Divertor detachment and reattachment with mixed impurity seeding on ASDEX Upgrade

While there have been a number of published scaling laws for the detachment threshold [16–20] and comparisons to high fidelity SOL models [21, 22], they typically depend on the electron separatrix density, $n_{\textrm{e,sep}}$, which is hard to measure due to uncertainty in the separatrix position. This paper focuses on a parameter, called the detachment qualifier q_det, which indicates the likelihood of the outer divertor reaching partially detached conditions. Pronounced detachment is predicted at $q_{\textrm{det}}\lt 1$, while partial detachment occurs at $q_{\textrm{det}}\approx1$. The equation for q_det was derived by comparing a 1D SOL model with an experimental scaling of the detachment threshold for N-seeded plasmas on AUG. For more details, refer to equation (9) in the study by Kallenbach et al [6]. The equation for q_det is as follows:

$$\begin{align} q_{\textrm{det}} &= 1.3\frac{P_{\textrm{sep}}}{R_{\textrm{maj}}}\frac{5~\textrm{mm}}{\lambda_{\textrm{int}}}\left(\frac{1.65~\textrm{m}}{R_{\textrm{maj}}}\right)^{0.1}\left( \left( 1+\sum_Z{f_Zc_Z}\right)p_0\right)^{-1} \nonumber\\[-12pt] \end{align} \tag{ 1 }$$

This equation replaces the dependency of $n_{\textrm{e,sep}}$ with the divertor neutral pressure, $p_{\textrm{0}}$, which is directly measured on AUG. P_sep is the difference between the absorbed heating power and the main chamber radiation. $\lambda_{\textrm{int}} = \lambda_q+1.64S$ is the broadened SOL power width due to the effect of power spreading. Since equation (1) was derived based on measurements from 1 MA scenarios, the same as used in the scenarios presented here, it is assumed that 5 mm$/\lambda_{\textrm{int}}\approx1$. $f_Zc_Z$ is the product of the radiation efficiency and impurity concentration, respectively. Note the inclusion of the summation over all impurities is introduced in this paper and is not part of the original model. It is noted that this simple detachment threshold formula assumes the effect of the divertor pressure and the impurities to be linear or independent. In reality, there may be second order cross terms, for example concerning the effect of charge exchange on the impurity radiation, and this is not addressed in this paper.

The radiation efficiency, f_Z , for the different impurities is a parameter representing the stronger radiation capability of the impurity in comparison to D. In principle, this parameter can be derived from atomic data. However, a non-coronal parameter $n_{\textrm{e}}\tau$ is usually needed to represent the enhancement to the radiation caused by impurity transport in the SOL. A smaller value of $n_{\textrm{e}}\tau$ tends to enhance the ion abundance towards hotter temperatures, effectively increasing the impurity radiation efficiency. Since $n_{\textrm{e}}\tau$ is difficult to predict, f_Z is effectively used as a fitting parameter between model and experiment. A value of $f_{\textrm{N}} = 18$ was gauged experimentally [23] and the 1D model reproduced this result using $n_{\textrm{e}}\tau = 0.5\times10^{20}$ ms m⁻³ [6]. The same $n_{\textrm{e}}\tau$ was then used for other impurities to show that partial detachment could be achieved with ${\approx}2\times$ more C, ${\approx}2.5\times$ less Ne, and $\approx5\times$ less Ar, respectively, compared with N implying that $f_{\textrm{C}} = 10$, $f_{\textrm{Ne}} = 45$, and $f_{\textrm{Ar}} = 90$. If $n_{\textrm{e}}\tau$ is different for each impurity or the fundamental atomic data is wrong, then these estimated radiation efficiency coefficients will also be wrong. The next subsections detail how f_Ne and f_Ar are gauged experimentally, with the results indicating values higher for Ar and lower for Ne.

The power crossing the separatrix is calculated using $P_{\textrm{sep}} = P_{\textrm{in}}-P_{\textrm{rad,core}}-\mathrm{d}W/\mathrm{d}t$, where P_in is the total input power (including Ohmic power), $P_{\textrm{rad,core}}$ is the core radiation measured using bolometry [24], and W is the core plasma stored energy. The divertor pressure is measured by a baratron connected by a pipe below the high field side divertor (for a simplified CAD view see figure 3 of [25]).

