Geophys. J. Int. (1995) 122,479-488
Movement of the lithosphere-asthenosphere interface in response to
erosion of thickened continental lithosphere: a moving boundary
approach
A. Manglik,'," A. 0. Gliko2 and R. N. Singhl
'
National Geophysical Research Insiitute, Uppal Road, Hyderabad 500 007, India
'Institute of Physics of the Earth, Bolshaya Gruzinskaya 10, Moscow, Russia
Accepted 1995 February 7. Received 1995 January 16; in original form 1994 February 7
SUMMARY
Models of retrograde metamorphism in many orogenic belts are based on crustal
thickening and erosion of continental lithosphere. The pressure and temperature
evolutions in such models are' generally obtained by solving the advection diffusion
equation with the lithosphere-asthenosphere boundary (LAB) either fixed at a
specified depth o r moving downwards at the same rate as erosion from the surface.
However, this boundary is defined as a solid-partial-melt boundary in many
geophysical interpretations. In the present work, we discuss a solution of the above
problem considering the LAB as a phase boundary that moves. This solution is
obtained by the Fourier series approach for a general case of surface erosion and
basal heat transport. The results obtained for different models of crustal thickening
show a significant movement of the LAB in response to erosion from the surface. A
corresponding variation in the lithospheric thickness is also a significant result of this
analysis. The earlier notion of a fixed-base lithosphere seems to be a good
approximation when analysing metamorphic data, as we obtain nearly the same
temperature profile in the crust as after including the LAB motion effect. However,
the erosion of thickened crust is found to affect the lithospheric growth. These
results indicate that metamorphic data (sampling the thermal structure of the upper
lithosphere) d o not preserve the signatures of such boundary motion.
Key words: Fourier series, heat conduction, lithosphere-asthenosphere
metamorphism, moving boundary, time-varying erosion
1 INTRODUCTION
Crustal thickening and subsequent erosion in many orogenic
belts has been discussed as a plausible mechanism of
retrograde metamorphism, and various models have been
proposed to analyse the pressure and temperature
evolutions (Clark & Jager 1969; England & Richardson
1977; Thompson 1981; Royden & Hodges 1984; Hubbard,
Royden & Hodges 1991). Clark & Jager (1969) proposed a
half-space model with equal rates of uplift and erosion in
order to explain metamorphic evolution. England &
Richardson (1977) used a numerical scheme to investigate
the effect of erosion on the temperature profile in different
mobile belts. A 1-D advection diffusion model with a
fixed-base lithosphere was used by England & Thompson
(1984) .and Royden & Hodges (1984) to model pressure-
* Now at: Institut fur Geophysik, Universitat Gottingen, 37075
Gottingen, Germany.
boundary,
temperature-time (P-T-t) paths. The mobility of the
lithosphere-asthenosphere boundary (LAB) in terms of the
downwards motion of the 1300 "C isotherm was included by
Hubbard et al. (1991).
These models of metamorphic evolution (P-T-t) in a
continental lithosphere undergoing erosion, however,
neglect the effect of solid-partial-melt transformation at the
lithosphere-asthenosphere interface (LAB) (Anderson &
Sammis 1970). In a thermal context, the LAB is defined as a
rheological boundary marked by a particular isotherm which
is 0.75-0.85 of the mantle solidus temperature, or as a phase
boundary marked by the solidus temperature (Spohn &
Schubert 1982). Since the experimental studies on the rocks
representative of the upper mantle composition suggest
an increase in the solidus temperature with pressure,
the solid-partial-melt transition temperature at the LAB
becomes a function of depth. If the LAB is considered as
a phase boundary, initially at thermodynamic equilibrium,
erosion from the surface of the lithosphere would disturb
479
480
A . Manglik, A . 0. Gliko and R . N . Singh
this equilibrium state. In the present work we develop a
mathematical solution for the simulation of LAB motion
and thermal evolution in a continental lithosphere
undergoing surface erosion by including the effect of phase
transformation.
