Resistive damping of an Alfvén wave (${\omega }^2+i{\eta }_m{k}_{\parallel }^2\omega -v_A^2{k}_{\parallel }^2=0$)

Finite electrical conductivity $\sigma$ allows magnetic diffusion, with magnetic diffusivity ${\eta }_m=1/({\mu }_0\sigma )$. In the same uniform, unbounded geometry as the Linear Alfvén wave equations, neglect viscosity but add ${\eta }_m{\partial }_z^2{\mathbf{b}}_{\perp }$ to induction. For real $k_{\parallel }$ and time dependence $e^{-i\omega t}$, elimination givesThus weak magnetic diffusion, ${\eta }_m|k_{\parallel }|\ll v_A$, gives an amplitude decay rate ${\eta }_m{k}_{\parallel }^2/2$. For ${\eta }_m|k_{\parallel }|>2v_A$, both temporal roots are purely decaying; they no longer describe an oscillating traveling mode. This temporal calculation at fixed real wavenumber is different from a driven vessel at fixed real frequency, where a complex spatial wavenumber describes penetration and attenuation. Viscosity also dissipates kinetic energy, and walls introduce transverse gradients and additional losses.

For an axial length $L$, comparing the magnetic diffusion time $L^2/{\eta }_m$ with the transit time $L/v_A$ gives the ratio $L{v}_A/{\eta }_m$. A large value favors propagation before diffusion, but a narrow vessel can introduce a shorter transverse diffusion scale. In the actual 1949 analysis, finite electrical conductivity and vessel dimensions restrict the useful frequency interval; above its upper limit the response becomes a skin wave.^[ref]