Hysteresis relation between turbulence andtemperature modulation during the heat pulse
propagation into a magnetic island in DIII-D
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言語: English
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公開日: 2021-07-05
キーワード (Ja):
キーワード (En):作成者: IDA, Katsumi, Ono, M., Evans, T.E., McKee,
G.R., Austin, M.E.
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http://hdl.handle.net/10655/00012552
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Attribution 3.0 International License.

Hysteresis relation between turbulence and temperature modulation during the heatpulse propagation into a magnetic island in DIII-D
K. Ida,1, 2 T. Kobayashi,1, 2 M. Ono,3 T.E. Evans,4 G.R. McKee,5 and M.E. Austin6
1

National Institute for Fusion Science, National Institutes of Natural Sciences, Toki, Gifu 509-5292, Japan
2
SOKENDAI (The Graduate University for Advanced Studies), Toki, Gifu 509-5292, Japan
3
National Institutes for Quantum and Radiological Science and Technology, Naka, Ibaragi, 311-0193, Japan.
4
General Atomics, San Diego, California 92186-5608, USA
5
University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
6
University of Texas, Austin, Texas 78712, USA(Dated: June 4, 2019)
The hysteresis relation between turbulence and temperature modulation during the heat pulse
propagation into a magnetic island is studied for the first time in toroidal plasmas. Lissajous curvesof the density fluctuation (ñ/n) and the electron temperature (Te ) modulation show that the (ñ/n)propagation is faster than the heat pulse propagation near the O-point of the magnetic island. Thisfaster ñ/n propagation is an experimental evidence of the turbulence spreading from X-point to the
O-point of the magnetic island.
PACS numbers: 52.55.Fa, 52.35.Ra, 52.35.Py

Magnetic islands are widely observed in both laboratory plasma and astrophysical plasma. The turbulence
inside the magnetic island is an important issue because
it has a significant impact on transport characteristics
in toroidal plasmas and turbulent reconnection of magnetic field in the solar flare [1, 2]. Therefore, the study ofthe hysteresis relation between the turbulence level andthe perturbation of the plasma parameter such as temperature is essential for understanding the dynamics andmechanism driving plasma turbulence. The hysteresis relation between the turbulence level and the radial electricfield has been studied in laboratory plasmas using limitcycle oscillation with the frequency of 2-5 kHz and thephase delay between the turbulence and the radial electric field is observed in the turbulence time-scale [3, 4].
Magnetic islands define a unique region in the plasma
because there are no temperature and density gradients,
which drive the turbulence. Since the heat propagation
inside the magnetic island is relatively slow [5, 6], turbulence could propagate faster than the heat pulse, if there
is a turbulence spreading [7] from X-point (or boundary)to the O-point of the magnetic island. If the turbulence
level is simply determined by the local temperature gradients or local radial electric field, the propagation of
the turbulence should track to the propagation of heatpulse within a turbulence time scale (< 10−4 sec) because the propagation of the heat pulse has a time scaleof 10−3 ∼ 10−2 sec. Therefore, the study of the dynamic
linkage (hysteresis relation) between the turbulence level
and temperature modulated inside the magnetic island
provides new and important information for the turbulence spreading and a new research area of non-local interaction between electron temperature (Te ) changes anddensity fluctuation (ñ). However, this hysteresis relation
has not been studied previously, in spite of its importance. In this letter, we show the hysteresis relation between the ñ/n and the Te modulation during the heatpulse propagation into a magnetic island in DIII-D in order to clarify the dynamic relation between the ñ/n andthe Te modulation inside the magnetic island.
For purposes of obtaining a better understanding offluctuation-driven transport inside a magnetic island,
repetitive heat pulses were injected into a magnetic island in the DIII-D tokamak. DIII-D is a tokamak device
with D-shape poloidal cross section, a major radius of 1.7m, and minor radius of 0.6 m for magnetic confinementof high temperature plasmas. In this experiment, theplasma current was 1.28 MA and the toroidal magnetic
field was 1.92 T with an inner wall limiter configuration
and a edge safety factor of qedge = 4.5 (q95 = 3.8). The
electron density (ne ) was 3.9-4.6 ×1019 m−3 and the Te
in the core region was 2.0-2.1 keV. RMP fields produced
by a non-axisymmetric magnetic field perturbation coil(C-coil) are used to produce magnetic islands at the resonant surface [12]. In this experiment, the perturbation
field has a resonance at a safety factor of q = 2 and thepoloidal and toroidal mode numbers (m, n) are (2,1). The
location of the magnetic island can be rotated toroidally
by 180◦ by changing the toroidal phase (ϕn=1 ) of C-coilfrom ϕn=1 = 5◦ to ϕn=1 = 185◦ or vice versa (phase flip).
As a result of this phase flip, the X-point and O-pointof the magnetic island appears between the electron cyclotron emission (ECE) measurement for Te at ϕ = 81◦
and at the beam emission spectroscopy (BES) measurement for ñ at ϕ = 150◦ . The electron cyclotron heating(ECH) power is deposited at ρ=0.42 with a modulation
frequency of 50 Hz and it heats the electrons√ and modifies
Te . Here, normalized minor radius ρ = ψN where ψN s
a normalized toroidal flux, such that it is 0 on the magnetic axis and 1 at the Last Closed Flux Surface (LCFS).
Figure 1 shows the time evolution of ECH power,
PECH , Te measured with ECE at ρ = 0.74, ñ/n measured
with BES at ρ = 0.74 near the O-point or the X-point of
the magnetic island, and the current of one of the C-coilsof C79. Here, ñ is the envelop of density fluctuation inte-

