# Direct optimization of neoclassical ion transport in stellarator reactors

B. F. Lee1, S. A. Lazerson2, H. M. Smith3, C. D. Beidler3, and N. A. Pablant4
1 Princeton University, Princeton, New Jersey, U.S.A.  2 Gauss Fusion GmbH,
Hanau, Germany  3 Max Planck Institute for Plasma Physics, Greifswald, Germany
4 Princeton Plasma Physics Laboratory, Princeton, New Jersey, U.S.A.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We directly optimize stellarator neoclassical ion transport while holding
neoclassical electron transport at a moderate level, creating a scenario
favorable for impurity expulsion and retaining good ion confinement.
Traditional neoclassical stellarator optimization has focused on minimizing
$\epsilon_{\mathrm{eff}}$, the geometric factor that characterizes the amount
of radial transport due to particles in the $1/\nu$ regime. Under expected
reactor-relevant conditions, core electrons will be in the $1/\nu$ regime and
core fuel ions will be in the $\sqrt{\nu}$ regime. Traditional optimizations
thus minimize electron transport and rely on the radial electric field
($E_{r}$) that develops to confine the ions. This often results in an inward-
pointing $E_{r}$ that drives high-$Z$ impurities into the core, which may be
troublesome in future reactors. In our optimizations, we increase the ratio of
the thermal transport coefficients $L_{11}^{e}/L_{11}^{i}$, which previous
work has shown can create an outward-pointing $E_{r}$. This effect is very
beneficial for impurity expulsion. We obtain self-consistent density,
temperature, and $E_{r}$ profiles at reactor-relevant conditions for optimized
equilibria. These equilibria are expected to enjoy significantly improved
impurity transport properties. We conclude by providing several directions of
future research that may help further improve the presented optimization
algorithm.

††:

captionUnknown document class (or package) monthyeardate[]

, , , ,

  *

Keywords: Nuclear Fusion, Stellarator, Optimization, Neoclassical, Impurity
Transport, Electron Root, Electric Field

##  1 Introduction

Stellarator reactors are expected to achieve relatively high temperatures in
their plasma cores, resulting in the electrons and fuel ions exhibiting
neoclassical transport characteristics of the long mean free path (lmfp)
regime. This regime may be divided into several pieces that can be visualized
in figure 12 of reference [1]. The $1/\nu$ regime corresponds to zero
normalized $\mathbf{E}_{r}\times\mathbf{B}$ drift velocity,
$v_{E}^{*}\equiv\left|E_{r}\right|/vB_{0}=0$, where
$\mathbf{E}_{r}=E_{r}(r)\nabla r$ is the radial electric field, $r$ is a flux
label, $v$ is particle speed, and $B_{0}$ is a reference magnetic field
strength. The radial particle and heat transport in the $1/\nu$ regime can be
characterized by the magnitude of the geometric factor
$\epsilon_{\mathrm{eff}}$ [2], which has been commonly used as a minimization
target for neoclassical stellarator optimizations. If $v_{E}^{*}>0$, two other
lmfp regimes exist: the $\nu$ regime at extremely low collisionalities and the
$\sqrt{\nu}$ regime at more moderate collisionalities. The thermal transport
coefficients for particle and heat fluxes are approximately proportional to
$\left|E_{r}\right|^{-2}$ in the former case and $\left|E_{r}\right|^{-3/2}$
in the latter [3]. Roughly speaking, $\mathbf{E}_{r}\times\mathbf{B}$ rotation
helps limit the radial excursions of particles from flux surfaces when
particle behavior is not entirely dominated by collisional effects. Since
$\epsilon_{\mathrm{eff}}$ characterizes transport levels when $v_{E}^{*}=0$,
lowering $\epsilon_{\mathrm{eff}}$ lowers the “ceiling” on radial transport
for populations in all lmfp regimes and thus reduces overall radial transport.
However, because $v_{E}^{*}\approx 0$ only for electrons under expected
reactor-relevant conditions, minimizing $\epsilon_{\mathrm{eff}}$ directly
minimizes only radial electron transport. One may think that radial ion
transport would be larger than radial electron transport — the larger ion
masses lead to larger radial excursion of the orbits and therefore larger
diffusion coefficients — but the charge separation immediately establishes the
aforementioned $\mathbf{E}_{r}$ to make the transport ambipolar [4]. The
electric field will point radially inward in this case ($E_{r}<0$). Because
the ambipolar $\mathbf{E}_{r}$ is determined by the condition that the net
radial current is zero and the ions are the primary contributor to the radial
current in (reasonably) low-$\epsilon_{\mathrm{eff}}$ configurations, the
$E_{r}<0$ solution of the ambipolarity condition is typically referred to as
the “ion root” ambipolar electric field.

In design studies, stellarator reactors have commonly targeted
$\epsilon_{\mathrm{eff}}$ values in the range $0.1\%$ – $1\%$ [5, 6], which is
sufficiently small to prevent neoclassical energy losses from being an
obstacle to ignition but not small enough to preclude other design objectives
from being taken into account. Such reactors are expected to have strong ion
root electric fields, so impurities (e.g., partially-ionized tungsten atoms
from the divertor and other plasma-facing components) would be driven into the
core and cause substantial radiative losses that would likely make continuous
reactor operation difficult. The presence of a neoclassical impurity pinch has
long been recognized in the literature [7, 8, 9, 10, 11], and several remedies
have been proposed. For instance, the temperature gradient of the fuel ions
may help counteract the pinch associated with a small ion root electric field
[12, 13], but this will likely not be effective if a strong ion root is
present throughout the plasma. Furthermore, tangential electric fields may
make this effect relatively unhelpful as they often enhance the impurity pinch
caused by the radial electric field, especially for high-Z impurities [14, 15,
16, 17]. Buller et al. attempted to directly optimize the tangential electric
field (through the electric potential) to prevent inward impurity flux but
were only able to reduce it [18].

Another possible solution exists for the impurity transport problem: if
$E_{r}>0$ in some region of the plasma, this will provide a thermodynamic
force that drives impurities (especially those with high $Z$) radially out of
that region. The case $E_{r}>0$ is referred to as the “electron root” solution
of the ambipolarity condition. Operating stellarators with an electron root
was historically considered superior to ion root operation since the larger
$\left|E_{r}\right|$ characteristic of the electron root is favorable for
confinement [19, 20]. On the other hand, experimental realizations of the
electron root typically involve substantial electron heating that causes the
electron temperature to be much higher than the fuel ion temperature [9, 21,
22, 23, 24], which contradicts the desire for reactors to have strong
collisional coupling such that plasma thermal energy efficiently drives fusion
reactions. This has lead to skepticism that stellarator reactors can feasibly
operate with a steady-state electron root.

In this work, we seek to show that the electron root may not be as elusive in
reactors as previously thought. Under the assumptions that transport processes
are radially local and the drift kinetic equation can be linearized about a
Maxwellian, the neoclassical particle flux may be written as

$\Gamma_{s}^{\mathrm{neo}}=-n_{s}L_{11}^{s}\left(\frac{n^{\prime}_{s}}{n_{s}}-\frac{q_{s}E_{r}}{T_{s}}+\left(\frac{L_{12}^{s}}{L_{11}^{s}}-\frac{3}{2}\right)\frac{T^{\prime}_{s}}{T_{s}}\right),$
(1)

where $q_{s}=Z_{s}e$ (with $e$ being the elementary charge), $n_{s}=n_{s}(r)$,
and $T_{s}=T_{s}(r)$ are the charge, density, and temperature of species $s$,
respectively, and the thermal transport coefficients are given by

$L_{ij}^{s}=\frac{1}{q_{s}^{2}\iota R_{0}B_{0}^{2}}\sqrt{\frac{\pi
m_{s}T_{s}^{3}}{8}}\int_{0}^{\infty}dK_{s}K_{s}^{2}e^{-K_{s}}D_{ij}^{*}(K_{s})h_{i}h_{j},$
(2)

with $\iota$ the rotational transform on a given flux surface, $R_{0}$ a
reference major radius, $m_{s}$ the mass of species $s$,
$K_{s}=\frac{m_{s}v_{s}^{2}/2}{T_{s}}=\left(\frac{v_{s}}{v_{s,\mathrm{thermal}}}\right)^{2}$,
$D_{ij}^{*}$ the monoenergetic transport coefficient evaluated for a given
$v_{E}^{*}$, $h_{1}=h_{3}=1$, and $h_{2}=K_{s}$ [3]. While it may initially
appear that $L_{11}^{e}/L_{11}^{i}\ll 1$ will always hold given the mass
scaling in equation 2, it has been previously noted (to help explain the
“impurity hole” phenomenon of the Large Helical Device [25, 26]) that
sufficiently large $L_{11}^{e}/L_{11}^{i}$ will cause $E_{r}>0$ [12].
Optimized configurations have also recently been discovered by Beidler and
coworkers with extremely small $D_{11}^{*}$ at fuel-ion-relevant $v_{E}^{*}$
and moderate values of $\epsilon_{\mathrm{eff}}$ [27], leading to a large
$L_{11}^{e}/L_{11}^{i}$ and therefore a strong electron root in the core under
reactor-relevant conditions. The theoretical foundations of this phenomenon
are explored in more detail in recent work by Helander et al. [28].

