# Energetic particle loss mechanisms in reactor-scale equilibria close to
quasisymmetry

E. J. Paul1, A. Bhattacharjee1, M. Landreman2, D. Alex3, J. L. Velasco4, and
R. Nies1 1Department of Astrophysical Sciences, Princeton University, 4 Ivy
Lane, Princeton NJ 08544, USA
2Institute for Research in Electronics and Applied Physics, University of
Maryland, College Park MD 20742, USA
3Department of Aerospace and Ocean Engineering, Virginia Tech, 460 Old Turner
Street, Blacksburg VA 24060, USA
4Laboratorio Nacional de Fusion, CIEMAT, Madrid, Spain<EMAIL_ADDRESS>
###### Abstract

Collisionless physics primarily determines the transport of fusion-born alpha
particles in 3D equilibria. Several transport mechanisms have been implicated
in stellarator configurations, including stochastic diffusion due to class
transitions, ripple trapping, and banana drift-convective orbits. Given the
guiding center dynamics in a set of six quasihelical and quasiaxisymmetric
equilibria, we perform a classification of trapping states and transport
mechanisms. In addition to banana drift convection and ripple transport, we
observe substantial non-conservation of the parallel adiabatic invariant which
can cause losses through diffusive banana tip motion. Furthermore, many lost
trajectories undergo transitions between trapping classes on longer time
scales, either with periodic or irregular behavior. We discuss possible
optimization strategies for each of the relevant transport mechanisms. We
perform a comparison between fast ion losses and metrics for the prevalence of
mechanisms such as banana-drift convection [1], transitioning orbits, and wide
orbit widths. Quasihelical configurations are found to have natural protection
against ripple-trapping and diffusive banana tip motion leading to a reduction
in prompt losses.

##  1 Introduction

Magnetic confinement reactors must confine both the thermal and energetic
particle populations as excessive losses of fusion-born alpha particles can
damage material structures and reduce the heating of the bulk plasma. When
three-dimensional (3D) magnetic fields are introduced, such as in a
stellarator or rippled tokamak, collisionless particle orbits are no longer
automatically confined and weakly collisional transport is generally enhanced.

Mitigating alpha particle losses is one of the critical issues in stellarator
reactor design. The alpha energy confinement impacts the power balance: an
alpha heating efficiency of at least 90% [2, 3] or 95% [4] is often assumed in
stellarator reactor studies. Unconfined alpha orbits can also lead to
localized heating of divertor plates or other plasma-facing surfaces, leading
to blistering, erosion, and degradation of mechanical properties [5]. Although
the alpha confinement was a target in the ARIES-CS design, the alpha losses
contributed 5 MW/m2 to the maximum divertor heat flux [6]. In a four-field-
period HSR4/18 Helias reactor study [7], the alpha particle energy loss was
reduced to 2.5% of the heating power [8]. While small enough not to impact
power balance significantly, these losses may still be a concern due to wall
loading. At fixed power, operating a stellarator reactor at higher field
strength requires a reduction in density and an increase in temperature [3].
While the decrease in beta may alleviate stability and equilibrium limits, the
increase in the slowing-down time necessitates stricter requirements on the
alpha confinement.

The confinement of fusion-born alpha particles is largely determined by
collisionless physics. Comparisons of collisionless with collisional alpha
particle losses have found that pitch-angle scattering effects are
insignificant for losses before the slowing-down time of $\approx 0.05$
seconds in a W7-X reactor configuration [9]. Generally, the difference between
collisionless and collisional calculations is configuration dependent [10].
Radial electric fields are not likely to have a large impact on the fusion-
born alpha particle dynamics long before their slowing down, given that
potential differences on the order of MV are required to change the particle
energy by order unity. While some calculations have accounted for the
ambipolar electric field [11], its calculation introduces additional
complications into the configuration comparisons through the choice of
profiles. Alpha particles can also be transported due to their interaction
with Alfvénic instabilities [12, 13]. We will not consider this mechanism in
this work, as this requires knowledge of the thermal and fast ion profiles and
more detailed modeling of the mode structure of the instabilities. In this
work we focus on the collisionless dynamics without electric fields or
perturbations in order to cleanly distinguish the orbital dynamics between
configurations without knowledge of the thermal and fast ion profiles.

Many potential collisionless loss mechanisms have been identified in 3D
magnetic fields. If 3D perturbations are large enough that local wells form
along magnetic field lines, local ripple trapping can occur, generally leading
to unconfined orbits. Particles trapped in the main toroidal field, analogous
to banana orbits in a tokamak, can exhibit banana diffusion and convection
[14, 15]. As trajectories transition between trapping in the ripple field and
the main toroidal field, collisionless trapping and detrapping leads to
substantial transport in both rippled tokamaks [16] and stellarators [17]. In
strongly 3D configurations where multiple trapping can occur, stochastic
transitioning behavior is present, leading to diffusive transport [18, 19,
20]. In-surface precessional drift motion can also alter the confinement
properties of fast ions. If a particle precesses sufficiently on a surface,
the perturbation to the radial drift is phase-mixed, reducing transport. On
the other hand, if a particle remains localized near a particular
perturbation, a directed radial drift may result, termed banana drift
convection transport [21]. Similarly, if the bounce or passing motion is in
resonance with a perturbation in the equilibrium, a directed radial drift can
occur [22, 23].

Previous studies have discussed some aspects of fast ion loss mechanisms in 3D
configurations, such as the distribution of losses on the first wall [19, 24,
25] or in velocity space [26, 27, 10]. Comparisons of guiding center losses
with banana drift convection metrics [10, 1] and theoretical predictions from
diffusion coefficients [18] have been made. However, a more quantitative
analysis of the dominant loss channels has not been performed.

In this work, we perform a detailed analysis of guiding-center confinement in
3D magnetic fields to quantify the relevant loss mechanisms on prompt and non-
prompt time scales. We focus our attention on configurations close to
quasisymmetry as it eases the classification of orbit topology. (For example,
the mirror points of trapped orbits in quasi-isodynamic configurations may
differ from those in a quasisymmetric configuration.) By highlighting the most
pertinent loss mechanisms in each configuration, we can identify avenues for
improving collisionless confinement. Furthermore, a deeper understanding of
fast ion orbits in 3D magnetic fields may indicate possible resonant
interactions with external perturbations [28, 23], such as MHD modes or error
fields. Finally, more effective optimization strategies can be elucidated by
isolating the most critical loss channels. Recent work has indicated that
magnetic fields with precise quasisymmetry exist [29, 30], yielding excellent
guiding center confinement. However, relaxing the requirement of precise
symmetry may ease the accommodation of other constraints, such as engineering
requirements or microstability.

Section 2 provides an overview of fast ion loss mechanisms in 3D fields and
classification strategies. In Section 3 we describe the guiding center
calculations and the configurations under consideration. Section 4 describes
the classification results for the configurations close to quasiaxisymmety and
quasihelical symmetry, including relevant example trajectories. In light of
these observations, in Section 5 we discuss equilibrium properties that reduce
the identified loss mechanisms. In Section 6 we summarize the results, and in
Section 7 we conclude.

##  2 Loss mechanisms and classification

We analyze the lost particles from guiding-center calculations to elucidate
the loss mechanisms. The positions of all turning points (where $v_{\|}=0$)
along the trajectory are identified. We define a half-bounce segment as the
segment of the entire trajectory between two consecutive turning points and a
bounce segment as two consecutive half-bounce segments such that the particle
returns to approximately the same position to form a closed orbit. Given the
position of each turning point, we categorize the orbit type for each half-
bounce segment as banana trapped (Section 2.1), ripple trapped (Section 2.2),
or barely trapped (Section 2.3). If a particle never reflects, it is
categorized as passing (Section 2.4). Below we discuss the classification of
each orbit type and their corresponding loss mechanisms.

(a)

(b)

(c)

Figure 1: Example ARIES-CS bounce segments in the (a) banana, (b) ripple, and
(c) barely-trapped classes are shown in the poloidal cross-section. Black
stars indicate bounce points. Black dashed lines indicate the initial magnetic
surface and plasma boundary.

###  2.1 Banana trapping

We use the term banana to refer to a particle trapped in the primary well
along a field line. This orbit type is analogous to the trapped particles in a
tokamak whose trajectories are banana-shaped in the
$(\sqrt{s}\cos\theta,\sqrt{s}\sin\theta)$ plane. Here $s=\psi/\psi_{0}$ is the
normalized toroidal flux and $\theta$ and $\zeta$ are the Boozer poloidal and
toroidal angles, respectively. Similarly, for a quasisymmetry magnetic field
$B(s,\chi)$ where $\chi=M\theta-N\zeta$, analogous banana motion appears in
the $(\sqrt{s}\cos\chi,\sqrt{s}\sin\chi)$ plane. We assume throughout that the
minimum of the field strength occurs at $\chi=0$. In perfect quasisymmetry,
the banana motion closes in the $(\sqrt{s}\cos\chi,\sqrt{s}\sin\chi)$ plane
without any net radial drift. Deviation from quasisymmetry or omnigeneity
introduces perturbations that allow for transport similar to that in a rippled
tokamak.

A typical banana orbit is centered about the midplane (has an average value of
$\chi$ close to zero), and the trajectory does not make a full transit through
the field strength variation (the change in $\chi$ between bounces,
$\Delta\chi$, is less than $2\pi$). We therefore compare the mean value of
$\chi$ along the half-bounce segment, $\mathrm{mean}(\chi)\in[-\pi,\pi)$, to
the position of the field minimum, $\chi=0$. We classify banana half-bounce
segments as those not satisfying the conditions for ripple trapping or barely
trapping as described in Sections 2.2 and 2.3. See Figure 1(a) for an example
of a banana orbit segment from ARIES-CS. Throughout the text, banana segments
are indicated by green.

####  2.1.1 Wide bananas

If the width of banana orbits is sufficiently large, particle orbits may
intersect the wall even without symmetry breaking. The timescale of this
transport mechanism, $\tau_{\mathrm{loss}}$, scales with the bounce time,
$\tau_{\mathrm{b}}\sim 10^{-5}$ seconds. With symmetry breaking, wide orbits
can lead to rapid diffusion, given that the orbit width sets the typical step
size for banana diffusion. Even in a precisely quasisymmetric field, a small
number of fast ions may be unconfined due to these effects [29].

####  2.1.2 Banana-drift convection

We will use the term banana-drift convection to refer to transport on
trajectories that do not precess rapidly compared to their radial drift. Since
these particles remain confined in toroidal angle, they may experience a
directed radial drift and be quickly lost, see Figure 17. (The term
superbanana has been used in the literature to denote this drift convective
motion [1], although we avoid this term since it can also be used to refer to
particles that are ripple-trapped during some segment of their orbit [31].)
The timescale for this transport is associated with the bounce-averaged radial
drift frequency, $\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla s}$, as
$\tau_{\mathrm{loss}}\sim(1-s_{0})/\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}$, where $1-s_{0}$ is the distance in normalized flux over which the
particle is transported from initial surface $s_{0}$. This type of transport
can also occur for ripple-trapped and barely-trapped particles. In the case of
a strong radial electric field, the electric and magnetic precession drifts
can balance each other, leading to the banana-drift convection condition [32].
However, the electric precession drift is typically not significant near the
fusion-born alpha birth energy.

