Download Terrestrial Heat Flow and the Mantle Convection Hypothesis

Geophys. J . R. astr. SOC.
(1967) 14, 413428.
Terrestrial Heat Flow and the Mantle Convection Hypothesis
M. H. P. Bott
Summary
The paper aims to show that the distribution of terrestrial heat flow is
consistent with the mantle convection hypothesis provided that certain
restrictions are placed on allowable patterns of convection. Some
objections to convection are first discussed.
Numerical experiments have been carried out on the loss of heat from
the upper surface of a mantle convection cell, assuming a simplified flow
pattern. The results are compared with observed heat flow. Subject to
the assumptions of the models, the results suggest that sub-oceanic
mantle convection currents, if they exist, are overlain by a layer 50- 100km
thick which is stationary or moves much less rapidly. This appears to
rule out the mechanism of continental drift by ocean floor spreading as
suggested by Dietz; continental drift affecting the overlying layer must
occur at a much lower velocity than the convection current. If thermal
diffusivity remains approximately constant with depth the velocity near
the upper surface of the convection cell needs to be at least about 20cm/y
to explain the uniformity of heat flow, but if radiative conduction becomes
dominant a lower velocity would be acceptable.
The experiments have been extended to a convection cell flowing
beneath a continental margin. The results suggest that the approximate
equivalence of oceanic and continental mean heat flow can best be explained
if convection currents are generally present beneath oceans but absent
beneath continents, unless the continental crust has a much lower radioactivity than is normally supposed. The anomalous low heat flow of
Pre-Cambrian shields suggests absence of convection in the mantle
beneath since the Pre-Cambrian.
Conditions particularly favourable to large-scale partial fusion of
the upper mantle occur in the topmost section of a rising convection
current. This should rejult in abundant igneous activity and local areas
of anomalously high heat flow. This supports the view that ocean ridges
overlie uprising convection currents. Belts of relatively low heat flow
adjacent to the ocean ridges are difficult to explain if the convecting mantle
behaves as a Newtonian viscous fluid, but are explicable on the nonNewtonian model of convection recently suggested by Orowan.
1. Introduction
There are two opposing hypotheses relating to the distribution of continents and
oceans through geological time. One hypothesis postulates the permanence of continents and oceans. The other postulates that the continents have moved relative to
each other during geological time, and appeals to mantle convection currents as
mechanism.
413
4 14
M. €1. P. Bott
In a series of recent papers MacDonald (e.g. 1963) has treated the problem of
terrestrial heat flow assuming the permanence of continents and oceans. This paper
aims t o show that the major features of terrestrial heat flow can also be understood if
convection currents occur in the mantle, especially beneath oceans.
Following a brief statement of the main features of terrestrial heat flow, the relevant
features of the convection hypothesis are stated including a discussion on the objections and different patterns suggested. This leads to the main part of the paper, which
is a quantitative investigation of the heat loss from the upper surface of a convection
cell by thermal condiction through an overlying layer, supplemented by magmatic
activity.
2. The pattern of terrestrial heat flow
The features of terrestrial heat flow:# which are discussed are as follows:
(1) The mean heat flow for oceanic regions is 6.70 x 10-6J/cm2s. This does not
differ significantly from the continental mean (6.00 x lom6)on the basis of present
observations (Lee & Uyeda 1965);
(2) local belts 200-300km wide with irregular heat flow up to eight times the
average are associated with the mid-Atlantic ridge (Nason & Lee 1962, Vacquier &
Von Herzen 1964) and the East Pacific rise (Von Herzen & Uyeda 1963);
(3) these belts of high heat flow are sometimes flanked by broad belts several
hundred kilometres wide where the heat flow is significantly lower than average
(McBirney 1963, Vacquier & Von Herzen 1964);
(4) the Precambrian Shields have a significantly lower heat flow than the average
for continents of 3.85 x
J/cm* s (Lee & Uyeda 1965).
Bullard, Maxwell & Revelle (1956) and Von Herzen & Uyeda (1963) have discussed
possible sources of oceanic heat flow and conclude that primary sources other than
radioactive decay within the underlying mantle can be discounted as major contributors. The situation for continents is different. If the average continental crust
has radioactivity intermediate between granite (7.00-9.5 x 10- l 3 J/g s) and basalt
(1.6-2.5 x
(McDonald 1963) a 33 km thick crust provides over 65% of the
average continental heat flow; it is possible that this is an over-estimate but it would
be unrealistic on present information to reduce the estimate below about 35%. It
follows that the heat flow in the topmost mantle is markedly greater beneath oceans
than continents.
This is the major problem of terrestrial heat flow. It can be met by postulating
different radioactivity for sub-oceanic and sub-continental upper mantle, an explanation which assumes the permanence of continents and oceans. Alternatively, mantle
convection currents may provide a mechanism by which heat is more effectively
transferred through the sub-oceanic mantle: this explanation may be linked with the
hypothesis of continental drift. The second mechanism is investigated in some detail
in this paper.
