External RMP effect on locked-mode-likeinstability in helical plasmas
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公開日: 2021-12-28
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キーワード (En):作成者: TAKEMURA, Yuki, WATANABE, Kiyomasa,
SAKAKIBARA, Satoru, OHDACHI, Satoshi,
NARUSHIMA, Yoshiro, IDA, Katsumi, YOSHINUMA,
Mikiro, LHD, Experiment Group
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EXTERNAL RMP EFFECT ON LOCKEDMODE-LIKE INSTABILITY IN HELICAL
PLASMAS
Y. TAKEMURA1,2, K.Y. WATANABE1,3, S. SAKAKIBARA1,2, S. OHDACHI1,4,
Y. NARUSHIMA1,2, K. IDA1,2, M. YOSHINUMA1,2 and LHD EXPERIMENTGROUP
1 National Institute for Fusion Science, National Institutes of Natural Science, Toki, Gifu 509-5292,

Japan
2 The Graduate University for Advanced Studies, SOKENDAI, Toki, Gifu 509-5292, Japan
3 Nagoya University, Graduate School of Engineering, Chikusa, Nagoya 464-8603, Japan
4 The University of Tokyo, Graduate School of Frontier Sciences, Kashiwa, Chiba 277-8561, Japan
E-mail: takemura.yuki@nifs.ac.jp

Abstract
The slowing-down mechanism of the locked-mode-like instabilities with and without the island structure is investigatedthrough the effects of an external RMP (Resonant Magnetic Perturbation) on the instabilities. For both instabilities, the slowingdown duration time decreases with the increase of the external RMP, and the RMP dependence is consistent with the braking
model by the j × B force due to the interaction between the instabilities and the external RMP. Moreover, the relationshipbetween the amplitude and the frequency of both locked-mode-like instabilities during the slowing-down is consistent with theforce balance model between the j × B force due to the external RMP and a viscous force. These results suggest that the slowingdown of both locked-mode-like instabilities with the finite external RMP occurs due to the j × B force driven by the external
RMP.
Keywords: MHD instability, rotation, RMP, helical plasma, LHD

1. Introduction
In the LHD (Large Helical Device), a reactor-relevant
plasma with a volume-averaged beta value of 5 % can bestably maintained for more than ten times as long as theconfinement time [1]. On the other hand, collapse events due
to various MHD instabilities observed in different operational
regimes cause serious degradation of confinement property:
the torus-outward magnetic axis shift configuration (Rax > 3.75m) with a peaked pressure profile [2] , the torus-inward shifted
configuration (Rax < 3.60 m) [3] and the standard magnetic
axis configuration (Rax = 3.60 m) with a low magnetic shear.
In particular, in low magnetic shear discharges, the frequencyof the precursor decreases, and the collapse occurs. This event
is called the locked-mode-like instability [4]. In high beta
discharges of the LHD, plasma pressure gradient is maintained
without the collapse events even if linear MHD analysis

