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Let the layer depth be $d$, its imposed temperature drop $\Delta T$, and its dimensional vertical field $B_0\hat{\mathbf{z}}$. Use length $d$, thermal time $d^2/\kappa$, velocity $\kappa /d$, temperature $\Delta T$ and field $B_0$ as units. Write the total dimensionless field as $\hat{\mathbf{z}}+\mathbf{b}$, and temperature as the conductive profile plus $\theta$. The Boussinesq equations coupled to the resistive induction equation areThe dimensionless parameters in this normalization areHere ${\alpha }_T$ is the thermal expansion coefficient, $\nu$ is kinematic viscosity, $\kappa$ is thermal diffusivity, and $\eta$ is magnetic diffusivity. The first parameter is the Rayleigh number and $\sigma$ is the Prandtl number.

The displayed dispersion relation fixes the magnetic normalization: its $Q$ must be the thermal-diffusion magnetic-field parameter above. The conventional Chandrasekhar number is instead $Q_{\mathrm{Ch}}=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$, so $Q=\zeta Q_{\mathrm{Ch}}$. If $Q$ is reserved for that conventional definition, the magnetic term in the printed polynomial requires an extra factor $\zeta$. We keep $Q$ for the printed polynomial throughout the following calculation.

Linearize about the conductive state. Absorbing linear magnetic pressure into $p$, the magnetic force becomes $\sigma Q{\partial }_z\mathbf{b}$. The pressure-eliminated vertical momentum equation isThe normal mode implicit in the given relation has $w,\theta \propto \sin \pi z$ and $b_z\propto \cos \pi z$, as for stress-free, fixed-temperature boundaries with the corresponding vertical-field magnetic conditions. Put $x=k^2$, $p_0={\pi }^2$ and $m=x+p_0={\beta }^2$. Taking growth proportional to $e^{st+i\mathbf{k}\cdot {\mathbf{x}}_h}$ gives the amplitude equationsTheir determinant isThis is the stated vertical-field magnetoconvection dispersion relation after accounting for its field-strength convention. The determinant derivation remains valid even at values where an intermediate division by $s+m$ or $s+\zeta m$ would not be allowed.

For steady marginality set $s=0$. One obtainsFor oscillatory marginality write $s=i\omega$ with $\omega \ne 0$. Dividing the polynomial by $m$ gives a real cubic $s^3+A{s}^2+Bs+C_0$, whereIts real and imaginary parts imply ${\omega }^2=B$ and $C_0=AB$. Solving for the oscillatory convection threshold givesandA physical oscillatory branch therefore requires $0<\zeta <1$ and positive frequency squared. In particular an algebraic continuation of $R_o$ into ${\omega }^2<0$ is not an oscillatory onset.

The steady-Hopf merger in vertical-field magnetoconvection occurs at zero frequency:Substitution verifies $R_s=R_o$. Equivalently the merged horizontal wavenumber satisfiesA finite-wavenumber intersection exists only when this expression is positive. At the intersection the growth polynomial has a double zero root: the nonzero-frequency neutral branch ends there.

For the requested merger at a magnetoconvection neutral-curve minimum, first note that a curve of the form $R=[A_{\ast }{m}^3+C_{\ast }m]/x$ has derivative zero exactly whenIts unique positive minimum has $x>p_0/2$, because the left side increases there from zero to infinity. At this minimum $R_c=2A_{\ast }{m}^3/p_0$.

Merger at the steady minimum. For $R_s$, the stationarity condition is $m_s^2(2x_s-p_0)=Q{p}_0^2/\zeta$. Combining it with the merger equation givesThe corresponding critical Rayleigh number is $R_{s,c}=2(x_s+p_0{)}^3/p_0$. For fixed positive $\sigma$ and large $Q$, set $\zeta =1-{\delta }_s$ and retain the leading powers. ThenMore generally, before taking $\zeta \to 1$, the steady minimum obeys $k_s^2\sim [Q{\pi }^4/(2\zeta ){]}^{1/3}$.

Merger at the oscillatory minimum. Define $A_o=(\sigma +\zeta )(1+\zeta )/\sigma$. The stationarity condition for $R_o$ isCombining it with the zero-frequency relation instead givesHere $R_{o,c}=2A_o(x_o+p_0{)}^3/p_0$. For fixed $\sigma >0$ and $Q\gg 1$, setting $\zeta =1-{\delta }_o$ givesThese two merger conditions are distinct parameter surfaces, not simultaneous assertions about the same minimum. Both have the strong-field wavenumber selection in magnetoconvection scaling $k=O(Q^{1/6})$.

If the first subleading Rayleigh corrections are desired, let $X_s=[Q{p}_0^2/(2\zeta ){]}^{1/3}$ and $X_o=[\sigma Q{p}_0^2/(2(1+\sigma )(1+\zeta )){]}^{1/3}$. With parameters bounded in the above regime,These retain the $O(Q^{2/3})$ corrections, including the effect of the $O(Q^{-1/3})$ diffusivity deficit at a merger.