An improved model for the accurate calculation of parallel heat fluxes at the JET bulk tungsten outer divertor

The two different arrangements shown in figure 3 have been tested and analysed. The first one corresponds to the 2013 melting experiment (figure 3 upper), for which three different geometries were installed:

• Leading edge lamella, with its front face exposed to the plasma with almost perpendicular incidence. The height of the protruding edge varies from 2.5 mm at the high-field side to 0.24 mm at the low-field side.
• Standard lamella, is the standard geometry used for all 9260 individual lamellas constituting the bulk W outer divertor target. These standard units have a carefully shaped top surface to optimize power deposition and avoid leading edges (see figure 1 and [26]). The one selected for analysis here is located on the downstream side of the leading edge lamella.
• Flat lamella, located in front (upstream side) of the leading edge lamella. These special unshaped lamellas were included to ensure sufficient field line penetration from the upstream side onto the leading edge lamella [5].

Zoom In Zoom Out Reset image size

The 2015–16 experiment was performed with a different geometrical configuration in which special flat lamellas are no longer required. In addition to the standard lamella, the following elements are studied (figure 3, lower):

• Sloped lamella, similar to a standard lamella, but with a poloidally 20 mm wide sloped section. The angle with respect to the tile reference plane is 15°.
• Shadowed lamella, geometrically identical to a standard lamella, but with its first 20 mm of length shadowed by the sloped lamella.

The temperature profiles are measured at the top surface of the tiles using a high speed medium-wavelength infrared (MWIR) camera, known locally on JET as KL9A [15]. This camera views the horizontal divertor target from above with a pixel field of view of approximately 1.7 mm/pixel, observing black body radiation from the tile in the wavelength range 3.1–4.7 μm. The recorded intensity is calibrated to give surface temperature using data from in-vessel calibrations performed in 2015, and the known wavelength and temperature dependent emissivity of bulk tungsten. The maximum camera frame rate was 3 kHz and 7.8 kHz for L-mode and H-mode shots, respectively. Such a high frame rate is achieved by reducing the field of view of the IR camera, covering half the width and length of the tile. 1D temperature profiles in the radial direction along the W lamellas of interest are extracted from each video frame. The locations of the relevant line profiles used for the previous and current melt experiments are shown in figure 4. In both cases, the signal for the lamella M (SM) corresponds to the profile on the protruding lamella.

Zoom In Zoom Out Reset image size

The local coordinate system has its origin s_a  =  0 at the inboard (high field side) edge of the lamella, with s_a increasing with device major radius (towards the low-field side). There is a 14.55° angle—shown in figure 5—in the vertical-poloidal plane between the tile surface and the viewing plane of the camera, which defines the transformation between the radial and local coordinate systems.

Zoom In Zoom Out Reset image size

The angle θ (see equation (1)) is defined by the intersection of the 3D field line trajectory and the divertor tile surface. As was noted in previous studies [6], the variation of this angle along the 58 mm poloidal width of the lamella is of  ∼1°. The dot product of the magnetic field vector and the lamella surface normal defines the cosine of the impinging angle, which depends on the poloidal coordinate as shown in figure 6 for the L-mode shot JPN 84514.

Zoom In Zoom Out Reset image size

The inverse thermal analysis of the IR temperatures provides the normal heat flux on the divertor tiles. In JET, this is automatically calculated and stored using the THEODOR code [18]. To double check this analysis stage, a new code ALICIA [16] has been developed and applied for the surface load reconstruction. The 2D geometries used for both codes are equivalent and based on the midplane rectangular section of the lamellas represented in figure 7. For the case of the sloped lamella, two independent rectangles are modelled at each side of the keyhole—one for the sloped section and another for the standard section. It has been checked that this approximation has negligible effect on the surface heat flux density calculation, compared to other sources of error. Instead of the α-parameter [17] and the explicit integration rule of the present version of THEODOR used at JET, ALICIA applies augmented Lagrangian and implicit integration schemes to bound the error to 5 °C on the surface of the lamella. The implicit integration rule allows the application of an extremely fine finite element (FE) mesh in the plasma facing surface, which leads to a much better ELM capture while substantially improving the reconstruction of the L-mode heat flux.

Zoom In Zoom Out Reset image size

Validation cases are presented in appendix A.2 using a synthetically generated heat flux profile to compare the corrected 2D inverse model to the 3D lamella behaviour.

