Download constraints on heat production and thickness of continental plates

Geophys. J. R . astr. SOC. (1982) 69,623-634
Old continental geotherms: constraints on heat
production and thickness of continental plates
Geoffrey F. Davies and John W.Strebeck Department ofEarth
and Planetary Sciences, and McDonnell Center for the Space Sciences,
Washington University, St Louis, Missouri 63130, USA
Received 1981 August 5 ; in original form 1981 March 12
Summary. Absolute and mutual constraints on radioactive heat productivity,
plate thickness and steady state continental geotherms are demonstrated for a
model in which heat productivity comprises a near-surface, exponentially
decreasing component and a component independent of depth. For a given
surface heat flux there are separate upper limits on each component and a
limiting trade-off curve between them. Below these limits there are linear
trade-offs imposed if either the plate thickness or the temperature at a given
depth are known, and, in principle, tighter bounds are imposed on each
component. Preliminary application of the model shows that a kimberlite
pyroxene geotherm requires a plate at least 200km thick and that lower
plate (upper continental mantle) heat productivities may be (but are not
required to be) as great as about 0.1 yW m3, which is about ten times the heat
productivity of chrondrites.
The estimation of continental geotherms depends strongly on assumptions about the vertical
distribution of radioactive heat sources in the continental plates and on whether or not a
steady state is assumed to have been achieved. The thickness of the plate is also directly
involved, since the base of the plate can be defined as the depth at which a characteristic
temperature is reached. It is the purpose of this paper to demonstrate, using a simple but
plausible model, the relationships between temperature, heat production and plate thickness, including some absolute limits on heat production. The relationships can be used to
evaluate the implications of various independent estimates of these quantities, and some
examples are discussed here.
An obvious limit on heat production is that in the steady state no more heat can be
generated in the plate than emerges at the surface. A constraint with less obvious implications is that the geotherm must reach the mantle temperature at a reasonable depth.
These constraints will be demonstrated here using a simple model of heat source distribution which is nevertheless plausible enough to give quantitatively useful results.
Attention will be confined here to steady state models. Although the possibility of
indefinitely continued cooling and thickening of the continental plates has been discussed
(Pollack & Chapman 1977), a severe problem with this hypothesis is that the associated
thermal subsidence should be accompanied by lOkm or more of crustal thickening, a
624
G. F. Davies and J. W. Strebeck
prediction contradicted by the exposed Precambrian shields (Jordan 1978, 1982; Oxburgh
& Parmentier 1978). The steady state models are presumed to be applicable to Precambrian
plates, in which there is no evidence for the thermal transients associated with tectonism
which are observed in Phanerozoic regions (Sclater, Jaupart & Galson 1980).
The questions of continental plate thickness and radioactive heat production in the subcontinental mantle have assumed considerable significance in recent years. Jordan (1978,
1982) has proposed that continental cratons are underlain by a thick (> 150 km) chemically
distinct mantle root which, together with the overlying crust, comprises the ‘tectosphere’.
The tectosphere controls the thermal boundary layer thickness and its formation may have
been intimately related to the stabilization of the cratons (Jordan 1978, 1982; Davies 1979).
On the other hand, others have proposed that there is no intrinsic difference between continental and oceanic lithosphere (e.g. Sclater et al. 1980). It will be demonstrated below that
the existence of a thick thermal boundary layer may require significant heat generation
within it, so it is significant that a growing body of geochemical and petrological evidence
indicates that the subcontinental mantle has undergone extensive secondary reenrichment
by volatiles and large-ion lithophile (LIL) elements (e.g. Lloyd & Bailey 1975; Frey, Green &
Roy 1978; Menzies & Murthy 1980). Jordan (1982) estimates a heat production rate of
about 0.15pW mF3 for an ‘average continental garnet lherzolite’, sufficient to produce a
substantial contribution to surface heat flow. Such LIL enrichments in the subcontinental
mantle could also comprise a significant reservoir for radiogenic Nd, and would substantially
increase estimates of the fraction of the mantle which is LIL-depleted (Davies 1981). Thus
these questions are important to our understanding of the evolution of the Earth and the
origin and stabilization of continental crust.
