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Alfvén waves are magnetohydrodynamic (MHD) waves that propagate in magnetized plasmas and have been the subject of intense study in laboratory, space and astrophysical plasmas in the last decades. In this study, we will investigate the mechanism of their excitation during Ohmic discharges in magnetically confined fusion plasmas with focus on reversed-field pinch (RFP) and tokamak configurations. The nonlinear 3D MHD cylindrical code SpeCyl will be used to analyze magnetic configurations with increasing level of complexity. Numerical solutions from these simulations will be analyzed by comparing them with the approximate analytical solutions obtained with the linearized ideal MHD model and cold plasma approximation. Alfvénic waves such as shear Alfvén wave (SAW), compressional Alfvén eigenmodes (CAEs) and global Alfvén eigenmode (GAE) will be identified and characterized, alongside phenomena like phase mixing of SAW and resonant absorption of CAEs. Modelling results are in good quantitative agreement with the experimentally observed Alfv´en waves in the RFX-mod device.

The understanding of transport phenomena is one of the most challenging tasks in fusion plasmas. Among transport mechanisms, chaotic transport of magnetic field lines acquires a crucial importance due to the strong anisotropy that governs the plasma transport. Despite their apparent "randomness", chaotic magnetic fields lines display a remarkable amount of structures. These structures have a fundamental impact on the plasma transport. For the study of magnetic field dynamics, we exploit the so-called Lagrangian Coherent Structures (LCSs)[1].

Recent progress about helical self-organization studies is reported. Extensive exploita5on of 3D nonlinear visco-resistive modeling, SpeCyl code, which describes current-driven dynamics typical of pinch configura5ons in cylindrical geometry. Magnetic topology studies are based on the Field Line Tracing code NEMATO and a new refined tool to detect Lagrangian Coherent Structures is compared with results from a temperature equa/on solver. The following Physics Issues in helical self-organization are addressed: o Boundary Conditions and dimensionless parameters impact, (RFP, and circular Tokamak) o Forma5on of internal transport barriers, (RFP) o Temporary loss of opera5onal point, reconnec5on events, (RFP and circular Tokamak), o Alfvén waves excitation (RFP and circular Tokamak), RESULTS show: o Reasonable comparison (validation) with RFP experimental observations, o Similarities between RFP and Tokamak-like configuration.

An improvement of the boundary conditions scheme of the 3D nonlinear MHD numerical code SpeCyl is presented. Boundary conditions have been shown to play a key role in the helical selforganization both in Reversed Field Pinch and tokamak plasmas. Two different sets of boundary conditions have been extensively tested against ubiquitous relaxation phenomena induced by plasma current in toroidal devices: ideal kinks and tearing modes. The role of wall position and resistivity on linear perturbations profiles and their exponential growth rates was tested, motivating the need for a reformulation of fluid boundary conditions as well. Preliminary results of such new boundary conditions are also presented, along with a summary of the relevant theoretical framework underlying linear MHD instabilities.

We will briefly recall the features of the basic reconnection process within viscoresistive 2D nonlinear numerical MHD approximation for both the Reversed Field Pinch (RFP) and Tokamak pinch configurations. After the current sheet formation, plasmoid formation is observed [Fig.1(a,b)], when proper numerical resolution is used, which speeds up the process. Then, the full three dimensional reconnection processes will be revisited for the experimental-like RFP quasi helical regimes [1, 2]. The 3D RFP simulations show much larger and abrupt magnetic energy conversion into kinetic one, with respect to the 2D reconnection processes, overriding the fine structures observed in 2D approximation. Relaxation-reconnection events cyclically interrupt the formation of quasi helical regime, featuring 3D current sheets formation (Fig.2), mode-mode- phase locking, excitation of Alfvèn waves (Fig.3) [3, 4]. The process reflects in several respects the features highlighted in the past, including typical scaling laws, within the fully developed 3D MHD turbulence obtained in low collisional ideal boundary conditions [5, 6]. Such reconnection processes have been envisaged to provide direct ion heating in the RFP [7]. We will describe how we manage to suitably pace the relaxation cycle in simulation and experiments [8].

