- Email: [email protected]
Conceptual design of the island divertor coils on the J-TEXT tokamak
Fusion Engineering and Design xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Conceptual design of the island divertor coils on the J-TEXT tokamak ⁎
Song Zhoua, Nengchao Wanga, , Bo Raoa, Zhuo Huanga, Da Lia, Mao Lia, Ruo Jiaa, Wei Zhanga, Shiyi Penga, Zebao Songa, Ying Hea, Kexun Yua, Yonghua Dinga, Yunfeng Lianga,b,c a International Joint Research Laboratory of Magnetic Confinement Fusion and Plasma Physics, State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China b Forschungszentrum Jülich GmbH, D-52425, Jülich, Germany c Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Island divertor Resonant magnetic perturbation Magnetic island Coil design Tokamak
A set of new resonant magnetic perturbation (RMP) coils is designed for the J-TEXT tokamak to form the island divertor (ID) configuration [F. Karger, Phys Lett A, 61 (1977) 385]. This design is aiming at investigating the effect of ID on the peak heat load reduction in a tokamak. The RMP field generated by these new coils can produce magnetic island in the plasma boundary. The field line of this magnetic island is opened by divertor targets, so as to form the island divertor configuration. Therefore, the edge island should be a vacuum island, which can be described by an analytical mode with assuming linear superposition of the plasma equilibrium field and the magnetic perturbation in a vacuum approximation. Numerical calculations have been carried out to calculate several parameters for the island divertor configuration such as the perturbed magnetic field generated by the coils, its spectrum, the field line tracing and the Poincaré plot. Conceptual design is based on the m/ n = 4/1 magnetic island, which can be generated by either 16 out-vessel saddle coils or 8 in-vessel S-type modular coils. Here, m and n are the poloidal and toroidal mode numbers respectively. The ID configuration can be then established by the boundary 4/1 island intersecting with proper divertor targets.
1. Introduction The high heat load on divertor target is one of the essential issues that must be addressed for next-step high-power steady-state fusion devices [1,2]. The heat load on the ITER divertor is expected to be above 10 MW m−2 in steady state or up to 0.6 ˜ 10 GW m−2 during disruption and ELMs [3]. An essential task of the divertor is to reduce/ disperse this power to an acceptable level, preferably less than 5 MW m−2 [4]. The high steady-state heat load is due to the significant reduction of the power decay length λq with increasing poloidal magnetic field Bp [5]. It is found in the heuristic drift-based model [6] that λq is proportional to the connection length L||. In the scrape-off layer (SOL) of a tokamak, L|| is inversely proportional to the poloidal magnetic field Bp. Many improvements of the poloidal divertor have been made to reduce the heat load on the target, e.g. snow-flake [7] or Super-X configuration [8] and radiation divertor [9,10]. An alternative concept is using the island divertor configuration, which has been proposed in 1977 for tokamak [11]. It has been established successfully in tokamak, e.g. TEXTOR [12], and stellarators, such as W7-AS [13], LHD [14] and recently in W7-X [15]. The island ⁎
divertor configuration not only has a long L|| in the SOL, but also forms a three-dimensional stochastic layer around the boundary magnetic island. It is beneficial to increase the equivalent radial transport, and consequently reduce the peak heat load on the divertor target [16,17]. Especially, the full stable detachment with island divertor configuration for 28 s has been achieved in W7-X [18]. Therefore, applying the island divertor configuration to tokamak is of great significance. The island divertor can be realized in different ways. In low-shear stellarators like W7-AS and W7-X, one makes use of the intrinsic islands. In moderate to high shear heliotrons like CHS and LHD, one can employ additional field perturbation coils to create externally imposed islands [17]. In the J-TEXT tokamak, the external RMP coils are designed to generate the boundary magnetic island and then to form the island divertor configuration. This article focuses on the design of the island divertor coils. 2. Parameters of the J-TEXT island divertor configuration The J-TEXT tokamak [19] is equipped with several RMP coils, which are shown in Fig. 1. It is convenient to investigate whether these coils could be used as the island divertor coils.
Corresponding author. E-mail address: [email protected] (N. Wang).
https://doi.org/10.1016/j.fusengdes.2019.01.109 Received 8 October 2018; Received in revised form 19 January 2019; Accepted 21 January 2019 0920-3796/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Zhou, S., Fusion Engineering and Design, https://doi.org/10.1016/j.fusengdes.2019.01.109
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
Fig. 3. Workflow of the coils design. Fig. 1. Layout of RMP coils on J-TEXT. The RMP system consists of 4 group and 12 saddle coils.
