 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 $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis 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.