The theory of kinetic effects on resistive wall mode stability in tokamaks

Tokamak fusion plasmas generate energy most efficiently when the ratio of plasma stored energy to magnetic confining field energy is high. This ratio can be characterized by the quantity beta-normal: β_N, defined as the ratio of the mean plasma pressure, $⟨ p ⟩$⁠, to the magnetic pressure $B 0 / 2 μ 0$⁠, normalized by $I p / a B 0$⁠, where I_p is the plasma current, a is the plasma minor radius, and B₀ is the toroidal magnetic field on axis. When a plasma reaches high β_N, a magnetohydrodynamic (MHD) kink-ballooning mode of instability can begin to grow. These MHD modes are macroscopic distortions of the plasma, the magnetic field lines balloon outward or kink in certain places, and if stabilizing effects do not oppose the motion, the growth continues unabated. This can lead to a disruption^1,2 of the plasma current necessary to create a component of the confining magnetic field of a tokamak, and therefore a complete loss of magnetic confinement of the plasma. Plasma physics, however, deals not only with the macro-scale of fluids, but also the micro-scale of individual charged particle motions in magnetic fields. It turns out that certain collective motions of individual particles can provide stabilizing effects to modes of instability on the macro-scale.

Theoretically, ideal MHD modes grow on the relatively short Alfvén timescale (which relates to the magnetic field strength, mass density of the particles, and characteristic size of the plasma). However, the growth rate of these modes can be slowed quite considerably by the presence of a close-fitting wall around the plasma (“wall” here being loosely defined as any conducting structure in the vicinity, like a vaccuum vessel, coil cases, structure underneath plasma-facing components, etc…). This forces the magnetic perturbations to penetrate the wall in order to grow, and the timescale for that penetration is much longer than the Alfvén timescale. In fact, if the wall is perfectly conducting, the time is infinite and the plasma is completely stabilized. In the real-world case where the wall has some resistance, the timescale is characterized by a wall time, τ_w. When the mode is converted to the more slowly growing mode in this way, it is called the resistive wall mode (RWM).^3,4 Here, “resistive” refers to the conducting structure, the wall itself, not to be confused with the resistivity of the plasma.

Resistive wall modes have been observed experimentally in high beta plasmas as exponentially growing signals on magnetic field sensors. An example, with an exagerated calculated plasma distortion for the National Spherical Torus Experiment (NSTX⁵), is shown in Fig. 1.⁶ Often the RWM will lead to a disruption of the plasma.

Theoretically, with “ideal” or “fluid” theory, plasmas are stable without considering the “kinetic” effects of particle motions up to a value of $β N , n = 1 no − wall$⁠, for toroidal mode number n = 1. Above this “no-wall” limit, the plasma is unstable to ideal kink-ballooning modes when no wall is present, or the resistive wall mode when a wall is present (see Fig. 2). There also exists a theoretical limit above which no amount of wall stabilizing effects can stop the ideal kink mode from growing, called the with-wall limit, $β N with − wall$⁠.

The RWM grows on a much slower timescale, but it is still fast compared to the duration of usual plasma discharges. Therefore, it is necessary to stabilize this mode as well. Originally it was thought that the presence of a resistive wall could slow down the kink-ballooning mode, but that the RWM itself could not be stabilized unless plasma rotation could make the resistive wall act as a conducting wall⁹ (and even this possible mechanism was not agreed upon¹⁰) Experiments soon found, however, that tokamaks could be stably operated above $β N no − wall$⁠,^6,11 even without active control (in which magnetic perturbations are applied from external coils to purposefully attempt to counteract the unstable plasma perturbation).^12,13 It was then postulated theoretically that the RWM can be stabilized by a combination of plasma rotational inertia and an energy dissipation mechanism.^14–16 

At first, researchers investigated whether considering certain energy dissipation mechanisms could explain RWM stability, i.e., if the energy of the growing mode could be dissipated into some other channel, thereby damping the mode growth. These included ideal MHD formulations with sound wave continuum damping^14,15,17 or shear Alfvén resonance damping,^14,18 or non-ideal MHD with resistive layer damping^19,20 or viscous boundary layer damping.¹⁶ 

Simple models proved to be insufficient to explain experimental results,^6,21–23 however, and recently theoretical investigation has turned to the kinetic effects on plasma stability.^{7,8,13,23–66} Here, we will explain the theoretical model for those kinetic stabilizing effects in detail, expanding greatly upon previous efforts that outlined derivations (for example, Refs. 28 or 58).

