Measurements of a cylindrical plasma have been conducted at different plasma cross-sections using three sets of tomography in the plasma device, PANTA (Plasma Assembly for Nonlinear Turbulence Analysis) [1]. The plasma structures of each cross-section have been analyzed using orthonormal function series expansion [1, 2]. However, the function expansion technique has not been developed for pursuing relations between structures at different cross-sections. This paper proposes such a simple and direct technique using function expansion methods, called structural correlation analysis, to analyze spatiotemporal relations between plasma images from two tomography systems.
In the analysis of plasma structures and mode patterns, a tomography image ε(r,θ) is expanded as
|
ε
(
r
,
θ
)
=
∑
n
α
0
,
n
ψ
0
,
n
(
r
)
+
∑
m
,
n
[
α
m
,
n
cos
(
m
θ
)
+
β
m
,
n
sin
(
m
θ
)
]
ψ
m
,
n
(
r
)
,
| (1) |
where ψm,n(r), m and n indicate the radial basis function (Bessel, Rectangular, Zernike function, and so on), azimuthal and radial mode numbers, respectively.
The structural correlation analysis can be applied to any choice of radial basis function. The analysis directly uses the expansion coefficients, cm,n(f)=F[cm,n(t)]=αm,n(f)+iβm,n(f), whose absolute value and argument represent the strength and inclination of the corresponding pattern of each basis function. The analysis defines complex vectors whose elements consist of these expansion coefficients as C→=(C→1,⋯,C→m,⋯,C→M) and C→m=(c→m1,⋯,c→mn,⋯,c→mN), which are called the general and modal complex vectors, respectively, where M and N denote the maximum numbers. Then overall and modal structural coherences, Γ and Γm, are defined as
|
Γ
=
⟨
C
→
X
⋅
C
→
Y
∗
⟩
⟨
|
C
→
X
|
⟩
⟨
|
C
→
Y
∗
|
⟩
,
Γ
m
=
⟨
C
→
m
X
⋅
C
→
m
Y
∗
⟩
⟨
|
C
→
m
X
|
⟩
⟨
|
C
→
m
Y
∗
|
⟩
,
| (2) |
where |⋅| and ⟨⋅⟩ represent the absolute value and the ensemble average, respectively. Their absolute values and the arguments indicate the overall (or modal) coherence and inclination angle between two structures represented by X and Y. In addition, simple algebra yields their relation, Γ=∑mwmΓm, where the weighting factor is defined as wm=⟨|C→mX|⟩⟨|C→mY∗|⟩/⟨|C→X|⟩⟨|C→Y∗|⟩.
Figure 1 presents the tomography images of the argon plasma to which the analysis is applied. The plasma (length: 4 m, radius: a = 5 cm) is produced at a gas pressure of 3 mTorr, a magnetic field of 0.09 T, and an RF power of 3 kW. The images observed at axial positions z = 0.35 and 0.85 m were expanded using the Fourier-Bessel Function (FBF) to obtain the corresponding expansion coefficients. Figure 1 shows the FBF images of the plasma emission and fluctuation at two cross-sections. The fluctuation images, denoted by ε~(r,θ,t), are calculated as follows: ε~(r,θ,t)=ε(r,θ,t)−ε¯(r,θ), where ε¯(r,θ) represents the time-averaged emission.
The structural correlation analysis is applied to the tomography images in the discharge. Figure 2 illustrates the results of the structural coherence between images at 0.35 and 0.85 m, along with the fluctuation powers of the FBF coefficients. The calculation uses an ensemble of 80 for a time window from 0.3 to 0.5 s. Note that a negative frequency indicates waves propagating in the electron diamagnetic direction. The results of the fluctuation power analysis show that the m = 1 fluctuations are dominant, and that coherent modes exist at frequencies of f = −4, −2, 2, and 4 kHz. High structural coherence, suggesting stiffness of the structure along the axial direction, is mainly observed at the negative-frequency peaks, f = −2 and −4 kHz. These results suggest the presence of coherent modes propagating in the electron diamagnetic direction. Moreover, it should be natural that the general structural coherence is lower than that of the m = 1 and 2 modes, since the general coherence includes many other incoherent contributions.
Fig. 2.
Total power of the FBF coefficients from m = 0 to 4, Ptotal, defined as Ptotal=∑m∑n|cm,n(f)|2, at (a) 0.35 and (b) 0.85 m, and (c) general structural coherence (m = 0–4) and modal coherences for the m = 1 and 2 modes.
Figure 3 shows the results of the phase difference, expressed as histogram diagrams. Note that the phase difference represents the real inclination between the two images. Each phase data point is obtained from an individual ensemble, and the ensemble average is indicated by the solid line. The histograms at f = −2 and 3 kHz are shown as examples on the right side. It is evident in every diagram that the high-coherence frequency at f = −2 kHz shows a sharp distribution, while the low-coherence frequencies exhibit broad ones. In other words, at a high-coherence frequency, the two images change while keeping a constant inclination angle, whereas at a low-coherence frequency, the relation remains undetermined. Therefore, care must be taken, as the inclination between the structures is significant only when the coherence is sufficiently high.
Fig. 3.
(a) Histogram diagrams of general phase differences for m = 0–4, (b) those of modal phase difference for the m = 1 mode, and (c) the m = 2 mode. A solid red line represents the ensemble average of general phase differences, while the red dashed line indicates a linear fit of the phase difference. The histograms at f = −2 and 3 kHz are shown on the right-hand side for each diagram.
Assuming that the inclination angle is linearly related to the frequency as ϕ=ωτ (see Fig. 3(a)), a rough evaluation using least squares fitting gives τ = 16.4 μs as the coefficient of proportionality. Using the axial distance between the two cross-sections dz(= 0.5 m), the axial wavenumber can be calculated from the relation kz=ϕ/dz. Note that deviations from the linear dispersion relation are observed at high-coherence frequencies around f = −2 and −4 kHz in the m = 1 and 2 diagrams, suggesting their own propagation properties.
On the other hand, the dispersion relation obtained from an individual realization can be written as ω=kV=kzvz+kθvθ=kzVζ, where the apparent propagation velocity is defined as Vζ=vz+(kθ/kz)vθ=dz/τ ∼ 30.5 km/s. Then, a simple estimation using the above linear relation and azimuthal mode number gives kz=ω/Vζ≲(2π×10)/30.5 ∼ 2m−1 and kθ∼m/r≳1/a ∼ 20 m−1, respectively. Thus, kθ≫kz, and the dispersion relation is expressed approximately as ω∼kθvθ. At f = −2 kHz, where the m = 1 mode is dominant, for example, vθ ∼ V ∼ 0.6 km/s, which is found to be comparable to the electron diamagnetic drift velocity in this discharge.
Finally, the proposed analysis is useful for investigating the spatiotemporal relationship between overall and modal fluctuation structures. The present results indicate, for example, that the observed inclination of the coherent mode structure should be caused by the azimuthal rotation of a stiff helical structure, and that the apparent propagation velocity, Vζ should correspond to the one with which the stiff structure is assumed to move in the axial direction.
This work is supported by JST SPRING Japan Grant Number JPMJSP2136, JSPS KAKENHI Grant Number JP25H00619, and RIAM Joint Research 2025EC-NU-2.
