Download Lithospheric necking: a dynamic model for rift morphology

Earth and Planetary Science Letters, 77 (1986) 373-383
373
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
[61
Lithospheric necking: a dynamic model for rift morphology
M.T. Z u b e r * and E.M. P a r m e n t i e r
Department of Geological Sciences, Brown Uniuersity, Providence, RI 02912 (U.S.A.)
Received August 26, 1985; revised version received January 3, 1986
Rifting is examined in terms of the growth of a necking instability in a lithosphere consisting of a strong plastic or
viscous surface layer of uniform strength overlying a weaker viscous substrate in which strength is either uniform or
decreases exponentially with depth. As the lithosphere extends, deformation localizes about a small imposed initial
perturbation in the strong layer thickness. For a narrow perturbation, the resulting surface topography consists of a
central depression and uplifted flanks; the layer thins beneath the central depression. The width of the rift zone is
related to the dominant wavelength of the necking instability, which in turn is controlled by the layer thickness and the
mechanical properties of the lithosphere. For an initial thickness perturbation with a width less than the dominant
wavelength, deformation concentrates into a zone comparable to the dominant wavelength. If the initial perturbation is
wider than the dominant wavelength, then the width of the zone of deformation is controlled by the width of the initial
perturbation; deformation concentrates in the region of enhanced thinning and develops periodically at the dominant
wavelength. A surface layer with limiting plastic (stress exponent n = ~ ) behavior produces a rift-like structure with a
width typical of continental rifts for a strong layer thickness consistent with various estimates of the m a x i m u m depth of
brittle deformation in the continental lithosphere. The width of the rift is essentially independent of the layer/substrate
strength ratio. For a power law viscous surface layer (n = 3), the dominant wavelength varies with layer/substrate
strength ratio to the one-third power and is always larger than for a plastic surface layer of the same thickness. The
great widths of rift zones on Venus may be explained by unstable extension of a strong viscous surface layer.
1. Introduction
Rift zones are areas of localized lithospheric
extension characterized by a central depression,
uplifted flanks, and thinning of the underlying
crust. These features have been identified on many
of the planets and their satellites; on the earth,
rifts are found both on the continents and in
ocean basins, and represent the initial stage of
continental breakup and seafloor spreading. Topographic profiles illustrating the general morphology of rifts on continents and on the surface
of Venus are shown in Fig. 1. High heat flow,
broad regional uplift, and local magmatism are
often associated with rifts, which suggests that
thermal as well as mechanical effects play a role in
determining their morphology. In this study, we
investigate the mechanical aspects of rifting by
developing a simple model in which the dynamical
consequences of flow in an extending plastic or
* Now at: Geodynamics Branch, N A S A / G o d d a r d Space Flight
Center.
0012-821X/86/$03.50
© 1986 Elsevier Science Publishers B.V.
viscous lithosphere are evaluated. For simple rheological stratifications we calculate the pattern of
near-surface deformation which arises due to
horizontal extension and compare the results to
major morphological features of rifts, such as those
shown in Fig. 1.
The width and morphology of rift zones was
first explained by Vening Meinesz [1], who suggested that rifts form in an extending elastic-brittle
layer which fails by normal faulting. Flexure of the
layer occurs in response to motion on a normal
fault and a second normal fault forms where the
bending stresses in the layer are a maximum, thus
defining the width of the rift. Flanking highs form
by isostatic upbending of the elastic layer in response to graben subsidence. The elastic properties
of the layer thus determine the width and morphology of the rift. Artemjev and Artyushkov [2]
qualitatively considered rifting due to necking in a
ductile crust which is strong near the surface and
weaker at depth. A perturbation in lower crustal
thickness localizes during uniform extension and a
corresponding stress concentration in the brittle
374
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Fig. 1. (a) Topographic profiles across several continental rift zones. From Buck [15]. (b) Map of radar bright and dark linear
features, interpreted as faults, and topographic profiles across rift zone in Beta Regio, Venus. Note the presence of a central trough
bounded by uplifted flanks. The arrows represent the approximate locations of major bounding faults shown on map. Vertical lines
mark edge of the map. The locations of Rhea and Theia Mons, interpreted as volcanic shields, are shown for reference. From
Campbell et al. [251.
375
upper crust results in a narrow zone of normal
faulting. Bott [3] developed a model for rifting by
crustal stretching in response to external tension
on the basis of the elastic model of Vening Meinesz.
In this model, the elastic upper crust responds to
tension by normal faulting and graben subsidence
while the ductile lower crust undergoes thinning
due to horizontal flow beneath the subsiding
graben.