1.25 cm < w < 5 cm. In that case, the required 4/1 perturbation magnetic field Br = mq′Bθ wc2 16rq should be larger than 3 G on the J-TEXT tokamak. 3. Design methods The key to the application of the island divertor on tokamaks is to form a stable magnetic island with a sufficient size at the plasma boundary. According to the above parameters design, the 4/1 islands, as the main part of the island divertor, should be located at r ≅ a. Therefore, we first need to select and construct a plasma equilibrium with edge safety factor qa slightly larger than 4. Since the field line of this magnetic island will be opened by a divertor target, the island should be a vacuum island. Furthermore, the plasma response to the magnetic perturbation has been monitored in the TEXT tokamak. The measurements showed that the response is less than 10% of the RMP field [23]. Therefore, the vacuum assumption is sufficient during the design calculation progress. Fig. 3 shows the workflow of the coils design. The first step in the series of calculations is to calculate the vacuum magnetic field of the island divertor coils, which is done by using the Biot–Savart law shown in the Eq. (2):
Fig. 2. Spectrum of the radial magnetic field Br generated by the RMP coils.
Previously, the 3/1 island in the edge plasma has been formed successfully by using these RMP coils [20], which could produce a dominant 3/1 RMP field with small 2/1 component, shown in Fig. 2. However, it is found that the 2/1 locked mode could be easily excited after the formation of a 3/1 locked island, especially at low density and high RMP amplitude. However, for an island divertor, a sufficient width of the boundary island is needed, which requires a large RMP amplitude. This increases the risk of generating 2/1 locked mode in low qa ( = 3) plasmas, which tends to trigger the disruption. As a result, it leads to large thermal loads on in-vessel components and strong electromagnetic forces on surrounding conductors [21]. Hence, the 3/1 island is not suitable for the island divertor configuration on J-TEXT. The above 2/1 locked mode is mainly caused by the toroidal mode coupling, thus the poloidal number of island divertor coils should be larger than or equal to 4. But magnetic fields with higher poloidal number decay faster, i.e. Br ∝ r|−m1| . In addition, magnetic shear is stronger on the rational surface with higher q-values, which makes it more difficult to form large magnetic islands. Therefore, the best choice is to select 4/1 as the main perturbation component. The current RMP coils can not produce a dominant 4/1 RMP, so the new island divertor coils are required to be designed. There is a critical island width wc, for the island width w is 0.5wc < w < 2wc, the enhanced effective radial heat diffusivity due to the parallel transport will be increased rapidly. Hence, the critical island width is selected as a clear and quantitative criterion to judge whether or not the island divertor coils can produce the optimum or sufficient magnetic field. Besides, the stochastic layer is expected to be beneficial, allowing to obtain an adequate radial transport even for islands marginally below wc. wc is calculated by Eq. (1) [22].
wc = a (χ⊥ χ∥ )1 4 (εan/8Lq)−1/2
B (r ) =
μ0 4π
∫C
coils
Idl × → r r3
(2)
Where dl is a vector along the path Ccoils whose magnitude is the length of the differential element of the wire in the direction of conventional → current I, r is the full displacement vector from the wire element dl to the point at which the field is being computed, and μ0 is the magnetic permeability of the vacuum. The second step is to calculate and analyze the radial-like component of the magnetic perturbations. At the beginning of this calculation step, we should construct the field-aligned coordinate (s, φ, θ*) by using the selected equilibrium, where s=ψ1/2 (ψ is the normalized poloidal magnetic flux), φ is the toroidal angle and θ* is the field-aligned poloidal angle. And then we project the magnetic perturbations in the direction perpendicular to the equilibrium flux surface. Finally, we calculate the Fourier spectrum of the radial-like magnetic perturbations with respect to the toroidal angle φ and intrinsic poloidal angle θ* [24]. 1 The resonant harmonics b˜res is calculated by the Eq. (3): 1 1 b˜res = 2 |b˜m, −n0 | sin (mθ∗ − n 0 (φ − φ0))
(3)
1 Where b˜m, −n0 is defined as the Eq. (4): 1 b˜m, −n0 (s ) =
→→
∗
s −i (mθ − n φ) dθ ∫φ=0 ∫θ =0 →B ⋅∇ → ⋅e 2π 2π
2π
∗
B ⋅∇ φ
∗
0
dφ 2π
(4)
The third step is to perform the field line tracing and Poincaré plots. In the vacuum hypothesis, we neglect any current that could appear in the plasma in response to the magnetic perturbations. The total background vacuum magnetic field is produced by the linear superposition of the equilibrium field and the external RMP field generated by the island divertor coils. The field line tracing is calculated in cylindrical coordinates (R, φ, Z), because the field-aligned poloidal angle θ* is undefined outside the last closed flux surface (LCFS) [25]. The Poincaré plots are done by numerically integrating the field line equations,
(1)
Where Lq = q q′ and ε = a R . For typical parameters of the J-TEXT tokamak (magnetic field B˜2 T, major radius R0 =1.05 m, minor radius a˜0.25 m, safety factor q = 4 with the shear q′q = 4 = dq dR = 45m−1, χ⊥ χ∥ ≈ 10−6 in the edge plasma), the island width should be 2
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
shown in Eqs. (5) and (6):
dR =
BR dφ Bφ
(5)
dZ =
BZ dφ Bφ
(6)
In order to perform the numerical integration, the 4th order RungeKutta algorithm is applied. The field lines are started at one toroidal angle. And the radial and poloidal positions can be recorded when the field lines complete one toroidal turn. These positions can be plotted as the Poincaré plots, which provides a good way to visualize the island geometry. The design of the island divertor coils is an iterative process, in which the spectrum analysis and magnetic field line tracing are two important methods. By controlling the coils structure, the island geometry can be modified. The ultimate goal of the coils design is to give the best parameters for the creation of an effective divertor plasma, like island size, connection length and X-point height above the target [17].
