Rayleigh-Taylor instability, lithospheric dynamics, surface topography at convergent mountain belts, and gravity anomalies

Authors

  • Peter Molnar,

    Corresponding author
    • Department of Geological Sciences and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado, USA
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  • Gregory A. Houseman

    1. Institute of Geophysics and Tectonics, School of Earth and Environment, University of Leeds, Leeds, UK
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Corresponding author: P. Molnar, Department of Geological Sciences and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309, USA. (molnar@colorado.edu)

Abstract

[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.

1 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 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.

Figure 1.

Experimental setup and definitions of parameters and boundary conditions. Symbols are defined in Table 1.

Table 1. List of Symbols
SymbolsParameters
DFlexural rigidity
EYoung's modulus
fdFlexural factor see equation (4)
gGravity
GNewton's gravitational constant
hThickness of lithosphere
kWave number
k′ = khDimensionless wave number
LABLithosphere-asthenosphere boundary
mCrustal thickness
m′ = m/hDimensionless crustal thickness
pPressure
qGrowth rate
q′ = qTDimensionless growth rate
tTime
t'=t/TDimensionless time
inline imageTime scale
TeEquivalent elastic thickness
ui = (u,w)Velocity
uHorizontal component of velocity
UHorizontal component of velocity eigenfunction
wVertical component of velocity
WVertical component of velocity eigenfunction
W′ = WT/hDimensionless velocity eigenfunction
xHorizontal coordinate
xjCoordinate (x, z)
zVertical coordinate
ZAdmittance
Z′ = Z/2πG(ρm − ρa)Dimensionless admittance
δijKronecker delta
δhVertical component of displacement of the LAB
δh′ = δh/hDimensionless vertical component of displacement of the LAB
δmVertical component of displacement of the Moho
δm′ = δm/hDimensionless vertical component of displacement of the Moho
δsVertical component of surface displacement
δs′ = δs/hDimensionless vertical component of surface displacement
ΔgGravity anomaly
inline imageStrain rate tensor delta
ηViscosity
η′ = ηc/ηmDimensionless viscosity (viscosity ratio)
ηcViscosity of crust
ηmViscosity of mantle lithosphere
νPoisson's ratio
ρDensity
inline imageDimensionless density contrast at the Moho
inline imageDimensionless density contrast at surface
ρaDensity of asthenosphere
ρa = ρa/(ρm − ρa)Dimensionless density of asthenosphere
ρcDensity of crust
ρmDensity of mantle lithosphere
σijStress tensor
τijDeviatoric stress tensor

[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:

display math(1)

where inline image and inline image are Fourier transforms of the gravity anomaly, Δg(x), and surface topography, δs(x). Obviously, both inline image and inline image 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 π, 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 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.

2 Rayleigh-Taylor Instability With a Crustal Layer

[7] Following Neil and Houseman [1999], we consider a stratified lithosphere of thickness h, consisting of a dense lower layer, with density ρm and viscosity ηm (mantle lithosphere), overlain by less dense layer with density ρc and viscosity ηc (crust), and underlain by an inviscid, slightly less dense, ρa < ρm, 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 lithosphere-asthenosphere 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 = 2ηm/[(ρm − ρa)gh], they are as follows: the fraction of the lithosphere that is crust, m′ = m/h, where m is crustal thickness; the ratio of viscosities of crust to mantle lithosphere, η′ = ηc/ηm; the density contrast at the Moho, ρ′ = (ρc − ρa)/(ρm − ρa) scaled to that at the LAB, (ρm − ρa); the similarly scaled density contrast at the Earth's surface, ρs = − ρc/(ρm − ρa); and the dimensionless wave number of perturbations to the thickness of the lithosphere, k′ = 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 (100 km), m′ = 0.33, but, like Neil and Houseman [1999], we carried out analysis with m′ = 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 × 10−5 K−1, ρm − ρa = 30 kg/m3. So, for ρc = 2.8 × 103 kg/m3, ρ′ = −16.7 and ρs = −93.3. We also consider values of ρm − ρa = 20 and 50 kg/m3, which yield ρ′ = −25 and ρs = −140, and ρ′ = −10 and ρs = −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 η′ = ηc/ηm unwise. Hence, following Neil and Houseman [1999] and Hoogenboom and Houseman [2006], we consider a range of values: 0.01 ≤ η′ ≤ 100. Finally we consider a wide range of values of k′. Thus, most of what we present addresses the effects of η′ and k′ on deflections of interfaces and gravity anomalies.

