Quasistatic magnetic response of a helical shearing wave ($\mathcal{E}\simeq -B_{0}H/(\eta |\mathbf{k}{|}^2)$)

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 ${\int }_{-\infty }^{\infty }(1+z^2{)}^{-2}dz=\pi /2$.