Influence of zonal vorticity on the cross phase

The study of the evolution of the cross phase $\delta$ between the electron pressure and the potential in the resistive-drift-Afv\'{e}n turbulence under the sheared magnetic field lines are reported. The linear cross phase is established by both the drift-wave dynamics and the Ohm's law and it is vanishingly small in the modes with low $k_\parallel$. The nonlinearly saturated cross phase is correspondingly different from the linear $\delta$ mainly due to the $E\times B$ flux. Numerical computations are performed in order to investigate the roles of $E\times B$ nonlinearity when the self-consistent zonal flow exists on $\delta$ by using BOUT++ platform. In this work, the influence of the non-uniformity of the zonal flow, the gradient and the curvature, on $\delta$ and transport is presented and the departure of $\delta$ from the linear state is particularly highlighted.