Proceedings of ITC/ISHW2007

Temporal evolution of the pressure profile and mode behavior
during internal reconnection events in the MAST sphericaltokamak
H. Tojo1) , M. Gryaznevich2) , A. Ejiri1) , A. Sykes2) , Y. Takase1)
1)

Graduate School of Frontier Sciences, The University of Tokyo Kashiwa 277-8561, Japan
2)
EURATOM/UKAEA Fusion Association, Culham Science Centre, OX14 3DB, UK(Received 15 October 2007 / Accepted)

Mode behaviors including non-linear development and pressure profiles during internal reconnection eventsare clarified in the MAST spherical tokamak. In a q0 > 1 discharge, tearing mode is a trigger but non-linearity ofmodes is not confirmed. On the other hand, in a q0 < 1 discharge, harmonics of a m/n = 4/1 mode of ∼22kHz
are confirmed. Method for identification of poloidal mode in ST configuration is also given.
Keywords: IRE, spherical tokamak, bicoherence, poloidal mode, reconnection
DOI: 10.1585/pfr.1.001

1Introduction

Internal Reconnection Events (IREs) have often been observed in spherical tokamaks (STs), limiting performance.
IREs are MHD instabilities, which show mode growth ofprecursor and flatten pressure profile. They result in disruption in the worst case. Three-dimensional MHD simulations have shown that linear growth rate and mode number of pressure-driven modes depend on q0 (safety factor
at the magnetic axis) and βt (toroidal beta), and that thenon-linear coupling of linear modes leads to a large plasma
deformation [1]. In this situation, magnetic reconnection
occurs and core plasma energy is lost along the reconnected magnetic line. The relationship between the mode
number (and its growth rate) and q0 and βt have not beenclarified experimentally. Moreover non-linear coupling ofmodes just before IRE haven’t been investigated in detail.
In MAST [2], IREs appear in both q0 > 1 and q0 < 1
discharges and they often limit the achievable plasma beta
in high-β regimes. Objectives of this study are to clarify
mode number and its dependence on such parameters, andto confirm the existence of non-linear coupling.

2

Experimental setup

2.1

Mirnov coil array and identification ofpoloidal mode number

Mirnov coils (magnetic probes) are sensitive to external
mode behavior and are employed to identify mode numbers. In MAST, magnetic coils are located along thetoroidal direction (12ch at the most), and along the centercolumn (40ch) and along the outboard (18ch at the most).
author’s e-mail: tojo@fusion.k.u-tokyo.ac.jp

In STs, the determination of poloidal mode numbers (m)is not easy because of the eﬀects of non-circular magnetic flux surfaces and asymmetric magnetic field pitch angle. To overcome these problems we developed a method
to evaluate the poloidal mode number and its intensity.
Firstly, we assume helical filamentary currents, which generate magnetic fluctuations, on their resonant surface calculated from EFIT [3]. The poloidal positions of these filaments are decided by tracing a magnetic field line. The
number of alternating positive and negative filament pairs
coincides with the mode number. Figure 1 shows trajectories of magnetic lines at the surfaces q = 1, 1.5, 2, and3. In this case, the initial poloidal angle (θ = 0) is locatedat the outboard midplane. The poloidal location of eachfilament (i.e., marker filament) can be obtained from theintersection of the curve and an integer or a half integer
toroidal turns (Fig. 1). The tangents of the lines becomesteep at θ ∼ π because of the strong toroidal field at the
inboard side. Therefore, poloidal distribution of the filaments is asymmetric, and most of them tend to locate at
the high field side. Number of filaments (and their rationalsurface), current amplitude and the poloidal location of theinitial filament were obtained from the best fit to the measured Mirnov coil signals. Performing fittings at each timing, we can derive poloidal rotation and amplitude growthof the mode. However in this report we used band-passed
signals and assume single mode distortion located at a rational surface to study the role of the dominant mode. Notethat this model is appropriate for localized rational modes,
such as tearing modes.

