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 9622889 研究人员和他的同事开发了数值方法来研究地幔对流问题。 计算数学、流体动力学和地球物理学领域之间的合作努力的主要目标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，以及科学可视化。 他们研究的地球物理问题需要探索高瑞利数状态下的对流。 特别感兴趣的是研究非平衡条件下流动所经历的各种转变，因为系统从非常高的瑞利数的初始状态演变，并且系统的对流活力由于冷却而随着时间的推移而降低。 这些转变包括（1）由相变引起的齐平不稳定性，以及（2）可变粘度三维对流中增强的环形表面速度场的出现。 开发并实施了大涡模拟方法，以准确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、调整和使用的数学工具包括基于克雷洛夫子空间的迭代技术，用于在谱变换方法的背景下处理变粘性对流、有效空间分辨率的域分解方法、以及适当的正交分解和小波变换技术以便对结果进行高效的后处理。 过去几年出现的一个重要问题是，由于内部相变，地球内部可能会出现全球重力不稳定。 这种不稳定性导致下地幔的超羽流间歇性喷发以及与之相关的地表强烈的火山活动。 通过地震层析成像推断，过去的海沟位置与下地幔冷异常之间的相关性越来越多，表明全球范围内的这种不稳定性可能发生在过去一亿年里。 这为恐龙的灭绝提供了可能的解释，但这种重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还表明，在剧烈对流下，可以产生具有可变粘度的集中剪切和绕垂直轴旋转的局部斑块。 这是对表面板块和地幔之间的相互作用进行自洽解释的重要一步。 随着剧烈对流的年轻地球随着时间的推移而冷却，该项目将理想平衡条件下的这些最新发现扩展到更现实的非平衡条件。 现代数值技术和数学方法的最新发展的结合对于成功研究这些复杂现象至关重要。 最后，研究这些不稳定性对地球和类地行星的长期热演化的影响是很有意义的。