Two geometrical corrections are required to match the behaviour of the approximate inverse models to the real lamella geometries: one related to the shape of the plasma-facing surface, and a second accounting for the 3D geometry of the bulk lamella. Each correction is applied using a corresponding factor, leading to an improved interpretation of the time-averaged heat load, $q_{\rm 2D}$:

• Load factor f_q relates the uniform load assumed by the inverse codes with the normal flux defined in figure 8 for the irregular lamella geometry q_n, and
• Mass factor f_m is a correction factor which accounts approximately for the 3D thermal mass distribution of the thin tiles.

Zoom In Zoom Out Reset image size

As shown in figure 8, the top surface of a standard lamella is far from flat. Its geometry is a combination of shadowed areas at the front and rear, a flat 1.5 mm surface in its central section, and a parabolic surface to the front. The thickness of each lamella is 5.5 mm, and they are arranged toroidally with a separation of 1 mm, defining a 6.5 mm cyclic symmetry within the stack.

The load factor f_q is defined as the ratio between the normal load on the flat region of a lamella, q_n, and the output of the 2D inverse codes, q_2D. The key condition to be met is that the toroidally integrated power for the tile under study is equal to the power integrated in an equivalent continuous slab. Although the loading pattern of the standard lamellas is complex, the cyclic assembly symmetry allows this factor to be calculated straightforwardly as the ratio of the lamella thickness to the distance between the symmetry planes:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_{q} = \frac{q_{\rm n}}{q_{{\rm 2D}}} = \frac{5.5~{\rm mm}}{6.5~{\rm mm}} = 0.85. \nonumber \end{align} \tag{ 7 }$$

This can be simply interpreted as the additional power that the lamellas receive due to the discontinuity produced by the gaps. Its consideration as a power factor facilitates its application to lamella geometries with no cyclic symmetry. It needs to be noted that the use of this factor assumes that the toroidal temperature profile produced by the complex surface heat flux has already formed. For the lamellas analyzed, this is true approximately 0.5 s after the strike point is fixed. Once this toroidal profile is stable, the net heat flow inside the lamella has a vertical direction, and the temperature rise at the surface is equivalent for all points located at the same poloidal coordinate (i.e. subjected to the same q_2D).

For the protruding lamellas, the factor is calculated by the ratio of the total power in the real geometry to the power on a flat 5.5 mm long slab loaded by q_n. As opposed to the standard lamella, the flat surfaces of the leading edge and sloped lamellas have a well defined solid angle, which is used for the analytical integration of the heat load. The loading scheme represented in figure 9 yields the following expression for the load factor of the protruding lamellas:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f_q = \frac{l_{{\rm tile}}}{l_{{\rm flat}}+l_{{\rm slope}}\ \frac{\sin(\theta+\alpha)}{\sin(\theta)}}, \nonumber \end{align} \tag{ 8 }$$

where α is an angle defining the geometrical shape of each of the special lamellas, and parameters l refer to the size of the various geometrical lengths in figure 9. The factor f_q is constant for the sloped lamella, but variable for the leading edge geometry since the height of the exposed edge changes linearly along its length:

• f_q  =  0.085–0.65 for the leading edge lamella, using:
  • $l_{\rm flat} = 5.5$ mm,
  • $l_{\rm slope} = 0.25$–2.4 mm,
  • $\alpha = 90$°.
• •f_q  =  0.18 for the sloped lamella, using:
  • $l_{\rm flat} = 1.0$ mm,
  • $l_{\rm slope} = \frac{4.2}{\cos(\alpha)}$ mm,
  • $\alpha = 15$°.

Zoom In Zoom Out Reset image size

A second adjustment is needed to match the 2D rectangular models to the real 3D geometry thermal response. As previously introduced, the inverse analysis 2D model is equivalent to a 3D block, which differs from a real lamella (see figure 7) in the following aspects:

• The real lamella thickness is not constant, with the plasma exposed section being thicker than the lower part.
• There is a central hole required for their assembly and packing by a tension chain.
• The 15° lamella has a keyhole separating the sloped section from the standard section.

The thermal mass factor, f_m, is proposed to account for these differences. Forward simulation using Abaqus [19] has been used to estimate the magnitude of this 3D solid geometry effect through a series of tests varying the peak heat flux and the location of the at which the peak heat flux strikes the lamella (corresponding approximately to the magnetic strike point position in the real experiment). Although the behaviour of the real geometries relative to the equivalent block depends slightly on these two parameters, the following values of f_m have been found for each of the lamellas which limit the maximum error to less than 5%:

• Std. lamella: f_m  =  0.9.
• Sloped 15° lamella: f_m  =  0.95.
• Leading edge lamella: f_m  =  0.95.
• Flat lamella: f_m  =  0.93.

