Counterintuitive Particle Confinement in a Helical Force-Free Plasma

Abstract

The force-free magnetic field solution formed in a high-aspect ratio cylinder is a non-axisymmetric ($m=1$), closed magnetic structure that can be produced in laboratory experiments. Force-free equilibria can have strong field gradients that break the usual adiabatic invariants associated with particle motion, and gyroradii at measured conditions can be large relative to the gradient scale lengths of the magnetic field. Individual particle motion is largely unexplored in force-free systems without axisymmetry, and it is unclear how the large gradients influence confinement. To understand more about how particles remain confined in these configurations, we simulate a thermal distribution of protons moving in a high-aspect-ratio force-free magnetic field using a Boris stepper. The particle loss is logarithmic in time, which suggests trapping and/or periodic orbits. Many particles do remain confined in particular regions of the field, analogous to trapped particles in other magnetic configurations. Some closed flux surfaces can be identified, but particle orbits are not necessarily described by these surfaces. We show examples of orbits that remain on well-defined surfaces and discuss the statistical properties of confined and escaping particles.

1. Introduction

Approximately force-free magnetohydrodynamic (MHD) equilibria are relatively simple to produce in a laboratory and are intriguing because of their self-organized nature and topological properties. Although the large-scale structure is well-studied, individual particle motion remains largely unexplored. Pioneered by Woltjer [1] and Taylor [2,3], among others, so called “Taylor states” are structures whose magnetic energy is a minimum for a given magnetic helicity. The magnetic field structure of these plasmas is described by the eigenvalue equation

$$\nabla \times \overrightarrow{B}=\lambda \overrightarrow{B},$$

(1)

derived from the magnetohydrodynamic equations (MHD). The plasma current is parallel to the field (no $\overrightarrow{J}\times \overrightarrow{B}$ force), so no pressure gradient is required for equilibrium. In the laboratory, plasma objects with initial helicity injected into a volume with a conducting boundary form Taylor states in a manner consistent with selective relaxation [4].

Similar relaxed plasma structures are hypothesized to be relevant to astrophysical structures, including jets, stellar eruptions, and radio lobes [5,6,7]. Processes in the solar corona are often modeled using force-free fields because gravity and pressure are negligible to lowest order [8]. Simulations have used both linear (constant $\lambda$) [9,10,11] and nonlinear (spatially varying $\lambda$) [12,13,14] force-free states to model solar eruptions. Understanding how particle motion is and is not described by MHD can help illuminate the processes involved in coronal evolution.

Force-free configurations are also of interest as targets for magneto-inertial fusion (MIF) [15,16,17] because they are typically self-organized. MIF combines aspects of compression schemes (inertial confinement fusion) and magnetic schemes (magnetic confinement fusion) to relax the requirements of each [18]. Targets in MIF schemes are typically warm, stable, small magnetized plasmas such as field-reversed configurations (FRCs) and spheromaks [19]. Understanding particle confinement in these target plasmas is necessary in order to choose the appropriate strategies and timescales for compression.

The most familiar of the Taylor states is perhaps the spheromak [20], which has been studied extensively in both fusion [21] and astrophysical contexts [22]. Spheromaks are axisymmetric ($m=0$) and become unstable in prolate cylinders with $L/a\gtrsim 1.67$ [23,24], where L and a are the length and radius of the boundary. If the aspect ratio of the boundary is larger than $L/a=1.67$, the lowest energy state is no longer the spheromak, but the non-axisymmetric ($m=1$) Taylor helix [2], “plectonemic Taylor state” [25], or simply plectoneme. Work on the Swarthmore Spheromak eXperiment has investigated plectonemes in the laboratory and the measured magnetic structure is very well described by MHD [26,27,28].

Axisymmetric MHD confinement schemes exhibit closed surfaces of constant magnetic flux or “flux surfaces” that are largely predictive of confinement properties [29]. Non-axisymmetric schemes can also exhibit flux surfaces, but identification is more difficult. The plectoneme provides a unique perspective on non-axisymmetric confinement, since the magnetic structure is self-organized rather than externally imposed as in a stellarator. Despite lacking axisymmetry, plectonemic Taylor states have shown slower cooling than anticipated and relatively long lifetimes that agree well with inductive decay [30]. The MOCHI experiment [31] recently demonstrated that even the conducting boundary need not be externally imposed; a cylindrical plasma jet also effectively confined a stable $m=1$ plectoneme [25].

