Sawtooth oscillation behavior with varying amounts of applied stellarator rotational transform

Sawteeth are periodic relaxations of the plasma temperature profile, and sometimes the plasma density profile, which were first observed in the ST tokamak in 1974.¹ They are primarily a core phenomenon usually consisting of three stages: (i) the ramp phase, (ii) the precursor phase, and (iii) the crash phase. The ramp phase of the sawtooth oscillation covers the time when the core of the plasma is being heated. Increasing core temperatures peak the current profile (due to reduced resistivity in the core) which lowers the central safety factor, q, below one, inducing a MHD instability within the plasma. During the precursor phase, the instability grows large enough to cause a rapid crash, leading to the re-organization of flux and expelling energy from the plasma core. Soft X-Ray (SXR) diagnostics¹ and two-dimensional images of the electron temperature evolution² confirm that thermal energy is transported from the core to outside the inversion radius, r_inv, during this crash phase. This expulsion of thermal energy from the plasma core increases the temperature out to the so-called mixing radius which is typically 25%–50% of the plasma minor radius, a, in a tokamak plasma.³ The sawtooth oscillation repeats itself as nested magnetic flux surfaces are restored in the core of the plasma after the crash phase.

Sawtooth oscillations are primarily a core phenomenon and do not, by themselves, typically lead to a termination of plasma confinement. Depending on the size of the inversion radius and core temperature drop, sawtooth oscillations are not inherently bad for a confined plasma. Small sawteeth can be beneficial by flushing impurities and helium ash from the core of the plasma.^4,5 Conversely, sawteeth with large temperature perturbations and inversion radii can have deleterious effects on tokamak discharges. Plasmas with large sawteeth are more susceptible to Edge-Localized Modes⁶ (ELMs) and may lead to the degradation of core confinement. Long period sawteeth can seed Neoclassical Tearing Modes (NTMs), leading to confinement degradation, possible mode locking, and discharge disruption.⁷ Therefore, the control of sawteeth is an important issue for tokamaks with large stored energies, such as ITER.^3,8 Control of the sawtooth oscillation can be achieved through active means including heating the electrons near the q = 1 surface, modifying the shear of the q–profile, changing the toroidal flow, and heating ions in the plasma.³ 

Studies have been performed to determine the effects of plasma shaping on the sawtooth oscillation. The existing studies focus on two-dimensional shaping such as increasing the elongation or the triangularity of the q = 1 flux surface in an axisymmetric plasma. Typically, higher elongation destabilizes the m = n = 1 mode, leading to smaller sawteeth,⁹ while increasing the triangularity stabilizes the sawtooth oscillation.¹⁰ Studies on RFX-mod and DIII-D also show that externally applied, n = 1 fields reduce the sawtooth period and amplitude.¹¹ 

Sawteeth and sawtooth-like behavior are observed in non-axisymmetric machines such as the Large Helical Device,¹² Compact Helical System,¹³ and Heliotron E.¹⁴ However, the sawteeth observed in these machines are not tokamak-like and are either associated with the q = 2 rational surface and the n = 1, m = 2 mode, or with q–profiles having the values of q = 1 at multiple radial locations.

This work explores the effect of the increasing levels of three-dimensional magnetic field shaping from external stellarator coils on the sawtooth oscillation in the Compact Toroidal Hybrid (CTH). Simulations using NIMROD¹⁵ for CTH-like plasmas with similar experimental parameters and three-dimensional magnetic fields found that the sawtooth period decreases as the three-dimensional shaping increases,¹⁶ consistent with the experimental results presented in this paper.

This paper is organized as follows: Sec. II describes the CTH device and diagnostics, as well as the computational techniques used to determine the plasma properties. Section III discusses the techniques used to extract the sawtooth properties, the experimental procedure used to systematically vary the applied fractional transform, and the similarity between the sawteeth observed in CTH and typical tokamak sawteeth. The effects of the application of three-dimensional magnetic fields on the sawtooth dynamics are presented in Sec. IV. Section V summarizes the main conclusions of this study and includes a discussion of the results.

