2. Computational procedure and initial conditions

In the approach of Nemov et al. (Reference Nemov, Kasilov and Kernbichler2014) a sample of 1000 particles (trapped plus passing) is followed with random starting points, as well as random values of pitch angles, on an initial magnetic surface. The guiding centre drift equations are solved within a relevant time interval. Every particle orbit is followed until the particle reaches some limiting surface outside the plasma confinement region. From the general sample of particles, trapped particles give the principal contribution to the collisionless particle losses. All classes of trapped particles are taken into account, i.e. particles trapped in one magnetic field ripple or in several adjacent ones.

Calculations are performed for the life time of 3.5 MeV ${\it\alpha}$ -particles in the stellarator magnetic configurations indicated above. They are adapted to reactor plasma parameters choosing $a=1.6$  m and $B=5$  T (see Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992), $a$ is the plasma average radius, $B$ is the magnetic field strength).

To accelerate computations, a Lagrange polynomial interpolation of the magnetic field is applied. The influence of an ambipolar radial electric field is not taken into account because it has only a negligible effect on ${\it\alpha}$ -particle motion.

Results are presented in plots of the collisionless time evolution of the trapped ${\it\alpha}$ -particle fraction. A decrease of this time corresponds to an increase of particle losses.

3. Magnetic configurations of type of U-2M and U-3M

The U-2M device (Pavlichenko Reference Pavlichenko1993) is an $l=2$ torsatron with $4$ helical field periods along the torus. Two different magnetic configurations of U-2M are considered here. For both configurations, the magnetic field parameters are chosen to provide magnetic surfaces which are well centred with respect to the vacuum chamber. For the first configuration, the toroidal magnetic field is chosen in such a way that the rotational transform ${\it\iota}$ is within $1/3<{\it\iota}<1/2$ ( $k_{{\it\varphi}}=0.31$ , with $k_{{\it\varphi}}=B_{th}/(B_{th}+B_{tt})$ where $B_{th}$ and $B_{tt}$ are the toroidal components of the magnetic field produced by the helical winding and the toroidal field coils, respectively (see, Pavlichenko Reference Pavlichenko1993)). The second configuration ( $k_{{\it\varphi}}=0.295$ ) has a slightly larger toroidal magnetic field and ${\it\iota}$ within $0.31<{\it\iota}<0.383$ . For this configuration, the resonant magnetic surfaces with ${\it\iota}=1/3$ are inside the confinement region. For both configurations the magnetic field is calculated using the Biot–Savart law code, where current feeds and detachable joints of the helical winding are taken into account.

Due to the asymmetric arrangement of these elements, the stellarator symmetry of the magnetic field of U-2M is broken and considerable island magnetic surfaces arise, which correspond to ${\it\iota}=1/3$ ( $k_{{\it\varphi}}=0.295$ ) and ${\it\iota}=2/5$ ( $k_{{\it\varphi}}=0.31$ ), see figure 1.

Figure 1. Magnetic surfaces for U-2M and U-3M; (a) for U-2M, $k_{{\it\varphi}}=0.31$ , island surfaces corresponding to ${\it\iota}=2/5$ are present; (b) for U-2M, $k_{{\it\varphi}}=0.295$ , island surfaces corresponding to ${\it\iota}=1/3$ are present; (c) for U-3M, island surfaces corresponding to ${\it\iota}=1/4$ are present. A circle with a radius of 34 shows the inner boundary of the vacuum chamber for U-2M.

The U-3M device (Lesnyakov et al. Reference Lesnyakov, Volkov, Georgievskij, Zalkind, Kuznetsov, Ozherel’ev, Pavlichenko, Pogozhev, Schwörer and Hailer1992) is an $l=3$ torsatron with $9$ helical field periods along the torus. A peculiarity of its standard configuration is that it is an outward shifted configuration to improve MHD stability. It is stated in Lesnyakov et al. (Reference Lesnyakov, Volkov, Georgievskij, Zalkind, Kuznetsov, Ozherel’ev, Pavlichenko, Pogozhev, Schwörer and Hailer1992) that small errors in the U-3M magnetic system exist. They are equivalent to a small eccentricity in the vertical field coils. The corresponding perturbations of the U-3M magnetic field lead to the appearance of large magnetic islands corresponding to ${\it\iota}=1/4$ inside the confinement region. The magnetic field for the U-3M vacuum configuration is calculated using the helical winding field decomposition into toroidal harmonic functions (with $33$ harmonics) and using complete elliptic integrals of the first and second kind for the field of the vertical field coils. Figure 1(c) shows the cross-sections of magnetic surfaces and island structures of U-3M.

