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The History and Dynamics of Global Plate Motions , GEOPHYSICAL MONOGRAPH 121, M. Richards, R. Gordon and R. van der Hilst, eds., American Geophysical Union, pp5–46, 2000
The Relation Between Mantle Dynamics and Plate Tectonics:
A Primer
David Bercovici
, Yanick Ricard
Laboratoire des Sciences de la Terre, Ecole Normale Sup´erieure de Lyon, France
Mark A. Richards
Department of Geology and Geophysics, University of California, Berkeley
Abstract. We present an overview of the relation between mantle dynam-
ics and plate tectonics, adopting the perspective that the plates are the surface
manifestation, i.e., the top thermal boundary layer, of mantle convection. We
review how simple convection pertains to plate formation, regarding the aspect
ratio of convection cells; the forces that drive convection; and how internal
heating and temperature-dependent viscosity affect convection. We examine
how well basic convection explains plate tectonics, arguing that basic plate
forces, slab pull and ridge push, are convective forces; that sea-floor struc-
ture is characteristic of thermal boundary layers; that slab-like downwellings
are common in simple convective flow; and that slab and plume fluxes agree
with models of internally heated convection. Temperature-dependent vis-
cosity, or an internal resistive boundary (e.g., a viscosity jump and/or phase
transition at 660km depth) can also lead to large, plate sized convection
cells. Finally, we survey the aspects of plate tectonics that are poorly explained
by simple convection theory, and the progress being made in accounting for them.
We examine non-convective plate forces; dynamic topography; the deviations
of seafloor structure from that of a thermal boundary layer; and abrupt plate-
motion changes. Plate-like strength distributions and plate boundary formation
are addressed by considering complex lithospheric rheological mechanisms. We
examine the formation of convergent, divergent and strike-slip margins, which
are all uniquely enigmatic. Strike-slip shear, which is highly significant in plate
motions but extremely weak or entirely absent in simple viscous convection, is
given ample discussion. Many of the problems of plate boundary formation remain
unanswered, and thus a great deal of work remains in understanding the relation
between plate tectonics and mantle convection.
1. INTRODUCTION
In the late 1930’s, following the introduction of Alfred Wegener’s theory of Continental Drift [Wegener, 1924], sev- eral driving mechanisms for the Earth’s apparent surface mo- tions were proposed. While Wegener himself favored tidal and pole fleeing (i.e., centrifugal) forces, Arthur Holmes and others hypothesized that thermal convection in the Earth’s mantle provided the necessary force to drive continental mo- tions [Holmes, 1931; see Hallam, 1987]. Even with the
Permanently at Department of Geology and Geophysics, School of Ocean and Earth Science and Technology, University of Hawaii
spurning and demise of the theory Continental Drift, and it’s resurrection and revision 30 years later in the form of “Plate Tectonics” [e.g., Morgan, 1968], mantle convection is still widely believed to be the engine of surface motions, and, in- deed, for many good reasons as we shall see in this review [see also review by Oxburgh and Turcotte, 1978]. However, there is still no complete physical theory which predicts how plate tectonics in its entirety is driven, or, more appropri- ately, caused by mantle convection. There is little doubt that the direct energy source for plate tectonics and all its attendant features (mountain building, earthquakes, volcanoes, etc.) is the release of the man-
tle’s gravitational potential energy through convective over- turn (and of course that radiogenic heating and core cooling continue to replenish the mantle’s gravitational potential en- ergy). However, the precise picture of how plate motions are caused by convection is far from complete. With the recog- nition of the importance of slab pull, and that subducting slabs are essentially cold downwellings, it is becoming more widely accepted that the plates are an integral part of mantle convection, or more to the point they are mantle convec- tion [e.g., Davies and Richards, 1992]. However, the idea that the plates arise from or are generated by mantle con- vection yields new complications and questions. Perhaps most difficult of these questions concerns the property of the mantle-lithosphere system which permits a form of convec- tive flow that is unlike most forms of thermal convection in fluids (save perhaps lava lakes; see Duffield, 1972), i.e., a convection which looks like plate tectonics at the surface.
In this paper we review progress on the problem of how plate tectonics itself is explained by mantle convection the- ory. We begin by surveying the aspects of basic convection in simple fluid systems which are applicable to plate tecton- ics. We next discuss those features of plate tectonics which are reasonably well explained by basic convection theory. We then examine the numerous features of plate tectonics which are poorly (or not at all) explained by simple convec- tion, and, in that regard, we discuss the possible new phyiscs which is needed to improve (and in some cases radically al- ter) present theories of mantle convection. This paper is not only a review and tutorial on the aspects of convection rel- evant to plate tectonics; it is also an exposition of how lit- tle we know about the causes of tectonic motions, and how much work remains in the important problem of unifying the theories of plate tectonics and mantle convection.
2. BASIC CONVECTION
Basic thermal convection in fluids is perhaps the most fundamental paradigm of self-organization in nonlinear sys- tems [see Nicolis, 1995]. Such self-organization is char- acterized by the forcing of simple homogeneous yet per- turbable or mobile systems (i.e., whose particles can move) far from equilibrium (e.g. by heating or imposition of chem- ical disequilibrium); in many cases, the systems can develop complex patterns and oscillatory or chaotic temporal behav- ior. In basic thermal convection, for example, a layer of fluid is heated uniformly and subsequently develops orga- nized polygonal patterns and cells of cold and hot thermals. Indeed, the formation of tectonic plates is invariably a form of convective self-organization.
Although the phenomenon of convection was first recog- nized by Count Rumford [1870] and James Thomson [1882] (the brother of William Thomson, Lord Kelvin), the first sys- tematic experimental study of basic convection was carried out by Henri B´enard on thin layers of spermaceti (whale fat) and paraffin [B´enard, 1900, 1901]. B´enard’s exper- iments yielded striking images of honey-combed patterns and vertical cellular structure. After John William Strutt,
Lord Rayleigh, failed to theoretically explain the experi- ments with hydrodynamic stability theory [Rayleigh, 1916], it was eventually deduced that B´enard had witnessed not thermal convection but (because his thin layers of spermaceti were exposed to air) surface-tension driven convection, also known as Marangoni convection [Pearson, 1958; see Berg et al., 1966]. When the thermal convection experiments were carried out correctly, they were found to be very well pre- dicted by Rayleigh’s theory [see Chandrasekhar, 1961, ch.2, 18]. Nevertheless, convection in a thin layer of fluid heated along its base is still called Rayleigh-B´enard (and often just B´enard) convection.
2.1. Convective instability and the Rayleigh number
Rayleigh’s stability analysis of convection predicted the conditions necessary for the onset of convection, as well as the expected size of convection cells (relative to the layer thickness). Without delving into the mathematics of this analysis, suffice it to say that at the heart of the theory is a search for the infinitesimal thermal disturbance or pertur- bation most likely to trigger convective overturn in a fluid layer that is gravitationally unstable (i.e., hotter, and thus less dense, on the bottom than the top). This most destabi- lizing perturbation is often called the least-stable or most- unstable mode; its prediction and analysis is fundamental to the study of pattern-selection in convection, as well as other nonlinear systems. Depending on the nature of the binding top and bottom surfaces of the layer (i.e., whether they are freely mobile, or adjacent to immobile walls), the most unstable mode leads to convection cells which have a width approximately equal to (though as much as 50% larger than) the depth of the fluid layer. This result is quite relevant to plate tectonics since it is the first basic step in predicting the size of convection cells, and thus the size of plates. The perturbation which leads to this form of convection is deemed most unstable because, of all the possible perturba- tions, it will induce overturn with the least amount of heat- ing. This heating however is measured relative to the proper- ties of the system and as such defines the Rayleigh number. For example, say the layer is heated by basal heating only; i.e., the bottom boundary is kept isothermal at temperature hotter than the top boundary (which is also kept isother- mal). The vigor of convection caused by this temperature drop
depends on the properties of the system, in par- ticular the fluid’s density , thermal expansivity , dynamic viscosity , thermal diffusivity , the layer’s thickness , and the gravitational field strength (acting normal to the layer and downward). These properties reflect how much the sys- tem facilitates convection (e.g., larger , , and allow more buoyancy) or impedes convection (e.g., larger and imply that the fluid more readily resists motion or diffuses ther- mal anomalies away). The combination of these properties in the ratio gives a temperature that
must exceed in order to cause convection. The Rayleigh number is the
height
0.0 0.5 1.
