Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.     

The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere

Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 10²⁰P. The viscosity of the mantle below the low-viscosity channel is kept at 10²²P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 10¹⁸P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.     

Post-seismic relaxation following the great 2004 Sumatra-Andaman earthquake on a compressible self-gravitating Earth

The   M _w γ 9.0  2004 December 26 Sumatra-Andaman and   M _w = 8.7  2005 March 28 Nias earthquakes, which collectively ruptured approximately 1800 km of the Andaman and Sunda subduction zones, are expected to be followed by vigorous viscoelastic relaxation involving both the upper and lower mantle. Because of these large spatial dimensions it is desirable to fully account for gravitational coupling effects in the relaxation process. We present a stable method of computing relaxation of a spherically-stratified, compressible and self-gravitating viscoelastic Earth following an impulsive moment release event. The solution is cast in terms of a spherical harmonic expansion of viscoelastic normal modes. For simple layered viscoelastic models, which include a low-viscosity oceanic asthenosphere, we predict substantial post-seismic effects over a region several 100s of km wide surrounding the eastern Indian Ocean. We compare observed GPS time-series from ten regional sites (mostly in Thailand and Indonesia), beginning in 2004 December, with synthetic time-series that include the coseismic and post-seismic effects of the 2004 December 26 and 2005 March 28 earthquakes. A viscosity structure involving a biviscous (Burgers body) rheology in the asthenosphere explains the pattern and amplitude of post-seismic offsets remarkably well.     

Post-seismic rebound of a spherical Earth: new insights from the application of the Post–Widder inversion formula

The post-seismic response of a viscoelastic Earth to a seismic dislocation can be computed analytically within the framework of normal-modes, based on the application of propagator methods. This technique, widely documented in the literature, suffers from several shortcomings; the main drawback is related to the numerical solution of the secular equation, whose degree increases linearly with the number of viscoelastic layers so that only coarse-layered models are practically solvable. Recently, a viable alternative to the standard normal-mode approach, based on the Post–Widder Laplace inversion formula, has been proposed in the realm of postglacial rebound models. The main advantage of this method is to bypass the explicit solution of the secular equation, while retaining the analytical structure of the propagator formalism. At the same time, the numerical computation is much simplified so that additional features such as linear non-Maxwell rheologies can be simply implemented. In this work, for the first time, we apply the Post–Widder Laplace inversion formula to a post-seismic rebound model. We test the method against the standard normal-mode solution and we perform various benchmarks aimed to tune the algorithm and to optimize computation performance while ensuring the stability of the solution. As an application, we address the issue of finding the minimum number of layers with distinct elastic properties needed to accurately describe the post-seismic relaxation of a realistic Earth model. Finally, we demonstrate the potentialities of our code by modelling the post-seismic relaxation after the 2004 Sumatra–Andaman earthquake comparing results based upon Maxwell and Burgers rheologies.     

Post-seismic reloading and temporal clustering on a single fault

Geological studies show evidence for temporal clustering of large earthquakes on individual fault systems. Since post-seismic deformation due to the inelastic rheology of the lithosphere may result in a variable loading rate on a fault throughout the interseismic period, it is reasonable to expect that the rheology of the non-seismogenic lower crust and mantle lithosphere may play a role in controlling earthquake recurrence times. We study this phenomenon using a 2-D, finite element method continuum model of the lithosphere containing a single strike-slip fault. This model builds on a previous study using a 1-D spring-dashpot-slider analogue of a single fault system to study the role of Maxwell viscoelastic relaxation in producing non-periodic earthquakes. In our 2-D model, the seismogenic portion of the fault slips when a predetermined yield stress is exceeded; stress accumulated on the seismogenic fault is shed to the viscoelastic layers below and recycled back to the seismogenic fault through viscoelastic relaxation. We find that random variation of the fault yield stress from one earthquake to the next can cause the earthquake sequence to be clustered; the amount of clustering depends on a non-dimensional number, W , called the Wallace number defined as the standard deviation of the randomly varied fault yield stress divided by the effective viscosity of the system times the tectonic loading rate. A new clustering metric based on the bimodal distribution of interseismic intervals allows us to investigate clustering behaviour of systems over a wide range of model parameters and those with multiple viscoelastic layers. For models with   W ≥ 1  clustering increases with increasing W , while those with   W ≤ 1  are unclustered, or quasi-periodic.     