Simulation of LAB motion in response to erosion requires
the problem to be solved as a moving phase-boundary
problem. Solution of moving phase-boundary problems is
complex because of the non-linearity involved in the system
(Crank 1984), but the significance of these problems in
understanding many geotectonic phenomena has led to
various analytical, semi-analytical and numerical schemes of
solution. O'Connell & Wasserburg (1967, 1972) developed
simple solutions to analyse uplift and subsidence problems.
The lithospheric thickening and thinning problem has been
solved by Spohn & Schubert (1982), Crough (1983) and
Kono & Ogawara (1989). Mareschal & Gangi (1977)
obtained a linear approximation to phase-boundary motion
in response to a change in surface pressure and used these
results to explain the effect of erosion and deposition of
sediments. A non-linear asymptotic solution for phaseboundary motion problems has been given by Gliko &
Mareschal (1989). Mareschal & Gliko (1991) applied the
results to model lithospheric thinning in the East African
rift. Hamdani, Mareschal & Hamed (1991) incorporated the
effect of an instantaneous temperature change at the LAB
on its motion in the modelling of intracratonic sedimentary
basins.
These solutions, based on the non-linear integral equation
approach, give good results for a small-amplitude phaseboundary motion and neglect the effect of the surface
boundary (Gliko & Mareschal 1989). A solution of the
problem of LAB motion including the effect of the surface
boundary has been developed by Gliko & Rovensky (1985)
and Manglik et al. (1992) following the Fourier series
approach and employing the modified finite-difference
scheme developed by Melamed (1958) to overcome the
problem of stiffness in the system. We use the Fourier series
approach to solve the problem of LAB motion in regions of
crustal thickening and erosion, and obtain the pressure and
temperature evolutions.
normalized with respect to T,, the solidus temperature at
the LAB of a normal-thickness continental lithosphere. V ( t )
and [ ( t ) are the normalized positions of the upper and lower
moving boundaries respectively.
The boundary and initial conditions are specified as
2
2-7
Y=-
MATHEMATICAL FORMULATION
The LAB movement and thermal evolution in an eroding
continental lithosphere are obtained by solving the 1-D
transient heat conduction equation with an internal heat
source term. This heat conduction equation is first
non-dimensionalized and then expressed in terms of a
thermal perturbation v ( z , t ) , where u(z, t ) = u(z, t ) - u,(z).
Here u ( z , t ) is the normalized temperature distribution in
the lithosphere and u&) is the steady-state temperature
distribution. In perturbation form, the heat conduction
equation is given as
where z , the space coordinate (positive downwards) and t ,
the time, are .normalized by the coefficients I , (the thickness
of the lithosphere) and T (the characteristic time),
respectively. The characteristic time is defined to be I ; / K ,
where K is the thermal diffusivity. The temperature is
where u,(((r)) is the pressure-dependent normalized solidus
temperature and is expressed in terms of the inverse slope of
the Clapeyron curve. u,(z) and s are the initial temperature
distribution in the lithosphere and the total crustal
thickening, respectively. For the case of erosion of a normal
lithosphere, v ( z , 0) and s are equal to zero. In the case of
thickening of continental crust by a convergence process, for
example by homogeneous compression or by thrusting of a
continental sheet, the initial temperature is perturbed from
the steady-state condition and v ( z , 0) is non-zero.
The Stefan condition at the moving LAB is given as
(3)
where q ( t ) is the heat flux below the phase boundary. This is
normalized with respect to the initial heat flux Qo. The
other coefficients, R and a , are expressed in terms of various
other lithospheric parameters as
(4)
where K is the thermal conductivity of the lithosphere, C the
specific heat, and A the latent heat.
