RADIOFREQUENCY WAVES, HEATING AND CURRENT DRIVE IN MAGNETICALLY CONFINED PLASMAS
6.2. THEORY OF RF WAVE PROPAGATION IN A MAGNETIZED PLASMA The theory of wave propagation in magnetized plasmas has been
6.3.1. ICRF heating and current drive experiments
The declassification of fusion research at the Geneva Conference in 1958 opened up information from the Princeton Stellarator RF heating programme of Stix and coworkers in the late 1950s (1958 and beyond). Simultaneously, Stix was preparing his famous text “The Theory of Plasma Waves”, which was published in 1962 [6.2]. This early work concentrated on understanding wave propagation and absorption in the “magnetic beach” geometry, which was then applied to some of the smaller stellarators (the B-65 device) in Princeton with various degrees of success by Stix and coworkers (Fig. 6.17).
The idea was that shear Alfvén waves with dispersion relationship v (1A / ci)1/2
k propagating at frequencies below the ion cyclotron frequency by a few tens of per cent would propagate into a region with lower magnetic field (the “magnetic beach”) and would dramatically increase their wavenumber, thus experiencing strong ion cyclotron absorption at 1 / ci 0, thereby transferring the wave energy to kinetic particle energy. This ingenious idea based on propagation of the left hand polarized slow Alfvén wave and its absorption at the ion cyclotron frequency (Fig. 6.18) was effective and produced a strong enhancement of neutrons at the resonant field [6.107]. This work was
followed by more extensive studies by Hooke and coworkers on the B-66 device with longer scale lengths, better suited to basic studies of wave propagation and absorption satisfying the WKB conditions on wave propagation [6.108]. Here the dispersion relationship and wave damping decrement were verified in detail, laying a solid foundation to the theory of “slow wave” propagation in the Alfvén wave regime in relatively small plasma columns. However, this wave had to be abandoned in future larger tokamak plasmas that followed in the 1970s and 1980s, in particular PLT in Princeton and TFR in France, since launching the slow wave efficiently became impractical in higher density plasmas (ne10 m19 –3) owing to the high values of Nrequired. Thus, exploration of fast wave heating started, initially at the deuterium harmonic cyclotron frequency, as well as at the ion-ion hybrid frequency.
FIG. 6.18. Strong neutron enhancement followed the resonance condition as the magnetic field was varied. Taken from [6.107]. Reprinted from Ref. [6.107]. Copyright (2011), American Institute of Physics.
FIG. 6.17. The B-65 stellarator and an early version of the Stix coil [6.107]. Reprinted from Ref. [6.107]. Copyright (2011), American Institute of Physics.
It was discovered early on by experiment and theory that ion cyclotron absorption in a single ion species plasma would not work due to the nearly complete right hand polarization of the fast wave at the fundamental cyclotron resonance, at least in the relatively cold (~1 keV) initially ohmically heated plasma. One of the earliest theoretical treatments of this picture discussed mode conversion of wave energy into IBWs, and electron heating, both for low and high field side launch, to explain the experiments in the early TFR tokamak and PLT results [6.41]. Some of the early results are depicted in Fig. 6.19 from the TFR tokamak, showing the effectiveness of heating and its spatial profile as a function of minority concentration. The heating is maximum around 25%
minority concentration when the ion-ion hybrid (mode conversion) layer is located near the axis of the plasma radius at R = 98 cm. This is in agreement with theoretical predictions given earlier. Unfortunately, only ions were measured and it is likely that electron heating was more substantial.
Fig. 6.19. (a) Profiles of the ion hybrid (mode conversion) layer in the TFR tokamak; (b) heating effectiveness as a function of nH/nD ratio and radius with Bϕ = 4.6 T. Here HYB is the ion-ion hybrid frequency, Ppl is the RF power absorbed by the plasma and TDO is the change in the central deuterium temperature. Taken from Ref. [6.109].
When carrying out experiments in “pure” deuterium plasmas at the deuterium cyclotron harmonic, better heating was achieved than one would have expected from theory. As it turns out, the actual absorption mechanism was minority absorption on the H minority species, since a few per cent of H is always present in deuterium plasmas. This was discovered by charge to mass resolved data from the escaping neutral atoms in the early TM-1-VCH device in the Soviet Union [6.110]. Energetic ion tails were also discovered in the early ATC experiments in the USA, but the H minority ion species was not
identified [6.111]. To optimize minority heating one has to reduce the minority species (H in this case) concentration to below 10% so as to move the mode conversion layer near the H minority resonance layer within the Doppler width
vth H, / cH
r k R w
D = , where cH is the hydrogen cyclotron frequency. At higher concentrations the mode conversion layer moves far away from the cyclotron resonance layer and mode conversion into IBWs, and electron Landau absorption would dominate. However, because of reflection from the L cut-off layer, single pass absorption efficiency by mode conversion is not as efficient as minority absorption, which can be 90% or better in even medium size tokamaks. The kinetic theory of minority species absorption that was formulated by Stix [6.43]
was verified first in the elegant PLT experiments with charge to mass discrimi- nation by Hosea and co-workers [6.112] (see Fig. 6.20).
FIG. 6.20. Measured distribution of energetic minority ion species in the PLT tokamak and comparison with the Stix theory [6.112]. Here Fh is the hydrogen minority species distribution function.
It is clear that for efficient bulk plasma heating, the energetic ions have to be well confined until they slow down by collisions with the bulk plasma.
