= Solution
Write $\mathbf u=\mathbf u'$ and $\theta=-z+\theta'$. Dropping quadratic perturbation terms gives the <Linearized Boussinesq equations>
$$
\partial_t\mathbf u'+\lambda\widehat{\mathbf z}\times\mathbf u'+\nabla p'
=\sigma Ra\,\theta'\widehat{\mathbf z}+\sigma\nabla^2\mathbf u',
\qquad
\nabla\mathbin\cdot\mathbf u'=0,
$$
and
$$
\partial_t\theta'-W=\nabla^2\theta',
\qquad W=\mathbf u'\mathbin\cdot\widehat{\mathbf z}.
$$
The fixed temperatures give $\theta'=0$ at $z=0,1$. Impermeable <stress-free boundary conditions> give
$$
W=D^2W=0,
\qquad
D\omega=0,
\qquad
\omega=\widehat{\mathbf z}\mathbin\cdot\nabla\times\mathbf u'.
$$
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