Scrape-off layer transport and filament characteristics in high-density tokamak regimes

AUG is a medium-sized divertor tokamak with major and minor radii of R  =  1.65 m and a  =  0.5 m, respectively, and equipped with a fully tungsten-coated wall [18]. The experimental investigation was carried out in the lower single null configuration, with the ion $\mathbf{B}\times\nabla B$ drift pointing towards the X-point. The plasma shape was tailored to follow the shape of the outer limiter (in a configuration dubbed the edge optimized configuration), and at the midplane up to 45 mm from the separatrix the field lines are connected to the divertor target. In figure 1(a) the layout of the principal diagnostics used for the present analysis is shown in the poloidal cross-section. The SOL density profiles are obtained from the lithium beam (LiB) diagnostic, observing the plasma at a vertical position approximately 30 cm above the midplane. Density profiles are obtained from the light emission profile, which is sampled with 200 kHz [19], and evaluated within the probabilistic Bayesian framework [20] with 1 kHz in the radial region spanning approximately $0.9\lesssim \rho \lesssim 1.05$, $\rho$ being the normalized poloidal flux. In the core and at the edge, line densities are obtained with the standard interferometer system shown in figure 1 as well. The principal diagnostic used for investigating the fluctuations is the midplane manipulator (MEM) equipped with a tungsten-coated carbon probe head designed to characterize turbulence and simultaneously withstand the high heat flux observed in H-mode. The probe features 16 pins distributed between three terraces at different radial positions (radial separation is 4 mm), whereas the different arrays are aligned in the bi-normal direction as described in [21] with typical distances of 6 mm. The pins are arranged to measure both the floating potential and ion saturation current. In particular the ion saturation current pins are distributed in order to have the measurements spaced both in the radial and in the poloidal direction, allowing a proper estimate of the radial and bi-normal velocities using the 2D cross-correlation algorithm described in the appendix of [21]. Several plunges can be done within a single pulse. For the L-mode shots presented hereafter, up to five plunges were performed with the probe sitting in a fixed position for up to 130 ms. For H-mode operation the time duration of the fixed position was reduced to 70 ms to limit the heat load deposition on the probe head. The acquisition sampling rate for all the shots was set to 2 MHz, which gives us a relatively long time series for turbulence analysis. In addition, one pin is run in swept mode in order to infer the local estimate of density and electron temperature. The information on divertor conditions is primarily obtained by two arrays of fixed flush mounted triple Langmuir probes, shown in figure 1 by red circles, sampled at low sampling rate (33 kHz).

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The Tokamak à configuration variable (TCV) is a medium-sized tokamak located at the Swiss Plasma Center, Switzerland [22]. It is a conventional aspect ratio tokamak (major and minor radii R  =  0.88 m and a  =  0.25 m, respectively) with a highly elongated vacuum vessel and a completely open divertor. TCV features 16 independently powered poloidal field coils, resulting in unique shaping capabilities, with the ability to accommodate highly elongated plasmas (up to 2.8) and triangularity in the range $-0.7\leqslant \delta \leqslant 1$. The near-complete coverage of TCV surfaces with graphite tiles allows extreme flexibility in the power load deposition, making TCV an ideal test-bed for the study of different magnetic configurations and divertor geometries.

In recent years great effort has been devoted to increasing the amount of diagnostic equipment in TCV for divertor studies. A new 32-chord divertor spectroscopy system [23] has been installed for extracting information on recombination and electron temperature from the balmer series spectra. This information can be combined with radiation measurements estimated from bolometry [24]. An array of wall-mounted Langmuir probes (LPs) [25] covers the inner and outer wall as well as the floor. The cylindrical tips have diameters of 4 mm and are embedded into the tiles, except at the floor where they have a dome-shaped head protruding from the tile shadow by 1 mm. For the present experiment the probes have been operated in swept mode to obtain the density and temperature profiles at the wall. The set of LPs has been extended by the installation of a fast reciprocating probe (RCP) [26]. The probe head, described in detail elsewhere [27], is equipped with ten graphite electrodes 1.5 mm in diameter. The electrodes are arranged in such a way to provide a double probe for the density and temperature profiles, two Mach probes for parallel flow investigations, a pin for fast ion saturation current measurements, with the remaining probes collecting the floating potential in order to infer radial and poloidal electric field fluctuations from the local floating potential gradient. The radial separation between the floating potential pins is 1.57 mm, whereas the poloidal separation between the pins is 4 mm and 10 mm respectively. The fast movement of the probe head is 20 cm which is reached within 90 ms with a maximum speed of 2.2 m s⁻¹. The ion saturation current and floating potential electronics have a bandwidth from 0.1 to 10 MHz with anti-aliasing filters at the Nyquist frequency. The acquisition frequency was set between 1–5 MHz for the different discharges used throughout the paper. The profiles in the SOL were obtained combining the data from RCP with that from the Thomson scattering diagnostic obtained in adjacent time instants. For TCV the profiles shown throughout the paper have been fitted using a Gaussian process regression technique (details can be found in [28] together with the link to the available software tool). The method allows for a proper determination of the fit and corresponding errors, as well as of the density gradient with the corresponding errors. This will be used throughout the paper to compute the e-folding length profile shown in the following figures. The line of sight (LOS) of bolometry, the location of the Langmuir probes and of the reciprocating manipulator used throughout the paper, and the LOS of the vertical far infrared interferometer for the edge and central chord, are shown in figure 2 to provide an idea of the spatial resolution of the diagnostic setup. The combined information provides a comprehensive set of measurements suited for the divertor investigation.

