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Boundary plasma response in RFX-mod to 3D magnetic field perturbations
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Nuclear Materials and Energy 0 0 0 (2017) 1–5
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Boundary plasma response in RFX-mod to 3D magnetic field perturbations P. Scarin∗, M. Agostini, L. Carraro, G. Spizzo, M. Spolaore, N. Vianello Consorzio RFX, C.so Stati Uniti 4, Padova, Italy
a r t i c l e
i n f o
Article history: Received 6 June 2016 Revised 2 February 2017 Accepted 6 March 2017 Available online xxx Keywords: RFX RFP PWI THB MPs Stochastic layers Magnetic chaos
a b s t r a c t The edge of the RFX-mod (R = 2 m, a = 0.46 m) Reversed Field Pinch device is characterized by weak magnetic chaos affecting ion and electron diffusion. Edge particle transport is strongly influenced by a toroidal and poloidal asymmetry caused by magnetic islands and an ambipolar radial electric field ensures local neutrality. At higher plasma current (Iφ >1MA) a spontaneous resonant dominant mode m/n = 1/7, slowly rotating, develops in the inner region. The edge electron pressure Pe and floating potential Vf show a shape of the Plasma Wall Interaction (PWI) with the same toroidal periodicity, which follows the edge local ideal magnetic displacement 1,7 . Detailed measurements along the poloidal direction of Pe and Vf have been undertaken newly, their time behavior present respect to 1,7 a time lag, which depends on the poloidal angle θ . The mode analysis in terms of helical angle reveals a role of the 0/7 mode in determining a poloidal phase lag respect to 1,7 of Pe and Vf . Since the 0/7 is the largest toroidal sideband of the 1/7 mode, this work suggests a role of toroidal coupling in determining the plasma response to a MP. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction The study of the three dimensional (3D) kinetic response to edge magnetic stochastic layers is of growing importance in the fusion community. As a matter of fact the need for a 3D, chaotic magnetic field in present and future fusion devices to modify plasma wall interaction, divertor loads, transport and MHD stability in the edge is widely recognized [1]. Nevertheless, the action of magnetic perturbations (MPs) on the edge plasma is far from being well understood, and to date there is no obvious relationship between electron pressure gradient ∇ P and the MP: this is an outstanding issue for ELM control and suppression in ITER, since edgelocalized modes are sensitive to the ∇ P and not directly to the MP. A chaotic edge layer can be spontaneous, as in the reversed-field pinch [2] and the Stellarator [3], or externally applied via MPs as done in both Tokamaks [1] and Stellarators [4]. In this respect, the RFX-mod experiment is suitable for studying the kinetic response of the edge plasma to 3D magnetic fields. In fact, the RFX-mod Reversed Field Pinch (RFP) is characterized by the presence of magnetic modes with different helicity, spontaneously present or externally induced, the amplitude and phase of the MP are well known, and dynamically controlled via a state-of-
∗
Corresponding author. E-mail address: [email protected] (P. Scarin).
the-art active coil feedback system [5]. In particular, operating at high plasma current (Iφ >1MA), the innermost resonant magnetic mode m/n = 1/7 becomes dominant with an almost monochromatic spectrum [6]. This monochromatic spectrum is not peculiar of RFXmod, but it has been observed in all RFP devices, namely MST [7], TPE-RX [8], EXTRAP T2R [9], and, more recently, in the low-aspect ratio RELAX device [10]. In this m/n = 7 helical state, the RFX-mod plasma is shaped as an helix: in particular, the edge plasma shows a 3D stochastic layer which, in terms of connection lengths and typical electron loss times [2], is comparable to the Tokamak edge when MPs are applied. Distributed diagnostics both in toroidal and poloidal directions can measure floating potential, edge electron density and temperature and neutral influxes, thus characterizing the space and time evolution of the edge plasma in great detail. Experimental observations in RFP discharges have shown a clear kinetic response to the spontaneous dominant perturbation and the edge electron pressure Pe , the particle influxes and the floating potential Vf show the same 1/7 modulation [11]. These results are reminiscent of edge modifications induced by MPs in Tokamak as seen in Textor [12], in DIII-D [13] and in Asdex [14]. A detailed study of the poloidal structure of Pe , and Vf has todate never been undertaken in RFX-mod. The present study tackles this issue, and reveals that, even if a strong m = 1 modulation is present in the magnetic topology, the plasma does not follow exactly the helicity of the MP: different poloidal harmonics are present.
