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
6.3.5. Applications of electron cyclotron heating and current drive
Electron cyclotron heating (ECH) and electron cyclotron current drive (ECCD) have many practical applications due to their unique characteristics, especially the ability to deposit power and/or current in a highly localized, robustly controllable location in the plasma. Another important feature is that EC waves propagate in the vacuum between the launcher and the plasma, unlike other wave heating techniques which have an evanescent region between the launcher and the plasma, and this means that the launcher can be far from the plasma boundary. There is little sensitivity of the wave coupling to the plasma edge conditions and no tendency to introduce impurities. These characteristics have led to the use of ECH/ECCD for such diverse applications as plasma pre-ionization and discharge startup, plasma heating, support and maintenance
of the desired current profile in tokamaks and control of some MHD modes that otherwise limit performance.
The propagation, absorption and effect on the electron distribution of EC waves is well described by theory based on first principles as described in Section 6.2.1. This body of theoretical work has been encapsulated in practical predictive computer codes, so that experiments can provide the empirical validation of the models. The validation of these models Refs [6.66, 6.192, 6.193]
and the references therein) supports the ability to predict with high confidence the EC power and current profile sources in plasma confinement devices.
6.3.5.1. Wave propagation and absorption experiments
In most situations of interest in fusion research, the propagation of EC plane waves is well described by the cold plasma dispersion relation as described in Section 6.1.1. The trajectory and wave vector evolution of the plane wave can be found, in the WKB approximation, as described in Section 6.2.7.1. In practical cases, however, plane waves are not applied; rather the power is launched as a narrow cone. There are two common ways of modelling the cone. One way is to propagate an array of independent, non-interacting rays with a power and angle distribution that simulates a Gaussian beam. This technique is relatively simple and in many cases is adequate, and it is embodied in codes like TORAY [6.194] and GENRAY [6.78, 6.195]. In cases where the EC beam is focused or has astigmatism, Gaussian beam codes must be used to address interference and diffraction effects. Representative Gaussian beam codes are TORBEAM [6.196], GRAY [6.197] and TRAVIS [6.198].
All ray tracing and beam codes make the approximations that the plasma is changing on a spatial scale large compared to the wavelength, that the imaginary part of the wavevector is small compared to the real part, and that a parallel wavenumber k can be defined over each part of the wavefront for calculation of the resonance condition and the absorption. These conditions are usually but not always satisfied in fusion applications. In cases where these conditions are not met, the only alternative is the full wave approach. Given the very large ratio of plasma dimension to wavelength, this is very challenging [6.199].
The ray tracing and beam codes calculate the beam absorption and current drive along the wave trajectories. The contributions to the heating and current are summed in radial bins and provide a calculation of the heat and current profiles in practical cases. The ray tracing and beam codes have been benchmarked for a representative ITER case and found to produce very similar profiles of heating and current density [6.200]. An example of ray tracing using the TORAY-GA code in a DIII-D equilibrium is shown in Fig. 6.49 for three different densities.
In the lowest density case the refraction is negligible, since the density is well below the cut-off density (that is satisfying the approximate condition for the
second harmonic x-mode 2pe 2/ 2, which for 110 GHz requires a density below 7.4 × 1019 m–3). Note that the effective density limit is usually smaller than cut-off density, sometimes significantly smaller, due to effects that affect how the beam approaches the resonance and the toroidal plasma geometry. A toroidal component to the beam, for example, enhances the tendency for the beam to glance off the plasma before it reaches the actual cut-off density, even after decreasing the cut-off density due to a no-zero value of k. In the intermediate density case, refraction is significant enough to change the radial heating profile measurably and in the highest density case the rays are wildly refracted.
FIG. 6.49. Ray tracing calculations by the TORAY-GA code. The blue lines are ray trajectories for X-mode rays and the red crosses are where the absorption peaks for each ray. The vertical line is the second harmonic resonance for 110 GHz. The toroidal field is 1.7 T, the rather peaked density profile is scaled to a central density of (left) 3 × 1019 m–3, (centre) 8 × 1019 m–3 and (right) 11 × 1019 m–3.