The term ‘plate’ is used here to mean the crust and that part of the mantle which moves
as a unit with the crust, as in the original meaning. It is necessary to distinguish plate from
lithosphere because lithosphere can be defined mechanically and thermally in several distinct
ways, and also because Jordan (1978, 1982)‘has pointed out that the plate might be stabilized not only by mechanical strength but also by neutral buoyancy. ‘Plate’ thus means the
same as Jordan’s ‘tectosphere’ but avoids the suggestion of sphericity. Since, by definition,
there are no internal motions in the plate, heat transport through it must be by conduction,
and it therefore comprises the conductive thermal boundary layer overlying the convecting
mantle.
Model
The heat source distributions considered here are assumed to comprise a component which
decreases exponentially with depth and a component which is independent of depth. Thus
S =Aoexp(-z/D) t B,,
(1)
where S is heat production rate per unit volume,Ao and Bo are surface values of the two
components, z is depth and D is a constant. The exponential component has been deduced
from the observation that in several provinces there is linear correlation between surface
heat flux, qo, and surface heat production, So (Roy, Blackwell & Birch 1968; Lachenbruch
1968). It should be emphasized that the exponential form depends on assuming that the
variations in So and 40 are due to differential erosion, and on assuming steady state conduction. Nevertheless, the exponential form can be viewed simply as a reasonable representation of the very general tendency of the heat producing elements (U, Th, K) to concentrate
in the upper crust (e.g. Lachenbruch & Sass 1977; Taylor 1979; Shaw 1980).
The constant term, Bo, in (1) is intended to represent the small amount of radioactivity
expected to be present in the lower crust and upper mantle. Although lower crustal heat
Continen tal geo therms
625
generation may be significantly greater than in the upper mantle, any such excess over the
value of Bo can be accomodated by increasing the value of A , without strongly affecting
sub-crustal temperatures, as will be shown later. It is being more explicitly acknowledged at
present that the lower crust is probably very heterogeneous (e.g. Taylor 1979; Shaw 1980),
so that estimates of mean heat production are difficult. Such estimates have tended to be
reduced in the last decade: e.g. Heier & Thoresen (1971) estimated a mean U content of
0.5 pg g-' , while Shaw (1980) estimated 0.4pg g-' and Taylor (1979) estimated 0.25 pg g-' .
Table 1 summarizes estimates of heat productivity based on these and other abundance
estimates.
Surface heat production, A , and mantle heat production, B o , are treated as variable parameters here. For reference, it may be noted thatAo varies in the range of about 1-7pW m-3,
with a mean of about 2pW m-3 (Sclater et al. 1980). Mantle heat productivities have long
been thought to be very small, in the range 0.01 -0.03 pW m-' (e.g. Clark & Ringwood 1964;
O'Nions er al. 1978; Davies 1980a), but Jordan (1982) has suggested that the productivity
may be as high as 0.15pW m-3 in the lower continental plate.
The steady state conduction equation in the absence of material motion is (e.g. Officer
1974)
d2T
K--+S=Q,
dz2
where K is the conductivity and T is temperature. It will be assumed here that K is constant.
Some of the data of Schatz & Simmons (1972) show variations in K by about a factor of
two with minima between about 1000 and 15OO0C, but others show little variation.
Shankland, Nitsan & Duba (1979) showed that the radiative contribution, which accounts
for the high temperature increase in K , may have been overestimated. Variations in K
between materials were also 50 per cent or more (Schatz & Simmons 1972). These uncertainties make it premature to adopt a more elaborate functional form for K , and the
constant value of 3 W(mK)-' will be used for most of the present calculations. The effect
of the uncertainties will be indicated by repeating some calculations with K = 4.5 W(m K)-'.
The boundary conditions on Tare
T = T o at z = O ,
(3)
Table 1. Estimates of uranium concentrations and heat production in the continental crust.
Source
Upper
Lower
Mean
-
0.5
Uranium concentration ( pg g-')
Heier & Thoresen (1971)
Taylor (1977)
Taylor (1979)
Shaw (1980)
-
-
2 .5
2.5
0.25
0.4
1.o
1 .o
1.5
Heat production (pW m-3)
Sclater & Francheteau (1970)
Taylor (1979)*
Shaw (1980)*
1.2
2 .o
2.0
0.24
0.2
0.3
0.8
1.2
-
*Calculated from U abundance assuming Th/U = 4 g g-' and K/U = 104g g-' (e.g.
