Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas

S S symmetry

Review Evolution of Heavy Ion Beam Probing from the Origins to Study of Symmetric Structures in Fusion Plasmas

Alexander Melnikov 1,2

1 Kurchatov Center for Thermonuclear Energy and Plasma Technologies, National Research Center Kurchatov Institute, 123182 Moscow, Russia; [email protected] 2 Plasma Physics Department, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia

Abstract: The overview discusses development of the unique fusion plasma diagnostics—Heavy Ion Beam Probing (HIBP) in application to toroidal magnetic plasma devices. The basis of the HIBP measurements of the plasma electric potential and processing of experimental data are considered. Diagnostic systems for probing plasma in tokamaks TM-4, TJ-1, TUMAN-3M and T-10, stellarators WEGA, TJ-II and Uragan-2M are presented. Promising results of the HIBP projects for various existing modern machines, such as TCV, TCABR, MAST, COMPASS, GLOBUS-M2, T-15 MD and W7-X and the international fusion tokamak reactor ITER are given. Results from two machines with similar size and plasma parameters, but with different types of the magnetic con-figuration: axisymmetric tokamak T-10 and helically symmetric stellarator TJ-II are compared. The results of studies of stationary potential profiles and oscillations in the form of quasimonochromatic and broadband fluctuations, turbulent particle flux, fluctuations of density and poloidal magnetic field

are presented. The properties of symmetric structures—zonal ﬂows and geodesic acoustic modes of plasma oscillations as well as Alfvén Eigenmodes excited by fast particles from neutral beam

Citation: Melnikov, A. Evolution of injection heating are described. General trends in the behavior of electric potential and turbulence in Heavy Ion Beam Probing from the magnetized fusion plasmas are revealed. Origins to Study of Symmetric Structures in Fusion Plasmas. Keywords: plasma symmetric structures; magnetic conﬁnement; heavy ion beam probing; electric Symmetry 2021, 13, 1367. https:// potential; geodesic acoustic mode; Alfvén eigenmode; broadband turbulence; zonal ﬂow doi.org/10.3390/sym13081367

Academic Editor: Alexander Kukushkin 1. Preface In recent decades, diagnostics of high-temperature plasma has become an independent Received: 27 May 2021 scientiﬁc discipline that has grown out of the physics of fusion plasma. Fusion plasma is Accepted: 9 July 2021 Published: 27 July 2021 a very unusual area of science. Unlike other areas of physics that study the properties of objects existing on Earth, plasma physics ﬁrst creates an object of its research in terrestrial

Publisher’s Note: MDPI stays neutral conditions—plasma with thermonuclear fusion temperatures higher than the temperature with regard to jurisdictional claims in in the Sun core. Superconducting magnets, creating magnetic fields for plasma confinement, published maps and institutional affil- operate near the absolute minimum temperature, in conditions of space cold. The vacuum iations. conditions in fusion machines for magnetic confinement of plasma are approaching to the conditions of cosmic space. The creation of such a complex and un-stable object as a fusion plasma in a terrestrial laboratory is in itself a formidable physical and technical issue. An equally difficult issue is plasma diagnostics—the development of tools and methods for studying plasma, reliable techniques and tools for measuring its properties and parameters, Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. such as the temperature of its electrons and ions, density, stored energy, effective charge, This article is an open access article and many others. The listed plasma parameters are not measured directly, which is distributed under the terms and obviously impossible in such a hot medium, but by their indirect signs, such as, for example, conditions of the Creative Commons plasma radiation in different energy ranges. From the point of view of mathematics, Attribution (CC BY) license (https:// the determination of the required plasma parameters from the quantities available for creativecommons.org/licenses/by/ measurement is a classic example of inverse problems that require the development of 4.0/). special regularization methods for their solution [1,2]. In the last decades of the 20th

Symmetry 2021, 13, 1367. https://doi.org/10.3390/sym13081367 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1367 2 of 46

century, one of the hottest areas of research has become the study of the electric field in the plasma of fusion machines with magnetic confinement. The reasons for this were the anomalous transport of particles and energy across the confining magnetic field, inherent in hot turbulent plasma; the identification of complex relationships between the electric field, plasma rotation, properties of turbulence and confinement. Over the past three decades, methods for diagnosing the potential of the electric fields in plasmas, as well as theoretical and numerical models of the transport processes associated with them, have been developed in the USSR and Russia. The present work is a brief overview of results of many years of research carried out by large teams under the leadership and with the active participation of the author at several fusion machines—tokamaks and stellarators.

2. Introduction The study of electric fields is one of the urgent problems of modern physics of fusion plasma. These studies are related to both the fundamental problem—the emergence of an electric field in a quasineutral plasma [3], and an applied problem—the understanding of the electric field effects on the plasma transport processes in closed magnetic configura- tions [4]. It is known that the anomalous mechanisms linked with plasma turbulence are dominated in the particle and energy transport across confining electric field [5]. Currently, the hypothesis that the plasma turbulence and anomalous transport are stabilized by shear (radial inhomogeneity) of poloidal plasma rotation in crossed radial electric Er and toroidal magnetic Bt fields is generally accepted in magnetic confinement studies [6]. However, this hypothesis has both agreements and disagreements with experiment, so, the effect of Er to plasma confinement still remains unclear. At present, it is believed that the problems associated with the electric field in tokamaks are becoming the principal ones in fusion research, and that for the success of the ITER project, it is necessary to change from studying the role of the electric field to controlling the anomalous transport with electric fields [7]. Measurement of the electric field in the plasma of modern fusion machines is a complex experimental issue [8]. At the plasma edge, the electric potential is measured with Langmuir probes. In the hot plasma core, the electric field, as a rule, is not measured directly, but it is determined from the measured plasma rotation velocity using spectroscopy or correlation reflectometry. The only direct method to find the electric potential in the plasma interior is the Heavy Ion Beam Probe (HIBP) diagnostics [9]. Diagnostic techniques for toroidal plasmas always have limitations concerning the access to the plasma. These constrains are dictated by the complexity of the toroidal devices and by the trend to reduce the ripple of the confining magnetic field by an increase in the number of toroidal field coils. This makes the diagnostic ports narrow in toroidal direction. Finally, most of the diagnostics observe just a part of the plasma vertical cross-section. They have to assume the symmetry of the measured parameter to complete the whole picture of its spatial distribution. The typical example of such an approach is the multi-chord plasma density measurement with a microwave interferometer as a function of the plasma radius that then assumes a poloidal symmetry of the density distribution. On top of that, there are theoretical predictions of various asymmetries for plasma parameters. For example, there could be the so-called ballooning effect, which deforms the contour lines of plasma parameters at the Low-Field Side (LFS) of the torus vertical cross-section, breaking the in-out symmetry. Gyrokinetic simulations predict a strong increase of the broadband electrostatic turbulence in the LFS in respect to the High Field Side (HFS). Therefore, the issue of symmetry is an open one in many aspects of plasma studies. Resolving this issue is highly demanding for the diagnostic techniques. This overview describes the HIBP diagnostics of toroidal plasma and its development over the past four decades for studying various phenomena in plasma including issues of their symmetry. Symmetry 2021, 13, x FOR PEER REVIEW 3 of 49

This overview describes the HIBP diagnostics of toroidal plasma and its development Symmetry 2021, 13, 1367 over the past four decades for studying various phenomena in plasma including issues3 of of 46 their symmetry.

2.1. Heavy Ion Beam Probe—A Tool for Measuring Electrical Potential and Plasma Turbulence 2.1. Heavy Ion Beam Probe—A Tool for Measuring Electrical Potential and Plasma Turbulence The Heavy Ion Beam Probe (HIBP) is a unique method for plasma studies in fusion The Heavy Ion Beam Probe (HIBP) is a unique method for plasma studies in fusion machines [10]. HIBP was first implemented in the late 1960s by Hickok and Jobs on the machines [10]. HIBP was first implemented in the late 1960s by Hickok and Jobs on the arc arc discharge and later on the ST tokamak [11]. For the first time in Russia (USSR), HIBP discharge and later on the ST tokamak [11]. For the first time in Russia (USSR), HIBP was was implemented in the early 1980s by Krupnik and Nedzelskiy at the TM-4 tokamak implemented in the early 1980s by Krupnik and Nedzelskiy at the TM-4 tokamak [12,13]. [12,13]. Currently, HIBP is the only non-disturbing method for direct measurements of the Currently, HIBP is the only non-disturbing method for direct measurements of the electric electric potential in a hot plasma core [14]. In this paper, we consider the mathematical potential in a hot plasma core [14]. In this paper, we consider the mathematical aspects of aspects of diagnostics, equipment and methods for measuring plasma parameters availa- diagnostics, equipment and methods for measuring plasma parameters available for the ble for the modern HIBP [15]. modern HIBP [15]. To Todate, date, there there are are only only two two HIBP HIBP diagnostics diagnostics operating operating in intokamaks tokamaks in inthe the world: world: Tuman-3MTuman-3M [16] [16 (Ioffe] (Ioffe Institute Institute of Physics Physics and and Technology, Technology, St. Petersburg, St. Petersburg, Russia) Russia) and ISTTOK and ISTTOK(Lisbon, (Lisbon, Portugal) Portugal) [17]. The [17]. T-10 The tokamak T-10 tokamak [18] (Kurchatov [18] (Kurchatov Institute, Institute,Moscow, Russia) Moscow, was Russia)terminated was terminated in 2018 due in to 2018 creation due of to the creation next generation of the next tokamak generation T-15MD. tokamak In addition, T-15MD. HIBP In operatesaddition, on HIBP the world’soperates largest on the stellarator world’s largest LHD (National stellarator Institute LHD (National for Fusion Institute Science, forToki, FusionJapan) Science, [19–21 ], Toki, on the Japan) TJ-II [19–21],stellarator on (CIEMAT, the TJ-II stellaratorMadrid, Spain) (CIEMAT, [22,23 ], Madrid, as well asSpain) on the [22,23],reversed as well field as pinch on the MST reversed [24] (Madison field pinch University, MST [24] Madison,(Madison WI, University, USA) and Madison, on the tandem WI, USA)mirror and GAMMA-10on the tandem [25 mirror] (Tsukuba GAMMA-10 University, [25] Tsukuba, (Tsukuba Japan). University, Tsukuba, Japan).

2.2.2.2. Physical Physical Principles Principles of HIBP of HIBP measurements Measurements To Toprobe probe the the plasma plasma with with the the beam beam of ofheavy heavy ions, ions, the the beam beam is injected is injected across across the the confiningconfining magnetic magnetic field. field. When When the the beam beam particles particles fly flythrough through the the plasma, plasma, some some of ofthem them collidecollide with with plasma plasma particles particles (mainly (mainly electrons) electrons) and and lose lose one one or ormore more electrons. electrons. As Asa result, a result, a fana fan of ofsecondary secondary ionized ionized particles particles with with higher higher charges charges is formed. is formed. The The HIBP HIBP diagnostic diagnostic schemescheme and and an anexample example of ofits itsrealization realization in inthe the T-10 T-10 tokamak tokamak are are shown shown in inFigure Figure 1.1 .

(a) (b)

FiguFigurere 1. ( 1.a) (HIBPa) HIBP experimental experimental set-up set-up at at T-10 T-10 [15]: [15]: (1) (1) accelerator, accelerator, (2) (2) primary primary beamline, (3)(3) secondarysecondary beamline, beamline, (4) (4) analyzer, ana- + ++ lyzer,(5) trajectory(5) trajectory of primaryof primary Tl+ Tlionsions (red (red curve), curve), (6) (6) trajectory trajectory of of secondary secondary Tl Tl++ ionsions (yellow(yellow curve),curve), (7)(7) ionizationionization point or or sample volume (SV). ( b) Photo of the the HIBP HIBP diagnostic diagnostic hardware. hardware. Reproduced Reproduced courtesy courtesy of of IAEA. IAEA. Copyright Copyright 2017 2017 IAEA. IAEA.

A small-apertureA small-aperture detector detector is usually is usually placed placed outside outside the the magnetic magnetic field ﬁeld that that allows allows the the secondarysecondary ion ion beam beam (part (part of ofthe the fan), fan), formed formed at atthe the secondary secondary ionization ionization point point (sample (sample volumevolume or orSV) SV) on onthe the probe probe beam beam trajectory, trajectory, to toenter enter the the detector. detector. The The detected detected secondary secondary ions carry information about the plasma parameters in the SV. The spatial resolution of the method λ is determined by the dimensions of the sample volume and mainly depends on the size of the aperture and the position of the detector. In real experiments on different machines λ = 0.2–2 cm. Symmetry 2021, 13, 1367 4 of 46

HIBP allows us to simultaneously measure several plasma parameters: electric potential ϕ, electron density ne, and magnetic potential A (or poloidal magnetic field Bp or current density j). The position of the sample volume can be moved along the plasma cross section by varying the energy of the probing beam Eb or the injection angle of the beam into the plasma α. Thus, the set of points observed by the detector forms a two-coordinate (Eb, α) grid called the detector grid. The trajectories of primary and secondary ions in a low-current tokamak lie in the meridional plane between the confining field coils or near it. For tokamaks with a high current density and for stellarators, the trajectories of the probing particles are shifted in the toroidal direction, and the detector grid becomes three-dimensional. This essential feature of the motion of probing particles should be taken into account during development projects of placing diagnostics on new machines installations, and in the analysis of measurement results. Thus, a specific feature of HIBP diagnostics is that the values of plasma parameters are determined from measuring the characteristics of the secondary beam: intensity, energy, etc., while the position of the measurement point can only be found from calculations of the trajectories of probing particles. Therefore, before placing HIBP diagnostics on a new machine installation, not only the development of diagnostic equipment is required, but also preliminary calculations. The position of the sample volume in the meridional plane depends on a large number of probing parameters, including: (i) geometric parameters: the coordinates of the injection xi, yi and detection points xD, yD, and the injection angle α (ii) the physical parameters: the toroidal magnetic field Bt, the energy Eb, the mass m, and the electric charge q of the probing particles. Probing particles move in the magnetic field of the machine along a Larmor circle with a radius: c p RL = 2mEb. (1) qeBt

It is necessary to determine not the value of RL itself, but the values of the parameters on the right hand side of Equation (1), which are subject to independent restrictions. The equality RL(q, m, Eb, Bt) = const reduces the number of physical parameters from 4 to 3. So, for 2D optimization in the meridional plane, there are eight parameters: xI, yI, xD, yD, α, Bt, Eb, q. The need to pass the trajectories of particles through the ports of the vacuum chamber and the structural elements of the machine impose serious constraints on these parameters. These constraints are related to each other because the allowable tolerances for each parameter depend on the others. Such relations defy analytical description due to the complexity of the trajectory behavior. The configuration of the vacuum chamber in the meridional plane approximately determines their tolerances. The problem of optimizing experimental conditions includes the choice of optimal probing parameter values in the sense of the following goal functions: 1. Pass the beam through the existing vacuum vessel ports; 2. Find the detector line from the core to the edge of the plasma; 3. Find a detector grid covering the maximum part of the plasma cross section; 4. Optimize the range of beam energies. In high-current tokamak discharges, as well as in stellarators, the magnetic field has a noticeable poloidal component Bp, which shifts the orbits of the probing particles out of the meridional plane and transforms them into spatial curves [26]. In this case, additional parameters appear that affect the sample volume. These are the toroidal coordinates of the injector and detector, ZI and ZD, and the injection angle between the initial velocity vector and the meridional plane β. This gives us a 12-dimensional optimization problem with implicit connections between the limits of parameters and with non-formalized goal functions. Figure2 shows an example of solving this problem by ‘the shooting’ method in T-10. Examples of solving the optimization problem in other machines are given in Sections7 and8. The first goal function was achieved in all considered machines, Symmetry 2021, 13, x FOR PEER REVIEW 5 of 49

vector and the meridional plane β. This gives us a 12-dimensional optimization problem with implicit connections between the limits of parameters and with non-formalized goal Symmetry 2021, 13, 1367 functions. Figure 2 shows an example of solving this problem by ‘the shooting’ method5 of 46in T-10. Examples of solving the optimization problem in other machines are given in Sec- tions 7 and 8. The first goal function was achieved in all considered machines, as the chosen probing schemes ensured the passage of trajectories through the vacuum vessel ports as the chosen probing schemes ensured the passage of trajectories through the vacuum and allowed measurements. Maximization of the second and third goal functions pro- vessel ports and allowed measurements. Maximization of the second and third goal vides the maximum possible radial and angular sizes of the observation area in plasma. functions provides the maximum possible radial and angular sizes of the observation The fourth function is not always so essential, because the energy range of modern accel- area in plasma. The fourth function is not always so essential, because the energy range erators is quite large. However, for large machines, such as the W-7X stellarator and the of modern accelerators is quite large. However, for large machines, such as the W-7X stellaratorconstructed and ITER the tokamak constructed reactor, ITER this tokamak task is reactor, of great this importance. task is of great As will importance. be shown As in willSection be shown7, the chosen in Section injection7, the chosenschemes injection allow one schemes to connect allow the one core to connect and the the edge core of and the theplasma edge with of the a detector plasma withline using a detector the acceptable line using beam the acceptable energy. beam energy.

Figure 2. Detector grid for T-10T-10 withwith thethe ﬁeldfield BBtt = 1.55 T: lines of equal beam energy ((EEbb)) are markedmarked in green,green, lineslines ofof equalequal injectioninjection angleangle αα,, labeled labeled with with scanning scanning voltage voltageU Uscanscan areare blue.blue. TheThe asterisksasterisks denote the nodes of the detector grid availableavailable forfor observationobservation withwith allall slitsslits ofof multi-slitmulti-slit detector.detector.

3. Mathematical ProblemsProblems of DeterminingDetermining PlasmaPlasma ParametersParameters Using HIBPHIBP 3.1. Determination the Spatial Distribution of Electric Potential 3.1. Determination the Spatial Distribution of Electric Potential The principle of HIBP measurements is based on the potentiality of the electric ﬁeld, The principle of HIBP measurements is based on the potentiality of the electric field, the conservation of the total energy of the charged particles in the plasma electric ﬁeld the conservation of the total energy of the charged particles in the plasma electric field (Figure3). The probing beam, hereinafter referred to as the primary beam, enters the (Figure 3). The probing beam, hereinafter referred to as the primary beam, enters the plasma with an initial energy Eb. At the ionization point, which is the plasma region of plasma with an initial energy Eb. At the ionization point, which is the plasma region of interest (SV), a particle of the primary beam loses an electron with potential energy −eϕSV interest (SV), a particle of the primary beam loses an electron with potential energypl whichSV is acquired by the secondary ionized probe ion. The total energy of secondary ions  e pl which is acquired by the secondarySV ionized probe ion. The total energy of sec- leaving the plasma is Ed = Eb + eϕ . Therefore, the local potential in SV is equal to the pl SV ondary ions leaving the plasma is Ed E b  e pl . Therefore, the local potential in SV is Symmetry 2021, 13, x FOR PEER REVIEWenergy difference: 6 of 49 SV equal to the energy difference: ϕpl = (Ed − Eb)/e. (2) SV  pl()/E d  E b e . (2)

FigureFigure 3. 3. PrinciplePrinciple of of plasma plasma potential potential measurement. measurement.

Thus,Thus, the the problem problem of of measuring measuring the the plasma plasma electric electric potential potential does does not notpose pose serious seri- mathematicalous mathematical difficulties. difﬁculties. However, However, from the from point the of point view of of view conducting of conducting physical physical measurements it is a very difficult task, as the dimensions of modern plasma machines (scale of 15 m) and the magnitude of the magnetic field (scale of 1–3 T) impose high requirements on the Larmor radius of probing particles, which must exceed the plasma size. Ac- cording to (1), this requires the use of probing ions with the minimum charge number (q =1), maximum possible mass (Cs+, Tl+, Au+) and energies (on a scale of several hundred keV—several MeV). On the other hand, the measured plasma potential has the scale of the electron temperature. In the plasma core, it ranges from several hundred eV to keV. At the plasma edge, it has a scale from several eVs to hundreds of eV. Thus, according to (2), the measurement of the potential claims the computation of a small difference of two large quantities. Such measurements require high accuracy of each quantity or highly stabilized high-voltage sources and the use of precision instruments to measure the energy of the secondary beam.

3.2. Determination of the Spatial Distribution of Plasma Density The local values of the plasma density n(l) can be found from the relation:

i2 i 1 n()() l  l , (3)

where i1 and i2 are the intensities of the primary and secondary beams in the SV;

 e/  b is the effective ionization cross section by electron impact, <συe> is the ioni-

zation rate averaged over the Maxwellian distribution of the electron velocity υe, itself a function of the electron temperature Te; υb is the initial velocity of the probing beam, υe > υb; l = l(x, y) is the coordinate on the detector line; and λ(l(E)) is the longitudinal dimension of SV, the length of the arc along the primary trajectory, from which the secondary ions reach the detector. The electron density n(l) is averaged over the length λ. If the density is low and the dimensions of the machine are small, then we can assume that the intensities i1 and i2 are the intensities of the primary beam at the exit from the ion injector and the secondary beam at the detector. In this case, Relation (3) can be directly used to determine the local values of the plasma density. However, in modern fusion machines installations, both the primary and secondary beams are significantly attenuated, when passing through the plasma. Methods for determining plasma density using HIBP in the case of strong attenuation are described in [27,28], and they were used for interpretation of experimental data from the TM-4 tokamak. The expression for the current of secondary ions to the detector It, taking into account the attenuation of along the beam trajectory, caused by elementary processes, including collisions of beam particles with plasma particles, can be written as follows: Il()2 I12 ()()()exp lnll   n  12 () sds  n  23 () sds , t b  Il  lD  (4)

Symmetry 2021, 13, 1367 6 of 46

measurements it is a very difﬁcult task, as the dimensions of modern plasma machines (scale of 1–5 m) and the magnitude of the magnetic ﬁeld (scale of 1–3 T) impose high requirements on the Larmor radius of probing particles, which must exceed the plasma size. According to (1), this requires the use of probing ions with the minimum charge number (q = 1), maximum possible mass (Cs+, Tl+, Au+) and energies (on a scale of several hundred keV—several MeV). On the other hand, the measured plasma potential has the scale of the electron temperature. In the plasma core, it ranges from several hundred eV to keV. At the plasma edge, it has a scale from several eVs to hundreds of eV. Thus, according to (2), the measurement of the potential claims the computation of a small difference of two large quantities. Such measurements require high accuracy of each quantity or highly stabilized high-voltage sources and the use of precision instruments to measure the energy of the secondary beam.

3.2. Determination of the Spatial Distribution of Plasma Density The local values of the plasma density n(l) can be found from the relation:

i2 = i1σn(l)λ(l), (3)

where i1 and i2 are the intensities of the primary and secondary beams in the SV; σ = hσυei/υb is the effective ionization cross section by electron impact, <συe> is the ionization rate averaged over the Maxwellian distribution of the electron velocity υe, itself a function of the electron temperature Te; υb is the initial velocity of the probing beam, υe > υb; l = l(x, y) is the coordinate on the detector line; and λ(l(E)) is the longitudinal dimension of SV, the length of the arc along the primary trajectory, from which the secondary ions reach the detector. The electron density n(l) is averaged over the length λ. If the density is low and the dimensions of the machine are small, then we can assume that the intensities i1 and i2 are the intensities of the primary beam at the exit from the ion injector and the secondary beam at the detector. In this case, Relation (3) can be directly used to determine the local values of the plasma density. However, in modern fusion machines installations, both the primary and secondary beams are signiﬁcantly attenuated, when passing through the plasma. Methods for determining plasma density using HIBP in the case of strong attenuation are described in [27,28], and they were used for interpretation of experimental data from the TM-4 tokamak. The expression for the current of secondary ions to the detector It, taking into account the attenuation of along the beam trajectory, caused by elementary processes, including collisions of beam particles with plasma particles, can be written as follows:

Z Z 12 12 23 It(l) = 2Ibσ (l)n(l)λ(l) exp − nσ (s)ds − nσ (s)ds , (4) Il lD

12 23 where Ib is the ion beam current leaving the injector, n is the electron density, σ and σ are the known effective cross sections for collisional ionization of the probing beam (e.g., Cs in the cases of TM-4 and TJ-II) by plasma electrons.

e + Cs+ → Cs2+ + 2e , e + Cs2+ → Cs3+ + 2e. (5)

The unknown function in this nonlinear integro-functional equation is the plasma electron density n(l); functional dependence It(l) is determined experimentally; Ib is a known constant. The integration lines Il and lD, and the quantities λ(l) are found by calculating the trajectory and the detector line. Equation (4) is an example of a nonlinear inverse problem with a nonclassical operator structure: the integral of the desired function is in the exponent. The limits of integration are variable; they depend on the coordinates of a point on the detector line l = l (x, y). This equation may be solved by a quasi-solution method: parametrization the desired function and ﬁnding the optimal values of the parameters by minimizing the discrepancy functional. Symmetry 2021, 13, x FOR PEER REVIEW 7 of 49

where Ib is the ion beam current leaving the injector, n is the electron density, σ12 and σ23 are the known effective cross sections for collisional ionization of the probing beam (e. g. Cs in the cases of TM-4 and TJ-II) by plasma electrons. 2  2 3  eCs  Cs  2 e , eCs  Cs  2 e . (5) The unknown function in this nonlinear integro-functional equation is the plasma electron density n(l); functional dependence It(l) is determined experimentally; Ib is a known constant. The integration lines Il and lD, and the quantities λ(l) are found by calculating the trajectory and the detector line. Equation (4) is an example of a nonlinear inverse problem with a nonclassical operator structure: the integral of the desired function is in the exponent. The limits of integra- Symmetry 2021, 13, 1367 7 of 46 tion are variable; they depend on the coordinates of a point on the detector line l = l (x, y). This equation may be solved by a quasi-solution method: parametrization the desired function and finding the optimal values of the parameters by minimizing the discrepancy Thefunctional. application The application of this method of this to themethod analysis to the of experimentalanalysis of experi datamental from TM-4data from is shown TM- in4 is Figure shown4. in We Figure see that 4. We the see reconstructed that the reconstructed density proﬁle density is consistentprofile is consistent with the with data the of otherdata of diagnostics, other diagnostics, so the method so the worksmethod well works for TM-4well for conditions. TM-4 conditions. More recently, More therecently, same techniquethe same technique was applied was forapplied the TJ-II for the data. TJ- TheII data. results The showresults that show the that reconstructed the reconstructed HIBP densityHIBP density proﬁle profile lies within lies within the error the barserror of bars the Thomsonof the Thomso scatteringn scattering data [29 data]. [29].

