Density dependence of ion cyclotron emission from deuterium plasmas in the large helical device

Ion cyclotron emission (ICE) is widely observed from magnetically confined fusion (MCF) plasmas [4]. In addition to historical observations from deuterium–tritium plasmas in JET [5, 6] and TFTR [7], since 2017 ICE has been reported and analysed from the KSTAR [8, 9], DIII-D [10], ASDEX-Upgrade [11–13], TUMAN-3M [14] and EAST [15] tokamaks, and the large helical device (LHD) heliotron–stellarator [16–18]. ICE spectra typically comprise a succession of narrow, strongly suprathermal, peaks at sequential cyclotron harmonics of an energetic ion species. These species include fusion-born ions [5], neutral beam injected (NBI) ions [19], and ions energised by ion cyclotron resonant heating [20]. The emitting region is identified by matching spectral peak frequencies to local magnetic field strength. While in most cases it corresponds to the outer midplane edge plasma, ICE from the plasma core has recently been reported from the TUMAN-3M [14], ASDEX-Upgrade [12] and DIII-D [21] tokamaks. ICE has also been observed in NSTX and NSTX-U where it was found to be spatially collocated with internal transport barriers [22]. It is clear that the plasma physics process responsible for ICE is the magnetoacoustic cyclotron instability (MCI) [23]. This excites waves on the fast Alfvén-cyclotron harmonic branch, by drawing on the free energy of fast ions whose distribution in velocity-space incorporates a population inversion. The linkage between ICE and the MCI was established by early analytical studies [24–27], and reinforced in the past decade by first principles computational studies using the particle-in-cell (PIC) approach [3, 8, 9, 20, 28–31]. In outline, the MCI involves the excitation, through wave-particle cyclotron resonance, of quasi-perpendicular propagating waves on the fast Alfvén-cyclotron harmonic branches. For these waves, the perpendicular phase velocity ω/k_⊥ ≃ V_A, with V_A the Alfvén speed, implying that energetic ions with v_⊥ ≃ V_A will dominate any wave-particle resonant instability drive. Such drive is, of course, only possible at spatial locations where the energetic ion velocity-space distribution near v_⊥ ≃ V_A exhibits a population inversion, that is, ∂f/∂v_⊥ > 0 locally. This intuitively attractive physics was identified in the pioneering formulation of the MCI by Belikov and Kolesnichenko [23] and in subsequent analytical studies [24–27]. It also emerges naturally from PIC-based studies of the relaxation of energetic ion distributions in majority thermal plasmas [3, 8, 9, 20, 28–31]. In these, the collective instability, and indeed the normal modes themselves, are not prior assumptions; instead they emerge from the summation, over tens of millions of macroparticles, of self-consistent gyro-resolved kinetics and field evolution.

The notion that the strongest drive comes from v_⊥ ≈ V_A originates in the mathematical analysis of the linear MCI. Specifically the kinetic dielectric tensor elements are rich in Bessel functions whose arguments k_⊥ v_⊥/Ω, see for example equations (11) to (13) of reference [26], become k_⊥ v_⊥,0/Ω for the driving energetic ion population, with Ω and v_⊥,0 its cyclotron frequency and speed respectively.

Since Ω also approximates to k_⊥ V_A for the fast Alfvén wave at its fundamental cyclotron resonance, the Bessel function argument approximates to v_⊥,0/V_A. In addition, since the most significant contributions (involving sign reversals, etc) from the Bessel functions and their derivatives concentrate around where their argument equals unity, linear theory implies in broad terms that the dominant contribution to driving the MCI comes from energetic ions that have v_⊥ close to V_A, irrespective of their v_∥. This line of reasoning, first developed by Belikov and Kolesnichenko [23] in the 1970s, also works for nth cyclotron harmonics and corresponding nth order Bessel functions and derivatives, hence the broadly comparable linear growth rates calculated across a range of low cyclotron harmonics shown in e.g. figure 3 of reference [26]. The central role of Bessel functions in linear MCI theory carries over into the sub-Alfvénic regime, see for example equations (9), (10) and (16) of reference [32]. The analytical theory of the MCI, which contributes to explaining many aspects of ICE observations, has recently undergone a renaissance, see for example references [33, 34]. The supposition, derived from linear MCI theory, that ICE observations at proton cyclotron harmonics in KSTAR deuterium plasma is mainly driven by energetic ions with v_⊥ ∼ V_A turned out to have predictive power, see reference [3].

