Application of Numerical Bifurcation Analysis to Geophysical Fluid Systems

数值分岔分析在地球物理流体系统中的应用

• 批准号: RGPIN-2014-06257
• 负责人: Lewis, Gregory
• 金额: $ 0.8万
• 依托单位: University of Ontario Institute of Technology
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588902
• 关键词: Application; Numerical; Bifurcation; Analysis; Geophysical

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.

我们建议研究简单的物理系统的数学模型，隔离的基本因素，确定大规模的地球物理流体系统的动态。 通常这些简单的（或模型）系统来自实验室实验。例如，一个特别感兴趣的模型系统，通常被称为差动加热旋转环，对应于一个实验，该实验包括观察旋转圆柱形环中包含的流体，同时旋转速率和环的内壁与外壁之间的温差变化。 由于差异加热和旋转在确定大规模地球物理流体（如地球大气层）的动力学方面起着核心作用，因此对这些简单系统的动力学的理解可以导致对大规模流动性质的深入了解。此外，与大规模流动的现实数学模型不同，简单系统的数学模型可以是易处理的，并且分析结果可以通过与实验观察的比较来定量验证。 我们将使用动力系统的方法来探索这些简单系统的动力学。 特别是，我们将研究流体流动的过渡进行分歧分析的各种数学模型的差分强迫旋转流体系统，包括模型的差分加热旋转流体在一个圆柱形的环和一个球壳，以及一个电强迫近晶A相液晶薄膜跨越环形间隙。 差分加热旋转流体系统的模型使用Boussinesq近似中的三维Navier-Stokes方程，而液晶膜的性质允许其运动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建模。 研究内容包括：（1）研究这些模型中的二次跃迁和不稳定性，即研究从旋转波到调制波的跃迁（2）对在液晶的电对流中观察到的空间局部化溶液的性质的研究;（3）发展和分析一个具有辐射强迫的球壳层中分层差热旋转流体的模型，这是一个非常简单的大气模型;（4）扩展以前对这些模型中观测到的初级转捩的分析。 对于感兴趣的模型，它是不可能计算的解析解，因此必须实施数值方法。感兴趣的模型由偏微分方程组成，因此，离散化导致必须用适当的数值方法求解的大型系统。例如，在许多情况下，将有必要使用切削刃无矩阵连续方法。 这样的方法能够计算各种类型的特殊解决方案，而不需要明确形成过大的定义系统和相应的矩阵。 该方法本身已被证明是有用的，在许多其他情况下，因此，它有望发现新的基本性质的过渡，在这些模型中，和不稳定性，在动态中发挥重要作用。