http://dx.doi.org/10.1016/j.nme.2017.03.006 2352-1791/© 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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These experimental results could be useful also for other toroidal magnetic confinement devices (RFP, Tokamak, Stellarator) that use MPs on the edge to control the 3D kinetic response of edge layers. In this paper, the edge effects concerning the spontaneous MP will be shown: the results of edge plasma pressure response will be presented in Section 2, while in Section 3 the observation of floating potential response will be discussed; Section 4 will be dedicated to the investigation of measurement results and Section 5 to the summary. 2. Edge plasma pressure response In RFX-mod the spontaneous innermost dominant mode 1/7 is coupled with the Shafranov shift, which is equivalent to a 1/0 mode, and the consequence is the development of a chain of 0/7 islands resonating on the q = 0 surface in the edge region, typically at r/a∼0.95, strongly modifying the edge parallel current map [15]. The resulting magnetic topology shows the m = 0 O-point (OP) aligning toroidally with the m = 1 X-point (XP) in the low-field side (LFS), and the opposite in the high-field side (HFS) [16]. In this magnetic topology the helical angle um,n =mθ -nφ +ϕ m,n (t) is often used as a reference frame [17]; in fact, it puts measurements performed at different poloidal and toroidal angles in the same helical frame of reference, similarly to what is done in stellarators. In the definition of um,n , θ and φ are the poloidal and toroidal angles of the diagnostic, respectively, ϕ m,n (t) is the phase of the slowly rotating mode (f∼40 Hz) and m and n are the poloidal and toroidal mode numbers. In the reference frame of this helical angle, the OP of the island is at um,n =π /2 and the XP at um,n = 3π /2 for each mode number (m,n), by construction. In a previous analysis [17], the measurements in the LFS (θ = 0°) and HFS (θ = 180°) of floating potential, plasma flow, neutral influx and electron pressure (the latter in the LFS only), have shown a clear m = 1 modulation. This has suggested a helical Plasma Wall Interaction (PWI) with a poloidal structure that closely follows u1,7 , and that is characterized by an increase of electron density, neutral influx and more negative Vf in front of the OP of the m = 1 islands, corresponding to u1,7 =π /2. More recently, edge electron pressure Pe measurements with the Thermal Helium Beam diagnostic (THB) [18] in two poloidal locations at θ = 0° and in UP position at θ = 90° (both at toroidal position φ = 262° and r/a = 0.98) have shown a more complicated behavior than the helical u1,7 . To show this, we will take as a proxy of the magnetic helical pattern, the ideal displacement (at r=a) of the dominant mode 1/7, 1,7 (t) [19]. This displacement is approximately 1,7 (t) = 0 sin(u1,7 ) (0 is the displacement amplitude, see Fig. 1 in Ref. [11]). This means that the following three conditions are equivalent: a) 1,7 is maximum; b) u1,7 =π /2; c) the measurement corresponds to the island OP. The cross correlation C(1,7 ,Pe ) between the local displacement 1,7 (t) and Pe (t) (during the plasma flat-top phase), in the two poloidal positions, is shown in Fig. 1 for a typical plasma discharge. In the equatorial plane Pe (t) and 1,7 (t) oscillate almost in phase and, since the correlation is maximum at zero time lag, the electron pressure increases when 1,7 > 0 (or, equivalently, u1,7 ∼π /2); but this does not happen at θ = 90°, where there is a finite time lag τ between the two quantities of about 4 ms, that is about T/6, where T is the period of spontaneous MP (1/T∼40 Hz). This means that Pe (t) does not follow exactly the dominant m = 1 magnetic modulation. To highlight this result the local fluctuations defined as δ Pe /Pe =(Pe −Pe )/Pe , where Pe is the time average during the plasma flat-top phase, have been analysed in terms of the helical angles u1,7 and u0,7. If δ Pe /Pe depended on the helical deformation 1,7 only, it should be δ Pe /Pe ∼ sin(u1,7 ) for all poloidal positions, with no θ -dependence (as it was found in the past on RFX-mod, when only LFS and HFS were diagnosed, see Ref. [11] and [17]). On the contrary Fig. 2a displays (for the two poloidal positions)
Fig. 1. Cross correlation between the edge local ideal magnetic displacement 1,7 , due to the dominant mode and the electron pressure Pe in two poloidal positions (θ = 0°, θ = 90°).
the Pe fluctuations as function of the helical angles u1,7 with a θ dependence. The measurements have been done every 2 ms and the triangles in the figure represent the average over the mode rotation periods for several discharges, with plasma current in the range 1.4÷1.7MA. Fig. 2a shows that at θ = 0° the maximum value of δ Pe /Pe is in front of the 1/7 OP, but this is not true for the θ = 90° where the maximum of Pe fluctuations is between the OP and the XP of the 1/7 mode: thus there is not a unique relation between the maximum of δ Pe /Pe and the 1/7 mode. Conversely using u0,7 as a reference angle (Fig. 2b), the maximum of Pe fluctuations is around the 0/7 XP for both the polidal positions, that would hint an m = 0 helicity in the poloidal region θ = 0°÷90°. This experimental result shows a connection between the dominant mode m = 1 that originates the perturbation and its toroidally coupled sideband m = 0. It has also been observed that the level of Pe fluctuations is lower at θ = 90° (∼10%) with respect to θ = 0° (∼20%) and this means a lower PWI in the upper portion of the helix. 3. Floating potential response A more complete picture of the 3D PWI in RFX is obtained by measuring the floating potential with toroidal and poloidal Langmuir probe arrays embedded in the graphite tiles covering the first wall. The toroidal probe array has 72 pins, and the Vf behavior in terms of helical angle u1,7 has been studied in Ref. [11]. The detailed poloidal structure of the floating potential is instead studied here. To accomplish this, we use the poloidal array of electrostatic probes (5 probes at θ = 32°, 84°, 186°, 238°, 341°) at a fixed toroidal angle φ ∗ = 249° [20]. The cross correlations C(1,7 ,Vf ) between 1,7 (t,θ ∗ ,φ ∗ ) and Vf (t,θ ∗ ,φ ∗ ) (during plasma flat-top phase) for the five poloidal positions θ ∗ is shown in Fig. 3a for a typical discharge: the time lag τ of the minimum cross correlation between the two signals depends on the poloidal position. This result extends that of Section 2, which was limited to two poloidal angles, only (θ = 0° and 90°). Again, not all of the poloidal angles are equivalent, and the kinetic response deviates from a perfect helix along θ . The outcome of the analysis over many shots (about 30) is shown in Fig. 3b, where the relative time lag τ /T of the minimum of the cross correlation is reported as a function of poloidal positions. Though data are characterized by some dispersion, it is possible to observe different average values of the time lag τ /T. The probe at θ = 341° presents an average relative time lag τ /T ∼ −0.1 (the minus sign means that the minimum of Vf is seen after the maximum of the ideal displacement 1,7 ) and at θ = 84° the largest one τ /T∼ −0.25. This means that while the 1/7 island OP rotates poloidally from 341° to 84° in the LFS, the PWI, identified
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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Fig. 2. Relative fluctuations of Pe for two poloidal positions as function of helical angle u1,7 (Fig. 2a) and u0,7 (Fig. 2b). The measurements have been done every 2 ms and the triangles in the figure represent the average over the mode rotation periods relative to several discharges.
