D.2 DIAGNOSING AE WITH MIRNOV COILS

It was shown in the description of TAE, Eqs. (6.36) and (6.37) in Chapter 6, that only magnetic perturbations perpendicular to the equilibrium magnetic field, SB, and SB,,. are significant in TAE, and SB, «: SB,) if the mode structure satisfies d/dr»m/r. Similar relations are valid for a majority of shear Alfven eigenmodes (EAEs, NAEs, ACs). For detecting AE perturbations, magnetic coils mounted just outside plasma are employed on a majority of magnetic fusion devices. Figure D.2 shows the geometry of Mirnov coils on JET, which measure an oscillatory magnetic flux induced by SB0 passing through the coil axis. The coils provide the value of

Owing to the high values of TAE frequency, (0= 106s"', which is a factor in front of the perturbed field SBdge in (D.IO), the coils are sensitive enough to detect the perturbed magnetic fields as low as |5B^dse / Z?„| = 10“8. The data acquisition system of the coils typically uses the sampling rate of 1 MHz on JET, so measurements of AEs up to 500 kHz can be made. The Mirnov coils are calibrated, that is, they provide the same amplitude and phase response to the same test signal, and hence, the cross-analysis of the coils separated in toroidal direction provides an accurate information regarding the toroidal mode numbers.

To determine the toroidal mode number n of a mode, two or more toroidally separated Mirnov coils are used and the phase shift is measured between them as Figure D.3 illustrates. Figure D.3 shows a toroidal plasma with the directivities indicative of the magnetic fields, plasma current, the beam injection, and the toroidal projections of electron and ion diamagnetic frequencies. Let us assume we have two toroidally separated magnetic pick-up coils at toroidal angles (p2. If both coils measure at the same time t0, the same sinusoidal wave of frequency w0, then a phase shift a exists between the measurements due to the finite toroidal angle, Дt/>, distance between the

D.2 JET cross-section showing the position and poloidal directivity of five high-frequency Mirnov coils H301 ... H305 separated in the toroidal angle

FIGURE D.2 JET cross-section showing the position and poloidal directivity of five high-frequency Mirnov coils H301 ... H305 separated in the toroidal angle.

D.3 Sinusoidal signals measured at different toroidal angles at the same time and at same frequency are shifted in phase by «

FIGURE D.3 Sinusoidal signals measured at different toroidal angles at the same time and at same frequency are shifted in phase by «.

coils. For a known Acp and the measured a, one can calculate what part of the wavelength fits within the toroidal distance between the probes, and an extrapolation of this length to a complete toroidal circle gives the toroidal mode number n.

D.3 FURTHER ADVANCES IN DIAGNOSING ALFVEN INSTABILITIES

Further advances in diagnosing Alfven instabilities are associated with recent expansion of tools and techniques for the detection and identification of unstable modes. Reflectometry, interferometry, ECE, and SXR measurements of perturbed electron density and temperature associated with AEs were successful alternatives to magnetic sensors.

In toroidal geometry, the perturbed electron density caused by AEs is found from the continuity equation:

where Sn, n„ are the perturbed and equilibrium densities, % is the plasma displacement caused by the perturbed electric field via -/со!; = (с/Ат)[8£х/}0], and L„ is the radial scale length of the density profile. The first term in the right-hand side of (D.ll) describes the usual convection of plasma proportional to the gradient of equilibrium density. The second term =•= 1 /R in (D.ll) is caused by the toroidicity and causes a non-zero 5n even if the profile of n„ is flat. This term also causes an anti-ballooning structure of the density perturbations 8n even when 8В has no significant in-out asymmetry [D.9].

