Relationship of the cross phase and the zonal flow in electrostatic ion-temperature-gradient turbulence

The cross phase $\delta $, that is the phase difference between the electric potential and the ion pressure, is examined in the electrostatic ion-temperature-gradient fluid turbulence. It is important to study the cross phase because the thermal transport $\Gamma $ is roughly proportional to sin$\delta $. Three-dimensional numerical simulations are performed in the BOUT$++$ platform with the shifted metric coordinate system. The cross phase seems to show an interesting feature such that it is almost constant when the zonal flow $V$is in the direction of the electron diamagnetic drift and the time evolution of the cross phase oscillates near zero at low poloidal wave number and $\Gamma $ is small. These phenomena are closely correlated with the fluctuation level and depend closely on the curvature $V''$ of the zonal flow.