2.1 Inclusion of multiple low-Z impurities

As in the original method (Sertoli et al. Reference Sertoli, Flanagan, Maslov, Maggi, Coffey, Giroud, Menmuir, Carvalho, Shaw and Delabie2018), light elements are included in the analysis assuming that their concentration profile is flat and symmetric on the flux surface $n_{Z_{1}}(\unicode[STIX]{x1D70C};t)=c_{Z_{1}}(t)\cdot n_{e}(\unicode[STIX]{x1D70C};t)$ . Due to their low mass and in ion temperature gradient (ITG) turbulence regimes such as those usually encountered at JET, these assumptions are supported by the current theoretical understanding (Angioni et al. Reference Angioni, McDermott, Fable, Fischer, Pütterich, Ryter and Tardini2011; Bonanomi et al. Reference Bonanomi, Mantica, Giroud, Angioni, Manas and Menmuir2018; Bourdelle et al. Reference Bourdelle, Camenen, Citrin, Marin, Casson, Koechl and Maslov2018; McDermott et al. Reference McDermott, Dux, Pütterich, Geiger, Kappatou, Lebschy, Bruhn, Cavedon, Frank and den Harder2018; Kappatou et al. Reference Kappatou, McDermott, Angioni, Manas, Pütterich, Dux, Viezzer, Jaspers, Fischer and Dunne2019).

The concentration is free to evolve in time and is calculated from the visible Bremsstrahlung LoS-averaged $Z_{\text{eff}}$ measurement (Behringer et al. Reference Behringer, Carolan, Denne, Decker, Engelhardt, Forrest, Gill, Gottardi, Hawkes and Källne1986) accounting for contributions of all the other impurities to the LoS integral,

(2.2) $$\begin{eqnarray}c_{Z_{1}}(t)=\frac{L\cdot (Z_{\text{eff}}(t)-1)-\displaystyle \mathop{\sum }_{s\neq Z_{1}}\displaystyle \int _{L}\frac{n_{s}(\unicode[STIX]{x1D70C},R;t)}{n_{e}(\unicode[STIX]{x1D70C};t)}\left[\langle q_{s}(\unicode[STIX]{x1D70C};t)\rangle ^{2}-\langle q_{s}(\unicode[STIX]{x1D70C};t)\rangle \right]\,\text{d}l}{\displaystyle \int _{L}\left[\langle q_{s}(\unicode[STIX]{x1D70C};t)\rangle ^{2}-\langle q_{s}(\unicode[STIX]{x1D70C};t)\rangle \right]\,\text{d}l}\end{eqnarray}$$

$n_{s}$ and $n_{e}$ are the impurity and electron densities respectively, $\langle q_{s}\rangle$ the impurity average charge (in units of the elementary charge), $L$ the LoS length and the sum is performed over all impurities excluding $Z_{1}$ . Differently from Sertoli et al. (Reference Sertoli, Flanagan, Maslov, Maggi, Coffey, Giroud, Menmuir, Carvalho, Shaw and Delabie2018), the contribution from each impurity is thus modelled self-consistently along the LoS accounting for poloidal asymmetries of the mid-/high-Z impurity densities $n_{s}(\unicode[STIX]{x1D70C},R;t)$ . In this and later equations, quantities assumed symmetric on the flux surface (impurity average charge, electron temperature and density, factors, etc.) will have a radial dependence $(\unicode[STIX]{x1D70C})$ , while those which may exhibit poloidal asymmetries are estimated in two dimensions on the $(R,z)$ plane and will have a radial dependence $(\unicode[STIX]{x1D70C},R)$ . The flux-surface label $\unicode[STIX]{x1D70C}$ is retained instead of $z$ in order not to lose the information on the parent flux surface. The flux-surface label $\unicode[STIX]{x1D70C}$ used in from now on corresponds to the normalized poloidal flux coordinate $\unicode[STIX]{x1D70C}_{pol}=\sqrt{(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{a})/(\unicode[STIX]{x1D713}_{s}-\unicode[STIX]{x1D713}_{a})}$ , where indices $s$ and $a$ refer to the separatrix and the magnetic axis respectively.

