Stability studies and the origin of the n=1 mode in the SPHEX spheromak experiment

Abstract

Oscillations with toroidal mode number n=1 are ubiquitous in helicity injected spheromaks and spherical tokamaks, and play a crucial role in current drive. It has been proposed that these arise from a current driven instability of the open flux tube. Stability calculations are presented to confirm this, and they are compared with experimental data from the Spheromak Experiment (SPHEX) [M. Rusbridge et al., Plasma Phys. Control. Fusion 39, 683 (1997)]. The equilibria are modelled as piece-wise constant μ profile force-free plasmas with different values for the μ in the open (μc) and closed (μa) flux regions. A stability map in μc,μa space is then calculated. The SPHEX experimental data is also reduced to the same space both as a culmination of direct single point measurements of μ and as a time history of the reconstructed equilibrium from a particular shot. The results show a favorable comparison of the stability map with experiment, both in magnitude and shape. The effect of inserting a central current-carrying rod on the stability is also discussed.

Coronal heating by magnetic reconnection in loops with zero net current

“…Indeed, a direct association between the flattening of the α profile and fluctuations with mode number n = 1 has been experimentally demonstrated in spheromaks (Willett et al 1999). Furthermore, relaxation associated with helicity-injected current drive in spheromals and spherical tokamaks has been shown to be associated with the onset of an ideal kink instability in the open flux region (Brennan et al 1999). In fact, a oscillation of the α profile across the kink instability boundary has been observed in a spheromak (Willett et al 1999;Brennan et al 1999), which is very reminiscent of the repeated relaxation events predicted by Browning & Van der Linden (2003) -although in a spheromak, a fully relaxed state is never attained due to the continual external driving and resistive dissipation.…”

“…Furthermore, relaxation associated with helicity-injected current drive in spheromals and spherical tokamaks has been shown to be associated with the onset of an ideal kink instability in the open flux region (Brennan et al 1999). In fact, a oscillation of the α profile across the kink instability boundary has been observed in a spheromak (Willett et al 1999;Brennan et al 1999), which is very reminiscent of the repeated relaxation events predicted by Browning & Van der Linden (2003) -although in a spheromak, a fully relaxed state is never attained due to the continual external driving and resistive dissipation. 3D numerical simulations of spheromak formation also demonstrate the role of ideal kink instabilities in the onset of relaxation (Sovinec et al 2001).…”

Solar coronal heating by relaxation events

“…We wish to determine the normal mode eigenfunctions and growth rates γ for such linear ideal MHD modes, which will have m = 1 for the most unstable modes. This is achieved using a numerical code CILTS, similar to the code MALTS described in , and which was also previously used to investigate instabilities in spheromaks (Brennan et al 1999). In this code, a bicubic Hermite finite element method is used to discretise the r and z dependencies.…”

“…This two parameter model, previously used with some extra constraints to model coronal loops (Melrose et al 1994) and spheromaks (Brennan et al 1999), represents the wider space of α distributions; as well as smoothly twisted profiles it allows also distributions such as an inner core twisted one way surrounded by a layer of reverse twist. The field components, for α 1 , α 2 > 0, are given by the well known Bessel function solutions to (2)…”

Achieving high flux amplification in a gun-driven, flux-core spheromak

“…Modeling of the column mode in SPHEX [20] suggests that the coupling to the spheromak limits its amplitude and can even stabilize the mode intermittently, but full stabilization has not been observed in SSPX or its simulations. In any event, it is clear that the gun voltage and thus the helicity injection rate,…”