Hall MHD 湍流中能量级串的数值研究

NUMERICAL INVESTIGATION OF ENERGY CASCADES IN HALL MHD TURBULENCE

周康成

11849187

硕士

力学

万敏平

2020-05-28

2020-07-08

哈尔滨工业大学

深圳

磁流体在宇宙空间与地球内部等自然环境中广泛存在。磁流体动力学 (Magnetohydrodynamics, MHD) 是描述导电流体与周围磁场相互作用下两者演化的物理模型。考虑 Hall 效应的 MHD 模型将离子惯性长度以下离子与电子的相对运动通过Hall 项的形式加入到磁感应方程中，控制方程组对小尺度等离子体流动行为的描述更为准确。然而，以往的研究工作对 Hall MHD 湍流的研究相对比较缺乏。本文对 Hall 效应强度对 MHD 湍流中的全局统计量、间歇性以及尺度间能量传递的影响开展了数值模拟研究。在保证其他条件不变的前提下，选择五组 Hall参数 (ϵ = 0.00, 0.01, 0.05, 0.10, 0.20)，采用 Fourier 伪谱法，对有外力施加的三维不可压缩 Hall MHD 湍流进行直接数值模拟。本文的主要研究内容包含以下三个部分。第一部分，我们对数值模拟的结果进行了统计分析。我们发现，湍动能能谱对 Hall 参数改变的敏感度不高，而磁能能谱则在高 Hall 参数的算例中出现小尺度能量聚集现象。概率分布函数、积累分布函数与考虑尺寸效应的平坦度分析表明，Hall 效应抑制了小尺度的电流场相干结构生成，进而导致电流场间歇性随 Hall 参数的提升而下降。同时，上述分析也表明 Hall 参数的提高对涡量场的间歇性没有呈现单调的影响。第二部分，我们在 Fourier 谱空间上对 Hall MHD 湍流中尺度间能量传递行为进行了分析。基于谱空间大尺度能量演化方程，发现 Hall 效应引起的磁能通量在不同尺度上呈现逆向传递与正向传递两种传递模式。对 Hall 项进行分解得到非线性项 b · ∇j 与 j · ∇b，由能量通量与 shell-to-shell 能量传递分析可知，两种不同的物理过程在能量传递方式与尺度上均存在差异。第三部分，我们基于空间滤波的方法在物理空间对 Hall MHD 湍流中尺度间能量传递行为进行了分析。推导了物理空间中 Hall MHD 湍流中大尺度能量演化方程，通过数值模拟计算探究亚格子能量通量的统计特性。分析了 Hall 效应强度对不同滤波尺度亚格子能量通量的影响，发现了 Hall 项引起的亚格子能量通量均有强间歇性。观察发现亚格子能量通量与物理场等值面之间的关联，并通过计算互相关系数将上述关联性实现量化。

Magnetohydrodynamic turbulence is a common phenomenon in natural environments such as the space and the interior of the Earth. Magnetohydrodynamics, or MHD, is a physical model describing the evolution of conducting fluid and the surrounding magnetic field while both physical fields interact with each other. The effect induced by the relative motions between ions and electrons can be depicted by MHD with the addition of Hall term. However, previous studies of Hall MHD turbulence remain rare.In this thesis, numerical investigations of global statistics, intermittency, and interscale energy transfer in Hall MHD turbulence have been conducted. Without changing other computational parameters, we performed direct numerical simulations of three dimensional forcing homogeneous isotropic Hall MHD turbulence using Fourier pseudospectral method with five different Hall parameters,ϵ = 0.00, 0.01, 0.05, 0.10, 0.20. The content of this thesis mainly consists of three parts.In the first part, we analyzed the numerical data from a statistical viewpoint. Our results show that, the kinetic energy spectrum of Hall MHD turbulence remains insensitive to the change of the Hall parameter while there exists an accumulation of magnetic energy in small scales with the increasing Hall parameter, revealed by the magnetic energy spectrum of Hall MHD turbulence. Based on the results of probability distribution functions, cumulative distribution functions, and scale-dependent kurtosis of vorticity and current density field in Hall MHD turbulence, we can conclude that Hall effect restrains the formation of small-scale current density structures, resulting in the decline of intermittency of current density. Meanwhile, the same analysis also shows that the increasing Hall parameter has no monotonical influence on the intermittency of vorticity.In the second part, we analyzed the inter-scale energy transfer of Hall MHD turbulence in Fourier spectral space. By computing spectral energy fluxes in the large-scale energy equations of Hall MHD, we found that both forward and inverse transfers can be observed in the inter-scale magnetic energy transfer induced by the Hall effect. Then, we decomposed the Hall term into the following nonlinear terms:b · ∇j and j · ∇b. Spectral energy flux and shell-to-shell energy transfer reveal that there exist two kinds of energy transfers with different directions and scales owing to the different physical processes behind these nonlinear terms.In the third part, we analyzed the inter-scale energy transfer in Hall MHD turbulence in physical space using a space-filtering technique. We derived the physical-space largescale energy equations in Hall MHD and investigated the statistical property of subgridscale(SGS) fluxes numerically. Dependence of SGS energy fluxes on the strength of the Hall effect shows that SGS flux induced by the Hall effect has very strong intermittency. Comparing the isosurfaces of SGS energy fluxes and physical fields, we observed a potential connection between these variables, which was quantified by computing their cross-correlation functions.

Hall 效应 磁流体湍流 能量级串 湍流间歇性

Hall effect MHD turbulence energy cascade intermittency in turbulence

中文

联合培养

南方科技大学

周康成. Hall MHD 湍流中能量级串的数值研究[D]. 深圳. 哈尔滨工业大学,2020.