On the interplay between MHD instabilities andturbulent transport in magnetically confinedplasmasメタデータ

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On the interplay between MHD instabilities and turbulent transport in magneticallyconfined plasmas
K. Ida1, 2
1National Institute for Fusion Science, National Institutes of Natural Sciences, Toki, Gifu 509-5292, Japan
2
The Graduate University for Advanced Studies, SOKENDAI, Toki, Gifu 509-5292, Japan(Dated: September 18, 2019)
Interplay between MHD and turbulence is an interesting topic in magnetically confined plasmaand solar plasma. Recently experimental observations below suggest the coupling and interplay
between MHD and turbulence in magnetically confined toroidal plasmas. 1) Turbulence spreading
into the magnetic island, 2) self-organized change in topology and turbulence in magnetic island, 3)
flow damping by stochastic magnetic field, 4) trigger mechanism for the MHD bursts, 5) impact of
MHD bursts on the ion velocity distribution and potential, and 6) turbulence exhausts at the MHD
burst event. In this paper, experimental evidence for the interplay between MHD and turbulencein toroidal plasmas is reviewed. The physics mechanism of the interplay and a possible link toastrophysical plasma physics are also discussed.
PACS numbers:

I.

INTRODUCTION

The interplay between MHD and turbulence is an interesting topic in magnetically confined plasma and solar
plasma research. Fast reconnection of magnetic field in
solar flares is well known. However the mechanism is
not fully understood. Turbulence in the current sheet
is a strong candidate to explain fast reconnection in solar flares [1, 2]. In magnetically confined plasmas, MHD
instabilities and electrostatic turbulence have been investigated independently. No coupling between the electrostatic turbulence and MHD instability is assumed.
However, recent experimental observations below suggest the coupling and interplay between MHD and turbulence in magnetically confined toroidal plasmas [3],turbulence spreading into the magnetic island [4], selforganized change in topology and turbulence in magnetic
island [5], flow damping by stochastic magnetic field [6],trigger mechanism for the MHD bursts [7], impact of
MHD bursts on the ion velocity distribution and potential [8, 9], and turbulence exhausts at the MHD burst
event [10]. Evidence for the interplay between MHD andturbulence in toroidal plasma is reviewed and the physics
mechanism of the interplay is also discussed in section
II. For example, in the heat pulse experiment in DIIID tokamak plasma, the turbulence increase occurs after
the arrival of the heat pulse with a significant delay at
the X-point, while it occurs before the heat pulse radial propagation at the O-point of the magnetic island.
This is due to the turbulence spreading from the X-point
to the O-point of the magnetic island faster than theheat pulse determined by the transport time scale inside
the magnetic island. In D-IIID tokamak, two states ofmagnetic islands are observed. The ’high-accessibility’state is when the heat pulse propagating from outside amagnetic island can penetrate inside the magnetic island,

while the ’low-accessibility’ state is when penetration of
the heat pulse into the magnetic island is prevented at
the boundary. These oscillations between these states are
self-regulated and the change in magnetic topology suchas stochastization of the magnetic field at the boundaryof the magnetic island is one of the possible mechanisms
for this self-regulated oscillation. A self-regulated oscillation of the topology and transport inside the magnetic island is observed owing to the interplay between the MHD
and turbulence. In Large Helical Device (LHD) helicalplasma, anomalous flow damping is observed related tothe stochastic magnetic field in the plasma core, whichsuggests the interplay between MHD and turbulence inplasma.
More recently, a strong MHD burst is found to havebeen triggered by the non-modal (no toroidal and no
poloidal periodic number) localized plasma deformation.
Here, non-modal means that there is no clear helicalstructure characterized by the low order toroidal andpoloidal periodic number usually expressed by n/m at
the rational surface. The non-modal perturbation has
solitary structure and is highly localized in toroidal andpoloidal direction. The perturbation with solitary structure is observed near the plasma edge when the edge
localized mode (ELM) activities are excited, and is also
known as a finger structure [11, 12]. This MHD burst
causes the distortion of ion velocity distribution and large
potential due to energetic ion loss. This MHD burst also
causes the rapid radial propagation of turbulence fromplasma core to SOL region. This is also clear evidence
for the interplay between MHD and turbulence. A newexperimental result on the impact of an MHD burst on
plasma flow and turbulent transport as evidence for interaction between MHD and turbulence is discussed in
section III.
Figure 1 shows a diagram of physics elements of interactions between topology and MHD and interactions
between turbulence and plasma flow. The interaction between flow shear of parallel, E × B flow, and zonal flow

2

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INTERPLAY BETWEEN TURBULENCEAND TOPOLOGYA.

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FIG. 1: Diagram of physics elements of interactions between
plasma flow, turbulence, topology, and MHD.

has been discussed since the strong E × B flow shear
was found in the pedestal region in the H-mode plasma.
The flow shear contributes to the suppression of turbulence shearing eﬀect, and flow shear has been recognized
to be a key element in the feedback loop of formationof edge and internal transport barrier. Recently, workon the interaction between topology and MHD where
the conclusion is that the plasma self-organizes by triggering an internal bifurcation of magnetic island chains[13, 14]. However, there are few experiments reported
on the interactions between turbulence and topology or
MHD and the interaction between topology and MHD in
the toroidal plasma.