(a) (b)

19 −319 −3 FigureFigure 4.4. PlasmaPlasma density density profiles profiles in in TM TM-4:-4: (a ()a Ohmic) Ohmic heated heated (OH) (OH) plasma, plasma, n̅e =n e1.5= × 1.5 10× m10 ; dashm ;line dash and line × is and a solution× is a solutionof Equation of Equation (3), without (3), attenuation; without attenuation; solid line solidand ○ line is solution and is of solution Equation of (4) Equation with a quasi (4) with-solution a quasi-solution method. (b method.) OH (n̅e = 1.3 × 1019 m−3) and ECR19 heated−3 plasmas; ○ is HIBP, + is interferometer; □ are probes and Δ is Thomson scaering data. (b) OH (ne = 1.3 × 10 m ) and ECR heated plasmas; is# HIBP, + is interferometer; are probes and ∆ is Thomson scattering data. # 3.3. Determination of the Plasma Magnetic Potential (Field of Plasma Current) 3.3. DeterminationThe plasma current of the Plasma profile Magnetic mainly determines Potential (Field the ofequilibrium Plasma Current) of the plasma column, its stability,The plasma energy current balance, profile and mainlythe characteristics determines of the fusion equilibrium plasma. of At the present, plasma for column, meas- itsurements stability, of energythe poloidal balance, magnetic and the field, characteristics there is a very of fusion limited plasma. set of diagnostics At present, [30 for], measurementswhich do not have of the a poloidalhigh temporal magnetic and field, spatial there resolution. is a very Therefore, limited set the of diagnosticsproblem of [cre-30], whichating diagnostics do not have for a poloidal high temporal fields in and toroidal spatial plasma resolution. is still Therefore,urgent. In thethis problemSubsection of, creatingwe consider diagnostics a method for for poloidal determining fields inthe toroidal plasma plasma current is field still from urgent. measurements In this Subsection, of the weshift consider of the probing a method beam for determiningcaused by this the field plasma [31,32 current]. field from measurements of the shiftLet of the us probingconsider beam the si causedmplest bymodel this fieldof a [symmetric31,32]. torus (without ripple), in which the toroidalLet us consider magnetic the moment simplest is model conserved of a symmetric for a probing torus charged (without particle. ripple), After in which integra- the toroidaltion the Lagrangian magnetic moment for the probing is conserved particle for along a probing the trajectories, charged particle. we obtain After the integration following theexpression Lagrangian for the for desired the probing function particle A, the along toroidal the component trajectories, of we the obtain magnetic the followingpotential: expression for the desired function Aϕ, the toroidal component of the magnetic potential: h i e R si Aϕ R sd Aϕ (ϕd − ϕ0) + q ds + (q + k) ds = mcv s0 r si r , (6) 2 . qe 1 R sd ds ke R sd ds = (r ϕ + Aϕ r ) 2 + Aϕ(s ) 2 r(s ). 0 0 mc 0 0 v s0 r mcv i si r i

Here s is the injection point, s is the ionization point, and s is the detection point, 0 i d q and q + k are dimensionless charge numbers of primary and secondary beam particles. Equation (6) connects the unknown function Aϕ with the angular displacement of the beam in the detector respect to the one of injector ϕd − ϕ0, that is, with the measured quantity. Equation (6) is a nonlinear Volterra-type integral equation of the second kind. As the integration limits are variable, they depend on the coordinates of the ionization point in the plasma (points on the detector line). In contrast with the equation of the ﬁrst kind, solving such equations is a well-posed problem. The function Aϕ is unknown only inside Symmetry 2021, 13, x FOR PEER REVIEW 8 of 49

e siAA s d   (  ) q ds  ( q + k ) ds   d 0 s  s mcv 0 ri r ,   (6) s s 2 qe 1 dds ke d ds (r  A r )  A  ( s ) r ( s ). 0 0 0 0 s2i  s 2 i mcv0 r mc v i r

Here s0 is the injection point, si is the ionization point, and sd is the detection point, q and q + k are dimensionless charge numbers of primary and secondary beam particles. Equation (6) connects the unknown function Aφ with the angular displacement of the beam in the detector respect to the one of injector φd − φ0, that is, with the measured quantity. Equation (6) is a nonlinear Volterra-type integral equation of the second kind. As the Symmetry 2021, 13, 1367 integration limits are variable, they depend on the coordinates of the ionization point8 of 46 in the plasma (points on the detector line). In contrast with the equation of the first kind, solving such equations is a well-posed problem. The function Aφ is unknown only inside ext the plasma column, for r < a. Outside the plasma, where r > a, Aextφ is determined by its the plasma column, for r < a. Outside the plasma, where r > a, Aϕ is determined by its values inside the plasma as follows: values inside the plasma as follows: 1j 1 () A Aext dV   dV  A ext () A ,     (7) 1 Z jϕ c R1 Z4(∆A R)ϕ Aext = dV = dV = Aext(A ), (7) where the integrationϕ is carriedc R out over4π the entireR region ϕof currentϕ flow, and R is the distance from the volume element dV to the position, where Aφext is computed. Note that where the integration is carried out over the entire region of current ﬂow, and R is the the integrals in (6–7) are curvilinear integrals over 3D trajectories. Theext arc element ds along distance from the volume element dV to the position, where Aϕ is computed. Note the particle trajectory depends on the unknown function Aφ: ds = ds(Aφ), therefore, Equa- that the integrals in (6–7) are curvilinear integrals over 3D trajectories. The arc element ds tion (7) is a nonlinear integral equation. along the particle trajectory depends on the unknown function A : ds = ds(A ), therefore, The solving of this equation is based on the iteration method.ϕ Figureϕ 5 shows the Equation (7) is a nonlinear integral equation. results of the numerical solution of Equation (7) for the conditions of the TUMAN-3M The solving of this equation is based on the iteration method. Figure5 shows the results tokamak. of the numerical solution of Equation (7) for the conditions of the TUMAN-3M tokamak.

(a) (b)

FigureFigure 5. 5.( a()a Toroidal) Toroidal shift shift of of trajectories trajectories as as function function of of the the path path traversed traversed by by the the probing probing particle particle in TUMAN-3M in TUMAN-3M (Bt =(B 0.5t = T;0.5 T; Ipl = 100 kA; R = 54 cm; a = 24 cm), with parabolic profile of current; sα1 and sα2 are distances from the injector to the Ipl = 100 kA; R = 54 cm; a = 24 cm), with parabolic profile of current; sα1 and sα2 are distances from the injector to the plasma plasma boundary; sd is the distance to the detector point; curves with different colors shows trajectories, born in different boundary; sd is the distance to the detector point; curves with different colors shows trajectories, born in different sample sample volumes, marked with black rectangles. (b) Shift of beam particles in the detector as a function of energy at a total volumes, marked with black rectangles. (b) Shift of beam particles in the detector as a function of energy at a total plasma plasma current Ipl = 100 kA with different profiles of current density, marked with numbers. The width of the beam track I currentfor eachpl current= 100 kA profile with corresponds different profiles to error of current in definition density, of markedthe beam with position numbers. relatively The width the meridional of the beam plane track of for 1 cm. each current profile corresponds to error in definition of the beam position relatively the meridional plane of 1 cm. 4. Application of the HIBP to Fluctuation Measurements 4. Application of the HIBP to Fluctuation Measurements The HIBP is a multi-purpose diagnostic tool capable to measure the local potential φ The HIBP is a multi-purpose diagnostic tool capable to measure the local potential using the secondary beam energy, plasma density ne from the beam current It, and the ϕ using the secondary beam energy, plasma density ne from the beam current It, and the poloidal magnetic field Bp from the toroidal angular displacement of beam φd or the linear poloidal magnetic field Bp from the toroidal angular displacement of beam ϕd or the linear beam shift ζ = φd R, where R is the distance from the detection point to the origin of the beam shift ζ = ϕ R, where R is the distance from the detection point to the origin of the cylindrical coordinated system [33]. cylindrical coordinate system [33]. High temporal resolution (up to 0.5 µs) allows measurements of fluctuations of the parameters under study. Expressions for the links between the relative fluctuations of the measured quantities and the parameters under study are following: For the variable component of the potential:

SV ϕepl (t) = 2Uan F δi(t), (8)

where δi is the normalized difference between the beam currents across the separated detector plates, Uan is the voltage applied to the analyzer, and F is its dynamic coefﬁcient. SV Thus, there is a simple linear relationship between the desired value δϕpl and the measured value δi(t). Symmetry 2021, 13, 1367 9 of 46

If the spatial correlation length of density ﬂuctuations is much less than the length of the beam trajectory, then the nonlocal terms in Equation (4) may be omitted due to integration of the oscillatory component δne over the beam trajectory. In this case:

δne(ρ, t) = eIt(ρ, t)/It(ρ). (9)

For large-scale density ﬂuctuations δne, for which the radial correlation length is large and comparable to the plasma radius, the local data on oscillations in the sample volume can be contaminated by attenuation effects accumulating along the entire trajectory. For the experimental data, considered in this paper the large-scale modes are not typical, so the density oscillations are measured locally [34]. The oscillating component of the toroidal shift of the beam in the detector ζd, is described by the following expression [35]:

δψ Z dl 1 Z dl 2 Z dl Pζinj Z dl δζ = SV − δψ − δψ + δψ + ζ . (10) d m R2 m R2 m R2 m R2 inj L2 L1 L2 L1

Here, ψ = Aϕ/R, Pζ inj is the toroidal magnetic moment at the injection point, known constant. Note that the integrals in (10) are taken along the total paths L1 and L2 from the injector to the position of the sample volume (SV), and from SV to the detector. Equation (10) shows that the local term δψSV input the main contribution to the oscillations of beam shift. The contribution of integral terms is much smaller for oscillations with shorter correlation length (or radial wavelength) due to the path integration. It was shown in Reference [36] that the integral terms do not strongly affect the local terms from MHD oscillations, therefore HIBP provides practically local measurements of magnetic fluctuations. However, a nonlocal contribution to ζd from oscillations with a large correlation length, comparable to the length of the beam trajectory in plasma, is also possible. In conclusion, we note that, despite the path integral effect, HIBP provides fairly local measurements of density and magnetic field fluctuations, taking into account the conditions specified above, as well as purely local measurements of electric potential fluctuations. All local measurements are averaged over the sample volume (SV), which size determines the spatial resolution of the HIBP diagnostic. The sample volume typically looks like the inclined elliptical disk with diameter equal to the diameter of the beam, and its thickness corresponds to the entrance slit of the analyzer. The typical radial dimension of SV ranges from 0.1 to 1 cm.

5. Mathematical Problems of Experimental Data Processing for Fluctuations of Plasma Parameters 5.1. Spectral Fourier Analysis of Oscillations To analyze the oscillating component of an arbitrary signal x(t), we introduce the Fourier transform as Sx(f ) and Sx*(f ) as its complex conjugate. Thus the power spectral density (PSD) of oscillations is:

∗ Sxx( f ) = Sx( f ) · Sx( f ). (11)

The cross-spectral oscillation power density (CSPD) for signals x(t) and y(t) is the following product of their Fourier transforms:

∗ Sxy( f ) = Sx( f ) · Sy( f ). (12)

An example of calculating the PSD of oscillations of plasma parameters determined in a certain time domain is shown in Figure6a. As a rule, when digitizing an experimental signal with a frequency of 1 µs, the PSD is calculated over a period of 0.25–1 ms, depending on the characteristic time of the change in the frequency of the processes under study. Symmetry 2021, 13, x FOR PEER REVIEW 10 of 49

The cross-spectral oscillation power density (CSPD) for signals x(t) and y(t) is the following product of their Fourier transforms: * Sxy()()() f S x f  S y f . (12) An example of calculating the PSD of oscillations of plasma parameters determined in a certain time domain is shown in Figure 6a. As a rule, when digitizing an experimental Symmetry 2021, 13, 1367 10 of 46 signal with a frequency of 1 μs, the PSD is calculated over a period of 0.25–1 ms, depending on the characteristic time of the change in the frequency of the processes under study.

(a) (b)

FigureFigure 6. 6.(a ()a PSD) PSD of of thethe potential and and plasma plasma density density fluctuations fluctuations measured measured with witha heavy a heavy ion beam, ion beam,and the and PSD the of mag- PSD of netic oscillations measured with a magnetic probe in the typical T-10 ohmic discharge [15]. (b) Spectrogram of plasma magnetic oscillations measured with a magnetic probe in the typical T-10 ohmic discharge [15]. (b) Spectrogram of plasma potential oscillations measured in the T-10 discharge with additional ECR heating from 610 to 800 ms. Reproduced cour- potentialtesy of oscillationsIAEA. Copyright measured 2017 inIAEA. the T-10 discharge with additional ECR heating from 610 to 800 ms. Reproduced courtesy of IAEA. Copyright 2017 IAEA. The power spectra of characteristic peaks of monochromatic fluctuations corresponding Theto the power Geodesic spectra Acoustic of characteristic Mode (GAM) peaks [37,38 of monochromatic] and magnetohydrodynamic fluctuations correspond- (MHD) ingtearing to the mode Geodesic with a Acousticpoloidal mode Mode number (GAM) m [=37 2, are38] seen. and magnetohydrodynamicIn contrast to the solitary peak (MHD) tearingof the m mode = 2 mode, with the a poloidal frequency mode GAM number peak is accompaniedm = 2 are seen. by a Inhigh contrast-frequency to the satellite solitary peakpeak. of the m = 2 mode, the frequency GAM peak is accompanied by a high-frequency satelliteFigure peak. 7 presents the physical picture of the zonal flows as symmetric torsional plasmaFigure oscillations7 presents [37] the. While physical zonal picture flow ofby the itself, zonal also flows named as symmetricresidual zonal torsional flow, plasmatypi- oscillationscally has low [37-to].-near While zero zonal frequency, flow by GAM itself, as alsoa higher named frequency residual branch zonal of flow,zonal typicallyflows hashas low-to-near a typical frequency zero frequency, of around GAM20 kHz. as Both a higher GAM frequency and tearing branch modes ofpresent zonal examples flows has a typicalof symmetric frequency structure of around in the 20plasma kHz. toroidal Both GAM magnetic and tearing configuration. modes GAM present is poloidally examples of symmetricand toroidally structure symmetric in the torsional plasma plasma toroidal oscillation magnetic with configuration. n = m = 0, where GAM n is is the poloidally toroi- anddal toroidallymode number symmetric [39]. MHD torsional tearing plasmamode is a oscillation chain of magnetic with n =islands,m = 0, that where haven heli-is the toroidal mode number [39]. MHD tearing mode is a chain of magnetic islands, that have Symmetry 2021, 13, x FOR PEER REVIEWcally symmetric structure, located on magnetic surface with rational value of safety11 factorof 49 helicallyq = m/n in symmetric the toroidal structure, magnetic located configuration on magnetic [40]. surface with rational value of safety factor q = m/n in the toroidal magnetic configuration [40].

Figure 7. 7. ConceptualConceptual scheme scheme of a ofzonal a zonal flow, flow,a radially a radially localized, localized, poloidally poloidally and azimut andhal azimuthallyly sym- symmetricmetric (m = (0,m n= = 0, 0)n electrostatic= 0) electrostatic potential potential structure, structure, along with along its with resulting its resulting radial electric radial electricfield Er, field and turbulence stabilizing poloidal E × B flow. Er, and turbulence stabilizing poloidal E × B flow. This spectrogram, describing a process lasting 500 ms, consists of a sequence of elementary spectra, each of which is calculated in 1 ms. The potential spectrogram is dominated by the GAM peak, accompanied by a satellite peak [41]. Application of additional ECR heating in the time interval from 600 to 800 ms leads to an increase in the frequency of both the GAM and the satellite due to growth of the electron temperature [42].

5.2. Fourier Analysis of Coherency and Cross-Phase of Oscillations Time evolution of the correlations between signals x(t) and y(t) are determined by spectrograms of their coherency Cohxy and cross-phase θxy, as following:

S xy xy Cohxy (,) f t  1/2 , 0 < Coh < 1, (15a) SSxx yy

1 Im(Sxy )  xy (f , t ) tan   , −π < θxy < π (15b) Re(Sxy )  Coherence and cross-phase are modulus and argument of the complex function CPSD. Figure 8 presents an example of calculating the coherence spectrogram. Values of statistically significant coherence indicate the Alfvén eigenmodes (AEs), which is another type of MHD mode. AEs are electromagnetic oscillations, propagating along the magnetic field line with Alfvén speed and possessing helically symmetric structure with finite m and n. In the vicinity of the magnetic flux surface with rational q value, equals to m/n for a specific mode, the dispersion relation for AE contains also a GAM component, causing GAM frequency to be a lower limit for AE frequency [43,44]. It presents a beautiful link between helically symmetric structure AE and toroidally symmetric structure GAM. Fig- ure 7 shows that the frequency of AE decreases in time, caused by the density increase in 1/2 accordance with Alfvén law fAE~ B t n e . The circles indicate different branches of the Alfvén eigenmodes with different poloidal mode numbers m = 4 and m = 3, calculated from the data of a set of magnetic probes.

Symmetry 2021, 13, 1367 11 of 46

This spectrogram, describing a process lasting 500 ms, consists of a sequence of elementary spectra, each of which is calculated in 1 ms. The potential spectrogram is dominated by the GAM peak, accompanied by a satellite peak [41]. Application of additional ECR heating in the time interval from 600 to 800 ms leads to an increase in the frequency of both the GAM and the satellite due to growth of the electron temperature [42].

Sxy Coh ( f , t) = , 0 < Coh < 1, (13a) xy 1/2 xy SxxSyy −1 Im(Sxy) θxy( f , t) = tan , −π < θxy < π (13b) Re(Sxy) Coherence and cross-phase are modulus and argument of the complex function CPSD. Figure8 presents an example of calculating the coherence spectrogram. Values of statistically significant coherence indicate the Alfvén eigenmodes (AEs), which is another type of MHD mode. AEs are electromagnetic oscillations, propagating along the magnetic field line with Alfvén speed and possessing helically symmetric structure with finite m and n. In the vicinity of the magnetic flux surface with rational q value, equals to m/n for a specific mode, the dispersion relation for AE contains also a GAM component, causing GAM frequency to be a lower limit for AE frequency [43,44]. It presents a beautiful link between helically symmetric structure AE and toroidally symmetric structure GAM. Figure7 shows that the frequency of AE decreases in time, caused by the density increase f ∼ B n−1/2 Symmetry 2021, 13, x FOR PEER REVIEWin accordance with Alfvén law AE t e . The circles indicate different branches12 of of49 the Alfvén eigenmodes with different poloidal mode numbers m = 4 and m = 3, calculated from the data of a set of magnetic probes.

FigureFigure 8. Spectrogram of the coherency between the total HIBP current (plasma density) oscillations, measuredmeasured in the plasma plasma core core at at r/ra/ =a 0.16= 0.16 and and a MP a MP signal signal [45]. [ 45Reproduced]. Reproduced courtesy courtesy of IAEA. of IAEA. Cop- yright 2010 IAEA. Copyright 2010 IAEA.

FigureFigure9 9shows shows an an example example of of calculating calculating the the cross-phase cross-phase spectrogram. spectrogram. WeWe see see that that thethe magnitudemagnitude of the cross cross-phase-phase does does not not change, change, remaining remaining equal equal to to −π−/2,π/2, although although the themode mode frequency frequency decreases decreases twice twice with with time, time, due dueto an to increase an increase in the in thedensity density in this in thisdis- discharge.charge. This This constancy constancy of the of cross the cross-phase-phase values values between between the signals the signals of the plasma of the plasmadensity densityand the andpoloidal the poloidal magnetic magnetic field indicates ﬁeld indicates that the that cross the-phase cross-phase is a stable is a stableindividual individual char- characteristicacteristic for each for each Alfvén Alfv eigenmode.én eigenmode.

Figure 9. Spectrogram of the cross-phase between signals It (~ne) and poloidal magnetic field Bp (~ζd) measured in the same SV in discharge with AEs. Insets show the histograms of the cross-phase computed over the areas marked by rectangles; the insets show the histograms for the cross-phases [45]. Reproduced courtesy of IAEA. Copyright 2010 IAEA.

5.3. Bispectral Fourier Analysis of Plasma Oscillations Bispectral analysis is widely used to study nonlinear three-wave interaction processes. The quadratic coefficient of cross-bicoherence for three independent quantities x(t), y(t) and z(t) is defined as follows: 2 * Sx()()() f1 S y f 2  S z f 1  f 2 b2 (,) f f  , 0 < b2 < 1, x,y,z 1 2 2 2 (16) Sx()()() f1 S y f 2  S z f 1  f 2

Symmetry 2021, 13, x FOR PEER REVIEW 12 of 49

Figure 8. Spectrogram of the coherency between the total HIBP current (plasma density) oscillations, measured in the plasma core at r/a = 0.16 and a MP signal [45]. Reproduced courtesy of IAEA. Cop- yright 2010 IAEA.

Figure 9 shows an example of calculating the cross-phase spectrogram. We see that the magnitude of the cross-phase does not change, remaining equal to −π/2, although the mode frequency decreases twice with time, due to an increase in the density in this discharge. This constancy of the cross-phase values between the signals of the plasma density Symmetry 2021, 13, 1367 12 of 46 and the poloidal magnetic field indicates that the cross-phase is a stable individual characteristic for each Alfvén eigenmode.

FigureFigure 9. 9. SpectrogramSpectrogram of ofthe the cross cross-phase-phase between between signals signals It (~Int e(~) andne) andpoloidal poloidal magnet magneticic field B ﬁeldp (~ζBd)p measured(~ζd) measured in the insame the SV same in discharge SV in discharge with AEs. with Insets AEs. show Insets the show histograms the histograms of the cross of the-phase cross- computedphase computed over the over areas the marked areas marked by rectangles; by rectangles; the insets the show insets the show histograms the histograms for the cross for the-phases cross- [45].phases Reproduced [45]. Reproduced courtesy courtesy of IAEA. of Copyright IAEA. Copyright 2010 IAEA. 2010 IAEA.