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Figure 3 reports the main plasma parameters concerning the current scan at a constant toroidal field both for AUG and TCV. Three levels of plasma current for the two devices as well as the edge line integrated density and the fuelling levels used for the discharges are shown. The parallel connection lengths $L_{\parallel}$, shown in figures 3(a) and (g), are the connection lengths from the outer target up to the X-point height, and clearly increase with decreasing current for both devices. The TCV discharges are purely ohmic plasmas, whereas in AUG an additional 0.5 MW of neutral beam injection (NBI) heating was added in order to keep a similar heating power through the separatrix during the scans. Still in figures 3(e) and (m), the power crossing the separatrix, estimated as $P_{\rm sep} = P_{\Omega} + P_{\rm heat} - P_{\rm rad, core}$, is reported for the various current levels for both devices. Clearly the additional heating allows us to keep much more comparable power levels for AUG. The divertor pressure on TCV, as seen from figure 3(n) measured by baratrons, does not exhibit differences between the various current levels. In AUG, where fast gauges located closer to the vessels are used, slightly higher pressure is achieved at a higher current (see figure 3(f)) [21].

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The first relevant information can be derived by considering the response of the divertor to different density levels at different current. In figure 4 the total integrated ion fluxes to the outer divertor are shown as a function of edge density and the edge Greenwald fraction (i.e. the edge density, obtained from interferometric edge chord, normalized to the Greenwald density value $n_e^e/n_G$) for AUG (a) and (b) and TCV (e) and (f ), respectively. The choice of normalization (edge density normalized to the Greenwald density) is done in order to consider possible effects due to the different density peaking obtained at a different edge safety factor [29]. As already observed, the integrated ion flux in TCV increases almost linearly with the density up to a threshold followed by a smooth rollover [30], the latter is assumed in the following as a proxy for plasma detachment. Unlike in AUG the increase of ion flux is faster than linear up to the threshold with a more pronounced rollover. These differences are likely due to the different degree of divertor closure [31]. In both devices we clearly see that ion flux rollover occurs early in the edge density at lower current. In AUG, where the power crossing the separatrix is a factor of two higher, the behaviour at different currents is reconciled in terms of the edge Greenwald fraction. The same observation holds for the two lower currents in TCV, whereas a higher edge Greenwald fraction is needed for the higher plasma current. In the scan shown in figure 3, where the density increase was kept similar at different values of current, indeed no detachment of the outer leg has been achieved for the higher current case. Conversely whenever the density is increased even further, as in the case of shot # 52065, reported as well in figures 4(e) and (f), clear outer leg detachment has been obtained even at a higher density. In the same figure 4 the behaviour of the inner divertor legs for both devices is shown. In AUG the integrated ion flux is much lower even though the behaviour in terms of edge density resembles what is observed in the outer divertor. In TCV the integrated ion flux to the inner divertor is larger and exhibits a less pronounced rollover at the lower current levels explored, whereas no sign of rollover is observed at a higher current. It is worth noting that unlike AUG, the inner target generally detaches later than the outer one in TCV [30, 32], and this has been observed in K-STAR as well [33]. It is worth noting that both K-STAR and TCV are carbon machines, with the inner strike point on a vertical target and a short poloidal distance to the X-point and an outer strike point on an horizontal plate with a much longer poloidal distance from the X-point. In both machines the presence of carbon impurities was able to increase the pressure and power dissipation with respect to the pure D case. In particular, carbon radiation is a strong power loss channel, but an additional indirect effect in the pressure balance may happen: the carbon-induced cooling and consequent reduction of the electron temperature can indirectly increase the momentum loss and increase the volumetric reaction rates, which are responsible themselves for momentum loss. Work is in progress within a 2D fluid SOL modelling effort for TCV to properly account for these contributions in similar plasmas [34]. In any case a close comparison of the detachment operational space between the two devices is beyond the scope of the present paper and will be eventually addressed in further investigations.