In this study, we examine rifting as the growth
of a necking instability. To nucleate a rift, a small
thickness perturbation is imposed at the base of a
strong layer which overlies a weaker substrate. For
a range of rheological parameters we evaluate the
conditions for which the initial disturbance will
amplify as the lithosphere extends, and we determine the associated pattern of near-surface deformation. As shown later (see Fig. 5), the pattern
of deformation consists of a central depression
beneath which the strong layer thins due to necking, and flanking uplifts, all of which are characteristic of rift zones. Horizontal extension of the
lithosphere, which drives the instability, could occur in response to remotely applied forces or
horizontal forces due to regional doming on a
scale much larger than the width of the rift zone.
The model results are thus applicable to both
passive and active rifting (e.g. [4]). The initial
thickness perturbation could correspond to a preexisting structural weakness such as a tectonic
suture or a thermal anomaly due to an igneous
intrusion, diapiric upwelling, or convective transport of heat to the base of the lithosphere. All of
these have been suggested in association with terrestrial rifts. A thickness perturbation due to diapirism or convective upwelling may be much
wider than the rift, while that due to igneous
intrusion or a pre-existing weakness may be narrower than the rift. As discussed later, differences
in the width of the perturbation may have implications for the character of rifting.
2. Model formulation
2.1. One-dimensional extension
The simplest formulation for the growth of an
instability in an extending layer treats the flow as
locally one-dimensional with a uniaxial stress Oxx
and strain rate c** [5]. Consider a layer of thick-
ness h ( x ) which has a viscosity much higher than
its surroundings. To satisfy the condition of equilibrium in the layer, the horizontal force F where:
(1)
F=axx h
must be independent of x. To satisfy the condition
of incompressibility:
Cxx = - h - I
Oh/Ot
(2)
where the minus sign indicates that the layer thins
as it extends. The constitutive relationship between stress and strain rate is:
,xx =AoL
(3)
where n is the stress exponent of the layer and A
is a constant. By combining (1) to (3), the rate of
change of layer thickness can be written:
Oh/Ot = - A F " h 1-"
(4)
For later comparison with our linearized two-dimensional formulation, h can be expressed:
h = n ( t ) - 8 ( x , t)
(5)
where H ( t ) is the layer thickness for uniform
thinning and 8 is a small (<< h) thickness variation. By substituting (5) into (4), expanding, and
integrating with respect to time, the amplitude of
the layer thickness variation may be written:
~(x, t) = 6 ( x , 0) e x p [ ( n - 1),xxt ]
(6)
where 8(x, 0) describes the shape of the initial
perturbation and Cxx is independent of time. The
stress exponent n defines the growth rate of a
thickness perturbation. If n > 1 an initial disturbance will grow with time, while if n < 1 a
disturbance will decay and the layer will thin
uniformly. Equation (6) demonstrates that n > 1 is
necessary for extensional instability and that the
horizontal layer thickness variation retains the
same shape as the perturbation grows in amplitude.
2.2. Linearized two-dimensional extension
In a two-dimensional formulation, the total flow
can be expressed as the sum of a mean flow or
basic state of uniform horizontal extension and a
perturbing flow which arises due to instability.
The lithosphere is represented as a strong layer of
thickness h overlying a weaker substrate. Two
strength stratifications, illustrated in Fig. 2, are
considered. For creep deformation, the strain rate
376
d-Model
C-Model
viscous fluid:
-fh
+
R=Ga,d
oij = 21acij - p3ij
(8)
where # is the dynamic viscosity and p is the
pressure. Substituting the stresses and strain rates
from (7) into (8) and linearizing about the basic
state gives:
6,, X = ( 2 ~ t / n ) g x x - ~
;/h
G =
6xz = 2~gxz
(9)
Fig. 2. Strength stratification of strength jump (J) and continuous strength (C) models. The former is described in terms of
the ratio of layer/substrate strength R = #~)/#~2), while the
latter is described by the ratio a = ~/h of the viscosity decay
depth in the substrate to the strong layer thickness. The layer
and substrate are described by uniform densities (O> P2) and
power law exponents (nl, n2).
where ~ is the viscosity evaluated at the stress or
strain rate of the basic state. For a single Fourier
harmonic, perturbing velocities in the vertical and
horizontal directions:
(c) is proportional to stress (o) to a power n I = 3.
Deformation by distributed faulting can be idealized using a perfectly plastic material with a stress
exponent of n I = m. The substrate is assumed to
deform by creep with n 2 = 3. In the strength j u m p
(J) model, the layer and substrate both have a
uniform strength with Oxx~'(1)> Uxx~'(2),where the superscripts 1 and 2 refer to the layer and substrate,
respectively. In the continuous strength (C) model,
the strength is uniform in the layer, continuous
across the layer-substrate interface, and decays
exponentially in the substrate with an e-folding
depth ~'. Since we are interested primarily in dynamic, stress-supported topography and numerous
previous studies have considered the effect of isostatically compensated crustal thickness variations,
the layer and substrate are each represented by a
uniform density (p) with I)1 = 02.
