Main

The main magnetic field configurations studied for the confinement of toroidal fusion-relevant plasmas are the tokamak3, the stellarator6 and the reversed-field pinch4,5 (RFP). In the tokamak, a strong magnetic field is produced in the toroidal direction by a set of coils approximating a toroidal solenoid, and the poloidal field generated by a toroidal current flowing into the plasma gives the field lines a weak helical twist. This is the configuration that has been most studied and has achieved the best levels of energy confinement time. Thus, it is the natural choice for the International Thermonuclear Experimental Reactor, which has the mission of demonstrating the scientific and technical feasibility of controlled fusion with magnetic confinement.

The RFP, like the tokamak, is axisymmetric and exploits the pinch effect due to a current flowing in a plasma embedded in a toroidal magnetic field. The main difference is that, for a given plasma current, the toroidal magnetic field in a RFP is one order of magnitude smaller than in a tokamak, and is mainly generated by currents flowing in the plasma itself. This feature is underlying the main potential advantage of the RFP as a reactor concept, namely the capability of achieving fusion conditions with ohmic heating only in a much simpler and compact device. In the past, this positive feature was overcome by the poorer stability properties, which led to the growth and saturation of several magnetohydrodynamic (MHD) instabilities, eventually downgrading the confinement performance. These instabilities, represented by Fourier modes in the poloidal and toroidal angles θ and φ as exp[i(m θn φ)], with m=1 and n>3R/2a (ref. 7), were considered as an unavoidable ingredient of the dynamo self-organization process4,8,9, necessary for the sustainment of the configuration in time. The occurrence of several MHD modes resonating on different plasma layers gives rise to overlapping magnetic islands, which result in a chaotic region, extending over most of the plasma volume10, where the magnetic surfaces are destroyed and the confinement level is modest. This condition is generally dubbed as the multiple-helicity state.

An efficient strategy for chaos suppression in RFPs is based on inductive control of the current profile, which was successfully proven, although transiently11,12. A different strategy, potentially stationary and relying on self-organization, is based on the theoretical prediction that the RFP could exist in the chaos-free single-helicity condition, where only one of the core-resonant resistive modes would provide the dynamo effect13,14,15. To now, the best approximation of single helicity were quasi-single-helicity (QSH) states: here, the innermost resonant m=1 mode dominates over the others, the so-called secondary modes, which maintain a finite amplitude. QSH states have been observed in several RFP devices5,16,17 and predicted in numerical simulations18. They are an intermediate state between the multiple-helicity and the theoretical single-helicity ones, because the secondary modes are still present.

Here, we present experimental findings obtained in the RFX-mod device19 that show a clear evolution of the plasma towards the single-helicity state. This has been made possible by the exploration of a wide range of plasma current levels, which for the first time in a RFP have reached values higher than 1 MA, and by the control of the radial magnetic field at the plasma boundary through an advanced feedback system20. The novel results are that, as the current is raised, the plasma shows an increasing preference for the QSH state, and that this state becomes purer (the normalized secondary mode amplitude is reduced). Furthermore, above a threshold of about 4% in the dominant mode normalized amplitude, a topological change in the magnetic configuration is observed: the separatrix X-point of the dominant mode magnetic island merges with the main magnetic axis, and the two disappear. The former island O-point becomes the only magnetic axis and a helical plasma column is obtained in an axisymmetric device. This new single-helical-axis (SHAx) state21 is theoretically predicted22 to be more resilient to the magnetic chaos induced by the secondary modes.

Here, we show that this new equilibrium features strong electron transport barriers21, consistent with the mean helical magnetic configuration associated with the dominant mode. The SHAx state is characterized by the emergence of almost invariant magnetic surfaces, although in the presence of a small residual magnetic chaos, giving rise to improved confinement properties. These surfaces are interpreted as ghost surfaces23,24.

Figure 1a shows a plasma current waveform for a 1.5 MA discharge carried out in RFX-mod. The amplitude of the innermost resonant m=1 mode (n=7) and the total amplitude of the secondary m=1 modes are shown, for the flat-top phase, in Fig. 1b. The graph shows that the plasma preferentially stays in a QSH state where the n=7 mode is dominant and the other modes are reduced. These phases can last ten times longer than the typical energy confinement time (3 ms). Occasional back-transitions to the multiple-helicity state occur, as observed also in numerical simulations18. In the QSH phases, the normalized edge amplitude of the dominant mode reaches values of about 4%, and a SHAx state is obtained.