Wave absorption in the various regimes is discussed at length in Section 6.2.1 and the references there. Many code routines have been developed to calculate absorption; for a summary see Ref. [6.200]. These routines use either linear absorption models, for example the model by Mazzucato [6.201]
used in the results shown in Fig. 6.49, or models that solve the Fokker–Planck equation in the quasi-linear limit (for an excellent review see Ref. [6.202]). The linear models may be either weakly relativistic or fully relativistic, depending on how the resonance condition is evaluated. The quasi-linear models are also fully relativistic in that sense, and they may have a more general model for the collision operator. In particular, the collision model in quasi-linear Fokker–Planck codes may avoid using the high speed limit of the collision operator and may use a model that conserves momentum in electron–electron collisions [6.203]
Recently, works by Marushchenko [6.204, 6.205] and by Smirnov [6.179] have generalized the collision operator in linear codes as well.
The calculated absorption profile location and width are partly determined by the geometry of the intersection of the beam with the flux surfaces in the neighbourhood of the resonance, with the width also affected by the Doppler and relativistic broadening and the location by the Doppler shift. Under a representative condition of off-axis ECH in projected ITER conditions, the codes produced similar results [6.200]. An example is shown in Fig. 6.50.
FIG. 6.50. (a) Profile of ECH power density deposited in electrons (qe), and (b) profile of ECCD current density j for some linear and quasi-linear (CQL3D, OGRAY) codes. Reprinted from Ref. [6.200].
In practice the absorption profile can be inferred by using the technique of modulating the EC power and measuring the changes in the electron temperature profile at the modulation frequency. The electron cyclotron emission (ECE) diagnostics has sufficiently low noise, high temporal response and high spatial resolution for this purpose. Such a measurement is shown in Fig. 6.51.
FIG. 6.51. Experiment to measure ECH location in DIII-D. (a) The blue lines represent the ECH beam and the vertical magenta line is the second harmonic resonance at 110 GHz. The ECE channels shown in light blue lie near the midplane. (b) ECH power (MW) modulated at 50 Hz. (c–e) Temperature response in keV of the ECE channels at of 0.50, 0.64 and 0.83.
(f) Fourier amplitude at 50 Hz of the ECE response. (g) Fourier phase of the ECE response relative to the ECH as a function of normalized radius.
The process illustrated in Fig. 6.51 can be applied to study systematic behaviour of the measured ECH location relative to the calculated location. The location can be measured with great precision using this technique, as shown in Fig. 6.52 [6.206] for the ASDEx Upgrade tokamak. This figure shows that by increasing the modulation frequency to 500 Hz the broadening of the response profile by transport on the timescale of the modulation period can be minimized, and extremely narrow profiles can be resolved and brought nearly into agreement with the TORBEAM calculations. Figure 6.52 shows that over a broad range of conditions the radial location is also well predicted by the TORBEAM code.
FIG. 6.52. (a) Fourier amplitude Te amp, of the ECE measurements of Te for ECH modulated at 500 Hz, as a function of normalized minor radius pol0, for high and low diffusivity eHPcases in ASDEX Upgrade (where this is the diffusivity determined from heat pulse propagation).
The high diffusivity is obtained by increasing the temperature gradient by applying additional ECH near the axis, while the low diffusivity case has ECH near = 0.5 in order to reduce the gradient. (b) Normalized minor radius of the peak plasma response pol0(pol/ )a versus the theoretical value from TORBEAM calculations for a variety of conditions. Reprinted from Ref. [6.206]. Copyright (2011), IOP Publishing Ltd.
For most applications to plasmas of interest in fusion research, the fundamental O-mode and the second harmonic x-mode are effectively fully absorbed in the plasma, so if the ray and beam codes get the location and width right, then the power density is also right. The high power density that results from the narrow deposition profile can be used for many purposes. An example is strong high central heating, which in the JT-60U tokamak has led to central temperatures of 23 keV [6.207], as shown in Fig. 6.53.
FIG. 6.53. Experiments on JT-60U showed that strong electron transport barriers, high central temperature and large ECCD could be generated by 0.6 MW of central ECH and 2.3 MW of central ECCD power, using fundamental O-mode and 110 GHz frequency. (a) Plasma current and edge loop voltage; ECH and neutral beam power; and central electron temperature and line-averaged electron density. (b) Electron temperature profile. (c) Deposition location for ECH (red) and co-ECCD (blue). (d) Components of the current density. Here, jBS is the bootstrap current density, jBD is the current density due to neutral beam injection, and jEC is the current density due to ECCD. The negative jOH is due to the induced electric field that arises to conserve the magnetic flux when the ECCD is applied. Taken from Ref. [6.207].
