Mechanisms of energetic-particle transport in magnetically confined plasmas

Superthermal particles occur frequently in both natural and laboratory plasmas. Although all plasma particles are energetic by ordinary standards, in this tutorial, an “energetic particle” (EP) has two properties: (1) the energy is substantially greater than the bulk plasma temperature and (2) Coulomb collisions cause negligible deflections on the timescale of a single orbit in the confining magnetic field. In nature, EPs are produced when a rapidly drifting plasma merges with a colder plasma, for example, when the solar wind collides with the magnetosphere. Instabilities that accelerate ions or electrons to high energies are another common source in natural plasmas. In fusion plasmas, there are four common sources. Because Coulomb drag decreases with energy, a DC electric field that is parallel to the magnetic field can create “runaway” electrons that continuously gain energy if the electric field acceleration exceeds Coulomb drag. For ions, injection of energetic neutral beams is one important source. Acceleration by radio frequency waves at the fundamental ion cyclotron frequency or its harmonics is a second important source of fast ions. Charged fusion reaction products, such as the ⁴He alpha particles created by deuterium–tritium reactions, are the third important source of fast ions in fusion plasmas.

All of these EPs share common properties that distinguish them from thermal plasma. The distribution function of a thermal species is described by a Maxwellian, possibly a drifting Maxwellian or a Maxwellian with small distortions. As a result, fluid equations derived by taking velocity moments of the underlying kinetic equations are a sensible starting point for transport theory. This is not the case for EPs. Because the energies are high and EP densities are generally low, inter-species EP collisions are rare. Consequently, EP distribution functions have complicated dependencies on energy and direction. Because different EP velocities behave quite differently, a single-particle picture is the appropriate starting point for EP transport theory.

The purpose of this review is to introduce the key ideas of single-particle transport theory for EPs. Many of these mechanisms also apply to thermal particles but they are particularly important for EPs. The review is tutorial in nature, not comprehensive. No attempt is made to cite the first or most seminal work on a particular topic, or to reference every relevant paper. Rather, examples are selected for their clarity.

Detailed discussion of the instabilities that cause transport is beyond the scope of this review. From the perspective adopted here, DC electromagnetic fields $E 0$ and $B 0$ govern the unperturbed equilibrium EP orbits. An instability produces electric and magnetic fields $E 1$ and $B 1$ that perturb the orbits. Static perturbations associated with field errors or an additional field coil can also be considered a perturbing field. The perturbing fields have different frequency, spatial structure, and polarization but, regardless of origin, they may cause EP transport.

The perturbing fields cause transport in both velocity space and configuration space. For magnetic fusion, cross field spatial transport is of paramount concern but, for this review, motion in any phase-space direction is considered “transport.”

Calculation of the EP distribution function is also outside the scope. The focus here is on processes that alter EP orbits on a relatively short timescale. On a longer timescale, the distribution function is shaped by sources, sinks, collisions, and the wave-particle interactions considered here. A Fokker–Planck equation is often used to describe these processes. Well-known examples of fast-ion distribution functions derived from Fokker-Planck equations can be found in Ref. 1 for neutral-beam injection and in Ref. 2 for EP tails created by RF acceleration. Although Coulomb and other collisions play important roles in shaping the distribution function, they are only briefly discussed here (Sec. III B). Another barely discussed topic is calculation of “prompt losses” (losses that occur in the first full orbit of an EP in a confinement device). More details on all of these topics appear in Secs. 3 and 4.1 of Ref. 3.

The important topic of diagnostic techniques that enable measurements of EP transport is also omitted.

The review begins with a discussion of equilibrium orbits, particularly their description using constants-of-motion and the importance of topological boundaries (Sec. II). Section III introduces the general framework for considering the effect of field perturbations, including the distinction between reversible and irreversible motion. Criteria for modification of a constant-of-motion are given. Section IV is about the wave-particle phase and the distinction between resonant and non-resonant perturbations. For non-resonant particles, orbit averaging dramatically reduces cross field transport (Sec. V). Section VI discusses the convective transport that occurs when a resonant particle stays in phase with the perturbing field. Multiple resonances cause stochastic diffusive transport (Sec. VII). Large perturbations introduce new effects, including fractional resonances (Sec. VIII).

We assume the existence of equilibrium electromagnetic fields that (in the absence of perturbations and collisions) confine EPs in much of phase space. The confining fields may be electric, magnetic, or both, for simplicity, consider magnetic confinement. The magnetic configuration could have open field lines (as in a solenoid), or be a toroidal system where the field lines trace out two-dimensional flux surfaces (as in an axisymmetric tokamak), or be a fully three dimensional toroidal system with regions without well-defined flux surfaces (as in a stellarator). The orbital motion consists of relatively fast gyromotion superimposed upon a drifting guiding center. Gyromotion is described by the magnitude of the perpendicular velocity $v ⊥$ and the rapidly varying gyroangle; the velocity vector of the drift orbit is described by $v ∥$ (the component of the velocity parallel to $B 0$⁠) and the perpendicular drift $v d$⁠. The “pitch” of the orbit is $v ∥ / v$⁠.

Because of the perpendicular drifts, to confine particles, magnetic field lines in toroidal geometries must twist toroidally as they advance poloidally, a property called “rotational transform.”⁴ 

Different timescales describe different aspects of the orbital motion. In all magnetic configurations, the gyromotion sets the fastest timescale, occurring at the gyrofrequency ω_c. In toroidal systems, two other frequencies describe periodicities of the drift orbit. One of these describes the frequency of motion in the toroidal direction $ω ϕ$⁠; the other describes the frequency of motion in the poloidal direction $ω θ$⁠. Normally the ratio of these periods is irrational, so the orbit covers a two dimensional drift surface; it is not periodic. A prototypical orbit is the orbit of a charged particle in a dipole field, representative of orbits in the radiation belts (Fig. 1). For this orbit, the fastest motion is the gyromotion at ω_c, followed by the north–south motion at bounce frequency $ω θ$⁠, followed by the east–west precession around the earth at frequency $ω ϕ$⁠.

The large deviations have two important consequences. First, unlike for thermal particles, an orbit cannot be meaningfully linked to one particular field line or flux surface in the plasma. As a result, EP orbits are most efficiently described by their constants-of-motion (Sec. II A).

The second consequence concerns orbit classification. Orbit classification is also useful for thermal particles. Perhaps the most familiar example is a magnetic mirror. In a mirror device, particles with sufficiently large $v ⊥ / v$ reflect off the high-field region in the mirror throat and remain confined but particles with low values of $v ⊥ / v$ escape [Fig. 2(a)]. In $( v ∥ , v ⊥ )$ velocity space, there is a boundary that separates the “loss cone” from the confinement region [Fig. 2(b)]. In a tokamak, a similar $( v ∥ , v ⊥ )$ boundary separates “passing” particles that circulate in a single direction toroidally around the device from “trapped” (also called “banana”) orbits that reverse toroidal direction due to mirror trapping in regions of high magnetic field. Similar topological boundaries occur for EPs but, for them, the large drifts create new orbit types that do not exist for thermal particles. Orbit classification and topological boundaries are discussed in Sec. II B.

A complete description of any orbit is given by its position r and velocity v as a function of time. Although accurate, this description requires six coordinates, three for velocity space and three for configuration space. Identification of invariants of the motion reduces the number of coordinates needed for unique designation of a particular orbit. These invariants are of two types: exact invariants and adiabatic invariants.