2.2

Measurements of soft X-ray profile

Soft X-ray radiation (SXR) is useful to study MHD phenomena and it includes information on internal mode struc-Proceedings of ITC/ISHW2007
Toroidal angle [Turns]

3.0

q=3/1
Positive filaments
Negative filaments

2.0

where f3 = f1 + f2 and X( f ) is a Fourier components ofsignal x(t). To evaluate the degree of nonlinear couplingfor each frequency component, following normalized auto
bispectrum is commonly used as

q=2/1
q=3/2
q=1/1

1.0

0.0

0.2
0.4
0.6
0.8
Poloidal angle [Turns]

b2 ( f1 , f2 ) =1.0

Fig. 1 Trajectory of the magnetic field lines at the rational surfaces q =1, 1.5, 2, 3. Poloidal angles of filaments are
obtained from the intersections between the traces andhorizontal lines with intervals of half integer turns.
Z [m]2#18547 t=0.245

f1 , f2

which represents quantitative measure of the non-linear
coupling.

1Horizontal
SXR camera(32ch)

HCAMU#1 0
HCAML#1

TCAM#16-1-20.0

TCAM#1
Tangential
SXR camera (16ch)

0.5

1.0
R [m]

1.5

2.0

Fig. 2 Configuration of horizontal (sight lines) and tangential(asterisk) SXR cameras.

ture. Horizontal (32ch) and tangential (16ch) SXR cameras are employed and their sight lines are shown in Fig. 2.
To compare behavior of SXR, q (calculated from EFIT)
and (electron) pressure profile (measured by Thomsonscattering), sight lines of each SXR channel are labeledby the minimum poloidal flux (ψ) obtained by the EFIT. In
other words, along each sight line we determine the minimum of the ψ (symbols in Fig. 2) and corresponding radial
position R.

2.3

(2)

If there is a strong non-linear relation among f1 and f2 ,b2 becomes large (nearly 1). We analyzed Mirnov coils
data for understanding non-linear development in IRE precursor. To obtain accurate bicoherence, many ensembles
should be averaged. In MAST, there are twelve (at themost) toroidal Mirnov coils around the center column atsame poloidal position. We employed these coil data toperform the ensemble averaging. Total bicoherence is defined as∑
B2tot =
B2 ( f1 , f2 )(3)

HCAMU#16

HCAML#16|B2 ( f1 , f2 )|2h|X( f1 )X( f2 )|2 ih|X ∗ ( f3 )|iAnalysis using bispectrum to show nonlinear coupling

To clarify non-linear development of the IRE precursors
we performed bispectrum analysis which is a Fourier analysis for three wave interaction and used frequently in turbulence analysis [4] [5]. Generally, auto bispectrum is defined asB( f1 , f2 ) = hX( f1 )X( f2 )X ∗ ( f3 )i(1)3IRE discharges

3.1

IRE with q0 > 1

A typical discharge with q0 > 1 and βt ∼ 9% is shown inFig. 3. Increase of the plasma current (I p ) and drop of theelectron density from interferometer were observed. Theincrease on the emission of Dα indicates interaction between the plasma and the vacuum vessel due to the loss ofthe plasma. One of unique characteristics of the events isthat SXR radiation profile shows propagation of the drop
starting from the area of fluctuations (precursor). ThisIRE has precursor not only in SXR radiation, but also in
the Mirnov coil signals. The position of reconnection isclose to HCAMU]12 (R ∼1.25) because HCAMU]13-15
increase, while HCAMU]3-11 decrease after t ∼ 0.2545(see arrows in Fig. 3). We executed the fitting code to determine the mode number. Here we applied band-pass (15kHz) filtering to the time integrated Mirnov signals. An
example of the fitting is shown in Fig. 4. The squares represent the Mirnov coil data at t = 0.24720 and the line showsa fitted curve assuming only 2/1 mode (filaments) at the rational surface q = 2. They show good agreement. Figure 4
indicates time evolution of the calculated helical filamentcurrent (a), poloidal angle of a filament (b) and, fitting error (c). As described in 2.1, filament current reflects mode
intensity and it can be used to derive the growth time. Fromt ∼ 0.241 the filament current increases exponentially, indicating a linear growth. Time constant (τ f il ) of this growthis about 6 ms. The phase of the filament decreases continuously, showing a poloidal rotation of the filaments. Theerror of the fitting is represented by the normalized residual error χ2 . When χ2 is less than 10, the quality of thefitting is acceptable in the present analysis. We preformed

