Global effects

If the equilibrium scale length become sizable when measured in terms of the perturbation wavelength, global effects lead to important modifications of the local dispersion properties. Take for example the shear-Alfvén wave ({ $\omega^*,\rho_i\} \rightarrow 0$ in eq.1) in a tokamak, use a Fourier decomposition toroidally $\exp(in\varphi)$ and poloidally $\exp(im\theta)$ in order to obtain an algebraic representation of the parallel wave vector $k_\Vert=(n+m/q)/R$. Fig.1 illustrates how different harmonics get coupled and yield so-called BAE, TAE, EAE gaps where the plasma beta, the toroidicity and ellipticity prevent shear-Alfvén waves of any frequency from propagating over large portions of the minor radius. Global (radially extended, $k_\perp\neq 0$) solutions however exist within these gaps, the so-called Alfvén eigenmodes (AEs): they often have a mixed T/EAE character and require a global calculation to determine the mode structure $k_\perp(s,\theta)$ and the corresponding damping / drive.

  

Figure 1: Global effects on the shear-Alfvén wave dispersion.

Global effects modify also the kinetic-Alfvén and the drift waves, which finally get combined with AE wavefields into what we call kinetic AEs (KAEs) [14] and drift-kinetic AEs (DKAEs) [15]. In this sense, Mett & Mahajan's KTAEs [16] are a special type of KAEs that involve only the kinetic-Alfvén wave. Finally, when the energetic particle character dominates the ion response, global solutions exist also for the resonant wavefield and are generally referred to as energetic particle modes (EPMs) [17].