Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 4 / a

Write $\alpha =\pi /\sqrt{3}$ and $L={\alpha }^2+{\pi }^2=4{\pi }^2/3$. The primary streamfunction mode describes a single roll spanning the box, while the first magnetic mode describes the distortion of the vertical field by that roll. The second magnetic mode is independent of height and has twice the horizontal wavenumber, as generated by interaction of the first two modes. TakeThese satisfy all the stated boundary conditions: the sine factors vanish on the vertical walls, $\psi$ and its Laplacian vanish on every wall, and $A_z$ vanishes at the horizontal walls.

Let $J(f,g)=f_x{g}_z-f_z{g}_x$. Since ${\nabla }^4\psi =L^2\psi$, $J(\psi ,{\nabla }^4\psi )=0$. The magnetic Jacobians areandProjecting onto the three retained Fourier modes and discarding the generated third horizontal harmonic yields, in the original time,Here $4{\alpha }^2=L$, so the second magnetic mode has the same linear diffusion rate as the first.

Choose $\tau =Lt$ and normalize the amplitudes byFor example the coefficient of $a$ in $b^{\prime }$ is $(\pi /L)(4\sqrt{2})/(3\sqrt{2}/\pi )=1$, and that of $ab$ in $c^{\prime }$ is $3$. The linear momentum coefficients become $r=R{\alpha }^2/L^3=9R/(64{\pi }^4)$ and $q=Q{\pi }^2/L^2=9Q/(16{\pi }^2)$. ThereforeThis is three-mode porous magnetoconvection. Because the nonlinear magnetic advection generates a discarded harmonic, it is a Galerkin approximation, not an exact three-dimensional invariant subspace of the partial differential equations.