Depth-dependent Viscosity Research Articles

EFFECTS OF COMBINED TEMPERATURE- AND DEPTH-DEPENDENT VISCOSITY AND HALL CURRENT ON AN UNSTEADY MHD LAMINAR CONVECTIVE FLOW DUE TO A ROTATING DISK

The present study investigates the effects of mixed temperature- and depth-dependent viscosity and Hall current on an unsteady flow of an incompressible electrically conducting fluid on a rotating disk in the presence of a uniform magnetic field. We assume that the fluid viscosity strongly depends on temperature and depth, which may be directly applicable to the earth's mantle and a uniform mid-ocean ridge basalt reservoir in whole mantle flow. The system of axial symmetric nonlinear partial differential equations governing the unsteady flow and heat transfer is written in cylindrical polar coordinates and reduced to nonlinear ordinary differential equations by introducing suitable similarity parameters. Solutions for the flow and temperature fields are obtained numerically assuming large Prandtl number by using Runge-Kutta and shooting methods. The nature of radial, tangential, and axial velocities and temperature in the presence of a uniform magnetic field is presented for changing various nondimensional parameters at different layers of the medium. The coefficients of skin frictions and the rate of heat transfer are calculated at different parameters. Comparison has been made for steady flow (C = 0) and shows excellent agreement with Sparrow and Gregg (1959), hence encouragement for the use of the present numerical computations.

Influence of temperature- and depth-dependent viscosity structures on postseismic deformation predictions for the large 1946 Nankai subduction zone earthquake

To investigate the influence of spatial change of viscosity on postseismic deformation associated with the interplate 1946 Nankai earthquake (M 8.0) at the Nankai Trough, southwest Japan, we newly constructed a realistic viscoelastic structure model, taking into account temperature- and depth-dependent viscosity of materials. For this purpose, we first compiled leveling and triangulation data during postseismic periods and clarified characteristics of the amount and spatial patterns of postseismic vertical displacement and principal strain fields. Then, we calculated the spatial distributions of viscosity from temperature and flow fields, which were obtained from 2D subduction models. By incorporating the obtained viscosity structure into 3D viscoelastic finite element models, we constructed a temperature- and depth-dependent viscosity structure model (MODEL P2). Based on MODEL P2, we constructed a viscoelastic structure model, taking into account Poisson's ratio for the oceanic plate and low-velocity regions and the existence of low-viscosity materials beneath the Shikoku and Chugoku districts (MODEL P3), which were revealed from seismic tomography. We also constructed a conventional layered viscoelastic structure model (MODEL L1) and plate subduction model (MODEL P1) with constant viscosity for each region and evaluated the effects of different viscoelastic structures on postseismic surface deformations, using the coseismic slip distribution obtained by inversion analyses of geodetic data. We also compared the calculated surface deformations with the observed postseismic crustal deformations in and around Shikoku. The results show that postseismic surface deformation fields for the newly constructed MODEL P2 are rather different from those for MODELs L1 and P1. Landward horizontal displacements for MODEL P2 are smaller than those for MODELs L1 and P1, seaward horizontal displacements are negligible, and vertical displacement is characterized by small subsidence over Shikoku. The postseismic horizontal principal strain field for MODEL P2 is characterized by contractions in the N–S to NW–SE directions at amounts smaller than those for MODELs L1 and P1. Postseismic surface deformations for MODEL P3 are almost the same as those for MODEL P2. The observed postseismic vertical displacement and horizontal principal strain fields could not be explained by the viscoelastic response for the realistic viscoelastic structure models P2 and P3. This indicates that the effects of elastic and viscoelastic responses due to interplate coupling on the plate interface, after-slip at the extension of the coseismic slipped region, and poroelasticity should be taken into account to precisely estimate postseismic surface deformation. This also suggests that, in order to evaluate postseismic crustal deformations derived from a large interplate subduction zone earthquake, it is essential to use realistic temperature- and depth-dependent viscoelastic structure models.

