Resonant and non resonant particle dynamics in Alfv\'{e}n Mode excitations

\newcommand{\gaeq}{\raisebox{-.7ex}{$\stackrel{\textstyle>}{\sim}$}} The resonant excitations of Alfv\'{e}n Modes in toroidal plasmas by fast ions is considered in the present work, revisiting previous analyses that demonstrated in general the existence of two types of modes; {\it i.e.}, a discrete shear Alfv\'{e}n gap mode, or Alfv\'{e}n Eigenmode (AE), and an Energetic Particle continuum Mode (EPM)~[1]. Here, we demonstrate that the resonant fast ion dynamics in toroidal plasmas is dominated by the magnetic drift curvature coupling in the vorticity equation, while the non-resonant response has various contributions, whose relative weight depends on the ratio of the characteristic fast ion orbit width, $\rho_{LE}$, to the perpendicular mode wavelength, $\lambda_\perp$. Specifically, we demonstrate that the optimal wavelength ordering for analyzing energetic ion transport in burning plasmas is $\lambda_\perp\gaeq \rho_{LE}$, considered in [1], for which both resonant and non-resonant fast ion behaviors are dominated by the magnetic drift curvature coupling, that fully accounts for the energetic ion dynamics, including the {\it charge uncovering} effect~[2]. \newline --------------------------------- \newline \noindent [1] L. Chen, Phys. \ Plasmas {\bf 1}, 1519, (1994).\newline \noindent [2] M. N. Rosenbluth, Physica Scripta T2/1, 104 (1982).