Introduction

During the disruption scenario in plasma, eddy and halo currents were considered to be the most important sources that came out as a result of this phenomenon [1]. The components which affect primarily are diverter, First Wal (FW) and other main components, because plasma is unstable in vertical displacement. In this scenario, plasma moved upward and downward resulting into plasma disruption, and as a result, halo current has been generated helically. This halo current was produced in SOL component and flows into the vacuum vessel through in-vessel components that may give rise to large force acting on the vessel and in-vessel components [2]. The number of research articles has been published to investigate the halo current in other Tokamak devices such as JET [3], JT-60U [4], and NSTX [5]. Like in Experimental Advanced Superconducting Tokamak (EAST) reactor [6,7,8], there are some failures of the feedback control in VD event caused by disruption. The EAST reactor is designed by Institute of Plasma Physics, Chinese Academy of Sciences, P.R. China which is considered to be the best superconducting and advanced Tokamak in the world.

Recently, EAST has been updated and had achieved longer pulse generations at high current mode. Main parameters of EAST reactor are given in Table 1 [9].

Table 1 Parameters of EAST [9]
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The EAST has been demonstrated as long-pulsed plasma operations with toroidal field Bt ≤ 3.5 T and plasma current Ip ≤ 1 MA [10,11,12]. The EAST upper diverter has been upgraded with W/Cu plasma facing components (PFCs) with ITER like W-monoblack [13]. The lower diverter has not been upgraded using graphite tiles for the first wall. It has a central dome, but the upper diverter does not equip with it. In addition, an EAST disruption database has been built [14] and is useful for quickly selecting disruptive discharges and their relevant parameters. Nearly, 27% of discharges terminated in a disruption [15].

In this reactor, some sensors have been installed for calculating the halo currents at different locations, as given in Table 2. In these experiments, it was observed that the halo current first spread out on outer baffle plate then moves to dome and finally return back to plasma. At first, we have developed model (2) to calculate magnetic field produced by circular current loop. Furthermore, we considered two cases to analyze the plasma when it flows up/down direction as observed by model (2) calculation data. Second, we have theoretically calculated horizontal/vertical forces connected with model (2). It was observed that generation of halo current was huge and it has strong field at the middle and weak or cancelling effects appear at the sideways positions. Third, during disruption, magnetic field appears at the conducting points consisting of large number of magnetic flux. Therefore, halo current model (18) has been developed using Eqs. (9 and 15) with specific parameters (rp, z, zp, ushift, \(\phi\)). A Matlab program has been developed to calculate halo current and magnetic field calculation points. The achievable maximum halo current was about 0.4 times of the plasma current and its maximum TPF values was 0.65 as estimated by set of sensors. EAST halo current is 10 KA for one cassette and total estimation is 400 KA by model (18).

Table 2 EAST halo diagnostics [17]
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Some of the work that has already been published for halo current is given as under

  • Strauss with his team carried out heavy simulation using M3D code for calculating the halo current and plasma disruption conditions [16].

  • Long et al. experimentally calculated the values of eddy and halo current in EAST reactor while conducting set of VDE experiments [17].

  • Other impressive contributions by Strauss and his team for assessing and simulation of halo current in ITER through VDE instabilities and developed 3D models for plasma disruption conditions [18].

  • Fantechi and Crutzen modeled the plasma FW contact under vertical instabilities with a 3D eddy current simulation. The simulation worked for transferring the electric current (halo current) to FW component through VDE [19].

