A novel method of studying the core boron transport at ASDEX Upgrade

The experiments in this work were conducted at AUG, which is a medium-sized, tungsten walled, divertor tokamak situated at the Max-Planck Institute for Plasma Physics in Garching, Germany [17]. The major and minor radii of the machine are 1.65 m and 0.5 m, respectively, and it is equipped with a comprehensive suite of diagnostics as well as flexible and powerful heating systems including neutral beam injection (NBI), electron cyclotron resonance heating (ECRH), and ion cyclotron resonance frequency (ICRF) heating [18]. At AUG, it was observed that a modulation of the power of the boron- and tungsten-coated ICRF antennas results in a modulation of the boron density in the plasma. This can be clearly seen in figure 1, where the ICRF heating power is presented in blue and the resultant boron modulation signal in red for one CXRS channel at ρ_tor = 0.76. The boron modulation is stronger at the edge than in the core and the edge also responds more quickly than the core to the change in the ICRF heating power, which indicates a change in the boron source from the edge plasma rather than a change in core transport, as the measurement of the boron influx at the limiters also increases when the power of the antennas is modulated. Therefore, this technique can be used for core boron transport studies. The exploration of this possibility is the subject of this work.

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It was discovered already in the 1980s that the operation of ICRF antennas can cause an increased influx of metallic impurities [19, 20], which arises due to the RF sheath potentials in the Faraday screen gaps [21, 22]. The ICRF concept in this work is similar to the work in [23], in which the tungsten transport at the edge was studied by modulating the power of the ICRF antennas. The ICRF heating system at AUG has been updated and is now comprised of four antennas in pairs of two: two 2-strap boron-coated and two 3-strap tungsten-coated antennas [24]. In this study, all the ICRF antennas are operated in phase and the ICRF heating scheme used is the hydrogen minority heating with a frequency of the ICRF generators of 36.5 MHz.

A feasibility study was conducted to better understand the boron source and if the technique can be used for transport studies. At AUG, boron is considered to be an intrinsic impurity due to the regularly performed boronizations, during which the vessel wall is covered with a thin layer of boron [25, 26]. It has been observed that the boron modulation signal is strongest when the experiments are performed in a freshly boronized machine. This suggests that the ICRF heating power modulation affects the boron coating on the antennas which originates from the boronization and not the boron of the antenna itself. This hypothesis is strengthened by the fact that a strong modulation signal is also achieved when only modulating the tungsten-coated ICRF antennas. Furthermore, only a modulation of the ICRF antennas have an impact on the boron density, since attempts at modulating the ECRH at various power levels did not result in a boron modulation. To enable boron CXRS measurements of good quality at AUG, a minimum of 2.5 MW of NBI power is necessary when performing these experiments. The feasibility study also demonstrated that this technique is applicable in a broad variety of H-mode, lower single null plasma conditions, although only one experimental dataset is shown in this paper. Apart from that the method is insensitive to the current, magnetic field, and plasma parameters. So far, the ICRF modulation has only been observed to affect the boron content in the plasma; helium, carbon, and nitrogen show no modulation behavior. On the other hand, the concentrations of these impurities were very low, making a possible modulation perhaps beyond detection. All in all, this method may be applicable in other machines which have carbon walls or also utilize boronization as a wall conditioning technique.

A requirement for the feasibility of this technique is a steady plasma background, which means keeping the electron density, the ion, and the electron temperatures constant such that the transport coefficients D and v are not time-dependent during the modulation. From the feasibility experiments it is clear that the amplitude of the measured boron density modulation scales with the ICRF heating power, but increasing the ICRF heating power also causes a bigger modulation of the ion and electron temperatures. Therefore, one has to choose an ICRF power level at which a clear modulation of the boron density is observed, while the modulations of the other quantities are kept as low as possible. From the study it was concluded that a power level of ∼1 MW of ICRF heating is sufficient to modulate the boron density up to 10% at the edge while keeping the modulation of the ion and electron temperatures to less than 1%–4%. Different modulation frequencies were tested and also here a balance between a clear boron modulation signal and the perturbation to the plasma background has to be maintained. The modulation frequency chosen for the ICRF heating power is 8–10 Hz and this frequency is directly translated to the perturbation seen in the boron density, which is clearly visible in figure 1.