The internal transport barrier seen in the electron temperature profile of Fig. 6.53 was also found in the LHD, where it was shown that there is a power threshold for generation of the barrier [6.208], as shown in Fig. 6.54.
Another example of the application of the highly localized heating charac- teristic of ECH is control of the electron temperature gradient for studies of turbulence and validation of turbulence models. Figure 6.55 shows the ECH heating profile as a function of time for a case in which the modulated heating is out of phase at two nearby radial locations, one location of which is shown in Fig. 6.51. The contours of electron temperature from the ECE show periodic spatial modulation, resulting in a periodic modulation of the temperature gradient. The frequency and amplitude of the low level turbulence are modulated as well, and these characteristics can be compared with predictions of turbulence codes.
FIG. 6.54. Electron temperature profiles in LHD for 177 kW and 282 kW incident ECH power, showing creation of an electron transport barrier at the higher power. Reprinted from Ref. [6.208]. Copyright (2011), IOP Publishing Ltd.
FIG. 6.55. Modulated ECH power density in two radial locations modulated out of phase for turbulence studies on DIII-D. (Top) Power density. (Bottom) Contours of electron temperature from the ECE diagnostics. As the contours become closer together, the Te gradient increases.
The cut-off density may limit the role ECH may play in heating applications.
In tokamaks, the density is empirically found to be limited to the Greenwald density, nGW[10 m ]20 -3 Ip[MA] /a2[m]. From Section 6.1, propagation of EC waves is limited to 2pe 2 for the fundamental ordinary (O1) mode and by 2pe 2/ 2 for the second harmonic extraordinary (x2) mode, where is the applied angular frequency. Then using q 2 2RB I / [MA]p , where q is the safety factor, is the inverse aspect ratio, is the elongation and R is the major radius, and Bϕ is the toroidal field, we get
co 2 GW
n qRB n
(6.149)
where nco is the EC cut-off density. This expression shows that the cut-off density becomes less restrictive for tokamaks with large major radius and large toroidal field.
Tokamaks of moderate size and field, like ASDEx Upgrade and DIII-D, have nco/nGW of order 0.3 to 0.5 for the O1-mode but of order unity for the x2-mode. For this reason, the x2-mode was implemented on those devices when high power gyrotrons with frequency above 100 GHz were developed. Then for the x2-mode, the cut-off density plays only a small role in limiting operating parameters. Larger, higher field tokamaks, like JT-60U and especially ITER, are not at all limited by the cut-off density. Small and/or low field tokamaks like TCV and spherical tokamaks like MAST are strongly limited in density range (see Ref. [6.209]). This limit has been circumvented in two different ways in these devices.
The limitation on density in TCV placed by the Greenwald limit has motivated installation of a higher frequency system that operates at the third harmonic of the extraordinary mode [6.210]. For harmonic numbers l2 of the x-mode, the propagation limit is 2pe l l( 1) ce2 or using lce,
2 [ ( 1) / ]2 2
pe l l l
or
19 3 2
2
[10 m ] 0.00122l l( 1) [GHz]
n f
l
(6.150)
From Eq. (6.150), by going from the x2-mode to the x3-mode by using 118 GHz power sources in place of 82.7 GHz sources, the limiting density was increased by a factor of 3.
Absorption is lower at higher harmonics, however. Generally, increasing the harmonic number of the x-mode by one introduces a factor Te/mc2 in the absorption coefficient [6.66]. (The absorption of the O1-mode and the x2-mode are comparable as discussed earlier.) To counter the weaker absorption in TCV, the waves were launched from the top of the plasma nearly parallel to the resonance. By this means the waves were made to travel a relatively long
distance in the immediate vicinity of the resonance and thereby the net absorption is increased, typically to 70% in ohmic plasmas and to nearly full absorption in plasmas with pre-heating at lower density using the x2 power which generated a seed non-thermal component to the electron distribution function [6.211, 6.212].
The geometry of the TCV experiment is shown in Fig. 6.56.
FIG. 6.56. Left: Electron temperature as a function of mirror steering angle for the X3 mode in TCV. Note the comparison between experimental measurements (blue curve) and predictions by the TORAY code (red). The yellow line is the average of the experimental signal for ease of comparison with the theoretically predicted value. Right: Ray bundles for three launch angles, illustrating typical trajectories. Taken from Ref. [6.211].
The second way to avoid the density limit, other than going to a higher harmonic, is to apply the electron Bernstein wave (EBW; see Fig. 6.2).
