Relation between energetic and standard geodesic acoustic modes

Geodesic Acoustic Modes (GAMs) are electrostatic, axisymmetric modes which are non-linearly excited by turbulence. They can also be excited linearly by fast-particles; they are then called Energetic-particle-driven GAMs (EGAMs). Do GAMs and EGAMs belong to the same mode branch? Through a linear, analytical model, in which the fast particles are represented by a Maxwellian bump-on-tail distribution function, we find that the answer depends on several parameters. For low values of the safety factor q and for high values of the fast ion energy, the EGAM originates from the GAM. On the contrary, for high values of q and for low values of the fast ion energy, the GAM is not the mode which becomes unstable when fast particles are added: the EGAM then originates from a distinct mode, which is strongly damped in the absence of fast particles. The impact of other parameters is further explored: ratio of the ion temperature to the electron temperature, width of the fast particle distribution, mass and charge of the fast ions. The ratio between the EGAM and the GAM frequencies was found in experiments (DIII-D) and in non-linear numerical simulations (code GYSELA) to be close to 1/2: the present analytical study allows one to recover this ratio