Fig. 3. Cross correlation between 1,7 (t) and floating potential Vf (t) at the different poloidal positions of the probes for a typical discharge (Fig. 3a). The relative time lag τ /T of the minimum of the cross correlation are reported in Fig. 3b as function of poloidal positions for many shots (blue points) and their average is shown (red line). The error bar depicts the rms of blue point for each poloidal position. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by Vf , lags behind of −0.25·2π =−π /2 (it means at the 0/7 island XP). Afterwards, when the 1/7 OP has reached the HFS, the PWI jumps in the direction of the 1/7 OP, with a decrease of relative time lag and then the PWI behaviour follows the 1/7 mode until the OP reaches θ = 341° near the LFS. This odd behaviour is in no way accounted for by a simple description in terms of 1,7 as done in the past in RFX-mod. To underline this, δ Vf =Vf -Vf (where Vf is the time average during the plasma flat-top phase) has been analysed as a function of the helical angles u1,7 and u0,7 at each poloidal position. The fluctuation δ Vf of the five probes has been evaluated every 2 ms during several MP rotation periods, in many plasma discharges. The average values of these fluctuations (normalized for each poloidal position) are shown in Fig. 4a and Fig. 4b as a function of the angles u1,7 and u0,7 , respectively. Again, similarly to the above discussion on Fig. 2a, if δ Vf depended on 1,7 only, it should be δ Vf ∼ sin(u1,7 ) for all poloidal positions, without θ -dependence. On the contrary, Fig. 4a shows that between the negative burst of Vf and the island OP at u1,7 =π /2 (maximum 1,7 ) exists phase lag δ u1,7 , which depends on the poloidal position. The lowest value δ u1,7 ∼ π /6 is found near the equatorial plane (θ = 341°) and the highest one δ u1,7 ∼ π /2 at θ = 86°. This means that Vf does not follow the helicity of m = 1and the space delay between Vf and the magnetic shift 1,7 reflects the time lag shown in Fig. 3b. The result of Fig. 4 also justifies earlier results in RFX-mod (Ref. [11] and [17]) which reported a helical pattern of PWI, since those measurements were usually performed on the equatorial plane, where the phase
lag δ u1,7 between the PWI and 1,7 is reduced. A similar analysis has been done as a function of the helical angle u0,7 and Fig. 4b shows the result: the negative bursts of Vf are localized close to u0,7 ∼3π /2 (XP of the 0/7 mode) for the probes in the poloidal region θ = 0°÷90° (similarly to the maximum δ Pe /Pe in Fig. 2b) and at u0,7 ∼3π /4 (π /4 from the OP of 0/7 mode) for the probes near the HFS . In this figure it is highlighted that the PWI, identified by Vf , completes about a full π -phase shift of u0,7 in ∼100 poloidal degrees,with two relevant poloidal zones: in the first one, PWI is determined by the m = 0 island XP (θ ∼0°÷90°); in the second one PWI is dominated by the OP (θ ∼180°÷270°). 4. Investigation of measurement results To deeper investigate the connection between local magnetic topology and the floating potential the ORBIT code [21] has been used. Fig. 5 shows the Poincaré plot obtained with the 1/7 and 0/7 modes only, for a typical high plasma current discharge. In the Fig. 5 the sequence a), b), c) and d) corresponds to the positions LFS, UP, HFS and DOWN, respectively. The plots show a chain of 1/7 islands in the core (purple points), plus a chain of 0/7 islands resonating at the reversal surface q = 0 (which corresponds to the green horizontal line at r = 44 cm). The 0/7 island arises due the physical fact that a pure helix exists in a cylinder, not in a torus. Let us imagine to measure an (m,n) mode at constant φ on the poloidal plane, sin(m θ ): while the m harmonic is pure in a cylinder, it is always a sin(m’ θ ) in a torus, with m’ = m ± k, k = 1,2… The first order correction, k = 1, is of or-
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Fig. 4. Fluctuations δ Vf for the five polidal probes as function of helical angle u1,7 (Fig. 4a) and u0,7 (Fig. 4b); δ Vf has been evaluated every 2 ms for the MP cycles of many plasma discharges and the average values are given, normalized for each poloidal position.