3. The mantle convection hypothesis
3 . 1 . Objections to the convection lzypothesis
If the mantle is assumed to be a uniform Newtonian viscous fluid, the initiation
of mantle-wide convection depends on the non-dimensional Rayleigh number
R = uflgd4/kv exceeding about 1500; 01 is the coefficient of expansion (- 2 x 10-5/0C),
p is the temperature gradient in excess of adiabatic, g is the gravity (- 103cm/s2),
* Throughout the paper the term hear flow per unit urea is abbreviated to heat flow.
Terrestrial heat flow and mantle convection
415
d is the thickness of the spherical shell (3 x 108cm), k is the thermal diffusivity
(- 10-'-10-2 cm2/s) and v is ths kinematic viscosity. If v is assumed to be loz2poises
as suggested for the upper mantle by the Fennoscandian post-glacial uplift, /3 needs
to be 10-3-10-4"C/km to initiate marginal convection, This shows that convection
is feasible on the Newtonian viscous model for quite a small temperature gradient
in excess of the adiabatic gradient.
There are, however, two main objections to the convection hypothesis arising
from (1) stress differences and (2) phase changes in the mantle. These are now
discussed.
Surface and satellite gravity measurements show that there are low-degree harmonics
of the Earth's gravity field which deviate from the gravity field predicted for hydrostatic equilibrium. These are caused by density anomalies within the mantle which
require stress differences of 100-200 bars in the mantle. These stress differences could
be caused either by viscous flow in a fluid oiit of hydrostatic equilibrium or by elastic
deformation of a mantle with finite strength, or both.
One interpretation of the stress differences (MacDonald 1963) related to the seconddegree zonal harmonic is that the mantle has a viscosity of
poises which causes a
lag in the attainment of the equilibrium figure as tidal friction slows the Earth's rate
of rotation. If this interpretation is correct it could provide an insuperable barrier
to mantle-wide convection since the initiation of convection against a viscosity of
loz6poises would need a superadiabatic temperature gradient of 1-lO"C/km.
Smaller-scale convection in a low-viscosity upper part of the mantle would not
necessarily be ruled out.
Another interpretation based on viscous flow has been given by Runcorn (1964),
who has shown qualitatively that the flow pattern associated with mantle convection
currents could cause the stress differences, thus suggesting one answer to the first
objection at least for harmonics above the second degree.
S
S
. . . . .
. . . :
1.".
:)
. . . . . .
I
I . . . . .
. . . . .
. . . . . .
FIG.1. A simple model of plastic convection caused by a central prism of hotter
material (after Orowan 1964). This shows that the upper surface of the convecting material is relatively deep above the centre of the upwelling current (T),
and shallow~rbetween the uprising and sinking currents ( S) .
416
M. H. P. Bott
If alternatively the stress differences arc caused by elastic deformation of a mantle
with finite strength, mantle convection currents would need to overcome this strength.
This difficulty has been at least partly met by a model of plastic convection recently
described by Orowan (1964). Orowan suggests that Andradean viscous behaviour
provides the most realistic rheological model for the hot crystalline material of the
mantle; this gives a finite ‘ creep strength’ above which the strain rate increases quasiexponentially with increasing stress. It approximates better to ideal plastic material
than to a Newtonian viscous fluid.
Orowan has constructed a simple model of plastic convection which demonstrates
its feasibility (Fig. I). Taking a hot rising column with vertical and horizontal dimensions of 1000km, a temperature difference of 100°C between the rising and sinking
columns and a coefficient of expansion of 10-5/T, he showed that convection can
be maintained provided the yield stress is less than 40 bars. If either the cell dimension
o r the coefficient of expansion is larger convection could occur against a greater yield
stress.
Orowan’s model of plastic convection has two further characteristics not shown by
Newtonian viscous convection which are relevant to this paper: (1) plastic flow means
that the velocity gradients are concentrated in a narrow marginal zone, a feature which
may be further accentuated if partial fusion reduces the yield stress near the upper
margin of the convection cell; and (2) Orowan’s diagram also shows that if there is
a n overlying non-convecting layer the upper surface of the convection cell is not at a
uniform depth but is deeper above the centre of the upwelling (Fig. 1 , T) and sinking
currents than between these (S).
The second objection to convection arises from the phase transitions such as the
olivine-spinel transition between 400 and 1000km depth in the mantle. Vening
Meinesz (1962) has concluded that the convection currents can cross this zone. But
Knopoff (1964), assuming that the phase change takes place to completion as the
upward flowing current crosses a surface (at 600km depth), comes to the opposite
conclusion that the transition provides an insuperable barrier to mantle-wide convection since no adequate source of heat is available to drive the transition to completion,
Knopoff’s argument, however, is unsatisfactory thermodynamically (see also
Verhoogen 1965). The true situation can be seen by thermally isolating a unit mass
of material and allowing it to rise through the transition with the upwelling current
‘E
I
PRESSURE
FIG.2. Temperature-pressure relations as a single component upwelling convection
current crosses a phase boundary EE‘, showing (1) the course for a thermally isolated mass carried up with the current (abcd), and (2) the course for material in
thermal contact with its surroundings (a’ bcd’).
