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Simulation study on a merging core fueling technique for an advanced fuel fusion spherical tokamak reactor
Fusion Engineering and Design xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Simulation study on a merging core fueling technique for an advanced fuel fusion spherical tokamak reactor ⁎
Shintato Koikea, , Toshiki Takahashia, Naoki Mizuguchib, Osamu Mitaraic a
Division of Electronics and Informatics, Gunma University, Kiryu, Gunma, 376-8515, Japan National Institute for Fusion Science, Toki, Gifu, 509-5292, Japan c Institute for Advanced Fusion and Physics Education, kita-ku, Kumamoto, 861-5525, Japan b
A R T I C L E I N F O
A B S T R A C T
Keywords: Advanced fusion D-3He reaction Spherical tokamak Core fueling Translation and merging MHD Particle trajectory simulation
With respect to a particle fueling method in a spherical tokamak reactor with an advanced fuel, we investigate the availability of a merging process following the translation of a small torus plasma through which a plasma current flows in the same direction as the main plasma. A three-dimensional resistive MHD model is used to examine the particle fueling effect by reproducing the translation process and simultaneously performing an ion trajectory calculation. The simulation shows the merging process between the translated plasma and the main plasma and exhibits potential as a particle fueling method. The results of the study indicate that up to 30% of the ions in the secondary plasma are supplied to the main plasma.
1. Introduction Nuclear fusion power generation uses advanced fuel that can suppress neutron generation and tritium-handling, and thus it could be superior in safety. Therefore, even when the construction of ITER progresses, it is necessary to investigate a D-3He fusion reactor in the category of advanced fuel fusion reactors. A spherical tokamak (ST) is a possible candidate for a D-3He fusion reactor due to its high-beta value [1]. However, fuel particles supplied from outside are ionized in the plasma edge region due to high temperature and lead to a D-3He fusion reaction, and thus it is very difficult to fuel into the plasma core. For example, it is difficult to supply fuel into the plasma core in conventional particle fueling methods by using gas puffing, supersonic gas jet, or pellet injection. Compact torus plasma (CT) injection [2] is also studied as a method of deep fueling to the fusion plasma core. Currently, studies attempted to solve several problems such as the technical complexity of an injector [3]. Conversely, Watanabe et al. investigated whether accelerated CT is re-neutralized to generate high-speed gas flow for fueling [4,5]. The results indicated that CT plasma with injection velocity of approximately 100 km/s is neutralized by almost 100%. Nevertheless, it is clear that particle fueling is difficult because neutral particles are ionized on the plasma surface. Hence, particle fueling into the core is a very difficult task and especially in an advanced fusion reactor. However, it is an important research subject that should be investigated. To overcome this situation, a novel fueling method is proposed
⁎
wherein a secondary plasma produced in the upper region of the main plasma is translated and then merged into a burning main plasma [6]. The conceptual diagram is shown in Fig. 1. The secondary plasma current is produced inductively with fresh fuel gas puffing and translated in the axial direction to the main plasma current with the same direction. The particle fueling method is feasible if particles in the secondary plasma are supplied to the core region of main plasma. Recent experimental studies also successfully merged ST plasmas [7,8] and high-beta field-reversed configuration plasmas [9]. Therefore, the main focus of this study is to use the merging process as a particle fueling method in a feasible situation. Thus, it is extremely meaningful to quantitatively obtain detailed data required for reactor design such as fueling efficiency. In this study, an MHD simulation and particle trajectory calculation are combined to investigate the merging process of ST plasmas and the ion motion at that time to clarify the feasibility of the merging fueling method and quantitatively evaluate fueling efficiency. 2. Simulation model In this study, we set the time when the generation of secondary plasma is completed as zero and perform a three-dimensional resistive MHD simulation on translation and merging processes. To focus on those processes, we do not calculate the generation process of secondary plasma here. The initial state of plasma is based on the device design [6] parameters shown in Table 1. We solve the Grad-Shafranov
Corresponding author at: Division of Electronics and Informatics, Gunma University, Kiryu, Gunma, 376-8515, Japan. E-mail address: [email protected] (S. Koike).
https://doi.org/10.1016/j.fusengdes.2018.01.003 Received 24 September 2017; Received in revised form 23 December 2017; Accepted 1 January 2018 0920-3796/ © 2018 Published by Elsevier B.V.
Please cite this article as: Koike, S., Fusion Engineering and Design (2018), https://doi.org/10.1016/j.fusengdes.2018.01.003
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Koike et al.
developed by Todo et al. is used. However, the MIPS was modified such that the time change in the external magnetic field for translation control is considered, and the function of simultaneous ion trajectory calculation that is described later is added. In MIPS, the resistive MHD equations (1–6) are spatially discretized in each direction of the threedimensional cylindrical coordinate system (r, θ, z). The number of grid points is 112 × 128 × 224 for r, θ, and z respectively. In terms of time, the temporal evolution of each parameter is calculated by performing time integration by using the 4-th order Runge-Kutta method. The MHD equations are as follows:
∂ρ = −∇⋅(ρu) ∂t
Fig. 1. Conceptual diagram of the merging fueling technique for the ST fusion reactor.