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A similar current scan has been performed varying the toroidal field together with the current in order to keep q₉₅ constant. The corresponding main plasma parameters are shown in figure 5 for both devices. From (a) and (g) we noticed that $L_{\parallel}$ was kept constant throughout the current scan, even though slight variations are observed in TCV at the lower current level. For the sake of completeness we need to underline that at the lower current level we had to operate TCV at an unusually low toroidal field ($B_t\approx 0.8$ T). The time evolution of the aforementioned $P_{\rm SOL}$ and of the divertor neutral pressure are shown in figure 5 for both devices. Again, the heating scheme used (pure ohmic discharges for TCV and an additional 0.5 MW of NBI heating in AUG) allows us to keep the $P_{\rm SOL}$ approximately constant in AUG. The pressure in the divertor is constant throughout the scan in TCV whereas a slightly higher pressure is achieved at a higher current in AUG. The evolution of total ion flux at the outer and inner divertor as a function of edge density and edge Greenwald fraction are shown in figure 6. For AUG, in analogy to the observations obtained during the scan at constant toroidal field, the ion flux rollover is observed at lower density for lower current but the behaviour is reconciled in terms of the edge Greenwald fraction for both the inner and outer divertor. On the other hand, the comparison of shots at a similar current but different toroidal field (e.g. # 34104 and 34106) reveals that the rollover density threshold is essentially unmodified by the variation of $L_{\parallel}$, but a more robust and faster reduction of ion flux just after the rollover is obtained at larger $L_{\parallel}$. In TCV, instead no sign of detachment was observed either at the inner or at the outer divertor, even though the achieved density was sufficient to guarantee plasma detachment whenever the same density ramp was run at a higher toroidal field (compare figure 6(e) with figure 4(e)). This is an interesting observation which does not seem to be due to a change in the power crossing the separatrix since the values of $P_{\rm sep}$ at the same current are comparable at least around the time where higher field discharge exhibits ion flux rollover, as can be inferred by comparing figures 3 and 5. Understanding the differences at lower toroidal fields is presently under investigation.

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Figures 7(i) and (ii) report on the evolution of the upstream profiles for different current levels at the same value of edge density and toroidal field for AUG and TCV respectively. In the same figures we have also included the density profiles at the outer target ((c) and (d) in figures 7(i) and (ii)) as well as the profiles of the normalized collisionality $\Lambda$ defined as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Lambda = \frac{L_{\parallel}\nu_{ei}}{c_s}\frac{\Omega_i}{\Omega_{e}}. \label{eq:lambda} \nonumber \end{align} \tag{ 1 }$$

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This quantity represents an effective collisionality, or equivalently the ratio between the parallel transit time divided by the inverse of the electron–ion collision frequency. This was originally introduced in [35] and adopted in [7] as a parameter to identify enhanced filamentary transport transition in high-density regimes. In particular in [7] it has been suggested that filamentary transport enhancement is regulated by the effective collisionality in the divertor, $\Lambda_{\rm div}$, which depends on the parallel connection length, evaluated as the length from the X-point height to the outer target, and on the values of density and temperature in the divertor region, estimated in the present paper from the target density and temperature given by Langmuir probes at the outer target. In the following, all data will be presented as a function of the divertor collisionality $\Lambda_{\rm div}$. The error bars on $\Lambda_{\rm div}$ shown throughout the paper have been estimated propagating the errors on the target density and temperature. It can be easily observed that flatter profiles are obtained at a lower current for the same density levels for both devices even though some differences may be noted. In AUG the different upstream profiles are associated with different target density profiles and clearly different divertor collisionality. All the cases where the profiles in AUG are flatter at the outer midplane are associated with higher values of divertor collisionality. In contrast, the outer divertor collisionality does not represent a proper metric for TCV since very different upstream profiles are obtained with comparable $\Lambda_{\rm div}$ profiles, even well above the threshold of $\Lambda_{\rm div}\gtrapprox 1$ identified in [7] for increased filamentary transport regimes. This confirms previous observations [9] where a variation of divertor collisionality obtained through the modification of target flux expansion shows little impact on the evolution of upstream profiles. On the other hand we have already observed that, apart from the TCV case at higher current, the edge Greenwald fraction can reconcile the target profile evolution. Figures 8(i) and (ii) compare again the upstream and target profiles, as well as divertor collisionality at a comparable edge Greenwald fraction, but different current levels for both the devices. We clearly recognize that the upstream profile evolution behaviour at a different current is better reconciled in terms of the edge Greenwald fraction, with a similar shoulder developed at comparable $n^e_e/$n_G.

