Low edge safety factor operation and passive disruption avoidance in current carrying plasmas by the addition of stellarator rotational transform

Disruptions and their consequences have long been an active area of study in tokamak physics¹ and can occur for a variety of reasons. Well known among these are the operational limits associated with the plasma pressure and density, and with high plasma current or low edge safety factor operation.^2,3 These catastrophic loss of confinement events pose significant design and plasma operational challenges for burning plasma experiments such as ITER⁴ and future reactor scale tokamaks.

The disruption free operational space of tokamak plasmas due to high current operation is typically cast in terms of the magnitude of the magnetohydyrodynamic (MHD) edge safety factor, q(a). This high current, low-q(a) boundary usually limits the operation of tokamaks to an edge safety factor of greater than or equal to two, $q(a)≥2$⁠.⁵ In many cases, disruptive behavior associated with kink or tearing mode instabilities are observed to occur at even higher values of q(a) prior to reaching the value of two.^6–8 Recent experiments on circular cross-section,⁹ and shaped tokamaks^10,11 have shown operation at values of q(a) < 2 in circular, or $q95<2$ for shaped discharges with the use of active feedback control techniques. This feedback-assisted low-q operation in an axisymmetric tokamak magnetic field, B₀, uses small non-axisymmetric or three dimensional (3D) magnetic fields of relative magnitude, $B3D/B0∼10−3$⁠, to suppress unstable MHD mode activity as the q(a) = 2 operational boundary is approached.

Experiments on current carrying stellarators have shown evidence of disruption avoidance and suppression of MHD activity with the application of increasing levels of stellarator helical coil supplied rotational transform in the past.^12–14 Recently, there has been a renewed interest in understanding how the addition of strong non-axisymmetric shaping can be used to further optimize tokamak performance and decrease the susceptibility to disruptive loss of confinement.^15–17 Quasi-axisymmetric perturbing fields that could enable the use of this magnitude of 3D shaping without the poor confinement associated with helical fields have also been studied.¹⁸ 

The objective of this work is to investigate to what extent strong non-axisymmetric shaping with $B3D/B0∼0.1$ can be used to enable high current operation at low edge safety factor and passively avoid disruptive processes that occur when the edge safety factory approaches or is below q(a) = 2. The use of such strong 3D fields has recently been shown to lead to suppression of unstable vertical motion due to plasma elongation.¹⁹ The 3D shaping is adjusted using the rotational transform of applied helical fields within the Compact Toroidal Hybrid (CTH) device.²⁰ Current is driven within a fully established stellarator equilibrium, and the fractional transform, defined as the ratio of the imposed external rotational transform from the stellarator coils to the total rotational transform, $f=ι̷vac(a)/ι̷tot(a)$⁠, is varied along with the plasma total edge transform or safety factor. The total transform is given by the sum of the transform due to the external coils and that contributed by the plasma current, $ι̷tot=ι̷vac+ι̷p=1/q$⁠. The vacuum transform is computed for the equilibrium surfaces of the vacuum magnetic field configuration resulting from the stellarator coil currents. The discharges discussed in this study with a vacuum transform of 0.02 represent the closest CTH is able to operate to that of a standard tokamak with no 3D shaping. This limitation is due to the need for fast vertical field control of the plasma position, at low vacuum transform, during discharge startup presently not available on the device. We employ the value of the vacuum transform at the plasma edge, $ι̷vac(a)=1/qvac(a)$ along with the fractional transform, f, as a proxy for the amount of 3D shaping supplied by the helical fields. This letter is organized as follows: we briefly overview the experimental details of CTH relevant to the current study, discuss the basic phenomenology of low-q(a) disruptions observed on CTH, comment on observations of the MHD mode structures associated with the low-q(a) disruption process, and delineate the effects of strong 3D shaping by addition of vacuum transform on low-q(a) disruptivity.