Table 1 Parameters of the calculation model (initial condition). Machine major radius (Ro)
6.7[m]
Machine minor radius Main plasma length Secondary plasma length Peak of plasma temperature Plasma density (Uniform) Toroidal field Alfven time(τA) Main plasma beta Secondary plasma beta
2.5[m] 15[m] 10[m] 118[keV] 1.97 × 1020[m−3] 3.06[T] 2.25[μs] 0.20 0.20
{
∂u ∂t
=−
+
ν 4 (∇⋅u ) ∇ρ ρ 3
1 ∇⋅ 2
(1)
}
1
u2 + (∇ × u ) × u − ρ (∇p − j × B )
{
− ∇ × (ρ∇ × u )
}
(2)
∂p 4 = −(∇⋅(pu) + (γ − 1) p∇⋅u) + ν (γ − 1) ρ ⎛j2 + ∇⋅u⎞ ∂t 3 ⎝ ⎠
(3)
∂B = −∇ × E ∂t
(4)
E = − u × B + ηj
(5)
∇×B j= μ0
(6)
Here, eight scalar variables are obtained in which ρ, u, p, B denote the mass density, flow velocity, plasma pressure, and magnetic field, respectively, and ν, γ, η, μ0 denote the viscosity coefficient, specific heat ratio, electric resistivity, and permeability of the vacuum, respectively. Fig. 3 shows the vacuum region set in the MHD calculation of this study and the external coil that triggers translation of the secondary plasma. The vacuum region as indicated by solid black is assumed as a sufficiently small plasma density or is defined as a region outside the device. In this blackened part, we assume that only the magnetic field changes. In Fig. 3, the placement of the assist coil for translation of the secondary plasma is illustrated, and the coil current is set to reach the peak after a linear increase exceeding 100 Alfvén time. The magnetic field lines show the state after 100 Alfvén time elapses. By superimposing the magnetic field generated by the coil current on the magnetic field in the conventional MHD equation, we successfully reproduced the translation. Here the secondary plasma is pushed out by the coil magnetic pressure on the left side of Fig. 3. The trajectory calculation is simultaneously performed (from the equation of motion of ions) with the MHD simulation as follows:
Fig. 2. 2D profile of plasma pressure and contour line of poloidal flux at an initial equilibrium state.
equation to obtain the equilibrium of the main ST plasma. We determine the initial pressure and magnetic field distribution shown in Fig. 2 by separately determining the equilibrium of a smaller plasma in which lines of magnetic force are connected smoothly. The initial distribution of the mass density is assumed as uniform as typically adopted in a general MHD simulation. A simulation with a non-uniform density distribution will be explored in a future study. Therefore, the present study focuses on investigating the effect of particle fueling by using a trajectory calculation. Table 1 shows the plasma parameters of the simulation in addition to the device parameters. The parameters are based on a D-3He nuclear fusion design, and thus the plasma temperature exceeds 100 keV. For both Alfvén time and plasma beta, the magnetic field is evaluated with a toroidal component, and the value on the magnetic axis is used in this evaluation. However, these initial setting of the parameters slightly differ from the proposed D-3He ST reactor [10] to avoid a calculation discrepancy in the MHD calculation. It gradually approaches similar parameters. With respect to the MHD simulation in this study, MIPS code [11]
q dvi dr = i (E + vi × B ) , i = vi dt mi dt
(7)
where qi and mi denote the charge and mass of the ion, respectively. The electromagnetic fields in Eq. (7) are obtained by the MHD simulation. To secure the accuracy of the trajectory calculation, the time step of the trajectory calculation is set to be short by 10−5 with respect to the time
Fig. 3. 2D profile of poloidal field generated by external coil. The assist coils and vacuum region (shadowed area) are also illustrated.