Implications of a non-adiabatic density gradient for the Earth's viscoelastic response to surface loading

The response of a viscoelastic Earth to the melting of the Late Pleistocene ice sheets has been the subject of a number of investigations employing PREM. In PREM, a non-adiabatic density gradient (NADG) exists in the upper mantle, and to understand the implications of this model it is thus important to examine the effects of this NADG on the Earth's response to surface loads. This paper is based on the assumption that the contribution to the depth dependence of the density that is not due to self-compression is due to compositional change. This contribution is referred to as 'non-adiabatic'. We evaluate the effects of a non-adiabatic density jump (NADJ) for the 670  km discontinuity and the NADG in the upper mantle by adopting a compressible earth model with both a compositional density gradient and a density jump. Numerical calculations based on these models indicate that the magnitude of the Earth's response associated with the NADG is much smaller than that associated with the NADJ at 670  km depth. It is also confirmed that the higher modes associated with the NADJ and the NADG are much more sensitive to the existence of an elastic lithosphere than the fundamental modes associated with the density jumps at the surface and core–mantle boundary.     

Material versus local incompressibility and its influence onglacial‐isostatic adjustment

We present an analytical form of the layer propagator matrix for the response of a locally incompressible, layered, linear‐viscoelastic sphere to an external load assuming that the initial density stratification ϱ ₀( r ) within each layer is parametrized by Darwin's law. From this, we show that the relaxation of a sphere consisting of locally incompressible layers is governed by a discrete set of viscous modes. The explicit dependence of the layer propagator matrix on the Laplace transform variable allows us to determine the amplitudes of the viscous modes analytically. Employing Darwin's parametrization, we construct three simplified earth models with different initial density gradients that are used to compare the effects of the local incompressibility constraint, div ( ϱ ₀ u )=0, and the material incompressibility constraint, div  u =0, on viscoelastic relaxation. We show that a locally incompressible earth model relaxes faster than a materially incompressible model. This is a consequence of the fact that the perturbations of the initial density are zero during viscoelastic relaxation of a locally incompressible medium, so that there are no internal buoyancy forces associated with the continuous radial density gradients, only the buoyancy forces generated by internal density discontinuities. On the other hand, slowly decaying internal buoyancy forces in a materially incompressible earth model cause it to reach the hydrostatic equilibrium after a considerably longer time than a locally incompressible model. It is important to note that the approximation of local incompressibility provides a solution for a compressible earth model that is superior to the conventional solutions for a compressible earth with homogeneous layers because it is based on an initial state that is consistent with the assumption of compressibility.     

Effect of the viscoelastic lithosphere on polar wander speed caused by the Late Pleistocene glacial cycles

Previous studies of the wander of the rotation pole associated with the Late Pleistocene glacial cycles indicate that the predicted polar wander speed is sensitive to the density jump at the 670 km discontinuity, the thickness of the elastic lithosphere, and the lower mantle viscosity. In particular, the M1 mode related to the density jump at 670 km depth has been shown to contribute a dominant portion of predicted polar wander speed for sufficiently small lower mantle viscosities. In this study, we examine the sensitivity of polar wander to variations in the viscosity of the viscoelastic lithosphere using simplified compressible Maxwell viscoelastic earth models. Model calculations for earth models with a viscoelastic lithosphere of finite viscosity indicate that the contribution of the M1 mode is similar to those associated with the density discontinuity at the core–mantle boundary (C0 mode) and the lithosphere (L0 mode). We speculate that this is due to the interaction between the M1 mode and the transient mode associated with the viscoelastic lithosphere, which reduces the magnitude of polar wander rates. Therefore, the M1 mode does not contribute a dominant portion of the predicted polar wander speed for earth models with a viscoelastic lithosphere of finite viscosity. In this case, predictions of polar wander speed as a function of lower mantle viscosity exhibit the qualitative form of an 'inverted parabola', as predicted for the J ˙₂ curve. We caution, however, that these results are obtained for simplified earth models, and the results for seismological earth models such as PREM may be complicated by the interaction between the M1 mode and the large set of transient modes.     

