Temporal characteristics of ELMs on the COMPASS divertor

The divertor heat loads during ELMy H-mode scenarios have been studied at different tokamaks [1–4] and also predicted by different models [2]. All these studies are essential for the lifetime or damage prediction of the plasma facing components for ITER [5, 6] and other large fusion reactors including liquid metal divertor concept [7–9]. However, further study of individual edge localised mode (ELM) temporal characteristics is important because the total power delivered by each ELM must be absorbed by the material of the plasma facing components during the ELM duration (τ_ELM). This time is of critical importance since the maximum surface target temperature rise caused by an ELM (which should not exceed the melting point of the divertor plasma-facing components) is approximately proportional to 1/$\sqrt {{\tau _{{\text{ELM}}}}}$ [10–13]. In previous works [4, 14], the ELM temporal evolution is typically described by two characteristic times, rise (τ_rise) and decay (τ_decay) times. The value of τ_rise is defined as the time duration for the divertor heat load to increase from 10% to 100% of its maximum and the τ_decay represents the characteristic exponential decay time after its maximum as shown by figure 2 in section 3. The recent work from HL-2A tokamak [15] shows that the values of τ_rise is nearly equal to the time of ELM parallel propagation (τ_||) defined as τ_|| = L_||/c_s, where c_s and L_|| represent the sound speed and the connection length between the upstream and corresponding divertor location, respectively. In [15] the connection length was obtained using a simple formula L_|| = 2πq₉₅ R_sep (with R_sep as outer midplane separatrix position and q₉₅ as a safety factor at the 95% poloidal flux surface), corresponding to its inter-ELM value. Previous studies on JET, AUG, JT-60U, MAST tokamaks (figure 4 in [14], figure 4 in [4], figure 12 in [16]) show that the values of τ_rise are mostly above the ones of τ_||, but still well correlated with τ_||.

However, the MHD modelling of the ELM crash indicates formation of the ergodic magnetic fields and islands leading to a strong enhancement of the parallel connection length inside the (pre-ELM) separatrix up to tens of kms in case of ITER [17, 18]. Other simulation within the SOL region demonstrate that the connection length is actually not enhanced significantly and τ_|| seems to be still correlated with the unperturbed connections length (L_||); e.g. in [19] effective connection length was found ∼5 × L_||. All the above-mentioned experimental results are based on the conditionally averaged ELM analysis using infrared (IR) measurements with relatively low temporal resolution in order of few kHz. The conditionally averaged ELM technique is also affected by irregular ELM filamentary structures [20, 21] and thus can provide only averaged values of the ELM temporal characteristics.

However, on the COMPASS tokamak divertor, the fast probe diagnostic system [3] was used to study these ELM temporal characteristics with high temporal resolution (∼1 μs). It allows us to provide ELM temporal characteristics based on a single ELM analysis without conditionally averaging. Such a high resolution is also necessary assuming small τ_rise values due to a short connection length (L_|| ∼ 5 m on the low field side (LFS)) on COMPASS.

The COMPASS tokamak with graphite plasma facing components is one of few devices with an ITER-like plasma shape [22] with major radius R = 0.56 m, minor radius a = 0.23 m, toroidal magnetic field B_T = 0.9–1.7 T, plasma current up to I_P = 350 kA, pulse duration <500 ms and with line averaged electron density up to n_e = 1.2 × 10²⁰ m⁻³. On COMPASS, a system of probes, combination of ball-pen (56 probes) and roof-top shaped Langmuir probes (two arrays of  55 probes each) [3] has been installed in the divertor region to systematically investigate the electron temperature as well as parallel heat flux during L-modes [23] and ELMy H-modes [3, 24, 25] plasmas with high temporal (∼1 μs) and good poloidal spatial (∼3.5 mm) resolutions.

