Download Whole-mantle convection and plate tectonics

Geophys. J. R . astr. Soc. (1977) 49,459-486
Whole-mantle convection and plate tectonics
Geoffrey F . Davies Department o j Geological Science, university oj'
Rochester, Rochester. N e w York 1462 7, USA
Received 1976 September 27; in original form 1976 July 8
Summary. It seems unlikely that the lower mantle is not involved in motions
related to plate tectonics. The evidence relating to several alleged obstacles to
lower-mantle convection is reviewed. The evidence for a chemical composition difference between the upper and lower mantle is not compelling.
There now seems to be good evidence that the viscosity of the mantle is fairly
uniform, contrary to a widely-held view, except possibly in the oceanic upper
mantle. Phase changes may locally enhance or retard thermal convection, but
are unlikely to prevent it. It is not necessary to assume that descending lithospheric slabs cannot penetrate below 700 km depth in order to explain the
distribution and source mechanisms of deep earthquakes. Other recent
seismic evidence suggests deep-mantle heterogeneities which may be related
to large-scale flow.
The behaviour of some simple layered-fluid models is analysed to show the
effect of viscosity contrasts. The results indicate that the lower mantle would
have to be 10000 times more viscous than the upper mantle in order to
confine thermal convection to the upper mantle. The lower mantle would
have to be at least 1000 times more viscous than the upper mantle in order to
exclude viscous flow entrained by the moving plates. Alternatively, a shallow
low-viscosity hyer would have to be about 10 000 times less viscous than the
underlying mantle for the 'return' flow to be confined to it. These and other
results indicate that only extreme viscosity contrasts (arising from non-linear
or temperature-dependent rheology, for instance, as well as from different
physical states) significantly affect large-scale flow patterns, and that flow
patterns are strongly affected by moving boundaries.
A version of whole-mantle thermal convection is proposed which seems to
be capable of explaining the main features of plate tectonics: the buoyancy
forces are concentrated in, but not confined to, the descending lithospheric
slabs. Descending slabs would then cause their attached plates to move
rapidly, as in the Elsasser model. The flow entrained by the fast plates would
dominate the flow pattern in the mantle: convection cells under fast plates
would be amplified; those under other plates would have smaller amplitudes
and would fit between the fast cells. This model can qualitatively account for
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the large horizontal scale of the plates, the imperfect correlation between
plate speed and the amount of attached descending slab, the motion of plates
with 110 attached slab, and the opening of the Atlantic in particular. The
model may provide a mechanism for the cyclic aggregation and dispersal of
continents. The large-scale flow probably has smaller scale complications,
especially in the upper mantle, and may be intrinsically unsteady. Narrow
ascending thermal plumes could probably be incorporated into this model.
Introduction
Despite considerable work by many people since the kinematic theory of plate tectonics was
formulated in the 1960’s, the question ‘What makes the plates go?’ still has not been
definitively answered. The main contention of this paper is that this lack of success is due to
the fact that most people have assumed that only the upper mantle, or possibly the upper
mantle and transition zone, is involved in plate dynamics. The main problem engendered by
this assumption has been the difficulty of explaining the large horizontal scale of the plates
(the order of 5000 km). The most likely mechanism of plate dynamics is thermal convection
driven by vertical temperature differences, and a characteristic of this type of convection is
that the width of convection cells is about the same as the depth of fluid involved. For
convection excluded from the lower mantle, this restricts the width of cells to less than
1000 km. Thus several such cells would be present under the major plates, and, since
neighbouring cells flow in opposite directions, their effects would tend to cancel, making
it difficult to account for the speed and persistence of plate motions.
If it is assumed that the lower mantle is involved in plate dynamics, the discrepancy
between plate widths and cell depths (almost 3000 km) is much less. An explanation for the
remaining discrepancy is provided by a model of mantle dynamics proposed here, and the
implications of the model for other features of plate tectonics are explored.
The idea of whole-mantle convection is certainly not new ( e g Holmes 1931; Pekeris
1935; Hales 1936; Griggs 1939; Hess 1962; Vening Meinesz 1947; Runcorn 1962, 1972;
Wilson 1963; Tozer 1972), although it has always been controversial. Before and during the
time that the theory of plate tectonics was being formulated, several lines of evidence
converged to suggest that the lower mantle is not involved in thermal convection these
included the inference that the Earth’s non-hydrostatic equatorial bulge is maintained by
the viscosity of the lower mantle, the observation that deep earthquakes occur down to,
but not in, the lower mantle, the expected effect ,Of phase changes on convection, and the
possibility of chemical composition differences between the upper and lower mantle. Thus,
ironically, when the mobility of the Earth’s interior was being most dramatically demonstrated, there was a reluctance to assume that the lower mantle was involved.
In this paper, the current status of the evidence that the lower mantle is immobile is
reviewed, and it is concluded that none of the evidence is compelling. In particular, recent
studies of the effects of the redistribution of surface loads due to continental glaciation and
sea level changes during and after the last ice age indicate that the viscosity of the lower
mantle is comparable to that of the upper mantle (Dicke 1969; O’Connell 1971; Cathles
1975).
It is assumed in this paper that plate motions are caused ultimately by thermal convection in the mantle which is in turn caused by heating of the mantle from within and below.
Others have adequately reviewed the likely sources of energy ( e g McKenzie, Roberts 6;
Weiss 1974; Turcotte & Oxburgh 1972).
It is convenient to classify ‘vertical’ thermal convection in terms of two extreme cases.
~
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46 1
In the first, the buoyancy forces are distributed fairly uniformly throughout the fluid; this
is the type most commonly encountered in other situations, and most often studied. In the
second, the buoyancy forces are concentrated in a small part of the fluid; for example, in a
cold descending ‘slab’ or a hot ascending plume. The cold descending slab model is similar
to that proposed by Elsasser (1969): the descending lithospheric slab sinks into the mantle,
and the attached lithosphere, acting as a stress guide, is effectively ‘pulled’ down behind the
slab. The hot ascending plume model was suggested by Morgan (197 1). Gravitational sliding
of lithosphere off oceanic rises (e.g. Hales 1969; Artyushkov 1973) might also be regarded
as resulting from a concentraion of buoyancy forces at the surface (although, of course,
other buoyancy is required to elevate the rise). These examples of the ‘concentrated
buoyancy’ type have the property that most of the fluid can be treated as a neutrally
buoyant, viscous fluid dragged by the active portion. Since Elsasser’s model, in particular,
seems to explain a number of features of plate tectonics, it will be considered in the models
discussed below, along with the conventional ‘distributed buoyancy’ model.
Distributed buoyancy models are considered first. Analysis of convective flow patterns
in some simple two-layer fluid models described here shows that viscosity contrasts must
be extreme in order to make one of the layers immobile. These models were motivated by
a paper by Takeuchi & Sakata (1970) which has received little notice. They analysed the
stability of a two-layer fluid heated from below in which the lower layer occupied 90 per
cent of the total thickness and was 1000 times more viscous. The flow pattern they found
is reproduced in Fig. 1. Larger horizontal velocities occur in the thin upper layer, but the
‘return’ flow is entirely contained in the lower, more viscous layer: its much higher viscosity
is compensated by its greater thickness. Also, as Takeuchi & Sakata pointed out, the width
of the cell which is the least stable to the adverse temperature gradient is still about the same
as the total depth of the fluid. This result, and others described below, are sufficient to cast
doubt on the belief that the lower mantle is not involved in plate dynamics, even without
the evidence for a moderate lower-mantle viscosity.
Layered fluid models, with implicit ‘concentrated’ buoyancy, are also used to elucidate
the important effect on flow patterns produced by moving upper boundaries. The ‘Pacific’
model of Richter (1973), in which the fluid is heated from below and the upper boundary is
assumed to be moving, shows that the viscous flow entrained by the upper plate overwhelms
the convective flow pattern which would occur with a stationary upper boundary. Some
simple solutions are presented here, with no heating of the fluid, for the flow in a layered
viscous fluid with a moving boundary.
1 Geophysical evidence concerning lower-mantle mobility
1.1 C H E M I C A L COMPOSfTION D I F F E R E N C E S
Probably the most potent obstacle to whole-mantle convection would be a chemical composition gradient involving a change in mean atomic weight (Richter & Johnson 1974).