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The investigation of the target evolution with the edge density in the current scan performed at constant q₉₅, shown in figure 6, has already revealed differences between AUG and TCV, in particular showing that a reduction of the toroidal field prevents the plasma from reaching a detached condition in TCV at a lower current. This also reflects the different upstream behaviour, as shown in figures 9(i) and (ii) respectively for AUG and TCV. Indeed while for AUG the behaviour at constant q₉₅ reflects what was observed at a constant toroidal field, with a substantial overlap of target and upstream profiles at equivalent values of edge Greenwald fraction, the behaviour of TCV is scattered and a clear shoulder only develops at high current, where the broader target profile is also recognized. We can also confirm that the evolution of the upstream profile is independent of the evolution of $\Lambda_{\rm div}$ as computed at the outer divertor. Investigation is in progress in TCV in order to eventually understand the dependence of shoulder formation on the divertor collisionality computed at the inner divertor, $\Lambda_{\rm div}^{\rm inn}$. Indeed, the inner divertor, with its shorter connection length $L_{\parallel}$ with respect to that of the outer divertor, was able to exhibit a greatly different value of $\Lambda_{\rm div}^{\rm inn}$, with lower values at similar edge densities.

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In order to properly understand the relation between the upstream and downstream condition we have monitored the evolution of the upstream profile with respect to the detachment condition. This can be done within a single shot in AUG, due to the availability of time-resolved SOL density profiles through LiB, whereas for TCV we had to rely on the repetition of similar shots, namely the 330 kA L-mode plasma with the same density ramp, where the reciprocating probe head has been plunged at different times. The result of this analysis is shown in figure 10 where upstream profiles, normalized to the value at the separatrix are shown for different levels of detachment. The obtained profiles suggest that the modification of the upstream profiles for both of the devices begins close to the ion rollover and further develops as soon as the ion-flux starts decreasing. For completeness it should be pointed out that the chosen scenario for shot repetition in TCV at 330 kA is the one which requires a higher edge Greenwald fraction to exhibit ion-flux rollover. A more refined and detailed analysis of the relation between the ion-flux rollover and upstream profile modification in TCV, including cases where detachment is achieved by pure N₂ seeding, will be addressed in a forthcoming paper [36]. As far as TCV is concerned, from the present dataset we can confirm that if detachment is achieved by pure fuelling and intrinsic impurity radiation, the upstream profile variation is only observed close to and after ion-flux rollover.

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Even though the increase of filamentary convective transport has been recognized since the beginning [4] to play a fundamental role in the process of shoulder formation, the role of other mechanisms is presently under consideration. Among them, the influence of neutrals in the divertor region, which has been theoretically proposed [37] and experimentally suggested in [38], is the subject of intense study. On the other hand, the role of neutrals in the main chamber, which can modify the ionization rate in the outer midplane (OMP), has also been proposed as a possible candidate in the process of shoulder formation [15], even though it is still being debated [8].