Modulation ECH

A

A

2

2

3 (a)
2
1
0
(b)
0.5
0.4
10-5
-6
10
0.3
-7
10
-8
0.2
10
10-9
0.1
0
0.0
(c) 4 10-750100
f (Hz)~n/n (%)

1

2 10

-7

0
0

50
100
f (Hz)

A2

A

2

10-5
10-6
10-7
10-8
10-9
0

4 10-72e

T (keV) P

ECH

(MW)

2

2 10-750 100f (Hz)

0
0

50 100f (Hz)

I

C79

(kA)

noise level0
4 (d) O-point phase
X-point phase
2
Ó
0
Ó
!
=
185
! =5
-2
n=1
n=1
-4
4.0
4.2
4.4
4.6
4.8
time (sec)
90
(e)
(f)
45
0
-45
-90
90
(g)
(h)
45
0
-45
-90
0.4
0.6 # 0.8
1 0.4
0.6 # 0.8

" (deg)" (deg)5.0

1# 165221

FIG. 1: Time evolution of (a) power of ECH, PECH (b) Te
measured with the ECE at ρ = 0.74, (c) ñ/n integrated from
10 kHz to 50 kHz at ρ = 0.74, and (d) current of C-coil andPoincaré map at the poloidal cross section of Te measurementsin (e) O-point and (f) X-point phase and Poincaré map at thepoloidal cross section of ñ/n measurements in (g) O-point and(h) X-point phase. The frequency spectrum of Te modulationand envelop of density fluctuation are also plotted in Figure(b) and (c).

grated in the frequency range of 10 - 50 kHz. The sign flipof the current in C79 seen in figure 1 (d) indicates the flipof phase of n = 1 perturbation with the toroidal angle of185◦ and 5◦ . In this paper, the period with perturbation
field of 185◦ and 5◦ is called O-point phase and X-point
phase, respectively. The Poincaré map at the poloidal
cross section of the Te and the ñ/n measurements in the
O-point phase and the X-point phase are also plotted,
where the x-axis is the normalized minor radius (ρ) andthe y-axis is the poloidal angle (θ). The Poincaré map

