Wave–particle interactions in tokamaks

Transport consequences of the wave–particle interactions in the quasilinear plateau (QP) regime are presented. Eulerian approach is adopted to solve the drift kinetic equation that includes the physics of the nonlinear trapping (NT) and QP regimes. The localization of the perturbed distribution simplifies the test particle collision operator. It is shown that a mirror force like term responsible for the flattening of the distribution in the NT regime is subdominant in the QP regime, and controls the transition between these two regimes. Transport fluxes, flux-power relation, and nonlinear damping or growth rate are all calculated. There is no explicit collision frequency dependence in these quantities; however, the width of the resonance does. Formulas that join the asymptotic results of these two regimes to facilitate thermal and energetic particle transport, and nonlinear wave evolution of a single mode are presented.


Introduction
Wave-particle interactions are pervasive in tokamaks.Because plasmas are confined in a bounded space, particle orbits must be periodic.A plethora of characteristic frequencies emerges, and provides ample opportunities for resonating with waves [1][2][3][4][5][6][7][8][9][10][11][12][13].Since confining fusion-born energetic particles is a must, the relevance of the effects of the thermal particles' finite gyro-radius to the physics of fusion grade tokamak plasmas is much reduced for a simple reason that ρ i /a ∼ 10 −3 , where ρ i is the gyro-radius of the fuel ions, and a is the minor radius.The effects of the finite banana width on the physics of the drift kinetic equation become important.When the wave frequency is of the order of the drift frequency, the physics is governed by the orbit averaged kinetic equation; analytic expressions for transport fluxes including the finite banana width effects Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
for both electrostatic and electromagnetic perturbations are obtained [14,15].In the bounce-transit frequency range, an Eulerian approach has been developed to treat the complications resulting from the radial drift of the equilibrium trapped and circulating particles, and from the poloidal mode coupling [15][16][17][18].The fundamental thrust of the approach is to recast the drift kinetic equation in a set of unconventional independent variables.Specifically, the radial coordinate is chosen to be p ζ , the toroidal component of the canonical momentum, together with a new set of the angle like variables.Several hitherto unknown results are obtained [17].It is demonstrated that the distribution function is flattened in the nonlinearly trapped region.It is also shown that there exists a relation, the flux-power relation, between a part of the particle flux, and the energy transfer rate caused by the wave-particle interactions.In addition, a new transport scaling insensitive to the equilibrium magnetic field strength that sets an upper limit on the normalized perturbed radial magnetic field strength to Br /B ∼ 10 −4 so as not to exceed neoclassical losses is obtained [17].
The theory is now extended to the quasilinear plateau (QP) regime with the emphasis on the mechanism for the flattening of the distribution.The Eulerian approach is adopted to solve the drift kinetic equation so that the poloidal mode coupling term is expunged from the equation to make analytical treatment possible.To obtain an accurate description of the distribution the test particle collision operator is adopted; and the results qualitatively differ from those obtained from the Krook operator.As is well known, the transport fluxes do not depend on the collision frequency in this regime.However, the resonant width in the phase space does, and depends on the details of the collision operator.It should be noted that in terms of the theory of Landau damping, QP regime is analogous to the linear phase of that theory.It is shown that the mechanism responsible for the flattening is the mirror-like force that pulls nonlinearly trapped particles back from the turning points.In the QP regime, the mirror-like force is subdominant, and thus the flattening of the distribution does not occur.In analogy to the theory of Landau damping, the flattening of the distribution occurs only in the nonlinear phase [17].Thus, the mechanism for the flattening could be the phase space diffusion.This should shed light on some of the nonlinear phenomena in wave-particle interactions.The demarcation between the nonlinear trapping (NT) and QP regimes in the collision frequency domain is also obtained.
The energy exchange rate between waves and particles can be either sign, i.e. the waves can either grow or damp, depending on the slopes of the distribution in the (E, p ζ ) space [17].
Here, E = v 2 /2 + eΦ /M is the particle energy, v is the particle speed, e is the charge, M is the mass, and Φ is the electrostatic potential including both the equilibrium and wave potential.Thus, not only the slope of the distribution in the energy E coordinate matters, but also that in the p ζ coordinate.The fluxpower relation follows naturally from the constraint relation between E and p ζ implied by the drift kinetic equation.
To facilitate the modeling of the transport losses of the energetic and thermal particles, and the modeling of the nonlinear evolution of the waves, formulas for the relevant physics quantities that join the asymptotic limits of the NT and QP regimes are constructed.
The rest of the paper is organized as follows.Magnetic coordinates and field representations are given in section 2. In section 3, results of the NT regime are summarized.The drift kinetic equation is solved in the QP regime in section 4. It is demonstrated that the mirror force term is subdominant in this regime.The transport fluxes, wave-particle energy exchange rate and the damping or growth rate are all calculated.In section 5, formulas that connect the asymptotic limits of the NT regime and the QP regime are presented to facilitate the modeling of the energetic and thermal particle transport, and of the nonlinear wave evolution.Concluding remarks are given in section 6.