As a step towards understanding the confinement properties of the plectonemic Taylor state, we simulate thermal proton trajectories in the computed fields of 10:1 aspect ratio plectoneme ($\lambda a=3.15$) relevant to SSX experiments [27]. In Section 2, we describe the simulation method, the magnetic structure of the plectoneme, and the validation strategy. In Section 3, we give an overview of the statistical confinement of a sample of simulated particles, discuss specific classes of orbits that exhibit unexpected characteristics, and investigate the role of strong gradients. In Section 4, we discuss our interpretation of the results and remaining questions we hope to address.

2. Materials and Methods

2.1. Orbit Simulation Algorithm

The Lorentz equation of motion for a charged particle in an electromagnetic field can be written as follows (in MKS units):

$$m{\displaystyle \frac{d\overrightarrow{v}}{dt}}=q(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}),$$

(2)

where m is the mass of the particle, q is the electric charge of the particle, v is the velocity of the particle, and $\overrightarrow{E}$ and $\overrightarrow{B}$ represent the electric and magnetic field values at the particle position, respectively. After re-casting in dimensionless form, normalizing charge and mass to proton values, and setting $\overrightarrow{E}=0$ for our magnetostatic field, we write this as

$${\displaystyle \frac{d\overrightarrow{v^{\prime }}}{d{t}^{\prime }}}=\overrightarrow{v^{\prime }}\times \overrightarrow{B^{\prime }}.$$

(3)

Time is scaled to the proton cyclotron time for thermal protons, magnetic field is scaled to a maximum of one in the eigenvalue calculation ($B_0=1$), and distances are scaled to thermal proton Larmor radii. From this point on, we drop the primes for clarity but continue to use normalized variables. Thermal protons thus have a velocity of unity and a Larmor radius of unity in a unit magnetic field: ${\rho }_0=m{v}_{\mathrm{th}}/q{B}_0=1$. Lengths are normalized to this fiducial orbit radius and times are normalized to the thermal proton orbit time in a unit magnetic field: ${\tau }_0=2\pi {\rho }_0/v_{\mathrm{th}}$. For scale, typical SSX parameter values are $B_0\sim 0.25\mathrm{T}$, $T_i\sim 10\mathrm{eV}$, ${\rho }_0\sim 1\mathrm{mm}$, and $f_{ci}=1/{\tau }_0\sim 4\mathrm{MHz}$.

We use the Boris stepper to integrate particle orbits [32], a standard method used for pushing particles in magnetic fields. The stepper corrects the error from the nth iteration in the $n+1$th iteration, resulting in stability. Furthermore, the algorithm is volume-preserving, and while not symplectic, conserves energy within fixed error bounds [33]. We reproduced orbits in canonical magnetic configurations like the spheromak and dipole fields to validate our code. The level of accuracy is very high for our chosen time step: variation in the radius of the orbit in a uniform field was at most $0.004\%$ using a time step of $\Delta t=0.01{\tau }_0$. All orbit integrations described in this work use $\Delta t=0.01{\tau }_0$.

2.2. Field Structure of the Taylor State in a Long Cylinder

The field structure is calculated by finding the lowest eigenvalue of Equation (1) using the PSI-Tet code [34,35]. This code, which is now available as the Marklin package as part of the open-source Open FUSION Toolkit [36], uses a high-order H(Curl) finite element method to solve for force-free uniform $\lambda$ equilibrium in general 3D geometries. Although a series solution to Equation (1) can be written down using Bessel and trigonometric functions, the non-axisymmetric solutions require an infinite number of terms [23]. The finite-element computational solution is therefore expedient.

We use a perfectly conducting, 10:1 aspect ratio ($L/a=10$) cylinder as the boundary condition, similar to the physical flux conserver of the SSX device [27]. We choose a simulation volume of radius $100{\rho }_0$ and length $1000{\rho }_0$ to model SSX, based on the fact that a thermal proton at the maximum detected field in the machine is of order $1\mathrm{mm}$ and the cylindrical flux-conserver on the machine has a radius of $10\mathrm{cm}$ [28].

