Parameters
| Value
|
Toroidal field
| <0.9 T
|
Plasma current
| <40 KA
|
Discharge time
| <35 ms
|
Electron density
| 0.7-1.5× 
|
Electron temperature
| 150 – 230 eV
|
Pressure before discharge
| 2.5-2.9 Torr
|
TheAdditionally, the tokamak also possessed a resonant helical field (RHF) that influenced plasma confinement via an external magnetic field generated by two conductors wound around the chamber with a givenspecified helicity. The minor radius for helical windings was estimated to be 21 cm (l = 2) and 22 cm (l = 3), whereas the major radius was 50 cm.
3. Methods
3.1. The biorthogonal decomposition
In the present study, the biorthogonal decomposition (BD) technique was applied to Chronos (temporal orthogonal modes) and Topos (spatial orthogonal spatial modes) to examine the space-time evolution of a complex signal. Time-space symmetry provides access to the signal structure and deterministic dynamics, regarded as and is considered a deterministic instrumenttool for a fully developed complex signal [17&19]. From this perspective, it is important to note that the dynamics produced signals in certain spatial phases. Each spatial mode belongs to one coherent structure with temporal evolution. One can conceive ofAssume a scalar quantity, such as y(x,t) (e.g., the magnetic field component), whose temporal evolution can be measured simultaneously at M spatial locations. The signal is simulated at M spatial locations and N moments of time to generate the 
matrix as follows:
; i= 1,…, N and j= 1,…, M (1)
y (
is expanded into a set of orthogonal modes through BD expansion as:
y(
(2)
and
(3)
The decomposition can also be calculated via SVD as follows:
(4)
where V and U denote N× M 
and M × M (
U=
matrixes, respectively, and 
is a unitary matrix. In addition, the diagonal M × M matrix S comprises elements 
with 
0, which are singular values of
placed in descending order. Furthermore, the singular values in S are square roots of the eigenvalues of 
or 
and are always real numbers. The eigenvectors of 
, which are the column vectors of V, are the signal’ssignal's temporal evolution known as components (PCs) that are equal to the projection of along U.
The column vectors of the U matrix, also known as principal axes, resemble the signal's fixed image and can estimate the spatial composition of the signal. In addition, the spatial eigenmodes 
(
and 
are termed Chronos and Topos, respectively. The Ak (weight)
is the diagonal values S matrix in Eq. (2). The signal's spatiotemporal properties are extracted via BD from Eq. (2). This section will define energy and entropy among the three essential signal parameters (dimension, energy, and entropy). Global space and time functions must be employed to extract temporal and spatial information. Additionally, BD can be derived through singular value decomposition.
An important advantage of biorthogonal is its weight distribution analysis, which provides usefulvaluable information about linear combinations and spatiotemporal symmetries. As demonstrated below, SVD factorization estimates the energy of signals as the sum of the squared Ak:
(5)
According to Eq. (5), a set of normalized squared singular values (dimensionless energy), 
/E is evaluated, which are the properties of a probability distribution 
and 
, and is considered a useful tool for indicating the existence mode discussed in the following section.
Each Mirnov coil has a signal due to its single-mode perturbation expressed as 
cos (
- 2πʋt), where (phase shift) is dependent on the topology (mode number) and position of the Mirnov coil, and ʋ is the temporal frequency of mode rotation. For M Mirnov coils in a poloidal cross-section and equally spaced positions, 
holds, where m is the poloidal mode number.
We consider Mirnov coils with known locationspositions and a signal matrix with isochronous samples and Mirnov channels as rows and columns, respectively. We assume that all Mirnov coils are equally sensitive to magnetic perturbations. A theoretical mode (n,m), where n and m denoterepresent toroidal and poloidal mode numbers sampled by M coils at (ϕ,ө) (toroidal and poloidal angles), respectively, and a rotating frequency of ʋ that was sampled N times, yields the following expression:
(6)
Notably, due to the spatial symmetry and dimensions of the IR-T1 tokamak, we use n=1 and ignore the toroidal effects. If the matrix 
is factorized and trigonometric sum formulas are applied, Eq. (6) can be rewritten as follows:
(7)
and 
are the first two principal axes obtained by factorization of the 
matrix. At this stagepoint, m is determinedcalculated by drawing the principal axes, counting their zero-crossings, and then dividing by 2. Due to noise, plasma's principal axes and their theoretical counterparts are not identical. Considering the nature of principal axes and a single SVD principal axis, it is determined that the signal is generated over orthogonal basis vectors. Consequently, a mode likelihood is computed as follows:
= (| .