Fig. 5. The 8 S-type modular coils are plotted in the θ-φ plane with thick solid line. The thin dotted and solid lines are two set of the 4/1 field lines.
4. Conceptual design 4.1. Coils type and spectrum Considering the features of J-TEXT tokamak, two types of island divertor coils are preliminarily designed, i.e. the out-vessel saddle coils and the in-vessel S-type modular coils. The saddle coils and spectrum are shown in Fig. 4, which are the upgraded version of the existing RMP coils on J-TEXT. In the poloidal direction, we increase the number of coils from 3 to 4 saddle loops and increase the poloidal angular span between coils. But the space on the high field side is still too narrow to install the new island divertor coils inside or outside the vacuum vessel. The magnetic field of the coils contain strong 3/1, 4/1 and 5/1 components. The S-type modular coils are illustrated in the θ-φ plane, as shown in Fig. 5. Here the 4 red dotted lines and 4 blue solid lines are two set of m/n = 4/1 field lines, which are calculated by field line tracing in the situation only with the equilibrium magnetic field. The 8 Gy thick solid lines are the S-type modular coils, covering 8 toroidal regions with the same angular span. The conductors are located at a common coil radius, rc. In the design, one of the most critical point is that the poloidal sections of the coils are all along the 4/1 magnetic field lines. The current in each coil is equal. As a result, the 4/1 component of the coils can be maximized and the sidebands are also very small when we analyze the magnetic field spectrum of these coils in the field-aligned coordinates. Considering the free space that can be used to install coils within the J-TEXT tokamak, the maximum toroidal angular span of each S-type coil is 10°. In order to obtain a stronger magnetic field and reduce harmonic components, the number of coils is 8. Fig. 6 shows the coils arrangement and its spectrum.
Fig. 6. Layout of the S-type modular coils and its spectrum of the radial magnetic field Br.
4.2. Field line tracing The field lines tracing is a process of solving the field line equations and recording the radial and poloidal locations of field lines. It is usually used to calculate various properties of the magnetic field structure. The Poincaré plot obtained for 0.5 kA (left plot) and 1 kA (right plot) in the S-type modular coils is shown in Fig. 7. The 4/1 islands are excited in the edge of the plasma. The widths of them are about 2 cm, which agree with the analytical calculations. If the current in the coils are increased larger, for example, from 0.5 kA to 1 kA, the plasma boundary becomes more stochastic. The stochastic magnetic field lines span up a whole volume, which will enhance the radial transport of particles and energy. It helps to increase the thermal deposition area and so as to reduce the peak heat load on the target plate. The final verification of the island divertor performance has to be assessed with more sophisticated codes (e.g. EMC3-EIRENE). 4.3. Island divertor configuration After the excitation of the 4/1 edge islands, the divertor targets can be set at the O-point of the islands. The boundary of the confinement region is defined by the separatrix of a 4/1 island intersected by target plates. Fig. 8 shows an example of a J-TEXT island divertor configuration. The divertor targets are not toroidally symmetric, which is
Fig. 4. Layout of the saddle coils and its spectrum of the radial magnetic field Br.