In a layered medium, any small perturbation
along an interface will cause deviatoric stresses
proportional to the amplitude of the perturbation
resulting in the growth of an instability. The total
velocities (u, w), stresses (oo), and strain rates
( q j) are written:
where D = d / d z and k ( = 27r/)~) is the wave
number, satisfy the incompressibility condition.
Within any layer in which the viscosity varies
exponentially with depth, the equations of equilibrium are satisfied if:
u=~+fi
w=~+~
% = o-,j + a,j
(7)
Eij = Eij + Eij
where a bar represents the mean flow and a tilde
the additional or perturbing flow. For an isotropic
= Wcos kx
=-k
(10)
1DWsin k x
D4W + 2~-lD3W+
-2k2~-'(2/,-
[~-2 _ 2 k 2 ( Z / n _ 1)] D 2 W
1 ) D W + k 2 [ k 2 + ~,-2] W = 0
(11)
Solutions of (11) are given in Fletcher and Hallet
[6]. The flow in any layer with uniform strength or
viscosity can be obtained from (11) by taking the
limit as ~"+ ~ .
The velocities u and w and the stress components %, and ozz must be continuous across each
interface. At the surface, the vertical normal stress
must equal the weight of the surface topography
and the shear stress must vanish. Expressed in
terms of the basic and perturbing stresses, the
stress continuity conditions at each interface give:
5x~(x, d i ) = [~x~/-') - ~ i ' ] d 3 , / d x
G(x,
(12)
4) = (p,-, - 0,)ga,
where d i is the depth of the interface. These
equations have been linearized for small distortions (3i) of the interface (cf. [7,8]). Vertical velocities arising from these perturbing stresses amplify
initial deformation of the interfaces further enhancing the perturbing stresses resulting in instability. At a given time, the shape of the ith inter-
377
face can be represented by the superposition of
Fourier harmonics:
6i(x, t ) = ~ A i ( k ,
t) sin
kx
(13)
k
where i = 1, 2 refer to the free surface and the
layer-substrate interface, respectively, and 8 and
A are the spatial and wave number domain representations of the interface shapes. In terms of the
perturbing vertical velocity, the time rate of change
of amplitude A is:
A, = W(k,
d~) - ~x~A,
(14)
where A = d A / d t . The system of equations consisting of (11), (14) and that obtained by substituting (13) into (12) has solutions of the form Acc
exp[(q - 1)~xt ], where q is the growth rate factor
which measures the degree of instability of the
disturbance at a particular wave number. The
value of unity subtracted from q represents the
kinematic interface distortion from the second
term on the right-hand side of (14). Complete
~olutions of this system of equations can be written:
Al(k, t) = A,1 e x p [ ( q , -
1)exx ]
+A12 exp[(q2 - 1)exx]
(15)
A2(k, t) =A21 exp[(q 1 - 1)exx ]
+A22 exp[(q2 - 1)exx]
where ql and q2 are eigenvalues, (An, A12 ) and
(A21 , A22) are the eigenvectors corresponding to
ql and q2, respectively, and exx ( = i x x t) is the
total horizontal extension at a time t. The length
of the eigenvectors are determined from the prescribed initial conditions. The surface topography
is assumed to be initially flat [81(x, 0 ) = 0], and
the initial shape of the layer-substrate interface is
given by a prescribed function. The Fourier transform of the initial interface shapes is substituted
into (15), and the perturbed surface topography is
found from the inverse Fourier transform of
Al(k, t). Patterns of deformation in the layer and
substrate can be determined by calculating the
displacements due to the perturbing flow at points
on a coordinate grid.
The number of growth rate factors at each wave
number (in this case two) equals the number of
interfaces. In general, only one value of q is
greater than unity and thus contributes to instability. While both values of q are required to completely describe the evolution of interface shapes,
solutions of (15) which retain only the positive
value of q yield a good approximation to the
perturbed interface shape. Note the correspondence of (15) to the one-dimensional growth rate
equation (6) with the substitution of q for n. As
subsequent results for the two-dimensional problem show, q varies directly with the stress exponent in the layer, but unlike the one-dimensional
growth rate, q is a function wave number.
The J and C models are each expressed in terms
of two dimensionless parameters. In both models,
S = ( P a - Po)gh/[a~lx)/2] where P0 is the density
above the layer, determines the relative importance
of surface topography which stabilizes the deformation and stresses due to surface distortion which
help drive the instability. In the J model, R =
~1)/7,(2)
the layer/substrate strength ratio, conx/Uxx,
trols the relative magnitude of stresses due to
distortion of the surface and the layer-substrate
interface. In the C model, the layer-substrate interface does not help to drive the instability because
the strength is continuous across it. The amount of
viscous resistance to deformation in the substrate
is determined by the ratio of the viscosity decay
depth to the layer thickness a = ~/h.