The edge of magnetically confined plasmas in toroidal configurations is characterized by the presence of various magnetic perturbations (MPs), appearing spontaneously as tearing modes in the Reversed Field Pinch (RFP) [1] or as peeling ballooning modes (ELM) in the tokamak. In the RFX-mod device during high-current discharges (R=2m, a=0.46m, IP>1MA) an almost monochromatic tearing mode (TM) spectrum spontaneously develops: this is the so-called quasisingle helicity (QSH) [2, 3], characterized by the presence of a single mode with helicity m/n, with (m=1, n=7) the poloidal and toroidal mode numbers respectively. However, the presence of secondary modes (m=1,n>7), with amplitudes one order of magnitude smaller than the dominant one, results in a local pattern of constructive interference (phase locking) and in a radial displacement of the plasma edge surface [4]. The intensity of the deformation can be comparable to that of the dominant mode, appearing as a sharp decrease ("hole") of the connection length to the wall at the locking angle, as shown by simulations with the ORBIT code [5,6]. An upgrade of RFX-mod device, RFX-mod2 [7], will be assembled in the near future. It will be characterized by a copper shell as continuous conductor nearest to the plasma and by a shellplasma proximity reduction from b/a=1.11 to b/a=1.04, likely improving feedback coils action. 3D MHD non-linear visco-resistive simulations show that secondary TM amplitude and the edge deformation due to phase locking will decrease by a factor 2 [8, 9]. Simulations with ORBIT show that in RFX-mod2 the average parallel connection length to the wall is expected to increase by a factor 8 with respect to RFX-mod, with no "hole" at the locking angle [6]. Having virtually cancelled the effect of TMs at r=a with a front end which behaves like an ideal wall, plasma wall interaction in RFX-mod2 could arise only due to the residual error fields at the gaps [8]: these upgrades are expected to lead to an optimized edge transport, with a well-formed SOL and to an improvement of the global plasma performance.

We will review key results from the 3D nonlinear MHD numerical modeling of Reversed-Field Pinch helical self-organization processes. Magnetic transport-barrier formation and nearly periodic reconnection events are found to be at play [1-4], akin to the experimental observation of thermal transport barriers and residual "back-transition" cycles when approaching helical regimes in high current discharges [4-7]. Similarities with Tokamak visco-resistive MHD snake/sawtooth-like phenomena will be discussed, as well as aspects at odds with the original Taylor's relaxation theory for the Reversed Field Pinch [8]. We will describe the recent successful technique to "channel" the system towards chosen "stimulated" macroscopic helical shapes by applying suitable (either Resonant or Non-Resonant) Magnetic Perturbations at the edge of the plasma, as predicted by nonlinear MHD modeling and observed in recent RFX-mod experiments [2]. In so doing, we are able to modify the transport properties of the configuration, with the two-fold objective of developing "handles" for the understanding of transport barrier formation processes and exploring new routes for optimization of pinch configurations. We have found that the magnetic chaos healing effect by helical structure development [9] appears to be more robust in the case of Non-Resonant helical regimes [2]. This line of research will be further explored in the upgraded RFX-mod2-device in Padova-Italy, expected to start operation in 2021.

The reversed-field pinch (RFP) is a configuration for the magnetic confinement of fusion plasmas, in which most of the toroidal field is generated by the plasma itself through a self-organized dynamo process, instead of being produced by external coils as in the tokamak. In the RFP, the nonlinear saturation of resistive-kink/tearing modes brings to the spontaneous emergence of helical states with improved confinement. This is observed both in nonlinear magnetohydrodynamics (MHD) modelling [1] and in RFP devices, especially at high current [2,3]. A major advance in the predictive capability of nonlinear MHD modelling for RFP plasmas was made possible by allowing helical perturbations of the radial magnetic field at the plasma boundary, which was suggested by analytical study of helical equilibrium equations [4]. A proper use of helical magnetic perturbations (MPs) in MHD modelling allowed to obtain experimental-like helical states [5] and to predict new helical states with chosen helical twist, successfully produced in RFX-mod [6]. The amplitude of the helical component and sawtooth frequency can be "tuned" as well, thus opening new routes to configuration optimization in several respects (ion heating, thermal and fast particle transport processes). Here, we discuss the effect of a more realistic magnetic boundary recently included in the modelling: a thin resistive shell and a vacuum layer between the plasma and the ideal shell are considered, similarly as in [7,8]. We present two main results: 1) The decrease of secondary modes by increased shell-plasma proximity (see Figure 1). This is of interest in view of the upgraded RFX-mod2 device (starting operation in 2021), in which the shell-plasma proximity will change from b/a=1.11 to b/a=1.04 [9]. 2) With a proper choice for the resistivity of the thin shell at the plasma boundary, helical states do emerge in a self-consistent way, as in the experiment, without the need to impose a fixed helical MP (see Figure 2). Finally, further extensions of the realistic boundary implementation, in order to take into account a double resistive shell and an active feedback control system, will be discussed.