Because collisions are negligible on the timescale of an orbit, the energy is an exact invariant of the equilibrium orbit. If a cross field electric potential $Φ$ exists, the conserved energy is the sum of the kinetic energy and the electrostatic potential energy. Since the potential energy is often much smaller than the kinetic energy for EPs, often the kinetic energy alone can be considered the conserved quantity. If there is a component of the electric field parallel to $B 0$⁠, the particle will accelerate and change energy. This happens, for example, with runaway electrons in a tokamak but typically the energy gained in a single orbit is $O ( 10 − 6 )$ of the kinetic energy; so parallel acceleration can be neglected in the orbital description.

The third adiabatic invariant is associated with cross field drifts such as the precession around the earth illustrated in Fig. 1(b). The full expression for J₃ includes a contribution from the mechanical momentum but it is often the case that v_d is sufficiently small that the magnetic contribution to the canonical momentum dominates. In that case, J₃ is proportional to the magnetic flux enclosed by the precessing orbit.

Exact and adiabatic invariants are not necessarily independent. For example, in an axisymmetric torus, the toroidal canonical angular momentum $P ϕ$ is related to the third adiabatic invariant J₃.

Once an invariant of the motion has been identified (whether exact or adiabatic), it can be used to reduce the dimensionality of the system. For example, the constancy of μ implies that the gyrophase does not impact the trajectory of the orbit. Neglecting the gyrophase reduces the six coordinates needed to describe arbitrary motion to five coordinates. Axisymmetry in a mirror device implies that the azimuthal angle is an ignorable coordinate that does not influence the orbital trajectory.

For example, in an axisymmetric tokamak, three quantities are constant: energy, magnetic moment, and toroidal canonical angular momentum. These three quantities are often used to enumerate the possible orbits. An advantage of these coordinates is that, in the presence of perturbations, a relationship often holds between changes in energy and changes in momentum; see Sec. IV. On the other hand, for numerical work, other coordinates are more convenient. (The toroidal canonical angular momentum is multi-valued depending on the sign of $v ϕ$ so a fourth coordinate is needed.) One convenient set of coordinates is the energy, the maximum major radius of the orbit, and the value of $v ∥ / v$ at the maximum radius.

In other configurations, other coordinates are favored. However, in all configurations, it is advisable to utilize constants-of-motion to reduce the dimensionality of the system.

The existence of two different orbit types and the corresponding topological map that separates them has already been illustrated for a simple magnetic mirror (Fig. 2). Other configurations have their own orbit types and topological maps. Figure 3 shows examples for tokamak and FRC geometry. Owing to the large drifts, EP orbits exist that have no counterpart for thermal particles. Consider the tokamak. For low-energy thermal particles, there are three orbit types: co-passing, trapped, and counter-passing. (Here, co- and counter-passing refer to the toroidal direction of the circulating particles with respect to the plasma current.) These also exist for EPs but there are also new orbit types [Figs. 3(a) and 3(b)]. One of these is a “loss” orbit that collides with the wall. Another is a “stagnation” orbit. A stagnation orbit circulates toroidally around the torus while scarcely moving poloidally. The phenomenon occurs because, for an EP, the poloidal component of parallel motion along the field line can be canceled by the vertical grad-B and curvature drifts. (This cancelation can also occur for a thermal particle but only in a negligibly small portion of phase space.) Additional non-standard orbits exist for EPs as well (Sec. 3.3 of Refs. 9 and 10).

Important orbit types in an FRC are “drift,” “figure-8,” and “betatron” orbits [Fig. 3(c)]. Figure-8 and betatron orbits exist because the large orbit sample regions of quite different magnetic field. The corresponding topological map is shown in Fig. 3(d). More details of FRC orbits can be found in Refs. 11 and 12.

Three dimensional stellarator configurations support even more orbit types for both thermal and energetic particles. One important example is a “super-banana” that becomes helically trapped between regions of high magnetic-field strength; this prevents the particle from precessing around the torus. This phenomenon also occurs in realistic tokamaks because the finite number of toroidal field coils causes “ripple” in the toroidal field that breaks the toroidal symmetry. In describing stellarator orbits and topology, convenient variables are energy, μ, and $J *$⁠, the latter being a generalized version of the second adiabatic invariant.¹³ 

The existence of topological boundaries has important implications for EP transport. If a Coulomb collision or field perturbation causes an EP to cross a topological boundary, the particle can take a large transport step. An example is shown for a tokamak in Fig. 3(a). The difference in velocity between the illustrated “counter-passing” orbit and the “lost” orbit is very slight (⁠ $Δ v ∥ / v < 0.3 %$⁠) but one orbit is well confined and the other hits the wall. Measurements with loss detectors often measure loss orbits near a topological boundary (Fig. 4). In this example, measurements of the velocity vector of the lost particles at the detector enable the experimenters to follow the lost orbit backward in time in the equilibrium fields. The observed losses correspond to an orbit that is at the boundary between counter-passing and lost orbits.

The equilibrium orbits shown in Fig. 3(a) illustrate a general feature of orbits in magnetic configurations: Orbits that circulate parallel to the plasma current are better confined than orbits that circulate opposite to the plasma current. In Fig. 3(a), for all three illustrated orbits, there is a location along the orbit where the vertical grad-B and curvature drifts nearly cancel the vertical component of the parallel drift $v θ$⁠. For the stagnation orbit, this cancelation occurs outside of the magnetic axis; for the counter-passing and lost orbits, this near-cancelation occurs inside of the magnetic axis. The two points of exact cancelation are very different, however.¹⁵ For the stagnation orbit, the exact cancelation occurs at an “O-point” in orbit topology space, so nearby orbits satisfy an elliptic equation and are well confined. For the counter-passing and lost orbits, the exact cancellation occurs at an “X-point” in orbit topology space, so nearby orbits satisfy a hyperbolic equation and deviate rapidly from the equilibrium point. (“O-points” and “X-points” in orbit phase space are illustrated in Sec. IV B.) Similar phenomena occur in other magnetic configurations. Modeling of a cylindrical astrophysical current-carrying jet shows that co-current orbits satisfy elliptic equations, while countercurrent orbits satisfy hyperbolic equations and are poorly confined.¹⁶ Even in stellarators where the current is carried by external conductors, beam ions injected in the co-current direction are better confined than beam ions injected in the countercurrent direction.

In analyzing magnetic field topology and particle orbits, one often draws phase-space maps called Poincare (or puncture) plots. These are made by plotting positions in phase space at regular intervals. For example, in tracing a magnetic field line in toroidal geometry, one can plot the (r, z) position of the field line every time it passes a particular toroidal angle. Figure 5 shows an example for two configurations in the W7-X stellarator.

In Poincare plots, a distinction is made between quasi-periodic and chaotic trajectories. In Fig. 5, field lines in the plasma interior always remain on the same two-dimensional surface and are on quasi-periodic trajectories. (If the Poincare plot is tracing field lines, these are the flux surfaces.) In contrast, under some conditions, traced field lines with nearly identical initial conditions diverge exponentially, even though the underlying B-field equations are deterministic. The field lines in the upper region of the plasma in Fig. 5 are stochastic. (Except near the edge, stochastic field lines are an undesirable property in a magnetic confinement device.) When a stochastic orbit is traced in a Poincare plot, it wanders ergodically throughout the stochastic region. Regions with stochastic trajectories are often limited by Kolmogorov–Arnold–Moser (KAM)¹⁸ surfaces of quasi-periodic orbits that prevent a stochastic field line from wandering through all of phase space.

Poincare plots play an important role in the analysis of EP orbits (Sec. IV E).

Although in common parlance “chaotic” and “stochastic” orbits are often used interchangeably, there is a technical distinction between “chaos” and “stochasticity.” Strictly speaking, stochastic motion is random at all times and distances, while chaotic motion is predictable on a short timescale but appears random for longer periods. Since collisionless orbits are deterministic, when they diverge exponentially, they are properly termed chaotic.