Fig. 7. Poincaré plot of the magnetic field structure with magnetic perturbation from the S-type modular coils. Left: with 0.5 kA. Right: with 1 kA. 3
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
References [1] H.Y. Guo, et al., Developing and validating advanced divertor solutions on DIII-D for next-step fusion devices, Nucl. Fusion 56 (2016) 126010. [2] V.P. Budaev, Results of high heat flux tests of tungsten divertor targets under plasma heat loads expected in ITER and tokamaks, Phys. Atom. Nucl. 79 (2016) 1137. [3] A. Loarte, et al., Power and particle control, Nucl. Fusion 47 (2007) S203. [4] C.S. Pitcher, et al., Experimental divertor physics, Plasma Phys. Controlled Fusion 39 (1997) 779. [5] T. Eich, et al., Inter-ELM power decay length for JET and ASDEX Upgrade: measurement and comparison with heuristic drift-based model, Phys. Rev. Lett. 107 (2011) 215001. [6] R. Goldston, Heuristic drift-based model of the power scrape-off width in low-gaspuff H-mode tokamaks, Nucl. Fusion 52 (2011) 013009. [7] D. Ryutov, Geometrical properties of a “snowflake” divertor, Phys. Plasmas 14 (2007) 064502. [8] P. Valanju, et al., Super-X divertors and high power density fusion devices, Phys. Plasmas 16 (2009) 056110. [9] C.S. Pitcher, et al., Experimental divertor physics, Plasma Phys. Controlled Fusion 39 (1997) 779. [10] S. Krasheninnikov, et al., Divertor plasma detachment, Phys. Plasmas 23 (2016) 055602. [11] F. Karger, et al., Resonant helical divertor, Phys. Lett. A 61 (1977) 385. [12] K. Finken, et al., Modelling of the field line penetration and force transfer by the dynamic ergodic divertor of TEXTOR, Nucl. Fusion 44 (2004) S55. [13] P. Grigull, et al., First island divertor experiments on the W7-AS stellarator, Plasma Phys. Controlled Fusion 43 (2002) 175. [14] T. Morisaki, et al., Local island divertor experiments on LHD, J. Nucl. Phys. Mater. Sci. Radiat. Appl. 337 (2005) 154. [15] Y. Feng, et al., On the W7-X divertor performance under detached conditions, Nucl. Fusion 56 (2016) 126011. [16] Y. Feng, et al., Comparison between stellarator and tokamak divertor transport, Plasma Phys. Controlled Fusion 53 (2011) 024009. [17] R. König, et al., The divertor program in stellarators, Plasma Phys. Controlled Fusion 44 (2002) 2365. [18] T. Pedersen, First divertor physics studies in Wendelstein 7-X, 27th IAEA Fusion Energy Conference Nov. 22nd – 27th (2018) EX-9-1. [19] G. Zhuang, et al., Overview of the recent research on the J-TEXT tokamak, Nucl. Fusion 55 (2015) 587. [20] Q. Hu, et al., Plasma response to m/n = 3/1 resonant magnetic perturbation at JTEXT Tokamak, Nucl. Fusion 56 (2016) 092009. [21] P.C.D. Vries, et al., Statistical analysis of disruptions in JET, Nucl. Fusion 49 (2009) 055011. [22] Q. Yu, Numerical modeling of diffusive heat transport across magnetic islands and local stochastic field, Phys. Plasmas 13 (2006) 062310. [23] S. McCool, et al., Electron thermal confinement studies with applied resonant fields on TEXT, Nucl. Fusion 29 (1989) 547. [24] E. Nardon, Ecole polytechnique, Edge Localized Modes Control by Resonant Magnetic Perturbations, (2007) PhD Thesis. [25] P. Denner, Mitigating Edge-localized Modes on the Mega-Ampere Spherical Tokamak Using Resonant Magnetic Perturbations, PhD Thesis University of York, 2012.
Fig. 8. Schematic diagram of the island divertor on J-TEXT.
similar to W7-AS or W7-X. 5. Conclusion A set of coils for generating island divertor configuration has been designed on J-TEXT tokamak. In this paper, we develop a conceptual design for the new island divertor coils. The most suitable mode number of the island divertor coils is the m/n = 4/1 and the required perturbation magnetic field should be larger than 8 G to excite the 4/1 boundary island. The numerical calculations have been carried out to calculate the perturbed magnetic field generated by the coils, its spectrum and the Poincaré plot. Finally, two types of the island divertor coils have been designed, i.e. 16 out-vessel saddle coils and 8 in-vessel S-type modular coils. The ID configuration can be established by the boundary 4/1 island intersecting with proper divertor targets. To verify the design, we need to measure the magnetic field value and structure next step. Besides, the engineering realization remain to be considered in detail. Acknowledgments The authors are very grateful for the help of J-TEXT team. This work is supported by the National MCF Energy R&D Program of China (Contract No. 2018YFE0309100).
