Large, spontaneous m/n =1/1 helical cores are predicted in tokamaks with extended regions of low- or reversed-magnetic shear profiles in a region within the q =1 surface and an onset condition determined by constant (dp/dρ)/Bt2 along the threshold. These 3D modes occurred frequently in Alcator C-Mod during ramp-up when slow current penetration results in a reversed shear q-profile. The onset and early development of a helical core in C-Mod were simulated using a new 3D time-dependent equilibrium reconstruction, based on the ideal MHD equilibrium code VMEC. The reconstruction used the experimental density, temperature, and soft-X-ray fluctuations. The pressure profile can become hollow due to an inverted, hollow electron temperature profile caused by molybdenum radiation in the plasma core during the current ramp-up phase before the onset of sawteeth, which may also occur in ITER with tungsten. Based on modeling, it is found that a reverse shear q-profile combined with a hollow pressure profile reduces the onset condition threshold, enabling helical core formation from an otherwise axisymmetric equilibrium.

Sawtooth instabilities occur periodically in tokamaks when the safety factor falls below 1 in the core. These not only have the beneficial effect of flushing impurities, such as metals and helium ash out of the core1 but also have been shown to trigger ELMs, neoclassical tearing modes, and disruptions when the amplitude is sufficiently large.2–4 All of which could preclude steady state operation. While targeted injection of electron cyclotron current drive in the core may be sufficient to mitigate the detrimental effects of sawteeth in ITER,5 its effectiveness is uncertain,4 and a more robust solution is desirable requiring less active control and recirculating power.

Steady state 3D equilibria with helical cores have been shown to self-organize and stabilize sawtooth instabilities through an MHD dynamo effect.6–8 This dynamo effect acts to redistribute poloidal magnetic flux further off-axis, the so-called “flux pumping,” which prevents the mode from destabilizing and maintains the 3D MHD equilibrium in a long-lived steady state.

These 3D equilibria with helical cores, sometimes referred to as snakes, have been previously observed experimentally in many tokamak devices. In ASDEX and DIII-D, helical cores have been driven by applied 3D magnetic perturbation fields in the hybrid scenario9 as well as in the ITER baseline QH-mode scenario in DIII-D.10 It is often found that helical cores can be triggered by either core impurity accumulation, as shown, e.g., in ASDEX and Alcator C-Mod,11,12 or fueling pellet injection, like in Jet, Tore Supra, and EAST.13–15 Theoretically, the helical core has been studied for decades and is essentially characterized as a saturated ideal internal kink instability or interchange mode.16–18 It has been modeled based on ad-hoc plasma profiles and shapes in JET and MAST.19 

Using the VMEC/V3FIT code suite,20–22 helical cores have recently been reconstructed in DIII-D23,24 and Alcator C-Mod,12,24,25 based on measurements of experimentally observed helical cores. Extrapolating this physics to ITER, it has been shown that an equilibrium with the predicted pressure and current profiles would be prone to developing helical cores.24 Whether these helical cores would exist as stable equilibria or develop into sawteeth is the focus of ongoing work, but the fact that they have been observed for time periods much longer than the energy confinement time in existing experiments suggests that they should be studied for their potential beneficial and detrimental impacts on confinement and stability. It is critical for the operation of ITER and future devices to understand the onset conditions of the helical core mode so as to either target it as a possible operation scenario or avoid it.

This paper is organized as follows. In Sec. II, the experimental observations of the C-Mod helical core are shown and the 3D equilibrium reconstruction is introduced. Emission measurements show different time evolution during the mode, which leads to a separation of electron density and electron temperature; this is discussed in Sec. III. The impact of the pressure profile shape on the helical core onset is analyzed in Sec. IV. Section V describes the overall time evolution of the reconstructed helical core. The final section summarizes the conclusions of this paper.

Snake features in soft-X-ray (SXR) emission data indicative of helical cores, like the one shown in Fig. 1, have been observed in C-Mod Ohmic discharges during the plasma current ramp-up phase or early in the plasma current flat-top. The mode spontaneously forms and remains active for several energy confinement times. Typically, the mode starts out as an ideal displacement of flux surfaces in the plasma core, as has been measured by tomographic inversion. After about a third to half of its duration, the mode smoothly transitions into a crescent shape accompanied by the onset of non-ideal MHD and sawteeth.12 Here, we focus on such a typical case in C-Mod discharge 1120208028.