predicts the interchange mode is unstable. In a helical plasma,
investigating the physical mechanism of the collapse and therelationship between the index of linear MHD instability andthe collapse is useful for reflecting MHD stability
characteristics of the high beta LHD discharges on the designof a helical fusion plasma.
Two types of locked-mode-like instabilities whose
precursors have different radial mode structures are observedin different regimes of a magnetic shear and a beta value ofthe LHD [4–6]. Regarding the radial profile of internal
fluctuations, the precursor of the first type has the odd-type
structure respect with the resonant surface. This structure issimilar to the tearing mode which is considered to have thelarge magnetic island (type-I, tearing-type locked-mode-likeinstability). The other instability has the even-type structure
similar to the interchange mode which is considered to nothave the large magnetic island (type-II, interchange-type
locked-mode-like instability).
In locked mode discharges of tokamaks [7] and RFPs [8],it is demonstrated that the disruption is escaped bymaintaining the rotation of a precursor of the locked mode byusing a rotating external RMP [9] and/or tangential NBIs.
These results suggest that the slowing-down of the precursor
could be related to the trigger mechanism of the collapse.
In the LHD, in order to establish the method of avoiding
the collapse, the slowing-down mechanism of the MHD
fluctuations due to the locked-mode-like instability has been
investigated. From the comparison between the frequency ofthe precursor and the E × B flow at the resonant surface during
the slowing-down phase, the precursor’s frequency is almost
the same as the E × B flow around the resonant surface [5]. It
is found that the decrease of the E × B flow around theresonant surface occurs through the following two stages [5].
The slowing-down of the first stage (tfirst) is caused by the
movement of the resonant surface to the plasma core region
with a small E × B flow due to increasing the plasma current
in the co-direction because the plasma current in the codirection enhances the increase of the core rotationaltransform. The slowing-down of the second stage (tsecond)
occurs by the decrease of the plasma flow around the resonantsurface but the resonant surface moves slightly. The above twostages are observed in both of the two types of instabilities [6].
Figure 1 shows the time evolution of (a) amplitude of themagnetic fluctuation due to the m/n = 1/1 mode measured bya magnetic probe, (b) mode frequency, (c) radial magnetic
fluctuation amplitude measured by saddle coils, (d) plasma
current and (e) the radial profile of an E × B flow and the radial
location of the /2 = 1 resonant surface in the interchangetype locked-mode-like instability discharge. Here, m and ndenote the poloidal and the toroidal mode number,respectively. Furthermore, it is found that the duration time ofthe slowing-down phase (tfirst + tsecond) decreases with theincrease of the external RMP (Resonant Magnetic
Perturbation) amplitude in interchange-type locked-mode-likedischarges [6]. However, the reason for the decrease of an E
× B flow in tsecond is unclear.
On the locked mode of tokamaks, several slowing-down
models are proposed [10]. The well-known slowing-down
force is the j × B force. Two types of the j × B forces are
considered depending on what induces B: Frw and FRMP. The
former is the slowing-down force due to the interaction
between the toroidal perturbed current (jt) due to theinstability and the perturbed magnetic field due to the eddy
current flowing in the resistive wall of the vacuum vessel. The
latter is the force due to the interaction between jt and theperturbed magnetic field due to the external RMP coils. In the
JT-60U tokamak, the relationship between the magnetic
fluctuation amplitude and frequency of the precursor during
the slowing-down phase is consistent with the prediction of

the Frw model, suggesting that the contribution of Frw to theslowing-down is large [11].
In this paper, when an amplitude of the imposed external
RMP is changed, the duration time of the slowing-down phase,and the relationship between the magnetic fluctuation
amplitude and frequency of the precursor are obtained inlocked-mode-like instability with different internal mode
structures. The experiment results are compared with the Frw
and FRMP models.
This paper is organized as follows. In Sec. 2, theexperimental setup is explained. In Sec. 3, the experimentresults regarding the locked-mode-like instability are analyzed.
In Sec.3.1, the effects of the external RMP on the slowingdown duration time of the interchange-type instability are
shown. In Sec. 3.2, the RMP dependence of the slowing-down
duration time is compared with the slowing-down models
considered in tokamaks. In Sec. 3.3, the experiment results ofthe tearing-type instability are shown and the effect of theinternal structure of the instability on the slowing-down isdiscussed. In Sec 3.4, the relationship between the magnetic
fluctuation amplitude and the frequency of a precursor of bothinstabilities is shown. The summary and discussion are shownin Sec. 4.

