Kevin Prieur
email: kevin.prieur@centralesupelec.fr

Daniel Durox

Guillaume Vignat

Thierry Schuller

Sébastien Candel

Experimental determinations of Flame Describing Functions of swirling spray flames

Keywords: Combustion dynamics, Flame Transfer Function, Flame Describing Function, Spray flames

Studies of the dynamics of spray swirling flames are of importance from a fundamental point of view and also because these flames are used in many practical systems like aero-engine combustors. Under certain conditions these flames give rise to various types of undesirable combustion instabilities. This article focuses on the measurement of the combustion dynamics of a system comprising a single injector designated as SICCA-Spray and reports novel Flame Describing Function (FDF) data on spray swirling flames. The test configuration comprises a tubular plenum fed with air by a swirling injector, a simplex atomizer that injects the liquid fuel (n-heptane) within a hollow cone spray and a cylindrical combustion chamber where the air and the fuel spray mix and burn. The air flow can be excited with two driver units plugged on the plenum, creating a flow perturbation which acts on the flame. The amplitude of fluctuation u is controlled with a hot wire in the plenum. The flame response to incoming perturbations is then examined by recording the OH* signal with a photomultiplier equipped with an interferometric filter. It is assumed that fluctuations of the light emission from this excited radical is proportional to heat release rate perturbations Q . The acoustic pressure p at the backplane of the chamber is also recorded with a cooled waveguide microphone. Two different FDFs are defined and compared: Q / Q = f (p /p) and Q / Q = g (u /ū). The data are then used to analyze self-excited instabilities observed in the SICCA-Spray burner and to interpret some recent observations of strong self-sustained azimuthal instabilities in the MICCA-Spray annular combustor, equipped with sixteen injectors identical to that used in the SICCA-Spray set-up.

Introduction

Advanced combustion systems developed for aircraft engines may give rise to combustion instabilities. These instabilities are one of the major problems encountered in the design of the new generation of low-emission modern engines. Since most practical combustors in aero-engines or in gas turbines feature an annular geometry an increasing number of investigations has been devoted to the analysis of thermoacoustic coupling in such configurations [START_REF] Krüger | Proceedings of ASME Turbo Expo[END_REF][START_REF] Schuermans | ASME Conference Proceedings Paper[END_REF][START_REF] Krebs | [END_REF][START_REF] Moeck | ASME Conference Proceedings, Paper GT[END_REF][START_REF] Worth | Proceedings of the Combustion Institute[END_REF][START_REF] Worth | [END_REF][START_REF] Bourgouin | Proceedings of ASME Turbo Expo 2013[END_REF]. There are yet very few works reported on annular systems powered by liquid fuels ,with the exception of LES calculations like [START_REF] Boileau | [END_REF] and experiments carried out in [START_REF] Prieur | Proceedings of the Combustion Institute[END_REF][START_REF] Prieur | Proceedings of the Combustion Institute[END_REF].

Recent experiments in the MICCA-Spray annular combustion chamber equipped with sixteen swirling injectors and powered by fuel spray revealed regimes with strong self-sustained combustion oscillations [START_REF] Prieur | Proceedings of the Combustion Institute[END_REF]. These instabilities were found to appear in the form of periodic bursts coupled to the first azimuthal 1A1L mode of the combustion chamber with acoustic pressure levels up to 5% of the chamber static pressure. The frequency of these instabilities was also shown to feature an intriguing pattern with large variations between 700 and 800 Hz during their growth and decay. One motivation of the present study is to gather data on the Flame Describing Function (FDF) of a single spray injector of the MICCA-Spray chamber to help unravel this complex dynamics.

In many theoretical analysis of combustion instabilities coupled to azimuthal modes [START_REF] Schuermans | 44th AIAA Aerospace Sciences Meeting and Exhibit[END_REF][START_REF] Noiray | [END_REF][13], the flame response is modeled by a relationship linking the heat release rate oscillations Q directly to pressure oscillations p in the chamber. The modeling effort carried out in parallel has emphasized the fundamental role of the flame response to mass flowrate perturbations and a large number of studies have focused on the determination of Flame Transfer Functions (FTF) linking heat release rate fluctuations in the flame to incoming flow-rate disturbances [14][15][START_REF] Candel | Proceedings of the Combustion Institute[END_REF][START_REF] Schuller | [END_REF]. The case of swirling flames of considerable technical interest was considered more recently [18,19]. An extension of the transfer function concept in the form of the Flame Describing Function (FDF) framework was developed to account for nonlinearities by representing the flame response in terms of frequency and amplitude of the impinging perturbation [20,21].

F (ω, u ) = Q / Q u /u = G (ω, u ) e iϕ(ω,u ) (1)
This nonlinear framework was successfully applied to determine the oscillation frequencies and oscillation levels of unstable modes in a premixed swirling combustor equipped with a single injector [19]. Analysis of the flame response to forced flow disturbances indicates that the flame shape defines to a large extent its linear [22] and nonlinear [START_REF] Durox | Proceedings of the Combustion Institute[END_REF] dynamical response.

Many of the available investigations concern premixed flames while practical systems are often fed with liquid fuel. The later injection systems are also prone to thermo-acoustic instabilities [START_REF] Giuliani | Proceedings of the Combustion Institute[END_REF][START_REF] Gajan | [END_REF][26][27][28][START_REF] Mirat | Proceedings of the Combustion Institute[END_REF]. A few studies report characterizations of the limit-cycle oscillations with OH* or CH* chemiluminescence measurements correlated with the spray dynamics [START_REF] Eckstein | [END_REF]27,28]. In most cases there is a sizable pressure drop in the liquid fuel feed line providing a large impedance to impinging pressure perturbations so that one only needs to consider fluctuations in the air feeding manifold and injection unit. Laboratory scale experiments reported in the literature explore the sensitivity of the flame to these air flow rate modulations but are limited to just a few forcing frequencies at fixed amplitudes [31,[START_REF] Giuliani | Proceedings of the Combustion Institute[END_REF][START_REF] Gajan | [END_REF].

The acoustic response and FDF of swirling spray flames to a broad frequency range of air modulation was recently considered in [START_REF] Mirat | Proceedings of the Combustion Institute[END_REF] where the spray flames investigated feature a distinct response compared to those obtained for premixed swirling flames. In particular, the low frequency limit of the FDF gain differs from that found in the premixed case and tends to zero in that limit. The FDF phase lag was also found to depend on the perturbation level, a feature already observed in partially premixed systems [32]. Measurements in [START_REF] Mirat | Proceedings of the Combustion Institute[END_REF] were carried out in the case of steam assisted liquid fuel injectors. This short review of the literature indicates that the dynamical response of spray flames and more specifically swirling spray flames is not well documented. The present study tries to reduce this lack of knowledge.

The next section briefly describes the experimental set up (section 2). The acoustic response to external modulations is discussed in section 3. The system features natural, self-excited instabilities that are characterized in section 4. The forcing of the flame is compared to the self-excited case (section 5) to validate the methodology. The response of swirling spray flames is then examined using the FDF framework for nonlinear combustion instability analysis (section 6). The frequency response of the flame to acoustic perturbations is characterized by comparing the chemiluminescence signal generated by combustion and assumed to represent the heat release rate in the flame to the velocity signal measured upstream the injector. The present investigation aims at identifying parameters that control the sensitivity of spray flames to incident velocity perturbations. It is also indicated that the knowledge obtained and theoretical results can be used to predict the behavior of such injection systems when they are placed in an annular configuration.

Experimental configuration

The experimental configuration represented schematically in Figure 1 comprises a plenum and a combustion chamber. The chamber is formed by a cylindrical quartz tube with a diameter d c = 70 mm and of variable length l c . Two driver units are plugged on the 300 mm cylindrical plenum to modulate the air flow. A swirled injection unit links the plenum to the combustion chamber. The injector is composed of an air circuit passing through a swirler and of a liquid atomizer to feed liquid fuel and form spray of droplets.

Air is injected at ambient temperature at the bottom of the plenum. It then reaches the injector where it passes through a radial swirler with six 4.5 mm holes and exits the injector with a measured swirl number of 0.63. The head loss through the injector is of the order of 4% of the atmospheric pressure at the nominal operating conditions investigated. Fuel (n-heptane) is injected using a simplex atomizer supplied pressurized at 9 bar. The atomizer generates a hollow cone and is positioned in the middle of the swirler. The droplet spray formed by this system has a mean diameter d 10 = 8 µm and a SMD d 32 = 25 µm. The injector is terminated by a convergent section.

Nominal injection conditions, used throughout this article, are as follows. Air flow rate is fixed at ṁair = 2.59 g s -1 and fuel flow rate is ṁfuel = 0.144 g s -1 . The global equivalence ratio is φ = 0.85 and the global power P = 6.5 kW. The bulk velocity of the air at the exit of the injector is U b = 43 m s -1 . Figure 2 presents the Abel transform of the CH* chemiluminescence of the flame using a Princeton Instrument PiMax 4 intensified camera. The flame takes a well defined M-shape, slightly lifted 10 mm above the chamber backplane. Detailed characterizations of the spray and of the airflow with laser optical diagnostics can be found in [START_REF] Prieur | Proceedings of the Combustion Institute[END_REF].

The Flame Describing Functions (FDF) based on the velocity perturbation u and on the acoustic pressure fluctuation p in the chamber are respectively defined by :

F(ω, |u |) = Q / Q u /u (2)
F(ω, |p |) = Q / Q p /p (3)
These two complex functions are measured with the diagnostics indicated in Figure 1. The gaseous flow is modulated by the two driver units of 150 W and 8 Ω each plugged in the plenum. Due to the relatively high head loss induced by the swirlers, these two units are enclosed in cavities linked to the plenum to balance the pressure on both sides of the driver unit membrane. The frequency of the oscillations ranges from 250 to 900 Hz. A sine-wave generator and an amplifier actuates the loudspeakers over this range. The amplitude u of the perturbation generated in the upstream flow is determined using a hot wire placed 100 mm upstream the combustion chamber backplane. This region is chosen in order to have a nearly laminar velocity profile. Measured u /u range from 0.1 to 1.3. Two microphones (MP1 and MP2) are placed around the same position to evaluate u using a second method and check the hot wire measurements. A third microphone (MC1) measures the acoustic pressure p in the chamber, pressure which appears in Eq. ( 3). It is flush mounted on a cooled waveguide directly plugged on the chamber backplane. A photomultiplier (PM) with an OH* filter is placed on the side of the combustion chamber to record the flame emission. The PM sensor is tilted towards the injector position in order to capture the entire flame luminosity whatever its shape. It is known that the OH* emission signal is proportional to the heat release rate in premixed flames. It is assumed in what follows and this is admittedly an approximation that there is a link between the two quantities in the present case where the flame is fed by a spray of droplets and it is considered that I /I Q / Q. It was shown in [START_REF] Mirat | Proceedings of the Combustion Institute[END_REF] that this approximation yielded reasonable results for instability predictions for globally lean injection conditions (φ < 1).

