in crust-free regions is implicit in the constant surface temperature boundary condition, which implies that material above the domain has effectively infinite thermal conductivity. The presence of a crustal accumulation means that, locally, convecting mantle is bounded by material

with a conductivity equal to its own. This imposes a laterally variable thermal condition on the mantle: a constant temperature condition in the crust-free region and an imperfect one in the region of thickened crust [26,27]. Fig. 2 shows, using a 2-D simulation, how such a heterogeneous thermal condition can induce a large temperature gradient at the edge of a crustal accumulation. This, combined with the ability of the crust to buffer the stress-free surface condition imposed on the mantle, allows the edge region to become the formation site of small-scale thermal boundary layer instabilities. Implicitly, 2-D experiments assume that such instabilities never break down in the third dimension. Thus, instabilities in the 2-D simulation represent slices through their perfectly tube-like equivalents in an imaginary 3-D run. Fig. 1 shows the instabilities can, in fact, break

Fig. 1. Evolution of a 3-D simulation with Ra = 5X 10’ and a

mild initial step in crustal thickness. Shown in lightest gray is the

C = 0.5 surface, which marks the center of the crust-mantle

interface zone. and two constant temperature surfaces; the darkest

gray is the 0.33 isotherm and intermediate gray is the 0.67

isotherm. Time is dimensionalized assuming d = 670 km and K = 10m6 m’/s. The sense of motion in the original thermal

convection model is indicated by the arrow on the axes.

L.-N. Moresi. A. Lenardic / Earth and PlanetaT Science Letters 150 (1997) 233-243 231

along strike as they travel below the accumulation, thereby generating pronounced lateral variation in the mantle. The drip-like thermals associated with this variation are secondary structures in that they form within, and are swept along by, a predominantly roll-like cell. Thus, linear convergence contin- ues to dominate in the region of crustal thickening. It is worth noting that laboratory experiments have also shown that secondary drip-like thermals can readily co-exist within broader convective cells for cases in which the upper thermal condition imposed on the convecting fluid is an imperfect one [28], which is

locally the case below the crustal accumulation in Fig. 1.

The spacing of the secondary thermals that form in the second column of Fig. I should not be taken as geologically meaningful because it is effected by the limited dimension of the modeling domain. Thus, no statement can be made about the preferred wavelength for the thermals. What can be said is that the existence of such thermals potentially allows significant 3-D variations in mantle structure to exist below what, at the surface, would appear to be a largely 2-D zone of mantle convergence.

125Hyr 147fiyr

670 km

134km

1

Fig. 2. Results from a 2-D simulation with a uniform initial crustal depth of 0.04, a chemical to thermal Rayleigh number ratio of 2, and

Ra = 5 x 10’. The frames in (a) show isotherms over composition shade plots for the full domain (gray tone indicates crust). Those in (b)

show isotherms spanning the cold half of the temperature field over composition shade plots for the upper 0.2 of the domain. The simulation

was preformed using the ConMan finite element code [37] modified to allow for thermal/chemical convection [24].

238 L-N. Moresi, A. Lenardic/Earth and Planetary Science Letters 150 (1997) 233-243

Although drip-like thermals remained persistent

for the simulation of Fig. I, this was not the case for

a less vigorous convection (Ra = 105) simulation

with the same initial crustal structure. For this latter

simulation, the component of 3-D flow generated by

the step in crustal thickness rapidly decayed and the

roll pattern of the thermal field re-established itself.

This can be seen in Fig. 3, which plots the y component rms velocity divided by the full rms

velocity over time for several simulations. This quan-

tity is zero for the roll structure which was used as

the thermal starting condition, since the axis lies

exactly along the y direction. For all simulations, the

initiation of a y component in the velocity field

reflects the symmetry breaking introduced by the

finite step in crustal thickness. For the low Ra

simulation with a mild initial crustal step (simulation

a, Fig. 3), the decay of the y component reflects the

fact that formation of drip-like thermals is rapidly

damped. Fig. 3 also includes results from simulations simi-

lar to the ones discussed so far but with a larger

initial crustal step (simulations b and d, Fig. 3). The

evolution of these simulations was similar to that of

Fig. 1, with drip-like thermals repeatedly forming

Growth of velocity along axis of convection roll

, - -

/

1 !

a00 _

0 200 400 600

Elapsed Time (Myr)

Fig. 3. Time evolution of the y component rms velocity divided by the full rms velocity for several 3-D simulations. For simula-

tions (a) and (b) Ra= 10”; for (c) and (d) Ra= 5X 105. Initial

crustal thicknesses across the right and left half of the domain are

0.035d and 0.045d for (a) and Cc). while for (b) and (d) the initial thicknesses are 0.02d and 0.06d. The time axis has been com-

pressed by a factor of 10 for runs (a) and (b).

