= Horizontal-field magnetoconvection dispersion relation
{title2=$\Lambda(s+\Lambda)[(s+\sigma\Lambda)(s+\zeta\Lambda)+\sigma\zeta Qm^2]-\sigma Ra_h^2(s+\zeta\Lambda)=0$}
For a unit-depth <Boussinesq> layer with <stress-free boundary conditions>, fixed boundary temperatures, imposed field along $y$, and horizontal magnetic perturbations at the boundaries, put $a_h^2=k^2+m^2$ and $\Lambda=a_h^2+\pi^2$. Poloidal vertical velocity, temperature and vertical field proportional to $e^{st+ikx+imy}\sin\pi z$ have amplitudes satisfying
$$
\Lambda(s/\sigma+\Lambda)W=Ra_h^2\Theta+\zeta Qim\Lambda C,\qquad
(s+\Lambda)\Theta=W,\qquad(s+\zeta\Lambda)C=imW.
$$
The <determinant> gives the stated <dispersion relation>. Here $\sigma$ is the <Prandtl number>, $\zeta$ the <magnetic-to-thermal diffusivity ratio>, $Q$ the <Chandrasekhar number> and $R$ the <Rayleigh number>. The steady neutral curve is $R=\Lambda^3/a_h^2+Qm^2\Lambda/a_h^2$.
// Target: fluid-mechanics.bigb
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