Fig. 3 Left: Acoustic response of the system for a range of frequencies up to 900 Hz under cold conditions with flowing air at ṁair = 2.59 g s -1 . The voltage of the excitation system is kept constant. Right: Measured velocity fluctuation ratio u /u obtained with the hot wire.

Frequency response of the system

In these first experiments there is no combustion and the system operates under cold flow conditions. The driver units are used to modulate the flow with increasing frequency from 100 to 900 Hz at a fixed voltage V 0 and the air flow is maintained constant at ṁ = 2.59 g s -1 . The resulting acoustic signals recorded using MP1, MP2 and MC1 are plotted in Figure 3, left. Two resonance peaks are found, one at 260 Hz and the other at 490 Hz. The first resonance corresponds to a pure plenum mode whereas the second is a coupled chamber-plenum mode. The right plot in Figure 3 shows the computed velocity fluctuation ratio u /u with u the mean velocity, when the frequency f is increased at fixed voltage V 0 . It is easy to see that u /u is not constant over the frequency range. This is due to an acoustic resonance near 490 Hz increasing the velocity ratio and the nonlinear response of the loudspeaker. It is important to underline this feature as it will be the method used to evaluate the FDF in section 6.

A Helmholtz solver is used to calculate the mode shape in the system at the frequency of f h = 543 Hz corresponding to one of the self-sustained combustion oscillation observed in the setup and analyzed in section 4 when the chamber length is varied. In this simulation, the temperature in the plenum is considered to be constant T 1 = 300 K while the temperature in the combustion chamber is T 2 = 1200 K. The length of the chamber is augmented to l tot = l c + l end = 320 mm in the simulation with l c = 280 mm the combustion chamber length and l end = 40 mm the end-correction which has been measured (not shown here). For simplicity, the complex injector system has been replaced by an single hole of equivalent surface with the same length. Note that the convergent end-piece of the injector is taken into account in this simulation. The mode shape is shown in Figure 4 through a cut of the system. One can see that the mode is longitudinal, with a maximum pressure near the backplane. The cylindrical pipe where the hot wire is located (see Figure 1), has only a slightly lower amplitude level, but with the same sign.

Self-excited instabilities

For certain geometrical lengths l c of the combustion chamber, the system exhibits well defined self-excited longitudinal instabilities. Frequency f , velocity fluctuation rate u /u, heat release rate Q / Q ratios and acoustic level in the chamber p are given in Table 1 for different lengths l c . One finds that a critical chamber length is necessary to observe thermo-acoustic modes and, in the case of the SICCA-Spray burner, l c needs to be greater than 245 mm to observe combustion instabilities. All self-excited instabilities are longitudinal modes developing in the chamber.

An example with l c = 280 mm is now examined. Combustion instabilities appear in this case at 530 Hz. This frequency can be retrieved by considering a quarter-wave mode of the combustion chamber cavity. The mean temperature in the system is of the order of T = 1200 K giving a speed of between the velocity and the heat release signals. The heat release ratio Q / Q fluctuates around 0.25 while the acoustic pressure in the chamber p ch reaches a peak value of 1700 Pa. Acoustic pressure and heat release rate are oscillating perfectly in phase, at the same frequency, indicating that the Rayleigh criterion is satisfied. This implies that p ch and u at the injector outlet are in phase quadrature because iρω = ∇p. Note that the velocity u which is plotted in Figure 5 is measured by the hot wire 100 mm upstream the combustion zone, where Q is measured by the photomultiplier. Between these two points, the information may not always travel at the acoustic velocity c ∼ 340 m s -1 all the way. It is known for example from [19] that the swirler induces a vorticity mode that propagates at the flow velocity in the injector unit. One may have to consider the energy transfer from the acoustic to the convective propagation mode which would indicate a major shift in the propagation velocity. LES calculations, not presented here, have demonstrated that the velocity perturbation u is transported at the acoustic speed for approximately 80 mm, corresponding to the beginning of the injector unit in the system. The velocity then evolves in a complex manner in the injector.

Phased chemiluminescence images are recorded using a PiMax 4 camera equipped with a CH* filter. Images are then Abel transformed to retrieve the contribution of a slice to the flame emission. As the flame is axisymmetric, only half images of the right part of the flame are shown in Figure 6 corresponding to phases from 0 to 330 • . One may note the strong flame displacement in the axial direction and a flame brush curving during the peak of the cycle around 180 • . The flame length is also notably affected. The luminosity of the flame varies along its length. Note that the algorithm used to extract the flame slice allows a quantitative comparison of luminosity between the different images. The anchoring point of the flame is essentially fixed in space at a distance of 10 mm from the injector exhaust.

Forced flame response

In this section, the combustion chamber is equipped with a shorter flame tube l c = 165 mm to avoid self-excitation of instabilities. As in section 4, the dynamics of the flame is recorded using a photomultiplier and three microphones. Loudspeakers depicted in Figure 1 are actuated by a function generator and an amplifier to force the flame motion at a chosen frequency. The amplitude is modulated in order to get the desired velocity fluctuation rate u /u at the hot wire location.

In Figure 7, the flame is forced at f = 530 Hz and at an amplitude u /u = 0.45 to match the conditions corresponding to the self-excited instability with the l c =280 mm flame tube. Velocity fluctuations are sinusoidal and at the same frequency and with a difference in amplitude of 10% in order to match with the self-excited case. The heat release rate fluctuations Q / Q in the modulated case are nearly the same as those recorded in the naturally excited case. The main difference is found in the phase ϕ uQ = -π/2 between velocity fluctuations and heat release rate fluctuations which is the opposite of that found in the self-excited case. The second difference is in the acoustic pressure in the combustion chamber p ch which is much lower in the forced case. The pressure is around 200 Pa peak whereas the pressure in the self-excited flame is around 2000 Pa. This is the consequence of the length of the combustion chamber cavity (165 mm) that is not scaled to resonate at this frequency. Considering, as in the last section, the quarter-wave mode of the 165 mm tube, the resonant frequency is 850 Hz which is much higher then the excitation at 530 Hz. Thus the acoustic level is not amplified by the system resonance. This is an important result as many theoretical models take p as an input to deduce Q . The present experiment shows that, in the case of a forced flame, p is not an input but a result of Q . An alternative is to consider that p ∼ d Q/dt as for flames radiating noise in the free field, implying in this case a phase quadrature between the signals. These models might help to shed light on certain phenomena but they will always face the problem of their experimental validation. It is then more natural to take u as the input of Q but problems still rise as seen here. Where ϕ uQ = π/2 when the flame naturally oscillates, ϕ uQ = -π/2 when the flame is forced. This is the consequence of the velocity measurement method, as mentioned before. These observations will be examined in section 6 when looking at different Flame Describing Functions. It is worth noting that numerical results using LES exhibits the same behavior concerning the phase difference ϕ uQ in the case of the self-excited flame or in the forced case. Results are not included here but should be presented in another article.

CH* chemiluminescence images for the forced case are now compared with the images corresponding to the self-excited case. In Figure 8, the forced case is on the left hand side of each image while the self-excited case is on the right hand side. One can identify similar shapes in the flame motion as described before. The luminosity, the flame brush and the anchoring point vary in the same fashion.

In conclusion, the previous data indicate that the flame dynamics under external forcing and under self-excited oscillations at one particular frequency have similar characteristics. The flame shapes at different phases during the cycle are the same and the relative velocity and heat release rate levels have the same amplitude levels but the phase difference ϕ uQ between velocity and heat release rate in the forced and in the self-excited case do not match. It is +π/2 in the first case and -π/2 in the second and this difference needs to be interpreted.

Flame Describing Function measurement

In this section, the Flame Describing Fuction (FDF) of the spray flame is evaluated using the SICCA-Spray burner shown in Figure 1. As described in section 3, if the frequency is increased with the voltage of the driver units kept constant, the resulting u /u is not constant. A twomicrophone method is also used to avoid signal aliasing when the velocity fluctuations become large enough to induce a bias in the hot wire measurements. Microphones MP1 and MP2 in the plenum are used to reconstruct the acoustic field. They are spaced by a distance ∆x = 50 mm and placed on the sides of the hot wire (see Figure 1). The velocity fluctuation u at the hot wire point can be deduced from the following equation:

u = 1 iρ 0 ω p 2 -p 1 ∆x (4)
where p 2 and p 1 are measured by microphones MP2 and MP1 respectively. The air density is taken ρ 0 = 1.2 kg/m 3 and ω = 2πf denotes the angular frequency. This method has been validated on the bench for several conditions (not shown here). Calculated u values with Eq. (4) will then be used in the rest of this article for consistency. To obtain an FDF, one has then to sweep the excitation frequency from f 1 = 100 Hz to f 2 = 900 Hz for several output voltages V 0 of the function generator. Frequencies lower than 100 Hz are not explored for several reasons. First, the excitation system using loudspeakers has a low efficiency at frequencies under 200 Hz, making so that high amplitudes perturbations are difficult to reach. Secondly, the frequencies of interest are in the higher spectral range, around 500 Hz for both the single burner SICCA-Spray and the annular combustor MICCA-Spray. A frequency ramp from f 1 to f 2 at a constant rate of 1 Hz s -1 is then used to modulate the system. Signals are recorded at a sampling frequency f s = 32,768 Hz during 2 s, ensuring that the forcing signal frequency only increases by 2 Hz and is essentially constant during the acquisition time. The transfer function is then estimated using the Welch's averaged periodogram method to calculate the Cross-Power Spectral Density (CPSD). The 2 s signals are divided into eight segments Fig. 9 FDF based on the velocity u and the heat release rate Q . Left: Phase ϕ and gain G of the FDF between the velocity and the heat release rate ratio and their fit. Right: Linear interpolation of the FDF between u /u = 0.1 and 0.9 based on the results presented in the left hand side. Note that the frequency axis of the two figures are inverted.

with Hamming windows and a 50% overlap. The gain G (f ) and phase ϕ (f ) of the transfer function are then deduced at the modulation frequency f . This processing is done to evaluate the transfer functions between u /u and Q / Q and p /p and Q / Q. This gives a large dataset of Flame Transfer Functions for different u /u between f 1 and f 2 . Data are finally processed to only keep specific values for u /u ranging from 0.1 to 1.3 with a spacing of 0.1. For each u /u value, a 5% error in the estimation of the velocity fluctuation ratio is admitted. Data are then interpolated for u /u < 0.9.

