Magnetic perturbations as a viable tool for edge turbulence modification

RFX-mod [1] is the largest RFP device presently in operation with a major radius R = 2 m and a minor radius a = 0.459 m. It is equipped with an advanced system for MHD control, composed of 192 independently fed saddle coils [14]. Through the amelioration of the plasma boundary and reduction of edge radial magnetic field, high current operation with plasma current up to 2 MA has been possible. At high plasma current (I_p ⩾ 1 MA) the RFP exhibits a transition from a globally chaotic magnetic topology with a broad spectrum of kink-tearing modes, towards helical equilibrium set by a single mode (usually the innermost resonant one). The amplitude of the mode is sufficient to modify the flux surfaces, creating a helical core surrounded by a quasi-symmetric boundary. The appearance of these helically states is accompanied by an amelioration of plasma performance, and in some cases by the appearance of an internal electron transport barrier [15]. In these helical regimes the helical ripple at the edge is relatively small, of the order of 1% of the total field, but nevertheless it is sufficient to modulate edge kinetic properties and plasma wall interaction (PWI) [16, 17], consistently with what has been reported in [18] in the case of distorted plasma boundary in presence of saturated MHD modes. Some preliminary indication of the effects of helical boundary on small scale turbulence has been already reported [19] but in order to allow probe insertion and turbulence analysis the helical plasma has been forced at lower current (I_p ≲ 450 kA) with an externally applied helical perturbation [20]. The insertion of probe head, extensively described elsewhere [21, 22], allows for a proper characterization of the perpendicular flow, density, temperature and fluxes, plus the determination of the electromagnetic properties of small scale blobs/filaments [21]. The probe head indeed consists of a 2D array of electrostatic pins, arranged in a five-pin triple balanced probe [23] in order to provide high frequency estimate of plasma potential, electron density and electron temperature plus. The arrangement of the pins allows also the estimate of the plasma vorticity as the $\nabla _{\bot}^{2}{{V}_{f}}$ [24] and the local current filaments fluctuations from the direct measurement of the $\nabla \times \overset{}{{b}}$ obtained from a 2D array of tiny pick-up coils. In particular data reported hereafter have been taken in discharges with I_p ≲ 450 kA, normalized greenwald fraction n/n_G ≃ 0.3–0.4 and edge safety factor q(a) ≈− 0.004. A helical perturbation of helicity (m, n) = (1, −7), rotating with a frequency 10 Hz, has been applied throughout the entire flat-top.

In addition to RFP operation RFX-mod can now routinely operate as a low current ohmic circular tokamak with the first wall representing the limiter, as already reported (see for example [25]). The maximum plasma current achieved is I_p = 0.15 MA, with a maximum toroidal magnetic field B_ϕ = 0.55 T, corresponding to an edge safety factor q(a) = 1.7. This low-q operation was made possible only by means of a feedback control on the (m, n) = (2, 1) mode, which otherwise grows too large and disrupts the discharge. The discharges presented in the paper exhibit a controlled variation from q(a) ≃ 2.1 to q(a) > 3 obtained by decreasing the plasma current. The flexibility of the MHD control system has allowed to vary the helical boundary condition within the same discharge with a (2, 1) radial perturbation during the q < 3 phase, whereas in the q(a) > 3 a magnetic perturbation on modes with the same toroidal periodicity (both n = 1 and n = 2) and poloidal mode numbers m = − 1, 0, 1, 2 has been applied. This produces a perturbation of the magnetic field with a bulging of the plasma towards the external equatorial plane, similar to the one obtained with the so-called C-Coils in DIII-D [26]. Data presented hereafter refer to the first part of the discharge with a single applied harmonic.

In figure 1, examples of the discharges analysed in the present paper are shown: they represent both High and Low current RFP discharges, respectively with a spontaneous helical perturbation and an imposed helical boundary, plus a tokamak discharges.

