 Steady-Hopf merger in vertical-field magnetoconvection

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.