Onset and Saturation of the Kink Instability in a Current-Carrying Line-Tied Plasma

An internal kink instability is observed to grow and saturate in a line-tied screw pinch plasma. Detailed measurements show that an ideal, line-tied kink mode begins growing when the safety factor $q=\frac{4{\pi }^2{r}^2{B}_z}{{\mu }_0{I}_p(r)L}$ drops below 1 inside the plasma; the saturated state corresponds to a rotating helical equilibrium. In addition to the ideal mode, reconnection events are observed to periodically flatten the current profile and change the magnetic topology.

Figure 1

A schematic of the line-tied pinch experiment. The solenoidal magnets produce a nearly uniform magnetic field between the two ends. The anode plate is on the left and drawn to highlight its the electrical connectivity. The hexagonally packed cathode array of plasma guns (shown on the right) breaks into three independently controllable regions of plasma represented by shadings in gray. The flux loops are shown in relation to the machine.

Figure 2

The current flowing to each segment of the anode (light gray, gray, black correspond to measurements at $r=8.1$, 5.2, and 2.3 cm, respectively) as well as total current ($B_z=350\mathrm{\text{Gauss}}$). The values of $q$ at the three radii corresponding to (a) are shown in (b). $B_r$ as measured by integrating the voltage from a external flux loop is plotted in (c).

Figure 3

(a) $q$ profile evolution near marginality ($B_z=270\mathrm{\text{Gauss}}$). (b) The time history of $B_r$ for a single flux loop and the best fit exponential. In (c) the logarithm of the $m=1$ mode amplitude is plotted with an $e^{\gamma t^{3/2}}$ curve and the simple exponential. The horizontal line across the bottom in (c) is the noise level of the signal and indicates that the mode first becomes detectable at 1.22 ms.

Figure 4

In (a) [$B_z=270\mathrm{\text{Gauss}}$, $I=5000\mathrm{\text{amps}}$] and (b) [$B_z=270\mathrm{\text{Gauss}}$, $I=7000\mathrm{\text{amps}}$] the frequency spectrum of a single flux loop from the saturated phase with and without reconnection. The spectra are taken for a time interval from 5 to 9 ms and correspond to the saturated phase. In (c) and (d) the solid and hashed lines are the instantaneous axial profiles of $B_r$ at fixed poloidal angle, band pass filtered, and then plotted at two different times separated by half a period. The standing wave structure shown in (c) corresponds to the mode at 25 kHz in (a), while the non-line-tied mode in (d) corresponds to the 10 kHz mode in (b).

Figure 5

(a) The current density measured by the segmented anode (solid and dashed lines correspond to before and after the relaxation event at 3.7 ms in Fig. 2). (b) The $q$ profile corresponding to the current density is plotted along with the density profile as measured by a Langmuir probe.