Using a similar formalism to that described in Gliko
& Rovensky (1985) and Manglik er al. (1992), the
transformation
V(Z3 t ) = u(z, t ) - y d J ( t ( t ) )- (1 - Y ) + ( 1 7 ( f ) ) ,
(51
where
5-
17'
is used to obtain a simplified form of the boundary
conditions. After the transformation, the heat conduction
equation becomes
After transformation, the boundary and initial conditions
(eq. 2) reduce to the following form:
(7)
The lithosphere-asthenosphcre iritrrfiice
The Stefan condition (eq. 3) reduces to
x [4(5(0) - IcI(v(t))l.
(8)
The variable V ( z ,t ) can now be expanded in terms of an
infinite set of orthogonal functions which satisfies the
boundary conditions (eq. 7) as
V ( z ,t ) =
2 A,(t) sin ( m y ) .
(5 - 17) n = i
(9)
The unknown coefficients A,(t) of this Fourier series are
obtained by substituting the expressions for V ( z , t ) (eq. 9)
into eq. (6) and applying the property of orthogonality. This
gives an ordinary differential equation in time for A,,(t).
which can be solved by using the prescribed initial
conditions. The equation for A,(t) is
48 1
inverse slope of the Clapeyron curve, the average density of
lithosphere and the acceleration due to gravit? . respectively.
The first-order ordinary differential equations (eqs 10 and
1 1 ) are solved iteratively to obtain the position o f the
moving interface and the thermal evolution for any given
time t. For dv/dt equal to zero, these equations reduce to
the system of equations obtained by Manglik et ul. (1992) to
analyse the problem of lithospheric thinning due to basal
heat flux perturbation. Therefore. the system discussed here
gives a more general case of the specific problem of
phase-boundary motion due to increased basal heat flux. An
important step at this stage is to prescribe initial values to
the Fourier coefficients. which are non-zero in the case of
continental thickening. The procedure for computation of
these. for the two types of model considered in this analysis,
is discussed in the appendix.
3
COMPUTATIONAL D E T A I L S
The system of ordinary differential eqs given by equations
(10) and (11) is complex, and is solved by using the
finite-difference scheme discussed in Melamed ( 1958). In
finite-difference form. the equations are
-
drl
[- A1J t )
dt 5-17
2n
IIK
IIK
- __
5-17
17111
+
I
17p11
~
+ /] d 17
.
(12)
dr
(lo)
["I
+I
h
=
+
- R!,,
a
where
where the coefficients R'," and R:' are given by
R;" = - d E
__I-
dt
[ f "2n- 9'"
( - I)"
+-
I1 a
( - 1 )"
A;"P,,, - ___
h
na
I
1
4(t'"')- ___
,'"
,rt
(
1
A::']
5"' - 9'''
The rate of lithosphere-asthenosphere boundary motion
for a specified rate of upper boundary movement is
computed from eq. (S), which is expressed, after
transformation, as
+-- IcIC1(')
(5- 17)
The variables
+ -1[ ( F '
Rq(t).
6,4and u,
dlCI
d17
u,(z) = 1 - 4 1 -
na
-
?ff')Ijl(7y)+ i+ !J (T )"' )]-,d 9
dt
are
(i/(v)=-!
5 + v),
where u = ypgI,,/T,,. Here y , p and g are the normalized
where m denotes the mth time-step and h denotes the
discretization interval in time.
In the above equations, sums of infinite series of
orthogonal functions are truncated after the first N
482
A . Mririglik. A . 0. GIiko and R . N . Singh
coefficients in the actual computations. This number was
chosen after performing a relative error analysis. This was
done by computing the results with N equal to 25 and 100.
Since the deviation in the results was found to be less than
1.5 per cent. we have chosen 2.5 Fourier coefficients in our
subsequent computations.
4
RESULTS A N D DISCUSSION
The solution developed here can be used to analyse the
effect of any general type of erosion-rate model on the
evolution of a lithosphere that has undergone instantaneous
crustal thickening, either by crustal thrusting or by
homogeneous compression of crust. The rate of erosion
varies over a geological time period, and it is different for
different geological regions (Burbank & Beck 1991).