The FDF based on the velocity u /u and the heat release rate Q / Q ratios is displayed in Figure 9. The left figures represent 2D-plots of phase (top) and gain (bottom). Data are fitted for u /u up to 0.8. This value is limited on purpose because higher values cannot be applied in various frequency ranges due to the difficulty to excite high levels of fluctuations over a wide range of frequencies. On the right hand side, a linear interpolation is applied to u /u to give a better understanding of the flame response. One observes that the phase increases quasi linearly across the frequency range. The impact of the velocity ratio u /u ratio is small except in the band 600 700 Hz where one can find a difference of 1 rad between the FDF phase lags for u /u = 0.1 and 0.8. Based on the phase ϕ, the time lag τ = φ/(2πf ) can be estimated. Considering that the phase slope is rather constant across the frequency range, τ = 1.8 ms.

In the gain plot (bottom left), the amplitudes at low frequencies are scattered. The trend in gain is clearer when the frequency reaches 300 Hz. One can then observe a peak at 640 Hz. Note that this value is far from the natural resonant frequency of the system, indicating that this is the result of the flame response. The gain strongly decreases when u /u is increased, demonstrating the nonlinearity of the phenomenon and the saturation of the flame response at high amplitudes of modulation.

Theory provides possible bands of instability. These bands plotted in gray in the following phase diagrams are approximately bounded by horizontal lines corresponding to {π, 2π}. When π < ϕ < 2π [mod. 2π], the configuration is potentially unstable, a result already reported in [START_REF] Schuller | [END_REF]33,34]. This first consideration does not take into account the possible role of the injector on the stability. The admittance of the injector has to be considered because the velocity measurement, made with the hot wire, is carried out on the upstream side of the injector. Thus the measured transfer function is not only linked to the flame contribution but also to the impact of the injector. To include this admittance, one has to shift the phase of the FDF by the value measured in Figure 10. It is now possible to estimate the bands of instability of the system by analysing the phase diagram with the following expression:

π < ϕ + δ < 2π [mod.2π] (5)
with the flame contribution to the instability between {π, 2π} and the injector admittance δ, shifting the phase. To estimate δ, a velocity perturbation at variable frequency is applied to the system using the excitation system under cold conditions. The velocity fluctuation is first measured with the hot wire and, during another experiment, with a Laser Doppler Anemometry (LDA)

Fig. 11 Top: Phase ϕ of the FDF and burner admittance coupled with the instability band in grey. Bottom: Gain G with the deduce position of the instability bands.

system at the exit of the burner, in the air jet. The phase between the velocity fluctuation u and the reference signal from the function generator is calculated in both cases. One can then deduce the phase between the velocity fluctuation at the hot wire and at the exit of the burner. The measurement of δ is presented in Figure 10 for several frequencies. One can see that it is quite constant with the frequency and may be considered to be close to -π/2. Figure 11, top, shows the position of the two instability bands in grey. It is then possible to calculate the frequency boundaries of the potential unstable modes. One can see that the phase enters the first instability band around 200 Hz and up to 580 Hz for the highest levels of u /u. It enters the second band at 720 Hz until the end of the FDF estimation at 900 Hz. This explains why the SICCA-Spray bench exhibits powerful naturally excited instabilities when the different length l c of the walls is increased. When the resonant frequency of the system becomes lower than 550 Hz, the system is potentially unstable. The more the resonant frequency decreases, the more the system becomes unstable. This is confirmed by the increase in Q / Q observed in Table 1 when l c increases. Note that the forced flame with l c = 165 mm is not naturally unstable. As indicated before, the resonant frequency would be 850 Hz. This value is in a potential unstable band but the gain is very low in this region with G < 0.08, probably much lower than the damping of the system. One may infer from the previous considerations the stability or instability of a combustion chamber equipped with multiple swirl spray injectors. Considering an annular combustor like the MICCA-Spray system investigated at EM2C lab where the fundamental frequency of the first azimuthal mode (1A1L) is around 750 Hz (experiments and a modal analysis are available in [START_REF] Prieur | Proceedings of the Combustion Institute[END_REF]).

Fig. 12 FDF based on the acoustic pressure in the chamber p and the heat release rate Q . Top: Phase ϕ and gain G of the FDF for the velocity and the heat release rate ratio and their fit.

The instability intervals in Figure 11 contains the eigenfrequency f = 750 Hz of the 1A1L mode. One may note however that this criterion is not sufficient to ensure that the system will be linearly unstable. This will be the case if the growth rate at low amplitude is greater than the damping rate. For this to be the case, the gain of the FDF must be sufficiently large, but one also notices that this gain is relatively low at the frequency of the 1A1L mode (G < 0.4). An issue that is not considered in the previous analysis is that of the possible influence of mutual interactions between adjacent flames and that of a suitable representation of the thermal conditions of the annular combustor. One then can expect that interactions between flames might modify the FDF determined on a single burner, but this is not yet quantified.

Note that the SICCA-Spray bench is equipped with a pressure sensor in the chamber (MC1), giving access to the acoustic pressure p at the flame base during the forcing of the flame. This can be used to determine an FDF based on the pressure p and on the heat release rate Q as shown in Figure 12. The behavior of this FDF is quite different from that of the standard FDF based on the velocity u . The phase does not evolve in a linear fashion with increasing frequency, it is nearly equal to π/2 over most of the frequency range. The measured p ch corresponds to the combustion noise induced by the flame in the combustion chamber. This radiated noise is in phase quadrature with the heat release rate fluctuation because p ch ∝ d Q /dt [35,36]. The gain has also a different shape as it decreases with frequency. No peak can be detected but there is a small bump around 450 Hz probably due to a coupling with a resonant mode of the system (see Figure 3). One may note that the gain and the phase barely change when the forcing amplitude is increased. These measurements confirm the observations made on the forced case in section 5 where it has been seen that the acoustic pressure in the chamber p has a low amplitude that is not representative of the self-excited flame. The conclusion on the phase ϕ pQ also holds as it is constant over most of the frequency range of the FDF.

Conclusion

This article is concerned with the determination of the Flame Describing Functions of swirling spray flames. These functions are obtained by making use of a single injector combustion system comprising a set of driver units. The system is used in two modes, the first of which corresponds to external modulation while the second pertains to self-excited oscillations. FDFs are determined in the first mode and the flame dynamics is observed using phase average imaging. In the second mode, the oscillation corresponds to a resonance of the system and is coupled by a longitudinal acoustic mode of the system involving the plenum and the combustion chamber. It is then possible to compare the flame response under external modulation to the response observed when the system is self-excited. Potential bands of instability are inferred from an examination of the measured FDF phase response measured in this single injector configuration and the possibility of instability in the MICCA-Spray annular system is deduced from these data. Finally, the FDF measured considering the acoustic pressure radiated by the flame in the chamber is investigated. It is shown that this quantity is not a proper input for the FDF as it does not provide a suitable portray of the of the dynamics of the combustion region. In effect the measured pressure is more of an output of the unsteady flame and cannot be taken as an input. This illustrates the difficulty of experimental determinations of relations linking the heat release rate fluctions to the pressure signal. It is finally shown that the classical FDF linking the heat release rate to the velocity fluctuation can be used effectively to predict instability trends of the single injector system and to predict possible unstable operation of the MICCA-Spray annular combustor.

Fig. 1

 1 Photomultiplier

Fig. 2

 2 Fig. 2 Image in false colors of the Abel transform of the CH* chemiluminescence of the flame with φ = 0.85, P = 6.5 kW, U b = 43 m s -1 . For symmetry, the Abel transform of one branch of the M-shape flame is deconvoluted using the Abel transform and replicated to generate the entire flame. The image is cropped in order to exhibit the exit of the burner on the bottom of the image.

Fig. 4

 4 Fig.4Acoustic pressure p at 543 Hz calculated using a Helmholtz solver and assuming a constant temperature of 1200 K in the combustion chamber.

Fig. 5

 5 Fig.5Time series of velocity, acoustic pressure in the chamber and heat release rate fluctuations for the selftriggered flame with lc = 280 mm. Data are normalized by their mean value except for the acoustic pressure. Signals are low pass filtered with a cut-off frequency fc =1000 Hz.

0°30°60°90°120°150°180°210°240°270°300°330°Fig. 6

 6 Phased CH* chemiluminescence of the flame during one cycle of oscillation. Images are averaged over 600 cycles. sound c = √ γrT = 694 m s -1 in the medium. The natural resonant frequency is thus f = c/4L = 540 Hz with L = l c + l end and l end = 40 mm the measured end-correction. This calculation is confirmed by the Helmholtz analysis presented in Figure 4. Figure 5 presents the filtered velocity, acoustic pressure and heat release rate signals during the instability. The velocity fluctuations at the hot wire, in the top figure, are plotted along with the heat release rate fluctuations. One can see a clear sinusoidal response of the velocity with a fluctuation of u /u = 0.5 stable over the 2 s of recording. The heat release rate oscillates at the same frequency and with a phase ϕ uQ of π/2

Fig. 7

 7 Fig.7Time series of velocity, acoustic pressure in the chamber and heat release rate fluctuations for the forced flame at 530 Hz. Data are normalized by their mean quantity except for the acoustic pressure. Signals are filtered with a low-pass filter at 1000 Hz.

0°60°120°180°240°300°Fig. 8

 8 Phased average CH* chemiluminescence of the flame during one cycle of oscillation for the forced flames (left branch of the flame) and for the self-triggered flames (right branch of the flame). Images are averaged over 600 cycles.

