Page 7 - Characterization and calibration of the Thomson scattering diagnostic suite for the C-2W field-reversed configuration experiment

Nucl. Fusion 58 (2018) 126026
profiles derived from actual diagnostic data of shot 43628 [7]
are utilized.
3.2. Comparison with existing theories
Although the FRC operation regime with comparable fast ion and bulk plasma pressure has never been achieved exper- imentally in the past, the stability of this scenario has been analysed theoretically before. Early theory based on an energy principle has predicted that modes with phase velocity parallel to the ion ring current are destabilized resonantly [13]. This directional condition is consistent with the observed micro- burst mode rotation direction. Recently, the fast ion-plasma resonance condition in small S* (ratio of plasma radius over plasma ion skin depth c/ωpi) kinetic FRCs is derived analyti- cally as nΩ − ωr = ω, where Ω and ωr are the fast ion toroidal frequency and the radial bounce frequency respectively, n is the mode number, and ω is the mode frequency [14]. The left- hand side of the resonance condition can be evaluated from the particle orbit calculation shown in figure 15. In a longer time scale than the ion cyclotron motion the fast ion orbit motion in the azimuthal direction can be regarded as the pre- cession rotation of a n = 2 structure (peanut- shaped structure to be exact) with a precession frequency Ωn. For a toroidal mode number of 2, nΩ − ωr can be interpreted as twice the precession frequency Ωn. This is easily seen by noticing that the time it takes the ellipse to rotate the angle of BOC (figure 15) is dt = 2(2π/ωr) and the total particle azimuthal rotation is Ωdt = 2π + ∠BOC, and Ωn = ∠BOC/dt.
As mentioned above, the measured micro-bursts have the elliptical n = 2 mode structure which matches the structure of the orbit precession, therefore, it is probably the precession mode predicted by Finn and Sudan [15]. The stability condi- tion from energy principle is that the mode frequency be less than n times the bulk plasma rigid rotation frequency, which is approximately the ion diamagnetic rotation frequency. This can explain the observation that micro-burst amplitude starts to decay when the frequency down chirps to fp, and fp is about twice the bulk plasma rigid rotation frequency as shown in figure 6. Also the theory predicts that the axial wave number is zero for the unstable mode, consistent with the data shown in figure 11.
To understand the beam energy dependence of micro- bursts, the fast ion orbit azimuthal precession frequency is plotted in figure 16. It depends sensitively on the fast ion energy perpendicular to the magnetic field (E⊥) and less sen- sitively on the particle angular momentum (represented by the ratio of the minimum to the maximum orbit radius, ε). The measured micro-burst unstable frequency range is between the two horizontal dashed lines in figure 16. It is seen that the instability condition is best met for particles with E⊥ between 12 and 15keV in the C-2U equilibrium, as their precession frequency best matches the unstable frequency range with most severe n = 2 deformation (small ε). This is probably why maximum micro-burst activity is observed in this energy range. For higher beam energy the precession mode may be unstable, however the orbit is more circular, and therefore the growth rate is smaller. This can be seen from figure 15, where
B.H. Deng et al
Figure 16. Calculated fast ion orbit azimuthal precession frequency 2Ω − ωr in C-2U equilibrium magnetic field as a function of the orbit ellipticity ε. The perpendicular beam particle energy (E⊥)
is denoted above each curve. The shaded region between the horizontal dashed lines indicates the measured unstable micro-burst frequency range.
the perturbed radial beam current (δjb) due to orbit ellipticity will enhance the elliptical deformation. When the orbit is more circular at higher beam energy, δjb is smaller, and there- fore, less instability. For small beam energy when the preces- sion frequency is below 2 times the plasma rotation frequency the mode is stable.
3.3. About the dynamics of micro-bursts
In the previous sub-section, the growth and decay and the spatial mode structure of micro-burst are explained by the theory of precession mode, and the stability condition agrees quantitatively with [15]. However, there is no existing theory that can resolve the detailed nonlinear micro-burst process, for examples, the frequency down chirping (figure 3), and the observation that higher amplitude burst leads to propor- tionally longer time interval to the next burst (figure 2). One heuristic model to understand the onset of the instability is shown in figure 15. With the accumulation of fast ion popula- tion, they will partly take over the role from bulk plasma ions as the current carriers. According to Lenz’s law, an inductive electric field in the opposite direction of the plasma current will be induced to slow down fast ions, which will force the fast ion orbits to phase bunch, showing organized collective orbits. Due to the elliptical structure of the fast ion orbits, each fast ion will carry radial current component as it bounces radi- ally. The net radial current is zero when the phase between each fast ion orbit is randomly distributed. When the fast ion orbits are phase bunched, net radial current results. As shown in figure 15, the Lorentz force due to this net radial current can lead to the enhancement of elliptical deformation, thus instability occurs. The inductive electric field will also lead to slowing down of fast ion orbit azimuthal precession and frequency down-chirping, as observed (figure 3). When the fast ion orbit precession frequency is down to the bulk plasma rotation frequency, the fast ions will resonantly pass on the angular momentum to the bulk plasma, leading to the decay phase of the micro-bursts. If the burst amplitude is small, less fast ions are scattered out of resonance, and the next micro- burst will occur sooner; while when the burst amplitude is
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