Download Molnar, P., and G. A. Houseman (2013), Rayleigh

JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, 2544–2557, doi:10.1002/jgrb.50203, 2013
Rayleigh-Taylor instability, lithospheric dynamics, surface
topography at convergent mountain belts, and gravity anomalies
Peter Molnar1 and Gregory A. Houseman2
Received 8 December 2012; revised 22 April 2013; accepted 25 April 2013; published 24 May 2013.
[1] Surface topography and associated gravity anomalies above a layer resembling
continental lithosphere, whose mantle part is gravitationally unstable, depend strongly on the
ratio of viscosities of the lower-density crustal part to that of the mantle part. For linear
stability analysis, growth rates of Rayleigh-Taylor instabilities depend largely on the wave
number, or wavelength, of the perturbation to the base of the lithosphere and weakly on this
viscosity ratio, on plausible density differences among crust, mantle lithosphere, and
asthenosphere, and on ratios of crustal to total lithospheric thicknesses. For all likely densities,
viscosities, and thicknesses, the Moho is drawn down (pushed up) where the base of the
lithosphere subsides (rises). For large viscosities of crust compared to mantle lithosphere
(ratios > ~30), a sinking and thickening mantle lithosphere also pulls the surface down. For
smaller viscosity ratios, crustal thickening overwhelms the descent of the Moho, and the
surface rises (subsides) above regions where mantle lithosphere thickens and descends
(thins and rises). Ignoring vertical variations of viscosity within the crust and mantle
lithosphere, we find that the maximum surface height occurs for approximately equal
viscosities of crust and mantle lithosphere. For large crust/mantle lithosphere viscosity ratios,
gravity anomalies follow those of surface topography, with negative (positive) free-air
anomalies over regions of descent (ascent). In this case, topography anomalies are smaller
than those that would occur if the lithosphere were in isostatic equilibrium. Hence, flow-induced
stresses—dynamic pressure and deviatoric stress—create smaller topography than that
expected for an isostatic state. For small crust/mantle viscosity ratios (< ~10), however,
calculated surface topography at long wavelengths is greater than it would be if the lithospheric
column were in isostatic equilibrium, and at short wavelengths local isostasy predicts surface
deflections of the wrong sign. For the range of wavelengths appropriate for convergent
mountain belts (~150–600 km), calculated gravity anomalies are negative over regions of
lithospheric thickening, especially when allowance for flexural rigidity of a surface layer is
included. Correspondingly, calculated values of admittance, the ratio of Fourier transforms of
surface topography and free-air gravity anomalies, are also negative for wave numbers relevant
to mountain belts. For essentially all mountain belts, however, measured free-air anomalies
and admittance are positive. Whether gravitational instability of the lithosphere affects the
structure of convergent belts or not, its contribution to the topography of mountain belts seems
to be small compared to that predicted for isostatic balance of crustal thickness variations.
Citation: Molnar, P., and G. A. Houseman (2013), Rayleigh-Taylor instability, lithospheric dynamics, surface
topography at convergent mountain belts, and gravity anomalies, J. Geophys. Res. Solid Earth, 118, 2544–2557,
doi:10.1002/jgrb.50203.
1.
1
Department of Geological Sciences and Cooperative Institute for
Research in Environmental Sciences, University of Colorado, Boulder,
Colorado, USA.
2
Institute of Geophysics and Tectonics, School of Earth and
Environment, University of Leeds, Leeds, UK.
Corresponding author: P. Molnar, Department of Geological Sciences
and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309, USA. ([email protected])
©2013. American Geophysical Union. All Rights Reserved.
2169-9313/13/10.1002/jgrb.50203
Introduction
[2] In recent years, geodynamicists have focused increasingly
on “dynamic topography” associated with flow in the mantle,
or mantle dynamics [e.g., Braun, 2010; Hager, 1984; Hager
et al., 1985; McKenzie, 1977; Morgan, 1965a, 1965b; Parsons
and Daly, 1983; Richards and Hager, 1984]. To 0 order, all
topography above sea level is affected by flow and deformation
of the mantle; otherwise, erosion would have eventually
destroyed it. Flow within the mantle moves crust so as to
thicken or thin it, and it moves hot and cold material in
response to a balance of stress given by the Navier-Stokes
equation. Where isostasy prevails, lateral differences in density
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MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
λ/2
Surface (z = 0)
σxz = 0, σzz + ρcgδs = 0
u=0
σxz = 0
Moho (z = – m)
LAB (z = – h)
Lithosphere
Density = ρm
Viscosity = ηm
Mantle
lithosphere
u = 0, σxz = 0
Density = ρc
Viscosity = ηc
Crust
σxz = 0, σzz - (ρm - ρa) gδh = 0
Asthenosphere
Density = ρa
Viscosity = 0
Table 1. List of Symbols
Figure 1. Experimental setup and definitions of parameters
and boundary conditions. Symbols are defined in Table 1.
associated with lateral variations in both crustal thickness and
temperature determine lateral differences in surface height.
By definition, isostasy pertains to a static state, however, and
consequently the following questions arise: If much of the
topography is isostatically compensated, can the signature of
mantle flow be detected by topography associated with
dynamic processes? and What fraction of topography is
supported by dynamic stresses—pressure and deviatoric
stress—induced by viscous flow?
[3] Analytic solutions for convective flow in a viscous
medium [e.g., Hager, 1984; McKenzie, 1969, 1977; Morgan,
1965a; Parsons and Daly, 1983] and countless numerical
calculations show that the surface rises over upwelling flow
and subsides over downwelling flow. For regions where the
lithosphere deforms, however, and in particular where crust
thickens or thins, the surface topography might not follow
the sublithospheric flow [e.g., Hoogenboom and Houseman,
2006; Neil and Houseman, 1999]; convergent flow and
downwelling may lead to sufficient crustal thickening that
the surface will stand high and form a mountain range.
Accordingly, we might expect topography over regions where
lithospheric deformation occurs to show a different signature
from those regions affected by sublithospheric flow.
[4] We consider a simple problem for which Neil and
Houseman [1999] obtained semi-analytical solutions:
Rayleigh-Taylor instability of a stratified lithosphere
consisting of a low-density crust over a more dense mantle
lithosphere that overlies an inviscid, slightly less dense asthenosphere (Figure 1). Gravity anomalies provide a test of
the extent to which isostatic balance accounts for surface
topography, and therefore the extent to which topography, both
at the surface and on internal density interfaces, is supported by
nonlithostatic stresses associated with flow [e.g., Morgan,
1965b]. So, we extend Neil and Houseman’s [1999] results to
describe the impact of Rayleigh-Taylor instability on free-air
gravity anomalies at the surface.
[5] The admittance, Z(k), the ratio of Fourier transforms
of gravity and topography as a function of wave number,
provides a scaling of gravity to surface deflection that is
useful in assessing the role of nonlithostatic stress in
supporting topography:
Z ðk Þ ¼
Δe
gðk Þ
desðk Þ
where Δe
g ðk Þ and desðk Þ are Fourier transforms of the gravity
anomaly, Δg(x), and surface topography, ds(x). Obviously,
both Δe
gðk Þ and desðk Þ should be complex, and hence so
should Z(k), but studies show that, except for large values
of k, the phase of Z(k) is essentially 0 for relevant data sets
[e.g., Fielding and McKenzie, 2012; McKenzie and Bowin,
1976; Watts, 1978]. In the simple model that we consider,
boundary conditions ensure that the phase is 0 (or p, which
is equivalent to multiplying the admittance by 1).
