Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus

We develop a method for spatio-spectral localization of harmonic data on a sphere and use it to interpret recent high-resolution global estimates of the gravity and topography of Venus in the context of geodynamical models. Our approach applies equally to the simple spatial windowing of harmonic data and to variable-length-scale analyses, which are analogous to a wavelet transform in the Cartesian domain. Using the variable-length-scale approach, we calculate the localized RMS amplitudes of gravity and topography, as well as the spectral admittance between the two fields, as functions of position and wavelength. The observed admittances over 10 per cent of the surface of Venus (highland plateaus and tessera regions) are consistent with isostatic compensation of topography by variations in crustal thickness, while admittances over the remaining 90 per cent of the surface (rises, plains and lowlands) indicate that long-wavelength topography is dominantly the result of vertical convective tractions at the base of the lithosphere. The global average crustal thickness is less than 30 km, but can reach values as large as 40 km beneath tesserae and highland plateaus. We also note that an Earth-like radial viscosity structure cannot be rejected by the gravity and topography data and that, without a mechanical model of the lithosphere, admittance values cannot constrain the thickness of the thermal boundary layer of Venus. Modelling the lithosphere as a thin elastic plate indicates that at the time of formation of relief in highland plateaus and tesserae, the effective elastic plate thickness, T_e , was less than 20 km. Estimates of T_e at highland rises are consistently less than 30 km. Our inability to find regions with T_e > 30 km is inconsistent with predictions made by a class of catastrophic resurfacing models.     

Isostatic compensation on a continental scale: local versus regional mechanisms

Summary. Using the techniques of linear and quadratic programming, it can be shown that the isostatic response function for the continental United States, computed by Lewis & Dorman (1970), is incompatible with any local compensation model that involves only negative density contrasts beneath topographic loads. We interpret the need for positive densities as indicating that compensation is regional rather than local. The regional compensation model that we investigate treats the outer shell of the Earth as a thin elastic plate, floating on the surface of a liquid. The response of such a model can be inverted to yield the absolute density gradient in the plate, provided the flexural rigidity of the plate and the density contrast between mantle and topography are specified.
If only positive density gradients are allowed, such a regional model fits the United States response data provided the flexural rigidity of the plate lies between 10²¹ and 10²² N m. The fit of the model is insensitive to the mantle/ load density contrast, but certain bounds on the density structure can be established if the model is assumed correct. In particular, the maximum density increase within the plate at depths greater than 34 kin must not exceed 470 kg m⁻³; this can be regarded as an upper bound on the density contrast at the Mohorovicic discontinuity.
The permitted values of the flexural rigidity correspond to plate thicknesses in the range 5–10 km, yet deformations at depths greater than 20 km are indicated by other geophysical data. We conclude that the plate cannot be perfectly elastic; its effective elastic moduli must be much smaller than the seismically determined values. Estimates of the stress-differences produced in the earth by topographic loads, that use the elastic plate model, together with seismically determined elastic parameters, will be too large by a factor of four or more.     

Differential thermal stresses in the Earth

Summary. Rocks have a significant thermal expansion coefficient and a high modulus of elasticity. Small thermal changes can therefore generate very substantial stresses. A convenient idealized example is the mechanically free, cooling thermal boundary layer in a material that suddenly turns from plastic to rigid in cooling through a 'rigidus' temperature. This simplified case can be solved analytically for both a linear and an error function temperature profile. A compressed layer develops near the surface, with a logarithmic singularity of stress at the boundary. At a depth of 0.2 of the elastic thickness, the stress changes to tensile, and peaks at 15 per cent of the confined shrinkage stress in the middle of the rigid layer. The situation is similar to the stresses in one half of a piece of tempered glass. Like the tempered glass, both the cooling lithosphere and a plated boundary to a magma chamber are subject to breakage when cracks are induced in the tensile layer. Positive pore (magma) pressure is required to overcome confining pressure at depth in the lithosphere, but cracking could become catastrophic in the plated boundary to a near-surface magma chamber.     

