Exact analytical solutions of the stokes equation for testing the equations of mantle convection with a variable viscosity

Presently, the study of the mantle flow structure is mainly based on numerical modeling. The most important stage of the development of a computer program is its testing. For this purpose, results of various test models of convection flows with a given set of parameters are compared. The solution of the Stokes equation, involving the derivative of viscous stresses, is most difficult. Exact analytical solutions of the Stokes equation are obtained in this work for various cases of special loads. These solutions can be used as benchmarks for testing programs of numerical calculation of viscous flows in both geophysics and engineering. The advantage of this testing technique is the exceptional simplicity of the solution form, the admissibility of any spatial viscosity variations, and the fact that solutions can be compared not for a narrow set of the solution parameters but for any distributions of velocities, viscous stresses, and pressures at all points of the space.     

Parameterized thermal model of a mixed mantle convection

IntroductionTectonicevolutionisinfluencedbythermalhistoryoftheEarth.TheEarthhasabout4.6Gahistory.ThermalenergyfromtheinterioroftheEachprovidesthemainpowerfortectonicevolution.ItnotonlycontrolstheformationofthelayeredstructuresinsidetheEarth,butalsopromotesthetectonicmovementsofthesurfaceplatesduringthegeologicalera.ThestudyofthethermalhistoryoftheEarthhaspassedseveralstages.Inearlystudies,onlyconductivemechanism(Lubimova,1958)isdiscussedinthethermalevolution.However,theimpotalceofthermalco…     

横向黏度变化的全地幔对流应力场初步研究

将地幔地震波速度异常转换为地幔横向黏度变化(达到3个数量级),在球坐标系下计算了瑞雷数为106、上边界为刚性、下边界为应力自由等温边界条件下的岩石层底部的地幔对流极型和环型应力场.结果表明,地幔对流极型应力场与地表大尺度构造具有良好的对应关系:俯冲带和碰撞带的应力呈现挤压状态,而洋中脊处的应力则呈现拉张状态.地幔对流环...     

A new conceptual model for whole mantle convection and the origin of hotspot plumes

A new conceptual model of mantle convection is constructed for consideration of the origin of hotspot plumes, using recent evidence from seismology, high-pressure experiments, geodynamic modeling, geoid inversion studies, and post-glacial rebound analyses. This conceptual model delivers several key points. Firstly, some of the small-scale mantle upwellings observed as hotspots on the Earth's surface originate at the base of the mantle transition zone (MTZ), in which the Archean granitic continental material crust (TTG; tonalite-trondhjemite-granodiorite) with abundant radiogenic elements is accumulated. Secondly, the TTG crust and the subducted oceanic crust that have accumulated at the base of MTZ could act as thermal or mechanical insulators, leading to the formation of a hot and less viscous layer just beneath the MTZ; which may enhance the instability of plume generation at the base of the MTZ. Thirdly, the origin of some hotspot plumes is isolated from the large low shear-wave velocity provinces (LLSVPs) under Africa and the South Pacific. I consider that the conceptual model explains why almost all the hotspots around Africa are located above the margins of the African LLSVP. Because a planetary-scale trench system surrounding a “Pangean cell” has been spatially stable throughout the Phanerozoic, a large amount of the oceanic crustal layer is likely to be trapped in the MTZ under the Pangean cell. Therefore, under Africa, almost all of the hotspot plumes originate from the base of the MTZ, where a large amount of TTG and/or oceanic crusts has accumulated. This conceptual model may explain the fact that almost all the hotspots around Africa are located on margins above the African LLSVP. It is also considered that some of the hotspot plumes under the South Pacific thread through the TTG/oceanic crusts accumulated around the bottom of the MTZ, and some have their roots in the South Pacific LLSVP while others originate from the MTZ. The numerical simulations of mantle convection also speculate that the Earth's mantle convection is not thermally double-layered at the ringwoodite to perovskite + magnesiowüstite (Rw → Pv + Mw) phase boundary, because of its gentle negative Clapeyron slope. This is in contrast with some traditional images of mantle convection that have independent convection cells between the upper and lower mantle. These numerical studies speculate that the generation of stagnant slab at the base of the MTZ (as seismically observed globally) may not be due to the negative Clapeyron slope, and may instead be related to a viscosity increase (i.e., a viscosity jump) at the Rw → Pv + Mw phase boundary, or to a chemically stratified boundary between the upper and the lower mantle, as suggested by a recent high-pressure experiment.     

