Geophys. J. R. astr. SOC. (1981)66, 535-552
Uplift by thermal expansion of the lithosphere
Jean-Claude Mareschal
School of Geophysical Sciences, Georgia Institute of
Technology, Atlanta. Georgia 30332, USA
Received 1980 October 24;in original form 1980 July 14
Summary. Green’s function solutions are derived for the surface heat flow
and the uplift caused by conductive heating when (axially symmetric)
variations in heat flow occur at the base of the lithosphere. The solutions
determined for different space-time dependence of the heat flux into the
lithosphere illustrate that: (1) When the lateral variations in heat flux occur
over short distances (in comparison to the lithospheric thickness), the
amplitude of the uplift and of the anomalous surface heat flow are attenuated
in comparison to the changes that would follow uniform variations; this
effect is slightly more important for the surface heat flow which spreads out
over larger distances. (2) There is a time lag between the uplift which starts
instantly and the change in the surface heat flow.
It is suggested that the model could be applied to the study of plateaux
uplift of moderate amplitude such as the Black Hills or the Adirondacks, and
possibly to the heating event and uplift that may have preceded the
subsidence of intracontinental basins.
1 Introduction
The major topographic features of the seafloor and the variations in oceanic heat flow have
been explained within the framework of plate tectonic theory. Following analyses by
McKenzie (1967) and by Turcotte & Oxburgh (1967), theoretical models of the cooling and
subsidence of the oceanic lithosphere as it moves away from the oceanic ridges have been
examined by many authors (e.g. Sclater & Francheteau 1970; Parker & Oldenburg 1973;
Davis & Lister 1974; Oldenburg 1975; Parsons & Sclater 1977). The heat flow values and
the topography of the oceanic floor predicted by these various models are in relatively good
agreement with the observations and these models provide a successful test of quantitative
predictions based on plate theory.
The situation within the continents is far from being as simple as in the oceans and, at
this point, the global features of intracontinental heat flow and topography are not
explained by a simple physical model. Basin subsidence (such as occurred in the Michigan
basin) or plateau uplift (such as the Colorado plateau or, on a smaller scale, the Black Hills
or the Adirondack uplift) are examples of simple tectonic events that appear to involve only
536
J. - C Mareschal
vertical crustal motion. There is no universally accepted explanation for such events,
although there is general agreement that isostatic equilibrium is achieved and that the
mechanism of uplift and subsidence is related to changes in the thermal regime of the
lithosphere.
One of the older explanations depends on the presence of phase transitions in the lithosphere (e.g. Lovering 1958; Kennedy 1959). Phase changes could provide the mechanism of
subsidence and uplift of the Earth surface with the amplitude and time history observed in
sedimentary basins (e.g. OConnell & Wasserburg 1967, 1972; Mareschal & Gangi 1977;
Mareschal 1978). However these models are based on the assumption that the Moho is a
phase boundary and they have not been widely accepted; because of objections raised by
petrologists (e.g. Ringwood & Green 1966) and despite some dissenting opinions (e.g.
Ito & Kennedy 1970), few geophysicists accept the hypothesis that the Moho is a phase
change (eg. McKenzie 1969). An alternate model involving phase changes has been suggested
by Haxby, Turcotte & Bird (1976).
Another class of models assumes that the uplift and subsidence are the result of thermal
expansion and contraction of the lithosphere. Models of subsidence of marginal basins by
cooling of an initially hot lithosphere have been examined by Sleep (1971, 1973), Turcotte
& Ahern (1977) and Turcotte & McAdoo (1979). A model of the subsidence of the Michigan
basin caused by cooling and thermal contraction of the initially hot lithosphere has been
examined by Sleep & Snell (1976) who assumed that the initial heating event had been
caused by bulk replacement of the lower lithosphere by hot asthenospheric material.