To illustrate that the simulated field structure closely matches experiment, we reproduce Figure 2 of Reference [26] in Figure 1a,b. In that work, a lower-aspect-ratio boundary was used, but the lowest-energy equilibrium is a similar $m=1$ plectoneme. Our aspect ratio was chosen for convenience, with the $\lambda a$ value within 0.5% of more recent experimental work [27]. A rendering of the force-free field structure (lowest $\lambda$) in the 10:1 cylinder ($\lambda a=3.15$) is shown in Figure 1c.

There are two degenerate states with this minimum eigenvalue, one of each handedness. We have chosen the left-handed solution for the simulations described in this work. The structure resembles a closed flux rope twisted into a helix. Field magnitudes are strongest near the center of the configuration and weaken significantly towards the ends of the cylinder.

We sample the calculated plectoneme solution on a grid and use tri-linear interpolation to find the field values at particle locations. Particle orbits simulated in a spheromak field generated from analytic expressions were compared with those in a gridded and interpolated spheromak field to validate the interpolation. Higher-order schemes were investigated and found to be unnecessary in the current study.

3. Results

3.1. Flux Surfaces

Axisymmetric magnetic fields are guaranteed to have closed flux surfaces [37], but fields that lack axisymmetry may or may not have closed flux surfaces. Previous work with similar objects has hinted at the presence of closed flux surfaces in the Taylor helix [25,26]. We do not attempt a comprehensive description of flux surfaces in the present work, but in order to make a comparison with particle trajectories, we illustrate some flux surfaces identified by field-line tracing. Starting with a Poincaré puncture plot for the magnetic field, we traced representative field lines where the punctures were most dense. Figure 2a shows the Poincaré puncture plot for field lines crossing the x-y plane (perpendicular to the cylinder axis) at $z=500{\rho }_0$ (halfway between the ends of the cylinder). Figure 2b shows three representative flux surfaces, one corresponding to each major clump of dense punctures for $x>0$. The colormaps are chosen arbitrarily to differentiate the three surfaces.

In contrast to the nested flux surfaces of more familiar magnetic geometries, the flux surfaces of the $m=1$ plectoneme are interlinked flux tubes. In future work, we hope to understand what distinctive properties these non-nested flux surfaces may have.

3.2. Statistical Properties of Proton Orbits in the $m=1$ Plectoneme

To obtain a sense of the general confinement properties of the plectoneme, we traced particles starting from two different grids of positions whose velocities are chosen from a Maxwellian distribution. To gain insight into any non-axysymmetric behavior at a granular level, we truncate an 8 by 8 by 7 rectangular grid to sample from $z=10{\rho }_0$ to $z=500{\rho }_0$ across the diameter of the cylinder (308 positions total). No particle starts within $5{\rho }_0$ of a boundary or the cylinder axis. We also set up a cylindrical grid in order to sample evenly in radius, with five radial positions, nine azimuthal positions, and six axial positions (270 total). At each position 100 particles were simulated, drawing a new pseudorandom velocity (magnitude and direction) from the thermal distribution for each. Simulations were carried out for a total of ${10}^4$ fiducial orbits at a time step of $\Delta t=0.01{\tau }_0$. Figure 3 shows our rectangular grid of initial points superposed on the field structure, with each initial point colored by the percentage of particles that remain confined for the duration of the simulation. Particles starting close to the edge almost always escape, while particles close to the center are much more likely to remain confined.

The fraction of particles confined as a function of time for both sets of initial positions is shown in Figure 4a. The loss of particles is initially rapid and slows down as the simulation progresses. Aside from a vertical offset, the confinement curve for the two grids of initial positions is nearly identical. We attribute this offset to the rectangular grid’s oversampling of large radii, where particles are more likely to escape quickly. In the rest of the paper, confinement is assessed using the cylindrical dataset, unless otherwise noted. In order to separate the rapid initial decay from the subsequent longer decay, we define a stabilization time, ${\tau }_s$, where the confinement fraction has dropped by $1/e$. We choose this convention because the average energy of particles that escape in any given time interval stabilizes by approximately ${\tau }_s$ (see Figure 5).

Plotting on a logarithmic time axis shows that the loss fraction grows logarithmically, at about 20% per decade of fiducial orbits, as shown in Figure 4b. We do not yet have a model for this behavior, but some work in dynamical systems suggests that logarithmic growth or decay can be a result of trapping. Random walks, where actors preferentially revisit sites they have already occupied (“reinforced walks” [38]), can exhibit logarithmic diffusion [39,40]. This suggests that particles confined for long times are trapped to subregions, which they traverse more than once.