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It is worthwhile placing the RFX-mod database in the classic operational plane for describing ELM suppression, featuring normalized pedestal collisionality against Greenwald density fraction. Figure 2 shows this operational space as taken from [4]: three type of RFX-mod discharges are shown on top of the existing database. In the case of RFX-mod, lacking an appropriate definition of a pedestal region, we will consider values at r/a ≈ 0.95. It is also worth to remember that RFX-mod tokamak operation is limited to ohmic L-mode where ELMs are not present. For the RFP case, due to the different magnetic topology, an adaptation of the standard formula for bounce frequency [27] has been derived, where the maximum bounce frequency can be written as

$$\begin{eqnarray} &&{{\omega}_{\text{max}}} =\frac{{{v}_{\text{th}}}}{\sqrt{2}R}\sqrt{\frac{R}{rf}}\sqrt{\frac{r}{r-\Delta \prime}}\end{eqnarray} \tag{ 1 }$$

being $f={{B}_{{\hat{\theta}}}}/{{B}_{0}}$. Corrections are needed in order to take into account the vanishing q = 0 surface and the fact that magnetic field is mainly poloidal towards the edge. The operational space covered by the present experiments enlarge the established one at higher collisionality, in particular at lower plasma current. On the other side the RFP is also known to generally operate at higher beta with respect to Tokamaks and for the present experiments a local electron β of about 1% may be estimate. This is important as a β dependence is expected to modify the behavior of magnetic field penetration [28], and in general to enhance the electromagnetic nature of plasma turbulence [29, 30], exploring ranges interesting for future devices but generally not covered by present machines. To complete the characterization of local plasma parameters relevant for transport studies, we can mention that the typical values of ρ^* = ρ_s/a are in the range of 0.002 for the RFP plasmas up to 0.006 for the RFX-mod tokamak discharges, thus in the same range, in terms of a-dimensional parameters, as larger machines [31]. We can consequently conclude that information presented hereafter are in the correct framework for the comprehension of the interaction between magnetic perturbation and kinetic properties. The already reported commonalities between tokamaks and RFPs in terms of ambipolar electric field [32], anomalous transport [33], electromagnetic structure characteristics [21, 24], suggest that a comparison of the effects of the topology on flow and transport on these two configuration could allow to disentangle the inherent fundamental mechanisms which regulates this physical problem, allowing a deeper comprehension of the topic.

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High current RFP operation reveals the tendency of the plasma to produce an helical equilibrium featuring ordered magnetic surfaces where a single m = 1 mode (generally the most internally resonating n = − 7) becomes dominant. This RFP state, dubbed quasi single helicity state (QSH) is substantially different with respect to what is generally observed at lower current, where a wide spectrum of m = 1 modes produces magnetic chaos in the plasma core. The chaotic case is dubbed multiple helicity (MH) state. It has been widely reported [16, 17, 19] that although the residual edge helical ripple is of the order of 1% this is sufficient to modulate all the plasma kinetic quantities. In particular density, pressure and flow are all found to exhibit an oscillation correlated with the toroidally rotating dominant magnetic perturbation. In order to properly handle the phase relation with the magnetic perturbation, the helical angle u_{m, n} has been introduced [17]. This is defined as:

$$\begin{eqnarray}&&{{u}_{m,n}}(t)=m\theta -n\phi +{{\varphi}^{m,n}}(t)\end{eqnarray} \tag{ 2 }$$

