Geophys. J. R. astr. SOC.(1977) 50,723-738
Isostatic rebound and power4aw flow in the
asthenosphere
S. Thomas Crough
Department o f Geophysics, Stanford university,
Stanford, California 94305, USA
Received 1977 February 2; in original form 1976 November 15
Summary. Laboratory experiments indicate that the asthenosphere probably
deforms as a power-law fluid. The experimental flow law for high-temperature peridotite and olivine is Cd"" = u, where d is strain-rate, u deviatoric
stress, n the power (observed to be about 3), and C a proportionality constant which is a function of composition, temperature, and confining pressure.
However, most theoretical treatments of isostatic rebound have assumed that
the asthensophere deforms as a Newtonian fluid. In this paper, numerical
solutions are found for the relaxation of a sinusoidal surface ,deflection above
a power-law medium. These solutions are applied to an analysis of isostatic
rebound data. First, following Post & Criggs (1973), it is shown by dimensional analysis that the rate of maximum uplift of a surface depression should
be proportional to the maximum amount of remaining depression raised to
the power n, where n is the power of the flow law. The rebound data from
Fennoscandia and Canada yield in-situ estimates of n between 2 and 4, in
good agreement with the experimental results. Second, using the finitedifferenm method, the proportionality constant is determined which relates
the rate of uplift to the amount of remaining depression. Using this constant
and assuming n = 3, the rebound data from Fennoscandia, Canada, and Lake
Bonneville yield in-situ estimates of about 10'0Ns'/3m-2 for the creep
coefficient C in the flow law. This is consistent with laboratory measurements
for the creep of dry olivine at 1200°C. However, all three rebound areas give
different values for C. Reasonable lateral temperature and compositional
differences in the asthenosphere can explain the observed variations, but it
is noticed that the estimates of C increase with increasing width of the
rebound area. This suggests that the value of C increases with depth in the
asthenosphere. Two models of increase of C with depth are examined and it is
found that both can explain the rebound data without invoking lateral variations in the asthenosphere. First, if the asthenosphere has a rigid base, that
base is at a depth of about 170km below the lithosphere. Second, if C
increases exponentially with depth, it increases by a factor of e every 135 km.
Present address: Geophysical Fluid Dynamics Laboratory, Princeton University, Princeton, New Jersey
08540, USA.
S. T. Oough
724
An exponential increase of this magnitude is expected from the increase in
confining pressure with depth.
Introduction
Most analyses of isostatic rebound (Crittenden 1963; McConnell 1968; Walcott 1973; Peltier
1974; Cathles 1975) have used the assumption that, over timgperiods of thousands of years,
the asthenosphere deforms as a Newtonian fluid, which m e e s that strain-rate is assumed to
be linearly related to deviatoric stress. Studies of this type are able to derive estimates of the
viscosity of the asthenosphere. Most investigators agree that the rebounds of Fennoscandia
and Canada, after the melting of the large continental ice sheets, give a viscosity of about
10" poise, while the adjustment of Lake Bonneville to filling and emptying gives a viscosity
about an order of magnitude smaller. However, it is difficult to interpret these viscosity
values in terms of composition or temperature in the asthenosphere.
Laboratory deformation studies of olivine and peridotite indicate that, at steady-state and
at the strain-rates typical of isostatic rebounds, the asthenosphere probably flows as a power
law fluid, where
Cpn=
U
(1)
u is differential stress, d is strain-rate, C is a proportionality constant which will be referred
to as the creep coefficient, and n is the power-law index, which is experimentally observed
to be around 3 (Weertman & Weertman 1975). The special case where n = 1 corresponds to
Newtonian flow. Relation (1) can be rewritten in a form which looks like the Newtonian
flow law
(&l
-m>
e=
(2)
where (Ce(' - n ) / n ) corresponds to viscosity in the Newtonian relation. It is more properly
termed an effective viscosity since it depends on strain-rate.
By estimating the strain-rate in the asthenosphere, the measurements of effective viscosity made using the Newtonian assumption could possibly be converted into estimates of the
creep coefficient, C. These estimates of C could then be compared with the laboratory
results. However, since it is difficult to estimate the strain-rate in the asthenosphere, it seems
more promising to start from the beginning and develop mathematical solutions for isostatic
rebound in a power-law fluid. These new relations can then be used to analyse the rebound
data directly to get estimates of the creep coefficient in the asthenosphere. These estimates
are important because if they are in reasonable agreement with the laboratory results, they
provide additional evidence that power-law flow is important in the asthenosphere and they
may provide information on the lateral and vertical variations in the properties of the
asthenosphere. Knowing the rheology of the asthenosphere is interesting in its own right,
but is also important as a step towards understanding the dynamics of the upper mantle.
Post & Griggs (1973) made what is probably the most important step in analysing the
rebound data in terms of power-law flow. They showed, by dimensional arguments, that if
relation (1) is the flow law then the maximum rate of uplift, dh/dt, will be proportional to
the maximum remaining depression, h, raised to the nth power. This assumes that the geometry of the uplift remains constant in time and that there are no large ambient strain-rates
in the asthenosphere. Furthermore, they stated that the rebound data from Fennoscandia
and Canada were consistent with a value of n between about 3 and 4, in good agreement
with the experimental results.
