Strong-field steady magnetoconvection varying along the field ($m_c^4\sim {\pi }^6/Q,R_c={\pi }^{2}Q+2{\pi }^3\sqrt{Q}+3{\pi }^4+o(1)$)

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