5.1.1. Major disruption cold vertical displacement event

In this subsection we map the eddy current load distribution to the ARC vessel using the two MD cold VDE design disruptions (§ 3) as input to the ThinCurr electromagnetics code (Hansen et al. Reference Hansen, Battey, Braun, Miller, Lagieski, Stewart, Sweeney and Paz-Soldan2025). In these simulations the CQ is represented using a set of 1600 toroidal filaments, arranged in a uniform 2-D grid, which approximate the time-varying plasma current distribution. The vacuum vessel (VV) is represented as three axisymmetric shells in the ThinCurr model: (i) 0.5 cm tungsten ( $\eta = 330$ nΩm) and (ii) 1 cm vanadium ( $\eta = 600$ nΩ m) shells for the inner VV and (iii) a 3 cm vanadium ( $\eta = 600$ nΩ m) shell for the outer VV. Each shell is located at the innermost surface of the corresponding volumetric region. Over 2 years of fusion power operation the resistivity of vanadium is expected to change by ${\lt}10$ % which will not significantly affect the results presented herein.

Figure 7.

Results from ThinCurr simulations of MD cold VDEs using the reduced filament model: (a) filament current distribution and force density in the outer VV for the mitigated case at $I_p/I_{p,0} = 20$ % and time histories of the (b) major radial force and (c) vertical force in the inner and outer VV for mitigated ( $\tau _{CQ} = 14.7$ ms) and ITPA ( $\tau _{CQ} = 4.89$ ms) cases. The colour bar corresponds to the vessel forces and not the current filament distribution.

The resulting net forces on the inner and outer VV are shown in figure 7 together with the distribution of force in the outer VV resulting from the controlled, mitigated CQ with a characteristic exponential time scale of $\tau _{cq,exp} = 14.7$ ms corresponding to an equivalent linear CQ duration of 34 ms. Note that the net radial force is an outward/inwardly directed hoop force, not a sideways force (see § 5.1.3 for the sideways force). The loading characteristics of the two VDE cases only yield differences on the fastest time scales during the quench and quickly converge to similar loads by 100 ms. The peak radial loads are not significantly different between the two CQ cases and both exhibit a significant imbalance between the inner and outer VV with total forces of approximately 35 and 65 MN respectively. Vertical loads generated during the late stage VDE are minor at less than 5 MN.

Simulations were also performed with representative port openings on the outboard midplane to assess the impact of toroidal asymmetry in available current paths on forces. These simulations showed only small changes ( $\lt 5$ %) in the total peak force on the inner and outer VVs, relative to the axisymmetric model, for all ThinCurr VDE simulations. Toroidal variation of the poloidal force distribution is similarly modest compared with the axisymmetric model, giving confidence that additional structures will not significantly affect the analysis of eddy-associated forces.

The mitigated MD cold VDE is expected to be the dominant disruption type in ARC and the one for which rapid restart is important. Considerable hoop forces are imparted on the vessel which will drive the vessel thickness, creating a design tension between heat conduction (which is optimised for thinner vessels) and structural integrity. The peak loads, their distribution, and a loading frequency of ${\sim}\,1$ mitigated MD cold VDE per day will inform the ARC vessel design.

5.1.2. Hot vertical displacement event

In this subsection we first assess the maximum vertical force using a simple, conservative approach that ignores the low-pass filtering effect of the vessel. We then complement this conservative estimate by inputting the hot VDE design disruption into ThinCurr which accounts for the vessel filtering effects and provides eddy current load distributions.

To provide a conservative upper bound on the maximum vertical force, we calculate the force between a displaced plasma with $q_{\textit{edge}}=1$ and the PF coils, referred to as $F_\textrm{pc}$ in Miyamoto (Reference Miyamoto2011). While empirical studies on AUG find disagreements with the full Miyamoto model, the quantity $F_\textrm{pc}$ is shown to bound the measured vertical forces (Pautasso & Fable Reference Pautasso and Fable2025).

An algorithm is developed to estimate $F_\textrm{pc}$ in a tokamak given an initial equilibrium and a first-wall limiter contour and is described in Appendix C. The maximum vertical force in ARC using this algorithm is 120 MN. This is conservatively the highest vertical force that eddy currents, halo currents or a combination of the two could impart on the ARC vessel (Miyamoto Reference Miyamoto2011; Clauser et al. Reference Clauser, Jardin and Ferraro2019). With a 0-D max vertical force now defined, we seek profiles to distribute this force on the VV, and in addition we will explore a higher fidelity and less conservative estimate of the vertical force that takes credit for the vessel screening.

The hot VDE design disruption is input to ThinCurr to calculate the vessel eddy current response and forces. The resulting net forces on the inner and outer VV are shown in figure 8 together with the distribution of $J \times B$ force in the outer VV when $I_\textrm{p}$ reaches 20 % of its initial value. As for the MD cold VDE, the net radial force is an outward/inwardly directed hoop force, not a sideways force, and the force distribution is given in terms of force per unit length in the toroidal direction.

Due to the high current through the vertical motion, the hot VDE yields significant vertical force, peaking at approximately 25–30 MN on both the inner and outer VV. Radial forces are also significant, although somewhat lower than the MD cold VDE case at approximately 38 and 52 MN in the inner and outer VV respectively. Additionally, the inner/outer loading imbalance is significantly lower for the hot VDE compared with MD cold VDEs. A pronounced valley in the vertical and radial loads is seen at the start of the CQ, due to partial cancellation of eddy currents driven by the VDE. However, this feature is expected to be artificial as this model does not include halo currents, which would act to ‘fill in’ these regions and increase the total force.

While the ThinCurr model does not include halo currents directly, the net force they produce on the structure can be approximated using the force on the toroidal plasma filaments in ThinCurr. In the true system, the net force on the plasma will be approximately zero during the CQ due to rapid force balance time scales in the plasma and the lack of significant stored energy following the TQ. Similarly, while forces can be quickly eliminated within the plasma, the net force within the VV cannot change faster than fields can diffuse through the wall. As a result, in this approximate picture, this ‘residual load’ on the plasma must be carried by the VV itself through modifications to the eddy currents and halo currents.

Figure 8(d–e) shows the impact of this residual plasma force on the total force over the combined inner and outer VV for the hot VDE as modelled by ThinCurr. As expected, the additional force contributed during the CQ by this term results in a smooth evolution of the radial and vertical forces through the CQ without the valley generated by eddy currents alone. The peak force from the combined effects is modestly higher (89 and 66 MN for the radial and vertical force respectively) than that predicted by the eddy currents alone (86 and 53 MN for the radial and vertical force respectively). The maximum vertical force of 66 MN found here is in good agreement with the 69 MN found with M3D-C1, despite the latter simulation using an earlier ARC version (see § 2). This method provides an assessment of the total vertical force that accounts for the shielding effect of the vessel. Next we discuss a method for finding the halo current distribution.

Figure 8.

Results from ThinCurr simulations of a hot VDE using the reduced filament model: (a) filament current distribution and force density in the outer VV at $I_p/I_{p,0} = 20$ % and time histories of the (b) major radial force and (c) vertical force in the inner and outer VV. The impact of halo currents on the total net (d) major radial force and (e) vertical force on the combined VV is also shown, using a reduced model based on residual plasma load (red) to approximate halo currents effects. Note that the colour bar in panel (a) corresponds to the vessel forces and not the current filament distribution.

The force distribution resulting from halo currents can be captured using the analytic model presented in Appendix E, and in the future, will be captured in higher resolution by imposing current sources and sinks to relevant first-wall components in Ansys, as was done for SPARC. Figure 9 shows the resulting halo current force distribution at the maximum vertical displacement of the hot VDE design disruption. This model takes as inputs the halo current source location, the sink location and the total vertical force $F_v$ and then calculates the force distribution, the net radial force, and the halo current magnitude consistent with $F_v$ . The source and sink locations in figure 9 are shown by the green and red markers and the input vertical force is 66 MN to drive the maximum force predicted by ThinCurr. It is interesting to note that the maximum vertical force is driven by $I_H=-0.9 \,\text{MA}$ corresponding to a relatively low halo current fraction of $f_H = 0.08$ . A similar result is found for SPARC when the halo current path transits the entire divertor region. It is speculated that the highest halo current fractions approaching $f_H=0.7$ reported in the ITPA Disruption Database (Eidietis et al. Reference Eidietis, Gerhardt, Granetz, Kawano, Lehnen, Lister, Pautasso, Riccardo, Tanna and Thornton2015) might occur when the halo current source and sink points are both on the outboard (or inboard) side. Equation (E4) shows that the total vertical force imparted is a function of the radial distance spanned by the source and sink, and thus for a large halo current fraction of $f_H=0.7$ to be consistent with the maximum vertical force, the radial span must be considerably smaller than that shown in figure 9.

The chosen source and sink points in figure 9 are consistent with the plasma configuration that leads to the highest vertical force, but many other possible halo current source and sink configurations are possible. Halo current magnitudes up to $f_H=0.7$ , as reported in the IDDB, will be driven through the structures in ARC to ensure the design is structurally robust. Further, as the 3-D design matures, the HEAT code (Looby et al. Reference Looby, Reinke, Wingen, Menard, Gerhardt, Gray, Donovan, Unterberg, Klabacha and Messineo2022) will be used to simulate halo current concentrations resulting from field lines that intersect protruding components.

The loads from the hot VDE are the dominant load driving the VV structural design. These loads are provided to the engineering teams to design resilient structures.

Figure 9.

