Proceedings of ITC/ISHW2007

Bispectral analysis of harmonic oscillations measured using beamemission spectroscopy and magnetic probes in CHS
T. Oishi, S. Kadoa, M. Yoshinuma, K. Ida, T. Akiyama, T. Minami,
K. Nagaoka, A. Shimizu, S. Okamura, and CHS group
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japana

High Temperature Plasma Center, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, JapanThe coherent MHD oscillation which has the fundamental frequency of several kilohertz and its higher
harmonics (harmonic oscillation: HO) has been observed in the compact helical system (CHS). HO consists of
two pairs of harmonic series. One locates in the core region near ι = 0.5 rational surface, the other locates in theedge region near ι = 1.0 rational surface. In the present study, bispectral analysis is applied to the fluctuation data
having HO measured using beam emission spectroscopy (BES) and magnetic probes. It has revealed that
fundamental mode of HO in both the magnetic fluctuation and core density fluctuation have phase correlation withthe harmonics including fundamental oscillation, while HO in edge density fluctuation does not have such phasecorrelation. Mode numbers of HOs are same between harmonic components having difference frequencies. It
suggests that the generation of harmonics can not be interpreted simply as the mode coupling, even thoughbicoherence value is significant. The bicoherence value and relative amplitude of higher harmonics correlate witheach other, which suggests that bicoherence means the degree of distortion of the signals.
Keywords: bispectral analysis, bicoherence, fluctuation, magnetohydrodynamics, edge harmonic oscillation,compact helical system, beam emission spectroscopy, edge transport barrier

1. Inroduction
Fluctuations of plasma parameters have beenrecognized to correlate with the properties of plasma
confinement. Characteristics of the fluctuations includes
the amplitude and level, frequency and wavenumber
spectrum, spatial structure represented by radial locality or
poloidal/toroidal mode number m/n, and so on. In
addition to these characteristics, recently, attention hasbeen paid to the nonlinear characteristics such as thecorrelation between spectral components having differentfrequencies.
In order to investigate the nonlinear
characteristics of the fluctuations, bispectal analysis isconsider to be useful [1]. The bispectral analysis has been
applied to several tokamaks [2], helicals [3] and othertypes of plasma confinement devices to investigate theinteraction among turbulence, magnetohydrodynamic(MHD) oscillations, and zonal flows [4-6].
Recently, a coherent MHD mode which consists ofthe fundamental frequency of several kilohertz and its
higher harmonic (harmonic oscillation: HO) [7,8] has been
observed in the edge transport barrier (ETB) discharges of
the compact helical system (CHS), which is a
low-aspect-ratio, middle-sized heliotron (major radius =1.0 m, minor radius = 0.2 m, toroidal period number = 8,
polarity = 2) [7,9-13].
The characteristics of theoscillation have been investigated in the viewpoint of theauthor’s e-mail: oishi@nifs.ac.jp

comparison with edge harmonic oscillation (EHO) [14-17],
which is recognized to have a role enhancing the particle
transport in the quiescent H-mode of tokamaks.
Fluctuation measurement using the beam emission
spectroscopy (BES) [18,19] and magnetic probe (MP)array [20] have revealed that HO in CHS consists of twopairs of harmonic series located in different radialposition. One locates in the core region of normalizedminor radius ρ = 0.4~0.5 near ι = 0.5 rational surface(denoted as “HO (core))”, the other locates in the edge
region of ρ = 0.9~1.0 near ι = 1.0 rational surface(denoted as “HO (edge)). All harmonic components ofboth HO (core) and HO (edge) have same mode numbers,
namely, m/n = -2/1 for HO (core) and m/n = -1/1 for HO(edge), even though the frequency of pth harmonic is p
times larger than that of 1st harmonic for both HO (core)and HO (edge), where p is positive integer [21].
In the present study, bispectral analysis is applied tothe fluctuation data having HO measured using BES andMP in order to investigate the correlation between theharmonic components.

2. Experimental Setup
BES has been developed in CHS to simultaneously
measure both local density fluctuations and gradients.
The BES method detects emissions from the collisionallyProceedings of ITC/ISHW2007
excited neutral beam atoms (denoted as "beam emission")[22]. The spatial channels used in BES consist of 16
optical fibers with an object lense. In the present
experiment, Hα emission from the neutral hydrogen atomic
beam for heating in the intersection of the beam line andthe sightline for each fiber channel was measured as thebeam emission. One can select the arrangement of thefibers in the radial or poloidal direction to measure eachstructure of the fluctuations.
Magnetic probe array was also used to measure boththe poloidal and toroidal structures of the magneticfluctuations. The poloidal magnetic probe array consistsof sixteen probes covering the poloidal angle ∆θ = 235.5
deg. We define the absolute poloidal coordinate as θ = 0
deg. and 180 deg. at the midplane outside and inside torus,
respectively [23]. Five toroidal magnetic probes covering
the toroidal angle ∆φ = π were located at every π/4 section
at the major radius R = 1.2 m.