The load and mass factors are applied for the interpretation of L-mode and H-mode time averaged heat fluxes. In the case of ELM heat loads, the geometrical factors do not apply since the transient heat loads they produce are much faster than the characteristic diffusion time of the lamella. The calculation of the ELM $q_{\parallel}$ is thus performed using the incident angle of the projected pixel for each of the profiles without any other geometrical consideration. Inter-ELM heat loads are also calculated using the load and mass factors, but might be affected by slightly larger error since the time between ELMs is shorter than the characteristic diffusion time of the lamellas in the toroidal direction (∼0.2 s). Nevertheless, for the typical ELM frequency of  ∼25 Hz used in the experiments discussed here, the validation results provided in appendix A.2, show that this toroidal transient effect between ELM and inter-ELM heat loads is of the order of 0.01 s.

The background heat flux, q_BG, is measured when the strike point is moved away from the lamella. It has been checked that its value is similar for all geometries with different surface temperatures, independently of whether or not they are magnetically shadowed. This confirms the accuracy of the IR measurements and minimizes any source of errors affecting a specific lamella.

The non-parallel component of the heat flux is measured at the shadowed lamella before moving the strike point away. The poloidal length of the shadowed region is only 20 mm, with only 10 mm of these unaffected by the heat flux discontinuities between shadowed and non-shadowed regions, along with the one produced by the low-field side edge of the lamella. The modular transfer function (MTF) of the optical system smoothes the sudden jump of temperature along these discontinuities, preventing proper characterization of the profile of the non-parallel component. Nevertheless, its peak value can be derived from the measurements by removing the background:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_{\nparallel} = q_{n,{\rm sh}} - q_{\rm BG}, \nonumber \end{align} \tag{ 9 }$$

where $q_{n, {\rm sh}}$ is the surface heat flux in the shadowed region.

In the case of ELMs striking a misaligned lamella, the heat load may be affected by the increased gyroradius, which leads to Larmor smoothing and a reduced ELM load. This effect is dependant on the local lamella geometry, and can only be estimated by simulations of irregular surfaces [20]. For the protruding edges, PIC modelling [21] reports small adjustments required of $\Delta q \simeq (0.2\pm 0.1) q_{\rm n}$. This has been taken into account via an additional 20% reduction of the ELM parallel heat flux projected to the protruding lamellas, leading to an excellent match of the ELM temperature increase measured at the sloped lamella surface.

The thermal simulation loop—i.e. the inverse reconstruction of heat flux and forward simulation of temperatures at the lamella—also needs to be consistent with other measurements not related to the IR diagnostic system.

The melting of the leading edge surface is captured by a high definition visible light camera inbetween shots. The thermal simulation of each of the experiments' first melting pulses should be able to reproduce the shape of the melted area. In each of the two special lamella experiments (leading edge and sloped), the thermal simulation should be able to capture the shape of the melt region observed optically immediately following the first of the pulses in which melting began. In the case of the sloped lamella, the melted zone is also observed directly by the IR system. The emissivity of the material is reduced instantly when melting starts, leading to an apparent reduction of the temperature measurements [22]. The comparison between simulated temperatures and IR measurements should therefore diverge at the exact moment of melting onset.

Two twin pulses have been chosen from the previous (JPN 84514) and new melting (JPN 89162) experiments. Both have the same parameters: toroidal field $B_{\rm t}=2.6$ T, plasma current $I_{\rm p}=2.45$ MA, neutral beam injection power $P_{\rm NBI}=$ 2.0 MW, and resistive heating power $P_{\rm ohmic}=1.5$ MW. The pulses begin with the outer divertor strike point located on Stack B, followed by a 2 s excursion onto Stack A where directly strikes the special lamellas. The only slight difference between the two pulses is the position of the strike point. As the 2015–16 melting experiment sloped lamella section is just 20 mm wide poloidally, the strike point has been displaced 12 mm inboard during the excursion with respect to the 2013 experiment.

At the beginning of the transition to Stack A, the inverse codes give a slight overshoot of the reconstructed heat flux history, shown in figure 10 along the local poloidal coordinate s_a. This effect was noted in the validation tests presented in appendix A.2, and is produced by the irregular toroidal heat distribution. Once the toroidal temperature profile is formed, the calculated heat flux stabilizes. This transient effect lasts between 0.2 and 0.5 s, and is therefore avoided when selecting the time interval used for the characterization of the plasma load.

Zoom In Zoom Out Reset image size

The transient response for the ALICIA model is represented in figure 10. The smoothness of the output facilitates the identification of the apparent overshooting period. The stable phase on Stack A—where the lamella heat flow is dominated by 2D diffusion—corresponds to the interval between 16.5–18.0 s.

3.1.1. 2015–16 melting experiment.