Computational results for the collisionless time evolution of the trapped ${\it\alpha}$ -particle fractions obtained for all three configurations are presented in figure 2. The results relate to particles started on magnetic surfaces corresponding to $A\approx 40$ for U-2M ( $A$ is the aspect ratio of the magnetic surface). The limiting surface is the inner surface of the vacuum chamber. For U-3M particles are started on magnetic surfaces corresponding to $r/a\approx 0.25$ and $r/a\approx 0.5$ ( $r$ is a mean radius of the magnetic surface). Here, a toroidal surface including the confinement region is considered as the limiting surface.

Figure 2. Collisionless evolution of the trapped ${\it\alpha}$ -particle fractions for the configurations of the U-2M and U-3M types adapted to reactor parameters. 1: for U-2M, $k_{{\it\varphi}}=0.295$ , with islands, 2: for U-2M, $k_{{\it\varphi}}=0.295$ , without islands, 3: for U-2M, $k_{{\it\varphi}}=0.31$ , with islands, 4: for U-2M, $k_{{\it\varphi}}=0.31$ , without islands, 5: for U-3M, $r/a\approx 0.25$ , without islands, 6: for U-3M, $r/a\approx 0.5$ , without islands, 7: for U-3M, $r/a\approx 0.25$ , with islands, 8: for U-3M, $r/a\approx 0.5$ , with islands; for U-2M particles are started on the magnetic surfaces corresponding to $A\approx 40$ .

If current feeds and detachable joints of the helical winding in U-2M, and the eccentricity of the vertical field coils in U-3M, are not taken into account, the corresponding magnetic surfaces become nested regular surfaces. For such magnetic configurations without islands the life time of ${\it\alpha}$ -particles is also calculated and those additional results are presented in figure 2.

It follows from the results that there is no noticeable difference for the particle life time in both configurations of U-2M. For the U-3M configuration, the loss time is approximately the same for $r/a=0.25$ and $r/a=0.5$ . Also, the presence of magnetic islands caused by a violation of the stellarator symmetry in U-2M or a small eccentricity in the vertical field coils in U-3M has no noticeable influence on the life time of ${\it\alpha}$ -particles.

Figure 3. Collisionless evolution of the trapped ${\it\alpha}$ -particle fractions for the quasi-helically symmetric stellarator adapted to reactor parameters. Particles are started on the magnetic surfaces corresponding to $r/a\approx 0.25$ (1) and $r/a\approx 0.5$ (2) with $r$ being a mean radius of the magnetic surface.

For all three configurations, the approximate ${\it\alpha}$ -particle loss time is $10^{-4}~\text{s}$ . This is approximately the same value as it was found in Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992) for the standard $l=2$ stellarator adapted to reactor parameters. This value is unacceptably low for a nuclear fusion reactor and, therefore, the standard stellarators need to be optimized with respect to particle confinement time and transport (see, e.g. Grieger et al. Reference Grieger, Lotz, Merkel, Nührenberg, Sapper, Strumberger, Wobig, Burhenn, Erckmann, Gasparino, Giannone, Hartfuss, Jaenicke, Kühner, Ringler, Weller and Wagner1992).