[10] Because, as discussed below, growth rates of perturbations are greatest for values of k′ = kh of 1 to 4, we focus on this range. As k = 2π/λ, where λ is the wavelength of the perturbation, with h = 100 km, this range corresponds to 150 km < λ < 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.

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 ρm − ρa, with the thickness of the unstable layer, with gravity, and inversely with the viscosity of the unstable layer. The nondimensionalization, q′ = q(ρm − ρa)gh/2ηm, removes those dependences, so that influences of the remaining parameters can be examined.

[12] For viscosity ratios η′ greater than about 1, the growth rate reaches a maximum near k′ = 3 (Figure 2). For smaller η′, the maximum growth rate shifts to k′ near 2 (η′ = 0.1) or near 1 (η′ = 0.01). Calculations with a range of values of m′ and ρ′ show that a reduction in the buoyancy of the crust, either because of smaller m′ or less negative ρ′, will make growth rates higher, because less buoyant crust offers less resistance to growth of the instability for small k′ [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 k′ = 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 k′.

Figure 2.

Dimensionless growth rates versus k′, for m′ = 0.333, ρ′ = −16.7, and ρs = −93.3. Colors distinguish η′ = 100 (magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red).

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 k′, viscosity ratio η′, and other parameters, m′ and ρ′. 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 W′(z′), the vertical component of velocity. For each value of η′, and in the range ~ 1 < k′ < ~ 4, profiles of W′(z′) resemble one another. So, we show them for only one value of k′ and for five values of η′ (Figure 3).

Figure 3.

Eigenfunctions of the vertical component of dimensionless velocity, W′(z′), as a function of depth (a) from the surface at z′ = 0 to the LAB at z′ = −1 and (b) zoom on the upper part of the layer, from z′ = 0 to z′ = −0.35. For Figures 3a and 3b, k′ = 2, m′ = 0.333, ρ′ = −16.7, and ρs = −93.3, and five values of η′: η′ = 100 (magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red) are shown.

[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 W′(z′), where the mantle lithosphere thickens and the LAB descends; hence, at the bottom, at z′ = − 1, the dimensionless value of vertical speed is W′(−1) = − 1. (Where the mantle lithosphere thins and the LAB rises, the velocity eigenfunctions W′(z′) should be multiplied by −1.) These velocity profiles are diagnostic of a 2-D velocity field that varies harmonically in the horizontal direction; where W′(z′) is minimum or maximum, horizontal components of velocity are 0, but horizontal compressive strain rates are maximum or minimum.

[15] In all cases, the region below the Moho, z′ = − m′ (= −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 dW′/dz′, is nearly constant through the mantle lithosphere. Differences in eigenfunctions for different values of η′ are most apparent in the crust, above z′ = − m′. For η′ = 100, the top surface (z′ = 0) descends where the LAB descends (Figure 4a), and W′(z′) 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.

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 η′ = ηc/ηm: 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.

[16] For η′ ≤ ~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 η′ = 10, W′(z′) 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 W′(z′) is positive in the uppermost part of the crustal layer (Figure 3b).

[17] For η′ ≤ 1, strain rates (|dW′/dz′|) within the crust are inversely related to η′. 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 η′ ~ 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 η′ decreases, strain rates within the crust increase (Figure 3b), but for η′ < ~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 W′(z′ = 0) and W′(z′ = − m′), changes little with η′ for η′ < ~1.

