= Exact critical Rayleigh relation for vertical-field magnetoconvection
{title2=$Q\pi^2=R_c-R_c^{2/3}R_0^{1/3}$}
At the minimum of the steady <vertical-field magnetoconvection dispersion relation>, stationarity gives $R_c=2(k_c^2+\pi^2)^3/\pi^2$. With nonmagnetic value $R_0=27\pi^4/4$, eliminating the critical <wavenumber> gives the stated relation and $k_c^2=\pi^2[\tfrac32(R_c/R_0)^{1/3}-1]$. These hold for every nonnegative <Chandrasekhar number>, including its zero-field limit.
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