= Conducting-square Darcy convection
{title2=$R_c=8\pi^2$}
In a unit square with prescribed conductive boundary <temperatures> and impermeable boundaries, the <stream function> convention $\mathbf u=(-\psi_z,0,\psi_x)$ gives linear equations $\Delta\psi=R\theta_x$, $\theta_t=\psi_x+\Delta\theta$, with $\psi=\theta=0$ on all sides. The <velocity> equation follows by taking the vertical-plane <curl> of <Darcy law>, and the <temperature> equation by linearizing the <heat equation> about $1-z$. Unlike a laterally unbounded Darcy layer, the conducting side boundaries produce a two-dimensional real marginal <eigenspace> at $R_c=8\pi^2$.
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