A secondary low-Z impurity $Z_{2}$ can be added with the additional constraint that the concentration is constant in time $n_{Z_{2}}(\unicode[STIX]{x1D70C},t)=c_{Z_{2}}\cdot n_{e}(\unicode[STIX]{x1D70C},t)$ . For the case of JET-ILW discharges without impurity seeding, Be is used as primary low-Z ( $Z_{1}$ ). If instead seeding takes place, Be is set as the constant background ( $Z_{2}$ ) and the seeding gas is set as primary and left free to evolve in time. The constant Be concentration is calculated from the $Z_{\text{eff}}$ measurement prior to the seeding phase assuming it is the only other low-Z impurity. Typical concentrations are of the order of 0.5 %–2.0 %.

After having applied (2.1) with the chosen low-Z impurities, if the main impurity $Z_{0}$ is tungsten the first guess of the impurity density is re-scaled comparing the result with the independent measurement from VUV spectroscopy (Schwob et al. Reference Schwob, Wouters, Suckewer and Finkenthal1987; Pütterich et al. Reference Pütterich, Neu, Dux, Whiteford and O’Mullane2008),

(2.3) $$\begin{eqnarray}n_{Z_{0}}(\unicode[STIX]{x1D70C},R;t)=\langle c_{W}(t)/c_{W}^{SXR}(t)\rangle _{t}\cdot n_{Z_{0}}^{SXR}(\unicode[STIX]{x1D70C},R;t),\end{eqnarray}$$

where $c_{W}$ is the VUV spectrometer measurement and $c_{W}^{SXR}$ the simulated one integrating $n_{W}^{SXR}$ from (2.1) along the LoS of the VUV spectrometer and weighing it on the fractional abundance envelope of the measured ionization stages. This provides a first consistency check of the absolute W density with the assumption that it is the major contributor to the SXR emissivity, but accounting for the fact that contributions from other impurities might be present. For further details, see § II A of Sertoli et al. (Reference Sertoli, Angioni and Odstrcil2017) and references therein.

This density is then used to evaluate a re-calibration factor for the SXR emissivity,

(2.4) $$\begin{eqnarray}\displaystyle M & = & \displaystyle \left\langle \frac{\unicode[STIX]{x1D716}^{SXR}(\unicode[STIX]{x1D70C},R;t)}{\unicode[STIX]{x1D716}_{exp}^{SXR}(\unicode[STIX]{x1D70C},R;t)}\right\rangle \nonumber\\ \displaystyle & = & \displaystyle \left\langle \frac{n_{e}(\unicode[STIX]{x1D70C};t)\;\left[n_{I}(\unicode[STIX]{x1D70C};t)\;L_{I}^{SXR}(\unicode[STIX]{x1D70C};t)+\displaystyle \mathop{\sum }_{s}n_{s}(\unicode[STIX]{x1D70C},R;t)\;L_{s}^{SXR}(\unicode[STIX]{x1D70C};t)\right]}{\unicode[STIX]{x1D716}_{exp}^{SXR}(\unicode[STIX]{x1D70C},R;t)}\right\rangle _{(\unicode[STIX]{x1D70C},R;t)},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{exp}^{SXR}$ is the result of the deconvolution routine and the average is performed over the whole time range of interest and over the plasma radius within the range of validity of the analysis, and contributions of all other secondary impurities $s$ are accounted for. The process is iterated a few times until convergence (two iterations have been found to suffice) after which the re-scaling versus the W concentration measurement is abandoned and the SXR multiplication factor is the only correction that is retained. If the main impurity is not W, then this multiplication factor is set to $M=3$ , a value previously estimated by comparing the SXR emissivity with bolometry in plasmas with a dominant low-Z impurity and negligible contributions from mid-/high-Z elements (Pütterich et al. Reference Pütterich, Dux, Beurskens, Bobkov, Brezinsek, Bucalossi, Coenen, Coffey, Czarnecka and Giroud2012). As a confirmation of this previous estimate, multiplication factors of the order of $2.5{-}3.5$ have been found for all those cases that show good consistency with other diagnostic measurements, where the main impurity is W and the W concentration measurements from VUV was available. This re-calibration factor $M$ is JET-specific and its origin is still not understood. It could be tied to uncertainties in the Be-filter thickness, in the atomic data or in the diagnostic acquisition, but the fact that it is constant over time and independent of plasma scenario legitimizes treating it as a re-calibration factor.

After having calculated the multiplication factor, equation (2.1) is re-applied and the whole procedure repeated until convergence (two more times are typically enough). Consistency checks with independent measurements are then applied to falsify or confirm the results obtained so far.