Surface vertical displacements, potential perturbations andgravity changes of a viscoelastic earth model induced by internal point dislocations

Using the viscoelastic correspondence principle, we utilize the surface coseismic spheroidal deformation fields (i.e. vertical displacements, potential perturbations and gravity changes) of SNREI earth models caused by four typical types of point dislocation, derived by Sun & Okubo (1993 ), to deduce the fundamental formulas for spheroidal fields relevant to viscoelastic earth models. In computations, we employ a strike-slip dislocation on a vertical plane buried at the bottom of the lithosphere to estimate the maximal viscous relaxation responses to this kind of source that possibly exist on the surface of the earth. We take the seismic moment as 10²²  N  m, which is characteristic of an average large earthquake. The numerical results demonstrate that, if we take the viscosity as 10¹⁹  Pa  s in the asthenosphere, and 10²¹  Pa  s in the other mantle layers, the rates of surface vertical displacements and gravity changes within about 2.5° for the 10 postseismic years are respectively 1.5–8.1  cm  yr⁻¹ and 4.0–14.9  μgal  yr⁻¹ : the viscous relaxation for this mantle viscosity profile proceeds much faster than for a constant mantle viscosity of 10²¹  Pa  s.     

Formation and growth of deep mantle plumes

Summary. If whole mantle convection occurs in the Earth's mantle, then the core–mantle boundary constitutes the lower boundary layer for mantle convection. This boundary layer appears to be unstable on a small scale, and thus may be a source of plumes of hot matter which penetrate the mantle and occasionally even the lithosphere (producing hot spots). A finite-amplitude numerical code is used to study the formation of such plumes and their growth through the mantle. The plumes are restricted to being two-dimensional sheets rather than cylinders. The initial conditions consist of a steadily convecting mantle, and plumes are produced by introducing a perturbation in the form of either a pulse or a steady stream of heat into the bottom of the mantle. Two main results are obtained: (1) A critical perturbation size has been found for a mantle with a Rayleigh number of 10⁷. Small perturbations produce plumes which fail to penetrate the mantle, and instead are swept up by the pre-existing convective pattern, while large perturbations succeed in penetrating the mantle and reaching the lithosphere. The critical perturbation size is shown empirically to be proportional to the effective bouyancy and to a factor related to the shape of the perturbation. A perturbed region 150km wide and 60 km deep should produce a successful plume when the temperature perturbation is 200K or more. (2) Deep mantle plumes appear to require on the order of 50–100Myr to penetrate the mantle; episodic plumes on shorter time-scales appear unlikely. A similar time is required for plumes forming in an initially static, uniform temperature mantle.     

Core–mantle boundary deformations and J2 variations resulting from the 2004 Sumatra earthquake

The deformation at the core–mantle boundary produced by the 2004 Sumatra earthquake is investigated by means of a semi-analytic theoretical model of global coseismic and postseismic deformation, predicting a millimetric coseismic perturbation over a large portion of the core–mantle boundary. Spectral features of such deformations are analysed and discussed. The time-dependent postseismic evolution of the elliptical part of the gravity field ( J ₂) is also computed for different asthenosphere viscosity models. Our results show that, for asthenospheric viscosities smaller than 10¹⁸ Pa s, the postseismic J ₂ variation in the next years is expected to leave a detectable signal in geodetic observations.     

Viscoelastic relaxation and long-lasting after-slip following the 1997 Umbria-Marche (Central Italy) earthquakes

We combine Global Positioning System (GPS) measurements with forward modelling of viscoelastic relaxation and after-slip to study the post-seismic deformation of the 1997 Umbria-Marche (Central Apennines) moderate shallow earthquake sequence. Campaign GPS measurements spanning the time period 1999–2003 are depicting a clear post-seismic deformation signal. Our results favour a normal faulting rupture model where most of the slip is located in the lower part of the seismogenic upper crust, consistent with the rupture models obtained from the inversion of strong motion data. The preferred rheological model, obtained from viscoelastic relaxation modelling, consists of an elastic upper crust, underlain by a transition zone with a viscosity of 10¹⁸ Pa s, while the rheology of deeper layers is not relevant for the observed time-span. Shallow fault creep and after-slip at the base of the seismogenic upper crust are the first order processes behind the observed post-seismic deformation. The deep after-slip, below the fault zone at about 8 km depth, acting as a basal shear through localized time-dependent deformation, identifies a rheological discontinuity decoupling the seismogenic upper crust from the low-viscosity transition zone.     

