Non-linear Scenarios of the Mode Evolution Described by (8.16)
One of the most important cases to consider in numerical modelling of (8.16) is when an effective diffusion replenishes the unstable distribution of energetic particles. In this case, (8.16) has no drag or Krook effects, «=/?=0, and the non-linear solutions are only determined by the diffusion term that depends on v. Figure 8.2 shows various solutions of (8.16) for the mode amplitude as the effective diffusion collisionality v changes. The amplitude solutions shown are the envelopes of the high- frequency waves, with the characteristic times determined by the inverse growth rate of the mode.
FIGURE 8.2 Numerical solutions for |A| of Eq. (8.16) with diffusion only for different values of effective normalised collisionality v. Elere, t = yt.
We can see from Figure 8.2 that for high values of v, v=6.69 and ^=2.71, non-linear solutions are established with a steady-state mode amplitude after the initial linear exponential growth and some transient oscillatory phase. The steady-state solutions are expected as there is a “minus” sign in front of the non-linear term, which tends to compensate the linear growth rate term (first in the right-hand-side of (8.16)). The “history exponent” provides a very narrow time window at high v, so the time integration should not cause surprises.
For lower values of u, v = 2.03 and v=2, the non-linear solution exhibits amplitude modulation [8.2]. Such a scenario is much less obvious than the steady-state one. The oscillatory character of the mode evolution means that the non-linear term in (8.16) changes its sign quasi-periodically with time. This becomes possible because the “history exponent” at lower v significantly expands the time window, and the product of three complex amplitudes taken at different times, which can generate both positive and negative values, becomes essential in the time integral.
We can also see from Figure 8.2 that at v=2 the amplitude modulation becomes very deep, that is, comparable to the amplitude itself. This causes the non-linear system to go through the zero amplitudes because of two different mechanisms: the high-frequency solution has zeroes itself, and the envelope in the form of the amplitude modulation provides another set of zeroes. Under these conditions, the non-linear system mixes the phase information around the zero values of the amplitude and picks up the phase randomly, as shown in Figure 8.3. This results in a “chaotic” mode evolution [8.3] that starts at v<2 and becomes very well-determined at v= 1.44 (see Figure 8.2).
For yet lower value of the diffusion coefficient, v= 1.24, the solution becomes explosive, that is, the mode amplitude becomes infinite during finite time, as shown in Figure 8.2. This means that the theory limited by the low-order cubic nonlinearity (8.16) becomes insufficient, and fully non-linear approach should be employed for describing the wave-particle coupled system. It was found in Ref. [8.4] that a fully non-linear approach gives, beyond the explosive scenario, a spontaneous creation of frequency-sweeping holes and clumps in the energetic particle distribution (to be discussed later).
In summary, four essentially different non-linear scenarios were found for (8.16) controlled by the parameter of the diffusive collisionality v
b. periodically modulated;
Another important case to consider in numerical modelling of (8.16) is when drag replenishes the unstable distribution of energetic particles. In this case, Eq. (8.16) has no diffusion or Krook effects, v=/]=0, and the non-linear solutions are only determined by the drag term that depends on a.
Only explosive solutions of (8.16) were found in this case [8.5] in contrast to the diffusion. The inclusion of a drag term introduces an oscillatory dependence to the integral in (8.16), instead of
FIGURE 8.3 Zoom of numerical solution of (8.16) for wave with a deep amplitude modulation. The phase flip is seen leading to a random phase pick-up at the maximum of the modulation causing chaotic mode evolution on a longer time scale.
FIGURE 8.4 Displays the boundaries in parameter space that give stable, unstable, and no steady-state solutions to (8.16) with drag and diffusion. The unstable solution lies in between the solid and dashed lines.
the exponent in the case of diffusion. The oscillatory dependence has a profound effect on the nonlinear evolution of the mode amplitude as the integral in (8.16) can easily change sign. The contributions to the integral from the time intervals with the sign “plus” exceed those with the sign “minus” so that the amplitude cannot saturate. Further description of the explosive scenario requires a fully non-linear approach.
