= Supercritical saturation of Darcy convection rolls
For $R=4\pi^2+\epsilon^2$, fixed roll phase, $T=\epsilon^2t$ and normalization $\psi=\epsilon A(T)\cos(\pi x)\sin(\pi z)+\cdots$, the <Landau amplitude equation> is
$$
A_T=\frac12A-\frac{\pi^2}{8}A^3.
$$
The <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 $\pm2/\pi$ within the chosen phase, giving a supercritical <pitchfork bifurcation> in this real-amplitude reduction.
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