Figure 1: Typical 1.5 MA plasma discharge in RFX-mod.
figure 1

a, Plasma current as a function of time. The vertical dashed lines delimit the so-called flat-top phase of the discharge. b, The black curve shows the amplitude of the m=1/n=7 dominant mode during the flat-top phase of the discharge, and the red curve shows the amplitude of the secondary modes (m=1/n=8–23), defined as the square root of the sum of their squared amplitudes. The amplitudes are those of the toroidal magnetic field component at the plasma surface, as obtained with a Fourier transform in space of the signals given by a system of 48×4 pick-up coils located outside the vacuum vessel, and are normalized to the average poloidal magnetic field measured at the same location. It is clearly seen that the system oscillates between two states, one where all of the modes have similar amplitudes (multiple helicity) and another one where there are a dominant mode and secondary ones (QSH).

The QSH properties are found to be related to the Lundquist number S, a dimensionless parameter relevant to the MHD dynamics, defined as the ratio of the resistive diffusion time to the Alfvén time3. Figure 2a shows the dependence of the QSH persistence with S, showing that higher S values, which correspond to hotter plasmas obtained at higher current levels, lead to longer persistence, reaching values up to 90%. The amplitude of the dominant and secondary modes during the QSH phase is also found to depend on the Lundquist number. This is shown in Fig. 2b, from which it is apparent that secondary modes decrease steadily as S rises, whereas the dominant one shows at first a fast rise, then followed by a slower increase. As the transition to a SHAx state occurs when a threshold in the dominant mode amplitude is exceeded, this implies that at high S the QSH states are indeed SHAx. It is important to remark that the increase of S shown in Fig. 2 is due to the increase of the flat-top current and the consequent rise in temperature, and that no extra sources of plasma heating are used.

Figure 2: Mode amplitudes versus the Lundquist number.
figure 2

QSH properties plotted versus the Lundquist number S for a wide database of discharges with currents ranging between 0.3 and 1.6 MA. a, QSH persistence, quantified as the ratio between the total duration of the QSH during the current flat-top and the flat-top duration, plotted as a function of S. The persistence is longer at higher S. b, Amplitude of the dominant (black) and secondary (red) modes during the QSH phase plotted as a function of S. As the Lundquist number increases, the dominant mode amplitude grows, whereas the amplitude of the secondary ones is reduced.

To assess the quality of magnetic equilibrium, we have developed a simple, yet effective method for computing the helical flux χ(r) and therefore the position and shape of magnetic surfaces for the SHAx state. When these surfaces are not destroyed by magnetic chaos, flux functions (that is, any function f(r) such that B · f=0) are constant on them1. In particular, this applies to kinetic pressure and, owing to the fast parallel transport, to temperature and density. A typical electron temperature profile measured in a SHAx state is shown in Fig. 3a. Figure 3b shows the same profile plotted as a function of the effective radius ρ=(χ/χ0)1/2, where χ0 is the helical flux at the plasma boundary. It can be seen that the two half-profiles measured on the two sides of the magnetic axis, marked by different colours, collapse onto each other only when plotted as a function of ρ. In particular, the strong gradients marked by the shaded region, which indicate the formation of a transport barrier, turn out to be the same, showing that they appear different in cylindrical coordinates only because of the different spacing of the flux surfaces on the two sides. A reconstruction of the two-dimensional (2D) temperature map on the poloidal plane is shown in Fig. 3c. The hot region is bean-shaped, and centred off the geometric axis, on the resonant surface of the m=1/n=7 mode.

Figure 3: Mapping of the temperature profile on the helical flux surfaces.
figure 3

The electron temperature at a given time is measured in 84 points aligned along a horizontal diameter of the vacuum vessel with the Thomson scattering technique28. a, Typical electron temperature profile measured in 1.5 MA discharges when the plasma is in a SHAx state. It can be seen that a high-temperature (0.8–0.9 keV) region is present in the core plasma and extends on both sides of the geometric axis, a typical feature of SHAx states. The profile is asymmetric with respect to the geometric axis, and the strong gradient regions (shaded) show substantially different gradients on the two sides. On the same graph, the continuous line gives the profile of the effective radial coordinate ρ. Red and blue refer to the two opposite sides with respect to the helical magnetic axis. A typical profile in multiple helicity is shown in green for reference. b, Temperature profile plotted as a function of ρ, showing the collapse of the two half-profiles one onto the other. The error bars in a and b are given by the confidence limit on the temperature resulting from the gaussian fit of the scattered light spectrum. c, Reconstruction of the full 2D map of the temperature on the poloidal plane, obtained from the dependence of χ on the geometric coordinates. The χ contour lines are also shown.

