= Mean-temperature correction in weakly nonlinear Darcy convection
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 $\psi_2=0$ and
$$
\theta_2=-\frac1{16\pi}\sin(2\pi z).
$$
This modifies the vertical temperature gradient. Its interaction with the critical roll produces the cubic term in the <amplitude equation>.
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