 Oscillatory marginality in vertical-field magnetoconvection

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.