A parameter that has been used in the literature for classifying drift
convection is [33],

$\gamma_{\mathrm{c}}=\frac{2}{\pi}\mathrm{arctan}\left(\frac{\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}}{|\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla\alpha}|}\right).$ (1)

Here $\alpha=\theta-\iota(s)\zeta$ is a field-line label,
$\textbf{v}_{\mathrm{m}}$ is the magnetic drift,

$\textbf{v}_{\mathrm{m}}=\frac{v_{\perp}^{2}}{2\Omega}\hat{\textbf{b}}\times\nabla\ln(B)+\frac{v_{\|}^{2}}{\Omega}\hat{\textbf{b}}\times\bm{\kappa},$
(2)

$v_{\|}$ is the parallel velocity, $v_{\perp}$ is the perpendicular velocity,
$\Omega=qB/m$ is the gyrofrequency, $q$ is the charge, $m$ is the mass,
$\hat{\textbf{b}}=\textbf{B}/B$ is the unit vector in the direction of the
magnetic field, and $\bm{\kappa}=\hat{\textbf{b}}\cdot\nabla\hat{\textbf{b}}$.
In (1), overlines indicate averages of the lowest-order bounce motion along a
field line,

$\overline{A}=\frac{\oint\frac{dl}{v_{\|}}\,A}{\oint\frac{dl}{v_{\|}}},$ (3)

where $l$ measures length along a field line and integration is performed
along the closed orbit between bounce points. The quantity
$\gamma_{\mathrm{c}}$ takes values in $[-1,1]$, with large positive values
indicating outward convective transport. Small values of
$|\gamma_{\mathrm{c}}|$ promote closure of the contours of the adiabatic
invariant,

$J_{\|}=\oint dl\,v_{\|}.$ (4)

Since $\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}/\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla\alpha}=\left(\partial
J_{\|}/\partial\alpha\right)/\left(\partial J_{\|}/\partial s\right)$, small
values of $|\gamma_{\mathrm{c}}|$ indicate small angles between $J_{\|}$
contours and flux surfaces.

To identify banana-drift convection among lost trajectories, we estimate the
$\gamma_{\mathrm{c}}$ parameter (1) as,

$\gamma_{\mathrm{c}}\approx\frac{2}{\pi}\tan^{-1}\left(\frac{\Delta
s}{|\Delta\theta-\mathrm{mean}(\iota)\Delta\zeta|}\right),$ (5)

where $\Delta$ represents the difference in the coordinate along the bounce
segment and $\mathrm{mean}(\iota)$ is the mean value of $\iota$ along the
bounce segment. Each bounce segment is classified as drift convective (DC) if
it satisfies $|\gamma_{\mathrm{c}}|>0.2$ [1]. The choice of this critical
value is somewhat arbitrary, as small positive values of $\gamma_{\mathrm{c}}$
may allow for convection along $J_{\|}$ contours, although on a longer
timescale. A DC segment can be in the ripple, banana, or barely-trapped class.
However, we will focus on the distinction between DC and non-DC banana bounce
segments due to the prevalence of DC transport among ripple-trapped
trajectories and the small number of barely-trapped trajectories. A bounce
segment classified as a non-DC banana indicates that diffusive banana tip
motion is dominant, resonant banana motion is present, or $J_{\|}$ contours
align well with flux surfaces and convective radial transport is relatively
slow.

####  2.1.3 Resonant banana-drift convection

On the drift timescale, convective loss of bananas can be driven by the
resonance between the bounce motion and toroidal precession [15, 34], see
Figure 13. Here the net drift in the toroidal angle, $\Delta\zeta$, between
successive bounces (an almost-closed orbit) satisfies the condition,

$l\Delta\zeta=k\frac{2\pi}{N_{P}},$ (6)

where $N_{P}$ is the number of field periods and $l$ and $k$ are integers such
that after $l$ full bounce orbits, the bounce tip returns to the same location
in toroidal angle. Since the same perturbation phase is applied upon
successive bounce orbits, a directed radial drift can occur if a perturbation
with the same mode numbers is present in the equilibrium. Similar to banana-
drift convection, the timescale for this transport is associated with the
bounce-averaged radial drift frequency,
$\tau_{\mathrm{loss}}\sim(1-s_{0})/\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}$. This transport mechanism has been suggested to be present in stellarator
equilibria [35], although there has been little numerical or experimental
evidence. A special case of this resonance condition with $l=1$, $k=0$
corresponds with banana-drift convective orbits, discussed in Section 2.1.2.
If the resonance condition is satisfied for $l\neq 1$, then the shape of the
magnetic field strength well may change substantially on successive bounce
orbits, leading to possible non-conservation of $J_{\|}$.

####  2.1.4 Banana-drift diffusion

When several resonances in the banana-drift motion begin to overlap, diffusive
transport of banana tips begins to occur [14, 15], see Figure 8. Here the
criterion for resonant overlap occurs when the radial displacement of the
banana tips, $\Delta s$, changes the net toroidal drift, $\Delta\zeta$, during
a full bounce orbit,

$\left(\Delta\zeta\right)^{\prime}(s)\Delta s\gtrsim\frac{2\pi}{N_{P}}.$ (7)

This resonant overlap condition is more likely to occur for wide banana
orbits. If strong overlap is present such that diffusive motion arises, the
timescale for loss scales as
$\tau_{\mathrm{loss}}\sim\left(1-s_{0}\right)^{2}/D$. Using the radial
magnetic drift $\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla s}$ and the
bounce time $\tau_{\mathrm{b}}$ as characteristic scales, we can postulate the
diffusion coefficient to scale as
$D\sim\left(\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}\right)^{2}\tau_{\mathrm{b}}$.

While banana-drift convection arises due to trajectories that follow $J_{\|}$
contours that are misaligned with surfaces, banana-drift diffusive yields
stochastic motion due to non-conservation of $J_{\|}$. This feature will be
discussed further in Section 5.2. To distinguish this mechanism, we evaluate
the parallel adiabatic invariant (4) if the particle remains within the same
trapping class for the full bounce orbit. Integration is performed along the
GC trajectory rather than along a field line between successive bounces.

###  2.2 Ripple trapping

If a given perturbation to quasisymmetry is large enough, local ripple wells
in the field strength can form along a field line. Since ripple-trapped
particles tend to be localized in poloidal and toroidal angles, the averaged
radial drift typically does not vanish over a bounce period. For a
configuration close to QS, the dominant contribution to the magnetic drift
comes from the unperturbed quasisymmetric field, leading to
$\textbf{v}_{\mathrm{m}}\cdot\nabla s\propto\partial B(s,\chi)/\partial\chi$.
If the ripple well is localized away from the field maxima or minima on a
surface, this can lead to rapid radial convection. Due to their localization,
ripple-trapped trajectories often have features in common with banana-drift
convection trajectories, including large values of $|\gamma_{\mathrm{c}}|$,
see Figure 6. Similar to banana-drift convection, the timescale for this
transport is associated with the bounce-averaged radial drift frequency,
$\tau_{\mathrm{loss}}\sim(1-s_{0})/\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla
s}$.

A ripple-trapped segment is characterized by localization in $\chi$ or small
values of $\Delta\chi$ along a half-bounce segment. To distinguish a ripple-
trapped segment from a deeply-trapped banana segment, we also compare the mean
value of $\chi$ along the half-bounce segment,
$\mathrm{mean}(\chi)\in[-\pi,\pi)$, to the location of the field minimum,
$\chi=0$. Specifically we classify a ripple-trapped half-bounce segment as
satisfying the inequalities $\mathrm{mean}(\chi)/(2\pi)>0.04$ and
$\Delta\chi/(2\pi)<1/8$. (In ITER, we instead use the condition
$\Delta\chi/(2\pi)<1/30$ or ($\mathrm{mean}(\chi)/(2\pi)>0.04$ and
$\Delta\chi/(2\pi)<1/8$) due to the prevalence of deeply-trapped banana
trajectories.) See Figure 1(b) for an example of a ripple-trapped bounce
segment in ARIES-CS. Throughout the text, ripple-trapped segments are
indicated by blue.

###  2.3 Barely trapping

We use the term barely trapped to refer to a trapped-particle trajectory that
can pass through a full period of the field strength variation,
$\Delta\chi=2\pi$, without mirroring. This behavior occurs because of the
variation of the field strength maxima with respect to the field line label or
magnetic surface label. We classify barely-trapped half-bounce segments as
satisfying the inequality $\Delta\chi/(2\pi)>1.25$ to prevent the
misclassification of banana-trapped segments as barely-trapped segments. See
Figure 1(c) for an example of a barely-trapped half-bounce segment in ARIES-
CS. Throughout the text, barely-trapped segments are indicated by red.

Many barely-trapped trajectories nearly close in the
$(\sqrt{s}\cos\chi,\sqrt{s}\sin\chi)$ plane during a bounce segment, see
Figure 18. If a barely-trapped trajectory makes few toroidal transits between
bounces, the radial drift may not vanish under an orbit average. This
mechanism is similar to banana-drift convection. An example can be seen in
Figure 18.

###  2.4 Passing trajectories

Passing trajectories do not mirror. If the radial width of such a passing
orbit is sufficiently large, it may intersect the plasma boundary [36],
similar to the case of wide bananas. Even if the orbit width is small, passing
orbits can be transported due to resonant phenomena discussed in Section
2.4.1.

####  2.4.1 Resonant passing trajectories

While passing particles on irrational field lines at low energy stay confined
since the net radial drift vanishes on a time average, passing particles on
rational trajectories may experience a net drift due to perturbations that
resonate with the orbit periodicity [37, 38, 39, 28]. The condition for such a
resonance to occur is,

$l=k\frac{\Delta\theta}{2\pi},$ (8)

where $\Delta\theta$ is the change in the poloidal angle after a transit
through one toroidal field period, $2\pi/N_{P}$. After a transit through $k$
field periods, $l$ full poloidal turns occur. At low energy, resonance will
occur for passing particles near a rational surface with $\iota=N_{P}l/k$. At
higher energy, such resonances can persist in the neighborhood of a rational
surface due to the magnetic drifts, see Figure 13. In analogy with banana-
drift diffusion, stochastic transport can occur if these resonances begin to
overlap [38, 39, 23].

###  2.5 Transitions between classes

We use the term class to refer to the collection of particles trapped within a
given well of the magnetic field strength. Since the local maximum of the
field strength may vary between field lines when symmetry is broken, a
particle may become collisionlessly detrapped or transition between trapping
classes through its cross-field drifts. For example, banana-trapped particles
can become entrapped in ripple wells or locally detrapped and make a transit
through a full period of $\chi$. Since these entrapping and detrapping
processes are associated with a radial displacement, it has been suggested
that this leads to diffusive transport [19, 18]. This irregular transitioning
behavior can be seen in Figure 11. The characteristic timescale for this
diffusive loss scales as $\tau_{\mathrm{loss}}\sim\left(1-s_{0}\right)^{2}/D$.
Using the radial step size associated with the transition, $\delta s$, and the
timescale associated with the separatrix crossing, $\tau_{\mathrm{cross}}$
(which scales with the precession frequency
$\overline{\textbf{v}_{\mathrm{m}}\cdot\nabla\alpha}$), as characteristic
scales, the characteristic diffusion coefficient scales as $D\sim(\delta
s)^{2}P/\tau_{\mathrm{cross}}$. Here $P$ is the transition probability.

A trajectory is classified as ripple-trapped, banana-trapped, or barely-
trapped between two bounce points. Transitioning trajectories have segments
that pass through more than one of these three classes or between barely
trapped segments that pass through a different number of field periods. The
number of transitions is evaluated along each trajectory. Transitioning
behavior can also be distinguished by jumps in the parallel adiabatic
invariant (4) associated with crossing a separatrix.

