Nonlinear Wave Problems in Fluid Flows

0071939

Milewski

The 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.