Novel internal measurements of ion cyclotron frequency range fast-ion driven modes

Novel internal measurements and analysis of ion cyclotron frequency range fast-ion driven modes in DIII-D are presented. Observations, including internal density fluctuation ( n~ ) measurements obtained via Doppler backscattering, are presented for modes at low harmonics of the ion cyclotron frequency localized in the edge. The measurements indicate that these waves, identified as coherent ion cyclotron emission (ICE), have high wave number, k ⊥ ρ fast ≳ 1, consistent with the cyclotron harmonic wave branch of the magnetoacoustic cyclotron instability, or electrostatic instability mechanisms. Measurements show extended spatial structure (at least ∼1/6 the minor radius). These edge ICE modes undergo amplitude modulation correlated with edge localized modes (ELM) that is qualitatively consistent with expectations for ELM-induced fast-ion transport.


Introduction
Waves at harmonics of the ion cyclotron frequency ( f = l f ci , |l| = 1, . . .) are commonly excited in fusion research plasmas by fast ions produced by neutral heating beams, acceleration by radio-frequency heating, and fusion reactions (see review in [1] and references therein). These waves, referred to as coherent ion cyclotron emission (ICE) since historically they have been detected as 'emission' from the plasma via external antennae, are thought to be attributable [1] to the magnetoacoustic cyclotron instability (MCI) [2][3][4][5], in which drive comes from Doppler-shifted cyclotron resonance between the waves and fast ions. Electrostatic instabilities have also been proposed as a potential explanation for observations of ICE at harmonics of edge values of f ci . (See e.g. [4] and references therein.) These modes potentially cause transport and anomalous heating [6] and may be exploited as a tool to diagnose the fast-ion population [1]. This paper presents novel internal measurements and analysis of edge localized coherent ICE modes in DIII-D which exhibit amplitude modulation correlated with edge localized modes (ELMs) that may potentially advance understanding of ELM-induced fast-ion transport. Observations of internal density fluctuation (ñ) measurements obtained via Doppler backscattering (DBS) [7] are presented for high-k waves (k ⊥ ρ fast 1) at low harmonics of f ci localized to the edge in Hmode plasmas. (The symbol k ⊥ refers to the component of the wave number vector perpendicular to the equilibrium magnetic field.) These waves are identified as ICE, with features consistent with cyclotron harmonic wave branch of the MCI instability [3] or electrostatic waves (e.g. [4]). (Note that the cyclotron harmonic wave branch [8] is sometimes referred to as the ion Bernstein branch [3,4].) The spectrum of edge ICE measured via DBS is shown to have interesting differences from the spectrum measured by edge magnetics [9], potentially explained by differences in the sensitivities of the two instruments. The structure of this paper is as follows. Section 2 describes the high frequency fluctuation diagnostics used to measure the modes, including in particular recent modifications to the DBS system that enable measurements of ICE. Section 3 presents and discusses the observations of ICE. Section 4 concludes the paper with a summary of the results.

High frequency fluctuation measurement technique
Two instruments are used to obtain the fluctuation measurements reported in this paper. The primary instrument is an eight channel DBS system for measuringñ [7]. This system was recently modified to expand its fluctuation measurement range up to ∼75 MHz by splitting the output signal of each channel into low frequency (LF) and high frequency (HF) components using a diplexer, which are then separately digitized. The HF electronics have a combined pass band of ∼16-75 MHz. The HF signals are recorded with a digitizer at 100 Msamples/sec. A DC-47 MHz anti-alias filter is used since signal components above the 50 MHz Nyquist frequency would alias to lower frequency upon digitization. The other instrument is a set of magnetic sensing loops, or antennae, positioned on the vacuum vessel wall near the plasma edge, with sensitivity to toroidal magnetic fluctuations in the 1-200 MHz range [9,10].
