 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 ${\psi }_2=0$ andThis modifies the vertical temperature gradient. Its interaction with the critical roll produces the cubic term in the amplitude equation.