Proceedings of ITC/ISHW2007
1.18 (a) Ip [MA]1.12
8 2180.0 (b) nel (interferometer) [10 /m ]120.0dB/dt (Mirnov coil: Z~0m) [a.u.]
0.
2450.
250

1180

1160

1140

1120

1100202.
0•
10ANE_DENSITY

201.
8•
1020
1.
6•
1020
1.
4•
1020
1.
2•
10(c)

XMC_CC/MV/220

1.
00.
50.
0-0.5-1.0191.
8•10(d) Dα[a.u.]
Upper

Filament Current [kA]

0.
25
5

AMC_PLASMA CURRENT

191.
6•
10

AIM_DA/HM10/T*
HCAMU#15 (R~1.36m)(e) SXR[a.u.]

191.
2•
1019
1.
0•
1018
8.
0•
100.
00
30.
00
20.
00
10.
00
80.
00
60.
00
40.
00
20.
00
0-0.002

0.
00
8HCAMU#13 (R~1.29m)(a)linear growth (τ㨪6ms)

19
1.
4•
100.
00
0HCAMU#14 (R~1.32m)

1000
100
1010.
00
60.
00
40.
00
20.
00
0-0.002

0.
01
20.
01
0rotation
0-100-200-300-400

Poloidal Position (marker filament) [deg.]
(b)
χ20.
00
8

HCAMU#12 (R~1.25m)

0.
00
60.
00
40.
00
20.
00
00.
02
50.
02
00.
01
5

HCAMU#11 (R~1.22)

SXR[a.u.]

0.
0100.
00
50.
00
00.
03
00.
02
50.
02
0HCAMU#9 (R~1.14m)

0.
01
50.
01
00.
00
50.
00
00.
03

HCAMU#7 (R~1.11m)

0.
02

0.
01

0.
00

0.
04

HCAMU#5 (R~1.01m)

0.
03

0.
02

0.
01

0.
00

0.
05

HCAMU#3 (R~0.95m)

30
20
1000.240

(c)
0.242

0.244

0.246
0.248
Time [s]

0.250

0.252

0.
04

0.
03

0.
02

0.
01

Center

0.
00

0.245

0.250
Time [s]0.255

Fig. 3 IRE in discharge with q0 > 1. (a): Plasma current [kA](b): Line integrated electron density [/m2 ] (c): dB/dt
from Mirnov coil [a.u.] (d) Emission of Dα [a.u.] (e):SXR radiation [a.u.] on horizontal SXR camera from upper to center side of the plasma.
0.0010Mirnov coil signals

Intensity [T]0.00050.0000-0.0005
fitted line
-0.0010-0.0015

-200

-100

0100
Poloidal angle [deg.]

200Fig. 4 Integrated Mirov coils signal (1-5kHz) and fitted curve of
the model assuming only m/n =2/1 mode for ]18547 att = 0.24720s. The origin of poloidal angle is the magnetic axis determined by the EFIT.

the bispectrum calculation on the toroidal Mirnov signals (φ = 10◦ , 50◦ , 70◦ , 110◦ , 130◦ , 170◦ , 230◦ , 290◦ ) with
250kHz sampling rate in order to check non-linearity. The
number of ensemble is 8 and data points per one ensemble is 750. There is strong bicoherence around ( f1 , f2 ) ∼(40kHz, 40kHz) indicating harmonic coupling with itself
but these modes are localized near the centre of the plasma.
Therefore the 2/1 mode was localized near the reconnection position and can be a candidate for the trigger of thisIRE.