Brennen (1974) derived an approximate solution for flow in a power-law half-space
where the surface vertical velocity is a sine function for the two-dimensional case or a zero-
Isostatic regound and power law flow in the asthenosphere
725
order Bessel function for the axisymmetric case. With the displacement of the surface given
b y h sin kx and the vertical surface velocity given by (dhldf)sinkx, his solution can be
rewritten as
- (dh/dt)= (pg/CK)" h"/k
(3)
where n is the power-law index, p is the density of the flowing medium,g is the acceleration
of gravity, and K is a numerical constant whose value is a function of both n and the geometry with depth. This result is more complete than that of Post & Griggs, but is in agreement with their work. As part of his solution, Brennen gave formulas for the value of K.
Thus, although the isostatic rebound areas are not exactly sinusoidal in form, equation (3)
can be used to make in-situ determinations of the creep coefficient, C.
However, Brennen's approximate solution may be inaccurate because it assumes that
strain-rate decreases exponentially with depth. For the case of Newtonian flow, it is known
that the strain-rate invariant J2 is zero at the surface of the deforming medium and increases
with depth for a while before it eventually begins to decrease. It seems likely that this will
also happen for values of n near unity.
As Brennen (1974) mentidns, with power-law flow one cannot combine the flow solutions of varying wavelength sinusoidal deflections to obtain a correct solution for flow with
an arbitrary surface geometry. Instead, one must solve the flow problem again for each new
geometry. Since it is unlikely that analytic solutions or even approximate solutions will be
found for power-law flow with the actual isostatic rebound geometries, the power-law flow
problem must probably be solved numerically. As a first step towards getting these complete numerical solutions, the fmitedifference method is used here to solve the twodimensional, plane-strain case of the relaxation of a sinusoidal surface deflection above a powerlaw medium. In order to calculate these numerical solutions, several assumptions are made.
The following assumptions may limit the applicability of the results to an analysis of Earth
deformation, so they should be kept in mind.
(1) It is assumed that steady-state creep of the type in equation (1) is the only process
of deformation in the medium. Viscoelastic effects and the transient creep which occurs
before steady-state is reached are ignored. Peltier's (1974) calculations for a viscoelastic
Earth suggest that, at times later than a few hundred years after the removal of a surface
load, the response is mainly viscous. For both post-glacial uplifts which will be examined
here, only data for times after the complete melting of the ice will be used. Since the melting
process took some 7000 yr (Cathles 1975), elastic effects will probably be negligible. Since
there is little information available on the transient creep of peridotite, there is no way to
justify ignoring this creep process. However, it may be reasonable to expect that 70CO yr
is enough time to move from transient to steady-state creep.
(2) It is assumed that the value of n in the flow law, equation (I), is the same throughout
the deforming medium. At extremely small strain-rates it is thought that deformation in the
mantle will be Newtonian not power law with n = 3 (Weertman & Weertman 1975). Thus
even if the flow immediately beneath rebound areas is power law, it may be Newtonian at
greater depths where the strain-rate is smaller. However, the rheology in the area of maximum strain-rate controls the uplift process, so it is probably all right to ignore this possible
transition.
Peltier (1974) has shown that extremely wide uplift areas can cause significant flow
beyond the depth of the Earth's core. Thus the deformation of the Earth's core must sometimes be considered, in which case the assumption of uniform n would probably be invalid.
The post-glacial uplift of Canada, which is the widest rebound which will be considered in
this study, has a wavelength comparable to a zonal harmonic of degree 5 . Peltier's calcula-
726
S. T. Crough
tions show that for a uniform Newtonian mantle uplifts of this degree are just barely
affected by the core; higher degrees (shorter wavelengths) are unaffected. As will be shown
later, uplifts above a uniform power law mantle with n = 3 cause flow to occur to only half
the depth that occurs in a uniform Newtonian mantle. Thus, if the asthenosphere deforms
by power-law flow as the laboratory experiments indicate, the core probably has no significant influence on the uplifts studied here.
(3) It is assumed that there is now flow in the deforming medium except that caused by
the relaxation of a surface deflection. Applying these solutions to an analysis of the asthenosphere thus requires assuming that strain-rates caused by plate motion or convection are
small compared to the strain-rates caused by the uplift process. This assumption is difficult.
to justify but can be tested. If preexisting strain-rates are significantly larger than the uplift
strain-rates, then the effective viscosity in relation (2) will be determined by the size of the
preexisting strain-rates and will be essentially independent of the uplift rates. In t h i s case
the asthenosphere will deform like a Newtonian fluid for the small additional flow required
for the uplift, even if the actual rheology is power law.
(4) An elastic layer corresponding to the lithosphere is not included above the deforming
medium. Cathles (1975) has shown that bending stresses within the lithosphere have a negligible effect on the long-term uplifts of Canada, Fennoscandia, and Lake Bonneville. However,
these numerical solutions may need to be corrected for bending effects before they can be
applied to an analysis of short-wavelength deflections in areas where the lithosphere is very
thick. It is also assumed that there is no lateral flow at the top of the deforming medium.