Numerical models of the conductive heating and uplift caused by thermal expansion of the
lithosphere have been presented by Crough & Thompson (1976). Lithospheric thinning and
uplift in rift zones following the convection of heat by magma intrusions in the lithosphere
have been examined by Withjack (1979). McKenzie (1978) and Jarvis & McKenzie (1981)
propose that mechanical stretching of the continental lithosphere is the initiating mechanism
in basin formation; slow subsidence would follow the conductive cooling and thermal
contraction. Alternatively, Bird (1979) suggested that a gravitational instability is present
between the lower lithosphere and the asthenosphere and that replacement of the cold dense
lithospheric mantle by hot asthenospheric material could occur in a short time with as
immediate effects uplift of the Earth surface and increase in heat flow. This might explain
the Colorado plateau uplift. It could also provide the mechanism that had brought the initial
conditions assumed by Sleep and Snell(l976) in their Michigan basin subsidence model.
This paper examines the uplifts caused by thermal expansion following the conductive
heating of the lithosphere when the heat flow varies at the base of the lithosphere.
Figure 1. The geometry of the problem treated: the coordinate system, basic equations and boundary
conditions.
Thermal expansion of the lithosphere
537
Analytical solutions describing the uplift and the change in surface heat flow are derived in
the form of Green’s functions, under the assumption that the problem is axially symmetric.
The major features of the solutions are: (1) The lateral variation effects are important only
when the ‘wavelength’, of the heat flux anomaly at the base of the lithosphere is smaller
than or of the order of the lithospheric thickness. (2) The uplift starts instantly but the
surface heat flux is not affected before a time of the order of the thermal time constant of
the lithosphere. Some solutions with specified heat flux history at the base of the
lithosphere are examined in detail to illustrate these effects. The possible geological
applications of the solutions presented here are discussed; it is this author’s opinion that
they could be usefully applied to the modelling of some small-scale plateau uplifts and
possibly to the history of basin subsidence.
2 Determination of uplift and surface heat flow
2.1
BASIC E Q U A T I O N S , I N I T I A L A N D B O U N D A R Y C O N D I T I O N S
The geometry of the problem considered is sketched in Fig. 1. Axial symmetry is assumed
because plateau uplift and basin subsidence often exhibit circular symmetry. A cylindrical
coordinate system ( r , z ;z positive downward) is used. The temperature at the Earth surface,
z =O, is constant; a heat flow perturbation 4a(r, t ) is specified at the depth z = a (that we
can identify with the base of the lithosphere). It is assumed that, in the absence of that
perturbation, the lithosphere is in thermal equilibrium. The problem is thus to determine the
non-equilibrium part of the temperature field, T ( z , r, t ) , which, in the slab 0 G z G a , satisfies
the heat equation (Carslaw & Jaeger 1959):
aT
az 1 a az \
- = K V ~ T = K - +-- tT(z,r,t)
at
lar’ r ar a z z J
where K is the thermal diffusivity assumed uniform throughout the lithosphere.
The boundary conditions verified by the temperature perturbation are:
(1) the temperature is constant on the surface z = 0:
T(z = 0 , r, t ) = 0,
(2) the heat flow perturbation is specified on the lower boundary, z = a :
aT
k -(z = a , r , t ) = qa(r, t ) ,
az
where k is the thermal conductivity. It is assumed that initially, the temperature field is not
perturbed:
T ( z ,r, t = 0) = 0.
(Id)
If it is assumed that the thermal expansion due to heating of the slab 0 G z G a is entirely
relieved by vertical movement, the surface uplift, h ( r , t), is given by:
h(r, t ) =
J:
a!T(r,z , t ) dz
where a! is the coefficient of thermal expansion of the lithospheric material. As long as the
surface z = O is above sea-level and no erosion takes place, isostatic equilibrium is
maintained. If those conditions are not met, lithospheric flexure would take place (e.g.
Walcott 1970) and it would have to be introduced; this problem has been analysed by
J. - C Mareschal
538
Beaumont (1978) who used the same geometry and formalism as the one developed in this
paper for the thermal effects.