##  3 Guiding center calculations

###  3.1 Numerical considerations

We perform integration of the guiding center (GC) trajectories of fusion-born
alpha particles (3.52 MeV) for 0.2 seconds, which corresponds with about three
slowing-down times based on the density and temperature profiles of ARIES-CS
[24]. The GC approximation is likely appropriate for the parameters under
consideration, given that the value of
$\rho_{*}=v_{\mathrm{th}}/(aqB/m)\approx 0.03$ for ARIES-CS fusion-born alpha
parameters, where $v_{\mathrm{th}}=\sqrt{2E/m}$ is the thermal velocity,
$E=3.52$ MeV is the energy, and $a$ is the effective minor radius.

The GC equations obtained from the Littlejohn Lagrangian [40] have been
implemented in the SIMSOPT code [41]. The equations of motion are integrated
in Boozer coordinates, $(s,\theta,\zeta)$ [42]. A benchmark between the
guiding center integration routines in SIMSOPT, BEAMS3D [43], and ANTS [27]
has been performed. Despite slight differences in the GC equations adopted in
the three codes (BEAMS3D and ANTS integrate in cylindrical coordinates and
neither uses the Langrangian formalism), the loss fractions computed for the
three codes typically agree within 1%. Integration is performed with the
Runge-Kutta Dormand-Prince 5 method provided by the BOOST library.

Given an initial energy $E_{0}$ and coordinates
($v_{\|,0},s_{0},\theta_{0},\zeta_{0}$), the initial magnetic moment
$\mu_{0}=(E_{0}/m-v_{\|,0}^{2}/2)/B(s_{0},\theta_{0},\zeta_{0})$ is computed
and passed to the integration routine as a parameter held fixed throughout.
The error in energy conservation at any point in time can be evaluated by
comparing $E=\frac{1}{2}mv_{\|}^{2}+m\mu_{0}B(s,\theta,\zeta)$ with $E_{0}$.
Alternatively, the error in magnetic moment conservation at any point can be
evaluated by comparing $\mu=(E_{0}/m-v_{\|}^{2}/2)/B(s,\theta,\zeta)$ with
$\mu_{0}$. Energy and magnetic moment are conserved to within a fractional
error of $\approx 10^{-3}$ throughout the timescales of interest.

Integration in Boozer coordinates as opposed to cylindrical coordinates eases
the analysis of the dynamics with respect to magnetic surfaces. One
complication of integration in flux coordinates is the presence of the
coordinate singularity at $s=0$. (While a similar complication also exists in
cylindrical coordinates, the singularity typically lies outside the domain of
interest.) Therefore, we do not analyze the orbits that pass through the
$s=0.01$ surface. The number of particles that are discarded for this reason
are tabulated in Tables 2 and 3.

###  3.2 Configurations

We consider a set of equilibria (Table 1) which are scaled to the same volume-
averaged field strength (5.86 T) and volume (444 m3) as ARIES-CS. We use the
same scaling protocol described in [10]. Throughout the text, we use the terms
QA and QH to refer to configurations close to quasiaxisymmetry and
quasihelical symmetry, respectively, but may have finite deviations from exact
quasisymmetry. All of the equilibria are fixed-boundary except for the ITER
equilibrium. In the case of ITER, the free-boundary equilibrium is computed
without ferritic inserts or test blanket modules. The ITER profiles correspond
to a steady-state scenario [44], which features a smaller rotational transform
than high-performance scenarios [45]. The HSX fixed-boundary equilibrium
includes the effects of coil ripple since a fixed-boundary equilibrium
predating the free-boundary equilibrium was not available. The ARIES-CS
configuration is the n3are equilibrium, and the NCSX configuration is the
li383 equilibrium. The collisionless and collisional confinement of the ARIES-
CS, Wistell-A, Ku5, and NCSX equilibria were discussed in [10]. In Figure 3 we
show the rotational transform and quasisymmetry error in each configuration,
defined by

$f_{QS}(s)=\frac{\sqrt{\sum_{Mn\neq
Nm}\left(B_{m,n}^{c}(s)\right)^{2}+\left(B_{m,n}^{s}(s)\right)^{2}}}{\sqrt{\left(B_{00}^{c}(s)\right)^{2}+\left(B_{00}^{s}(s)\right)^{2}}},$
(9)

where the field strength in Boozer coordinates is
$B(s,\theta,\zeta)=\sum_{m,n}B_{m,n}^{c}(s)\cos(m\theta-n\zeta)+B_{m,n}^{s}(s)\sin(m\theta-n\zeta)$.
All configurations except for ITER are stellarator symmetric, so their
$B_{m,n}^{s}$ coefficients vanish. The contours of the magnetic field strength
on the plasma boundary are shown in Figure 2.

(a) ARIES-CS

(b) NCSX

(c) ITER

(d) HSX

(e) Wistell-A

(f) Ku5

Figure 2: Contours of the magnetic field strength are plotted as a function of the Boozer angles on the plasma boundary. Configuration | $N_{P}$ | Symmetry | Aspect ratio | Avg. $\iota$
---|---|---|---|---
ARIES-CS [46, 24] | 3 | QA | 4.55 | 0.57
NCSX [47, 48] | 3 | QA | 4.37 | 0.55
ITER [49] | 1 | QA | 2.31 | 0.28
HSX [50] | 4 | QH | 9.97 | 1.06
Ku5 [51] | 5 | QH | 10.01 | 1.18
Wistell-A [52] | 4 | QH | 6.72 | 1.08
Table 1: Quasiaxisymmetric and quasihelically symmetric equilibria under
consideration. The number of field periods of the nearby quasisymmetric field
is $N_{P}$. The aspect ratio is computed with the same definition used in the
VMEC code, given in [53]. The average of $\iota$ is computed with respect to
the toroidal flux.

In each configuration, $10^{4}$ particles are initialized with a uniform
distribution in $v_{\|}/v_{\mathrm{th}}$ and position on the $s=0.3$ surface
as described in [54], which corresponds to an isotropic distribution in 3D
velocity space. Particles are considered lost if they pass through the $s=1$
surface. The loss fractions as a function of time are presented in Figure 3.
While the quasisymmetry errors in the ARIES-CS and NCSX configurations are of
similar magnitudes, ARIES-CS has significantly fewer prompt losses than NCSX
due to its further optimization. While the Wistell-A and Ku5 configurations
have quasisymmetry errors comparable to ARIES-CS and NCSX, the loss fractions
in these two QH configurations are significantly reduced. HSX has reduced
$f_{QS}$ compared with Wistell-A and Ku5, but the loss fraction is elevated.
The rippled ITER configuration has the smallest quasisymmetry error but a
substantial number of losses compared to the Wistell-A and Ku5 configurations.
(With ferritic inserts and test blanket modules in addition to the coil
ripple, there are no collisionless losses from ITER [10].) We will analyze the
loss mechanisms and configuration characteristics responsible for these
differences in confinement.

(a) QA configurations

(b) QH configurations

Figure 3: Collisionless loss fractions (left), quasisymmetry error (center)
defined in (9), and rotational transform profile (right) for the QA and QH
configurations under investigation.

##  4 Summary of data

###  4.1 Quasiaxisymmetric configurations

Configuration | Total losses | Axis ($s<10^{-2}$) | Prompt | Transitioning
---|---|---|---|---
ARIES-CS | 2470 | 542 | 257 | 1693
NCSX | 2343 | 211 | 1298 | 1745
ITER | 796 | 170 | 20 | 388
Table 2: Number of losses for QA configurations. For reasons discussed in
Section 3.1, axis trajectories which pass through $s<10^{-2}$ are not included
in the classification analysis. Prompt losses leave the plasma boundary in
less than $10^{-3}$ seconds. Transitioning losses pass through several
trapping classes before leaving the plasma boundary.

The raw number of losses in the QA configurations is given in Table 2. Here
axis losses indicate trajectories that come close to the magnetic axis
($s<10^{-2}$) and are discarded from the analysis since the magnetic
coordinates break down. Prompt losses indicate particles that pass through the
plasma boundary in less than $10^{-3}$ seconds.

An overview of the statistics for the QA configurations is shown in Figures 4
and 5. Figure 4 displays the distribution of the mean value of
$|\gamma_{\mathrm{c}}|$ over bounce segments, the value of
$|\gamma_{\mathrm{c}}|$ for the final bounce segment before the particle exits
the plasma boundary, and the number of class transitions for prompt and non-
prompt losses. The distributions are normalized by the total number of prompt
or non-prompt losses in each configuration such that the sum is unity. In
Figure 5 we display the distribution of orbit segment types for each
configuration. For a given prompt or non-prompt loss, the fraction of orbit
indicates the fraction of bounce segments that fall into each trapping
category. For each trapping class in each configuration, the distribution is
normalized by the total number of prompt or non-prompt losses such that the
sum is unity. For example, in all of the histograms displayed for the QA
configurations, we see a distribution of barely-trapped orbit segments
centered at an orbit fraction near zero with a fraction of particles close to
unity, indicating that almost all losses spend almost none of their orbit in
the barely-trapped class. On the other hand, in ARIES-CS, about 20% of the
prompt losses spend close to 100% of the orbit in the non-DC banana class,
indicating that diffusive or resonant banana transport is a substantial prompt
loss channel. On the right, we show the distribution of orbit types for the
final bounce segment before the trajectory is lost. Here sub-bounce indicates
orbits that mirror one or fewer times before exiting such that the segment
type cannot be classified. The sub-bounce orbits are not included in the
histograms on the left. No passing trajectories are lost from the equilibria.
Several example trajectories are shown in the following Sections. Animations
corresponding to some of these trajectories are published in an accompanying
Zenodo archive [55].

Prompt losses

The distribution of the mean and final $|\gamma_{\mathrm{c}}|$ are shifted
toward zero for ARIES-CS in comparison with NCSX. This behavior is indicative
of the optimization of the ARIES-CS configuration to reduce prompt losses
compared to NCSX, as can also be seen from Table 2. We can also see from
comparing Figures 5(a) and 5(b) that the fraction of prompt losses arising
from ripple and drift-convective transport is diminished in ARIES-CS in
comparison with NCSX. However, a few ripple-trapped trajectories remain in
ARIES-CS, as seen in Figure 6. Ripple trapping is substantial in NCSX, and 10
of the prompt losses from NCSX did not complete a full bounce segment due to
rapid ripple losses. As only a small fraction of the initialized particles
will be born on ripple trajectories, the trajectories that transition onto
ripple segments can greatly impact the transport. This can be seen in Figure
10, an NCSX orbit that transitions to a ripple-trapped segment near
$\theta=-\pi/2$ and is subsequently lost. In contrast, in ARIES-CS the largest
ripple wells are localized near $\theta=\pi$ where the radial drift is
reduced, resulting in eventual detrapping (Figure 12). These differences in
ripple trapping partially account for the prompt confinement characteristics
of ARIES-CS and NCSX and will be discussed further in Section 5.3.

Ripple trapping is also quite substantial among prompt losses in ITER. The
coil perturbation in ITER is localized on the outboard side, resulting in
ripple wells near the outboard midplane. As the radial drift is relatively
small near $\theta=0$, the particles often precess and collisionlessly
transition onto deeply-trapped banana orbits. In Figure 9 we show an example
of such behavior.

On prompt timescales, all three configurations have a significant number of
transitioning orbits. These often correspond with transitions to ripple loss
channels (Figure 9 and 10). There are also prompt NCSX trajectories featuring
many separatrix crossings with somewhat irregular behavior (Figure 10). In the
ARIES-CS configuration, the transitions are often due to periodic ripple
detrapping and entrapping, similar to Figure 12.

