Improving fast-particle confinement in quasi-axisymmetric stellarator optimization

A method to improve fast-particle confinement during quasi-axisymmetric stellarator optimization has been identified. Quasi-axisymmetric (qa) stellarator designs have improved neoclassical transport due to their special symmetry of the magnetic field strength. Previously, it has been shown that, in general, quasi-symmetry can only be obtained on one single flux surface (Garren and Boozer 1991 Phys. Fluids B 3 2805–21). Even though quasi-symmetry can be a crucial property of stellarator design, there is no established convention for choosing the flux surface on which this should be optimized. To address this question, the flux surface on which quasi-axisymmetry is optimized has been varied in a qa configuration. The optimal location was found to lie between half radius and the plasma edge, since this allows for two beneficial features: it increases the number of flux-surfaces with improved quasi-axisymmetry and it increases the volume enclosed by the flux surface with the best qa quality.


Introduction
With the successful design, construction and first operation of Wendelstein 7-X [1], optimized stellarators have been shown to be a feasible alternative to tokamaks. However, stellarators do not automatically exhibit good particle confinement and therefore can drastically benefit from optimization.
One way to reduce neoclassical transport in stellarators is to make the magnetic field strength, rather than the magnetic field itself, independent of one of the Boozer coordinate angles [2]. This property implies the conservation of the canonical momentum conjugate to the angle in question since in these coordinates the guiding center Lagrangian only depends on the magnetic field strength and not on its direction. One thus obtains a third invariant, besides the energy and the magnetic moment, which leads to good particle confinement. This type of magnetic field is said to be quasi-symmetric [3,4]. In this paper, we focus on quasi-axisymmetric (qa) equilibria in which magnetic field strength is symmetric in the toroidal Boozer coordinate [5,6], but the results can be extended to other symmetries.
Currently, there are two ways to develop quasi-symmetric stellarator designs. The analytic approach is to derive them by large aspect ratio expansion [7]-mainly by employing a nearaxis expansion [8][9][10]. This approach is valuable as it allows physical insight into this class of stellarators and can serve as a starting point for the second approach: numerical optimization. Optimizing stellarators numerically is the standard method for obtaining optimized designs such as Wendelstein 7-X [11], the quasi-helical stellarator HSX [12,13], or the qa designs NCSX [14], CHS-qa [15], and ESTELL [16]. One reason for the numerical optimization is that the list of desired properties of a stellarator often involve many qualitatively different aspects such as confinement, stability, bootstrap current, which are difficult to combine into a single criterion. One particularly important property for future fusion reactors is the fast-particle confinement. Fast, high-energy (3.5 MeV) alpha particles are created in the fusion reaction and need to be confined for at least a slowing-down time in order that they heat the plasma and keep the fusion process running and therefore this is a crucial property for the future success of stellarator reactors.
One can optimize this property directly by following particle orbits, but this can be numerically expensive. Instead, as mentioned above, this property can be improved by simply enhancing quasi-symmetry.
It has been shown analytically that in general one can only achieve quasi-axisymmetry on one flux-surface, but not necessarily on the neighboring flux-surfaces due to additional constraints on the magnetic field, e.g. the requirement that it should be divergence-free [7,8]. When optimizing qa configuration one has the freedom to choose on which flux-surface one improves the quasi-axisymmetry [17]. Therefore the following questions arise naturally: does it matter for fast-particle confinement on which flux surface one enhances quasi-axisymmetry? If yes, are there better locations to optimize this property compared to other locations based on fast-particle losses?
To the best of the authors' knowledge, these questions have not been investigated in depth before. If one could obtain perfect qa-symmetry at only one flux surface, one could assume that it would be preferable at the plasma edge, since all the particles would have to pass this flux surface to leave the plasma. However, perfect quasi-axisymmetry is normally not achieved and deteriorates away from the optimized surface. To maximize the region of improved quasi-axisymmetry, one would expect the best qa-symmetry to be closer to the half radius.