5.3.5.3. Bispectral Bispectral Fourier Fourier Analysis Analysis of of Plasma Plasma Oscillations Oscillations BispectralBispectral analysis is is widely widely used used to to study study nonlinear nonlinear three-wave three-wave interaction interaction processes. processes.The quadratic The quadratic coefﬁcient coefficient of cross-bicoherence of cross-bicoherence for three for independent three independent quantities quantitiesx(t), y(t) x and(t), yz((tt)) and is deﬁned z(t) is defined as follows: as follows: 2 ∗* 2 Sx(Sf x)()()()· f1Sy( Sf y) f 2· S z S( zf f 1+  f f 2) 2 2 1 2 1 2 2 2 b ( f1b, x,y,zf2)(,) = f1 f 2  , 0, 0< b< < b1, < 1, (14) x,y,z D 2E2 D 2 2E (16) Sx( fS1)x()()()· fS1y( Sf2 y) f 2· |S Sz z( f f1 1+  ff 22)|

where brackets denote the time averaging. Similar to the cross-phase, biphase is deﬁned as an argument of the complex function:

∗ −1 Im Sx( f1) · Sy( f2) · Sz( f1 + f2) θx,y,z( f1, f2) = tan ∗ , −π < θ < π. (15) Re Sx( f1) · Sy( f2) · Sz( f1 + f2)

Auto-bicoherence for random variable x(t) is deﬁned as:

Statistically significant bicoherence b points to three-wave interaction between oscillations at frequencies f 1, f 2 and f 3 = f 1 + f 2. An example of calculating bicoherence and biphase is shown in Figure 10. The excess of the bicoherence coefficient over the background at the GAM frequency indicates a significant three-wave interaction between the GAM and the broadband turbulence of potential [46]. Note that, at the GAM frequencies the biphase has finite values, while outside them it has a stochastic character. Thus, analysis of HIBP data shows that GAM in T-10 linked with broadband turbulence [47,48], and is not excited by other mechanisms, such as, e.g., the effect of energetic particles. Symmetry 2021, 13, x FOR PEER REVIEW 13 of 49

where brackets denote the time averaging. Similar to the cross-phase, biphase is defined as an argument of the complex function: ImS ( f ) S ( f )  S ( f  f )*  (f , f ) tan 1 x1 y 2 z 1 2 , −π < θ < π. x,y,z 1 2 * (17) ReSx ( f1 ) S y ( f 2 )  S z ( f 1  f 2 ) Auto-bicoherence for random variable x(t) is defined as: 2 S()()() f S f  S* f  f 2 x1 x 2 x 1 2 b(,) f f  , 0 < bx2 < 1. x 1 2 2 2 (18) Sx()()() f1 S x f 2  S x f 1  f 2

Statistically significant bicoherence b points to three-wave interaction between oscillations at frequencies f1, f2 and f3 = f1 + f2. An example of calculating bicoherence and biphase is shown in Figure 10. The excess of the bicoherence coefficient over the background at the GAM frequency indicates a significant three-wave interaction between the GAM and the broadband turbulence of potential [46]. Note that, at the GAM frequencies the biphase has finite values, while outside them it has a stochastic character. Thus, analysis Symmetry 2021, 13, 1367 of HIBP data shows that GAM in T-10 linked with broadband turbulence [47,48], and13 of 46is not excited by other mechanisms, such as, e.g., the effect of energetic particles.

(a) (b)

Figure 10. AutoAuto-bicoherency-bicoherency ( aa)) and and biphase ( (bb)) for for the the plasma plasma potential potential in in the the ohmic ohmic discharge discharge of of the the T T-10-10 tokamak tokamak [46] [46].. Reproduced courtesy of IAEA. Copyright 2017 IAEA.

6.6. Measurements Measurements by by Several Several Spatial Spatial Channels Channels 6.1.6.1. Measurement Measurement of of Turbulent Turbulent Particle Flux on the TJ-IITJ-II Stellarator AA funda fundamentallymentally new new element element in in the the HIBP operation is the use of a multichannel energyenergy analyzer, analyzer, which allows one to measure both the potential and density in several samplesample volumes. volumes. Figure Figure 11 11 shows shows a cross a cross-sectional-sectional view view of a of multichannel a multichannel analyzer analyzer in- stalledinstalled in inT- T-10.10. On On the the TJ TJ-II-II stellarator, stellarator, where where two two diagnostic diagnostic systems systems HIBP1 HIBP1 and and HIBP2 areare currently currently operating operating (see (see Section Section 7),7), on on the the second second HIBP2 HIBP2 system system,, the the same same analyzer analyzer is installed,is installed, somewhat somewhat reduced reduced in size. in size. Let Letus consider us consider the operation the operation of a multichannel of a multichannel ana- Symmetry 2021, 13, x FOR PEER REVIEW 14 of 49 lyzanalyzerer using using the simplest the simplest example example of a two of a-channel two-channel analyzer analyzer with two with slots two and slots two and detec- two tors.detectors. Such an Such analyzer an analyzer is installed is installed on the onHIBP1 the HIBP1 complex complex of the TJ of- theII stellarator. TJ-II stellarator. The detec- The tordetector line for line the for TJ the-II two TJ-II-channel two-channel analyzer analyzer is shown is shown in Figure in Figure 12. 12.

FigureFigure 11. 11. MultiMulti-slit-slit energy energy analyzer analyzer for for direct direct measurements measurements of of particle particle flux: ﬂux: 5 5--SS——ﬁvefive entrance entrance slits;slits; D D—detector;—detector; G G—grid;—grid; GP GP—ground—ground plate; plate; HVP HVP—high-voltage—high-voltage plate; plate; WW——adjustmentadjustment window. window.

The probing parameters are adjusted in such a way that both sample volumes lie on the same magnetic surface, and they are shifted poloidally by ∆x ~1 cm, as shown in Figure 13. This method allows us to directly measure the poloidal component of the electric field Ep by the potential difference, Ep = (φ1 − φ2)/∆x. The finite distance between the investigated volumes limits the poloidal wave vector of oscillations kp = kθ < 3 cm−1. Direct measurement of Ep allows us to define the radial velocity of E  B drift Vr = Ep/Bt. As oscillations of the total particle current or plasma density are measured simultaneously with the potential oscillations, we may directly calculate the radial turbulent particle flux:   r()t nV e r  1/Bt n e ( t ) E p ( t )   E B , (19)

Such use of HIBP on stellarators was first applied on the TJ-II. To find ΓEB ()t , the  density fluctuations ne must be measured simultaneously with E p in the same spatial region. To analyze the frequency spectrum of the flux and its temporal dynamics, it is  sufficient to measure the relative density fluctuations ne()()/ t  I t t I t . In the low-den-

sity case, to estimate the absolute values of the flux EB ()t we may use the measured  value It () t as ne .

Symmetry 2021, 13, x FOR PEER REVIEW 14 of 49

Symmetry 2021, 13, 1367 14 of 46 Figure 11. Multi-slit energy analyzer for direct measurements of particle flux: 5-S—five entrance slits; D—detector; G—grid; GP—ground plate; HVP—high-voltage plate; W—adjustment window.

FigureFigure 12. 12. DetectorDetector line line for for the the two two-slit-slit energy energy analyzer analyzer [15] [15. ].The The upper upper slit slit is ismarked marked in in blue, blue, the the lowerlower one one—in—in red red.. Reproduced Reproduced courtesy courtesy of ofIAEA. IAEA. Copyright Copyright 2017 2017 IAEA. IAEA. As oscillations of the total particle current or plasma density are measured simul- The probing parameters are adjusted in such a way that both sample volumes lie on taneously with the potential oscillations, we may directly calculate the radial turbulent the same magnetic surface, and they are shifted poloidally by ∆x ~1 cm, as shown in Figure particle flux: 13. This method allows us to directly measure the poloidal component of the electric field Γr(t) = neVer =1/Btne(t)Eep(t) = ΓE×B, (17) Ep by the potential difference, Ep = (φe1 − φ2)/∆x. Thee finite distance between the investigated −1 volumesSuch limits use the of poloidal HIBP on wave stellarators vector o wasf oscillations first applied kp = onkθ < the 3 cm TJ-II.. Direct To find measurementΓE×B(t), the ofdensity Ep allows fluctuations us to definene ethemust radial be velocity measured of E simultaneously  B drift Vr = Ep with/Bt. Eep in the same spatial region.As oscillations To analyze of the the frequencytotal particle spectrum current of or the plasma flux anddensity its temporalare measured dynamics, simulta- it is Symmetry 2021, 13, x FOR PEER REVIEWneouslysufficient with to the measure potential the relativeoscillations, density we may fluctuations directlyδ calculatene(t) = eI tthe(t) /radialIt. In turbulent the low-density15 par-of 49

ticlecase, flux: to estimate the absolute values of the ﬂux ΓE×B(t) we may use the measured value eIt(t) as nee.   r()t nV e r  1/Bt n e ( t ) E p ( t )   E B , (19)

sity case, to estimate the absolute values of the flux EB ()t we may use the measured  value It () t as ne .

Figure 13. Scheme of the poloidal electric field measurements in the plasma using the double-slit Figure 13. Scheme of the poloidal electric field measurements in the plasma using the double-slit analyzer. The potential ϕ and ϕ are measured by the first and second slit correspondingly; ∆x is analyzer. The potential φ1 1and φ2 2are measured by the first and second slit correspondingly; ∆x is distancedistance between between the the points points of of measure measurements;ments; EEpp isis poloidal poloidal electric electric field; field; BBt tisis toroidal toroidal magnetic magnetic field,field, VVr risis the the velocity velocity of of radial radial drift drift in in crossed crossed EEp p× ×Bt Bfields.t fields.

InIn the the high high-density-density case, case, the the beam beam attenuation attenuation should should be be accountedaccounted.. The The density density should be normalized as nee =eItot/Itot · ne, where the oscillating density componenteIt/It should be normalized as ne I tot/ I tot  n e , where the oscillating density component IIt/ t is measured with HIBP, and the normalized factor ne is measured by another diagnostic, ise.g., measured by the interferometrywith HIBP, and or the by normalized Thomson scattering.factor ne is measured by another diagnostic, e.g., byFourier the interferometry transform for or the by time-dependent Thomson scattering flux Γ.E ×B(t), or the spectral function of flux is calculatedFourier transform as follows for [49 ]:the time-dependent flux ΓEB ()t , or the spectral function of flux is calculated as follows [49]: ΓE×B( f ) = −2/Bt · Re(SnE) (18) E譈 (f )   2/ Bt  Re( S nE ) (20) Here SnE is the Fourier cross-spectrum between density and poloidal electric field Ep oscillations.Here SnE is the Fourier cross-spectrum between density and poloidal electric field Ep oscillations. Figure 14 shows an example of the calculated turbulent particle flux in discharge with NBI heating. The flux, linked with broadband turbulence, demonstrates the intermittent nature. It consists of a stochastic sequence of controversially directed bursts, which are mostly directed outwards. The spectrograms show Alfvén eigenmodes in the form of a family of quasi-monochromatic oscillations with a frequency decreasing in time. We see that AEs may contribute both to outward and inward flux, or also could produce no flux, depending on the phase relations between Ep and the density oscillations [50]. Usually, AE AEs contribute to the turbulent particle flux ΓEB , making up an appreciable fraction of the AE total flux ΓEB . Figure 14 shows that ΓEB can be much larger than the broadband turbu- BB lent flux ΓEB in the same frequency range [51]. It was shown that the value of the turbulent flux is comparable to the value of the total particle flux; therefore, ГE×B plays an essential role in the particle balance. It was shown that the transition to the improved confinement mode is characterized by suppression of broadband fluctuations of the plasma potential and density, as well as by a decrease ГE×B both at the plasma edge and in the hot core [52].

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Figure 14 shows an example of the calculated turbulent particle flux in discharge with NBI heating. The flux, linked with broadband turbulence, demonstrates the intermittent nature. It consists of a stochastic sequence of controversially directed bursts, which are mostly directed outwards. The spectrograms show Alfvén eigenmodes in the form of a family of quasi-monochromatic oscillations with a frequency decreasing in time. We see that AEs may contribute both to outward and inward flux, or also could produce no flux, depending on the phase relations between Ep and the density oscillations [50]. Usually, AE AEs contribute to the turbulent particle flux ΓE×B, making up an appreciable fraction of AE the total flux ΓE×B. Figure 14 shows that ΓE×B can be much larger than the broadband BB turbulent flux ΓE×B in the same frequency range [51]. It was shown that the value of the turbulent flux is comparable to the value of the total particle flux; therefore, ΓE×B plays an essential role in the particle balance. It was shown that the transition to the improved confinement mode is characterized by suppression of broadband fluctuations of the plasma Symmetry 2021, 13, x FOR PEER REVIEW 16 of 49 potential and density, as well as by a decrease ΓE×B both at the plasma edge and in the hot core [52].

Figure 14.14. TheThe timetime evolutionevolution of of the the NBI-sustained NBI-sustained discharge discharge with with AEs AEs in TJ-II. in TJ (a-)II. Frequency (a) Frequency re- −−11 resolvedsolved particle particle flux flux ГΓE×BE× B(in(in a. a. u.) u.) measured measured by by HIBP atat ρρ= =− −0.5, kkθθ < 2 cm ;; red red color meansmeans positive positiveoutward outward flux, and flux, blue and means blue meansnegative negative inward inward flux. flux.(b) Cross (b) Cross-phase-phase θEp-neθ Ep-nebetweenbetween Ep andEp ne oscil- andlations.ne oscillations. Only the points Only the with points high with Coh highne- Ep Coh> 0.3ne- are Ep > shown. 0.3 are shown.The color The bar color is in bar radians. is in radians. Three chosen Threebranches chosen of the branches AE are of marked the AE areby markedcolor ovals by color. (c) The ovals. histograms (c) The histograms of the cross of the-phase cross-phase for each branch foris marked each branch with is corresponding marked with corresponding colors, indicating colors, the indicating flux direction; the flux left direction; box (red): left boxθEp-ne (red): = −3/4π, cor- θrespondsEp-ne = −3/4 to πthe, corresponds outward flux to the; central outward box flux; (green): central θ boxEp-ne (green): = −π/2θ, Ep-necorresponds= −π/2, correspondsto zero flux; to right box zero(blue): flux; θEp right-ne ~0 box corresponds (blue): θEp-ne to~0 the corresponds inward flux to the. (d inward) PSD flux.of the (d turbulent) PSD of the particle turbulent flux, particle taken at some flux,typical taken time at instant some typical, averaged time instant,over 1 ms. averaged Three overfrequency 1 ms. Threepeaks frequency related to peaks the AE related branches to the identified AEabove branches are marked identified with above corresponding are marked withcolors. corresponding Adapted from colors. [51] Adapted Copyright from 201 [512] IAEA. Copyright 2012 IAEA. 6.2. Measurement of Turbulent Particle Flux in the T-10 Tokamak In contrast to the TJ-II stellarator, the geometry of the T-10 machine gives more uncertainties in determining the flux. The scheme of the experiment on measuring the flux with the 5-slit analyzer is shown in Figure 11. An example of calculating trajectories for the standard discharge: Bt = 2.4 T, and energy Eb = 220 keV is shown in Figure 15. We can see that the potential difference φk − φj corresponds to the electric field E, directed along the segment kj. To measure the poloidal field, it is necessary to align the segment in the poloidal direction. This is achieved in the point of deepest penetration for the detector line of equal energy. [53]. Thus, we can measure the flux for each energy in one spatial point.

Symmetry 2021, 13, 1367 16 of 46

6.2. Measurement of Turbulent Particle Flux in the T-10 Tokamak In contrast to the TJ-II stellarator, the geometry of the T-10 machine gives more uncertainties in determining the flux. The scheme of the experiment on measuring the flux with the 5-slit analyzer is shown in Figure 11. An example of calculating trajectories for the standard discharge: Bt = 2.4 T, and energy Eb = 220 keV is shown in Figure 15. We can see that the potential difference ϕk − ϕj corresponds to the electric field E, directed along Symmetry 2021, 13, x FOR PEER REVIEWthe segment k . To measure the poloidal field, it is necessary to align the segment in the 17 of 49 j poloidal direction. This is achieved in the point of deepest penetration for the detector line of equal energy. [53]. Thus, we can measure the flux for each energy in one spatial point.

FigureFigure 15. SchemeScheme for for measuring measuring the the electric electric field field in plasma in plasma using using the 5-slit the analyzer. 5-slit analyzer. The potential The potential ϕ S k E φkk isis measured measured in in the the point point Sk byby slit slit with with number k = 1 1–5;–5; Ep isis the the poloidal poloidal component of of the the electric V E × B E fieldelectric required field required to calculate to calculate Vr; the rradial; the radial Ep × Bpt driftt driftvelocity velocity in the in thecrossed crossed fields, fields, and and Er risis its radial component.its radial component. Figure 16 presents an example of the turbulent flux measurements in the standard T-10Figure discharge 16 withpresents ohmic an and example ECR heating. of the turbulent Only two flux frequency measurements ranges with in non-zerothe standard T- 10flux discharge are seen inwith the ohmic spectral and flux ECR function: heating. the quasi-coherent Only two frequency (QC) mode ranges and thewith geodesic non-zero flux areacoustic seen mode in the (GAM). spectral In the flux remaining function: frequency the quasi ranges,-coherent the flux (QC) is stochastic. mode and The the QC geodesic acousticmode dominates mode (GAM). during theIn the entire remaining discharge, frequency allowing particles ranges, to the flow flux outward. is stochastic. The flux The QC GAM modein the GAMdominates frequency during range theΓ Eentire×B is bydischarge, the factor allowing of 5–6 less particles than the fluxto flow of the outward. QC mode The flux ΓQC GAM inE the×B [GAM54]. frequency range ΓEB is by the factor of 5–6 less than the flux of the QC mode QC ΓEB [54].

Figure 16. Time evolution of the T-10 discharge with additional ECR heating. (a) Flux spectral function ГE×B (rel. un.) measured at ρ = 0.78, kθ < 2; the red color corresponds to the outward flux, positive, blue—to the inward flux, negative. The time-frequency regions of the quasi-coherent mode (QC) and geodesic acoustic mode (GAM) are marked; (b) the total particle fluxes averaged over GAM QC GAM ( ΓEB ) and QC ( ΓEB ) frequency ranges, and the line-average density ne [54]. Reproduced with permission IoP.

Symmetry 2021, 13, x FOR PEER REVIEW 17 of 49

Figure 15. Scheme for measuring the electric field in plasma using the 5-slit analyzer. The potential φk is measured in the point Sk by slit with number k = 1–5; Ep is the poloidal component of the electric field required to calculate Vr; the radial Ep × Bt drift velocity in the crossed fields, and Er is its radial component.

Figure 16 presents an example of the turbulent flux measurements in the standard T- 10 discharge with ohmic and ECR heating. Only two frequency ranges with non-zero flux are seen in the spectral flux function: the quasi-coherent (QC) mode and the geodesic acoustic mode (GAM). In the remaining frequency ranges, the flux is stochastic. The QC mode dominates during the entire discharge, allowing particles to flow outward. The flux Symmetry 2021, 13, 1367 GAM 17 of 46 in the GAM frequency range ΓEB is by the factor of 5–6 less than the flux of the QC mode QC ΓEB [54].

Figure 16. 16. TimeTime evolution evolution of ofthe the T-10 T-10 discharge discharge with with additional additional ECR ECR heating. heating. (a) Flux (a) spectral Flux spectral func- tionfunction ГE×BΓ (rel.E×B un.)(rel. measured un.) measured at ρ = at 0.78,ρ = k 0.78,θ < 2;k θthe< 2;red the color red corresponds color corresponds to the tooutward the outward flux, posi- flux, tive,positive, blue blue—to—to the theinward inward flux, flux, negative. negative. The The time time-frequency-frequency regions regions of of the the quasi quasi-coherent-coherent mode (QC) andand geodesicgeodesic acoustic acoustic mode mode (GAM) (GAM) are are marked; marked; (b) the(b) totalthe total particle particle fluxes fluxes averaged averaged over GAM over GAM QC QC GAMGAM ( Γ ) and QC ( Γ ) frequency ranges, and the line-average density ne [54]. Reproduced (ΓE×B ) andEB QC (ΓE×B ) frequencyEB ranges, and the line-average density ne [54]. Reproduced with withpermission permission IoP. IoP.

Another example of the flux calculation for GAM is shown in Figure17. Here we consider the ohmic T-10 discharge with slow density ramp-up. The density reaches the critical value at t = 600 ms, then the MHD instability develops, which leads to the disruption at t = 750 ms. Figure 17a shows that GAM dominates on the spectrogram of potential fluctuations, but its amplitude decreases with density increase. The spectrogram of fluctuations in the potential difference (Figure17b) completely contrasts with Figure 17a, as in Figure 17b GAM does not dominate, but it is completely invisible due to the background noise. This observation agrees with the well-known fact that GAM on the potential has poloidal symmetry. Additionally, GAM was not observed on the spectrogram of ﬂux [55]. An important conclusion follows from the above experiments: the radial turbulent particle ﬂux at GAM frequencies is on the noise level or only slightly exceeds it due to the poloidal symmetry of potential oscillations at GAM frequencies. Symmetry 2021, 13, x FOR PEER REVIEW 18 of 49

Another example of the flux calculation for GAM is shown in Figure 17. Here we consider the ohmic T-10 discharge with slow density ramp-up. The density reaches the critical value at t = 600 ms, then the MHD instability develops, which leads to the disruption at t = 750 ms. Figure 17a shows that GAM dominates on the spectrogram of potential fluctuations, but its amplitude decreases with density increase. The spectrogram of fluctuations in the potential difference (Figure 17b) completely contrasts with Figure 17a, as in Figure 17b GAM does not dominate, but it is completely invisible due to the background noise. This observation agrees with the well-known fact that GAM on the potential has poloidal symmetry. Additionally, GAM was not observed on the spectrogram of flux [55]. Symmetry 2021, 13, 1367 An important conclusion follows from the above experiments: the radial turbulent18 of 46 particle flux at GAM frequencies is on the noise level or only slightly exceeds it due to the poloidal symmetry of potential oscillations at GAM frequencies.

FigureFigure 17. 17. TimeTime evolution evolution of ofOhmic Ohmic T- T-1010 discharge discharge with with slow slow density density ramp ramp-up.-up. (a ()a Power) Power spectral spectral density (PSD) of potential oscillations measured by the central slit φ3, SV is located at r/a = 0.57; (b) density (PSD) of potential oscillations measured by the central slit ϕ3, SV is located at r/a = 0.57; PSD for difference of potentials between the central and edge slits φ1 − φ3 = Ep Δx; (c) the frequency (b) PSD for difference of potentials between the central and edge slits ϕ1 − ϕ3 = Ep ∆x;(c) the resolved spectral function of flux ГE×B. The frequency domain of GAM is marked by rectangle [54]. frequency resolved spectral function of ﬂux Γ × . The frequency domain of GAM is marked by Reproduced with permission IoP. Copyright IoP E B rectangle [54]. Reproduced with permission IoP. Copyright IoP.

6.3.6.3. Measurement Measurement of ofthe the GAM GAM Poloidal Poloidal Symmetry Symmetry in in the the T- T-1010 Tokamak Tokamak InIn the the previous previous Subsection, Subsection, we we considered considered correlation correlation measurements measurements of of potential, potential, whichwhich are are physically physically corrected corrected as as the the potential potential measurements measurements are are completely completely local. local. In In the the lowlow-density-density case, case, when when the the attenuation attenuation factors factors in in the the analysis analysis of of the the variable variable component component ofof the the density density may may be be neglected, neglected, the the correlations correlations of of the the density density fluctuations fluctuations may may be be also also considered.considered. The The issue issue of of the the locality locality of of density density measurements measurements is isdiscussed discussed in in detail detail in in the the Reference [34]. Finally, as shown in Section4, the density oscillations measurements are localized in the sample volume, so the density coherence is free from the non-local effects. An example of a two-point measurement technique [56,57] is shown in Figure 18. The figure shows that the sample volumes are separated in the poloidal direction by 1.67–1.74 cm, but also have some radial shift of 0.81 cm. The measurement results are shown in Figure 19. Time traces of the main discharge parameters are shown in Figure 19a. Figure 19b,c shows the low-frequency part of the power spectrograms of the potential oscillations measured using the center ϕ3 (Figure 19b) and edge ϕ5 (Figure 19c) slits. Figure 19d shows that the coherence coefficient between ϕ3 and ϕ5 in the GAM frequency domain reaches unity. Figure 19e shows that the phase shift between these poloidally shifted signals is zero, indicating that GAM is poloidal symmetric, i.e., m = 0. Symmetry 2021, 13, x FOR PEER REVIEW 19 of 49

Reference [34]. Finally, as shown in Section 4, the density oscillations measurements are localized in the sample volume, so the density coherence is free from the non-local effects. An example of a two-point measurement technique [56,57] is shown in Figure 18. The Symmetry 2021, 13, 1367 figure shows that the sample volumes are separated in the poloidal direction by 1.67191.74 of 46 cm, but also have some radial shift of 0.81 cm.

Symmetry 2021, 13, x FOR PEER REVIEW 20 of 49 FigureFigure 18. 18. TwoTwo-point-point technique technique of of correlation correlation measurements measurements of of the the plasma plasma electric electric potential potential using using thethe 5 5-slit-slit analyzer. analyzer.