2
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For the simulation of the two cases in which the merging process is completed, we verified the effect of particle fueling by performing a simulation without translation merging (without changing the external magnetic field). Fig. 5 shows the time evolution of the number of ion particles in each plasma in the simulations. In both cases in which the viscosity coefficient is ν = 10−4 and ν = 10−5 , both the main and secondary plasmas exhibit ballooning instability, and the number of particles commences decreasing when 100 Alfvén time elapses after the occurrence of instability. As shown in Fig. 5, the ion number in the main plasma once increases and begins to decrease in the acceleration phase. The observed change is similar to the time change in the volume of the main plasma. Therefore, ions in the peripheral region of the main plasma do not follow the sudden change in magnetic field, and the time change in Fig. 5 in the acceleration phase simply reflects the change in volume of the main plasma. Similarly, the number of particles in the secondary plasma decreases in the acceleration phase because its volume decreases when it is suddenly pushed from behind due to the change in the external magnetic field change. In Fig. 5(a) corresponding to the case of ν = 10−5 , the results indicate that the particle retention rate of the main plasma exceeds that when the secondary plasma is not translated. The results reveal that the number of particles in the main plasma increases after merging although the increment due to merging is 9.2% of the initial particles in the secondary plasma. Fig. 5(b) shows the one order larger normalized viscosity of ν = 10−4 , and the total retention rate for translation merging is slightly lower than that in the non-translation case. Conversely, in the merging process, the number of ions in the main plasma significantly increased. During the merging process, the ion increment corresponds to 30.5% of the initial number of particles in the secondary plasma, and thereby exhibits a certain particle supply as shown in Fig. 5(b). The above results clearly indicate that particles of secondary plasma are supplied to the main plasma by the merging process. However, abrupt particle loss occurred due to the occurrence of ballooning instability. Hence, in our present case, it is impossible to obtain a state that exceeds the initial density of the ST plasma by merging the fueling method. Additionally, pressure flattening due to occurrence of ballooning instability may promote merging, and it is necessary to clarify the possibility of realizing a merging fueling method in a stable plasma in the future by performing a fine simulation. In this calculation, different merging processes were reproduced by simulation when the viscosity coefficient is artificially given. According to Braginskii [12], the viscosity changes as follows:
step of the MHD simulation. The initial distribution of the particles is uniformly arranged by using random numbers. This is the same as the setting of the initial uniform density profile in the MHD simulation. The initial ion velocity is determined as a Maxwellian distribution by using normal random numbers. Thus, the temperature is spatially distributed corresponding to the initial pressure distribution of the plasma. The objective of the ion trajectory calculation is to quantitatively evaluate the supply of particles from the secondary plasma to the main plasma. The trajectory calculation makes it possible to accurately discuss the particle supply effect by calculating the ion behavior near the X-point of the magnetic reconnection region. In this study, the simulation was performed by changing the viscosity coefficient ν in Eqs. (2) and (3) of the MHD equation to clarify the effect of particle fueling by simulating the collision/merging process under multiple circumstances. This is because the viscosity coefficient ν affects the translation speed of secondary plasma, occurrence of instability, and resultant fueling efficiency. Similarly, the electrical resistivity η is a parameter that determines the speed of the magnetic reconnection process. However, the dependence of the merging process on the electrical resistivity will be explored in a future study. 3. Results and discussion The results of the simulation confirm the translation of the secondary plasma and collision/merging to the main plasma. Fig. 4 shows the pressure and magnetic field integral values (upper figure at the respective times) and ion trajectory calculation results (lower figure shows the respective times) at initial (Fig. 4, top) and after the 420 Alfvén time (Fig. 4, middle) and 755 Alfvén time (Fig. 4, bottom). Here, the normalized viscosity coefficient is 10−4 in which the normalization is defined as follows:
ν =
ν , ρ0 R o VA
(8)
where the value of reference quantity ρ0RoVA = ρ0Ro2/τAis 16.4 [m−1 kg s−1] in our case, and VA denotes the Alfvén velocity. The calculation is three-dimensional, and it is not always possible to define the magnetic surface. Therefore, in this study, the following expression is obtained by integration for the ease of understanding:
ψ (r , 0, z ) =
∫ Bz (r , 0, z ) r dr
(9)
It thereby defines the separatrix of the main plasma and secondary plasma. We hereby distinguish between the existence region of ions by ψ(r, 0, z). In the figure shown in the ion trajectory calculation result, red represents ions initially arranged in the main plasma, yellow green represents ions inside the secondary plasma, and blue green represents ions initially arranged that do not correspond to the first two categories. After 420 Alfvén time elapses since the start of the translation (Fig. 4 upper figure), the separatrices of both plasmas collide. Thereafter, the magnetic axes of the two plasmas gradually approach each other, and a sudden merging phenomenon occurs after 750 Alfvén time. Finally, after 755 Alfvén time elapses, merging is completed as shown in the lower part of Fig. 4. In contrast, as shown in the pressure distribution, it is observed that the ballooning instability has already occurred in the two plasmas at the time of collision, and it significantly attenuates when compared with the peak value of the initial pressure. To investigate the collision/merging process and fuel supply effect of the two ST plasmas, several simulations are performed for three cases of the viscosity coefficient. Table 2 shows the time elapsed until occurrence of instability, collision, and merging observed in the simulations. It is observed that occurrence of the instability is delayed when the viscosity coefficient increases. Conversely, when ν = 10−3 with a high viscosity coefficient, the secondary plasma stops in the middle of translation due to the resistance force from the peripheral plasma, and collision of the two plasmas ceases by the end of calculation.
ν ∝ Ti5/2 .
(10)
where Ti denotes the ion temperature. Therefore, the viscosity of the high temperature D- 3He nuclear fusion plasma is high, and instability and merging will be less likely to occur according to the results of the study. However, in general MHD models, a simplified model for viscosity is typically used, and thus it may be necessary to adopt a more detailed MHD model for viscosity in the future. The merging-compression concept verified on MAST produces 400 kA of plasma current within 20 ms [7]. In a manner similar to this experiment, our simulation result also indicates a sharp increase in the plasma current of the main plasma. The occurrence of instability prior to merging is not mentioned in both MAST [7] and UTST [8] experimental reports. The calculation focuses on advanced fuel nuclear fusion, and thus it is difficult to perform a direct comparison with the experiment. However, a future study should advance the comparative study by setting the calculation to the experiment that is currently performed.