3. Results
In the limit of large layer/substrate viscosity
contrast, the one-dimensional growth rate agrees
with the growth rate of the dominant wavelength
for the two-dimensional formulation, as noted by
Emerman and Turcotte [9]. However, Fig. 3a shows
that the growth rate spectrum for one-dimensional
extension is independent of wave number so this
model fails to predict a dominant wavelength.
Thus, to explain the regular development of geologic structures using one-dimensional theory, initial thickness perturbations along layer interfaces
must be distributed periodically. In the absence of
a process that defines a natural length scale, the
occurrence of periodic initial disturbances seems
unlikely.
Fig. 3 also shows growth rate spectra for twodimensional models with limiting plastic (n 1 = 10 4)
and power law viscous (nl = 3) surface layers
plotted as a function of dimensionless wave number k' (=2~rh/)Q. The wave number (k~) at
378
a)
b}
r, ~..-3
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/~=m
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a)
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......... ~ .
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.........
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~
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0
_
2
b)
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0.5
o
!
d model
\ k'
-2C
r~:~/[
-3o
[ l
C model
Fig. 3. (a) Growth rate spectra as a function of wave number
non-dirnensionalized by the strong layer thickness k ' ( = kh).
The growth rate for one-dimensional flow in a layer with stress
exponent n = 3 is simply equal to n over the entire range of k'.
Thus the one-dimensional model fails to predict a dominant
wave number. For two-dimensional flow, the solid and dashed
lines refer to cases with limiting plastic (n] = 104) and power
law viscous (n 1 = 3 ) surface layers, respectively, for the J
model. For both cases S = 1.2, R = 50 and n 2 = 3. The peaks
in the growth rate spectra define the dominant wave numbers.
(b) Two-dimensional growth rate spectra for the C model for
plastic and power law viscous surface layers with S = 1.2, a = 5
and n 2 = 3. In this model the growth rate for a viscous surface
layer is always less than unity and therefore the layer is stable
with respect to necking.
which the growth rate factor is maximized ( q d )
defines the dominant wavelength )kd ( = 2 ' n ' / k d ) .
For the J model (Fig. 3a), the magnitude of the
growth rate for a lithosphere with a plastic surface
layer is approximately two orders of magnitude
greater than that for a power law viscous layer
with the same value of R. This indicates that a
lithosphere in which the near-surface deforms
plastically is much more unstable in extension
than a lithosphere which deforms viscously
throughout. The growth rate spectrum for the C
model is shown in Fig. 3b for both plastic and
viscous surface layers. For n I = 3 the growth rate
is everywhere less than unity, which indicates that
the lithosphere is stable with respect to necking.
An initial disturbance in strong layer thickness
will decay with time, and deformation of the
medium will be manifest as uniform thinning.
This holds for any reasonable range of lithosphere
mechanical properties.
Topographic profiles for the two-dimensional
growth rate spectra in Fig. 3 and an initial thickness perturbation o f the form ~2(X, O)(X e (~/d)2
where d << h are shown in Fig. 4. The topography
in each case consists of a central depression and
flanking uplifts which are dynamically supported
I
Fig. 4. (a) Rift morphology for limiting plastic (n 1 = 104, solid
line) and power law viscous (nl = 3, dashed line) surface layers
for the strength j u m p (J) model. For both cases S = 1.2, R = 50
and n 2 = 3. For comparison, the surface topography in each
case is normalized by the depth of the central depression. (b)
Rift morphology for the C model with plastic surface layer and
with S = 1 . 2 , c t = 5 and n 2 = 3 .
and arise in response to the perturbing flow induced as the layer necks. For the J model, a plastic
surface layer (n] = 10 4) results in deformation
localized into a region overlying the initial thickness perturbation. With a power law viscous
surface layer (hi = 3) deformation is more broadly
distributed. For the C model, the relative amount
of uplift of the flanks is much greater than in the J
model, and small surface depressions occur outside
the flanking highs. A variety of initial perturbation
shapes including Gaussian, boxcar, and triangular
functions have been examined. The topography is
independent of the shape of the initial perturbation.
The amplitude of the deformation is dependent
on the amplitude of the initial disturbance, the
amount of horizontal extension exx, and the magnitude of the growth rate factor. In the linearized
theory, the amplitude of deformation for a given
horizontal extension is exactly proportional to the
amplitude of the initial disturbance. Since for actual rift zones the amplitude of this disturbance is
unknown, only relative amplitudes, which depend
on the growth rate spectrum and horizontal extension, are shown in Fig. 4. Fig. 4a shows that the
flanking uplifts are narrower and have a greater
amplitude relative to the depth of the central
depression for the J model with large n t. For the
same initial disturbance amplitude, the absolute
magnitude of the topography for the plastic surface
layer is greater than that for the viscous layer,
reflecting the larger growth rate for the former
case as shown in Fig. 3a.
