Drift kinetic theory of alpha transport by tokamak perturbations
UUnder consideration for publication in J. Plasma Phys. Drift kinetic theory of alpha transport bytokamak perturbations
Elizabeth A. Tolman † and Peter J. Catto Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA02139, USA Institute for Advanced Study, Princeton, NJ 08540, USA(Received xx; revised xx; accepted xx)
Upcoming tokamak experiments fueled with deuterium and tritium are expected tohave large alpha particle populations. Such experiments motivate new attention tothe theory of alpha particle confinement and transport. A key topic is the interactionof alphas with perturbations to the tokamak fields, including those from ripple andmagnetohydrodynamic modes like Alfvén eigenmodes. These perturbations can transportalphas, leading to changed localization of alpha heating, loss of alpha power, and damageto device walls. Alpha interaction with these perturbations is often studied with singleparticle theory. In contrast, we derive a drift kinetic theory to calculate the alpha heatflux resulting from arbitrary perturbation frequency and periodicity (provided these canbe studied drift kinetically). Novel features of the theory include the retention of a largeeffective collision frequency resulting from the resonant alpha collisional boundary layer,correlated interactions over many poloidal transits, and finite orbit effects. Heat fluxesare considered for the example cases of ripple and the toroidal Alfvén eigenmode (TAE).The ripple heat flux is small. The TAE heat flux is significant and scales with thesquare of the perturbation amplitude, allowing the derivation of constraints on modeamplitude for avoidance of significant alpha depletion. A simple saturation conditionsuggests that TAEs in one upcoming experiment will not cause significant alpha transportvia the mechanisms in this theory. However, saturation above the level suggested by thesimple condition, but within numerical and experimental experience, which could beaccompanied by the onset of stochasticity, could cause significant transport.
1. Introduction
Next generation tokamak experiments fueled with deuterium and tritium, includingARC (Sorbom et al. et al. et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] N ov E. A. Tolman and P.J. Catto
Ripple is a periodic, stationary magnetic perturbation of the plasma equilibrium due tothe discrete nature of toroidal field coils. † MHD modes are fluctuations of the plasmawhich include Alfvén eigenmodes (AEs) (Cheng & Chance 1986; Heidbrink 2008), drivenby energetic particles, neoclassical tearing modes (NTMs) (La Haye 2006), destabilizedby tokamak bootstrap current, and fishbones (Betti & Freidberg 1993), also destabilizedby energetic particles. A major mechanism of such transport is the anomalous radialmotion introduced by the perturbation via drifts and changed magnetic field direction.When an alpha particle’s motion is in phase with the periodicity of the perturbation, thisradial motion results in transport. Transport of this nature has sometimes been describedas bounce harmonic resonant transport (Linsker & Boozer 1982; Mynick 1986; Park et al. et al. et al. et al. (2013) and Collins et al. (2017)], using orbit following codes, including ASCOT(Hirvijoki et al. et al. et al. single particle . Foundational works on transport identify the onset of significantenergetic particle redistribution with individual particle trajectories becoming stochasticin phase space (Goldston et al. et al. et al. et al. et al. et al. et al. † Information on ripple in ITER can be found in Kurki-Suonio et al. (2009); information onripple in SPARC in Scott et al. (2020). rift kinetic theory of alpha transport by tokamak perturbations † The transport is akin to a quasilinear evaluation with acollisionally broadened overlap of the isolated mode surfaces spaced within a collisionalstep size of each other. Indeed, as our results will ultimately prove to be independent ofcollision frequency, it may be that they are even valid for mild resonance overlap, as longas the slowing down distribution is not significantly distorted. This "sea" of perturbationsof various toroidal mode numbers is reduced to a tractable problem by focusing on asingle toroidal mode centered at a flux surface. The transport associated with the firstfew bounce and transit harmonics and the most resonant poloidal mode is then evaluated,because these contributions dominate the overall transport. The transport due to otherpoloidal mode numbers and higher bounce and transit harmonics will be smaller becauseof their phase mismatch.The techniques developed herein are kept in fully general form so that they can be usedfor tokamak perturbations of any frequency, toroidal periodicity, poloidal periodicity,and radial structure (provided that the perturbation is treatable with drift kinetics).However, two perturbations are given as examples. First, alpha particles are found to havemany resonances with ripple, but these resonances drive negligible transport. Second,alpha particles resonate with toroidal Alfvén eigenmodes (TAEs). These resonances drivesignificant alpha heat flux that increases with the square of mode amplitude. The heatflux expression can be used to derive a constraint on mode amplitude for avoidance ofsignificant alpha depletion. A simple model for TAE saturation is developed to estimateTAE amplitude. The saturation estimate is consistent with numerical results presentedelsewhere (Slaby et al. et al. (1992). As an example, the constraints and the saturation amplitude areevaluated for SPARC-like parameters. The estimate suggests that the TAE amplitude inSPARC will not cause significant depletion. However, a TAE saturation amplitude abovethat suggested by the simple model, but within experimental and modelling experience,could cause alpha particle depletion.The structure of this paper follows. Section 2 introduces the mathematical descriptionof the tokamak equilibrium and the drift kinetic equation which is used to describe alphatransport. It provides parameters for a SPARC-like tokamak which will be used as anexample tokamak equilibrium throughout the paper. Then, the section and appendix Aderive the slowing down distribution, which describes the unperturbed alpha population.Section 3 introduces the form of the perturbations which we study in this paper, anddescribes the particular parameters that describe ripple and TAE, the perturbations † Such transport is analogous to the situation illustrated in figures 3 and 4 of Berk et al. (1995).
E. A. Tolman and P.J. Catto we use as examples. Section 4 discusses the plasma response to these perturbations,first giving the equation that governs how the alpha distribution function respondsto perturbed fields, then solving this equation for the response as a function of theparameters describing the perturbation. Section 5 develops general formulae for thealpha particle heat flux from both trapped and passing alpha particle populations. Theseformulae are the key results of this paper. This section also considers the nature of aresonance function and phase factor that are key components of the formulae. Section 6considers this heat flux for the case of ripple, and shows that it is very small. Section 7derives compact expressions for trapped and passing fluxes for the TAE. These fluxes areused to understand what amplitude of TAE causes significant alpha depletion. A simplesaturation condition is developed and the amount of transport caused by this saturatedamplitude is considered. The conclusion, section 8, summarizes the paper results, givesimplications for future experiments, and presents avenues for future work.
2. Equilibrium and governing equations
In this section, we first describe the tokamak magnetic equilibrium in which thealpha particle transport occurs and the parameters that describe alpha particles in thisequilibrium. Then, we introduce the drift kinetic equation which is used to study alphabehavior. Next, we calculate the equilibrium alpha particle slowing down distributionfunction. The section concludes by giving specific tokamak parameters which will beused as examples throughout the paper.2.1.
Equilibrium and phase space coordinates
The coordinates describing the tokamak equilibrium are 𝜓 (the poloidal flux function), 𝜗 (the poloidal angle) † , and 𝛼 , defined by 𝛼 ≡ 𝜁 − 𝑞 ( 𝜓 ) 𝜗. (2.1)Here, 𝜁 is the toroidal angle. The safety factor is 𝑞 , which characterizes the twist of thebackground magnetic field by 𝑞 ( 𝜓 ) ≡ ˆ 𝑏 · ∇ 𝜁/ (︁ ˆ 𝑏 · ∇ 𝜗 )︁ with ˆ 𝑏 giving the direction ofthe equilibrium magnetic field. ‡ We study the behavior of alpha particles confined by amagnetic field which is axisymmetric and stationary except for the perturbation. Such afield is stated to zeroth order in the perturbation via the Clebsch representation as ⃗𝐵 = ∇ 𝛼 × ∇ 𝜓 = 𝐼 ( 𝜓 ) ∇ 𝜁 + ∇ 𝜁 × ∇ 𝜓, (2.2)with 𝐼 ( 𝜓 ) characterizing the strength of the toroidal magnetic field by 𝐵 𝜁 = 𝐼 ( 𝜓 ) /𝑅 ,where 𝑅 is the major radius coordinate. We do not consider background electric fieldsbecause they do not affect alpha particle trajectories as strongly as the magnetic fields † At this point, 𝜗 is a fully general straightened field line poloidal coordinate which must onlybe chosen such that 𝑞 ( 𝜓 ) = (^ 𝑏 · ∇ 𝜁 ) / (^ 𝑏 · ∇ 𝜗 ) . This allows for shaping and finite aspect ratio.Later, in section 5, a circular-cross-section, high-aspect-ratio approximation is used to obtainanalytic expressions for the flux. ‡ Later in the paper, an approximate form of this definition, 𝑞 ( 𝜓 ) ≈ (︁ 𝑅 ^ 𝑏 · ∇ 𝜗 )︁ − , with 𝑅 the major radius coordinate, will be used to simplify expressions. rift kinetic theory of alpha transport by tokamak perturbations ¶ As noted previously, the unit vector corresponding to this field is ⃗𝐵𝐵 ≡ ˆ 𝑏. (2.3)The poloidal component of the field is denoted 𝐵 𝑝 , and can be found from 𝐵 𝑝 ≈ 𝜖𝐵/𝑞 ,with the inverse aspect ratio 𝜖 ≈ 𝑟/𝑅 , where 𝑟 is local the minor radius. [The fluxcoordinate and the minor radius are related by 𝜕/𝜕𝜓 = (1 /𝑅𝐵 𝑝 ) 𝜕/𝜕𝑟 .] The shear of thefield is given by 𝑠 ≡ ( 𝑟/𝑞 ) 𝜕𝑞/𝜕𝑟. (2.4)The total magnetic and electric fields affecting the alpha particles will include boththe equilibrium magnetic field and magnetic and electric fields resulting from the per-turbations, which are discussed in section 3. The total magnetic field from all of thesesources is denoted ⃗𝐵 𝑡𝑜𝑡 (with unit vector ˆ 𝑏 𝑡𝑜𝑡 ) and the total electric field is ⃗𝐸 𝑡𝑜𝑡 . Severalparameters can be defined in terms of the total field or the unperturbed field; the totalquantities are in general only used before the perturbation analysis is carried out.Alpha particles in these fields are characterized by their velocity, 𝑣 (or, equivalently,energy normalized to mass, ℰ ≡ 𝑣 / ), the sign of 𝑣 ‖ , their velocity parallel to theequilibrium magnetic field, and their pitch angle, which may be defined relative to thetotal magnetic field, 𝜆 𝑡𝑜𝑡 = 𝐵 𝑣 ⊥ ,𝑡𝑜𝑡 𝐵 𝑡𝑜𝑡 𝑣 , (2.5)with 𝐵 the equilibrium on-axis magnetic field strength, or relative to the unperturbedfield, 𝜆 = 𝐵 𝑣 ⊥ 𝐵𝑣 . (2.6)(Here, 𝑣 ⊥ ,𝑡𝑜𝑡 is the speed perpendicular to total field and 𝑣 ⊥ is the speed perpendicularto the unperturbed field.)The alpha particle mass and charge are given by 𝑀 𝛼 and 𝑍 𝛼 , respectively; 𝛺 𝑡𝑜𝑡 ≡ 𝑍 𝛼 𝑒𝐵 𝑡𝑜𝑡 /𝑀 𝛼 𝑐 is the gyrofrequency in the total field and 𝛺 ≡ 𝑍 𝛼 𝑒𝐵/𝑀 𝛼 𝑐 the gyrofre-quency in the unperturbed field. The alpha particle poloidal gyrofrequency is 𝛺 𝑝 ≡ 𝑍 𝛼 𝑒𝐵 𝑝 /𝑀 𝛼 𝑐 . The alpha particle gyroradius is 𝜌 𝛼 ≡ 𝑣 ⊥ /𝛺 , and the poloidal gyroradius 𝜌 𝑝𝛼 ≡ 𝑣 ⊥ /𝛺 𝑝 .2.2. Drift kinetic equation and equilibrium alpha particle distribution
We study the effect of perturbations on the energetic alpha particle distributionfunction 𝑓 using Hazeltine’s drift kinetic equation (Hazeltine 1973). This formalism isappropriate when 𝜌 𝛼 (which at the alpha particle birth speed is typically a couple ofcentimeters) is much less than the length scales relevant to the problem, including theperpendicular scale length of the perturbation and the scale length of the alpha particle ¶ This follows from the expression for the particle drifts, given in (2.13), where we can findthat for alphas the tangential component of the ⃗𝐸 × ⃗𝐵 drift is given by ⃗𝑣 ⃗𝐸 × ⃗𝐵 · ∇ 𝛼 = − 𝑐𝜕𝛷 /𝜕𝜓 ,with 𝛷 the background potential, while the ∇ 𝐵 drift is given by ⃗𝑣 ∇ 𝐵 · ∇ 𝛼 ∼ − 𝜌 𝛼 𝑣 / ( 𝑎𝑅 ) , with 𝜌 𝛼 the alpha gyroradius and 𝑣 the alpha birth speed. Typically, 𝜕𝛷 /𝜕𝜓 ∼ − ( 𝑇 𝑖 /𝑒𝑛 𝑖 ) 𝜕𝑛 𝑖 /𝜕𝜓 ,with 𝑇 𝑖 and 𝑛 𝑖 the background ion temperature and density, respectively. Taking the scale lengthof the ion population to be the device minor radius 𝑎 gives ⃗𝑣 ⃗𝐸 × ⃗𝐵 · ∇ 𝛼 ∼ 𝜌 𝑖 𝑣 𝑖 /𝑎 , with 𝜌 𝑖 theion gyroradius. The tangential component of the ∇ 𝐵 drift is then larger than the tangentialcomponent of the ⃗𝐸 × ⃗𝐵 drift by a factor of 𝜖𝜌 𝛼 𝑣 / ( 𝜌 𝑖 𝑣 𝑖 ) , which is large. E. A. Tolman and P.J. Catto density, 𝑎 𝛼 ≡ − 𝑛 𝛼 𝑅𝐵 𝑝 𝜕𝑛 𝛼 /𝜕𝜓 , (2.7)with 𝑛 𝛼 the alpha particle density. The drift kinetic equation reads: 𝜕𝑓𝜕𝑡 + (︁ 𝑣 ‖ ,𝑡𝑜𝑡 ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝑓 + [︂ 𝑍 𝛼 𝑒𝑀 𝛼 (︁ 𝑣 ‖ ,𝑡𝑜𝑡 ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ⃗𝐸 𝑡𝑜𝑡 + 𝜆 𝑡𝑜𝑡 𝑣 𝐵 𝜕𝐵 𝑡𝑜𝑡 𝜕𝑡 ]︂ 𝜕𝑓𝜕 ℰ = 𝐶 { 𝑓 } + 𝑆 𝑓𝑢𝑠 𝛿 ( 𝑣 − 𝑣 )4 𝜋𝑣 . (2.8)Here, 𝑆 𝑓𝑢𝑠 is the source rate due to fusion, 𝑣 is the alpha particle birth speed ( . × cm s − ), and ⃗𝐸 𝑡𝑜𝑡 is the total electric field (equal to the contribution from the per-turbations, since we neglect the background electric field). The velocity parallel to thetotal field is 𝑣 ‖ ,𝑡𝑜𝑡 . The energetic alpha collision operator represents collisions with thebackground plasma electrons and ions, and is given by (Cordey 1976; Catto 2018) 𝐶 { 𝑓 } = 1 𝜏 𝑠 𝑣 𝜕𝜕𝑣 [︀(︀ 𝑣 + 𝑣 𝑐 )︀ 𝑓 ]︀ + 2 𝑣 𝜆 𝐵 𝜏 𝑠 𝑣 𝐵 𝑣 ‖ 𝑣 𝜕𝜕𝜆 (︂ 𝜆 𝑣 ‖ 𝑣 𝜕𝑓𝜕𝜆 )︂ . (2.9)(Note that the distinction between the total and the unperturbed field is unimportant inthe collision operator.) The first term represents electron and ion drag while the secondrepresents pitch angle scattering off of bulk ions. Here, the alpha slowing down time isgiven by 𝜏 𝑠 ( 𝜓 ) = 3 𝑀 𝛼 𝑇 / 𝑒 ( 𝜓 )4 (2 𝜋𝑚 𝑒 ) / 𝑍 𝛼 𝑒 𝑛 𝑒 ( 𝜓 ) ln 𝛬 𝑐 , (2.10)with ln 𝛬 𝑐 , 𝑇 𝑒 , 𝑛 𝑒 , and 𝑚 𝑒 , the Coulomb logarithm, the electron temperature, density, andmass, respectively. The critical speed at which alpha particles switch from being mainlydecelerated by electrons to being mainly decelerated by ions is found by summing overbackground ions, 𝑣 𝑐 ( 𝜓 ) = 3 𝜋 / 𝑇 / 𝑒 ( 𝜓 )(2 𝑚 𝑒 ) / 𝑛 𝑒 ( 𝜓 ) ∑︁ 𝑖 𝑍 𝑖 𝑛 𝑖 ( 𝜓 ) 𝑀 𝑖 , (2.11)with 𝑍 𝑖 , 𝑛 𝑖 , and 𝑀 𝑖 the charge, density, and mass of each of the background species.This is of similar size to 𝑣 𝜆 , the speed at which pitch angle scattering is important to thebehavior of the equilibrium energetic alpha population, which does not have the narrowboundary layers encountered later: 𝑣 𝜆 ( 𝜓 ) ≡ 𝜋 / 𝑇 / 𝑒 ( 𝜓 )(2 𝑚 𝑒 ) / 𝑛 𝑒 ( 𝜓 ) 𝑀 𝛼 ∑︁ 𝑖 𝑍 𝑖 𝑛 𝑖 ( 𝜓 ) . (2.12)The drift velocity is given by ⃗𝑣 𝑑,𝑡𝑜𝑡 = 𝑐𝐵 𝑡𝑜𝑡 ⃗𝐸 𝑡𝑜𝑡 × ⃗𝐵 𝑡𝑜𝑡 + 𝜆 𝑡𝑜𝑡 𝑣 𝐵 𝛺 𝑡𝑜𝑡 ˆ 𝑏 𝑡𝑜𝑡 × ∇ 𝐵 𝑡𝑜𝑡 + 𝑣 ‖ 𝛺 𝑡𝑜𝑡 ˆ 𝑏 𝑡𝑜𝑡 × (︁ ˆ 𝑏 𝑡𝑜𝑡 · ∇ ˆ 𝑏 𝑡𝑜𝑡 )︁ , (2.13)where the different terms represent, respectively, the ⃗𝐸 × ⃗𝐵 drift, the ∇ 𝐵 drift, and thecurvature drift. We use ⃗𝑣 𝑑 to refer to the zeroth order drift in the presence of only theunperturbed field.The alpha particle distribution, 𝑓 , consists of an equilibrium component, 𝑓 , and aresponse to any electromagnetic perturbations, 𝑓 . Using (2.8), 𝑓 can be calculated tobe the familiar slowing down distribution: rift kinetic theory of alpha transport by tokamak perturbations Parameter Value 𝐵 . × G 𝑅 cm 𝑛 𝑒 , 𝑛 𝑖 × cm − 𝑇 𝑒 keV 𝑣 . × cm s − Table 1: Example tokamak parameters used in this paper, similar to those plannedfor SPARC (Creely et al. et al. 𝑣 , even though this parameter is the same inany tokamak. 𝑓 ( 𝜓, 𝑣 ) = 𝑆 𝑓𝑢𝑠 ( 𝜓 ) 𝜏 𝑠 ( 𝜓 ) 𝐻 ( 𝑣 − 𝑣 )4 𝜋 [ 𝑣 + 𝑣 𝑐 ( 𝜓 )] . (2.14)The derivation of this expression for suprathermal alphas is given in Appendix A.2.3. Example tokamak parameters
When numerical examples are needed in this paper, we use the equilibrium parametersgiven in table 1, which are similar to those planned for SPARC, a DT tokamak experimentcurrently under development (Creely et al. et al.
