Design of Passive and Structural Conductors for Tokamaks Using Thin-Wall Eddy Current Modeling

Large currents can be induced in tokamak conducting structures in the presence of time-varying magnetic fields, which can be produced by the plasma or external coils. For example, significant induced magnetic fields are generated in the majority of tokamak discharges during the plasma start-up and breakdown phase when currents are ramped quickly in the equilibrium field coils [1]. Changing magnetic fields are also present in tokamak disruptions during the current quench (CQ) as the plasma current decays rapidly due to a sudden increase in the plasma resistivity after the thermal quench phase[2]. These changing magnetic fields have the ability to generate large electric fields which when coupled with non-axisymmetric features in nearby conducting structures have the potential to drive large non-asymmetric currents and therefore produce large non-axisymmetric magnetic fields. Non-axisymmetric magnetic fields can have a variety of negative consequences from affecting the null in the equilibrium poloidal field[3], needed for plasma breakdown, to inducing large forces on structures, and potentially driving dangerous magnetohydrodynamic (MHD) instabilities[4, 5, 6]. However, if properly accounted for, these three-dimensional currents can also be beneficial by slowing the growth rates of MHD instabilities [7, 8, 9, 10, 11, 12] and deconfining runaway electrons (REs) in the case of runaway electron mitigation coils (REMCs), which are being designed for both the SPARC [13, 14, 15, 16, 17, 18, 19, 20] and DIII-D [21, 22, 23, 24, 25, 26, 27] tokamaks. It will be demonstrated in this work that these eddy currents can also significantly affect the response time and efficacy of these REMCs. These potential negative consequences are expected to become more significant in future fusion-relevant machines which will have larger equilibrium magnetic fields and plasma currents, and will therefore be more prone to runaway electrons (REs) and large vessel forces. Therefore, in order to accurately and reliably design and predict the behavior of future devices, it is crucial to be able to simulate the three-dimensional behavior of currents induced in device conducting structures [28, 29, 30, 31, 32].

The remainder of this paper will be split into five sections. In Section 2, the physics and numerical methods employed by ThinCurr are discussed. A study of the proposed REMCs for both the DIII-D and SPARC devices is then presented in Section 3 before a discussion of disruption-induced forces in Section 4. Conclusions and ideas for future work are the discussed in Section 5.

This work utilizes a newly-developed 3D electromagnetic modeling code (ThinCurr), that is based on the PSI-Tet 3D MHD code [33, 34, 35], and solves a thin-wall eddy current formulation, similar to that used in the VALEN [36] and STARWALL [37] codes, that has proven useful for a variety of electromagnetic applications in fusion [11, 10, 40, 41, 8, 9, 7, 36, 42, 38, 39, 12, 35]. However, ThinCurr leverages the modern features of PSI-Tet, such as a CAD interface, the use of unstructured (triangular) grids, parallelization, and interfaces to scalable direct and iterative solvers, to provide additional power and flexibility over prior tools used to study eddy currents in fusion devices. This section provides a brief overview of the code and methods, sufficient to support the presented results. A separate publication will follow with additional information on details of numerical methods and features beyond the scope of this work.

Due to its flexible unstructured finite-element representation, it is straightforward to develop detailed models for 3D conducting structures based on high fidelity CAD models of existing and future machines. The work presented in this article will highlight analysis work completed for both the DIII-D [21] and SPARC tokamaks [17, 20]. An example of the effect of 3D features on calculated current decay time-scales is highlighted in Figure 1 which shows the current distribution (eigenvector) corresponding to the largest eigenvalue ($L\dot{I}=\tau_{W}RI$) for the SPARC vacuum vessel model computed by ThinCurr. The eigenvector pictured on the left corresponds to the longest lived current pattern, which for a simple torus corresponds to an axisymmetric toroidal current localized on the outboard midplane. Without accounting for three-dimensional features, this current decay time-scale is significantly overestimated by $49.4\%$ when compared to the more accurate 3D model (118ms (2D) vs 79ms (3D)) shown in figure 1 (left). The shorter time-scale in the 3D case is a consequence of currents being forced to take longer and narrower, and therefore more resistive, current paths as they maneuver around 3D features such as vessel ports.

This difference between the 2D approximation and the full 3D model can be less significant for particular eigenvectors such as the top-bottom anti-symmetric mode shown in Figure 1 (right). This difference (60ms (2D) vs 51ms (3D)) is less than the largest eigenvalue case as the majority of the current flows in the top and bottom of the vessel. These current paths are therefore not forced to change as significantly in the presences of vessel ports, which are primary located at the midplane, and therefore experience comparable resistivities to the 2D approximation. This particular current pattern is important as its decay affects the growth rate of the vertical instability and limits the time-response of external coils in toroidal devices.