This motivated activity to determine the emission of neutral deuterium in the divertor region in AUG, based on the evaluation of signals collected by two cameras with two different filters for $D_{\alpha}$ (656 nm) and $D_{\gamma}$ (434 nm) lines, respectively. The two absolutely calibrated cameras [39] are both located just below the midplane (Z  =  −0.27) at slightly different toroidal angles (the centre of the two cameras is shifted by 6 toroidal degrees and they both observe the same divertor region). A tomographic algorithm has been developed in order to infer the 2D map of the emissivity $\newcommand{\e}{{\rm e}} \epsilon$. Assuming toroidal symmetry, the emission is a function of the radial and vertical positions only, i.e. $\newcommand{\e}{{\rm e}} \epsilon = \epsilon(R,Z)$. This allows us to to project each line of sight (LoS) corresponding to each camera pixel in the plane $(R,Z)$ [40]. In this plane the different LoSs intersect each other, allowing for the development of a proper tomographic reconstruction. In order to introduce as few a priori constraints as possible, the pixel method is used, and the inversion is performed with an iterative algorithm. Two assumptions are made: $D_\alpha$ and $D_\gamma$ emissions do not depend on the toroidal angle, and they are restricted to the region outside the separatrix. Thus the divertor region outside the separatrix is divided into rectangular pixels and the emissivity $\newcommand{\e}{{\rm e}} \epsilon$ in each pixel is considered constant. Using this method, the link between the intensity measured by each LoS of the camera and the emissivity is a linear system of equations, that can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mathbf{I} = \mathbf{A} \cdot \mathbf{\epsilon} \label{eq:eq1} \nonumber \end{align} \tag{ 2 }$$

where I_j is the line-integrated signal measured by the LoS j ; $\newcommand{\e}{{\rm e}} \epsilon_i$ is the unknown emissivity of the pixel i; and the matrix element a_ij is the length of the LoS j  inside the pixel i. This matrix is evaluated only once, and it depends on the geometry of the LoS and the pixel division. In equation (2) there are $\approx 40000$ LoSs and $\approx 200$ pixels, so the system is overdetermined. To invert it and obtain the emissivity of each pixel, the simultaneous algebraic reconstruction technique is used [41, 42]. This is an iterative technique which solves the linear system of equation (2) via an iterative error-correcting procedure, which can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \epsilon_i^{k+1} = \epsilon_i^{k} + \frac{\Sigma_j \left[ a_{ij} \frac{\mathbf{I}_j-\mathbf{a}_j \cdot \mathbf{\epsilon}^{(k)}}{\Sigma a_{ij}} \right]}{\Sigma_j a_{ij}} \label{eq:sart} \nonumber \end{align} \tag{ 3 }$$

where $\newcommand{\e}{{\rm e}} \epsilon_i^{k}$ is the emissivity of pixel i after k iterations. The initial estimate $\newcommand{\e}{{\rm e}} \epsilon_i^{0}$ is set to 0. Since the emissivity of each pixel is positive, this constraint is enforced in each iteration by setting to zero the coefficients that are less than zero after an iteration step. The convergence is quite rapid, and after about 20 iterations it is reached.

The results of this analysis are shown in figure 11 where the inversions for both $D_{\alpha}$ and $D_{\gamma}$ are shown at three different values of the edge-normalized Greenwald density. At the beginning, emission is strongly localized in the inner divertor, consistent with the presence of a high field side high-density region [43, 44]. During the fuelling ramp the divertor moves into a high recycling regime and $D_{\alpha}$ and $D_{\gamma}$ radiation move towards the low field side (LFS) region, initially in the private flux region and then in the main SOL, moving upstream once the target density rolls over. This strongly resembles the observation in JET in the horizontal target [38]. In this configuration indeed the JET upstream profile develops a clear shoulder whenever fuelling is raised, and this is accompanied by a clear $D_{\alpha}$ radiation front moving into the main LFS SOL. On the other hand whenever it is run in the vertical target configuration no shoulder is observed in JET at the same level of fuelling, and $D_{\alpha}$ radiation was confined to a narrow region along the divertor leg. This observation seems to reconcile the behaviour of JET and ASDEX Upgrade despite the different divertor configurations.

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Effort is ongoing to use the 2D map of the emissivity of the two Balmer lines in the divertor to infer the neutral density [45].

