Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations

We address a “multi-reflection” approach to model Dirichlet and Neumann time-dependent boundary conditions in lattice Boltzmann methods for arbitrarily shaped surfaces. The multi-reflection condition for an incoming population represents a linear combination of the known population solutions. The closure relations are first established for symmetric and anti-symmetric parts of the equilibrium functions, independently of the nature of the problem. The symmetric part is tuned to build second- and third-order accurate Dirichlet boundary conditions for the scalar function specified by the equilibrium distribution. The focus is on two approaches to advection and anisotropic-dispersion equations (AADE): the equilibrium technique when the coefficients of the expanded equilibrium functions match the coefficients of the transformed dispersion tensor, and the eigenvalue technique when the coefficients of the dispersion tensor are built as linear combinations of the eigenvalue functions associated with the link-type collision operator. As a particular local boundary technique, the “anti-bounce-back” condition is analyzed. The anti-symmetric part of the generic closure relation allows to specify normal flux conditions without inversion of the diffusion tensor. Normal and tangential constraints are derived for bounce-back and specular reflections. The bounce-back closure relation is released from the non-physical tangential flux restriction at leading orders. Solutions for the Poisson equation and for convection–diffusion equations are presented for isotropic/anisotropic configurations with specified Dirichlet and Neumann boundary conditions.     

Multiple anisotropic collisions for advection–diffusion Lattice Boltzmann schemes

This paper develops a symmetrized framework for the analysis of the anisotropic advection–diffusion Lattice Boltzmann schemes. Two main approaches build the anisotropic diffusion coefficients either from the anisotropic anti-symmetric collision matrix or from the anisotropic symmetric equilibrium distribution. We combine and extend existing approaches for all commonly used velocity sets, prescribe most general equilibrium and build the diffusion and numerical-diffusion forms, then derive and compare solvability conditions, examine available anisotropy and stable velocity magnitudes in the presence of advection. Besides the deterioration of accuracy, the numerical diffusion dictates the stable velocity range. Three techniques are proposed for its elimination: (i) velocity-dependent relaxation entries; (ii) their combination with the coordinate-link equilibrium correction; and (iii) equilibrium correction for all links. Two first techniques are also available for the minimal (coordinate) velocity sets. Even then, the two-relaxation-times model with the isotropic rates often gains in effective stability and accuracy. The key point is that the symmetric collision mode does not modify the modeled diffusion tensor but it controls the effective accuracy and stability, via eigenvalue combinations of the opposite parity eigenmodes. We propose to reduce the eigenvalue spectrum by properly combining different anisotropic collision elements. The stability role of the symmetric, multiple-relaxation-times component, is further investigated with the exact von Neumann stability analysis developed in diffusion-dominant limit.     

A lattice BGK model for advection and anisotropic dispersion equation

This paper presents a lattice Boltzmann model (LBM) for 2-D advection and anisotropic dispersion equation (AADE) based on the Bhatnagar, Gross and Krook (BGK) model. In the proposed model, the particle speed space is discretized using a rectangular lattice that has four speeds in nine directions, and the single relaxation time is assumed to be directionally dependent. To ensure that the collision is mass-invariant when the relaxation time is directionally dependent, the concentration is calculated from a weighted summation of the particle distribution functions. The proposed model was verified against benchmark problems and the finite difference solution of solute transport with spatially variable dispersion coefficients and non-uniform velocity field. The significant results are that it conserves mass perfectly and offers accurate and efficient solutions for both dispersion-dominated and advection-dominated problems.     

Effects of anisotropic diffusion on onset of rotating magnetoconvection in plane layer; stationary modes

