Effects of toroidally-distributed-divertor biasing on scrape-off-layer (SOL) current drive, divertor particle flux and fast electron confinement in the QUEST spherical tokamak

A novel divertor biasing by four biasing plates distributed toroidally (TDDB) on the upper divertor target plate is applied to low density tokamak plasmas started-up by 28 GHz 2nd harmonic electron cyclotron current drive (ECCD) in a quasi-double null configuration of the QUEST spherical tokamak (ST). In the ST plasmas of line averaged electron density < ne > ∼0.7–1 × 1018 m−3, about 20%–40% of the current Ibias driven by ∼85 V sawtooth bias voltage reaches the lower divertor plate along the biased scrape-off-layer (SOL) flux tube as the SOL current ISOL . The fact ISOL is noticeably lower than Ibias  indicates an appreciable leakage of parallel current from the biased SOL flux tube. The leakage currents in the toroidal and radial directions are confirmed by detection of them using the unbiased plates in the TDDB experiments. From the ion saturation current density profile obtained by a divertor Langmuir probe array, the fall-off lengths of divertor particle flux are estimated together with the strike line position. Total particle flux to the upper divertor, evaluated by the integrated ion saturation current density profile is reduced by up to 45% during positive biasing of the TDDB, depending on the position of the strike line to the biased plate. In addition, the TDDB also induces a noticeable loss of fast electrons produced by ECCD, leading to an ∼2% reduction in the maximum toroidal current of the ST plasma compared to a shot without the TDDB. Reduction of the divertor particle flux and enhancement of the fast electron losses are thought to be dominantly caused by E ×B drift induced by the TDDB. In the present experimental conditions, the effects of magnetic perturbations produced by the SOL currents on the fast electron losses can be neglected because of a too small SOL current.

A novel divertor biasing by four biasing plates distributed toroidally (TDDB) on the upper divertor target plate is applied to low density tokamak plasmas started-up by 28 GHz 2nd harmonic electron cyclotron current drive (ECCD) in a quasi-double null configuration of the QUEST spherical tokamak (ST). In the ST plasmas of line averaged electron density <n e > ∼0.7-1 × 10 18 m −3 , about 20%-40% of the current I bias driven by ∼85 V sawtooth bias voltage reaches the lower divertor plate along the biased scrape-off-layer (SOL) flux tube as the SOL current I SOL . The fact I SOL is noticeably lower than I bias indicates an appreciable leakage of parallel current from the biased SOL flux tube. The leakage currents in the toroidal and radial directions are confirmed by detection of them using the unbiased plates in the TDDB experiments. From the ion saturation current density profile obtained by a divertor Langmuir probe array, the fall-off lengths of divertor particle flux are estimated together with the strike line position. Total particle flux to the upper divertor, evaluated by the integrated ion saturation current density profile is reduced by up to 45% during positive biasing of the TDDB, depending on the position of the strike line to the biased plate. In addition, the TDDB also induces a noticeable loss of fast electrons produced by ECCD, leading to an ∼2% reduction in the maximum toroidal current of the ST plasma compared to a shot without the TDDB. a Present address: Japan Coast Guard Academy, Kure 737-8512, Japan. b Present address: National Institutes for Quantum Science and Technology, Naka, Japan. * Author to whom any correspondence should be addressed.
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Introduction
In future large fusion devices, a large and steady heat load to the divertor should be controlled to avoid serious damage to the divertor targets [1]. A candidate scenario is detached divertor operation by introducing impurity gases to the divertor region. A main concern is that the detached divertor operation often degrades the pedestal performance of H-mode plasmas. It is challenging to maintain good H-mode confinement over longer time scales, since the detached divertor operation tends to increase upstream density leading to degradation of the H-mode pedestal. Injected impurity for detached divertor operation can potentially increase the impurity level of the core plasma and bring about dilution of fuel ions in long-pulse plasmas. Compatibility of divertor heat load control and high core plasma confinement in detached divertor operation is intensively investigated in large tokamaks [2][3][4][5][6][7]. Some promising detached divertor operations are achieved, but so far sustained for short time scales less than 10 s. As an approach to avoid undesirable confinement degradation in detached divertor operation, new advanced divertor magnetic configurations are proposed theoretically [8][9][10] and tested experimentally [11,12]. A geometrical structure of the divertor dome is also being improved toward long pulse advanced tokamak operation [13]. The other approach is to explore a possibility of enhanced radial particle and heat transport in the scrape-off layer (SOL), of which enhanced transport could broaden the divertor heat load profiles and mitigate severe divertor heat load. This approach might relax the requirements for the detached divertor operation. An interesting alternative approach is an idea of toroidally asymmetric divertor realized by geometrical target structures, surface layer composites, electrical biasing, or controlled gas puffing through slots of the targets [14,15]. Toroidally asymmetric divertor biasing was tested in two tokamaks, and some promising results were reported [16,17]. Apart from the novel divertor biasing experiments, the effects of toroidally symmetric divertor biasing were intensively studied experimentally and theoretically, with particular focus on generation of plasma rotation and the associated electric field in the SOL, the L-H confinement transition and divertor exhaust/pumping of fuel and impurity gases [18][19][20][21][22][23][24][25][26].
