Strong-field wavenumber selection in magnetoconvection ($k_{\ast }\propto Q^{1/6}$)

Both steady and oscillatory neutral curves have the form $R=C(k^2+p{)}^3/k^2+DQp(k^2+p)/k^2$, with $p={\pi }^2$. Their exact minimizing condition is $x^2(2x+3p)=p^3+(D/C)Q{p}^2$, where $x=k^2$. For fixed positive coefficients and large Chandrasekhar number, $x\sim [D{p}^{2}Q/(2C){]}^{1/3}$ and $R_{\min }=DpQ+3C^{1/3}(D{p}^{2}Q/2{)}^{2/3}+O(Q^{1/3})$. The optimum balances transverse diffusion against the magnetic penalty, leading to increasingly narrow rolls.