Ductile shear zones: some aspects of constant slip velocity and constant shear stress models

Summary The thermomechanical differential equations governing deformation in viscous shear zones have been solved for both constant velocity and constant stress boundary conditions. The solutions show that the inertial term in these equations can be neglected everywhere.
The starting condition of the constant velocity model has been shown to be a constant velocity gradient and not a Heaviside function. The temperature anomaly produced by shear heating at the centre of the shear zone is shown to increase gradually and continuously with time, not reaching an asymptotic value. Conclusions for the constant velocity boundary condition are otherwise generally similar to those presented by Yuen et al , and agree with Fleitout & Froidevaux. The temperatures reached by constant velocity shears are sufficient for partial melting.
Constant stress boundary condition shear zone models show an initially broad shear zone with uniform shear velocity gradient. Depending on the level of applied shear stress and ambient temperature, localized intense shear heating may develop followed by thermal runaway. At lower ambient temperatures relatively high stresses are required to produce thermal runaway.
The broadening of the constant velocity shear zone proceeds more rapidly with increased ambient temperature. This can be used to show that shear zones broaden with depth. The merging of parallel shear zone pairs has been investigated and shear zones separated by distances of less than 10km coalesce to form a single shear zone within 3 Myr. Only shear zones separated by 50km or more remain distinct over periods of tens of millions of years.     

A dislocation model of microplate boundary ruptures n the presence of a viscoelastic asthenosphere

Summary. A microplate is modelled as an elastic plate with two long strike-slip boundaries, lying over a Maxwell-type viscoelastic asthenosphere. The microplate is subjected to a constant and uniform shear strain rate by the opposite motions of two adjoining larger plates. After the occurrence of an earthquake at one of the microplate boundaries, the time evolution of shear stress at the other boundary is studied. It is found that stress build-up at the second boundary is delayed due to stress diffusion governed by the asthenosphere relaxation. Earthquake occurrence at this latter boundary would be delayed depending upon both the microplate width and the ratio between the Maxwell relaxation time of the asthenosphere and a characteristic time required for tectonic strain to recover rupture conditions. It turns out that the parameters which determine the occurrence of seismic activity along the microplate boundaries are more strictly constrained in the presence of a viscoelastic asthenosphere than in the case of an elastic half-pace model.     

On the stability of frictionally heated shear flows in the asthenosphere

Summary . Frictional heating in upper mantle shear flows may lead to localized thermal runaway and partial melting in the asthenosphere, but only as the result of a finite-amplitude disturbance. A rigorous two-dimensional stability analysis shows that asthenospheric shear flows are stable to small-amplitude perturbations whether such flows are supercritical (shear stress decreases with increasing plate velocity) or subcritical (shear stress increases with increasing plate velocity). Disturbances which maintain a shear stress larger than the critical value for sufficiently long will lead to runaway. The response of the asthenosphere to events which do not satisfy this criterion must be determined by a non-linear analysis. Reasonable models of flow in the asthenosphere could be driven to runaway, at a superexponential growth rate, by sudden increases in shear stress of less than 10 bar. Disturbances resulting from plate collisions may maintain large enough stresses for sufficiently long times to initiate runaways, while stress changes associated with large earthquakes probably occur too rapidly to do so.     

Implications of stress concentration on a strike-slip fault in an elastic plate subject to basal shear stress

Summary We consider a long strike-slip fault in a lithosphere modelled as an elastic slab. To the base of the slab a shear stress distribution is applied which simulates the viscous drag exerted by the asthenosphere. The resulant stress on the fault plane may directly fracture the lithosphere in its brittle upper portion; alternatively it may give rise at first to a stable aseismic sliding in the lower portion. In the latter case, stress concentration due to the deep aseismic slip is the relevant feature of the pre-seismic stress acting on the upper section of the lithosphere. The two cases are examined by use of dislocation theory and their observable effects compared. Different depths of the aseismic slip zone and the presence or absence of a uniform friction on the seismic fault are allowed for. If the model is applied to the San Andreas fault region, where a steady sliding condition actually seems to be present at shallow depth, it turns out that the slip amplitudes commonly associated with large earthquakes are consistent with average basal stress values which can be substantially lower than a few bars, a value often quoted as the steady state basal stress due to a velocity gradient in the upper asthenosphere.     