Uplift and heat flow following the injection of magmas into the lithosphere

Summary. The thermal effect of a rapid injection of hot magmas into the lower part of the lithosphere is modelled as an increase in heat production through the invaded region. The change in surface heat flow and the uplift resulting from the thermal expansion are determined in three-dimensional axially symmetric geometry: they are expressed as the space time convolutions of a Green's function with the anomalous heat production.
The anomalies with shorter wavelength (compared to the lithospheric thickness) are attenuated. This filtering affects the surface uplift more than the heat flow anomaly; the attenuation effect is larger when only the lower part of the lithosphere is invaded.
The uplift time constant is of the same order as the heat conduction time if the lower lithosphere is invaded by magmas at a moderate rate (i.e. the rate of injection does not exceed the equivalent of 0.1 per cent of the lithospheric volume in 10⁶yr). Fifty per cent of the total uplift takes place in about 80 × 10⁶yr for a lithosphere 100 km thick. The uplift is slightly faster when the whole lithosphere is invaded. The heat flow anomaly is delayed when the lower part of the lithosphere is invaded.
The spatial extent and the timing of the uplift and heat flow anomalies are critical in determining the mechanism's feasibility. Magma injections explain rapid uplifts [> 100 m (10⁶ yr)⁻¹] only if the magma is supplied at a very high rate (i.e. at least 10 per cent of the lithosphere volume per 10⁶yr). It is a feasible mechanism for uplifts that occur over longer periods of time (≊ 30 × 10⁶yr) such as those that seem to have occurred when the African plate came to rest with respect to the mantle.     

Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets

Planetary topography can either be modelled as a load supported by the lithosphere, or as a dynamic effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However, local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this paper, we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young's modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. While being constrained by the static assumption, degree-one loads can deform the shell and generate stresses. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.     

Small-scale heterogeneities below the Gräfenberg array, Germany from seismic wavefield fluctuations of Hindu Kush events

Small-scale elastic heterogeneities (<5  km) are found in the upper lithosphere underneath the Gräfenberg array, southeast Germany. The results are based on the analysis of broadband recordings of 17 intermediate-depth (201–272  km) events from the Hindu Kush region. The wavefront of the first P arrival and the following 40  s coda are separated into coherent and incoherent (scattered) parts in the frequency range from 0.05 to 5  Hz. The frequency-dependent intensities of the mean and fluctuation wavefields are used to describe the scattering characteristics of the lithosphere underneath the receivers. It is possible to discriminate a weak-fluctuation regime of the wavefield in the frequency range below approximately 1.5–2.5  Hz and a strong-fluctuation regime starting at 2.0–2.5  Hz and continuing to higher frequencies. In order to explain the observed wavefield fluctuations, an approach with seismic scattering at random media-type structures is proposed. The preferred model contains heterogeneities with 3–7 per cent perturbations in seismic velocity and correlation lengths of 0.6–4.8  km in the crust. This is compatible with models from active seismic experiments. Scattering in the lithospheric mantle is not required, but cannot be excluded at weak velocity contrasts (<3 per cent).     

Scattering parameters of the lithosphere below the Massif Central, France, from teleseismic wavefield records

Teleseismic P -wave recordings are analysed in the frequency range 0.3–6  Hz to derive structural (statistical) parameters of the lithosphere underneath the French Massif Central. For this we analyse differences in frequency-dependent intensities of the mean wavefield and the fluctuation wavefield. It is possible to discriminate a weak fluctuation regime of the wavefield in the frequency range below 1  Hz and a strong fluctuation regime starting above 1  Hz and continuing to higher frequencies. The observed wavefield fluctuations in the frequency range 0.3–3  Hz can be explained by scattering of the teleseismic P wave front at elastic inhomogeneities in the lithosphere. A statistical distribution of the inhomogeneities is assumed and the concept of random media is applied. The lithospheric structure under the Massif Central can be described as a 70  km thick heterogeneous layer with velocity fluctuations of 3–7 per cent and correlation lengths of the heterogeneities of 1–16  km.     

