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从Braginskii冷离子流体模型出发，利用准环坐标系，导出了轴对称(环向模数为零的)扰动密度和扰动电位所服从的两个二维线性偏微分本征波方程。采用WKB法及Langer变换，解出了测地声模过转向点一致有效的整体模结构和分立本征频率谱；论述了不存在连续谱解的原因；指出Schrödinger意义下的径向高激发态对应了近零频模；论证了与测地声模定级分析的自洽性将导致“蝌蚪”定域化；并讨论了“蝌蚪”定域化理论与实验上所观察到的“多个测地声模共存”现象之间的关係。

The "tadpole" localization in radial structure of geodesic acoustic mode

Based on the Braginskii’s cold ion fluid model，using the quasi-toroidal coordinate system, the two partial differential equations of perturbed density and electrostatic potential describing axial-symmetric (zero toroidal mode number) sound waves have been derived. The coupled equations are then solved for the geodesic acoustic mode (GAM) by making use of the WKB method and Langer transform to obtain the uniformly valid radial mode structure over WKB turning point, and the eigen-frequencies are determined. It is found that no continuum solution for GAM is possible. The near-zero mode frequencies are identified to be associated with the “high energy” excited states in a sense of Schrödinger equation. It is also argued that the radial structure consistency with GAM ordering demands the “tadpole” localization. The implication of the “tadpole” localization to “multi-GAM co-existence” observed is experimentally discussed.