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
6.2.8. Wave propagation in toroidal geometry, full-wave treatment
In order to simulate wave fields in the ICRF regime it is necessary to employ full-wave electromagnetic field solutions for several reasons. First, the wavelength of the fast magnetosonic wave can be of the order of the device size, which violates the conditions necessary for the validity of the geometric optics approach and ray tracing. Second, in the minority heating scheme that is often used to heat plasma with ICRF power, an incoming fast wave reaches a cyclotron resonance layer where it may be only partially absorbed. The wave can then propagate beyond this layer to a cut-off where the wave can tunnel to a mode conversion layer where it can convert to either a short wavelength ion Bernstein wave (IBW) or ion cyclotron wave (ICW). Ray tracing techniques are conceptually difficult to apply in these cases because it is not straightforward to match the incident fast wave fields to the short wavelength mode converted fields. Instead this process is described using Maxwell’s equations combined with a conductivity relation to form a Helmholtz wave equation
2 2 0
0 p ant
E E i j i j
c
w wà
wε
-∇ × ∇ × + + = -
(6.125) where jant
is the externally driven antenna current and the plasma current density jp
is in general a non-local, non-linear, integral operator on the electric field with a conductivity kernel that is given by
( 0 )
( , ) ( ), , , , ( , )
t
p s
s
j r t dr dtσ f E r r t t E r t
-∞
′ ′ ′ ′ ′ ′
=∑∫ ∫ ⋅
(6.126)
where s refers to the plasma species, either electrons e or ions i, σ is the conductivity operator, r´ is the position, and t´ is time. Full-wave field solvers with varying levels of physics detail have been used to solve Eqs (6.125) and (6.126) [6.85–6.94]. One example of an advanced full-wave solver is the 2-D and 3-D “All Orders Spectral Algorithm” AORSA2D and AORSA3D [6.85], which is spectral in all three dimensions (R, Z, f). Here (R, Z, f) are cylindrical coordinates where ‘R’ lies in the equatorial plane of the tokamak, ‘Z’ is the vertical dimension, and ‘f’ is the toroidal angle. The code includes all cyclotron harmonics in the evaluation of the plasma conductivity and no assumption is made about the perpendicular wavelength () relative to the ion gyroradius (i). The plasma response formulated in this manner leads to an integral relation and the resulting
matrices that must be inverted are large, dense, non-symmetric, indefinite and complex [6.85]. Another sophisticated full-wave solver is the 2-D TORIC code [6.86, 6.87], which employs a mixed spectral, finite element representation for the electric field in ( , , ), where ( , , ) are the usual pseudo-toroidal coordinates. The conductivity relation is truncated at second order in ( / )i 2 and at the second cyclotron harmonic. These assumptions result in a closed form for the plasma response and a matrix system to invert that is sparse and banded with large dense blocks. Both the AORSA and TORIC matrix inversion algorithms take advantage of the resulting dense block structures by using scalable libraries of parallelized linear algebra routines to efficiently invert their matrices.
The semi-spectral basis set used in the TORIC solver has also been employed in the TASK-WM code [6.88, 6.93], which has been used to study ICRF wave propagation in 3-D configurations such as stellarators as well as in 2-D tokamak geometry. Finally, a relatively new approach has been employed by Dumont in the EVE code [6.89], which uses a Hamiltonian description for the particle dynamics in order to evaluate the conductivity operator in terms of action-angle variables. An example of minority hydrogen cyclotron damping simulated with the AORSA2D solver for Alcator C-Mod is shown in Fig. 6.14. The parameters used were (0) 5.4 TB = , nH /ne7% and f0 80 MHz. This calculation would not be possible without the massively parallel architectures required for the global-wave solver. A spatial mesh of 256 × 256 cells in (R, Z) was used in the solver. The need for high spectral resolution even in the case of ICRF heating at low minority concentration where mode conversion is weak can be seen upon examining the left circularly polarized electric field computed by AORSA in Fig. 6.14 left. Mode converted IBW can be seen clearly on the high field side near the midplane and mode converted ICW [6.44] can be seen propagating back towards the low field side above the midplane. Although very little power flows into mode converted waves during minority heating (<10%), the presence of these short wavelength modes does alter the polarization of the incoming fast wave in the mode conversion region. Although the minority cyclotron resonance layer passes through the plasma centre in this case, the ICRF power dissipation is clearly peaked off-axis. This feature can be seen more prominently in Fig. 6.14 right where the 2-D (R, Z) minority power dissipation has been plotted. There is a clear peak in the 2-D absorption along the resonance chord at about 0.07 m above the midplane. The ICRF wave fields shown in Fig. 6.14 left have been used to evaluate the RF quasi-linear diffusion coefficient in the bounce averaged Fokker–Planck code CQL3D [6.80]. In this way a non-thermal distribution for the minority ion species is evolved. The quasi-linear diffusion coefficient was then recomputed by the full-wave solver (AORSA) using the new minority ion distribution function in the plasma conductivity and the process was repeated until convergence was achieved [6.52, 6.95, 6.96].