Fig. 4. Final large-scale flow state of a simulation with Ra = IO’ and a large initial crustal step (left) and a simulation with Ra = 5

X 10’ and a mild initial crustal step (right). A square cell mantle

flow pattern is observed in both with the higher Ra case having a

smaller cell aspect ratio.

below the region of thickened crust. The only difference between the Ra = lo5 cases of Fig. 3 (a and b)

is the initial crustal configuration. All formal system

control parameters (Ra, Ra,, system size, and crustal volume) are equal. That they evolve to such different

states is a testament to the fact that convecting

systems can have multiple solutions for the same range of control parameters, as already established

for purely thermal mantle convection models 1291. Each solution is associated with its own fixed do-

main of attraction and the domain that a system falls

into depends on initial conditions. Thus, for equal

control parameters, the potential of multiple steady states, or a combination of steady and time-depen-

dent states, exists. For simulation a of Fig. 3 the

perturbation exerted on the mantle by crust is not

sufficient to alter its long-term flow morphology, which begins and ends in the form of a roll. For

simulation b, the initial finite amplitude perturbation exerted on the system by the crust is sufficient to induce permanent 3-D flow structure. Clearly, this implies a highiy non-linear system response and makes it impossible to say, based on a limited model set, that the generation of 3-D structure below zones

of linear crustal convergence is the preferred system behavior. What can be said at this stage is that it is a realizable one and whether it is realized depends on crustal structure, as well as the convective state of

the mantle. For Ra = 5 X lo5 simulations, evolution to a 3-D

state could be induced with a mild initial crustal thickness step as opposed to the larger one required at the lower Ra (Fig. 3). This suggests that the degree of lateral crustal variation required to initiate

L.-N. Moresi, A. Lmardic / Earth and Planetary Science Letters 150 (1997) 233-243 239

persistent sinking plume-like structures below a zone

of linear convergence is a decreasing function of Ra. This is intuitively expected. As Ra increases the thickness of the upper thermal boundary layer de-

creases. Thus, the effect of a fixed step in crustal thickness on determining boundary layer buoyancy

increases with increasing Ra. Notice also in Fig. 3

that the Ra = 5 X 10’ simulations suggest that, be-

yond a critical value, further increases in the initial

crustal step amplitude do not drastically affect long-

term system evolution.

All models that entered evolution phase 2 (Fig. 1, column 2) eventually entered a third and final phase

in which the large-scale thermal roll became unstable

and was replaced by a square cell pattern (Fig. 4). This final state is not altogether surprising. Theoreti-

cal studies have shown that, at slightly super-critical

Ra, the preferred pattern of thermal convection in a

3-D Cartesian domain is in the form of rolls, if the horizontal boundaries of the domain are maintained

at constant temperature, but can be in the form of square cells if the domain is bounded by slabs of

finite conductivity [30,3 11. Studies, theoretical or

otherwise, of convecting systems with a laterally

variable thermal coupling condition between a con-

vetting fluid and material that bounds it from above

are rare. However, based on the studies cited above,

one could well imagine that such systems might be

characterised by a non-linear pattern competition

between roll and square cell morphology. The simulations of this paper do seem to bear this out because

they evolve to a final square or roll pattern and go

through an intermediate phase in which plume-like

thermals exist within a large-scale roll.

G=mgil)

-50.0 50.0

Topo (km) ,?L^,.

-10.0 7.5

Fig. 5. Snapshot of a simulation with Ra = lo5 taken after 2 Byr when a neat-periodic shedding of small-scale instabilities is present.

Topography in (b) is computed using the vertical normal stresses obtained from the velocity solution. The topography surface is shaded by

the strength of the gravity anomaly. Contours of the topography are shown on the base plane at 750 m intervals. In (c) topography computed

using only the crustal thickness information and density difference between crust and mantle was subtracted from the topography of (b).

Thus, dark shading indicates where the ‘Airy’ prediction overestimates the ‘true’ topography (overcompensation). Plot (d) is similar to (c)

but the density anomalies between z = 0.8 and z = 1.0 (surface) were integrated to obtain a density anomaly that was then used to calculate an ‘isostatic’ topography.

