Geophysical Journal (1988) 94, 559-565
RESEARCH NOTE
Rheological modelling and deformation instability of lithosphere
under extension-11. Depth-dependent rheology*
Gianna Bassi? and Jean Bonnin$
t Atlantic Geoscience Centre, Bedford Institute of Oceanography, Box 1006, Dartmouth, N.S. 32Y 4A2, Canada
$ Laboratoire de Gtodynamique, Institut de Physique du Globe, 5, rue Rent Descartes, 67084 Strasbourg Cedex, France
Accepted 1988 March 9. Received 1988 March 8; in original form 1987 November 16
SUMMARY
Bassi & Bonnin (1988) have recently discussed the stability of continental lithosphere in
extension. They examine the hypothesis that thickness inhomogeneities can be amplified
during extension and be responsible for lithospheric necking. We carry our analysis one step
further by accounting for the increase of strength with depth in brittle layers, namely the
upper crust and, in some cases, the upper part of the mantle. This variation had been
neglected in the previous study to simplify the mathematical formulation. Rather than a
constant, the equivalent viscosity is now a linear function of depth, as predicted by Byerlee’s
friction law which is postulated to control the mechanical behavior of brittle layers. This more
realistic rheological model confirms the results of the original analysis, specifically that the
interfaces between layers tend to flatten during extension unless they correspond to an
inversion of density. Therefore, extension of continental lithosphere appears to be stable if
there is no other initial defect susceptible of localizing the deformation.
Key words: deformation, lithosphere, rheology
INTRODUCTION
Bassi & Bonnin (1988) (hereafter referred to as Paper I)
have recently addressed the problem of stability of
continental lithosphere under extension by discussing the
possible amplification of small-scale variations in layer
thicknesses. This process would localize the deformation,
and eventually lead to lithospheric necking. Several possible
variations of lithospheric rheology with depth were
examined, based on assumptions that are recalled below.
These rheological models do not favour the growth of
mechanical instabilities in the lithosphere except for a
gravitational instability of the lithosphere-asthenosphere
boundary if an inversion of density is assumed at this
interface.
Brace & Kohlstedt’s (1980) model of lithospheric
rheology has been widely used in the literature, because this
approach, even though it is approximate, seems to enclose
the main characteristics of the evolution of rock rheology
with depth. In Paper I, we have used Brace & Kohlstedt
(1980) as a reference model, but we have introduced several
approximations to render the mathematical formulation
more tractable. In the ductile layers, where power-law creep
is the dominant deformation mechanism, the strong
temperature dependence of stresses at a given strain-rate is
* Geological Survey of Canada Contribution No. 34587.
approximated by an exponential depth dependence. In
addition, the brittle layers, where friction along pre-existing
faults controls strength, are described by an average
constant strength, rather than the linear increase predicted
by Byerlee’s law. This last simplification is one of the major
shortcomings of the rheological approach adopted in Paper I
and could critically influence the conclusions, first because it
applies to the layers that are potentially unstable with
respect to tension, and second because a high sensitivity of
the results to the strength stratification was observed in this
previous analysis.
The constant-strength brittle layer restriction can be
relaxed without requiring the use of a numerical solution,
and the analysis can be carried out for the theoretical,
linear, curve with the same analytical method that has been
used previously. We present here the results of this more
accurate rheological description and show that they
essentially confirm our previous conclusions.
FORMULATION
Mechanical model
The rheological and mechanical assumptions of the model
are discussed extensively in Paper I, therefore we do not
justify these in detail here. Instead, we recall the main
characteristics of the approach and emphasize the particular
aspects that are modified.
559
560
G. Bassi and J . Bonnin
base of
lithosphere
where the vectors V ( q i ) are the natural modes of
deformation of the system. The associated eigenvalues,
q&,
determine the rate of amplification of the
corresponding mode. The largest eigenvalue, generally
called the growth rate factor and labelled q, must be
positive to observe an increase of the topography, which is
then necessarily unstable because of the exponential in (1).
