= Solution
The full perturbation equations are
$$
\nabla^2\psi'+R\theta'_x=0,\qquad
\nabla^2\theta'-\psi'_x=\theta'_t+J(\psi',\theta'),
$$
where the <streamfunction advection bracket> is $J(f,g)=f_zg_x-f_xg_z$. Put $R=R_c+\epsilon^2$ and introduce the <slow time> $T=\epsilon^2t$, so $\partial_t=\epsilon^2\partial_T$ on the amplitude-dependent fields. Then
$$
\boxed{\nabla^2\psi'+R_c\theta'_x=-\epsilon^2\theta'_x,\qquad
\nabla^2\theta'-\psi'_x=\epsilon^2\theta'_T+J(\psi',\theta').}
$$
The <weakly nonlinear expansion> balances the small linear growth against the cubic saturation on this slow time scale.
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