= Solution
Linearization about $w=0$ removes the quadratic term and gives
$$
w_t=\Delta^3w-R\Delta_1w.
$$
The three homogeneous boundary conditions select the vertical <normal modes>
$$
\boxed{W_j(z)=A_j\sin(j\pi z)},
\qquad j=1,2,\ldots.
$$
Take a horizontal <Laplacian eigenfunction> satisfying
$$
\Delta_1f+a^2f=0.
$$
On $W_jf$, the full Laplacian has eigenvalue $-(a^2+j^2\pi^2)$. The <growth rate> is consequently
$$
\boxed{s(a,R,j)=Ra^2-(a^2+j^2\pi^2)^3}.
$$
Neutrality occurs at
$$
R_{c,j}(a)=\frac{(a^2+j^2\pi^2)^3}{a^2}.
$$
For fixed nonzero $a$, this increases strictly with $j$, so the first unstable vertical mode is $j=1$ and
$$
\boxed{R_c(a)=\frac{(a^2+\pi^2)^3}{a^2}}.
$$
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