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\begin{document}

\title{Grad-Shafranov equation: MHD simulation of the new solution obtained
from the Fadeev and Naval models}



\author[1]{Arian Ojeda González}%
\author[1]{Leandro Nunes dos Santos}%
\author[2]{José Juan González-Avilés}%
\author[3]{Victor De la Luz}%
\author[4]{Pablo Rubén Muñoz-Gutberlet}%
\affil[1]{Universidade do Vale do Paraíba}%
\affil[2]{CONACyT-Servicio de Clima Espacial México-Laboratorio Nacional de Clima Espacial, SCiESMEX-LANCE,}%
\affil[3]{Escuela Nacional de Estudios Superiores, Unidad Morelia, Universidad Nacional Autónoma de México}%
\affil[4]{Departamento de Física e Astronomía, Universidad de La Serena}%


\vspace{-1em}



  
  \date{May 21, 2024}


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\begin{abstract}
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\textgreater{}0\$ and the
other two at \$z\textless{}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.%
\end{abstract}%



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