The starting point, ideal MHD, is a framework for describing plasmas as fluids using low-frequency Maxwell's equations without resistivity, so that the magnetic field lines are “frozen in” to the conducting fluid. As such, any perturbations that arise to the plasma pressure are treated as perturbations to the fluid pressure, not to the distribution of particles that make up the fluid. Though plasma resistivity is potentially an important factor that is being neglected,^59,67,68 it is beyond the scope of the present work and will not be considered here.

It was then recognized, however, that ideal MHD was not adequate in describing motion of particles parallel to magnetic fields, first for fast rotating plasmas. Later, these kinetic effects were understood to be important for the RWM also at slow plasma rotation (“slow” here meaning below the ion diamagnetic drift frequency). The reasons for this are, first, being a low-frequency mode, the RWM is basically “locked” to the wall, hence, it rotates with respect to the plasma, and second, because of the global mode structure (displacement of magnetic field lines), the RWM has a global kinetic interaction with particles. Early efforts in the direction of kinetic stability theory included the parallel viscous force model for sound wave damping by Chu et al.⁶⁹ and the semi-kinetic model including mode resonance with bounce motion of thermal ions of Bondeson et al.^70–72 Unlike the ideal fluid MHD model, the pressure tensor in kinetic MHD is calculated by taking the velocity space integral of the perturbed drift-kinetic distribution function. By doing this, one recovers all the fluid terms and in addition one gets the kinetic contributions related to the trapped and passing (or circulating) particles. Figure 3 shows the trajectories of trapped particles in a tokamak that do not have enough energy to make it from the low field (outer) to high field (inner) side and so literally bounce back, and the circulating particles that do. The circulating particles of course have a frequency of their motion, and the trapped particles have a frequency of the bounce motion, but also a precession frequency because the bounces do not return the particle to the same place as it started, so after many bounces these particles precess around the device.

The kinetic stabilizing effects are through resonances between the plasma rotation and these thermal particle drift motions: magnetic precession (very slow rotation), trapped particle bounce motion (faster rotation), and passing particle transit motion (even faster rotation). The particles in a plasma have a distribution of energies, and therefore of frequencies, but what is meant by resonance in this context is that a certain amount of particles undertaking these motions will closely oppose the fluid rotational motion of the bulk plasma. Therefore, these particles look to be essentially stationary with respect to the RWM perturbation. It then becomes easy for the energy of the mode (displacing magnetic field lines) to transfer to kinetic energy of those particles, thus sapping the mode of energy to continue its growth. It is through these resonances that motions of individual particles affects the macroscopic stability of the plasma.

Energetic particles from neutral beam injection or alpha particles from fusion reactions have frequencies of motion that are typically very high and so are not in resonance with the plasma rotation. These particles can provide a stabilizing effect in a different way, however, by acting to make the magnetic flux more rigid and resistant to change by the RWM.⁷ It is difficult for a magnetic field line to move when all particles that are moving along it at high speeds must also move with it.

In this paper, we will outline the two general approaches to calculating RWM stability. The most basic way to understand whether a mode of instability in a plasma is stable or unstable is to consider some small pertubation from an equilibrium state and to determine whether it grows or is damped. Let us consider all plasma quantities to be perturbed in time by a mode of instability from their equilibrium states with the following form:⁷⁴  $x = x 0 + x ̃ e − i ω t − i n ϕ$⁠. Here, x is any quantity such as position, velocity, pressure, etc. We use the notation $ω = ω r + i γ$ for the complex mode frequency, so that ω_r is the real mode rotation frequency, and γ is the growth rate. Also, $ϕ$ is the toroidal angle, $ω ϕ$ is the plasma toroidal rotation frequency, and n is the toroidal mode number. That is to say, that if the displacement (for example, in Fig. 1) was broken down into Fourier harmonics, n would be the periodicity of each in the toroidal direction. The n = 1 component, it turns out, is usually the fastest growing (least stable) for the RWM.