The measurement results are shown in Figure 19. Time traces of the main discharge parameters are shown in Figure 19a. Figure 19b,c shows the low-frequency part of the power spectrograms of the potential oscillations measured using the center φ3 (Figure 19b) and edge φ5 (Figure 19c) slits. Figure 19d shows that the coherence coefficient between φ3 and φ5 in the GAM frequency domain reaches unity. Figure 19e shows that the phase shift between these poloidally shifted signals is zero, indicating that GAM is poloidal symmetric, i.e., m = 0.

FiguFigurere 19. Correlational measurements of of electric potential with with 5-slits analyzer. Traces of plasma currentcurrent IIpp andand line-averaged line-averaged density density nnee inin dischargedischarge withwith BBtt = 2 2.3.3 T ( a); power spectral densities of potential oscillations measured in two most distant SVs by upper slit ϕφ11 (b); and by lower slit φϕ5 ((cc);); theirtheir coherency coherency ( d); and cross-phase ((ee)) atat rr = 21 cm. GAM GAM dominates in all PSDs. Adapted from [58] [58] CopyrightCopyright IAEA IAEA 2015.

FiFiguregure 20 shows thethe resultsresults of of similar similar measurements measurements carried carried out out in in the the same same mode mode on onthe the radial radial interval interval of of 18–26 18–26 cm. cm. Figure Figure shows shows that that for for closely closely located located pointspoints 22 andand 4, the cross-phase at the GAM frequency is close to zero over the entire measurement interval. The averaged error of cross-phase measurement, determined by the signal-to-noise ratio, is about 0.1 radians, increasing towards the edge to 0.2 radians. This gives the poloidal mode number m = 0 c with error ±1, and ±2 at the edge. Figure 21 shows the results of the cross-phase measurements for the most distant points 1 and 5 in the same shot, but in a wider radial range. We see that the cross-phase between the signals from the 1st and 5th points at the GAM frequency is also close to zero over the entire measurement interval. The error of phase measurement is a constant of about 0.1 radians on average, and the error in determining the poloidal mode number m = 0 decreased due to an increase in the distance between points S1 and S5 on average to ±0.5, except for the edge, where it remained at a level of ±2 [58]. Thus, phase measurements of the potential confirmed the theoretically predicted poloidal symmetry of potential oscillations for GAM. Symmetry 2021, 13, 1367 20 of 46

cross-phase at the GAM frequency is close to zero over the entire measurement interval. The averaged error of cross-phase measurement, determined by the signal-to-noise ratio, is about 0.1 radians, increasing towards the edge to 0.2 radians. This gives the poloidal mode number m = 0 c with error ±1, and ±2 at the edge. Figure 21 shows the results of the cross-phase measurements for the most distant points 1 and 5 in the same shot, but in a wider radial range. We see that the cross-phase between the signals from the 1st and 5th points at the GAM frequency is also close to zero over the entire measurement interval. The error of phase measurement is a constant of about 0.1 radians on average, and the error in determining the poloidal mode number m = 0 decreased due to an increase in the distance between points S1 and S5 on average to ±0.5, except for the edge, where it remained at a Symmetry 2021, 13, x FOR PEER REVIEW 21 of 49 Symmetry 2021, 13, x FOR PEER REVIEWlevel of ±2 [58]. Thus, phase measurements of the potential conﬁrmed the theoretically21 of 49

predicted poloidal symmetry of potential oscillations for GAM.

FigureFigure 20. 20. TheThe phase phase shift shift between between the potential the potential oscillations oscillations and the and poloidal the poloidal mode number mode number m meas-m Figure 20. The phase shift between the potential oscillations and the poloidal mode number m meas- t e 19 −3 uredmeasured by intermediate by intermediate slits of slitsthe analyzer of the analyzer in Ohmic in Ohmicdischarge discharge with B with= 2.3 T,Bt n= 2.3= (0.9 T,–n2.4)e = (0.9–2.4)× 1019 m−3×, ured by intermediate slits of the analyzer in Ohmic discharge with Bt = 2.3 T, ne = (0.9–2.4) × 10 m , Ip = 19220 kA.−3 Ip 10= 220m kA., Ip = 220 kA.

Figure 21. The phase shift between the potential oscillations measured by most distant slits 1 and 5 FigureFigure 21. 21. TheThe phase phase shift shift between between the the potential potential oscillations oscillations measured measured by by most most distant distant slits slits 1 1and and 5 5 of the analyzer for the plasma scenario shown in Figure 19. Adapted from [58] Copyright IAEA 2015. ofof the the analyzer analyzer for for the the plasma plasma scenario scenario shown shown in in Figure Figure 19. 19 Adapted. Adapted from from [58] [58 Copyright] Copyright IAEA IAEA 2015. 2015. 7. HIBP Diagnostics and Their Outcome for Plasma Symmetry Study 7.7. HIBP HIBP Diagnostics Diagnostics and and Their Their Outcome Outcome for for Plasma Plasma Symmetry Symmetry Study Study 7.1.7.1. TM TM-4-4 HIBP HIBP 7.1. TM-4 HIBP TheThe first ﬁrst HIBP HIBP complex complex in in Russia Russia (USSR) (USSR) was was on on the the TM TM-4-4 tokamak, tokamak, created created early early in in The first HIBP complex in Russia (USSR) was on the TM-4 tokamak, created early in thethe 1980s 1980s by by Krupnik, Krupnik, Bondarenko Bondarenko and and Nedzelskiy Nedzelskiy [59 [59].]. The The scheme scheme of of the the complex complex is is the 1980s by Krupnik, Bondarenko and Nedzelskiy [59]. The scheme of the complex is shownshown in in Figure Figure 22. 22 . shown in Figure 22.

Symmetry 2021, 13, x FOR PEER REVIEW 21 of 49

Figure 20. The phase shift between the potential oscillations and the poloidal mode number m measured by intermediate slits of the analyzer in Ohmic discharge with Bt = 2.3 T, ne = (0.9–2.4) × 1019 m−3, Ip = 220 kA.

Figure 21. The phase shift between the potential oscillations measured by most distant slits 1 and 5 of the analyzer for the plasma scenario shown in Figure 19. Adapted from [58] Copyright IAEA 2015.

7. HIBP Diagnostics and Their Outcome for Plasma Symmetry Study 7.1. TM-4 HIBP

Symmetry 2021, 13, 1367 The first HIBP complex in Russia (USSR) was on the TM-4 tokamak, created early21 of in 46 the 1980s by Krupnik, Bondarenko and Nedzelskiy [59]. The scheme of the complex is shown in Figure 22.

Figure 22. Scheme of the TM-4 HIBP complex: 1—source power unit; 2—high-voltage source for accelerator; 3—ion source; 4—accelerating tube; 5—high-voltage divider; 6—ion beamline; 7— vacuum valves; 8—primary beam pick-up; 9—plasma; 10—tokamak chamber; 11—grid analyzer for secondary ions; 12—secondary-electron multiplier. Adapted from [13] Copyright IAEA 1985.

The energy of secondary particles was determined by a multigrid ion analyzer. On TM-4, the first measurements of the potential profiles were carried out at a fixed position of the sample volume (SV), depending on the discharge parameters, such as the average chord density, plasma current and toroidal magnetic field. These measurements were carried out in a series of reproduced shots, in which the SV scans plasma from the center to the edge. It was shown that the plasma potential at the core has a negative sign and a scale about several hundred volts. For the first time, the link between the potential and the average plasma density was established: a higher density corresponds to a larger absolute value of negative potential. The measurements of the plasma density profile with a HIBP are described in Section 3.2. The TM-4 tokamak was terminated in 1982 due to the construction of the T-15 tokamak. Creation and scientific exploitation of TM-4 HIBP complex made the physical and technological basis for further development of the HIBP diagnostics towards the tool to study symmetric structures in toroidal plasmas.

7.2. T-10 HIBP The HIBP complex on the T-10 tokamak was created in the mid-1980s by Krupnik, Bondarenko, Khrebtov and Nedzelskiy [18,60]. In contrast with the TM-4 complex, it used a Cs ion beam accelerated up to an energy of 160 keV, and a ‘flat mirror’ analyzer [61] with a higher energy resolution than the grid analyzer of TM-4. As the complex was implemented into the existing machine, there were no convenient ports to conduct the beam through the plasma. For injection we use a standard wide vertical port, and to pull out the secondary beam, we have to cut the vacuum chamber and weld an additional port to it [62]. Such a significant reconstruction of the machine became possible due to the successful HIBP operation on the TM-4 and the obtaining of important physical results— first measurements of plasma potential profile. This important decision was adopted with support by Dnestrovskij, Razumova and Sokolov. The initial version of the HIBP complex is schematically shown in Figure 23. Symmetry 2021, 13, x FOR PEER REVIEW 22 of 49

Figure 22. Scheme of the TM-4 HIBP complex: 1—source power unit; 2—high-voltage source for accelerator; 3—ion source; 4—accelerating tube; 5—high-voltage divider; 6—ion beamline; 7—vacuum valves; 8—primary beam pick-up; 9 − plasma; 10—tokamak chamber; 11—grid analyzer for secondary ions; 12—secondary-electron multiplier. Adapted from [13] Copyright IAEA 1985.

The energy of secondary particles was determined by a multigrid ion analyzer. On TM-4, the first measurements of the potential profiles were carried out at a fixed position of the sample volume (SV), depending on the discharge parameters, such as the average chord density, plasma current and toroidal magnetic field. These measurements were carried out in a series of reproduced shots, in which the SV scans plasma from the center to the edge. It was shown that the plasma potential at the core has a negative sign and a scale about several hundred volts. For the first time, the link between the potential and the average plasma density was established: a higher density corresponds to a larger absolute value of negative potential. The measurements of the plasma density profile with a HIBP are described in Section 3.2. The TM-4 tokamak was terminated in 1982 due to the construction of the T-15 tokamak. Creation and scientific exploitation of TM-4 HIBP complex made the physical and technological basis for further development of the HIBP diagnostics towards the tool to study symmetric structures in toroidal plasmas.

7.2. T-10 HIBP The HIBP complex on the T-10 tokamak was created in the mid-1980s by Krupnik, Bondarenko, Khrebtov and Nedzelskiy [18,60]. In contrast with the TM-4 complex, it used a Cs ion beam accelerated up to an energy of 160 keV, and a ‘flat mirror’ analyzer [61] with a higher energy resolution than the grid analyzer of TM-4. As the complex was implemented into the existing machine, there were no convenient ports to conduct the beam through the plasma. For injection we use a standard wide vertical port, and to pull out the secondary beam, we have to cut the vacuum chamber and weld an additional port to it [62]. Such a significant reconstruction of the machine became possible due to the successful HIBP operation on the TM-4 and the obtaining of important physical results—first measurements of plasma potential profile. This important decision was adopted with sup- Symmetry 2021, 13, 1367 port by Dnestrovskij, Razumova and Sokolov. The initial version of the HIBP complex22 of 46is schematically shown in Figure 23.

Figure 23. Initial scheme of the heavy ion beam probe onon thethe T-10T-10 tokamak.tokamak.

Setting up up the the T T-10-10 complex complex and and conducting conducting the the ion ionbeam beam into intothe analyzer the analyzer encoun- en- counteredtered with withan unexpected an unexpected issue— issue—aa strong strong toroidal toroidal displacement displacement of the of beam, the beam, which which ham- hamperedpered conducting conducting the theion ionbeam beam through through the the long long narrow narrow ports ports of of the the tokamak. tokamak. To To solve it, for the first time in the HIBP operation, the toroidal correction of secondary particles was applied and a secondary ion beamline was installed outside the vacuum chamber. The passage of the beam into the analyzer through narrow diagnostic ports of the tokamak became possible thanks to the assistance of the T-10 leading experimentalist Dr. Bobrovskiy. At the initial stage of the HIBP operation, the measurements were carried out in a series of repeated tokamak shots, in which the SV position moved from the plasma edge towards the center. The limitations on the beam energy and ion mass allowed us to carry out measurements at a reduced magnetic field Bt = 1.5 T in the range 0.6 < ρSV < 1 [63–65]. Later on, the T-10 HIBP complex has permanently evolved in the following direc- tions [66]: 1. The Cs ions were replaced by heavier Tl ions. The energy range was consequently increased to 220, then to 280, 300, and finally to 330 kV. These changes allow to study discharges with microwave heating (ECRH) at 2.08 < Bt < 2.5 T. In discharges with Bt < 2.17 T, the radial region 0.2 < ρSV < 1 was investigated; 2. The ion beamlines were modernized in the following way: • Two serial pairs of steering plates were installed into the primary beamline. The first plates deflect the beam “counter-current” to the extreme toroidal position near the second plates. Thus, the beam approaches the second plates not axially, but maximally shifted. The second plates deflect the beam “co-current”. As a result, a zigzag trajectory is formed, and it became possible to use the full toroidal width of the port, not half of it, as in the axial injection of the beam, and maximize the toroidal correction angle β. • The secondary beamline was significantly expanded. The toroidal correction plates were placed inside the vacuum chamber to increase the angle of toroidal correction of the secondary trajectories β3. Correcting plates were equipped with a continuous baking system that maintains their operating temperature at the level of 220–250 ◦C. This allows us to completely avoid the deposition of hydrocarbon films on the surface of the plates, and to exclude the appearance of high-voltage breakdowns between the plates or from the plate to the grounded T-10 wall. Immediately before the working shot of the tokamak, the baking was turned off and the baking circuit was disconnected in order to exclude the possibility of the interaction of the closed loop with the current and the confining magnetic fields of the machine. The electromagnetic forces resulting from this Symmetry 2021, 13, 1367 23 of 46

interaction could deform the plates, making it impossible for the secondary beam to pass into the analyzer. The described upgrades made it possible to expand the operational limits of measurements up to the operational limits of the tokamak 140 kA < Ipl < 330 kA. 3. The technique of scanning the entry angle of the particle beam into the plasma was implemented. Additionally, the vertical correction plates for secondary trajectories were installed to optimize the particle entry angle into the analyzer. It allowed us to obtain fragments of the radial profile per discharge; 4. A control system for the beam was created, which allows us to select a complete set of control voltages in the primary and secondary ion beamlines in each subsequent tokamak shot based on the analysis of the beam position in the previous shot (injection and sweeping angles in primary and secondary beamlines α1, α2, α3, β1, β2, β3). Its implementation radically simplified and accelerated the process of selecting control voltages, to significantly increase the accuracy of beam positioning, thereby providing systematic measurements both with a fixed SV position or with radial scan. Besides these two traditional HIBP operating modes, the developed system made it possible to operate in new non-standard modes. The most popular among them: • Periodic variation of the SV between two spatial positions (“colon”), • Periodic change of the SV between several positions (“multiple points”), • Alternation of the scan and at point measurements during one shot (“scan + point”). 5. A feedback for toroidal displacement was created, which adjust the toroidal correction voltage in the secondary beamline (toroidal angle β3), depending on the measured toroidal displacement in the detector ζd. This allows us to automate the selection of the toroidal correction voltage from shot to shot, as well as during one shot, especially in shots with changing plasma current (ramp-up, ramp-down) as well as the start of the plasma discharge with sharp increase of the plasma current. The feedback system for toroidal displacement is described in detail in the Reference [67]. 6. The initial single-channel energy analyzer was first replaced by a two-channel and then a five-channel one. As a result, we can carry out correlation measurements over several spatial channels. In particular, in the region of maximal beam penetration into the plasma, it became possible to measure the poloidal potential correlations or Ep, a turbulent particle flux ΓE×B, as well as the poloidal density correlations or rotation of turbulence. Closer to the plasma edge, it is possible to measure the velocity of radial correlations or the radial propagation of potential and density perturbations. 7. An emitter-extractor unit has been developed, which makes it possible to significantly increase the primary beam current from 2–20 µA to 100–130 µA. Its use has expanded the allowable density limit for HIBP towards both high densities (plasma core) and also ultra-low densities at the plasma edge and in the scrape-off layer (SOL) [68]. The final version of the complex and its photo are shown in Figures1 and2. The T-10 tokamak was terminated in March 2018 due to the construction of the T-15MD tokamak. Creation and development of the T-10 HIBP complex provides comprehensive study of Geodesic Acoustic Mode (GAM)—the toroidally symmetric structures in tokamak plasmas, see Sections 5.1 and 6.3.

7.3. TJ-I HIBP The HIBP complex for the TJ-I tokamak in the CIEMAT Science Center in Madrid, Spain, was created early in the 1990s by Krupnik, Bondarenko, Khrebtov and Nedzel- + skiy [69]. It operated with a Cs ion beam, accelerated up to energy Eb = 60 keV. The scheme of the complex is shown in Figure 24. Continuing the T-10 experiment, the TJ-I used the correcting toroidal plates for the secondary beam. As well as in T-10, the analyzer of the ‘flat mirror’ type was used on the TJ-I. The TJ-I tokamak had rectangular toroidal field coils. The TJ-I was the first, where the radial scanning method was used. The results Symmetry 2021, 13, 1367 24 of 46

of its application for Eb = 20 keV are shown in Figure 25. We see that the plasma density has a bell-shaped proﬁle with a pronounced maximum in the center, which shifts outward along the major radius during the discharge. The distribution of the plasma potential along the radius is inhomogeneous and also evolves with the density: the density decrease is accompanied by potential evolution towards the positive values. The TJ-I tokamak was terminated in 1995 due to the construction of the TJ-II stellarator. Symmetry 2021, 13, x FOR PEER REVIEW 25 of 49 Symmetry 2021, 13, x FOR PEER REVIEWCreation and scientiﬁc exploitation of TJ-I HIBP complex made the basis for the next-step25 of 49 TJ-II HIBP diagnostics, the most fruitful one in studies of the symmetric structures in toroidal plasmas.

Figure 24. Scheme of the heavy ion beam probe on the TJ-I tokamak. (a) Machine and diagnostics; FigureFigure 24. 24. SchemeScheme of of the the heavy heavy ion ion beam beam probe probe on on the the TJ TJ-I-I tokamak. tokamak. (a ()a )Machine Machine and and diagnostics; diagnostics; (b) detector grid with lines of equal injection angle from −12 to +12 (dashed), and the equal beam (bb) detector grid with lines of equal injection angle from −−12 ◦to +12 ◦(dashed), and the equal beam ( )energy detector Eb grid from with 6 to lines 60 keV of equal(solid injection lines). angle from 12 to +12 (dashed), and the equal beam energy Eb from 6 to 60 keV (solid lines). energy Eb from 6 to 60 keV (solid lines).

(a) (b) (a) (b) Figure 25. The profiles of the plasma potential (a) and density (b) for three spatial scans in TJ-I. FigureFigure 25. 25. TheThe profiles profiles of of the the plasma plasma potential potential (a ()a )and and density density (b (b) )for for three three spatial spatial scans scans in in TJ TJ-I.-I. 7.4.7.4. TUMAN-3M TUMAN-3M HIBP HIBP 7.4. TUMAN-3M HIBP TheThe HIBP HIBP complex complex on on the the small small aspect aspect ratio ratio tokamak tokamak TUMAN-3M TUMAN-3M in thein the A.F. A.F. Ioffe Ioffe The HIBP complex on the small aspect ratio tokamak TUMAN-3M in the A.F. Ioffe InstituteInstitute in in St. St. Petersburg Petersburg was was created created inin thethe latelate 1990s by by Krupnik, Krupnik, Bondarenko Bondarenko and and Ko- Institute in St. Petersburg was created in the+ late+ 1990s by Krupnik, Bondarenko and Ko- Komarovmarov [[70].70]. ItIt operatesoperates withwith aa beam beam of of K K ions ions accelerated accelerated up up to to the the energy energyEb E=b 60= 60 keV. keV. marov [70]. It operates with a beam of K+ ions accelerated up to the energy Eb = 60 keV. TheThe scheme scheme of theof the HIBP HIBP complex complex is shownis shown in Figurein Figure 26. 26. Figure Figure shows shows that that the the detector detector The scheme of the HIBP complex is shown in Figure 26. Figure shows that the detector lineline connects connects the the center center and and edge edge of theof the plasma. plasma. It isIt suitableis suitable for for determination determination of radialof radial line connects the center and edge of the plasma. It is suitable for determination of radial profilesprofiles over over a total a total radial radial interval. interval. profiles over a total radial interval. Figure 27 shows the equatorial projection of the calculated trajectories of the probing particles. We see that the plasma current shifts the trajectories, as well as that variation of the injection toroidal angle β effectively changes the displacement of the trajectories. Choosing β, we can place the detector line close to the meridian plane of the torus, z = 0. On the TUMAN-3M, it is possible to measure the evolution of the potential and plasma density at fixed spatial points.

Symmetry 2021, 13, 1367 25 of 46 Symmetry 2021, 13, x FOR PEER REVIEW 26 of 49

Figure 26. Scheme ofof thethe HIBPHIBP on on the the TUMAN-3M TUMAN-3M tokamak, tokamak, the the major major axis axis of theof the torus torus is on is theon left.the left.Thin Thin curves curves mark mark the trajectoriesthe trajectories of probing of probing particles, particles,I—injector, I—injector,D—detector. D—detector. Asterisks Asterisks indicate indi- Symmetry 2021, 13, x FOR PEER REVIEWcate points of the detector line available for observation. The calculation was carried out for27 ofa beam49 + points of the detector line available for observation. The calculation was carried out for a beam of Cs of Cs+ ions with the beam energies Eb = 4–13 keV, the point with Eb = 10 keV is located in the plasma ions with the beam energies Eb = 4–13 keV, the point with Eb = 10 keV is located in the plasma center. center.

Figure 27 shows the equatorial projection of the calculated trajectories of the probing particles. We see that the plasma current shifts the trajectories, as well as that variation of the injection toroidal angle β effectively changes the displacement of the trajectories. Choosing β, we can place the detector line close to the meridian plane of the torus, z = 0. On the TUMAN-3M, it is possible to measure the evolution of the potential and plasma density at fixed spatial points.

Figure 27. Numerical study study of of the the sensitivity sensitivity of of the the toroidal toroidal shift shift of of the the trajectories trajectories to tothe the plasma plasma currentcurrent and the initial toroidal toroidal angle angle ββ ofof the the TUMAN TUMAN-3M.-3M. Projections Projections of ofthe the trajectories trajectories onto onto the the equatorialequatorial plane of the the torus torus ( (xx, ,zz)) for for different different values values of of the the plasma plasma current current I andI and the the initial initial toroidal toroidal angle β (a–d); XI and XD injector and detector coordinates; asterisks mark the points of the detector angle β (a–d); XI and XD injector and detector coordinates; asterisks mark the points of the detector line available for observation for energies Eb = 4, 5, 6, 8, and 10 keV. line available for observation for energies Eb = 4, 5, 6, 8, and 10 keV. 7.5. WEGA HIBP Stellarator WEGA is a two-period torsatron with a low toroidal magnetic field and rotational transform ι/2π. The HIBP complex on the WEGA stellarator in the Max Planck Institute for Plasma Physics in Greifswald, Germany was created early in the 2000s by Krupnik, Bondarenko and Komarov [71–73]. It operates with a beam of Na+ ions accelerated up to the energy Eb = 50 keV. The scheme of the HIBP complex is shown in Figure 28, and the detector grid is shown in Figure 29.

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7.5. WEGA HIBP Stellarator WEGA is a two-period torsatron with a low toroidal magnetic ﬁeld and rotational transform ι/2π. The HIBP complex on the WEGA stellarator in the Max Planck Institute for Plasma Physics in Greifswald, Germany was created early in the 2000s by Krupnik, Bondarenko and Komarov [71–73]. It operates with a beam of Na+ ions acceler- Symmetry 2021, 13, x FOR PEER REVIEW 28 of 49 Symmetry 2021, 13, x FOR PEER REVIEWated up to the energy Eb = 50 keV. The scheme of the HIBP complex is shown in Figure 2828, of 49

and the detector grid is shown in Figure 29.

FigureFigure 28. 28. SchemeScheme of thethe HIBPHIBP on on the the WEGA WEGA stellarator. stellarator.Bt = B 0.5t = 0.5 T, ι /2T,π ι/2=π 0.18. = 0.18. Figure 28. Scheme of the HIBP on the WEGA stellarator. Bt = 0.5 T, ι/2π = 0.18.