At AUG two core CXRS systems regularly measure the boron content in the plasma. In total these systems have 72 lines-of-sight (LOS) and a typical integration time of 10 ms [27, 28]. The two core systems are located on opposite sides of the torus and, hence, view two different NBI sources, with beam energies of 60 keV (box I) and 93 keV (box II), respectively. The spatial and maximal temporal resolution of the system utilizing the 60 keV beam are 1.0–2.5 cm and 3.5 ms [27], while for the system utilizing the 93 keV beam they are 1–1.5 cm and 2.5 ms [28]. However, the systems are often signal limited to longer integration times. The experimental data shown in this paper is from the core system viewing the 93 keV beam (box II), but as a consistency check the density profiles are cross-checked and validated against the other core system viewing the 60 keV beam. The measurements presented in this paper were performed with an integration time of 10 ms.

The CXRS diagnostics measure the spectral line emission from the charge exchange interaction of the the neutral beam particles with the impurities in the plasma [29, 30]. The plasma rotation can be deduced from the measured Doppler shift and the ion temperature from the Doppler broadening of the spectral line. From the measured line intensities I_CX,Z the impurity density n_Z can be calculated in the following way:

$$\begin{eqnarray}&&{n}_{{Z}}=\displaystyle \frac{4\pi }{h\nu }\displaystyle \frac{{I}_{\mathrm{CX},{Z}}(\lambda )}{{\int }_{{\rm{LOS}}}{\displaystyle \sum }_{i}{\displaystyle \sum }_{j}\langle \sigma v{\rangle }_{i,j}{n}_{0,i,j}{\rm{d}}{l}},\end{eqnarray} \tag{ 3 }$$

where $\langle \sigma v{\rangle }_{i,j}$ is the effective CX emission rate coefficient for a given spectral line, ${n}_{0,i,j}$ are the neutral atoms from the beam for a given energy component i and a given excited state j of the beam. The calculation of the impurity density has been performed with the in-house CHICA code, which stands for charge exchange impurity concentration analysis. To calculate the impurity density, CHICA takes the electron density, electron temperature, ion temperature, the measured CXRS line intensity, and the plasma equilibrium as inputs. The resultant impurity density profile depends strongly on these input profiles, especially the electron density, hence they have to be well known. Another important factor for the calculations is the geometry of the beams. The beam geometries have been constrained using beam emission spectroscopy (BES) measurements with different LOS as well as thermal images of the beam impact points on the inner wall. The cross-check of the calculated impurity density profiles between the CXRS systems on box I and box II, which have different geometries and energy components, provides confirmation that the beam geometries are correctly calculated. Differences in absolute magnitude of 10%–20% are often seen between the diagnostics and are attributed to calibration uncertainties and unavoidable deterioration of the in-vessel optical components during the experimental campaign. However, for determining the transport coefficients it is the gradient, e.g. the profile shape, that is important and, as can be seen in figure 2 and [28], both systems reproduce the same profile shapes. Note that in figure 2, a correction factor of 0.88 was applied to the density profile of the NBI box I CXRS system to match the absolute value of the impurity density profiles. Linear fits of the two datasets in the radial window ρ_tor = 0.4–0.6 are performed and the correction factor is determined from the ratio of the radial averages of the two linear fits. This is a standard method for combining the two datasets.

