DOI: 10.5061/dryad.5qfttdz1k
title: Grad-Shafranov equation: MHD simulation of the new solution obtained from the Fadeev and Naval models

description: This article aims to obtain a new analytical solution of a specific form
 of the Grad-Shafranov (GS) equation using Walker's formula. The new
 solution has magnetic field lines with X-type neutral points, magnetic
 islands and singular points. The singular points are located on the
 x-axis. The X-points and the center of the magnetic islands do not appear
 on the x-axis an island appears at $z&gt;0$ and the other two at
 $z&lt;0$.  The aforementioned property allows us to use this
 solution as an initial condition at $t=0$ s in an magnetohydrodynamic
 (MHD) numerical simulation by excluding the singular points of the
 solution, i.e., the x-axis, and maintaining the magnetic structure of the
 islands, as well as the X-type neutral points.  For this, we
 numerically solve the equations of the classical ideal MHD in two
 dimensions using the Newtonian CAFE code.  The code is based on
 high resolution shock capturing methods using the Harten-Lax-van
 Leer-Einfeldt (HLLE) flux formula combined with MINMOD
 reconstructor. The MHD simulation shows a very fast dissipation
 in less than one second of the magnetic islands present in the initial
 configuration.  Almost all structures left the integration region
 at $13.2$ s, and the magnetic field vector reverses its polarity very
 quickly.  In addition, our simulation allows us to observe the
 fast temporal evolution of the magnetic islands turning into elongated
 current sheets.  As a limitation of the model, the difficulty in
 relating it to a physical system because of fast temporal evolution is
 considered.
description: We numerically solve the equations of classical ideal MHD in two
 dimensions using the Newtonian CAFE code (J. J. Gonz ́alez-Avil ́es et
 al., 2015; J. Gonz ́alez-Avil ́es & Guzm ́an, 2018). In
 particular, the ideal MHD equations are solved on a single uniform
 cell-centered grid using the method of lines with a third-order
 Runge-Kutta time integrator. In order to use the method of lines, the MHD
 equations are discretized using a finite volume approximation with high
 resolution shock capturing methods. For this, we first reconstruct the
 variables at cell interfaces using the MINMOD limiter (Harten et al.,
 1997). On the other hand, the numerical fluxes are built with the
 Harten-Lax-van Leer-Einfeld (HLLE) approximate Riemann solver formula
 (Harten et al., 1983; Einfeldt, 1988). The numerical
 evolution of initial data involving Maxwell equations leads to the
 violation of the divergence free constraint equation, developing as a
 consequence unphysical results like the presence of a magnetic net charge.
 Among the several methods available for controlling the growth of the
 constraint violation (T ́oth, 2000), in our simulation we use the extended
 generalized lagrange multiplier (EGLM) method (Dedner et al.,
 2002). The MHD equations are solved as an initial value
 problem. For this reason we define the set of initial conditions
 corresponding to the variables derived from
  \begin{equation}<br>  {\scriptstyle \Psi(X,\;Z)= \ln
 \left(\frac{\left(1+f_p^2\right) \left(\cosh ^2(b Z) +\sin ^2(b
 X)\right)-2 f_p \sqrt{1+f_p^2} \cos (b X) \sinh (b Z)}{2 b
 \sqrt{\left(1+f_p^2\right) \left(\cosh ^2(b Z)-\sin ^2(b
 X)\right)}}\right)}.<br>   \end{equation}
description: The dataset is used to generate the figures 2 and 3 of the
 article.