[18] Calculations with different crustal thickness m′ or density ρ′ show patterns that are easily understood. For η′ = 100, where the crust is less buoyant, whether it is thinner (smaller m′) or denser (less negative ρ′), the surface is pulled down faster than shown in Figure 3. For η′ = 10 and for less buoyant (i.e., denser) crust, the crust thickens more rapidly. For smaller values of η′ (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, η′. For values of η′ > ~10 (depending on density ratios and crustal thickness), sinking (rising) of mantle lithosphere draws the surface down (up) (Figure 4a). For smaller values of η′, sinking (rising) of mantle lithosphere induces thickening (thinning) of the crust and a rising (subsiding) surface (Figures 4b and 4c).

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 δm′/δh′ = (dm′/dt′)/(dh′/dt′), where dm′/dt′ and dh′/dt′ 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 W′(−m′)/W′(−1) plot downward. For essentially all wave numbers, the Moho descends most rapidly, relative to the sinking LAB, for η′ ~ 1 (Figures 5 and 6 and Table 2). We can understand this as follows. For large values of η′, the more viscous crust simply resists deformation, including deflection of the Moho. For small values of η′, although the crust deforms easily, the internal buoyancy generated at the Moho induces internal crustal deformation that resists deflection of the Moho.

Figure 5.

Ratio of Moho to LAB displacements as a function of wave number, for m′ = 0.333, ρ′ = −16.7, and ρs = −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 η′: η′ = 100 (magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red).

Figure 6.

Ratio of Moho to LAB displacement rate versus log10(η′), for m′ = 0.333, ρ′ = −16.7, ρs = −93.3, and four values of k′: k′ = 1 (green), 2 (black), 3 (red), and 4 (blue). Positive ratios indicate downward movement of the Moho where the LAB moves downward.

Table 2. Summary of Vertical Movements and Admittance for Different Viscosity Ratios
Viscosity RatioSurfaceMohoLABAdmittance
η′ = ηc/ηm Displacements  
0.01up (small)downdownnegative
0.1updowndownnegative
1.0up (largest)down (largest)downsmall
10.0updowndownnegative
100.0downdown (small)downpositive

[21] Calculations with different crustal thicknesses m′ and different density differences ρ′ 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 > ρ′ > −50 [Neil and Houseman, 1999].

2.4 Surface Uplift or Subsidence

[22] As for Moho displacements, we scale surface displacements to LAB displacements using δs′/δh′ = (ds′/dt′)/(dh′/dt′) = W′(0)/W′(−1). Again, we invert the vertical scale (Figures 7 and 8) so that if the surface goes up over the sinking LAB, the associated negative values of δs′/δh′ 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 δe that the entire lithospheric column must be raised or lowered for it to be in isostatic equilibrium. With signs that make displacements (δs, δm, δh, and δe) positive upward, isostasy requires the following:

display math(2)
Figure 7.

Ratio of surface to LAB displacement rate for m′ = 0.333, ρ′ = −16.7, and ρs = −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 η′: η′ = 100 (magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red).

Figure 8.

Ratio of surface to LAB displacement rate as a function of log10(η′), for m′ = 0.333, ρ′ = −16.7, and ρs = −93.3, and four values of k′: k′ = 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.

[23] Normalizing displacements to δh and nondimensionalizing using ρa = ρa/(ρm − ρa), ((2)) becomes the following:

display math(3)

[24] Based on our calculated values of δs′/δh′ and δm′/δh′, we then use ((3)) to estimate what the dimensionless surface displacement, (δs′ + δe′)/δh′, 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 η′ considered in Figure 7, only η′ = 100 (for k′ > ~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 equilibrium would cause the surface to be pulled down yet more (dashed magenta curve in Figure 7).

