is the complex frequency of the zonal flow. The frequencies couple as $\omega_{\pm} = \Omega \pm \omega$.

The equation for the time evolution for the zonal flow, that is assumed to be Hasegawa-Mima type, is expanded using the above fields, $$\begin{aligned}
-\frac{\partial}{\partial t} \nabla_x^2 \phi_{ZF} -\mu \nabla_x^4 \phi_{ZF} = \left[\phi, \nabla_{\perp}^{2} \phi\right].\end{aligned}$$

In addition the sidebands are to be determined. The sidebands evolve according to Eq. 1 and also identify the coupling between the drift wave and the zonal flow as the sideband component. This gives a problem with four coupled waves where the zonal flow seed may be unstable and grow exponentially. This gives the equation system, $$\begin{aligned}
(i\Omega q_x^2 - \mu q_x^4) \phi_{ZF} & = & k_y q_x^2 [a_{+} \phi_{DW}^{*} \phi_{+} + a_{-}\phi_{DW} \phi_{-}] \\
\phi_{+} & = &  i\beta q_x k_y k_{\perp -}\omega_{+}\frac{\phi_{DW} \phi_{ZF}}{\omega_{+}(\omega_{+}-k_y+\epsilon_n g_i \beta k_y) - k_{\parallel}^2 \beta} \\
\phi_{-} & = & - i\beta q_x k_y k_{\perp -}\omega_{-}\frac{\phi_{DW}^{*} \phi_{ZF}}{\omega_{-}(\omega_{-} + k_y - \epsilon_n g_i \beta k_y) - k_{\parallel}^2 \beta}.\end{aligned}$$ In deriving these equations the FLR effects are considered to be small. Here $$\begin{aligned}
a_{\pm} & = & 2 k_x \pm q_x \\
k_{\perp -} & = & k_x^2 + k_y^2 - q_x^2 \\
\beta 