The structure of 2D mantle convection and stress fields: Effects of viscosity distribution

Spatial fields of temperature, velocity, overlithostatic pressure, and horizontal stresses in the Earth’s mantle are studied in two-dimensional (2D) numerical Cartesian models of mantle convection with variable viscosity. The calculations are carried out for three different patterns of the viscosity distribution in the mantle: (a) an isoviscous model, (b) a four-layer viscosity model, and (c) a temperature- and pressure-dependent viscosity model. The pattern of flows, the stresses, and the surface heat flow are strongly controlled by the viscosity distribution. This is connected with the formation of a cold highly viscous layer on the surface, which is analogous to the oceanic lithosphere and impedes the heat transfer. For the Rayleigh number Ra = 10⁷, the Nusselt number, which characterizes the heat transfer, is Nu = 34, 28, and 15 in models with constant, four-layered, and p, T-dependent viscosity, respectively. In all three models, the values of overlithostatic pressure and horizontal stresses σ _xx in a vast central region of the mantle, which occupies the bulk of the entire volume of the computation domain, are approximately similar, varying within ±5 MPa (±50 bar). This follows from the fact that the dimensionless mantle viscosity averaged over volume is almost similar in all these models. In the case of temperature- and pressure-dependent viscosity, the overlithostatic pressure and stress σ _xx fields exhibit much stronger concentration towards the horizontal boundaries of the computation domain compared to the isoviscous model. This effect occurs because the upwellings and downwellings in a highly viscous region experience strong variations in both amplitude and direction of flow velocity near the horizontal boundaries. In the models considered with the parameters used, the stresses in the upper and lower mantle are approximately identical, that is, there is no denser concentration of stresses in the upper or lower mantle. In contrast to the overlithostatic pressure field, the fields of horizontal stresses σ _xx in all models do not exhibit deep roots of highly viscous downwelling flows.     

Lagging mantle convection,the geoid and mantle structure

In numerical models of convection incorporating migration of a simulated subduction zone, the main descending flow lags far behind the migrating trench, and a geoid low is associated with the main descending flow. This provides physical plausibility for the suggestion by Chase and Sprowl that present very long-wavelength (degree 2–4) geoid lows are associated with Mesozoic trench locations, and suggests further that the present long-wavelength geoid, deep mantle structure and hotspot distribution may be straightforward consequences of plate evolution since the Upper Paleozoic.     

Penetrative convection in the planetary mantle

Abstract

The hydrodynamic equations for thermal convection in a plane layer of viscous, heat conducting fluid are scaled using the normalization of Ostrach (1965) in which the magnitude of the non-dimensional group τ = gαd/c_p determines the importance of compression work and viscous dissipation in the energy balance of the flow. A linear asymptotic theory valid in the limit τ → ∞ is constructed for the Bénard problem and this is shown to be analogous to Couette flow between contra-rotating cylinders. For sufficiently large τ the flow becomes penetrative. This fact is illustrated for homogeneous fluids by the numerical integration of a set of coupled 1st order differential equations, both for the Bénard and internally heated configurations. The effect of viscosity and thermal conductivity in-homogeneity on the depth of penetration of the main cell in the circulation pattern are assessed and it is concluded that such interactions may be sufficient to effectively limit the depth extent of mantle convection. Finally a discussion of the effect of phase transitions is given following the technique of Busse and Schubert (1971).     