Both ions and electrons are heated, and the slower particles heat ions while the faster ones heat electrons. While in the early experiments (C-stellarator, ST tokamak, etc.) energetic particle confinement was poor, in modern tokamaks (PLT, etc.) with at least 0.5 MA of plasma current the energetic particles were well confined, and therefore we would expect efficient heating of the plasma.
This fundamental understanding was followed by impressive heating results in PLT and later TFTR, JT-60 and JET. One of the earliest multi-MW, multi-keV experiments was carried out on the PLT tokamak and the results are illustrated in Fig. 6.21 [6.113] and Fig. 6.22 [6.114] for the minority fundamental and majority second harmonic heating, respectively. Note the substantial second harmonic ion
tail in a pure H plasma in order to avoid mix up with minority heating in the earlier experiments [6.111]. A review of these early experiments has been given by Porkolab [6.1] and Colestock [6.115]. Because of the complexity of these heating regimes, historically the name ICRF or ion cyclotron resonance range of frequencies was adopted. Later, efficient beam ion acceleration was demonstrated in combined NBI and 2nd and 3rd harmonic ICRF injection experiments in JT-60 [6.116].
FIG. 6.21. Ion heating results from PLT with 3He minority heating in a D plasma with B = 3.2 T, Ip = 600 kA, ne = 3.7 × 1019 m–3, PRF = 4.3MW [6.113].
We should briefly mention here attempts to heat plasmas with directly launched IBWs in both PLT [6.117] and Alcator-C [6.118, 6.119]. While there was difficulty to inject large amounts of power owing to the parallel antenna straps required to launch E,which ultimately resulted in significant impurity injection, there was clear evidence of particle confinement improvement (by factors up to 3 in Alcator-C) and peaking of density profiles. In today’s terminology this may be interpreted as formation of a thermal barrier, at least for particles (which also led to impurity accumulation). While concomitant ion heating was also observed, this was typically limited at most to 30% of the initial temperature. The results were hard to understand and there was strong evidence that non-linear phenomena played an important role [6.120]. A large scale experiment at the MW level on DIII-D [6.121] failed to produce the expected core plasma heating and there was strong evidence of edge parametric decay phenomena, and these experiments were abandoned in the USA. However, there was a continuing programme on FT-U in Frascati at high frequencies (0.5 GHz) to reduce non-linear effects, with some success.
FIG. 6.22. Proton energy distribution in the presence of second harmonic heating in PLT.
“Before” means before application of the ICRF power, and “During” means during application of the high power RF. B = 1.4 T, PRF = 2.8 MW [6.114]. Reprinted from Ref. [6.114]. Copyright (2011) by the American Physical Society.
Following the pioneering work in the 1980s, 1990s and 2000s, large amounts of ICRF power were installed in JET and TFTR, to test burning plasma relevant heating scenarios, including startup plasmas. The results were excellent and a large number of publications resulted (see Refs [6.122] for TFTR and [6.123, 6.124] for JET and related references). Some of these results included the following: second harmonic tritium heating was demonstrated in TFTR [6.122], and fundamental D minority, 3He minority, fundamental T minority and T majority second harmonic heating, were all demonstrated in JET [6.123] (see Fig. 6.23).
At the same time it was shown that the RF driven fusion power with 6 MW of ICRF was comparable to that achieved with NBI power. Other references from recent research in JT-60U, ASDEx-Upgrade and Alcator C-Mod may be found in Refs [6.125, 6.126] and [6.79], respectively. For example, in JT-60, 5 MW of ICRF power was coupled successfully in a 4 MA hot ion H-mode plasma with 0.15 m antenna–separatrix gap, which meets ITER requirements [6.125].
We note that in a two ion species plasma mode conversion current drive up to 130 kA was demonstrated in TFTR [6.127]. Effective mode conversion heating was also found in ASDEx-Upgrade [6.126] and Alcator C-Mod [6.128]. While the heating was believed to be due to IBW mode conversion, as shown by theory off-axis, the magnetic shear significantly up-shifts the launched Nspectrum and coupling to kinetic ion cyclotron waves dominates IBW [6.44]. This effect was discovered in Alcator C-Mod by phase contrast imaging diagnostics that verified the dominant role played by the mode converted ICW, as opposed to the IBW [6.46, 6.129]. Recent experiments also confirmed mode conversion heating in ASDEx-Upgrade [6.126] and Alcator C-Mod [6.128].
FIG. 6.23. Bulk ion heating was achieved on JET with D minority heating in T majority plasmas, yielding Q = 0.22 [6.123].
Other exciting ICRF physics, besides various forms of heating, include fast wave current drive including ICCD, or ion cyclotron current drive, on the JET tokamak [6.130]. Another form of ICRF current drive is due to Landau absorption of the directionally launched fast wave (FW) as demonstrated in DIII-D and Tore-Supra. As shown in Fig. 6.24, in DIII-D the current drive efficiency is in good agreement with theoretical predictions when no other competing absorption mechanism is present [6.131].
FIG. 6.24. The measured fast wave current drive efficiency FW in DIII-D is in excellent agreement with theoretical predictions [6.131]. CURRAY is the ray tracing and Fokker–
Planck combined code that calculates the current drive efficiency. The symbols 1 T and 2 T are the magnetic fields where the experiments were carried out, and are distinguished by the red and blue colours. The green symbols correspond to H mode plasmas.