Studies of the dynamics of sawtooth oscillations with applied three-dimensional magnetic fields are performed in the CTH tokamak/stellarator hybrid device.¹⁷ CTH can be operated as a current-free torsatron, where the external magnetic fields generate the required rotational transform for confinement, or in a hybrid mode, where a plasma current is ohmically driven in a pre-existing plasma with magnetic surfaces. CTH is equipped with a set of toroidal field coils which can be used to either accentuate or reduce the toroidal field produced by the ℓ = 2, m = 5 helical field coil. The ratio of currents within the toroidal and helical field coils can be adjusted to vary the rotational transform profile. This flexibility of magnetic field configurations makes CTH an ideal platform to study the effects of varying amounts of the external stellarator transform on the sawtooth instability.

Figure 1(a) demonstrates the range of vacuum rotational transform profiles, $ι−vac(s)$⁠, used within this study made possible by adjusting the ratio of currents within the toroidal and helical field coils. Here, s is the toroidal flux normalized to that at the last closed flux surface. It is a radial-like coordinate with a value of s = 0 at the magnetic axis and a value of s = 1 at the last closed flux surface. These reconstructions used fifteen equally spaced normalized toroidal flux coordinates since we are primarily concerned with the edge value. Vacuum rotational transform is the term used for the applied three-dimensional stellarator transform that exists prior to driving the plasma current ohmically. The discharges presented in this paper have vacuum rotational transform profiles with values at the edge, $ι−vac(a)=ι−vac(s=1)$⁠, varying by approximately an order of magnitude from 0.02 to 0.15.

During ohmic operation, the rotational transform generated by the plasma current provides up to 97% of the total rotational transform. The total rotational transform is defined as the sum of the rotational transform from the ohmically driven plasma current, $ι−p$⁠, and the vacuum rotational transform, $ι−(s)=ι−p(s)+ι−vac(s)$⁠. Figure 1(b) shows the applied vacuum rotational transform (dashed line) and the corresponding total rotational transform after a plasma current of 35 kA is induced in the plasma (solid line). The transform profiles are calculated from three-dimensional reconstructions using V3FIT¹⁸ with the Variational Moments Equilibrium Code^19,20 (VMEC) for the MHD equilibrium solver.

We define the fractional rotational transform as $f=ι−vac(a)/ι−(a)$⁠. The fractional transform is used as a proxy for the amount of three-dimensional shaping applied to the plasma and is directly correlated with the mean elongation, discussed in Sec. IV B. The fractional transform of CTH magnetic configurations can vary from 2% to 100%, where a value of f = 0% would represent an axisymmetric tokamak and f = 100% represents a current-free stellarator. For the discharges presented in this paper, the fractional transform is varied from 4% to 42%.

The sawtooth properties are measured using three two-color SXR cameras²¹ viewing the plasma at the same toroidal location, $ϕ=252°$⁠. Each two-color camera has a bandwidth of 55.5 kHz and an acquisition frequency of 500 kHz. The chords for each camera are shown in Fig. 2 with a typical last closed flux surface plotted in black. The camera at the mid-plane, SXR_M, is used to measure the sawtooth inversion radius due to the up-down symmetry of the flux surfaces at this toroidal location. Other sawtooth instability properties, such as the sawtooth period, are determined from the measured values averaged over all three cameras.

Sawtoothing plasmas are observed during the ohmic phase of CTH discharges. The characteristics of a typical sawtoothing discharge are shown in Fig. 3. The plasma current, Fig. 3(a), peaks at approximately 30 kA, about 30 ms into the discharge. The line-integrated electron density, Fig. 3(b), is approximately constant during the discharge at 2 × 10¹⁹ m⁻³. Figure 3(c) shows two SXR signals from the SXR_M two-color camera. An expanded view of the SXR signals shown in Fig. 3(d) is over the grey-boxed region shown in Figs. 3(a)–3(c). Standard sawtoothing behavior is observed in the SXR signal from a chord viewing the center of the plasma, while inverted sawteeth are seen in the SXR signal from a chord that views the plasma just outside the inversion radius.

