Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 60 / 3

Write $s={\alpha }^2+{\pi }^2$ and use modes satisfying all boundary conditions,The quadratic temperature distortion $c$ is retained for the saturation calculation; it does not enter the linear onset. Substitution of the primary modes givesThe determinant vanishes whenThe trace is $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Thus no conjugate eigenvalues can cross the imaginary axis away from zero, and the initial instability is stationary, not oscillatory. Indeed for $R\ge 0$ the discriminant is $[\sigma (s+Q{\pi }^2/s)-s{]}^2+4\sigma R{\alpha }^2/s>0$. This is quasistatic vertical-field magnetoconvection: there is no independent magnetic evolution mode that could produce a magnetic-thermal Hopf bifurcation instability.

For a fixed box the other allowed horizontal wavenumbers are $n\pi /L$, and the actual first threshold is the minimum of their corresponding $R_c$ values. Vertical index $m$ replaces ${\pi }^2$ by $m^2{\pi }^2$ and strictly increases the threshold at fixed horizontal wavenumber, so $m=1$ is selected. The displayed formula describes the prescribed fundamental roll; part (b) optimizes its width.

Put $x={\alpha }^2$ and $p={\pi }^2$. ThenThe minimizing condition isIts left side increases strictly for $x>0$ and the neutral curve diverges at both ends, so there is a unique global minimum. For large $Q$, $x\to \infty$ and the leading balance is $2x^3\sim Q{p}^2$. ThereforeThis is the strong-field wavenumber selection in magnetoconvection scaling. In a box of fixed width one compares the two admissible horizontal wavenumbers bracketing the continuum optimum.

The primary vorticity Jacobian determinant vanishes because ${\nabla }^2\psi =-s\psi$. The quadratic temperature Jacobian determinant isThe mean temperature correction therefore obeys $\dot{c}=-4{\pi }^{2}c-\alpha \pi ab/2$. Its Jacobian determinant with the primary streamfunction isProjecting on the retained primary modes gives, with $D=s+Q{\pi }^2/s$,For a nonzero steady roll, $b=(s^2+Q{\pi }^2)a/(R\alpha )$ and $c=-\alpha ab/(8\pi )$. Substitution in the second equation gives, within this truncation,The positive denominator ensures branches only on the supercritical side for every $Q\ge 0$. To verify dynamical stability as well as existence, the critical right eigenvector is $(1,\alpha /s)$; normalizing the left eigenvector gives linear eigenvalue derivativeQuadratic slaving gives $c=-{\alpha }^2{a}^2/(8\pi s)$, and projection of its feedback gives the cubic amplitude equationIts nonzero branches are stable in the center direction, while all other eigenvalues remain negative at a simple onset. Reflection about the mid-width sends $a\to -a$, so the bifurcation is a supercritical pitchfork bifurcation for all $Q$, in the generic case of a single critical roll mode.

The cubic saturation of quasistatic magnetoconvection result is identical in Fourier truncation and perturbation theory at the orders needed here. At quadratic order the only forced mode is precisely the retained horizontally uniform $\sin 2\pi z$ temperature mode. At cubic order its feedback projects on the primary mode with the same coefficient. The generated third vertical harmonic is orthogonal to that primary projection and contributes only to higher-order corrections. Thus both methods give the same linear threshold and cubic solvability condition; the truncated finite-amplitude branch is not an exact solution of the full PDE at all amplitudes.