This is equivalent to assuming that there is no flow in the lithosphere, since the no-slip
condition at the base of the lithosphere then requires no horizontal flow at the top of the
asthenosphere underneath it.
(5) The power-law flow relation (1) which is observed in uniaxial compression tests is
generalized to apply to twodimensional flow (Appendix). Although the generalization
appears to be reasonable, it needs to be tested by experiment.
Dimensional analysis
In order to make the numerical results as general as possible, tne IIOW problem must first be
nondimensionalized. The boundary conditions to be imposed at the top of the deforming
medium are that the vertical velocity, V, equals Vosinkx and that the horizontal velocity,
U,equals zero. At the base of the asthenosphere both V and U are required to be zero. With
these boundary conditions it is natural to scale all quantities with Voand k, the variables in
the boundary conditions, plus C, the scaling factor in the flow law (1). The new dimensionless quantities are
v' = v/vo
x'=kx
P' = d/Vok
u' = u/C( Vok)'/".
(4)
The flow law becomes d = e"" and the surface boundary conditions become V' = sinx'
and U ' = O .
After obtaining a numerical solution for the dimensionless velocity field of the flow, a
dimensionless stress field can be calculated from the flow law. In particular, the dimensionless normal stress at the top of the flow can be determined. It can be assumed with reasonable confidence that this normal stress will have the form K sinx', where K is some constant.
It is known that for n = 1 the stress will be a sine function, so for values of n near 1 the
deviation from a sine function will probably be small. Using the stress scale factor (4), the
actual surface normal stress caused by the flow will be CK (V0k)"" sin kx. Isostatic rebound
flows are driven by the normal stresses created by the displacement of the surface from its
Isostatic rebound and power law flow in the asthenosphere
727
equilibrium position. For a surface deflection of the form h sinkx, the resultant surface
normal stress is pgh sinkx. Balancing this stress, which drives the flow, with the surface
stress resulting from the flow and setting V , = dh/dt gives equation (3). Thus dimensional
analysis alone gives the dependence of the uplift velocity on the important variables h, k ,
and C. The numerical calculations simply determine the value of the constant K and provide
a check that the surface stress is approximately sinusoidal in form.
Rebound data
While Post 8c Griggs (1973) have already stated that the post-glacial rebounds of Canada and
Fennoscandia are consistent with power-law flow in the asthenosphere, there is a more convincing way of showing this result than the method they used. Taking the logarithm of both
sides of equation (3), it is found that
log Idh/dt I = log [(pg/CK)"/k]t n log h.
Rebound data are commonly presented on a graph of height, the elevation of former shore
lines above present sea level, versus time, the age of the shorelines. From gravity data the
amount of depression remaining at the present time can be estimated. Adding this value to
the elevations above present sea level converts the shoreline data into an estimate of the
actual depression at various times. The velocity of uplift can be determined by simple difif the log of the uplift velocity is plotted
ferencing of this data. According to equation (3,
versus the log of the remaining depression, then all the data points should fall on a straight
line with a slope of n.
Walcott's (1972) data from the Canadian uplift are presented in this manner in Fig. 1.
Hudson Bay is the area of maximum uplift, so the data from three sites in this area are
shown along with the maximum values taken from Walcott's contour maps of the uplift.
Walcott has drawn smooth curves through the site data. The points in Fig. 1 are taken from
I
Contour maps
Cape Henrietta Maria
C hurc hill
Keewat in
0.6
6
+/
.
E, 0.4
h
L
>,
.Q
9
I
U
0.0
1 '+
I
I
I
r
2.60
2.50
LOG H (m)
Figure 1. Postglacial rebound data from Hudson Bay, Canada. The log of the uplift velocity is plotted
against the log of the amount of remaining depression.
728
S. T. Gough
these curves. Thus the points show less scatter than if the’iaw data itself had been used.
Although Cathles (1975) has argued that the presence of negative free-air gravity anomalies
over Canada and Fennoscandia is merely coincidental, Walcott’s (1973) estimate that the
anomalies indicate 300 and 120 m of remaining depression respectively is used in the present
work.
The solid line in Fig. 1 is the predicted relation for n = 3, the value consistent with the
laboratory results. This fits the data quite well, although the data from the contour maps
are more consistent with n = 4. A line with the slope of unity (dashed line) provides a poor
representation of the data.
In Fig. 2 the Fennoscandian uplift data of Liden (1920) from Angermanland, the area of
maximum rebound, are presented on a log-log plot. Since Liden’s data are so complete,
velocities are calculated directly from his points, not from smooth curves drawn through
the points. To give some idea of the possible error in the plotted values, velocities are calcuiated assuming that the age of the strandlines has been determined correctly but that the
elevations are only correct to It 2.5 m. This gives the error bars shown in the figure. The line
in the figure is drawn by eye as a best fit to the data; it determines n to be 2.6. The figure is
plotted assuming 160 m of remaining depression. When 120 m are assumed, the best value
of n decreases to 2.2.
-10. !
2.2
2.3
LOG
2.4
H (rn)
2.5
Figure 2. Postglacial rebound data from Fennoscandia.