Thermochemical convection in the mantle with oceanic crust recirculation

Basalts of mid-ocean ridges are depleted in incompatible elements that have passed into the continental crust. Basalts of hot spots (oceanic islands and igneous provinces) have a chemical composition close to the primary uniform mantle and are even somewhat enriched in incompatible elements. At present, for explaining the reason for this difference, there are different qualitative schemes of differentiation and mixing of substance in the mantle. In the present work, the results of numerical modeling of the two-component thermochemical convection in the mantle are given. They quantitatively demonstrate with which parameters in the mantle the layers of different chemical composition can remain unchanged. Models with different density contrasts and with variable viscosity are examined. The times of the partial mixing of layers depending on the values of these parameters are calculated. For retaining the stratified mantle for two Ga, the density contrast must be more than 2%. If the layer D″ contains a substance of the primary composition, then, its upper boundary can be the place of origin of the plumes that feed the hot spots of the Earth. The enrichment in the incompatible elements and the variety of the chemical composition of hot spots can be explained by the mixing of the substance of the slowly eroded D″ layer and the oceanic crust accumulated in it.     

A three-dimensional numerical sea model with eddy viscosity varying piecewise linearly in the vertical

The solution of the three-dimensional linear hydrodynamic equations which describe wind-driven flow in a homogeneous sea are solved using the eigenfunction method. The eddy viscosity is taken to vary piecewise linearly in the vertical over an arbitrary number of layers. Using this formulation the eigenfunctions are given in terms of Bessel functions. The coefficients of integration as well as the eigenvalues are determined accurately such that the boundary conditions are satisfied. Values of the eigenfunctions at any depth can then be determined very fast and to a high degree of accuracy.Current profiles at any position can hence be computed accurately. The expansion of the horizontal component of current converges very fast at all depths.     

Upper mantle convection driving by density anomaly and a test model

Introduction Richter and Mckenzie (1978) supposed that there is a small-scale convection system in the mantle. For a long time lots of research provides observational data to infer the possibility of a small-scale convection in the upper mantle. For example, Haxby and Weissel (1986) discussed the relationship between SEASAT map and small-scale convection. Baudry and Kroenke (1991), Maia and Diament (1991) found that the geoid and bathymetry exhibit peaks in the 400~650 km range in the Pa…     

Origin of tectonic stresses in the Chinese continent and adjacent areas

Based on data of principal stress orientation from focal mechanism and of geological features in China, we made pseudo-3D genetic algorithm finite element (GA-FEM) inversion to investigate the main forces acting on the Chinese continent and adjacent areas which form the Chinese tectonic stress field. The results confirm that plate boundary forces play the dominant role in forming the stress field in China, as noticed by many previous researchers. However, we also find that topographic spreading forces, as well as basal drag forces of the lower crust to the upper crust, make significant contribution to stresses in regional scale. Forces acting on the Chinese continent can be outlined as follows: the collision of the India plate to the NNE is the most important action, whereby forces oriented to the NW by the Philippine plate and forces oriented to the SWW by the Pacific plate are also important. Topographic spreading forces are not negligible at high topographic gradient zones, these forces are perpendicular to edges of the Tibetan Plateau and a topographic gradient belt running in the NNE direction across Eastern China. Basal drag forces applied by the ductile flow of the lower crust to the base of upper crust affect the regional stress field in the Tibetan Plateau remarkably, producing the clockwise rotation around the eastern Himalaya syntax.     

三维地震波速结构约束下的变黏度地幔对流及其动力学意义

建立了三维黏度扰动下的变黏度地幔对流模型,并提供了在引入地幔的三维地震波速度结构下相应的求解方法. 依此反演了瑞利数Ra = 106时,两种不同边界条件下的极、环型场对流图像,这有助于深化对地幔物质流动和大地构造运动的深部动力学过程的认识和理解. 研究结果表明,不但地幔浅部的极型场对流图像显示出了与大地构造运动的相关性并揭示了其深部动力学过程,更重要的是,地幔浅部的环型场对流图像首次为我们认识和理解板块构造的水平与旋转运动提供了重要的信息：环型场速度剖面中在赤道附近存在一条大致南东东—北西西向的强对流条带,可能与环赤道附近大型剪切带的形成相关,进而表明可能是该带强震发生的深部动力学背景;在南北半球存在的旋转方向相反的对流环表明它们整体上可能存在差异旋转.     

