The mantle convection model with non-Newtonian rheology and phase transitions: The flow structure and stress fields

The mantle convection model with phase transitions, non-Newtonian viscosity, and internal heat sources is calculated for two-dimensional (2D) Cartesian geometry. The temperature dependence of viscosity is described by the Arrhenius law with a viscosity step of 50 at the boundary between the upper and lower mantle. The viscosity in the model ranges within 4.5 orders of magnitude. The use of the non-Newtonian rheology enabled us to model the processes of softening in the zone of bending and subduction of the oceanic plates. The yield stress in the model is assumed to be 50 MPa. Based on the obtained model, the structure of the mantle flows and the spatial fields of the stresses σ_xz and σ_xx in the Earth’s mantle are studied. The model demonstrates a stepwise migration of the subduction zones and reveals the sharp changes in the stress fields depending on the stage of the slab detachment. In contrast to the previous model (Bobrov and Baranov, 2014), the self-consistent appearance of the rigid moving lithospheric plates on the surface is observed. Here, the intense flows in the upper mantle cause the drift and bending of the top segments of the slabs and the displacement of the plumes. It is established that when the upwelling plume intersects the boundary between the lower and upper mantle, it assumes a characteristic two-level structure: in the upper mantle, the ascending jet of the mantle material gets thinner, whereas its velocity increases. This effect is caused by the jump in the viscosity at the boundary and is enhanced by the effect of the endothermic phase boundary which impedes the penetration of the plume material from the lower mantle to the upper mantle. The values and distribution of the shear stresses σ_xz and superlithostatic horizontal stresses σ_xx are calculated. In the model area of the subducting slabs the stresses are 60–80 MPa, which is by about an order of magnitude higher than in the other mantle regions. The character of the stress fields in the transition region of the phase boundaries and viscosity step by the plumes and slabs is analyzed. It is established that the viscosity step and endothermic phase boundary at a depth of 660 km induce heterogeneities in the stress fields at the upper/lower mantle boundary. With the assumed model parameters, the exothermic phase transition at 410 km barely affects the stress fields. The slab regions manifest themselves in the stress fields much stronger than the plume regions. This numerically demonstrates that it is the slabs, not the plumes that are the main drivers of the convection. The plumes partly drive the convection and are partly passively involved into the convection stirred by the sinking slabs.     

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...     

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.     

Plume-induced geoid anomalies from 2D axi-symmetric temperature- and pressure-dependent mantle convection models

A collection of numerical simulations of 2D axi-symmetric thermal convection is presented here. The aim is to investigate the shape of geoid anomalies and dynamic topography above a plume. The simulation is based on the Boussinesq approximation and infinite Prandtl number and is carried out in the spherical shell with strongly temperature- and depth-dependent Arrhenius-type viscosity. According to the Arrhenius law, plume models with purely depth-dependent rheology are unphysical and should be taken with care. The strongly coupled temperature- and depth-dependent viscosity enables us to better understand the plume's behavior inside the Earth.The topography and geoid anomalies produced from plumes are sensitive to rheology of the mantle and rheology of the plume; both have effects on shape and amplitude of the geoid anomalies. We determined different categories of the geoid which are related to various rheology. Depth-dependent viscosity models show a geoid with a negative sign above the plume, and temperature-dependent viscosity models depict a bell-shaped geoid. We identified different behaviors in the combined model with temperature- and pressure-dependent viscosity.     

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.     

The geoid and single-cell mantle convection

The geoid shows an antisymmetric departure from the spheroid of best fit. A single zero-elevation contour divides its surface into nearly equal strips in one of which the elevation is everywhere positive and in the other everywhere negative. These two areas are interleaved roughly like the strips covering a tennis ball. This pattern may indicate global single-cell convection in the mantle. It is argued that on this convection hypothesis, the upcurrents underlie the low-geoid strip, although the opposite view could be supported. No simple relation is to be expected between the proposed whole-mantle convection and plate motions, because other constraints act on plates and because the asthenosphere will partially decouple the whole-mantle motions from the lithosphere. However, the proposed whole-mantle convective system is consistent with rapid northwestward motion of the Pacific plate, with fast spreading of the East Pacific Rise and with slow spreading of the North Atlantic Ridge. Seismological velocity anomalies in the mantle, while highly relevant to whole-mantle convection, do not at present decide for or against the hypothesis here advanced.     

地幔对流对全球岩石圈应力产生与分布的作用

利用动力学模拟方法研究地幔对流对于大尺度岩石圈内部应力场形成的作用. 地幔物质内部的密度横向非均匀及表面板块运动引起地幔流动,并在岩石圈底部产生一个应力场. 该应力场作为面力将造成岩石圈本身变形,从而产生岩石圈内部的应力分布. 模拟计算结果表明,大部分俯冲带及大陆碰撞带区域应力均呈现挤压特征,如环太平洋俯冲带及印度－欧亚碰撞带等;而东太平洋洋脊、大西洋洋脊及东非裂谷处应力状态均表现为拉张;并且绝大多数热点位置处于应力拉张区域,这与目前对全球构造应力状态的理解是一致的. 计算的岩石圈内部最大水平主压应力的方向与观测表现出相当的一致,其结果总体上吻合得较好,然而在局部区域（例如西北太平洋的俯冲带、青藏高原等地区）存在着较大的差异. 研究表明,地幔对流是造成岩石圈内部大尺度应力状态及分布的一个重要因素.     