The possibility of chemical gradients in the mantle has been discussed since the first detailed
density models of the mantle were developed (e.g. Bullen 1940, 1963). Density and the
elastic velocities were found to increase unusually rapidly with depth in the ‘transition zone’,
between about 400 and 900 km depth. Birch (1952,1961) argued strongly that most of the
unusual behaviour must be due to pressure-induced phase changes in the mantle material,
and that the velocity-density correlation in the mantle was consistent with a mean atomic
weight which did not deviate much from a value of 21. Shock-wave and static compression
studies have confirmed the existence of such phase transitions in silicates and related compounds (e.g. McQueen, Marsh & Fritz 1967; Ringwood 1975). Since then there has been a
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lively discussion about the possibility that the mean atomic weight increases by about one
unit downwards through the transition zone (e.g. Anderson 1968; Anderson & Jordan 1970;
Press 1972; Wang & Simmons 1972; Ringwood 1975). The argument in favour was essentially that the ratio of the elastic velocity increase to density increase through the transition
zone is less than would be expected if the mean atomic weight is constant.
Recent work indicates that this behaviour can be explained without invoking a change
in mean atomic weight. New measurements of the elasticity of high-pressure polymorphs
(Liebermann & Ringwood 1973) and new calculations of the properties of mixtures of dense
oxides (Davies 1974a) showed that some phase transitions deviate considerably from the
empirical velocity-density trend for constant mean atomic weight established by Birch
(1961). In particular, phase transitions involving increases in the coordination of silicon
atoms (or the corresponding atoms in related compounds) from 4 to 6 follow velocitydensity trends with characteristically low slopes (Davies 1974a, 1976; Liebermann 1974).
This is just the type of phase transition believed to occur in the lower part of the transition zone of the mantle (Ringwood 1975).
Mao (1974) proposed an alternative to Birch’s (1961) velocity -density correlation which
incorporates Poisson’s ratio. The correlation is at least as good as Birch’s, and it has the
interesting property of not requiring an increase in mean atomic weight in the transition
zone (Ma0 1974; Watt, Shankland & Mao 1975).
Comparison of the properties of the lower mantle with equations of state of some
relevant materials at high pressures and temperatures (Davies 1974b), and comparisons of
the properties of the lower mantle extrapolated to low pressure and temperature with
measurements of relevant materials (Davies & Dziewonski 1975) do not require the mean
atomic weight of the lower mantle to be significantly different from that of the upper
mantle.
To summarize, presently available information does not require the mantle to be
chemically non-uniform, although this possibility cannot be completely ruled out. Henceforth, this paper explores the consequences of the assumption that the mantle is chemically
uniform.
1.2
P H A S E TRANSITIONS
The effect of the mantle phase transitions on convection in the mantle was the subject of
controversy for some time (Vening-Meinesz 1962; Knopoff 1964; Verhoogen 1965;
Ringwood 1972, 1975), but more recent, more detailed calculations have led to more of a
consensus. Two approaches to the problem have, been taken. The first is to determine the
conditions for the marginal stability of a fluid layer heated from below (Schubert, Turcotte
& Oxburgh 1970; Busse & Schubert 1971; Schubert & Turcotte 1971; Schubert, Yuen &
Turcotte 1975). The second is to calculate the effects for finite velocity flow. The latter has
been done for descending lithospheric slabs (Hasebe, Fujii & Uyeda 1970; Turcotte &
Schubert 1971, 1973; Toksoz et al. 1973; Schubert er al. 1975), and for a ‘distributed
buoyancy’ model (Richter 1973).
The most thorough analysis of the effect of phase transitions on the conditions for
marginal stability is that by Schubert et al. (1975). They considered both univariant and
divariant transitions, and found that the transition could enhance or retard the instability of
a layer with initially constant temperature gradient, depending on the relative effects of the
latent heat of the transition and the vertical displacement of the phase boundary by the
perturbed temperature field. The parameters for the phase transition used by Schubert
et al. in reaching this cqnclusion were those measured or estimated for the olivine-spinel
Whole-mantle convection and plate tectonics
4(1 3
transition and the spinel--mixed oxides transition. These are probably not the transitions
actually occurring in the mantle (e.g. Liu 1975; Ringwood 1975), and some of the paiameter values may be revised (Jackson, Liebermann & Ringwood 1974), but their values are
probably quite representative. The most uncertain of the quantities used by Schubert er al.
was the kinematic viscosity, which they took t o be in the range lo2*-1026cni2/s.It is likely
that values in the lower end of this range are appropriate for the mantle (Cathles 1975; see
discussion below), in which case the results of Schubert et al. indicate that the combined
effects of latent heat and thermal expansion dominate those of the phase boundary displacement. If this were true, then a transition like the olivineespinel transition (which has a
positive phase boundary slope, d P / d T ) would enhance the stability of the layer. The results
of Schubert et al. indicate that for a kinematic viscosity of 1OZ2cm2/s,the critical (initially
constant) super-adiabatic temperature gradient necessary for convection through the phase
transition zone would be several times that necessary for separate convection cells above and
below the transition zone. Thus the marginal stability analysis suggests that phase transitions
may significantly affect the flow pattern of convection.
The finite amplitude calculations of Richter (1973) give results in good agreement with
those of Schubert et al. (1975): in the parameter range where the stability analysis indicates
that the olivine spinel transition is destabilizing, Richter finds that the convective velocities
are about twice those which would occur without the phase transition. Unfortunately.
Richter does not give any results for the parameter range where the olivine- spinel transition
would be stabilizing. He does give results for the spinel --oxides transition with parameters
which would be mildly stabilizing, according t o the stability analysis, and finds that the flow
structure is hardly affected and the convective velocities are lower than if the transition were
not present. These results apparently shed n o light on the question of what the flow
structure is at finite amplitudes when, according t o the marginal stability analysis, separate
convection cells above and below the transition zone is a less stable mode than convection
through the transition zone.
Calculations of the effects of an olivine-spinel-type transition on the descending lithospheric slabs all seern t o agree that the presence of the phase change increases the downward
buoyancy force on the slab - the upward displacement of the phase boundary dominates
the combined effects of latent heat release and thermal expansion ( e g Schubert & Turcotte
1971 ; Turcotte & Schubert 1971 ; Toksoz, Minear & Julian 1971; Toksoz, Sleep & Smith
1973; Griggs 1972; Schubert et al. 1975). The variation of the buoyancy force down a slab
has been calculated explicitly by Schubert et al. (1975), on the basis of the thermal regime
of the slab calculated by Turcotte & Schubert (1973), and their results show that the
upward displacement of the divine-spinel phase boundary gives a major contribution t o
the total downward buoyancy force.
Although the calculations reviewed above lead to apparently conflicting conclusions, the
following reasoning suggests that phase transitions will not greatly affect flow patterns in
the mantle. The effect of latent heat and associated thermal expansion is quite important if
the initial temperature gradient is constant. Material descending through an olivineespineltype transition zone will release latent heat and its temperature will rise as a result. The
effect of the latent heat release is t o increase the adiabatic temperature gradient in the transition zone t o a value corresponding t o the Clapeyron slope, dT/dP, of the phase boundary
(e.g. Ringwood 1972, 1975). If the initial temperature gradient is constant and equal t o the
adiabatic gradient outside the transition zone, then material descending through the transition zone will end up hotter than the surrounding material, and the motion will be
inhibited. If, on the other hand, the initial temperature gradient is everywhere equal to the
local adiabatic gradient, then the material will be everywhere neutrally buoyant -- the phase
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I+: Davic~s
transition will have no effect on the motion. Now consider an initial temperature gradient
which is greater than the critical gradients necessary for both ‘double cell’ convection (i.e.
separate cells above and below the transition zone) and ‘single cell’ convection (through the
transition zone), with the double-cell critical gradient less than that of the single cell. Then,
presumably, the iess stable double-cell mode will grow more rapidly and be initially
dominant. Thermal boundary layers, with locally steeper temperature gradients, will form
at the top and bottom of the tluid, and at the transition zone, which is between the upper
and lower cells. The temperature gradient in the transition zone will thus increase towards
the local adiabatic gradient, and the instability of the single-cell mode will be progressively
enhanced. Only a detailed calculation will determine the final balance between the modes,
but this argument shows that th,e stability analysis of Schubert et d.(197.5) probably overestimates the importance of the stabilizing effect of latent heat.
