Flux Rope Dynamics: Experimental Study of Bouncing and Merging

We show experimentally for the first time that two mutually attracting flux ropes may bounce back instead of merging together, leading to a variety of dynamics not expected from a two-dimensional model. Attraction forces due to flux rope currents compete with repulsion from field line bending of in-plane and out-of-plane magnetic fields and elastic plasma compression. Bouncing dynamics occurs if the line-bending force due to an out-of-plane field dominates. Otherwise, the ropes merge. Further reduction in the field line-bending force results in violently erratic magnetic states.

Figure 1

Schematic of the RSX showing plasma guns, kinking flux ropes, the conical external anode that enables choice of the axial boundary condition [6], the coordinate system, and the geometry. 3D probe positioners are located at several vessel ports. The radial scale is exaggerated to highlight the plasma structure.

Figure 2

Magnetic data in the $x\mathrm{\text{−}}y$ cut plane at $z=46\mathrm{cm}$, showing magnetic field arrows (max $B_{xy}=8\mathrm{G}$) and contours $J_z=\nabla \times B_{xy}/{\mu }_0$ for (a) $t=1.202\mathrm{ms}$ [red circles indicate where $\alpha (z)$ pierces the $x\mathrm{\text{−}}y$ plane], (b) $t=1.212\mathrm{ms}$, (c) $t=1.221\mathrm{ms}$, and (d) $t=1.225\mathrm{ms}$. For the definition of $\alpha (z)$, see Fig. 4a. The two flux ropes start out vertically aligned and rotate clockwise as mutual flux rope attraction plus kink instability drive the column displacement into two gyrating helices. $B_{z0}=200\mathrm{G}$, and $I_p$ ramps up 50% to 200 A, approximately 140% of the kink threshold. The device axis is at $(x=-15.5,y=-1.5)\mathrm{cm}$.

Figure 3

The time evolution of distance $\delta (t)$ and deformation $\epsilon (t)$ between the two flux ropes, as they approach each other and finally bounce apart. Shown are model predictions for the best fit to the data $\chi =20$ (solid curve) and excessive ($\chi =100$, dash-dotted curve) and very small ($\chi =5$, dash-dashed curve; merging reconnection) repulsion force.

Figure 4

(a) $B_z$ field line shape $\alpha (z)$ from the theoretical kink eigenfunction (solid line), distorted by plasma flow from the gun at the left to an external anode at the right, with data points (triangles and diamonds), and the bouncing flux rope perturbed shape $\gamma (z)=\alpha (z)+\beta (z)$. (b) Three data points (diamonds) at $z=3,43,46\mathrm{cm}$ from the gun and curve fit $\beta (z)$ with parameters.

Figure 5

Magnetic data in the $x\mathrm{\text{−}}y$ cut plane at $z=46\mathrm{cm}$ for (a) $t=1.202\mathrm{ms}$, (b) $t=1.212\mathrm{ms}$, (c) $t=1.221\mathrm{ms}$, and (d) $t=1.225\mathrm{ms}$. $B_{z0}=100\mathrm{G}$, and $I_p$ is 220 A.

Figure 6

Magnetic data in the $x\mathrm{\text{−}}y$ cut plane at $z=46\mathrm{cm}$ for (a) $t=1.202\mathrm{ms}$ and (b) $t=1.212\mathrm{ms}$. $B_{z0}=30\mathrm{G}$. and $I_p$ is 200 A.