2. Properties of Infinity Two stellarator design

In this work, we evaluate alpha-particle transport and confinement in the Infinity Two FPP baseline plasma physics design (Hegna et al. Reference Hegna2025). Infinity Two is a quasi-isodynamic (QI) (Helander Reference Helander2014) four-field-period stellarator designed for reactor conditions. Plasma parameters used for this evaluation (Guttenfelder et al. Reference Guttenfelder2025) include the peak electron density and temperature at the magnetic axis $n_{e,0} = 2.5\times 10^{20}$ m $^{-3}$ and $T_{e,0}=17.5$ keV, a 50–50 D–T mix, and an assumed 5 % He ash. This corresponds to a volume-average thermal plasma $\langle \beta \rangle = 1.6\,\%$ and alpha-particle beta $\langle \beta _{\alpha } \rangle = 0.31\,\%$ , with an alpha-particle power of $P_{\alpha }=158$ MW corresponding to a fusion power of $P_{fus}\approx 790$ MW. The major and minor radii of this configuration are $R_0 = 12.5$ m and $a = 1.25$ m. However, for this study, we describe a confinement zone (with topologically toroidal flux surfaces) for a somewhat smaller plasma (with $a_{eff} = 1.1$ m) from a free boundary VMEC calculation and use BMW to describe the fields outside this region which includes the presence of an $N/M = 4/5$ divertor magnetic island.

Figure 1.

Plasma profiles of Infinity Two. Electron and ion density and temperature profiles using constant values at the SOL, $n_{SOL}=n_{LCFS}$ and $T_{SOL}=T_{LCFS}$ , are shown with solid lines. Profiles using an exponential radial decay at the SOL, $n_{SOL}=n(\rho )$ and $T_{SOL}=T(\rho )$ , are shown with dashed lines. We only show deuterium density profiles since tritium profiles are the same in this 50–50 D–T plasma. Radial profiles of alpha-particle power density, $\mathcal {P}_\alpha$ , and cumulative alpha-particle power, $P_\alpha$ , are the same for both flat and radially decaying profiles at the SOL. The slowing-down time of alpha particles, $\tau _{S_\alpha }$ , spans several time scales from milliseconds to seconds.

In figure 1, we show the plasma profiles as function of the normalised radius, $\rho$ . Here, $\rho =r/a=\sqrt {\psi /\psi _{LCFS}}$ , with $\psi$ the toroidal magnetic flux and $\psi _{LCFS}$ the toroidal magnetic flux at the LCFS. This plasma has a relatively flat density profile at the core and concentrated alpha-particle power at $\rho \leq 0.7$ from fusion reactions following thermal plasma profiles. Additionally, $P_{\alpha }$ is computed using Bosch–Hale reactivity of D–T plasmas (Bosch & Hale Reference Bosch and Hale1992). Note that the plasma profiles beyond the LCFS at $\rho =1$ , that is, in the SOL, are kept constant in this simplified model. In other words, density, temperature and other plasma parameters at the SOL use the constant value found at the LCFS. This means constant collisionality for alpha particles from the LCFS (magenta) up to the location of the wall, shown in figure 2. During operation, the fast-ion particles will actually travel through a very cold and dense region in the magnetic islands before passing into the far SOL. In this work, we did not attempt to self-consistently simulate the helically non-uniform edge structure but instead, we analyse the sensitivity of the results to the SOL conditions by varying the plasma conditions.

Figure 2.

Poincaré sections of (a) vacuum and (b) finite- $\beta$ magnetic fields of Infinity Two at half-field period, $\phi =45^\circ$ . The magenta line and red cross represent the LCFS and magnetic axis location, respectively, as obtained from the free-boundary VMEC equilibrium. The red dashed line show the location of the standard Infinity Two wall. The blue dashed lines show the location of the walls obtained from three-dimensional extensions of the LCFS. We observe a small Shafranov shift in the finite- $\beta$ magnetic field, seen on this poloidal section as a displacement of flux surfaces (black markers) towards the outboard side of Infinity Two. This is visible by comparing the location of flux surfaces in both cases with respect to the LCFS and the magnetic axis.

For analysing the sensitivity of our results on wall loads to SOL conditions, we generate an additional model for the plasma that includes an exponential radial decay of the plasma profiles in the SOL, see figure 1. That is, density and temperature profiles in the SOL have the following radial dependence:

(2.1) \begin{eqnarray} T(\rho ) & = & T_{LCFS}\exp {\left ( -\rho /\lambda '\right ) }, \end{eqnarray}

(2.2) \begin{eqnarray} n(\rho ) & = & n_{LCFS}\exp {\left ( -\rho /\lambda '\right ) }, \end{eqnarray}

where $\lambda ' = \lambda /a$ is the normalised radial decay length of the SOL and $\lambda$ is chosen to be of the order of $\lambda \sim 10$ cm. This value allows us to assess both alpha-particle transport and wall loads in a different collisionality regime (with respect to flat profiles at the SOL) and to obtain sensible values of plasma density $n\sim 10^{18}$ m $^{-3}$ and temperature $T\sim 1$ eV at the location of the farthest wall (outermost dashed line in figure 2). Notice that this radial decay of the plasma profiles is uniform along the helical direction, in contrast to the actual configuration which has a high degree of helical dependence due to the magnetic island.