|+ | .
|) /2 (8)
where .
is the scalar multiplication between vectors of experimental and theoretical principal axes, and the experimental principal axes are projected on their theoretical ones. In addition, the mode likelihood is estimated as the absolute projection value of the first and second couple, resulting in a limited likelihood. According to Eq. (8), the likelihood value ranges from zero to one. The first case is orthogonally complete, and the second represents the identity case between the experimental and theoretical principal.
3.2. Indicators of MHD instability indicators
Since the voltage across the coils is proportional to the derivative time of magnetic flux, the frequency determines the sensitivity of Mirnov coils to magnetic field perturbations. It is assumed that allAll Mirnov coils are assumed to have the same sensitivity to magnetic perturbations to improve theoretical principal axes. ConsequentlyThus, the theoretical principal axes are computed by assuming A=1 in Eq. (6), and RMS normalization is performed. AfterThe results must be post-processed after signal normalization and SV decomposition, the results must be post-processed to generate suitable MHD indicators.
The first post-processing is presented in Eq. (8). The likelihood of each mode is calculated and categorized using mode topology. Having more than one active mode, they They can be separated into couples by SV factorization, if they have more than one active mode, whereas singular values, which are valid in the case of equispaced Mirnov coils (such as the IR-T1 tokamak), are described in detail in Ref. [5]. SV factorization converts amplitude data to singular values. Since the degree of order/disorder is a crucial signal parameter, it is essential to specify the corresponding entropy, also known as singular values entropy. Here, theThe complete space-time signal structure is described by the global signal quantity (the global entropy):) as follows:
(9)
Furthermore, the normalizing factor 
is introduced to compare various signal entropies as the signal is obtained from different locations. A good indicator of instability is the entropy of singular values. With RMS signal normalization, the quantity of entropy is constrained between 0 and 1. H only becomes zero if a single eigenvalue is a nonzero (when all signals are concentrated in the first structure). Alternatively, H becomes 1 if all eigenvalues are equal (when energy is equidistributed in all structures). Singular entropy is suitable for having a Boolean instability indicator [2617]. Other markers are the relative squared magnitude of the first two pairs of principal axes:
(10)
(11)
Both quantities are limited between 0 and 1. If the value of Eq. (10) is closer to 1, the first mode of the signal is dominant; if the value of Eq. (11) is closer to 1, the second mode of the signal is dominant; and if either value is closer to 1, plasma performance is reduced.
4. Experimental results
In this section, an investigation is conducted using the BD method to analyze the magnetic fluctuation data from the Mirnov coils. PureTo this end, pure hydrogen is injected into the tokamak at a pressure of 2.5 torrs. Plasma current and poloidal magnetic field fluctuations profile for tokamak plasma for two shots 9611011-8 (Fig. 1 (a, b)) and 9611011-11-1 (Fig. 1 (c, d)) are depicted in Fig. 1. Fig. 2 (a) shows the eigenvalues corresponding to the principal axes (PAs) resulting from the BD method's analysis of Mirnov coils data for a typical shot of 9611011-8 in a time range of 24-26 ms.
The results indicateshow that the value of the first pair of singular values ishave a significantly greaterhigher value than that ofthe other pairs, indicating that singular values have low entropy. The significant variation between the first pair of singular values and the remaining ones indicatessuggests that the energy of the first pair of principal axes (PAs) energy is substantialsignificant, indicating the presence of an active magnetohydrodynamics (MHD) mode. Fig. 2 (b, c, d) depicts theillustrates this shot's first three pairs of PAs for this shot in the 24-26 ms time range. The poloidal mode numbers for m = 2, 3, and 4 are determinedfound by counting the graphs with the number 0 at their crossingintersection and dividing by 2. Due to the multimode MHD activities, thisThis method is ineffective infor determining the poloidal mode number forof the IR-T1 tokamak due to the multimode MHD activities.