The role of enhanced convective filamentary transport in the formation of the SOL density shoulder has already been suggested [3, 4, 7, 9], even though reduced parallel losses can also influence the process. The relation between the profile evolution and blob sizes has been investigated in the present scan using properly designed probes. In both the devices the blob size is determined as $\delta_b = \frac{\tau_b}{2}v_{\perp}$ with $\tau_b$ estimated as the FWHM of the conditionally averaged ion saturation current signal, where peaks 2.5 standard deviation $\sigma$ higher than the average values have been detected. Due to the different experimental setup of the probe head, the estimate of the perpendicular velocity is done differently in the two devices. For TCV we have used the method extensively described in [27] and the perpendicular velocity is estimated as $v_{\perp} = \sqrt{v_{r, \mathbf{E}\times\mathbf{B}}^2 + v_{\rm pol, cross-corr}^2}$. $v_{r, \mathbf{E}\times\mathbf{B}}$ has been estimated from the conditionally averaged sample (CAS) $E_{\theta}$ using as a condition the detection of the peaks on the ion saturation current. The fluctuating electric field is computed from the local floating potential gradients. The poloidal velocity component is estimated from the 2D cross-correlation of poloidally and radially spaced floating potential measurements as detailed in [27]. For AUG the estimate of $v_{\perp}$ is done following the method described in the appendix of [21], based on the 2D cross-correlation using a conditionally sampled ion saturation current structure measured by pins spaced both in the radial and poloidal direction. For a better comparison with previous results we underline that $\tau_b$ is estimated differently with respect to [27], since in the present manuscript the asymmetric shape of the ion saturation current CAS is kept, thus considering the trailing wake neglected in the aforementioned paper [27]. Furthermore in previous AUG papers [7, 15, 21] $\tau_b$ was approximated by the auto-correlation time, and all the data shown were actually the blob diameter, rather than the blob radii, which will be shown in the present paper. In AUG the filaments are detected during the fixed positions of the probe. This implies a long signal length (up to 130 ms at 2 MHz for the L-mode cases) where several hundreds of filaments are detected and used to infer the CAS waveforms. In TCV conversely the probe is continuously moving: the peaks on $I_{\mathrm{sat}}$ exceeding the chosen threshold are sampled from a 3 ms window, as done in [27, 46], after removing the slow time evolution caused by the motion of the reciprocating probe. Several windows are sampled for each plunge at different radial insertions of the probe. Typical examples of the results obtained from the CAS technique for the two devices are shown in figure 12. The yellow bar indicates the estimate of the FWHM: for TCV we have also shown the typical waveform of the poloidal electric field fluctuations from the CAS. The data presented refers to a shot at low density: the shape of the ion saturation current, presented in units normalized to the rms value $\sigma$, is asymmetric for both devices, as already seen [6, 47–49], even though the total pulse length is different between the two machines. The pattern observed for AUG on the pins displaced in the radial and poloidal directions clearly recognizes the intermittent structure whereas from the time-delay a proper determination of the movement in the two directions can be inferred. The observed fluctuation of the poloidal electric field for TCV is of the order of 1.5–1.8 kV m⁻¹ giving a radial velocity of the order of 1.5 km s⁻¹ in agreement with the previous estimate [27].

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As mentioned, the recent results ascribe the broadening of the SOL density profile to a transition in the filamentary regime from a connected regime to a disconnected or inertial one [7, 15, 50]. A threshold has been identified with the transition to inertial regimes occurring for $\Lambda_{\rm div} \gtrsim 1$. The experimental results supporting this hypothesis have clearly been obtained in AUG and in JET, though for the latter device only when run in horizontal target configuration [7, 38]. However, the dependence on divertor collisionality is not universally recognized: for example $\Lambda_{\rm div}$ fails to describe the operation of JET in the vertical target configuration [38] as well as in TCV, where very similar profiles are obtained with largely different outer target $\Lambda_{\rm div}$ values, even though, as mentioned, the evaluation of the influence of the inner divertor $\Lambda_{\rm div}$ influence is still under investigation.