A locally steeper, approximately adiabatic temperature gradient in a transition zone will
tend to be maintained by fluid motions, and reduced by thermal conduction. Turcotte &
Schubert (1971) have considered these competing effects, and their results show that motion
through a transition zone will dominate if the vertical velocity is of the order of 1 nim/yr or
greater (for parameter values appropriate to the mantle). Thus a transition zone may act as
a cutoff, preventing flow through the zone with vertical velocities less than a critical value,
but offering little hindrance if the vertical velocity is greater than the critical value. The
effect of this kind of behaviour would be to narrow the zones of upwelling and downwelling through the transition zone. Whether it is an important effect in the mantle depends
on the relative importance of several other effects (see later discussion).
To conclude this section, it seems that phase transitions will not prevent whole-mantle
convection, although they may modify the flow pattern somewhat by preventing some flow
paths from penetrating the transition zone.
1.3
RHPOLOGY O F THE MANTLE
Speculations on and models of mantle flow associated with plate tectonics have been dominated by the concepts of an elastic, high-viscosity lithosphere, a relatively low-viscosity
asthenosphere, and a high-viscosity mesosphere. The concepts of an elastic lithosphere and
an asthenosphere, or low-viscosity channel, have been relatively non-controversial (e.g. Daly
1934; Heiskanen & Vening Meinesz 1958; Crittenden 1963; McConnell 1968; Walcott
1970a; Van Bemmelen & Berlage 1935; Haskell 1937; Artyushkov 1971; Parsons 1972;
Cathles 197.5). Cathles has reviewed these and other studies which inferred mechanical
properties of the mantle from geological evidence of uplift following the removal of lakes
and ice sheets from the Earth’s surface, and has done some more detailed calculations.
Cathles concludes that there is reasonably strong evidence for a low-viscosity channel
(0.4-1 .O x lo2’P) of the order of 100 km thick underlying an elastic lithosphere of the
order of 100 km thick. The evidence for these features includes the uplift of the basin
formerly occupied by Lake Bonneville (Crittenden 1963), post-glacial uplifts in Greenland
and the Canadian Arctic, and the behaviour of peripheral bulges of the Fennoscandian and
Canadian ice sheets.
The asthenosphere has been variously defined. In its original geological connotation, it
referred to the region where surface loads are compensated. It has also been identified with
the seismic low-velocity zone of the upper mantle, with the whole upper mantle (down to
about 400km depth) and with the upper mantle and transition zone (down to 7001000 km depth). Mantle viscosity below the low-viscosity channel discussed above can be
inferred from the post-glacial uplift of Fennoscandia (see Cathles 1975, for review). The
Wholc-mantle convection and platci tectonics
465
Fennoscandian uplift can be modelled by either a uniform upper-mantle viscosity of
P,
or by a more substantial low-viscosity channel, 100 kxn thick and 1.3 x 1020P (Cathles
1975). For consistency with North American post-glacial uplift and peripheral bulge data,
Cathles prefers the former. The viscosity of the remainder of the upper mantle is thus
probably substantially higher than that of the low-viscosity channel, but not substantially
less than that of the lower mantle (see below). If the term ‘asthenosphere’ is to be useful.
it should probably refer to the low-viscosity channel.
Artyushkov (1973) suggests that the low-viscosity channel may be much more strongly
developed under oceans than under continents, because of the higher inferred temperatures
in the oceanic upper mantle. He estimates one or two orders of magnitude differential in
viscosity, assuming 1 OO---200°C temperature differential. Based on a viscosity of about
1OZoP under continents (see above), he thus estimates a viscosity of 10’8--IO’9P in a
(presumably) 100 km thick oceanic low-viscosity layer. Observational constraints on the
viscosity of the oceanic upper mantle seem to be few. Loads from oceanic volcanoes
(Walcott 1970b) and lithospheric subduction at trenches (Hanks 1971: Watts & Talwani
1974) are too small scale, and probe only the lithosphere. Loads from sea-levcl changes are
too large scale, and average to great depth in the mantle (Cathles 1975). Thus a substantial
low-viscosity layer under oceans cannot be discounted, especially since the seismic lowvelocity zone is well developed under oceans (e.g. Forsyth 1975).
The viscosity of the lower mantle can be inferred from North American post-glacial uplift
data, from the accompanying adjustment of the ocean basins to the increased load of the
oceans, as reflected in world-wide sea-level changes, and from the changing rate of rotation
of the Earth inferred from ancient records of eclipses (Dicke 1969; O’Connell 1971 ; Cathles
1975). These have all been modelled by a mantle with a uniform viscosity of about lOz2P.
This value is substantially lower than many previous estimates. The belief in a higher
viscosity lower mantle arose partly from a preference for low-viscosity channel models of
the Fennoscandian uplift, discussed above. It received support from the belief that the Earth
has a large non-hydrostatic equatorial bulge which was interpreted as being a ‘fossil’ bulge
from a time when the Earth was spinning faster (Munk & MacDonald 1960). This would
require most of the mantle to have a viscosity of at least loz6P . It also received support from
estimates of the pressure and temperature dependence of deformation mechanism i n
silicates (Gordon 1967; McKenzie 1967), but the latter estimates are subject to great
uncertainties (e.g. Weertman 1970; O’Connell 1977). In particular, O’Connell (1977) has
shown that a reasonable estimate of the effect of pressure on activation volume can yield
an estimated viscosity which is constant or decreases with depth after temperature effects
are included. The ‘equatorial%ulge’ has been shown to be partly an artifice of the spherical
harmonic analysis of the Earth’s shape, which magnifies the amplitude of the
component, and partly a misinterpretation of the Earth’s shape, which is more nearly triaxial
in its deviations from hydrostatic shape (Goldreich & Toomre 1969).
To summarize, recent studies, notably that of Cathles (1975), indicate that the viscosity
of the mantle is probably fairly uniform and about 1022P,except for a low-viscosity channel
in the uppermost mantle with a viscosity of about 0.04 x 102’P. The low-viscosity channel
may be more or less pronounced in some regions, especially under oceans. and may be
closely related to the seismic low-velocity zone.
Cathles (1975) points out that a high-viscosity layer a t depth. a non-Newtonian flow law
(in which effective viscosity decreases with increasing stress) and sub-adiabatic density
gradients would all tend to restrict flow to nearer the surface. By concluding that deep flow
is required to explain the relevant observations, all three of these possibilities is excluded (or
their probability is reduced), and an adiabatic, uniform Newtonian viscosity mantle is justi-
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tied a posteriori. This is positive evidence in support of the possibility, discussed earlier, that
the temperature profile in the mantle is adiabatic, and, in particular, that phase changes in
the mantle transition zone are not a barrier to flow in the mantle.
1.4
SEISMOLOGICAL EVIDENCE
TWOcharacteristics of deep earthquakes have been widely held to support the idea that the
lower mantle is not involved in plate dynamics. These are the sharp drop to zero in the
frequency of earthquakes at about 700 km depth and the preponderance of compressive
stresses down the dip of descending lithospheric slabs between depths of about 300 and
700 km, in contrast to the preponderance of tensile stresses above 300 km depth (e.g.
Isacks, Oliver & Sykes 1968; Isacks & Molnar 1969, 1971).
The maximum depth of earthquakes (700 km) coincides approximately with the depth
of one of the major seismic transition zones (650-700 km; e.g. Johnson 1967). Hypotheses
which have been proposed to explain the depth range of earthquakes include (e.g. Isacks
et al. 1968): (1) the lower mantle mechanically resists penetration by the descending slabs,
(2) earthquakes cease when the slabs are heated by the surrounding mantle to temperatures
above their range of brittle behaviour, the time constant for heating being of the order of
10 Myr, ( 3 ) the current episode of plate motions has been in progress for only about the
past 10 Myr. Hypothesis ( 3 ) is not supported by subsequent determinations of the history
of plate motions (e.g. Pitman & Talwani 1972; Maxwell et al. 1970). Hypothesis (2) is
supported by the observed correlation between down-dip length of slab and rate of subduction (Isacks et al. 1968) and by calculations of the thermal structure of the slabs
(McKenzie 1969; Griggs 1972). Hypothesis (1) receives some additional support from the
contortions observed in some of the seismic Benioff zones at depths below 300 km, and in
the distribution of stresses, discussed further below (Isacks & Molnar 1971).