The Langmuir probes are made of graphite protruding 1.5 mm into the plasma and provide floating potential or ion saturation current measurements. The ball-pen probes provide plasma potential measurements and in combination with floating Langmuir probes (LPs) also the electron temperature. The LPs have a 20° chamfer with a total exposed area above the divertor target of  22 mm² and projected area (on one side) of S_LP⊥  = 2.8 mm². First achieved results on the analysis of the ELM energy fluence (_||) were published in [3, 15] and compared with Eich's model prediction [2]

$$\begin{align}{\varepsilon _{||,{\text{model}}}} = {\Delta _{{\text{equi}}}} \cdot 2\pi\ \! {a_{{\text{geo}}}}\sqrt {\frac{{1 + {\kappa ^2}}}{2}} \times \frac{3}{2}n_{{\text{e,ped}}} \cdot {k_B} \cdot T_{{\text{e,ped}}} \cdot \frac{{{\text{B}_{{\text{tor}}}}}}{{{B_{{\text{pol}}}}}}\end{align} \tag{ 1 }$$

for the corresponding values of pedestal density n_e,ped in [10²⁰ m⁻³], temperature T_e,ped in [keV], plasma elongation κ and minor radius a_geo [m]. The values B_tor and B_pol in [T] on the outer midplane as well as geometry factor Δ_equi ∼ 1.5 for COMPASS are obtained using the magnetic equilibrium reconstruction.

On the other hand, the experimental ELM energy fluence _|| is the time integral of the parallel heat flux (q_||) over the duration of an ELM (τ_ELM), ${\varepsilon _{||}} = \mathop \smallint \nolimits_{{\tau _{{\text{ELM}}}}} {q_{||}}\left( {t,\,R} \right){\text{dt}}$. This comparison has demonstrated that the fast probe measurements based on a single ELM analysis are in good agreement with the model prediction, as well as with previous IR measurements based on conditionally averaged ELM analysis from AUG, JET and MAST tokamaks. During the years 2019–2020, an extended analysis of the _|| using COMPASS measurements on the LFS was performed and the resulting maximum values of _|| (new 23 points from LFS probes (2019), 6 new points from IR (2019), 7 points of nitrogen seeding experiments (2020)) are shown in figure 1, together with AUG, JET and MAST results [2]. This latest analysis also includes probe measurements of _|| during impurity seeding experiments, in which nitrogen was introduced at variable rates (1.5–3 × 10²⁰ s⁻¹) in the vicinity of the outer strike point (for more details of impurity injection at COMPASS see [26]). The impurity seeding was capable of influencing the plasma scenario (e.g. enforcing termination of ELMy H-mode) but did not produce clear signs of ELM buffering (power dissipation during ELM transport in the SOL) and also did not result in inter-ELM detachment at the outer target. Note that the parallel heat flux (q_|| = γ·T_e·I_sat ⁺/S_LP⊥), which is calculated by using the electron temperature (T_e in eV), ion saturation current of the Langmuir probe (I_sat ⁺) and heat transmission coefficient (γ), is newly calculating using γ = 11. The value γ = 11 was achieved by comparing divertor probes and IR camera measurements in L- and H-mode plasmas [23, 27] and from theoretical prediction [28] for non-ambipolar conditions (grounded tiles, T_i/T_e = 1, no secondary electron emission). Moreover, the first COMPASS IR measurements of _|| on LFS are also included in figure 1. Fast infrared thermography system measurements were performed using the Telops FAST-IR 2K camera equipped with an InSb detector sensitive to medium wavelength IR radiation (3–5 μm) [29]. This system provided time evolution of the full radial profile of the divertor temperature with a temporal resolution of 33 kHz and a spatial resolution of 0.6 mm px⁻¹ at the divertor surface for the studied discharge #18240. Incident plasma heat flux was evaluated using the THEODOR code [30].

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In general, all COMPASS results shown in figure 1 are in good agreement, within the (3:1–1:1) boundaries, with the model prediction, as well as with the previous measurements from AUG, JET, MAST and recently also from the HL-2A tokamak [15]. Note that these COMPASS points at low energy fluence make the (linear) extrapolation towards ITER more reliable. In addition, they seem to indicate that the hypothesis of the Eich's model is valid, i.e. collisionless plasma and ELM transport dominated along magnetic field lines. However, the limitations of H-mode scenarios and heating systems do not allow us for precise determination of ELM Types at COMPASS, e.g. by observing changes of ELM frequency with changes of input power.