Figure 5 shows the initial particle energy versus confinement time for all escaping particles. We sort particle initial conditions by confinement time and then use a Gaussian window of width $\sigma ={\tau }_0$ (1 fiducial orbit) to calculate the moving average (blue solid line).

The average energy of escaping particles quickly approaches the initial average energy, ${\displaystyle \frac{3}{2}}kT$ (blue solid line), while the average energy of particles confined for the duration of the simulation have an average energy of ${\langle U_0\rangle }_{\mathrm{conf}}\sim 1.1kT$ (horizontal dashed line). The first 37% of the particles lost are much more energetic than the remaining particles (${\tau }_s$ is indicated by the vertical dotted line), but otherwise there is no strong trend. We interpret this as evidence that differences in energy distributions between escaped and confined particles are not an artifact of too short a simulation time.

The energy distributions for escaped and confined particles are shown in Figure 6. The escaped particles are separated into those that escaped before ${\tau }_s$ and those that escaped after that. Figure 6a shows the stacked particle distributions, while Figure 6b shows the percentages of particles with each energy in each class.

As might be expected, lower energy particles are more likely to be confined, and the average escaping particle is higher in energy than the average confined particle. Although the total particle loss will likely continue to increase if the simulation is run longer, the statistical properties of the different classes of particles are stable by the end of the simulation (i.e., the total simulation time is sufficient).

One interesting aspect of the confinement is that particles that start out parallel to the magnetic field are very unlikely to remain confined. Figure 7 shows distributions of initial pitch angle for each class of particle. As above, Figure 7a shows the stacked fractions of particles in each class, while Figure 7b shows the fractions of particles with each initial pitch angle. Because the initial velocities are selected at random from the entire sphere of possible directions, very few particles start out parallel to the field.

Confined particles are most likely to have initial pitch angles near $\pi /2$, while particles that escape rapidly are more likely to have pitch angles close to 0 or $\pi$. This suggests that confinement in the plectoneme may have aspects similar to confinement in a magnetic mirror, where particles with pitch angles too close to parallel (inside the loss cone) escape the magnetic field [41]. However, as we show in Section 3.3, many of the trajectories that exhibit long-term confinement are inconsistent with mirror forces.

We might expect that particles with large initial gyroorbits might escape more quickly, which is true to an extent but not absolute. Figure 8 shows distributions of initial gyroradii for escaped and confined particles. As above, Figure 8a shows the stacked fraction of particles in each confinement class. Figure 8b shows the distributions of particles with each initial gyroradius.

The simulation domain has a radius of $100{\rho }_0$, so most particles that remain confined start with gyromotion on a scale below 5% of the domain size. The tail of the ‘escaped-early’ distribution continues well past the edge of the plots, as the maximum initial gyroradius in the sample of particles is $160{\rho }_0$. Since these are not the particles of interest for this study, we truncate the escaped-early distribution in Figure 8. The average gyroradius versus confinement time (not shown) shows a trend similar to that of the average particle energy (Figure 5).

Examining the locations where particles hit the boundary of the simulation volume indicates that there are preferred pathways for escaping particles. Figure 9 shows the exit positions of all of the escaping particles (rectangular grid), where the color of each position indicates the number of iterations that particle remained confined before hitting the boundary. Particles that escape very early in the simulation, which are statistically at much higher energies and larger gyroradii (c.f. Figure 5, Figure 6 and Figure 8), travel ballistically through the field structure and hit the walls. After that, particles escape preferentially in the “gaps” between turns of the helix where the field has a large radial component.

3.3. Observed Trajectories

Two categories of confined particle orbits are particularly salient in the simulation data. Both categories of particle remain confined to ribbon-like regions that do not conform to flux surfaces. We call these regions “confinement surfaces”. Category 1 trajectories are confined predominantly between the twists of flux rope that make up plectoneme. Category 2 trajectories are confined predominantly inside the flux rope. Despite remaining confined on well-defined surfaces, neither category of particle exhibits the guiding center motion we would expect when adiabatic invariance holds. An example of a Category 1 orbit is shown in Figure 10.

In Figure 10a, the particle trajectory (yellow/lighter value) is shown with one of the flux surfaces identified above (blue/darker value). Although the trajectory seems to follow the helical pitch of the flux surface, the particle’s path is removed from the surface itself and in fact passes through several different flux surfaces. The particle trajectory, along with all three flux surfaces from Figure 2, is shown in Figure 10b. The view is rotated ${90}^{\circ }$ about the long axis of the cylinder to show the intersections of the particle orbit with various flux surfaces.