+ b Figure 29. Detector grid for Na + ions in the WEGA stellarator. The beam energy E and the injection FigureFigure 29. 29. Detector gridgrid for for Na Na+ ions ions in in the the WEGA WEGA stellarator. stellarator. The The beam beam energy energyE and E theb and injection the injection angle (expressed in scanning voltage Uscan) are indicated. b scan angleangle (expressed (expressed inin scanningscanning voltage voltageU scanU ) are) are indicated. indicated. Stellarator WEGA was terminated in 2006 due to the construction of the W7-X stel- StellaratorStellarator WEGAWEGA was was terminated terminated in in 2006 2006 due due to theto the construction construction of the of W7-X the W7 stel--X stellarator. The scientific exploitation of WEGA HIBP brings an additional experience of the larator.larator. The scientiﬁcscientific exploitation exploitation of of WEGA WEGA HIBP HIBP brings brings an additional an additional experience experience of the of the HIBP operation in stellarators, so contributes to the next-step TJ-II HIBP diagnostics, the HIBPHIBP operation operation inin stellarators,stellarators, so so contributes contributes to theto the next-step next-step TJ-II TJ HIBP-II HIBP diagnostics, diagnostics, the the most fruitful one in studies of the symmetric structures in toroidal plasmas. mostmost fruitful fruitful oneone inin studies studies of of the the symmetric symmetric structures structures in toroidal in toroidal plasmas. plasmas. 7.6.7.6. URAGAN URAGAN-2M-2M HIBPHIBP 7.6. URAGAN-2M HIBP TheThe HIBP HIBP complex onon URAGAN-2M URAGAN-2M stellarator stellarator at theat the Institute Institute of Plasma of Plasma Physics Physics of of the KharkovThe HIBP Institute complex of Physics on URAGAN and Technology-2M stellarator of the Academy at the ofInstitute Sciences of of Plasma Ukraine Physics was of the Kharkov Institute of Physics and Technology of the Academy of Sciences of Ukraine the Kharkov Institute of Physics and Technology of the Academy of Sciences of Ukraine was created in the mid-2010th by Krupnik, Kozachek, Komarov, Zhezhera and Chmyga was created in the mid-2010th by Krupnik, Kozachek, Komarov, Zhezhera and Chmyga [74,75]. It operates with a Cs+ ion beam accelerated to energies up to Eb = 70 keV. The [74,75]. It operates with a Cs+ ion beam accelerated to energies up to Eb = 70 keV. The scheme of the HIBP complex is shown in Figure 30a, and the equatorial projection of tra- scheme of the HIBP complex is shown in Figure 30a, and the equatorial projection of trajectories is shown in Figure 30b. URAGAN-2M was recommissioned in 2006 after a long jectories is shown in Figure 30b. URAGAN-2M was recommissioned in 2006 after a long pause caused by the reconstruction. It is currently undergoing the stage of adjustment and pause caused by the reconstruction. It is currently undergoing the stage of adjustment and reaching the design parameters. reaching the design parameters.

Symmetry 2021, 13, 1367 27 of 46

created in the mid-2010th by Krupnik, Kozachek, Komarov, Zhezhera and Chmyga [74,75]. + It operates with a Cs ion beam accelerated to energies up to Eb = 70 keV. The scheme of Symmetry 2021, 13, x FOR PEER REVIEW the HIBP complex is shown in Figure 30a, and the equatorial projection29 ofof trajectories49 is

shown in Figure 30b. URAGAN-2M was recommissioned in 2006 after a long pause caused by the reconstruction. It is currently undergoing the stage of adjustment and reaching the design parameters.

(a) (b)

Figure 30.FigureScheme 30. ofScheme the HIBP of the on theHIBP URAGAN-2M on the URAGAN stellarator.-2M Thestellarator. optimal The detector optimal line isdetector marked line by blueis marked asterisks, the green solidby blue line isasterisks, the trajectory; the greenI is injector, solid lineD is is detector, the trajectory; and P is I plasma;is injector, (a) verticalD is detector, plane, ( band) equatorial P is plasma; plane. (a) vertical plane, (b) equatorial plane. 7.7. TJ-II HIBP—The Most Advanced Diagnostics to Study Plasma Symmetric Structures 7.7. TJ-II HIBP—The MostThe TJ-IIAdvanced stellarator-heliac Diagnostics has to beenStudy successfully Plasma Symmetric operating Structures from the early 2000s to the The TJ-II stellaratorpresent at-heliac the National has been Fusion successfully Laboratory operating of the CIEMAT from the Science early Center 2000s into Madrid,the Spain. present at the NationalFrom the Fusion start of its Laboratory operation till of the the start CIEMAT of the world Science largest Center stellarator in Madrid, W7-X in 2016, the TJ-II was the world’s second largest stellarator after the LHD and the largest in Europe. Spain. From the start of its operation till the start of the world largest stellarator W7-X in Now it is the third stellarator after W7-X and LHD. 2016, the TJ-II was the world’s second largest stellarator after the LHD and the largest in The initial version of the HIBP complex on the TJ-II was created in the early 2000s Europe. Now it isin the collaboration third stellarator with Krupnik, after W7 Nedzelskiy,-X and LHD. Khrebtov and Bondarenko from the Kharkov The initial versionInstitute of of the Physics HIBP and complex Technology on the (Ukraine) TJ-II was and created the National in the Research early 2000s Center in Kurchatov collaboration withInstitute Krupnik, (Russia) Nedzelskiy, [76,77]. Khrebtov It operates and with Bondarenko a Cs+ ion beam from accelerated the Kharkov up In- to the energy stitute of PhysicsE andb = 150Technology keV. (Ukraine) and the National Research Center Kurchatov Institute (Russia) [76,77Photos]. It operates of the HIBP with complex a Cs+ ion on thebeam TJ-II accelerated stellarator areup shownto the energy in Figure E b31 = . Figure 32 150 keV. presents the scheme of the HIBP complex, and the magnetic conﬁguration with detector Photos of theline, HIBP along complex which theon the measurements TJ-II stellarator were performed,are shown isin shown Figure in 31. frame. Figure 32 presents the scheme of the HIBP complex, and the magnetic configuration with detector line, along which the measurements were performed, is shown in frame.

(a) (b)

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(a) (b) Figure 30. Scheme of the HIBP on the URAGAN-2M stellarator. The optimal detector line is marked by blue asterisks, the green solid line is the trajectory; I is injector, D is detector, and P is plasma; (a) vertical plane, (b) equatorial plane.

7.7. TJ-II HIBP—The Most Advanced Diagnostics to Study Plasma Symmetric Structures The TJ-II stellarator-heliac has been successfully operating from the early 2000s to the present at the National Fusion Laboratory of the CIEMAT Science Center in Madrid, Spain. From the start of its operation till the start of the world largest stellarator W7-X in 2016, the TJ-II was the world’s second largest stellarator after the LHD and the largest in Europe. Now it is the third stellarator after W7-X and LHD. The initial version of the HIBP complex on the TJ-II was created in the early 2000s in collaboration with Krupnik, Nedzelskiy, Khrebtov and Bondarenko from the Kharkov In- stitute of Physics and Technology (Ukraine) and the National Research Center Kurchatov Institute (Russia) [76,77]. It operates with a Cs+ ion beam accelerated up to the energy Eb = 150 keV. Photos of the HIBP complex on the TJ-II stellarator are shown in Figure 31. Figure 32 Symmetry 2021, 13, 1367 presents the scheme of the HIBP complex, and the magnetic configuration with 28detector of 46 line, along which the measurements were performed, is shown in frame.

Symmetry 2021, 13, x FOR PEER REVIEW 30 of 49

(a) (b)

FigureFigure 31. 31. PhotoPhoto of of the the HIBP HIBP in in TJ TJ-II:-II: ( (aa)) ion ion energy energy analyzer; analyzer; ( (bb)) ion ion beam beam injector. injector.

FigureFigure 32. 32. SchemeScheme of of the the HIBP HIBP in inTJ- TJ-II.II. Inset Inset shows shows plasma plasma cross cross-section-section and anddetector detector line for line E forb = 127Eb = keV, 127 keV,scanning scanning voltage voltage limits limits are indicated: are indicated: −6 kV−6 at kV HFS at HFS(ρ = (−1ρ )= and−1) +4 and kV +4 at kV LFS at ( LFSρ = +1). (ρ = +1).

TheThe TJ TJ-II-II stellarator turnedturned outout to to be be the the unique unique machine, machine, for for which which the the HIBP HIBP complex com- plexwas plannedwas planned at the at stage the stage of machine of machine design, design, rather rather than its than operation, its operation, unlike unlike other HIBPs. other HIBPs.For this For reason, this HIBP reason, design HIBP was design optimized was optimized in advance in aiming advance for aiming the unique for the properties: unique properties:the ability tothe obtain ability full to profiles obtain offull plasma profiles parameters of plasma by parameters scanning voltage by scanning variation voltage in one variationdischarge. in one discharge. AnAn advanced advanced HIBP HIBP in in TJ TJ-II-II have have passed passed several several steps steps to to become become a a diagnostics diagnostics capable capable toto resolve resolve the the issue issue of of the the symmetry/asymmetry symmetry/asymmetry of of the the toroidal toroidal plasma plasma parameters. parameters. TheThe first first step step was was done done in in resolving the issue of LFS LFS–HFS–HFS asymmetry. The detector line, started with radial interval −1 < ρ < 0 in the HFS and producing many important line, started with radial interval −1 < ρ < 0 in the HFS and producing many important results in the characterization of the potential profile in biasing experiments [78,79], electron results in the characterization of the potential profile in biasing experiments [78,79], elec- confinement study [80,81] and transition to the improved confinement regime [82,83], was tron confinement study [80,81] and transition to the improved confinement regime [82,83], then extended towards the LFS, 0 < ρ < 1, see the inset to Figure 32. This way the first was then extended towards the LFS, 0 < ρ < 1, see the inset to Figure 32. This way the first experimental test for the potential profile symmetry was performed. The results are shown experimental test for the potential profile symmetry was performed. The results are in Figure 33. This observation was the first experimental prove of the potential symmetry shown in Figure 33. This observation was the first experimental prove of the potential LFS–HFS in the toroidal devices. In this experiment, plasma density was varied by an symmetry LFS–HFS in the toroidal devices. In this experiment, plasma density was varied order of magnitude, from 0.3 to 3 × 1019 m−3. It was shown that plasma potential evolves by an order of magnitude, from 0.3 to 3 × 1019 m−3. It was shown that plasma potential dramatically with the density raise, changing the shape and the sign [84]. In the whole evolves dramatically with the density raise, changing the shape and the sign [84]. In the whole observed domain of plasma parameters (density, temperature, collisionality, way of heating: ECRH, NBI) the evolving plasma potential is symmetric in terms of LFS–HFS. Later on, the symmetric structure of plasma potential profiles was proven for more extended domain of plasma parameters [85,86].

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Symmetry 2021, 13, x FOR PEER REVIEWobserved domain of plasma parameters (density, temperature, collisionality, way of31 heating:of 49 ECRH, NBI) the evolving plasma potential is symmetric in terms of LFS–HFS. Later on, the symmetric structure of plasma potential proﬁles was proven for more extended domain of plasma parameters [85,86].

Figure 33. Plasma potential profile evolution in TJ-II. The first proof of the LFS–HFS symmetry [85]. Figure 33. Plasma potential profile evolution in TJ-II. The first proof of the LFS–HFS symmetry [85]. ReproducedOperating courtesy with of the IAEA. full Copyright detector 2011line IAEA.−1 < ρ < 1, on the TJ-II, the first measurements Reproduced courtesy of IAEA. Copyright 2011 IAEA. in stellarators were carried out, in which the dependence of the potential on the line-av- eragedOperating density withfor a thefixed full position detector of linethe −sample1 < ρ < volume 1, on the (SV) TJ-II, was the studied, first measurements as well as the Operating with the full detector line −1 < ρ < 1, on the TJ-II, the first measurements inpotential stellarators profiles were and carried their dependence out, in which on theheating dependence methods ofand the plasma potential parameters on the line-were in stellarators were carried out, in which the dependence of the potential on the line-av- averagedmeasured density [87,88]. for At a this fixed stage position there of w theas made sample a volumesubstantial (SV) contributio was studied,n to as the well study as the of eraged density for a fixed position of the sample volume (SV) was studied, as well as the potentialhelically profilessymmetric and structure their dependence—Alfven onEigenmodes. heating methods Figure and 34 shows plasma an parameters example of were the potential profiles and their dependence on heating methods and plasma parameters were measuredHelicity Induced [87,88]. AtAlfven this stageEigenmode there was (HAE) made detected a substantial and identified contribution by HIBP to the combined study of measured [87,88]. At this stage there was made a substantial contribution to the study of helicallywith the symmetricMHD-code structure—Alfven AE3D [51]. The next Eigenmodes. principally Figure important 34 shows step anin the example widening of the of helicallyHelicity symmetric Induced Alfvenstructure Eigenmode—Alfven Eigenmodes. (HAE) detected Figure and 34 identified shows an by example HIBP combined of the the diagnostic capabilities of HIBP was the measurement of the two-dimensional (2D) Helicitywith the Induced MHD-code Alfven AE3D Eigenmode [51]. The (HAE) next principally detected and important identified step by in theHIBP widening combined of the spatial distribution of the plasma potential and turbulence. This 2D map was obtained by withdiagnostic the MHD capabilities-code AE3D of HIBP[51]. The was next the measurementprincipally important of the two-dimensional step in the widening (2D) spatial of the movement of the detector line across the plasma with the Eb variation [89,90]. thedistribution diagnostic capabilitiesof the plasma of potential HIBP was and theturbulence. measurement This of 2D the map two- wasdimensional obtained (2D) by the spatialmovement distribution of the of detector the plasma line acrosspotential the and plasma turbulence. with the ThisEb variation 2D map was [89,90 obtained]. by the movement of the detector line across the plasma with the Eb variation [89,90].

(a) (b)

FigureFigure 34.34. 3D structure of of HAE HAE mode mode mm/n/ n= 4=/7; 4/7; (a) (topa) topview; view; (b) poloidal (b) poloidal cut [51] cut. [Reproduced51]. Reproduced courtesy courtesy of IAEA. of IAEA.Copy- Copyrightright 2012 2012IAEA. IAEA. (a) (b) Figure 34. 3D structure of HAE mode m/n = 4/7; (a) top view; (b) poloidal cut [51]. Reproduced courtesy of IAEA. Copy- right 2012 IAEA.

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Symmetry 2021, 13, 1367 30 of 46

To construct a 2D map, the systematic measurements were performed in a series of the reproducibleTo construct shots. a 2D map, The theresults systematic are presented measurements in Figure were 35. performed This 2D map in a series of plasma of the potentialreproducible is the shots. first experimental The results are prove presented of the in potential Figure 35 poloidal. This 2D symmetry map of plasma in the potentialtoroidal devices.is the ﬁrst Figure experimental 35a shows prove that ofthe the contours potential of poloidalthe plasma symmetry potential in (equipotentials) the toroidal devices. are wellFigure in 35linea showswith the that magnetic the contours flux ofsurfaces the plasma [91]. potentialIn contrast, (equipotentials) the 2D map areof the well plasma in line potentialwith the magneticand density ﬂux turbulence surfaces [91 is]. not In contrast,poloidally the symmetric, 2D map of it the has plasma higher potential level at andthe HFS,density as presented turbulence in is Figure not poloidally 35b. symmetric, it has higher level at the HFS, as presented in Figure 35b.

(a) (b)

FigureFigure 35. 35. TheThe spatial spatial distribution distribution of of the the plasma plasma potential potential ( (aa)) and and its its oscillations oscillations ( (bb)) over over the the vertical vertical cross cross-section-section in in TJ TJ-II.-II. ((aa)) Potential Potential contours contours cover cover about about one one half half of of the the plasma plasma poloidal poloidal cross cross-section:-section: the the maximum maximum of of 2D 2D potential potential distribution distribution (~1000 V) is located at the magnetic axis, equipotentials are consistent with vacuum magnetic flux surfaces, and 2D poten- (~1000 V) is located at the magnetic axis, equipotentials are consistent with vacuum magnetic flux surfaces, and 2D potential tial distribution is symmetric (LFS–HFS and up–down). (b) The maxima of 2D RMS distribution are located at the magnetic distribution is symmetric (LFS–HFS and up–down). (b) The maxima of 2D RMS distribution are located at the magnetic axis (~30 V) and at the edge (~3540 V), equipotentials are basically consistent with vacuum magnetic flux surfaces, 2D axis (~30 V) and at the edge (~35–40 V), equipotentials are basically consistent with vacuum magnetic flux surfaces, 2D potential distribution is not fully symmetric: at the mid-radius ~15 V at LFS vs. ~20 V at HFS, i.e., 1.3 times higher. [91]. potential distribution is not fully symmetric: at the mid-radius ~15 V at LFS vs. ~20 V at HFS, i.e., 1.3 times higher. [91]. It is worthwhile to highlight that these observations are contrasting to the gyrokinetic It is worthwhile to highlight that these observations are contrasting to the gyrokinetic expectations for toroidal devices, always predicting the contrary: higher level of turbu- expectations for toroidal devices, always predicting the contrary: higher level of turbulence lence at the LFS [92]. This experimental test should help the theorists to look deeper at the at the LFS [92]. This experimental test should help the theorists to look deeper at the theoretical basis for the modelling. theoretical basis for the modelling. The final step of the paramount importance is a duplication of the HIBP diagnostics The final step of the paramount importance is a duplication of the HIBP diagnostics inin one one machine machine and and creation creation of of the the Dual Dual HIBP HIBP [93,94 [93,94].]. This This system system is is aimed aimed for for the the study study ofof the the long long-range-range toroidal toroidal correlations correlations in in plasma plasma parameters, parameters, and, and, more more specifically, specifically, z zonalonal fflowlow [95,96 [95,96].]. It It consists consists of of the the former former complex, complex, now now called called HIBP HIBP-1-1 and and the the new new HIBP HIBP-2,-2, separatedseparated toroidally toroidally in in a a quarter quarter of of the the torus torus as as presented presented in in Figure Figure 36.36. The The photo photo of of the the DualDual HIBP HIBP is is shown shown in in Figure Figure 37.37.

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(a) (b) (a) (b) Figure 36. Dual HIBP on TJ-II—the tool to study plasma symmetric structures. (a) Top view of the TJ-II diagnostics layout. FigureFigure 36.36. DualDual HIBPHIBP onon TJ-II—theTJ-II—the tooltool toto studystudy plasmaplasma symmetricsymmetric structures.structures. (aa)) TopTop viewview ofof thethe TJ-IITJ-II diagnosticsdiagnostics layout.layout. (b) Side view of HIBP-2. ((bb)) Side Side view view of of HIBP-2. HIBP-2.

Figure 37. Photo of the dual HIBP on TJ-II. FigureFigure 37.37. PhotoPhoto ofof thethe dualdual HIBPHIBP onon TJ-II.TJ-II. Long-range toroidal correlations were found with the Dual HIBP in low-frequency Long-rangeLong-range toroidaltoroidal correlationscorrelations werewere foundfound withwith thethe DualDual HIBPHIBP inin low-frequencylow-frequency plasma potential oscillations. Their radial distribution is presented in Figure 38. Figure 38 plasmaplasma potentialpotential oscillations.oscillations. TheirTheir radialradial distributiondistribution isis presentedpresented inin FigureFigure 38 38.. FigureFigure 38 38 shows that there is a statistically significant coherence between plasma potential oscilla- showsshows thatthat there there is is a statisticallya statistically significant significant coherence coherence between between plasma plasma potential potential oscillations, oscillations, measured by HIBP1 and HIBP2 [97,98]. The only low-frequency (<30 kHz) oscilla- measuredtions, measured by HIBP1 by HIBP1 and HIBP2 and [HIBP297,98]. [97,98 The only]. The low-frequency only low-frequency (<30 kHz) (<30 oscillations kHz) oscilla- are tions are coherent and having a zero cross-phase. Therefore, the observed phenomenon coherenttions are andcoherent having and a zerohaving cross-phase. a zero cross Therefore,-phase. Therefore, the observed the phenomenonobserved phenomenon presents presents toroidally symmetric (n = 0) low-frequency potential perturbations, which is, by toroidallypresents toroidally symmetric symmetric (n = 0) low-frequency (n = 0) low-frequency potential perturbations, potential perturbations, which is, by which definition, is, by definition, the zonal flow [99]. thedefinition, zonal flow the [z99onal]. flow [99].

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FigureFigure 38.38.Long-range Long-range toroidaltoroidal correlationscorrelations inin plasmaplasma potentialpotential oscillationsoscillations ininTJ-II TJ-II[ 97[97]].. PlasmaPlasma po- potential long-range-correlations radial distribution in the ECRH discharge #39894, PECRH = 2  240 tential long-range-correlations radial distribution in the ECRH discharge #39894, PECRH = 2 × 240 kW. kW. (a) Potential coherency spectrogram coh(φ1(ρSV1), φ2(ρSV2)) vs. radius ρSV1 by HIBP-1; ρSV1 scans (a) Potential coherency spectrogram coh(ϕ1(ρSV1), ϕ2(ρSV2)) vs. radius ρSV1 by HIBP-1; ρSV1 scans from −1 at HFS to +1 at LFS, while HIBP-2 measures at fixed position ρSV2 = −0.63; (b) spectrogram from −1 at HFS to +1 at LFS, while HIBP-2 measures at fixed position ρSV2 = −0.63; (b) spectrogram of the cross-phase between φ1(ρSV1) and φ2(ρSV2), calculated for the coherency, exceeding noise of the cross-phase between ϕ (ρ ) and ϕ (ρ ), calculated for the coherency, exceeding noise level. level. (c) Radial distribution1 of SV1potential coherency2 SV2 averaged by 0 < f < 20 kHz. Reproduced cour- (tesyc) Radial of IoP distribution. Copyright of2018 potential IoP. coherency averaged by 0 < f < 20 kHz. Reproduced courtesy of IoP. Copyright 2018 IoP. In addition to study zonal flows [100], the double HIBP was also focused on the long- In addition to study zonal flows [100], the double HIBP was also focused on the long- range correlations driven by helically symmetric structures—Alfvén eigenmodes [101– range correlations driven by helically symmetric structures—Alfvén eigenmodes [101–105] 105] and induced by pellet injection [106,107]. and induced by pellet injection [106,107]. The development and scientific exploitation of TJ-II dual HIBP have shown an im- The development and scientific exploitation of TJ-II dual HIBP have shown an importance of the symmetry and symmetry braking studies at the flux surface in toroidal portance of the symmetry and symmetry braking studies at the flux surface in toroidal devices. The results obtained erects the interest to widen such studies in other tokamaks devices. The results obtained erects the interest to widen such studies in other tokamaks and stellarators. The next section describes the HIBP projects under consideration for var- and stellarators. The next section describes the HIBP projects under consideration for ious devices in operation or in construction. various devices in operation or in construction.

8.8. FutureFuture ProspectsProspects toto StudyStudy ofof SymmetricSymmetric StructuresStructures inin ToroidalToroidal Plasmas—Concep- Plasmas—Conceptualtual Design of the HIBP Design Diagnostics of the HIBP for V Diagnosticsarious Toroidal for VariousDevices Toroidal Devices TheThe technicaltechnical demandsdemands toto thethe futurefuture HIBPHIBP projectsprojects areare basicallybasically formulatedformulated inin Sec-Sec- tionstions2 –24–.4. An An experience experience of of the the operation operation and and physical physical results results obtained obtained during during the the HIBP HIBP operation,operation, discusseddiscussed in in SectionsSections5 –57–,7, form form the the new new and and more more challenging challenging level level for for the the new new projects.projects. ItIt consistsconsists ofof thethe followingfollowing items:items: (a)a) SimultaneousSimultaneous measurementsmeasurements of of the the all all three three signals signals for for potential, potential, density, density, magnetic magnetic oscillationsoscillations for for comprehensive comprehensive analysis analysis of the of plasmathe plasma phenomena, phenomena, including includingturbulence; turbu- (b) Multichannellence; measurements for correlation studies, including plasma turbulence b) poloidalMultichannel rotation measurements and radial propagation, for correlation and studies, turbulent including particle plasma ﬂux; turbulence po- (c) Maximalloidal rotation extension and radial of the propagation, detector grid and for turbulent 2D mapping particle and flux; study of the poloidal c) symmetry/symmetryMaximal extension of braking;the detector grid for 2D mapping and study of the poloidal (d) Creationsymmetry of/symmetry the dual HIBP braking; system for study of toroidal/helical symmetry. d) TheCreation projects of brieﬂythe dual presented HIBP system below for are study oriented of toroidal/helical to address the criticalsymmetry. issues of HIBP compatibilityThe projects to the briefly machines, presented and HIBP below capability are oriented to reply to toaddress the new the challenges critical issues (a)–(d). of HIBP compatibility to the machines, and HIBP capability to reply to the new challenges (a)–(d).

