Investigating the occurrence and predictability of pitch angle scattering events at ADITYA-Upgrade tokamak with the electron cyclotron emission radiometer

1.   Introduction

Electron runaway, a fundamental concept in plasma physics that occurs when electrons acquire relativistic energies due to a rapidly increasing mean free path with energy, is critical for understanding high-energy phenomena in both laboratory and astrophysical plasmas [1–4]. This phenomenon is triggered when the applied electric field accelerates thermal electrons and surpasses their critical velocity to be runaway. In low-density tokamak discharges, a significant number of runaway electrons can reach energies of the orders of tens of MeV and damage vacuum vessel components [5, 6]. However, they also contribute to plasma current and may enhance confinement [6–8]. This scenario, known as the slide-away regime, predominates in low-density discharges, leading to distortions in electron distribution and potentially causing kinetic instabilities above thermal energies [5, 9].

Theoretical and experimental research is being done globally to identify the circumstances that can reduce or limit the energy of these runaway electrons to mitigate their destructive impact [10, 11]. Several techniques can reduce/limit the energy of the runaway electrons. Some of these are synchrotron radiation loss [12–15], bremsstrahlung radiation loss [16, 17], massive gas injection (MGI) [18–20], supersonic molecular beam injection (SMBI) [21], shattered pellet injection (SPI) [22], etc. Also, any method that limits the runaway energy through pitch angle scattering (PAS) can do so by increasing the energy perpendicular to the magnetic field and increasing the power radiated by the electron [23, 24]. Runaway electrons can undergo a variety of resonant interactions that might result in PAS, including interactions with lower hybrid waves [6, 24–28], magnetic field ripples [29], magneto hydro dynamics (MHD) modes in a stochastic magnetic field, etc [30].

To study plasma electron temperature evolution, tokamak research often relies on electron cyclotron emission (ECE) diagnostics [31–39]. Kirchhoff’s black body equation states that for optically thick plasma in thermal equilibrium, the cyclotron radiation from the resonance region correlates with local electron temperature. In low-density discharges, the ECE signal reflects cyclotron radiation from supra-thermal electrons rather than bulk electron temperature due to the plasma’s optically thin nature. Unlike hard X-rays, which mainly rely on electron energy, ECE power depends on both energy and the pitch angle of runaway electrons, akin to synchrotron emission (SE) [7]. SE exhibits strong forward beaming while ECE demonstrates a wide-angle distribution. Analyzing the ECE signal can offer insights into PAS events, particularly in cases like ADITYA-U [40, 41], where lower-energy runaway emissions are anticipated. PAS in tokamaks involves particle deflection due to magnetic field variations, potentially blocking runaway energy increase [12, 23–28, 30]. While PAS events are studied in a few tokamaks, further exploration is warranted to assess their viability as a runaway electron mitigation strategy.

This article delves into the investigation of various types of PAS events observed in the medium-sized ADITYA-U tokamak validated through ECE radiometer signals. The paper is structured as follows: section 2 outlines the experimental setup for the ADITYA-U tokamak and the diagnostic tools utilized. Section 3 delves into the conceptual understanding, observation, and analysis of PAS events along with their types. It also presents threshold conditions for pre-estimation of the repetitive PAS events and puts forward a novel proposal for using PAS events as an alternate and/or complementary runaway electron mitigation strategy. Section 4 explores attempts to simulate PAS events using the PREDICT code and confirm their occurrence.

2.   Experimental arrangement

To achieve shaped-plasma operations with an open diverter in single- and double-null configurations, the first Indian tokamak, ADITYA, which ran for more than two decades with a circular poloidal limiter, was modified to the ADITYA-U tokamak.

The ADITYA-U is a medium-sized tokamak with major and minor radii of R = 0.75 m and a = 0.25 m, respectively. Using 20 magnetic field coils placed symmetrically in the toroidal direction, a maximum toroidal magnetic field of B_T = 1.5 T is created, as shown in figure 1. ADITYA-U has produced ohmically heated circular plasmas with the following plasma characteristics: plasma current I_p $\leqslant$ 200 kA, discharge duration t $\leqslant$ 500 ms, chord averaged electron density ${n}_{\mathrm{e}}$ ~ (2$-$3)$\times$10¹⁹ ${\mathrm{m}}^{-3},$ and electron temperature ${T}_{\mathrm{e}}$ ~ 300$-$500 eV.

Figure  1.  Top view of the ADITYA-U tokamak with ECE radiometer mounted radially at port 7 and hard X-ray (HXR) detectors mounted tangentially at port 3.

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The ADITYA-U ECE radiometer is a 16-channel (16-ch.) super-heterodyne receiver that measures second-harmonic X-mode electron cyclotron emission across a broad range of toroidal magnetic fields (0.75–1.5 T) [42]. This newly designed system provides an extended dynamic range, allowing wideband measurements at different toroidal magnetic fields. Various RF units, each dedicated to a fixed set of second-harmonic frequencies owing to a fixed toroidal magnetic field, can be integrated at a time (as per requirement) with the single, 16-channel, fixed frequency, 1$-$ 20 GHz intermediate frequency (IF) receiver. As the toroidal magnetic field varies, the corresponding RF unit will change, while the 16-channel IF receiver remains fixed, providing ECE measurements at 16 radial locations at different B_T. On the radial axis, it provides a spatial resolution of 1.2 cm and a temporal resolution of 10 μs for estimating the electron temperature for typical ohmic discharges at the ADITYA-U tokamak. Figure 2 represents the typical characteristics at ADITYA-U for a plasma discharge #35219 with plasma current of ~ 120 kA and discharge duration of > 250 ms with other diagnostics signatures (i.e., ECE, soft X-ray (SXR), HXR, figures 2(e)–(f).