Fig. 5. The sequence of Poincaré plots in a), b), c) and d) corresponds to the positions LFS, UP, HFS and DOWN, respectively. In these plots has been pointed out the phase lag δ φ between the ideal displacement 1,7 (OP of 1/7 mode) and the outstanding point of effective local deformation of magnetic surfaces in the edge eff (marked in black line) estimated with ORBIT code. Yellow and green lines represent the wall and q = 0 surface, respectively and the blue band the edge stochastic layer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
der ε =a/R: in the RFX case, this means that a 1/7 mode is always coupled to its toroidal sidebands, (m,n) = 0/7 and 2/7. This mechanism is well explained in Ref. [19]. The nonlinear interaction between the neighbouring 1/7 and the 0/7 modes creates a stochastic layer (blue band in Fig. 5) separating the spatial domains pertaining to the two resonances [22]. From Fig. 5, it is possible to define an “effective” local deformation of magnetic surfaces in the edge eff (black line), as the convolution of the outermost points in the Poincaré map, before touching the wall. The behaviour of eff is mostly m = 1 but it is also deformed by the m = 0 islands and this deformation depends on the poloidal angle. The deviation of eff from the ideal displacement 1,7 (which is equivalent to the OP of the m = 1 island) is indicated by the phase lag δ φ between the maximum of 1,7 and that of eff . The phase lag δ φ (drafted in Fig. 5) is negligible at LFS, it is maximum in the UP position (Fig. 5b) and decreasing in the HFS and the DOWN position. The shape and amplitude of eff depends on the topology of the stochastic layer, in particular on its radial displacement relative to the q = 0 resonance (green line in Fig. 5). The details are determined by the competition of magnetic structures closest to the wall: the stochastic layer, or the m = 0 island OP’s. To highlight this point, we calculated the radial position of the last closed orbit of the stochastic layer, which is shown in Fig. 6 together
Fig. 6. Radial position of the last closed orbit belonging to the stochastic layer together with the radial position of the m = 0 island OP, for a given poloidal section.
with the radial position of the m = 0 island OP.. The layer radius is maximum at θ = 0° and minimum at θ = 180°: conversely, the OP’s are closer to the wall at θ = 180° and displaced further inside at θ = 0. Therefore, PWI will be dominated by the stochastic layer
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the connection length Lc on the θ -φ plane, which was shown earlier in the RFX-mod edge (see bottom frame of Fig. 9 in Ref. [6]). 5. Summary
Fig. 7. Cross correlation between 1,7 (φ ) and eff (φ ) at a fixed time as function of polidal positions. The phase lag δ φ presents a poloidal trend similar at that of time lag τ shown in Fig. 3a.
in the poloidal region −60°< θ < 80°, while the OP’s will prevail in the range 80°< θ < 300°, and the behaviour of eff will follow the same rule, accordingly. In order to quantify the difference between the effective local deformation of magnetic surface eff and the ideal displacement 1,7 the cross correlation C(1,7 ,eff ) along φ between the 1,7 (φ ,θ ∗ ,t∗ ) and eff (φ ,θ ∗ ,t∗ ) has been evaluated at a fixed time t∗ for different poloidal positions θ ∗ . The result is shown in Fig. 7: the phase lag δ φ which corresponds to the maximum cross correlations between 1,7 and eff ranges between the positive values of 0.1 rad at θ ∼0° and −0.2 rad at θ ∼90°. Dividing δ φ for the toroidal period 2π /7 of dominant mode one obtains the correspondent period fraction that goes from 0.1 to −0.2. The difference between eff and 1,7 is mainly due to the non-linear interaction between the 1/7 and its toroidal sideband 0/7, since if we evaluate eff with all the modes, m = 0,1 and 1 ≤ n ≤ 24, the result shown in Fig. 7 remains unchanged. The poloidal behavior of the phase lag δ φ between 1,7 and eff , as function of poloidal position, presents a similar trend to that seen for the time lag between 1,7 and Vf (see Fig. 3a): in both cases the delay ranges are comparable (relative values are 0.1÷−0.2 and −0.1÷−0.25 respectively) and the maximum is around θ = 90°. This comparison of the two cross correlations C(1,7 ,eff ) and C(1,7 ,Vf ), despite the different domains, suggests a relationship between the effective local deformation eff and the edge phenomenology of Vf poloidal behaviour. These analyses explain in detail also the peculiar “zig-zag” plot of
The analysis of the edge measurements of Pe and Vf along the poloidal angle in RFX-mod reveals that the PWI does not simply follow the rotation of the spontaneous m = 1 MP, but there is the presence of the m = 0 mode, which has been identified with the analysis in the reference frame of the helical angles u1,7 and u0,7 . A good indicator of the presence of m = 0 mode is the “effective” local deformation of magnetic surfaces in the edge eff , obtained thanks to the simulations of the coupling of the 1/7 and 0/7 modes. The cross correlations C(1,7 ,eff ) and C(1,7 ,Vf ) prompt to a close connection between the effective local deformation of magnetic surface and the edge floating potential. So the eff behaviour can be at the origin of the poloidal phase lag between Vf and the 1/7 island OP. This means that the kinetic plasma response to a monochromatic magnetic perturbation depends on the toroidally coupled sidebands, which are not negligible. This result could be useful when one uses a monochromatic MP for ELM control, especially in elongated plasmas. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
T.E. Evans, Plasma Phys. Control. Fusion 57 (2015) 123001. G. Spizzo, et al., Nucl. Fusion 52 (2012) 054015. Y. Feng, et al., Nucl. Fusion 48 (2008) 024012. S.N. Pandya, et al., Nucl. Fusion 56 (2016) 046002. L. Marrelli, et al., Plasma Phys. Control. Fusion 49 (2007) B359. E. Martines, et al., Nucl. Fusion 50 (2010) 035014. L. Marrelli, et al., Phys. Plasmas 9 (2002) 2868 W. F. Bergerson et al., Phys. Rev. Lett. 107 (2011) 255001. L. Frassinetti, et al., Phys. Rev. Lett. 97 (2006) 175001. L. Frassinetti, et al., Phys. Plasmas 14 (2007) 112510. R. Ikezoe, et al., Plasma Fusion Res 3 (2008) 029. N. Vianello, et al., Nucl. Fusion 53 (2013) 073025. M.W. Jakubowski, et al., Phys. Rev. Lett. 96 (2006) 035004. R.A. Moyer, et al., Nucl. Fusion 52 (2012) 123019. M. Willensdorfer, et al., Plasma Phys. Control Fusion 58 (2016) 114004. M. Spolaore, et al., Nucl. Fusion 55 (2015) 063041. G. De Masi, et al., Nucl. Fusion 53 (2013) 083026. P. Scarin, et al., J. Nucl. Mat. 463 (2015) 467–470. M. Agostini, et al., Rev. Sci. Instrum. 86 (2015) 123513. P. Zanca, et al., Plasma Phys. Control Fusion 46 (2004) 1115. G. Serianni, et al., Rev. Sci. Instrum. 74 (2003) 1558–1562. R.B. White, et al., Phys. Fluids 27 (1984) 2455–2467. A.J. Lichtenberg, M.A. Lieberman, in: Regular and Chaotic Dynamics, 2nd Ed., Springer, 1992, p. 51.
