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

Citation

Ojeda Gonzalez, Arian et al. (2020), Grad-Shafranov equation: MHD simulation of the new solution obtained from the Fadeev and Naval models, Dryad, Dataset, https://doi.org/10.5061/dryad.5qfttdz1k

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 $z>0$ and the other two at $z<0$.  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.

Methods

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}
 {\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)}.
  \end{equation}

Usage Notes

The dataset is used to generate the figures 2 and 3 of the article.

Funding

Conselho Nacional de Desenvolvimento Científico e Tecnológico, Award: 431396/2018-3

CONACyT, Mexico, Award: Ciencia Básica 254497