2.2 Consistency checks versus independent measurements

On top of the consistency checks already used in the original method (Sertoli et al. Reference Sertoli, Flanagan, Maslov, Maggi, Coffey, Giroud, Menmuir, Carvalho, Shaw and Delabie2018) which are listed below as (i–iii), a few more have been implemented to provide further robustness to the methodology:

1. (i) if the main impurity $Z_{0}$ is tungsten, compare its time evolution with the independent concentration measurement from VUV spectroscopy;

2. (ii) from the observed LFS-high-field-side (HFS) asymmetry of $Z_{0}$ evaluate the expected toroidal rotation and compare with CXRS measurements;

3. (iii) including contributions from all ions, calculate the total radiation (in two dimensions) and compare with estimates tomographic reconstructions of the bolometry diagnostic;

4. (iv) integrate the estimated total radiation along the LoS of the bolometry diagnostic and compare with the experimental values;

5. (v) if a seeding gas is included, compare the time evolution and absolute value of the independent concentration measurement from CXRS.

While checks (i)–(iii) and (v) are relatively straightforward, comparing with LoS integrals of bolometry requires an extrapolation of the results beyond the range of validity of the analysis. This is a delicate part and will be explained in the next section.

2.3 Inclusion of a secondary mid-/high-Z impurities

If the solution of (2.1) including only low-Z impurities yields estimates that do not match the consistency checks within the estimated experimental uncertainties, one more impurity $Z_{3}$ can be added to the mix. This can be a mid-/high-Z element and its time evolution, profile shape and asymmetry are modelled on the basis of $n_{Z_{0}}$ with the following assumptions:

1. (i) both impurities $Z_{0}$ and $Z_{3}$ are at equilibrium, the impurity sources and sinks do not change during the time range of analysis and the time averaging is long enough (typically 20 ms) such that the net particle flux through the separatrix is zero. If this is not the case, the sources and sinks of both impurities are assumed to behave identically;

2. (ii) the central peaking of $Z_{0}$ and $Z_{3}$ is driven by neoclassical processes;

3. (iii) poloidal asymmetries are driven mainly by centrifugal effects.

If the peaking of the impurity density profiles in the plasma centre is neoclassically driven, as it typically is in the centre of JET H-mode plasmas (Angioni et al. Reference Angioni, Casson, Mantica, Pütterich, Valisa, Belli, Bilato, Giroud and Helander2015; Casson et al. Reference Casson, Angioni, Belli, Bilato, Mantica, Odstrcil, Pütterich, Valisa, Garzotti and Giroud2015), the radial impurity flux can be modelled following Breton et al. (Reference Breton, Casson, Bourdelle, Angioni, Belli, Camenen, Citrin, Garbet, Sarazin and Sertoli2018b , equation (7)). If the particle flux is zero and the profiles are in equilibrium, the normalized impurity density gradient can then be calculated as,

(2.5) $$\begin{eqnarray}\left\langle \frac{R}{L_{n_{s}}}\right\rangle _{FSA}\propto q_{s}\left[\frac{R}{L_{n_{I}}}+\frac{R}{L_{T_{I}}}\left(\frac{H}{K}+\frac{H_{0}f_{c}}{K}\frac{P_{B}}{P_{A}}\right)\right],\end{eqnarray}$$

where $R$ is the tokamak major radius, $L_{A}=A/\unicode[STIX]{x1D735}A$ the gradient length of quantity $A$ , $m_{I}$ the mass of the main ion $I$ and $T_{I}$ its temperature, $q_{s}$ the impurity charge, $H$ , $H_{0}$ and $K$ factors that depend on the collisionality, $f_{c}$ the fraction of circulating particles, $P_{A}$ and $P_{B}$ the geometrical factors related to poloidal asymmetries (Casson et al. Reference Casson, Angioni, Belli, Bilato, Mantica, Odstrcil, Pütterich, Valisa, Garzotti and Giroud2015; Breton et al. Reference Breton, Casson, Bourdelle, Angioni, Belli, Camenen, Citrin, Garbet, Sarazin and Sertoli2018b ). All quantities are flux-surface averages (FSA), including the left-hand side of the equation. The subscript $FSA$ will be neglected from now on, but the symbol $\langle \,\rangle$ will retain the meaning of flux-surface-averaged quantity.

