Nonlinear Wave Problems in Fluid Flows

0071939 Milewski该项目将包括研究流体力学和非线性波中的三个问题。第一个项目涉及了解色散波湍流的某些方面，即大量相互作用的色散波的统计描述，例如海洋表面上的色散波。首先，将使用一个简化的模型，其中包含基本的非线性过程，并可以产生能量传递机制的缩放。第二，频谱的二维和三维海浪与减少有限深度和深水有效的方程将被计算和比较，从减少的模型的结果。第二个项目涉及研究三维孤立波的制度，其中表面张力是一个重要组成部分的动态。例如，薄流体层在小障碍物上的流动可以产生这些波。在这里，建议使用已经计算的解决方案，以找到物理感兴趣的制度，如增加深度的额外的解决方案。第三个项目是研究存在空间不均匀性的反应扩散方程的动力学，例如，在某些化学反应的模型中，反应物浓度在空间上不均匀。在空间均匀的情况下，人们在反应中获得各种相干图案。这些模式和它们的边界是如何修改的不均匀性将被研究。本研究的目标是了解几个方面的波动力学在流体中使用的理论和先进的计算相结合。将研究三种不同的现象。首先，将研究波湍流的演化：不同波长和不同方向传播的许多波叠加的物理情况。最简单的例子是海洋表面上波浪的明显随机混合。目标是预测不同波的相对能量以及不同大小的波交换能量的机制。这些都是重要的预测，其应用范围从了解卫星遥感数据到气候动态。第二，我们将研究一类称为块孤子的水波：沿特定方向传播的局域相干波。我们的目标是获得这些波可以存在的物理情况的范围。这项工作的影响，在各种薄膜和涂层的应用。最后，将研究反应物浓度在空间变化的生物和化学反应系统的组分的动力学。将研究用于反应的催化剂不均匀分布，因此反应在不同位置进行不同的特定情况。我们的目标是了解反应如何从一个地方到另一个地方变化，以及在反应改变特征的边界处发生了什么。

0071939MilewskiThe project will consist of the study of three problems in fluid mechanics and nonlinear waves. The first project involves understanding certain aspects of dispersive wave turbulence, that is, the statistical description of a large number of interacting dispersive waves, such as those on the ocean surface. First, a reduced model will be used which contains the fundamental nonlinear processes and can yield the scaling for the energy transfer mechanisms. Second, spectra of two-- and three--dimensional ocean waves with a reduced equation valid for finite depth and deep water will be computed and compared with results from the reduced model. The second project involves the study of three-dimensional solitary waves in regimes where surface tension is an important part of the dynamics. These are waves that can be generated, for example, by flow of a thin fluid layer over a small obstacle. Here, it is proposed to use solutions that have already been computed to find additional solutions in regimes of physical interest, such as increasing depth. The third project is to study the dynamics of reaction-diffusion equations in the presence of spatial inhomogeneities, as for example, in models of certain chemical reactions where the reactant concentration is not uniform in space. In the spatially homogeneous case, one obtains various coherent patterns in the reaction. How these patterns and their boundaries are modified by the inhomogeneities will be studied.The goal of this research is to understand several aspects of wave dynamics in fluids using a combination of theory and advanced computation. There are three distinct phenomena that will be studied. First, the evolution of wave turbulence will be studied: the physical situation in which many waves of different wavelengths and traveling in different directions are superposed. The simplest example is the apparent random mix of waves on the surface of the ocean. The goal is to predict the relative energy in the different waves and the mechanisms by which waves of different sizes exchange energy. These are important predictions whose applications range from understanding satellite remote sensing data to climate dynamics. Second, a class of water waves called lump solitons will be studied: localized coherent waves that travel in a particular direction. The goal is to obtain the range of physical situations in which these waves can exist. This work has implications in a variety of thin film and coating applications. Lastly, the dynamics of the components of biological and chemical reacting systems where the concentration of the reactants vary in space will be studied. The particular case where a catalyst for the reaction is not distributed uniformly and therefore the reaction proceeds differently in different places will be studied. The goal is to understand how the reaction varies from place to place and what happens at the boundaries where the reactions change character.