Excitation of Alfvén eigenmodes by fusion-born alpha-particles in D-³He plasmas on JET

In a thermonuclear fusion reactor, the reaction D(T,n)⁴He between deuterium (D) and tritium (T) will be the main source of energy. ⁴He-ions (alpha-particles), which are born with an energy of 3.5 MeV, must be well confined to transfer their energy to the plasma particles during their collisional slowing-down and thus to provide the power for a self-sustained D-T burning plasma. It is equally important that additional effects associated with the presence of the alpha-particles should not lead to a detrimental degradation of the plasma confinement. Among these, the possible excitation of magnetohydrodynamic (MHD) modes by fusion-born alpha-particles is of particular importance. Indeed, Alfvén instabilities can significantly influence the fast-ion transport and plasma thermal confinement. This topic is thus of considerable interest for high-Q operation in ITER (e.g. [1] and references therein). Therefore, dedicated alpha-particle studies were undertaken in past D-T experiments on the Tokamak Fusion Test Reactor and the Joint European Torus (JET) [2, 3]. Further alpha-particle physics studies are planned in the forthcoming D-T campaign in JET with the ITER-like wall (JET-ILW) [4]. An important effort of physics understanding is being carried out by performing analyses and modelling of JET experiments with D-plasmas towards an improvement of the understanding of the D-T plasma and fusion-born alpha particle physics [5]. In D-T plasmas, Alfvén eigenmodes (AEs) can be driven by the fusion-born alpha-particles, inducing a significant redistribution and loss of fast-ions. Recent experiments have been conducted in JET D-plasmas in order to prepare scenarios aimed at observing alpha-driven toroidal Alfvén eigenmodes (TAEs) in JET D-T experiments [6].

However, using the newly developed three-ion scenario, we are able to study effects of fusion-born alpha-particles in plasmas without tritium well before the sophisticated full-scale D-T experiments characterised by harsh radiation conditions for diagnostics, using the aneutronic D-³He fusion reaction

$$\begin{align}{\text{D }}{ + ^{\text{3}}}{\text{He }}{ \to ^{\text{4}}}{\text{He}}\left( {3.6{\text{ MeV}}} \right) + { }p\left( {14.7{\text{ MeV}}} \right)\end{align}$$

which generates alpha-particles with an energy very close to those produced in the D-T reaction, albeit with a lower reaction rate. The D-³He plasma is still a source of 2.5 MeV neutrons due to the D-D reaction. For deuteron energies up to ∼4 MeV, both T(D,n)⁴He and ³He(D,p)⁴He reactions have non-monotonic cross-sections $\left( \sigma \right)$ with a maximum for deuteron energies ${E_{\text{D}}}$ ≈ 0.11 MeV and ≈0.44 MeV, whereas the D(D,n)³He reaction has a monotonic cross-section [7]. In order to maximise the fusion-born alpha-particle source rate relative to the neutron rate from the D-D fusion, deuteron energies should be in the range ${E_{\text{D}}}$ ≈ 0.15–1 MeV, where ${\sigma _{{\text{D3He}}}} \gtrsim 3{\sigma _{{\text{DD}}n}}$.

We carried out fast-ion studies in mixed D-³He plasmas using the three-ion D-(D_NBI)-³He radio frequency (RF) scenario [8–10] to accelerate deuterons from the neutral beam injection (NBI) energy (≈100 keV) to higher energies resulting in a highly efficient generation of fusion alpha-particles from D-³He reactions [11]. The experiments were undertaken at a toroidal magnetic field on axis B₀ = 3.7 T, plasma current I_p = 2.5 MA, central electron densities n_e0 ≈ 5–7 × 10¹⁹ m⁻³, at an ICRH frequency f = 32.2–33.0 MHz using dipole antenna phasing. The cyclotron resonances for thermal D- and ³He-ions are located off-axis at the high- and low-magnetic field sides, respectively. Rather large ³He-concentrations, ${n_{{\text{3He}}}}/{n_{\text{e}}}$ ≈ 20%–25%, were purposely chosen to position the ion-ion hybrid (IIH) layer in the plasma core, such that energetic D-ions are generated in the plasma centre. Figure 1(a) shows the RF power deposition computed with the TORIC code [12], which accounts for the poloidal magnetic field and its impact on the propagation and absorption of RF waves in the plasma, illustrating that most of RF power is indeed absorbed in a small region in the plasma core by fast D-NBI ions. Figure 1(b) shows several waveforms of JET pulse #95679 (P_NBI ≈ 6.9 MW, P_RF ≈ 5.8 MW), in which the central electron temperature increased from 3.6 keV during the NBI-only phase to 7.6 keV in the combined ICRH + NBI phase. The efficient generation of high-energy deuterons with the three-ion scenario resulted in a strongly enhanced neutron rate, increasing from ∼5 × 10¹⁴ s⁻¹ to ∼1.0 × 10¹⁶ s⁻¹. The D-D neutron and D-³He alpha-particle rates shown in figure 1(b) were simulated by TRANSP [13]. The bottom panel in figure 1(b) shows the time evolution of the ³He concentration, X[³He] = ${n_{{\text{3He}}}}/{n_{\text{e}}}$ ≈ 22%, controlled by the real-time feedback system in JET.