horizontally averaged temperature
Figure 2. Temperature profiles (i.e., horizontally averaged temperature versus depth) for a basally heated, plane layer of fluid undergoing ther- mal convection when its viscosity is constant (solid curve) and temperature- dependent (dashed). The profiles show that most of the temperature change across the fluid occurs in relatively narrow thermal boundary layers near the top and bottom surfaces. In between the two boundary layers, most of the fluid is stably stratified or (if very well mixed) homogeneous. The fluid with temperature-dependent viscosity develops a stiffer upper thermal boundary layer which acts as a heat plug (i.e., it reduces convection’s ability to eliminate heat), causing most of the rest of the fluid to heat up to a larger average temperature. (After Tackley [1996a].)
the top and bottom boundaries are isothermal, the heatflow through the layer is, in theory, unbounded; i.e., convection can be so vigorous as to give the fluid layer the appearance of being a material with infinite thermal conductivity. How- ever, the same is not true for convection with only internal heating (see below) since the heatflow is limited by the net rate of heat production in the layer.)
As mentioned, the thermal boundary layers are where the gravitationally unstable material is confined, and thus fluid in these layers must eventually either sink (if in the cold top boundary layer) or rise (if in the hot bottom boundary layer), thereby feeding the vertical convective currents, i.e., down- wellings and upwellings, respectively. The feeding of ver- tical currents induces motion of the boundary layers toward convergent zones (e.g., a downwelling for the top bound- ary layer) and away from divergent zones (e.g., over an up- welling). Such flow naturally causes the boundary layers to thicken in the direction of motion (see Fig. 1). For ex- ample, in the top cold thermal boundary layer, fluid from hotter depths newly arrived at the divergent zone heats and thus thins the boundary layer; but as the fluid moves toward the convergent zone it cools against the surface, and the boundary layer gradually thickens before eventually grow- ing heavy enough to sink into the downwelling. We will re- visit boundary layer thickening again when discussing sub- duction zones and seafloor topography.
2.2.3. The size of convection cells As mentioned pre- viously, stability theory predicts that convection initiates
with cells that are approximately as wide as the layer is thick. For the most part, this also holds true in fully developed ba- sic convection. One of the things determining the lateral ex- tent of a single convection cell is the length of the horizontal boundary layer currents. For example, as material in the top boundary layer current flows laterally it loses heat and buoy- ancy (being adjacent to the cold surface) and thus can travel only a certain distance before it becomes so cold and nega- tively buoyant that it sinks and feeds the cold downwelling. In the end, the lateral distance that material in the top (or bot- tom) boundary layer current can traverse before sinking (or rising) essentially determines the size of convection cells, and by inference (in a naive sense) the size of the tectonic plates themselves. 2.2.4. Thermal boundary layer forces What force makes the thermal boundary layer currents flow laterally? While it may seem intuitively obvious that boundary layer fluid feeding a vertical current must flow horizontally (such as fluid going toward a drain), the forces behind these cur- rents are of significance in regard to plate forces. It should be clearly understood that buoyancy does not drive the bound- ary layer currents directly; buoyancy only acts vertically while boundary layer currents move horizontally (buoyancy or gravity does eventually deflect these currents back into the convecting layer, but it cannot drive their lateral flow). In- deed, horizontal pressure gradients are the primary driving force for thermal boundary layers. For example, when hot upwelling fluid impinges on the top surface, it is forced to move horizontally away from a high pressure region which is centered above the upwelling itself; the high pressure re- gion results from a force exerted by the surface on the fluid to stop the vertical motion of the upwelling thermal. As the top cold thermal boundary layer moves away from an upwelling to its own downwelling it thickens, gets heavier and acts to pull away from the surface; this induces a suc- tion effect and thus, because of the boundary layer’s grow- ing weight, increasingly lower pressures in the direction of motion, eventually culminating in a concentrated low pres- sure zone where the downwelling separates from the surface. Thus the horizontal boundary layer current flows from the induced high pressure over the upwelling to the low pressure over the downwelling, i.e., it flows down the pressure gradi- ent. Invariably plate driving forces are related to these pres- sure highs and lows, i.e., pressure gradient effects. Ridge push is the gradual pressure gradient going from the pres- sure high at a ridge outward; slab pull is effectively due to a concentrated pressure low caused by slabs pulling away from the surface (and the concept of a slab stress-guide sim- ply means that the pressure low is kept concentrated thus leading to sharp pressure gradients). We will discuss plate forces in greater detail later. Though in this paper we are mainly concerned with the top boundary layer current of the mantle (i.e., the plates), suffice it to say that identical dynamics occurs at the bottom boundary layer current: the cold downwelling induces a high pressure region when it hits the bottom boundary, thereby forcing fluid away from it, etc.
Figure 3. Laboratory experiments on convection in viscous liquids in a plane layer, bounded between two horizontal rigid glass plates, and heated from below. All images shown employ the shadowgraph techniqe (see text for discussion) and thus dark regions are relatively hot, while light regions are colder. The roll pattern (a) is also called 2-D convection; this pattern becomes unstable to the bimodal pattern (b) at moderate Rayleigh numbers, which eventually gives way to the spoke pattern (c) at yet higher Rayleigh numbers (around * (^) ). The roll, bimodal and spoke pattern are typical for convection in isoviscous fluids, although they also occur in fluids with temperature-dependent viscosity (the photographs shown for figures (a)-(c) are in fact for weakly temperature-dependent viscosity fluids). More unique to convection in temperature-dependent viscosity fluids are squares (d), hexagons (e) and even triangles (f). (After White [1988].)
2.3. Patterns of convection
The two-dimensional convection described above is a special type of convective flow also known as convection with a roll planform; i.e., the 2D convection cells – when extended into three-dimensions – are infinitely long counter- rotating cylinders or rolls. This pattern of convection is typ- ically unstable at moderately high Rayleigh numbers (i.e., somewhat greater than the critical ) at which point con- vection becomes three dimensional (3D) [Busse and White- head, 1971]; i.e., roll-like convection cells break down into more complicated geometrical shapes. The study of convec- tive planforms and pattern selection is a very rich and fun- damental field in itself, although a full quantitative discus- sion is beyond the scope of this paper; the interested reader should read Busse [1978]. However, we will attempt a brief qualitative discussion of convective patterns.
