Numerical model for the generation of the ensemble of lithospheric plates and their penetration through the 660-km boundary

In the kinematic theory of lithospheric plate tectonics, the position and parameters of the plates are predetermined in the initial and boundary conditions. However, in the self-consistent dynamical theory, the properties of the oceanic plates (just as the structure of the mantle convection) should automatically result from the solution of differential equations for energy, mass, and momentum transfer in viscous fluid. Here, the viscosity of the mantle material as a function of temperature, pressure, shear stress, and chemical composition should be taken from the data of laboratory experiments. The aim of this study is to reproduce the generation of the ensemble of the lithospheric plates and to trace their behavior inside the mantle by numerically solving the convection equations with minimum a priori data. The models demonstrate how the rigid lithosphere can break up into the separate plates that dive into the mantle, how the sizes and the number of the plates change during the evolution of the convection, and how the ridges and subduction zones may migrate in this case. The models also demonstrate how the plates may bend and break up when passing the depth boundary of 660 km and how the plates and plumes may affect the structure of the convection. In contrast to the models of convection without lithospheric plates or regional models, the structure of the mantle flows is for the first time calculated in the entire mantle with quite a few plates. This model shows that the mantle material is transported to the mid-oceanic ridges by asthenospheric flows induced by the subducting plates rather than by the main vertical ascending flows rising from the lower mantle.     

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.     

Displacements of ridges and subduction zones in the models of mantle convection with lithospheric plates

In the existing kinematic theory of the tectonics of lithospheric plates, the position and the parameters of plates are assigned a priori in the initial and boundary conditions. However, in the self-consistent dynamic theory, the properties of oceanic plates (as well as the structure of the mantle’s convection) should appear automatically as the solution of the differential equations of energy, mass, and momentum transfer for a viscous fluid. In this case, the viscosity of the mantle’s substance as a function of temperature, pressure, shear stress, and chemical composition must be taken from the data of laboratory measurements. In the present work, the results of the numerical solution of the equations of convection are presented in the problem formulation mentioned above on a simple model of heated viscous fluid with properties that correspond to the mantle’s substance. In this case, to reveal the main reason for the generation of plates and their influence on the convection, a number of simplifications are introduced; in particular, temperature variations in the viscosity in the mantle are disregarded. In spite of the undertaken simplifications, the models show how the rigid lithosphere can be split into separate plates immersed in the mantle, how in the course of evolution the sizes of plates and their number can change, and how in this case the ridges and subduction zones can be displaced.     

How valid are dynamic models of subduction and convection when plate motions are prescribed?

A detailed comparison between fully dynamic and kinematic plate formulations has been made in models of mantle convection. Plate velocity is computed self-consistently from fully dynamic plate models with temperature- and stress-dependent viscosity and preexisting mobile faults. In fully dynamic models, the flow is driven solely by internal buoyancy, while in kinematic models the flow is driven by a combination of the prescribed surface velocity and internal buoyancy. Only a temperature-dependent viscosity, close to the effective viscosity determined from the fully dynamic models, is used in the kinematic models. The two types of models give very similar temperature structures and slab evolutionary histories when the effective viscosity and surface velocity are nearly identical. In kinematic plate models, the additional work introduced by the prescribed velocity boundary condition is apparently dissipated within the lithosphere and has little influence on the convection under the lithosphere. In models with periodic lateral boundary conditions, slabs sink into the lower mantle at an oblique angle and this contrasts with the vertical sinking which occurs with reflecting boundary conditions. Models show that we can simulate fully dynamic models with kinematic models under either periodic boundary conditions or reflecting boundary conditions.     

A quantitative kinematic theory of convection currents in the earth's mantle

Summary Based on a system of structurally simple postulations the kinematics of mantle convection is derived. With regard to the strain-stress relations valid in the mantle and the energy source of the convection the theory is without any presumptions. In compliance with recent hydrodynamic investigations the flow is introduced rather as roll currents than as a hexagonal cell pattern. From the feasible types of current a theoretical topography is derived which is in quantitative agreement with the observed one. Also the distribution of the seismic discontinuities substantiates the validity of the expression. Finally, some suggestions are given for a hydrodynamic theory of mantle convection. This paper contains that part of the theory which is necessary for testing the calculations, while the relationship of the theory to other geological and geophysical problems are dealt with by the author [26]²).     

