Mathematical Sciences: "RUI: Magnetohydrostatic Problems Relevant to Current Sheets and Heating of the Solar Corona"

9622923 Stredulinsky这个项目的主要目标是对E.N.Parker关于太阳日冕加热的一个理论进行仔细的数学分析。根据帕克的说法，日冕中的主要热源由非常薄的强电流区域(电流片)组成，这些电流片是日冕等离子体松弛到平衡的趋势和某些拓扑约束之间的中介。产生这种限制的原因是日冕等离子体的近乎完美的导电性，以及磁力线在其足点或末端锚定在光球层的稠密气体中，足点由光球层的湍流运动所吸引。在试图理解Parker模型中的电流片形成现象时，将使用与能量最小化相关的变分方法，该方法依赖于磁场的分解作为描述和规定场线拓扑的一种手段。我们将仔细研究哪些场线拓扑能够经受住与等离子体向最小能量驰豫相关的微弱的索波列夫空间限制。将发展一个一般的二维理论，并将其应用于分析天体物理学文献中著名的电流片形成的例子。一个全三维理论的开端将被描述，特别强调定域矢势类似于等离子体在二维中的通量函数表示。还将探索涉及螺旋度的拓扑约束，以及更精细的场线拓扑测量高阶连杆的措施。太阳天体物理学中一个突出的悬而未决的问题是太阳大气(日冕)中存在着约200万华氏度的极高温度，这一问题几代人以来一直困扰着天体物理学家。虽然已经探索了一些合理的日冕加热模型，但这个问题仍然引起了激烈的争论。在这个项目中，由于E.N.Parker的缘故，日冕加热的主要模型之一将在严格的数学水平上进行分析。希望这将有助于解决不确定性，并澄清日冕加热涉及的基本物理机制。由于与太阳黑子和太阳耀斑(将等离子体和相关磁场推向地球的太阳日冕的剧烈破坏)的关系，日冕加热问题直接与地球气候变化和电磁干扰等实际问题有关。***

9622923 Stredulinsky The main goal of this project is to give a careful mathematical analysis of a theory due to E.N. Parker on solar coronal heating. According to Parker the main source of heat in the solar corona consists of very thin regions of intense current (current sheets) which arise as a mediation between the tendency of coronal plasma to relax to equilibrium, and certain topological constraints. Such constraints arise due to the near perfect conductivity of the coronal plasma, and the anchoring of magnetic field lines at their footpoints or ends in the dense gas of the photosphere, the footpoints being convected by the turbulent motion of the photosphere. In attempting to understand the phenomenon of current sheet formation in Parker's model a variational approach associated with energy minimization will be used which relies on decompositions of magnetic fields as a means of characterizing and prescribing field line topology. A careful study will be made of which field line topologies withstand the weak Sobolev space limits associated with the relaxation of a plasma towards minimum energy. A general two-dimensional theory will be developed and applied to the analysis of well known examples of current sheet formation in the astrophysics literature. Beginnings of a fully three-dimensional theory will be described with special emphasis on localized vector potential analogs of the flux function representation of plasmas in two dimensions. Also topological constraints involving helicity, and more refined measures of field line topology measuring higher order linkage will be explored. %%% One of the outstanding open problems in solar astrophysics is the existence of enormously high temperatures in the sun's atmosphere (corona), on the order of two million degrees Fahrenheit, which have baffled astrophysicists for generations. Though a number of reasonable models of coronal heating have been explored the issue still engenders intense debate. In this project one of the leading models of coronal heating, due to E.N.Parker, will be analyzed on a rigorous mathematical level. It is hoped that this will help resolve uncertainties, and clarify the basic physical mechanism involved in coronal heating. Due to the relationship with sun spots and solar flares(violent disruptions in the solar corona which propel plasma and associated magnetic fields toward the earth), the issue of coronal heating is directly tied to practical issues of variations in the earth's climate and electromagnetic interference. ***