[6] Where stresses associated with lithospheric deformation
affect surface elevations, free-air gravity anomalies can be
negative where the surface rises (and crust thickens), as
(1)
Symbols
Parameters
D
E
fd
g
G
h
k
k0 = kh
LAB
m
m0 = m/h
p
q
q0 = qT
t
t’¼t/T
m
T ¼ ðr 2
m ra Þgh
Te
ui = (u,w)
u
U
w
W
W0 = WT/h
x
xj
z
Z
Z0 = Z/2pG(rm ra)
dij
dh
dh0 = dh/h
dm
dm0 = dm/h
ds
ds0 = ds/h
Δg
e_ ij
0 = c/m
c
m
n
r
0
a
r ¼ rrc r
r
m
a
c
r s ¼ r r
m ra
ra
r0 a = ra/(rm ra)
rc
rm
sij
tij
2545
0
Flexural rigidity
Young’s modulus
Flexural factor (see equation (4))
Gravity
Newton’s gravitational constant
Thickness of lithosphere
Wave number
Dimensionless wave number
Lithosphere-asthenosphere boundary
Crustal thickness
Dimensionless crustal thickness
Pressure
Growth rate
Dimensionless growth rate
Time
Dimensionless time
Time scale
Equivalent elastic thickness
Velocity
Horizontal component of velocity
Horizontal component of velocity eigenfunction
Vertical component of velocity
Vertical component of velocity eigenfunction
Dimensionless velocity eigenfunction
Horizontal coordinate
Coordinate (x, z)
Vertical coordinate
Admittance
Dimensionless admittance
Kronecker delta
Vertical component of displacement of the LAB
Dimensionless vertical component of displacement of
the LAB
Vertical component of displacement of the Moho
Dimensionless vertical component of displacement of
the Moho
Vertical component of surface displacement
Dimensionless vertical component of surface
displacement
Gravity anomaly
Strain rate tensor delta
Viscosity
Dimensionless viscosity (viscosity ratio)
Viscosity of crust
Viscosity of mantle lithosphere
Poisson’s ratio
Density
Dimensionless density contrast at the Moho
Dimensionless density contrast at surface
Density of asthenosphere
Dimensionless density of asthenosphere
Density of crust
Density of mantle lithosphere
Stress tensor
Deviatoric stress tensor
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
Dimensionless growth rate
0.4
η = ηc / ηm
= 0.01
0.3
0.1
0.2
1
10
0.1
η = ηc / ηm
0.0
0
1
2
= 100
3
4
5
Dimensionless wave number
Figure 2. Dimensionless growth rates versus k0 , for m0 =
0.333, r0 = 16.7, and r0 s = 93.3. Colors distinguish 0 = 100
(magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red).
Hoogenboom and Houseman [2006] showed for calculations
of finite amplitude deformation with radial symmetry. Thus,
unlike most cases with sublithospheric flow, negative
admittance is possible. The conditions under which it may
be negative motivate this study.
(100 km), m0 = 0.33, but, like Neil and Houseman [1999],
we carried out analysis with m0 = 0.25 and 0.5. With a mean
temperature difference between mantle lithosphere and
asthenosphere of approximately 300 K (corresponding to a
maximum difference of 600 K), a density of asthenosphere
of 3.3 103 kg/m3, and a coefficient of thermal expansion
of 3 105 K1, rm ra = 30 kg/m3. So, for rc = 2.8 103
kg/m3, r0 = 16.7 and r0 s = 93.3. We also consider values
of rm ra = 20 and 50 kg/m3, which yield r0 = 25 and
r0 s = 140, and r0 = 10 and r0 s = 56, respectively. The
assumption of constant viscosity, of course, is a gross approximation. Moreover, our ignorance of which of the crust
and mantle lithosphere provides the greater resistance to deformation makes assigning values to 0 = c/m unwise.
Hence, following Neil and Houseman [1999] and
Hoogenboom and Houseman [2006], we consider a range
of values: 0.01 ≤ 0 ≤ 100. Finally we consider a wide range
of values of k0 . Thus, most of what we present addresses
the effects of 0 and k0 on deflections of interfaces and gravity anomalies.
(a)
0.0
2. Rayleigh-Taylor Instability With a
Crustal Layer
η
= η c / η m = 0.01
0.1
1
-0.4
-0.6
10
-0.8
η = ηc
-1.0
-1.0
-0.8
-0.6
-0.4
/ η m = 100
-0.2
0.0
(b)
0.0
Dimensionless depth
[7] Following Neil and Houseman [1999], we consider a
stratified lithosphere of thickness h, consisting of a dense lower
layer, with density rm and viscosity m (mantle lithosphere),
overlain by less dense layer with density rc and viscosity
c (crust), and underlain by an inviscid, slightly less dense,
ra < rm, half-space (asthenosphere) (Figure 1). Neil and
Houseman [1999] calculated how deflections of the two
interfaces, analogous to those at the Earth’s surface and at
the Moho, respond to deflections of the lithosphereasthenosphere boundary (hereafter, LAB) for a variety of
conditions and for different wave lengths of harmonic
perturbations to the LAB. We extend these results to
consider the extent to which these calculated deflections
differ from those that would exist if the entire layered structure
were in isostatic equilibrium, and we examine how gravity
anomalies depend on the various assumed parameters.
[8] We summarize in section Appendix A the basic equations
for Rayleigh-Taylor instability of a three-layer structure that
resembles crust, mantle lithosphere, and asthenosphere (Figure 1).
With the assumptions of constant viscosity and constant
density in each layer, five key dimensionless numbers govern solutions (see Table 1 for definitions of all symbols).
With distances scaled by the thickness of the lithosphere,
h, and times by T = 2m/[(rm ra)gh], they are as follows:
the fraction of the lithosphere that is crust, m0 = m/h, where
m is crustal thickness; the ratio of viscosities of crust to
mantle lithosphere, 0 = c/m; the density contrast at the
Moho, r0 = (rc ra)/(rm ra) scaled to that at the LAB,
(rm ra); the similarly scaled density contrast at the Earth’s
surface, r0 s = rc/(rm ra); and the dimensionless wave number of perturbations to the thickness of the lithosphere, k0 = kh.
[9] Plausible ranges of some of these dimensionless
quantities are sufficiently small, that the behavior of the
system does not require numerical experimentation with a
wide range of values. For instance, with typical values of
crustal thickness (33 km) and lithospheric thickness
Dimensionless depth
-0.2
-0.1
-0.2
-0.3
-0.10
-0.05
0.00
0.05
-0.10
Dimensionless wave number
Figure 3. Eigenfunctions of the vertical component of dimensionless velocity, W0 (z0 ), as a function of depth (a) from
the surface at z0 = 0 to the LAB at z0 = 1 and (b) zoom on
the upper part of the layer, from z0 = 0 to z0 = 0.35. For
Figures 3a and 3b, k0 = 2, m0 = 0.333, r0 = 16.7, and r0 s =
93.3, and five values of 0 : 0 = 100 (magenta), 10 (blue),
1 (black), 0.1 (green), and 0.01 (red) are shown.
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MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
Figure 4. Examples of deflections of the lithosphere-asthenosphere boundary (LAB) (red), Moho
(green) and surface (brown) induced by a growing Rayleigh-Taylor instability, for three values of 0 =
c/m: 100 (left), 1 (center), and 0.01 (right). All deflections are scaled to that at the LAB, but relative
to it, those for the Moho and surface are vertically exaggerated 25 times.
2.1. Growth Rates
[11] For linear stability, perturbations grow exponentially
with time, as exp(qt), where q is the growth rate. The growth
rate scales with the density anomaly, in this case that at the
LAB rm ra, with the thickness of the unstable layer, with
gravity, and inversely with the viscosity of the unstable
layer. The nondimensionalization, q0 = q(rm ra)gh/2m,
removes those dependences, so that influences of the
remaining parameters can be examined.
[12] For viscosity ratios 0 greater than about 1, the growth
rate reaches a maximum near k0 = 3 (Figure 2). For smaller
0 , the maximum growth rate shifts to k0 near 2 (0 = 0.1) or
near 1 (0 = 0.01). Calculations with a range of values of m0
and r0 show that a reduction in the buoyancy of the crust,
either because of smaller m0 or less negative r0 , will make
growth rates higher, because less buoyant crust offers less
resistance to growth of the instability for small k0 [Neil and
Houseman, 1999]. For a free top surface boundary condition
and without a low-density layer, dimensionless growth rates
reach a maximum of 0.5 at k0 = 0 [e.g., Whitehead and
Luther, 1975]. Accordingly, as the crustal layer becomes
thinner or less buoyant, the lithosphere becomes increasingly
unstable, and the maximum growth rate not only approaches
0.5 but also is found at lower values of k0 .
