Mathematical Sciences: Nonlinear Phenomena in Fluid Dynamicsand Related P.D.E.'s with Applications to Atmosphere-Ocean Science

[9301094] Majda研究者继续他在流体动力学偏微分方程方面的研究，涉及建模、数学理论、渐近方法和科学计算的结合。目前的项目围绕着不可压缩流中湍流的各种理论问题，并有一个新的方向涉及到大气-海洋科学的潜在应用。本提案包含以下领域的具体研究项目：(a)湍流输运中的涡流扩散、重整化和蒙特卡罗模拟；(b)不可压缩流动中小尺度三维结构的生成；(c)稳定分层流场内重力波与旋涡结构的统计相互作用。理解大气或海洋中流体流动所需的物理尺度范围是巨大的。与分子扩散有关的过程涉及小于厘米的尺度，而对天气预报和其他环境问题具有重要意义的大尺度模式涉及数百至数千公里的尺度。即使是最大的超级计算机处理全球观测的网格间距也不小于500公里。人们如何评估许多尺度的平均影响，这些尺度不能详细描述对环境意义的大规模过程？提案讨论了这个问题。这项工作涉及当前的大气和海洋学问题。* * *

9301094 Majda The investigator continues his research on the partial differential equations of fluid dynamics, involving a combination of modelling, mathematical theory, asymptotic methods, and scientific computing. The present project centers around various theoretical issues regarding turbulence in incompressible flows, with a new direction involving potential applications to atmosphere-ocean science. This proposal contains specific research projects in the following areas: (a) Eddy diffusivity, renormalization, and Monte Carlo simulation in turbulent transport; (b) The generation of small scale three-dimensional structures in incompressible flow; (c) The statistical interaction of internal gravity waves and vortical structure in stably stratified flow fields. The range of physical scales necessary for understanding fluid flow in the atmosphere or ocean is enormous. The processes associated with molecular diffusion involve scales smaller than centimeters while the large scale patterns of significance for weather prediction and other environmental issues involve scales of hundreds to thousands of kilometers. Even the largest supercomputers process observations over the globe at a mesh spacing with each side no smaller than 500 kilometers. How can one assess the mean impact of the many scales, which cannot be described in detail on the large-scale processes of environmental significance? The proposal takes up this question. The work bears on current issues in atmospherics and oceanics. ***