Document [original]

International Atomic Energy Agency

Nuclear Fusion

Nucl. Fusion 60 (2020) 106021 (11pp) https://doi.org/10.1088/1741-4326/aba72b

ECCD-induced sawtooth crashes at

W7-X

M. Zanini1, H.P. Laqua1, H. Thomsen1, T. Stange1, C. Brandt1, H. Braune1,

K.J. Brunner1, G. Fuchert1, M. Hirsch1, J. Knauer1, U. Höfel1, S. Marsen1,

E. Pasch1, K. Rahbarnia1, J. Schilling1, Y. Turkin1, R.C. Wolf1,2,

A. Zocco1and W7-X team1,a

1Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany

2Zentrum für Astronomie und Astrophysik,Technische Universit¨

at Berlin, 10623 Berlin, Germany

E-mail: [email protected]

Received 30 March 2020, revised 11 June 2020

Accepted for publication 17 July 2020

Published 9 September 2020

Abstract

The optimised superconducting stellarator W7-X generates its rotational transform by means of

external coils, therefore no toroidal current is necessary for plasma confinement. Electron

cyclotron current drive experiments were conducted for strikeline control and safe divertor

operation. During current drive experiments periodic and repetitive crashes of the central

electron temperature, similar to sawtooth crashes in tokamaks, were detected. Measurements

from soft x-ray tomography and electron cyclotron emission show that the crashes are preceded

by weak oscillating precursors and a displacement of the plasma core, consistent with a

(m,n)=(1, 1) mode. The displacement occurs within 100µs, followed by expulsion and

redistribution of the core into the external part of the plasma. Two types of crashes, with

different frequencies and amplitudes are detected in the experimental program. For these

non-stationary parameters a strong dependence on the toroidal current is found. A 1-D heuristic

model for current diffusion is proposed as a first step to explain the characteristic crash time.

Initial results show that the modelled current diffusion timescale is consistent with the initial

crash frequency and that the toroidal current rise shifts the position where the instability is

triggered, resulting in larger crash amplitudes.

Keywords: stellarator, sawtooth, ECCD

(Some figures may appear in colour only in the online journal)

1. Introduction

The superconducting optimised stellarator Wendelstein 7-X

(W7-X) [1,2] is mainly heated by a flexible electron cyclotron

resonance heating (ECRH) system, composed of 10 gyrotrons,

able to deliver up to 7.5 MW of power into the plasma [3].

aSee Klinger et al 2019 (https://doi.org/10.1088/1741-4326/ab03a7) for the

W7-X team.

Original Content from this work may be used under the

terms of the Creative Commons Attribution 3.0 licence. Any

further distribution of this work must maintain attribution to the author(s) and

the title of the work, journal citation and DOI.

The region where the energy transfer from the microwaves to

the electrons occurs is well localised and can be changed by

steering the launcher mirrors, thus making ECRH a versatile

tool. When the injection is oblique to the magnetic axis in the

toroidal direction, a selective heating of electrons is possible

and a net toroidal current (electron cyclotron current drive,

ECCD) is generated [4,5]. In principle, a toroidal current is not

needed for plasma confinement in stellarators, since the rota-

tional transform is generated by means of external coils. How-

ever, since the W7-X divertor concept relies on the existence

of magnetic island chains at the plasma edge, which occur at

resonant values of the rotational transform, ECCD is one of the

principal tools for strike lines control [6–9], by compensating

or counter-acting the intrinsic plasma currents [10]. ECCD

experiments have been conducted at W7-X in OP1.1 and

1741-4326/20/106021+11$33.00 1 © EURATOM 2020 Printed in the UK

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

OP1.2 campaigns. In such experiments, repetitive crashes of

the central electron temperature were observed: these crashes

have similar characteristics as sawtooth oscillations observed

in tokamaks [11] and, under certain conditions, in current car-

rying stellarators [12–14]. While small sawteeth can be bene-

ficial, countering or avoiding impurity accumulation [15,16],

large amplitude crashes can be detrimental for plasma per-

formances. The sawtooth instability is generally associated

with an unstable (m,n)=(1, 1) mode [17], where mand nare

respectively the poloidal and toroidal mode numbers. The first

explanation of this phenomenon was proposed by Kadomtsev

[18]. The core is displaced by the formation and growth of an

m=n=1 magnetic island around it. The temperature drop is

caused by the expulsion of the hot core, resulting in a flatten-

ing of the pressure profile inside the new plasma core, which in

turn reduces the central current density removing the instabil-

ity drive again. This model, however, turned out to be unable

to explain certain experimental observations, such as charac-

teristic timescales, measured post-crash rotational transform

profiles (ι-=ι/2π=1/q, where qis the safety factor) or the

existence of (1, 1) postcursors, hence other models have been

proposed [19].

In W7-X, operational magnetic configurations have been

designed to avoid major resonances in the rotational transform

profile. During ECCD experiments, the driven current itself

can locally modify ι-, thus making the plasma susceptible to

MHD instabilities.