Figure  2.  Temporal evolutions of few plasma parameters of discharge #35219. (a) Plasma current ${I}_{\mathrm{p}}$, (b) loop voltage ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$(V), (c) central chord averaged density ${n}_{\mathrm{e}}$, (d) ECE amplitude and ratio $\omega _{\mathrm{p}\mathrm{e}}/{\omega }_{\mathrm{c}\mathrm{e}}$, (e) hard X-ray (HXR) emission, (f) soft X-ray (SXR) emission, and (g) Mirnov oscillations.

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A waterfall plot of the radiation temperature for discharge #35219 is obtained from the ECE radiometer diagnostic, depicting its spatial as well as temporal distribution in figure 3. The maximum temperature, located centrally, exceeds 350 eV, while the minor disruption around 96.8 ms results in a decrease in radiation temperature by 15%–20%. In this discharge, optical thickness conditions (τ $\gg$ 1) are not fully met. Calculations show that optical thickness (τ) exceeds 1 only in the central plasma region. Consequently, the displayed temperature reflects the radiation temperature (${T}_{\mathrm{r}\mathrm{a}\mathrm{d}}$) from optically thin plasma.

Figure  3.  Temporal and spatial evolutions of radiation temperature (${T}_{\mathrm{r}\mathrm{a}\mathrm{d}}$) as measured by the ECE radiometer diagnostic for discharge #35219.

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3.   PAS events – theory and experimental findings

Electron cyclotron radiation is generated by gyrating electrons in the presence of a magnetic field. In general, for a tokamak, the observed frequency is given as ${\omega }_{\mathrm{c}\mathrm{e}}$ [31], i.e.,

where n is the harmonic number, e and m_e are electron charge and electron rest mass, $\gamma$ is the relativistic factor, ${\beta }_{\parallel }={v}_{\parallel }/ c$ is the parallel component of the normalized electron velocity in the direction of toroidal magnetic field, c is the light velocity, $\theta$ is the angle between line of observation and toroidal magnetic field, and ${B}_{\mathrm{T}}$ is the toroidal magnetic field strength. ${v}_{\perp }$ is the electron velocity perpendicular to the toroidal magnetic field and ${\rho }_{\mathrm{L}}$ is the relativistic Larmor radius. Equation (1) represents the full resonance condition for cyclotron frequencies, i.e., this single equation incorporates the three major broadening effects that affect the cyclotron resonance wherein the frequency of the electromagnetic wave matches the modified cyclotron frequency of the electron in a tokamak environment considering: the relativistic effects ($\gamma$) and Doppler broadening effects $(1-{\beta }_{\parallel }\mathrm{c}\mathrm{o}\mathrm{s}\theta )$ apart from the usual toroidal magnetic field, i.e., ${B}_{\mathrm{T}}$ field effects (${B}_{\mathrm{T}}\propto 1/R$).

For optically thick plasma, the ECE measurements provide localized thermal estimations, while for the optically thin cases, the non-thermal ECE radiation is broad-spectrum and non-localized. The component $\gamma$, which is responsible for cyclotron frequency downshift and its harmonics, undergoes significant changes in the presence of runaway electrons. However, their narrow pitch angles may limit their ability to emit and absorb electron cyclotron waves [7].

3.1   Conceptual framework on the PAS events

Due to the constant acceleration from the applied electric field, the initial Maxwellian electron distribution starts to develop a runaway tail. Various instabilities can be triggered as a result of the anisotropic distribution. The resonant electrons and plasma oscillations can interchange the free energy of the Maxwellian component [1]. One such instability, known as Parail Pogutse (PP) instability, is triggered as a consequence of which a broad spectrum of waves become unstable, including the lower hybrid waves [43]. Experimentally, it is observed that once the runaway electrons’ energy reaches the value of ${W}_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}$, Langmuir waves are excited, given as [28, 44]

where ${\omega }_{\mathrm{c}\mathrm{e}}$ is the electron cyclotron frequency and ${\omega }_{\mathrm{p}\mathrm{e}}$ is the electron plasma frequency. The calculated beam energy for runaways at ADITYA-U should be greater than 0.66 MeV. ${W}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ is the minimum kinetic energy that the runaway electrons shall have, and it is defined as:

For electrons going faster than the critical velocity ${v}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ (velocity above which the collisional drag force is less than the applied electric field force), these accelerated electrons are known as the “runaway electrons”. The critical velocity [25] is given as

where ${n}_{{\mathrm{e}}}$ is the electron density, $\mathrm{l}\mathrm{n}\mathrm{\Lambda }$ is the coulomb logarithm, ${Z}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective ion charge, $\varepsilon_{0}$ is the permittivity of free space, and E is the applied electric field. For low plasma density conditions, often observed during the density and/or current ramp-down instances of a discharge, the anomalous doppler resonance (ADR) will cause pitch angle scattering of these runaway electrons on the excited lower hybrid waves, satisfying the resonance criteria given as:

where ${\omega }_{k}$ is the wave frequency, ${\omega }_{\mathrm{c}\mathrm{e}}=e{B}_{\mathrm{T}}/{m}_{\mathrm{e}}$ is the cyclotron frequency for non-relativistic plasma, i.e., γ = 1, n is the resonance number, ${k}_{\parallel }$ is the parallel wave number, ${B}_{\mathrm{T}}$ is the toroidal magnetic field, and ${v}_{\parallel }$ is the parallel velocity of resonant electrons. Additionally, empirical thresholds exist, such as the ratio of electron plasma frequency to electron cyclotron frequency (${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}< 1$), which serve as markers for the discharge entering the slide-away domain through ADR. This phenomenon is validated by low-density discharges observed in TEXTOR, EAST, TFR, HT-7, etc. [24, 25, 45, and 46]. The ADR resonance conditions as stated in equation (5) were modified as

