Mathematical Sciences: RUI Problems in Magnetohydrostatic Equlilbrium Arising in the Study of the Solar Corona

数学科学：日冕研究中出现的磁流体静力平衡中的 RUI 问题

• 批准号: 9406573
• 负责人: Edward Stredulinsky
• 金额: $ 3.8万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9406573&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Problems; Magnetohydrostatic

9406573 Stredulinsky One of the outstanding open problems in solar astrophysics is the existence of enormously high temperatures in the sun's corona, on the order of a million degrees Kelvin, which have baffled astrophysicists for generations. The main object of this project is to give a careful mathematical analysis of a theory due to E.N. Parker which claims that coronal heating is primarily due to the formation of certain violent disruptions in the suns magnetic field (the formation of current sheets) associated with solar flares. Due to the relationship with sun spots and solar flares the issue of coronal heating is directly tied to the practical issues of variations in the earth's climate and electromagnetic interference. The main focus of the analysis will be the study of tube like loops of magnetic field lines with ends anchored in the suns surface or photosphere. It is proposed that current sheets form when the ends of such a tube are twisted more than a certain critical amount. In attempting to understand the phenomenon of current sheet formation in Parker's model of coronal heating, the equations of ideal 3D magnetohydrodynamics will studied in an open flux tube geometry. Existence of solutions will be considered subject to prescription of the twist in the magnetic field lines from one end of the tube to the other. It is conjectured that smooth solutions will exist up to a certain critical twist, at which point current sheets will form i.e. the curl of the magnetic field (the current) will become a singular measure, the singular part supported on a set of finite two dimensional measure. The transition from subcritical to critical values of the twist is conjectured to correspond to a critical exponent in a geometric type of Sobolev inequality linked to the topology of the field line structure. The main emphasis will be placed on the constant pressure or force free case of ideal MHD. Existence of solutions will be studied through the use of an iteration scheme which avoids the lack on compactness inherent in many variational approaches to the problem.

小行星9406573 太阳天体物理学中一个悬而未决的问题是日冕中存在着极高的温度，大约在一百万开氏度，这让几代天体物理学家感到困惑。这个项目的主要目的是对E.N.帕克提出，日冕加热主要是由于与太阳耀斑有关的太阳磁场中某些剧烈破坏的形成（电流片的形成）。由于与太阳黑子和太阳耀斑的关系，日冕加热问题直接关系到地球气候变化和电磁干扰的实际问题。分析的主要焦点将是研究末端固定在太阳表面或光球层中的磁场线的管状环。有人提出，电流片形成时，这样的管的两端扭曲超过一定的临界量。 为了理解帕克日冕加热模型中的电流片形成现象，我们将在开放的磁通管几何中研究理想三维磁流体力学方程。解的存在性将被认为是受从管的一端到另一端的磁场线的扭曲的处方。证明了光滑解将存在到某一临界扭曲，在该临界扭曲处将形成电流片，即磁场（电流）的旋度将成为奇异测度，奇异部分支撑在一组有限的二维测度上。据推测，扭曲从亚临界值到临界值的转变对应于与场线结构拓扑结构相关的索博列夫不等式几何类型中的临界指数。重点将放在恒压或力自由的情况下，理想磁流体。将通过使用迭代方案，避免了在许多变分方法的问题固有的紧凑性缺乏解决方案的存在性进行研究。