[26] For η′ < ~10 and for all k′, the surface rises over regions where the LAB and Moho descend (Figure 7). The value of η′ that separates rising and subsiding surfaces over downwellings depends on k′ (Figure 8) and is larger for thicker (large m′) than thinner crust and for less dense (more negative ρ′) 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 k′ (>2 or 3), subsidence is predicted when isostatic balance is imposed (Figures 7 and 8). Thus, unlike η′ = 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 k′ < ~2 and subsidence for larger values of k′. 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 short-wavelength, large-k′ topography. For a plate of flexural rigidity inline image, where E is Young's modulus (3.3 × 1010 Pa in the crust), Te is the equivalent elastic thickness, and ν (= 0.25) is Poisson's ratio, the dimensionless surface uplift is given by

display math(4)

g is gravity, and Δρ is density contrast across the deflected interface, which in this case is the Earth's surface and hence Δρ = ρc (= 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 k′ < ~1, but depending on the equivalent elastic thickness, Te, surface deflections can be suppressed substantially for large values of both k′ and Te (Figure 9).

Figure 9.

The effect of an elastic surface layer on surface topography for calculations using Rayleigh-Taylor instability (solid lines) with m′ = 0.333, ρ′ = −16.7, ρs = −93.3, and η′ = 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, δs′/δh′, 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.

[28] For small k′ (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 k′, 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 k′, 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: k′ = 1 to 4. Relative to the topography computed for local isostasy, at k′ = 1 or 2, the amplitude is increased relative to that for regional isostasy (by 100 s of meters for k′ = 2 and typical scaling constants), but at k′ = 3 or 4, the sign is flipped, and the amplitude of the topography is attenuated.

3 Gravity Anomalies and Admittance

[29] If surface topography δs(x) is sufficiently small that we may consider surface density anomalies (mass/area) given by ρcδs, the gravity anomaly is Δg(x) = 2πGρcδs(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 inline image [e.g., Parker, 1973]. In local “Airy” isostatic equilibrium, the admittance would therefore be as follows:

display math(5)

Z(k) is positive, rises from 0 at small k, and approaches 2πGρc at large k. The range of k that marks the transition from Z(k) approximately proportional to k at small wave numbers to 2πGρc 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, (ρm − ρa): Δg′ = Δg/2πG(ρm − ρa)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 Δg′/δh′ = Δg/2πG(ρm − ρa)δh, which we calculate using the following:

display math(6)

(The minus sign on the left-hand side reflects positive gravity anomalies over regions where the LAB descends, δh′ < 0.) In considering the effect of flexure, which we discuss below, we use the value of δs′/δh′ given by ((4)) and assume that flexure affects only that interface, not the Moho or LAB.

Figure 10.

Dimensionless gravity anomalies as a function of wave number, k′, for different values of η′, with m′ = 0.333, ρ′ = −16.7, and ρs = −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 η′: η′ = 100 (magenta), 10 (blue), 1 (black), 0.1 (green), and 0.01 (red).

Figure 11.

Dimensionless gravity anomalies versus log10(η′), for m′ = 0.333, ρ′ = −16.7, ρs = −93.3, and four values of k′: k′ = 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)).

[31] From ((6)), the dimensionless admittance becomes

display math(7)

where inline image and inline image are the Fourier transforms of Δg′(x′)/δh′ from ((6)) and δs′(x′)/δh′, respectively. We compare values of gravity inline image and admittance Z′(k′) 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 δs′/δh′, 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

display math(8)

[33] Again, with ρs < 0, and the surface typically rising over a descending limb so that W′(0)/W′(−1) < 0, gravity is positive. The admittance for the “Airy isostatic” calculation is, from ((5)),

display math(9)

(which is positive because ρs < 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 δe′/δh′ (short dashed lines in Figure 11):

display math(10)

3.1 Gravity Anomalies as a Function of Viscosity Ratio and Wave Number

[35] For extreme values of η′ (0.01 and 100), gravity anomalies are markedly negative over a deepening LAB for essentially all relevant values of k′ (Figure 10). For η′ = 100 (and also for η′ = 10 with thin crust, m′ = 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 η′ = 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.