4
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Conceptual design of the island divertor coils on the J-TEXT tokamak ⁎
Song Zhoua, Nengchao Wanga, , Bo Raoa, Zhuo Huanga, Da Lia, Mao Lia, Ruo Jiaa, Wei Zhanga, Shiyi Penga, Zebao Songa, Ying Hea, Kexun Yua, Yonghua Dinga, Yunfeng Lianga,b,c a International Joint Research Laboratory of Magnetic Confinement Fusion and Plasma Physics, State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China b Forschungszentrum Jülich GmbH, D-52425, Jülich, Germany c Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Island divertor Resonant magnetic perturbation Magnetic island Coil design Tokamak
A set of new resonant magnetic perturbation (RMP) coils is designed for the J-TEXT tokamak to form the island divertor (ID) configuration [F. Karger, Phys Lett A, 61 (1977) 385]. This design is aiming at investigating the effect of ID on the peak heat load reduction in a tokamak. The RMP field generated by these new coils can produce magnetic island in the plasma boundary. The field line of this magnetic island is opened by divertor targets, so as to form the island divertor configuration. Therefore, the edge island should be a vacuum island, which can be described by an analytical mode with assuming linear superposition of the plasma equilibrium field and the magnetic perturbation in a vacuum approximation. Numerical calculations have been carried out to calculate several parameters for the island divertor configuration such as the perturbed magnetic field generated by the coils, its spectrum, the field line tracing and the Poincaré plot. Conceptual design is based on the m/ n = 4/1 magnetic island, which can be generated by either 16 out-vessel saddle coils or 8 in-vessel S-type modular coils. Here, m and n are the poloidal and toroidal mode numbers respectively. The ID configuration can be then established by the boundary 4/1 island intersecting with proper divertor targets.
1. Introduction The high heat load on divertor target is one of the essential issues that must be addressed for next-step high-power steady-state fusion devices [1,2]. The heat load on the ITER divertor is expected to be above 10 MW m−2 in steady state or up to 0.6 ˜ 10 GW m−2 during disruption and ELMs [3]. An essential task of the divertor is to reduce/ disperse this power to an acceptable level, preferably less than 5 MW m−2 [4]. The high steady-state heat load is due to the significant reduction of the power decay length λq with increasing poloidal magnetic field Bp [5]. It is found in the heuristic drift-based model [6] that λq is proportional to the connection length L||. In the scrape-off layer (SOL) of a tokamak, L|| is inversely proportional to the poloidal magnetic field Bp. Many improvements of the poloidal divertor have been made to reduce the heat load on the target, e.g. snow-flake [7] or Super-X configuration [8] and radiation divertor [9,10]. An alternative concept is using the island divertor configuration, which has been proposed in 1977 for tokamak [11]. It has been established successfully in tokamak, e.g. TEXTOR [12], and stellarators, such as W7-AS [13], LHD [14] and recently in W7-X [15]. The island ⁎
divertor configuration not only has a long L|| in the SOL, but also forms a three-dimensional stochastic layer around the boundary magnetic island. It is beneficial to increase the equivalent radial transport, and consequently reduce the peak heat load on the divertor target [16,17]. Especially, the full stable detachment with island divertor configuration for 28 s has been achieved in W7-X [18]. Therefore, applying the island divertor configuration to tokamak is of great significance. The island divertor can be realized in different ways. In low-shear stellarators like W7-AS and W7-X, one makes use of the intrinsic islands. In moderate to high shear heliotrons like CHS and LHD, one can employ additional field perturbation coils to create externally imposed islands [17]. In the J-TEXT tokamak, the external RMP coils are designed to generate the boundary magnetic island and then to form the island divertor configuration. This article focuses on the design of the island divertor coils. 2. Parameters of the J-TEXT island divertor configuration The J-TEXT tokamak [19] is equipped with several RMP coils, which are shown in Fig. 1. It is convenient to investigate whether these coils could be used as the island divertor coils.
Corresponding author. E-mail address: [email protected] (N. Wang).
https://doi.org/10.1016/j.fusengdes.2019.01.109 Received 8 October 2018; Received in revised form 19 January 2019; Accepted 21 January 2019 0920-3796/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Zhou, S., Fusion Engineering and Design, https://doi.org/10.1016/j.fusengdes.2019.01.109
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
Fig. 3. Workflow of the coils design. Fig. 1. Layout of RMP coils on J-TEXT. The RMP system consists of 4 group and 12 saddle coils.