where ${\beta }_{\parallel }$ is the normalized velocity of electrons and ${N}_{\parallel }={k}_{\parallel }c/ {\omega }_{\mathrm{L}\mathrm{H}}$ is the parallel refractive index. The excited LH wave frequency (${\omega }_{\mathrm{L}\mathrm{H}}$) [25] can be calculated as

where A is the atomic mass number, ${Z}_{\mathrm{e}\mathrm{f}\mathrm{f}}=3$ is the effective ion charge, ${n}_{\mathrm{e}}$ = 0.5 $\times$ 10¹⁹ ${{\mathrm{m}}}^{-3}$ is the electron density at the time of the PAS event, ${\omega }_{\mathrm{p}\mathrm{i}}$ is the ion plasma frequency, ${n}_{\mathrm{i}}$ is the ion plasma density, e is electron charge, ${m}_{\mathrm{i}}$ is the ion mass, ${\mathrm{\varepsilon }}_{0}$ is the permittivity of free space, and Z is the atomic number of ions. The calculated value of the excited LH frequency (${\omega }_{\mathrm{L}\mathrm{H}}$) for ADITYA-U is 5 GHz. The energy of the resonant electrons is given as

Using typical ADITYA-U tokamak parameters and a refractive index (${N}_{\parallel }$) of 4, the calculated resonant electron energy is 7 MeV. These calculations apply to low densities ${n}_{\mathrm{e}}\leqslant 1\times {10}^{19}\;{\mathrm{m}}^{-3}$ and different toroidal magnetic field strengths (${B}_{\mathrm{T}})$, offering a range of LH frequencies for ADITYA-U. This resonant interaction prevents runaway electrons from surpassing W_res, potentially limiting their energy [25, 26]. A comparative study of the previous work carried out on this phenomenon in other tokamaks with machine parameters, along with those of ADITYA-U, sharing a similar aspect ratio as that of ITER, is portrayed below in table 1.

Table  1.  Comparative study of machine parameters, corresponding radiometer specifications, and LH frequencies in a few other tokamaks wherein investigation on PAS is carried out and ITER projections given for comparison.

No.	Tokamak	R/a	BT (T)	Radiometer specifications	${\mathit{N}}_{\parallel }$	${\mathit{\omega }}_{\rm{L}\rm{H}}$ (GHz)	Wres (max) (MeV)
1	HT-7	4.51	1.7$-$2.5	16-channels, frequency range = 98$-$126 GHz, temporal resolution = 4 μs	4	3.5	21
2	TEXTOR	3.8	< 2.9	11-channels, frequency range = 98$-$180 GHz, temporal resolution = 50 μs (variable)	4	2.9	29
3	EAST	3.88	< 3.5	32-channels, frequency range = 104$-$168 GHz, temporal resolution = 10 μs	5	4.6	17
4	HL-2A	4.12	< 2.8	60-channels, frequency range = 60$- 90$ GHz, temporal resolution = 3 μs	1.25–3.45	2.45	41.7
5	ITER	3.1	5.3	58-channels, frequency range = 122$-$230 GHz, temporal resolution = 10 ms (variable)	4*	5.03	31
6	ADITYA-U	3	$\leqslant$ 1.5	16-channels, frequency range = 64$-$86 GHz, temporal resolution = 10 μs	4	5	8.86
* Assumed refractive index values for ITER

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Table 1 shows that only a few tokamaks with higher magnetic fields have reported and investigated this phenomenon, thus emphasizing the need to study this event. Since the ECE is highly dependent on the pitch angle of the electrons, kinetic instabilities like the PP instability can have a striking impact on ECE radiation [6]. In contrast to the usual growth in runaway presence, a sudden, abrupt rise in the ECE amplitude is seen, signaling that the discharge has entered the slide-away regime due to PP instability. The short time scales rule out the possibility of the increase being attributed to an energy gain or an increase in the number of runaways [25]. Given the absence of significant changes in bulk parameters, PAS emerges as the sole phenomenon responsible for the rapid rise in ECE amplitude. In low-density slide-away discharges, the plasma is optically thin, resulting in contributions from wall reflections that affect measurements along with a frequency downshift. While this fact is not dismissed, the intensity increase from wall reflections should not manifest as a sharp spike in ECE signatures; instead, its contribution will be spread across a broad spectrum. Moreover, the wall reflections will affect the intensity throughout the temporal evolution of the ECE signal and not as a sudden, steep rise.

3.2   Experimental findings on PAS events and their types

A distribution of fast pitch angle scattering (FPAS) events at varied B_T is represented in the form of a pie chart in figure 4(a). These discharges shall be further investigated for obtaining threshold conditions in the next section. PAS events are categorized based on observed steps in ECE amplitude, illustrated in figure 4(b). A PAS event may manifest as a single substantial step (FPAS), a series of steps (multiple fast pitch angle scattering, MFPAS), or as a gradual linear rise (slow pitch angle scattering, SPAS).

Figure  4.  (a) Pie chart showing the distribution of considered discharges that observed FPAS at varied B_T, (b) types of PAS events.

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3.2.1   Fast pitch angle scattering

A sudden, single jump is observed in the ECE signatures, as depicted in figure 5, with a 30% rise in amplitude in a few μs. This figure illustrates a low-density discharge (#32810) at ADITYA-U, B_T = 1.28 T, with a maximum plasma current (${I}_{\mathrm{p}}$) of 130 kA and discharge duration of 147 ms, featuring a single FPAS event observed in the ECE radiometer central channel.