Terrestrial heat flow and mantle convection
417
(Fig. 2, abcd). In the pressure-temperature field the two phases are separated by a
phase boundary having a gradient related to the change in density and the latent heat
L, by the Clausius-Clapeyron equation. The phase boundary is steeper than the
adiabatic gradient. The thermally isolated material follows the adiabatic gradient
until it reaches the phase boundary (Fig. 2, ab). It then loses one degree of freedom
and is constrained to follow the phase boundary until the transition is completed (bc),
when it resumes the adiabatic gradient in the field of the low-pressure phase (cd).
The heat of transition is provided by the steeper cooling along the phase boundary.
Lack of thermal isolation (Fig. 2, a’bcd’) and presence of other phases and solidsolution complicates the argument but does not alter the main principle that the phase
change does not inhibit convection provided the temperatures beneath the transition
are raised by an additional amount somewhat greater than LJc where c is the appropriate specific heat, and the rate of reaction is sufficiently fast to keep close to the
two-phase boundary.
It is thus concluded that (1) the stress differences within the mantle are consistent
with convection either if they arise from the viscous flow of the convection pattern
or if plastic convection breaks through a finite strength, and (2) Knopoff‘s objection
concerning phase transitions is not valid.
3.2. The pattern of mantle convection currents
Until recently it has been usual to assume that convection, if it exists, is mantle
wide. The possibility of a prohibitively high viscosity in the lower mantle and Knopoff’s
argument on the phase transition zone have led to the suggestion by many authors
that convection is restricted to the upper mantle. The computations presented in this
paper mainly assume a horizontal cell dimension of the order of the thickness of the
mantle, but most of them could also be applied in approximate form to cells with a
smaller horizontal dimension. The vertical thickness of the cell should not affect
the argument provided it is a few hundred kilometres o r more.
Pekeris (1935) and Vening Meinesz (Heiskanen & Vening Meinesz 1958) have,
among others, suggested that convection currents normally rise beneath continents
and sink beneath oceans, but this view is difficult to reconcile with the approximate
equality of continental and oceanic heat flow. Runcorn (1962a, b) and many others
have suggested alternatively that convection currents rise beneath the oceans, particularly beneath ocean ridges, and sink either beneath continents or near continental
margins. Runcorn has attributed the changing pattern of convection through geological time to progressive growth of the core.
If thermal convection does take place then it would be expected to contribute
significantly to the Earth’s heat flow pattern. In particular, an acceptable pattern of
convection must be capable of explaining (1) the high heat flows associated with the
ocean ridges and the adjacent marginal lows, (2) the relative uniformity of oceanic
heat flow elsewhere, (3) the approximate equivalence of the mean oceanic and continental heat flow, and (4) the low heat flow of the Shields.
In the remaining part of the paper the major features of the observed pattern of
heat flow are compared with numerical prediction for various theoretical models of
convection, including (1) whether or not they reach the Earth’s surface and (2) whether
they underlie continents and oceans alike or are mainly restricted to the sub-oceanic
mantle as the writer has suggested elsewhere (Bott 1964).
4. The calculation of the heat loss from the upper surface of a mantle convection current
In this section an approximate method of calculating the heat flow from the upper
surface of a mantle convection current is derived. In later sections the results from
applying the method are compared with observations of terrestrial heat flow.
The temperature and velocity distribution in a convecting system are governed
418
M. H. P. Bott
A 0
C(5mY)
(80rn9E
D(20rny)
0
4. . . . . . . .
.
1
. . . . . . . . OVERLYING LAYER' . ' . ' . . * . ' , ' . ' .
Z.(bOkm) . . . . . . .. . . . . . conductivity K.' . . . . . . . . . . . . . . . . . . .
. . . . . . . . h;KJKze . . . . . . . . . .
:
g.'
r
-
'
____,
CONVECTION CELL
r
r
conductivity K
diffusivity I(
velocity v
FIG. 3. The simplified model of convection assumed in Figs. 5 and 6. Uniform
horizontal velocity is assumed. The horizontal scale depends on the velocity.
The initial temperature distribution of Figs. 5 and 6 lies vertically below A , and
the calculated temperature distributions correspond to vertical sections beneath
B, C, D and E.
by (1) the equation of heat conduction in a moving medium, (2) the equation of rheological flow (not necessarily Newtonian viscous flow), (3) the equation of continuity,
and (4)an equation of state. In general these cannot be solved independently, but
as an approximation if a velocity field may be assumed then the equation of heat
conduction can be solved independently.