FIG. 1.

Overview of C-Mod discharge 1120208028 with (a) SXR brightness data from the vertical array, (b) schematic of the SXR arrays, and (c) Fourier spectrum of a core chord.

FIG. 1.

Overview of C-Mod discharge 1120208028 with (a) SXR brightness data from the vertical array, (b) schematic of the SXR arrays, and (c) Fourier spectrum of a core chord.

Close modal

The mode onset is between 0.31 s and 0.32 s into the discharge. The SXR diagnostic clearly shows the snake in the plasma core, as shown in Fig. 1(a). The displacement of the magnetic axis is well resolved by the SXR diagnostic due to the molybdenum impurity radiation in the core and has a magnitude of about 1.5 cm after the mode onset. The layout and coverage of the SXR arrays are shown in Fig. 1(b). The Fourier spectrum for an SXR core channel for the total 150 ms duration of the mode is shown in Fig. 1(c). The mode rotates with the plasma at about 4 kHz. Note that the signals around 8 and 12 kHz in the Fourier spectrum are just the higher harmonics of the fundamental mode. After about 0.45 s into the discharge, the discharge starts to exhibit sawtooth crashes.

The high temporal and spatial resolution of the vertical and horizontal SXR arrays allows for high fidelity tomographic inversions. Two such inversions are shown in Fig. 2. The inversions around 0.33 s indicate that the flux surfaces are displaced ideally and not broken by reconnection. Using the SXR data around 0.33 s, the helical core equilibrium was reconstructed by the V3FIT code,21 following the same procedure as in the DIII-D tokamak.24 The details of the reconstruction are discussed in Subsection II A.

FIG. 2.

Tomographic inversion of the SXR data around 0.33 s. The two inversions show the maximum core displacement to the high-field side (poloidal angle θ = π) and low-field side (θ = 0), respectively.

FIG. 2.

Tomographic inversion of the SXR data around 0.33 s. The two inversions show the maximum core displacement to the high-field side (poloidal angle θ = π) and low-field side (θ = 0), respectively.

Close modal

Figure 3 compares the reconstructed 3D equilibrium (shown in black) with the axisymmetric EFIT equilibrium at the same time (red). The cross section in Fig. 3(a) clearly shows the displacement of the flux surfaces near the magnetic axis with an axis displacement of 1.42 cm. The pressure profile in Fig. 3(b) matches between the EFIT and 3D equilibria, since it is given as a fixed input to the reconstruction. V3FIT reconstructs the current density profile in Fig. 3(c) between ρ = 0 and ρ = 0.7, while then, VMEC computes the flux surfaces and q-profile in Fig. 3(d) self-consistently. The q-profile shows a significant reversed shear with its minimum qmin = 1.005 at the location ρqmin = 0.4 and the value on axis q0 = 1.63. Note that qmin is larger than 1, which agrees with the sawteeth-free progression of the discharge. The current density is significantly different between EFIT and the reconstruction. Unfortunately, C-Mod does not have a direct current density measurement for this shot to locally constrain the current density profile and therefore the q-profile during the reconstruction. Nevertheless, in agreement with numerical predictions,24 a reversed shear q-profile is needed to match the helical core in the equilibrium to the measured data.

FIG. 3.

Helical core reconstruction at 0.33 s. (a) Cross-section, (b) pressure profile, (c) parallel current density, and (d) q-profile.

FIG. 3.

Helical core reconstruction at 0.33 s. (a) Cross-section, (b) pressure profile, (c) parallel current density, and (d) q-profile.

Close modal

The vertical (array 1) and horizontal (array 3) SXR arrays [as shown in Fig. 1(b)] are used in the reconstruction. These arrays were chosen because they operate with an identical setup, are perpendicular, and collect excellent datasets in this experiment. In order to provide 3D data to V3FIT, four evenly spaced time slices within a single rotation period of the helical core around 0.33 s are chosen. These four time slices are used to reconstruct a single helical core equilibrium. As the helical core rotates in time, it also rotates toroidally in the vessel. So, the chosen four time slices can be identified as an equivalent of four evenly spaced toroidal locations throughout the vessel, turning the measured data from the respective slices into measurements of four identical virtual diagnostic sets.