As Post & Griggs (1973) and Cathles (1975) have mentioned, the exact value of n for
each uplift is quite sensitive to the value assumed for the amount of depression remaining at
present. In fact, Cathles (1975) has shown that if the gravity anomalies are unrelated to the
amount of remaining depression, one can choose small values of remaining depression such
that the uplift data are consistent with Newtonian flow, n = 1 . While this is possible, it seems
worthwhile to pursue the assumption that the gravity and uplift are related and to see if this
assumption leads to reasonable results. Given the uncertainty in the gravity data, however,
it is not clear that the difference in the best values of n for Canada and Fennoscandia is significant. Probably all that can be claimed with confidence’is that the rebound data are
consistent with power law flow in the asthenosphere with n = 3.
However, even if Walcott’s (1973) estimates of remaining uplift are accepted, the data in
Figs 1 and 2 only require that the flow be power law if the geometry of the uplift area (primarily the width) does not change during the uplift process. The geometrical factor is
included in the parameters k and K of the uplift relation (3). If these parameters change
during the uplift, the data might resemble power law flow with n = 3 even if n really equals
1 . For example, in plane-strain, two-dimensional flow Z Gaussian surface depression
[h(x) = ho exp(-x2/202)] above a Newtonian asthenosphere with a rigid base at shallow
depth will rebound such that dh,/dt is proportional to hi (Lliboutry 197 1). This is because
Isostatic rebound and power law flow in the asthenosphere
729
t (1000 y r )
60
40
20
0
H (m)
-100
Figure 3. Predicted maximum depression of the Lake Bonneville crust as a function of time. Solid line
is Crittenden's (1963) calculation assuming an inviscid asthenosphere. This is equivalent to a value of CK
less than 4 x 109N~ " ~ mDashed
- ~ . line (a) is the prediction assuming CK = 4.3 x 10"N ~ . " ~ m -the
* , value
found for Fennoscandia. Dashed line (b) assumes CK = 4.3 X 1 0 r o N s * ~ 3 m - 2 .
the width of such a depression will increase dramatically during the uplift. As Post & Griggs
(1973) have emphasized, the available post-glacial uplift data show little to no change in the
width of the areas as they rise.
Assuming that n = 3 in the asthenosphere, in-situ estimates of the product CK in relation
(3) can be made for the rebounds of Canada, Fennoscandia, and Lake Bonneville. For
Canada, using p = 3.3 g ~ m - h~ =, 300 m, dh/dt = 2.0 cm yr-', and k = n/2500km-' (Walcott
1973), CK = 1.O x 10'' N ~"~m-'. For Fennoscandia, using h = 120 m, dh/dt = 1.O cm yr-',
and k = n/1350km-' (Walcott 1973), CK = 4.3 x 10" N s ' / ~ ~ - ' .The Lake Bonneville data
is less complete, so the method of Crittenden (1963) is used. The loading and unloading
history of the lake is known -with some accuracy. The response of the asthenosphere to this
loading is calculated for varying values of the product CK using k = n/250km-' (Fig. 3).
The maximum acceptable value is the one which causes 64-m of uplift from the time of the
most recent maximum load to the present. Thus, for Lake Bonneville, CK is roughly
4 x 10'0Ns"3m-z. It should be emphasized that there is nothing in the Lake Bonneville
data which requires power-law flow. However, if a value of n is assumed, the value of the
product CK can be determined.
Notice that the three different areas give three different values for the product CK. There
are two diametrically opposed ways of interpreting this result. The first is to assume that the
value of the numerical constant K is the same for all uplifts. Then the data indicate how the
creep coefficient, C, of the asthenosphere varies according to region. The second is to assume
that C is the same for all uplifts. Then the data indicate how K varies. In the next two
sections both methods will be tried.
Properties of the asthenosphere - uniform half-space
The asthenosphere is assumed to be so thick that it appears to be a half-space to all the
uplifts. For simplicity, C is assumed to be constant with depth. The numerical calculations
(Appendix) for this case give K as a function of n as shown in Fig. 4. Brennen's approximate
results are also shown. Both types of solution agree for n = 1, where the analytic solution
provides a check, but the predictions diverge for larger n.
In Fig. 5 the numerically calculated depth profiles of maximum vertical velocity are
plotted for various values of n. For n = 1 the maximum values of strain-rate occur in the
depth interval n/6 < y' < n/2. For higher values of n relation (2) indicates that this area of
high strain-rate becomes an area of low effective viscosity. To minimize the energy
730
S. T. Crough
30 -
Numerical
S olu t i on
0
1
1
2
3
4
5
n
Fmre 4. The value of K in equation (3) for different values of the power law index n. The results for the
numerical and approximate solutions are strikingly different.
expended, the highest strain-rates will preferentially occur in this low-viscosity zone. Thus,
as n increases the flow becomes confined within a smaller distance of the surface. The
approximate solution predicts a greater shallowing effect than the numerical results.
While there is always the possibility of numerical error, the numerical results are probably
correct. As mentioned previously, the approximate solution assumes that the strain-rate
V'max
Figure 5. The dimensionless maximum vertical velocity as a function of dimensionless depth, y', for
different values of n. The dotted line is the approximate solution for n = 3.
