# The Grad–Shafranov Reconstruction of Toroidal Magnetic Flux Ropes: Method Development and Benchmark Studies

The Grad–Shafranov Reconstruction of Toroidal Magnetic Flux Ropes: Method Development and... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Solar Physics Springer Journals

# The Grad–Shafranov Reconstruction of Toroidal Magnetic Flux Ropes: Method Development and Benchmark Studies

, Volume 292 (9) – Aug 15, 2017
23 pages

Publisher
Springer Netherlands
Subject
Physics; Astrophysics and Astroparticles; Atmospheric Sciences; Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics)
ISSN
0038-0938
eISSN
1573-093X
D.O.I.
10.1007/s11207-017-1134-z
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Solar PhysicsSpringer Journals

Published: Aug 15, 2017

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