Figures 4(a) and 4(b) show the measured signal with error bars in each channel of arrays 1 and 3, respectively, for the four time slices identified with toroidal locations (solid lines). The vertical dashed line shows the axisymmetric location of the magnetic axis just before the helical core onset. The location of peak emission clearly oscillates around the vertical dashed line. During the reconstruction, V3FIT adjusts the current density profile as well as the toroidal phase of the helical core with respect to the diagnostic locations to minimize the difference between measured and synthetic signals, given by

χ2=i(SimeasuredSisynthetic)2σi2,
(1)

with the respective measurement uncertainty σ. After each modification, the 3D equilibrium is reconverged and the synthetic signals are evaluated. The final result is the minimum in χ2 and therefore, within error bars, the best possible match between model and data.

FIG. 4.

(a) SXR emission for array 1 (vertical array): (solid line) measured and (dashed line) V3FIT at different toroidal angles. The vertical dashed line shows the n = 0 magnetic axis location. (b) Same as (a) but for array 3 (horizontal array).

FIG. 4.

(a) SXR emission for array 1 (vertical array): (solid line) measured and (dashed line) V3FIT at different toroidal angles. The vertical dashed line shows the n = 0 magnetic axis location. (b) Same as (a) but for array 3 (horizontal array).

Close modal

In order to calculate the synthetic SXR signals, an emissivity profile needs to be specified, which gives the emission as a function of the radial coordinate ρ. Assuming that the SXR emission is constant on a flux surface, the synthetic signal can be evaluated as the line integral along the line of sight of each channel. The emissivity profile can be reconstructed beforehand, either using axisymmetric measurements just before the mode onset or using time averaged signals, which are made axisymmetric by averaging of one or more helical core rotation periods. The emissivity profile is then kept fixed during the 3D equilibrium reconstruction.

In Figs. 4(a) and 4(b), the colored dashed emission profiles represent the synthetic emission, calculated by V3FIT, for the final reconstructed helical core equilibrium. An excellent match is found for the peak emissions and their respective locations. Further towards the plasma edge, there are minor deviations, which could further suggest that small adjustments to the current density profile at these locations might be needed. But without local current measurements, these adjustments cannot be resolved.

The reconstruction procedure can be outlined as follows:

  1. A 2D kinetic EFIT equilibrium is converted into an axisymmetric VMEC equilibrium, which allows for non-axisymmetry.

  2. If magnetic perturbations or error fields are present, they can be included via a free boundary VMEC equilibrium, resulting in a perturbed 3D last closed flux surface (LCFS). For C-Mod, no such fields are included, and so this step is skipped, continuing on with an axisymmetric LCFS.

  3. The VMEC equilibrium from step (2) is used to reconstruct the emissivity profile.

  4. The final 3D equilibrium reconstruction requires an initial guess, which can be either the VMEC equilibrium from step (2) or an otherwise modeled VMEC equilibrium. The closer the initial guess is to the final result, the faster the V3FIT converges. Since a helical core equilibrium is expected, the equilibrium from step (2) is modified with a reversed shear q-profile, based on a parameter scan of ρqmin and q0.24 

  5. Using the initial guess from step (4) and the emissivity profile from step (3), the 3D equilibrium is reconstructed with V3FIT by keeping the LCFS fixed and triggering a possible bifurcation into the helical state by an initial random magnetic axis displacement. Multiple knots with ρ < 0.7 of the spline representation for the current density profile are reconstructed, while the flux surfaces and the q-profile are solved self-consistently. The pressure profile is kept fixed.

Using this procedure, a total of four equilibria at times between 0.31 s and 0.34 s were reconstructed [see Fig. 13(a)]. While the helical core has not yet formed at 0.31 s, which results in an axisymmetric reconstructed equilibrium, the other time slices reconstruct to 3D helical core equilibria, similar to the one shown in Fig. 3 for 0.33 s.

