= Solution
Use the <Galerkin method>: substitute the retained trigonometric modes and project onto them with the $L^2$ <inner product> over the rectangle. Each mode satisfies the printed <boundary conditions>. Write $X=\alpha x$, $Z=\pi z$, and abbreviate the four mode prefactors by $C=2\sqrt2\beta/\alpha$, $D=2\sqrt2/\beta$, $V_1=2\sqrt2\sigma\pi R_\Omega/(\alpha\beta)$ and $V_2=\sigma R_\Omega/\alpha$. The <Jacobian determinant> in the advection terms is $J(f,g)=f_xg_z-f_zg_x$.
The <stream function> is a single <Laplacian eigenfunction>, so $\nabla^2\psi=-\beta^2\psi$ and $J(\psi,\nabla^2\psi)=0$ exactly. For the temperature modes, direct multiplication gives
$$
\begin{aligned}
J(\psi,Db\cos X\sin Z)&=4\pi ab\sin2Z,\\
\operatorname{proj}J(\psi,-c\sin2Z/\pi)&=2\sqrt2\beta ac\cos X\sin Z.
\end{aligned}
$$
For the transverse-velocity modes, the retained products are
$$
\begin{aligned}
J(\psi,V_1d\sin X\cos Z)&=-\frac{4\sigma\pi^2R_\Omega}{\alpha}ad\sin2X,\\
\operatorname{proj}J(\psi,V_2e\sin2X)&=\beta^2V_1ae\sin X\cos Z.
\end{aligned}
$$
For example, $\sin Z\cos2Z=(\sin3Z-\sin Z)/2$ and $\sin X\cos2X=(\sin3X-\sin X)/2$ supply the signs of the two retained $ac$ and $ae$ couplings. The third harmonics are orthogonal to the retained modes and are discarded by this <Galerkin method>; they are not claimed to vanish in the full <partial differential equations>.
Before changing time, the five coefficient equations are
$$
\begin{aligned}
\frac{da}{dt}&=-\sigma\beta^2a+\frac{\sigma R\alpha^2}{\beta^4}b-\frac{\sigma^2\pi^2R_\Omega^2}{\beta^4}d,\\
\frac{db}{dt}&=\beta^2(a-b-ac),\\
\frac{dc}{dt}&=4\pi^2(-c+ab),\\
\frac{dd}{dt}&=\beta^2(a-\sigma d-ae),\\
\frac{de}{dt}&=-4\sigma\alpha^2e+4\pi^2ad.
\end{aligned}
$$
Dividing by $\beta^2$ gives the stated <five-mode rotating-convection truncation>, with
$$
\boxed{r=\frac{\alpha^2}{\beta^6}R,\qquad r_\Omega=\frac{\pi}{\beta^3}R_\Omega,\qquad h=\varpi=\frac{4\pi^2}{\beta^2}.}
$$
The squared rotation parameter is $q=r_\Omega^2\ge0$. The printed $R_\Omega$ is proportional to rotation rate, so it is a square-root-type rotation parameter rather than the more usual <Taylor number> proportional to squared rotation rate; the algebra uses the coefficient as printed. At zero rotation the physical transverse velocity modes vanish; the normalized $d,e$ equations can still be understood through their continuous zero-rotation limit.
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