[36] The smallest gravity anomalies associated with the dynamic instability are for η′ ≈ 1 (Figures 10 and 11); for short or long wavelengths (k′ < ~1 and k′ > ~3), they are even positive. Among the five values of η′ for which we studied a dependence on k′, surface uplift is greatest for η′ ≈ 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 k′ > ~3, small positive gravity anomalies are predicted in a limited range of η′, given by ~0.3 < η′ < ~10, for regions where mantle lithosphere thickens and sinks (Figure 11).

[37] For η′ = 1 or 10, and for sufficiently small k′, 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 k′, they become negative over subsiding LAB (Figure 10), because surface uplift is small (Figure 7).

[38] The buoyancy of crust (affected by its thickness m′ or by density difference ρ′) can shift these patterns. In general, gravity anomalies are least negative (and even positive for η′ = 1 or 10) for buoyant crust (e.g., m′ = 0.5 or ρ′ = −25), and most negative for thin, dense crust (e.g., m′ = 0.25 or ρ′ = −10), but for η′ = 0.01, gravity anomalies become less negative as m′ 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 ≤ k′ ≤ 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 η′ = 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 k′, 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 k′ > ~ 1 (compare Figures 10 and 12). With increasing Te, calculated gravity anomalies become increasingly negative.

Figure 12.

Dimensionless gravity anomalies for lithospheric structures calculated for Raleigh-Taylor instability (solid lines) with η′ = 1, m′ = 0.333, ρ′ = −16.7, and ρs = −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, δs′/δh′, using ((4)).

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 2πGρc = 117 mGal/km [e.g., McKenzie and Fairhead, 1997].

[42] For large values of the crustal viscosity ratio (e.g., η′ = 100), admittance is also positive in our calculations for k′ > 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 k′ = ~0.5 and is negative at smaller wave number.

Figure 13.

Dimensionless admittance, ratio of Fourier transforms of topography and gravity, for calculations with m′ = 0.333, ρ′ = −16.7, ρs = −93.3, and different values of η′, as a function of log10 k′. Dashed line shows admittance for Airy isostatic equilibrium, as given by ((5)). To obtain dimensional values of admittance, in mGal/km, multiply by 2πG(ρm − ρa) = 1.25 for parameters used here.

[43] In the range of wave numbers of interest here, 1 < k′ < 4, the admittance is close to 0 for η′ ≈ 1 but becomes markedly negative for crust that is stronger or weaker by a factor of 10 or so, with maximum amplitude around k′ = 3 (Figure 13 and Table 2). Positive dimensionless admittances between about 0 and 30 (~40 mGal/km for sensible crustal densities) are observed for k′ = 4 over a limited range of viscosity contrasts: ~0.5 < η′ < ~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 η′, the admittance goes through a singularity and re-appears with large positive values (Figure 14).

Figure 14.

Dimensionless admittance, ratio of Fourier transforms of topography and gravity, for calculations as a function of log10 η′, with m′ = 0.333, ρ′ = −16.7, ρs = −93.3, and different values of k′. Dashed lines show admittance for Airy isostatic equilibrium, as given by ((5)).

Figure 15.

Dimensionless admittance, ratio of Fourier transforms of topography and gravity, for calculations for η′ = 1, m′ = 0.333, ρ′ = −16.7, and ρs = −93.3, corrected for the effect of flexure with different values of the equivalent elastic thickness, Te, as a function of log10 k′. Dashed line shows admittance for Airy isostatic equilibrium.

