Beam-driven chirping instability in DIII-D

During neutral-beam injection into the DIII-D tokamak, instabilities with frequencies that 'whistle' down a factor of two in a single 2 ms burst are observed between 50 and 200 kHz. The instabilities have toroidal mode numbers n=1-8 and cause the loss of beam ions from the plasma. In contrast to the usual Alfven modes, which are fluid modes of the background plasma, these instabilities seem to be beam modes that are nearly stationary in the plasma frame.


Introduction
'Fast' ions with speeds considerably greater than thermal speeds are common in tokamak plasmas [l]. Substantial fast-ion populations are produced by neutral-beam injection, by radiofrequency heating, and by fusion reactions. Large fast-ion populations sometimes destabilize instabilities that cause anomalous transport of the fast ions. Anomalous losses degrade the plasma performance and can damage internal vacuum-vessel hardware, so it is important to understand and control fast-ion driven instabilities.
The usual theoretical framework for understanding fast-ion driven instabilities assumes that the fast-ion population is a minor perturbation to the background plasma. If a weakly damped normal mode of the background plasma interacts resonantly with the fast ions, the fast ions can give energy to the wave and drive the wave unstable. Thus, the real part of the frequency w, is essentially the frequency of the normal mode, while the fast ions only affect the imaginary part of the frequency y . For electrostatic modes, this picture is valid if the fast-ion density nf is much smaller than the density of the background plasma ne (a condition that is almost always satisfied in practice). For the fast-ion population to have a negligible effect on electromagnetic modes, the fast-ion beta must be small relative to the total beta @ / p << 1). This condition is sometimes violated in low-density tokamaks with strong auxiliary heating.
p, new modes are possible [2]. For example, for the fishbonc instability, there is a fluid branch with frequency near the diamagnetic frequency w,i that is a normal mode of the background plasma [3]. The fast-ion mode occurs on a branch with fiequencies characteristic of the precessional motion of the fast ions wpR [4]. Both branches have the II = 1 mode structure characteristic of the internal kink. Both branches seem to occur experimentally [I]. Usually, the instabilities occur in bursts of -1 ms duration. In some cases (particularly when ww.i). the mode frequency changes 5 20% during a burst [5], suggesting identification as a fluid mode. In other cases [6,7], the frequency is comparable to and the mode frequency drops a factor of two during a burst, suggesting identification as a fast-ion mode. For example, in PDX [6], the frequency 'whistled' down from 25 to 10 kHz during a fishbone burst. Alfvin waves can be driven unstable by a fast-ion population. In a tokamak, the most unstable Alfvin waves are eigenmodes with frequencies in 'forbidden' gaps of the Alfvin continuum. The gaps are created by geometrical effects such as toroidal curvature (the toroidicity-induced Alfvkn eigenmode or TAE [SI) or geodesic curvature and plasma compressibility (the beta-induced Alfvin eigenmode or BAE [9] [19]. On TFTR, reflectometer diagnostics observe modes that 'chirp' up and down in frequency (as well as the usual TAE modes) [18].
In this paper, we report new observations 'of instabilities with rapidly changing ('chirping') frequencies in the range of 5&200 kHz. More information on.the mode structure is given than in previous studies. In addition, the relationship between the chirping modes and the fluid Alfvtn modes is clarified. It is found that chirping occurs when the fast-ion beta Br, the Alfvin speed U*, and the plasma rotation are all relatively large.