Now, we consider what happens when the plasma is displaced perpendicular to the magnetic field lines a small distance from its equilibrium position of $ξ 0 = 0$⁠, so, $ξ = ξ ̃ e − i ω t − i n ϕ$⁠. The goal is to find out whether this small displacement is stable or unstable, i.e., whether it will damp $( γ < 0 )$ or grow exponentially in time (⁠ $γ > 0$⁠). In order to do this, a system of equations that describes the plasma must be used, and solved for γ.

First, the self-consistent approach, outlined in Sec. II, is to write a set of equations for ω in terms of known quantities and then to solve the system. This approach has the advantage of self-consistency between the calculation of the mode frequency ω and the mode displacement $ξ$⁠.

The second approach, which is the subject of the rest of the work, is to write an expression for ω in terms of changes in potential energy (δW) called a dispersion relation, and then solve for the δW terms. This approach has the advantage of clarity in distinguishing the various stabilizing and destabilizing effects.

In Sec. III, we begin with a conservation of energy equation and decompose it into constituent kinetic and potential energy terms. This equation becomes the basis of the dispersion relation for the RWM that we outline in Sec. IV. In this work, we will concentrate on the change of potential energy that arises from the perturbed kinetic pressure and is written in terms of the perturbed distribution function of the various species of particles in the plasma. This perturbed distribution function comes from the drift kinetic equation, and in Sec. V, it is used to find a form of $δ W K$⁠, the kinetic term. Heuristic dependencies of the kinetic term are discussed in Sec. VI, and finally, the paper is concluded with a discussion of the theory and insight of kinetic effects on RWM stability.

At this point, the problem naturally separates into the fluid and kinetic approaches. In the fluid approach, the perturbed pressures are given in terms of macroscopic quantities. In the kinetic approach, $p ̃ ⊥$ and $p ̃ ∥$ are defined by using the perturbed distribution function of particles, $f ̃$⁠.

If the following assumptions are made: isotropic equilibrium pressure (⁠ $p ∥ − p ⊥ = 0$⁠), $ρ = constant$ in Eq. (21) to get a new Eq. (29), rotation gradient $∇ ω ϕ$ in the perpendicular direction, and if we substitute Eq. (4) (⁠ $v 0 = R 2 ω ϕ ∇ ϕ$⁠) for $v 0$ and use $ρ ̃ = − ∇ · ( ρ 0 ξ )$ from Eq. (17) in Eq. (24) [eliminating Eq. (27)], then the result is the set of equations used in the MARS-K code^3,28,29,84 to self-consistently solve for the stability of the RWM. Note that this set of equations makes no specific reference to the wall surrounding the plasma. The dependence of the RWM displacement, $ξ$⁠, on the geometry of the device arises self-consistently from the specification of $j 0$ and $B 0$⁠.

A different approach is to change the force balance equation into an equation in terms of changes of kinetic and potential energies (δW), and then to write a dispersion relation for the complex mode frequency ω in terms of these δW terms.^81,85 This approach has been called an energy principle⁸⁶—the principle being that if any small displacement, $ξ$⁠, from the equilibrium can be found that causes the potential energy to decrease, the kinetic energy to increase, and the displacement to grow exponentially in time, then that equilibrium is unstable. Specifically applicable to the RWM is the so-called “low-frequency” energy principle,^{78,79,81,85,87} which requires the inclusion of the particle drift frequencies, as these can not be considered to be much lower than the mode frequency.