Fig. 10

 10 Fig.10Evaluation of the phase of the injector admittance δ for frequencies between 500 and 900 Hz.

Table 1

 1 Characterization of the self-excited instabilities as a function of the combustion chamber length lc in the SICCA-Spray bench.

	lc (mm)	f (Hz)	u /u	Q / Q p (Pa)
	215	595	0.06	0.01	19
	230	605	0.08	0.07	175
	245	565	0.36	0.34	1370
	265	535	0.49	0.21	1170
	280	530	0.38	0.16	890
	295	455	0.63	0.28	1550
	315	415	0.76	0.48	3020

Acknowledgements This work was supported by Safran Tech and by CNRS. The authors wish to thank J. Beaunier, Y. Le Teno and E. Jean-Bart for the technical support provided to this research.


D F Escande

Emergence of the helical ohmic state in the reversed field pinch

Introduction

As exemplified by the story of the ZETA machine, which revealed the existence of the RFP state after being shut down, the story of the RFP is a series of strokes of fate and of strokes of luck. The emergence of the helical ohmic state is typical of this story. Indeed, during three decades, the attractive features of the ohmic RFP as a fusion reactor were handicapped by terrible confinement properties: weak confinement resulted from chaotic magnetic field lines and from strong plasma-wall interaction due to a slinky, a toroidally localized large helical deformation of the plasma. This was meant to be intrinsic to the ohmic configuration. The emergence of the helical ohmic state broke through this stereotype, but it was initially unnoticed. Since 1990, MHD numerical simulations had revealed a bifurcation from a toroidally localized to a toroidally uniform helical deformation of the plasma coming with magnetic order. However, this bifurcation was overlooked, even though since 1993, all RFP's had progressively revealed the existence of transient states with an almost regular helical deformation of the plasma, later called quasi single helicity (QSH) states. Revisiting the 1990 simulations and extending them with a different perspective was an incentive to scrutinize the RFX data base in 1999. Surprisingly, many shots with long lasting QSH states were found to have been present for quite a time in the data base, but the emergence of these helical ohmic states had been overlooked, because it was out of the traditional paradigm of the configuration.

The RFX data base revealed that increasing the current led to better QSH states, but strong plasma-wall interaction prevented from operating the machine above 1 MA. In December 1999, a stroke of fate, the toroidal power supplies of RFX were destroyed by a fire and the machine had to be shut down for five years. This eventually turned out to be a stroke of luck, since it was the occasion to upgrade RFX into RFX-mod, which provided a good control of the edge radial magnetic field. This control enabled overcoming the previous 1 MA operational bound, and eventually reaching the 2 MA nominal current of the machine. A new stroke of luck, at about 1.5 MA, a topological bifurcation to a single helical magnetic axis was found to occur, coming with a broad hot helical central domain bounded by an electron internal transport barrier (eITB). This was a change of paradigm for the ohmic RFP, as was symbolically exhibited by the cover story of the August 2009 issue of Nature Physics: ``Reversed-field pinch gets self-organized" (Lorenzini et al 2009b). The possibility of this helical state and of a corresponding improvement in confinement had been already theoretically predicted in 2000. After the discovery of a positive isotopic effect, there is now the emergence of driven helical ohmic states with a hot helical domain covering most of the non reversed region.

The emergence of the helical ohmic states occurred in three stages: a first one where it was almost unnoticed (section 2), a theoretical reinterpretation (section 3), and an experimental stage (section 4). Section 5 is devoted to the corresponding vision of the dynamo, and section 6 to the single helical axis version of these states. Section 7 describes such states when driven, section 8 the confinement properties of the helical ohmic states, section 9 provides further theoretical results, and section 10 concludes.

First stage

Because of the limited power of computers, the initial MHD numerical simulations of the RFP might have been the occasion to trigger an interest in the helical ohmic states of the RFP. Indeed, some indication of its existence was already present in the first numerical simulation of the RFP [START_REF] Sykes | Proceedings of the 8th EPS Conference on Controlled Fusion and Plasma Physics[END_REF]. Furthermore, since 1983, in order to save computer power and memory, numerical simulations had 0   , and worked with a forced single helicity (SH): only one ratio / mn for the Fourier harmonics was retained beyond (0,0) (Caramana et al 1983) [START_REF] Aydemir | Sustained Self-Reversal in the Reversed-Field Pinch[END_REF] (Holmes et al 1985) [START_REF] Schnack | Three-dimensional magnetohydrodynamic studies of the reversed-field pinch[END_REF] (Kusano and Sato 1986). These simulations used an elementary visco-resistive compressible nonlinear MHD model in the constant-pressure, constant-density approximation, whose equations are

( ) ( ), t         B v B J (1) 2 ( ) ( ), t          v v v J B v (2)
where  and  are the radial distributions of respectively the resistivity and of the kinematic viscosity, and   JB and 0   B

.

Here

B is normalized to the value 0 B of the axial magnetic field on axis, time and velocity are normalized to the Alfvén time

A  and velocity

A v respectively computed with 0 B , and the position to the plasma radius a . In these units  is the inverse Lundquist number,   where the averages are done over the toroidal and poloidal angles. This type of model has been widely used for other RFP simulations (see [START_REF] Ortolani | Magnetohydrodynamics of plasma relaxation (Singapore: World Scientific) Paccagnella R 2016a Aspect ratio scaling of the single helical states in the reversed field pinch plasmas[END_REF]) [START_REF] Cappello | Bifurcation in the MHD behaviour of a self-organizing system: the reversed field pinch (RFP) Plasma[END_REF]) and references therein). Usually, numerical simulations start from a paramagnetic pinch state with a slightly positive () z Ba. For 1.55  ‰ the system which is (resistive) kink unstable, relaxes toward a RFP state where the reversal parameter ( ) / zz F B a B    is in the range 0 0.5 F    . From 2008 on, the simulations with the SpeCyl and PIXIE3D codes were performed with a constant loop voltage instead of a constant  However, the initial numerical 2D SH states were deemed too far from the multiple helicity (MH) experimental ones, and computational physicists took advantage of the increase of the power of computers to deal with the full 3D problem. The parameters of initial numerical simulations led them into the MH state, an unfortunate convergence with the then dominant state of ohmic plasmas. Furthermore, Taylor relaxation was meant as the explanation of the RFP state, and this further prevented RFP physicists from focussing on SH physics. In 1990, 3D numerical simulations revealed a bifurcation of the magnetic configuration from the MH state to the SH state when viscosity is increased [START_REF] Cappello | Proc. Workshop on Theory of Fusion Plasmas[END_REF]) [START_REF] Cappello | Nonlinear plasma evolution and sustainment in the reversed field pinch[END_REF] [START_REF] Finn | Single and multiple helicity Ohmic states in reversed-field pinches[END_REF] [START_REF] Cappello | Reconnection processes and scaling laws in reversed field pinch magnetohydrodynamics[END_REF].

On the experimental side, since 1993, plasmas with transient QSH states were detected in all large RFP's [START_REF] Brunsell | Coherent magnetic field fluctuations and locked modes in a reversed-field pinch[END_REF] [START_REF] Nordlund | Nonlinear dynamics of kink-tearing modes and their interaction with the current profiles in a reversed-field pinch[END_REF] [START_REF] Hirano | Self-organization and its effect on confinement in a reversed field pinch plasma Plasma[END_REF] (Sarff et al 1997) [START_REF] Martini | Spontaneous and driven reduced turbulence modes in the RFX reversed field pinch Plasma[END_REF] (Martin 1999), and found to last for several energy confinement times. However, neither these results, nor the 90-92 simulations triggered more interest in SH.

Theoretical reinterpretation

The helical ohmic states fully emerged by revisiting the 90-92 simulations. An appropriate rescaling of time and velocity in equations (1-2) shows that, for given radial distributions of  and  , the dynamics is ruled by the magnetic Prandtl number , a classical dimensionless number in conductive fluid MHD; however, dissipation is the dominant parameter when the inertia term becomes negligible, which happens for a large range of simulation parameters [START_REF] Cappello | Bifurcation in Viscoresistive MHD: The Hartmann Number and the Reversed Field Pinch[END_REF]) [START_REF] Cappello | Bifurcation in the MHD behaviour of a self-organizing system: the reversed field pinch (RFP) Plasma[END_REF]). These simulations reveal that d (or H ) is the right control parameter independently of P and  . Therefore, the previous ``high viscosity" simulations turned out to be in reality ``high dissipation" or "small H " simulations. As usual in nonlinear dynamics, an increase of dissipation (decrease of H ) is favourable to a laminar behaviour of the system. Depending on the initial conditions, two nearby different helicities were found to be selected by the plasma when relaxing to SH (Cappello and Escande 2000) [START_REF] Cappello | Bifurcation in the MHD behaviour of a self-organizing system: the reversed field pinch (RFP) Plasma[END_REF]).

When d decreases, the pure helix develops a toroidally (axially) localized bulging, and longer and longer intermittent slinky phases interrupt this bulged helix, up to the point where it bulges enough to be called a slinky. Concomitantly, secondary modes with other helicities show up in the Fourier spectrum. The bulged helix corresponds to a QSH state where secondary modes are smaller than the original one, and the slinky corresponds to a MH state where at least two modes have similar amplitudes. When d decreases, the duration of the QSH phases decreases, as well as the percentage of time where QSH dominates, and after a transition region corresponding to 34 10 10 H ˆˆ, the system reaches a non-stationary MH regime [START_REF] Cappello | Bifurcation in Viscoresistive MHD: The Hartmann Number and the Reversed Field Pinch[END_REF]. This is shown in figure 1 where the energy of the 0, 0 mn  modes is used as an order parameter for the SH-MH transition; indeed these modes vanish in the SH state, since they result from the beating of at least two different helicities. Furthermore, magnetic chaos sets in progressively during the transition.

Emergence in experiments

As told in the introduction, the numerical scenario of the transition from MH to SH led to the discovery that shots with long lasting QSH states had been present for quite a time in the RFX data base, and were lasting over the whole flat top at about 1 MA (Escande et al 2000b). The dominant magnetic mode had an amplitude 1 to few percents of the central magnetic field, and several times that of the other modes [START_REF] Martin | Quasi-single helicity states in the reversed field pinch: Beyond the standard paradigm[END_REF] (Escande et al 2000b) [START_REF] Piovesan | Measurements of the MHD Dynamo in the Quasi-Single-Helicity Reversed-Field Pinch[END_REF] [START_REF] Martin | New insights into MHD dynamics of magnetically confined plasmas from experiments in RFX[END_REF]. On top of the one found in the reversal region, a strong temperature gradient was found at the edges of a magnetic island with the dominant helicity: an electron internal transport barrier (eITB) was found to exist [START_REF] Alfier | Electron temperature profiles in RFX-mod Plasma[END_REF] [START_REF] Puiatti | Internal and external electron transport barriers in the RFX-mod reversed field pinch[END_REF]; with a ``bean"-like hot structure corresponding to the magnetic island [START_REF] Martin | Quasi-single helicity states in the reversed field pinch: Beyond the standard paradigm[END_REF] (Escande et al 2000b) [START_REF] Piovesan | Measurements of the MHD Dynamo in the Quasi-Single-Helicity Reversed-Field Pinch[END_REF] [START_REF] Martin | Overview of quasi-single helicity experiments in reversed field pinches[END_REF] (Marrelli et al 2002) (Franz et al 2006). For such states, termed Double Axis (DAx) states, the magnetic field displays two magnetic axes: the unperturbed axis-symmetric one and the one related to the island O-point.