The NBI ions driving ICE are not necessarily super-Alfvénic, even marginally. An early example of sub-Alfvénic ICE arose in TFTR DT supershots [35]. In the LHD cases considered here, v_⊥,0 = v_NBI is less than V_A by factors of around two, with v_NBI the speed of the NBI ions. This deeply sub-Alfvénic ICE regime is underexplored in terms of published experimental data. Here we report three instances from LHD, which we examine using first principles studies of the nonlinear regime of the MCI, using PIC-type approaches. In addition, the path taken here differs from that of reference [18] for which both the density and magnetic field were changed in order to vary v_NBI/V_A, whereas only the density was changed here. For example, if one adjusts the value of v_NBI/V_A by changing the injection energy of the fast ions (hence the value of v_NBI), in principle the bulk plasma parameters could remain invariant. Whereas if one adjusts the value of v_NBI/V_A by changing the plasma density (hence the value of V_A), this results also in a change of the value of key bulk plasma ratios such as plasma frequency to cyclotron frequency, and in the value of the lower hybrid cutoff. These are, of course, important for fast wave physics in plasmas. PIC code algorithms [36] solve the Maxwell–Lorentz system of equations, typically for tens or hundreds of millions of interacting charged macroparticles—energetic ions, thermal ions, and electrons—together with the self-consistent electric and magnetic fields. Collective instabilities thus emerge at the level of particle kinetics and field dynamics. The PIC approach can retain full gyro-orbit resolution, and is thus particularly suitable for phenomena that incorporate gyroresonance, such as the MCI. In PIC studies for ICE interpretation, a minority energetic ion population is initialised with a distinct, physically motivated, velocity-space inversion. This population then relaxes under the Maxwell–Lorentz system, while coupled to the thermal ions and electrons, and interacting with, and generating, self-consistent fields. Typically, the frequency and wavenumber dependence of the saturated amplitudes of the MCI-excited fields determine the simulated ICE spectra, which are then compared to the observed ICE spectra. Our PIC computational approach, using the hybrid-PIC code [29], is the same as used in reference [18], to which we refer for further detail. The foregoing approach was recently applied to observations of ICE from hydrogen plasmas in LHD, driven by NBI protons injected perpendicular to the confining magnetic field [18]. A focus of reference [18] is the dependence of observed ICE properties on whether the NBI protons are super-Alfvénic or sub-Alfvénic in the emitting region of the plasma; that is, whether v_NBI/V_A is larger or smaller than unity, where v_NBI is the speed of the NBI ions and V_A the local Alfvén speed. The present computational study addresses measurements of ICE from significantly sub-Alfvénic populations of energetic NBI ions, under conditions where the local density differs significantly between LHD plasmas. At fixed NBI energy and local field strength, the value of V_A is governed by the value of the density. In recent PIC studies of ICE from the KSTAR tokamak, ICE chirping on sub-microsecond timescales during the ELM crash was interpreted [3] in terms of fast density dependence arising from filament motion. Understanding the density dependence of ICE phenomenology is thus topical, both experimentally and in relation to theory and interpretation. In particular, it will assist the exploitation of ICE measurements to infer the key features of the velocity-space distribution of energetic ion populations in MCF plasmas.

