= Insulated square-column Darcy convection onset
{title2=$\Gamma_c=\pi^2\kappa/(Ba^2)$}
A tall porous column with square width $a$, insulated impermeable sidewalls and a maintained upward temperature decrease $\Gamma$ admits nearly vertical, zero-net-flux overturning modes. Let $B=\rho_0g\alpha K/\mu$ be buoyancy mobility and $\kappa=k_e/(\rho_0c_f)$ the thermal transport coefficient accompanying <Darcy flux>. Neumann transverse modes have $k_\perp^2=\pi^2(m^2+n^2)/a^2$. In the long-column limit the lowest nonconstant modes have growth rate proportional to $B\Gamma-\kappa\pi^2/a^2$. They are warm ascending and cold descending columns with return flow at distant ends. The two lowest modes, varying along either side direction, are degenerate; linear theory alone does not select their nonlinear superposition.
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