For peaked profiles, the pinch term proportional to $R/L_{n_{I}}$ will dominate over both the ion temperature screening term proportional to $R/L_{T_{I}}$ as well as turbulence. Neglecting at first order the temperature screening term, the normalized impurity density gradient becomes directly proportional to the main ion gradient multiplied by the impurity charge $q_{s}$ . Using the previously estimated density profile of impurity $Z_{0}$ , the gradients of the secondary impurity $Z_{3}$ can be calculated as,

(2.6) $$\begin{eqnarray}\left\langle \frac{R}{L_{n_{Z_{3}}}}\right\rangle \approx \frac{q_{Z_{3}}}{q_{Z_{0}}}\left\langle \frac{R}{L_{n_{Z_{0}}}}\right\rangle .\end{eqnarray}$$

Inverting this equation will then yield the shape of the impurity density profile,

(2.7) $$\begin{eqnarray}\langle n_{Z_{3}}\rangle =\exp \left[\displaystyle \int _{\unicode[STIX]{x1D70C}}\left\langle \frac{1}{L_{n_{Z_{0}}}}\right\rangle \frac{q_{Z_{3}}}{q_{Z_{0}}}\,\text{d}r\right].\end{eqnarray}$$

Since this re-adaptation of the flux-surface-averaged impurity density gradient is not performed in absolute terms with respect to $R/L_{n_{I}}$ , but relative to the primary impurity $Z_{0}$ whose gradient is determined experimentally, second-order effects mitigating the impurity peaking coming from neoclassical temperature screening and turbulence are still somewhat retained.

In regions of the plasma where the profile of impurity $Z_{0}$ is flat or hollow, then neoclassical temperature screening or turbulence will dominate over the neoclassical pinch term. In JET-ILW plasmas the former is usually responsible for effects close to the plasma centre, while the latter is typically the dominating transport mechanism for radii $\unicode[STIX]{x1D70C}>0.3{-}0.4$ and will lead to profiles $n_{s}\propto n_{e}$ (Angioni et al. Reference Angioni, Casson, Mantica, Pütterich, Valisa, Belli, Bilato, Giroud and Helander2015; Casson et al. Reference Casson, Angioni, Belli, Bilato, Mantica, Odstrcil, Pütterich, Valisa, Garzotti and Giroud2015; Bourdelle et al. Reference Bourdelle, Camenen, Citrin, Marin, Casson, Koechl and Maslov2018). In this case, the flux-surface-averaged impurity density gradient of $Z_{3}$ will be set equal to that of the primary high-Z impurity $L_{n_{Z_{3}}}=L_{n_{Z_{0}}}$ .

For the determination of the absolute value and time evolution, assumption $(i)$ provides the necessary framework by imposing that the flux-surface-averaged density profile of $Z_{3}$ follows closely that of $Z_{0}$ . The result of (2.7) is thus normalized by setting the total number of $Z_{3}$ ions in the confined region equal to that of $Z_{0}$ at each time point,

(2.8) $$\begin{eqnarray}\langle n_{Z_{3}}\rangle =\langle n_{Z_{3}}\rangle \frac{\displaystyle \int \langle n_{Z_{0}}\rangle \,\text{d}V}{\displaystyle \int \langle n_{Z_{3}}\rangle \,\text{d}V}\end{eqnarray}$$

and finally multiplying by a constant $C$ which is a single number for the whole time range of analysis

(2.9) $$\begin{eqnarray}\langle n_{Z_{3}}\rangle =C\cdot \langle n_{Z_{3}}\rangle .\end{eqnarray}$$

This constant has the meaning of the fraction of the total impurity density of $Z_{3}$ with respect to $Z_{0}$ . This is evaluated applying consistency check (d) comparing the LoS integrals of the bolometric diagnostic with its synthetic value estimated from the two-dimensional (2-D) maps of the total radiation calculated including contributions from the main ion and all impurities,

(2.10) $$\begin{eqnarray}\frac{1}{C}=\mathop{\left\langle \frac{1}{N}\displaystyle \mathop{\sum }_{k=1}^{N}\frac{\displaystyle \int _{L_{k}}\left[n_{e}(\unicode[STIX]{x1D70C};t)\left(n_{I}(\unicode[STIX]{x1D70C};t)L_{I}^{tot}(\unicode[STIX]{x1D70C};t)+\mathop{\sum }_{s}n_{s}(\unicode[STIX]{x1D70C},R;t)L_{s}^{tot}(\unicode[STIX]{x1D70C};t)\right)\right]\text{d}l}{B_{k}^{\text{bolo}}(t)}\right\rangle }\nolimits_{\!t},\end{eqnarray}$$