Halo current force distribution resulting from the highest vertical load of $F_v=66$ MN as predicted by ThinCurr. The force here has been integrated over the full toroidal angle assuming axisymmetry. Forces are reported per unit length along the vessel $RZ$ contour. The green and red markers denote the source and sink points for the halo current.

5.1.3. Asymmetric VDE

With the vast majority of disruptions mitigated, asymmetric VDEs are likely a rare event. If a RE mitigation coil (see § 4.2) is included in ARC, it might be that the coil increases the likelihood that a VDE becomes asymmetric. This will be learned on SPARC to better refine our expectations for ARC. We evaluate the loads on ARC to ensure the device can handle a few sideways loading events. It is shown here following the same analysis methodology as SPARC (Riccardo Reference Riccardo2022) that the sideways force associated with the asymmetric VDE is 21 MN and dominantly reacted by the toroidal field (TF) coil, and the resulting vessel displacement is ${\lt}6$ mm owing to an eddy current braking force induced by the vessel movement which is also reacted by the TF.

The sideways force results from the kinking of a plasma that has displaced vertically until its edge safety factor reduces to near $q_{\textit{edge}}=1$ (Riccardo et al. Reference Riccardo, Walker and Noll2000b ). The sideways force has been the focus of many theoretical works discussed in Pustovitov (Reference Pustovitov2022) and fewer simulation works as these simulations are computationally expensive (Jardin, Clauser & Sovinec Reference Jardin, Clauser and Sovinec2020; Artola et al. Reference Artola, Sovinec, Jardin, Hoelzl, Krebs and Clauser2021b ). The Noll force (Riccardo et al. Reference Riccardo, Noll and Walker2000a ) provides a method for estimating the sideways force using a tilted current ring interacting with predominantly the TF, and despite the notable inconsistency that the plasma is not force free in this model, the Noll force matches the measured sideways force in JET. The Noll force approach is not applicable to very conductive vessels where magnetic diffusion delays field changes from being communicated to the external coils which react the force.

Figure 10.

(Left) COMSOL model of ARC V3A used for sideways force modelling. Two sections of the vessel and the titled current filament are shown. (Right) Integral force on the vessel (blue) and Maxwell stress tensors integrated over the outer (green) and inner (red) surfaces of the vessel.

Here, we use a plasma model based on the Noll Force with the inclusion of a realistic vessel modelled in COMSOL (Reference Cosmol2023) to capture the magnetic diffusion physics. This approach has been used to model JET and due to the relatively resistive vessel, it does not significantly change the Noll Force predictions for JET and thereby good agreement is again found with experimental data.

A model of the ARC VV is built in COMSOL and shown in figure 10(a). The double-walled VV is simplified as a single wall with an exaggerated thickness of 20 cm for numerical convenience. The actual wall thicknesses are 3 cm for the outer wall, 1 cm for the inner wall (both made of vanadium) and 0.5 cm of tungsten for the plasma facing layer. At operating temperature, vanadium has a resistivity of $\eta _\textrm{v} = 600 \, \textrm{n} \Omega \textrm{m}$ and the first-wall tungsten operating at 1000 C has a resistivity of $\eta _\textrm{w} = 330 \, \textrm{n} \Omega \textrm{m}$ . Treating a wall section as a set of parallel resistors we find an effective uniform resistivity of $\eta _{\textit{eff}} = 45 \eta _\textrm{w} \eta_\textrm{v} / (5 \eta _\textrm{v} + 40 \eta_\textrm{w}) = 550 \,\textrm{n} \Omega \textrm{m}$ . Finally, to account for the modelled 20 cm thick vessel we scale $\eta _{\textit{eff}}$ to find $\eta _{\textit{eff},20} = \eta _{\textit{eff}} 20 / 4.5 = 2.44 \, \mu \Omega \textrm{m}$ , and this value is input to COMSOL.

Current filaments are used to model the plasma magnetic field evolution. For numerical convenience we do not model the plasma vertical displacement and instead kink a plasma that is centred in the machine. This simplification will move the current distribution in the vessel from being up–down asymmetric to having some symmetry about the midplane, but is not expected to grossly change the vessel diffusion time and the resulting force. The current starts on-axis at the full $I_\textrm{p0}=12$ MA. At 5 ms the CQ begins and current is then shared with a filament with an axis tilted by 40 % of the minor radius (i.e. the maximum deviation from the untilted ring is $0.4a$ ). The slowest $\tau _\textrm{cq}=142$ ms CQ duration is used here to maximise the flux diffusion through the vessel and thereby maximise the sideways force. The current in the kink rises linearly from zero and peaks at $0.4I_\textrm{p0}$ over a duration of $0.2\tau _\textrm{cq}$ . After reaching $0.4I_\textrm{p0}$ , the current in the kink decays linearly to zero over the remainder of the CQ (i.e. over $0.8 \tau _\textrm{cq}$ ). An untilted filament (not shown) carries $I_\textrm{p0}$ at time zero and then time evolves so as to complement the tilted filament, generating a net toroidal current that decays linearly to zero over 142 ms (i.e. the untilted filament decays linearly from $I_\textrm{p0}$ to $0.6I_\textrm{p0}$ over $0.2\tau _\textrm{cq}$ , and then decays linearly from $0.6I_\textrm{p0}$ to zero over the remaining $0.8\tau _\textrm{cq}$ ). The TF is modelled by a single vertical current filament located at $R=0$ carrying the total TF coil current, and all PF coils are included with nominal currents from the end of the plasma flattop. The Maxwell stress tensor is then evaluated at all times over the outer surface of the VV, resulting in the net electromagnetic force on the vessel. This force is shown by the green curve in figure 10(b). The force rises over ${\sim}\,100$ ms reaching a peak sideways force of 21 MN.

A model is developed to capture the vessel dynamics during this sideways loading and is described in detail in Appendix D. The basic concept is the vessel inboard wall is represented by a cylinder moving ‘sideways’ in a TF and induced eddy currents and the resultant $J_\textrm{eddy} \times B_\textrm{T}$ are solved for. The system resembles a damped harmonic oscillator with the exception that the centre of the ‘potential well’ can move when the vessel motion occurs on a time scale comparable to the eddy current decay. The simplicity of the model is complemented by COMSOL simulations that tune parameters in the model, those being the cylinder height (related to the total force) and the effective resistivity (related to the vessel diffusion time). The fluid FLiBe is expected to damp motions of the vessel; the FLiBe is not included here.

The same COMSOL model is used to tune the equation of motion by measuring the stiffness of the vessel moving in the TF as well as the L/R decay of the eddy currents. For simplicity, the single central TF filament is moved by 10 cm in 1 ms, instead of moving the vessel. A time scale of 1 ms is chosen to well separate the displacement and eddy current decay time scales for the purposes of sampling the stiffness and should not be interpreted as a physical expectation of the actual dynamics. The artificial force on the vessel is calculated and peaks at ${\sim}\,600$ MN as shown in figure 11(a), and thereby the magnetic stiffness is $k_{mag} = 6$ GN m⁻¹. The TF filament is then held fixed and the force is allowed to decay providing a measure of the vessel diffusion time. The vessel displacement of 10 cm in 1 ms is also simulated in the reduced model. The cylinder height is tuned to match the 6 GN/m and the result is $h=4.5$ m which is a reasonable length scale that is comparable to the height of the inboard vessel wall. Next the resistivity is adjusted to approximately match the decay of the force. Increasing the resistivity by 60 % (i.e. $\eta = 880$ nΩ m) is found to reasonably match the decay curve (see figure 11(a)). The residual discrepancy in the decay of the force is likely attributable to the COMSOL model exhibiting a multi-mode behaviour with different exponential decay times superimposed whereas the reduced model assumes only a single mode. For the purpose of this initial sideways motion scoping the agreement is sufficient.

The reduced model is then subject to a half-sine sideways force peaking at 21 MN and with a time evolution that is comparable to the COMSOL modelling. The force and resulting vessel motion are shown in figure 11(b–c). The maximum sideways displacement is $\lt\,6$ mm which is relatively small in comparison with the size of the device. Interestingly, the vessel does not return to $x=0$ mm but rather returns to $x\approx 3$ mm and oscillates at the $\sqrt {k_{mag}/m}$ frequency, where $m$ is the vessel mass. As the vessel will be immersed in fluid FLiBe, it is expected that oscillations will be further damped than shown here. The vessel mass is chosen to be $m=150$ metric tonnes for the simulation shown in figure 11(c). All but the oscillation frequency is found to be relatively insensitive to the vessel mass (values of 50 and 300 metric tonnes also result in $\lt\,6$ mm total displacement). An additional structural stiffness $k_{struct}$ is added to return the vessel back to $x=0$ mm over a longer time scale. The value of $k_{struct}$ is chosen to be smaller than $k_{mag}$ to prevent significant sideways load transfer to the structure providing the stiffness, but large enough to effectively re-centre the vessel. In the simple modelling here a value of $k_{struct} = k_{mag} / 10 = 0.6$ GN/m is used.

While the TF is responsible for driving the plasma sideways, and through eddy currents, driving the vessel sideways, it is also responsible for the force that resists the vessel movement. The net effect is a relatively small displacement that can be handled by ensuring adequate compliance in the vessel mounting. Attention will be paid to ensure the vessel supports can provide a sufficient re-centring force following a sideways loading event. The sideways force distributions on the vessel as well as the total displacement of ${\sim}\,6$ mm will be translated into requirements for the ARC VV.