Orbit calculations are foundational for an understanding of EP transport. Although the focus of this tutorial review is on the physics of EP transport, not diagnostic or computational techniques, a few brief remarks are appropriate here. One type of orbit calculation involves solving the Lorentz force law; these are called “full-orbit” or “particle” calculations. The disadvantage of full-orbit codes is that resolving the gyromotion usually increases the computational expense and numerical error. In systems without high frequency perturbations where the magnetic moment is an adiabatic invariant, one can average over the cyclotron motion and maintain only the average particle motion in space. The elimination of the rapid cyclotron motion is computationally efficient. A code that follows the guiding center is a “guiding-center” code.

In addition to computational speed, a Hamiltonian formalism has a further advantage. In all orbit calculations, it is important to minimize numerical errors that cause constants-of-motion to diverge from their true values. A Hamiltonian formalism has favorable conservation properties.⁷ In that regard, it is also important to select a favorable numerical method, such as 4th order Runge–Kutta or an accurate symplectic integrator such as Boris integration.²⁰ 

An advantage of EP calculations is that the dilute EPs do not interact among themselves. Consequently, efficient orbit-following codes utilize parallel processing.

This section establishes the general framework for the discussion of EP transport. Transport occurs when perturbations cause changes in the constants of motion. Section III A explains when constants-of-motion are conserved and when they are broken. Section III B explains why collisions cause minimal transport. Section III C discusses the distinction between reversible and irreversible transport and the relationship between microcsopic kicks imparted by perturbations and macroscopic transport.

The theory of adiabatic invariants is well established within classical mechanics. Within plasma physics, the monograph written by Northrop over 50 years ago²¹ still provides a useful introduction and the textbook by Bellan uses pendulum motion to illustrate the key ideas.²² To summarize, three criteria must be satisfied for an adiabatic invariant to be conserved.

1. The orbit must experience gradual variations on the timescale of the periodic motion.

2. The perturbations cannot resonate with a periodicity of the motion.

3. The perturbation must be small.

These criteria are explained in more detail below.

The criterion for gradual variation is most easily explained for gyromotion. The first adiabatic invariant μ is the constant-of-motion associated with this periodicity of the orbit. The magnetic field may be changing in time, may have a gradient in space, or both. The gyrating particle “feels” these changes along its actual trajectory. The meaning of “gradual” change in time is that B changes slowly compared to a cyclotron period, i.e., $( ∂ B / ∂ t ) / ( ω c B ) ≪ 1$⁠. The meaning of gradual change in space is that the particle experiences small changes in B as it traverses its gyro-orbit, i.e., $ρ c ∇ B / B ≪ 1$⁠, where ρ_c is the Larmor radius. When both these conditions are satisfied, μ is conserved. In practice, in many actual calculations, these inequalities do not need to be particularly small for μ conservation to hold. (For example, 25% variations are often tolerable.)

Analogous criteria hold for the second and third adiabatic invariants.

The second general criterion is that the perturbations cannot resonate with a periodicity of the motion. This concept is easily grasped. Imagine pushing a child on a swing. If small pushes are synchronized with the natural frequency of the swing, the child gains energy. Randomly timed small pushes only yield an exasperated child! Resonance is discussed in detail in Sec. IV.

The third general criterion could be viewed as a corollary of the first but it is convenient to state it separately. Even if a perturbation is stationary in time and has a small gradient, if the amplitude is sufficiently large, it may “kick” the particle onto an entirely new orbit by displacing it into a region of altered equilibrium field.

Although there is a theoretical distinction between exact and adiabatic invariants, the distinction is unimportant in practice. For example, the toroidal canonical angular momentum is an exact invariant in an ideal tokamak but real tokamaks invariably have field errors that cause deviations from perfect axisymmetry. In practice, one still needs to know when $P ϕ$ is a good constant of motion and when it is not. The general criteria listed above are useful for this purpose.

As an example, consider the effect of toroidal field ripple on fast-ion confinement in a tokamak. In a real tokamak, the toroidal field is corrugated because of the finite number of toroidal field coils. From the standpoint adopted here, the toroidal field ripple is a periodic $B 1$ perturbation of zero frequency. If $B 1$ is sufficiently large, the invariant $P ϕ$ is broken.

The effect is most pronounced for trapped particles. Since $∇ B 1$ is small on the scale of the gyromotion, μ is conserved. At the turning point of a trapped ion, the parallel velocity vanishes and $μ = W ⊥ / B = W / B$⁠. Since the energy of the particle W is another constant of the motion in static magnetic fields, small changes in the magnitude of B associated with field ripple cause perturbations in the position of the turning point. Two effects on the orbit are distinguished. The one discussed here is called “ripple trapping;” the other is called “stochastic ripple diffusion.”²³ Both are explained together with many experimental examples in Sec. 4.2 of Ref. 3. An additional theoretical explanation appears in Sec. 8.7 of Ref. 9.

In ripple trapping, the toroidal field ripple creates a secondary magnetic well. An ion trapped in the secondary well executes an orbit known as a “super-banana” and begins to drift vertically because the $∇ B$ drift is no longer compensated by rotational transform. Since EPs are collisionless, they keep drifting vertically until they are lost. Figure 6 shows an example of a lost super-banana orbit. Two tokamaks have performed experiments where the number of energized toroidal field coils was intentionally halved, thereby increasing the toroidal field ripple an order of magnitude. In both cases, large reductions in the number of confined EPs on trapped orbits were observed.^24,25

Minimization of the number of super-bananas is a key aspect of stellarator design. A number of optimization strategies have been developed that reduce transport associated with ripple trapping.^27,28 Equilibria with superior confinement have EP orbits with large poloidal drifts but small radial drifts. Large poloidal drift velocities promote the formation of poloidally closed contours of the second adiabatic invariant J₂, resulting in better trapped-particle confinement.²⁹ 

Collisions are a key topic in the discussion of thermal transport. Coulomb collisions cause “classical” transport in a uniform magnetic field.³⁰ In magnetic confinement geometries, a finite drift-orbit width and transitions between different orbit types cause increased transport relative to classical levels; this is called “neoclassical” transport.³¹ In some configurations, neoclassical transport is the dominant thermal transport mechanism, exceeding transport caused by unstable waves.

This is not the case for EPs. Because of their large velocities, collisions are rare. Appreciable transport occurs almost exclusively near topological boundaries. Near a boundary, the slight change in velocity vector associated with pitch-angle scattering can kick the EP onto an entirely different orbit, resulting in a large spatial step or even particle loss. Figures 2, 3(a), 3(b), and 4 show examples of loss orbits that differ only slightly from a confined orbit.

Figure 7 shows a concrete example for the ITER tokamak.³² In the calculation, an orbit-following code that includes Coulomb collisions and realistic ITER fields follows alpha particles until they are lost or thermalized. Particles born in a loss region escape on their first full orbit. Particles born in a region where stochastic ripple diffusion and ripple transport are operative are lost after $∼ 10$ orbits. Away from these regions, losses are very slight; for example, over 50% of particles with a pitch within 0.1 of a boundary remain confined for thousands of orbits. Most particles are thermalized before they are scattered into a loss channel, which is why the fraction of particles lost decreases rapidly as one moves away from the loss channels. As another example, quantitatively, the neoclassical diffusion coefficient of beam ions in contemporary tokamaks is D < 0.1 m²/s, which implies minimal spatial transport in a slowing-down time. Accordingly, collisionless transport is the focus of this review.

