be found, $\omega_{Slab}^2 \approx - k_{\parallel}^2 \tau (\eta_i - 2/3)$, in analogy with Ref. [@a27]. In comparison with the model in Ref. [@a27], the present model yields a positive threshold value $\eta_{i th} = 2/3$ for the slab like ITG mode. In the toroidal ITG mode branch approximate solutions are close to those found without parallel ion motion e.g. Eqs 8 - 10 in Ref. [@a23] (given here for convenience), $$\begin{aligned}
\omega_r & \approx & \frac{k_y}{2\left( 1 + k_{\perp}^2\right)} \left( 1 - \left(1 + \frac{10\tau}{3} \right) \epsilon_n g_i - k_{\perp}^2 \left( \alpha_i + \frac{5}{3} \tau \epsilon_n g_i \right)\right)  \\
\gamma & \approx & \frac{k_y}{1 + k_{\perp}^2} \sqrt{\tau \epsilon_n g_i\left( \eta_i - \eta_{i th}\right)} \\
\eta_{i th} & \approx & \frac{2}{3} - \frac{1}{2 \tau} + \frac{1}{4 \tau \epsilon_n g_i} + \epsilon_n g_i\left( \frac{1}{4 \tau} + \frac{10}{9 \tau}\right)\end{aligned}$$ where $\omega = \omega_r + i \gamma$. The origins of the slab and toroidal ITG mode instabilities are inherently different: the slab like branch is driven unstable by parallel acoustic waves; while the other branch is driven by toroidicity [@a27].

The model for zonal flow generation
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In this section the derivation of the zonal flow dispersion relation will be explained briefly. The method 