2.2. Profiles of Velocity Through the
Lithosphere (Eigenfunctions)
[13] As described in section Appendix A, the velocity field
of a perturbation that grows exponentially has a characteristic
dependence on depth that depends on wave number k0 , viscosity
ratio 0 , and other parameters, m0 and r0 . Following traditional
approaches [e.g., Chandrasekhar, 1961], we determine the
eigenvectors for the matrix of coefficients, Ac, Bc, . . . Dm, in
(A8), for which the determinant is 0. From those eigenvectors,
we then determine the resulting eigenfunctions of W0 (z0 ), the
vertical component of velocity. For each value of 0 , and in
the range ~ 1 < k0 < ~ 4, profiles of W0 (z0 ) resemble one
another. So, we show them for only one value of k0 and for five
values of 0 (Figure 3).
[14] Linear stability defines the depth variation of downward speed but not its absolute magnitude. Accordingly, we
scale the velocity profiles in Figure 3 to the rate of descent
(or ascent) of the lithosphere-asthenosphere boundary
(LAB). Thus, normalized profiles are invariant during the
period of exponential growth. The plots in Figure 3 show
W0 (z0 ), where the mantle lithosphere thickens and the LAB
descends; hence, at the bottom, at z0 = 1, the dimensionless value of vertical speed is W0 (1) = 1. (Where the
mantle lithosphere thins and the LAB rises, the velocity
eigenfunctions W0 (z0 ) should be multiplied by 1.) These
velocity profiles are diagnostic of a 2-D velocity field that
varies harmonically in the horizontal direction; where W0
(z0 ) is minimum or maximum, horizontal components of
velocity are 0, but horizontal compressive strain rates are
maximum or minimum.
0.00
Ratio of deflections of interfaces
Moho to LAB
[10] Because, as discussed below, growth rates of perturbations are greatest for values of k0 = kh of 1 to 4, we focus
on this range. As k = 2p/l, where l is the wavelength of
the perturbation, with h = 100 km, this range corresponds
to 150 km < l < 625 km, which spans the widths of most
active mountain ranges where lithospheric shortening
occurs today, e.g., the Tien Shan, Mongolian Altay, eastern
Alps, or Apennines.
0.04
0.08
100
10
0.12
1
0.1
0.16
η’ = ηc / ηm = 0.01
0.20
0
1
2
3
4
5
Dimensionless wave number
Figure 5. Ratio of Moho to LAB displacements as a function
of wave number, for m0 = 0.333, r0 = 16.7, and r0 s = 93.3.
Positive ratios indicate downward movement of the Moho
where the LAB moves downward. Colors show dependence
on wave number for different values of 0 : 0 = 100 (magenta),
10 (blue), 1 (black), 0.1 (green), and 0.01 (red).
2547
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
[18] Calculations with different crustal thickness m0 or density
r show patterns that are easily understood. For 0 = 100, where
the crust is less buoyant, whether it is thinner (smaller m0 ) or
denser (less negative r0 ), the surface is pulled down faster than
shown in Figure 3. For 0 = 10 and for less buoyant (i.e., denser)
crust, the crust thickens more rapidly. For smaller values of
0 (1, 0.1, and 0.01), more buoyant (i.e., less dense) crust
results in greater strain rates throughout both crustal and
mantle layers at a comparable stage of growth.
[19] To summarize the key results in this section, we
observe two different behaviors of the surface, depending
on the viscosity ratio, 0 . For values of 0 > ~10 (depending
on density ratios and crustal thickness), sinking (rising) of
mantle lithosphere draws the surface down (up) (Figure 4a).
For smaller values of 0 , sinking (rising) of mantle lithosphere
induces thickening (thinning) of the crust and a rising (subsiding)
surface (Figures 4b and 4c).
0
0.02
Moho to LAB
Ratio of deflections of interfaces
0.00
0.04
k’ = kh = 4
k’ = 3
k’ = 2
0.06
0.08
k’ = 1
0.10
-2
-1
0
1
2
log10(η’= ηc / ηm)
Figure 6. Ratio of Moho to LAB displacement rate versus
log10(0 ), for m0 = 0.333, r0 = 16.7, r0 s = 93.3, and four
values of k0 : k0 = 1 (green), 2 (black), 3 (red), and 4 (blue).
Positive ratios indicate downward movement of the Moho
where the LAB moves downward.
[15] In all cases, the region below the Moho, z0 = m0
(= 0.3333 for Figure 3), descends, with downward speeds
increasing with depth almost linearly (Figure 3a). Thus, the
mantle lithosphere thickens, and the dimensionless strain rate,
given by dW0 /dz0 , is nearly constant through the mantle lithosphere. Differences in eigenfunctions for different values of
0 are most apparent in the crust, above z0 = m0 . For 0 =
100, the top surface (z0 = 0) descends where the LAB descends (Figure 4a), and W0 (z0 ) increases in magnitude
almost linearly through the crust (Figure 3b). Hence, the
strain rate within the crust, which is the much more viscous
layer, is nearly constant and much smaller than that in the
mantle lithosphere.
[16] For 0 ≤ ~10, the upper surface rises where the LAB
descends (Figure 3b), but the rate at which it rises is much
smaller than the rate of descent of the LAB [Neil and
Houseman, 1999]. Whereas viscous processes within the
mantle lithosphere limit the speed at which the LAB can
descend, gravity acting on the density contrast at the surface
limits the rate at which the surface can rise. For 0 = 10, W0
(z0 ) is negative and increases in magnitude with depth through
nearly all of the crust so that the strain rate within the crust is
nearly constant. Despite this extensional vertical strain rate in
the crust, the buoyancy of the crust pushes the top surface
upward slightly, and W0 (z0 ) is positive in the uppermost
part of the crustal layer (Figure 3b).
[17] For 0 ≤ 1, strain rates (|dW0 /dz0 |) within the crust are
inversely related to 0 . The top surface rises, the Moho
subsides, and the maximum upward speed occurs within
the crust, not at the surface (Figure 3b). As shown below,
the maximum rate of surface uplift occurs for 0 ~ 1; for the
wave numbers that we consider, this rate is between 0.1%
and 1% of the rate of descent of the LAB. As 0 decreases,
strain rates within the crust increase (Figure 3b), but for
0 < ~1, crustal thinning in the upper part of the crust
works against crustal thickening in the lower part, and
the net rate of crustal thickening, given by the difference
between W0 (z0 = 0) and W0 (z0 = m0 ), changes little with 0
for 0 < ~1.
2.3. Displacement of the Moho
[20] As shown by Neil and Houseman [1999] and by (A10),
the ratios of deflections of interfaces are proportional to ratios
of displacement rates on these interfaces. Scaling by the
displacements, or respectively by rates of displacement, of
the LAB, we plot dm0 /dh0 = (dm0 /dt0 )/(dh0 /dt0 ), where dm0 /dt0
and dh0 /dt0 are the dimensionless vertical speeds of the Moho
and LAB. If both move in the same direction, the ratio is
positive, as it is in all cases that we consider (Figure 5).
Because we focus on regions where the LAB and Moho
descend, however, we invert plots like Figure 5 so that
positive values of W0 (m0 )/W0 (1) plot downward. For
essentially all wave numbers, the Moho descends most
rapidly, relative to the sinking LAB, for 0 ~ 1 (Figures 5
and 6 and Table 2). We can understand this as follows.
For large values of 0 , the more viscous crust simply resists
deformation, including deflection of the Moho. For small
values of 0 , although the crust deforms easily, the internal
buoyancy generated at the Moho induces internal crustal
deformation that resists deflection of the Moho.
[21] Calculations with different crustal thicknesses m0 and
different density differences r0 show, as expected, that more
buoyant crust resists crustal thickening more effectively. The
wave number of maximum growth increases from about 2.1
to about 2.7, and the Moho deflection ratio decreases from
~0.215 to ~0.025 as the buoyancy increases in the following
range: 1 > r0 > 50 [Neil and Houseman, 1999].
