= Vertical-field magnetoconvection dispersion relation
{title2=$(\lambda+\sigma s)(\lambda+s)(\lambda+\zeta s)+\sigma Q\zeta\pi^2(\lambda+s)-\sigma Rk^2(\lambda+\zeta s)/s=0$}
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 $R^{(e)}=[s^3+Q\pi^2s]/k^2$.
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