Phase changes within the slabs may also affect the distribution of earthquakes, in several
ways. The volume reduction associated with a phase change may directly cause the earthquake, or, more likely, trigger the release of stresses arising from other sources. The change
in crystal structure may be associated with an increase in mechanical resistance (e.g.
McKenzie & Weiss 1975). The phase transition will release or absorb latent heat, and the
resulting change in temperature will affect the mechanical properties. The association of
earthquakes with phase changes may explain the relative increase in the frequency of earthquakes below 300 km depth (Isacks er al. 1968). If the phase transition believed to be
associated with the seismic discontinuity near 650 km depth released latent heat, the accompanying increase in temperature might remove the slab material from its range of brittle
behaviour and thus account for the relatively sudden cessation of seismicity near 700 km
depth. On the other hand, the change in crystal structure might extend the range of brittle
behaviour. The nature and properties of the phase transition near 650 km depth are
currently quite uncertain. Anderson (1967) proposed that it is the transition from the spinel
structure to a mixture of simple oxides, or similar phase, and Ahrens & Syono (1967) estimated that the spinel-oxides transition would absorb latent heat. Jackson ef af. (1974)
re-examined the evidence and concluded that the latent heat released by the reaction might
be zero or slightly positive. More recent evidence (Liu 1975; Anderson 1976) suggests that
the transition may be to a perovskite-like structure, the thermodynamic properties of which
are not known.
The change from tensile to compressive stresses down the slabs (hacks & Molnar 1969,
1971) does not necessarily imply that the slabs cannot penetrate below a depth of 700 km.
All that is required to explain this stress distribution is that the net downward force per unit
Whole-mantle convection and plate tectonics
467
length of the slab reach a maximum between the zones of tension and compression. The
decreasing net downward force associated with the zones of compression might arise from a
decreasing negative buoyancy, or an increasing resistance from the surrounding mantle, or
both. The negative buoyancy calculated by Schubert et al. (1975) in their descending slab
model, including the effects of phase boundary displacements in the slab, has exactly the
kind of distribution to explain the observed stress distribution.
The Tonga slab, and possibly the Honshu slab, seems to be in compression at all depths
(Isacks & Molnar 1971), which implies that the maximum net downward force on the slab
occurs near the surface. The origin of such a force distribution is puzzling, but. again, there
is no necessity to assume a barrier to the slabs near 700 km depth. W e t h e r the decreasing
downward net force actually reverses sign and becomes an upward force is, of course, an
important question to consider, but it is not required by the observed stress distribution.
As mentioned above, the observed contortions of the deep seismic zones might support this
possibility.
At present, it seems that our understanding of deep earthquakes is sufficiently uncertain
that neither of hypotheses (1) and (2), above, can be confidently excluded.
Recent seismic evidence suggests more directly that the lower mantle is not static, and
that at least some phenomena are directly related to plate dynamics. Several studies have
produced evidence for localized seismic velocity anomalies beneath seismic Benioff zones
(Jordan & Lynn 1974; Jordan 1975; Engdahl 1975; Powell 1976). The anomalies have
higher seismic velocities, and range in depth from 700 to 1400 km or more. They have been
interpreted as the result of the cooling effect of the descending slab persisting to these
depths (Jordan 1975). This would be consistent with the thermal structure of the descending
slab calculated by Schubert et at. (1975). Other studies have indicated the presence of very
large-scale seismic velocity anomalies throughout the mantle, the largest amplitude anomalies
being in the upper mantle and near the core-mantle boundary (Niazi 1973; Julian &
Sengupta 1973; Wright & Lyons 1975; Dziewonski, Hager & O’Connell 1977). These largescale anomalies correlate with long-wavelength components of the gravity field (Dziewonski
et al. 1977), which suggests that they are directly associated with motion of the mantle
(O’Connell 1977). Finally, a recent series of studies has convincingly reinterpreted seismic
precursors to core phases, which were previously interpreted as arising from a transition zone
between the inner and outer core, as arising from seismic waves scattered in the lowermost
mantle (eg. Haddon 1972; Cleary & Haddon 1972; Husebye, King & Haddon 1976). The
scattering requires small-scale (1 0 km) random heterogeneities in seismic velocities, with
variations of the order of 1 per cent in velocities, in the lower few hundred lulometres of the
mantle (Husebye et al. 1976). ’
To conclude thls section, the distribution of seismicity and stress in descending lithospheric slabs does not clearly distinguish whether or not the slabs penetrate below 700 km
depth. Other seismic evidence suggests that they do, and that the whole mantle may be in
motion, with the most pronounced effects on seismic velocities being concentrated in the
uppermost and lowermost few hundred kilometres of the mantle.
2 Flow patterns with layered viscosity and moving boundaries
2.1
FORMULATION
The preceding review attempts to demonstrate that whole-mantle convection should at least
be considered as a viable alternative. Nevertheless, allowing for continuing uncertainty and
scepticism, it seems worthwhile to further explore the conditions required to exclude fluid
G. 1:. Davies
468
motions from the lower mantle, or, as recently suggested (McKenzie & Weiss 1975), to
separate lower mantle flow from the upper mantle. The viscosity of the mantle has been the
most uncertain and contentious parameter relevant t o mantle flow, and the models discussed
here explore flow patterns in fluids with layers of contrasting viscosity. The physical
rationalization for considering layered viscosity is that it is a way of representing the effects
of physical changes in mantle material produced by phase changes or changing pressure or
temperature. The layers are supposed not t o be chemically distinct: thus fluid can pass
through a boundary into another layer, but the boundary is not carried along with the fluid.
In the Introduction it was suggested that rnoving upper boundaries (i.e. plate motions)
might strongly affect mantle flow patterns. Richter’s (1973) calculations of fluid flow with
moving boundaries assume dimensions appropriate for upper mantle convection and include
the effects of heating from below. He notes the correspondence between his calculated flow
patterns and those calculated by Pan & Acrivos (1967) for a viscous fluid in a twodimensional box with a moving boundary, but i t is difficult to extrapolate from these results
t o other situations. Solutions are presented below which show the effect of a moving
boundary on a viscous fluid half-space, with n o heating. The effects of viscosity layering can
be easily explored with this model.
Although, ultimately. the effects of heating, moving boundaries and viscous layering may
have to be simultaneously considered. considerable insight can be obtained by considering
these factors in simpler combinations, since solutims are then much more easily obtainable.
The models discussed below, then. are ( a ) a layered viscous lluid heated from below, with
stress-free upper and lower boundaries, and (b) a layered viscous half-space with no heating,
and with a moving boundary.
The object of these calculations is t o explore the gross properties of flow patterns. Thus.
extreme simplifications will be assunied. Two-dimensional flow in rectangular geometry will
be considered. The fluid is assumed t o be linear-viscous (Newtonian), incompressible and
Boussinesy (the effect of temperature on density is included only in the gravitational body
force). The non-heated flow is assumed t o be steady-state and ‘slow’. Only the stability of
the heated fluid layer will be considered, i.e. only infinitesimal perturbations from the static
state: steady state, finite-amplitude motions are not calculated. Perhaps the most important
factor neglected is the extreme temperature dependence of the viscosity expected in the
mantle (e.g. McKenzie 1967; Weertman 1970; O’Connell 1976). This factor considerably
distorts convective flow patterns, causing narrow rising currents and broad sinking currents
(Torrance & Turcotte 1971), but probably not enough t o invalidate the main conclusions
reached here.
\
The relevance of these approximations to mantle flow, and the derivation of the
equations governing the flow, have been discussed elsewhere (Chandrasekhar 1961 ;
McKenzie 1968, 1969; Turcotte & Oxburgh 1972; Richter 1973; Peltier 1973; McKenzie
er al. 1974). A stream function
can be defined, such that the fluid velocity, u, is
~
u=
vx $,
(1)
and where
V.$=O.