Completely independent calculations of the parallel heat flux density have been carried out using the IR temperature for the standard and leading edge geometries, respectively. The application of the procedure detailed in the previous section to each geometry leads to the following factors for the heat flux interpretation:

• Std. lamella: f_q  =  0.85, f_m  =  0.9.
• Special 15° lamella: f_q  =  0.17, f_m  =  0.95.

The time evolution of the reconstructed surface heat load averaged every 0.5 s, $\bar{q}_{\rm 2D}$, is represented in figure 11, showing a stable peak power value and an almost constant profile. The results for the shadowed lamella shown in figure 12 give the two measurements required for adjusting the plasma heat load correction parameters:

• During the strike point excursion onto the slope, the value measured for the heat flux in the shadowed region is $q_{n, {\rm sh}} = (900 \pm 100)\ f_q \ f_m = 690 \pm 77$ kW m⁻².
• After the excursion, the background heat is evaluated. Its magnitude of $q_{\rm BG}=(50 \pm 40)\ f_q \ f_m = 38 \pm 30$ kW m⁻² is found to be of the order of the radiated power calculated by the code TorusMC using bolometry data [23], which is negligible compared to the heat measured during the excursion.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The difference between the two gives the value for the non-parallel heat $q_{\nparallel} = (650 \pm 107)$ kW m⁻². It is important to note that this represents a large fraction of the standard lamella surface heat flux density, $q_{\nparallel} \simeq 0.20 \max(q_{\rm n})$, even though the applied correction only corresponds to a small fraction of the parallel heat flux, $q_{\nparallel} = 1.2\% \ q_{\parallel}$.

The discontinuity between shadowed and non-shadowed regions, as well as that separating the sloped and standard sections of the sloped lamella are affected by the MTF of the optical system. Instead of a discrete step in the heat flux, the smoothing of the temperatures produced by the IR diagnostic system gives an apparent transition in the the shadowed region, and a poloidal overshoot in the non-shadowed part, affecting the calculated $q_{\parallel}$ in a similar way as was noted in [14]. An additional interpolated profile has been added to the analysis, in the case of the sloped lamella experiment, which skips the irregularity using a linear approximation between the two sides of the keyhole. Results for the computed plasma parallel heat density are compared in figure 13, with and without the plasma heat load correction.

Zoom In Zoom Out Reset image size

A difference of 15% in peak power is found between the standard and exposed lamellas when using the strict optical projection, which disappears when the non-parallel heat correction is applied. The poloidally integrated power is also very well matched for the latter, with a difference of just 3% when the discontinuity of the profile is corrected, and negligible without any interpolation.

3D forward analyses have been run for the sloped lamella using both the strict optical, and corrected optical profiles. Figure 14 shows how the IR temperatures are only matched for the sloped lamella when the corrected interpretation is used. Note that the projection model is irrelevant when modelling the same geometry from which the parallel heat flux was calculated. This is why there is no difference for the standard side of lamella (s_a  >  20 mm).

Zoom In Zoom Out Reset image size

3.1.2. 2013 melting experiment.

An equivalent procedure has been performed for the re-interpretation of the old experimental results in L-mode. The power factors used for the leading edge differ with respect to the 2015–16 sloped lamella, as detailed in section 2. Here, a third lamella geometry is now introduced, corresponding to a flat lamella installed upstream of the leading edge in order to allow field line penetration to the leading edge (see figure 3 upper). The analysis is performed using the following factors:

• Flat lamella: f_q  =  0.85, f_m  =  0.93.
• Std. lamella: f_q  =  0.85, f_m  =  0.9.
• Leading edge lamella: f_q  =  0.085–0.65, f_m  =  0.95.

In the absence of a local shadow, as the plasma parameters are essentially identical for both experiments, the measured $q_{\nparallel}$ from the new experiment has been used to correct the parallel heat flux. The agreement between the profiles calculated using each of the geometries improves when this correction is applied (see figure 15), with a very good match of peak power density and integrated power along the lamella. The maximum difference of just 5% in these two measurements suggests that the correction effect was equally relevant for the leading edge geometry of the 2013 experiment.

Zoom In Zoom Out Reset image size

Figure 16 shows the time evolution of the maximum temperatures for the 3D forward simulation. The thermal model leads to unrealistic results when the strict optical approximation is used, while an excellent match to the IR temperatures has been found when using the corrected optical projection.

Zoom In Zoom Out Reset image size

Figure 17 compiles the three $q_\parallel$ profiles extracted at $t=[18, 18.5]$ s when the surface temperature is maximum at 500 °C, illustrating the differences between the new approach with ALICIA (both strict optical and corrected optical) and the old analysis procedure using the THEODOR code. The latter grossly overestimates $q_\parallel$ across the whole poloidal length of the lamella.