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The energy distribution of the RF-accelerated D-NBI ions in these experiments was controlled by varying the ratio P_RF/P_NBI. One can see that with P_RF ≈ 6 MW and P_NBI ≈ 7–11 MW, the TRANSP-simulated D-³He alpha-particle rate R_α ∼ 1.7 × 10¹⁶ s⁻¹ was achieved, corresponding to a fusion power P_D3He ≈ 50 kW.

The central electron temperature T_e0 shows recurrent changes known as sawtooth oscillations. It is well-known that the presence of fast-ions in the plasma has a stabilising effect on sawtooth oscillations [14]. However, after long sawteeth a steep and fast drop in T_e0 occurs—a so-called sawtooth crash—when a certain stability threshold is crossed [15]. This then leads to a redistribution of the fast-ions in the plasma. Depending on P_ICRF and P_NBI and other operational parameters, the sawtooth period $\Delta {t_{{\text{ST}}}}$ in our experiments varied from ∼150–300 ms to ∼3.9 s, the so-called 'monster' sawtooth.

A large variety of fast-ion driven MHD modes such as TAEs, elliptical Alfvén eigenmodes (EAEs) and reversed shear Alfvén eigenmodes [16] with different toroidal mode numbers n, including axisymmetric n = 0 AEs, were observed in these JET-ILW experiments. The excitation of most of these AEs can be explained by resonant wave–particle interactions with a population of ICRF-generated fast D-ions. However, the simultaneous observation of n = ‒1 and n = 0 modes in the EAE frequency range implies the presence of a fast-ion population in the plasma with a large fraction of highly energetic counter-passing ions (parallel velocity ${v_\parallel }$ < 0), together with a positive gradient in the fast-particle energy distribution function $\partial f/\partial E$ > 0. A combined fast-ion and MHD analysis shows that the ICRF-generated fast D-ions are not capable to excite n = ‒1 EAEs. Instead, we show that these AEs with toroidal numbers n = 0 and n = −1 are driven by fusion-born alpha-particles in the JET plasma [17].

The paper is organised as follows. Diagnostics used for fast-ion measurements are described in the next section. The experimental evidence for the observed AEs instabilities driven by fusion-born alpha-particles are presented in section 3. Results of the linear stability analysis and Fokker-Planck calculations are summarised in sections 4 and 5. In the summary and conclusions section, we discuss an extension of our results to D-T plasmas.

The fast-ion diagnostics set on JET allows us to study confined and escaping fast-ions, including fusion-born alpha-particles. Confined fast-ions are diagnosed with gamma-ray spectrometry [18, 19]. This diagnostic detects gamma-rays that are born in nuclear reactions between confined fast-ions and fuel ions or beryllium, which is the main intrinsic low-Z impurity in JET-ILW plasmas. The gamma-ray diagnostic provides information on the spatial distribution of fast-ions and the MeV range fast-ion energy tail. The gamma-ray energy spectra were recorded by a collimated LaBr₃ scintillation detector with a tangential LoS in the midplane of the torus [20].

For the fast D-ion energy assessments we used a high-resolution gamma-ray spectrometer based on the high-purity germanium detector (HpGe). This spectrometer views the plasma along a vertical LoS, through the plasma centre (R = 2.96 m) perpendicular to the toroidal field. It has a relative efficiency of ≈95% and an energy resolution of ≈0.2%, ($\Delta {E_{{\text{FWHM}}}}$ ≈ 2.6 keV at ${E_\gamma }$ = 1332.5 keV). This high resolution allows measurements of the Doppler gamma-ray line broadening due to kinematics of nuclear reactions between fast-ions and low-Z impurities and thus to infer an effective fast-ion energy [21, 22] and even a measurement of the two-dimensional (2D) fast-ion velocity distribution function by integrated data analysis [23].

Losses were observed with the fast-ion loss detector (FILD) located below the mid-plane (Z ≈ −0.28 m) [24]. The light from the FILD scintillator plate (the characteristic scintillator decay-time of ≈0.5 μs), transmitted with a coherent optical fibre-bundle is equally split between a charge-coupled device (CCD camera) and a 4 × 4 array of photo-multiplier tubes (PMTs). The PMT array delivers fast signals digitized with 2 MHz rate. The CCD camera provides time-resolved information (Δt ≈ 20 ms) on the lost-ion pitch-angle, $\theta = {\text{ }}{\cos ^{ - 1}}({v_\parallel }/v)$, between 35° and 85° with a resolution ∼5% and gyro-radius, ρ_gyr, in the range 3–14 cm with a resolution ∼15%. The major radius at the ion bounce reflection and the pitch-angle θ of the grid are related by $R\left( \theta \right) = { }{R_{{\text{FILD }}}}{\text{si}}{{\text{n}}^2}\theta$, where R_FILD = 3.825 m is the radial position of the FILD scintillator in the vessel. The energy of ions $\left( {{Z_{\text{i}}},{ }{A_{\text{i}}}} \right)$ is related to the gyro-radius as ${E_{\text{i}}} = {\left[ {{\rho _{{\text{gyr}}}}\left( {{\text{cm}}} \right){B_{{\text{FILD}}}}\left( {\text{T}} \right)/14.45} \right]^2}(Z_{\text{i}}^{\text{2}}/{A_{\text{i}}})$, where ${B_{{\text{FILD}}}}$ is magnetic field at the location of the FILD scintillator plate, which in the presented case, 2.874 T. The grid, (ρ_gyr, θ), used to interpret measurements from the scintillator plate is calculated from the EFIT equilibrium.