2. Experimental setup
The locked-mode-like instability typically occurs in low
magnetic shear plasmas of the LHD. When the plasma aspectratio Ap increases, the magnetic shear in the whole region of aplasma decreases [12]. In this experiment, Ap is set to 7.1,
which is higher than the configurations with Ap = 5.8~6.6,
where the reactor-relevant high beta discharges are achievedin the LHD.
For production and heating of plasmas, two tangential NBIs
are used. The plasma current increases during discharges dueto two tangential NBIs with the same direction and the corerotational transform increases, leading the decrease of themagnetic shear. Perpendicular NBIs are modulated formeasurement of an E × B flow by a charge exchange
spectroscopy.
The toroidal and poloidal mode number of magnetic
fluctuations are identified by a toroidal array with 6 magnetic
probes and a helical array with 15 probes outside a plasma [13].
For evaluation of the amplitude and frequency of a magnetic
fluctuation, one of the toroidal array probes located ~ 0.3 m
away from the /2 = 1 surface is used. The slowly changing
radial magnetic fluctuation amplitude is measured by twoarrays of saddle loops with large cross-section area. The
effective area size through which the radial magnetic flux
passes is ~ 0.4 m2. The averaged distance between the saddle
loop and the /2 = 1 surface is ~ 0.5 m.
The definition of tfirst is the duration time when the /2 =1 surface largely decreases, and that of tsecond is when the /2
= 1 surface does not largely change but the E × B flow velocity

at the /2 = 1 surface decreases. Therefore, accurate
evaluation of the radial location of the /2 = 1 surface isimportant. During the slowing-down phase in the lockedmode-like instabilities, the small flattering region in the radial
electron temperature profile appears due to the m/n = 1/1
precursor. The Thomson scattering system with high spatial
resolution can observe the time evaluation of the flattening.
The RMP coil system in the LHD, which has 10 verticalpairs of coils at the top and the bottom, can calibrate theintrinsic error field. According to measurement of themagnetic surface mapping in vacuum [14], there is an m/n =1/1 magnetic island due to the intrinsic error field, which isalmost corrected by the external RMP with RMP coil current(IRMP/Bt) of 110 A/T. Namely, the magnetic island due to theintrinsic error field shrinks to a smaller size. In this paper,
IRMP/Bt is changed from 0 to 100 A/T. As IRMP/Bt increases, thecorrected error field amplitude decreases, which means that apositive sign of IRMP/Bt corresponds to the opposite phase to
the error field. It should be noted that IRMP/Bt and the error
field amplitude have a negative correlation.

3. Experiment results
3.1 External RMP effect on interchange-type lockedmode-like instability
Figure 1 shows a typical waveform of the interchange-type
locked-mode-like instability discharges where the intrinsic
error field is almost cancelled by imposing an external static
RMP of IRMP/Bt = 100 A/T. Next, when the amplitude of theimposed external RMP is changed, behaviours of the m/n =1/1 mode as the precursor are explained. Figure 2 shows
typical waveforms with IRMP/Bt of 30 and 15 A/T: (a) (e) radial
magnetic fluctuation amplitude measured by saddle loops, (b)(f) mode frequency, (c) (g) magnetic field signal and (d) (h)radial location of the /2 = 1. Here, the blue and red hatchedregions correspond to the first and second stages of theslowing-down phase, respectively. It should be noted that thetiming/amount of gas puffing, the magnetic configuration andthe heating condition are almost the same with those in Fig. 1
except for IRMP/Bt. In Figs. 1 and 2, the error field amplitude
increases when IRMP/Bt decreases. From Figs. 1 and 2, it isfound that tfirst is not largely changed, but tsecond decreases
as the error field increases.
Figure 3 shows the external RMP dependence of (a) tfirstand (b) tsecond. Circles display several discharges of the same
IRMP/Bt and cross symbols correspond to the averaged value ofthe discharges of each IRMP/Bt. There is no clear dependenceof tfirst on the amplitude of the external RMP in the 0 to 100
A/T region. On the other hand, tsecond decreases as IRMP/Bt
decreases. It is found that the external RMP dependence of theduration time of the slowing-down phase (tslowingtfirst +tsecond) as reported in ref.[6] reflects the external RMP

dependence of tsecond. It is interesting that tfirst for IRMP/Bt =100 A/T is different from the other cases. The duration time
tfirst is determined by the change rate of the current profileand the time when the radial movement of the resonant surface
stops. The effect of the external RMP on the change rate andthe time as shown in the above is outside the scope of thispaper and is a topic for future research.