Singular value decomposition (SVD) analysis²² is used to find the inversion radius during the sawtoothing segment highlighted in grey in Fig. 3. Figures 4(a)–4(f) show the largest three modes of the SVD analysis. The spatial components are on the left and the temporal modes multiplied by the weight of the mode are on the right with subscripts referring to the mode number. The first spatial–temporal pair [v₁, W₁u₁; (a) and (d)] illustrates the time evolution of the sawtooth rise and crash, while the second temporal mode [W₂u₂; (e)] exhibits clear inverted sawtooth behavior. The third spatial mode, Fig. 4(c), reveals the m = 1 radial fluctuation. The corresponding temporal mode component, Fig. 4(f), shows the m = 1 mode oscillating in time prior to reaching its maximum amplitude at the sawtooth crash. The m = 1 mode observed for sawteeth in CTH plasmas is discussed further in Sec. IV A.

Figure 4(g) shows a contour plot of reconstructed signals combining the first and second modes of the SVD. A linear fit of the signal is subtracted from each channel to suppress the overall SXR emissivity increase during the course of the discharge and to accentuate the sawtoothing behavior. Four sawtooth oscillations are evident in Fig. 4(g), each illustrating a rise in core emissivity and an abrupt crash. The central SXR channels (channels 8–13) show an increase in core emissivity for approximately 0.5 ms followed by an abrupt emissivity decrease that lasts on the order of 0.1 ms. The inversion radius of the sawteeth is clearly evident from the increase in emissivity measured by the outer SXR channels (channels 4–7 and 14–17) relative to the core channels. The dashed white lines show the channel positions that delineate the core and outer channels determined through SVD analysis by where the second spatial mode crosses zero. The sawtooth inversion radius is determined through a weighted average of the impact parameters, p, for the channels adjacent to the calculated inversion channel. As an example, for a calculated inversion channel of 10.3, the inversion radius would be determined by $rinv=(11−10.3)*p10+(10.3−10)*p11$⁠, where p₁₀ and p₁₁ are the impact parameters for SXR channels 10 and 11.

A total of 155 sawtoothing discharges with varying levels of the fractional transform, density, and total rotational transform are analyzed for this study. Due to the changing values of the density, plasma current, and their radial profiles over the course of a discharge, the analysis of the sawteeth is divided into time segments containing three to five sawteeth. Only time segments near the peak plasma current are selected for the database as is illustrated by the grey region in Fig. 3(a). Each data point in the scatter plots shown in this paper represents a separate CTH sawtoothing discharge. Electron densities for times near peak plasma current range from 0.6 to 3.5 × 10¹⁹ m⁻³. Each line integrated density measurement,²³ $∫nedl$⁠, is divided by the reconstructed path length to find the estimated electron density. The plasma current is systematically varied from 15.7 kA to 60.8 kA to scan a range of total rotational transform values. The edge vacuum rotational transform, $ι−vac(a)$⁠, is found during the pre-ohmic phase of the discharge, while the total transform, $ι−(a)$⁠, is found at the center of the sawtoothing window.

In tokamak devices, the normalized inversion radius (the sawtooth inversion radius divided by the plasma minor radius) is proportional to the total rotational transform at the edge of the plasma.²⁴ Figure 5(a) shows the normalized inversion radius of the sawtooth discharges in this CTH study plotted as a function of the total edge rotational transform. It is evident that the volume of core plasma affected by the sawtooth oscillation scales with the total rotational transform, similar to the behavior observed in tokamaks. In Fig. 5(b), the normalized inversion surface radius is instead plotted as a function of the fractional transform or the relative amount of vacuum transform compared to the total transform. From this figure, it can be seen that the normalized inversion radius does not scale strongly with the amount of three-dimensional shaping. We note that there is a relatively broad range of inversion radii at a low fractional transform, but this range decreases as more fractional transform is added to the plasma equilibrium.