shows the 3D vacuum perturbation field from the C-coilsuperimposed on the axisymmetric DIII-D equilibrium
field reconstructed using the EFIT code. The perturbation field of the C-coils has various Fourier components
δB(m, n) that have a resonance at q = 2, 3, 4. Here,the perturbation fields are calculated using the Fourier
analysis module in the trip3d code [12]. There is no q =1 rational surface because the qmin > 1 in this plasma.
The n = 1 poloidal mode strength for this shot is 8.1, 7.7,
7.4 gauss for the poloidal mode number of m = 2, 3, 4,
respectively. There are also high-n modes produced but
these are more than an order of magnitude smaller than
the n = 1 modes. The m/n = 2/1, 3/1 and 4/1 magnetic islands appear at the normalized minor radius ρ of
0.74, 0.88, and 0.96, respectively. In this experiment, thewidth of the 2/1 magnetic island is enlarged comparedto the vacuum islands seen in figure 1(e-h). At the Opoint phase, the poloidal angle of O-point, θO = +20◦
(the poloidal angle of X-point, θX = -110◦ , +110◦ ) for
Te measurements and θO = -40◦ (θX = -130◦ , +110◦ ) forñ/n measurements. At the X-point phase, θX = +20◦
(θO = -110◦ , +110◦ ) for Te measurements and θX = 40◦ (θO = -130◦ , +90◦ ) for ñ/n measurements. As seenin figure 1(a), the modulation frequency of ECH power
is set to 50 Hz, which is low enough to investigate theslow heat pulse propagation inside the magnetic island.
The modulation amplitude of Te in the O-point phase(t < 4.52 sec), where the O-point of the magnetic island
is located near the poloidal cross section of the Te andthe ñ/n measurements, is much smaller than that in the
X-point phase (t > 4.52 sec). Here, the X-point of themagnetic island is located near the poloidal cross sectionof the Te and the ñ/n measurements. The ñ/n level isalso lower in the O-point phase than that in the X-point
phase. The frequency spectrum of Te modulation and envelope of density fluctuation is also plotted in Figure.1.
It is interesting that the frequency spectrum of Te modulation shows narrower peak at the modulation frequency(50Hz) with X-point, while the envelope of density fluctuation shows narrower peak at the O-point of the magnetic
island. This observation shows a strong correlation between the Te modulation at the X-point and the density
fluctuation at the O-point of the magnetic island.
In order to improve the signal to noise ratio, we
perform a conditional average of ECE and BES signal
as function of relative time of τ . This is defined as(1/N )ΣN
i=1 Ψ(ti + τ ), for an arbitrary variable Ψ, where
ti indicates the i-th time of MECH turn-on and N is thenumber of modulation. Figure 2 shows the patterns ofconditionally sampled signals, radial profiles of mean Te ,amplitude and delay time of Te modulation, ñ/n levels,and modulation amplitude of ñ/n in the O-point phaseand the X-point phase. The modulation amplitude is defined from coeﬃcient of Fourier component of ECE andBES signal at the frequency of ECH modulation (50 Hz).
The Te profile shows the flattening at ρ = 0.62-0.77 in
the O-point phase, while there is no Te flattening observed in the X-point phase. As seen in figure 2(d), themagnetic
islandX-point
O-point
e
e

2
1
0
-1
-2
2
1
0
-1
-2

"T /T (%) "T /T (%)
e e

e

2.0
(a)
1.5
1.0
0.5
0.0
(b)
5
0
X-point
-5
-10
(c)
5
0
O-point-5-10
10 (d)

X-point~ (%)"(n/n)~n/n (%)

dt (ms)ee"T /T (%)! (ms)! (ms)_T (keV)

3

50 Hz

1

O-point

10 (e)
86
4
2
0
10 (f)
1O-point
X-point

X-point
O-point

1 (g) 50 Hz
0.1
X-point
0.01
0.4

O-point
0.5

0.6

0.7 # 0.8

0.9

1#165221 #165216

FIG. 2: Radial profiles of (a) mean Te , the patterns of conditionally sampled signals in (b) X-point and (c) O-point phase,(d) normalized Te modulation amplitude, δTe /Te , (e) delay
time of Te modulation (td ) given by the phase shift between
ECH pulse and Te modulation, (f) ñ/n levels measured with
BES, and (g) ñ/n modulation amplitude, δ(ñ/n) levels, in the
O-point phase and X-point phase.

