A new technique for the determination of coronal magnetic fields: a fixed mesh solution to Laplace's equation using lineofsight boundary conditions.
Abstract
A new method for computing potential magnetic field configurations in the solar atmosphere is described. A discrete approximation to Laplace's equation is solved in the domain R _{☉} ≤ r ≤R _{1}, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π (R _{1}being an arbitrary radial distance from the solar center). The method utilizes the measured lineofsight magnetic fields directly as the boundary condition at the solar surface and constrains the field to become radial at the outer boundary, R _{1}. First the differential equation and boundary conditions are reduced to a set of twodimensional equations in r, θ by Fourier transforming out the periodic φ dependence. Next each transformed boundary condition is converted to a Dirichlet surface condition. Then each twodimensional equation with standard DirichletDirichlet boundary conditions is solved for the Fourier coefficient it determines. Finally, the solution of the original three dimensional equation is obtained through inverse Fourier transformation. The primary numerical tools in this technique are the use of a finite fast Fourier transform technique and also a generalized cyclic reduction algorithm developed at NCAR. Any extraneous monopole component present in the data can be removed if so desired. The code was developed for the HAO solarinterplanetary modeling effort in response to the following specific requirements:
(1)
High resolution.
(2)
Speed in computation.
(3)
Sufficiently accurate solutions of Laplace's equation at all heights. The spatial resolution of the present code is such that measured surface lineofsight magnetic fields to a resolution of 2.8° in both latitude and longitude can be adequately treated. Comparisons with the AltschulerNewkirk code shows that the two methods are moreorless equivalent for computing largescale fields at ≈2R _{☉}but that the fixed mesh method is capable of much greater accuracy close to the Sun.
 Publication:

Solar Physics
 Pub Date:
 January 1976
 DOI:
 10.1007/BF00157566
 Bibcode:
 1976SoPh...46..185A
 Keywords:

 Boundary Conditions;
 Dirichlet Problem;
 Laplace Equation;
 Magnetic Field Configurations;
 Potential Fields;
 Solar Corona;
 Solar Magnetic Field;
 Boundary Value Problems;
 Convergence;
 Dimensional Analysis;
 Fourier Transformation;
 Legendre Functions;
 Solar Physics