We use the above models for the plasma to assess core confinement of alpha particles and to obtain estimates of wall loads from alpha particles leaving the core and ultimately hitting any of the walls shown in figure 2. Note that in these walls, the divertor components were not included. In figure 2, we also show Poincaré sections of the vacuum (panel a) and finite- $\beta$ (panel b) magnetic fields for this configuration at half-field period, that is, at $\phi =45^\circ$ . The vacuum magnetic field is computed using currents in the coils of Infinity Two. The finite- $\beta$ fields include the response from the plasma in addition to the vacuum magnetic fields. This is computed via the Biot–Savart Magnetic VMEC Vector-potential (BMW) code (Cianciosa Reference Cianciosa2024; Frerichs Reference Frerichs2024). The BMW code uses Biot–Savart integration of the equilibrium current density to obtain a continuous magnetic vector potential everywhere. By combining the vacuum and plasma portions of the vector potential, the resulting total magnetic field can be evaluated from $\nabla \times \textbf {A}=\textbf {B}$ at any point in space (even beyond the VMEC boundary) and ensures divergence-free fields.

SIMPLE simulations are collisionless, guiding-centre calculations used to provide a first estimate of alpha-particle losses in optimised configurations. Once a particular configuration is found to have promising alpha-particle confinement, a more detailed and computational-expensive analysis follows using other set of codes such as ASCOT5 and KORC-T. The magnetic field for SIMPLE simulations is provided by a VMEC equilibrium. Particles are sourced on a single flux surface by selecting random locations along a field line. Each particle is given an isotropic starting velocity. Particles are followed until they either leave the LCFS indicating a lost particle or until a set amount of time has elapsed. We follow these alpha particles in time for 0.1 s, which is enough time for analysing prompt losses. SIMPLE simulations showed very low alpha-particle losses. Only 0.4 % of the launched particles reached the LCFS.

SIMPLE is extremely quick to run and therefore can provide results for a vast number of equilibria. However, while useful to verify a baseline of good confinement, these simulations are insufficient to characterise how a configuration will perform in realistic conditions. A more detailed analysis, including a proper 3-D birth distribution of alphas, the effect of collisions and full-orbit effects, is conducted to obtain more robust estimates of particle and energy losses of fusion-born alpha particles as well as estimates of wall loads. This is presented in the following sections.

3. Core particle and energy confinement

In this section, we present an assessment of particle and energy confinement of fusion-born alpha particles in Infinity Two core plasmas.

In figure 3, we show the initial location in the $\theta \phi$ -plane in VMEC coordinates of lost alpha particles in a SIMPLE simulation of Infinity Two. These alpha particles are initialised at $\rho =0.55$ . As can be seen from this figure, all lost particles are located in a region of relatively low magnetic field, $|B|$ , indicating the poor orbits are due to deeply trapped particles. One of the design principles of Infinity Two is the alignment of $B_{min}$ (and $B_{max}$ ) along the field as this provides robust collisionless guiding centre orbit confinement. As shown in figure 3(b ), which shows a plot of $|B|$ along the field, this design goal is largely produced. However, not all of the local $B_{min}$ locations are perfectly aligned. In these regions, particles can be locally trapped with a non-zero bounce averaged radial drift.

Figure 3.

(a) Isosurfaces of the magnitude of the magnetic field $|B|$ at the flux surface $\rho =0.55$ and an example of a magnetic field line starting in the well region (white trace). Initial location of simulated lost particles in Infinity Two are shown with magenta crosses. (b) Same magnetic field line as in panel (a) as a function of the toroidal angle (VMEC $\phi$ coordinate). Red and blue dashed lines show the global maximum and minimum values of $|B|$ at $\rho =0.55$ .

To further illustrate the role of bad deeply trapped particle orbits on the confinement, we use results from collisionless guiding-centre theory to classify particle orbits in SIMPLE simulations based on the magnitude of the magnetic field at which they are expected to be reflected (bounce), $B_{mirror} = \mathcal {E}_0/\mu$ , where $\mathcal {E}_0$ is the initial (birth) energy of alpha particles at 3.5 MeV and $\mu$ is the magnetic moment. Because orbits are collisionless, both quantities are assumed to be constant. In figure 4, we show the classification of an ensemble of 5000 simulated alpha particles according to their value of $B_{mirror}$ . Particles with $B_{mirror}$ between the global minimum of $|B|$ at that flux surface, $B_{min}$ , and the global maximum of $|B|$ at the same flux surface, $B_{max}$ , are considered to be trapped alpha particles. Those with $B_{mirror}\gt B_{max}$ are passing alpha particles. After verifying that no passing particles are ever lost in the SIMPLE calculations, we only follow trapped particles. From these SIMPLE simulations, we observe that it is deeply trapped particles ( $B_{mirror}\approx B_{min}$ ) that drive collisionless alpha-particle losses, see figure 4(b). In these simulations, we consider that an alpha particle is lost to the SOL when it crosses the LCFS as defined by the free boundary VMEC equilibrium. Most of the losses occur prior to 10 milliseconds. Particles lost in this time region are expected to exit with most of their energy.

Figure 4.

Orbit classification of simulated collisionless GC alpha particles using the SIMPLE code.