Sheared radial electric field is an important parameter in regulating cross-field transport in fusiondevices. The edge and scrape-off layer (SOL) turbulence coupling, as well as the parallel dissipation,are responsible for the SOL width maintenance, which, in turn, is a critical parameter for particle andenergy exhaustion in tokamaks \cite{Grenfell:2019}. This coupling is usually attributed to the trans-port of streamers or blobs (also called filaments). They are mostly created in the plasma edge, in highgradient free energy regions, and transported radially by ExB drift. The turbulent filaments prop-erties, e.g. velocity and size, and ultimately their penetration into the SOL, are defined by the typeof instability that generates them as well as how they interact with the background \cite{Myra:2013}.High beta filaments, on the other hand, tend to be more resilient to the background. This is thecase of the Edge-Localized Mode filaments in H-mode regimes, one of the current threats for thefeasibility of future fusion reactors. In this work, we study the filament or blob cross-field transportin the RFX-mod operating as tokamak in different plasma regimes by using a set of electrostatic andmagnetic probes \cite{Grenfell:2018}. It has been shown recently that H-mode ELM-free and ELMyare induced in the RFX-mod tokamak by electrode biasing technique \cite{Spolaore:2017}. We showthat the properties of filaments in these regimes, as well as in L-mode, are distinct. While in L-modeblobs propagate almost freely into the SOL scaling as sheath-connected filaments \cite{Manz:2013},in the ELM-free H-mode, filaments are trapped (their cross-field velocity approaches to zero) and dis-sipated around the plasma boundary, where the ExB shearing rate is marginally comparable to theireffective convective time \cite{Grenfell:2018}. ELMs filaments propagate further into the SOL carry-ing a larger parallel current density associated with the edge transport barrier relaxation.

The reversed-field pinch (RFP) is a configuration for the magnetic confinement of fusion plasmas, in which most of the toroidal field is generated by the plasma itself through a self-organized dynamo process, instead of being produced by external coils as in the tokamak. In the RFP, the nonlinear saturation of resistive-kink/tearing modes brings to the spontaneous emergence of helical states with improved confinement. This is observed both in nonlinear magnetohydrodynamics (MHD) modelling [1] and in RFP devices, especially at high current [2]. A major advance in the predictive capability of nonlinear MHD modelling for RFP plasmas was made possible by allowing helical perturbations of the radial magnetic field at the plasma boundary [3]. A proper use of helical magnetic perturbations (MPs) in MHD modelling allowed to obtain experimental-like helical states [4] and to predict new helical states with chosen helical twist, successfully produced in RFX-mod [5].Here, we describe a further refinement of the magnetic boundary modelling. Instead of applying fixed helical MPs, we study the helical self-organization in the presence of a thin resistive shell and a vacuum layer between the plasma and the ideal shell. Two main results are discussed. On the one hand, by varying the distance between the plasma and the ideal wall it is possible to provide a nonlinear estimate for the decrease of secondary modes by increased shell-plasma proximity. This is of interest in view of the upgraded RFX-mod2 device (starting operation in 2020), in which the shell-plasma proximity will change from b/a=1.11 to b/a=1.04 [6]. On the other hand, it is observed that with a proper choice for the resistivity of the conducting shell at the plasma boundary, experimental-like helical states do emerge in a self-consistent way, as in the experiment, without the need to impose a fixed helical MP. Finally, further extensions of the realistic boundary implementation, in order to take into account a double resistive shell and a feedback control system, will be discussed.

