AMC-SS: Stochastic Waves and Wave-Mean Interactions

研究人员对数学流体动力学中的强随机效应进行理论和数值研究，特别是在波平均相互作用理论领域。这里的“强”是指与标准确定性设置相比，实质上改变流体系统整体动力学的效果。一般来说，波-平均相互作用理论描述了小尺度波与大尺度平均流之间的非线性相互作用。存在大量基于确定性波的经典波-均值相互作用，但在问题的随机版本上做得很少。目前的工作结合了一些旨在在这个方向上扩展经典理论的项目。例子包括将随机理论用于波浪动力学和非线性破波，用于二维波涡湍流，以及用于推导受高频振荡强迫影响的分层流动的有效平均流动方程。波浪和平均海流之间的相互作用包括：海滩上的破波驱动的长岸流和涡旋运动，如激流（对土木工程和海军作战很重要），平流层的破波产生的晴空湍流（对航空和环境很重要），以及60千米以上的中层的破波驱动的全球空气环流（对气候演化很重要）。这些过程在空间尺度上太小，无法在大气-海洋动力学的数值模型中进行解析，这意味着在这些模型中，它们的影响必须基于理论和观测来手工计算。这被称为无法解决过程的“参数化”问题，即使在最大的超级计算机上，它在未来几十年仍将是一个瓶颈问题。现在，在这一领域的经典“决定论”理论中，普遍的条件被认为是简单和完全已知的。然而，在现实中，普遍的情况往往是复杂和鲜为人知的。随机理论通过考虑不确定或随机的情况因素来解决这个问题。该项目将现代随机理论应用于与大气-海洋科学相关的波浪-平均相互作用问题。有两个主要的项目目标：第一，发现确定性理论遗漏的新效应；其次，为在气候预测、航空和近岸土木工程中使用的数值模型中更真实地表示波浪-平均相互作用奠定基础。

BuhlerDMS-0604519 The investigator undertakes a theoretical and numericalstudy of strong stochastic effects in mathematical fluiddynamics, specifically in the area of wave-mean interactiontheory. Here "strong" refers to effects that substantially changethe overall dynamics of the fluid system when compared to thestandard deterministic setting. In general, wave-meaninteraction theory describes the nonlinear interactions betweensmall-scale waves and the large-scale mean flows on which thewaves are propagating. There exists a substantial body ofclassical wave-mean interaction that is based on deterministicwaves, but very little has been done on the stochastic version ofthe problem. The present work combines a number of projectsaimed at extending the classical theory in this direction. Examples include the use of stochastic theory for wave dynamicsand nonlinear wave breaking, for two-dimensional wave-vortexturbulence, and for the derivation of effective mean-flowequations for stratified flows subject to high-frequencyoscillatory forcing. Interactions between waves and mean currents include thedriving of long-shore currents and vortical motions such as ripcurrents by breaking waves on a beach (important for civilengineering and naval operations), the creation of clear-airturbulence by breaking waves in the stratosphere (important foraviation and the environment), and the driving of the global aircirculation by breaking waves in the mesosphere, above 60kmaltitude or so (important for climate evolution). Theseprocesses are far too small in spatial scale to be resolvable innumerical models for atmosphere-ocean dynamics, which means thatin these models their impact must be put in by hand based ontheory and observations. This is called the "parametrization"problem for unresolvable processes, and even on the biggestsupercomputers it will remain a bottleneck problem for decades tocome. Now, in the classical "deterministic" theory in this area,the prevailing conditions are assumed to be simple and perfectlyknown. However, in reality, the prevailing conditions are oftencomplex and poorly known. Stochastic theory addresses this byallowing for uncertain, or random, components of the situation. This project brings modern stochastic theory to bear on the kindof wave-mean interaction problems that are relevant toatmosphere-ocean science. There are two principal project aims:first, finding new effects that are missed by the deterministictheory; and second, laying the foundation for a more realisticrepresentation of wave-mean interactions in numerical models foruse in climate prediction, aviation, and near-shore civilengineering.