On the role of filaments in perpendicular heat transport at the scrape-off layer

As explained in the previous section, I_sat measurements from the reference pin at the MPM were used to characterize turbulence in the outer midplane. In figure 3, the profiles of three relevant parameters are displayed for the three collisionality ranges: First, in figure 3(a), the relative amplitude of the fluctuations is displayed. This is defined as the standard deviation of the fluctuations divided by the mean value, $\sigma/\mu$. As can be seen, the amplitude tends to grow with distance to the separatrix, and also seems to increase for intermediate and large $\Lambda_{\rm div}$. In figure 3(b), the profiles of the auto-correlation time, $\tau_{\rm AC}$ are presented. $\tau_{\rm AC}$ is calculated using the standard definition found in literature (as the half width at half maximum of the auto-correlation function). In the first 20–25 mm of the SOL, it can be seen how it is substantially increased after the shoulder begins to form, while further away from the separatrix the difference is not as clear. Both trends are consistent with previous observations of turbulence around the shoulder formation [4].

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Figure 3(c) shows the filament packing fraction, $f_{{\rm fil}}$, defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{{\rm fil}}=\nu_{\rm detect}\tau_{\rm AC}, \label{eq4} \nonumber \end{align} \tag{ 4 }$$

where $\nu_{\rm detect}$ is the filament detection frequency. This magnitude indicates the fraction of the total sample duration in which a filament is being detected by the probe. As with $\tau_{\rm AC}$, $f_{{\rm fil}}$ increases over a factor of 2 after the shoulder is formed in the first 25 mm of the SOL. By conditional analysis, the averaged properties of the filaments can be obtained for each data sample. In particular, using the relative delays of the filament detection times in several pins separated radially and poloidally, an estimation of the average radial and binormal velocity can be obtained (see the appendix of [15] for further details). Then, the average size of filaments in both directions can be obtained as the product of these velocities and the filament passage duration, $\tau_{\rm AC}$. The evolution of filament size and velocity with $\Lambda_{\rm div}$ was analyzed in previous work [11] and is out of the scope of the present study.

As already explained in section 2, the main diagnostic for measuring temperature in the midplane was the MPM, either equipped with a sweeping pin in the case of T_e, or with an RFA in the case of T_i. In both cases, the amount of data is limited and MPM measurements had to be combined with additional diagnostics in order to include the near SOL. Even more, no profiles could be provided for the whole range of collisionalities discussed in this paper. Instead, a careful characterization was carried out for the low and high collisionality cases, and then the profiles corresponding to intermediate $\Lambda_{\rm div}$ were interpolated between the two limit cases.

Characteristic low and high collisionality T_e profiles are displayed in figure 4: Thomson scattering provides measurements in the region around the separatrix and ECE channels cover mostly the confined plasma edge. The measurements from both diagnostics are included in the IDA profile calculation (also displayed in the figure). However, neither of them is considered reliable for the far SOL, where the MPM is to be used instead. In order to define profiles for each case, an exponential curve is used in the near SOL, defined here as the first 10 mm in front of the separatrix (the region between the separatrix and the dash-dotted line in the figure). The corresponding e-folding lengths are taken from TS data. According to recent L-mode measurements [40], this means $\lambda_{T_e} \simeq 12$ and ≃20 mm for the low and high collisionality cases, respectively. As can be seen in the picture, this fit (represented by thick solid lines) provides a reasonable link between the IDA and the MPM data. For $R-R_{\rm sep} > 10$ mm, typical probe measurements are taken instead. In order to define a different T_e for filaments and background, only MPM data can be used. The $T_{e, {\rm fil}}/T_{e, {\rm back}}$ ratio, measured by the MPM as explained in section 2 is displayed for each case in the inserts of figure 4. As can be seen, before the shoulder is formed, fluctuations seem to have a similar T_e values as the background. Therefore, $T_{e, {\rm fil}}\simeq T_{e, {\rm back}}$ is assumed. For higher collisionalities, although data show some dispersion, filaments seem to have higher T_e. Therefore, taking an average of the ratio, $T_{e, {\rm fil}}\simeq 1.25 T_{e, {\rm back}}$ is assumed for this case.

