The grand-challenge of magnetic fusion energy (MFE) research is the development of a predictive capability for the evolution of macroscopic plasma profiles on long time scales, comparable with the energy confinement time or even longer. On these long time scales, it is reasonable to expect that fluctuation-induced plasma transport plays an important role, and must enter on the same footing of scattering processes due to Coulomb collisions as well of particle and energy sources and sinks, such as plasma fueling and radio-frequency (RF) additional heating/current-drive. In the view of investigating these physics and developing a whole device modeling capability, the generally adopted approach is to extend first-principle-based gyrokinetic simulations to long time scales1–3, including progressively more sophisticated and articulated physics ingredients, which are too demanding for present days peta-flop computers and are challenging even for the new class of high-performance computing systems foreseen by the US Exascale Computing Project (ECP)4. Thus, there is room and need for reduced descriptions, which preserve all the necessary physics ingredients. Theoretical approaches exist in the literature5–7, which adopt a multi-scale gyrokinetic description of plasma transport, coupled with evolution equations for the fluctuation spectra. They are based on a systematic separation of micro-scale instabilities underlying plasma turbulence from the evolution of plasma profiles on the macro-scales. Fusion plasmas, however, are characterized by the unique role of energetic particles (EPs) as mediators of cross-scale couplings1–3,8–10; i.e., as crucial actors in the interplay between micro-scale instabilities and macro-scale plasma profiles, which takes place on intermediate spatiotemporal scales (the meso-scales). Thus, a systematic separation of scales cannot be assumed, and a novel approach must be developed for taking these new physics into account, which are extensively reviewed in Ref. [10]. This is the aim of the present project proposal, which will leverage the results and knowledge of two former Enabling Research projects: “Theory and simulation of energetic particle dynamics and ensuing collective behaviors in fusion plasmas” (NLED)11, and “Nonlinear interaction of Alfvénic and turbulent fluctuations in burning plasmas” (NAT)12.
The state of the art of MFE research in this area can be summarized as follows. EP transports in burning plasmas due to shear Alfvén wave (SAW) fluctuations as well as of lower frequency MHD modes have been typically addressed by test-particle methods13,14, neglecting the self-consistent feedback of EP dynamics on the mode structures. As Alfvénic fluctuations, e.g. Alfvén Eigenmodes (AEs), are local in nature and have generally small intensity15, EP redistributions by AEs are expected to be typically small, unless stochastization threshold of EP motions in phase-space is reached in the presence of many modes. The near marginal stability Ansatz is often used for computing EP profiles in the simplest applications to ITER16, based on the assumptions that EP profiles are stiff17. This approach has been recently extended18, focusing on applications to DIII-D19, and is expected to capture only the averaged EP density profiles on sufficiently long spatiotemporal scales. More detailed investigations are needed to predict fluctuations about averaged profiles of EP density, temperature, etc., and to describe nonlinear dynamics of corresponding transport events10. It was proposed that the onset of phase-space stochasticity may be described by a “line-broadened” quasilinear model20, accounting for a discrete spectrum of overlapping modes in the case of multiple AEs and which was recently extended and applied to the analysis of beams interacting with AEs in DIII-D21,22. However, while quasilinear theory is suited for explaining EP transport due to drift wave turbulence (DWT)10, realistic predictions of test particle transport and stochastization threshold in ITER is very sensitive to details of the underlying physics and adopted model23,24. In particular, there is increasing experimental25–27 and numerical simulation1–3 evidence that transport depends on phase-space features of the EP distribution function28,29. Meanwhile, analyses of EP transport in ITER reveal that quasilinear theory fails in capturing details of spatial redistribution and evolution of the fluctuation intensity spectrum30. This is consistent with the nonlinear gyrokinetic analysis of EP transport9,10, showing the crucial role played by phase-space zonal structures (PSZS)8, which are due to resonant wave-EP interactions and are undamped by collisionless processes. For example, phase locking and phase bunching are both important processes in EP transport, and they fall outside the realm of quasilinear analysis8,10; and fluctuations of the linear stable spectrum play crucial roles30,31.
