= 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 obeys
$$
(\partial_t+u\cdot\nabla)e^{ik(t)\cdot x}=i(\dot k\cdot x+u\cdot k)e^{ik\cdot x}=0,
$$
because $\dot k_y=-k_0/T$ and $u\cdot k=k_0y/T$. Moreover $\dot p=(k_0/T,0,0)$, and magnetic stretching is $b\cdot\nabla u=(Rk_0/T,0,0)=R\dot p$. Thus the changing basis exactly supplies the shear-stretching term. Finally,
$$
k\times p=k^2\hat z,\qquad k\times\hat z=-p,\qquad
\nabla\times(be^{ik\cdot x})=i(k^2R\hat z-Sp)e^{ik\cdot x},
$$
and $\nabla^2$ acts as $-k^2$. Equating coefficients verifies
$$
\boxed{\dot R=-i\alpha S-\eta k^2R,\qquad \dot S=i\alpha k^2R-\eta k^2S.}
$$
Take $k_0,T>0$ and first $\alpha\ne0$. Remove <magnetic diffusion> by writing $R=e^{-D}G$, $S=e^{-D}H$, where
$$
D(t)=\eta\int_0^tk^2(s)ds=\eta k_0^2\left(t+\frac{t^3}{3T^2}\right).
$$
Then $\dot G=-i\alpha H$, $\dot H=i\alpha k^2G$, and $\ddot G=\alpha^2k^2G$. Equivalently, rescaling time by $|\alpha|$ and replacing $H$ by $\operatorname{sgn}(\alpha)H$ gives the pair used in the question. Its growing <WKB approximation> is
$$
G\sim Ck^{-1/2}\exp\left(|\alpha|\int_0^tk(s)ds\right),\qquad
H\sim i\operatorname{sgn}(\alpha)kG.
$$
The 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 component
$$
E\sim C_1k\exp\left[2|\alpha|\int_0^tk(s)ds-2\eta\int_0^tk^2(s)ds\right].
$$
At $t\gg T$, $k\sim k_0t/T$, so its leading exponential is
$$
\boxed{E\ \propto\ t\exp\left(\frac{|\alpha|k_0}{T}t^2-\frac{2\eta k_0^2}{3T^2}t^3+\text{subleading terms}\right).}
$$
The positive quadratic term permits strong amplification before the negative cubic diffusion term dominates. The leading maximum occurs near $t_{\rm peak}=|\alpha|T/(\eta k_0)$. Put $\varepsilon=\eta k_0/|\alpha|$ and $b_0=|\alpha|k_0T$. A well-separated late-time growing interval requires
$$
\boxed{\varepsilon\ll1,\qquad \varepsilon^2\ll b_0.}
$$
The first ensures $t_{\rm peak}\gg T$; the second ensures the WKB condition at the peak, since $|\dot k|/(|\alpha|k^2)\sim\varepsilon^2/b_0$. For fixed nonzero $b_0$ both follow by taking sufficiently small positive $\eta$. The peak's leading exponent is $b_0/(3\varepsilon^2)$, 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 $S(0)=i\operatorname{sgn}(\alpha)k_0R(0)$ with $R(0)\ne0$. The initial logarithmic energy derivative is $2(|\alpha|k_0-\eta k_0^2)>0$ when $\varepsilon<1$. 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.
Back to article page