The objective of our work is to follow the suggestion of reference [27] and
optimize stellarator magnetic configurations to increase
$L_{11}^{e}/L_{11}^{i}$. We control $L_{11}^{e}$ through
$\epsilon_{\mathrm{eff}}$ and $L_{11}^{i}$ through an array of $D_{11}^{*}$
evaluated at fuel-ion-relevant $v_{E}^{*}$ and $\nu^{*}\equiv R_{0}\nu/\iota
v$ (normalized collisionality) values. Monoenergetic transport coefficients
should, in principle, provide a more general target than thermal transport
coefficients because they are characteristic of a given magnetic field and do
not depend on plasma profiles. Our transport optimizations are entirely
neoclassical — because turbulence is often the dominant transport mechanism in
optimized stellarators such as Wendelstein 7-X [29], the techniques presented
here may only be useful when coupled with turbulence optimization techniques.
Indeed, lowering the overall level of turbulence in a reactor may well reduce
the outward impurity flux provided by the turbulence and necessitate the
presence of another impurity removal mechanism, such as an electron root. We
note that large $E_{r}^{\prime}$ (and thus large
$\mathbf{E}_{r}\times\mathbf{B}$ shear), which is common in electron root
scenarios, may help reduce radial turbulent fluxes [30, 31]. Large
$E_{r}^{\prime}$, usually in the ion root region near the last closed flux
surface, is known to allow access to improved confinement modes of operation
somewhat similar to H-modes in tokamaks [9, 32, 33].

##  2 General Formulation

STELLOPT [34] is the primary optimization software used in this work. STELLOPT
attempts to optimize finite-$\beta$ MHD equilibria by minimizing an objective
function of the form

$\chi^{2}=\sum_{i}\chi_{i}^{2},$ (3)
$\chi_{i}^{2}=\frac{\left(f_{i}-f_{i}^{\mathrm{target}}\right)^{2}}{\sigma_{i}^{2}},$
(4)

where $f_{i}$ is any quantity that is callable by the STELLOPT routines,
$f_{i}^{\mathrm{target}}$ is the desired value for $f_{i}$, and $\sigma_{i}$
is a user-specified inverse weight. VMEC [35] is used as the MHD equilibrium
solver in all cases. Differential evolution [36] and a Garabedian boundary
representation [37] are utilized to find global minima more reliably except
where noted in section 3.1.

The optimizations begin from a fixed-boundary configuration based on the W7-X
high-mirror configuration [38] whose volume and on-axis magnetic field
strength have been scaled to the reactor-relevant values of roughly 1756
$\mathrm{m^{3}}$ and 5 T, respectively [39]. Stellarator symmetry and five
field periods are assumed for all optimizations. The initial value of $\beta$
varies between optimizations as detailed in the next section. Zero bootstrap
current is assumed during the MHD calculations within the optimization loop.
The core feature of the optimizations is the minimization of an array of
$D_{11}^{*}$ with fuel-ion-relevant $v_{E}^{*}$ and $\nu^{*}$. The
$D_{11}^{*}$ values are calculated by DKES [40, 41], and a $3\times 3$ array
consisting of $\left|E_{r}\right|/v=\left\\{5.00\times 10^{-4},2.74\times
10^{-3},1.50\times 10^{-2}\right\\}~{}\mathrm{T}$ and
$\nu/v=\left\\{2.50\times 10^{-6},1.58\times 10^{-5},1.00\times
10^{-4}\right\\}~{}\mathrm{m^{-1}}$ is used. (These quantities, rather than
their normalized counterparts, are specified as targets in STELLOPT.) The
exact values of $\left|E_{r}\right|/v$ and $\nu/v$ seem relatively unimportant
— the upper and lower values for each quantity are chosen to be approximately
reactor-relevant bounds, and the middle values are the logarithmic means of
the upper and lower values. Notably, modifications of $B_{0}$, $R_{0}$, and
$\iota$ are relatively small in the optimizations, so explicitly targeting
$\left|E_{r}\right|/v$ and $\nu/v$ rather than $v_{E}^{*}$ and $\nu^{*}$ is
acceptable. Even nearly doubling $R_{0}$ at the beginning of the Configuration
1 optimization (see section 3.1) made no noticeable difference to the output
quality. In addition to $D_{11}^{*}$, we target $\epsilon_{\mathrm{eff}}=1\%$
(calculated by NEO [2]) in all optimizations except where noted in section
3.1. All neoclassical objectives are calculated for the
$\rho=r/a=\\{0.25,0.5,0.75\\}$ flux surfaces. MHD objectives such as plasma
aspect ratio, elongation, and others are also included in the optimizations.
Section 3 gives tabulated objective information for each optimization
presented.

Once optimized, the transport characteristics of each configuration are
analyzed using DKES and NTSS [42]. NTSS performs flux-surface-averaged, time-
dependent transport calculations (including momentum correction techniques
[43]) to determine self-consistent profiles for density, temperature, radial
electric field, bootstrap current, and other quantities on 51 radial grid
points for a given equilibrium. It should be emphasized that the equilibrium
itself is not updated during the transport calculations, but NTSS calculates
bootstrap current using the $D_{31}^{*}$ database provided by DKES and updates
the rotational transform profile by utilizing susceptance matrices [44].
Reactor volumes are rescaled to 1450 or 1900 $\mathrm{m^{3}}$ and on-axis
magnetic field strengths are rescaled to 5 or 6 T to achieve an alpha power of
600 MW because this approximately corresponds to a 3 GWth reactor. (We note
that the electron root is expected to be more difficult to achieve as the
magnetic field strength increases [3, 31], which may need to be taken into
account if our methods are used alongside those for turbulence optimization.)
Deuterium and tritium density profiles are fixed with on-axis values for each
in excess of $10^{19}$ $\mathrm{m^{-3}}$ and core gradients relatively flat in
$\rho$ — see figure 4 for details. NTSS uses the D-T cross-section to
calculate the birth rate of alpha particles, which for simplicity are assumed
to slow down on the flux surface of their birth. This determines the helium
density, and quasineutrality then determines electron density. We assume there
are no heavy impurities in the plasma. The temperature profiles for each
species are initialized with roughly parabolic shapes in $\rho$, and the
initial on-axis temperatures vary between analyses as described in the next
section. A very simple turbulent transport model is included based on fits to
W7-AS data. It takes the form [45]

$Q_{s}^{\mathrm{turb}}=-\chi_{\mathrm{turb}}n_{s}T^{\prime}_{s},$ (5)
$\Gamma_{s}^{\mathrm{turb}}=-D_{\mathrm{turb}}n^{\prime}_{s},$ (6)

where $Q_{s}^{\mathrm{turb}}$ is the radial energy flux of species $s$ due to
turbulence. Similarly to reference [27], we set
$\chi_{\mathrm{turb}}=0.0065(P/\mathrm{MW})^{3/4}/\left(n_{s}/10^{20}~{}\mathrm{m^{-3}}\right)~{}\mathrm{m^{2}s^{-1}}$
and
$D_{\mathrm{turb}}=0.0003(P/\mathrm{MW})^{3/4}/\left(n_{s}/10^{20}~{}\mathrm{m^{-3}}\right)~{}\mathrm{m^{2}s^{-1}}$
where $P=P_{\alpha}-P_{\mathrm{Br}}$, $P_{\alpha}$ is the power deposited by
fast ions, and $P_{\mathrm{Br}}$ is the power radiated due to Bremsstrahlung.
No external heating is considered. NTSS determines the full profile of the
radial electric field by solving a diffusion-like differential equation with
hysteretic behavior [46] that minimizes the generalized heat production rate
of the plasma [47]. This is necessary because the ambipolarity condition
generally produces either one or three possible solutions for $E_{r}$ [19, 48,
49] and a discontinuous $E_{r}$ profile is nonphysical. In the case of three
solutions, one is the ion root, one is the electron root, and one is an
unstable root that is not physically realized. We choose the “diffusion
coefficient” in the diffusion-like equation to be 2 $\mathrm{m^{2}/s}$ such
that the transition region from the electron root to the ion root is roughly 5
cm wide, which is reasonable in light of global neoclassical simulations [31].
Specifying a detailed startup scenario is beyond our scope, so we initialize
NTSS runs by instructing the software to “select” the electron root solution
if it exists. This situation is relevant for reactors that primarily utilize
electron heating for startup. NTSS steps forward in time until it converges on
a steady-state, self-consistent solution for the plasma profiles. We focus
only on making the electron root realizable from a neoclassical standpoint and
ensuring the optimized configurations can support the electron root during
steady-state operation. (Configurations with insufficiently optimized ion
confinement, such as scaled W7-X high-mirror, may be artificially seeded with
an electron root during high-density operation but will transition to an ion
root relatively quickly.)