1.25 cm < w < 5 cm. In that case, the required 4/1 perturbation magnetic field Br = mq′Bθ wc2 16rq should be larger than 3 G on the J-TEXT tokamak. 3. Design methods The key to the application of the island divertor on tokamaks is to form a stable magnetic island with a sufficient size at the plasma boundary. According to the above parameters design, the 4/1 islands, as the main part of the island divertor, should be located at r ≅ a. Therefore, we first need to select and construct a plasma equilibrium with edge safety factor qa slightly larger than 4. Since the field line of this magnetic island will be opened by a divertor target, the island should be a vacuum island. Furthermore, the plasma response to the magnetic perturbation has been monitored in the TEXT tokamak. The measurements showed that the response is less than 10% of the RMP field [23]. Therefore, the vacuum assumption is sufficient during the design calculation progress. Fig. 3 shows the workflow of the coils design. The first step in the series of calculations is to calculate the vacuum magnetic field of the island divertor coils, which is done by using the Biot–Savart law shown in the Eq. (2):
Fig. 2. Spectrum of the radial magnetic field Br generated by the RMP coils.
Previously, the 3/1 island in the edge plasma has been formed successfully by using these RMP coils [20], which could produce a dominant 3/1 RMP field with small 2/1 component, shown in Fig. 2. However, it is found that the 2/1 locked mode could be easily excited after the formation of a 3/1 locked island, especially at low density and high RMP amplitude. However, for an island divertor, a sufficient width of the boundary island is needed, which requires a large RMP amplitude. This increases the risk of generating 2/1 locked mode in low qa ( = 3) plasmas, which tends to trigger the disruption. As a result, it leads to large thermal loads on in-vessel components and strong electromagnetic forces on surrounding conductors [21]. Hence, the 3/1 island is not suitable for the island divertor configuration on J-TEXT. The above 2/1 locked mode is mainly caused by the toroidal mode coupling, thus the poloidal number of island divertor coils should be larger than or equal to 4. But magnetic fields with higher poloidal number decay faster, i.e. Br ∝ r|−m1| . In addition, magnetic shear is stronger on the rational surface with higher q-values, which makes it more difficult to form large magnetic islands. Therefore, the best choice is to select 4/1 as the main perturbation component. The current RMP coils can not produce a dominant 4/1 RMP, so the new island divertor coils are required to be designed. There is a critical island width wc, for the island width w is 0.5wc < w < 2wc, the enhanced effective radial heat diffusivity due to the parallel transport will be increased rapidly. Hence, the critical island width is selected as a clear and quantitative criterion to judge whether or not the island divertor coils can produce the optimum or sufficient magnetic field. Besides, the stochastic layer is expected to be beneficial, allowing to obtain an adequate radial transport even for islands marginally below wc. wc is calculated by Eq. (1) [22].
wc = a (χ⊥ χ∥ )1 4 (εan/8Lq)−1/2
B (r ) =
μ0 4π
∫C
coils
Idl × → r r3
(2)
Where dl is a vector along the path Ccoils whose magnitude is the length of the differential element of the wire in the direction of conventional → current I, r is the full displacement vector from the wire element dl to the point at which the field is being computed, and μ0 is the magnetic permeability of the vacuum. The second step is to calculate and analyze the radial-like component of the magnetic perturbations. At the beginning of this calculation step, we should construct the field-aligned coordinate (s, φ, θ*) by using the selected equilibrium, where s=ψ1/2 (ψ is the normalized poloidal magnetic flux), φ is the toroidal angle and θ* is the field-aligned poloidal angle. And then we project the magnetic perturbations in the direction perpendicular to the equilibrium flux surface. Finally, we calculate the Fourier spectrum of the radial-like magnetic perturbations with respect to the toroidal angle φ and intrinsic poloidal angle θ* [24]. 1 The resonant harmonics b˜res is calculated by the Eq. (3): 1 1 b˜res = 2 |b˜m, −n0 | sin (mθ∗ − n 0 (φ − φ0))
(3)
1 Where b˜m, −n0 is defined as the Eq. (4): 1 b˜m, −n0 (s ) =
→→
∗
s −i (mθ − n φ) dθ ∫φ=0 ∫θ =0 →B ⋅∇ → ⋅e 2π 2π
2π
∗
B ⋅∇ φ
∗
0
dφ 2π
(4)
The third step is to perform the field line tracing and Poincaré plots. In the vacuum hypothesis, we neglect any current that could appear in the plasma in response to the magnetic perturbations. The total background vacuum magnetic field is produced by the linear superposition of the equilibrium field and the external RMP field generated by the island divertor coils. The field line tracing is calculated in cylindrical coordinates (R, φ, Z), because the field-aligned poloidal angle θ* is undefined outside the last closed flux surface (LCFS) [25]. The Poincaré plots are done by numerically integrating the field line equations,
(1)
Where Lq = q q′ and ε = a R . For typical parameters of the J-TEXT tokamak (magnetic field B˜2 T, major radius R0 =1.05 m, minor radius a˜0.25 m, safety factor q = 4 with the shear q′q = 4 = dq dR = 45m−1, χ⊥ χ∥ ≈ 10−6 in the edge plasma), the island width should be 2
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
shown in Eqs. (5) and (6):
dR =
BR dφ Bφ
(5)
dZ =
BZ dφ Bφ
(6)
In order to perform the numerical integration, the 4th order RungeKutta algorithm is applied. The field lines are started at one toroidal angle. And the radial and poloidal positions can be recorded when the field lines complete one toroidal turn. These positions can be plotted as the Poincaré plots, which provides a good way to visualize the island geometry. The design of the island divertor coils is an iterative process, in which the spectrum analysis and magnetic field line tracing are two important methods. By controlling the coils structure, the island geometry can be modified. The ultimate goal of the coils design is to give the best parameters for the creation of an effective divertor plasma, like island size, connection length and X-point height above the target [17].