where the sum is performed over $N$ lines of sight of length $L_{k}$ and $B_{k}^{\text{bolo}}$ are the experimental LoS-integrals. Since the method for the equation can be applied above $T_{e}(\unicode[STIX]{x1D70C}_{\max })=1.5~\text{keV}$ , the densities $n_{Z_{0}}$ and $n_{Z_{3}}$ must be extrapolated from $\unicode[STIX]{x1D70C}_{\max }$ up to the separatrix. This can be done in a number of ways, including scaling the electron density over the whole region $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D70C}_{\max }$ or extrapolating the value or the derivative up to a specified radius $\unicode[STIX]{x1D70C}_{\text{interp}}$ and then following the shape of the electron density scaled to match the impurity density at $\unicode[STIX]{x1D70C}_{\text{interp}}$ . All of these methods are performed on the mid-plane values of the 2-D impurity densities, independently for LFS and HFS, to maintain the observed asymmetry at $\unicode[STIX]{x1D70C}_{\max }$ . For the pedestal region, in order to artificially compensate for the underestimated cooling functions calculated neglecting transport, the impurity density is slightly enhanced by increasing the pedestal value to $n_{Z_{0}}(\unicode[STIX]{x1D70C}>0.9)=(n_{Z_{0},n_{e}}+n_{Z_{0}}(\unicode[STIX]{x1D70C}>0.9))/2$ .

The last point to clarify is how to evaluate the poloidal asymmetry of $Z_{3}$ . This is straightforward if assumption (iii) is valid and an estimate of the toroidal rotation profile is available. Following equation (9) of Odstrcil et al. (Reference Odstrcil, Pütterich, Angioni, Bilato, Gude, Odstrcil and Team2018) and neglecting the asymmetries driven by fast particles (first term on the right-hand side), the impurity density on a flux surface $\unicode[STIX]{x1D70C}$ at a generic mayor radius $R$ can be calculated with respect to the value at a reference major radius $R_{0}$ ,

(2.11) $$\begin{eqnarray}n_{s}(\unicode[STIX]{x1D70C},R;t)\approx n_{s}(\unicode[STIX]{x1D70C},R_{0};t)\exp \left[\unicode[STIX]{x1D706}_{s}(\unicode[STIX]{x1D70C};t)(R(\unicode[STIX]{x1D70C};t)^{2}-R_{0}(\unicode[STIX]{x1D70C};t)^{2})\right],\end{eqnarray}$$

where the asymmetry parameter $\unicode[STIX]{x1D706}_{s}$ is fully determined by knowing the main ion and impurity masses ( $m_{I}$ and $m_{s}$ ), the average impurity charge $\langle q_{s}\rangle$ , the effective charge $Z_{\text{eff}}$ , the electron and ion temperatures $T_{e,I}$ and the toroidal rotation frequency $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}$ ,

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{s}(\unicode[STIX]{x1D70C};t)\approx \frac{m_{s}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70C};t)^{2}}{2T_{I}(\unicode[STIX]{x1D70C};t)}\left(1-\frac{\langle q_{s}(\unicode[STIX]{x1D70C};t)\rangle }{m_{s}}\frac{m_{I}Z_{\text{eff}}(\unicode[STIX]{x1D70C};t)T_{e}(\unicode[STIX]{x1D70C};t)}{T_{I}(\unicode[STIX]{x1D70C};t)+Z_{\text{eff}}(\unicode[STIX]{x1D70C};t)T_{e}(\unicode[STIX]{x1D70C};t)}\right).\end{eqnarray}$$

The impurity density at the chosen reference radius $R_{0}$ on the LFS mid-plane can then be estimated by inverting (2.11) and calculating the flux-surface-averaged value of the exponent,

(2.13) $$\begin{eqnarray}n_{s}(\unicode[STIX]{x1D70C},R_{0};t)=\frac{n_{s}(\unicode[STIX]{x1D70C};t)}{\langle \exp [\unicode[STIX]{x1D706}_{s}(\unicode[STIX]{x1D70C};t)(R(\unicode[STIX]{x1D70C};t)^{2}-R_{0}(\unicode[STIX]{x1D70C};t)^{2})]\rangle }.\end{eqnarray}$$

All other quantities in (2.12) are already included in the analysis or self-consistently calculated using the available profile data.

