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.

For ${\psi }_1=\cos (\pi x)\sin (\pi z)$, the first temperature mode is ${\theta }_1=(2\pi {)}^{-1}\sin (\pi x)\sin (\pi z)$. The streamfunction advection bracket is $J({\psi }_1,{\theta }_1)=(\pi /4)\sin (2\pi z)$, independent of $x$. The second-order equations therefore have ${\psi }_2=0$ andThis modifies the vertical temperature gradient. Its interaction with the critical roll produces the cubic term in the amplitude equation.