where φ^{m, n} is the proper phase velocity of the rotating magnetic perturbation. The helical angle assumes, by construction, the value of π/2 whenever the O-point (OP) of the magnetic island passes in front of the observer, whereas assumes value of 3π/2 at the X-point (XP). As an example of the effect of the helical boundary on edge plasma the plasma pressure as a function of the helical angle and radius is shown in figure 3. The pressure pattern is superimposed to the Poincaré map of the magnetic perturbation, where only the dominant mode (m, n) = (1, 7) and its toroidally coupled m = 0 component (m, n) = (0, 7) are retained. Poincaré plot has been obtained with the field line tracing code FLiT [34]. The eigenvalues given as input to FLiT are obtained by solving Newcomb's equations in toroidal geometry through the code NCT [35]: constraints are the edge magnetic field measurements of B_r, B_θ, B_ϕ. Although much richer, we have limited the spectrum included in the computation of the Poincaré map to highlight the space relation with respect to the dominant periodicity. Plasma pressure is observed to oscillate as a function of the helical angle, with increased pressure located at u_1, − 7 ≈ π/2 at the maximum of the radial displacement induced by the dominant mode [17], mainly due to an increasing of the local plasma density confirmed also by microwave reflectometer measurements [36]. It is worth recalling [19] that the modulation is also observed on the entire edge profile, with a reduction of the pressure characteristic length ${{L}_{p}}=-{{P}_{e}}/\nabla {{P}_{e}}$ at the OP of the (m, n) = (1, −7) island, u ≈ π/2. Investigation obtained through the Gas Puff Imaging diagnostic [36] revealed that the modification of L_p and in general of the edge properties determine the modification also of local small scale fluctuations. The edge region of RFX-mod is indeed characterized by the presence of localized fluctuations, dubbed as blobs/coherent structures, which are filaments elongated in the parallel direction featuring a parallel current, parallel vorticity and local density accumulation [21, 24]. The perpendicular dimensions of these blobs are found to exhibit a modulation consistent with the helicity of the existing perturbation as shown for example in figure 4 where the dimensions have been normalized to the local sound gyroradius ρ_s = c_s/Ω_i. A clear reduction of both dimensions is observed in the region 3/2π ≲ u ≲ 2π, close to the location of the XP of the (1, −7) islands, or correspondingly in the region of inward plasma shift.

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It has also been reported that perpendicular flow is modulated by the helical angle, and this has been confirmed independently by the helicity of the perturbation [37]. Owing to the small contribution coming from the diamagnetic flow the observed velocity modulation reflects the modification of the radial electric field, which has been theoretically explained in terms of ambipolar constraints [37, 38]. Indeed species behave differently in presence of a magnetic ripple, or magnetic islands, with electrons strictly following magnetic field lines due to their smaller drift. In some of cases, in high density regimes, it has been shown that the modulation of the flow causes a reversal in a toroidal sector of the machine, which determines the appearance of a point of density accumulation causing a radiative collapse of the discharge [2]. This was observed as due to a macroscopic (m, n) = (0, 1) magnetic island characterizing those high density regimes.

In order to properly investigate flow and turbulence as a function of the magnetic perturbation, QSH states have been forced at lower plasma current (I_p ⩽ 450 kA), imposing a rotating helical boundary with a (1, −7) perturbation. The lower current allows insertion of probes, providing detailed information on both perpendicular components of the E × B flow, high frequency electromagnetic turbulence, plus a direct estimate of the parallel current and parallel vorticity fluctuations [24]. The rationale behind these experiments is the possibility to provide a detailed map of the perpendicular flow perturbation caused by the magnetic perturbation, and to investigate how the global changes affect the turbulence responsible for anomalous transport. A first important result is shown in figure 5: in this plot we report in color-code, as a function of radius and helical angle, the value of density as measured simultaneously by an array two five-pins balanced triple probes. Superimposed with arrows, the streamline plot of the E × B flow resulting from the combination of spatially resolved measurements of the two perpendicular components of the velocity is reported (light blue in the on-line color version). To properly compare with the underlying topology the Poincaré map of the magnetic field, reconstructed similarly to figure 2 is also shown. As observed in the high current regimes (see figure 2), density is increased around u ≈ π/2, at the maximum of the helical radial displacement (OP). What is far more important is the observation that pattern of the underlying flow follows rather strictly the magnetic field line, creating sort of vortex-like pattern around the m = 0 island. On the same plot, in the top panel the particle influx estimated from H_α emission as a function of the helical angle is shown. One can clearly seen that the emission is enhanced at u ≈ π where the m = 0 island is touching the wall. On the other side following the pattern of the flow, particles injected in the region of higher emission are convected and tend to accumulate around π/2 where the X-point of the (0, 7) island is located. Thus the observed modulation of the density, which is not homogeneous along the helical angle seems motivated by a modulation of the perpendicular components of the velocity and the already observed trapping effect caused by the presence of a magnetic XP [38]. Still in figure 5 in red dots the particle flux resulting from electrostatic fluctuation, $\Gamma =\langle {{\tilde v}_{r}}{\tilde n} \rangle$ is shown, exhibiting an increase in the region π/2 ≲ u ≲ π. Particle flux has been derived from density and radial velocity fluctuations both of the quantities sampled at 5 MHz. Computation has been done using standard technique in the frequency domain [39] with an estimate every 10 ms. The data refer to the position r = 0.446 or r/a = 0.97. As a matter of fact, the modulation of the magnetic topology provides a modulation of the particle losses due to small scale electrostatic fluctuations, with the same periodicity. It is worth remembering that the particle fluxes due to electrostatic fluctuations, account for almost all the particle losses in the edge region of an RFP plasma, despite the high level of magnetic fluctuations. A large fraction of these losses can be imputed to the presence of blobs [40]. In order to verify the possible effect of the magnetic topology on blob properties, the detection frequency of the blob f_b, defined as the number of blobs detected for second, has been computed as a function of the helical angle and reported in figure 6(a). We observe that increasing of the particle fluxes coincides with an increase of the frequency of the observed blobs, as also clearly demonstrated in figure 6(b), thus strongly relating the enhanced transport to a substantial increase of blob-driven particle losses.