Much of the work on convective planforms was motivated by laboratory experiments in convection in thin layers [e.g., Busse and Whithead, 1971; Whitehead, 1976; Whitehead and Parsons, 1978; see Busse, 1978]. The planforms are most readily (albeit qualitatively) observed through a simple experimental technique called the shadowgraph [Busse and Whitehead, 1971]. Most laboratory observations of convec- tive planforms are done through this method, so it deserves a brief mention. In the laboratory model of convection, the
convecting fluid is bounded between two glass plates; the bottom is heated from below by hot water flushing along its outer surface, while the top is cooled by cold water flushed along the top. Collimated light (i.e., with parallel rays) is shown vertically up through the tank. In the convecting fluid, the index of refraction of light is dependent on temperature such that light rays diverge away from hot regions and con- verge toward colder regions. The projection of the resulting rays on a white screen thus show hot zones as dark shadows, and cold zones as concentrations of brightness. In the end, the convection patterns observed with ex- perimental (and computational) techniques are quite varied, though we will attempt to provide some order and logic to their presentation. As discussed in the previous sec- tion, purely basally-heated, plane-layer isoviscous convec- tion is naturally a very symmetric system. Thus, even when convection is three-dimensional, the upwelling and down- welling currents retain some basic symmetry; e.g., in bi- modal convection, the upwellings assume the geometry of two adjacent sides of a square, while the downwellings form the other two sides of the square (Fig. 3). However, the convection planform can also become highly irregular as with the spoke pattern which shows several nearly linear upwellings joined at a common vertex (and likewise with downwellings) (Fig. 3); even in this irregular planform the
terior of the fluid layer. Thus, in purely internally heated convection, there are only concentrated downwelling cur- rents; to compensate for the resulting downward mass flux, upwelling motion occurs, but it tends to be a broad back- ground of diffuse flow, rising passively in response to the downward injection of cold material (Plate 1). Therefore, since the downwellings descend under their own negative buoyancy they are typically called active currents, while the backgrond upwelling is deemed passive.
If we now allow for convection with both internal and basal heating (i.e., the bottom boundary is hot and isother- mal – instead of insulating – and thus permits the passage of heat) we have a form of convection a bit more complex than with either purely basal heating or purely internal heat- ing. However, the nature of the resulting convection can be understood if one realizes that the bottom thermal boundary layer must conduct in the heat injected through the bottom while the top thermal boundary layer must conduct out both the heat injected through the bottom as well as the heat gen- erated internally. Thus to accomodate this extra heat flux, the top thermal boundary layer develops a larger tempera- ture drop (to sustain a larger thermal gradient) than does the bottom boundary layer. In this way, the top boundary layer has a larger thermal anomaly than the bottom one, leading to larger, more numerous and/or more intense cold down- wellings than hot upwellings. Invariably, internal heating breaks the symmetry between upwellings and downwellings by leading to a preponderance of downwellings driving con- vective flow (Fig. 4). In the case of the Earth, whose net surface heat flux is thought to be 80% or more due to radio- genic sources, the mantle is predominantly heated internally, and only a small amount basally. Thus, one can expect a top thermal boundary layer with a large temperature drop across it feeding downwellings which dominate the overall convec- tive circulation; active upwellings from the heated bottom boundary provide only a small amount of the net outward heat flux and circulatory work. Although we have yet to dis- cuss many of the complexities leading to the plate-tectonic style of mantle convection, this simple picture of internally heated convection is in keeping with the idea that the large scale circulation is driven by downwellings (slabs) fed by an intense thermal boundary layer (the lithosphere and plates), while active upwellings (mantle plumes) are relatively weak and/or few in number [see Bercovici et al., 1989a; Davies and Richards, 1992].
2.5. Influence of temperature-dependent viscosity
Mantle material is known to have to have a highly tem- perature dependent viscosity for subsolidus, or solid-state, flow. Whether such subsolidus flow occurs by diffusion creep (deformation through the diffusion of molecules away from compressive stresses toward tensile stresses) or dislo- cation power-law creep (dislocations in the crystal lattice propagate to relieve compression and tension), the mobil- ity of the molecules under applied stresses depends strongly on thermal activation; i.e., the atom’s thermal kinetic energy
Figure 4. Laboratory experiment for internally heated convection in a plane layer. Images shown employ the shadowgraph technique (see Fig. 3 and text for discussion). Apparent internal heating is accomplished by ini- tiating basally heated convection and then steadily decreasing the tempera- ture of the top and bottom surfaces; this causes the average fluid temperature to be greater than the average temperature of the two boundaries. The fluid thus loses its net heat and the rate of this bulk cooling is a proxy for inter- nal heating. The frames show the convective pattern when the layer is all basally heated (a) and thus upwellings (dark zones) and downwellings (light zones) are comparable; when bulk cooling (internal heating) is stronger than basal heating, the downwellings (light) are dominant (b); when bulk cooling is much stronger than basal heating the upwellings are not distinct enough to register in the shadowgraph (c). (After Weinstein and Olson [1990].)
determines the probablity that it will jump out of a lattice site [Weertman and Weertman, 1975; Evans and Kohlstedt, 1995; Ranalli, 1995]. The viscosity law for silicates there-
fore contains a quantum-mechanical probablity distribution
in the form of the Arrhenius factor ^ where is
the activation enthalpy (basically the height of the energy potential-well of the lattice site out of which the mobilized atom must jump), is the gas constant,
is temperature, and thus
represents the average kinetic energy of the atoms in the lattices sites. Because of this factor, a few hun- dred degree changes in temperature can cause many orders of magnitude changes in viscosity. Moreover, with the in- verse dependence on
in the Arrhenius exponent, viscos- ity is highly sensitive to temperature fluctuations at lower temperatures (i.e., the viscosity versus temperature curve is steepest at low
). Thus, in the coldest region of the man- tle, i.e., the lithosphere, viscosity undergoes drastic changes: mantle viscosity goes from $(% '
) in the lower part of the upper mantle, to as low as % '
) in the asthenosphere [see King, 1995] to $ '
) [Beaumont, 1976; Watts et al., 1982] or potentially higher in the lithosphere. Thus the viscosity may change by as much as 7 orders or magnitude in the top 200 hundred kilometers of the mantle. In the end, the effect of temperature-dependent viscosity on mantle con- vection is to make the top colder thermal boundary – i.e. the lithosphere – much stronger than the rest of the man- tle. This phenomenon helps make thermal convection in the mantle plate-like at the surface in some respects, but it can also make convection less plate-like in other respects.
A strongly temperature dependent viscosity can break the symmetry between upwellings and downwellings in much the same way as internal heating. This occurs because the top cold thermal boundary layer is mechanically much stronger and less mobile than the hotter bottom thermal boundary layer. The less mobile upper boundary induces something of a heat plug that forces the fluid interior to warm up; this in turn causes the temperature contrast between the fluid interior and the surface to increase, and the contrast be- tween the fluid interior and underlying medium (the core in the Earth’s case) to decrease. This effect leads to a larger temperature jump across the top boundary layer (which par- tially reduces the boundary layer’s stiffness by increasing its average temperature), and a smaller one across the bot- tom boundary layer (see Fig. 2). Thus, the temperature- dependent viscosity can lead to an asymmetry in the thermal anomalies of the top and bottom boundary layers much as we see in the Earth.
Temperature dependent viscosity can also cause a signif- icant change in the lateral extent of convection cells. Be- cause material in the top thermal boundary must cool a great deal and thus for a long time to become negatively buoyant enough to sink against its cold, stiff surroundings, it must travel horizontally a long distance while waiting to cool suf- ficiently, assuming it travels at a reasonable convective ve- locity. This can cause the upper thermal boundary layer and thus its convection cell to have excessively large lateral ex- tents relative to the layer depth (i.e., large aspect ratios ). This effect has been verified in laboratory and numerical ex- periments (Fig. 5) [e.g., Weinstein and Christensen, 1991; Giannandrea and Christensen, 1993; Tackley, 1996a; Rat-
Figure 5. Lab experiment for convection with temperature-dependent viscosity fluid and mobile top thermal boundary layer. The mobilitity of the top boundary layer is facilitated by a free-slip upper boundary, accom- plished by inserting a layer of low-viscosity silicon oil between the working fluid (corn syrup) and the top, cold glass plate. Without the oil, the rigidity of the top glass plate tends to help immobilize the cold and stiff top bound- ary layer. Frame (a) shows, for comparison, the experiment when the top boundary is rigid, yielding a predominantly spoke-like pattern. Frame (b) shows the planform when the oil is used to make the top boundary free-slip, causing a dramatic increase in the convection cell size to what is deemed the spider planform. (After Weinstein and Christensen [1991].)