##  3 Results

(a)

(b)

(c)

(d)

Figure 1: Toroidal cross-sections of the last closed flux surface equally
spaced in the toroidal angle across one half-field-period for (1(a))
Configuration 1 (section 3.1), (1(b)) Configuration 2 (section 3.2), (1(c))
Configuration 3 (section 3.3), and (1(d)) scaled W7-X high-mirror.

(a)

(b)

(c)

(d)

Figure 2: Scans of $D_{11}^{*}$ versus $\nu^{*}$ with various $v_{E}^{*}$ at
$\rho\approx 0.35$ for (2(a)) Configuration 1 (section 3.1), (2(b))
Configuration 2 (section 3.2), (2(c)) Configuration 3 (section 3.3), and
(2(d)) W7-X high-mirror. The upper and lower variational bounds on the
$D_{11}^{*}$ values are calculated by DKES and bracketed by the “$+$” symbols,
but these “error bars” are too small to see in most cases. The dotted lines
indicate $D_{11}^{*}$ for a tokamak with the same aspect ratio, rotational
transform, and (average) elongation as the given configuration for the same
$\rho$. Note that the light blue and purple curves, roughly corresponding to
fuel-ion-relevant $v_{E}^{*}$, are far more depressed in the optimized
configurations (2(a) – 2(c)) than in W7-X high-mirror. The characteristic gap
between the (fuel-ion-relevant) high-$v_{E}^{*}$ curves and (electron-
relevant) $v_{E}^{*}=0$ curves is also much larger in the optimized
configurations than in W7-X high-mirror. Figure 3: The geometric factor
$\epsilon_{\mathrm{eff}}$ for the configurations outlined in section 3 and
W7-X high-mirror. These calculations are performed on the optimized equilibria
without taking effects from the NTSS modeling (such as modifications to
bootstrap current, $\beta$, etc.) into account. The large values of
$\epsilon_{\mathrm{eff}}$ in Configuration 1 as $\rho\rightarrow 1$ are likely
numerical error — the high elongation of this configuration means many Boozer
modes are required to fully describe the outermost surfaces, leading any codes
that rely on Boozer coordinates to have questionable results there.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 4: NTSS results for the steady-state density and temperature profiles of Configuration 1 (4(a)+4(d)), Configuration 2 (4(b)+4(e)), and Configuration 3 (4(c)+4(f)). The profiles of radial electric field (4(g)), $L_{11}^{e}/L_{11}^{i}$ (4(h)), bootstrap current (4(i)), and $\iota$ (4(j)) are also included. Note that the results for Configuration 1 are suspect for $\rho\gtrsim 0.8$ — see the caption of figure 3 or the text in section 3.1 for an explanation. Table 1: Plasma parameters of the configurations described in section 3. The “Initial $P_{\alpha}$” and “Initial $P_{Br}$” entries refer to the alpha power and bremsstrahlung power, respectively, at the beginning of the NTSS simulation for the given configuration. The mirror ratio does not change throughout a simulation as the field strength is a user-specified parameter. All other entries refer to the end of the relevant NTSS simulation. $\tau_{E}$ is the energy confinement time, and $\tau_{E}^{\mathrm{ISS04}}$ is calculated using the ISS04 scaling expression [50]. Quantity | Configuration 1 | Configuration 2 | Configuration 3
---|---|---|---
Mirror Ratio (%) | 42.1 | 27.8 | 26.9
$\left<\beta\right>_{\mathrm{Vol}}$ (%) | 4.94 | 4.43 | 3.14
Initial $P_{\alpha}$ (MW) | 482.6 | 777.0 | 495.6
Final $P_{\alpha}$ (MW) | 489.2 | 628.9 | 601.5
Initial $P_{Br}$ (MW) | 43.8 | 38.3 | 33.5
Final $P_{Br}$ (MW) | 72.7 | 73.7 | 72.7
$\tau_{E}$ (s) | $2.13=1.30\tau_{E}^{\mathrm{ISS04}}$ | $2.25=1.40\tau_{E}^{\mathrm{ISS04}}$ | $2.34=1.23\tau_{E}^{\mathrm{ISS04}}$

###  3.1 Configuration 1

Table 2: Optimization objectives used to produce Configuration 1 (section 3.1). Note that stages 1 and 2 are performed using STELLOPT, while SciPy optimization routines coupled with VMEC and SFINCS are used for stage 3. Note also that the $L_{ij}^{s}$ values listed in stage 3 refer to the direct inputs for SFINCS, so they are unitless and have a different sign convention than the $L_{ij}^{s}$ in equations 1 and 2. All targets except the aspect ratio are evaluated on the $\rho=\\{0.25,0.5,0.75\\}$ flux surfaces. Objective | Target Value | Weight $\left(1/\sigma\right)$
---|---|---
Stage 1
Aspect Ratio | 20 | $10^{1}$
QP Symmetry Breaking | 0 | $10^{2}$
Stage 2
Aspect Ratio | 20 | $10^{1}$
QP Symmetry Breaking | 0 | $10^{1}$
$\epsilon_{\mathrm{eff}}$ | $1\%$ | $10^{2}$
Fuel-Ion-Relevant $D_{11}^{*}$ | 0 | $10^{2}$
Stage 3
$L_{11}^{e}$ | $\\{-1.65\times 10^{-4},-5.16\times 10^{-4},-1.51\times 10^{-3}\\}$ | $10^{0}$
$L_{11}^{i}$ | 0 | $10^{0}$
$L_{31}^{e}$, $L_{31}^{i}$ | 0 | $10^{0}$

This optimization begins with a volume-averaged $\beta$ of roughly $3.5\%$.
The $R_{00}$ Fourier component of the configuration is increased such that the
initial aspect ratio is approximately 20. (Note that this increases the
initial volume from the 1756 $\mathrm{m^{3}}$ quoted in section 2.) $R_{00}$
is then removed from the optimization space to keep the aspect ratio roughly
constant.

Unlike the other presented configurations, Configuration 1 is optimized in
several stages. In stage 1, quasipoloidal (QP) symmetry is targeted. This was
found to bring the configuration to a more favorable region of optimization
space (to a point), likely because lowering the value of the Boozer $B_{10}$
harmonic, the principle poloidal variation of $B$, tends to raise
$L_{11}^{e}/L_{11}^{i}$ [31]. In stage 2, the weight on QP symmetry is lowered
in favor of targets for $\epsilon_{\mathrm{eff}}$ and fuel-ion-relevant
$D_{11}^{*}$. This is where the majority of the $D_{11}^{*}$ optimization
occurs. In stage 3, SFINCS [51] is used to “polish” the magnetic configuration
by directly targeting thermal transport coefficients using differential
evolution implemented in SciPy [52]. The optimizer is allowed to alter only
the Fourier harmonics used by VMEC to describe the plasma, except for
$R_{00}$. SFINCS is directed to perform calculations for electrons and a
deuterium-tritium “hybrid” species. Density and temperature profiles are
chosen to be generically reactor-relevant and consistent with the choice of
$\beta$ for the configuration. The radial electric field is arbitrarily set to
$15~{}\mathrm{kV/m}$ on each of the three considered flux surfaces. The
$L_{11}^{e}$ values from the end of stage 2 are used as targets in an attempt
to hold radial electron transport roughly constant while driving $L_{11}^{i}$
(and therefore radial ion transport) to zero. The bootstrap current thermal
transport coefficients $L_{31}^{e}$ and $L_{31}^{i}$ are also targeted and
driven toward zero to minimize the bootstrap current of the final
configuration. Details of the targets used for this configuration can be found
in table 2.