A striking feature of the Class 1 trajectories is the apparent reflection of the trajectory near the outer radius of the volume, which is not consistent with the mirror force at all. Mirror confinement and other processes that depend on adiabatic invariance break down at many locations in the volume, as we explore further in Section 3.4. Breakdown of adiabatic invariance is to be expected in a configuration with significant curvature, like the plectoneme. However, it is interesting that the turning points are precisely where the field strength, $B=|\overrightarrow{B}|$, is decreasing towards the edge. Figure 11 shows the same trajectory with isosurfaces of B at 10%, 40%, and 70% of the maximum field strength.

The radial turning points of this motion do coincide with regions of strong gradients in the field strength, but in the opposite direction from traditional mirroring. Figure 12 shows the same trajectory as in Figure 10 and Figure 11 along with an isosurface of $|\nabla B|$ at 75% of its maximum.

Our working theory is that a combination of the strong gradient between the flux ropes and the increase in gyroradius as the particle approaches the boundary allows particles to cross from the edge of one piece of flux rope to the edge of the neighboring piece. The radial motion of the guiding center is reversed during this crossing so that the particles remain confined. Only a subset of the simulated trajectories experience this type of “reflection”. We do not yet know whether the process is effectively statistical or deterministic and more work is needed to determine what conditions must be met for it to take place.

An example of a Category 2 orbit is shown in Figure 13, showing that it appears to be contained in the flux rope itself. At first glance, this trajectory exhibits more typical guiding center motion, consistent with the usual drifts, including analogs of the azimuthal, and bounce motion seen in a dipole field. However, the turning points for the motion in the z-direction (cylinder axis) in Figure 13 are in regions of lower field: the field strength at labeled points A and C is lower than at B.

Despite the appearance of confinement within the flux rope, Category 2 orbits do not conform to flux surfaces any more than do Category 1 orbits. Figure 14 shows this second example trajectory along with the flux surfaces from Figure 2. As above, Figure 14a shows the particle trajectory with a single flux surface, while Figure 14b shows the particle trajectory with all three flux surfaces in a rotated view.

Both of these example trajectories seem to lie on surfaces that are well-defined and match the pitch of the magnetic structure. However, both trajectories pierce the identified flux surfaces, which rules out the existence of an unidentified flux surface coincident with the confinement surface.

3.4. Role of Strong Gradients

Because the experimental plasma is so well-described by MHD, we might expect any confinement surfaces to be determined by adiabatic invariants. On the other hand, the orbit sizes are often large compared to the gradient scales in the field and the usual adiabatic invariants are broken in many regions. As shown above, none of the initial parameters predict confinement in a deterministic way but the turning points of many orbits are in regions of high magnetic field gradient. A further piece of evidence that the mechanism is not a usual guiding center effect is the variation of the particle magnetic moment along the trajectory. Figure 15 shows a rendering of the Category 1 trajectory in Figure 10, colored by the instantaneous value of the particle magnetic moment, $\mu$. The value of $\mu$ changes by more than a factor of 10 during the particle’s motion, and the changes in some regions occur over lengthscales smaller than the local gyroradius. The time evolution of $\mu$ for a small slice of the same trajectory (approximately two bounces across the diameter) is shown in Figure 16.

Figure 16a shows the instantaneous value of $\mu \left(t\right)$, colored by the elapsed simulation time. A gyro-averaged value is also shown, calculated by a Gaussian moving average with width ∼ average orbit time for this section of the trajectory. There are rapid changes in $\mu$ on the timescale of the gyromotion, as well as slower changes as the guiding center executes the bounce motion. Figure 16b shows the slice of the trajectory in the relevant portion of the simulation volume. The trajectory is colored by elapsed time in both panels so that the reader may identify the location in space with the location in time. The magnetic moment is clearly not invariant on the relevant spatial or temporal scales. The same is true for the other two typical mirror invariants corresponding to the slower bounce and drift motions: the longitudinal or “second” invariant and the azimuthal or “third” invariant [41].