While ripple and DC orbit segments are present in the ARIES-CS and NCSX prompt
losses, the non-DC banana segments dominate the orbit fraction. Evidence for
diffusive banana tip motion associated with non-conservation of $J_{\|}$ can
be seen in both NCSX and ARIES-CS (Figure 8). Here the normalized change in
$J_{\|}$ is computed as $\Delta J_{\|}/\langle J_{\|}\rangle=(J_{\|}-\langle
J_{\|}\rangle)/\langle J_{\|}\rangle$ where the average
$\langle\dots\dot{\rangle}$ is performed over the bounce segments. We note
that $J_{\|}$ is also relatively poorly conserved for drift-convective orbits
in these configurations (Figure 7), resulting in a small amount of non-secular
radial motion of the banana tips. The addition of diffusive motion to
convective transport along $J_{\|}$ contours may enhance losses. This behavior
is in contrast with DC banana orbits in QH configurations, which better
conserve $J_{\|}$ and whose banana tips drift radially in a secular fashion
(Figure 17). We see a similarly significant presence of non-DC banana segments
in ITER. In addition to diffusive banana tip motion, some of the non-DC banana
transport in ITER is due to resonant orbits. In Figure 13 we see an example of
such a barely-trapped banana trajectory in ITER which exhibits a bounce-
transit resonance with $k=-3$ and $l=2$ in the notation of (6).

Considering the orbit types on the last bounce in Figure 5, most of the prompt
losses in the NCSX and ITER equilibria terminate on ripple orbit segments. In
contrast, in the ARIES-CS equilibria, the number of DC banana and non-DC
banana orbit segments become comparable to ripple segments. Furthermore, in
all three equilibria, we see that the distribution of $|\gamma_{\mathrm{c}}|$
on the last bounce is shifted toward larger values compared to the
distribution of the mean. This behavior indicates many trajectories that spend
most of their orbit time in the non-DC class but are eventually lost due to
ripple or DC mechanisms.

Non-prompt losses

For each configuration, the distribution of the mean $|\gamma_{\mathrm{c}}|$
is shifted toward zero on non-prompt timescales in comparison with prompt
timescales (Figure 4), indicative of non-DC transport mechanisms that dominate
on longer timescales. Similar behavior is seen in the distribution of the last
$|\gamma_{\mathrm{c}}|$, except for in the NCSX equilibrium. The distinct
behavior in NCSX reflects orbits which stay confined for a long period of time
and are eventually lost on a ripple trajectory, as can be seen from the orbit
segment type on the last bounce (Figure 5(b)) and final value of
$|\gamma_{\mathrm{c}}|$ (Figure 4). A population of trajectories terminates on
DC segments in all three configurations, indicating that these loss channels
remain important even on longer time scales.

The ARIES-CS equilibrium features many trajectories with hundreds of
transitions. For example, Figure 11 shows a trajectory that irregularly
changes classes before being lost on a ripple orbit. Sometimes trajectories
featuring many transitions can exhibit more regular behavior. For example, in
Figure 12 we see a trajectory from ARIES-CS which transitions periodically
between banana and ripple trapping classes and is eventually lost when the
separatrix crossings become more irregular. We see similar periodic
transitions between ripple and banana classes in the ITER equilibrium before
being lost on a resonant barely-trapped orbit. However, irregular
transitioning behavior does not appear among the ITER lost trajectories.

Some non-prompt lost trajectories feature other types of periodic behavior.
For example, a periodic DC banana orbit in NCSX is shown in Figure 14 which is
eventually lost on a ripple orbit. A similar periodic DC banana orbit is
observed in ARIES-CS, eventually lost on a diffusive banana orbit. These can
be understood as drift-convective trajectories whose orbits are healed due to
global features of the orbit given the closure of $J_{\|}$ contours within the
plasma boundary [17]. Eventual loss of such an orbit can occur due to slight
non-conservation of $J_{\|}$.

Many of the non-prompt losses in ITER terminate on a barely-trapped orbit
(Figure 5(c)). An example of a transition from a banana orbit segment to a
resonant barely-trapped orbit segment is shown in Figure 13. The barely-
trapped segment is close to the $k=3$, $l=-1$ resonance in the notation of (8)
given the $\iota$ profile which crosses through $1/3$ (Figure 3). This
periodicity gives rise to fluctuations in the radial coordinate concurrent
with each toroidal transit and a secular radial drift due to a resonance in
the equilibrium. As the loss of a resonant barely-trapped orbit requires many
toroidal transits, this mechanism is more prevalent on long timescales.

(a) Prompt

(b) Non-prompt

Figure 4: Trajectory statistics for QA configurations. Prompt losses leave the
plasma boundary in less than $10^{-3}$ seconds. The left histograms display
the distribution of the mean value of $|\gamma_{\mathrm{c}}|$ over bounce
segments while the middle histograms display the value of
$|\gamma_{\mathrm{c}}|$ for the final bounce segment before being lost (see
(5)). The distribution of the number of class transitions along the lost
trajectories is shown on the right. Each histogram’s counts are scaled by the
number of losses so that the distributions can be displayed on the same scale
for each configuration.

(a) ARIES-CS

(b) NCSX

(c) ITER

Figure 5: Distribution of orbit segment types for QA configurations on prompt
($<10^{-3}$ seconds) and non-prompt timescales. For each half-bounce segment,
a trajectory is categorized into the ripple, banana, or barely-trapped
categories. Drift-convective (DC) transport is indicated by
$|\gamma_{\mathrm{c}}|>0.2$ (see (5)). Each histogram’s counts are scaled by
the number of losses so that the distributions can be displayed on the same
scale for each configuration. The bar charts on the right display the
distribution of orbit types on the bounce segment before the trajectory is
lost. Here sub-bounce indicates a particle that does not complete a full
bounce orbit before being lost.

(a)

(b)

(c)

(d)

(e)

Figure 6: ARIES-CS prompt ($3.1\times 10^{-5}$ seconds) loss on a ripple
trajectory. Black stars indicate bounce points. (a) Poloidal cross-section.
Black dashed lines indicate the initial magnetic surface and plasma boundary.
(b) Radial coordinate $s=\psi/\psi_{0}$ as a function of time. (c) Drift-
convection parameter evaluated for each bounce segment indicative of banana
drift-convective transport. (d) The normalized change in the parallel
adiabatic invariant indicates that $J_{\|}$ is well conserved along this
trajectory. (e) Overhead view of orbit indicative of toroidal localization.
The magnetic axis is indicated by a black dashed line.

(a)

(b)

(c)

(d)

(e)

Figure 7: ARIES-CS prompt ($2.5\times 10^{-4}$ seconds) loss on a banana
drift-convective trajectory. Black stars indicate bounce points. (a) Poloidal
cross-section. Black dashed lines indicate the initial magnetic surface and
plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a function of
time. (c) Drift-convective parameter evaluated for each bounce segment
indicative of significant banana drift-convective transport. (d) The
normalized change in the parallel adiabatic invariant indicates that $J_{\|}$
is poorly conserved. (e) Overhead view of orbit indicating toroidal
localization. The magnetic axis is indicated by a black dashed line.

(a)

(b)

(c)

(d)

(e)

Figure 8: ARIES-CS prompt ($3.6\times 10^{-4}$ seconds) loss featuring
diffusive banana tip motion. Black stars indicate bounce points. (a) Poloidal
cross-section, demonstrating the diffusive motion of banana tips. Black dashed
lines indicate the initial magnetic surface and plasma boundary. (b) Radial
coordinate $s=\psi/\psi_{0}$ as a function of time. (c) The drift-convective
parameter is evaluated for each bounce segment, indicating that banana drift-
convective transport is insignificant. (d) The normalized change in the
parallel adiabatic invariant indicates poor conservation along the trajectory.
(e) Overhead view of orbit indicating the absence of toroidal localization.
The magnetic axis is indicated by a black dashed line.

(a)

(b)

(c)

Figure 9: ITER prompt ($2.0\times 10^{-4}$ seconds) loss featuring a deeply-
trapped banana orbit (green) and transition to a ripple orbit (blue). (a)
Poloidal cross-section. Black dashed lines indicate the initial magnetic
surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a
function of time. (c) Parallel adiabatic invariant normalized by the velocity
magnitude, evaluated for each bounce segment that the trajectory remains in
the same trapping class.

(a)

(b)

Figure 10: NCSX prompt ($8.2\times 10^{-4}$ seconds) loss featuring several
transitions between banana (green) and barely-trapped (red) classes before
being lost on a ripple orbit segment (blue). Black stars indicate bounce
points. (a) Poloidal cross-section. Black dashed lines indicate the initial
magnetic surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$
as a function of time. Irregular transitioning behavior is observed.

(a)

(b)

Figure 11: ARIES-CS non-prompt ($1.4\times 10^{-2}$ seconds) loss featuring
many transitions between banana (green), barely-trapped (red), and ripple
(blue) classes before being lost on a diffusive banana orbit. Black stars
indicate bounce points. (a) Poloidal cross-section. Black dashed lines
indicate the initial magnetic surface and plasma boundary. (b) Radial
coordinate $s=\psi/\psi_{0}$ as a function of time. Irregular transitioning
behavior is observed.

(a)

(b)

(c)

Figure 12: ARIES-CS non-prompt ($3.4\times 10^{-2}$ seconds) loss featuring
many transitions between banana (green) and ripple (blue) classes. The
particle is eventually lost on a more irregular transitioning orbit (not
shown). (a) Poloidal cross-section. Black dashed lines indicate the initial
magnetic surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$
as a function of time. (c) Overhead view of orbit. The magnetic axis is
indicated by a black dashed line.

(a)

(b)

(c)

(d)

(e)

Figure 13: ITER non-prompt ($2.8\times 10^{-2}$ seconds) resonant banana orbit
(green) which transitions to a ripple (blue) and resonant barely-trapped orbit
(red). (a) Poloidal cross-section. Black dashed lines indicate the initial
magnetic surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$
as a function of time. (c) The normalized change in the parallel adiabatic
invariant is evaluated for each bounce segment that the trajectory remains in
the same trapping class. The secular change in $J_{\|}$ indicates a resonant
perturbation. (d) The upper bounce points, $\zeta_{b}$, are plotted as a
function of the number of bounce segments, $b$, to demonstrate the $l=2$,
$k=-3$ periodicity in the notation of (6). (e) The final barely-trapped orbit
segment is plotted to demonstrate the $l=-1$, $k=3$ periodicity in the
notation of (8).

(a)

(b)

(c)

(d)

(e)

Figure 14: NCSX non-prompt ($1.3\times 10^{-1}$ seconds) loss featuring
periodic banana drift-convective motion, eventually lost on a ripple orbit
(not shown). Black stars indicate bounce points. (a) Poloidal cross-section.
Black dashed lines indicate the initial magnetic surface and plasma boundary.
(b) Radial coordinate $s=\psi/\psi_{0}$ as a function of time. (c) Drift-
convective parameter evaluated for each bounce segment. (d) The normalized
change in the parallel adiabatic invariant indicates that $J_{\|}$ has
deviations of a few percent. (e) Overhead view of orbit, indicative of the
toroidal localization. The magnetic axis is indicated by a black dashed line.

###  4.2 Quasihelical configurations

Configuration | Total losses | Axis ($s<10^{-2}$) | Prompt | Transitioning
---|---|---|---|---
HSX | 2825 | 696 | 741 | 1441
Ku5 | 332 | 0 | 0 | 318
Wistell-A | 120 | 14 | 14 | 94
Table 3: Number of losses for QH configurations. Axis trajectories pass
through $s<10^{-2}$ and are not included in the analysis. Prompt losses leave
the plasma boundary in less than $10^{-3}$ seconds. Transitioning losses pass
through several trapping classes before leaving the plasma boundary.