The nonlinear SAW physics in fusion plasmas can be discussed and understood along two main routes32: (i) wave-wave interactions and the resultant spectral energy transfer; (ii) nonlinear wave-particle interactions of SAW instabilities with EPs. As different codes are based on different physics models and approaches, the benchmark activity combined with comparisons with the reduced model description will allow identifying the relevance of various elements in the multi-level description of the nonlinear processes involved in EP transport.
Based on the interpretation and the understanding of present day experiments, tools for predictive EP transport in burning plasmas have to be developed. The research activities carried out within the NLED11 and NAT12 ER Projects provide the necessary general/theoretical understanding and numerical simulation capability of the single processes that play an important role10, from wave-wave interactions to nonlinear wave-particle interactions of SAW instabilities with EPs32. The comprehensive nonlinear gyrokinetic and hybrid codes involved in the present project proposal (cf. Sec. 3) can simulate EP transport from the solution of the nonlinear gyrokinetic equation36,37 for the fluctuating particle distribution function accurate up to second order in the asymptotic expansion parameter. Using the gyrokinetic theory for EP transport38,39 based on the nonlinear dynamic description of PSZS8,10, it is possible to explicitate EP fluxes in phase space in terms of fluctuating fields and perturbed particle distribution function. In particular, it is possible to demonstrate that EP redistribution on the transport time scale40 can be computed by means of the perturbed particle distribution function accurate up to first order in the asymptotic expansion parameter. In turn, perturbed particle distribution function accurate up to second order can be used, by means of explicit expressions, to compute EP fluxes in phase space that allow predicting flux surface averaged particle redistribution on time scales even longer than the transport characteristic time38,39. The explicit expression of EP fluxes in phase space, using numerical fluctuation spectrum and perturbed particle distribution function, can be verified by direct comparison of the predicted corresponding flux surface averaged EP redistribution with EP transport obtained directly from the flux surface averaged solution of the nonlinear gyrokinetic equation. This is the initial objective in the development of the reduced description of EP transport. Note that, different from turbulent transport, the phase space features of EP fluxes, i.e. PSZS, are important due to the resonant character of EP transport8,10. Similarly, structure formation on the meso-scales, including zonal structures, and deviation of the particle reference state from local thermodynamic equilibrium are important elements that will be analyzed in this way38,39.
The inclusion of appropriate EP sources/sinks and collision operators are necessary for long time-scale realistic simulations. Some of the numerical codes involved in the present project proposal (cf. Sec. 3) can take these physics into account and, thus, can be compared with the predictions from EP fluxes is the phase space due to PSZS nonlinear evolution in the presence of sources/sinks. As the inclusion of progressively more sophisticated physics in first principle gyrokinetic and hybrid simulations on the long time scales is very demanding (cf. Sec. 1), the development of reliable and satisfactory collision models as well as EP sources/sinks in the reduced description for nonlinear evolution of PSZS is another scientific objective of this proposal.
The expected outcome of achieving the aforementioned goals is the implementation of a framework for numerically solving PSZS transport equations, computing flux surface averaged fluxes in general geometry from either numerical solutions or simplified expressions, with the possibility of accounting for Coulomb collisions as well as EP sources/sinks. This approach will be formulated in 2D tokamak geometry first and then extended to 3D stellarator equilibria. These results can be used to extend the EP transport predictive capability of gyrokinetic/numerical codes that have simplified or no capability of handling Coulomb collisions and EP sources/sinks. Furthermore, these results will pave the road to the further development and validation of the reduced EP transport model to be possibly used as tool for whole device modeling and/or predictive interpretation of experimental observations.