The most unstable initial configuration is the mode of
deformation associated with q. Our intent is to determine
the growth rate spectrum q ( k ) ( k being the wavenumber)
and the corresponding mode of deformation for a range of
model parameters.
base of
asthenosphere
(fixed)
Figure 1. Model of three-layer lithosphere under extension. The
initial geometry of the interfaces is a sinusoid (wavelength L ) about
a mean horizontal position. Underlying astenosphere is a
Newtonian fluid and perturbing velocity is assumed to vanish at the
base of this layer.
Brace & Kohlstedt's (1980) model approximates lithospheric structure using either three or four layers, depending
on the geothermal gradient and rheological parameters.
From the surface down they are a brittle upper crust, a
lower ductile crust, in some cases a brittle upper mantle
region (although small changes in ductile flow-law
parameters can shift the depth of brittle-ductile transition
and eliminate this region of brittle deformation), and a
ductile upper mantle region. We assume that the various
interfaces are initially deformed with respect to the
horizontal (Fig. l ) , and we wish to determine whether these
disturbances can be amplified when the lithosphere is
stretched.
As long as the amplitudes of the interfaces remain small
with respect to the layer thicknesses, the mechanical
problem can be solved by using a perturbation method. This
method, which has been extensively applied to this kind of
analysis (Smith 1977; Fletcher & Hallet 1983; Ricard &
Froidevaux 1986; Zuber, Parmentier & Fletcher 1986),
considers that all the physical quantities of the problem can
be written as the sum of a mean value and a perturbing
value. In this particular case, the mean or basic flow consists
of a uniform extension at constant strain-rate ixx.This
formulation enables us to linearize a problem which is
otherwise typically non-linear, as each layer in the
lithosphere is approximated by a power-law fluid rheology
(see next section). The perturbing flow can thus be
determined independently of the basic flow. Moreover, each
Fourier component of these perturbations can be analysed
separately because the conditions of continuity of stresses
and velocities at the interfaces, when expanded to the first
order, are h e a r with respect to the amplitudes of the
disturbances (equations 17-20 of Paper I).
The position of the various interfaces through time (at
least in the incipient stages of extension) can be determined
by performing an eigenanalysis: if H is the vector of the
deflections at the rn deformable interfaces (rn = 4 or S), then
(as in Paper I):
Rheology
Brace & Kohlstedt (1980) use experimental rock mechanics
data for quartz and olivine to construct the strength
envelope, on the assumption that properties of these
minerals control the stress level in the crust and upper
mantle, respectively. The upper mantle is composed
primarily of olivine, which is thus likely to represent its
rheology adequately. The structure of continental crust is far
more complex, and involves polycrystalline aggregates for
which flow laws are only beginning to be determined.
Available data for quartzite may reflect the upper crustal
response to the stress environment, but feldspathic and
dioritic rocks are more typical of mid-crustal levels and
granulitic rocks of lower crustal levels (Carter & Tsenn
1987), and information concerning these materials is still
scarce. Laboratory studies of quartz are therefore used for
an estimate of the rheology of continental crust at any
depth. Since quartz is more ductile than feldspathic or
dioritic rocks at lower crustal temperatures, this assumption
leads to an underestimation of the lower crustal strength.
Ductile deformation of both quartz and olivine in typical
lithospheric conditions is postulated to occur by dislocation
creep (Brace & Kohlstedt 1980; Kirby 1983). This
mechanism corresponds to a power-law fluid behaviour so
that the ductile layers are described by the following
constitutive relations:
I
Q
2q=Aexp-JJ,
RT
(1-,,)/2
,
(3)
where uij and E, are the stress and strain-rate tensors,
respectively, and p is the pressure. The viscosity r] depends
on the material constants A , Q and II, and on the second
invariant of the deviatoric stress J,.
