Geophys. J . Int. (1990) 102, 15-24
The thermal structure of mantle plumes: axisymmetric or triplejunction?
G. A. Houseman
Department of Earth Sciences, Monash University, Clayton, VIC 3168, Australia
Accepted 1989 December 12. Received 1989 December 11; in original form 1989 September 18
SUMMARY
A mantle plume is probably a complex 3-D thermal structure that possesses
approximate axisymmetry as it approaches the base of the lithosphere from below,
but followed down towards the base of the layer, probably consists of a
triple-junction or quadruple-junction of connected hot sheets. A relatively weak hot
sheet rising only part way through the layer probably connects two neighbouring
mantle plumes. These conclusions are suggested by numerical experiments on a 3-D
constant-viscosity, plane layer with stress-free boundaries, which detail the gradational change in the planform of a convecting layer from the top of the layer to its
base. The planform of a convecting layer is a map in the horizontal plane of the
principal thermal anomalies in the layer. These anomalies are the main sources of
positive (for hot fluid) or negative (for cold fluid) buoyancy, and therefore they
drive the convective flow. They may appear in cross-section as structures with either
axial symmetry (columns), planar symmetry (sheets) or some complex asymmetric
form. When convection is driven at least partially by basal heating, the planform
near the top of the layer may be described as a network of cold sinking sheets and
isolated hot columns, while near the base of the layer it appears as a network of hot
rising sheets and isolated cold columns. The hot columns near the upper surface
arise from the vertices or nodes of the network of hot sheets on the lower surface,
and similarly the cold columns at the base of the layer form below the vertices of the
network of cold sheets near the upper surface. Near the upper surface, the apparent
planform of this experiment is analogous to that of mantle convection, the cold
sheets compared to subduction zones and the hot columns compared to mantle
plumes. The hot plumes impinging on the upper surface produce approximately
axisymmetric temperature anomalies, surface uplift and extensional stress fields.
However, the relatively minor deviations from axisymmetry of surface observables
reflect the deep structure of the mantle plume, formed by the junction of three or
four hot sheets on the base of the layer. It seems likely that the commonly occurring
triple-junction form of continental rifts may reflect an underlying structure that is
implicit in the convective circulation of the mantle beneath.
Key words: convection, mantle plumes.
1 INTRODUCTION
The group of structures known as mantle plumes is one of
the major elements in any exposition of the theory of plate
tectonics (e.g. Turcotte & Schubert 1982). These plumes are
usually conceived of as axially symmetric thermal structures
that arise at a thermal boundary layer on the base of the
convecting mantle layer, span the entire thickness of that
layer, and impinge on the base of the lithosphere, thus
producing the range of consequences that we describe as
hotspot activity (Wilson 1965; Morgan 1971; Burke &
Wilson 1976; Crough 1983; White & McKenzie 1989). These
consequences may include elevated heat flow, volcanism,
elevated topography, gravity and geoid anomalies, an
extensional stress field and possibly active extension. Major
features on the oceanic lithosphere, such as the HawaiianEmperor seamount chain, Iceland and the Kerguelen
plateau, are interpreted as the direct products of hotspot
15
16
G. A. Houseman
volcanism. These features indicate that the mantle plumes
may be relatively long lived (of order 100Myr) and that,
compared to the sometimes rapid motions of the overlying
lithospheric plates, they move only slowly (Molnar & Stock
1987).
Many models of the mantle plume-lithosphere interaction assume that the effect of the plume on the lithosphere
may be modelled by means of some axially symmetric
thermal or stress anomaly acting on the base of the
lithosphere (e.g. Nakiboglu & Lambeck 1985; Courtney &
White 1986; Olson & Nam 1986; Griffiths, Gurnis &
Eitelberg 1989; White & McKenzie 1989; Richards, Duncan
& Courtillot 1989). In general this is probably a very good
assumption, because the near-surface structure of a plume
(as illustrated by the numerical experiments described
below) has a high degree of axial symmetry, and the
structure of the plume at greater depth may well have
negligible effect on the lithosphere. However, the
plume-like features observed in the numerical experiments
show complex horizontal and vertical structure. Horizontal
cross-sections through the plumes show that their geometry
changes dramatically (but systematically) from near
axisymmetry at the top of the convecting layer to a branched
sheet structure at the base of the layer. Near the base of the
layer, thermal plumes stand out only as the nodes (or
connection points) of a branched network of hot vertical
sheet-like structures. Thus if mantle plumes are basically
thermal in origin, their location at any given time is
determined by the cellular structure of the convection
pattern.
Whitehead & Parsons (1978) described the spoke-like
opposing networks of hot ascending sheets and cold
descending sheets observed in high Rayleigh number
convection in an experimental tank of high-viscosity fluid
heated from below. But while their experimental set-up
enabled them to see qualitatively how the flow in the two
sets of sheet was accommodated, detailed measurement of
the 3-D structure was limited by the technical difficulty of
inserting a probe in the layer. Numerical experiments
completely avoid this measurement problem, enabling any
part of the solution to be interrogated in detail without
affecting the flow pattern, and thus facilitating the
description of the complex 3-D geometry.
Bercovici, Schubert & Glatzmaier (1989a,b) have
obtained solutions for the compressible convection problem
in a spherical annular layer representing the entire thickness
of the Earth’s mantle. They describe their solutions as
showing only cylindrical upwellings and planar downwellings, as opposed to the complex structures described below.
The differences between the results of Bercovici et al.
(1?89a,b) and those described below might be attributed
partially to the lower Rayleigh number used by them (about
100 times critical) compared to that used below (about loo0
times critical), or they may be due entirely to the difference
between spherical and planar geometry. Both sets of
experiments are of intrinsic physical interest but the
relevance of either set to the discussion of mantle
convection depends entirely on whether convection in the
Earth’s mantle is layered or not.