The SXR data are primarily used in the equilibrium reconstruction because they have a good signal to noise ratio and a large gradient in the region of interest. Nevertheless, other diagnostics clearly show the helical core as well. SXR emission scales with Zeffne2/Te, and so, it has not only a strong electron density dependence but also an electron temperature and impurity dependence. To better compare density and temperature oscillations, a Two-Color-Interferometer (TCI), which measures ne, and the Electron Cyclotron Emission (ECE) from the FRCECE diagnostic,26 which measures Te, are used. Also an Absolute Extreme Ultra-Violet (AXUV) detector observes oscillations with the mode and provides a measure of the total radiated power.

The Fourier spectra for those three diagnostics are shown in Fig. 5. While the TCI and AXUV spectra are quite similar to the SXR spectrum in Fig. 1(c) (of which all have strong ne dependencies), the ECE spectrum shows some differences. The temperature oscillations appear the strongest right after the mode onset, while the density oscillations rise to their peak amplitude only after the temperature oscillations have dropped, around 0.35 s. Such a decoupling between electron density and electron temperature oscillations around 0.35 s suggests a change in the mode.

FIG. 5.

Fourier spectra of (a) TCI, (b) AXUV, and (c) ECE diagnostics. The red inset in (c) shows the total electron temperature as measured by the same ECE channel.

FIG. 5.

Fourier spectra of (a) TCI, (b) AXUV, and (c) ECE diagnostics. The red inset in (c) shows the total electron temperature as measured by the same ECE channel.

Close modal

For a better direct comparison, the ne and Te mode spectra can be integrated over frequency from 2 kHz to 15 kHz to obtain the integrated power spectrum (IPS) of each diagnostic. Figure 6 shows the IPS for ne (red), Te (blue), and SXR (green), compared with the magnetic axis displacement δH (black)

δH=(RR0)2+(ZZ0)2/2,
(2)

with R and Z being the axis location as measured by SXR and R0 and Z0 being the average over one helical core period. At the mode onset, at about 0.315 s, the axis displacement jumps up to about 1.5 cm along with all of the IPS signals. Until about 0.34 s, the mode remains rather unchanged. Then, the axis displacement and the SXR IPS start to increase. Around 0.35 s, the Te IPS quickly drops to about 30% of the initial peak value, while the ne IPS doubles compared to the mode onset. After 0.35 s, the axis displacement starts to show more and more oscillations, while rising steadily. Around 0.45 s, the ne and SXR IPS signals drop, while the axis displacement becomes erratic. Total Te measurements, as shown by the red inlay in Fig. 5(c), show clear signs of sawteeth at this time. At about 0.48 s, the mode finally collapses, leaving the discharge sawtoothing strongly. As has been observed previously, the helical core transitions smoothly from an ideal kink to a crescent structure, similar to a 1/1 island, during its evolution.12 This transition is also observed here around 0.35 s. The onset of strong oscillations in the axis displacement around 0.45 s suggests the onset of sawteeth, which then grow in time and eventually become large when the mode finally collapses. Since the mode transitions from an ideal kink to a crescent shape, we limit the VMEC/V3FIT 3D equilibrium reconstruction to the early phase of the mode with t <0.35 s, because these codes are limited to ideal MHD with nested flux surfaces.

FIG. 6.

Integrated power spectrum for ne (red), Te (blue), and SXR (green) fluctuations, respectively, compared with the magnetic axis displacement δH (black), measured by SXR.

FIG. 6.

Integrated power spectrum for ne (red), Te (blue), and SXR (green) fluctuations, respectively, compared with the magnetic axis displacement δH (black), measured by SXR.

Close modal

Reconstructed 3D equilibria can be used to determine the optimum toroidal phase of the helical core to match the SXR data or the ECE data at a given time. Figure 7 shows the χ2 dependence on the helical core phase angle ϕoff using a reconstructed helical core equilibrium at 0.32 s for both SXR and ECE data each independently. The respective minima are in very good alignment. The phase difference, defined by

δϕ=ϕECEϕSXR,
(3)

is only δϕ = 0.15 rad = 8.6°. So, we can conclude that at 0.32 s, right after the mode onset, the electron density and electron temperature displacements are in phase as one would expect from an ideal MHD 1/1 kink mode. The phase difference remains constant through 0.33 s and then rises to about 33° at 0.34 s. This reconstructed shift is caused by the observed decoupling of ne and Te oscillations and suggests that when the mode starts to transition, ne and Te oscillations move out of phase. Such a decoupling is beyond the limitations of the VMEC code, which can only consider total pressure. Simulations with the nonlinear 3D MHD code M3D27 have shown that such a decoupling can be explained if temperature and density are evolved separately.25 

FIG. 7.