2.4. Surface Uplift or Subsidence
[22] As for Moho displacements, we scale surface displacements to LAB displacements using ds0 /dh0 = (ds0 /dt0 )/
(dh0 /dt0 ) = W0 (0)/W0 (1). Again, we invert the vertical scale
(Figures 7 and 8) so that if the surface goes up over the sinkTable 2. Summary of Vertical Movements and Admittance for
Different Viscosity Ratios
Viscosity Ratio
Surface
0
= c/m
0.01
0.1
1.0
10.0
100.0
2548
Moho
LAB
Admittance
down
down
down
down
down
negative
negative
small
negative
positive
Displacements
up (small)
up
up (largest)
up
down
down
down
down (largest)
down
down (small)
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
η ’ = η c / η m = 0.01
0.1
1
10
-0.010
-0.005
100
0.000
0.005
0.010
0
1
2
3
4
5
Dimensionless wave number
Figure 7. Ratio of surface to LAB displacement rate for
m0 = 0.333, r0 = 16.7, and r0 s = 93.3. Positive ratios indicate
downward movement of the surface where the LAB moves
downward. Solid lines show those calculated from linear
stability. Dashed lines show calculated values for isostatic
compensation of both crust and mantle lithosphere thickened
by the amount calculated for linear stability. Colors show
results for different values of 0 : 0 = 100 (magenta), 10 (blue),
1 (black), 0.1 (green), and 0.01 (red).
ing LAB, the associated negative values of ds0 /dh0 plot above
the 0 line. In Figure 7, we also show surface displacements that
we would expect if local isostatic equilibrium applied to the
same amounts of thickening of crust and mantle lithosphere
obtained in dynamical calculations. For this, we calculate an
amount de that the entire lithospheric column must be raised
or lowered for it to be in isostatic equilibrium. With signs that
make displacements (ds, dm, dh, and de) positive upward, isostasy requires the following:
rc ðds þ deÞ þ ðrm rc Þðdm þ deÞ
þðra rm Þðdh þ deÞ ¼ 0
equilibrium would cause the surface to be pulled down
yet more (dashed magenta curve in Figure 7).
[26] For 0 < ~10 and for all k0 , the surface rises over regions
where the LAB and Moho descend (Figure 7). The value of
0 that separates rising and subsiding surfaces over downwellings depends on k0 (Figure 8) and is larger for thicker
(large m0 ) than thinner crust and for less dense (more
negative r0 ) than more dense crust. When the assumption
of local isostatic equilibrium is applied to the calculated thicknesses of both crust and mantle lithosphere, however, surface
heights over thickening mantle lithosphere are generally
lower than those obtained from full dynamic calculations
(Figures 7 and 8). Moreover, for sufficiently large k0 (>2 or 3),
subsidence is predicted when isostatic balance is imposed
(Figures 7 and 8). Thus, unlike 0 = 100, for smaller values
of the viscosity ratio, dynamic stresses create positive
topography, or surface uplift, over regions of thickening
mantle lithosphere, whereas the same thicknesses of crust
and mantle lithosphere in isostatic equilibrium would induce
less surface uplift for k0 < ~2 and subsidence for larger values
of k0 . In all cases that we investigated, dynamic stresses cause
the surface above a zone of lithospheric thickening to have
a higher elevation (or less subsidence) than it would if
the surface elevation were in isostatic equilibrium with crustal
and lithospheric thickness.
[27] These calculations so far ignore the effect of flexural
rigidity of the lithosphere, which suppresses shortwavelength, large-k0 topography. For a plate of flexural rigidity
D ¼ ETe3 =½12ð1 n2 Þ , where E is Young’s modulus
-0.01
Ratio of deflections: Surface to LAB
Ratio of deflections: Surface to LAB
-0.015
(2)
[23] Normalizing displacements to dh and nondimensionalizing using r0 a = ra/(rm ra), (2) becomes the
following:
0
0
0
dm0
de
1
0 ds
¼
rs 0 þ r 1
þ1
dh0 r0 a
dh
dh0
(3)
[24] Based on our calculated values of ds0 /dh0 and dm0 /dh0 ,
we then use (3) to estimate what the dimensionless surface
displacement, (ds0 + de0 )/dh0 , would be if isostatic equilibrium
were assumed, and we compare it (Figure 7, dashed lines)
to the relative surface displacements obtained from the full
dynamic calculations (Figure 7, solid lines). Below we discuss
the effect of flexural rigidity of an elastic plate superimposed
on the viscous lithosphere used here.
[25] In the absence of an elastic plate, the crust thickens
where the LAB descends (Figure 3), but among the five values
of 0 considered in Figure 7, only 0 = 100 (for k0 > ~0.5)
produces downward surface displacement (Table 2). Because
the rate of crustal thickening is small, the thickened lithosphere
pulls both the Moho and the surface down. Moreover, the
surface is pulled down more for thinner/denser crust than for
thicker/less dense crust [Neil and Houseman, 1999]. Without
the viscous stresses induced by the flow, however, isostatic
k’ = kh = 1
k’ = 2
0.00
k’ = 3
0.01
-2
-1
k’ = 4
0
1
2
log10(η’ = ηc /ηm)
Figure 8. Ratio of surface to LAB displacement rate as a
function of log10(0 ), for m0 = 0.333, r0 = 16.7, and r0 s =
93.3, and four values of k0 : k0 = 1 (green), 2 (black), 3
(red), and 4 (blue). Positive ratios indicate downward movement of the surface where the LAB moves downward. Solid
lines show those calculated from linear stability. Dashed
lines show calculated ratios of surface to LAB displacements
when isostatic compensation is assumed for the amounts of
thickening of both crust and mantle lithosphere calculated
for linear stability. The isostatic calculation neglects shear
stresses on vertical planes, deviatoric stresses associated
with viscous deformation, and the perturbation to pressure
due to flow.
2549
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
Ratio of deflections: Surface to LAB
-0.015
Te = 0
10
20
30
40
-0.010
-0.005
Te = 40
30
20
10
Te = 0
0.000
0.005
0.010
0
3.
km
km
km
km
km
[29] If surface topography ds(x) is sufficiently small that
we may consider surface density anomalies (mass/area)
given by rcds, the gravity anomaly is Δg(x) = 2pGrcds(x).
If this density anomaly were compensated by a mass deficit
at the base of a layer of thickness m, the Fourier transform of
gravity anomalies associated with the compensating
masses would be simply Δe
gðk Þ ¼ 2pGrc desðk Þ expðkmÞ
[e.g., Parker, 1973]. In local “Airy” isostatic equilibrium, the
admittance would therefore be as follows:
km
km
km
km
km
1
Z ðk Þ ¼ 2pGrc 1 ekm
2
3
4
5
Dimensionless wave number
Figure 9. The effect of an elastic surface layer on surface
topography for calculations using Rayleigh-Taylor instability
(solid lines) with m0 = 0.333, r0 = 16.7, r0 s = 93.3, and
0 = 1, and lithospheric isostasy, using deflections of the
Moho and LAB calculated for Rayleigh-Taylor instability, as
in Figure 7 (dashed lines). Positive ratios indicate downward
movement of the surface where the LAB moves downward.
Surface deflections, ds0 /dh0 , have been modified using (4) to
take into account the effect of an elastic plate at the surface
with different values of the equivalent elastic thickness, Te.
where fd ¼
1=4
1 D
h gΔr
0
0
0
dm0
0 0
0
Δg
0 ds
r 1
þ exp k
0 ¼ r s
0 þ
0 exp k m
dh
dh
dh
(6)
(4)
g is gravity, and Δr is density contrast across the deflected
interface, which in this case is the Earth’s surface and hence
Δr = rc (= 2.8 103 kg/m3). For Te = 10 km, D = 2.93 1021
N m, and with a lithospheric thickness h = 100 km, fd = 0.18.
The impact of flexure on topography associated both with
dynamics and with isostasy is negligible for k0 < ~1, but
depending on the equivalent elastic thickness, Te, surface deflections can be suppressed substantially for large values of
both k0 and Te (Figure 9).