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8.1. HIBP Design for the TCABR Tokamak 8.1. HIBPThe Design conceptual for the design TCABR of Tokamakthe HIBP complex for TCABR tokamak at the Plasma Phys- ics Laboratory of the University of São Paulo in Brazil was developed in the mid-2000s by The conceptual design of the HIBP complex for TCABR tokamak at the Plasma Physics Kuznetsov and Krupnik on the initiative of Severo and Nasimento [108]. It is assumed Laboratory of the University of São Paulo in Brazil was developed in the mid-2000s by that the diagnostics will operate with a Tl ion beam accelerated up to the energy Eb = 105 Kuznetsov and Krupnik on the initiative of Severo and Nasimento [108]. It is assumed that keV. The TCABR tokamak has rectangular toroidal field coils. The scheme of the HIBP the diagnostics will operate with a Tl ion beam accelerated up to the energy Eb = 105 keV. Thecomplex TCABR is tokamak shown in has Figure rectangular 39a, and toroidal the detector ﬁeld coils. grid The is scheme shown of in the Figure HIBP 39b. complex At the isTCABR shown intokamak, Figure 39thea, polarization and the detector of the grid plasma is shown edge (biasing) in Figure and 39b. heating At the by TCABR Alfvén tokamak,waves [109 the] polarizationare studied. For of the both plasma experimental edge (biasing) programs, and heating2D data by on Alfv the plasmaén waves potential [109] areand studied. its fluctuations For both are experimental required. programs, 2D data on the plasma potential and its ﬂuctuations are required.

(a) (b)

FigureFigure 39. 39.(a )(a Scheme) Scheme of of the the HIBP HIBP at at TCABR TCABR tokamak; tokamak; (b ()b Detector) Detector grid grid of of the the TCABR TCABR tokamak tokamak for for Tl Tl ions ions with with beam beam energy Eb = 55106 keV and the injection angle 65–76◦ . The grid calculated with a step of 5 keV and ◦1. energy Eb = 55–106 keV and the injection angle 65–76 . The grid calculated with a step of 5 keV and 1 .

8.2.8.2. HIBP HIBP Design Design for for the the Globus-M2 Globus-M2 Tokamak Tokamak TheThe conceptual conceptual design design of of the the HIBP HIBP complex complex for for the the Globus-M2 Globus-M2 spherical spherical tokamak tokamak withwith an an extremely extremely small small aspectaspect ratioratio ((RR/aa == 0 0.36.36 m/0.24 m/0.24 m m= 1 =.5), 1.5), operating operating in the in theA.F. A.F. Ioffe IoffePhysical Physical Institute, Institute, St. Petersburg, St. Petersburg, was basedwas based at the at initial the initial design design studies studies for the for earlier the earlierversion version Globus Globus-M-M [70,110 [70]. ,It110 is]. assumed It is assumed that the that diagnostics the diagnostics will operate will operate with a withTl ion abeam Tl ion accelerated beam accelerated up to E upb ≤ to45 EkeV.b ≤ 45Figure keV. 40 Figure shows 40 the shows scheme the schemeof the HIBP of the complex. HIBP complex.Figure 40a Figure shows 40a both shows the both vertical the vertical and the and equatorial the equatorial projections projections of the of calculated the calculated trajec- trajectoriestories of the of pro thebing probing particles. particles. Figure Figure 40b shows 40b shows the detector the detector grid aiming grid aiming for the forstudy the of studythe 2D of structures the 2D structures at the outer at the half outer of halfplasma of plasma radius in radius the LFS. in the LFS.

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(a) (b)

FigureFigure 40. 40. (a()a )Scheme Scheme of of the the HIBP HIBP at at Globus Globus-M2-M2 tokamak. tokamak. HIBP HIBP primary primary (black) (black) and and secondary secondary (red) (red) trajectories trajectories for for a a detector line of the probing scheme using 78°◦ input port, Eb = 40 keV, with the secondary beamline. Primary beamline detector line of the probing scheme using 78 input port, Eb = 40 keV, with the secondary beamline. Primary beamline angles are α = 45°, β = −5°. Secondary beamline angles are α = 0°, β = 20°. Top: side view, bottom: bottom view. Green star angles are α = 45◦, β = −5◦. Secondary beamline angles are α = 0◦, β = 20◦. Top: side view, bottom: bottom view. Green star denotes the detection point, blue diamond denotes plasma center. (b) Detector grid for the probing scheme shown in (a). denotes the detection point, blue diamond denotes plasma center. (b) Detector grid for the probing scheme shown in (a).

AtAt the the Globus Globus-M2-M2 tokamak, tokamak, studies studies of of plasma plasma confinement conﬁnement under under conditions conditions of of an an extremelyextremely small small aspect aspect ratio ratio and and plasma plasma heating heating with with NBI NBI and and lower lower hybrid hybrid waves waves are are conductedconducted with with focus focus to to Alfvén Alfvén eigenmodes. eigenmodes. Most Most of of experimental experimental programs programs require require data data onon the the plasma plasma potential potential and and turbulence. turbulence.

8.3.8.3. HIBP HIBP D Designesign for for the the COMPASS COMPASS Tokamak Tokamak TheThe small small aspect aspect ratio ratio COMPASS COMPASS tokamak tokamak,, operating at thethe InstituteInstitute ofof PlasmaPlasma PhysicsPhys- icsof of the the Czech Czech Academy Academy of of Sciences Sciences in in Prague, Prague, is usedis used to studyto study the the L-H L- transitionH transition in dis-in dischargescharges with with ohmic ohmic and NBIand plasma NBI plasma heating. heating. The studies The are studies focused are to focused Alfvén eigenmodes to Alfvén eigenmodesand GAMs [ and111, 112 GAMs]. These [111,112 experimental]. These programsexperimental require programs data on require the plasma data potential on the plasmaand its potential fluctuations. and its fluctuations. ConceptualConceptual design design of of the the HIBP HIBP for for the the COMPASS COMPASS tokamak tokamak (R (=R 0.56= 0.56 m, m,a = a0.23= 0.23 m, I m,p < I360p < 360kA, kA, Bt ≤B t2.1,≤ 2.1,PNBIP (atNBI 40(at keV) 40 keV) = 2 =× 20.3× 0.3MW) MW) with with a, was a, was created created on on the the initiative initiative of of Prof.Prof. J. J.Stökel. Stökel. COMPASS COMPASS has has a asingle single-null-null divertor divertor plasma plasma configuration configuration with with elongation elongation k k= =1.8, 1.8, triangularity triangularity ∆ =∆ 0.2,= 0.2,and andhorizontal horizontal plasma plasma size of size 16 cm. of 16 The cm. detector The detector line, calcu- line, latedcalculated in the classical in the classical probing probing scheme scheme for and for rectangular and rectangular toroidal toroidal field coils, field does coils, not does pass not throughpass through the plasma the plasma center. center. Therefore, Therefore, we consider we consider another another possible possible combination combination of prob- of ingprobing ports, ports,with the with injection the injection of primary of primary particles particles through through the lower thelower inclined inclined port and port reg- and istrationregistration of secondary of secondary particles particles through through the upper the upper inclined inclined port. port. The calculation The calculation and de- and signdesign results results are areshown shown in Figure in Figure 41. 41 We. We see see that that the the detector detector line line connects connects the the center center and and thethe plasma plasma edge. edge. Thus, Thus, this this probing probing scheme scheme is is preferable. preferable. However, However, fo forr realization realization of of this this scheme,scheme, the the direction direction of of the the magnetic magnetic field field BBt tshouldshould be be reversed. reversed.

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FigureFigure 41. 41. DesignDesign of of the the heavy heavy ion ion beam beam probing probing scheme scheme for for thethe COMPASSCOMPASS tokamak.tokamak.

FigureFigure 42 showsshows aa detectordetector gridgrid correspondingcorresponding toto thethe schemescheme inin FigureFigure 4141.41..

Figure 42. Detector grid for the Tl ions at B = 2.1 T and beam energies E = 130–250 keV. The grid is FigureFigure 42.42. DetectorDetector gridgrid forfor thethe TlTl ionsions atat BBtt == 2.12.1 TT andand beambeam energiesenergies EEbb == 130130––250250 keV.keV. TheThe gridgrid isis calculatedcalculated withwith aaa step stepstep of ofof 10 1010 keV. keV.keV. The TheThe line linelineE b EE=bb = 210= 210210 keV keVkeV allows allowsallows us usus to to connectto connectconnect the thethe center centercenter and andand edge edgeedge of the ofof theplasma,the plasma,plasma, 0 < 0r0/ <

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scheme of the HIBP complex is shown in Figure 43a, and the detector grid is shown in Figure 43b. scheme of the HIBP complex is shown in Figure 43a, and the detector grid is shown in Figure 43b.

(a) (b)

Figure 43.Figure(a) Schematic 43. (a) Schematic of the HIBP of complex the HIBP on thecomplex TCV tokamak:on the TCVBt = tokamak: 1.07 T, Ipl =Bt 135 = 1.07 kA; (T,b )I detectorpl = 135 kA; grid ( forb) Tl ions, detector grid for Tl ions, Eb = 120–250 ◦keV and the injection angle is 65–80. The grid◦ is drawn with Eb = 120–250 keV and the injection angle is 65–80 . The grid is drawn with a step of 10 keV and 1 . a step of 10 keV and 1. 8.5. HIBP Design for the MAST Tokamak 8.5. HIBP Design for theThe MAST MAST Tokamak (Mega-Ampere Spherical Torus) tokamak at the Culham Science Center in The MAST Culham,(Mega-Ampere UK, is used Spherical to study Torus) the plasma tokamak confinement at the Culham in the configurationScience Center of a spherical in Culham, UK, tokamak,is used to L-Hstudy transition the plasma under confinement conditions ofin ohmicthe configuration and NBI plasma of a spherical heating, and divertor tokamak, L-H transitionphysics. Particularunder conditions attention of is paidohmic to and the studyNBI plasma of the Alfv heating,én eigenmodes and divertor and Edge Local- physics. Particularized attention Modes (ELMs). is paid Theseto the experimentalstudy of the Alfvén programs eigenmodes require data and on Edge the plasma Lo- potential calized Modes (ELMs).and its fluctuations.These experimental programs require data on the plasma potential and its fluctuations. The conceptual design of the HIBP complex for MAST spherical tokamak was developed in on the initiative of Winsor and Sharapov [114]. It is assumed that the diagnostics The conceptual design of the HIBP complex for MAST spherical tokamak was devel- will work with the Tl ion beam accelerated to E = 105 keV. The MAST tokamak has pa- oped in on the initiative of Winsor and Sharapov [114]. It is assumedb that the diagnostics rameters R = 0.85 m, a = 0.65 m, Bt = 0.6 T, Ipl = 1 MA, an extremely small aspect ratio will work with theand Tl rectangular ion beam toroidalaccelerated field to coils. Eb = The 105 vertical keV. The section MAST of thetokamak tokamak has vacuum pa- chamber rameters R = 0.85is m, also a = rectangular. 0.65 m, Bt = The0.6 T, scheme Ipl = 1 ofMA, HIBP an extremely is shown insmall Figure aspect 44a ratio [115], and and one of the rectangular toroidalprobing field options coils.and The the vertical detector section grid is of shown the tokamak in Figure vacuum44b. Note chamber that several is trajectories also rectangular.(blue) The s havecheme intersections of HIBP is that shown complicate in Figure interpretation 44a [115], and of HIBP one of measurements, the probing while other options and the trajectoriesdetector grid (red) is haveshown not in intersections. Figure 44b. TheNote similar that several situation trajectories was in COMPASS (blue) (Figure 41). have intersections that complicate interpretation of HIBP measurements, while other trajectories (red) have not intersections. The similar situation was in COMPASS (Figure 41).

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140 360 220 400 380 78 76 With

120 240

180 intersection 260

280 74

300 75 73.66 100 73.33

sample volume 80 E = const 73  = const 60 73 220 180 Z, cm Z, 260 360 40 280 73.33 380 73.66 74 140 20 75

120 Without 76 intersection 100 90 70 78 80 80 82 0 84

-20 30 50 70 90 110 X, cm

(a) (b)

t pl FigureFigure 44. 44.( a(a)) Schematic Schematic of of the the HIBP HIBP complex complex on on the the MAST MAST spherical spherical tokamak: tokamak:Bt =B 0.6 = 0.6 T, IT,pl =I 1= MA;1 MA I—injector;; I—injector; A, BA, and B and C inC insetin inset are are various various detector detector positions. positions. Inset Inset demonstrates demonstrates intersection intersection of secondary of secondary trajectories; trajectories; (b) detector (b) detector grid for grid Cs for ions. Cs ions. Areas without intersection the secondary trajectories are marked in red, with the crossing—in blue. Lines of equal Areas without intersection the secondary trajectories are marked in red, with the crossing—in blue. Lines of equal angle are angle are solid, lines of equal energy are dashed. solid, lines of equal energy are dashed.

8.6.8.6. HIBPHIBP Designdesign for the T-15MDT-15MD Tokamaktokamak TheThe T-15T-15 MD MD tokamaktokamak inin thethe NationalNational ResearchResearch CenterCenter KurchatovKurchatov Institute,Institute, Moscow,Moscow, Russia,Russia, isis commissionedcommissioned in 2021. T T-15-15 MD MD intended intended to to study study the the plasma plasma confinement confinement in conditions of a strong magnetic field (Bt  2 T) combined with a small aspect ratio (R/a = in conditions of a strong magnetic field (Bt ≤ 2 T) combined with a small aspect ratio (R/a1.48= m/0.67 1.48 m/0.67 m). It m).is planned It is planned to use to ohmic, use ohmic, ECRH, ECRH, ICRH ICRH and andNBI NBIplasma plasma heating heating and andcurrent current drive drive (CD). (CD). Particular Particular attention attention will will be be paid paid to to the the study study of of the role ofof thethe electric electric fieldfield inin L-H L-H transitions, transitions, Alfv Alfvénén eigenmodes, eigenmodes, and and GAMs. GAMs.These These experimental experimental programs programs requirerequire 2D 2D data data on on the the plasma plasma potential, potential, density, density, magnetic magnetic field field and and its its fluctuations. fluctuations. Currently,Currently, conceptual conceptual design design of of the the HIBP HIBP complex complex for for T-15 T-15 MD MD is is developed developed as as a a part part ofof thethe diagnosticdiagnostic complexcomplex of the machine [ [116116].]. The The HIBP HIBP will will use use the the Tl Tl ion ion beam beam with with Eb E=b 400= 400 keV. keV. T T-15-15 MD MD magnetic magnetic system system was was taken taken into account inin thethe detaileddetailed trajectorytrajectory calculationscalculations [117[117].]. TheThe schemescheme ofof thethe HIBPHIBP complexcomplex is is shown shown in in Figure Figure 45 45aa [ 118[118],], and and the the detectordetector gridgrid is is shown shown in in Figure Figure 45 45b.b. In the future, it is planned to upgrade diagnostics and create a double HIBP, consisting of two identical complexes, shifted along the torus by about 90◦. Thus, the dual HIBP on the T-15 MD tokamak will be basically similar to the dual HIBP on the TJ-II stellarator. The main task of the dual HIBP on T-15 MD is to study zonal flows and long-range correlations of potential and density at the frequencies of GAMs [119,120] and Alfvén eigenmodes [121] including the effects of the ECRH and CD [122]. On top of that, the special attention will be dedicated to the ICRH and CD [123,124].

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(a) (b)

t pl FigureFigure 45.45. ((aa)) SchematicSchematic ofof thethe HIBPHIBP trajectoriestrajectories onon thethe T-15 T-15 MD MD tokamak: tokamak:B Bt = = 22 T,T,I plI == 22 MA;MA; ((bb)) partpart ofof thethe poloidalpoloidal sectionsection ofof T-15T-15 MD MD chamber chamber and and the the detector detector grid. grid. TheThe calculationscalculations werewere performedperformed forfor TlTl ionion beambeam inin reducedreduced magnetic magnetic field Bt = 1 T. Lines of equal energy Eb = 80–220 keV are green. ﬁeld Bt = 1 T. Lines of equal energy Eb = 80–220 keV are green.

8.7. HIBPIn the Design future, for it the is planned W7-X Stellarator to upgrade diagnostics and create a double HIBP, consisting of two identical complexes, shifted along the torus by about 90. Thus, the dual HIBP The conceptual design of the HIBP complex for the world’s largest stellarator W7-X on the T-15 MD tokamak will be basically similar to the dual HIBP on the TJ-II stellarator. was developed in the early 2000s. This installation has the following design parameters: The main task of the dual HIBP on T-15 MD is to study zonal flows and long-range corre- Bt = 3 T, R = 5.5 m, a = 0.53 m. W7-X was commissioned at the end of 2015 in the Max lations of potential and density at the frequencies of GAMs [119,120] and Alfvén Planck Institute for Plasma Physics in Greifswald, Germany. W7-X intended to study the eigenmodes [121] including the effects of the ECRH and CD [122]. On top of that, the plasma confinement in the optimized 3D magnetic configuration. It uses microwave and NBIspecial for plasmaattention heating. will be Particulardedicated attention to the ICR isH focused and CD on [123,124 study the]. role of the electric field at L-H transitions and Alfvén eigenmodes. These experimental programs require data on Symmetry 2021, 13, x FOR PEER REVIEW8.7. HIBP Design for the W7-X Stellarator 41 of 49 the plasma potential and its fluctuations. A preliminary calculation of trajectories for the proposedThe conceptual W7-X plasma design probing of the scheme HIBP iscomplex shown infor Figure the world 46[125’s largest,126]. stellarator W7-X was developed in the early 2000s. This installation has the following design parameters: Bt = 3 T, R = 5.5 m, a = 0.53 m. W7-X was commissioned at the end of 2015 in the Max Planck Institute for Plasma Physics in Greifswald, Germany. W7-X intended to study the plasma confinement in the optimized 3D magnetic configuration. It uses microwave and NBI for plasma heating. Particular attention is focused on study the role of the electric field at L-H transitions and Alfvén eigenmodes. These experimental programs require data on the plasma potential and its fluctuations. A preliminary calculation of trajectories for the proposed W7-X plasma probing scheme is shown in Figure 46 [125,126].

FigureFigure 46.46.Scheme Scheme of of HIP HIP for for the the W7-X W7 stellarator.-X stellarator. (1)—primary (1)—primary trajectories trajectories injected injected through through the port the port A11 at the bottom of the figure; (2)—fan of secondary trajectories directed to the analyzer A11 at the bottom of the ﬁgure; (2)—fan of secondary trajectories directed to the analyzer through through the port N11; 3—total fan of secondary trajectories. the port N11; (3)—total fan of secondary trajectories. 8.8. HIBP Design for the International Experimental Tokamak Reactor ITER The ITER, which is currently being assembled in the city of Cadarache in the south of France, aims to achieve the fusion power amplification factor Q = 5. The ITER diagnostics are subordinated to this engineering task. They are primarily focused to achieving the design parameters of the machine and provide the safety of its operation. So, HIBP is not included in the list of top-priority diagnostics for ITER, but it may be used in studying the physics of the L-H transition. Therefore, the HIBP allows one to study the periphery of the ITER plasma. Estimations have shown that HIBP will operate with a Tl ion beam accelerated to the energy of Eb = 3.5 MeV [127]. This is a standard energy range for large- scale machines in operation [19]. ITER has ovoid-shaped toroidal field coils. The scheme of the HIBP complex is shown in Figure 47, and the detector grid is shown in Figure 48a. In the standard ITER mode, the nominal plasma current will lead to a noticeable toroidal shift of the trajectories of the probing particles, shown in Figure 48b. We see that with the initial toroidal shift zI = 0 at the injection point I and the value of the toroidal angle β = 7, the toroidal coordinate of the detection point D, zD = 10 cm, and easily fits into the dimensions of the horizontal port. Because in ITER the discharge will transit to the H-mode that is always accompanied by change in the electric field [52,128,129], the measurements of the electric potential and turbulence will be an important issue for the physics of ITER. Recently, it was first shown by HIBP in the T-10 tokamak that in the discharges with low collisionality—which mimic the ITER plasma condition—the core plasma potential has positive values and an enhanced level of turbulence, so the prediction for the ITER plasma was made [130]. Verification of this prediction could be a very important and challenging task for physics studies in ITER.

Symmetry 2021, 13, 1367 39 of 46

8.8. HIBP Design for the International Experimental Tokamak Reactor ITER The ITER, which is currently being assembled in the city of Cadarache in the south of France, aims to achieve the fusion power amplification factor Q = 5. The ITER diagnostics are subordinated to this engineering task. They are primarily focused to achieving the design parameters of the machine and provide the safety of its operation. So, HIBP is not included in the list of top-priority diagnostics for ITER, but it may be used in studying the physics of the L-H transition. Therefore, the HIBP allows one to study the periphery of the ITER plasma. Estimations have shown that HIBP will operate with a Tl ion beam accelerated to the energy of Eb = 3.5 MeV [127]. This is a standard energy range for large- scale machines in operation [19]. ITER has ovoid-shaped toroidal field coils. The scheme of the HIBP complex is shown in Figure 47, and the detector grid is shown in Figure 48a. In the standard ITER mode, the nominal plasma current will lead to a noticeable toroidal shift of the trajectories of the probing particles, shown in Figure 48b. We see that with the initial toroidal shift zI = 0 at the injection point I and the value of the toroidal angle ◦ β = 7 , the toroidal coordinate of the detection point D, zD = 10 cm, and easily fits into the dimensions of the horizontal port. Because in ITER the discharge will transit to the H-mode that is always accompanied by change in the electric field [52,128,129], the measurements of the electric potential and turbulence will be an important issue for the physics of ITER. Recently, it was first shown by HIBP in the T-10 tokamak that in the discharges with low collisionality—which mimic the ITER plasma condition—the core plasma potential has Symmetry 2021, 13, x FOR PEER REVIEWpositive values and an enhanced level of turbulence, so the prediction for the ITER plasma 42 of 49 was made [130]. Verification of this prediction could be a very important and challenging task for physics studies in ITER.

FigureFigure 47.47. SchematicSchematic of of the the HIBP HIBP complex complex on the on fusion the fusion reactor reactor ITER: B ITER:t = 6 T, BItpl == 6 15T, MA.Ipl = The15 MA. inset The inset showsshows thethe detectordetector grid grid and and fragments fragments of ion of beamion beam lines. lines. The asterisks The asterisks mark the mark detector the detector line, which line, which allowsallows one one to to study study the theradial radial proﬁles profiles of the plasma of the potential; plasma potential;I—injection I point,—injection D—detection point, point, D—detection point,dimensions dimensions in cm. in cm.

(a) (b)

Figure 48. (a) The detector grid for ITER. Calculated for Eb = 2.6 − 3.4 MeV with steps of 0.1 MeV and angle 0.5. The line for α = 13 allows one to probe a fragment of the potential profile at the plasma periphery, 0.76 < r/a < 0.98, by changing the energy from shot to shot. The line Eb = 3.3 MeV allows one to obtain a fragment of the potential profile 0.76 < r/a < 1 by changing the injection angle β. (b) Toroidal shift in the equatorial projection of the detector line with α = 14.

9. Summary Over more than 30 years of development, diagnostics of fusion plasma using a beam of heavy ions has become a powerful and multifunctional tool for physical research at the magnetic confinement devices. Starting with a time-averaged potential signal measured in the single spatial point, it becomes the diagnostic for studying 2D spatial distributions of plasma potential and turbulence, and even 3D characteristics of turbulence. On modern operating medium-scale machines, this diagnostic makes a significant contribution to re- veal the role of electric fields and turbulence in confinement and the transport of plasma energy and particles, physics of fast particles and Alfvén eigenmodes. These advances have allowed HIBP to study the spatial and frequency characteristics of toroidally symmetric structures: geodesic acoustic modes and zonal flows, and helically symmetric

Symmetry 2021, 13, x FOR PEER REVIEW 42 of 49

Figure 47. Schematic of the HIBP complex on the fusion reactor ITER: Bt = 6 T, Ipl = 15 MA. The inset shows the detector grid and fragments of ion beam lines. The asterisks mark the detector line, which allows one to study the radial profiles of the plasma potential; I—injection point, D—detection Symmetry 2021, 13, 1367 40 of 46 point, dimensions in cm.