In recent years the use of dynamical techniques to investigate the transport features in magnetized plasmas assumed an important role, especially in plasmas with low collisionality. In this regime transport is highly anisotropic, collisions are no longer the main actors and, as it has been shown, e.g. in Reversed Field Pinch (RFP) and Stellarator studies1,2, the magnetic topology plays a very important role: neglecting the finite Larmor radius and drifts, in this regime thermal particles move essentially along magnetic field lines. The transport properties in such systems are usually analyzed drawing the Poincaré map of the magnetic field. Unfortunately, especially when strong chaos affects the system, the Poincaré map gives only a general picture of the transport neglecting that there exist coherent patterns governing the transport process. Moreover, the Poincaré map can be applied only under some circumstances: the system has to be periodic and thus, for an evolving 3D magnetic configuration, this means to study the confinement properties at fixed time instant. The goal of the present work is to go beyond these limits applying Lagrangian Coherent Structures (LCS) technique3, borrowed from the study of Dynamical Systems, to magnetic field configurations in order to underline coherent patterns and thus regions of the system having different transport characteristics. In our work the LCS technique has been applied to carry on three studies. The first study focuses on a simplified model that allows us to consider explicitly the case where the magnetic field evolves in time on timescales comparable to the particles transit time through the configuration4,5. In contrast with previous works on this topic6, this analysis requires that a system that is aperiodic in time be investigated. The second study, expanding previous works7, extends our analysis to realistic numerical reproduction of a RFP configuration. In particular we focus in two different situations with resonant and non resonant dominant mode. In this two frame, a further distinction regards the amplitude of the dominant mode respect to the others: two time instants, with different level of field line chaos, are analyzed. Finally in the third part, starting from the cases of previous study, an effective magnetic field8,9 for non-thermal particles is constructed and analyzed. This allows us to show the different coherent patterns that determinates the transport of thermal and non-thermal particles and thus how different energy particles obey to different transport.

A variety of Alfvénic eigenmodes have been observed in the magnetic spectra of reversed-field pinch (RFP) plasmas in past years, in machines like RFX-mod [1] and MST [2]. In particular, in the RFX-mod device two branches of long-wavelength Alfvén eigenmodes have been observed. On the one hand, a high-frequency branch (in the range 500?1500 kHz) with two discrete eigenmodes is present during the whole discharge and interpreted as global Alfvén eigenmodes [1]. On the other hand, a lower frequency branch (f < 500 kHz) with three coherent peaks is detected only during helical RFP states, which spontaneously emerge at high plasma current (above 1 MA) and are associated with improved confinement [3].Here, we report the investigation on the modelling study of Alfvén waves in reversed-field pinch plasmas, and the comparison with experimental findings in the RFX-mod device. The nonlinear 3D MHD cylindrical code SpeCyl [4,5] has been used to analyze configurations with increasing level of complexity. First of all, numerical solutions have been compared with analytical ones in the simplest case of a uniform axial magnetic field: an excellent agreement is obtained for both the shear Alfvén wave (SAW) and the compressional Alfvén eigenmodes (CAEs). Then, the RFP configuration has been studied by assuming perturbations with a single space periodicity. Phenomena such as phase mixing of SAW, resonant absorption of CAEs and the appearance of the global Alfvén eigenmode (GAE) are reported. Finally, the fully 3D RFP case with realistic magnetic reconnection events [6] has been investigated, showing for the first time in nonlinear RFP simulations the excitation of Alfvén wavesby magnetic reconnection. Modelling results are in good quantitative agreement with the experimental characterization of the Alfvénic activity observed in RFX-mod [1]. In particular, the two high-frequency eigenmodes observed experimentally are reproducedin nonlinear MHD modelling, and identified as a global and a compressional Alfvén eigenmode. Overall, these findings suggest that reconnection processes can destabilize global Alfvén eigenmodes, as well as compressional Alfvén eigenmodes (normally disregarded in Tokamak plasmas, because of the associated very high frequency).

Nonlinear MHD modeling of toroidal pinch configurations (reversed-field pinch and tokamak) for hot plasma confinement describes several features of the helical self-organization, observed in both reversed-field pinches and tokamaks. It can also give a hint on why transport barriers are formed, by far one of the more interesting observations in experiments. The research work will be summarized in two points. The first one regards the recent successful technique to "channel" reversed-field pinches into a chosen macroscopic helical shape through the use of small edge magnetic perturbations. Helical self-organization is predicted in nonlinear MHD modeling, successfully tested in RFX-mod experiment in Padova and awaits further exploration in the updated device RFX-mod2. The upgraded machine, soon starting operation, is characterized by a shell-plasma proximity reduction, likely improving feedback coils action, thus favoring the study of the beneficial properties of helical states built upon helixes not resonating with the safety factor profile. The second point regards the MHD description of the process of formation of transport barriers by macroscopic magnetic chaos healing, which allows the emergence of hidden Lagrangian structures acting as barriers to the transport. The radial position of such structures is observed to correspond to higher gradients of magnetic field lines connection length to the edge and to higher temperature gradients, computed by numerically solving the anisotropic heat flow equation. Comparison with thermal measurements in RFX-mod will also be offered.

