This work package has two distinctive components: one (WP2.1) focused on practical applications and calculations based on the theoretical framework of WP1; and another one (WP2.2) devoted to a more speculative analysis of EP transport.

WP2.1 Numerical implementation of PSZS transport equations: The numerical solution of PSZS transport equations is 1D in space (flux surface average) and 2D in velocity space (nonlinear gyrokinetic) and is referred to, here, as general PSZS transport module. Its implementation in general geometry is the first objective of WP2.1. The novel aspect of this sub-work-package is the implementation of the explicit form of fluxes in the EP phase space and of corresponding evolution of the fluctuation spectrum, with the various level of approximations discussed in WP1. This analysis will illuminate the underlying physics and produce a hierarchy of validated reduced models.
As a starting point, the PSZS transport module will be solved assuming that the fluctuation spectrum and perturbed EP particle distribution function are given numerically (cf. WP3). This step can be considered as verification of the PSZS transport module, as it will eventually show that the explicit EP fluxes computed in this way are consistent with the evolution of the flux surface averaged EP distribution function.
The next step will be the verification of the small amplitude expansion in the explicit expression of EP fluxes (first level approximation in WP1). At this level, only the fluctuation spectrum will be assumed from numerical simulations, while the perturbed particle distribution will be calculated analytically. Both equations for the parallel mode structure, e.g. δφ̂_n(r, θ), and radial envelope A_n(r) are fully nonlinear (time dependence is implicitly assumed) and could be used for an independent numerical evaluation but, at this stage of the proposed multi-level approach, it is outside our intended scope.
The further step (second level approximation in WP1) is based on the assumption that the parallel mode structures can be computed using linear theory. Therefore, the explicit expression of EP fluxes will be calculated from the radial envelope A_n(t, r) only. This can be either provided numerically or by solution of the nonlinear envelope equations, projected over the linear parallel mode structures used as basis functions. For the computation of the parallel mode structure, we will adopt the AWECS code⁴³, an open source code that will be adapted to general geometry. To this aim, we will leverage the international partnership with the QST Rokkasho Fusion Institute, Aomori JP, in the person of Andreas Bierwage, one of the developers of the AWECS code. Meanwhile, for the computation of the ''matrix coefficients'' involved in the nonlinear envelope equations, we will work in synergy with the ongoing Drift Alfvén Energetic Particle Stability (DAEPS) project, funded by the National Magnetic Confinement Fusion Energy Research Program in China, and, in particular, with some of the key researchers involved in that project, Prof. L. Chen, Prof. Z. Qiu and Prof. Y. Xiao of the Institute for Fusion Theory and Simulation (IFTS) at Zhejiang University (ZJU).
The final step (third level approximation in WP1) will be neglecting the non-diagonal matrix coefficients involved in the nonlinear envelope equations. As discussed in WP1, this method is more general than quasilinear description, but admits it as a limit; and will allow verifying the conditions under which EP transport can be correctly described in this way. More

WP2.2 Transport analysis: The aim is to understand the relaxation dynamics of beam-plasma systems beyond the oft-used quasi-linear approximation. Here, we reconsider previous findings about the ballistic-like relaxation of beam particles³¹ in the view of nonlinear wave-particle interaction with a focus on the asymptotic properties of the transport. We will investigate the conditions motivating the possible breakdown of the diffusive style paradigm and a pathway to non-Gaussian, non-diffusive transport in velocity space after the transient relaxation processes have been accomplished, abandoning the familiar quasi-linear picture of velocity-space diffusion on the asymptotic time scales. The dynamical and basic physics conditions permitting this for beam-plasma systems are presently not clear. To solve this conundrum, we would assess two competitive approaches, which may be mutually complementary. One is based on the fractal time random walk scheme adapted to velocity space transport⁴⁴, and may be combined with the inclusion of explicit nonlinearities owed to a back-reaction of very energetic particles on the underlying turbulence pattern. Another approach is to relax the condition that the turbulence is ''weak'' and to look at the very peculiar impact of ''strong'' fluctuations on the particle random walk. We expect this type of approach to lead to a Lévy reduced model of velocity space transport starting essentially from the turbulence microscopic fluctuation level. Unifying the two approaches would result in a new understanding of complex plasma phenomena in phase space likely upraising the idea of Zaslavsky⁴⁵ that Gaussian diffusion is a characteristic yet very special dynamical process occurring in ''sample systems for simple settings''. More

Milestones:
WP2.1-M1 - Numerical implementation of general form of phase space transport equations (Dec. 2019).
WP2.2-M1 - Practical basic understanding of conditions permitting non-diffusive scenarios of velocity space transport for beam plasma systems (Dec. 2019).
WP2.1-M2 - Numerical implementation of phase space transport equations, taking into account the multi-level approximation approach (Dec. 2020).
WP2.1-M3 - Implementation of general geometry in the AWECS code for computation of parallel mode structures (Dec. 2020).

Deliverables:
WP2.1-D1 - Verified transport module with explicit expression of EP fluxes and assuming fluctuating fields and particle distributions from numerical simulation (2019).
WP2.1-D2 - Verified transport module with explicit expression of EP fluxes taking into account the multi-level approximation approach (2020).
WP2.1-D3 - Numerical computation of the ''matrix coefficients'' involved in the non-linear envelope equations using AWECS output (2020).
WP2.2-D1 - Construction of an effective transport model of velocity space transport beyond the classic bump-on-tail paradigm (2020).

Bibliography