Ring-beam distributions in velocity-space, as adopted in the present paper, are appropriate for representing freshly ionised populations of NBI ions, and were used in initial studies of the excitation of electrostatic modes to interpret probe measurements of ICE in TFTR plasmas with deuterium and tritium NBI in reference [27]. This electrostatic instability (as distinct from the fully electromagnetic treatment now prevalent, including in the present paper) did not rely on the fast Alfvén wave, and could be driven by significantly sub-Alfvénic fast ions, given a very narrow distribution of speeds parallel to the magnetic field. Computed growth rates for tritium (T) harmonics degenerate with background D were relatively lower, providing a link with the observed reduced intensity of ICE spectral peaks at the third and sixth T harmonics in some TFTR experiments, see figure 2 of reference [27]. Subsequently in TFTR, ICE was observed at cyclotron harmonics of the fusion products (³He in DD supershots, and both ⁴He and ³He in T and in DT supershots) and of the sub-Alfvénic NBI-injected D and T. As with ICE in JET, the emission originated from the outer midplane edge plasma, but there were significant differences with respect to JET, particularly in the time evolution of the TFTR ICE signals. Typically, during the first 50 to 200 ms following the NBI trigger, cyclotron frequencies of the fusion products dominated the measured ICE spectrum, but then died out. They were replaced by ICE spectral peaks at multiple cyclotron harmonics of the injected D and T, until the NBI injectors were turned off. In subsequent TFTR experiments, Helium was puffed [35], forcing the plasmas to transit from typical supershots to pure L-modes. This changed the electron critical density and hence the local value of the ratio v_α /V_A ≶ 1, where v_α is the alpha-particle birth velocity and V_A is the local Alfvén speed in the ICE-emitting region at the plasma edge. The transient nature of ICE in TFTR supershots could be explained by the fact that the alpha-particle fusion products were sub-Alfvénic in the edge plasma. A plasma shifting from supershot to L-mode leads to early alpha-particle-driven ICE generation lasting for ≈200 ms which then soon extinguishes, typical of the supershot phase, see figure 7 of reference [35]. The linear MCI treatment [7, 27, 32] distinguishing the sub-Alfvénic behaviour of TFTR from super-Alfvénic regimes typical of JET lead to higher growth rates for the former compared to the latter. This phenomenology follows from orbits and velocity-space considerations. In the observations from DIII-D (reference [10]) and ASDEX-Upgrade (references [11–13]) the NBI ions are also sub-Alfvénic, but the observational focus is on core ICE. For core ICE, the MCI physics is more complex and less certain: partly because fusion-born ions may additionally contribute to the drive; and partly because there are no core ICE counterparts to the very high-resolution measurements of plasma conditions for edge ICE, as shown for LHD in e.g. figure 4 of the present paper. The distinctive ICE measurements from NSTX and NSTX-U [22] appear to be driven, near transport barriers, by ions that are locally mildly super-Alfvénic. ICE from the core plasma in the small (major radius 0.557 m, minor radius 0.256 m) TUMAN-3M tokamak was observed in association with sub-Alfvénic NBI using a mixture of hydrogen and deuterium using magnetic probes [14]. The density-dependence of ICE spectra driven by substantially sub-Alfvénic NBI ions has not previously been explored, and especially not in otherwise similar plasmas, as here. Reference [18] addresses two cases of ICE, on either side of the boundary between sub-Alfvénic and super-Alfvénic in LHD. References [3, 8] for KSTAR include density-dependence, but this is continuously and rapidly varying on microsecond timescales in the ELM crash, which is a very different context from the quasi-stationary conditions in the present paper. Furthermore the driving ions (3 MeV fusion-born protons) in references [3, 8] are strongly super-Alfvénic with v_⊥ close to V_A. The NBI deuterons driving the ICE in KSTAR, reference [9], are mildly sub-Alfvénic. As noted in the caption to figure 5 there, the action of the MCI is concentrated in (ω, k) space in the fairly narrow wedge between v_NBI and V_A. Among other things, this motivates the drive to 'open the wedge' in the present paper; as noted in the main text and captions, we go down to v_NBI/V_A = 0.36 in LHD plasma 138 439.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the LHD heliotron–stellarator, the ICE driven by perpendicular NBI deuterium, and the distinctive ICE driven by tangential NBI, have previously been observed using loop antennas [1]. Moreover, radiofrequency emission has been observed in the lower hybrid range, with frequency dependent on plasma density. Here we address recent measurements of radio-frequency emission at multiple deuteron cyclotron harmonics from deuterium plasmas in LHD, which show substantial variation in spectral character, between otherwise similar plasmas that have different local density in the emitting region. These measurements are obtained with the dipole antenna described below. We show that this ICE is driven by perpendicular NBI deuterons, freshly ionised near their injection point in the outer midplane edge of LHD. These NBI deuterons undergo collective sub-Alfvénic relaxation, which we follow deep into the nonlinear phase of the MCI. The variation with density of the spectral character of the simulated ICE corresponds well with that of the observed ICE from LHD. These results from heliotron–stellarator plasmas complement recent studies of density-dependent ICE from tokamak plasmas in KSTAR [2, 3], where the spectra vary on sub-microsecond timescales after an ELM crash. Taken together, these results confirm the strongly spatially localised character of ICE physics, and reinforce the potential of ICE as a diagnostic of energetic ion populations and of the ambient plasma.