The sawteeth in CTH, for plasmas with monotonically decreasing rotational transform profiles dominated by the plasma current, have characteristics of those observed in tokamaks despite the three-dimensional confining magnetic fields. The observed properties discussed previously include (1) a central emissivity rise and then a sudden crash with a well-defined inversion radius, (2) the presence of an m = 1 emissivity fluctuation, and (3) the normalized inversion radius scales with the total edge rotational transform. These properties make the observed sawteeth more tokamak-like compared to sawteeth observed in other non-axisymmetric machines.

Three discharges are selected to highlight the effects of varying the applied three-dimensional field on the sawtooth oscillation prior to a discussion and further confirmation that the behavior observed within these three discharges is seen throughout the sawtoothing database. The three discharges, shown in Fig. 6, have similar peak plasma currents (a), line integrated electron densities (b), and central SXR signal currents (c). The applied three-dimensional field varies between the shots with fractional transforms of 0.11 (shown in black), 0.21 (blue), and 0.33 (orange). Figure 6(d) shows the same central SXR signals for the discharge over a time window of a few milliseconds, but the signals are shifted in time so that the first sawtooth oscillation for each discharge begins at t = 0 ms and offset at the signal level for clarity. The initial time, t = 0 ms, in Fig. 6(d) corresponds to approximately 15 ms in Figs. 6(a)–6(c). The average signal over the sawtoothing window is subtracted from each of the three SXR signals, and a diamond is plotted on each of the SXR signals after the crash of the sixth sawtooth oscillation. It is clear that the time required for six sawtooth oscillations is the longest for the low fractional transform case and increasing the fractional transform decreases the sawtooth period. However, the sawtooth crash time is not dependent on the amount of the fractional transform applied. The average sawtooth crash time for these cases is 85.6 μs for the lowest fractional transform, 106.0 μs for f = 0.21, and 102.3 μs for f = 0.33. Additionally, with the lowest fractional transform applied, the amplitude of the sawtooth oscillation is approximately 13 nA in contrast to the 8 nA amplitude for the high fractional transform discharge.

The dynamics of the m = 1 mode during sawtooth oscillations are different for the three-dimensional shaped plasmas compared to axisymmetric tokamaks. In axisymmetric tokamaks, the m = 1 mode typically exhibits a growing sinusoidal-like oscillation, reaching the maximum amplitude prior to the crash. Immediately following the sawtooth crash event, it is often the case that no m = n = 1 mode is detected. However, there have been observations where the m = n = 1 mode is present throughout the sawtooth cycle in axisymmetric tokamaks.^25,26

The third temporal mode from SVD analysis, u₃, extracted through SXR emissivity measurements for the discharges presented in Fig. 6 is shown in Fig. 7. Above the third temporal mode for each discharge is the corresponding central SXR signal. Each signal is taken approximately 10 to 14 ms into the discharges shown in Fig. 6 to ensure that the sawtooth instability is established. A linear fit is subtracted from each SXR signal, and the signals are shifted in time to start at 0 ms. The u₃ mode represents the magnitude of the m = 1 oscillation with time. The magnitude of the u₃ mode peaks during each sawtooth crash event, is not equal to zero for an extended amount of time, and does not exhibit a clear growing sinusoidal pattern (contrary to the mode behavior in axisymmetric tokamaks). This behavior suggests that a full reconnection²⁷ does not take place during the sawtooth crash events.