(a) Prompt

(b) Non-prompt

Figure 15: Trajectory statistics for QH configurations. Prompt losses leave
the plasma boundary in less than $10^{-3}$ seconds. The left histograms
display the distribution of the mean value of $|\gamma_{\mathrm{c}}|$ over
bounce segments while the middle histograms display the value of
$|\gamma_{\mathrm{c}}|$ for the final bounce segment before being lost (see
(5)). The distribution of the number of class transitions along the lost
trajectories is shown on the right. Each histogram’s counts are scaled by the
number of losses so that the distributions can be displayed on the same scale
for each configuration.

The raw number of losses in the QH configurations is given in Table 3.
Compared with the QA equilibria, we note that the total number of prompt
losses is relatively small in the Ku5 and Wistell-A configurations.
Transitioning orbits remain prominent in all three configurations. An overview
of the statistics for the QH configurations are shown in Figures 15 and 16. No
passing trajectories are lost.

Prompt losses

The small number of prompt losses in Wistell-A have higher values of the mean
and final $|\gamma_{\mathrm{c}}|$, indicating the prevalence of drift-
convective transport. The histograms confirm this behavior in Figure 16, which
suggests that non-DC banana transport is not significant for Wistell-A prompt
losses. An example of a DC banana loss in Wistell-A is shown in Figure 17. HSX
similarly has a significant population of DC banana orbits in addition to
losses on non-DC banana orbits. An example of a non-DC banana trajectory can
be seen in Figure 21, which features diffusive banana tip motion. In this
example trajectory, we note that $|\gamma_{\mathrm{c}}|$ increases as it is
transported outward in radius, indicative of convection along in addition to
diffusion across $J_{\|}$ contours.

In addition to DC banana losses, DC barely-trapped orbits contribute toward
the transport in Wistell-A (Figure 18). Due to the large variation of the
field strength maximum on the surface (Figure 29), there exists a significant
population of trajectories that can traverse through a full period of the
field strength variation before bouncing, leading to the presence of barely-
trapped orbits among the final orbit types (Figure 16). There are also a small
number of resonant barely-trapped orbits In HSX. In Figure 19 we show a
barely-trapped orbit close to the $l=4$, $k=4$ resonance in the notation of
(8), as the rotational transform lies close to $\iota=1$ with low shear.

Non-DC banana and ripple losses are not as prominent among the QH prompt
losses as they are among the QA losses (Figures 5 and 16). The physical
mechanisms behind these differences will be discussed further in Section 5.
Ripple losses are present among the HSX prompt losses (Figure 20): many
trajectories transition to ripple-trapped segments and are rapidly lost. In
contrast, Wistell-A features minimal ripple trapping.

As shown in Figure 15, prompt transitions are common in both Wistell-A and HSX
as they were in the QA configurations. An example from HSX is shown in Figure
20, which features transitions between the banana and barely-trapped classes
before being lost on a ripple orbit.

Non-prompt losses

Transitions become much more prominent on non-prompt timescales for all three
configurations, especially for Wistell-A and Ku5 (Figure 15). An example of
irregular transition-driven transport in Ku5 is shown in Figure 22. Other
losses feature regular periodic transitions, as can be seen in Wistell-A in
Figure 23. Due to the variation of the field strength maximum on the surface,
transitions between banana and barely-trapped segments are prevalent. In the
Ku5 equilibrium, only one lost orbit does not undergo transitions but exhibits
periodic banana tip motion.

Similar to the QA non-prompt losses, we note that the QH non-prompt losses
feature a distribution of the mean $|\gamma_{\mathrm{c}}|$ that is shifted
toward zero in comparison with the prompt losses. This behavior indicates that
non-DC mechanisms are prominent along the lost trajectories on long
timescales. However, the final value of $|\gamma_{\mathrm{c}}|$ remains
substantial, indicating that a trajectory may be confined for an extended
period before it is lost on a DC orbit segment. An example Wistell-A
trajectory is shown in Figure 23, which exhibits periodic transitioning
behavior before being lost on a DC banana orbit segment.

Similar to the QA configurations (Figure 14), there remain DC orbits that
exhibit periodic motion and are confined for long time scales. In the example
HSX trajectory shown in Figure 24, the trajectory is eventually lost on a
ripple orbit.

(a) HSX

(b) Wistell-A

(c) Ku5

Figure 16: Distribution of orbit segment types for QH configurations on prompt
($<10^{-3}$ seconds) and non-prompt timescales. For each half-bounce segment,
a trajectory is categorized into the ripple, banana, or barely-trapped
categories. Drift-convective (DC) transport is indicated by
$|\gamma_{\mathrm{c}}|>0.2$. Each histogram’s counts are scaled by the number
of losses so that the distributions can be displayed on the same scale for
each configuration. The bar charts on the right display the distribution of
orbit types on the bounce segment before the trajectory is lost. Here sub-
bounce indicates a particle that does not complete a full bounce orbit before
being lost.

(a)

(b)

(c)

(d)

(e)

Figure 17: Wistell-A prompt ($2.2\times 10^{-4}$ seconds) loss which features
a banana drift-convective orbit. Black stars indicate bounce points. (a)
Poloidal cross-section. Black dashed lines indicate the initial magnetic
surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a
function of time. Secular radial banana tip motion is observed. (c) Drift-
convective parameter evaluated for each bounce segment. Drift-convective
transport is evident throughout the trajectory. (d) The normalized change in
the parallel adiabatic invariant indicates that $J_{\|}$ is well conserved.
(e) Overhead view of orbit indicating toroidal localization. The magnetic axis
is indicated by a black dashed line. Localization in toroidal angle is
evident.

(a)

(b)

(c)

Figure 18: Prompt loss ($1.7\times 10^{-4}$ seconds) from Wistell-A featuring
a DC banana orbit (green) which transitions to a DC barely-trapped orbit
(red). Black stars indicate bounce points. (a) Poloidal cross-section. Black
dashed lines indicate the initial magnetic surface and plasma boundary. (b)
Radial coordinate $s=\psi/\psi_{0}$ as a function of time. (c) Drift-
convective parameter evaluated for each bounce segment. DC transport is
evident during both the banana and barely-trapped segments.

(a)

(b)

(c)

Figure 19: Prompt loss ($6.3\times 10^{-4}$ seconds) from HSX featuring a
resonant barely-trapped orbit. Black stars indicate bounce points. (a)
Poloidal cross-section. Black dashed lines indicate the initial magnetic
surface and plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a
function of time. (c) The first barely-trapped trajectory segment is plotted
to demonstrate the $l=4$, $k=4$ resonance in the notation of (8).

(a)

(b)

Figure 20: HSX prompt ($3.4\times 10^{-4}$ seconds) loss featuring transitions
between the banana (green) and barely-trapped (red) classes before being lost
on a ripple orbit (blue). Black stars indicate bounce points. (a) Poloidal
cross-section. Black dashed lines indicate the initial magnetic surface and
plasma boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a function of
time.

(a)

(b)

(c)

(d)

Figure 21: HSX prompt ($2.9\times 10^{-4}$) loss from featuring diffusive
banana tip motion. Black stars indicate bounce points. (a) Poloidal cross-
section. Black dashed lines indicate the initial magnetic surface and plasma
boundary. Diffusive banana tip motion is evident. (b) Radial coordinate
$s=\psi/\psi_{0}$ as a function of time. (c) Drift-convective parameter
evaluated for each bounce segment. DC transport is evident as it nears the
boundary. (d) The normalized change in the parallel adiabatic invariant
demonstrates that $J_{\|}$ is not well conserved, indicative of diffusive
banana tip motion.

(a)

(b)

Figure 22: Ku5 non-prompt ($3.0\times 10^{-2}$ seconds) loss featuring
transition-driven transport between the banana (green) and barely-trapped
(red) classes before being lost on a diffusive banana orbit. Black stars
indicate bounce points. (a) Poloidal cross-section. Black dashed lines
indicate the initial magnetic surface and plasma boundary. (b) Radial
coordinate $s=\psi/\psi_{0}$ as a function of time. Irregular transitioning
behavior is evident.

(a)

(b)

(c)

(d)

Figure 23: Wistell-A non-prompt ($2.7\times 10^{-2}$ seconds) loss featuring
periodic transitions between the banana (green) and barely-trapped (red)
classes. The particle is eventually lost on a DC banana segment. Black stars
indicate bounce points. (a) Radial coordinate $s=\psi/\psi_{0}$ as a function
of time for a long segment where the periodic behavior is observed. (b) Radial
coordinate $s=\psi/\psi_{0}$ as a function of time for one period. (c) The
poloidal cross-section for one period. Black dashed lines indicate the initial
magnetic surface and plasma boundary. (d) Parallel adiabatic invariant
normalized by the velocity magnitude, evaluated for each bounce segment that
the trajectory remains in the same trapping class for a long segment where the
periodic behavior is observed. Jumps in $J_{\|}$ associated with separatrix
crossings are evident.

(a)

(b)

(c)

(d)

(e)

Figure 24: HSX non-prompt ($1.6\times 10^{-1}$ seconds) loss featuring a
periodic DC banana, eventually lost on a ripple-trapped segment (not shown).
Black stars indicate bounce points. (a) Poloidal cross-section. Black dashed
lines indicate the initial magnetic surface and plasma boundary. (b) Radial
coordinate $s=\psi/\psi_{0}$ as a function of time. Periodic behavior is
evident. (c) Drift-convective parameter evaluated for each bounce segment. DC
transport is evident. (d) The normalized change in the parallel adiabatic
invariant indicates that $J_{\|}$ is well conserved. (e) Overhead view of
orbit, indicative of toroidal localization. The magnetic axis is indicated by
a black dashed line.

##  5 Optimization techniques

Having identified the relevant transport channels for the given equilibria, we
discuss the impact of the equilibrium magnetic field and possible optimization
techniques. Here we consider a magnetic field which is close to a
quasisymmetric magnetic field, $B=B(s,\chi)$ with $\chi=M\theta-N\zeta$ and a
model for the perturbation from quasisymmetry with mode numbers $m_{\epsilon}$
and $n_{\epsilon}$,

$\delta B(s,\theta,\zeta)=\overline{B}\epsilon(s)\cos(m_{\epsilon}\theta-
n_{\epsilon}\zeta),$ (10)

where $\overline{B}$ is a characteristic field strength of the equilibrium. We
will use the notation $\chi_{\epsilon}=m_{\epsilon}\theta-n_{\epsilon}\zeta$.

###  5.1 Orbit-width effects

The conserved canonical momentum of guiding center motion in a quasisymmetric
magnetic field is,

$p_{\zeta}(s,\chi)=\frac{mv_{||}\left(MG(s)+NI(s)\right)}{B(s,\chi)}+q\left(Ns\psi_{0}-M\psi_{P}(s)\right),$
(11)

where $2\pi\psi_{P}(s)$ is the poloidal flux and $q$ is the charge. At the
inflection points where $d\psi/d\chi=0$, the field strength is minimized along
the orbit, corresponding to $\chi=0$. Labeling the surface where $s$ is
maximized/minimized along the banana orbit as $s_{\pm}$, we Taylor expand
$p_{\zeta}$ about $s_{-}$ and use conservation of $p_{\zeta}$ to obtain the
expression,

$\Delta
s=\frac{2mv_{\|}(s_{-},0)\left(MG(s_{-})+NI(s_{-})\right)}{B(s_{-},0)q\psi_{0}(N-\iota(s_{-})M)},$
(12)

dropping higher order terms in $\rho_{*}$. We can define a metric for the
orbit width,

$f_{\mathrm{orbitwidth}}=\left|\frac{MG+NI}{M\iota-N}\right|.$ (13)

From this expression we note the impact of the symmetry helicity on orbit
width. For typical stellarator configurations $|I(s)|\ll|G(s)|$ such that
quasi-poloidal configurations ($M=0$) have the smallest orbit width.
Quasihelical configurations ($M\neq 0$, $N\neq 0$) also enjoy reduced orbit
width since typically $|N-\iota M|>1$. Quasiaxisymmetric configurations
($N=0$) typically have the widest orbits, although the width can be reduced by
increasing $\iota$. Finally, in both QA and QH configurations, the orbit width
decreases with decreasing $G$. The expression for $G$ on the magnetic axis is
$G_{0}=LB_{0}/(2\pi)$, where $B_{0}$ is the field strength on the axis and $L$
is the length of the magnetic axis [56]. As we compare equilibria scaled to
the same volume and field strength, smaller aspect ratio configurations enjoy
reduced orbit widths.