The DBS system launches a beam of millimetre waves into the plasma from an antenna consisting of 8 fixed frequencies from 55-75 GHz (specifically 55, 57.5, 60, 62.5, 67.5, 70, 72.5 and 75 GHz). As illustrated in figure 1, the beam approaches from the low field side, near the midplane, obliquely incident to the plasma at a small angle to normal. The beam can be poloidally steered to control the angle of incidence. (For the results reported in this paper, the pivot point for steering is at a major radius of R = 3.01 m, 0.09 m below the tokamak geometric midplane. An upward tilt of ∼3.0 • is typically required for normal incidence, with the exact angle depending on plasma shape and position.) As millimetre waves propagate through plasma, they scatter fromñ and some of the scattered waves propagate directly back to the antenna, carrying information aboutñ. An electronic circuit measures the amplitude (A) of the received waves and their phase (φ) relative to the launched waves, encoding these in 'in-phase' (I = A cos φ) and 'quadrature' (Q = A sin φ) signals. The received radiation is typically Doppler-shifted, which manifests as a continually accumulating relative phase. Results presented in this paper include analysis of the 'quadrature electric field' (E = Ae iφ = I + iQ) signal.
ICEñ can influence the scattered millimetre wave spectrum in at least two ways that are potentially relevant to understanding the observations presented in section 3. Scattering from a short wavelength ICE wave may contribute to the received spectrum, or a long wavelength ICE wave may modulate the spectrum of millimetre waves scattered from turbulence by displacing the plasma (and thus the turbulence) in the scattering region or by modulating the index of refraction along the beam path. Millimetre waves at different frequencies separate into individual beams upon entering the plasma, which refract poloidally along separate curved trajectories as each approaches its respective cutoff. At their closest approach to cutoff, the waves at each frequency reach a minimum in wave number and propagate approximately in the poloidal direction, as illustrated in figure 1. The ray trajectory and cutoff location, as well as wave number along the ray, can be determined from a ray tracing calculation. (For the results reported in this paper, the GENRAY ray tracing code was used [11].) The direction of launch and of the magnetic fields in the tokamak ensure that the millimeter wave number vector is approximately perpendicular to the magnetic field everywhere along the ray. Millimetre waves scatter fromñ along the path according to the Bragg scattering relations, ω s = ω i + ωñ and k s = k i + kñ, where ω and k correspond to frequencies and wave number vectors and the subscripts s, i andñ correspond to the incident and scattered millimetre waves and theñ causing the scattering. Since ωñ ω s,i , |k s | ≈ |k i | and the waves that scatter fromñ with kñ ≈ −2k i propagate back to the antenna, Doppler-shifted by a frequency ωñ. For direct scattering from an ICE wave at frequency ω ICE , the scattered spectrum will have peaks at +ω ICE or −ω ICE , probably with unequal power in the two peaks, as illustrated in figure 2(a). The dominant peak depends on the direction of the ICE wave propagation and the amplitude is determined by the amplitude of the ICE wave. (In principle, a synthetic diagnostic model could be used to infer the ICE wave amplitude from the measurement, but development of this model is left to future work.) Scattering can take place anywhere along the path. The location can in principle be determined with an understanding of ICE wave dispersion characteristics since k i varies along the path. (For dispersion characteristics of electromagnetic ICE waves, see e.g. [3]. Electrostatic instabilities, e.g. [4], have also been proposed to produce ICE waves.) For modulation of the turbulent scattering spectrum, the effect of ICEñ is somewhat simpler to interpret because the spectrum reaching the antenna is expected to be dominated by scattering near cutoff. The scattering crosssection is proportional to the turbulenceñ amplitude, which tends to be largest for turbulence near cutoff, since |kñ| reaches its smallest value there. The modulation creates equal amplitude sideband peaks in the turbulent scattering spectrum at ω turb ± ω ICE , as illustrated in figure 2(b), where ω ICE is the frequency of the ICE wave and ω turb is the Doppler shift of the turbulent scattering spectrum. The sideband amplitude is determined by the ICE wave amplitude. (The Doppler shift, or peak frequency of the LF spectrum, is set by the poloidal velocity of the turbulent fluctuations causing the scattering, which is typically approximately the E × B velocity. In particular, ω turb ≈ k θ,ñ v E×B .) (The symbol k θ,ñ refers to the k θ component of the density perturbation wave number vector, where k θ is the component in the binormal direction, perpendicular to the equilibrium magnetic field and the flux surface normal. The binormal direction is typically approximately in the poloidal direction, which motivates the use of the subscript θ.) It may sometimes be possible to distinguish between the two cases in figure 2 via simple inspection of the measured spectra, but in cases where the Doppler shift is small it may be difficult to discern with sufficient precision whether or not the frequencies of the peaks for a given mode in the positive ( f + ) and negative ( f − ) frequency ranges match each other (i.e. f − = − f + ). Fortunately, it possible to use a combination of bandpass filtering and coherence analysis to make a strong determination even in this case. In particular, the quadrature electric field signal can be bandpass filtered to isolate signals corresponding to the positive and negative frequency peaks of a mode. These signals take the form The sensitivity of this test will, of course, depend on details such as the amount of data used and the relative amplitude of the noise. For the measurements reported here, this analysis was used for each mode.