There has been considerable debate in recent years about
whether mantle convection occurs in a single layer 2900 km
thick, or in multiple layers with an upper layer 700 km thick.
Table 1. Physical parameters used for normalization of the dimensionless calculations.
The value for t ) is the approximate geometric
mean of the range of acceptable values quoted
by Lambeck (1988). The value of pb is appropriate for isostatically compensated elevation of the ocean floor. Both t ) and pbare only
used here in the scaling of Fig. 6(b).
parameter
symbol
value
dew
thermal diffusivity
surface hear flow
thermal conductivity
dynamic viscosity
gravity
overburdendensity
Because of the conflicting geophysical and geochemical
interpretations (e.g. Davies 1984; Creager & Jordan 1986;
Heinz & Jeanloz 1987; Ringwood & Irifune 1988; De Paolo
1988; Davies 1988b; Anderson 1989; Jeanloz & Knittle
1989) it seems best to treat this question as not yet settled,
and numerical experiments should therefore be used to
investigate both possible cases. In the event that the mantle
is layered, then the spherical curvature of the upper 700 km
thick layer is minor and can probably be neglected. The
study reported here is aimed specifically at this case, so
planar geometry is assumed for simplicity, and the
dimensionless calculations described below have been scaled
using a set of parameters appropriate to an upper mantle
layer (Table 1).
It is clear from many previous experiments (e.g.
McKenzie, Roberts & Weiss 1974; Carrigan 1982;
Houseman 1988) that the generation of thermal plumes
requires some component of basal heating, because
concentrated hot upwelling structures are not observed in
convection when all the heat is generated internally. Hot
upwelling structures would play a mechanically more
important role in the upper layer of a two-layer system than
in a single-layer system. Because of substantial heat
generation in the lower mantle (e.g. Schubert, Stevenson &
Cassen 1980), an upper mantle layer would be primarily
heated from below, while a single-layer mantle would
probably be primarily heated from within (although thermal
plumes remain possible if there is a significant outflow of
heat from the core). This difference provides a possible basis
for discriminating between the two earth models using
observations at the Earth’s surface (e.g. Davies 1988b).
However, it is first necessary to understand the 3-D
geometry of the hot thermal structures that drive the flow
and carry the heat, and the experiments described below
make a step in that direction.
The term ‘plume’ in current usage is applied to describe
both transient and steady-state vertical flow driven by
internal buoyancy, often implying axisymmetry about a
vertical axis. The class of transient structures generated for
Thermal structure of mantle plumes
example by instability of the hot thermal boundary layer and
characterized by the ascent of an approximately spherical
(or circular in 2-D) diapir pushing its way through
undisturbed material above (e.g. Olson et al. 1988; Sleep,
Richards & Hager 1988; Griffiths et al. 1989; Richards et al.
1989) is not the subject of the present investigation.
Although the transient phenomena are no doubt important
in the mantle, the longevity of the Hawaiian mantle plume
indicates that the steady-state structures, consisting essentially of a continuing stream of hot vertically flowing mantle,
must also be important. The plume structures described
below are essentially steady-state phenomena, although they
are continually modified by the time-dependent convective
circulation within which they are embedded.
Table 2. Scale factors used to convert from dimensionless results of
calculations to quoted physical
values, using nominal parameter
values given in Table 1.
2 FORMULATION OF THE CALCULATIONS
An approximate model of convection in the Earth’s mantle
is defined by the three equations that specify incompressible
flow, conservation of momentum (infinite Prandtl number
and constant viscosity) and conservation of energy (e.g.
Richter 1973; McKenzie et al. 1974):
v*u=o,
qv2u + p g = v p ,
-
d T / d t + u V T = KV’T
+H/(pC),
(3)
where u is the velocity field, p is the density, p is pressure, g
is acceleration due to gravity, q is dynamic viscosity, T is
temperature, K is thermal diffusivity, H is heat generation
per unit volume and C is heat capacity at constant pressure.
The Boussinesq approximation is assumed; the physical
properties are constant except that where the density
appears in (2), it depends linearly on temperature:
P =Po(1-
(4)
where pa is the density at T = 0 and (Y is the coefficient of
thermal expansion. Nominal parameter values are given in
Table 1.
The set of Boussinesq equations is only a first
approximation when dealing with flow in the Earth’s mantle.
The rheology of the silicate mantle is strongly temperaturedependent and under high differential stress or low
temperature conditions is also non-linear (e.g. Ranalli
1987). Thermal conductivity also depends on temperature,
and if the depth scale is taken to be the thickness of the
mantle, the fluid is relatively compressible. Although
previous simulations have included such non-Boussinesq
effects (e.g. Torrance & Turcotte 1971; Jarvis & McKenzie
1980; Buck & Parmentier 1986; Davies 1988a; Bercovici et
al. 1989a,b), the Boussinesq approximation is an appropriate place to start in an attempt to understand the 3-D
geometry of convection-generated structures if we bear in
mind that extrapolations to the Earth’s mantle may be
uncertain because of the above factors.
The momentum equation may be formulated as a
biharmonic equation and efficiently solved (Houseman 1987,
1990) if the velocity field is expressed in terms of a
solenoidal vector potential function A,
u=VxA.
(5)
11
physical quantity
s a l e fxm
lenglh
d
time
d21K
velocity
K/d
swin me
dd2
temperature
(F+HdW
SUCSS
Wd2
surface uplift
?W2pPtJ
The vertical component of A is zero in this problem. The
solutions described below are obtained using standard finite
difference techniques with an isotropic mesh of grid points,
in a box whose horizontal width (in x and y directions) is
four times its depth. With 32 mesh intervals in the vertical
direction, there are 549 153 nodal temperature and velocity
values. The side boundaries of the box are assumed to be
reflecting boundaries, while the horizontal boundaries are
assumed to be stress-free, with constant temperature on the
top and constant heat flow on the base.