Several experiments have been conducted in order to eval-

uate the impact of sawtooth oscillations on W7-X plasmas.

In this work we present the analysis of a characteristic dis-

charge, in order to elucidate the main features of sawtooth-

like crashes in W7-X. These crashes are often preceded by

a fast displacement of the plasma core, consistent with a

(m,n)=(1, 1) structure, but sometimes weak precursors can

also be observed. The crash evolution is unchanged during

the discharge, but the amplitude and the time interval between

two events is found to be not constant in time. Comparing the

amplitude and the location of the crashes, we identified two

types of crashes: one affecting the central part of the plasma

and another, stronger, extending the crashed volume to about

50 % of the effective radius.

Possible mechanisms for W7-X sawteeth have been

recently proposed, whether purely current-density-gradient-

driven [20,21] or current and pressure driven [22]. In both

cases, non-ideal corrections to Ohm’s law (collisional or kin-

etic) and ι-≈1 are the key excite instabilities with low mode

numbers.

The paper is structured as follows: in section 2, a descrip-

tion of W7-X and the main diagnostics is provided. In sec-

tion section 3we present a typical ECCD experimental pro-

gram, where sawtooth-like crashes were detected. In section

section 4we discuss the relation between crash parameters

as a function of time and the main differences between the

two observed types of crashes. Finally, in section section 5we

discuss the effects of current drive on the rotational transform

profile and the relation between the current and the crash

parameters.

2. Experimental setup

The magnetic field of W7-X is generated by 50 superconduct-

ing non-planar coils and 20 planar coils. The current in each

coil type can be indipendently set, thus allowing a wide range

of magnetic field configurations. The minor radius varies from

0.49 m to 0.55 m and the major radius is 5.5 m, resulting in an

aspect ratio of the W7-X stellarator of around 10. The mag-

netic field on the axis in the bean-shaped plane is generally

around 2.5 T. The rotational transform depends on the choosen

magnetic configuration, but for the analysed discharge, the so-

called standard configuration was used, which has a rotational

transform value of 0.85 in the core and 0.97 at the last closed

flux surface (LCFS).

W7-X is a low shear stellarator which relies on the natural

island chain at the edge. Therefore the control of the total tor-

oidal current is necessary in order to prevent component dam-

ages due to a misalignment of the strikelines on the divertor

plates. Plasma heating is mainly provided by ECRH. Up to 10

microwave beams are injected with a frequency of 140GHz.

The gyrotrons are able to selectively operate in O-mode or

in X-mode. ECRH launchers are installed in two modules in

the proximity of the so-called bean-shaped planes, where the

magnetic field has the maximum strength. The flexibility of

the ECRH system allows to deposit the power at different

poloidal locations of the plasma cross-section. It is also pos-

sible to inject microwave beams in toroidal direction to create

(ECCD). The current drive efficiency decreases as the elec-

tron collisionality increases, therefore the experiment presen-

ted here was conducted at low density (about 2 ·1019 m−3).

The electron temperature was about 5 keV in the core for

1.2−1.8MW of injected power. Due to low coupling between

electrons and ions, the ion temperature was about 1 keV. Co-

ECCD was applied to the plasmas presented here, i.e. the cur-

rent was driven in a direction such that the rotational transform

is increased.

This work mainly focuses on the experimental results

obtained from electron cyclotron emission (ECE, with a

sampling rate between 200 kHz and 1MHz) [23] analysis,

which is capable of radially resolving the fast and local

changes of the electron temperature. A 1-D electron temperat-

ure (Te) profile can be reconstructed by the measurement along

the line of sight of the ECE diagnostic, viewing the plasma

from the outboard side and crossing the plasma centre. The

magnetic field increases with the distance to the receiver optics

such that the ECE spectra can be ideally interpreted in terms of

a local electron temperature, if the plasma behaves like a black

body, i.e. the optical thickness is large. The ECE channels are

mapped to the effective radius reff =√⟨A⟩/π using the fol-

lowing definition: channels detecting at the outboard region of

the plasma, namely the low field side (LFS), are labelled with

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 1. Left: Typical Teprofile, mapped to the radial position. Right: ECE profiles (blue for HFS and green for LFS) against a Thomson

scattering profile (red).

negative radial position, while inboard (high field side, HFS)

channels with a positive value. The mapping is performed

taking into account diamagnetic effects, the Shafranov shift

and the finite emission layer. These effects would introduce

a significative shift from the HFS to the LFS if not taken into

account. The Tespectrum is estimated with a Bayesian model-

ling [24]. In figure 1(a) a typical Teprofile is plotted, while in

figure 1(b) ECE data are compared to the Thomson scattering

data. Both sides of the ECE profile agree with the Thomson

scattering profile, especially at lower temperatures. At higher

temperatures the HFS and LFS channels present some discrep-

ancies which are however within the errorbars. The dominant

uncertainty measurement is given by calibration errors, which

are plotted in figure 1(a) along with the temperature profile

itself. Calibration errors are much higher than the noise con-

tribution, which is relatively small (less than 6 %) therefore

only the absolute value of the measurement is affected. In this

work ECE data are mainly used to study the relative signal

drop due to a crash, therefore calibration errors do not affect

the measurement quality.