2.2
METHOD O F SOLUTION
The heat equation with its boundary and initial conditions can be formally solved with a
double integral transform (e.g. Sneddon 1972). The Laplace transform of the temperature
field is defined by:
exp (- s t ) T ( z , r, t ) dt
L { T ( z ,r, t ) ) =
(3)
10-
and the Hankel transform is defined by:
H [ ~ ( zr,,t ) ]=
1;
~ ( zr,, t )J,-,(ur)rdr
(4)
where J o is the Bessel function of order 0.
The transformed temperature field, T(z, u, s), is a solution of the ordinary differential
equation (which contains the initial condition):
with the boundary conditions:
T(z = 0, u, s) = 0
(5b)
dT
k - (z = Q , u, s) = Ju(u, s)
dz
where Qa(u, s) is the double integral transform of the specified anomalous heat flow on the
lower boundary. The solution of equation (5a) satisfying the boundary conditions (Sb, c) is:
sinh ( ( s / K + U ~ ) ~ / * Z }
q z , u, s ) =
(S/K
t u ) ” ~cosh {(s/Kt u)’”a}
(7a(u, s)
k
(6)
’
The transform of the anomalous surface heat flow, Q(u, s), is directly obtained as:
G(u, s) = k
- (z = 0, u, s) = &(u, s) sech [(s/K+
(7)
dz
Replacing the temperature field given by equation (6) into the equation (2) yields the
transform of the surface uplift:
(u, s) = a!
T ( z , u, s ) dz
The surface uplift and the surface heat flow are thus the space-time convolution of the lower
boundary heat flux with the corresponding Green’s function. The convolutions for radially
symmetric functions can formally be written as (e.g. Churchill 1972):
Thermal expansion of the lithosphere
539
Reciprocally, if either the history of the uplift h(r, t) or of the surface heat flow q(r, t)
are known, equations (7) or (8) could be inverted to determine the heat flow anomaly
at the base of the lithosphere.
It can be seen from the equations (7) and ( 8 ) that both the surface uplift and the surface
heat flow Green’s functions filter out rapid time variations or short ‘wavelength’ space
variations of the lower boundary heat flux. Consequently, the inversion of the surface heat
flow or the surface uplift amplifies short wavelengths and rapid time variations. This
inversion would yield stable results when considering distances of the order of or larger than
the lithospheric thickness and time-scales of the order of the time constant for heat
conduction through the lithosphere. It could certainly not be applied without filtering out
the relatively rapid oscillations of the Earth’s surface that have been reported by several
workers (e.g. Hinze et al. 1980).
2.3
T H E O N E - D I M E N S I O N A LU P L I F T A M P L I T U D E
In the case of a uniform heat flow variation on the lower boundary, the one dimensional
solution for the uplift is given by:
ks
For a stepwise time change in heat flow (i.e. Qa(s) = 4&), the final uplift would be
(Doetsch 1963):
This uplift amplitude could have been determined directly by multiplying the coefficient
of thermal expansion by the average temperature change in the slab O < z < a . In what
follows, the uplift will be normalized to a fictitious ‘one-dimensional amplitude’ defined as:
wu(r)a2 or
ad, (u)a2
2k
2k
which is the uplift that would be observed if heat was transported only vertically. The onedimensional heat flux change should obviously be equal to the heat flux anomaly on the
lower boundary; thls can be verified by performing the same analysis with the equation (7).