(a) (b)

◦ FigureFigure 48. (a 48.) The (a) detector The detector grid for grid ITER. for Calculated ITER. Calculated for Eb = 2.6–3.4 for EbMeV = 2.6 with − 3.4 steps MeV of with 0.1 MeV steps and of angle0.1 MeV 0.5 .and The line for α = 13angle◦ allows 0.5 one. The to line probe for a α fragment = 13 allows of the one potential to probe proﬁle a fragment at the plasma of the periphery, potential 0.76 profile < r/a

energyperiphery, from shot 0.76 to shot. < r/a The < 0.98, line byEb =changing 3.3 MeV allowsthe ener onegy to from obtain shot a fragment to shot. ofThe the line potential Eb = 3.3 profile MeV 0.76 allows < r/ a < 1 by changingone to the obtain injection a fragment angle β.( bof) Toroidalthe potential shift in profile the equatorial 0.76 < r projection/a < 1 by changing of the detector the injection line with αangle= 14◦ β. . (b) Toroidal shift in the equatorial projection of the detector line with α = 14. 9. Summary 9. Summary Over more than 30 years of development, diagnostics of fusion plasma using a beam Over more thanof 30 heavy years ions of hasdevelopment, become a powerful diagnostics and multifunctional of fusion plasma tool forusing physical a beam research at the magnetic confinement devices. Starting with a time-averaged potential signal measured in of heavy ions has become a powerful and multifunctional tool for physical research at the the single spatial point, it becomes the diagnostic for studying 2D spatial distributions of magnetic confinementplasma devices. potential Start anding turbulence, with a time and-averaged even 3D characteristicspotential signal of turbulence. measured On modern in the single spatialoperating point, it becomes medium-scale the diagnostic machines, for this study diagnosticing 2D makes spatial a significantdistributions contribution to of plasma potential revealand turbulence, the role of electric and even fields 3D and characteristics turbulence in of confinement turbulence. and On the modern transport of plasma operating medium-energyscale machines, and particles, this physics diagnostic of fast makes particles a andsignificant Alfvén eigenmodes. contribution These to re- advances have veal the role of electricallowed fields HIBP and to turbulence study the spatial in con andfinement frequency and characteristics the transport of of toroidally plasma symmetric energy and particles,structures: physics geodesic of fast acousticparticles modes and Alfvén and zonal eigenmodes. flows, and helicallyThese advances symmetric structures: have allowed HIBPAlfv to studyén Eigenmodes. the spatial The and future frequency studies characteristics will be focused of on toroidally the further sym- development of operating HIBP capabilities and the installations to the other considered toroidal devices. metric structures: geodesic acoustic modes and zonal flows, and helically symmetric Funding: This research was funded by Russian Science Foundation, project number 19–12-00312 and supported in part by Competitiveness Program of MEPhI. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable. Acknowledgments: I warmly acknowledge the long-term collaboration with my colleagues from HIBP group and T-10 tokamak in Kurchatov Institute, Moscow, Russia, HIBP group of Kharkov Institute of Physics and Technology, Kharkov, Ukraine, leaded by L.I. Lrupnik and A.S. Kozachek, and also colleagues from CIEMAT, Madrid, Spain, leaded by Carlos Hidalgo. Special words of thanks come to my teacher, Yu. N. Dnestrovskij and Sergey E. Lysenko for fruitful discussions and helpful comments. Conﬂicts of Interest: The author declares no conﬂict of interest. Symmetry 2021, 13, 1367 41 of 46

References 1. Tikhonov, A.N.; Arsenin, V.V. Solutions of Ill Posed Problems; John, F., Translator, V.H., Eds.; Winston and Sons: Silver Spring, MD, USA; Wiley: New York, NY, USA, 1977. 2. Dnestrovskij, Y.N.; Kostomarov, D.P.; Melnikov, A.V. Mathematical problems of active corpuscular diagnostics of plasmas. Proc. Moscov State Univ. Ser. 15 Appl. Math. Cybern. 1986, 3, 20–28. (In Russian) 3. Melnikov, A.V. Applied and fundamental aspects of fusion science. Nat. Phys. 2016, 12, 386–390. [CrossRef] 4. Van Oost, G.; Bulanin, V.V.; Donne, A.J.H.; Gusakov, E.; Kraemer-Flecken, A.; Krupnik, L.I.; Melnikov, A.; Nanobashvili, S.; Peleman, P.; A Razumova, K.; et al. Multi-machine studies of the role of turbulence and electric fields in the establishment of improved confinement in tokamak plasmas. Plasma Phys. Control. Fusion 2007, 49, A29–A44. [CrossRef] 5. Kadomtsev, B.B. Collective Phenomena in Plasmas; Pergamon Press: Oxford, UK, 1982; ISBN 10-0080250182/13-978-0080250182. 6. Burrell, K.H.; Carlstrom, T.N.; Doyle, E.J.; Finkenthal, D.; Gohil, P.; Groebner, R.J.; Hillis, D.L.; Kim, J.; Matsumoto, H.; Moyer, R.A.; et al. Physics of the L-mode to H-mode transition in tokamaks. Plasma Phys. Control. Fusion 1992, 34, 1859. [CrossRef] 7. Tendler, M. Major achievements and challenges of fusion research. Phys. Scr. 2015, 90, 98002. [CrossRef] 8. Melnikov, A.V. The study of electric potential of magnetically confined fusion plasmas. Comput. Nanotechnol. 2017, 2, 13–23. Available online: https://www.elibrary.ru/item.asp?id=29368049 (accessed on 1 January 2021). (In Russian) 9. Melnikov, A.V. The evolution of the mathematical and technical aspects of heavy ion beam probe plasma diagnostics. Comput. Nanotechnol. 2018, 1, 31–57. Available online: https://www.elibrary.ru/item.asp?id=32775187n (accessed on 1 January 2021). (In Russian) 10. Dnestrovskij, Y.; Melnikov, A.; Krupnik, L.; Nedzelskij, I. Development of heavy ion beam probe diagnostics. IEEE Trans. Plasma Sci. 1994, 22, 310–331. [CrossRef] 11. Jobes, F.; Hickok, R. A direct measurement of plasma space potential. Nucl. Fusion 1970, 10, 195–197. [CrossRef] 12. Bugarya, V.I.; Gorshkov, A.V.; Grashin, S.A.; Ivanov, I.V.; Krupin, V.A.; Krupnik, L.I.; Nedzel’skij, I.S.; Mel’nikov, A.V.; Razumova, K.A.; Sokolov, Y.A.; et al. Electric potential and toroidal and poloidal rotation velocities of a tokamak plasma. JETP Lett. 1983, 38, 404–408. 13. Bugarya, V.; Gorshkov, A.; Grashin, S.; Ivanov, I.; Krupin, V.; Mel0nikov, A.; Razumova, K.; Sokolov, Y.; Trukhin, V.; Chankin, A.; et al. Measurements of plasma column rotation and potential in the TM-4 tokamak. Nucl. Fusion 1985, 25, 1707–1717. [CrossRef] 14. Donné, A.J.H.; Melnikov, A.V.; Van Oost, G. Diagnostics for radial electric field measurements in hot magnetized plasmas. Czechoslov. J. Phys. 2002, 52, 1077–1096. [CrossRef] 15. Melnikov, A.; Krupnik, L.; Eliseev, L.; Barcala, J.M.; Bravo, A.; Chmyga, A.; Deshko, G.; Drabinskij, M.; Hidalgo, C.; Khabanov, P.; et al. Heavy ion beam probing—diagnostics to study potential and turbulence in toroidal plasmas. Nucl. Fusion 2017, 57, 072004. [CrossRef] 16. Askinazi, L.G.; Kornev, V.A.; Lebedev, S.V.; Tukachinsky, A.S.; Zhubr, N.A.; Dreval, N.B.; Krupnik, L.I. Heavy ion beam probe development for the plasma potential measurement on the TUMAN-3M tokamak. Rev. Sci. Instrum. 2004, 75, 3517–3519. [CrossRef] 17. Cabral, J.A.C.; Malaquias, A.; Praxedes, A.; Van Toledo, W.; Varandas, C. The heavy ion beam diagnostic for the tokamak ISTTOK. IEEE Trans. Plasma Sci. 1994, 22, 350–358. [CrossRef] 18. Bondarenko, I.S.; Gubarev, S.P.; Krupnik, L.I.; Mel’nikov, A.V.; Nedszel’skii, I.S.; Utkin, A.A.; Khrebtov, S.M. Plasma diagnostics using a heavy-ion beam in the T-10 tokamak. Sov. J. Plasma Phys. 1992, 18, 110. 19. Fujisawa, A.; Iguchi, H.; Taniike, A.; Sasao, M.; Hamada, Y. A 6 MeV heavy ion beam probe for the Large Helical Device. IEEE Trans. Plasma Sci. 1994, 22, 395–402. [CrossRef] 20. Ido, T.; Shimizu, A.; Nishiura, M.; Kato, S.; Nakano, H.; Ohshima, S.; Yokoyama, M.; Murakami, S.; Wakasa, A.; Nakamura, S.; et al. Development of 6-MeV Heavy Ion Beam Probe on LHD. Fusion Sci. Technol. 2010, 58, 436–444. [CrossRef] 21. Melnikov, A.; Hidalgo, C.; Ido, T.A.; Shimizu, A.; Fujisawa, K.; Dyabilin, S.; Lysenko, S.E. Plasma Potential in Toroidal Devices: T-10, TJ-II, CHS and LHD. Plasma Fusion Res. 2012, 7, 2402114. [CrossRef] 22. Bondarenko, I.S.; Chmuga, A.A.; Dreval, N.B.; Khrebtov, S.M.; Komarov, A.D.; Kozachok, A.S.; Krupnik, L.I.; Coelho, P.; Cunha, M.; Gonçalves, B.; et al. Installation of an advanced heavy ion beam diagnostic on the TJ-II stellarator. Rev. Sci. Instrum. 2001, 72, 583–585. [CrossRef] 23. Melnikov, A.; Dyabilin, K.; Eliseev, L.; Lysenko, S.; Dnestrovskij, Y. Measurement and modelling of electric potential in the TJ-II stellarator. Probl. At. Sci. Technol. Ser. Thermonucl. Fusion 2011, 34, 54–73. (In Russian) [CrossRef] 24. Lei, J.; Shah, U.; Demers, D.R.; Connor, K.A.; Schoch, P.M. Calibration and initial operation of the HIBP on the MST. Rev. Sci. Instrum. 2001, 72, 564–567. [CrossRef] 25. Yoshikawa, M.; Sakamoto, M.; Miyata, Y.; Aoyama, M. Potential fluctuation study from the core plasma to end region in GAMMA 10. Nucl. Fusion 2013, 53, 073031. [CrossRef] 26. Dreval, N.; Krupnik, L.; Chmyga, A.A.; Khrebtov, S.M.; Komarov, A.D.; Kozachek, A.S.; Hidalgo, C.; Melnikov, A.V.; Eliseev, L.G. Features of HIBP diagnostics application to stellarator-like devices. Probl. At. Sci. Technol. Ser. Plasma Phys. 2005, 2, 223–225. 27. Dnestrovskii, Y.N.; Krupnik, L.I.; Mel’nikov, A.V.; Nedzel’skii, I.S. Density determination by probing with heavy-ion beams. Sov. J. Plasma Phys. 1986, 12, 130. 28. Schwelberger, J.G.; Aceto, S.C.; Connor, K.A.; Zielinski, J.J.; Baylor, L.A.; England, A.C.; Isler, R.; Ma, C.H.; Murakami, M.; Uckan, T.; et al. Electron density profile measurement with a heavy ion beam probe. Rev. Sci. Instrum. 1992, 63, 4571–4573. [CrossRef] Symmetry 2021, 13, 1367 42 of 46

29. Khabanov, P.; Eliseev, L.; Melnikov, A.; Drabinskiy, M.; Hidalgo, C.; Kharchev, N.; Chmyga, A.; Kozachek, A.; Pastor, I.; De Pablos, J.; et al. Density profile reconstruction using HIBP in ECRH plasmas in the TJ-II stellarator. J. Instrum. 2019, 14, C09033. [CrossRef] 30. Levinton, F.M. The motional Stark effect: Overview and future development (invited). Rev. Sci. Instrum. 1999, 70, 810–814. [CrossRef] 31. Dnestrovskii, Y.N.; Mel’nikov, A.V. Determination of the current profile in a plasma by probing with heavy-ion beams. Sov. J. Plasma Phys. 1986, 12, 393. 32. Weisen, H.; Melnikov, A.V.; Perfilov, S.; Lysenko, S. On the Possibility of Using a Heavy Ion Beam Probe for Local Poloidal Flux Measurements in a Tokamak. Fusion Sci. Technol. 2011, 59, 418–426. [CrossRef] 33. Melnikov, A.V. Electric Potential in Toroidal Plasmas; Springer Nature Switzerland AG: Basel, Switzerland, 2019; p. 240. ISBN 978-3-030-03480-1. 34. Khabanov, P.O. Computational analysis density fluctuations measurements with heavy ion beam probe. Comput. Nanotechnol. 2018, 1, 25–30. Available online: https://www.elibrary.ru/item.asp?id=32775186 (accessed on 1 January 2021). (In Russian) 35. Melnikov, A.V.; Eliseev, L.G.; Jiménez-Gómez, R.; Ascasibar, E.; Hidalgo, C.; Chmyga, A.A.; Ido, T.; Khrebtov, S.M.; Könies, A.; Komarov, A.D.; et al. Study of Alfvén eigenmodes in the TJ-II stellarator. Plasma Fusion Res. 2010, 5, S2019. [CrossRef] 36. Simcic, V.J.; Crowley, T.P.; Schoch, P.M. Internal magnetic and electrostatic fluctuation measurements of MHD modes in the TEXT tokamak. Phys. Fluids B 1993, 5, 1576. [CrossRef] 37. McKee, G.R.; Fonck, R.J.; Jakubowski, M.; Burrell, K.H.; Hallatschek, K.; A Moyer, R.; Nevins, W.; Rudakov, D.L.; Xu, X. Observation and characterization of radially sheared zonal flows in DIII-D. Plasma Phys. Control. Fusion 2003, 45, A477–A485. [CrossRef] 38. Zenin, V.N. Geodesic acoustic modes in the tokamaks. Comput. Nanotechnol. 2018, 1, 108–113. Available online: https: //www.elibrary.ru/item.asp?id=32775193 (accessed on 1 January 2021). (In Russian) 39. Zenin, V.N.; Eliseev, L.G.; Kozachek, A.S.; Krupnik, L.I.; Lysenko, S.E.; Melnikov, A.V.; Perfilov, S.V. Study of poloidal structure of geodesic acoustic modes in the T-10 tokamak with heavy ion beam probing. Probl. At. Sci. Technol. Ser. Plasma Phys. 2014, 6, 269–271. Available online: http://vant.kipt.kharkov.ua/TABFRAME.html (accessed on 1 January 2021). 40. Eliseev, L.G.; Ivanov, N.V.; Kakurin, A.M.; Lysenko, S.E.; Maltsev, S.G.; Melnikov, A.V.; Perfilov, S.V.; Zenin, V.N. Study of the large-scale MHD mode and its effect on GAM in the T-10 tokamak. In Proceedings of the 42nd EPS Conference on Plasma Physics, Lisbon, Portugal, 22—26 June 2015; ECA. Volume 39E, p. P5.159. Available online: http://ocs.ciemat.es/EPS2015PAP/Pdf/P5.1 59.pdf (accessed on 1 January 2021). 41. Melnikov, A.V.; Eliseev, L.G.; Lysenko, S.E.; Perfilov, S.V.; Shelukhin, D.A.; Vershkov, V.A.; Zenin, V.N.; Krupnik, L.I.; Kharchev, N.K. Correlation properties of Geodesic Acoustic Modes in the T-10 tokamak. J. Phys. Conf. Ser. 2015, 591, 012003. [CrossRef] 42. Melnikov, A.V.; Eliseev, L.G.; Gudozhnik, A.V.; Lysenko, S.E.; Mavrin, V.A.; Perfilov, S.V.; Zimeleva, L.G.; Ufimtsev, M.V.; Krupnik, L.I.; Schoch, P.M. GAM evolution with ECRH Investigation of the plasma potential oscillations in the range of geodesic acoustic mode frequencies by heavy ion beam probing in tokamaks. Czech. J. Phys. 2005, 55, 349–360. [CrossRef] 43. Breizman, B.N.; Pekker, M.S.; Sharapov, S.E. Plasma pressure effect on Alfvén cascade eigenmodes. Phys. Plasmas 2005, 12, 112506. [CrossRef] 44. Eliseev, L.G.; Melnikov, A.V.; Ascasibar, E.; Cappa, A.; Drabinskiy, M.; Hidalgo, C.; Khabanov, P.O.; Kharchev, N.K.; Kozachek, A.S.; Liniers, M. Experimental observation of the geodesic acoustic frequency limit for the NBI-driven Alfvén eigenmodes in TJ-II. Phys. Plasmas 2021, 28.[CrossRef] 45. Melnikov, A.V.; Eliseev, L.G.; Jiménez-Gómez, R.; Ascasibar, E.; Hidalgo, C.; Chmyga, A.A.; Komarov, A.D.; Kozachok, A.S.; Krasilnikov, I.A.; Khrebtov, S.M.; et al. Internal measurements of Alfvén eigenmodes with heavy ion beam probing in toroidal plasmas. Nucl. Fusion 2010, 50, 084023. [CrossRef] 46. Melnikov, A.; Eliseev, L.; Lysenko, S.E.; Ufimtsev, M.; Zenin, V. Study of interactions between GAMs and broadband turbulence in the T-10 tokamak. Nucl. Fusion 2017, 57, 115001. [CrossRef] 47. Gryaznevich, M.; Stockel, J.; Van Oost, G.; Del Bosco, E.; Svoboda, V.; Melnikov, A.; Kamendje, R.; Malaquias, A.; Mank, G.; Miklaszewski, R.; et al. Contribution of joint experiments on small tokamaks in the framework of IAEA coordinated research projects to mainstream fusion research. Plasma Sci. Technol. 2020, 22, 055102. [CrossRef] 48. Riggs, G.A.; Nogami, S.; Koepke, M.; Melnikov, A.; Eliseev, L.; Lysenko, S.; Khabanov, P.; Drabinskij, M.; Kharchev, N.; Kozachek, A.; et al. Bispectral analysis of broadband turbulence and geodesic acoustic modes in the T-10 tokamak. J. Plasma Phys. 2021, 87. [CrossRef] 49. Powers, E. Spectral techniques for experimental investigation of plasma diffusion due to polychromatic fluctuations. Nucl. Fusion 1974, 14, 749–752. [CrossRef] 50. Melnikov, A.V.; Eliseev, L.G.; Alonso, A.; Ascasiba, E.; Cappa, A.; Chmyga, A.A.; Estrada, T.; Hidalgo, C.; Jiménez-Gómez, R.; Krasilnikov, I.; et al. Plasma potential profiles and oscillations in ECRH and NBI plasmas in the TJ-II stellarator. In Proceedings of the 37th EPS Conference on Plasma Physics, Dublin, Ireland, 21–25 June 2010; ECA. Volume 34A, p. Rep. P1.066. Available online: http://ocs.ciemat.es/EPS2010PAP/html/ (accessed on 1 January 2021). 51. Melnikov, A.V.; Eliseev, L.G.; Ascasibar, E.; Chmyga, A.A.; Hidalgo, C.; Ido, T.; Jiménez-Gomez, R.; Komarov, A.D.; Kozachek, A.S.; Krupnik, L.I.; et al. Alfvén eigenmode properties and dynamics in the TJ-II stellarator. Nucl. Fusion 2012, 52, 123004. [CrossRef] Symmetry 2021, 13, 1367 43 of 46

52. Melnikov, A.; Eliseev, L.; Estrada, T.; Ascasibar, E.; Alonso, A.; Chmyga, A.; Hidalgo, C.; Komarov, A.; Kozachek, A.; Krupnik, L.; et al. The changes in plasma potential and turbulent particle flux in the core plasma measured by heavy ion beam probe during L-H transitions in the TJ-II stellarator. Nucl. Fusion 2013, 53, 092002. [CrossRef] 53. Eliseev, L.G.; Lysenko, S.E.; Melnikov, A.V.; Krupnik, L.I.; Kozachek, A.S.; Zenin, V.N. Study of GAMs and related turbulent particle flux with HIBP in the T-10 tokamak. Probl. At. Sci. Technol. Ser. Plasma Phys. 2017, 107, 241–243. 54. Eliseev, L.G.; Zenin, V.N.; Lysenko, S.E.; Melnikov, A.V. Measurement of geodesic acoustic modes and the turbulent particle flux in the T-10 tokamak plasmas. J. Phys. Conf. Ser. 2017, 907, 12002. [CrossRef] 55. Eliseev, L.G.; Melnikov, A.V.; Lysenko, S.E.; Khabanov, P.O.; Zenin, V.N.; Drabinskij, M.A.; Kharchev, N.K.; Kozachek, A.S.; Krupnik, L.I.; Team, H. Evaluation of Turbulent Particle Flux by Heavy Ion Beam Probe in the T-10 Tokamak. Plasma Fusion Res. 2018, 13, 3402106. [CrossRef] 56. Eliseev, L.; Melnikov, A.; Perfilov, S.; Hidalgo, C. Two point correlation technique for the measurements of poloidal plasma rotation by heavy ion beam probe. Plasma Fusion Res. 2012, 7, 2402064. [CrossRef] 57. Drabinskiy, M.A.; Eliseev, L.G.; Khabanov, P.O.; Melnikov, A.V.; Khartchev, N.K.; Sergeev, N.S.; Grashin, S.A. Radial structure of quasi-coherent mode in ohmic plasma of the T-10 tokamak. J. Phys. Conf. Ser. 2019, 1383, 012004. Available online: https://iopscience.iop.org/article/10.1088/1742-6596/1383/1/012004 (accessed on 1 January 2021). [CrossRef] 58. Melnikov, A.; Eliseev, L.; Perfilov, S.; Lysenko, S.; Shurygin, R.; Zenin, V.; Grashin, S.; Krupnik, L.; Kozachek, A.; Solomatin, R.; et al. The features of the global GAM in OH and ECRH plasmas in the T-10 tokamak. Nucl. Fusion 2015, 55, 063001. [CrossRef] 59. Bondarenko, I.S.; Krupnik, L.I.; Nedzel’skii, I.S.; Mel’nikov, A.V. Plasma probing with a heavy ion beam in the TM-4 tokamak. Sov. J. Tech. Phys. 1986, 31, 1390. 60. Melnikov, A.V.; Bondarenko, I.S.; Efremov, S.L.; Kharchev, N.K.; Khrebotv, S.M.; Krupnik, L.I.; Nedzelskij, I.S.; Zimeleva, L.G.; Trofimenko, Y.V. HIBP diagnostics on T-10. Rev. Sci. Instrum. 1995, 66, 317. [CrossRef] 61. Melnikov, A.V.; Kharchev, N.K.; Eliseev, L.G.; Bondarenko, I.S.; Krupnik, L.I.; Khrebtov, S.M.; Nedzelskij, I.S.; Trofimenko, Y.V. Calibration of the heavy ion beam probe parallel plate analyzer using the gas target and reference beam. Rev. Sci. Instrum. 1997, 68, 308–311. [CrossRef] 62. Krupnik, L.; Melnikov, A.; Nedselskij, I. Development of beam probe diagnostics and recent measurements on the TJ-1 and T-10 tokamaks. Fusion Eng. Des. 1997, 34–35, 639–644. [CrossRef] 63. Melnikov, A.V.; Krupnik, L.I.; Nedzelskij, I.S.; Kharchev, N.K. Plasma potential measurements in the T-10 tokamak. In Proceedings of the 18th EPS Conference on Control. Fusion and Plasma Physics, Berlin, Germany, 3–7 June 1991; Volume 15C, p. 221. 64. Melnikov, A.V.; Zimeleva, L.G.; Eliseev, L.G.; Bondarenko, I.S.; Krupnik, L.I.; Khrebtov, S.M.; Nedzelskij, I.S.; Kharchev, N.K. HIBP measurements of the plasma electric potential on T-10. In Proceedings of the 23rd EPS Conference on Controlled Fusion and Plasma Physics, Kiev, Ukraine, 24–28 June 1996; Volume I, p. 434. 65. Melnikov, A.V.; Tarasyan, K.N.; Vershkov, V.A.; Dreval, V.A.; Sushkov, A.V.; Lysenko, S.E.; Sannikov, V.V.; Gorshkov, A.V.; Krupnik, L.I.; Nedzelskij, I.S. Space and time evolution of plasma potential in T-10 under variation of main gas influx. IEEE Trans. Plasma Sci. 1994, 22, 363–368. [CrossRef] 66. Melnikov, A.; Drabinskiy, M.; Eliseev, L.; Khabanov, P.; Kharchev, N.; Krupnik, L.; De Pablos, J.; Kozachek, A.; Lysenko, S.; Molinero, A.; et al. Heavy ion beam probe design and operation on the T-10 tokamak. Fusion Eng. Des. 2019, 146, 850–853. [CrossRef] 67. Kharchev, N.K. Correction of toroidal displacement of diagnostic heavy ion beam. Comput. Nanotechnol. 2018, 1, 58–61. Available online: https://www.elibrary.ru/item.asp?id=32775188 (accessed on 1 January 2021). (In Russian) 68. Drabinskiy, M.A. The study of the electric potential of magnetically confined fusion plasmas. Comput. Nanotechnol. 2018, 1, 62–70. Available online: https://www.elibrary.ru/item.asp?id=32775189 (accessed on 1 January 2021). (In Russian) 69. Bondarenko, I.S.; Khrebtov, S.M.; Krupnik, L.I.; Nedzelskij, I.S.; Gordeev, O.A.; Kharchev, N.K.; Melnikov, A.V.; Tarasyan, K.N.; Zimeleva, L.G.; Hidalgo, C.; et al. Heavy ion beam probe diagnostics on TJ-1 tokamak and the measurements of the plasma potential and density profiles. Rev. Sci. Instruments 1997, 68, 312–315. [CrossRef] 70. Melnikov, A.V.; Zimeleva, L.G.; Krupnik, L.I.; Nedzelskij, I.S.; Trofimenko, Y.V.; Minaev, V. Heavy ion beam probe systems for tight aspect ratio tokamaks. Rev. Sci. Instrum. 1997, 68, 316–319. [CrossRef] 71. Krupnik, L.I.; Deshko, G.N.; Zhezhera, A.I.; Chmyga, A.A.; Komarov, A.D.; Kozachek, A.S.; Melnikov, A.V.; Perfilov, S.V.; Otte, M.; Shubert, M. The heavy ion beam probing development for WEGA stellarator. Fusion Sci. Technol. 2006, 50, 276–280. [CrossRef] 72. Podoba, Y.; Otte, M.; Wagner, F.; Bondarenko, I.; Chmyga, A.; Deshko, G.; Komarov, A.; Kozachek, A.; Krupnik, L.; Khrebtov, S.; et al. First HIBP results on the WEGA stellarator. AIP Conf. Proc. 2008, 993, 235–238. [CrossRef] 73. Bondarenko, I.S.; Chmyga, A.A.; Deshko, G.N.; Komarov, A.D.; Kozachok, A.S.; Krupnik, L.I.; Khrebtov, S.M.; Zhezhera, A.I.; Podoba, Y.Y.; Otte, M.; et al. HIBP results on the WEGA stellarator. Probl. At. Sci. Technol. Ser. Plasma Phys. 2009, 15, 28–30. 74. Krupnik, L.I.; Mel’nikov, A.V.; Nedzel’skii, I.S.; Samokhvalov, N.V.; Schwelberger, J.; Connor, K.A. Plasma diagnostics using a heavy-ion beam in the U-2M stellarator. Plasma Phys. Rep. 1994, 20, 173–175. 75. Krupnik, L. 50 years of hot plasma diagnostic with heavy ion beam probing (HIBP) at the Kharkov institute of physics and technology. Probl. At. Sci. Technol. Ser. Plasma Phys. 2021, 154–162. [CrossRef] 76. Bondarenko, I.S.; Chmyga, A.A.; Dreval, N.B.; Khrebtov, S.M.; Komarov, A.D.; Kozachok, A.S.; Krupnik, L.I.; Melnikov, A.V.; Yudina, O.A.; Coelho, P.; et al. Installation of the advanced heavy ion beam probing diagnostic on the TJ-II stellarator. Czechoslov. J. Phys. 2000, 50, 1397–1412. [CrossRef] Symmetry 2021, 13, 1367 44 of 46

