Increasing-gradient functional for poorly conducting convection ($\dot{V}=⟨T_t^2⟩\ge 0$)

For the conserved-mean convection amplitude equations before amplitude reduction, the functional $V=⟨\mu |\nabla T{|}^2/2-(\Delta T{)}^2/2-|\nabla T{|}^4/4⟩$ generates a gradient flow with increasing $V$. Periodic integration by parts identifies its variational gradient with the evolution equation. Completing the square gives $V={\mu }^2/4-⟨(|\nabla T{|}^2-\mu {)}^2/4+(\Delta T{)}^2/2⟩\le {\mu }^2/4$. Hence ${\int }_0^{\infty }⟨T_t^2⟩dt<\infty$ along a globally smooth solution. Under compactness of the orbit, its limiting states are stationary; a nonstationary periodic orbit is impossible.