The second main diagnostic is the x-ray tomography sys-

tem. Plasma emission in the soft x-ray range (1 −12 keV) is

measured by an array of 20 pinhole cameras, each with 18 lines

of sight, with a sample rate of 2MHz. X-ray tomography is

capable of reconstructing the evolution of the plasma emissiv-

ity in a poloidal plane [25].

Other major diagnostics used for this work are a single

dispersion interferometer [26] which yields line integrated

density measurement (sample rate of 100 kHz), continuous

Rogowski coils for toroidal plasma current (sample rate of

50 kHz) and diamagnetic loops for the stored plasma energy

(sample rate of 50 kHz) [27]. No direct measurements of

the current density profile are available at the moment, there-

fore the ECCD profile is calculated using the ray-tracing code

Table 1. Main experimental program parameters for the discharge

20171206.025.

Tor. Magnetic field (axis) B=−2.52 T

EC Heating Power P=1.2MW

Major radius R0=5.5 m

Minor radius a=0.5 m

Electron Temperature (axis) Te=5 keV

Ion Temperature (axis) Ti=1 keV

Plasma energy Wdia =170 kJ

Line int. density <ne>=2·1019 m−3

Toroidal current (max) Itor =−12kA

L/Rtime τ≈5−10 s

Resistive time τη=62 s

Central β βmax ≈0.7%

TRAVIS [28]. It calculates ECRH propagation and absorbtion

into the plasma, taking in account a pre-calculated magnetic

equilibrium, for instance, using VMEC (Variational Moments

Equilibrium Code [29]) and electron pressure profile. The cur-

rent density profile evolution is then calculated resolving a dif-

fusion equation, as discussed in section 5.

3. ECCD and sawtooth-like crashes at W7-X

Several current drive experiments have been conducted. A typ-

ical experiment is presented in detail in figure 2. The elec-

tron temperature in the core is around 5 −6 keV and line

integrated density about 2 −2.5·1019 m−3. At these Teand

nevalues, the coupling between ions and electrons is low,

therefore the ion temperature is about 1 keV. In the experi-

mental program 20171206.025 the plasma was initially heated

by one ECRH beam in the centre (600kW) and by two

ECRH beams obliquely injected, leading to generate current

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 2. Overview plot of experimental program 20171206.025.

Sawtooth-like crashes are visible in the ECE channels, plotted in the

second panel (core ECE channel in red and the ECE channel

detecting at reff ≈0.3 m in blue). Bigger crashes can be seen in the

diamagnetic energy (third plot) and in the toroidal current (fourth

plot). The line integrated density is plotted in the fifth plot, where

density rise due to pellet injections is visible from t=23 s.

drive. Because of the Doppler shift, the power deposition is

slightly off-axis and calculations using TRAVIS show that the

deposition is located at around 15 % of the effective radius,

for approximately −14kA of driven toroidal current. After

some seconds the gyrotron providing the core ECRH beam

dropped and the plasma was sustained until the end of the

discharge by the two obliquely injected beams. Besides the

heating power drop at 1.3 s and a phase with pellet injections

from t=23 s, density and plasma energy are relatively con-

stant. The toroidal current is given by the sum of bootstrap

current and the generated ECCD. Since the pressure gradi-

ent is relatively low, no strong contribution from the boot-

strap current is expected (about 2 −3kA, broadly distributed).

The plasma behaves like an inductor which opposes changes

of the magnetic flux, therefore the whole current does not

develop suddenly, but evolves on the timescale of the L/R

time (τL/R), where L=R0µ0(ln(8R0/a)−2)) ≈18µH and

R=2πR0(´a

0σ2πrdr)−1≈2µΩare the plasma inductance

and resistance, with σbeing the parallel Spitzer conductivity

[30]. The toroidal current evolution can be described by the

following equation:

Itor(t) = Itor(t→ ∞)(1−e−t/τ).(1)

τL/Ris estimated to be between 5 s and 10 s. Finally, it can

be noticed that the toroidal plasma current saturates around

15 s. In this phase, the main current component is created

by ECCD and the experimental value is in good agreement

with the one obtained by the ray-tracing code TRAVIS. Cent-

ral temperature crashes started to appear after hundreds of

milliseconds after the start of the plasma heating. While the

dynamic of every crash shows the same features (i.e. strong

Tedecrease in the centre, along with a slight increase on the

external plasma region), the main crash parameters, such as

frequency and amplitude change in time. An example, further

discussed in section 4, can be seen in figure 3. In the selected

time window (from 16 s to 17 s), it is possible to distinguish

two main types of temperature crashes, according to their amp-

litude: in ECE timetraces, two big crashes can be distinguished

between the minor crashes. The central temperature drops gen-

erally between 30–50 % and recovers in about 5–20 ms. Cor-

responding to the strong temperature crashes, a drop in the

stored plasma energy (about 10 kJ, corresponding to 5 % of the

whole energy) is detected by diamagnetic loops and the time

for the energy to recover is on the order of 200 ms, one order

of magnitude slower than the central temperature recovering

time mentioned above.