The above physical variables may be non-dimensionalized
(Table 2) using a length scale d (the layer thickness), a
thermal time constant d 2 / K and a temperature scale
(F H d ) d / k where k is the thermal conductivity and F is
the prescribed basal heat flow. The dimensionless equations
are then completely specified by the Rayleigh number
+
R = [pga(F + H d ) d 4 ] / ( k ~ q )
(6)
and the heating mode number
p =Hd/(F
+Hd).
(7)
The values of R and p in the Earth are not well
constrained and depend critically on the layer depth. For the
experiment discussed below we assume values of R =
5.87 x lo5 (approximately 1000 times critical) and p = 1/2,
and the results of the calculations are scaled using the
parameters of Table 1. Note however that the results quoted
below can be rescaled using Table 2 if scaling parameters
different to those given in Table 1 are required. These
values of R and p may be approximately correct for an
upper mantle layer, but the value of R would be increased
(perhaps by a factor of 300) and p would be closer to 1 for a
whole mantle layer.
3
THERMAL STRUCTURES
For the convection experiment shown in Fig. 1 half of the
heat is generated within the layer by a uniformly distributed
heat source, while half of the heat is input by means of a
uniform heat flow imposed OD the base of the layer. Figs 1
and 2 show the state of the temperature field after
18
G . A . Houseman
approximately 5 x lo9 yr. The significance of the elapsed
time is that the convection has had time to evolve into some
sort of equilibrium with its heat sources after being started
from an arbitrary temperature distribution [the result of
another experiment, as described by Houseman (19SS)l.
The flow remains strongly time-dependent, as shown (Fig. 3)
by the dependence of its mean square flow velocity
(MSFV = volume integral of u u divided by total volume)
on time.
Figure 1 shows three horizontal temperature crosssections in perspective view, with colour indicating
temperature, ranging from blue through red to yellow as
temperature increases. The interior of the box (the
mid-plane) is at a relatively uniform temperature with
localized hot and cold anomalies. Near the top of the
convecting layer the cold structures appear relatively
-
It
kt
it
..
J
h
/
i
th
ti
I
b
Figure 2. (continued)
sheet-like while the hot structures are almost axial. The
converse is true near the base of the layer. The network of
cold sinking sheets near the top is comparable with the
subduction zones in the Earth's mantle. At the mid-level of
the layer, most of this network can still be traced as a weak
C
0.35-
Mean
square velocity
-
(cm/yrf -
0.30-
Figure 2. Contours of temperature in the horizontal planes (a)
z = 0, (b) z = 8h, lc) z = 16h, id) z = 24h, (e) z = 28h, (f) z = 30h,
and in the vertical planes (9) y = 0, (h) y = a h , (i) y = 74h, (j)
x = 50h, (k) x = 72h and (I) x = 92h (where h = 1/32) for the same
temperature field shown in Fig. 1. The arrows adjacent to (a), (8)
and (j) show the intersection coordinates of the orthogonal sections.
Planes (a), (c) and (f) are the same as shown in Fig. 1; the bottom
left comer of these drawings corresponds to the distant comer in
the 3-D projection of Fig. 1. The contour interval is 300"C, the
T = 1500 "C contour is shown dashed, and temperatures hotter than
1500°C occur in the stippled areas.
I
2 .o
I
l
l
l
l
I
l
3.0
l
4.0
l
l
l
I
3
Time (IO'yrs)
Figure 3. Mean square flow velocity (MSFV) of the solution region
(proportional to kinetic energy per unit volume) versus time, for the
experiment with R = 5.87 X lo5 and p = 1/2. The temperature
distribution shown in Figs 1 and 2 occurs at the endpoint of this
time series. Time zero for this experiment was an arbitrary initial
temperature distribution obtained from an internally heated
convection planform at the same Rayleigh number.
= 5.87 x 1 6 and p = 1/2 after about
5 X lo9 yr. Temperature increases as the colour changes from blue, through red to yellow, and is shown on the three planes z = 0 (the basal
plane), z = 16h (the mid-plane) and L = 3Oh (just below the top), where h = 1/32. The fluid sinks where the temperature is cold and rises
where it is hot. Note the vertical exaggeration; the box is actually four times as wide as it is deep. The solution continues beyond each of the
four side boundaries of the box by reflection in that boundary.
Figprr 1. Colour-coded view of the temperature field for a time-dependent convection flow with R
Thermal structure of mantle plumes
cold temperature anomaly (mauve on the colour projection), but the most prominent features at that level (blue)
are relatively localized cold sinkers located directly beneath
those locations where the cold sheets are joined (the vertices
of the network of sheets). On the basal plane the only
indications of the network of subduction zones above are
approximately circular areas of depressed temperature
beneath those same vertex points.
The basal plane is in fact dominated by a network of hot
rising sheets (yellow) whose geometry at first glance seems
unrelated to that of the cold sinking sheets. At the
mid-plane level the main branch of this hot network is still
clearly visible, but the hot temperature anomaly is
noticeably concentrated in three or four separate locations
directly above those places on the base where the hot sheets
are joined (the vertices of the network). Near the top of the
layer the hot structures appear as isolated, approximately
circular, hot thermal anomalies such as those that are
generally inferred to exist from the evidence of hotspot
traces. Thus, the features that we see on the Earth's surface
and interpret as hotspots are possibly related to an
interconnected network of hot rising sheets on the base of
the layer. The hotspots appear directly above the vertex
nodes of this network, where the vertical flow is most
intense. Adjacent hotspots are usually connected by a hot
sheet that only rises part way through the layer, and are
sometimes separated by a cold sinking sheet that is
approximately orthogonal to the hot sheet. The hot and cold
sheets apparently meet in a stagnation zone where they
cross. The orthogonal hot and cold sheets are similar to the
spokes described by Whitehead & Parsons (1978) in high
Rayleigh number convection heated from below. They are
also evident in the numerical experiments of Craig &
McKenzie (1987) where a relatively low Rayleigh number
permitted a regular grid-like distribution of hot and cold
sheets.