The D-D fusion neutrons are measured with the TOFOR spectrometer [25] and the 2D neutron camera (a neutron profile monitor) [26]. TOFOR is a time-of-flight spectrometer for measurements of the neutron emission of the deuterium plasma with different auxiliary heating scenarios, which views the plasma centre (≈2.74–3.02 m) vertically. The 2D neutron camera consists of two collimated fan-shaped arrays of NE213 detectors with ten horizontal and nine vertical lines of sight.

Gamma-ray measurements provide strong evidence for fusion alpha-particle production in these experiments. Figure 2(a) depicts the gamma-ray spectrum recorded by the tangential spectrometer in discharge #95679. It was identified that events recorded in the energy range 12–17 MeV originate from the ³He(D,γ)⁵Li reaction, which is a weak proportional branch (∼10⁻⁵) of the ³He(D,p)⁴He fusion reaction. Also, the ⁹Be(D,nγ)¹⁰B and ⁹Be(D,pγ)¹⁰Be reactions give rise to gammas, which are clearly seen in the rescaled spectrum of figure 2(b). Hence, we can conclude that there is a D-ion population with energies in the range ${E_{\text{D}}} \gtrsim \,$0.5 MeV, where the reaction cross-sections are soaring up [7].

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The high-resolution gamma-ray HpGe-spectrometer provides further information on the effective energy of the accelerated D-ions in the plasma centre, which is in the field of view of this collimated detector. The HpGe gamma-ray spectrum recorded for t = 8.5–11.5 s in this discharge is presented in figure 3(a). The gamma-ray peaks related to the ⁹Be(D,nγ)¹⁰B and ⁹Be(D,pγ)¹⁰Be reactions are broadened due to the Doppler effect. In addition, a broadened gamma-ray peak at ${E_\gamma }$ = 4439 keV, which is related to the ⁹Be(α,nγ)¹²C reaction, indicates the presence of confined fusion-born alpha-particles [18] in the plasma centre. Note that the line at ${E_\gamma }$ = 2223 keV is rather narrow; this gamma-ray line does not originate from nuclear reactions in the plasma but is related to background gamma-rays from the polyethylene neutron-attenuator bricks placed in front of the detector.

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The energy of gamma-rays emitted by the recoil compound nucleus in a nuclear reaction can be expressed as ${E_\gamma } \approx E_\gamma ^{\,0}\left[ {1 + \left( {{V_{\text{R}}}/c} \right)\cos {\theta _\gamma }} \right]$, where $E_\gamma ^{\,0}$ is energy of the gamma-ray transition in the recoil nucleus, ${V_{\text{R}}}/c \approx 0.046\sqrt {{m_{\text{f}}}{E_{\text{f}}}\left( {{\text{MeV}}} \right)} /{M_{\text{R}}}$ is the ratio of the recoil velocity of the excited compound recoil nucleus with mass number ${M_{\text{R}}}$ to the speed of light $c$, ${\theta _\gamma }$ is the angle between the velocity of fast-ion with mass number ${m_{\text{f}}}$ and the axis of the collimated detector. Figure 3(b) shows a fragment of the experimental spectrum with the 2872 keV line and the best Gaussian fit to the data. Note that this line appears in the spectrum due to deexcitation of the ¹⁰B* nucleus excited in the ⁹Be(D,nγ)¹⁰B* reaction. Taking into account the energy resolution of the spectrometer of ≈5 keV, the obtained Gaussian broadening width (2σ) of ≈29 ±3 keV, results in $\langle {E_{\text{D}}}\rangle \, \gtrsim$ 0.7 MeV for the energy of the fast D-ions averaged over the sawtoothing period. This represents a lower limit for $\langle {E_{\text{D}}}\rangle$ considering that the fast D-ions are trapped, i.e. their pitch ${v_\parallel }/v$ = 0. Note that this is a first order assessment of the average fast D-ion energies and a much more detailed Doppler shape analysis will be published in the future.

Discharge #95679 was simulated by TRANSP [27] coupled with the external heating modules NUBEAM [28] and TORIC and prepared with the OMFIT integrated modelling platform [29]. As input to TRANSP we used electron density and temperature profiles obtained from high resolution Thomson scattering and electron cyclotron emission data. Supported by charge exchange measurements we set the ion temperature profile equal to the electron temperature profile. The plasma equilibrium and safety factor used in the simulations was provided by an EFIT++ equilibrium reconstruction, constrained by magnetics and realistic pressure profiles, i.e. including kinetic profiles as well as the contribution from the RF accelerated fast-ion population. The same equilibrium was used for mapping of the profile data. The concentration of ³He in the plasma, prescribed as a fraction of the electron density, was based on real-time measurements of the JET tokamak control system. The impurity content was assumed to be 0.9% Be with the remainder ascribed to Ni based on a quasi-neutrality calculation constrained by the measured effective-charge, Z_eff, from visible spectroscopy.

The TRANSP simulations show that most of the fast deuterons in our experiments have a pitch-angle θ ≈ 50°–60°, or pitch parameter $\lambda = {v_\parallel }/v \approx$ 0.5–0.65, in the energy range ${E_{\text{D}}}\, \approx \,$ 0.5–1.25 MeV. These results are consistent with the Doppler broadening measurements, i.e. for the fast D-ion pitch angles predicted by TRANSP the energy $\langle {E_{\text{D}}}\rangle \, \approx$ 0.9–1.1 MeV. The radial profile of the alpha-particle density after slowing-down computed with TRANSP confirms the strong core localisation of the fusion-born alpha-particles. This simulation and the Doppler broadening measurements agree rather well with neutron spectrometry results, i.e. the TOFOR spectra analysis (figure 4(c)) indicates that the energy of the fast D-ions in the plasma centre below 2.7 MeV in this discharge.