2.4
T H E UPLIFT GREEN’S FUNCTION: UPLIFT VELOCITY FOR STEPWISE H E A T
FLOW C H A N G E S
The Green’s function for the surface uplift is given by the double inversion integral:
g(r, t ) =
10
‘s
+i-
J~ (ur)u du 2ni -i-
exp ( s t ) { l - sech [ ( s / K
(S/K
+ u ~ ) ” ~ ucls] }
4- U 2 )
( 1 2)
Considered as a function of the complex variable s, the transform is a single valued
function with poles in s = - K u2 - (2n + l ) ’ n 2 ~ / 4 a(n
Z = 0 , 1 ,. . .). The inversion of the
Laplace transform can thus be performed by closing the contour of integration in the left
half of the complex plane; the inverse transform is then represented by the series of residues:
J.- C Mareschal
540
The Hankel transform (13) can be inverted directly (Erdelyi 1954) and it yields the
Green’s function for the surface uplift:
The normalization factor, go is the one-dimensional uplift amplitude (aqUa2/2k).The
Green’s function is plotted as a function of radius at different times on Fig. 2. This Green’s
function represents the uplift that would follow a change in heat flow through the lower
boundary qa(r,t ) = qa6 (r/u)S ( t / ~(where
)
6 is the Dirac delta function). The radius is normalized to the lithospheric thickness, a, and the time to the characteristic heat conduction time
constant for the lithosphere T (7 = U’/K).
From the Green’s function, it is possible to determine the uplift velocity for a stepwise
change in lower boundary heat flux. If the anomalous heat flux on the lower boundary has a
Gaussian spatial dependence:
0
8
-t
0
9
hl
\
M
\
T=.025
0
s
c\I
\
0
9
(D
7
0
9
m
__
x
0
T=25
T=
10
I
5
0.20
I
0.40
I
0.60
I
0.80
7
1 .oo
UPLIFT GREEN ‘ S FUNCTION
Figure 2. The uplift Green’s function plotted as a function of distance at different times: the surface
uplift when the heat flow anomaly on the lower boundary is qa6(r/a)6( t / s ) (6 is the Dirac delta
function). The Green’s function is normalized to aq&*/2k. The radial distance is normalized to the
lithospheric thickness a and the time is normalized to the heat conduction time constant for the
lithosphere. T = u ~ / K .
0
9
7
0
2
0
x
0
t
0
0
2
x
0
(
UPLIFT VELOCITY
UPLIFT VELOCITY
Figure 3. (a) The uplift velocity as a function of distance at different times after a stepwise change in the
lower boundary heat flux qa exp(-r2/c2)H(t).(b) The uplift velocity as a function of distance at
different times after a stepwise change in the lower boundary heat flux qa exp (- r2/25c 2 ) H ( f ) .For both
figures the velocity is normalized to (aq,a'/kr), initial velocity at the centre of the anomaly. Distance
and time are normalized to lithospheric thickness and conduction time constant.
J. - C Mareschal
542
(H(t) is the Heaviside unit function), the uplift velocity is obtained by introducing into the
equation (13) the Hankel transform of the lower boundary condition and inverting. It
yields :
u(r, t ) - 4b’ exp [-r’/(b’ t 4~ t ) ]
--
(b’
UO
4K t)
c-
(-In
exp [- ( 2 n t 1 ) ’ n ’ k t / 4 ~ ]
(2n t 1)
fl=0
(16)
The normalization factor is given by: uo = (uqaa2/k7 (e.g. u(r = 0, f = 0) = 1). The uplift
velocities for different ‘widths’ of the heat flux anomaly (i.e. b = a and b = Sa) are plotted
in Fig. 3. The initial velocity at the centre of the anomaly is independent of the ‘width’, but
it decreases more rapidly with time for narrow anomalies (i.e. small b/a ratios).
2.5
THE H E A T FLOW GREEN’S FUNCTION
The same analysis yields the surface heat flow Green’s function which can be written as:
OD
40(r’ t , =
exp [- r2/4t ]
4a
t
2
(-)“(2n t 1) exp { - (2n
+ 1)’n’
t/4~).
(17)
n=O
This heat flow Green’s function as a function of radial distance (normalized to the lithospheric thickness) is plotted for different times (normalized to the lithospheric heat
conduction time constant) in Fig. 4 . The comparison between the surface heat flow and the
0
In
dl
0.00
I
0.20
I
0.40
I
0.60
0.80
1 .oo
HEAT FLOW G R E E N ’ S FUNCTION
Figure 4. The heat flow Green’s function plotted as a function of radial distance (normalized to the lithospheric thickness, a ) for different times (normalized to the lithospheric heat conduction time constant
T = u * / K ) : the surface heat flow anomaly when the heat flow at the base of the lithosphere varies as
qa6 @/a)&( t / ~ )The
. Green’s function is normalized to q a / T .