The above model assumes that the plasma kink is stationary, ignoring the rotation that is often observed empirically, and the potential for a structural resonance that follows. Should an REMC be installed in ARC, it is possible that the $n=1$ field locks the asymmetric VDE. Given the uncertainty in REMC installation and its interaction with the asymmetric VDE, here we consider the rotation frequency. This rotation can resonate with the vessel and further increase loads, and is expected to be responsible for the worst case mechanical loads on ITER (Schioler et al. Reference Schioler, Bachmann, Mazzone and Sannazzaro2011). The asymmetric VDE rotation frequency expected in ARC is calculated using an empirical scaling law (Saperstein et al. Reference Saperstein, Levesque, Mauel and Navratil2022, Reference Saperstein, Tinguely, Granetz, Levesque, Mauel and Navratil2023) which gives $f_{rot} = 292 / B_T (S/\pi ) \sim 12$ Hz (where $B_T$ is the toroidal magnetic field and $S$ is the poloidal cross-section). This rotation frequency is similar to that predicted for ITER (Myers et al. Reference Myers, Eidietis, Gerasimov, Gerhardt, Granetz, Hender and Pautasso2017; Saperstein et al. Reference Saperstein, Levesque, Mauel and Navratil2022). This scaling law has an error of approximately a factor of 2, so rotation frequencies between 5 and 30 Hz are possible. The resonant frequencies of the ARC vessel have not been calculated to date, but we expect to find resonances in this 5–30 Hz range. These predictions will be used to inform a dynamic amplification factor for use in the design of the vessel and the mounting structure.

Figure 11.

(a) Tuning of the reduced model to the COMSOL model by displacing the vessel by 10 cm in 1 ms in both models. (b) Input half-sine wave forcing to the reduced model. (c) Resulting vessel motion for a 150 metric tonne vessel.

5.2.1. Mitigated disruption thermal loads

Massive material injection distributes the plasma thermal energy across the first wall and controls the rate of magnetic energy dissipation and its deposition on the first wall. Parameterised synthetic disruptions describing the time evolution of the stored thermal and magnetic energy are used to derive radiated powers by assuming that radiation is the sole loss channel (i.e. the worst-case radiation flash). The radiation poloidal peaking is captured by two radiation structures that are ray traced to the first wall using the HEAT code (Looby et al. Reference Looby, Reinke, Wingen, Menard, Gerhardt, Gray, Donovan, Unterberg, Klabacha and Messineo2022) producing a radiation intensity map. The toroidal peaking is applied by enhancing the HEAT intensities as a function of the toroidal angle. A nonlinear 1-D heat equation using temperature-dependent material properties for tungsten is then used to assess the temperature rise from a realistic starting surface temperature of 1000 C. The initial tungsten surface temperature is informed by estimated flattop plasma heat fluxes and thermo-hydraulic modelling of the FLiBe coolant behind the first wall. Using the parameterised synthetic disruptions and 12 MGI valves, a Monte Carlo study finds that within realistic ranges of uncertain parameters, the probability of flash melting at the highest loaded location (i.e. the outboard midplane) is almost as likely as not melting. A mitigation strategy is then presented demonstrating how the mitigated disruption might be tuned to ensure melt prevention.

Recent studies of JET mitigated disruptions leveraging 3-D tomographies (Stein-Lubrano et al. Reference Stein-Lubrano2024) provide insight into the radiation structure evolution during mitigated disruptions (Stein-Lubrano et al. Reference Stein-Lubrano2025). As the impurity gas or shattered pellet fragments enter the plasma, a radiation plume local to the injection with a toroidal span of ${\sim}\,\pm\,45^\circ$ is observed together with a lower emissivity helical structure with a larger toroidal span. During this pre-TQ phase many tens of per cent of the plasma thermal energy is radiated. The TQ phase occurs next characterised by the highest radiated power emitted from a structure with greatly reduced toroidal peaking and a less well defined shape, likely explained by the intersection of a $m/n=1/1$ convective heat flux with the impurity distribution in the vicinity of the $q=2$ surface (Izzo et al. Reference Izzo2015, Reference Izzo, Stein-Lubrano, Battey, Sweeney, Hansen and Tinguely2025). The CQ follows with a relatively uniform volumetric radiation from the plasma.

Informed by these empirical JET studies we choose to model two radiation structures; one for the pre-TQ, and a second for both the TQ and CQ. Note that while the same structure is used for the TQ and CQ, different toroidal peaking factors will be applied. The pre-TQ structure is a toroidal ring just within the last closed flux surface and with a circular spot size with a minor radius scaled from JET by the ratio of major radii which gives $0.3\,\textrm{m} \,(4.62 / 3.0)=0.46\,\textrm{m}$ . The toroidal span of the plume feature observed in JET is ${\sim}\,\pm 45^\circ$ or ${\sim}\,5$ m, and is conservatively assumed to remain the same in ARC (i.e. 5 m). If we use 12 injectors spaced equidistant toroidally and ignore the helicity of the plumes, the resulting radiation structure is approximately toroidally uniform. This pre-TQ structure is shown in figure 12(i). Note that half of the toroidal extent of the distribution is shown, but it is taken to be toroidally continuous when calculating the heat flux. The TQ and CQ structure is a simple volumetric plasma radiation source as shown in figure 12(ii), and similarly, half of the structure is pictured.

Figure 12.

(i) Pre-TQ radiation structure modelled in HEAT. (ii) TQ and CQ radiation structure modelled in HEAT. (iii) Radiation intensity resulting from ray tracing a 1 MW total radiated power source from the pre-TQ structure. (iv) Radiation intensity resulting from ray tracing a 1 MW total radiated power source from the TQ/CQ structure. (a–f) Synthetic plasma disruption. (a) Plasma current and internal inductance. (b) Thermal energy, magnetic energy and total radiated energy. (c) Total radiated power and outboard midplane HEAT couplings. (d) Radiation intensity and toroidal peaking factor. (e) Heat-flux factor (blue) and the tungsten melt limit (dashed red) and energy fluence. (f) Tungsten surface temperature (blue) compared with the melt limit (dashed red).

HEAT is then run to perform ray tracing from these two structures to the ARC first wall. The radiation intensity measured at the wall for a total radiated power in each structure of 1 MW is shown in figure 12(iii–iv). The pre-TQ structure irradiates the wall with a peak intensity of 10 kW m⁻² at the outboard midplane. The TQ and CQ structure produces a peak of 4 kW m⁻² and this peak also occurs at the outboard midplane. The radiated power in an ARC mitigated disruption is in the tens of GW and thus we can expect the radiation intensity to be in the hundreds of MW m⁻². The outboard midplane is the most loaded location on the first wall and the most likely location for flash melting to occur.

Parameterised synthetic disruptions are produced by forward modelling the thermal energy and plasma current decay. One parameterised disruption is shown in figure 12(a–d). The thermal energy starts decaying at a given rate representing the pre-TQ and then accelerates during the TQ phase (figure 12(b), solid blue). The plasma current remains constant during the pre-TQ and TQ and shortly thereafter exhibits a plasma current spike of 20 % coincident with a drop in the internal inductance $l_i$ (figure 12 a). The magnetic energy is assumed to reduce by 10 % during the current spike, and from both the imposed energy reduction and current spike increase, the internal inductance is calculated self-consistently, dropping from 1.0 to 0.55. The final internal inductance is not strongly affected by the 10 % choice; choosing 0 % results in a final value of 0.64. The dissipation of the thermal and magnetic energy is summed to produce the time evolution of the radiated energy (figure 12(b), dotted orange), and the time derivative of this trace provides the radiated power (figure 12(c), solid blue). The radiated power is first attributed to the pre-TQ structure and then transitions to the second structure during the TQ as shown by the time evolution of the outboard midplane HEAT couplings (figure 12(c), dotted blue). Taking the product of the radiated power, the HEAT couplings and the toroidal peaking factor gives the radiation intensity at the highest loaded toroidal position on the outboard midplane (figure 12 d).

With the time evolution of the peak radiation intensity now defined for a given disruption, a heat-flux factor (HFF) that is proportional to the temperature rise can be found by solving the integral in equation 4 of Hu et al. (Reference Hu, Artola, Nardon, Lehnen, Kong, Bonfiglio, Hoelzl and Huijsmans2024). The critical HFF for tungsten starting from room temperature is often quoted as 50 MJ m⁻² s $^{-0.5}$ . When starting at the elevated temperature of the ARC first wall the critical HFF reduces to ${\sim}\,32$ MJ m⁻² s $^{-0.5}$ . A Monte Carlo study is performed by randomly sampling the disruption parameters from Gaussian distributions modified to have hard limits on the left and on the right. The parameters and their distributions are shown in table 2. It is found that 55 % of the disruptions exceed ${\sim}\,32$ MJ m⁻² s $^{-0.5}$ at the outboard midplane and are therefore expected to melt. We conclude that avoiding flash melting in ARC mitigated disruptions will require some optimisation of the mitigation recipe which can be learned in SPARC; see §§ 6.2 and 6.3.

Table 2.

Parameters defining the synthetic disruptions used in the radiation flash analysis. Each is described by a truncated Gaussian with specified centre, standard deviation $\sigma$ and hard limits.

