Wavenumber selection in oblique-field magnetoconvection ($k_c\to -\cot \alpha (Q\to \infty )$)

For fixed vertical wavenumber one, write the stationary threshold as $F(k)+QG(k)$, with $F=(1+k^2{)}^3/k^2$ and $G=(1+k^2)(\cos \alpha +k\sin \alpha {)}^2/k^2$. The nonmagnetic minima are $k=\pm 1/\sqrt{2}$ with $F=27/4$; a small positive field selects the negative one. Expanding the stationarity condition gives $k_c=a-Q{G}^{\prime }(a)/F^{\prime \prime }(a)+O(Q^2)$, $a=-1/\sqrt{2}$, andFor large $Q$, the magnetic penalty forces $k$ toward its unique zero $b=-\cot \alpha$. Since $G^{\prime \prime }(b)>0$, $k_c=b-F^{\prime }(b)/[Q{G}^{\prime \prime }(b)]+O(Q^{-2})$ and $R_c={\csc }^6\alpha /{\cot }^2\alpha -F^{\prime }(b{)}^2/[2Q{G}^{\prime \prime }(b)]+O(Q^{-2})$. Thus strongly magnetized rolls align their wavevector perpendicular to the imposed field, rather than continuing to raise their stationary threshold proportionally to field strength.