3.2

IRE with q0 < 1In the case of q0 < 1, IRE should be distinguished fromsawtooth oscillation, which is instability at q = 1 rational surface. In this study, IRE is defined as the instability
with mode couplings and/or often accompanied by a collapse at a rational surface except for q = 1 surface. Figure 6 shows an IRE discharge with q0 < 1. This IRE
doesn t terminate the plasma but reduces βt significantly.
The position of the reconnection is around HCAMU]14(R ∼1.45m). However, the area with strong oscillation isclosed to HCAMU]12(R ∼1.2-1.3m) and the phase diﬀer-Fig. 5 Time evolution of the fitting parameters for ]18547. (a):Filament current [kA] (b): Poloidal angle of the one chosen marker filament to understand poloidal rotation. (c):
χ2 (error of the fittings)

ence between HCAMU]9 and 11 is significant. Therefore, this is tearing mode and the resonant surface is at
R ∼ 1.24 − 1.28m. The fitting code doesn’t work wellon this shot because the calculated position of the resonantsurface (from EFIT) is presumably not good. However, as
is indicated in section 2.1, the number of filaments (modelof the mode) on outboard side is very few because of low
aspect ratio configuration. Therefore the poloidal mode
number can be estimated roughly by counting peaks along
the centre stack and adding one for outboard filament. In
this way m/n =2/1 is confirmed for this q0 <1 discharge.
Note that it is diﬃcult to derive the growth time and rotation w/o the fitting. The width of 2/1 mode seems increasing, as shown by SXR radiation from outer side, for example, HCAMU]14 from t ∼ 0.392s. In addition, Mirnov
coils show another high frequency mode (∼22kHz). Unfortunately, there is no strong cross correlation between
this high frequency mode and any of SXR channels, andthe position of this mode is not clear from SXR signals.
However, poloidal mode number (m) is presumably 4 andtoroidal mode (n) is 1. These mode numbers are determined by the Mirnov coil arrays. Therefore, this mode islocalizes at the edge. To clarify the relationship between
these modes (i.e., 3kHz, and 22kHz), bicoherence spectrum is calculated. For t = 0.389 − 0.392 there is no significant coupling between the two modes. Instead of that, 4/1mode (∼22kHz) shows second and third harmonic mode(∼45kHz, ∼68kHz). Figure 7 shows the result in later period t = 0.392 − 0.395s. These frequency are localized on
the frequency space and sharp coherence disappears, but
broad bicoherence in the region f1 + f2 <50kHz appears,
suggesting non-linear coupling among a lot of modes. Thisis consistent with the non-linear phase of the MHD simulation in the reference [1]. The total squared bispectrum(duration of ensemble is∼3ms) at these two periods show asiginificant increase of about 50 times. However, we cannot decide which mode is a trigger of this IRE because theposition of the high frequency mode has not been identified.

Proceedings of ITC/ISHW2007
740.0(a) Ip [kA]
8 2725.0
140.0 (b) nel (interferometer) [10 /m ]125.0 (c) dB/dt (Mirnov coil: Z~0m) [a.u.]
0.
39
10.
39
20.
393

0.
39
40.
3950.
396740

AMC_PLASMA CURRENT

7350.00
51.2•1073072
5ANE_DENSITY

1.
40
•1020

1.
35
•1020

1.
30
•1020

1.
25
•1020

0.12

0.25

0.37

0.49

0.61

0.74

0.86

0.98

0.
8

XMC_CC/MV/220

0.
6

0.
4

0.
2

0.
0

-0.2

-0.4

-0.6

19
1.
6•
10

AIM_DA/HM10/T*(d) Dα[a.u.]

19
1.
4•
1019
1.
2•
105

19
1.
0•
1018
8.
0•
10

1.0•1018
6.
0•
1018
4.
0•
100.
00
6UpperHCAMU#16 (R~1.5)(e) SXR[a.u.]0.
00
40.
00
20.
00
0-0.002

f2 (frequency [Hz])0.
00
8HCAMU#15 (R~1.49)

0.
00
60.
00
40.
00
20.
00
0HCAMU#14 (R~1.45)

0.
00
80.
00
60.
00
40.
00
20.
00
0

HCAMU#13 (R~1.41)

0.
02
00.
01
50.
01
00.
00
50.
02
5HCAMU#12 (R~1.37)

SXR[a.u.]