On LHD, the acquisition system of the ICE measured at a dipole antenna located in the 10-O port inside the vacuum vessel was developed in partnership with KSTAR [37–40]. A fast digitizer performs direct sampling of the radiofrequency measurements at a frequency of 1.25 GSa/s. The time evolution of the RF spectral intensity is determined from this signal using a 14-channel filter bank spectrometer in the range of 70 MHz to 2800 MHz, with intermediate spectral resolution and with μs time resolution; for a duration spanning the whole plasma discharge [39]. This dipole antenna measuring the ICE is located close to NBI #5 and toroidally opposite to NBI #4, see figure 1.

Zoom In Zoom Out Reset image size

During perpendicular deuterium NBI experiments in LHD, it was observed that the spectral character of the ICE changes significantly between plasmas whose parameters in the emitting edge region differ only in their electron density. The left panel of figure 2 shows that ICE at frequencies 30 MHz and 60 MHz was observed only when NBI #5 was operated without NBI #4 in LHD plasmas 138 439 and 138 458. Conversely, in these plasmas, ICE was not observed when only NBI #4 was operated without NBI #5. Moreover, as shown in the right panel of figure 2 , ICE was continuously observed during continuous operation of NBI #5 in LHD plasma 138 433. These results support the interpretation below that ICE measured at the dipole antenna is driven locally by perpendicular-NB injected deuterons from NBI #5. Figure 3 shows the ICE power spectra observed at the time indicated by the vertical blue dashed line in figure 2 for LHD plasmas 138 439, 138 458 and 138 433 which differ primarily in their edge plasma densities. The profiles of the electron density, electron temperature, and ion temperature are shown in figure 4. These were taken at instants which are very close to the times at which the ICE spectra, shown in figure 3, were observed. Figure 3 shows that, as the electron density increases, the amplitudes of the ICE spectral peaks typically increase, in the frequency range of interest to us below 120 MHz; the number of harmonics increases; and the width of spectral peaks increases. If we identify the 12.05 MHz spacing between the ICE spectral peaks in figure 3 with the local deuterium cyclotron frequency, this suggests that the local background magnetic field value is 1.581 T and hence the emission location is at R = 4.651 m, as illustrated in figure 5. Table 1 summarises the plasma parameters at the inferred emission location where the deuteron cyclotron frequency matches 12.05 MHz, with reference to figure 4, together with values for the velocity of freshly injected ions v_NBI, and the local Alfvén speed V_A. We note for future analysis that, since the Alfvén speed is given by

