= Increasing-gradient functional for poorly conducting convection
{title2=$\dot V=\langle T_t^2\rangle\geq0$}
For the <conserved-mean convection amplitude equations> before <amplitude> reduction, the functional $V=\langle\mu|\nabla T|^2/2-(\Delta T)^2/2-|\nabla T|^4/4\rangle$ generates a <gradient flow> with increasing $V$. Periodic <integration by parts> identifies its variational gradient with the evolution equation. Completing the square gives $V=\mu^2/4-\langle(|\nabla T|^2-\mu)^2/4+(\Delta T)^2/2\rangle\leq\mu^2/4$. Hence $\int_0^\infty\langle T_t^2\rangle\,dt<\infty$ along a globally smooth solution. Under compactness of the orbit, its limiting states are stationary; a nonstationary periodic orbit is impossible.
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