3. Form of electromagnetic perturbations
The aim of this paper is to calculate the response of the tokamak background describedpreviously to a magnetic field perturbation, an electric field perturbation, or a combina-tion thereof. In this section, we describe the form of these perturbations and discuss theexample cases used in this paper.3.1.
General description of perturbations
As discussed in 2.2, the drift kinetic treatment bounds the scale length of the per-turbation to be much longer than the alpha particle gyroradius, but the form of theperturbation is otherwise flexible. In general, perturbations to the tokamak backgroundfield can be represented by a perturbed vector potential, ⃗𝐴 = ⃗𝐴 ⊥ ( 𝜓, 𝜗, 𝛼, 𝑡 ) + 𝐴 ‖ ( 𝜓, 𝜗, 𝛼, 𝑡 ) ˆ 𝑏, (3.1)and a perturbed electric potential 𝛷 ( 𝜓, 𝜗, 𝛼, 𝑡 ) . These correspond to a perturbed electricfield ⃗𝐸 = −∇ 𝛷 − (1 /𝑐 ) 𝜕 ⃗𝐴 /𝜕𝑡 and a perturbed magnetic field ⃗𝐵 = ∇ × ⃗𝐴 . We referto the parallel perturbed magnetic field resulting from ⃗𝐴 ⊥ as 𝐵 ‖ . The overall magneticfield unit vector ˆ 𝑏 𝑡𝑜𝑡 is related to the perturbation and the background field by ˆ 𝑏 𝑡𝑜𝑡 ≈ ˆ 𝑏 + 𝐵 − (︁ ∇ 𝐴 ‖ × ˆ 𝑏 )︁ . (3.2)A perturbation can be characterized by a frequency 𝜔 , a toroidal mode number 𝑛 ,a poloidal mode number 𝑚 , and an effective radial wave number 𝑘 𝜓 , or by a sumof components each characterized by these quantities. These fully-general forms aremaintained in the paper until specific examples are computed in sections 6 and 7. E. A. Tolman and P.J. Catto
Example perturbations: ripple and TAE
The first example perturbation considered in this paper is ripple, which results fromthe discrete nature of tokamak field coils. Ripple is composed of a perpendicular vectorpotential perturbation and corresponding parallel magnetic field 𝐵 ‖ and is time inde-pendent. It is characterized by a high 𝑛 ≈ and low 𝑚 in most devices; its radial scalelength is roughly determined by the device size or some fraction thereof, so 𝑘 𝜓 ∼ /𝑎 (note that 𝑘 𝜓 depends on several other parameters, including 𝑛 , but we require onlyknowledge of the rough size of 𝑘 𝜓 for ripple). These characteristics mean that the driftkinetic formalism is appropriate. Ripple is typically of strongest magnitude on the lowfield side in regions of high safety factor 𝑞 , high minor radius normalized to major radius 𝜖 , and high shear 𝑠 . [Information on ripple in ITER can be found in Kurki-Suonio et al. (2009); information on ripple in SPARC in Scott et al. (2020).] A summary of thesetypical values, as well as the specific example values of key ripple parameters used todemonstrate the evaluation of alpha heat flux, is given in table 2.The second example perturbation is the TAE. The TAE is a special class of Alfvénwave that exists in a tokamak; its structure is described by magnetohydrodynamics(Van Zeeland et al. † They are represented by a vector potential with ⃗𝐴 ⊥ = 0 and an electric potentialthat obeys − 𝑐 𝜕𝐴 ‖ 𝜕𝑡 − ˆ 𝑏 · ∇ 𝛷 = 𝐸 ‖ = 0 . (3.3)[This representation of the modes is used frequently in orbit-following codes (Hirvijoki et al. 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) , with 𝑣 𝐴 the on-axis Alfvénspeed, 𝑣 𝐴 ≡ 𝐵 / √ 𝜋𝑛 𝑖 𝑚 𝑖 , with 𝑚 𝑖 the average ion mass (Heidbrink 2008). TAEs mostlyexist in the core and outer core of the tokamak, where 𝑞 , 𝜖 , and 𝑠 have low to moderatevalues (Rodrigues et al. 𝑛 ranging from √ 𝜖𝑅𝛺 𝑝 / ( 𝑞𝑣 𝐴 ) to 𝑅𝛺 𝑝 / ( 𝑞𝑣 𝐴 ) , depending on if TAEs are driven primarily by trapped or passing particles(Fu & Cheng 1992; Breizman & Sharapov 1995; Fülöp et al. et al. ‡ Alpha-driven TAEs interact with both trapped and passing particle populations, suchthat the most destabilized values of 𝑛 will be intermediate between these values. However,for the specific value of 𝑛 we use to compute numerical examples in this paper, we selecta value near the bottom of the range in order to strictly obey the drift kinetic ordering.(Higher 𝑛 modes are narrower and their width may approach the alpha gyroradius,such that drift kinetics is not a good description of their behavior.) The poloidal modenumber 𝑚 is given by 𝑛𝑞 − / (Heidbrink 2008) and the radial wave number 𝑘 𝜓 is givenby 𝑘 𝜓 = 𝑛𝑞/ ( 𝜖𝑅 ) (Breizman & Sharapov 1995; Rodrigues et al. et al. 𝛷 = 𝑣 𝐴 𝐴 ‖ /𝑐 . Asummary of the typical TAE characteristics and the specific values used as examples inthis paper are given in table 2. † Inclusion of finite ion gyroradius effects introduces a parallel electric field, but these effectsare small and are outside the scope of the MHD model often used to describe TAE structure. ‡ This estimate results from balancing the width of the mode /𝑘 𝜓 ∼ 𝜖𝑅/ ( 𝑛𝑞 ) , with thewidth of the particle orbit of resonant alpha particles (which scales with 𝑣 𝐴 √ 𝜖/𝛺 𝑝 for trappedparticles and with 𝑣 𝐴 /𝛺 𝑝 for passing particles). Various variations of this balance exist in theliterature. rift kinetic theory of alpha transport by tokamak perturbations Parameter Typical ripple val. Example ripple val. Typical TAE val. Example TAE val. 𝑞 high ∼ . 𝜖 high . medium . 𝑠 ≡ 𝑟𝑞 𝜕𝑞𝜕𝑟 high low 𝐴 ‖ 𝐴 ‖ N/A 𝛷 𝑣 𝐴 𝐴 ‖ /𝑐 N/A 𝐵 ‖ 𝐵 ‖ N/A 𝑛 ≈
18 18 √ 𝜖𝑅𝛺 𝑝 / ( 𝑞𝑣 𝐴 ) to 𝑅𝛺 𝑝 / ( 𝑞𝑣 𝐴 ) 10 𝑚 𝑛𝑞 − / 𝜔 𝑣 𝐴 / (2 𝑞𝑅 ) 𝑣 𝐴 / (2 𝑞𝑅 ) 𝑘 𝜓 𝑎 𝑎 𝑛𝑞/ ( 𝜖𝑅 ) 𝑛𝑞/ ( 𝜖𝑅 ) Table 2: Defining characteristics of this paper’s two demonstration perturbations andtypical equilibrium parameters where these perturbations exist. The value 𝑛 is, for ripple,a common ripple periodicity, and, for TAEs, the range of strongly-driven toroidal modenumbers. However, other values of 𝑛 are also sometimes of interest for these perturbations.The specific values of these quantities used for numerical examples in this paper are alsoincluded. "N/A" denotes parameters for which specific example values are not needed.
4. Plasma response to perturbations
In this section, we calculate the plasma response to the perturbation described in theprevious section. Doing so requires developing a phenomenological understanding of thetransport and the role of the collision operator. Later, in section 5, this plasma responseand an expression for the radial velocity resulting from the perturbation will be used tocalculate the alpha heat flux caused by the perturbation.4.1.
Initial equation setup
To first order in the perturbation, the drift kinetic equation (2.8) is 𝜕 𝑡 𝑓 + (︁ 𝑣 ‖ ˆ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝑓 + [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ 𝜕𝑓 𝜕𝜓 + [︂ 𝑍 𝛼 𝑒𝑀 𝛼 (︁ 𝑣 ‖ ˆ 𝑏 + ⃗𝑣 𝑑 )︁ · ⃗𝐸 ]︂ 𝜕𝑓 𝜕 ℰ = 𝐶 { 𝑓 } , (4.1)where 𝑓 is the distribution function response to the perturbation and [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ is the component of [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ resulting from the perturbation. Furtheranalysis of this equation is simplified by working in a coordinate that follows the alphaparticle orbit, which has a finite departure from a flux surface. This variable is thefollowing constant of the alpha particle’s motion: † 𝜓 ⋆ ≡ 𝜓 − 𝐼𝑣 ‖ 𝛺 . (4.2) † This drift kinetic angular momentum is a constant of the alphaparticle’s motion because the left-hand side of (A 1) operating on it is zero: (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝜓 ⋆ = ⃗𝑣 𝑑 · ∇ 𝜓 − 𝑣 ‖ ^ 𝑏 · ∇ (︀ 𝐼𝑣 ‖ /𝛺 )︀ − ⃗𝑣 𝑑 · ∇ (︀ 𝐼𝑣 ‖ /𝛺 )︀ . It can be shown (Helander &Sigmar 2005; Parra & Catto 2010) that ⃗𝑣 𝑑 · ∇ 𝜓 = 𝑣 ‖ ^ 𝑏 · ∇ (︀ 𝐼𝑣 ‖ /𝛺 )︀ and ⃗𝑣 𝑑 · ∇ (︀ 𝐼𝑣 ‖ /𝛺 )︀ = 0 , suchthat (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝜓 ⋆ = 0 . E. A. Tolman and P.J. Catto
The coordinate 𝛼 is modified from (2.1) in accordance: 𝛼 ⋆ ≡ 𝜁 − 𝑞 ⋆ 𝜗, (4.3)with 𝑞 ⋆ = 𝑞 ( 𝜓 ⋆ ) (4.4)the safety factor evaluated at 𝜓 ⋆ instead of 𝜓 . Consistent with the approximation (A 10),we can take that 𝜕𝑓 /𝜕𝜓 ⋆ ≈ 𝜕𝑓 /𝜕𝜓 .The perturbed distribution 𝑓 has two parts, such that 𝑓 = ℎ + ( 𝑍 𝛼 𝑒𝛷/𝑀 𝛼 ) 𝜕𝑓 /𝜕 ℰ .The second term is an adiabatic response to the perturbation, which does not participatein transport (it is exactly out of phase with the perturbed velocity that causes transport).Evaluation of [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ gives that the equation governing ℎ is 𝜕 𝑡 ℎ + (︁ 𝑣 ‖ ˆ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ ℎ − 𝐶 {︂ ℎ + 𝑍 𝛼 𝑒𝛷𝑀 𝛼 𝜕𝑓 𝜕 ℰ }︂ = − 𝑍 𝛼 𝑒𝑀 𝛼 (︂ 𝜕𝛷𝜕𝑡 − 𝑣 ‖ 𝑐 𝜕𝐴 ‖ 𝜕𝑡 + 𝜆𝑣 𝑐𝛺 𝜕𝐵 ‖ 𝜕𝑡 )︂ 𝜕𝑓 𝜕 ℰ − 𝑐 (︂ 𝜕𝛷𝜕𝛼 ⋆ − 𝑣 ‖ 𝑐 𝜕𝐴 ‖ 𝜕𝛼 ⋆ + 𝜆𝑣 𝑐𝛺 𝜕𝐵 ‖ 𝜕𝛼 ⋆ )︂ 𝜕𝑓 𝜕𝜓 ⋆ . (4.5)Note that the collision operator acting on the adiabatic component is not zero, as itsometimes is in similar calculations.The perturbed fields 𝛷 , 𝐴 ‖ , and 𝐵 ‖ are functions of 𝜓 , 𝜁 , and 𝜗 and can be Fourieranalyzed in those coordinates, then rewritten in terms of 𝜓 , 𝛼 , and 𝜗 . For example, theelectric potential is given by 𝛷 ( 𝜓, 𝜗, 𝛼 ) = ∑︁ 𝑚,𝑛,𝜔 𝛷 𝑚𝑛𝜔 cos [︃ 𝑛𝛼 − 𝜔𝑡 + ( 𝑛𝑞 − 𝑚 ) 𝜗 + ∫︁ 𝜓 𝑑𝜓 ′ 𝑘 𝜓 𝑅𝐵 𝑝 ]︃ = ∑︁ 𝑚,𝑛,𝜔 ℜ [︂ 𝛷 𝑚𝑛𝜔 𝑒 𝑖𝑛𝛼 − 𝑖𝜔𝑡 + 𝑖 ( 𝑛𝑞 − 𝑚 ) 𝜗 + 𝑖 ∫︀ 𝜓 𝑑𝜓 ′ 𝑘𝜓𝑅𝐵𝑝 ]︂ . (4.6)Moreover, for use in (4.5), the fields can also be written in terms of the starred coordi-nates, so that, for example, 𝛷 ( 𝜓, 𝜗, 𝛼 ) = ∑︁ 𝑚,𝑛,𝜔 ℜ [︂ 𝛷 𝑚𝑛𝜔 𝑒 𝑖𝑛𝛼 ⋆ − 𝑖𝜔𝑡 + 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖 ∫︀ 𝜓⋆ 𝑑𝜓 ′ 𝑘𝜓𝑅𝐵𝑝 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ]︂ . (4.7)Due to the dominance of parallel streaming over drifts, we may take that ⃗𝑣 𝑑 · ∇ 𝜗 ≪ 𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 . Also, the form of the magnetic field introduced in (2.2) can be used to show 𝑣 ‖ ˆ 𝑏 · ∇ ℎ = 𝑣 ‖ ˆ 𝑏 · ∇ 𝜗𝜕ℎ/𝜕𝜗 . Then, (4.5) becomes 𝜕 𝑡 ℎ + 𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 𝜕ℎ𝜕𝜗 + 𝜔 𝛼 ⋆ 𝜕ℎ𝜕𝛼 ⋆ = 𝐶 {︂ ℎ + 𝑍 𝛼 𝑒𝛷𝑀 𝛼 𝜕𝑓 𝜕 ℰ }︂ − 𝑖 ∑︁ 𝑚,𝑛 𝐷 𝑛𝜔 𝑆 𝑚𝑛𝜔 𝑒 𝑖𝑛𝛼 ⋆ − 𝑖𝜔𝑡 + 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖 ∫︀ 𝜓⋆ 𝑑𝜓 𝑘𝜓𝑅𝐵𝑝 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 , (4.8) rift kinetic theory of alpha transport by tokamak perturbations † 𝜔 𝛼 ⋆ ≡ ⃗𝑣 𝑑 · ∇ 𝛼 + 𝑣 ‖ ˆ 𝑏 · ∇ (︂ 𝜗 𝐼𝑣 ‖ 𝛺 𝜕𝑞𝜕𝜓 )︂ = 𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 𝜕𝜕𝜓 (︂ 𝐵𝑣 ‖ 𝛺 ˆ 𝑏 · ∇ 𝜗 )︂ (4.9)and 𝑆 𝑚𝑛𝜔 is given by 𝑆 𝑚𝑛𝜔 ( 𝜗, 𝑣, 𝜆 ) = 𝛷 𝑚𝑛𝜔 − 𝑣 ‖ 𝑐 𝐴 ‖ 𝑚𝑛𝜔 + 𝜆𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 . (4.10)(The 𝜗 dependence displayed in 𝑆 𝑚𝑛𝜔 is due to the dependence of 𝑣 ‖ on this quantity.)Also, we define 𝐷 𝑛𝜔 ( 𝑣, 𝜓 ⋆ ) ≡ 𝑐𝑛 𝜕𝑓 𝜕𝜓 ⋆ (︂ − 𝜔𝑛𝜔 ⋆ )︂ (4.11)with 𝜔 ⋆ ≡ 𝑐𝑀 𝛼 𝜕𝑓 /𝜕𝜓 ⋆ 𝑍 𝛼 𝑒𝜕𝑓 /𝜕 ℰ . (4.12)Note that the drifts on the left side of (4.8) are caused by the unperturbed fields. Thus,we can employ the following form of ℎ : ℎ ( 𝜓 ⋆ , 𝜗, 𝛼 ⋆ , 𝑡, 𝑣, 𝜆, 𝜎 ) = ∑︁ 𝑛,𝜔 ℎ 𝑛𝜔 ( 𝜗, 𝑣, 𝜆, 𝜎 ) 𝑒 𝑖𝑛𝛼 ⋆ − 𝑖𝜔𝑡 + 𝑖 ∫︀ 𝜓⋆ 𝑑𝜓 𝑘𝜓𝑅𝐵𝑝 . (4.13)Here, we have introduced the variable 𝜎 , defined as follows: 𝜎 = {︃ , trapped particles 𝑣 ‖ | 𝑣 ‖ | , passing particles . (4.14)4.2. Role of collisionality, phenomenological description of transport, and form ofcollision operator