Fig. 5. The 8 S-type modular coils are plotted in the θ-φ plane with thick solid line. The thin dotted and solid lines are two set of the 4/1 field lines.
4. Conceptual design 4.1. Coils type and spectrum Considering the features of J-TEXT tokamak, two types of island divertor coils are preliminarily designed, i.e. the out-vessel saddle coils and the in-vessel S-type modular coils. The saddle coils and spectrum are shown in Fig. 4, which are the upgraded version of the existing RMP coils on J-TEXT. In the poloidal direction, we increase the number of coils from 3 to 4 saddle loops and increase the poloidal angular span between coils. But the space on the high field side is still too narrow to install the new island divertor coils inside or outside the vacuum vessel. The magnetic field of the coils contain strong 3/1, 4/1 and 5/1 components. The S-type modular coils are illustrated in the θ-φ plane, as shown in Fig. 5. Here the 4 red dotted lines and 4 blue solid lines are two set of m/n = 4/1 field lines, which are calculated by field line tracing in the situation only with the equilibrium magnetic field. The 8 Gy thick solid lines are the S-type modular coils, covering 8 toroidal regions with the same angular span. The conductors are located at a common coil radius, rc. In the design, one of the most critical point is that the poloidal sections of the coils are all along the 4/1 magnetic field lines. The current in each coil is equal. As a result, the 4/1 component of the coils can be maximized and the sidebands are also very small when we analyze the magnetic field spectrum of these coils in the field-aligned coordinates. Considering the free space that can be used to install coils within the J-TEXT tokamak, the maximum toroidal angular span of each S-type coil is 10°. In order to obtain a stronger magnetic field and reduce harmonic components, the number of coils is 8. Fig. 6 shows the coils arrangement and its spectrum.
Fig. 6. Layout of the S-type modular coils and its spectrum of the radial magnetic field Br.
4.2. Field line tracing The field lines tracing is a process of solving the field line equations and recording the radial and poloidal locations of field lines. It is usually used to calculate various properties of the magnetic field structure. The Poincaré plot obtained for 0.5 kA (left plot) and 1 kA (right plot) in the S-type modular coils is shown in Fig. 7. The 4/1 islands are excited in the edge of the plasma. The widths of them are about 2 cm, which agree with the analytical calculations. If the current in the coils are increased larger, for example, from 0.5 kA to 1 kA, the plasma boundary becomes more stochastic. The stochastic magnetic field lines span up a whole volume, which will enhance the radial transport of particles and energy. It helps to increase the thermal deposition area and so as to reduce the peak heat load on the target plate. The final verification of the island divertor performance has to be assessed with more sophisticated codes (e.g. EMC3-EIRENE). 4.3. Island divertor configuration After the excitation of the 4/1 edge islands, the divertor targets can be set at the O-point of the islands. The boundary of the confinement region is defined by the separatrix of a 4/1 island intersected by target plates. Fig. 8 shows an example of a J-TEXT island divertor configuration. The divertor targets are not toroidally symmetric, which is
Fig. 4. Layout of the saddle coils and its spectrum of the radial magnetic field Br.
Fig. 7. Poincaré plot of the magnetic field structure with magnetic perturbation from the S-type modular coils. Left: with 0.5 kA. Right: with 1 kA. 3
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Zhou et al.