For the DIII-D REMC, ThinCurr was used to determine the effect of coil radius, and therefore distance from the center-post, on the coil’s performance. The effect of coil resistivity was then explored for the REMCs on both devices. The design robustness of both coils was also verified for the case a varying CQ duration and plasma vertical position during the CQ which could vary during a ‘hot’ vertical displacement event (VDE).

ThinCurr was also leveraged to explore the effectiveness of proposed runaway electron mitigation coils (REMCs) for both the DIII-D and SPARC tokamaks. These coils are passive structures which are designed to prevent and mitigate runaway electrons (REs) which are produced by large loop voltages driven during the CQ phase of tokamak disruptions [31, 47]. These large electric fields are capable of producing REs through two primary mechanisms [30]. Electrons flowing through a hot plasma experience a collisional drag force which is a decreasing function of velocity ($1/v_{e}^{2}$) for electrons moving faster than the electron thermal velocity and this force asymptotes for ultra-relativistic electrons at:

where $n$ is the electron density, $m$ is the electron mass, $e$ is the electron charge, $c$ is the speed of light, and $\ln{\Lambda}$ is the Coulomb logarithm. Therefore, application of an electron field in excess of the Connor-Hastie field,

will eventually produce runaway electrons with unimpeded acceleration of electrons with $v>c\sqrt{E_{c}/E}$ [48, 49, 50]. Only a small fraction of the entire electron population falls immediately into the runaway energy range when the electric field is less than the Dreicer field,

unless there is a significant hot tail on the electron distribution function.[51, 52, 53, 54] This hot tail is commonly formed in tokamak disruption plasmas due to an influx of impurities which causes cooling of the hot electrons through collisional drag. Since the higher energy electrons experience fewer collisions they are cooled at a slower rate resulting in surviving “hot-tail” of the initial electron Maxwellian which is strongly susceptible to runaway. The combination of this hot-tail effect and Dreicer production are the primary mechanism for runaway production with these effects dominating in current tokamaks.

The secondary or avalanche mechanism occurs when a thermal electron experiences a hard collision with an existing runaway electron and is bumped into the runaway population while the initial runaway remains in the relativistic population. This effect results in a runaway population which grows exponentially with time as opposed to a linear growth induced by the Dreicer effect [55]. The efficiency of this mechanism scales as an exponential function of the predisruption plasma current and will therefore dominate along with the hot-tail mechanism for future devices like ITER and SPARC [16, 32, 56, 57, 58, 59, 60]. In fact this mechanism is expected to be so efficient that it is expected that a considerable fraction of plasma current will be converted directly to RE current unless avoidance methods are implemented.

These RE beams will potentially carry MAs of current, composed of electrons in the MeV range of energies, in future large plasma current devices such as ITER and SPARC. This will pose a serious risk to plasma facing components in future devices. At high plasma currents in ITER RE impacts on the Be limiters could lead to water leaks [61, 62]. SPARC is designed for shorter 10s plasma flattops and does not have active cooling of the tungsten first wall tiles, limiting the worst RE impact failure mode to significant melting [17]. Work remains to assess the space of possible melt damage. Proposed REMCs must be designed to couple to the large disruption-induced loop voltages and apply significant non-axisymmetric fields capable of safely deconfining REs and preventing further RE seed generation. Screening from induced eddy currents, as well as the forces on the overall system, are a crucial factors to consider when evaluating the performance of a potential REMC.

A large effort is therefore underway to determine a method of avoiding REs and, if avoidance is not possible, a method by which the deleterious effects of REs can be safely mitigated [63, 29, 64, 65]. An exciting candidate for both RE avoidance and mitigation is a passive REMC which takes advantage of the large loop voltage generated by the disruption CQ by placing a non-axisymmetric conductor inside the tokamak vacuum vessel. If this coil is designed properly this loop voltage will drive large currents in the passive conductor which will in turn produce significant non-axisymmetric magnetic fields. Similar 3D fields have been shown both numerically and experimentally to deconfine relativistic electrons by destroying magnetic surfaces [66, 67, 68, 69], driving large MHD instabilities [70, 19, 71, 72, 73, 74, 75], and broadening the pitch angle distributions [29, 64, 76, 31, 47, 77, 78, 66]. These mechanisms are capable of both preventing the formation of a RE-beam by deconfining seed populations and destroying magnetic surfaces as well as terminating an existing population. Therefore, the effectiveness of the coil is dependent on applying a large 3D field early in the CQ. As a consequence of this, it is crucial to understand potential screening current induced in conductors near the coil which can significantly reduce the magnitude of the applied field at this critical stage.

