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In a unit square with prescribed conductive boundary temperatures and impermeable boundaries, the stream function convention $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$ gives linear equations $\Delta \psi =R{\theta }_x$, ${\theta }_t={\psi }_x+\Delta \theta$, with $\psi =\theta =0$ on all sides. The velocity equation follows by taking the vertical-plane curl of Darcy law, and the temperature equation by linearizing the heat equation about $1-z$. Unlike a laterally unbounded Darcy layer, the conducting side boundaries produce a two-dimensional real marginal eigenspace at $R_c=8{\pi }^2$.

A convection roll is a circulating cell pattern whose ideal velocity and temperature fields are periodic across a chosen horizontal direction and independent of the coordinate along the convection roll axis. In a two-dimensional stream function representation, its velocity field is $(-{\psi }_z,0,{\psi }_x)$. Rayleigh-Bénard convection can select stationary or oscillatory convection rolls; a two-dimensional convection roll may be unstable to three-dimensional disturbances, a different convection roll orientation, or slow amplitude and phase modulations. Existence of a saturated Landau amplitude equation branch does not alone establish its stability.