Local and global stability

Following the local analysis from sect.2.1, a harmonic oscillation $\exp(-i\omega t +\gamma t)$ is locally unstable if the drive exceeds the damping

$$\frac{\gamma}{\omega} = \left.\frac{\gamma}{\omega}\right\v... ...} + \left.\frac{\gamma}{\omega}\right\vert _\mathrm{damp} > 0.$$ (3)

This criterion is extended to global modes by measuring the total power transfer from the particles to the wavefield P_tot=P_f+P_e+P_i: normalizing with respect to the wave reactive power $\omega W$, a global instability occurs if

$$\frac{\gamma}{\omega} = \frac{P_f+P_e+P_i}{\omega W} > 0$$ (4)

i.e. when a net power flows from the particles to the wavefield and amplifies an initial perturbation. This criterion remains valid in the presence of mode conversion as seen for example with the DKAE instabilities in DIII-D [15], where power flows from the AE wavefield to an electromagnetic drift wave that is Landau damped locally by the electrons, and simultaneously channels from the fast ions driving this drift wave back to the global wavefield to be driven in fact through the conversion layer.