The structure and the time evolution of the crashes are

mainly studied using electron cyclotron emission, which

allows high temporal (200 kHz for this experimental program)

and spatial resolution. Soft x-ray tomographic reconstructions

have been performed for the strongest events. Smaller events,

such as the intermediate crashes in figure 3are too weak and

too localised to be detected by soft x-ray emission so far.

The crash is occasionally preceded by short living precursor

oscillations (2 −6 kHz) and in most of the cases by a displace-

ment of the plasma core. Precursors have not been identified

in x-ray data, while ECE data suggest the signal perturbation

has an odd poloidal number. Timetraces of four ECE channels

are plotted in figure 4. Two channels are high field side (HFS)

channels and two are low field side (LFS) channels. It is pos-

sible to observe small amplitude precursors from t=2.7785 s

to t=2.7791 s. The two core channels (channels 11 and 13)

show Teoscillations 180◦out of phase, consistent with an odd

poloidal number. An example of a crash without oscillating

precursors is plotted in figure 5. In figure 5(a), the timetraces

of the ECE channels show that no strong oscillating precursors

are present before the crash. Instead, the plasma core under-

goes a displacement, as plotted in red dashed line in figure

5(b) or depicted in the contour plot of figure 6, where the rapid

movement of the core to the low field side is visible before the

crash.

The core displacement evolves on a timescale of 100µs.

The analysis of different crashes shows that the core move-

ment can happen in every poloidal direction. Having ECE only

a 1-D horizontal line of sight, only horizontal movements of

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 3. Zoom of the overview plot of figure 2, between 16 and

17s. Two types of crashes can be distinguished. The strong events at

16.4 and 16.9 s are also visible in the diamagnetic energy (third

panel) and toroidal current (fourth panel).

the core can be detected, nonetheless soft x-ray tomography

confirms the existence of vertical displacements. No relev-

ant differences have been detected so far in these cases. The

ECE radiometer and the x-ray camera array are placed almost

180 degrees apart in toroidal direction. We observed that an

horizontal movement in the ECE from the core to the LFS

(HFS) corresponds, in the x-ray tomography, to a movement

from core to HFS (LFS). Comparing the movement directions

of x-ray signals with respect to ECE signals (figures 6and

7), we conclude that the toroidal mode number nhas to be

odd. Regarding the poloidal structure of the core movement,

it creates a strongly asymmetric structure, typical for an m=1

mode, as plotted in figure 5(b), time step II. The crash phase

happens in 20µs, in which core energy is redistributed outside

the inversion radius (defined in figure 5(c)) and symmetry is

restored (time step III in figure 5(a) and (b)). The Teprofile

inside the crash region is almost flat and Teis higher out-

side the inversion radius, with respect to the pre-crash pro-

file. The re-heating phase (time step IV) depends on the amp-

litude of the crash and the involved region: for the crashes in

this experimental program the typical time is about 5 −20 ms,

Figure 4. Precursor activity in 20171206.025 detected by ECE. The

oscillation amplitude is small and generally relatively short. After

the oscillating precursors, the core temperature displacement is

visibile. Before the crashes, a strong increase of temperature is

visible in the two low field side channels (ch08 in red and ch11 in

dark red), while a decrease is detected in the high field side central

channel (ch13, dark blue), indicating a fast movement of plasma

centre from the core to the LFS.

faster than the time interval between two events. The soft x-ray

tomographic reconstruction (4µs of temporal resolution) dis-

played in figure 7confirms the m=1 structure detected by

ECE and allows to better visualise the crash dynamics. The

data presented here have been filtered by applying singular

value decomposition (SVD) [31] to remove the dominant com-

ponent of the emissity which, for this case, consisted of a con-

stant value. After the displacemet phase, the tomograms show

a poloidal propagation of the expelled core. The redistribution

is not isotropic and occurs in the direction of the electron dia-

magnetic drift. Eventually, once the core expulsion has been

completed, the SVD-filtered emissivity profile results in an

hollow shape, followed by a recover of the electron temper-

ature profile after the crash.

4. Crash pattern evolution

The characterisation of crash parameters, in particular amp-

litude and frequency, is crucial for future applications of

ECCD in W7-X: the localisation and the amplitude of the crash

are related to the released energy during this event. Therefore

it is necessary to understand how these parameters change dur-

ing the discharge. Looking at figure 2, apart from some minor

changes in the density, the main changing parameter seems

to be the toroidal current, which starts to saturate between

15 s and 20 s, reaching a value of −12kA, consistent with

TRAVIS calculations. We studied the evolution of the three

main parameters: the crash frequency, the inversion radius and

the maximum amplitude during a crash (figure 8). An example

of how these parameters are defined can be seen in figure 5(c).