3.2 Comparison between slowing-down models andexperimental observation of locked-mode-likeinstability
In tokamaks, the slowing-down of the precursor of thelocked mode is considered by j × B forces [10], as shown insection 1. There are two types of j × B forces, Frw and FRMP,
due to a perturbed magnetic field induced by eddy current on
resistive wall of a vacuum vessel (Beddy) and by the externalcoils (BRMP), respectively. Assuming that Beddy is proportional
to a perturbed current due to the precursor, jt, and its rotation
angular frequency ,
𝜔𝜏w
𝜔𝜏w
(1)
𝐹rw ∝ 𝛿𝑗𝑡2
~𝛿𝑏r2
.
2
2
2
1 + 𝜔 𝜏w
1 + 𝜔 2 𝜏w
Here, jt is assumed to be proportional to the radial magnetic
fluctuation amplitude br measured outside a plasma and w
is the wall time constant. Assuming that BRMP is proportional
to Ierr/Bt - IRMP/Bt and the penetration of the external RMP
into a plasma is shielded due to the rotation of the external
RMP in the frame of the precursor, FRMP is expressed as
𝜔0′ 𝜏rec
𝐹RMP ∝
𝛿𝑗t × (𝐼err ⁄𝐵t − 𝐼RMP ⁄𝐵t )2√1 + (𝜔0′ )2 𝜏rec
𝜔0′ 𝜏rec~𝛿𝑏r × (𝐼err ⁄𝐵t − 𝐼RMP ⁄𝐵t ).
(2)
2√1 + (𝜔0′ )2 𝜏rec
Here, IRMP/Bt of 110 A/T corresponds to the compensationcoil current for the intrinsic error field, as shown in section 2.
In addition, 𝜔0′ is the slip frequency on the external RMP
frame and 𝜏rec is the typical reconnection time-scale [10]. In
this study, 𝜔 = 𝜔0′ because the external RMP does not rotate
in the laboratory frame.
As mentioned above, br plays an important role on both of
Frw and FRMP. Therefore, the dependence of IRMP/Bt on brtime-averaged during tsecond of the interchange-type lockedmode-like instability is shown in Figure 4. It is clearly foundthat br increases with IRMP/Bt in the 0 to 100 A/T region. Notethat br is the amplitude of the minor radial component of themagnetic fluctuation due to the m/n =1/1 mode measured by amagnetic probe and the time-averaged value of br duringtsecond cannot be evaluated if tsecond = 0.
Figure 5 shows relationships between tsecond and (a) Frw
and (b) FRMP. The forces are the time-averaged value during
the second stage of the slowing-down phase. When a force
strongly contributes to the slowing-down, the slowing-downtime decreases with the increase of the force. In other words,

the correlation between the force and the slowing-down timeis negative. From Fig.5, tsecond increases with Frw, but tsecond
decreases with the increase of FRMP. These results suggest thecontribution of FRMP to the slowing-down is large against Frw.
Here, w is assumed to be 1 ms. Since w is the resistive skintime of the vacuum chamber, it is not changed by the
instability. If a different value for w is selected, the change
does not affect the above qualitative behaviour.

additional data and more systematic analyses are required infuture.

