J. Yang et al.

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INTERNAL STRUCTURE OF MHD FLUCTUATIONS

FOR VARIOUS CURRENT DENSITY PROFILES

DURING CURRENT RISE PHASE OF

OHMIC DISCHARGE IN VEST

J. YANG, S.C. KIM, Y.S. HWANG

Seoul National University

Seoul, Republic of Korea

Email: yhwang@snu.ac.kr

Abstract

The internal magnetic probe array is used to validate the equilibrium reconstruction and the fluctuation measurement

using the external magnetic diagnostics in characterization of the internal structure of the MHD activity, i.e. the magnetic

islands, during the plasma current ramp up phase of the Ohmic discharge in VEST. The details of the commissioning and

implementation of the magnetic island diagnostics is presented. Then, the classical tearing mode theory on the magnetic

island onset and suppression is tested with the dedicated experiment and the developed magnetic island diagnostics.

1. INTRODUCTION

In VEST (Versatile Experiment Spherical Torus) [1], the magnetohydrodynamic (MHD) activity is routinely

excited during a plasma current ramp up phase of the Ohmic discharge as shown in Fig. 1, due to the fast plasma

current ramp up of up to 20 MA/s. The MHD activity is related to the tearing modes, which limits the maximum

achievable plasma current in VEST and even leads to the plasma disruption in other devices. The equilibrium

reconstruction and the fluctuation measurement using the magnetic sensors placed at the vacuum vessel may

characterize the MHD activity. However, the direct picture of the magnetic islands, which compose the internal

structure of the MHD activity, is not provided from the external magnetic diagnostics, which casts doubt on the

validity of the measurements.

In this paper, the internal magnetic probe array is used to validate the external magnetic diagnostics

based characterization of the MHD activity. Upon the validation, the magnetic island dynamics at the various

current density profiles formed during a current rise phase of an Ohmic discharge in VEST is discussed, in the

classical tearing mode perspective. Section 2 presents the experimental setupm and Section 3 presents the

comparative measurement of the magnetic island diagnostics. Section 4 discusses the magnetic island dynamics

when the current profile is varied. Section 5 concludes the paper.

FIG. 1. Routine MHD activity excited during a plasma current ramp up phase of the Ohmic discharge in VEST. (a)

Spectrogram of the outboard midplane Mirnov coil (b) Plasma current (c) Poloidal mode number (d) Mode frequency (e-f)

Mode amplitudes. The negative signs in 𝑛 = −1 (black) and 𝑛 = −2 (red) modes indicate the mode rotation in the

opposite direction of the plasma current. The mode numbers marked for assistance in (e-f) are based on (c).

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2. EXPERIMENTAL SETUP

The internal magnetic probe array is used for the measurement of the magnetic islands, which verifies the

equilibrium reconstruction and the fluctuation measurements of the internal structure of the MHD activity. The

implementation of each measurement tools are presented in the first half of this section. In the second half of

this section, the internal magnetic probe array signal from the poloidally rotating magnetic islands is estimated,

to provide the reference of interpretation.

2.1. Magnetic diagnostics in VEST

The magnetic diagnostics is used to perform the equilibrium reconstruction and the fluctuation measurement and

measure the magnetic islands. A set of magnetic sensors, including the flux loops, the magnetic probes and the

Mirnov coils, is placed around the vacuum vessel, along with a movable array of the internal magnetic probes,

as shown in Fig. 2. All magnetic probes and the internal magnetic probes are calibrated using a Helmholtz coil.

The signals from the magnetic probes / Mirnov coils (flux loops) are then digitized by PXI-6133 modules

manufactured by National Instruments featuring the resolution of 16 bits within ±5 V, at the sampling rate of

250 kS/s (25 kS/s), and the signals are heavily processed numerically using the offset remover, the integrator,

the amplifier, the filters and the phase shifter, for the simultaneous equilibrium and fluctuation measurements.