The next step in our analysis of alpha-particle confinement in the core plasma is the use of the ASCOT5 code (Särkimäki Reference Särkimäki2019), which can be used to follow the collisionless and collisional dynamics of alpha particles. For these simulations, the BMW code is used to convert the magnetic field components from the VMEC equilibrium onto the rectangular cylindrical grid that ASCOT5 uses. BMW includes both the vacuum magnetic fields from the coils of Infinity Two and the plasma contribution to the magnetic field. We have performed a benchmark test between SIMPLE and collisionless GC ASCOT5 simulations to verify that the different representations of the magnetic field do not result in different core alpha-particle confinement properties. In this test, we use the same initial isotropic distribution in pitch angle of 3.5 MeV alpha-particles in both codes. Also, simulated alpha particles are initialised at the same flux surface $\rho =0.55$ . We find good agreement between estimates of alpha-particle losses from SIMPLE and ASCOT5 simulations, being 0.4 % for the former and 0.6 % for the latter. As in SIMPLE simulations, ASCOT5 simulations predict that particle losses are driven by deeply trapped alpha particles in Infinity Two, and are lost on the same time scales of hundreds of micro-seconds to tens of milliseconds, as shown in figure 5. This benchmark test provides confidence that the different representations of the magnetic field do not influence results of core alpha-particle confinement significantly.

Figure 5.

Orbit classification of simulated collisionless GC alpha particles using the ASCOT5 code.

We now simulate the dynamics of collisional alpha particles in Infinity Two using ASCOT5. These simulated alpha particles are initialised with an isotropic distribution for their initial pitch angle and initial birth energy of $\mathcal {E}_0 = 3.5$ MeV, and are evolved in time until either they become locally thermalised with the background plasma via Coulomb collisions or are lost to the SOL, that is, when they cross the LCFS. Their initial spatial distribution follows the D–T fusion-reactivity distribution computed using the built-in AFSI code in ASCOT5, which applies the Bosch–Hale model (Bosch & Hale Reference Bosch and Hale1992) of D–T fusion reactions to thermal plasma profiles of Infinity Two. We observe that most alpha particles are born in the core plasma at $\rho \leq 0.7$ , with very few being born at the plasma edge. In addition, importance sampling is used to efficiently sample the initial distribution function of fusion-born alpha particles. For this, each numerical particle is weighted according to the initial local alpha-particle density. In figure 6, we show an example of the initial spatial distribution of simulated alpha particles in ASCOT5, generated as described above.

Figure 6.

Initial spatial distribution of simulated alpha particles in ASCOT5. Simulated alpha particles are coloured by their initial radial location ( $\rho$ ) as show in the colour bar on the right. The black lines show the location of the LCFS of the Infinity Two plasma at various toroidal angles.

We also perform some FO simulations to study finite Larmor effects on alpha-particle confinement in Infinity Two. For this, we use the ASCOT5 and KORC-T codes. Both codes use the same initial conditions for the simulated alpha particles, representation of the magnetic field, plasma profiles, and model for collisions between alpha particles and the background plasma, which includes electrons, D, T and He. Also, they use the same type of spline interpolations for the magnetic field components. They mainly differ in the algorithm for advancing in time the particles subject to the relativistic Lorentz force. ASCOT5 uses a volume-preserving algorithm (VPA) (Zhang et al. Reference Zhang, Liu, Qin, Wang, He and Sun2015) while KORC-T uses a modified Boris algorithm (Vay Reference Vay2008). Performing FO simulations with these two codes allows us to assess the effect of using different algorithms for solving the FO orbits on estimates of alpha-particle confinement.

In table 1, we show estimates of core particle and energy losses from GC and FO ASCOT5 simulations as well as FO KORC-T simulations. As expected for the Monte Carlo approach to estimate particle confinement, we note that as we increase the number of markers, we improve statistics of these estimates. Simulations using 10k markers already provide accurate particle and energy loss estimates in these simulations. From these collisional simulations, we conclude that core alpha-particle energy losses are approximately 2 % of the initial energy according to GC and FO ASCOT5 and approximately 4 % according to FO KORC-T simulations. Particle losses are higher, approximately 8 % of alpha-particles traced from the initial distribution according to GC and FO ASCOT5 simulations and 12 % according to KORC-T simulation. Due to collisions with the background plasma, the simulated alpha particles transfer some of their initial energy to the background plasma before they are lost to the SOL. We stress that the main difference between ASCOT5 and KORC-T simulations is the algorithm used to evolve the orbits of the particles. The discrepancy between ASCOT and KORC-T FO results highlight the difficulty involved in obtaining accurate and robust results for full-orbit calculations.

Table 1.

Simulated particle and energy losses in core Infinity Two plasma.

In figure 7, we show an attempt to classify lost GC and FO alpha particles from simulations in table 1 based on their initial value of $B_{mirror}$ . Because collisions do not conserve the energy and magnetic moment of particles, those that start on confined orbits, such as passing particles, can be lost. However, these are lost after many collisions and thus long time scales. Most lost particles prior to 10 ms arise from the deeply trapped orbits, as was seen before in the collisionless GC case above, see figure 5. There is a particular increase of lost particles near the trapped–passing boundary ( $B_{mirror} \approx B_{max}$ ) with respect to collisionless SIMPLE and ASCOT5 simulations relative to what was seen in figures 4 and 5. Also, a small population of initially passing particles ( $B_{mirror} \gt B_{max}$ ) are observed to be lost in these simulations, which is likely due to collisions modifying their pitch angle, eventually turning them into trapped particles that leave the plasma.

Figure 7.

Orbit classification of simulated alpha particles in ASCOT5: (a) initial condition of all particles; (b) initial condition of lost particles in collisional GC simulation; and (c) initial condition of lost particles in FO simulation.

