Fusion Programme - ENEA - Fusione

Fusion ProgrammeShear Alfvén wavecontinuous spectrumThe work was done incollaboration with thePhysics Department of theUniversity of Pisa and thein the presence of aDepartment of Physics andmagnetic island Astronomy at UCI. Theshear Alfvén wavecontinuous spectrum iscalculated for finite-β tokamak equilibria in thepresence of a finite-size magnetic island, adopting aslab model and constant-ψ near the correspondingrational surface. The beta-induced Alfvén eigenmodecontinuum accumulation point (BAE-CAP) [1.22-1.24]is found to be shifted in space from the rationalsurface of the island to the separatrix flux surfaceposition (labelled ψ=ψ sx in fig. 1.11), while theaccumulation points at the O pointfrequency f BAE remains the same. Splitting betweenthe frequencies of the odd and even modes becauseof the non-uniformity of the magnetic field along the field lines was also discussed. The most remarkablefeature is the presence of new continuum accumulation points at the O-point of the island (ψ 0 ), whichdepend on the toroidal mode number n, and thereby give rise to gaps in the continuous spectrum andregions free of continuum damping. This fact could make the existence of new magnetic-island-inducedAlfvén eigenmodes (MiAEs) possible, excited via wave-particle resonances, provided the island size besufficiently wide with respect to the mode radial localisation. The MiAE-CAP frequencies are given byf 2MiAE3MiAE2MiAE1BAEn=5n=4n=3MiAE-CAPsn=2n=1n=3n=2n=1BAE-CAP0 0.5 1 1.5ψ/ψ sxFig. 1.11 - Continuous spectrum frequencybranches for several n-modes (M=10-2). TheBAE continuum accumulation point isshown at the separatrix with the new MiAEf MiAE−CAP = f BAE1+ q 0 s B 2isl,0 f A n 2B 2 pol f BAE, (1)where q 0 , s, and B pol are respectively the values of the safety factor, shear and poloidal magnetic fieldcalculated at the rational surface of the island, B isl,0 is the magnetic island amplitude and f A is the Alfvénfrequency: f A =v A /(2πqR) with v A =B/√4πρ and ρ the mass density. For small-amplitude magnetic islandsthe scaling is linear with the amplitude, and the approximate value is2007 Progress Report20f MiAE−CAP ≅ f BAE + q 0 s B 2 isl,0 f A n 22 B pol f BAE. (2)The regime of validity of the linear approximation is given byB isl,0n 2 √β/M) even forfinite-β plasmas, where M=q 0 sB isl,0 /B pol . In the case of low-βplasmas, it is worthwhile noting that the order of magnitude of[1.22] M.S. Chu et al., Phys. Fluids B4, 3713(1992)[1.23] A.D. Turnbull et al., Phys. Fluids B5,2546 (1993)[1.24] F. Zonca, L. Chen and R.A. Santoro,Plasma Phys. Control. Fusion 38, 2011(1996)[1.25] P. Buratti et al., Nucl. Fusion 45, 1446(2005)[1.26] P. Smeulders et al., Proc. of the 29 thEPS Conference on Plasma Physicsand Controlled Fusion (Montreaux2002), on line at http:// epsppd.epfl.ch/Montreux/ pdf/D5_016.pdf[1.27] S.V. Annibaldi, F. Zonca and P. Buratti,Plasma Phys. Control. Fusion 49, 475(2007)[1.28] O. Zimmermann et al., Proc. of the32 nd EPS Conference on Plasma