We start by combining the first two equations of the self-consistent set, eliminating $v ̃$⁠. Then, to convert the force balance to an energy balance, we multiply both sides of Eq. (7) by $ξ * / 2$ (where $ξ *$ is the complex conjugate of $ξ$⁠), integrate with respect to volume, and sum over all species j. In the energy principle approach, $ξ$ is taken as an input, not solved self-consistently with ω.

This equation can be written $δ I = − δ W$⁠, where the left-hand side is the kinetic energy, also known as the inertial term, and the right-hand side is the negative of the change in the potential energy. Alternatively, we can write $L = δ I + δ W = 0$⁠, where $L$ is the MHD Lagrangian (a function summarizing the dynamics of the system).⁸² 

One can see that there are various rotational terms in Eq. (30). These are distinct from the rotational resonance effects that will appear through consideration of kinetic effects through the perturbed pressure term of Eq. (30). These rotational effects have been previously considered^89,90 and have been shown to be important modifications to the ideal wall limit.⁵⁴ The term including $z ̂ × ξ$ has sometimes been called a “dissipation” energy integral and is actually the Coriolis force.⁵⁴ The term containing $ξ ⊥ · ∇ ω ϕ$ is a Coriolis force dependent on the rotation gradient.⁵⁴ Note that rotation shear has been identified as potentially important for RWM stability.^91–94 Finally, the last term is the centrifugal force.⁵⁴ Note that the contribution to the centrifugal force from energetic particles may be as important, or more important, than that of thermal particles.⁸³ 

There are various simplifications that can be made considering the rotational terms. For example, in the case of uniform rotation, $ω ϕ$ may be taken out of the volume integrals and also $∇ ω ϕ = 0$ in the third term on the right-hand side of Eq. (30). Then, a dispersion relation can be written for the quantity $γ + i ( n ω ϕ − ω r )$⁠, as in Ref. 69.

We can consider all these rotational terms together as $δ W rot = δ K i + δ K 2 + δ W Ω + δ W d Ω + δ W cf$⁠, and it is easily seen that this is a complex quantity. Additionally, some of these terms include the quantities to be solved for, γ and ω_r, which will lead to non-linearities if these terms are kept. For the purpose of the present paper however, which focuses on kinetic effects, let us now continue without further consideration of these rotational terms.

The difference between the fluid and kinetic approaches to δW is that in the kinetic approach, the perturbed pressure is solved by taking moments of the perturbed distribution function. There is an intermediate approach, however, which relaxes the assumptions of isotropic equilibrium and perturbed pressures by specifying them through some model (but not going fully through the kinetic route). The most common approach along these lines is the previously mentioned CGL model, which results from taking two adiabatic equations. Incidentally, the CGL perturbed pressures are also a limiting case of the kinetic approach with high mode rotation.⁵⁸ 

Trapped particles were seen to provide a non-resonant contribution (called the Kruskal–Oberman term from the high frequency limit¹⁰⁰) as well as resonant contributions with the precession and bounce frequencies. For trapped particles and low frequencies, the kinetic MHD model was also formulated as an energy principle by Antonsen and Lee to include the kinetic effects of the thermal species (not just energetic particles).⁷⁸ 

Hu and Betti later extended the Haney–Freidberg formulation of the MHD energy principle for RWM¹⁰¹ to include the kinetic terms of Antonsen and Lee⁷⁸ into the plasma contribution to the energy principle.^24,25 They derived a modified Haney–Freidberg energy principle for RWMs that includes the kinetic contributions of trapped particles. This was later extended by Berkery et al. to include the kinetic contributions of circulating ions and electrons,³⁴ equilibrium pressure anisotropy,⁵⁸ energetic particles,⁷ and more (as encapsulated in the MISK code).

The so-called perturbative approach assumes the kinetic terms can be ordered as small compared to the fluid terms and uses the ideal MHD eigenfuction to compute both the fluid and the kinetic terms. Liu et al. used the same starting equations (as shown in Sec. II) without constructing an energy principle but instead directly solving the eigenvalue problem (in the MARS-K code).²⁸ This self-consistent approach is a more complicated numerical solution of the problem. Later, a benchmarking exercise between MARS-K and MISK indicated that the two approaches can give comparable solutions.⁵⁷ 

Before we get into the details of the $δ W K$ term itself, let us consider the dispersion relation it contributes to, and the effect that the real and imaginary parts have. We wish to solve for the complex mode frequency of the RWM in terms of the δI and δW terms outlined in Sec. III through a dispersion relation. In this section, we will derive the dispersion relation and discuss the impact of the various δW terms on the mode's stability.