Components of the stress tensor, P, are
where
P is the hydrostatic stress (pressure), tjjj is the Kronecker delta, v is the viscosity and
Whole-mantleconvection and plate tectonics
469
x is a position vector. In the gravitational body force, the density, p , is assumed to depend
on temperature, T , according to
~ = ~ o [ l - - a (7'011
T
(4)
where (Y is the volume coefficient of thermal expansion and po is the density at a reference
temperature To.
Foi the fluid layer heated from below, combining these equations with equations
expressing conservation of mass, niomentum and energy in the present approximation yields
the following equation for the marginally stable infinitesimal perturbation flow (in dimensionless quantities)
Ilere, V 2 is the Laplacian operator, V4J, = V2(V2jl), etc. I?, the Kayleigh number, is the
dimensionless combination
g is the acceleration due to gravity, is the average vertical temperature gradient across the
fluid layer, M is the thickness of the fluid layer, K is the thermal diffusivity of the fluid
(K = k/pC,, where k is the thermal conductivity), and i and j are unit vectors in the x and J'
directions. With appropriate boundary conditions, equation ( 5 ) specifies an eigenvalue
problem. Solutions exist a t specific values of R , and they describe the modes by which the
fluid becomes convectively unstable (Chandrasekhar 196I ).
If there is n o heating, so that the temperature is constant in the present approximation,
then R = 0 and the energy equation decouples. leaving the fourth-order equation
V4$ = 0 .
(7)
This is the so-called biharmonic equation.
Now let the spatial coordinates be denoted by ( x , y . z ) , with z measured vertically
upwards. Assuming u , =~0 and that all variables are independent of y specifies that the
motion is two-dimensional. In that case, $x and & d o not enter the y-components o f (5)
and (71, and only GY need b e considered. Henceforth, $ y will be denoted as $.
The dependence of )I on the spatial variables can be separated, and the horizontal (x)
dependence taken to b e harmonic in b o t h ( 5 ) and (7). Thus, 3, can be written
Icl(x, 2 ) = Hz) e x p (dx),
(8)
where A is the horizontal wa?e number; A = 2n/L, where L is the horizontal wavelength.
Substitution of (8) into (5) and (7) yields
(D2
(9 1
= -RA2f,
and
( 0 2 - AZ)2{
= 0,
(10)
where D = d/dz. Solutions to (9) and (10) will now be discussed separately.
2.2
CONVECTIVE FLOW WITH L A Y E R E D VISCOSITY-STABILITY
ANALYSIS
Chandrasekhar (1961) has discussed the derivation of equation (9) and its solution for a
uniform fluid layer. The solution corresponding t o the smallest eigenvalue, R , , is the mode
by which the fluid first becomes unstable as the temperature gradient, p (and hence R ) is
increased from zero.
G. F. Davies
470
I
J
~
Figure 1. Flow pattern obtained by Takeuchi & Sakata (1970) for the first unstable convective mode in a
two-layered viscous fluid heated from below. Upper layer depth is 1/10 of total depth, upper layer
viscosity is 1/1000 of lower layer viscosity. Upper boundary is free-slip, lower boundary is no-slip.
The stability analysis of Takeuchi & Sakata (1970; see Fig. 1) was for a two-layer fluid
with v = vo from z = 0 to 0.9 and v = 10-3v0 from z = 0.9 to 1 .O. They assumed a free-slip
(zero shear-stress) upper boundary and a no-slip (zero velocity) lower boundary, and that
heating was confined to the lower boundary. Peltier (1973) did a similar calculation, but
with a no-slip upper boundary and heating distributed uniformly through the fluid. His
results are quite similar to those of Takeuchi & Sakata (Fig. 1).
In this study, both boundaries are assumed to be free-slip. For the lower boundary, this
choice is obvious, since the boundary is to be identified with the core--mantle boundary
in the Earth, and the core is liquid with a viscosity many orders of magnitude less than that
of the mantle (eg. Cans 1972). A free upper boundary amounts to assuming that a lithospheric plate would be free to move with the underlying mantle. The choice of upper
boundary condition is not critical (cf: Chandrasekhar 1961) - a free boundary simplifies
the calculation, and in fact allows the flow to be shallower in the layer, so it is the conservative choice in the present context. Thus the boundary conditions applied to equation (9)
are as follows (see, e.g. Takeuchi & Sakata 1970). At the top and bottom boundaries, the
vertical velocity, shear stress and temperature perturbation are zero
z=O,l:
f=O,
( D 2 + A 2 ) {= 0 ,
(D2-k)Z
{ = 0.
At the interface between layers, z = H , say, the vertical and horizontal velocities, shear and
normal stresses, temperature and heat flow should all be continuous
z=H:
c>
DC>
v(DZ+A2)5',
v(D2-- 3 A 2 )DC,
(0' - A 2 ) 2{,
(D2- A 2 ) 2 DC, all continuous.
Whole-mantleconvection and plate tectonics
47 1
Takeuchi & Sakata (1970) and Peltier (1973) evidently solved this problem numerically.
Itowever, Chandrasekhar (1961) describes the general analytic solution to (9) for a uniform
layer, consisting of six terms of the form exp(qz), where 4 is complex. The six values of
4 are
where
The general solution for a uniform layer can thus be written
{ = a l sinqoz + a 2 sinhq,z cosq,z +a3 coshqlz siny2z +a4 cosqoz + a s cosliylz C O S Q ~ Z
(15)
+a6 sinhylz sinq2z.
The boundary conditions at z = 0 can be satisfied by taking only the odd terms in (15), i.e.
u4 = a5 = u6 = 0. Similarly, the boundary conditions at z = 1 can be satisfied by an analogous
function which is odd in the variable (1 z). The six boundary conditions (12) at z = H then
yield six equations for the remaining six parameters. These equations have a non-trivial
solution only if the matrix of coefficients is singular. The values of R for which the deter~
minant of coefficients is zero are the eigenvalues of the problem.
This problem was solved for a series of values of viscosity in the upper layer, and with
H = 0.8. The latter choice corresponds to a depth in the mantle of about 600 kni (total
mantle depth is about 2900 km), i.e. near the top of the lower mantle. The problem solved
by Takeuchi & Sakata was also solved by a method analogous to that described above to
check the consistency of the calculations, with positive results.
As defined in equation ( 6 ) , R is actually variable through the fluid, so it is useful to
define reference values, which can be taken t o be either of
i.e. R' is defined in terms of the viscosity, v ' , in the lower layer, and RU in terms of v',
the viscosity in the upper layer.
For a given value of horizontal wave number, A , there is a series of eigenvalues Rf,,
corresponding t o different numbers of nodes in the function c(z). The smallest eigenvalue,
Rb, corresponds t o the mode (with no nodes) which first becomes unstable as R' is increased.
Rb varies with A , and has one minimum value, Rh, for a particular value, A , , o f A . A , specifies the horizontal wavelength of the least stable mode of motion of the fluid, and RE (the
critical Rayleigh number) the physical conditions necessary to destabilize the fluid, i.e. to
initiate convection. (See Chandrasekhar 1961, for a more detailed discussion of these
considerations.)
The values of RL, R,U and A , found for a series of values of vU/v' are given in Table 1.
The corresponding solutions for {(z) are illustrated in Fig. 2. The streamlines of the flow are
lines of constant 4 . These are illustrated in Figs 3 and 4 for the cases vU/v' =
and lo-'.
472
G. F. Davies
Table 1. Critical parameters for stability of a two-layer fluid of
depth H : upper/lower viscosity ratio, u'/d, horizontal wavenumber,
A , , and wavelength, L,, Rayleigh numbers, RL and RY, referred to
lower and upper viscosities, respectively.
uyu'
A,
R:
R:
L,IH
1
0.5
lo-'
1w 2
10-3
2.22
2.3
2.3
1.8
1.3
1.3
12
13.4
657
537
355
215
169
153
5.96
65 7
1073
3550
2.15X l o 4
1.69X 10'
1.53XlO'
5.96 X 10'
6.88 X 10'
2.83
2.6
2.6
3.0
4.8
4.8
0.52
0.47
10-3
0
1.0
0.0
0
0.2
0.4
06
0
Z
Figure 2. Vertical variation of stream function for the first unstable convective mode in a two-layered
viscous fluid heated from below. Vertical velocity is proportional to <, horizontal velocity to d</dz.
