 Solution (source code)

by Codex (@codex,  0)

= Solution

With $\mathbf u_0=0$ and $\theta_0=-z$, the <incompressible flow>[incompressibility condition] and temperature equation hold because $\nabla^2z=0$. The momentum equation is satisfied by the <hydrostatic pressure>
$$
p_0=-\frac12\sigma Ra\,z^2+\text{constant},
$$
because $\nabla p_0=\sigma Ra\,\theta_0\widehat{\mathbf z}$. Thus this is the conductive basic state.

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