1.60
I
2.00
R/A
2 00
4 00
6 00
8 00
10 00
Figure 5. (a) The time rate of change of the surface heat flow at different times after a stepwise change in
lower boundary heat flux (qa(r,t ) = q a H ( t ) exp ( - r z / a 2 ) . (b) The rate of change of the surface heat flow
after a stepwise variation in the lower boundary heat flow qa(r, t ) = q a H ( t ) exp (- r2/25a'). For both
figures, the rate of change is normalized to qa/z.Time and distance are normalized as before.
(b!
1.20
(a)
0.80
RATE OF CHANGE OF SURFACE HEAT FLOW
0.40
RATE OF CHANGE OF SURFACE HEAT FLOW
0.00
0
J. - C Mareschal
544
uplift Green’s functions illustrate differences in behaviour between the two anomalies. While
the uplift Green’s function is maximum for t = 0, the heat flow Green’s function reaches its
maximum after a time of the order of 7/4. The uplift anomaly is also slightly sharper than
the heat flow which spreads out further away from the centre.
After a stepwise change in heat flux at the lower boundary, qa(r, t) = qa exp (- rz/bz)
H(t), the rate of change of the surface heat flux is given by:
7
aqa -
qa a t
nbz
c
m
exp[-rz/(bz t 4 ~ t ) ]
b2 t 4 K t
(-)“(2n t l ) e x p { - ( 2 n tl)’nzt/47}.
(18)
n=O
Although the series is not defined for t = 0, it converges for any t > 0 and its limit is 0 for
t+0.
The time derivative of surface heat flux is plotted for different widths of the anomaly on
Fig. 5(a) (b = a) and Fig. 5(b) (b = 5a). It can be observed that, in opposition to the uplift
velocity, the rate of change of the surface heat flow is not maximum for t = 0 but reaches its
maximum for t 7/4.
-
2.6
U P L I F T A F T E R A STEPWISE C H A N G E I N H E A T FLOW A T L O W E R B O U N D A R Y
When a stepwise time change in heat flow occurs at the lower boundary, i.e.
qa(r. t) = q,(r)H(t)
(1 9a)
or
4u@,
(19b)
s) = 4 a @ ) / s
the transform, i ( u , s), of the surface uplift is obtained by introducing the heat flow change
(equation 19b into equation 8).
The Hankel spectrum of the surface uplift as a function of time can be determined by
inverting the Laplace transform as above. (The only difference is the presence of an
additional pole at s = 0 which yields the final iplift.) The Hankel spectrum of the uplift as
a function of time is:
&u, t)
32
- 2
-[ 1- sech ua] -- exp [- u2aZt/r]
( c ~ 4 ~ ( u ) ak~)/ 2 uza’
R
For one-dimensional problems (i.e. when the heat flow varies uniformly on the lower
boundary), the surface uplift is given by
h(t)
-=1-7
ho
32
c (-y exp [-(2n(2n
n =O
t 1)’n’t/~]
t 113
The uplift corresponding to different dimensionless ‘wavenumbers’, ua, (a = lithospheric
thickness) is plotted in Fig. 6 as a function of time (normalized to the time constant for heat
conduction through the lithosphere).
It can be observed that the uplift starts instantly and that about 1/2 of the total uplift
takes place in a time of the order of 7/4 after the initiation of the heat flow anomaly. The
attenuation of the uplift amplitude for larger values of the ‘wavenumber’, w, is a twodimensional effect. (The Hankel transform variable u plays a role similar to the wavenumber
Thermal expansion of the lithosphere
545
0
0
__
-1-
I
O.O:!
0.80
0.40
V k S .
-
7
7.