[28] For small k0 (long wavelength), the thickening of
mantle lithosphere and deflection of the LAB dominate the
stress balance, and the dynamic stresses are small enough
that surface deflection is similar for isostatic and dynamic
calculations (Figures 7 and 8). At large k0 , the topographic
signal tends toward 0 in the dynamic case because the
short-wavelength stress variations are attenuated through
the buoyant resistive crust. In the calculation for local isostasy, relatively large deflections are permitted at large k0 ,
but they will be suppressed by elastic flexure (Figure 9).
Variations in stress associated with lithospheric deformation
are thus most significant in the intermediate range of
wave numbers: k0 = 1 to 4. Relative to the topography
computed for local isostasy, at k0 = 1 or 2, the amplitude
is increased relative to that for regional isostasy (by
100 s of meters for k0 = 2 and typical scaling constants),
but at k0 = 3 or 4, the sign is flipped, and the amplitude
of the topography is attenuated.
0.3
Dimensionless gravity anomalies
0
(5)
Z(k) is positive, rises from 0 at small k, and approaches
2pGrc at large k. The range of k that marks the transition
from Z(k) approximately proportional to k at small wave
numbers to 2pGrc at large wave numbers clearly depends
on the depth of compensation, which for Airy isostasy is
the crustal thickness, m.
[30] Because gravity anomalies grow exponentially in time
along with the deflections of surfaces (at least for small perturbations and linear stability), it makes sense to rescale them
using the same reference thickness, h, and density, (rm ra):
Δg0 = Δg/2pG(rm ra)h. The summed contributions to the
gravity anomaly from the three density interfaces can then
be scaled relative to the displacement of the LAB
(Figures 10 and 11) using Δg0 /dh0 = Δg/2pG(rm ra)dh,
which we calculate using the following:
(3.3 1010 Pa in the crust), Te is the equivalent elastic thickness, and n (= 0.25) is Poisson’s ratio, the dimensionless surface uplift is given by
ds
W ’ð0Þ
1
0 ¼
W ’ð1Þ 1 þ ð f d k’Þ4
dh
Gravity Anomalies and Admittance
η’ = 1
10
0.2
0.1
0.0
-0.1
-0.2
0 .1
0 .01
η’ = ηc /ηm = 100
-0.3
-0.4
0
1
2
3
4
5
Dimensionless wave number
Figure 10. Dimensionless gravity anomalies as a function of
wave number, k0 , for different values of 0 , with m0 = 0.333,
r0 = 16.7, and r0 s = 93.3. Solid lines show calculations
using (6), and dashed lines show values for cases where the
calculated surface elevation is balanced by crustal thickening
so as to maintain Airy isostasy using (8). Color indicates
the value of 0 : 0 = 100 (magenta), 10 (blue), 1 (black),
0.1 (green), and 0.01 (red).
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MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
[33] Again, with r0 s < 0, and the surface typically rising over
a descending limb so that W0 (0)/W0 (1) < 0, gravity is positive.
The admittance for the “Airy isostatic” calculation is, from (5),
Dimensionless gravity anomalies
0.4
k’ = kh = 1, 2, 3, 4
0.2
0.0
Z
0
Airy
0
0 0
0
k ¼ r s 1 ek m
(9)
-0.2
(which is positive because r0 s < 0).
[34] Second, as above, we used (3) to calculate the surface
height that would exist if the entire column were isostatically
balanced. We then computed the gravity anomaly for
“lithospheric isostasy” using the adjusted form of (6) where
all three interfaces are raised by de0 /dh0 (short dashed lines in
Figure 11):
-0.4
-0.6
Airy isostasy
Dynamic topography
Lithospheric isostasy
-0.8
-1.0
-2
-1
0
1
2
0
log10(η’ = ηc /ηm)
Figure 11. Dimensionless gravity anomalies versus log10
(0 ), for m0 = 0.333, r0 = 16.7, r0 s = 93.3, and four values
of k0 : k0 = 1 (green), 2 (black), 3 (red), and 4 (blue). Solid lines
show calculations based on (6), long dashed lines show values
for cases where the calculated surface elevation is balanced by
crustal thickness that maintains Airy isostasy from (8), and
short dashed lines show anomalies evaluated when the full
density structure derived from the dynamic case is adjusted
vertically to be in isostatic equilibrium from (10).
(The minus sign on the left-hand side reflects positive gravity anomalies over regions where the LAB descends, dh0
0.) In considering the effect of flexure, which we discuss
below, we use the value of ds0 /dh0 given by (4) and assume
that flexure affects only that interface, not the Moho or
LAB.
[31] From (6), the dimensionless admittance becomes
0 0
Δge0 Airy
W ð0Þ 0
¼rs 0
1 ek m
0
W ð1Þ
dh
(10)
(7)
0.3
0
0
0
0
where Δge0 ðk Þ=dh and dse0 ðk Þ=dh are the Fourier transforms
of Δg0 (x0 )/dh0 from (6) and ds0 (x0 0 )/dh0 , respectively. We
compare values of gravity Δge0 ðk Þ and admittance Z0 (k0 )
calculated for the developing instability using (6) and (7)
with those expected for local isostatic equilibrium, using
two different assumptions for local isostasy.
[32] First, because one often assumes that topography is in
local isostatic equilibrium, compensated simply by thick
crust (Airy isostasy), we made that assumption using the
surface topography ds0 /dh0 , computed from dynamic calculations. We then computed the gravity anomaly associated
with the corresponding deflections of surface and Moho
(ignoring both the gravitational and the isostatic impact of
the deflected LAB). We refer to this as the “Airy isostasy”
gravity anomaly (long dashed lines in Figures 10 and 11),
so that
0
0
3.1. Gravity Anomalies as a Function of Viscosity Ratio
and Wave Number
[35] For extreme values of 0 (0.01 and 100), gravity
anomalies are markedly negative over a deepening
LAB for essentially all relevant values of k0 (Figure 10).
For 0 = 100 (and also for 0 = 10 with thin crust, m0 = 0.25),
a negative gravity signal arises because the surface subsides
over the deepening LAB, and the mass deficit at the surface
dominates the gravity signal. For 0 = 0.01, the negative
anomalies develop because the surface rises little over the
deepening LAB and Moho (Figure 7), and the deficit of
mass associated with a deep Moho makes the dominant
contribution to gravity anomalies.
(8)
Dimensionless gravity anomalies
0 0
0 Δge0 k =dh
0 0
Z k ¼
dse0 k =dh
0
0
Δg lithospheric
0 ds þ de
¼rs
0
dh
dh0
0
dm0 þ de0
0 0
þ r 1
exp k m
0
dh
0
0
de
þ 1 þ 0 exp k
dh
Te = 0 km
10 km
20 km
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
Te = 30 km
40 km
1
2
3
4
5
Dimensionless wave number
Figure 12. Dimensionless gravity anomalies for lithospheric
structures calculated for Raleigh-Taylor instability (solid lines)
with 0 = 1, m0 = 0.333, r0 = 16.7, and r0 s = 93.3, using (6)
and for Airy isostasy (dashed lines) using (8). In both cases,
corrections for flexure of an elastic plate of the equivalent
elastic thickness, Te, were made by adjusting displacements
of the surface, ds0 /dh0 , using (4).
2551
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
Dimensionless admittance
Airy isostasy
η’ = 100
0
η’ = 1
-100
η’ = 10
-200
η’ = 0.1
η’ = ηc /ηm = 0.01
-300
-1
-0.5
0
0.5
1
log10(k’ = kh)
Figure 13. Dimensionless admittance, ratio of Fourier
transforms of topography and gravity, for calculations with
m0 = 0.333, r0 = 16.7, r0 s = 93.3, and different values of
0 , as a function of log10 k0 . Dashed line shows admittance
for Airy isostatic equilibrium, as given by (5). To obtain dimensional values of admittance, in mGal/km, multiply by
2pG(rm ra) = 1.25 for parameters used here.