77. Krupnik, L.; Bondarenko, I.; Chmyga, A.; Dreval, M.; Khrebtov, S.; Komarov, A.; Kozachok, A.; Hidalgo, C.; García-Cortés, I.; Rodriguez-Rodrigo, L.; et al. The first operation of the advanced heavy ion beam probing diagnostic on the TJ-II flexible heliac. Fusion Eng. Des. 2001, 56–57, 935–939. [CrossRef] 78. Melnikov, A.V.; Hidalgo, C.; Chmyga, A.A.; Dreval, N.B.; Eliseev, L.; Khrebtov, S.M.; Komarov, A.D.; Kozachok, A.S.; Krupnik, L.I.; Pastor, I.; et al. Plasma potential measurements by the heavy ion beam probe diagnostic in fusion plasmas: Biasing experiments in the TJ-II stellarator and T-10 tokamak. Fusion Sci. Technol. 2004, 46, 299–311. [CrossRef] 79. Hidalgo, C.; Pedrosa, M.A.; Dreval, N.; McCarthy, K.J.; Eliseev, L.; Ochando, M.A.; Estrada, T.; Pastor, I.; Ascasíbar, E.; Calderón, E.; et al. Improved confinement regimes induced by limiter biasing in the TJ-II stellarator. Plasma Phys. Control. Fusion 2003, 46, 287–297. [CrossRef] 80. Estrada, T.; Krupnik, L.; Dreval, N.; Melnikov, A.; Khrebtov, S.M.; Hidalgo, C.; van Milligen, B.; Castejón, F.; Ascasíbar, E.; Eliseev, L.; et al. Electron internal transport barrier formation and dynamics in the plasma core of the TJ-II stellarator. Plasma Phys. Control. Fusion 2003, 46, 277–286. [CrossRef] 81. Estrada, T.; Alonso, A.; Chmyga, A.A.; Dreval, N.; Eliseev, L.; Hidalgo, C.; Komarov, A.D.; Kozachok, A.S.; Krupnik, L.; Melnikov, A.V.; et al. Electron internal transport barriers, rationals and quasi-coherent oscillations in the stellarator TJ-II. Plasma Phys. Control. Fusion 2005, 47, L57–L63. [CrossRef] 82. Estrada, T.; Happel, T.; Eliseev, L.; López-Bruna, D.; Ascasíbar, E.; Blanco, E.; Cupido, L.; Fontdecaba, J.M.; Hidalgo, C.; Jiménez- Gómez, R.; et al. Sheared flows and transition to improved confinement regime in the TJ-II stellarator. Plasma Phys. Control. Fusion 2009, 51.[CrossRef] 83. Van Milligen, B.P.; Pedrosa, M.A.; Hidalgo, C.; Carreras, B.A.; Estrada, T.; Alonso, J.A.; de Pablos, J.L.; Melnikov, A.; Krupnik, L.; Eliseev, L.G.; et al. The dynamics of the formation of the edge particle transport barrier at TJ-II. Nucl. Fusion 2011, 51, 113002. [CrossRef] 84. Melnikov, A.V.; Alonso, A.; Ascasíbar, E.; Balbin, R.; Chmyga, A.A.; Dnestrovskij, Y.N.; Eliseev, L.; Estrada, T.; Fontdecaba, J.M.; Fuentes, C.; et al. Plasma potential evolution study by HIBP diagnostic during NBI experiments in the TJ-II stellarator. Fusion Sci. Technol. 2007, 51, 31–37. [CrossRef] 85. Melnikov, A.; Hidalgo, C.; Eliseev, L.; Ascasibar, E.; Chmyga, A.; Dyabilin, K.; Krasilnikov, I.; Krupin, V.; Krupnik, L.; Khrebtov, S.; et al. Plasma potential and turbulence dynamics in toroidal devices (survey of T-10 and TJ-II experiments). Nucl. Fusion 2011, 51. [CrossRef] 86. Krupnik, L.I.; Melnikov, A.V.; Hidalgo, C.; Eliseev, L.G.; Chmyga, A.A.; Komarov, A.D.; Kozachok, A.S.; Perfilov, S.V.; Lysenko, L.G.; Zhezhera, A.I. Recent measurements of the electric potential profile and fluctuations in ECRH and NBI in TJ-II stellarator. Probl. At. Sci. Technol. Ser. Plasma Phys. 2009, 1, 31–33. 87. Krupnik, L.; Alonso, A.; Ascasibar, E.; Estrada, T.; Hidalgo, C.; Van Milligen, B.; Ochando, M.; Pedrosa, M.; De Pablos, J.L.; Tribaldos, V.; et al. Radial electric fields and confinement in the TJ-II stellarator. Czechoslov. J. Phys. 2005, 55, 317–339. [CrossRef] 88. Melnikov, A.V.; Eliseev, L.G.; Grashin, S.A.; Gudozhnik, A.V.; Lysenko, S.E.; Mavrin, V.A.; Perfilov, S.V.; Vershkov, V.A.; Krupnik, L.I.; Chmyga, A.A.; et al. Study of the plasma potential evolution during ECRH in the T-10 tokamak and TJ-II stellarator. Czechoslov. J. Phys. 2005, 55, 1569–1578. [CrossRef] 89. Sharma, R.; Khabanov, P.O.; Melnikov, A.V.; Kharchev, N.K.; Sánchez, E.; Chmyga, A.A.; Deshko, G.N.; Eliseev, L.G.; Hidalgo, C.; Khrebtov, S.M.; et al. Poloidal 2D scans to investigate potential and density profiles in the TJ-II stellarator using heavy ion beam probe. In Proceedings of the 45th EPS Conference on Plasma Physics, Prague, Czech Republic, 2–6 July 2018; ECA. Volume 42A, p. P5.1061. Available online: http://ocs.ciemat.es/EPS2018PAP/pdf/P5.1061.pdf (accessed on 1 January 2021). 90. Sharma, R.; Khabanov, P.O.; Melnikov, A.V.; Hidalgo, C.; Cappa, A.; Chmyga, A.; Eliseev, L.G.; Estrada, T.; Kharchev, N.K.; Kozachek, A.S.; et al. Measurements of 2D poloidal plasma profiles and fluctuations in ECRH plasmas using the heavy ion beam probe system in the TJ-II stellarator. Phys. Plasmas 2020, 27, 062502. [CrossRef] 91. Eliseev, L.G.; Melnikov, A.V.; Drabinskiy, M.A.; Khabanov, P.O.; Kharchev, N.K.; Lysenko, S.E.; Hidalgo, C.; Barcala, J.M.; Cappa, A.; Liniers, M.; et al. 2D distributions of potential and density mean-values and oscillations in the ECRH and NBI plasmas at the TJ-II stellarator. In Proceedings of the 28th IAEA Fusion Energy Conference (FEC 2020), 10–15 May 2021; EX/P6-17. Available online: http://iaea.org/events/fec-2020 (accessed on 1 January 2021). 92. Garbet, X.; Idomura, Y.; Villard, L.; Watanabe, T.H. Gyrokinetic simulations of turbulent transport. Nucl. Fusion 2010, 50, 043002. [CrossRef] 93. Melnikov, A.; Krupnik, L.; Barcala, J.M.; Bravo, A.; Chmyga, A.; Deshko, G.; Drabinskij, M.; Eliseev, L.; Hidalgo, C.; Khabanov, P.O.; et al. Heavy ion beam probing—a tool to study geodesic acoustic modes and Alfven eigenmodes in the T-10 tokamak and TJ-II stellarator. Probl. Atom. Sci. Technol. Ser. Plasma Physics 2017, 107, 237–240. 94. Chmyga, O.O.; Ascasibar, E.; Barcala, J.; Drabinskiy, M.A.; Eliseev, L.G.; Hidalgo, C.; Khabanov, P.O.; Khrebtov, S.M.; Komarov, O.D.; Kozachok, O.S.; et al. A dual heavy ion beam probe diagnostic on the TJ-II stellarator. Probl. Atom. Sci. Technol. Ser. Plasma Phys. 2019, 1, 248–251. 95. Fujisawa, A.; Itoh, K.; Iguchi, H.; Matsuoka, K.; Okamura, S.; Shimizu, A.; Minami, T.; Yoshimura, Y.; Nagaoka, K.; Takahashi, C.; et al. Identification of zonal flows in a toroidal plasma. Phys. Rev. Lett. 2004, 93, 165002. [CrossRef] 96. Fujisawa, A.; Ido, T.; Shimizu, A.; Okamura, S.; Matsuoka, K.; Iguchi, H.; Hamada, Y.; Nakano, H.; Ohshima, S.; Itoh, K.; et al. Experimental progress on zonal flow physics in toroidal plasmas. Nucl. Fusion 2007, 47, S718–S726. [CrossRef] Symmetry 2021, 13, 1367 45 of 46

97. Melnikov, A.V.; Krupnik, L.I.; Ascasíbar, E.; Cappa, A.; Chmyga, A.A.; Deshko, G.N.; Drabinskij, M.A.; Eliseev, L.G.; Hidalgo, C.; Khabanov, P.O.; et al. ECRH effect on the electric potential and turbulence in the TJ-II stellarator and T-10 tokamak plasmas. Plasma Phys. Control. Fusion 2018, 60, 084008. [CrossRef] 98. Zhezhera, A.I.; Chmyga, A.; Deshko, G.N.; Eliseev, L.G.; Hidalgo, C.; Khrebtov, S.M.; Komarov, A.D.; Kozachek, A.S.; Krupnik, L.I.; Melnikov, A.V.; et al. New capabilities of plasma potential and density measurements using a dual heavy ion beam probing (HIBP) diagnostic in the TJ-II stellarator. Probl. Atom. Sci. Techn. Ser. Plasma Phys. 2017, 107, 261–264. 99. Fujusawa, A. A review of zonal flow experiments. Nucl. Fusion 2009, 49, 013001. [CrossRef] 100. Hidalgo, C.; Chmyga, A.; Chercoles, J.; Deshko, G.N.; Eliseev, L.G.; Khabanov, P.O.; Khrebtov, S.M.; Komarov, A.G.; Kozachek, A.S.; Krupnik, L.I.; et al. On the influence of ECRH on neoclassical and anomalous mechanisms using a dual heavy ion beam probe diagnostic in the TJ-II stellarator. In Proceedings of the 26th IAEA Fusion Energy Conference, Kyoto, Japan, 17–22 October 2016; Rep. EXC/P7-44. Available online: https://nucleus.iaea.org/sites/fusionportal/Shared%20Documents/FEC%202016/fec2 016-preprints/preprint0209.pdf (accessed on 1 January 2021). 101. Nagaoka, K.; Ido, T.; Ascasíbar, E.; Estrada, T.; Yamamoto, S.; Melnikov, A.V.; Cappa, A.; Hidalgo, C.; Pedrosa, M.A.; van Milligen, B.P.; et al. Mitigation of NBI driven Alfven eigenmodes by electron cyclotron heating in the TJ-II stellarator. Nucl. Fusion 2013, 53, 072004. [CrossRef] 102. Jiménez-Gómez, R.; Könies, A.; Ascasíbar, E.; Castejón, F.; Estrada, T.; Eliseev, L.G.; Melnikov, A.V.; Jiménez, J.A.; Petty, D.G.; Jiménez-Rey, D.; et al. Alfvén eigenmodes measured in the TJ-II stellarator. Nucl. Fusion 2011, 51, 033001. [CrossRef] 103. Melnikov, A.; Eliseev, L.; Castejón, F.; Hidalgo, C.; Khabanov, P.; Kozachek, A.; Krupnik, L.; Liniers, M.; Lysenko, S.; De Pablos, J.L.; et al. Study of NBI-driven chirping mode properties and radial location by the heavy ion beam probe in the TJ-II stellarator. Nucl. Fusion 2016, 56, 112019. [CrossRef] 104. Garcia-Munoz, M.; Sharapov, S.E.; Van Zeeland, M.A.; Ascasibar, E.; Cappa, A.; Chen, L.; Ferreira, J.; Galdon-Quiroga, J.; Geiger, B.; Gonzalez-Martin, J.; et al. Active control of Alfvén eigenmodes in magnetically confined toroidal plasmas. Plasma Phys. Control. Fusion 2019, 61, 054007. [CrossRef] 105. Yamamoto, S.; Nagasaki, K.; Nagaoka, K.; Varela, J.; Cappa, A.; Ascasibar, E.; Castejón, F.; Fontdecaba, J.M.; Regaña, J.M.G.; González-Jerez, Á.; et al. Effect of ECH/ECCD on energetic-particle-driven MHD modes in helical plasmas. Nucl. Fusion 2020, 60, 066018. [CrossRef] 106. Alonso, J.A.; Sánchez, E.; Calvo, I.; Velasco, J.L.; McCarthy, K.J.; Chmyga, A.; Eliseev, L.G.; Estrada, T.; Kleiber, R.; Krupnik, L.I.; et al. Observation of oscillatory radial electric field relaxation in a helical plasma. Phys. Rev. Lett. 2017, 118, 185002. [CrossRef] 107. McCarthy, K.J.; Panadero, N.; Combs, S.K.; Tamura, N.; Ascasíbar, E.; Calvo, M.; Chmyga, A.; Estrada, T.; Fontdecaba, J.M.; García, R.; et al. The impact of fast electrons on pellet injection in the stellarator TJ-II. Plasma Phys. Control. Fusion 2018, 61, 014013. [CrossRef] 108. Chmyga, A.A.; Deshko, G.N.; Dreval, N.B.; Melnikov, A.V.; Perfilov, S.V.; Galvao, R.M.; Kuznetsov, Y.K. Heavy ion beam probe design study for TCABR. Probl. At. Sci. Technol. Ser. Plasma Phys. 2003, 1, 160–162. 109. Kuznetsov, Y.; Nascimento, I.; Silva, C.G.; Figueiredo, H.; Guimarães-Filho, Z.; Caldas, I.; Galvão, R.; Severo, J.; Toufen, D.; Ruchko, L.; et al. Long-distance correlations in TCABR biasing experiments. Nucl. Fusion 2012, 52.[CrossRef] 110. Melnikov, A.V.; Perfilov, S.V. Heavy Ion Beam Probe (HIBP) diagnostics design study for GLOBUS-M tokamak. In Proceedings of the 26th EPS Conference on Plasma Physics and Controlled Fusion, Maastricht, The Netherlands, 14–18 June 1999; ECA, 23J. p. P4.116. 111. Melnikov, A.V.; Markovic, T.; Eliseev, L.; Adamek, J.; Aftanas, M.; Bilkova, P.; Boehm, P.; Gryaznevich, M.; Imrisek, M.; Lysenko, S.E.; et al. Quasicoherent modes on the COMPASS tokamak. Plasma Phys. Control. Fusion 2015, 57.[CrossRef] 112. Seidl, J.; Krbec, J.; Hron, M.; Adamek, J.; Hidalgo, C.; Markovic, T.; Melnikov, A.; Stockel, J.; Weinzettl, V.; Aftanas, M.; et al. Electromagnetic characteristics of geodesic acoustic mode in the COMPASS tokamak. Nucl. Fusion 2017, 57, 126048. [CrossRef] 113. Siegrist, M.; Tonetti, G.; Weisen, H.; Melnikov, A.; Perfilov, S.V.; Krupnik, L.; Karpushov, I. The Heavy Ion Beam Diagnostic Project for the TCV Tokamak. In Proceedings of the 21st IEEE/NPS Symposium on Fusion Engineering SOFE 05, Knoxville, TN, USA, 26–29 September 2005; pp. 1–4. [CrossRef] 114. Melnikov, A.V.; Perfilov, S.V.; Sharapov, S.E. Heavy ion beam probe for studying potential and turbulence on MAST. In Proceedings of the 37th EPS Conference on Plasma Physics, Dublin, Ireland, 21–25 June 2010; ECA. Volume 34A, p. P5.120. Available online: http://ocs.ciemat.es/EPS2010PAP/html/ (accessed on 1 January 2021). 115. Melnikov, A.V.; Perfilov, S.V. Effect of the sample volume splitting in heavy ion-beam probing (HIBP) for spherical tokamaks (by the example of HIBP for mega ampere spherical tokamak). Rev. Sci. Instrum. 1999, 70, 1402–1406. [CrossRef] 116. Melnikov, A.V.; Sushkov, A.V.; Belov, A.M.; Dnestrovskij, Y.N.; Eliseev, L.G.; Ivanov, D.P.; Kirneva, N.A.; Korobov, K.V.; Krupin, V.A.; Lysenko, S.E.; et al. Physical program and conceptual design of the diagnostics of the T-15 upgrade tokamak. Fusion Eng. Design 2015, 96, 306–310. [CrossRef] 117. Ilin, A.M.; Khabanov, P.O.; Melnikov, A.V. Probing ion trajectory simulations for the HIBP diagnostics at the T-15MD tokamak. J. Phys. Conf. Ser. 2019, 1383, 012006. [CrossRef] 118. Drabinskiy, M.; Melnikov, A.; Khabanov, P.; Eliseev, L.; Kharchev, N.; Ilin, A.; Sarancha, G.; Vadimov, N. Conceptual design of the heavy ion beam probe diagnostic for the T-15MD tokamak. J. Instrum. 2019, 14, C11027. [CrossRef] Symmetry 2021, 13, 1367 46 of 46

119. Melnikov, A.; Eliseev, L.; Lysenko, S.; Perfilov, S.; Shelukhin, D.; Vershkov, V.; Zenin, V.; Krupnik, L.; Kozachek, A.; Kharchev, N.; et al. Long-distance correlations of geodesic acoustic modes in T-10. Probl. At. Sci. Technol. Ser. Thermonucl. Fusion 2015, 38. [CrossRef] 120. Melnikov, A.V.; Vershkov, V.A.; Eliseev, L.; Grashin, S.A.; Gudozhnik, A.V.; Krupnik, L.I.; Lysenko, S.E.; Mavrin, V.A.; Perfilov, S.V.; Shelukhin, D.A.; et al. Investigation of geodesic acoustic mode oscillations in the T-10 tokamak. Plasma Phys. Control. Fusion 2006, 48, S87–S110. [CrossRef] 121. Melnikov, A.V.; Eliseev, L.G.; Ascasíbar, E.; Cappa, A.; Castejón, F.; Hidalgo, C.; Ido, T.; Jiménez, J.A.; Kozachek, A.S.; Krupnik, L.I.; et al. Effect of magnetic configuration on nonlinear evolution of NBI-driven Alfvén modes in TJ-II. Nucl. Fusion 2016, 56, 076001. [CrossRef] 122. Cappa, Á.; Varela, J.; Lopez-Bruna, D.L.; Ascasíbar, E.; Liniers, M.; Eliseev, L.; Fontdecaba, J.; García-Regaña, J.; González-Jerez, A.; Kharchev, N.; et al. Stability analysis of TJ-II stellarator NBI driven Alfvén eigenmodes in ECRH and ECCD experiments. Nucl. Fusion 2021, 61, 066019. [CrossRef] 123. Ongena, J.; Messiaen, A.; Melnikov, A.; Ragona, R.; Kazakov, Y.O.; Van Eester, D.; Dnestrovskii, Y.N.; Khvostenko, P.; Roy, I.; Romannikov, A. Conceptual study of an ICRH system for T-15MD using traveling wave antenna (TWA) sections. Fusion Eng. Des. 2019, 146, 787–791. [CrossRef] 124. Melnikov, A.V.; Ongena, J.; Messiaen, A.M.; Ragona, R.; Sushkov, A.V.; Kazakov, Y.O.; Van Eester, D.; Dnestrovskii, Y.N.; Khvostenko, P.P.; Roy, I.N. Conceptual study of an ICRH traveling wave antenna (TWA) for T-15MD at 60 MHz. AIP Conf. Proc. 2020, 2254, 070007. [CrossRef] 125. Perfilov, S.; Melnikov, A.; Krupnik, L.; Hartfuss, H.J. Applicability of Heavy Ion Beam Probing for Stellarator W7-X. AIP Conf. Proc. 2006, 812, 199–202. [CrossRef] 126. Perfilov, S.; Melnikov, A.; Krupnik, L.; Hartfuss, H.J. Applicability of Heavy Ion Beam Probing for Stellarator W7-X. Fusion Sci. Technol. 2007, 51, 38–45. [CrossRef] 127. Melnikov, A.V.; Eliseev, L.G. Optimized heavy ion beam probing for International Thermonuclear Experimental Reactor. Rev. Sci. Instrum. 1999, 70, 951–954. [CrossRef] 128. Melnikov, A.V.; Eliseev, L.G.; Razumova, K.A.; Krupnik, L.I.; Chistyakov, V.V.; Myalton, T.B.; Dnestrovskij, Y.N.; HIBP Team. On the sensitivity of the plasma electric potential to the edge transport barrier formation. Jpn. J. Plasma Fusion Res. Series 2000, 4, 203–208. 129. Alikaev, V.V.; Borshchegovskij, A.A.; Vershkov, V.A.; Volkov, V.V.; Gorshkov, A.V.; Gott, Y.V.; Grashin, S.A.; Dremin, M.M.; Eliseev, L.G.; Esipchuk, Y.V.; et al. Investigation of the H-mode during ECRH in the T-10 tokamak. Plasma Phys. Rep. 2000, 26, 917. [CrossRef] 130. Melnikov, A.V.; Eliseev, L.G.; Grashin, S.A.; Drabinskiy, M.A.; Khabanov, P.O.; Kharchev, N.K.; Krupin, V.A.; Lysenko, S.E.; Ryzhakov, D.A.; Soloviev, N.A.; et al. Evolution of the electric potential and turbulence in OH and ECRH low-density plasmas at the T-10 tokamak. In Proceedings of the 28th IAEA Fusion Energy Conference (FEC 2020), 10–15 May 2021; Conference was held online. EX/5-6 (Nucl. Fusion under review). Available online: https://conferences.iaea.org/event/214/contributions/17104/ (accessed on 1 January 2021).