cliff et al., 1997] As we will discuss later (
3.6), large as- pect ratio convection cells are considered to occur in the Earth, especially if one assumes that the Pacific plate and its subduction zones reflect the dominant convection cell in the mantle. Thus, temperature-dependent viscosity can be used to explain the large aspect convection cells of mantle convection, but with some serious qualifications. With very strongly temperature dependent viscosity, the top thermal boundary layer can also become completely im- mobile and the large aspect ratio effect vanishes. The immo- bile boundary layer happens simply because it is so strong that it cannot move. As a result, the top boundary layer ef- fectively imposes a rigid lid on the rest of the underlying fluid, which then convects much as if it were nearly in iso- viscous convection with a no-slip top boundary condition.
convective solution to be stable (i.e. steady or statistically steady) the total heat input through the bottom boundary must be equal to the heat output through the top. However, because of sphericity, the bottom boundary has significantly less surface area than the top boundary through which heat passes; to compensate for this smaller area, the bottom ther- mal boundary layer generally has a larger temperature drop (and thus larger thermal gradients) across it than does the top boundary layer. This asymmetry leads to upwellings with larger temperature anomalies and velocities than the downwellings, which is opposite to the asymmetry between upwellings and downwellings thought to exist in the Earth. Thus, effects like internal heating and temperature depen- dent viscosity are even more important in order to overcome the asymmetry imposed by sphericity, and to give a more Earth-like asymmetry.
2.7. Poloidal and toroidal flow
Convective motion, with its upwelling and downwelling currents, and the associated divergent and convergent zones at the surface and lower boundary, is also called poloidal flow. Basic convection in highly viscous fluids essentially has only poloidal flow. However, while a great deal of the Earth’s plate motion is also poloidal (speading centers and subduction zone), much of it – with perhaps as much as 50% of the total kinetic energy – also involves strike-slip motion and spin of plates, which is called toroidal motion [Hager and O’Connell, 1978, 1979, 1981; Kaula, 1980; Forte and Peltier, 1987; O’Connell et al., 1991; Olson and Bercovici, 1991; Gable et al., 1991; Cadek and Ricard, 1992; Lithgow- Bertelloni et al., 1993; Bercovici and Wessel, 1994; Du- moulin et al., 1998]. The existence of toroidal flow in the Earth’s plate-tectonic style of mantle convection is a major quandary for geodynamicists and is at the heart of a unified theory of mantle convection and plate tectonics. Therefore, the reason that toroidal motion does not exist in basic con- vection deserves some explanation.
Most models of mantle convection treat the mantle as nearly incompressible, i.e., as if it has constant density. In fact, convection models must allow for thermal buoyancy, thus they really treat the mantle as a Boussinesq fluid; this means that while density is a function of temperature (and thus actually not constant), the density fluctuations are so small that the fluid is still essentially incompressible ex- cept when the density fluctuations are acted on by gravity. Thus the fluid acts incompressible, but can still be driven by buoyancy. Moreover, even without thermal density anoma- lies, the mantle is still not really incompressible; due to in- creases in pressure its density changes by almost a factor of 2 from the top to the bottom of the mantle (e.g., from the IASP91 reference Earth model [Kennett and Engdah, 1991]). However, mantle flow occurs on a much slower time scale than compressional phenomena – in particular acoustic waves – and thus the mantle is really considered anelastic for which the separation of flow into poloidal and toroidal parts still applies [Jarvis and McKenzie, 1980; Glatzmaier, 1988;
Bercovici et al., 1992]. However, even the influence of this anelastic component of compressibility is rather small com- pared to other effects in thermal convection [Bercovici et al., 1992; Bunge et al., 1997]. The incompressibility or Boussinesq condition requires that the rate at which mass is injected into a fixed volume must equal the rate at which the mass is ejected, since no mass can be compressed into the volume if it is incompress- ible; i.e., what goes in must equal what goes out. Math- ematically, when considering infinitesimal fixed volumes – or, equivalently, individual points in space (since a point is essentially an infinitesimal volume) – this condition is called the continuity equation and is written as (1)
where
is the velocity vector and
is the gradient opera- tor. This equation says that the net divergence of fluid away from or toward a point is zero; i.e., if some fluid diverges away from the point, an equal amount must converge into it in order to make up for the loss of fluid: again, what goes in must balance what goes out. The most general velocity field that automatically satisfies this equation has the form (in Cartesian coordinates) (2)
(where
is the unit vector in the vertical direction) since of any vector is zero. (Note that the velocity field in (2) uses only two independent quantities, namely and
, to account for three independent velocity components
!
and
!"
. This is allowed because equation (1) enforces a de- pendence of one of the velocity components on the other two and thus there are really only two independent quantiti- ties; i.e., in the parlance of elementary algebra, one equation with three unknowns means that there are really only two unknowns.) The quantity is called the poloidal potential which represents upwellings, downwellings, surface diver- gence and convergence and (as we will show below) is typi- cal of convective motion. The variable
is the toroidal po- tential and involves horizontal rotational or vortex-type mo- tion, such as strike-slip motion and spin about a vertical axis; toroidal flow is not typical of convective motion (Fig. 7). What actually forces the velocity fields in a highly vis- cous fluid (where fluid particles are always at terminal veloc- ity, i.e., acceleration is negligible) is determined by the bal- ance between buoyancy (i.e., gravitational) forces, pressure gradients and viscous resistance; in simple convection with a constant viscosity fluid, this force balance is expressed as
$#% '
(^)
$
#
(^) (3)
where ' is pressure, is viscosity, is density, and is gravitational attraction (& ' (*)+
) $). (The viscous resistance term, proportional to , in (3) expresses that viscous forces are due to imbalances or gradients in stress, while stress is due to gradients in velocity – i.e., shearing, stretching and squeezing – imposed on a fluid with a certain stiffness or
poloidal
toroidal
Figure 7. Cartoon illustrating simple flow lines associated with toroidal and poloidal motion.
viscosity.) Taking
of (3) leads to
^ $ $#% $
(^) (4)
where
$
^
(^) is the 2D horizontal Laplacian. Equation (4) shows that the poloidal motion (left side of the equation) is directly driven by buoyancy forces (right side). However, if we take
of (3) we only obtain $
$
(5)
which shows that there are no internal driving forces for toroidal flow. (Note, an identical analysis also exists for spherical geometry [e.g., see Chandrasekhar, 1961; Busse, 1975; Bercovici et al., 1989b].) Thus, for isoviscous con- vection, toroidal motion does not occur naturally and the only way to generate it is to excite it from the top and/or bottom boundaries; otherwise, toroidal motion does not ex- ist. (The same argument holds even if viscosity is a func- tion of height ; see
4.8.2.) To have convective flow drive toroidal motion itself, i.e., from within the medium, requires a forcing term on the right side of (5); as discussed later (
4.8.2), this can only be done if the viscosity varies laterally, i.e., (^) (see Plate 2 for a qualitative explanation). The problem of how to get buoyancy driven poloidal mo- tion to induce toroidal flow, either through boundary condi- tions or through horizontal viscosity variations, is called the poloidal-toroidal coupling problem. This will be discussed in more detail later in the paper (see
4.8).
3. WHERE DOES BASIC CONVECTION
THEORY SUCCEED IN EXPLAINING
PLATE TECTONICS?