In figure 8, we show the end states of energy of simulated collisional GC alpha particles in ASCOT5. We stress that in these simulations of core alpha-particle confinement, we follow the simulated particles in time until they either become thermalised with the background plasma (THERMAL) or until they get lost to the SOL by crossing the LCFS (RHOMAX). In the same figure, we show the corresponding end states of simulated time of these alpha particles. As it can be seen from this figure, most alpha-particle losses occur in time scales shorter than hundreds of milliseconds, $t_{sim}\leq 100$ ms, and mostly correspond to alpha particles with relatively high energy, $\mathcal {E}\geq 100$ keV. FO ASCOT5 simulations show very similar results.

Figure 8.

(a) Energy and (b) simulated time end states of GC alpha particles in ASCOT5 simulation of core Infinity Two plasma. The sharp increase of $f_{\alpha }(\mathcal {E}$ at energies $\mathcal {E}\sim 10^4$ eV is expected, since confined alpha particles thermalise via Coulomb collisions with the background plasma, where temperature is of the order of $T\sim 10^4$ eV.

When we analyse the initial parameters of the distribution of lost alpha particles in these simulations, we find that their mean initial radial position is approximately $\rho =0.54$ with very small mean pitch angle ( $v_\parallel /v \approx 0$ ), initially located in the region of low magnetic field of Infinity Two ( $\phi \approx 45^\circ$ ), corresponding to deeply trapped particles, see figure 9.

Figure 9.

Initial spatial distribution of lost, trapped alpha particles in ASCOT5. Consistent with the analysis of figure 3, most lost, trapped alpha particles are located in the region of low magnetic field of Infinity Two, around $\phi =45^\circ$ . Simulated alpha particles are coloured by their initial radial location ( $\rho$ ) as shown in the colour bar on the right. The black lines show the location of the LCFS of the Infinity Two plasma at various toroidal angles as a visual aid.

4. Power wall loads

Now, we analyse power loads on the Infinity Two wall due to alpha-particle losses. For this analysis, we use the same initial set-up for ASCOT5 simulations described in § 3. In this analysis, we only use GC simulations of alpha particles. The number of particles required to obtain good statistics of power wall loads was too large to be simulated including FO effects. Furthermore, the loss characteristics between ASCOT GC and ASCOT FO were very similar, justifying the use of GC for this calculation. The ASCOT5 code has been extensively used in the past to study power wall loads due to energetic ions on different tokamak devices, including reactor-relevant scenarios in ITER (Kurki-Suonio et al. Reference Kurki-Suonio, Kiviniemi, Sipilä, Heikkinen, Fundamenski, Matthews and Riccardo2002; Hynönen et al. Reference Hynönen, Kurki-Suonio, Suttrop, Dux and Sugiyama2007; Kurki-Suonio et al. Reference Kurki-Suonio2009; Shinohara et al. Reference Shinohara2011; Snicker et al. Reference Snicker, Sipilä and Kurki-Suonio2012; Kurki-Suonio et al. Reference Kurki-Suonio2017).

For this analysis, we use four models for the wall that do not include any divertor structures. The first of these models corresponds to the standard Infinity Two wall (ITW). This wall is generated by computing a conformal envelope of the plasma that is outside the magnetic islands that are used for the island divertor. The average distance from this wall to the separatrix of the magnetic island chain is 10 cm and its total area is 1034 m $^2$ . The other three models are obtained from three-dimensional extensions of the LCFS as defined by the free-boundary VMEC equilibrium described previously, using the methodology suggested in the appendix of Landreman (Reference Landreman2017). These three walls have different separation between the wall and the LCFS which we will refer to as the ‘wall gap’, $\Delta _{w}$ . We consider three values of $\Delta _w$ in our analysis: 30 cm, 50 cm and 70 cm, corresponding to a total wall area of 1046 m $^2$ , 1153 m $^2$ and 1263 m $^2$ , respectively. These walls are transformed into a triangulation (tri-mesh) which is then passed to ASCOT5 for the simulations, each triangulation consisting of 50 552 triangles. These models for the wall are intended to assess the sensitivity of the wall loads on the wall structure and to inform the later design of divertor structures and shielding of first wall components.

Table 2.

Peak wall loads, particle and energy losses to the wall, and wetted area as function of number of markers in GC ASCOT5 simulations of wall loads in Infinity Two. All simulations reported in this table use a wall with $\Delta _w=30$ cm, simulations using other wall model show the same trend.

Table 3.

Simulated peaked wall loads, particle and energy losses to the wall, and wetted area in Infinity Two plasmas. Results from GC ASCOT5 simulations using 100k markers.

In table 2, we show peak wall loads, particle and energy losses to the wall, and wetted area in Infinity Two plasmas for GC ASCOT5 simulations using different numbers of markers. Here, wetted area is defined as the sum of the areas of triangles of the discrete wall that have been hit by at least one alpha particle. Particle and energy losses refer to alpha particles being lost to the wall, unlike core losses in § 3, which were defined as those alpha particles crossing the LCFS ( $\rho =1.0$ ). The implications on the values of particle and energy losses from using this different definition is discussed below. From this table, we note that estimates of particle and energy losses are well described by simulations with as few as 1k markers. However, peak wall loads are not correctly described by simulations with fewer markers than 100k. Therefore, all results in the rest of this section refer to GC ASCOT5 simulations using 100k markers. Regarding the wetted wall area, this is a standard quantity used in past reactor-relevant analysis (Kurki-Suonio et al. Reference Kurki-Suonio2017; Scott et al. Reference Scott, Kramer, Tolman, Snicker, Varje, Särkimäki, Wright and Rodriguez-Fernandez2020). However, we observe that this quantity depends on both the number of markers and area of triangles (via the number of triangles) used in the simulations. We do not observe convergence towards a value as for particle and energy losses, and peak wall loads. Rather, the wetted wall area keeps increasing as we increase the number of markers, although these new wetted areas with more markers correspond to wall loads of the order of kW m^- $^2$ or lower. Thus, we report the wall wetted area in our simulations of Infinity Two only for completeness.