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Fig. 4. Color contours of the plasma pressure and contour line of magnetic field integral values ψ(r, 0, z) and ion distribution at the initial time, time of collision and the time of merging. In the distribution map of the ions, the red markers represent the ions initially placed in the separatrix of the main plasma, and the yellow markers represent the ions initially placed in the separatrix of the secondary plasmas. Green color denotes the ions initially placed outside the separatrix of both the main and secondary plasma. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion
axial translation of small torus plasma to ST plasma. The particle trajectory analysis was also simultaneously performed with the MHD simulation to discuss the particle supply effect more accurately by considering the complex ion motion near the X-point in the magnetic
In this study, a three-dimensional resistive MHD simulation was performed to verify the effectiveness of new particle fueling method by 4
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leading to the collision/merging. Specifically, in the study, simulation was performed by changing the viscosity coefficient for the purpose of investigating particle transfer from secondary plasma to main plasma in various translation merging processes. When the viscosity coefficient is low, the translation speed is fast and the time required for merging decreases. Conversely, an increase in the viscosity coefficient stops the secondary plasma during the translation, and collision does not occur. Between the viscosity coefficients, the results indicated that approximately 30% of secondary plasma is supplied to the main plasma by merging. The results revealed that the fueling efficiency depends on the viscosity of the plasma, and thus it is important to further examine viscosity in the MHD simulation. We also note that the ballooning instability is more likely to occur when the viscosity coefficient decreases, and the particle loss increases with the occurrence of instability as clarified in the ion trajectory calculation. The initial trial of MHD simulation on merging fueling indicates the variety of the phenomena, and thus it should be investigated in a future study.
Table 2 Generation time and supply effect of each event in simulations with different viscosity coefficients. Viscosity coefficient ν
10−3
10−4
10−5
Occurrence of instability [τA] Collision [τA] Merging [τA] Supply rate from secondary plasma [%]
1150 N/A N/A N/A
200 415 750 30.5
100 250 640 9.2
Acknowledgments This work is performed on “Plasma Simulator” (FUJITSU FX100) of NIFS with the support and under the auspices of the NIFS Collaboration Research program (NIFS17KNXN356 and NIFS15KNST087). And the authors would like to thank Enago (www.enago.jpwww. enago.jp) for the English language review. References [1] O. Mitarai, H. Matsuura, Y. Tomita, Aspect ratio dependencies of D-3He fueled tokamak reactors, Fusion Eng. Des. 81 (2006) 2719–2724. [2] N. Fukumoto, et al., Compact toroid injection system for JFT-2 M, Fusion Eng. Des. 81 (2006) 2849–2857. [3] A. Loarte, et al., Progress in the ITER physics basis, Chapter 4: Power and particle control, Nucl. Fusion 47 (2007) S203–S263. [4] T. Watanabe, et al., Computation of neutral gas flow generation from a CT neutralization fuel-injector, Plasma Fusion Res. 7 (2012) 2405042-1-4. [5] T. Watanabe, et al., Hybrid simulation of neutral gas flow generation from a CT neutralization fuel-injector, Trans. Fusion Sci. Technol. 63 (2013) 358–360. [6] O. Mitarai, et al., Comparative studies of inner and outer divertor discharges and a fueling study in QUEST, Fusion Eng. Des. 109–111 (2016) 1365–1370. [7] T. Yamada, et al., Merging startup experiments on the UTST spherical tokamak, Plasma Fusion Res. 5 (2010) S2100-1-4. [8] K. Gi, et al., Merging experimental study of high-beta ST formation for non-inductive plasma start-up assisted by NBI in TS-4, Plasma Fusion Res. 8 (2013) 1402023-1-5. [9] M.W. Binderbauer, et al., Dynamic formation of a hot field reversed configuration with improved confinement by supersonic merging of two colliding high-β compact toroids, Phys. Rev. Lett. 105 (2010) 4 045003-1-4. [10] O. Mitarai, et al., Ignition studies of D-3He spherical tokamak reactor, ISFNT Conference, Kyoto, 2017 in this. [11] Y. Todo, et al., Simulation study of ballooning modes in the Large Helical Device, Plasma Fusion Res. (2010) S2062-1-4. [12] S.I. Braginskii, M.A. Leontovich (Ed.), Transport Process in a Plasma, Reviews of Plasma Physics, Vol. 1 Published by Consultants Bureau, New York, 1965, p. 205.