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The neutral beam density n₀ in equation (3) can be measured with BES or calculated with various beam attenuation codes. In this work, a beam attenuation code has been utilized. The beam has three different energy components as well as a beam halo making i = 1, 2, 3, 4 in equation (3). Furthermore, the beam and the beam halo can be in different excited states j. The beam halo is a cloud of deuterium neutrals around the injected neutral beam, which are produced from charge exchange interactions between the injected neutral particles and the thermal plasma deuterium ions. The effective charge exchange emission rates $\langle \sigma v\rangle$ for a given spectral line are derived from the charge exchange cross-sections found in the ADAS database [31]. The effective rate $\langle \sigma v\rangle$ is obtained by integrating σv over the velocity space of the beam neutrals and the impurity. The product of the neutral beam density n₀ and effective charge exchange emission rate $\langle \sigma v\rangle$ is integrated along the LOS and then summed over the different energy components and excited states. Note that in equation (3) the boron density is assumed to be constant along the intersection of the LOS and the neutral beam volume. This assumption is well met in the plasma core where the gradient lengths are small, but is not valid for the CXRS LOS that intersect in the edge pedestal inside of the beam volume. This is the case for the outermost LOS of both core systems (ρ_tor > 0.85). For these LOS, special analysis is required that has not been performed here and to avoid any effect of uncertain gradient on the profiles, the analysis region is restricted to ρ_tor < 0.70.

Solving equation (1) for the impurity density n_Z given a D and v is called the forward problem. However, the situation at hand is the opposite: the measured boron density n_B is known and the corresponding D and v profiles should be deduced. This is the inverse problem. This task, unlike the simple forward problem, is non-trivial since the problem itself is ill-posed, thus small errors in the measured data are greatly amplified in the solution. One way of regularizing the problem is to impose smooth D and v profiles. The boron density n_B is measured at discrete radial locations ${({r}_{l})}_{l=1,\ldots ,{N}_{{\rm{los}}}}$ and time points ${({t}_{k})}_{k=1,\ldots ,{N}_{t}}$, where r_l denotes the radial and t_k the time measurement positions. t_k depend on the integration time of the CXRS diagnostic and r_l on the geometry of the LOSs and their intersection volumes with the NBI. This yields the measured data points ${n}_{B}^{l,k}={n}_{B}({r}_{l},{t}_{k})$, such that there is no continuous representation of n_B(r, t) available from which the v and D profiles can be calculated directly as in [32]. Therefore, smooth v and D profiles have to be found such that the resulting simulated local densities n_S evaluated at the measurement points n_S(r_l, t_k) are in good agreement with the measured data ${n}_{B}^{l,k}$. This inverse problem can be cast in the formalism of a minimization. By inserting some initial D and v profiles in equation (1) a forward calculation is performed yielding a simulated density n_S(r, t). The simulated density is evaluated at the discrete measurement points resulting in n_S(r_l, t_k) such that the difference between the measured and simulated density at the respective points can be expressed by $| {n}_{S}({r}_{l},{t}_{k})-{n}_{B}^{l,k}|$. By varying the input D and v profiles a new value of $| {n}_{S}({r}_{l},{t}_{k})-{n}_{B}^{l,k}|$ is obtained and this process is iterated until a minimum of the difference is found. In this way, the D and v profiles corresponding to the measured boron density n_B^l,k is acquired. This minimization procedure can mathematically be written as:

$$\begin{eqnarray}&&\mathop{\min }\limits_{D,v,s}\displaystyle \frac{1}{2}\displaystyle \sum _{{\bf{l}}}\displaystyle \sum _{{\bf{k}}}{\left|\displaystyle \frac{{n}_{S}({r}_{l},{t}_{k})-{n}_{B}^{l,k}}{{\sigma }_{B}^{l,k}}\right|}^{2}\quad {\rm{w}}.{\rm{r}}.{\rm{t}}\ \mathrm{equation}\ (1),\end{eqnarray} \tag{ 4 }$$

where σ_B^{l, k} = σ_B (r_l, t_k) are the standard deviations of the densities at the measurement points r_l and t_k. Since the measured boron signal is periodic (see figure 1), a robust ansatz for the reconstructed density n_S(r, t) as well as the boundary condition s(t) which fits the measured data particularly well is

$$\begin{eqnarray}{n}_{S}(r,t) & = & {n}_{0}(r)+a(r)\cos (\omega t)+b(r)\sin (\omega t)\\ s(t) & = & {s}_{0}+{a}_{0}\cos (\omega t)+{b}_{0}\sin (\omega t).\end{eqnarray} \tag{ 5 }$$