After the fire and the upgrade of RFX into RFX-mod, QSH states were rapidly recovered [START_REF] Paccagnella | Active-Feedback Control of the Magnetic Boundary for Magnetohydrodynamic Stabilization of a Fusion Plasma[END_REF]. At currents of 0.6 MA or above, and at shallow reversal, QSH states with m=1, n=7 dominate: a QSH persistency up to 85% of the discharge flat top is obtained (Piovesan, P. and al 2008), and the duration and amplitude of the dominant helical mode grows with the plasma current (Marrelli et al 2007) [START_REF] Piovesan | Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control[END_REF]; the ratio of this amplitude to the central magnetic field tends to become a constant (Piovesan, P. andal 2008) (Valisa et al 2008) [START_REF] Piovesan | Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control[END_REF]. The probability of obtaining longlasting QSH spectra is enhanced when forcing a sudden decrease of the edge toroidal magnetic field [START_REF] Cravotta | A statistical analysis of pulsed poloidal current drive in the Reversed Field eXperiment[END_REF] [START_REF] Puiatti | Analysis and modelling of the magnetic and plasma profiles during PPCD experiments in RFX[END_REF]. The RFX results were an incentive to investigate the data bases of other RFP's. The same phenomenology was found to exist in MST (Marrelli et al 2002), in TPE-RX [START_REF] Piovesan | Measurements of the MHD Dynamo in the Quasi-Single-Helicity Reversed-Field Pinch[END_REF], in EXTRAP T2R [START_REF] Frassinetti | Spontaneous quasi single helicity regimes in EXTRAP T2R reversed-field pinch[END_REF], and in RELAX (Ikezoe et al 2012). In TPE-RX, long-lasting highly reproducible QSH states were obtained by applying a delayed reversal of the toroidal field at the edge [START_REF] Hirano | Quasi-single helicity state at shallow reversal in TPE-RX reversed-field pinch experiment[END_REF], and were systematically triggered by applying a small positive pulse in the initially weakly reversed edge magnetic field [START_REF] Hirano | Quasi-single helicity state by a small positive pulse of toroidal magnetic field in TPE-RX reversed field pinch experiment[END_REF]. The end of the QSH phase came with a distortion of the m = 0 magnetic island chain, and the occurrence of a chaotic region, expelling energy at the location of the slinky [START_REF] Frassinetti | Toroidally localized soft x-ray expulsion at the termination of the improved confinement regime in the TPE-RX reversed-field pinch experiment[END_REF]. QSH could also be found at low current and deep reversal, but it was more intermittent than in the high current and shallow reversal case [START_REF] Piovesan | Measurements of the MHD Dynamo in the Quasi-Single-Helicity Reversed-Field Pinch[END_REF]. In RELAX, QSH states lasting above 30 % of the flat-top can be obtained [START_REF] Oki | Characterization of Quasi-Single-Helicity States in a Low-Aspect-Ratio RFP Plasma[END_REF]. The good features of QSH in RELAX suggest an advantage of a low aspect ratio configuration to reach QSH, as expected from the scarcer density of the central m = 1 tearing modes in this case [START_REF] Oki | Characterization of Quasi-Single-Helicity States in a Low-Aspect-Ratio RFP Plasma[END_REF] [START_REF] Oki | Dependence of Properties of Quasi-Single-Helicity States on Field Reversal Parameter in a Low-Aspect-Ratio Reversed Field Pinch Fusion[END_REF].

The results obtained in the various RFPs have common features: QSH is favoured by shallow reversal and the dominant mode resonates near the magnetic axis [START_REF] Hirano | Quasi-single helicity state by a small positive pulse of toroidal magnetic field in TPE-RX reversed field pinch experiment[END_REF] [START_REF] Oki | Characterization of Quasi-Single-Helicity States in a Low-Aspect-Ratio RFP Plasma[END_REF] [START_REF] Oki | Dependence of Properties of Quasi-Single-Helicity States on Field Reversal Parameter in a Low-Aspect-Ratio Reversed Field Pinch Fusion[END_REF]) [START_REF] Sarff | Overview of results from the MST reversed field pinch experiment[END_REF]. The latter point is in contrast with the outcome of numerical simulations with a vanishing edge radial magnetic field [START_REF] Cappello | Bifurcation in Viscoresistive MHD: The Hartmann Number and the Reversed Field Pinch[END_REF]. There is a higher probability to obtain QSH spectra with higher plasma current (Marrelli et al 2002) [START_REF] Bolzonella | Magnetic fluctuation spectra and non-linear MHD mode interaction in RFX Plasma[END_REF] (Martin et al 2003). Therefore, the current plays the role of a bifurcation parameter analogous to dissipation in MHD simulations. Numerical simulations with an edge magnetic perturbation show the scaling with S at fixed P is closer to the experimental trend, which indicates a hidden viscosity effect in the experimental dependence of secondary modes on S [START_REF] Bonfiglio | Experimental-like Helical Self-Organization in Reversed-Field Pinch Modeling[END_REF].

Ohmic dynamo

Theoretically, the RFP dynamo is a mere consequence of ohmic magnetic relaxation, and is analogous to the one occurring in the saturated tearing mode: the helical displacement of magnetic surfaces causes a modulation of the parallel current density along each flux tube, which requires the build-up of a helical electrostatic potential producing the dynamo flow as an electrostatic drift [START_REF] Bonfiglio | Dominant Electrostatic Nature of the Reversed Field Pinch Dynamo[END_REF] [START_REF] Cappello | Magnetohydrodynamic dynamo in reversed field pinch plasmas: Electrostatic drift nature of the dynamo velocity fielda[END_REF] [START_REF] Cappello | Equilibrium and transport for quasi-helical reversed field pinches[END_REF]. In the plasma core, the contribution from the loop voltage is larger than that from the mean parallel current density, but it is smaller in the edge. The difference is balanced by the electrostatic term, which provides an anti-dynamo contribution in the core and a genuine dynamo contribution in the edge (see figure 5 of [START_REF] Cappello | Magnetohydrodynamic dynamo in reversed field pinch plasmas: Electrostatic drift nature of the dynamo velocity fielda[END_REF]). It is worth noticing that a recent theory explains the hybrid mode of operation of the tokamak by invoking a dynamo explicitly stated to be analogous to the RFP one (Jardin et al 2015). This type of dynamo is also present in a flux rope configuration susceptible to the kink instability (Lapenta and Skender 2017).

As told above, the MH slinky is the result of a continuous deformation of the SH uniform helical shape. This continuity bears on the dynamo as well. Indeed, though a fluctuating inductive electric field be present in the MH state, the velocity field is still dominated by the electrostatic contribution; the velocity field in the SH state is topologically equivalent to the radial pinch velocity field of an axis-symmetric paramagnetic pinch [START_REF] Bonfiglio | Dominant Electrostatic Nature of the Reversed Field Pinch Dynamo[END_REF] (Bonfiglio et al 2006) [START_REF] Cappello | Magnetohydrodynamic dynamo in reversed field pinch plasmas: Electrostatic drift nature of the dynamo velocity fielda[END_REF]. The pinch velocity leads to a build-up of the plasma density on the helical axis in MHD simulations where density is free to evolve [START_REF] Onofri | Compressibility Effects in the Dynamics of the Reversed-Field Pinch[END_REF]. This does not occur in a genuine plasma where the central density profile is flat, which proves the existence of an outward transport mechanism. This build-up was avoided in [START_REF] Delzanno | Electrostatic mode associated with the pinch velocity in reversed field pinch simulations[END_REF] by adding a diffusive term in the continuity equation for the density. The pinch flow present in this case and in the case with constant density can generate an electrostatic instability related to angular momentum conservation for some boundary conditions at the wall; this instability can be stabilized by using homogeneous Dirichlet no-slip boundary conditions [START_REF] Delzanno | Electrostatic mode associated with the pinch velocity in reversed field pinch simulations[END_REF].

The QSH dynamo was first measured in MST, and showed: (i) the magnetic and velocity fluctuation spectrum to have the same dominant wave number; (ii) the corresponding component of the velocity to extend throughout the plasma volume and to couple with magnetic fluctuations, which produces a significant MHD dynamo electric field; (iii) the radial profile of this component to be consistent with the one obtained in QSH runs of the SpeCyl MHD code [START_REF] Piovesan | Measurements of the MHD Dynamo in the Quasi-Single-Helicity Reversed-Field Pinch[END_REF]. A similar study was performed in RFX-mod [START_REF] Bonomo | Flow measurements and modelling in helical RFX-mod equilibria[END_REF]: a helical plasma flow was observed to form a m = 1 convective cell creating a localized sheared flow outside the eITB; however, the experimental pattern revealed the maximum flow shear to be external to the null of the magnetic shear, in contrast with the predictions of 3D MHD simulations; this might be due to an ambipolar component of the helical electric field on top of the MHD one. Even weak external 3D fields can modify significantly the flow profile and in particular its shear [START_REF] Piovesan | Influence of external 3D magnetic fields on helical equilibrium and plasma flow in RFX-mod Plasma[END_REF]. In RFX-mod, the edge flow displays a QSH modulation [START_REF] Scarin | Topology and transport in the edge region of RFX-mod helical regimes[END_REF]. Probes may be used in low current discharges (up to 0.45 MA) of RFX-mod; this enabled to exhibit in forced QSH states a modulation of the perpendicular components of the flow (Vianello et al 2015) concomitant with the density modulation along the toroidal angle, first found in [START_REF] Puiatti | Interaction between magnetic boundary and first wall recycling in the reversed field pinch Plasma[END_REF].