In the theoretical models below it is assumed that the upper part of the convection
cell flows with a uniform horizontal velocity, and that it may be overlain by a stationary
layer (see Fig. 3). The justification for this simple model comes from the zone of
apparently low viscosity in the upper mantle, and the likelihood that the flow may be
plastic or Andradean; both of these factors would favour the existence of a relatively
narrow marginal zone in which the steep velocity gradient is concentrated. Thus the
relatively simple model adopted is probably more realistic, applied to mantle convection, than the Rayleigh-Boussinesq model of convection. It is acknowledged that the
abrupt transition from vertical to horizontal flow is unrealistic, but this assumption
only affects the solution in the immediate vicinity of the uprising and sinking currents.
We now investigate the cooling of the upper part of a convection cell through a
stationary overlying layer of vertical thickness zo. The equation of heat conduction
in the upper part of the convection cell, neglecting heat sources, is
kv2e-v.ve-
ae =o,
aT
where k is the thermal diffusivity which is assumed to be uniform, 8 is the temperature,
v (ox, vy, v,) is the velocity and z is the time. Neglecting the Earth's curvature, we
take orthogonal Cartesian coordinate axes with the z-axis pointing vertically downwards (Fig. 3). z=O forms the horizontal plane boundary between the convection
cell and the overlying layer, and the x-axis points in the direction of flow with x=O
at the abrupt transition from vertical to horizontal flow (see Fig. 3). We assume
ae
(1) a two-dimensional model such that - =uy=O for all (x, z); (2) the convection
aY
current has been flowing for a sufficiently long time for steady state conditions to
ae
aZ
have been established giving - = O (of the order of 100my for a 60km thick over-
Terrestrial heat flow and mantle convertion
419
a2
o can be neglected;
-2
(4) the convection current rises vertically at
ax
x = O where it abruptly turns to flow horizontally with constant velocity v,, independent of position (see Fig. 2). Equation (1) simplifies to give
lying layer); (3)
a20
k - -v,a22
ae
ax
=O.
To solve (2) we need to know O(x, 0) for all x > 0 (the boundary condition). The
boundary condition is obtained by solving the steady state equation of heat conduction
in the stationary overlying layer. Neglecting the relatively small term
this equation is
E)
2-( K O
8Z
+ A (s,z)= 0,
(3)
where both the thermal conductivity Ko(.u,z ) and the radioactive heat produced
per unit volume per unit time A(x, z ) may vary with position. The Earth's surface
forms the upper surface of this layer at z= -zo and we assume that 0(-zo)=O for
all x> 0. In considering sub-oceanic convection put A ( x , z)=O and integrate twice
giving:
0
- i"
a0
where Q = K -=constant
0
az
for fixed x (the heat flow per unit area). Since the
heat flow and temperature are continuous across the plane z=O, equation (4) provides
the required boundary condition for equation ( 2 ) as follows:
--ae +h8=0
aZ
where
(at z=O for all x>,O),
n
K being the thermal conductivity in the convection cell at z=O. If the radioactive
heat sources in the overlying layer are included, (5) takes the form:
where 8, can be readily calculated for specific distributions of A and KO. Substituting
t=x/v, in (2), it becomes equivalent to the unidimensional equation of heat flow
subject to what is commonly known as the ' radiation ' boundary condition (5) or (6).
A solution of (2) subject to conditions ( 5 ) and given initial conditions can be
obtained analytically for simple cases by integrating an expression given by Carslaw &
Jaeger (1959, p. 359). However, a more versatile approach adaptable to complicated
boundary and initial conditions is to solve by computer using finite difference approximations. The computer approach has been used for most of the calculations included
in this paper although the stability of the solutions has been checked analytically.
If the convection current reaches the Earth's surface, then zo =O. Assuming the
vertical temperature distribution at x = O is given by (0, + Tz) the surface heat flow
at (x,0) is given by { T K +BoJ(ku,/nx)}.
420
M. H. P. Bott
5. Oceanic heat flow in the presence of underlying mantle convection
In this section it is assumed that convection currents underlie the normal oceanic
crust, possibly rising beneath the ocean ridges. It is investigated under what conditions the relative uniformity of oceanic heat flow is consistent with the convection
hypothesis.
If there is an overlying layer, its thickness (z,) needs to be estimated. Linear
extrapolation of the geothermal gradient suggests that fusion occurs at 60-70 km
depth; if the uniform gradient extended to 100 km depth there would be wholesale
melting. This suggests that the overlying layer is probably about 60-100 km thick.
In the following calculations zo has been taken as either 60 km or zero.
For initial conditions (at x=O) it is assumed that the temperature at the top of
the convection cell is 1300°C (or 1200°C for model l), that the fusion gradient of
1.3"C/km applies to the topmost 50 km of the convection cell, and that beneath this
the initial gradient is 0.3"C/km, which is slightly in excess of the adiabatic gradient.
