We develop an approach of the Grad–Shafranov (GS) reconstruction for toroidal structures in space plasmas, based on in situ spacecraft measurements. The underlying theory is the GS equation that describes two-dimensional magnetohydrostatic equilibrium, as widely applied in fusion plasmas. The geometry is such that the arbitrary cross-section of the torus has rotational symmetry about the rotation axis, Z $Z$ , with a major radius, r 0 $r_{0}$ . The magnetic field configuration is thus determined by a scalar flux function, Ψ $\Psi$ , and a functional F $F$ that is a single-variable function of Ψ $\Psi$ . The algorithm is implemented through a two-step approach: i) a trial-and-error process by minimizing the residue of the functional F ( Ψ ) $F(\Psi)$ to determine an optimal Z $Z$ -axis orientation, and ii) for the chosen Z $Z$ , a χ 2 $\chi^{2}$ minimization process resulting in a range of r 0 $r_{0}$ . Benchmark studies of known analytic solutions to the toroidal GS equation with noise additions are presented to illustrate the two-step procedure and to demonstrate the performance of the numerical GS solver, separately. For the cases presented, the errors in Z $Z$ and r 0 $r_{0}$ are 9 ∘ $9^{\circ}$ and 22%, respectively, and the relative percent error in the numerical GS solutions is smaller than 10%. We also make public the computer codes for these implementations and benchmark studies.
Solar Physics – Springer Journals
Published: Aug 15, 2017
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