Small-scale mantle flows and induced lithospheric stress near island arcs

Summary. This paper explores the middle ground between complex thermally-coupled viscous flow models and simple corner flow models of island arc environments. The calculation retains the density-driven nature of convection and relaxes the geometrical constraints of corner flow, yet still provides semianalytical solutions for velocity and stress. A novel aspect of the procedure is its allowance for a coupled elastic lithosphere on top of a Newtonian viscous mantle. Initially, simple box-like density drivers illustrate how vertical and horizontal forces are transmitted through the mantle and how the lithosphere responds by trench formation. The flexural strength of the lithosphere spatially broadens the surface topography and gravity anomalies relative to the functional form of the vertical flow stresses applied to the plate base. I find that drivers in the form of inclined subducting slabs cannot induce self-driven parallel flow; however, the necessary flow can be provided by supplying a basal drag of 1–5 MPa to the mantle from the oceanic lithosphere. These basal drag forces create regional lithospheric stress and they should be quantifiable through seismic observations of the neutral surface. The existence of a shallow elevated phase transition is suggested in two slab models of 300 km length where a maximum excess density of 0.2 g cm⁻³ was needed to generate an acceptable mantle flow. A North New Hebrides subduction model which satisfies flow requirements and reproduces general features of topography and gravity contains a high shear stress zone (75 MPa) around the upper slab surface to a depth of 150 km and a deviatoric tensional stress in the back arc to a depth of 70 km. The lithospheric stress state of this model suggests that slab detachment is possible through whole plate fracture.     

Thickening model of the continental lithosphere

Summary. The thickening plate theory proposed by Yoshii and Parker & Oldenburg for the oceanic lithosphere is extended to include the continental lithosphere. The theory is based on the assumption that the lithosphere—asthenosphere boundary is a solidus and that as a result solidification of the top of the asthenosphere is occurring. Observational data imply that the relationship between the plate thickness and basement age for the North American continent is y = 1.7 √ t + (50 ± 10), where y (km) is the plate thickness and t (Myr) is the basement age.
The theory is tested against changes with basement age of the observed surface heat-flow and seismic estimate of plate thickness. The following conclusions are inferred:
(1) The changes both of the observed heat flow and plate thickness with basement age are explained by this theory.
(2) The surface erosion and vertical distribution of radiogenic heat sources are important factors in controlling the thickening process of the continental lithosphere.
(3) The equality of the average surface heat-flow over the oceans and over the continents is a consequence of a faster release of latent heat at the lithosphere—asthenosphere boundary under the oceans, instead of a higher heat production in the continental crust.     

The lithosphere–asthenosphere boundary beneath the western United States

S receiver functions from 67 broad-band seismic stations in the western United States clearly reveal the existence of a mantle discontinuity with velocity reduction downward, which we interpret as the lithosphere–asthenosphere boundary (LAB). The average depth of the LAB is ∼70 km. The boundary is relatively sharp with an overall sharpness of less than 20 km. The boundary is more prominent south of the Mendocino Triple Junction, where the Farallon Plate has completely subducted. This may indicate partial melts at the base of the lithosphere caused by the upwelling of the asthenospheric flow through the slab window. A double low velocity zone is observed at base of the lithosphere beneath southern Sierra Nevada, implying a second melting zone at a depth of ∼100 km, well correlated with previous studies of lithospheric delamination in the area.     

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.     

A study of shallow global mantle flow due to the accretion and subduction of lithospheric plates

Summary. Models of shallow, global mantle circulation due to the accretion and subduction of lithospheric plates are formulated as potential theory problems on a sphere. Subducting and accreting plate boundaries represent sources and sinks respectively for the sublithospheric flow. Solutions, which are obtained by finite difference approximations, give the instantaneous flow velocities within the asthenosphere compatible with plate boundaries and relative plate motions. Results are presented for present-day plate boundaries and relative plate motions for the case of a uniform viscosity asthenosphere and for that of a low viscosity zone at the base of the lithosphere. These results are discussed in terms of available geophysical data. Some of the implications of a shallow, mantle-wide circulation are also considered.     

Rheology of the Earth and a thermoconvective mechanism for sedimentary basin formation

A power-law non-Newtonian fluid is usually assumed to model slow flows in the mantle and, in particular, convective flows. However, the power-law fluid has no memory, in contrast to a real material. A new non-linear integral (having a memory) model is proposed to describe the rheology of rocks. The model is consistent with the theory of simple fluids with fading memory and with laboratory studies of rock creep. The proposed model reduces to the power-law fluid model for stationary flows and to the Andrade model for flows associated with small strains. Stationary convection beneath continents has been studied by Fleitout & Yuen (1984 ), who used the power-law fluid model and obtained the cold immobile boundary layer (continental lithosphere). In a stability analysis of this layer, the Andrade model must be used. The analysis shows that the lithosphere is overstable (the period of oscillation is about 200  Ma). In the present study, it is suggested that these thermoconvective oscillations of the lithosphere are a mechanism for sedimentary basin formation. The vertical crustal movement in sedimentary basins can be considered as a slow subsidence on which small-amplitude oscillations are superimposed. The longest period of oscillatory crustal movement is of the same order of magnitude as the period of convective oscillation of the lithosphere found in the stability analysis. Taking into account the difference between depositional and erosional transport rates we can explain the permanent subsidence as well as the oscillations.     