The initial conditions are shown as the topmost curves of Figs. 5 and 6. It is assumed
that radioactive heat sources are uniformly distributed throughout the mantle, and
therefore that they contribute negligibly to the cooling of the upper surface of the
convection cell (if not, they would supply a contribution to heat flow which is independent of x). The pattern of surface heat flow as a function of x, the distance from
the uprising current, has been computed for three models as follows, based on the
above assumptions:
Model 1: zo = 0, k = 0-01cm2/s, K = 0.032 J/cm2 s, 6 , = 1200"C,u, = 20 and 4 cm/y,
and the initial temperature gradient at x = 0 is taken as the fusion gradient at 1.3 "C/km.
This model represents a convection current reaching the solid Earth's surface beneath
the ocean.
Model 2: zo = 60 km, k =0.01 cm2/s, K = KO=0.032J/cm2s, 0, = 1300 "C,
u,=20 cm/y. This assumes an overlying non-convecting layer 60km thick with the
same thermal properties as the convection cell.
ModeZ 3: zo = 60 km, k = 0.1 cm2/s, 0-1K = KO=0.032 J/cm2s, 8, = 1300"C,
ux= 5 cm/y. This is similar to Model 2 except that the convection cell has a higher
thermal conductivity owing to the radiative heat transfer and the velocity is lower.
- - - -
l
o
o
r
20 0
r
CONVECTION CELL
h
FIG.4. Surface heat flow as a function of distance from an uprising sub-oceanic
mantle convection cell, computed for models 1-3 (seetext).
42 1
Terrestrial heat flow and mantle convection
TEMP
(OC)
1500
-
1000
-
h = 0.016671km
k = 0.070 cm2/s
500 -
0
K = K.
/
-overlying layer- + c
- - - - - - - - - - -convection cell- - - - - - - - - - -
0
100
50
150
DEPTH (krn)
FIG.5. Temperaturedepth distribution in the upper part of convection cell and
overlying layer at various times after commencement of horizontal flow (see Fig. 3).
Uniform thermal conductivity has been assumed and the initial temperature distribution is shown as the uppermost curve, with a fusion zone of 50 km. For
assumptions see text.
For ail three models the progressive fall-off in surface heat flow as a function of x ,
the distance from the uprising current, is shown in Fig. 4. The curves are valid for
other values of u, provided the horizontal distance is scaled by the same factor as
the velocity. To show the effect of cooling on the convection cell itself, temperature
depth profiles for Models 2 and 3 at positions A to E of Fig. 3 are shown in Figs, 5 and 6.
These models can be applied to the problem of heat flow associated with suboceanic mantle convection, subject to the validity of the assumptions. Oceanic heat
1900
-
1000
h * 0.001667Ihm
h a 0.10 cm*/s
500
-
c- overlying layerO
60
K m 10Ke
- - - - - __ - - - -- -convection cell- - - ._- __ - - _ _ - - -
+4
I
0
50
100
150
200
422
M. H. P. Bott
flow is fairly uniform apart from the anomalies associated with ocean ridges (where
convection currents may rise). Any acceptable hypotheses need to be able to explain
this uniformity. The main conclusions from the models are as follows:
(1) If the convection cell reaches the Earth's surface, the surface heat flow would
be expected to fall off approximately as x-) from the uprising current as shown in
Fig. 4 (Model 1). Except in the immediate vicinity of the ridge, observed heat flow
does not fall off in this way. Furthermore, for a realistic velocity of sea floor spreading
of 4cm/y, the heat flow falls off to less than half the observed value at distances over
about 1000km from the uprising current. Thus this hypothesis appears to be inconsistent with heat flow observations unless either convection is restricted to the immediate vicinity of the uprising current or other sources of heat are supplying most of
the observed oceanic heat flow.
(2) The effect of a stationary (or nearly stationary) overlying layer is to make the
pattern of heat flow much more uniform, as shown in Fig. 4 (Models 2 and 3). The
fall-off in heat flow with increasing x is reduced by (i) increasing vx, or (ii) increasing
the thermal diffusivity k . For a 60km thick overlying layer, a velocity of 20cm/y
as shown in Fig. 4 would probably give an acceptable comparison with observations,
but if the diffusivity is a factor of ten higher from radiative heat transfer, a velocity
of 5 cm/y would be acceptable.
(3) Comparison of the computations for Models 2 and 3 (Figs. 4-6) emphasizes
the possible importance of radiative heat transfer in cooling the upper part of the
convection cell. For a given velocity of convection, it makes the surface heat flow
much more uniform and it allows cooling to penetrate much deeper into the convection cell.
Model 1 is probably a reasonably realistic representation of convection reaching
the Earth's surface. Models 2 and 3 show an extreme type of flow pattern, but provided v, is taken as the velocity near the upper surface of the convection cell, more
realistic velocity fields would give rise to an even more uniform pattern of heat flow.
Thus the conclusion that an overlying layer of about 60 km thick is consistent with
observed heat flow would probably be valid for realistic velocity fields. The effect
of non-steady convection which changes pattern at intervals of less than about 100 my
would be to make the heat flow less uniform.