3. Data Analysis Method
When the wave having the frequency f is phasecorrelated with the spectral components with thefrequencies of f1 and f2 satisfying the summation rule f = f1
± f2, we can use the bicoherence value as an indicater ofphase correlation of these three waves [1]. Bicoherenceb2(f1, f2) for the waves having the frequency f, f1 and f2 isdefined asb 2 ( f1 , f 2 ) =|< X ( f1 ) X ( f 2 ) X * ( f ) >|2<| X ( f1 ) X ( f 2 ) |2 ><| X ( f ) |2 >(1)

Additional gas was puffed as a means to increase density.
The standard magnetic field configuration of CHS wasapplied, in which the vacuum magnetic axis position Rax
was 92.1 cm from the torus center, the toroidal magnetic
field strength at the magnetic axis Bax was 0.95 T, and themagnitude of the quadrupole field Bq denoting the degreeof cancellation of the intrinsic quadrupole component
formed by the helical coils was -50 %. In the cases inwhich the heating power exceeds a certain threshold, ETB
characterized by a sudden drop in the Hα intensity signal as
shown in Fig. 1(b) is formed. It has been found that theline-averaged density, the edge density gradient, and thestored energy begin to increase at the ETB transition[7,9-13]. In the latter half of the phase having ETB, HO
which consists of a 1st harmonic with the frequency of 4.0
kHz and a 2nd harmonic appears in the magnetic
fluctuation as shown in Fig. 1(c) and (d). It is known that
this oscillation is identical to HO (core) [21].
We applied the bispectral analysis to the fluctuation
data having harmonic oscillation measured using themagnetic probes and BES.
Figure 2 shows thebicoherence plane of (a) magnetic fluctuation, (b) core (ρ =0.42) density fluctuation, and (c) edge (ρ = 0.95) density
fluctuation. b2(f1, f2) of both the magnetic fluctuation andcore density fluctuation have a clear peak having thefrequencies (f1, f2) = (4.0 kHz, 4.0 kHz). This indicatesthe 1st component of HO (core) having the frequency 4.0
kHz is phase correlated with both the 1st component itself
and the 2nd component having the frequency 8.0 kHz = f1
+ f2, where f1 = f2 = 4.0 kHz. On the other hand, b2(f1, f2)
edge density fluctuation does not have such a peak, even

where X(f) is the Fourier transform of the time trace x(t)and < > means ensemble averaging.
Wave-wave coupling, namely which is the wave
having the frequency f is generated by the coupling
between the waves having the frequencies f1 and f2, is therepresentative case of which the bicoherence value
becomes significant. However, the wave-wave coupling
is not necessary the only mechanism responsible of thephase correlation revealed by bicoherence analysis.
Summation rule about both frequency f = f1 ± f2 andwavenumber k = k1 ± k2 must be obeyed to generate theanother wave by wave-wave coupling.

4. Experimental Results
Figure 1 shows Typical experimental conditions andtemporal evolution of plasma parameters for a discharge inwhich the harmonic oscillation appears: (a) heating
condition, (b) Hα intensity, (c) magnetic fluctuation, and(d) frequency spectrum of magnetic fluctuation. Plasma
was initiated by electron cyclotron heating and was further
heated by two neutral beam injection (NBI) systems.

Fig.1

Typical experimental conditions andtemporal evolution of plasma parameters fora discharge in which HO appears: (a) heating
condition, (b) Hα intensity, (c) magneticfluctuation, and (d) frequency spectrum ofmagnetic fluctuation.