Increasing time scales and reducing peaking can prevent melting. The parameterised disruption shown in figure 12(a–f) provides an example of a mitigated disruption that remains below melt thresholds. The pre-TQ in this case is pessimistically 3.8 ms, consistent with § 4.1, while the TQ duration is somewhat optimistic at 2 ms, although supported here by DREAM modelling with a reduced Ne quantity and using the flux limited TQ heat-flux model in § 5.3.1. Empirical studies support that reducing the neon content does in fact lengthen the radiation flash time scales (Bodner et al. Reference Bodner2025). The full 1-D nonlinear heat equation is solved for this case and shows that the surface temperature reaches a max temperature of 3110 K and does not melt (figure 12 f). The toroidal peaking here is pessimistic at 1.5 in the TQ and 1.2 during the CQ. Lower TQ peaking factors of 1.2 have been observed with a single injector in JET (Piron et al. Reference Piron2024). We anticipate that 12 injectors in ARC will keep peaking low, and further, TQ peaking is observed to decrease with higher thermal energy (Lehnen et al. Reference Lehnen2011), which extrapolates favourably to ARC.

In the above analysis, it is assumed that the fast ion population is thermalised during the pre-TQ and TQ, as the fast ion energy is included in the total thermal energy. Simulations from ASCOT5, a Monte Carlo orbit-following code which has previously been used to predict alpha particle transport and losses in ITER (Snicker et al. Reference Snicker2012), SPARC (Scott et al. Reference Scott, Kramer, Tolman, Snicker, Varje, Särkimäki, Wright and Rodriguez-Fernandez2020) and others, are used to provide a first estimate of the fusion alpha particle slowing down time in the pre-TQ and TQ plasma conditions. In these simulations, 8192 α particles born at 3.5 MeV, randomly distributed within $\rho \in [0,0.4]$ , were followed for 0.1 ms. The magnetic equilibrium and TF were kept the same as the base ARC scenario, but an additional $1.4\times 10^{22}$ neon ions and $2.5\times 10^{23}$ deuterium ions, consistent with the predictions in § 4.1, were evenly distributed throughout the plasma volume. Three simulations were performed, with spatially uniform ion and electron temperatures of 10 eV, 100 eV and 1 keV, and corresponding neon charge states of 2, 7 and 8. The alpha particle slowing down times, calculated from the ASCOT5 results by averaging over the alphas, showed a decrease from 360 ms for the baseline scenario to 0.24, 0.19 and 0.43 ms for the three simulations. These time scales are shorter than the pre-TQ duration, and they are comparable to or shorter than the TQ duration. Given the core localisation of the fast ion population and the poor core fuelling efficiency of MGI during the pre-TQ, it might be necessary to complement the MGI system with a core-directed pellet injection to ensure sufficient time for thermalisation prior to the TQ when the magnetic field becomes stochastic. Thermalisation of the fast ion population during mitigated disruptions will be studied on SPARC.

It is likely that full melt prevention during mitigated disruptions is possible in ARC, an expected requirement for the achievement of the rapid restart strategy.

5.2.2. Unmitigated disruption thermal loads

Unmitigated thermal loads present a significant risk to rapid restart of ARC, and therefore these events need to be rare. Disruption prediction is out of the scope of this paper as it depends on the diagnostics and the control system, which are not described in this physics basis. The Discussion section offers some perspectives on the achievement of the prediction performance that is needed to ensure unmitigated disruptions are rare. The primary thermal loading events are the TQ and halo current heating. In this subsection we explore these unmitigated loads. Loads from non-thermal populations of electrons will be addressed in the following section.

Projecting the unmitigated TQ divertor loads to future devices is typically done by calculating a HFF (Hender et al. Reference Hender2007; Sweeney et al. Reference Sweeney2020) informed by the plasma equilibrium and empirical scalings (ITER et al. 1999; Loarte et al. Reference Loarte2005; Arnoux et al. Reference Arnoux, Loarte, Riccardo, Fundamenski and Huber2009). This formulation is empirical and lacks a strong physics basis as noted in Hender et al. (Reference Hender2007) where sheath limits are highlighted as important physics to consider. Together CFS and IPP Garching have undertaken the development of a cross-machine TQ heat-flux scaling that is revealing a correlation between the TQ divertor heat-flux duration and the divertor energy fluence in AUG and JET, consistent with a heat flux limit (Schramm et al. Reference Schramm, Eich, Faitsch, Maraschek, McDermott, Sweeney, Zohm, Team and Contributors2025). In this subsection we will follow the standard approach to derive a TQ HFF for ARC, not including flux limits, and then discuss how this prediction might change as a result of the ongoing cross-machine study.

ARC has a thermal energy of $W_{th}=231$ MJ and a fast ion energy of $W_{fast}\approx 30$ MJ which we will assume is thermalised during the TQ. While in a mitigated TQ ASCOT5 simulations support the possibility of thermalising the fast ions (see § 5.2.1), thermalisation of the fast ions in unmitigated thermal quenches is not well justified; we leave the study of non-thermalised fast ion losses during the TQ for future work. With a minor radius of $a=1.18$ m the IPB99 (ITER Physics Expert Group 1999) core collapse TQ scaling predicts a duration of $\tau _{core}\approx 200$ ${\unicode{x03BC}}$ s, and the same temporal broadening of 1.5–15 used in Hender et al. (Reference Hender2007) gives a heat-flux duration at the divertor of $\tau _{div}= 0.3-3$ ms. The steady-state divertor wetted area is estimated to be $A_{div,dis} = 4 \pi R_{div} \lambda _q f_x (S / \lambda_q) f_{dis} / \sin (\alpha ) = 4.4$ m $^2$ where $R_{div}\approx 5$ m is the radius of the outboard divertor, $\lambda _q\approx 1$ mm is the characteristic heat-flux width, $f_x\approx 6$ is the flux expansion, $(S / \lambda_q) \approx 2$ is the divertor broadening, $f_{dis}=7$ is the disruption broadening (Riccardo & Loarte Reference Riccardo and Loarte2005; Hender et al. Reference Hender2007), $\alpha =10$ deg is the poloidal angle of incidence and an additional factor of 2 appears to account for power sharing between the upper and lower outboard divertors. It is conservatively assumed here that the inboard divertor does not contribute any area, and the poloidal angle of incidence is assumed to be comparable to SPARC. The resulting TQ HFF is $U=1100{-}3400$ MJ m⁻² s $^{-0.5}$ , which is well over the 32 MJ m⁻² s $^{-0.5}$ tungsten melt limit when starting from 1000 C. Accounting for the usual energy degradation prior to the final collapse (Riccardo & Loarte Reference Riccardo and Loarte2005) and radiation losses would reduce these numbers, whereas some disruptions exhibit less disruption broadening (Riccardo & Loarte Reference Riccardo and Loarte2005; Schramm et al. Reference Schramm, Eich, Faitsch, Maraschek, McDermott, Sweeney, Zohm, Team and Contributors2025) which would increase these numbers.

Sheath physics are well studied in the plasma boundary community (Stangeby Reference Stangeby1986; Fundamenski Reference Fundamenski2005) and have been applied to plasma transients including edge-localised modes (ELMs) (Eich et al. Reference Eich, Kallenbach, Fundamenski, Herrmann and Naulin2009) and halo current heat loading (Kiramov et al. Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016; Artola et al. Reference Artola2021a). Previous divertor infrared studies suggest that the TQ heat fluxes are comparable to ELMs (Riccardo & contributors Reference Riccardo2003) and thus they might share similar transport physics. The sheath limited heat flux $q_{\parallel }$ is given by

(5.1) \begin{equation} q_{\parallel } = n_i c_s \gamma k_B T_e, \end{equation}

where $n_i$ and $T_i$ are the ion density and temperature, $c_s$ is the ion-acoustic wave velocity and $\gamma$ is the sheath transmission factor that is $\leqslant 7$ (Stangeby Reference Stangeby1986). For the purpose of comparing the heat flux between machines, we can take $T_i=T_e$ and use the Greenwald fraction $f_G\approx 1\propto n_i a^2 / I_p$ to swap $n_i$ for $I_p$ so that the heat-flux expression reduces to

(5.2) \begin{equation} q_{\parallel } \propto I_p T_i^{1.5}/a^2 . \end{equation}

Assuming sheath limits do apply to the TQ heat flux, the shortest value of $\tau _{div} = 300$ ${\unicode{x03BC}}$ s predicted above seems unlikely. The AUG and JET study finds a maximum divertor heat flux $q_{\perp }$ at the target of hundreds of MW/m $^2$ . The heat flux implied for ARC using the standard approach and with $\tau _{div} = 300$ ${\unicode{x03BC}}$ s gives $q_{\perp } = W_{th} / A_{div,dis} \tau _{div} = 27$ GW m $^{-2}$ which is nearly two orders of magnitude larger than what is observed in AUG and JET. Scaling from JET to ARC gives a factor of ${\sim}\,4$ in plasma current, ${\sim}\,3$ in ion temperature, and a 20 % larger minor radius for a parallel heat flux increase of ${\sim}\,15$ . Assuming the poloidal angle of incidence in AUG and ARC are similar during the TQ, this implies that $q_\perp$ is approximately one order of magnitude larger in ARC, and thus we expect order GW/m $^2$ which is consistent with $\tau _{div}\sim O$ (ms). This reduces the expected heat-flux factor to many hundreds or low thousands, but does not change the conclusion that melting is expected for unmitigated TQs. However, this does have an impact on the impurity requirements for thermal load mitigation. With a reduced divertor heat flux, the impurity density required to mitigate this heat flux is reduced which is beneficial for both radiation flash loads and for RE generation by increasing the CQ duration. Studying the TQ heat flux in SPARC will better inform expectations for ARC and will refine the mitigation system injection requirements, namely (4.1).

