Horizontal-field magnetoconvection dispersion relation ($\Lambda (s+\Lambda )[(s+\sigma \Lambda )(s+\zeta \Lambda )+\sigma \zeta Q{m}^2]-\sigma R{a}_h^2(s+\zeta \Lambda )=0$)

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 satisfyingThe 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 $R={\Lambda }^3/a_h^2+Q{m}^2\Lambda /a_h^2$.

// Target: fluid-mechanics.bigb

Restrict the horizontal-field magnetoconvection dispersion relation to $k=0$, $x=m^2>0$. The steady neutral curve becomes $R={\pi }^{2}Q+(Q+3{\pi }^2)x+x^2+3{\pi }^4+{\pi }^6/x$. Its unique minimum satisfies $x^2(Q+3{\pi }^2+2x)={\pi }^6$, since $R^{\prime \prime }(x)=2+2{\pi }^6/x^3>0$. Expanding at large Chandrasekhar number gives $x\sim {\pi }^3{Q}^{-1/2}$ and the stated asymptotic expansion. Unlike field-aligned rolls, these disturbances bend the field, and their stationary threshold grows with field strength.

// Target: fluid-mechanics.bigb

At a fixed horizontal wavenumber magnitude, the steady neutral curve in the horizontal-field magnetoconvection dispersion relation is minimized by $m=0$. The disturbance is then constant along the imposed magnetic field; it induces no field bending and incurs no magnetic tension penalty. Minimizing $(k^2+{\pi }^2{)}^3/k^2$ gives the displayed stationary threshold for every $Q\ge 0$. These convection rolls have axes parallel to the imposed field. This assertion concerns steady onset.

// Target: fluid-mechanics.bigb