Wasserburg et al. 1964; O'Nions et al. 1978; Shaw 1980). This gives 0.82rW m-3
for a U concentration of 1 Mg g-'.
G. F. Davies and J. W. Strebeck
626
and an auxiliary condition defines the bottom of the plate:
T = T m at z = H .
Here 4 is the conducted heat flux, measured positive downwards, T , is the temperature of
the mantle under the plate, and H is the thickness of the plate. Since 40 is a negative quantity for an upward surface heat flux, it is convenient to define the positive quantity
qo*= - 4 0 .
Note that, in principle, any pair of the three equations (3-5) could be used as boundary
conditions. Equations (3) and (4) are chosen because To and 40 are observed directly at the
surface, so that solutions are thereby selected which are consistent with these surface observations. Equation ( 5 ) then serves to define H , for a given value of T,, but the resulting
value of H will depend on the distribution heat sources, i.e. on A . and Bo. An explicit
relationship between H, A . and Bo will be derived below (equation 12), and this will be the
basis for the subsequent discussion.
The general solution to ( 2 ) , subject to equations (1, 3 and 4), is
Values of parameters used here are summarized in Table 2. The values of qz and D are from
Sclater et al. (1980) for a typical shield. The value of the mantle temperature, T,, is based
on an estimate of mid-ocean ridge magma temperature of about 1350°C km-'(Stacey 1977).
Special and limiting cases
In this section the effects of the two components of the heat source distribution (1) are
demonstrated separately as a preliminary to examining their combined effects.
U N I F O R M HEAT SOURCE
(A,=())
Geotherms are quadratic in this case (equation 6), and examples for various values ofBo are
illustrated in Fig. 1. Of course, regions with negative temperatures or negative gradient have
no geophysical significance, and must correspond to where heat is transported by convection, rather than conduction, as assumed here. With a fixed surface heat flux, larger values of
Bo yield lower maximum temperatures, and the maxima occur at shallower depths. This
corresponds with the fact that there is zero vertical heat flux at the temperature maximum
and a thinner layer above this point is required to generate the surface heat flux when Bo is
larger.
There is an upper limit imposed on Bo by requiring T to reach T , . This limit, B, , occurs
when the maximum T is equal to T , , and from ( 6 ) it is
B, =q$/2KT,
(7)
Table 2. Parameter values used in the model.
Symbol
Quantity
Value
To
Tm
K
Surface temperature
Mantle temperature
Conductivity
Surface heat flux
Depth scale of heat sources (equation 1)
0"C
1400°C
3 W (m K)-'
46 mW m-'
7km
q:
D
Continental geotherms
627
DEPTH (km)
Figure 1. Geotherms for the special case where the heat source distribution is uniform: A , = 0 in equation
(1). Curves are labelled by the value of the heat source concentration, B,, in pW m-3. In all cases the
surface heat flux is 46mW m-' (Table 2). Regions with negative temperature or gradient have no
geophysical significance (see text).
In this limiting case all of the surface heat flux is generated in the plate, which has a thickness H,,
=q:/B,. In the other limiting case, B o = 0, all of the surface heat flux comes
For the values in
from below the plate, which has a thickness Hmh =KTm/q,*=H,,,/2.
Table 2 , B , = 0.252yW m-3,Hmi, = 91 km andH,,,
= 182 km.
EXPONENTIAL HEAT SOURCE
(Bozo)
In this case geotherms become linear for z D , and examples for various values of A , are
illustrated in Fig. 2 . There is an upper limit imposed on A , by requiring that the geotherm
never has a negative slope. From ( 6 ) , the limit is
Am
=q3D
(8)
but in this case the plate thickness is unbounded. In this limit all of the surface heat flux is
generated by the exponential component, concentrated near the surface, and there is no heat
flux from below. The lower limit, A , = 0, is identical to that for Bo = 0. For the values in
Table 2 , A , = 6.57yW m-3.
