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Magnetohydrodynamics modelling successfully predicts new helical states in reversed-field pinch fusion plasmas

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Nuclear Fusion

International Atomic Energy Agency Nucl. Fusion 57 (2017) 116029 (7pp)

https://doi.org/10.1088/1741-4326/aa7f46

Magnetohydrodynamics modelling successfully predicts new helical states in reversed-field pinch fusion plasmas Marco Veranda1 , Daniele Bonfiglio1, Susanna Cappello1 , Dominique Franck Escande2, Fulvio Auriemma1, Dario Borgogno3, Luis Chacón4, Alessandro Fassina1, Paolo Franz1, Marco Gobbin1, Daniela Grasso5 and Maria Ester Puiatti1 1

  Consorzio RFX (CNR, ENEA, INFN, Università di Padova, Acciaierie Venete SpA) Corso Stati Uniti 4, Padova, Italy 2   Aix-Marseille Univ, CNRS, PIIM, Marseille, France 3  Université Côte d’Azur, Nice, France 4   Los Alamos National Laboratory, Los Alamos, NM, United States of America 5   Istituto dei Sistemi Complessi-CNR, Politecnico di Torino, Torino, Italy E-mail: [email protected] Received 24 February 2017, revised 8 July 2017 Accepted for publication 12 July 2017 Published 9 August 2017 Abstract

Nonlinear fluid modelling predictions of qualitatively new self-organized helical states in the reversed-field pinch configuration are confirmed by experiments in the RFX-mod device. The new states are realized by using a seed edge magnetic field, which can impose its helical pitch to the whole plasma. In simulations, we show increased magnetic order and reduced transport of magnetic field lines in regimes with the twist of a non-resonant mode. We reveal the existence of Cantori, encompassing the region characterized by conserved magnetic surfaces, which act as barriers to transport of magnetic field lines. This opens a new research line for transport studies in hot magnetized plasmas and highlights a path towards reversed-field pinches with high confinement at high current. Keywords: numerical modelling, MHD, reversed-field pinch, validation of numerical results (Some figures may appear in colour only in the online journal)

1. Introduction

The stronger helical self-organization in the RFP results from the fact that most of the magnetic field is produced by cur­ rents flowing in the plasma and not by external coils as in the tokamak, and from the natural tendency of intense currents to bend into a helix. Historically, helical self-organization in the RFP was first anticipated within 3D nonlinear viscoresistive MHD modelling [13–17]. It was identified as the nonlinear saturation of a symmetry breaking MHD instability, directed towards an ohmic equilibrium state, and sustained by an electrostatic dynamo mechanism [18–21]. Plasmas with a transient helical deformation were indeed occasionally observed in the past in all RFP experiments (see for example [22–26]), but only high-current experiments revealed their self-organized

Helical self-organization is an ubiquitous feature in astrophysical plasmas [1, 2] and in current-carrying toroidal plasmas for magnetic fusion, such as the tokamak and the reversedfield pinch (RFP). In a tokamak [3], helical states can form in the core, associated with strong 3D magnetohydrodynamics (MHD) activity. Examples are the ‘snake’ phenomenon [4, 5] or the hybrid scenarios that allow bypassing both the confinement degradation and the disruption risk observed during the ‘sawtooth’ cycles [6–8]. In the RFP configuration [9], a global self-organization process systematically occurs at high plasma current, impressing a helical shape to the whole plasma column [10, 11], with beneficial effects on confinement [12]. 1741-4326/17/116029+7$33.00