In table 3, we show peak wall loads, particle and energy losses to the wall, and wetted area in Infinity Two plasmas using both flat plasma profiles at the SOL, see figure 1, and plasma profiles with exponential radial decay at the SOL, see figure 1. First, we note that both particle and energy losses are lower than the core alpha-particle losses discussed in § 3. This is because once alpha particles leave the core plasma and enter the SOL plasma, they experience more collisional plasma. These collisions stop (thermalise) some of these energetic alpha particles in our ASCOT5 simulations before they hit the wall. This can be seen in figure 10, where we show the end state of energy and simulated time of alpha particles for both the flat and plasma profiles with exponential radial decay at the SOL. While we observe that particle losses to the wall occur on a similar time scale as core losses, see figure 8(b), the final distribution of simulated alpha particles shows a larger population of particles in the range of energies from $\mathcal {E}=1$ eV to $\mathcal {E}=100$ eV due to alpha particles interacting via Coulomb collisions with the SOL plasma.

Figure 10.

(a,c) Energy and (b,d) simulated time end states of GC alpha particles in ASCOT5 simulation using (a,b) flat and (c,d) radially decaying plasma profiles at the SOL. The sharp increase of $f_{\alpha }(\mathcal {E}$ at energies $\mathcal {E}\sim 10^4$ eV is expected, since confined alpha particles thermalise via Coulomb collisions with the background plasma, where temperature is of the order of $T\sim 10^4$ eV.

Regarding estimates of peak wall loads, we observe lower values of this quantity for the wall with $\Delta _w=30$ cm. Note that we only perform simualtions of the ITW with flat SOL profiles. Given the proximity of the ITW to the LCFS, there is very little difference in the analysis when different SOL profiles are used. We note that for these cases with lower peak wall loads, we obtain the largest wetted areas. This suggests that wall loads are reduced when alpha-particle losses to the wall are spread over wider wall areas.

In figure 11, we show the power wall loads on the $\theta \phi$ -plane for simulations with $\Delta _w=30$ cm, $\Delta _w=70$ cm and the ITW for flat SOL plasma profiles. In these figures, the positions of wall elements (triangles) are mapped onto the $\theta \phi$ -plane and are coloured according to the power load computed by ASCOT5. In all simulations, we observe higher power loads at approximately $\phi =60^\circ$ , with different poloidal locations. From the analysis of § 3, we observe that most lost alpha particles are deeply trapped particles with $B = B_{mirror}\approx 7.5$ T, initially at radial location $\rho \approx 0.5$ and toroidally localised around $\phi =45^\circ$ , the region of low magnetic field of Infinity Two, see figure 9. We compute the location of the reflection points of these particles by calculating the angle $\phi$ at which $\bar {|B|}(\phi )=B_{mirror}$ at $\rho =0.5$ . We find that these occur at $\phi \approx 30^\circ$ and $\phi \approx 60^\circ$ . This suggests that these particular alpha particles radially drift outwards around those toroidal locations, crossing the LCFS and escaping the core plasma around the poloidal location of the x-points. Also, we observe the four-field-period periodicity of the wall loads patterns, but stellarator symmetry (Dewar & Hudson Reference Dewar and Hudson1998) is broken, as observed in previous works (Lazerson et al. Reference Lazerson, Leviness and Lion2021b , Reference Lazersonc ). Our analysis shows that this asymmetry is caused by $\nabla B$ drifts, which for positive ions and the toroidal field pointing clockwise in Infinity Two effectively points downwards, according to our simulations. Thus, peak wall loads are localised around $\phi =60^\circ$ and $\theta \lt 0^\circ$ . A test ASCOT5 simulation of Infinity Two using negatively charged alpha particles confirm this, showing the same four-field-period periodicity, similar wall load values, but with peak wall loads now localised at $\phi =30^\circ$ and $\theta \gt 0^\circ$ , as expected from $\nabla B$ drifts. In figure 12, we show power wall loads in 3-D for the ASCOT5 simulation using the ITW. Peak wall loads are shown with an arrow, other hot spots in the MW/m $^2$ range are visible in other locations, but these are lower than the peak value of 2.51 MW m^- $^2$ for this wall.

Figure 11.

Power wall loads on the $\theta \phi$ -plane for flat SOL profiles for wall with (a) $\Delta _w=30$ , (b) $\Delta _w=70$ and (c) for the ITW. Only the first field period ( $0^\circ \leq \phi \leq 90^\circ$ ) is shown given the four-field-period periodicity of wall loads in ASCOT5 simulations. (d) Poincaré sections of Infinity Two at $\phi =60^\circ$ show the approximate poloidal locations, $\theta$ , where peak wall loads occur.

Figure 12.

Power wall loads of ASCOT5 simulation using the ITW in 3-D geometry. The edges of the triangles used in ASCOT5 simulations are shown in black. Reddish triangles show wall elements with higher values of power loads. Grey triangles show wall elements with negligible (below 1 kW m^- $^2$ ) power loads.

