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

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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. ***