DEPTH (km)
Figure 2. Geotherms for the special case where the heat source concentration decreases exponentially
with depth: B , = 0 in equation (1). Curves are labelled by the value of the surface heat production, A , , in
pW m - 3 .
628
G. F, Davies and J. W. Strebeck
General case
The limits (7) and (8) can be viewed as arising from the requirement that no more heat be
generated in the plate than emerges at the surface, or from the requirement that T reaches
T,,, at a finite and real value of z. These concepts illuminate the combined case, with S given
by equation (1).
It is useful t o separate the heat flux from below the plate, q b , from the heat generated
within the plate, q p .For H B D (the usual case),
q p = A o D + BoH.
(9 )
In the steady state,
q o * = q p' q b
and the upper limit on q p occurs when qb = 0:
(10)
qo* ' q p = A o D + B o H ( A o , B o )
(1 1)
where the dependence of H on A , and Bo (through the definition 5) is shown explicitly to
emphasize that A . and Bo enter this relationship non-linearly. Applying condition (5) to the
general solution (6) yields (again for H * D )
T, = T o+ A o D 2 / K+ H ( q & A o D ) / K - H 2 B o / 2 K .
Eliminating H between (12) and the limiting condition (1 1) yields
Bo = ( 4 : - A 0 D ) 2 / 2 (KT,
-
AoD2).
(13)
Figure 3. Trade-offs between the uniform (B,) and exponential (A,) contributions to the total heat
source concentration (equation 1) under various constraints. Upper curve corresponds to all of the surface
heat flux (46mW m-*) coming from within the plate: points above are not permitted in the steady state,
points below correspond to some heat coming from below the plate. Solid straight lines are trade-offs for
a specified plate thickness (given next to line in kilometres). Dashed lines are trade-offs when geotherms
are required to pass through 1000°C at a specified depth (given next t o line in kilometres). Shaded region
shows estimates of chronditic heat source concentrations (Davies 1980a) for comparison.
Continental geotherms
I
I
I
I
629
I
DEPTH (km)
Figure 4. Geotherms illustrating three types of trade-off between A , and B , : each curve is labelled by the
values of ( A , , B , ) in pW m-3. (a) Trivial case, A , = B , = 0, all heat comes from below. (b) Range of
possible geotherms when the trade-off line intersects both axes of Fig. 3: some heat always comes from
below. (c) Range of geotherms from the limit when no heat comes from below (upper) to the limit
B , = 0; note large temperature range.
This is the relationship between A , and Bo at the limit where all of the surface heat flux is
generated within the plate and it can be viewed as a tradeoff curve. It can be seen that (13)
Ieduces to the special limiting cases (7) and (8). Fig. 3 shows the relationship (13) in a plot
of A . versus Bo. All points above this curve exceed the limit (1 I), while for all points below
the curve, 4 b is greater than zero.
The implications of points in the allowed region in Fig. 3 can be clarified by noting that
for a given value of H , equation (12) yields a linear relationship between A, and Bo: such
lines are shown (solid) in Fig. 3 for various values of H . The lines can be grouped into three
types.
The first type is the trivial case A , =Bo=O,which yields the minimum value Hmin
discussed above and in which all of the heat flux comes from below the plate. The unique,
linear geotherm for this case is shown in Fig. 4 (for the values given in Table 2).
The second type of line comprises those which do not intersect the limiting curve given
by (13), corresponding to values of H between 91 and 182 km. A trade-off between A, and
Bo is possible for a given value of H , and a corresponding range of geotherms is possible.
There are separate upper limits o n A o and Bo where the line intersects the axes. The range of
geotherms for H = 125 km is included in Fig. 4: q b is non-zero throughout this range.
The third type comprises those lines which intersect the limiting curve. For these cases
there is an upper limit on Bo at which all of the surface heat flux is generated in the plate
( q p = q:, 4 b = O), and A , reaches a lower limit at this point. There is also an upper limit on
A0 when Bo = 0. The range of permissible geotherms when H = 300 km is shown in Fig. 4:
the extreme values of ( A , , B , ) are (2.7, 0.09)pW mW3and (4.7, 0)pW m-3. Note that the
lines crowd together for larger values of H , consistent with the fact that H becomes infinite
at A . = 6.57pW m-3.