Figure 1 shows the cross-sections for Configuration 1, which are highly
elongated (as expected when QP optimization is used). The elongation creates
some difficulties in performing neoclassical calculations for the outer flux
surfaces since the Boozer spectra required to describe these flux surfaces are
very large — the numerical values of neoclassical quantities are likely
imprecise for $\rho\gtrsim 0.8$ in Configuration 1. Figure 2 shows that in the
core, the fuel-ion-relevant $D_{11}^{*}$ are substantially reduced compared to
W7-X high-mirror while the electron-relevant $D_{11}^{*}$ are somewhat
elevated. The latter point is corroborated by the $\epsilon_{\mathrm{eff}}$
calculation shown in figure 3. These factors suggest this configuration will
tend to have a relatively large $L_{11}^{e}/L_{11}^{i}$ in the core. For the
NTSS simulation shown in figure 4, the reactor volume is scaled to
$1450~{}\mathrm{m^{3}}$ and the on-axis magnetic field is scaled to
$5~{}\mathrm{T}$. The simulation is initialized with an electron core
temperature of $15.60~{}\mathrm{keV}$ and a fuel ion core temperature of
$14.85~{}\mathrm{keV}$. The configuration has $L_{11}^{e}/L_{11}^{i}>1$ for
$0\leq\rho\lesssim 0.4$ and consequently a strong electron root in this
region. The bootstrap current is also small, and the configuration is
coincidentally Mercier stable [53]. However, $\iota^{\prime}<0$ throughout,
which makes the configuration vulnerable to neoclassical tearing modes [54],
the aspect ratio is likely too large to be reactor-relevant, and the mirror
ratio is very high (see table 1). Additionally, a final $P_{\alpha}$ of
$600~{}\mathrm{MW}$ could not be achieved due to the high elongation of the
configuration causing an errant divergence in the electric field near the last
closed flux surface when the reactor volume was raised above 1450
$\mathrm{m^{3}}$. While Configuration 1 is not a fully developed reactor
design, it demonstrates the conceptual effectiveness of separately optimizing
neoclassical ion transport (through $D_{11}^{*}$) and electron transport
(through $\epsilon_{\mathrm{eff}}$).

###  3.2 Configuration 2

Table 3: Optimization objectives used to produce Configuration 2 (section 3.2). All targets except aspect ratio are evaluated on the $\rho=\\{0.25,0.5,0.75\\}$ flux surfaces. Objective | Target Value | Weight $\left(1/\sigma\right)$
---|---|---
Aspect Ratio | 10.89 | $10^{1}$
$\epsilon_{\mathrm{eff}}$ | $1\%$ | $10^{2}$
Fuel-Ion-Relevant $D_{11}^{*}$ | 0 | $10^{2}$

This optimization begins with a volume-averaged $\beta$ of roughly $4.3\%$.
The $R_{00}$ Fourier harmonic is removed from the optimization space to
prevent large changes in the aspect ratio through the course of the
optimization. The philosophy of this optimization is to use a one-stage
approach simpler than that of Configuration 1; consequently, only
$\epsilon_{\mathrm{eff}}$, fuel-ion-relevant $D_{11}^{*}$, and aspect ratio
are targeted, as can be seen in table 3. The first two targets are the most
important and provide neoclassical optimization of the electrons and ions,
respectively. The aspect ratio target is less important since $R_{00}$ is not
included in the optimization space — it serves only to keep the aspect ratio
in the neighborhood of the initial configuration by regulating relatively
small fluctuations due to higher boundary modes.

From figure 1, Configuration 2 is significantly less elongated than
Configuration 1, with cross-sections much more similar to scaled W7-X high-
mirror. However, as seen in figure 2, the reduction of fuel-ion-relevant
$D_{11}^{*}$ was even more substantial than in Configuration 1. From figure 3,
$\epsilon_{\mathrm{eff}}$ was somewhat lower than in W7-X throughout the
volume, staying below the targeted $1\%$ for $0\leq\rho\lesssim 0.85$. This
results in lower electron-relevant $D_{11}^{*}$ values which compete with the
small fuel-ion-relevant $D_{11}^{*}$ to make $L_{11}^{e}/L_{11}^{i}$ a bit
smaller. For the NTSS calculations shown in figure 4, the reactor volume is
scaled to $1900~{}\mathrm{m^{3}}$ and the on-axis magnetic field is scaled to
$5~{}\mathrm{T}$. The initial core electron temperature is
$24.39~{}\mathrm{keV}$ and the initial core fuel ion temperature is
$23.21~{}\mathrm{keV}$. The figure shows that $L_{11}^{e}/L_{11}^{i}$ is only
slightly larger than 1 for Configuration 2 in the region $0\leq\rho\lesssim
0.35$. An electron root appears in the same region with a peak $E_{r}$ value
roughly half that of Configuration 1. The electric field profile for
$\rho\gtrsim 0.35$ has a somewhat unconventional shape, likely because the
$\epsilon_{\mathrm{eff}}$ profile is non-monotonic. While Configuration 2 has
a favorable $E_{r}$ profile and a lower mirror ratio than Configuration 1, it
is Mercier unstable, has $\iota^{\prime}<0$ for most flux surfaces, and has a
relatively large bootstrap current. It should also be noted that, as shown in
table 1, the input power required to achieve the NTSS results shown is higher
than the final output power of the configuration — while the displayed
profiles are self-consistent, steady-state solutions to the transport
equations, the system “relaxes” into them from profiles requiring extremely
high input power. This suggests that while it is possible to obtain an
electron root by optimizing fuel ion transport such that the ions (rather than
electrons) are the rate-determining species for the radial neoclassical flux,
it is very challenging to do so. The challenge may be attributed to the
$\sqrt{m_{s}}$ factor in equation 2.

###  3.3 Configuration 3

Table 4: Optimization objectives used to produce Configuration 3 (section 3.3). The neoclassical targets are evaluated on the $\rho=\\{0.25,0.5,0.75\\}$ flux surfaces. The rotational transform is evaluated on the $s=\rho^{2}=\\{0,0.5,1\\}$ flux surfaces. Objective | Target Value | Weight $\left(1/\sigma\right)$
---|---|---
Aspect Ratio | 10.89 | $4\times 10^{-2}$
Average Boundary Elongation | 1.87 | $10^{-1}$
Volume | $1755.76~{}\mathrm{m^{3}}$ | $10^{-2}$
$\epsilon_{\mathrm{eff}}$ | $1\%$ | $10^{2}$
Fuel-Ion-Relevant $D_{11}^{*}$ | 0 | $10^{2}$
Rotational Transform | $\\{8.14\times 10^{-1},8.70\times 10^{-1},9.47\times 10^{-1}\\}$ | $10^{0}$

This optimization begins with a volume-averaged $\beta$ of roughly $5.0\%$.
Unlike Configurations 1 and 2, $R_{00}$ is included in the optimization space
to allow for (small) changes to the aspect ratio, which is again one of the
optimization targets. Volume and boundary elongation targets are also included
in an attempt to prevent strong shaping like that found in Configuration 1.
Values of these parameters from the initial configuration are used as targets.
Rotational transform targets from the initial configuration are also included
to help keep $\iota^{\prime}>0$ as this helps simultaneously improve MHD and
neoclassical tearing mode stability. The same neoclassical targets were
included as in section 3.2 and were the most important ingredients in the
optimization. See table 4 for details.

Figure 1 shows that the last closed flux surface of Configuration 3 is very
similar to that of Configuration 2, and figure 2 shows similar $D_{11}^{*}$
optimization between Configurations 2 and 3. But unlike the other optimized
configurations, Configuration 3 has an $\epsilon_{\mathrm{eff}}$ of almost
exactly $1\%$ throughout the core. Because the fuel-ion-relevant $D_{11}^{*}$
are well-optimized and $\epsilon_{\mathrm{eff}}\approx 1\%$, we expect
$L_{11}^{e}/L_{11}^{i}>1$ in the core. For the NTSS simulation, the reactor
volume is scaled to $1900~{}\mathrm{m^{3}}$ and the on-axis magnetic field is
scaled to $6~{}\mathrm{T}$. The initial core electron temperature is
$21.46~{}\mathrm{keV}$ and the initial core fuel ion temperature is
$18.57~{}\mathrm{keV}$. Figure 4 shows that $L_{11}^{e}/L_{11}^{i}$ is
slightly greater than 1 for $0.15\lesssim\rho\lesssim 0.45$. Configuration 3
consequently has a weak ion root for $0\lesssim\rho\lesssim 0.05$ and an
electron root for $0.05\lesssim\rho\lesssim 0.45$. The peak strength of
$E_{r}$ in the electron root region is intermediate between that of
Configurations 1 and 2. The shape of the electric field profile in the ion
root region is similar to that of W7-X for the same values of $\rho$. The
difficulty in achieving an electron root deep in the core (which was observed
several times in our testing) can perhaps be explained by the fact that flux
surface shapes always tend to become elliptical as $\rho\rightarrow 0$ [55].
Confinement (omnigenity) of electrons places constraints on the contours of
$B_{\mathrm{min}}$, whereas confinement of fuel ions places constraints on the
contours of $B_{\mathrm{max}}$ [56]. If cross-sections are almost purely
elliptical, as is the case near the magnetic axis, it is difficult to optimize
the magnetic field such that the latter constraints are (approximately)
respected while the former are violated, which is necessary for the production
of an electron root. Far from the axis, where there is more freedom to shape
flux surfaces, this task becomes easier. In addition to an electron root in
most of the core, Configuration 3 has a relatively small bootstrap current,
nearly monotonically positive magnetic shear, and a reasonable power balance
(see table 1). However, it is Mercier unstable. Again, we find that optimizing
neoclassical ion and electron transport separately enables the production of a
steady-state electron root at reactor-relevant collisionalities, but the task
is challenging.