MHD instability causes the change of magnetic topology in the plasma. The magnetic island and the stochastic magnetic field are topology eﬀects commonly observedin a toroidal plasma. The oscillations are observed in themagnetic probe signal or temperature due to the plasma
rotation. The oscillation frequency is finite in the laboratory frame but zero in the plasma frame. Therefore, when
the plasma does not have rotations, oscillations owing tothis topology eﬀect disappear. In contrast, the MHD
has a characteristic frequency of the oscillation of themagnetic field in the plasma frame and sometimes grows
non-linearly and triggers the collapse of plasma. Energetic particle driven modes, Edge localized mode (ELM),
the sawtooth crash, and disruptions are included in thiscategory.

In the local transport model, the turbulence level in theplasma is assumed to be determined by the local plasma
parameters such as temperature and density, their gradients, and the magnetic field and potential structure(magnetic shear and radial electric field shear). However, various non-local phenomena that cannot be explained by local transport model have been observed inexperiments in toroidal plasma [15]. These experimental
results are evidence that the local closure of flux-gradient(turbulence-gradient) relation is violated. Several of themechanisms causing the violation of these relations havebeen proposed in theory. Coupling between the turbulence of meso- or long-range correlation and micro turbulence have been proposed to explain the strong coreedge coupling of turbulence and transport observed inexperiment. In avalanche model, a strong nonlinearity of
the growth rate of the micro-scale turbulence can cause
a simultaneous fast radial propagation of both micro
turbulence and gradients, and consequences in ballistictransport and supper diﬀusion. The turbulence coupling
and turbulence spreading are the most likely mechanisms
causing the non-local transport. Recently, turbulence
spreading from the pedestal region to the scrape oﬀ layer(SOL) is expected to be beneficial in broadening the SOL
power decay length [16–21]. Therefore, the experimental
identification of turbulence spreading becomes one of the
crucial issues in edge transport.
In spite of the importance of the turbulence spreading, there are only a few measurements on turbulence.
This is because the separation of the turbulence origin(spreading turbulence and locally driven turbulence) is
impossible in the usual turbulence measurements. The
novel analysis of turbulence to distinguish the origin of
the turbulence with the modulation of bias was reported
from TJ-II. The time evolution of free turbulent energy⟨ñ2 ⟩ (where ñ = n - ⟨n⟩) is obtained from the radial partof the continuity equation by neglecting cross-field coupling and damping as well as the eﬀect of backgroundflows as [22]

1 ∂ 2
1 ∂
∂n
⟨ñ ⟩ = −⟨ ⟩⟨ṽr ñ⟩ −⟨ṽr ñ2 ⟩
2 ∂t
∂r
2 ∂r

(1)

The angular brackets ⟨...⟩ indicate a running averageover a time interval much larger than the turbulence correlation time. First term is a product of the turbulent
particle flux and density gradient and the second term is
a diﬀusion of turbulent energy. With the normalizationby the local turbulence amplitude, ⟨ñ2 ⟩, two rates of turbulence are defined. One is the rate of turbulence drive
due to the density gradient as

3

(a)

(b)

(c)

(d)

FIG. 2: Rates of (a)(c) local turbulence drive and (b)(d) the
turbulence spreading during the edge biasing. (from figure 8
in [22]).

ωD = −2⟨ ∂n∂r ⟩⟨ṽr ñ⟩⟨ñ2 ⟩(2)and the other is the rate of turbulence spreading due
to the turbulence intensity gradient as∂

ωS = − ∂r⟨ṽr ñ2 ⟩⟨ñ2 ⟩

FIG. 3: Time evolution of perturbation of electron temperature at (a) X-point and (b) O-point of magnetic island, density
fluctuation intensity at (c) X-point and (d) O-point of magnetic island, and (e) radial profiles of time delay of density
fluctuation perturbation with respect to the electron temperature perturbation in DIII-D (from figure 3 in [4]).

(3)These two quantities have a quite diﬀerent response to
the edge biasing as seen in Figure 2. The negative edge
bias of the voltage of -380 V and the modulation frequency of 40 Hz during the discharge (bias is oﬀ-on-oﬀon-oﬀ about 1180 - 1240 ms). The rate of the turbulence
locally driven, ωD inside LCFS (r < r0 ), is significantly
reduced during the on phase of edge biasing, while thereduction of the rate outside LCFS (r > r0 ) is moderate. In contrast, the rate of the turbulence spreading,ωS is significant outside LCFS (r > r0 ), although there
is almost no impact inside LCFS (r < r0 ). The experimental data clearly shows the turbulence locally driven is
suppressed at the plasma edge by E × B shear producedby edge biasing, while the turbulence spreading observedin the SOL region is suppressed by E × B shear at theplasma boundary. This experiment suggested that theE × B flow shear suppress the turbulence spreading fromcore to SOL.
The other approach to identify the spreading turbulence is the measurement of turbulence response in theregion where the local driven turbulence is expected tobe small. The magnetic island is an ideal region for the

spreading turbulence because there should be no driven
turbulence inside the large magnetic island in which thetemperature and the density flattening region are larger
than the correlation length of micro turbulence. Magnetic islands define a unique region in the plasma because the temperature and density gradients which drive
the turbulence are much weaker inside than that outside the magnetic island [23]. Since the heat propagation inside the magnetic island is relatively slow [24, 25],
turbulence could propagate faster than the heat pulse
if there is a turbulence spreading [26, 27] from X-point(or boundary) to the O-point of the magnetic island. Inorder to identify the spreading turbulence, the phase relation between micro turbulence intensity measured withbeam emission spectroscopy (BES [28, 29]) and electron
temperature measured with electron cyclotron emission(ECE) was investigated in the X-point and the O-pointof the magnetic island in the heat pulse experiment produced by modulated electron cyclotron heating (MECH)
in DIII-D [4]. Figure 3(a)-(d) shows time evolution of
perturbation of electron temperature and density fluctuation intensity at X-point and O-point of magnetic is-