In this respect we have investigated the evolution of the e-folding length of the SOL density profile, defined as $\lambda_n = \left(\frac{|\nabla n_e|}{n_e}\right){}^{-1}$ as a function of the previously defined divertor collisionality $\Lambda_{\rm div}$ and as a function of the blob size $\delta_b$. As previously mentioned, the e-folding length is not computed from the local exponential fit of the profiles, but rather by estimating the local density gradient as a result of the fitting procedure which provides an estimate of the error on the gradient as well, and is used then to propagate the error on the e-folding length estimate. The e-folding length is computed at the same radial location as the estimate of the blob size, namely for $\rho\approx 1.02{{\rm \mbox{--}}}1.03$ in AUG and $1.025\lesssim \rho \lesssim 1.045$ in TCV. Blob dimensions are normalized to the local ion–sound gyroradius $\rho_s = \sqrt{\frac{T_e+\gamma T_i}{m_i}}$ where the assumption T_i  =  T_e has been assumed for TCV. In AUG instead, where detailed investigation of the behaviour of ion temperature in the SOL at different levels of density has been performed [10], we have assumed T_i  =  3T_e for $\Lambda_{\rm divs} \lesssim 1$ and T_i  =  T_e for larger collisionality. In order to get rid of possible uncertainties due to the scattered points, in TCV we have used the evaluation of the profile from the Gaussian-process regression fit shown in the previous figures (see figures 7–9), which also provides an evaluation of the error on the radial derivative of the fit. The results of this analysis are shown in figure 13 respectively. The original trend observed in AUG [7] is confirmed in both series of discharges with a clear increase of the e-folding length as soon as $\Lambda_{\rm div}$ increases and $\lambda_n$ strongly depending on the blob size. It is worth noting that within the different current levels no distinction is clearly observed. This is in agreement with the observation obtained from LiB diagnostics reported in [51], where no sensible dependence of blob size on the toroidal field strength was observed either scaling at constant I_p or constant q₉₅. The corresponding blob values for the comparable Greenwald fraction are compatible with the observation obtained using LiB in AUG [51] or GPI [52]. In TCV, limited to the dataset with the same toroidal field, the e-folding length increases with blob size (see figure 13(g)), whereas we confirm that no dependence is observed on the outer target $\Lambda_{\rm div}$ (see figure 13(c)). On the other hand, whenever the current is scanned, keeping q₉₅ constant in TCV, the e-folding length is constant despite the two orders of magnitude of difference in $\Lambda_{\rm div}$ (see figure 13(d)), but at the same time small variations of blob size are observed (see figure 13(h)). From the observation of figure 5 we have already suggested a possible link between the upstream profile variation and detachment condition: during the scan at constant q₉₅ no signature of detachment or ion saturation rollover was observed (as seen in figure 6), and this analysis confirms that no variations of the upstream e-folding length and a small variation of blob size are observed if no ion-flux rollover occurs.

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In order to check the dependence of the blob size on the divertor detachment status we consider a similar set of discharges used in figure 10 and evaluate the blob size at different edge Greenwald density fractions. The result is shown in figure 14. In AUG the number of plunges in a single shot prevents a detailed analysis close to the rollover. Nevertheless we can substantially confirm that the blob sizes are small with the plasma in the attached condition, and then the size increases close to and after the rollover. In TCV a sharp increase of blob size is observed around $n^e_e/n_G \approx 0.3$ with a further saturation at a larger edge Greenwald fraction. The same information can be obtained on a statistical basis considering all the databases available for both devices. For the sake of clarity we have included only the shots at the same toroidal field in TCV and for currents below 300 kA, where a clearer location of the ion-flux rollover is observed. The result is shown in figure 15(a): in AUG a sharp increase is observed around $n_e/n_g \approx 0.3$, which coincides—as seen in figure 4—with the transition to a high-recycling regime. For TCV we observe an increase of the blob size up to the edge Greenwald fraction $0.25 \lesssim n_e/n_G \lesssim 0.3$ where again ion-flux rollover at the outer target is observed for currents below 330 kA. Afterwards the blob size seems to saturate. The behaviour of the two devices is reasonably well reconciled, with an increase during the increase of the ion-flux to the outer target up to saturation after the rollover. For the sake of completeness it is worth mentioning that the smaller blobs in AUG have, as estimated, a diameter which is at the limit of our diagnostic and method capabilities. Nevertheless the observed trend, which is consistent with past results even though it was obtained with a different probe head, made us confident of the reliability of the drawn conclusions. The blob size estimates from the probes are based on the evaluation of the characteristic time $\tau_b$ and on the binormal velocity $v_{\perp}$ which can be projected into the radial and vertical motion. In figures 15(b)–(d), the different contributions are still shown as a function of edge density normalized to the Greenwald fraction. We clearly see that the larger variations are observed for $\tau_b$, which increase with $n^{e}_e$/n_G in agreement with the results reported in [9], with the increase of the auto-correlation time observed in AUG [21] and with similar analyses based on the LiB diagnostic in AUG as well [53]. It is worth noting that these results seems to contradict observations obtained using GPI on C-Mod [48] or probe measurements still in TCV [47] where the pulse shapes of the intermittent structures were insensitive to the changes in density. So far no clear explanation of these differences has been determined. On the other hand, in the same figures we report both the radial and poloidal velocities normalized to the local ion-sound speed. For AUG a modest increase with the density is observed confirming what was shown previously using probes [21] or LiB [53]. The behaviour in TCV exhibits a different trend with an almost constant value for the poloidal component and a modest decrease, if any, at higher densities in TCV: this behaviour confirms previous observations [9]. On the other hand previous measurements of the effective radial velocity in TCV [47] show a small variation with density in the far SOL and a larger variation in the near SOL which is still compatible with present results.