From this analysis, we conclude that the lowest peak wall loads due to alpha-particle losses to the wall are found to be approximately 2.17 MW m^- $^{2}$ , for the wall with $\Delta _w=30$ , with localised poloidal position close to the x-point of the magnetic island chain outside the plasma volume of Infinity Two, see figure 11(d). Other power loads in these simulations are in the range of tens of kW m^- $^2$ to hundreds of kW m^- $^2$ . It is expected that these peak power loads can be accommodated by existing helium-cooled plasma-facing component technologies, either through the use of specialty FW tiles (Arbeiter et al. Reference Arbeiter, Chen, Ghidersa, Klein, Neuberger, Ruck, Schlindwein, Schwab and von der Weth2017) or helium-cooled limiters (Norajitra et al. Reference Norajitra, Basuki, Giniyatulin, Hernandez, Kuznetsov, Mazoul, Richou and Spatafora2015). Further, this suggest that there might be an optimal shaping and separation between the wall and plasma volume that can reduce peak power loads with respect to the standard ITW. Different collisional conditions at the SOL seem to modify peak wall loads by modifying slightly the wetted area but preserving the same patterns on the $\theta \phi$ -plane.

5. Stability of Alfvén eigenmodes

In this section, we assess the Alfvén eigenmode (AE) activity driven by alpha particles in Infinity Two. Alfvén waves are ubiquitous collective modes supported by thermal plasma in the presence of a magnetic field. The frequency at which Alfvén waves occur depends on local plasma conditions. In the case of toroidal plasmas with radial plasma profiles and sheared magnetic fields, the wave frequency follows a radial dependence, too. This is known as the Alfvén continuum. In these plasmas, coupling between counter-propagating Alfvén waves might occur due to the toroidal and/or poloidal periodicity of the magnetic field, leading to regions in frequency where these Alfvén waves cannot exist. This is the result of destructive interference between coupled waves. These regions are known as Alfvén gaps (Heidbrink Reference Heidbrink2008). In these gaps, radially extended, weakly damped modes might be driven unstable by alpha particles. We aim to identify these modes in Infinity Two, if any, that may affect the confinement of fusion-born alpha particles and cause inefficient plasma heating and enhancement of wall loads.

For this, we use the STELLGAP code to compute the Alfvén continuum and Alfvén gaps of Infinity Two. The FAR3d code is used to assess the stability of the dominant and sub-dominant Alfvén eigenmodes existing in these gaps. The FAR3d code solves a reduced set of equations for high-aspect ratio plasmas and moderate $\beta$ -values (of the order of the inverse aspect ratio), retaining the toroidal angle dependency in an exact three-dimensional VMEC equilibrium. In these simulations, we include the effects of finite Larmor radius (FLR) damping effects of thermal and energetic ions as well as electron–ion Landau damping. The free-energy source of alpha-particle destabilisation of AE is provided to FAR3d via moments of the gyro-kinetic distribution function of the alpha particles, specifically, the alpha-particle density and their average parallel velocity. The correct model calibration requires performing gyro-kinetic simulations to calculate the Landau closure coefficients in the gyro-fluid simulations, matching the analytic TAE growth rates of the two-pole approximation of the plasma dispersion function with a Lorentzian energy distribution function for the energetic particles. This calibration has been done for DIII-D tokamak geometry by matching the gyro-fliud response function with its kinetic analogue for a parallel propagating Alfvén wave (Spong Reference Spong2013). The lowest-order Lorentzian can be matched either to a Maxwellian or to a slowing-down distribution by choosing an equivalent average energy. In our assessment, we use a Maxwellian distribution for the alpha particles using a temperature that matches the moment-averaged temperature of the simulated alpha-particle distribution, see discussion in § 2.3 of Reference Varela, Spong, Garcia, Ghai and OrtizVarela et al. (2024d). Previous benchmarking studies validated FAR3d code results with the gyro-kinetic codes EUTERPE, GEM, GTC, GYRO and ORB, with the hybrid code MEGA, and with the perturbative eigenvalue NOVA-K, see Taimourzadeh et al. (Reference Taimourzadeh2019) for details. The FAR3d code has been validated against experimental measurements in helical plasmas such as LHD (Varela et al. Reference Varela, Spong and Garcia2017b , Reference Varela, Spong, Garcia, Ohdachi, Watanabe and Seki2019a , Reference Varela, Nagasaki, Nagaoka, Yamamoto, Watanabe, Spong, Garcia, Cappa and Azegami2020a , Reference Varela, Spong, Garcia, Ohdachi, Watanabe, Seki and Ghai2021, Reference Varela, Spong, Todo, Garcia, Ghai, Ortiz and Seki2022b , Reference Varela2024a,Reference Varelab), TJ-II (Varela et al. Reference Varela, Spong and Garcia2017a ; Cappa et al. Reference Cappa2021; Eliseev et al. Reference Eliseev2021) and Heliotron J (Yamamoto et al. Reference Yamamoto2020; Varela et al. Reference Varela, Nagasaki, Nagaoka, Yamamoto, Watanabe, Spong, Garcia, Cappa and Azegami2020b , Reference Varela2022a ). Similarly, validation against experimental measurements in tokamak plasmas, such as DIII-D (Pace et al. Reference Pace2018; Varela et al. Reference Varela2018, Reference Varela, Spong, Murakami, Garcia, D’Azevedo, Van Zeeland and Munaretto2019b ; Huang et al. Reference Huang2020; Spong et al. Reference Spong, Van Zeeland, Heidbrink, Du, Varela, Garcia and Ghai2021; Ghai et al. Reference Ghai, Spong, Varela, Garcia, Van Zeeland and Austin2021), EAST (Wang et al. Reference Wang2023; Sun et al. Reference Sun2024) and JET (Garcia et al. Reference Garcia2024), has also been done.