4

(a)

(b)

face and has poloidal asymmetry. In the normal nested
magnetic flux surface, the radial propagation speed of
the heat pulse is equal to that of density fluctuation.
However, near the X-point of the magnetic island, theeﬀective radial propagation speed of heat pulse (the projection of parallel heat pulse propagation to the radial direction) becomes much higher than the speed of density
fluctuation propagating across the magnetic flux surface.
Therefore, the heat pulse propagates to the X-point of
the magnetic island before turbulence. In the O-point,
the heat pulse propagates from the boundary of the magnetic island of the O-point slowly due to the low level offluctuation inside the magnetic island (due to the lack of
locally driven turbulence). However, turbulence propagating from the X-point to the O-point is faster than theheat pulse and causes the negative delay time of turbulence perturbation (turbulence increases before the heatpulse). This is clear evidence of turbulence propagationfrom the X-point to the O-point of the magnetic island.

B.
FIG. 4: Poincaré map of the magnetic field at (a) X-point and(b) O-point of the magnetic island. (from figure 4(d), 4(h) in[5] modified).

land. In this experiment, the heat pulse propagates into
magnetic islands. The modulation frequency is 50 Hz
and the density turbulence intensity measured with BES
is integrated in the frequency range of 10 - 50 kHz. At
the X-point of the magnetic island, the delay time ofdensity fluctuation modulation with respect to the electron temperature modulation is positive. The heat pulse
perturbation reaches to the X-point before the density
fluctuation perturbation. In contrast, the delay time ofdensity fluctuation modulation with respect to the electron temperature modulation is negative at the O-pointof the magnetic island, which indicates that the density
fluctuation perturbation reaches to the O-point before
the heat pulse perturbation. As seen in figure 3(e), thenegative delay time is observed in the low field side of the
O-point of the magnetic island (R > RO−point , where theheat pulse propagates inward from the boundary of magnetic island to the O-point of magnetic island.
Poincaré maps of magnetic field at X-point and Opoint of the magnetic island with the direction of heatpulse propagation and density fluctuation propagation
indicated with arrows are plotted in figure 4. Here, theperturbation fields are calculated using the Fourier analysis module in the trip3d code [30]. In general, the electron temperature becomes constant on a magnetic flux
surface because the propagation speed of heat pulse parallel to the magnetic field is an order of the electron thermal velocity and much higher than the propagation speed
perpendicular to the magnetic field. In contrast, the density fluctuation is not constant on a magnetic flux sur-

Bifurcation of turbulence spreading at theboundary of the magnetic island

The amount of heat flux propagating through X-point
or O-point depends on the velocity of heat pulse propagation at X-point or O-point region. As the heat pulse
propagation becomes slow inside the magnetic island (Opoint), the modulation amplitude decreases. Because theheat pulse propagates at the boundary of O-point, the delay time of the heat pulse is peaked at the O-point of themagnetic island[5]. There is a clear relation between the
propagation speed (delay time) and reduction of modulation amplitude[31]. A large reduction of modulation
amplitude is a result of gradual heat pulse propagation
due to low turbulence level. Therefore, by measuring themodulation amplitude at O-point of the magnetic island,
one can estimate the level of transport and turbulence
inside the magnetic island.
The time evolutions of electron temperature measured
with ECE at the O-point phase and the X-point phase ofmagnetic island in DIII-D are plotted in figure 5. In thisexperiment, the toroidal phase of the resonant magnetic
field perturbation (RMP) is flipped by 180 degrees during the discharge. Here O-point (X-point) phase stands
for the phase when the O-point (X-point) of the magnetic
island locates at the toroidal location of ECE measurements. The heat pulse is produced by the modulation
ECH and the modulation amplitude of the heat pulse isevaluated by ECE measurements. As seen in figure 5,
the modulation amplitude at the X-point phase is larger
than the modulation amplitude at the O-point phase.
In the O-point phase, the flattening of electron temperature is observed near 0.4 keV. There is periodic change inmodulation amplitude in the O-point phase which showsthat the reduction of modulation amplitude is oscillating.
One is the phase with moderate reduction of modulation
amplitude and the other is the phase with a significant

5

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FIG. 6: Possible image of electron temperature profile at Xopint and O-point of magnetic islnad to explain the bifurcation of high and low accessibility state of magnetic island.

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FIG. 5: (a) Time evolution of electron temperature measured
with ECE at the O-point phase and the X-point phase of magnetic island and contour of modulation amplitude of electron
temperature in time and space at the transition (b) from highto low accessibility states and from (c) low to high accessibility states in DIII-D. (from figure 6 in [5])reduction of modulation amplitude. The former is called
high-accessibility state of the heat pulse and the latter iscalled low-accessibility state of the heat pulse.
Figure 5 (b) (c) shows the contour of modulation amplitude of electron temperature in time and space at thetransition (b) from high to low accessibility states andfrom (c) low to high accessibility states. In the highaccessibility states, the modulation amplitude gradually
decreases from the boundary of the magnetic island (ρ=0.64 and ρ =0.8) to O-point (ρ =0.72) of the magnetic
island. In contrast, the modulation amplitude sharply
drops to the low level (1 % of temperature). The timescale of the transition from high accessibility states to
low accessibility states is 4 ms and the back transition
from high to low accessibility states is 7 ms.

