Asymptotic Analysis for Magnetostrophic Turbulence

地球磁场对地球上的生命至关重要。它的作用是保护地球免受太阳风带电粒子的影响，否则这些粒子会剥离高层大气。磁场的变化对气候变化有重要影响。地球磁场的来源是在地球内部深处，一个小的固体铁中心被一个巨大的球形导电液态铁包围。这种液体中的对流是通过发电机效应产生地球磁场的起源：这是旋转、对流的熔融铁维持地磁场对抗欧姆衰减的过程。这个过程的数学描述是一个高度复杂的三维系统，非线性偏微分方程（PDE），管理对流磁流体动力学。计算机模型已经被用来模拟实际的地球发电机;然而，还没有三维模型在空间分辨率上运行，以涵盖地球流体核心中肯定存在的广泛的湍流。研究人员和合作者研究了一个PDE模型的核心，其中驱动地球发电机的随机力被解释为一个源，不断再生一个统计稳定的浮力分布在整个流体。对于与地核相关的尺度，这个偏微分方程系统有许多小参数。该团队通过分析随机强迫PDE在极小参数极限下的渐近性来利用这一特性。他们的目标是建立偏微分方程系统维持各态历经统计稳定状态，从而为湍流地球发电机提供严格的基础。一名研究生参与了这项研究。控制流体动力学的偏微分方程是出了名的具有挑战性。挑战的一个方面是方程的奇异行为，因为由问题的物理学产生的某些参数消失了。研究这类问题对于阐明奇异极限所描述的物理过程和促进新的数学技术的创造都很重要。研究人员和合作者通过研究地球流体核心中磁旋湍流的数学模型为这一奋进做出了贡献。 分析需要详细了解的非线性和随机强迫之间的微妙的相互作用。该小组计划应用最近的发展，在一个理论的hypoellipticity随机偏微分方程与空间退化强迫证明存在独特的遍历不变的措施偏微分方程模型磁旋湍流。

The Earth's magnetic field is vitally important to the existence of life on our planet. It serves to protect the Earth from the charged particles of the solar wind that would otherwise strip away the upper atmosphere. Changes in the magnetic field have significant implications for climatic changes. The source of the Earth's magnetic field is deep inside the Earth where a small solid iron center is surrounded by a huge spherical mass of electrically conducting liquid iron. Convection in this liquid is at the origin of Earth's magnetic field through a dynamo effect: this is the process by which the rotating, convecting, molten iron maintains the geomagnetic field against ohmic decay. The mathematical description of this process is a highly complex system of three-dimensional, nonlinear partial differential equations (PDE) that govern convective magnetohydrodynamics. Computer models have been used to simulate the actual geodynamo; however, no three-dimensional model has yet been run at the spatial resolution required to encompass the broad spectrum of turbulence that surely exists in the Earth's fluid core. The investigator and collaborators study a PDE model for the core where a stochastic force driving the geodynamo is to be interpreted as a source that is continuously regenerating a statistically steady buoyancy distribution throughout the fluid. For scales relevant to the Earth's core, this PDE system has many small parameters. The team exploits this feature by analyzing the asymptotics of the stochastically forced PDE in the limit of vanishingly small parameters. They aim to establish that the PDE system sustains ergodic statistically steady states, thus providing a rigorous foundation for a turbulent geodynamo. A graduate student is involved in the research. The partial differential equations that govern fluid dynamics are notoriously challenging. One aspect of the challenge is the singular behavior of the equations as certain parameters, which are produced by the physics of the problem, vanish. Studying such problems is important both to illuminate the physical processes described by the singular limits and to advance the creation of new mathematical techniques. The investigator and collaborators contribute to this endeavor by studying mathematical models for magnetostrophic turbulence in the Earth's fluid core. The analysis requires a detailed understanding of the delicate interaction between the nonlinearity and the stochastic forcing. The team plans to apply recent developments in a theory of hypoellipticity for stochastic PDE with spatially degenerate forcing to prove the existence of unique ergodic invariant measures for the PDE that model magnetostrophic turbulence.