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

数学科学：地幔中的强混沌热对流研究：分析、计算和可视化视角

• 批准号: 9201042
• 负责人: Sivaramakrishna Balachandar
• 金额: $ 4万
• 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: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201042&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Study; Strongly; Chaotic

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.

使用分析，数值和可视化技术，应用于应用数学（流体动力学）和地球物理动力学（地球物理流体动力学）之间的这种协作努力的主要力量是对强烈混乱的三维对流的重点研究。 研究者和他的同事戴维·尤恩（David Yuen）将常规的恒定性能模拟的通常规范公式扩展到包括逼真的深度依赖性热膨胀性和粘度导热率以及多个内部相变和内部热量产生。 相图中存在三重点和更准确的温度依赖性粘度，也将其纳入数学公式中。 研究人员使用新开发的正交分解（POD）和线性随机估计（LSE）技术来压缩大量三维数值模拟产生的大量数据。 POD和LSE技术还用于表征和了解相关连贯结构的时空动力学，例如热上升的热羽和冷下沉板。 研究人员研究了相关的地球物理问题，例如地幔中大规模循环的潜在原因，来自深度依赖性材料特性的上升层相对平稳性，由多个相变引起的重力不稳定性以及温度依赖粘度对地幔动力学的影响。 他们还在底部进行了具有时间依赖时间的边界条件的模拟，以解释核心的冷却，以便通过以非常高的雷利数（例如10 ** 8）开始研究热史历史，并慢慢降低瑞利（Rayleigh）通过冷却数字。 研究人员使用现代的数学理论和数值技术来研究地球内部的三维动力学。 去年引起的一个重要问题是，由于相变而导致的重力不稳定性在地幔中发展。 这种不稳定性导致超级膨胀和相关强烈的火山活性的周期性喷发。 合作者是最早在三个维度上对此现象进行建模的小组之一。 他们计划通过纳入更现实的流量定律和热力学来进一步研究这一点。 这种引力不稳定的现象仍然有许多方面可以探索，因为从过去的沟槽部位与下层中的冷异常之间的相关性越来越多，这是从地震层析成像中推断出来的，这表明这种地幔不稳定可能在过去100次发生。百万年。 对这些不稳定性的认识可能会改变稳态过程作用的传统观点。 他们还研究了地震学揭示的下层中相干大规模流量结构的性质，以及这些不稳定性对地球和地球样行星的长期热演化产生了什么影响。 仅几年前，在地球物理学中引入了羽毛碰撞中的热吸引子的想法。 他们预计，通过地幔对流动态维持的大规模连贯结构的工作也可能影响传统的地球物理行为观点。