= Stress-free convection growth-rate polynomial
{title2=$(\sigma+\mathrm{Pr}a^2)(\sigma+a^2)=\mathrm{Ra}\,\mathrm{Pr}\lambda^2/a^2$}
For a fixed-temperature layer with <stress-free boundary conditions>, a vertical <Fourier sine series> has $a^2=\lambda^2+n^2\pi^2$, where $\lambda$ is horizontal <wavenumber>. The coupled vertical <velocity> and temperature amplitudes satisfy the displayed quadratic for their <growth rate> $\sigma$. Its roots are real for positive <Rayleigh number>, while both roots have negative real parts for negative <Rayleigh number>. Setting $\sigma=0$ gives the <free-slip convection neutral curve>.
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