Observations of ion cyclotron harmonic range modes
Unique observations of modes with frequencies above f ci ( f ∼ 20-40 MHz ∼2-3 f ci ) test aspects of theory. Measurements of ICEñ are obtained in a variety plasma of conditions via DBS inside the plasma edge across a significant spatial range, giving mode amplitude and structure. In one instance, a highk plasma wave at a frequency of 2 f ci is observed to extend from the plasma mid-radius to the top of the pedestal of an Hmode plasma. In another instance, spatially extended high-k plasma waves at frequencies of 2 f ci and 3 f ci are observed in the scrape-off layer (SOL) of an H-mode plasma. The plasma waves in both cases are identified as cyclotron harmonic waves excited by the MCI or electrostatic waves. In both cases, the waves exhibit amplitude modulation during ELMs consistent with stability modification by ejection of fast-ions. Figure 3 shows DBS measurements of high-k ICEñ at ∼2 f ci in a deuterium H-mode plasma (shot 180203). Measurements are obtained with eight channels having cutoffs ranging from near the magnetic axis to the top of the pedestal. As shown in figure 4, at t = 2340 ms, the DBS system probes wave numbers k θ = 2.9 cm −1 − 1.   related to the millimetre wave k ⊥ ( figure 4(b)) by k θ = 2k ⊥ . During the period shown, the plasma has current I P = 820 kA, and electron temperature, electron density, safety factor and magnetic field at the magnetic axis of T e0 = 3 keV, n e0 = 7.0 × 10 19 m −3 , q 0 = 4.4 and B 0 ≈ 1.59 T, respectively. (The magnetic field is actually slowly ramping down, decreasing <2% within the period shown in figure 3, and ∼18% over the period t = 1400-3500 ms.) The magnetic axis is at R 0 = 1.81 m and the last closed flux surface outer boundary is R midout = 2.23 m. (These locations remain approxi-mately fixed over most of the period t = 1400-3500 ms.) Equilibrium reconstruction is performed using external magnetic measurements, as well as internal magnetic field pitch measurements from motional Stark effect spectroscopy [12]. The toroidal magnetic field is oriented in the same direction as the plasma current. Deuterium neutral beam power varies between P NB ∼ 5 MW and 9 MW as various beams are modulated on and off, including the more tangential (R TAN = 1.15 m) on-axis and more perpendicular (R TAN = 0.76 m) on-axis beams, as well as off-axis beams. The on-axis beams, in particular, are rapidly switched on and off with on-and off-times as short at 10 ms, while the off-axis beams remain on except for occasional short (∼10-20 ms) off periods. All beams inject in the plasma current direction, with E beam = 73-81 keV. The fast-ion gyro-radius is ρ fast ∼ 3-4 cm for fast ions deposited on the low field side by the on-axis beams. The off-axis beamlines were tilted, leading to more parallel injection and smaller ρ fast than the on-axis beams because of the direction of the toroidal field [13].