As mentioned previously, plasma disruptions are not the only time where large loop voltages are present during a tokamak plasma discharge such as those applied during plasma start-up or from the vertical stability coils (VSCs), which are expected to be located very near the SPARC REMC. There is therefore a risk of the proposed REMC reacting to these various loop voltages and apply unwanted 3D fields. For this reason the REMCs will be able to be placed into an ‘open-circuit’ mode which will prevent current from flowing in the coils. When desired the coils can be returned to their nominal states, where current does not flow until a threshold voltage is reached. This allows passive activation of the coil, while ensuring non-axisymmetric fields are only applied when desired. The REMCs will also be equipped with a variable resistor which will allow both the time-response and maximum voltage to be tailored for a specific research goal.

The ThinCurr code was leveraged to study the effect of screening currents induced in conducting structures near both the DIII-D and SPARC REMCs as well as the ability to inductively couple to the changing plasma current. These screening current effects can vary significantly between devices due to substantial differences in the vacuum vessel resistivities. While both coils are similar in that they are designed to drive a primarily n=1 field structure, they differ significantly because the DIII-D coil will be mounted on the high-field (inboard) side near the midplane while the SPARC coil will be attached to the low-field (outboard) side near the Vertical Stability Coils (VSCs) as shown in Figure 4. These different coil shapes also result in significantly different applied field structures with the radial and vertical fields for the DIII-D and SPARC REMCs shown in Figure 5 and 6, respectively. As a result of these design choices this paper will also serve as a comparison between these two potential mounting locations. Since the performance of the coil depends crucially on its ability to couple to changing currents within the plasma, special attention was paid to how these changing currents were modeled within ThinCurr.

A ThinCurr model can be driven by prescribing either a voltage or current waveform to a set of predefined filaments. These filaments can be defined by a major radius and vertical position pair or by a collection of points in Cartesian space. The latter provides greater flexibility and the potential to define a coil with 3D features. The most straightforward application involves using a single filament to represent an external magnetic field coil as pictured in Figure 7. As implemented in the SPARC ThinCurr model for the central solenoid coils, driven filaments can also be grouped to represent coils with non-negligible height and width. In this case the resolution in both height and width is set by the user with the total coil current shared evenly across the filament set.

A more sophisticated driver configuration is used to represent currents flowing in the plasma which was used for all disruption related studies described in this article. In this configuration, circular filaments are laid out in an even rectangular (R-Z) grid with the desired resolution. This grid is them trimmed according to the size and shape of the last closed flux-surface from a reference plasma equilibrium. These filaments are then independently driven to recreate the desired current distribution. These current distributions typically correspond to to equilibrium reconstructions from historical discharges for the DIII-D cases and a target equilibrium for the SPARC cases.

The currents in the filaments can then be scaled down to simulate a plasma current quench. While the current is scaled the profile can be held rigid or a series of profiles can be provided to allow the structure of the currents to change with time. An example of this technique is shown in Figure 8 for a typical SPARC equilibrium. Note that due to the peakedness of this profile there is significantly less coupling to the wall than a flat profile with identical plasma current and significantly more coupling than using a single filament at the equilibrium current centroid.

Initially the REMC coils were modeled as a single three-dimensional filament; however, in order to capture the effects of the rectangular coil cross-section the coil was instead modeled as a sheet and meshed with a 10cm resolution. This allowed for the cross-sectional height to be accurately captured, and the thickness of the coil is then accounted for by modifying the modeled resistivity. These design choices enabled the coil geometry to accurately match the latest versions of both the SPARC and DIII-D REMCs.

Due to ThinCurr’s ability to calculate both the eddy current induced in conducting structures and the magnetic fields produced from external coil sets in a fully 3D geometry it is uniquely suited to calculate the $\vec{j}\crossproduct\vec{B}$ forces resulting from tokamak disruptions. This is of particular interest for REMCs which will conduct 100s of kAs of current in the presence of large toroidal magnetic fields. The results of these calculations for the DIII-D REMC and vacuum vessel are shown in Figure 14(top), respectively. As expected, the largest forces induced in the horizontal plane are a result of currents flowing in the longest vertical leg of the coil where the currents are orthogonal to the toroidal magnetic field. This is evident by the angle of the force which points directly at this segment of the coil. Interestingly the largest force on the vacuum vessel during a disruption results from the eddy currents formed in response to the REMC. This force points in the opposite direction to the REMC force but with a significantly lower magnitude. With less than a $1MN$ of force on both the REMC and vacuum vessel these transient forces should be manageable when properly accounted for.