The toroidal rotation and ion temperature are taken from CXRS measurements on impurity ions (e.g. $Ne\,X$ ) and are assumed equal for all impurities. This assumption is consistent with previous findings (Baylor et al. Reference Baylor, Burrell, Groebner, Houlberg, Ernst, Murakami and Wade2004; Lebschy Reference Lebschy2018; Grierson et al. Reference Grierson, Chrystal, Haskey, Wang, Rhodes, McKee, Barada, Yuan, Nave and Ashourvan2019) and can be confirmed here for consistency using the radial force balance equation $E_{r}=\unicode[STIX]{x1D735}p/qn+v_{\unicode[STIX]{x1D719}}B_{\unicode[STIX]{x1D717}}-v_{\unicode[STIX]{x1D717}}B_{\unicode[STIX]{x1D719}}$ . Since the radial electric field will be equal for all species, with the assumption that their temperatures are the same, that their peaking follows (2.6) and that the poloidal flow term is negligible at first order, the difference between the toroidal velocities of two ions will be,

(2.14) $$\begin{eqnarray}v_{\unicode[STIX]{x1D719},s_{1}}-v_{\unicode[STIX]{x1D719},s_{2}}\approx -\frac{\unicode[STIX]{x1D735}T}{B_{\unicode[STIX]{x1D717}}}\left(\frac{q_{s_{2}}-q_{s_{1}}}{q_{s_{2}}q_{s_{1}}}\right),\end{eqnarray}$$

with $\unicode[STIX]{x1D735}T$ given in $(\text{eV}/\text{m})$ and $B_{\unicode[STIX]{x1D717}}(T)$ . This leads to small differences for impurity ions, and slightly larger values when comparing impurities and main ion rotation. For the example discharge shown in the next section, the toroidal rotation difference between the main ion and the measured neon is expected to be of the order of a few tens of $\text{km}~\text{s}^{-1}$ , while the difference between tungsten or nickel turns out to be a only a few $\text{km}~\text{s}^{-1}$ , i.e. less than a per cent of the measured toroidal rotation and therefore negligible.

3.1 Effects on Zeff

The different assumptions for the impurity content lead to noticeable changes in the contribution to the $Z_{\text{eff}}$ LoS integrated measurement (figure 5 a). The low-Z impurities (Be + Ne) are in general the major contributors (short dashes). Tungsten contributes little because of its low concentration (continuous lines in figure 5 c), reaching a maximum of $\unicode[STIX]{x0394}Z_{\text{eff}}\sim 0.2$ if its density is re-scaled to fill in for the missing radiated power (case (ii)). When nickel is included (case (iii)), it becomes a major contributor towards the end of the accumulation process, accounting for approximately half of the measured LoS-integrated value. The neon concentration (continuous red line in figure 5 b) is therefore lowest in case (iii) (approximately a factor 2), having to account for less $Z_{\text{eff}}$ than in the other two cases. When not including nickel, the match of the estimated neon concentration with the radially averaged CXRS data (dashed line) is way outside the confidence interval of the CXRS measurement which has been estimated propagating a $15\,\%$ error with the standard deviation of all radial points. With the contribution of nickel (case (iii)), the match is instead extremely good for the whole time range of analysis.

3.2 Effects on total radiation

The flux-surface-averaged W concentration (continuous lines in figure 5 c) reaches a maximum of ${\sim}3{-}4\times 10^{-4}$ in the centre (red) when normalized to the VUV concentration measurement (cases (i) and (iii)) changing slightly if nickel is included in the mix (case (iii)). Double that amount is needed to improve the match with bolometric measurements (figure 5 d) as in case (ii). It is clear from these latter plots that the match of the back LoS-integrals (dashed lines) with the experimental data for three LoS (colour coded) over the whole time range of interest is best for case (iii). The total amount of Ni needed to match the bolometer signals is approximately one order of magnitude larger than that of W (dashed lines, case (iii) figure 5 c).

The consistency check with bolometry is further enhanced comparing all available LoS not viewing the divertor (figure 6 a). This comparison clearly shows that increasing the W density to match the central LoSs does not provide a satisfactory match neither for horizontal LoSs viewing the top and bottom of the plasma, nor for the vertical LoS viewing the HFS. The situation improves considerably when including nickel, with an almost perfect match for all LoS of the vertical camera and a very good agreement for the horizontal camera. The reasons for are that nickel exhibits a lower peaking factor and lower centrifugal asymmetry due to its lower mass and charge (equations (2.6) and (2.11)). This can be seen in the plots of the W and Ni density profiles at mid-plane at the time points of interest for case (iii) (figure 7 a,d). Including also effects from the shape of the cooling factor, with nickel radiating less at higher electron temperatures than tungsten, a global reduction in radiation peaking and an increased radiation on the HFS will take place, improving the match with the bolometer LoSs viewing away from the plasma centre.