On the relative importance of mineral phase transitions and viscosity stratification in controlling the sinking rates of detached slab remnants

[1] We employ a two-dimensional control-volume model of convection in a cylindrical shell to study sinking rates of cold parcels of material in the Earth's mantle. Our model uses periodic boundary conditions, at 0 and 360 degrees, to eliminate side-wall effects and includes mineral phase transitions at 410 km and 660 km depths and a depth-dependent viscosity. We compare the rates of sinking of cold material with and without phase boundaries and with and without viscosity increases with depth. When the viscosity increase is sufficiently large we find the sinking rate is relatively insensitive to the presence or absence of phase boundaries; the effects of viscosity dominate. If recent estimates of large viscosity increases in the lower mantle are correct, our model could account for mid-mantle slab remnants of Jurassic age independent of the existence of phase transition boundaries in the mantle.

Dynamics of cratons in an evolving mantle

The tectonic quiescence of cratons on a tectonically active planet has been attributed to their physical properties such as buoyancy, viscosity, and yield strength. Previous modelling has shown the conditions under which cratons may be stable for the present, but cast doubt on how they survived in a more energetic mantle of the past. Here we incorporate an endothermic phase change at 670 km, and a depth-dependent viscosity structure consistent with post-glacial rebound and geoid modelling, to simulate the dynamics of cratons in an “Earth-like” convecting system. We find that cratons are unconditionally stable in such systems for plausible ranges of viscosity ratios between the root and asthenosphere (50–150) and the root/oceanic lithosphere yield strength ratio (5–30). Realistic mantle viscosity structures have limited effect on the average background cratonic stress state, but do buffer cratons from extreme stress excursions. An endothermic phase change at 670 km introduces an additional time-dependence into the system, with slab breakthrough into the lower mantle associated with 2–3 fold stress increases at the surface. Under Precambrian mantle conditions, however, the dominant effect is not more violent mantle avalanches, or faster mantle/plate velocities, but rather the drastic viscosity drop which results from hotter mantle conditions in the past. This results in a large decrease in the cratonic stress field, and promotes craton survival under the evolving mantle conditions of the early Earth.

Topography and geoid anomalies due to density heterogeneities at the base of the thermal lithosphere: application to oceanic swells and small-wavelength geoid lineations

SUMMARY The surface topography, gravity and geoid associated with loads situated at the base of the thermal lithosphere are computed. The model lithosphere is composed of a viscous lower layer with depth-dependent viscosity overlain by an elastic lid. When the viscosity varies strongly with depth, the amplitude of the surface topography decreases rapidly as the wavenumber characterizing the mass anomaly increases. The response for this two-layer lithosphere differs substantially from the response computed for the loading from below of an elastic lid. If the 200km wavelength geoid anomalies observed in the oceans are due to convective instabilities at the base of the lithosphere, a positive geoid anomaly with an amplitude reaching 30 to 50cm is expected over a cold downwelling for ages greater than 10 Myr. No detectable topography associated with the lineations should be present for ages greater than 20 Myr. Our model predicts that some topographic anomalies might be visible for younger ages. If 1000 km broad swells are due to light material at a depth of about 100 km, the associated geoid-over-topography ratios are quite reduced compared to those predicted by the long-wavelength linear relationship usually employed for estimating the apparent compensation depth.

Importance of lateral viscosity variations in the whole mantle for modelling of the dynamic geoid and surface velocities

Abstract We investigate the effect of lateral viscosity variations (LVV) in the mantle on dynamic geoid and near-surface mantle velocities. Contrary to the previous studies, we analyse the effect not only of the lithospheric keels and asthenosphere but also of the whole mantle. A 3D global viscosity model is constructed using (a) the S20 seismic tomography model converted to temperature and (b) assumptions about homologous temperature in the mantle Paulson et al. [Paulson, A., Zhong, S., Wahr, J., 2005. Modeling post-glacial rebound with lateral viscosity variations. Geophys. J. Int. 163, 357–371]. The conversion parameters provide a depth-dependent viscosity profile, which is generally consistent with the existing studies. Therefore, the horizontal variations are produced self-consistently within this approach. We estimate them as ‘conservative’, thus providing a minimum limit for possible horizontal changes; consequently the obtained results show a low limit for dynamic topography, geoid and surface velocities’ disturbances. To estimate the effect of LVV we have employed the perturbation method of Zhang and Christensen [Zhang, S., Christensen U., 1993. Some effects of lateral viscosity variations on geoid and surface velocities induced by density anomalies in the mantle. Geophys. J. Int. 114, 547–551], which is modified to implement mantle compressibility. It has been found that the impacts of the upper and lower mantle are comparable in amplitude. The difference between the initial (only radial viscosity) dynamic geoid and the geoid with implemented LVV reaches −27 to +19 m for the whole mantle LVV, while the effects of the upper and lower mantle are equal to −24 to +16 and −20 to +16 m correspondingly. In general these effects are not correlated. The difference in the geoid response leads to substantially altered velocity-to-density scaling factors (up to 30% compared to the initial ones) obtained in least squares inversion. The effect of the lower mantle LVV on surface velocities is also substantial, in particular with respect to the toroidal component, which does not exist in the initial model.

