Finite Length Effects on Collisional Damping of Plasma Waves in Single-Species Plasmas

A recent paper\footnote{M.W. Anderson and T.M. O'Neil, Phys. Plasmas {\bf 14}, 112110 (2007).} analyzed the collisional damping of a plasma wave propagating on a single-species plasma column of infinite length. For high-phase-velocity $\omega/k_z$ and weak collisions $\nu_{\perp \|}$, the predicted damping rate is $\gamma \cong - \nu_{\perp \|} ( k_z \mathrm{v}_{\mathrm{th}} / \omega )^2$, where $\mathrm{v}_{\mathrm{th}} \equiv \sqrt{T/m}$. Measurements of the $k_z = \pi / L_p$ mode on Mg$^+$ plasmas corroborate the temperature and density scaling implicit in this formula; however, the measured damping rates are about 40$\times$ greater than predicted. Here we investigate finite-length effects as a possible source of this discrepancy. The ends of a plasma column couple higher $k_z$ components to the fundamental mode;\footnote{S.A. Prasad and T.M. O'Neil, Phys. Fluids {\bf 26}, 665 (1983).}; and these high-$k_z$ components should enhance collisional damping. Motivated by this intuitive picture, we derive a generalized integral expression for the collisional damping rate that allows for arbitrary $z$-dependence in the waveform. We find that small amplitude high-$k_z$ components can provide the dominant contribution to the mode damping, bringing theory and measurements into better accord.