Magnetic coordinates and field representations
We adopt Hamada coordinates in the development of the theory.In these coordinates, the magnetic field B is commonly expressed as [19] where prime denotes d/dV, V is the volume enclosed by a magnetic surface divided by 4π 2 , 2π ψ is the toroidal flux, 2π χ is the poloidal flux, θ is the poloidal angle, ζ is the toroidal angle, ζ 0 = qθ−ζ is the field line label, and q = ψ ′ /χ ′ is the safety factor.The corresponding covariant representation is [20] where G and F are, respectively, the toroidal current inside and poloidal current outside a magnetic flux surface divided by c/2, and c is the speed of light.The function φ satisfies , where angular brackets denote the flux surface average.We use the Fourier components of the electrostatic potential ϕ , and the parallel component of the vector potential A ∥ for the perturbed wave fields.For a (m, n) mode, where ω is the mode frequency, the subscript m and n indicate the poloidal and toroidal mode number respectively.The perturbed magnetic field It has been shown that Eulerian approach can be employed to solve drift kinetic equation including the effects of finite banana width in the radial, poloidal and toroidal directions in the sub-cyclotron frequency range.This is accomplished by choosing a set of independent variables: t, p ζ , η, ζ 0 , E, µ , i.e. the radial variable is p ζ , the poloidal angle is η which normalizes the conventional poloidal angle θ for the equilibrium trapped and circulating particles, and µ = v 2 ⊥ / (2B) with v ⊥ , the perpendicular particle speed.The new toroidal angle ζ 0 is chosen to remove the poloidal angle dependence in the toroidal drift frequency so as to expunge the poloidal mode coupling in the drift kinetic equation when used in conjunction with η.Explicitly [17], where Ω = eB/ (Mc) is the gyro-frequency, ⟨•⟩ b,t denotes the bounce or transit average over the equilibrium particle orbits, v db • ∇ζ 0 includes the equilibrium and perturbed drifts and is The indefinite integral in equation ( 4) is well defined because the secular terms are subtracted and their contributions are made explicit in equation ( 4).The averaged toroidal drift fre-quency for the equilibrium quantities can be found in [21].The definition for η is [17] for the equilibrium trapped particles, and [17] for the equilibrium circulating particles, where the subscript m indicates the quantities are evaluated at the minimum of B, and θ t1 and θ t2 are tuning points of the equilibrium trapped particles.The corresponding bounce frequency ω b and transit frequency ω t are, respectively, For a (m, n) mode, there are many (l, n) modes in terms of the new set of the variables t, p ζ , η, ζ 0 , E, µ , where l is an integer.Only one particular (l, n) mode, i.e.only a pair of ϕ ln , A ∥ln is treated here, i.e.
where the wave phase y = ωt + [l − δ t nq (p ζ )] η + nζ 0 , and δ t = 1 for equilibrium circulating particles and 0 for equilibrium trapped particles.The wave amplitudes in equation ( 8) can also include the effects of the finite orbit width of equilibrium particles in the radial, toroidal and poloidal directions.In the QP regime, modes are independent from each other and results are additive.In the NT regime, it is assumed that modes do not overlap.When overlapping occurs, orbits become chaotic and QP regime persists.

Summary of the results in the NT regime
When the effective collision frequency is smaller than the bounce frequency of the nonlinearly trapped particles, NT in the phase space becomes relevant.The NT occurs in the vicinity of the resonances.The resonance conditions derived from the stagnation point of the wave phase are, respectively, for the equilibrium circulating particles, and for the equilibrium trapped particles, where σ is the sign of the parallel particle speed, and is the toroidal drift frequency.All the characteristic frequencies of the particle motion in equations ( 9) and ( 10) are properly averaged and depend neither on the poloidal nor on the toroidal angle variables.The results of the NT regime are summarized as follows.

Flattening of the distribution in the nonlinearly trapped regime
The frequency ω = ω + [l − δ t nq (p ζ )] σω b,t + nω d plays a role here similar to the parallel particle speed v ∥ in the neoclassical theory, where σ = 1 for ω b .In the vicinity of the resonance where ω ≈ 0, ω can be expressed in a pendulum form where σ ω = ±1 indicating the sign of ω, ω is the typical magnitude of ω of the nonlinearly trapped particles and is the pitch angle like parameter k2 that characterizes the NT state is For nonlinearly trapped particles 0 < k2 < 1, and for nonlinearly circulating particles k2 > 1.
The distribution function f in the nonlinearly trapped region is where f 0 is the equilibrium distribution, (p ζ0 , E 0 ) are the reference coordinates over which transport fluxes are calculated, h is ν eff is the effective collision frequency of the nonlinearly trapped particles, and ω b,nl is the bounce frequency of the nonlinearly trapped particles.For a given µ, ω0 is ω evaluated at (p ζ0 , E 0 ).In this regime, ν eff /ω b,nl ≪ 1, and terms of that order are neglected in equation ( 14).