The previous discussion concerns the dominant collisional process in fully ionized plasma, small-angle Coulomb scattering. In special circumstances, other collisional processes can be important. When measuring EPs at a pitch far from their birth pitch, large-angle nuclear scattering³³ can make the dominant contribution to the signal. For runaway electrons, the pitch-angle scattering rate is modified by synchrotron losses;³⁴ also, the conventional Coulomb scattering rate is modified for collisions with impurities that retain bound electrons.³⁵ For fast ions, charge-exchange reactions^5,36 with injected or edge neutrals can play an important role in evolution of the distribution function.

Not all waves cause macroscopic transport. Consider a small amplitude, oscillating azimuthal electric field $E θ$ in a solenoid with uniform axial magnetic field $B = B 0 z ̂$⁠. The electric field causes radial $E × B$ drifts of the particles. However, if the wave amplitude gradually increases to a maximum then gradually decreases, after the perturbation has past, the particles will all return to their initial position. No radial transport has occurred.

Our EPs are nearly collisionless. This implies that, on a short timescale, their motion is completely reversible. One could watch a video of the motion over a few wave periods without being able to discern if the video was running forward in time or in reverse.

In contrast, imagine watching a video of a group of similar particles on a longer timescale. Now one could easily tell the direction of time: if the particles spread radially as the video advances, time is in the forward direction. If the particles begin at different positions and cluster together at nearly identical positions, the video has run in reverse. Irreversible motion appears when one considers groups of particles on an intermediate timescale. In this context, “intermediate” means longer than several wave periods but shorter than the collisional timescale on which the entire distribution function evolves. If δt is the timescale of the reversible motion, $Δ t$ is the intermediate timescale, and t is time in a Fokker–Planck equation, our ordering is $δ t ≪ Δ t ≪ t$ (Fig. 8). The statistics of the microscopic kicks determines the type of macroscopic transport that occurs on the intermediate timescale. Similarly, the statistics of intermediate-timescale transport determines the evolution of the distribution function on the long collisional timescale.

To achieve irreversibility, something is needed to disrupt the reversibility of the microscopic motion. There are several possibilities. One possibility is multiple uncorrelated perturbations, such as those that occur when there is a turbulent spectrum of waves. Another possibility occurs in deterministic chaos (Sec. II C). A third possibility is the almost negligible collisions we usually neglect.

Different distributions of microscopic kicks produce different macroscopic spreading. In general, the relationship between the microscopic probabilities and the macroscopic behavior is governed by a “master” equation that relates the probability that the system is found in a given state to the transition (microscopic kick) probabilities.³⁷ Several simple limits exist, however. If the microscopic kicks are all of comparable magnitude in the same direction, the mean position of the macroscopic distribution grows as $⟨ Δ x ⟩ = ( δ x / δ t ) t$⁠. Motion of this type is called “convective.” Examples of convective transport for EPs appear in Sec. IV. If the kicks have a distribution of sizes but a preferred direction, a combination of convective and diffusive transport can occur.

Even in the absence of convection, the spreading need not be diffusive. A well-known example is the distribution of kick sizes adopted by many foraging animals. They often take many small steps while feeding at a particular plant, and then take a long step to a different part of the landscape. These produce a distribution of microscopic steps that is a type of “Lévy flight” distribution. The resulting spreading deviates from diffusive scaling [Eq. (9)]; instead, the population spreads as $t γ$ with $γ > 1$⁠. When $γ > 1$⁠, the spreading is called “super-diffusive.” When $γ < 1$⁠, the spreading is called “sub-diffusive.” Sub-diffusive transport occurs when large-amplitude kicks are rarer than in the normal distribution of Eq. (11). Figure 9 illustrates the relationship between microscopic kick probabilities and macroscopic transport for several simple cases that are relevant for EP transport.

Sub-diffusive, diffusive, and super-diffusive spreading are all observed for EPs and analyzed theoretically.³⁹ A particularly clear example was measured on the TORPEX toroidal device. Beams of energetic lithium ions with different energies were launched through transient turbulent electrostatic fluctuations called “blobs” and their mean square displacement was measured at various distances from the source (Fig. 10).³⁸ Simulations explained the different behavior that was observed.⁴⁰ On occasion, the blobs imparted a large $E × B$ kick to lower-energy fast ions. The resulting distribution of kicks resembling Lévy flights produced super-diffusive transport with $γ > 1$⁠; this is like the case illustrated in Figs. 9(e) and 9(f). In contrast, for higher energies, the increased grad-B drift caused the fast ions to drift vertically through the electrostatic structures so rapidly that the radial displacement was reduced, causing a truncated kick distribution like the one illustrated in Fig. 9(g); the resulting macroscopic transport was sub-diffusive [Fig. 9(h)].

The concept of random, irreversible kicks is widely employed in theories of EP transport. For example, in cyclotron heating, owing to the resonant interaction described in Sec. IV, particles receive a kick in energy when their orbits traverse locations where $ω = ω c$⁠, causing diffusion in velocity space.² Random kicks are readily incorporated into codes that use Monte Carlo markers to represent energetic particles. For example, randomly applied kicks in different portions of phase space successfully model EP interactions with Alfvén waves.⁴¹ 

This section begins by explaining the fundamental conditions for wave-particle resonance, Eq. (13). Next (Sec. IV B), the relationship between the amplitude of the perturbing field and the width of the resonance is explained. Section IV C explains why collisionless wave-particle trapping becomes irreversible. The relationship between wave-particle trapping and mode stability is briefly discussed in Sec. IV D. Section IV E describes the emergence of chaotic orbits caused by particle trapping in wave-particle resonances. Finally, Sec. IV F summarizes the main points of this section.

Whether a particle is resonant or non-resonant depends upon the phase Θ between the particle and the wave. Consider the simple case of an ion gyrating in a uniform magnetic field in the presence of an electric field of frequency ω that is spatially uniform and linearly polarized. If the wave frequency matches the cyclotron frequency, the ion always sees an electric field that points in the direction of its horizontal motion [Fig. 11(a)]. The ion is continuously accelerated and its perpendicular energy steadily increases [Fig. 11(e)]; consequently, the first adiabatic invariant $μ = W ⊥ / B$ is not conserved. The perturbation has violated the second criterion for conservation of an adiabatic invariant (Sec. III A).

Note that whether the ion gains or loses energy depends upon its initial phase with respect to the wave. If the initial phase is flipped $180 °$⁠, the ion continuously loses energy rather than gains energy. Nevertheless, $W ⊥$ has changed and μ is not conserved.

Now consider a wave where $ω ≠ ω c$⁠. Now the ion gains energy for a while but subsequently loses any gained energy when the ion sees E of opposite phase [Fig. 11(d)]. If the electric field is sufficiently small, the time-averaged change in $W ⊥$ is zero and the time-averaged magnetic moment μ is conserved.

Note that, for resonance to occur, the electric field need not accelerate the ion on the entire orbit. Imagine that the electric field of Fig. 11(a) only existed in the upper half of the orbit. If $ω = ω c$⁠, the ion would still gain energy on every orbit; if $ω ≠ ω c$⁠, the ion would still gain energy for a few cycles, and then lose an equal amount of energy on subsequent cycles. One can imagine an even more complicated spatial structure, where the ion gains energy on 2/3 of the orbit but loses energy on 1/3. If $ω = ω c , W ⊥$ would still change every cycle and μ would not be conserved.

It is not essential that the ion completes its orbit in a single cycle of the wave; the wave can oscillate multiple times in a cyclotron period and still exchange an increment of energy every orbital period. This occurs, for example, in high harmonic cyclotron heating, where finite values of $k ⊥ ρ c$ yield non-zero values of $∮ E · d l$ after a cyclotron orbit. (⁠ $k ⊥$ is the perpendicular component of the wavevector.) As long as $Δ W$ is non-zero and remains the same every orbital period, resonance still occurs. In other words, the resonance condition is $ω = l ω c$⁠, where l is an integer.