The reversed-field pinch (RFP) is a configuration for the magnetic confinement of fusion plasmas, in which most of the toroidal field is generated by the plasma itself through a self-organized dynamo process, instead of being produced by external coils as in the tokamak. In the RFP, the nonlinear saturation of resistive-kink/tearing modes brings to the spontaneous emergence of helical states with improved confinement. This is observed both in nonlinear magnetohydrodynamics (MHD) modelling [1] and in RFP experiments, especially at high current [2,3]. A major advance in the predictive capability of nonlinear MHD modelling for RFP plasmas was made possible by allowing helical perturbations of the radial magnetic field at the plasma boundary, assuggested by analytical calculations based on helical equilibrium equations[4]. A proper use of helical magnetic perturbations (MPs) in MHD modelling allowed to obtain experimental-like helical states [5] and to predict new helical states with chosen helical twist, successfully produced in RFX-mod [6]. Here, we describe a further refinement of the magnetic boundary modelling. We study the helical self-organization in the presence of a thin resistive shell at the plasma boundary r=a, surrounded by a vacuum layer and an ideal shell at r=b. The new magnetic boundary is implemented in the SpeCyl code [7] in a similar way as in Refs. [8,9]. Two main results are discussed. On the one hand, by varying the distance between the plasma and the ideal shell it is possible to provide a nonlinear estimate for the decrease of secondary modes by increased shell proximity to the plasma. This is of interest in view of the upgraded RFX-mod2 device (starting operation in 2021), in which the shell proximity will change from b/a=1.11 to b/a=1.04 [10]. Based on nonlinear MHD modelling,a factor of 2 reduction of the edge radial magnetic field is expected going from RFX-mod to RFX-mod2, with the beneficial consequence of a milder plasma-wall interaction. On the other hand, it is observed that with a proper choice for the resistive diffusion time of the thin shell at r=a, helical states do emerge in a spontaneous and systematic way, as in the experiment, without the need to impose a fixed helical MP. Finally, further extensions of the realistic boundary implementation, in order to take into account a double resistive shell and a feedback control system, will be discussed.

Self-organized helical states are a ubiquitous feat ure in astrophysical and magnetic confinement curre nt carrying plasmas. In the reversed-field pinch toroi dal plasmas quasi-helical states are observed both in high current experiments [1] and in nonlinear magne tofluid numerical simulations [2]. In the tokamak helical self-organization is an essential part of t he dynamics in advanced regimes [3]. In this work w e show two main advancements in our 3D nonlinear visc o-resistive magnetohydrodynamic (MHD) studies. First, we study the effect of a macroscopi c plasma rotation on helical states, to model both reversed-field pinches and tokamaks: in particular, we analyze the interaction between an external momentum source and/or seed magnetic perturbations (MPs), both static and rotating. We find the expected interplay between plasma rotation and appl ied MPs in tokamaks [4] confirming MPs screening over a threshold in normalized plasma rot ation. We then focus on the reversed-field pinch case: starting from previous works [5,6,7], where i t is shown that a mean flow arises from and interac ts with nonlinearly coupled kink/tearing modes, and th en introducing a momentum source, we analyse the impact of mean plasma flow on self-organized helica l regimes. Our modelling indicates that an external momentum source of sufficiently high intensity can prevent the formation of a helical state, while a moderate one is compatible with a slight enhancemen t of the intensity of the helical state. We present a novel study in the realistic cases [2] of experimen tal-like helical states stimulated by MPs: the pres ence of a plasma flow, previously not considered, introd uces quantitative changes to their properties. As a final novelty, we here also show (for the first tim e) that the typical intermittency with reconnection events displayed in the RFP helical self-organizati on process is accompanied by the excitation of both compressional and shear Alfvén waves, in reasonable agreement with experimental measurement [8].