In equation (5), n₀ is the steady-state density and ω = 2πf, where f is the frequency of the modulation. The boron source term s(t) is assumed to have the same time dependence as the density n_S. This ansatz is used when solving the inverse problem (4) and inserting the ansatz in equation (4) yields

$$\begin{eqnarray}&&\mathop{\min }\limits_{D,v,{s}_{0},{a}_{0},{b}_{0}}\displaystyle \frac{1}{2}\displaystyle \sum _{{\bf{l}}}\displaystyle \sum _{{\bf{k}}}{\left|\displaystyle \frac{({n}_{0}({r}_{l})+a({r}_{l})\cos (\omega {t}_{k})+b({r}_{l})\sin (\omega {t}_{k}))-{n}_{B}^{l,k}}{{\sigma }_{B}^{l,k}}\right|}^{2}.\end{eqnarray} \tag{ 6 }$$

By also inserting the ansatz in the radial transport equation (1), three equations are obtained: one for the steady-state n₀(r)

$$\begin{eqnarray}&&\displaystyle \frac{1}{r}\displaystyle \frac{\partial }{\partial r}[r(D(r){n}_{0}^{\prime} (r)-v(r){n}_{0}(r))]=0\end{eqnarray} \tag{ 7 }$$

and two for the modulation part $a(r)\cos (\omega t)+b(r)\sin (\omega t)$

$$\begin{eqnarray}&&\omega a(r)r+\displaystyle \frac{\partial }{\partial r}[r(D(r)b^{\prime} (r)-v(r)b(r))]r=0,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&-\omega b(r)r+\displaystyle \frac{\partial }{\partial r}[r(D(r)a^{\prime} (r)-v(r)a(r))]=0.\end{eqnarray} \tag{ 9 }$$

Inserting the boundary condition s(r) in the transport equations yields

$$\begin{eqnarray}&&n^{\prime} (0)=0,\,n({r}_{\max })={s}_{0},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&a^{\prime} (0)=0,\,a({r}_{\max })={a}_{0},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&b^{\prime} (0)=0,\,b({r}_{\max })={b}_{0},\end{eqnarray} \tag{ 12 }$$

where r_max is the last data point of the experimental data. Finally, the Neumann boundary conditions for D and v, which resolve the singularity at r = 0, can be expressed as follows:

$$\begin{eqnarray}&&v(0)=0,\,v^{\prime} (0)=0,\,v^{\prime} ({r}_{\max })=0,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&D^{\prime} (0)=0,\,D^{\prime} ({r}_{\max })=0,\,D\geqslant 0.\end{eqnarray} \tag{ 14 }$$

Equations (6)–(14) represent the complete mathematical description of the minimization problem. The model can thus be reduced to the steady-state and the modulation at the frequency ω, which corresponds to a Fourier transform in time at the modulation frequency. Thus, no time discretization is necessary.

