Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 65 / 1 / Solution

Consider the complex field before taking its real part. Put $p=\hat{z}\times k=(-k_y,k_x,0)$ and $b=Rp+S\hat{z}$. Both components are perpendicular to $k$, so the solenoidal magnetic-field constraint holds. The time-dependent phase obeysbecause ${\dot{k}}_y=-k_0/T$ and $u\cdot k=k_{0}y/T$. Moreover $\dot{p}=(k_0/T,0,0)$, and magnetic stretching is $b\cdot \nabla u=(R{k}_0/T,0,0)=R\dot{p}$. Thus the changing basis exactly supplies the shear-stretching term. Finally,and ${\nabla }^2$ acts as $-k^2$. Equating coefficients verifies

Take $k_0,T>0$ and first $\alpha \ne 0$. Remove magnetic diffusion by writing $R=e^{-D}G$, $S=e^{-D}H$, whereThen $\dot{G}=-i\alpha H$, $\dot{H}=i\alpha k^{2}G$, and $\ddot{G}={\alpha }^2{k}^{2}G$. Equivalently, rescaling time by $|\alpha |$ and replacing $H$ by $\mathrm{sgn}(\alpha )H$ gives the pair used in the question. Its growing WKB approximation isThe complex relative phase follows directly from $H=i\dot{G}/\alpha$; the printed $H\sim kG$ is sufficient for magnitudes but omits that phase.

The phase-averaged magnetic energy is $(k^2|R{|}^2+|S{|}^2)/(4{\mu }_0)$, hence for a nonzero growing-mode componentAt $t\gg T$, $k\sim k_{0}t/T$, so its leading exponential isThe positive quadratic term permits strong amplification before the negative cubic diffusion term dominates. The leading maximum occurs near $t_{\mathrm{peak}}=|\alpha |T/(\eta k_0)$. Put $\epsilon =\eta k_0/|\alpha |$ and $b_0=|\alpha |k_{0}T$. A well-separated late-time growing interval requiresThe first ensures $t_{\mathrm{peak}}\gg T$; the second ensures the WKB condition at the peak, since $|\dot{k}|/(|\alpha |k^2)\sim {\epsilon }^2/b_0$. For fixed nonzero $b_0$ both follow by taking sufficiently small positive $\eta$. The peak's leading exponent is $b_0/(3{\epsilon }^2)$, whereas the eventual exponent tends to minus infinity. This is mean-field shearing-wave transient amplification, not sustained dynamo action.

Growth from the initial instant is possible explicitly: choose $S(0)=i\mathrm{sgn}(\alpha )k_{0}R(0)$ with $R(0)\ne 0$. The initial logarithmic energy derivative is $2(|\alpha |k_0-\eta k_0^2)>0$ when $\epsilon <1$. Growth is not asserted for every initial polarization, for example an identically zero field. If $\alpha =0$, the amplitudes simply diffuse and the energy is proportional to $[k^2|R_0{|}^2+|S_0{|}^2]e^{-2D}$; shear can still give algebraic transient amplification of an in-plane field for small diffusivity, but a purely vertical field only decays. Positive diffusivity is essential to the final decay.