[36] The smallest gravity anomalies associated with
the dynamic instability are for 0 1 (Figures 10 and 11);
for short or long wavelengths (k0 < ~1 and k0 > ~3), they
are even positive. Among the five values of 0 for which
we studied a dependence on k0 , surface uplift is greatest for
0 1 (Figure 7). The positive gravity anomalies associated
with this surface uplift virtually cancel the negative gravity
signal produced by the thickening crustal root. For k0 > ~3,
small positive gravity anomalies are predicted in a limited
range of 0 , given by ~0.3 < 0 < ~10, for regions where
mantle lithosphere thickens and sinks (Figure 11).
[37] For 0 = 1 or 10, and for sufficiently small k0 , gravity
anomalies become positive over subsiding LAB (Figure 10),
because the surface rises sufficiently (Figure 7) to compensate for the subsidence of the Moho (Figure 4); for larger
k0 , they become negative over subsiding LAB (Figure 10),
because surface uplift is small (Figure 7).
[38] The buoyancy of crust (affected by its thickness m0 or
by density difference r0 ) can shift these patterns. In general,
gravity anomalies are least negative (and even positive for
0 = 1 or 10) for buoyant crust (e.g., m0 = 0.5 or r0 = 25),
and most negative for thin, dense crust (e.g., m0 = 0.25 or
r0 = 10), but for 0 = 0.01, gravity anomalies become less
negative as m0 increases.
[39] As is clear from (5) or (8), gravity anomalies calculated assuming local Airy isostatic equilibrium of elevated
or subsided surfaces are positive (negative) over regions
of surface uplift (subsidence) (Figures 10 and 11, long
dashed lines). By contrast, those calculated assuming local
isostatic re-balance of the entire column (“lithospheric
isostasy”) are uniformly negative for values of 1 ≤ k0 ≤ 4
(Figure 11, short dashed lines). The weight of the thickened
mantle lithosphere pulls the Moho down to greater depth
than that calculated for the dynamic flow (Figure 7), which
in all cases results in a more negative free-air gravity
anomaly. Flexure of a thin elastic plate will not alter these
patterns, because it affects only the relatively small deflections of the surface, and the negative gravity anomalies result
from deflections of deeper interfaces.
[40] Flexure has its most interesting effect for 0 = 1. In the
absence of flexure, almost no gravity anomaly is produced
because the mass excesses associated with surface uplift
and downward deflection of the LAB are canceled by the
mass deficit in the deepened Moho (Figures 10 and 11). If
the upper crust behaves like an elastic plate, however, it
suppresses the surface uplift at relatively large values of
k0 , which reduces its positive contribution to the gravity
anomalies (over a subsiding LAB). Thus, gravity anomalies
for all of the dynamical calculations and both forms of
isostatic compensation are more negative than they would
be without flexure, for k0 > ~ 1 (compare Figures 10 and 12).
With increasing Te, calculated gravity anomalies become
increasingly negative.
3.2. Admittance
[41] First, note that for topography resulting from
sublithospheric flow, analytic solutions for flow produced
by internal buoyancy [e.g., McKenzie, 1977, 2010; Morgan,
1965a], and related numerical calculations of convective
flow with stratified or temperature dependent viscosity, the
calculated admittance is positive. Exceptions to this pattern
can arise where an elastic plate and a thermally insulating
lid overlie a convecting fluid [e.g., McKenzie, 2010]. In
general, however, at small wave number, Z(k) may have values
on the order of 50 mGal/km attributable to sublithospheric
convective flow (e.g., the Hawaiian swell [Watts, 1978]). For
large k (short wavelength), the flexural rigidity of the
lithosphere can redistribute the compensation of surface
loads so that admittance values approach the value for
uncompensated topography of 2pGrc = 117 mGal/km
[e.g., McKenzie and Fairhead, 1997].
200
Dimensionless admittance
100
Airy isostasy
100
0
k’ = kh = 4
3
2
1
-100
-200
-2
-1
0
log10(η’ = ηc /ηm)
1
2
Figure 14. Dimensionless admittance, ratio of Fourier
transforms of topography and gravity, for calculations as a
function of log10 0 , with m0 = 0.333, r0 = 16.7, r0 s =
93.3, and different values of k0 . Dashed lines show admittance for Airy isostatic equilibrium, as given by (5).
2552
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
admittance (Figures 13 and 14), the inclusion of flexure
makes calculated admittance large and negative for values
of ~1 < k0 < ~4, depending on the value of Te (Figure 15).
100
Te = 0 km
0
4. Discussion: Relevance to Convergent
Mountain Belts
1
Figure 15. Dimensionless admittance, ratio of Fourier
transforms of topography and gravity, for calculations for
0 = 1, m0 = 0.333, r0 = 16.7, and r0 s = 93.3, corrected
for the effect of flexure with different values of the equivalent elastic thickness, Te, as a function of log10 k0 . Dashed
line shows admittance for Airy isostatic equilibrium.
[42] For large values of the crustal viscosity ratio
(e.g., 0 = 100), admittance is also positive in our calculations
for k0 > 0.5 (Figure 13 and Table 2). For such cases, the
surface descends (rises) where the LAB descends (rises). This
positive admittance resembles that for topography caused by
sublithospheric flow. The ratio becomes singular, however,
where the surface displacement changes sign near k0 = ~0.5
and is negative at smaller wave number.
[43] In the range of wave numbers of interest here, 1 < k0
4, the admittance is close to 0 for 0 1 but becomes
markedly negative for crust that is stronger or weaker by a
factor of 10 or so, with maximum amplitude around k0 = 3
(Figure 13 and Table 2). Positive dimensionless admittances
between about 0 and 30 (~40 mGal/km for sensible crustal
densities) are observed for k0 = 4 over a limited range of
viscosity contrasts: ~0.5 < 0 < ~5 (Figure 14), beyond which
the peak admittances become negative and increase rapidly in
amplitude as surface deflections tend to 0. (Dimensionless
admittances in Figures 13–15 can be dimensionalized by
multiplying them by 1.25 (= 117/93.3) mGal/km.) When
the surface deflection changes sign with further increase of
0 , the admittance goes through a singularity and re-appears
with large positive values (Figure 14).
[44] For essentially all values of k0 that are relevant to
lithospheric instability, 1 < k0 < 4, the admittance is positive
but small in the range ~ 0.5 < 0 < ~ 5 (Figure 14), which
implies that an increasing gravity anomaly tracks the rising
surface and deepening LAB. For ~ 0.5 < 0 < ~ 5, the
admittance is much smaller than it would be for local Airy
isostasy (Figure 14) and smaller than it would be for
sublithospheric flow modulated by an elastic plate
[McKenzie, 2010].
[45] Because flexure makes calculated gravity anomalies
more negative as Te increases (Figure 12), it does the
same for admittance (Figure 15). Even for 0 = 1, for which
gravity anomalies are small (Figures 10–12), and so is
Wavelength (km)
0.03
0.04
100
0.5
150
0
log10 (k’ = kh)
200
-0.5
[46] The line between dynamic topography and isostatic
topography is easily confused. One perspective is that all
topography is a consequence of the dynamic equilibrium that
results from a given density distribution in the crust and
mantle. Yet for historical and pedagogic reasons, there is great
utility in the concept of isostasy. Like Orth and Solomatov
[2011], we tested the validity of the isostatic assumption in
comparison with a fully dynamic calculation, albeit of a
simplified geological model, in which stresses associated with
lithospheric deformation are clearly important. We find that in
these calculations the assumption of isostatic balance leads to
estimates of the amplitude of topography associated with
crustal thickening that are in error by several hundred meters
(and often not even of the right sign).