At this point, the treatment of the wall itself must be considered. Generally two approaches have been used: the assumption of a thin¹⁰¹ or thick^102–105 wall.

Let us now also define $δ W ∞ = δ W F + δ W V ∞$⁠, the sum of the plasma fluid and vacuum perturbed potential energies when the wall is placed at infinity (the no-wall condition), and $δ W b = δ W F + δ W V b$⁠, the sum of the plasma fluid and vacuum δW terms when the wall is placed at a specific location b. These two contributions to the energy principle have been theoretically developed for years,¹⁰¹ and computer codes have been written to solve for them, such as PEST,^95, DCON,¹⁰⁷ and the VACUUM code⁹⁷ for the vacuum region. Note that setting $τ w = 0$ leads to the internal kink mode dispersion relation,²⁶ starting from $δ I + δ W K + δ W ∞ = 0$⁠.

The above expression is valid in the range where $δ W b > 0$⁠, that is where $β < β b$⁠, the “with-wall” or “ideal” limit. Also, in order for the problem to be considered ideal, not resistive,⁵⁹ the wall time should be less than the ideal-wall tearing mode growth time.

We have already seen that the $δ W K , δ W rot$⁠, and δI terms also include γ and ω_r in their formulations, so the above expression is non-linear. In fact, in general, there are three possible roots of the RWM from this dispersion relation,^{16,28,32,38,63} but we will put that complication aside for the moment. Further discussion of the multiple roots of the RWM can be found in Ref. 38.

Note that if the complex kinetic term, $δ W K$⁠, is also neglected, the result is the fluid growth rate for resistive wall modes neglecting plasma inertia and kinetic effects, which is written:^101,103 $γ F τ w = − δ W ∞ / δ W b$⁠. This expression, from ideal theory, does not provide any means for the plasma to exert a torque on the resistive wall mode.¹⁰ 

At this point, heuristic approximations can be made to examine the stability space of the RWM, such as those made in Ref. 24: $δ W ∞ ∼ ( β ∞ − β ) , δ W b ∼ ( β b − β )$⁠, and $δ W K ∼ β ( x + i y )$⁠, where x and y representing the size of the nonresonant and resonant particle contributions. This results in the stability diagram shown in Fig. 4, where the RWM is seen to be stabilized or destabilized in various ways depending on the levels and ratio of real and imaginary parts of $δ W K$⁠. The upper left diagram of Fig. 4, for example, is equivalent to the cartoon presented in Fig. 2, with some additional stabilization between the no-wall and with-wall limits.

Some examples of such diagrams can be found in Refs. 7, 33, 34, 36, 58, and 63 and one is shown in Fig. 5, where it can also be seen how additional real fluid effects such as anisotropy in the examples, change the stability contours. Additional complex quantities such as the rotational effects should be included in $δ W K$ (on the axes), changing the position of the plasma equilibrium on the stability diagram.

Therefore, for a given $δ W ∞$ and $δ W b$⁠, the plasma is stable if the calculated $R e ( δ W K )$ and $I m ( δ W K )$ lie outside of a circle centered at $[ − 1 2 ( δ W b + δ W ∞ ) , 0 ]$ with radius $1 2 ( δ W b − δ W ∞ )$⁠. Once the values of $δ W ∞$ and $δ W b$ are known, it can be predicted what values of $I m ( δ W K )$ and $R e ( δ W K )$ will be necessary to provide stabilization. If $R e ( δ W K ) > − 1 2 ( δ W b + δ W ∞ )$⁠, which is usually the case, then increasing $R e ( δ W K )$ will decrease the growth rate. Increasing $| I m ( δ W K ) |$ always decreases the growth rate.