In general, a magnetic island is surrounded by alayer of stochastic magnetic field. Stochastization of themagnetic-field, i.e., magnetic braiding or appearance ofthe secondary magnetic island [34–37] can occur when
the island size reaches a critical value. The width of themagnetic island in DIII-D is 15% of minor radius andlarge enough to expect the appearance of stochastic layer
surrounding the magnetic island. When the stochastic
layer surrounding the magnetic island appears, the second derivative of temperature decreases due to the enhancement of the eﬀective radial transport. Since theE × B flow shear is roughly proportional to the second
derivative of temperature, E × B flow shear decreases by
the appearance of stochastic layer. Because the E × B
flow shear plays an important role to prevent the turbulence spreading, the stochastisation of the field outside
the island should increase the rate of spreading of turbulence into the island. Therefore, change of stochastic layer surrounding the magnetic island can be one of
the candidates for the bifurcation of turbulence spreading state inside the magnetic island. Figure 6 shows thepossible image of electron temperature profile at X-point
and O-point of magnetic island to explain the bifurcation of high and low accessibility state of magnetic island by a simultaneous change in E × B flow shear andturbulence spreading at the boundary of the magnetic
island. Turbulence measurements in bias experiment in
TJ-II suggest that the turbulence spreading is reduced by
E × B flow shear[22]. In theory, external E × B shearing
is expected to hinder turbulence spreading [32] and theturbulence spreading is also proposed to be one of thecandidates for non-local transport mechanism [15, 33].
These theoretical and experimental results support the6working hypothesis that the transition from a high to a
low accessibility state is due to the change in suppressionof turbulence spreading by the E × B flow shear at theboundary of magnetic island where the second derivativeof electron temperature becomes significantly large.
The width of the stochastic region at the boundaryof the magnetic island except for the X-point region issmaller than the correlation length of turbulence. However, at the X-point, the width of the stochastic region
can be larger than the correlation length of turbulence
and also should have an impact on the turbulence. The
suppression of turbulence spreading results in increasing
the second derivative of temperature (sharp boundaryof magnetic island) and the narrow width of stochastic region (Ws ) at X-point. The increase of the second
derivative of temperature (∂ 2 T /∂r2 ) contributes to theenhancement of E × B flow and causes positive feedback.
The reduction of the stochastic region width at the Xpoint of the magnetic island contributes to the reduction of turbulence spreading (narrow turbulence spreading window) from X-point to O-point of the magnetic
island. Therefore, the low accessibility state is characterized by strong E × B shear, sharp boundary (large∂ 2 T /∂r2 ), and narrow stochastic region (small Ws ) at
X-point of magnetic island (turbulence shielding state).
In contrast, the high accessibility state is characterized
by weak E × B shear, broad boundary (small ∂ 2 T /∂r2 ),and wide stochastic region (large Ws ) at X-point of magnetic island (turbulence penetration state). These twostates can be bifurcated because of the feedback process
through the interplay between E × B shear, stochastic
magnetic field and turbulence shielding.

C.