In axisymmetric tokamaks, increasing the mean elongation is found to destabilize the m = n = 1 mode, leading to smaller, more frequent sawteeth.⁹ Plasmas within CTH are non-axisymmetric, and the average elongation, κ, varies with the amount of the fractional transform imposed. The elongation typically varies from 1.76 to 1.15 for the full and half field periods at a low fractional transform, $f(a)≤0.11$⁠, while at a high fractional transform, f(a) > 0.3, the elongation is around 2.39 at the full field period and 1.34 at the half field period.

A mean elongation, $κ¯$⁠, of the outermost flux surface is then used to compare the three-dimensional elongated shaping of CTH to axisymmetric tokamak plasmas. The mean elongation is calculated at the last closed flux surface with V3FIT reconstructions from the volume, the toroidally averaged cross-sectional area, and the surface area of the plasma²⁸ using only external magnetic diagnostics. More comprehensive equilibrium reconstructions utilizing SXR emissivity measurements along with the magnetics²⁹ were performed on the three discharges presented in Fig. 6 to verify if the elongation found at the outermost flux surface can be used as a proxy for the elongation at the q = 1 surface. The more comprehensive reconstructions show that the mean elongation at the q = 1 surface increases with the increasing fractional transform, 1.23 for f = 0.11, 1.28 for f = 0.21, and 1.37 for f = 0.33. The same trend is observed in the reconstructions using only external magnetic diagnostics for the mean elongation at the outermost flux surface, 1.57 for f = 0.11, 1.83 for f = 0.21, and 1.94 for f = 0.33. Additionally, the mean elongation between the two reconstructions is scaled down by a similar amount for each fractional transform. The mean elongation from the more comprehensive reconstruction is 78% of the mean elongation found through the magnetic diagnostic only reconstructions for the lowest rotational transformation. Similarly, the reconstructions are scaled down by 70% for f = 0.21 and 71% for f = 0.33. A similar scaling factor and the increasing trend show that the elongation of the outermost flux surface found through reconstructions using only external magnetic diagnostics can be used as a proxy for the amount of elongation at the q = 1 surface of CTH plasmas.

Figure 8(a) shows the sawtooth period as a function of the edge fractional transform, color coded with ranges of mean elongation. There is a significant decrease in the sawtooth period, approximately a factor of two, as the fractional transform increases from 0.04 to 0.42. The sawtooth amplitude in Fig. 8(b) shows a wide range of amplitudes at a low fractional transform. However, large amplitude sawteeth are not observed as the level of fractional transform increases. This observed decrease in the sawtooth period and the absence of large amplitude sawteeth are also correlated with increasing mean elongation since the mean elongation increases with the fractional transform.

Despite the strong correlation between the sawtooth period and fractional transform (and mean elongation) presented in Sec. IV B, it is important to eliminate other possible systematic experimental effects that may also contribute to changes in the sawtooth period. The sawtooth period is the combination of the sawtooth rise and crash time, and the overall decrease in the sawtooth period is largely driven by a decrease in rise time. The observed crash time within this database ranges from 0.09 to 0.13 ms and does not have a clear correlation with the fractional transform, indicating that the final reconnection dynamics causing the crash are not significantly affected by the three-dimensional shaping.

There is a relatively large variation in electron densities and ohmically driven plasma currents within the database, raising the question as to whether the variation of ohmic heating power and electron densities could have a systematic effect on the sawtooth rise time through changes in the ohmic heating rate inside the q = 1 surface. To investigate this possibility, we estimated the ohmic heating rate within the q = 1 surface through two different methods since a precise measurement is not possible with the current diagnostic suite installed on CTH. First, the ohmic heating power is estimated through $POH=ηsJ2$⁠, where the Spitzer resistivity, η_s, is calculated from the estimated electron temperature measured by the two-color camera measurement.²¹ Second, an additional estimation of the ohmic heating rate is found through the sawtooth ramp rate (sawtooth amplitude rise divided by the rise time) following Jahns.³⁰ Neither of these estimates show any clear correlation of the estimated ohmic heating power inside the q = 1 surface and the fractional transform or sawtooth period. However, these estimates require considerable approximations such as assuming that the plasma is a straight cylinder, ignoring profile-flattening effects, and that the electron-ion energy transfer is low. The validity of these estimates is difficult to determine; therefore, we cannot eliminate the possibility that a change in the ohmic heating rate is also a contributing factor to the decrease in the sawtooth period as the fractional transform increases.