Moreover, in future fusion devices transient heat load induced by edge localized modes (ELMs) can be superimposed to the steady divertor heat load [1]. Various ELM control and mitigation methods are intensively studied and developed in many tokamaks [27][28][29][30][31][32][33] and a helical device [34]. The most promising method is to apply resonant magnetic perturbations (RMPs) to H-mode plasmas for a tailoring of the steep pedestal pressure gradient minimizing confinement degradation. Installation of internal and external RMP coils having large area sizes and ampere-turns may not be easy in future fusion devices, ensuring a certain time response. It should be noted that the toroidally and poloidally localized RMP coils can generate appreciable magnitude of magnetic field perturbations in the plasma core region, of which field perturbations might induce appreciable damping of toroidal rotation and locked mode disruptions. An alternative method of RMP generation is to use several parallel currents flowing helically in the SOL just outside the pedestal region [35,36]. The helical current filaments in the SOL can also be produced by a kind of toroidally asymmetric divertor using electrical biasing as mentioned above. In this paper, this biasing method is called toroidally distributed divertor biasing (TDDB). Magnetic field perturbations by SOL currents automatically resonate with the safety factor in the pedestal region. Recently, it is confirmed by numerical simulations that RMPs produced by SOL current localize only near the pedestal edge without enhancing error field perturbations in the plasma core [37]. The TDDB is advantageous for fundamental studies of both particle transport responses to and parallel current drive in the SOL from the applied electrical biasing.
The divertor heat load problem is investigated by evaluating the fall-off length of the heat load profile on the divertor targets and projecting to the width of the SOL. The heat load profile is usually measured by an infra-red camera viewing divertor targets, for instance in [38,39]. The divertor heat load profile is also derived from the product of ion saturation current density and the electron temperature measured by divertor Langmuir probe (DivLP) arrays at the targets, with the assumed sheath heat transmission factor γ s = 7, for instance [40][41][42]. However, the electron temperature measurement by DivLP sometimes gives unreliable results due to various nonideal effects. Broadening of the divertor particle flux profile is often discussed using the ion saturation current density profile, since the measurement is simple and more reliable [43].
Initial TDDB experiments [44] have been conducted for fundamental understanding of SOL plasma behaviors and generation of SOL current in the Q-shu University Experiment with steady-state Spherical Tokamak (QUEST) [45]. In this paper, SOL current drive efficiency by TDDB is evaluated experimentally by the direct measurements. Then, effects of TDDB on divertor particle flux and core plasma behaviors are discussed, based on detailed analysis of divertor particle flux profiles obtained by DivLP. This paper is organized as follows. In section 2, the experimental setup such as the magnetic configuration and divertor biasing system is described. Details of a newly developed nonlinear least-square (NLSQ) method to derive the radial fall-off lengths of divertor fluxes and strike line position is described. Experimental results of TDDB are presented in section 3. Leakage of bias-driven current in the toroidal and radial directions from the SOL biased flux tube is shown, and the SOL current drive efficiency by TDDB is evaluated, based on the experimental data. By the NLSQ method, the radial fall-off lengths and strike line position are simultaneously obtained for a plasma shot with and without TDDB. TDDB effects on divertor particle flux are discussed for biasing scenarios such as the number of biased plates and the biasing polarity. The effects on confinement of fast electrons produced by 28 GHz electron cyclotron current drive (ECCD) are also discussed. In section 4, mechanisms of the divertor particle flux reduction and fast electron losses are discussed and results of the first TDDB experiments are summarized.

Biasing plates and divertor biasing system
Four biasing plates (#302, #304, #306 and #308) are arranged every 90 • toroidally on the upper divertor plate which consists of 16 fan-shape segments, as shown in figure 1(a) [44]. The short and long toroidal widths of the stainless biasing plate having a trapezoidal shape are respectively 0.15 m and 0.19 m. The expansion of the plate in the major radius direction is from R = 0.42 m to R = 0.52 m. One of the biasing plates, #308 is divided into four pieces (A, B, C and D) with a 2 mm gap, to study leakage currents from the biased flux tube (figures 1(a) and (d)). Each biasing plate is electrically insulated from the upper divertor plate by alumina-ceramic insulators, of which material also ensures good heat conduction from the biasing plate to the upper divertor plate. The front surface of the biasing plate is adjusted to the vertical position Z = +0.954 m, which is 4 mm behind the molybdenum (Mo-) guard limiter at Z = +0.950 m, where the equatorial-plane of the torus is at Z = 0 m (figure 1(b) [44]). Note that the biasing plate is away from the shadow of the limiter along the SOL field line. The high-field side mouth and the right and left sides of each biasing plate are covered with alumina insulator plates, to avoid the loss of the bias driven currents to the vacuum vessel via tenuous plasma in the rear side of the biasing plate. Each biasing plate (#302, #304, #306 or #308A) is respectively connected to a bipolar power supply, as shown in figure 1(c). The maximum output voltage and current are 100 V and 5 A, respectively. Various voltage waveforms such as a sawtooth up to 20 kHz generated by a function generator are input to four bipolar power supplies and amplified. In the present experiment, a sawtooth waveform voltage of 85 V and 50 Hz (200 Hz in a few shots) is applied to each biasing plate. Each biasing plate can be biased in-phase or out-of-phase with respect to other biasing plates. The toroidal mode number n of the generated electrostatic potential or SOL current pattern is specified as n = 1, 2 or 4 with n = 0 components by phase adjustment between four bias plates which are separated every 90 • by the grounded diverter plate, as shown in figure 1(a). Note that the poloidal electric field or radial magnetic field produced by the potential or current is dominated by n = 1, 2 or 4 components without the n = 0 one. The MHD equilibrium configuration of a target plasma for TDDB is also shown in figure 1(c).