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quasi-helical nature [10, 11, 27, 28]. More recently, a clear statistical trend of increasing helical state persistence with increasing plasma current has been shown to characterize the two major RFP devices, RFX-mod [29] and MST [30]. In the upgraded RFX-mod device [31], good control of the edge radial magnetic field reduced plasma-wall interaction and allowed plasma currents up to 2 MA and temperatures above 1 keV. In experiments above 1 MA, systematic quasi-helical regimes were observed, characterized by a dominant toroidal periodicity n = 7. More importantly, a beneficial topological change in the magnetic field lines was observed in high cur­ rent experiments, whereby they spontaneously wind around a single helical axis, which is in turn linked to the formation of a hot helical central region bounded by internal transport barriers (ITBs) [12, 32–34]. In recent years, the use of edge magnetic perturbations (MPs) with the helicity of the spontaneous n = 7 dominant mode allowed experiments with increased persistence of such states [35, 36]. The use of MPs in 3D nonlinear MHD simulations of the RFP resulted in a decisive improvement in the descriptive capability of the typical experimental intermittency [37–40]. This paper reports a milestone in MHD modeling of RFP plasmas. First, we show that the MHD prediction of new helical states is confirmed by experiments in RFX-mod. The key element is the use of small edge magnetic perturbations with a helical periodicity different from the one observed in standard discharges: qualitatively new helical states can be obtained with a non-resonant helical twist (i.e. the twist of a resistive wall mode). Second, in fluid modeling we find a higher magnetic order when MPs are applied with the non-resonant helical twist just discussed: we indeed observe a broader zone where magnetic field lines lie on conserved magnetic surfaces. Third, a novel technique reveals the presence of Cantori hidden in the magnetic field topology, which act as barriers to the transport of magnetic field lines. We begin this study with a description of the modeling tools. Next, we describe the MHD dynamics stimulated by helical MPs and the corresponding experimental achievement in the RFX-mod device. We then characterize the magnetic topology properties of the numerical states, also using the above mentioned technique for the numerical detection of magnetic field lines transport barriers. As a point for discussion, we show experimental electron temperature profiles that display a broader central region encompassed by strong gradients in the novel non-resonant helical states. Final remarks conclude the paper.

a model with two dimensionless parameters: resistivity η and viscosity ν. The equations written in dimensionless units are: (1a) ∂t v + v · ∇v = J × B + ν∇2 v ∂t B = ∇ × (v × B − ηJ) (1b) ∇×B=J (1c) ∇ · B = 0. (1d)

Models adopting similar approximations have been extensively used for compressible laboratory plasmas, capturing major physical effects observed in experiments (see for example [43–51]) such as the characteristic RFP dynamo effect, provided by current-driven resistive kink/tearing modes. Finite pressure, two-fluid effects and toroidal geometry, included in recent computations, lead to similar macroscopic RFP dynamics in simulations [19, 52–55], though quantitative differences can be present. Numerical simulations are performed in cylindrical geometry with aspect ratio R0 /a = 4 , where a = 1 is the cylinder radius and 2πR0 the periodicity of the cylinder in the axial direction. Simulations reported in this study employ resistivity increasing towards the edge, η(r/a) = η0 (1 + 20(r/a)10 ) with η0 = 10−6, and uniform viscosity ν = 10−4 . The relevant time scales of the system are the Alfvén time τA = a(µ0 ρ)1/2 B−1 0 with B0 on axis 2 field, and the resistive time τR = µ0 a σ0 , with σ0 on-axis 1/2 electrical conductivity. S = τR /τA = aB0 µ0 ρ−1/2 σ0 = η0−1 is the Lundquist number. SpeCyl solves the equations using finite differences in the radial direction (mesh with 101 points) and a spectral decomposition in the poloidal and axial p­ eriodic coordinates (with a wide spectrum of 225 modes, previously validated [42, 56]). Magnetic boundary conditions are defined as follows. The edge radial magnetic field is either zero (the so-called ‘ideal conducting wall’), or helically modulated through MPs. The MP helical twist is defined by nMP with poloidal periodicity m = 1, and MP intensity is measured by the quantity MP% = br (a)/Bθ (a)%. For simplicity, we show only cases with nMP = 6, 7, 8. The simulations start from an axisymmetric, non-reversed, unstable Ohmic equilibrium [57] with Bθ (a) = 1.6. An uniform induction elecpinch parameter Θ = B z tric field E = E0ˆz is imposed to sustain the plasma current, with E0 /η0 = 4.2. A slight perturbation nonlinearly induces the reversal and starts the MHD dynamics, typically driven by modes of resistive-kink/tearing nature [37]. The axisymmetric z safety-factor profile q(r/a) = RrB is plotted in figure 1 both 0 Bθ at the beginning of the simulation (dotted) and during the nonlinear phase, indicating that the m = 1, n = 7 is the first resonant mode in the nonlinear phase.