Figure 2 shows contour line drawings in Cartesian
projection of the thermal anomalies at six horizontal levels
(including the same levels as Fig. 1) and six vertical sections.
This presentation format conveys quantitative information
on the magnitude of particular temperature anomalies, but
it does not so easily convey the large-scale structural
features as does the colour projection of Fig. 1. Both
presentation formats are valuable in attempting to
understand this complex 3-D structure. Vertical sections like
those of Fig. 2 (g,h and i) differ from familiar 2-D solutions
due to the major component of flow in and out of the plane
of the section. Fig. 2(h) is dominated by the upwelling of the
two major hotspots and the basal sheet structure, while Fig.
2(g) is dominated by the downwelling of the cold boundary
layer, which is sheet-like near the top but concentrated in
local downwellings near the base. Fig. 2(i) is removed from
the main areas of vertical flow, and the major feature
evident is the upper thermal boundary layer with some sort
of boundary layer instability in progress. In the orthogonal y
direction, two of the vertical sections that cross the hot
structure (Fig. 2kJ) look rather similar to analogous 2-D
calculations, although the intensity of the hot upwelling
varies with the x coordinate of the section. Fig. 2(h and 1)
(sections that intersect at the approximate location of one of
the hotspots) illustrates the lack of symmetry of the hot
19
structure by its narrow extent in the y direction (Fig. 21) and
its broad extent in the x direction (Fig. 2h).
Most of the cold structures in this solution are arranged
along the side boundaries of the box (planes of reflection
symmetry), suggesting that the box is too small. Clearly the
solution is affected by the side boundaries, and the
calculations should be redone in a larger box (8 X 8 X 1 for
example) when sufficient computing resources are available.
However, the type of structure present on the boundary is
observed also in the interior of the box and appears to be
stable to repeated disturbances caused by thermal boundary
layer instabilities. In particular, the main hotspot structures
are at some distance from the boundaries, and I think it
unlikely that the major characteristics of their structure are
significantly influenced by these boundaries.
The time-dependence of the solution is manifested, first,
by gravitational instabilities of the cold upper thermal
boundary layer, which generally get swept up by the main
flow before having the opportunity to form new vertical
structures. Second, there is a gradual rearrangement of the
opposing networks of hot and cold sheets, with new,
relatively weak, branches appearing and disappearing under
the influence of these instabilities. In detail the flow appears
chaotic, and manifests a complex time-dependence which
will be discussed in detail elsewhere. However the overall
thermal structure of the layer is relatively unchanging, and
the characteristic networks of hot sheets on the base and
cold sheets on the top appear to be permanent features of
the convection solution, even if the configuration details
vary with time.
#en R is reduced by a factor of three (with p = 1/2) the
same types of structure are observed, with the difference
that the thermal boundary layers are thicker and the
time-dependent perturbations in the MSFV are relatively
smaller. When the experiment is repeated with p = O
(heated from below) at R = 5.87 X 105, the same basic
thermal structure is found, with hotspots separated by cold
sheets near the top of the layer and hot sheets separating
cold spots near the base of the layer. The major differences
are that, first, there is more symmetry between hot and cold
thermal structures because the same amount of heat flows
through cold upper and hot lower thermal boundary layers,
and second, the large-scale structures of the flow appear to
be more stable and less subject to disturbance by boundary
layer instabilities. With p = 1 (internally heated) at the same
Rayleigh number (Houseman 1988), the thermal structure is
fundamentally different. The flow is strongly timedependent and dominated by cold sinking columns. There
are no concentrated hot rising structures and the hottest
parts of the fluid are found in haloes surrounding, and near
the top of, the cold columns.
4
VELOCITY STRUCTURES
For 2-D calculations it is conventional to show instantaneous
velocity fields by means of contoured streamfunction values,
where the velocity at any point is parallel to the contours.
For the 3-D potential function this method is no longer
possible and arrow plots, as shown in Fig. 4, seem to be the
most useful alternative. The arrow consists of the two long
sides of an isosceles triangle with apical angle of 15". The
20
G. A. Houseman
a
b
Figure 5. Contours of vertical strain rate for the same temperature
field for (a) z = O and (b) z = 1. The contour interval is
1.O x lO-’’s-’, the zero contour is dashed and regions of positive
strain rate (vertical stretching) are stippled.
Figure 4 shows the horizontal velocity distribution on the
basal and top surfaces. Because these surfaces are
stress-free, the maximum horizontal velocities tend to occur
thereon. If the two arrow plots (Fig. 4a,b) are overlain one
sees that generally the horizontal velocity at z = 1 is about
equal and opposite to the horizontal velocity at z = 0.
However, there are significant departures from this rule with
regard to both flow direction and magnitude. The flow
direction on one surface differs by as much as 45” from what
would be predicted by taking the opposite of the velocity
vector on the opposite surface. Because the flow is
incompressible, those locations where the arrows indicate
convergence of the flow are places where there is significant
movement in the vertical direction away from the surface.
Conversely, divergence of the flow indicates movement of
the fluid towards the plane. The vertical strain rate ( a w / d z ,
where w is vertical velocity component) at any point is a
measure of the horizontal divergence or convergence. Fig. 5
shows contours of vertical strain rate on top and bottom
planes for comparison with the arrow plots in Fig. 4.