However, England & Thompson (1984) have used a
finite-duration, constant-erosion-rate model in which erosion
starts 20 Ma after the instantaneous thickening event,
continues for a finite duration of time, and ceases when the
total additional crust is eroded. They introduced a time
delay of 20 Ma between the crustal thickening event and the
onset of erosion to simulate the conditions prevailing in
some areas of the Alps, where erosion was relatively low
during the first 30 Ma (Richardson & England 1979). W e use
this finite-duration erosion-rate model in the present
computations. For a crustal thickening of 35 km, an erosion
rate of 0.3.5 mm yr-l is assumed which continues for 100 Ma.
This model of erosion is used to compare the results
obtained by the present mathematical solution (FS model)
with those given by England & Thompson (1984) ( E T
model) on the basis of finite-difference modelling. However,
the mathematical formulation discussed in this paper can be
used to simulate the conditions where erosion is
simultaneous with the crustal thickening event.
The P-T-t analysis in the E T model is carried out under
the assumption of a fixed basal heat flux condition. In the FS
model, the temperature (pressure-dependent solidus) at the
Table 1. Values of different parameters used in the computations
of LAB motion and P-T-t paths.
Variable
Radiogenic Heat Source
Thickness of radiogenic layer
Thermal Conductivity
Normal Crustal Thickness
Thickness of Thrusted Sheet
Crustal Compression Factor
Lithospheric Thickness
Mantle Solidus
Density of Lithosphere
Inverse Slope of
Clapeyron curve
Gravity Acceleration
Heat flux a t LAB
Thermal Diffusivity
Erosion R a t e
Latent Heat
Surface value of Solidus
Temperature
Numerical Value
2.0 ~ k V r n - ~
15 km
2.5 Wm-'I<-'
35 km
35 km
2.0
125 km
1311°C
3.2 g/cm3
@.08"C/MPa
9.8rn~-~
30m Win1 . 2 ~ 1 0 -nz's-'
~
0.35 k m / M a
334 bJkg-'
1000 " C
base of the lithosphere is prescribed and the upper surface
moves downwards as a result of erosion. In the present
computations, the temperature at the LAB, placed at a
depth of 150 km, is fixed at 1302 "C. This temperature is
obtained by constructing a steady-state geotherm with a
thermal conductivity of 3.0 W m - ' K-l in the lithosphere,
the radiogenic sources distribution parameters given in
Table 1, and a flux of 30 m W m-' in the subcrustal mantle
(as used in the ET model). The solidus temperature gradient
is applied at this depth. The results of the ET model and
those obtained by the FS method are shown in Fig. 1. ET
temperature profiles are shown by solid squares at times 0,
4.5, 9.0, 18, 36 and 72 Ma. The thermal structure obtained
by the FS method is replotted with the z = O surface as a
reference level (the Earth's surface), and is shown by solid
lines. These results match well for all times. A small
discrepancy shown by the two curves at 72Ma, however,
may be attributable to numerical errors. These results
indicate that the thermal structure obtained by a model
having a constant flux condition at the base of the
lithosphere with uplift and erosion terms in the heat
conduction equation is almost same as that obtained by a
model having a constant temperature condition at the base
of the lithosphere and a moving erosional surface.
In the above results, the effect of L A B motion in response
to surface erosion has not been included. In further
Temperature
0
250
500
750
8
(OC
10 0 1 2
to
ET Model
FS Model
n
60-
€
Y
r
80-
U
Q
Q) 100n
120140-
\.
160
Figure 1. A comparison of the results of thermal evolution
computed by the Fourier spectral method (solid lines) with the
results of the England & Thompson (1984) model (solid squares). In
this case, continental lithosphere is thickened by crustal-scale
thrusting and a finite-duration erosion rate is used.
The lithosphere-asthenosphere interface
computations the thermal parameters as given in Table 1 are
used. The value of thermal conductivity is reduced to
2.5 W m-l K-' from the value of 3.0 W m-' K-' used above
for the purpose of comparison of results with the ET model.