The BD analysis method is recommended due to the simultaneous operation of several multiple MHD modes (degenerate modes) in the magnetohydrodynamics activities of the IR-T1 tokamak plasma. A usefulThe MHD instabilities are studied using a helpful index namedcalled the entropy of singular values is used to investigate the MHD instabilities, and an effective relationship will be addressed between the amplitude of the poloidal magnetic field fluctuations and the entropy value. is described. For instance, based on Fig.3(a,b), the entropy increasesrises with increasing magnetic field fluctuations in the time range between 24.4 ms and 24.6 ms, (as shown in Fig. 3(a, b)) and it isfalls to zero between 25.2 and 25.7 ms (when there are no magnetic field fluctuations in the time range between 25.2 ms and 25.7 ms.occur). Fig. 3(c) illustrates two instability indicators P1 and P2, i.e., the relative energy value for the first two PA pairs between 24 and 26 ms.
The energy value of the first PA pair P1 is greater than that of the second PA pair P2 when instabilities occur in a range of times, such as between 24.4 and 24.6 ms. Multiple positive and negative peaks are depicted on the graphs for P1 and P2 to represent the two active modes operating concurrently. In addition, using , the dominant modes can be identified. using . Fig. 3(e, d) depicts the probability of the presence of two active modes (n=1, m=2) and (n=1, m=3) that are active with energy close to one another. Fig. 3(f) shows the normalized probability-weighted quantity P1 obtained by multiplying by P1, as well as the simultaneous activity for two (n=1, m=2) and (n=1, m=3) modes.
The probability-weighted value P1 can be introduced as a weighted factor to search the mode presence probability, which can classify the modes according to their activity rate. The mode n=1/m=2 is the primary mode, and the mode n=1/m=3 is the secondary mode, as shown in Fig. 3(Ff). Moreover, Figs. 4 and 5 illustrate the magnetic field fluctuations dB/dt, entropy indicators H, P1, P2, Lnm3n1, Lnm2n13n1, 2n1, and Ln n,m((t)*P1(t) for the 9611011-11-1 and 9611011-8 shots, respectively. For 9611011-11-1 and 9611011-8, the resonance helical magnetic fields (RHF) are (l=2, n=1) and (l=3, n=1), respectively. In addition, the results reveal the magnetic field fluctuations, the entropy of singular values, and relative energy values for the initial two PA pairs. Furthermore, the probability of the present modes is altered in accordance with the normal case using RHF. The data graphs of shots No. 961011-1 and 961011-8 are drawn in Figures 4 and 5. The comparison of Fig 4(a & b) and Fig 5(a & b ) with Fig 3(a & b ) in the time interval of 25 to 25.2 milliseconds shows that with the application of the RHF magnetic fluctuations and the entropy of the singular values increase. The comparison of figures 4(f) and 5(f) with Fig 3(f) in the interval time from 25 to 25.2 shows that the weighted probability of the p1 index increases with the application of the RHF and the simultaneous presence of two active modes m=2 n=1 and m=3 n=1 increases, but the probability of the presence of the mode m=2 n =1 is more than the probability of presence of mode m=3 n=1, that is, mode m=2 n=1 is the dominant mode. The results show that the application of resonant helical field strengthens the presence of active modes. By comparing Fif3(d,e) , Fig 4(d,e) and 5(d,e) , these results can be seenIn addition, using RHF, the probability of the present modes is modified in accordance with the normal case.
ShotFigs. 4 and 5 depict the data graphs for shots No. 9611011-11-1 and 961011-8, respectively. Comparing Fig. 4(a, b) and Fig. 5(a, b) with Fig. 3(a, b) in the time interval of 25 to 25.2 ms reveals that magnetic fluctuations and the entropy of singular values increase with the application of RHF magnetic fields.