240 L.-N. Moresi, A. Lenardic / Earth and Planetary Science Letters 150 (I 997) 233-243

For cases in which the square cell pattern comes to dominate, hot plumes become cell-defining upflows. Cell-defining downflows remain in the form of sheets that can break along-strike as local drip-like instabilities develop. The surface expression of dominant thermal downflows is well reflected by the morphology of a crustal accumulation and its associated topography. For both the roll and square cell patterns, crustal accumulations display large-scale linear trends. In both cases topography in a zone of crustal thickening is supported by active convergence in the mantle and is not isostatic in an Airy sense. The balance of forces is between the tractions applied by the mantle at the moho and the viscous collapse of the thickened crust. However, away from a thermal downwelling, a crustal accumulation does appear to be close to isostatically compensated, if the density of the mantle thermal boundary layer is factored into an isostatic balance (Fig. 5d).

Relative to cell-defining downflows, the signature of secondary thermals is not as clearly expressed in topography (Fig. 5). These secondary instabilities in the mantle lithosphere, the 3-D drips, alter the local topography only slightly by creating mild local highs above themselves. They are far more visible in terms of their effect on local compensation mechanisms. As a drip forms, moho deflection precedes that of the upper surface and the first signature of the instability is a mild local gravity high. The mature drip remains associated with a moderate local gravity and topo- graphic high which is ‘over-compensated’ (topography is lower than would be expected based on the thickness of the crustal root and the assumption of Airy isostacy). As Fig. 5 shows, the variation in apparent compensation along the strike of a region of linearly thickened crust that results is a relatively strong surface signature of the drips.

The exact evolution of topography and gravity must be controlled by the relative rates of drip formation and crustal flow. This cannot be well explored in constant viscosity cases. However, as the lower crust is thought to be relatively weak, and therefore should respond to drips at least as rapidly as the mantle lithosphere, we believe that local topography associated with mature drips should appear overcompensated. An exception would occur if upper mantle lithosphere were strong enough to shield mantle flow stresses from the crust. If there are

places where the mantle Iithosphere is this strong, then drips could still be detectable through their gravity signature because their negative density will not be partially compensated by deflection of the crust-mantle boundary.

3. Discussion

The most obvious lateral heterogeneity within the solid Earth is associated with chemically buoyant continents. A fundamental geodynamic issue is un- derstanding how this near surface heterogeneity effects mantle convection. Although idealized, the simulations of the previous section provide considerable insight. That the lateral heterogeneity imposed on the mantle by the presence of continents is likely to have an important effect on mantle flow is borne out by two principal observations from our simulations: first, that the presence of a model continent within a mantle convection cell can introduce time-dependent flow patterns not observed in its absence; and, second, that it can cause a change in the fundamental planform of a mantle convection cell. It should be borne in mind that model continents of the previous section are composed solely of crust. We do not make the added assumption that continents are also composed of thick keels of depleted mantle-melt residuum [32]. Although the effects of continental keels on mantle convection have often been specu- lated upon, the effects of crust are usually neglected as being of higher order, solely due to the small relative volume of crust. In our simulations, crust comprises less than 5% of the model volume yet it can exert first-order control on mantle flow patterns. This is because it resides within the upper thermal boundary layer and can determine the surface thermal condition seen by convectively overturning por- tions of the solid Earth.

Simulations in this paper were designed such that their initial evolution stage allows them to be useful for exploring another geodynamic issue: the effects of crustal thickening on lithospheric stability. The connection between the two has been discussed prin- cipally in the context of continental mountain belts and plateaus. It has been suggested that lithospheric thickening should accompany tectonic crustal thickening and that thickened mantle lithosphere could

L.-N. Moresi, A. Lmardic/Earth and Planetary Science Letters 150 (1997) 233-243 241

become convectively unstable and sink into the deep can induce finite buoyancy variations in thickened

mantle inducing synchronous uplift, volcanism, and lithosphere. The potential effect on lithospheric sta-

changes in the stress state of the overlying crust [33]. bility is hinted at by simulations a and b of Fig. 3.

