= Growth-rate solvability for conducting-square Darcy convection
{title2=$\lambda=(2q_1/q_c)\|\nabla\theta_0\|_2^2/\|\theta_0\|_2^2$}
For $q=q_c+\epsilon q_1$, $\theta_t=\epsilon\lambda\theta$, the first-order equations are $\Delta\psi_1-q_c^2\theta_{1x}=2q_cq_1\theta_{0x}$ and $\psi_{1x}+\Delta\theta_1=\lambda\theta_0$. Multiply them by $\psi_0,q_c^2\theta_0$ and integrate. <Integration by parts> and the leading equations cancel the homogeneous correction terms, giving $2q_cq_1\langle\psi_0\theta_{0x}\rangle+q_c^2\lambda\langle\theta_0^2\rangle=0$. Since $\langle\psi_0\theta_{0x}\rangle=-\langle|\nabla\theta_0|^2\rangle$, the displayed title formula follows. The reciprocal norm ratio is incompatible with this <solvability condition>. Reflection parity diagonalizes the first-order splitting of the two marginal modes.
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