= Solution
Write $s=\alpha^2+\pi^2$ and use modes satisfying all <boundary conditions>,
$$
\psi=a(t)\sin\alpha x\sin\pi z,\qquad
\theta=b(t)\cos\alpha x\sin\pi z+c(t)\sin2\pi z+\cdots.
$$
The quadratic <temperature> distortion $c$ is retained for the saturation calculation; it does not enter the linear onset. Substitution of the primary modes gives
$$
\frac d{dt}\begin{pmatrix}a\\b\end{pmatrix}
=\begin{pmatrix}-\sigma(s+Q\pi^2/s)&\sigma R\alpha/s\\\alpha&-s\end{pmatrix}
\begin{pmatrix}a\\b\end{pmatrix}.
$$
The <determinant> vanishes when
$$
\boxed{R_c(Q,\alpha)=\frac{s(s^2+Q\pi^2)}{\alpha^2}.}
$$
The <trace> is $-[s+\sigma(s+Q\pi^2/s)]<0$. Thus no conjugate <eigenvalues> can cross the imaginary axis away from zero, and \b[the initial instability is stationary, not oscillatory]. Indeed for $R\ge0$ the <discriminant> is $[\sigma(s+Q\pi^2/s)-s]^2+4\sigma R\alpha^2/s>0$. This is <quasistatic vertical-field magnetoconvection>: there is no independent magnetic evolution mode that could produce a magnetic-thermal <Hopf bifurcation> instability.
For a fixed box the other allowed horizontal <wavenumbers> are $n\pi/L$, and the actual first threshold is the minimum of their corresponding $R_c$ values. Vertical index $m$ replaces $\pi^2$ by $m^2\pi^2$ and strictly increases the threshold at fixed horizontal <wavenumber>, so $m=1$ is selected. The displayed formula describes the prescribed fundamental roll; part (b) optimizes its width.
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