4. Quasi-helically symmetric stellarator

Quasi-helically symmetric stellarators (QHS stellarators) represent one way to improve the particle and energy confinement in stellarators. In Nührenberg & Zille (Reference Nührenberg and Zille1988) the possibility for the existence of quasi-helically symmetric toroidal stellarators has been shown in magnetic coordinates. Calculations of ${\it\alpha}$ -particle losses in some quasi-helically symmetric configurations performed in magnetic coordinates in Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992) have demonstrated a very good confinement for particles, which are started in the inner half of the device. A real-space realization of a zero-beta variant of the quasi-helically symmetric configuration (Nührenberg & Zille Reference Nührenberg and Zille1988) (with 6 field periods along the torus) has been considered in Nemov et al. (Reference Nemov, Kasilov and Kernbichler2014) with a magnetic field presented as a decomposition in toroidal harmonic functions containing the associated Legendre functions. The decomposition coefficients of the superposition have been obtained by minimizing the magnetic field component normal to the boundary magnetic surface given by the three-dimensional Variational Moments Equilibrium Code (VMEC) (Hirshman & Betancourt Reference Hirshman and Betancourt1991) equilibrium in Nührenberg & Zille (Reference Nührenberg and Zille1988) (the Neumann boundary condition at the VMEC boundary).

Figure 4. Magnetic surface cross-sections of the quasi-helically symmetric stellarator. (a) Bean shaped and after a half of the field period; (b) with increased scale for magnetic islands. 1: VMEC boundary, 2: island surfaces ${\it\iota}=3/2$ , 3: the magnetic surface just inside ${\it\iota}=3/2$ , 4: $r/a$ about 0.5, 5: $r/a$ about 0.25.

Computational results for the collisionless time evolution of the trapped ${\it\alpha}$ -particle fraction obtained in Nemov et al. (Reference Nemov, Kasilov and Kernbichler2014) for QHS are given in figure 3. The ${\it\alpha}$ -particle loss time is essentially larger than in all variants of Uragan. Figure 3 shows that during 1 s trapped particles initiated at $r/a\approx 0.25$ are not lost, however, some fraction of particles initiated at $r/a\approx 0.5$ is lost. Results for $r/a\approx 0.25$ are in good agreement with the results of Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992) using magnetic coordinates, however, results for $r/a\approx 0.5$ suffer from higher losses (about 6 % of trapped plus passing particles) than reported in Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992) (about 3 %).

To understand the possible reason for such a difference in particle losses, calculations of ${\it\alpha}$ -particle confinement for QHS are also performed for particles launched at positions with $r/a$ somewhat smaller than 0.5.

As a result, particle losses of 3 % are found for starting positions at $r/a\approx 0.4$ . For further analysis let us consider cross-sections of the magnetic surfaces for the considered QHS configuration which are shown in figure 4. Magnetic islands are seen in the near boundary region of the magnetic configuration. These islands correspond to resonant perturbations of the real-space magnetic field with ${\it\iota}=3/2$ . The formation of island magnetic surfaces instead of low-order rational surfaces, which are present in magnetic coordinates, leads to some deformation of adjacent non-island surfaces in the optimised VMEC equilibrium where particle losses have been minimised. Therefore, for some inner real space magnetic surfaces, which are not very far from the plasma boundary, the optimisation conditions will be somewhat violated. Due to this phenomenon, particle drift across magnetic surfaces as well as particle losses can be increased as compared to those found in magnetic coordinates and therefore the level of optimization may be reduced. Thus, the phenomenon related to the near boundary resonant-magnetic surfaces can be responsible for the higher level of particle losses at $r/a\approx 0.5$ in the real-space magnetic field as compared to those for the field given in the magnetic coordinates.

Note that the perturbation of the real-space magnetic field is a blend of resonant and non-resonant components. We focus on the analysis of the effect of the resonant component of the perturbation of the magnetic field, because the distortion of the magnetic field magnitude (obtained in magnetic coordinates) by the non-resonant magnetic field perturbation is proportional to the amplitude of this perturbation, while the distortion of this magnitude by the resonant magnetic field perturbation in regions adjacent to the resonant flux surface is proportional to the square root of the amplitude of the perturbation (see, e.g. Shaing Reference Shaing2002), i.e. is much greater than that for a non-resonant component. Configurations of Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992), which are used for comparison with QHS, are somewhat different from the configuration of Nührenberg & Zille (Reference Nührenberg and Zille1988). The configuration in Nührenberg & Zille (Reference Nührenberg and Zille1988) has 6 field periods along the torus, whereas the configurations of Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992) have 4 and 5 such periods. Besides that, the configuration with 4 field periods has a smaller aspect ratio than the configuration with 5 field periods. Nevertheless, the particle losses for these two configurations are the same. Therefore one can conclude that the level of particle losses calculated in magnetic coordinates for the configuration of Nührenberg & Zille (Reference Nührenberg and Zille1988) will not be larger than that for the configurations in Lotz et al. (Reference Lotz, Merkel, Nührenberg and Strumberger1992).

