Alfvénic fluctuations in burning plasmas have typically low amplitudes, ε_δ≡|δB_⊥|/B₀≤ 5×10⁻⁴, where the ⊥ subscript stands for perpendicular to the equilibrium magnetic field direction b≡B₀/B₀. Therefore,resonant EPs, more than non resonant ones,are expected to play crucial roles in transport processes¹⁰. Transport equations in the phase space can be generally obtained by taking a proper flux surface average of the nonlinea rgyrokinetic equation^8,10. Here, in particular,we consider transport processes in two-dimensional (2D) magnetized plasma equilibria^38,39, and will generalize the existing approach to 3Dstellarator geometry. By this procedure, we will derive the nonlinear evolution equation of PSZS (cf. Secs. 1,2)^8,10; that is, we will obtain explicit expressions of fluxes (particle and energy) in the phase-space^38,39, and generalize them to include a suitable bounce averaged collision operator⁴² in the presence of sources and sinks. While bounce and flux surface averaged nonlinear gyrokinetic equation exists in the literature³⁷,explicit expressions of phase space fluxes that can be used for actual computations and comparisons with nonlinear simulations are not provided. Providing such explicit expressions, valid in general geometry, is the first objective of WP1; meanwhile, the evolution equations for the fluctuation spectrum will be provided self-consistently. 

The next objective of WP1 is providing the theoretical framework for the various levels of approximation to be adopted, based on the former general description. For collective SAW fluctuations excited by EPs as well as DWT, the mode structure can be decomposed in Fourier series in toroidal direction, labeled by the toroidal mode number n. While in tokamaks the different n’s are linearly independent, in stellarators they are coupled. For each toroidal mode number n, the Fourier components can be further decomposed as a parallel (tob) mode structure, e.g. δφ̂_n(r, θ) for the scalar potential, and a radial envelope A_n(r) (time dependence is implicitly assumed). Correspondingly, nonlinear interactions can take the following three forms: mode coupling between twons, modulation of the radial envelope; and distortion of the parallel mode structure^8–10. These three processes also have their own characteristic spatiotemporal scales.The parallel mode structure is typically formed on a time scale proportional to the inverse characteristic frequency |ω|⁻¹; while distortion of the radial envelope and coupling between two ns takes place on a nonlinear time scale,τ_NL, which,for Alfvénic fluctuations, can be assumed of the order of the inverse linear growth rate of the considered instability, γ⁻¹.The three levels of approximation are obtainedasfollows.

• The first level of approximation assumes a weak amplitude expansion, based on the ordering ε_δ≡|δB_⊥|/B₀≤ 5×10⁻⁴ introduced above. This will result in a weak turbulence reduction of the general framework.
• Since, typically, τ_NL ∼γ⁻¹∼|ω|⁻¹, the parallel mode structure remains essentially unmodified on the shortest time scale. At the second level of approximation, thus, the parallel mode structure is set by the linear theory, while nonlinear dynamics canbe under stood as coupling of the remaining relevant degrees of freedom [n and A_n(r)] on a time scale τ_NL∼γ⁻¹.
• The third level of approximation assumes that the fluctuation spectrum evolves neglecting the coupling among different ns; thus, nonlinearity is all accounted for by the EP transport processes in the phase. This, in the limit of multiple overlapping resonances and short de/auto-correlation time,will reduce to the usual quasi-linear description, but generally has the capability of addressing phase-locking and phase-bunching, which are important processes in nonlinear dynamics of Alfvénic fluctuations and EP transport¹⁰.

Milestones:
WP1-M1 - Derivation of phase space transport equations in 2D tokamak equilibria, including sources and collisions, and taking into account the multi-level approximation approach (Dec. 2019).
WP1-M2 - Derivation of nonlinear evolution equations for Alfvénic fluctuations in 2D tokamak equilibria taking into account the multi-level approximation approach (Dec. 2019).
WP1-M3 - Derivation of phase space transport equations in 3D equilibria, extending WP1-M1 (Dec. 2020).
WP1-M4 - Derivation of nonlinear evolution equations for Alfvénic fluctuations in 3D equilibria, extending WP1-M2 (Dec. 2020).

Deliverables:
WP1-D1 - Explicit expressions of EP fluxes in phase space as input to transport code (2019).

Bibliography