= Transverse energy instability of nonconstant temperature rolls
{title2=$V[T_0(x)+\delta T_0(y)]-V[T_0(x)]=\frac{\delta^2}{2}\operatorname{Var}(T_0'^2)+O(\delta^4)$}
A nonconstant smooth periodic roll obeys $\mu\langle T_0'^2\rangle-\langle T_0''^2\rangle-\langle T_0'^4\rangle=0$. The orthogonal-roll perturbation raises the <increasing-gradient functional for poorly conducting convection> by the displayed amount. The <variance> is strictly positive: a nonconstant periodic function has a derivative that vanishes somewhere and is nonzero somewhere. The <Hessian> of $V$ therefore has a positive direction. Its self-adjoint linearized evolution operator has a positive <Rayleigh quotient> and hence a positive <eigenvalue>, proving transverse <linear instability> wherever a nonconstant roll exists.
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