The high-k ICE in figure 3 has features indicating that the peaks at | f | ∼ 2 f ci in the spectra are the result of the millimetre waves scattering from a high-k plasma wave. In particular, figures 3(a) and (b) show the positive and negative frequency ranges of the DBS quadrature E spectra (∝ñ) for millimetre waves with a cutoff at ρ = 0.87. Both feature a peak at | f | ∼ 20.5 MHz. This is similar to the value of 2 f ci (∼20.7 MHz) at ρ = 0.87 on the low field side midplane, and it matches 2 f ci just outside ρ = 1. (Notably, this peak appears in the spectrum throughout most of the time period t = 1400-3500 ms during which B T ramps down, changing frequency strictly in proportion to B T . This proportionality ensures that the mode frequency matches 2 f ci very close to the boundary during the entire ramp.) The frequencies of the peaks in the positive ( f + ) and negative ( f − ) frequency ranges match each other ( f − = − f + ). Furthermore, the amplitude of the peak at positive frequency is greater than that at negative frequency. This agreement in frequency is expected for scattering from a highk plasma wave and the asymmetric distribution of power is also consistent. (For the alternative case where the peaks were turbulence sidebands caused by modulation of the millimeter waves by a long wavelength plasma wave, the peak in the negative frequency range would be expected to have frequency f − = 2 f turb − f + , where f turb ∼ 100-300 kHz is the Doppler shift observed in the LF spectrum for this plasma, and the two peaks would have equal power.) The millimetre wave beam probes a minimum k at cutoff of k θ = 1.2 cm −1 , so the plasma wave has k ⊥ 1.2 cm −1 . The high k and near-harmonic frequency of the plasma wave causing the scattering are consistent with a cyclotron harmonic wave driven by the MCI instability or an electrostatic wave. The MCI instability involves excitation of electromagnetic waves on the fast Alfvén and cyclotron harmonic wave branches [3]. A fast Alfvén wave would have k = ω/v A ∼ 0.4 cm −1 for a 20.5 MHz mode (n e ∼ 3.75 × 10 19 m −3 and v A ∼ 3.4 × 10 6 m s −1 at ρ = 0.87), which is sufficiently low that the mode cannot be a fast Alfvén wave. However, the value of k ⊥ for the cyclotron harmonic wave is expected to be significantly larger (k ⊥ ρ fast 1). The high k may also be consistent with an electrostatic wave (e.g. [4]). Distinguishing between these possibilities will require careful consideration of what each theory predicts for these conditions. The ∼20.5 MHz mode is radially extended and exhibits amplitude modulation correlated with ELM activity. Peaks at the same frequency of | f | ∼ 20.5 MHz are visible in the positive and/or negative frequency ranges of the spectra of the four outermost channels, which probe k θ = 1.5 cm −1 -1.2 cm −1 at cutoffs at ρ = 0.63-0.87. Figures 3(c) and (d) show examples at ρ = 0.87 and 0.63. (Note that the negative frequency range of the spectrum is shown for ρ = 0.63 in figure 3(d) because the power distribution at | f | ∼ 20.5 MHz is very asymmetric and a peak at +20.5 MHz is difficult to discern above the background level, in keeping with the mode being high-k.) These cutoffs span a radial distance of ΔR ∼ 7 cm ≈ a/6, (using a = R midout − R 0 ), or ΔR ∼ 2ρ fast . However, the radial extent of the mode is potentially greater. While the four innermost channels do not see the mode, placing a limit on the extent of the mode toward the magnetic axis, there are no channels closer to the edge to show the limit of the mode's radial extent in that direction. A radially extended mode would be expected for the MCI instability since emission from the resonant fast ions can propagate