The inter ELM measurements from the Langmuir probes operated as triple probes [26] were used to assess the detachment state, providing the target electron temperature, $T_{\textrm{e,LP}}$, parallel heat flux, $q_{\textrm{par,LP}}$, and electron density, $n_{\textrm{e,LP}}$. Strike-point sweeps were not performed in this database, and therefore the target quantities were derived by averaging all available data points measuring within 4 cm of the strike-point (typically consisting of two probes).

The probe measurements are also used in the next section to determine the momentum loss factor, $f_{\textrm{mom}} = 2n_{\textrm{e,LP}}T_{\textrm{e,LP}}/T_{\textrm{e,sep}}n_{\textrm{e,sep}}$, providing an additional assessment of the divertor conditions. The upstream separatrix electron temperature, $T_{\textrm{e,sep}}$, and density, $n_{\textrm{e,sep}}$, are estimated under the assumption of Spitzer–Härm electron conduction and with a scaling of the divertor neutral pressure [25].

The real-time outer divertor target temperature, $T_{\textrm{e,RT}}$, is derived from the measurement of the inter ELM shunt current, I_shunt, obtained on the outer divertor tile [27]. When the inner divertor is fully detached and the outer divertor is hotter than ≈3 eV, I_shunt is dominated by the thermocurrent and exhibits an approximate scaling relationship with the outer divertor temperature, as $T_{\textrm{e,RT}} = 0.02I_{\textrm{shunt}}\approx T_{\textrm{e,LP}}$ [23, 26]. However, when the outer divertor becomes detached, this assumption no longer holds valid, and other residual currents that may flow in the opposite direction (or connect with a hot main chamber wall) become dominant in the shunt current measurement. While it is clear that negative values of $T_{\textrm{e,RT}}$ are non-physical as a temperature, experimental observations demonstrate that $T_{\textrm{e,RT}}$ becomes more negative as the degree of detachment increases or as the XPR position moves further above the X-point. Therefore, negative values of $T_{\textrm{e,RT}}$ still serve as a valuable measurement for assessing the level of detachment or extent of an XPR. Figure 3(a) presents a comparison between $T_{\textrm{e,LP}}$ and $T_{\textrm{e,RT}}$ from this study.

Confinement $H_{98(\textrm{y},2)}$ scaling factors are shown in figure 3(b) as a function of $T_{\textrm{e,RT}}$. The analysed seeded scenarios typically show $H_{98(\textrm{y},2)}\gt 0.9$ when $T_{\textrm{e,RT}}\gt 10$ eV. As $T_{\textrm{e,RT}}$ drops to negative values associated with pronounced detachment, the confinement tends to drop to $H_{98(\textrm{y,}2)}\approx0.8$ which is consistent with recent results from JET [28]. A more significant drop to $H_{98(\textrm{y},2)}\approx0.65$ was found in an Ar seeded scenario with 16.5 MW of heating power and ≈2% Ar concentration (AUG #37493) which resulted in an XPR. The database also shows that the mixed Ar+N scenarios typically had higher $H_{98(\textrm{y},2)}$ compared to Ar-only scenarios.

Impurity concentrations, c_Z , are determined using divertor spectroscopy, using the line intensity measured through the red LOS indicated in figure 2(a) (i.e. LOS 5). The spectroscopic model for c_N has already been developed [11, 29, 30] and the equivalent models for c_Ne and c_Ar are described in appendix. Comparisons of the divertor c_Z with the ratio of gas valve fluxes (i.e. $c_Z = \frac{\Gamma_Z/Z}{\Gamma_Z/Z+\Gamma_{\textrm{D}}}$ where Γ is the valve flux in elec s⁻¹) are shown in figure 4, though the latter is only valid after at least ≈0.5 s of constant seeding. However, to avoid this transient phase in the analysis, steady state averaging windows were chosen at least 0.5 s after the seeding begins. Generally, there is agreement within ≈20% between the two measurements during the steady state inter-ELM phases. The c_Ne inter-ELM measurement is considerably lower than the intra-ELM signal driven mainly by the increase in Ne II line intensity. For this to occur, there needs to be a significant population of neutrals in the vicinity of the sightline that are ionised during the ELM. Conversely, c_Ar shows little difference between intra-ELM and inter-ELM measurements. Previous AUG measurements showed a stronger difference between the inter-ELM and intra-ELM measurements for c_N measured through a sightline viewing closer to the target (e.g. figure 4 of [29]). This result is consistent with Ne neutrals having a longer ionisation mean free path in comparison to N or Ar due to the FIP effect. The neutral atom ground state ionisation energies taken from NIST [31] are ≈15 eV for N and Ar and ≈22 eV for Ne. Collisional-radiative effects may alter the effective ionisation rate for each impurity, however high quality data to assess this does not yet exist for Ar (although work towards this is underway [32]).