1.20
GO
2.Ciil
I
UPLIFT IiANKEL SPECTRUIV1
Figure 6. The surface uplift as a function of time for lower boundary heat flow anomalies of different
dimensionless wavenumbers, w. Time and distance are normalized as before (i.e. wavenumber is
multiplied by lithospheric thickness). The uplift is normalized to aqaa2/2k;the maximum uplift that
would be observed if heat was transported only vertically.
in Fourier transform theory.) It can be observed that for ua > 1 (i.e. at 'wavelengths' shorter
than the depth to the lower thermal boundary), the amplitude of the uplift is attenuated.
But for ua < 1 (i.e. for wavelengths larger than the depth to the lower thermal boundary),
the amplitude of the uplift will be comparable to what would be determined by a strictly
one-dimensional analysis.
2.7
S U R F A C E H E A T FLOW AFTER A STEPWISE C H A N G E IN LOWER B O U N D A R Y
FLOW
For the stepwise heat flux variation at the lower boundary, the Hankel spectrum of the
surface heat flow at different times, co(u, t ) , can be determined by inversion of the Laplace
transform as above. It yields:
0
- n exp (-u2aZ t/T)
C (-ln
n=O
(2n +l)exp[-(2n
+1)'nzt/4~]
uZa2+ (2n + I)'n2/4
When the heat flow variation is uniform on the lower boundary, the one-dimensional
solution for the anomalous surface heat flow is given by:
qO(t)
- =I-4a
19
4
OD
nn=o
(-)n
exp [- (2n t i ) 2 n 2 t / 4 ~ ]
(2n + 1)
J.- C Mareschal
546
0
N
0
1
0.00
6.40
6.80
I
1.20
I
1.60
I
2.00
T
HEAT FLOW HANKEL SPECTRUM
Figure 7. The surface heat flow as a function of time for a lower boundary heat flow anomaly of specified
wavenumber, ua. Time and distance normalizations are the same as in Fig. 5. The heat flow is normalized
to the lower boundary heat flux, q,,.
On Fig. 7, the anomalous surface heat flow for different normalized ‘wavenumbers’ is
plotted as a function of time. (The normalization conventions are the same as defined
above for Fig. 6.)
The surface heat flow does not change significantly before a time of the order of 7/8.
This time lag between the surface uplift and the change in surface heat flow, which confirms
the numerical results of Crough & Thompson (1976), is expected since the uplift starts as
soon as the temperature in the lower lithosphere changes, well before the thermal effects
reach the surface. For large values of the transform variable, attenuation of the surface
heat flow is also apparent and the two-dimensional attenuation effect is slightly more
important for the heat flow anomaly than for the uplift. The uplift averages the temperature
changes between the upper and lower boundaries, while the surface heat flow is determined
by the temperature gradient on the surface.
2.8 E X A M P L E S
In order to illustrate those effects, uplift and surface heat flow anomaly have been computed
for a lower boundary heat flow variation corresponding to equation (15a); the width of the
anomaly is varied by varying the parameter b. Two different techniques could be used to
SURFACE UPLIFT
y-
2.00
4.00
6.00
8.00
10
SURFACE UPLIFT
Figure 8. (a) The surface uplift as a function of distance at different times after a stepwise change in lower
boundary heat flux 4&, t) = 4a exp (- r Z / u z ) H ( t ) .(b) The surface uplift as a function of distance at
different times after a stepwise change in lower boundary heat flux, 4&, t ) qa exp (- r z / 2 5 a ’ ) H ( t ) .
Distance and time are normalized as above. The uplift is normalized to the final uplift that would be
observed at the centre of the anomaly if heat was transported only vertically (i.e. q & / 2 k ) .
