Mathematical Sciences: Large-eddy Simulation & Mathematical Analysis of Non-equilibrium & Non-linear Processes in Mantle Convection

巴拉钱达尔，9622889，研究员和他的同事发展了数值方法来研究地幔对流问题。计算数学、流体力学和地球物理学领域合作的主旨是研究地幔动力学中的一类具有挑战性的问题，其中涉及将几种现代数学和计算技术应用于大规模数值模拟、数据处理和科学可视化。他们研究的地球物理问题需要探索高瑞利数区域的对流。当系统从非常高的瑞利数的初始状态演化时，以及当系统的对流能量因冷却而随时间降低时，流动在非平衡条件下所经历的各种转变的研究尤其令人感兴趣。这些转变包括：(1)相变引起的冲洗不稳定性；(2)变粘性三维对流中出现增强的环面速度场。发展并实现了一种大涡模拟方法，以精确地模拟年轻地球的极高瑞利数复杂动力学。研究人员在这些问题中开发、适应和使用的数学工具包括：基于Krylov子空间的迭代技术，用于在谱变换方法的背景下处理变粘性对流；区域分解方法，用于有效的空间分辨率；以及适当的正交分解和小波变换技术，用于有效的后处理结果。最近几年出现的一个重要问题是，由于内部相变，全球引力不稳定在地球内部发展的可能性。这种不稳定性导致了来自下地幔的超强羽流的间歇性喷发，以及与之相关的地表强烈的火山活动。越来越多的证据表明，过去的海沟位置与从地震层析成像推断的下地幔冷异常之间的相关性，表明这种全球尺度的不稳定可能发生在过去1亿年。这为恐龙的灭绝提供了一个可能的解释，但这种引力不稳定的许多方面仍然需要探索。最近的大规模高性能模拟也表明，在强烈对流下，可以产生粘性可变的围绕垂直轴的集中剪切和旋转的局部斑块。这是朝着自洽地解释表面板和地幔之间的相互作用迈出的重要一步。该项目将这些最新发现在理想化的平衡条件下扩展到更现实的非平衡条件，因为强烈对流的年轻地球随着时间的推移而冷却。结合现代数值技术和数学方法的最新发展是成功研究这些复杂现象的关键。最后，研究这些不稳定性对地球和类地行星长期热演化的影响是有意义的。

Balachandar 9622889 The investigator and his colleague develop numerical methods to study problems of mantle convection. The main thrust of this collaborative effort between the areas of computational mathematics, fluid dynamics and geophysics is investigating a class of challenging problems in mantle dynamics, which involves the application of several modern mathematical and computational techniques to large-scale numerical simulation, data-processing, and scientific visualization. The geophysical problems they investigate entail the exploration of convection in the high Rayleigh number regime. Particular interest is in the investigation of various transitions the flow undergoes under non-equilibrium conditions as the system evolves from its initial state of very high Rayleigh number and as the convective vigor of the system decreases over time due to cooling. Among these transitions are (1) the flush instabilities induced by phase transitions, and (2) the appearance of enhanced toroidal surface velocity fields in variable viscosity three-dimensional convection. A large eddy simulation methodology is developed and implemented to accurately simulate the very high Rayleigh number complex dynamics of the young Earth. Mathematical tools that the investigators develop, adapt and employ in these problems include iterative techniques based on Krylov subspace for treating the variable viscosity convection in the context of spectral transform method, domain decomposition methodology for efficient spatial resolution, and proper orthogonal decomposition and wavelet transform techniques for efficient post-processing of the results. An important question that has arisen in the last few years is the possibility of global gravitational instability that develops in the Earth's interior due to internal phase transition. This instability results in episodic eruption of superplumes from the lower mantle and associated intense volcanic activity at the surface. Th ere are increasing evidences from correlation between past trench sites and cold anomalies in the lower mantle, inferred from seismic tomography, that such instabilities on global scale could have occurred in the past 100 million years. This provides a possible explanation for the extinction of the dinosaurs, but there are many aspects of this gravitational instability still needs to be explored. Recent large-scale high performance simulations have also revealed that localized patches of concentrated shear and rotation about a vertical axis can be generated with variable viscosity under vigorous convection. This is an important step towards a self-consistent explanation of the interaction between the surface plates and mantle. This project extends these recent findings under idealized equilibrium conditions to more realistic non-equilibrium conditions, as the vigorously convecting young Earth cools over time. Incorporation of modern numerical techniques and recent developments in mathematical methods are essential for the successful investigation of these complex phenomena. Finally, it is of interest to investigate what the effects of these instabilities are on the long-term thermal evolution of the Earth and Earth-like planets.