Seismic evidence for a sharp lithospheric base persisting to the lowermost mantle beneath the Caribbean

Broad-band data from South American earthquakes recorded by Californian seismic networks are analysed using a newly developed seismic wave migration method—the slowness backazimuth weighted migration (SBWM). Using the SBWM, out-of-plane seismic P -wave reflections have been observed. The reflection locations extend throughout the Earth's lower mantle, down to the core–mantle boundary (CMB) and coincide with the edges of tomographically mapped high seismic velocities. Modelling using synthetic seismograms suggests that a narrow (10–15 km) low- or high-velocity lamella with about 2 per cent velocity contrast can reproduce the observed reflected waveforms, but other explanations may exist. Considering the reflection locations and synthetic modelling, the observed out-of-plane energy is well explained by underside reflections off a sharp reflector at the base of the subducted lithosphere. We also detect weaker reflections corresponding to the tomographically mapped top of the slab, which may arise from the boundary between the Nazca plate and the overlying former basaltic oceanic crust. The joint interpretation of the waveform modelling and geodynamic considerations indicate mass flux of the former oceanic lithosphere and basaltic crust across the 660 km discontinuity, linking processes and structure at the top and bottom of the Earth's mantle, supporting the idea of whole mantle convection.     

Electromagnetic core–mantle coupling—I. Explaining decadal changes in the length of day

Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.     

Radial resolving power of far-field differential sea-level highstands in the inference of mantle viscosity

For two decades leading to the late 1980s, the prevailing view from studies of glacial isostatic adjustment (GIA) data was that the viscosity of the Earth's mantle increased moderately, if at all, from the base of the lithosphere to the core–mantle boundary. This view was first questioned by Nakada & Lambeck , who argued that differential sea-level (DSL) highstands between pairs of sites in the Australian region preferred an increase of approximately two orders of magnitude from the mean viscosity of the upper to the lower mantle, in accord with independent inferences from observables related to mantle convection. We use non-linear Bayesian inference to provide the first formal resolving power analysis of the Australian DSL data set. We identify three radial regions, two within the upper mantle (110–270 km and 320–570 km depth) and one in the lower mantle (1225–2265 km depth), over which the average of viscosity is well constrained by the data. We conclude that: (1) the DSL data provide a resolution in the inference of upper mantle viscosity that is better than implied by forward analyses based on isoviscous regions above and below the 670 km depth discontinuity and (2) the data do not strongly constrain viscosity at either the base or top of the lower mantle. Finally, our inversions also quantify the significant bias that may be introduced in inversions of the DSL highstands that do not simultaneously estimate the thickness of the elastic lithosphere.     

Material versus isobaric internal boundaries in the Earth and their influence on postglacial rebound

Most previous earth models used to calculate viscoelastic relaxation after the removal of the Late Pleistocene ice loads implicitly assume that there is no exchange of mass across the mantle density discontinuities on periods of tens of thousands of years (the material boundary formulation). In the present study, simple incompressible models are used to determine the Earth's behaviour in the case where the density discontinuity remains at a constant pressure rather than deforming with the material (the isobaric boundary formulation). The calculation of the movement of the boundary is more rigorous than in earlier studies and uses the local incremental pressure calculated at the depth of the boundary and allows for the vertical deformation caused by the change in volume as material changes phase. It is shown that the buoyancy modes associated with the density discontinuities decrease in strength and increase in relaxation time analogous to what results when the density contrast is reduced. Also, two viscoelastic modes arise from an isobaric boundary, which is also predicted when there is a contrast in rigidity or viscosity across a material boundary. The difference in predicted radial deformation between the isobaric boundary model and the material boundary model is largest for long-wavelength loads for which the material incremental pressure at depth is largest. If the isobaric boundary model is appropriate for the treatment of the mineral phase changes in the mantle on glacial rebound timescales, then previous inferences of the deep-mantle to shallow-mantle viscosity ratio based on large-scale deformation (spherical harmonic degree < 10) of the Earth and including data from the early part of the glacio-isostatic uplift are too small.     

Coseismic and post-seismic signatures of the Sumatra 2004 December and 2005 March earthquakes in GRACE satellite gravity