Viscous mantle tangential stresses in a numerical model of a supercontinental cycle

The values of maximum tangential stresses, as well as their spatial orientations, are calculated at each point of the mantle. These calculations are performed for successive stages of the supercontinental cycle in terms of a numerical 2-D model of assembling and diverging continents driven by mantle flows; in turn, the flows themselves are formed under the thermal and mechanical action of continents. Zones of maximum tangential stresses, within which their values are equal to or exceed 30 MPa (300 bar), are shown to be located in places where ascending or descending mantle flows approach the horizontal mantle boundaries. These zones have the shape of a bowl with the broad side toward the boundary, and their maximum tangential stresses are oriented at an angle of about 45° to the horizontal. The dimensions of these zones amount approximately to 300 km laterally and 300 km in depth. Their subhorizontal and subvertical tangential stresses are close to zero. It is also shown that the stress values vary significantly during the evolution of the suboceanic mantle under a new opening ocean (after the breakup of a supercontinent). Thus, the maximum tangential stresses in this vast mantle region at the stage of fast ocean opening are, on average, more than two times higher than at the subsequent slow stage. Note that, at all stages, the axes of the maximum tangential stresses are oriented nearly horizontally or vertically almost throughout the aforementioned area.     

Viscosity distribution in the mantle convection models

Viscosity is a fundamental property of the mantle which determines the global geodynamical processes. According to the microscopic theory of defects and laboratory experiments, viscosity exponentially depends on temperature and pressure, with activation energy and activation volume being the parameters. The existing laboratory measurements are conducted with much higher strain rates than in the mantle and have significant uncertainty. The data on postglacial rebound only allow the depth distributions of viscosity to be reconstructed. Therefore, spatial distributions (along the depth and lateral) are as of now determined from the models of mantle convection which are calculated by the numerical solution of the convection equations, together with the viscosity dependences on pressure and temperature (PT-dependences). The PT-dependences of viscosity which are presently used in the numerical modeling of convection give a large scatter in the estimates for the lower mantle, which reaches several orders of magnitude. In this paper, it is shown that it is possible to achieve agreement between the calculated depth distributions of viscosity throughout the entire mantle and the postglacial rebound data. For this purpose, the values of the volume and energy of activation for the upper mantle can be taken from the laboratory experiments, and for the lower mantle, the activation volume should be reduced twice at the 660-km phase transition boundary. Next, the reduction in viscosity by an order of magnitude revealed at the depths below 2000 km by the postglacial rebound data can be accounted for by the presence of heavy hot material at the mantle bottom in the LLSVP zones. The models of viscosity spatial distribution throughout the entire mantle with the lithospheric plates are presented.     

Planform of convection with strongly temperature-dependent viscosity

Abstract

This paper experimentally investigates the convective planform near critical in a fluid layer whose temperature-dependent viscosity varies from top to bottom by up to a factor of 1500. Convection occurs in three different planforms: rolls, hexagons and squares. The square planform, which appears only for fluids with viscosity variation greater than about 50, replaces the hexagonal convection pattern as the Rayleigh number increases much above critical. The large amplitude of hexagonal convection with strong viscosity variation precludes studying the hexagon-square transition with perturbation methods of the type used to study the hexagon-roll transitions at smaller viscosity variations.     