ABSTRACT

Turbulence in the Earth's outer core not only increases all diffusive coefficients, but it can lead to their anisotropic properties. Therefore, the model of rotating magnetoconvection in horizontal plane layer rotating about vertical axis and permeated by homogeneous horizontal magnetic field, influenced by anisotropic diffusivities, viscosity and thermal diffusivity, is advanced by considering the magnetic diffusivity as anisotropic too. The case of full anisotropy, i.e. all coefficients anisotropic, is compared with both the case possessing isotropic diffusion coefficients and the case of partial anisotropy, i.e. mixed case with isotropic and anisotropic diffusive coefficients (viscosity and thermal diffusivity anisotropic and magnetic diffusivity isotropic). The existence and preference of instabilities is sensitive to all non-dimensional parameters, as well as on anisotropic parameter, the ratio of horizontal and vertical diffusivities. Two types of anisotropy, BM (introduced by Braginsky and Meytlis) and SA (stratification anisotropy) are studied. BM as well as SA were applied by ?oltis and Brestenský to the study of the partial anisotropy; this study is extended, in this paper, to full anisotropy cases (full SA and full BM) and it is shown that the style of convection given by the onset of stationary modes is more affected by anisotropic diffusivities in BM than in SA anisotropy. The important influence of strong anisotropies in the Earth's core dynamics is stressed.     

Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically-conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the evolution of weakly nonlinear mean perturbations are derived under the assumption that the α-effect is insignificant in the leading-order (e.g. due to a symmetry of the system). The mean-field equations generalise the standard equations of hydromagnetic convection: New terms emerge – a second-order linear operator representing the combined eddy diffusivity and quadratic terms associated with the eddy advection. If the perturbed CHM regime is nonsteady and insignificance of the α-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a nonlocal pseudodifferential operator. If the perturbed CHM state is almost symmetric, α-effect terms appear in the mean-field equations as well. Near a point of a symmetry-breaking bifurcation, cubic nonlinearity emerges in the equations. All the new terms are in general anisotropic. A method for evaluation of their coefficients is presented; it requires solution of a significantly smaller number of auxiliary problems than in a straightforward approach.     

岩石圈流变强度与中国大陆构造运动关系的探讨

以GPS观测资料和地震学研究成果为约束，针对不同流变参数的中国大陆岩石圈模型，数值模拟了岩石粘度与中国大陆板块边界作用强度的关系，探讨了陆-陆碰撞对中国大陆分层岩石圈运动的驱动机制．给出了陆-陆碰撞驱动力、附加地形与山根浮力及热浮力对中国大陆构造运动的驱动特点．印度板块、太平洋板块和菲律宾板块对中国大陆驱动的边界作用强度之比约是4：1．25：1，所引起的水平主压应力主要集中在坚硬岩石层；而附加地形等垂直方向作用力在水平方向产生的最大主压应力则主要集中在软弱岩石层．这种垂直方向上的作用力在高原南部地区阻碍陆-陆碰撞向北的推挤运动，在高原东北部增加对其它块体的推挤作用。     

Anisotropic turbulent thermal diffusion and thermal convection in a rapidly rotating fluid sphere

Estimates of the molecular values of magnetic, viscous and thermal diffusion suggest that the state of the Earth’s core is turbulent and that complete numerical simulation of the geodynamo is not realizable at present. Large eddy simulation of the geodynamo with modelling of the sub-grid scale turbulence must be used. Current geodynamo models effectively model the sub-grid scale turbulence with isotropic diffusivities larger than the molecular values appropriate for the core. In the Braginsky and Meytlis (1990) picture of core turbulence the thermal and viscous diffusivities are enhanced up to the molecular magnetic diffusivity in the directions of the rotation axis and mean magnetic field. We neglect the mean magnetic field herein to isolate the effects of anisotropic thermal diffusion, enhanced or diminished along the rotation axis, and explore the instability of a steady conductive basic state with zero mean flow in the Boussinesq approximation. This state is found to be more stable (less stable) as the thermal diffusion parallel to the rotation axis is increased (decreased), if the transverse thermal diffusion is fixed. To examine the effect of simultaneously varying the diffusion along and transverse to the rotation axis, the Frobenius norm is used to control for the total thermal diffusion. When the Frobenius norm of the thermal diffusion tensor is fixed, it is found that increasing the thermal diffusion parallel to the rotation axis is destabilising. This result suggests that, for a fixed total thermal diffusion, geodynamo codes with anisotropic thermal diffusion may operate at lower modified Rayleigh numbers.     

