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

这一合作努力的主要目标是应用 数学（流体动力学）和地球物理学（地球物理流体 动力学）是一个重点研究强混沌 应用于地幔的三维对流， 使用分析、数值和可视化技术。 的 调查人员和他的同事大卫袁延长了通常的 恒定特性模拟的规范公式，包括 真实的深度相关的热膨胀系数和粘度 多内相沿着导热系数 转变和内部发热。 的存在 相图中的三重点和更准确的 温度依赖性粘度也被引入到 数学公式 研究人员使用新的 提出了正交分解（POD）和线性 随机估计（LSE）技术来压缩巨大的 大量的数据产生于大规模的三维 数值模拟 POD和LSE技术也被使用 来描述和理解 相关的相干结构，如热上升热 羽流和冰冷的下沉层 研究人员研究相关 地球物理问题，如大规模的根本原因， 地幔环流的相对稳定性 与深度相关的材料特性不同， 多重相变引起的引力不稳定性 以及温度依赖性粘度对地幔的影响 动力学 他们还进行模拟与强加的 底部的时变边界条件， 核心的冷却，为了研究热历史， 从很高的瑞利数开始，比如10**8， 通过冷却降低瑞利数。 研究人员使用现代数学理论， 数值技术来研究三维动力学 地球内部 一个重要的问题出现在最后 年是引力不稳定性的可能性， 由于相变而在地幔中形成。 这种不稳定性 导致超级羽流的周期性喷发， 强烈的火山活动 合作者是第一批 小组在三个维度上模拟这种现象。 他们计划 为了进一步研究这一点， 和热力学。 这里面还有很多方面 引力不稳定现象的探索，因为越来越多的 从过去的海沟遗址和 下地幔冷异常，由地震推断 断层扫描表明，这种地幔不稳定性可能 发生在过去的一亿年里 承认这些 不稳定性可能会改变传统的观点， 稳态过程。 他们还调查了 下地幔相干大尺度流动结构， 以及这些不稳定性的影响 地球和类地行星的长期热演化 星球 就在几年前， 吸引子的碰撞羽流被引入， 物理学。 他们希望在大规模的相干 由地幔对流动态维持的结构也可能 影响了地球物理行为的传统观点。

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.