Because vertical velocity is zero on top and bottom planes,
the vertical strain rate values are proportional to vertical
velocity values at a level just inside (one mesh increment in
the calculations) the boundary plane. Fig. 6(a) shows the
distribution of the vertical velocity component in the middle
of the box ( z = 1/2) for comparison with the vertical strain
rate distribution near the upper and lower boundaries.
a
height of the triangle is proportional to the velocity at the
mid-point of its baseline and the azimuth of the velocity is
the direction in which the arrow points. Only the velocity
components in the plane of the section are shown; the
orthogonal velocity component is also generally non-zero.
b
Figure 6. (a) Contours of vertical velocity in the midplane
(z = 1/2), with contour interval of 0.5cmyr-I. (b) Contours of
surface uplift with contour interval 50 m. The zero-contour is
dashed and regions of negative velocity (a) and subsidence below
the mean level (b) are stippled. The dots shown in (b) identify those
points on the contours where the radius of curvature has a local
minimum (see text for explanation).
Thermal structure of mantle plumes
The velocity field associated with the temperature
structures described above can be briefly summarized as
follows. On the upper surface of the box the fluid flow is
approximately radially outwards from the hotspots. There
are relatively large areas of the surface where the velocity
field is relatively uniform (i.e. plate-like), and linear zones
of convergent flow are associated with the cold sheet
structures. Associated with the convergent flow into a cold
sheet there is often a relatively large component of flow
parallel to the strike of the sheet. The major differences
between this picture and the reality of plate tectonics can be
attributed to the absence in these calculations of the rigid
tectonic plates. On the basal plane approximately radial
divergence of the flow field is associated with the cold spots,
and convergence is associated with the hot sheets. The area
of most intense convergence [largest vertical strain rates,
Fig. 5(a)] on the basal plane is the branch of the hot-sheet
network that connects the two main hotspots. The
horizontal velocities on this branch are very small (Fig. 4a).
The continuation of the hot sheet beyond the two main
hot-spots also shows convergence of the velocity field in the
direction perpendicular to the sheet, but at the same time
shows a relatively large component of horizontal velocity
parallel to the rising sheet. Similar lateral flow along
spoke-like thermal sheets was also observed, by means of
dye injection and trace particles, by Whitehead & Parsons
(1978).
In grasping the complexity of this flow field, it may be
useful to picture the local flow field in the vicinity of a
horizontal boundary as the sum of two flow fields: an
approximately radial flow associated with (or driven by) the
nearest hot (or cold) spot, plus an approximately 2-D flow
associated with the nearest cold (or hot) sheet. The location
of the spots is of course determined by the intersections of
the sheets on the opposite boundary.
5
SURFACE DEFORMATION
Flow impinging on a horizontal boundary exerts a normal
stress on that boundary which must be matched by an
external load in order for the condition of zero normal
velocity to be met on the boundary (McKenzie 1977;
Parsons & Daly 1983; Lin & Parmentier 1985; Craig &
McKenzie 1987). For convection in the mantle this external
load may be provided by variation in topographic load at the
Earth’s surface. The normal surface stress
a,, = - p
+2q(dw/dz)
(8)
has two components: p is the perturbation to the lithostatic
pressure field, while the second term is the viscous normal
stress. The stress at a given time is obtained from the two
stream function components A, and A, using a method
described by Parsons & Daly (1983), here generalized to
3-D flow fields. In the Fourier domain, one may write for a
stress-free boundary
s,
= - i q { [ d 2 / d z Z- 3(kz
+ k:)]
x @,a, - kp,))/(kf + k:)
d/dz
(9)
where s,,, a, and a,,are the horizontal Fourier transforms
of a,,, A, and A,, and k, and k, are the x - and
y-component Fourier wavenumbers. Equation (9) is applied
21
to the horizontal Fourier transforms of A, and A, (with the
vertical derivatives obtained by finite difference methods).
The variation of stress on the boundary is then obtained by
taking the inverse Fourier transform of s,,. To facilitate
interpretation the stress may be described as an equivalent
topographic height in metres for a given overburden density,
assuming that the vertical stress is locally balanced (scale
factor given in Table 2). Because maximum flow velocity
scales approximately as ( K / ~ ) R(McKenzie
”~
et al. 1974),
the magnitude of the maximum normal stress on the surface
scales roughly as ( q ~ / d ‘ ) R ~ ’ ’i.e.
, proportional to the
square root of the viscosity of the convecting fluid. As there
is perhaps an order of magnitude uncertainty in the effective
viscosity of the upper mantle (Lambeck 1988; Nakada &
Lambeck 1989), the absolute magnitudes of the uplift are
correspondingly uncertain.
Figure 6(b) shows elevation/subsidence contours for the
upper surface of the experiment under discussion.
Comparison with Figs 2(f) (near-surface temperatures) and
5(b) (vertical strain-rate at surface) shows that essentially
the same geometrical features are evident in all three plots.
The evident difference between the geometrical distribution
of near-surface temperature anomalies and that of vertical
normal stress is that the elongated cold features result in a
less elongated, more rounded, surface depression, giving
greater prominence to the cold spots where sheets join. The
uplift above the hotspots is to first order axisymmetric and
there is, at first sight, little in the surface deformation to
suggest the existence of a basal network of hot sheets.
Predicted gravity anomalies would show a similar geometrical distribution in the horizontal plane.
However, although the surface uplift above the hot
structures is nearly axisymmetric, the axial symmetry is
perturbed by second-order components to the structure that
arise from the underlying network of hot sheets. The
contours of surface uplift for both of the major hotspots in
this temperature distribution deviate from the circular in a
way that reflects the branched network of hot sheets
presently beneath, and its recent evolution. The following
approximate graphical analysis allows this structure to be
roughly estimated from these contours. Each uplift contour
associated with the hotspot has a radius of curvature that
varies with distance along the contour. Using a set of circles,
and moving along each contour, local minima in the radius
of curvature can be approximately located and marked [the
dots on Fig. 6(b)]. If the dots on neighbouring contours are
then joined up in a radial direction, the various branches of
the underlying network of hot sheets are approximately
identified and located [compare Figs 2(a) and 6(b)].