Fig. 5. Time change in the number of ions in main plasma and secondary plasma. The solid line shows the result of translation/merging, and the broken line shows the result of non-translation. Red color represents the number of ions in the main plasma, light blue represents the number of ions in the secondary plasma, and light green represents the number of ions in the main + secondary plasma. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
reconnection process. In the MHD simulation, it was possible to calculate the process of translation of secondary plasma and the process
5
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Simulation study on a merging core fueling technique for an advanced fuel fusion spherical tokamak reactor ⁎
Shintato Koikea, , Toshiki Takahashia, Naoki Mizuguchib, Osamu Mitaraic a
Division of Electronics and Informatics, Gunma University, Kiryu, Gunma, 376-8515, Japan National Institute for Fusion Science, Toki, Gifu, 509-5292, Japan c Institute for Advanced Fusion and Physics Education, kita-ku, Kumamoto, 861-5525, Japan b
A R T I C L E I N F O
A B S T R A C T
Keywords: Advanced fusion D-3He reaction Spherical tokamak Core fueling Translation and merging MHD Particle trajectory simulation
With respect to a particle fueling method in a spherical tokamak reactor with an advanced fuel, we investigate the availability of a merging process following the translation of a small torus plasma through which a plasma current flows in the same direction as the main plasma. A three-dimensional resistive MHD model is used to examine the particle fueling effect by reproducing the translation process and simultaneously performing an ion trajectory calculation. The simulation shows the merging process between the translated plasma and the main plasma and exhibits potential as a particle fueling method. The results of the study indicate that up to 30% of the ions in the secondary plasma are supplied to the main plasma.
1. Introduction Nuclear fusion power generation uses advanced fuel that can suppress neutron generation and tritium-handling, and thus it could be superior in safety. Therefore, even when the construction of ITER progresses, it is necessary to investigate a D-3He fusion reactor in the category of advanced fuel fusion reactors. A spherical tokamak (ST) is a possible candidate for a D-3He fusion reactor due to its high-beta value [1]. However, fuel particles supplied from outside are ionized in the plasma edge region due to high temperature and lead to a D-3He fusion reaction, and thus it is very difficult to fuel into the plasma core. For example, it is difficult to supply fuel into the plasma core in conventional particle fueling methods by using gas puffing, supersonic gas jet, or pellet injection. Compact torus plasma (CT) injection [2] is also studied as a method of deep fueling to the fusion plasma core. Currently, studies attempted to solve several problems such as the technical complexity of an injector [3]. Conversely, Watanabe et al. investigated whether accelerated CT is re-neutralized to generate high-speed gas flow for fueling [4,5]. The results indicated that CT plasma with injection velocity of approximately 100 km/s is neutralized by almost 100%. Nevertheless, it is clear that particle fueling is difficult because neutral particles are ionized on the plasma surface. Hence, particle fueling into the core is a very difficult task and especially in an advanced fusion reactor. However, it is an important research subject that should be investigated. To overcome this situation, a novel fueling method is proposed
⁎
wherein a secondary plasma produced in the upper region of the main plasma is translated and then merged into a burning main plasma [6]. The conceptual diagram is shown in Fig. 1. The secondary plasma current is produced inductively with fresh fuel gas puffing and translated in the axial direction to the main plasma current with the same direction. The particle fueling method is feasible if particles in the secondary plasma are supplied to the core region of main plasma. Recent experimental studies also successfully merged ST plasmas [7,8] and high-beta field-reversed configuration plasmas [9]. Therefore, the main focus of this study is to use the merging process as a particle fueling method in a feasible situation. Thus, it is extremely meaningful to quantitatively obtain detailed data required for reactor design such as fueling efficiency. In this study, an MHD simulation and particle trajectory calculation are combined to investigate the merging process of ST plasmas and the ion motion at that time to clarify the feasibility of the merging fueling method and quantitatively evaluate fueling efficiency. 2. Simulation model In this study, we set the time when the generation of secondary plasma is completed as zero and perform a three-dimensional resistive MHD simulation on translation and merging processes. To focus on those processes, we do not calculate the generation process of secondary plasma here. The initial state of plasma is based on the device design [6] parameters shown in Table 1. We solve the Grad-Shafranov
Corresponding author at: Division of Electronics and Informatics, Gunma University, Kiryu, Gunma, 376-8515, Japan. E-mail address: [email protected] (S. Koike).
https://doi.org/10.1016/j.fusengdes.2018.01.003 Received 24 September 2017; Received in revised form 23 December 2017; Accepted 1 January 2018 0920-3796/ © 2018 Published by Elsevier B.V.
Please cite this article as: Koike, S., Fusion Engineering and Design (2018), https://doi.org/10.1016/j.fusengdes.2018.01.003
Fusion Engineering and Design xxx (xxxx) xxx–xxx
S. Koike et al.
developed by Todo et al. is used. However, the MIPS was modified such that the time change in the external magnetic field for translation control is considered, and the function of simultaneous ion trajectory calculation that is described later is added. In MIPS, the resistive MHD equations (1–6) are spatially discretized in each direction of the threedimensional cylindrical coordinate system (r, θ, z). The number of grid points is 112 × 128 × 224 for r, θ, and z respectively. In terms of time, the temporal evolution of each parameter is calculated by performing time integration by using the 4-th order Runge-Kutta method. The MHD equations are as follows:
∂ρ = −∇⋅(ρu) ∂t
Fig. 1. Conceptual diagram of the merging fueling technique for the ST fusion reactor.