Figure  5.  Temporal evolutions of few plasma parameters of discharge #32810. (a) Plasma current ${I}_{\mathrm{p}}$, (b) loop voltage ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$(V), (c) central chord averaged density ${n}_{\mathrm{e}}$, (d) ECE amplitude, (e) hard X-ray (HXR) emission, (f) soft X-ray (SXR) emission and ratio ${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$, (g) Mirnov oscillations, and (h) bolometer signals.

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The loop voltage declines at ~ 85 ms, indicating entry into the slide-away regime. As density decreases further during the current ramp-down phase of the discharge, SXR and bolometer signals also decrease. However, a sudden single jump occurs in ECE signals at 117.1 ms, with the ratio dropping to 0.30. While other plasma parameters remain largely unchanged, hard X-ray (HXR) significantly decreases.

Further, figure 6 shows the temporal evolution of diamagnetic energy, W_dia and ${\beta }_{\mathrm{p}}\;(\beta -\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{l}$) of the discharge #32810, which shows a slight rise with the occurrence of FPAS events distinguished by a single, vertical line. The increase in diamagnetic energy confirms the rise in perpendicular energy of the runaway beam [25]. However, this increase is not significantly observed in all discharges with such events. This could be because the change in runaway beam perpendicular energy is not so significant to be visible in the diamagnetic signals. This increase is observed in about 30%$-$40% of the discharges that were investigated for this rise. As mentioned in [25], a sudden single increase in ECE amplitude may stem from the initial interaction of waves depleting their energy density, hindering them from re-entering the resonance region, and so a single step is seen instead of multiple ones. Additionally, radiation losses via PAS may constrain the energy of runaway electrons, leaving them with limited energy for further interaction.

Figure  6.  Temporal evolutions of steep rise in ECE signal co-related to the diamagnetic energy (${W}_{\mathrm{d}\mathrm{i}\mathrm{a}}$) and ${\beta }_{\mathrm{p}}$ measurements.

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The steep amplitude rise is visible in all the channels of the radiometer within the existing radial coverage of 2.2 cm (low field side, LFS) to $-$14 cm (high field side, HFS) for B_T = 1.2 T, as shown in figure 7. It can be seen that the “dome shape”, as observed in the ECE signals, is different for different channels. This variation can be considered as a signature for an optically thin plasma due to non-thermal existence, frequency downshift contribution, plasma position movement, or a combination of all of these. At this moment, it is difficult to quantify their contribution, but it may be considered for future investigation.

Figure  7.  Temporal evolutions of PAS events seen as steep amplitude rise in all 16 channels of radiometer for discharge # 33392.

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3.2.2   Multiple fast pitch angle scattering (MFPAS)

The MFPAS event, a repetitive stepwise increase in ECE signal akin to FPAS, aligns with the perpendicular energy rise of runaway electrons. As demonstrated in figure 8, it depicts the temporal evolutions of various plasma parameters for low-density discharge #33392 at ADITYA-U with B_T = 1.28 T.

Figure  8.  Temporal evolutions of few plasma parameters of discharge #33392. (a) Plasma current ${I}_{\mathrm{p}}$, (b) loop voltage ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$ (V), (c) central chord averaged density ${n}_{\mathrm{e}}$, (d) ECE amplitude, (e) hard X-ray (HXR) emission, (f) soft X-ray (SXR) emission and ratio ${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$, (g) Mirnov oscillations and a zoomed view of the ECE amplitude showing the jumps.

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Figure 8(a) illustrates a current decline from 93.7 ms, mirrored by a drop in loop voltage in figure 8(b), signifying entry into the slide-away regime. In figure 8(c), density decreases in a usual manner. At 90.19 ms, when the ADR condition is met (ratio ~ 0.37), the ECE signal amplitude sharply rises, almost doubling within μs, continuing as steps until saturation. Zoomed ECE emission (figure 8(d)) reveals initial prominence, consistent with other tokamaks. There are several experimental data points between the start and the end of the jump in the ECE signal. HXR emission (figure 8(e)) fluctuates but lacks a significant correlation with the ECE jump. SXR (figure 8(f)) reduces due to density fall, persisting until discharge ends, suggesting that this is not a bulk phenomenon. The lack of notable Mirnov oscillations (figure 8(g)) during the PAS event also suggests that this event is not due to any magnetic oscillations. Unlike the single step case, there is a possibility that the waves of higher energy density might have been excited so that they get a chance to enter the resonance region multiple times.

3.2.3   Slow pitch angle scattering (SPAS)

Apart from the discharges wherein there is a steep jump in ECE amplitude, there are other discharges as well, wherein there is a sudden linear rise instead of a steep jump. Depicted in figure 9 is discharge #36544 which has a maximum plasma current of 100 kA and a discharge duration of 100 ms for a toroidal magnetic field of 1.28 T.

Figure  9.  Temporal evolutions of few plasma parameters of discharge #36544. (a) Plasma current ${I}_{\mathrm{p}}$, (b) loop voltage ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$(V), (c) central chord averaged density ${n}_{\mathrm{e}}$, (d) ECE amplitude, (e) hard X-ray (HXR) emission, (f) soft X-ray (SXR) emission and ratio ${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e} }$, and (g) Mirnov oscillations.

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As the discharge proceeds to an end with a decrease in current and density, the ECE amplitude shows a linear increase, depicting the penetration of non-thermals, which is also an indication of the possibility of pitch angle scattering. This exponential rise in ECE amplitude is understood as a slow pitch angle scattering event that requires a time scale of up to 2–5 ms, unlike the steep rise in the fast pitch angle scattering event, which is within $\leqslant$ 100 μs.