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In figure 5, T_i measurements for both regimes are plotted. Up to the separatrix, NBI blips were used in order to obtain CXRS data, while the RFA was used to measure $T_{i, {\rm fil}}$ and $T_{i, {\rm back}}$ in the far SOL. As has already been discussed elsewhere [15], for $\Lambda_{\rm div} \simeq 0.01$, high ion temperatures are observed in the SOL, with a slow radial decay (in the order of $\lambda_{T_i} \simeq 30$ mm) and hot filaments featuring $T_{i, {\rm fil}} \simeq 2T_{i, {\rm back}}$. Instead, T_i displays a strong reduction after the shoulder is formed, $T_{i, {\rm fil}}$ becomes similar to $T_{i, {\rm back}}$ and the e-folding length drops to $\lambda_{T_i} \simeq 8$ mm for $\Lambda_{\rm div} \simeq 10$. Similar evolutions have been reported in Alcator C-mod and MAST [22, 41]. The fits used for each case are also displayed in the figure: for the background in the low $\Lambda_{\rm div}$ case, a linear fit is found to be the most appropriated to account for the two sets of data points. As in the case of T_e, the average $T_{i, {\rm fil}}/T_{i, {\rm back}} \simeq 2$ ratio has been kept for the $T_{i, {\rm fil}}$ curve. In the high collisionality case, an exponential has been fitted and $T_{i, {\rm fil}}/T_{i, {\rm back}} \simeq 1$ has been assumed. The IDA T_e profiles have also been included in figure 5 in order to compare the temperature profiles of both species. As can be seen, T_i departs from T_e a few cm before reaching the LCFS. This is a well known fact in AUG [42], but also in other tokamaks such as MAST, Tore-Supra, Alcator C-mod, or JET (see [43] and references therein), where typically $T_{i, {\rm sep}} > T_{e, {\rm sep}}$. In particular, for AUG $T_{i, {\rm sep}} \simeq 3T_{e, {\rm sep}}$, regardless of the state of shoulder formation.

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Finally T_i and T_e profiles were obtained both for filaments and background using $\log(\Lambda_{\rm div})$ to interpolate between the two collisionality values, $\Lambda_{\rm div}^{\rm low} = 0.05$ and $\Lambda_{\rm div}^{\rm high} = 10$, in which the temperature profiles, $T^{\Lambda_{\rm low}}$ and $T^{\Lambda_{\rm high}}$, were measured:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_\alpha=T^{\Lambda_{\rm low}}_\alpha + \frac{\log({\Lambda_{\rm div}/\Lambda_{\rm div}^{\rm low}})}{\log{(\Lambda_{\rm div}^{\rm high}/\Lambda_{\rm div}^{\rm low}})}(T^{\Lambda_{\rm high}}_\alpha - T^{\Lambda_{\rm low}}_\alpha), \nonumber \end{align} \tag{ 5 }$$

where α stands for each of the electron/ion, background/filament combinations. The result, displayed in figure 6, is a $T_\alpha(R-R_{\rm SOL}, \Lambda_{\rm div}$) function for each of the four parameters, which will allow to estimate the SOL temperature for shots in which it was not directly measured. It must be taken into account that, since no filament temperature data is available in the near SOL neither for electrons nor ions, the values assumed there are simply an extrapolation of the trends observed in the far SOL. This will be noted when appropriate during the analysis.

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As explained in section 2, divertor conditions were monitored using the flush mounted triple probes in the target. This system provides direct measurements of I_sat and T_e, which can be used [44] to estimate plasma density, n_e,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle n_e= 2 \frac{I_{\rm sat}}{Z_iec_sA_{\rm eff}}, \label{eq:ne} \nonumber \end{align} \tag{ 6 }$$

where A_eff is the effective collection area of the probe, $c_s = \sqrt{(T_e+T_i)/m_i}$ is the sound speed, and $Z_i \simeq 1$ can be assumed for the described experiments. As well, using the sheath transmission theory, the perpendicular heat load on the target plates, q_w, can be derived from the fixed probes measurements. Indeed, following the development in [45],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_{w}=\frac{I_{\rm sat}}{A_{\rm eff}Z_ie}(T_e[\gamma_i\tau_i(1-R_E)+\gamma_e-R_E\frac{eV_S}{T_e}]+E_{\rm rec}), \label{eq:qw} \nonumber \end{align} \tag{ 7 }$$