Small-scale mantle convection system and stress field under the Pacific plate

The existence of nonhydrostatic high-degree harmonics in the gravitational field of the earth has recently been determined using satellite and gravity observations. In this paper, we have applied the Goddard Space Flight Center GEM-8 gravity field model to calculate the small-scale mantle flow system under the Pacific plate. The resulting tectonic forces or stresses exerted by the flow currents show tensional forces under the Hawaiian Island chain and a system of latitudinal convection rolls under the East Pacific plate and are in agreement with geophysical theories.     

The effect of spatial variations in viscosity on the structure of mantle flows

According to an opinion widespread in the literature, high viscosity regions (HVRs) in the mantle always affect the structure of mantle flows, changing it in both the HVR itself and the entire mantle. Moreover, a simplified relation is often adopted according to which the flow velocity in the HVR decreases in inverse proportion to viscosity. Therefore, in order to treat a smoother value, some authors introduce a new variable equal to the product of the flow velocity and the viscosity value in a given place. On the basis of numerical modeling, this paper shows that HVRs of two types should be distinguished in the mantle. If an HVR is immobile, mantle flows actually do not penetrate it. If the viscosity increase is more than five orders, the HVR behaves as a solid and flow velocities within it almost vanish. However, if an HVR is free, it moves together with the mantle flow. Then, the general structure of flows changes weakly and flow velocities within the HVR become approximately equal to the average velocity of flows in the absence of the HVR. Horizontal layers and vertical columns differing in viscosity from the mantle behave as regions of the first type, whose flow velocities can differ by a few orders. However, even such large-scale regions as the continental lithosphere, whose viscosity is four to five orders higher than in the surrounding mantle, float together with continents at velocities comparable to mantle flows, i.e., behave as regions of the second type.     

The long wavelength component of mantle convection

Theoretical models of mantle convection predict that the major temperature fluctuations within the mantle are confined to narrow horizontal boundary layers and vertical plumes. These fluctuations result in heterogeneities in seismic body wave velocities which could, in principle, be detected by seismic tomographic techniques. However, recent tomographic images of the spatial variations of temperature in the mantle are considerably “out of focus” in that only the longest wavelength components can be resolved. To assess this partial recovery of the total tomographic image, theoretical temperature fields have been generated with a numerical model of high Rayleigh number mantle convection and then Fourier analysed in two dimensions. Upon re-synthesizing the model temperature fields, the Fourier series expansions were truncated at various levels of resolution. The truncated expansions, containing only the long wavelength components of the model temperature fields, are compared to both the complete field and the tomographic images of the mantle. At the current level of resolution it seems unlikely that seismic tomography could distinguish between layered and whole-mantle convection. Estimates, based on current tomographic data, of long wavelength fluctuations of temperature and surface topography are predicted, in the case of whole-mantle convection, to represent approximately 10% of the total temperature variation, and approximately 50% of the total topographic relief. Thus topography at the core-mantle boundary may be more accurately inferred from seismic tomography than may the characteristic lateral temperature fluctuation which drives the convective circulation.     

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.     

The generation of plate tectonics from mantle convection

In the last decade, significant progress has been made toward understanding how plate tectonics is generated from mantle dynamics. A primary goal of plate-generation studies has been the development of models that allow the top cold thermal boundary layer of mantle convection, i.e. the lithosphere, to develop broad and strong plate-like segments separated by narrow, weak and rapidly deforming boundaries; ideally, such models also permit significant strike-slip (toroidal) motion, passive ridges (i.e. pulled rather than pried apart), and self-consistent initiation of subduction. A major outcome of work so far is that nearly all aspects of plate generation require lithospheric rheologies and shear-localizing feedback mechanisms that are considerably more exotic than rheologies typically used in simple fluid-dynamical models of mantle flow. The search for plate-generating behavior has taken us through investigations of the effects of shear weakening (‘stick-slip’) and viscoplastic rheologies, of melting at ridges and low-viscosity asthenospheres, and of grain-size dependent rheologies and damage mechanics. Many such mechanisms, either by themselves or in combination, have led to self-consistent fluid-mechanical models of mantle flow that are remarkably plate-like, which is in itself a major accomplishment. However, many other important problems remain unsolved, such as subduction intiation and asymmetry, temporal evolution of plate geometry, rapid changes in plate motion, and the Archaean initiation of the plate-tectonic mode of convection. This paper presents a brief review of progress made in the plate-generation problem over the last decade, and discusses unresolved issues and future directions of research in this important area.     