To summarize, the relative uniformity of oceanic heat flow is explicable on the
convection hypothesis provided there is an overlying layer of the order of 60 km thick,
but is difficult to explain simply if the convection current reaches the Earth's surface.
The heat flow anomalies associated with the ocean ridges are not adequately explained
on the simple models of this section, but it will be shown in Section 7 that these can be
explained in terms of physical processes associated with the rising convection current.
6. Comparison of continental and oceanic heat flow on the convection hypothesis
6.1. Model of convection currents passing beneath continental margins
Numerical models of steady state heat flow in mantle convection currents passing
beneath a continental margin are examined in this section. The continental crust is
taken to differ significantly from the oceanic crust in its radioactive content, which is
assumed to raise the sub-crustal temperature under steady state conditions by 200 "C.
Taking the crust as 35 km thick with a uniform thermal conductivity of 0.032 J/cm s "C
and a uniform distribution of heat sources, the resultant contribution to the surface
heat flow from crustal radioactivity would be 3.67 x
J/cm2 s. The same assumptions are adopted as for the oceanic models 2 and 3 of Section 6 with the following
additions:
(1) The upper surface of the convection current has a uniform depth below sea
level (65 km);
423
Terrestrial heat flow and mantle convection
(2) the overlying layer is consequently 60 km beneath oceans and 65 km beneath
continents;
(3) the transition from oceanic to continental crust takes place uniformly over a
horizontal distance of 200 km.
Two extreme models have been studied with k=0.01 and 0.1 cm2/s, the corresponding velocities of the convection current being taken as 20 and 5 cm/y respectively.
The results are shown in Fig. 7 which gives the surface heat-flow distribution and the
heat loss from the upper surface of the convection current.
The two models give similar results. For both, the mean continental heat flow
is significantly higher than the oceanic mean by about 35%, the change taking place
abruptly as the continental margin is crossed. The rate of heat loss from the convection
cell falls by about 20% as the margin is crossed. Both models show a slight rise in the
temperature within the topmost part of the convection cell and a reduction in the
rate at which the heat flow falls off after crossing the boundary.
The calculations shown in Fig. 7 have been repeated for different assumptions of
radioactive content of the continental crust. The results (which are not presented here)
show that both the increase in surface heat flow and decrease in heat loss from the
convection cell as the margin is crossed are approximately proportional to the radioactive content of the continental crust, assuming it to be uniformly distributed.
The most important conclusions arising from these models is that if convection
currents pass beneath continental margins, then (1) there should be an abrupt increase
in heat flow on the continental side of the margin as a result of the radioactivity of the
continental crust, and (2) there should be a corresponding reduction in the rate of loss
00
h = 0 001667/krn
-
-
-
,"
60 -
-6
.A
k = 0 10 cm2/s
K = 10Ko
\
8.
-
7
40
b*
**- - - - - - -m B
-
,, . . .
.
_
.
I
_
II
,
. _ . ,_ . - . * . , . . . . . . .
OVERLYING LAYER
60krn
--+
CONVECTION CELL
--+
,,
-
--I
-+
.
'
-
1
CONTINENTAL CRUST
--+----+
FIG.7. Models showing the distribution of surface heat flow above a sub-oceanic
convection current which flows beneath a continent (curves A) and the heat flow
crossing the upper surface of the convection cell (curves B). The lower graph is
based on uniform thermal conductivity, and a convecting velocity of 20cm/y.
The upper graph assumed the thermal conductivity of the convection cell to be
greater than the overlying layer by a factor of ten, taking the convecting velocity
as 5 cm/y. Assumptions are stated in the text.
424
M. H. P. Bott
of heat from the convection current as it passes beneath a continent. These conclusions
are not sensitive either to the velocity pattern or the thermal properties of the convection cell and are probably generally valid provided the radioactivity of the continental
crust has not been greatly overestimated.
6 . 2 . Explanation of continental heat flow on the convection hypothesis
According to Lee & Uyeda (1965) the arithmetic mean values of oceanic and continental heat flow do not differ significantly. It has usually been considered that this
close agreement is unlikely to be fortuitous. Thus the problem needs discussion in
relation to the convection hypothesis.
The numerical experiments of the last section show that if the average radioactivity
of the continental crust contributed more than about 2.5 x
J/cmZs to the heat
flow, then if convection currents are equally active beneath continents and oceans the
mean continental heat flow should exceed the mean oceanic by more than 20%. The
difference would be further accentuated if the currents flowed from continents towards
oceans.
A difference of more than 20% between the computed mean oceanic and mean
continental heat flow is not in statistical agreement with the observed data. This can
be interpreted in either of two ways:
(1) the continental radioactivity has been overestimated by a factor of two or
more, or
(2) convection currents are primarily active beneath oceans.
The second explanation is adopted here.