where $V_S$ is the voltage drop at the sheath, $eV_S/T_e \simeq 2.85$, E_rec is the combination of the ionization energy and the Franck–Condon dissociation energy per recycled ion ($E_{\rm rec} = E_{\rm ion} + E_{{\rm FC}, D_2}/2= 13.6 + \frac{5.5}{2}$ eV), and $\gamma_i$ and $\gamma_e$ are the ion and electron sheath transmission coefficients, $\gamma_i \simeq 2$ and $\gamma_e \simeq 5$. Finally, R_E is the wall energy reflection coefficient which can be calculated using the empirical formula in [46] for light ion reflection and taking the atomic number and mass of deuterium and tungsten corresponding to the incident and target materials, respectively. This approach is valid for ion impact energies $E_0 < 10^5$ eV, where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_0 = \gamma_iT_i+eV_S \simeq T_e (\gamma_i\tau_i +2.85), \nonumber \end{align} \tag{ 8 }$$

meaning that all data in this work falls within such range. When calculating q_w in the AUG divertor, $\tau_i = 1$ has additionally been assumed. This is not only consistent with available measurements of T_i in the divertor region [47], but also has been proved a right approach for AUG in [45], where equation (7) was used to calculate heat fluxes onto the targets under several plasma conditions and compare it with independent infrared thermography measurements, showing remarkable agreement between the two.

For each of the points in the database, a divertor profile of n_e, T_e and q_w has been measured. In figure 7, the average values for each collisionality range are displayed. As discussed in a previous work [15], changes in collisionality are associated to the onset of detachment, which transits from an attached state for $\Lambda_{\rm div} \ll 1$ to a partially detached one for $\Lambda_{\rm div} \simeq 5$. This can be seen in the figure: as $\Lambda_{\rm div}$ increases, the points of maximum particle and heat flux move outwards, T_e drops from around 20 eV to less than 5 eV and a clear drop in the power to the wall can be seen in the strike point region, although not in the divertor region corresponding to the far SOL ($\rho_p > 1.01$), where the density remains high enough to sustain q_w despite the lower temperature.

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As explained in section 2, bolometry measurements are used to obtain the amount of power radiated inside the separatrix, P_rad, in order to calculate P_SOL. Unfortunately, for the kind of low power, L-mode discharges studied in the present work, the intensity of the radiated power is barely over the detection threshold of the system (this is particularly the case for the low-density plasmas where $\Lambda_{\rm div} \ll 1$). As well, most radiation comes from the X-point region, where the distinction between radiation originated inside or outside the LCFS becomes particularly difficult. As a result, no detailed characterization of P_SOL as a function of $\Lambda_{\rm div}$, input power, density, etc, was possible. Instead, based on the available data for this set of shots, it was roughly estimated that $P_{\rm rad} \simeq 0.15 P_{\rm tot}$ for $\Lambda_{\rm div} < 1$ and $P_{\rm rad} \simeq 0.25 P_{\rm tot}$ for $\Lambda_{\rm div} > 1$, where $P_{\rm tot} = P_{\rm ECH}+P_{\rm ohm}$ is the total input power.

Once T_e and T_i are defined for the different $\Lambda_{\rm div}$ and the whole $R-R_{\rm sep}$ range, it is now possible to obtain density values from MPM I_sat measurements using equation (6). In this case, $A_{\rm eff} = 3.2$ mm² is used, corresponding to the effective pin surface. Also, for each point, the I_sat, T_e and T_i values associated to filaments and background have been used separately in order to produce the corresponding n_e values.

The resulting density profiles are displayed in figure 8 for both filaments and background, with the latter in the same color/marker scheme as usual, but with lighter tones of red/black/blue. For comparison, density profiles from the lithium beam diagnostic corresponding to high and low $\Lambda_{\rm div}$ values are displayed as solid lines. As can be seen, a very good agreement exists between the two diagnostics: the high/low density curves from the LiB fall between the background and filament density levels measured by the probe for the corresponding $\Lambda_{\rm div}$ discharges. A clear increase for both filament and background density can be seen when $\Lambda_{\rm div} \simeq 1$ is surpassed, indicating the shoulder formation. This result is properly reproduced by data points measured at high collisionality discharges. It is interesting to note that the fact that $T_{e, {\rm fil}}$ and $T_{i, {\rm fil}}$ are extrapolated in the near SOL does not seem to affect strongly the comparison with the LiB data. Although the influence of the temperature in this measurement is reduced ($n_e \propto I_{\rm sat}(T_e+T_i){}^{-1/2}$), this indicates that the extrapolation is at least qualitatively correct.