In this work, equation (4) is solved with sequential least squares programming [33], which is a quasi-Newton method, yielding the reconstructed density n_S by solving equation (1) for various D and v profiles as well as the source term s, which is composed of the coefficients s₀, a₀, and b₀. The implementation of the problem has been set up in Python using the Scipy minimization library [34]. Second order finite differences are used for solving the transport equation. The D and v profiles are represented with arbitrary order B-splines, which enforce smoothness of the solution and is therefore a way of applying a regularization to the ill-posed problem. The minimization is thus performed over the B-spline knots of D and v as well as the coefficients s₀, a₀, and b₀. An example of the measured (left) and reconstructed (right) boron density is presented in figure 3. This shows very good agreement. This becomes even clearer when studying the left-hand side of figure 4, which displays the difference between the simulated and measured boron density $| {n}_{S}-{n}_{B}|$. No additional modulation at the modulation frequency of 8.33 Hz or any other frequencies can be seen and hence, the ansatz from equation (5) is, for this problem, very well suited. This claim is further confirmed by computing a Fourier decomposition of the measured boron density n_B at ρ_tor = 0.21 (blue curve on the right-hand side of figure 4), which only displays a peak at the modulation frequency at 8.33 Hz, hence no higher harmonics are present in the central region of the plasma. In orange the same quantity at ${\rho }_{{\rm{t}}{\rm{o}}{\rm{r}}}\,=\,0.76$ is shown. At this location near the plasma edge, additional smaller peaks are visible at the second and third harmonics, indicated by the black dash-dotted lines. These peaks are, however, one order of magnitude smaller than the peak at the first harmonics, i.e. 8.33 Hz, and barely distinguishable from the noise. Furthermore, the Fourier decompositions of the differences between the simulated and measured boron density $| {n}_{S}-{n}_{B}|$ at ${\rho }_{{\rm{tor}}}\,=\,0.21$ and ${\rho }_{{\rm{tor}}}\,=\,0.76$ are plotted with dashed green and red lines, respectively, and this quantity shows no additional frequency peaks outside of the noise.

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In the dataset presented here, sawteeth are present with a frequency of 16 Hz and an inversion radius of ρ_tor ∼ 0.25. That corresponds to ∼1.6 sawtooth cycle for every CXRS integration time. The experimental data inside of ρ_tor < 0.25 is, hence, a sawtooth average, and therefore, the sawtooth frequency cannot be seen in the right plot of figure 4.

With the new ICRF modulation technique multiple modulation cycles can be measured and analyzed together in an otherwise constant background plasma, see figure 3. This is a big advantage compared to other techniques such as laser blow-off and gas puffing, which commonly determine the transport coefficients by analyzing only one single cycle, for example, one individual laser blow-off. The method presented here reduces the relative noise and has a lower statistical uncertainty.

Before using the numerical simulation tool to predict the transport coefficients, it is important to build trust in its reliability and make sure the solver has been implemented correctly. This can be done by checking whether the simulation tool accurately reproduces an analytical solution. This approach is called the method of manufactured solutions [35, 36]. The procedure is straight-forward: suppose the D and v profiles have been found. The transport equation can then be solved analytically for the steady-state n₀(r) by integrating equation (2) from r to r_max:

$$\begin{eqnarray}&&{\displaystyle \int }_{r}^{{r}_{\max }}\displaystyle \frac{v(r^{\prime} )}{D(r^{\prime} )}{\rm{d}}r^{\prime} ={\displaystyle \int }_{r}^{{r}_{\max }}\displaystyle \frac{1}{{n}_{0}(r^{\prime} )}\displaystyle \frac{\partial {n}_{0}(r^{\prime} )}{\partial r^{\prime} }{\rm{d}}r^{\prime} \\ &&\quad =\,{[\mathrm{log}({n}_{0}(r^{\prime} ))]}_{r}^{{r}_{\max }}=\mathrm{log}({n}_{0}({r}_{\max }))-\mathrm{log}({n}_{0}(r)).\end{eqnarray} \tag{ 15 }$$

Expanding both sides yields:

$$\begin{eqnarray}&&\Rightarrow \,\exp \left(-{\int }_{r}^{{r}_{\max }}\displaystyle \frac{v(r^{\prime} )}{D(r^{\prime} )}{\rm{d}}r^{\prime} \right)=\displaystyle \frac{{n}_{0}(r)}{{n}_{0}({r}_{\max })}=\displaystyle \frac{{n}_{0}(r)}{{s}_{0}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{n}_{0}(r)=\exp \left(-{\int }_{r}^{{r}_{\max }}\displaystyle \frac{v(r^{\prime} )}{D(r^{\prime} )}{\rm{d}}r^{\prime} \right){s}_{0}.\end{eqnarray} \tag{ 17 }$$

By inserting analytical D and v profiles in equation (17), an analytical expression for n₀(r) can be calculated. The analytical n₀(r) is compared to the the n₀(r) obtained by inserting the same analytical D and v profiles into the second order finite difference solver for the transport equation. If the two different approaches give the same n₀(r), the radial transport solver has been correctly implemented. This test was performed and the normalized analytical steady-state n₀(r) (blue) and the steady-state n₀(r) obtained from the solver (orange) are presented in figure 5. The agreement is perfect and it can, thus, be concluded that the radial transport solver has been correctly implemented.

