SPONTANEOUS GENERATION OF HOLES AND CLUMPS IN ENERGETIC PARTICLE DISTRIBUTION BEYOND THE EXPLOSIVE SCENARIO AND FREQUENCY-SWEEPING MODES
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It was noted above that the cubic non-linearity in (8.16) is insufficient for describing the explosive scenario. In a long-time fully non-linear modelling [8.4], long-living holes and clamps in the energetic ion distribution were observed for the first time. Then, to accommodate all types of the effective collisionality restoring the distribution function, a fully non-linear ID model ВОТ was developed for the ВОТ instability including the effects of dynamical friction (drag) and velocity space diffusion on the energetic particles driving the wave. The ВОТ results show that in the early non-linear phase of the instability, the drag facilitates the explosive scenario of the wave evolution. Later, the drag effect leads to the creation of phase space holes and clumps moving away from the original eigenfrequency. The combined effect of drag and diffusion produces a diverse range of non-linear scenarios, including “hooked” frequency chirping and undulating regimes.
Fully Non-linear 1D Bump-on-Tail (ВОТ) Model for Long- Time Non-linear Scenarios beyond the Explosive Phase
The fully non-linear system consists of a purely electrostatic wave in a plasma of three species. The first two are the thermal plasma ions and electrons and the third is a low density population of fast electrons that are subject to weak collisions (much less than the background species) and whose distribution function is treated kinetically.
In the normalised units of the Berk-Breizman theory, the starting set of coupled non-linear equations has the form [8.6]:
where the collisionality terms are given by (8.8), ^ = kx- cot, and u> = e kE jme is the non-linear bounce frequency of the electrons trapped in the field of the wave.
For describing holes and clumps identified in Ref. [8.4], we represent the frequency in the form iо = a)pe 8+a»(r). As the holes and clumps evolve, the electric field becomes a sum of non-interacting BGK modes with time-dependent frequencies, so the envelope takes the form
and the distribution function can be written in a similar manner.
To fully explore the non-linear system (Eq. 8.17 and 8.18), a numerical scheme similar to the one developed in Ref. [8.7] was employed for long-term evolution of the system. More specifically, the Fourier series representation of F in space transformed (8.17) into a set of coupled partial differential equations in t and u. By Fourier transforming in velocity, a set of advection equations was obtained for numerical processing with the ВОТ code. A fixed near-threshold parameter, l/J/ft^O.9 will be used for ВОТ simulations throughout this section, as Figure 8.8. shows a satisfactory agreement with more demanding case of |xrf|/yi=0.99.
Figure 8.9 shows the result of the ВОТ code for long-time evolution of the non-linear system (8.17) and (8.18) in the collision-less limit.
The holes and clumps move away from the original resonance, as shown schematically in Figure 8.10. This motion is almost adiabatic and preserves the value of the distribution function for
FIGURE 8.8 Comparison of the bump-on-tail simulation with the cubic equation (8.16) in the near-threshold regime with diffusion for two near-threshold parameters yd //, .
FIGURE 8.9 Spectrogram of the electric field amplitude £, for the collision-less case close to the threshold. The white line is the best l'n fit passing through the upper and lower frequency-sweeping structures*.
FIGURE 8.10 Cartoon illustrating the motion of holes and clumps and the wake dotted line that steepens the distribution function, creating a favourable environment for instability*.
FIGURE 8.11 Spectrogram of the electric field amplitude showing chirping asymmetry for the pure drag case*.
Reproduced from [M.K. Lilley et al„ Phys. Plasma 17 (2010) 092305]. with the permission of AIP Publishing.
FIGURE 8.12 Spectrogram of the electric field amplitude showing the “hooked” frequency spectrum with drag and diffusion.
the particles trapped by the wave. We note that within this simplified theory there is no limit to the extent of the chirp, which is indeed supported by the simulation. The chirping behaviour shows the correct t112 scaling [8.4].
The drag introduces a preferred direction of particle flow into the system (from high-energy source to low-energy sink). Consequently, one can expect the hole-clump symmetry observed in Figure 8.9 to be broken when drag is introduced, which is indeed the case as Figure 8.11 shows.
In the general case of both drag and diffusion affecting the distribution function at the resonance, more sophisticated patterns of the frequency sweeping are seen. One of this is the “hooked” frequency sweep shown in Figure 8.12.
Non-linear Frequency-Sweeping Scenarios Observed Experimentally for TAEs Excited with Energetic Ions Resulted from the Drag Relaxation
Among all machines exhibiting the frequency-sweeping spectra, spherical tokamaks (START, MAST, NSTX) with super-Alfvenic NBI are the best test beds for investigating such phenomena, as discussed in Section 7.3.
The exceptional conditions on STs are determined by the low magnetic fields of STs and low values of T„ at the beginning of NBI, so that the following ordering is valid for the beam ions:
where V^t is the beam speed corresponding to the critical energy (3.3), and Vu is the initial speed of the injected beam. Under the conditions (8.20), the beam passes through the principal resonance V||(, = VA because of the beam slowing-down due to the electron drag. Figure 8.13 schematically shows that a short blip of NBI moves first uni-directionally from the injection high-energy range to low energy due to the drag, without changing its shape because the diffusion effect is small. Only at the low beam energy comparable to £clit, the shape of the beam blip diffuses, and this diffusion is not uni-directional anymore. Because the resonance area E^ comes in this case to the region of the dominant beam relaxation due to the drag, we have a perfect drag scenario similar to that investigated in the ID ВОТ model. Frequency-sweeping modes associated with the hole-clump generation are inevitable in such a scenario, which is what we observe in STs with super-Alfvenic NBI.
Figure 8.14 shows magnetic spectrogram from MAST experiment, in which super-Alfvenic NBI drives Alfven instability when the resonance Vj|tKam = VA is in phase space region dominated by electron drag of the beam ions. It is seen that FS modes dominate the spectrum, with some modes sweeping in frequency to a very long range of 8(i> / ft)| = 0.5. Modelling with the HAGIS code [8.8]
FIGURE 8.13 Schematic illustration of the beam blip evolution with time in the case of ordering (8.20) typical for STs.
FIGURE 8.14 Spectrogram showing FS Alfven modes driven by NBI in MAST discharge #27177.
FIGURE 8.15 Non-linear HAGIS simulation of Alfven instability in MAST #27177.
FIGURE 8.16 Top: Mirnov coil data; Middle: magnetic spectrogram; Bottom: Time evolution of tangential energetic neutral spectrum. The NPA viewing angle is set to 0°.
and with the beam pure drag-like relaxation model installed was performed to compare non-linear wave-particle dynamics with the experimentally observed in Figure 8.14. Figure 8.15 shows that the HAGIS modelling reproduces pretty well the characteristic spectrum observed in experiments, although the range of the frequency sweeping is not as large as this observed on MAST.
Finally, we note that the dominant transport mechanism for non-linear FS modes is convection of particles trapped in the wave field. Experimentally, the hole-clump formation and transport were observed for the first time with an NPA diagnostic on stellarator LHD [8.9]. Figure 8.16 shows how the flux of energetic beam ions sweeps in energy together with the chirping modes.