As for the topography discussed above, the
379
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z']']l']'~'l I',li
.. ~ l l l l l l l l l l l l
IIIIIIlIIl]l
Illlllllllllllllll~
_/x
"~4.UIIIIIIIIIIIIIIIII
VE-Zx
Fig. 5. Deformation of an extending plastic surface layer with
an initial thickness perturbation at its base. The width of the
disturbance is determined by the growth rate spectrum in Fig.
3a. The shape of the initial perturbation of width d (<< h) is
shown schematically by the dashed line. The central depression
and uplifted flanks produced due to the unstable growth of the
initial perturbation are characteristic topographic features of
rift zones (cf. Fig. 1).
pattern of strong layer deformation shown in Fig.
5 reflects the shape of the growth rate spectrum.
While the absolute amplitude is arbitrary, the relative amplitudes are everywhere properly represented. Hence if the topography at the surface or
another interface is known, the amplitude of deformation at any point in the layer or substrate is
determined. For example, Fig. 5 shows that the
amount of upwarping at the base of the strong
layer is approximately four times the depth of the
central depression. Likewise, the amount of relative uplift of the flanks is about half the depth of
the central depression.
The width of the rift zone is controlled by the
dominant wavelength. For the J model, the dominant wave numbers shown in Fig. 3 correspond to
wavelength to layer thickness ratios of Xd/h = 4.0
and 10.6, respectively, for plastic and viscous
surface layers. In the J model, the width of the rift
from flank-to-flank is almost exactly equal to the
dominant wavelength. In the C model for n t 10 4,
the dominant wave number corresponds to ~d/h
= 3.7, which is very similar to the result in the J
I0°
K~
102
R
i0~
I0
V3
I/6
V9
VI2
I/f5
Va
Fig. 7. (a) Dominant growth rate factor qd vs• R for the J
model• (b) qd VS. 1/a for the C model• For r/1 = 3 , q is always
less than unity and the lithosphere extends uniformly. S = 0
and n 2 = 3 for both cases.
model with a plastic surface layer. However, the
flank-to-flank width of the rift zone for the C
model is about twice the dominant wavelength. In
this case the width of the central depression is
approximately equal to the dominant wavelength.
The relationships between rift zone width and the
dominant wavelength demonstrated by the above
examples hold for the complete range of cases we
have examined for the J and C models.
Fig. 6 shows the variation of k~ with R for the
J model. For power law viscous near-surface behavior k d varies a s R -]/3. This relationship was
first determined for flexural buckling of a layered
medium [10] and was later shown to be valid for
boudinage and folding of a layered viscous medium
in the limit of large viscosity contrast [7,8,11]. In
the plastic limit k~ is essentially independent of
R. For this case the width of the rift will be
=
1.8 '~'- -~v
........
-' -' -" -' ~,6o'8°
15
~0
~k'
1.2t q ~ ' X , ~
nl=~o
%[
ioo
- - n,'lO'~
- - n,-3
120
%
,oo
~
0.9[
80
0.6
SO
40
•
~ - --Tkd.
. . .
0
IOI
102
R
IOa
Fig. 6. D o m i n a n t wave number k~ as a function of strength
contrast R for the J model with S = 0 and n 2 = 3.
2
4
20
. . . . . . . . . .
6
8
I0
S
12
14
Fig. 8. Dominant wave number k~l and dominant growth rate
factor qd as a function of S for n] =104 (solid) and n 1 = 3
(dashed) with R = 50 and n 2 = 3.
380
primarily a function of the strong layer thickness.
The dominant wavelength decreases with increasing stress exponent in the layer (n 1), which is
consistent with results on folding and boudinage
of a non-Newtonian layer embedded in a viscous
medium [11].
The variation of the growth rate at the dominant wave number with R for the J model and
with 1/a for the C model is shown in Fig. 7. Note
that qd increases with R and l / a , indicating
enhanced instability for large strength contrasts
and small e-folding depths. Increased instability is
a result of the relative decrease in viscous resistance in the substrate for both cases. In the plastic
layer case for the J model the magnitude of qa is
generally proportional to R, indicating that the
strength contrast at the layer-substrate interface
provides a significant contribution to driving the
instability.
Fig. 8 summarizes the effects of the b u o y a n c y /
strength parameter S for the J model and shows
that k~ and qa decrease with increasing S for
both large and small n~. As would be expected, an
increase in density with depth across an interface
stabilizes the perturbing flow. The decrease in
dominant wave number with increasing S is consistent with results for folding of a layered viscoelastic medium under the influence of gravity [12].