At this point, a discussion of the role of collisionality in the transport reveals theappropriate treatment of the collision operator in (4.8). This role is directly analogous tothat played by collisionality in superbanana plateau tokamak transport [see discussionin Shaing (2015); Calvo et al. (2017); Catto (2019 a )]. It is also similar to the role playedby collisionality in the plateau regime of neoclassical transport [see Helander & Sigmar(2005)], in the damping of plasma echos [see Su & Oberman (1968)], and in other wave-particle resonance processes (Duarte et al. ‡ A small boundary layer of finite width in pitch † The quantity 𝜔 𝛼 ⋆ is found as follows: (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝛼 ⋆ = (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · ( ∇ 𝜁 − 𝑞 ⋆ ∇ 𝜗 )= (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · [︀ ∇ 𝜁 − 𝑞 ∇ 𝜗 + (︀ 𝐼𝑣 ‖ /𝛺 )︀ ( 𝜕𝑞/𝜕𝜓 ) ∇ 𝜗 ]︀ = (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · [︀ ∇ 𝛼 + 𝜗 ∇ 𝑞 + (︀ 𝐼𝑣 ‖ /𝛺 )︀ ( 𝜕𝑞/𝜕𝜓 ) ∇ 𝜗 ]︀ ≈ ⃗𝑣 𝑑 · ∇ 𝛼 + 𝑣 ‖ ^ 𝑏 · ∇ [︀ 𝜗 (︀ 𝐼𝑣 ‖ /𝛺 )︀ ( 𝜕𝑞/𝜕𝜓 ) ]︀ . In the first equality, we have used that (︁ 𝑣 ‖ ^ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝜓 ⋆ = 0 ; in the last equality, we have used that 𝑣 𝑑 · ∇ 𝜓 = 𝑣 ‖ ^ 𝑏 · (︀ 𝐼𝑣 ‖ /𝛺 )︀ .More information about the evaluation of ⃗𝑣 𝑑 · ∇ 𝛼 can be found in Catto (2019 b ). ‡ Here, we refer only to behavior existing within the equation we solve: a linearized driftkinetic equation with a perturbation of zero or real frequency. Physically, it is possible forother effects to resolve the resonance. For instance, resonant particles will dephase from the E. A. Tolman and P.J. Catto λ - λ res h n ω Figure 1: A schematic representation of the boundary layer around the resonance. Aspecific value of pitch angle, 𝜆 = 𝜆 𝑟𝑒𝑠 , is resonant with the mode, and the non-adiabaticalpha response ℎ 𝑛𝜔 is largest for this value of 𝜆 . Pitch angle scattering collisions broaden ℎ 𝑛𝜔 about this resonant value. The sharp variation of ℎ near the resonant 𝜆 𝑟𝑒𝑠 enhancesthe importance of the pitch angle scattering operator [given by the second term in (2.9)],which contains a second derivative with respect to 𝜆 .angle space and velocity space will collisionally disrupt resonance with the mode, and theperturbed distribution function will vary sharply near this layer. Pitch angle scattering,represented by the second term in (2.9), rather than drag, is the important collisionalbehavior because the second derivative of the pitch angle scattering operator is strongernear the resonance boundary layer than the first derivative of the drag. A schematicpicture of ℎ about a resonant location in pitch-angle space is given in figure 1. A possiblephysical interpretation of this resonance structure is that particles drift while resonantwith the wave until they pitch angle scatter, giving rise to gradual diffusion (Helander &Sigmar 2005).This understanding yields a phenomenological estimate of the heat flux which shouldresult from the transport being considered. The heat flux will be given by 𝑄 𝛼 ∼ 𝑀 𝛼 𝑣 𝑟𝑒𝑠 𝛿𝜆 {︁[︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ 𝛥𝑡 }︁ 𝛥𝑡 − 𝜕𝑛 𝛼 𝜕𝜓 , (4.15)with 𝑀 𝛼 the alpha particle mass, 𝑣 𝑟𝑒𝑠 a characteristic alpha resonant velocity, 𝛿𝜆 thefraction of particles that are resonant with the mode, [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ the radialvelocity caused by the perturbation, 𝛥𝑡 the time that a particle moves before becomingdecorrelated by a collision (such that [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ 𝛥𝑡 is the step a particletakes before becoming decorrelated). The fraction of particles affected by the mode isestimated from the width of the boundary layer representing the broadened resonance,given by balancing the left side of (4.8) (we use 𝜔 𝛼 ⋆ to include pitch angle dependence) perturbation as they are moved by it, resulting in nonlinear resolution of the resonance throughthe formation of phase space islands. The addition of a small imaginary component to theperturbation frequency is also able to resolve the resonance. However, it is not clear that phasespace island formation (in the absence of resonance overlap) or an imaginary component of thefrequency provide the decorellation and dissipation necessary for transport. The relationshipbetween these mechanisms and pitch angle collisions in resolving the resonance and in causingtransport is an interesting avenue for future work. One possibility is that resonance resolutionby these mechanisms is similar enough to the collisional resolution that fluxes calculated in thispaper are valid in cases where collisional resolution is less important than other mechanisms. rift kinetic theory of alpha transport by tokamak perturbations 𝑛𝜔 𝛼 ⋆ 𝛿𝜆 ∼ 𝜈 𝑝𝑎𝑠 𝛿𝜆 , (4.16)with 𝜈 𝑝𝑎𝑠 representing the coefficient of the second term of (2.9). [Note that the effect ofthe collision operator on the adiabatic component, ( 𝑍 𝛼 𝑒𝛷/𝑀 𝛼 ) 𝜕𝑓 /𝜕 ℰ , is negligible incomparison to the sharp variation of ℎ in the resonant boundary layer.] Thus, we have 𝛿𝜆 ∼ (︂ 𝜈 𝑝𝑎𝑠 𝑛𝜔 𝛼 ⋆ )︂ / . (4.17)The decorrelation time 𝛥𝑡 is given by 𝛥𝑡 ∼ 𝛿𝜆 /𝜈 𝑝𝑎𝑠 , such that estimated flux is 𝑄 𝛼 ∼ 𝑀 𝛼 𝑣 𝑟𝑒𝑠 [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ 𝑛𝜔 𝛼 ⋆ 𝜕𝑛 𝛼 𝜕𝜓 . (4.18)Note that this flux is independent of the collision frequency 𝜈 𝑝𝑎𝑠 because 𝛿𝜆 times 𝛥𝑡 is independent of the collision frequency. The role of the collisions is crucial to thetransport, but collisionality does not appear in the final expression for the flux. Thus,in the following steps we use a Krook collision operator (Bhatnagar et al. 𝐶 { ℎ } = − 𝜈ℎ . The Krook operator 𝜈 is related to 𝜈 𝑝𝑎𝑠 by 𝜈 ∼ 𝜈 𝑝𝑎𝑠 /𝛿𝜆 [recall that the pitch angle scattering operator (2.9) depends onthe second derivative of ℎ with respect to 𝜆 ]. We also ignore the effect of the collisionoperator on the adiabatic response because the sharp variation of ℎ is more important.Whenever collisions play a role in the calculation, they allow the convergence of a sumor resolve a singularity, actions which are not qualitatively sensitive to the full form ofthe collision operator. 4.3. Solution for ℎ with the Krook operator Inserting the Krook operator and the form (4.13) into (4.8) reveals the equation to besolved for each ℎ 𝑛𝜔 : 𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 𝜕ℎ 𝑛𝜔 𝜕𝜗 − 𝑖 ( 𝜔 − 𝑛𝜔 𝛼 ⋆ + 𝑖𝜈 ) ℎ 𝑛𝜔 = − 𝑖𝐷 𝑛𝜔 ∑︁ 𝑚 𝑆 𝑚𝑛𝜔 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 . (4.19)Now, recall the variable 𝜏 , which characterizes the progression of particles along themagnetic field [see (A 6)]. Then, defining 𝛬 ( 𝜏 ) ≡ 𝜔 − 𝑛𝜔 𝛼 ⋆ − 𝑣 ‖ ˆ 𝑏 · ∇ [︂ ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑘 𝜓 𝐼𝑣 ‖ 𝑅𝐵 𝑝 𝛺 ]︂ , (4.20)(4.19) may be written 𝜕𝜕𝜏 [︁ ℎ 𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ ( 𝜔 − 𝑛𝜔 𝛼⋆ + 𝑖𝜈 ) ]︁ = − 𝑖𝐷 𝑛𝜔 ∑︁ 𝑚 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑒 𝑖𝑘𝜓𝐼𝑣 ‖ ( 𝜗 =0) 𝑅𝐵𝑝𝛺 . (4.21)Here, we have chosen 𝜏 such that 𝜗 ( 𝜏 ) = 0 . The perturbed distribution function ℎ 𝑛𝜔 can then be found by integrating from 𝜏 = −∞ , where ℎ 𝑛𝜔 = 0 , forward to the "present"time 𝜏 = 0 , with 𝜗 ( 𝜏 = 0) = 𝜗 , ℎ 𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ ( 𝜔 − 𝑛𝜔 𝛼⋆ + 𝑖𝜈 ) ⃒⃒⃒ 𝜏 =0 = − 𝑖𝐷 𝑛𝜔 𝑒 𝑖𝑘𝜓𝐼𝑣 ‖ ( 𝜗 =0) 𝑅𝐵𝑝𝛺 ∑︁ 𝑚 ∫︁ −∞ 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏. (4.22)4 E. A. Tolman and P.J. Catto
Moving the exponential on the left side to the right side gives gives ℎ 𝑛𝜔 = − 𝑖𝐷 𝑛𝜔 ∑︁ 𝑚 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ∫︁ −∞ 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏. (4.23)The preceding expression represents the response of the plasma distribution to thepresence of a perturbation of the tokamak fields. Next, we work to write this expressionin a form which can be more easily evaluated and understood. The expression (4.23) canbe split into contributions from each complete bounce (for a trapped particle) or poloidaltransit (for a passing particle) of duration given by 𝜏 𝑏 ≡ ∮︁ 𝑑𝜏, (4.24)such that ℎ 𝑛𝜔 = − 𝑖𝐷 𝑛𝜔 ∑︁ 𝑚 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ∞ ∑︁ 𝑗 =0 [︃∫︁ − 𝑗𝜏 𝑏 − ( 𝑗 +1) 𝜏 𝑏 ]︃ 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏. (4.25)Note that at the trapped-passing boundary, the value of 𝜏 𝑏 for a trapped particle is twicethat for a passing particle. This complication will not affect the calculation that followsas velocity space integrations account for both signs of 𝑣 ‖ . Each portion of the integralmay be modified with the change of variables 𝜏 → 𝜏 + 𝑗𝜏 𝑏 , such that the entire integralreduces to the geometric series ℎ 𝑛𝜔 = − 𝑖𝐷 𝑛𝜔 ∑︁ 𝑚 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ∞ ∑︁ 𝑗 =0 𝑒 𝑖𝑗 ∮︀ 𝑑𝜏 [ 𝛬 ( 𝜏 )+ 𝑖𝜈 ] ∫︁ − 𝜏 𝑏 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏. (4.26)This transformation requires that ∮︀ 𝑑𝜏 𝛬 ( 𝜏 ) be the same (up to a factor of 𝜋 ) for eachbounce or transit, which corresponds to the existence of periodicity. Trapped particles,which traverse the same values of 𝜗 on each orbit, automatically fulfill this condition,because ∮︀ 𝑑𝜏 𝛬 ( 𝜏 ) is trivially the same for each bounce. Passing particles do not reversecourse and traverse values of 𝜗 ranging from negative infinity to infinity. For them, thesplit is only possible when the particle motion is resonant with the wave, such that thecontributions to transport from each transit are the same. When the particle is not inphase with the wave, however, (4.23) will approach zero integrated over the entire passingparticle trajectory because of the rapid oscillation of the phase factor, unmitigated byperiodicity. The following analysis is appropriate for all trapped particles and for resonantpassing particles. The analysis will reveal in (4.29) onwards that transport comes fromresonant particles only, such that the formulation is appropriate for all transport causingparticles, passing and trapped.Because of the small imaginary part provided by the collisionality, the series (4.26) isconvergent and may be summed as ℎ 𝑛𝜔 = ∑︁ 𝑚 − 𝑖𝐷 𝑛𝜔 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ∫︀ − 𝜏 𝑏 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏 − 𝑒 𝑖 ∮︀ 𝑑𝜏 [ 𝛬 ( 𝜏 )+ 𝑖𝜈 ] . (4.27)Expanding the denominator about its zeros using 𝑒 𝑖 𝜋𝑙 = 1 , with 𝑙 any integer, denotingthe bounce or poloidal transit harmonic, gives ℎ 𝑛𝜔 = ∑︁ 𝑚,𝑙 𝐷 𝑛𝜔 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 ∫︀ − 𝜏 𝑏 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ [ 𝛬 ( 𝜏 ′ ) + 𝑖𝜈 ] 𝑑𝜏 ∮︀ 𝑑𝜏 [ 𝛬 ( 𝜏 ) + 𝑖𝜈 ] − 𝜋𝑙 . (4.28) rift kinetic theory of alpha transport by tokamak perturbations 𝜈 ∮︀ 𝑑𝜏 ≪ ∮︀ 𝛬𝑑𝜏 − 𝜋𝑙 . In this limit, analogous to thelimit that occurs in the plateau regime of neoclassical transport [see Helander & Sigmar(2005)], (4.28) approaches ℎ 𝑛𝜔 = − 𝑖𝜋𝐷 𝑛𝜔 ∑︁ 𝑚,𝑙 𝛿 ( 𝑄 𝑙 ) 𝑒 𝑖 ( 𝑛𝑞 ⋆ − 𝑚 ) 𝜗 + 𝑖𝑘𝜓𝐼𝑣 ‖ 𝑅𝐵𝑝𝛺 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) ∮︁ 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) 𝑑𝜏. (4.29)Here, we have also split the integral in the exponential into two parts which both havelimits at 𝜏 , where 𝜗 ( 𝜏 ) = 0 . In addition, we have introduced a delta function 𝛿 ( 𝑄 𝑙 ) with a resonance function as its argument, 𝑄 𝑙 ( 𝑣, 𝜆 ) ≡ ∮︁ 𝛬𝑑𝜏 − 𝜋𝑙. (4.30)Only particles for which 𝑄 𝑙 is very close to zero participate in transport, and the strengthof pitch angle scattering is accentuated for them from the normal weak level. Thesharp variation in ℎ 𝑛𝜔 between resonant particles and non-resonant particles (visibleschematically in figure 1) leads to this accentuation.These resonances, discussed at greater length in section 5.4, are bounce harmonicresonances, which occur when the mode frequency, the particle bounce frequency, and thetangential precession frequency resonate. Such resonances are familiar from treatments ofneoclassical toroidal viscosity of bulk tokamak plasmas (Linsker & Boozer 1982; Mynick1986; Park et al. et al. et al. et al. ℎ in terms of the unstarred coordinatesgives the expression needed to form the heat flux, ℎ = − ∑︁ 𝑚,𝑛,𝜔,𝑙 {︂ 𝑖𝜋 [︂ 𝑒 𝑖𝑛𝜁 − 𝑖𝜔𝑡 − 𝑖𝑚𝜗 + 𝑖 ∫︀ 𝜓 𝑑𝜓 ′ 𝑘𝜓𝑅𝐵𝑝 − 𝑖 ∫︀ 𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) ]︂ 𝐷 𝑛𝜔 𝛿 ( 𝑄 𝑙 ) × ∮︁ 𝑆 𝑚𝑛𝜔 𝑒 − 𝑖 ∫︀ 𝜏𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) 𝑑𝜏 }︂ . (4.31)