Throughout the present biasing experiments, the toroidal drift direction of the ions is upward, i.e. the toroidal field is directed counterclockwise. The toroidal plasma current direction is clockwise. Of the bias driven currents, the current reaching the lower diverter plate along the SOL, that is, the SOL current, is directly measured by a toroidal ring electrode placed on the lower divertor plate at Z = −1.09 m for the co-axial helicity injection (CHI) experiments [46], as shown in figure 1(c). The electrode is grounded through a 1 Ω resister for the current detection. A DivLP array consisting of ten sets of triple probes is installed for monitoring electron density, floating potential and electron temperature at various major radii on the upper divertor plate, where the bias voltage is −54 V which is expected to be much higher than the electron temperature there. The standby position of the DivLP is shown in figures 1(a) and (b). In TDDB experiments, it is set at Z = 0.945 m, which is 5 mm inside the Mo limiter. The divertor particle flux profile is obtained as an ion saturation current (or current density) profile. The divertor heat flux profile can be obtained by a product of the ion saturation current density and the electron temperature with the assumed sheath heat transmission factor, for instance γ = 7. However, the thus estimated heat flux profile often exhibits a physically peculiar shape due to appreciable errors caused by some non-ideal effects in the electron temperature measurement by the DivLP. In this paper, we focus on behaviors of the divertor particle flux instead of the heat flux. In the TDDB experiments, a guard limiter installed at the outer mid plane near the vacuum vessel wall is grounded via a 1 Ω resister and used to detect charged particle losses during biasing, as shown in figure 1(c). Note that large in-vessel structures are installed for the hot-wall experiment [47], by which lost charged particles may be captured before reaching the above-mentioned loss detector.

A simple model of TDDB
A simple model of the TDDB in QUEST is shown in figure 2, where the thermoelectric current effect is ignored. The thermoelectric current carried by electrons can flow from the hotter toward the colder SOL ends, i.e. the divertor target plates [48,49]. It is usually limited by ion saturation current at the colder target. In the present TDDB experiments, the electron temperature measured near the divertor plate by the DivLP is about 10 eV or less so that the temperature difference between the biased plate and the CHI electrode will be less than ∼20 eV. Since a bias voltage of ∼85 V is applied, the thermoelectric current can be neglected for the bias driven current. Here, we discuss the SOL current drive by a simplified model of a large-sized Langmuir probe. As mentioned above, the biasing plate has a large area contacting the plasma, compared with a usual small-sized Langmuir probe pin. Accordingly, the biased region can be extended along the SOL flux tube. The length of the biased flux tube along collisionfree SOL, L ∥f will be determined by the balance between particle fluxes parallel and perpendicular to the field line in the SOL depending on the cross field particle diffusion coefficient D ⊥ [50,51], that is, where The parameters d and h are respectively a density decay length in the SOL and the projection length of the toroidal width of the biased plate w tor on the field line. The random particle velocity υ ∥j of the particle species j corresponds to the thermal velocity. The inclination angle of the SOL field line to the plate is θ in , i.e. h = w tor sinθ in . The density decay length of the SOL, d is expressed as where L c is the connection length of the field line, i.e. L c = 2π q SOLRSOL (q SOL : effective safety factor of the SOL,R SOL : averaged major radius of the SOL). The other parameter C s is the sound velocity in the SOL. From equations (1) and (2), the length of the SOL current path, in other words, the length of the biased flux tube is expressed for the particle species j, i.e. electron or ion It should be noted that the length of the electron flow driven by positive biasing L ∥ef is much longer than that of the ion flow driven by negative biasing L ∥if because of υ ∥e ≫ υ ∥i . If L ∥jf is larger than L c , the length of the biased flux tube extends to the grounded divertor plate at the other end of the SOL. The biasdriven current reaches the grounded divertor plate, which corresponds to the SOL current evaluated at the end of the biased flux tube. This case is shown in figure 2(a). On the other hand, if L ∥jf is much shorter than L c , the SOL current driven by the biased plate disappears on the way from the biased plate to the grounded divertor plate, as shown in figure 2(b). If the mean free path of electrons or ions in the SOL λ e or λ i (∼λ e ) is shorter than L ∥jf on the condition of L ∥jf ≫ L c , the current path length in this collisional SOL will be estimated as, Although the above estimation (3) or (4) of the length of the biased flux tube is fairly crude, it is useful to setup the TDDB experiments for the SOL current drive and divertor flux control.

Evaluation of the fall-off lengths in SOL and PFR from a divertor particle flux profile
The shape of the heat flux profile on the divertor targets is often discussed to assess the divertor heat load. The profile shape depends on the heat diffusion in the SOL and private flux region (PFR) in the divertor configuration [38,39]. For the reasons stated in section 1, we focus on the divertor particle flux profile, i.e. the profile of the ion saturation current density at the divertor targets. The divertor particle flux profile j s is expressed as a function of the distance of each divertor pin position from the strike line position R sp along the targets in [52] and is a similar functional form as the heat flux profile given in [38]. That is, where s = R − R sp . R and R sp are the positions of each divertor probe and the strike line, respectively. The fall-off lengths in the SOL and the PFR are respectively expressed as λ SOL and λ PFR . Here, the background signal of j s is ignored. The parameter f x is the poloidal flux expansion rate from the outer mid plane to the target surface and is given from the MHD equilibrium calculation. The erfc(x) is the complementary error function: π´x 0 e −t 2 dt. By fitting the above expression equation (5), with experimentally obtained divertor particle flux profiles λ SOL and other unknown parameters are determined. We introduce a nonlinear least square (NLSQ) method for the fitting. An important point of our fitting method is to include the strike line position R sp also, as an unknown variable X(3) of the NLSQ method in addition to the other three unknown variables, λPFR and X(2) = λ SOL f x . Usually, R sp is given by the EFIT analysis [53] as a known parameter. In our method, the strike line position R sp which is a key quantity in divertor flux control is straightforwardly obtained by the NLSQ profile fitting with j s data alone. Moreover, our method can capture fast time evolutions of R sp induced by transient events such as ELMs and minor collapses. On the other hand, EFIT may not reflect the transient effects on R sp , since it does not directly include the image currents induced in vacuum vessel and plasma facing components (PFCs).