Flux-power relation and nonlinear damping or growth rate
It has been shown using the explicit expressions for Ė and .
p ζ for a single n mode, the rate of the energy transfer in the waveparticle interactions is proportional to the particle flux driven by the components that break toroidal symmetry, i.e. [17], where )/ (8π )⟩, the angular brackets denote the period, radial and magnetic surface averaged wave energy, E is the wave electric field, B is the wave magnetic field, Γ an,ζ is defined as [22] Γ N is plasma density, e ζ = ∇V × ∇θ, and V ∥ is the parallel mass flow.The parallel gradients of the perturbed fields to Γ an is neglected in the definition of the flux-power relation here for its straightforwardness.It is obvious that flux-power relation shown in equation ( 16) is valid for all collisionality regimes.The nonlinear damping or growth rate γ NL is [17] which can be used to model the nonlinear evolution of a single mode.

A new scaling of the transport coefficients
The transport fluxes in the NT regime are [17] where q an,ζ is the radial heat flux driven by the components that break toroidal symmetry, T is the temperature, x = v/v t , ν D is the deflection frequency, ω2 ν is the second derivatives of ω with respective to the dimensionless velocity space variables, and I k is a numerical number resulting from an integral over the variable k.
One of the notable results in equation ( 19) is new transport coefficient scalings.For example, for magnetic perturbations, the heat conductivity χ Br scales as [17] where s is the magnetic field shear parameter, ε is the inverse aspect ratio, and r is the local minor radius.This scaling results from the equilibrium circulating particles that are in resonance; this is basically the Landau theory in tokamaks.It should be noted that the scaling shown in equation ( 20) depends neither on the gyro-radius nor on the poloidal gyro-radius; thus, increasing the equilibrium magnetic field strength will not reduce the energy loss rate.

Quasilinear plateau regime
We employ the drift kinetic equation that is used to calculate the physics consequences of the NT for the QP regime when plasmas are more collisional.This allows us to examine the transition between these two regimes.In the process, we illustrate that a mirror force like term that is responsible for the flattening of the distribution is subdominant in this regime.Thus, the mechanism for flattening the distribution can be the phase space diffusion.The relevant drift kinetic equation is [17] ω ∂f 1 ∂y where f 1 is the perturbed distribution resulting from the wave fields, and C (f 1 ) denotes collision operator.As noted in [17], even though f 1 < f 0 , the slopes of which in the phase space are comparable to those of the equilibrium.
It follows that for a single mode where H is a constant of motion; the constraint is dictated by the drift kinetic equation under the approximation shown in [17], and is consistent with that in [22,23].Here, we remark that the relation between Ė and .p ζ has been employed and reflected in the thermodynamic forces in the quasilinear theory for a Maxwellian equilibrium distribution [22].Another constant of motion Z for the NT in the vicinity of the resonance is In terms of (Z, H, y, µ), equation (21) becomes That equation is employed to calculate the physics consequences in the NT regime.Here, we use the same equation to calculate physics quantities in the QP regime.The independent variables for the QP regime are (Z, H, y, ω) and equation ( 25) becomes The second term on the left of equation ( 26) is mirror force like that is responsible for the NT, and can flatten the distribution.The localization of f 1 simplifies the collision operator to [14,24] where ω2 ν is defined in [14,15].In the QP regime, the mirror force like term is subdominant because which also marks the transition from the NT regime to the QP regime as collision frequency increases.Thus, equation ( 26) reduces to in this regime.The width of the resonance layer can be estimated to be which cannot be derived from the Krook model, and thus shows the inadequacy of the approach of the integration along the unperturbed orbit.The upper bound of the QP regime in the collision frequency domain is The solution to equation ( 29) is where π Hi (z) = ´∞ 0 dtexp zt − t 3 /3 , and Z± = ∓iω/ ν D ω2 ν 1/3 [25].The solution in equation ( 32) satisfies the condition that f 1 → 0 as |z ± | → ∞.The perturbed distribution f 1 is even relative to ω = 0, and thus the distribution is not flattened in the vicinity of the resonance.The reason is that the mirror force like term is subdominant in this regime.It must rely on other mechanisms, such as phase space diffusion, to flatten the distribution.
The transport fluxes are The energy integral is performed over the resonance domain.
The loss mechanism in this regime is similar to numerical results obtained in [26,27] in which collisions are not involved.Employing flux-power relation shown in equation ( 16), we obtain the power transfer between wave and particles during the interactions.The nonlinear damping or growth rate can also be calculated using equation (18) for a given resonance.When ∂f 0 /∂E > 0, the wave amplitude will grow; otherwise, it will be damped.This is consistent with the theory of Landau damping.The slope of ∂f 0 /∂p ζ can either stabilize or de-stabilize the wave depending on the charge.In addition, the ambipolarity also plays a role.