The inversion radius is calculated for both HFS and LFS of

the plasma and it is defined as the radial position where the

temperature remains constant before and after the crash. The

maximum amplitude, ∆Te, is defined as the maximum relative

change between the profile before and after the crash. Finally,

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

(a)

(b) (c)

Figure 5. (a): Teevolution profile during a crash. Four ECE channels are plotted. Red-shaded timetraces correspond to LFS channels, while

blue-shaded to HFS channels. Vertical lines represents the time points which have been used to reconstruct Teprofiles in (b). The

timewindow for the profile average is 15µs. I) represents the equilibrium profile, II) the profile during the displacement, III) the profile right

after the crash and IV) during the re-heating phase. In (c), the relative difference of Tebefore (t=15.6710 s) and after (t=15.6727 s) the

crash is displayed. The radial positions where the difference is zero are defined as inversion radii.

the crash frequency is defined as the temporal interval between

two consecutive crashes of the same type. Crashes are detec-

ted applying to ECE data a ridge detection algorithm, based on

continuous wavelet transform (CWT) [32]. Since crashes of

different amplitude are detected, we sort them into two types,

according to their amplitude. Type A events are those over-

coming a certain amplitude threshold, while type B are those

events with amplitude below the choosen threshold and occur-

ring between two type A events. The crash parameters, i.e.

inversion radii, the temporal intervals and the amplitudes of

both crash types exhibit a strong time dependent behaviour

analysed in the following.

4.1. Crash classification: Type A-crash

Figure 8shows the evolution of the crash parameters for the

two crash types. The inversion radius (first plot of figure 8) and

the amplitude (third plot) increase in time, with crashes in the

latter stage of the discharge reaching an amplitude up to 60 %

and causing a temporary decrease of the diamagnetic energy.

The temporal interval (second plot of figure 8) between type

A crashes increases from 0.05 s to 0.5 s and saturates after 8 s.

Temperature oscillations caused by sawtooth precursors are

detected in most crashes at the beginning of the discharge. The

oscillating precursor phase becomes shorter as the discharge

evolves, until the oscillating precursors become not recognis-

able anymore from the noise and only the core displacement

is observed before the temperature crash. The amplitude and

the expulsion region increase with current (see figures 2(a)

and (b)). One can assume that this is related to an outward

shift of the position where the resonance is reached. Since

Te,neand the beam deposition region do not change dur-

ing the experimental program, the increase of rinv has to be

primarly caused by a change in the position of rι

-=1, caused

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 6. Contour plot of a crash, using ECE data. No precursors

are visible before the crash. The core dispacement, from centre to

low field side, is visible at 15.6725 s, before the crash. The crash

phase is highlighted by the two dashed lines and occurs within

25µs. The temperature flattening observed at the same time at

r/a≈0.3 may represent the development of a magnetic island.

in turn by an increase of the toroidal current enclosed in this

plasma volume. This increase can be explained by assuming

that the current density is more intense in the proximity of the

deposition region, which is inside the inversion radius, and,

therefore, most of the current be inside this region, thus shift-

ing rι-=1outwards. Presently this hypothesis can not be con-

firmed with experimental data, since direct measurements of

the current density are not yet available at W7-X, however this

hypothesis can be supported by the relation between the meas-

ured toroidal current and the crash amplitude evolution (com-

pare figures 2(b) and (d)). The instantaneous onset of sawtooth

oscillations suggests that, in principle, even small values of

peaked and localised toroidal current can cause it, through a

mild distortion of the vacuum rotational transform profile.

4.2. Crash classification: Type B-crash

The presence of the smaller crashes, which occur between the

type A events, can be detected after some seconds, as the tem-

poral interval between main crashes has increased. In these

intermediate events the main features of the big crashes are

reproduced: fast crash of the temperature in the core and a

moderate increase in the external part. The crash amplitude

varies between 10 % and 35 %, while the temporal interval

between these crashes is about 0.01 s. The mode analysis car-

ried out in section 3is now more difficult: these events are

hardly detected by the soft x-ray tomography diagnostic and

being well localised in a narrower radial region than the type A

crashes, as seen in figure 8. Consequently it was not possible

to detect a relevant displacement of the core before the crash

itself. If we consider the time window where the current and

the crash amplitude have saturated, we notice that the pattern

of type B-crashes is very regular: after every type A crash,

small crashes with small amplitudes are detected. Between

two type A crashes the amplitude and the temporal interval

between type B crashes increase and saturates, untill the netx

type A crash occurs (figure 3).