To avoid the need for a fully numerical solution, in Paper
I we approximated the viscosity f j for the basic, pure shear,
flow by an exponential function as suggested by Fletcher &
Hallet (1983):
f j = qoeYz
(4)
which takes into account the sharp decrease of strength and
viscosity with depth.
Brittle deformation in the upper crust and in the
uppermost mantle is thought to be controlled by friction
along faults rather than fracture. If the fault spacing is
smaller than the length scale of the structure, and if faults of
all orientations exist in the medium, then a perfectly-plastic
Rheology and extension of lithosphere-II
solid can be a reasonable rheological model for these layers,
provided that yield strength varies with depth according to
Byerlee’s law. Obviously, this model relies on a number of
assumptions that are questionable, particularly the hypothesis that the crust is extensively and randomly fractured at
depth. However, because there is no acceptable theory
which adequately predicts fracture strength under general
states of stress (Brace & Kohlstedt 1980; Carter & Tsenn
1987), we retain this model and Byerlee’s law as the most
plausible quantitative relation for the variation of strength
of the brittle layers. Furthermore, as mentioned in Paper I,
a perfectly-plastic rheology is closely approximated using a
power-law fluid behavior with a large stress exponent
(n = 1000). As a consequence, relation (2) holds in the
brittle layers also, together with the following expressions of
Byerlee’s law: First, in the crust,
axx- Uzz = 22.4d + 20,
(6)
where d, is the depth of the Moho discontinuity. The
constants 22.4 and 26.4 result from the choice of a
2800kgm-3 density for the crust, 3300kgm-3 for the
mantle lithosphere and a zero pore pressure, a condition
that probably exaggerates the strength contrasts in the crust.
Note that relations (5) and (6) differ slightly from the
corresponding equations B-8 of Paper I in as far as the
resistance to friction does not vanish at the Earth’s surface.
A finite value, which is in fact more realistic, had to be
introduced to avoid divergence of the solution at the
surface. As indicated by Fletcher & Hallet (1983), values of
surface resistance (uxx- uz,)/2, based on estimates of
coherent rock strength, range from 9 to 43 MPa (Zoback &
Zoback 1980). However, at low temperature and pressure,
fracture strengths exceed the stress needed for slip on a
pre-existing fault (Goetze & Evans 1979) so that 10MPa
seems to be a reasonable estimate of strength at the Earth’s
surface.
The effective viscosity can be defined by combining (2)
and (5) or (6):
axx- azz= 4qZxx.
(7)
The effective viscosity (in Pa s) in the crustal brittle layer
and mantle brittle layer, respectively, is given by:
ij = y--(22.4d + 20)
and:
4Exx
1000
1500
crust
mantle
1
E
4:
50
5a
Q)
0
100
150
Figure 2. Models of continental geotherm used to constrain
rheological models for the lithosphere. No radiogenic heat
production is assumed in the mantle where the geotherm is
described by qmrthe heat flow at Moho, which ranges from 30 to
50mWm-* (Morgan & Sass 1984). In the crust, the heat
production rate is an exponential function of depth, with a depth
decay length b of 10 km and a surface value H, varying between 2
and 6 pW m-3. The base of the lithosphere is defined by the 1300 “C
is0therm.
parameters, unless specified, are those indicated in Table 1
and are identical to the ones used in Paper I. Fig. 3
compares the effective viscosity used in the calculations and
the value predicted by Brace & Kohlstedt’s (1980) approach
for two particular models of geotherm. In both cases, the
two curves are clearly in good agreement. The results should
therefore provide important insight on the behavior of the
lithosphere, assuming the Brace & Kohlstedt’s (1980) model
is a good representation of its rheology. The input
parameters (Table 2) are basically the same as in Paper I,
except that there is no viscosity discontinuity at the
brittle-ductile transition and the surface strength must now
be specified.
Table 1. Parameters of the strength envelope.