We report a recent modeling study on Alfvén waves in reversed-field pinch plasmas, and the comparison with experimental findings in the RFX-mod device. The nonlinear 3D MHD cylindrical code SpeCyl [1] has been used to analyze configurations with increasing level of complexity. First of all, numerical solutions have been compared with analytical ones in the most simple case of a uniform axial magnetic field: an excellent agreement is obtained for both the shear Alfvén wave (SAW) and the compressional Alfvén eigenmodes (CAEs). Then, the RFP configuration has been studied by assuming perturbations with a single space periodicity. Phenomena such as phase mixing of SAW, resonant absorption of CAEs and the appearance of the global Alfvén eigenmode (GAE) are reported. Finally, the fully 3D RFP case with realistic magnetic reconnection events [2] has been investigated, showing for the first time in nonlinear RFP simulations the excitation of Alfvén waves by magnetic reconnection. Modeling results are in good quantitative agreement with the experimental characterization of Alfvénic activity observed in RFX-mod [3]. [1] S. Cappello and D. Biskamp, Nucl. Fusion 36 (1996) 571 [2] D. Bonfiglio et al., Phys. Rev. Lett. 111 (2013) 085002 [3] S. Spagnolo et al., Nucl. Fusion 51 (2011) 083038

The magnetic topology of the stochastic edge of a helical reversed-field pinch in the quasisingle helicity state (QSH)\footnote{R. Lorenzini et al., Nature Phys., \textbf{5}, 570574, (2009).}, which is characterized by a dominant tearing mode with helicity m/n=1/7, shows to be deeply influenced by higher harmonics (m±1)/n, with the same n=7, due to the toroidal coupling of the dominant mode with the Shafranov shift. Recent analyses\footnote{M. Agostini et al., Nucl. Fusion, \textbf{57}, 076033, (2017).} show that even a modest amount (~5~mT total) of higher 77 in the modified RFX-mod2 experiment could result in a dramatic reduction of PWI in this device.

A variety of Alfvén eigenmodes has been observed in the magnetic spectra of reversed field pinch (RFP) plasmas. In the RFX-mod device, long wavelength high frequency (>400kHZ) modes have been detected in ohmically heated plasmas, whose amplitude is found to be enhanced during the spontaneous reconnection processes characterizing the RFP dynamics [1]. The Alfvénic nature of such fluctuation is deduced by the linear dependence of the associated frequency on the Alfvén velocity. More recently, a lower frequency branch of discrete coherent Alfvén eigenmodes has been observed to be destabilized in RFX-mod when the spontaneous transition from an axysimmetric to a single helical equilibrium occurs at high plasma current levels (above 1MA). In the MST experiment, energetic particle modes, part of them lying in the Alfvèn continuum, characterized by a bursty behavior in the 50-200 kHz frequency range, have been observed to be destabilized by the injection of 25 keV particles from a 1MW NBI system [2]. A newly discovered higher frequency, more continuous in time, Alfvenic mode induced in MST by NBI will be here presented, whose spectral properties are found to be strongly influenced by the beam characteristics, in particular, by the fast particle velocity and modulated by the low frequency MHD behavior associated with tearing mode activity. Moreover, it will be shown that in the MST spectra, a long wavelength mode family has been recognized in ohmically heated plasmas, which resembles those observed in the RFX-mod experiment when operated in standard axisymmetric equilibria. The spectral properties of such instabilities will be here compared to those predicted by numerical simulations performed with the 3D MHD nonlinear visco-resisitive code SpeCyl [3], which suggest that reconnection processes in the RFP may destabilize Global Alfvén eigenmodes, as well as Compressional modes, normally disregarded in Tokamak plasmas, because of the associated very high frequency.

