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At first order, ${\nabla }^2{\psi }_1+R_c{\theta }_{1x}=0$ and ${\nabla }^2{\theta }_1-{\psi }_{1x}=0$. Since ${\nabla }^2{\psi }_1=-2{\pi }^2{\psi }_1$ and $R_c=4{\pi }^2$, the first equation fixes ${\theta }_{1x}={\psi }_1/2$. The homogeneous temperature boundary conditions and the second equation remove any horizontally uniform term, givingAt second order,The forcing is horizontally uniform. Taking no additional critical-mode component, a solution satisfying the homogeneous boundaries isThis mean-temperature correction in weakly nonlinear Darcy convection modifies the vertical temperature gradient and provides the feedback that saturates the convection.

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.