In Figure 25 we compare the banana orbit widths for the same initial condition
in Boozer coordinates and pitch angle, corresponding to moderate trapping in
the main well, across the six QA and QH configurations. The non-symmetric
modes are artificially removed such that the orbits close in the poloidal
plane. The Wistell-A and Ku5 configurations have the smallest orbit widths
given their helical symmetry and small aspect ratio in comparison to HSX; see
Table 1. The widest orbit widths are found in the ARIES-CS and NCSX
configurations, which feature a larger aspect ratio than ITER.

Reducing the orbit width can prevent the prompt loss of wide banana orbits
that intersect the boundary. These orbits can be present, for example, in QA
equilibria with relatively small values of the rotational transform [29].
Furthermore, smaller orbit width reduces the transport due to diffusive banana
tip motion. Since the transition to stochasticity occurs due to the resonant
overlap experienced in an orbit width, wider banana orbits can contribute more
strongly to banana tip diffusion [14], as discussed in the following Section.

(a)

(b)

(c)

(d)

Figure 25: (a) and (c) A banana trapped particle is followed in each
configuration with the same initial condition, and its position is shown in
the poloidal plane to compare the orbit widths across configurations. The
orbits in ARIES-CS and NCSX and in Wistell-A and Ku5 nearly overlap. (b) and
(d) Distribution of the normalized variation in $J_{\|}$ over non-
transitioning lost guiding center orbits. The QA configurations show more non-
adiabatic behavior than the QH configurations.

###  5.2 Diffusive banana transport

In analyzing transport mechanisms in QA and QH configurations, we have found
that non-DC banana mechanisms are dominant among the lost trajectories.
Furthermore, a significant number of lost orbits terminate on a non-DC banana
orbit, especially in QA configurations (Figure 5 and 16). One such non-DC
banana mechanism that emerges is diffusive banana tip motion (Figure 8).

The diffusive motion of banana tips arises due to the non-conservation of
$J_{\|}$. If $J_{\|}$ is conserved in addition to the magnetic moment and
energy, the guiding center motion in 3D magnetic fields is (nearly)
integrable. Deviations of $J_{\|}$ arise due to the change in the shape of the
magnetic field strength well during one bounce period. If the shape of the
effective potential well changes on a timescale comparable to the periodic
bounce motion, the assumption of adiabaticity breaks down. Thus, this effect
is enhanced for particles with increased in-surface magnetic drifts and more
significant deviations from quasisymmetry. While drift-convective transport
arises due to the misalignment of $J_{\|}$ contours with magnetic surfaces and
is present in the collisionless dynamics as long as $J_{\|}$ is roughly
conserved, diffusive banana transport becomes more substantial as energy
increases due to the increase in the precessional magnetic drift.

For a given quasisymmetric magnetic field $B(s,\chi)$, the maximum bounce time
is,

$\tau(s)=\int_{0}^{2\pi}\frac{d\chi}{v_{\|}\hat{\textbf{b}}\cdot\nabla\chi}=\frac{G(s)+\iota(s)I(s)}{M\iota(s)-N}\int_{0}^{2\pi}\frac{d\chi}{B(s,\chi)v_{\|}(s,\chi)}.$
(14)

The time to precess through one period of the perturbation is,

$\tau_{\epsilon}=\int_{0}^{2\pi}\frac{d\chi_{\epsilon}}{\textbf{v}_{\mathrm{m}}\cdot\nabla\chi_{\epsilon}},$
(15)

where $\chi_{\epsilon}=m_{\epsilon}\theta-n_{\epsilon}\zeta$ (see (10)). For
smaller values of,

$f_{\mathrm{diffusive}}=\frac{\epsilon\tau}{\tau_{\epsilon}},$ (16)

a given perturbation with amplitude $\epsilon$ (see (10)) will not break
adiabaticity as easily, thus minimizing the diffusive banana response. In the
vacuum limit, the above expression scales as
$f_{\mathrm{diffusive}}\sim\epsilon m_{\epsilon}\left|\partial B/\partial
s\right|/(M\iota-N)$. The inverse scaling with $(M\iota-N)$ explains the
reduction of banana-diffusive transport in QH (in comparison with QA)
configurations seen in Section 4. However, some non-adiabaticity is present in
HSX due to the higher mode number ripple perturbations (Figure 2). The scaling
with $|\partial B/\partial s|$ implies that less compact configurations may be
advantageous [56].

In addition to the in-surface precession, rapid radial drift can change the
well shape and cause the breaking of adiabaticity. This behavior can be seen
in a DC trajectory in ARIES-CS (Figure 7), which features $\approx 15\%$
deviations in $J_{\|}$ due to the significant radial drift although the orbit
remains toroidally localized. As a result, the banana tips wander slightly in
this example.

In Figure 25 we present the distribution of the relative change in $J_{\|}$
along guiding center trajectories for the configurations. The distribution is
only presented for trajectories that do not transition between classes. The
Ku5 configuration has only one lost trajectory that does not transition,
having $\sqrt{\left\langle\Delta J_{\|}^{2}\right\rangle}/\langle
J_{\|}\rangle=0.08$ and the Wistell-A configuration only has 12 non-
transitioning orbits with a maximum value of $\sqrt{\left\langle\Delta
J_{\|}^{2}\right\rangle}/\langle J_{\|}\rangle=0.03$. There are significant
deviations from adiabaticity for each configuration (except for Wistell-A), as
had been anticipated for stellarator configurations [57]. The spread in
deviation can also be significant, which is accounted for by the difference in
precession velocities among trajectories. For example, this can be seen in
comparing the evolution of $J_{||}$ between a DC trajectory (Figure 6) and
non-DC trajectory (Figure 8) in the same configuration. The adiabaticity is
generally improved in the QH configurations compared to the QA configurations,
consistent with (16). We note that the non-adiabaticity is especially
noticeable in the rippled ITER configuration. In this case, the resonant
banana-drift mechanism (Figure 13) is present in addition to banana diffusion.

Figure 26: Rotational transform and quasisymmetry error (9) as a function of
normalized flux for the set of configurations optimized for quasisymmetry on
the boundary. The black dashed line represents the surface on which
trajectories are initialized for the $J_{\|}$ conservation studies.

(a)

(b)

(c)

Figure 27: Prompt ($1.0\times 10^{-3}$ seconds) loss featuring diffusive
banana tip motion. Black stars indicate bounce points. (a) Poloidal cross-
section. Black dashed lines indicate the initial magnetic surface and plasma
boundary. (b) Radial coordinate $s=\psi/\psi_{0}$ as a function of time.
Diffusive banana tip motion is evident. (c) The normalized change in the
parallel adiabatic invariant indicates that $J_{\|}$ is not conserved well,
featuring some secular increase in addition to periodic oscillations.  Figure
28: The variation in the parallel adiabatic invariant is shown for a diffusive
banana orbit in the three configurations optimized for quasiaxisymmetry on the
boundary (26). The same initial conditions are used with varying energy and
configuration. At a small perturbation amplitude, periodic oscillations in
$J_{\|}$ emerge. In equilibria closer to quasiaxisymmetry and at lower energy,
$J_{\|}$ is better conserved.

To further study the mechanisms for the deviations in $J_{\|}$, we perform GC
integration in several vacuum equilibria that are optimized for
quasiaxisymmetry on the boundary [58] in addition to having a value of
$\iota=0.52$ on the boundary and aspect ratio 6. For the base equilibrium, the
optimization is terminated when the quasisymmetry on the boundary is
$f_{QS}=1.8\times 10^{-2}$, see (9). We compare with two other equilibria, for
which the optimization is terminated at $f_{QS}=1.8\times 10^{-3}$ and
$f_{QS}=1.3\times 10^{-4}$. The quasisymmetry error and rotational transform
profiles for these equilibria are shown in Figure 26.

We consider initial conditions for a 3.52 MeV fusion-born alpha particle on
the $s=0.3$ surface corresponding to a trajectory that is lost due to
diffusive banana transport in the base equilibrium, see Figure 27. In the base
equilibrium we note that $J_{\|}$ has periodic oscillations on top of a
secular increase up to $15\%$ above the mean value. In Figure 28 we compare
the conservation of $J_{\|}$ in the two other equilibria with reduced $f_{QS}$
and for an equilibrium where the non-symmetric modes of the field strength are
artificially removed, $f_{QS}=0$. As the quasisymmetry error is reduced, the
secular increase is eliminated, and periodic oscillations in $J_{\|}$ remain.
The periodicity hints at the existence of a higher-order adiabatic invariant
[59, 60]. As the symmetry-breaking is further reduced, $J_{\|}$ is well
conserved. Similar behavior is seen as the particle energy is reduced in the
base equilibrium.

The periodic oscillations of $J_{\|}$ about a conserved mean value at low
perturbation amplitudes are reminiscent of oscillations in $\mu$, which has
been termed super-adiabaticity in the spherical tokamak [61, 62] and mirror
[63] literature. The periodic oscillations of $J_{\|}$ are consistent with
transport due to a single island chain in the banana tip motion. At larger
perturbation amplitudes, islands may begin to overlap, leading to stochastic
transport.

###  5.3 Ripple trapping

In all of the QA configurations (Figure 5) in addition to HSX (Figure 16(a)),
ripple trapping makes up a significant fraction of the losses, especially on
prompt timescales. Considering the model for a magnetic field close to
quasisymmetry (10), the criterion for the perturbation to be large enough to
form a local minimum at ($s_{r},\theta_{r},\zeta_{r}$) along field lines is,

$\epsilon(s_{r})\gtrsim\Bigg{|}\frac{M\iota-N}{(m_{\epsilon}\iota-
n_{\epsilon})\sin(m_{\epsilon}\theta_{r}-n_{\epsilon}\zeta_{r})}\frac{\partial
B(s,M\theta_{r}-N\zeta_{r})}{\partial\chi}\displaystyle\Bigg{|}.$ (17)

An equilibrium will be less susceptible to ripple wells for smaller values of,

$f_{\mathrm{ripple}}=\left|\frac{m_{\epsilon}\iota-
n_{\epsilon}}{M\iota-N}\left(\frac{\partial
B(s,\chi)}{\partial\chi}\displaystyle\right)^{-1}\right|.$ (18)

Given the factor $(M\iota-N)^{-1}$, this presents a potential advantage for QH
configurations in avoiding ripple formation for a given perturbation.
Increasing $\iota$ may also reduce the ripple formation for small values of
$m_{\epsilon}/M$. Given the factor $\left(\partial
B/\partial\chi\right)^{-1}$, more compact configurations are advantageous.
Furthermore, we can note from (17) that higher mode number, $m_{\epsilon}$ and
$n_{\epsilon}$, perturbations form ripple more easily. Similar to the
diffusive mechanisms (16), perturbations with smaller mode number
$m_{\epsilon}$ are less deleterious. We see that the factor $1/(M\iota-N)$ is
in common with features that reduce diffusive (16) and orbit width (13)
effects. However, the factor $\left(\partial B/\partial\chi\right)^{-1}$ is in
conflict with features that reduce non-adiabaticity. From the GC statistics,
ripple-trapping is less prevalent among the lost orbits in QH configurations
(16), except for in the HSX equilibrium, which features higher mode number
ripple perturbations (Figure 2).