It is possible that other adiabatic invariants exist and describe the confinement surfaces that we observe. Especially for Category 1 trajectories, the turning points often occur near field-reversals. This means that we may see features like Speiser orbits [42] and we can draw on the large body of work describing particle motion in reconnection layers and other highly curved field geometries. For example, Büchner and Zelenyi [43] identified an adiabatic invariant associated with motion across a 2D magnetic reversal. In their slab geometry, the field away from the reversal is primarily in the $\pm x$ direction and the crossing motion is in the z direction. For particles with fast oscillations in z, slow motion along the field lines is governed by an invariant $I_A={\displaystyle \frac{1}{2\pi }}\oint \dot{z}dz$. Future work will investigate whether their result holds in our particles in this 3D-reversal geometry.

In the meantime, we have attempted to explain the effects of the strong gradients using a “collisional” model. Wherever a particle passes through a region where the magnetic field changes rapidly, we record a “gradient collision”, which could change the particle trajectory abruptly. For our model, “rapid” change means that the scale length for changes in the magnitude or direction of the magnetic field is smaller than the gyroradius of the particle at that location, $\rho$. In order to detect rapid changes in the field, we evaluate the scale lengths for the components of the tensor $\nabla \overrightarrow{B}$:

$$\nabla \overrightarrow{B}={\displaystyle \frac{\partial B_i}{\partial x_j}}={\partial }_j{B}_i=\left[\begin{array}{ccc}{\partial }_x{B}_x& {\partial }_y{B}_x& {\partial }_z{B}_x\\ {\partial }_x{B}_y& {\partial }_y{B}_y& {\partial }_z{B}_y\\ {\partial }_x{B}_z& {\partial }_y{B}_z& {\partial }_z{B}_z\end{array}\right]$$

(4)

For each column of the tensor, we divide the field component $B_i$ by the above elements to construct a matrix of gradient scale lengths:

$$\mathit{L}=\left[\begin{array}{ccc}{\displaystyle \frac{B_x}{{\partial }_x{B}_x}}& {\displaystyle \frac{B_x}{{\partial }_y{B}_x}}& {\displaystyle \frac{B_x}{{\partial }_z{B}_x}}\\ {\displaystyle \frac{B_y}{{\partial }_x{B}_y}}& {\displaystyle \frac{B_y}{{\partial }_y{B}_y}}& {\displaystyle \frac{B_y}{{\partial }_z{B}_y}}\\ {\displaystyle \frac{B_z}{{\partial }_x{B}_z}}& {\displaystyle \frac{B_z}{{\partial }_y{B}_z}}& {\displaystyle \frac{B_z}{{\partial }_z{B}_z}}\end{array}\right]=\left[\begin{array}{ccc}{L}_{xx}& L_{xy}& L_{xz}\\ L_{yx}& L_{yy}& L_{yz}\\ L_{zx}& L_{zy}& L_{zz}\end{array}\right]$$

(5)

Wherever the minimum length scale is less than the local gyroradius, $\min \left(\mathit{L}\right)<\rho$, we say a gradient collision is occurring.

As a first look at this framework, Figure 17a shows histograms of the fraction of time particles spend in gradient collisions (large gradient regions).

Particles are grouped as above according to their confinement status: escaped before $t_s$, escaped after $t_s$, confined for the duration of the simulation. Although the distributions are clearly distinct, all three still overlap. There is also no apparent correlation between the number of gradient collisions and the confinement time, as shown in Figure 17b. The strong gradients in field magnitude do not seem to be an obvious means of improving nor degrading confinement and more work is needed to understand the directionality of the gradients relative to the particle motion.

4. Discussion

Although plectoneme fields are very well described by MHD, gyroradii can be large relative to the gradient scale lengths of the magnetic field. The usual adiabatic invariance does not hold, so understanding the confinement properties involves investigating full particle trajectories. The particle loss is logarithmic in time, which indicates that there is some localized trapping. Some closed flux surfaces can be identified, but particle orbits need not lie on these surfaces. Instead, some particles remain confined to open geometrical surfaces that intersect with multiple flux surfaces and are not indexed by a single-particle property. Future work will investigate the possibility of new adiabatic invariants, as in Reference [43], and explore the role of non-adiabatic MHD invariants. Just as the total magnetic helicity in a volume is conserved in ideal MHD, perhaps the observed confinement surfaces are described by a volume-integrated quantity rather than a local quantity. Our next step is to approach particle confinement from the perspective of canonical vorticity [44] in order to ask questions about global and topological invariants.