Proceedings of ITC/ISHW2007
though HO (edge) observed in BES signal consists of the
1st frequency of 4.0 kHz and its 2nd harmonic. It
indicates that phase correlation between harmoniccomponents seems to be different between HO (core) andHO (edge).
We have found that b2(f1, f2) which indicates the phasecorrelation between harmonic components has a poloidal
asynmetry. Figure 3 shows the frequency spectrum ofmagnetic fluctuation having up to 3rd harmonic of HO.
The 1st, 2nd, and 3rd frequency f(1st), f(2nd), and f(3rd)is 3.0 kHz, 6.5 kHz, and 10.0 kHz, respectively. The
bicoherence planes of magnetic fluctuations (a) outside
torus and (b) inside torus are shown in Fig. 4. b2(f1, f2)outside torus has peaks with (f1, f2) = (f(1st), f(1st)), (f(2nd),

f(1st)), (f(3rd), f(1st)), (f(2nd), f(2nd)) and so on whichindicates the phase correlation between each harmoniccomponent. On the other hand, b2(f1, f2) inside torus also
has peaks located at similar frequencies as those outside
torus, however the values are small. Figure 5 shows (a)b2(f1, f2) at (f1, f2) = (f(1st), f(1st)) and (f(2nd), f(1st)) and(b) amplitudes of the 2nd (f(2nd) = f(1st) + f(1st)) and 3rd(f(3rd) = f(2nd) + f(1st)) harmonic of HO divided by that ofthe 1st harmonic with respect to poloidal angles. One cansee that relative amplitudes of higher harmonics decrease
with decreasing corresponding b2(f1, f2) in the poloidaldirection.
We should pay attention to the physical interpretationof the result of the bispectral analysis. In the present
study, b2(f1, f2) has peaks at f(1st) + f(1st) = f(2nd) andf(2nd) + f(1st) = f(3rd). However, the poloidal/toroidal
mode number m/n does not satisfy the summation rule,
namely, they show m(1st) = m(2nd) = m(3rd) = -2, n(1st) =n(2nd) = n(3rd) = 1 [23]. Therefore, generation ofharmonics can not be interpreted simply as the mode
coupling. b2(f1, f2) can be regarded as the degree of“distortion” of the signals in our case, because the poloidal
asymmetry of b2(f1, f2) and relative amplitude of higher
harmonics indicates that they have a positive correlation.
Mechanism to distort signals has not been clarified yet,which remains as a future study.

5. Summary
HO having the fundamental frequency of 3~5 kHz
and its higher harmonics has been observed in CHS. It
consists of two pairs of harmonic series. One locates incore region near ι = 0.5 rational surface, the other locatesin edge region near ι = 1.0 rational surface.
We applied the bispectral analysis to the fluctuation
data having HO measured using BES and MP. It
revealed that fundamental mode of HO in both themagnetic fluctuation and core density fluctuation havephase correlation with the harmonics including
fundamental oscillation, while HO in edge density
fluctuation does not have such phase correlation.
Generation of harmonics can not be interpreted
simply as the mode coupling, even though bicoherence

Fig.2

Bicoherence plane of (a) magnetic
fluctuation, (b) core (ρ = 0.42) density
fluctuation, and (c) edge (ρ = 0.95) density
fluctuation. b2(f1, f2) of both the magnetic
fluctuation and core density fluctuation havea clear peak having the frequencies (f1, f2) =(4.0 kHz, 4.0 kHz), while that of edge density
fluctuation does not have such a peak.

Fig.3

Frequency spectrum of magnetic fluctuation
having up to 3rd harmonic of HO.

Proceedings of ITC/ISHW2007
value b2(f1, f2) is significant, because mode numbers ofHOs are same between harmonic components having
difference frequencies. Correlation between b2(f1, f2)
and relative amplitude of higher harmonics suggests thatb2(f1, f2) means the degree of distortion of the signals.
The reason why the signals are distorted and all
harmonic components have same mode numbers should
be investigated in the future.

Acknowledgement
The work was supported in part by the National
Institute for Fusion Science Collaborative Research
Program (NIFS02KZPD003) directed by the secondauthor.

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Fig.4

Bicoherence plane of magnetic fluctuations(a) outside torus and (b) inside torus having
frequencies of HO f(1st) = 3.0 kHz, f (2nd) =6.5 kHz, and f (3rd) = 10.0 kHz. b2(f1, f2)outside torus has peaks with (f1, f2) = (f(1st),f(1st)), (f(2nd), f(1st)), (f(3rd), f(1st)), (f(2nd),
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Fig.5 (a) b2(f1, f2) at (f1, f2) = (f(1st), f(1st)) and(f(2nd), f(1st)), (b) amplitudes of the 2nd(f(2nd) = f(1st) + f(1st)) and 3rd (f(3rd) =f(2nd) + f(1st)) harmonic of HO devided bythat of the 1st harmonic with respect topoloidal angles. Relative amplitudes ofharmonics
decreasewithdecreasingcorresponding b2(f1, f2) in the poloidal
direction.