Densities (kg m-3): crust: pc = 2800
mantle lithosphere: p,, = 3300
asthenosphere: pa = 3250
4 L
fi = y
[22.4dc + 26.4(d - d,) + 201.
“C-
-
No pore pressure
10“
lo6
500
(5)
where the depth d is positive down from the surface and is
expressed in km, yielding (axx- aZz)in MPa; similarly, in
the mantle:
aXx- aZz= 22.4dC+ 26.4(d - dc)+ 20,
Temperature
0
561
(9)
The solution of the Navier-Stokes equation for such a
linear viscosity law can be obtained by expressing it as a
power series in z (Appendix A).
We consider the implications for stretching instabilities of
a series of rheological models based on the approximations
described above. Each is characterized by a different model
of the geotherm (Fig. 2). The values of the other
Crustal thickness: d, = 35 km
Mean strain-rate: iS,, = 1 0 - ’ 5 s ~ 1
Olivine flow law (Goetze & Evans 1979):
Quartz flow law (Christie et a[., in preparation):
= 2.4 lO-’(u,,
[
- uzz)3
exp -
(stress in MPa)
1-
149.103
RT
562
G. Bassi and J . Bonnin
THREE-LAYER LITHOSPHERE
FOUR-LAYER LITHOSPHERE
E f f e c t i v e V i s c o s i t y (10" Pas)
E f f e c t i v e V i s c o s i t y (10'' Pa.s)
0
10
5
0
15
10
Y
Moho
....
n
E
3
40
30
Moho
........
40
._
x
W
5
3
E
20
W
60
5
a
60
a
80
a,
!3
'""I
Base of Lithosphere
.......................................
120
too
Surface Heat Production Rate : 4 pW.nY3
Heat Flow a t Moho : 40 r n W . r n - '
Base of Lithosphere
.......................................
Surface Heat Production Rate : 4 ~ W . r n - ~
Heat Flow a t Moho : 30 rnW.m-'
Brace & Kohlstedt
Model
p1
-
Figure 3. Effective viscosity versus depth for a three-layer model of lithosphere (surface heat flow 80 mW m-') and a four-layer model (surface
heat flow 70 mW m-'). The dotted area defines the viscosity curve predicted by Brace & Kohlstedt's (1980) description of lithospheric
rheology; the solid line is the viscosity law used in this study.
RESULTS
Figures 4 and 5 show the growth rate factor q as a function
of the dimensionless wavenumber k = 2 n d c / L ( L is the
disturbance wavelength) for three-layer and four-layer
lithospheric models, respectively. They are almost identical
to Figs 4 and 5 of Paper I, with a maximum rate of
amplification around 15 for an approximate wavelength of
50 km.
Table 2. Dimensionless parameters required for the determination
of the perturbing flow.
Parameter
Meaning
-k = -2ndc
-
Wavenumber
- A h
-r* = 22xx,,77s
Mean position of the interfaces
Stress exponent in each layer
Inverse of viscosity decay length in
the crust and mantle respectively
Viscosity discontinuity at Moho
Surface strength
L
-zi/dc (i = 1, 4 or 5 )
-ni (i = 1, 3 or 4)
- ~ c d c ,Ymdc
The results are also comparable for the modes of
deformation, as the only amplified mode at any wavelength
corresponds to an instability of the lithosphereasthenosphere boundary, with no deformation of any other
interface. This kind of behaviour has been interpreted in
Paper I as a gravitational instability, due to the lower
density of the asthenosphere with respect to the mantle
lithosphere. The evidence for this is shown again on Fig. 6
which indicates that the growth rate factor increases when
the parameters S,, S, and S, (see Table 2) increase. As
those parameters measure the intensity of gravitational
versus extensional stresses, their influence suggests that
gravity rather than extension is the instability driving
mechanism. A straightforward test to confirm this
interpretation is to set S, = 0 (Fig. 6), i;e. a uniform mantle
density; in this latter case, no amplification is observed at
any wavelength and any interface topography will be
damped progressively as the lithosphere is stretched.
