First-order smoothing approximation

First-order smoothing neglects the fluctuating nonlinear velocity-magnetic-field product in the fluctuation resistive induction equation, but retains $⟨\mathbf{u}\times \mathbf{b}⟩$ in the mean-field electromotive force. For a constant test field it gives ${\partial }_t\mathbf{b}-\eta {\nabla }^2\mathbf{b}=({\mathbf{B}}_0\cdot \nabla )\mathbf{u}$. Its validity requires the omitted terms to be small compared with the retained forcing and diffusion terms.

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.