Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 77 / Section III

Use the same length, thermal-time, velocity and temperature scales as in thermal convection. Write the imposed magnetic field as $B_0\hat{\mathbf{z}}$ and measure its perturbation in units of $B_0$. The physical linear magnetohydrodynamic momentum equation contains magnetic tension $(B_0^2/{\mu }_0){\partial }_z\mathbf{b}$; the gradient part of the Lorentz force density is absorbed into magnetohydrodynamic total pressure.

Divide the momentum equation by the viscous force scale ${\rho }_0\nu \kappa /d^3$. Its acceleration coefficient is $\kappa /\nu =1/\sigma$ and its magnetic coefficient isHere $Q$ is the Chandrasekhar number, comparing magnetic and viscous-diffusive effects, and $\zeta$ is the ratio of magnetic diffusivity to thermal diffusivity. The induction scale is $B_0\kappa /d^2$, so linearization givesComplete the linear system with $\dot{\theta }=w+{\nabla }^2\theta$, $\nabla \cdot \mathbf{u}=0$ and the solenoidal magnetic-field constraint $\nabla \cdot \mathbf{b}=0$. The omitted advective and perturbed-field tension terms are quadratic in the disturbances. Magnetic tension resists bending of the vertical field, while diffusion allows that bending to relax.

Interpret perfect conductivity here as the perfectly conducting thermal boundary condition $\theta =0$; impose the stated magnetic verticality separately. Together with the stress-free boundary condition, admissible vertical profiles areeach multiplied by $e^{i\mathbf{k}\cdot \mathbf{x}+\lambda t}$. Solenoidality requires $i\mathbf{k}\cdot {\mathbf{U}}_h=-\pi W$ and $i\mathbf{k}\cdot {\mathbf{B}}_h=\pi Z$. The resistive induction equation preserves both profile choices. In particular ${\mathbf{b}}_h=0$ at each boundary and ${\partial }_z{b}_z=0$ there.

Set $s=k^2+{\pi }^2$ and $p={\pi }^2$. The pressure-projected vertical momentum, thermal and vertical induction equations reduce toTheir determinant gives the vertical-field magnetoconvection dispersion relationIt is best kept as a polynomial rather than canceling a factor that might vanish at a special eigenvalue. For $\lambda =0$,For an oscillatory marginal mode $\lambda =i\omega$ with $\omega \ne 0$, write the polynomial as ${\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3$. Its coefficients areSeparating real and imaginary parts gives $a_3=a_1{a}_2$ and ${\omega }^2=a_2>0$. Solving yieldsTherefore oscillatory marginality in vertical-field magnetoconvection requiresThe equality has zero frequency and is not an oscillatory marginal mode. For some positive horizontal wavenumber to satisfy this condition, it is necessary and sufficient that $Q>\zeta (1+\sigma ){\pi }^2/[\sigma (1-\zeta )]$, although the associated threshold must still be minimized over admissible modes. When ${\omega }^2>0$, $R^{(o)}<R^{(e)}$ at the same wavenumber; this also follows from $R^{(e)}-R^{(o)}=(\sigma +1+\zeta )s{\omega }^2/(\sigma \zeta k^2)$.

Both neutral curves can be written with $x=k^2$ and $p={\pi }^2$ asFor steady convection, $(C,D)=(1,1)$; for oscillatory convection,The oscillatory result is physically relevant only with positive frequency; for fixed $0<\zeta <1$ and fixed positive $\sigma$, that holds at the minimizing wavenumber for sufficiently large $Q$.

Differentiation gives the exact minimization conditionIts left side is strictly increasing for $x>0$, so the minimum is unique. Define $h=[D{p}^{2}Q/(2C){]}^{1/3}$. An asymptotic expansion givesSubstituting back, the leading magnetic term is order $Q$, while the two balanced terms $C{x}^2$ and $DQ{p}^2/x$ sum to $3C{h}^2$. Hence, through order $Q^{2/3}$,In particular,while the same boxed general formulas with $(C_o,D_o)$ give the oscillatory minimum and critical wavenumber. Thus strong-field wavenumber selection in magnetoconvection has $k\propto Q^{1/6}$ for both types, with different parameter-dependent constants.

For the exact steady result at arbitrary $Q\ge 0$, the stationarity equation with $C=D=1$ lets us eliminate $Q$ from $R^{(e)}$. It givesAt $Q=0$, $x=p/2$ and $R_0=27p^2/4$. ConsequentlyThis exact critical Rayleigh relation for vertical-field magnetoconvection includes the nonmagnetic minimum, with $R_c\ge R_0$ and $k_c^2\ge {\pi }^2/2$.