Active Scalar Equations and a Geodynamo Model

主动标量方程和地球发电机模型

• 批准号: 1207780
• 负责人: Susan Friedlander
• 金额: $ 19万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207780&HistoricalAwards=false
• 关键词: Active; Scalar; Equations; Geodynamo; Model

Active scalar equations describe a number of physical phenomena that arise in fluid dynamics. Because of their physical importance and challenging mathematical nature there is a very extensive literature on such equations. However, many problems connected with the nonlinearity of the equation remain open, particularly when the drift velocity U is nonlocal. This project addresses some of these questions with particular emphasis on examples where the operator M that encodes the physics of the problem relating U to the active scalar is singular and unbounded. It is proposed to study the interplay of diffusive effects and the nonlinearity in singular active scalar equations for a range of fractional powers of the Laplacian. For certain powers it will be shown that the system is nonlinearly unstable: for lower powers the system is Lipschitz ill-posed. These results will be obtained via the construction of unstable eigenvalues for the evolution equation linearised about an appropriate equilibrium. Techniques of continued fractions will be employed to produce an explicit lower bound on the growth rate of the unstable eigenvalues as a function of the physical parameters. The project will also study the highly singular limit of vanishing diffusivity. This limit is often physically appropriate and may be closely connected with turbulent phenomena modeled by weak solutions of the active scalar equations. The project addresses general classes of active scalar equations: particular physical examples include the incompressible porous media equation and the magnetogeostrophic equation. In both examples the Fourier multiplier symbol for the relevant operator M is even with respect to the wave number vector. This prevents certain cancellations in energy estimates arising in the nonlinearity. Hence the problem is more subtle than some frequently studied equations, such as the surface quasigeostrophic equation, where the analogous Fourier multiplier symbol is odd and commutator estimates can be employed.The mathematical study of the partial differential equations that model fluid motion forms an essential foundation for many applications. These equations are highly complex and challenging. A subset of these equations are so-called "active scalar equations" where the evolution in time of a scalar quantity such as the density of the fluid is governed by the motion of the fluid where the velocity itself varies with this scalar field. This feedback produces an intricate nonlinearity. Friedlander uses techniques of "hard" analysis to study general classes of such equations. A particular example is a model for the geodynamo. This is the process by which the Earth's magnetic field is created and sustained through the motion of the fluid core which is composed of a rapidly rotating, density stratified, electrically conducting fluid. Friedlander will prove that the model can indeed produce dynamo action by demonstrating the existence of a strong instability whose growth rate can be bounded from below by an explicit expression that depends on the physical parameters of the Earth's fluid core.

主动标量方程描述了流体力学中出现的许多物理现象。由于它们的物理重要性和具有挑战性的数学性质，关于这类方程的文献非常广泛。然而，与方程的非线性有关的许多问题仍然悬而未决，特别是当漂移速度U是非局部的时候。这个项目解决了其中的一些问题，特别强调了一些例子，其中编码U与活动标量相关问题的物理的运算符M是奇异的和无界的。研究了拉普拉斯分数次幂范围内奇异有源标量方程的扩散效应和非线性的相互作用。对于某些幂，系统将被证明是非线性不稳定的：对于较低的幂，系统是Lipschitz不适定的。这些结果将通过构造关于适当平衡线性化的发展方程的不稳定本征值来获得。我们将使用连分式技术来给出不稳定本征值增长率作为物理参数的函数的显式下界。该项目还将研究消失扩散系数的高度奇异极限。这个极限通常在物理上是适当的，并且可能与由活动标量方程的弱解模拟的湍流现象密切相关。该项目涉及一般类型的活动标量方程：特殊的物理例子包括不可压缩的多孔介质方程和磁地转方程。在这两个例子中，相关运算符M的傅里叶乘数符号相对于波数向量是偶数。这防止了在非线性中引起的能量估计中的某些取消。因此，这个问题比一些经常研究的方程更微妙，例如表面准地转方程，其中类似的傅立叶乘子符号是奇数，可以使用换位器估计。对模拟流体运动的偏微分方程组的数学研究为许多应用奠定了重要的基础。这些方程式非常复杂，极具挑战性。这些方程的一个子集是所谓的“活动标量方程”，其中标量(如流体密度)的时间演化由流体的运动决定，其中速度本身随标量场变化。这种反馈产生了一种复杂的非线性。弗里德兰德使用“硬”分析技术来研究这类方程的一般类别。一个特别的例子是地球发电机的模型。这是通过流体核心的运动产生和维持地球磁场的过程，流体核心由快速旋转的、密度分层的、导电的流体组成。弗里德兰德将通过证明强不稳定性的存在来证明该模型确实可以产生发电机作用，其增长率可以通过依赖于地球流体核心的物理参数的显式表达式来从下限定。