The CTH experiment is a low aspect ratio (⁠$R=0.75 m,a∼0.2 m, R/a∼4$⁠), five–field period torsatron with a continuous $ℓ=2$ helical winding.²⁰ Fully established stellarator equilibria with plasmas having line-averaged densities of $ne=0.1–0.2×1019 m−3$ and electron temperatures of $Te<20 eV$ are obtained with Electron Cyclotron Resonance Heating (ECRH) using a magnetic field strength of $⟨|B|⟩=0.5 T$⁠. The vacuum transform of the stellarator equilibrium is adjusted by varying the ratio of currents in the helical and toroidal field coils. At the plasma edge, or last closed flux surface, the value of the vacuum transform can be varied an order of magnitude in the range of $0.02≤ι̷vac(a)≤0.4$⁠. Fig. 1 shows the vacuum magnetic flux surface configuration only due to the external coils for a representative value of edge vacuum transform, $ι̷vac(a)=0.05$⁠, used in this study. The helical modulation of the magnitude of the magnetic field shown in Fig. 1 is approximately $0.15 T$ about the average field value.

Ohmic currents are driven on the pre-existing stellarator vacuum surfaces by energizing a central solenoid. Typical operating parameters for the ohmic phase of the discharge are $ne≤5×1019 m−3$ and $Te<150 eV$⁠, but for the ensemble of discharges in this study, the line averaged density was kept in the range of $ne=0.5–0.9×1019 m−3$ to ensure the plasmas are not near the Greenwald density limit. In CTH discharges with plasma current, up to $∼95%$ of the total rotational transform can be provided by the plasma current, with the resultant q-profiles similar to that of a typical tokamak. Yet, the overall magnetic surface structure clearly retains the 3D nature of the underlying stellarator equilibrium.

The evolution of a typical low-q discharge on CTH is shown in Fig. 2. The vacuum transform for this discharge is $ι̷vac(a)=0.024$ with a fractional transform at peak current of f = 0.04. The time history of the plasma current and the edge safety factor are shown in Figs. 2(a) and 2(b). The reconstructed equilibria are calculated using the V3FIT code.^21,22 The output of one B-dot probe is shown in Fig. 2(c). Fig. 2(d) shows the plasma density, which was kept essentially constant over the quiescent portion of the discharge. The initial current rise phase of the discharge from $t=20 ms$ to $30 ms$ has a strong current ramp, with $dIp/dt≈3 MA/s$⁠, giving rise to broad current profiles with steep edge gradients. Bursts of magnetic fluctuations with $m/n=4/1$ and 3/1 are visible in Fig. 2(c) as the q = 4 and q = 3 surfaces exit the edge of the plasma. After the initial current rise, a plasma current ramp rate of $dIp/dt≈1 MA/s$ is maintained from approximately $t=30 ms$ to $50 ms$⁠, again leading to relatively strong edge current gradients in the discharges studied, as the q = 2 surface moves into the vacuum region outside the last closed flux surface. Periods when the observed magnetic fluctuations exhibit helicities of $m/n=2/1$ and 3/2 are indicated in Fig. 2(c). Note that the q = 1.5 surface does not enter the vacuum region before peak plasma current is reached. The plasma current subsequently decreases prior to the current disruption. The loss of confinement occurring at $t=70 ms$ is accompanied by the usual tokamak disruption signatures of a positive plasma current spike, negative loop voltage spike (not shown), and rapid decay of the total plasma current. The sudden rise of the plasma density just after the disruption begins is conjectured to be due to particle and heat flow to plasma limiting surfaces, caused by the loss of confinement, liberating neutral hydrogen from the limiters which is then briefly ionized as the plasma current quenches.