$$\begin{equation}{V}_{\mathrm{A}}=\frac{{B}_{0}}{\sqrt{{\mu }_{0}{n}_{\mathrm{e}}{m}_{\mathrm{D}}}},\end{equation} \tag{ 1 }$$

with n_e and m_D the electron density and the deuteron mass respectively, the key dimensionless ratio of the NBI deuteron speed to the Alfvén speed, v_NBI/V_A, also increases as n_e increases. We observe that in all three plasmas, the NBI deuterons are in the sub-Alfvénic regime. This regime was previously investigated in earlier ICE experiments carried out in TFTR, involving gas puffing [35, 41]. This changed the edge plasma density, transitioning the fusion-born alpha-particles from a sub-Alfvénic to a super-Alfvénic regime in that region. The consequences for the TFTR ICE spectrum are discussed in figures 7–9 of reference [35]. We note that the Alfvén speed definition equation (1) includes D only: the ratio D/(D + H) remains constant and very close to unity during the plasmas studied. Our simulations incorporate the effect of both the thermal deuterons and the energetic deuterons NBI population as explained in the next section.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

5.1. Basis of the computational approach

5.2. Simulated ICE spectra compared to observations

In our PIC-hybrid simulations, the energetic minority deuteron population is initialised with a ring-beam distribution in velocity space, of the form ${f}_{\text{NBI}}\left({v}_{\perp },{v}_{{\Vert}}\right)=1/\left(2\pi {v}_{\text{NBI}}\right)\delta \left({v}_{{\Vert}}\right)\delta \left({v}_{\perp }-{v}_{\text{NBI}}\right)$, where v_NBI is the injection speed resulting from the NBI #5b beam energies given in table 1. These NBI deuterons are initially distributed randomly and uniformly in gyroangle while the thermal deuterons are loaded by mean of a quiet start [46, 47], in both positions and velocities. The relaxation of this energetic ion population under the Maxwell–Lorentz system of equations gives rise to self-consistently excited electric and magnetic fields; see references [3, 28, 29] for details of this approach. Fourier transformation of the excited fields yields spectral information that is compared to the LHD observations. We focus on the lower cyclotron harmonics, for which the distribution of energy between the ICE spectral peaks shown in figure 3 displays significant changes between the three LHD plasmas. Figure 6 shows the simulated power spectra, alongside the measured spectra extracted from figure 3. The red, green and blue traces correspond to LHD plasmas 138 439, 138 458 and 138 433 respectively. In the middle and bottom panels of figure 6, these traces are obtained by taking the spatiotemporal fast Fourier transform of the values of δB_z output from the PIC-hybrid computations in the time interval between t = τ_D and t = 9τ_D (where τ_D = 2π/Ω_D is the deuteron gyroperiod), which is summed over wavenumbers up to k = 40Ω_D/V_A. The agreement between observed and simulated ICE spectra appears good quantitatively. In both the top (experimental ICE) and middle (simulated spectra) panels in figure 6, the number of cyclotron harmonic spectral peaks increases as v_NBI/V_A increases, as does their amplitude. These trends can be examined in greater detail for the three plasmas separately in figure 7. Here our simulated power spectra are shown as dark traces, together with the measured ICE spectra which are plotted as coloured traces for comparison: red, LHD plasma 138 439; green, LHD plasma 138 458; blue, LHD plasma 138 433. The spectra are vertically offset in comparison with figure 6 such that the ordinate of the peak with largest amplitude is zero in each panel. In the left panel of figure 7, for which v_NBI/V_A = 0.36, the strongest spectral peaks in the simulation are at relatively low deuteron cyclotron harmonics, second to fourth. In the middle panel, for which v_NBI/V_A = 0.44, the middle order harmonics (fourth and fifth) become more pronounced. In the right panel of figure 7, for which v_NBI/V_A = 0.60, the amplitudes of the spectral peaks are substantially larger, and the simulation captures the experimental spectrum in extending strongly across the middle order harmonics (fifth to seven). In all three plasmas of figure 7, the strongest harmonic peaks in the experiment and in