0
-
9
0
2
0
x
0
2
0
2
0
8
SURFACE HEAT FLOW
0
9
-1
.oo
R/A
(b)
SURFACE HEAT FLOW
Figure 9. (a) The surface heat flow as a function of distance at different times after a stepwise change in
heat flow at the lower boundary qa(r, t) = qa exp (- r 2 / u 2 ) H ( t ) (b)
. The surface heat flow as a function
of distance at different times after a stepwise change in heat flow at the base of the lithosphere
qa(r, t ) = qa exp (- r’/25a2)H(t).The surface heat flow is normalized to the maximum heat flow change
at the lower boundary, qa. Time and distance are normalized as before.
Themal expansion of the lithosphere
549
determine the surface uplift: numerical integration of the uplift rate given by the equation
(16) and numerical inversion of the Hankel transform given by the equation (20) (Anderson
1979).
The uplift as a function of distance to the centre of the anomaly at different times has
been plotted on Fig. 8. The distance is normalized to the lithospheric thickness, a, and the
time is normalized to the heat conduction time constant, T = a2/K. In the case shown on
Fig. 8(a), the anomaly width is equal to the lithospheric thickness: the final uplift amplitude
is only 1/2 the one-dimensional uplift. In the case, shown on Fig. 8(b), of a much wider
anomaly (b/a = 5), the final uplift at the centre of the anomaly is slightly less than the one
dimensional uplift (- 94 per cent). In both cases, about one-half of the total uplift takes
place in a time of the order of 7/4.
The surface heat flow anomaly is plotted on Fig. 9(a) for a heat flow anomaly of width
equal to the lithospheric thickness and on Fig. 9(b) for a width equal to 5 times the
lithospheric thickness. The normalization conventions for time and distance are the same as
above. The effects mentioned above appear clearly through a comparison of Figs 8 and 9.
First, the attenuation of the anomaly amplitude is more important for the heat flow than
for the uplift (- 40 per cent the one-dimensional heat flow anomaly versus 50 per cent the
one-dimensional uplift in the case where the anomaly width is equal to the lithospheric
thickness). Secondly, the time lag between uplift and heat flow can also be noticed. Almost
1/2 of the uplift takes place in a time of the order of 7/4; at that time the heat flow change
is only 1/3 of the total change. Thirdly, the spreading out of the heat flow anomaly can be
perceived near the edges of the figures. Although near the centre, the uplift amplitude is
larger than the heat flow amplitude, near the edges the heat flow is slightly larger than the
uplift. Finally, it can be noted that both for uplift and heat flow, the approach to
equilibrium will be slightly more rapid in the case of a narrow anomaly where twodimensional effects are not completely negligible.
-
-
3 Discussion and conclusions
The method outlined here provides a simple way of determining the uplift and the heat flux
anomaly caused by the conductive heating of the lithosphere following changes in heat flow
at the lithosphere-asthenosphere boundary. Whether this mechanism can explain any of the
observed uplifts remains an open question.
The major restrictions in applying this model are related to the amplitude and time-scale
of the uplift. The coefficient of thermal expansion, Q, of lithospheric material is estimated
to be 3 x lo-' R' (Skinner 1966); the average thermal conductivity in the lithosphere is of
the order of 2.4 W/m/K (Clark 1966). If the depth of the lower thermal boundary is 80 km,
an upper bound to the uplift amplitude may be set by assuming that the anomalous heat
flow should not exceed a value of the order of 12mWm-2 in order to avoid melting of the
lithosphere or exceedingly large horizontal temperature gradients at the base of the
lithosphere. Under these assumptions the uplift amplitude would not exceed 960 m, about
one-half of which would take place in a time of the order of 50 x lo6 yr. Very large
amplitude uplifts, such as the Colorado plateau, would thus be difficult to account for
within this model. On the other hand, although some early heat flow measurements
indicated that there is a heat flow anomaly associated with the Colorado Plateau (Reiter,
Mansure & Shearer 1979) more recent measurements (Chapman, private communication)
suggest that no large heat flow anomaly is present in that region; this could be in agreement
with the time lag between uplift and surface heat flow variations discussed above. However,
550
J.- C Mareschal
in the Colorado plateau an uplift of more than 1000m took place in 30 x lo6yr. The average
uplift velocity is too large to be explained by this present model. It would take more than
150 x lo6 yr, to produce such an uplift if the heat flow anomaly at the base of the lithosphere is not to exceed 12 mW m-'. Alternatively, an uplift of 1000 m in 30 x lo6yr would
require an exceedingly large heat flow change at the base of the lithosphere (of the order of
60mW1n-~). If the heat flow data contradict a model in which the thermal anomaly is
caused by bulk replacement of the lower lithospheric material such as the continental
delamination model (Bird 1979), the uplift data cannot be explained by the conduction
mechanism examined in this paper without making unreasonable assumptions. In this
author's opinion, the presence of phase changes in the lower lithosphere (which would be
affected by conductive heating of the lithosphere) would provide a more satisfactory
alternative. McCetchin & Silver (1972) suggested that eclogite may be present in large
proportions (- 25 per cent) in the upper mantle for the Colorado plateau region; its
transformation into gabbro may have been responsible for the Colorado plateau uplift
(Thompson & Zobback 1979).