In the NLSQ method, the following objective function F(X) is minimized for four variables, X(0), X(1), X(2) and X(3) using the Levenberg-Marquardt algorithm. The weighting factors w n (s n ) n = 0, 1, …, N−1 are set to be unity for all data analysis of the present TDDB experiments. In this paper, the above-mentioned fitting method is applied to the data of ion saturation current density j s obtained by the DivLP. The signal of the innermost channel locates in the PFR throughout a plasma shot is artificially set to be zero, because the channel is obviously affected by divertor particle flux to the high field side strike line.

Bias-driven current and measurement of the SOL current
Time evolutions of toroidal plasma current and line-integrated electron density of a target plasma of the TDDB experiments are shown in figure 3(a), where the toroidal field is 0.5 T at the major radius R =0.3 m and ∼0.25 T at the plasma center R ∼ 0.59 m. The n e L ∼ 1 ×10 18 m −2 corresponds to the line averaged electron density <n e > ∼0.7 ×10 18 m −3 . The target tokamak plasma is started up by an oblique injection of ∼130 kW, 28 GHz second harmonic electron cyclotron waves (ECWs) to the plasma pre-ionized with ∼30 kW, 8.2 GHz fundamental ECW [54]. The 28 GHz ECW power is dominantly absorbed by fast electrons. Power flow from thus generated fast electrons to bulk electrons is very low because of low electron density and high temperature fast electrons of ∼50 keV [54]. The bulk plasma is mainly heated by ∼30 kW, 8.2 GHz fundamental ECWs. In this shot, the four plates are biased by applying a 50 Hz, 85 V sawtooth voltage. The voltage phase among the plates is set to be the same for measurement of the total current reaching the CHI ring electrode as the SOL current. The toroidal mode number of poloidal electric field generated by the potential perturbations is expected to be dominated with n = 4 component. Time evolutions of total bias-driven current I bias and the SOL current detected by the CHI electrode I CHI are shown in figure 3(b), where the applied bias voltage is shown in figure 3(c). Figure 3(d) shows a cross-sectional image of the tokamak plasma taken at t = 2.7 s by a high-speed TV camera with a 1 kHz framing rate. The image clearly indicates a typical spherical tokamak (ST) plasma started up by 28 GHz 2nd harmonic ECCD. The currents I bias and I CHI are strongly suppressed after t ∼ 2.75 s in the raised density phase by additional gas puff. Due to a large current disruption just after the switch-off of the gas puffing at t = 2.8 s, plasma current is mostly replaced by the runaway current. The temporal evolution is explained by the whole area of the biased plates moving into the PFR due to a noticeable deviation of R sp from the target position. The MHD equilibrium configuration calculated for the plasma at t = 2.7 s is shown in figure 3(e). Major and minor radii of the core plasma are respectively R o = 0.59 m, a = 0.35 m, respectively. The plasma cross-section is characterized by the elongation of κ = 1.1 and triangularity δ = 0.18. The obtained magnetic configuration deviates from a standard double-null configuration, that is, a core plasma is bounded by the inboard guard limiter in a double-null divertor configuration. A core plasma region 'I' is surrounded by the 'inner' SOL (marked 'III') defined by the guard limiter on the center post, and the 'inner' SOL is further surrounded by the 'outer' SOL (marked 'II') defined by the low-field-side magnetic separatrix. This configuration has two X-points, so that the 'outer' SOL has a character of the double-null configuration. Hereafter, we call the 'outer' SOL simply 'SOL' as long as there is no need to distinguish between the two. Depending on the position of the outer and upper divertor leg intersecting with the biased plate, the TDDB can drive currents also in the PFR marked 'IV' of the upper divertor as well as the outer SOL. In the quasi-double null configuration, most of the input power is transported into the inner SOL and only a small portion of input power will flow into the outer SOL, because the distance between regions I and II in the outer mid-plane is about two times of the density decay length in the inner SOL. Although this plasma condition is unique, fundamental studies of the SOL and divertor responses to TDDB are possible using the outer SOL region 'II' linked with the PFR 'IV'.