Flow damping due to stochastic magnetic field

Because the interplay between the stochastic magnetic
field and plasma flow is important, it is an interesting issue how the stochastic magnetic field aﬀects the plasma
flow. A heliotron plasma has a significant advantage forthis study because a large stochastic region up to 60% of
the plasma minor radius can be produced in the plasmacore without a disruption. In general, the magnetic flux
surface becomes nested when the magnetic shear is large
enough, while the magnetic flux surface becomes dominated by the lowest order magnetic island (ex. 2/1 forι = 0.5)) when the magnetic shear is small. Therefore,
the control of the magnetic shear is a key to producing a large region with a stochastic magnetic field neara rational surface. Tangential neutral beam has been
widely used as a convenient tool of non-inductive current
drive in toroidal plasma and called neutral beam current
drive (NBCD). Because of the conservation of poloidal
flux linking the plasma, the total current can only change
on a resistive time scale. Therefore, when the NBI drives
current at the plasma edge, this must induce a loop voltage that drives a compensating current in the opposite
direction in the plasma core. In the high temperatureplasma, this compensating current can exceed the noninductive current by NBCD locally especially in the caseof the oﬀ-axis NBCD. In LHD, this compensating current
in the opposite direction becomes large enough to change
the rotational transform, ι, in the direction opposite to
NBCD (e.g. decrease ι for co-NBCD and increase ι forcounter-NBCD) near the magnetic axis in the discharge
where the direction of the oﬀ-axis NBCD is exchanged in
the middle of the discharge. The exchange of NBCD in
the opposite direction with each other (NBCD direction
switch) has been used to change the magnetic shear at
half radius of the plasma in LHD [38]. When the directionof the NBCD is switched from co-direction (parallel toequivalent plasma current direction) to counter-direction(anti-parallel to equivalent plasma current direction), theedge ι decreases due to counter NBCD but the central ι
increases due to the inductive current. Then the magnetic shear at the half of the plasma minor radius decreases after the beam switch from co to counter NBCD.
Toroidal flow velocities at reﬀ /a99 = 0, 0.25, 0.5 show
the abrupt drop to almost zero after the magnetic shear
at the rational surface of ι = 0.5 at reﬀ /a99 decreases to∼ 0.5 as seen in figure 7 [6]. The magnetic shear at therational surface of ι = 0.5 decreases from 1.2 to 0.5 after
the beam switch from co-NBI to counter-NBI at t = 5.3
sec. The magnetic shear is kept constant at 0.5 after t= 5.8 sec and abrupt drop of toroidal rotation velocity
is observed at t = 6.0 sec. The heat pulse propagation
technique is used to determine the magnetic topology.
Monotonic increase in delay time of heat pulse at t =5.75 sec shows that the magnetic flux surface is nested
before the abrupt drop of toroidal rotation velocity (flowdamping). During the phase of flow damping the delay
time of heat pulse shows very large flattening region flat
up to reﬀ /a99 = 0.6, which indicates that the magnetic
flux surface in the core becomes stochastic[39, 40]. Laterin the discharge, the toroidal rotation velocity begins toincrease after t = 6.7 sec and partially recovers. The
radial profile of delay time of heat pulse is peaked at
the rational surface (reﬀ /a99 = 0.5), which indicates thedisappearance of stochastic magnetic field region and theappearance of magnetic island.
Figure 7(c) shows the radial profile of onset of stochastization of the magnetic field. The stochastization of themagnetic field is initiated at the rational surface of ι =0.5 located at reﬀ /a99 = 0.5. The region of the stochastic magnetic field expands both inward and outward andthe region of the stochastic magnetic becomes 20 % of theminor radius in 10 ms, which is called a partial stochastization. The outward propagation of stochastic magnetic
field stops at reﬀ /a99 = 0.6 due to the higher magnetic
shear towards the plasma periphery. However, the inward propagation of stochastic magnetic field continues
until the stochastic magnetic field region expands to themagnetic axis in 40 ms, which is called a full stochastization.
Figure 7(d) shows the radial profile of radial electricfield before the stochastization (with nested magnetic

7

(a)

(b)

(c)

(d)

stochastization of the magnetic field causes the strong
flow damping. The plasma flow is also expected to playa role to prevent the magnetic field to become stochastic.
Therefore, once the plasma flow starts to decrease due tothe enhanced damping due to stochastization, the regionof flow damping and stochastization expands due to thepositive feedback until the stochastic region reaches tothe plasma axis. In contrast, when the stochastic region starts to shrink, the plasma flow recovers due to theexternal torque input. The stochastic region disappears
and the magnetic island appears or is completely healed(stochastization healing [42]).

III.

The energetic particle driven MHD instabilities havebeen studied intensively in nuclear fusion research [43–46]. MHD instability driven by energetic particle often
shows the non-linear characteristic and causes a minor
collapse of the helical plasma [47, 48]. In LHD, various MHD instability driven by energetic particle is observed when the high power neutral beams (∼ 30 MW)
are injected into the low density (∼ 1 × 1019 m−3 ) target plasma [49, 50]. The minor collapse of the plasma
associated with the burst of oscillation of magnetic fieldobserved with magnetic probe arrays is characterized by
the sudden decrease of neutron emission rate, central ion
temperature, and kinetic energy of the plasma [51, 52].

A.

FIG. 7: Time evolution of (a) toroidal flow velocity at reﬀ /a99
= 0, 0.25, 0.5 and (b) magnetic shear at the rational surfaceof ι = 0.5 at reﬀ /a99 ∼ 0.5 and (c) radial profile of (c) onset ofstochastization of magnetic field and (d) radial electric fieldin LHD (from figure 1(a)(d) and figure 4(b)(d) in [6]).

flux surface) and after the stochastization of magnetic
field. The positive radial electric field in the core region (reﬀ /a99 < 0.4) decreases and the negative radial
electric field outside this region increases after stochastization of the magnetic field. The change in the radial
electric field is more significant near the rational surface, where the dominant modes are resonant, while theflattening of the electron temperature is observed in thewhole core region. This observation is consistent with thefact that the transport enhanced due to the stochastization of magnetic field is ambipolar except for the region
near locations where the dominant modes are resonant[41].
The radial propagation of stochastic magnetic field isconsidered to be due to the consequence of interplay between stochastization and plasma flow damping. The

INTERPLAY BETWEEN TURBULENCE
AND MHD

Energetic particle driven MHD collapse

Toroidal Alfven Eigenmodes (TAEs) driven by the energetic particle are widely observed in tokamak and helical plasma when the fast ion pressure gradient is large
enough [53–55]. Recently, the abrupt onset of the perturbation with the tongue-shaped topology (localized inpoloidal and toroidal directions) which leads the sudden redistribution of energetic ion and MHD burst associated with the sudden increase of plasma rotation by
the change in the radial electric field. This new type of
MHD burst is characterized by a unique trigger mechanism. It is triggered by the tongue shaped deformationthat appears between the low-order of a rational surfaces.
MHD instability causing the tongue shaped deformation
is highly non-linear and grows within one cycle and triggers the re-distribution and loss of energetic trapped ions,which are indicated by the sharp jump of RF intensity.
The plasma starts to rotate after the loss of ions and
MHD burst starts due to the resonance between the mode
frequency and the precession frequency of the trapped
particle.
As seen in figure 8(a)(b)(c), the repeated burst ofmagnetic field perturbation with large amplitude are observed with magnetic probes at toroidal angle of 90, 198,and 270 degrees [56], and are called MHD burst in LHD.