The hypothesis that convection currents rise beneath the oceanic ridges and sink at
continental margins is thus supported by the observed heat flow pattern. If continental drift has occurred, it is likely that many continental regions overlie portions of
the mantle where convection was active prior to the last episode of drift but has now
ceased. A contribution to the continental heat flow would come from the progressive
cooling by conduction of the underlying non-convecting upper mantle. The magnitude
of this contribution can be studied by allowing the motion of the convection current
shown in Fig. 7 to die out and investigating the subsequent cooling at the point marked
X (Fig. 8). On this model, in which the continental crust initially contributes slightly
50
0
I
100
TIME (my)
2GO
FIG.8. The surface heat flow at point X of Fig. 7 as a function of time after the
convection current has ceased to flow. Uniform thermal conductivity has been
assumed.
300
Terrestrial heat flow and mantle convection
425
over half of the heat flow, the continental heat flow exceeds the mean oceanic by 20%
after 60 my and is approximately equivalent after 240 my.
If this model is realistic it shows that the distribution of terrestrial heat flow is
consistent with the hypothesis that mantle convection currents are at present active
beneath oceans but are absent beneath most continental regions: and that the continents are in general overlying parts of the mantle which ceased convecting between
60 and 500my ago. This is consistent with major continental drift between the
Carboniferous and the early Tertiary.
There are some continental regions, such as eastern Australia (Sass & Le Marne
1963), where the heat flow is significantly and systematically higher than the mean
continental value. These regions may possess a crust or mantle with an anomalously
high content of radioactive heat sources, but an alternative explanation is that they
overlie active (or recently active) mantle convection currents. It would be of interest
to measure the heat flow in the adjacent ocean.
The continental shield areas show anomalously low heat flow. It is possible that
these overlie portions of the mantle which have not been subject to convection since
the Pre-Cambrian, and have thus cooled more than the average sub-continental mantle.
7. Partial fusion in relation to mantle convection
1.1. The mechanism of partial fusion
Basaltic igneous activity in oceanic regions shows conclusively that some partial
fusion does take place in the upper mantle. The convection hypothesis requires that
the most widespread fusion occurs in the rising current, since temperatures at a given
depth are highest here.
Melting relations of eclogite (Yoder & Tilley 1962) and the melting point gradients
of silicate minerals appropriate to ultrabasic rocks suggest that the fusion gradient
in the upper mantle is about 1.3-3*O°C/km; this is an order of magnitude greater than
the adiabatic gradient (about O.l-O.4"C/km) and an order of magnitude less than
the near surface conduction gradient. Partial fusion in the rising current can thus
only occur above the depth at which the fusion curve intersects the near adiabatic
temperature distribution holding at greater depths. Above this depth the temperature
is controlled by the fusion gradient (more strictly, lies within the fusion range) until
partial fusion is complete.
Taking the minimum estimate of the difference between adiabatic and fusion
gradients as l.O"C/km, the specific heat as 1*2J/gand the heat of fusion as 400J/g,
the amount of fusion is 0.3%/km, or 10% fusion in 33 km rise. This is obviously a
powerful mechanism for magma production on a large scale.
A zone of partial fusion at the upper surface of the convection cell has further
significance in that it provides a zone of low strength favourable as a contact between
the cell and the overlying layer as previously mentioned.
7.2. Origin of belts of high heat-pow
The source of most anomalously high heat flow values associated with ocean ridges
can be shown to be of shallow origin because of the sharpness of many of the anomalies
and because otherwise unrealistically high temperatures would exist in the upper mantle
beneath. There is now almost unanimous agreement that they are caused by near
surface magmatic intrusions (Bullard 1963, McBirney 1963, Von Herzen & Uyeda
1963).
The hypothesis outlined in the previous paragraph provides a mechanism for the
development of magma on an enormous scale beneath the ocean ridges (Bott 1965b).
Since partial fusion involves a reduction in density of the order of 10% for the fused
fraction, the low density magma is gravitationally unstable and tends to rise when a
sufficiently large fraction has been produced. This magma rises into the overlying
topmost mantle and crust as dykes, magma chambers, etc. where it causes the high
426
M. H. P. Bott
heat flow values associated with the ridges. A fraction of the magma may remain
trapped in the convection current as it turns to flow horizontally.
It has been suggested (Bott 1965b) that magma in a rising convection current also
causes the present uplift of the oceanic ridges. It can be shown that development of
a 10% magma fraction produces a mass deficiency of about - 0.03 g/cm3. If such a
mass deficiency has a vertical extent of 100-200 km it causes a gravity anomaly of
100-250mga1, which is the right order of size to account for the isostatic uplift of
Iceland (Bott 1965a) and the oceanic ridges.
7 . 3 . Belts of relatively low heat flow adjacent to ocean ridges
These belts are inexplicable if the uniform Newtonian viscous model for the mantle
is assumed, since the highest temperatures correspond to the upwelling current. If,
however, a plastic mantle is assumed, Orowan’s model shows that the upper surface
of the convecting material is deeper over the upwelling current than over the horizontal
flowing part (Fig. 1, T and S).