We now bound the worst case melt depths using a simple energy balance approach that ignores heat of vaporisation and all other loss channels like heat conduction into the tungsten bulk and radiation. We take the initial temperature of the tungsten to be $T_0 = 1273\,\text{K}$ . The average temperature dependent specific heat of tungsten between 1273 K and the melting temperature of 3695 K is $\langle c_{p,w} \rangle = 200$ J kg⁻¹ K. The heat of fusion is 50 kg mol⁻¹ (Tolias Reference Tolias2017) and with 0.18 kg mol⁻¹ we have $H=280$ kJ kg⁻¹. We form the following equation that assumes all the thermal energy goes into heating and melting tungsten

(5.3) \begin{equation} m_W = \frac {W_{th}}{\langle c_{p} \rangle (T_{melt} - T_0) + H}. \end{equation}

With the full $W_{th}=261$ MJ (assuming thermalisation of fast ions) we find $m_W=340$ kg. Using the density of tungsten at $T_{melt}$ of $\rho =18\,000$ kg m⁻³ and a wetted area of $A_{div,dis}=4.4$ m $^2$ we find a maximum melt depth of 4.3 mm. Including heat of vaporisation, radiation losses and heat conduction would reduce this melt depth, while peaking would increase it. We conclude that a single worst-case unmitigated TQ is highly unlikely to breach FLiBe cooling channels, and melt layers of hundreds of microns to millimetres might be expected. Rapid restart is almost certainly precluded by such an unmitigated TQ due to mobilised tungsten that may interfere with the breakdown of the next plasma, emphasising the importance of mitigating the vast majority of disruptions in ARC.

Following the TQ and during the CQ, the magnetic energy is dissipated into the plasma, into a region of open field lines surrounding the plasma that carries halo currents, and into surrounding conductors that share mutual inductance with the plasma. The ohmic heating in the plasma is expected to be dissipated by impurity radiation and by perpendicular heat transport across the last closed flux surface. The energy mutually coupled to conductors is responsible for imparting high electromagnetic forces, but typically the associated resistive heating is negligible. Halo currents on the other hand are thought to have been responsible for macroscopic melting of the beryllium first wall in JET (Matthews et al. Reference Matthews2016). Guided by the JET experience we will assess the halo current heating during the CQ in ARC.

Halo current heat loads have been investigated for ITER using high fidelity 3-D MHD workflows (Artola et al. Reference Artola, Pitts, Carvalho, Simic, Loarte, Gerasimov, Redl and Kong2025) and using a 2-D disruption simulator coupled with field-line tracers and melt dynamics codes (Coburn et al. Reference Coburn2021). Recently, a model similar to Kiramov et al. (Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016) was used to do time-dependent halo current heat load modelling for EU-DEMO (Pautasso & Fable Reference Pautasso and Fable2025). Here, we perform a simple time-independent scoping that provides the heat flux over a wide range in the dominant free parameters to get a sense for the melt risk should one of these events go unmitigated. The model used here is based on the physics in Kiramov et al. (Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016) and uses the ARC hot VDE design disruption. Power balance in a halo region flux tube in the absence of radiation is given by (Kiramov et al. Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016)

(5.4) \begin{equation} q_{\perp , halo}(T_e,Z) A_{wet} = \eta (T_e,Z) j_{h\parallel }^2 V_h, \end{equation}

where $q_{\perp , halo}$ is the perpendicular heat-flux incident on the wall, $A_{wet} = A_{in} + A_{out}$ is the total footprint area, $V_h$ is the volume of the halo region in the scrape off layer (SOL), $\eta$ is the Spitzer resistivity and $j_{h\parallel }$ is the halo current density following field lines in the SOL. Using the IDDB we choose an upper bound for the total poloidal halo current $I_{h\theta }=0.5 I_p = 6$ MA. From Coburn et al. (Reference Coburn2021) we estimate a minimum halo current width of $w_h = 0.2$ m, and we can now solve for the maximum poloidal halo current density $j_{h\theta ,max} = 0.5 I_p / (2 \pi R_c w_h) = 1.1$ MA/m $^2$ . Halo currents follow field lines in the SOL with a pitch angle $\alpha$ that is related to the equilibrium fields $\sin (\alpha ) = B_\theta (a)/ \sqrt {B_\theta (a)^2 + B_T^2}$ . Enforcing charge conservation at the wall and assuming cylindrical geometry with dimensions of $R_c$ and $a_c$ gives $j_{h\parallel } = j_{h\theta } /\sin (\alpha ) = j_{h\theta } \sqrt {1 + (R_c q_{\textit{edge}}/a_c)^2}$ and a maximum of $j_{h\parallel , max} = 6.3$ MA m $^{-2}$ (parameters used here are $R_c=4.3$ m, $a_c=0.77$ m, $q_{\textit{edge}}=1$ ).

The electron temperature and the charge state are free parameters in (5.4). We can constrain the electron temperature by requiring that the heat flux given by (5.4) is consistent with the sheath limited heat flux (Kiramov et al. Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016)

(5.5) \begin{equation} q_{\parallel ,sheath}(n_i, T_e, m_i) = \gamma n_i c_s(T_e,m_i) k_B T_e . \end{equation}

To map from the parallel heat flux to that incident on the wall we must account for the field pitch which gives a factor of $\sin (\alpha )$ . We will model a representative halo current flux tube with an average incident angle on the first wall in the RZ plane that reduces the perpendicular heat flux by $2 A_{\perp } / (A_{wet})$ (recall that $A_{wet}$ is the sum of the inboard and outboard wetted areas, and hence the factor of 2)

(5.6) \begin{equation} q_{\perp ,sheath}(n_i, T_e,m_i) A_{wet} = \gamma n_i c_s(T_e,m_i) k_B T_e \sin (\alpha ) 2 A_{\perp }, \end{equation}

where $q_{\perp , sheath}$ is a monotonically increasing function of $T_e$ and $q_{\perp ,halo}$ is a monotonically decreasing function of $T_e$ . Their intersection gives the stable $T_e$ that we expect the halo region to quickly attain and maintain, owing to the relatively low stored thermal energy in the halo flux tube. Equation (5.4) ignores radiation and therefore we choose $Z=1$ and $m_i=2.5$ amu for consistency. Similar to the TQ sheath limited heat flux presented earlier in this section and consistent with (Stangeby Reference Stangeby1986), we take $\gamma = 7$ . In combining (5.4) and (5.6), together with the geometry shown in figure 13(a), the ion density $n_i$ is a free parameter, and we will be interested in considering not only the maximum halo current density but also a range. Figure 13(b–c) shows the electron temperature and the halo current heat flux across a wide range of $n_i$ and $j_{h\parallel }$ . In a given disruption we expect the plasma to move in this 2-D space, starting at $j_{h\parallel }=0$ and rising to some value before falling again, and the ion density also likely evolves.

This space can be further constrained by invoking the ion saturation current limit that was proposed in C-Mod (Tinguely et al. Reference Tinguely, Granetz, Berg, Kuang, Brunner and LaBombard2017), measured in COMPASS experiments (Adamek et al. Reference Adamek2022), and imposed in JOREK simulations (Artola et al. Reference Artola2021a ). Taking the ion saturation current to be $J_{sat}^+ = Ze n_i c_s$ with $Z=1$ , contours of $J_{sat}^+$ , and twice its value, to provide some uncertainty margin, are plotted in red in figure 13(b–c). The grey regions are expected to be inaccessible due to the ion saturation current limit.

The black contours show the time scale in milliseconds to melt given by $(HFF_{crit}/q_{\perp ,halo})^2$ , where $HFF_{crit}\approx 32$ MJ m $^{-2}$ s $^{-0.5}$ for an initial tungsten temperature of 1000 C. The plasma would need to remain at or above a given black contour for the duration indicated for melt to occur. Note that these contours correspond to the most ideal situation where the field-line incident angle does not increase anywhere within the wetted area. Refinement of the first-wall design is likely to introduce features that lead to heat-flux concentration; unmitigated halo current heating will be a risk to consider in the design process.

From figure 13 it is clear that the halo current melt risk depends on the maximum $j_{\parallel h}$ and the maximum $n_i$ expected in a disruption. In Artola et al. (Reference Artola, Schwarz, Gerasimov, Loarte and Hoelzl2024) a maximum $j_{\parallel h}$ is proposed based on the assumption of a cylindrical plasma with a flat $q=1$ profile throughout the plasma, including the halo region. This prediction gives $j_{\parallel h} = 4.2$ MA m⁻² and $n_i = 5.5\times 10^{20}$ m $^{-3}$ , and is plotted as the blue square in figure 13. At this predicted density and maximum current, and assuming perfect uniformity of the first wall, the melt risk of the halo current heating in isolation is low. However, the TQ heat flux is not included here and is expected to exceed melt limits on its own, which would then be followed in time by the halo current heat flux. Further, if higher densities can be attained, the blue point moves horizontally, off of the ion saturation curve and allows accessing higher heat flux. The highest density plotted in figure 13 corresponds to putting all of the plasma ions into half the halo volume considered above (i.e. $w_h=0.1$ m and the density is calculated by $\langle n_e \rangle V_p / V_h$ ). The maximum current density plotted comes from $I_{h\theta }=6$ MA, $w_h=0.1$ m, and the same $\sin (\alpha )$ as used throughout this subsection.

Figure 13.