Layers meet at elevation z = 0.8. Upper and lower boundaries are free-slip. Curves are labelled by the ratio
of upper to lower layer viscosities, uu/u'.
These results show, as inight be expected, that the flow is progressively displaced into the
less viscous layer as the viscosity contrast is increased, but it is remarkable that fully five
orders of magnitude contrast is required t o exclude the flow from the more viscous layer.
If the upper layer were thinner, this contrast would have t o be even greater.
The normalized horizontal wavelength, L,/H (Table 1 , Figs 3 and 4). beconies quite
large for viscosity contrasts of
before {ecreasing to about l / S of its initial value.
The latter value is to be expected, since in the limiting case the flow is confined t o l / S of the
total layer thickness. The liiniting case should, in fact, correspond to a fluid with one free
Figure 3. Flow pattern for the case Y"/u' =
of
2.
Whole-mantle convection and plate tectonics
Figure 4. Flow pattern for the case
U"/V'
473
= l o - <of I;ig. 2.
and one rigid boundary, for which the critical parameters (scaled t o the appropriate layer
thickness) are R , = 1101, A , = 2.68 (Chandrasekhar 1961). These values, scaled to the
present problem, are given in the last line of Table 1. It can be seen frotn Table 1 that the
critical Rayleigh number and horizontal wavelength obtained in these solutions d o indeed
approach the limiting case.
The larger horizontal wavelength obtained for intermediate viscosity contrasts (Fig. 3,
Table 1) evidently is possible because dissipation in the higher viscosity layer is reduced by
increasing the scale of the flow, and the resulting distortion of the flow can be acconimodated by the less viscous layer. The persistence of this behaviour t o viscosity contrasts
is the most remarkable feature of the solutions obtained by Takeuchi &
of the order of
Sakata (1970), Peltier (1973) and in the present work.
2.3
I:LOW
WlT'lI M O V I N G HOtlN1)AKIISS
Solutions for slow viscous flow, as described by equation (7),
will be discussed here for a
fluid halfspace (z G 0). If the conditions o n the boundary of the halfspace are harmonic in
the x-direction, then the solu\tion can be written in the form (8), and (7) reduces t o (10).
The horizontal velocity at the boundary will be taken to be harmonic in x. Thus, the
appropriate boundary conditions are
z=o:
u, = 0, u, = U exp ( i A x ) ,
i.e. ( = 0, D{ = U :
z = -00:
u, = u, = 0
i.e. ( = D< = 0.
At the interface between layers of different viscosity, the horizontal and vertical velocity
and the shear and normal stresses should be continuous. These reduce to the first four of the
boundary conditions (12) of the last section.
The general solution t o (10) is
( = b lexp(Az)+b2exp(-Az)+b3z exp(Az)+b4z exp(-Az)
16
474
G. F. Davics
For a halfspace of uniform viscosity, the solution satisfying conditions (1 7) and (18) is just
j- = Uz exp ( A z ) .
(20)
This function is illustrated in Fig. 5 (with U = I ) , as the curve labelled 1 , and the corresponding flow pattern is illustrated in Fig. 6. A notable, and not very surprising, feature of
the flow pattern is that it penetrates to a depth comparable to the horizontal scale of the
imposed boundary velocity. The stagnation point of the flow is at z = L/27-r, i.e. at a depth
0 4,
I
I
I
I
I
I
I
I
Z/L
Figure 5. Vertical variation of stream function for flow in a two-layer viscous half-space with an applied
horizontal velocity o n the surface with harmonic horizontal variation. Layers meet at depth z = 0.2L,
where I, is horizontal wavelength. Curves are labelled with ratio, u l / u u , of lower layer to upper layer
viscosities.
~
Figure 6. Flow pattern for the case ul/uu = 1 of Fig. 5 , i.e. a uniform half-space with an applied horizontally harmonic horizontal surface velocity.
475
Whole-mantle cotzvection and plate tectonics
of about 1/3 of the half-wavelength. The results of Pan & Acrivos (1967) show a similar
effect
in a rectangular box with one shorter side moving, a cell of about unit aspect ratio
is induced next to the moving boundary. Thus, since lithospheric plates have horizontal
dimensions of the order of 5000 km, we should expect a moving plate to induce flow
throughout the whole depth of the mantle (3000 km), if the mantle is effectively uniform.
If the mantle is not uniform, then the flow pattern will obviously be modified. Solutions
with two layers of different viscosity, with an interface at z = H = --0.2L, are also illustrated
in Fig. 5. The viscosity of the lower layer must be about 100 times that of the upper layer in
order to substantially exclude the flow. Two-layer solutions with H = --0.11, are illustrated
in Fig. 7: the viscosity contrast must be about 1000 to confine the flow to the upper layer.
The flow pattern for the latter case is illustrated in Fig. 8. If the upper layer in the latter case
~
0 I5
0 IC
-5
0 05
p
\\
I
1
I
C
I
-0 I
Figure 7. As in Fig. 5 , b u t with layers meeting at z
5
=
.
0.1L
Figure 8. Flow pattern for the case v'/vu = 1000 of Fig. 7 .
476
G. F. Davies
is identified with the upper mantle and transiton zone, say 700 km thick, the corresponding
half-wavelength is 3500 km: quite modest compared to the average lithospheric plate.
A bigger plate, or a thinner upper layer, would require a larger viscosity contrast to confine
the flow to the upper layer.
It has recently been suggested (McKenzie & Weiss 1975) that there may be a relatively
thin layer of higher viscosity at about 700 km depth in the mantle which would have the
effect of separating any flow in the upper and lower mantle. Such a layer could be caused by
a sharp increase in viscosity associated with phase transitions believed to occur in that depth
range, and a steep temperature gradient in a thermal boundary layer at the top of the lower
mantle, such that the viscosity is substantially reduced again at greater depths. McKenzie &
Weiss (1975) consider a layer thickness of about 100 km as plausible. The effect of such a
layer on flow induced by a moving boundary can be tested with a three-layer halfspace
model. Fig. 9 illustrates results with interfaces at HI = -0.2L and H 2 = --0.25L. The
viscosity of the middle layer is varied, and that of the lower layer is assumed to be the same
as for the upper layer. The flow in the lower layer is actually increased for viscosities of the
middle layer up to about 100 times that of the other layers. Thereafter it decreases, with the
flow being confined to the upper layer for a viscosity of the middle layer about lo5 times
that of the other layers. The initial behaviour may seem surprising: it apparently occurs
because the dissipation in the thin, more viscous layer is minimized if the fluid flow through
it is as near vertical as possible. Hence, the coupling between the upper and lower layers is
initially increased. The dimensions assumed in this example are actually quite conservative.
If HI corresponds to 700 km depth, then the middle layer thickness is 140 km and the horizontal half-wavelength is 1750 km.The required viscosity contrast is already very large.
0
0
-5
0
(
z/
1
Figure 9. Vertical variation of stream function for flow in a three-layer viscous half-space with an applied
horizontally harmonic horizontal surface velocity. Layers meet at z = -0.2L and z = -0.25L, where L is
horizontal wavelength. Viscosities of upper and lower layers are equal. Curves are labelled with ratio,
um/u", of middle to upper layer viscosities.
Whole-niantlr convection and plate tectonics
477
A larger horizontal scale or a thinner layer would require extreme viscosity contrasts. It thus
seems unlikely that such a laver could separate upper mantle flow from the lower mantle, or
vice versa. The results of the previous section indicate that incluhng the effects of heat
sources would only n u k e the problem worse, since even larger viscosity contrasts seem to be
required t o aff‘ect the How in that case.
A more effective way t o decouple upper nnntle flow froin the lower mantle might be
with a lower viscosity layer. Such a layer has already been suggested for the upper mantle,
roughly in the depth range 100--200km, as was discussed in the review above. Another thin
low-viscosity layer (or several) might possibly occur in the transition zone (400 700 kni
depth) because of ‘transfhrrnational superplasticity’: solids will deform much more readily
if they are undergoing a phase transformation (Sammis & Dein 1974; Dein & Sammis 1976).
Perhaps the situation suggested by McKenzie & Weiss (1975) could also produce a complementary low-viscosity layer -- a steep temperature gradient across a phase boundary could
produce lower viscosities abovc the phase boundary where temperatures might be relatively
high.
