Quasistatic vertical-field magnetoconvection ($R_c=\frac{s(s^2+Q{\pi }^2)}{{\alpha }^2},s={\alpha }^2+{\pi }^2$)

When the magnetic field supplies instantaneous damping rather than an evolving magnetic perturbation, the roll amplitudes obey a two-variable linear system with trace $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Its determinant vanishes at the displayed stationary threshold. Therefore its initial instability cannot be a Hopf bifurcation. This reduced model must not be confused with a three-variable magnetoconvection model carrying finite magnetic diffusivity and an additional oscillatory mode.

The primary roll and temperature modes force only a horizontally uniform temperature mode proportional to $\sin (2\pi z)$ at quadratic order. Its amplitude is $c=-\alpha ab/(8\pi )$. Its feedback on the primary temperature mode saturates magnetoconvection, giving the displayed positive squared amplitude above threshold. Thus a simple roll onset is a supercritical pitchfork bifurcation for every $Q\ge 0$. The three-mode Fourier truncation and full perturbation calculation agree through the cubic solvability condition because all forced quadratic modes are retained; this is not an assertion that the truncation gives an exact finite-amplitude PDE solution.

// Destination: dynamical-systems.bigb