Nonlinear Mechanics for Energy Transfer in the Atmosphere and the Ocean

塔巴克9701751在这份职业补助金中，研究人员对大气和海洋的非线性动力学进行了两个方向的研究：一个侧重于色散系统中的湍流级联，特别是海洋中能量的蒙克-加雷特尺度分布，另一个侧重于赤道动力学。这两个项目都在非常不同的尺度上研究地球物理波之间的共振：前者考虑通过自相似惯性范围的能量转移，从系统被迫超过的长波到耗散机制接管的短波，而后者集中在与地球半径相当的超长波上，强迫在其中发挥重要作用。事实上，赤道长波之间的共振可以被认为是共振能量向短波转移的起点。一旦尺度变得相对较小，关于力的性质的详细信息就会丢失，一个统计上的自相似机制就会出现，进一步将能量从大范围的尺度带到非常短的波。这项提议的教育部分涉及开发一门融合应用数学和地球物理流体动力学的课程。特别致力于将实验和数值工作整合到研究生和本科教育中。库兰特研究所的流体动力学实验室计划于1997年春开放，从而促进了这项工作。开发了两门新的地球物理波数学建模研究生课程，一门是入门课程，另一门是高级课程，后者侧重于地球物理波之间能量传递的非线性机制。数值建模和桌面实验被整合到高级本科数学课程中，目的是向本科生介绍跨学科工作的诱惑力和潜力。海洋和大气科学已经达到了成熟的程度，因此对天气甚至更长期的气候变化的准确预测似乎是触手可及的。在过去的几十年里，人们已经对盛行的风和洋流背后的许多基本过程有了基本的理解；计算机革命带来的计算能力使在相对精细的网格中运行全球模型成为可能。然而，天气和气候的动态具有巨大的复杂性，各种现象发生在广泛的空间和时间尺度上。这种复杂性使得如果不诉诸强有力的简单化假设就无法解决所有相关现象。应用数学提供了强大的工具，可以帮助澄清各种假设的有效性，并阐明许多尚未完全理解的现象。这项研究的两个拟议主题是这一贡献应特别富有成果的领域的例子。众所周知，赤道波的动态强烈影响全球天气和气候；厄尔尼诺南方涛动等现象及其全球影响就是例证。为了研究这类现象，人们需要超越波动的天数量级的时间尺度，到积累微小影响的月份或年份，以产生波动的幅度和行为的实质性变化。渐近多尺度分析是实现这一目标的理想应用数学工具。至于另一种研究，了解海洋和大气中不同尺度之间的能量转移对于预测强迫变化--例如大气和海洋中释放的化学物质所带来的变化--可能对我们的天气产生的长期影响是至关重要的。让学生参与这些领域的研究，并将与研究相关的成果和方法带入他们的课堂教育，也有助于培养一代更有能力理解大气和海洋相互作用的困难和重要问题的科学家。

Tabak 9701751 In this CAREER grant, the investigator pursues two lines of research on the nonlinear dynamics of the Atmosphere and the Ocean: one focused on turbulent cascades in dispersive systems, particularly the Munk-Garret scale distribution of energy in the Ocean, and the other focused on equatorial dynamics. Both projects study resonance among geophysical waves, in very different scales: the former considers energy transfer through the self-similar inertial range, from the long waves beyond which the system is forced, to the short waves below which dissipative mechanisms take over, and the latter concentrates on the very long waves, comparable to the radius of the Earth, where forcing plays an important role. In fact, the resonance between long equatorial waves could be considered as a starting point for the resonant energy transfer toward shorter waves. Once the scales become relatively small, detailed information on the nature of the forces is lost, and a statistically self-similar regime arises, further carrying the energy through a wide range of scales toward the very short waves. The educational component of this proposal involves developing a curriculum which blends applied mathematics and geophysical fluid dynamics. Particular efforts are devoted to integrate experimental and numerical work into both graduate and undergraduate education. This is facilitated by the building of a Laboratory for Fluid Dynamics at the Courant Institute, scheduled to open in the Spring of 1997. Two new graduate courses in the mathematical modeling of geophysical waves are developed, one introductory and the other more advanced, the latter focused on nonlinear mechanisms of energy transfer among geophysical waves. Numerical modeling and desk-top experimentation are integrated into advanced undergraduate mathematical courses, with the goal of introducing undergraduate students to the lure and potential of interdisciplinary work. The Ocean and Atmospheric Sciences have reached a degree of maturity such that the accurate prediction of the weather and even of longer term climatological changes appears to be within reach. A basic understanding of many of the fundamental processes underlying the prevailing winds and currents has been developed over the last few decades; and the computational power brought about by the computer revolution makes it possible to run global models with relatively fine grids. However, the dynamics of the weather and climate has a tremendous complexity, with phenomena taking place in a wide range of spatial and temporal scales. This complexity makes it hopeless to resolve all the relevant phenomena without appealing to strong simplifying assumptions. Applied Mathematics provides powerful tools that may help clarify the validity of the various assumptions and shed light on many phenomena not yet fully understood. The two proposed subjects of this research are examples of fields where this contribution should be particularly fruitful. The dynamics of equatorial waves is known to strongly affect the global weather and climate; phenomena such as the El Nino Southern Oscillation and its global effects exemplify this. In order to study phenomena of this kind, one needs to go beyond the time scale of the order of days of the waves, to the months or years where small effects accumulate to yield substantial changes in the amplitude and behavior of the waves. Asymptotic multiple-scale analysis is an ideal applied mathematical tool to achieve this. As for the other line of research, understanding the transfer of energy among scales in the Ocean and Atmosphere is fundamental to predict the long-time effects that a change in forcing, such as the one brought about by the release of chemicals in the Atmosphere and the Ocean, may produce on our weather. Involving students in these lines of research, and bringing research-related results and methods into their classroom education, also helps develop a generation of scientists better equipped to understand the difficult and important problems of atmosphere and ocean interactions.