The entire database of sawtoothing discharges discussed in this study has a peak plasma current ranging from 15.7 to 60.8 kA, while the core line integrated electron density varies from 0.6 to 3.5 × 10¹⁹ m⁻³. Since the plasma current is the main contributor to the total rotational transform (up to 97%), there could be a correlation between the peak plasma current and the sawtooth period decrease with the fractional transform. Figure 9 shows the entire database as a function of peak plasma current and density, color coded based on the sawtooth period. As can be seen, there appears to be a lack of high sawtooth period discharges (>0.45 ms; blue squares) for plasma currents less than 20 kA. However, there is no strong correlation of the sawtoothing period for discharges greater than 20 kA. Additionally, there does not appear to be any strong correlation between the sawtooth period and the plasma current despite having a large distribution of plasma currents.

The electron density has been shown on the TCV tokamak to increase the sawtooth period by approximately a factor of two as the electron density increased from 1 to 5 × 10¹⁹ m⁻³.¹⁰ However, the increase in the sawtooth period with increasing density was less pronounced at lower electron densities (<3.1 × 10¹⁹ m⁻³), which encompasses the majority of discharges presented here. There is a slight increase in the sawtooth period (or lack of low sawtooth period discharges) in the plasmas presented in this study for densities greater than 2.5 × 10¹⁹ m⁻³ shown on the right side in Fig. 9. The vast majority of the discharges scattered between 1 and 2 × 10¹⁹ m⁻³ do not appear to have any correlation with the factor of two decrease in the sawtooth period observed with the edge fractional rotational transform.

The effects of three-dimensional shaping on the sawtooth instability are explored by varying the density, the total rotational transform, and the amount of the three-dimensional field applied (fractional transform) from the external stellarator coils. Each of the 155 sawtoothing discharges analyzed has a monotonically decreasing rotational transform profile and exhibits properties similar to those observed in tokamaks. Similarities include (1) a central emissivity rise and then sudden crash with a well-defined inversion radius, (2) the presence of an m = 1 emissivity fluctuation, and (3) the normalized inversion surface radius scales with the total edge rotational transform.

Experimentally, the measured sawtooth period is observed to decrease by a factor of approximately two with the increasing levels of the fractional transform from 0.04 to 0.42. The measured crash time of the sawtooth oscillation is not correlated with the fractional transform, indicating that the decrease in the sawtooth period is predominantly due to a decrease in the rise time of the oscillation. The lack of dependence between the sawtooth crash time and the fractional transform implies that the final stage nonlinear reconnection dynamics of the MHD kink-tearing instability are not significantly affected. Measurements of the sawtooth amplitude showed a lack of high-amplitude sawteeth at a high fractional transform and a wide range of amplitudes at a low fractional transform.

The observed decrease in the sawtooth period and only low-amplitude sawteeth are correlated with the fractional transform and, by extension, the mean elongation of the last closed flux surface, which is used as a proxy for the elongation of the q = 1 surface, rather than being due to core equilibrium changes which would cause changes in the central ohmic heating rate. Given that the kink-tearing mode is well known to be destabilized by elongation in tokamak plasmas, this observation supports the interpretation that the reduced sawtooth period results from a change in the linear stability properties of the kink-tearing mode responsible for the crash.

The authors would like to thank John Dawson for his help in keeping the CTH experiment operational and for his work in the design and construction of the SXR amplifiers and filters. This work was supported by a U.S. DOE 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.