 Insulated square-column Darcy convection onset

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 $B={\rho }_{0}g\alpha K/\mu$ be buoyancy mobility and $\kappa =k_e/({\rho }_0{c}_f)$ 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.