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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Nuclear Materials and Energy 0 0 0 (2017) 1–5
Contents lists available at ScienceDirect
Nuclear Materials and Energy journal homepage: www.elsevier.com/locate/nme
Boundary plasma response in RFX-mod to 3D magnetic field perturbations P. Scarin∗, M. Agostini, L. Carraro, G. Spizzo, M. Spolaore, N. Vianello Consorzio RFX, C.so Stati Uniti 4, Padova, Italy
a r t i c l e
i n f o
Article history: Received 6 June 2016 Revised 2 February 2017 Accepted 6 March 2017 Available online xxx Keywords: RFX RFP PWI THB MPs Stochastic layers Magnetic chaos
a b s t r a c t The edge of the RFX-mod (R = 2 m, a = 0.46 m) Reversed Field Pinch device is characterized by weak magnetic chaos affecting ion and electron diffusion. Edge particle transport is strongly influenced by a toroidal and poloidal asymmetry caused by magnetic islands and an ambipolar radial electric field ensures local neutrality. At higher plasma current (Iφ >1MA) a spontaneous resonant dominant mode m/n = 1/7, slowly rotating, develops in the inner region. The edge electron pressure Pe and floating potential Vf show a shape of the Plasma Wall Interaction (PWI) with the same toroidal periodicity, which follows the edge local ideal magnetic displacement 1,7 . Detailed measurements along the poloidal direction of Pe and Vf have been undertaken newly, their time behavior present respect to 1,7 a time lag, which depends on the poloidal angle θ . The mode analysis in terms of helical angle reveals a role of the 0/7 mode in determining a poloidal phase lag respect to 1,7 of Pe and Vf . Since the 0/7 is the largest toroidal sideband of the 1/7 mode, this work suggests a role of toroidal coupling in determining the plasma response to a MP. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction The study of the three dimensional (3D) kinetic response to edge magnetic stochastic layers is of growing importance in the fusion community. As a matter of fact the need for a 3D, chaotic magnetic field in present and future fusion devices to modify plasma wall interaction, divertor loads, transport and MHD stability in the edge is widely recognized [1]. Nevertheless, the action of magnetic perturbations (MPs) on the edge plasma is far from being well understood, and to date there is no obvious relationship between electron pressure gradient ∇ P and the MP: this is an outstanding issue for ELM control and suppression in ITER, since edgelocalized modes are sensitive to the ∇ P and not directly to the MP. A chaotic edge layer can be spontaneous, as in the reversed-field pinch [2] and the Stellarator [3], or externally applied via MPs as done in both Tokamaks [1] and Stellarators [4]. In this respect, the RFX-mod experiment is suitable for studying the kinetic response of the edge plasma to 3D magnetic fields. In fact, the RFX-mod Reversed Field Pinch (RFP) is characterized by the presence of magnetic modes with different helicity, spontaneously present or externally induced, the amplitude and phase of the MP are well known, and dynamically controlled via a state-of-
∗
Corresponding author. E-mail address: [email protected] (P. Scarin).
the-art active coil feedback system [5]. In particular, operating at high plasma current (Iφ >1MA), the innermost resonant magnetic mode m/n = 1/7 becomes dominant with an almost monochromatic spectrum [6]. This monochromatic spectrum is not peculiar of RFXmod, but it has been observed in all RFP devices, namely MST [7], TPE-RX [8], EXTRAP T2R [9], and, more recently, in the low-aspect ratio RELAX device [10]. In this m/n = 7 helical state, the RFX-mod plasma is shaped as an helix: in particular, the edge plasma shows a 3D stochastic layer which, in terms of connection lengths and typical electron loss times [2], is comparable to the Tokamak edge when MPs are applied. Distributed diagnostics both in toroidal and poloidal directions can measure floating potential, edge electron density and temperature and neutral influxes, thus characterizing the space and time evolution of the edge plasma in great detail. Experimental observations in RFP discharges have shown a clear kinetic response to the spontaneous dominant perturbation and the edge electron pressure Pe , the particle influxes and the floating potential Vf show the same 1/7 modulation [11]. These results are reminiscent of edge modifications induced by MPs in Tokamak as seen in Textor [12], in DIII-D [13] and in Asdex [14]. A detailed study of the poloidal structure of Pe , and Vf has todate never been undertaken in RFX-mod. The present study tackles this issue, and reveals that, even if a strong m = 1 modulation is present in the magnetic topology, the plasma does not follow exactly the helicity of the MP: different poloidal harmonics are present.
