Oscillatory marginality in vertical-field magnetoconvection (${\omega }^2=\sigma \zeta (1-\zeta )Q{\pi }^2/(1+\sigma )-{\zeta }^2{s}^2$)

A nonzero imaginary root of the vertical-field magnetoconvection dispersion relation requires $0<\zeta <1$ and $Q>\zeta (1+\sigma )s^2/[\sigma (1-\zeta ){\pi }^2]$. Its marginal curve is $R^{(o)}=C{s}^3/k^2+DQ{\pi }^{2}s/k^2$, with $C=(\sigma +\zeta )(1+\zeta )/\sigma$ and $D=\zeta (\sigma +\zeta )/(1+\sigma )$. A formal minimum of that algebraic curve is a physical oscillatory threshold only if the frequency squared there is positive.

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.

At a steady-curve minimum, differentiation and the steady-Hopf merger in vertical-field magnetoconvection give the first displayed wavenumber. At an oscillatory-curve minimum they give the second. For fixed positive Prandtl number $\sigma$ and large $Q_{\kappa }$, the corresponding merger diffusivity deficits are $1-{\zeta }_s\sim [(1+\sigma )/\sigma ]({\pi }^2/(4Q_{\kappa }){)}^{1/3}$ and $1-{\zeta }_o\sim [(1+\sigma ){\pi }^2/(16\sigma Q_{\kappa }){]}^{1/3}$. The two parameter surfaces are distinct: coincidence of neutral curves at one wavenumber does not imply coincidence of their minimizing wavenumbers.