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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 often treated as electrostatic oscillations, GAMs intrinsically possess an electromagnetic component, manifested as magnetic field perturbations. This component is essential for GAM's interaction with electromagnetic turbulence and 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 in the radial and poloidal magnetic field perturbations and dominant $m=1$ component in the toroidal magnetic field perturbation, where $m$ denotes the poloidal wavenumber. In contrast, most gyro-kinetic studies, adopting the conventional parallel vector potential approximation ($\delta\vec{A} \approx \delta A_\|\vec{b}$), are restricted to describing only the $m=2$ poloidal component while systematically neglecting the radial and parallel (toroidal) components. This limitation has created a theoretical gap, 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. The 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 $\delta\phi$, the parallel vector potential $\delta A_\|$, and the parallel magnetic perturbation $\delta B_\|$ (associated with the perpendicular vector potential $\delta A_\perp$). By retaining the full perturbed magnetic vector potential $\delta\vec{A}$, the framework naturally incorporates both parallel current perturbations (linked to $\delta A_\|$) and diamagnetic effects (linked to $\delta B_\|$). Analytical solutions are obtained in the long-wavelength limit for a large-aspect-ratio, 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 ($\delta B_r$) and poloidal ($\delta B_\theta$) perturbations exhibit a dominant $m=2$ standing-wave structure, while the parallel perturbation ($\delta B_\|$) exhibits 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 reach of single-fluid MHD. The analytical expressions reveal distinct roles of ions and electrons: the $m=2$ radial and poloidal magnetic field perturbations, associated with parallel currents, are more strongly influenced by the ion thermal pressure, whereas the $m=1$ parallel magnetic field perturbation, linked to diamagnetic effects, receives a relatively larger contribution from the electron thermal pressure. These results not only unify the theoretical description of GAM magnetic perturbations but also advance our understanding of their kinetic physics, offering a more accurate foundation for experimental diagnostics and numerical simulation.-
Keywords:
- Gyro-kinetic equation /
- Electromagnetic geodesic acoustic mode /
- Magnetic field perturbation
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