(a) Optimum toroidal phase of the helical core at 0.32 s for measured SXR data. (b) Same as (a) but for measured ECE data.

FIG. 7.

(a) Optimum toroidal phase of the helical core at 0.32 s for measured SXR data. (b) Same as (a) but for measured ECE data.

Close modal

From previous studies,24 we found that the helical core onset threshold is given by

max(dp/dρ)/B02=C,
(4)

where dp/ is the pressure gradient, B0 is the toroidal magnetic field, and C is a constant. The latter gives the location of the onset threshold within the operating space of pressure gradient and toroidal field. The smaller the C, the lower the onset threshold, which makes accessing the helical state easier. C can depend on various parameters, like the shape of the q-profile, which has been analyzed in detail in previous work. In this section, we will discuss the dependence of C on the shape of the pressure profile.

Strong core radiation due to the accumulation of high-Z impurities, usually metals like tungsten or molybdenum, can lead to a local cooling in the core, resulting in an inverted, hollow electron temperature profile. This can be observed in C-Mod. Figure 8(a) shows the electron temperature measurements from multiple diagnostics during the helical core (at 0.33 s). All diagnostics, Thomson scattering in black and Electron Cyclotron Emission (ECE) in red, show a rollover towards the core and a temperature peak off-axis around R =0.73 m. Note that the two diagnostics measure about 36° toroidally apart, which does not affect the profiles, because the data points are averaged over a 10 ms window which covers about 40 periods of the rotating helical core.

FIG. 8.

(a) Measured electron temperature from Thomson scattering (black) and ECE (red). (b) Pressure profile at 0.33 s: (black) standard monotonic profile and (red) hollow profile.

FIG. 8.

(a) Measured electron temperature from Thomson scattering (black) and ECE (red). (b) Pressure profile at 0.33 s: (black) standard monotonic profile and (red) hollow profile.

Close modal

Using the kinetic EFIT equilibrium reconstruction, we can generate two distinct pressure profiles. First, using only Thomson scattering data, a more typical peaked, monotonic pressure profile can be enforced, which is shown in Fig. 8(b) by the black curve. Including also the ECE data and releasing the monotonic profile constraint result in an inverted, hollow pressure profile as given by the red line in Fig. 8(b).

As mentioned above, the pressure profile is a fixed input to the reconstruction. So, for the analysis, the monotonic profile is continuously morphed into the hollow one by

p=(1x)pmonotonic+xphollow,
(5)

using the dimensionless parameter x. For x =0, the pressure profile is monotonic, and for x =1, it becomes hollow. Keeping every other input fixed from the reconstructed helical core equilibrium, we can model helical core equilibria with VMEC by only changing the pressure profile in this way.

Figure 9 shows the evolution of the magnetic axis displacement δH with x. The displacement monotonically increases by about 34% with the pressure profile becoming hollow, which indicates a significantly stronger drive of the kink. So, we would expect that the onset threshold for helical core formation drops, which is confirmed by the calculation of the onset threshold criterion in Fig. 10.

FIG. 9.

Change in the helical core size with continuous change in the pressure profile from monotonic (x =0) to hollow (x =1).

FIG. 9.

Change in the helical core size with continuous change in the pressure profile from monotonic (x =0) to hollow (x =1).

Close modal
FIG. 10.

Helical core onset thresholds for monotonic (black) and hollow (red) pressure profiles. The dashed line case is modeled, while the solid line cases are reconstructed. The dots indicate the discharge conditions at 0.33 s for the monotonic (black) and hollow (red) profiles, respectively.

FIG. 10.

Helical core onset thresholds for monotonic (black) and hollow (red) pressure profiles. The dashed line case is modeled, while the solid line cases are reconstructed. The dots indicate the discharge conditions at 0.33 s for the monotonic (black) and hollow (red) profiles, respectively.