radially. It should be noted that in this scenario the location where the emission originates could be near, but not at, the location where the frequency matches the harmonic because of a non-vanishing Doppler shift contribution to resonance. As can be seen in figure 3(e), the mode amplitude is transiently suppressed during spikes in D α emission corresponding to ELMs. Any candidate instability must be able explain the radial extent and amplitude sensitivity to ELMs. The amplitude of the mode is not obviously correlated over the long period of the B T ramp with the modulation of any the beams, including in particular the on-axis beams, which are rapidly modulated. Figure 5 shows DBS measurements of high-k ICEñ at ∼2 f ci and ∼3 f ci in the SOL of an H-mode deuterium plasma (shot 179437). For the time period shown, the plasma has T e0 = 6.5 keV, q 0 = 0.7 and B 0 = 2.0 T. Plasma current is I P = 1.5 MA, while electron density ramps up from n e0 = 5.5 × 10 19 m −3 at t = 1650 ms to 6.5 × 10 19 m −3 at t = 1880 ms and then I P ramps quickly down as plasma termination begins, passing through 1.35 MA by t = 1900 ms. The deuterium neutral beam power is initially P NB = 6.5 MW and then steps down to 4.5 MW at t = 1700 ms, finally shutting off at t = 1880 ms. All beams are injecting in the plasma current direction, opposite the direction of the toroidal field, with energies of E beam = 71-79 keV and a fast-ion gyro-radius of ρ fast ∼ 2 cm for fast ions deposited on the low field side. Some on-axis beams are rapidly modulated off and on, while others remain on until t = 1880 ms. The off-axis beams are modulated more slowly before t = 1700 ms, completely shutting off afterwards (contributing to the step-down in total power). Notably, while the beams are all off after t = 1880 ms, the total D-D neutron rate drops only by a factor of ∼2 by the end of the time window in figure 5, t = 1900 ms. The magnetic axis is R 0 = 1.73 m and the last closed flux surface outer boundary is R midout = 2.27 m. Equilibrium reconstruction is performed using external magnetic measurements [14].
The high-k ICE in figure 5 has features indicating that the peaks at | f | ∼ 2 f ci and 3 f ci in the spectra are the result of millimetre waves scattering from a high-k plasma wave. In  identical positive and negative frequencies implies that this is a high-k wave, not millimetre waves scattered by turbulence and the asymmetry in amplitude is consistent with this interpretation. (The Doppler shift of the turbulent spectrum is f turb ∼ 100 kHz for this case.) The millimetre wave beam probes a minimum k at cutoff of k θ = 2.2 cm −1 , so the plasma wave has k 2.2 cm −1 . The high-k and near-harmonic frequency of the plasma wave causing the scattering are consistent with a cyclotron harmonic wave or electrostatic wave. The value of k is sufficiently high that it cannot be a fast Alfvén wave, which would have k = ω/v A ∼ 0.1 cm −1 for the 23.5 MHz peak (n e ∼ 4 × 10 18 m −3 , and v A ∼ 12 × 10 6 m s −1 ), too small for the probed k.
These modes at | f | ∼ 2 f ci and 3 f ci are spatially extended and exhibit amplitude modulation correlated with ELM activity. In particular, figures 5(c) and (d) show DBS quadrature E spectra (∝ñ) from millimetre-wave beams probing k θ = 2.2 cm −1 and k θ = 3.0 cm −1 at cutoffs in the SOL at ρ = 1.06 and 1.05. These cutoffs are separated by ΔR ∼ 3/4 cm. Only two channels are available for measurements in this case, so the full spatial extent cannot be determined. Figure 5(e) show that, in contrast to the previous example, ELMs transiently enhance the emission for these SOL measurements. The emission is not obviously correlated with the modulation of any the beams.
