Mathematical Sciences: Study of Strongly Chaotic Thermal Convection in the Earth's Mantle: Analytical, Computational and Visualization Perspectives

应用数学(流体动力学)和地球物理学(地球物理流体动力学)之间的这种协作努力的主要目的是利用分析、数值和可视化技术，重点研究应用于地幔的强混沌三维对流。这位研究人员和他的同事David袁扩展了通常的常物性模拟的正则公式，以包括真实的随深度变化的热膨胀系数和粘性导热系数，以及多个内部相变和内部热产生。相图中三相点的存在和更准确的温度相关粘度也被纳入数学公式中。研究人员使用新开发的本征正交分解(POD)和线性随机估计(LSE)技术来压缩大规模三维数值模拟产生的海量数据。POD和LSE技术也被用来描述和理解相关相干结构的时空动力学，例如热升热羽和冷沉降板。研究人员研究了相关的地球物理问题，如地幔大规模环流的根本原因，依赖深度的物质性质引起的上升流的相对稳定性，多个相变造成的重力不稳定性，以及依赖温度的粘度对地幔动力学的影响。他们还在底部用强加的依赖于时间的边界条件进行模拟，以说明核心的冷却，以便通过从非常高的瑞利数开始研究热历史，例如10**8，然后通过冷却慢慢降低瑞利数。研究人员使用现代数学理论和数值技术来研究地球内部的三维动力学。去年出现的一个重要问题是，由于相变，地幔中可能会出现重力不稳定。这种不稳定性导致了超级烟柱的周期性喷发和相关的强烈火山活动。合作者是第一批对这种现象进行三维建模的小组之一。他们计划通过结合更现实的流动定律和热力学来进一步研究这一点。这种重力不稳定现象还有很多方面需要探索，因为越来越多的证据表明，过去的海沟地点与从地震层析成像推断的下地幔冷异常之间的相关性表明，这种地幔不稳定性可能发生在过去1亿年。认识到这些不稳定性可能会改变对稳态过程作用的传统看法。他们还研究了地震学揭示的下地幔相干大尺度流动结构的性质，以及这些不稳定对地球和类地行星的长期热演化有什么影响。仅仅在几年前，地球物理学才引入了羽流碰撞产生的热吸引子的概念。他们预计，由地幔对流动态维持的大规模相干结构的工作也可能影响对地球物理行为的传统看法。

The main thrust of this collaborative effort between applied mathematics (fluid dynamics) and geophysics (geophysical Fluid dynamics) is a focussed study of strongly chaotic three-dimensional convection as applied to the Earth's mantle, using analytical, numerical and visualization techniques. The investigator and his colleague David Yuen extend the usual canonical formulation of constant property simulation to include realistic depth-dependent thermal expansivity and viscosity thermal conductivity along with multiple internal phase transitions and internal heat generation. The presence of a triple point in the phase diagram and a more accurate temperature-dependent viscosity are also incorporated into the mathematical formulation. The investigators use the newly developed proper orthogonal decomposition (POD) and linear stochastic estimation (LSE) techniques to compress the enormous amount of data resulting from large-scale three-dimensional numerical simulations. The POD and LSE techniques are also used to characterize and understand the spatio-temporal dynamics of the relevant coherent structures, such as hot rising thermal plumes and cold sinking sheets. The investigators study relevant geophysical problems such as the underlying causes of large-scale circulation in the mantle, the relative stationarity of upwellings from depth-dependent material properties, gravitational instabilities caused by multiple phase transitions and the effect of temperature dependent viscosity on the mantle dynamics. They also conduct simulations with an imposed time-dependent boundary condition at the bottom to account for the cooling of the core, in order to study the thermal history by starting at very high Rayleigh number, like 10**8, and slowly lowering the Rayleigh number via cooling. The investigators use modern mathematical theories and numerical techniques to study the three-dimensional dynamics of the Earth's interior. An important issue arising in the last year is the possibility for gravitational instabilities to develop in the mantle due to phase transitions. This instability results in periodic eruption of superplumes and associated intense volcanic activity. The collaborators are among the first groups to model this phenomenon in three dimensions. They plan to study this further by incorporating more realistic flow laws and thermodynamics. There are still many aspects in this phenomenon of gravitational instability to explore, as increasing evidence from the correlation between past trench sites and the cold anomalies in the lower mantle, inferred from seismic tomography, suggests that such mantle instabilities could have occurred in the past 100 million years. Recognition of these instabilities may change traditional views of the role of steady-state processes. They also investigate the nature of coherent large-scale flow structures in the lower mantle as revealed by seismology, and what effects these instabilities have on the long-term thermal evolution of the Earth and Earth-like planets. It was only a few years ago that the idea of a thermal attractor from the collisions of plumes was introduced in geophysics. They expect that work on large-scale coherent structures maintained dynamically by mantle convection may also affect traditional views of geophysical behavior.