The orbital frequencies in Eq. (13) are averaged over the relevant orbital motion. For example, in a tokamak, the cyclotron frequency ω_c is a function of major radius and therefore varies along the drift orbit; when considering a resonance involving the poloidal orbit, ω_c in Eq. (13) represents the orbit-averaged cyclotron frequency.

When the frequency of the perturbation is small compared to the cyclotron frequency, μ is conserved and the cyclotron term in the resonance condition [Eq. (13)] is omitted, $ω = m ω θ + n ω ϕ$⁠. Naively, for a perturbation with toroidal mode number $n ′$ and poloidal mode number $m ′$⁠, one might expect that the strongest resonance (i.e., the one with the largest value of $∮ E · d l$⁠) would occur for $n = n ′$ and $m = m ′$ but, because of the large EP orbits, that is not generally the case. To take this effect into account, some authors define a “kinetic resonance”⁴² or perform “orbit-based resonance analysis”⁴³ that explicitly distinguishes the helicity of the strongest resonance from the helicity of the perturbing mode. In an axisymmetric device, resonance does require that $n = n ′$ but energy exchange can be appreciable for multiple values of m. For example, for passing particles in a toroidal device, the orbit shift due to drift is primarily m = 1, a shift outward of a co-moving particle and a shift inward of a counter-moving particle. This shift coupled with the $m ′$ value of the perturbation often leads to strong resonances at $m = m ′ ± 1$⁠. In a stellarator, appreciable energy change can also occur for $n = n ′ + ν N$⁠,⁴⁴ where ν is an integer and N is the number of periods of the helically symmetric stellarator coils. In general, multiple values of l, m, and n can contribute important resonances.

Examples of multiple important resonances for a low-frequency tokamak perturbation appear in Fig. 12. The orbital frequencies $ω ϕ$ and $ω θ$ are functions of the constants of motion, so orbits that satisfy the resonance condition appear in different parts of phase space. For the examples of Fig. 12, l = 0 and n is equal to the toroidal mode number of the Alfvén wave $n ′$⁠, so the multiple resonances occur at harmonics of the poloidal bounce frequency $ω θ$⁠.

In practice, evaluation of the resonance condition [Eq. (13)] is readily performed, as only the equilibrium orbits and mode frequency are needed for the calculation. However, to tell if a resonance is important, one needs detailed measurements or modeling of the perturbation in order to evaluate $∮ E · d l$ for each resonance. Figure 12(a) is an example of the simpler evaluation, while Fig. 12(b) is an example of a calculation of actual energy exchange.

A simple demonstration of cyclotron resonance was performed for EPs in a long solenoidal machine, the Large Plasma Device (LAPD). Alfvén waves that produced azimuthal electric fields were launched at one end of the device and a beam of lithium ions on helical orbits orbited through the wave field. When the Doppler shifted wave frequency $ω − k ∥ v ∥$ matched the EP cyclotron frequency ω_c, large spreading of the beam was observed. A secondary peak with reduced spreading was also observed when $∮ E · d l$ accelerated the ions in one direction for 2/3 of a cycle and in the opposite direction for 1/3 of the cycle.^48,49

In general, the orbital frequencies ω_c, $ω θ$⁠, and $ω ϕ$ are complicated functions of phase-space coordinates so the resonance condition [Eq. (13)] is only satisfied on a narrow curve in phase space, as in Fig. 12(a). In reality, however, resonances are broadened to span a region in phase space. [This broadening is apparent in Fig. 12(b).] Particles with a slight mismatch between their orbital frequencies and the wave frequency get trapped in the wave, as in nonlinear Landau damping. This section examines the relationship between the resonance width and mode amplitude, showing that the broadening of the resonance is typically proportional to the square root of the perturbation amplitude [Eq. (19)].

Equation (16) can also be obtained from a quantum-mechanical perspective. The wave and particle exchange both energy and momentum. The exchanged energy is $Δ W = ℏ ω$ and the exchanged momentum is $Δ P ϕ = r ℏ k ϕ$⁠. Since the toroidal wavenumber is $k ϕ = n / r$⁠, it follows that $ω Δ P ϕ = n Δ W$⁠.

For this Hamiltonian, an EP whose motion in θ and $ϕ$ is related by the ratio n/m satisfies the resonance condition, Eq. (13). Particles that satisfy the resonance condition exactly are at the elliptic or hyperbolic point of the resonance.

The motion of particles that are trapped in a wave-particle resonance becomes ergodic.⁵¹ Eventually, on average, all particles trapped by the wave adopt the constants-of-motion of the exact resonance. This section explores the origin of irreversibility for the example of Fig. 13.

In Fig. 14 is shown an actual Poincare plot of a 50 kHz mode resonating with 25 keV particles in a tokamak. The circulation of particles around the hyperbolic point is shown at the right, with snapshots at increasing time intervals. The rotation is most rapid near the elliptic point, and goes to zero at the separatrix. Because of the variation of the rotation rate with the distance from the elliptic point, with increasing time, the mixing of different energies and values of $P ϕ$ occurs at smaller and smaller distances, until it finally reaches interparticle scales. At this point, even an infinitesimal collision rate is sufficient to guarantee irreversibility and the average value of energy and $P ϕ$ of the entire trapped population is the same as that of the elliptic point.

Equation (19) states that the resonance width in phase space is proportional to the square root of the perturbation amplitude. From the perspective adopted in the majority of this review, the perturbation amplitude is an independent parameter that will determine the consequent EP transport. It could be experimentally measured. It could be externally controlled by coils or antennas. It could be the result of an instability whose amplitude is governed by interaction with the thermal plasma, such as an MHD instability. Or it could be an EP-driven instability. In the latter case, there is an interplay between the wave-particle trapping, the properties of the EP distribution function, and the mode amplitude that is briefly explored here.

As discussed in Sec. IV C, all EPs that are trapped in the resonance eventually adopt the constants-of-motion of the exact resonance. As in Landau damping, whether these nonlinearly trapped particles deliver or extract energy from the wave depends upon the slope of the distribution function f in the vicinity of the resonance. In Landau damping, the relevant slope is $∂ f / ∂ v ∥$⁠; more generally, the relevant slope is across the resonance in constants-of-motion space and involves terms such as $∂ f / ∂ W$ and $∂ f / ∂ P ϕ$⁠. If more particles gain energy than lose energy, the wave damps. If the energy gained by the wave exceeds any intrinsic damping, the wave grows.

If there is an imbalance between wave growth and damping, the amplitude of the perturbation changes. There are many possible scenarios for the subsequent evolution of the mode. One possibility is that a balance is achieved between energy extracted from the distribution function and replenishment of the driving gradients; an example of a possible steady-state scenario appears in Ref. 52. If mode growth depletes driving gradients, a burst of transient growth followed by decay can occur. If the flattening of local phase-space gradients causes steepening of neighboring gradients, the mode may chirp in frequency $d ω / d t$⁠, so that the nonlinear trapping region sweeps through new portions of phase space as the mode evolves. In the Berk-Breizman model (reviewed in Refs. 53 and 54), the relative values of linear drive, mode damping, trapping frequency, and scattering rates determine which of these scenarios occur. A comprehensive discussion of the many possibilities including the role of the mode structure appears in Ref. 55.

In the presence of multiple resonances, islands may overlap. When they do, orbits are usually chaotic.

