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; 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调查员和他的同事开发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这种协作努力的主要力量是研究地幔动力学中的一系列具有挑战性的问题，其中涉及将几种现代数学和计算技术应用于大型数值模拟，数据处理和科学可视化。 他们研究的地球物理问题需要在高瑞利数字制度中探索对流。 在对各种过渡的调查中，特别感兴趣的是，由于系统从其初始状态的雷利数非常高，并且由于系统的对流活力随着时间的流逝而随着时间的流逝而降低，因此流量在非平衡条件下发生。 这些过渡中有（1）相变引起的冲洗不稳定性，以及（2）在可变粘度三维对流中增强的环形表面速度场的出现。 开发并实施了大型涡流模拟方法，以准确模拟年轻地球的非常高的瑞利数量复杂动力学。 研究人员在这些问题中开发，适应和采用的数学工具包括基于Krylov子空间的迭代技术，用于在光谱转换方法的背景下处理可变的粘度对流，域分解方法，用于有效的空间分辨率以及正交分解和波动式变换技术，以高效的后处理后处理结果。 在过去几年中，出现的一个重要问题是，由于内部相变而导致地球内部发展的全球重力不稳定的可能性。 这种不稳定性导致表面下地幔和相关强烈的火山活性的超斑点发作。 从地震层析成像中推断出的是，过去的地幔中的沟渠站点与寒冷异常之间的相关性的证据越来越大，即在过去的1亿年中，这种不稳定性可能发生在全球范围内。 这为灭绝恐龙的灭绝提供了一个可能的解释，但仍需要探索这种引力不稳定的许多方面。 最近的大规模高性能模拟还显示，在剧烈对流下，粘度可产生垂直轴的局部浓缩剪切和旋转。 这是朝着对表面板和地幔之间相互作用的自愿解释的重要一步。 该项目将这些最新的发现在理想化的平衡条件下扩展到更现实的非平衡条件，因为随着时间的推移，剧烈对流的年轻地球会冷却。 现代数值技术和数学方法中的最新发展融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似地球的行星的长期热演化的影响是什么。