 Critical Rayleigh number for an isothermal layer

For fixed temperatures and stress-free impermeable boundaries, linear marginal Rayleigh-Bénard convection with dimensionless horizontal wavenumber $q$ obeys $\mathrm{Ra}(q)=(q^2+{\pi }^2{)}^3/q^2$. Differentiation gives $q_c=\pi /\sqrt{2}$ and the displayed minimum. The threshold depends on the mechanical boundary conditions: two rigid no-slip boundaries instead give approximately $1708$.