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The heat flux deposited by the plasma onto the MPM probe head can be calculated by adapting the method discussed in section 3.3 for the divertor to the midplane conditions: filament and background contributions, which could not be separated in the divertor, must now be combined in order to obtain a total heat flux equivalent to the one derived from the IR data. In order to do this, the following assumption is made: for a given period of time t_sample, turbulence properties in the SOL, such as density, temperature, etc correspond to those of filaments for $f_{{\rm fil}}t_{\rm sample}$, and to those of the background for $(1-f_{{\rm fil}})t_{\rm sample}$. Therefore, the heat flux onto the wall associated to filaments, $q_{w}^{{\rm fil}}$, and to the background, $q_{w}^{{\rm back}}$, are calculated using equation (7) but taking the corresponding I_sat, T_e and T_i values from the database (meaning that $\tau_i = 1$ generally no longer applies), and the total heat flux onto the probe wall is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_{w}= f_{{\rm fil}} q_{w}^{{\rm fil}} + (1-f_{{\rm fil}}) q_{w}^{{\rm back}}. \label{eq:sheath_trans} \nonumber \end{align} \tag{ 9 }$$

The results are displayed in figure 9: two clear tendencies can be observed before and after the transition: starting from a common value of $q_{w} \simeq 5$ MWm⁻² at $R-R_{\rm sep} \simeq 20$ mm, a much slower decay is observed after the shoulder formation, with intermediate collisionality points falling nicely between the two other. This observation is in remarkable agreement with the results obtained from the IR camera, $q_{w}^{IR}$: In figure 9, the $q_{w}^{IR}$ values calculated at the radial strip highlighted in red in figure 2 are also presented. The color code indicates the $\Lambda_{\rm div}$ value at the time of the corresponding plunge and also indicates a clear change in the parallel heat flux with the shoulder formation: In good agreement with previous work [4], the e-folding length at the probe wall, $\lambda_q$ goes from ≃5 mm up to ≃18 mm over the transition. In this case, since no IR data is available in the near SOL for comparison, the fact that $T_{e, {\rm fil}}$ and $T_{i, {\rm fil}}$ have been extrapolated (see section 3.2) does not have any impact.

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First, the particle transport associated to filaments can be obtained from the magnitudes discussed in section 3, as the number of particles traveling radially in an average filament, factored by the fraction of the time in which a filament is present at a given point of the far SOL:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Gamma_{r,{\rm fil}}=v_rn_{{\rm fil}}\,f_{{\rm fil}} \label{eq:G}. \nonumber \end{align} \tag{ 10 }$$

The results are displayed in figure 10, where the values of $\Gamma_{r, {\rm fil}}$ are displayed as a function of $R-R_{\rm sep}$. As can be seen, two clear profiles emerge for the low and high collisionality cases, the second almost one order of magnitude higher than the first. It must be pointed out that this is not just the trivial consequence of the density increase associated to the shoulder formation (which can be seen in figure 8): as can be seen in the bottom plot of figure 10, where the profile of the $n_{e, {\rm fil}}/n_{e, {\rm back}}$ ratio is presented, the amplitude of density fluctuations is also greatly increased after the transition. This is in good agreement with equivalent measurements carried out over the shoulder formation at TCV [48].

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It must be noted that this definition of $\Gamma_{r, {\rm fil}}$ only accounts for the transport caused by large convective structures (at least large enough to be detected using the $2.5\sigma$ threshold), and neglects diffusive transport and convective transport associated to smaller structures. In principle, this seems to be a reasonable approximation as the general consensus in literature [2] points in the direction of far SOL transport being dominated by large convective cells. Besides, density profiles flatten for high collisionalities (when the shoulder is formed), thus reducing the relevance of the diffusive transport. However, a detailed comparison of diffusive versus convective perpendicular transport is left for future work.