Upper Mantle Velocity Structure Estimated From Ps-Converted Wave Beneath the North-Eastern Japan Arc

Summary. The upper boundary of the descending oceanic plate is located by using PS -waves (converted from P to S at the boundary) in the Tohoku District, the north-eastern part of Honshu, Japan. the observed PS-P time data are well explained by a two-layered oceanic plate model composed of a thin low-velocity upper layer whose thickness is less than 10 km and a thick high-velocity lower layer; the upper and lower layers respectively have 6 per cent lower and 6 per cent higher velocity than the overriding mantle. the estimated location of the upper boundary is just above the upper seismic plane of the double-planed deep seismic zone. This result indicates that events in the upper seismic plane, at least in the depth range from 60 to 150 km, occur within the thin low-velocity layer on the surface of the oceanic plate.     

Gravity anomalies and flexure of the lithosphere at Ascension Island

Ascension Island, in the northern South Atlantic, forms the summit of a volcanic edifice 60 km in diameter which places a substantial load on the underlying young oceanic lithosphere. An analysis of a combined data set of recent and historical surface-gravity and bathymetry measurements on and around the island suggests that the lithosphere responds to this load by flexure equivalent to that of an elastic plate only ≈ 3 km thick, and that the mean density of the volcanic edifice is ≈ 2500 kg m^-3. A steep gravity gradient across the island cannot be explained by a simple flexural model and must be attributed to lateral density variations within the volcano itself. The effective elastic thickness is considerably less than the expected ≈ 12 km mechanical thickness of the ≈ 6 Ma lithosphere loaded by the volcano, and less even than zero-age elastic thicknesses commonly observed at slow-spreading ridges with axial rift valleys. The unusually small elastic thickness may be attributed to the combined effects of the high curvature beneath the island, which produces bending stresses that are limited by the yield stress envelope, localized heating of the lithosphere during emplacement of the island, and crustal thickening. When these factors are taken into account, the observed flexure is consistent with rheological models based on experimental rock mechanics.     

On the elastic-perfectly plastic bending of the lithosphere under generalized loading with application to the Kurd Trench

Summary. The bending of the lithosphere as it approaches an oceanic trench is modelled using an elastic-perfectly plastic rheology. The rock within the elastic lithosphere is assumed to behave elastically until a yield stress is reached; for larger strains the stress is assumed to remain at the yield value (no strain hardening). The deflection of the lithosphere is obtained numerically for a range of applied moments, vertical forces and horizontal forces at the trench axis. The results of the analysis are compared with a series of topographic and gravity profiles across the Kuril Trench. In order to model the large observed curvatures of the lithosphere on the outer trench slope plasticity is required. The location of the region of high curvatures gives a value for the compressive axial load. One conclusion is that the lithosphere is under compression seaward of the Kuril Trench.     

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.     

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.     

A simple model for thermal instability in the asthenosphere

Summary. Motion of the lithosphere over a low viscosity asthenosphere concentrates shear and thus energy dissipation in the asthenosphere. This heat source warms the asthenosphere and, in extreme circumstances, may lead to thermal instabilities. The conditions for thermal stability have been investigated by Melosh who supposed that constant stress acted on the plate, and by Yuen & Schubert who assumed constant velocity boundary conditions. In this paper we investigate a simple analytical model which behaves qualitatively like the more complex systems. This model reproduces the results of Melosh for constant stress and of Yuen & Schubert for constant velocity. The velocity—shear stress characteristic curve for this model shows three branches. The stability of solutions on each branch is a function of the boundary conditions, whether constant stress or constant velocity. The simplicity of the model allows us to investigate stability when neither constant stress nor constant velocity apply and to study the structure of the solutions as these limits are approached. A relation between the velocity of a plate and the driving force is constructed. A loading-line analysis specifies the actual stress and velocity of the plate. Although the solutions are unique for many combinations of the loading-line parameters, there is a region of multiple solutions. These solutions exhibit the characteristics of a 'cusp catastrophe' both a low velocity and a high velocity state are stable, while an intermediate state is unstable. Continental lithosphere may lie in this region, leading to epirogenic movements when the plate changes its velocity with respect to the mantle. Oceanic lithosphere almost certainly moves in the low velocity state.     