DISCUSSION
The results of this stability analysis do not appear to be
affected by the more realistic rheology of the brittle layers
adopted in this paper compared to Paper I. Qualitatively,
the absence of mechanical instability was not predictable in
view of previous studies, as the model happened to be very
sensitive to the strength stratification. However, once this
Rheology and extension of lithosphere-11
563
L
0
2"
I
nt----r
0
~
,
I
6
4
2
I
8
, I
10
12
2
0
20
4
6
8
,
1 0 1 2
01
qrn= 40
L d = 50 k m
3
3
e
e
0
0
10
1
I
O
h
0
L
2
4
A
6
8
I
1
I
/
,
i
5!/
O
'
I/
10
12
0
I
,
I
I
I
I
2
4
6
8
10
12
li
5-1
I
I
'/
0
first result is established, the close quantitative correspondence between this study and the previous one is physically
reasonable. Because the only interface which will eventually
be unstable under extension is the lithosphereasthenosphere boundary, one can expect the corresponding
growth rate factor to depend on the layer thicknesses and on
the mechanical behaviour of the layers concerned, but not on
the details of rheological modelling of the upper crust or
uppermost mantle. Because the former parameters are
unchanged with respect to our previous work, it is sensible
to observe a very similar growth rate curve.
Other authors (Ricard & Froidevaux 1986; Zuber et al.
1986) have developed the same approach on the basis of
different rheological assumptions, and concluded that
stretching instabilities are predicted by their models. As a
consequence, they have interpreted the alternate occurrence
of basins and ranges in the Basin and Range province of the
United States in terms of boudinage-like instabilities of the
upper crust. From the analysis of the factors that control this
discrepancy (Paper I), it appears that a critical parameter is
the strength contrast between the unstable layer (or layers)
and those lying immediately above and below it. More
precisely, the studies mentioned above show evidence for
mechanical instabilities because they assume that the
viscosities in the upper crust and the region of high strength
in the mantle (the unstable layers of the model) are typically
100 times higher than elsewhere. This importance of a high
strength ratio between adjacent layers is not really a new
result since Smith (1977) stated that the unstable layer has
to be more competent than the surrounding medium to
allow the growth of an instability. This requirement is even
stronger if the structure is submitted to stabilizing gravity
forces. With our particular rheological models, for example,
even increasing the surface strength to the point where
qrn=
Hs= 43 0 ~
2
4
6
8
1
0
Dimensionless Wavenumber k
U 1I ne 11\ 10 n lesi \1 ve n11 mi b r r k
Figure 4. Growth rate factor q as a function of dimensionless
wavenumber for three-layer lithospheric models. t,,refers to the
wavelength for which q is maximum. The diagram on each curve is
a schematic representation of the corresponding mode of
deformation of the lithosphere. Note the similarity of the results for
the different models.
'
+---A.
Figure 5. Same plot as in Fig. 4 but here the uppermost layer of the
mantle is brittle and the lithosphere behaviour is approximated using
four layers.
strength of the brittle crust is almost uniform (Fig. 7) is not
sufficient to initiate an instability ( q = 0). The gravitational
forces must be considerably reduced with this rheology to
observe a different mechanical behavior. In the limiting case
S,=S,=O,
two peaks appear on the growth rate factor
spectrum: a long wavelength peak around 220km,
comparable to the wavelength observed by Ricard &
Froidevaux (1986) and Zuber et al. (1986), and a short
wavelength peak around 32km which corresponds to a
boudinage-like instability of the upper crust. Note that the
latter wavelength is approximately four times the layer
thickness, as predicted by other analyses (Smith 1977;
Fletcher & Hallet 1983).