Finally, we note that ripples form more easily near the minima and maxima of
the field strength. Indeed, although the ripple perturbation amplitude is
relatively small on the inboard side of ITER, ripple trapping can occur near
the maximum of the field strength (Figure 13). However, ripple wells localized
near the field maxima or minima may not be as deleterious, as the radial drift
scales as $|\partial B/\partial\chi|$. Particles trapped in these wells mainly
drift parallel to surfaces and may eventually detrap as the ripple
perturbation amplitude diminishes. This behavior can be seen in the ARIES-CS
trajectories (Figure 12), which periodically detrap and entrap in ripple
wells. On the other hand, the quasisymmetry deviations in NCSX lead to ripple
wells away from the maxima and minima [46], which causes transitions from
banana trapping to prompt ripple losses (Figure 10).

###  5.4 Transitioning particles

While the GC motion in perfectly quasisymmetric magnetic fields is free from
class transitions, even small perturbations can introduce chaotic regions in
phase space for trajectories trapped near the field maximum [64]. We have
found that a significant fraction of lost trajectories in both the QA and QH
configurations undergo transitions between classes (Figures 4 and 15). In some
cases, these class transitions are associated with irregular behavior,
characterized by many separatrix crossings without periodic structure (Figure
11 or 22). While the separatrix crossings appear to be accompanied by small
radial displacements, the radial transport is often not rapid compared to the
diffusive banana tip or drift-convective mechanisms. In both trajectory
examples, relatively little radial transport occurs during the irregular
transition behavior, but the particle is lost due to diffusive banana tip
motion. In other cases, irregular transitioning appears responsible for the
eventual loss (Figure 12).

Furthermore, transitions are not always irregular but display periodic
behavior (Figures 12 and 23). In the case of the trajectory shown in Figure
12, we see a ripple-trapped particle that drifts vertically, allowing it to
detrap from the ripple localized on the inboard side. The trajectory then
transitions to a banana segment. The trajectory can entrap and repeat the
cycle as the banana-trapped segment nears the inboard side. While $J_{\|}$ is
not conserved along such trajectories, the periodicity indicates the presence
of a higher-order adiabatic invariant. In analogy with the context of banana
diffusion, if the periodic orbit is associated with a large radial excursion,
such as in Figure 23, slight wandering of the super-orbit may occur due to the
breaking of higher-order adiabaticity. Thus transitions associated with wide
super-orbits should especially be avoided.

Several equilibrium characteristics prevent class transitions.

  1. 1.

Ensuring that the maxima of the field strength along a field line are the same
for all field lines on a magnetic surface prevents transitions due to in-
surface precession [57, 27, 65]. In Figure 29 we compare the variation of
$B_{\mathrm{max}}$ on surfaces for the set of QA and QH equilibria. We see
that the two equilibria with the largest fraction of transitioning lost
trajectories (Tables 2 and 3), Ku5 and Wistell-A, feature the largest
variation of $B_{\mathrm{max}}$. As this metric does not account for
transitions due to ripple wells away from the field maxima, it does not
explain the behavior in ARIES-CS and NCSX, the two equilibria with the largest
total number of transitioning lost trajectories.

  2. 2.

Pseudosymmetry [66] is defined as the quantity,

$\displaystyle f=\frac{\textbf{B}\times\nabla\psi\cdot\nabla
B}{\textbf{B}\cdot\nabla B},$ (19)

being non-zero everywhere such that the radial magnetic drift vanishes at
field maxima and minima. Pseudosymmetry implies that particles trapped near
the field maxima cannot transition by drifting radially and ensures the
confinement of deeply-trapped particles.

  3. 3.

Isodrasticity [67] is formulated as a condition on the surface $\Sigma^{-}$,
defined by the local maxima of the field strength along a field line
($\hat{\textbf{b}}\cdot\nabla B=0$ and
$\hat{\textbf{b}}\cdot\nabla\left(\hat{\textbf{b}}\cdot\nabla B\right)<0$). On
$\Sigma^{-}$, the isocurves of $J_{\|}$ and $B$ must coincide. Isodrasticity
prevents transitions due to the bounce-averaged radial or in-surface drifts as
the transition probability is proportional to [68, 18],

$D\propto\frac{\partial
J_{\|}(s,\alpha,\lambda)}{\partial\alpha}\displaystyle\frac{\partial
B(s,\alpha)}{\partial s}\displaystyle-\frac{\partial
J_{\|}(s,\alpha,\lambda)}{\partial s}\displaystyle\frac{\partial
B(s,\alpha)}{\partial\alpha}\displaystyle,$ (20)

where $\alpha$ is a field line label and $D$ is evaluated on $\Sigma^{-}$. If
condition (i) is satisfied, then $\partial B/\partial\alpha=0$ on the
intersection of $\Sigma^{-}$ with a magnetic surface. If condition (ii) is
satisfied, then $\partial J_{\|}/\partial\alpha=0$ on $\Sigma^{-}$. If both
(i) and (ii) are satisfied, then the transition probability vanishes. However,
the condition of isodrasticity is more general than enforcing (i) and (ii).

Figure 29: Normalized variation across field lines of the maximum field
strength along the field line. On each surface, $\Delta B_{\mathrm{max}}$ is
the maximum deviation of $B_{\max}$ from the average over field line label.
This value is normalized by the average $B_{\max}$ over the field line label.

###  5.5 Drift-convective transport

A significant fraction of trajectories is lost on drift-convective orbits,
especially in the QH configurations (Figure 16). Drift-convective transport
arises for a class of particles that do not precess quickly in comparison to
their radial drift, quantified by large values of $|\gamma_{\mathrm{c}}|$, see
(1). While the bounce-averaged radial drift will depend on the specific
perturbation applied to a given equilibrium, the bounce-averaged poloidal
drift will be largely driven by the equilibrium field [69]. Under the vacuum
field assumption, the poloidal magnetic drift,
$\textbf{v}_{\mathrm{m}}\cdot\nabla\theta$, is proportional to $\partial
B/\partial s$ while $\textbf{v}_{\mathrm{m}}\cdot\nabla\zeta=0$. As $|\partial
B/\partial s|$ scales inversely with the aspect ratio, more compact
configurations may be advantageous [56] for avoiding DC transport. Increasing
$|\partial B/\partial s|$ can also be accomplished by peaking the pressure
profile in some configurations. The resulting increased precession leads to
improved fast ion confinement with increasing pressure in some W7-X
configurations [27], while fast ion confinement decreases in high-beta
extrapolated LHD configurations [70]. As the QA configurations under
consideration are more compact, we observe that proportionately more losses
terminate on a DC orbit in the QH than the QA configurations (Figures 5 and
16). As the precession drift decreases with increasing normalized flux, more
DC trajectories are likely to be found near the plasma boundary. This can be
seen in the DC trajectories (Figures 6, 17, and 18) which demonstrate
increasing values of $|\gamma_{\mathrm{c}}|$ as the particle drifts radially.

Many techniques have been discussed in the literature to prevent DC transport
[71]. Alignment of $J^{*}$, an approximation of $J_{\|}$, with flux surfaces
has improved the collisionless confinement in compact configurations [57].
Other techniques based on the alignment of $J_{\|}$ with surfaces improved
confinement of marginally-trapped particles in the W7-X configuration space
[27]. The $\Gamma_{\mathrm{c}}$ metric averages $|\gamma_{\mathrm{c}}|^{2}$
over velocity space [21, 33] in order to quantify the misalignment of flux
surfaces and $J_{\|}$ surfaces. Optimization including $\Gamma_{\mathrm{c}}$
has successfully improved confinement in QH [54] and QA [72] configurations.
However, the existence of DC orbits does not always indicate prompt losses.
For example, we find DC orbits that exhibit periodic motion that are confined
for long time scales (Figure 14 and 24). To this end, the metric
$\Gamma_{\alpha}$ improves over the $\Gamma_{\mathrm{c}}$ metric. Some degree
of misalignment between $J_{\|}$ surfaces and flux surfaces is not penalized,
and the possible closure of $J_{\|}$ contours is considered. The
$\Gamma_{\alpha}$ model also accounts for the in-surface precession drift onto
unconfined orbits [1].

In Figure 30, we compare our classification of DC orbits with the prediction
of DC losses based on $\Gamma_{\mathrm{c}}$ and $\Gamma_{\alpha}$ evaluated
with the KNOSOS code [73] on the $s=0.3$ surface. Here we define DC banana
losses as trajectories which are classified as banana trapped with
$|\gamma_{\mathrm{c}}|>0.2$ for at least one bounce segment. In the
computation of $\Gamma_{\alpha}$, the threshold $|\gamma_{\mathrm{c}}|>0.2$ is
also used to identify drift-convective losses. While both metrics should be
correlated with the DC banana loss fraction, only $\Gamma_{\alpha}$ provides a
quantitative estimate of the loss fraction. For this reason in Figure 30 we
include the line $y=x$ in the right plot in order to compare the magnitude of
the DC banana losses with $\Gamma_{\alpha}$. We see that $\Gamma_{\alpha}$ is
indeed a better predictor of the magnitude of DC banana losses than
$\Gamma_{\mathrm{c}}$. However, we see little correlation between either
metric and the DC loss fraction. Both the $\Gamma_{\mathrm{c}}$ and
$\Gamma_{\alpha}$ models are limited as they are radially local. Furthermore,
they do not account for class transitions and possible unconfined trajectories
due to the variation of the minimum of the field strength along a field line.

If considering configurations within the same class, $\Gamma_{\alpha}$ and
$\Gamma_{\mathrm{c}}$ appear to have better correlations with GC confinement.
Good correlation has been observed for a set of QA equilibria close to li383
[72] as well as for W7-X configurations [1]. These results suggest that these
metrics may perform well as optimization criteria, although the correlation is
not as good when comparing across configuration types.

Figure 30: Comparison of banana drift-convection (DC) loss fraction with the
(left) $\Gamma_{\mathrm{c}}$ and (right) $\Gamma_{\alpha}$ metrics.

##  6 Summary

We have developed classification methods to investigate the relevant
collisionless guiding center loss mechanisms of fusion-born alpha particles in
equilibria close to quasiaxisymmetry (QA) and quasihelical (QH) symmetry.

###  6.1 Prompt loss mechanisms

In each configuration, the dominant fraction of the orbit is spent in the
banana trapping class without drift convection (DC) characteristics. These
characteristics indicate that $J_{\|}$ contours align well with flux surfaces
and convective radial transport is small for much of the orbit. DC banana
transport is also significant throughout the trajectories in each
configuration except Ku5. For the QA configurations and HSX, ripple trapping
additionally makes up a substantial fraction of the orbits. Many barely-
trapped orbit segments, which can pass through at least one field period of
the field strength variation, are present in the QH configurations. Most
trajectories transition among these classes on short and long timescales.
Promptly lost orbits often terminate on a DC orbit, indicating that proxies
that align the contours of $J_{\|}$ with flux surfaces may reduce prompt
losses.

In the ITER and NCSX configurations, most promptly lost trajectories terminate
on a ripple-trapped segment. On the other hand, the prominence of ripple
transport is reduced in ARIES-CS due to the localization of ripple wells on
the inboard side [46]. Instead, many ARIES-CS losses terminate on a banana
orbit with DC or non-DC characteristics. Some non-DC orbits appear to exhibit
diffusive banana tip motion (Figure 8). A fraction of HSX losses also
terminates on non-DC banana orbit segments, which appear consistent with
diffusive banana tip motion (Figure 21). This mechanism is associated with
non-conservation of the parallel adiabatic invariant, $J_{\|}$.