Energy geodynamo parameters compatible with analytical,numerical, paleomagnetic models and observations

A hydromagnetic dynamo is only possible at a sufficiently powerful convection. In the Earth’s core, it is probably the nonthermal convection very much in excess of its critical level with the molecular transporr coefficients. However, in the case of medium- or large-scale fields, the critical energy level caused by the turbulent tranport coefficients is likely to be slightly below the actual level. This probably explains both the 22-year success of this type of simplified geodynamo models and the energy scaling laws for hydromagnetic fields, which generalize these models. Also the review of energy-dependent analytical and observational estimates of vortex fields, hydromagnetic scale sizes, and velocities in the core is presented. These typical parameters are partly in a new way linked to the observed and more ancient magnetic variations. New, albeit, simplified and self-evident, substantiation is given to the paleomagnetic hypothesis about the predominance of the axial dipole under a certain time averaging. In (Pozzo et al., 2012) and more recent works, it is shown that the adiabatic heat flow and electrical conductivity in the Earth’s core are severalfold higher than the generally accepted estimates. Here, the dynamo supporting Braginsky’s convection (Braginsky, 1963) (under the crystallization of the heavy fraction of a liquid onto the solid core) started less than 1 Ga ago, whereas the more ancient geodynamo was supported by the compositional convection of another type. The known mechanisms implementing this convection, which differ by the scenarios of magnetic evolution, are reviewed. This may help identify the sought mechanism through the most ancient paleomagnetic estimates of the field’s intensity and through the numerical models. The probable mechanisms of generation and their absence for the primordial and recent magnetic field of the studied terrestrial planets are discussed.     

Numerical studies of non-Newtonian mantle convection

Numerical model computations have been carried out to determine how the stress-dependence of non-Newtonian viscosity affects the flow structure of thermal convection. The viscosity laws have been chosen in accordance with present knowledge of upper mantle rheology, based on the diffusion and dislocation creep laws of olivine. The results show that there are important differences between the structures of Newtonian and non-Newtonian convection. While the Newtonian models are insufficient in some respects, the non-Newtonian solutions can explain the characteristics of the real mantle flow. However, this may require a faster plastic deformation than power law dislocation creep, at least in the high-stress regions of the mantle, e.g. at the active plate margins.     

Geophysical flows under location uncertainty,Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction, an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.     

Wave propagation through ionospheric plasma in presence of time-varying random irregularities

Summary The dielectric tensor of a time-varying anisotropic electron-plasma has been studied with the consideration of the effect of collision. The tripple splitting of the electromagnetic waves is obtained from the dispersion relation which gives the two usual magnetoionic modes in the limiting case. The randomness of electron number density has been evaluated by the help of field variables and also by the nature of fluctuation of dielectric tensor.     

A weakly nonlinear stability of centrally symmetric magnetohydrodynamic systems to perturbations involving large scales

The problem of weakly nonlinear stability of 3-D centrally symmetric magnetohydrodynamic systems to perturbations involving large scales is considered. It is assumed that large space-time scales are absent in the magnetohydrodynamic state under study, which is stable with respect to perturbations whose scales are as small as those of the state itself. Equations derived by asymptotic methods for average fields of perturbations generalize the Navier-Stokes and magnetic induction equations. They include a combined eddy diffusion operator, generally anisotropic and not necessarily negative definite, and additional quadratic terms. An effective method is proposed for the calculation of coefficients of eddy diffusion and advection in equations governing average fields.     

Non-isothermal soil water transport and evaporation

A detailed model was formulated to describe the non-isothermal transport of water in the unsaturated soil zone. The model consists of the coupled equations of mass conservation for the liquid phase, gas phase and water vapor and the energy conservation equation. The water transport mechanisms considered are convection in the liquid phase, and convection, diffusion and dispersion of vapor in the gas phase. The boundary conditions at the soil–atmosphere interface include dynamical mass flux and energy flux that accounts for radiation transport. Comparison of numerical simulations results with published experimental data demonstrated that the present model is able to describe water and energy transport dynamics, including situations of low and moderate soil moisture contents. Analysis of field studies on soil drying suggests that that dispersion flux of the water vapor near the soil surface, which is seldom considered in soil drying models, can make a significant contribution to the total water flux.     