Horizontal stresses in the mantle and in the moving continent for the model of two-dimensional convection with varying viscosity

Numerical experiments on studying the spatial fields and evolution of viscous overlithostatic horizontal stresses and pressure in the mantle and in the moving continent are carried out. The continent moves consistently with time-dependent forces, which act from the viscous mantle. By introducing the varying viscosity, we gain the possibility for taking into account the oceanic lithosphere and the difference between the viscosity of the upper and the lower mantle in the context of a purely viscous model. The typical overlithostatic horizontal stresses in the main part of the mantle are ±(7–9) MPa (70–90 bar); in the highly viscous regions and, particularly, in the subduction zones they are at least three times larger. The descending mantle flows in the depth interval from approximately 50 km to about 300 km are more sharply pronounced in the pressure field than in the field of horizontal stresses. At the considered stages of motion and in different parts, the continent is characterized by the following typical values of stresses: the overlithostatic pressure ranges from ?5 to +15 MPa; the horizontal overlithostatic tensile stress amounts up to ?4MPa (?40 bar); and the compressive stress in case of the overriding of the subduction zone attains +35 MPa (350 bar).     

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.     

横向黏度变化对球层中热对流的影响

本文在常黏度的基础上,加上横向黏度变化,纯粹研究了横向黏度变化对球层中热对流的影响.结果表明,横向黏度变化可以加速或阻碍球层中的热对流,并对球层中热传输方式有调节作用,但是横向黏度变化对球层中热对流的影响会限制在一定的强度范围内,并且难以改变球层中热对流的格局.横向黏度变化所产生的环型场速度占总速度场的比例最多为十几个百分点,难以解释观测到的地表板块运动中的环型场能量大小.     

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(10³) to O(10⁴), 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.     

Lateral variation in upper mantle viscosity: role of water

Differences in the viscosity of the earth's upper mantle beneath the western US (∼10¹⁸-10¹⁹ Pa s) and global average values based on glacial isostatic adjustment and other data (∼10²⁰-10²¹ Pa s) are generally ascribed to differences in temperature. We compile geochemical data on the water contents of western US lavas and mantle xenoliths, compare these data to water solubility in olivine, and calculate the corresponding effective viscosity of olivine, the major constituent of the upper mantle, using a power law creep rheological model. These data and calculations suggest that the low viscosities of the western US upper mantle reflect the combined effect of high water concentration and elevated temperature. The high water content of the western US upper mantle may reflect the long history of Farallon plate subduction, including flat slab subduction, which effectively advected water as far inland as the Colorado Plateau, hydrating and weakening the upper mantle.     

Hotspots,mantle convection and plate tectonics: A synthetic calculation

A model is developed that unifies vigorous hotspots with global-scale mantle convection and plate tectonics. The convection dynamics are assumed to generate flow patterns that emerge as closely packed polygonal cells in approaching the asthenosphere, and whose geometry is completely determined by a defining set of vigorous hotspots. Overlying viscously coupled rigid plates are driven with unique velocities (Euler vectors) at which the area integral of the shear forces is zero; these velocities are dynamically stable. The computed plate velocities, resulting from convection based on 15 hotspots, are compared with the velocities of plate motion models AM1-2 (Minster andJordan, 1978) and HS-NUVEL1 (Gripp andGordon, 1990), which combine transform fault geometries, magnetic anomalies and seismic data. The comparison shows a striking agreement for a majority of the plates. Geophysical implications of this numerical exercise are discussed.     

Plate motions and the viscosity structure of the mantle — Insights from numerical modelling

We use sulfur (S) isotope signatures within ancient sediments and a photochemical model of sulfur dioxide (SO₂) photolysis to interpret the evolution of the atmosphere over the first half of Earth's history. A decrease in mass-independent sulfur isotope fractionation has been reported in Archean rocks deposited between ~ 2.7 Ga and ~ 3.2 Ga, and is reinforced by new S isotope data that we report here. This pattern has been interpreted by some as evidence that atmospheric oxygen (O₂) was elevated during this time. In this paper, we argue against that conclusion, and show that it is inconsistent with other geochemical data. In its place, we propose a new model that can explain the sulfur isotope record that can also avoid conflicts with independent constraints on O₂ and account for concurrent glacial deposits. Specifically, we suggest that prior to the rise of O₂ excursions in the sulfur isotope record were modulated by the thickness of an organic haze. This haze would have blocked the lower atmosphere from the UV fluxes responsible for the anomalous sulfur photochemistry and would have caused an anti-greenhouse effect that triggered the glaciations. We used a photochemical model to verify that a haze could have affected the isotopic signal in this manner, and to examine how changes in atmospheric methane (CH₄) and carbon dioxide (CO₂) concentrations could have controlled haze thickness. Finally, we combined the resulting relationships with climate models and sulfur isotope and glacial records to deduce a new evolutionary sequence for Archean climate, surface chemistry, and biology.