[44] For essentially all values of k′ that are relevant to lithospheric instability, 1 < k′ < 4, the admittance is positive but small in the range ~ 0.5 < η′ < ~ 5 (Figure 14), which implies that an increasing gravity anomaly tracks the rising surface and deepening LAB. For ~ 0.5 < η′ < ~ 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 η′ = 1, for which gravity anomalies are small (Figures 10-12), and so is admittance (Figures 13 and 14), the inclusion of flexure makes calculated admittance large and negative for values of ~1 < k′ < ~4, depending on the value of Te (Figure 15).

4 Discussion: Relevance to Convergent Mountain Belts

[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 (η′ ~ 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 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 2πGρc as k becomes large (for ρc = 2.8 × 103 kg/m3, 2πGρc = 117 mGal/km). In dimensionless units (Figures 13-15), this large-k′ admittance approaches Z′ → ρc/(ρm − ρa) = 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., ρc − ρwater = 1.8 × 103 kg/m3, 2πGρc = 75 mGal/km, as is again observed [e.g., McKenzie and Bowin, 1976; Watts, 1978].

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 km−1, shown here correspond to 0 < k′ < 7.

[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 η′ ~ 10 or η′ < ~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 Rayleigh-Taylor 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 ≤ η′ = ηc/ηm ≤ 100, growth rates of such a Rayleigh-Taylor 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, η′ = ηc/ηm > ~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 η′ = ηc/ηm < ~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 η′ ~ 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 < η′ < ~ 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 η′ < ~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 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.

Appendix A:: Basic equations and solutions

[56] As is common [e.g., Neil and Houseman, 1999], we solve the equation of dynamic equilibrium in two dimensions assuming plane strain:

display math(A1)

where σij is the stress tensor, i and j correspond to the x and z components, ρ is density, g is gravity, and δij is the Kronecker delta. The stress tensor includes devatoric stress, τij, and pressure, p:

display math(A2)

[57] We assume Newtonian viscosity, η, and a constitutive relationship between deviatoric stress and strain rate, inline image:

display math(A3)

[58] Strain rate is expressed in terms of the x and z components of velocity, ui = (u,w):

display math(A4)

[59] Finally, incompressibility ensures that

display math(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:

display math(A6a)
display math(A6b)
display math(A6c)

[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 U, one obtains for a layer of constant viscosity the following:

display math(A7)

[62] The solution of ((A7)) for each layer has the following form:

display math(A8)

[63] With two layers, crust and mantle lithosphere, for each wave number k′, 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 σxz at z = 0 and a normal stress σzz at the top, z = δs given by σzz = − ρcgδs, where δs is the small harmonic deflection of the surface. Continuity of all of u, w, and the two components of stress, σxz and σzz + (ρc − ρm)gδm (where δm 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, σxz at z = −h, and no normal stress relative to a hydrostatic state at the LAB, σzz + (ρm − ρa)gδh = 0 at z = − h + δh, where δh 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 W′(z′), 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:

display math(A9)

where δs′, δm′, and δh′ 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:

display math(A10)

[66] From ((A8)), for each value of k′, η′, m′, ρ′, and ρs, we obtain an eigenfunction that describes W′(z′) except for an arbitrary scale factor determined by setting W′(−1) = 1. Deflections of the top surface (z′ = 0) and Moho (z′ = − m′) are simply scaled to that of the LAB (z′ = − 1).

[67] Following substitution and algebraic manipulation, the boundary conditions described above then simplify to the following.

[68] No shear stress at z′ = 0:

display math(A11)

[69] No normal stress at z′ = δs′:

display math(A12)

[70] Continuity of w at z′ = − m′:

display math(A13)

[71] Continuity of u at z′ = − m′:

display math(A14)

[72] Continuity of shear stress at z′ = − m′:

display math(A15)

[73] Continuity of normal stress at z′ = − m′:

display math(A16)

[74] No shear stress at z′ = − 1:

display math(A17)

[75] No normal stress at z′ = − 1, relative to the background hydrostatic stress state:

display math(A18)

Acknowledgments

[76] 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.

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