The von Karman Sodium experiment exhibited a dynamo in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers, when at least one is made of a ferromagnetic material [START_REF] Monchaux | The von Kármán Sodium experiment: Turbulent dynamical dynamos[END_REF]. It is worth noting that he magnetic field averaged over a long enough time corresponds to a SH RFP magnetic state with a large dominant m = 0 mode (see Fig. 7 of [START_REF] Monchaux | The von Kármán Sodium experiment: Turbulent dynamical dynamos[END_REF]). The location of reversal is controlled by a copper wall surrounding the turbulent fluid. The reversed field is maximum on this wall, because the azimuthal component of the electric field must vanish there.

Single helical axis states

On top of the MHD bifurcation leading from MH to SH, there is another one suppressing the magnetic island and making the magnetic topology kink-like. Indeed, when the amplitude of the SH mode increases, the island X-point collides with the unperturbed axis of the configuration, which leads to a topology with a single helical axis (SHAx) corresponding to the former island O-point (Escande et al 2000c). This occurs because the inner loop of the separatrix becomes tighter and tighter about the former axisymmetric Opoint and vanishes when its inner area vanishes. As a consequence of the separatrix disappearance, the helical safety factor profile goes through a maximum located in the vicinity of the former separatrix (Gobbin et al 2011a) (Gobbin et al 2011b). This maximum corresponds to a minimum of the rotation frequency of the pendulum-like Hamiltonian describing the SH dynamics; when the amplitude of the SH mode increases further after the separatrix disappearance, a stronger and stronger amplitude of secondary modes is required to produce chaos about the maximum of the helical safety factor: there is magnetic chaos healing (Escande et al 2000c). As a result, SHAx states are typically more resilient to chaos than DAx ones (Escande et 3). Plasma properties such as electron temperature, SXR emissivity and electron density are constant on helical magnetic surfaces (Lorenzini et al 2009b) reconstructed with independent measurements (see figure 4), indicating that SHAx states have almost invariant magnetic surfaces, in contrast with the low current MH states. As plasma current increases, long-lasting QSH states occur more frequently and their total persistency increases; at currents above 1.5 MA the latter exceeds 90% of the plasma current flat-top [START_REF] Carraro | Advances in understanding RFX-mod helical plasmas[END_REF]. The amplitude of the dominant mode and that of the secondary modes scale in opposite ways with the current; when the dominant mode amplitude, normalized to the edge magnetic field, is larger than about 3-4 %, all QSH states become SHAx states (Lorenzini, R. et al 2016). In agreement with the theoretical description (Escande et al 2000c), the transition from narrow to wide eITBs occurs above the DAx/SHAx threshold (Lorenzini, R. et al 2016).

FIG. 3: 2D soft X-ray emissivity map in a SHAx state with a wide hot domain in RFX-mod, shot #23977, t=168ms [START_REF] Bonomo | 2D characterization of core thermal topology changes in controlled RFX-mod QSH states[END_REF] The best plasma performances at high current in QSH regimes have been reached with shallow reversal (-0.01 0 a q  , where a q is the edge safety factor), when the amplitudes of secondary modes is minimum [START_REF] Carraro | Advances in understanding RFX-mod helical plasmas[END_REF]. As yet, spontaneous SHAx states are observed at plasma densities  Greenwald density) (Puiatti et al 2015).

QSH periods end by crashes where the amplitude of secondary modes increases, inducing magnetic reconnection with the formation of poloidal current sheets (Zuin et al 2009). These crashes are spiky: an abrupt increase of the amplitude of secondary modes is immediately followed by an abrupt return to QSH.

The strongest crashes lead to MH (Lorenzini et al 2015).

There is a positive isotopic effect on the magnetics of SHAx states: in deuterium, only 55 % of the crashes lead to MH while this fraction is 75% in hydrogen (Lorenzini et al 2015), QSH periods last longer, the secondary mode amplitude is lower by about 20 % with respect to hydrogen [START_REF] Gobbin | Spontaneous versus induced hydrogen and deuterium helical shaped plasmas with electron internal transport barriers Plasma[END_REF].

SHAx states were also found in MST, at lower density and higher temperature, and thus similar Lundquist number S despite lower plasma current (about 0.5 MA) than in RFX-mod, but also with shallow toroidal field reversal [START_REF] Puiatti | Internal and external electron transport barriers in the RFX-mod reversed field pinch[END_REF]. The dominant n = 5 helical mode grows as large as 8 % of the axisymmetric field; when S increases, the secondary m = 1 modes decrease, which implies reduced magnetic chaos [START_REF] Sarff | Overview of results from the MST reversed field pinch experiment[END_REF]. A direct measurement of the internal magnetic field structure associated with SHAx states was obtained by using the Faraday Effect (Bergerson et al 2011). The steeper electron temperature gradient, which appears during the QSH state in RELAX suggests the existence of an eITB in this machine too [START_REF] Sanpei | Features of the Electron-Temperature Distribution in a Low-Aspect-Ratio Reversed Field Pinch Plasmas[END_REF].

Driven helical states

The SH ohmic state can be described analytically as a small helical perturbation of an axis-symmetric ohmic pinch with small edge axial magnetic field and conductivity. This description uses the pinchstellarator equation, which shows how the axial field of a cylindrical SH RFP evolves radially because of the pinch effect and of a stellarator contribution due to the helical deformation of the plasma (Pustovitov, V.D. 1982b) (Pustovitov, V.D. 1982a) (Finn et al 1992) [START_REF] Bonfiglio | Necessary criterion for magnetic field reversal in the reversed-field pinch[END_REF]. Reversal is due to the edge stellarator contribution, which is more efficient when the edge pinch contribution is weak. The latter condition corresponds to a resistive edge plasma, which contradicts the idea suggested by Taylor theory that edge resistivity prevents reversal. Furthermore, simulations performed with a flat resistivity profile display a reduction of the dynamo action, which brings to marginally-reversed or even non-reversed equilibrium solutions (Bonfiglio, D. . This occurs for the following reason: with a larger resistivity at the edge, the electrostatic field is larger too, in order to balance parallel Ohm's law ; then the electrostatic drift due to this larger electrostatic field provides an enhanced dynamo action, which is sufficient to sustain the reversed configuration (Bonfiglio et al 2006). The edge resistivity of real RFPs is high enough for the reversed axis-symmetric part of the toroidal field to be rather constant in the reversal domain after a steady decrease in the non reversed one (see Fig. 2 of 0. 0.2 -0.2 0.4 -0.4 r(m) [START_REF] Bodin | IAEA Fusion Energy Conference[END_REF]). The pinch-stellarator equation enables the derivation of a necessary criterion for the reversal of the edge axial field [START_REF] Bonfiglio | Necessary criterion for magnetic field reversal in the reversed-field pinch[END_REF]. This criterion and MHD simulations show that a finite edge radial magnetic field is favourable for field reversal. The criterion is found to be satisfied in RFX-mod with forced QSH states, but only marginally in spontaneous ones [START_REF] Bonfiglio | Necessary criterion for magnetic field reversal in the reversed-field pinch[END_REF]. In RFX-mod, the control of the radial edge magnetic field is unable to force it to vanish (Zanca 2010). The criterion shows this undesired feature to be useful, one more instance where an apparently negative fact turns into a positive one! Until recently, QSH states obtained in SpeCyl calculations had a temporal behaviour which was very far from the experimental one. However, the former had a vanishing radial magnetic field at the edge, while the latter correspond to a finite one, as just recalled. A temporal behaviour close to that of experimental QSH states was found in numerical simulations performed with a small m = 1, n = 7 resonant magnetic perturbation (RMP) with an amplitude similar to the experimental one (about 1.5 % of the central magnetic field) and with 7 10 S  [START_REF] Bonfiglio | Experimental-like Helical Self-Organization in Reversed-Field Pinch Modeling[END_REF]. A more recent work shows that the similarity with experiments bears on several features [START_REF] Bonfiglio | Helical self-organization in 3D MHD modelling of fusion plasmas Plasma[END_REF]: (i) the dependence of the temporal behaviour of QSH states on the RMP amplitude, (ii) the dependence of the amplitude of the dominant mode on the same parameter, (iii) and that of the position of shear reversal and maximum q, (iv) the amplitude of the dominant mode is independent of dissipation, whereas the amplitude of secondary modes decreases with increasing resistive and viscous Lundquist numbers. Numerically, a stationary state is reached for a RMP of 10%.

In agreement with numerical simulations, imposing a m = 1, n = 7 RMP to RFX-mod with shallow reversal, strongly increases the persistence of the helical equilibrium [START_REF] Piovesan | Influence of external 3D magnetic fields on helical equilibrium and plasma flow in RFX-mod Plasma[END_REF]. However, in hydrogen plasmas the resulting average eITB duration decreases from 2 ms to 1.5 ms, while in deuterium almost stationary eITBs are observed with record values of duration up to 18 ms [START_REF] Gobbin | Spontaneous versus induced hydrogen and deuterium helical shaped plasmas with electron internal transport barriers Plasma[END_REF]. Furthermore, QSH states can be obtained at higher densities ( G / 0.5 nn ) than in the spontaneous case [START_REF] Piovesan | RFX-mod: A multi-configuration fusion facility for three-dimensional physics studiesa[END_REF]. An increase of the QSH probability up to 10 % and a larger magnetic island occurred in EXTRAP T2R by applying a resonant magnetic perturbation (RMP) with the corresponding helicity (Frassinetti et al 2009b).

As predicted by numerical simulations (Veranda et al 2013(Veranda et al , 2017)), by providing a corresponding helical boundary condition in RFX-mod, it is possible to excite non resonant QSH n = 6 states in RFX-mod [START_REF] Cappello | Nonlinear Modeling for Helical Configurations in Toroidal Pinch Systems[END_REF], whose best ones have a hot region covering most of the non reversed domain.