Although this procedure may also detect weak branches
which are either being dissipated by the flow or only just
starting to grow, it usually also reveals the characteristic
triple- or tetrad-junctions at which hot sheets join together.
6 IMPLICATIONS FOR CONTINENTAL
EXTENSION MODELS
It has often been suggested that continental rifting, as for
example occurred during the break-up of Gondwana, or as
is presently occurring in East Africa, is the result of an
extensional stress field in the lithosphere induced by uplift
above some hot thermal structure in the mantle (e.g. Sleep
22
G. A . Houseman
1971; Crough 1979; Van Houten 1983; Houseman &
England 1986; Houseman & Hegarty 1987). The 3-D
numerical simulations described above validate the idea of
approximately axisymmetric mantle plumes pushing up on
the base of the lithosphere with stresses sufficient to cause
rifting. The simulations also suggest that, although a
network of hot sheet-like thermal structures probably exists
on the base of the convecting layer, only the nodes of this
network (the hotspots) are able to penetrate the thickness of
the layer and make p significant impact on the overlying
lithosphere. The exteht of this impact is likely to be highly
variable, depending on the local strength and thickness of
the lithosphere and the magnitude of the thermal anomaly
in the plume. The maximum impact would appear to be
the generation of large volumes (about 2 X 106km3) of
continental flood basalts (Richards et al. 1989), and rifting
of the continental lithosphere (White & McKenzie 1989). If
the lithosphere is sufficiently strong however, there may be
uplift without either extension or significant volcanism. The
generation of anomalously large volumes of magma in the
maximum impact cases has been attributed both to the
initial impact of a new plume head impinging on the base of
the lithosphere (Richards et al. 1989) and to the effect of
rapid continental extension causing decompression melting
above an existing plume (White & McKenzie 1989).
The characteristic triple-junction structure of many
continental rift settings has strong associations with domal
uplift and volcanism (Cloos 1939; Burke & Whiteman 1973;
Burke & Wilson 1976). Although one must look to the
mechanical properties of the lithosphere in order to explain
why extensional strain is typically concentrated in narrow
arms or rifts (e.g. Fletcher & Hallett 1983; Zuber,
Parmentier & Fletcher 1986), the characteristic geometry of
these structures may also indicate something of the stress
field imposed by the convection beneath. If we take as a
guide the numerical experiment described above, there are
several important features of continental rifting that we can
predict. First, the doming (Sahagian 1988) and radial arms
are consistent with an approximately circular uplift (Fig. 6).
Second, the departures from axisymmetry of this uplift
function provide a perturbation which could be amplified,
by a mechanical instability in the extending lithosphere (e.g.
Fletcher & Hallett 1983; Zuber et al. 1986), into the
characteristic radial multi-arm rift structure. Thus, while the
geometry of rifting is no doubt influenced by pre-existing
structure and heterogeneity in the crust, an explanation of
the commonly observed triple-junction structure of continental rifts (Burke & Dewey 1973; Burke & Wilson 1976)
may well be found in the branched structure of the network
of hot sheets on the base of the convecting layer. It seems
unlikely that the triple-junction geometry would be so
typical of rift environments if the geometry of rifting were
entirely attributed to pre-existing structure and
heterogeneity.
Third, if rifting continues to the point of formation of a
new ocean basin, the new passive margin typically develops
from two of the rift arms (which normally meet at an angle
between 120" and 180") while the third failed arm is
preserved as an aulacogen (Burke & Wilson 1976). But
typically, mantle plumes do not act in isolation; one expects
to find a number of hotspot structures separated by
distances of the order of the layer depth, but connected at
the base of the layer by a network of hot sheets. If active
continental extension is associated with two or more of these
structures (as in present-day East Africa), and continues to
the point of passive margin formation, then one predicts
that the resulting passive margin will inherit the geometrical
configuration of the basal hot sheet network active at the
time of continental break-up. Places where the azimuth of
the margin changes direction discontinuously, in association
with an aulacogen, probably mark the locations of the
hotspots at that time, and should be associated with
the most intense pre-break-up uplift and volcanism and the
earliest initiation of rifting. Rifting is likely to propagate out
from the centres of the hotspots along trends such as those
indicated by the dots in Fig. 6(b), resulting in approximately
linear sections of the margin that develop between
neighbouring hotspots. These regions would be associated
with minimal pre-break-up uplift and volcanism, as for
example has been documented for the Gulf of Suez
(Steckler 1985). Two basic modes of rifting are thus
expected, consistent with the observations of Sengor &
Burke (1978).
In contrast to the predicted close association between
mantle plume location and continental rifting, the
subsequent geometrical evolution of the mid-ocean ridge
(and any oceanic ridge-ridge-ridge
triple junctions)
following the onset of sea-floor spreading is determined
primarily by the rules of plate tectonics (McKenzie &
Morgan 1969). Whether a mantle plume will remain beneath
an ocean ridge depends on the relative horizontal motion of
ocean ridge and mantle plume. Numerical experiments by
Houseman (1983) determined that 2-D hot sheet structures
cannot follow a ridge migrating faster than about 1cm yr-'.
It may however be easier for a ridge to trap a mantle plume
because of the 3-D nature of the flow field around the
plume, and there is a suggestion that hotspots are associated
with the slower moving ridges such as the mid-Atlantic
ridge. Iceland and the Azores are two of the better known
examples. There also appear to be examples such as
Kerguelen where the mantle plume stayed with the ridge for
a time and then was eventually left behind. However, the
overall relatively low mobility of hotspots (Molnar & Stock
1987) is consistent with the mantle plumes being part of a
larger structure (the network of hot sheets) whose geometry
evolves only slowly compared to the plate velocities.