The primary objective of the present project proposal is the formulation of a multi-level reduced description of EP transport in the framework of the rigorous gyrokinetic analysis of PSZS nonlinear evolution introduced above8,10. The first level of simplification in the EP transport analysis will be based on the small amplitude expansion yielding a “weak turbulence” analysis of the EP fluxes in phase space. A second level of simplification will assume that parallel mode structures (to the ambient magnetic field) are weakly distorted and essentially linear, reducing nonlinearity to the distortion of the radial envelope of the considered fluctuations. Finally, a third level of simplification will consist in the quasilinear approximation for EP fluxes in the phase space. This final level of approximation is the most commonly adopted simplification used in the literature as state-of-the art predictive analysis of EP transport; while test-particle studies, which are typically adopted as they produce better agreement with experimental observations, need to be complemented by inputs from experimental measurements (cf. Sec. 1). By systematic comparison of the various levels of approximation in this multi-level reduced description with the numerical simulation results of gyrokinetic and hybrid codes involved in the project proposal (cf. Sec. 3), the expected outcome is a validated reduced model of EP transport, where the essential underlying physics is identified and understood. In general, we expect that these physics may change depending on the considered plasma scenarios. For this reason, in addition to the aforementioned ASDEX Upgrade based NLED reference case34, and in the same spirit (cf. Sec. 1), we will adopt reference cases based on JT60-SA, ITER, DEMO and the Divertor Tokamak Test (DTT) facility41, with the aim of “extracting” the relevant physics processes and investigating them in conditions of practical interest.
The conceptual approach is that of Refs. [9,10], where fluctuations that generally consist of magnetic field-aligned structures with the parallel characteristic length much longer than their transverse length are described as quasi-particles carrying energy and momentum. While particles are conserved in number, quasi-particles can be emitted and reabsorbed owing to the non-uniform particle distribution, which is relaxed via transport processes. Meanwhile, fluctuation nonlinear interaction takes place via spectral transfer and involves nonlocal processes. EPs can emit quasi-particles on the system-size macro-scales, because of the interplay of fusion reactivity profiles with equilibrium radial non-uniformities. These fluctuations are coupled via mode conversion to micro-scale fluctuations at the thermal ion Larmor radius. Furthermore, EPs can resonantly excite Alfvénic fluctuations at their gyroscales, i.e., meso-scales in between those of turbulence and macro-scales. In short, this is the reason why EPs are said to be mediators of cross-scale couplings in fusion plasmas9,10 and why, as noted in Sec. 1, it is important to develop reduced descriptions for EP transport in phase space that can introduce various levels of approximation in the physics, which can be verified against comprehensive numerical simulations, and validated via comparisons with experimental observations.
As anticipated in Sec. 1, the present project proposal is structured as follows: (i) the formulation of the reduced description of EP transport in fusion devices; (ii) the implementation of a numerical framework for solving transport equations coupled with the dynamic evolution of the fluctuation spectrum; (iii) the comparison of the reduced model description with numerical simulation results from comprehensive nonlinear gyrokinetic and hybrid codes in a number of reference/paradigm problems of practical interest adopted as benchmark cases. Each of these topics is described below and identified as a major work-package. Further to this, a fourth work-package is identified to include, in the present proposal, the analysis of the multi-scale processes underlying the interaction of EPs and MHD modes. In particular, we focus on the interaction of EPs and tearing modes.
3.1 Theoretical framework and phase space transport equations: WP1
3.2 Numerical implementation of PSZS transport equations and transport analysis: WP2
3.3 Comparison of reduced model description with numerical simulation results: WP3
3.4 Energetic ion transport induced by tearing modes: WP4
The project team is largely based on a long-lasting and successful collaboration of two Enabling Research projects funded by EUROfusion; i.e., “Theory and simulation of energetic particle dynamics and ensuing collective behaviors in fusion plasmas” (NLED)11, and “Nonlinear interaction of Alfvénic and turbulent fluctuations in burning plasmas” (NAT)12. The detailed manpower for each involved Beneficiary has been provided already. An international partnership with the Institute for Fusion Theory and Simulation (IFTS) at Zhejiang University (ZJU) at no cost for EUROfusion is foreseen. In particular, synergy is planned with the ongoing Drift Alfvén Energetic Particle Stability (DAEPS) project, funded by the National Magnetic Confinement Fusion Energy Research Program in China, and with some of the key researchers involved in that project, Prof. L. Chen, Prof. Z. Qiu and Prof. Y. Xiao. An international partnership is also foreseen with QST Rokkasho Fusion Institute, Aomori JP, in the person of Andreas Bierwage, one of the developers of the AWECS code to be used in WP2 and who will act as “consultant”. Confirmation of these international partnerships has been already provided.
Details on Work plan Milestones and Deliverables are given within each Work Package: WP1, WP2, WP3 and WP4.