solution $\Omega_0$ for $k_{\parallel} = 0$, where the dispersion relation is reduced to third order, with $\Omega = \Omega_0 + \delta \Omega$ and solving for $\delta \Omega$, $$\begin{aligned}
\Im (\delta \Omega) \approx \frac{k_{\parallel}^2 \beta}{5 (\Omega_0^{IM})^4}(K_{-}-K_{+})\gamma |\phi_{DW}|^2 \\
\Re (\delta \Omega) \approx \frac{k_{\parallel}^2 \beta}{5 (\Omega_0^{IM})^4}(K_{-}-K_{+})\omega_{r} |\phi_{DW}|^2.\end{aligned}$$ Here it is assumed that the zonal flow growth rate is much larger than the real frequency ($\Omega_0^{IM} >> \Omega_0^{RE}$). The important observation from Eqs. 24 - 25 is the rapid variation with the parallel wave number in the change in the zonal flow growth rate and the real frequency in the case of slab like ITG mode generated zonal flows. The slab like generated zonal flow growth rate and real frequency change in the same manner as the slab ITG mode. This also means that the zonal flow is driven by the acoustic waves through the slab like ITG mode instability. The Eqs. 24 - 25 also indicates a weak dependency of the parallel wave number on $\delta \Omega$ for the zonal flows generated from toroidal ITG mode turbulence.

The fifth order dispersion relation (Eq. 18) is solved numerically and special attention is given to the behavior of zonal flow generation as a function 