Due to the significantly larger toroidal field and plasma current in SPARC, it is crucial to accurately predict the forces induced by the REMC and its resulting eddy currents in order to ensure its safe operation. Similar to the DIII-D study the forces calculated by ThinCurr are shown in Figure 14(bottom) for both the REMC and vacuum vessel. These forces exceed those predicted for DIII-D by an order of magnitude with forces transiently approaching 10MN. Once again these forces are pointed towards the vertical legs where current flows orthogonal to the magnetic field. The toroidal legs of the coil also interact with the poloidal field (PF) coils, producing a net overturning moment on the vacuum vessel. This will be explored in a future work. An interesting difference from the DIII-D study is that the vessel eddy current and REMC forces are nearly equal and opposite, and since the REMC is supported by the vessel, the net force on the vessel is greatly reduced. This is most likely due to the substantially thicker and therefore more conducting vacuum vessel where again these forces are a result of eddy currents in response to the REMC fields. The magnitude and direction of these forces as well as any induced torques have to be carefully considered when designing the mounting structures for the coil.

The ability to accurately model currents induced in passive conducting structures is a crucial aspect of tokamak design. A newly developed, finite-element, electromagnetic inductive model ThinCurr has been leveraged for disruption studies for both the SPARC and DIII-D tokamaks. A highly detailed 3D model was developed for both SPARC and DIII-D within ThinCurr including varying material thickness, vessel ports, mounts for the REMC and VSCs (SPARC), and a passive REMC. The fundamental current decay time-scales in the vacuum vessel were found to vary significantly between the fully three-dimensional model and the simplified modeled created by sweeping the 2D cross-section. This further motivated the choice of fully three-dimensional analysis.

Particular attention was then placed on analyzing the effect of various parameters on the performance of the proposed REMC coils for both the SPARC and DIII-D devices. As a consequence of early design choices, this comparison serves as a unique study on the effect of coil location on several key performance parameters with the SPARC coil to be installed on the low-field side and the DIII-D coil on the high-field side. One consequence of the DIII-D REMC’s location on the centerpost is that it experiences significant eddy current screening which reduces its ability to apply large fields during the early half of the current quench. This is not expected to limit its effectiveness, but it will be important to offset the coil as much as is allowed by the DIII-D plasma facing tiles.

The performance of the coils was then evaluated as a function of coil resistance which will be set in practice using an external resistor. The range of resistances explored indicate that it will be possible to both reduce the current for the first commissioning phase and maximize the applied fields for RE mitigation and prevention. The robustness of the coils were then evaluated for the case of varying current quench duration and plasma vertical position. Both REMCs were shown to robustly apply significant fields at the midpoint of the CQ regardless of the $\tau_{CQ}$ with the somewhat surprising result of slightly larger fields at moderate CQ durations.

The robustness of both coils was also assessed for the case of a hot VDE which would result in the CQ occurring away from the midplane. For the low-field side SPARC coil it was determined that any reduction in coupling efficiency was more than compensated by an increase in $n=1$ vertical field. The high-field side DIII-D coil showed a much less encouraging result as both the ability to couple to the predisruption plasma current and drive $n=1$ vertical and radial field quickly diminished as a function of vertical position. This shows a distinct advantage for the low-field side coil in dealing with these types of VDEs.

ThinCurr was then used to calculate the forces induced on both the vacuum vessel and REMC during a plasma current disruption. For both coils this force is a primary consequence of current flowing in the vertical segments of the coil, while the forces on the vessels are a consequence of eddy currents induced in response to the REMC fields. The local forces reacted between the REMC mounts and the vessel were calculated to be less than 1MN for DIII-D and less than 10MN for SPARC. The net body force on the vessel is the sum of the coil force and the vessel eddy current force and these simulations find that it is much reduced relative to the local forces. Engineering efforts are underway to ensure these forces can be tolerated in both devices.

The work presented in this article was supported through a combination of public and private funds. The results pertaining to the DIII-D tokamak were supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award(s) DE-SC0022270, DE-SC0019239, DE-SC0014264, and DE-FC02-04ER54698, while the SPARC studies were supported by Commonwealth Fusion Systems. Some coauthors were supported through a combination of funding sources. A.F. Battey and C. Paz-Soldan were supported by DE-SC0022270 for their work relating to the DIII-D device and Commonwealth Fusion Systems for the work on the SPARC machine. Similarly, R. Sweeney was supported by DE-SC0014264 for collaboration on the DIII-D studies and by Commonwealth Fusion Systems for the SPARC REMC studies and C. Hansen was supported by DE-SC0019239 for work related to ThinCurr model development and DIII-D studies and by Commonwelath Fusion Systems for all SPARC research. D. Garnier, R.A.Tinguely, and A.J. Creely were supported entirely by Commonwealth Fusion systems while D. Weisberg was supported entirely through DE-FC02-04ER54698.

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