Here, we try to predict the length of the SOL parallel current or that of the biased flux tube along the SOL, referring to the discussions in section 2.2. From DivLP data, electron temperature and density at the divertor target are typically T et ∼ 10 eV and n et ∼ 9.5 × 10 15 m −3 at the position of the peak particle flux (R ∼ 0.52 m). Here, the ion gyro radius is ∼1.4 mm and is nearly half of the LP pin diameter. The collecting surface area of the LP assumed on high magnetic field approximation is adopted here. As mentioned above, the parallel heat flux to the upper target is very small (q ∥ = γesTejs 0.6 ∼ 4.7 kW m −2 on the assumption of the sheath heat transmission factor γ es = 7). Very low power flux to the divertor plate in the outer SOL is due to two reasons. As mentioned above, one is that a core plasma is surrounded by the inner SOL defined by the inboard guard limiter and most of the power flow from the core plasma goes into the inner SOL. Only a small portion of the power is transported beyond the low-field side separatrix to the outer SOL. The other is that the 2nd harmonic electron cyclotron waves obliquely launched to the low density plasma is dominantly absorbed by fast electrons carrying plasma current. Energy transfer from fast electrons to bulk plasma is too slow due to low density plasma, so that the divertor power flux by bulk plasma loss becomes very low. In this plasma condition, the electron temperature and electron density at the upstream are inferred to be T eu ∼ 10 eV and n eu ∼ 1.9 × 10 16 m −3 from a simple two-point model of the SOL bounded by the divertor target [55]. For the configuration shown in figure 3(e), the connection length is The fall-off length of particle flux in the SOL is estimated to be d = 0.027 m from equation (2). For these parameters, the length of electron SOL current is estimated by equation (3) as L ∥ef ∼ 51 m ≳ L c and L ∥ef < λ e . Here, the inclination angle of field line to the target is θ in ∼ sin −1 (B ⊥ /B t ) ∼ 3.8 • , so that the projection width of the biased plate to the SOL field line is h = w tor sin(θ in ) ∼ 0.012 m with w tor ∼ 0.17 m. Here, B ⊥ is the required vertical field for the MHD equilibrium. Since the length for the ion SOL current L ∥if is much less than L c , the ion current driven by negative biasing will quickly decay along the SOL and not reach the CHI electrode. As seen from figure 3(b), above prediction of the SOL current path length is consistent with finite current detected by the CHI electrode during positive biasing but not during negative biasing.
From the electrical probe characteristics of the biased plate, electron saturation current I es and ion saturation current I is are estimated for a large enough bias voltage of ∼85 V compared with electron temperature T e at the plate, where the electron temperature obtained by the DivLP is ∼10 eV in the high toroidal current phase of more than 40 kA. The ratio of the saturation currents I es /I is ∼ 12 is by a factor of about three smaller than that predicted by the Langmuir probe theory for a proton plasma without a magnetic field. This is due to a finite magnetic field and a small inclination angle of the field line to the biased plate [56]. About 10 times the current of the ion saturation current can be driven by positive biasing in future fusion machine, so that large bias driven current could be obtained to generate the necessary magnitude of the SOL current for edge and divertor plasma control.

Effects of the strike line position on the bias-driven and SOL currents
The NLSQ method mentioned above is successfully applied for best fitting of ion saturation current density j s profiles measured by the DivLP with the model equation (5) . Although the EFIT data is not available so far in the QUEST device, the temporal evolution of the strike line position was successfully obtained from the DivLP data alone by our NLSQ method. As mentioned above, the DivLP can provide important information of the strike line position in studies of heat and particle fluxes to divertor targets, even if the plasma suffers from transient events such as ELMs and minor plasma collapses. The j s -profile data are quite reliable for the determination of the strike line position. In conclusion, the time evolution of the strike line movements induced by various transient plasma events can be obtained with good accuracy.
Obvious correlation of the bias-driven current and the SOL current detected by the CHI electrode with the strike line position R sp is shown in figures 4(a) and (b). In the present TDDB experiments, R sp is not fixed in time due to appreciable deviation of MHD equilibrium control from a target configuration in the latter phase of the ST-tokamak shots. If R sp moves out beyond the outer edge of the biased plate, the TDDB no longer drives parallel current in the (outer) SOL as seen from figure 1(c) or figure 3(c). Moreover, the total current driven by the biased plates is also strongly suppressed. This indicates When the strike line position Rsp moves outward beyond the outer edge of the biased plate, the biasing plate is in very low density PFR and the SOL current is not driven in the outer SOL by the TDDB. When Rsp goes back on the biased plate just before the plasma current termination, the currents I bias and I CHI recover to a noticeable level. that electron density in the PFR must be extremely low due to isolation by inboard and outboard side separatrices forming the upper X-point. When R sp comes back on the biased plates just before the termination of plasma, the currents I bias and I CHI recover to appreciable levels, as seen from figure 4. In conclusion, the strike line position obtained from the j s -profile data by the newly developed NLSQ method is consistent with the time evolutions of I bias and I CHI .

Experimental evaluation of leakage currents from the biased SOL flux tube
In section 2.2, the possibility of SOL current drive by the TDDB is examined by discussing how long the biased flux tube extends along the SOL magnetic field lines relative to the connection length. However, even if the length of the biased flux tube in the SOL is much longer than the connection length, full conversion of the bias driven current to the SOL current is not obvious because electrons and ions carrying the SOL current would be lost to some extent from the biased flux tube. The current detected by the CHI electrode might be reduced appreciably with respect to the bias driven current. Time evolution of the current detected by the unbiased #302 plate is shown in figure 5(a), together with the bias-driven currents by the #304 and #306 plates. An inset in figure 5(a) indicates the configuration of the biased plates #304 and #306 (magenta) and unbiased ones #302 and #308 (light blue). Negative bias-driven current indicates the current flowing out of the positively biased #304 and #306 plates. The leakage current from the biased flux tubes is detected by the #302 plate as positive current. If the leakage rate between two plates away by 90 degrees in the toroidal direction is expressed as ξ φ , the leakage current detected by the #302 plate, I 302 ∼ ξ φ I 304 + 2ξ φ 2 I 306 is approximately expressed. The leakage rate is estimated as ξ φ ∼ 0.06 (∼6%) near the upper divertor region, using the observed data shown in figure 5(a).