The comparison of Figs. 4(f) and 5(f) to Fig. 3(f) in the time interval from 25 to 25. 2 demonstrates that the weighted probability of the P1 index increases with the application of RHF, as does the simultaneous presence of two active modes m=2 n=1 and m=3 n=1. However, the likelihood of the presence of mode m=2 n =1 is a continuation ofgreater than the probability of the presence of mode m=3 n =1, indicating that mode m=2 n =1 is the dominant mode. The results indicate that applying a resonant helical field improves the presence of active modes. This is evident when comparing Fig. 3(d, e), Fig. 4(d, e), and 5(d, e).
Shot No. 9611011-11-2 (Fig 6 )continues Shot No. 1-9611011-11-1. In this shot, the combination of l=2&3 fields has beenwas applied in thebetween a time interval of 30 toand 35 milliseconds .ms. The constant P2 index (Fig. 6(c)) indicates the absence of MHD modes, and the presence probability of the modes is almost zero in the figures (Fig. 6 (d) and (, e))) for the time interval 30 ms to 34 ms. In Fig. 6(c), the increase in P1 (Eq. (10)) indicates the onset of plasma instability. The P1 indicator remains quiterelatively high between 34 and 34.5 ms, resulting in a more robust mode indicator. Notably, several negative and positive peaks in P1 and P2 have confirmedconfirm the continuous development and disappearance of the two modes. However, Fig. 6(d, e, and f) demonstrateindicate that the presence probability of modesmode presence is low when combined RHF (l=2&3, n=1) is applied.
5. Conclusion
The present study examinedused the BD technique to examine data collected from an array of 12 Mirnov coils in the IR-T1 tokamak plasma using the BD technique. MHD instabilities were then investigated, and the corresponding mode number was assigned. The magnetic fluctuations corresponding to the rotating MHD modes were also examined by evaluating the SVD instability indices (H, P1, and P2).
The results revealed a strong correlation between the entropy of singular values (H), the relative square value for the first two PA pairs (P1 and P2), and plasma instabilities. In addition, the presence of two active (m=2, n=1) and (m=3, n=1) modes was determined, with the (m=2, n=1) mode being superior, by calculating the probability of the modes' presence in the tokamak. The activities of MHD modes were then evaluated with and without RHF. As a result, it was observed that the magnetic field fluctuations in the first case were lower than in the second, whereas the entropy of singular values roseincreased. Using RHF (l=2, n=1) and RHF (l=3, n=1), the likelihoods () of the (n=1, m=2) and (n=1, m=3) modes increased, while utilizing RHF (l=2&3, n=1) decreased the likelihoods of both modes.
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Figure 1: plasma current and magnetic fluctuations (dB/dt) for IR-T1shot 9611011-8 & 9611011-11-1
Figure 2: SVs(a) and PAs (b,c & d ) of the Mironov coil signals obtained by BD Shot 9611011-8
Figure 3. (a): Mirnov coil data , (b) :H Entropy (c): 
and 
indicators (d): Ln (n =1,m =2) Likelihood, ( e ) : Ln (n =1,m =3) Likelihood and ( f ): weighted likelihoods for IR-T1 shot no .9611011-5 without RHF in the time during 24-26 ms .
Figure 4. (a): Mirnov coil data plot , (b) :H Entropy (c): 
and 
indicators (d): Ln (n =1,m =2) Likelihood, ( e ) : (n =1,m =3) Likelihood and ( f ): weighted likelihoods for IR-T1 shot no .9611011-11-1 with RHF (L=2, n=1 ) in the time during 24-26 ms .
Figure 5: (a): Mirnov coil data plot , (b) :H Entropy (c): and indicators (d): Ln(n =1,m =2) Likelihood,( e ) : Ln(n =1,m =3) Likelihood and ( f ): weighted likelihoods for IR-T1 shot no .9611011-8 with RHF (L=3 , n=1) in the time during 24-26 ms .
Figure 6: (a) Mirnov coil data plot (b): H Entropy H (c) p1, p2 indicators (d) Ln n=1,m=2 Likelihood (e) Ln n=1,m=3 Likelihood , and (f) p1 weighted mode likelihoods for IRT-1 shot no 9611011-11-2 with RHF (L=2&L=3 n =1) in the time during 30-35 ms
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, where m denotes the number of poloidal modes, and n represents the toroidal mode number [6].