= Wavenumber selection in oblique-field magnetoconvection
{title2=$k_c\to-\cot\alpha\quad(Q\to\infty)$}
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 $k=\pm1/\sqrt2$ with $F=27/4$; a small positive field selects the negative one. Expanding the stationarity condition gives $k_c=a-QG'(a)/F''(a)+O(Q^2)$, $a=-1/\sqrt2$, and
$$
R_c=\frac{27}{4}+3Q(\cos\alpha-\sin\alpha/\sqrt2)^2-\frac{G'(a)^2}{2F''(a)}Q^2+O(Q^3).
$$
For large $Q$, the magnetic penalty forces $k$ toward its unique zero $b=-\cot\alpha$. Since $G''(b)>0$, $k_c=b-F'(b)/[QG''(b)]+O(Q^{-2})$ and $R_c=\csc^6\alpha/\cot^2\alpha-F'(b)^2/[2QG''(b)]+O(Q^{-2})$. Thus strongly magnetized rolls align their wavevector perpendicular to the imposed field, rather than continuing to raise their stationary threshold proportionally to field strength.
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