[47] Admittance based on free-air gravity anomalies over
terrain that is in isostatic equilibrium (or supported by
sublithospheric dynamics) is expected to be positive for all
wave numbers. Where gravitational instability of the lithosphere governs deformation and surface elevations and for
a modest viscosity contrast (0 ~ 1) between crust and mantle,
admittance is also expected to be positive, but small, for
wavelengths of interest in mountain building. For a significant
viscosity contrast at the Moho or when allowance for flexural
rigidity of a surface layer is included, our calculations suggest
300
-300
-1
Te 20 km
Te 30 km
40 km
500
400
-200
Te = 10 km
2000
1000
-100
Admittance (mgal km-1)
Dimensionless admittance
Airy isostasy
120
80
40
0
0
0.01
0.02
0.05
0.06
0.07
Wavenumber, k (km-1)
Eastern Siberia
Central Mongolia
Western USA
Longmenshan
Figure 16. Measurements of admittance, ratio of Fourier
transforms of topography to gravity, for selected regions:
Eastern Siberia and Western U.S. [McKenzie and Fairhead,
1997], Central Mongolia [Bayasgalan et al., 2005]; and the
Longmenshan and adjacent Sichuan Basin [Fielding and
McKenzie, 2012]. To avoid cluttering the figure, estimated
uncertainties in values, typically 5–20 mGal/km, have been
omitted. Note that for a lithospheric thickness of 100 km,
the values of wave number, 0 < k < 0.07 km1, shown here
correspond to 0 < k0 < 7.
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MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
that negative gravity anomalies may be expected where
surface uplift occurs over regions of crustal thickening and
mantle downwelling (Figures 10–12), implying negative
admittance (Figures 13–15).
[48] From (5), for local isostatic compensation, the admittance should approach 0 as k goes to 0, and 2pGrc as k becomes
large (for rc = 2.8 103 kg/m3, 2pGrc = 117 mGal/km). In
dimensionless units (Figures 13–15), this large-k0 admittance
approaches Z0 ! rc/(rm ra) = 93.3. Measurements from
continental regions give the 117 mGal/km asymptote for large
k (e.g., Figure 16) [e.g., Bayasgalan et al., 2005; Fielding and
McKenzie, 2012; McKenzie and Fairhead, 1997], if for
some regions, where sublithospheric flow is interpreted,
the admittance remains finite and positive at small values
of k [e.g., McKenzie, 2010]. For submarine topography, for
which a density difference between crust and water applies,
e.g., rc rwater = 1.8 103 kg/m3, 2pGrc = 75 mGal/km,
as is again observed [e.g., McKenzie and Bowin, 1976;
Watts, 1978].
[49] In every region that we have found where admittance
has been measured, it is positive (e.g., Figure 16) [e.g.,
Bayasgalan et al., 2005; Fielding and McKenzie, 2012;
McKenzie and Bowin, 1976; McKenzie and Fairhead,
1997; Watts, 1978]. The failure to observe negative admittance stands in stark contrast with the large negative values
of admittance calculated for either 0 ~ 10 or 0 < ~0.1
(Figures 13 and 14). Thus, such viscosity ratios may not
apply to the Earth, and the crust and mantle seem to offer
comparable amounts of resistance to lithospheric deformation
as Houseman et al. [2000] inferred using other arguments for
the Transverse Ranges of California.
[50] Of course, the ultra-simple nature of the model, two
layers of constant density and constant viscosity over an
inviscid halfspace, prevent strong conclusions about the
Earth from being drawn. In particular, we neglected the
likely strong variations of viscosity with depth in the crust
and in the mantle, not to mention that the crust deforms in
part by brittle failure. Including these factors might change
admittance calculations sufficiently to invalidate conclusions
about the relevance of these viscosity ratios to the earth.
Thus, although it seems hard to ignore the markedly
negative admittance for viscosity ratios that are much different from 1 (Figure 13), we refrain from offering quantitative
limits on the ratio of strength between crust and
mantle lithosphere.
[51] An alternative explanation for why we do not see an
obvious tectonic signature of Rayleigh-Taylor instability
beneath active mountain belts is that gravitational instability
dynamics is superimposed on a regime of overall crustal
shortening and associated crustal thickening, as Billen and
Houseman [2004] assumed in an analysis of the Western
Transverse Ranges of California. The crustal thickening
induced by shortening is associated with approximate isostatic
equilibrium, and the consequent topography and admittance
signals dominate those produced by a growing RayleighTaylor instability (Figures 13 and 14). Thus, gravity anomalies
might be only mildly sensitive to lithospheric instability. As an
example, although free-air gravity anomalies are positive over
the Tien Shan in Central Asia [e.g., Steffen et al., 2011], they
are also ~100 mGal smaller than they would be for isostatic
equilibrium, from which Burov et al. [1990] suggested that
the deficit of mass implied by such a difference from isostasy
could be due to downwelling flow that has drawn the Moho
down beneath the belt.
4.
Conclusions
[52] We considered perturbations to the thickness of a
lithosphere, consisting of a low-density crustal layer and a
mantle lithosphere that overlie an inviscid asthenosphere
that is slightly less dense than the mantle lithosphere. For a
range of ratios of crustal to mantle lithospheric viscosity,
0.01 ≤ 0 = c/m ≤ 100, growth rates of such a RayleighTaylor instability are largest for wavelengths of ~1.5 to 6
times the lithospheric thickness, h. For a representative
lithospheric thickness of h = 100 km, these correspond to
half-wavelengths of ~75–300 km, the approximate dimensions of active mountain belts. Thus, we might expect such
gravitational instabilities to be associated with the development of convergent mountain belts.
[53] Two different manifestations of such instability can be
distinguished by whether the surface subsides where the
LAB subsides or the surface rises over a subsiding LAB
(Figure 4). For relative large viscosity ratios, 0 = c/m > ~30,
crustal thickening is modest, and the surface is pulled down
where mantle lithosphere thickens and subsides (Figure 7
and 8). If such viscosity ratios pertained to mountain belts,
we must conclude that other processes, like externally forced
convergence [e.g., Billen and Houseman, 2004], create the
high topography.
[54] For most viscosity ratios 0 = c/m < ~10, the surface
rises where the LAB descends, but calculated free-air gravity
anomalies over regions of thickening crust are generally
negative, or positive but small in the case of 0 ~ 1
(Figure 10). By contrast, free-air anomalies over most
convergent mountain belts are positive [e.g., Bayasgalan
et al., 2005; Fielding and McKenzie, 2012; Hatzfeld and
Molnar, 2010; McKenzie and Bowin, 1976; McKenzie and
Fairhead, 1997]. Similarly, measurements of admittance,
which is the ratio of Fourier transforms of free-air gravity
anomalies and topography, are positive for all wavelengths
over mountain belts that have been studied (Figure 16)
[e.g., Bayasgalan et al., 2005; Fielding and McKenzie,
2012; McKenzie and Bowin, 1976; McKenzie and
Fairhead, 1997; Watts, 1978], but admittance calculated for
gravitational instability of the lithosphere is typically negative
or, if not negative, smaller than that expected for isostatic
equilibrium (Figures 13 and 14). Only for viscosity ratios in
the range ~ 0.5 < 0 < ~ 5 are free-air gravity anomalies
(Figures 10 and 11) and admittance (Figures 13 and 14)
positive, but they are smaller than what would be found if
surface topography were in Airy isostatic equilibrium. Moreover, when allowance for flexure is included, both gravity
anomalies (Figure 12) and admittance (Figure 15) are negative. Thus, either this treatment of Rayleigh-Taylor instability
with constant viscosity in two layers and with a range of
viscosity ratios of 0 < ~10 is irrelevant to the Earth, or other
processes, like forced convergence and crustal thickening in
isostatic equilibrium, generate gravity anomalies that dominate those associated with lithospheric instability.
[55] Although the stresses associated with lithospheric
instability generally cause positive topography above regions
of mantle downwelling (Figures 7 and 8), they do so without
producing sizable free air gravity anomalies. The positive
2554
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
free-air anomalies over mountain belts suggest that lithospheric instability cannot account for more than a small
fraction of the mean heights of mountain ranges, hundreds of
meters, not kilometers.