Model of magnetic field calculation points for Tokamak

In this research work, a new model for calculating different aspects of Tokamak reactor has been designed. In the recent past, number of work has been published already about the shaping and geometrical description of plasma [20, 21]. One of the important problem is to identify and simulate the plasma shape and control including the numerical calculations on elongated and shaped equilibria [22]. In this paper, a new model was developed on the basis of an applied mathematical approach [23], that is

$$\begin{aligned} R^{\prime} &= R_{0} + a_{\text{p}} \cos (\theta + \delta \sin \theta ) \\ Z^{\prime} & = Z_{0} + ka_{\text{p}} \sin \theta \end{aligned}$$
(1)

where “k” is elongation, “δ” is triangularity, “a” is minor radius, and “R ”0 is the major radius. Using the algebraic techniques, as the magnetic field is B = BX + BY, we obtained Bx and Bz using shape model (1) in energy integral [24]. The calculated points are given as under

$$\begin{aligned} B_{x} & = B_{r} \cos \alpha - B_{\theta } \cos \left( {\frac{\pi }{2} - \alpha } \right) = B_{r} \cos \alpha - B_{\theta } \sin \alpha \\ B_{z} & = B_{r} \sin \alpha + B_{\theta } \sin \left( {\frac{\pi }{2} - \alpha } \right) = B_{r} \sin \alpha + B_{\theta } \cos \alpha . \\ \end{aligned}$$
(2)

Magnetic field produced by a circular current loop [25] is

$$\begin{aligned} B_{r} & = \frac{{\mu_{0} I}}{2\pi r}\frac{z - b}{{\left[ {(a + r)^{2} + (z - b)^{2} } \right]^{{\frac{1}{2}}} }}\left[ { - K(k) + \frac{{a^{2} + r^{2} + (z - b)^{2} }}{{(a - r)^{2} + (z - b)^{2} }}E(K)} \right] \\ B_{\theta } & = \frac{{\mu_{0} I}}{4\pi r}\frac{1}{{\left[ {a + r)^{2} + (z - b)^{2} } \right]^{{\frac{1}{2}}} }}\left[ {K(k) + \frac{{a^{2} - r^{2} - (z - b)^{2} }}{{(a - r)^{2} + (z - b)^{2} }}E(K)} \right] \\ K(k) & = \int\limits_{0}^{\pi /2} {\frac{{{\text{d}}\alpha }}{{\sqrt {1 - K^{2} \sin^{2} } \alpha }}} ,E(k) = \int\limits_{0}^{\pi /2} {\sqrt {1 - K^{2} \sin^{2} } \alpha {\text{d}}\alpha } , \\ \end{aligned}$$
(3)

where R0 = 1.7 –1.8, B0 = 3.5T, plasma current Ip ≤ 1 MA.

Plasma orientation and development of halo current

In case of plasma, it was observed that some balancing and unbalancing forces acted upon, like the case of plasma as torus which is very close to outboard wall. At this stage, poloidal field has different radial locations and have variable sideways forces as well. In this case, one side of torus has higher values than other side and poloidal field has different values at different locations.

Case 1

In this case, we considered the plasma at a position (R, Z) from the origin and assumed that during the disruption occurs (Fig. 1), the plasma moves horizontally [26, 27]. The governing equations are given as under

$$F_{\text{h}} = F_{1} \cos \alpha + F_{1} \sin \alpha ,\;\alpha = 0,180,360 \ldots$$
(4)
$$F_{\text{h}} = F_{1} \cos \alpha .$$
(5)
Fig. 1
figure 1

Plasma orientation (case 1)

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Since

$$F_{\text{h}} = BI\Delta x\cos \alpha ,$$
(6)

where

$$\Delta x = \cos \alpha \delta x_{\text{p}} .$$

Therefore

$$F_{\text{h}} = BI_{\text{p}} \left( {\cos \alpha \delta x_{\text{p}} } \right)\cos \alpha .$$
(7)

Since the plasma is in the range of 0 → 2π, therefore

$$F_{\text{h}} = \int\limits_{0}^{2\pi } {BI_{\text{p}} \cos^{2} \alpha \delta x_{\text{p}} } {\text{d}}\alpha .$$
(8)
$$F_{\text{h}} = BI_{\text{p}} \delta x_{\text{p}} \int\limits_{0}^{2\pi } {\cos^{2} \alpha {\text{d}}\alpha } = \pi BI_{\text{p}} \delta x_{\text{p}} .$$
(9)

