Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 60 / 3 / Solution

For constant alpha tensor and positive magnetic diffusivity $\eta$, the steady anisotropic alpha-squared dynamo equation isFor a nonzero Fourier mode with wavevector $\mathbf{K}=(k,l,m)$, this becomes $\eta K^2\mathbf{B}=i\mathbf{K}\times (\mathbf{\alpha }\mathbf{B})$. Rotational symmetry of the alpha tensor in the horizontal plane lets us set $\mathbf{K}=(q,0,m)$, with $q^2=k^2+l^2$. The component equations areSubstitute the first and third into the second. A nonzero steady amplitude requires ${\eta }^2{K}^4={\alpha }_0^2{m}^2+{\alpha }_0{\alpha }_1{q}^2$. Conversely, this relation supplies a nonzero amplitude through the same component equations, and the solenoidal magnetic-field constraint $q{B}_x+m{B}_z=0$ is automatically satisfied. Thus the steady-mode condition isFor the first form, no division by $m^2+\delta q^2$ is needed. The quotient applies when that denominator is nonzero, in particular for $q>0$ and $0<\delta <1$.

Put $Q=q^2>0$ and $r=m^2\ge 0$. The derivative of $F(r)=(Q+r{)}^2/(r+\delta Q)$ isFor $0<\delta <1/2$, the derivative changes from negative to positive at $r=(1-2\delta )Q$. For $1/2\le \delta <1$ it is nonnegative throughout the allowed half-line and the minimum is at $r=0$. Therefore the optimized uniaxial alpha dynamo threshold isBoth expressions agree at $\delta =1/2$. If the horizontal boundary conditions permit $q=0$, a nonzero mode instead has ${\gamma }^2=m^2$ and the infimum is zero as $m\to 0$; it is not attained by a nonzero wavevector. The exactly uniform mode $q=m=0$ has no diffusive or alpha curl term and must be treated separately.