Isostatic rebound and power law flow in the asthenosphere
73 1
decreases exponentially with depth and hence, from relation (2), that the effective viscosity
increases exponentially. Both the numerical results and the approximate solution itself
indicate that this assumption is incorrect. The numerically calculated strain-rate invariant J2
increases with depth for a while before decreasing. The strain-rate invariant calculated from
the approximate solution also increases before decreasing. As a check on the numerical
results, the grid spacing normally used in the finitedifference calculations was halved for
one set of computations. The results were essentially unchanged. For another computation
Brennen's half-space solution for n = 3 (Fig. 5) was used instead of the Newtonian solution
as the starting point for the perturbation technique. Again the results were unchanged, the
flow quickly readjusted to the pattern for n = 3 shown by the solid line in Fig. 5.
Brennen's assumption of an exponentially decreasing strain-rate is excellent in that it
greatly simplifies the mathematics and allows the derivation of an approximate solution.
This assumption, dthwgh incorrect, might have led to accurate results. However, the
difference between the nimerical and approximate solutions seems to indicate the assumption bads to significant inaccuracies. Therefore, in the analysis which follows, only the
numerical results will be used.
Assuming n = 3, the value o f K for a half-space is 4.5. If a half-space is a good representation of the asthenosphere, then the creep coefficient under Canada, Fennoscandia, and
Bonneville is respectively 23, 9.5, and 1 x 10'0Ns''3m-2. The estimates for the two shield
areas differ by a factor of 2, which is probably large enough to be significant. The estimate
for the Basin and Range area is about an order of magnitude smaller than the estimates for
the shields.
The differences in the creep coefficient values could be caused by differences in mantle
composition in the three areas. Experimentally, the creep Coefficient is known to be sensitive to the water content of peridotite (Weertman & Weertman 1975). Wet samples creep at
about three orders of magnitude greater strain-rate than dry samples at the same stress. Using
n = 3, this indicates that the creep coefficient is changed by about an order of magnitude by
the presence or absence of water (presumably by other volatiles as well). Since this is about
the magnitude of the observed change between the shield and Basin and Range areas, it is
quite possible that the low value of creep Coefficient observed in the Basin and Range area
is caused by a greater amount of water in the asthenosphere there than under the shields.
The seismic measurements in these two types of area provide some support for this interpretation. The low-velocity zone is well-developed beneath the Basin and Range province.
There is a large velocity decrease at depth. Beneath the shield areas this decrease is much
smaller, if it exists at all (Biswas & Knopoff 1974). Since a small amount of melt can cause a
large velocity decrease, it has been argued that the low-velocity zone is a region of partial
melt (Nur & Simmons 1969). It is known that the solidus of peridotite is lowered by an
increase in water pressure (Kushiro, Syono & Akimoto 1968). Thus, if the asthenosphere is
at the same temperature beneath the shield areas and the Basin and Range, an additional
amount of water beneath the Basin and Range could cause additional melting and generate
the observed seismic velocity decrease.
Lateral temperature differences in the asthenosphere could also cause the variation in the
creep Coefficient. The experimentally observed temperature dependence of C at constant
pressure is compatible with the relation
where T is absolute temperature and Q is the activation energy for diffusion (Weertman &
Weertman 1975). Using n = 3, Q = 135 kcalmole-' (Goetze & Kohlstedt 1973), and the
deformation results of Kohlstedt & Goetze (1974) for dry olivine at 1400°C and 100 bar
732
S. T. Crough
differential stress, relation (6) gives the values listea in Table 1. Converting the rebound estimates of C into temperatures, it appears that the asthenosphere beneath Canada, Fennoscandia, and Bonneville could be respectively at 950,1000, and 1200°C. These temperatures
appear to be reasonable. The melting point of basalt is about 1200°C. The recent basaltic
vulcanism in the Basin and Range area suggests that the asthenosphere in that region should
be at about that temperature. The poorly developed low-velocity zone beneath the shield
areas may be an indication of sub-solidus temperatures.
Table 1. Temperature dependence of the creep coefficient, C.
Temperature ("C)
C (10"N s'"rn-')
1400
1300
1200
1100
1000
900
800
0.15
0.37
0.98
3 .O
11.0
51.0
310.0
Properties of the asthenosphere - increase of C with depth
It is interesting to note that the rebound estimates of the product CK are related to the
width of the rebounding depression. Lake Bonneville has the smallest width and yields the
smallest value for CK. Canada has the largest width and yields the largest value for CK. In a
uniform half-space asthenosphere, flow occurs to a depth comparable to the width of the
surface depression (Fig. 5). Thus the Canadian uplift would cause flow almost as far down
as the core-mantle boundary, while the Bonneville uplift would cause flow in only the
upper few hundred kilometres of the mantle. The uplifts of different width thus provide a
way of examining the possible variation of the creep coefficient with depth.
If the observed variation of the product CK is to be explained this way, there must be an
increase in the value of C with depth. The asthenosphere appears to be stiffer under the
wider uplifts because the flow in these areas is more influenced by the stiffness of the deeper
mantle. There are, of course, an infinite number ofways in which such an increase in Cwith
depth could occur. For simplicity, only two models will be examined here. In the first, the
asthenosphere is assumed to have a finite thickness, D,and to have a uniform value of C. The
material below the asthenosphere is assumed to be rigid. In the second, the asthenosphere is
assumed to be a half-space, but the value of the creep coefficient is assumed to be exponentially depth-dependent, C b ) = Coexp(y/D).
