Application of Numerical Bifurcation Analysis to Geophysical Fluid Systems

点击翻译按钮获取中文摘要

We propose to study mathematical models of simple physical systems that isolate the fundamental factors that determine the dynamics of large-scale geophysical fluid systems. Often these simple (or model) systems are derived from laboratory experiments. For example, a model system of particular interest, which is often referred to as the differentially heated rotating annulus, corresponds to an experiment that consists of observing a fluid contained in a rotating cylindrical annulus while the rotation rate and the temperature difference between the inner and outer walls of the annulus are varied. Because differential heating and rotation play a central role in determining the dynamics of large-scale geophysical fluids, such as Earth’s atmosphere, an understanding of the dynamics of these simple systems can lead to insight into the nature of the large-scale flows. Furthermore, unlike realistic mathematical models of large-scale flows, mathematical models of the simple systems can be tractable and the results of the analysis may be quantitatively verified by comparison with experimental observations. We will use methods from dynamical systems to explore the dynamics of these simple systems. In particular, we will study fluid flow transitions by performing a bifurcation analysis of a variety of mathematical models of differentially forced rotating fluid systems, including models of differentially heated rotating fluids in both a cylindrical annulus and in a spherical shell, as well as an electrically forced smectic-A phase liquid crystal film spanning an annular gap. Models of the differentially heated rotating fluid systems use the three-dimensional Navier-Stokes equations in the Boussinesq approximation, while the nature of the liquid crystal film allows its motion to be modelled using the two-dimensional Navier-Stokes equations for a conducting fluid together with equations for the charge density and the three-dimensional electric potential. The proposed research includes (1) an investigation of the secondary transitions and instabilities that occur in these models, i.e. we study the transition from rotating waves to modulated waves (also called vacillating flow); (2) an investigation of the nature of a spatially localized solution that has been observed in the electroconvection of the liquid crystal; (3) development and analysis of a model of a stratified differentially heated rotating fluid in a spherical shell with radiative forcing, representing a very simple model of the atmosphere, and (4) an extension of previous analysis of the primary transition that is observed in these models. For the models of interest, it is not possible to compute solutions analytically, and therefore numerical methods must be implemented. The models of interest consist of partial differential equations, and thus, discretization leads to large systems that must be solved with appropriate numerical methods. For instance, in many cases, it will be necessary to use cutting edge matrix-free continuation methods. Such methods are able to compute various types of special solutions without the need to explicitly form prohibitively large defining systems and the corresponding matrices. The approach itself has proven useful in many other contexts, and thus it is expected to uncover new fundamental properties of the transitions in these models, and of the instabilities that play an important role in the dynamics.