Magnetically confined plasmas of fusion interest display phenomena that find several analogies in astrophysics and other complex systems. This is the case of toroidal pinches where magnetic self-organization molds the plasma into a peculiar helical shape when the ratio of plasma current to toroidal magnetic flux exceeds the socalled Kruskal-Shafranov limit. In this case, a core kink instability ("sawtoothing" and/or "snake") tends to develop in Tokamaks 1 . Similarly, a global helical shape (so-called Quasi Single Helical, QSH, regimes) forms in Reversed Field Pinch experiments (RFP)2,3 , helically modulating the plasma up to the edge 4,5 . MHD modeling has been largely successful in capturing the basic features of such a phenomenon 6,7,8 , quite evading the famous Taylor's relaxation theory for RFP 9,10 . Nearly periodic relaxation events involving current sheet reconnection can be observed 6,9 , together with a magnetic chaos healing effect when the helical states are robustly achieved11,8 . The latter effect is presently the best candidate to explain the formation of internal electron transport barriers2 observed in RFP helical regimes. In particular, "hidden" magnetic field lines transport barriers have been recently detected in experimental-like numerical simulations, which are associated with "fine" topological structures like Cantori sets or Lagrangian Coherent Structures 12 . After summarizing these general features, we here discuss the recent successful MHD prediction of alternative helical regimes, obtained by seed edge magnetic perturbations with suitable choice of helical pitch. A first set of RFXmod experiments substantially confirms modeling predictions 13 . The new helical regimes obtained as plasma response to edge Magnetic Perturbations are predicted to favor magnetic chaos healing in the case of non-resonant seeds. After first indications obtained in RFX-mod experiment 13 , we expect to validate modeling predictions concerning transport properties in the modified device RFX-mod2 starting operation in 2020. The device, characterized by reduced plasma-wall/feedback coils distance, will provide efficient control action in order to negotiate at best with RFP helical magnetic self- organization.

Magnetic self-organization in toroidal pinches takes a peculiar helical shape when the ratio between toroidal current and toroidal magnetic flux (Pinch Parameter) is pushed above a characteristic threshold. This is clearly seen in the form of core kink instability in Tokamaks (so called Kruskal-Shafranov instability) and as a global helical shape in Reversed Field Pinch experiments (RFP). In this last case the nonlinear saturation approaching an ohmic helical equilibrium can be recognized. In fact, MHD modeling has been largely used to successfully capture such a basic phenomenon, quite evading the famous Taylor's theory for RFP relaxation. The globally helical RFP has been shown to bring interesting confinement properties, featuring internal transport barriers, which might be understood as a result of magnetic topological structures like Cantori sets or Lagrangian Coherent Structures, and magnetic chaos healing. Together with summarizing these general features, we here discuss the recent successful MHD prediction of alternative helical regimes, achievable by using edge magnetic perturbations with suitable choice of amplitude and helical pitch. In this way, typical features like helical amplitude and dithering frequency can be tuned almost at will, with impact on transport properties. A first set of RFX-mod experiments substantially confirms modeling predictions.

Magnetic chaos healing in 3DMHD modeling of helical regimes, achievements and open issues

Magnetic islands are spontaneously present in the reversed-field pinch (RFP) and the stellarator edge, and they are produced via magnetic perturbations (MP) in the tokamak edge for various purposes (ideal mode stabilization, ELM suppression, etc.). In a chaotic edge, the ion and electron drifts yield a predominantly electron driven radial diffusion when approaching the island X-point, while ion diffusivities are generally an order of magnitude smaller. This results in a strong, positive radial electric field Er > 0 pointing outward the island O-point. This mechanism has been often neglected on the grounds that electron drifts are small [1], but this is not the case in a chaotic environment. The electric field is associated with a convective cell, which can dominate transport within the island with respect to the ordinary chaotic radial diffusion. The convective cell has, at a first approximation, the same symmetry m/n as the parent island. An analytical model for the plasma potential is implemented in the Hamiltonian, guiding-centre code ORBIT [2], and analyses of the ambipolar flow/radial electric field (in RFX-mod [3] and TEXTOR devices [4]) show that both ion- and electron-dominated transport regimes can exist, known as ion and electron roots in stellarators. This gives the hint that localized ECRH/ICRH heating can modify the ambipolar solution and therefore greatly influence transport near the island. In addition to this, recent results [5] show that the electrostatic response to the island has symmetry m/n only at a first approximation: higher harmonics (m?k)/n, k=1,2..., with the same n, toroidally couple to the base mode, influencing the electrostatic and kinetic response to the MP. We will show that this phenomenon is deeply rooted in the magnetic topology itself, by using the standard analysis of connection lengths, plus a more sophisticated metric, such as the Poincaré recurrence time analysis.

The kinetic response of the edge plasma to a magnetic perturbation (MP) is studied in the RFX-mod device, both in RFP and tokamak discharges. It is shown that the helicity of the edge magnetic topology and also of the kinetic properties of the plasma edge is quite different from the one of the dominant MP. When a mostly monochromatic MP is present, the connection length to the wall of the magnetic field has a complex poloidal structure, with a rich poloidal spectrum, and both the floating potential and the electron pressure measured in the plasma edge reflect this behaviour.