Causes of spatially variable tectonic subsidence in the Miocene Bermejo Foreland Basin, Argentina

ABSTRACT Tectonic subsidence in the 20–9 Ma Bermejo basin resulted from spatially variable crustal loading on a lithosphere of spatially variable strength (e.g. elastic thickness). Reconstruction of the crustal loads added between 20 and 9 Ma, and assessment of the effects of these loads on an elastic, isotropic lithosphere confirm this hypothesis. Elastic models effectively explain tectonic subsidence east of the Iglesia–Calingasta basin, but west of it crustal loads were locally compensated. Elastic models also prove that the 20–9 Ma Frontal Cordillera loading is of no importance in the mechanical system of the Bermejo basin. 2D and 3D elastic models of a uniformly strong lithosphere under 20–9 Ma crustal loads corrected for post‐9 Ma erosion successfully replicate the 9 Ma Bermejo basin's proximal palaeotopography. However, they fail to replicate the 9 Ma basin's medial and distal palaeotopography. A 3D finite element model of a lithosphere with bimodal strength (weak below the Bermejo basin and west of the Precordillera, and strong below the Precordillera and east of the Valle Fértil lineament) successfully replicates the 9 Ma basin's palaeotopography. That variable strength model introduces a southward decrease in the wavelength of flexural deformation, which results in a basin that narrows southward, consistent with the 9 Ma Bermejo basin. The preferred 9 Ma lithospheric strength distribution is similar to the present lithospheric strength field derived from gravity data, suggesting that the bimodal strength signature was retained throughout the entire basin's evolution. Late Miocene flattening of the subducting slab, tectonic change to a broken foreland, or deposition of a thick (～8–10 km) sedimentary cover did not affect the strength of the lithosphere underlying the Bermejo basin. The long‐term bimodal strength field does not correlate with the documented thickness of the seismogenic crust.     

Crustal structure and isostatic compensation near the Kane fracture zone from topography and gravity measurements—I. Spectral analysis approach

Summary. Six gravity and bathymetry profiles perpendicular to the Kane fracture zone, each more than 300 km long, were gathered to study the variation in crustal structure in the vicinity of a major fracture zone and the gravitational edge effect at the contact between lithosphere of two different ages. A spectral analysis of the gravity and bathymetric series as a function of wavelength shows that the gravitational edge effect is only significant at the longest wavelengths. For remaining wavelengths the admittance, the ratio of the amplitude of the gravity anomaly to the amplitude of the bathymetry, is best explained by a model of isostasy in which topographic loads are partially supported by the flexural rigidity of an elastic plate, about 6 km in thickness. After subtracting the gravitational attraction of the bathymetry and its compensation, substantial isostatic anomalies remain. We interpret these anomalies as being caused by variations in crustal thickness which have little correlation with surface topography, except at very long wavelengths. The apparent crustal thickness varies by as much as a factor of 2, but there is no evidence indicating systematic thinning of the crust beneath the fracture zone. Our data do suggest that such density variations within the plate are also compensated by the isostatic response of an elastic plate but with very different effect from those at the surface. This indicates that there are two different modes of crustal formation with different gravity and topographic signatures: effusive volcanism which loads the surface of the elastic plate producing both topographic relief and coherent gravity anomalies, and intrusive volcanism or underplating producing gravity anomalies but little topographic relief.     

Constraints on the frequency–magnitude relation and maximum magnitudes in the UK from observed seismicity and glacio-isostatic recovery rates