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The velocity-space sensitivity of fusion product measurements can be assessed with weight functions [30, 31]. For TOFOR the observed velocity space for the time-of-flight t_ToF = 46 ns is shown in figure 4(b), where all detected neutrons for the given time-of-flight originate from the coloured region. We did not consider neutron detections with times-of-flight less of 42 ns, the range where a random coincidence background is dominant. For the TOFOR setup energy of detected neutrons related to ${t_{{\text{ToF}}}}$ as ${E_{\text{n}}}\left( {{\text{MeV}}} \right) \approx {\left[ {100/{t_{{\text{ToF}}}}\left( {{\text{ns}}} \right)} \right]^2}$. Energetic deuterons resulting in a detection of a neutron with a time-of-flight of 46 ns have energies of at least 1.5 MeV. The energy sensitivity of the different times-of-flight can be estimated by integration of the weight functions over pitch. Here, we integrated over pitches parameter $\lambda = {v_\parallel }/v$ from −0.55 to +0.55, where sensitivity of the TOFOR weight functions is rather high. Figure 4(c) shows a comparison of the energy sensitivity of the different times-of-flight. One can see that the maximum deuteron energies is situated in the range 1.5–2.7 MeV.

Gamma-ray emission from the ⁹Be(D,nγ)¹⁰B and ⁹Be(D,pγ)¹⁰Be reactions recorded with tangential spectrometer indicates significant changes in the energy distribution of the fast D-ions that could be attributed to sawtooth crashes in the reported discharges. A solid variation of the gamma-ray line intensities in discharge #95679 can be explained by the fast-growing cross-section of the ⁹Be(D,nγ)¹⁰B reaction in the energy range ${E_{\text{D}}}$ ∼ 0.2–2 MeV.

This considerable energy redistribution of the fast D-ion population due to sawtooth crashes lead to changes of the alpha-particle source because the ³He(D,p)⁴He reaction cross-section (see figure 5) is highly non-monotonic around ${E_D} \approx$ 0.4 MeV. Indeed, the intensity of the 17 MeV gamma-rays from the ³He(D,γ)⁵Li reaction shows clear variations that are correlated with the sawtooth crashes, from which we infer a modulation in the production rate of the alpha-particles. Figure 6 demonstrates relative changes of intensity of the 17 MeV gammas as function of time in discharge #95679. These variations are correlated with changes in the integral neutron emissivities recorded by both the central and outer channels of the neutron camera. Indeed, for the first four sawtooth crashes in the figure, the D-³He fusion rate increases quite strongly (indicated by the black full lines), while at the same time the relative D-D fusion rate slightly drops. For the following three sawteeth, the D-D reaction rate is growing with increase of the sawtooth period, $\Delta {t_{{\text{ST}}}}$, but the ³He(D,γ)⁵Li reaction rate, i.e. the D-³He fusion rate, drops after crashes. This effect could be explained by both changes in the energy distribution of the fast D-ions and the specific energy dependence of the reaction cross-sections shown in figure 5. During the long sawteeth with $\Delta {t_{{\text{ST}}}}$ ≈ 0.46 and ≈0.73 s, D-ions are accelerated by RF to higher energies than during sawteeth with shorter periods $\Delta {t_{{\text{ST}}}}$ ≈ 0.20–0.25 s. For energies ${E_D} \gtrsim$ 0.5 MeV the D(D,n)³He cross-section is monotonically increasing with energy, while the cross-section for the ³He(D,p)⁴He reaction (and similar for ³He(D,γ)⁵Li reaction) is decreasing. The ³He(D,p)⁴He cross-section significantly exceeds that of the D(D,n)³He reaction in the energy range 0.2 $\lesssim \,{E_{\text{D}}}\, \lesssim$ 1 MeV. The ratio of the cross-sections of the ³He(D,p)⁴He and D(D,n)³He reactions changes dramatically with $\langle {E_{\text{D}}}\rangle$, as clearly shown by the ratios (0.5 MeV):(1 MeV):(1.9 MeV):(2.5 MeV) ≈ 11:3:1:0.7. Hence, the 17 MeV gamma-ray modulation (and the alpha-particle source rate) depends on the variations in the energies of the fast D-ions modulated by the sawteeth. Depending on $\Delta {t_{{\text{ST}}}}$ and the slowing down time, ${\tau _{\text{D}}}$ of the fast D-ions, the sawtooth crashes lead to different modulation amplitudes of the ³He(D,γ)⁵Li (and ³He(D,p)⁴He) reaction rate.

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Together with variations in the energy of the fast D-ions, the sawtooth crashes also lead to a significant spatial redistribution of fast-ions, resulting in an expulsion of the fast D-ions from the plasma centre to the outer region. This explains the drops in the neutron rate in the central channels of the 2D camera and the increase in the outer channels in figure 6.

The tomographic reconstruction of the line-integrated emissivity signals from the 2D neutron camera clearly demonstrates that the strongly localized D-D fusion source before the sawtooth crash (figure 7(a)) is broadened after the crash (figure 7(b)). Thus, the post-crash central neutron emission is significantly reduced in the plasma centre.