On the aspect ratios of two-layer mantle convection

Some dynamic implications of separate convection systems in the upper and the lower mantle of the Earth are investigated. It is shown that the horizontal scale of convection cells in the lower mantle is likely to determine the scale of flow in the upper mantle. This does not preclude the nearly independent realization of convection cells with horizontal dimensions comparable to the depth of the upper mantle.     

利用地震层析成像数据计算地幔对流新模型的探讨

假设地幔地震层析成像数据对应的地幔横向不均匀结构是地幔热对流的结果. 将地震层析成像数据转化为地幔温度(或密度)不均匀分布，考虑热流体动力学的三个基本方程，顾及热输运方程中的非线性项，直接将地震层析成像转化的地幔温度不均匀分布作为内部荷载引入基本方程, 反演计算地幔对流. 本文在利用地震层析成像数据计算地幔对流模型的新理论和方法的基础上，用SH12WM13地震层析成像模型数据，计算了全球地幔对流格局. 结果表明，对流格局不仅依赖地震层析成像数据，而且在很大程度上受地幔动力学框架、热动力参数和边界条件的所确定的系统响应函数的影响. 显示了地幔中复杂的对流格局，特别是区域性层状对流以及多层对流环可能在地幔中存在的现象.     

The Myasnikov global theory of the evolution of planets and the modern thermochemical model of the Earth’ls evolution

The thermochemical model of the authors is shown to be naturally related to the general theory of V.P. Myasnikov. A heterogeneous modification of this homogeneous theory is described in light of the present ideas on the differentiation of the mantle substance at the boundary with the core and its eclogitization during submersion from the outer boundary and at the endothermic phase transition at a depth of 670 km. The Earth’ls evolution from an initial hot state is numerically modeled. The evolution is shown to start with an abrupt mantle overturn followed by a long period of steady evolution. Global mantle overturns recur a few times, gradually weaken, and are transformed into regional avalanches. The spatial configuration of overturns is represented by a predominant funnel-shaped sink and a few (three to five) ascending superplumes, which convincingly explains the causes of the formation of supercontinents, the opening of oceans, and the observed asymmetry of the planet. The times of overturns remarkably correlate with geological data on the existence of supercontinents. The processes of core growth, mantle cooling, and crust formation exhibit a clearly expressed stepwise behavior. The supplementation of the endothermic phase transition by chemical transformations favors the overcoming of the phase barrier between the upper and lower mantle, enhances the nonlinearity of mantle convection, and imparts a heterocyclic pattern to the process of evolution. It is shown that the lower mantle plume of chemical origin is fragmented by the phase transition into parts that, interacting with the thermal convection, generate a system of upper mantle plumes. This modeling provides an explanation of the coeval systems of oceanic plateaus and continental traps observed on the surface.     

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.     

Plume Mode of Thermal Convection in the Earth’s Mantle

In our previous works, based on numerical models, it was shown that under certain conditions a hot material can rise in portions in the tails of thermal mantle plumes. The spectrum of these pulsations can correspond to the observed spectra of catastrophic hotspot eruptions. Since most of the existing numerical models of thermal convection for the mantle of the present Earth do not reveal these pulsations, in this work, we analyze the physical cause and initiation conditions of pulsations of thermal plumes. The results of a numerical solution of the thermal convection equations for a material with varying parameters in the extended Boussinesq approximation are presented. It is shown how the structure of the convection is transformed with the increase of convection intensity. At the Rayleigh numbers Ra > 10⁶, convection becomes unsteady, and the configuration of the ascending and descending flows changes. The new flow emerging at the mantle bottom acquires a mushroom shape with a head and a tail. After the rise of the plume’s head to the surface, the tail remains in the mantle in the form of a quasi-stationary hot steam. It turns out that at Ra ~ 5 × 10⁷, the thermal mantle plume becomes pulsating and its tail is in fact a heated channel through which the hot material rises in successive portions. At the Rayleigh numbers Ra > 5 × 10⁸, the tail of the thermal plume breaks and the plume becomes a regular conveyor of separate ascending portions of the hot material, which are referred to as thermals. Thus, thermal convection with pulsating plumes takes place at the transitional stage from the regime of quasi-stationary plumes to the regime of thermals.     