Therefore, determining the stability of a plasma equilibrium to resistive wall modes involves the calculation of $δ W ∞ , δ W b$⁠, and $δ W K$ (and any other terms that are not neglected). Let us now turn our attention to the solution for the $δ W K$⁠, which we have already formulated up to the point of dependence on $ℙ K$⁠.

Theoretically, one can use different distribution functions of particles in the solution of Eq. (57), but for thermal particles typically a Maxwellian distribution is used.

Now that the frequency resonance fraction is defined, it becomes possible to look again heuristically at how the stabilization of the RWM depends on various important terms. Obviously, in any tokamak plasma, there is a large collection of particles with a spread of energies and pitch angles, and the codes that calculate kinetic RWM stability do the full integration over these terms, but it is useful to do some simpler exercises. The resonances that happen between various frequencies in the denominator are the primary focus. The simple picture is that when ω_E, which again is a proxy for plasma rotation, is close to equal and opposite to one of the other frequencies, it makes the denominator small and therefore the kinetic stabilization term large.

It should be noted as a caveat that the stabilization mechanism being described is the wave–particle interaction extracting energy from the mode, which comes from the real part of $δ W K$⁠. However, the imaginary contributions of $δ W K$ cause the mode to rotate with respect to the wall (real frequency ω_r). The mode rotation causes AC wall stabilization (skin effect in the wall) and suppression of the RWM because the rotating mode cannot penetrate the wall that effectively acts as a superconductor. It is difficult to truly decouple the two effects (energy transfer via wave-particle resonance and AC wall stabilization) to isolate the physics of the RWM kinetic stabilization, especially when $ω r τ w ≳ 1$⁠.³⁸ For the purpose of this descriptive summary, though, we will consider $ω r ∼ 0$ and now examine some specific cases in turn.

First, energetic particles tend to have frequencies of motion that are considerably larger than the plasma rotation. This means that changing the plasma rotation within an experimental range does not change much about the effect of energetic particles. They are not in resonance with the rotation, but they do provide some stabilizing effect that is generally proportional to their share of the particle population. Their effect can be thought of as a stiffening of the field lines when many energetic particles are moving at high speeds along them.⁷ 

An effective collision frequency appears also in the denominator. One can get a sense of the effect of collisions by lumping all the other terms in the denominator into a term $Ω 2$⁠.³⁶ If the plasma rotation is close to a resonance with another frequency of particle motion, $Ω 2$ can be small (⁠ $≪ ν eff$⁠). Then, in simplest terms, large collisionality decreases the effectiveness of the resonant stabilization by washing it out, while small collisionality allows the resonances to more effectively stabilize the plasma. When the rotation is away from resonances, $Ω 2 ≥ ν eff$⁠, and the expectation is that collisions will have less effect on the weaker kinetic stabilization. In reality, calculations show that changing collisionality can effectively change the rotation level at which the resonances occur.

Finally, what of the plasma rotation itself? Because of the distribution of particles in velocity space mentioned before, there is no rotation profile that causes a singular resonant response. Rather, at high rotation there are many particles with circulating or bounce motions that can interact with the mode and provide stabilization. As the rotation profile is decreased, however, these effects decrease, and there can be a point at an “intermediate” plasma rotation where the kinetic effects are not strong enough, and the RWM can go unstable.³⁴ At lower rotations, resonance between the plasma rotation and the precession drift can be important. Again, resonance is not in a narrow range of rotation, but rather the stabilizing effects can be approximated as a Gaussian distribution of strength vs plasma rotation, as has been done in a reduced model of the effect.⁸ Finally, at the lowest rotations, the plasma can be unstable again. Note that the E × B frequency differs from the plasma rotation by the particle diamagnetic frequency, so there is a possible low frequency stabilizing resonance left that makes $ω E ∼ 0$ in Eq. (57), but in practicality, low rotation is typically unstable.