Solution of the transport equations using a moving coordinate system

A convection-diffusion equation arises from the conservation equations in miscible and immiscible flooding, thermal recovery, and water movement through desiccated soil. When the convection term dominates the diffusion term, the equations are very difficult to solve numerically. Owing to the hyperbolic character assumed for dominating convection, inaccurate, oscillating solutions result. A new solution technique minimizes the oscillations. The differential equation is transformed into a moving coordinate system which eliminates the convection term but makes the boundary location change in time. We illustrate the new method on two one-dimensional problems: the linear convection-diffusion equation and a non-linear diffusion type equation governing water movement through desiccated soil. Transforming the linear convection diffusion equation into a moving coordinate system gives a diffusion equation with time dependent boundary conditions. We apply orthogonal collocation on finite elements with a Crank-Nicholson time discretization. Comparisons are made to schemes using fixed coordinate systems. The equation describing movement of water in dry soil is a highly non-linear diffusion-type equation with coefficients varying over six orders of magnitude. We solve the equation in a coordinate system moving with a time-dependent velocity, which is determined by the location of the largest gradient of the solution. The finite difference technique with a variable grid size is applied, and a modified Crank-Nicholson technique is used for the temporal discretization. Comparisons are made to an exact solution obtained by similarity transformation, and with an ordinary finite difference scheme on a fixed coordinate system.     

大地电磁二维各向异性反演及其在青藏高原北部的应用

地球介质的电性特征在不同深度上都可能表现出各向异性,识别电各向异性有助于深入理解地球内部物质状态及变形环境,促进对动力学模型的理解.本研究在已实现的任意电各向异性大地电磁有限差分正演算法的基础之上,通过构建常规的各向异性反演目标函数,完成对目标函数及其梯度的计算.借助成熟的非线性共轭梯度反演技术,实现了对大地电磁各向异性介质的反演过程.在此基础上,构建了二维方位各向异性理论模型,来验证反演过程的稳定性.理论模型的反演结果直观地显示了大地电磁各向异性反演对模型参数中垂直电导率恢复上的局限性;在各向异性结构分析上需要将电导率张量作为整体考虑.对青藏高原北部东昆仑—柴达木盆地西段的实测大地电磁数据进行反演,并对比早期二维各向同性反演结果发现,各向同性结构中位于祁漫塔格山脉下方上地幔顶部存在低阻异常带,在相同的位置,各向异性结构中表现出明显的方位各向异性,其各向异性低阻主轴电阻率指示的剪切带走向与地表造山带走向不一致,这暗示该处的应力环境复杂,除了受印度—欧亚板块碰撞控制外,还可能受到邻近阿尔金走滑断裂带左行剪切运动的影响.

     

地震震源运动学参数获取方法研究进展

本文基于从简单到复杂的各向同性点源模型、质心矩张量模型、有限矩张量模型和有限震源滑动分布模型4种震源运动学模型,概述了相应的研究进展.     

New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity

 Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy, that is, stochastic processes with variograms that have direction dependent correlation lengths and direction independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models can be separated into a product of one-dimensional functions oriented along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy, i.e., the ratio(s) of the correlation lengths. This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the covariance tensor. The procedure is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions. In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its implementation with numerical simulations.     

Small-scale magnetic buoyancy and magnetic pumping effects in a turbulent convection

We determine the nonlinear drift velocities of the mean magnetic field and nonlinear turbulent magnetic diffusion in a turbulent convection. We show that the nonlinear drift velocities are caused by three kinds of the inhomogeneities; i.e., inhomogeneous turbulence, the nonuniform fluid density and the nonuniform turbulent heat flux. The inhomogeneous turbulence results in the well-known turbulent diamagnetic and paramagnetic velocities. The nonlinear drift velocities of the mean magnetic field cause the small-scale magnetic buoyancy and magnetic pumping effects in the turbulent convection. These phenomena are different from the large-scale magnetic buoyancy and magnetic pumping effects which are due to the effect of the mean magnetic field on the large-scale density stratified fluid flow. The small-scale magnetic buoyancy and magnetic pumping can be stronger than these large-scale effects when the mean magnetic field is smaller than the equipartition field. We discuss the small-scale magnetic buoyancy and magnetic pumping effects in the context of the solar and stellar turbulent convection. We demonstrate also that the nonlinear turbulent magnetic diffusion in the turbulent convection is anisotropic even for a weak mean magnetic field. In particular, it is enhanced in the radial direction. The magnetic fluctuations due to the small-scale dynamo increase the turbulent magnetic diffusion of the toroidal component of the mean magnetic field, while they do not affect the turbulent magnetic diffusion of the poloidal field.