Such variation in the depth to the upper surface of the convection cell in the vicinity
of the rising current provides an explanation of the belts of low heat flow adjacent
to the ocean ridges. The temperature of the upper surface should follow the fusion
gradient approximately; thus in the absence of magmatic activity the heat flow through
the overlying layer should be nearly inversely proportional to its thickness. In the
absence of magmatic activity the heat flow would be a minimum over the centre of
the rising current and low values should extend for a considerable distance on either
side. As postulated earlier, heat from near-surface magmatic sources substantially
increases the heat flow above the rising current, but in the adjacent lateral belts where
magmatic activity is less conspicuous below average heat flow is to be expected (Fig. 9).
A further implication is that partial fusion should continue to take place until the
flow becomes horizontal. This helps to keep the subsequent heat flow more uniform.
’
2000
0
1500
1000
OVERLYING LAYER
I .
100
-2
1
. . . . .
. . . . . . .
. . . . .. .. .. .. .
I
I-
:200
n
*
..
. .. .. .. .
. .
v
km
0
500
.
I
.
.
. .
.
. .
I . . . . . .
’
adidbatic
. partial
fusion
. .,
. .
gradlent
300
I
I
FIG.9. Illustration of the hypotheses advanced to account for the anomalously
high heat flow of oceanic ridges and for below average values in the adjacent belts.
Terrestrial heat flow and mantle convection
427
8. Conclusions and discussion
This paper has shown that the distribution of terrestrial heat flow is consistent
with the mantle convection hypothesis, but, subject to the assumptions of the model,
it places limitations on the allowable patterns of convection, as outlined below.
It has been shown that convection currents almost certainly do not reach the
Earth’s surface at the present time. Otherwise a less uniform pattern of oceanic heat
flow would be observed. Convection cells need to be overlain by an effectively
stationary layer which (on magmatic grounds) is probably about 60 km thick. The
relative uniformity of oceanic heat flow can be understood provided the velocity of
the convection current is sufficiently high; a velocity of about 20cm/y is allowable
if thermal diffusivity remains constant with depth, and about 4cm/y if it increases
by a factor of ten below 60 km depth.
This conclusion conflicts with the suggestion made by Dietz (1961) that continental
drift takes place by ocean floor spreading, with the ocean floor forming the upper
surface of the convection cell. This mechanism is outruled as a present-day process.
Continental drift is more likely to take place as a result of the shear stress exerted by the
convection cell on the overlying layer, which causes tension over the uprising current
and compression between sinking currents. Vertical tension fractures occur above the
uprising current, which are filled with magma; at the other end the compression is
taken up either by crustal shortening (and mountain building) or by compressive
fracture with part of the crust being carried down with the sinking current. It is envisaged that drift according to this mechanism takes place at an order of magnitude slower
than the velocity of the convection current.
It has been shown that widespread partial fusion is likely to occur in the top section
of an uprising current. This is fully consistent with the idea that convection currents
rise beneath oceanic ridges causing their abundant magmatic activity and providing
also a mechanism for their uplift. The local high heat flow is explained by near-surface
magma chambers. The belts of relatively low heat flow adjacent to ocean ridges can
be explained by the convection pattern expected in material with a finite creep strength,
causing a deeper upper surface of the convection cell above the rising and sinking
currents.
Unless the radioactivity of the continental crust has been overestimated by a factor
of two, numerical experiments show that convection currents cannot be equally active
beneath oceans and continents. They may be present beneath most of the oceans,
rising beneath oceanic ridges and sinking near the continental margins. They appear
to be absent beneath most continental areas, excluding anomalous regions of high heat
flow such as the western United States and East Africa, where the ocean ridge system
passes into a continent. They may also flow beneath south-east Australia, explaining
the anomalously high heat flow there.
The magnitude of average continental heat flow is consistent with the suggestion
that convection currents were active in most of the mantle at present underlying
continents until about 60-500my ago, when they ceased. The continental heat flow
is thus made up from the radioactive contribution from the crust supplemented by the
progressive cooling of parts of the mantle which ceased convecting during the late
Palaeozoic or Mesozoic. This is consistent with continental drift, the continents
moving onto parts of the mantle which were previously sub-oceanic and causing convection to die out as suggested by Bott (1964). The anomalously low heat flow of the
Canadian and Ukrainian shields can be understood if the underlying mantle has not
been subject to convection since the Pre-Cambrian, causing the contribution to the
heat flow from mantle cooling to be unusually small.
It is concluded that the pattern of terrestrial heat flow is in harmony with largescale mantle convection cells underlying a layer about 60 km thick, and rising beneath
ocean ridges and sinking near continental margins. Convection is mainly absent
beneath continents, apart from a few anomalous regions.
2a
428
M. H. P. Bott
9. Acknowledgments
I am grateful to Sir Edward Bullard, Mr J. Tanner, Dr D. C . Tozer and Dr D. J.
Tritton for reading the manuscript and for making helpful suggestions. All calculations have been performed by the Durham University Elliott 803 computer and
acknowledgment is made for the facilities provided.
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