Objective magnetic vortex detection

Magnetic coherent vortical structures are ubiquitous in space and astrophysical plasmas and their detection is key to understanding the nature of the intrinsic turbulence in those conducting fluids. A recently developed method to detect magnetic vortices is explored in problems of two- and three-dimensional magnetohydrodynamic simulations. The integrated averaged current deviation, the normed difference of the current density at a point and the mean current density in the domain, integrated along a magnetic field line, is proved to be objective, i.e., invariant under rotations and translations of the observer. The method is shown to detect accurately the boundary of magnetic vortices in two-dimensional simulations, as well as magnetic flux ropes in three dimensions.

Figure 1

Magnetic field lines computed from a vector field measured by two different observers: (a) an observer $\mathcal{O}$ detects a saddle point structure and (b) another observer $\overline{\mathcal{O}}$ rotated and translated in relation to $\mathcal{O}$ detects a magnetic vortex.

Figure 2

“Lagrangian” and spatial coordinates of a deformable body.

Figure 3

Orszag-Tang vortices at $t_0$: (a) the magnetic field vectors, (b) an enlargement of the central region of (a), and (c) line integral convolution plot showing the magnetic field lines. The large arrows indicate the location of an $X$ point.

Figure 4

Orszag-Tang vortices at $t=1$: (a) the magnetic field vectors and (b) line integral convolution plot showing the magnetic field lines.

Figure 5

Orszag-Tang current density at $t_0$: (a) the normalized magnitude of the current density and the vortices detected as convex closed curves (black) from the contour plot of $\left|\mathit{J}\right|$ and (b) the line integral convolution plot of the magnetic field lines (dark gray) and the field lines starting from initial conditions (light gray [yellow online] circles) on the $\mathit{J}$ vortex boundary shown in (a) (red).

Figure 6

Orszag-Tang IACD at $t_0$ with $\xi =100$: (a) the normalized IACD field and the vortices detected as convex closed curves from the contour plot of IACD and (b) the line integral convolution plot of the magnetic field lines (dark gray) nearby one of the vortices. The circles are initial conditions on the vortex detected by IACD and the yellow lines are the corresponding magnetic field lines obtained by solving Eq. (2).

Figure 7

Orszag-Tang current density at $t=1$: the normalized magnitude of the current density and the small vortices detected as convex closed curves (rings) from the contour plot of $\left|\mathit{J}\right|$.

Figure 8

Orszag-Tang IACD at $t=1$, with $\xi =25$: (a) the normalized IACD field and the vortices detected as convex closed curves from the contour plot of IACD and (b) the line integral convolution plot of the magnetic field lines (dark gray) and the field lines starting from initial conditions (light gray [yellow online] circles) on an IACD vortex boundary shown in (a).

Figure 9

Random vortices Eulerian fields at $t=300$ (upper panel) and $t=386$ (lower panel): (a, d) magnetic vectors, (b, e) magnetic field streamlines, and (c, f) current density.

Figure 10

Random vortices current density field at $t=300$: (a) surface plot of the $\left|\mathbf{J}\right|$ field with rings denoting the outermost convex closed level curves surrounding selected local maxima; (b) the same as (a) but in a 2D color plot.

Figure 11

Random vortices IACD at $t=300$ from $s_0=0$ to $s_0+\xi$, with $\xi =100$: (a) surface plot of the IACD field with rings denoting the outermost convex closed level curves surrounding selected local maxima; (b) the same as (a) but in a 2D color plot.

Figure 12

Random vortices magnetic field lines at $t=300$ from $s_0=0$ to $s_0+\xi$, with $\xi =100$. The white lines denote the field lines starting from initial conditions (white circles) on an IACD vortex; the light gray (green online) lines denote field lines starting from surrounding initial conditions.

Figure 13

Random magnetic vortices merging: magnetic field vectors at (a) $t=362.34$, (b) $t=372.32$, (c) $t=384.54$, and (d) $t=386.82$, respectively; (e–h) the line integral convolution plots corresponding to the magnetic field lines of (a–d). The arrow points to two vortices that coalesce at $t=386.82$.

Figure 14

Random magnetic vortices merging in the IACD field: magnetic vortices detected from the IACD field at (a) $t=362.34$, (b) $t=372.32$, (c) $t=384.54$, and (d) $t=386.82$, respectively; (e–h) surface plots of IACD corresponding to (a–d).

Figure 15

Interlocked magnetic flux rings at $t=0$.

Figure 16

Magnetic field lines from random initial conditions at $t=100$: (a, b) two views of the same data from different angles.

Figure 17

Horizontal magnetic field components on $z=0$ at $t=100$: (a) horizontal components $(B_x,B_y)$ of the magnetic field vectors, (b) two-dimensional streamlines of the magnetic field computed using only the horizontal components, and (c) the IACD field with two magnetic vortices (black lines).

Figure 18

Magnetic flux-rope reconstruction at $t=100$ with $\xi =2000$: (a) magnetic field lines (dark gray [red online] lines) emanating from the small vortex boundary (black line), (b) magnetic field lines from the large vortex boundary; (c, d) the same as in (a) and (b), but with the inclusion of magnetic field lines emanating from random initial conditions (light gray [green online]).

Figure 19

Magnetic flux-rope reconstruction at $t=100$ with $\xi =500$: (a) the IACD field on the midplane $z=0$, with one magnetic vortex (black line), and (b) magnetic field lines (red lines) emanating from the vortex boundary (black line).