Numerical Study of Alfv\'{e}n Eigenmodes in a High-Beta Toroidal Plasma

\def\aTAE{$\alpha$TAE} \def\aTAEs{$\alpha$TAEs} Discrete toroidal Alfv\'{e}n eigenmodes trapped in $\alpha$-induced potential wells (so-called {\aTAE}) and their interaction with trapped energetic ions is studied numerically. Here, $\alpha = -q^2R\beta'$ is a measure for the pressure gradient. Previous investigations using positive magnetic shear ($s \equiv rq'/q > 0$) are extended to the $s<0$ negative shear regime. It is found that {\aTAEs} exist as bound states (radially and along the field line) regardless of the sign of $s$. While for $s>0$ {\aTAEs} tend to be localized in the bad-curvature region ($|\vartheta| < \pi/2$), for $s<0$ {\aTAEs} tend to peak at the top and bottom ($\vartheta \sim \pm \pi/2$), and also have larger amplitudes in the good-curvature region. These quasi-marginally stable modes can be excited by trapped energetic ions through resonance with the precessional drift or bounce-precession resonances, whereby excitation is easier for $s>0$. Extensions to regimes with low magnetic shear and the inclusion of thermal ions are currently underway in order to study the properties of {\aTAEs} near the minimum of the safety factor $q$ in reversed-shear configurations and near the second ballooning stability boundary. Corresponding results will be reported as they become available.