Upper mantle heterogeneities beneath Central Europe

Summary. An inversion of ISC travel-time data from selected earthquakes in the distance range 30°-90° to 53 stations in Central Europe has been used to model velocity down to 600 km depth. The model explains 0.1–0.2s of the residuals, as for other array studies, leaving 0.5 s unexplained as noise. The uppermost 100 km of the mantle and crust contains inhomogeneities that correlate remarkably well with the geology. This may be due to deep-seated thermal anomalies or, in some areas, to delays introduced by passage of the rays through sedimentary cover. The deeper anomalies are smaller and unrelated to those in the lithosphere, which suggests that the asthenosphere is decoupled from the rigid lithosphere. The structure at 600 km depth is again quite inhomogeneous and might be due to undulations of the 650 km discontinuity. The models show some suggestion of a high velocity slab trending from east to west beneath the Alps.     

Lithospheric structure of the continent–continent collision zone: eastern Turkey

We infer the lithospheric structure in eastern Turkey using teleseismic and regional events recorded by 29 broad-band stations from the Eastern Turkey Seismic Experiment (ETSE). We combine the surface wave group velocities (Rayleigh and Love) with telesesimic receiver functions to jointly invert for the S -wave velocity structure, Moho depth and mantle-lid (lithospheric mantle) thickness. We also estimated the transverse anisotropy due to Love and Rayleigh velocity discrepancies. We found anomalously low shear wave velocities underneath the Anatolian Plateau. Average crustal thickness is 36 km in the Arabian Plate, 44 km in Anatolian Block and 48 km in the Anatolian Plateau. We observe very low shear wave velocities at the crustal portion (30–38 km) of the northeastern part of the Anatolian Plateau. The lithospheric mantle thickness is either not thick enough to resolve it or it is completely removed underneath the Anatolian Plateau. The shear velocities and anisotropy down to 100 km depth suggest that the average lithosphere–asthenosphere boundary in the Arabian Plate is about 90 and 70 km in Anatolian block. Adding the surface waves to the receiver functions is necessary to constrain the trade-off between velocity and the thickness. We find slower velocities than with the receiver function data alone. The study reveals three different lithospheric structures in eastern Turkey: the Anatolian plateau (east of Karliova Triple Junction), the Anatolian block and the northernmost portion of the Arabian plate. The boundary of lithospheric structure differences coincides with the major tectonic boundaries.     

Analytical approach for the toroidal relaxation of viscoelastic earth

This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.     

A New Model for Heat Flow in Extensional Basins: Radiogenic Heat,Asthenospheric Heat,and the McKenzie Model

The McKenzie model proposed in 1978, which is widely used in calculating the thermal history of rift basins and other extensional basins, incorrectly assumes that all heat passing through the lithosphere originates below the lithosphere. In reality, heat from radiogenic sources within the lithosphere, especially in the upper crust, may represent more than half the heat flow at the top of basement. Thinning of the lithosphere during extension does indeed result in an increase of heat flowing from the asthenosphere, but this thinning also reduces the radiogenic heat from within the lithosphere. Because these two effects cancel to a large degree, the direct effects of lithospheric extension on heat flow at the top of basement are smaller than those predicted by the McKenzie model. Because of permanent loss of radiogenic material by lithospheric thinning, the heat flow at the top of basement long after rifting will be lower than the pre-rift heat flow.The McKenzie model predicts an instantaneous increase in heat flow during rifting. The Morgan model proposed in 1983, however, predicts a substantial time delay in the arrival of the higher heat flow from the asthenosphere at the top of basement or within sediments. Using the Morgan model, heat flow during the early stages of rifting will actually be lower than prior to rifting, because the time delay in the loss of radiogenic heat is less than the time delay in arrival of new heat from the asthenosphere.     

Orogenic volcanism caused by thermal runaways?