Table 1 Parameters of the calculation model (initial condition). Machine major radius (Ro)
6.7[m]
Machine minor radius Main plasma length Secondary plasma length Peak of plasma temperature Plasma density (Uniform) Toroidal field Alfven time(τA) Main plasma beta Secondary plasma beta
2.5[m] 15[m] 10[m] 118[keV] 1.97 × 1020[m−3] 3.06[T] 2.25[μs] 0.20 0.20
{
∂u ∂t
=−
+
ν 4 (∇⋅u ) ∇ρ ρ 3
1 ∇⋅ 2
(1)
}
1
u2 + (∇ × u ) × u − ρ (∇p − j × B )
{
− ∇ × (ρ∇ × u )
}
(2)
∂p 4 = −(∇⋅(pu) + (γ − 1) p∇⋅u) + ν (γ − 1) ρ ⎛j2 + ∇⋅u⎞ ∂t 3 ⎝ ⎠
(3)
∂B = −∇ × E ∂t
(4)
E = − u × B + ηj
(5)
∇×B j= μ0
(6)
Here, eight scalar variables are obtained in which ρ, u, p, B denote the mass density, flow velocity, plasma pressure, and magnetic field, respectively, and ν, γ, η, μ0 denote the viscosity coefficient, specific heat ratio, electric resistivity, and permeability of the vacuum, respectively. Fig. 3 shows the vacuum region set in the MHD calculation of this study and the external coil that triggers translation of the secondary plasma. The vacuum region as indicated by solid black is assumed as a sufficiently small plasma density or is defined as a region outside the device. In this blackened part, we assume that only the magnetic field changes. In Fig. 3, the placement of the assist coil for translation of the secondary plasma is illustrated, and the coil current is set to reach the peak after a linear increase exceeding 100 Alfvén time. The magnetic field lines show the state after 100 Alfvén time elapses. By superimposing the magnetic field generated by the coil current on the magnetic field in the conventional MHD equation, we successfully reproduced the translation. Here the secondary plasma is pushed out by the coil magnetic pressure on the left side of Fig. 3. The trajectory calculation is simultaneously performed (from the equation of motion of ions) with the MHD simulation as follows:
Fig. 2. 2D profile of plasma pressure and contour line of poloidal flux at an initial equilibrium state.
equation to obtain the equilibrium of the main ST plasma. We determine the initial pressure and magnetic field distribution shown in Fig. 2 by separately determining the equilibrium of a smaller plasma in which lines of magnetic force are connected smoothly. The initial distribution of the mass density is assumed as uniform as typically adopted in a general MHD simulation. A simulation with a non-uniform density distribution will be explored in a future study. Therefore, the present study focuses on investigating the effect of particle fueling by using a trajectory calculation. Table 1 shows the plasma parameters of the simulation in addition to the device parameters. The parameters are based on a D-3He nuclear fusion design, and thus the plasma temperature exceeds 100 keV. For both Alfvén time and plasma beta, the magnetic field is evaluated with a toroidal component, and the value on the magnetic axis is used in this evaluation. However, these initial setting of the parameters slightly differ from the proposed D-3He ST reactor [10] to avoid a calculation discrepancy in the MHD calculation. It gradually approaches similar parameters. With respect to the MHD simulation in this study, MIPS code [11]
q dvi dr = i (E + vi × B ) , i = vi dt mi dt
(7)
where qi and mi denote the charge and mass of the ion, respectively. The electromagnetic fields in Eq. (7) are obtained by the MHD simulation. To secure the accuracy of the trajectory calculation, the time step of the trajectory calculation is set to be short by 10−5 with respect to the time
Fig. 3. 2D profile of poloidal field generated by external coil. The assist coils and vacuum region (shadowed area) are also illustrated.