Figure 2 Location of the magnetic sensor set of the flux loops, the magnetic probes, the Mirnov coils and the internal

magnetic probe array. (a) Poloidal cross sectional view (b) Top view. The sections K(J), b and f host the outboard, inboard

magnetic probes / Mirnov coils and the internal magnetic probe array, respectively.

The internal magnetic probe array utilizes the printed circuit board to mount both the chip inductors and

the Hall sensors, the chip inductor 1008CS-472XJLB manufactured by Coilcraft and the Hall sensor WSH315-

XPCN2 manufactured by Winson, in the aim to acquire both the equilibrium and fluctuation information. The

magnetic sensors are machine soldered (surface mounted) to a PCB for the reliability in the alignment. The total

8 groups of sites are printed, 50 mm apart and spanning 350 mm, with 3 sites at each group scattered within 10

mm. The three sites within each group mounts one Hall sensor and two chip inductors, oriented to measure the

slow varying 𝐵𝜙 with the Hall sensor, and the fast varying 𝐵𝑍 and 𝐵𝑅 with the chip inductors as shown in

Fig. 3. The array is movable up to 𝑅 > 0.3 m in a shot by shot manner with the step size of 5 cm. With the

insertion of the array, the plasma current is degraded by up to 10%. The in-situ calibration and the build of the

internal magnetic probe array enclosures are the same as presented in [3].

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Figure 3 Orientation and index of the internal magnetic probes. The sensors and the PCB are drawn schematically.

The equilibrium reconstruction in VEST is performed by a delegated code VFIT, an EFIT [4] like code

with the filament modeled wall eddy current estimation and the consideration of the “open field currents”. The

equilibrium reconstruction is converged only when the sum of the “open field current” is subtracted from the

total plasma current, but the physical presence of the “open field current” is being analyzed. The pressure and

poloidal flux functions 𝑝′ and 𝐹𝐹′ are modelled by the polynomial up to order 𝑛𝑃 = 3 and 𝑛𝐹 = 2 to cover

the rather hollow current density profiles expected in the current ramp up as fast as 20 MA/s in VEST. The

tolerance for the convergence is set at 10-3, small enough for the result to be used as an input in the stability

codes. The safety factor profile is post processed from the reconstruction result in the flux surface grid with

𝑛𝜓 = 50 and 𝑛𝜃 = 180 points obtained by the field line following method. The singularity is easily

encountered in the safety factor evaluation near the magnetic axis, and therefore

𝑞0 ≈ 𝐹𝐴 𝑅𝐴⁄ √det(𝐻(𝜓𝐴)) (1)

is used to approximate the on axis safety factor, and the whole safety factor profile is linear interpolated. The

internal magnetic probe array data is used as the extra constraint for the equilibrium reconstruction.

The fluctuation measurement in VEST is performed by a similar method with [5], where the Fourier

analysis and the singular value decomposition is combined due to the complex vacuum vessel geometry where

the constant phase connecting method may be inaccurate. The poloidal array of Mirnov coils are shown in Fig. 2,

where the toroidal array of two additional off-plane Mirnov coils with the toroidal separation of 30° are installed

but not shown.

2.2. Estimated internal magnetic probe signal

When the tearing mode is unstable, the magnetic surfaces of a plasma are torn and rejoined to form a chain of

magnetic islands. Each magnetic island is a flux tube carrying a small helical current and having its own

magnetic axis. Considering the toroidal component of the helical island current, the poloidal fields of opposite

directions are produced at the either side of the island magnetic axis, giving rise to a phase reversal.

Consider an island modelled by the toroidal ring current rotating in the poloidal direction, with the

current flowing clockwise when seen from the above, as shown in Fig. 4 (a). An internal magnetic probe located

at the midplane, between the outboard wall and the ring current, at the location A in Fig. 4 (a), will always

measure the negative vertical field (going into the paper), while the sensor at location B in Fig. 4 (b) will always

measure the positive vertical field (coming out of the paper). As the island approaches (coming to the paper

surface from beneath), both fields are increased, and as the island departs (coming out of the paper), both fields

are decreased. The vertical field change is then negative/positive for the approaching island and

positive/negative for the departing island at the location A/B. Thus the phase reversal occurs centering around

the island magnetic axis. The schematic of the vertical field change near the passing by magnetic island is

shown in Fig. 4 (b).