8
dicated by the sharp increase of the RF signal. The frequency of the MHD oscillation and MHD burst is ∼ 5kHz, which is lower than the typical MHD frequency of
Toroidal Alfven Eigenmodes (TAEs) in LHD (∼ 70 kHz[55]). Diﬀerent from the MHD burst driven by TAEs,
the MHD burst starts associated with the plasma rotation driven by loss of energetic ions [7]. MHD burst is
not the cause of energetic ions loss but the result of theplasma rotation driven by energetic ion loss. The oscillation of the magnetic field at 90 and 270 degree probes is
out-of-phase and shows the n = 1 structure. Therefore,
the mode between MHD burst is n = 1 stationary mode.
In contrast, MHD burst (0 - 2ms after the collapse) showsphase delay in toroidal direction and indicates that thisis n=1 rotating mode. It should be noted that the perturbation of the magnetic field 0.1ms before the collapse
indicated by the vertical dashed line does not have n =1 structure and the perturbation of the magnetic field is
localized poloidally and toroidally.

(d)

(e)

(f)

FIG. 8: (a) Time evolution of magnetic field perturbation
measured with magnetic probes at toroidal angle of (a) 90,(b) 198, and (c) 270 degrees and expanded view of magnetic
field perturbation (from 4ms before to 4ms after the collapse)
at toroidal angle of (d) 90 and (e) 270 and contour of equitemperature surface evaluated from ECE signal in LHD (fromfigure 1(a)(b)(c) in [7] and figure 1(a)(c) in [8] modified).

The similar magnitude of the oscillation amplitude inthese three probes indicates that the MHD perturbations
are rotating toroidally. However, the magnetic perturbation amplitude at 198 degree probes is much smaller
than these at 90 and 270 degree probes between the MHD
burst. This indicates that the MHD oscillation between
the MHD burst is caused by the standing wave. Figure
8(d) (e) are expanded plots of magnetic perturbations
from 4ms before to 4ms after the collapse, which is in-

Figure 8(f) shows the contour of displacement of equitemperature surface evaluated from δTe /∇Te measured
with ECE signal. Here the positive value (red) stands
for the outward displacement, while the negative value(blue) stands for the inward displacement. The positive displacement indicated with red starts between two
rational surfaces of ι = 0.5 at reﬀ /a99 = 0.5 and ι =1 at reﬀ /a99 = 0.9. The positive displacement propagates outward and then propagates inward. The negative displacement (blue) is also observed just after theoutward displacement. The outward propagation of positive displacement is called tongue formation, and the inward propagation of positive and negative displacement
is called tongue collapse [7].

These displacements at the tongue formation and collapse are much more significant than the displacement
at the rational surfaces (reﬀ /a99 = 0.5 and 0.9.) andare not localized but are radially propagating. The displacement of the tongue-shaped deformation reaches up
to 2cm at the end of the formation, which is much larger
than the displacement (0.2 ∼ 0.5cm) observed at the two
rational surfaces of ι = 0.5 and 1.0 [57]. The tongueshaped deformation typically quickly decays within one
or a few cycles, while the displacement at the rationalsurface continues to oscillate. It should be noted thatthe perturbation of the magnetic field is correlated tothe oscillation localized in the rational surface of ι = 1
at reﬀ /a99 = 0.9 as seen in the few cycles in t = -3.5 ms.
The perturbation of magnetic field due to tongue-shaped
deformation has comparable magnitude to that of MHD
oscillation localized at reﬀ /a99 = 0.9, although the displacement of tongue-shaped deformation is 4 times thedisplacement localized at ι = 1.0 rational surface.

9

(a)

(b)

(c)

(d)

(e)

FIG. 9: (Time evolution of (a) perturbation of the magnetic
field (eﬀective ion temperature, (c) eﬀective toroidal rotation
velocity and the radial profile of (d) the eﬀective ion temperature and (e) eﬀective toroidal rotation velocity in LHD.

B.

FIG. 10: Time evolution of (a) 1st moment (M1 ), (b) squareroot of 2nd moment (M2 ), (c) 3rd moment (M3 ), and (d)4th moment (M4 - 3) of ion velocity distribution of carbon at
reﬀ /a99 = 0.8, at the tongue event. Here τ is the relative timeof the event with respect to the time of the sharp increase of
RF intensity in LHD (from figure 4 in [9]).

Impact of tongue collapse on ion velocity
distribution

The MHD burst triggered by the tongue collapse has
a strong eﬀect on plasma velocity distribution and alters
the eﬀective ion temperature and eﬀective toroidal rotation velocity. At each MHD burst, the abrupt increase ofeﬀective ion temperature and increase of eﬀective toroidal
rotation velocity in the counter-direction is observed as
seen in figure 9(a)(b)(c). The change of eﬀective ion temperature and eﬀective toroidal rotation velocity is within
1 ms and much faster than the transport time scale. Thisis due to the redistribution of ions in space, where thehot ions and counter-traveling ions in the core region(reﬀ /a99 < 0.4) move outward in a short time associatedwith the tongue formation and collapse. The increase ofeﬀective ion temperature and eﬀective toroidal rotation

velocity shows the relaxation in the ion-ion collision timescale.
Radial profiles of eﬀective ion temperature and eﬀective toroidal rotation velocity 4 ms before and 1 ms after
the tongue collapse are plotted in figure 9(d)(e). The
increase of eﬀective ion temperature is 0.5 keV and relatively constant in space and ion temperature gradientis almost unchanged. Before the tongue collapse the effective toroidal rotation velocity is co-direction (parallelto equivalent plasma current direction) and it is in the
counter-direction after the tongue collapse. The rapidchange in the eﬀective toroidal rotation velocity is theconsequence of outward movement of counter-travelingparticles, because the time scale of the change is much
faster than the time scale of the momentum transport.