http://dx.doi.org/10.1016/j.nme.2017.03.006 2352-1791/© 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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These experimental results could be useful also for other toroidal magnetic confinement devices (RFP, Tokamak, Stellarator) that use MPs on the edge to control the 3D kinetic response of edge layers. In this paper, the edge effects concerning the spontaneous MP will be shown: the results of edge plasma pressure response will be presented in Section 2, while in Section 3 the observation of floating potential response will be discussed; Section 4 will be dedicated to the investigation of measurement results and Section 5 to the summary. 2. Edge plasma pressure response In RFX-mod the spontaneous innermost dominant mode 1/7 is coupled with the Shafranov shift, which is equivalent to a 1/0 mode, and the consequence is the development of a chain of 0/7 islands resonating on the q = 0 surface in the edge region, typically at r/a∼0.95, strongly modifying the edge parallel current map [15]. The resulting magnetic topology shows the m = 0 O-point (OP) aligning toroidally with the m = 1 X-point (XP) in the low-field side (LFS), and the opposite in the high-field side (HFS) [16]. In this magnetic topology the helical angle um,n =mθ -nφ +ϕ m,n (t) is often used as a reference frame [17]; in fact, it puts measurements performed at different poloidal and toroidal angles in the same helical frame of reference, similarly to what is done in stellarators. In the definition of um,n , θ and φ are the poloidal and toroidal angles of the diagnostic, respectively, ϕ m,n (t) is the phase of the slowly rotating mode (f∼40 Hz) and m and n are the poloidal and toroidal mode numbers. In the reference frame of this helical angle, the OP of the island is at um,n =π /2 and the XP at um,n = 3π /2 for each mode number (m,n), by construction. In a previous analysis [17], the measurements in the LFS (θ = 0°) and HFS (θ = 180°) of floating potential, plasma flow, neutral influx and electron pressure (the latter in the LFS only), have shown a clear m = 1 modulation. This has suggested a helical Plasma Wall Interaction (PWI) with a poloidal structure that closely follows u1,7 , and that is characterized by an increase of electron density, neutral influx and more negative Vf in front of the OP of the m = 1 islands, corresponding to u1,7 =π /2. More recently, edge electron pressure Pe measurements with the Thermal Helium Beam diagnostic (THB) [18] in two poloidal locations at θ = 0° and in UP position at θ = 90° (both at toroidal position φ = 262° and r/a = 0.98) have shown a more complicated behavior than the helical u1,7 . To show this, we will take as a proxy of the magnetic helical pattern, the ideal displacement (at r=a) of the dominant mode 1/7, 1,7 (t) [19]. This displacement is approximately 1,7 (t) = 0 sin(u1,7 ) (0 is the displacement amplitude, see Fig. 1 in Ref. [11]). This means that the following three conditions are equivalent: a) 1,7 is maximum; b) u1,7 =π /2; c) the measurement corresponds to the island OP. The cross correlation C(1,7 ,Pe ) between the local displacement 1,7 (t) and Pe (t) (during the plasma flat-top phase), in the two poloidal positions, is shown in Fig. 1 for a typical plasma discharge. In the equatorial plane Pe (t) and 1,7 (t) oscillate almost in phase and, since the correlation is maximum at zero time lag, the electron pressure increases when 1,7 > 0 (or, equivalently, u1,7 ∼π /2); but this does not happen at θ = 90°, where there is a finite time lag τ between the two quantities of about 4 ms, that is about T/6, where T is the period of spontaneous MP (1/T∼40 Hz). This means that Pe (t) does not follow exactly the dominant m = 1 magnetic modulation. To highlight this result the local fluctuations defined as δ Pe /Pe =(Pe −Pe )/Pe , where Pe is the time average during the plasma flat-top phase, have been analysed in terms of the helical angles u1,7 and u0,7. If δ Pe /Pe depended on the helical deformation 1,7 only, it should be δ Pe /Pe ∼ sin(u1,7 ) for all poloidal positions, with no θ -dependence (as it was found in the past on RFX-mod, when only LFS and HFS were diagnosed, see Ref. [11] and [17]). On the contrary Fig. 2a displays (for the two poloidal positions)
Fig. 1. Cross correlation between the edge local ideal magnetic displacement 1,7 , due to the dominant mode and the electron pressure Pe in two poloidal positions (θ = 0°, θ = 90°).
