Tara Ahmadi et al. HIGH-FLUX FRC FORMATION AND ITS FLOW DRIVE BY SPHEROMAK MERGING AND CENTER-SOLENOID FLUX INJECTION T. Ahmadi, Y. Ono, Y. Cai, H. Tanabe, Y. Doke, and T. Tanaka Graduate school of frontier sciences, University of Tokyo Tokyo, Japan Email: [email protected] Abstract By merging opposing toroidal magnetic fields (Bt) within spheromak plasmas and increasing magnetic flux through a central solenoid (CS) coil, we achieved a breakthrough in generating high-flux Field-Reversed Configurations (FRCs). In TS-4 experiments, we produced a record-high poloidal flux of 15 mWb using just 250 kJ of capacitor bank energy, significantly improving efficiency. Ion Doppler spectroscopy revealed that merging opposite helicity toroidal fields created a stable bi-directional toroidal shear flow in the resulting FRC. Even in TS-3's inherently unstable equilibrium with an n=1 mode, this instability remained controlled as long as toroidal shear persisted. Investigating intermittent merging of spheromak plasmas to manipulate toroidal flow, supported by 3D Magnetohydrodynamic (MHD) simulations and experiments, shows promise for controlling n=1 instability and enhancing FRC stability and performance. Field-reversed configurations (FRCs), spheromaks, and spherical tokamaks (STs) all belong to the category of compact and/or spherical torus plasma systems (CT/ST). These systems possess nearly equal aspect ratios, close to 1. Their distinctive geometries allow for axial translation[1, 2] and merging [3] process. In this manuscript, we aim to provide a comprehensive examination of the intricate processes involved in the gradual formation of an FRC. This achievement is realized through the magnetic reconnection of spheromaks, each of which carries opposing toroidal magnetic fields. The precise coordination of this merging technique is demonstrated within the experimental platforms of TS-3[4] and TS-4[1]. A crucial aspect of this merging process involves the phenomenon of magnetic reconnection, a complex interaction that transforms a portion of the toroidal and poloidal magnetic energies from the merging spheromaks into ion thermal energy, thus giving rise to the emerging FRC structure. The conventional approach of utilizing the field-reversed theta-pinch (FRTP) technique[5] has demonstrated the capability to generate plasmas with elevated density (ne ~ 10^21 m^-3) and significant temperature (Ti + Te ~ 500 eV). However, the limitations of this method are apparent in its low efficiency in plasma formation and its dependence on swift capacitor banks. These constraints pose challenges in achieving larger magnetic fluxes for FRCs and sustaining them through the utilization of central solenoid (CS) current drive/amplification and heating techniques. The efficiencies observed in the merging formation of FRCs through flux core and the subsequent flux amplification achieved via Central Solenoid (CS) coils are notably several times superior compared to the conventional theta-pinch method employed, as seen in devices like C-2W [6]. Capitalizing on the present TS-4 spheromak merging experiment, an exceptional opportunity emerges to illustrate the interplay encompassing scenarios. A distinctive stride forward is taken by concurrently executing both the merging formation and the amplification of flux via the CS coil, a juxtaposition demonstrated for the first time and showcased in Figure 1a. The architectural blueprint of the spheromak design is meticulously constructed, bearing an equilibrium between the toroidal and poloidal magnetic fields—labeled as Bt and Bp, respectively. This calculated equilibrium draws inspiration from the Taylor magnetic-energy-minimum state, an elegant construct that guarantees global magnetohydrodynamic stability. Intriguingly, the Taylor state finds its equilibrium within a force-free domain, where thermal pressure can be trivialized. Although the FRC plasma has the potential for instability across various Magnetohydrodynamic (MHD) modes, such as the internal tilt mode, it is important to acknowledge that empirical observations from multiple experiments have consistently shown a significant level of stability in FRCs, especially concerning low-n modes.