5. Expression for flux
In this section, we show how to use the alpha distribution perturbation ℎ derived in theprevious section to obtain the resulting alpha heat flux. This heat flux is the quantity ofpractical use in understanding and predicting tokamak discharges. In addition, we discusskey parts of the expression for this flux.5.1. Setup of flux expression
To find the heat flux, the perturbation to the distribution function, ℎ , found in (4.31),is multiplied by the perturbed radial velocity that also results from the electromagneticperturbation, [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ . † The result is weighted by energy and integrated † Some techniques for computing energetic particle transport, like the use of orbit followingcodes, focus only on the perturbed radial velocity. However, consistent consideration of bothcomponents is critical. An experimental demonstration of energetic particle transport caused bythe product of the distribution function perturbation and the perturbed radial velocity is given E. A. Tolman and P.J. Catto over the distribution function. Then, this quantity is averaged over the flux surface: 𝑄 𝛼 = ⟨∫︁ 𝑑 𝑣 (︀ 𝑀 𝛼 𝑣 / )︀ ℎ [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ ⟩ . (5.1)Here, the flux surface average is defined by ⟨ 𝐴 ⟩ ≡ ∮︀ 𝐴𝑑𝜗𝑑𝜁⃗𝐵 ·∇ 𝜗 ∮︀ 𝑑𝜗𝑑𝜁⃗𝐵 ·∇ 𝜗 . (5.2)To enable analytic evaluation of the flux, we now adopt the following approximateexpression for the strength of the axisymmetric magnetic field: 𝐵 = 𝐵 [1 − 𝜖 ( 𝜓 ) cos 𝜗 ] , (5.3)with 𝜖 ≈ 𝑟/𝑅 (as introduced in section 2) small. Then, ⟨ 𝐴 ⟩ ≈ 𝐵 𝜋 𝑞𝑅 ∮︁ 𝐴𝑑𝜗𝑑𝜁⃗𝐵 · ∇ 𝜗 . (5.4)The perturbation to the distribution function ℎ may be found in (4.31). The perturbedradial velocity [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ may be read off from (4.5), giving [︁(︁ 𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ]︁ = ∑︁ 𝑚,𝑛,𝜔 𝑖𝑛𝑐𝑆 𝑚𝑛𝜔 𝑒 𝑖𝑛𝜁 − 𝑖𝜔𝑡 − 𝑖𝑚𝜗 + 𝑖 ∫︀ 𝜓 𝑑𝜓 ′ 𝑘𝜓𝑅𝐵𝑝 . (5.5)The flux has contributions from both trapped and passing particles, and will be evaluatedin slightly different ways for each of these populations. In general, we illustrate thetrapped particle techniques first, then modify them for passing particles later. One of thekey differences between the trapped and the passing calculations is the treatment of theparticle pitch angle 𝜆 (2.6). For trapped particles, integration in pitch angle is carriedout in terms of the trapping variable 𝜅 , which is defined in terms of 𝜆 , 𝜅 = 1 − (1 − 𝜖 ) 𝜆 𝜖𝜆 . (5.6)This variable is defined such that, for a trapped particle, the angle of the turning pointin the magnetic field (5.3) is given by − ( 𝜅 ) . Deeply trapped particles have 𝜅 = 0 and barely trapped particles have 𝜅 = 1 . For passing particles, the analogous variable is 𝑘 ≡ 𝜅 , (5.7)where a value of 𝑘 = 1 corresponds to a barely passing particle and a value of 𝑘 =0 corresponds to a fully passing particle. [A pedagogical introduction to the trappingparameter can be found in Helander & Sigmar (2005).] With the definitions of 𝜆 (2.6)and 𝜅 (5.6) we can write the differential in (5.1) as 𝑑 𝑣 = 𝜋𝐵𝑣 𝑑𝑣𝑑𝜆𝐵 𝑣 ‖ ; 𝑑𝜆 = − 𝜖𝜅𝑑𝜅 (1 − 𝜖 + 2 𝜖𝜅 ) ; (5.8)an equivalent expression in terms of 𝑘 holds for passing particles. in Nagaoka et al. (2008). Todo (2019) discusses these results and the importance of developingenergetic particle transport theories that self consistently treat both of these components. rift kinetic theory of alpha transport by tokamak perturbations Discussion of synergistic transport from different 𝑚 To evaluate the flux, (4.31), (5.4), (5.5), and (5.8) are inserted into (5.1). Notethat the resulting flux includes sums over two different sets of 𝑚 and 𝑛 : one set forthe perturbed distribution function (4.31), and one for the perturbed radial veloc-ity (5.5). The complex quantities are written as their real equivalent and the 𝜁 integral isevaluated using the identities ∮︀ 𝑑𝜁 sin ( 𝑛𝜁 − 𝜑 𝑎 ) sin ( 𝑛 ′ 𝜁 − 𝜑 𝑏 ) = 𝜋𝛿 𝑛𝑛 ′ cos ( 𝜑 𝑎 − 𝜑 𝑏 ) and ∮︀ 𝑑𝜁 cos ( 𝑛𝜁 − 𝜑 𝑎 ) sin ( 𝑛 ′ 𝜁 − 𝜑 𝑏 ) = 𝜋𝛿 𝑛𝑛 ′ sin ( 𝜑 𝑎 − 𝜑 𝑏 ) , with 𝜑 𝑎 and 𝜑 𝑏 some phase and 𝛿 𝑛𝑛 ′ the Kronecker delta. Using these expressions, and stating the 𝜗 integral in terms of 𝜏 (A 6), we find for trapped particles 𝑄 𝛼,𝑡 = ∑︁ 𝑙,𝑚,𝑚 ′ 𝑛,𝑛 ′ 𝛿 𝑛𝑛 ′ 𝑐𝑀 𝛼 𝜖𝜋 𝑞𝑅 ∫︁ 𝑣 𝑑𝑣 ∫︁ 𝑑𝜅 ∮︁ 𝑑𝜏 𝑣 𝛿 ( 𝑄 𝑙 ) 𝜅𝑛 ′ 𝑆 𝑚 ′ 𝑛 ′ 𝜔 ′ 𝐷 𝑛𝜔 (1 − 𝜖 + 2 𝜖𝜅 ) × ⎧⎨⎩ sin ⎡⎣ ( 𝑚 − 𝑚 ′ ) 𝜗 + ( 𝜔 − 𝜔 ′ ) 𝑡 − ∫︁ (︁ 𝑘 𝜓 − 𝑘 ′ 𝜓 )︁ 𝑅𝐵 𝑝 𝑑𝜓 ⎤⎦ (︂ S 𝑚𝑛𝜔 cos ∫︁ 𝜏𝜏 𝛬𝑑𝜏 ′ − C 𝑚𝑛𝜔 sin ∫︁ 𝜏𝜏 𝛬𝑑𝜏 ′ )︂ − cos ⎡⎣ ( 𝑚 − 𝑚 ′ ) 𝜗 + ( 𝜔 − 𝜔 ′ ) 𝑡 − ∫︁ (︁ 𝑘 𝜓 − 𝑘 ′ 𝜓 )︁ 𝑅𝐵 𝑝 𝑑𝜓 ⎤⎦ (︂ C 𝑚𝑛𝜔 cos ∫︁ 𝜏𝜏 𝛬𝑑𝜏 ′ + S 𝑚𝑛𝜔 sin ∫︁ 𝜏𝜏 𝛬𝑑𝜏 ′ )︂⎫⎬⎭ (5.9)(the subscript 𝑡 indicates this is a trapped particle expression). Here, we have defined C 𝑚𝑛𝜔 ( 𝑣, 𝜅 ) ≡ ∮︁ 𝑆 𝑚𝑛𝜔 cos [︂∫︁ 𝜏𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) ]︂ 𝑑𝜏 (5.10)and S 𝑚𝑛𝜔 ( 𝑣, 𝜅 ) ≡ ∮︁ 𝑆 𝑚𝑛𝜔 sin [︂∫︁ 𝜏𝜏 𝑑𝜏 ′ 𝛬 ( 𝜏 ′ ) ]︂ 𝑑𝜏. (5.11)Equivalent expressions in terms of 𝑘 instead of 𝜅 apply to passing particles. Considerationof (5.9) makes clear that there is no coupling between perturbation components ofdifferent toroidal mode numbers: the perturbed distribution function of one 𝑛 will notinteract with the perturbed radial velocity of another 𝑛 ′ . However, perturbations (orcomponents of one perturbation) with the same toroidal mode number, but differentpoloidal mode numbers, can couple via the cos [( 𝑚 − 𝑚 ′ ) 𝜗 . . . ] and sin [( 𝑚 − 𝑚 ′ ) 𝜗 . . . ] terms. For perturbations of very different 𝑚 , these terms are highly oscillatory, such thatthe integral over 𝜗 ( 𝜏 ) will be small, as required by the Riemann-Lebesgue lemma. Theflux resulting from this coupling approaches zero.For perturbations of similar poloidal mode number, the coupling is significant and theperturbations cause synergistic transport in which the perturbed distribution functionfrom one perturbation interacts with the perturbed drift velocity of the other to causetransport. Note that this synergistic transport will oscillate with time at a frequencyequal to the difference between the two perturbations, as is typical for beat frequencyphenomena. Transport at such a beat frequency has been observed experimentally instudies of transport in the presence of NTMs and field perturbations caused by resonantmagnetic perturbations (Snicker et al. 𝑛 and 𝜔 , but differentvalues of 𝑚 , the relationship 𝑛𝑞 − 𝑚 = 1 / (see table 2), ensures that the harmonics willhave different values of 𝑞 and different radial profiles. The resulting rapid sinusoidal8 E. A. Tolman and P.J. Catto oscillations tend to reduce their contributions to transport. Thus, from here on, weevaluate the flux for a single perturbation, characterized by one value of 𝑚 , 𝑛 , and 𝜔 , with the knowledge that if a perturbation of a similar poloidal mode number is alsopresent in the device at the same radial location, transport may be altered in a synergisticmanner. We denote the single 𝑚 , 𝑛 , and 𝜔 flux 𝑄 𝛼,𝑚𝑛𝜔 .5.3. Assumption of 𝑞 ⋆ ≈ 𝑞 and final statement of flux The flux 𝑄 𝛼,𝑚𝑛𝜔 is a function of flux 𝜓 ; the perturbation is also a function of thisvariable. However, 𝜓 ⋆ , defined in (4.2), not 𝜓 , is a constant of the alpha particle’s motion,such that the alpha particle response is a function of 𝜓 ⋆ . In the flux expression (5.9) 𝜓 ⋆ appears only in the variable 𝑞 ⋆ through the dependence of 𝑄 𝑙 and 𝛬 on this quantity.Recalling (4.4) and (4.2), we have 𝑞 ⋆ ≈ 𝑞 ( 𝜓 ) − 𝜕𝑞𝜕𝜓 𝐼𝑣 ‖ 𝛺 = 𝑞 − 𝑞𝑠𝑣 ‖ 𝜖𝑅𝛺 𝑝 (5.12)[note that here we have provided 𝑞 ⋆ in terms of both 𝜕𝑞/𝜕𝜓 and shear 𝑠 , (2.4)]. Because 𝑣 ‖ is a function of 𝜗 , this quantity varies poloidally in a complicated manner. Suchcomplication introduces significant challenge in the evaluation of the flux, in particularmaking the term that enforces resonance, 𝛿 ( 𝑄 𝑙 ) , dependent on 𝜗 .In this work, we thus choose to neglect the difference between 𝑞 and 𝑞 ⋆ , leaving adetailed study of the effect of the difference between these quantities to future work.This simplification places constraints on the values of shear 𝑠 for which our work is valid.Because 𝑞 ⋆ appears in the combination ( 𝑛𝑞 ⋆ − 𝑚 ) , we can compare 𝑞 ⋆ to the other termsin this combination to develop the following condition for the validity of setting 𝑞 ⋆ = 𝑞 : 𝑠 ≪ 𝜖𝑅𝛺 𝑝 ( 𝑛𝑞 − 𝑚 ) 𝑛𝑞𝑣 . (5.13)The degree of restriction that this condition places on the magnitude of the shear dependson the characteristics of the perturbation in question. For ripple, the condition is 𝑠 ≪ 𝜖𝑅𝜌 𝑝𝛼 , (5.14)a generous condition which allows the approximation to be valid even in the high shearregions near the tokamak edge where the ripple is important. For the TAE, the equivalentexpression is 𝑠 ≪ 𝜖𝑅𝑛𝑞𝜌 𝑝𝛼 , (5.15)which, given the fairly high values of 𝑛 characterizing TAEs, is a more restrictivecondition, though it should still hold in much of the low-shear core plasma where TAEactivity is most common.With the replacement 𝑞 ⋆ → 𝑞 , the outer 𝜏 integral in (5.9) can be commuted past 𝛿 ( 𝑄 𝑙 ) . In addition, using (2.14), we can insert 𝜕𝑓 𝜕𝜓 ≈ 𝜕𝑛 𝛼 /𝜕𝜓 𝜋 ( 𝑣 + 𝑣 𝑐 ) ln ( 𝑣 /𝑣 𝑐 ) ≈ 𝜕𝑛 𝛼 /𝜕𝜓 𝜋𝑣 ln ( 𝑣 /𝑣 𝑐 ) (5.16)into the definition of 𝐷 𝑛𝜔 (4.11). (Recall that we have also taken 𝜕𝑓 /𝜕𝜓 ⋆ ≈ 𝜕𝑓 /𝜕𝜓 .) rift kinetic theory of alpha transport by tokamak perturbations Quantity Trapped particles Passing particles 𝜏 𝑏 𝑞𝑅𝐾 ( 𝜅 ) 𝑣 √ 𝜖 𝑞𝑅𝑘𝐾 ( 𝑘 ) 𝑣 √ 𝜖𝜆 𝜔 𝛼 ⋆ 𝑣 { 𝐸 ( 𝜅 ) − 𝐾 ( 𝜅 )+4 𝑠 [ 𝐸 ( 𝜅 ) − ( − 𝜅 ) 𝐾 ( 𝜅 ) ] } 𝛺 𝑝 𝑅 𝐾 ( 𝜅 ) 𝑣 [ 𝐸 ( 𝑘 ) − ( − 𝑘 ) 𝐾 ( 𝑘 )+4 𝑠𝐸 ( 𝑘 ) ] 𝛺 𝑝 𝑅 [ (1 − 𝜖 ) 𝑘 +2 𝜖 ] 𝐾 ( 𝑘 ) Table 3: Quantities involved in resonance condition. The complete elliptic integral of thefirst kind is denoted by 𝐾 ( 𝜅 ) ; the complete integral of the second kind is 𝐸 ( 𝜅 ) .This gives, for trapped particles, 𝑄 𝛼,𝑚𝑛𝜔,𝑡 = − ∑︁ 𝑙 𝜖𝑀 𝛼 𝑐 𝑛 𝜕𝑛 𝛼 /𝜕𝜓 𝑞𝑅 ln ( 𝑣 /𝑣 𝑐 ) ∫︁ 𝑣 𝑑𝑣 ∫︁ 𝑑𝜅 𝜅𝑣 𝛿 ( 𝑄 𝑙 ) (︁ − 𝜔𝑛𝜔 ⋆ )︁ (︀ C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 )︀ (1 − 𝜖 + 2 𝜖𝜅 ) . (5.17)The equivalent expression for passing particles is below, adding a subscript 𝑝 to makeclear the expression refers to passing particles: 𝑄 𝛼,𝑚𝑛𝜔,𝑝 = − ∑︁ 𝑙 𝜖𝑀 𝛼 𝑐 𝑛 𝜕𝑛 𝛼 /𝜕𝜓 𝑞𝑅 ln ( 𝑣 /𝑣 𝑐 ) ∫︁ 𝑣 𝑑𝑣 ∫︁ 𝑑𝑘 𝑘𝑣 𝛿 ( 𝑄 𝑙 ) (︁ − 𝜔𝑛𝜔 ⋆ )︁ (︀ C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 )︀ [(1 − 𝜖 ) 𝑘 + 2 𝜖 ] . (5.18)These expressions are central results for this paper and can be used to evaluate thealpha flux caused by a wide range of tokamak perturbations. In later sections, theywill be applied to the flux resulting from ripple and TAEs as examples. Already, theexpressions display clear and expected trends. That the flux is driven by the alpha densitygradient is shown by the expression’s dependence on 𝜕𝑛 𝛼 /𝜕𝜓 . That the flux increaseswith the perturbed amplitude squared is also clear. The following subsections examinethe elements of these fluxes that remain opaque: the term 𝛿 ( 𝑄 𝑙 ) , which enforces theresonance condition, and the term C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 , which we refer to as a "phase factor."5.4. Resonance condition