Combined geothermal-geophysical models of the earth's crust and upper mantle for the European continent

An attempt is made to obtain a combined geophysical model along two regional profiles: Black Sea— White Sea and Russian Platform—French Central Massif. The process of the model construction had the following stages: 1. The relation between seismic velocity (V_p, km/s) and density (σ, g/cm³) in crustal rocks was determined from seismic profiles and observed gravity fields employing the trial and error method. 2. Relations between heat production HP (μW/m³), velocity and density were established from heat flow data and crustal models of old platforms where the mantle heat flow HF_M is supposed to be constant. The HF_M value was also determined to 11 ± 5 mW/m². 3. A petrological model of the old platform crust is proposed from the velocity-density models and the observed heat flow. It includes 10–12 km of acid rocks, 15–20 km of basic/metamorphic rocks and 7–10 km of basic ones. 4. Calculation of the crustal gravity effects; its substraction from the observed field gave the mantle gravity anomalies. Extensively negative anomalies have been found in the southern part of Eastern Europe (50–70 mgal) and in Western Europe (up to 200 mgal). They correlate with high heat flow and lower velocity in the uppermost mantle. 5. A polymorphic advection mechanism for deep tectonic processes was proposed as a thermal model of the upper mantle. Deep matter in active regions is assumed to be transported (advected) upwards under the crust and in its place the relatively cold material of the uppermost mantle descends. The resulting temperature distribution depends on the type of endogeneous regime, on the age and size of geostructure. Polymorphic transitions were also taken into account.     

The structure of convection in the spherical mantle with internal heating

Thermal convection in the mantle is caused by the heat transported upwards from the core and by the heat produced by the internal radioactive sources. According to the data on the heat transfer by the mantle plumes and geochemical evidence, only 20% of the total heat of the Earth is supplied to the mantle from the core, whereas most of the heat is generated by the internal sources. Along with the models that correctly allow for the internal heat sources, there are also many publications (including monographs) on the models of mantle convection that completely ignore the internal heating or the heat flux from below. In this study, we analyze to what extent these approximations could be correct. The analytical distributions of temperature and heat flux in the case of internal heating without convection and the results of the numerical modeling for convection with different intensity are presented. It is shown that the structure of thermal convection is governed by the distribution of the heat flux in the mantle but not by the heat balance, as it is typically implicitly assumed in most works. Heat production by the internal sources causes the growth of the heat flux as a function of radius. However, in the spherical mantle of the Earth, the heat flux decreases with radius due to the geometry. It turned out that with the parameters of the present Earth, both these effects compensate each other to a considerable extent, and the resulting heat flux turns out to be nearly constant as a function of radius. Since the structure of the convective flows in the mantle is determined by the distributions of heat flux and total heat flux, in the Cartesian models of the mantle convection the effective contribution of internal heating is small, and ignoring the heat flux from the core significantly distorts the structure of the convective currents and temperature distributions in the mantle.     

Evidence for low viscosity garnet-rich layers in the upper mantle

The rheological properties of upper mantle rocks play an important role in controlling the dynamics of the lithosphere and mantle convection. Experimental studies and microstructures in naturally deformed mantle rocks usually imply that olivine controls the upper mantle rheology. Here we show for the first time evidence from the geometry of folded compositional layers in mantle rocks from Western Norway that garnet-rich rocks can have lower solid-state viscosities than olivine-rich rocks. Modeling of melt-free and dry rheology of garnet and olivine confirms that the reversed viscosity contrast between garnet-rich and olivine-rich layers for this folding event can be achieved over a relatively wide range of temperatures at low stress conditions when the fine-grained garnet deforms by diffusion creep while the coarse-grained olivine deforms by dislocation creep and/or diffusion creep.In general, modeling of the fold viscosity contrast shows that in the stable subcontinental lithospheric mantle or convecting mantle such a reversed viscosity contrast can be formed due to diffusion creep processes in fine-grained garnets in a dry mantle environment or at conditions where the garnet-pyroxene layer is partially molten, i.e. close to solidus–liquidus conditions in the upper mantle. Alternatively in cold plate tectonic settings, e.g. in subduction zones, some water-weakening is a feasible mechanism to create the reversed viscosity contrast between garnet and olivine.