 Oscillatory neutral curve of rotating convection

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe 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.