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Both neutral curves can be written with $x=k^2$ and $p={\pi }^2$ asFor steady convection, $(C,D)=(1,1)$; for oscillatory convection,The oscillatory result is physically relevant only with positive frequency; for fixed $0<\zeta <1$ and fixed positive $\sigma$, that holds at the minimizing wavenumber for sufficiently large $Q$.

Differentiation gives the exact minimization conditionIts left side is strictly increasing for $x>0$, so the minimum is unique. Define $h=[D{p}^{2}Q/(2C){]}^{1/3}$. An asymptotic expansion givesSubstituting back, the leading magnetic term is order $Q$, while the two balanced terms $C{x}^2$ and $DQ{p}^2/x$ sum to $3C{h}^2$. Hence, through order $Q^{2/3}$,In particular,while the same boxed general formulas with $(C_o,D_o)$ give the oscillatory minimum and critical wavenumber. Thus strong-field wavenumber selection in magnetoconvection has $k\propto Q^{1/6}$ for both types, with different parameter-dependent constants.

For the exact steady result at arbitrary $Q\ge 0$, the stationarity equation with $C=D=1$ lets us eliminate $Q$ from $R^{(e)}$. It givesAt $Q=0$, $x=p/2$ and $R_0=27p^2/4$. ConsequentlyThis exact critical Rayleigh relation for vertical-field magnetoconvection includes the nonmagnetic minimum, with $R_c\ge R_0$ and $k_c^2\ge {\pi }^2/2$.