The presence of smaller crashes occuring in a narrower

region close to the magnetic axis may be caused by a crossing

of ι-=1 close to the magnetic axis. This hypothesis can not

be confirmed, but it is not contraddicted by the calculations

discussed in section 5.1.

5. Heuristic iota evolution model

At W7-X the so-called standard configuration (standard mag-

netic field configuration) has a ι-ρ=0=0.85 and ι-ρ=a=0.97,

so major resonances are avoided. Nevertheless, if a toroidal

current is present, it can be written as a combination of two

terms:

ι-=ι-vac +ι-curr (2)

where ι-vac is the rotational transform created by magnetic coils

only and ι-curr is the contribution to the rotational transform

given by toroidal currents. ι-(r)curr ∝I(r)/r2, with I(r) the tor-

oidal current enclosed in the plasma volume of radius r. The

1/r2dependence, together with W7-X being a low shear stel-

larator, makes the rotational transform very sensitive to local

changes of toroidal current, such as ECCD. Examples of mod-

ified ι- profiles are plotted in figure 9.

In section 5.1 we introduce a 1-D cylindrical model in order

to estimate the typical timescale for the onset of the sawtooth

instability. A 1-D cylindrical model is insufficient to describe

the complexity of the stellarator geometry but can be used to

provide a first quantitative guess of the current drive effects

on the rotational transform profile. This approach is justified

by the fact that the ECCD is driven in the proximity of the

axis, inside reff/a<0.25. This fact, along with being W7-X a

large aspect stellarator (R/a≈10) allows for using a 1-D cyl-

indrical approximation as a first step to evaluate the current

profile evolution. In 5.2 we estimate the role of the toroidal

current in determining the position where the instability can

be triggered.

5.1. 1-D current diffusion model

The current density jtot(r,t)is composed of three terms: jBS(r),

jECCD(r)and jind(r,t):jBS(r)is the bootstrap current (calcu-

lated by the NTSS code [33]), jECCD(r)is the ECCD (cal-

culated by the ray-tracing code TRAVIS.) and jind(r,t)is the

shielding current. The plasma, due to Lenz’s law, generates the

latter to oppose magnetic flux changes. For the TRAVIS calcu-

lation of jECCD(r)the magnetic equilibrium reconstruction has

been calculated with VMEC. No additional recalculations of

the magnetic equilibrium due to toroidal current changes have

been performed. jECCD(r)and jind(r,t)are plotted in figure 10a

(dashed lines). ECCD is strongly peaked and generated inside

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 7. SVD-filtered soft x-ray tomographic reconstruction of a sawtooth crash at W7-X. The black dashed curve represents an estimation

of the LCFS. Tomographic data have been SVD-filtered by removing the dominant component, which for this case consisted of a constant

value. Plot (I) shows the plasma at the equilibrum. In II and III the development of the displacement (from core to high field side) is

depicted. In IV the core is expelled and rotates (V, VI, VII) in anti-clockwise direction. In VIII and IX the poloidal symmetry is restored.

r/a< 0.25, while the boostrap current has a very broad profile.

If jECCD(r)is constant in time, jind(r,t)evolves as:

µ0

∂jind(r,t)

∂t=1

∂

∂r(r∂

∂r(ηsp(r)jind(r,t))) (3)

with µ0being the vacuum permeability, ηsp(r)the Spitzer

plasma resistivity ([30]) and jind(r,0) = −(jECCD(r) + jBS(r))

. In figure 10 the numerical solution of equation (3) is plotted

for three different timesteps (solid lines). The shielding cur-

rent diffuses mainly towards the centre, allowing for a strong

and localised current density increase on a timescale faster

than the skin time. In the right plot of figure 10 the sum of

currents is displayed for different timesteps.

In 1-D cylindrical geometry, the rotational transform pro-

duced by the toroidal current can be written as:

ι-curr(r) = µ0R0Itor(r)

2πBtorr2.(4)

Therefore, because the toroidal current and the rota-

tional transform are connected, it is possible to express

equation (3) as:

∂ι-ind

∂t=1

µ0r

∂

∂r[ηsp

∂

∂r(r2ι-ind)].(5)

Equation (5) describes the resistive evolution of the

rotational transform. Taking this into account, considering

neoclassical corrections, as discussed in [34], and renorm-

alising equation (3) with respect to the resistive time τη=

µ0l2/η0, where η0is the Spitzer resistivity in the core and l

is a characteristic length:

∂ι-ind

∂τ =1

∂

∂ˆ

r[ˆη

∂

∂ˆ

r(ˆ

r2ι-ind)](6)

with τ=t/τη,ˆ

r=r/l, and ˆη=ηsp/η0. The rotational trans-

form of the system is finally given by equation (2),

with ι-curr(r,t) = ι-ind(r) + ι-ECCD(r) + ι-BS(r), the latters being

respectively the rotational transform modifications due to sta-

tionary ECCD and stationary bootstrap current, i.e. the modi-

fications of ι- after several L/Rtimes. At t=0, the initial con-

dition is ι-ind(r,t=0) = −(ι-ECCD(r) + ι-BS(r)).