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Analysis of the neutron camera profiles indicates the differences in the spatial redistribution of the fast D-ions caused by the crashes after short and long sawteeth. Indeed, the broadening of the neutron emissivity profiles is much less for crashes after short sawteeth (figure 8(a)) than after long sawteeth (figure 8(b)). We conclude that the impact of the spatial redistribution of the fast D-ions on the measured ³He(D,γ)⁵Li reaction rate, proportional to the alpha-particle generation rate, is quite significant. Also, fast D-ions can appear in the loss cone region due the spatial redistribution caused by sawtooth crashes.

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Massive sawtooth induced losses of the RF-accelerated D-ions and D-³He alpha-particles have been observed in these experiments. Figure 9(a) represents a typical FILD footprint of the first-orbit losses of fusion-born alpha-particles in discharge #94698 just before the monster sawtooth crash at 10.53 s. The gyro-radius of the lost ions exceeds ≈8 cm, corresponding to energies for the lost alpha-particles ${E_\alpha }\, \gtrsim$ 2.5 MeV. This agrees with the alpha-particle birth energy spectra shown in figure 10(b) calculated with the Monte-Carlo code GENESIS [32], which used as input the distribution function for the fast D-ions calculated by TRANSP (see figure 10(a)) and extended D-³He fusion cross-sections [33, 34]. These calculations reveal that the RF-accelerated D-beam ion distribution function responsible for the broad alpha-particle birth spectrum, with energies inbetween 2 and 6 MeV, depending on the value for the pitch parameter λ. This justifies attributing the measured fast ion losses with large gyro-radii to fusion-born alpha-particles. So, the energy distribution of alpha-particle losses with pitch-angle $\theta \, \approx$ 57.5° presented in figure 9(b) corresponds closely to the calculated alpha-particle energy spectra at $\lambda \, =$ +0.5 (see figure 10(b)).

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Figure 9(c) depicts the loss footprint during the monster sawtooth crash. The FILD loss signal is very strong and distributed along the pitch-angle $\theta \, \approx$ 52.5° that is related to the trapped-passing boundary in the orbit phase-space. This suggests indeed that the substantial spatial redistribution of the fast D-ions can explain the losses observed. Using the FILD data we obtained the energy distribution of the lost D-ions shown in figure 9(d). The energy range, 0.7–2.4 MeV, of the lost D-ions is consistent with the gamma-ray and neutron measurements as well as the TRANSP simulations (see figure 10(a)). Orbits of lost fast D-ions and alpha-particles, calculated backward-in-time from hot-spots on the FILD scintillation plate, are represented in figure 11. The figure shows that lost alpha-particles originate from the plasma centre, whereas the lost D-ion orbits intersect the sawtooth inversion radius q = 1 at the radius $R\, \approx$ 3.25 ± 0.05 m (obtained with ECE diagnostics).

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We observed a rich variety of fast-ion driven AEs, including TAEs, EAEs as well as Alfvén cascades. Like past fast-ion experiments with ICRF on JT-60U [35], the behaviour of TAEs and EAEs was correlated with sawtooth dynamics. In our experiments EAEs were mostly observed during phases with short sawteeth. In each of these discharges (see main plasma parameters in the table 1), a monster sawtooth crash facilitated the excitation of EAEs; under these conditions, the delay between the start of EAE activities after the monster sawtooth crash was typically ∼80–100 ms (see figure 12).

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The observed EAEs are located at the q = 1 surface, as is inferred from correlation reflectometer measurements. They consist of two counter-propagating poloidal modes with mode numbers m₁ = n − 1 and m₂ = n + 1.

The destabilization of AEs is only possible when the mode drive by fast ions is sufficiently large and overcomes mode damping. Alfvénic instabilities are usually driven unstable by the free energy in the spatial gradient of the distribution function of energetic ions. The peaked pressure profile of fast D ions provides a positive term for the destabilization of modes with n > 0. Yet, the same mechanism provides damping, rather than a drive, for modes with n < 0. Thus, other drive mechanisms are required for the destabilization of the observed n = −1 mode. Possible mechanisms include the anisotropy of the distribution function with respect to the pitch angle and the bump-on-tail mechanism, as discussed further.

In the JT-60U experiments mentioned above, only EAEs with positive toroidal mode numbers n = 3–6 were observed [35]. However, in the JET D-³He experiments described here, EAEs with lower positive n, as well as n = 0 modes and even EAEs with negative toroidal mode numbers were detected. In what follows, we focus our analysis on the n = ‒1 EAEs and axisymmetric n = 0 AEs [36] as shown in the magnetic spectrograms of figure 12. The n = 0 AE can only be excited if a fast-ion population is present in the plasma with an energy distribution where $\partial f/\partial E$ > 0 (so-called 'bump-on-tail distribution) at rather high energies for ${v_\parallel } = {v_{\text{A}}}$ [17]. The simultaneous observation of these modes requires a large number of highly energetic counter-passing ions with such a 'bump-on-tail' distribution [37].

Note that fundamental ICRF minority heating leads to a Maxwellian energy distribution function $f\left( E \right) \propto {\text{ exp}}\left( { - E/T} \right)$ with an energetic tail characterised by $\partial f/\partial E < 0$. The three-ion D-NBI RF heating scenario used in the experiments discussed here creates a population of accelerated D-ions with ${v_\parallel }/v > 0$ in the range ${E_{\text{D}}}\, >$ 0.8 MeV, which is different from a Maxwellian distribution, but it is also characterised by $\partial {f_{\text{D}}} / \partial E < 0$.