FIG. 6.14. Left: Electric field contours computed by the AORSA solver for minority hydrogen heating in Alcator C-Mod. Right: Power density contours of hydrogen ion cyclotron absorption from AORSA for the Alcator C-Mod minority heating case. Contour levels follow the rainbow, with red, orange and yellow being the highest values. Taken from Ref. [6.97].
One limitation with this approach is that when the ICRF generated minority ions are highly energetic or when the ICRF waves are interacting with energetic ions from neutral beam injection or fusion processes (alpha particles), the zero ion orbit width assumption used in the Fokker–Planck solver is no longer a valid approximation. A number of simulation models have been developed to address this problem. One such combined model, the SELFO code [6.98], has been used extensively to simulate finite ion orbit width effects during minority ICRF heating in the JET device. In this approach, electric fields from a full-wave solver (LION code [6.90]) were used to accelerate resonant particles in the FIDO Monte Carlo orbit code [6.99]. Non-thermal particle distributions were then reconstructed from the orbit code results and used in the full-wave solver. Finite ion orbit effects during ICRF heating have also been simulated in 3-D configurations such as the LHD stellarator using wave fields from the TASK WM solver [6.93] in the 5-D drift kinetic equation solver GNET [6.88]. More recently the AORSA solver [6.85] has been combined with the 5-D Monte Carlo code ORBIT RF [6.100]
and with the Monte Carlo code sMC (Green [6.101]) to study ICRF minority heating in the Alcator C-Mod device and high harmonic fast wave heating in the DIII-D and NSTx devices. The coupling between AORSA and the sMC code is quite rigorous in the sense that particle “lists” or statistics from the Monte Carlo code are used to directly re-evaluate the plasma response in the full-wave solver.
Also the 4D RF diffusion coefficient (two velocity and two spatial dimensions)
calculated form the electric fields in AORSA is used directly in the Monte Carlo code to accelerate resonant particles.
6.2.8.2. Full-wave simulations of mode conversion in the ICRF
Full-wave simulations of mode converted ion Bernstein waves (IBWs) and ion cyclotron waves (ICWs) in tokamak plasmas have also been performed using full-wave solvers [6.87, 6.102]. These calculations are especially challenging because of the disparate spatial scale lengths between the incoming fast wave and short wavelength mode converted waves. Mode converted ICRF waves are of particular interest for plasma profile control because they can be used to generate localized current and plasma rotation. Efficient mode conversion can be realized by increasing the minority ion concentration so that the cyclotron resonance is well separated spatially from the mode conversion layer. Furthermore, owing to the high minority ion concentration the maximum ion tail energies are not sufficient to overlap the mode conversion layer with the Doppler broadened cyclotron resonance layer. Figure 6.15 is an example of an ICRF mode conversion simulation in Alcator C-Mod.
FIG. 6.15. TORIC simulation of ICRF mode conversion in the Alcator C-Mod device. Contours are plotted for the real part of the parallel electric field Re(EZ). Contour levels shown on the right are logarithmic and in units of kV.m–1. B0 = 5.8 T, 33% H, 23% 3He, 21% D, Te ằ Ti ằ 1.5 keV. Reprinted from Ref. [6.79]. Copyright (2011), American Nuclear Society.
It can be seen that a short wavelength mode (the ICW) is excited to the low field side of the usual IBW mode conversion layer. Longer wavelength ICWs are also apparent in the simulation above and below the tokamak midplane and can
be seen propagating back to the low field side of the mode conversion layer at 0.02 cm X 0 cm
. The TORIC simulation used 255 poloidal modes and 240 radial elements to represent the electric field in its semi-spectral basis set.