The reversed-field pinch configuration for the magnetic confinement of fusion plasmas is characterized by the emergence of self-organized quasi-helical states found both in magnetohydrodynamic (MHD) simulations and in high current experiments 1, 2, 3. Numerical modeling can reproduce with high fidelity the MHD dynamics of high current RFP experiments, which are characterized by the formation of helical states 4 . Such helical states are quasi-periodically interrupted by transient relaxation to 3D states associated with strong reconnection events. Seed helical perturbations at the magnetic boundary are predicted to affect the helicity and the intermittency of the emergent helical states and first experimental tests have indeed confirmed this capability. Of particular interest is the possibility to stimulate non-resonant helical states, which feature better topological properties with respect to magnetic chaos healing 5,6 . This opens a new path to configuration optimization. Simulation results and first experimental evidence of alternative helical states formation in RFX-mod will be presented, together with a discussion of the different magnetic topology properties and their impact on transport barrier formation.

Recent theoretical studies of the reversed-field pinch (RFP) have demonstrated the possibility of stimulating new quasi-single helicity (QSH) regimes by allowing a small helical deformation of the magnetic boundary. In particular, QSH states based on non-resonant helicities are predicted to be more resilient to magnetic stochasticity induced by secondary modes. This has motivated a series of successful experiments in the RFX-mod device with applied magnetic perturbations (MPs). We present here recent developments in nonlinear MHD modeling of RFP, highlighting aspects which share similarities between the RFP helical regimes, the tokamak sawtoothing and the stellarator. RFP helical configurations are obtained by applying small helical MPs with resonant (n 7) and non-resonant (n < 7) toroidal periodicities. MPs turn out to be a powerful tool for varying the helical equilibrium properties, like the helical safety factor or the magnetic well profile. The width of the region of conserved magnetic surfaces turns out to be the largest for the non-resonant case. By using a method to compute Lagrangian Coherent Structures, barriers to the diffusion of magnetic field lines are diagnosed in the weakly stochastic region surrounding the conserved helical core in numerical QSH states, providing a new tool for the study of Internal Transport Barriers formation at the transition to helical regimes in RFP experiments. Similar to the tokamak configuration, the sawtoothing dynamics of RFP plasmas is mitigated with the application of helical MPs. As for the edge region, the RFP is characterized by m = 0 island chains at the q = 0 reversal surface, which play a role similar to edge islands in tokamak and stellarator plasmas.

The understanding of fusion plasmas benefits from investigating the same phenomena in different magnetic configurations such as the tokamak, the stellarator, and the reversed - field pinch (RFP) [1 - 4] . Here, the same approach is applied i n the framework of nonlinear 3D MHD modelling . We first focus on sawtoothing, i.e. , the quasiperiodic magnetic relaxation of RFP and tokamak plasma s . Sawtooth ing manifests itself together with magnetic reconnection events , and with the formation of current sheets and of dynamo electric fields . Such processes are relevant to laboratory and astrophysical plasmas as well. In both tokamak and RFP configurations , sawtoothing is quenched leading to long - l a sting helical states whenever visco - resistive dissipation is increased [ 5 ,6 ] or small helical mag netic perturbations (MPs) are applied [ 7 ,8 ]. A similar effect of MPs has been observed in the RFX - mod device [ 9 ]. We secondly consider the topological properties of partially ergodic magnetic field s , which characterize helical states in the RFP [7] . Barriers to the stochastic diffusion of field lines (the so - called cantori also studied in the tokamak configuration [10] ) are diagnosed in the weakly stochastic core of the helical RFP [ 1 1 ] , which could explain experimentally observed internal transport barriers. C hains of magnetic island s naturally occur at the RFP edge [1 2 ] . Such chains may be exploited to realize a so - called island divertor, which may alleviate plasma - wall interaction s like in the stellarator [3]. Numerical results are provided by two MHD codes , successfully benc hmarked against one another [ 6 ].

With this internal note we want to make available to the RFX colleagues the study performed for the invited talk "Effect of helical magnetic perturbations on the 3D MHD self-organization of fusion plasmas" given by D. Bonfiglio at the 7th IAEA Technical Meeting on "Theory of Plasmas Instabilities" held in Frascati, Italy on 4-6 March 2015. The proceeding papers of invited and contributed talks, including a version of this internal note, will be published on the conference website http://www.iaeatm-inst2015.enea.it/