2.  Numerical model The simulations presented here are performed with the SpeCyl code [16], recently verified numerically against the PIXIE3D code [41] with excellent results [42]. SpeCyl deals with the simple visco-resistive approximation which combines Faraday law, momentum equation  (viscous Navier–Stokes) and the single fluid resistive Ohm’s law (in equations  (1a)– (1c)) to evolve the magnetic B and velocity v fields in time t. Plasma density is assumed to remain uniform and constant (ρ = 1 in the equation (1)). Pressure is neglected. This yields

3.  Stimulated helical states The first result of this work is that the MHD modelling predictions are confirmed by experiments: in fact a small helical field produced with the saddle coils system at the edge of RFX-mod 2

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Figure 1.  Axisymmetric safety factor profiles in nonlinear MHD modelling. The dotted line shows the safety factor at the beginning of numerical simulations. The colored profile represents the average safety factor during the nonlinear phase.

Figure 2.  Small seed MPs can shape the plasma with their helicity, both in MHD simulations (panel a) top) and in the RFX-mod experiment (panel b) bottom). We show the temporal evolution of the dominant m  =  1 harmonics of the magnetic field. r = rp is the radial position at 90% of the distance between the core and where MPs are imposed. (a): SpeCyl with MP% = 1% and Lundquist number S = 106. (b) Experimental discharge #30932 in RFX-mod: MP% = 1.4%. Plasma current is 1.5 MA until t = 0.2, and it slowly decreases afterwards to 1MA at the end. τR ∼ 4s and S ∼ 2 · 107 in this discharge. In the final part of the discharge, the n = 7 MHD mode spontaneously emerges, with an average edge radial field br (rp )% = 1.3%. When MPs are imposed experimentally on n = 7, the behavior of the mode can be less oscillatory than in this example, see [35, 36].

can shape the plasma column with its helical periodicity, and the behavior reproduces the one observed in SpeCyl. The result is exemplified in figure 2, where the MHD modes evo­ lution demonstrates that, by using MPs, helical states can be produced with flexibility both in simulation, panel (a) and in experiment, panel (b). In both cases, we impose successively MPs with helicity n = 6, n = 8 and n = 7, and we achieve three quasi-helical states with the same helical twist of MPs. We observe, both in nonlinear MHD and in experiment, that when a given nMP is applied (starting from the times marked by dotted lines) the corresponding MHD mode grows to become the dominant one in the whole plasma, and the other modes are strongly tamed, creating a quasi-helical state. The results of figure 2(a) are typical of numerical simulations [38] and of a wide set of recent dedicated experiments. In both cases the plasma response to the imposed MPs depends on their amplitude and helicity. We find that the most responsive modes are those with a helicity close to 7, the one usually observed in RFX-mod. Simulations, like the ones made for the standard n = 7 case in [40], show that above a certain threshold in MP intensity (MP%  10%), the temporal fluctuations of the magnetic field are mitigated and a nearly quiescent global quasi-helical state can be obtained with low-amplitude secondary modes also for each new externally stimulated helicity. We find that, in nonlinear MHD simulations of the RFP, MPs are a necessary condition to observe QSH states. We also find that when visco-resistive dissipation is reduced, a given small MP intensity is more effective in stimulating a QSH state. In fact, when reducing dissipation, the intensity of the dominant MHD mode does not change (because it is proportional to MP%) but secondary perturbations diminish (we find that bsec ∝ (ην)0.15 , with a slight dependence on the stimulated MHD mode). Therefore, we can expect that in high-current plasmas (with higher temperature and thus lower resistive dissipation) small MPs/error fields may have sufficient amplitude to trigger a quasi-helical state. This is supported by the observation that helical states spontaneously emerge in RFX-mod when increasing current, and always with the presence of a small n = 7 modulation of the edge magnetic field [35].