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To investigate what features can be resolved in the transport coefficient profiles with the AUG CXRS diagnostics, further method validation was performed. Synthetic D and v profiles were given to the radial transport solver and the steady-state n₀ density, phase and amplitude profiles were calculated (see section 4 on how the phase and amplitude are calculated). In the following examples, the frequency of the modulation was 10 Hz.

First, the D and v profiles (blue lines called True in figure 6) were scaled with a factor of 1.3 (dashed orange lines in figure 6). Additionally in red, example experimental data points with typical error bars are plotted for the steady-state density, phase, and amplitude to display what kind of features we are able to measure with our CXRS systems. It should be noted that our CXRS diagnostic has three times as many data points, but for the sake of clarity only a few are shown. As already discussed in the introduction, scaling both the D and v profiles with the same factor does not influence the steady-state density profile, since it depends on the ratio v/D, but it does affect the phase and amplitude profiles. When increasing the transport coefficients, the phase shift decreases meaning the modulation propagates more quickly from the edge into the core. In this example, multiplying the transport coefficients by a factor of 1.3 results in a change of the phase shift from ∼40 to ∼30 ms. Additionally, the amplitude of the modulation is less damped compared to the case with the smaller D and v profiles. Hence, scaling the transport coefficients with a larger factor would result in a faster inward propagation and less damping. Judging from the error bars on the phase and amplitude in figure 6, a change resulting from a smaller factor than 1.3 in the transport coefficients cannot be distinguished outside of the error bars with the CXRS diagnostic.

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The second task was to investigate how a potential MHD mode could affect the profiles. Such a mode could, for example, enhance the transport locally and, thus, give rise to a sharp localized peak in the D profile. The resultant steady-state density, phase and amplitude profiles from such hypothetical D profiles (keeping the v profile constant) are shown in figure 7. The blue lines corresponds to the same profiles shown in figure 6. For the dashed magenta lines, a sharp Gaussian peak in D has been added around ρ_tor ∼ 0.33, to simulate a 3/2 neoclassical tearing mode [37]. This sharp rise in the D profile leads to a flattening of the steady-state profile n₀ at ρ_tor ∼ 0.33 as well as in the phase and the amplitude profiles. This flattening is still within the experimental error bars for the steady-state and the amplitude, but on the borderline for the phase. It is also interesting to investigate how the sharpness of such a peak affects the profiles. The result of such an investigation is displayed in figure 8. The blue lines are, again, the original profiles and the width of the added Gaussian peak has consequently been increased for the dashed green, magenta, and orange lines. One can see that increasing the width also leads to a flattening of the steady-state, phase, and amplitude profiles. Here, the green case is well within the experimental error bars, the magenta case is borderline for the steady-state and phase, but the orange case is outside the error bars for the steady-state and phase. It is obvious to see that such a small feature in the dashed green steady-state profile can probably not be distinguished from the blue case within the error bars, whereas it should not be a problem to distinguish between the orange and the blue cases. However, given the width of the peak in the magenta and orange case, the peak cannot be classified as localized anymore. The fact that a very large change in the D or v profile only leads to small changes in the experimental data, that is in the steady-state, phase, and amplitude, stems from the ill-posedness of the inverse problem. This further strengthens the argument that when solving such an inverse problems, some kind of regularization must be applied in order to heal the ill-posedness. To conclude, it would be difficult to distinguish a mode outside of the error bars with our CXRS systems, since such a mode has to cause a peak with a very high diffusion and/or be very broad, in which case the mode would not be localized anymore.