The flux expressions (5.17) and (5.18) include the delta function 𝛿 ( 𝑄 𝑙 ) . This deltafunction enforces the resonance given in equation (4.30) and determines which particlesare able to be transported by the mode and to contribute to the alpha flux. The expressionfor the resonance (4.30) can be evaluated to read [setting 𝑞 ⋆ = 𝑞 , as discussed insection 5.3] 𝑄 𝑙 ( 𝑣, 𝜅 ) = 𝜔𝜏 𝑏 − 𝑛𝜔 𝛼 ⋆ 𝜏 𝑏 − 𝜋𝜎 ( 𝑛𝑞 − 𝑚 ) − 𝜋𝑙, (5.19)where the bounce or transit time 𝜏 𝑏 is defined in (4.24), 𝜎 is defined in (4.14), and thetransit average drift is defined by 𝜔 𝛼 ⋆ ≡ 𝜏 − 𝑏 ∮︁ 𝜔 𝛼 ⋆ 𝑑𝜏. (5.20)The terms in (5.19) can be evaluated for trapped and passing particles to give theexpressions in terms of 𝑣 and 𝜅 or 𝑘 in Table 3. † The term 𝛿 ( 𝑄 𝑙 ) in the flux expression † This evaluation follows from using the definitions of 𝜏 (A 6) and 𝜔 𝛼 ⋆ (4.9). In a magneticfield with strength described by (5.3), the parallel velocity appearing in 𝜔 𝛼 ⋆ and 𝑑𝜏 is 𝑣 ‖ = ± 𝑣 √︀ [1 − (1 − 𝜖 ) 𝜆 ] − 𝜖𝜆 sin 𝜗/ . The changes of variable given later on in (5.21)and (5.28) are needed. E. A. Tolman and P.J. Catto
Figure 2: A banana orbit, with legs 1 and 2 defined. Note that banana orbits alwaysmove in the sense indicated (i.e. with positive velocity on the outer side of the banana)for plasma current in the direction of positive 𝜁 because of the conservation of 𝜓 ⋆ (4.2).thus relates the pitch angle and speed of particles that participate in transport andcontribute to the flux. 5.5. Evaluation of phase factor
The term C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 in (5.17) and (5.18) depends on integrals defined in (5.10)and (5.11). These integrals are evaluated along particle trajectories, which advance with 𝜏 (A 6). This complex term, which we refer to as a "phase factor," determines howeffective a given resonance is at creating a perturbation to the alpha distribution function.In this subsection, we write the phase factor for trapped and passing particles in a formthat can be integrated numerically or approximated.We begin with trapped particles. Trapped particles follow banana orbits, representedschematically in figure 2, with two legs on which the particle velocity is in oppositedirections. Our evaluation assumes that the particle orbit is in resonance at a particularharmonic 𝑙 with the mode via (5.19). Only such resonant particles are able to contributeto the flux. The integrals in the phase factor are evaluated along each of these legs, whichcontribute in distinct ways that depend on 𝑙 and finite orbit effects to the overall phasefactor.We begin by introducing the variable 𝑥 , which is the following function of 𝜗 : 𝜅 sin 𝑥 ≡ sin ( 𝜗/ . (5.21)The definition of 𝜅 (5.6) can be used to show that the banana bounce points occur at 𝑥 = ± 𝜋/ , so that the use of 𝑥 standardizes the necessary integration. Now, to write C 𝑚𝑛𝜔 and S 𝑚𝑛𝜔 explicitly, we must write various quantities that appear in 𝜏 (A 6) or 𝛬 (4.20) in terms of 𝑥 . The parallel velocity in the magnetic field (5.3) is given by 𝑣 ‖ = ± 𝜅𝑣 √ 𝜖𝜆 cos 𝑥 ≈ ± 𝜅𝑣 √ 𝜖 cos 𝑥, (5.22)where the last expression exploits that for trapped particles 𝜆 differs from by at most ± 𝜖 . The drift term appearing in 𝛬 , (4.9), is given by 𝜔 𝛼 ⋆ = 𝑣 𝛺 𝑝 𝑅 (︀ − 𝜅 sin 𝑥 + 4 𝑠𝜅 cos 𝑥 )︀ . (5.23) rift kinetic theory of alpha transport by tokamak perturbations Expression Value 𝑎 𝑡 ( 𝑥 ) 𝑞𝑅𝑣 √ 𝜖 ∫︀ 𝑥 𝑑𝑥 ′ [︂ 𝜔 − 𝑛𝑣 𝛺𝑝𝑅 ( − 𝜅 sin 𝑥 ′ +4 𝑠𝜅 cos 𝑥 ′ ) ]︂ √ − 𝜅 sin 𝑥 ′ 𝑏 𝑡 𝑘 𝜓 √ 𝜖𝑣𝜅𝛺 𝑝 𝑐 𝑡 ( 𝑥 ) 𝑏 𝑡 cos 𝑥𝑑 𝑡 ( 𝑥 ) 2 ( 𝑛𝑞 − 𝑚 ) sin − ( 𝜅 sin 𝑥 ) Table 4: Definition of quantities used in the trapped particle phase factor.We will also use ˆ 𝑏 · ∇ 𝜗 ≈ / ( 𝑞𝑅 ) to simplify our expressions.To make our presentation of the phase factor more compact, we define a set ofexpressions [ 𝑎 𝑡 ( 𝑥 ) , 𝑏 𝑡 , 𝑐 𝑡 ( 𝑥 ) , and 𝑑 𝑡 ( 𝑥 ) ] given in table 4. Then, the contribution of theouter leg , which has velocity in the direction of positive 𝜗 , to C 𝑚𝑛𝜔 (5.10) can be shownto equal (where we have again taken 𝜆 ≈ ) C 𝑚𝑛𝜔,𝑜𝑢𝑡𝑒𝑟 = 2 𝑞𝑅 (︁ 𝛷 𝑚𝑛𝜔 + 𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos [ 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 − 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 − 𝐴 ‖ 𝑚𝑛𝜔 𝑞𝑅𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 𝜅 cos 𝑥 cos [ 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 − 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 . (5.24)Meanwhile, the inner leg is given by C 𝑚𝑛𝜔,𝑖𝑛𝑛𝑒𝑟 = 2 𝑞𝑅 (︁ 𝛷 𝑚𝑛𝜔 + 𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖 ∫︁ − 𝜋/ 𝜋/ 𝑑𝑥 − cos [ 𝜋𝑙 − 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 + 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 − 𝐴 ‖ 𝑚𝑛𝜔 𝑞𝑅𝑐 ∫︁ − 𝜋/ 𝜋/ 𝑑𝑥 𝜅 cos 𝑥 cos [ 𝜋𝑙 − 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 + 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 . (5.25)Here, our use of 𝑙 follows from the recognition that only particles which are in resonanceaccording to (5.19) contribute to the flux, and that the quantity 𝜋𝑙 accumulates from 𝜔 and 𝑛𝜔 𝛼 ⋆ integrated over the entire orbit starting from 𝑥 = 0 . The inner leg value canbe restated by applying the relationship cos ( 𝑥 + 𝜋𝑙 ) = ( − 𝑙 cos 𝑥 and flipping the limitsso they agree with those given for the outer leg: C 𝑚𝑛𝜔𝑙,𝑖𝑛𝑛𝑒𝑟 = 2 𝑞𝑅 ( − 𝑙 (︁ 𝛷 𝑚𝑛𝜔 + 𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos [ − 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 + 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 + 𝐴 ‖ 𝑚𝑛𝜔 𝑞𝑅 ( − 𝑙 𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 𝜅 cos 𝑥 cos [ − 𝑎 𝑡 ( 𝑥 ) + 𝑏 𝑡 + 𝑐 𝑡 ( 𝑥 ) − 𝑑 𝑡 ( 𝑥 )] √︀ − 𝜅 sin 𝑥 . (5.26)Equivalent expressions hold for S 𝑚𝑛𝜔 . Forming and evaluating the quantity C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 reveals that the dependence on 𝑏 𝑡 vanishes, as is to be expected given that thisparameter depends on the parallel velocity evaluated at 𝜗 ( 𝜏 ) = 0 , an arbitrary location.Suppressing the 𝑥 dependence of 𝑎 𝑡 , 𝑐 𝑡 , and 𝑑 𝑡 for compactness, the full phase factor2 E. A. Tolman and P.J. Catto reads C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 𝑞 𝑅 = {︃ 𝛷 𝑚𝑛𝜔 + 𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 𝑣 √ 𝜖 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos ( 𝑎 𝑡 − 𝑐 𝑡 − 𝑑 𝑡 ) + ( − 𝑙 cos ( − 𝑎 𝑡 + 𝑐 𝑡 − 𝑑 𝑡 ) √︀ − 𝜅 sin 𝑥 − 𝐴 ‖ 𝑚𝑛𝜔 𝜅𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos 𝑥 [︁ cos ( 𝑎 𝑡 − 𝑐 𝑡 − 𝑑 𝑡 ) − ( − 𝑙 cos ( − 𝑎 𝑡 + 𝑐 𝑡 − 𝑑 𝑡 ) ]︁√︀ − 𝜅 sin 𝑥 ⎫⎬⎭ + {︃ 𝛷 𝑚𝑛𝜔 + 𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 𝑣 √ 𝜖 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 sin ( 𝑎 𝑡 − 𝑐 𝑡 − 𝑑 𝑡 ) + ( − 𝑙 sin ( − 𝑎 𝑡 + 𝑐 𝑡 − 𝑑 𝑡 ) √︀ − 𝜅 sin 𝑥 − 𝐴 ‖ 𝑚𝑛𝜔 𝜅𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos 𝑥 [︁ sin ( 𝑎 𝑡 − 𝑐 𝑡 − 𝑑 𝑡 ) − ( − 𝑙 sin ( − 𝑎 𝑡 + 𝑐 𝑡 − 𝑑 𝑡 ) ]︁√︀ − 𝜅 sin 𝑥 ⎫⎬⎭ . (5.27)Though this expression is daunting, in practice many simplifications are possible. First,most modes have only a subset of 𝛷 𝑚𝑛𝜔 , 𝐵 ‖ 𝑚𝑛𝜔 , and 𝐴 ‖ 𝑚𝑛𝜔 , and these quantities maybe proportional to each other. In addition, it is sometimes possible to neglect some ofthe terms in the arguments of the cosine. The integrals can also be approximated in avariety of ways.Passing particles are simpler to treat than trapped particles. For passing particles thevariable 𝑥 is defined as: 𝑥 ≡ 𝜗 . (5.28)Recalling the variable 𝑘 defined in (5.7), the parallel velocity in the magnetic field (5.3)for passing particles is given by 𝑣 ‖ = ± 𝑣 √ 𝜖𝜆𝑘 √︀ − 𝑘 sin 𝑥. (5.29)Note that 𝜆 can be very small for passing particles and thus cannot be set to 1 as it wasfor the trapped particles. For passing particles, the drift (4.9) is: 𝜔 𝛼 ⋆ = 𝜆𝑣 𝛺 𝑝 𝑅 𝑘 [︀ 𝑘 − 𝑘 sin 𝑥 + 4 𝑠 (︀ − 𝑘 sin 𝑥 )︀]︀ . (5.30)We again define certain expressions to make the presentation of the phase factor morecompact in Table 5. A passing particle orbit goes around the entire device, from 𝑥 = − 𝜋/ to 𝑥 = 𝜋/ . Its C 𝑚𝑛𝜔 is given by C 𝑚𝑛𝜔 = 2 𝑘𝑞𝑅 (︁ 𝛷 𝑚𝑛𝜔 + 𝜆𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖𝜆 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 √︀ − 𝑘 sin 𝑥 cos ( 𝑎 𝑝 + 𝑏 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) − 𝜎𝑞𝑅𝐴 ‖ 𝑚𝑛𝜔 𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos ( 𝑎 𝑝 + 𝑏 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) . (5.31) rift kinetic theory of alpha transport by tokamak perturbations Expression Value 𝑎 𝑝 ( 𝑥 ) 𝑞𝑅𝑘𝜎𝑣 √ 𝜖𝜆 ∫︀ 𝑥 𝑑𝑥 ′ [︃ 𝜔 − 𝑛𝑣 𝜆 ( 𝑘 − 𝑘 𝑥 ′ +4 𝑠 ( − 𝑘 𝑥 ′ )) 𝑘 𝛺𝑝𝑅 ]︃ √ − 𝑘 sin 𝑥 ′ 𝑏 𝑝 𝜎𝑘 𝜓 √ 𝜖𝜆𝑣𝑘𝛺 𝑝 𝑐 𝑝 ( 𝑥 ) 𝑏 𝑝 √︀ − 𝑘 sin 𝑥𝑑 𝑝 ( 𝑥 ) 2 ( 𝑛𝑞 − 𝑚 ) 𝑥 Table 5: Definition of quantities used in the passing particle phase factor. Note that 𝜎 isthe parameter introduced (4.14), which characterizes the direction of a particle’s motionin 𝜗 .An analogous expression holds for S 𝑚𝑛𝜔 . Forming the sum of the two terms gives C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 𝑞 𝑅 = ⎧⎨⎩ 𝑘 (︁ 𝛷 𝑚𝑛𝜔 + 𝜆𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖𝜆 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos ( 𝑎 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) √︀ − 𝑘 sin 𝑥 − 𝜎𝐴 ‖ 𝑚𝑛𝜔 𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 cos ( 𝑎 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) ⎫⎬⎭ + ⎧⎨⎩ 𝑘 (︁ 𝛷 𝑚𝑛𝜔 + 𝜆𝑣 𝑐𝛺 𝐵 ‖ 𝑚𝑛𝜔 )︁ 𝑣 √ 𝜖𝜆 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 sin ( 𝑎 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) √︀ − 𝑘 sin 𝑥 − 𝜎𝐴 ‖ 𝑚𝑛𝜔 𝑐 ∫︁ 𝜋/ − 𝜋/ 𝑑𝑥 sin ( 𝑎 𝑝 − 𝑐 𝑝 − 𝑑 𝑝 ) ⎫⎬⎭ . (5.32)Again, in practice this complicated expression can be simplified in a variety of ways.