Zoom In Zoom Out Reset image size

In the course of the 2013 experiment, the leading edge lamella melted during pulse JPN 84779, with $B_{\rm t}=2.88$ T, $I_{\rm p}=3.0$ MA, $P_{\rm NBI}=21.0$ MW, and ELM frequency of 32 Hz. To compare the H-mode response between the old and new experiments, pulse number JPN 89445 for the sloped lamella has been selected as the discharge with highest power density before melting of the sloped surface. This pulse had $B_{\rm t}=2.88$ T, $I_{\rm p}=3.0$ MA, $P_{\rm NBI}=14.8$ MW, $P_{\rm ICRH}=2.6$ MW, and an ELM frequency of 25 Hz [22]. Selecting a shot without melting minimizes the uncertainty on the emissivity produced by the phase change, which is required for comparing the parallel heat profiles between the sloped and standard lamellas. For completeness, the additional pulse 91965 has also been included in the forward analysis for which melting of the sloped surface did occur. This latest discharge used the highest values of B_t, $I_{\rm p}$, and heating power [22], with $B_{\rm t}=3.03$ T, $I_{\rm p}=3.23$ MA, $P_{\rm NBI}=24.9$ MW, and $P_{\rm ICRH}=2.8$ MW.

3.2.1. 2015–16 melting experiment.

The new exposed lamella design allows direct measurement of the temperature evolution including ELM transients, represented in figure 18.

Zoom In Zoom Out Reset image size

Figure 19 completes a series of time averaged heat flux profiles (averaging period 0.5 s) across the standard and sloped lamella. The value for the averaged peak heat flux including ELM and inter-ELM is stable during heating in both geometries. In the case of the shadowed lamella, the equivalent time-averaged profiles are shown in figure 20. It is clear that the L-mode observation of heat deposition into the shadowed region also persists in H-mode.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The heat measured at the shadowed lamella is $q_{n, {\rm sh}} \simeq (3.0 \pm 0.2)\ f_q \ f_m = 2.3 \pm 0.15$ MW m⁻², corresponding to a fraction of up to 35% of the standard lamella heat flux, q_n. The background effect is measured at the standard and shadowed lamella after t  =  15.0 s (when the strike point has moved away from the slope), giving the same result for both, $q_{\rm BG}=(0.6 \pm 0.3)\ f_q \ f_m = 0.46 \pm$ 0.23 MW m⁻², which leads to a value of the non-parallel heat of $q_{\nparallel} = (1.84 \pm 0.38)$ MW m⁻². In this case, its fraction of the normal heat, $q_{\nparallel} \simeq 0.3 \max(q_{\rm n})$, is even larger than in the case of the L-mode pulses.

The increased heat flux density measured on the shadowed lamella surface in H-mode compared to L-mode can be explained by the contribution of the ELMs. Figure 21 compiles the heat flux profiles for the ELM and inter-ELM heat loads, compared with the time-averaged profile. The heat flux in the shadowed region is much larger during ELMs, which might be related to an increased ion temperature. Further simulation of the Larmor effects during ELMs would be required to test this hypothesis. It should be noted, however, that the evaluation of the inter-ELM profile on this shadowed lamella tile is prone to a large error, since the toroidal heat redistribution due to ELMs affects the measurement of q_2D for at least 0.01 s after an ELM event. On the other hand, the dominant ELM effect has a practical benefit, as it allows using the same value for $\frac{q_{\nparallel}}{q_{\parallel}}$ for both ELM and inter-ELM loads. It will be shown in the forward analysis that this assumption leads to a very accurate prediction of both the ELM temperature rise and the global lamella behaviour during 3D forward simulation.

Zoom In Zoom Out Reset image size

The results for $q_{\parallel}$ including ELM and inter-ELM are shown in figure 22. These are calculated independently for the sloped and standard lamella, in a similar way as performed for the L-mode case (see figure 13). When the profiles are extracted using the strict optical approach, a large difference is observed. If the background and non-parallel contributions are used in correcting each of the profiles, the ratio between the calculated parallel profiles approaches unity within an error margin of 10%. It is worth noting that the value for $q_{\nparallel}$ corresponds just to 1.9% of the maximum $q_{\parallel}$, even after the latter is corrected (see 1.2% in the L-mode case).