For the larger values of H , especially the third type, the range of possible geotherms
permits a large range of temperatures in the middle and lower plate. In such cases, or in
cases where the thickness of the plate is unknown, it is desirable to use other constraints on
630
G. F. Davies and J. W. Strebeck
the geotherm such as those derived from petrology. For example, Boyd's (1973) kimberlite
geothermometry and geobarometry data can be approximately represented by requiring the
geotherm to pass through 1000°C at 150km depth. Such a constraint is analogous to the
condition (5): we require
T=Tdatz=d.
(14)
Applying this to the general solution (6) yields a linear A . - Bo relation analogous to (12),
and another family of straight lines in Fig. 3 : the dashed lines are for Td = 1000°C and
various values of d . The role of these lines is analogous to the solid lines. Fig. 5 shows the
range of permitted geotherms for d = 150 km. In this example, A . is confined to the range
3.0-3.9pW m-3, while Bo is confined to the range 0-0.8 pW me3.
In principle, knowledge of both (d, T d ) and (If,T,) pairs would allow A,,, Bo and the
geotherm to be uniquely defined, but of course they are unlikely to give very strong constraints in practice because of uncertainties in the data. Nevertheless, their combined constraints may be stronger than either one taken individually. It should be noted also that the
result would be unique only within the limits of the simple form of the distribution assumed
in equation (1).
Limitations of the model
This type of model is only useful if its results are not too sensitive to the assumptions upon
which it is based. The results may be most sensitive to variations or uncertainties in the
conductivity, to deviations from the assumed uniform mantle heat source distribution, and
to the total crustal heat productivity, but not to the distribution within the crust. These
will be discussed in reverse order.
The total crustal heat productivity determines the temperature gradient at the Moho and
so will obviously have a substantial influence on the depth at which T reaches T , . However
the vertical distribution of crustal heat sources has only a weak influence on Moho temperature. Consider the example A . = 3pW mW3,D = 7 km. If Bo = 0, then the temperature at
I
I
I
I
I
I
1
DEPTH (km)
Figure 5. Range of geotherms consistent with the constraint that T = 1000°C at 150km depth. Curves are
labelled with values of ( A , , B , ) in W m-3.
Continental geotherms
63 1
35 km depth is, from equation (6), 341°C. If the same crustal productivity (21 mW m-') is
distributed uniformly through the crust (B, = 0.6pW m-3,Ao = 0), the Moho temperature is
415"C, 74°C higher than the first case. Thus the mantle geotherm is unlikely to be shifted
by more than about 70°C by this kind of uncertainty.
A greater concern is that the total crustal heat productivity may be inaccurately
estimated if it is based on surface productivity measurements. Since the model assumes that
lower crustal productivity is equal to the mantle productivity, which is about 0.1 pW m-3 or
less (Fig. 3), it is likely to underestimate the total crustal productivity (see Table 1). If, for
example, A , is constrained by surface measurements, and the lower crustal productivity is
underestimated by 0.2 pW m-3, the total productivity would be underestimated by about
6 mW m-2. The effect of adding this amount would be similar to that of increasing A , by
about 1pW m-3 (for D = 7 km). The effect of changing A , from 2 to 3 pW m-3 with Bo =
0.05pW m-3 can be seen from Fig. 3: the calculated plate thickness is changed from about
140 to 200 km, a substantial change.
It is clearly an important assumption that the mantle heat productivity is uniformly
distributed. Concentrating the same total heat production near the top or the bottom of
the lower plate would change the geotherm by hundreds of degrees. This must be regarded
as a prime hypothesis to be tested by future refinements of the observations.
Uncertainties in conductivity are important, although the general character of the results
of this model is unaffected. Fig. 6 shows the A , - Bo plot which results from assuming
A,
(yW/rn3)
Figure 6. As in Fig. 3 , but with conductivity K = 4.5 W (m K)-I rather than 3 W (mK)-'.