In order to discuss changes in the perpendicular to parallel transport balance, measurements from the Mach probe are presented in figure 11: in it, the radial profile of the Mach number in the direction parallel to the magnetic field, M_||, at the probe position in the midplane is displayed for the different collisionality regimes (M_||  >  0 indicates a flow towards the HFS divertor). The low collisionality regime features higher values, in the range of $M_{||} \simeq 0.3$. Instead, for $\Lambda_{\rm div} > 3$ M_|| drops roughly by a factor of three, with intermediate collisionalities sitting nicely between the other two clouds of points. This may be seen as the result of a transition between a sheath-limited divertor regime for low $\Lambda_{\rm div}$, in which the main particle source in the SOL is still the separatrix, to a high recycling and detached regime for high collisionalities, in which recycling in front of the target becomes the main particle source in the SOL, thus drastically reducing the parallel velocity upstream. Again, if the filamentary transport is assumed to dominate perpendicular particle transport, these measurements allow for the comparison between the parallel and perpendicular convection terms,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\Gamma_{r,{\rm fil}}}{\Gamma_{||,{\rm conv}}}=\frac{n_{e,{\rm fil}}}{n_{e,{\rm mean}}}\frac{f_{{\rm fil}}v_r}{c_{s,{\rm mean}}M_{||}}, \nonumber \end{align} \tag{ 11 }$$

where $n_{e, {\rm mean}}= f_{{\rm fil}} n_{e, {\rm fil}} + (1-f_{{\rm fil}}) n_{e, {\rm back}}$ and $c_{s, {\rm mean}}= f_{{\rm fil}} c_{s, {\rm fil}} \,+$ $(1-f_{{\rm fil}}) c_{s, {\rm back}}$, since only an average M_|| is measured. This is displayed in the bottom plot of figure 11, where it can be seen how, even if the parallel term dominates in both cases, a clear change is observed in the far SOL, where the ratio increases by almost an order of magnitude for $\Lambda_{\rm div} >3$. This is consistent with the hypothesis that the filamentary transition plays a role in the formation of the shoulder, as the perpendicular to parallel particle balance is clearly changed for high collisionalities. Since $n_{e, {\rm fil}}$ includes a $(T_e+T_i){}^{-1/2}$ factor, the set of the points in the near SOL could be affected by the extrapolation of T_i values. Nevertheless, they seem to follow the same trend as the rest of points and $n_{e, {\rm fil}}$ values seem qualitatively correct in figure 8, so probably the picture in figure 11 is at least roughly correct.

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Finally, once $\Gamma_{r, {\rm fil}}$ is known from equation (10), temperature measurements can be used to calculate the radial heat flux associated to filaments [49],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_{r,{\rm fil}}^{\rm total}=q_{r,{\rm fil}}^{\rm electron}+q_{r,{\rm fil}}^{\rm ion}=\frac{5}{2}\Gamma_{r,{\rm fil}}T_{e,{\rm fil}}\nonumber\\ +\frac{1}{2}\Gamma_{r,{\rm fil}}[5T_{i,{\rm fil}}+m_i(c_sM_{||})^2], \nonumber \end{align} \tag{ 12 }$$

where $m_e \ll m_i$ has been taken into account. In order to compare this flux with the total power flowing into the SOL, P_SOL, the total power crossing the corresponding flux surface can be defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P_{{\rm fil}}=q_{r,{\rm fil}}^{\rm total}S_{{\rm fil}}, \label{eq:P fil} \nonumber \end{align} \tag{ 13 }$$

where a characteristic surface $S_{{\rm fil}}$ is defined for each data point. This surface, displayed in figure 12, represents the region of the flux surface where the MPM measurement is made, in which there is filamentary transport. Given the ballooning nature of the filaments, this is only expected to happen where the curvature is unfavorable [50–52], meaning that only a poloidal region around the low field side midplane should be considered. Considering the axial symmetry of a tokamak, $S_{{\rm fil}}$ can then be defined in cylindrical coordinates as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S_{{\rm fil}} = \int_{z(-\theta_{\rm max})}^{z(\theta_{\rm max})}{2\pi R(z){\rm d}z}, \label{eq:Sfil} \nonumber \end{align} \tag{ 14 }$$