(a) Halo currents flowing in the open field-line region at the start of the CQ in the hot VDE design disruption with $w_h=0.2$ m. Red limiter surfaces mark the wetted areas which are $A_{out}=22.6$ m $^2$ and $A_{in}=21.7$ m $^2$ . The grey shaded region has a cross-sectional area of $A_{halo} = 0.69$ m $^2$ and a volume of $V_{halo} = 19.1$ m $^3$ . (b) Electron temperature of the halo region as a function of the halo current density $j_{\parallel h}$ and the ion density $n_i$ . (c) Halo current heat-flux incident on the wall plotted as a function of the same variables as (b). Black contour lines in (b–c) show the time scale in milliseconds where tungsten melt is expected if the plasma remains at or above a given contour, assuming a starting temperature of 1000 C. Red contours in (b–c) show where the ion saturation current $J_{sat}$ and twice the value are reached. The blue square marks the prediction using the scaling in Artola et al. Reference Artola, Schwarz, Gerasimov, Loarte and Hoelzl(2024).

Using (5.3) and swapping $W_{th}$ for 50 % of the total magnetic energy (354 MJ) and using the wetted area of $A_{out} + A_{in} = 44.3$ m $^2$ , we find a melt depth of up to 0.9 mm. Like the TQ melt depth, this estimate is pessimistic in that it assumes perfect energy conversion to melting, and is optimistic in that it does not account for asymmetries resulting from the plasma and from local features on the first wall that might increase the incidence angle.

Halo current heating events are expected to be more rare than unmitigated thermal quenches owing to the long time scale of the hot VDE that provides ample time to mitigate. In the temperature range shown in figure 13, neon is an efficient radiator and therefore it is expected that MGI will be effective at mitigating these heat fluxes. SPARC is expected to experience higher halo current heat fluxes than ARC (see § 6.2), and therefore SPARC operation will provide critical data on the mitigated and unmitigated loads.

5.3.1. Mitigated runaway electron loads

Should runaway generation and impact models validated on SPARC suggest that these events will happen in ARC and result in melting, mitigation will likely be required to achieve rapid restart. Full prevention of REs might be possible using the REMC approach, and in this case there are negligible RE loads to consider. The REMC is passive, driven by the disruption-generated loop voltage, and therefore may prove to be a robust runaway prevention system. Scoping for an ARC REMC system is presented in § 4.2. Present alternatives to the REMC either have lower probability of success or are unexplored. Alternatives are briefly discussed at the end of this subsection.

It is unlikely that MMI can sufficiently prevent REs, but it is likely that the maximum beam amplitude can be reduced. We now discuss modelling of runaway generation following the MMI primary injection.

The MMI that is delivered prior to the TQ, referred to as the primary injection, will have a strong effect on the resulting runaway beam current. DREAM (Hoppe, Embreus & Fülöp Reference Hoppe, Embreus and Fülöp2021) modelling is used to characterise the runaway production as a function of the quantity of injected neon and deuterium while simultaneously attempting to satisfy TQ and CQ mitigation targets. The same plasma model was used for a similar study for SPARC, described by Ekmark et al. (Reference Ekmark, Hoppe, Tinguely, Sweeney, Fülöp and Pusztai2025). This study should be seen as bounding what might be possible with some MMI system, and not necessarily specific to the baseline ARC disruption mitigation system (DMS) discussed in § 4.1. Assumptions that are inconsistent with the baseline ARC DMS will be noted.

We employ the pitch-angle-averaged kinetic model for superthermal electrons in DREAM, to accurately capture the seed generation mechanisms (Ekmark et al. Reference Ekmark, Hoppe, Fülöp, Jansson, Antonsson, Vallhagen and Pusztai2024). To explore the dynamics of different MMI compositions and find the qualitative correlations of interest for mitigation, we use multi-objective Bayesian optimisation of the densities of the injected material. The maximum runaway current (targeting $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ ), the transported fraction of the heat loss ( $\eta _{\textrm {tr}}\lt\,{10}$ %), and the CQ time ( ${11.3}\,\textrm {ms}\lt\,\tau _{\textrm {CQ}}\lt\,{142}\,\textrm {ms}$ ) are accounted for in the cost function, as described by Ekmark et al. (Reference Ekmark, Hoppe, Fülöp, Jansson, Antonsson, Vallhagen and Pusztai2024). The runaway current target of 150 kA should be interpreted as an order of magnitude estimate, motivated by limits set for the ITER tungsten divertor (Lehnen Reference Lehnen2021).

The disruption simulation is started by a uniform and instantaneous deposition of D and Ne across the plasma volume. Note this is an optimistic assumption, particularly for a MGI system; shattered pellet injection or electromagnetic particle injection might get closer to this uniform distribution assumption. The initial plasma temperature, density and current density profiles correspond to the nominal ARC profiles (Hillesheim et al. Reference Hillesheim2026). To emulate the effect of magnetic field stochastisation during the TQ, a Rechester–Rosenbluth-type diffusive transport (Rechester & Rosenbluth Reference Rechester and Rosenbluth1978) of heat and superthermal/REs is commenced simultaneously with the material deposition, employing spatially and temporally constant magnetic perturbations. The transport is active until the end of the TQ, defined here by the mean temperature falling below ${20}\,\textrm {eV}$ . We use a relative magnetic perturbation amplitude of $\delta B/B = 0.3 $ % in the diffusion coefficients $D\propto |v_\|| R_0 (\delta B/B)^2$ , yielding TQ durations of ${\sim}\,0.1$ – ${2}\,\textrm {ms}$ , depending on the injected quantities. This is consistent with the ITPA scaling (ITER Physics Expert Group 1999), but this range might shift toward longer durations when accounting for heat flux limits, and most importantly might preclude the sub-millisecond cases – see § 5.2.2.

The parameter regions where mitigation is successful with respect to the various targeted figures of merit are colour shaded in figure 14. Apart from the uninteresting grey-shaded lower left corner, characterised by an incomplete TQ and CQ, there is only a small region close to the lower right corner of the parameter space where the $I_{\textrm {re}}$ target is met (red contour and shading). Here, the electron density is sufficiently high (due to the high $n_{\textrm {D}}$ ) to elevate the effective critical electric field, while the fraction of total-to-free electron density is sufficiently low (due to the low $n_{\textrm {Ne}}$ ) to avoid strong avalanching. However, note that this requires a density rise by a factor of ${\sim}\,150$ above the pre-disruption density, which is unprecedented in existing experiments with any MMI technology, where electron densities typically rise by up to a factor of ${\sim}\,7$ (Granetz et al. Reference Granetz, Whyte, Izzo, Biewer, Reinke, Terry, Bader, Bakhtiari, Jernigan and Wurden2006; Sheikh et al. Reference Sheikh2020), and in some cases up to ${\sim}\,50$ , but not uniformly, and peaking at an absolute density of $2\times 10^{21}$ m $^{-3}$ (Sweeney et al. Reference Sweeney2021).

For the full region corresponding to $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ , the $\tau _{\textrm {CQ}}$ target is met (yellow). However meeting the $\eta _{\textrm {tr}}$ target requires both $n_{\textrm {D}}$ and $n_{\textrm {Ne}}$ to be high, and the corresponding region is around the upper right corner, such that all targets cannot be met simultaneously. Indeed, the optimal sample located at $n_{\textrm {D}}={5.6\times 10^{22}}{{\textrm {m}}^{-3}}$ and $n_{\textrm {Ne}}={3.7\times 10^{17}}{\textrm {m}^{-3}}$ has an $I_{\textrm {re}}={950}\,\textrm {kA}$ and $\tau _{\textrm { CQ}}={25}\,\textrm {ms}$ , but $\eta _{\textrm {tr}}={26}$ %.

Table 3.

Different optimal samples, depending on weighting. The first row corresponds the optimal sample under the constraint of $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ . The second row corresponds to the optimal sample with all figures of merit weighted equally. The third row corresponds to the optimal sample under the constraint of $\eta _{\textrm {tr}}\lt\,{10}{\,\%}$ . These samples are marked by a diamond, a star and a square, respectively, in figures 14 and 15. The columns present the D density, Ne density, runaway current including ( $I_{\textrm {re}}^{\textrm {w}}$ ) and excluding ( $I_{\textrm { re}}^{\textrm {w/o}}$ ) the delayed Compton seed, the transported fraction of the heat loss, the CQ time and the TQ time.

Figure 14.

Regions of successful mitigation (colour shaded areas) with regards to the targets of $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ (red), $\eta _{\textrm {tr}}\lt\,{10}$ % (blue) and ${11.3}\,\textrm {ms}\lt\,\tau _{\textrm {CQ}}\lt\,{142}\,\textrm {ms}$ (yellow). The red dashed contour indicates where $I_{\textrm {re}}={1}\,\textrm {MA}$ , and there is a small shaded red region of $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ below this curve. This region is more visible in the left window, which is a zoom-in of the parameter area in the lower right corner. The optimal sample with all figures of merit equally weighted is indicated by a star, the optimal sample while constraining $I_{\textrm {re}}\leqslant {150}\,\textrm {kA}$ by a diamond, and the optimal sample while constraining $\eta _{\textrm {tr}}\leqslant {10}$ % by a square. The grey area covers the region of incomplete TQ and CQ.

Figure 15.

(a, b) The runaway current, and (c) the ratio of the runaway currents obtained with and without accounting for delayed Compton scattering, as functions of the injected densities. (b) Is a zoom-in of the parameter area in the vicinity of the optimal sample, bounded by black solid lines in panel (a). The optimal sample with all figures of merit equally weighted is indicated by a star, the optimal sample while constraining $I_{\textrm {re}}\lt\,{150}\,\textrm {kA}$ by a diamond, and the optimal sample while constraining $\eta _{\textrm {tr}}\lt\,{10}$ % by a square. The grey area covers the region of incomplete TQ and CQ.