These parameters are used to obtain the temperature at the
LAB and the thickness of the normal continental
lithosphere. This temperature is consistent with the solidus
temperature obtained from the solidus curve, which has the
inverse slope of the Clapeyron curve ( y ) , i.e. 0.08 K MPa-'
(Fig. 2). Since the value of thermal conductivity is reduced
to 2.5 Wm-'K-',
we obtain an LAB temperature of
1313°C for a normal continental lithosphere of 125km
thickness. The results of P-T-t paths for the two models of
crustal thickening, i.e. the crustal thrusting and the crustal
compression models, are computed for the above
parameters.
The results of thermal evolution and P-T-t paths
obtained from the crustal thrusting model are shown in Fig.
3. These results have been obtained for the model of erosion
discussed above for 35 (Fig. 3a) and 15 km (Fig. 3b) crustal
thickening. Since temperature profiles obtained in the crust
with the LAB as a moving boundary are not significantly
different from those obtained by assuming the LAB to be a
fixed boundary, we discuss only the results obtained by
including the effect of LAB motion. For 35 km thickening,
temperature profiles are shown in Fig. 3(a) for 0, 10, 20, 40,
100 and 200Ma with the P-T-t paths of rocks buried at
0
0
50
100
150
Figure 2. Computation of LAB equilibrium temperature. This is
obtained by the intersection of the geotherm (solid line) with the
solidus temperature curve (thick line). The initial steady-state and
the initial perturbed (thrusting model) temperature profiles are
shown by thin solid lines.
483
30, 40, 50 and 60 km after the thrusting event. These results
are the same as those obtained by using the E T algorithm.
For the 15km thickening model, the results of thermal
evolution and P-T-t paths for rocks initially buried at 10,
20, 30 and 40 km depth are shown in Fig. 3(b).
The results of LAB motion and lithospheric thickness
variation for both values of crustal thickening are shown in
Fig. 4. Fig. 4(a) shows the position of the LAB at different
times for different degrees of partial melting after the
thrusting of a 35 km thick crustal layer. The solid curve
corresponds to the 100 per cent solid-melt case, the
solid-square curve is for 50 per cent partial melt, and the
solid-triangle curve represents the 10 per cent partial
melting case. The rectangular curve shows the erosion rate
with time, and the dashed curve marks the position of the
LAB in a normal continental lithosphere (125 km). Crustal
thrusting changes the thermodynamic equilibrium at the
LAB. Since erosion starts after 20Ma, the LAB position
remains fixed for the first 20Ma. At the onset of erosion,
this boundary first moves downwards for a very short time
and then starts moving upwards. This behaviour is a result
of pressure reduction and an insignificant increase in
temperature in the initial stage. After the erosion ceases, the
LAB starts moving downwards again and approaches its
original position at large times. In the above case,
computations up to 400 Ma show this trend (approaching the
dashed curve). The corresponding results on lithospheric
thickness variation are shown in Fig. 4(b). The thickness of
lithosphere at any time is obtained by subtracting the
position of the upper boundary (erosion surface) from the
position of the lower moving boundary (LAB). The notation
is the same as in Fig. 4(a). An instantaneous addition of a
35 km crust increases the thickness of lithosphere by this
amount, which is shown by the downwards movement of the
curves at initial time. The thickness of lithosphere remains
the same for the first 20Ma because no erosion is applied.
With the start of erosion, the lithosphere grows in thickness
for a very short time. This growth, however, is negligible. As
the erosion continues, the thickness of lithosphere starts
decreasing with time and becomes less than the thickness of
normal lithosphere. Results for these values of thermal
parameters suggest a maximum thinning of 24km if the
LAB is defined as a solid-partial melt (10 per cent)
boundary. This amount of erosion is significant and suggests
that erosion of a thickened continental lithosphere might
result in lithospheric thinning. The lithosphere starts
growing after the completion of erosion and attains its
normal thickness at large times. For a 15 km crustal sheet,
the results of LAB motion and lithospheric thickness
variation are shown in Figs 4(c) and (d), respectively. In this
case, the lithosphere thins beyond its normal thickness by
8.8 and 6 km for 10 and 100 per cent melting cases,
respectively.