6. Ripple flux
In this section, we consider the trapped and passing fluxes (5.17) and (5.18) for our firstexample perturbation: ripple, which has characteristic parameters described in table 2.We will find that the contribution of bounce harmonic resonances to the transport forthis perturbation is small because 𝑛𝑞 is very high. Because of this realization, we do notprovide an analytic expression for the flux.First, we consider the structure of the resonances between alphas and ripple in atokamak with parameters given in table 1. These are plotted in figures 3, 4, and 5, fortrapped particles with 𝜎 = 0 , for passing particles with 𝜎 = 1 , and for passing particleswith 𝜎 = − . Specifically, these figures show, for typical values of 𝑙 , the values of 𝑣 and 𝜅 (or 𝑘 ) for which 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 and the delta function in (5.17) or (5.18) has support. Thelines in the plot identify the areas of phase space which could contribute to transport.The resonance structure is only shown up to the alpha birth velocity 𝑣 because alphaparticles are unable to interact with perturbations through resonances above this velocity.Trapped ripple resonances are shown in figure 3. Trapped resonances can occur at mostvalues of 𝜅 for negative values of 𝑙 of intermediate magnitude ( 𝑙 = − and 𝑙 = − areshown in the plot). An alpha particle is born at 𝑣 and a specific value of 𝜅 . As the particleslows down, it will move left along the 𝑥 -axis while maintaining the same value of 𝜅 .During this process, the alpha interacts with the perturbation when it crosses a resonanceline. Note that in this case, the lines representing some harmonics are multivalued in 𝜅 E. A. Tolman and P.J. Catto
Figure 3: Trapped particle ( 𝜎 = 0 ) resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] forripple with parameters given in table 2 in a tokamak described by the values in table 1. / v k l =-
60 l =-
65 l =- Figure 4: Passing, 𝜎 = 1 , particle resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] for ripplewith parameters given in table 2 in a tokamak described by the values in table 1. / v k l =
40 l =
45 l = Figure 5: Passing, 𝜎 = − , particle resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] forripple with parameters given in table 2 in a tokamak described by the values in table 1.at some velocities. This simply means that at that velocity, multiple pitch angles are ableto interact with the mode.A series of resonances near the trapped-passing boundary ( 𝜅 = 1 ) also exists. These rift kinetic theory of alpha transport by tokamak perturbations 𝑙 < and all of 𝑙 = 0 , † shown in the plot. There are alsoseveral more resonances, corresponding to 𝑙 > , at 𝜅 ≈ , which are not displayed inthe plot for clarity. Because the resonances at 𝜅 ≈ are very close to each other inpitch angle and very near to the trapped-passing boundary layer where other boundarylayer physics enters (Catto 2018), the use of the Krook operator to represent pitch anglescattering is not well justified, making the treatment in this paper unsatisfactory for thisregion. A more elaborate boundary layer theory is necessary to satisfactorily treat theseresonances, so in this paper we focus on the lower 𝜅 parts of the trapped resonances with 𝑙 < .For passing particles, shown in figures 4 and 5, which have 𝜎 ̸ = 0 in (5.19), the largevalue of 𝑛𝑞 characterizing ripple means that resonances occur at values of 𝑙 of largemagnitude. At alpha birth (the right side of the plot), particles near trapped-passingboundary ( 𝑘 = 1 ) resonate with ripple, while at lower speeds, freely passing particles(small 𝑘 ) resonate with ripple.After understanding the resonance structure, the next step in the evaluation of flux isto consider the phase factor C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 , discussed in section 5.5. However, this phasefactor is very small for ripple resonances. As an example, figure 6 shows the numericallyevaluated phase factor for the 𝑙 = − trapped harmonic displayed in figure 3. Specifically,this plot shows the quantity (5.27) as a function of 𝑣 , where (5.27) is numerically evaluatedat the value of 𝜅 that is resonant at that value of 𝑣 . The small values of the phase factorevident in figure 6 are typical of those for all ripple resonances.These small values result from the highly oscillatory nature of ripple phase factor,which we demonstrate using the original definition of C 𝑚𝑛𝜔 , (5.10), and of S 𝑚𝑛𝜔 , (5.11).In particular, the argument of the sinusoids in (5.10) and (5.11) can be roughly approx-imated by ∫︁ 𝜏𝜏 𝛬 ( 𝜏 ′ , 𝑣, 𝜅 ) 𝑑𝜏 ′ ≈ − 𝑛𝜔 𝛼 ⋆ ( 𝜏 − 𝜏 ) − 𝑛𝑞𝜗 ( 𝜏 ) (6.1)for values of 𝑣 and 𝜅 satisfying the resonance condition. [Here, 𝜔 𝛼 ⋆ is defined in (5.20).]To obtain this expression, we have considered the definition of 𝛬 , (4.20), and inserted 𝑚 = 0 , 𝑞 ⋆ ≈ 𝑞 , and 𝜔 = 0 , as appropriate for ripple. Then, we have treated 𝜔 𝛼 ⋆ as thoughit were constant and independent of 𝜏 [we note that a similar approximation was usedto evaluate related phase factors in previous studies of the effect of non-axisymmetrieson bulk plasma (Park et al. ∫︁ 𝜏𝜏 𝛬 ( 𝜏 ′ , 𝑣, 𝜅 ) 𝑑𝜏 ′ ≈ {︃ − 𝜋𝑙 ( 𝜏 − 𝜏 ) 𝜏 𝑏 − 𝑛𝑞𝜗 ( 𝜏 ) , trapped particles − 𝜋𝑙 ( 𝜏 − 𝜏 ) 𝜏 𝑏 , passing particles . (6.2)(For the passing expression, we have taken the additional approximation 𝜎 ( 𝜏 − 𝜏 ) /𝜏 𝑏 ≈ 𝜗/ 𝜋 .) These expressions indicate that the arguments of the sinusoids in (5.10) and (5.11)change very quickly if, for trapped particles, | 𝑙 + 𝑛𝑞 | is large or, for passing particles, | 𝑙 | islarge. The corresponding phase integrands in the phase factor are highly oscillatory. TheRiemann-Lebesgue lemma states that such integrals must approach zero as | 𝑙 + 𝑛𝑞 | − fortrapped particles and | 𝑙 | − for passing particles.Ripple is characterized by 𝑛𝑞 >> . Then, consideration of the harmonics 𝑙 character- † Note that the 𝑙 = 0 resonance is so close to 𝜅 = 1 because we selected as an exampleparameter 𝑠 = 1 (see table 2). For lower values of shear, the 𝑙 = 0 resonance occurs at lowervalues of 𝜅 and does not suffer from the resonance crowding described in this paragraph. However,ripple does tend to be strongest in regions of high shear, so 𝑠 = 1 is an appropriate examplevalue. E. A. Tolman and P.J. Catto
Figure 6: Exact, numerically evaluated trapped phase factor for the 𝑙 = − resonanceof ripple. (Note that the phase factor is evaluated for the lower 𝜅 branch ofthe resonance only, whenever two branches exist. Our approach is not capable oftreating the higher branch, where resonances are bunched very closely together.) Thenormalization is determined using the coefficients of 𝐵 ‖ 𝑚𝑛𝜔 in (5.27); the sum ofintegrals (︁∫︀ 𝜋/ − 𝜋/ 𝑑𝑥 [cos . . . ] / √︁[︀ − 𝜅 sin 𝑥 ]︀)︁ + (︁∫︀ 𝜋/ − 𝜋/ 𝑑𝑥 [sin . . . ] / √︁[︀ − 𝜅 sin 𝑥 ]︀)︁ is normalized to 𝜋 .izing resonances that exist in a typical tokamak, illustrated in figures 3, 4, and 5, revealsthat for trapped particles | 𝑙 + 𝑛𝑞 | is always large and for passing particles | 𝑙 | is large. ‡ Based on Eq. (6.2) and the discussion in the previous pragraph, this means the phasefactors for ripple perturbations are extremely small, as exemplified in figure 6.With such a small phase factor, the flux described by (5.17) and (5.18) is negligible.Therefore, bounce harmonic resonant transport is very weak for ripple perturbations andwill not create noticeable alpha heat flux. Alpha transport from ripple may, however,occur via other mechanisms, including stochastic ripple diffusion and ripple trapping(Goldston et al. et al. † In addition, further work shouldconsider the impact of externally-applied 3D magnetic field perturbations, such as thoseused for ELM control, which could cause significant alpha transport (Sanchis et al.
7. TAE flux
In this section, we evaluate the trapped and passing fluxes (5.17) and (5.18) for theexample case of the TAE, which has characteristic parameters described in table 2. We ‡ The reader might ask if it is possible to construct any tokamak in which | 𝑙 + 𝑛𝑞 | is smallfor trapped particle ripple resonances and | 𝑙 | is small for passing particle ripple resonances.For trapped particles, this would require 𝑛𝑞 ∼ − 𝑙 in (5.19). This reduces to the requirement 𝑣 ∼ 𝛺 𝑝 𝑅 , which requires unrealistically small machine size or magnetic field. A similar conditionis found for passing particles. † Some of these mechanisms are collisionless. One might at first expect collisionless processesto be unimportant in steady state, where refilling of loss regions is necessary for sustainedtransport. However, alpha particles are continuously introduced to the tokamak via fusion,which serves to refill loss regions. Therefore, consideration of collisionless processes in tokamakdesign is critical to achieving satisfactory steady state behavior. rift kinetic theory of alpha transport by tokamak perturbations / v κ l = = = Figure 7: Trapped particle ( 𝜎 = 0 ) resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] for aTAE with parameters given in table 2 in a tokamak described by the values in table 1. / v k l = = Figure 8: Passing, 𝜎 = 1 , particle resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] for aTAE with parameters given in table 2 in a tokamak described by the values in table 1.find that the TAE can cause significant alpha heat flux, and develop a constraint theTAE amplitude must obey to prevent significant alpha depletion. At the end of thesection, we develop a simple saturation model for TAE. This model suggests that TAEsin SPARC-like tokamaks will saturate below the level at which bounce harmonic resonanttransport can cause significant alpha depletion. However, saturation amplitudes abovethose suggested by our model, but within computational and experimental experience,could cause depletion. 7.1. Evaluation of flux
The resonance structures for the TAE are shown in figures 7, 8, and 9. Trapped particlesare able to resonate with the TAE at low values of the harmonic 𝑙 , as seen in figure 7.Near the alpha particle birth speed (the right edge of the plot) the mode resonates witha small subset of pitch angles nearer the trapped-passing boundary ( 𝜅 = 1 ). As the alphaparticles slow down (moving left along the x-axis), a wider range of pitch angles canresonate with the mode. For a given value of 𝜅 , lower values of 𝑙 are able to resonate withhigher speed particles.Passing particle resonances (with both 𝜎 = 1 and 𝜎 = − ) with the TAE are similar,with alpha particles near birth resonating with barely passing particles and those thathave slowed down resonating at a wider variety of pitch angles. These resonance structures8 E. A. Tolman and P.J. Catto / v k l = = Figure 9: Passing, 𝜎 = − , particle resonance structure [i.e., where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 ] for aTAE with parameters given in table 2 in a tokamak described by the values in table 1.can be seen in figures 8 and 9. Again, lower values of 𝑙 tend to resonate with higher velocityparticles for a given value of 𝑘 . As the drift 𝜔 𝛼 ⋆ is negative, an 𝑙 = 0 resonance is onlypossible for 𝜎 = 1 , but we will soon find the transport associated with this sign of 𝜎 isvery small. Note that fully passing particles (with 𝑘 = 0 ) resonate at the Alfvén speedand one third of the Alfvén speed. This aligns with the resonance structure described insimplified treatments of TAE resonance which focus on freely passing particles (Heidbrink2008; Betti & Freidberg 1992).Now, we move to the evaluation of the TAE flux, starting with trapped particles. Again,this procedure begins with consideration of the phase factor (5.27). Full evaluation ofthis quantity must be done numerically. However, a tolerable approximation of its valuecan be made for low values of 𝑙 . At low 𝑙 for typical TAE parameters, the dominantterm in the sinusoids of the phase factor is the finite orbit width term 𝑐 𝑡 . This can beverified numerically. After neglecting 𝑎 𝑡 and 𝑑 𝑡 , the terms in the phase factor which areproportional to 𝐴 ‖ can be neglected as higher order in 𝜖 . Then, the terms cos ( 𝑐 𝑡 ) =cos ( 𝑏 𝑡 cos 𝑥 ) (for even 𝑙 ) and sin ( 𝑐 𝑡 ) = sin ( 𝑏 𝑡 cos 𝑥 ) (for odd 𝑙 ) can be written in termsof Bessel functions using the Bessel generating function 𝑒 𝑖𝑏 𝑡 cos 𝑥 = ∑︀ 𝑛 = ∞ 𝑛 = −∞ 𝑖 𝑛 𝑒 𝑖𝑛𝑥 𝐽 𝑛 ( 𝑏 𝑡 ) .Here, 𝐽 𝑛 represents the Bessel function of the first kind of order 𝑛 . Higher values of 𝑛 in the sum are more oscillatory and do not contribute much to the phase factor integral.So, only the lowest 𝑛 are maintained. This process gives: C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 𝑞 𝑅 ≈ ⎧⎨⎩ 𝛷 𝑚𝑛𝜔 𝐽 ( 𝑏 𝑡 ) 𝐾 ( 𝜅 ) 𝜖𝑣 , 𝑙 even 𝛷 𝑚𝑛𝜔 𝐽 ( 𝑏 𝑡 ) ( sin − 𝜅 ) 𝜖𝜅 𝑣 , 𝑙 odd . (7.1)The approximation in (7.1) is compared to a full numerical evaluation of (5.27) for 𝑙 = 0 , 𝑙 = 1 , and 𝑙 = 2 in figure 10. This comparison shows that, although the approximation isnot exact, it captures key trends in the phase factor for low values of 𝑙 . The approximationis less good for higher values of 𝑙 , where 𝑎 𝑡 becomes more important and