Global Geodesic Acoustic Modes in Tokamak Plasmas

Global Geodesic Acoustic Modes (GGAM) in Tokamak Plasmas are investigated in the framework of reduced ideal MHD. The axisymmetric eigenvalue problem for perturbed pressure and electrostatic potential is formulated as a recurrent set of equations for poloidal Fourier harmonics. For uniform safety factor q and temperature profile with a maximum at radius $r=r_0 \neq0$ the analytical solution of this eigenvalue problem is obtained for a truncated set of equations taking into account the $m=0$ and $m=2$ poloidal harmonics of potential and the $m=1 $ harmonic of pressure. This solution exists in wide range of $\beta q^2$. It is shown both analytically and numerically that the higher harmonics of pressure ($m=3$) and electrostatic potential ($m=4$) reduce the range of the parameters, in which GGAM exist, due to the resonance with continuum spectrum. The domain of GGAM existence in the ($\beta q^2$, $r_0$)-plane is represented. Higher poloidal harmonics ($m>4$) are shown to weakly affect the GAM spectrum and do not lead to the appearance of other global eigenmodes. The work is supported in part by grant RBRF 10-02-01302 and by Ministry of Education and Science of the RF, contract 1.5-508-008-045.