SPAS events were also observed at HT-7 tokamak as normal pitch angle scattering (NPAS), wherein it was found that NPAS occurs at lower plasma current values compared to FPAS events, while the latter is triggered at lower ratio values [46, 47]. Unfortunately, the number of SPAS events obtained at ADITYA-U is not sufficient compared to the FPAS events, so it would be difficult to provide something conclusive about such events at the moment. However, certain observations from the few SPAS discharges are that their ECE amplitude rise is slow and linear, unlike the steep jump in FPAS. It is observed that SPAS discharges terminate soon upon their occurrence, with a discharge duration of 90$-$110 ms, unlike FPAS discharges, which terminate with a discharge length of 120$-$150 ms. Also, SPAS events are found to occur earlier in the discharge duration than FPAS events. However, further investigation into such events is still underway.

3.3   Threshold conditions for pre-estimation of PAS

It has been proposed that with this mechanism, lower hybrid waves under proper conditions can be used to disrupt runaway electron beams, and that this is particularly useful for disruption-generated runaway electrons [25, 26]. Henceforth, an attempt has been made, for the first time, to explore this possibility by primarily investigating the relation of discharge parameters with the trigger of a PAS event due to their pertinent redundancy. Based on the experiments conducted on different machines, it has been observed that the discharge enters the slide-away regime when the condition: ratio, i.e., ${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$< 1, is satisfied. This criterion shows its dependence on the toroidal magnetic field and electron density. A preliminary investigation was done to confirm this dependence through experimental parameters obtained from a small database of 80 FPAS discharges at ADITYA-U. The considered ohmic discharges have a plasma current I_p $\leqslant$ 160 kA, discharge duration of < 300 ms, and varied magnetic fields. The experimental values of plasma current, discharge length, and ramp rate of plasma current, during the descent phase of the discharge, are plotted as a function of the ratio (${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$) that provides insight into the possible trigger conditions for the instability to take place.

Figure 10(a) shows the inverse dependence of ratio (${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$) on toroidal magnetic field (B_T), which was evaluated for four different values: 0.89, 1.07, 1.2, and 1.28 T. Figure 10(b) depicts the linear dependence of electron density (n_e) on the ratio for the four different B_T values considered. For lower density values (n_e $\leqslant$ 1$\times$10¹⁹ ${\mathrm{m}}^{-3}$), the ratio mostly lies below 1. It is yet to be explored as to why, at B_T = 1.28 T, the values are spread out compared to others. Nevertheless, they do follow linearity.

Figure  10.  Variation of ratio ${\omega }_{\mathrm{p}\mathrm{e}}/{\omega }_{\mathrm{c}\mathrm{e}}$ with (a) toroidal magnetic field (B_T), (b) electron density ${n}_{\mathrm{e}}$, (c) plasma current I_p, (d) discharge duration, and (e) rate of plasma current fall ${\mathrm{d}}{I}_{\mathrm{p}}/ {\mathrm{d}}t$.

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Figure 10(c) illustrates a linear relationship between plasma current (I_p) and toroidal magnetic field (B_T), implying that initiating PAS at higher B_T and I_p values requires lowering electron density to below 0.5 $\times$10¹⁹ ${\mathrm{m}}^{-3}$. Additionally, in figure 10(d), repetitive occurrences of PAS are observed for discharges that last for over 100 ms. Nonetheless, these findings are preliminary, and further investigation is ongoing. Furthermore, figure 10(e) indicates that discharges with plasma current ramp rates of 3$-$5 MA/s frequently experience such events in the ADITYA-U tokamak, suggesting that adjusting the current ramp rate could trigger PAS. Based on figure 10, the trigger conditions are tabulated below.

Notably, primary conditions such as discharge length are crucial for confirming the efficacy of PAS in limiting runaway energy increase. In the case of the ADITYA-U tokamak, where the discharge length is small and the runaway electron energy range is low, detection through ECE diagnostic signals is feasible. However, in large-size tokamaks like ITER, where the discharge length is much higher and the runaway electron energy range is on the order of tens of MeV detection through IR and/or visible cameras would be more viable due to emissions falling within detectable ranges. Nevertheless, we explore the possibility of triggering PAS events using their repeatability. We utilize a methodology involving the following steps experimentally:

• Obtain a database of discharges from the ADITYA-U tokamak that have experienced PAS events.

• Determine the ADR satisfying condition, i.e. ratio ${\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}$< 1, and confirm its satisfaction.

• Obtain the correlation of this ratio with other discharge parameters like I_p, dI_p/dt, n_e, B_T, etc. as shown in table 2.

Table  2.  Threshold conditions for PAS occurrence.

Parameter	Specification
BT (T)	1.28	1.2	1.07	0.89
Ratio	0.25–0.7	0.4–0.7	0.6–0.8	0.7–0.8
ne ($\times$1019 ${\mathrm{m}}^{-3}$)	0.4–0.7	0.35–0.7	0.5–0.8	0.4–0.9
Ip (max) (kA)	100–150	115–132	95–125	65–100
dIp/dt (kA/ms)	3.6–4.8	2.5–5.08	1.8–4.6	1–4

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• For the discharge parameters as stated above, to confirm if these threshold values trigger PAS experimentally, the following methodology can be applied: (i) for given B_T, decrease the electron density such that the ADR condition (ratio < 1) is satisfied and PAS event is triggered, (ii) ramp down the plasma current at a rate such that ADR condition (ratio < 1) is satisfied and PAS is triggered.

This novel yet preliminary study suggests that for a given discharge, once the trigger conditions are satisfied, PAS can be expected. However, whether fulfillment of these experimentally obtained trigger conditions results in PAS excitation requires further investigation. This can be confirmed either experimentally or through modeling. For this manuscript, the PREDICT code is used to model the occurrence of PAS if the given experimentally obtained threshold parameters are met, and the same is discussed further. In the subsequent section, for larger-sized tokamaks such as ITER, the potential energy range of runaway electrons may exceed tens of MeV and could be detected using visible and/or IR cameras within synchrotron emission ranges, further emphasizing the universality of these findings. Similar discharge parameters along with a few others can be investigated for their pre-estimation.