Lower mantle dynamics with the post-perovskite phase change, radiative thermal conductivity, temperature- and depth-dependent viscosity

The new post-perovskite phase near the core–mantle boundary has important ramifications on lower mantle dynamics. We have investigated the dynamical impact arising from the interaction of temperature- and depth-dependent viscosity with radiative thermal conductivity, up to a lateral viscosity contrast of 1 0 4 , on both the ascending and descending flows in the presence of both the endothermic phase change at 670 km depth and an exothermic post-perovskite transition at 2650 km depth. The phase boundaries are approximated as localized zones. We have employed a two-dimensional Cartesian model, using a box with an aspect-ratio of 10, within the framework of the extended Boussinesq approximation. Our results for temperature- and depth-dependent viscosity corroborate the previous results for depth-dependent viscosity in that a sufficiently strong radiative thermal conductivity plays an important role for sustaining superplumes in the lower mantle, once the post-perovskite phase change is brought into play. This aspect is especially emphasized, when the radiative thermal conductivity is restricted only to the post-perovskite phase. These results revealed a greater degree of asymmetry is produced in the vertical flow structures of the mantle by the phase transitions. Mass and heat transfer between the upper and lower mantle will deviate substantially from the traditional whole-mantle convection model. Streamlines revealed that an overall complete communication between the top and lower mantle is difficult to be achieved.

The importance of radiative heat transfer on superplumes in the lower mantle with the new post-perovskite phase change

A new post-perovskite phase has been identified recently from both X-ray observations and first-principles calculations. This novel phase change occurs at a depth about 200 km above the core–mantle boundary for temperatures of around 2800 K. Using a two-dimensional, extended Boussinesq model, we have studied the dynamical consequences of this deep-mantle phase transition on mantle convection with particular emphasis on the effects on lower mantle plume structures. We have employed a depth-dependent viscosity with a viscosity maximum in the mid-lower mantle and two phase transitions, one at 670 km depth and the other at a depth of 2650 km. The phase transition at 670 km is endothermic, while we have examined both exothermic and endothermic possibilities for the new phase transition. Our results show the following situations favorable for the development of superplumes: (1) endothermic phase transition for both constant and radiative thermal conductivities and (2) exothermic phase transition and radiative thermal conductivity. Smaller unstable plumes are found for exothermic phase transition and constant thermal conductivity. Extremely high lateral temperature increases, exceeding 1500 degrees, are found inside the lower mantle plume for constant thermal conductivity. This lateral temperature contrast is mitigated by the presence of radiative thermal conductivity. In order for superplumes to prevail in the lower mantle with a deep exothermic phase change, we must invoke some form of radiative thermal conductivity for stabilizing and promoting large upwellings with a width exceeding at least 500 km. Strong lateral heterogeneities in both the seismic velocity and density fields can be produced in the plumes and the surrounding mantle for this new phase transition with a large Clapeyron slope, close to 10 MPa/K.

The structure and the surface manifestation of mantle plumes in depth-dependent three-dimensional models

Thermal convection has been modeled using a 3D Cartesian model, in order to study the structure of mantle plumes and their surface manifestation. Hotspots can be regarded as the surface manifestation of mantle plumes. The main characteristic features of the hotspots are their volcanism, topographic, geoid and heat flow anomalies. The following characteristics of the hotspot-generator plumes are studied: their geometrical size, temperature distribution, the horizontal extent and maximum amplitude of the surface features: geoid, topographic and heat flow anomalies. The aim of the model calculations is to investigate the effect of depth-dependent viscosity and internal heating in case of scaling the non-dimensional result for upper and whole mantle convection. When upper mantle convection is supposed, the calculated amplitudes of the anomalies fit well to the observed values, but their lateral extent is too small. The extent of the anomalies suggests whole mantle plume. Taking into account the high viscosity of the lithosphere, the low viscosity of the asthenosphere and the D″ layer at the CMB and using internal heating helps to gain better fit to the observations. The most complex whole mantle model predicts quite realistic features, except the topographic height.