Optimization
The code ROSE [18] was used to perform the optimization of the ideal-MHD equilibrium properties by varying the plasma boundary used as in input in the fixed boundary equilibrium version of VMEC [19]. We used the standard optimization method of ROSE, which employs Brent's algorithm [20]. Within the optimization routine, the cost function f is defined as the weighted sum: where w i are weights which are usually adapted for obtaining various optimal configurations. An optimal configuration is characterized by the feature that one cannot improve one criterion without diminishing at least another one. The set of all optimal points is called the Pareto frontier. If the Pareto frontier is non-convex, there are points on it that cannot be found by the weighted sum method. F i is the value for the criterion, i, calculated for each plasma boundary and F ĩ is the corresponding target value. For this study, we kept the following properties fixed: number of field periods, N=2, the volume-averaged plasma beta, = , where p is the plasma pressure and μ 0 is the vacuum permeability, angular brackets denote a volume average, and the aspect ratio where R=r 0,0 and = a r z 1,0 1,0 with the r mn and z mn given by the VMEC Fourier series describing the plasma boundary: where r and z are cylindrical coordinates, u the VMEC poloidal and v the VMEC toroidal angle [19]. Both VMEC angles vary from 0 to 1 for one rotation. Additionally the pressure profile is proportional to µ - where s is the normalized toroidal flux and the current density profile is given by where I(s) is the total toroidal current enclosed by the flux surface s. The targeted equilibrium properties for the optimization are: • Rotational transform, ι, at the magnetic axis and at the plasma boundary, where the rotational transform is the change of the poloidal flux χ with respect to the toroidal flux ψ: where F¢ is the specific volume given by Δρ is a small distance away from the axis. • The integrated absolute value of the Gaussian curvature of the plasma boundary, where κ 1 and κ 2 are the two principle curvatures: where the integral is over the plasma boundary d. • And at a particular flux surface s=s qa , the qa error where the magnetic field strength is given by We performed a scan by varying the flux surface s qa on which the qa error was minimized. All the other input parameters of the optimization were fixed.

Fast-particle losses
The confinement of fast alpha particles with energies of 3.5 MeV was investigated without collisions using the full-f Monte-Carlo code ANTS [21].
First, the designs were scaled to reactor size with a volume of 1900 m 3 , and with a volume averaged magnetic field of 5 T. Next, one thousand test particles were evenly distributed on each evaluated flux surface and then launched with uniformly distributed pitch angles similar to a fusionproduced fast-particle population. The particles were traced for half a second. Every particle which crosses the plasma boundary is counted as lost.

Dependence of fast-particle losses on the choice of qa-location
The qa error E qa , defined in equation (9), was minimized at different radial locations s qa for eight different optimization runs 1 . This led to small changes in some of the other targeted quantities, see table 1.
The minimum value of E qa after the optimization is approximately located where it was minimized s qa , see figure 1(a) which makes it possible to examine how the location s qa of a minimum value of E qa changes the properties of the design. The qa quality deteriorates away from that location as one would expect from [7,8]. The smallest value for the error (E qa =0.102%) is found for the smallest flux surface investigated: s qa =0.06. Towards s qa =1 there is a slight increase in the the minimum qa-error: E qa =0.388%.
This suggests that it is more difficult to optimize for the qaerror at the plasma edge with the additional targets of the optimization presented in this paper.
Interestingly, the location of the minimum values of the effective ripple, which is a measure of neoclassical transport [22], does in general not coincide with s qa , see figure 1(b). The   ( ), the minimum is approximately located at s qa =0.5. Therefore, the volume averaged values of the qa error and the effective ripple are minimized for s qa =0.5.
As is clear from figure 2, for particles launched on flux surfaces on s=0.5 or farther inside the designs with s qa =0.4 and s qa =0.5 are clearly better than the others, but for larger flux surfaces this advantage disappears. For comparison, the volume-average of the fast-particle losses given by equation (11) was evaluated, see figure 3, and the lowest averaged losses are also found for s qa =0.5 and = s 0.4 qa . This result might seem surprising at first. If one could reach perfect qa symmetry at only one flux-surface, then one would perhaps choose the edge s qa =1, since all the particles of the confined plasma would have to cross this flux surface to leave the plasma. However, in practice perfect quasi-axisymmetry is not achieved anywhere in general and the quality of qa-symmetry deteriorates in both directions away from its    minimum value at s qa . Choosing s qa somewhat between halfradius and the edge requires a large number of particles to pass through this surface and also reduces the maximum value of the qa-error E qa . In addition, while there exists a minimum at only one location (approximately at s qa ), nearby flux surfaces will also have a relatively low value for E qa . By locating s qa at the edge, one loses around half the nearby, good flux surfaces. Conversely, optimizing at the axis only allows a small fraction of particles to be confined in the highly-optimized region. For these reasons, the s qa =0.5 design has the smallest averaged E qa value of all designs.