##  4 Conclusions

We have shown through several examples that neoclassical electron and fuel ion
transport can be optimized separately to raise the ratio
$L_{11}^{e}/L_{11}^{i}$ and produce an outward-pointing ambipolar electric
field in the core under reactor-relevant conditions. These configurations are
expected to expel heavy impurities from the core during steady-state
operation. However, we were unable to generate any completely satisfactory
configurations despite substantial effort. One configuration we produced had
stronger fuel-ion-relevant $D_{11}^{*}$ suppression than any of the
configurations presented here, but effects related to the slope of the
$D_{11}^{*}$ vs $\nu^{*}$ curves (like those in figure 2) and the resonant
electric field prevented an electron root from being realized. Further
development is needed before the approach outlined in this work is ready for
mainstream use.

A possible extension of this work involves coupling a modern, fast
neoclassical transport code such as KNOSOS [57, 58] to a modern optimization
framework such as SIMSOPT [59] and performing optimizations similar to those
presented in section 3. This will likely avoid some of the numerical
difficulties we encountered while attempting to improve the optimizations and
allow for consideration of additional neoclassical effects. Notably, KNOSOS
includes the tangential magnetic drifts (unlike DKES), which can be
substantial for some configurations [60] and alter the confinement of fuel and
impurity ions [61]. The tangential drifts can also make the electron root more
difficult to achieve in maximum-$J$ configurations [62]. Because the
maximum-$J$ property helps mitigate turbulence [63], it may be a useful
optimization target for future devices. A code that takes the tangential
drifts into account would therefore be helpful if an electron root is to be
targeted as well. Due to its speed, KNOSOS can also calculate thermal
transport coefficients very quickly. The final stage of the optimization in
section 3.1 successfully improved a configuration with optimized monoenergetic
transport coefficients by targeting thermal transport coefficients using the
(relatively slow) SFINCS code. This suggests that targeting thermal transport
coefficients throughout the optimization may be helpful — Lascas Neto and
coworkers [64] have recently used SFINCS to do so and obtained promising
results.

Another possible extension of this work involves developing a more theoretical
approach to ensuring excellent neoclassical confinement of fuel ions and
moderate confinement of electrons. Helander and coworkers [28] exploited the
fact that in the formulation of omnigenity, there are separate conditions for
zero average radial drift of marginally trapped particles (which pertains to
fuel ions) and deeply trapped particles (which pertains to electrons). They
derive an expression that allows the former conditions to be satisfied while
the latter are violated and optimize a stellarator configuration using this
expression as an objective. We suggest that further refinement of the
analytical proxy, or perhaps a blend of the analytical and computational
approaches, may provide the most reliable results for future stellarator
optimization efforts.

Once a more reliable algorithm for electron root optimization has been
devised, it will likely be beneficial to include a reasonably sophisticated
turbulence model in the post-optimization transport simulations. This will
indicate whether the large $E_{r}^{\prime}$ in the core reduces turbulent
fluxes. Recently devised simulation frameworks [65] should provide this
capability. Exploring the Pareto fronts between the electron root objective
and other physics and engineering objectives will also be helpful for future
design studies.

## Acknowledgments

We thank Stefan Buller, José Luis Velasco, Matt Landreman, Per Helander,
Michael Drevlak, Alan Goodman, Eduardo Lascas Neto, Eduardo Rodríguez, and
Katia Camacho Mata for their insights and assistance.

This work was supported by a Fulbright grant from the German-American
Fulbright Commission. This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research, under Award Number DE-SC0024386. This work has been
carried out within the framework of the EUROfusion Consortium, funded by the
European Union via the Euratom Research and Training Programme (Grant
Agreement No. 101052200 – EUROfusion). Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union or the European Commission. Neither the European Union nor
the European Commission can be held responsible for them. Computations were
performed on the Cobra and Raven HPC systems at the Max Planck Computing and
Data Facility.