Experiment
Chirping instabilities are rarely observed on the magnetic probes in DII-D: of -1000 discharges examined for evidence of Alfvtn activity, only six discharges have chirping modes. All six discharges are from the same day of low density, high-power operation. These discharges are double-null divertor deuterium plasmas (figure 1). To obtain clean, low-density plasmas (typically 6, = 2.5 x IO" and 2 , 2: lS), the boron-coated walls are baked to -300°C the day before the experiment and glow discharge cleaning is employed after each discharge. Near tangential (tangency radius Rgn Y 1.10 m), -75 keV deuterium neutrals are injected in the direction of the plasma current. The plasma current is 1, = 0.6 MA and the toroidal field is varied between ET = 1.0 and 1.7 T. The plasma shape and q profile are obtained from the magnetic configuration and eight central motional Stark effect (MSE) polarimetry measurements 1201 using the fitting code EFIT [21]. Electron temperature and density profiles are measured by Thomson scattering [22]. Ion temperamre and toroidal rotation speed profiles are obtained from charge exchange recombination spectroscopy, utilizing one of the heating beams [23]. The neutron flux is measured with plastic and ZnS scintillators [24] that are cross-calibrated to a set of absolutely calibrated neutron counters [25]. Fluctuation diagnostics include extensive poloidal and toroidal mays of magnetic probes mounted inside the vacuum vessel (figure 1). Toroidal mode numbers n are obtained from the best fit to the phase differences of a toroidal array of eight probes. Radial information about the fluctuations is obtained from an m a y of soft x-ray detectors (figure l), reflectometer channels [26], and a far-infrared laser scattering diagnostic [27]. vertical 'sausage' at 1663 ms in this contour plot). Figure 3 is typical: the burst cycle is highly variable both in amplitude and period. Even the mode number of the chirping mode differs on successive bursts. Generally, there are two types of chirping modes. One is a 'pure' mode with a single toroidal mode number, such as the burst shown in figure 2. There are also 'multiple' chirping modes, such as the burst shown at 1619 ms in figure 3. In a multiple chirping mode, the n = 1 amplitude is initially dominant (figure 4). As the burst evolves, the magnetic waveform becomes increasidgly distorted (from a sinusoidal shape) and higher harmonics appear in the spectrum. Then. as the burst begins to decay, something unusual happens. The n = 1 mode decays more rapidly in amplitude than its 'harmonics' so that, at the end of the burst, only the higher mode numbers remain (cf the n = 4 mode at 1685 ms in figures 3 and 4). In the example of figure 4, the ratio of n = 4 amplitude to n = 1 amplitude is only &/E, = 0.02 at the peak of the instability, but = 4 at the time of maximum amplitude of the n = 4 mode. Apparently an initially unstable n = 1 mode excites an n = 4 mode, which then decays more slowly than the 'fundamental' instability. Another example of a multiple mode appears in figure.5..
For both pure and multiple chirping modes the frequency drops a factor of two during a burst. The observed mode frequency is affected by the Doppler shift [28]. Approximately, the frequency in the laboratory frame fiub is related to the frequency in the plasma frame where n is the toroidal mode number. Because of the large angular momentum of the background plasma, it is unlikely that the plasma rotation profile ha(r) changes appreciably during a burst. The rotation profile is sheared, however, so fro&) decreases with increasing minor radius. Thus, the frequency could drop for one of two reasons: fpl may decrease or the mode may shift to larger radius where the Doppler shift nfml is smaller.
The available measurements of the spatial structure are somewhat inconclusive, but suggest little shift to larger radii. In a pure chirping mode, the toroidal mode number remains constant throughout a burst. Reflectometer channels detect the instability in the scrapeoff plasma (possibly due to fast ions that are expelled from the plasma). The far-infrared laser scattering diagnostic also detects the chirping instability, but the spatial resolution is too coarse to detect a shift in spatial structure. The fluctuations measured by six soft x-ray channels during a multiple chirping mode are shown in figure 5. Because the soft x-ray measurement is a chordal measurement, edge fluctuations may contribute to all of the signals. Nevertheless, it is apparent that the instability has a substantial amplitude in the  interior of the plasma. There may be some shift outward in radius as the burst evolves, but the evolution of the various signals is fiirly similar. The poloidal m a y of magnetic probes (figure 1) suggests the spatial sbtllcture barely changes during a burst. Figure 6 shows the data at three times during the multiple burst of figure 5. No significant alteration in strncture is detected. However, since the probe signals are most sensitive to fluctuations at the plasma edge, these data do not exclude the possibility of internal rearrangement. Similar probe results are obtained for pure chirping modes.