Zoom In Zoom Out Reset image size

To further confirm the observed differences between the strict and corrected projections, 3D forward analysis has been performed using the $q_{\parallel}$ reconstructed from the standard lamella measurements. A time step of 0.13 ms has been used, corresponding to the 7.8 kHz rate of the IR cameras. Such a small time step is needed to capture the fast transients of the H-mode, and explains the improved reconstruction of the temperatures compared to the simulations performed for the L-mode shots of the sloped lamella (figure 14), which used a larger time step of 34 ms. Figure 23 shows the results for each projection, along with the IR measurements. This very close match between the simulated surface temperature for the corrected projection of $q_{\parallel}$ and the IR measurement is a good demonstration of consistency of the approach. In addition, it is important to note that the simulation results using the strict optical projection would predict generalized melting of a large portion of the sloped lamella, which was not observed experimentally in this particular discharge.

Zoom In Zoom Out Reset image size

In order to further test the accuracy of the corrected projection, two of the pulses for each experiment (leading edge and sloped lamella) where melting occurs have also been analysed. In the case of the 2015–16 experiment, the simulated temperatures are compared to the IR measurements in figure 24, showing again a perfect match until the onset of melting. The change of emissivity once the melting begins affects the IR measurements, and from that point on the match is lost.

Zoom In Zoom Out Reset image size

Finally, figure 25 compares the melted profile obtained by the forward simulation with the optical image of the lamella taken in situ immediately after the first pulse in which melting was observed. Both the location and the shape are very well captured. The minor discrepancies between the actual melt profile and that inferred from the thermal analysis are likely due to the limited applicability of the formulation used for the forward analysis, which does not take into account any latent heat of fusion, thermionic emission, or vapour shielding effect.

Zoom In Zoom Out Reset image size

3.2.2. 2013 melting experiment.

The correction presented for the H-mode heat flux derived from the sloped lamella in the previous section is now applied to the leading edge geometry in the older experiment. The pulse under study, 84779, corresponds to the first discharge in which melting was observed.

The profiles for $q_{\parallel}$ are obtained following the same procedure and correction values used for the L-mode study (see figure 15). Figure 26 compares the time-averaged parallel heat flux (including both ELM and inter-ELM contributions) between the previous analysis using the THEODOR code [5] and the new ALICIA calculation using the corrected models. For the latter, both the strict optical and fully corrected projection are shown. In this case, the two inverse codes differ only by 10%, so that the main differences observed correspond to the procedure for extracting the parallel heat flux component. The factor of 0.4 between the corrected optical profile (green curve in figure 26) and the original THEODOR analysis without any correction (red curve) was precisely the value required by MEMOS code to match the observed melt profile during the leading edge lamella exposure [7].

Zoom In Zoom Out Reset image size

Post-mortem analysis of the leading edge lamella concluded that the observed topological damage was a consequence of ELM-induced flash melting [5]. A forward analysis using full 7.8 kHz IR signal has been run for the new profiles of the parallel heat flux density projected with and without correction. The maximum temperature is achieved for all cases on the edge of the lamella, about 1/3 of the lamella length from the inboard side. The time evolution of this maximum temperature is shown in figure 27. When no correction is applied, the temperatures are much higher than the tungsten melting temperature (3400 °C). Only the corrected heat flux is consistent with the occurrence of transient melting during the ELMs.

Zoom In Zoom Out Reset image size

The application of the methodology developed in the previous section has led to an explanation for the mismatch observed for protruding lamellas in the JET tungsten outer divertor target. The original apparent reduction factors referred to the ratio $\frac{q_{\parallel\ {\rm forward}}}{q_{\parallel\ {\rm IR}}}$, representing how much the parallel heat flux density measured onto a standard lamella needed to be reduced in order to obtain the correct behaviour of the forward FEM thermal models of the special lamellas. The JET 2013 experiment required ratios of 0.2 in L-mode and 0.4 in H-mode [6].

For the L-mode case, the analysis of the new pulses along with the reinterpretation of the previous experiment has increased the ratio between the required forward load and the previous interpretation of $q_{\parallel}$ to  ∼0.5. Figure 28 represents the corrections applied to the standard lamella heat flux measurements, and shows how the impact of the corrected optical projection explains almost all of this observed difference. The parallel heat flux is similar for the twin pulses 89162 and 84514, and at the same time agrees with the results presented in [14] within an error margin of 10%. What is more, all the forward analyses are consistent with the corrected reconstruction of the parallel heat flux density using the IR temperatures, confirming the overestimation of $q_{\parallel}$ using the previous interpretation.

Zoom In Zoom Out Reset image size

The new H-mode analysis reveals a higher influence of the non-parallel heat component. The breakdown between inter-ELM and ELM contributions in the shadow region behind the sloped lamella concludes that ELMs are responsible for the increase of the non-parallel heat flux density component with respect to that calculated for L-mode shots.

