= Oscillatory neutral curve of rotating convection
For $s=i\omega$, real-imaginary separation of the rotating-convection cubic gives
$$
\omega^2=P^2\left[\frac{1-P}{1+P}\frac{\mathrm{Ta}\,n^2\pi^2}{a}-a^2\right],\qquad R_o=\frac{2(1+P)a^3+2P^2\mathrm{Ta}\,n^2\pi^2/(1+P)}{k^2}.
$$
The curve is admissible only if $\omega^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive <angular frequency> squared is not a <Hopf bifurcation>. At zero <angular frequency> this curve meets the stationary curve for that fixed <wavenumber> in a double-zero limit.
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