Increasing S results in deepening of the rift and
damping of the bounding flanks.
4. Discussion
as a mechanical response to localized extension. In
the present model the flanks are solely a consequence of viscous or plastic flow.
A thermal anomaly that is broader than the
region of crustal thinning may also explain flanking uplifts. Buck [15], Keen [16], and Steckler [17]
suggest that thermally-produced uplifted flanks
form as a result of horizontal heat transfer due to
small-scale convection induced by the rift temperature structure. Steckler [17] interpreted the
amount of uplift in the Gulf of Suez to be indicative of lithospheric heating greatly in excess of
that expected due to uniform lithosphere extension. Buck [15] showed that the magnitude and
distribution of uplift associated with continental
rifts may be explained by small-scale convection.
The relative contributions of thermal and mechanical mechanisms should be reflected by the
timing of uplift on flanks. If uplift is dynamic,
then deepening of the central trough and uplift of
the flanks should occur simultaneously. If it is a
thermal effect, then uplift should lag the formation of the central depression. Flanking uplifts
along the Gulf of Suez formed during the main
phase of rift development [17], but the relative
timing of the formation of the rift valley and the
flanks is 'not yet clearly defined. Thermal uplift,
unless it is frozen in by a thickening elastic lithosphere, should decay due to cooling after extension ceases. Uplift due to viscous or plastic flow,
produced while the extending lithosphere is at
yield, must also be frozen-in as stresses fall below
the yield stress at the cessation of extension.
4.1. Flanking uplifts
4.2. Continental rifts
Uniform stretching of the lithosphere resulting
in crustal thinning and subsequent conductive
cooling may account for the subsidence of rift
basins (e.g. [13]) but does not explain the flanking
highs characteristic of rift zones. Flanking uplifts
in our necking model form in response to dynamic
upwarming produced by the unstable flow. Since
there are no density contrasts at depth, the flanks
are supported entirely by stresses in the layer. As
in the models of Vening Meinesz [1] and Bott [3]
which treat the near-surface as an elastic layer, the
flanking uplifts regionally compensate the rift depression. Finite element solutions [14] incorporating elastic as well as plastic and viscous behavior have also shown that flanking uplifts can occur
The width of a rift zone formed by necking is a
function of the thickness of the strong layer of the
lithosphere. The strength stratification of the lithosphere is determined by brittle behavior at shallow
depths and creep at greater depths (cf. [18]). The
brittle strength increases linearly with depth to a
maximum value determined by a creep strength
which decreases with depth as temperature increases. Neither the J nor C model is an exact
representation of this strength stratification since
in both models the linear increase of brittle strength
with depth is approximated by a layer of uniform
strength. In addition, in the C model the strength
falls to zero at depth while in the J model the
381
strength is discontinuous at the base of the strong
layer. For the purposes of the present study, the
strong layer thickness will be taken to correspond
to the depth of the brittle-ductile transition. In a
model lithosphere with a plastic surface layer, the
layer-substrate interface corresponds to a change
in power law exponent, indicating the change in
the mode of deformation from faulting to ductile
flow. Strength decreases rapidly with depth below
the brittle-ductile transition. If estimated from this
strength stratification, the strong layer thickness
would not be significantly greater than the depth
to the brittle-ductile transition.
As shown in Fig. 4, a rift formed in a lithosphere with a plastic surface layer has a flank-toflank width = 4h. If the strong upper crustal
region of the continental lithosphere is best described by a plastic theology, then a continental
rift with a typically observed width in the range
35-60 km [19] requires a strong layer thickness of
9-15 km. An extending continental lithosphere in
which a quartz rheology approximates flow in the
crust undergoes the transition from brittle to
ductile behavior at a depth of about 15-20 km
depending on the geothermal gradient and strain
rate (cf. [18]). This is comparable to the depth to
which earthquakes are observed in the continental
lithosphere and to which faulting penetrates in
continental rifts as shown in seismic reflection
profiles.
For a strong viscous surface layer with nl = 3,
the dominant wavelength, and therefore the rift
zone width, depends on the ratio of layer to substrate strength as shown in Fig. 6. For a rift zone
60 km wide, typical of the well-studied East African Rift or Rhinegraben, a strong layer thickness
of 15 km requires an R only slightly greater than
unity.