In RFX-mod, the experimental confirmation of the numer­ ical results is possible because the device is equipped with an advanced active-feedback control system of the magnetic boundary [31, 58–60]. A set of experiments with applied MPs on m = 1 modes with toroidal periodicity from n = 5 to n = 11 has been performed in RFX-mod. In the experiment of figure  2(b), we exploited the possibility to feedback control with external saddle coils the amplitude of a prescribed harmonic, closely mimicking the boundary condition of numerical simulations of figure 2(a). Figure 2(b) displays the experimental magnetic field dynamics that can be divided into three parts, depending on the dominant MHD mode. In the first part of the experiment (until t = 0.045 τR ), the control system is tuned to impose a MP with non-resonant (nMP = 6) helicity. We observe that the corresponding MHD mode becomes dominant, as found in simulations. In the second part of the experiment, MPs with nMP = 8 are imposed, and again at the MPs helicity transition the corresponding MHD mode becomes the dominant one. Then after t = 0.090 τR the control system is tuned to aim at an ideal wall: as a result, the usual resistive-kink/tearing mode n = 7 spontaneously emerges as the dominant one, with a finite average edge radial field. We note that the n = 6 helicity corresponds to the first non-resonant MHD mode. In fact both in our simulations (see figure 1) and in experiments [34, 61] the axisymmetric safety factor has a core value 17 < q(0) < 16. In RFX-mod the corresponding MHD modes would grow on the resistive time scale without a proper feedback control [60], hence their names ‘resistive wall mode’ [62, 63]. These 3

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double-axis to a single-helical-axis state, when the mode amplitude dynamically grows above a threshold [27, 32, 64] like the one depicted in panel (b) of figure  3. In the n = 7 case, the black helical flux surface in panel (b) marks the core region with a reversed-shear in the helical safety factor plotted in panel (e). The n = 8 case is usually characterized by the presence of a magnetic separatrix (black flux surface in panel (c)), typically a region that represents the seed for the onset of magnetic chaos [27]. Correspondingly, the helical safety factor profiles, second column of figure 3, decreases monotonically for the non-resonant case (panel (d)), whereas a region of flat or reversed magnetic shear appears in resonant configurations [39], similarly to what is obtained in experimental reconstructions [34], where it is associated with the formation of internal transport barriers. We examine the topology of the total 3D magnetic field through Poincaré plots obtained with the NEMATO field line tracing code [65] (successfully benchmarked against the ORBIT code [66]). The typical magnetic topology of stimulated helical states is highlighted on a θ = 0 surface of the cylindrical domain in figure 4. The topology of a n = 6 state is plotted in panel (a): there are two sets of conserved magnetic surfaces, the ones in the core associated with the helical deformation induced by the dominant helical mode in green and the ones at the edge in pink. This non-resonant state clearly exhibits a wider conserved core region than the resonant states in panels (b) and (c), and embraces the geometrical axis of the cylinder at r/a = 0 . Panel (b) represents the typical topology of the resonant/spontaneous states with n = 7, whose conserved surfaces are painted in orange. Panel (c) depicts the typical topology of the n = 8 dominant mode (in blue). The edge structures are related to m = 0 magnetic islands forming around the zero of the safety factor: their width is not significantly influenced by the MP [66] and they are useful for limiting field-line diffusion [50] and controlling plasma-wall interaction [36, 67]. In addition to the conserved magnetic surfaces, there is a hidden order in the seemingly stochastic part of the magnetic field. In fact, we detect magnetic structures acting as barriers to the stochastic transport, and we show that the order induced by these structures is higher in the non-resonant n = 6 case. This is the third main result of this study. We use a method to reveal a hidden skeleton of Lagrangian coherent structures (LCS), following the techniques developed in [68–70], then applied to a schematic RFP case [71], and finally applied in this work to a numerical realistic RFP case. We determine the field of finite time Lyapunov exponents for the magnetic field lines (the finite time in this case is the arc-length of the field line) and identify coherent structures with its ‘ridges’, which can be intuitively defined in the same way as one would determine the ridges of a mountain chain. Figure 5(a) displays in black the ridges of the FTLE field computed from a 3D MHD simulation of a non-resonant helical state, like the ones shown in figure  2(a) and figure  4(a). The proof of their effectiveness in limiting the diffusion of magnetic field lines is tested by analyzing magnetic field lines starting from the colored