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Finally, in order to properly compare the filamentary features between the two devices we consider all the filaments detected in the $\Theta-\Lambda_{\rm div}$ plane, where $\Lambda_{\rm div}$ has already been described, the parameter $\Theta$ is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Theta = \left(\frac{\delta_b R^{1/5}}{L_{\parallel}^{2/5}\rho_s^{4/5}}\right)^{5/2} \label{eq:theta} \nonumber \end{align} \tag{ 4 }$$

with an obvious meaning for the different quantities. Specifically, for both devices both $\Theta$ and $\Lambda_{\rm div}$ are computed considering $L_{\parallel}$ as the parallel connection from the outer target up to the X-point height, whereas the ion-sound gyroradius is locally computed at the position of the filament measurements. These two invariants were originally introduced in the so-called two-region model [35]. The second invariant $\Theta$ is a typical spatial blob scale. The proposed model allows the distinction of different regimes for filaments where different velocity scalings with respect to the filament size and different current closure schemes are identified. While a detailed description can be found in the original paper [35], for the present one it is sufficient to distinguish between the filaments electrically connected to the target (in the sheath-connect (C_s) or in the connected ideal interchange (C_i) regimes) on one side and those electrically disconnected as the resistive X-point (RX) or the resistive ballooning (RB) regimes on the other. These different regimes occupy different portions in the $\Lambda-\Theta$ plane, as pointed out in [27, 35]. The evaluation of $\Lambda$ and $\Theta$ for all filaments detected in both devices is shown in figure 18. In the same plot we have also marked the regions for the different regimes. Actually, the boundary of the connected ideal interchange regime C_i depends on the magnetic fanning parameter $\newcommand{\e}{{\rm e}} \epsilon_x$, which is a measure of the elliptical distortion of the flux surfaces. In analogy to what was done in [27] for the plot shown in figure 16 we have chosen a value of $\newcommand{\e}{{\rm e}} \epsilon_x\approx 0.3$. As pointed out in [27] TCV blobs have features that are mainly consistent with a RB type and a radial velocity which should scale as the square root of the blob size. For AUG the points at very low collisionality ($\Lambda_{\rm div} \lesssim 6\times 10^{-2}$) appear to be close to a connected regime, whereas the transition to a clearly disconnected regime is achieved whenever higher collisionality is reached. Thus while in AUG different filamentary regimes seem to have been attained, in TCV in the explored collisionality region no regime variations have been observed and this could be responsible for the lack of dependence on collisionality. This could also be the reason for the lack of dependence of the e-folding length from $\Lambda_{\rm div}$, although further investigations are indeed mandatory. Presently, work is in progress to compare velocity estimates with independent measurements, such as fast camera velocimetry measurements, which will give us additional data to eventually support these considerations. Besides the determination of the filamentary regime, investigations are also under way to quantify the level of transport associated with the filaments at different levels of shoulder formation, in analogy to what was done in [15], since increased convective losses are ultimately responsible for the enhanced radial transport.

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As already mentioned, increasing investigatory effort has been devoted to the role of neutrals in determining the SOL profile flattening at high density, both from an experimental [8] and a numerical point of view [54–57]. In this respect an attempt to distinguish the behaviour of the upstream SOL e-folding length with respect to the neutral pressure as measured at the midplane or in the divertor region in AUG has been carried out, and the results are shown in figure 19. In figures 17(a) and (b) the e-folding length, still computed around $\rho\approx 1.02{{\rm \mbox{--}}}1.03$, is shown as a function of divertor collisionality and of blob size respectively with a colour code representing the measurement of the subdivertor neutral pressure, whereas in (c) and (d) the same plot is shown with the colour code proportional to the pressure as measured by midplane gauges. The plot suggests, even though not perfectly, a general tendency to develop flatter profiles in the SOL at higher neutral pressure, with a weaker dependence on the divertor values with respect to the midplane ones. This is actually interesting, and in partial contradiction with the observation in JET [38] and MAST [8]. We will address a similar analysis in TCV as well, where more robust and reliable measurements from gauges both in the divertor region and midplane will be available in the near future. What the results up to now point out clearly is that global comprehension, which unifies the experimental observations obtained from a variety of devices, must retain the physics of interaction between plasmas as well as its fluctuations and neutrals.

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