The effect of a low-viscosity layer on flow induced by a moving boundary can also be
tested with three-layer models. Results are illustrated in Fig. 1 0 for H I = - 0.2L,
H 2 = 0.25L, and in Fig. 1 I for H , = 0.02L and H 2 = 0.04L. Viscosities lower by
factors of 1 0-3 and 1 O-’, respectively, in the middle layers are required to decouple the flow
from the bottom layer. The dimensions in Fig. 10 would correspond, for instance, to a lowviscosity layer between 600 and 720 km depth in the mantle, and a half-wavelength of
1500 km. The dimensions in Fig. 1 1 would correspond, for instance, to a low-viscosity layer
between 100 and 200 kin depth and a half-wavelength of 2500 km. Again, these dimensions,
especially the horizontal dimensions, are conservative. Thinner low-viscosity layers, or bigger
plates, would require considerably greater viscosity contrasts.
It seems clear that a shallow (100 km) low-viscosity layer could not decouple the flow
entrained by lithospheric plates from the rest of the mantle and still be consistent with
-
-
0.4
0:
5
0.2
0.1
C
Z/L
Figure 10. As in Fig. 9 for the case of a less viscous middle layer. Curves ?re labelled with ratio, um/vU,
of middle to upper layer viscosities.
478
G. F. Davies
Z/L
Figure 11. As in Pig. 10, with layers meeting at z
=
--0.02L and z = -0.04L. Curves are labelled with
ratio, v"/vu, of middle to upper layer viscosities.
observed deformations of the Earth's surface. Cathles (1975), for instance, favours a lowviscosity layer about 75 km thick and with a viscosity a factor of about 25 less than the rest
of the mantle, a much smaller contrast than the factor of 104-105 required (Fig. 11).
Schubert & Turcotte (1972), using a more realistic rheology in a one-dimensional model,
concluded that unrealistic pressure gradients, topography and gravity anomalies would be
produced if the 'return flow' were confined to a shallow 'asthenosphere'. A very pronounced
low-viscosity layer under the oceans, such as suggested by Artyushkov (1973), might substantially modify the flow under oceans. However, even a viscosity as low as 10'8-1019P,
over a mantle of viscosity lOZZP,would probably not completely contain the return flow
under oceans. Also, the coupling to the deeper mantle under continents must be stronger.
Forsyth & Uyeda (1975) estimate that a decoupling of the lithosphere from the mantle
sufficient to be consistent with their model of the forces acting on plates would require a
layer between 80 and 300 km depth with a viscosity of 5 x 1019P under oceans and
1- 4 x lOZ0P under continents. They note that such viscosities have been inferred to occur
under continents, but a layer of this viscosity could not be much more than 75 km thick
without becoming inconsistent with the observations?according to Cathles (1975).
Deeper low-viscosity layers in the mantle (400-700 km depth) might be more effective
in decoupling upper mantle flow from the lower mantle, but their existence is more
speculative. A layer of sufficient thickness and/or with low enough viscosity would probably
produce observable effects on seismic wave propagation, although interpretation of velocity
and attenuation structure in the transition zone is difficult. Several reported shear velocity
profiles have included low-velocity regions (Ibrahim & Nuttli 1967; Anderson & Julian 1969;
Jordan & Anderson 1974), but the reality of these features is questionable (e.g. Helmberger
& Engen 1974).
To conclude this section, it seems that even if the mantle plays a passive role in plate
dynamics, the flow induced by moving plates is likely to penetrate deep into the mantle.
A lower mantle with a viscosity at least 1000 times more viscous than the upper mantle
would be required to exclude this flow, or a thinner layer of extremely high viscosity, or a
layer of substantially lower viscosity. The only possibility which seems to be allowed by the
present observational constraints is a moderately pronounced low-viscosity layer (or possibly
Whole-mantleconvection and plate tectonics
479
more than one), and it is doubtful that this would be sufficient to decouple the flow. It
should be noted that the role of a low-viscosity layer is quite different if heat sources are
present: the forces driving convective motions are vertical, and the motion of the fluid
would be enhanced.
3 The form of mantle convection
The preceding results demonstrate some properties of the two extreme types of convection
referred to in the Introduction - the distributed buoyancy and concentrated buoyancy
models (buoyancy concentrated in descending slabs, or in the lithosphere over rises, is
implicit as the source of motion in the moving boundary calculations). They demonstrate
that even if the buoyancy is strongly concentrated in the lithosphere, the viscous flow
entrained by moving plates is likely to penetrate the lower mantle. Any additional buoyancy
present in the mantle away from the lithosphere will amplify the flow in the mantle. At the
other extreme, in the distributed buoyancy model, without imposed moving boundaries,
the flow would be centred in the lower mantle if, as seems likely, the mantle does not have
major heterogeneities.
It has been often remarked that it was difficult to understand how deep-mantle convection, as it was invoked to explain continental drift, could be the source of plate motions,
since the surface pattern of plate motions does not resemble that to be expected from a
relatively small number of large convective cells. In particular, ridges would not be expected
to migrate, jump or coalesce with trenches. The calculations by Richter (1973), and recently
by Parmentier (1976), make it clear that the plate motions will have a strong influence on
mantle flow patterns. Parmentier (1976) finds that even a plate moving as slowly as 3 cm/yr
over a mantle with Rayleigh number l o 6 dominates the flow, so that a single, large-aspectratio, upper-mantle cell would occur (both Richter and Parmentier assumed that flow was
confined to the upper mantle). If the whole mantle is involved in the flow, the aspect ratio
of a single cell under a plate does not have to be as large, but it would still be as large as 3
for the largest plates. Thus the results of Richter and Parmentier are still relevant to wholemantle convection, and they indicate that it is quite reasonable for there to be a single cell
under a large fast plate such as the Pacific plate.
Although some aspects of plate motion (e.g. Elsasser 1969; Harper 1975; Forsyth &
Uyeda 1975) and calculations of the thermal structure of descending slabs (e.g. Schubert
et al. 1975) indicate that bdoyancy forces are strongly concentrated in descending slabs, it
is clear that other driving forces are also involved in plate motions; most notably, there are
several plates with very little slab attached which are moving relative to each other (e.g.
McKenzie 1969; Forsyth & Uyeda 1975). This motion might be explained by other interor intraplate forces (see below) or by coupling to underlying convection cells. A major
problem with convection models restricted to the upper mantle has been their failure to
satisfactorily explain these motions. Richter (1973) has thoroughly explored upper mantle
convection driven by vertical temperature gradients, ‘vertical’ convection modified by
horizontal temperature gradients, and convection modified by an adjacent descending slab,
and found that the resulting motions of plates would be too slow, too short-lived, or both,
to explain the motion of the American and African plates.
With whole-mantle convection, there is an obvious mechanism for the movement of such
plates. Only one convection cell would fit under such plates, or perhaps two cells under the
larger ones. The motion of these cells would be complementary to the motion of any
adjacent cells under plates driven by slabs, but with smaller velocities, the actual velocity
480
G. F. Davies
achieved depending on the available thermal power. The South American plate would seem
to be a good example of a plate which might be driven by a single such cell.
McKenzie & Weiss (1975) have noted that the number of plates will tend to decrease with
time because the symmetry of ridges and asymmetry of trenches causes relative motion
between them, so that they will ultimately merge. If plate tectonics is a long-term process
on the Earth, there must be ways of breaking up the larger plates. McKenzie & Weiss (1975)
have suggested that spreading centres behind island arcs may sometimes evolve into major
ridges, but there is little direct evidence to support this hypothesis. Harper (1975) has
suggested that pull from descending slabs may have split some plates, starting at re-entrant
corners. The trouble with such mechanisms, which involve only plate interactions, is that
they depend on the chance occurrence of the appropriate situation. If the whole mantle is
actively involved, this problem is avoided -- any plate which becomes large enough to cover
two or three whole-mantle cells will automatically be subjected to stresses tending to break
it up. The question to be answered is whether these stresses are sufficient to cause breakup.