In the presence of both drag and diffusion, Eq. (8.16) exhibits a variety of non-linear scenarios. Figure 8.4 shows the summary of non-linear scenarios found within the framework of (8.16) with the drag and diffusion effective collisionality terms.
Non-linear Scenarios Observed Experimentally for TAEs Excited with Energetic Tail Ions Resulted from the RF-Diffusion
TAEs excited by ICRH-accelerated ions is the best case for demonstrating the effect of a diffusive-type replenishment of the energetic ion distribution on the non-linear evolution of TAE. The energetic ion tail in ICRH case is built up by RF-diffusion given by Eq. (3.8), which has an effective frequency v replenishing the distribution about ten times higher than that of Coulomb collisions [8.2]. All but one explosive non-linear scenarios of wave-particle interaction predicted by the Berk-Breizman near-threshold theory were validated experimentally in plasmas of JET.
In specific JET experiment [8.3], TAEs were excited by energetic H-minority tail ions accelerated by ICRH with gradually increasing ICRH power, as shown in Figure 8.5 (top). Due to the increase in ICRH power, the associated net growth rate of TAE makes the mode to go through the sequence of the non-linear saturated scenarios displayed in Figure 8.5 (bottom). First, we see a steady-state saturation of the TAE amplitude in the magnetic spectrogram. Then, from ~44.28 to ~44.4s, the fine “pitchfork” splitting of the TAE spectral lines is observed. Finally, the modes go into the “chaotic” non-linear scenario, with very noisy spectral lines, widths of which remain the same as the “pitchfork” splitted TAE widths.
During the “chaotic” phase of the TAE excited, the phase analysis becomes complicated for measuring phase shift between the signals seen in toroidally separated Mirnov coils for determining toroidal mode numbers n. This phenomenon shown in Figure 8.6 results from the phase flips illustrated in Figure 8.3.
The different non-linear scenarios observed in Figure 8.5 follow one-by-one in time showing each of the four scenarios during relevant time intervals. Figure 8.7 shows the raw data from Mirnov coils corresponding to these time intervals in the magnetic spectrogram.
FIGURE 8.5 Top: the wave-forms of ICRH and NBI auxiliary heating power in JET discharge #49447. Bottom: Amplitude spectrogram of magnetic fluctuations showing the various non-linear scenarios of TAE evolution. The TAEs have toroidal mode numbers ranging from n = 3 to 6.
FIGURE 8.6 Phase magnetic spectrogram for determining toroidal mode numbers of TAEs in JET pulse #49447. The phase analysis becomes difficult during the chaotic TAE evolution.
A direct comparison between Figures 8.5 and 8.7 and the theoretical Figure 8.2 shows a remarkable similarity in the types of the non-linear evolution of the modes, one of which is driven by the ВОТ in the Berk-Breizman theory, and the other one is TAE driven by ICRH-accelerated ions in JET. Such a similarity could be explained by the structure of the wave-particle resonance for TAE and ICRFI-accelerated ions. Because ICRH generates energetic trapped ions with their banana tips at the vertical line co=ojbh(R), all such ions have their pitch-angles A=const (A=l for on-axis ICRH). The wave-particle resonance at a fixed value of A takes the form of a ID line in the (£,
FIGURE 8.7 Raw data for JET pulse #49447 showing amplitude of TAE magnetic perturbations for different time intervals corresponding to TAE non-linear scenario of (a) steady-state type, (b) the amplitude modulation, (c) deep amplitude modulation, and (d) chaotic amplitude evolution.
Pv)-space, and the physics of the resonant interaction is determined entirely by the motion of energetic particles across the resonance. This mechanism is similar to the ID Vlasov ВОТ problem solved in the Berk-Breizman theory.
The explosive scenario was not achieved in the experiment shown in Figures 8.5-8.7. This is difficult to achieve with ICRH-driven TAEs dominated by the RF-diffusion at the resonance due to a strong RF-driven pitch angle scattering. However, the explosive scenario is much easier to obtain for TAEs driven by NBI-produced energetic ions when a dominant drag replenishment of the beam distribution is achieved.