Multigrid iterative algorithm using pseudo-compressibility for three-dimensional mantle convection with strongly variable viscosity

A numerical algorithm for solving mantle convection problems with strongly variable viscosity is presented. Equations for conservation of mass and momentum for highly viscous and incompressible fluids are solved iteratively by a multigrid method in combination with pseudo-compressibility and local time stepping techniques. This algorithm is suitable for large-scale three-dimensional numerical simulations, because (i) memory storage for any additional matrix is not required and (ii) vectorization and parallelization are straightforward. The present algorithm has been incorporated into a mantle convection simulation program based on the finite-volume discretization in a three-dimensional rectangular domain. Benchmark comparisons with previous two- and three-dimensional calculations including the temperature- and/or depth-dependent viscosity revealed that accurate results are successfully reproduced even for the cases with viscosity variations of several orders of magnitude. The robustness of the numerical method against viscosity variation can be significantly improved by increasing the pre- and post-smoothing calculations during the multigrid operations, and the convergence can be achieved for the global viscosity variations up to 10 10.

Possible effects of lateral viscosity variations induced by plate-tectonic mechanism on geoid inferred from numerical models of mantle convection

In a traditional analytical method, the convective features of Earth’s mantle have been inferred from surface signatures obtained by the geodynamic model only with depth-dependent viscosity structure. The moving and subducting plates, however, bring lateral viscosity variations in the mantle. To clarify the effects of lateral viscosity variations caused by the plate-tectonic mechanism, I have first studied systematically instantaneous dynamic flow calculations using new density–viscosity models only with vertical viscosity variations in a three-dimensional spherical shell. I find that the geoid high arises over subduction zones only when the vertical viscosity contrast between the upper mantle and the lower mantle is O ( 1 0 3 ) to O ( 1 0 4 ) , which seems to be much larger than the viscosity contrast suggested by other studies. I next show that this discrepancy may be removed when I consider the lateral viscosity variation caused by the plate-tectonic mechanism using two-dimensional numerical models of mantle convection with self-consistently moving and subducting plates, and suggest that the observed geoid anomaly on the Earth’s surface is significantly affected by plate-tectonic mechanism as a first-order effect.

A dynamic model for the Iceland Plume and the North Atlantic based on tomography and gravity data

SUMMARY The North Atlantic around Iceland is characterized by a large geoid high and an anomalous shallow ocean floor; both observations are presumably due to upwelling of hot material in the mantle. We extracted this region from a global tomography data set to study the structure of the mantle in more detail. The tomography data reveal a confined low-velocity structure from the core–mantle boundary (CMB) to the upper mantle, stretching from a location at the CMB below southwest Greenland towards the upper mantle in a strongly eastward inclination. In the present study we compare the observed gravity potential field in the North Atlantic with a modelled field based on mantle temperature variations estimated from tomography. Seismic traveltime residuals are converted to temperature variations assuming a linear relation between seismic velocity and density and a pressure-dependent thermal expansivity. We found a maximum excess temperature in the plume conduit at the CMB of ∼250 °C, weakly decreasing towards the phase transition zone (PTZ). In the PTZ a temperature rise of ∼50–70 °C is found, which is in agreement with the latent heat release by the olivine phase transitions. The 3-D temperature field is then used as the driving force for viscous flow in a Cartesian convection model. The model dimensions are chosen four times as large as the tomographic section to allow resolution for long wavelengths and convective return flow. For a number of constant viscosity and temperature- and depth-dependent viscosity cases the dynamic topography, gravity anomaly and geoid undulations are calculated and compared with the EGM96 potential field coefficients in the wavelength range of 400 to 4000 km. The observation data were also corrected for ocean lithosphere cooling and isostatic compensation of continental crust. The best agreement between observation and modelled data (78 per cent fit for geoid and 47 per cent for gravity) is obtained for a temperature-dependent viscosity of about one order of magnitude for 500 °C temperature variation and an increase of viscosity with depth by no more than a factor of 50 from the upper to the lower mantle. The generally good spatial agreement supports the tomographic model, at least for the upper mantle, and indicates that the East Greenland margin as well as the outer Faroer Ridge are dynamically supported. Low-density material west of the Kolbeinsey Ridge might be linked to low-density anomalies below the Greenland Shield. The presence of lower mantle anomalies causes a large-scale geoid high of ∼3 to 5 m in agreement with observations, but our approach cannot further constrain the spatial distribution of anomalies in the lower mantle.