4.   PAS events – modeling through the PREDICT code

Attempts were made to model the ADITYA-U tokamak PAS event using the PREDICT numerical tool [48], a 0-D code in real space and a 2-D code in momentum space. It tracks the temporal evolution of RE parameters (energy, density, pitch angle) for given plasma scenarios. Based on the relativistic test particle model, it describes RE trajectories in the momentum space and considers various RE generation mechanisms like primary and secondary generation, hot-tail, tritium decay, and Compton scattering [48, 49]. Test particle equations estimate RE energy evolution, including electric field acceleration and losses from collisions and synchrotron radiation. It also accounts for pitch-angle scattering from interactions with magnetic field ripple resonance, LH waves, and stochastic magnetic fields. The PREDICT code has been applied on a few tokamaks for runaway electron modeling [49–53]. Normalized test particle equations describing the resonant interaction between the RE’s and lower hybrid waves generated within the plasma due to the ADR effect are written as [26]:

where ${q}_{\parallel }$, ${q}_{\perp }$, and q are the parallel, perpendicular, and total electron momenta normalized to ${m}_{\mathrm{e}}$c and $\mathrm{\gamma }$ is the relativistic gamma factor. ${\alpha }_{{Z}}$ = 1+Z_eff and ${F}_{\mathrm{g}\mathrm{c}}$, ${F}_{\mathrm{g}\mathrm{y}}$ are contributions due to the guiding center motion and electron gyromotion respectively. Equations (9) and (10) represent the rates of change of parallel and perpendicular normalized momentum on LHS while the first term on the RHS is the acceleration term due to the normalized electric field ($D={E}_{\left|\right|}/{E}_{{\mathrm{C}}}$), and the second term describes the collision effect with plasma density. The third term includes the effect of synchrotron radiation losses. D_LH is the normalized diffusion coefficient that includes the effect of LH wave resonant interaction as provided by equation (10). A detailed description of each parameter in equations (9)–(13) can be found in reference [26]. While individual wave–particle interactions occur on the gyro-motion timescale, the broader resonance process, influenced by plasma collisions and simulated as a diffusion-like behavior in phase space, evolves on a longer timescale. Hence, for modeling the resonance process, simulating the test particle equations of a longer timescale compared to the gyration timescale is still valid. This TPM model [26] is valid for the conditions ${k}_{\perp }{\rho }_{\mathrm{L}}\ll 1$, and considering the energy range of REs in typical ADITYA-U plasma scenarios is expected from 1 to 20 MeV. Here, ${k}_{\perp }$ is the perpendicular wave vector. The contour plot in figure 11 shows two elements in the runaway electron energy (E_RE) versus wave frequency space (parametric regime), namely (i) resonance condition: $|n{\omega }_{\mathrm{c}\mathrm{e}}+{k}_{\parallel }{v}_{\parallel }-\omega | < \Delta \omega$ and (ii) the product of perpendicular wave-vector and the relativistic Larmor radius (${k}_{\perp }{\rho }_{\mathrm{L}}$).

Figure  11.  Theoretically expected range of REs in ADITYA-U tokamak and corresponding LH-wave frequency according to the resonance condition in equation (6). Different black lines show different contour levels for ${k}_{\perp }{\rho }_{\mathrm{L}}$, confirming the validity of the model in the region where resonance is most likely.

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For element (i), the left side (LHS) of the resonance condition |$n{\omega }_{\mathrm{c}\mathrm{e}}+{k}_{\parallel }{v}_{\parallel }-\omega$| is plotted as a surface plot whose value is indicated by the color bar. For the resonance condition to be satisfied, the value of the LHS (|$n{\omega }_{\mathrm{c}\mathrm{e}}+{k}_{\parallel }{v}_{\parallel }-\omega$|) needs to be lower than the RHS of the equation, Δω, i.e. the resonance width. For element (ii), the dash-dotted, dotted, and dashed black lines are contours that represent the constant ${k}_{\perp }{\rho }_{\mathrm{L}}$ corresponding to the contour levels 1, 0.1, and 0.01, respectively. These contours are shown to check the applicability of the pitch-angle scattering diffusion coefficient model used in the present simulations which requires ${k}_{\perp }{\rho }_{\mathrm{L}} \ll1$. Figure 11 shows that the resonance condition is satisfied (the dark area on the surface plot) close to the contour of the level ${k}_{\perp }{\rho }_{\mathrm{L}}$ ~ 0.1 (dotted line) and establishes the validity of the model and range of LH waves from 1 to 60 GHz that can cause the PAS event for a different product of the ${k}_{\perp }{\rho }_{\mathrm{L}}$ values for the ADITYA-U tokamak.

Figure 12(a) shows the simulation of a test particle momentum evolution in the phase space (${q}_{\perp }^{2},{q}_{\left|\right|}$) using equations (9) and (10). Here, three different cases were studied assuming the following constant plasma parameters for the duration of 5 s: n_e = 0.5$\times$10¹⁹ ${\mathrm{m}}^{-3}$, Z_eff = 2.5, N_|| = 4, n = −1, $\mathrm{l}\mathrm{n}\mathrm{\Lambda }$ = 15, R = 0.75 m, B_T = 1.28 T, E = ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$/$2 {\text{π}} R$ = 0.05 V/m, $\omega$_LH ~ 2 GHz, ${E}_{\perp }$ = 300 V/m, with initial condition of test RE momenta ${q}_{\parallel }$ = 0.2 and ${q}_{\perp }$ = 0.012 (corresponding to RE energy ~ 50 keV, just above the critical energy to become RE). The choice of these plasma parameters is motivated by the typical plasma parameters expected in the ADITYA-U tokamak except for the duration of the plasma discharge ($\leqslant$ 0.5 s). However, although the parameters used in figure 12 mimic those of an ADITYA-U-like tokamak during the ramp-down phase, the initial electric field is much lower than that in ADITYA-U, resulting in a prolonged duration to attain sufficient beam energy. The longer time scales of the discharge duration, on the order of seconds, also demonstrate that RE energy limitation in an ADITYA-U-like tokamak can occur when the discharge duration is extended.

