= Stokes temperature-slaving operator
{c}
{title2=$\mathcal W_R=Rk^2(D^2-k^2)^{-2}$}
For a nonzero horizontal <Fourier mode> $k$, solve $(D^2-k^2)^2w=Rk^2\theta$, $w=w''=0$, to obtain $w=\mathcal W_R\theta$. Under homogeneous <temperature> <Dirichlet boundary conditions>, the sine modes diagonalize this <self-adjoint operator>, with <eigenvalues> $Rk^2/(k^2+n^2\pi^2)^2$. The <temperature> <linear operator> is $\mathcal L_R=D^2-k^2+\mathcal W_R$. At a marginal mode $g=\sin(n\pi z)$ it has $\mathcal W_Rg=(k^2+n^2\pi^2)g$.
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