Simultaneous measurements ofb with the toroidal loop on the vacuum vessel wall show distinct differences from the spectrum of high-k ICE modes observed by DBS in figure 5. Figures 6(a) and (b) show time slices of the spectra at t = 1716.5 ms from figures 5(c) and (d), while figure 6(c) shows theb spectrum at the wall from the same time. Theb spectrum exhibits peaks at f ∼ 2 f ci and 3 f ci for a value of f ci in the edge, as does the DBS spectrum, but the frequencies of those peaks are lower than those in the DBS spectrum by ∼ 1 2 MHz. (This difference cannot be attributed to differences in the time bases for the magnetic and DBS diagnostics. Testing with a crystal oscillator showed that the time bases differ by less than 10 Hz.) In contrast, the peaks in both DBS spectra match each other, in keeping with the spatially extended nature of the mode. The explanation for this difference is under investigation; one possible explanation is that, owing to wave attenuation in the SOL, the tile loops are more sensitive to low-k waves than to the high-k waves detected by DBS. Fast Alfvén waves, which would have a long wavelength of λ = v A / f 30 cm (using f 35 MHz, v A 12 × 10 6 m s −1 ), may potentially more readily penetrate across the vacuum to the tile loop.
The high-k edge ICE modes depicted in figures 3 and 5 both show a sensitivity to ELM activity that is potentially explained by fast ion ejection by ELMs [15] if the modes are cyclotron harmonic waves driven by the MCI. The pedestal-top mode in figure 3 exhibits sudden transient drops in amplitude during spikes in D α line emission (figure 3(e)), while the SOL modes in figure 5 are in contrast transiently destabilized when D α spikes (figure 5(e)) due to ELMs. (A transient excitation of edge ICE by ELMS in DIII-D was also reported in [16].) The transient stabilization of the pedestal-top mode is consistent with a transient depletion of fast ions responsible for exciting the mode [17]. The transient excitation of the SOL modes is consistent with the passage of ejected fast ions transiently providing drive for the modes [18]. Unfortunately, it is not possible from existing data to discriminate between fast ions from beams and fusion products as the source of drive. ICE observed by the toroidal loops on the vacuum vessel wall has been shown in other cases in DIII-D to be excited by ions from beams based on an observed correlation between modulation of beam power and ICE amplitude [16]. Although there is some modulation of some beams for the two cases, there is no obvious correlation of the emission with the beam modulation. Following the reasoning in [16], this would argue against at least some beams as the source of the fast ions exciting the ICE, but not all. The final burst of mode activity after t ∼ 1890 ms in figure 5 is interesting since it occurs ∼10 ms after all beams have shut off and persists for ∼10 ms. The total D-D neutron rate has dropped by a factor of ∼2 by t ∼ 1900 ms, so there clearly remains a significant population of fast ions throughout this period, as well as continued production of D-D fusion products, to be transported by the associated ELM. In addition to causing fast-ion transport, ELMs also cause abrupt changes in edge density and temperature profiles which must also be accounted for. These profile changes could play a role in the stability and characteristics of an edge-localized ion cyclotron harmonic wave driven by fast ions. They could also destabilize electrostatic waves, potentially offering an alternate explanation these observations. Examples include ion-cyclotron drift waves [19] or drift-cyclotron loss-cone waves [20]. Of course, the stability of such gradient driven electrostatic instabilities may potentially be altered by the presence an anisotropic fast-ion population. Also, the sudden ejection of fast ions may indirectly affect wave stability by, for instance, transiently changing the local plasma velocity profile [21]. Further analysis and modelling will be required to evaluate these possibilities.

Conclusions
In conclusion, novel measurements are presented for high wave number (k ⊥ ρ fast 1 ICE in the edge of DIII-D. These measurements are shown to be consistent with the cyclotron harmonic wave branch of the MCI instability or an electrostatic instability and not consistent with fast Alfvén waves, which would have much lower wave number. Measurements of ICE spatial structure show significant spatial extent (at least ∼1/6 the minor radius). These observations of ICE may potentially advance our understanding of ELM-induced fast-ion transport. Comparisons of experiment with theory show areas of agreement, building confidence in its predictive power. These results strengthen the physics basis for predicting fast-ion driven mode activity and concomitant fast-ion transport in future burning plasmas.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.