Having reviewed some of the essential aspects of sim- ple viscous convection, we now examine those features of plate tectonics which are reasonably well explained as being
Plate 2. Cartoon illustrating the need for variable viscosity to obtain toroidal flow. Clear viscous fluid is drained through a hole in the bottom of a tank; on the clear fluid floats a uniformly cold congealed oil (e.g., butter or lard), shown in blue. The flow of clear fluid down the drain is predominanty poloidal, i.e., it only involves vertical motion and convergence toward the drain. The overlying oil is forced by the motion of the clear fluid to con- verge over the drain, and as long as the oil remains at one temperature it converges over the drain symmetrically, thus its flow is also poloidal. How- ever, once one side of the oil is heated up (bottom frame), the warmer and softer side of the oil converges over the drain more readily. The differen- tial motion between the warm, soft oil and the colder, stiffer oil appears as shear or toroidal motion (e.g., compare the light-blue and adjacent yellow flow vectors). Note that toroidal shear concentrates over the largest vis- cosity gradient in the oil (i.e., in the red region, at the transition between the warmest (yellow) and coldest (blue) oil); moreover, the toroidal motion only occurs because the viscosity gradient is perpendicular to the conver- gent poloidal flow.
characteristic of basic convective flow. Subsequently (
4), we discuss those aspects of plate tectonics which are poorly explained by basic convection.
3.1. Convective forces and plate driving forces
3.1.1. Slab pull and convective downwellings If we are to compare simple viscous convection to mantle flow and plate tectonics, then we wish to identify subducting slabs with the downwelling of a cold upper thermal boundary layer. As demonstrated by Forsyth and Uyeda [1975], the correlation between the connectivity of a plate to a slab (i.e., the percent of its perimeter taken by subduction zones) and the plate’s velocity argues rather conclusively for the domi- nance of slab pull as a plate driving force. Therefore, if slabs are simply cold downwellings, then the descent velocity of a downwelling in simple convection should be comparable to that of a slab and by inference the velocity of a plate (in particular a fast slab-connected or active plate). (As men- tioned earlier (
2.2), the pull of a slab on a plate is in fact a horizontal pressure gradient acting in the cold upper ther- mal boundary layer and caused by the low pressure associ- ated with a slab pulling away from the surface; invariably the pressure gradient is established so that the boundary layer or
then there is a compensation depth beneath the lithosphere at which every overlying, infinitely high, column of mate- rial with equal cross-sectional area weighs the same; this also means that hydrostatic pressure (i.e., weight per area) is the same at every horizontal position along the compen- sation level. However, columns that do not extend to the compensation depth do not weigh the same; in particular the column centered on the ridge axis weighs the most since off-axis columns still include too much light material (like water) and not enough heavy material (like lithosphere) to equal the weight of the column of hot asthenosphere beneath the ridge. Thus, for depths less than the isostatic compensa- tion depth, pressure is not horizontally uniform and is in fact highest beneath the ridge axis, thereby causing a pressure gradient pushing outward from the ridge [see Turcotte and Schubert, 1982]; however, this pressure gradient exists ev- erywhere within the lithosphere (i.e., where ever sea floor depth changes laterally), and not as a line-force at the ridge axis.
How does this ridge-force pressure gradient relate to con- vective forces? As discussed previously (
2.2), the horizon- tal pressure gradient in the convective thermal boundary lay- ers is what drives the lateral motion of the boundary lay- ers. If we permitted the upper surface to be deformable we would find that pressure highs at divergent zones would push up the surface and the pressure lows at convergent zones would pull down on the surface. Thus the pressure gradi- ent in the thermal boundary layer would be manifested as surface subsidence from a divergent to convergent zone. In fact, as we shall see in the next section, the resulting surface subsidence of a simple convective boundary layer is nearly identical to that observed for oceanic lithosphere. Thus, the ridge-push pressure gradient is essentially indistinguishable from the pressure gradient in a convective thermal boundary layer.
3.2. Structure of ocean basins
Large scale variations in bathymetry, or sea-floor topora- phy, can be related to simple convective processes, in partic- ular the cooling of the top thermal boundary layer. Although we have already discussed the cooling-boundary-layer con- cept in the previous section, we will re-iterate the arguments in slightly more detail in order to examine the validity of the underlying assumptions.
As mentioned above, when the top thermal boundary layer of a convection cell moves horizontally, it cools by heat loss to the colder surface. In simple convection, the surface is assumed impermeable, i.e., no mass crosses it and thus heat loss out of the top thermal boundary layer is assumed entirely due to thermal diffusion; i.e., since the boundary layer abuts the impermeable surface, vertical heat loss due to ejection of hot mass or ingestion of cold mass across the surface is assumed negligible. While this assumption is rea- sonable for most of the extent of the thermal boundary layer, we will later see that it causes convective theory to mispre- dict some details of ocean-basin topography and heat flow
age (Myrs)
age (Myrs)
depth relative to ridge axis (km)
heat flow (hfu)
(^0 50 100 )
Figure 9. Bathymetry – i.e., seafloor topography – (top frame) and heat- flow (bottom frame) versus age for typical ocean floor. Bathymetry is for the North Pacific (circles) and North Atlantic (squares) oceans; heatflow (in (^) (^) ) shows average global values (circles), and associ- ated error bars. Both frames show the predictions from convective boundary layer theory (solid curves); i.e., a ^ law for bathymetry and a law for heatflow. The theoretical curves fit the observations reasonably well, except near and far from the ridge axis. Adapted from Turcotte and Schu- bert [1982] after Parsons and Sclater [1977] and Sclater et al., [1980]. See Stein and Stein [1992] for more recent analysis.
4.2). As discussed above, the assumption of diffusive heat loss means that thickening of the boundary layer is controlled by
thermal diffusivity , with units of ) $+
) ; thus, again by di- mensional homogeneity (see
3.1.1 and Furbish [1997]), the
boundary layer thickness
. As the thermal bound- ary layer thickens it weighs more and thereby pulls down on the surface with increasing force; if the surface is de- formable, it will be deflected downward (until isostasy is established). The surface subsidence thus mirrors the in- creasing weight of the boundary layer; the weight increases
only because grows, and thus surface subsidence goes as
. Therefore, sea floor subsidence is predicted by convec- tive theory (more precisely convective boundary layer theory [Turcotte and Oxburgh, 1967]) to increase as the square-root
of lithospheric age; this is called the law. Moreover,
this type of boundary layer theory predicts that the heatflow (units of
$ ) out of the ocean floor should obey a
+ law, which can be inferred by the fact that the heat-
flow across the boundary layer is mostly conductive and thus
goes as ,
(where , is thermal conductivity and
is the temperature drop across the thickening thermal bound- ary layer, as also defined in
3.1.1). Profiles, perpendicular to the spreading center, of sea floor bathymetry and heatflow versus age (Fig. 9) show that, to first order, sea floor subsi- dence and heatflow do indeed follow boundary layer theory
(i.e., follow and + laws, respectively) [Parsons and Sclater, 1977; Sclater et al., 1980; see also Turcotte and Schubert, 1982, ch.4; and Stein and Stein, 1992], further emphasizing that the oceanic lithosphere is primarily a con- vective thermal boundary layer. However, at a greater level
of detail, the laws fail; this will be discussed later in
the paper.
3.3. Slab-like downwellings are characteristic of 3D convection
In simple, purely basally heated plane-layer convection, the convective pattern is often characterized by intercon- nected sheets of downwellings. For example, the hexago- nal pattern is characterized by one upwelling at the center of each hexagon, surrounded by a hexagonal arrangement of downwelling walls (Fig. 3). A similar situation occurs for spherical systems [Busse, 1975; Busse and Riahi, 1982, 1988; Bercovici et al., 1989b]. When internal heating of the fluid is included, the upwellings and downwellings can be so imbalanced as to destroy any trace of symmetry; even so, the downwellings can persist as linear, albeit no longer intercon- nected, sheet-like downwellings at low to moderate Rayleigh numbers [Bercovici et al., 1989a]. The linear downwellings can desist and give way to downwelling cylinders and blobs at high Ra [Glatzmaier, et al., 1990; Bunge et al., 1996]; however, even at high Ra, the linearity of the downwellings is recovered by allowing viscosity to increase with depth, as is thought to occur in the Earth [Bunge et al., 1996] (see
and Plate 1). Thus, although sheet-like downwellings do not always occur in simple models of thermal convection, they are a very common feature. In that sense, the occurence of slab-like downwellings in mantle convection is characteris- tic of basic thermal convection. However, as we will discuss later, there are many other characteristics of subducting slabs that are unlike simple convective flow.