Figure  12.  (a) The evolution of a test particle momentum in the phase space (${q}_{\perp }^{2},{q}_{\left|\right|}$) and (b) corresponding temporal evolution of runaway electron energy (E_RE) assuming following constant plasma parameters for the duration of 5 s: n_e = 0.5$\times$10¹⁹ ${\mathrm{m}}^{-3}$, Z_eff = 2.5, N_|| = 4, n = −1, $\mathrm{l}\mathrm{n}\mathrm{\Lambda }$ = 15, R = 0.75 m, B_T = 1.28 T, E = ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$/$2{\text{π}}R$ = 0.05 V/m, ${\omega }_{{\mathrm{LH}}}$ = 2 GHz, ${E}_{\perp }$= 300 V/m with initial condition of test RE momenta q_|| = 0.2 and ${q}_{\perp }$ = 0.012.

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The first case is the evolution of RE momentum without considering resonance, where the RE attains a maximum energy of ~ 35 MeV at 5 s. The second case shows a PAS of the test RE when the resonance condition is satisfied at ${q}_{\left|\right|}$ = 37, corresponding to $\omega$_LH = 2 GHz, at t ≈ 1.4592 s. The RE experiences PAS, increasing its pitch angle ($\theta\mathrm{_p}=\mathrm{t}\mathrm{a}\mathrm{n}^{-1}(q_{\perp}/q_{\left|\right|}$)) and leading to enhanced synchrotron radiation. The balance between the electric field acceleration and enhanced synchrotron radiation results in the blocking of RE energy evolution at E_RE =19 MeV throughout the simulation duration. The third case shows the evolution of RE momentum when the accelerating electric field (E) is increased by 10%. In this case, the RE first experiences PAS momentarily when the resonance condition is satisfied at t ≈ 1.4592 s, but it eventually escapes from the resonance layer within ~ 64 ms due to the stronger electric field and accelerates again. It should be noted that these representative cases have a duration longer than the typical plasma discharges at ADITYA-U (discharge duration $\leqslant$ 0.5 s) and are only shown to demonstrate the possibility of RE energy blocking as a result of PAS. Regardless, the RE can gain sufficient energy in experimental discharges to satisfy the resonance condition as discussed below.

The same model has been applied to the ADITYA-U plasma discharge scenario where FPAS events have been observed. Time-varying plasma parameters, namely plasma density (n_e), loop voltage (${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$), plasma current (I_p), magnetic field (B_T), and calculated temperature (T_e) are taken as input in the PREDICT code as described in detail [48] to simulate RE parameters. Figure 13 shows temporal evolutions for these input parameters for the representative plasma discharge #32535, where the PAS event has been observed. In this discharge, the B_T = 1.28 T was constant throughout the discharge.

Figure  13.  Time evolutions of the plasma parameters, namely plasma current ${I}_{\mathrm{p}}$, loop voltage ${V}_{\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$(V), and plasma density ${n}_{\mathrm{e}}$ for discharge #32535.

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Figure 14 shows the temporal evolution of test RE energies generated at different phases of the discharge. The test RE energy evolution is calculated using equations (9), (10), and (14) taking into account the time-varying input parameters of the electric field, plasma density, and electron temperature. Figure 14 indicates that REs generated at different times of the discharge can have different energies, leading to the existence of a wide range of RE energies from 0.1 to 20 MeV.

Figure  14.  Temporal evolutions of test RE energies generated at different phases of the discharge #32535.

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Here, the parameter q is the total momentum ($q=\sqrt{{{q}_{\parallel }^{2}+q}_{\perp }^{2}}$) and is normalized to ${m}_{\mathrm{e}}c$.

To model the PAS event at the time of the ECE signal jump, it is assumed that the LH waves in the frequency range from $\omega$_pi to $\omega$_pe have been excited as suggested in [25].

In the present simulations, an attempt has been made to determine the occurrence of PAS events by considering the existence of LH waves in the frequency range of ω_LH ≈ 1−60 GHz when the following trigger conditions (table 2) have been satisfied. These inputted conditions for the code are:

(i) Plasma current is in the decay (ramp-down) phase.

(ii) Rate of change of plasma current is dI_p/dt $\geqslant$ −2.5 MA/s.

(iii) Threshold ratio is $\omega$_pe/$\omega$_ce$<$1 depending upon the applied toroidal magnetic B_T.

(iv) The minimum energy of REs $\geqslant {W}_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}$ as per equation (2). These conditions are shown in figures 15(a)−(d) for the ADITYA-U plasma discharge #32535.

Figure  15.  Temporal evolutions of the (a) rate of change of plasma current (dI_p/dt), (b) ratio $\left({\omega }_{\mathrm{p}\mathrm{e}}/ {\omega }_{\mathrm{c}\mathrm{e}}\right)$, (c) ECE signal and logic level to trigger LH-wave in the simulation, and (d) critical beam energy $\left({W}_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}\right)$.