Numerical simulations have been used to argue that That instability is suppressed for a but not for b

this is indeed likely [16] and to argue that it is suggests that it is of finite amplitude type. Such

unlikely, owing to lithospheric strength [17]. Simula- instabilities, unlike those more often discussed in

tions used to argue the former position mimicked fluid dynamics, require a finite perturbation to in-

lithospheric thickening by artificially stretching duce their growth. The idea that lithospheric instabil-

near-surface geotherms in 2-D, isoviscous numerical ities might fall into this category is not new. McKen-

convection calculations. Those used to argue the zie [35] suggested that subduction initiation may be a

latter followed the same procedure but allowed for finite amplitude instability. He thought this might be

temperature-dependent mantle viscosity. Both then so because lithospheric strength seemed to prevent

tracked the amplitude growth of infinitesimal litho- its wholesale foundering into the mantle unless a

spheric thickness variations to determine if thickened finite negative density could be generated below a

lithosphere was convectively unstable. Neither in- would-be subduction zone. For the problem at hand,

cluded crust; that is, lithospheric stability was as- the finite amplitude kick that allows for instability

sumed to depend solely on mantle strength and growth below a region of linear crustal convergence

thermal structure. Collectively, they showed how the is pre-existing crustal structure (i.e., structure not

stability of a thickened mantle boundary layer could generated directly by the convergence itself). This

depend on the local Rayleigh number of the bound- suggests a strong history dependence to issues of

ary layer itself. The local thermal Rayleigh number mantle stability below regions of continental conver-

expresses the ratio of negative boundary layer buoy- gence because pre-existing crustal heterogeneity,

ancy, which drives instability, to boundary layer generated from previous tectonic and/or magmatic

viscosity, which suppresses it [34]. If boundary layer episodes, can have significant effects. What this

viscosity exceeds a critical value, instability is says, simply, is that the density of the lithosphere is

damped [17]. Despite debate as to the plausibility the determined by composition as well as temperature

instability, it has been suggested that, in a region of and that it is important to consider both when mak-

linear crustal thickening, the instability of thickened ing inferences of lithospheric stability. With this in

mantle lithosphere may be associated with along- mind, simulations in this paper suggest that the idea

strike break-up. This, it is argued, could lead to of linear mountain belt formation generating signifi-

pronounced variations in deep mantle structure be- cant mantle variations below itself [ 151 is, at the very

low a largely 2-D mountain chain [151. least, fluid dynamically plausible.

The thermal Rayleigh number dependence of

lithosphere stability below a region of thickened crust is illustrated by simulations a and c of Fig. 3.

The rapid decay of 3-D instabilities in a suggests

that, if the thermal boundary layer Rayleigh number of the mantle is comparable to that of the simulation, then such instabilities would be unlikely in nature.

However. simulations a and b illustrate the fact, for thermal/chemical convection, boundary layer stability is not simply a function of thermal structure.

Application of thermal convection models to issues of lithosphere stability has proved deceptive in giv- ing the impression that buoyancy variations in me

lithosphere are controlled solely by temperature. In zones of tectonic convergence it is likely that pre-existing crustal structure will be present. Such structure

The potential of 3-D mantle instability below

mountain belts has been investigated only under the most idealized of conditions. However, the existence

of finite amplitude effects suggests that ruling out

the plausibility of such an instability using more

complex simulations will be a difficult task, given the history dependence that may be involved and the tendency of ongoing continental tectonics to over-

print past history. A more direct application may lie in using simulations to predict the surface signature of mantle instabilities and comparing predictions to

observations [36] to see if there is consistency; if there is, then the case for such instabilities existing in the Earth becomes stronger. Fig. 5 represents a step in this direction. It suggests that, if drip-like mantle instabilities do exist below mountain chains,

242 L.-N. Moresi. A. Lenardic/Earth and Planetary Science Letters 150 (1997) 233-243

then they should be visible in variations in the apparent mode of compensation along a chain. Be- cause we have presented only constant viscosity calculations, this inference is limited to instances where the crust and mantle lithosphere are relatively weak: an end-member. Clearly, the effects of rheo- logic variations need to be explored to see if they can alter the inference that regions of local subcontinental downflow should appear over-compensated to the degree that this might provide a signature of the downflow itself.

4. Conclusion

Numerical simulations of mantle convection with a buoyant crustal component suggest that the presence of continents within the Earth’s lithosphere has a first-order effect on large-scale mantle flow patterns. The presence of a continent in the simulations leads to time-dependent flow not associated with purely thermal mantle convection and can induce a change of convective planform within the mantle. The simulations suggest that the morphology of subcontinental mantle flow may differ from that below oceanic regions, being more time-dependent and more prone to the formation of shorter wavelength features.