3.3.1. Cylindrical upwellings and mantle plumes are also characteristic of 3D convection Although mantle plumes and hotspots are not, strictly speaking, part of plate tectonics, they play an important role in our ability to mea- sure absolute plate motions (e.g., because of assumed hotspot fixity), and in understanding the nature of mantle convection and the relative sizes of its thermal boundary layers.
As with sheet-like downwellings, cylindrical or plume like upwellings are prevalent in simple thermal convection. (It is possible to get different shaped – e.g. sheet-like – up- wellings, but they tend to break down into more columnar features before reaching the surface.) However, such up- wellings always require a significant bottom thermal bound- ary layer heated from below by a relatively hot reservoir, such as the Earth’s outer core; this boundary layer pro- vides a source region for plumes and its thickness there- fore determines their size [Loper and Eltayeb, 1986; Chris- tensen, 1984b; Bercovici and Kelly, 1997]. In basic con- vection with significant internal heating, the bottom ther- mal boundary layer typically has a much smaller tempera- ture drop than does the top thermal boundary layer. In this
case, upwellings still retain some plume-like quality though they are fewer in number and/or quite weak. Nevertheless, the existence of cylindrical upwelling plumes in the man- tle – indirectly inferred from hotspot volcanism, geochemi- cal analyses [see Olson, 1990; Duncan and Richards, 1991; White and McKenzie, 1995] and perhaps most directly in seismic analysis [Nataf and VanDecar, 1993; Wolfe et al., 1997; Chen et al., 1998] – is consistent with basic models of thermal convection. However, it should be understood that even the weak plumes seen in complete 3D models of inter- nally heated convection are an integral part of the convective flow [see Bercovici et al., 1989a]; i.e., while they may not have much influence on the dynamics of the top cold bound- ary layer, they will position themselves in accordance with the whole convective pattern, typically as far from down- wellings as possible [Weinstein and Olson, 1989]. Plumes well integrated into the global convective circulation are very unlikely to establish themselves where they must pass through adverse conditions (e.g., beneath a downwelling or across a strong “mantle wind”), and even if they do, they can only exist as evanescent features [c.f. Steinberger and O’Connell, 1998].
3.4. Relative fluxes of plumes and slabs are in agreement with the mantle’s presumed heating mode
The heat flux transported by mantle plumes, relative to that transported by the cooling lithosphere and slabs, is also well predicted by convection models using the Earth’s pre- sumed proportion of internal heating. In basic convection, the net plume heat flux is approximately the same as the heating injected through the bottom boundary; i.e., plumes essentially carry only heat input from below [see Davies and Richards, 1992]. The Earth’s mantle is thought to be be- tween 80 and 90% heated internally by radiogenic sources [see Turcotte and Schubert, 1982]. Temperatures at the core- mantle boundary are not well known enough [see Boehler, 1996; also discussion by Davies and Richards, 1992] to definitively constrain the basal heat flux (i.e., to determine the temperature drop across the bottom thermal boundary layer, which is possibly the D” seismic layer; see Loper and Lay, 1995). However, the energy output necessary to power the geodynamo (i.e., the Earth’s magnetic field) places the heat flux out of the core at approximately 10% of the net terrestrial flux [see Gubbins, 1977; Verhoogen, 1981, ch.4]. Thus, by the convective picture, plumes should transport on the whole between 10-20% of the total heat flux. Plume heat flux can be estimated by calculating the buoyancy flux necessary to keep hotspot swells inflated [Davies, 1988b; Sleep, 1990]. These calculations suggest that plumes trans- port roughly ^ % of the net heat flow which is very con- sistent with our picture of basic, predominantly internally heated convection.
3.5. Mantle heterogeneity and the history of subduction So far we can construct a simple view of the connection between plate tectonics and mantle convection in which sub-
buoyancy to penetrate a moderately resistive boundary. This “reddening” of the convective spectrum (thus creating large- aspect ratio, nearly-Pacific size cells) has been well doc- umented in three-dimensional models of spherical convec- tion at reasonably high Rayleigh number for both the phase change [Tackley et al., 1993, 1994] and the viscosity-jump effects [Bunge et al., 1996; see also Tackley, 1996b].
In the end we see that with temperature-dependent vis- cosity, or an internal boundary resistive to downwellings, basic convection can generate Earth-like, large-wavelength (or long aspect ratio) plate-sized convection cells. However, while these effects move us closer to obtaining plate-like- scales in convection models, none actually generate plates.
4. WHERE DOES BASIC CONVECTION
THEORY FAIL IN EXPLAINING PLATE
TECTONICS, AND WHAT ARE WE DOING
(OR MIGHT WE BE DOING) TO FIX IT?
Although models of basic thermal convection in the Earth’s mantle have made significant progress in explaining many features of plate tectonics, there remain a great number of unsolved problems. While we make no claim to present an exhaustive list of such unresolved issues, here we sur- vey some of the large-scale problems remaining as well as progress being made to solve them.
4.1. Plate forces not well explained by basic convection
While slab pull and to a large extent ridge push (really, distributed ridge push) are manifestations of simple convec- tion, other forces are not. Most of the forces unaccounted for by convection are related to some feature of plate tec- tonics that convection does not (or does not easily) gener- ate. One such force is transform resistance, as this involves strike-slip motion which cannot be readily generated, if gen- erated at all, in simple convection. The lack of excitation of strike-slip motion is a major short-coming of basic con- vection theory, thus we will defer discussion of this until later (see
4.8). The precise distribution of the ridge push force also cannot be predicted in simple convection models since they do not generate narrow passive divergent zones, i.e. thin ridges fed by shallow nonbuoyant upwellings. The lack of formation of passive but focussed upwellings is also a signficant shortcoming of basic convection theory and will be discussed later as well (see
4.7). Finally, collision re- sistance does not explicitly occur in basic convection since most convection models do not include continents that re- sist each other’s motion at their contact points. Collision resistance must in the end be a manifestation of chemical segregation, i.e., placement of chemically light continental material at the surface which retains a buoyancy that resists being drawn into a downwelling [e.g., Lenardic and Kaula, 1996].
4.2. Structure of ocean basins (deviations from the
law)
As discussed previously, the classic bathymetry and heat-
flow profiles (Fig. 9) show a predominant depen-
dence. However, bathymetry and especially heatflow devi-
ate significantly from the curve near the ridge axis
itself and far from the ridge axis. One must recall that the
prediction assumes that only diffusive transport of
heat occurs in the upper thermal boundary layer. This as- sumption likely becomes invalid at and far from the ridge axis. At the ridge axis, heat transport by magma migration [see Spiegelman, 1996 and references therein] and cooling by hydrothermal circulation are highly significant [see Tur- cotte and Schubert, 1982, ch.4; Stein and Stein, 1992, and references therein]. It is likely that supression of the bathy- metric and, in particular, the heatflow highs at the ridge is due to cooling not predicted by the diffusive thermal bound- ary layer theory; the most likely candidate for this extra cooling is indeed hydrothermal circulation. Incorporation of realistic or more complete transport phenonema relevant to ridges (e.g., magmatism and/or water ingestion) into convec- tion models is also related to the problem of how to generate focussed but passive ridges (see
4.7).
The flattening of bathymetry, relative to the law,
far from the ridge axis has been the subject of some con- cern. It has been proposed that such flattening is the result of extra heating of the lithosphere (causing effective reju- venation) brought on by secondary small-scale convection [Parsons and McKenzie, 1978], viscous heating [Schubert et al., 1976] and mantle plumes [Davies, 1988c]. This flatten- ing has also been attributed to active, pressure-driven, as- thenospheric flow that lifts up the lithosphere as the flow is forced into an increasingly constricted asthenospheric channel (the constriction being due to the cooling, thicken- ing lithosphere) [Phipps Morgan and Smith, 1992]. How- ever, since three-dimensional convection itself differs sig- nificantly from the two-dimensional boundary-layer theory
from which the law is derived [Turcotte and Oxburgh,
1967], such flattening and other deviations in the far-axis bathymetry are perhaps not altogether surprising from a mantle convection perspective.