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All these conditions have been satisfied at t ~ 0.0901 s (figure 15) and only then does the code assume the presence of LH waves, while its possibility to lead to PAS shall be discussed later. Although these findings are preliminary and based on a small database of discharges, it is a novel attempt to investigate PAS as a complementary mitigation plan by exploring its repeatability. This also emphasizes the need for further investigations to validate the triggering of PAS events under dynamic tokamak conditions, i.e., experimentally. Further, by adjusting the simulation input parameters and trigger conditions, SPAS and MPAS can be modeled as well using this model.

In figure 15(d), there is a time delay observed between the logic level attaining value 1 in the simulation to trigger the LH waves which take place at t ≈ 0.0900 s, and experimental observation of step-like jump in the ECE signal, which is at t ≈ 0.0958 s. The difference is about ≈ 5.8 ms. This time delay is attributed to the input parameters taken into account to trigger the LH waves in the simulations, namely, the ratio $\omega$_pe/$\omega$_ce $\leqslant$ 0.5 inferred from the average value of the database for B_T = 1.28 T (figure 10(a)) and the similarly negative slope of the rate of change of plasma current (dI_p/dt) $\geqslant$ −3 MA/s (figure 10(e)), which is a lower bound of dI_p/dt for which significant occurrence of the PAS event has experimentally been observed. Due to the finite statistical spread in the experimental data of $\omega$_pe/$\omega$_ce and dI_p/dt to trigger the LH waves in the simulation, minor deviation in the simulated trigger time can be expected.

Results of the simulated PAS event for the REs are in figures 16(a) and (b) for #32535 after t ~ 0.0900 s. Figure 16(a) shows a sudden rise of perpendicular momentum (${q}_{\perp }$) of the representative REs shown in figure 14 at t ~ 0.0900 s. The total radiated power (synchrotron radiation, integrated over emission angle and wavelength range) emitted in the synchrotron radiation process by the RE fractions during the discharge is calculated using equation (14) and depicted in figure 16(b) [49].

Figure  16.  Temporal evolutions of (a) sudden rise of the normalized perpendicular momentum (${q}_{\perp }$) of REs at t ~ 0.901 s that are born at each time step, dt ≈ 5 ms, during the discharge due to wave-particle interaction and (b) a qualitative comparison between the total radiated power (P_tot) by RE fractions and the ECE signal.

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where ${\beta }_{\perp }$ = ${v}_{\perp }$/c and r_e = ${\mathrm{e}}^{2}/ 4 {\text{π}} {\varepsilon }_{0}{m}_{{\mathrm{e}}}{c}^{2}$ is classical electron radius.

A qualitative comparison is made between the total synchrotron radiated power by RE fractions and the ECE signal, as shown in figure 16(b). This shows reasonable consistency between a step-like jump in the ECE signal and the simulated PAS of REs. It should be noted that the total synchrotron radiated power in figure 16(b) is calculated per runaway electron and summed over all fractions born throughout the discharge. For better comparison, this total synchrotron radiated power should be calculated with the weighted RE density, considering the loss of these RE density fractions during the discharge as well as the spectral range of the ECE detectors and emission angles based on the spatial configuration of the ECE diagnostic. Thus, the present analysis is a preliminary investigation wherein the experimental trigger conditions for PAS occurrence are set as inputs to model the occurrence of these events and provide reasonable consistency with the experimental observations.

The experimental plasma scenario discussed in this section is such that the RE energy does not increase after ~ 90 ms even without the resonance phenomenon, as shown in figure 17(a), inferred from the simulations. Additionally, the discharge terminates soon after the PAS condition is satisfied. So, the effect of the PAS event is minimal on the energy evolution for the present discharge. However, a significant difference can be seen in the temporal evolution of ${q}_{\perp }$ shown in figure 17(b). Hence, while the presented experimental discharge simulation does not indicate the PAS event as the primary energy-blocking mechanism, the occurrence of the PAS event can still be simulated.

Figure  17.  Time evolutions of (a) the maximum RE energy with and without LH wave resonance and (b) normalized perpendicular momentum (${q}_{\perp }$) with and without LH wave resonance.

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These attempts demonstrate that under specific tailored trigger conditions, the PAS event can indeed be initiated like for a given magnetic field, manipulating a specific current ramp rate or reducing the density to meet the PAS trigger conditions. As a future scope of work, experimental discharges have been planned to improve and isolate the effect of the PAS event as an energy-blocking mechanism. Additional improvements in the simulation are also planned taking into account the RE density to obtain a quantitative estimate of the synchrotron radiated power.

5.   Conclusion

Different forms of pitch angle scattering events: fast (single) pitch angle scattering, multiple fast pitch angle scattering, and slow pitch angle scattering have been observed through ECE radiometer diagnostics, with a steep rise in its amplitude in $\leqslant$100 μs for low-density discharges at ADITYA-U tokamak. These are proposed to occur due to a resonance between the runaway electrons and the excited lower hybrid waves for discharges that satisfy the anomalous Doppler resonance threshold conditions. PAS being a repetitive phenomenon, a database of such discharges is obtained, and various discharge parameters are investigated to understand the predictability of these events and pre-estimate their occurrence by ramping down the plasma current at a specific rate or the electron density at given values of plasma current and toroidal magnetic field, which, according to the author, is a first-of-its-kind investigation. As these events are favorable for limiting runaway energy, their pre-estimation can aid in reducing the possible damage caused to machines. Experimentally obtained trigger conditions are set as input to the PREDICT code to explore the favorability of PAS occurrence. Preliminary modeling investigations confirm a reasonable consistency with the ECE observations. These findings reported herein are preliminary and based on a small database of discharges, emphasizing the need for further investigations to validate the triggering of PAS events under dynamic tokamak conditions experimentally as well as through the simulations.