An example appears in Fig. 15(a) for the case of an ion that streams along a magnetic field in the $z ̂$ direction in the presence of an electrostatic wave that propagates obliquely to the field. The plot shows where ions with three different initial conditions lie in $( z , v z )$ space at regular intervals. These three ions satisfy the Doppler-shifted resonance condition [Eq. (14)] $ω − k z v z = ℓ ω c$ with $ℓ = − 1 , 0 , 1$⁠. For the case shown in Fig. 15(a), the amplitude of the wave is small, so the three orbits lie on well-defined curves.

Figure 15(b) shows an example of chaotic orbits that appear when the perturbing electrostatic wave is large. Often some orbits remain periodic even when most orbits are chaotic. The solid curves in Fig. 15(b) are examples of regular orbits that persist even when most of phase space is stochastic.

Experts in nonlinear dynamics have extensively analyzed the conditions that result in stochasticity. One helpful concept is the Chirikov or “island overlap” criterion. The labels $w − 1$⁠, w₀, and w₁ in Fig. 15 mark the widths of the three different islands. As discussed in Sec. IV B, the widths grow in size when the amplitude of the perturbation increases. The Chirikov criterion states that when the widths of the islands exceed the distance between islands, stochasticity ensues. Although this criterion is not quantitatively accurate in most situations, it is a useful qualitative guide to ascertain conditions that disrupt confinement.

Stochasticity leads to increased transport, but the nature of the transport varies in different circumstances, including subdiffusive, diffusive, and superdiffusive transport.³⁹ 

1. Resonance occurs when the particle and the wave return to the same initial phase after a multiple of an orbital period.

2. The initial phase determines whether the particle loses or gains energy but has no impact on whether resonance occurs.

3. The energy exchange is determined by $∮ E · d l$ integrated over the orbit. If this quantity is zero, even if the resonance condition [Eq. (13)] is satisfied, there is no energy exchange. Important resonances have non-zero values of $Δ W = q ∮ E · d l$ or $Δ P = ∮ F d t$⁠.

4. Particles that do not satisfy the resonance condition [Eq. (13)] are weakly affected by the wave.

5. For a finite amplitude wave, the particle need not match the resonance condition exactly. Nonlinear trapping captures particles that are slightly out of resonance. The width of this resonance-broadened region increases with the square root of the amplitude of the perturbation [Eq. (19)].

6. Nonlinearly trapped particles ultimately adopt the constants-of-motion of the exactly resonant particles.

7. When resonances overlap, orbits become chaotic and transport becomes large.

When EPs are non-resonant, the large EP orbits often reduce transport below the level experienced by thermal particles. This reduction occurs whenever the spatial structure of the perturbations is comparable to or smaller than the orbit size. The mechanism responsible for this reduction is called “phase averaging.”

Consider the simple example illustrated in Fig. 16(a). An EP executes a helical orbit in a solenoid filled with plasma. The plasma contains a low-frequency (⁠ $ω ≪ ω c$⁠) electrostatic wave with very long parallel wavelength (⁠ $k ∥ ≃ 0$⁠), so the situation is essentially two-dimensional. The electrostatic potential fluctuations are described by $Φ = Φ 0 cos k y y$⁠, where k_y is the vertical wavenumber. The electric field associated with this potential causes horizontal $E × B$ displacement of the EP orbit. If the Larmor radius is much smaller than the spatial structure of the electrostatic wave (⁠ $k y ρ c ≪ 1$⁠), the electric field is upward throughout the entire orbit, so the particle always drifts to the right. However, if the Larmor radius is large, the orbit samples regions of both upward and downward electric field. The particle drifts to the right on some of its orbit and to the left on others. Consequently, the net drift is reduced. Mathematically, $∮ E · d l ≃ 2 E Δ y$ for the small gyroradius but $∮ E · d l ≃ ∫ E cos ( k y y ) d y < 2 E Δ y$ for the large gyroradius. The usual $E × B$ drift is reduced by phase averaging by a factor of $J 0 ( k y ρ c )$⁠.⁵⁷ (J₀ is the Bessel function of the first kind.)

An experiment very similar to the simple example of Fig. 16(a) was conducted in the Large Plasma Device (LAPD).⁵⁸ A rectangular obstacle placed in the plasma created a sharp density gradient that generated electrostatic fluctuations on the scale of the thermal ion Larmor radius. A beam of energetic lithium ions of variable gyroradius passed through the fluctuations and the spreading of the beam was measured. As expected, beam spreading decreased monotonically with increasing gyroradius [Fig. 16(b)].

In a related experiment on the LAPD,^59,60 the EP gyroradius was held fixed while the scale length of the electrostatic fluctuations was varied. As expected, beam spreading was smallest for small-scale, highly coherent fluctuations where phase averaging is particularly effective. When the scale of the fluctuations was larger or the collection of small-scale fluctuations was more turbulent, the EP transport was increased. A similar example from the TORPEX device was already presented in Fig. 10. Owing to orbit averaging, the highest energy ions suffer far less transport than lower energy EPs.

This is a general result observed in many experiments: when EPs are non-resonant, they generally have much better confinement than thermal particles. As long as the scale length of the perturbation is smaller than characteristic EP orbit sizes, the transport associated with any type of perturbation is reduced.

A similar reduction in radial transport by fluctuating electromagnetic fields was observed in the MST reversed field pinch, where, due to orbit averaging, energetic beam ions were much better confined than thermal ions.⁶⁵ 

The TFTR experiment of Fig. 4 provides yet another example. In that experiment, the rate at which counter-passing fusion products crossed a loss boundary was measured. The diffusion coefficient inferred from the data was D < 0.1 m²/s, much less than typical thermal coefficients of $D ∼ 1$ m²/s.¹⁴ 

It can also occur that small perturbations lead to long term complex traps for particles, with the associated Lévy flights producing either subdiffusion or superdiffusion. Subdiffusion is regularly observed in the stochastic field of RFX.³⁹ 

As a final example, consider the transient fields produced in a tokamak when a “sawtooth” internal reconnection event occurs. A sawtooth event generally is initiated by a growing n = 1, m = 1 internal kink mode that triggers the n = 0 “sawtooth crash.” (Here, n is the toroidal mode number of the perturbation and m is the poloidal mode number.) Normally, the electron density and temperature profiles flatten at a sawtooth crash, as the scrambled magnetic field lines associated with the reconnection event enable rapid parallel electron transport. This is not necessarily the case for energetic ions, however. In a theory by Kolesnichenko,^66,67 only lower energy ions of a given orbit type experience significant radial transport; due to orbit averaging, higher energy ions are only weakly affected. Additionally, because their orbit widths are larger, trapped particles of a given energy are less affected than passing particles. The qualitative features of this prediction have been confirmed by many experiments.^68–73 

Although this section has emphasized non-resonant interactions, phase averaging can also reduce the impact of a resonant perturbation. The key requirement for phase averaging is that the orbital size exceeds the scale length of the perturbations, making $∮ E · d l$ small.

Under some circumstances, an EP always receives a kick in the same direction, resulting in strong convective transport [Figs. 9(c) and 9(d)]. A famous example of this type of transport is the “fishbone” instability observed in tokamaks. This is a large global mode with an n = 1 toroidal structure. The wave produces a poloidal electric field that pushes the particles radially through the $E × B$ drift in the strong toroidal equilibrium magnetic field. For reasons discussed below, the resonant fast ions preserve their wave-particle phase on every wave cycle. After 4–6 kicks outward, the EP is lost [Fig. 18(a)].