The simulations also show how crustal thickening associated with the formation of predominantly 2-D mountain belts can induce convective instability of the lithosphere below. The simulations suggest that instability growth from infinitesimal perturbations can be suppressed for a low thermal Rayleigh number associated with a strong mantle lithosphere but that, under the same condition, finite amplitude instabilities can be initiated by pre-existing crustal thickness variations. The morphology of secondary mantle instabilities is in the form of sinking drips that circulate within a large-scale thermal roll associated with initial linear crustal thickening. The preliminary simulations suggest that secondary mantle drips below a continental mountain range can potentially lead to variations in apparent compensation across the range, with regions above a drip appearing over- compensated.

Acknowledgements

This work was done while LNM was a Postdoc- toral Fellow at RSES. Computing was carried out at the Australian National University and the San Diego Supercomputer Center. We thank two anonymous referees for helpful reviews leading to an improved paper. [RVI

References

[II

[21

[31

[41

[51

[61

171

[s1

[91

[lOI

1111

[121

[131

R.D. van der Hilst, S. Widiyantoro, E.R. Engdahl, Evidence

for deep mantle circulation from global tomography, Nature

386 (19971578-584.

D. Bercovici, G. Schubert, G.A. Glatzmaier, Three-dimen-

sional spherical models of convection in the Earths mantle,

Science 244 (1989) 950-955.

M.A. Richards, D.C. Engebretson, Large-scale mantle con-

vection and the history of subduction. Nature 355 (1992)

437-440.

T.N. Clifford. African structure and convection, Trans. Leeds

Geol. Assoc. 7 (1968) 291-301.

D. McKenzie, N. Weiss, Speculations on the thermal and

tectonic history of the Earth, Geophys. J. R. Astron. Sot. 42

(19751 131-174.

M.A. Ethridge, R.W.R. Rutland, L.A.I. Wybom, Orogenesis

and tectonic process in the early to middle Proterozoic of

Northern Australia, in: A. Krdner (Ed.). Proterozoic Litho-

spheric Evolution, Am. Geophys. Union Geodyn. Ser. 17

(1987) 131-147.

J.W. Elder, The evolution of the African thermal sublayer,

Trans. Leeds Geol. Assoc. 7 (1968) 303-309.

L.A.I. Wybom. Petrology, geochemistry, and origin of a

major Australian 1880- 1840 Ma felsic volcano-plutonic suite:

a model for intracontinental felsic magma generation, Pre-

cambrian Res. 40/41 (1988) 37-60.

St. Mueller, Deep-reaching geodynamic processes in the

Alps, in: M.P. Coward, D. Dietrich, R.G. Park (Eds.1. Alpine

Tectonics, Geol. Sot. London Spec. Publ. 45 (1989) 303-328.

E.D. Jones, H. Kanamori, S.W. Roecker. Missing roots and

mantle “drips”: Regional P, and teleseismic arrival times in

the southern Sierra Nevada and vicinity. California, J. Geo-

phys. Res. 99 (1994) 4567-4601.

E.D. Humphreys, R.W. Clayton, Tomographic image of the

southern California mantle, J. Geophys. Res. 95 (1990)

19725-19746.

D. Seber, M. Barazangi, A. Ibenbrahim, A. Demnati, Geo-

physical evidence for lithospheric delamination beneath the

Alboran Sea and Rif-Betic mountains, Nature 379 (1996)

785-790.

L.I. Makeyeva, L.P. Vinnik. S.W. Roecker, Shear-wave split-

ting and small-scale convection in the continental upper

mantle. Nature 358 (1992) 144-147.

L.-N. Moresi, A. Lenardic/ Earth and Planetan Science Letters 1.50 (1997) 233-243 243

1141 S.W. Roecker, T.M. Sabitova, L.P. Vinnik, Y.A. Burmakov,

MI. Golvanov, R. Mamatkanova. L. Munirova, Three-di-

mensional elastic wave velocity structure of the western and

central Tien Shan. J. Geophys. Res. 98 (1993) 15779-15795.

[ 151 P. Molnar. Crust in mantle overdrive. Nature 358 (1992)

105- 106.

[16] G.A. Houseman. D.P. McKenzie, P. Molnar, Convective

instability of a thickened boundary layer and its relevance for

the thermal evolution of continental convergent belts, J.