We next discuss the results of erosion of a continental
lithosphere thickened by uniform crustal compression. In
this case we consider the same parameters and the same
erosion model as used in earlier cases, but thicken the crust
by a compression factor f , . Two values off, are chosen such
that thickenings of 35 and 15km are achieved. The two
models of crustal thickening differ, however, in the
distribution of radiogenic sources. In the compression
model, radiogenic sources are confined to the uppermost
484
A . Manglik, A. 0. Gliko and R. N . Singh
Temperature ( " C )
200
0
400
1( 0
800
600
0
01,
Temperature ( " C )
200
400
600
8 0
10
20
Y
30
v
c
x
4
0
a,
cf!
50
60
\
70
\
Figure 3. Evolution of temperature in continental crust after a crustal thrusting of (a) 35 km and (b) 15 km. Temperature is shown by solid
lines. The erosion starts after 20 Ma and continues to 120 Ma. The pressure evolution is shown by the solid triangles.
part of the thickened crust, whereas in the case of the crustal
thrusting model radiogenic sources are concentrated in the
upper part of the thrusting slice and in the upper part of the
underlying continental crust. The results of thermal
evolution after crustal thickening of 35 km are shown in Fig.
5(a) for times of 0, 10, 20, 40, 100 and 200Ma with the
P-T-t paths of rocks initially buried at 30,40,50 and 60 km.
In this case, the rise in temperature is less than that in the
case of the thrusting model. The results of thermal evolution
for 15 km thickening of continental crust in the compression
model are shown in Fig. 5(b). These results show a smaller
increase in temperature than that in the case of 35 km
thickening.
Corresponding moving LAB positions and lithospheric
thickness variations for 35 km thickening are shown in Figs
6(a) and (b) for the three values of melting. The LAB
initially moves downwards (Fig. 6a) for a short time because
of the dominant role of pressure reduction over thermal
evolution. As thermal relaxation becomes prominent, the
LAB starts moving upwards. Total upwards movements of
14 km from the normal position (dashed curve) for 10 per
cent partial melting and 9 km for total melting are obtained
-0.3
too
200
rl
300
+5
0
4 00
Time (Mo)
-In.
.n
-- I
I -
4
-20,
(c>
100
200
l i m e (Mo)
X30
1
t
, -0.1
400
,o
4
-0 3
-0.1
-00
fb)
\
t40
160
200
Time (Ma)
300
,
400
Figure 4. (a) Movement of the LAB for crustal thrusting of 35 km and finite-duration erosion model for 10, 50 and 100 per cent partial melting,
and (b) corresponding variation in the thickness of lithosphere with time. (c) LAB motion after thrusting of 15 km thick crustal layer, and (d)
corresponding lithospheric thickness variation. The dashed line represents the normal position of the LAB and the rectangular curve is the
erosion-rate model.
The lithosphere-asthenosphere interface
Temperature
200
0
400
600
("C)
800
1( I0
0
0
Temperature 6 ° C )
200
40
485
E 0
10
10
20
n
E 20
A
E 30
Y
Y
v
v
-c
5
z
4
0
$30
9
n
c3
50
40
60
50
70
Figure 5. Evolution of temperature in continental crust after a crustal compression of (a) 35 km and (b) 15 km. This is shown by solid lines for
0, 10, 20, 40, 100 and 200Ma. The erosion starts after 20Ma and continues to 120Ma. The pressure evolution is shown by the solid triangles.
for the present model parameters. The LAB moves
downwards after the cessation of erosion and reaches a
depth below its normal position (dashed curve) in this case.