The leakage rate from the biased flux tube to the radial direction was investigated by the #308 biasing plate which is divided into two pieces toroidally and two pieces radially as shown in figure 1(a). The leakage rate in the radial direction is evaluated by measuring the current flows into the unbiased #308C plate when the #308A plate is biased. The gap between the #308A and the #308C is 2 mm, and electron and ion gyro radii are ∼0.033 mm and ∼1.4 mm, respectively. Charged particles would not be lost from the biased flux tube without radial transport. The waveforms of these currents are shown in figure 5(b) together with an inset showing the configuration of the #308 plate. From the ratio of the observed currents I 308C and I 308A , the leakage rate is estimated to be ∼7% to ∼14% near the upper divertor region. The leakage rate in the radial direction is comparable or double to that in the toroidal direction. The leakage currents in the toroidal and radial directions will appreciably decrease the SOL current detected at the CHI electrode with respect to the bias-driven current at the upper divertor region. However, it is difficult to quantitatively estimate the conversion rate of the bias-driven current to the SOL current, because the leakage currents will depend on cross field particle diffusion and various particle drifts along the biased flux tube.

SOL current drive efficiency
As mentioned above, the bias-driven currents, I bias , measured at the biased plates gradually decrease along the biased flux tube in the SOL, so that the SOL current would be noticeably smaller than I bias . In this experiment, the SOL current I CHI is directly detected at the lower divertor by using the CHI ring electrode grounded through a low resistance of 1 Ω. Time evolutions of the SOL current drive efficiency η SOL = I CHI /I bias evaluated at the lower end of the biased flux tube are shown in the cases of four plates, three plates and two plates biasing with the same phase among the plates in figures 6(a)-(c), where #302, #304, #306 and #308A in four plates biasing, #302, #304 and #306 in three plates biasing, and #304 and #306 in two plates biasing are activated, respectively. In all cases, when the strike line position R sp is moving from ∼40 cm to ∼50 cm, the efficiency η SOL is in the range of 0.30-0.35. Note that the inner and outer edges of the biased plate are respectively R = 42 cm and R = 52 cm. It should be noted that the efficiency is kept nearly the same even if R sp ∼ 50 cm which is ∼2 cm inside of the outer edge of the biased plate. This suggests that the radial width of the SOL current is thought to be of the order of 2 cm. This is by a factor of ∼3 or 4 times smaller than the SOL fall-off length of the divertor particle flux as seen from the following section 3.5. The SOL current drive efficiency does not depend on toroidal plasma current in the range of ∼40 kA to ∼65 kA. When R sp exceeds the outer edge of the biased plates, the TDDB drives current only in PFR but not in the outer SOL. In a shot where R sp was sustained in the range of ∼42 cm to ∼45 cm, the highest η SOL ∼ 0.40-0.45 was obtained by three plates biasing ( figure 6(d)). In this shot, the biasing frequency is set to be 200 Hz. Highest η SOL does not depend on the bias frequency 50 Hz or 200 Hz. From these results, it is expected to be the larger value of ∼0.65-0.73 at the outer mid-plane. The results suggest that the control of R sp around a certain position on the biased plates is important for getting the highest SOL current drive efficiency by the TDDB. The efficiency is evaluated at the position of the CHI ring electrode as the ratio I CHI / |I bias |, where I bias is the total current driven by the TDDB. Blue and white time intervals indicate positive and negative biasing phases, respectively. (a) In-phase biasing by four plates #302, # 304, #306 and #308A in a shot #42077, (b) in-phase biasing by three plates #302, #304 and #306 in a shot #42088, (c) in-phase biasing by two plates #304 and #306 in a shot #42092. In these three shots, the biasing voltage frequency is 50 Hz. The η SOL in positive biasing phase is typically in the range of 0.30-0.35. Note that the efficiency evaluated in negative biasing phase is not reliable, because of too small I bias and I CHI . (d) Highest η SOL of 0.40-0.45 is observed by three plates biasing in a shot #42022 where Rsp is maintained at Rsp ∼ 42 cm. In this shot, the biasing frequency is set to be 200 Hz. Highest η SOL is not due to the increased frequency from 50 Hz to 200 Hz. The inner and outer edges of the biased plates are indicated by two horizontal single-pointed lines.

Effects of TDDB on the fall-off lengths of particle flux in SOL and PFR
Good agreement between the experimentally obtained j s profile and the profile calculated by the NLSQ method described in section 2.3 is seen from figure 7(a). In figure 7(b), the relative fitting error σ/(j smax ) is shown near the plasma current flat top, where σ = √ F (X) estimated from equation (6) with an equal weight w(n) = 1. The NLSQ fitting is applied every 0.2 ms in this shot. The relative fitting errors are about 15 %, except those increase around 20% in very low-density runaway discharge phase from t ∼ 2.83 s. It is concluded that the NLSQ method is successfully applied to the analysis of the TDDB experimental data.
