Spatial damping of a driven Alfvén wave ($k=\frac{\omega }{v_A}(1-iϵ{)}^{-1/2},ϵ={\eta }_m\omega /v_A^2$)

For fixed real positive angular frequency $\omega$, a transverse Alfvén wave proportional to $e^{i(kz-\omega t)}$ in a uniform fluid has $k^2={\omega }^2/(v_A^2-i{\eta }_m\omega )$ when magnetic diffusivity ${\eta }_m$ is retained and viscosity is neglected. Choose the branch with positive real and imaginary parts for propagation and attenuation along $+z$. For $ϵ\ll 1$, $k=(\omega /v_A)(1+iϵ/2+O({ϵ}^2))$. The amplitude attenuation length is $L_A=1/\mathrm{Im}k$, whereas the energy attenuation length is $L_E=1/(2\mathrm{Im}k)$. With wavelength $\lambda =2\pi /\mathrm{Re}k$, $L_A/\lambda \sim 1/(\pi ϵ)$ and $L_E/\lambda \sim 1/(2\pi ϵ)$. Fixed-real-wavenumber temporal Resistive damping of an Alfvén wave is a different boundary-value prescription.