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The presence of a magnetic perturbation modifies the entire profile of the E × B flow. This can be observed for example in figure 7 where the E × B shear at r = 0.446 is shown as a function of helical angle. Velocity profile is found profoundly modified, with an inversion of the value of the shear around u ≈ π/2. On the other side shear is well known to act on turbulent eddies. In particular we are interested in the process of eddies shear selection which is a know feature in both fluids [41] and plasmas [42]. Indeed in plasmas there have been observed blobs with both positive and negative vorticity [42], where positive and negative are defined as parallel and antiparallel with respect to the guiding magnetic field, although with different abundance [42]. If we now compute the ratio, along the helical angle, between blobs with vorticity antiparallel to B₀ with respect to the total number of structures (shown in figure 7 with orange dots), we observe that it is not homogeneous and the fraction is peaked whenever the shear change sign, in accordance to the shear selection rule.

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In order to establish the universality of the mechanism described so far, we have exploited the versatility of RFX-mod device to study the same phenomenology in the tokamak configuration. As shown in figure 1, the operational space of RFX-mod in terms of edge safety faction q(a) is enlarged, thanks to the possibility to operate with q(a) ≳ 2 with MHD control system mitigating the (m, n) = (2, 1) perturbation. In terms of collisionality, RFX-mod tokamak is roughly positioned at the same values of KSTAR or DIII-D at high ν^*. The experiments presented hereafter refer to the part of the discharge where q(a) ≈ 2.1, during which an helical radial magnetic perturbation (m, n) = (2, 1) rotating with a frequency of 10 Hz has been applied. As shown in the Poincaré map in figure 8, the perturbation creates a large (m, n) = (2, 1) island whose external convolution touches directly the first wall shown in a thick blue line. The map is the result of the already described field line tracing code FLiT [34], where only the eigenfunctions for perturbation m = 1 with toroidal periodicity n = 2, 3 are kept. The eigenfunctions are still calculated with code NCT [35] extended to the tokamak configuration. The measurements presented hereafter are obtained at the radial location indicated by the red dashed line. Assuming that the magnetic island is rigidly rotating with the applied perturbation, we can infer the poloidal map of the various kinetic quantities. The evolution along the perturbation of the radial electric field estimated by plasma potential measurements is shown in the same figure 8. The electric field exhibits an m = 2 periodicity and it is strongly reduced at the locations of the X-points of the (2, 1) island. It is worth noting that these are measurement that according to the reconstruction lies inside the magnetic island itself. A plausible interpretation of the profile observed is the following: the presence of the island touching directly the material wall creates a wide scrape off layer region (SOL), with the internal separatrix of (2, 1) mode representing the separatrix between confined plasma and SOL. The positive radial electric field is consistent with an higher electron mobility (generally observed in the far SOL, see for example [43]). On the other side at the X-point the separatrix position is moving towards the wall and the reduction of the E_r may be interpreted as a reduction of the difference in electron-ion mobilities and is equivalent to provide a measurement of the electric field in a region closer to the separatrix, where also in standard discharges the E_r approaches the value of 0 and than changes sign (e.g. [43]). This is consistent with measurements performed in LHD [44], where the change of sign of the E_r is suggested to provide a valid measurement of the effective plasma boundary in presence of a 3D magnetic field. Within the same assumption of rigid body rotation we can obtain the poloidal distribution of the electron density shown in figure 9 after a low pass filtering: a clear depletion at the location of the X-points and an enhancement in correspondence to the O-point of the m = 2 island can be distinguished. In the same figure we also show the spectrogram of the density fluctuations up to 200 kHz which suggest a strong damping in all the spanned frequency range at the location of the two X-point of the islands. Also in the tokamak configuration the turbulence modification has influence on the particle losses due to electrostatic turbulence. In figure 10 the poloidal profile of the radial particle flux due to electrostatic turbulence is shown. A clear m = 2 periodicity can be derived and reduction of the flux is observed in correspondence to the damping of the fluctuations shown in figure 9. To confirm the observation drawn in RFP configuration on the effects of the presence of magnetic island on small scale electrostatic turbulence figure 11 shows the poloidal modulation of the E × B shear as derived from the electric field profile obtained by radially resolved plasma potential measurement. Consistently, m = 2 periodicity is observed also on the shear, with an abrupt reduction at the location of the O-points of the island, which confirms flat radial electric field profile at the O-point of the island as observed for example in [45]. On the same plot also the fraction of vortices with vorticity anti-parallel to B₀ is shown, and in analogy to the observation obtained in the RFP configuration, a strict correlation with the shear profile is confirmed.