A conservative form of the deep convection equations and the efficiency of heat energy conversion in a polytropic mantle model

A wide class of equations is defined for a high pressure and subcritical temperature range of a fluid state whose thermodynamic properties enable the construction of a polytropic model of the mantle. A variant of deep convection equations of the Ogura and Phillips type is substantiated in terms of the polytropic mantle model. The proposed system of the deep convection equations includes fluctuation of the generalized potential temperature, has a quasi-incompressible form, and is transformed into Mihaljan’s system of shallow convection equations with a decrease in the layer depth. This circumstance is of great importance because it validates the use of the same dimensionless parameters as in the shallow convection model. The advantage of the proposed variant of the deep convection equations is its complete conservatism, which allows one to gain constraints on the efficiency of energy conversion in deep mantle processes and the thermal energy power expended on the generation rate of the convection kinetic energy and associated processes. This power is shown to be of the order of half the geothermal flux measured on the Earth’s surface.     

Convection of a fluid with strongly temperature and pressure dependent viscosity

Plate tectonics on the Earth is a surface manifestation of convection within the Earth’s mantle, a subject which is as yet improperly understood, and it has motivated the study of various forms of buoyancy-driven thermal convection. The early success of the high Rayleigh number constant viscosity theory was later tempered by the absence of plate motion when the viscosity is more realistically strongly temperature dependent, and the process of subduction represents a continuing principal conundrum in the application of convection theory to the Earth. A similar problem appears to arise if the equally strong pressure dependence of viscosity is considered, since the classical isothermal core convection theory would then imply a strongly variable viscosity in the convective core, which is inconsistent with results from post-glacial rebound studies. In this paper we address the problem of determining the asymptotic structure of high Rayleigh number convection when the viscosity is strongly temperature and pressure dependent, i.e. thermobaroviscous. By a method akin to lid-stripping, we are able to extend numerical computations to extremely high viscosity contrasts, and we show that the convective cells take the form of narrow, vertically-oriented fingers. We are then able to determine the asymptotic structure of the solution, and it agrees well with the numerical results. Beneath a stagnant lid, there is a vigorous convection in the upper part of the cell, and a more sluggish, higher viscosity flow in the lower part of the cell. We then offer some comments on the possible meaning and interpretation of these results for planetary mantle convection.     

Long wavelength thermal convection between non-conducting boundaries

Non-linear Rayleigh-Bénard convection in a fluid layer is considered as a model of convection in the Earth's upper mantle. Previous studies have shown that when the temperature is held fixed at one of the boundaries of the layer, convection takes place in cells of width of the order of the layer depth or less. We investigate the effects of a different thermal boundary condition, in which the flux of heat is held fixed on both layer boundaries; then if this flux is just greater than that required for the onset of convection, motion takes place on horizontal scales much greater than the layer depth. An analytical treatment of the equations, based on an expansion in the depth-to-width ratio of the cells, shows that cells of a definite horizontal scale are the fastest growing according to linearised theory, but that these cells are unstable to ones of larger wavelength than themselves. Thus the dominant wavelength lengthens with time. The results hold whether the heat flux is generated internally of comes from beneath the layer. These results produce flow patterns similar to those found when the heat flux is much greater than the critical value. The results have important consequences for the understanding of mantle convection.     