Summary. Quartz eclogite as source rock seems capable of explaining the chemistry of calc-alkaline volcanics. This model requires partial melting of quartz eclogite in the depth range 100–200 km. Shear heating of the subducting crust is one of the most debated models. The problem may be treated with two different boundary conditions: constant strain rate or constant shear stress. In the former case an increase in temperature tends to reduce the heat production thus stabilizing the shear flow; the result is a moderate increase in temperature which will remain constant during flow. The latter case may lead to a thermal feedback instability and thus temperature and strain rate may suddenly grow to very high values. This phenomenon is termed a runaway and will be discussed in this paper using an adiabatic approximation. It is shown that for a runaway to occur the local energy density must amount to a common value independent of the rheology. In contrast to the constant strain rate case, shear heating is negligible until just before the instability occurs. When the melting point is reached shear stress will break down but the stored local energy will be set free and supply the latent heat of melting. The possibility of a runaway occurring is strongly dependent on the ambient temperature. In subduction shear zones neither shear stress nor strain rate are likely to be constant throughout, but if the former is constant or changes little on a 1–10 km scale a runaway is liable to occur at a depth of around 150 km thus possibly being the cause of calcalkaline volcanism.     

Asthenospheric counterflow: a kinematic model

Summary. Present-day plate motions imply that about 240 km³ of oceanic lithosphere is created by sea-floor spreading and destroyed by subduction per year. A greater volume of asthenosphere will be dragged along by plate motions. Given the fluxes generated at plate boundaries, the horizontal direction and net rate of counterflow required to maintain mass balance is determined globally by a simple analytical model. Time-dependent calculations indicate that the motions are approximately valid in the hotspot reference frame over the past 5 Myr. Under most plates, the model return flow is opposite to the lithospheric motion in the hotspot frame. The counterflow dominates the resisting stresses to plate motion, so driving force models based on plate drag alone are not valid where the directions of plate motion and counterflow differ. The most marked departure of the two directions is under the North American plate. The model counterflow directions indicate that the sources of mantle hotspots are not located within the asthenosphere. Model flux balances demonstrate exchange of material between asthenospheric reservoirs located beneath different plates. Suggestions of southward asthenospheric motion under the North Atlantic, based on physical features around Iceland and strontium isotope geochemistry, are consistent with the direction of flow predicted by the model.     

A critical assessment of viscous models of trench topography and corner flow

Summary. We obtain stresses for Newtonian viscous flow in simple geometries (e.g. corner flow, bending flow) in order to study the effect of imposed velocity boundary conditions. Stress for a delta function velocity boundary condition decays as 1/ r ²; for a step function velocity, stress goes as 1/ r ; for a discontinuity in curvature, the stress singularity is logarithmic. For corner flow, which has a discontinuity of velocity at a certain point, the corresponding stress has a 1/ r singularity. However, for a more realistic circular-slab model, the stress singularity becomes logarithmic. Thus the stress distribution is very sensitive to the boundary conditions, and in evaluating the applicability of viscous models of trench topography it is essential to use realistic geometries.
Topography and seismicity data from northern Honshu, Japan, were used to construct a finite element model, with flow assumed constant speed and tangent to the top of the grid, for both Newtonian and non-Newtonian flow (power law 3 rheology). Normal stresses at the top of the grid are compared to the observed trench topography. There is poor agreement. Purely viscous models of subducting slabs with simple, geometrically consistent velocity boundary conditions do not predict normal stress patterns compatible with observed topography. Elasticity and plasticity appear to be important in determining trench topography.     

Massif Central (France): new constraints on the geodynamical evolution from teleseismic tomography

The Massif Central, the most significant geomorphological unit of the Hercynian belt in France, is characterized by graben structures which are part of the European Cenozoic Rift System (ECRIS) and also by distinct volcanic episodes, the most recent dated at 20 Ma to 4000 years BP. In order to study the lithosphere-asthenosphere system beneath this volcanic area, we performed a teleseismic field experiment.
During a six-month period, a joint French-German team operated a network of 79 mobile short-period seismic stations in addition to the 14 permanent stations. Inversion of P -wave traveltime residuals of teleseismic events recorded by this dense array yielded a detailed image of the 3-D velocity structure beneath the Massif Central down to 180 km depth. The upper 60 km of the lithosphere displays strong lateral heterogeneities and shows a remarkable correlation between the volcanic provinces and the negative velocity perturbations. The 3-D model reveals two channels of low velocities, interpreted as the remaining thermal signature of magma ascent following large lithospheric fractures inherited from Hercynian time and reactivated during Oligocene times. The teleseismic inversion model yields no indication of a low-velocity zone in the mantle associated with the graben structures proper. The observation of smaller velocity perturbations and a change in the shape of the velocity pattern in the 60–100 km depth range indicates a smooth transition from the lithosphere to the asthenosphere, thus giving an idea of the lithosphere thickness. A broad volume of low velocities having a diameter of about 200 km from 100 km depth to the bottom of the model is present beneath the Massif Central. This body is likely to be the source responsible for the volcanism. It could be interpreted as the top of a plume-type structure which is now in its cooling phase.     

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.