= Rotating-convection growth-rate polynomial
For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal <wavenumber> $k$ and vertical index $n$, put $a=k^2+n^2\pi^2$ and $P=\operatorname{Pr}$. Vertical <velocity>, <temperature> and vertical <vorticity> amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=Rk^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta=W$. Taking their determinant gives
$$
(s+a)[a(s/P+a)^2+\mathrm{Ta}\,n^2\pi^2]-Rk^2(s/P+a)=0.
$$
This is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.
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