Geophys. Res. 86 (1981) 6115-6132.

[17] W.R. Buck, M.N. Toksoz, Thermal effects of continental

collision: Thickening a variable viscosity lithosphere,

Tectonophysics 100 (1983) 53-69.

[18] H. Schmeling, G. Marquart, The influence of second-scale

convection on the thickness of continental lithosphere and

crust, Tectonophysics 189 (1991) 281-306.

[I91 A. Lenardic, W.M. Kaula, More thoughts on convergent

crustal plateau formation and mantle dynamics with regard to

Tibet. J. Geophys. Res. 100 (1995) 15193-15203.

[20] L.-N. Moresi. VS. Solomatov, Numerical investigations of

2D convection with extremely large viscosity variations,

Phys. Fluids 7 (1995) 2154-2162.

[21] J. Cahouet. J.-P. Chabard, Some fast 3D finite element

solvers for the generalized Stokes’ problem. Int. J. Numer.

Methods Fluids 8 (1988) 869-895.

[22] S. Zhong. M. Gurnis. G. Hulbert, Accurate determination of

surface normal stress in viscous flow from a consistent flux

method. Phys. Earth Planet. Inter. 78 (1993) l-8.

[23] L. Moresi. S. Zhong, M. Gumis, The accuracy of finite

element solutions of Stokes’ flow with strongly varying

viscosity. Phys. Earth Planet. Inter. 97 (1996) 83-94.

[24] A. Lenardic. W.M. Kaula, A numerical treatment of geody-

namic viscous flow problems involving the advection of

material interfaces, J. Geophys. Res. 98 (1993) 8243-8269.

[25] A. Lenardic, W.M. Kaula. Near surface thermal/chemical

houndary layer convection at infinite Prandtl number: Two-

dimensional numerical experiments. Geophys. J. Int. 126

(1996) 689-7 II.

[26] E.M. Sparrow, R.J. Goldstein, V.K. Jonsson. Thermal insta-

bility in a horizontal layer: effect of boundary conditions and

non-linear temperature profile, J. Fluid Mech. 18 (1964)

5 13-528.

[27] L. Guillou, C. Jaupart. On the effect of continents on mantle

convection, J. Geophys. Res. 100 (1995) 24217-24238.

[28] C.R. Carrigan, Multiple-scale convection in the Earth’s man-

tle: A three-dimensional study, Science 215 (1982) 965-967.

[29] U. Hansen, A. Ehel. Time-dependent thermal convection -

a possible explanation for a multi-scale flow in the Earth’s

mantle. Geophys. J. 94 (1988) 181-191.

1301 F.H. Busse, N. Riahi, Nonlinear convection in a layer with

nearly insulating boundaries, J. Fluid Mech. 96 (1980) 243-

256.

[31] N. Riahi, Nonlinear thermal convection with finite conducr-

ing boundaries. J. Fluid Mech. 152 (1985) I l3- 123.

[32] T.H. Jordan, Composition and development of the continen-

tal tecrosphere. Nature 274 (1978) 544-548.

[33] P. Molnar. P. England, J. Martinod. Mantle dynamics. the

uplift of the Tibetan Plateau, and the Indian monsoon, Rev.

Geophys. 3 1 ( 1993) 357-396.

[34] L.N. Howard. Convection at high Rayleigh number, Applied

Mechanics, in: H. Gortler (Ed.), Proc. 1 Ith Congr. of Ap-

plied Mech., Munich, Germany, 1966. pp. 1109-I 115.

[35] D. McKenzie. The initiation of trenches: A finite amplitude

instability, in: M. Talwani, W.C. Pitman III (Eds.). Island

Arcs, Deep Sea Trenches, and Back-Arc Basins, Maurice

Ewing Ser.. vol. I. AGU, Washington. D.C.. 1977, pp.

57-61.

1361 E.V. Burov. M.G. Kogan, H. Lyon-Caen, P. Molnar, Gravity

anomalies, the deep structure, and dynamic processes be-

neath the Tien Shan. Earth Planet. Sci. Lett. 96 (1990)

367-383.

1371 S.D. King, A. Raefsky. B.H. Hager. ConMan: Vectorizing a

finite element code for incompressible two-dimensional con-

vection in the Earth’s mantle, Phys. Earth Planet. Inter. 59

(1990) 195-207.

Documents

Three-dimensional numerical simulations of crustal deformation and subcontinental mantle convection