For both models, the value of K has been calculated numerically for n = 3 as a function
of the dimensionless depth scale, D' = kD (Fig. 6). For the exponential increase model, the
values of creep coefficient used in the calculations are normalized by Co. Thus, for this
model, Coshould replace C in the uplift relation (3) and the scaling relations (4). The factor
K is essentially the geometrical stiffness of the medium. As expected, the thinner the
asthenosphere or the larger the increase in C with depth, the stiffer the medium appears to
be. The results in Fig. 6 are probably accurate. As part of each power-law numerical solution, the Newtonian flow solution has also been calculated. For the variable thickness case,
the numerical Newtonian values of K are always within 5 per cent of the analytic values as
given by Ramberg (1967).
Using the results in Fig. 6, values of D are searched for such that the rebound data can be
explained by a laterally uniform asthenosphere. It is found that both models of creep coefficient increase can explain the observed relation between the product CK and the width of
Isostatic rebound and power law flow in the asthenosphere
733
1
0
n
nl2
3nl2
D‘
Figure 6. The value of K as a function of the dimensionless depth scale, D’, for two models of creep
coefficient increase with depth. For curve (a), the asthenosphere has a rigid base at a depth D’.For curve
@), the creep coefficient increases as exp (y’/D’).
the rebound area. With the channel model of the asthenosphere, the thickness of the
asthenosphere beneath shield areas is found to be about 170 km. This gives a dimensionless
thickness, D’, of n/15 and n/8 for Canada and Fennoscandia. Seismic, magnetic, and heat
flow data all indicate that the lithosphere beneath Lake Bonneville is about 80 km thinner
than under the shield areas (Crough & Thompson 1976). Thus the asthenosphere is assumed
to be 80 km thicker under Bonneville than under the shields. This gives D’= n for Bonneville. From Fig. 6 it can be seen that the values of K for Canada, Fennoscandia, and
Bonneville are 120, 50, and 4.5. These yield values of Cof 0.83,0.86, and 6.89 x 101oNslfl
m-’.
With the exponentially increasing creep coefficient model, the depth scale, D , is found to
of n/18.5, n/10, and nll.8 for Canada,
be 135 km. This gives a dimensionless depth scale, D’,
Fennoscandia, and Bonneville. These values of D’give values of K of 120,65, and9.5. The
asthenosphere comes closer to the surface beneath Bonneville than beneath the shields. If C
depends on depth then the value of C immediately below the lithosphere will be less in the
Bonneville area than in the shield areas. To get comparable results, this effect must be
removed. Using D = 135 km, the 80 km decrease in lithosphere thickness beneath Bonneville
lowers the value of Coby a factor of 1.8. The Bonneville estimate of Co should agree with
the shield estimates only if it is increased by a factor of 1.8. This is equivalent to lowering
the value of K by a factor of 1.8, so K should be taken as 5.3 for Bonneville. These values
beneath
indicate that at about 100 km depth Cequals 0.83,0.66, and 0.75 x 1010Ns”3m~2
Canada, Fennoscandia, and Bonneville respectively.
Of course, the rebound data are not accurate enough to determine the value of C to two
significant digits. It can only be said that both models of creep coefficient increase with
depth indicate that C equals about 1010Ns’’3m-Z at the base of shield lithosphere. This
yields a temperature estimate for the top of the asthenosphere of about 1200°C. This is
consistent with the values used in thermal models of the lithosphere-asthenosphere system
(Parker & Oldenburg 1973; Crough 1975).
The value of the depth scale, D, is not well determined. It depends critically on the value
of the product CK for Lake Bonneville and this is the least well determined value from any
of the three rebounds. Conceivably, the given estimates of D might be inerror by a factor of
three. It should also be remembered that the estimates of D are based on the assumption of
734
S. T. Crough
lateral uniformity in the asthenosphere. The more one allows the Basin and Range asthenosphere to have a higher temperature or volatile content than shield asthenosphere, the larger
the estimate of D will become.
However, suppose that the asthenosphere is laterally uniform and that the estimate of D
for the exponential model is correct. What causes the increase in the creep coefficient with
depth? The simplest cause is hydrostatic pressure. It is known that at constant temperature
Cis given as a function of pressure, P,by
C(P) = Co exp(fi/nRT)
(7)
where u is the activation volume for diffusion (Weertman & Weertman 1975). Since pressure
increases with depth, if the activation volume is large enough there will be a noticeable
increase in Cwith depth, even if temperature remains constant.
Kohlstedt & Goetze (1974) have noted that their deformation experiments at atmospheric confining pressure show a factor of 3 greater strain-rate than the experiments of
Carter & Ave'Lallement (1970) at the. same differential stress but at 15 kbar confrning
pressure. If this difference is caused by the pressure effect of relation (7), then the size of
the pressure effect in the mantle can be estimated. An increase in confining pressure of
45 kbar will decrease the strain-rate by a factor of 27. Using n = 3, this is equivalent to
increasing C by a factor of 3. A 45 kbar increase in pressure is caused by a 135 km increase
in depth. Thus, from the pressure effect alone, the creep coefficient should depend on depth
approximately as C ( y )= C, exp [y(km)/l35]. This is almost exactly what the rebound data
indicate.