## References

  * [1] Per Helander.  Theory of plasma confinement in non-axisymmetric magnetic fields.  Reports on Progress in Physics, 81(9):087001, 2018.
  * [2] V. V. Nemov, S. V. Kasilov, W. Kernbichler, and M. F. Heyn.  Evaluation of $1/\nu$ neoclassical transport in stellarators.  Physics of Plasmas, 6(12):4622–4632, 1999.
  * [3] C. D. Beidler, K. Allmaier, M. Yu Isaev, S. V. Kasilov, W. Kernbichler, G. O. Leitold, H. Maaßberg, D. R. Mikkelsen, S. Murakami, M. Schmidt, D. A. Spong, V. Tribaldos, and A. Wakasa.  Benchmarking of the mono-energetic transport coefficients - Results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS).  Nuclear Fusion, 51(7):076001, 2011.
  * [4] Darwin D.-M. Ho and Russel M. Kulsrud.  Neoclassical Transport in Stellarators.  Physics of Fluids, 30(2):442–461, 1987.
  * [5] C. D. Beidler, E. Harmeyer, F. Herrnegger, Yu. Igitkhanov, A. Kendl, J. Kisslinger, Ya. I. Kolesnichenko, V. V. Lutsenko, C. Nührenberg, I. Sidorenko, E. Strumberger, H. Wobig, and Yu. V. Yakovenko.  The Helias reactor HSR4/18.  Nuclear Fusion, 41(12):1759–1766, 2001.
  * [6] L. P. Ku, P. R. Garabedian, J. Lyon, A. Turnbull, A. Grossman, T. K. Mau, M. Zarnstorff, and the ARIES Team.  Physics Design for ARIES-CS.  Fusion Science and Technology, 54(3):673–693, 2008.
  * [7] P. H. Rutherford.  Impurity transport in the Pfirsch-Schlüter regime.  The Physics of Fluids, 17(9):1782–1784, 1974.
  * [8] W VII-A Team and NI Group.  Impurity transport in the wendelstein VII-A stellarator.  Nuclear Fusion, 25(11):1593–1609, 1985.
  * [9] M. Hirsch, J. Baldzuhn, C. Beidler, R. Brakel, R. Burhenn, A. Dinklage, H. Ehmler, M. Endler, V. Erckmann, Y. Feng, J. Geiger, L. Giannone, G. Grieger, P. Grigull, H.-J. Hartfuß, D. Hartmann, R. Jaenicke, R. König, H. P. Laqua, H. Maaßberg, K. McCormick, F. Sardei, E. Speth, U. Stroth, F. Wagner, A. Weller, A. Werner, H. Wobig, S. Zoletnik, and the W7-AS Team.  Major results from the stellarator Wendelstein 7-AS.  Plasma Physics and Controlled Fusion, 50(5):053001, 2008.
  * [10] J. M. García-Regaña, R. Kleiber, C. D. Beidler, Y. Turkin, H. Maaßberg, and P. Helander.  On neoclassical impurity transport in stellarator geometry.  Plasma Physics and Controlled Fusion, 55(7):074008, 2013.
  * [11] A. Mollén, M. Landreman, H. M. Smith, S. Braun, and P. Helander.  Impurities in a non-axisymmetric plasma: Transport and effect on bootstrap current.  Physics of Plasmas, 22(11):112508, 2015.
  * [12] J. L. Velasco, I. Calvo, S. Satake, A. Alonso, M. Nunami, M. Yokoyama, M. Sato, T. Estrada, J. M. Fontdecaba, M. Liniers, K. J. McCarthy, F. Medina, B. Ph Van Milligen, M. Ochando, F. Parra, H. Sugama, and A. Zhezhera.  Moderation of neoclassical impurity accumulation in high temperature plasmas of helical devices.  Nuclear Fusion, 57(1):016016, 2017.
  * [13] P. Helander, S. L. Newton, A. Mollén, and H. M. Smith.  Impurity Transport in a Mixed-Collisionality Stellarator Plasma.  Physical Review Letters, 118(15):155002, 2017.
  * [14] J. M. García-Regaña, C. D. Beidler, R. Kleiber, P. Helander, A. Mollén, J. A. Alonso, M. Landreman, H. Maaßberg, H. M. Smith, Y. Turkin, and J. L. Velasco.  Electrostatic potential variation on the flux surface and its impact on impurity transport.  Nuclear Fusion, 57(5):056004, 2017.
  * [15] S. Buller, H. M. Smith, P. Helander, A. Mollén, S. L. Newton, and I. Pusztai.  Collisional transport of impurities with flux-surface varying density in stellarators.  Journal of Plasma Physics, 84(4):905840409, 2018.
  * [16] Iván Calvo, Felix I. Parra, José Luis Velasco, J. Arturo Alonso, and J. M. Garcia-Regana.  Stellarator impurity flux driven by electric fields tangent to magnetic surfaces.  Nuclear Fusion, 58(12):124005, 2018.
  * [17] A. Mollén, M. Landreman, H. M. Smith, J. M. García-Regaña, and M. Nunami.  Flux-surface variations of the electrostatic potential in stellarators: impact on the radial electric field and neoclassical impurity transport.  Plasma Physics and Controlled Fusion, 60(8):84001, 2018.
  * [18] S. Buller, H. M. Smith, and A. Mollén.  Recent progress on neoclassical impurity transport in stellarators with implications for a stellarator reactor.  Plasma Physics and Controlled Fusion, 63(5):054003, 2021.
  * [19] Harry E. Mynick.  Effect of collisionless detrapping on nonaxisymmetric transport in a stellarator with radial electric field.  The Physics of Fluids, 26(9):2609–2615, 1983.
  * [20] D. E. Hastings, W. A. Houlberg, and K. C. Shaing.  The ambipolar electric field in stellarators.  Nuclear Fusion, 25(4):445–454, 1985.
  * [21] H. Maaßberg, C. D. Beidler, U. Gasparino, M. Romé, K. S. Dyabilin, N. B. Marushchenko, and S. Murakami.  The neoclassical “electron root” feature in the Wendelstein-7-AS stellarator.  Physics of Plasmas, 7(1):295–311, 2000.
  * [22] M. Yokoyama, H. Maaberg, C. D. Beidler, V. Tribaldos, K. Ida, T. Estrada, F. Castejon, A. Fujisawa, T. Minami, T. Shimozuma, Y. Takeiri, A. Dinklage, S. Murakami, and H. Yamada.  Core electron-root confinement (CERC) in helical plasmas.  Nuclear Fusion, 47(9):1213–1219, 2007.
  * [23] T. Klinger, A. Alonso, S. Bozhenkov, R. Burhenn, A. Dinklage, G. Fuchert, J. Geiger, O. Grulke, A. Langenberg, M. Hirsch, G. Kocsis, J. Knauer, A. Krämer-Flecken, H. Laqua, S. Lazerson, M. Landreman, H. Maaßberg, S. Marsen, M. Otte, N. Pablant, E. Pasch, K. Rahbarnia, T. Stange, T. Szepesi, H. Thomsen, P. Traverso, J. L. Velasco, T. Wauters, G. Weir, T. Windisch, and The Wendelstein 7-X Team.  Performance and properties of the first plasmas of Wendelstein 7-X.  Plasma Physics and Controlled Fusion, 59(1):014018, 2017.
  * [24] N. A. Pablant, A. Langenberg, A. Alonso, C. D. Beidler, M. Bitter, S. Bozhenkov, R. Burhenn, M. Beurskens, L. Delgado-Aparicio, A. Dinklage, G. Fuchert, D. Gates, J. Geiger, K. W. Hill, U. Höfel, M. Hirsch, J. Knauer, A. Krämer-Flecken, M. Landreman, S. Lazerson, H. Maaßberg, O. Marchuk, S. Massidda, G. H. Neilson, E. Pasch, S. Satake, J. Svennson, P. Traverso, Y. Turkin, P. Valson, J. L. Velasco, G. Weir, T. Windisch, R. C. Wolf, M. Yokoyama, D. Zhang, and the W7-X Team.  Core radial electric field and transport in Wendelstein 7-X plasmas.  Physics of Plasmas, 25(2):022508, 2018.
  * [25] M. Yoshinuma, K. Ida, M. Yokoyama, M. Osakabe, K. Nagaoka, S. Morita, M. Goto, N. Tamura, C. Suzuki, S. Yoshimura, H. Funaba, Y. Takeiri, K. Ikeda, K. Tsumori, and O. Kaneko.  Observation of an impurity hole in the Large Helical Device.  Nuclear Fusion, 49(6):062002, 2009.
  * [26] K. Ida, M. Yoshinuma, M. Osakabe, K. Nagaoka, M. Yokoyama, H. Funaba, C. Suzuki, T. Ido, A. Shimizu, I. Murakami, N. Tamura, H. Kasahara, Y. Takeiri, K. Ikeda, K. Tsumori, O. Kaneko, S. Morita, M. Goto, K. Tanaka, K. Narihara, T. Minami, I. Yamada, and the LHD Experimental Group.  Observation of an impurity hole in a plasma with an ion internal transport barrier in the Large Helical Device.  Physics of Plasmas, 16(5):056111, 2009.
  * [27] C. D. Beidler, M. Drevlak, J. Geiger, P. Helander, H. M. Smith, and Y. Turkin.  Reduction of neoclassical bulk-ion transport to avoid helium-ash retention in stellarator reactors.  Submitted to Nuclear Fusion, 2024.
