= Steady-Hopf merger in vertical-field magnetoconvection
{title2=$Q_\kappa\pi^2=\zeta^2(1+\sigma)(k^2+\pi^2)^2/[\sigma(1-\zeta)]$}
For thermal-time <magnetoconvection>, a physical oscillatory neutral mode has $\omega^2=\sigma(1-\zeta)Q_\kappa\pi^2/(1+\sigma)-\zeta^2(k^2+\pi^2)^2>0$. The steady and oscillatory neutral curves meet when this frequency tends to zero. Thus $0<\zeta<1$ is necessary and the displayed relation locates the merger. A formal oscillatory-curve minimum with negative frequency squared is not a physical onset.
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