Geodesic acoustic mode in tokamaks: local consideration and eigenvalue analysis

A set of magnetohydrodynamic equations describing the geodesic acoustic mode (GAM) in tokamak plasmas is derived. The obtained equations take into account the presence of the energetic ions and allow to study energetic-ion-driven GAM instability perturbatively or non-perturbatively (EGAM mode). They are applicable to plasmas with (beta) over barq(2) less than or similar to 1, where (beta) over bar = beta(s)/(1 + beta(s)), beta(s) = c(s)(2)/v(A)(2), c(s) is the sound velocity, v(A) is the Alfven velocity, q is the tokamak safety factor. Using these equations, GAM/EGAM instability is studied in a local approach and by means of the eigenvalue analysis. It is shown that beta-coupling (the coupling of Fourier harmonics of the perturbation due to finite beta-ratio of the plasma pressure to the magnetic field pressure-and the curvature of the field lines) can be responsible for the radial structure of the GAM-mode. A conclusion is drawn that conditions for the GAM/EGAM instability to arise are mildest in the case of counter-injection of energetic ions with pitch angles chi(2) < 0.6 and large ratio of Larmor radius of the energetic ions to a characteristic length of inhomogeneity of these ions. A numerical code solving the derived equations is developed. Specific calculations are carried out for tokamaks with a non-monotonic safety factor. On the other hand, it is found that due to the presence of the energetic ions the GAM/EGAM continuum can have an extremum even when the safety factor q(r) is monotonic, which indicates that global modes can exist also in this case.