Figure 13(a), shows the rotational transform of Infinity Two, which is optimised to avoid major resonant surfaces in the core and to resonate with the $n/m = 4/5$ surface at the edge. Figure 13(b) shows the radial profile of alpha-particle density and energy used for this assessment. These distributions are taken from a collisional GC ASCOT5 simulation at $t=50$ ms. At this time in the simulation, the alpha particles have redistributed in the core plasma following drifts and start to slow down, with peak energy of approximately 2.0 MeV at the magnetic axis and mean average energy of the entire distribution of 1.48 MeV. These alpha particles have an average velocity of $v_\alpha /V_A \sim 0.7$ at the core and $v_\alpha /V_A \sim 0.3$ at the edge, suggesting that alpha particles can potentially drive instabilities. Thus, the importance of performing this stability analysis. Here, $V_A$ is the local Alfvén velocity. In figure 13(c), we analyse the contribution of alpha particles of the slowing-down distribution (Alonso et al. Reference Alonso, Calvo, Carralero, Velasco, García-Regaña, Palermo and Rapisarda2022) with different energies (red filled circles) to the total $\langle \beta _\alpha \rangle =0.31\,\%$ (black star). These $\langle \beta _\alpha \rangle$ for different energy ranges of the alpha-particle distribution are the ones used in the analysis in § 5.1, where the stability limit of Infinity Two is explored. The geometry of Infinity Two is included in STELLGAP and FAR3d simulations via Boozer coordinates of the free-boundary VMEC equilibrium. The normalised radius is discretised using 200 radial points. Our sensitivity analysis varying the number of radial points show this is already enough resolution to compute the Alfvén continuum with STELLGAP and to perform the stability analysis of FAR3d.

Figure 13.

(a) Iota profile of Infinity Two. The dashed coloured vertical and horizontal lines indicate the radial location of the main rational surfaces. (b) Alpha-particle density (black line) and energy (blue) radial profiles in FAR3d simulations. (c) Total alpha-particle beta $\langle \beta _\alpha \rangle$ (black star) and $\langle \beta _\alpha \rangle$ for alpha particles with different ranges of energies (red filled circles) according to a theoretical slowing-down distribution function (Alonso et al. Reference Alonso, Calvo, Carralero, Velasco, García-Regaña, Palermo and Rapisarda2022). Here, the total $\langle \beta _\alpha \rangle$ is the sum of the $\langle \beta _\alpha \rangle$ for alpha particles with different energies. Note that $\langle \beta _\alpha \rangle$ is very small ( $\lt 10^{-3} \,\%$ ) for alpha-particle energies $\mathcal {E}\lt 1.0$ MeV.

In figure 14, we show the Alfvén gap structure of Infinity Two computed by STELLGAP for $n=0$ , $1$ and $2$ mode families. Here, a given mode family $n=k$ is defined as $n=k\ \mbox {mod}\ N$ , where $N=4$ is the magnetic field periods of Infinity Two. The Alfvén gaps are calculated for the modes $n=0,4,8,12,16,20,24,28$ , $n=1,3,5,7,9,11,13,15,17,19,21,23,25,27$ and $n=2,6,10,14,18,22,26$ , including the coupling with the sound wave (BAE gap). In figure 14(a), we show the complete Alfvén gap structure including all three mode families. In figure 14(b), we show the Alfvén gap structure only for the $n=0$ mode family. In figures 14(c)–14(e), we separate the spectra to show the details of panel (b), and to label some of the Alfvén gaps as reference. In figures 14(f)–14(i), we do the same for mode family $n=1$ , and in figures 14(j)–14(m) for mode family $n=2$ . Wide BAE gaps cover all the minor radius ( $\rho = 0.0{-}0.75$ ) in the frequency range of $5 {-} 50$ kHz. There are TAE gaps from the inner to outer plasma region in the frequency range of $60 {-} 100$ kHz. In addition, broad EAE gaps are observed above $100$ kHz.

Figure 14.

Alfvén gap structure of Infinity Two. (a) Complete spectra of Alfvén modes. (b), (f) and (j) Same spectra but separating mode families $n=0$ , $n=1$ , $n=2$ , respectively. (c)–(e) Break down of the spectra of panel (b) so we can label some of the computed Alfvén gaps as reference. (g)–(i) Details of panel (f) for the mode family $n=1$ , and (k)–(m) details of panel (j) for the mode family $n=2$ .

In figure 15, we show the expected helical gaps in Infinity Two as computed by STELLGAP. These gaps are located in the plasma periphery where no significant alpha-particle density gradient is observed from ASCOT5 simulations. Thus, destabilisation of helical AEs by alpha particles in these gaps is not likely to occur.

Figure 15.

Helical Alfvén gaps in Infinity Two. Orange dashed oval indicates the radial location and frequency range of the helical gaps.

The stability assessment of AE that might occur in the Alfvén gaps obtained from STELLGAP simulations is now performed with FAR3d. The stability assessment includes the modes $n=1,3,5,7,9,11,13$ and $n=2,6,10,14$ .