The activation volume can be estimated by equating the inferred depth dependence with
the predicted dependence (7). Using p = 3.3 g ~ m - n~ =, 3, and T = 1500 K, the activation
volume for the asthenosphere is about 9 cm3mole-'. This appears to be reasonable. Kirby
& Raleigh (1973) have used 11 cm3mole-' as a likely value. Goetze & Brace (1972) have
suggested that 15-20 cm3mole-' is proper for the mantle. Future experiments will clearly
help refine these estimates of activation volume. At the moment, the increase in hydrostatic
pressure with depth can explain all of the possible increase of the creep coefficient with
depth.
Conclusions
The isostatic adjustments of Canada, Fennoscandia, and Lake Bonneville indicate that the
value of the creep Coefficient, C,in the power-law flow relation, Ce''3 = u, is approximately
10'0Ns''3m-2 for the asthenosphere. The available experimental data on the deformation
of peridotite reveal that this is the creep coefficient expected of a dry asthenosphere at a
temperature of about 1200°C. This temperature estimate is consistent with existing estimates made by other methods. This consistency provides additional evidence that the
asthenosphere does deform by power-law flow.
Assuming that the asthenosphere is infinitely thick and uniform with depth, the rebounds
in the two shield areas, Canada and Fennoscandia, give estimates of the creep coefficient
which are an order of magnitude larger than the estimate given by the rebound of Lake
Bonneville. The experimental data indicate several ways to explain this difference. The
asthenosphere beneath the shield areas could have a lower temperature or a lower volatile
content than the asthenosphere beneath Lake Bonneville.
The preferred explanation of the difference, however, is that it indicates an increase in
the creep coefficient with depth. The three rebounds are consistent with a laterally uniform
asthenosphere which has either a thickness of 170 km or a creep coefficient which increases
Isostatic rebound and power law flow in the asthenosphere
735
with depth by a factor of 3 every 135 km.This latter increase is the preferred explanation.
The experimental data indicate that the increased hydrostatic pressure with depth would
cause just such an increase.
It should be remembered that the isostatic rebound areas have been modelled as sinusoidal
surface deflections of infinite horizontal extent. Only the behaviour of the central part of
each uplift has been examined. Clearly the next step is to develop solutions for mi-symmetric
surface deflections with more realistic shapes. With these new solutions it will be possible to
examine the data from all parts of the rebound areas. Hopefully, when more data are
analysed it will be possible to better resolve the lateral and vertical variations in the properties of the asthenosphere.
Acknowledgments
Bill Brace suggested that this work be done. Amos Nur showed me the perturbation technique
for solving power-law flow problems and acquainted me with the experimental data. The
finitedifference computer program used is a modification of a program originally developed
by Lee Bell to analyse antiplane flow. Steve Kirby provided useful criticisms of an earlier
version of this paper. I am grateful to the National Science Foundation, the Amoco Foundation, and the Education Office of Woods Hole Oceanographic Institution for fellowship
support while this work was accomplished.
References
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dispersion of Rayleigh waves, Geophys. J. R. asfr. SOC.,36,515-539.
Brennen, C., 1974. Isostatic recovery and the strain rate dependent viscosity of the earth’s mantle,J. geophys. Res., 79,3993-4001.
Carter, N. & Ave’Lallemant, H., 1970. High temperature flow of dunite and peridotite, Geol. SOC.Am.
BUN., 81,2181-2202.
Cathles, L., 1975. The viscosify o f f h e eurfh’s mantle, Princeton University Press, New Jersey.
Crittenden, M., 1963. Effective viscosity of the earth derived from isostatic loading of Pleistocene Lake
Bonneville, J. geophys. Res., 68,5517-5530.
Crough, S. T., 1975. Thermal model of oceanic lithosphere, Nufure, 256,388-390.
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Kirby, S . H. & Raleigh, C. B., 1973. Mechanisms of high-temperature, solid-state flow in minerals and
ceramics and their bearing on the creep behaviour of the mantle, Tecfonophys.. 19,165-194.
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Kushiro, I., Syono, Y. & Akimoto, S., 1968. Melting of a peridotite nodule at high pressures and high
water pressures, J. geophys. Res., 73,6023-6029.
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Foren. Stockholm Foch., 60,397-404.
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Peltier, W. R., 1974. The impulse response of a Maxwell earth,Rev. Geophys. Space Phys., 12,649-669.
