Mathematical Sciences: Hydrodynamics and Magnetohydrodynamics

提出的研究方向是流体和磁流体中的一些湍流模型。湍流中相干结构的确定性和统计平衡理论将使用解析和数值方法进行探讨。这些理论补充了驱动耗散系统的面向级联概念，为由保守进化方程控制的流和场的长期、大规模行为提供了模型。它们的平衡解以变分原理为特征，例如由控制动力学的守恒定律决定约束的最大熵原理。将开发理论工具和计算算法来解决由这些一般原理产生的约束优化问题。从具体的物理应用中得到的各种基本的、原型的问题将用这些方法来处理；例如，它们包括剪切层中有组织的漩涡结构，以及最可能的磁约束等离子体状态。这类可处理的问题基本上都是二维的；然而，类似的三维问题的某些方面也将使用相同方法的扩展来解决。这项研究的主要目的是发展对湍流物理学的基本认识。紊流介质可以是一种普通的流体，例如在管道中流动的水或流过机翼的空气；或者，它可以是磁化等离子体，比如热核聚变反应堆中的电离气体。这些复杂物理问题的简化数学模型是为了研究它们所表现的现象的一些基本性质而制定的。特别关注的是在动荡无序的存在下出现并持续存在的连贯或组织结构。这些结构是通过推导和求解方程来定量研究的，这些方程反映了系统的粗粒度行为，同时部分地忽略了它的细粒度波动。通过这种方式，可以对具有大量自由度的物理系统进行有用的预测，否则这些预测将超出当前数学分析或计算机模拟方法的掌握范围。将采用跨学科的方法，将抽象的分析工具、现代计算技术和坚实的物理推理统一运用。

The proposed research is directed toward some models of turbulence in fluids and magnetofluids. Deterministic and statistical equilibrium theories of coherent structures in turbulence will be explored using both analytical and numerical methods. These theories, which complement the cascade-oriented concept of a driven and dissipative system, provide models for the long-time, large-scale behavior of flows and fields governed by conservative evolution equations. Their equilibrium solutions are characterized by variational principles, such as a maximum entropy principle with constraints dictated by the conservation laws for the governing dynamics. Theoretical tools and computational algorithms will be developed to solve the constrained optimization problems that arise from these general principles. A variety of fundamental, prototype problems derived from specific physical applications will be treated with these methods; they include, for instance, the organized vortex structures in a shear layer, and the most probable state of a magnetically-confined plasma. Most of the tractable problems of this kind are essentially two-dimensional; however, some aspects of the analogous three-dimensional problems will be also tackled using extensions of the same methods. The principal aim of this research is to develop a basic understanding of the physics of turbulence. The turbulent medium can be an ordinary fluid, such as water flowing in a pipe or air streaming past a wing; or, it can be a magnetized plasma, such as ionized gas in a thermonuclear fusion reactor. Simplified mathematical models of these complex physical problems are formulated in order to investigate some fundamental properties of the phenomena they exhibit. Particular attention is focussed on the coherent or organized structures that emerge and persist in the presence of turbulent disorder. These structures are studied quantitatively by deriving and then solving equations that captu re the coarse-grained behavior of the system while partially ignoring its fine-grained fluctuations. In this way, useful predictions can be made of physical systems having a very large number of degrees of freedom, which otherwise would be beyond the grasp of current methods of mathematical analysis or computer simulation. An interdisciplinary approach will be adopted in which abstract analytical tools, modern computing techniques, and solid physical reasoning are employed in a unified manner.