Zoom In Zoom Out Reset image size

Finally, it is noted that c_Ne increases by a factor 2 compared to the estimated value from the gas valve flux ratio in phases of deep detachment which is due to the assumption of constant electron temperature in the model becoming less accurate (see appendix). Therefore, in these deeply detached phases, the gas valve flux ratio is used to estimate c_Ne.

The experimental q_det is inferred from measurements described in the previous section averaged over 300 ms in steady state phases of each pulse. The results are shown as a function of $q_{\textrm{par,LP}}$ and $T_{\textrm{e,LP}}$ in figures 5(a) and (b), respectively. Although there is no prior evidence to suggest that these quantities should vary linearly with q_det, a best-fit dashed line with linear dependence is shown for reference and shows reasonable agreement to the data, more so for $T_{\textrm{e,LP}}$. Figures 5(e) and (f) show a roll-over in $n_{\textrm{e,LP}}$ and sudden drop in f_mom once the divertor temperature falls below ≈6 eV.

Zoom In Zoom Out Reset image size

The experimental data in figures 5(a) and (b) show that the estimated values of $f_{\textrm{Ne}} = 45$ and $f_{\textrm{Ar}} = 90$ tend to marginally under-predict the c_Ne and over-predict the c_Ar required to reach partial detachment compared to a best-fit line with linear dependence combining data from all impurity mixtures. The data is now used to find values of f_Ne and f_Ar that provide better consistency between the different impurity mixtures. This is done by finding values of f_Ne and f_Ar which satisfy a criteria gauged on the N-only data:

$$\begin{eqnarray} &&q_{\textrm{det}}[T_{\textrm{div,LP}} \lt 6~\textrm{eV}] \lt 2.5 \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&q_{\textrm{det}}[T_{\textrm{div,LP}} \gt 6~\textrm{eV}] \gt 1.9 \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&q_{\textrm{det}}[T_{\textrm{div,LP}} \gt 10~\textrm{eV}] \gt 2.0 . \end{eqnarray} \tag{ 4 }$$

Values of f_Ne and f_Ar satisfying the criteria were $f_{\textrm{Ne}} = 15\text{-}50$ and $f_{\textrm{Ar}} = 175\text{-}200$. The results using the average value of the acceptable ranges are shown in figures 5(c) and (d). These values improve the goodness of fit R² between the data and the best-fit line, and moves the best-fit prediction closer to unity at $T_{\textrm{e,LP}}\approx1$ eV. Notably, this also opens the possibility of using q_det more generally to predict the target temperature and heat loads. However, such a model would have some uncertainty due to the probe measurements below 2–3 eV being typically unreliable.

The most probable solution to achieve the detachment point at $q_{\textrm{det}}\approx1$ in Ne-seeded cases mixed with N is by lowering the Ne radiation efficiency. However, there is an alternative solution involving lower radiation efficiency for N instead of Ne. On the other hand, this alternative solution would contradict the observed N radiation efficiency in both the N-only seeding database presented in this paper and the previous study by Kallenbach et al [1]. Therefore, this alternative solution is not considered likely. Furthermore, the model in equation (1) assumes a constant impurity concentration in the SOL and therefore, since the upstream SOL impurity concentration is not available, it is assumed that the divertor impurity concentration is equal to the average SOL concentration. This assumption could impact the analysis of the radiation efficiency, but it is not expected to change the overall conclusion.