[7, 8]. To investigate spheromak merging and the FRC formation, we created a 2.5D MHD model that emulates the TS-3 spherical tokamak configuration depicted in Figure 1b. This model includes representations of the poloidal field (PF) and toroidal field coil (TF) waveforms used in the experimental setup. Additionally, this numerical framework will be employed to explore an innovative approach aimed at prolonging the stability of the FRC. This entails the sequential ejection and merging of multiple spheromak plasmas into the FRC structure, to achieve extended stabilization periods. FIG. 1. a: Cartoon view of two spheromaks merging with opposite Bt; b: Cross-sectional view of TS-3 spheromak merging device; c: 2.5D MHD simulation model; d: Coil currents waveform in TS-3 experiment; and e: Coil currents waveform of MHD simulation for intermittent merging scenario. The BOUT++ framework employs a high-order finite difference approach for MHD simulations[9, 10]. In this simulation, a finite element domain of NR x Ntheta x NZ dimensions is established. This method utilizes adaptive order and adaptive time steps to achieve a specified user-defined tolerance, with a maximum time step imposed to ensure result accuracy. When normalized to a reference mass density rho0, an Alfven timescale tau0 is obtained, where we adopt a reference length scale of 1 meter. Consequently, the normalized Hall-MHD equations take the following form: dA||/dt = [psi, phi]/R - eta * Jpar - epsilon * [Btheta, psi]/R + omega * d(A||)/dtheta + A0 * cos(omega * time) * (cos(n * theta) + sin(m * zeta)) - Vtheta . grad||(A||); (1), (2) dBtheta/dt = [Btheta, phi]/R + [Vtheta, psi]/R + eta * grad_perp^2 Btheta + epsilon * [J||, psi]/R + omega * d(Btheta)/dtheta + B0 * cos(omega * time) * (cos(n * theta) + sin(m * zeta)) - Vtheta . grad||(Btheta); (3), (4) dVtheta/dt = [Vtheta, phi]/R + [Btheta, psi]/R + nu * grad_perp^2 Vtheta + omega * d(Btheta)/dtheta + C0 * cos(omega * time) * (cos(n * theta) + sin(m * zeta)); (5), (6), (7) Where J|| = mu0 J_hat_||, A|| = A_hat_||, and phi = tauA phi_hat are the normalized toroidal current density, magnetic flux, and magnetic potential and Btheta, Vtheta are the toroidal magnetic field and velocity. The dissipation terms are the kinematic viscosity nu = tauA nu_hat, resistivity eta = (tauA/mu0) eta_hat, and epsilon = c/omega_p is the Hall coupling factor. In the context of these equations, the toroidal and poloidal perturbations undergo evolution about the theta-toroidal and zeta-poloidal angles. Here, n and m denote the toroidal and poloidal mode numbers, while the subscript 0 pertains to the amplitude value. Our simulation was set up on a 3D toroidal domain with a limited number of grids in the toroidal direction (128 x 32 x 256) with the initial parameters set as follows: an initial temperature value prior to merging denoted as T0 and equal to Ti is established at 10 electronvolts (eV). The plasma density is defined as n0 = 5 x 10^18 particles per cubic meter (m^-3). We employ normalized values for resistivity and viscosity, specifically 10^-6 and 10^-4 respectively. To standardize our major and minor radii, we introduce a typical length scale of L0 = 1m. Consequently, the toroidal Alfven speed becomes v0 = 21 x 10^6 m/s, and the characteristic time scale is tau0 = L0/v0 = 0.05 us. In our simulation, we impose a boundary condition throughout the domain characterized by a perfectly conducting wall. Under this condition, the normal component of the magnetic field denoted as Bn, is required to approach zero. It is important to emphasize that, within the toroidal geometry, we strictly enforce the condition that the velocity (v) equals zero at the boundary. This constraint is essential to ensure the absence of a convective electric field or any normal Poynting flux across the boundary. The poloidal and toroidal magnetic fields are generated by the internal flux cores, driven by the currents illustrated in Figure 1(d). These flux cores are positioned at R = 0.22 meters, with an axial placement at Z = +/-0.2 meters, resembling the configuration of the flux cores in TS-3 as depicted in Figure 1(b). This model incorporates the entire process of spheromak formation, including the current ramp-down phase and the merging process, to replicate the precise experimental setup for FRC formation. This approach distinguishes the current results from previous numerical investigations[11, 12, 13]. At T = 0 us, we observe a gradual decrease in the poloidal current over time, concurrent with an increase in the toroidal current. When we reach T = 10 us, which is the pinch-off time, the poloidal current diminishes to zero while the toroidal current attains its maximum value. This leads to the formation of a pair of spheromaks with oppositely directed toroidal magnetic fields, indicated by the reversal of the toroidal current within the flux cores. These spheromaks, driven by the mutual attraction of the parallel currents, subsequently move toward each other, ultimately merging at the mid-plane. This merging process results in the formation of an FRC. The temporal evolution of poloidal flux, represented by contour lines, unfolds although the FRC plasma has the potential for instability across various Magnetohydrodynamic (MHD) modes, such as the internal tilt mode. Flux contours parallel with the amplitude of the toroidal magnetic field (Bt), depicted through dynamic color