Project Details
Description
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.
Status | Finished |
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Effective start/end date | 7/15/00 → 7/31/03 |
Funding
- National Science Foundation: $137,601.00