The birth energy distribution of fusion products, i.e. D-³He alpha-particles, can be approximated by a Gaussian distribution function [38] with a low energy wing characterized by $\partial {f_\alpha } / \partial E$ > 0. The alpha-particle birth distribution function, ${f_\alpha }\left( {E,{\text{ }}\lambda } \right)$ also depends on the pitch $\lambda = {v_\parallel }/v$ as already discussed above and illustrated in figure 10. The relaxation of the birth alpha-particle distribution to the steady-state distribution, which does not have a 'bump' in energy space, i.e. $\partial {f_\alpha }/\partial E$ < 0, occurs on the time scale of the characteristic slowing-down time, $\tau _\alpha ^* = \left( {{\tau _{{\text{se}}}}/3} \right)\ln \left( {1 + {{\left( {{E_0}/{E_{\text{c}}}} \right)}^{3/2}}} \right)$, where ${\tau _{{\text{se}}}}$ is the Spitzer slowing-down time, ${E_0}{\text{ }}$ ≈ 3.6 MeV and ${E_{\text{c}}} \approx 41{ }{T_{\text{e}}}$ is the initial and the critical energy for the alpha-particles. Thus, the ratio $\Delta {t_{{\text{ST}}}}/\tau _\alpha ^*$ could be used as a parameter to characterise the time evolution of the distribution function of the fusion-born alphas [39]. In our experiments in D-³He plasmas, it varies between $\Delta {t_{{\text{ST}}}}/\tau _\alpha ^*$ < 1 and $\Delta {t_{{\text{ST}}}}/\tau _\alpha ^*$ ≫ 1 at $\tau _\alpha ^*$ ≈ 400 ms.

One of the most significant observations in our experiments is a periodic modulation of the D-³He fusion source during short-period sawtooth phases (see figure 6), when the redistribution and the losses of the fast D-ions prevent the formation of a steady-state slowing down distribution for the fusion-born alpha-particles. One can compare this to a series of alpha-particle birth blips that lead to an energy distribution with $\partial {f_\alpha }/\partial E$ > 0, i.e. the 'bump-on-tail' energy distribution. Also, the observation of strong alpha-particle losses is crucial for the understanding of the destabilisation of the axisymmetric $n = 0$ AEs. Indeed, figure 13 shows the waveform of the FILD photomultiplier, which represents first orbit losses of alpha-particles with energy E_α > 4.5 MeV. The spike in the alpha-particle losses followed by the post-sawtooth decrease in the losses indicates a relaxation of the alpha-particle energy distribution similar to that before the sawtooth crash. This sawtooth crash strongly disturbs the alpha-particle distribution function, making a sort of notch in the established monotonic distribution function and building up the non-monotonic phase with $\partial {f_\alpha }/\partial E$ > 0. A detailed analysis of the condition $\partial {f_\alpha }/\partial E$ > 0 for ${v_{\parallel ,\alpha }} = {v_{\text{A}}}$, which is required to excite the axisymmetric n = 0 modes, is discussed in section 5.

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AEs are destabilized by energetic ions through the energy transfer between resonant particles and AEs [16]. The necessary condition for a resonant wave-particle interaction is given by $\omega = n{\omega _\varphi } - p{\omega _\theta }$, where $\omega$ and $n$ are the AE frequency and the toroidal mode number; ${\omega _\varphi }$ and ${\omega _\theta }$ are the toroidal and poloidal orbital frequencies; ${\text{ }}p$ is an integer. The destabilization of AEs is only possible when the mode drive by the fast-ions is sufficiently large and overcomes mode damping by other mechanisms.

Figure 14(a) shows the computed efficiency of the resonant wave-particle interaction for the observed n = ‒1 EAE at f ≈ 555 kHz in discharge #95679 (see figure 12(b)). The efficiency is obtained by calculating the variance of test-particle energy due to work done by the eigenmode electric field over many wave periods. The computations have been carried out using the HAGIS [40] and MISHKA [41] codes for a typical fixed mode amplitude δB/B = 1 × 10⁻⁵ supposing that the n = ‒1 EAE is interacting with energetic D-ions.

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From the plotted efficiency for the wave-particle interaction in figure 14(a), the drive of the mode with a negative toroidal mode number would require the presence of a large population of energetic counter-moving D-ions (ν_|| < 0) in the plasma. The energy range for fast D-ions has been limited to ≈2.5 MeV, reflecting the experimental observations from the fast-ion diagnostics.

The two most prominent power transfer lines in figure 14(a) have been labelled to show the corresponding wave-particle resonances identified for different p-numbers in the equation, confirmed by computing the toroidal and bounce frequencies of the test-particles. The computations show that the observed n = ‒1 EAE receives energy most effectively from fast counter-passing ions via the $p$ = 0 $\left( {\omega = - {\omega _\varphi }} \right)$ and the $p$ = +1 $\left( {\omega = - {\omega _\varphi } - {\omega _\theta }} \right)$ resonances. As the toroidal and poloidal orbital frequencies for passing fast-ions are given by ${\omega _\varphi } = {v_{\parallel {\text{ }}}}/R$ and ${\omega _\theta } = {v_{\parallel {\text{ }}}}/qR$ [16], n = ‒1 modes interact effectively with these fast-ions at parallel velocities ${v_\parallel } = - \omega R$ and ${v_\parallel } = - \omega R/\left( {1 + 1/q} \right) \approx - \omega R/2$.