modulation amplitude of Te inside the magnetic island(O-point) is much smaller than that outside the magnetic island by a factor of 5 ∼ 6. This is in contrast to
no reduction of modulation amplitude of Te at the Xpoint of the magnetic island. The reduction of modulation amplitude is due to the slow heat pulse propagation
inside the magnetic island as observed in the peaked td
inside the magnetic island in figure 2(e). The peaked
td indicates that the heat pulse induced by the modulated ECH power propagates faster across the X-point
and then slowly propagates towards the O-point of themagnetic island. Heat pulse propagation speed inside
the magnetic island is much slower than the speed out-side the magnetic island and at the X-point. The delay
time increases monotonically in the X-point but not in
the outer region (ρ > 0.8) during the O-point phase as
seen in Figure 2(b)(c)(d)(e). This is due to the eﬀectof the higher harmonics components of the heat pulse,which have a faster speed and longer decay length thanthat predicted by a diﬀusive model [8] Therefore, a simple heat pulse propagation analysis with a fundamental
component only gives an apparent roll over of the delay
time especially near the plasma edge, where the eﬀectof higher harmonics component becomes relatively large
particularly in the O-point phase.
As seen in figure 2(f), the magnitude of the ñ/n measured with the BES shows a sharp decrease in the plasmacore from the plasma edge. The magnitude of the ñ/nat the O-point is smaller than that at the X-point by afactor of 2 ∼ 3, while the magnitude of the ñ/n outside
magnetic island region ( ρ = 0.82-1.0) is almost identical between the O-point phase and the X-point phase.
These results are consistent with the previous results observed in the Doppler back scattering (DBS) [9, 10] anda significant reduction of thermal diﬀusivity inside themagnetic island observed in JT-60U [11]. The lower levelof the ñ/n in the O-point seen in Figure 2(f) results in
the reduction of transport, which is consistent with theslower heat pulse propagation as seen in Figure 2(e). The
modulation amplitude of the ñ/n is also reduced inside
the magnetic island as seen in figure 2(g). The small
modulation amplitude of ñ/n is attributed to the small
modulation amplitude of Te . The diﬀerence in the widthof the magnetic island between the Te measurements andthe ñ/n measurements is due to the diﬀerence in toroidal
angle between the two diagnostics as seen in figure 1 (e)and (g). The reduction of ñ/n level is observed in theentire region near the magnetic island at the O-point asseen in figure 2(f). In contrast, as seen in figure 2(g),the reduction of the 50 Hz modulation amplitude of ñ/ndriven by the Te modulation is observed deep inside themagnetic island where the Te modulation is significantly
reduced. This wider width of the reduced ñ/n observedin this experiment is interpreted as the reduction of ñ/ndue to the flow shear that often appears at the boundaryof the magnetic island [13]. The small drop of the fluctuation and the modulation amplitude at ρ ∼ 0.9 implies
the existence of m/n=3/1 magnetic island.
The hysteresis relation between the ñ/n and Te modulation is investigated in order to study the causal relation
between the propagation of the ñ/n and the heat pulse.
The Te profile inside the magnetic island is modulated
between slightly peaked and slightly hollow by the heatpulse propagation. However, the Te is expected to be
isothermal on magnetic flux surface and the Te gradient at the O-point is also expected to be zero because of
the topology of the magnetic island. Therefore, the hysteresis relation between local turbulence and local Te isdiscussed rather than the relationship between the turbulence and the Te gradient, which is not a good measureof hysteresis in a magnetic island. Figure 3 shows the

4

0.6

0.9

X-point~n/n(%)

0.7

e

-5

e

e

! = 0.74 (d)O-point

0
-1

"#

"#
~n/n(%)

0.8~n/n (%)

e

1

2

e

_ _

e

e

X-point0-1 _0 _ 1
(T -T )/T (%)

e! = 0.74 (c)5! = 0.74

0.3
-2

8

e

e

e

(T -T )/T (%)

e

0.7

0.45

0.40

-10

-5

6

0
# (ms)

ECH
5

10

-10

-5

ECH
0
5
# (ms)

X-point

4"# (ms)0.4! = 0.74-4 _ 0 _ 4(T -T )/T (%)(b)

O-point
0.5(T -T )/T (%)~
n/n(%)

0.8

0.6
-8

_ _

(a)

2

T risee~n/n rise0-2-4

magnetic island-6
0.60#165221

0.65

10

(e)
~
n/n rise
T riseeO-point
0.70
!

0.75

0.80

FIG. 3: Hysteresis of amplitude between ñ/n and Te modulation at ρ = 0.74 in the (a) X-point and (b) O-point, time
evolution of Te and ñ/n at ρ = 0.74 in the (c) X-point and(d) O-point phase with the relative time (τ ) respect to theonset of ECH, and (e) radial profiles of delay time diﬀerence
between modulation of Te and ñ/n (δτ ) at X-point phase andthe O-point phase.

results of conditional average sampled aforementioned.
As seen in figure 3 (a) and (b), there is a clear diﬀerence in the hysteresis relation between the O-point and
the X-point. In the X-point, the hysteresis relation shows
counter clockwise (CCW) direction, while it shows clockwise (CW) direction in the O-point. These results showthat the change in Te precedes the change in ñ/n in the
X-point, while the change in ñ/n precedes the change in
Te at the O-point.
The delay time diﬀerence between modulation of Te
and ñ/n (δτ ) is evaluated from the phase delay of sinusoidal wave function which gives the best fit to themeasurements. As seen in figure 3 (c) and (d), the δτ
is 2 ∼ 3 ms at the X-point, while this δτ becomes negative ( -2 ∼ -3 ms) at ρ = 0.74 at the O-point. It should
be noted that density fluctuations in the X-point region