10
In order to investigate the mechanism for the rapidchange of eﬀective ion temperature and eﬀective toroidal
velocity, the distortion of ion velocity distribution fromMaxwell-Boltzmann distribution is evaluated by the
moment analysis of ion velocity distribution measured
with charge exchange spectroscopy viewing the plasma
toroidally [58]. The 0th to the 4th moment of ion velocity distributions are defined as

(a)

∫
M0 =

f (v)dv

M3 =

M4 =

(4)

∫

1
M1 =
M0
1
M2 =
M0

(b)

(c)

vf (v)dv(5)(v − M1 )2 f (v)dv

(6)

∫
(d)

∫

1
3/2
M0 M2

1
M0 M22(e)(v − M1 )3 f (v)dv(7)(v − M1 )4 f (v)dv.

(8)∫

When the ion velocity distribution is MaxwellBoltzmann distribution, M1 = Vϕ , M2 = Vth , M3 =0, and M4 - 3 = 0, where Vth is an ion thermal velocity. When the ion velocity distribution is distorted fromMaxwell-Boltzmann distribution, M3 and M4 −1 become
non-zero.
The time evolutions of 1st moment (M1 ), square rootof 2nd moment (M2 ), 3rd moment skewness (M3 ), and4th moment kurtosis (M4 - 3) of ion velocity distributionof carbon at reﬀ /a99 = 0.8 are plotted in figure 10. Here
τ is the relative time with respect to the tongue collapse
indicated by the sharp increase of RF intensity. Afterthe tongue collapse, 1st moment (M1 ) and square root of
the 2nd moment show the rapid change, which is consistent with the rapid increase of eﬀective toroidal rotation
velocity in counter-direction and the rapid increase of effective ion temperature. Before the tongue collapse, the3rd moment is close to zero, which shows that the ion distribution is Maxwell-Boltzmann distribution. After the
tongue collapse, M3 value drops to -0.15 and recovers tozero within 1 - 2 ms. The change is much larger than theerror bar of the measurements. The M4 - 3 also shows
the significant drop to -0.2 after the tongue collapse, The
negative M4 - 3 indicates the increase of epithermal ionsdue to the outward shift of hot ions towards the plasmaedge. This negative M4 - 3 also recovers to zero within
1 - 2 ms due to the thermalization of ion distribution
through ion-ion collisions.
The outward shift of ions causes the intense negative
radial electric field in the core region. The change in

FIG. 11: Time evolution of (a) perturbation of the magnetic
field, (b) RF intensity, and (c) poloidal rotation velocity, andradial profiles of (d) poloidal rotation velocity and (d) radial
electric field in LHD.

the radial electric field is measured with charge exchange
spectroscopy, while the timing of loss of ion is monitoredby the RF probe. RF intensity measured with RF probe
is a good time indicator of the ion loss, because the highfrequency instability is excited when the iron in the hotcore is exhausted to the plasma edge. Therefore, theintensity of RF radiation probes is used as a timing indicator for the energetic ion loss from the plasma[59] inthis work because the RF probe has high time resolution
and is sensitive to the high frequency RF signals excitedby the loss of energetic ions at the plasma edge [60, 61].
Large poloidal flow in the electron diamagnetic direction
and sharp increase of RF intensity is observed at each
MHD burst as seen in figure 11(a) (b) (c).
The radial profile of poloidal rotation velocity and radial electric field shows the large negative electric field
well at reﬀ /a99 = 0.9, while the radial electric field before
the tongue collapse is positive as seen in figure 11(d)(e).
The change in radial electric field near the plasma edge
(reﬀ /a99 > 0.8) is mainly contributed by the change inpoloidal flow in electron diamagnetic direction, while thechange in radial electric field further inside the plasma

11FIG. 12: (a) Time evolution of (a) perpendicular velocity atthe edge stochastic region (reﬀ /a99 > 0.95) and inside plasma(reﬀ /a99 < 0.95) and contour of fluctuation level integrated(b) low frequency (30 - 150kHz) turbulence and (c) high frequency (150 - 490 kHz) turbulence and frequency spectrumat the edge stochastic region and inside plasma in LHD (fromfigure 3(b)(c)(g) and figure 4(c)(d) in [10]).

is contributed by increase of toroidal flow in counterdirection. The radial electric field outside plasma boundary (reﬀ /a99 = 1) is almost unchanged. The change ofthe radial profile of the radial electric field to more negative clearly shows the loss of bulk ions associated withthe tongue collapse.

C.