Figure 3.  Helical magnetic field features from numerical simulations of stimulated RFP regimes. First column: surfaces of constant helical flux function χ varying the twist of the stimulated quasi-helical states. Second column: helical safety factor profiles. Figure 4 reports the topology in the same cases, analyzed using the total 3D magnetic field.

modes are controlled by feedback in RFX-mod [60] and by the nearby ideal boundary in the simulations. 4.  Magnetic topology and Cantori The second result of this study is that a non-resonant MP-driven helical RFP configuration displays a larger area of conserved magnetic surfaces. To illustrate this result, first we examine the properties of the 2D helical field Bh, obtained by retaining only the axisymmetric component of the total magnetic field plus the harmonics with the dominant helicity. From such field it is possible to define a helical flux  function χ such that ∇χ · Bh = 0 ,

an effective radius ρ = dΨtor (ρ) dΨpol (ρ) :

χ−χmin χmax −χmin

and a helical safety factor

q(ρ) = Ψpol (ρ) and Ψtor (ρ) represent the poloidal and toroidal flux across a helical flux surface labelled by ρ, and q(ρ) gives the number of toroidal turns that field lines perform for one poloidal turn around the helical magnetic axis. Figure 3 shows the helical flux function and the q(ρ) profiles from three MHD simulations with different stimulated helical states n = 6, 7, 8 (the same snapshots are used in figure 4 to describe the 3D magnetic topology). The shape of the helical flux surfaces (constant χ surfaces) varies from the helically displaced but nearly circular surfaces in the non-resonant n = 6 case to the bean-shaped surfaces in resonant cases n = 7, n = 8. Considering the n = 6 case, the ­non-resonant nature of the dominant MHD mode avoids the formation of a magnetic island from the beginning of the dynamical evo­ lution and just causes a kinking of the plasma column around an helical axis. In the case with n = 7, a magn­etic island forms but its separatrix is systematically expelled by a trans­ ition that changes the topology of the flux surfaces from a

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Figure 5.  Ridges point out a barrier for the diffusion of magnetic field lines. Panel (a) shows the skeleton of LCS in black, at z = 0 and the four sets of magnetic field lines initialized at the colored dots, placed at opposite sides of two different ridges. Panel (b) shows how distinct sets of field lines do not leak through the ridges. This case comes from a numerical simulation of a nonresonant helical state with n = 6. Figure 4.  The magnetic field topology shows a larger area of conserved magnetic surfaces in the n = 6 case, panel (a), compared to the cases of panel (b) (n = 7) and (c) (n = 8). Poincaré maps are computed at the θ = 0 surface of section, considering three snapshots of MHD simulations with MP% = 2%. We find that better topological properties in non-resonant states can be detected during the whole nonlinear evolution of MHD modes. The solid lines in panels (b) and (c) correspond to the helical safety factor maximum, highlighted in the corresponding panels of figure 3.

circles placed at opposite sides of ridges and followed up to a finite arc-length L = 80Lz , Lz = 2πR0 . Figure 5(b) proves that ridges determine the sharp division between colors: they set the limits of a series of detached regions (which cannot be easily located by simpler diagnosis methods like Poincaré plots) and define a barrier. The key to understand the structure of the seemingly stochastic field is to recognize that ridges can be identified with Cantori. Cantori are the invariant sets remaining after KAM tori are destroyed by chaos [72, 73]. They are known to be the most important barrier to field-line transport [74, 75] for a finite time. In our case, such a time, ¯t , is longer than the other relevant time scales in MHD and in experiment. In fact, a test particle in RFX-mod ( R0 = 2 m) with a typical temperature of T ∼ 700 eV and with the mass of an electron would cover the length L = 80Lz ∼ 103 m in  a time scale ¯t = L/vth = L/ 2T/me that is longer than the typical dynamical scales (¯t/τA ∼ 103), longer than the typical collisional time between electrons and ions (¯t/τei ∼ 10) and comparable with the energy confinement times measured in RFX-mod [76]. The n = 7 and n = 8 cases show a similar ridge structure, i.e. a bundle of LCS located inside the stochastic region surrounding the core conserved KAM tori. A similar evaluation of ¯t in the n = 7 and n = 8 cases yields a value at least a factor of two smaller than in the n = 6 case.