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…     

海沟后退对地幔对流的影响

本文分析了有海底扩张无海沟后退、有海沟后退无海底扩张以及海底扩张与海沟后退共存等３种情况下，俯冲板片运动与海沟迁移的关系．用幂律流体有限元方法计算海沟后退对地幔对流的影响．地幔有效粘度除依赖应力偏张量的第二不变量以外，尚与温度、压力（含流体静压力和流动压力）有关．计算表明，对流环、高流动负压区以及低粘区的个数和位置，均受控于海沟是否后退以及海底是否扩张；温度场与海沟后退无明显关系．流动压力对形成洋中脊和弧后火山、驱动地幔对流以及维持板片的倾斜角度都具有重要意义．     

上地幔密度异常驱动小尺度对流及实验模型

建立了由密度异常驱动上地幔小尺度对流的数学 物理模型, 发展了利用地震层析成像数据反演上地幔小尺度对流的基本理论和方法. 该模型建立在三维直角坐标系框架上, 假设地震层析成像所显示的地震波速度异常对应于上地幔物质密度异常, 而该密度异常反映了上地幔小尺度热对流系统的温度异常场. 模型首先将地震层析成像确定的地震波速度异常转换为密度异常, 并视其为对流的驱动力; 进而利用三维傅立叶变换, 在波数域内, 在给定的边界条件下, 求解控制流体行为的运动方程和连续性方程, 最后求得对流的流场. 为检验本研究提出的理论和方法的有效性, 本文使用了两个简单的实验模型: 热体和冷体模型; 俯冲断离( break off)板片模型, 计算了其驱动的地幔流场. 结果表明, 本文提供的理论和方法, 可以直接应用于与区域岩石层构造动力学相关的上地幔小尺度对流的研究.      

Pervasion degree of convective mixing and investigations of mantle mixing

Geodynamic studies have shown that mantle convection is like a giant blender to make the original heterogeneous mantle mixing and homogenizing. However,some models,especially from geochemical data show that the modern mantle may still contain a number of reservoir bodies with different chemical composition. Then,the modern mantle is homogeneous? Authors have defined a box replacement degree of convective mantle mixing and pervasion degree of convective mantle mixing (that equals to initial density of tracin...     

利用多种地球物理观测资料直接反演地幔对流模型

假定地幔为一个均匀的、粘滞系数为常数、同时均匀分布放射性热源的流体球层,其内部存在的对流则由流体力学3个基本方程:运动方程、能量方程和连续性方程确定.如果假定地幔处于低瑞利数的状态（临界瑞利数1.5倍左右）,那么上述方程中的非线性项可以忽略不计.作为一类可能的模型,本文计算一组用6个边界条件确定6个未知数的线性方程组.这些条件包括板块绝对运动极型场、地球大地水准面异常和地震层析结果提供的地幔密度分布横向不均匀相应的“刚性地球”水准面异常等.模型计算表明:1.地幔中流体运动格局不仅受地幔热动力学参数（瑞利数）控制,而且强烈地受边界条件的影响.2.若不限定下边界为等温边界,则上、下地幔之间并不呈现出活动性明显差异;但是在模型瑞利数加大到一定值时,核-幔边界附近将出现一些局部的小尺度对流环.3.当模型瑞利数从很小增加时,对流格局将发生变化,这些格局可能反应由地幔热动力学参数决定的地幔固有特性.4.当瑞利数为50000和80000时,核-幔边界形变与PcP波得到的结果吻合较好.     

A benchmark comparison of numerical methods for infinite Prandtl number thermal convection in two-dimensional Cartesian geometry

Abstract

A comparison is made between seven different numerical methods for calculating two-dimensional thermal convection in an infinite Prandtl number fluid. Among the seven methods are finite difference and finite element techniques that have been used to model thermal convection in the Earth's mantle. We evaluate the performance of each method using a suite of four benchmark problems, ranging from steady-state convection to intrinsically time-dependent convection with recurring thermal boundary layer instabilities. These results can be used to determine the accuracy of other computational methods, and to assist in the development of new ones.