  * [28] Per Helander, Alan G. Goodman, Craig D. Beidler, Michal Kuczyński, and Håkan M. Smith.  Optimised stellarators with a positive radial electric field, May 2024.  arXiv:2405.07085.
  * [29] D. Carralero, T. Estrada, E. Maragkoudakis, T. Windisch, J. A. Alonso, M. Beurskens, S. Bozhenkov, I. Calvo, H. Damm, O. Ford, G. Fuchert, J. M. García-Regaña, N. Pablant, E. Sánchez, E. Pasch, J. L. Velasco, and the Wendelstein 7-X Team.  An experimental characterization of core turbulence regimes in Wendelstein 7-X.  Nuclear Fusion, 61(9):096015, 2021.
  * [30] P. W. Terry.  Suppression of turbulence and transport by sheared flow.  Reviews of Modern Physics, 72(1):109–165, 2000.
  * [31] M. D. Kuczyński, R. Kleiber, H. M. Smith, C. D. Beidler, M. Borchardt, J. Geiger, and P. Helander.  Self-consistent, global, neoclassical radial-electric-field calculations of electron-ion-root transitions in the W7-X stellarator.  Nuclear Fusion, 64(4):046023, 2024.
  * [32] F. Wagner.  A quarter-century of H-mode studies.  Plasma Physics and Controlled Fusion, 49(12B):B1–B33, 2007.
  * [33] T. Estrada, T. Happel, L. Eliseev, D. López-Bruna, E. Ascasíbar, E. Blanco, L. Cupido, J. M. Fontdecaba, C. Hidalgo, R. Jiménez-Gómez, L. Krupnik, M. Liniers, M. E. Manso, K. J. McCarthy, F. Medina, A. Melnikov, B. van Milligen, M. A. Ochando, I. Pastor, M. A. Pedrosa, F. L. Tabarés, D. Tafalla, and the TJ-II Team.  Sheared flows and transition to improved confinement regime in the TJ-II stellarator.  Plasma Physics and Controlled Fusion, 51(12):124015, 2009.
  * [34] Samuel Lazerson, John Schmitt, Zhu Caoxiang, Joshua Breslau, and All STELLOPT Developers.  STELLOPT, 2020.
  * [35] S. P. Hirshman and J. C. Whitson.  Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria.  The Physics of Fluids, 26(12):3553–3568, 1983.
  * [36] Rainer Storn and Kenneth Price.  Differential Evolution – A Simple and Efﬁcient Heuristic for Global Optimization over Continuous Spaces.  Journal of Global Optimization, 11:341–359, 1997.
  * [37] Paul R. Garabedian.  Quasi-axially symmetric stellarators.  Proceedings of the National Academy of Sciences, 95(17):9732–9737, 1998.
  * [38] A. Dinklage, C. D. Beidler, P. Helander, G. Fuchert, H. Maaßberg, K. Rahbarnia, T. Sunn Pedersen, Y. Turkin, R. C. Wolf, T. Andreeva, S. Bozhenkov, B. Buttenschön, Y. Feng, J. Geiger, M. Hirsch, U. Höfel, M. Jakubowski, T. Klinger, J. Knauer, A. Langenberg, H. P. Laqua, N. Marushchenko, A. Mollén, U. Neuner, H. Niemann, E. Pasch, L. Rudischhauser, H. M. Smith, T. Stange, G. Weir, T. Windisch, D. Zhang, S. Äkäslompolo, A. Ali, J. Alcuson Belloso, P. Aleynikov, K. Aleynikova, J. Baldzuhn, M. Banduch, C. Beidler, A. Benndorf, M. Beurskens, C. Biedermann, D. Birus, M. Blatzheim, T. Bluhm, D. Böckenhoff, M. Borchardt, H. S. Bosch, L. G. Böttger, R. Brakel, Ch Brandt, T. Bräuer, H. Braune, K. J. Brunner, B. Brünner, R. Burhenn, R. Bussiahn, V. Bykov, A. Card, A. Carls, S. Degenkolbe, C. P. Dhard, M. Drevlak, P. Drewelow, A. Dudek, P. van Eeten, G. Ehrke, M. Endler, E. Erckmann, N. Fahrenkamp, J. H. Feist, J. Fellinger, O. Ford, B. Geiger, D. Gradic, M. Grahl, H. Grote, O. Grulke, P. Hacker, K. Hammond, H. J. Hartfuß, D. Hartmann, D. Hathiramani, B. Hein, S. Henneberg, C. Hennig, H. Hölbe, A. Hölting, H. Jenzsch, J. P. Kallmeyer, U. Kamionka, C. Killer, R. Kleiber, A. Könies, M. Köppen, R. König, P. Kornejew, R. Krampitz, J. Krom, M. Krychowiak, G. Kühner, S. Kwak, R. Laube, S. Lazerson, L. Lewerentz, J. F. Lobsien, J. Loizu Cisquella, A. Lorenz, S. Marsen, M. Marushchenko, P. McNeely, O. Mishchenko, B. Missal, T. Mönnich, D. Moseev, M. Nagel, D. Naujoks, C. Nührenberg, J. Nührenberg, J. W. Oosterbeek, M. Otte, N. Panadero, A. Pavone, V. Perseo, D. Pilopp, S. Pingel, G. Plunk, A. Puig Sitjes, F. Purps, J. Riemann, K. Riße, A. Rodatos, P. Rong, K. Rummel, T. Rummel, A. Runov, N. Rust, J. Schacht, F. Schauer, G. Schlisio, M. Schneider, W. Schneider, M. Schröder, T. Schröder, R. Schroeder, B. Shanahan, P. Sinha, C. Slaby, H. Smith, A. Spring, U. Stridde, J. Svensson, H. Thomsen, H. Trimino Mora, I. Vakulchyk, S. Valet, H. Viebke, R. Vilbrandt, F. Warmer, L. Wegener, T. Wegner, J. Wendorf, U. Wenzel, F. Wilde, E. Winkler, P. Xanthopoulos, J. Zhu, A. Zocco, F. Köster, A. Alonso, E. Ascasibar, E. Blanco, I. Calvo, A. Cappa, F. Castejon, A. de la Pena, H. Esteban, T. Estrada, J. Fontdecaba, J. García Regaña, M. Garcia-Munoz, C. Guerard, J. Hernandez Sanchez, C. Hidalgo, K. McCarthy, L. Pacios Rodriguez, N. Panadero Alvarez, M. Sanchez, B. Milligen, J. L. Velasco, B. Blackwell, J. Howard, A. Wright, A. Czarnecka, M. Kubkowska, W. Figacz, T. Fornal, A. Galkowski, M. Gruca, S. Jablonski, J. Kacmarczyk, N. Krawczyk, G. Pelka, L. Ryc, M. Scholz, J. Wolowski, F. Effenberg, O. Schmitz, T. Barbui, H. Frerichs, J. Green, T. Kremeyer, L. Stephey, V. Winters, G. Kocsis, T. Szepesi, G. Anda, G. Cseh, T. Ilkei, G. Náfrádi, S. Récsei, V. Szabó, T. Szabolics, Z. Szökefalvi-Nagy, S. Tulipán, S. Zoletnik, A. Krämer-Flecken, J. Assmann, W. Behr, O. Bertuch, W. Biel, V. Borsuk, S. Brezinsek, Y. Cai, A. Charl, G. Czymek, P. Denner, T. Dittmar, M. Dostal, Ph Drews, A. Freund, Y. Gao, N. Gierse, X. Han, F. Harberts, K. P. Hollfeld, D. Höschen, M. Knaup, A. Knieps, R. Koslowski, Th Krings, M. Lennartz, Y. Liang, Ch Linsmeier, S. Liu, O. Marchuk, Ph Mertens, O. Neubauer, G. Offermanns, A. Panin, M. Rack, D. Reiter, G. Satheeswaran, H. Schmitz, B. Schweer, A. Terra, J. Thomas, B. Unterberg, M. Vervier, E. Wang, N. Wang, Y. Wei, N. Pablant, P. Bolgert, D. Gates, P. Heitzenroeder, B. Israeli, S. Langish, D. Loesser, M. Mardenfeld, D. Mikkelsen, J. Mittelstaedt, H. Neilson, A. Reiman, M. Sibilia, P. Titus, M. Zarnstorff, H. Zhang, G. A. Wurden, G. Wurden, M. Landreman, F. Durodie, A. Goriaev, Y. Kazakov, J. Ongena, M. Vergote, T. Wauters, R. Alzbutas, G. Dundulis, T. Kaliatka, R. Karalevicius, M. Povilaitis, S. Rimkevicius, E. Urbonavicius, S. G. Baek, J. Maisano-Brown, J. Terry, M. Barlak, G. Gawlik, J. Jagielski, R. Koziol, P. Kraszewsk, K. Baumann, G. Dammertz, W. H. Fietz, G. Gantenbein, M. Huber, H. Hunger, S. Illy, J. Jelonnek, Th Kobarg, R. Lang, W. Leonhardt, M. Losert, A. Meier, D. Mellein, D. Papenfuß, A. Samartsev, T. Scherer, A. Schlaich, W. Spiess, M. Thumm, S. Wadle, J. Weggen, J. Boscary, M. Czerwinski, B. Heinemann, P. Junghans, B. Mendelevitch, R. Nocentini, S. Obermayer, G. Orozco, R. Riedl, P. Scholz, R. Stadler, B. Standley, J. Tretter, A. Vorkörper, M. Zilker, I. Abramovic, H. Brand, S. Paqay, J. Proll, B. Cannas, F. Pisano, L. Carraro, M. Zuin, B. Carvalho, A. da Silva, H. Fernandes, B. Goncalves, F. Chernyshev, M. Cianciosa, M. J. Cole, J. H. Harris, J. Lore, A. Lumsdaine, D. A. Spong, R. Citarella, V. Giannella, Ciupiński, G. Krzesinski, P. Marek, G. Claps, F. Cordella, A. Czermak, L. Haiduk, Z. Sulek, K. Czerski, A. da Molin, D. Ennis, D. Maurer, J. Schmitt, P. Traverso, M. Führer, T. Funaba, X. Huang, K. Ida, H. Kasahara, N. Kenmochi, S. Masuzaki, T. Morisaki, K. Ogawa, B. Peterson, S. Ryosuke, R. Sakamoto, S. Satake, Y. Suzuki, N. Tamura, K. Toi, H. Tsuchiya, T. Tsujimura, H. Yamada, I. Yamada, R. Yasuhara, M. Yokoyama, A. Gogoleva, L. Vela, A. Grosman, M. Houry, V. Moncada, T. Ngo, S. Renard, J. M. Travere, V. Huber, D. Kinna, S. Schmuck, C. P. von Thun, M. Turnyanskiy, A. Ivanov, A. Khilchenko, I. V. Shikhovtsev, W. Kasparek, M. Krämer, C. Lechte, R. Munk, B. Plaum, F. Remppel, H. Röhlinger, B. Roth, K. H. Schlüter, S. Wolf, A. Zeitler, M. Keunecke, F. Köchl, Y. Kolesnichenko, V. Lutsenko, J. Koshurinov, L. Lubyako, I. Ksiazek, F. Musielok, E. Pawelec, T. Kurki-Suonio, S. Sipliä, A. Lücke, A. Pieper, H. Schumacher, B. Wiegel, A. Zimbal, T. Mizuuchi, S. Murakami, F. Sano, W. Pan, X. Peng, J. Preinhaelter, J. Urban, J. Zajac, M. E. Puiatti, M. Romé, H. J. Roscher, J. Skodzik, D. Timmermann, M. Stoneking, N. Vianello, and R. Schrittwieser.  Magnetic configuration effects on the Wendelstein 7-X stellarator.  Nature Physics, 14(8):855–860, 2018.