Figure 7.

Final profile plots for case (iii) including nickel: (a) W density at mid-plane (positive $\unicode[STIX]{x1D70C}$ corresponding to LFS, negative to HFS); (b) W density asymmetry (dashed line with confidence band) compared with the estimated value using CXRS measured toroidal rotation; (c) flux-surface-averaged total radiated power and contributions from the different impurities (see label); (d) Ni density at mid-plane; (e) Ni density asymmetry (dashed line with confidence band) compared with the estimated value using CXRS measured toroidal rotation; (f)  $Z_{\text{eff}}$ profile and contributions from the various impurities (see label).

3.3 Match with CXRS Ne concentration

The comparison of the estimated Ne concentration with the CXRS profiles (figure 6 b) confirms the good match already seen in the time evolution, but at the same time reveals a peaking in the centre not included in the analysis because of the flat concentration constraint. This experimental point could be an outlier and two independent considerations corroborate this opinion.

The error estimation of the CXRS rotation and temperature measurements relies only on the fit of the spectral lines, while the estimation of the impurity concentration requires knowledge of the beam characteristics including the beam attenuation whose uncertainty is very large close to the plasma centre (Giroud et al. Reference Giroud, Meigs, Negus, Zastrow, Biewer and Versloot2008). Moreover, the most central CXRS radial measurement at $\unicode[STIX]{x1D70C}\sim 0.05$ (not shown here) has be independently excluded from the analysis due to extremely noisy ion temperature and toroidal rotation result. This means that the beam attenuation in this region is already very large and errors in its estimation could lead to large changes in the calculated impurity density.

The second point is that the experimental peaking is much larger than that expected when calculating the Ne peaking using (2.6). Normalizing the profiles to mid-radius, the expected central value should be $c_{Ne}(\unicode[STIX]{x1D70C}=0.15)\sim 0.76$ , $0.75$ and $0.77$ for the three time points respectively. While the difference for the first time point (black) is within the error bars, for the latter two the experimental values are well beyond the $2\unicode[STIX]{x1D70E}$ boundary.

3.4 Match with VUV

Calculating the expected W concentration measurement from VUV spectroscopy and comparing it with the experimental values (figure 6 c), the best match with the quasi-continuum measurement $c_{W,qc}$ (triangles) is obtained only when accounting for nickel contributions (case (iii)). The reason for this is that the presence of nickel accounts for part of the SXR emissivity that was initially ascribed to W, changing the shape of the final W density profile, especially in the later phase of the analysed time range. The match with the spectral line measurement $c_{W,l}$ is instead not optimal for any of the three cases. This is most probably caused by uncertainties in the equilibrium reconstruction: errors in the Shafranov shift will have a large impact on this measurement, but will be more forgiving on the quasi-continuum value which measures further out.

3.5 Match with toroidal rotation

The final consistency check is between the measured CXRS toroidal rotation of $Ne\,X$ and the value estimated from the W density asymmetry modelled inverting (2.11) to extract $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}$ . This match is relatively good for all cases (figure 6) and this consistency check alone would not be sufficient to distinguish between the three. This is apparent when considering the uncertainties in the W density asymmetry and viewing the problem in the opposite direction comparing the measured W density asymmetry with the expected one estimated using the CXRS toroidal rotation measurements (figure 7 b). It is clear from this plot that the slight differences in asymmetry are all within the confidence interval and therefore cannot be sufficiently discriminated. The errors have been estimated propagating the error of the W density, calculated by propagating the uncertainties of the deconvoluted SXR emissivity ( $5\,\%$ error from the raw SXR data and deviation of the back fit with respect to the data) and those of the cooling factors (estimated from upper and lower estimates of the electron temperature given the diagnostic uncertainties and the standard deviation of data points if time averaging of the raw data is performed). This is clearly a lower estimate, since it does not include uncertainties in either the W atomic data or those of other elements included in the analysis.

Figure 8.

Time evolution and profile consistency for case (iv) (Ni only): (a) experimental $Z_{\text{eff}}$ measurement $ZEFH$ (red) compared with the back LoS-integrated value (thick black line) and single contributions from Be+Ne and Ni; (b) radially averaged experimental Ne concentration from CXRS (dashed line with confidence interval) compared with the estimated value (continuous line); (c) Ni density asymmetry (dashed line with confidence band) compared with the estimated value using CXRS measured toroidal rotation; (d) experimental bolometric LoS-integrated measurements (points with error bars) for all LoS shown in figure 1 compared with the back LoS-integrated values (continuous line).

