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托卡马克等离子体中的测地声模（Geodesic acoustic mode，GAM）及其伴随的电磁场扰动在湍流调控与约束改善中发挥着重要作用．然而，现有的动理学理论与磁流体力学（Magnetohydrodynamics，MHD）在描述GAM扰动磁场的三维结构上存在一个显著差异，即动理学描述通常采用平行磁矢势近似，而无法自洽给出GAM的径向与环向磁场扰动．为弥合理论上的这一差异，本文在线性电磁回旋动理学框架下，摒弃了传统的平行磁矢势近似，保留完整的扰动磁矢势，并结合准中性条件及安培定律，自洽地推导了GAM的电磁扰动特性．推导结果首次在动理学层面自洽给出了GAM磁场扰动在径向、极向与平行（环向）方向上的完整三维结构：径向与极向磁场扰动呈现m=2（m是极向波数）的驻波形式，而平行磁场扰动则呈现m=1的结构．该结果在定性上与理想MHD理论的预测高度一致，从而弥合了长期以来两种理论在电磁GAM描述上的分歧．此外，动理学模型能够清晰区分电子与离子的贡献，进一步分析表明：离子热压对径向和极向磁场扰动的作用更为显著，而电子热压在平行磁场扰动中的贡献相对更大．这展现了动理学效应对GAM电磁特性的细致修正，为相关实验诊断与数值模拟研究提供了更加精确的理论依据．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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