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

数学科学：大涡模拟

• 批准号: 9622889
• 负责人: Sivaramakrishna Balachandar
• 金额: $ 8万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622889&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Large; eddy; Simulation

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.

Balachandar 这位研究者和他的同事发展了研究地幔对流问题的数值方法。 计算数学、流体动力学和地球物理学领域之间的这一合作努力的主要目标是研究地幔动力学中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。 他们调查的地球物理问题需要在高瑞利数制度的对流勘探。 特别感兴趣的是在调查的各种过渡的流动经历的非平衡条件下，系统从其初始状态的非常高的瑞利数的演变和系统的对流活力随着时间的推移而减少，由于冷却。 这些转变包括（1）相变引起的齐平不稳定性，以及（2）在变粘度三维对流中出现增强的环形表面速度场。 大涡模拟方法的开发和实施，以准确地模拟非常高的瑞利数复杂的年轻地球动力学。 数学工具，研究人员开发，适应和雇用在这些问题中，包括基于Krylov子空间的迭代技术，用于治疗的变粘度对流的背景下，频谱变换方法，区域分解方法，有效的空间分辨率，和适当的正交分解和小波变换技术，有效的后处理的结果。 在过去几年中出现的一个重要问题是，由于内部相变，在地球内部发展的全球引力不稳定性的可能性。 这种不稳定性导致下地幔的超级地幔柱的间歇性喷发，以及与之相关的地表强烈火山活动。 通过地震层析成像技术，从过去的海沟位置和下地幔冷异常之间的相关性，越来越多的证据表明，这种全球范围的不稳定性可能发生在过去的1亿年中。 这为恐龙的灭绝提供了一个可能的解释，但这种引力不稳定性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，局部补丁集中剪切和旋转的垂直轴可以产生剧烈对流下的可变粘度。 这是一个重要的一步，走向一个自洽的解释之间的相互作用的表面板块和地幔。 该项目将理想化平衡条件下的这些最新发现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调查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很有意义的。