Close modal

In the figure, the black line gives the onset threshold for spontaneous helical core formation for the monotonic pressure profile. Below the threshold, the pressure gradient that drives the instability is too small to excite the helical core spontaneously. C-Mod does not apply 3D fields which could drive a helical core to finite size in this region, as has been seen on DIII-D.9 Above the threshold, a small one time 3D kick, like a local fluctuation in the magnetic field, can trigger the helical core. The mode then grows and saturates at a finite size. The dots in the figure give the operational conditions of the discharge for the respective profile. Both are above the threshold, so the axisymmetric equilibrium is unstable and a helical core forms, as seen experimentally. By changing the pressure profile from monotonic to hollow, the onset threshold drops. The dashed line is for the modeled case, while the solid line is for a 3D equilibrium reconstruction using the hollow pressure profile from the start.

The drop in the onset threshold means that less drive is required for the axisymmetric equilibrium to become kink unstable. The hollow pressure profile therefore makes it easier for helical cores to appear. This seems counter intuitive since the maximum pressure gradient and the gradient at the location of qmin are slightly smaller than in the monotonic profile. But for both cases, the drive is already above the threshold. The threshold position, on the other hand, is determined by the constant C in Eq. (4).

In Fig. 11, the q-profiles for both equilibrium reconstructions, monotonic and hollow (red) pressure profiles, are compared. The differences are minimal. qmin and q0 are the same, while ρqmin is shifted slightly radially inwards to 0.35 compared to 0.4 for the monotonic profile. The reconstructed helical core has a slightly larger magnetic axis displacement of 1.6 cm compared to 1.42 cm for the monotonic profile, but still close to the experimental value of about 1.5 cm. Note that the modeled case, used for the dashed red line in Fig. 10, has a q-profile identical to the monotonic case, but a magnetic axis displacement of 1.9 cm. So, we conclude that C also depends on the pressure profile shape, while the changes in the shape of q raise the onset threshold a little so that the helical core size is in better agreement with the experimental helical core. A hollow pressure profile shape supports helical core formation better than a monotonic shape.

FIG. 11.

Change in q-profile for reconstruction at 0.33 s with monotonic (black) and hollow (red) pressure profiles.

FIG. 11.

Change in q-profile for reconstruction at 0.33 s with monotonic (black) and hollow (red) pressure profiles.

Close modal

In Sec. III, the time evolution of various fluctuation measurements has been discussed. Here, we examine the time evolution of the reconstructed equilibria, limiting the analysis to the early stage of the mode with t <0.35 s, where the mode can be considered ideal and therefore represented appropriately by VMEC. 3D equilibria were reconstructed for 0.31 s, which is before the mode onset, as well as 0.32 s, 0.33 s, and 0.34 s, which are within the ideal phase. In all four cases, the helical core is triggered in the simulation by an initial axis displacement, a one time kick, of about 2 cm, which enables the bifurcation into the helical state. But as has been shown in previous work, the size of the initial axis displacement does not affect the size of the resulting helical core.

Even though the simulation allows for 3D and the initial kick was applied in the same way as in all the cases, the equilibrium at 0.31 s returns to axisymmetry during the reconstruction. The other three cases bifurcate into a helical core equilibrium. The onset thresholds for each case were determined. In order to compare the cases, the threshold constant C [see Eq. (4)] of each case is normalized so that all thresholds coincide. The operational points are scaled, respectively, so that their relative positions with respect to the threshold are maintained. Figure 12 shows the comparison. The equilibrium at 0.31 s is below the onset threshold, so the axisymmetric state is stable, while the other equilibria are above the onset threshold, resulting in helical cores. As expected, the helical core onset of the reconstructed equilibria represents the experimentally observed onset, i.e., 0.31 < t <0.32 as seen in Fig. 1(a).

FIG. 12.

Onset threshold for times 0.31, 0.32, 0.33, and 0.34 s.

FIG. 12.

Onset threshold for times 0.31, 0.32, 0.33, and 0.34 s.

