= Oscillatory magnetic response in first-order smoothing
{title2=$a=\eta k^2,\quad d=a^2+\Omega^2$}
For a velocity <Fourier mode> with wavenumber $k$, define $a=\eta k^2$, $d=a^2+\Omega^2$ and $\mathbf C=i(\mathbf B_0\cdot\mathbf k)\hat{\mathbf u}$. A cosine-forced mode has periodic response coefficients $(\mathbf p,\mathbf q)=(a\mathbf C,\Omega\mathbf C)/d$, while a sine-forced mode has $(-\Omega\mathbf C,a\mathbf C)/d$. Homogeneous transients decay as $e^{-at}$. The periodic response is a particular solution, not arbitrary initial data.
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