2
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For the simulation of the two cases in which the merging process is completed, we verified the effect of particle fueling by performing a simulation without translation merging (without changing the external magnetic field). Fig. 5 shows the time evolution of the number of ion particles in each plasma in the simulations. In both cases in which the viscosity coefficient is ν = 10−4 and ν = 10−5 , both the main and secondary plasmas exhibit ballooning instability, and the number of particles commences decreasing when 100 Alfvén time elapses after the occurrence of instability. As shown in Fig. 5, the ion number in the main plasma once increases and begins to decrease in the acceleration phase. The observed change is similar to the time change in the volume of the main plasma. Therefore, ions in the peripheral region of the main plasma do not follow the sudden change in magnetic field, and the time change in Fig. 5 in the acceleration phase simply reflects the change in volume of the main plasma. Similarly, the number of particles in the secondary plasma decreases in the acceleration phase because its volume decreases when it is suddenly pushed from behind due to the change in the external magnetic field change. In Fig. 5(a) corresponding to the case of ν = 10−5 , the results indicate that the particle retention rate of the main plasma exceeds that when the secondary plasma is not translated. The results reveal that the number of particles in the main plasma increases after merging although the increment due to merging is 9.2% of the initial particles in the secondary plasma. Fig. 5(b) shows the one order larger normalized viscosity of ν = 10−4 , and the total retention rate for translation merging is slightly lower than that in the non-translation case. Conversely, in the merging process, the number of ions in the main plasma significantly increased. During the merging process, the ion increment corresponds to 30.5% of the initial number of particles in the secondary plasma, and thereby exhibits a certain particle supply as shown in Fig. 5(b). The above results clearly indicate that particles of secondary plasma are supplied to the main plasma by the merging process. However, abrupt particle loss occurred due to the occurrence of ballooning instability. Hence, in our present case, it is impossible to obtain a state that exceeds the initial density of the ST plasma by merging the fueling method. Additionally, pressure flattening due to occurrence of ballooning instability may promote merging, and it is necessary to clarify the possibility of realizing a merging fueling method in a stable plasma in the future by performing a fine simulation. In this calculation, different merging processes were reproduced by simulation when the viscosity coefficient is artificially given. According to Braginskii [12], the viscosity changes as follows:
step of the MHD simulation. The initial distribution of the particles is uniformly arranged by using random numbers. This is the same as the setting of the initial uniform density profile in the MHD simulation. The initial ion velocity is determined as a Maxwellian distribution by using normal random numbers. Thus, the temperature is spatially distributed corresponding to the initial pressure distribution of the plasma. The objective of the ion trajectory calculation is to quantitatively evaluate the supply of particles from the secondary plasma to the main plasma. The trajectory calculation makes it possible to accurately discuss the particle supply effect by calculating the ion behavior near the X-point of the magnetic reconnection region. In this study, the simulation was performed by changing the viscosity coefficient ν in Eqs. (2) and (3) of the MHD equation to clarify the effect of particle fueling by simulating the collision/merging process under multiple circumstances. This is because the viscosity coefficient ν affects the translation speed of secondary plasma, occurrence of instability, and resultant fueling efficiency. Similarly, the electrical resistivity η is a parameter that determines the speed of the magnetic reconnection process. However, the dependence of the merging process on the electrical resistivity will be explored in a future study. 3. Results and discussion The results of the simulation confirm the translation of the secondary plasma and collision/merging to the main plasma. Fig. 4 shows the pressure and magnetic field integral values (upper figure at the respective times) and ion trajectory calculation results (lower figure shows the respective times) at initial (Fig. 4, top) and after the 420 Alfvén time (Fig. 4, middle) and 755 Alfvén time (Fig. 4, bottom). Here, the normalized viscosity coefficient is 10−4 in which the normalization is defined as follows:
ν =
ν , ρ0 R o VA
(8)
where the value of reference quantity ρ0RoVA = ρ0Ro2/τAis 16.4 [m−1 kg s−1] in our case, and VA denotes the Alfvén velocity. The calculation is three-dimensional, and it is not always possible to define the magnetic surface. Therefore, in this study, the following expression is obtained by integration for the ease of understanding:
ψ (r , 0, z ) =
∫ Bz (r , 0, z ) r dr
(9)
It thereby defines the separatrix of the main plasma and secondary plasma. We hereby distinguish between the existence region of ions by ψ(r, 0, z). In the figure shown in the ion trajectory calculation result, red represents ions initially arranged in the main plasma, yellow green represents ions inside the secondary plasma, and blue green represents ions initially arranged that do not correspond to the first two categories. After 420 Alfvén time elapses since the start of the translation (Fig. 4 upper figure), the separatrices of both plasmas collide. Thereafter, the magnetic axes of the two plasmas gradually approach each other, and a sudden merging phenomenon occurs after 750 Alfvén time. Finally, after 755 Alfvén time elapses, merging is completed as shown in the lower part of Fig. 4. In contrast, as shown in the pressure distribution, it is observed that the ballooning instability has already occurred in the two plasmas at the time of collision, and it significantly attenuates when compared with the peak value of the initial pressure. To investigate the collision/merging process and fuel supply effect of the two ST plasmas, several simulations are performed for three cases of the viscosity coefficient. Table 2 shows the time elapsed until occurrence of instability, collision, and merging observed in the simulations. It is observed that occurrence of the instability is delayed when the viscosity coefficient increases. Conversely, when ν = 10−3 with a high viscosity coefficient, the secondary plasma stops in the middle of translation due to the resistance force from the peripheral plasma, and collision of the two plasmas ceases by the end of calculation.
ν ∝ Ti5/2 .
(10)
where Ti denotes the ion temperature. Therefore, the viscosity of the high temperature D- 3He nuclear fusion plasma is high, and instability and merging will be less likely to occur according to the results of the study. However, in general MHD models, a simplified model for viscosity is typically used, and thus it may be necessary to adopt a more detailed MHD model for viscosity in the future. The merging-compression concept verified on MAST produces 400 kA of plasma current within 20 ms [7]. In a manner similar to this experiment, our simulation result also indicates a sharp increase in the plasma current of the main plasma. The occurrence of instability prior to merging is not mentioned in both MAST [7] and UTST [8] experimental reports. The calculation focuses on advanced fuel nuclear fusion, and thus it is difficult to perform a direct comparison with the experiment. However, a future study should advance the comparative study by setting the calculation to the experiment that is currently performed.