Another approach to measuring the radiation efficiency is to assess the total radiation in the SOL (including divertor, XPR, and main chamber SOL) measured using bolometry. For clarity, the measured $P_{\textrm{rad,SOL}}$ is defined as $P_{\textrm{rad,SOL}} = P_{\textrm{rad,tot}}-P_{\textrm{rad,sepmain}}$, using the same nomenclature as shown in figure 5 of [24]. A simple model is proposed to assess the SOL radiation:

$$\begin{equation} P_{\textrm{rad,SOL}} = \left(\frac{P_{\textrm{sep}}}{6~\textrm{MW}}\right)\left(C\sum_Z{f_Zc_Z} + P_{\textrm{back}}\right) \end{equation} \tag{ 5 }$$

where C is a free parameter with units in MW and P_back is the background radiation estimated using the radiation measured before seeding occurs. The normalised P_sep is included to account for the relatively linear dependence with $P_{\textrm{rad,SOL}}$. The spectroscopic c_Z measurements described previously are again used in this model.

The model first assumes a set of f_Z and then tunes C to achieve the best fit to the data from one shot. This value of C is then used consistently among other shots. In this example, the model is tuned on AUG #39520 which has Ar+N seeding and then compared to an Ar-only (AUG #39519) and Ne+N (AUG #41033) scenario. To simplify the analysis, these discharges were chosen due to their similar P_sep. The results from the model are shown in figure 6. The red and green lines show the results using $f_{\textrm{N,Ne,Ar}} = 184\,590$ and $f_{\textrm{N,Ne,Ar}} = 1830\,200$, respectively. The red curves give worse agreement overall, and values of f_Z similar to those found in the analysis of q_det agree better with the bolometer data proving that the new parameters for f_Z provide better consistency overall with experimental data than previous estimates.

Zoom In Zoom Out Reset image size

The NBI power was increased during phases with partial divertor detachment and full divertor detachment (XPR). Figures 7(a)–(d) shows the resulting power crossing the separatrix, real-time divertor temperature, probe divertor temperature, and XPR position for the cases with ≈2.5 MW increase (orange and green lines) and ≈5 MW increase (black and blue lines) in power. The position of the localised intense radiation front above the X-point is measured by AXUV diodes [14] and is shown in figure 7(d) to indicate the evolution of the radiation front following the power transient. In the analysis below, the XPR position is used as a proxy for the ionisation front. The partially detached divertor cases where an XPR is not present (black and green lines) show an immediate rise in temperature and take ≈100 ms to reach peak temperature. For comparison, the core confinement time in these plasmas is ≈60 ms. The fully detached divertor cases with XPR (blue and orange lines) show an immediate change in the XPR position, taking ≈100 ms before the XPR reaches a location where the radiation peak is in the inner divertor and not representing an XPR anymore. This is consistent with the temperature only showing a rise ≈100 ms after the power increase. Overall, the fully detached scenario takes ≈250 ms to reach peak temperature—significantly longer than the partially detached scenario. Since the power is kept at a higher level for a timescale longer than that required for reattachment, the peak divertor temperature reaches the steady state value expected for the associated value of q_det.

Zoom In Zoom Out Reset image size

There are published models describing the threshold and stability of the XPR [33, 34] but these do not directly predict the location of the XPR inside the confined region. To complement these studies, a simple scalable model is proposed to predict the time required to move the ionisation front from the X-point back to the target following a change in power entering the SOL. While the XPR position represents the extent of the radiation front within the confined plasma, rather than the location of the ionisation front between the X-point and target in the SOL, the movements of both are closely coupled. Furthermore, as shown in figure 7, the target temperature generally increases once the XPR position measurement reaches the noise level. Therefore, the XPR position serves as a useful proxy for the ionisation front position. To match the functional form of the measured XPR position decay after the power increase, an exponential form is selected for the model. The model assumes that the timescale is driven by the evolution of the power over time, which spans tens or hundreds of milliseconds, coupled with the timescales of ionisation and recombination particle reservoirs, which are several milliseconds. The chosen form of the model is as follows:

$$\begin{equation} \mathrm{XPR}_{\textrm{pos}}(t) = \mathrm{XPR}_{\textrm{peak}}e^{\frac{-tP_{\textrm{sep}}(t)}{Vn_{\textrm{0}}E_{\textrm{ion}}\genfrac{}{}{0.3pt}{3}{\tau_{\textrm{resid}}}{\tau_{\textrm{ionis}}}}} \end{equation} \tag{ 6 }$$