632
G. F. Davies and J. W. Strebeck
K = 4.5 W (m"C)-', SO per cent greater than for Fig. 3 and at about the upper limit of the
measurements of Schatz & Simmons (1972). The main effects are to increase the calculated
plate thickness and to decrease the permissible range ofBo (as is required by equation 7). If,
as is likely, K is also a function of temperature (Shatz & Simmons 1972), then results would
probably be intermediate in character between those of Figs 3 and 6. Since apparently such
high-temperature data on relevant materials are still sparse, and high-pressure data are nonexistent, more detailed models were judged to be not justified at present.
Preliminary constraints on plate thickness and heat production
Perhaps the most direct constraints on these models come from petrological estimates of
mantle geotherms. For example, the range of pyroxene geotherm points based on mantle
inclusions in kimberlites (e.g. Boyd 1973; Danchin & Boyd 1976) is indicated in Fig. 5. As
discussed earlier, these data, and the parameters of Table 2 , would require A . to be in the
range 3-41.1W m-3 and Bo in the range 0-0.1 pW md3 (Fig. 3). This range for A . is rather
high compared with the estimates in Table 2: if either K , D or lower crustal heat production
has been underestimated, then it should be reduced. Note also that the product A o D
(21-28mW m-') is comparable to the heat production of a 12km layer with the heat
productions of Table 1. In any case, the plate is required to be at least 200 km thick (Figs 3 ,
5 and 6) and may be 300 km thick (Figs 3 and 5). Also in any case, Bo is permitted to range
up to 0.1 pW m-3. For comparison, the estimated range of heat productivity of chondrites,
assuming K/U = 104g g-', is 0.012-0.018pW m-3 (e.g. Davies 1980a), and this range is
shown in Figs 3 and 6. It is clear that lower plate heat productivities up to as much as ten
times chondritic are permitted by this model.
Jordan (1982) has summarized seismological and geochemical evidence supporting the
existence of a plate at least 150 km thick with a heat productivity in the mantle part of the
order of 0.15pW m-3. The values are clearly quite compatible with the results found here,
and would in fact require A . to have rather small values and most of the surface heat flux to
be generated within the plate.
On the other hand, Sclater et al. (1980) have suggested that both continental and oceanic
plates approach an equilibrium thickness of 125-150 km. This requires the assumption of
low values of B o , but it is equally compatible with the constraints deduced here. However,
the hypothesis would seem to have more difficulty contending with the seismological
evidence for differences between continental and old oceanic plates (Sipkin & Jordan 1981;
Jordan 1982), and with the fact that there is no direct evidence that oceanic lithosphere
approaches a steady thermal state (Davies 1980b).
Conclusions
The simple model of vertical heat source distribution used here, comprising an upper crustal
component and a uniform component, permits a relatively simple demonstration of the
relationships between heat source concentrations, geotherms and the thickness of old (i.e.
steady state) continental plates. The A . -Bo plots (Figs 3 and 6) show the mutually constraining effects of observations of surface heat flux, surface heat productivity, plate thickness and geothermometry . These characteristics will be retained by more elaborate models,
although numerical values will differ.
A prime assumption of the model is the uniform heat productivity of the lower plate, and
this must be regarded as a first approximation to be refined by further observation. The
other main source of uncertainty in the results is probably the conductivity value assumed,
including the assumption that it is constant.
Contin en tal geo th erm s
633
A preliminary application of observational constraints shows that geotherms deduced
from kimberlite pyroxenes require the plate to be at least 200 km thick and allow lower
plate heat productivity (Bo) to range from essentially zero up to about ten times chrondritic
values. The preferred parameter values used here lead to the higher Bo values, but this can be
avoided by assuming higher conductivity or higher crustal heat production.
It is important that high values of Bo are permitted because, although mantle inclusions
in deep-seated volcanic rocks also indicate high Bo values, it is far from clear that these are a
representative sample of the lower continental plate. If high Bo values could be established
with some confidence, it would imply that such enrichments of heat sources are widespread.
This in turn would suggest that other incompatible elements are enriched to a comparable
degree, so that the lower continental plate would be a significant reservoir of mantle
differentiates (Jordan 1982; Davies 1981).
Acknowledgments
We thank Henry Pollack for constructive comments. This research was supported in part by
National Science Foundation grant EAR80-08835.
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