The width of the rift is controlled by the dominant wavelength as long as the initial perturbation
width d < ?~d. However, if d > ?~a the width of the
rift is governed by the width of the initial disturbance. If an initial perturbation is much wider
than the dominant wavelength, corresponding perhaps to broad-scale lithospheric thinning, extension localizes in the thinned region and deformation within that region develops periodically at the
dominant wavelength [6,20]. The vast regional extent of the Basin and Range Province ( > 103 km)
may reflect an initial strong layer thickness per-
turbation much wider than the spacing of individual basins and ranges ( -~ 30 km). The depth distribution of seismicity [21], high heat flow [22], and
low Pn velocities [23] in this area suggest that an
upper mantle thermal anomaly of large horizontal
extent could be the cause of the initial perturbation.
4.3. Rifting on Venus
A number of features on Venus with the morphology of rift zones have been recognized from
Pioneer Venus radar altimetry [24]. High-resolution earth-based radar images have recently been
obtained for a major rift in Beta Regio [25]. As
shown in the across-strike profiles in Fig. lb, the
topography consists of a central depression and
bounding highs which may represent uplifted
flanks. The width of the rift varies along strike but
averages about 150 km from flank to flank. Two
volcanic shields occur along the rift, Rhea Mons in
the north and Theia Mons in the south. While
volcanic construction could contribute to the
flanking highs, they are best developed where the
central depression is deepest and in areas not
associated with the volcanoes (profiles G and H in
Fig. lb).
The high surface temperature of Venus, approximately 700 K, suggests that ductile deformation will occur at shallower depths than on the
earth. On the basis of estimates of rock strength,
surface temperature, and the spacing of features of
presumed tectonic origin in the banded terrain,
Solomon and Head [26] suggest that the elastic
lithosphere thickness on Venus is in the range
1-10 km, and therefore too thin to explain the
width of the Beta Regio rift by elastic flexure,
which would require an elastic-brittle layer thickness in excess of 60 km. Schaber [24] estimated a
brittle layer thickness in the range 47-69 km on
the basis of the widths of other presumed rift
valleys on Venus. Radar bright lineaments in Beta
Regio, interpreted as faults [25], suggest the presence of a brittle surface layer. On the basis of Fig.
4, the width of the Beta Regio rift would require a
plastic layer thickness of about 40 km. If the
near-surface of Venus is weaker than the earth, S
will be greater than that assumed in Fig. 4. The
limit of large S corresponds to a rift zone width
= 6h and a plastic layer thickness of 25 km. This
382
is greater than the depth to the brittle-ductile
transition on Venus estimated by Solomon and
Head [26] and would require a geothermal gradient less than that which they assume (20 K k m - l ) .
The 10-20 km spacing of bright lineaments where
they are most well-developed is comparable to the
width of radar bright bands in the banded terrain.
If the radar bright features are due to faulting
with a spacing comparable to the brittle-elastic
layer thickness, then this thickness in Beta Regio
should be similar to that in areas of banded terrain and too thin to explain the rift zone width. If
the plastic region of near-surface faulting is thin,
then lithospheric necking may be controlled by a
thicker viscous layer deforming by creep. In this
case the width of the rift zone is determined by
the strong layer thickness and the layer/substrate
strength contrast. On the basis of Fig. 6, 10 and 20
km thick strong layers require values of R = 100
and 10, respectively, to explain the width of the
Beta Regio Rift.
5. Summary
Unstable extension of a lithosphere consisting
of a strong surface layer overlying a weaker viscous
substrate results in a pattern of deformation that
is consistent with the major morphological characteristics of rift zones. The surface topography
consists of a central depression, beneath which the
layer thins by necking, and flanking uplifts. The
rift is nucleated by introducing a small amplitude
thickness perturbation at the base of the layer. For
an initial perturbation narrower than the dominant wavelength, deformation concentrates in a
zone of width comparable to the dominant wavelength. For an initial thickness perturbation wider
than the dominant wavelength, deformation develops periodically at the dominant wavelength in the
region above the perturbation.
The dominant wavelength is controlled by the
layer thickness and by the growth rate spectrum of
extensional instability, which is a function of the
layer/substrate strength contrast, the density
stratification, and the stress exponents describing
flow in the layer and substrate. Extension of a
strong surface layer that deforms by distributed
faulting, idealized by plastic behavior (n 1 = oo),
results in a rift zone width approximately four
times the plastic layer thickness. If the base of the
strong layer corresponds to the maximum depth of
brittle deformation, determined either by experimental flow laws or the observed depth of earthquakes, then the width predicted by this model is
generally consistent with that of a typical continental rift zone. Dominant wavelength and therefore rift zone width for a plastic surface layer is
essentially independent of the layer/substrate
strength contrast. For a strong viscous surface
layer (n~ = 3), dominant wavelength varies as the
strength ratio to the one-third power. Extension of
a viscous surface layer always produces a rift zone
that is wider than that for a plastic surface layer.
The great width of rift zones on Venus, relative to
the spacing of other features of presumed tectonic
origin, may be explained by ductile necking of a
strong viscous layer.