changes from red to blue were shown in figure 2. A significant peak is observed at T = 30 us, immediately following the conclusion of the spheromak merging process, resulting in a maximum poloidal flux magnitude of 15mWb. This noteworthy achievement, accomplished with a 250 kJ capacitor bank, surpasses the poloidal flux values achieved in the C-2W[6] device, which employed an MJ capacitor bank and achieved values ranging from 5 to 8 mWb. This represents an enhancement by a factor of 2 to 3. Following this, we turned our attention to the critical issue of n = 1 instability, which is inherent in scenario(a) of figure 1. We addressed this concern by conducting experiments using the TS-3 facility, known for its susceptibility to n = 1 tilt instability in the equilibrium field. The results of these investigations are presented in Figure 2(a), where we show the contours of poloidal flux (indicated by lines) along with the amplitude of the toroidal magnetic field (Bt), displayed in a color spectrum from red to blue. These visuals depict the phase involving the merging of two spheromaks, each having opposing toroidal magnetic fields (Bt), within the TS-3 setup. Additionally, we provide radial profiles of toroidal velocity (Vt), obtained through a combination of data from 2D magnetic probe arrays and ion Doppler tomography techniques. Initially, the ion toroidal velocity remains stationary, only to be set into motion due to the counter-helicity merging. This collision of spheromaks results in a bidirectional toroidal shear flow, with velocities of approximately -15 km/s in regions where R < 0.2 m and approximately +10 km/s in regions where R > 0.2 m. This complex interplay of velocities becomes apparent at around T = 15 us, reaching its peak in the middle of the merging process, and gradually subsiding about 15 us after the merging is completed. Surpassing typical velocities, the peak of this bidirectional flow reaches a high 16 km/s, representing a significant 80% of the Alfven speed. The mechanisms driving this surge in flow can be understood through the insights provided in Figure 2(a). FIG. 2. a: Poloidal flux contour, toroidal magnetic field (background), and radial evolution of toroidal velocity measured in TS-3 spherical torus merging experiment and b: Similar parameters calculated by 2.5D MHD simulation. Figure 2(a) displays the toroidal field Bt and 2D poloidal flux Psi contours, obtained through the utilization of a 2D magnetic probe array. In contrast, Figure 2(b) exhibits the results derived from the MHD model, along with the corresponding radial evolution of toroidal velocity at the axial location of the X-point, specifically at Z ~ 0m. During the time interval from T = 10 us to T = 30 us, we observe the merging of two spheromaks, each possessing opposing toroidal magnetic fields (Bt). These spheromaks are formed inductively via the action of poloidal field coils (PF) situated at R = 0.22m on both sides of the vacuum vessel, located at Z = +/-0.2m. The injection of the toroidal magnetic field into the formed spheromak occurs precisely at the pinch-off moment[3, 14]. The snapshots provided correspond to a specific time when an equivalent amount of flux has been reconnected in both the experimental and simulation scenarios. Through a direct comparison between the numerical and experimental outcomes, it becomes evident that the MHD model offers a comprehensive explanation for the spheromak merging process within fusion devices, along with the subsequent formation and equilibrium of FRC. Initially, the toroidal velocity experiences a simultaneous increase on both edges, reaching its peak at 27 +/- 3 us in both the experimental and MHD model data. Currently, the toroidal velocity exhibits a negative value at R < 0.17m and a positive value at R > 0.17m, with the X-point located precisely at 0.17m. Based on the findings from both frameworks, it becomes evident that the two halves of the interconnected magnetic field lines accelerate the plasma in opposite directions along the toroidal axis. The maximum attained toroidal velocity registers at 12km/us at R = 0.22m, which, notably, falls below the calculated Alfven velocity for the specified plasma parameters. As time progresses, at T = 22 us, the sheared velocity gradually diminishes. Beyond T = 25 us, we observe a reduction in the toroidal magnetic field (Bt), signifying a decrease in bi-directional toroidal shear velocity due to the diminishing motion of field lines moving in opposite directions. Finally, the Bt profile relaxes to zero indicating the formation of an FRC with no toroidal magnetic field present. The relatively slower progression of the simulation, compared to the merging process observed in the experiment, is likely attributed to the rapid reconnection induced by anomalous resistivity. Indeed, comparisons