Carlevaro, N.

Falessi, M.

Lauber, P.

Montani, G.

English

MPG.PuRe

Energetic particle transport:diffusion vs convection and phase-space barriersN. Carlevaro1, G. Montani1,2, M.V. Falessi1, Ph. Lauber31 ENEA, FNS Department, C.R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy2 Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy3Max-Planck-Institut fuer Plasmaphysik, Boltzmannstrasse 2, D-85748 Garching, GermanyAbstract Energetic particle (EP) redistribution [1, 2] in the presence of multiple Alfvén eigen-modes (AEs) is analyzed in Ref.[3] for the ITER 15MA baseline scenario: non-linear hy-brid simulations (within their well known limits) point out that transport can be dominatedby avalanches under certain conditions. These phenomena are properly reproduced by the 1Dreduced description of Ref.[4]. Here, using this simplified 1D model, we define the transportcharacter (convective/diffusive) of self-consistent EP redistribution. Transport barriers in phasespace are studied using the Lagrangian Coherent Structures (LCS) technique [5].Reduced model for EP transport In Ref.[4], a mapping procedure has been set up to trans-late the evolution of the radial profile of fast ions (interacting with multiple AEs) into the dy-namics of an equivalent beam-plasma system (BPS). The mapping consists in a one-to-one linkbetween the radial coordinate (s) of the realistic 3D scenario and the 1D BPS velocity (u). It isdefined around a single chosen resonance, resulting in a linear relation, and then extended to themultiple mode case. For the reference resonance, the mapping is formally constructed locallyfrom the resonance condition and reads as: u = (1− s)/`∗ (where `∗ is an arbitrary constantdefining the spectral features and the periodicity length of the 1D slab).The BPS setup can be closed by determining the beam to plasma density ratio parameter (η).We note that the EP/AE system has an higher dimensionality with respect to the BPS and, thus,the non-linear BPS transport results more efficient than the realistic radial transport, in the caseof equal drive [4]. η can be set to obtain optimized EP relaxation in the s direction: we fix suchparameter by performing a scan in order to match the relaxed QL flattening width of the twoschemes directly integrating QL equations for the BPS [6]. Multiple modes can be included bymeans of the resonance conditions written in the radial space and, to preserve the asymptoticmode decay, we impose equal dampings (normalized to the effective mode frequencies ω forthe BPS and Ω for the 3D system) between the two systems. Mode frequencies and drives of thereduced scheme can be in fact evaluated using the linear BPS dispersion relation consideringthe initial averaged 3D EP distribution mapped in u. The dispersion relation also provides, asa result, the drives γD j of the reduced model. Due to the reduced dimensionality and to the48th EPS Conference on Plasma Physics P5a.113effective wave-particle power exchange, γD j will result smaller than the normalized linear AEdrives ΓD j. Introducing γD j/ω j = α j ΓD j/Ω j, we can evaluate the important parameter α j(≤ 1)indicating, roughly, the fraction of the most resonant particles. In fact, α ' 1 when comparingonly EPs which maximize the energy exchange in the realistic scenario. In Ref.[4], it also tracedthe methodology for estimating the parameter α by invoking the phase space splitting.EP redistribution We analyze the ITER 15MA beseline scenario described in Ref.[3] byrunnig simulations of the BPS mapped back into the radial coordinate s. The addressed scenarioincludes the least damped 27 toroidal AE: n ∈ [12,30] for the main branch and n ∈ [5,12] forthe low branch (the role of the poloidal harmonics is discussed in Ref.[4]). n = 21 is chosen asreference resonance for the mapping since it is characterized by average value of radial positionand essentially also of the growth rate of the main branch. QL evolution presented in Ref.[3]outlines a flattening width of the distribution function fH between s' 0.32 and s' 0.58. Aftera scan by integrating the BPS version of the QL equations, this width is found to be reproducedfor η ' 0.007 (see Fig.1) corresponding to α21 = 0.4 (we have set `∗ = 400).Figure 1: Gradient of the EP redistribution from QL theory. Left-hand panel: results of Ref.[3] for late times.Right-hand panel: reduced BPS evolution. Blue arrows indicate the flattening width.The reduced self consistent evolution of the distribution function is now plotted the left-handpanel of Fig.2. While a first peak is well localized around s ' 0.35 (excitation of the highbranch), a second peak, related to the linearly stable low branch, is shifting in time towards' 0.65, outlining an avalanche-like transport toward the edge. This sheds light on the relevanceof the stable (sub dominant) spectrum component destabilization due to domino EP dynamics.Diffusion vs convection We now study the transport character during the self-consistent re-laxation by means of test particle evolution. The tracers are initialized around a reference radialvalues s0 to characterize different portions of the distribution function and the transport featuresare defined by the mean square path as a function of time (average 〈...〉 