Group velocity in spatiotemporal representation of collisionless trapped electron mode in tokamak

Abstract

The multiple scale derivative expansion method is used to manipulate the electron drift kinetic equation, following the theoretical framework of drift wave–zonal flow system developed by Zhang et al. [Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1063/1.4995302"> 2017 <i>Phys. Plasmas</i> <b>24</b> 122304 </ext-link>]. At the zeroth order it is the linear eigenmode equation describing the trapped electron mode on a mirco-scale. At the first order it is the envelop equation for trapped electron mode modulated by the zonal flow on a meso-scale. The eigenmode equation has been solved by Xie et al. [Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1063/1.5048538"> 2019 <i>Phys. Plasmas</i> <b>26</b> 022503 </ext-link>] to obtain the eigenvalue and two-dimensional mode structure of trapped electron mode. These are essential components in calculating group velocities contained in the envelop equation. The radial group velocity arises from the geodesic curvature of magnetic field in tokamak. The poloidal group velocity stems from the normal curvature and diamagnetic drift velocity, which yields the mapping between the poloidal angle and time. Since the radial group velocity is also a function of poloidal angle, it is mapped to a periodic function of time with a period of milliseconds. The numerical results indicate the rapid zero-crossing, which is significant in the drift wave – zonal flow system and provides a sound foundation for studying zonal flow driven by trapped electron mode.