For f_EAE = 555 kHz as in discharge #95679 (figure 12(b)), one gets ν_|| ≈ ‒1.1 × 10⁷ m s⁻¹ and ν_|| ≈ ‒5.4 × 10⁶ m s⁻¹, respectively. For fast D-ions, these velocities correspond to energies of ≈1.2 MeV and ≈0.3 MeV. We should note that weak higher-order wave-particle resonances for p = +2, +3, ... are also visible in figure 14(a), but only at negative pitches and at high D-ion energies. An important remark is that while the p = +2 resonance can be formally identified for particles with a positive pitch, the energy transfer between these D-ions with ν_|| > 0 and the n = ‒1 mode is several orders of magnitude less efficient and thus not visible in figure 14(a).

It is important to note that the three-ion D-(D_NBI)-³He ICRF scenario, which is used to accelerate fast D-NBI ions in the vicinity of the IIH layer in the mixed D-³He plasmas, is strongly selective on the sign of the parallel velocity for the resonant D-NBI ions. This is supported by TRANSP modelling confirming the preferential acceleration of fast D-ions with ${v_\parallel }/v > 0{\text{ }}$ and clearly showing the absence of counter-passing energetic D-ions with ${E_{\text{D}}}\left( {{\text{MeV}}} \right) \gtrsim 0.8$ and ${v_\parallel }/v < 0$ required to excite n = −1 EAEs under these conditions. In contrast, the birth velocity distribution of D-³He fusion-born alpha-particles shown in figure 10, has a rather large broadening in energy space $2 \lesssim {E_\alpha }\left( {{\text{MeV}}} \right) \lesssim 7$ with a pitch parameter ${v_\parallel }/v$ covering the full range from −1 to +1. Thus, fusion-born counter-passing alphas with ${v_\parallel }/v \cong - 1$, rather than ICRF-accelerated fast D-ions, are a natural choice for the fast-ion species that drive the observed EAEs with n = ‒1.

As follows from the analysis presented above, the energy transfer between the counter-passing fast-ions with ν_|| ≈ ‒1.1 × 10⁷ m s⁻¹ and ν_|| ≈ ‒5.4 × 10⁶ m s⁻¹ and n = ‒1 EAEs is the most efficient. The map showing the efficiency of the resonant wave-particle interaction for alpha-particles as a function of their energy and pitch, is shown in figure 14(b). Note that it is similar to the one for fast D-ions (figure 14(a)), but the corresponding wave-particle resonances are shifted towards higher fast-ion energies as a result of the higher mass of the alpha-particles. As follows from the figure, counter-passing alphas with energies E_α ≈ 2 MeV provide the strongest drive for the n = ‒1 EAE mode. These particles are naturally present in the plasma, originating from the D-³He fusion reactions. As a side remark, note that in these fast-ion experiments we obtained alpha-particle production rates comparable to or even higher than the D-D neutron rates. For example, a neutron rate of ∼1 × 10¹⁶ s⁻¹ was achieved in JET pulse #95679, while the D-³He fusion rate (and thus alpha production rate) was estimated to be ∼1.5 × 10¹⁶ s⁻¹ (see figure 1(b)).

Both the sawtooth redistribution of fast D-ions and the massive prompt losses of the fusion-born alpha-particles after a sawtooth crash, lead to a periodic modulation of the fusion alpha-particle source, in turn leading to a 'bump-on-tail' energy distribution function with $\partial {f_\alpha }/\partial E$ > 0.

Considering the slowing-down of fusion-born alphas in the MeV-energy range ${ }\left( {2 \lesssim {E_\alpha }\left( {{\text{MeV}}} \right) \lesssim 7} \right)$, the electron friction is the dominant term in the Fokker–Planck equation, describing the relaxation of the alpha-particle energy distribution. Since the alpha-particle energies are significantly larger than the critical energy (${E_{\text{c}}} \simeq { }$ 0.3 MeV), the fast-ion drag, the pitch-angle scattering and energy diffusion are negligible. At these high energies, the fusion-born alphas with different pitch values are gradually decelerated in collisions with electrons without changing their ${v_\parallel }/v$, and this relaxation can be described by the equation $\partial {f_\alpha }/\partial t = {\tau _\alpha }^{ - 1}{v^{ - 2}}\partial \left[ {{v^3}{f_\alpha }} \right]/\partial v + {S_\alpha }\left( {v,t} \right)$, where ${\tau _\alpha }$ is the Spitzer slowing-down time; ${S_\alpha }$ is the source term for alphas. Using the alpha-particle birth spectrum shown in figure 10 as an initial condition for the above equation, the deceleration process of alphas due to the electron drag can be easily solved numerically.

Using this simplified model, we calculated the temporal evolution of the alpha-particle distribution function, ${f_\alpha }\left( E \right)$, just after sawtooth crashes in discharges #95675 and #95679 (see figure 12). To create a modulation of the alpha-particle source due to the sawtooth crashes, both blips and notches with a duration of 50 ms were triggered exactly at the time of the sawtooth crash, as soon as the steady-state distribution function with $\partial {f_\alpha }/\partial E$ < 0 was established. The counter-passing alpha-particle source spectrum with ${v_\parallel }/v$ = −1 was used (see figure 10). For clarity, we assumed that both the blip amplitude and depth of the notch wells are equal to ½ of the source rate. Figure 15 shows ${f_\alpha }\left( E \right){\text{ }}$ distributions for the case of discharge #95675 computed at $t\, =$ 0.25, 0.4 and 1 s after the blip/notch. One can see that in both cases the perturbed alpha-particle distribution functions develop a characteristic 'bump-on-tail' shape with $\partial {f_\alpha }/\partial E$ > 0.

