Application of Numerical Bifurcation Analysis to Geophysical Fluid Systems

我们建议研究简单物理系统的数学模型，这些模型分离出决定大规模地球物理流体系统动力学的基本因素。通常，这些简单(或模型)系统是从实验室实验中得出的。例如，一个特别令人感兴趣的模型系统，通常被称为差热旋转环空，对应于一个实验，该实验包括观察旋转的圆柱形环空中包含的流体，同时改变环空内壁和外壁之间的转速和温差。由于差异加热和旋转在确定地球大气等大尺度地球物理流体的动力学方面起着核心作用，因此了解这些简单系统的动力学可以帮助我们洞察大尺度流动的本质。此外，与实际的大尺度水流数学模型不同，简单系统的数学模型是易于处理的，分析结果可以通过与实验观测的比较进行定量验证。 我们将使用动力学系统的方法来探索这些简单系统的动力学。特别是，我们将通过对差力旋转流体系统的各种数学模型进行分叉分析来研究流体流动转变，这些模型包括圆柱环空和球壳中的差热旋转流体模型，以及跨越环隙的电强迫近晶-A相液晶薄膜。差热旋转流体系统的模型使用Boussinesq近似下的三维Navier-Stokes方程，而液晶薄膜的性质允许使用导电流体的二维Navier-Stokes方程以及电荷密度和三维电势的方程来模拟其运动。 所提出的研究包括(1)研究这些模型中发生的二次相变和不稳定性，即我们研究从旋转波到调制波(也称为摇摆流)的转变；(2)研究在液晶电对流中观察到的空间局域解的性质；(3)发展和分析具有辐射强迫的球壳中分层的差热旋转流体的模型，它代表了一个非常简单的大气模型；以及(4)在这些模型中观察到的一次相变的扩展分析。 对于感兴趣的模型，不可能以解析的方式计算解，因此必须采用数值方法。感兴趣的模型由偏微分方程组成，因此，离散化导致了必须用适当的数值方法求解的大系统。例如，在许多情况下，有必要使用前沿的无矩阵延拓方法。这种方法能够计算各种类型的特解，而无需显式地形成大得令人望而却步的定义系统和相应的矩阵。该方法本身已被证明在许多其他情况下是有用的，因此它有望揭示这些模型中的转变以及在动力学中扮演重要角色的不稳定性的新的基本性质。

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.