Non-linear Evolution of Coupled Energetic Particle Populations and Energetic Particle-Driven Modes
GENERIC BERK-BREIZMAN THEORY ON THE NEAR-THRESHOLD WAVE EXCITATION BY ENERGETIC PARTICLES
In a typical plasma heating scenario with energetic ions, a gradual build up of energetic ion pressure occurs, so that the energetic ion drive of an AE, yL, increases in time at unchanged AE damping, yd. At the AE linear instability threshold, an exact balance between the AE drive and damping is achieved,
As soon as the drive overcomes the instability threshold (8.1), the AE gets excited with the net linear growth rate determined by the difference between yL and yd. In a theory developed in Ref. [8.1], excitation scenarios and early non-linear evolution of any energetic particle-driven modes are of a universal type in the near-threshold regime:
In this case, the small net growth rate becomes comparable to the effective collisional frequency restoring the energetic particle distribution,
and the electric field of the perturbation that tends to flatten the distribution function at the resonance competes with the source of the energetic particles that replenishes the unstable distribution function.
The Berk-Breizman near-threshold non-linear theory [8.1] was developed for one-dimensional (ID) bump-on-tail (ВОТ) instability of a single electrostatic mode when this mode gets excited via
Landau resonance due to the positive gradient, — > 0, at the resonance (as Figure 8.1 shows),
between the wave and fast electron population. The mode damping is an essential part of the theory, and the damping rate yd was kept constant in Ref. [8.1] throughout the mode evolution.
The non-linear scenarios [8.1] were found in many various types of wave-particle resonance systems with an excitation threshold, making it clear that the Berk-Breizman theory is very generic. Here, we present the key points of the Berk-Breizman near-threshold theory, and illustrate its validity via some experiment-to-theory comparison examples.
FIGURE 8.1 Schematic illustration of a distribution function of with the bump-on-tail in the high-energy range close to the wave resonance velocity calk.
In order to accommodate the collisions, a ID Fokker-Planck equation is used for describing energetic particle distribution F(x, v, t) coupled to the electric field of the mode
The Fokker-Planck equation to be solved has the form
together with Maxwell’s equations for electric field (8В = 0 for this problem):
where jf is the energetic particle contribution to perturbed current produced by the wave. Only resonant particles contribute to the wave evolution, so the relevant part of the collisional operator can be taken in the vicinity of the resonance as
and «, v, ft are coefficients of drag, diffusion, and the Krook operator, respectively. For the Krook operator, the coefficient is constant, but for drag and diffusion these are taken at the resonant point. The equation with both the perturbed electric field of the wave and the collisions is:
We represent the distribution function as a Fourier series
so that the wave equation relating the field and the fast particle current becomes
Consider time scales shorter than non-linear bounce period of the wave. With the distribution function being not too significantly perturbed, that is, within the ordering
where f admits a power series in E(t)
which allows the first-order (cubic) nonlinearity to be captured by the following truncated Fourier expansion:
The wave amplitude equation near the threshold is obtained then assuming the smallness of the net growth rate (8.2), and it takes the form of cubic non-linear integral equation
Here, the time integration corresponds to the “memory” of the system of coupled energetic particles and the wave because the temporal mode evolution at some time slice depends on the previous mode evolution integrated over the entire time of the instability. In the absence of the drag, «=0, the exponent in (8.16) represents an effective window back in time, during which the information on the history of the mode evolution is essential. The argument of the “history exponent” is determined by v and [5, and it determines how far back in time the history matters for the solution.
For different values and the three different types of the effective collisionality, (8.16) exhibits both “soft” and “hard” early non-linear scenarios of excitation. In the “soft” scenario, the evolution of the unstable system tends to return to the original state by flattening the distribution function in the resonance area. The mode amplitude evolves to a low level reflecting the closeness to the threshold.
In the “hard” excitation scenario, the instability pushes the system further away from the original state, and the mode amplitude “explodes” in a finite time. In this case, the low-order cubic nonlinearity becomes insufficient soon after the mode excitation, and one needs a fully non-linear model for describing the long-term mode evolution. We now consider non-linear solutions of (8.16) in some important cases and illustrate theory-to-experiment comparison.