References [1] H.Y. Guo, et al., Developing and validating advanced divertor solutions on DIII-D for next-step fusion devices, Nucl. Fusion 56 (2016) 126010. [2] V.P. Budaev, Results of high heat flux tests of tungsten divertor targets under plasma heat loads expected in ITER and tokamaks, Phys. Atom. Nucl. 79 (2016) 1137. [3] A. Loarte, et al., Power and particle control, Nucl. Fusion 47 (2007) S203. [4] C.S. Pitcher, et al., Experimental divertor physics, Plasma Phys. Controlled Fusion 39 (1997) 779. [5] T. Eich, et al., Inter-ELM power decay length for JET and ASDEX Upgrade: measurement and comparison with heuristic drift-based model, Phys. Rev. Lett. 107 (2011) 215001. [6] R. Goldston, Heuristic drift-based model of the power scrape-off width in low-gaspuff H-mode tokamaks, Nucl. Fusion 52 (2011) 013009. [7] D. Ryutov, Geometrical properties of a “snowflake” divertor, Phys. Plasmas 14 (2007) 064502. [8] P. Valanju, et al., Super-X divertors and high power density fusion devices, Phys. Plasmas 16 (2009) 056110. [9] C.S. Pitcher, et al., Experimental divertor physics, Plasma Phys. Controlled Fusion 39 (1997) 779. [10] S. Krasheninnikov, et al., Divertor plasma detachment, Phys. Plasmas 23 (2016) 055602. [11] F. Karger, et al., Resonant helical divertor, Phys. Lett. A 61 (1977) 385. [12] K. Finken, et al., Modelling of the field line penetration and force transfer by the dynamic ergodic divertor of TEXTOR, Nucl. Fusion 44 (2004) S55. [13] P. Grigull, et al., First island divertor experiments on the W7-AS stellarator, Plasma Phys. Controlled Fusion 43 (2002) 175. [14] T. Morisaki, et al., Local island divertor experiments on LHD, J. Nucl. Phys. Mater. Sci. Radiat. Appl. 337 (2005) 154. [15] Y. Feng, et al., On the W7-X divertor performance under detached conditions, Nucl. Fusion 56 (2016) 126011. [16] Y. Feng, et al., Comparison between stellarator and tokamak divertor transport, Plasma Phys. Controlled Fusion 53 (2011) 024009. [17] R. König, et al., The divertor program in stellarators, Plasma Phys. Controlled Fusion 44 (2002) 2365. [18] T. Pedersen, First divertor physics studies in Wendelstein 7-X, 27th IAEA Fusion Energy Conference Nov. 22nd – 27th (2018) EX-9-1. [19] G. Zhuang, et al., Overview of the recent research on the J-TEXT tokamak, Nucl. Fusion 55 (2015) 587. [20] Q. Hu, et al., Plasma response to m/n = 3/1 resonant magnetic perturbation at JTEXT Tokamak, Nucl. Fusion 56 (2016) 092009. [21] P.C.D. Vries, et al., Statistical analysis of disruptions in JET, Nucl. Fusion 49 (2009) 055011. [22] Q. Yu, Numerical modeling of diffusive heat transport across magnetic islands and local stochastic field, Phys. Plasmas 13 (2006) 062310. [23] S. McCool, et al., Electron thermal confinement studies with applied resonant fields on TEXT, Nucl. Fusion 29 (1989) 547. [24] E. Nardon, Ecole polytechnique, Edge Localized Modes Control by Resonant Magnetic Perturbations, (2007) PhD Thesis. [25] P. Denner, Mitigating Edge-localized Modes on the Mega-Ampere Spherical Tokamak Using Resonant Magnetic Perturbations, PhD Thesis University of York, 2012.
Fig. 8. Schematic diagram of the island divertor on J-TEXT.
similar to W7-AS or W7-X. 5. Conclusion A set of coils for generating island divertor configuration has been designed on J-TEXT tokamak. In this paper, we develop a conceptual design for the new island divertor coils. The most suitable mode number of the island divertor coils is the m/n = 4/1 and the required perturbation magnetic field should be larger than 8 G to excite the 4/1 boundary island. The numerical calculations have been carried out to calculate the perturbed magnetic field generated by the coils, its spectrum and the Poincaré plot. Finally, two types of the island divertor coils have been designed, i.e. 16 out-vessel saddle coils and 8 in-vessel S-type modular coils. The ID configuration can be established by the boundary 4/1 island intersecting with proper divertor targets. To verify the design, we need to measure the magnetic field value and structure next step. Besides, the engineering realization remain to be considered in detail. Acknowledgments The authors are very grateful for the help of J-TEXT team. This work is supported by the National MCF Energy R&D Program of China (Contract No. 2018YFE0309100).
4