Five-mode vertical-field magnetoconvection

A Galerkin method approximation can retain one roll streamfunction mode, two temperature modes and two magnetic-flux perturbation modes in vertical-field magnetoconvection. Projection onto these five modes, rather than pointwise equality of the truncated fields, gives a closed system of five ordinary differential equations. Products generate higher spatial harmonics that are discarded by orthogonal projection. The system has symmetry $(a,b,c,d,e)\mapsto (-a,-b,c,-d,e)$ and hence admits paired steady roll branches.

The conduction linearization has two negative eigenvalues and a cubic factor $s^3+A{s}^2+Bs+C$, where $A=\sigma +1+\zeta$, $B=\sigma (1-r)+\sigma \zeta (1+q)+\zeta$, $C=\sigma \zeta (1+q-r)$. A double zero requires $B=C=0$, giving the displayed Bogdanov–Takens bifurcation location for $0<\zeta <1$. The remaining cubic eigenvalue is $-A$. The imaginary-pair condition $AB=C$ yields $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$ and ${\omega }^2=\sigma \zeta (1-\zeta )q/(\sigma +1)-{\zeta }^2$. Thus the oscillatory onset exists precisely for $q>q_{\ast }$ when $0<\zeta <1$.

For positive diffusivity ratio $\zeta$ and $0<\phi <4$, define $\mu =(4-\phi ){\zeta }^2/\phi$. The nonzero steady branch has $b=a/(1+a^2)$, $c=a^2/(1+a^2)$, $d=\mu a/[\zeta (\mu +a^2)]$, and $e=a^2/(\mu +a^2)$. Substitution yields the displayed branch through $r=1+q$. Its initial slope in $a^2$ has the sign of $\phi q(2-\phi )+(4-\phi ){\zeta }^2(1+q)$. Positive slope is a supercritical pitchfork bifurcation direction, negative slope a subcritical pitchfork bifurcation direction. This geometric criterion alone does not assert stability against every perturbation.

Put $m=(4-\phi ){\zeta }^2/\phi$. The derivative of the stationary branch of five-mode magnetoconvection at $a^2=0$ is $r_2=1+q+q(2-\phi )/m$. For fixed positive $q$ and small $\zeta$, it changes from a forward to a backward pitchfork bifurcation direction at the displayed value. Expanding the exact numerator $q\phi (2-\phi )+(4-\phi ){\zeta }^2(1+q)$ gives this result. The expansion in amplitude requires $a^2\ll m$; branch direction alone does not determine stability against all other modes.