Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 51 / 2 / a

Use the Galerkin method, rather than pointwise equality: products generate harmonics outside the retained five-dimensional space. Put $X=\alpha x$, $Z=\pi z$, and denote the five prefactors byThe retained streamfunction is $P\sin X\sin Z$, the retained temperature perturbation is $T\cos X\sin Z+C\sin 2Z$, and the retained magnetic-flux perturbation is $D\sin X\cos Z+E\sin 2X$. These satisfy all the stated boundary conditions. Let $J(f,g)=f_x{g}_z-f_z{g}_x$, the streamfunction advection bracket. Orthogonality of the sine and cosine modes gives the following coefficients of the retained harmonics:For example the $C$ contribution to the temperature bracket is $\alpha \pi PC[\cos X\sin 3Z-\cos X\sin Z]$. The magnetic bracket follows by writing $A=A_D+A_E$: $J(A,\Delta A)=({\beta }^2-4{\alpha }^2)J(A_D,A_E)$ and $J(A_D,A_E)=\alpha \pi DE(\sin 3X-\sin X)\sin Z$. These identities exhibit both the retained terms and discarded harmonics.

Before rescaling time, the projected coefficient equations areSubstitute the prefactors and divide time derivatives by ${\beta }^2$. With $\tau ={\beta }^{2}t$, $\varpi =4{\pi }^2/{\beta }^2$ and $q={\pi }^{2}Q/{\beta }^4$, the resulting five-mode vertical-field magnetoconvection equations areHere primes denote differentiation in $\tau$. The factor $3-\varpi$ is $(3{\alpha }^2-{\pi }^2)/{\beta }^2$, which checks the magnetic nonlinear sign. There is a terminology qualification in the source: on this thermal diffusion time scale, its $\zeta$ is the magnetic-to-thermal diffusivity ratio $\eta /\kappa$. The conventional magnetic Prandtl number is $\nu /\eta =\sigma /\zeta$, where $\sigma$ is the Prandtl number. The displayed equations and subsequent calculation use the source's $\zeta$ unchanged.