The above mentioned results are important to improve the predictions of the ELM heat loads for ITER and other fusion facilities. However, it is also crucial to understand the temporal evolution and the time scales of the ELM heat load patterns on the divertor to estimate the material limit and lifetime of the plasma facing components. For this purpose, we used an extensive ELMy H-mode database (including ELMs shown in figure 1) to analyze the rise and decay times of the total ELM power incoming to the outboard divertor. The database consists of 142 single ELM events within 29H-mode discharges with single null configuration and deuterium plasma. The major parameters were I_p ∼ 220 kA, B_T = 1.15 or 1.5 T, 2 < n [10¹⁹ m⁻³] < 9, q₉₅ ∼ 3 with/without neutral beam injector (NBI) heating. Figure 2 shows an example of the temporal evolution of the total (perpendicular) ELM power (P_ELM = 2π∫q_||sin(α)RdR, α—magnetic field line incident angle) integrated within the outboard area (all LFS probes) during a single ELM event (starting at t₀ ∼ 1097 ms) in shot # 18447 (B_T = 1.15 T, I_P = 200 kA, n_e = 5 × 10¹⁹m⁻³, q₉₅ = 2.9 with P_NBI = 0.2 MW). The blue line shows the ELM power P_ELM with high temporal resolution (∼1 μs). Its values are strongly fluctuating with clear evidence of the inner filamentary structure highlighted by the smoothed red line. This filamentary structure is not visible when the ELM conditionally averaged technique is applied typically for IR measurements over many ELM events. Then, we use a formula (equation (5)) in [31] based on the vacuum free-streaming model [32] to fit the whole temporal evolution of P_ELM and to define a representative maximum, which is otherwise difficult to estimate from the fluctuating values. This technique might still underestimate or overestimate the peak values of P_ELM, but it provides a reasonable value less affected by the presence of the fluctuations and ELM filamentary structures. The value of τ_rise is then defined as the time duration for P_ELM to go from 10% to 100% of its maximum (subtracting the pre-ELM power value), as is done similarly in [14, 15]. The second part of the temporal evolution of P_ELM, after it reaches its maximum, is fitted by an exponential decay function (see the orange line in figure 2) with characteristic decay time τ_decay also used in [14, 15]. The error bars of both quantities are based on the fitting of the P_ELM by the free-streaming model (FSM) function (error of the maxima values) or the exponential decay (error on the characteristic decay time parameter).

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Then, the rise time values can be compared with τ_|| in order to study the connection length L_|| during the ELM propagation phase. The value of τ_|| is given by τ_|| = L_||/c_s with the sound speed c_s calculated for equal electron and ion pedestal temperatures, T_e ^ped = T_i ^ped. The values of T_e ^ped were obtained from T_e profiles measured by high resolution Thomson scattering (HRTS) system [33] using the so-called two-line fitting technique [24, 34]. Note that the measurements of the HRTS system with a constant repetition frequency allowed us to get reasonable measurements of the T_e ^ped in the last 30% of the previous ELM cycle (corresponding to fully developed H-mode pedestal) only in a few cases. In total, we have obtained 20 ELMs with corresponding T_e ^ped and for which τ_|| can be calculated. The final comparison of τ_rise and τ_|| values is plotted in figure 3. Here, we have used two different methods to obtain the corresponding L_|| on the LFS before the ELM (inter-ELM phase). First, we have calculated L_|| using a field line tracing software (Pleque [35]) and the EFIT magnetic reconstruction during the inter-ELM period. In this case, the L_|| calculation starts at 1 mm outside the last closed flux surface at the outer midplane until the outboard divertor. Second, the value L_|| (appr.) is found by using a simple formula L_|| (appr.) = 2πq₉₅ R_sep/3. This is again for the case when the ELM energy released location is at the outer midplane and propagates to the outboard divertor [36]. It is seen in figure 3 that the resulting values τ_rise are slightly higher than τ_|| values. The similar trend was observed also during previous measurements on JET, AUG, JT-60U, MAST, HL-2A tokamak [4, 14, 15] using IR measurements or simulations [37–39, 40]. It is also clear that both methods of L_|| calculation lead to a similar resulting τ_|| values.