Earthquake populations have recently been shown to have many similarities with critical-point phenomena, with fractal scaling of source sizes (energy or seismic moment) corresponding to the observed Gutenberg–Richter (G–R) frequency–magnitude law holding at low magnitudes. At high magnitudes, the form of the distribution depends on the seismic moment release rate M˙ and the maximum magnitude m _max . The G–R law requires a sharp truncation at an absolute maximum magnitude for finite M˙ . In contrast, the gamma distribution has an exponential tail which allows a soft or 'credible' maximum to be determined by negligible contribution to the total seismic moment release. Here we apply both distributions to seismic hazard in the mainland UK and its immediate continental shelf, constrained by a mixture of instrumental, historical and neotectonic data. Tectonic moment release rates for the seismogenic part of the lithosphere are calculated from a flexural-plate model for glacio-isostatic recovery, constrained by vertical deformation rates from tide-gauge and geomorphological data. Earthquake focal mechanisms in the UK show near-vertical strike-slip faulting, with implied directions of maximum compressive stress approximately in the NNW–SSE direction, consistent with the tectonic model. Maximum magnitudes are found to be in the range 6.3–7.5 for the G–R law, or 7.0–8.2 m _L for the gamma distribution, which compare with a maximum observed in the time period of interest of 6.1 m _L . The upper bounds are conservative estimates, based on 100 per cent seismic release of the observed vertical neotectonic deformation. Glacio-isostatic recovery is predominantly an elastic rather than a seismic process, so the true value of m _max is likely to be nearer the lower end of the quoted range.     

Lithospheric flexure due to prograding sediment loads: implications for the origin of offlap/onlap patterns in sedimentary basins

Abstract Simple elastic plate models have been used to determine the stratigraphic patterns that result from prograding sediment loads. The predicted patterns, which include coastal offlap/onlap and downlap in a basinward direction, are generally similar to observations of stratal geometry from Cenozoic sequences of the U.S. Atlantic and Gulf Coast margins. Coastal offlap is a feature of all models in which the water depth and elastic thickness of the lithosphere, T _e (which is a measure of the long-term strength of the lithosphere), are held constant, and is caused by a seaward shift in the sediment load and its compensation as progradation proceeds. The coastal offlap pattern is reduced if sediments prograde into a subsiding basin, since subsidence causes an increase in the accommodation space and loading landward of a prograding wedge. The stratal geometry that results is complex, however, and depends on the sediment supply, the amount of subsidence, and T _e. If the sediment supply to a subsiding basin proceeds in distinct 'pulses' (due, say, to different tectonic events in a source region) then it is possible to determine the relationship between stratal geometry and T _e. Coastal offlap and downlap are features of most models where the lithosphere either has a constant T _e slowly increases T_e with time, or changes T _e laterally; however, in the case where sediments prograde onto lithosphere that rapidly increases T _e with rime, the offlap can be replaced by onlap. Lithospheric flexure due to prograding sediment loads is capable of producing a wide variety of stratal geometries and may therefore be an important factor to take into account when evaluating the relative role of tectonics and eustatic sea-level changes in controlling the stratigraphic record.     

Free-surface formulation of mantle convection—II. Implication for subduction-zone observables

Viscous and viscoelastic models for a subduction zone with a faulted lithosphere and internal buoyancy can self-consistently and simultaneously predict long-wavelength geoid highs over slabs, short-wavelength gravity lows over trenches, trench-forebulge morphology, and explain the high apparent strength of oceanic lithosphere in trench environments. The models use two different free-surface formulations of buoyancy-driven flows (see, for example, Part I): Lagrangian viscoelastic and pseudo-free-surface viscous formulations. The lower mantle must be stronger than the upper in order to obtain geoid highs at long wavelengths. Trenches are a simple consequence of the negative buoyancy of slabs and a large thrust fault, decoupling the overriding from underthrusting plates. The lower oceanic lithosphere must have a viscosity of less than to²⁴ Pa s in order to be consistent with the flexural wavelength of forebulges. Forebulges are dynamically maintained by viscous flow in the lower lithosphere and mantle, and give rise to apparently stiffer oceanic lithosphere at trenches. With purely viscous models using a pseudo-free-surface formulation, we find that viscous relaxation of oceanic lithosphere, in the presence of rapid trench rollback, leads to wider and shallower back-arc basins when compared to cases without viscous relaxation. Moreover, in agreement with earlier studies, the stresses necessary to generate forebulges are small (∼ 100 bars) compared to the unrealistically high stresses needed in classic thin elastic plate models.     

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.     

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.