Figure 6.  As the helical twist is reduced, the hot temperature central region gets broader. Radial electron temperature profiles are measured with a Thomson scattering diagnostics. The plotted data represent the moving average of local temperature values. The three examples, taken from RFX-mod database, have comparable plasma parameters: plasma current  ∼1.5 MA, density  ∼3 · 1019 m−3.

RFX-mod using a Thomson scattering diagnostic [77, 78], during the experiments related to the novel helical states. Such profiles display rather different shapes when the pitch of the induced helical state is varied. A clear trend in the width of the hot core is observed: as the helical twist of the plasma column is reduced the hot central region gets larger, as shown in figure 6. Figure 6 shows that RFP states built upon controlled RWM can sustain broad temperature profiles. Interestingly, the experimental trend of increasing width of the hot helical core when going to non-resonant helicities correlates well with the increasing width of the regular domains plus the area occupied by Cantori observed in numerical simulations of non-resonant helical states. However, it should be noted that a further step towards a more quantitative interpretation of this and other features of the experimental thermal properties, like the lower central temperature in the n = 6 case, or the relative steepness of the temperature gradients, would require the inclusion of physics beyond the MHD model used in this work: this will be the subject of future numerical and experimental activity. We also note that additional transport

5.  Discussion and final remarks A direct comparison with thermal properties of exper­ imental helical RFP states is beyond our present capabilities. Nevertheless, as a starting point for discussion we present examples of electron temperature profiles, measured in 5

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mechanisms, such as microturbulence transport via microtearing [79, 80] or ion-temperature-gradient modes [81–84], could play a significant role. In conclusion, this study provides three major contrib­ utions. The first is that the predictions of stimulated helical RFP regimes made by 3D nonlinear visco-resistive MHD are confirmed by new experiments in RFX-mod. We have presented results from RFX-mod showing that the introduction of a small helical magnetic perturbation at the plasma boundary can stimulate global helical states with the chosen helical twist. The second contribution is that we have described an increase of magnetic order in Poincaré plots from MHD simulations with non-resonant MPs. In fact, qualitatively new and effective non-resonant helical states are successfully obtained in experiments by actively exploiting Resistive Wall Modes. This represents a new approach: instead of cancelling RWMs out as in ordinary operations we exploit them to form a new configuration with externally chosen pitch. RFX-mod electron temperature profiles display the signature of the different topological properties of resonant and non-resonant states, which we speculate to be in relation with a broader area of reduced magnetic chaos. These findings suggest that linearly unstable MHD modes, previously deemed dangerous, can and should be beneficially harnessed nonlinearly. As a final contribution of this study, by using the ridge technique we have detected Cantori structures hidden in the weakly stochastic region surrounding the conserved KAM surfaces. Such structures increase magnetic field lines confinement and can explain the presence of ITBs in experiments. Thus, following previous studies [74, 75, 85], we speculate that the ridges technique can be useful also in the analysis of the experimental measurements of transport barriers in tokamaks and stellarators, in addition to the already established mechanisms of microturbulence suppression by magnetic shear and sheared flows [86, 87] or confinement improvement by resonant MP applications [88] and the corre­ sponding results in two-fluid modeling [89]. As an additional remark, we note that the novel ability of changing the global helical twist of the plasma through edge MP can be effectively used to test the theoretical interpretation of the mechanisms involving transport barriers formation, in particular the role of the safety factor. Most importantly we showed that macroscopic magnetohydro-dynamics numerical modelling, though resulting from elementary underlying equations, can drive promising experimental outcomes.