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It is interesting to note that a notch produces a 'bump-on-tail' shape in the slowing-down distribution function with a delay compared to a blip. This is shown in figure 16, where the temporal evolution of the derivative $\partial {f_\alpha }/\partial E$ is given for 'bump-on-tail' distribution functions resulting from blips and notches distribution functions. Note that the maxima for the positive derivatives are linked to counter-passing alpha-particles decelerated to resonance energies ${E_\alpha } = {E_{\text{A}}}$ ≈ 1.3 MeV for ${v_{\parallel ,\alpha }} = {v_{\text{A}}}$. This modelling can explain the systematic periodic observations of axisymmetric n = 0 AEs in the experiments in D-³He plasmas discussed here.

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To understand the dynamics of the n = 0 AEs, figure 17 shows the temporal dependency of the calculated blip/notch derivatives ($\partial {f_\alpha }/\partial E$ > 0) together with the waveforms of the central temperature ${T_{e0}}$, the FILD-PMT currents (the same as in figure 13(b)) and the AE amplitudes during sawteeth for the discharges #95675 and #95679. The mode amplitudes were obtained calculating a root-mean-square deviation in the time-dependent window (≈3 ms) for smoothing the mode amplitude noise. To separate the modes from other fluctuations, a band-pass 500–540 kHz filter (2nd order Butterworth filter) was used for discharge #95675 and 565–600 kHz for discharge #95679.

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From figure 17 it follows that the positive part of the blip derivatives $\partial {f_\alpha }/\partial E$ coincides with the onset of the n = 0 modes mimicking the delay between the sawtooth crash and the AE excitation observed in the experiments. In addition, the positive part of the notch derivatives coincides with the n = 0 modes providing additional energy to stabilise the mode. In fact, this is a synergetic effect between the substantial redistribution of the fast D-ions and prompt losses of fusion-born alpha-particles due to sawtooth crashes with short periods $\Delta {t_{{\text{ST}}}}/\tau _\alpha ^*$ < 1. These results, obtained with a simplified model of the alpha-particle source modulation, are fully consistent with our observations. We conclude that the conditions in our experiments create indeed a self-sustained 'bump-on-tail' fusion-born alpha-particle energy distribution.

In this paper we show that fusion-born alpha-particles from the D-³He reaction are able to excite AEs with toroidal mode numbers n = ‒1 and n = 0. The D-³He reaction, with a birth energy and kinematics for alpha-particles very similar to those released in D-T reactions but with relatively low neutron production, allows the use of fast-ion diagnostics that are incompatible with the high neutron flux environment of D-T experiments.

In our experiments we demonstrate that only the fusion-born α-particles can effectively excite n = ‒1 EAEs, confirmed by a linear stability analysis.

We propose a theoretical mechanism and provide experimental evidence for the formation of a bump-on-tail distribution for the fusion alphas required to excite n = 0 AEs, which is self-sustained by a periodic modulation of the fusion source due to sawtooth crashes. The mechanism for a persisting bump-on-tail distribution of the resonant fast-ions exciting AEs is very similar to the one applied in recent NBI experiments in DIII-D, where short beam modulation periods were applied to create transiently a 'bump-on-tail' velocity distribution for the beam ions [42]. However, in the JET experiments in D-³He plasmas reported here, such a mechanism is self-sustained since the source rate of fusion-born alpha-particles was modulated by the sawtooth crashes.

The Alfvén instability is a significant issue for high-Q operation in ITER. Indeed, the excitation and interaction of fusion-born alpha-particles with n = ‒1 and n = 0 modes in the EAE frequency range studied in these experiments is important for the development of magnetic fusion reactors since the consequences on fast-ion transport and confinement can be substantial.

Based on the experimental and theoretical analysis of fast-ion experiments in D-³He plasmas presented here, we are planning a special scenario for demonstrating alpha-driven AEs in the forthcoming JET D-T plasma studies. The principal idea is to realize the conditions at which the energy distribution function of fusion-born alphas is far from the steady-state distribution and has a 'bump-on-tail' in the Alfvénic energy range. As demonstrated in this paper, this can be achieved by modulating the alpha-particle source rate on a time scale shorter than their characteristic slowing-down time, $\Delta {t_{{\text{ST}}}}/\tau _\alpha ^*$ < 1. Thus, we propose an experiment, in which NBI power in D-T plasmas is modulated with a period shorter than $\tau _\alpha ^*$, a similar experiment as the one in deuterium plasmas on DIII-D [42]. Demonstrating alpha-particle driven AEs with a periodic modulation of NBI power in D-T plasmas with NBI-only heating is the most straightforward way as it rules out any other possibilities for fast-ion drive.

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053 and from the RCUK Energy Programme (Grant No. EP/T012250/1). To obtain further information on the data and models underlying this paper please contact PublicationsManager@ukaea.uk. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

The data that support the findings of this study are available upon reasonable request from the authors.