  * [39] Samuel A. Lazerson, Alexandra Leviness, and Jorrit Lion.  Simulating fusion alpha heating in a stellarator reactor.  Plasma Physics and Controlled Fusion, 63(12):125033, 2021.
  * [40] S. P. Hirshman, K. C. Shaing, W. I. van Rij, C. O. Beasley, and E. C. Crume.  Plasma transport coefficients for nonsymmetric toroidal confinement systems.  Physics of Fluids, 29(9):2951–2959, 1986.
  * [41] W. I. van Rij and S. P. Hirshman.  Variational bounds for transport coefficients in three-dimensional toroidal plasmas.  Physics of Fluids B, 1(3):563–569, 1989.
  * [42] Y. Turkin, H. Maassberg, C. D. Beidler, J. Geiger, and N. B. Marushchenko.  Current control by ECCD for W7-X.  Fusion Science and Technology, 50(3):387–394, 2006.
  * [43] H. Maaßberg, C. D. Beidler, and Y. Turkin.  Momentum correction techniques for neoclassical transport in stellarators.  Physics of Plasmas, 16(7):072504, 2009.
  * [44] P. I. Strand and W. A. Houlberg.  Magnetic flux evolution in highly shaped plasmas.  Physics of Plasmas, 8(6):2782–2792, 2001.
  * [45] H. Ringler, U. Gasparino, G. Kühner, E. Maaßberg, H. Renner, F. Sardei, the W VII-AS Team, and the NBI Team.  Confinement studies on the Wendelstein VII-AS stellarator.  Plasma Physics and Controlled Fusion, 32(11):933–948, 1990.
  * [46] D. E. Hastings.  A differential equation for the ambipolar electric field in a multiple-helicity torsatron.  Physics of Fluids, 28(1):334–337, 1985.
  * [47] K. C. Shaing.  Stability of the radial electric field in a nonaxisymmetric torus.  Physics of Fluids, 27(7):1567–1572, 1984.
  * [48] E. F. Jaeger, D. A. Spong, and C. L. Hedrick.  Neoclassical Transportation in the ELMO Bumpy Torus.  Physical Review Letters, 40(13):866–869, 1978.
  * [49] D. E. Hastings.  Bifurcation phenomena and the radial electric field in a nonaxisymmetric plasma.  Physics of Fluids, 29(2):536–543, 1986.
  * [50] H. Yamada, J. H. Harris, A. Dinklage, E. Ascasibar, F. Sano, S. Okamura, J. Talmadge, U. Stroth, A. Kus, S. Murakami, M. Yokoyama, C. D. Beidler, V. Tribaldos, K. Y. Watanabe, and Y. Suzuki.  Characterization of energy confinement in net-current free plasmas using the extended International Stellarator Database.  Nuclear Fusion, 45(12):1684–1693, 2005.
  * [51] M. Landreman, H. M. Smith, A. Mollén, and P. Helander.  Comparison of particle trajectories and collision operators for collisional transport in nonaxisymmetric plasmas.  Physics of Plasmas, 21(4):042503, 2014.
  * [52] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul Van Mulbregt, SciPy 1.0 Contributors, Aditya Vijaykumar, Alessandro Pietro Bardelli, Alex Rothberg, Andreas Hilboll, Andreas Kloeckner, Anthony Scopatz, Antony Lee, Ariel Rokem, C. Nathan Woods, Chad Fulton, Charles Masson, Christian Häggström, Clark Fitzgerald, David A. Nicholson, David R. Hagen, Dmitrii V. Pasechnik, Emanuele Olivetti, Eric Martin, Eric Wieser, Fabrice Silva, Felix Lenders, Florian Wilhelm, G. Young, Gavin A. Price, Gert-Ludwig Ingold, Gregory E. Allen, Gregory R. Lee, Hervé Audren, Irvin Probst, Jörg P. Dietrich, Jacob Silterra, James T Webber, Janko Slavič, Joel Nothman, Johannes Buchner, Johannes Kulick, Johannes L. Schönberger, José Vinícius De Miranda Cardoso, Joscha Reimer, Joseph Harrington, Juan Luis Cano Rodríguez, Juan Nunez-Iglesias, Justin Kuczynski, Kevin Tritz, Martin Thoma, Matthew Newville, Matthias Kümmerer, Maximilian Bolingbroke, Michael Tartre, Mikhail Pak, Nathaniel J. Smith, Nikolai Nowaczyk, Nikolay Shebanov, Oleksandr Pavlyk, Per A. Brodtkorb, Perry Lee, Robert T. McGibbon, Roman Feldbauer, Sam Lewis, Sam Tygier, Scott Sievert, Sebastiano Vigna, Stefan Peterson, Surhud More, Tadeusz Pudlik, Takuya Oshima, Thomas J. Pingel, Thomas P. Robitaille, Thomas Spura, Thouis R. Jones, Tim Cera, Tim Leslie, Tiziano Zito, Tom Krauss, Utkarsh Upadhyay, Yaroslav O. Halchenko, and Yoshiki Vázquez-Baeza.  SciPy 1.0: fundamental algorithms for scientific computing in Python.  Nature Methods, 17(3):261–272, 2020.
  * [53] Claude Mercier.  A necessary condition for hydromagnetic stability of plasma with axial symmetry.  Nuclear Fusion, 1(1):47–53, 1960.
  * [54] C. C. Hegna and J. D. Callen.  Stability of bootstrap current-driven magnetic islands in stellarators.  Physics of Plasmas, 1(9):3135–3137, 1994.
  * [55] Matt Landreman and Wrick Sengupta.  Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates.  Journal of Plasma Physics, 84(6):905840616, 2018.
  * [56] John R. Cary and Svetlana G. Shasharina.  Omnigenity and quasihelicity in helical plasma confinement systems.  Physics of Plasmas, 4(9):3323–3333, 1997.
  * [57] J. L. Velasco, I. Calvo, F. I. Parra, and J. M. García-Regaña.  KNOSOS: A fast orbit-averaging neoclassical code for stellarator geometry.  Journal of Computational Physics, 418:109512, 2020.
  * [58] J. L. Velasco, I. Calvo, F. I. Parra, V. d’Herbemont, H. M. Smith, D. Carralero, T. Estrada, and the W7-X Team.  Fast simulations for large aspect ratio stellarators with the neoclassical code KNOSOS.  Nuclear Fusion, 61(11):116013, 2021.
  * [59] Matt Landreman, Bharat Medasani, Florian Wechsung, Andrew Giuliani, Rogerio Jorge, and Caoxiang Zhu.  SIMSOPT: A flexible framework for stellarator optimization.  Journal of Open Source Software, 6(65):3525, 2021.
  * [60] Keiji Fujita, S. Satake, M. Nunami, J. M. García-Regaña, J. L. Velasco, and I. Calvo.  Study on impurity hole plasmas by global neoclassical simulation.  Nuclear Fusion, 61(8):086025, 2021.
  * [61] J. L. Velasco, I. Calvo, E. Sánchez, and F. I. Parra.  Robust stellarator optimization via flat mirror magnetic fields.  Nuclear Fusion, 63(12):126038, 2023.
  * [62] Iván Calvo, Felix I. Parra, José Luis Velasco, and J. Arturo Alonso.  The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity.  Plasma Physics and Controlled Fusion, 59(5):055014, 2017.
  * [63] J. H. E. Proll, G. G. Plunk, B. J. Faber, T. Görler, P. Helander, I. J. McKinney, M. J. Pueschel, H. M. Smith, and P. Xanthopoulos.  Turbulence mitigation in maximum-J stellarators with electron-density gradient.  Journal of Plasma Physics, 88(1):905880112, 2022.
  * [64] E. Lascas Neto, R. Jorge, C. D. Beidler, and J. Lion.  Electron root optimisation for stellarator reactor designs, May 2024.  arXiv:2405.12058.
  * [65] A. Bañón Navarro, A. Di Siena, J. L. Velasco, F. Wilms, G. Merlo, T. Windisch, L. L. LoDestro, J. B. Parker, and F. Jenko.  First-principles based plasma profile predictions for optimized stellarators.  Nuclear Fusion, 63(5):054003, 2022.