0.
02
00.
01
50.
01
00.
00
50.
03
5

HCAMU#11 (R~1.33)

0.
03
00.
02
50.
02
00.
01
50.
01
00.
03
5

HCAMU#10 (R~1.28)

0.
03
04

8.0•1046.0•104

4.0•100.
02
50.
02
00.
01
50.
04
5HCAMU#9 (R~1.24)

0.
04
00.
03
50.
03
00.
02
50.
05
5HCAMU#8 (R~1.20)

0.
05
00.
04
52.0•10

0.
04
0Center

0.391

0.392

0.393 0.394
Time [s]

0.395

0.396

40Fig. 6 IRE in a discharge with q0 <1. (a): Plasma current [kA](b): Line integrated electron density [/m2 ] (c): dB/dt
from Mirnov coil [a.u.] (d): Emission of Dα [a.u.] (e):SXR radiation [a.u.] on horizontal SXR camera fromupper to center （SXR signals for midplane to upperchords.） side of the plasma.

4f1 (frequency [Hz])8400

2.0•104 4.0•104 6.0•104 8.0•104 1.0•105 1.2•10 5

Fig. 7 ]18501 Bicoherence spectrum (t = 0.392 − 0.395) of thetoroidal Mirnov coils (top Fig.) and power spectrum on
logarithmic scale (bottom Fig.).

IRE drive

54.5

5Summary and conclusion

In this paper, IREs in the MAST plasma are studied using afilament model, in which low aspect ratio eﬀects are taken
into account. In the case of q0 > 1 no mode coupling concerning to IRE are observed but the growth of the 2/1 tearing mode with τ ∼ 6ms is confirmed. The mode triggers acollapse. In the IRE with q0 < 1, slow growth of the 2/1
tearing mode is observed and 2nd and 3rd harmonic from a
4/1 mode, ( f1 , f2 ) ∼(22kHz, 22kHz) and ( f1 , f2 ) ∼(22kHz,44kHz), are confirmed in Mirnov coil signals. After that,these bicoherence peaks disappear and broad bicoherence
on a range of f1 + f2 <50kHz is observed. This is consistent
with the time evolution of each mode in [1]. Although wehaven’t found a clear evidence of non-linear developmentcausing an IRE followed by collapse, the 2/1 mode maytrigger the collapse. To find out the source of IRE traces

4
Magnetic shear

To understand the drive of IRE is important for the prevention. For a comparison with the reference [1], whichsuggested pressure-driven mode as the source of the instabilities, we evaluated pressure gradient for some IRE
discharges. Figure 8 shows time evolutions of a correlation between magnetic shear (S ) and pressure gradient(dP/dr) for thirteen shots, where magnetic shear definedas S = (r/q) · dq/dr. These two parameters are calculated by kEFIT which takes into account pressure profile
from the Thomson scattering profile. Red circles represent
the values just before each IRE. IRE occurs when pressure
gradient exceed critical value, which becomes high at highshear. Note that although there are many shots showing increase of the gradient by the time of IRE, some shots don’t
show such behavior under the result from the kEFIT.

3.5
32.5
21.5
10.5
0020000 40000 60000 80000 100000 120000 140000
dP/dr (kEFIT) [Pa/m]

Fig. 8 Trace of magnetic shear (S ) and pressure gradient fromthe kEFIT ressult at q = 2 rational surface before (0 ∼21ms) IRE. Same symbol shows same shot. The red
shaded circles indicate just before the IRE.

of pressure gradient and magnetic shear. Critical pressure
gradient at q = 2 increases with magnetic shear suggestingpressure-driven nature of IREs.

6Acknowledgment

This work was supported by JSPF Grant-in-Aid for Scientific Research 16106013.
[1] N. Mizuguchi et al., Phys. Plasmas 7 (2000) 940.
[2] B. Lloyd et al., Nucl. Fusion 43 (2003) 1665.
[3] L. Lao et al., Nucl. Fusion 25 (1985) 1611.
[4] Y.C. Kim and E.J. Powers, Phys. Fluids 21 (1978) 1452.
[5] Y. Nagashima et al., Plasma Phys. Control. Fusion 49 (2007)
1611.