Fig. 3 depicts the measured poloidal fluctuation amplitude, $δBθ(t)$⁠, of the $m/n=2/1$ and 3/2 modes as determined by biorthogonal decomposition²³ during the time windows indicated by the gray vertical bars in Fig. 2. Measurement of the magnetic fluctuations is accomplished with an array of 36 sensors distributed poloidally, and a toroidal array of 10 sensors. We expect that wall eddy currents and wall stabilization effects are not important factors in the observed MHD mode dynamics, given that the vacuum vessel wall radius is large with respect to the average plasma minor radius obtained from V3FIT reconstructions, $rwall/a=1.5$⁠. The measured $m/n=2/1$ fluctuations shown in Fig. 3(a) are small, relative to the 3/2 mode observed later in Fig. 3(b). The $m/n=2/1$ mode can be bursty, although the case depicted has a relatively coherent fluctuation in Fig. 3(a). The amplitude of the 2/1 mode depicted in Fig. 3(a) does not increase when the q = 2 surface passes into the vacuum outside of the last closed flux surface. Large $m/n=3/2$ oscillations are observed when the edge safety factor approaches $q(a)=1.5$⁠. This 3/2 mode along with an $m/n=4/3$ instability are implicated by the external magnetic sensors as the primary cause of the low-q(a) disruptions. Soft x-ray emission measurements indicate that a non-sawtoothing, periodic $m/n=1/1$ mode is also present in the core plasma at the time of disruption. There is still a substantial current ramp rate of $dIp/dt≈1 MA/s$ during the time window when the q = 2 surface moves into the vacuum region, yet the 2/1 mode remains at small amplitude for the ensemble of plasmas studied and does not appear to play a significant role in the disruption process.

The effect of applying varying amounts of vacuum transform on these disruptions is shown in Fig. 4. The evolution of the plasma current and edge safety factor is shown in Figs. 4(a) and 4(b) for edge vacuum transform values of $ι̷vac(a)=0.02$⁠, 0.03, and 0.05. The plasma current contribution to the total transform or q(a) evolution was programmed to be the same for the three cases. The slight differences in edge q(a) evolution are due primarily to the amount of vacuum field. The plasmas evolve from complete collapse of the plasma current at the lowest attainable values of vacuum transform to a partial collapse of the plasma current at intermediate values of $ι̷vac(a)$ and then cease to occur as the external helical transform is raised further. Those plasmas that do suffer a current disruption are observed to terminate with a vertical displacement event (VDE). The differing level of magnetic fluctuation activity for the $ι̷vac(a)=0.05$ and $ι̷vac(a)=0.03$ examples are shown next in Figs. 4(c) and 4(d).

The effect of stellarator transform on these low-q(a) disruptions has been studied for an ensemble of 526 discharges. For this study, the vacuum rotational transform is varied while keeping the plasma current ramp rates comparable. A plot of q(a) at peak plasma current versus $ι̷vac(a)$ is shown in Fig. 5(a) for the entire ensemble. The value of the edge rotational transform at peak plasma current has been used as a convenient parameter to compare both the disrupting and non-disrupting shots. The disruptivity of these low edge safety factor plasmas is observed to have three distinct regimes depending upon the level of external helical coil supplied vacuum transform, $ι̷vac(a)$⁠, imposed prior to initiation of the ohmic phase of the discharge. A regime of operation is observed where the plasma always experiences a fast current quench when the edge vacuum transform is, $ι̷vac(a)≲0.03$⁠, a regime with fast and partial quenches of the plasma current and the start of disruption suppression when the edge vacuum transform is in the range, $0.03≲ι̷vac(a)≲0.07$⁠, and disruption free operation at these low total edge safety factor values, q(a) < 2, when the edge vacuum transform is greater than $ι̷vac(a)≳0.07$⁠. Fig. 5(b) shows the same data set plotted as a function of the fractional transform, f, rather than $ι̷vac$(a). The fractional transform required for complete disruption avoidance is seen to be approximately $f≈0.1$ or 10% of the total transform at the edge of the plasma.

The detailed reasons for the lack of strong n = 1 kink mode activity after the current rise phase of the discharge will be the subject of continued study. The hypothesis that the observed disruption avoidance is due to the decoupling of the destabilizing current gradient from the low order n = 1 rational surface in the transform profile, invoked to explain previous disruption avoidance experiments,¹³ will be tested using a Hall probe to measure the edge current profile. We note, in closing, that the previous work on this topic quoted a value of the fractional transform, $f≳0.3$⁠,^13,14 to suppress predominately $m/n=2/1$ tearing mode driven disruptions.

The authors would like to thank John Dawson for technical assistance throughout the course of this work. This research was supported by U.S. Department of Energy Grant No. DE-FG-02-00ER54610. Data for the figures presented in this article may be found at http://www.auburn.edu/academic/cosam/departments/physics/research/fusion/publications.htm.