3.4 Relationship between amplitude and frequency of
precursor during slowing-down phase

Next, the relationship between the frequency (f) andmagnetic fluctuation amplitude (b/Bt) of the precursor during
the slowing-down phase is investigated. According to the JT60U tokamak's experiment study, the relationship between f
3.3 Effect of internal mode structure of locked-modeand b/Bt of the tearing mode, which is the precursor of thelike instabilities on slowing-down mechanism
locked mode, is almost consistent with the equation of the Frw
The previous sections show experiment results of the model [11]. This result suggests that the contribution of Frw
interchange-type locked-mode-like instability. From here, the to the slowing-down of the precursor of the locked mode isresults of the tearing–type locked-mode-like instability are large. The b/Bt-f relationship of the interchange-type
shown. The tearing-type instability appears in the region of a instability during the sum of tfirst and tsecond is alreadyhigher magnetic shear and a lower volume-averaged beta reported [6]. However, there was no conclusive result. Here,value than the parameter region where the interchange-type the b/Bt-f relationship only during tsecond is focused upon,
instability appears [6]. In order to clarify the slowing-down and is compared with the relationship based on the following
mechanism of the tearing-type locked-mode-like instability, force balance models.
the external RMP dependences of the slowing-down time of
In the force balance model of Frw and a viscous force (Fvc),
the tearing-type locked-mode-like instability are investigated. the relationship between the mode frequency and the magnetic
Figure 6 shows the external RMP dependences of tfirst and fluctuation amplitude of the mode is expressed in [10]. The
tsecond. Similar to the interchange-type locked-mode-like viscous force is expressed as(3)
𝐹vc ∝ (𝑓0 − 𝑓),instability, tfirst of the tearing-type locked-mode-likeinstability does not depend on IRMP/Bt, but tsecond increases where f0 corresponds to the rotation frequency due to theneoclassical flow. The force balance between Frw and Fvc iswith IRMP/Bt.
(4)
𝐹rw = −𝐹vc .
Figure 7 shows the dependence of IRMP/Bt on br of the
Substituting
Eqs.
(1)
and
(3)
into
Eq.
(4)
yieldstearing-type locked-mode-like instability, together with theexperimental data of the interchange-type instability as
𝑓 = 𝛽rw (1 + √1 − 𝛼rw (𝛿𝑏⁄𝐵t )2 ).
(5)
already shown in Fig.4. For the tearing-type instability as well In addition, the b/Bt-f relationship based on the force balanceas the interchange-type instability, br increases with IRMP/Bt. of FRMP and Fvc is also derived.
It is interesting that br of the interchange-type instability is(6)
𝐹RMP = −𝐹vc
larger than that of the tearing-type instability at the same𝑓 = −𝛼RMP (𝛿𝑏⁄𝐵t ) × (𝐼err ⁄𝐵t − 𝐼RMP ⁄𝐵t ) + 𝛽RMP . (7)IRMP/Bt. The external RMP drives the locked mode in Here, 𝜔𝜏rec ≫ 1 is assumed since 𝜔 is several krad/s and 𝜏rectokamaks, while it suppresses the tearing mode appearing ~ 0.2 s in the typical LHD plasma parameters.
before the mode locking. Similarly, in the locked-mode-likeOn the other hand, if the external RMP is completely
instability discharges, the suppression of the precursor during penetrated, FRMP,no-slip and the b/Bt-f relationship based on thethe slowing-down by the external RMP is also observed. FRMP,no-slip model is derived as follows[10].
However, the suppression mechanism is a future work.
𝐹RMP,no−slip ∝ 𝛿𝑏r × (𝐼err ⁄𝐵t − 𝐼RMP ⁄𝐵t ).
(8)
Figure 8 shows (a) Frw dependence and (b) FRMP 𝑓 = −𝛼
(δ𝑏
⁄
)
(𝐼
⁄
⁄
)
𝐵
×
𝐵
−
𝐼
𝐵
+
𝛽
RMP,no−slip
t
err
t
RMP
t
RMP,no−slip . (9)dependence of tsecond of the tearing-type locked-mode-like It should be noted that the b/B -f relationship based on the notinstability together with the interchange-type locked-mode- slip model of F
RMP has same dependence with that on based
like instability. Similar to the interchange-type locked-mode- on the slip model in the limit of 𝜔𝜏 ≫ 1. Here,  ,  ,
rw
rw
rec
like instability, tsecond increases as Frw increases, but tsecond  , 
RMP
RMP, RMP,no-slip and  RMP,no-slipare free parameters. Notedecreases as FRMP increases. This result suggests that FRMP has
that rw, RMP, RMP,no-slip (each ) are related witha large contribution to the slowing-down regardless of the
proportionality constant of Frw, FRMP and FRMP,no-slip (Fvc)internal mode structure of the instability, that is, the size of the
except b and f, respectively.
magnetic island of the instability. However, quantitatively, it
Figure 9(a) shows the b/Bt-f relationship of thecan be seen that tsecond of the tearing-type instability is shorter
interchange-type locked-mode-like instability with thethan that of the interchange-type instability at the same FRMP.
external RMP of IRMP/Bt = 50 A/T during tsecond. Black andIn order to understand the reason of the quantitative difference,
grey symbols correspond to the typical behaviour and atransient behaviour of b/Bt and f, respectively. The transient
behaviour is observed for a short time just after the precursor
suddenly appears. The intermittent appearance of the
precursor is often observed in discharges with external RMP
as shown in [6]. The blue dash-dotted lines display theequations of the Frw model with (rw, rw) = (1.0 × 108, 0.8)
and (2.5 × 107, 0.8), and the red dashed line displays theequation of the FRMP model with (RMP, RMP) = (6.6 × 101,
1.7). The b/Bt–f relationship is consistent with FRMP, but it is
not consistent with the Frw model. This result suggests that the
FRMP model is closer to the experimental data when comparing
between the two models.
Figure 9(b) shows the b/Bt-f relationship of the tearingtype instability with the external RMP of IRMP/Bt = 50 A/T,
which coincides with the FRMP model of (RMP, RMP) = (6.6 ×
101, 1.4). These results suggest that the FRMP model isconsistent with the b/Bt-f relationship during tsecond of thelocked-mode-like instability regardless of the internal mode
structure of the instability, supporting the result in section 3.3.