If the crash is triggered by the resistive diffusion of ι-, the

diffusion has to occur on a timescale of ∆τη≈∆tcrash ≈0.2 s

on a characteristic spatial scale l, that in this case repres-

ents the skin depth. For the plasma parameters of the dis-

cussed discharge in this paper, ∆τηcan be expressed as

∆τη≈500l2s/m2, therefore l≈2 cm. The numerical solu-

tion of equation (6) at different times is plotted in figure 11.

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 8. Temporal evolution of the inversion radii (rinv), crash

interval (∆t) and crash amplitude (∆Tmax) as a function of time

from t=1.3 s to t=23.5 s (dashed lines in the bottom plot). In the

first plot the rinv of both crash types is displayed. It is possible to

notice a relevant outward shift for type A crash. In the second plot,

the temporal interval between two crashes of the same type is

plotted. ∆tfor type A crashes strongly increases in the first 10 s of

the discharge, from 100 ms to around 400 ms. In the third plot,

∆Tmax is plotted. For type A crashes the amplitude goes from

−35 % to −70 %. For type B crashes, the amplitude increases and

saturates until the next type A crash. (Also see figure 3).

The ECCD modifies the rotational transform and a low order

rational value is reached on the order of 0.0012τη≈74 ms,

which is consistent with the measured time for sawtooth activ-

ity to start. The red curve represents the case where almost the

whole shielding current is dissipated and no current redistri-

bution has been taken into account. Ideally a double cross-

ing of ι-=1 is possible. The dominant contribution is given

by ECCD, being it deposited close to the axis with a stronly

peaked profile. For the calculated current profiles, the contri-

bution of the bootstrap current inside rinv i.e. for r/a< 0.4 is

less than 10 %.

The presented model presents two main limits. Since the

sawtooth crash can introduce significant changes on the cur-

rent profile and since no current redistribution has been intro-

duced yet, we presently use this model to estimate the ι- profile

at the beginning of the discharge only. The strong dependence

Figure 9. Sketch of rotational transform modification due to a

toroidal current. In (a) , three different current density profiles are

plotted: a flat current profile (red) , a parabolic profile (blue) and a

Guassian profile (green). The three profiles yield the same toroidal

current when integrated on the whole plasma volume. In (b) the

toroidal current is plotted as a function of the radius. In (c) the

modified ι- profiles are plotted. The dashed line indicates the

resonance ι-=1.

Figure 10. Modelled current density profiles for different timesteps.

On the left image, jECCD(r),jBS(r)(dotted lines) and jind(r,t)are

plotted. jind(r,t)represents the numerical solution of equation (3),

without taking into account any current redistribution after a crash.

The shielding current jind(r,t)diffuses towards the centre and a net

current is created. On the right image the sum of the two

components is displayed. The dashed vertical line represents the

position of rinv at the beginning of the discharge.

of ι-curr(r)∝I(r)/r2makes our model no longer valid in the

axis, where the rotational transform is strongly affected. In

the axis a strong current can change also the toroidal flux

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

Figure 11. Modelled rotational transform profile modified by ECCD

for different times. Plotted curves represent the solution of

equation (5).

[35] and therefore the crossing of ι-=0.5 is not taken into

consideration.

5.2. Estimation of rinv

Here we present an estimation of rι-=1for type-A crashes based

on two major observations. The first assumption is that the

driven toroidal current is mainly generated in the proximity

of the magnetic axis, therefore it is deposited inside the meas-

ured rinv. The second observation concerns the bootstrap cur-

rent, which has been calculated to be about 3kA in the sta-

tionary phase, with a broad profile. Combining these assump-

tions, we can assume the toroidal current density to be mainly

distributed inside rinv and therefore that the measured tor-

oidal current yields a good estimation of the toroidal current

inside the inversion radius, i.e. Itor(r=a)≈Itor(rι-=1), where

Itor(rι-=1) = 2π´rι-=1

0jtor(r)rdr, with rι-=1being the radius

where ι-=1 is crossed. The assumption that the majority of

the current is distributed close to the magnetic axis allows us to

estimate ι-=1 despite the lack of information about the real j(r)

profile. As plotted in figure 10, the crossing position depends

only on the integrated value of j(r), which corresponds

with this assumption to the experimentally measured toroidal

current.