cannot beneglected.However, higher values of 𝑙 will have minimal contribution to the flux for three reasons.First, the resonance structures in figures 7, 8, and 9 make clear that, for a given value of 𝜅 , higher values of 𝑙 correspond to lower values of 𝑣 . These values of 𝑙 contribute less heatflux, which depends on 𝑣 . Second, the second term in − 𝜔/ ( 𝑛𝜔 ⋆ ) , which appears inthe flux expression (5.17), scales with velocity like /𝑣 [as we will see later on in (7.3)],further reducing the contribution of lower velocities to the flux. Finally, similar arguments rift kinetic theory of alpha transport by tokamak perturbations (a) 𝑙 = 0 (b) 𝑙 = 1 (c) 𝑙 = 2 Figure 10: Exact, numerically evaluated phase factor for trapped particles interactingwith the TAE compared to the approximate form given in (7.1), for 𝑙 = 0 , , . Theapproximation reproduces overall trends well for low values of 𝑙 . For higher values of 𝑙 ,the neglect of 𝑎 𝑡 in the approximation is inappropriate; however, higher values of 𝑙 havevery small contributions to the flux. The normalizations in these plots are found in thesame way as those in figure 6.0 E. A. Tolman and P.J. Catto to those used in the consideration of the ripple phase factor show that the sinusoids inthe phase factor will become highly oscillatory for high values of 𝑙 , such that the valueof the phase factor must shrink by the Riemann-Lebesgue lemma. This can be verifiedcomputationally. In figure 10, the phase factor for 𝑙 = 2 is already smaller than for 𝑙 = 0 and 𝑙 = 1 , which is an example of this effect. Because higher 𝑙 harmonics have such smallcontributions to the flux, in this section we compute the contributions from 𝑙 = 0 , , only.We now proceed with the evaluation of (5.17). To lowest order in 𝜖 , the denominatorof (5.17) is 1. The delta function enforcing the resonance is 𝛿 ( 𝑄 𝑙 ) , with 𝑄 𝑙 as evaluatedin (5.19) with 𝜎 = 0 for trapped particles. This delta function is used to evaluate thevelocity integral, recalling the general property that 𝛿 [ 𝑓 ( 𝑥 )] = ∑︁ 𝑖 𝛿 ( 𝑥 − 𝑎 𝑖 ) ⃒⃒⃒ 𝑑𝑓𝑑𝑥 ( 𝑎 𝑖 ) ⃒⃒⃒ (7.2)for any function 𝑓 ( 𝑥 ) with roots 𝑎 𝑖 . Furthermore, we can write that − 𝜔𝑛𝜔 ⋆ = 1 − 𝛺 𝑝 𝜔𝑅𝑎 𝛼 𝑛𝑣 , (7.3)with 𝑎 𝛼 the alpha scale length defined in (2.7). For typical values of 𝑎 𝛼 , 𝑎 𝛼 /𝑅 ∼ 𝜖 ;inserting this scaling and the resonant value of 𝑣 where 𝑄 𝑙 ( 𝑣, 𝜅 ) = 0 gives that − 𝜔𝑛𝜔 ⋆ = 1 − 𝒪 ( 𝜖 ) , (7.4)such that − 𝜔/ ( 𝑛𝜔 ⋆ ) may be set to to lowest order in 𝜖 (at higher values of 𝑙 thanthose considered here, where the resonant 𝑣 becomes very small for many values of 𝜅 ,this replacement would not be appropriate). Then, the contribution to the flux from each 𝑙 reduces to a fairly compact integral over 𝜅 . This integral is different for 𝑙 even and 𝑙 odd: 𝑄 𝛼,𝑚𝑛𝜔,𝑡 = − 𝑀 𝛼 𝑛 𝑐 𝛷 𝑚𝑛𝜔 𝑞𝑅 ln ( 𝑣 /𝑣 𝑐 ) 𝜕𝑛 𝛼 𝜕𝜓 ∑︁ 𝑙 ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∫︀ 𝜅 𝑑𝜅 𝜅𝐽 (︂ √ 𝜖𝑘𝜓𝑣𝑟𝑒𝑠𝜅𝛺𝑝 )︂ 𝐾 ( 𝜅 ) ⃒⃒⃒ 𝜕𝑄𝑙𝜕𝑣 ( 𝑣 𝑟𝑒𝑠 ) ⃒⃒⃒ , 𝑙 even ∫︀ 𝜅 𝑑𝜅 𝐽 (︂ √ 𝜖𝑘𝜓𝑣𝑟𝑒𝑠𝜅𝛺𝑝 )︂ ( sin − 𝜅 ) 𝜅 ⃒⃒⃒ 𝜕𝑄𝑙𝜕𝑣 ( 𝑣 𝑟𝑒𝑠 ) ⃒⃒⃒ , 𝑙 odd . (7.5)Here, 𝑣 𝑟𝑒𝑠 is the resonant velocity at which 𝑄 𝑙 ( 𝑣 𝑟𝑒𝑠 , 𝜅 ) = 0 . The value of 𝜅 in resonanceat the birth velocity 𝑣 is called 𝜅 , i.e., using (5.19) and table 3, 𝑞𝑅𝐾 ( 𝜅 ) 𝜔𝑣 √ 𝜖 − 𝑛𝑞𝑣 [2 𝐸 ( 𝜅 ) − 𝐾 ( 𝜅 )] 𝛺 𝑝 𝑅 √ 𝜖 − 𝜋𝑙. (7.6)The integrals in (7.5) can easily be evaluated numerically for specific values of the relevantparameters.To obtain a closed expression, we now make another set of approximations for the valuesof 𝜅 . For 𝑙 = 0 , 𝜅 is low, and the resonance condition (7.6) is expanded about 𝜅 = 0 to 𝒪 (︀ 𝜅 )︀ and then solved for 𝜅 . For higher values of 𝑙 , with higher values of 𝜅 , differentapproximations are used. The quantity 𝐸 ( 𝜅 ) − 𝐾 ( 𝜅 ) vanishes at the point 𝜅 ≈ . and is small near that value. Resonances with 𝑙 > have values of 𝜅 in this region. For 𝑙 = 1 , we can simply approximate 𝜅 with this value. For resonances of higher 𝑙 , we canneglect the term proportional to 𝐸 ( 𝜅 ) − 𝐾 ( 𝜅 ) in the resonance condition (7.6). Also,we make the replacement 𝐾 ( 𝜅 ) → ln (︁ / √︀ − 𝜅 )︁ in the remaining terms [this is the rift kinetic theory of alpha transport by tokamak perturbations Expression Value 𝐶 ,𝑡 − 𝛺 𝑝 𝑅𝑣 𝐴 𝑛𝑞𝑣 𝐶 ,𝑡 . 𝑛 𝑞 𝑣 𝜖𝑅 𝛺 𝑝 (︃ − √︂ 𝑛𝑞𝑣𝐴𝜖𝑅𝛺𝑝 )︃ 𝐶 ,𝑡 (︂ − 𝑒 − 𝜋𝑣 √ 𝜖𝑣𝐴 )︂ (︃ − √︂ 𝑛𝑞𝑣𝐴 𝜖𝑅𝛺𝑝 )︃ Table 6: Coefficients of harmonics of contributions to flux used in Eq. (7.8). Only valuesfor 𝑙 = 0 , , are given because higher values of 𝑙 have negligible contributions to theflux. We have inserted 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) to simplify the coefficients.limit of 𝐾 ( 𝜅 ) as 𝜅 → ]. This process gives, where we have inserted 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) tosimplify the expressions, † 𝜅 ≈ ⎧⎪⎪⎨⎪⎪⎩√︁ − 𝑣 𝐴 𝛺 𝑝 𝑅𝑛𝑞𝑣 , 𝑙 = 00 . , 𝑙 = 1 √︁ − 𝑒 − 𝜋𝑙𝑣 √ 𝜖𝑣𝐴 , 𝑙 > . (7.7)The integrands can be expanded about 𝜅 = 0 to lowest non-trivial order in 𝜅 [ 𝒪 ( 𝜅 ) for 𝑙 even and 𝒪 (︀ 𝜅 )︀ for 𝑙 odd]. Prior to making the expansion for odd 𝑙 , the velocity in theargument of 𝐽 is replaced with the birth velocity. Comparison of the resulting approxi-mate integrands to the numerically evaluated integrands shows that the approximationsare good throughout the integration interval. Then, we calculate 𝑄 𝛼,𝑚𝑛𝜔,𝑡 ≈ − √ 𝜖𝑀 𝛼 𝑛𝜋𝑐 𝛷 𝑚𝑛𝜔 𝛺 𝑝 𝑅 √ 𝑣 /𝑣 𝑐 ) 𝜕𝑛 𝛼 𝜕𝜓 ∑︁ 𝑙 𝐶 𝑙,𝑡 , (7.8)where 𝐶 𝑙,𝑡 is a parameter given in table 6 for the most important low 𝑙 harmonics.This expression, giving the trapped particle alpha heat flux from a TAE of a givenmagnitude, is a central result of this paper. For the example parameters in this paper, ∑︀ 𝑙 𝐶 𝑙,𝑡 ≈ . .Now, we consider passing particles, with flux given by the expression in (5.18). The firststep is to approximate the phase factor (5.32). The dominant term in the phase factorsinusoids is again 𝑐 𝑝 . However, it is not possible to use the Bessel generating function forthe passing particles, so instead we simply take cos 𝑐 𝑝 ≈ cos 𝑏 𝑝 and sin 𝑐 𝑝 ≈ sin 𝑏 𝑝 . Thenthe phase factor becomes, for 𝑙 even or odd, C 𝑚𝑛𝜔 + S 𝑚𝑛𝜔 𝑞 𝑅 = 𝜋 𝑘 𝛷 𝑚𝑛𝜔 𝜖𝜆𝑣 (︃ 𝐾 ( 𝑘 ) 𝜋 − 𝜎𝑣 √ 𝜖𝜆𝑘𝑣 𝐴 )︃ . (7.9)This approximation is compared to the full numerical evaluation of the phase factor intable 7.Notably, the phase factor is very small for 𝜎 = 1 in both the numerically evaluated andthe approximated forms. This results from the cancellation between the contribution to † The 𝑙 = 0 approximation, and the fluxes that follow from it, only provide physical valueswhen 𝑣 𝐴 < 𝑣 𝑞𝑛/ ( 𝛺 𝑝 𝑅 ) . When device Alfvén speed violates this inequality, the 𝑙 = 0 resonancevanishes, and its contribution should be neglected. E. A. Tolman and P.J. Catto 𝜎 = 1 𝜎 = − 𝑙 = 0 No resonance 𝑙 = 1 𝑙 = 2 Very low contribution to flux
Table 7: Exact, numerically evaluated phase factor for passing particles interacting withthe TAE compared to the approximate form given in (7.9). The approximation reproducesoverall trends well for low values of 𝑙 . Note that the phase factor is much smaller for 𝜎 = 1 than for 𝜎 = − . The normalizations in the plots in this figure are different from thoseused in the analogous plots for trapped particles, found in figure 6 and figure 10. Inaddition to being in terms of 𝑘 and 𝜆 , the normalization is a factor of higher, whichreflects that passing phase factors are intrinsically larger.the phase factor from the electric potential and that from the parallel vector potential.Thus, moving forward we evaluate only the contribution from 𝜎 = − .The denominator in (5.18) cannot be set to as it was for the evaluation of thetrapped flux (5.17); however, the delta function is applied in the same way as for the rift kinetic theory of alpha transport by tokamak perturbations Expression Value 𝐶 ,𝑝 − 𝑣 𝐴 𝑣 𝐶 ,𝑝 𝑣 𝐴 √ 𝜖𝑣 ⎡⎣ − 𝑒 − 𝑣 ( √ 𝜖𝛺𝑝𝜋𝑅 +4 𝑛𝑞𝑣 ) 𝑛𝑞𝑣 𝛺𝑝𝑅𝑣𝐴 ⎤⎦ Table 8: Coefficients of harmonics of contributions to the flux used in Eq. (7.12).trapped particles, using (5.19), and − 𝜔𝑛𝜔 ⋆ is again to lowest order in 𝜖 . We set 𝜎 = − because, as discussed previously, this is the only significant contribution to the flux. Thenthe integral to be evaluated is 𝑄 𝛼,𝑚𝑛𝜔,𝑝 = − 𝑀 𝛼 𝑛 𝜋 𝑐 𝛷 𝑚𝑛𝜔 𝑞𝑅 𝑣 /𝑣 𝑐 ) 𝜕𝑛 𝛼 𝜕𝜓 ∑︁ 𝑙 ∫︁ 𝑘 𝑑𝑘 𝑘 (︁ 𝐾 ( 𝑘 ) 𝜋 + 𝑣 𝑟𝑒𝑠 √ 𝜖𝜆𝑘𝑣 𝐴 )︁ [(1 − 𝜖 ) 𝑘 + 2 𝜖 ] ⃒⃒⃒ 𝜕𝑄 𝑙 𝜕𝑣 ( 𝑣 𝑟𝑒𝑠 ) ⃒⃒⃒ , (7.10)where 𝑣 𝑟𝑒𝑠 is the resonant velocity at which 𝑄 𝑙 ( 𝑣 𝑟𝑒𝑠 , 𝑘 ) = 0 . The upper limit of integrationis the value of 𝑘 in resonance at the birth velocity 𝑣 , which can be found for each valueof 𝑙 using (5.19) and the passing quantities in table 3.For 𝑙 = 1 , a good approximation for 𝑘 is found from expanding the resonance conditionabout 𝑘 = 0 to 𝒪 (︀ 𝑘 )︀ . In contrast, for 𝑙 = 2 , a good approximation is found byexpanding about 𝑘 = 1 to 𝒪 [ln (1 − 𝑘 )] . This process gives, with 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) usedto simplify the expressions: 𝑘 = ⎧⎨⎩ √ 𝜖 √︁ 𝑣 𝑣 𝐴 − , 𝑙 = 11 − 𝑒 − 𝑣 ( √ 𝜖𝛺𝑝𝜋𝑅 +4 𝑛𝑞𝑣 ) 𝑛𝑞𝑣 𝛺𝑝𝑅𝑣𝐴 , 𝑙 > . (7.11)The integrand can again be expanded about 𝑘 = 0 to lowest non-trivial order in 𝑘 . Then,we find, where we have inserted 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) 𝑄 𝛼,𝑚𝑛𝜔,𝑝 ≈ − 𝑀 𝛼 𝑛 𝜋𝑣 𝑐 𝛷 𝑚𝑛𝜔 𝑞𝑅 ln ( 𝑣 /𝑣 𝑐 ) 𝜕𝑛 𝛼 𝜕𝜓 ∑︁ 𝑙 𝐶 𝑙,𝑝 , (7.12)where 𝐶 𝑙,𝑝 are parameters given in table 8. Again, we include only harmonics 𝑙 (cid:54) .Higher harmonics have small contribution to the flux. This expression, giving the passingparticle alpha heat flux from a TAE of a given magnitude, is another central result ofthis paper. For the example parameters in this paper, ∑︀ 𝑙 𝐶 𝑙,𝑝 ≈ . . The passing flux islarger than the trapped flux. This results from multiple factors. For instance, there aremore passing particles than trapped particles in the isotropic alpha population. Also, thepassing phase factor tends to be larger than the trapped phase factor.The expressions for the trapped, (7.8), and passing, (7.12), fluxes can be comparedto the phenomenological flux, (4.18). In particular, for the trapped flux, (7.8), setting 𝑛𝜔 𝛼 ⋆ ∼ 𝑛𝑣 𝑟𝑒𝑠 / (︀ 𝛺 𝑝 𝑅 )︀ and (︁ ⃗𝑣 ‖ ˆ 𝑏 𝑡𝑜𝑡 + ⃗𝑣 𝑑,𝑡𝑜𝑡 )︁ · ∇ 𝜓 ∼ 𝑛𝑐𝛷 𝑚𝑛𝜔 reproduces the dependen-cies seen in (7.8). For the passing flux, setting 𝑛𝜔 𝛼 ⋆ ∼ 𝜔 ≈ 𝑣 𝐴 / (2 𝑞𝑅 ) , 𝑣 𝑟𝑒𝑠 ∼ 𝑣 𝐴 ∼ 𝑣 and making the same replacement as for the trapped for the radial velocity reproducesthe dependencies seen in (7.12). As intuition suggests, the heat flux scales with the squareof the perturbation amplitude, 𝛷 𝑚𝑛𝜔 , and increases with the periodicity of the mode, 𝑛 .4 E. A. Tolman and P.J. Catto
Interpretation of flux and development of saturation condition
The magnitude of the alpha flux caused by the TAE is understood by comparing therate of the diffusion it causes to the rate of slowing down. Slowing down (described inappendix A) removes alpha particles from the energetic population and transfers theirenergy to the local bulk plasma, leaving the alpha particles as helium ash. So, with 𝐷 the diffusion coefficient describing the TAE flux and 𝑎 the device minor radius, if 𝐷𝜏 𝑠 𝑎 ≪ , (7.13)TAE diffusion is weak and does not remove alpha particles before they can give theirenergy to the background plasma and transition to ash. When 𝐷𝜏 𝑠 𝑎 ∼ , (7.14)TAE diffusion is very strong. It would be capable of diffusing alpha particles across thewhole poloidal tokamak cross section (with an area that scales as 𝑎 ) in a slowing downtime if mode activity of similar strength existed across that area. (Note that the width ofan individual mode is significantly smaller than 𝑎 .) At this level, alpha particles diffusesufficiently fast to cause significant modification to the alpha heat deposition profile. (Atthis point the perturbative approach used in this paper becomes inappropriate. For lossesat this level, a quasilinear treatment which retains the radial flux losses in (A 1) for 𝑓 isneeded.) The diffusion coefficient is related to the flux by 𝐷 ≡ − 𝑄 𝛼 𝑅 𝐵 𝑝 𝜕𝑛 𝛼 𝜕𝜓 𝑀 𝛼 𝑣 , (7.15)such that the condition for avoiding strong TAE diffusion, (7.13), becomes √ 𝜖𝛺 𝑝 𝜋𝑣 𝐴 𝑅 ∑︀ 𝑙 𝐶 𝑙,𝑡 𝑛𝑣 (︂ 𝐵 𝑚𝑛𝜔 𝐵 )︂ 𝜏 𝑠 𝑎 ≪ (7.16)for trapped particles and 𝜋𝑞𝑣 𝐴 𝑅 ∑︀ 𝑙 𝐶 𝑙,𝑝 𝑣 (︂ 𝐵 𝑚𝑛𝜔 𝐵 )︂ 𝜏 𝑠 𝑎 ≪ (7.17)for passing particles. Here, we have stated the flux in terms of the strength of theperturbed magnetic field, 𝐵 𝑚𝑛𝜔 /𝐵 ∼ 𝑛𝑐𝛷 𝑚𝑛𝜔 / ( 𝑣 𝐴 𝑅𝐵 𝑝 ) , which is more commonlyseen in the literature than 𝛷 𝑚𝑛𝜔 . In addition, we have used ln ( 𝑣 /𝑣 𝑐 ) ≈ . For fixedtokamak equilibrium parameters and fixed radial location 𝜖 , these conditions constrainthe maximum amplitude for the mode to avoid serious alpha depletion. We plot the valueof 𝐷𝜏 𝑠 /𝑎 as a function of the normalized mode amplitude, 𝐵 𝑚𝑛𝜔 /𝐵 , in figure 11.Understanding how this constraint compares to typical TAE amplitudes requires arough estimate for mode saturation. We now develop such an estimate. TAE saturationhas been estimated using a variety of methods. Some works balance the linear growthrate of the instability with the rate at which particles get trapped by the wave andflatten the gradient driving the instability, sometimes including collisions which restorethe original gradient (Wu & White 1994; Fu & Park 1995; Wang & Briguglio 2016;Zhou & White 2016; Todo 2019). Other works consider the effects of zonal flows andnonlinear mode couplings (Spong et al. rift kinetic theory of alpha transport by tokamak perturbations × − to × − .of this gradient. However, we consider the removal of the drive to be due to flattening,rather than due to particle trapping.The mechanism of saturation is represented schematically in figure 12. As the TAEgrows, the non-adiabatic particle response ℎ reduces the alpha gradient responsiblefor mode drive. However, collisions concurrently act to restore the original particlegradient by counteracting this flattening. The balance of these processes results in modesaturation. (If the original drive were not restored fast enough by collisions, ℎ and themode amplitude would shrink, while if it were restored faster than it was reduced, ℎ andmode amplitude would grow.) Mathematically, this balance is stated 𝑐𝑛𝑆 𝑚𝑛𝜔 𝜕ℎ 𝑚𝑛𝜔 𝜕𝜓 ∼ 𝜈 𝑝𝑎𝑠 𝜕 ℎ 𝑚𝑛𝜔 𝜕𝜆 . (7.18)The left side is the rate at which the drive for ℎ 𝑚𝑛𝜔 , 𝑐𝑛𝑆 𝑚𝑛𝜔 𝜕𝑓 /𝜕𝜓 , is reduced by ℎ 𝑚𝑛𝜔 .The right side, 𝜈 𝑝𝑎𝑠 𝜕 ℎ 𝑚𝑛𝜔 /𝜕𝜆 , is the rate at which the original drive is restored bycollisions.This saturation condition can be evaluated in two different limits. The first, lowcollisionality, limit occurs when 𝜕ℎ 𝑚𝑛𝜔 /𝜕𝜓 ∼ ( 𝜕ℎ 𝑚𝑛𝜔 /𝜕𝜅 ) ( 𝜕𝜅/𝜕𝜓 ) ∼ ℎ 𝑚𝑛𝜔 / (︀ 𝑅 𝐵 𝑝 𝛿𝜆 )︀ ,with 𝛿𝜆 given by (4.17). (This expression is for trapped particles; the equivalent pass-ing expression is in terms of 𝑘 .) The second, high collisionality, limit occurs when 𝜕ℎ 𝑚𝑛𝜔 /𝜕𝜓 ∼ 𝑘 𝜓 ℎ 𝑚𝑛𝜔 / ( 𝑅𝐵 𝑝 ) , with 𝑘 𝜓 given by the value for the TAE given in ta-ble 2. The transition between the two limits occurs when 𝑘 𝜓 ∼ / ( 𝑅𝛿𝜆 ) . Inserting 𝜕 ℎ 𝑚𝑛𝜔 /𝜕𝜆 ∼ ℎ 𝑚𝑛𝜔 / ( 𝛿𝜆 ) , with 𝛿𝜆 given by (4.17) with 𝑛𝜔 𝛼 ⋆ ∼ 𝜔 , and 𝑐𝑛𝑆 𝑚𝑛𝜔 ∼ E. A. Tolman and P.J. Catto