Synthetic seismic signature of thermal mantle plumes

The first seismic images of mantle plumes have been a source of significant debate. To interpret these images, it is useful to have an idea of a plume’s expected seismic signature. We determined a set of dynamic thermal whole-mantle plumes, with parameters appropriate for the Earth’s mantle and shallow-mantle temperature contrasts compatible with surface observations. We explore the sensitivity of amplitude and width of thermal plume anomalies to model parameters. The conversion of thermal to seismic structure accounts for effects of temperature, pressure, an average mantle composition including phase transitions, and anelasticity. With depth-dependent expansivity and temperature- and depth-dependent viscosity, these relatively weak plumes have lower-mantle diameters of 300–600 km at one half of the maximum temperature anomaly. To attain the narrow upper-mantle plumes inferred from surface observations and tomography, viscosity reduction by a factor 30–100 is necessary, either as a jump or as a strong gradient. All model plumes had buoyancy fluxes ≥4 Mg/s and it seems difficult to generate whole-mantle thermal plumes with fluxes much lower. Due to changing seismic sensitivity to temperature with depth and mineralogy, variations in the plumes’ seismic amplitude and width do not coincide with those in their thermal structure. Velocity anomalies of 2–4% are predicted in the uppermost mantle. Reduced sensitivity in the transition zone as well as complex velocity anomalies due to phase boundary topography may hamper imaging continuous whole-mantle plumes. In the lower mantle, our plumes have seismic amplitudes of only 0.5–1%. Unlike seismic velocities, anelasticity reflects thermal structure closely, and yields plume anomalies of 50–100% in dln(1/ Q S).

Evolution of orogenic wedges and continental plateaux: insights from crustal thermal-mechanical models overlying subducting mantle lithosphere

The links between an early phase of orogenesis, when orogens are commonly wedge shaped, and a later phase, with a plateau geometry, are investigated using coupled thermal–mechanical models with uniform velocity subduction boundary conditions applied to the base of the crust, and simple frictional–plastic and viscous rheologies. Models in which rheological properties do not change with depth or temperature are characterized by growth of back-to-back wedges above the subduction zone. Wedge taper is inversely dependent on the Ramberg number (Rm; gravity stress/basal traction); increasing convergence velocity or crustal strength produces narrower and thicker wedges. Models that are characterized by a decrease in crustal viscosity from ηc to ηb with depth or temperature, leading to partial or full basal decoupling of the crust from the mantle, display more complex behaviour. For models with a moderate viscosity ratio, ηb/ηc∼ 10−1, the crustal wedges have dual tapers with a lower taper in the central region and a higher taper at the edges of the deformed crust. A reduction in the viscosity ratio (ηb/ηc∼ 10−2) is sufficient to cause a transition of the central wedge region to a plateau. This transition depends on the basal traction, therefore the thickness of the weak basal layer also affects the transition. Further reduction of the viscosity ratio (ηb/ηc∼ 10−4) leads to full basal decoupling and the development of plateaux in all cases considered. In most models, the plateaux grow laterally at constant thickness between characteristic edge peaks associated with the transitions from coupled to decoupled lower crust. Where the crust is fully decoupled, large-scale model geometries for both depth- and temperature-dependent rheologies are similar with gravity-driven flow concentrated in the low-viscosity region. However, strong lateral temperature gradients within these models, controlled by the interaction of horizontal and vertical thermal advection, diffusion and heterogeneous thickening of the radioactive crustal layer, lead to differences in the velocity and deformation fields between the two cases, particularly at the plateau margins. The results suggest that simple depth-dependent viscosity models may be reasonable approximations for describing the large-scale geometry of fully developed plateaux, but that they are not appropriate for describing the internal features of large orogenic systems or the transition from wedge to plateau geometry.