between computational and experimental data reveal that the developed model aligns excellently with the experimental results. This alignment suggests that a discrepancy of a few microseconds between the two is entirely reasonable. Throughout the entire reconnection process, the presence of a thin CS coil with a radius of approximately 3 cm played a crucial role in maintaining the stability of the merging plasmas[13]. It's important to note that during and after the reconnection, all toroidal modes exhibited growth. This growth was influenced by the removal of the central conductor and was contingent on the overall stability of the generated plasma. Figure 3 illustrates the temporal progression of n = 1 mode amplitudes during the merging of two spheromaks as depicted in Figure 2. FIG. 3. Time evolution of n = 1 toroidal mode amplitude measured in TS-3 spherical torus merging experiment and that in MHD model. In both scenarios, it was observed that the initial amplitude of the n = 1 mode was reduced by half during the counter-helicity reconnection process. The time evolution of the toroidal mode amplitude obtained from the MHD simulation closely aligns with the amplitude profile measured in the TS-3 experiment. However, it's noteworthy that the simulation results were consistently delayed by an average of 3 us due to the slower reconnection speed. In the TS-3 experiment, the reversed poloidal field (PF) coil current eventually drops to zero, whereas in the numerical framework, it is held constant. This difference in the treatment of the PF coil current accounts for part of the delay observed in the simulation results compared to the experimental data. The prominent toroidal shear flow generated during the counter-helicity reconnection process appears to be the primary factor contributing to the stabilization of the n = 1 mode. Initially, the toroidal flow is oriented exclusively toward the positive toroidal axis. However, as the reconnection progresses, a negative toroidal flow emerges, giving rise to a robust shear flow. Consequently, the n = 1 mode experiences damping when the shear flow reaches its maximum, with the toroidal flow exhibiting a positive sign for r > 0.17m and a negative sign for r < 0.17m. This bi-directional flow serves to stretch the unstable n = 1 mode toroidally, effectively contributing to the stabilization of the FRC. The study revealed that the sustained Alfvenic bi-directional flow plays a pivotal role in suppressing the n = 1 toroidal mode during thin Central Solenoid (CS) operation. TS-3 with its thin CS coil configuration is capable of preserving the stability of the n = 1 mode. However, when the FRC translates into the burning chamber without the presence of shear flow or CS coils, the n = 1 mode may potentially re-emerge. To address this issue, it is advisable to consider replacing the low-efficiency Neutral Beam Injection (NBI)[15] method with a more efficient flow injection technique, which could contribute to a more stable and controlled operation. This challenge prompted the development of a novel technique known as "intermittent merging," illustrated within the orange rectangle in Figure 1(a). In this approach, after each FRC formation, two newly created spheromaks are introduced to merge with the existing FRC. This process is designed to sustain bi-directional toroidal flow for extended durations, achieving quasi-steady toroidal shear flow. Following the successful integration of the newly formed spheromaks into the FRC, another pair of spheromaks is generated to continue the sequence. As depicted in Figure 1(c), an initial configuration for the 2.5D MHD model is presented to assess the effectiveness of intermittent merging. The first pair of spheromaks, responsible for initiating the FRC, is generated by two primary flux cores labeled as L/R1 and positioned radially at 0.37 meters on opposite sides of the vacuum vessel. Subsequently, after FRC formation, the secondary flux cores come into play. In the secondary flux cores L/R2, the poloidal field current experiences an initial ramp-up phase, which can influence the magnetic configuration of the FRC. To address this issue, two separation coils are strategically placed near the flux cores L/R2 to compensate for the gradual increase in current associated with the secondary flux cores. Figure 4 provides an informative overview of the process, including the poloidal flux contours, toroidal magnetic field distribution, and the radial profile of the toroidal velocity at Z = 0, both during standard spheromak merging and the intermittent spheromak merging technique. FIG. 4. Time evolution of poloidal flux contours, toroidal magnetic field (background), and radial profile of toroidal flow during the three cycles of the intermittent merging of two spheromaks