Substitution of Eqs. (2) into (9) gives the changes in magnetic field while applying the horizontal forces on the plasma giving plasma position in different points, as presented in Table 3. For static position, each cross section depends on a, b, r, and z. For one cross section, the peak values of r and z are (± 5.2, ± 4.37); therefore, (Br(max,up), Br(max,down)) = (0.3918, − 0.3847). Then, the current loop of magnetic field range was (Bx, Bz)max = (0.3862, 0.1698) for selected degree of cos (alpha) and plasma one cross-sectional area energy was calculated to be 8.0262e + 004. For tilting position, peak values of r and z were (± 5.4, ± 4.17); therefore, (Br(max,up), Br(max,down)) = (0.4225, − 0.4247). Hence, the current loop of magnetic field range was (Bx, Bz)max = (0.4221, 0.1974) for selected degree of cos(alpha) and calculated plasma one cross-sectional area energy was 8.0258e + 004 (see Table 3).

Table 3 Magnetic field with horizontal displacement by Eqs. (2 and 9)
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Case 2

In this case, we considered plasma at a position (R, Z) from the origin and assumed that during disruption occurs (Fig. 2), plasma moves vertically downward [26, 27]. The governing equations are given as under

$$F_{v} = F_{2} \cos \beta + F_{2} \sin \beta ,\,\beta = 90,270 \ldots$$
(10)
Fig. 2
figure 2

Plasma orientation (case 2)

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Then

$$F_{v} = F_{2} \sin \beta .$$
(11)

Since

$$F_{v} = BI\Delta z\sin \beta ,$$
(12)

Where

$$\Delta z = - \sin \beta \delta z_{\text{p}} .$$

Therefore

$$F_{v} = BI_{\text{p}} \left( { - \sin \beta \delta z_{\text{p}} } \right)\sin \beta .$$
(13)

Since the plasma is in the range of 0 → 2π:

$$F_{v} = - \int\limits_{0}^{2\pi } {BI_{\text{p}} \sin^{2} \beta \delta z_{\text{p}} } {\text{d}}\beta .$$
(14)

and

$$F_{v} = - BI_{\text{p}} \delta z_{\text{p}} \int\limits_{0}^{2\pi } {\sin^{2} \beta {\text{d}}\beta } = - \pi BI_{\text{p}} \delta z_{\text{p}} .$$
(15)

Changes in magnetic field can be described by substituting Eq. (2) into Eq. (15) to get the vertical forces applies to the plasma so as to calculates the plasma position at different points. Like in Case-1, each cross section depended upon a, b, r, and z. For one cross section, the peak values of r, z were (± 3.5, ± 2.3); therefore, (Br(max,up), Br(max,down)) = (0.1865, −0.4652). Then, the current loop of magnetic field range was (Bx, Bz)max = (0.4637, 0.1995) for selected degree of cos(alpha) and plasma one cross-sectional area energy was 8.0457e + 004. For tilting position, peak values of r and z were (± 3.8, ± 2.5); therefore, (Br(max,up), Br(max,down)) = (0.2762, − 0.3851). The current loop of magnetic field range was (Bx, Bz)max = (0.5747, 0.2525) for selected degree of cos(alpha) and the calculated values of plasma one cross-sectional area energy was 8.0484e + 004 (see Table 4).