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As an additional check, our radial transport solver was benchmarked against the impurity transport code STRAHL [12], which solves the forward problem. The final D and v profiles from our solver were given to STRAHL as inputs to cross-check the simulated steady-state density n₀ and the phase and amplitude of the modulation (see section 4 on how the phase and amplitude are calculated). As can be seen in figure 9, there is an excellent agreement in all quantities between the two codes. However, the radial transport solver developed in this work was able to perform one forward calculation in a few milliseconds, whereas STRAHL, which deploys an additional computational expensive time discretization using the Crank–Nicolson scheme, needs several seconds for the same task. In fact, in some cases the complete minimization procedure is finished in the same time it takes STRAHL to complete one forward calculation. The fact that the inverse problem can be solved in few seconds makes it possible to carry out a brute force uncertainty estimation, described in the next section, in a couple of hours. For such analysis STRAHL would require several days.

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As mentioned in section 2.1, the calculation of the experimental boron density via equation (3) is subjected to several different uncertainties. The first one is the error on the measured CXRS line intensity, which is well known and characterized. The second one is the uncertainty on the atomic data that goes into the effective charge exchange emission rates $\langle \sigma v\rangle$, which is estimated at ∼15%–20%. The changes in $\langle \sigma v\rangle$ are dominated by the interaction energy, which is constant in our case. Changes due to the electron density, ion, and electron temperatures along the beams are minimal, and, therefore, this uncertainty does not significantly impact the error on the overall profile shape, which is of interest in our case. The third and final one is the uncertainty in the determination of the neutral beam density. This in turn depends on the uncertainties of the beam stopping rates, the beam geometry as well as the measured electron density. The same argument as above applies to the beam stopping rates, i.e. they do not significantly affect the profile shape error bar and have, therefore, been neglected. The beam geometry is well diagnosed and benchmarked by the two different NBI systems as well as by cross-checking that the same answer is obtained when using different beam combinations. The electron density n_e has the biggest impact on the attenuation of the neutral beam. If the standard deviation on the electron density was known, this uncertainty could be added to the overall uncertainty on the neutral beam density. However, the electron density is deduced from an integrated data analysis framework [38], where the measured electron densities from several different diagnostics have been combined into one profile using Bayesian inference and this procedure does not produce a standard deviation, but rather confidence bands on the electron density, which cannot be used in a standard error propagation scheme. Therefore, this last source of error is not routinely included in the overall uncertainty on the impurity density, but can be tested by running the analysis with variations of the electron density.

The measurement data n_B(r_l, t_k) = ${n}_{B}^{l,k}$ is, therefore, subjected to an error such that each data point is normally distributed ${ \mathcal N }({n}_{B}^{l,k},{\sigma }_{B}^{l,k})$ with standard deviation ${\sigma }_{B}^{l,k}$. The relative error on the boron density is difficult to assess (see discussion above) and, therefore, the error on the boron intensity ${I}_{B}^{l,k}$ is used for ${\sigma }_{B}^{l,k}$. To assess the uncertainties on the measured transport coefficients, a full Monte Carlo approach is used, where this multi variate normal distribution of the experimental data ${ \mathcal N }({n}_{B}^{l,k},{\sigma }_{B}^{l,k})$ is assumed. N_S = 10 000 random samples are drawn from this distribution and every sample goes through the simulation procedure described in section 2.3, thus resulting in a posterior distribution. The posterior distribution itself, however, is not normally distributed even though the samples were drawn from a normal distribution. The reason for this is because the transport equation is nonlinear. The uncertainty bands of the D and v profiles shown in the next section are pointwise confidence intervals of 95% of the posterior distribution. These confidence intervals of the D and v profiles, therefore, only represent a statistical error. Other non-statistical uncertainties most certainly also play a role, but require a full Bayesian framework, which is outside the scope of this work.