with opposite Bt in MHD model. In the initial phase (0 - 20 us), the gradual reduction of poloidal coil currents leads to the creation of a pair of spheromaks with opposing toroidal magnetic fields (Bt). Once these spheromaks are disengaged from the flux-core coils, the mutual attraction of the parallel currents propels them toward each other, ultimately resulting in the magnetic reconnection process and the formation of an oblate FRC. During this merging phase, bi-directional toroidal flow is maximized at t = 48 us, effectively suppressing the n = 1 mode. Following the formation of the FRC, the bi-directional toroidal flow is sustained even during the ramp-up phase of the secondary flux cores L/R2. The bi-directional toroidal flow reaches its minimum when the second pair of spheromaks separates from the flux cores L/R2 at 73 us. Subsequently, it begins to rise again at 76 us as these spheromaks merge into the FRC. As the additional two spheromaks, each with opposing Bt, collide with the pre-existing FRC, the merging process causes the field lines of the FRC to stretch in the toroidal direction, resulting in the bi-directional toroidal flow. The time interval from the pinch-off moment of the second pair of spheromaks to their complete merging with the FRC is approximately 10 us. During this phase, the toroidal outflow from the reconnection drives the bi-directional flow shear. Another ramp-down of flux-core currents occurs at 95 us, separating the third pair of spheromaks from the coils. Just before the final relaxation of the FRC and the loss of negative toroidal flow, the third pair of spheromaks merge into the FRC, thereby extending the duration of the bi-directional shear flow. Throughout the three cycles of counter-helicity merging, the driving force behind the maintenance of quasi-steady bi-directional Alfvenic toroidal flow is the suppression of the amplitude of the n = 1 toroidal mode. Several key parameters come into play during intermittent merging, including the positioning of the secondary flux cores L/R2, the intensity and direction of the toroidal magnetic field, and the pinch-off time of the spheromaks. These parameters must be optimized to effectively drive the bi-directional toroidal flow. FIG. 5. Time evolution of n = 1 toroidal mode amplitudes in high-shear flow (13 x 10^4 1/s) and low-shear flow (6 x 10^4 1/s) operation of intermittent spheromak merging in MHD simulation. Figure 5 provides a valuable comparison of the temporal evolution of the n = 1 mode amplitude between two scenarios: one involving high flow shear intermittent merging (blue line) and another featuring low flow shear intermittent merging without optimization of the secondary flux coils (red line). In both instances, after the initial pair of counter-helicity spheromaks are pinched off, the n = 1 mode amplitude reaches its maximum. Subsequently, as the merging process commences, the toroidal mode amplitude decreases, reaching its lowest point at 48 us in the optimized case, where the maximum toroidal flow attains +/-13 km/us. After the initial merging cycle, as the bi-directional toroidal velocity rapidly decreases, the mode amplitude begins to increase. However, this increase is more pronounced when the bi-directional toroidal flow slows down to 20%, as indicated by the red curve. The mode amplitude experiences a decrease once again during the second merging cycle, and in the most favorable scenario, it is maintained at 25% of its initial amplitude. To achieve the lowest possible n = 1 mode amplitude, it is imperative to optimize the settings of the secondary coils to generate a stronger bi-directional flow. By replacing NBI with precisely controlled quasi-steady bi-directional toroidal flow, it becomes feasible to maintain a steady state with improved stability. In summary, our study has culminated in several key findings. Firstly, we have successfully developed a comprehensive MHD model that has undergone rigorous validation through direct comparisons with experimental magnetic measurements conducted during counter-helicity spheromak merging experiments within the TS-3 device, featuring a thin 6cm central solenoid coil. Throughout our investigations into counter-helicity spheromak merging, we have demonstrated the effective suppression of the n = 1 mode amplitude. This suppression is attributed to the stretching of magnetic field lines in the toroidal direction, a consequence of the presence of bi-directional toroidal shear flow. Notably, a direct correlation has been established between the toroidal velocity shear and the amplitude of the n = 1 mode. Furthermore, our study has introduced an innovative intermittent spheromak merging technique, which plays a pivotal role in driving the toroidal shear flow. 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