4.3. Dynamic topography Convective currents impinging on or separating from the top boundary of the mantle induce vertical stresses that de- flect the Earth’s surface. In particular, upwellings should be associated with topographic highs and downwellings with depressions. Of course the most obvious topographic signal is not related to these thermal anomalies but to the chemical differences between the thick, light, continental crust and the denser, thinner oceanic crust. However, one of the more distinct features of the Earth’s topography, namely the mid- oceanic ridges, is a direct consequence of mantle convection (see
3.2). What is referred to as dynamic topography is what re- mains of the observed topography when the topography due
to shallow, isostatic mass anomalies in the crust and litho- sphere (i.e., thickness variations in the crust and lithosphere) are removed. This remaining topography is thus some- times called non-isostatic topography. In fact, the distinc- tion between isostatic and dynamic or non-isostatic topogra- phy is somewhat nebulous, e.g., while ridges and subsiding seafloor are isostatically compensated, they are also related to dynamic convective processes and plate motion. How- ever, even in this case strict removal of isostatic topography is informative. In particular, the residual (i.e, dynamic or non-isostatic) topography of the sea floor (observed topog-
raphy minus subsidence) reveals, in theory, the pres-
ence of thermal anomalies below the lithosphere, especially
far from the ridge axis since deviations from the law
near ridges are presumably due to hydrothermal circulation (see
4.2). The dynamic topography associated with long-wave-length, tomographically inferred, deep-seated heterogeneities should be theoretically quite large, i.e., 1000 meters [e.g., Hager and Clayton, 1987] in order to explain undulations in the geoid (the gravitational equipotential surface) of the Earth [e.g., Ricard et al., 1984; Richards and Hager, 1984]. How- ever attempts to detect this dynamic topography have been unsuccessful, or have concluded that it cannot exceed
300 meters, which is in contradiction with theroretical expecta- tions [Colin and Fleitout, 1990; Cazenave and Lago, 1991; Kido and Seno, 1994; Le Stunff and Ricard, 1995]. On a more regional scale, Lithgow-Bertelloni and Silver [1998] have concluded that South-East Africa does indeed have anomalously high topography which correlates, as predicted by theory, with both lower-mantle heterogeneity and geoid undulation. However, many other lower-mantle hetero- geneities mapped by tomography are not reflected in the surface topography. The theoretical, convective dynamic to- pography and the observed topography might be reconciled by assuming that part of the vertical stresses of convective currents are balanced by deflection of internal boundaries (e.g., the phase transitions at 400 and 670 km depth) [Tho- raval et al., 1995; Le Stunff and Ricard, 1997; Thoraval and Richards, 1997].
Since mantle convection is likely quite vigorous and time- dependent then convectively generated dynamic topography is also a time-dependent phenomenon. This means that two continents should have significantly different relative sea- level histories. Thus, the sedimentological records of con- tinental platforms should provide rigorous constraints on time-dependent, convection based models of dynamic to- pography [Gurnis, 1990, 1993; Mitrovica et al., 1989]. That sea-level changes possibly reflect large-scale convective pro- cesses is still somewhat alien to the conventional sedimento- logical view that such changes reflect only the global eustatic signal [Vail et al., 1977]. However, Gurnis [1993] has shown that a qualitative agreement exists between possible varia- tions of dynamic topography during the Phanerozoic and the inundation history of continents. Clearly, a quantitative so- lution to the dynamic topography problem would be facil- itated by further interactions between sedimentologists and
geodynamicists.
4.4. Changes in plate motion
Plate motions evolve with various time-scales. Some are clearly related to mantle convection, such as those associ- ated with the so-called Wilson cycle [Wilson, 1966], i.e. the periodic formation and breakup of Pangea, approximately every 500 Myrs). Various hypotheses have been proposed to explain the dispersal of supercontinents. It is generally accepted that the presence of a supercontinent tends to insu- late and thus, with radiogenic heating, warm the underlying mantle, eventually inducing a hot upwelling which weakens and breaks up the overlying lithosphere [Gurnis, 1988]. The fact that Pangea was surrounded by subduction also facili- tates the thermal insulation of that portion of mantle trapped under the continent. Although this mechanism is physically sound, the present-day continents do not suggest that they stand above hotter than normal mantle. On the contrary, their basal heat flux seems very low [e.g., Guillou et al. 1994]. The effect of a super-continent on the mantle, how- ever, may have been different from that of the present-day normal-sized continents. Plate motion changes that have occurred on shorter time- scales are much more difficult to understand. The plate tec- tonic history presumably recorded in hotspot tracks consists of long stages of quasi-steady motions separated by abrupt reorganizations. During a stage of quasi-steady motion, our understanding of force balances on the plates, either in terms of plate or boundary forces (slab pull, ridge push, mantle drag, etc.) [e.g. Forsyth and Uyeda, 1975], or in terms of the buoyancy of large-scale, tomographically-inferred man- tle heterogeneities [e.g., Ricard and Vigny, 1989; Lithgow- Bertel-loni and Richards, 1995], is certainly one of the obvi- ous quantitative successes of geodynamics. This success is reinforced by the fact that the parameters entering the theory (e.g., mantle viscosity, lithospheric thickness) are in agree- ment with independent observations (e.g., post-glacial re- bound, sea-floor topography). However, the success in pre- dicting the direction of plate velocities is much less obvious, or is based on a tautology. For constant plate motion, the pat- tern of sea floor age and the location of slabs are obviously such that their associated forces push and pull the plates in the right direction. Thus, it is not surprising that, when start- ing from the observed positions of ridges and slabs, we pre- dict the correct directions of plate motions. As the seismic tomography is well correlated, at least in the upper mantle, with the distribution of ridges and trenches, the ability to predict the surface motion from mantle heterogeneity simply confirms that surface tectonics and mantle dynamics belong to the same unique convecting system. Abrupt changes in plate motion, however, are not eas- ily related to convective processes. The most dramatic plate motion change is recorded in the Hawaiian-Emperor bend, dated at 43Ma; this bend suggests a velocity change of a major plate of approximately 45+ during a period no longer than 5 Myrs, as inferred from the sharpness of the bend. It
4 5
1 2
3
0
1
0 0.1 0.2 0.3 0.4 0.
0
1
0 0.1 0.2 0.3 0.4 0.
strain rate
stress
Figure 10. Constitutive (stress versus strain-rate) laws for (1) plastic, (2) non-Newtonian power law, (3) Newtonian, (4) brittle stick-slip and (5) self-lubricating (also called continuous stick-slip) rheologies.
layer will tend to be weakest at the divergent zone (where the boundary layer is thinnest) which approaches plate-like behavior. However, the viscosity will be highest over the cold convergent zone, tending to immobilize it and eventu- ally the entire boundary layer, which is of course one of the root causes for the subduction initiation problem (see
4.6); in the end, this also leads to a thermal boundary layer with little plateness. Therefore, it is a reasonable assumption that plate-like strength distributions – or high plateness – require a lithospheric rheology that permits boundaries to be weak, regardless of temperature.