Diagnostics that are sensitive to edge losses measure a burst of signal each time the instability rotates past the detector [Fig. 18(b)]. Like a rotating lighthouse searchlight, these periodic bursts pass the observer once each cycle. As expected, the burst has a definite phase with respect to the instability [Fig. 18(b)], namely, the phase where the $E × B$ drift pushes the EP outward. (For the opposite initial phase, an EP is pushed inward, becoming better confined.) In Ref. 75, seven different edge diagnostics each measured a definite phase with respect to the mode, depending on their toroidal locations.

Equation (24) has important implications for the stability of low-frequency modes that are driven unstable by an EP population. To drive instability, the EPs must impart energy to the wave, so $Δ W$ must be negative for the majority of resonant particles that become trapped by the wave. Under some circumstances, the instability can adapt its frequency to preserve resonance with this clump of resonant particles as they are radially transported. [The wave frequency must change because, in general, ω_c, $ω ϕ$⁠, and $ω θ$ in Eq. (13) are functions of position.] A wave that obeys a dispersion relation associated with an EP population (rather than the dispersion relation of a normal mode of the background plasma) is called an “energetic particle mode” (EPM).⁷⁶ Because it preserves the wave-particle phase, an EPM is particularly effective in causing convective resonant transport. A sophisticated discussion of conditions that support an EPM appears in Sec. 4 of Ref. 55.

The predicted reduction in energy is observed experimentally. In the example of Fig. 18(a), the neutral beams were injected at 45 keV but the largest and most coherent loss signal was at $∼ 35$ keV.⁷⁷ A change of energy of $Δ W ≃ − 10$ keV is expected for beam ions that transport from the plasma center to the edge in a wave of the observed frequency and mode number.

A common feature of EP losses caused by convective resonant transport is a linear dependence on the mode amplitude. This is readily understood: The $E × B$ drift is linearly proportional to the amplitude of the electric field E. Examples abound. The losses caused by fishbones scaled linearly with mode amplitude.⁷⁸ A probe that directly measured the mode amplitude and EP losses was inserted into the CHS stellarator.⁷⁹ During an energetic particle mode, the portion of the signal that oscillated at the mode frequency (also called the coherent signal) scaled linearly with the mode amplitude, while the incoherent signal scaled quadratically with mode amplitude (Fig. 19). (The latter dependence is often observed for the diffusive resonant transport discussed in Sec. VII) Similarly, coherent losses measured at the edge of the ASDEX-Upgrade tokamak scaled linearly with the amplitude of a type of Alfvén wave known as a toroidal Alfvén eigenmode (TAE) but the incoherent losses scaled with the square of the mode amplitude.⁸⁰ 

Under some circumstances, losses caused by convective transport can occur even when an EP does not satisfy the usual resonance condition of Eq. (13). Convective losses of this type were caused by an Alfvén wave called a “reversed shear Alfvén eigenmode” (RSAE) on the DIII-D tokamak.⁸² The experiment was designed to detect the EP orbit after it made a single transit through a spatially localized RSAE [Fig. 20(a)]. When the wave-particle phase [Fig. 20(c)] stays nearly constant during the transit through the mode, a large $E × B$ drift occurs. Coherent losses are observed with inferred orbital displacements as large as 10 cm.^82,83 This occurs despite the fact that the resonance condition [Eq. (13)] is not satisfied by the measured orbit. Presumably, if the particle remained confined, it would regain the lost energy on subsequent transits through the mode but, because it crosses a loss boundary, the interaction is irreversible. This is an example of a constant-of-motion being broken by a large kick, the third criterion listed in Sec. III A.

In a finite amplitude wave, the resonance condition is broadened (Sec. IV B). However, if the perturbation is small the broadening is modest, so only a small portion of phase space is affected. As a concrete example, the effect of a single EP resonance with an Alfvén wave of amplitude $δ B / B = 10 − 4$ is essentially undetectable with available tokamak diagnostics. To achieve appreciable transport, multiple resonances must interact. From the perspective of Sec. III, the additional resonances provide randomizing elements that promote irreversible transport. Multiple resonances can occur because multiple waves cause different resonances, because large EP orbits produce multiple resonant harmonics with a single wave (as in Fig. 12), or both.

When multiple resonances intersect in phase space, diffusive transport often occurs. The associated losses often scale with the square of the perturbation amplitude. This amplitude dependence of the transport is readily explained.⁸⁴ As previously mentioned, for convective transport or losses of particles besides a loss boundary, linear scaling with amplitude is predicted⁸⁴ and observed, as in Fig. 19(a). For multiple resonances, each interaction imparts energy and momentum kicks [Eq. (24)]. If the kicks are random and small amplitude, $⟨ ( Δ P ϕ ) 2 ⟩ = 2 D t$ [Eq. (9)], with a diffusion coefficient D that is proportional to the square of the size of the kick [Eq. (10)]. Since the kick sizes are proportional to the wave amplitudes, D scales quadratically with mode amplitude. If, in addition, the transport follows Fick’s law (⁠ $Γ = − D ∇ n$⁠), then the flux Γ is proportional to (amplitude)². (Here, $∇ n$ is the EP density gradient in phase space.) Measured incoherent losses with quadratic scaling already appeared in Fig. 19(b). Another excellent example appears in Ref. 80.

Although diffusive transport is one possible response to multiple interacting resonances, another possibility is “stiff” transport. In stiff transport, transport is negligible up to a threshold in mode amplitude, then increases rapidly when the system is driven past the threshold. A condition with stiff resonant transport from multiple small-amplitude waves and multiple resonances per mode has been studied extensively in the DIII-D tokamak. In that experiment, neutral beam ions are the EPs and Alfvén eigenmodes of different types and mode numbers provide the small amplitude perturbing waves. Analysis of the EP orbits using measured mode structures and amplitudes shows the existence of multiple resonances that intersect in phase space (Fig. 21). Below a threshold in the number of modes and their amplitudes, negligible transport is measured but, above this threshold, the phase-space flux of EPs rises rapidly (Fig. 22). Note that, owing to the modest mode amplitude, the width of each resonance is modest. Nevertheless, the overlapping resonances produce appreciable transport. Experimental measurements show that, because the resonances differ in different parts of phase space, the threshold for stochastic transport differs in different parts of phase space.⁸⁹ The threshold correlates with the destruction of KAM surfaces (Fig. 22); in other words, EP transport becomes large when the orbits become chaotic. Further discussion of these examples appears in Sec. 5 of Todo’s review paper.⁵⁴ 

A single mode of sufficiently large amplitude can also produce stochastic transport. A relatively simple example was analyzed by Konovalov and Putvinskii⁹¹ and Mynick.⁹⁰ Consider circulating EPs with $v ∥ ≃ v$ in a tokamak that also contains a large helical perturbation with toroidal mode number n = 1 and poloidal mode number m = 2. (Perturbations of this type are produced by tearing modes.). The circulating EPs experience a large curvature drift. The curvature drift can be thought of as adding an n = 0, m = 1 perturbation to a particle orbit that would otherwise follow the field lines. When one plots the Poincare map for the orbit, an n = 1, $m = 2 + 1 = 3$ island appears associated with beating between the tearing mode and the curvature drift (Fig. 23). If the mode amplitude and EP energy are modest, the effect on the orbit is negligible. However, when the mode amplitude and EP energy are sufficiently large that the m = 2 and m = 3 islands begin to overlap, stochasticity ensues. For the example shown in Fig. 23, this occurs when the mode amplitude is doubled.

As predicted, experimental measurements show EP confinement is degraded for large amplitude tearing modes.^3,92–94 The degradation agrees qualitatively with the island overlap theory and quantitatively with numerical calculations.⁹⁵ Recent calculations show that the effect on trapped particles is even stronger than the effect on the passing particles of Fig. 23.^96,97 This is not surprising, as trapped particle displacements in tokamaks are even larger than passing-particle orbit shifts.