This downwards motion beyond the normal position is
obtained because of the removal of radiogenic sources,
which tend to be concentrated in the uppermost part of the
thickened crust in the crustal compression model. This
removal results in a further cooling of the lithosphere after
the completion of erosion.
-20
+ I04
0
- 10,
r 0 4
I
. .
100
200
Time
Time
(Mo)
(Ma)
The variation in lithospheric thickness is shown in Fig.
6(b) for all values of melting cases. The lithosphere first
thickens due to instantaneous thickening of the crust and
then, with the onset of erosion, it starts thinning. The
lithosphere becomes thinner than normal although the total
thinning obtained in this case is less than that obtained in
the case of crustal thrusting. Total thinning from its normal
thickness is 14 km for 10 per cent partial melting and 9 km
for 100 per cent melting. The lithosphere grows in thickness
10
300
-
rO 4
W
(c)
0
I00
2 00
Time (Mo)
300
- 101
460
-0 I
-a
loo
Time
200(Ma)
300
4
.L-0 1
400
Figure 6. '(a) Movement of the LAB in response to surface erosion at the rate of 0.35 mm yr-' after crustal compression of 35 km. The results
are shown for 10, 50 and 100 per cent partial melting. (b) Variation in the thickness of lithosphere with time for the above erosion-rate model.
(c) LAB motion after an increase of 15 km in crustal thickness. (d) Corresponding change in the lithospheric thickness. Dashed line represents
the normal position of the LAB and the rectangular curve is the erosion-rate model.
486
A. Manglik, A. 0. Gliko and R. N . Singh
after the thickened crust is eroded, and its thickness
becomes greater than the normal thickness because of the
removal of radiogenic sources. In the case of thickening of
15 km by compression mode (Figs 6c and d), the effect of
different degrees of partial melting is not significant.
Maximum thinnings of 4.5 and 3.5 km are obtained for 10
and 100 per cent melting, respectively. The lithosphere
becomes thicker than the normal pre-erosion episode
thickness in this case also. This behaviour can be used to
choose an appropriate model of crustal thickening in the
metamorphic data analysis.
5
CONCLUSIONS
In the present analysis we have obtained a solution for the
LAB motion in response to erosion from the surface of a
thickened continental lithosphere. This solution is based on
the Fourier series approach, a method used earlier by Gliko
& Rovensky (1985) and Manglik et al. (1992) to simulate the
effect of basal heat flux increase on the lithospheric thinning.
The solution includes the effect of the pressure-dependent
solidus temperature at the lithosphere-asthenosphere
boundary. The results of thermal evolution, when plotted by
taking the Earth’s surface as a reference level, give the same
numerical values as those obtained by uplift and constant
heat flux models.
The algorithm discussed in this paper is fairly general in
that it can be used for various general types of erosion
models, for example an exponentially decaying erosion-rate
model (England & Richardson 1977) and other time-varying
erosion-rate models (Burbank & Beck 1991). Other models
of erosion rate can also be used to simulate P-T-t paths and
LAB motion along with the lithospheric thickness variation.
A further generalization is in terms of the time-varying
nature of the heat flux supplied at the base of the
lithosphere. This algorithm can be used to simulate the
combined effect of thermal perturbation at the base and
erosion from the surface of the lithosphere. For the case of
no surface erosion, this algorithm reduces to the form
discussed by Manglik et al. (1992) to analyse the problem of
lithospheric thinning in response to various types of heat
flux increase models.
The results obtained for the two models of crustal
thickening show a significant movement of the lithosphereasthenosphere boundary in response to erosion from the
surface. The corresponding variation in the lithospheric
thickness is also a significant result of this analysis. The
earlier notion of a fixed-base lithosphere seems to be a good
approximation when analysing metamorphic data, as we
obtain nearly the same temperature profile in the crust even
after inclusion of the LAB motion effect. However, the
erosion of thickened crust is found to affect the lithospheric
growth. In this case we can conclude that metamorphic data
(sampling the thermal structure of the upper lithosphere) do
not preserve the signatures of such boundary motions.