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The similarity of both τ_rise and τ_|| can lead to the conclusion that the L_|| during ELM, or at least at the beginning of the ELM event, is similar to the inter-ELM value. Thus, the magnetic field lines might be only weakly ergodized during ELMs in the SOL region on COMPASS. Indeed, it was shown that the time rises of the electron temperature in 1D3V kinetic simulations of ELM propagation through the SOL on COMPASS with no ergodization are in very good agreement with fast ELM T_e measurements on divertor and the results of the free-streaming (vacuum) model [24]. However, it is also shown in [36] that the free-streaming model approach would define the relation between the empirical value τ_rise and the theoretical τ^FSP as τ_rise = 0.39 × τ^FSP. The theoretical value τ^FSP is the expected time between the beginning of the ELM event on midplane and the observed maximum of the heat flux on the outboard divertor. In this case, we would need to apply a correction of our calculated connection length L_||/0.39 during ELM to agree with the free-streaming model prediction. Nevertheless, even the value ∼L_||/0.39 (∼factor two longer connection length) leads to the similar conclusion that magnetic field lines are not significantly ergodized in the SOL region on COMPASS during ELMs.

Because the total ELM power is deposited during both periods, τ_rise and τ_decay, it is of interest to know if any relation between these two temporal characteristics exists. Both characteristic times of all 142 single ELM events are plotted in figure 4 versus each other. The two boundary lines represent the variation of the ratio τ_decay/τ_rise within the interval 1.5 up to 4, as also used in [15]. We clearly see that τ_decay and τ_rise values fit well to these borders. A similar behavior was observed, not only on HL-2A [15], but also on JET [4]. This agreement found across different devices can provide general and useful upper and lower limits of the ELM duration on divertor. Note, it is visible in figure 4 that these τ_rise appear to be only above roughly 50 μs. This is of course not a physical limit, but the analysis of the τ_rise values on COMPASS has some diagnostic limitation. Very small τ_rise values are typically expected for high c_s values (due to the large T_e ^ped) and therefore the ELM T_e peak values on the divertor are overloading the ion saturation current measurements of the negatively biased Langmuir probes. Although we do not see any clear dependence between τ_decay and τ_rise in figure 4 within the range of our results, it might be possible that some ELM or general plasma parameters affect the ratio τ_decay/τ_rise. In our case, this could not be checked with the major plasma parameters, like plasma current or the toroidal magnetic field, as they are nearly constant during a given discharge and even very similar within our 29H-mode shot database. On the other hand, the line averaged density is varying significantly within the range 2 < n [10¹⁹ m⁻³] < 9. Nevertheless, it is seen in figure 5 that the resulting ratio has no clear dependence on this parameter. We have found the same behavior of the ratio also with respect to the relative ELM energy ΔW/W (W is the total pre-ELM plasma energy), as seen in figure 6. The ratio was compared also to values of the turbulence parameter α_t = 3 × 10⁻¹⁸ R q² n Z_eff T⁻² (with major radii R [m], cylindrical safety factor q, electron density n[m⁻³], effective charge Z_eff and the electron temperature T [eV]), introduced in [41], with no clear dependence within the range 0.07 < α_t < 0.17. However, in this case we have used only ELM data plotted in figure 3 with Thomson scattering measurements at the separatrix position, which limits our statistics.