4 Summary and discussion

the second stage with the external RMP is not consistent withthat predicted by the Frw model, but it is consistent with the
FRMP model. This result supports the statement that thecontribution of FRMP to the slowing-down is larger than Frw.
The above results are on the experiment results of theinterchange-type locked-mode-like instability, which has amagnetic island with a small width. The same qualitative
results for the tearing-type locked-mode-like instability are
obtained. These results suggest that the slowing-down of the
precursor is mainly caused by the j × B force driven by theexternal RMP regardless of the size of a magnetic island of theinstability.
According to an early work, it is reported that thepenetration of the external RMP into a plasma is shieldedwhen the external RMP rotates in the frame of the precursor,
and that the BRMP is reduced at the resonant surface and FRMP
is smaller than that FRMP,no-slip in the complete penetration case.
The tsecond dependence on FRMP shown in Figures 5 (b) and 8(b) is almost the same against the FRMP,no-slip in the limit of𝜔𝜏rec ≫ 1 . The evaluation of experimental BRMP at theresonant surface during the shielding of the penetration of theexternal RMP is a future research topic.
From the quantitative aspect, the duration time of thesecond stage of the tearing-type locked-mode-like instability
looks shorter than that of the interchange-type instability at thesame FRMP. Here, jt is assumed to be proportional to brobserved outside a plasma. The interchange-type locked-mode
instability is expected to have an odd-function-type jt profile,which would have larger |jt| around the resonance surface
than in the tearing-type instability for the same br. The
improvement of the quantitative accuracy for FRMP of theinterchange-type locked-mode-like instability is a future task.
In this study, it is found that the contribution of theelectromagnetic force due to the external RMP to the slowingdown of the precursor is large from the dependence of theslowing-down time on the external RMP. On the contrary,even in the discharges with the small external RMP (IRMP/Bt =100 A/T), the precursor frequency decreases for finite tsecond.
According to a previous work of the external RMP effects on
the locked-like mode instability, the locked location of the
precursor is consistent with the phase of the error field when
the finite error field exists. On the other hand, the locked
locations of the precursor are not fixed when the error field isalmost compensated [15]. The behaviour suggests that theslowing-down mechanism on the precursor of the lockedmode like instability in the case without the error field isdifferent from that with the error field. The slowing-down
mechanism in the discharges with the small external RMP
should be resolved in future.