The crustal thrusting model predicts a normal lithospheric
thickness at very large time (>400 Ma for the present values
of parameters), but the crustal compression model results
indicate a lithosphere that is thicker than the normal
lithosphere at large times because of the removal of the
internal sources of heat during erosion of the crust. One
important consequence of this kind of erosion-induced
lithospheric thinning may be viewed in terms of the onset of
lithospheric instability, and, if the region is under an
extensional regime (Platt 1993), a re-breakup of continental
lithosphere might occur in such regions.
ACKNOWLEDGMENTS
The authors acknowledge the Department of Science and
Technology, India, and the Russian Academy of Sciences,
Russia, for their support under the program ILTP(B2.7).
AM and RNS are thankful to Dr Harsh K. Gupta, Director
NGRI, for his encouragement and permission to publish this
work.
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Initial values of the Fourier coefficients A,,(O) required to
solve eq. (10) are obtained from the initial conditions given
in eq. (7) by using the orthogonality property. This step
needs the initial perturbed temperature profile due to crustal
thickening and the steady-state temperature distribution in
the lithosphere. The thermal perturbation can therefore be
expressed as
u ( z , 0 )= u , ( z ) - U ( j ( Z ) .
487
conditions at the upper and the lower houndaries. This is
given as
“o(z)
=
where
1
a={u,,, + Pl,D(2( 1
1+T
A P P E N D I X A: INITIAL VALUES OF THE
FOURIER COEFFICIENTS
it 1rerfcicr
+S ) - (D + T)]}.
Substituting these expressions in eq. ( A l ) and applying
the propertv of orthogonality, the expression for the initial
values o f the Fourier coefficients is obtained iis
(All
In the case of erosion of a normal lithosphere, the two
profiles u i ( z ) and u o ( z ) are the same and A,,(O) becomes
zero for all values of n. If the crustal thickening is modelled.
these two thermal distributions depend on the mechanism of
thickening and the redistribution of internal sources of heat
generation. We have considered the crustal-scale thrusting
and the uniform compression models of crustal thickening
(England & Thompson 1984) and obtained the initial values
of A,(O).
(AJ)
where
UNIFORM CRUSTAL COMPRESSION
MODEL
CRUSTAL-SCALE THRUSTING MODEL
In this model, we consider a whole-crust thickening, with a
uniform distribution of radiogenic heat sources in the upper
radiogenic layer, that increases the number of radioactive
heat sources in the crust. A more general crustal thickening
model, having arbitrary thickness of the thrust sheet and a
general radiogenic heat source distribution, can also be
considered in this formulation by adequately defining the
radiogenic sources term. After the thickening of continental
lithosphere, the initial temperature profile is obtained
following the ET model as
Ui(Z) =
where T is the crustal thickness T,, and 4, and PI are
A&/KT, and QolO/KTm, respectively. Other parameters
are given in Table 1.
The steady-state temperature profile u , , ( z ) in the thickened lithosphere is obtained by solving the steady-state heat
conduction equation with specified temperature boundary
The uniform compression model of crustal thickening ditfers
from the thrusting model in terms of the nature o f the
redistribution of radiogenic sources. In this model, sources
are confined to the upper part of the crust. In this case, the
initial temperature profile after the thickening event has
been taken from England & Thompson (1984). and the
steady-state thermal profile u o ( z ) is obtained by solving the
steady-state heat conduction equation. These profiles are
488
A . Manglik. A. 0.Gliko and R. N . Singh
These equations are substituted into eq. ( A l ) .and the initial
values of the Fourier coefficients after the application of the
orthogonality condition are obtained as
A , , ( o )=
(nn
- Y ) - [ ~ o , @,)(sin4 , - 4I cos bI)
-
c$,=n&T/s;
8, = i W + " - a )
f c f ,