In conclusion, an attempt is made to reduce the sizes of the island at ${\it\iota}=3/2$ by a change in the spectrum of the pertinent toroidal harmonic functions (set of 465 harmonics). These islands correspond to the presence of a harmonic with $m=8$ and $n=-12$ in the spectrum ( $m$ is the poloidal number and $n$ is the toroidal number, 8 islands are seen in the island chain in figure 4). One can try to reduce the sizes of the islands reducing the amplitude of this harmonic. It should be kept in mind that, in addition to the influence of the above mentioned harmonic on the resonance magnetic surface corresponding to ${\it\iota}=3/2$ , this harmonic also participates in the formation of the entire magnetic configuration and changing this harmonic can lead to destruction of a part of the configuration (since the decomposition coefficients for the initial harmonic set are obtained by satisfying the Neumann boundary condition at the VMEC boundary).

First, a case was considered when the harmonic $m=8$ , $n=-12$ is fully eliminated. It turned out that in such a case the magnetic surfaces are fully disrupted outside the surface corresponding to $r/a=0.076$ ( $a$ is identified here with an average radius of the boundary surface, obtained with the aid of the VMEC code).

Since such a situation is unacceptable, a further study is performed where the amplitude of the $m=8$ , $n=-12$ harmonic is only slightly decreased. In figures 5 and 6 some results are presented for the case when the amplitude of this harmonic in the initial spectrum is decreased by 5 %. Figure 5 shows a cross-section of the magnetic surface by which the confinement region is now limited. Outside this surface magnetic surfaces are destroyed. It is seen that the confinement region now is noticeably smaller than in figure 4.

Figure 5. VMEC boundary (1) and outermos undestroyed magnetic surface for QHS with decreased amplitude of the resonant harmonic (2).

Figure 6. Collisionless evolution of the trapped ${\it\alpha}$ -particle fractions for QHS for decreased amplitude of the resonant harmonic (1). Particles are started on the magnetic surfaces corresponding to $r/a\approx 0.5$ . Curve (2) shows the corresponding result from figure 3.

Calculations of the collisionless time evolution of the trapped ${\it\alpha}$ -particle fraction are carried out for a tracing time of 0.1 s. The results of calculations performed for particles started at $r/a\approx 0.25$ practically coincide with those in figure 3. However, for particles started at $r/a\approx 0.5$ , the particle losses are now somewhat greater than those obtained before. The trapped particle evolution for this case is given in figure 6. The corresponding plot from figure 3 is also shown for comparison.

So, an attempt to suppress the islands near the boundary by some change of the harmonic spectrum led to a deterioration of the particle confinement. This can be understood from the difference between the modified spectrum and the spectrum obtain from VMEC using the Neumann boundary condition.

In addition to calculations with the set of 480 harmonics, calculations were also made for a set increased to 840 harmonics. For both sets, the results for the particle confinement turned out to be practically the same. In the calculations with the set of 840 harmonics, an attempt to suppress the influence of islands with ${\it\iota}=3/2$ was undertaken, but now the fundamental harmonics of the spectrum are not changed. Instead of this, a small harmonic is added which is not contained in the fundamental spectrum, namely a harmonic $m=8$ , $n=-11$ or a harmonic $m=8$ , $n=-13$ with toroidal number close to $n=-12$ . This is a multiple of the number of the field periods over the torus. The amplitude of the additional harmonic is set to 0.01 % of the amplitude of the fundamental harmonic $m=8$ , $n=-12$ .

From the computational results it follows that in this case the magnetic surfaces are destroyed outside the surface corresponding to $r/a=0.58$ ( $n=-11$ ) or $r/a=0.25$ ( $n=-13$ ).

Note that in Nemov et al. (Reference Nemov, Kasilov and Kernbichler2014) a comparison has been carried out for the particle confinement in QHS with that in the HSX device (practical realization of the quasi-helically symmetric stellarator). There, the possibility of an improvement of particle confinement in the HSX has been discussed.