/
S=
,
[/I
50
sL=
, 0
,
I
4
6
8
0
0
I
2
10
12
Dimensionless Wavenumber k
Figure 6. Influence of the intensity of gravitational versus
extensional stresses on the growth rate factor spectrum. When the
parameter S,,and consequently ,
S and S, (see Table Z), decreases,
the amplification becomes less efficient. It is zero in the limiting case
S, = 0, that is pa = pm.
564
G . Bassi and J . Bonnin
Effective Viscosity (10" Pas)
0
5
10
r
15
Moho
Base of Lithosphere
..~~.................~.....~...........
100
0
Brace & Kohlstedt
Model
n
2
4
6
8
10
12
Dimensionless Wavenumber k
-
Figure 7. Model of lithosphere rheology and corresponding growth rate factor. The layer thicknesses and ductile characteristics are the same as
for the qm= 40 mW . m-'- H, = 4 p W . m-3 model, but the surface layer strength has been increased so that the surface layer strength is
uniform (left-hand side figure). Moreover, all gravitational effects have been neglected (S, = S, = S, = 0). Note that the dominant wavelength
is about four times the surface layer thickness (9 km).
CONCLUDING REMARKS
This complementary study confirms that Brace &
Kohlstedt's (1980) description of lithospheric rheology does
not favour the growth of stretching instabilities in the
lithosphere, at least if these instabilities have to be initiated
by thickness irregularities alone. The physical explanation
for this behavior is that the strength contrasts predicted by
the model are not large enough to counteract the stabilizing
effect of gravity forces. Of course, this rheological model
may not be an accurate representation of lithospheric
rheology. The choice of a zero pore pressure and a linear
increase of frictional resistance with depth is believed to
overestimate the actual strength in the brittle regime. In
contrast, using quartz as a representative mineral of the
lower crust probably leads to an underestimation of its
strength, as mentioned earlier. Moreover, a semi-brittle
regime, for which quantitative laws are not available yet, is
likely to dominate at midrange depths of the continental
crust (Kirby 1983; Carter & Tsenn 1987). However,
incorporating these properties in the rheological model
would result in an even smoother strength-depth curve,
which is unlikely to produce conditions more favourable to
the existence of instabilities.
Rather than questioning the rheological aspect of the
problem only, other assumptions of the model can be
reconsidered. If, as seems to be the case, the conditions of
instability are not present initially in a continental
lithosphere with average crustal thickness, one may wonder
if continuing extension, by modifying the layer thicknesses,
thermal regime and therefore mechanical properties, can
create a more favorable environment. Furthermore, our
basic hypothesis, namely that thickness irregularities could
be responsible for necking of the lithosphere, is not
necessarily the most appropriate. Lateral heterogeneities in
the mechanical properties, for example, which are easy to
imagine because of the complex history of continental crust,
may be more efficient at localizing the deformation.
ACKNOWLEDGMENTS
We thank C. E. Keen and G . S. Stockmal for their
constructive reviews of an early draft of this paper.
Comments by R. Wortel, D. Praeg and an anonymous
reviewer are also gratefully acknowledged. G. Bassi is
funded by a SNEA(P) (SociktC Nationale Elf-Aquitaine)
post-doctoral fellowship.
REFERENCES
Bassi, G. & Bonnin, J., 1988. Rheological modelling and
deformation instability of lithosphere under extension,
Geophys. J., 93, 485-504.
Brace, W. F. & Kohlstedt, D. L., 1980. Limits on lithospheric stress
imposed by laboratory experiments, J. geophys. Res., 85,
6248-6252.
Carter, N. L. & Tsenn, M. C., 1987. Flow properties of continental
lithosphere, Tectonophys., 136, 27-63.
Fletcher, R. C. & Hallet, B., 1983. Unstable extension of the
lithosphere: a mechanical model for Basin-and-Range structure, .I
geophys.
.
Res., 88, 7457-7466.