6.2.8.3. Full-wave simulations of lower hybrid waves
Because of the millimeter perpendicular wavelengths of LH waves, ray tracing techniques have long been the preferred method for treatment of wave propagation in this regime. It is well known, however, that the geometrical optics approximation breaks down at caustics or focal points as well as at layers where waves reflect, such as wave cut-offs. In particular, important effects such as focusing and diffraction are neglected in the lowest order geometrical optics approximation [6.103] at caustic surfaces. It is possible to capture these effects to some extent by using beam tracing techniques [6.104]. However, the advent of massively parallel computing architectures has made it possible to perform full-wave electromagnetic field simulations of LH waves in present day sized tokamak devices [6.105]. An example of such a simulation using the TORIC-LH solver is shown in Fig. 6.16. These simulations were performed for the Alcator C-Mod parameters of Section 6.6.3 (f04.6 GHz, ne(0) 7 10 m 19 –3,
(0) 5.3 T
B = and Te(0) 2.33 keV ) and required 1023 poloidal modes and 980 radial elements in the TORIC spectral basis set. The electric field pattern in Fig. 6.16 left corresponds to a weak single pass damping regime where N= -1.55 initially and the field pattern on the right corresponds to a strong single pass case where N= -3.6 initially. A Maxwellian electron distribution was used in both cases to compute the electron absorption. The LH waves are coupled from four waveguides on the low field side (on the right of each figure). In both cases, focusing of the incident LH beam at a caustic surface near the centre is apparent.
In the weak damping case these beams undergo multiple reflections from cut-offs at the plasma edge while in the strong damping case on the right the incident beam undergoes a single reflection from the edge before the wave undergoes strong spectral broadening ink-space and damps. The full-wave fields shown above have also been employed to evaluate the RF diffusion coefficient in an iterative solution of the electron Fokker–Planck equation using CQL3D and TORIC-LH. A further exciting development in full-wave field analysis has been the application of a pure finite element method to solve the LH wave propagation problem from a waveguide launcher to the plasma core [6.106]. This approach benefits from the fact that in principle the waveguide fields, coupling at the plasma edge and propagation to the core can be treated simultaneously. Proper calculation of the dissipation of these waves in the different regions of the plasma is under development.
FIG. 6.16. Left: TORIC-LH simulation of LH waves in Alcator C-Mod in the weak damping regime (N= -1.55, T (0)= 2.33 keVe ). Right: TORIC-LH simulation of LH waves in Alcator C-Mod in the strong damping regime (N= -3.6, T (0) 2.33 keVe ). Reprinted from Ref. [6.105]. Copyright (2011), American Institute of Physics.
6.3. ExPERIMENTAL RESULTS ON RF HEATING AND CURRENT DRIVE A summary of RF heating and current drive results will be given in this section. Due to the extensive literature, we will confine our summary to toroidal devices, in particular tokamaks. There was a parallel evolution of applications of ECH and ICH in mirror machines in earlier years, but with the exception of a few cases, these machines have been shut down more than a decade ago.
The interested reader may want to look up additional references on this topic elsewhere. Historically, in the ion cyclotron range of frequencies (ICRF) regime, heating results in the 1960s and early 1970s were limited by poor confinement of energetic minority species ions in both stellarators and tokamaks. However, this was followed by spectacular heating results in the 1980s at the multi-MW power levels in larger machines with excellent confinement, such as PLT, TFTR, JT-60 and JET. Rapid progress was obtained because of the availability of relatively low cost amplifier tubes (tetrodes) at the 1–2 MW level in this frequency regime.
In the late 1970s and early to mid-1980s the theory of LHCD was developed (see previous section) and by the mid-1980s this drive was firmly established in experiments in the USA, in Europe and in Japan. Presently, multi-MW long pulse LHCD experiments are being carried out on Tore-Supra, JET and most recently Alcator C-Mod. Long pulse experiments are in preparation for the new superconducting tokamaks in China (EAST) and the Republic of Korea (KSTAR). Meanwhile, ECRH was progressing rapidly in the 1980s with the development of gyrotron tubes at 28–56 GHz at the 200 kW level, while in
the late 1990s MW level gyrotrons were developed at frequencies in the range 110–140 GHz. Recent multi-MW experiments in T-3, DIII-D, ASDEx-U, TCV and JT-60 have yielded excellent heating and current drive results in good agreement with theory. Recent tube development at the 1 MW level has been achieved at frequencies up to 170 GHz, and development of 2 MW long pulse tubes is under way for ITER applications. While RF wave propagation in the plasma core is now well established, the coupling to high amplitude RF waves around metallic antennas in the ICRF and LH regimes is still an area of active research, including possible non-linear effects. Steady progress is being made to identify these interactions by both theory and experiment and development of realistic antenna codes to model these interactions is under way. We expect that reliable RF wave techniques in all three frequency regimes should be available for implementation in ITER during the next decade as needed. Reliability of antenna operations at high RF voltages and long pulses will need further attention and novel antenna design concepts are being developed and tested in ongoing experiments. Finally, we should mention that the development of new diagnostics, namely phase contrast imaging (PCI), offers ways to measure the injected RF waves and mode converted waves inside the hot plasma [6.46].
In the future, using synthetic diagnostics in full-wave codes, a quantitative determination of power flow inside hot fusion plasmas should be feasible. We shall now discuss some of the experimental results in some detail. Appropriate references will be given whenever appropriate.