The GRACE satellite mission has been measuring the Earth's gravity field and its temporal variations since 2002 April. Although these variations are mainly due to mass transfer within the geofluid envelops, they also result from mass displacements associated with phenomena including glacial isostatic adjustment and earthquakes. However, these last contributions are difficult to isolate because of the presence of noise and of geofluid signals, and because of GRACE's coarse spatial resolution (>400 km half-wavelength). In this paper, we show that a wavelet analysis on the sphere helps to retrieve earthquake signatures from GRACE geoid products. Using a wavelet analysis of GRACE geoids products, we show that the geoid variations caused by the 2004 December ( M _w= 9.2) and 2005 March ( M _w= 8.7) Sumatra earthquakes can be detected. At GRACE resolution, the 2004 December earthquake produced a strong coseismic decrease of the gravity field in the Andaman Sea, followed by relaxation in the area affected by both the Andaman 2004 and the Nias 2005 earthquakes. We find two characteristic timescales for the relaxation, with a fast variation occurring in the vicinity of the Central Andaman ridge. We discuss our coseismic observations in terms of density changes of crustal and upper-mantle rocks, and of the vertical displacements in the Andaman Sea. We interpret the post-seismic signal in terms of the viscoelastic response of the Earth's mantle. The transient component of the relaxation may indicate the presence of hot, viscous material beneath the active Central Andaman Basin.     

Core precession: flow structures and energy

Experiments using a precessing liquid-filled oblate spheroid with ellipticity ( a − b )/ a =1/400 extend and clarify earlier research. They yield flow data useful for estimating flows in the Earth's liquid core. Observed flows illustrate and confirm a nearly rigid liquid sphere with retrograde drift and lagging a cavity (mantle) axis in precession. The similarities of the observed lag angle with that computed for a rigid sphere, and earlier energy dissipation research both support the use of a rigid sphere analytical model to predict energy dissipation and first-order flow within the core–mantle boundary (CMB). Second-order boundary layer and interior cylindrical flow structures also are photographed and measured. Interior flows are never turbulent or unstable at near-Earth parameters, although complex and transient flow patterns are observed within the boundary layer. Other mechanisms proposed to explain net heat loss from the Earth and maintenance of the geodynamo typically require acceptance of some critical but unproven premise. Precession and CMB configuration are known with certainty and precision. Analytical difficulties have been the obstacle. Experiments illustrate the consequences of precession and ellipticity, provide criteria for validating analytical and numerical models, and may yield direct knowledge of the Earth's deep interior with careful scaling.     

Numerical models of ductile rebound of crustal roots beneath mountain belts

Crustal roots formed beneath mountain belts are gravitationally unstable structures, which rebound when the lateral forces that created them cease or decrease significantly relative to gravity. Crustal roots do not rebound as a rigid body, but undergo intensive internal deformation during their rebound and cause intensive deformation within the ductile lower crust. 2-D numerical models are used to investigate the style and intensity of this deformation and the role that the viscosities of the upper crust and mantle lithosphere play in the process of root rebound. Numerical models of root rebound show three main features which may be of general application: first, with a low-viscosity lower crust, the rheology of the mantle lithosphere governs the rate of root rebound; second, the amount of dynamic uplift caused by root rebound depends strongly on the rheologies of both the upper crust and mantle lithosphere; and third, redistribution of the rebounding root mass causes pure and simple shear within the lower crust and produces subhorizontal planar fabrics which may give the lower crust its reflective character on many seismic images.     

Mantle convection with a brittle lithosphere: thoughts on the global tectonic styles of the Earth and Venus

Plates are an integral part of the convection system in the fluid mantle, but plate boundaries are the product of brittle faulting and plate motions are strongly influenced by the existence of such faults. The conditions for plate tectonics are studied by considering brittle behaviour, using Byerlee's law to limit the maximum stress in the lithosphere, in a mantle convection model with temperature-dependent viscosity.
When the yield stress is high, convection is confined below a thick, stagnant lithosphere. At low yield stress, brittle deformation mobilizes the lithosphere which becomes a part of the overall circulation; surface deformation occurs in localized regions close to upwellings and downwellings in the system. At intermediate levels of the yield stress, there is a cycling between these two states: thick lithosphere episodically mobilizes and collapses into the interior before reforming.
The mobile-lid regime resembles convection of a fluid with temperature-dependent viscosity and the boundary-layer scalings are found to be analogous. This regime has a well defined Nusselt number–Rayleigh number relationship which is in good agreement with scaling theory. The surface velocity is nearly independent of the yield stress, indicating that the 'plate' motion is resisted by viscous stresses in the mantle.
Analysis suggests that mobilization of the Earth's lithosphere can occur if the friction coefficient in the lithosphere is less than 0.03–0.13—lower than laboratory values but consistent with seismic field studies. On Venus, the friction coefficient may be high as a result of the dry conditions, and brittle mobilization of the lithosphere would then be episodic and catastrophic.