Goetze, C. & Evans, B., 1979. Stress and temperature in the
bending lithosphere as constrained by experimental rock
mechanics, Geophys. J. R. astr. SOC., 59,463-478.
Kirby, S. H., 1983. Rheology of the lithosphere, Rev. Geophys.
Space Phys., 21, 1458-1487.
Rheology and extension of lithosphere-II
565
Morgan, P. & Sass, J. H., 1984. Thermal regime of the continental
lithosphere, J. Geodynam., 1, 143-166.
Ricard, Y. & Froidevaux, C., 1986. Stretching instabilities and
lithospheric boudinage, J. geophys. Res., 91, 8314-8324.
Smith, R. B., 1977. Formation of folds, boudinage, and mullions in
non-newtonians materials. Bull. geol. SOC. Am., 88, 312-320.
a b a c k , M. C. & a b a c k , M. D., 1980. Faulting patterns in
North-Central Nevada and strength of the crust, J. geophys.
Res., 85, 275-284.
Zuber, M. T., Parmentier, E. M. & Fletcher, R. C., 1986.
Extension of continental lithosphere: a model for two scales of
Basin and Range deformation, J . geophys. Res., 91,
4826-4838.
It is easy to show that these functions are particular
solutions of (A-3) provided that the coefficients aij satisfy
the following recurrence-relations:
APPENDIX A
with:
We wish to determine the perturbing flow in a layer with
linear depth dependence of the effective viscosity:
rl
0 = z,, + -2,
P
f j =pz
+ 7.,
A40ai0 - 2&ail- 4&0ai2+ 12ai3+ 240ai4 = 0,
+ A 4 0 U i - 2&(i+ 1)2ai+l
- 2&0(i+ 2)(i + l)ui+2
+ (i + 3)(i + 2)2(i + l ) U i + 3
+ 0(i + 4)(i + 3)(i + 2)(i + l)ui+4= 0
A4ai-l
(A-1)
If 6 and 6 are the horizontal and vertical components,
respectively, of the perturbing velocity, the separable
solution:
k = W coshr
1dW.
6 = ---sinAx
A dz
(A4
satisfies Navier-Stokes equations if W is a solution of the
fourth-order differential equation:
d4W
d2W
&
=
(& 1)P.
Moreover, they constitute a fundamental set of solutions
if they are chosen in such a way that:
Wi(i--l)(Ztr)
= 1 (i = 1, 4)
if k # i - 1 (0 5 k 5 3).
=0
{ Wlk)(z,,)
The expressions of velocities and stresses are then simply
derived from relation (A-2) and constitutive relations (2).
This yields:
dz
(i
- 2 --1
dW
A2p-+A4fjW=0.
dz
(A-3)
This equation has no singular point in the depth range of
interest. It admits therefore for each value of z a solution
which is expressible as a power series in z. In order to write
the boundary conditions at the interface between the layer
and the surrounding ones, the sum of this series has to be
evaluated for these two depths; this is most easily done if
the series is developed about one of these two coordinates,
say z,,, the position of the brittle-ductile transition.
Actually, it is convenient, as will appear below, to express
the general solution of (A-3) as a linear combination of a
fundamental set of solutions, namely:
c AiW,(z).
4
W ( z )=
i=l
Assuming that:
m
w , ( z )=
c
j=O
a,(z - z,,)'.
(i = 1, 4)
k(x, z) =
6&,
"
A,W,(z)]cos hr,
i=l
cOsAxi
dW-
z ) = - Ai[(:-1)A2fj$
A2 , = I
d3K
d2w.
-fj:-p7-pA2w,..
dz
dz
1
With this formulation, the final determination of velocities
and stresses in the layer is very similar to the constant
viscosity or exponential viscosity case as these quantities are
expressed as a function of four integration coefficients.
These coefficients are determined by writing the boundary
conditions at the top and bottom of the layer. The rest of
the calculation is thus carried out as described in Paper I.