Plasma tomography is being developed at a linear magnetized plasma device PANTA. The capability of tomography to measure the entire plasma cross-section [1] has recently revealed that the pattern distortion of solitary wave oscillations should be caused by its nonlinear interaction with the background asymmetry [2]. The tomography shows that the fluctuation patterns of the solitary wave oscillations, as shown in Fig. 1, should rotate with suffering distortion. In this work, a method called moment vector analysis has been used to quantify the degree of the global pattern distortion of fluctuations. Here, the moment vector, (Am(t), Bm(t)), is defined for each azimuthal mode number, m, as
|
A
m
(
t
)
=
1
π
∫
ε
̃
(
r
,
θ
,
t
)
cos
(
m
θ
)
dS
,
B
m
(
t
)
=
1
π
∫
ε
̃
(
r
,
θ
,
t
)
sin
(
m
θ
)
dS
,
| (1) |
where ε̃(r,θ,t) represents the fluctuation of emission. In the analysis, the distortion is successfully expressed as the modulation in amplitude and phase of the vectors (see Fig. 1(b)), or deformation of a circle in its Lissajous. The Lissajous trajectory corresponds to the variable Cm(t)=Am(t)+iBm(t) in the complex plane. In this paper we present extended analysis, local moment vector analysis, using Fourier-Rectangular Function (FRF) expansion [3], which can quantify local property of the pattern distortion as a function of radius, with the results obtained after the application of the local method.
Fig. 1.
(a) The deterministic trend of solitary wave oscillation. (b) Lissajous trajectories of the global moment vector Am and Bm in azimuthal mode m = 1, 2, and 3.
The solitary wave oscillations are observed at PANTA under the operation condition: magnetic field strength, gas pressure, and RF power are set to 700 G, ~0.4 Pa, and 3 kW, respectively. The tomography system has totally 128 azimuthal channels installed to detect line-integrated emission. The local emission profile is reconstructed from the set of data using the Maximum Likelihood-Expectation Maximization (ML-EM) method.
The obtained two-dimensional image can be represented by FRF as follows,
|
ε
̃
(
r
,
θ
)
≃
∑
n
=
1
N
r
a
0
,
n
R
n
(
r
)
2
+
∑
n
=
1
N
r
∑
m
=
1
M
R
n
(
r
)
[
a
m
(
r
n
,
t
)
cos
(
m
θ
)
+
b
m
(
r
n
,
t
)
sin
(
m
θ
)
]
,
| (2) |
where Rn(r) is a rectangular function that has a finite value in a radial band from r=rn-δr to r =rn+δr. By substituting Eq. (2) to Eq. (1), the result is expressed as the summation of local moment vectors (am(rn,t), bm(rn,t)), which can evaluate the local moment distortion [3] as a function of radial position and azimuthal numbers. Figure 2(a) shows the examples of the application of the method to the solitary wave oscillation in Fig. 1: the moment vector elements, their amplitudes and phases for m = 1 ~ 3 in the radial band of r ~ 2.5 cm. The results clearly show that every azimuthal mode is modulated in amplitude and phase during a cycle of the oscillations. In addition, as shown in Fig. 2(b), the Lissajous trajectories of the complex variables, cm(rn,t)=am(rn,t)+ibm(rn,t), exhibit obvious deviation from circles, similarly to the global analytical results shown in Fig. 1(b), obviously deviating from a circle.
Fig. 2.
(a) The temporal evolution of the moment vector elements am and bm of azimuthal mode (red and blue dotted line) and its amplitude (black line) and phase (green line) at r ~ 2.5 cm. (b) Lissajous trajectories of moment vectors (am(rn, t), bm(rn,t)).
The degree of the distortion is quantified by introducing the Fourier coefficients, cm(rn,t), whose definition are
|
c
m
R
(
r
n
,
ω
)
=
a
̂
m
(
r
n
,
ω
)
+
i
b
̂
m
(
r
n
,
ω
)
,
c
m
L
(
r
n
,
ω
)
=
a
̂
m
*
(
r
n
,
ω
)
+
i
b
̂
m
*
(
r
n
,
ω
)
,
| (3) |
where the coefficients, cmR(rn,ω) and cmL(rn,ω) represent right-hand (clockwise) and left-hand (counter clockwise) rotating mode, respectively. Here, âm(rn,ω) and b̂m(rn,ω) are the Fourier coefficients of am(rn,t) and bm(rn,t), respectively. The coefficients satisfy the following relationships,
|
a
m
(
r
n
,
t
)
=
∑
a
̂
m
(
r
n
,
ω
)
e
i
ω
t
+
∑
a
̂
m
*
(
r
n
,
ω
)
e
−
i
ω
t
,
b
m
(
r
n
,
t
)
=
∑
b
̂
m
(
r
n
,
ω
)
e
i
ω
t
+
∑
b
̂
m
*
(
r
n
,
ω
)
e
−
i
ω
t
,
| (4) |
where the superscript asterisk (*) denotes the complex conjugate. Based on the above preparation, the degree of the distortion can be evaluated in terms of deviation from circles in the complex space, and modal polarization characteristics (standing or rotating). The deviation from circle is evaluated by an index,
|
ξ
(
r
n
,
M
)
=
∑
m
p
(
r
n
,
m
,
M
ω
)
-
p
(
r
n
,
M
,
M
ω
)
p
(
r
n
,
M
)
,
| (5) |
where p(rn,M)=∑mp(rn,m,Mω) means the total power of the mode at frequency Mω. The parameter can range from 0 to 1; if ξ = 0 it means that the oscillation is completely harmonic, or that the corresponding Lissajous is a complete circle. On the other hand, the degree of the polarization
|
η
(
r
n
,
M
)
=
∑
m
p
±
(
r
n
,
m
,
M
ω
)
γ
(
r
n
,
m
,
M
ω
)
p
(
r
n
,
M
)
,
| (6) |
where γ(rn,m, Mω) represents the power fraction of linear polarization to the total (see Ref. [4] for the details of ‘polarization’). The parameter also ranges from 0 to 1; if η = 0 and 1, it corresponds to complete circular and linear polarization, respectively. Figure 3 shows these local indices as a function of radius, revealing that the degree of distortion tends to decrease in both indices toward outer direction. In other words, the mode characteristics change from standing in center to almost circular rotating mode in the outer region.
Fig. 3.
(a) The radial profile of the degree of distortion parameter ξ(rn) of modes at harmonic frequencies. (b) The degree of linear polarization contamination η(rn) of harmonic modes.
In summary, moment vector analysis is extended to investigate local characteristics of distortion of fluctuation patterns using FRF expansion in terms of two aspects: contamination of non-harmonic modes and modal rotation (polarization) property. The proposed method succeeds in characterizing the local properties of the distortion observed in the solitary wave oscillations.
This work was partly supported by JST SPRING, Grant Number JPMJSP2136, JSPS KAKENHI Grant Numbers JP17H06089, JP20K14443, JP21K13898, JP22H00120, JP23KJ1702 and JP25H00619.
