 Vertical-field magnetoconvection dispersion relation

For a stress-free, fixed-temperature layer with vertical magnetic perturbation at its boundaries, take vertical velocity and temperature proportional to $\sin \pi z$, vertical magnetic perturbation proportional to $\cos \pi z$, and horizontal magnetic perturbations proportional to $\sin \pi z$. With $s=k^2+{\pi }^2$, the vertical velocity, temperature and magnetic amplitudes form a three-variable linear system whose determinant is the stated polynomial. The Prandtl number is $\sigma$, magnetic-to-thermal diffusivity ratio is $\zeta$, and field strength is the Chandrasekhar number $Q$. The steady marginal curve is $R^{(e)}=[s^3+Q{\pi }^{2}s]/k^2$.