U, one obtains
the following:
Appendix A: Basic Equations and Solutions
[62] The solution of (A7) for each layer has the
following form:
@
0
k
@z0
(A1)
where sij is the stress tensor, i and j correspond to the x and z
components, r is density, g is gravity, and dij is the
Kronecker delta. The stress tensor includes devatoric stress,
tij, and pressure, p:
sij ¼ tij þ pdij
(A2)
[57] We assume Newtonian viscosity, , and a constitutive
relationship between deviatoric stress and strain rate, e_ ij :
tij ¼ 2_eij
(A3)
[58] Strain rate is expressed in terms of the x and z components of velocity, ui = (u,w):
e_ ij ¼
1 @ui @uj
þ
2 @xj @xi
(A4)
[59] Finally, incompressibility ensures that
e_ xx þ e_ zz ¼ 0
(A5)
[60] We examine linear stability of such a stratified structure
(Figure 1). Because the mantle lithosphere is denser than the
asthenosphere, it is unstable to perturbations to its base, the
lithosphere-asthenosphere boundary (LAB), and such perturbations should grow because of the force of gravity acting on
them. Following the traditional approach of Chandrasekhar
[1961], we consider perturbations to the LAB that vary
harmonically in the x coordinate, and we assume an exponential growth of perturbations with time. These lead to
the following:
uðx; z; t Þ ¼ U ðzÞ sin kx expðqtÞ
(A6a)
wðx; z; t Þ ¼ W ðzÞ cos kx expðqtÞ
(A6b)
pðx; z; tÞ ¼ PðzÞ cos kx expðqt Þ
(A6c)
a
layer
2 @
0
þk
@z0
of
2
constant
0
W ¼0
viscosity
(A7)
0
0
0 0
0 0
0 0
0 0
W z ¼ A sinh k z þ B cosh k z þ Ck z sinh k z
[56] As is common [e.g., Neil and Houseman, 1999], we
solve the equation of dynamic equilibrium in two dimensions
assuming plane strain:
@sij
rgdiz ¼ 0
@xj
for
0 0
0
0
þ Dk z coshk z
(A8)
[63] With two layers, crust and mantle lithosphere, for
each wave number k0 , there are eight unknown quantities:
Ac, Bc, Cc, Dc, Am, Bm, Cm, and Dm, where subscripts refer
to the two layers. Eight boundary conditions are needed to
determine those coefficients. Two of them are no shear sxz
at z = 0 and a normal stress szz at the top, z = ds given by
szz = rcgds, where ds is the small harmonic deflection of
the surface. Continuity of all of u, w, and the two components of stress, sxz and szz + (rc rm)gdm (where dm is the
small harmonic deflection of the Moho), at the interface between layers, z = m, yield another four. Finally, for the last
two boundary conditions, we assume no shear stress, sxz at
z = h, and no normal stress relative to a hydrostatic state
at the LAB, szz + (rm ra)gdh = 0 at z = h + dh, where dh
is the small harmonic deflection of the Moho.
[64] Three possible growth rates emerge for the case we
consider. One is positive and reflects accelerating growth
of perturbations to the LAB. The other two are negative,
and they result from the stability of the Moho and the top
surface to perturbations; gravity will oppose such perturbations,
and in the absence of other processes, such perturbations will
decay. Following Neil and Houseman [1999], we consider only
the positive growth rate, but Molnar and Houseman [2004]
showed how a negative one can be important if thickening of
lithosphere is forced by horizontal shortening.
[65] The eight boundary conditions express linear relationships among the eight coefficients expressed by (A8)
(four for each layer), for which a nonzero solution exists when
the determinant of the 8 8 matrix of coefficients vanishes.
In that case, one of the eigenvalues of the matrix is 0, and
the corresponding eigenvector defines W0 (z0 ), except for a
scaling factor. Because the eigenfunctions grow exponentially
with time, so do the rates of change of surface elevation, and
the depths of the Moho and LAB:
0
0 ddm
dds
0
0
0
0
0
0
m ¼
¼ q dm ; W ð1Þ
0 ¼ q ds ; W
dt 0
dt 0
ddh
0
0
¼ 0 ¼ q dh
(A9)
dt
0
0
W ð0Þ ¼
where ds0 , dm0 , and dh0 are the (dimensionless) magnitudes
of the harmonic perturbations to the surface and the depths
of the Moho and LAB. It follows that the amplitudes of
the perturbations grow proportionally to one another:
[61] Here, q is the growth rate. With the nondimensionalization
discussed in section 2 and summarized in Table 1,
substituting (A6) into (A1)–(A4), taking the curl of the
new version of (A1), and applying (A5) to eliminate
2555
0
0
0
0
0
W m
ds
W ð0Þ
dm
¼
¼
and
W 0 ð1Þ
dh0 W 0 ð1Þ
dh0
(A10)
MOLNAR AND HOUSEMAN: LITHOSPHERIC DYNAMICS AND TOPOGRAPHY
[66] From (A8), for each value of k0 , 0 , m0 , r0 , and r0 s, we
obtain an eigenfunction that describes W0 (z0 ) except for an
arbitrary scale factor determined by setting W0 (1) = 1.
Deflections of the top surface (z0 = 0) and Moho (z0 = m0 )
are simply scaled to that of the LAB (z0 = 1).
[67] Following substitution and algebraic manipulation,
the boundary conditions described above then simplify to
the following.
[68] No shear stress at z0 = 0:
Bc þ Cc ¼ 0
0
0
0
0
0
Am sinh
þ Cm coshk þ k sinhk
k þ0 Bm coshk
0
0
Dm sinh k þ k coshk ¼ 0
(A17)
[75] No normal stress at z0 = 1, relative to the background hydrostatic stress state:
h
i
0
0
0
0
0 0
0
0
k q Am coshk Bm sinh k Cm k coshk þ Dm k sinhk
h
i
0
0
0
0
0
0
þ Am sinhk þ Bm coshk þ Cm k sinh k Dm k coshk ¼ 0
(A18)
(A11)
[69] No normal stress at z0 = ds0 :
0
0
0
kq
Ac Bc ¼ 0
r0 S
(A12)
[70] Continuity of w at z0 = m0 :
0
0
0
0
0
0
0
0
0
0
0
[76] Acknowledgments. We thank P. C England and an anonymous
reviewer for constructive comments on the manuscript. This research has
been supported in part by the National Science Foundation under grants
EAR-0909199 and EAR-1211378.
0
Ac sinhk m þ Bc coshk m þ Cc k m sinh k m Dc k m coshk m
0
0
0
0
0
0
0
0
0
0
þAm sinhk m Bm coshk m Cm k 0 m0 sinh k m þ Dm k m coshk m References
Bayasgalan, A., J. Jackson, and D. McKenzie (2005), Lithosphere rheology
¼0
(A13)
[71] Continuity of u at z0 = m0 :
0 0
0
0
0
0
0
0
0
0
Ac coshk m Bc sinh k m Cc k m coshk m þ sinhk m
0 0
0
0
0
0
0
0
þDc k m sinh k m þ coshk m Am coshk m
0 0
0
0
0
0
0
0
þBm sinhk m þ Cm k m coshk m þ sinh k m
0 0
0
0
0
0
Dm k m sinhk m þ coshk m ¼ 0
(A14)
[72] Continuity of shear stress at z0 = m0 :
h
0
0
0
0
0
0
0
0
0
0
0
Am sinhk m þ Bc coshk m þ Cc coshk m þ k m sinh k m
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0
0
0
0
0
0
Dc sinh k m þ k m coshk m
h
0
0
0
0
0
0
0
0
0
0
þ Am sinh k m Bm coshk m Cm coshk m þ k m sinhk m
i
0
0
0
0
0
0
þDm sinh k m þ k m coshk m
¼0
(A15)
[73] Continuity of normal stress at z0 = m0 :
h 0 0 0
i
0
0
0
0
0
Ac k q coshk m r sinhk m
h 0 0 0
i
0
0
0
0
0
Bc k q sinh k m r coshk m
h 0 0 0
i
0
0
0
0
0
0
0
Cc k m k q coshk m r sinhk m
h 0 0 0
i
0
0
0
0
0
0
0
þDc k m k q sinh k m r coshk m
h 0 0
i
0
0
0
0
Am k q coshk m sinh k m
h 0 0
i
0
0
0
0
þBm k q sinh k m coshk m
h 0 0
i
0
0
0
0
0
0
þCm k m k q coshk m sinhk m
h 0 0
i
0
0
0
0
0
0
Dm k m k q sinhk m coshk m ¼ 0
[74] No shear stress at z0 = 1:
(A16)
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