= Three-mode porous magnetoconvection
{title2=$\dot a=\sigma(r-1)a-\sigma\zeta qb,\quad\dot b=a-\zeta b-ac,\quad\dot c=-\zeta c+3ab$}
A <Galerkin method> using one convective <streamfunction> mode and two <Cartesian magnetic flux function> modes yields the displayed system. With $\alpha=\pi/\sqrt3$, use $\psi=4\sqrt2a\sin(\alpha x)\sin(\pi z)$ and $A=(3\sqrt2/\pi)b\sin(\alpha x)\cos(\pi z)+(\sqrt3/\pi)c\sin(2\alpha x)$, time $\tau=(4\pi^2/3)t$, $q=9Q/(16\pi^2)$ and $r=9R/(64\pi^4)$. The origin's planar trace is $\sigma(r-1)-\zeta$ and determinant $\sigma\zeta(1+q-r)$; the third <eigenvalue> is $-\zeta$. The stationary threshold is $r=1+q$, and the oscillatory threshold is $r=1+\zeta/\sigma$ when $q>\zeta/\sigma$. They meet in a symmetric double-zero point at $(q,r)=(\zeta/\sigma,1+\zeta/\sigma)$. Higher spatial harmonics generated by the quadratic terms are discarded, so this is a projected approximation rather than an exact invariant subspace of the original equations.
Back to article page