Figure 16.

The prompt (solid blue) and delayed (dashed black) photon flux spectra used for the RE generation from Compton scattering, plotted together with the data used to make the fits – the blue circles correspond to prompt gammas, and the black triangles to delayed gammas after ${0.5}\,\textrm{ms}$ .

Even though the $I_{\textrm {re}}$ target is only met in a small region of the parameter space, material injection can limit the RE current over a wide parameter region, as seen in figure 15(a), showing $I_{\textrm {re}}$ as function of the injected densities. Depending on the relative weighting of the $I_{\textrm {re}}$ and $\eta _{\textrm {tr}}$ figures of merit, one may find different optimal solutions. Of the sample simulations used in the optimisation, the lowest $I_{\textrm {re}}$ is ${2.2}\,\textrm {MA}$ when $\eta _{\textrm {tr}}\leqslant {10}$ %, while the lowest $\eta _{\textrm {tr}}$ is ${55}$ % for $I_{\textrm { re}}\leqslant {150}\,\textrm {kA}$ . Notably, all of the different optimal solutions occur at similar D densities, as clarified by table 3.

The prompt photon flux spectrum used during the TQ and the delayed photon flux spectrum used during the CQ in these disruption simulations are plotted in figure 16, proportionally scaling the Compton seed source. In a tokamak radiation environment, the gamma-ray field comprises two distinct components: ‘prompt’ gammas, which are emitted instantaneously from nuclear interactions (such as neutron capture) as the neutron flux impinges on machine components, and ‘delayed’ gammas, which originate from the subsequent radioactive decay of activated nuclides within these materials. The prompt spectrum is applicable to the pre-TQ and TQ when the plasma is still in fusion conditions, whereas the delayed spectrum is applicable to the CQ. Consequently, the Compton seed spectrum was derived for both the prompt $(n, \gamma )$ photon flux and the delayed activation gamma spectrum. To account for their distinct physical production mechanisms, separate computational workflows were employed. The prompt contribution was calculated directly via Monte-Carlo N-Particle (MCNP) (Rising et al. Reference Rising, Armstrong and Bolding2023), while the delayed contribution utilised a rigorous 2-step (R2S) method (MCNP $\to$ FISPACT-II $\to$ MCNP). In the R2S workflow, neutron spectra were obtained for each tokamak component to generate activation sources via FISPACT-II (Sublet et al. Reference Sublet, Eastwood, Morgan, Gilbert, Fleming and Arter2017). These sources were then transported in MCNP to tally the ultimate gamma spectrum in the VV at 0.5 ms post-shutdown, capturing the decay of short-lived isotopes.

Figure 17.

Time evolutions of the total (solid) and the runaway (dashed) currents for (a) a high $n_{\textrm {D}}$ and low $n_{\textrm {Ne}}$ scenario, (b) a high $n_{\textrm {D}}$ and high $n_{\textrm {Ne}}$ scenario and (c) a low $n_{\textrm { D}}$ and high $n_{\textrm {Ne}}$ scenario.

The importance of the delayed Compton seed is quantified by the ratio of $I_{\textrm {re}}$ with and without the delayed Compton source, as shown in figure 15(c). For most of the parameter space the delayed Compton seed is inconsequential. In the region where we see significant contributions from delayed Compton scattering (e.g. $\gt{25}{\,\%}$ ), we also observe relatively low runaway currents $\lt{2}\,\textrm{MA}$ . More specifically, the importance of the delayed Compton seed increases with decreasing runaway current, as exemplified by the three samples considered in table 3. The absolute effect on the runaway current when accounting for delayed Compton scattering in ARC is $\lesssim {600}\,\textrm {kA}$ , which is high compared with the runaway current target of 150kA, but small compared with the several MA predicted in the majority of the injected density space considered. The delayed Compton scattering source is the only RE seeding mechanism that cannot be observed in SPARC; these results suggest the risk of this science gap is low since it is not the dominant seed mechanism.

The current evolution of the optimal sample with $\eta _{\textrm {tr}}\leqslant {10}$ % (marked by a square in figures 14 and 15) is shown in figure 17(a), and it indicates that at high D and low Ne densities the RE beam is rather short lived, owing to the electric field becoming sub-critical quickly. When increasing the Ne density, the qualitative behaviour is retained, but the peak runaway current is significantly increased, as illustrated by figure 17(b). This can be explained by the enhanced avalanche caused by the increased ratio of total to free electron densities, both due to the higher Ne density and due to faster rate of D recombination. In contrast, a long-lived RE plateau is developed when decreasing the D density, as shown in figure 17(c). The plateau runaway current is of several MA, but is smaller than the peak runaway current with a higher D density due to reduced D recombination.

The case shown in figure 17(c) is closest to the ARC baseline injection discussed in § 4.1, which gives densities of $n_{\textrm{Ne}}=7.4\times 10^{19}$ m $^{-3}$ and $n_{\textrm{D}}=1.4\times 10^{21}$ m $^{-3}$ , assuming 10 % assimilation. This modelling predicts that the ARC baseline injection results in runaway beams of many MA. The modelling suggests that it is unlikely that any MMI system can prevent REs in ARC disruptions. Further, the results suggest that thermal mitigation with the baseline injection is below the 90 % target, however, we now discuss inconsistencies in the thermal transport model relative to the empirical findings reported in § 5.2.2.

Our baseline approach to evaluate transported electron heat losses in DREAM is based on parallel heat fluxes along perturbed field lines scaling as $q_\|\sim n_{\textrm {e}} v_{\textrm {te}} T_{\textrm {e}}$ , with the electron thermal speed $v_{te}$ . However, growing experimental evidence on current experiments suggests that the electron heat flux along the perturbed field lines in the edge/halo region may be flux limited, where the parallel heat flux scales with the sound speed instead of the electron thermal speed (see § 5.2.2). To emulate this, we use the electron heat transport boundary condition $\partial W/\partial t|_{r=a}=-2q_\parallel /3L_\parallel$ , with $q_\parallel =16n_{\textrm {e}}c_{\textrm {s}}T_{\textrm {e}}$ being the parallel heat flux and $L_\parallel =2\pi qR_0$ the length of the flux tube (Kiramov et al. Reference Kiramov, Lehnen, Khayrutdinov and Lukash2016; Pautasso & Fable Reference Pautasso and Fable2025), instead of using $W(r=a)=0$ as was done above. Using this boundary condition on the electron heat transport, we find that the transported fraction of the heat load is reduced significantly for all the samples considered in table 3. For the optimal sample found (second row of table 3), the transported fraction of the heat loss is reduced to $\lt\,{10}$ %. The reduced transport of heat allows for using lower Ne densities during the MMI, and this will both prolong the TQ and reduce the runaway current. As for the effect of the boundary condition on the TQ time, for the high Ne density case of table 3, the TQ time remains largely unchanged compared with when not accounting for this heat flux limit, since radiative losses dominate in this case. On the other hand, for the low Ne density case, the TQ is prolonged to ${2.0}\,\textrm {ms}$ .

From the DREAM results it appears that MMI is unlikely to prevent REs in ARC. If needed, the REMC is explored as an option for ARC in § 4.2. If the REMC does not fully prevent beams but rather only reduces their total current, it may be necessary to complement this technique with another such as benign termination (Paz-Soldan et al. Reference Paz-Soldan, Eidietis, Liu, Shiraki, Boozer, Hollmann, Kim and Lvovskiy2019b ; Reux et al. Reference Reux, Paz-Soldan and Aleynikov2021), or to seek alternative techniques. For benign termination to succeed, it is likely that the primary injection quantity needs to be reduced significantly from what is proposed here, and such a change would need to be demonstrated consistent with thermal and electromagnetic load mitigation. Sacrificial limiters have been explored for DEMO (Barrett et al. Reference Barrett, Chuilon, Kovari, Hernandez, Richiusa, Adame, Tivey, Vizvary, Xue and Maviglia2019; Vannini et al. Reference Vannini2025) and could allow rapid restart of ARC if the melt of such a limiter could be contained or directed away from plasma interaction surfaces, or a material chosen whose melt/ablation does not impact subsequent plasmas. Continuous injection of pellets to deplete runaway seeds via the stopping power has been studied numerically (Nardon et al. Reference Nardon, Matsuyama, Hu and Wieschollek2021), and a similar tungsten granule injection has been proposed for suppression of already formed beams (Lively et al. Reference Lively, Perez, Uberuaga, Zhang and Tang2024).

Early stage work has also demonstrated the ability of plasma waves to assist in RE mitigation. Recent experiments at DIII-D show that helicon waves can directly interact with the RE population via a wave–particle resonance to lower peak RE kinetic energies (Choudhury et al. Reference Choudhury2025). Alfvénic instabilities observed during current quenches in DIII-D correlate with failed RE plateau formation (Lvovskiy et al. Reference Lvovskiy, Heidbrink, Paz-Soldan, Spong, Molin, Eidietis, Nocente, Shiraki and Thome2019), and on AUG without a strong correlation (Heinrich et al. Reference Heinrich, Papp, Lauber, Pautasso, Dunne, Maraschek, Igochine and Linder2024). It is not known if this activity will be more pronounced on SPARC as compared with current tokamaks. If Alfvénic activity is driven by the remnant of the fast ion population (via fusion alphas (Lier et al. Reference Lier, Papp, Lauber, Embreus, Wilkie and Braun2021)), it may be more commonly observed, although increased density will also increase collisional damping of these instabilities. If the instability is driven by REs, then the magnitude of the instability may depend on the detailed generation profiles in space and time of the REs. Were these instabilities to occur in SPARC and ARC, they could also potentially mitigate RE formation. While a predictive understanding of Alfvénic phenomena is still not available, SPARC will provide a relevant platform to observe RE dynamics. SPARC will be used to identify both the inherent runaway risk and as needed, the runaway solution for ARC; see §§ 6.2 and 6.3 for more discussion.