taken over the tracers):〈δ s2〉= 〈[s(τ)−〈s(τ)〉]2〉 , 〈δ s2〉 ∝ τ ⇒ DIFFUSION, 〈δ s2〉 ∝ τ2⇒ CONVECTION.48th EPS Conference on Plasma Physics P5a.113Figure 2: Left-hand panel: evolution of EP profile gradient as function of s. Colored lines represent differenttimes (dashed black line is the initial distribution fH0). Right-hand panel: plot of 〈δ s2〉(τ) for tracers initializedaround s0 ' 0.34 (blue) and s0 = 0.6 (yellow). Dashed gray lines are guide for eyes underlining the two regimes.In Fig.2 (right-hand panel), we plot 〈δ s2〉 as a function of time for tracers initialized arounds0 = 0.34 (close to the first peak), and around s0 = 0.6 (second peak). Focusing on time scalesafter saturation (i.e., after the dynamics drive dominance by rotating phase space clumps), thediffusive behavior of tracers initialized in the first peak is evident during the post saturationphase. The late dynamics does not outline pure diffusive features due to the statistical weight ofsignificantly displaced tracers. Tracers of the second peak are instead globally characterized bya τ2 dependence during the whole considered time interval, underlining the convective characterof the transport due to the dominance of avalanches related to the low-n branch.Phase-space transport barriers The local dynamics is related to phase space rotating clumps.These can be identified by studying the regions where the mixing phenomena are faster andtheir borders which are called Lagrangian Coherent Structures (LCS) [5, 7]: the most repulsiveor attractive material line defined by a peaked profiles of the Finite Time Lyapunov Exponent(FTLE) fields. We adopt here a test particle approach by tracing trajectories of markers ini-tialized, at a given time τ , in two phase-space grids (the 1D BPS phase space (x̄,u), where x̄denotes the normalized “space” coordinate, is mapped into (x̄,s)): one defined by (x̄i,s j) and asecond radial displaced one at (x̄i,s′j), where s′j = s j +δi j with δi j = δ = const. At τ +∆τ , themarkers will be at a distance ∆i j and the FTLE field in the phase space (x̄,s) is defined asσ(x̄i,s j,τ) = ln [∆i j/δ ]/∆τ . (1)For ∆τ > 0 (< 0), the FTLE peaked profiles define repulsive (attractive) transport barriers.In Fig.3, we show the contour plots of σ(x̄i,s j) at fixed times implementing two grids, eachwith 1.6× 105 tracers, with ∆τ = ±100. For two snapshots, we overplot sample markers inorder to underline the early rotating clump dynamics and also to show the attractive/repulsivecharacter of the barriers. At early times, the dominance of the high branch modes (n ∈ [12,30])is consistent with the more peaked transport barriers in the corresponding resonance region atlow s. As time increases, the progressive avalanche excitation of the low branch (n ∈ [5,12])yields to the emergence of more robust barriers moving toward large s values.48th EPS Conference on Plasma Physics P5a.113Figure 3: Contour plots of σ(x̄i,s j) from Eq.(1) at fixed times as indicated over the graphs. We set ∆τ =±100 andwe use arbitrary units with white/brown corresponding to peaks, while green/blue denote valleys. For the snapshotsat τ = 400,600, we also overplot a sample of tracers (in black) to highlight attractive/repulsive barriers.Concluding remarks We showed how the BPS translation of a realistic 3D scenario for theinteraction of EP with AE be predictive in reproducing fundamental transport features. In par-ticular, we were able to characterize the self-consistent dynamics by determining diffusive andconvective transport regimes. The analysis on the LCS allowed a characterization of the trans-port barrier formation and disruption in the phase space. This analysis offers a valuable back-ground for future investigations concerning the mapping here considered, in view of a precisealgorithm to estimate EP fluxes from very fast reduced numerical codes.References[1] L. Chen, F. Zonca, Rev. Mod. Phys. 88, 015008 (2016)[2] B. Breizman, S. Sharapov, Plasma Phys. Contr. Fusion 53, 054001 (2011)[3] M. Schneller, Ph. Lauber and S. Briguglio, Plasma Phys. Control. Fusion 58, 014019 (2016)[4] N. Carlevaro et al., Plasma Phys. Control. Fusion 64, 035010 (2022)[5] M.V. Falessi et al., J. Plasma Phys. 81, 495810505 (2015)[6] G. Montani, F. Cianfrani, N. Carlevaro, Plasma Phys. Contr. Fus. 61, 078018 (2019)[7] N. Carlevaro, M.V. Falessi, G. montani, F. Zonca, J. Plasma Phys. 81, 495810515 (2015)This work has been carried out within the framework of the EUROfusion Consortium [ER Project ATEP - CfP-FSD-AWP21-ENR-03-MPG-01], funded by the European Union via theEuratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarilyreflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.48th EPS Conference on Plasma Physics P5a.113

Energetic particle transport: diffusion vs convection and phase-space barriers

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Energetic particle transport: diffusion vs convection and phase-space barriers

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