All of these dependencies taken together can be seen in a calculation of RWM growth rate in the space of ν vs $ω ϕ$ that was made for the NSTX device, shown in Fig. 6. Moving horizontally on the plot illustrates the different rotational resonances, as just described. Artificially reducing the energetic particle content would have the first order effect of deepening the red, unstable, regions and lightening the blue, stable, regions. One can see moving vertically upward on the plot, collisionality is increased. Near the dark blue precession resonance of $ω ϕ ∼ 0.7$ of the experimental profile for this case, the resonant stabilization decreases with collisionality. At the low rotation, away from any stabilizing resonances, changing collisionality has little effect. At intermediate rotation, changing collisionality effectively changes the rotation frequency that lies between the precession and bounce resonances.

A reduced model using the Gaussian representations of the resonances has been implemented in the DECAF^TM code¹¹⁰ and produces similar results.⁸ Additionally, machine learning tools for the reduced kinetic RWM stability model are also being investigated.^111,112

Finally, in future devices, the plasma rotation may be quite different from present-day beam-heated experiments, with implications for stability. First, while we have neglected poloidal rotation, in future devices with either no beams or low torque from the beams due to high inertia of large plasmas, the toroidal rotation may be quite low, so the inclusion of poloidal rotation in the theory^76,77 should be revisited. Second, though low toroidal rotation level experiments have been performed through non-resonant magnetic braking⁶ or balanced beam injection,²³ for confidence in theoretical projection to future devices, more comparison to experiments emulating future rotation levels would be welcome.

Theory shows that the stability of the resistive wall mode in tokamak fusion devices depends upon kinetic effects. Calculation of the complex mode frequency, ω, can either be performed by the solution of a set of self-consistent equations, or through the energy principle approach of calculating changes in potential energy (δW) due to various effects and inputting these into a dispersion relation for ω (or stability criteria for the growth rate γ). Inertial or fluid rotational effects may be important at higher plasma rotation. Various closures to the set of equations by specification of the equilibrium and perturbed pressures are possible. Here, we utilize the kinetic approach, in which the perturbed pressure is solved by taking moments of the perturbed distribution function.

Using the drift kinetic equation to find a general expression for the perturbed distribution function leads to general forms for the δW terms. We find that δW can be neatly divided into $δ W K$⁠, a kinetic term that depends upon the frequency resonance fraction, and fluid terms, which are strictly real.

The major physics insight for RWM stability that have been obtained over the years by the development of kinetic theory, the numerical tools for its computation, and the application of the theory to experimental results include the following. Energetic particle frequencies of motion are typically very high and so are not in resonance with the plasma rotation, but rather they provide a stabilizing effect by acting to make the magnetic flux more rigid and resistant to change by the RWM (in certain cases, energetic particles can also drive a fishbone-like external kink, with the mode frequency matching the precession drift frequency of EPs). The effect of energetic particles and alpha particles on RWM stability will be important to consider in low-rotation ITER plasmas. Particle collisions can have two competing effects: collisional dissipation of mode energy when rotational resonances are not present, or damping of those stabilizing resonances when they are. Further analysis using the theory showed new, positive ramifications for improved mode stability, albeit with a larger stability gradient, at low collisionality and slow plasma flow,³⁶ as expected in ITER and future devices. RWM stability can be increased by kinetic effects at low rotation through precession drift resonance and at high rotation by bounce and transit resonances, while intermediate rotation can remain susceptible to instability. Ultimately, the big-picture insight is that rotational resonances in the kinetic term between the particle motions and the plasma rotation can provide stabilization by dissipating the energy of a potentially growing mode.

This work was supported by the U.S. Department of Energy under Contract Nos. DE-AC02-09CH11466, DE-FC02-04ER54698, DE-FG02-95ER54309, DE-SC0018992, and DE-SC0014196.

The United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purpose.

The authors have no conflicts to disclose.

John William Berkery: Conceptualization (lead); Writing – original draft (lead). Riccardo Betti: Writing – review & editing (equal). Yueqiang Liu: Writing – review & editing (equal). S.A. Sabbagh: Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.