Turbulence exhaust driven by MHD collapse

It is interesting to investigate the impact of MHD collapse on turbulence. The Doppler reflectometer is a useful tool for measuring the density fluctuation near theplasma boundary. The frequency-hopping Doppler reflectometers have been installed in LHD and the frequency is
set to 30 GHz in this experiment[62, 63]. The location of

the measurement (reflection point) can be scanned fromthe edge stochastic region to the nested region inside
during the discharge where the edge density gradually
decreases in time. The analysis technique of the radial
scan of the reflection point of the Doppler reflectometerusing the density ramp up/down in time has been appliedin LHD experiment. This method has been recognized
to be a useful technique to measure the radial profile ofdensity fluctuation amplitude and perpendicular velocity
near the plasma edge[64].
Figure 12 shows time evolution of perpendicular velocity at the edge stochastic region (reﬀ /a99 > 0.95) andinside plasma (reﬀ /a99 < 0.95). The contour of fluctuation level integrated low frequency (30 - 150kHz) turbulence and high frequency (150 - 490 kHz) turbulence
are also plotted. When the reflection point is in theedge stochastic region, the positive spikes of intensityof high frequency (f = 150 - 490 kHz) micro turbulence
are observed at the tongue collapse event. In contrast,
the slight decrease of high frequency micro turbulence
is observed at the nested flux region inside the plasma(reﬀ /a99 < 0.95). It should be pointed out that there
is no change in turbulence intensity at lower frequency
range (f = 30 - 150 kHz).
The time evolution of perpendicular velocity of thestochastic region and the nested region inside shows thechange of perpendicular velocity in the electron diamagnetic direction. This data indicates the formation ofnegative radial electric field consistent with the negative poloidal flow measured with charge exchange spectroscopy. The formation of a negative electric field istransient and lasts only 2 ms. As seen in the spectrum
at stochastic region (reﬀ /a99 > 0.95) plotted in figure12(d)(e), the micro turbulences only in the frequency region above 150 kHz increase rapidly within 0.15 ms after
the collapse. This rapid increase of turbulence observed
is due to the rapid avalanche like radial propagation process [65–68] into the SOL rather than the change in locally driven turbulence associated with the changes intemperature or density gradients in the SOL.

IV.

DISCUSSIONAs the example of interplay between the turbulence
and the topology, the following experimental evidence
has been reported in a toroidal plasma. Turbulence
spreading from the X-point to the O-point of the magnetic island is identified from the phase delay between
heat pulse and turbulence response. The bifurcation ofmagnetic island states can be explained by the interplay
between turbulence spreading and stochastization at Xpoint of the magnetic island. The radial propagation of
the stochastic magnetic field towards the magnetic axisof helical plasma can be also explained by the interplay
between the plasma flow and stochastization of the magnetic field.
The experimental evidence for two states of a magnetic

12
island provides a new insight into understanding the radial flux of (surface-averaged) heat transport in the presence of magnetic islands during RMP experiments. This
understanding can be further improved and it may bepossible to achieve better control of the heat transport
in discharges with RMP fields by using the heat pulse
propagation technique as a tool to identify the state of
the magnetic island. Turbulence spreading observed in
the magnetic island also gives a new insight on turbulence
spreading physics which is necessary for the better understanding of turbulence in the SOL, where the spreading
turbulence dominates locally driven turbulence. Because
the turbulence in the SOL is beneficial for the broadening of the power decay length, deeper understandingof SOL turbulence is essential for better control of the
SOL decay length in future device. The impact of magnetic field stochastization on turbulence spreading andflow damping explore new research fields. The former isimportant in understanding the complicated plasma response of particle and heat transport to the RMP field
and the latter is indispensable to understand the responseof plasma flow to RMP fields, where partial stochastization is expected. Better understanding of the impact of
RMP fields on plasma flow is essential for the more reliable prediction of power threshold of the L to H-mode
transition in future device.
The strong impact of energetic particles driven MHD
instability has been observed. MHD bursts are triggered by the tongue formation and collapse. This tongue
collapse has an impact of ion velocity distribution andcauses the distortion from Maxwell-Boltzmann distribution. The distortion has been found in experiments witha significant transient change in skewness and kurtosisof ion velocity distribution. The tongue collapse alsoenhances the loss of bulk ions of epithermal ions andproduces the negative radial electric field and excite the

MHD instability at the plasma edge. The turbulence
exhaust by the tongue collapse is also observed in the
Doppler reflectometer. This interplay between MHD andturbulence is essential for the integrated understandingfor various phenomena in toroidal plasma.
Understanding the impact of MHD instabilities driven
by energetic particles on plasma flow and turbulence becomes more important for future devices, because a increased fraction of energetic particles is expected in burning plasmas. Because of the strong coupling between
plasma rotation and the plasma potential in toroidal
plasma, the plasma rotation can be driven by energetic
ion losses triggered by MHD instabilities. Once theplasma starts to rotate, another MHD instability can be
excited by the resonance between the mode frequency
and the precession frequency of the trapped particles.
This is an example where an MHD instability drives another MHD instability through the plasma flow which isdetermined by the momentum transport. Therefore, theinterplay between MHD and transport is a key physics inunderstanding the situation where various MHD modes
appear one after the other during the discharge in theplasma where the fraction of beam beta is high [16, 69].

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The author wishes to thank LHD Experiment Groupand DIII-D team for their excellent support of thiswork and Dr. T.E. Evans (GA) for his useful comments. This work is supported by the National Institute for Fusion Science grant administrative budgets (NIFS10ULHH021,NIFS17KLPH030) and JSPS
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