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Fig. 4. Color contours of the plasma pressure and contour line of magnetic field integral values ψ(r, 0, z) and ion distribution at the initial time, time of collision and the time of merging. In the distribution map of the ions, the red markers represent the ions initially placed in the separatrix of the main plasma, and the yellow markers represent the ions initially placed in the separatrix of the secondary plasmas. Green color denotes the ions initially placed outside the separatrix of both the main and secondary plasma. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion
axial translation of small torus plasma to ST plasma. The particle trajectory analysis was also simultaneously performed with the MHD simulation to discuss the particle supply effect more accurately by considering the complex ion motion near the X-point in the magnetic
In this study, a three-dimensional resistive MHD simulation was performed to verify the effectiveness of new particle fueling method by 4
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leading to the collision/merging. Specifically, in the study, simulation was performed by changing the viscosity coefficient for the purpose of investigating particle transfer from secondary plasma to main plasma in various translation merging processes. When the viscosity coefficient is low, the translation speed is fast and the time required for merging decreases. Conversely, an increase in the viscosity coefficient stops the secondary plasma during the translation, and collision does not occur. Between the viscosity coefficients, the results indicated that approximately 30% of secondary plasma is supplied to the main plasma by merging. The results revealed that the fueling efficiency depends on the viscosity of the plasma, and thus it is important to further examine viscosity in the MHD simulation. We also note that the ballooning instability is more likely to occur when the viscosity coefficient decreases, and the particle loss increases with the occurrence of instability as clarified in the ion trajectory calculation. The initial trial of MHD simulation on merging fueling indicates the variety of the phenomena, and thus it should be investigated in a future study.
Table 2 Generation time and supply effect of each event in simulations with different viscosity coefficients. Viscosity coefficient ν
10−3
10−4
10−5
Occurrence of instability [τA] Collision [τA] Merging [τA] Supply rate from secondary plasma [%]
1150 N/A N/A N/A
200 415 750 30.5
100 250 640 9.2
Acknowledgments This work is performed on “Plasma Simulator” (FUJITSU FX100) of NIFS with the support and under the auspices of the NIFS Collaboration Research program (NIFS17KNXN356 and NIFS15KNST087). And the authors would like to thank Enago (www.enago.jpwww. enago.jp) for the English language review. References [1] O. Mitarai, H. Matsuura, Y. Tomita, Aspect ratio dependencies of D-3He fueled tokamak reactors, Fusion Eng. Des. 81 (2006) 2719–2724. [2] N. Fukumoto, et al., Compact toroid injection system for JFT-2 M, Fusion Eng. Des. 81 (2006) 2849–2857. [3] A. Loarte, et al., Progress in the ITER physics basis, Chapter 4: Power and particle control, Nucl. Fusion 47 (2007) S203–S263. [4] T. Watanabe, et al., Computation of neutral gas flow generation from a CT neutralization fuel-injector, Plasma Fusion Res. 7 (2012) 2405042-1-4. [5] T. Watanabe, et al., Hybrid simulation of neutral gas flow generation from a CT neutralization fuel-injector, Trans. Fusion Sci. Technol. 63 (2013) 358–360. [6] O. Mitarai, et al., Comparative studies of inner and outer divertor discharges and a fueling study in QUEST, Fusion Eng. Des. 109–111 (2016) 1365–1370. [7] T. Yamada, et al., Merging startup experiments on the UTST spherical tokamak, Plasma Fusion Res. 5 (2010) S2100-1-4. [8] K. Gi, et al., Merging experimental study of high-beta ST formation for non-inductive plasma start-up assisted by NBI in TS-4, Plasma Fusion Res. 8 (2013) 1402023-1-5. [9] M.W. Binderbauer, et al., Dynamic formation of a hot field reversed configuration with improved confinement by supersonic merging of two colliding high-β compact toroids, Phys. Rev. Lett. 105 (2010) 4 045003-1-4. [10] O. Mitarai, et al., Ignition studies of D-3He spherical tokamak reactor, ISFNT Conference, Kyoto, 2017 in this. [11] Y. Todo, et al., Simulation study of ballooning modes in the Large Helical Device, Plasma Fusion Res. (2010) S2062-1-4. [12] S.I. Braginskii, M.A. Leontovich (Ed.), Transport Process in a Plasma, Reviews of Plasma Physics, Vol. 1 Published by Consultants Bureau, New York, 1965, p. 205.
Fig. 5. Time change in the number of ions in main plasma and secondary plasma. The solid line shows the result of translation/merging, and the broken line shows the result of non-translation. Red color represents the number of ions in the main plasma, light blue represents the number of ions in the secondary plasma, and light green represents the number of ions in the main + secondary plasma. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
reconnection process. In the MHD simulation, it was possible to calculate the process of translation of secondary plasma and the process
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