Time evolutions of the radial fall-off lengths of the particle flux profile in the SOL λ SOL and the PFR λ PFR together with the decay length of the integral flux λ int in two shots with four plates biasing and without biasing are respectively shown in figures 8(a) and (b). In figure 8, the strike line position R sp , the bias-driven and the SOL currents are also shown. Here, the decay length λ int [38] is estimated as where the integral range [s min , s max ] indicates the measurement range of the DivLP and j smax is the peak value of the j sprofile. These three lengths λ SOL , λ PFR and λ int show smaller values during positive biasing than those in the negative biasing phase, as seen from figure 8(a). Note that the negative bias-driven current phase corresponds to the positive biasing phase. However, the TDDB effects are not clear in comparison between two shots with and without the biasing, due to considerable temporal variations. Instead of focusing on these three lengths, we compare the total particle flux to the divertor Γ tot = (λ int f x )j smax in two shots with and without the TDDB. The total particle flux Γ tot corresponds to the particle flux integrated over the DivLP measurement zone in the upper divertor targets per unit length in the toroidal circumference. Time evolutions of Γ tot are shown in figure 9 for four typical shots with four plates (#42077, #42078), three plates (#42088) and two plates (#42092) biasing, compared with a shot without biasing (#42081). As seen from figure 9, Γ tot in shots of four plates and three plates biasing decreases down to ∼65% during the positive biasing phase, while the decrease in a shot with two plates biasing is slightly lower (down to ∼70%). The result shows a similar tendency to the result observed in the asymmetric divertor biasing experiment on MAST [17]. The reduction rate does not depend on the magnitudes of the bias driven current I bias and the SOL current I CHI . Obvious effects of plasma current and electron density of core plasma are also not recognized. The reduction rate of Γ tot is maximized in the course that the strike line position R sp travels from the inner edge of the biased plate to the outer edge, as seen from figure 10. Although the temporal behavior of Γ tot is not fully understood so far, the strike line position  in the divertor magnetic configuration is thought to be one of the key parameters in the QUEST for maximizing favorable effects of 'non-axisymmetric divertor biasing' proposed in [14,15].
Here, we have estimated the effective width of the particle flux at the outer midplane of the SOL from the value of λ SOL f x . The poloidal flux expansion factor f x is estimated to be f x ∼ 4.6 averaged over the range from R ∼ 0.45 m to R ∼ 0.49 m on the target and to be f x ∼ 3.0 in the range from R ∼ 0.45 m to R ∼ 0.59 m, from the MHD equilibrium shown in figure 3(e). If we adopt f x ∼ 4.6, the effective width of the particle flux profile at the outer midplane is estimated to be λ SOL ∼ 17 mm in the I p ∼ 65 kA flattop of shot #42077. In shot #42022 where the current flattop is I p ∼ 55 kA, λ SOL ∼ 26 mm is Figure 9. Effects of the TDDB on the total divertor particle flux per unit length of the toroidal circumference in four plates (#302, #304, #306 and #308A) biasing shots #42077 (a) and #42078 (b), in three plates (#302, #304 and #306) biasing shot #42088 (c) and 2 plates (#304 and #306) biasing shot #42092 (d), of which total particle flux is shown with a red curve, compared with that without the TDDB (#42081) of which one is shown with a blue curve. Note that the area size of #308A plate is about a quarter of the other biasing plates. In the figures, the bias-driven current I bias (green curve) and the SOL current I CHI (black curve) are also shown, where yellow shaded zone corresponds to the positive biasing phase. obtained. For the effective width of the integral particle flux, λ int ∼ 29 mm for the former shot and λ int ∼ 35 mm for the latter shot. These fall-off lengths of the SOL outer midplane are comparable to the density decay length of the SOL d ∼ 27 mm in section 3.1.

Effects of TDDB on fast electron confinement
In the QUEST, two guard limiters are installed in the outer midplane near the inner surface of the vacuum vessel. One of the limiters is used for detection of charged particle losses by connecting it to the ground via a low resistance of 1 Ω. In figure 11(a), time evolution of the guard limiter current I OGL is shown together with the bias-driven current I bias and the SOL current I CHI for two shots with four plates biasing (#42077) and without the biasing (#4281). A clear drop in I OGL accompanying multiple large negative spikes is observed during each positive biasing phase (i.e. negative I bias phase), while I OGL in the shot without biasing evolves monotonically having frequent small negative and positive spikes ( figure 11(b)). The drop in I OGL indicates lost fast electrons flowing into the limiter against the potential barrier of the order of ∼4T e due to the sheath and pre-sheath. Moreover, the fast electron losses lead to noticeably suppressed ramp-up rate of the plasma current I p in contrast to the smooth and fast ramp up in the shot without the TDDB. It should be noted that the observed fast electron losses do not have obvious correlation with the decrease in Γ tot . The decrease in Γ tot attributes to bulk plasma losses induced by potential perturbations of the TDDB, but not fast electron losses. Fast electron losses are also not correlated with the magnitude of the SOL current I CHI or the magnetic perturbations generated by I CHI , because fast electron losses are more significant in the decreasing phase of I CHI . Effects of the TDDB-induced electric field perturbations could play an important role in the observed fast electron losses near the flat top of the plasma current.