Smaller amplitude uplifts such as the Black Hills or slower uplifts such as the Adirondacks
could be more realistically accounted for by this model. However, if the size of each uplifted
region and the amplitude of doming appear to be of the right order of magnitude, before any
attempt is done to fit the available data to this model, erosion and lithospheric flexure
should be included in the analysis (e.g. Beaumont 1978, whose Hankel-Laplace transform
technique could be adapted without difficulty). This would be of major importance for the
deeply eroded Adirondacks where the recent history of uplift may be dominated by the
response of a viscoelastic lithospheric plate to a very long erosion history.
It was suggested that the subsidence of intracontinental sedimentary basins may be the
results of thermal contraction of the lithosphere following an episode of thermal expansion,
uplift and erosion; for instance, a major unconformity exists in Middle Ordovician time in the
Michigan basin (Sleep & Snell 1976). For these authors, thermal expansion and uplift was
the result of bulk replacement of the lower lithosphere by hot light asthenospheric material;
however no evidence was found for such an heating event in the Michigan basin (Sleep &
Sloss 1978). If the heating were to result from increased heat flow at the base of the lithosphere, no large temperature variation would occur in the upper crust. A relatively small
heat flow change (- 8mWm-') could have caused an uplift of the order of 800m and
erosion followed by isostatic adjustments would then occur. However, the geological data
do not indicate that the uplifterosion episode in the Michigan basin lasted as long as the
100 x lo6yr required by any conduction model. It should also be pointed out that, in order
to produce the uplift, the temperature variations at the base of the lithosphere would have
been as high as 300°C. It is thus not clear that this model could be applied to reproduce
the uplift-erosion episode that would have preceded basin subsidence.
Another approach that could illustrate the restrictions of this model is to consider the
inversion of the surface heat flow or uplift data to determine the heat flow anomaly at the
lower boundary (equations 7 and 8). This inversion amplifies both the short wavelengths (compared to the slab thickness) and rapid time variations (compared to the heat
conduction time constant) of the heat flow or uplift. Therefore, it is only when the uplift
duration is of the order of the time constant for heat conduction through the slab, that the
uplift amplitude will not require exceedingly large heat flow anomalies at the lower thermal
boundary.
Despite the obviously restricted range of applications for the mechanism examined here, a
method was outlined for analysing uplifts that are relatively slow and of moderate
amplitude. A more complete investigation requires erosion and isostatic adjustments to be
Thermal expansion of the lithosphere
551
included; such an investigation would determine whether and when conductive heating and
thermal expansion of the lithosphere is a feasible mechanism of uplift.
Acknowledgments
The author is very grateful to Mr Chang Kong Lee who helped with the computing and
plotting of the figures. The author appreciates the care with which two reviewers examined
the original version of this paper; an error was pointed out by one of the reviewers. A special
thanks to Dawn Heindselman for her patience and care in typing this paper. Thls work was
supported in part by NSF grant EAR 7918199.
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