5.3.2. Unmitigated runaway electron loads

Here, we scope the melt limits for the space of possible beam currents, energies and wetted areas, and show results from MEMENTO simulations of a moderate RE beam striking tungsten. The MEMENTO simulations were performed for a SPARC plasma facing component (PFC) geometry, but the resulting melt depth and erosion by vaporisation provides expectations for ARC.

The average electron kinetic energy, the total beam energy, and the wetted area are the dominant factors in determining the severity of RE impacts. A runaway beam stores both kinetic and magnetic energy. Assuming that all REs are moving at approximately the speed of light, the total kinetic energy is

(5.7) \begin{equation} W_k = \frac {2 \pi R_0 I_{RE} E_{av}}{ec}, \end{equation}

where $R_0$ is the major radius of the plasma, and $I_{RE}$ and $E_{av}$ are the RE current and the average electron energy. The magnetic energy of the beam depends on the runaway current and inductance, and can be calculated as

(5.8) \begin{equation} W_m = \frac {1}{2} L_{RE}I_{RE}^2, \end{equation}

where $L_{RE}$ is the inductance of the RE beam and is estimated as the inductance of a torus. Some reduction of the minor radius $a$ is expected due to radial force balance during the CQ; 70 % of the pre-disruption $a$ is used here. A smaller $a$ would increase the inductance, while using the pre-disruption value would reduce the magnetic energy by ${\sim}\,20$ %. Due to its dependence on $I_{RE}^2$ , magnetic energy dominates the kinetic energy of the beam above currents of ${\sim}\,1$ MA when the electron average energy is below 20 MeV, as expected.

In experiments it has been shown that magnetic energy can be converted into heating during RE strikes (Igitkhanov Reference Igitkhanov2014). Magnetic energy is deposited during the RE beam termination by avalanching more electrons into the runaway space which then impact the same area. The energy conversion efficiency of this process is estimated to be between 20 % and 60 % (Loarte et al. Reference Loarte, Riccardo, Martin-Solís, Paley, Huber and Lehnen2011). Plasma to RE current conversion factors above 50 % would lead to deposited energies in excess of 100 MJ.

The wetted area is an important factor in determining the severity of a strike. In terms of surface loading, for a beam of a given energy, a larger footprint would imply milder thermal response. However, REs deposit energy volumetrically, with the profile sensitive not only to RE energy but also to the impact angle, which is generally defined by the magnetic field inclination and the RE pitch angle. Under the assumption of an exponential in-depth profile based on the continuous slowing down approximation, an analytical solution exists that yields a threshold for tungsten melting from an initial temperature of 1000 C equal to 2–5 MJ m⁻² (Lehnen et al. Reference Lehnen, Abdullaev and Arnoux2009). The physics basis to extrapolate RE wetted areas from existing machines to ARC is lacking, likely owing to the final loss MHD which is difficult to predict a priori (Hollmann et al. Reference Hollmann2025; Ratynskaia et al. Reference Ratynskaia2025a ). Wetted areas vary by orders of magnitude across devices, and even across different strike events in the same device. Results from JET show RE beam wetted areas between tens of cm $^2$ (Matthews et al. Reference Matthews2016) and thousands of cm $^2$ (Lehnen et al. Reference Lehnen, Abdullaev and Arnoux2009; Reux et al. Reference Reux, Plyusnin and Alper2015b ). The strike wetted area is left as a free parameter in the analysis that follows.

Figure 18.

The deposited beam energy of a RE strike as a function of average RE energy and beam current. 50 % of the total energy (including both kinetic and magnetic) is assumed to be deposited. Contours represent the beams which exceed an energy density threshold of 3 MJ/m $^2$ for various wetted areas. Points on or to the right of a given contour would cause melting. The third DREAM optimum from table 3 ( $n_{\mathrm{D}} = 5.8 \times 10^{22}\,\mathrm{m}^{-3}$ and $n_{\mathrm{Ne}} = 2.1 \times 10^{18}\,\mathrm{m}^{-3}$ ) is shown by the black square.

Using (5.7) and (5.8) and assuming that 50 % of the RE beam’s total energy is deposited into the wall, deposited energies are plotted in figure 18 as a function of RE current and average energy. RE currents between 0 and 12 MA (i.e. the full plasma current) and average energies between 0 and 20 MeV are shown. Assuming that exceeding an energy density threshold of 2 MJ $\mathrm{m}^{-2}$ will cause melting, contours corresponding to various wetted areas where that threshold is met are plotted on the graph. Any point to the right of a contour would have enough energy to melt that wetted area or more. It is found that a 5 MA RE beam depositing 50 % of its energy would require a wetted area $\gt 10$ m $^2$ to prevent melting.

A point representing one of the optimums from the above DREAM simulations is now calculated and plotted. The average RE energy is estimated from a 0-D calculation of the total energy converted into the RE beam kinetic energy $E_k \approx 2\pi R_0 \int E_\parallel I_{RE} \, \text{d}t$ , where $E_\parallel (t)$ is the parallel electric field (Martín-Solís et al. Reference Martín-Solís, Loarte and Lehnen2015; Datta et al. Reference Datta, Clauser, Ferraro, Liu, Sweeney and Tinguely2025). For the $\eta _{tr} \lt\,10\,\%$ optimal case from the DREAM simulations reported in the previous subsection, we expect an average energy of roughly $E_{av} \sim E_k/N_{RE} \approx {18}\,\textrm {MeV}$ . This is comparable to typical JET RE beams which exhibit average energies of around 10 MeV (Loarte et al. Reference Loarte, Riccardo, Martin-Solís, Paley, Huber and Lehnen2011). This optimum point is plotted in figure 18.

The recently developed GEANT4-MEMENTO workflow (Ratynskaia et al. Reference Ratynskaia, Paschalidis and Rizzi2025b ) has been employed for modelling the thermal response and damage of SPARC PFCs accounting for the volumetric deposition and the geometry defined by the component and the incident field angle. While the simulations are run for a SPARC geometry, the results provide insight into ARC expectations. The simulations have been performed for a SPARC tungsten heavy alloy PFC (note that pure tungsten is planned for ARC). The initial temperature is taken to be 20 C, meaning these results are somewhat optimistic compared with ARC with initial temperatures near 1000 C.

Figure 19.

Temperature increase vs depth into a tungsten SPARC PFC tile for various impact angles with respect to the surface normal calculated using the GEANT4-MEMENTO workflow (Ratynskaia et al. Reference Ratynskaia, Paschalidis and Rizzi2025 b). The RE kinetic energy is 10 MeV and pitch angle is 0.2 rad. The total energy deposited is 100 kJ on ${\sim}$ 50 cm $^2$ over (a) 1 ms and (b) 10 ms. Note that temperatures above tungsten’s vaporisation temperature ( ${\sim }6200$ K) are more uncertain due to extrapolations of W thermophysical properties to high temperatures.

The modelled SPARC PFC is curved, hence the local magnetic field inclination angle varies over the surface and so do the RE impact angles. Averaging the inclination angle over four representative sections yields values of 86°, 78°, 71° and 65° with respect to the surface normal – see Ratynskaia et al. (Reference Ratynskaia2025a ) figure 32 for details. A total energy of 100 kJ is deposited on an area of ${\sim}$ 50 cm $^2$ . This is comparable to a 1–2 MA RE beam in ARC carrying 2.5–10 MJ of magnetic energy and a wetted area of ${\sim}$ 0.05–0.25 m $^2$ (the 50 % energy deposition assumption from above is again used here). From figure 18, melting is expected, consistent with the higher fidelity results below. A single RE kinetic energy of 10 MeV and pitch angle (with respect to the local magnetic field) of 0.2 rad and an energy deposition time of 1–10 ms are assumed. As seen below, the impacts are mostly insensitive to the order of magnitude change in the loading duration.

The corresponding in-depth temperature profiles for four panel sectors with representative values of the magnetic field inclination angles are shown in figure 19. As discussed in Ratynskaia et al. (Reference Ratynskaia2025a ,Reference Ratynskaia, Paschalidis and Rizzi b ), vaporisation cooling is responsible for the non-monotonic temperature even in cases with monotonic in-depth energy deposition profiles. Non-monotonic temperature profiles can result in a thermomechanical fracture failure mode, as has been empirically observed (Angeli et al. Reference Angeli2022; Ratynskaia et al. (Reference Ratynskaia2025c ). Given the moderately high surface temperatures and short load duration, the mass loss due to vaporisation is negligible, with an erosion depth of only ${\sim}25$ ${\unicode{x03BC}}$ m for the steepest angle case (65°). In the case of 1 ms loading, unavoidable extrapolations in the W thermophysical properties at temperatures above the boiling point of 6200 K leads to additional uncertainties.

Higher current beams can carry significantly more magnetic energy (going like $I_{RE}^2$ ), so despite the ${\sim}\,1$ mm melt depths shown in figure 19, this analysis does not preclude the possibility of breaching a FLiBe channel with a single strike. Future melt depth analysis will use updated RE kinetic distribution functions, first-wall geometries, and simulation boundary conditions specific to ARC.