Plasma elongation effects on temperature gradient driven instabilities and geodesic acoustic modes

Plasma shaping effects on temperature gradient driven instabilities and geodesic acoustic oscillations are investigated with gyrokinetic theory and a local magnetohydrodynamic equilibrium model. In particular, we focus on the effect of the elongation kappa, including its radial derivative s(kappa) = (r/kappa)(partial derivative kappa/partial derivative r), in the large aspect ratio limit. An analytical formula of the dependence of the geodesic acoustic mode (GAM) frequency on the elongation is given. It is found that the GAM frequency sharply decreases with increasing elongation by the dependence of [(2-alpha s(kappa))/(kappa(2) + 1)](1/2) with alpha = 0.5-1, which comes from the modification of ion classical polarization balanced by that of curvature drift polarization. The dependence of the critical threshold of the ETG/ITG instability on the elongation is numerically studied and a semi-analytical formula is given as (R-0/L-Tc)/(R-0/L-Tc)(s kappa=0,kappa=1) = (1 + 0.36s(kappa))[1 + 0.11(kappa - 1)], where R-0 is the major radius and L-Tc is the critical scale length of the temperature gradient.