Fast-particle losses with collisions
Up to this point, all fast-particle losses were calculated without considering collisions. There are two reasons why this is a sensible approach. First, if collisions are to be retained, one must assume specific density and temperature profiles, which are uncertain since the exact transport is not known, which creates an certain arbitrariness. Second, the most detrimental losses of alpha particles are the fast losses because those particles cannot transfer their energy to the bulk plasma. To evaluate those fast losses which happen without many collisions, one can neglect the collisions.
To determine whether collisions alter the location of the optimum flux surface s qa , we used the same profiles as in [23]. The temperature and radial electric field profiles are selfconsistent with the deuterium and tritium profiles. Ten thousand test particles were followed for each evaluated flux surface.
The smallest losses were again achieved with s qa =0.5 and s qa =0.4 for flux surfaces near the magnetic axis, see figure 4. Interestingly, the energy losses of the design with s qa =1.0 improved compared to the case without collisions but averaged over the entire volume it has more losses than the s qa =0.5 case, see figure 5. In general, the volume averaging favors the s qa =1.0 case as its best results compared to the other s qa lies closer to the plasma edge which encompasses the largest volume.
Thus, at least for this specific set of profiles, the approach described above remains valid even when collisions are taken into account. More generally, if reliable plasma density and temperature profiles are available one could try to minimize the energy losses of fast ions but simultaneously maximize their particle losses, in order to alleviate the problem of ashremoval.

Difference between the best and worst design
The best design with respect to the fast-particle loss fraction is now briefly compared with the worst design. Only major differences are pointed out since both designs are very similar to that with s qa =0.4, which was presented in greater depth in [23]. It is shown there that the design is stable to ideal magnetohydrodynamic instabilities and possesses low fastparticle losses. Additionally, the neoclassical transport coefficients are shown to be almost equivalent to those of a tokamak, with a clear banana regime at half-radius.
Despite the great difference in fast-particle confinement, the shape of flux surfaces only varies slightly, as shown in figure 6. However, since the non-qa components of the magnetic field strength at s=s qa was the only difference in the optimization procedure, Fourier spectra of the magnetic field strength, already visible in figure 1(a), differ, see figure 7. The greatest difference is in the mirror component B 01 . Only the components B mn with m=0 components are finite on axis. Therefore if quasi-axisymmetry is optimized near the axis these contributions tend to change the most. Other evident differences are the largest components at the edge: for s qa =0.5 there are three qa contributions B m0 out of the largest four components, but only two for the s qa =0.06 case.
We next compare the fast-particle losses with respect to time, see figure 8: the biggest relative change is for the innermost flux surface s=0.06. There are 14 times more losses for the s qa =0.06 than for the best design. The fastparticle losses launched on the flux surface s=0.25 only differ by a factor of 3.5. In the case with collisions, figure 9, the trends are similar as for the case without collisions. Both the particle and energy losses of the fast ions are lower in the configuration with better quasisymmetry.
It might again be surprising that the largest difference appears to be at s=0.06 since the s qa =0.06 design has its minimum value of both the qa-error E qa and effective ripple ò eff at this radial location. But this again supports the picture that the E qa of the entire plasma matters since any particle which is lost has to drift across all flux surfaces outside the one on which it was born.

Conclusion
It has been numerically shown that quasi-axisymmetry does not, in general, provide the best fast-particle confinement when