The chirping modes appear to satisfy two separate conditions simultaneously: the laboratory frequency fiab is comparable to the circulation frequency of the beam ions fcin = q / q 2~R ; and the frequency in the plasma frame fpl is nearly zero. To study the chirping modes, a database of 21 pure chirping modes, three multiple chirping modes, and six BAE modes was assembled. A representative sample of chirping modes from all six discharges that exhibit the instability was selected. One discharge on this day (shot 81410) had parameters similar to the discharges with chirping modes, but exhibited BAE activity instead. This discharge and the five discharges in the toroidal field scan of [I51 were selected as representative of plasmas with BAE activity. instabilities was obtained from the discharges described in [29].
In contrast to BAE modes, the modes with chirping frequencies tend to be stationary in the plasma frame ( figure 7). The ratio fiab/&(o) scales linearly with n ( figure 7(a)), suggesting that the laboratory frequency is determined primarily by the Doppler shift ( j a b -nLOt = fpI N 0). Indeed, the ratio j&,/(fZfot(o)) E 0.8 is nearly constant ( figure 7(b)), as expected for a mode that is stationary near the centre of the plasma. Both the frequency at maximum amplitude fg-(figure 7) and the frequency at the beginning of the burst scale linearly with nfrot(0). In all cases, the laboratory frequency is comparable to the beam circulation frequency f~i~, but the dependence is weak and, after taking into account the Doppler shift, there is no dependence of fiab/(nfmt(O)) on BT, 40, or 495 (the q at the flux surface that encloses 95% of the flux). Pure chirping modes with n = 3-8 are observed.
The chirping frequencies fall below the Alfven continuum of ideal MHD (figure 8), in the same gap where BAE modes are observed 1151. Although the frequencies in the laboratory frame are comparable to BAE modes, the frequency in the plasma frame is lower, because the chirping modes occur in plasmas that are rotating rapidly (compare figure 4 of [I51 with figure 8). The initial laboratory~frequency is comparable to the circulation frequency of~the beam ions f & = qjq2irR in the plasma centre. Interestingly, fiab = fCjm at approximately the same radius that fpl Y 0. As the burst evolves, fiab decreases, so the radius where fhb = fcjrc, increases, but the rotation profile is such that fpl N 0 at this position too (figure 8). In contrast, for BAE modes, fpl > 0 everywhere in the plasma.
Apparently, three conditions are required for chirping instability. F k t , the rotation frequency must be sufficiently large that the conditions fpl N 0 and fiabfci, are simultaneously satisfied. The most obvious difference in plasma parameters between plasmas with chirping modes and those with BAE activity is that frOt(O) is larger in discharges with chirping activity ( figure 7(b)). A second requirement is a large beam ion population ((Br) 2 l%), a condition that is also required for Alfvtn activity (fimre 9).

A possible third condition is a low value of u~~/ u A .
Theoretically, the fast-ion drive for TAE modes is an order of magnitude larger for U I I > UA than for u A / 3 < U I I < U A [311.
Finite-orbit width effects broaden this sharp resonance [32]. Empirically, in DIII-D plasmas with Zp N 0. The neutron, infrared camera, and MSE observations confirm that chirping modes cause the loss of some beam ions from the plasma. For pure chirping modes, the magnitude of the losses scales roughly linearly with the mode amplitude (figure 11). No significant dependence on toroidal mode number n or toroidal field (or q ) is observed. The correlation with B is similar to the correlation with E.
For a given amplitude at the edge, the chirping modes cause much smaller losses of beam ions than BAE or TAE modes but much larger losses than fishbone modes ( figure 11).