Thermochemical convection and helium concentrations in mantle plumes

Compositionally heterogeneous material may exist in the lowermost mantle [van der Hilst and Kárason, Science 283 (1999) 1885–1888]. Here we use a numerical model to investigate the dynamics of (i) the subducted oceanic crust and lithosphere, (ii) a deep layer chemically denser, relatively undegassed and enriched in radiogenic elements. Tracers carry U, Th, K, and He concentrations which vary due to radioactive decay and to partial melting and degassing processes. We investigate the stability of the denser layer and find that by considering a depth dependent thermal expansion coefficient and temperature dependent viscosity, a layer with a chemical density excess of 2.4% can remain stable and poorly mixed until present-day time. The calculated helium ratios are in good agreement with 3He/ 4He observed at ridges and hotspots and show that the large spectrum of helium ratios of OIB can be explained by mixing between undegassed material, recycled oceanic crust and lithosphere. For MORB, the sharp spectrum of helium ratios may be due to a degassed, homogeneous and well mixed shallow mantle.

Is the transition zone an empty water reservoir? Inferences from numerical model of mantle dynamics

Water is probably present everywhere in the Earth’s mantle today, with abundances ranging between scales of percent (%) to the parts per million (ppm). Mantle total water content is estimated to be between 10% to several times that of the present-day hydrosphere. Numerous studies have been devoted to the determination of water solubility in mantle material [D.R. Bell, G.R. Rossmann, Science 255 (1992) 1391–1397; J. Ingrin, H. Skogby, Eur. J. Mineral. 12 (2000) 543–570]. They all show strong solubility variations from one mineral phase to another. Principally, water partitioning has made the transition zone a probable trap for water from the Earth’s mantle [N. Bolfan-Casanova et al., Earth Planet. Sci. Lett. 182 (2000) 209–221; D.L. Kohlstedt et al., Contrib. Mineral. Petrol. 123 (1996) 345–357]. Nevertheless, water distribution within the mantle is still debated. We have studied the role of mantle dynamics in water distribution by modeling water transport and mantle convection in a two-dimensional (2-D) cartesian geometry. The model takes into account water partitioning between the mantle’s transition zone and the upper mantle of 10:1 and between the lower mantle and the transition zone of 1:100 (i.e. respectively between olivine and spinel and spinel and post-spinel). We have modeled the mantle temperature field using depth-dependent viscosity and plate-like surface conditions. Water injection at the trench has also been simulated. Our numerical experiments suggest that diffusivity of water has to be very high, at least two orders of magnitude higher than the one experimentally determined [D.R. Bell, G.R. Rossmann, Science 255 (1992) 1391–1397; J. Ingrin, H. Skogby, Eur. J. Mineral. 12 (2000) 543–570] to significantly influence water distribution in Earth’s mantle. In fact, the diffusion process is not efficient enough to balance the mixing due to mantle dynamics and to force water into the transition zone. We show that the distribution of water should be quite homogeneous throughout the mantle if advection and diffusion are the only processes involved in water transport in the mantle. This homogeneity implies that water below the transition zone could be in excess according to the lower mantle rocks solubility. This addresses the question of stability of free water in the lower mantle and its mobility by percolation process, which could be a very efficient transport process, previously unconsidered in this field of research.

Effects of depth-dependent viscosity and plate motions on maintaining a relatively uniform mid-ocean ridge basalt reservoir in whole mantle flow

[1] Mid-ocean ridge basalts (MORBs) exhibit relatively uniform and depleted rare earth element concentrations compared with ocean island basalts (OIBs). Previous researchers have focused on long-term (billion-year timescale) preservation of an enriched and heterogeneous OIB reservoir within the convecting mantle. Such studies commonly conclude that the OIB reservoir must exist in an area which remains isolated from convection, i.e., D″. Here we investigate the maintenance of MORB reservoir homogeneity over shorter timescales in the face of vigorous upper/lower mantle mass exchange (deep subduction), which may be due to two effects: (1) a high-viscosity lower mantle and/or (2) chaotic mixing due to toroidal flow generated by surface plate motions. We explore this conceptual model using three-dimensional spherical numerical models that include surface plate motions, radial viscosity variation, and a geophysically plausible model of mantle density contrasts. A correlation dimension method is used to characterize mixing of passive tracers. For a uniform viscosity mantle the upper and lower mantles mix on essentially the same timescales. A factor of 100 viscosity contrast results in a relative mixing time for the lower mantle only ∼30–60% longer than that of the upper mantle. Therefore neither a strong viscosity contrast nor toroidal mixing significantly affects the relative mixing times of the upper and lower mantle. We conclude that return flow from the lower mantle is of similar (depleted) composition and that the depleted MORB source reservoir constitutes most of the mantle, except for a convectively isolated OIB source region at the base of the mantle.

