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.

Three equal-amplitude transverse waves along orthogonal axes, with the same circular polarization, each satisfy $\nabla \times {\mathbf{u}}_j=\sigma k{\mathbf{u}}_j$, $\sigma =\pm 1$. Distinct spatial phases have zero cross averages. The periodic first-order smoothing response to a uniform test field satisfies $({\partial }_t-\eta \Delta ){\mathbf{b}}_j=(\overline{\mathbf{B}}\cdot \nabla ){\mathbf{u}}_j$. Writing $a=\eta k^2$, the response is ${\mathbf{b}}_j=k{\overline{B}}_j[a{\partial }_{\varphi }{\mathbf{u}}_j-\Omega {\mathbf{u}}_j]/(a^2+{\Omega }^2)$. Since ${\mathbf{u}}_j\times {\partial }_{\varphi }{\mathbf{u}}_j=-\sigma U^2{\mathbf{e}}_j$, summing yields the displayed isotropic alpha effect. It includes the resistive and temporal weighting absent from an unqualified instantaneous helicity rule.

For a locally uniform mean magnetic field, first-order smoothing solves ${\partial }_t\mathbf{b}-\eta \Delta \mathbf{b}=(\overline{\mathbf{B}}\cdot \nabla )\mathbf{u}$. Inserting its causal diffusion propagator into the mean-field electromotive force and taking the trace of the isotropic response tensor gives the displayed relation to kinetic helicity. When diffusion during a correlation time is negligible, it reduces to $\alpha =-{\tau }_H⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/3$. At small magnetic Reynolds number, a quasistatic monochromatic velocity instead gives $\alpha =-⟨\mathbf{u}\cdot \nabla \times \mathbf{u}⟩/(3\eta k_0^2)$. The effective correlation time and diffusion weighting should not be silently discarded.

For background shear $\Omega y\hat{\mathbf{x}}$ and uniform field $B_0\hat{\mathbf{x}}$, first-order smoothing gives the magnetic shearing-wave amplitude equationIf magnetic relaxation is fast compared with shear, $c_y,c_z\simeq i{k}_x{B}_0(v_y,v_z)/(\eta K^2)$ after free transients decay. Thus the doubled Fourier-mode electromotive quantity $\mathcal{E}=\mathrm{Re}(v_y{c}_z^{\ast }-v_z{c}_y^{\ast })$ is $-B_{0}H/(\eta K^2)$, where $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$. The spatially averaged mean-field electromotive force of real waves has an extra factor $1/2$. Using the helicity invariant of an inviscid shearing wave, and assuming the approximation holds uniformly in time with $\Omega k_x\ne 0$, givesIndeed $K^2=A+{\Omega }^2{k}_x^2{t}^2$, $A=k_x^2+k_z^2$, and substitution $z=|\Omega k_x|t/\sqrt{A}$ uses ${\int }_{-\infty }^{\infty }(1+z^2{)}^{-2}dz=\pi /2$.

For a steady forced Fourier mode with $|\mathbf{q}|=k$, coupled viscous and magnetic response gives $\hat{\mathbf{u}}=[\eta k^2\hat{\mathbf{f}}+i(\mathbf{B}\cdot \mathbf{q})\hat{\mathbf{g}}/({\mu }_0\rho )]/D$ and $\hat{\mathbf{b}}=[\nu k^2\hat{\mathbf{g}}+i(\mathbf{B}\cdot \mathbf{q})\hat{\mathbf{f}}]/D$. Their mean-field electromotive force contains both forcing helicities and a possible weighted cross-forcing correlation. Jointly isotropic forcing removes the latter; its vanishing at zero mean field alone is insufficient for anisotropic forcing.

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.