the Pe fluctuations as function of the helical angles u1,7 with a θ dependence. The measurements have been done every 2 ms and the triangles in the figure represent the average over the mode rotation periods for several discharges, with plasma current in the range 1.4÷1.7MA. Fig. 2a shows that at θ = 0° the maximum value of δ Pe /Pe is in front of the 1/7 OP, but this is not true for the θ = 90° where the maximum of Pe fluctuations is between the OP and the XP of the 1/7 mode: thus there is not a unique relation between the maximum of δ Pe /Pe and the 1/7 mode. Conversely using u0,7 as a reference angle (Fig. 2b), the maximum of Pe fluctuations is around the 0/7 XP for both the polidal positions, that would hint an m = 0 helicity in the poloidal region θ = 0°÷90°. This experimental result shows a connection between the dominant mode m = 1 that originates the perturbation and its toroidally coupled sideband m = 0. It has also been observed that the level of Pe fluctuations is lower at θ = 90° (∼10%) with respect to θ = 0° (∼20%) and this means a lower PWI in the upper portion of the helix. 3. Floating potential response A more complete picture of the 3D PWI in RFX is obtained by measuring the floating potential with toroidal and poloidal Langmuir probe arrays embedded in the graphite tiles covering the first wall. The toroidal probe array has 72 pins, and the Vf behavior in terms of helical angle u1,7 has been studied in Ref. [11]. The detailed poloidal structure of the floating potential is instead studied here. To accomplish this, we use the poloidal array of electrostatic probes (5 probes at θ = 32°, 84°, 186°, 238°, 341°) at a fixed toroidal angle φ ∗ = 249° [20]. The cross correlations C(1,7 ,Vf ) between 1,7 (t,θ ∗ ,φ ∗ ) and Vf (t,θ ∗ ,φ ∗ ) (during plasma flat-top phase) for the five poloidal positions θ ∗ is shown in Fig. 3a for a typical discharge: the time lag τ of the minimum cross correlation between the two signals depends on the poloidal position. This result extends that of Section 2, which was limited to two poloidal angles, only (θ = 0° and 90°). Again, not all of the poloidal angles are equivalent, and the kinetic response deviates from a perfect helix along θ . The outcome of the analysis over many shots (about 30) is shown in Fig. 3b, where the relative time lag τ /T of the minimum of the cross correlation is reported as a function of poloidal positions. Though data are characterized by some dispersion, it is possible to observe different average values of the time lag τ /T. The probe at θ = 341° presents an average relative time lag τ /T ∼ −0.1 (the minus sign means that the minimum of Vf is seen after the maximum of the ideal displacement 1,7 ) and at θ = 84° the largest one τ /T∼ −0.25. This means that while the 1/7 island OP rotates poloidally from 341° to 84° in the LFS, the PWI, identified
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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Fig. 2. Relative fluctuations of Pe for two poloidal positions as function of helical angle u1,7 (Fig. 2a) and u0,7 (Fig. 2b). The measurements have been done every 2 ms and the triangles in the figure represent the average over the mode rotation periods relative to several discharges.
Fig. 3. Cross correlation between 1,7 (t) and floating potential Vf (t) at the different poloidal positions of the probes for a typical discharge (Fig. 3a). The relative time lag τ /T of the minimum of the cross correlation are reported in Fig. 3b as function of poloidal positions for many shots (blue points) and their average is shown (red line). The error bar depicts the rms of blue point for each poloidal position. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by Vf , lags behind of −0.25·2π =−π /2 (it means at the 0/7 island XP). Afterwards, when the 1/7 OP has reached the HFS, the PWI jumps in the direction of the 1/7 OP, with a decrease of relative time lag and then the PWI behaviour follows the 1/7 mode until the OP reaches θ = 341° near the LFS. This odd behaviour is in no way accounted for by a simple description in terms of 1,7 as done in the past in RFX-mod. To underline this, δ Vf =Vf -Vf (where Vf is the time average during the plasma flat-top phase) has been analysed as a function of the helical angles u1,7 and u0,7 at each poloidal position. The fluctuation δ Vf of the five probes has been evaluated every 2 ms during several MP rotation periods, in many plasma discharges. The average values of these fluctuations (normalized for each poloidal position) are shown in Fig. 4a and Fig. 4b as a function of the angles u1,7 and u0,7 , respectively. Again, similarly to the above discussion on Fig. 2a, if δ Vf depended on 1,7 only, it should be δ Vf ∼ sin(u1,7 ) for all poloidal positions, without θ -dependence. On the contrary, Fig. 4a shows that between the negative burst of Vf and the island OP at u1,7 =π /2 (maximum 1,7 ) exists phase lag δ u1,7 , which depends on the poloidal position. The lowest value δ u1,7 ∼ π /6 is found near the equatorial plane (θ = 341°) and the highest one δ u1,7 ∼ π /2 at θ = 86°. This means that Vf does not follow the helicity of m = 1and the space delay between Vf and the magnetic shift 1,7 reflects the time lag shown in Fig. 3b. The result of Fig. 4 also justifies earlier results in RFX-mod (Ref. [11] and [17]) which reported a helical pattern of PWI, since those measurements were usually performed on the equatorial plane, where the phase
lag δ u1,7 between the PWI and 1,7 is reduced. A similar analysis has been done as a function of the helical angle u0,7 and Fig. 4b shows the result: the negative bursts of Vf are localized close to u0,7 ∼3π /2 (XP of the 0/7 mode) for the probes in the poloidal region θ = 0°÷90° (similarly to the maximum δ Pe /Pe in Fig. 2b) and at u0,7 ∼3π /4 (π /4 from the OP of 0/7 mode) for the probes near the HFS . In this figure it is highlighted that the PWI, identified by Vf , completes about a full π -phase shift of u0,7 in ∼100 poloidal degrees,with two relevant poloidal zones: in the first one, PWI is determined by the m = 0 island XP (θ ∼0°÷90°); in the second one PWI is dominated by the OP (θ ∼180°÷270°). 4. Investigation of measurement results To deeper investigate the connection between local magnetic topology and the floating potential the ORBIT code [21] has been used. Fig. 5 shows the Poincaré plot obtained with the 1/7 and 0/7 modes only, for a typical high plasma current discharge. In the Fig. 5 the sequence a), b), c) and d) corresponds to the positions LFS, UP, HFS and DOWN, respectively. The plots show a chain of 1/7 islands in the core (purple points), plus a chain of 0/7 islands resonating at the reversal surface q = 0 (which corresponds to the green horizontal line at r = 44 cm). The 0/7 island arises due the physical fact that a pure helix exists in a cylinder, not in a torus. Let us imagine to measure an (m,n) mode at constant φ on the poloidal plane, sin(m θ ): while the m harmonic is pure in a cylinder, it is always a sin(m’ θ ) in a torus, with m’ = m ± k, k = 1,2… The first order correction, k = 1, is of or-
Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006
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Fig. 4. Fluctuations δ Vf for the five polidal probes as function of helical angle u1,7 (Fig. 4a) and u0,7 (Fig. 4b); δ Vf has been evaluated every 2 ms for the MP cycles of many plasma discharges and the average values are given, normalized for each poloidal position.