The observed areal density of hotspots should, in
conjunction with the numerical simulations, provide a
constraint on the thickness of the convecting layer. For
d =700km, the two or three hotspots in this experiment
each occupy on average 3.9 or 2.6 X 10l2 km2 of the upper
surface. This estimate is quite consistent with the empirical
values obtained by Sahagian (1980) for the slowly moving
continental plates such as Africa, Eurasia and Antarctica. If
d were 2900 km, the area occupied by each hotspot in the
experiment would be increased by a factor of 17,
significantly greater than the estimate for any of the
continents. The apparent spacing of hotspots is thus more
consistent with a 700km thick layer than with a 2900km
thick layer, although the unknown influence of a non-linear,
temperature-dependent rheology, different boundary conditions, and a much higher Rayleigh number might change
this conclusion. In arriving at this conclusion it must also be
acknowledged that many of the hotspots are associated with
Thermal structure of mantle plumes
relatively minor volcanic activity, and it is yet unclear
whether the mantle circulation is dominated by a small
number of energetic hotspots as illustrated in the
simulations of Bercovici et al. (1989a,b), or whether the
large numbers of hotspots previously identified (e.g. Vogt
1981) are actually indicative of a more complex circulation
characterized by a relatively small horizontal length scale.
ACKNOWLEDGMENTS
The calculations described here were done using computer
time provided by the CSIRO (Cyber 205), the ANU
(Fujitsu VP-50) and Leading Edge Inc. (Cray X-MP, many
thanks t o Tom Kopp).
REFERENCES
Anderson, D. L., 1989. Where on Earth is the crust?, Physics
Today, 42, NO.3, 38-46.
Bercovici, D., Schubert, G. & Glatzmaier, G. A,, 1989a. Influence
of heating mode on three-dimensional mantle convection,
Geophys. Res. Lett., 16, 617-620.
Bercovici, D., Schubert, G. & Glatzmaier, G. A., 1989b.
Three-dimensional spherical models of convection in the
Earth’s mantle, Science, 244, 950-955.
Buck, W. R. & Parmentier, E. M., 1986. Convection beneath
young oceanic lithosphere: implications for thermal structure
and gravity, J. geophys. Res., 91, 1961-1974.
Burke, K. & Dewey, J. F., 1973. Plume generated triple junctions:
key indicators in applying plate tectonics to old rocks, J . Geol.,
81, 406-433.
Burke, K. & Whiteman, A. J., 1973. Uplift, rifting and the
break-up of Africa, in Implications of Continental Drift to the
Earth Sciences, vol. 2, pp. 735-755, eds Tarling, D. H. &
Runcorn, S. K., Academic Press, London.
Burke, K. C. & Wilson, J. T., 1976. Hot spots on the Earth’s
surface, Sci. A m . , 235, 46-57.
Carrigan, C. R., 1982. Multiple-scale convection in the Earth’s
mantle: a three-dimensional study, Science, 215, 965-967.
Cloos, H., 1939. Hebung-spaltung-vulcanismus, Geol. Rundschau,
30,405-527.
Courtney, R. C. & White, R. S., 1986. Anomalous heat flow and
geoid across the Cape Verde Rise: evidence for dynamic
support from a thermal plume in the mantle, Geophys. J . R.
astr. SOC., 87, 815-867.
Craig, C. H. & Mckenzie, D. P., 1987. Surface deformation, gravity
and the geoid from a three-dimensional convection model at
low Rayleigh numbers, Earth planet. Sci. Lett., 83, 123-136.
Creager, K. C. & Jordan, T. H. 1986. Slab penetration into the
lower mantle beneath the Manana and other island arcs of the
northwest Pacific, J. geophys. Rex, 91, 3573-3589.
Crough, S. T., 1979. Hotspot epeirogeny, Tectonophysics, 61,
321-333.
Crough, S. T., 1983. Hotspot swells, Ann. Rev. Earth planet. Sci.,
11, 165-193.
Davies, G. F., 1984. Geophysical and isotopic constraints on mantle
convection, J. geophys. Res., 89, 6017-6040.
Davies, G. F., 1988a. Role of the lithosphere in mantle convection,
J. geophys. Res., 93, 10 451-10 466.
Davies, G. F., 1988b. Ocean bathymetry and mantle convection 1.
Large-scale flow and hotspots, J. geophys. Res., 93, 10 46710 480.
De Paolo, D. J., 1988. Neodymium Isotope Geochemistry,
Springer-Verlag,Berlin.
Fletcher, R. C. & Hallett, B., 1983. Unstable extension of the
23
lithosphere: a mechanical model for Basin and Range
structure, J. geophys. Rex, 88,7457-7466.
Griffiths, R. W., Gurnis, M. & Eitelberg, G., 1984. Holographic
measurements of surface topography in laboratory models of
mantle hotspots, Geophys. J., 96, 477-495.
Heinz, D. L. & Jeanloz, R., 1987. Measurement of the melting
curve of M&,,Fe,.,SiO, at lower mantle conditions and its
geophysical implications, 1.geophys. Res., 92, 11 437-11 444.
Houseman, G. A,, 1983. The deep structure of ocean ridges in a
convecting mantle, Earth planet. Sci. Lett., 64, 283-294.
Houseman, G. A., 1987. TDPOIS, a vector-processor routine for
the solution of the three-dimensional Poisson and biharmonic
equations in a rectangular prism, Comp. Phys. Comm., 43,
257-267.
Houseman, G. A., 1988. The dependence of convection planform
on mode of heating, Nature, 332, 346-349.
Houseman, G. A., 19%. Boundary conditions and efficient solution
algorithms for the potential function formulation of the
three-dimensional viscous flow equations, Geophys. J. Int.,
100, 33-38.
