& = & 1 + \tau + \tau \delta \\
\delta & = & \frac{(\eta_i - \frac{2}{3})k_y + \frac{2}{3}\omega}{\omega + \frac{5}{3}\tau \epsilon_n g k_y}.\end{aligned}$$ It should be observed that if $k_{\parallel} \rightarrow 0$, Eqs. 11 - 13 are reduced to the corresponding equations derived without parallel ion motion [@a20]. It is also important to observe the fact that at $q_x = 0$ the Reynolds stress term vanishes and the drive for the secondary generated zonal flow disappears. Using Eqs. 11 - 13 it is now possible to derive a fifth order dispersion relation for the zonal flow component. $$\begin{aligned}
(\Omega + i \mu q_x^2) \bar{\Omega}_{+} \bar{\Omega}_{-} &  = &  [ K_{+}(\Omega + \omega) \bar{\Omega}_{-} + K_{-}(\Omega - \omega)\bar{\Omega}_{+} ] |\phi_{DW}|^2 \\
\bar{\Omega}_{+} & = & (\Omega + \omega)(\Omega +\alpha) - k_{\parallel}^2 \beta \\
\bar{\Omega}_{-} & = & (\Omega - \omega)(\Omega - \alpha) - k_{\parallel}^2 \beta\\
K_{+} & = & k_y k_{\perp -} \beta a_{+}\\
K_{-} & = & k_y k_{\perp -} \beta a_{-}\\
\alpha & = & \omega - k_y + \epsilon_n g_i k_y \beta\end{aligned}$$

A detailed study of the dispersion relation shows the explicit effect of parallel ion motion on the zonal flow. Expanding the dispersion relation around the 