Acknowledgements
This research was supported by NASA grant
NSG-7605. We thank Ray Fletcher for helpful
discussions and Philip England for a constructive
review of the manuscript.
References
1 F.A. Vening Meinesz, Les "graben" africains r6sultat de
compression ou de tension dans la croBte terrestre?, K.
Belg. Kol. Inst. Bull. 21, 539-552, 1950.
2 M.E. Artemjev and E.V. Artyushkov, Structure and isostasy
of the Baikal Rift and the mechanism of rifting, J. Geophys. Res. 76, 1971.
3 M.H.P. Bott, Formation of sedimentary basins of graben
type by extension of continental crust, Tectonophysics 36,
77-86, 1976.
4 A.H.C. SengSr and K. Burke, Relative timing of rifting and
volcanism on earth and its tectonic implications, Geophys.
Res. Lett. 5, 419-421, 1978.
5 E.W. Hart, Theory of the tensile test, Act. Metall. 15,
351-355, 1967.
6 R.C. Fletcher and B. Hallet, Unstable extension of the
lithosphere: a mechanical model for Basin and Range
structure, J. Geophys. Res. 88, 7457-7466, 1983.
7 R.C. Fletcher, Folding of a single viscous layer: exact
infinitesimal amplitude solution, Tectonophysics 39,
593-606, 1977.
8 R.B. Smith, Unified theory of the onset of folding,
boudinage, and mullion structure, Geol. Soc. Am. Bull. 86,
1601-1609, 1975.
9 S.H. Emerman and D.L. Turcotte, A back-of-the-envelope
approach to boudinage mechanics, Tectonophysics 110,
333-338, 1984.
10 M.A. Biot, Folding instability of a layered viscoelastic
383
11
12
13
14
15
16
17
18
19
medium under compression, Proc. R. Soc. London, Ser. A
242, 444-454, 1957.
R.B. Smith, Formation of folds, boudinage, and mullions in
non-Newtonian materials, Geol. Soc. Am. Bull. 88, 312-320,
1977.
M.A. Biot, The influence of gravity on the folding of a
layered viscoelastic medium under compression, J. Franklin
Inst. 267, 211-228, 1959.
D. McKenzie, Some remarks on the development of sedimentary basins, Earth Planet. Sci. Lett. 40, 25-32, 1978.
H.D. Lynch, Numerical Models of the Formation of Continental Rifts by Processes of Lithospheric Necking, pp.
164-216, Ph.D. Thesis, New Mexico State University, Las
Cruces, N.M., 1983.
W.R. Buck, Small-scale convection induced by passive rifting: the cause for uplift of rift shoulders, submitted to
Earth Planet. Sci. Lett., 1985.
C.E. Keen, The dynamics of rifting: Deformation of the
lithosphere by active and passive driving mechanisms, Geophys. J. R. Astron. Soc. 80, 95-120, 1985.
M.S. Steckler, Uplift and extension at the Gulf of Suez:
Indications of induced mantle convection, Nature 317,
135-139, 1985.
W.F. Brace and D.L. Kohlstedt, Limits on lithospheric
stress imposed from laboratory experiments, J. Geophys.
Res. 85, 6248-6252, 1980.
I.B. Ramberg and P. Morgan, Physical characteristics and
20
21
22
23
24
25
26
evolutionary trends of continental rifts, Proc. 27th Int.
Geol. Congr. 7, 165-216, 1984.
M.T. Zuber, E.M. Parmentier and R.C. Fletcher, Extension
of continental lithosphere: A model for two scales of Basin
and Range deformation, J. Geophys. Res., in press, 1985.
G.P. Eaton, The Basin and Range Province: origin and
tectonic significance, Annu. Rev. Earth Planet. Sci. 10,
409-440, 1982.
A.H. Lachenbruch and J.H. Sass, Models of an extending
lithosphere and heat flow in the Basin and Range Province,
in: R.B. Smith and G.P. Eaton, eds., Geol. Soc. Am. Mem.
152, 209-250, 1978.
R.B. Smith, Seismicity, crustal structure, and intraplate
tectonics of the interior of the western cordillera, in: R.B.
Smith and G.P. Eaton, eds., Geol. Soc. Am. Mem. 152,
111-144, 1978.
G.G. Schaber, Venus: limited extension and volcanism
along zones of lithospheric weakness, Geophys. Res. Lett.
9, 499-502, 1982.
D.B. Campbell, J.W. Head, J.K. Harmon and A.A. Hine,
Venus: volcanism and rift formation in Beta Regio, Science
226, 167-170, 1984.
S.C. Solomon and J.W. Head, Venus banded terrain:
tectonic models for band formation and their relationship
to lithospheric thermal structure, J. Geophys. Res. 89,
6885-6897, 1984.