The proposed correction of the parallel heat flux is based on the direct measurement of both the background and a non-parallel power density. The origin of this non-parallel fraction is unknown, and might be a combination of a number of possible contributors which this article does not attempt to unravel. Further physics studies, both experimental and theoretical, will be required to elucidate the potential mechanisms. Nevertheless, the correction has proven to work in modelling the behaviour of up to four different geometries in several plasma conditions. The relative values between the different measurements for each pulse are summarized in table 1.

Table 1. Ratio between the different measured and calculated components of $q_{\parallel}$, with q_n the heat flux density at the tile, measured at a standard lamella, and $q_{n, {\rm sh}}$ the heat flux density at the magnetic shadow.

Mode (pulse number)	$\frac{q_{n, {\rm sh}}}{q_{\rm n}}$	$\frac{q_{\nparallel}}{q_{n, {\rm sh}}}$	$\frac{q_{\nparallel}}{q_{\rm n}}$	$\frac{q_{\nparallel}}{q_{\parallel}}$
L-mode (89162)	0.22	0.94	0.20	0.012
H-mode (89445)	0.35	0.80	0.28	0.019

The ratios between the non-parallel and parallel heat flux may seem almost negligible—they are indeed close to the value of  ∼0.5% reported by much earlier studies on DIII-D [25]—but not when compared to the tile surface flux. Overall, it has been shown that the $q_{\nparallel}$ component plays a major role in the characterization of the heat flux density.

Once the errors found during the inverse analysis stage have been corrected, the impact of the proposed model on the divertor heat loading can be isolated. As shown in figure 28, a larger reduction of the parallel flux is found for the H-mode pulses. Figure 29 shows how the non-parallel fraction impacts the ratio between $q_{\parallel}$ and q_n at different solid angles. The relative effect with respect to the typical optical projection is more important as the impinging angle gets shallower. For typical high-power operations with an angle of 2°, the calculated L-mode parallel heat flux is 25% lower than expected (blue line in figure 29), while the reduction in H-mode goes up to 35% (green line in figure 29).

Zoom In Zoom Out Reset image size

The effects of the optical corrected projection for the JET divertor tile arrangement for a given SOL power are two-fold. First, the tile heat flux density during H-mode discharges is predicted to be 10% lower for the typical range of JET toroidal wetted fraction, TWF  =  $[0.6, 0.8]$, compared to the value given by the strict optical projection. Second, this corrected optical projection model leads to 1/3 less sensitivity to tolerance effects, as represented in figure 30 showing the effect of the impinging angle on the normalized heat flux density.

Zoom In Zoom Out Reset image size

Diagnostic issues cannot be ruled out in the measurement of $q_{\nparallel}$ in the shadowed lamella, but even if there were errors affecting the measurements at the shadowed region, it would not affect the important reduction of $q_{\parallel}$ reported here as the parallel heat flux is derived from the temperature measurements on the standard lamellas. These lamellas are monitored by several IR protection cameras from different angles—all of them agreeing within 50 °C. In addition, the reduction of the parallel heat flux is particularly required to explain the melting observations on the leading edge and sloped lamellas. The nature of $q_{\nparallel}$ will need to be confirmed in further experiments, since its effect on the reduction of divertor heat loads is currently not understood in terms of our knowledge of divertor physics and plasma transport.

The complicated parabolic profile of the standard lamella [26] shown in figure 8 is usually simplified either neglecting local shadowing effects [27], or—when including them—considering the load on the wetted surface as a constant heat flux [28]. For the present simulations, the non-flat surface shadow is modelled with a constant slope. Its behaviour is perfectly equivalent to the parabolic profile once the pixel size of the IR camera diagnostic measuring the surface temperature on JET is taken into account. To ensure consistency between this simplification and the real parabolic distribution, the toroidally integrated power density is matched between the two, also taking into account the variable shadow length produced by the poloidal variation of the impinging angle. Figure A1 shows the toroidal section of the 3D geometry used for forward analysis, along with its equivalent section on the inverse models. A direct integration of the power in both models leads to:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:fq} 5.5 q_{\rm 2D} = 1.5 q_{\rm n} + \frac{1.7}{\cos(\alpha)} q_{\rm slope}, \nonumber \end{align} \tag{ A.1 }$$

where a shadow length of 1.6 mm has been assumed.