Confinement properties in the helical ohmic state

In RFX, DAx states displayed a magnetic island with high temperature and soft X-ray emissivity (Escande et al 2000b). In RFX-mod, at high current (above 1.1 MA), this island comes with steep core gradients, peak electron temperatures exceeding 1 keV, and heat diffusivity inside the island between 6 and 35 m 2 / s falling in the tokamak range; the corresponding electron confinement time is higher than in MH states, with a best electron energy confinement time of about 1.3 ms [START_REF] Alfier | Electron temperature profiles in RFX-mod Plasma[END_REF]. Simulations of test ion and electron transport show the average diffusion coefficients inside the helical core to be about one order of magnitude lower than those found in MH plasmas [START_REF] Gobbin | Ion and electron local transport inside single helicity islands in the reversed field pinch[END_REF]. In TPE-RX, the transition from MH to QSH came with an enhancement of particle confinement of about 30 %, together with strong indications of a higher energy confinement [START_REF] Frassinetti | Toroidally localized soft x-ray expulsion at the termination of the improved confinement regime in the TPE-RX reversed-field pinch experiment[END_REF]. In EXTRAP T2R, the heat diffusivity inside the DAx island is one to two orders of magnitude lower than the diffusivity in the surrounding plasma; this diffusivity is divided by about 3 and the temperature increment inside the island is multiplied by about 4, when the current is doubled (Frassinetti et al 2009a).

When DAx states are stimulated by a helical edge toroidal magnetic field, there is a decrease of magnetic chaos involving also the plasma outside the island, and allowing for a global enhanced confinement with an improvement up to 50 % [START_REF] Terranova | Enhanced Confinement and Quasi-Single-Helicity Regimes Induced by Poloidal Current Drive[END_REF]. A similar improvement is found in self-similar current decay experiments, where an initial fast decrease in the mode amplitudes (about 40 % of the initial value) is observed (Zanca 2007).

The remaining of this section deals with the SHAx state. In MST, it comes with an improvement of the global energy confinement by about 50 %; a three-fold improvement in confinement is obtained by forcing a slow decay of the plasma current [START_REF] Sarff | Overview of results from the MST reversed field pinch experiment[END_REF].

We now summarize the SHAx results in RFX-mod. eITBs with high electron temperature gradients comparable to those achieved in tokamaks are characterized by an electronic thermal diffusivity lower than 20m 2 / s with values approaching 2 m 2 / s, still higher than neoclassical predictions (under 1m 2 / s), but much lower than outside the barrier or in MH states (about 100m 2 / s) (Gobbin et al 2013). Both the diffusivity and the electron temperature gradient length scale with the total amplitude of the secondary m = 1 modes [START_REF] Puiatti | Internal and external electron transport barriers in the RFX-mod reversed field pinch[END_REF][START_REF] Auriemma | Investigation of the particle transport properties in RFX-mod SHAx state[END_REF]. So does the inverse of the maximum electron temperature gradient [START_REF] Carraro | Advances in understanding RFX-mod helical plasmas[END_REF][START_REF] Auriemma | Investigation of the particle transport properties in RFX-mod SHAx state[END_REF]. These scalings indicate a strong link between the level of magnetic chaos and the strength of the barrier. However, this is not the whole story, since the electron temperature gradients are the highest (above keV/m) in the rising phase of the QSH state, and display oscillations during the flat top [START_REF] Carraro | Advances in understanding RFX-mod helical plasmas[END_REF] [START_REF] Franz | Experimental investigation of electron temperature dynamics of helical states in the RFX-Mod reversed field pinch[END_REF]. The minimum electron diffusivity is the same in DAx and SHAx states [START_REF] Fassina | Electron Pressure and Transport Analysis During QSH States in RFX-mod[END_REF].

The electron energy confinement time Ee  increases on average by a factor about 2 in DAx states with respect to MH states. In SHAx states an additional factor about 2 is gained, which leads to an overall gain by a factor about 4 with respect to standard RFP plasmas [START_REF] Piovesan | Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control[END_REF] [START_REF] Martin | Overview of RFX-mod results[END_REF]) [START_REF] Puiatti | Overview of the RFX-mod contribution to the international Fusion Science Program[END_REF]. Under stationary conditions at G / 0.3 nn , the largest Ee  's are around 2 ms [START_REF] Martin | Overview of RFX-mod results[END_REF]. If one assumes equal ionic and electronic temperatures, then the global energy confinement time is twice. A three-parameter fit with plasma current, average density and r b , the amplitude of the radial magnetic field of secondary modes measured at the plasma edge, yields a Ee  scaling like 0.88 0.03 r b  (Innocente et al 2009). This result is an incentive to decrease the amplitude of r b by moving the thin shell closer to the plasma edge, as in the presently proposed modification of the front-end for RFX-mod.

A strong improvement of particle confinement in QSH states was predicted by a numerical study of particle transport [START_REF] Predebon | Particle-Transport Analysis in Reversed Field Pinch Helical States[END_REF]. This improvement was progressively confirmed, in particular by using pellet injection (Terranova et al 2010). Inside the eITB, the particle diffusion coefficient is smaller by a factor of about 2 to 5 with respect to the MH regime, the pinch velocity is outwards, independent of the temperature gradient length, of the order of 10 m/s, and lower than the estimate based upon transport in a chaotic magnetic field; this rules out a dominant chaotic transport inside the eITB (Auriemma, F. et al 2012) [START_REF] Carraro | Advances in understanding RFX-mod helical plasmas[END_REF]. The eITB prevents penetration from outside too [START_REF] Puiatti | Internal and external electron transport barriers in the RFX-mod reversed field pinch[END_REF]. When passing from MH to SHAx, a pellet entering the central hot region triggers an enhancement of Ee  by a factor 2 to 3, and a higher density with a possibly peaked profile; a record particle confinement time of 12 ms was obtained at 1.5 MA (Terranova et al 2010). Particle confinement may be also estimated by an effective diffusivity with a vanishing pinch velocity [START_REF] Auriemma | Characterization of particle confinement properties in RFX-mod at a high plasma current[END_REF]: (i) in the core region of the SHAx state, this diffusivity is about 1 m 2 / s along the central 90 % of the radius, while it is about 50 m 2 / s in MH, (ii) however, transport is still anomalous, since neoclassical diffusivity is about 0.01 to 0.4 m 2 / s, (iii) in the external 10 % of the radius, the edge diffusivity is linked with the presence of pressure coherent structures. The edge temperature gradient scales almost linearly with plasma current, and increases when the secondary modes decrease (Vianello et al 2009).

Inside the core region of SHAx states, the confinement of Ne and Ni impurities does not increase, which makes unlikely their central accumulation [START_REF] Carraro | Improved confinement with internal electron transport barriers in RFX-mod[END_REF]. This was further confirmed in [START_REF] Menmuir | Impurity transport studies in RFX-mod multiple helicity and enhanced confinement QSH regimes Plasma[END_REF], which estimated a core diffusivity one order of magnitude higher than at the edge, and an outwardly directed pinch velocity over the whole plasma with a strong maximum around the core/edge transition point of the diffusivity. An external velocity barrier was found to exist also for light impurities [START_REF] Barbui | Light impurity transport studies with solid pellet injections in the RFX-mod reversed-field pinch Plasma[END_REF].The computation of single particle trajectories exhibits a different influence of helical states on main gas and impurity transport [START_REF] Gobbin | Ion and electron local transport inside single helicity islands in the reversed field pinch[END_REF]. It also shows that, while passing particles are confined for a very long time, particle trapping dominates transport across the helical structure at the rather low collisionality achieved in QSH regimes of RFX-mod: the neoclassical diffusion coefficient is one order of magnitude larger than the classical one [START_REF] Gobbin | Trapped Particles in the Reversed Field Pinch[END_REF]. When trapped particles drift out of the helical core, at r/a about 0.6 they become almost passing without being lost, because the helical distortion decreases when going outwards; this results in an almost total absence of superbanana particles at the experimental levels of helical magnetic fields (about 10 % of the total) [START_REF] Gobbin | Neoclassical Transport in the Helical Reversed-Field Pinch[END_REF]. Therefore, transport is proportional to the collision frequency, in contrast with unoptimized stellarators where the lower bound of transport scales inversely with this frequency [START_REF] Gobbin | Neoclassical Transport in the Helical Reversed-Field Pinch[END_REF]. The diffusion coefficients computed by this approach were confirmed by local neoclassical transport computations including the radial electric field; a comparison with power balance estimates shows that residual chaos due to secondary tearing modes and small-scale turbulence still drives anomalous transport in the barrier region. For an ion temperature of 0.7 keV, in the absence of chaos and of impurities, the radial electric ambipolar field near the eITB is about -2kV/m. The toroidal and poloidal flows have a magnitude of about 10-20 km/s and 2-8 km/s respectively, decreasing with lower ion temperature gradients (Gobbin et al 2011a).

There is a positive isotopic effect on the confinement of SHAx states similar to the one in tokamaks. For plasma currents about 1.5 MA, the energy confinement time scales like 0.3 i M , where i M is the ion mass; the particle influx is significantly reduced and the particle confinement time scales like 0.45 i M ; the effect on confinement is mainly due to the mitigation of transport at the edge (Lorenzini et al 2015). In deuterium plasmas, the electron temperatures are higher by about 0.2 keV over most of the plasma volume (r/a < 0.8); the numerical calculation of particle orbits shows that the loss time in QSH regimes is about 1.3 times higher in deuterium than in hydrogen for both spontaneous and induced helical states (Gobbin et al 2015).

Linear gyrokinetic calculations show that microtearing (MT) modes are unstable at the eITB; for the strongest eITB's, quasi-linear estimates of the associated transport show MT-driven thermal transport to be comparable to the experimental one (Predebon et al 2010b). MT modes were observed in RFX-mod [START_REF] Zuin | Experimental Observation of Microtearing Modes in a Toroidal Fusion Plasma[END_REF].

The stability of ion temperature gradient (ITG) modes was investigated by gyrokinetic calculations. In axisymmetric RFP plasma states, such instabilities are strongly stabilized compared with tokamak plasmas, due to the stronger Landau damping acting in low-q configurations such as the RFP (Predebon et al 2010a). However, their stability decreases in SHAx states, because of the enhancement of temperature gradients in the outer part of the helical deformation where magnetic surfaces are close-packed; therefore, ITG turbulence might be an important contributor to the total heat transport [START_REF] Predebon | Ion temperature gradient turbulence in helical and axisymmetric RFP plasmas[END_REF]. The stability of ITG modes might be further decreased in the presence of strongly outwardly peaked impurity profiles (Liu et al 2011). However, quasilinear and nonlinear three-species simulations of ITG turbulence prove that the inward impurity flux corresponding to a strong ITG would not be compatible with the measured outward flux [START_REF] Predebon | On the mutual effect of ion temperature gradient instabilities and impurity peaking in the reversed field pinch Plasma[END_REF].