The influence of tectonic plates on mantle convection patterns, temperature and heat flow

SUMMARY The dynamic coupling between plate motion and mantle convection is investigated in a suite of Cartesian models by systematically varying aspect ratios and plate geometries. The aim of the study presented here is to determine to what extent plates affect mantle flow patterns, temperature and surface heat flux. To this end, we compare numerical convection models with free-slip boundary conditions to models that incorporate between one and six plates, where the geometries of the plates remain fixed while the plate velocities evolve dynamically with the flow. We also vary the widths of the plates and the computational domain in order to determine what constraint these parameters place on the mean temperature, heat flux and plate velocity of mantle convection models. We have investigated the influence of plates for three whole-mantle convection cases that differ in their heating modes (internally heated and basally heated) and rheologies (isoviscous and depth-dependent viscosity). We present a systematic investigation of over 30 models that exhibit increasingly complex behaviour in order to understand highly time-dependent systems using the insight gained from simpler models. In models with aspect ratios from 0.5 to 12 we find that for the same heating mode, variations in temperature can be as much as 40 per cent when comparing calculations with unit-width plates to models incorporating plates with widths equal to five times the model depth. Mean surface heat flux may decrease by 60 per cent over the same range of plate widths. We also find that internally heated mantle convection models incorporating plates exhibit novel behaviour that, we believe, has not been described previously in mantle convection studies. Specifically, in internally heated models, plate motion is characterized by episodic reversals in direction driven by changes in the mantle circulation from clockwise to counterclockwise and vice versa. These flow reversals occur in internally heated convection and are caused by a build-up of heat in the interiors of wide convection cells close to mantle downwellings. We find that flow reversals occur rapidly and are present in both single-plate and multiple-plate models that include internal heating. This behaviour offers a possible explanation for why the Pacific plate suddenly changed its direction some 43 Ma.

Are mantle plumes adiabatic?

The issue concerning the state of adiabaticity of mantle plumes has been examined in a cartesian two-dimensional box with an aspect-ratio of six. We have investigated in the quasi steady-state regime high Rayleigh number convection with both depth-dependent viscosity and thermal expansivity for both the Boussinesq and the extended Boussinesq approximations. We have also assessed the influence of various forms of thermal conductivity and internal heating. We have generalized the classical Bullen’s parameter equation from one dimension to multidimensions. For assessing the local state of adiabaticity inside plumes and in their surroundings, we have extracted from the local geotherms and the local thermodynamic properties the corresponding Bullen’s parameter profiles and the two-dimensional maps portraying the state of adiabaticity in the mantle. Histograms characterizing the frequency of adiabaticity are also employed for quantification purposes. In general, superadiabatic thermal gradients are found inside the thick plume limbs and sometimes along the central part of the plume. The centers of plume heads are subadiabatic or nearly adiabatic, but the edges of the plume heads are strongly subadiabatic. Alternating strips of subadiabaticity and adiabaticity are found in the downwellings. The ambient mantle outside the plumes is generally adiabatic and is sometimes perforated with islands of marked deviations from adiabaticity.

The Number of Hotspots in Mantle Convection: Effect of Depth-Dependent Viscosity and Internal Heating in Two-Dimensional Models

Two-dimensional numerical models of mantle convection have been calculated for a range of high Rayleigh numbers, depth-dependent viscosity and basal plus internal heating. Large aspect ratio boxes have been used in the calculations in order to estimate the expected areal density of upwellings in infinite fluid layers. The results are analyzed with regard to the number of the Earth’s hotspots which are assumed to be surface imprints of cylindrical upwellings in the mantle. For a pure whole-mantle situation, 6–7 upwellings can be expected. If the upper mantle convects separately above the 660 km discontinuity (allowing a second convective layer below 660 km depth), the theoretically estimated number of upper-mantle plumes can be as high as 250. Given the number of real hotspots (42 to 117 according to different compilations), it is suggested that the flow regime of the mantle is intermediate between the pure whole-mantle or pure two-layer circulation.