At high stresses mantle rocks undergo non-Newtonian creep wherin deformation responds nonlinearly to the ap- plied stress; i.e., the stress–strain-rate constitutive law is nonlinear through a power-law relation such that
^
^ ^ (10)
where
is strain-rate, is stress, and is the power-law in-
dex (see Fig. 10); for mantle rocks,
, typically [Weert- man and Weertman, 1975; Evans and Kohlstedt, 1995]. The effective viscosity is
^ +
^
which means that for
viscosity decreases with in- creasing strain-rate. The power-law rheology causes regions of the material that are rapidly deformed to become softer, while slowly-deforming regions become relatively stiff. This rheology not only yields reasonable plateness, but also leads to a feedback effect: as the deformed parts of the fluid soften
they become more readily deformed, causing further strain to concentrate there, thereby inducing further softening, etc. (Comparable effects can be caused with other rheologies such as Bingham plastics and bi-viscous laws; however, these are just further mathematical models for essentially the same strain-softening effect represented by a power-law rhe- ology.) Such non-Newtonian rheologies have been incorpo- rated into numerous 2-D convection models for
and have found little plate like behavior [e.g., Parmentier et al., 1976; Christensen, 1984c]. Other models, concentrating on plate formation, placed a thin non-Newtonian “lithospheric” fluid layer atop a thicker convecting layer, or mathemati- cally confined the non-Newtonian behavior to a near-surface layer [Weinstein and Olson, 1992; Weinstein, 1996; Moresi and Solomatov, 1998; see also Schmeling and Jacoby, 1981] (Fig. 11). In these latter models, the non-Newtonian ef- fect does indeed yield an upper layer with reasonably high plateness for basally heated convection: the layer becomes weaker over both divergent zones (i.e. over upwellings) and convergent zones (downwellings), and is relatively immo- bile and strong in between the these two zones. Moreover, as noted by King et al. [1992], 2D models with non-Newtonian rheology are often indistinguishable in some respects from those with imposed plate geometries [e.g., Olson and Cor- cos, 1980; Davies, 1988a] and lithospheric weak zones [e.g., King and Hager, 1990; Zhong and Gurnis, 1995a]. How- ever, to obtain sufficient plateness with the non-Newtonian models, it is necessary to use a power-law index con- siderably higher than inferred from rheological experiments (typically
is needed). Thus, it seems that a general strain-weakening type rheology is a plausible and simple so- lution to generating high plateness , i.e. weak boundaries and strong plate interiors, in a convection model. However, when the convective flow is driven by internally heated con- vection, which has little or no concentrated upwellings, then the divergent zones in the lithospheric layer tend to be broad and diffuse, i.e., not at all like narrow, passively spreading ridges [Weinstein and Olson, 1992] (Fig. 11). It is undoubtedly naive to assert that plate-like behavior simply requires the generation of weak boundaries; in fact each type of boundary, i.e., convergent, divergent and strike- slip, develops under very different deformational and ther- mal environments. Obtaining sufficient plateness is invari- ably related to the the nature of how the different boundaries form. As each type of boundary is uniquely enigmatic, they warrant individual discussion. These will be the focus of the following sections.
4.6. Convergent margins: initiation and asymmetry of subduction
The downwelling currents in simple convection are for the most part very symmetric; i.e., one downwelling is com- posed of two cold thermal boundary layers converging on each other (Plate 1). In the Earth, the downwellings associ- ated with plate motions – i.e., subducting slabs – are highly asymmetric, i.e., only one side of the convergent zone – only
surface velocity
temperature
basally heated
surface velocity
temperature
internally heated
(a)
(b)
Figure 11. A lithospheric layer with non-Newtonian (power-law) rheol- ogy is driven by underlying 2D convection for both basal (a) and internal (b) heating. For both cases the convecting layers have ^ &^ &&^ and the overlying lithosphere has a rheology with a very large power-law index of ^. The bottom of segments of (a) and (b) show a cross section of a temperature field in the convecting layer. Above each of these is shown the horizontal velocity for the non-Newtonian lithospheric layer. The step-like transitions in velocities suggest plate-like motion (high plateness) since a segment of nearly constant velocity is moving as a contiguous block with little internal deformation (the plate-like segments are numbered in the top frame). The step-like character of the velocity field is much more distinct for the basally heated convection where there are both active upwellings and downwellings to define narrow divergent and convergent margins. The internal heating has broad passive upwellings which cause the overlying divergent zones to have smooth, wide and thus unplate-like changes in ve- locity. (After Weinstein and Olson [1992].)
one plate – subducts. The cause for this asymmetry remains one of the greater mysteries in geodynamics. A common hy- pothesis is that the asymmetry is caused by chemically light continents counteracting the negative buoyancy of one of the boundary layers approaching the convergent zone [e.g., see Lenardic and Kaula, 1996]; however this hypothesis is not universely valid since there are subduction zones entirely in the ocean.
The asymmetry may also reflect an inequality of pressure on either side of the subducting slab if it deviates slightly out of vertical; i.e., a more acute corner flow on one side of the subduction zone induces lower pressure than on the opposite side, thereby causing the slab to be torqued to the more acute side, i.e., to enhance its obliquity [see Turcotte and Schubert, 1982]. The oblique downwelling may then act to impede the motion and descent of the boundary layer approaching the acute angle, since it would have consider- able resistance to making a greater than & + downward turn, eventually leading to asymmetric convergence. This effect
is most significant for slabs that are effectively rigid rela- tive to the surrounding mantle, in order that they act as stiff paddles while being lifted up; thus if it occurs, the effect would theoretically be visible in basic convection calcula- tions with strongly temperature-dependent viscosity. How- ever, if the viscosity is so temperature-dependent that it in- duces such strong slabs then it will also likely place con- vection itself into the stagnant lid regime; i.e., to make the slabs strong enough will also lock up the lithosphere (see 2.5 and
3.6). To then adjust convection models to miti- gate this extra problem requires proper initiation of subduc- tion from cold, thick and strong lithosphere [see Mueller and Phillips, 1991; Kemp and Stevenson, 1996; Schubert and Zhang, 1997]. Models with imposed weak zones in the lithosphere show that cold rigid lithosphere subducts readily and can even assume a fairly oblique slab-like angle [Gurnis and Hager, 1988; King and Hager, 1990; Zhong and Gurnis, 1995a]. However, it is important to note that, based on seis- mically inferred deformation of slabs, it is not clear whether actual slabs are excessively stiff, or even stronger than the surrounding mantle [Tao and O’Connell, 1993]. The generation of the weak zone which permits subduc- tion is itself enigmatic. Recent models have suggested that the necessary weak zone occurs as the result of faulting and rifting [Kemp and Stevenson, 1996; Schubert and Zhang, 1997]. However, brittle failure of the entire lithosphere at its thickest point is problematic (see
4.5.1). Other weakening mechanisms at the subduction zones may be necessary [e.g., King and Hager, 1990], although the nature of these mech- anisms is still unclear, showing that the problem of down- welling asymmetry and subduction initiation remains one of the more elusive yet fruitful fields of geodynamical research.
4.7. Divergent margins: ridges, and narrow, passive upwellings Basic convection predicts that upwellings occur as either columnar plumes rising actively under their own buoyancy, or as a very broad background of upwelling ascending pas- sively in response to the downward flux of concentrated cold thermals. No where in basic convection theory or mod- elling does there occur concentrated but shallow and pas- sive upwellings (i.e., which rise in response to lithospheric spreading motion, not because of any buoyancy of their own) analagous to ridges [Lachenbruch, 1976; see Bercovici et al., 1989a]. That all mid-ocean ridges involve passive upwelling is not necessarily universal [Lithgow-Bertelloni and Silver, 1998]. However, the fastest and arguably the most signfi- cant ridge, the East Pacific Rise, is almost entirely devoid of a gravity anomaly, implying shallow, isostatic support and thus that there is no deep upwelling current lifting it up [see Davies, 1988b]. (Although the interpretation of gravity by itself is nonunique, the lack of a signficant free-air gravity anomaly over a topographic feature suggests that the gravity field of the topographic mass excess is being cancelled by the field of a nearby mass deficit; this deficit is most read- ily associated with a buoyant and shallow isostatic root on