The stability of AE modes in Infinity Two is mainly determined by the following factors: (1) the alpha-particle $\langle \beta _\alpha \rangle$ , driving the strength of the perturbation; (2) Alfvén continuum damping, dictated by the magnetic field structure and thermal plasma profiles; (3) FLR damping effects; and (4) electron–ion Landau damping. In Infinity Two, Alfvén continuum damping plays a relatively small role on reducing the radial extent of perturbations due to its weak magnetic shear (Varela et al. Reference Varela2024c ). Our FAR3d simulations with the alpha-particle density and energy profiles shown in figure 13 indicate the absence of unstable AE in the Alfvén gaps of Infinity Two, this is the result of its low alpha-particle beta, $\langle \beta _\alpha \rangle = 0.31\,\%$ , not being large enough to drive any AE unstable.

5.1. Stability limit of Infinity Two

Finally, we address the issue of finding the stability limit of Infinity Two to AE activity driven by fusion-born alpha particles. This analysis provides information about trends of AE stability of Infinity Two away from its nominal steady-state operational regime. For this, we performed a parametric analysis varying beta of the simulated alpha-particle population. We perform this analysis for $\langle \beta _\alpha \rangle =0.2\,\%$ , $0.5\,\%$ , $1.0\,\%$ and $2.0\,\%$ . We note that when setting a value of $\langle \beta _\alpha \rangle$ , we can either choose to keep fixed alpha-particle density and modify their energy, or to keep fixed energy and modify the alpha-particle density. In this analysis, we follow both approaches to mimic the effects of two populations of alpha particles: relatively low-density and high-energy alpha particles corresponding to alpha particles in the early slowing-down phase, and higher density and relatively low-energy alpha particles corresponding to alpha particles in the late slowing-down phase. The shape of the alpha-particle profile of figure 13 is kept fixed in this analysis, given that the profile shape is not observed to change significantly for alpha particles with energies ranging from 3.5 MeV down to 0.5 MeV in our ASCOT5 simulations. In a similar way to our analysis above, our FAR3d simulations for this parametric analysis use Maxwellian distributions with a temperature that matches the moment-averaged energy of the analysed alpha-particle population. Specifically, we analyse AE driven by alpha particles with $\mathcal {E} = 0.25$ , 0.50, 1.0, 1.5, 2.0, 2.5 and 3.0 MeV, that is, the same energies for which we calculate $\langle \beta _\alpha \rangle$ in figure 13(c). Also, we keep thermal plasma profiles and magnetic configuration fixed.

Our parametric analysis indicates that at high- $\langle \beta _\alpha \rangle$ particles in the early-slowing-down phase ( $\mathcal {E}=1.5$ MeV to 2.0 MeV) lead to destabilisation of dominant low-frequency AEs, between 30 and 40 kHz, falling into the frequency range of BAEs. The critical value of $\langle \beta _\alpha \rangle$ for the destabilisation of these AEs is $\langle \beta _\alpha \rangle =0.5\,\%$ , increasing to $\langle \beta _\alpha \rangle =1.0\,\%$ as we decrease the alpha-particle energy down to $\mathcal {E}=1.5$ MeV. These critical values of $\langle \beta _\alpha \rangle$ fall well above the operational $\langle \beta _\alpha \rangle =0.31\,\%$ of Infinity Two for beta values of alpha particles with $\mathcal {E}\geq 1.5$ MeV.

However alpha particles in the late slowing-down phase ( $\mathcal {E}\lt 1.5$ MeV) show the destabilisation of AEs above 70 kHz, corresponding to the frequency range of the TAE gap and lower bound of the EAE gap. Nevertheless, if the $\langle \beta _\alpha \rangle$ is large enough, BAEs are the dominant instability. The critical value of $\langle \beta _\alpha \rangle$ for destabilisation of these modes is $\langle \beta _\alpha \rangle =1.0\,\%$ for the $n=1$ mode family. The critical $\langle \beta _\alpha \rangle$ decreases to $\langle \beta _\alpha \rangle =0.5\,\%$ for alpha particles with $\mathcal {E} \lt 0.5$ MeV. The frequency of the dominant mode in this case is close to 120 kHz. We observe similar trends and mode structure for the unstable modes of the $n=2$ mode family. Importantly, GAEs are also identified in the parametric analysis, although their growth rates are smaller compared with BAE and TAE. Thus, GAEs are not the most limiting stability for the device performance.

In figure 16, we show the growth rates and frequency ranges of dominant modes identified from this analysis driven by alpha particles with different energies when their beta is varied. The observed trend for growth rates of these dominant modes is that these increase as we increase alpha-particle beta, this being the main mechanism for destabilisation of AEs in Infinity Two. The analysis of dominant AEs also indicates that the resonance induced by alpha particles in the early-slowing-down phase lead to the destabilisation of BAEs, while alpha particles in the late slowing-down phase can trigger higher frequency modes in the range of the TAE and EAEs. BAEs shows a larger growth rate compared with the TAE and EAE.

From this analysis, we conclude that the required alpha-particle betas required to destabilise AEs in Infinity Two are much higher than those from its base operational regime, showing strong stability against AE activity. Importantly, we note that from past validation of FAR3d against gyro-kinetic simulations, these trends of stability are expected to hold.

Figure 16.

(a) Growth rate and (b) frequency of dominant AE destabilised by the $n=1$ mode family. (c) Growth rate and (d) frequency of the dominant AE destabilised by the $n=2$ mode family.