Geodesic acoustic modes (GAMs), the high-frequency branch of zonal flows, play a crucial role in regulating turbulence and the associated anomalous transport in tokamaks. Although they are often treated as electrostatic oscillations, GAMs intrinsically possess an electromagnetic component, which is manifested as magnetic field perturbations. This component is essential for GAM’s interaction with electromagnetic turbulence and also for the existence of global GAM eigenmodes. However, a long-standing discrepancy exists between magnetohydrodynamic (MHD) and gyro-kinetic theories regarding the three-dimensional (3D) structure of these perturbations. MHD models consistently predict a full 3D structure, with dominant m = 2 components present in the radial and poloidal magnetic field perturbations and a dominant m = 1 component in the toroidal magnetic field perturbation, where m denotes the poloidal wavenumber. In contrast, most gyro-kinetic studies adopt the traditional parallel vector potential approximation ( \textδ\boldsymbolA \approx \textδ A_/ / \boldsymbolb ), and are limited to describing the m = 2 poloidal component while systematically neglecting the radial and parallel (toroidal) components. This limitation can result in a theoretical gap, thereby preventing a unified understanding of the electromagnetic nature of GAMs.

To address this issue, we employ a self-consistent electromagnetic gyro-kinetic model without invoking the parallel vector potential approximation. Starting from the linear electromagnetic gyro-kinetic equation, we describe the perturbed distribution functions of both ions and electrons. This model is closed with a self-consistent set of field equations—including the quasi-neutrality condition and both the parallel and perpendicular components of Ampère’s law—which determine the evolution of the electrostatic potential \textδ\phi , the parallel vector potential \textδ A_/ / , and the parallel magnetic perturbation \textδ B_/ / (associated with the perpendicular vector potential \textδ A_\perp ). By retaining the full perturbed magnetic vector potential \textδ\boldsymbolA , the framework naturally incorporates both parallel current perturbations (related to \textδ A_/ / ) and diamagnetic effects (linked to \textδ B_/ / ). Analytical solutions are obtained in the long-wavelength limit for a large-aspect-ratio and circular tokamak, including first-order finite-Larmor-radius (FLR) and finite-orbit-width (FOW) effects.

For the first time within a gyro-kinetic framework, our analysis yields the complete 3D magnetic perturbation structure of the electromagnetic GAM. The results explicitly demonstrate that the radial ( \textδ B_r ) and poloidal ( \textδ B_\theta ) perturbations exhibit a dominant m=2 standing-wave structure, while the parallel perturbation ( \textδ B_/ / ) displays a dominant m=1 structure. This spatial structure is in excellent qualitative agreement with the predictions of ideal MHD theory, thereby resolving the long-standing discrepancy between the two theoretical approaches. Moreover, the gyro-kinetic model provides a refined physical picture beyond the scope of single-fluid MHD. The analytical expressions reveal different roles of ions and electrons: the m = 2 radial and poloidal magnetic field perturbations, related to parallel currents are more strongly affected by the ion thermal pressure, whereas the m = 1 parallel magnetic field perturbation, related to diamagnetic effects, receives a relatively large contribution from the electron thermal pressure. These results not only unify the theoretical description of GAM magnetic perturbations but also deepen our understanding of their kinetic physics, thereby laying a more accurate foundation for experimental diagnostics and numerical simulation.