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Figure 4 Estimation of the internal magnetic probe signal. (a) Island model seen from the top of VEST (b) Schematic

of the vertical field change near the passing by magnetic island.

3. MAGNETIC ISLANDS IDENTIFICATION

The internal magnetic probe array measurement during the MHD activity is shown in Fig. 5. The fluctuation

𝑑𝐵𝑍 𝑑𝑡⁄ is measured by the array of calibrated chip inductors covering 0.35 m at 0.05 m interval (8 points). The

successive insertion of the magnetic probes cause the plasma current degradation and the MHD activity change

within 10% in amplitude, with the waveforms nearly unchanged. The structure predicted in Fig. 4 (b) is clearly

shown in Fig. 5 (a), repeated four times within the shown 0.4 ms time domain, indicating a frequency around 10

kHz, consistent with the measured mode frequency (similar to shot #18028 shown in Fig. 1). Two such

structures are shown in Fig. 5 (b), around the perceived 2/1 and 3/1 surfaces found from the equilibrium

reconstruction. Note that the two islands shown in Fig. 4 (b) are in phase. The consistency of the internal

magnetic probe array measurement with the other magnetic diagnostics is presented in the following paragraph.

Figure 5 Internal magnetic probe array measurement during the MHD activity. (a) Measurement of 0.30 – 0.65 m

(shot #18452) (b) Measurement of 0.50 – 0.85 m (shot #18457). Presumed island positions in white dashed lines are drawn

based on the equilibrium reconstruction result from VFIT of the respective shots.

It is easy to predict that the magnitude of the measured fluctuation |𝑑𝐵𝑍 𝑑𝑡⁄ | is maximized at the island

magnetic axis. The peak of the |𝑑𝐵𝑍 𝑑𝑡⁄ | profile then indicates the island position, which is expected to agree

with the rational flux surface corresponding to the identified mode numbers 𝑚 𝑛⁄ . The magnetic islands during

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the MHD activity is shown in Fig. 6. The magnitudes of the shot by shot inserted (log listed in Table 1) internal

magnetic probe measurements are accumulated at the overlapping positions and normalized to obtain a wide

coverage |𝑑𝐵𝑍 𝑑𝑡⁄ | measurement from 𝑅 = 0.30 m to the outboard wall at 𝑅 = 0.80 m. Two peaks of |𝑑𝐵𝑍 𝑑𝑡⁄ | are clearly visible in Fig. 6 (a), where the perceived 2/1 and 3/1 flux surfaces are overlaid in white

(and partially black) lines along with the plasma magnetic axis. Until around 0.307 s, the magnetic islands are

positioned inside the 2/1 surface and on the 3/1 surface. Then, the “inner” island near the magnetic axis moves

out to the 2/1 surface with the “outer” island still at the 3/1 surface. Finally at around 0.311 s, the “outer” island

moves out of the 3/1 surface. The evolution of the island positions is consistent with the evolution of the mode

numbers shown in Fig. 6 (c, e-f), where initially the 3/1+3/2 modes are excited, which change to 3/1+4/2 and

4/1+4/2 subsequently. The 3/2 surface is not drawn in Fig. 6 (a) since the equilibrium reconstruction failed to

detect the 3/2 surface, although identified as the helicity of the “inner” island until around 0.307 s. Considering

the evolution of the reconstructed safety factors of shot #18452 shown in Fig. 7, however, the error bar indicates

that the 3/2 surface may exist near the minimum 𝑞 surface (inside the 2/1 surface) until 0.307 s. The large

uncertainty is as expected for the central safety factor values.

Figure 6 Magnetic islands during the MHD activity. (a) Internal magnetic probe array measurements (b) Plasma

current (c) Poloidal mode number (d) Mode frequency (e-f) Mode amplitudes. The negative signs in 𝑛 = −1 (black) and

𝑛 = −2 (red) modes indicate the mode rotation in the opposite direction of the plasma current. The mode numbers marked

for assistance in (e-f) are based on (c).