Zoom In Zoom Out Reset image size

The real shadow depends on the impinging angle, and is variable along the lamella. Nevertheless, the 3D models used for the forward analyses have a constant shadow. The approximated geometry and shadow models have been validated, and are applicable whenever the thermal gradient inside the lamella is vertical. Since the heat flux density variation in time is smooth, this situation holds for all the L-mode forward simulations. For H-mode shots, the load in the sloped section is adjusted using the following rule:

when the exact—radially variable—length ($l_{n, {\rm shadow}}(r)$) is used, then the value of the load at a slope angle α can be calculated by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:real_shadow} q_{\rm slope} = \frac{5.5\ q_{\rm 2D} - 1.5\ q_{\rm n}}{3.3-l_{n,{\rm shadow}}} \cos(\alpha) = q_{\rm n} \frac{\sin(\theta+\alpha)}{\sin(\theta)}. \nonumber \end{align} \tag{ A.2 }$$

The assumption of a shadow with constant length (1.6 mm for the standard lamella was taken) is valid only if the power is matched between the two models. This condition allows the calculation of the equivalent load to be applied in the forward 3D models for the standard lamella:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle q_{\rm slope} = \frac{5.5\ q_{\rm 2D} - 1.5\ q_{\rm n}}{1.7}\ \cos(\alpha) = \frac{4.225}{1.7}\ q_{\rm 2D}\ \cos(\alpha). \nonumber \end{align} \tag{ A.3 }$$

Note that in order to satisfy the toroidal power balance, the exact length of the shadow needs to be calculated for complex heat flux profiles (i.e. using the real parabolic profile). It might be thought that these models would lead to a higher precision, but tests have shown that the numerical error due to the inaccurate application of the highly non-linear load leads to even poorer results. In the present case, where a constant slope and shadow length is used, the discretized model can be meshed to precisely match the shape of the loaded patch. If the real length of the shadow is required, its value can easily be obtained by substituting equation (A.1) into equation (A.2):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle l_{n,{\rm shadow}} = 3.3 - \frac{4.225}{0.85} \ \frac{\sin(\theta)\ \cos(\alpha)}{\sin(\theta+\alpha)}. \nonumber \end{align} \tag{ A.4 }$$

Synthetic comparisons between the 2D inverse models—with the geometrical factors—and the 3D forward analysis meshes have been performed to ensure the validity of the procedure used to extract the heat fluxes. An initial synthetic profile is defined for q_n which is projected onto a 3D finite element ANSYS [29] model of the lamella using variable magnetic impinging angles. The resulting 3D simulation temperature evolution is then plugged into ALICIA [16] to obtain $q_{\rm 2D}$, and the proposed geometrical factors are applied to calculate q_n.

It is worth clarifying the applicability of each model. Due to computational limitations, 2D models can have more refined meshes close to the surface of the lamella but neglect any toroidal heat flow inside the material. This initial consideration means that 2D models are in principle more suitable for capturing fast transients such as ELMs, but will lack the toroidal heat redistribution that appears on longer timescales. The element sizes required for the 2D ALICIA model to capture Heaviside functions representing worst-case ELMs are as low as 50 μm. This value also depends on the time step of the analysis, and is valid up to the maximum MWIR camera frequency, $f_{\rm max}=7.8$ kHz, used for H-mode shots.

For this validation case, the commonly employed convolution of a Gaussian with an exponential function [13] is used to represent the divertor poloidal heat flux density profile. An L-mode simulation is run for a 2 s duration. The H-mode consists of a shorter analysis but includes two ELM events of 1 ms duration at a frequency of 25 Hz.

The L-mode validation test shows that the 2D model suffers an initial overshoot as shown in figure A2 during the time when the toroidal temperature profile develops. The typical MTF spreading of the JET IR system [30] is simulated with a 5% smoothing of the temperatures. After a period of around 0.5 s, the heat flux is stabilized and the match between models is very good, confirming the validity of the geometric factors f_m and f_q. The main source of errors relates to the differences in shape at the high field side of the lamella, as shown in the comparison between the 2D model and the exact lamella geometry in figure 7.

Zoom In Zoom Out Reset image size

The H-mode validation has been also successful, with two different effects shown in figure A3 being worth explaining in detail:

• The overshoot in the L-mode validation due to the toroidal heat transient has a smaller effect, but its counterpart appears as an undershoot after an ELM which can lead to apparent negative fluxes of q_2D. Their effect is more cosmetic, since under- and overshooting compensate each other, leading to a consistent balance of energy.
• The ELM loading evolution does not match perfectly due to the limitations of the 3D model. It has been checked that the apparent damping of the load is due only to a numerical effect and not to any transient toroidal heat flow. Even with these limitations, the peak flux value is captured well over the expected time scale of JET ELMs. Since this effect appears at both the rise and fall of the ELM, the energy of the system is preserved in a similar fashion as in the case of overshoot, but on a much shorter timescale.

Zoom In Zoom Out Reset image size

Validation and comparison tests between 2D inverse codes can be found in [16] both for synthetic and real data.