Description
We analytically investigate geodesic acoustic modes (GAMs) in tokamak plasmas with up-down asymmetric and non-circular cross-sections using magnetohydrodynamics (MHD) and a Miller-like flux surface model. Explicit expressions for GAM frequency, magnetic field perturbations, and Lagrangian displacement are presented. Our results reveal that (I) up-down asymmetry ($\sigma$) slightly increases the GAM frequency and introduces additional sin or cos components (opposite to the dominant component) to the perturbations; (II) the inverse aspect ratio ($\epsilon$), the gradient of the Shafranov shift ($\Delta'$), triangularity ($\delta$), and its gradient ($s_\delta$) can induce additional subdominant components of perturbations. The poloidal mode numbers of the dominant and subdominant components differ, and in certain cases, the amplitude of the subdominant component can approach or even exceed that of the dominant component. These results provide analytical explanations for previous MHD and gyro-kinetic simulation
outcomes, and offer useful guidance for measuring multiple components of perturbations.