Houseman, G . & England, P., 1986. A dynamical model of
lithosphere extension and sedimentary basin formation, J .
geophys. Res., 91, 719-729.
Houseman, G. A. & Hegarty, K. A., 1987. Did rifting on
Australia’s southern margin result from tectonic uplift?,
Tectonics, 6, 515-527.
Jarvis, G. T. & McKenzie, D. P., 1980. Convection in a
compressible fluid with infinite Prandtl number, 1. Fluid Mech.,
96, 515-583.
Jeanloz, R. & Knittle, E., 1989. Density and composition of the
lower mantle, Phil. Trans. R. SOC. Lond., A, 328, 377-389.
Lambeck, K., 1988. Geophysical Geodesy, Clarendon Press,
Oxford.
Lin Jian & Parmentier, E. M., 1985. Surface topography due to
convection in a variable viscosity fluid: application to short
wavelength gravity anomalies in the central Pacific Ocean,
Geophys. Res. Lett., 12, 357-360.
McKenzie, D. P., 1977. Surface deformation, gravity anomalies and
convection, Geophys. J. R. astr. SOC.,48, 211-238.
McKenzie, D. P. & Morgan, W. J., 1969. Evolution of triple
junctions, Nature, 224, 125-133.
McKenzie, D. P., Roberts, J . M. & Weiss, N. O., 1974. Convection
in the Earth’s mantle: towards a numerical simulation, J. Fluid
Mech., 62, 465-538.
Molnar, P. & Stock, J., 1987. Relative motions of hot spots in the
Pacific, Atlantic and Indian Oceans since late Cretaceous time,
Nature, 327, 587-591.
Morgan, W. J., 1971. Convective plumes in the lower mantle,
Nature, 230, 42-43.
Nakada, M. & Lambeck, K., 1989. Late Pleistocene and Holocene
sea-level change in the Australian region and mantle rheology,
Geophys. J., 96,497-517.
Nakiboglu, S. M. & Lambeck, K., 1985. Thermal response of a
moving lithosphere over a mantle heat source, J. geophys.
Rex, 90, 2985-2994.
Olson, P. & Nam, I. S., 1986. Formation of seafloor swells by
mantle plumes, J . geophys. Res., 91, 7181-7191.
Olson, P., Schubert, G., Anderson, C. & Goldman, P., 1988.
Plume formation and lithosphere erosion: a comparison of
laboratory and numerical experiments, J. geophys. Res., 93,
15 065-15 084.
Parsons, B. & Daly, S., 1983. The relationship between surface
topography, gravity anomalies and temperature structure of
convection, J. geophys. Res., 88, 1129-1144.
Ranalli, G., 1987. Rheology of the Earth, Allen and Unwin,
Boston.
Richards, M. A., Duncan, R. A. & Courtillot, V. E., 1989. Flood
basalts and hot-spot tracks: plume heads and tails, Science,
246, 103-107.
24
G. A . Houseman
Richter, F. M., 1973. Dynamical models of sea-floor spreading,
Rev. Geophys. Space Phys., 11, 223-287.
Ringwood, A. E. & Irifune, T., 1988. Nature of the 650 km seismic
discontinuity: implications for mantle dynamics and
differentiation, Nature, 331, 131-136.
Sahagian, D., 1980. Sublithospheric upwelling distribution, Nature,
287, 217-218.
Sahagian, D., 1988. Epeirogenic motions of Africa as inferred from
Cretaceous shoreline deposits, Tectonics, 7 , 125-138.
Schubert, G., Stevenson, D. & Cassen, P., 1980. Whole planet
cooling and the radiogenic heat source contents of the Earth
and Moon, J. geophys. Res., 85, 2531-2538.
Sengor, A. M. C. & Burke, K., 1978. Relative timing of rifting and
volcanism on Earth and its tectonic implications, Geophys. Res.
Lett., 5 , 419-421.
Sleep, N. H., 1971. Thermal effects of the formation of Atlantic
continental margins by continental break-up, Geophys. J. R.
mtr. SOC.,24, 325-350.
Sleep, N. H., Richards, M. A. & Hager, B. H., 1988. Onset of
mantle plumes in the presence of preexisting convection, J.
geophys. Res., 93, 7627-7689.
Steckler, M. S., 1985. Uplift and extension at the Gulf of Suez:
indications of induced mantle convection, Nature, 317,
135-139.
Torrance, K. E . & Turcotte, D. L., 1971. Thermal convection with
large viscosity variations, J. Fluid Mech., 47, 113-125.
Turcotte, D. L. & Schubert, G . S., 1982. Geodynamics, John Wiley
& Sons, New York.
Van Houten, F. B., 1983. Sirte Basin, north-central Libya:
Cretaceous rifting above a fixed mantle hotspot?, Geology, 11,
115-118.
Vogt, P. R., 1981. On the applicability of thermal conduction
models to mid-plate volcanism: Comments on a paper by Gass
et al., J. geophys. Res., 86, 950-960.
White, R. & McKenzie, D., 1989. Magmatism at rift zones: the
generation of volcanic continental margins and flood basalts, 1.
geophys. Rex, 94, 7685-7729.
Whitehead, J. A. Jr & Parsons, B., 1978. Observations of
convection at Rayleigh numbers up to 760,000 in a fluid with
large Prandtl number, Geophys. Asfrophys. Fluid Dyn., 9,
201-217.
Wilson, J. T., 1965. Evidence from ocean islands suggesting
movement in the Earth, Phil. Tram. R . SOC. Lond., A, 258,
145- 167.
Zuber, M. T., Parmentier, E. M. & Fletcher, R. C., 1986.
Extension of continental lithosphere: a model for two scales of
Basin and Range deformation, 1. geophys. Res., 91,
4826-4838.