the simulations are the same: from left, second, second and fourth. The fundamental computational noise level in our simulations is addressed in appendix A, where it is found to lie at about −145 dB. The instrumental noise level in the ICE detection system on LHD imposes the noise floor that is visible in the top panel of figure 6. The question of where the instrumental noise floor should lie in our simulated spectra cannot be answered a priori. However the bottom panel of figure 6 shows that the agreement between the observed and simulated ICE spectra is greatest if we conjecturally assign a value of −133 dB to the instrumental noise floor when considering our simulated spectra. One possible source of difference with regard to the width of the measured and simulated power spectra is the time during which the time series is acquired. In the calculation, it corresponds to the duration of the simulation which is 10τ_D.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The spatiotemporal fast Fourier transforms of δB_z are plotted in figure 8. These show that the concentration of excited wave amplitude in $\left(\omega ,k\right)$ space lies on horizontal bands at successive cyclotron harmonics, and that each of these bands is approximately centered on a line which satisfies ω/k = v_NBI. This is represented by the dark line in each panel of figure 8, for which $\left(\omega /k\right)/{V}_{\mathrm{A}}={v}_{\text{NBI}}/{V}_{\mathrm{A}}=0.36$, 0.44 and 0.60 from top to bottom. The nearly horizontal structures are electromagnetic cyclotron harmonic waves supported by both the thermal deuterons (as a positive-energy wave) and the NBI deuterons (as a negative-energy wave). This is apparent from the linear analytical theory of the MCI: restricting attention to the strictly perpendicular case for mathematical clarity, let us consider for example the final three terms on the right-hand side of equation (67) of reference [24]. Importantly, the signs of the coefficients β₄ and β₅ in this equation, which are defined at equations (51) and (52), depend on the specific distribution of the energetic ions in velocity-space. This leads on to the negative-energy character of the energetic ion support to the corresponding cyclotron harmonic waves, as anticipated by Belikov and Kolesnichenko [23]. The electromagnetic generalisation of the foregoing, for finite k_∥, is at equation (20) of reference [26]. The dark line overplotted on the spatiotemporal Fourier transforms in figure 8, which delineates Ω_D/k = v_NBI, can be recast as k_⊥ v_NBI/Ω_D ≃ 1. At ω = Ω_D, this represents the Bessel function argument which we mentioned earlier in the introduction; excitation is concentrated at values of the argument close to 1. The steepest yellow–red feature in figure 8 corresponds to the linear fast Alfvén dispersion relation ω = kV_A and incorporates significant noise energy. Figure 9 plots the time evolution of the change in the different components of particle and field energy density. It shows that, in these simulations, the NBI beam deuterons relax and saturate faster, releasing more energy to the excited fields and the bulk plasma, as the value of v_NBI/V_A increases towards unity.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The computational approach adopted in this paper has involved solving the Maxwell–Lorentz equations for tens of millions of macroparticle ions with fully gyro-resolved orbits, together with fluid electrons, self-consistently with the electric and magnetic fields. This is carried out with the PIC code [36] used in references [29, 31], using a local (slab geometry) description, with one spatial and three velocity-space coordinates (1D3V), and retaining all three vector components of E and B. It transpires here that the self-consistent particle dynamics and field evolution, when summed over, correspond to the theory of the MCI [3, 8, 9, 20, 23–31], where appropriate. The Fourier transforms of the excited fields, in the saturated regime of the instability, constitute our simulated ICE spectra, presented in section 5. This approach has previously been successfully applied to interpret ICE observations from MCF plasmas, notably JET [20, 28–31], KSTAR [3, 8, 9] and NBI-heated hydrogen plasmas in LHD [18].