A launched microwave О-mode beam on JET with frequency above the cut-off frequency of О-mode was found to deliver detection of AEs far superior to that made with magnetic sensors [D. 10]. This “О-mode interferometry” shows many more unstable AEs in the plasma core, some

D.4 Spectrograms showing AEs with different toroidal and poloidal mode numbers

FIGURE D.4 Spectrograms showing AEs with different toroidal and poloidal mode numbers: Top: interferometry О-mode measurements with 45.2 GHz microwave beam. Bottom: measurements of same modes with Mirnov coils.

of which are not even detected with Mirnov coils as Figure D.4 illustrates. The particular settings of the О-mode system on JET does not allow measurements to be made above plasma densities of ~6-7 1019 nr3, so a higher frequency instrument had to be developed.

The standard far infra-red (FIR) JET interferometer was digitised to a high sampling rate, which enabled detecting AEs in plasmas of high density [D.4]. The FIR interferometer on JET has four vertical lines-of-sight as Figure D.5 shows, which provide measurements of the line-integrated perturbed density perturbations. The FIR interferometry often detects AEs deep in the plasma core, which are hardly seen with Mirnov coils, and the quality of the interferometry signal is quite

D.5 Geometry of JET interferometer with vertical lines-of-sights (counted from left to right as Channels 1 ... 4)

FIGURE D.5 Geometry of JET interferometer with vertical lines-of-sights (counted from left to right as Channels 1 ... 4)

D.6 Core-localised TAEs inside the q= 1 radius (tornado modes) detected with the vertical Channel 3 of the JET interferometer shown in Figure D.5 (line-of-sight passing through the magnetic axis)

FIGURE D.6 Core-localised TAEs inside the q= 1 radius (tornado modes) detected with the vertical Channel 3 of the JET interferometer shown in Figure D.5 (line-of-sight passing through the magnetic axis).

satisfactory as Figure D.6 shows. A similar FIR interferometry technique was employed for detecting AEs in DIII-D discharges. It was observed for the first time that a “sea of modes” exists in reversed-shear DIII-D, w'ith toroidal mode numbers up to n = 40 [D.ll], The interferometry technique has significantly increased the quality of AE detection and assures that all unstable AEs are detected even deeply in the plasma core. As the interferometry technique of detecting AEs requires only interferometers used for plasma density measurements, this method is a good candidate for ITER and DEMO.

The main limitation of using interferometry or Mirnov coils for detecting AEs is that the AEs cannot be localised from the measurements and the amplitudes of AEs cannot be found with precision. However, the successful development of ECE [D.5] and ECE imaging [D.6], beam emission spectroscopy (BES) [D.12], and phase contrast imaging (PCI) [D.13] have addressed the problem of measuring mode structure. Together with the existing SXR technique and X-mode reflectometry used for observing alpha-driven AEs in DT plasmas [D.2], the new' diagnostics provide measurements of the spatial structure of the modes to a degree required for an accurate experiment-to- theory comparison.

REFERENCES

[D.l] Progress in the ITER Physics Basis, Nucl. Fusion 47 (2007) SI. [D.2] R. Nazikian et al., Phys. Rev. Lett. 78 (1997) 2976.

[D.3] S. Hacquin et al.. Plasma Phys. Control. Fusion 49 (2007) 1371. [D.4] S.E. Sharapov et al., Nucl. Fusion 46 (2006) S868.

[D.5] M.A. Van Zeeland et al.. Phys. Rev. Lett. 97 (2006) 135001.

[D.6] S.J. Freethy et al.. Rev. Sci. Instrum. 87 (2016) 11E102.

[D.7] W.W. Heidbrink et al., Phys. of Plasmas 24 (2017) 056109.

[D.8] M.A. Van Zeeland et al.. Nucl. Fusion 46 (2006) S880.

[D.9] R. Nazikian et al.. Phys. Rev. Lett. 91 (2003) 125003.

[D. 10] S.E. Sharapov et al.. Phys. Rev. Lett. 93 (2004) 165001.

[D.ll] R. Nazikian et al., Phys. Rev. Lett. 96 (2006) 105006.

[D.12] R.D. Durst et al., Phys. Fluids B4 (1992) 3707.

[D. 13] E.M. Edlund et al.. Plasma Phys. Control. Fusion 52 (2010) 115003.

 
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