Including the effects resulting from the greater buoyancy of continental crust in this
picture suggests possible explanations for the slower velocities of ‘continental’ plates and for
the cyclic aggregation and dispersal of continents. Since the lighter continental crust
apparently resists subduction, convergence of continental plates will result in continental
aggregation, as with India and Asia. Also, continental plates will tend to evolve to have less
attached descending lithosphere than oceanic plates, and so will tend to have lower
velocities. Finally, large aggregations of slow-moving continents, bounded by subduction
zones, would be favourable to the evolution of a pair (or set) of convection cells which
would ultimately cause the aggregation to disperse again. This scenario should be regarded
as speculative, although promising. Even if other factors were not involved, the evolution of
such a system would be extremely complicated.
Several attempts have been made to use observed plate motions to constrain models of
the forces acting on lithospheric plates (e.g. Solomon & Sleep 1974; Harper 1975; Forsyth
& Uyeda 1975; Richardson, Solomon & Sleep 1976). Solomon & Sleep showed that
‘absolute’ plate velocities (i.e. relative to some kind of global average of velocities) discriminate poorly between various models of the ‘drag’ of the asthenosphere on various parts of
the lithosphere. The results of Forsyth & Uyeda demonstrate that more detailed consideration of plate motions still does not discriminate clearly between a wide range of possible
force models, although they preferred one solution to their problem as being more ‘physically reasonable’. Harper (1975) achieved a remarkably good fit to observed motions by
considering only a large ‘slab pull’, a smaller ‘ridge push’, and asthenospheric drag.
Richardson et al. (1976) showed that observation\ of intraplate stresses are significant
additional constraints on such force models. All such force models share a sensitivity to the
dependence of the forces on parameters such as the dimensions of and relative or absolute
velocities of plates and slabs, and so the uniqueness of solutions is a considerable problem.
The form of the interaction between plates and the asthenosphere (the ‘drag’ under plates)
is especially uncertain. Its magnitude is usually assumed to be proportional to absolute
velocity, and its direction parallel to or opposite to the absolute velocity.
If the whole mantle is convecting, the interaction between the lithosphere and the underlying mantle must be quite complex. The turnover time of a deep-mantle cell, assuming
dimensions of 3000 km and velocities of a few cm/yr, would be of the order of 108yr.
However, the surface boundary condition, i.e. the configuration of the plates, also changes
considerably on this timescale. While the timescale of viscous readjustment is effectively
instantaneous, the timescale of thermal readjustment will be of the order of the turnover
time. Thus the pattern of mantle flow will be a complicated function of the previous lo8 or
Whole-mantleconvection and plate tectonics
48 1
so years of plate motions. (This has also been discussed by Garfunkel 1975.) A striking
example of this may be the present motion of the North American plate, which may result
from a plate configuration which no longer exists. Specifically, the motion may be caused by
a convection cell which arose in response to subduction of oceanic plates at the western
margin of North America which apparently occurred through the Cenozoic, and probably
longer, but which ceased 10-20 Myr ago when the East Pacific Rise intersected the subduction zone (McKenzie & Morgan 1969; Atwater 1970).
Two particular points may be noted concerning the force models of Forsyth & Uyeda
(1975), Harper (1975) and Richardson et al. (1976). These all have substantial ‘ridge push’
forces, and small ‘asthenospheric drag’. The small drag might be explained as they suggest,
by a low upper-mantle viscosity, or, alternatively, by an ‘active’ mantle which is predisposed
to move and will readily conform to the movement of plates (as amplified in other parts of
this paper). Some ridge push must be expected, because of the elevation of ridges. In
addition, however, the rheology of the mantle is highly temperature sensitive, and this tends
to concentrate rising convective flow into narrow regions (e.g. Torrance & Turcotte 1971),
and in that case much of the stress transmission from the mantle to the lithosphere would
occur close to the zone of upwelling, i.e. near the ridge. Thus it is quite conceivable that a
strongly coupled active-plate, active-mantle model would be consistent with the observational constraints.
Conclusions
The results of calculations presented here, and the results of others discussed here, indicate
that it is quite likely that the flow associated with the moving lithospheric plates penetrates
the whole mantle. Observational evidence indicates that the viscosity of the lower mantle is
insufficient to prevent this, and that there are significant heterogeneities in seismic velocities
in the lower mantle which may be associated with mantle flow. Theoretical understanding of
the effects of phase changes is incomplete, but it seems unlikely that they would exclude the
flow from the lower mantle. Possibilities for which there is not clear observational support,
but which cannot be ruled out, are that the lower mantle has a higher mean atomic weight
than the upper mantle, or that there is a very pronounced low-viscosity layer under the
oceans; the first of these might confine the flow to the upper mantle, the second might
substantially reduce the flow reaching the lower mantle, but would probably not exclude all
of it. The horizontal dimensions of the lithospheric plates are consistent with those of
whole-mantle convection cells, modified somewhat by the strength and movement of the
plates. The velocities of the’plates are qualitatively consistent with those to be expected for
whole-mantle convection cells modified by the concentration of substantial (negative) buoyancy in the descending lithospheric slabs.
The models discussed here are extremely simplified, and only the gross flow pattern has
been discussed. The justification for this is the assumption that the flow pattern depends
most strongly on the boundary conditions, and less strongly on the properties of the fluid.
The validity of this assumption seems to be borne out by the fact that extreme viscosity
variations were required to strongly affect the flow pattern. The results of other studies
support this conclusion. Thus Parmentier, Turcotte & Torrance (1976) found that a stressdependent rheology had very little effect on the flow pattern, while flow patterns with
depth or temperature dependent viscosity calculated by Torrance & Turcotte (1971) and
Houston & De Bremaecker (1975) were significantly affected, but also had internal
variations of viscosity of two or more orders of magnitude. It is notable that Houston &
De Bremaecker (1975) calculated cells with large aspect ratios (width to depth) in situations
48 2
G‘. b-. Davies
with high viscosities near the upper boundary, which was free to niove. Coinparisoil with
calculations with imposed moving upper boundaries (Richter 1973; Parmentier 1976). which
also yielded large aspect ratio cells, indicates that the near surface viscous layer was acting
as a stress guide and approaching the behaviour of a rigid moving ‘lithosphere’.
Although many useful insights have resulted from other calculations of mantle flow
referred to in this paper, it seems that the establishment of correct boundary conditions is
of prime importance, especially the depth of fluid involved and the tangential velocity of
any no-slip boundaries. More complex models than those discussed here must of course be
investigated. Some of the more important factors to be included would be more realistic
velocity boundary conditions (Davies 1977), heat sources, temperature-dependent rheology
and vertical or lateral variations in rheology. Only with more complex models is it reasonable
to go beyond the most basic observational constraints (plate dimensions and velocities) and
use other geophysical constraints such as observed surface heat flow, topography, and
gravity field, and intraplate stresses. Ultimately, of course, a successful theory of plate
dynamics must be tested against a vast array of geophysical, geochemical and geological
data.
It is the thesis of this paper that plate motions are directly associated with a large-scale,
deep-mantle flow, but, of course, considerable complication can be expected in more
complex models, and in the Earth. Thus, for instance, the large-scale flow might have smaller
scale complications associated with any low-viscosity layer(s) in the upper mantle. These
might become evident only when three-dimensional motions are considered; for example,
the kind of longitudinal convective rolls described by Richter & Parons (1975) might occur
in a shear zone under a fast plate. There are anomalies in the gravity field of the North
Pacific which might be related to such upper-mantle convective rolls (Marsh & Marsh 1976),
and there are anomalies in the topography and gravity field of the North Atlantic which
might be related to more equidimensional upper mantle convection ‘cells’ (Sclater, Lawver
& Parsons 1975). Thermal plumes (Morgan 1971) might be incorporated into the large-scale
flow, and might occur in a model with temperature-dependent viscosity (Parmentier,
Turcotte & Torrance 1975; Yuen & Schubert 1976). The numerical experiments of
McKenzie et al. (1974) suggest that convection in a fluid with internal heat sources may be
intrinsically unsteady. Laboratory experiments (Busse & Whitehead 197 1) and calculations
(Foster 1971) show that even convection in a fluid heated from below may be unsteady at
sufficiently high Rayleigh number. This has been discussed in relation to the mantle by
Jones (1975).
Acknowledgments
I am grateful to R. J. O’Connell for many stimulating discussions. Computing time was
provided by the University of Rochester.
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