V is the volume of neutrals extending from the target to the X-point, $E_{\textrm{ion}}\approx30$ eV is the effective ionisation energy loss for deuterium including radiative losses [35] (assuming burn-through of impurities is not significant), $n_{\textrm{0}}$ is the average neutral atom density, τ_resid is the residence time of an ion, and τ_ionis is the ionisation timescale. A crude approximation is made to estimate the volume of neutrals as $V\approx2\pi RA_{\textrm{pol}}\approx0.4$ m⁻³, where $A_{\textrm{pol}}\approx0.04$ m⁻² is the estimated poloidal area of the neutrals (see blue shaded region in figure 2) and R is the average radius of the neutral volume. $n_{\textrm{0}}\approx p_{\textrm{0}}/k_{\textrm{B}}T\approx10^{21}$ m⁻³ assuming a molecular flux at T = 300 K at the baratron located underneath the high field side divertor. The inclusion of $P_{\textrm{sep}}(t)$ means that the temporal evolution of the power rise (i.e. mainly driven by the core energy confinement time) is accounted for. A lower limit for τ_resid is calculated assuming the ion velocity to be the ionic sound speed giving:

$$\begin{equation} \frac{\tau_{\textrm{resid}}}{\tau_{\textrm{ionis}}} = \frac{L_{{||}}n_{\textrm{e}}S_{\textrm{CD}}}{\sqrt{2eT_{\textrm{e}}/m_{\textrm{D}}}}. \end{equation} \tag{ 7 }$$

$L_{{||}}\approx12$ m is the approximate connection length between the X-point and the target, and $S_{\textrm{CD}}\approx1\times10^{-16}$ m³ s⁻¹ is the hydrogenic ionisation rate coefficient at $T_{\textrm{e}} = 2.2$ eV. Stark broadening measurements show that $n_{\textrm{e}}\approx3\times10^{20}$ m⁻³ which gives $\tau_{\textrm{resid}}/\tau_{\textrm{ionis}}\approx30$. This ratio $\tau_{\textrm{resid}}/\tau_{\textrm{ionis}}$ is also used by Janev and Reiter [36] to define the number of recombination cycles that a hydrogenic plasma ion can make during its residence time in the divertor.

The red dashed line in figure 7(d) represents the predicted decay of $\mathrm{XPR}_{\textrm{pos}}$ due to the power increase in #40250, indicated by the blue lines. Surprisingly, considering the simplicity of the model, there is remarkably good agreement with the measured decay of $\mathrm{XPR}_{\textrm{pos}}$. To provide a clearer understanding of the uncertainty, the parameters in equations (6) and (7) with the highest degree of error are divided into two groups: $N = Vn_{\textrm{0}}$ and $S_{\textrm{eff}} = S_{\textrm{CD}}(T_{\textrm{e}})/\sqrt{T_{\textrm{e}}}$. It is assumed that the remaining parameters are reasonably well constrained by measurements. S_eff effectively acts as a fitting parameter, providing that T_e falls within the likely range of 2–3 eV, given the significant variation of S_CD with T_e. By varying N by a factor ×5 and $\times1/3$, respectively, the same predicted decay of $\mathrm{XPR}_{\textrm{pos}}$ as shown in figure 7(d) can be achieved within the limits of T_e between 2 eV and 3 eV. Although a previous study [37] suggests approximate agreement between the fluxes measured in the outer divertor and sub-divertor under detached conditions, providing some confidence in the prediction of $n_{\textrm{0}}$, it is important to consider the crude estimation of V and the remaining uncertainties in the other parameters. In light of these uncertainties, it is expected that the specified range of N reasonably encompasses the overall uncertainty.