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It was shown that the temporal characteristic, ratio τ_decay/τ_rise, is changing significantly, which might affect the final temperature evolution of the divertor components during ELM. In general, you might find that the material surface temperature is peaking during τ_rise as seen on JET [12]. However, if the τ_decay ≫ τ_rise then τ_decay period, when the major part of _|| will be delivered, must play also an important role in the temperature increase. The average temperature increase during ELM depends on the material property like heat conductivity etc and is proportional to 1/$\sqrt {{\tau _{{\text{ELM}}}}}$ [10], τ_ELM = τ_rise + 3 × τ_decay. Of course, the final impact on the temperature increase during ELM is also given by the maximum of _|| and the wetted area [4]. The wetted area is commonly defined as a toroidally and radially integrated q_|| profile divided by the peak value of q_||. However, our fast divertor probe measurements have shown that the peak values of q_|| are strongly influenced by the ELM filamentary structure, with no well-defined maximum. Therefore, we will further use only _|| and its radial profile (see example, figure 7 in [3]). The radial integration of _|| along the outboard divertor surface (with distance s) divided by the maximum of _|| provides the integral _|| decay length λ = ∫_||ds/max(_||) or its value mapped to the midplane, λ ^mid. A similar definition is used in previous work for integral power decay length [4]. It is shown in figure 7 that these ELM space characteristic values (λ ^mid) are clearly linked to the ratio of the ELM temporal characteristic values (τ_decay/τ_rise). In principal, the ELM event with very sharp transition in time (τ_decay/τ_rise ∼ 1) will cause larger footprint on the divertor surface and vice versa. This actually helps to better distribute the ELM energy along the divertor for the short ELM duration (τ_ELM). It might be beneficial for the fusion devices if a similar trend is also found. Note, the typical values of the divertor power decay length mapped on the midplane (λ_q ^mid) on COMPASS during the inter-ELM period is within the range 0.5 < λ_q ^mid [mm] < 1.0 as shown in figure 8.17 [27].

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The first systematic study of the temporal characteristics of ELM events on the COMPASS outboard divertor with high temporal resolution probe measurements (∼1 μs) provides large statistics of rise (τ_rise) and decay (τ_decay) time values. The results are based on individual ELM analysis using divertor probe measurements. It was found that τ_rise values are in the range of about 50 μs–100 μs and comparable to the characteristic time of the ELM parallel propagation τ_|| = L_||/c_s. The values L_|| were obtained either by field line tracing software (Pleque) or by a simple approximation of the midplane—outboard connection length L_|| = 2πq₉₅ R_sep/3. We have also discussed the free-streaming model (FSM) prediction that τ_rise = 0.39 × τ^FSM. We might conclude that the magnetic field lines are only weakly ergodized during ELMs in the SOL region on COMPASS. This also implies that the peak ELM energy fluence (max (_||)) on the outboard divertor is dominated by the ELM parallel transport, which is confirmed by a reasonable agreement with model prediction. The values of the ratio of τ_decay and τ_rise for each ELM event fit very well to the boundaries 1.5 < τ_decay/τ_rise < 4, as already shown on JET [4] as well as on the HL-2A tokamak [15]. The general limits of the ratio across different tokamaks bring a reasonable prediction of the maximum and minimum ELM duration τ_ELM = τ_rise + 3 × τ_decay, which is associated with the increase of the temperature of the divertor material as ∼1/$\sqrt {{\tau _{{\text{ELM}}}}}$.

It was shown that the ratio has no clear dependence on the relative ELM energy or line averaged electron density. Moreover, we have found that values of the integral _|| decay length (mapped on midplane λ ^mid) are clearly linked to this ratio. It is for the first time shown on COMPASS data that the ELM events with a very sharp transition in time (τ_decay/τ_rise ∼ 1) will cause larger footprint on the divertor surface and vice versa. This mutual dependence actually helps to better distribute the ELM energy along the divertor for a short ELM duration (τ_ELM).

The authors would like to thank R. Pitts, T. Eich and J. Horacek for the valuable discussion on the ELM characteristic times. Part of the work presented in this article was realized in the frame of summer and winter schools (SUMTRAIC and EMTRAIC) held at the IPP Prague [42]. This work was supported by MYES Projects# LM2023045 and CZ.02.1.01/0.0/0.0/16_013/0001551 and IAEA CRP F13019—Research Contract No. 22727/R0 and the Czech Science Foundation within the project GACR 20-28161S.