We used equation (2) and equation (4) to estimate how the

rotational transform is modified by the toroidal current. For

every detected crash, we calculated rι-=1, which is the radial

position that solves equation (2) and compared it to the inver-

sion radius (rinv) of every crash as displayed in figure 12. The

temporal evolution of the calculated rι

-=1position follows the

trend of rinv increase. While at the beginning of the discharge

the two positions deviate approximately by 25% , the deviation

is reduced in time, until the two datasets differ by less than

10 %. This deviation indicates that rinv and rι

-=1might not coin-

cide. The initial bigger difference can have different explana-

tion. At the beginning of the dischage, due to the shielding

current jind(r,t), the toroidal current inside the inversion radius

does not represents the whole toroidal current measured by the

Figure 12. Comparison between the inversion radii rinv obtained by

experimental data and rι-=1, which represents the radial position that

satisfies ι-=1. In the top plot, the two quantities are plotted as a

function of time. In the bottom plot rinv is plotted against rι-=1. The

dashed grey line represents the line x=y.

Rogowski coils. Other issues are geometrical factors neglected

in the cylindrical approximation, which could have an higher

impact close to the axis. Although rinv and (rι-=1 do not coin-

cide, it is interesting to notice that they follow a similar trend

(bottom plot of figure 12) and their difference becomes con-

stant at the end of the discharge. This confirms that the increase

of the crash size is caused by the toroidal current increase

inside rinv.

6. Discussion and conclusions

In this work we presented a typical ECCD experiment with

sawtooth-like activity at the W7-X stellarator.

W7-X is a low shear stellarator which relies on the exist-

ence of magnetic island chains at the plasma edge. The

strikeline control on the divertor plates is necessary in order

to avoid component damages. However it was found that a

localised current, such as ECCD can induce local changes to

the rotational transform, thus making the plasma susceptible to

MHD instability. The presence of strong ECCD seems to be

fundamental for the trigger of the sawtooth crashes at W7-X.

No sawtooth crashes have been observed so far in experiments

without ECCD.

In typical ECCD programs, two crash types can be distin-

guished, whose parameters (such as crash amplitude, inversion

radius and temporal interval between them) change in time.

For stronger temperature crashes a mode structure, consist-

ent with a (1, 1) mode, is detected before the crash. In many

cases, the precursor is a pure displacement and no temperature

Nucl. Fusion 60 (2020) 106021 M. Zanini et al

oscillations are detected. Precursors are generally associated

to diamagnetic rotation: at the beginning of the discharge,

these crashes occur in a region with higher ∇p(and there-

fore, stronger diamagnetic rotation). As the crash localisation

moves outwards, ∇pdecreased the and rotation is strongly

reduced.

Considering the operational regimes and the characteristic

times for the instability to grow, a potential explanation for the

mechanism could be: 1) current drive drastically changes the

iota profile, eventually crossing resonant values. 2) An internal

(1, 1) non ideal kink is destabilised, as discussed in [20–22].

3) The fast movement of the core and the deformation of the

pressure profile creates a narrow region were ∇pis increased.

4) The point of the plasma between the displaced core and

the unperturbed external magnetic surfaces represents the X-

point were magnetic reconnection takes place, allowing for a

fast redistribution of pressure. In the analysed experiments the

core is expelled outside the ι-=1 region in about 20 µs, con-

sistent with the results of [21], and rotates in the direction of

the electron diamagnetic drift. Two different types of crashes

have been observed, with different amplitudes and temporal

intervals. This analysis has been conducted for bigger crashes:

smaller crashes are more difficult to resolve. The coupling

between these two types of crash is still unclear, but an initial

explanation assumes a double resonance crossing, on differ-

ent positions. The two modes, nevertheless, seem to develop

on distinct timescales. A 1-D model has been developed, to

estimate typical timescales for the ι- evolution. We found that

the modifications caused by the current drive is fast and strong

enough to produce resonant values consistent with experi-

mental results. The main contribution to the rotational trans-

form modifications is given by ECCD, which is stronly peaked

and localised close to the axis. The bootstrap current, having

a very broad profile, yields a negligible contribution to the

net toroidal current in the region where crashes are detected.

This model is valid only for the initial events, since no cur-

rent redistribution caused by the instabilities has been imple-

mented yet. Nonetheless, combining equation (2) and equa-

tion (4), we have been able to find a relation between rinv

and Itor, which explain the crash size increase. As the current

develops, the rι-=1 position moves outwards, thus leading to

a bigger crash amplitude. Further investigations, using MHD

codes, are still ongoing, in order to confirm experimental

observations about the nature and characteristic times of mode

developement.

Acknowledgments

We thank Valentin Igochine and Joachim Geiger for sev-

eral insightful discussions. This work has been carried out

within the framework of the EUROfusion Consortium and has

received funding from the Euratom research and training pro-

gramme 2014-2018 and 2019-2020 under Grant Agreement

No. 633053. The views and opinions expressed herein do not

necessarily reflect those of the European Commission.

ORCID iDs

M. Zanini https://orcid.org/0000-0002-8717-1106

C. Brandt https://orcid.org/0000-0002-5455-4629

K.J. Brunner https://orcid.org/0000-0002-0974-0457

J. Schilling https://orcid.org/0000-0002-6363-6554

R.C. Wolf https://orcid.org/0000-0002-2606-5289

A. Zocco https://orcid.org/0000-0003-2617-3658

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