Table 4 Magnetic field with vertical displacement by Eqs. (2 and 15)
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During VDE, plasma changes the area under vertical and horizontal forces and the magnetic field changes in radial direction at center of plasma with major radius (R), x–z displacement and plasma radial/vertical length (zp, rp). During this plasma change, magnetic field of each point can be calculated by model (2) (vertical and horizontal). The magnetic field appears at the conducting points consisting of large number of magnetic flux, which are given below as

$$\begin{aligned} \phi & = \phi_{1} + \phi_{2} + \phi_{3} + \phi_{4} \cdots + \phi_{n} \\ & = BA_{1} \cos \phi_{1} + BA_{2} \cos \phi_{2} + BA_{3} \cos \phi_{3} \cdots BA_{n} \cos \phi_{n} \\ & = B\left( {A_{1} \cos \phi_{1} + A_{2} \cos \phi_{2} + A_{3} \cos \phi_{3} \cdots A_{n} \cos \phi_{n} } \right) \\ \phi_{\text{total}} & = B\sum\limits_{i = 1}^{n} {A_{i} \cos \phi_{i} } . \end{aligned}$$
(16)

At the conducting points, generation of halo current was huge and it has strong field at the middle and weak forces or cancelling effects appeared at the sideways positions. Due to horizontal and vertical forces, some balancing and unbalancing forces appeared and it was expected that asymmetric plasma positions were along the toroidal coordinates. The poloidal halo current which balances the plasma vertical displacement was toroidally asymmetric. According to the plasma positions, these halo currents have different flow poloidal paths along with the toroidal coordinates. Similarly, due to the asymmetry of poloidal halo currents, the second sideways force occured and the sum of these forces gave sideways balancing forces as well:

$$u(\alpha ,\beta ) = c\delta x_{\text{p}} .\cos \alpha + d\delta z_{\text{p}} .\sin \beta .$$
(17)

The total forces includes horizontal and vertical and the summation of these forces acted upon the plasma VDE direction gives halo current as given below:

$$I_{\text{halo}} = \frac{2}{B}\left[ {\frac{{F_{\text{h}} }}{{(A_{\text{p}} - z_{\text{p}} )u_{1} }} + \frac{{F_{v} }}{{(A_{\text{p}} - x_{\text{p}} )u_{2} }}} \right] + \phi_{\text{total}} .$$
(18)

During the course of disruption, halo and eddy currents considered to be the main source of electro-mechanical loads that appeared. Consequently, halo current fraction and toroidal peaking factor (TPF) in vessel components depend upon the halo current density. In MHD simulation problems, plasma model comprises of three regions, namely, core, halo, and resistive wall region integrating plasma to external vacuum magnetic field. In EAST reactor, plasma was inherently unstable against vertical displacement and during upward and downward movement creates disruption along with large halo current generation. In this case, when the plasma flows into the vacuum vessel through in-vessel components, halo current produced large values of J × B forces acting on the vessel through in-vessel components. The production and movement of halo current is such that it first appeared on the outer plate in clockwise direction and maximum generation of halo current was estimated to be about 0.4 times of the plasma current. Figure 3 shows the evolution of halo current and filament. EAST halo current is 10 KA for one cassette and total 400KA recorded by model (18). In EAST, Rogowski coils have been designed for both the upper and lower diverters to measure the disruption of halo currents. EAST upper diverter was upgraded with a new tungsten diverter consisting of 80 cassettes in the toroidal direction. Four upper diverter, cassettes have been instrumented with a set of 10 small-cross sections. Rogowski coils to determine where the halo currents enter and exit the diverter, and how much current flows through the water cooling tubes. In this paper, we have successfully performed theoretical investigation between halo current-plasma deformation/displacement and theoretical calculation of total halo currents as well.

Fig. 3
figure 3

Evolution of halo current and filament

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Conclusion

The developed theoretical model calculates plasma cross sections by Bx and Bz magnetic field points and displacement subjected to the start of VDE and magnetic field flux variations. The developed two conducting points give an indication to halo current percentages as well. This model can calculate theoretically halo current during the disruption phases in a very short time. Furthermore, mathematical techniques have been developed successfully which shows the relation between halo currents and plasma displacement/deformation in EAST Tokamak. Computational program has been developed to calculate total halo current and magnetic field calculation points. Theoretical investigation of each cassette has been calculated by model. This model can be subjected to experimental data for other Tokamak devices as well.