In the previous examples, the temporal variation of the mode amplitude was gradual; however, in practice, the perturbing fields may rapidly cross a stochastic threshold. When this occurs, a domino effect may occur that causes a sandpile-like avalanche of EP transport, as the overlap of closely spaced modes or the growth of previously stable modes occurs.⁹⁸ In these situations, the connection between mode stability and fast-ion gradients causes a complicated interplay between mode amplitude and fast-ion transport that is best treated theoretically by a comprehensive simulation such as the one in Ref. 99. Nevertheless, from the standpoint of EP transport alone, the phenomenon can be understood as another example of enhanced transport when a stochastic threshold is crossed.

Figure 24 shows examples from the spherical tokamak NSTX and the conventional tokamak JT-60U. In both cases, neutral beams inject circulating ions with speeds greater than the Alfvén speed that drive Alfvén eigenmodes unstable. A sequence of repetitive bursts of moderate amplitude ensues [Figs. 24(a) and 24(b)]. At some point, amplitudes grow larger and trigger a major event that involves more toroidal mode numbers at larger amplitudes, called an “avalanche” at NSTX and an “abrupt large-amplitude event” (ALE) at JT-60U. For both devices, in both simulation^99,100 and experiment,^101,102 the avalanches cause EP transport that is an order of magnitude larger than the smaller preceding events [Figs. 24(c)–24(e)]. At an avalanche in NSTX, the pitch of lost fast ions spans a much wider range of values than normal, a signature of chaotic orbits.¹⁰³ 

If the perturbation becomes sufficiently large, new resonances are created. These resonances are sometimes called “fractional resonances” because the “integers” in the resonance condition of Eq. (13) now assume rational values such as 1/2 or 1/3. These fractional values occur when the phase of the perturbation is the same after the particle has completed two, three, or more transits of its orbit.⁵⁴ 

The mathematical origin of these new resonances is easily seen. The nonlinear production of additional resonances was studied by Fibonacci (1170–1250 A.D.) centuries ago. Consider two perturbations of equal amplitude α but different toroidal and poloidal mode numbers m/n and $m ′ / n ′$⁠. Simply multiplying the terms $α e i ( m θ − n ϕ )$ and $α e i ( m ′ θ − n ′ ϕ )$ gives $α 2 e i ( m + m ′ ) θ − i ( n + n ′ ) ϕ$⁠. In addition to the original resonances at m/n and $m ′ / n ′$⁠, a higher-order fraction $( m + m ′ ) / ( n + n ′ )$ has been created. Note that the higher order fraction is always bounded by the parent fractions. By continuing to multiply perturbations thus produced there results an infinite number of islands produced by any pair. Nevertheless, the KAM¹⁸ theorem guarantees that if the original perturbations are sufficiently small, the sum of all these island widths remains small, so that there are domains in which the original KAM surfaces are distorted but retain their original topology.

The physical origin of these new resonances is also easy to understand. The EP “sees” a wave phase $Θ = k · r − ω t$⁠. If the wave amplitude is sufficiently large to deflect the equilibrium orbit a distance $δ r$⁠, the wave phase is modified by an amount $k · δ r$⁠. This change modifies the original wave-particle resonant interaction and creates additional resonances. The experiment illustrated in Fig. 20 provides a readily understood example. In some cases, the EP orbit passes through two unstable Alfvén eigenmodes of appreciable amplitude. The modes are at different positions in the plasma but the orbit traverses both of them. When that happens, each mode gives the orbit a kick that modifies the phase at the other location. When the phase factor Θ is expanded and the orbital displacement is calculated, in addition to oscillating displacements at the primary mode frequencies ω₁ and ω₂, displacements also occur at the sum and difference frequencies $ω 1 + ω 2$ and $| ω 1 − ω 2 |$⁠. As expected, when the mode amplitudes are appreciable, losses at the sum and difference frequencies are observed experimentally.^105,106

A general theory of nonlinear resonances was recently published.¹⁰⁷ In generalized phase-space coordinates X, the equivalent of $k · r$ is $δ X · ∇ X$⁠. Integration over the unperturbed orbit results in an expansion in Bessel functions whose argument is the phase change. A nonlinear resonance becomes important whenever a phase change is O(1).

A tokamak instability called the energetic-particle-induced geodesic acoustic mode (EGAM) provides an example of this phenomenon. The EGAM is a low-frequency, global electrostatic perturbation of n = 0, m = 1 structure that can assume large amplitudes under some conditions. For this instability, the resonance condition reduces to $ω = m ω θ$⁠. An orbit-following code analyzed the effect on the orbits as a function of mode amplitude. At low amplitude, energy exchange occurred at the usual integer values of m [Fig. 25(a)] but, as the mode amplitude increased, appreciable energy exchange appeared at m = 1/2 [Fig. 25(c)]. Experimental evidence in support of the phenomenon was found in a loss-detector signal that observed coherent losses at $ω / 2$ in a plasma with a large-amplitude EGAM.¹⁰⁸ 

The ASDEX-Upgrade tokamak provides a second example. In this experiment, three-dimensional static field perturbations are superimposed upon the usual axisymmetric tokamak fields. Neutral beams are the source of EPs. Since the perturbing fields are static (ω = 0) and μ is conserved, the linear resonance condition [Eq. (13)] is $n ω ϕ = m ω θ$ for these conditions. For all mode amplitudes, changes in canonical angular momentum $P ϕ$ are observed at the linear resonances but, at higher amplitude, $P ϕ$ also changes at fractional resonances (Fig. 26). Measurements with fast-ion loss detectors are consistent with the calculations.¹⁰⁹ 

For fast ions in magnetic fusion devices, small EP transport is desirable. In the present devices, neutral beams or ion-cyclotron heated ions are used to heat the plasma or provide momentum and current. In a reactor, charged fusion products must transfer their energy to the bulk plasma before escaping; moreover, concentrated losses threaten the integrity of the walls. From the standpoint of fast ions, the large EP orbits are both a benefit and a curse. For non-resonant perturbations, orbit averaging significantly reduces transport to levels well below that of thermal ions. However, the large orbits increase the number of important resonances, enhancing the probability that fast-ion driven instabilities will cause appreciable transport.

For runaway electrons, the situation is reversed: poor confinement is desirable in order to minimize acceleration to high energies and the creation of additional runaways through an avalanche process. Non-resonant orbit averaging inhibits the ability of external perturbations to degrade runaway confinement. On the other hand, resonant interactions with internally excited or externally launched waves might prove useful in minimizing runaway damage.

Although the basic mechanisms of EP transport are known, much remains to be understood. We have treated the perturbations as given when, in truth, EP transport and mode stability are often tightly coupled in a feedback loop (Sec. IV D). For an EP-driven instability, the growth or damping of the wave amplitude depends upon EP gradients, which depend upon the EP transport caused by the mode, which depends upon the mode amplitude. Much remains to be understood and experimentally confirmed about this coupled system. Another artificial aspect of the material presented here is that each perturbation is treated independently when, in practice, different types of perturbations often act concurrently and synergistically on the EP population. The fundamentals presented here are the building blocks of a comprehensive understanding of the EP distribution function.

The many helpful suggestions on the tutorial presentation by students and colleagues are gratefully acknowledged, especially the review committee of E. Bass, C. Collins, X. Du, L. Stagner, and M. Van Zeeland and the figures contributed by L. Sanchis and M. García-Muñoz. A. Bierwage, M. Podestà, and the anonymous referees suggested many improvements to the manuscript. Both W.W.H. and R.B.W. are indebted to the U.S. Department of Energy for decades of support of EP research.