 Supercritical saturation of Darcy convection rolls

For $R=4{\pi }^2+{ϵ}^2$, fixed roll phase, $T={ϵ}^{2}t$ and normalization $\psi =ϵA(T)\cos (\pi x)\sin (\pi z)+\cdots$, the Landau amplitude equation isThe mean-temperature correction in weakly nonlinear Darcy convection supplies the cubic negative feedback. Projection of the third-order equations onto the critical eigenfunction via the Fredholm solvability condition for a self-adjoint operator fixes both coefficients. The stable nonzero amplitudes are $\pm 2/\pi$ within the chosen phase, giving a supercritical pitchfork bifurcation in this real-amplitude reduction.