Discussions and summary
As shown in figure 9, the total divertor particle flux Γ tot is clearly reduced during the positive biasing phase of the TDDB, compared with that in a shot without the TDDB. This indicates enhanced outward particle flux by the TDDB to the vacuum vessel wall and PFCs, but not to the DivLP measurement region in the upper divertor plate of the QUEST. Fluctuations of ion saturation current and floating potential are dominated in low frequency range less than 20 kHz. The turbulent fluctuations change modestly during positive biasing, so that the TDDB effects on fluctuations are not clear. Instead of plasma turbulence effects in the SOL, the radial drift by the TDDBinduced poloidal electric field E θ along the biased flux tube in the SOL extending from the upper plate to the lower plate can effectively enhance the outward particle flux in the SOL. Note that E θ in one side of the flux tube (i.e. counterclockwise side) contributes to the outward drift, and that in the opposite side (i.e. clockwise side) to the inward drift. Here, the toroidal field and plasma currents are directed counterclockwise and clockwise, respectively. The radial drift induced by poloidal electric field along the biased flux tube, i.e. the TDDB induced convective cell predicted in [14,15], is thought to be a possible candidate mechanism of the obvious reduction of the divertor particle flux. This idea would be supported by the observation of the convective cell structure in the past experiments where the SOL was biased by an inserted electrode [57] and by local divertor biasing [16]. This SOL biasing corresponds to the one plate biasing in the present TDDB experiments where one flux tube in the SOL is biased having a helical structure in the SOL. The outward E θ × B t drift along the SOL biased flux tube can eject the bulk plasma outward to the hot wall panels shown in figure 1(c), vacuum vessel and other PFCs, and lead to the reduced divertor particle flux, as seen in figures 9 and 10. Only one side of each biased flux tube contributes to enhancing the outward particle flux. It is expected that the TDDB effects can be maximized by appropriate modifications of the geometrical configuration of biasing plates, biasing scenarios such as the phasing among biasing plates and control of the strike line position on the biased plate. A reason why a reduction of Γ tot is not observed during the negative biasing phase is thought to be due to the shorter length of the negatively biased flux tube, of which length is determined by ion flow induced negative biasing as discussed in section 2.2. Although convective cell formation in the SOL is thought to be a most likely cause of the Γ tot reduction, effects of radial electric field produced also by the TDDB and its shear on turbulent transport in the SOL are left as a future important research topic.
Moreover, the TDDB also induces the enhanced losses of fast electrons in low density tokamak plasmas started up by an oblique injection of the 2nd harmonic 28 GHz ECWs, where trapped (accompanying toroidal precession motion) and passing fast electrons are generated. Deeply trapped fast electrons which exist near the core plasma edge and their orbits extending slightly beyond the magnetic separatrix to the biased SOL flux tube would experience a secular radial drift by the TDDB induced poloidal electric field E θ along the biased flux tube and be lost as shown in figure 11. Counter-moving passing fast electrons which carry toroidal plasma current and their orbits are outward-shifted appreciably for the magnetic surface could also experience a secular radial drift by the TDDB induced E θ and be lost. A part of these lost passing fast electrons may also be detected by the outer guard limiter, as shown in figure 11. It should be noted that several numbers of the SOL current filaments wrapping around the outer SOL helically can generate magnetic perturbations resonating with the safety factor in the core plasma edge, i.e. RMPs, and affect the fast electron confinement. In the present experiments, however, very low magnitude RMPs are predicted to be b/B t ≲ 1 ×10 −5 at 2 cm inside the separatrix. The experimental data indicates that no correlation between the fast electron losses and the observed SOL current is recognized, that is, the RMP effects by SOL currents are negligibly small. Study of RMP effects by the SOL currents in higher density plasma is interesting as future research, because the bias-driven current and the SOL current would be increased considerably. Detailed orbit analyses of fast electrons in ST plasmas produced in the QUEST are needed for clear understanding of the loss mechanisms and development of a new control knob of fast electron confinement.
In a novel divertor biasing, 'toroidally distributed divertor biasing (TDDB)' experiments in the QUEST ST, important and interesting results are obtained as follows.
(1) The SOL current is directly measured by the CHI ring electrode which is uniquely installed on the QUEST ST. Four biasing plates are activated in phase among these plates. A sawtooth waveform of 85 V amplitude and 50 Hz-1 kHz frequency is applied as a biasing voltage. For all biasing scenarios using four plates, three plates and two plates, the SOL current drive efficiency evaluated at the end of the biased flux tube is obtained to be 0.3-0.35 in the positive biasing phase. So far, the maximum drive efficiency of ∼0.4 to ∼0.45 was achieved in the case that the strike line position was maintained around a favorable position on the biased plates. The efficiency at the outer mid-plane of SOL is expected to be ∼0.65-0.75. The information of the SOL current drive efficiency is crucial for a feasibility study of RMP effects induced by the SOL currents. (2) The integral decay length λ int and the peak amplitude of the particle flux j smax are noticeably reduced during positive biasing of the TDDB with 4 and 3 biased plates. This indicates that the total particle flux Γ tot = λ int j smax is reduced clearly by about 35% during positive biasing. The reduced flux is thought to be transported to the outboard side of the vacuum vessel and various PFCs. The radial drift induced by a poloidal electric field along the biased flux tube, i.e. the TDDB induced convective cell, is thought to be a possible candidate mechanism of the obvious reduction of the divertor particle flux. In the present TDDB experiments, the phasing among biased plates was set to be in-phase. Appropriate phasing among the biased plates is expected to maximize the TDDB effects on the present geometrical configuration in the QUEST. A careful control of the strike line position on the biased plates is also thought to be a key factor to maximize the TDDB effects. (3) Enhanced losses of fast electrons generated by ECCD are observed near the plasma current flat-top during the positive biasing. Fast electron losses lead to a slight decrease in the toroidal plasma current. The fast electron losses have no correspondence with the enhanced total divertor flux Γ tot and the SOL current magnitude. Deeply trapped and counter-moving passing fast electrons generated by the 2nd harmonic ECCD could be lost due to the E θ × B t drift by the TDDB along the biased flux tubes.