exhibit higher harmonic oscillations, while there is little
higher harmonic oscillation in the density fluctuation in
O-point, which is also shown in figure 1(c). Radial profiles of δτ at the X-point and the O-point are plotted infigure 3 (e). The δτ is positive in the entire region at the
X-point, which indicates that the fluctuation responds
to the change in the Te gradient due to the heat pulse
propagation. The Te gradient across the x-point is larger
than the one across the O-point as shown in figure 2(a),which causes the thermal transport to lead the change in
the ñ/n resulting in the positive (CCW) hysteresis. This
result is consistent with the previous result that the ñ/nlevel increases as the Te gradient is increased [10]. However, the δτ in the O-point is positive in the inner region(ρ < 0.68) and negative in the outer region (ρ > 0.68).
The negative δτ observed shows that the propagation offluctuation is faster than that of the heat pulse, which issimilar to the observation in LHD [14].
The positive and negative δτ observed in this experiment imply the existence of the non-locality of the transport, because the existence of hysteresis is evidence of
non-locality in the transport [15, 16]. The negative δτis associated with the CW hysteresis in the O-point due
to the ñ/n leading the heat pulse in the island O-point.
A possible physics model here is that the heat pulse isshunted in the parallel direction around the good internal island flux surfaces while the perpendicular thermaltransport into the island O-point is slow, as seen in figure 2(c) and can only be enhanced once the ñ/n increase
suﬃciently to enhance the cross-field thermal transport.
On the other hand, the thermal transport at the x-pointis enhanced by the stochastic field lines which allows thefast parallel thermal transport to lead the ñ/n across that
region of the island compared to the O-point. The negative δτ indicates that the fluctuation propagates from the
X-point of the magnetic island by turbulence spreading
before the heat pulse propagates into the O-point of themagnetic island.
The enhancement of heat pulse propagation speed
due to the turbulence spreading could be one of thecandidates to explain the bifurcation between highaccessibility state (with larger amplitude and fast heatpulse propagation) and low-accessibility state (withsmaller amplitude and slower heat pulse propagation) of
the O-point of the magnetic island [17]. In the highaccessibility state, the ñ penetrates into the O-point
across the x-point by turbulence spreading and leads theenhancement of transport and heat pulse propagation.
Turbulence spreading can be shielded by the radial electric field shear (Er ) shear [18] which is localized at theboundary of the stationary magnetic island [11, 13, 19].
Therefore, if the Er shear becomes large enough to shield
the turbulence spreading, the ñ does not penetrate into
the O-point of the magnetic island and results in the
reduction of transport and slow heat pulse propagation
inside the magnetic island as seen in the low-accessibility
state. Once the turbulence spreading is shielded, theEr shear is expected to increases due to the reduction

5of viscosity. Then the bifurcation between the following
two states are possible; one is weak turbulence spreadingwith the large Er shear and the other is strong turbulence
spreading with the small Er shear. The slight change in
Er field shear at the boundary of the magnetic island
can cause the bifurcation between high-accessibility andlow-accessibility state.
In conclusion, the hysteresis relation between the turbulence and the Te modulation during the heat pulse
propagation into a magnetic island is studied in DIIID. The δτ is negative (fluctuation propagation is fasterthan the heat pulse propagation) in the low field sideof the O-point inside the magnetic island, while it ispositive (fluctuation propagation is slower than the heatpulse propagation) in the entire region near the X-pointof magnetic island. The observations of hysteresis andthe large diﬀerence in delay time between the propagation of fluctuation and heat pulse suggests the feedback
loop between the turbulence propagation and heat pulse
propagation, where the faster propagation of turbulence
enhances the speed of the heat pulse propagation. Understanding transport inside magnetic islands should have astrong impact on the prediction of the H-mode threshold
power in ITER, where the resonant magnetic perturbations (RMPs) is applied recognized to suppress the edge
localized mode (ELM) in order to avoid damage to thewall due to the transient heat load to the divertor intokamaks [20, 21].

The authors would like to thank the technical staﬀ
of DIII-D for their support of these experiments andDr.L.Bardóczi for useful discussions.
This work is
partly supported by JSPS KAKENHI Grant Number
JP15H02336. This work is supported by NIFS/NINS under the project of Formation of International ScientificBase and Network (KEIN1111, KEIN1113) and by the
US Department of Energy under DE-FC02-04ER54698,
DE-FG03-97ER54415 & DE-AC05-00OR22725.
This report was prepared as an account of work sponsored by an agency of the United States Government.
Neither the United States Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liabilityor responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinionsof authors expressed herein do not necessarily state orreflect those of the United States Government or any
agency thereof. DIII-D data shown in this paper canbe obtained in digital format by following the links athttps://fusion.gat.com/global/D3D DMP.

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