We note that our first-principles Maxwell–Lorentz 1D3V PIC approach necessarily omits a very large number (tens, perhaps hundreds) of physical effects that may contribute to the observed ICE spectra. These potentially include: toroidal magnetic field effects; background gradients, for example of magnetic field strength and density; dissipation effects, other than the cyclotron (and Landau) resonant effects that are captured by the PIC approach; wave propagation effects, other than $\left(\omega ,k\right)$ dispersion which is an emergent property at the Maxwell–Lorentz PIC level of description, together with slab geometry growth or damping which is also emergent; antenna characteristics; and signal processing effects.

Discrepancies between Maxwell–Lorentz PIC outputs and the observed ICE spectra are therefore to be expected, and will be due to physical effects, and combinations thereof, under the headings outlined in the preceding paragraph. The degree of agreement between our PIC outputs and the observed ICE spectra, reported here in section 5, suggests that it is possible that none of the additional physical effects play a major role in determining the overall ICE spectrum. The cost-benefit of trying to isolate which (perhaps several together) of these additional effects is primarily responsible for the relatively minor differences between PIC outputs and the observed ICE spectra may, or may not, be attractive for future studies. At present we are able to conclude that the plasma physics emission mechanism for this ICE is well understood, aided by the fact that ICE is evidently a highly spatially localised phenomenon. The results presented here further reinforce the scope for exploiting ICE for diagnostic purposes.

Specifically, we are able to explain the physical origin of, and difference between, the ICE spectra obtained from three deuterium plasmas in LHD that differ primarily in their edge density. The collective instability of deuterons at about 67 keV originating from NBI #5, and relaxing under the non linear MCI in the outer midplane edge region is shown to be responsible for the ICE. An important consequence of the different edge densities across the three plasmas is that the ratio of v_NBI/V_A for freshly ionised NBI deuterons near their injection point changes while v_NBI is kept constant. The NBI deuterons are significantly sub-Alfvénic: v_NBI/V_A = 0.36, 0.44 and 0.60. Our PIC-hybrid computations of the collective relaxation of these NBI deuterons, evolving self-consistently with the thermal plasma and the excited electric and magnetic fields under the Maxwell–Lorentz equations, show similar trends with the LHD ICE observations. In particular, as the value of v_NBI/V_A increases, the number of cyclotron harmonic peaks in the simulated ICE spectra increases, along with their amplitude. These results are consistent with our recent comparison of ICE from super-Alfvénic and sub-Alfvénic NBI proton populations in LHD [18]. They also appear broadly consistent with the results of earlier ICE experiments carried out in TFTR, involving gas puffing [35, 41].

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. The work received support from the RCUK Energy Programme [Grant No. EP/T012250/1], NIFS budget NIFS15KLPF045 and from NRF Korea Grant No. 2014M1A7A1A03029881. The views and opinions expressed herein do not necessarily reflect those of the European Commission. ROD acknowledges the hospitality of Kyushu University. BCGR acknowledges helpful discussion with Dr. Carbajal-Gomez.

We have checked the noise level of our simulation results by running PIC-hybrid calculations without any energetic minority population, retaining only the deuterium thermal background plasma, and with all other parameters kept equal, for each of three plasmas. The noise in the simulation is not compared to the noise of the experimental detection system. We simply aim to establish the significance of the spectral peaks in the PIC-based simulations that include NBI ions, with respect to the spectral peaks that are generated by thermal noise in the code. Figure 10 displays: the power spectra for computations that include both the NBI and thermal deuterons, shown by continuous lines; and the spectra obtained from the simulations that retain only the thermal deuterium plasma, shown by the dashed curves. The spectral peaks in the latter case result from the concentration of noise energy at normal modes—in this case, cyclotron harmonic waves supported by the thermal plasma—in line with the fluctuation dissipation theorem [48]. The dashed lines in figure 10 indicate that the fundamental computational noise level is similar in all our simulations, at −145 dB, although slightly higher for the simulation of LHD plasma 138 433.

Zoom In Zoom Out Reset image size