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Appendix: finitedifference solution of twodimensional, plane-strain power-law flow
problems
As shown in equation (2) of the text, it is possible to rewrite the power-law constitutive
relation in a form which looks like the Newtonian flow law. In tensor form equation (2)
should be written as
where J1,Jz, J3 are the first, second, and third invariants of the strain-rate tensor, f i s some
function which allows (8) to satisfy the experimental results, and the pressure, P, has been
introduced so o is now total stress not deviatoric stress. For simplicity, the deforming
material is assumed to be isotropic. With the further assumption that the material is incompressible for permanent deformation, J1vanishes. Because of the plane-strain condition, J3
equals 0 so f may be taken as a function of Jz alone. With J2 = (i:, t i:y),the simplest
choice for fis a square root function giving
The term C(e2, t e:y)(1-n)12nis the effective viscosity, 9,for general power-law flow.
A perturbation method of solution is used. The equivalent Newtonian flow problem with
constant viscosity is solved first. From this solution, a strain-rate field is calculated. These
strain-rates are then converted into an effective viscosity field using a particular value of n.
The Newtonian flow problem is then resolved using these new effective viscosities. The process is repeated until a stable solution is found.
The major difficulty in the solution procedure is finding the solution for Newtonian flow
with variable viscosity. Since this step must be performed several times for each value of n,
it is important that the numerical method used be relatively fast yet accurate. The following
technique has been found to be satisfactory.
A streamfunction, JI, is introduced such that
v = - a+lax
u =aGlay.
(10)
This satisfies the condition of incompressibility. The two stress equilibrium equations for
plane-strain and negligible inertial forces are
Substituting (10) into the constitutive relation (9), substituting the expressions for stress
into (1 l), and then differentiating and subtracting to eliminate the pressure gives the govern-
737
Isostatic rebound and power law flow in the asthenosphere
If q is constant, this reduces to the biharmonic equation.
A 13 point finitedifference approximation of (12) is used. For an interior point,
the ith row and jth column of the grid, the formulation is
(qi,j-l) $i,j-2
+(4qt-%,j-%
-4(qi-%,j-%
t(Qi-1.j)
+Vi+%,j-%)
in
p q i - 1 ~- ~ i , j - ~ $i-1,j-1
)
$i,j-l
$i-2,j-4(~i-%,j-s
+ ( 4 ~ i + % , j - % - ~ i + ~ ,- j~ i , j - 1 ) + i + l , j - l
+ T i - % , / + % )$ i - l , j + ( q i , j - l
+ ( 4 ~ i - ~ , j -+q ~i - ~ , j - q i , j + ~ ) $ i - l , j + l
+(4qi+%,/+%
$i,j,
- ~ i + l , j - ~ i , j + ~$i+1,/+1
)
-4(qi-%,j+%
+ ( q i , / + ~ )$ i , j + 2
+Vi,j+l
+qi-l,j+qi+l,j
+ q i + % , j + % ) Jli,j+l
=O.
The formulation is slightly different for points on or near the boundaries. Using relations
like (13) a set of M equations is created for the M unknown values of $ at the grid points.
These equations are solved directly for $ by Gaussian elimination. On an IBM 370 computer
one iteration, that is, forming the effective viscosity field, setting up the equations, and
solving by elimination, takes 0.3 s of CPU time for a grid with 144 unknowns. Since a stable
solution is usually found in less than five iterations, the method is quite fast and inexpensive.
To know the stresses throughout the deforming medium, a set of equations for the
pressure must be set up and solved. However, to determine the value of K in equation (3) it
is only necessary to know the normal stress at the upper surface of the flow. Since the horizontal velocity is required to be zero for all x at the surface, aV,,ay = 0 at the surface and
the normal stress is simply the pressure, P. From (9), ( l o ) , and ( 1 1 ) it can be shown that,
along the surface,
M a 2$lay2- az $/ax2)] .
ap/ax = -(a/ay)
(14)
The right-hand side of this relation can be evaluated from the flow solution, so the surface
pressure can be determined to within an additive constant by numerical integration. Setting
P = 0 where the vertical velocity equals zero completely determines the pressure.
Another way to calculate the surface pressure is to calculate the total rate at which
viscous work is done within the flowing medium. Since the pressure within the medium
does no work, the work rate can be determined without knowing the pressure field. The
total work rate throughout the medium must equal the total work rate of the driving normal
stresses at the upper surface of the flow. Assuming that the surface pressure has the
sinusoidal form K sin kx,the surface work rate,
1
V(X)P(X)
h
9
can be evaluated in terms of the unknown K. This is then set equal to the numerically calculated work rate throughout the medium and solved for K.
This latter method of determining the surface pressure gives more consistent results,
iteration after iteration, than relation (14). This is to be expected since relation (14) uses
only the near-surface values of $ while the total work rate method uses them all. For n = 3
738
S.
T.Czough
1-
PlP,
:
rrl2
0
X'
F i r e . 7. Dots indicate calculated surface pressure, P,a8 a function of dimensionless horizontal distance,
x'. Values are normalized by dividing by the maximum surface pressure,P,. Solid line is the assumed sine
function distribution.
the two methods give results which agree, on average, within 20 per cent. The values shown
in the text, however, are always the work rate values, since these are probably more accurate.
The pressure calculations using relation (14) indicate that it is reasonable to assume that
the surface pressure has a sinusoidal form. In Fig. 7 the normalized surface pressure is shown
for four randomly selected flow solutions using n = 3 and varying thicknesses for the
asthenosphere. There is some scatter, but the values are well represented by a sine function.