In order to clarify the slowing-down mechanism of the
precursor in the locked-mode-like instability of the LHD, theeffect of the external RMP on the slowing-down isinvestigated. The slowing-down phase can be divided into twostages depending on slowing-down processes. In the first stageof the slowing-down phase, the E × B flow at the resonantsurface decreases due to the movement of the resonancesurface to a small E × B flow region, and in the second stage,
the resonance surface is almost constant, but the E × B flow
around the resonant surface itself decreases. This
characteristic was already found in the previous research, but
it is unclear how the duration time of the second stage isdetermined. It is investigated how the duration time of thesecond stage changes by imposing the external RMP, and the
RMP dependence of the duration time of the second stage iscompared with the slowing-down models for a precursor of
the locked mode in tokamaks: Frw and FRMP. The former is thej × B force by the interaction between a perturbed current due
to the instability and a RMP field created by its eddy current,
and the latter is the j × B force by the interaction between theperturbed current and a RMP field due to the external coils at
the resonant surface. In addition, FRMP with the complete
penetration of the external RMP into the resonant surface(FRMP,no-slip) is also considered.
It is found that the duration time of the second stage
decreases as the effective external RMP amplitude increases.
The duration time of the second stage has a negative
correlation with FRMP, but a positive correlation with Frw. Thus,
the above results suggest that FRMP has a large contribution to
Acknowledgements
the slowing-down of the precursor.
The authors are grateful to the LHD Operation Group for
Furthermore, the relationship between the magnetic
fluctuation amplitude of the precursor and its frequency during their excellent technical support. One of the authors (Y.T.)

would like to thank Dr. S.Nishimura (Housei Univ.), Dr. A.
Isayama (QST) and Dr. M. Furukawa (Tottori University) for
useful discussion. This work was supported in part by NIFS
under constract NIFS07KLPH004 and by a dispatch grantfrom the Future Energy Research Association.

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Kamada Y. and Ozeki T. 2010 Effect of magnetic island
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Nucl.Fusion 53 043010

tfirst

slowing

const.

102

tsecond
locking5

b/Bt (10 )(a)#143218101
100
4 Frequency (kHz)
3
2
1
0
-1
40 r,m=1/Bt (10-5 Wb/T)
30
20
10
0
I /B (kA/T)
40 p t20
0

fmode

fE  B

e.d.
i.d.

(c)

(d)

(e) fEB
(kHz)

4.2
R (m)

(b)

2

4.1

e.d.

0

4.0
3.9
4.0

i.d.

-2

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Time (s)Figure 1. Typical waveform of the interchange-type locked-modelike instability with IRMP/Bt = 100 A/T. (a) amplitude of a magnetic
fluctuation due to the m/n = 1/1 mode measured by a magnetic
probe, (b) mode frequency (closed circles) and the E × B flow at
/2 = 1 (open circles), (c) radial magnetic fluctuation amplitude
measured by saddle loops, (d) plasma current, and (e) radial profile
of a E × B flow (contour) and the radial location of /2 = 1 (opencircles). Plus and minus of frequency and red and blue regions of
the contour correspond to frequencies in the electron and ion
diamagnetic direction, respectively [6].