Approximate formula for modeling
The results in the NT and QP regimes can be cast into approximate expressions that reproduce the proper asymptotic limits, shown in equations ( 19) and (33), in the respective regimes and they are where κ QP = π ν D ω2 ν I k , and κ NT = π 2 ω3 /32.Symbolically, the expressions in equation ( 34) is similar to those in [18]; the details are not of course.The transition between the NT and QP regimes indicated in equation (34) in the collision frequency domain is shown schematically in figure 1.The nonresonant and collisional regimes are also shown.In the nonresonant regime, ω ̸ = 0 and the fluxes are proportional to collision frequency.When the collision frequency dominates over all other frequencies, plasmas are in the collisional regime and fluxes scale inversely proportional to the collisional frequency.The formulas can be used to model the thermal and energetic particle transport losses and the nonlinear evolution of a single mode in tokamaks.

Discussions and concluding remarks
The physics addressed is by no means panacean.The transport losses associated with the phase space zonal structures [28] are not included.Our results are limited to the regime where γ NL < ω b,nl ; other relevant physics has been discussed in [29].The effects of the background turbulence [30,31] are not included either.The physics in a kick model presented in [32] is qualitatively similar to that in the QP regime in our theory.
Several versions of the collision operator have been employed in the context of the wave-particle interactions in tokamaks.Pitch angle scattering operator, which is part of the test particle operator, has been used to study the effects of collisions on toroidal Alfven eigenmodes [33].A model operator similar to the one in [34] is used to investigate the δf algorithm for simulations [35].A test particle operator in (p ζ E, µ) coordinates, that are relevant to our theory, has been implemented numerically [36] to model equilibrium distribution function.
We have summarized the results in the NT regime and developed transport consequences in the QP regime when the collisions become more frequent.It is shown that in the QP regime, the mirror force like term is subdominant, i.e. the perturbed distribution is even relative to ω, and thus it cannot flatten the distribution; other mechanisms such as phase space diffusion could be responsible for the flattening.The transport fluxes, flux-power relation, and nonlinear damping and growth rate are calculated.These quantities do not depend on the collision frequency.However, the width of the resonance does.When resonance width is wider than the mode separation in the frequency space, e.g.
transport losses will be greatly enhanced resulting from the resonance overlapping.Here, ω ln is the mode frequency in resonance with a wave with mode numbers (l, n).We also like to emphasize that flux-power relation is grounded on the constraint implied by the drift kinetic equation and is valid regardless of the collision frequency.
The nonlinear damping or growth rate depends not only on the slope ∂f 0 /∂E but also on the slope of ∂f 0 /∂p ζ ; the former is expected from the theory of Landau damping but the latter is beyond the scope of that theory.The physics implied will be investigated further elsewhere.
Effective collision frequency ν eff in the QP regime is also of interest.Using the collisional resonance width, we can estimate ν eff to be where ω here is the typical value and not the resonance value.This result cannot be obtained using a Krook model.The transition between NT and QP regimes is similar to the banana-plateau transition in the neoclassical theory [37][38][39].Thus, we again demonstrate the unification of the waveparticle interactions and neoclassical theory.The underlying physics and the mathematical procedure of resolving the resonance are the same.The unification is actually obvious.In terms of the physics of Landau damping, the NT regime corresponds to the NT of an electrostatic wave, and QP regime is similar to the linear Landau damping where the collision frequency does not appear.Some aspects of the theory have been observed in experiments.The Br /B 2 scaling in the transport fluxes in the QP regime has been seen in [40].There is also an observation of the Br /B scaling in the NT regime in [41].We must note, however, that the details of the results in the NT regime remain to be observed experimentally.
We also construct formulas that join the asymptotic limits of the NT and QP regimes to facilitate modeling of the thermal and energetic particle transport, and nonlinear evolution of a single mode in tokamaks.The transition between these two regimes depends on the parameters and occurs naturally.The formula can be employed to model energetic particle losses in ITER.
The theory will be extended to allow for compressible magnetic perturbations in a separate article.

Figure 1 .
Figure 1.Schematic collision frequency dependence of Γ an,ζ in log-log scale.When the resonances overlap in the nonlinear trapping regime (NT), nonlinear orbits become stochastic and quasilinear plateau regime (QP) scaling persists, which is indicated as a horizontal dotted line.The non-resonant transport, which is indicated as ν and ν −1 , is also shown in the high collision frequency domain.