Close modal

For each time slice, a set of parameters can be extracted from the reconstructed equilibria. Foremost is the size of the helical core, as given by the magnetic axis displacement δH. Figure 13(a) shows the temporal evolution of the helical core size in black, compared with the axis displacement measured by SXR in red. The good agreement within the error bars of the SXR measurements underlines the fidelity of the 3D reconstructions. As discussed earlier, the phase difference δϕ [see Eq. (3)] between optimum toroidal alignment of the helical core equilibrium with SXR and ECE measurements, respectively, increases in time, shown in Fig. 13(b). Figures 13(c)–13(e) show various parameters of the reconstructed q-profiles. q on axis, q0, is consistent and significantly higher than qmin; all q-profiles are strongly reversed in the core. The minimum of q along the radial profile, qmin, is very close to unity, but remains larger than 1, consistent with the sawtooth-free operation of the discharge at these times. The radial location of the minimum in q, ρqmin, tends to move slightly inward with time.

FIG. 13.

Time trace of reconstructed helical cores. (a) Helical core size; the red markers with error bars give the size measured by SXR; (b) difference in the optimum helical core phase between SXR and ECE data; (c) q on axis; (d) minimum of q; and (e) radial location of qmin. The dashed line indicates the mode onset based on SXR signals.

FIG. 13.

Time trace of reconstructed helical cores. (a) Helical core size; the red markers with error bars give the size measured by SXR; (b) difference in the optimum helical core phase between SXR and ECE data; (c) q on axis; (d) minimum of q; and (e) radial location of qmin. The dashed line indicates the mode onset based on SXR signals.

Close modal

The spontaneous excitement of rotating helical modes in the plasma core is observed during the ramp-up or early flattop phase of C-Mod discharges. During the early stage of the mode, the flux surfaces in the core are helically displaced and can be described by ideal MHD. Here, the onset of such a helical core in C-Mod has been reconstructed using 3D equilibrium reconstruction with VMEC/V3FIT. It is found that a reversed shear q-profile is needed to enable the helical core in the 3D equilibria. Such reversed shear q-profiles are typical for the ramp-up phase of the discharge, due to the slow current penetration from the edge into the core. Several time slices before and after the mode onset were reconstructed, and good agreement between the measured and modeled displacements of the magnetic axis is shown. The transition from axisymmetry to the helical core equilibrium during the discharge agrees with crossing the onset threshold for helical core formation in the model.

Analyzing time traces of fluctuating signal measurements by various diagnostics reveals that the mode transitions after about a third of its duration from an ideal kink, where electron density and electron temperature oscillations are in phase, to a different state, dominated by density oscillations and later the onset of sawteeth. The latter indicates magnetic reconnection, suggesting the transition from an ideal 1/1 kink to a resistive 1/1 island. This paper focuses only on the ideal stage of the mode.

Varying the pressure profile shape can shift the helical core onset threshold. Radiative cooling of the plasma core due to high-Z impurities leads to a hollow electron temperature profile, as measured by the ECE diagnostic. Including the ECE data in the initial 2D EFIT equilibrium reconstruction to constrain the electron temperature results in a hollow pressure profile. Comparing a hollow and a monotonic pressure profile, it is found that the reconstructed 3D equilibrium with a hollow profile has a significantly lower helical core onset threshold in ∇p than the one with a monotonic profile. Simply replacing the monotonic profile with the hollow one causes the helical core to grow in size by about 34%. We can conclude that radiative cooling of the core lowers the onset threshold and therefore enables the formation of helical cores.

Both conditions, radiative cooling of the core by high-Z impurities and reversed shear q-profiles with qmin close to unity are expected to occur in ITER. Combined with the results from Ref. 24, which show that ITER is highly susceptible to large helical cores and operates far above the onset threshold, we conclude that the ITER ramp-up phase is likely to experience helical cores. Whether such core modes are desirable or need to be avoided remains an open question and is part of ongoing research. Numerical studies indicate that helical cores could be used to avoid sawteeth.8 Experiments and modeling at MAST showed that helical cores can cause massive redistribution of neutral beam ions to the uncompressed region, resulting in a significant shift of heating and current drive away from the axis.28 Preliminary studies with ITER helical core equilibria indicate that the large aspect ratio and high toroidal field suppress this effect and that the helical core has only minor impact on the fusion alpha density distribution. This and other effects are part of future work.

Discussions with Amanda Hubbard, Jerry Hughes, Matt Reinke, and David Pfefferlé are gratefully acknowledged. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Nos. DE-AC05-00OR22725, DE-AC02-09CH11466, and DE-FC02-99ER54512. This manuscript was authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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