Ripple trapping is reduced in the QH configurations due to their reduced
connection lengths. The exception is HSX, where higher mode number
perturbations due to coil ripple are present (Figure 2), increasing the
propensity for local well formation. Overall, the number of prompt losses is
reduced in the QH configurations compared to the QA configurations, given the
reduction of diffusive banana tip motion and ripple trapping. A similar
observation of the reduction of losses in QH in comparison to QA
configurations has been observed in other studies [10, 74].

Orbits lost due to barely-trapped trajectory segments are observed in
Wistell-A and ITER. In the case of Wistell-A, the significant variation of the
field strength maximum on the surface allows for an increased number of
transitions between banana and barely-trapped orbit types. These barely-
trapped orbits can be in the DC class (Figure 18), leading to prompt losses.
In the case of ITER, periodic barely-trapped orbits can resonate with the
$N=18$ coil ripple. A small number of resonant barely-trapped orbits are also
observed in HSX (Figure 19). No losses of passing particles are observed.

###  6.2 Non-prompt losses

On non-prompt timescales, we find that many trajectories transition between
trapping classes, as has been anticipated by theory [18] and numerical studies
[25, 75]. However, transitioning trajectories do not always imply rapid radial
diffusion. Some configurations feature irregular transitioning behavior
(Figures 11 and 22) accompanied by relatively small radial displacements.
Transitioning orbits may be confined for a long time but are subsequently lost
due to other mechanisms, such as diffusive banana tip motion (Figure 11).

There exist a class of particles that perform complex periodic orbits (Figures
12 and 23). Although $J_{\|}$ is not conserved along transitioning orbits, the
presence of complex periodic orbits hints toward the existence of a higher-
order adiabatic invariant [59, 60]. This observation indicates that
optimization metrics that minimize class transitions may be overly
restrictive.

On non-prompt timescales we also observe DC trajectories whose orbits are
healed due to closure of the $J_{\|}$ contours within the plasma boundary
(Figures 14 and 24). The orbits feature periodic radial displacements and
toroidal localization. The effective width of the DC orbit may imply
additional diffusive motion when collisions are included.

###  6.3 Optimization considerations

While precise quasisymmetry can yield excellent GC confinement [29], relaxing
the requirements on the symmetry level may accommodate the optimization for
other metrics of interest, such as microstability. To this end, we consider
equilibrium properties that can reduce alpha losses. The geometric dependence
of the transport mechanisms is summarized in Table 4. The banana tip diffusion
due to non-conservation of $J_{\|}$ is associated with short timescales of
precession through one period of a given perturbation compared to the bounce
period. In contrast with DC losses, which are present in the collisionless
dynamics even at low energy, the prevalence of this loss channel increases
with particle energy. This transport mechanism is reduced in configurations
with large values of the effective transform, $M\iota-N$, increased aspect
ratio, and reduced field strength variation on a magnetic surface (see (13)).

The propensity for forming ripple wells along field lines is reduced in
configurations with significant effective transform and decreased aspect
ratio. Perturbations localized near the field maxima and minima form wells
more easily. On the other hand, since the radial drift is minimal near the
field maxima and minima, particles trapped in these regions often transition
to deeply-trapped or barely-trapped banana trajectories (Figures 9 and 12)
rather than being promptly lost. Higher mode number perturbations, such as
those arising from coil ripple, are more deleterious for both ripple formation
and breakdown of $J_{\|}$ conservation. Such perturbations have also been
found to increase flow damping substantially [76].

In addition to the considerations above, banana or barely-trapped orbits that
resonate with a perturbation in the equilibrium should be avoided. Resonant
barely-trapped orbits have been observed in the HSX (Figure 19) and rippled
ITER (Figure 13) configurations, which arise in the neighborhood of surfaces
with low-order rational values of the rotational transform. If a configuration
has low magnetic shear throughout the volume, as in the case of HSX (Figure
3), this can lead to the persistence of the orbit periodicity through the
volume [28].

As DC transport is present in all six equilibria, we compare the number of DC
losses with the $\Gamma_{\mathrm{c}}$ and $\Gamma_{\alpha}$ metrics based on
the radially-local bounce-averaged motion (Figure 30). Due to the prevalence
of class transitions and finite orbit width effects, the number of DC losses
can deviate significantly from the $\Gamma_{\alpha}$ metric. However, it
captures the order of magnitude and approximate scaling. The main limitation
of $\Gamma_{\mathrm{c}}$ and $\Gamma_{\alpha}$ is the assumption of radial
locality, which cannot be justified in a naturally radially-global problem.
Implementation of a radially-global drift kinetic equation in the KNOSOS code
is underway to overcome this limitation.

| Wide banana | Diffusive banana | Ripple | Drift convection
---|---|---|---|---
Aspect ratio | Increases | Decreases | Increases | Increases
$|M\iota-N|$ | Decreases | Decreases | Decreases |
$m_{\epsilon}$, $n_{\epsilon}$ |  | Increases | Increases |
Table 4: Summary of magnetic configuration impact on transport mechanisms.
“Increases”/“decreases” indicates that increasing the given parameter
increases/decreases the corresponding transport mechanisms. No entry implies
no apparent dependence. The vacuum assumption is assumed for a quasisymmetric
field with helicity $M$, $N$ subject to a perturbation with mode numbers
$m_{\epsilon}$, $n_{\epsilon}$ as described in (10).

##  7 Discussion and conclusions

We highlight the following conclusions from our analysis of loss mechanisms.

  * •

While the quasisymmetry levels in the six equilibria are of comparable
magnitudes (except for ITER), the GC confinement properties are markedly
distinct. This behavior is partially accounted for by differences in the
equilibria that provide natural protection against ripple wells and diffusive
transport. As has been observed in several studies [10, 74], QH configurations
enjoy reduced losses due to their short connection lengths. In addition, the
spatial localization of quasisymmetry errors is an important consideration.
Ripple localized near the field maximum and minima tends to promote
transitions between ripple and banana classes, while in other regions it can
more easily yield prompt ripple losses. This feature could be accounted for in
coil optimization problems or the definition of a spatially-localized
quasisymmetry metric.

  * •

The transport mechanisms responsible for both prompt and long-timescale losses
differ widely between equilibria. Therefore, analyzing the relevant loss
mechanisms should inform further optimization for improved confinement.

  * •

Non-conservation of $J_{\|}$ is prevalent among both prompt and long-timescale
losses due to class transitions, resonances, or significant deviations of the
field strength well shape along a trajectory. This can lead to additional
diffusive effects on DC orbits (Figure 7) and additional prompt losses due to
rapid banana tip diffusion (Figure 8). Diffusion across $J_{\|}$ surfaces may
contribute toward convective losses by enabling transport onto lost orbits.
Quantifying the impact of non-conservation of $J_{\|}$ will require a
comparison of GC with orbit-averaged calculations, which will be explored in
future work. Radially local metrics for fast ions based on the bounce-averaged
motion may not always be applicable. As the radially-global bounce-averaged
guiding center equations are valid in a $\rho_{*}\ll 1$ expansion, the
breakdown of $J_{\|}$ adiabaticity may indicate the eventual breakdown of the
GC approximation. Therefore, full-orbit simulations may eventually be required
to model the dynamics faithfully.

  * •

Many prompt losses are not lost on DC banana orbits but on diffusive banana,
ripple, or resonant orbits. Some of the non-DC banana losses can be mitigated
by improving the alignment of $J_{\|}$ surfaces with flux surfaces, while
other losses require different avenues for improving prompt loss metrics in 3D
configurations. One approach is integration over the full guiding center
motion to identify diffusive or DC mechanisms rather than relying solely on
bounce-averaged metrics. Similar objectives involving the full guiding center
motion have been employed [77], for example, in the design of ARIES-CS [24].
Another approach is directly targeting integrability in the banana tip map
[14]. Metrics similar to those used to eliminate magnetic islands in
stellarators, such as Greene’s residue [78], may be applicable.

  * •

It is commonly assumed that the diffusive transport associated with many
separatrix crossings is responsible for long-timescale losses [3, 75]. While
this mechanism is present, it is often the same mechanisms that lead to prompt
losses (ripple trapping, DC, or diffusive banana-drift motion) that result in
the eventual loss of the trajectory. Thus reducing these prompt loss channels
may consequently improve the long-timescale behavior. Furthermore, transitions
often appear in a periodic rather than irregular fashion.

  * •

Several features of GC confinement are common to stellarators close to QS and
rippled tokamaks, including the prevalence of ripple trapping, diffusive
banana tip motion, and collisionless detrapping processes. On the other hand,
resonant barely-trapped orbits are more prevalent in ITER and are responsible
for many non-prompt losses. In the stellarator equilibria, such resonant lost
orbits are not as common, although this may be a potential loss mechanism in
other equilibria [28]. The stellarator equilibria also uniquely feature
irregular transitioning behavior, while the rippled ITER equilibrium features
only periodic transitioning. Furthermore, the ITER configuration features
enhanced losses over several QH stellarators on long timescales. This behavior
is due to the high mode-number perturbations arising from coil ripple that
enables enhanced non-adiabatic losses, ripple trapping, and resonant barely-
trapped losses. To compare tokamaks and stellarators on equal footing would
require free-boundary stellarator equilibria close to quasisymmetry.

Several physical effects may introduce additional subtleties.

  * •

We have not included collisions and instead consider the dynamics on
timescales shorter than a collisional slowing-down time ignoring the pitch-
angle scattering. As collisions are more likely to modify the dynamics on long
timescales, additional diffusive mechanisms may arise due to wide periodic
transitioning orbits (Figure 23) or periodic DC orbits (Figure 24).

  * •

Throughout, we have assumed that continuously nested magnetic surfaces exist
in each equilibrium. However, fast particle losses can be enhanced when flux
surfaces are destroyed [26]. Additional transport mechanisms will likely arise
in this case due to the interaction of particle trajectories with a magnetic
island or chaotic magnetic field lines [79, 80].

  * •

We have assumed that a particle is considered lost at the plasma boundary.
This assumption may lead to overestimating the losses due to re-entering
trajectories. A more realistic quantification of the losses in a reactor
requires modeling of plasma-facing components [81, 4] as orbits can generally
re-enter the plasma volume. Such an analysis may provide insight into the
localization of losses on the first wall and possible damage to material
structures.

  * •

We have neglected the possible interaction of fast particles with
fluctuations, such as Alfvénic instabilities. It has been suggested that
because stellarators can operate at higher density, the relative alpha
particle pressure will be small enough that the instability drive will be
sufficiently small [82]. On the other hand, additional gaps in the continuum
spectrum arise in the presence of 3D fields [83]. Furthermore, there may be
different loss mechanisms not considered in this paper, caused by the
breakdown of adiabatic invariants due to wave-particle resonances. Further
analysis is needed to determine how these interactions will impact stellarator
reactors.

## Acknowledgements

The authors acknowledge fruitful conversations with R. White, H. Mynick, A.
Boozer, R. Mackay, and F. Parra. A. Bader provided the Wistell-A, Ku5, NCSX,
and ARIES-CS equilibria and assistance with benchmarking of the SIMSOPT code.

EJP was supported by the Presidential Postdoctoral Research Fellowship at
Princeton University. This research was also supported by the Simons
Foundation/SFARI (560651, AB and ML). Some calculations were performed using
computing resources at Princeton Plasma Physics Laboratory under DOE contract
DE-AC02-09CH11466.

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