Figure 12: Schematic representation of TAE saturation. The non-adiabatic particleresponse ℎ reduces the alpha gradient, 𝜕𝑓 /𝜕𝜓 and the drive to the TAE. Collisions(represented by 𝜈 ) restore the original particle gradient. The balance of these processes,as expressed in (7.18), results in mode saturation. 𝑐𝑛𝛷 𝑚𝑛𝜔 ∼ 𝑣 𝐴 𝑅𝐵 𝑝 𝐵 /𝐵 reveals the saturated amplitude of the TAE: 𝐵 𝐵 = ⎧⎪⎨⎪⎩ 𝑅𝜈 / 𝑝𝑎𝑠 𝜔 / 𝑣 𝐴 , 𝜈 𝑝𝑎𝑠 𝜔 < (︁ 𝜖𝑛𝑞 )︁ 𝜖𝑅𝜈 / 𝑝𝑎𝑠 𝜔 / 𝑛𝑞𝑣 𝐴 𝜈 𝑝𝑎𝑠 𝜔 > (︁ 𝜖𝑛𝑞 )︁ . (7.19)Here, 𝜈 𝑝𝑎𝑠 ∼ 𝑣 𝜆 / (︀ 𝑣 𝜏 𝑠 )︀ [from the second term in the collision operator (2.9)]. Similarscalings of saturated amplitude with collisionality are found elsewhere in the literature(Berk et al. et al. et al. 𝐵 /𝐵 ≈ . × − . This saturation estimate is comfortably below thestochastic threshold given in Berk et al. (1992), making it consistent with our assumptionthroughout the paper that the TAE amplitude is below the threshold for stochasticity.Saturated AE amplitudes observed experimentally (Nazikian et al. et al. et al. et al. et al. − to − across a wide range of machine parameters and types of energetic particlepopulations; in extreme cases, amplitudes of − have even been found (Todo et al. . × − , is indicated on the plot of diffusion strength, figure 11, with a purple dash-dotted line, revealing that at this amplitude, 𝐷𝜏 𝑠 /𝑎 is small for both the trapped andthe passing populations. At this amplitude, significant redistribution of alphas due tothis TAE is not expected, though some localized alpha gradient flattening may occur.(In addition, synergistic action with overlapping modes of different 𝑚 and 𝑛 could changethe transport; since TAEs consist of coupled poloidal harmonics, this change is likely to rift kinetic theory of alpha transport by tokamak perturbations − − − range commonly seen in the literature, could leadto significant redistribution. This significant activity is especially likely for the passingpopulation.
8. Conclusion
This work has developed the drift kinetic theory of transport that results from reso-nance between alpha particle motion and a perturbation to the tokamak magnetic andelectric fields. The theory allows the calculation of the alpha particle heat flux of trappedand passing particles that results from these perturbations, (5.17) and (5.18). Theseexpressions are examined for two specific perturbations: ripple and TAE. For ripple, thismechanism causes negligible alpha particle transport, as discussed in section 6. For TAE,the mechanism causes significant heat flux [given by (7.8) for trapped particles and (7.12)for passing particles] which allows the development of constraints on TAE amplitude toavoid significant depletion of the alpha population [given by (7.16) for trapped particlesand (7.17) for passing particles]. A simple saturation condition suggests that the TAEamplitude in a tokamak scenario similar to that which might occur in the next generationtokamak SPARC is below this limit. This is a reassuring conclusion because significantalpha transport due to TAEs has the potential to seriously impair tokamak operation.However, the presence of other mechanisms for TAE transport [such as the removal ofalphas due to their being born in stochastic regions as considered further in Sigmar et al. (1992) and Collins et al. (2016)] and the precedent for larger saturation amplitude in theexperimental and analytical literature suggest that caution is still important.Our expressions for the radial heat transport are independent of the collision frequencyeven though each wave-particle resonance path in velocity space is at the center of anarrow collisional boundary layer. Indeed, our resonant plateau model is reminiscent ofquasilinear treatments in the sense that the narrow collisional boundary layer results inand gives meaning to the delta function behavior that removes the collision frequencywhen the alpha heat flux is evaluated [see the diffusive limit in Duarte et al. (2019)and the pitch angle scattering treatment of Catto (2020)]. Our expressions for the alphaheat fluxes are most appropriate when the phase space islands about the wave-particleresonance curves do not overlap. However, the expressions may remain reasonable evenas the islands begin to overlap provided the TAE amplitude remains below the thresholdfor full stochasticity.Avenues for future work based on this paper fall into three categories: analytical,numerical, and experimental. On the analytical front, future work should adapt andimprove the formalism developed in this paper so that it can be used to study otherspecies of energetic particle, which have more complicated distribution functions than thealpha distribution function. This will enable comparison of this model’s predictions tocurrent experiments without alpha particles. In addition, analytical work could focus onthe calculation of the flux for perturbations for which the approximation 𝑞 ≈ 𝑞 ⋆ , discussedin section 5.3, is not possible. This includes the neoclassical tearing mode (La Haye 2006),for which 𝑛𝑞 ∼ 𝑚 , making it necessary to avoid the condition in (5.13). Evaluation of theflux from such perturbations requires modified techniques. Finally, additional analyticalwork should consider adaptation of the techniques developed in this paper for use witha more precise collision operator than the Krook operator. This would allow the studyof resonances, like some of the ripple resonances discussed in section 6, which are spacedtoo closely for the Krook operator to be appropriate.On the numerical front, effort should be made to determine the relative importance8 E. A. Tolman and P.J. Catto of the transport mechanisms studied in this paper and those that are the focus ofother works on energetic particle transport. Other mechanisms include overlapping phasespace islands from different resonances, the transfer of alpha particles across topologicalboundaries (for example, the trapped-passing boundary), and convective transport whichoccurs when an alpha particle near the edge stays in phase with a perturbation withoutscattering until it leaves the device (Heidbrink 2008). Simulations of alpha transport dueto tokamak perturbations using the orbit following codes discussed in the introduction canbe examined to distinguish transport resulting from these mechanisms. Such simulationsshould take care to properly resolve the boundary layer around the resonant velocityand pitch angles. We note that collisional boundary layer behavior was recently observedin White et al. (2019), but was attributed to an alternate mechanism.On the experimental front, the predictions in this paper should be validated againstestimates of alpha particle flux from measurements of actual experiments. Such validationwill be possible in next generation experiments which are dominantly alpha heated,like SPARC and ITER, and may be possible in "afterglow" scenarios of experimentswith some alpha heating, like the forthcoming JET DT campaign. [Afterglow scenarios,discussed in Sharapov et al. (1999), occur when external heating is suddenly turnedoff in order to transiently observe alpha particle effects.] If the formalism in this paperis adapted to distributions typical of energetic particles resulting from ICRF or beams,which are common in current day experiments, precise comparison to current experimentsis possible. Already, current experimental measurements can be examined for evidenceof general trends suggested by our work. For instance, part of the energetic particleflux resulting from energetic particle mode perturbations has been nicely demonstratedto scale with mode amplitude squared in Nagaoka et al. (2008). This agrees with ourpredictions.The authors thank Steven Scott, Ryan Sweeney, Abhay Ram, Ian Abel, Ian Hutchinson,Eero Hirvijoki, Antti Snicker, Konsta Särkimäki, Paulo Rodrigues, Earl Marmar, FelixParra, Iván Calvo, Vinícius Duarte, Lucio Milanese, and Martin Greenwald for helpfulconversations. The authors are especially grateful to Nuno Loureiro for suggesting P.J.C.become involved in the problem and for several insightful suggestions. The authors ac-knowledge support from the National Science Foundation Graduate Research Fellowshipunder Grant No. DGE-1122374, support from US Department of Energy award DE-FG02-91ER54109, and support from the Bezos Membership at the Institute for AdvancedStudy.
Appendix A. Derivation of equilibrium alpha particle slowing downdistribution
The equilibrium energetic alpha distribution, 𝑓 , is determined by (2.8) in the unper-turbed magnetic field (2.2): (︁ 𝑣 ‖ ˆ 𝑏 + ⃗𝑣 𝑑 )︁ · ∇ 𝑓 = 𝐶 { 𝑓 } + 𝑆 𝑓𝑢𝑠 𝛿 ( 𝑣 − 𝑣 )4 𝜋𝑣 . (A 1)This expression is solved by separating its terms by size and solving for the correspondingsize terms in the distribution function, with 𝑓 = 𝑓 + 𝑓 + ... . The largest term in theunperturbed drift kinetic equation (A 1) is the streaming term, so that 𝑣 ‖ ˆ 𝑏 · ∇ 𝑓 = 0 . (A 2) rift kinetic theory of alpha transport by tokamak perturbations 𝑓 is a flux function, i.e., 𝑓 = 𝑓 ( 𝜓, 𝑣, 𝜆 ) . (A 3)The next order expression includes the drift term (which can be shown to be smallcompared to the streaming term by the poloidal alpha gyroradius over the perpendicularscale length of 𝑓 ), the collision operator (which is small by the connection length 𝑞𝑅 over the alpha mean free path), and the source term (which must be the same size asthe collision operator for alpha particles to slow down before they are lost due to otherprocesses). This expression reads 𝑣 ‖ ˆ 𝑏 · ∇ 𝑓 + ⃗𝑣 𝑑 · ∇ 𝑓 = 𝐶 {︀ 𝑓 }︀ + 𝑆 𝑓𝑢𝑠 𝛿 ( 𝑣 − 𝑣 )4 𝜋𝑣 . (A 4)Because 𝑓 spatially depends only on 𝜓 , we have that ⃗𝑣 𝑑 · ∇ 𝑓 = ⃗𝑣 𝑑 · ∇ 𝜓 𝜕𝑓 𝜕𝜓 = 𝑣 ‖ ˆ 𝑏 · ∇ (︂ 𝐼𝑣 ‖ 𝛺 )︂ 𝜕𝑓 𝜕𝜓 , (A 5)where the final expression can be derived as explained in equation (8.14) of Helander &Sigmar (2005).We define the variable 𝜏 characterizing the progression of particles along the magneticfield, 𝑑𝜏 ≡ 𝑑𝑙𝑣 ‖ = 𝑑𝜗𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 ( 𝜏 ) , (A 6)with 𝑑𝑙 a unit of distance along the magnetic field. Integration over 𝜏 over a closed orbitannihilates the left side of (A 4), giving ∮︁ 𝑑𝜗𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 𝑣 ‖ ˆ 𝑏 · ∇ (︂ 𝑓 + 𝐼𝑣 ‖ 𝛺 𝜕𝑓 𝜕𝜓 )︂ = 0 = ∮︁ 𝑑𝜗𝑣 ‖ ˆ 𝑏 · ∇ 𝜗 [︂ 𝐶 {︀ 𝑓 }︀ + 𝑆 𝑓𝑢𝑠 𝛿 ( 𝑣 − 𝑣 )4 𝜋𝑣 ]︂ . (A 7)Because 𝑓 is not a function of 𝜗 [recall (A 3)], the integrand on the right side mustbe identically zero. Enforcing this condition using (2.9) (note that alphas are bornisotropically, so only the first term in 𝐶 is important here) gives the slowing downdistribution, 𝑓 ( 𝜓, 𝑣 ) = 𝑆 𝑓𝑢𝑠 ( 𝜓 ) 𝜏 𝑠 ( 𝜓 ) 𝐻 ( 𝑣 − 𝑣 )4 𝜋 [ 𝑣 + 𝑣 𝑐 ( 𝜓 )] . (A 8)Insertion of the slowing down distribution into (A 4) gives that 𝑣 ‖ ˆ 𝑏 · ∇ (︂ 𝑓 + 𝐼𝑣 ‖ 𝛺 𝜕𝑓 𝜕𝜓 )︂ = 0 , (A 9)which shows that 𝑓 + (︀ 𝐼𝑣 ‖ /𝛺 )︀ 𝜕𝑓 /𝜕𝜓 is a flux function. Evaluation of the drift kineticequation to the next order allows the derivation of neoclassical alpha transport drivenby the term (︀ 𝐼𝑣 ‖ /𝛺 )︀ 𝜕𝑓 /𝜕𝜓 , as discussed in Catto (2018). However, in this paper wefocus on transport caused by perturbations and take that 𝑓 = 𝑓 . This means that weassume 𝐼𝑣 ‖ 𝛺 𝑓 𝜕𝑓 𝜕𝜓 ≪ , (A 10)which is equivalent to stating 𝜌 𝑝𝛼 /𝑎 𝛼 ≪ , with 𝑎 𝛼 the alpha particle scale length definedin (2.7).0 E. A. Tolman and P.J. Catto
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