This approach has been applied to stellarators, which are not subject to the Greenwald density limit and therefore often operate at high values of 2pe /ce2, and to spherical tokamaks, since they have very low field. The EBW, being an electrostatic wave, does not propagate in vacuum, so it cannot be excited directly by launchers outside the plasma. Instead, the ordinary-extraordinary-Bernstein wave (O-x-B) mode conversion approach [6.213] is used to launch the EBW.
This work predicted that a critical window in the parallel index of refraction of the launched ordinary mode at the plasma boundary would result in effective mode conversion to the extraordinary mode at the surface where the ordinary mode is reflected. The outward travelling extraordinary mode is then converted to an inward propagating Bernstein wave at the upper hybrid layer. The dispersion relation of the EBW provides propagation to arbitrarily high density.
Pioneering work on the Wendelstein 7-AS stellarator validated the theory of the O-x-B process [6.214]. Figure 6.57 shows the plasma heating as a function of the parallel index of refraction of the launched wave. In this experiment the magnetic field was set to have no cyclotron resonance in the plasma, so the wave
damping was non-resonant, and the central electron density was more than twice the O-mode cut-off density. Excellent agreement with theory of O-x and x-B mode conversion was found, as shown in the figure. Comparison to NBI heating showed that more than 70% of the launched EC wave power was found in the plasma core, showing that EBW heating can be quite efficient. These results were further confirmed by experiments on the MAST spherical tokamak [6.215]. A recent review of EBW heating has been given by Laqua [6.216].
FIG. 6.57. Increase of the plasma energy content by O-X-B heating versus the longitudinal refractive index of the incident O-mode wave Nz. The solid line is the calculated transmission normalized to the maximum energy increase [6.214]. Here Nz,opt is the optimal value of Nz, the index of refraction parallel to the magnetic field, where maximum absorption takes place;
0 / ( 0 )
k L L 2 , where L is the path length of the microwave beam and 0 the free space wavelength. Reprinted from Ref. [6.214]. Copyright (2011) by the American Physical Society.
6.3.5.2. Electron cyclotron current drive (ECCD) experiments
Electron cyclotron waves can drive current in a toroidal plasma even without introducing significant toroidal angular momentum [6.60, 6.65]. Wave absorption can be highly localized in space near the intersection of the wave with the cyclotron resonance or its low harmonics, so the current may also be localized in a robustly controllable manner. This unique localization and control- lability offer the opportunity to apply ECCD to applications like controlling the growth of MHD instabilities in the neighbourhood of rational surfaces or driving broad off-axis currents for optimizing the equilibrium for improved confinement and stability.
In experiments, the ECCD net current can be measured by comparing the plasma toroidal resistance (the loop voltage measured with a flux loop divided by the total plasma current measured with a Rogowski coil) with the neoclassical resistance calculated from measurements of the Te and the Zeff profiles. Plasma resistance below the calculated resistance is indicative of
ECCD. This determination of the driven current is subject to a rather high degree of uncertainty, particularly if MHD activity is present. Greater clarity can be obtained if the ECCD is larger than the total plasma current, so that the loop voltage goes negative. This was done in experiments on the TCV tokamak [6.217], in which over 80 kA of ECCD was driven by 1.0 MW of EC power, driving the loop voltage to or below zero for 4 s, which is many resistive times (Fig. 6.58). Such experiments may be done with great sensitivity in stellarator geometry since stellarators have little net plasma current to compete with the ECCD. Definitive ECCD experiments were reported from the Wendelstein 7-AS stellarator [6.218].
FIG. 6.58. Steady state current fully supported by ECCD using 1.0 MW of power distributed over the plasma cross-section. Reprinted from Ref. [6.217].
The loop voltage method can determine the total driven current but not the current profile. The profile of ECCD can be inferred from measurement of the local magnetic field inside the plasma, for example by the motional Stark effect (MSE) diagnostics (process developed by C.B. Forest [6.219]). The information from the internal magnetic measurements can be used to constrain the reconstruction of the equilibrium to obtain an accurate calculation of the flux surfaces in the plasma. Since the local toroidal electric field can be found from the time derivative of the local flux, the ohmic current density can be found by multiplying by the neoclassical resistivity. Subtracting the ohmic current density from the total current density yields the sum of the non-inductive currents (i.e. bootstrap current, ECCD and possibly neutral beam driven current) (see Ref. [6.219]). However, this method can have difficulty resolving the very narrow current profiles that ECCD can produce.