Despite the uncertainties described above, the advantage of this model is that it provides a simple method for scaling the reattachment time in a reactor. If ${\approx}e^{-5}$ is used as an arbitrary threshold for the ionisation front having reached the target, then the time taken is

$$\begin{align} t = 0.09\left(\frac{p_{\textrm{0}}}{2~\textrm{Pa}}\right)\left(\frac{n_{\textrm{e}}}{3^{20}~\textrm{m}^{-3}}\right)\left(\frac{V}{0.4~\textrm{m}^3}\right)\left(\frac{L_{||}}{12~\textrm{m}}\right)\left(\frac{\lt P_{\textrm{sep}}\gt }{2~\textrm{MW}}\right)^{-1}, \end{align} \tag{ 8 }$$

where $\lt P_{\textrm{sep}}\gt$ is the average rise in power. An interesting application of the model can be made to the JET pulse #89244 documented by Field et al [38]. This had similar injected power of 8 MW and an XPR induced by N$_{\textrm{2}}$-seeding. The heating was increased to 15 MW over ≈2 s resulting in a ≈5 MW rise in P_sep. Using a simple upscaling of the above parameters to JET, namely $V/0.4$ m ≈ 7, $L_{{||}}/12$ m ≈ 1.75, and $\lt P_{\textrm{sep}}\gt \approx2$ MW, equation (8) predicts that the ionisation front moves back to the target within ≈1 s. While there is no measurement of the XPR position, there is a significant increase in the target Langmuir probe saturation current and N II emissivity near the target ≈1 s after the power increase (see figure 24 in [38]) consistent with the model prediction.

Reattachment could also be achieved by a cut of the impurity seeding instead of an increase in power. This was done by replacing the increases in NBI power in the scenario with cuts in the impurity gas to test the reattachment timescales. As shown in figure 7(e), cutting both impurity (Ar+N) gas flows causes P_sep to rise by ≈3 MW and the resulting evolution of the divertor temperature and XPR position are shown in figures 7(f)–(h). The divertor reaches its peak temperature at ≈200 ms after the gas cut. In figure 7(h), the red dashed line depicts the corresponding XPR_pos using the P_sep displayed in figure 7(e). The model successfully captures the temporal behaviour starting from 100 ms after the gas cut, but it does not account for the immediate movement of the XPR, which occurs around ≈30 ms after the gas cut. This is because there are two additional time scales to consider in this scenario: the delay between the gas valve switching off and the impurity concentration beginning to drop, and the residence time of each impurity. The former timescale is ≈30 ms indicated by the vertical dashed line in figures 7(e)–(h).

The impurity residence times for each impurity in figure 8 show that N decays fastest of the three impurities with an exponential decay rate of ≈40 ms, followed by Ar at ≈100 ms, and finally by Ne at ≈200 ms. The wall storage is likely the dominant factor influencing the decay rate of N in these plasmas, while the entrainment in the D₂ flow to the pump is believed to be the primary mechanism responsible for the decay rate of Ne and Ar. A lower impurity enrichment would result in reduced entrainment, indicating that Ar has a higher enrichment compared to Ne. This observation is consistent with the findings discussed in the previous Sections, which showed that the Ne II line intensities peak further from the target compared to N II and Ar II.

Zoom In Zoom Out Reset image size

While the XPR position is sensitive to the power crossing the separatrix, it is also affected by the power radiated in the SOL. Ar primarily influences P_sep, while N mainly impacts $P_{\textrm{rad,SOL}}$. Despite both gases being cut off simultaneously, N decays at a faster rate, resulting in a change in $P_{\textrm{rad,SOL}}$ within ≈40 ms, whereas the change in P_sep becomes noticeable after ≈100 ms according to the decay rate of Ar. The current formulation of equation (6) does not account for variations in $P_{\textrm{rad,SOL}}$. Quantifying the drop in $P_{\textrm{rad,SOL}}$ is challenging because, overall, the divertor radiation marginally increases due to the rise in P_sep. In summary, despite the absence of physics concerning the impurity pumping timescales, the current model still predicts the timescales for reattachment caused by changes in the power crossing the separatrix and may be applicable to larger devices as well.