Figure 7 Evolution of the reconstructed safety factors of shot #18452.

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4. DISCUSSION

In the previous sections, the characterization of the internal structure of the MHD activity by the equilibrium

reconstruction and the fluctuation measurement are validated using the internal magnetic probe array

measurement, which provided the direct picture of the magnetic islands. In this section, the internal structures at

the various current density profiles formed during a plasma current ramp up phase of an Ohmic discharge in

VEST are discussed.

In the classical tearing mode theory, the radial gradient of the equilibrium toroidal current density, i.e. the

small radial gradient of the safety factor, drives the instability. A dedicated experiment is conducted to elaborate

on the suggestion. The VEST PF01 coil is powered by the double swing circuit, where the coil current is ramped

up (C1), swung down to zero (C2) and then swung down further to reversed polarity (C3) [1]. The second swing

down of C3 phase initiates at 0.305 s when the switching changes the capacitance and charged voltage of the

PF01 circuit and the additional loop voltage is induced, namely the second current drive. The current density

profile evolution by the second current drive is useful to examine the relation between the onset of the instability

and the radial gradient of the safety factor 𝑑𝑞 𝑑𝑟⁄ .

The second current drive experiment is shown in Fig. 8, and the surface averaged profiles of the second

current drive experiment is shown in Fig. 9. The loop voltage shown in Fig. 8 (d) clearly indicates the second

current drive at 0.305 s. The current density profile evolution shown in Fig. 9 (b) is consistent with the loop

voltage drive: The initially peaked current density profile (blue) is made hollow by the additional voltage drive

(red) before the current diffusion to the center makes the current profile less and less hollow (yellow and purple).

The safety factor profile evolution shown in Fig. 9 (a) is closely related with the change in the MHD fluctuation

shown in Fig. 8 (b): At 0.304 – 0.305 s, the 3/1 mode is suppressed while the 𝑑𝑞 𝑑𝑟⁄ at 3/1 surface is increased

(blue and red lines); at 0.305 – 0.306 s the 2/1 mode is onset while the 𝑑𝑞 𝑑𝑟⁄ at the 2/1 surface is decreased

(red and yellow lines); At 0.306 – 0.308 s the 2/1 and 3/2 modes remain while the safety factor profile is only

slightly broadened. The behavior of the MHD fluctuation as the radial gradient of the safety factor profile is

changed verifies that the onset of the instability can be explained by the classical tearing mode theory. Note that

3/2 mode is onset while the 𝑑𝑞 𝑑𝑟⁄ at the 3/2 surface is rather unchanged, indicating the possibility that the

magnetic perturbation of 2/1 mode may have induced the 3/2 mode.

Figure 8 Second current drive experiment. (a) Plasma current (b-c) Mode amplitude (d) Loop voltage (e) Radial

gradient of safety factor at 𝑚 𝑛⁄ = 2 1⁄ surface. The colored dashed lines correspond to the colors of the profiles in Fig. 9.

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Figure 9 Surface averaged profiles of the second current drive experiment. (a) Safety factor (b) Current density. The

colors of the profiles correspond to the time of the colored dashed lines in Fig. 8.

5. CONCLUSION

The equilibrium reconstruction and the fluctuation measurement characterizes the internal structure of the MHD

activity, or the magnetic islands. The internal magnetic probe array is used to measure the magnetic islands

directly, validating the characterization of the equilibrium reconstruction and the fluctuation measurement. The

onset and suppression of the magnetic islands are closely related to the current density profile, or the safety

factor profile, as can be seen from the dedicated experiment where the profiles are varied within the plasma

current ramp up phase of an Ohmic discharge in VEST.

ACKNOWLEDGEMENTS

This research is supported by the Brain Korea 21 Plus Program (No. 21AA20130012821) and the National R&D

Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT &

Future Planning (No. 2014M1A7A1A03045374).

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