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It has been postulated [46] that the presence of eddies with opposite vorticity may change the nature of turbulent transport. This is basically related to the change of probability for particles leaving a given eddies to encounter other eddies with the same or opposite axial vorticity. On the other side it is also known that internal blob spin change their dynamics [47], enhancing their coherence and that for spinning blobs the interaction of the blob with the ambient velocity field depends on the sign of the external shear with respect to the spin. As an indication that this mechanism is likely to take place we report in figure 12 particle transport as a function of the relative abundance of eddies with vorticity anti-parallel to the guiding magnetic field. The analysis seems to indicate an enhancement of losses whenever the fraction increasing suggesting that the presence of eddies with opposite vorticity with approximately the same probability increases the particle losses. This represents an indirect confirmation of what theoretically suggested [46] and represents a confirmation of preliminary indication given in [42]. The mechanism is confirmed by the observation of similar results independently on the configuration.

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A detailed study of the effect of magnetic perturbation on edge flow and small scale turbulence has been done in RFX-mod. The unique versatility of the device offers the possibility to analyse the same phenomenology in two different configurations, Reversed Field Pinch and Tokamak, on the same machine and with the same diagnostic. Both the configurations reveal a modulation of the electron density which exhibits the same periodicity of the underlying topology. The radial electric field is modulated as well and it also exhibits the same symmetry as the magnetic topology. It has been demonstrated that the presence of magnetic island close to the wall modifies the pattern of E × B flow consistently with the geometry of the island, with the appearance of a convective-cell around the island. The observed flow acts as convective motion which contributes to the observed modulation of plasma density with local accumulation, for the RFP case at the X-point of the islands. Particle fluxes induced by electrostatic turbulence is found to be modulated as well, still exhibiting the same symmetry. The variation of transport is ascribed to a modification of small scale turbulence which is not homogeneous in presence of magnetic perturbation. Indeed both the appearence frequency of radially propagating blobs, and the intrinsic vorticity of these structures are modified, and particles losses increases whenever the frequency increase and the relative abundance of counter-rotating eddies are found to be approximately equivalent. The observations reported indicate that a complete understanding of the processes occurring during RMP experiments should take into account both the convective motion caused by modified electric field and enhanced particle losses due to small scale variation. Furthermore we provide indication that shear flow modification effects on transport is much richer than pure fluctuation amplitude suppression and a that a complete comprehension of transport events should take into account intrinsic properties of turbulent eddies. The unique versatility of RFX-mod, in term of operational space and diagnostic for turbulence analysis, has allowed for a configuration-independent physical description of detailed interaction between topology and transport.

This project has received funding from the European Unions Horizon 2020 research and innovation program under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.