3.6 Nickel as main high-Z impurity

As a final confirmation that including nickel in the impurity mix provides the best match to all measurements but that the presence of W is of utmost importance and cannot be neglected, figure 8 shows what happens if nickel is the only mid-/high-Z impurity $Z_{0}$ . Accounting for most of the SXR emission, the Ni contribution to $Z_{\text{eff}}$ dominates the measurement already half-way during the time range of interest (figure 8 a). The Ne concentration therefore decreases, matching relatively well the CXRS measurement up to ${\sim}5~(\text{s})$ , but going outside of the CXRS confidence interval for later time points. The match with the bolometer LoS-integrals (figure 8 d) is marginally good, but still worse than case (iii) (figure 6 a). Finally, the consistency check that clearly falsifies the assumption that Ni is the only mid-/high-Z impurity, is the comparison with toroidal rotation (figure 8 c) where it is clear that the asymmetry in the SXR emissivity is far too large to be ascribed to nickel.

3.7 Impact of other diagnostic uncertainties

Discrepancies between different diagnostics or LoS measurements are known to take place and it is sometimes unclear which one should be discarded and which one retained for the analysis. In this framework, the main discrepancies that have to be addressed are between LIDAR and HRTS for the electron profile diagnostic and between the vertical and horizontal lines of sight of the $Z_{\text{eff}}$ visible bremstrahlung measurement.

Using LIDAR instead of HRTS for the electron profile fits (figure 3 c), due to the slightly lower density, a lower background $Be$ concentration ( $1.5\,\%$ ) is necessary to match the initial measured $Z_{\text{eff}}$ (still using ZEFH). With this change only, the final results for Ne, W and Ni are all recovered to within a few per cent and, since the results are well within the final uncertainties, no relevant information can be provided to discard either of the profile diagnostics.

Figure 9.

Case (iii) analysed using $Z_{\text{eff}}$ measurement ZEFV instead of ZEFH. Radial profiles of (a) electron temperature and (b) density (data from HRTS shown as points, fits as continuous lines); time evolution of (c) radially averaged experimental Ne concentration from CXRS (dashed line with confidence interval) compared with the estimated value (continuous line) and (d) flux-surface-averaged W and Ni concentrations (continuous and dashed lines respectively) at three different radial positions (colour coded as in label). (e) W density asymmetry (dashed line with confidence band) compared with the estimated value using CXRS measured toroidal rotation; (f) experimental bolometric LoS-integrated measurements (points with error bars) for all LoS shown in figure 1 compared with the back LoS-integrated values (continuous line).

The situation is slightly more complicated when using the vertical LoS in place of the horizontal one (black and red traces in figure 2 h). The ZEFV measurement is higher by ${\sim}20\,\%$ with respect to ZEFH up to around 6 s (figure 2 h) and well outside of the statistical uncertainty ( ${<}5\,\%$ ). In order to match this $Z_{\text{eff}}$ value before the neon puff, a Be concentration of the order of $6\,\%$ is necessary. Using this value as a starting point leads to results that are very similar to the original case, for both the W and Ni concentrations (figure 9 d) as well as for the W density asymmetry (figure 9 e) and total radiated power (figure 9 f). The only consistency check that fails dramatically is that with the Ne concentration. The neon required to match the $Z_{\text{eff}}$ measurement is approximately a factor 2 higher than the one estimated by CXRS and well outside the confidence interval up to ${\sim}5.8~\text{s}$ . Beyond this time, when nickel starts contributing substantially to $Z_{\text{eff}}$ , the concentration drops, going well below the CXRS estimates. This suggests that the vertical measurement is affected by an unknown systematic uncertainty that is not constant in time. As stated earlier in the paper, reflections on the machine wall could lead to an increased value and could be a possible cause of this discrepancy.

These results therefore confirm the choice of ZEFH as the most trustworthy measurement for the effective charge and underline the importance of multiple consistency checks in an integrated framework for the calculation of the plasma composition. If the independent CXRS measurement of the neon concentration were not available, another way to assess the reliability of the background Be concentration would be to model the neutron rate (e.g. using TRANSP (Ongena et al. Reference Ongena, Voitsekhovitch, Evrard and McCune2012)) which would be affected by the dilution which would increase of $20\,\%$ when increasing the Be concentration from 2 % to 6 %.