Fig. 5. The sequence of Poincaré plots in a), b), c) and d) corresponds to the positions LFS, UP, HFS and DOWN, respectively. In these plots has been pointed out the phase lag δ φ between the ideal displacement 1,7 (OP of 1/7 mode) and the outstanding point of effective local deformation of magnetic surfaces in the edge eff (marked in black line) estimated with ORBIT code. Yellow and green lines represent the wall and q = 0 surface, respectively and the blue band the edge stochastic layer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
der ε =a/R: in the RFX case, this means that a 1/7 mode is always coupled to its toroidal sidebands, (m,n) = 0/7 and 2/7. This mechanism is well explained in Ref. [19]. The nonlinear interaction between the neighbouring 1/7 and the 0/7 modes creates a stochastic layer (blue band in Fig. 5) separating the spatial domains pertaining to the two resonances [22]. From Fig. 5, it is possible to define an “effective” local deformation of magnetic surfaces in the edge eff (black line), as the convolution of the outermost points in the Poincaré map, before touching the wall. The behaviour of eff is mostly m = 1 but it is also deformed by the m = 0 islands and this deformation depends on the poloidal angle. The deviation of eff from the ideal displacement 1,7 (which is equivalent to the OP of the m = 1 island) is indicated by the phase lag δ φ between the maximum of 1,7 and that of eff . The phase lag δ φ (drafted in Fig. 5) is negligible at LFS, it is maximum in the UP position (Fig. 5b) and decreasing in the HFS and the DOWN position. The shape and amplitude of eff depends on the topology of the stochastic layer, in particular on its radial displacement relative to the q = 0 resonance (green line in Fig. 5). The details are determined by the competition of magnetic structures closest to the wall: the stochastic layer, or the m = 0 island OP’s. To highlight this point, we calculated the radial position of the last closed orbit of the stochastic layer, which is shown in Fig. 6 together
Fig. 6. Radial position of the last closed orbit belonging to the stochastic layer together with the radial position of the m = 0 island OP, for a given poloidal section.
with the radial position of the m = 0 island OP.. The layer radius is maximum at θ = 0° and minimum at θ = 180°: conversely, the OP’s are closer to the wall at θ = 180° and displaced further inside at θ = 0. Therefore, PWI will be dominated by the stochastic layer
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the connection length Lc on the θ -φ plane, which was shown earlier in the RFX-mod edge (see bottom frame of Fig. 9 in Ref. [6]). 5. Summary
Fig. 7. Cross correlation between 1,7 (φ ) and eff (φ ) at a fixed time as function of polidal positions. The phase lag δ φ presents a poloidal trend similar at that of time lag τ shown in Fig. 3a.
in the poloidal region −60°< θ < 80°, while the OP’s will prevail in the range 80°< θ < 300°, and the behaviour of eff will follow the same rule, accordingly. In order to quantify the difference between the effective local deformation of magnetic surface eff and the ideal displacement 1,7 the cross correlation C(1,7 ,eff ) along φ between the 1,7 (φ ,θ ∗ ,t∗ ) and eff (φ ,θ ∗ ,t∗ ) has been evaluated at a fixed time t∗ for different poloidal positions θ ∗ . The result is shown in Fig. 7: the phase lag δ φ which corresponds to the maximum cross correlations between 1,7 and eff ranges between the positive values of 0.1 rad at θ ∼0° and −0.2 rad at θ ∼90°. Dividing δ φ for the toroidal period 2π /7 of dominant mode one obtains the correspondent period fraction that goes from 0.1 to −0.2. The difference between eff and 1,7 is mainly due to the non-linear interaction between the 1/7 and its toroidal sideband 0/7, since if we evaluate eff with all the modes, m = 0,1 and 1 ≤ n ≤ 24, the result shown in Fig. 7 remains unchanged. The poloidal behavior of the phase lag δ φ between 1,7 and eff , as function of poloidal position, presents a similar trend to that seen for the time lag between 1,7 and Vf (see Fig. 3a): in both cases the delay ranges are comparable (relative values are 0.1÷−0.2 and −0.1÷−0.25 respectively) and the maximum is around θ = 90°. This comparison of the two cross correlations C(1,7 ,eff ) and C(1,7 ,Vf ), despite the different domains, suggests a relationship between the effective local deformation eff and the edge phenomenology of Vf poloidal behaviour. These analyses explain in detail also the peculiar “zig-zag” plot of
The analysis of the edge measurements of Pe and Vf along the poloidal angle in RFX-mod reveals that the PWI does not simply follow the rotation of the spontaneous m = 1 MP, but there is the presence of the m = 0 mode, which has been identified with the analysis in the reference frame of the helical angles u1,7 and u0,7 . A good indicator of the presence of m = 0 mode is the “effective” local deformation of magnetic surfaces in the edge eff , obtained thanks to the simulations of the coupling of the 1/7 and 0/7 modes. The cross correlations C(1,7 ,eff ) and C(1,7 ,Vf ) prompt to a close connection between the effective local deformation of magnetic surface and the edge floating potential. So the eff behaviour can be at the origin of the poloidal phase lag between Vf and the 1/7 island OP. This means that the kinetic plasma response to a monochromatic magnetic perturbation depends on the toroidally coupled sidebands, which are not negligible. This result could be useful when one uses a monochromatic MP for ELM control, especially in elongated plasmas. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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Please cite this article as: P. Scarin et al., Boundary plasma response in RFX-mod to 3D magnetic field perturbations, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.006