Gyrokinetic simulations show that at a  of 4.5%, a transition between an ITG and a MT mode is observed through coupling to shear Alfvén waves, and that there is evidence for a collisionless MT mode [START_REF] Carmody | Gyrokinetic studies of microinstabilities in the reversed field pinch[END_REF].

Transport related to self-consistently generated vortical drift motions due to electrostatic turbulence was proposed to interpret the flat temperature profile of the hot region of SHAx states [START_REF] Sattin | Modelling the temperature plateau in RFX-mod single-helical-axis (SHAx) states Plasma[END_REF].

When 3D non-axisymmetric magnetic fields are applied in RFX-mod, at the edge the electrostatic turbulence induced flux is modulated by an induced m = 0 magnetic island, with an enhancement close to the O-point and a reduction at the X-point; there transport is due to fluctuations propagating in the electron diamagnetic drift direction, with a spectrum peaked at perp 0.1 i k  [START_REF] Rea | Comparative studies of electrostatic turbulence induced transport in presence of resonant magnetic perturbations in RFX-mod[END_REF].

Further theoretical results

A simple toy model makes magnetic field reversal intuitive: it consists in a current-carrying resistive wire initially placed on the axis of a cylindrical flux conserver, whose spontaneous kink leads to the reversal of the edge axial magnetic field, and not to a "disruption" (Escande, D. F. and Bénisti, D. 1997) (Escande et al 2000a). This effect is present in a genuine RFP, but most of the current corresponds to the paramagnetic pinch component of the configuration, as discussed in section 7.

When performed in toroidal geometry, 3D nonlinear visco-resistive MHD simulations show that toroidal coupling prevents the system from reaching a pure SH state when dissipation increases, but that magnetic chaos due to toroidal coupling stays limited in the close to SH states for the aspect ratios of the largest present RFP's [START_REF] Sovinec C R, Gianakon | NIMROD: A computational laboratory for studying nonlinear fusion magnetohydrodynamics[END_REF]. This is confirmed by toroidal simulations with the PIXIE3D code, also showing an m=0 islands chain induced by the toroidal coupling at the q=0 reversal surface [START_REF] Cappello | Nonlinear Modeling for Helical Configurations in Toroidal Pinch Systems[END_REF]. Incompressible 3D nonlinear visco-resistive MHD simulations show that, in contrast with the cylindrical case, the toroidal one presents a double poloidal recirculation cell with a shear localized at the plasma edge [START_REF] Morales | The effect of toroidicity on reversed field pinch dynamics Plasma[END_REF].

Analytical calculations in [START_REF] Cappello | Equilibrium and transport for quasi-helical reversed field pinches[END_REF] show that (i) the SH state is the same on using single or two-fluid Ohm's law, which backs up a numerical result (section VA of (King et al QSH states were obtained in simulations performed with an anisotropic thermal conductivity, and using a multiple-time-scale analysis to this end; the temperature distribution indicates the existence of closed magnetic surfaces, and there is a hot confined region (Onofri et al 2010a). When resistivity increases steeply at the edge, SH states are obtained for parameters similar to the case without thermal transport [START_REF] Onofri | Single-helicity states in compressible magnetohydrodynamics simulations of the reversed-field pinch with nonuniform resistivity[END_REF]) [START_REF] Onofri | Effects of the resistivity profile on the formation of a reversed configuration and single helicity states in compressible simulations of the reversed-field pinch[END_REF].

The magnetic topology of both SHAx and DAx states of RFX-mod can be reproduced by a one parameter fit of a minimally constrained equilibrium model using only five parameters, and resulting from a twodomain generalization of Taylor's theory. Both states appear as a consequence of the formation of a transport barrier in the plasma core, in agreement with experimental results [START_REF] Dennis | Minimally Constrained Model of Self-Organized Helical States in Reversed-Field Pinches[END_REF].

Recently, a new MHD approach was proposed to explain why high current is favourable to SHAx states, why the innermost resonant m = 1 mode is the spontaneous dominant mode, and why there are crashes of this mode. It invokes the shear stabilization associated with the 3D structure of the dominant mode to interrupt the nonlinear mode-mode coupling occurring in the MH regime (Terry and Whelan 2014), and to also suppress magnetic-fluctuation-induced thermal transport, producing temperature-gradient steepening in the strong shear region (McKinney and Terry 2017). This kind of ideas had been first introduced in (Kim and Terry 2012). The crashes can also be interpreted as the consequence of pressuredriven resistive modes, which introduce a feedback between transport and the MHD stability of the system [START_REF] Paccagnella | Pressure-driven reconnection and quasi periodical oscillations in plasmas[END_REF].

The aspect ratio scaling law for the toroidal mode number of experimental QSH states was shown to be close to the one corresponding to the optimal electromagnetic response of the toroidal shell surrounding the plasma (Paccagnella 2016a). The observed scaling with the aspect ratio and reversal parameter for the dominant mode in the Single Helical states can be obtained (Paccagnella 2016b) by minimizing the distance of the relaxed state described in [START_REF] Bhattacharjee | Energy Principle with Global Invariants for Toroidal Plasmas[END_REF] from a state which is constructed as a two region generalization of the Taylor's relaxation model [START_REF] Tassi | Saturation of tearing modes in reversed field pinches with locally linear force-free magnetic fields and its application to quasi-singlehelicity states[END_REF].

MHD simulations show increased magnetic order and reduced transport of magnetic field lines in regimes with the twist of a non-resonant mode, and reveal the existence of Cantori encompassing the region characterized by conserved magnetic surfaces, which act as barriers to transport of magnetic field lines (Veranda et al 2017).

Conclusion

The above results show that RFP physics moved away from the Taylor-chaotic RFP paradigm (see section 7 of [START_REF] Cappello | The Reversed Field Pinch toward magnetic order: a genuine self-organization[END_REF] for more information about this issue). The new paradigm stems from the existence of two bifurcations: that from multiple to (quasi) single helicity, and that from a double to a single helical axis. This new paradigm comes with other good news. First, no poloidal currents need to be driven in the resistive edge of the plasma. Second, the dynamo mechanism is no longer mysterious: it is a mere electrostatic drift due to the helical deformation of the current lines, as occurs for the nonlinear tearing mode.

However several mysteries are still present on top of those listed in the last paragraph of the previous subsection: How far can magnetic chaos recede when increasing the current and when improving magnetic boundary, particle fuelling, and plasma-wall interaction? What are the mechanisms limiting confinement in the eITBs?

MHD numerical simulations have played an important role to introduce relevant concepts, and to suggest experimental investigations. Until now, these simulations have been done with fixed resistivity profiles. However, there is an actual interplay between transport and resistivity through the temperature profile, which should be investigated numerically.

It is worth noting that the above results were obtained by a small community with a budget more than two orders of magnitude smaller than the tokamak one. Since December 2015 is operated the KTX RFP in

  is simulated as a straight periodic cylinder with axial periodicity 2 R  . The plasma current and the axial magnetic flux are taken as constant, which implies the constancy

FIG. 1 :

 1 FIG. 1: Transition diagrams at two values of the pinch parameter: a) Θ = 1:9, b) Θ = 1:6. E m=0 is the time averaged magnetic energy of the m = 0 modes and H is the Hartmann number. The symbol △ corresponds to S = 3:3 10 3 with P = [0:012; 50], to S = 3:0 10 4 with P = [1; 5000], and • to S = 10 5 with P = 10. The vanishing SH m = 0 modes energy is represented as a finite conventional value with different offsets associated to the different preferred helicities (reproduced by permission from (Cappello 2004) doi:10.1088/0741-3335/46/12B/027).

FIG. 2 :

 2 FIG.2: Temperature distribution in a SHAx state; data from Thomson scattering (diamonds), double filter (blue circles) and thermal helium beam (triangles) diagnostics[START_REF] Gobbin | Heat transport in helical RFX-mod plasmas by electron temperature dynamics from soft-x-ray diagnostics Plasma[END_REF]

  al 2000c)[START_REF] Bonfiglio | Magnetic chaos healing in the helical reversed-field pinch: indications from the volume-preserving field line tracing code NEMATO[END_REF] [START_REF] Cappello | Equilibrium and transport for quasi-helical reversed field pinches[END_REF] (Veranda et al 2013).This theoretical scenario turned out to occur in RFX-mod. SHAx states were first triggered by forcing an oscillation of the edge toroidal magnetic field (Lorenzini et al 2008): an initial small localized hot island is replaced by a large plateau which is then warmed up (Lorenzini et al 2009b)[START_REF] Bonomo | 2D characterization of core thermal topology changes in controlled RFX-mod QSH states[END_REF] (see figure2). At currents of the order of 1.4 MA or higher, SHAx states occur spontaneously(Lorenzini et al 2009a). The thermal content of the plasma increases and magnetic reconstructions display magnetic chaos healing (Lorenzini et al 2008)(Lorenzini et al 2009a). The helical q profile has a maximum(Terranova et al 2010) (Gobbin et al 2011a) (Gobbin et al 2011b). An eITB occurs in the vicinity of this extremum[START_REF] Puiatti | Helical equilibria and magnetic structures in the reversed field pinch and analogies to the tokamak and stellarator Plasma[END_REF] (Gobbin et al 2011a) (Gobbin et al 2011b), in analogy with tokamak eITB's. The region inside the eITB has a flat temperature profile, and spans a significantly bigger volume than in DAx states(Valisa et al 2008) (see figure

  et al 2006) (Onofri et al 2008) (Onofri et al 2009) (Onofri et al 2010b)

FIG. 4 :

 4 FIG. 4: Temperature and density profiles as a function of the helical flux. Black points: central Thomson scattering, red points: edge Thomson scattering (Alfier et al 2010)

  2011)); (ii) the SH mode amplitude is insensitive to S, which backs up experimental[START_REF] Piovesan | Magnetic order and confinement improvement in high-current regimes of RFX-mod with MHD feedback control[END_REF] and numerical results in a two-fluid context (King et al 2011). Adding gyroviscosity in the force balance equation enables to get the experimental ratio of secondary to primary modes, while it is twice too large otherwise (King et al 2011). More analytical results can be found in the course(Escande, D. F. 2015).

Holmes J A, Carreras B A, Hender T C, Hicks H R, Lynch V E, An Z G and Diamond P H 1985 Nonlinear interaction of tearing modes: A comparison between the tokamak and the reversed field pinch configurations Phys. Fluids 1958Fluids -1988