Within a burst, the loss rate is greatest when the mode amplitude is largest. For the bursts in our database, the maximum gradient in the neutron emission -dZ,/dt lags the peak amplitude of the burst Bm, by 0.07 f 0.20 ms. Similar behaviour was observed for fishbone bursts 1171 and TAE bursts 1341. Concurrently, the mode frequency drops most rapidly when the mode amplitude peaks. (The maximum value of -dfi,b/dt leads Bmu by 0.14f0.15 ms.) This is also similar to the temporal evolution of a fishbone burst [7,17,19].

Discussion
The experimentally observed chirping modes may be related to the energetic-particle continuum mode discussed by Tsai and Chen 1351 in their paper about kinetic ballooning modes. Tsai and Chen predict a mode frequencyof 0.83q/qZnR, which is comparable to the experimental observations (figure 8). They also predict a threshold condition for instability where czf = -q2Ro dgf/dr is propoaional to the fast-ion pressure gradient and s = rq'/q is the s h q parameter. Because of the relatively large value of BF far -0.2), the relatively low value of UII/UA (-0.3), and the modest value of s (-0.5) in these shaped DIU-D plasmas, equation ( 2 ) is satisfied in the plasma interior for the discharges with chirping modes. Qualitative considerations are also consistent with theory. One expects beam modes to occur at low values of u~~/ u A .
where the interaction with AlfvCn modes is relatively weak, and this is the experimental observation ( figure 9). The modes occur in plasmas where ,3~ is a significant fraction (2 lj3) of the total f3, as expected for an electromagnetic beam mode.
The theory of [35] neglects plasma rotation, but this is clearly important in the experiment ( figure 7). Experimentally, chirping modes seem to select values of n and fiilb that cause the conditions fpl N 0 and flabfeirc to be satisfied simultaneously. We speculate that the first condition minimizes the energy required to excite the mode, while the second condition is necessary to extract energy from the fast-ion population. Presumably, the second condition is less stringent than the first because the fast-ion distribution function spans a wide range in parallel velocity.
It is not clear why the mode frequency drops so dramatically. Figure 8 suggests that the mode 'tracks' a resonant population of fast ions as they move outward radially. (This idea was first proposed to explain the drop in frequency at a fishbone burst [4].) The chirping modes may act as a naturally occurring 'bucket' in phase space that transports beam ions to the plasma edgc 136,371. Alternatively, the resonant population may lose energy, causing fciro to decrease in time at a fixed spatial location. A third possibility is that, as the mode amplitude grows, magnetic interaction with the vessel or with uncorrected error fields exerts a torque on the mode, causing it to slow down.
The time evolution of the multiple chirping modes is particularly intriguing (figure 4). A possible explanation for the transition from n = 1 activity early in the burst to higher n activity later in the burst is that, as the ti = 1 frequency drops below -20 kHz, it becomes too small to interact effectively with the beam-ion population (i.e. fiab 1: nfrot becomes smaller than for n = 1). At this frequency, the n = 3 or n = 4 'harmonics' (at 60 or 80 Wz) resonate more strongly with the fast ions, so the higher harmonics persist after the n = 1 mode has decayed away. For a given mode amplitude, the chirping modes are much less effective in transporting beam ions than Alfv6n instabilities (figure 11). possibly suggesting a weaker resonant interaction. This may explain why Alfvin modes are normally observed in pla$mas with large values of pf, while chirping modes are a rarity.

Conclusion
Modes with frequencies that drop a factor of two during a burst are observed in the DIII-D tokamak. Many toroidal mode numbers occur (n = 1-8). The modes are nearly stationary in the plasma frame, but have frequencies that allow for resonant interaction with the circulating beam ions. The instabilitiescause fast-ion loss. The modes are rare compared to the fluid Alfvin modes, apparently requiring a combination of large p h m a rotation, large fast-ion beta, and low Katio of beam speed to Alfvin speed. The chirping modes seem to represent a 'beam' branch that lies in the frequency gap below the Alfvin continuum. A complete theoretical treatment of these modes will include the effects of plasma rotation.