Figure 2. Typical waveform of the interchange-type locked-modelike instability with (a)-(d) IRMP/Bt = 30 A/T and (e)-(h) 15 A/T. (a)(e) radial magnetic fluctuation amplitude measured by saddle loops,(b) (f) mode frequency, (c) (g) magnetic field signal measured by amagnetic probe, (d) (h) radial location of /2 = 1. The blue and redhatched regions correspond to the first and second stages of theslowing-down phase, respectively.

almost-reduced

non-reduced

0.4

tsecond (s)0.4tfirst (s)
0.3

(a)

0.3

Averaged value

0.2

0.2

0.1

0.1
interchange-type

0.0
0

20

40 60 80
IRMP/Bt (A/T)

0.0
0

1000

2000

0.4

tsecond (s)(b)
Ierr/Bt110A/T
interchange-type

0.3

0.4

tsecond (s)
0.3

3000

Averaged Frw (a.u.)

100 120

almost-reduced

non-reduced

(b)
0.2

Averaged value

0.2

0.1

0.1

0.0
0

200

400

600

800

Averaged FRMP (a.u.)

0.0
0

20

40

60

80

100 120

IRMP/Bt (A/T)
Figure 3. External RMP dependence on (a) tfirst and (b) tsecondof the interchange-type locked-mode-like instability. Cross
symbols correspond to the averaged value at each IRMP/Bt.

almost-reduced

non-reduced

br/Bt (10-5)
10

1

(a)
interchange-type

0

20

40

60

80

100 120

IRMP/Bt (A/T)
Figure 4. External RMP dependence on the radial magnetic
fluctuation amplitude time-averaged during tsecond of theinterchange-type locked-mode-like instability.

Figure 5. Dependence of (a) Frw and (b) FRMP time-averaged duringtsecond on tsecond.

almost-reduced

non-reduced

0.4

tsecond (s)0.4tfirst (s)
0.3

(a)

(a)
interchange-type
tearing-type

0.3

Averaged value

0.2

0.2

0.1

0.1
tearing-type

0.0

0.0
0

0

20

40 60 80
IRMP/Bt (A/T)

100 120

400

200

800

600

1000

Averaged Frw (a.u.)

0.4

0.4

tsecond (s)

0.3

(b)

Averaged value(b)
Ierr/Bt110A/T

0.3

interchange-type
tearing-type

0.2

0.2

0.1

0.1
tearing-type

0.0

0.0
0

200

400

600

800

Averaged FRMP (a.u.)

0

20

40

60

80

100 120

IRMP/Bt (A/T)
Figure 6. External RMP dependence on (a) tfirst and (b) tsecond of
the tearing-type locked-mode-like instability. Cross symbols
correspond to the averaged value at each IRMP/Bt.

almost-reduced

non-reduced

br/Bt (10-5)
10interchange-type
tearing-type1

tsecond (s)almost-reduced

non-reduced

0

20

40

60

80

100 120

IRMP/Bt (A/T)
Figure 7. External RMP dependence on the radial magnetic
fluctuation amplitude time-averaged during tsecond of theinterchange-type and tearing-type locked-mode-like instability.

Figure 8. Dependence of (a) Frw and (b) FRMP time-averaged duringtsecond on tsecond. Here, closed (open) circles correspond to theinterchange-type (tearing-type) locked-mode-like instability.

2.0

(a)

interchange-type
IRMP/Bt50 A/T

1.5

Frw modelf (kHz)rw  2.5107
Frw model

1.0rw 108

0.5
FRMP model

0.0
1

10

100
5

b/Bt (10 )2.0
tearing-type
IRMP/Bt50 A/T(b)

f (kHz)

1.5

1.0

0.5
FRMP model

0.0
1

10

100
5

b/Bt (10 )Figure 9. Relationships between the mode frequency and magnetic
fluctuation amplitude of (a) the interchange-type and (b) tearingtype locked-mode-like instability during tsecond with an external
RMP of the IRMP/Bt = 50 A/T. It should be noted that the magnetic
fluctuation amplitude is shown in log scale. Grey symbols indicate
data just after the mode suddenly appears. Blue dashed-dotted lines
correspond to the equation of the Frw model with different freeparameters. Red dashed line corresponds to the equation of the
FRMP model.