Centre (IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between EURATOM and Japan, implemented by Fusion for Energy and JAEA. The authors would like to thank the whole RFX-mod team. One author, M.V., would like to thank L. Marrelli and G. Spizzo for profitable comments and discussions. ORCID iDs Marco Veranda https://orcid.org/0000-0002-5821-2896 Susanna Cappello https://orcid.org/0000-0002-2022-1113 References [1] Meier D.L., Koide S. and Uchida Y. 2001 Science 291 84 [2] Gibson S.E., Fan Y., Török T. and Kliem B. 2007 Space Sci. Rev. 124 131 [3] Wesson J. 1997 Tokamaks 2nd edn (Oxford: Clarendon) [4] Weller A., Cheetham A.D., Edwards A.W., Gill R.D., Gondhalekar A., Granetz R.S., Snipes J. and Wesson J.A. 1987 Phys. Rev. Lett. 59 2303 [5] Delgado-Aparicio L. et al 2013 Phys. Rev. Lett. 110 065006 [6] Buratti P. et al 2012 Nucl. Fusion 52 023006 [7] Graves J.P. et al, The TCV Team and The MAST Team 2013 Plasma Phys. Control. Fusion 55 014005 [8] Chapman I. et al The MAST, Teams N. and Contributors E.J. 2014 Nucl. Fusion 54 083007 [9] Escande D.F. 2014 What is a reversed field pinch? Rotation and momentum transport in magnetized plasmas vol 2 ed P.H. Diamond, X. Garbet, P. Ghendrih and Y. Sarazin (Singapore: World Scientific) ch 9 p 247 [10] Escande D. et al 2000 Phys. Rev. Lett. 85 1662 [11] Martin P. et al 2003 Nucl. Fusion 43 1855 [12] Lorenzini R. et al 2009 Nat. Phys. 5 570 [13] Cappello S. and Paccagnella R. 1990 ISPP-6 Joint VarennaLausanne International Workshop on Theory of Fusion Plasmas Proc. of the Workshop on Theory of Fusion Plasmas (Varenna, Italy, 27–31 August 1990) ed E. Sindoni, F. Troyon and J. Vaclavik (Bologna: Compositori) p 595 [14] Finn J., Nebel R. and Bathke C. 1992 Phys. Fluids B 4 1262 [15] Cappello S. and Paccagnella R. 1992 Phys. Fluids B 4 611 [16] Cappello S. and Biskamp D. 1996 Nucl. Fusion 36 571 [17] Cappello S. and Escande D.F. 2000 Phys. Rev. Lett. 85 3838 [18] Bonfiglio D., Cappello S. and Escande D.F. 2005 Phys. Rev. Lett. 94 145001 [19] Cappello S. et al 2011 Nucl. Fusion 51 103012 [20] Jardin S.C., Ferraro N. and Krebs I. 2015 Phys. Rev. Lett. 115 215001 [21] Piovesan P. et al 2017 Nucl. Fusion 57 076014 [22] Brunsell P.R., Yagi Y., Hirano Y., Maejima Y. and Shimada T. 1993 Phys. Fluids B 5 885 [23] Nordlund P. and Mazur S. 1994 Phys. Plasmas 1 4032 [24] Hirano Y., Yagi Y., Maejima Y., Shimada T. and Hirota I. 1997 Plasma Phys. Control. Fusion 39 A393 [25] Sarff J.S., Lanier N.E., Prager S.C. and Stoneking M.R. 1997 Phys. Rev. Lett. 78 62 [26] Martini S., Terranova D., Innocente P. and Bolzonella T. 1999 Plasma Phys. Control. Fusion 41 A315 [27] Escande D., Paccagnella R., Cappello S., Marchetto C. and D’Angelo F. 2000 Phys. Rev. Lett. 85 3169 [28] Martin P. et al 2000 Phys. Plasmas 7 1984 [29] Piovesan P. et al 2009 Nucl. Fusion 49 085036 [30] Sarff J. et al 2013 Nucl. Fusion 53 104017 [31] Sonato P. et al 2003 Fusion Eng. Des. 66 161

Acknowledgments This work was supported by the Euratom Communities under the contract of Association between EURATOM - ENEA and partially performed under Task Agreement WP14-FRFENEA/Veranda. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. A part of this work was carried out using the HELIOS supercomputer system at Computational Simulation Centre of International Fusion Energy Research 6

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