These results are further confirmed by the high-space-resolution soft X-ray (SXR) tomographic diagnostic: SXR emissivity is, in fact, a strongly increasing function of the electron temperature. By assuming the SXR emissivity to be a simple function of the helical flux (as shown in Fig. 4a) and properly tuning the ρ dependence, it has been possible to well reproduce the line-integrated measurements (Fig. 4b). The 2D emissivity map (Fig. 4c) shows once again that the hot core region is bean-shaped.

Figure 4: Mapping of line-integrated emissivity and line-averaged density measurements on helical flux surfaces.
figure 4

The X-ray emissivity is measured by silicon photodiode sensors29 along 78 lines of sight organized in two sets, one nearly horizontal and one nearly vertical, lying on the same poloidal plane. Each measurement is a line integral along the corresponding chord giving the SXR brightness. The plasma density is measured along 8 chords by an interferometric technique30. Each measurement is a line integral along the corresponding chord. a, Emissivity profile parametrized using a simple three-parameter model of the form ɛ(ρ)=ɛ0(1−ρα)β. b, Measurements of line-averaged X-ray emissivity in red and values reconstructed using the ɛ profile in a in black. An excellent match between measurements and reconstructed values can be observed. c, 2D emissivity map resulting from the reconstructions. d, Density profile parametrized using a model of the form n(ρ)=a+b ρ2+c ρ3+d ρ4. e, Line-averaged density values measured by the interferometric system during a pellet injection and values reconstructed using the profile in d. The error bars are given by the standard deviations calculated at the end of the discharge. f, Reconstructed 2D density map. The straight lines represent the chords along which measurements are made.

Finally, the same approach has been applied to electron density measurement. Owing to the small particle source in the plasma core, being the plasma fuelled at the edge by wall recycling, the density profile inside the core is nearly flat25. To induce a density gradient in the core, solid hydrogen pellets26 have been launched into the plasma, so that their trajectories would pass through the centre of the bean-shaped region in SHAx states. One such case is illustrated in Fig. 4e, where measurements of chord-averaged plasma density are shown at the time when pellet ablation ends. The asymmetric profile can be explained only by a dependence of the density on the helical flux, as previously done for the electron temperature and SXR emissivity profiles. An analysis similar to that carried out for the tomographic data has been carried out. The reconstructed profile and the 2D density pattern are respectively shown in Fig. 4d and f.

The available ion temperature measurements (determined from Doppler broadening of the spectral line emitted by the H-like O7+ ion) are not sufficiently resolved in space and time to assess whether an ion transport barrier is also formed. The measured line-averaged ion temperature, ranging from approximately 0.5 to 0.7 of the electron temperature, is consistent with a classical collisional electron–ion energy transfer.

On the basis of the results shown in Figs 3 and 4, the assumption of good isobaric and isoemissive helical flux surfaces—a signature of MHD equilibrium—allows a consistent interpretation of the data. Although residual magnetic chaos is present, these good-quality surfaces lead to robust electron temperature gradients and are the likely signature of ghost surfaces at constant χ.

The spontaneous occurrence of a new self-organized helical equilibrium with a single helical axis, reduced magnetic fluctuations and strong transport barriers provides a change of paradigm for the RFP. As the persistence of these improved confinement phases increases with plasma current, the likelihood of achieving steady SHAx states in multi-mega-ampere devices can be inferred. This opens a promising path for this configuration, because the emergence of ghost surfaces is likely to be followed, at even lower secondary mode amplitudes, by the separation of the magnetic islands. Island separation would lead to the formation of regions of conserved surfaces, giving rise to transitions to further improved confinement regimes. Further improvements are also expected by optimization of magnetic boundary and fuelling, the latter corroborated by preliminary experiments with pellet injection. This new vision supports a reappraisal of the RFP as a low-external-field, non-disruptive, ohmically heated approach to magnetic fusion, exploiting both self-organization and technological simplicity. The RFP is thus confirmed as a flexible environment for developing fusion science and for contributing to fusion predictive capabilities.

Methods

The flux surfaces in the SHAx state have been computed assuming that the magnetic field is the superposition of an axisymmetric equilibrium and of the helical perturbation corresponding to the dominant m=1 mode eigenfunction. This eigenfunction is computed by means of the toroidal force-free Newcomb’s equation, supplemented by magnetic measurements at the plasma edge used as boundary conditions27. The helical flux χ(r) is given by χ=m Ψ0n F0+(m ψmnn fmn)exp[i(m ϑn ϕ)]+c.c., where Ψ0 and F0 are the poloidal and toroidal fluxes of the axisymmetric equilibrium, ψmn and fmn are those of the dominant mode and ϑ and ϕ are the flux coordinates defined in ref. 27. It can be shown that χ is constant along magnetic field lines (that is, B · χ=0). Therefore, its contour plots represent the nested helical flux surfaces of the SHAx state.