where θ is the poloidal coordinate (with $\theta = 0$ indicating the outer midplane), $R(\theta)$ is the radial coordinate of the corresponding flux surface and $\theta_{\rm max}$ is the poloidal angle limiting the region with filamentary transport. In order to calculate $R(\theta)$, a field line is traced from the MPM measurement point to the LFS divertor, keeping track of its $z(\theta)$ and $R(\theta)$ components. In order to decide the extent of $S_{{\rm fil}}$, $\theta_{\rm max} \simeq \pi/3$ has been assumed, resulting in typical values of $S_{{\rm fil}}$ in the range of 13 m². This must be regarded as an approximation: a constant value of $q_{r, {\rm fil}}^{\rm total}$ has been assumed in equation (13), corresponding to the one measured at the probe position (roughly $\theta_{\rm MPM} \simeq \pi/6$). In fact, the filamentary drive (and thus $\Gamma_{r, {\rm fil}}$) can be expected to increase closer to the equator and decrease further away. To test this definition of $P_{{\rm fil}}$, the simple hypothesis $q_{r, {\rm fil}}(\theta)=q_{r, {\rm fil}}^{\rm eq}\cos{\theta}$ can be made, where $q_{r, {\rm fil}}^{\rm eq}$ is the radial heat flux at the equator. In order to match the MPM measurements, $q_{r, {\rm fil}}^{\rm eq}=q_{r, {\rm fil}}^{\rm total}/\cos{(\theta_{\rm MPM})}$. In this case, equation (13) reads now $P_{{\rm fil}}= q_{r, {\rm fil}}^{\rm total}S'_{{\rm fil}}$, where

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle S'_{{\rm fil}} =& \int_{z(-\theta_{\rm max})}^{z(\theta_{\rm max})}{\frac{2 \pi}{\cos{(\theta_{\rm MPM})}} \biggl(1+(z-z_{\rm mag})^2/(R(z)-R_{\rm mag})^2\biggr)^{-1/2}}\nonumber\\ &\times {R(z) {\rm d}z} , \label{eq:Pfil2} \nonumber \end{align} \tag{ 15 }$$

and z_mag and R_mag are the coordinates of the magnetic axis. By integrating the new effective area, $S'_{{\rm fil}}$, it is observed that the total radial heat flux under this poloidal distribution remains unchanged close to the separatrix and decreases slightly (up to $5\%$) for the data points with higher $R-R_{\rm sep}$ values. Of course, this result depends on the specific function selected as $q_{r, {\rm fil}}(\theta)$, but as a first approximation, the error introduced by the poloidally constant flux hypothesis is expected to be small and a detailed discussion of this problem is left for future work.

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In figure 12, the $P_{{\rm fil}}/P_{\rm SOL}$ ratio is displayed. As can be seen, the perpendicular power flux associated to filaments may be a substantial fraction of P_SOL for radial positions in the near SOL (ideally reaching $P_{{\rm fil}}/P_{\rm SOL} = 1$ at $R-R_{\rm sep} = 0$ mm). Nevertheless, given the extrapolation involved in the filament temperatures these data points must be regarded with caution. In particular, the fact that high collisionality points near the separatrix show $P_{{\rm fil}}/P_{\rm SOL} > 1$ can be regarded as a overestimation of $T_{i, {\rm fil}}$ and/or $T_{e, {\rm fil}}$, but also as an overestimation of P_rad for those particular cases. In the far SOL, there seems to be a radial position around $R-R_{\rm sep} = 20$ mm where the heat fluxes are equal for low and high collisionalities. Beyond this point, both regimes decay at different rates, with the higher collisionality retaining higher values for equivalent radial positions. These results are reminiscent of those displayed in figure 9. This is not a trivial result, as the evolution of $\Gamma_{r, {\rm fil}}$ was not involved in the calculation of heat deposition on the MPM surface. As a general result, it can be seen how the flows associated to filaments still represent a significant fraction of P_SOL 20 mm in front of the separatrix (and beyond 25 mm in the medium and high collisionality cases).

It must be taken into account that $S_{{\rm fil}}$ is roughly the same for all data points. The same holds true for P_SOL, as the input power $P_{\rm ohm}+P_{\rm ECH}$ is approximately equal for all shots and P_rad does not change much with the transition, as discussed in section 3.4. This means that the values of $P_{{\rm fil}}/P_{\rm SOL}$ displayed in figure 12 are basically proportional to the ones of $q_{r, {\rm fil}}^{\rm total}$.

Finally, the different contributions to the perpendicular heat flow of ions and electrons are displayed in figure 13. As can be seen, a clear evolution takes place for all radial positions: at low collisionalities, $T_i \gg T_e$, meaning that the perpendicular energy transport is dominated by the ion channel. Instead, as the transition takes place ions cool down, and their relative contribution to transport decreases. Eventually, both components reach similar values as $T_i \simeq T_e$ for $\Lambda_{\rm div} \gg 1$.

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